ELECTRONIC, SPECTROSCOPIC AND ELASTIC PROPERTIES OF EARLY TRANSITION METAL COMPOUNDS
I. POLLINI, A. MOSSER, J. C. PARLEBAS INFM, Dipartimento di Fisica dell+Universita` di Milano, Via Celoria 16; I-20133 Milano, Italy IPCMS - CNRS (UM 7504), 23 rue du loess, F-67037 Strasbourg Cedex, France
AMSTERDAM } LONDON } NEW YORK } OXFORD } PARIS } SHANNON } TOKYO
Physics Reports 355 (2001) 1–72
Electronic, spectroscopic and elastic properties of early transition metal compounds I. Pollinia , A. Mosserb , J.C. Parlebasb; ∗ a
INFM, Dipartimento di Fisica dell’Universita di Milano, Via Celoria 16; I-20133 Milano, Italy b IPCMS, CNRS (UM 7504), 23 rue du loess, F-67037 Strasbourg Cedex, France Received December 2000; editor: D:L: Mills
Contents 1. Introduction 1.1. Basic remarks and a brief overview 1.2. Delocalized and localized models 1.3. The e1ects of electron correlations on spectroscopic properties 2. Pretransition and scandium compounds 2.1. Model formulation of the core level XPS 2.2. Tendencies at the very beginning of the series 3. Titanium compounds 3.1. Electron band structure and valence band XPS of Ti dioxides 3.2. Ti core level XPS spectra of TiO2 3.3. A few insights about Ti2 O3 4. Vanadium compounds 4.1. BIS, valence band and core level XPS of V2 O5 , VO2 and V2 O3
3 3 6 10 18 18 20 22 23 25 28 29 29
4.2. Resonant photoemission 4.3. A related problem: VCn clusters for V adsorption on graphite 5. Chromium compounds 5.1. Valence band XPS and BIS 5.2. Core level XPS 6. Systematic trends 7. Spectroscopic and elastic properties of sub-stoichiometric TiCx , and TiNx compounds 7.1. Core level XPS 7.2. Electron band structure and valence band XPS of Ti carbides and nitrides 7.3. Young’s modulus 8. Summary and outlook 9. Glossary of symbols and acronyms Acknowledgements References
∗ Corresponding author. Tel.: 00-33-3-88107072; fax: 00-33-3-88107249. E-mail addresses:
[email protected] (I. Pollini),
[email protected] (A. Mosser),
[email protected] (J.C. Parlebas).
c 2001 Elsevier Science B.V. All rights reserved. 0370-1573/01/$ - see front matter PII: S 0 3 7 0 - 1 5 7 3 ( 0 1 ) 0 0 0 1 8 - 7
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Abstract Photoemission experiments on early 3d transition metal compounds (TMC), involving both valence bands and core levels of the 3d elements, are reviewed. Extensive use is made of ab initio schemes as well as simple models and the emphasis is put on understanding the results of experiments more than on the experimental details and methods. Compounds of the TMs are Drst analyzed in terms of ab initio band structure calculations, which are shown to be usually suEcient as far as the interpretation of valence photoemission spectra is concerned. A discussion of ultraviolet valence band photoemission (UPS) and bremsstrahlung isochromat spectroscopy (BIS) is also made. Other theories involving conDguration interaction (CI) in the modelization are then shown to be necessary for an understanding of the core-level photoemission spectra and the observed satellite features. The electronic structure of a wide range of early TMCs, from Sc to Cr, is discussed by means of the CI cluster model analysis of the metal 2s-, 2p-, 3s- and 3p-level X-ray photoemission spectra (XPS). Early TMCs, like Ti, V, Cr oxides and halides (e.g. CrF3 , CrCl3 ) have been originally regarded as typical Mott–Hubbard (MH) systems. The MH model results from the electron correlations which dominate the inter-atomic overlaps that lead to bands. The concept of 3d-ligand orbital hybridization leads to the Zaanen–Sawatsky–Allen (ZSA) theory and to the charge transfer (CT) systems. Moreover, we discuss how the analysis of 3s XPS spectra can predict or not the formation of localized magnetic moments. The values of the charge transfer energy and d–d Coulomb repulsion energy U point to systematic trends for the early TM compounds as found in the case of late TM compounds. Simple and competing mechanisms for the excitation of photoemission satellites are presented and the systematic trends for the compounds of the early TM series are discussed. Finally, in addition to the study of the above stoichiometric compounds, we review recent results on the electronic properties of substoichiometric binary alloy (TiNx , TiCx ) by means of core and valence XPS spectra. Self-consistent ab initio calculations with empty spheres at the empty lattice ligand sites performed on these alloys provide the total densities of the occupied states to be compared with the observed valence XPS spectra. An extension of calculations to full potential methods is necessary c 2001 Elsevier Science B.V. All rights for interpreting the elastic properties, e.g. the bulk modulus. reserved. PACS: 79.60.−i; 71.20.−b; 61.72.−y Keywords: Valence and core XPS; BIS; Transition metal compounds; Electronic structure; Electron correlations; Cluster model; Substoichiometric compounds; Bulk modulus
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1. Introduction In this introducting section we would like to brieJy recall a few basic ideas concerning the electronic structure of transition metal compounds (Sections 1.1 and 1.2). The purpose is not to provide an exhaustive state of the art of the subject, but rather to introduce a few concepts, especially on the description of electron correlations (Section 1.3), which will be further developed in the following chapters. 1.1. Basic remarks and brief overview Transition metal compounds (TMC) have been a subject of study for many years because of their diverse physical properties. In spite of their apparently similar electronic structures, having unDlled 3d shells, their electrical conductivities vary widely from metallic to insulating behavior, and they show besides diverse magnetic properties. According to independent electron band theory most of TMCs should be metallic because of their unDlled 3d shells. Some of them (CuS, NiTe, CoS, VO2 ) are indeed metals, but many compounds are insulators (CuO, NiF2 , NiO, CrCl3 ) or subject to metal-to-nonmetal transitions (NiS, V2 O3 , Ti2 O3 ) depending on temperature or pressure. To solve this apparent contradiction Mott (1949, 1990) and Hubbard (1963, 1964a) considered the following charge Juctuations of interatomic type: din djn → din−1 djn+1 ;
(1.1)
where i and j label TM sites and n is the number of d electrons per site (Fig. 1.1a). They proposed that for electronic conduction to occur the energy required for the above process should be less than the 3d band (dispersional) bandwidth w (Fig. 1.2). That is, for some TMCs, the large value of the d–d Coulomb energy U makes it impossible for the charge Juctuation in Eq. (1.1) to occur, so that these compounds become Mott–Hubbard (MH) insulators (U ¿w) in spite of their unDlled d shells. However, when U ¡w, TMCs will be metallic, as predicted by elementary band theory. MH model was devised to explain the insulating nature of NiO (U ¿w), which in a one electron picture would be metallic due to the partial occupation of the Ni 3d shell (formally 3d 8 , Ni2+ ) (Feinleib and Adler, 1968). However, in its simplest form MH
Fig. 1.1. Schematic representation of an ionic lattice consisting of TM ions (d n ) and closed shell anions. Charge Juctuation excitations in MH (a) and CT (b) regimes are illustrated. In the CT regime the p–d hybridization (V) between TM and ligands (L) is also indicated.
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Fig. 1.2. Energy level diagrams corresponding to an ionic ground state and electron removal excitation spectra. The dispersion 3d bandwidth w and the anion (p-like) valence bandwidth W are shown. The Coulomb energy ∼ EG + W=2) are indicated for the excitations as shown in parameter U and the charge transfer energy parameter (= Fig. 1.1.
theory fails to account for the metallic behavior of NiS, although even in this case is U ¿w. First of all, the theory requires a wide range of U values (depending on anions), from higher values of about 8–10 eV in insulating compounds, such as NiO (Fujimori and Minami, 1984; Sawatzky and Allen, 1984; Hufner et al., 1984; Sawatzky, 1987; Hufner, 1985, 1994, 1995; Bengone et al., 2000) to lower values of the same order of magnitude as the 3d bandwidth w (less than 1 eV) in metallic compounds, such as CuS, NiS and CoS (Sawatzky and Allen, 1984; Hufner, 1985; Usuda and Hamada, 2000). Besides, the conductivity gap in late 3d TMCs appears to be directly related to anion electronegativity. Such a big change of the U parameter and the bandgap variation with anions are not likely to be described adequately by the MH model. Since TM 3d electrons are fairly localized in TMCs it is diEcult to explain why the correlation energy (U ) and bandgap (Eg) values are so much dependent upon the anions. The MH model can probably better explain the properties of some early TMCs (Sc, Ti, V, Cr) as compared to the case of late TMCs (Cu, Ni, Co, Fe). A comprehensive discussion of the properties of TMCs in the framework of the MH picture has been reviewed some years ago by Adler (1968), Wilson (1972, 1973, 1985) and Brandow (1977). In 1985 a theory due to Zaanen et al. (ZSA) was proposed to describe the band gap and electronic structure of TMCs consistently, by modifying the conventional MH model (Zaanen et al., 1985; Sawatzky, 1987). This theory considers two types of charge Juctuations for the conduction mechanism. The Drst one, described in Eq (1.1), is the “interatomic polar Juctuation” or MH-type Juctuation. The other type of charge Juctuation is the charge transfer (CT) process between ligand p and metal d levels: din → djn+1 L ;
(1.2)
where L denotes a hole in the anion p band (Fig. 1.1b). The CT process requires to take into account the charge transfer energy . The conductivity gap is always determined by the minimum energy for the creation of uncorrelated electrons and holes excited from the ground state (Fig. 1.2). ZSA theory makes use of the single impurity Anderson (1961) model (SIAM)
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Fig. 1.3. (a) One-electron diagram for TMCs (oxides). The di1erent nature of the band gap is shown in (b) the Mott–Hubbard diagram for a TMC, with Udd ¡ and EG (MH) = Udd ; (c) the charge transfer-diagram for a TMC, with Udd ¿ and EG (CT) = .
or its extension to the periodic Anderson model (PAM), the so-called “two hybridized band” MH model (e.g. Parlebas, 1990 and refs. therein). The signiDcant parameters of the theory are the hybridization strength V between the ligand (L) band and metal 3d states, the anion valence bandwidth W (∼5 eV), the Coulomb energy U , the d bandwidth w and the charge transfer energy . Thus, according to ZSA theory, TMCs can be classiDed as metals (U ¡w or ¡W ) and insulators in the MH regime (w¡U ¡) or insulators in the CT regime (W=2¡¡U ). Fig. 1.3 presents some typical diagrams used for the description of the electronic properties of transition metal oxides (TMOs) and TMCs. Progress in photoemission spectroscopy (PES) has allowed the direct determination of the parameters of the theory from the experimental data (Hufner, 1995). One of most reliable methods for investigating the electronic structures of TMCs is X-ray photoemission spectroscopy (XPS), particularly when it is supplemented by inverse photoemission and ultraviolet photoelectron spectroscopy (UPS). In fact, valence band XPS spectra (v-XPS) measure the energy of the Dlled states and bremsstrahlung isochromat spectroscopy (BIS) that of the empty conduction states, either for stoichiometric and substoichiometric compounds (see Section 7.2). Moreover, one cannot discuss high energy spectra, such as core level photoemission or photoabsorption spectra (c-XPS or XAS), without explicitly taking into account the hybridization e1ects between
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TM 3d orbitals and L orbitals, as it was shown in the case of late TMCs (van der Laan et al., 1981; Fujimori and Minami, 1984; Okada and Kotani, 1992a, 1993). Many TMCs also show weak satellite structures, which are often studied in TM 2p core level spectra. In insulating Cu halides, for instance, satellite features have been explained by a model which is based on the same physical picture and uses the same parameters (; U; V ) for the analysis of UPS and BIS spectra (van der Laan et al., 1981). In c-XPS spectra a new parameter Udc (=Q), the Coulomb attraction energy between the core hole and the 3d electrons, is introduced. By Dtting the energy positions and intensities of the main peak and satellites in 2p-XPS spectra, one can determine the parameters ; U; V and Udc , which in turn give information on the electronic structure of the valence band. The interpretation of PES satellite structures, as due to di1erent screening channels, was Drst proposed for La metal by Kotani and Toyozawa (Kotani and Toyozawa, 1973, 1974) on the basis of the Anderson model and subsequently applied to other metals and insulators by Gunnarsson and Schonhammer (1983, 1985) and Kotani et al. (1988). Neglecting the widths of TM 3d and L valence bands one obtains the so-called cluster model (CM), which is based on a molecular orbital description (Asada and Sugano, 1976; Larsson, 1975, 1976), and is equivalent to the CT model. Finally, it is worthwhile to recall that other powerful experimental techniques are used for the study of the electronic structure of TMCs, such as resonant photoemission spectroscopy (RPES), Raman scattering or soft X-ray emission spectroscopy (SXES). In particular, RPES (Davis, 1982, 1986) and SXES (Ederer and McGuire, 1996) can be useful for determining the nature of the band gaps and investigating hybridization e1ects in TMCs. In principle, a unique set of parameters (; U; V; : : :) should be valid in order to analyse the results from di1erent experimental techniques (XPS, XAS, XES; : : :). 1.2. Localized and delocalized models In this section we just recall a few basic concepts of the ligand (or crystal) Deld theory (LFT) (Section 1.2.1) and band picture (Section 1.2.2), as far as TMCs are concerned. 1.2.1. Ligand (crystal) <eld theory It is well known that 3d orbitals are split in energy by the surrounding ligand Deld (LF) in TMCs. This e1ect is not primarily due to the electrostatic repulsion, but comes from the bonding interactions of TM orbitals with L orbitals (Sections 3.2 and 4.3). The hybridized d states are really combinations of TM d and ligand p orbitals. The d orbitals which point directly towards ligand atoms form combinations with a stronger degree of overlap than combinations, formed by d orbitals pointing between the ligand atoms. The antibonding orbitals are higher in energy and this forms a major contribution to LF splitting. Sometimes the d state bandwidth may be less than LF splitting, giving a gap between lower t2g and upper eg bands. In this case, a compound with six d electrons can have a full t2g band and be non-metallic. However, quite a number of TMCs are metallic and have a d band a few eV wide. A simple approach for understanding the electronic structure in TMCs and interpreting photoemission and photoabsorption spectra is given by the localized CM. If one considers a TM ion surrounded by L ions with the assumption that electron correlation is so strong that electronic excitations are essentially atomlike, then only a small cluster needs to be considered.
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Many investigators have treated the CM with the LFT by including the multiplet e1ects in the TM ion. In this case the optical spectra have been interpreted in terms of forbidden transitions within the crystal Deld multiplets of the d n conDguration (Sugano et al., 1970). These ideas have provided a starting point for the discussion of UPS and XPS data Drst obtained in TMCs (Eastman and Freeouf, 1975; Hufner and Wertheim, 1975; Pollini, 1994). In this case, one expects that the photoemission intensity at the top of the valence band will be dominated by d emission because the TM d state cross section is larger than that of ligand p levels at most photon energies. The main parameters of LFT are the splitting between the eg and t2g levels (10Dq) and the Racah integrals A, B and C, which are linear combinations of the Slater integrals F0 , F2 and F4 . UPS measurements made on NiO, CoO, FeO, MnO and Cr 2 O3 (Eastman and Freeouf, 1975) have given the partial density of states (DOS) for the O2p and TM3d states. 6 e2 with symmetry The NiO (3d 8 ) compound, for example, has a ground state conDguration t2g g 3 A . The photoemission process would then correspond to a transition to the 3d 7 conDguration 2g 6 e1 (2 E) plus a continuum electron and the dipole selection rule will give the three Dnal states t2g g 5 2 4 2 and t2g eg ( T1g and T1g ) with relative intensities 2, 4 and 2, respectively. A similar analysis was also made for XPS data (Wertheim and Hufner, 1972). These studies have indicated that the 3d states are conDned to a narrow 3d band (1–2 eV) located at about 3 eV above the O2p band (Eastman and Freeouf, 1975; Uozumi et al., 1993). The same picture was later used for describing the main 3d emission in v-XPS spectra of trivalent Cr halides (Pollini, 1994, 1998). In Cr 3+ compounds (3d 3 ; CrCl3 and Cr 2 O3 ) all 3d-electrons are in a spin-up state and therefore, irrespective of which electron is photoemitted, the Dnal state is always a triplet and a single peak should be observed in the photoemission spectra of CrCl3 and Cr 2 O3 compounds (Section 5.1). In the same spirit, in Fig. 1.4, the UPS spectra of Ti2 O3 (3d 1 ), V2 O3 (3d2 ), Fe2 O3 (3d 5 ) and NiO (3d 8 ) are reported and compared to each other. With increasing 3d-shell occupation, the cation 3d and O2p bands move closer to each other, indicating an increasing hybridization. This fact has suggested that Ti2 O3 and V2 O3 might be considered MH compounds with a correlation gap between occupied and unoccupied d-states (lower and upper Hubbard bands), while Fe2 O3 and NiO should be more likely CT compounds, with a correlation gap containing the O2p states (Hufner, 1985, 1995). We recall however that this simple picture may break down in some cases, when one makes c-XPS or RPES experiments. As a matter of fact, it turns out to be necessary to include the e1ect of conDguration interactions (CI) in the CM in order to reach a better agreement with experiment (Fujimori and Minami, 1984; Davis, 1986; Parlebas et al., 1990; see also Sections 3.2 and 4.3). As recalled, many of these compounds are metals and present a 3d band 2–4 eV wide, as it occurs, for example, in TiO, TiC and TiN compounds (Ohring, 1992; Soriano et al., 1997; Guemmaz et al., 1997a and refs. therein). These compounds may also contain a large proportion of lattice vacancies, which make the lattice to contract and decrease the distance between metal atoms resulting in a large overlap between them. Section 7 will be devoted to the study of lattice e1ects observed in TiC and TiN. 1.2.2. Limitations of the band picture The band picture of electrons fails when the various interactions neglected in band calculations, such as non-periodic potentials, electron–phonon interaction or parts of the
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Fig. 1.4. UPS spectra of Ti2 O3 ; V2 O3 ; Fe2 O3 and NiO compounds (adapted from Hufner, 1995).
electron–electron interaction, become larger than the band width. Electron–electron interaction is taken into account to some extent in the usual band calculation (mean Deld theory) and, when the Hartree–Fock approximation is applicable, the one-electron excitation energies are determined from the calculated band structure. This is known as the Koopman’s theorem (Kittel, 1963). For example, the energy of a semiconductor is simply given as the energy di1erence between the bottom of the conduction band and the top of the valence band. It is the sum of the energies needed to extract an electron from the top of the valence band to inDnity and that of adding an electron to the bottom of the conduction band. When electron correlation is large, however, the energy gap cannot be obtained so simply, because the energy spectrum itself is altered at every stage of the process. In the Madelung description of an ionic solid core and valence state energies were simply obtained by correcting the free-ion energy levels by the electrostatic potential existing at the site of each ion. This approach was successful for core levels, but did not give a good description of the valence electrons, since it neglected the e1ects of wave-function overlap leading to band formation and hybridization. A better description of the electronic properties of TMCs requires a complete band structure calculation, where the inclusion of intra-atomic exchange can account for the insulating properties in some cases (Mattheiss, 1972). The reason for the failure of the band model in TMCs is the neglect of the electron repulsion, which gives a MH or CT insulator, although metallic properties can be observed when the TM orbital overlap is strong enough. The fact that TiO and VO are metallic, but MnO, FeO, CoO and NiO are not, can be understood in consideration of the large overlap of 3d orbitals in the early transition metal series, while the insulating properties of late TMOs are chieJy due to the contraction of 3d orbitals which occurs as the nuclear charge increases.
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Band structure calculations of TMCs in which exchange and electron correlations are replaced by an e1ective one-electron potential (Madelung, 1978) often predict metallic ground states or gaps which are an order of magnitude smaller than the ones observed experimentally (Terakura et al., 1984). These calculations are based on the density functional theory (DFT) (Hohenberg and Kohn, 1964), which applies strictly only to ground state properties. Here the interaction of electrons with the periodic crystal potential is fully included and leads to itinerant Bloch states. The Coulomb interaction between electrons is treated in a self-consistent way, so that electron correlation in the ground state is accounted for, at least approximately, as in the case of the local spin density approximation (LSDA) (Kohn and Sham, 1965). One shortcoming of DFT calculations is that they do not contain correlations e1ects relevant for excitations from the ground state. As a consequence the agreement or disagreement between calculated and experimental DOS results yields information on the e1ects of correlations on single-particle spectra, the so-called self-energy e1ects (Ziman, 1969; Hufner, 1995). It is well known that problems arise when the DFT-LSDA approach is applied to materials with metal ions that contain incomplete d- or f-shells, such as TMOs and rare earths compounds. For instance, Terakura et al. attempted a self-consistent treatment of the exchange and correlation e1ects with the LSDA. Their result pictured O2p bands deep and completely occupied, TM 4s and 4p states high in energy and completely unoccupied and TM3d bands in between, as shown in Fig. 7 of Terakura’s article. In MnO the calculated exchange splitting was large enough to split the 3d bands into spin-up and spin-down bands, with only the up-bands occupied, while in NiO an additional ligand Deld gap, due to the Ni3d–O2p hybridization, split the spin-down band into t2g and eg sub-bands with the former one Dlled. In MnO and NiO gaps were rather small (0.4 and 0:3 eV, respectively), while the insulating antiferromagnetic FeO and CoO were computed to be metallic. The physical origin of the LSDA failure can be traced back to the mean-Deld character of the Kohn–Sham (1965) equations, where the electron–electron correlation potential is understimated for strongly correlated electronic systems. Recently various forms of the generalized gradient approximation (GGA) have been introduced in order to take into account the inhomogeneity of the charge density in real systems. GGA corrections yield in general correct values of the cohesive energies and equilibrium distances for solids containing light atoms up to the 3d elements (metals). The version suggested by Perdew (1986a,b, 1991), Perdew and Yue (1986) and Perdew et al. (1997) has been also used for MnO and NiO and has given insulators with sizable gaps (although still too small), while FeO and CoO were still calculated to be metallic. This problem was partly overcome by using a di1erent GGA parametrization, which calculated the insulating ground states of FeO and CoO in qualitative agreement with experiment (Dufek et al., 1994; Blaka et al., 1996). Methods using DFT plus GGA approximations applied to solids have greatly changed the calculation approach giving a correct ground state energy and improved lattice parameters for many solids (Perdew et al., 1992; Becke, 1988). Another approach using the self-interaction correction (SIC) approximation was introduced by Svane and Gunnarsson (1990); Svane (1992); Szotek et al. (1993). The electronic structure of NiO, CoO, FeO and MnO was also obtained by Wei and Qi (1994), who have calculated magnetic moments, energy gaps and orbital character of the states near the Fermi energy in agreement with experiment. In conclusion it turns out that an improved agreement with experiments can be reached for TM monoxides by means of “LSDA +U ” approaches
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(Anisimov et al., 1991; Lichtenstein and Katsnelson, 1998; Ezhov et al., 1999; Bengone et al., 2000), where the 3d states have been treated separately through the on-site Coulomb interaction U . These results have indicated that this approach shows order-of-magnitude improvements over the conventional LSDA calculations, giving a correct description of the insulating ground state (antiferromagnetic) of TMOs (see also Usuda and Hamada, 2000). Finally we recall the important “Drst principles” approach predicting the quasi-particle properties of solids given by the GW approximation (Hedin, 1965 and 1999). GWA extends the HF approximation for the self energy (exchange potential), by replacing the bare Coulomb potential by the dynamically screened potential W and the one-electron Green function G. GWA can be said to describe an electron polaron, an electron surrounded by an electronic polarization cloud, which greatly resembles the ordinary polaron, dressed by a cloud of phonons. The dynamical screening adds crucial features beyond the HF approximation: with GWA not only band structures, but also spectral functions can be calculated, as well as charge densities, momentum distributions and total energies. The remarkable point is due to the inclusion of many-body e1ects in the calculation of the electronic properties without omitting the chemical bonding e1ects, unlike the parametrized Hubbard model. Hedin has recently focused the main physical concepts of the GW approximation (Hedin, 1999 and references therein) with the praiseworthy aim to extend these ideas to spectroscopies as photoemission (Lee et al., 1999), electron scattering and X-ray absorption, as typical cases in need of improvements, where the e1ects of electron correlation are in general treated in a rather incomplete way. Another example is given by NiO, where self-consistent GW calculations have given a much better description than the one obtained by a pure LDA approach (Aryasetiawan and Gunnarsson, 1995 and 1998). However the non-local Coulomb interactions make such calculations very much time consuming. In a more standard way, simple examples of band calculations using ab initio methods with LDA will be also reported later and discussed in the case of Ti dioxides (Section 3.1), carbides and nitrides (Section 7.2). The special case of electronic band structure calculations addressed to substoichiometric compounds will be presented in Section 7.2., while the need of using full potential methods to treat their elastic properties will be emphasized in Section 7.3. 1.3. The e?ects of electron correlations on spectroscopic properties Before discussing possible origins of satellites which show up in core level XPS spectra (Sections 1.3.3 and 1.3.4), we will introduce the Mott–Hubbard model (Section 1.3.1), the Zaanen–Sawatsky–Allen theory and the charge transfer model (Section 1.3.2). 1.3.1. The Mott–Hubbard model This model is perhaps the simplest description of the electron correlation e1ects in solids. It was Drst discussed by Gutzwiller (1963, 1964) in an attempt to describe the e1ect of correlations for d-electrons in transition metals and then extensively by Hubbard (1963, 1964b, 1965, 1966), especially in the context of metal–insulator transitions. The Hamiltonian is characterized by a kinetic energy term Tij denoting the hopping of an electron from a site i to its nearest site j, and by the extra energy cost U of putting two electrons (spin up and spin down) on the same site.
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The Hubbard Hamitonian is + H= Tij ci cj + U ni ni− ; ij
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(1.3)
i
+ where the operator ci creates an electron of spin in the Wannier state centered on Ri , and cj annihilates an electron of spin at the site Rj . Tij is the Fourier transform of the Bloch energies and the factor U is the Coulomb repulsion energy between two electrons with opposite spins on the same site. In the Wannier representation the term U is given by the matrix element arising from the four Wannier functions centered on the same site:
ii|g|ii = U :
(1.4)
The Hamiltonian contains the three parameters: T0 = Tii ; T1 = Tij and U . The T0 term represents the mean energy of the d band, the parameter T1 = Tij is zero unless i and j are nearest neighbors and is equal to half of the d band, i.e. 2T1 = w, and the matrix element Tij describes the electron hoppings between nearest neighbors. In the absence of U , this is just a tight-binding Hamiltonian with atomic energy levels i (k) at all sites being the same (zero) and with nearest neighbor overlap Tij leading to a band of one-electron states. The Mott–Hubbard correlation term U is, in the atomic limit, the di1erence between the ionization energy I and the electron aEnity A. The model is characterized by the dimensionless parameter (T=U ) and the electron density or average number of electrons per site n, which can range from zero to two. The Hubbard model is a “highly oversimpliDed model” for strongly interacting electrons in solids: there is little doubt that it is too simple to describe actual solids faithfully. It has no long-range Coulomb repulsion between electrons, no orbital degeneracy or many-band e1ects, no electron– lattice coupling and assumes a perfectly ordered lattice. Nevertheless, the Hubbard model is one of the most important historical models in theoretical physics. The parameters of the model Tij and U are in general poorly known. However, experimentally determined values for these parameters are essential for a meaningful discussion of the TMCs within the framework of the Hubbard Hamiltonian. The bandwidth can in principle be obtained directy from the experimental data and the magnitude of U is most directly determined by combining photoemission (PES) and bremsstrahlung isochromat spectroscopy (BIS). PES measures roughly the energies of the singly ionized electronic states, while BIS measures the energies of conDgurations with one-electron added. The di1erence in the energies d n−1 and d n+1 for a particular ground d electron conDguration d n is just the quantity I − A = U measured in situ. The values of U are in general considerably less than their atomic values (I − A) especially in metals. This point was emphasized quite early by Herring (1966) in his review on the magnetism in transition metals. The reduction is due to the screening by other electrons physically present in the unit cell of the solid (see for example, Parlebas et al., 1990). The parameter Tij is not determinable directly by experiment, but electron dispersion curves E(k) have been obtained from ARUPS experiments for several transition metals (Eastman and Himpsel, 1981; Himpsel, 1983). By Dtting to a tight-binding type model, the hopping integrals can be inferred. First principle band-structure calculations also lead to estimates of Tij for simple systems.
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The parameters which determine the metal–insulator transition are the d state bandwidth w and the Coulomb repulsion U . The transition occurs when w≈U , an insulator being characterized by wU , and a metal by wU . Hubbard worked out the special cases of a parabolic density of states at √ U = 0, and also presented a solution that exhibits “an insulator–metal transition” at w=U = 2= 3. For small w=U , the atomic limit is obtained and the two bands are separated by an energy gap, Eg . For large values of w=U , the Bloch limit results and one gets a single half Dlled band, with Fermi energy at T0 + U=2. The e1ect of the electron–electron repulsion is to make the partially Dlled band d insulating, when the interaction between TM atoms becomes small. The study of the Mott–Hubbard metal–insulator transition has recently attracted much interest again, being an example of how strong correlation drastically changes the properties of a system (Georges et al., 1996). As it is known, the MH transition was originally studied without orbital degeneracy for simplicity, however, for many real systems, the orbitals primarily involved in the transition are degenerate, and if the orbital degeneration Nd is taken into √ account, the ratio w=U , where the MH transition takes place is increased by roughly a factor Nd . In fact, the electron hopping reduces the MH gap in the limit of a large Coulomb interaction U as EG ≈U − w Nd ; (1.5) √ because the hopping contribution grows as w Nd (Gunnarsson et al., 1996, 1997). Moreover, metal–insulator transitions do not arise only in TMCs, but also in glasses, in amorphous semiconductors, in dilute solutions and in the impurity band conduction, where, for instance, one may Dnd abrupt leaps in conductivity, if a particular parameter, as the temperature or the defect concentration, is varied. Not all of these phenomena are of course, Mott transitions, and in many cases it is not even clear which is the resonable mechanism (Madelung, 1978; Mott, 1990). For a quantitative discussion of the Hubbard approximation, the possibility for its extensions and applications the classical articles by Adler (1968); Doniach (1969) and Mott and Zinamon (1970) are recommended and some more recent contributions (Tasaki, 1998; Lieb, 1998) should be consulted. Nevertheless, we should also add that the simple one-band Hubbard model is not appropriate to describe the insulating properties of a number of materials, since it turns out that many TMCs present a band gap strongly a1ected by the nature of the ligands (anions), whereas, according to this model, the energy gap only involves the d–d Coulomb interaction on the TM atom, and the magnitude of the gap is nearly independent of the nature of the anions. For instance, it would not be easy to explain why compounds like NiS and CuS are metallic (see for example Usuda and Hamada, 2000 and refs. therein), while NiO and CuO are insulators with similar values of U (see also Lombardo et al., 1998). Thus, in many cases, one is forced to consider a model which takes also into account the electronegativity of the anions. 1.3.2. The Zaanen–Sawatsky–Allen theory and the charge transfer model The basic problem with the interpretation of the photoelectron emission in late TMCs stems from the fact that the previously considered Hubbard–Mott model neglects the large 3d ionization energies of the TM ions and the hybridization of 3d electrons with ligand p electrons. Thus, another possibility is that there is a large gap in the d states due to electron correlations, but that the energy gap corresponds to charge transfer (CT) transitions from the anion p-valence
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band to the empty metal d states, a p–d gap. This also leads to insulating behavior, however in this case the energy gap will also depend on the electronegativity of the anions. This means that the d-like states of the valence band are in fact a mixture of ligand p electrons and metal 3d-electrons and they can be written as d n L. The valence band now leads to two Dnal states after photoemission, namely to d n−1 L and d n L states, where the Drst corresponds to the state already considered, but the second, not considered so far, has been produced by the ligand-to-metal charge transfer out of the photoionized state d n−1 L. The d n−1 L and d n L states contain the same number of electrons or holes and therefore tend to mix. For example, in NiO, the d 8 L state produces most of the weight near the Fermi energy, while the d 7 L state produces most of the ∼ 9 eV (Sawatzky and Allen, 1984; Hufner et al., 1984; weight in the satellite structure at EB = Hufner and Riesterer, 1986). Thus one realizes that there are two types of electronic excitations possible in the valence band of TMCs, that is the excitation of a 3d-electron from a TM ion and its transfer to another TM ion, governed by the energy U in the spirit of the Hubbard model, and, in addition, the excitation of a ligand electron onto a TM ion described by the CT energy . The relative magnitude of the energies U and determines whether a material is a MH or a CT compound, as shown in Fig. 1.3, where the various diagrams used for the discussion of the electronic properties of TMOs are reported (Sawatzky, 1987). 1.3.3. Satellite structure Excitation of valence electrons accompanying the core level photoemission of the 3d and 4f ions in solids causes the appearance of satellite features on the high binding energy side of the main emission in c-XPS spectra. The origin of such satellites has been the subject of a large number of investigations (Shirley, 1978) and is still matter of research. It was originally suggested that satellites could arise from the simultaneous excitations (shake-up) of outer electrons, although it was not clear which levels were involved. Strong TM 2p electron satellites in the iron group compounds were observed for ions with almost-Dlled shells (Cu2+ ; Ni2+ and Co2+ ) and half-Dlled shells (Mn2+ and Fe2+ ), but no satellite features were observed for the Dlled shell in Zn2+ (Rosencwaig et al., 1971). Intense “shake-up” satellites were also observed from Ti 2s and Ti 2p photoemission lines in TiF3 and TiO2 , and Sc 2s and Sc 2p lines in ScF3 and Sc2 O3 (Wallbank et al., 1973a, 1978). These satellites have been observed at an energy separation of 12–13 eV from the photoemission main lines. Their origin was attributed to excitations of electrons from ligand orbitals to metal 3d states, in analogy to the explanation Drst given by Kim for the satellites observed in Ni2+ and Cu2+ compounds (Kim, 1974). In Fig. 1.5 we report, as an example, the cases of TiF3 (Ti 2s XPS) and TiO2 (Ti 2p XPS). Further photoemission experiments in the 2p-XPS spectra of early TMCs, such as Sc trihalides and Ti tetrahalides, again showed satellites at about 10 –13 eV higher binding energy with respect to the main lines, but the origin of the satellite peaks was di1erently interpreted. As a matter of fact, an exciton mechanism (de Boer et al., 1984) was proposed for the origin of satellite peaks (Section 1.3.4), although it has been later shown that the CT mechanism can also explain satellite peaks, if the p–d hybridization strength is suEciently large (Parlebas, 1992). Nevertheless, in some systems we can expect competition between the charge transfer and the exciton mechanism, as formally discussed in the original de Boer’s article. For other systems, on the other hand, it is rather well established that the satellite structures seen in 2p-XPS of late
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Fig. 1.5. (a) Ti 2s XPS spectrum of TiF3 and (b) Ti 2p XPS spectrum of TiO2 (adapted from Wallbank et al., 1973b).
TMCs originate from the CT mechanism of valence electrons (van der Laan et al., 1981; Zaanen et al., 1986; Oh, 1988; Park et al., 1988; Okada and Kotani, 1992b). The importance of CT has been also conDrmed in the analysis of the 3d-XPS of rare-earth compounds (Gunnarsson and Schonhammer, 1983; Jo and Kotani, 1986; Kotani, 1987; Kotani et al., 1988; Kotani, 1999). Satellites observed in the spectra of the Drst row TMCs have been mostly explained by CT transitions, within the framework of the sudden approximation and molecular orbital theory (Hufner, 1995). Satellite transitions result from electron excitation in the valence shell as a result of the change in central potential caused by the sudden removal of an electron from the core shell. Through these studies, many late TM compounds, such as CuO and NiO, have been
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Fig. 1.6. Schematic representation of the exciton model. The Dlled circle (•) means an electron; the empty one (◦) represents a hole. Final state (0) corresponds to the main line and Dnal state (1) to the exciton satellite. Also shown is the wave function for the valence electron in the initial and the Dnal states.
classiDed as CT-type insulators instead of MH insulators, as originally made. Recently, Bocquet et al. (1992, 1996) have analyzed the 2p-XPS spectra for several Ti and V oxides with formal valence d 0 to d 6 and have classiDed them as intermediate compounds between CT and MH compounds. Moreover, these studies suggest that the CT theory developed within the cluster impurity Anderson model (CIAM) is applicable to the analysis of the high-energy spectra of early TMCs as well as of the late TM systems (Parlebas, 1992; Okada and Kotani, 1992a, 1993; Kotani et al., 1995; Parlebas et al., 1995). 1.3.4. Other mechanisms for core level XPS satellites In heavy TMCs the charge transfer mechanism from the TM 3d to ligand p levels accounts for the satellites observed in core XPS spectra (Okada and Kotani, 1992b). In light TMCs, there is however some evidence that other mechanisms may be necessary to explain the data (de Boer et al., 1984). For example estimates of the binding energies can be made from the measured optical gaps and molecular calculations and these satellite energies are larger than expected for the charge-transfer process. The chemical trends also disagree with the mechanism. In addition, weaker satellites at lower binding energies have also been observed, so that a new model has been introduced that tries to describe the low-energy satellites, as due to ligand-to-metal charge transfer processes, and the high-energy (¿7 eV) satellites as due to excitations of excitons. The mechanism for screening the core hole created by the photoemission is the polarization of the surroundings. In compounds with polarizable ions (halides and oxides), the screening of a core hole on a cation can be produced by the polarization of the ligands. As polarization corresponds to a mixing in of the higher states, one can expect satellites due to transitions to these states. The energy of excitons and satellites should then be close to the optical excitation energies. A simple version of this model is sketched in Fig. 1.6. The creation of a core hole on the metal atom polarizes the ligand atoms, mixing s and p orbitals. In the ground state, only the p orbital is occupied. The Dnal state with the lower-energy orbital occupied gives the main line. Excitation of electrons to higher orbitals originates the satellite. Electron excitation corresponds
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Table 1.1 Satellite separations (eV) for the Ti 2s, Ti 2p, Sc 2s and Sc 2p photoelectron lines in 3d 0 and 3d 1 compounds (de Boer et al., 1984) Compounds
2s
2p1=2
2p3=2
ScF3 Sc2 O3 TiO2 TiF3
12.0 12.0 13.4 12.6
12.0 10.8 12.6 12.7
12.2 11.3 13.3 12.4
Table 1.2 Measured satellite distance Es to TM 2p main peak, measured intensities I3=2 and I1=2 of satellites relative to the 2p3=2 and 2p1=2 main line, metal–ligand distance R, calculated polarizabilities Ep, and calculated intensity ratio I c = Ep=Es Compounds
ES
I3=2
I1=2
R
Ep
Ic
ScF3 ScCl3 ScBr 3 ScI3 Sc2 O3 TiF4 TiCl4 TiBr 4 TiI4 TiO2
12.3 9.6 8.6 7.4 11.2 13.4 9.6 8.7 7.3 13.4
0.25 0.2 0.15 0.15 0.3 0.19 0.08 0.14 0.18 0.25
0.35 0.4 0.3 0.1 0.4 0.43 0.28 0.04 0.25 0.40
2.01 2.58 2.76 2.97 2.11 1.90 2.22 2.31 2.55 1.96
2.4 2.1 2.2 2.4 3.2 2.2 2.7 3.0 3.2 4.0
0.20 0.22 0.25 0.32 0.29 0.17 0.28 0.34 0.44 0.30
to an electron–hole pair on the ligand atom and becomes an exciton in the solid. By means of the exciton model it is also possible to calculate the intensities of the satellites from the anion polarizability. Comparison of the theory using anion polarizabilities with experimental satellites intensities and energies is generally favorable, as one can see in Tables 1.1 and 1.2, adapted from de Boer’s article. The measured satellite distances to metal 2p main peaks and the experimental intensities are reported for Sc and Ti halides and oxides (Sc2 O3 and TiO2 ) to show the prevision of the model. The Sc 2p spectra of Sc trihalides are also presented in Fig. 1.7, where the so-called exciton satellites are shown. The question naturally arises whether the exciton satellite mechanism can explain the satellite structures of all early TMCs, as it seems from a perusal of vacuum ultraviolet spectra available for TiO2 and Ti2 O3 compounds, gaseous TiCl4 and crystalline CrCl3 and CrBr 3 solids (Guizzetti et al., 1976). All compounds have a region up to 9 eV in the oxides and up to 6 or 7:7 eV in the halides, which is ascribed to a charge transfer transition from the anion p to the metal 3d orbitals. The XPS satellites show a striking similarity with these spectra. Satellites are observed between 3 and 7 eV and at 14 eV in TiO2 and BaTiO3 (Sen et al., 1976; Chermette et al., 1980); at 4.0 and 9:4 eV in gaseous TiCl4 (Wallbank et al., 1978), and at 11 eV in CaF2 (Rublo1, 1972). For the layered compounds CrCl3 and CrBr 3 , recent data reported in v-XPS (Pollini, 1999, 2000) seem to indicate a competition
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Fig. 1.7. Sc 2p XPS spectra of ScF3 ; ScCl3 ; ScBr 3 and ScI3 (adapted from de Boer et al., 1984).
between the exciton mechanism and the charge transfer process. For example, in CrCl3 and CrBr 3 , the v-XPS and the 3s-, 3p- and 2p-XPS data show satellites located around 8 and 11 eV away from the main emission bands, which seems a large energy separation to be explained only by the charge-transfer model with reasonable parameters. In this sense, the interest for the exciton satellite mechanism may be considered, since the exciton mechanism appears to dominate over the CT model, when the CT energy is large and the Q (=Udc ) parameter is small and there are besides many empty 3d orbitals. In the following sections (2–5), putting main emphasis on CI–CT models, we will analyze and discuss some examples of core level spectra for pre- and early-TMCs, including the chromium compounds (Section 5), and consider the systematic trends observed along the TM series (Section 6). Among the various kinds of TMCs, the early transition metal oxides (TMOs) of the Drst TM row show a great variety of electronic and magnetic properties and will be largely considered here. Also in Section 4.3 we will calculate and discuss an other mechanism for the 3s-XPS satellite, i.e. the so-called multiplet splitting due to the presence of a localized magnetic moment (Shirley, 1978; Fadley, 1988). Section 7 will be devoted to review similar topics in substoichiometric titanium compounds and to the study of their elastic properties (Guemmaz et al., 1997, 1999, 2000).
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2. Pretransition and scandium compounds In this section we Drst consider the simplest mechanism responsible for the main line-versussatellite structure in the c-XPS of early 3d TMCs. In our simple model, as already mentioned in the preceding section, the physics of the c-XPS is described only in terms of a few pertinent parameters, such as the on-site Coulomb repulsion U , the charge-transfer energy , the hybridization energy V related to the metal-ligand transfer integrals and the core-hole-3d-electron Coulomb attraction energy Q (=Udc ) (Parlebas, 1992, 1993; Parlebas et al., 1995). In Section 2.1, we recall our model formulation of c-XPS in the case of a series of pre- and early-TM insulating compounds, nominally characterized by a 3d 0 conDguration in the ground state. Then, in Section 2.2, we present some results and a general discussion relevant to the very early TMCs as well as K and Ca compounds. As far as those last compounds are concerned let us just recall that several years ago a method was also presented for the calculation of the imaginary part of the dielectric function (Parlebas and Mills, 1978) when the transition involved a core level in an insulator (Fliyou et al., 1987): especially X-ray absorption spectra in alkali halides could be qualitatively understood in terms of a simple method used for a schematic model. 2.1. Model formulation of the core level XPS Here we consider early-TMCs characterized by the following band structure somewhat related to the previously presented CT model (Section 1.3.2): a completely Dlled ligand band (LB), mostly arising from anion states, and separated by an energy gap of a few eV from the empty conduction d bands, t2g and eg (often separated by another gap). Actually, in our simpliDed formulation we only keep in mind an electron system consisting of a Dlled LB and a degenerate 3d level as well as the corresponding core level (2s or 2p TM level). Similarly to the case of the 4f 0 La insulating compounds (La2 O3 ; LaF3 ) (Kotani et al., 1987), the 3d level in the ground state of the TMC is assumed to be well above the LB, so that it is practically empty; this corresponds to the nominally 3d 0 conDguration. However, due to conDguration interaction we have to consider a strong mixture of 3d 0 and 3d 1 L conDgurations where L means a ligand hole in the LB. See also the appreciable hybridization V found in Section 3.1 between the LB and the 3d states by Khan et al. (1991). Actually, we also consider the 3d 2 L2 conDguration in the ground state. In the Dnal state of the c-XPS, the 3d level is pulled down by the core hole potential—Udc , and a similar conDguration-interaction mixture takes place between c3d 1 L and c3d 2 L2 , where c labels the core hole. It is interesting to note the analogy of the present situation with what happens in copper dihalides (van der Laan et al., 1981) or in CuO (Mosser et al., 1991), for example, where 3d 10 L and 3d 9 conDgurations are strongly mixed. The Dlled band single impurity Anderson model (SIAM) Hamiltonian of our system is written as H = HL + HTM + Hmix ; where HL =
k;!
L (k)a+ L (k; !)aL (k; !) ;
(2.1) (2.2)
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HTM = d0
!
a+ d (!)ad (!) + Udd
!¿#
+ + c a+ c ac − (1 − ac ac )Udc
and
19
+ a+ d (!)ad (!)ad (! )ad (! )
!
a+ d (!)ad (!)
(2.3)
√ Hmix = (V= N ) [a+ L (k; !)ad (!) + h:c:] ;
(2.4)
k;!
where L (k); d0 and c are the energies of the LB states, 3d level, and core level, respectively, + + and a+ L (k; !); ad (!) and ac are the electron creation operators in the corresponding states. Here k denotes the index of energy level (k = 1; : : : ; N ) in the LB and ! speciDes both the spin and orbital degeneracies (! = 1; : : : ; Nd ). Actually Nd can be taken between 2 (only spin degeneracy) and 10 (full d spin–orbital degeneracy). However, the spin and orbital degeneracies for the core states are here disregarded as well as the corresponding spin–orbit coupling (see Section 3.2.1 where spin–orbit coupling has been included), since in our model we do not try to describe the full (core 2p) multiplet structure, but only one component (2p3=2 ). Similarly the cubic crystal Deld e1ect on the d0 level is ignored since it is not a basic parameter to calculate the satellite features in this case. The Hamiltonian used in (2.1) is based on a broken translational symmetry for the 3d states of the TM ion (3d single-site model). Such an approximation is expected to make sense if the dispersional 3d-bandwidth is small as usual in 3d insulating compounds. We could, of course, also break the anion translational symmetry ending up with the previously considered cluster calculation (van der Laan et al., 1981; Uozumi et al., 1997; Bocquet et al., 1992, 1996), but usually the LB width is quite large and therefore rather well taken into account with a band model. Eq. (2.1) simply treats the 3d states of the TM ion as impurity states (the core hole Udc acting like an impurity potential) in a lattice formed by itinerant ligand states. Let us recall that in the case of the present Dlled LB SIAM, it is easy to obtain the exact solutions of the preceding ‘two-electron’ Hamiltonian (2.1) (Kotani et al., 1988). The ground state |g, when the core level is occupied (a+ c ac = 1), is then expressed as a linear combination of the following basis states: |d 0 ; |d 1 ; L(k)
and
|d 2 ; L(k1 ); L(k2 )
(2.5)
with, respectively, 0, 1 and 2 holes in the LB. Also, each Dnal state |f of the c-XPS is expressed, apart from the emitted photoelectron, as a linear combination of the following states: |d 0 ; c; |d 1 ; L(k)c
and
|d 2 ; L(k1 ); L(k2 ); c
(2.6)
where c is a core hole. An important quantity is the ligand-to-metal charge transfer energy , deDned by the energy di1erence between the 3d 0 and 3d 1 L conDgurations: ≡ E[3d 1 L] − E[3d 0 ] = d0 − p ;
(2.7) d0
is the degenerate where p is the center of the LB (hereafter taken as the energy origin) and energy of the d states. In our model we treat the Dlled LB as a Dnite periodic system consisting of N discrete levels with equal spacing and in our numerical results, we take N = 6, which is enough for a good convergence (Kotani et al., 1988). We also take W ≈5:5 eV as a good
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order of magnitude for the width of the ligand band in typical early TMCs we are considering. Actually the precise value of 5:5 eV is taken from the LMTO band structure calculation by Khan et al. (1991) made on the O 2p-band of rutile TiO2 (Section 3.1). Finally, for simplicity, we put Nd = 2 in the following numerical applications. However, the essential features of the spectrum do not change much with Nd , provided that Nd V 2 is kept constant (Kotani et al., 1988). That means that, e.g., V = 6 eV with Nd = 2 is essentially equivalent to V ≈3:46 eV with Nd = 6 (like t2g ) or to V ≈2:68 eV with Nd = 10 (like t2g + eg ). Disregarding the interaction of the emitted photoelectron with other electrons (sudden approximation), the c-XPS is Dnally expressed as F(EB ) = |f|g|2 L(EB + Eg − Ef ) : (2.8) f
In Eq. (2.8) L(x) is a Lorentzian deDned as L(x) = 'c =[(x2 + 'c2 )], where EB is the electronic binding energy obtained by subtracting the kinetic energy of the photoemitted core electron from the energy ! of the incident photon (EB = ! − ), and 'c (=1 eV) labels the spectral broadening of our empirical Lorentzian function, the width arising from the core hole lifetime and experimental resolution. 2.2. Tendencies at the very beginning of the series Let us Drst remark that in Eq. (2.3), Udd and Udc are not really independent parameters. In our numerical calculation we take the parameter value Udd somewhat smaller (or at most equal) than Udc because the average distance between two 3d electrons is usually larger than the distance between a 3d- and a core-electron. In free atoms, the ratio Udd =Udc has been found to vary between 0.7, when c = 2p, and 0.9, when c = 3p, (de Boer et al., 1984). In Fig. 2.1, we have shown the c-XPS results for a range of values of the ratio Udd =Udc from 0.666 (Fig. 2.1a) to 0.875 (Fig. 2.1f), in reasonable agreement with the de Boer’s results. Actually, Fig. 2.1 summarizes a qualitative study of the e1ect of increasing the couple values (Udd ; Udc ) with respect to the Dxed parameter. The CT energy was taken as 4 eV, with a LB width of 5:5 eV, i.e. a bandgap of 1:25 eV between d0 and the highest LB state, before considering any e1ect of Udd or Udc on d0 . From Fig. 2.1, and quite generally, we can distinguish two regimes of parameter values (Zaanen et al., 1985; Bocquet et al., 1992): i) (Udd ; Udc )¡, (as in Fig. 2.1a), where the satellite is practically absent from the spectra; ii) (Udd ; Udc ) ¿ (as in Figs. 2.1c–f), where the satellite is clearly visible, even for small V values. For a simpliDed analytical model, restricted to two conDgurations, we refer to van der Laan et al. (1981): this model relates the position of the satellite and its intensity to ; V and Udc (but does not include Udd ). Veal and Paulikas (1985) reported 2p-XPS spectra for 2+; 3+, and 4+ cations in CaF2 ; ScF3 and TiF4 Juorides (see Fig. 7 of their article). For all these pre- and early-3d insulating compounds, where the cation conDguration is a nominally pure 3d 0 4s0 , satellites are separated from the main core line by 10 –15 eV. As far as Sc 2p XPS spectra of ScF3 compounds is concerned one should also see Fig. 1.7. The situation seems similar to that of CaO; Sc2 O3 , and TiO2 compounds (de Boer et al., 1984; Veal and Paulikas, 1985). For the calcium compounds (CaF2 , CaO), the satellite peaks are the weakest ones, but still visible. However, in the K 2p-XPS of potassium halides, like KF (Veal and Paulikas, 1985), where the cation conDguration corresponds again to 3d 0 4s0 ,
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Fig. 2.1. c-XPS spectra in 3d 0 oxides plotted against binding energy for various hybridization strengths V =1; 2; : : : ; 7 eV and with (Udd ; Udc )=(a) (2; 3); (b) (3; 4); (c) (4; 5); (d) (5; 6); (e) (6; 7); (f ) (7; 8) eV. The other parameters are = 4 eV; Nd = 2; 'c = 1 eV.
no satellite structure can be discerned and attributed to local screening states. In this case, the unDlled 3d states are located too far above the LB (wide-band-gap material) so that, even in the Dnal state of the 2p-XPS spectra, the 3d levels are still well above the LB and cannot participate in the local screening of the core hole. In our calculation (Fig. 2.2) we have shown that if the charge transfer energy becomes larger and larger, from 3 to 9 eV, then the satellite becomes smaller and smaller (as in Ca compounds) and Dnally disappears (in K monohalides) for (Udd ; Udc ). It is interesting to underline that, contrary to K + , the Ca2+ cation in the Dnal state of the c-XPS shows a low-lying 3d behavior, which is typical of the considered cation in the whole early TM series for insulating compounds. The important common feature is that, in the Dnal state, the Drst unoccupied allowed level is 3d like. The main c-XPS peak (at low binding energy) roughly corresponds to a locally well-screened state 3d 1 and the higher binding energy satellite corresponds to a locally poorly screened state 3d 0 . The fact that K and Ca ions show di1erent, but predictable satellite structures, gives support to the 3d-screening
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Fig. 2.2. c-XPS spectra in 3d 0 oxides plotted against binding energy for various charge-transfer energies = 3; 4; : : : ; 9 eV and with (Udd ; Udc ) = (3; 4) eV; (V; Nd ) = (5 eV; 2); ' = 1 eV.
model for the core hole. Moreover, the energy separation between the main and satellite peaks (van der Laan et al., 1981) is observed to increase in the series CaF2 ; ScF3 ; TiF4 (Veal and Paulikas, 1985) or CaO; Sc2 O3 ; TiO2 (de Boer et al., 1984), which is again in agreement with the increase of the hybridization parameter V in our model, going from Ca to Ti compounds (Fig. 2.1d–f). In conclusion in this section we have calculated the c-XPS spectra, which result from the creation of a core hole within the Dlled-band SIAM. Our simpliDed calculation seems to explain well the spectra for the series of the very early TM insulating compounds, characterized by a nominally 3d 0 conDguration in the ground state, including those compounds which involve pre-transitional elements like K or Ca. 3. Titanium compounds Now, we essentially focus on the Ti compounds with nominal 3d 0 conDgurations, with TiO2 as a typical example. Although Section 3 is essentially restricted to valence band (Section 3.1) and core XPS (Section 3.2), we would like to mention the X-ray emission spectra (XES) measured in the region of Ti 2p and O 1s excitation threshold by Tezuka et al. (1996) as well as the resonant XES analyzed in details by Finkelstein et al. (1999) and Matsubara et al. (2000). Also, the soft X-ray Raman scattering has been recently investigated at the Ti 2p absorption edge of TiO2 (Harada et al., 2000) and the corresponding Ti 1s absorption edge has been revisited (see Uozumi et al., 1992; Joly et al., 1999). Titanium dioxide is an important compound with applications in a number of areas, such as photocatalysis. Here, we limit ourselves to the case of the TiO2 rutile structure. However, a comparison between TiO2 rutile, anatase and
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brookite structures has been also performed with the help of reJection spectra measurements of TiO2 single crystals by Sekiya et al. (1998). The case of Ti2 O3 will be then mentioned in Section 3.3. 3.1. Electron band structure and valence band XPS of Ti dioxides As introduced in Section 1.2.2, we would like here to recall some band structure calculations and results (Section 3.1.1) as well as their comparison with experimental v-XPS spectra (Section 3.1.2) in the typical case of TiO2 compounds. 3.1.1. LMTO calculations and results W c = 2:89 A W and u = 0:31. TiO2 (rutile) crystallizes into tetragonal structure with a = 4:49 A; The unit cell contains 2 Ti and 4 O atoms. The core charge distribution for Ti and O was obtained through a self-consistent Dirac atomic potential. Electrons up to and including O 1s and Ti 3p were treated as frozen core electrons. We used two energy panels for the band structure calculation (Khan et al., 1991) within local density approximation (LDA), using a self-consistent linear muEn tin orbital (LMTO) method in the atomic sphere approximation (ASA), where the combined correction is properly included. The scalar-relativistic SchrXodinger equation with the von Barth–Hedin exchange correlation (von Barth and Hedin, 1974) was solved. We considered the muEn-tin orbitals of angular momentum s, p and d for O and Ti atoms and we diagonalized a matrix of 54 × 54 order. The eigenvalues and eigenvectors were calculated on a grid of 75 k-points in the irreducible 1=16th Brillouin zone. The one-electron energy bands have been calculated in the high symmetry directions '–M –X –'–Z–R–X –M –A–Z (Fig. 1 from Khan et al., 1991). In the present paper, we call as valence band (VB) the ligand band (LB), which lies between −0:72 and −0:1625 Ry (Fig. 3.1), and the Drst conduction band (CB) is placed over the forbidden gap (3:06 eV). According to this band scheme, TiO2 is a semiconductor with an indirect forbidden gap of 3:06 eV. In the calculation of Kazowski and Tait (1979) they also obtained an indirect forbidden gap, but with another minimum of the CB. Although they used the same LMTO method, they however performed their calculation without (i) the combined correction, (ii) with a small number of basis functions, and lastly (iii) without the exchange correlation. Fig. 3.1 shows the total DOS per unit cell. In the Ti Wigner–Seitz spheres (WSS) the d symmetry is so dominant that the presence of s and p symmetries is almost unobserved, although non zero (Khan et al., 1991). The number of states of di1erent symmetries and of di1erent chemical origins in LB and CB have been also tabulated in Khan’s article. Actually, all bands correspond to mixed states and in each case all di1erent symmetries are present. The LB is mainly of O 2p origin (80%), but it also contains 13% Ti 3d and 7% of the other symmetries. The Drst CB is 75% Ti ‘d’ (t2g ), about 11% each of O p and O d, and it also contains a small amount of Ti p. The Ti t2g band is about 2:3 eV wide. Beyond the forbidden gap of 2:7 eV starting from the lower band, we obtain the next conduction band mainly of Ti eg origin. This band is split into two bands because of the crystal structure. The lower band (1 eV wide) is mainly of x2 –y2 symmetry of Ti origin. Beside our calculation, another LMTO calculation has been performed for TiO2 by Poumellec et al. (1991), exhibiting slight di1erences as compared to our results (see inset in Fig. 3.1): (i) the width of the VB is smaller, (ii) the indirect forbidden gap is reduced to 2 eV and (iii) there is no additional
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Fig. 3.1. Total DOS per unit cell in TiO2 compounds according to Khan et al. (1991). The insert shows the same result, according to Poumellec et al. (1991).
forbidden gap within the conduction bands. However, both band calculations are deduced from LMTO methods which include the exchange corrections. Nevertheless, in Khan’s calculation, the convergence was considered achieved when the experimental gap (3:06 eV) was obtained, since it is known that any band structure calculation for insulators in the LDA with exchange corrections gives a band gap which is too small in general (see Section 1.2.2). In order to recover a realistic band gap it would be necessary to use the X. Slater potential, with . ¿ 1 (Khan and Callaway, 1980). See also Finkelstein et al. (1999) for the application to TiO2 of a recent full potential LMTO method (FPLMTO). 3.1.2. Experimental valence band XPS and discussion Valence photoemission spectra (v-XPS) and ultraviolet photoelectron spectra (UPS) on TiO2 (rutile) have been measured by Tezuka et al. (1994, 1996) together with bremsstrahlung isochromat spectra. Although the experimental resolution of the BIS was not as good as that of Beaurepaire et al. (1993), nevertheless Tezuka et al. were able to estimate the magnitude of the fundamental bandgap from the joint v-XPS-BIS spectra of TiO2 (see Fig. 1 of Tezuka et al., 1994). The estimated magnitude is in good agreement with that obtained in Khan’s energy band calculation. Moreover, the v-XPS gives information about the density of states (DOS) of occupied levels. Fig. 3.2 reports the experimental UPS (excitation energy at 80 eV) and v-XPS spectra of TiO2 which have been compared to the preceding DOS curves shown in Fig. 3.1, convoluted with a Gaussian broadening function (due to experimental resolution) of width 1:0 eV. The convoluted DOS curve based on Poumelec’s result shows a two-peak proDle in rough agreement with the experimental spectra, whereas the DOS curve based on Khan’s calculation exhibits a wider width and a more complicate shape when compared to the
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Fig. 3.2. Experimental UPS and v-XPS spectra of TiO2 compared to the convoluted DOS values of Khan et al. (1991) and Poumellec et al. (1991) (adapted from Tezuka et al., 1994).
experiment. The origin of the discrepancy is probably related to the problem of convergence (and fundamental gap) in semiconductor calculations, as mentioned before. As far as the dispersion curves E(k) are concerned, the calculated valence-band structure of TiO2 has been compared to ARUPS data measured with synchrotron radiation from the (1 0 0) and (1 1 0) crystal surfaces (Hardman et al., 1994). A reasonable agreement was found between the experimental data and the calculations made by Glassford et al. (1990) and Poumellec et al. (1991).
3.2. Ti core level XPS spectra of TiO2 In this section, in order to mimic a rough band structure of TiO2 , we consider a TiO6 cluster localized model with an adequate crystalline geometry, which essentially provides a Dnite number of 2p states of oxygen, hybridized with the 3d orbitals at one Ti photoexcited site. However, this cluster impurity Anderson model (CIAM) allows us to extend the SIAM Hamiltonian given by Eq. (2.1) (Section 3.2.1) and to calculate the corresponding core level XPS spectra (Section 3.2.2).
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3.2.1. Cluster impurity Anderson model for TiO6 (Oh symmetry) The following analysis is based on a MO6 octahedral CIAM (Okada et al., 1992, 1994) with M = Ti. For the calculation of TM c-XPS, the Hamiltonian was given by the adaptation and extension of Eq. (2.1): H = HL + HTi + Hmix ; where HL =
';
HTi =
p a+ p' ap' ;
!
+
d d!+ d! +
2
+ 0d
!1 ;!2
Hmix =
!
(3.2) 2p p+ ! p! +
!1 ;!2 ;!3 ;!4
gpd (!1 ; !2 ; !3 ; !4 ) d!+1 d!2 p+ !3 p!4
1 g (!1 ; !2 ; !3 ; !4 )d!+1 d!2 d!+3 d!4 2 ! ;! ;! ;! dd 1
and
(3.1)
';
3
4
(˜l · ˜s)!1 ;!2 d!+1 d!2 + 0p
!1 ;!2
+ V (')(d' ap' + a+ p' d' ) :
(˜l · ˜s)!1 ;!2 p+ !1 p!2
(3.3)
(3.4)
Here vi denotes the combined indices representing the spin () and orbital (') states, and ' runs over t2g and eg states. HL describes the ligand molecular orbitals, which couple to the 3d orbitals through the hybridization Hmix . The HTi describes the atomic 3d and 2p electron states of the TM ion. In Eq. (3.3) the gpd and gdd symbols represent the Coulomb interaction between 2p and 3d states and that between 3d states, respectively, and include the Slater integrals in their explicit forms; 0d and 0p are the spin–orbit coupling parameters. The Hamiltonian H is diagonalized within the basis states in the space of 3d n , 3d n+1 L; 3d n+2 L2 conDgurations, where n = 0 for TiO2 and L represents a ligand hole. We have deDned the key parameters, ; Udd and Udc , as follows: the CT energy is given by E[d n+1 L] − E[d n ] = , and the Coulomb interaction Udd is deDned by E[d n+2 L2 ] − E[d n+1 L] = + Udd , where E[d n+2 L2 ]; E[d n+1 L] and E[d n ] represent the conDguration average energy of d n+2 L2 ; d n+1 L and d n , respectively. The 2p core hole potential −Udc (2p) is deDned by E[2pd n+1 L] − E[2pd n ] = − Udc (2p). The spectrum of Ti c-XPS is expressed as in Eq. (2.8): |f(XPS)|g|2 L(EB + Eg − Ef (XPS)) ; (3.5) F(EB ) = f
where |g and |f(XPS) represent the initial and Dnal states with energy Eg and Ef (XPS), respectively, and EB is the binding energy. 3.2.2. Core-level XPS We analyze Ti 2p, 3s and 3p XPS of TiO2 with the full multiplet TiO6 cluster model (3.1). The symmetry of the cluster is approximately taken as Oh , because the deviation from Oh
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Fig. 3.3. Experimental (a) and theoretical (b) and (c) results of Ti 2p-XPS of TiO2 . In (b) the e1ect of the covalency hybridization is taken into account, while it is disregarded in (c).
symmetry in TiO2 is too small to a1ect the XPS spectra. According to experimental data (Sen et al., 1976) displayed in Fig. 3.3a (see also Fig. 1.5b), Ti 2p XPS consists of two main peaks, i.e. 2p3=2 and 2p1=2 core levels, each of which is accompanied by a satellite about 13 eV above the main peak. With the same parameter values as those obtained in the analysis of Ti 1s pre-edge XAS (Kotani et al., 1995), we can fairly well reproduce the Ti 2p XPS spectrum. After a slight readjustment of parameters [V (eg ) = 3:0 eV; Udd = 4:0 eV; Udc = 6:0 eV), the calculated result (Okada et al., 1992) is shown in Fig. 3.3b. As an origin of the satellite, an exciton mechanism has been proposed (de Boer et al., 1984, Section 1.3.4), where a simultaneous excitation of O 2p to O 3s states gives rise to the satellite. However, in the present calculation the satellite is caused by the charge transfer (CT) e1ect due to the strong covalency hybridization. If we disregard the CT e1ect by putting V (eg ) = V (t2g ) = 0, the satellite vanishes as shown in Fig. 3.3c. Another e1ect of the hybridization is to make the 2p1=2 main peak broader than the 2p3=2 one. The broadening of the 2p1=2 peak is caused by the multiplet splitting which is induced by the hybridization, so that the peak is narrowed in Fig. 3.3c. These facts are an evidence of the strong covalency hybridization in TiO2 . In the present calculation, the ground state consists of 3d 0 (39.5%), 3d 1 L (48.2%) and 3d 2 L2 (12.3%) conDgurations, resulting in the average 3d electron number equal to 0.73. The calculations of Ti 3s and 3p XPS were made with the same parameter values as those used for 2p XPS, except that Udc , for the 3s and 3p core states is taken to be smaller by 1 eV than the 2p state. Furthermore, in the calculation of the Ti 3s-XPS, we have taken into account the intra-atomic conDguration interaction (CI) between 3s1 3p6 3d n and 3s2 3p4 3d n+1 conDgurations. The result (Okada et al., 1992) is shown in Fig. 3.4, and compared with the corresponding experimental data (dots) (Tezuka et al., 1994). The agreement between the theoretical and
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Fig. 3.4. Experimental (dots) and theoretical (solid curve) results of (a) Ti 3s-XPS and (b) Ti 3p-XPS of TiO2 .
experimental results is fairly good. The 13 eV satellite of Ti 3p-XPS is still caused by a CT e1ect, while the complicated satellite structure of Ti 3s-XPS seems to come from the combined CT and CI e1ects. From the systematic analysis so far given, we can conclude that in TiO2 the e1ective hybridization 1 Ve1 (∼7 eV) is larger than the CT energy (∼4 eV), so that the ground state has a strongly mixed valent nature. Besides, the Coulomb interaction Udd (=4–5 eV) is as large as the parameter. It is to be mentioned that the Ti 2p-XAS as well as the v-XPS-BIS spectra of TiO2 can also be reasonably reproduced with these parameter values, by using the preceding TiO6 cluster model (Beaurepaire et al., 1993). 3.3. A few insights about Ti2 O3 Let us now shortly comment on Ti2 O3 sesquioxides, which were considered as “MH insulators” with a nominal “Ti3+ (3d 1 )” ground state. However, no detailed analysis of the high-energy spectroscopic data had been made until the analysis made by Uozumi et al. (1996) and Tezuka et al. (1996). Experimentally, they observed the Ti 2p XPS spectrum of Ti2 O3 as well as the corresponding resonant photoemission (RPES) at the Ti 2p excitation threshold, and then they interpreted their results within a TiO6 CIAM. The satellite structure of the 2p XPS was assigned to a CT satellite, while the RPES was explained through the e1ect of a strong covalency hybridization. Finally, Uozumi et al. (1996, 1997) classiDed the Ti2 O3 compounds as intermediate-type insulators between CT and MH insulators as well as the V2 O3 ; Cr 2 O3 ; Mn2 O3 sesquioxides (whereas they classiDed Fe2 O3 compounds as CT insulators).
1
Ve1 is the e1ective hybridization strength between 2p3d 0 and 2p3d 1 L conDgurations (or between 3d 0 and 3d 1 L conDgurations in the ground state), which is given by [6{V (t2g )}2 + 4{V (eg )}2 ]1=2 ; 6 and 4 being the number of empty places of the t2g and eg states, respectively (see Eqs. (4.3) and (5.1)).
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4. Vanadium compounds Let us now consider the cases of V2 O5 , VO2 and V2 O3 compounds, studied with PES, BIS and c-XPS techniques (Section 4.1), and with resonant photoemission (Section 4.2). Section 4.3 will be devoted to the special examples of VC2 and VC6 cluster models. 4.1. BIS, valence band and core level XPS of V2 O5 , VO2 and V2 O3 Here we consider the binary oxides V2 O5 , VO2 and V2 O3 , of vanadium with cation valence 5, 4 and 3, respectively, corresponding to the formal valence shell conDguration 3d 0 , 3d 1 and 3d 2 . The V2 O5 compound (e.g. Shin et al., 1993) is a diamagnetic insulator with a gap of EG = 2 eV, while VO2 undergoes an insulator-to-metal transition with a sharp conductivity increase of almost Dve orders of magnitude at 340 K: its gap in the insulating low-temperature phase amounts to EG = 0:5 eV (see for example, Blaauw et al. (1975) and Sawatzky and Post (1979)). The V2 O3 compound crystallizes in the corundum structure and exhibits an insulator-to-metal transition with a 6 –7 orders of magnitude jump in the conductivity at 160–170 K and a gap of 0:2–0:3 eV in the insulating phase. Despite the many e1orts towards obtaining an understanding of their electronic behavior, the vanadium oxides still pose many open questions. V2 O3 for example has often been regarded as a prototype for the spin 12 -MH compounds with U ¡ (Hufner, 1985, 1994, 1995; Shin et al., 1995, 1996). The metallic and insulating behaviors as well as the metal-to-insulator transitions have been analyzed in terms of these parameters (U; ) and their relation to the bandwidths W and w of the O 2p and TM 3d bands. However, the nature of the phase transition in V2 O3 and the inJuence of the antiferromagnetic ordering on it are still a matter of controversy (Thomas et al., 1994; Ezhov et al., 1999). VO2 is nominally a 3d 1 system (like Ti2 O3 ), while TiO2 (Section 3.2) is a nominally 3d 0 . When one goes from TiO2 to VO2 , one expects a decrease of the parameter in analogy with the behavior of late TMCs (Bocquet et al., 1992, 1996), while the change of other parameters should be much smaller. To calculate the PES (or v-XPS) it is possible to adjust Eq. (3.1), by considering the hole in the VB and not in the core level (Udc = 0), and using a similar VO6 cluster system (Okada and Kotani, 1993; Beaurepaire et al., 1993), as in the case of TiO6 . BIS spectra can also be calculated with the same scheme (always Udc = 0). In the Dnal state of BIS, when an incident electron has been absorbed into a 3d orbital with the emission of a photon, the ground state |g turns into the Dnal states f(BIS) with energies {Ef (BIS)} and the BIS spectrum is expressed as a summation over the f(BIS) states: + FBIS (E) = f(BIS)|d' |gL(E − Ef (BIS) + Eg ) ; (4.1) f
'
where the Lorentzian includes the spectral broadening formed by the lifetime of the Dnal states and the experimental resolution. In the case of VO2 , PES and BIS have been calculated for = 2 eV, and the same parameters used for TiO2 compounds. The result (Fig. 4.1) is in fairly good agreement with the available experimental data (Uozumi et al., 1993). In Fig. 4.1, the calculated PES essentially consists of a broad VB which mainly originates from the O 2p character. On the other hand, the calculated BIS shows a two peak structure which mainly correspond to the Ti t2g and eg orbitals. In addition to the experimental PES spectrum of VO2
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Fig. 4.1. Photoemission spectrum (PES) and bremsstrahlung spectrum (BIS) calculated for VO2 (adapted from Uozumi et al., 1993).
Fig. 4.2. VB-PES for VO2 compounds. The sum of the partial density components V2p-SXES and O1s-SXES gives the PES spectrum (adapted from Shin et al., 1998).
the experimental SXES spectrum compound is shown in Fig. 4.2, where the hybridization of O2p and V3d band is demonstrated experimentally. The VB-PES of VO2 consists mainly of O2p and V3d components: the O2p component, with two structures as calculated above, is dominant at higher binding energy and the V3d component is located just below the Fermi level (see also Fig. 4.1). The PES spectrum can be resolved into the V2p and O1s components
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Fig. 4.3. Experimental (dots) core-level V2p-XPS photoemission for V2 O3 compounds: in the theoretical spectrum (full line) two CT satellites are obtained, to which only one is visible in the experiment (S) (adapted from Uozumi et al., 1998).
by SXES: the V2p emission reJects the V3d partial density of states and the O1s emission reJects the O2p partial density of states (see the diagram of levels). One sees that the band B of Fig. 4.2 has a strong 3d component in the O2p band; on the other hand, there is also an intensity of the O2p component in the 3d band at the Fermi level. Thus, these results indicate how the two components are hybridized with each other (Shin et al., 1998). A systematic XPS study of the core levels within the CIAM framework has been recently performed for many early TMOs (Uozumi et al., 1993, 1997). In the case of V 2p-XPS of V2 O3 (3d 2 conDguration) the satellite structure related to the V 2p1=2 feature is not easily observed (Fig. 4.3) because of the superposition of a sharp O1s peak occurring at about the same excitation energy (Sawatzky and Post, 1979). The cluster model predictions for the metal 2p core-level XPS spectra of V oxides have been obtained by varying the parameter basis of the Ti oxide case in order to provide a good Dt to the experimental data. For example, Fig. 4.3 shows the calculated V2p-XPS and the experimental data in V2 O3 : the 2p1=2 and 2p3=2 peaks due to spin–orbit splitting of the 2p core-level are observed in the spectrum together with the O1s peak. In addition to the main peaks, satellite structures indicated by CT are seen in the calculated spectra. The satellites originate from the CT mechanism, which is similar as that in 2p-XPS of the late TMCs. The satellite associated with the 2p3=2 peak can be seen in the experimental data, whereas the satellite associated with the 2p1=2 peak is obscured by the O1s line as already mentioned. Both main peak and satellite are strongly mixed states between cd n and cd n+1 L conDgurations, as in the case of Ti2 O3 (and Cr 2 O3 compounds). As a characteristic feature of the experimental data of the TM 2p-XPS (Ti to Fe sesquioxides), one can see that the energy separation ES between the main peak and the corresponding CT satellite systematically changes from about 12 eV for Ti2 O3 to 6:5 eV for Fe2 O3 . This variation in ES is mainly caused by the p–d e1ective hybridization energy Ve1 : Es = |( − Udc )2 − 4(Ve1 )2 |1=2 ≈2Ve1 ;
(4.2)
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Fig. 4.4. VB-XPS spectra of V2 O3 ; VO2 and V2 O5 at room temperature, showing the O2p and V3d contributions (adapted from Zimmermann et al., 1998, 1999).
where Ve1 is deDned by Ve1 = [N (eg ){V (eg )}2 + N (t2g ){V (t2g )}2 ]1=2 :
(4.3)
Since V (eg ) itself decreases monotonically from 3:0 eV for Ti2 O3 to 1:9 eV for Fe2 O3 and N (t2g ) increases from 9 for Ti2 O3 to 5 for Fe2 O3 , Ve1 systematically decreases from 6:9 eV for Ti2 O3 to 3:2 eV for Fe2 O3 . It is important to note that Ve1 is rather large in early TMCs so as to give in Eq. (4.2) ES ≈2Ve1 , because | − Udc |Ve1 in most cases. In this way the systematic change in ES (main peak-satellite distance) is mainly caused by the hybridization energy Ve1 (Uozumi et al., 1997). Finally, an experimental comparison of the VB-XPS spectra of V2 O3 , VO2 and V2 O5 compounds is presented in Fig. 4.4. The VB are separated into a main part, covering the binding energy range from 3 to 9 eV and a separate small peak near the Fermi level, which is absent for V2 O5 , and then grows in intensity with formal d 1 and d 2 occupations in the other two oxides. Therefore, it would be tempting to interpret the VB spectra in these oxides in terms of a simple ionic picture, namely to assign the main part to the Dlled O2p shell and the structure at low binding energy to the increasing number of V3d electrons. However, it has been shown that there is a considerable hybridization which mixes the orbital character of these spectral features and modiDes the electronic properties in these oxides (Zimmermann et al., 1998, 1999). 4.2. Resonant photoemission Some RPES results will be now recalled in order to better understand the electronic properties of this interesting class of materials. RPES from atomic 3d states occurs when the energy of the ionizing photons coincides with the 3p–3d optical absorption threshold. As the photon energy is
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changed from below to above the threshold, the 3d photoelectron yield is observed to increase dramatically. This phenomenon has been extensively investigated in both atoms and solids and comprehensive reviews have been published about a decade ago (Davis, 1986; Parlebas et al., 1990). In large part, these studies have been concentrated on the phenomenon in the rare-earth metals and heavy transition metals (Z¿25, where Z is the atomic number). Far fewer studies of the resonance in the light transition metals have been undertaken. It was however remarked that the 3p–3d PES resonance in the light TMs, such as Ti, V and Cr, is very di1erent from that of heavier TMs (Bethel et al., 1983; Barth et al., 1985). For example, the resonance proDles (plots of the emission intensity versus photon energy) in the lighter metals do not display the characteristic Fano-type line shape observed for the heavier ones. These resonance were found to extend over 40–50 eV in photon energy as opposed to approximately 10 eV in the heavier metals. These studies have been also undertaken with the hope of using the resonance e1ect to identify the extent of d-state hybridization in the O 2p peaks in oxides of these metals. The enhancement of the 3d photoabsorption yield at photon energies close to the 3p–3d optical absorption threshold can be written in an atomic picture as 3p6 3d n + h! → 3p6 3d n−1 + e− ( ) ;
(4.4)
where a photon of energy h! ionizes the neutral atom and eject a 3d electron with kinetic energy , 3p optical absorption can be written as 3p6 3d n + h! → [3p6 3d n+1 ]∗ ;
(4.5)
where the photon energy h! is close to the 3p threshold and the asterisk denotes an excited state. The [3p6 3d n+1 ]∗ state consists of a hole in the 3p shell of a metal ion and an extra electron in the d shell of the same atom. This excited state can de-excite by a number of mechanisms. One such mechanism is the autoionization, [3p6 3d n+1 ]∗ → [3p6 3d n ]∗ + e− ( ) ;
(4.6)
followed by a super-Coster–Kronig (SCK) Auger decay, [3p6 3d n ]∗ → 3p6 3d n−2 + e− ( ) ;
(4.7)
where is the kinetic energy of the d electron emitted in the M23 M45 M45 Auger event. Since the energies of the emitted electron in this decay channel are di1erent from the energies of the photoemitted electron, no resonance occurs. However, in competition with this mechanism there is a direct recombination, [3p6 3d n+1 ]∗ → 3p6 3d n−1 + e− ( ) :
(4.8)
Here the Dnal state and the electron kinetic energy are identical to those following the conventional 3d photoemission of Eq. (4.4). Thus, the process of direct recombination can interfere with the 3d photoemission process and produces a characteristic narrow, asymmetrical line shape (the Fano line shape) for the resonance proDle (Fano, 1961). RPES experiments in VO2 , V2 O3 and V2 O5 have shown that both O 2p and V 3d bands present resonant enhancement through the vanadium 3p–3d threshold (Smith and Henrich, 1988; Zimmermann et al., 1998). For example, Smith and Henrich observed in Ti2 O3 and V2 O3 a strong angular anisotropy, probably related to the molecular-orbital structure of these oxides. Also the maximum in the cation 3d
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resonant emission occurred at the same photon energy in both metals and oxides, indicating a signiDcant degree of localization of the cation resonance. These results, together with the RPES determination of the VB electronic structure of Cr 2 O3 , were found similar to those observed in Ti2 O3 and V2 O3 , so that these authors claimed that all the compounds should be considered MH insulators (Li et al., 1992). Moreover, vacuum ultraviolet (VUV) reJectance and PES of V2 O5 , VO2 and V2 O3 have been measured to investigate the 3d-band structure, the electron correlation e1ects (Shin et al., 1990) and the metal–insulator phase transitions. Since the DOS at the Fermi level in V2 O3 was found to be rather low, even in the metallic phase, it was concluded that electron correlation e1ects were important in these oxides and it was thereby suggested that a Mott-type metal–insulator transition is induced in V2 O3 . However, recent vand c-XPS results (Bocquet et al., 1996; Zimmermann et al., 1998) have shown with a cluster model calculation that a considerable hybridization mixes the orbital character of the spectral features of the three vanadium oxides. These results reveal the importance of the covalency in vanadium oxides and seem thus to demand a reconsideration of the standard classiDcation of these oxides as simple MH compounds. 4.3. A related problem: VCn clusters for V adsorption on graphite In Section 4.3.1 we recall a preliminary study (Parlebas et al., 1997) for the calculation of 3d-metal core 3s-XPS spectra in the case of the dilute limit of vanadium adsorption on graphite, i.e. by considering only one 3d-metal atom adsorbed at a bridge position on the graphite surface (0001) and irradiated by an incident X-ray photon of energy h!. In Section 4.3.2 we then present various extensions shown in Section 4.3.1 of the preliminary calculation made by Kruger et al. (1997, 1999). 4.3.1. Bridge position in VC2 and appearance of localized magnetic moments Historically, what prompted us to perform the present calculation was the result of Binns et al. (1992). These authors found a satellite structure in the vanadium 3s-XPS which was ascribed to the presence of magnetic moments on V surface atoms of an islanded V Dlm. However, as far as the appearance of magnetism is concerned for a vanadium condensed system, it is well known that the interactions between nearest neighbors of vanadium atoms in general hinders the onset of localized magnetic moments (Parlebas and Gautier, 1973). Thus, in the present model, we only envision the most favorable case for V magnetism i.e. a single V atom, adsorbed on graphite at a bridge position, as it will become clear afterwards. For this reason we extend the SIAM and take into account both the Udc core-hole screening and the Jdc exchange term between the spins of the core electrons and 3d electrons in the Dnal state of the c-XPS process. Actually, the c-XPS spectra strongly reJect many-body e1ects arising from the outer electrons of an irradiated material. When a 3s core hole is produced at a 3d transition metal atom by an incident photon, the outer electron system is perturbed by the core hole charge and screens it: Udc is more precisely the Coulomb integral between a 3d and a 3s electron as already deDned in the preceding sections. In this process the core hole plays the role of a “test charge” which induces a many-body dielectric response of the outer electron system. However, since the 3s core hole has not only a charge but also a spin, it can also play the role of a “test spin”, thereby inducing a magnetic response of the outer electron system mediated by the exchange integral Jdc . In order
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to complete Eq. (2.1) and in addition to Udc already present, we had to keep Udd , the Coulomb integral between two d electrons, originally introduced in the pioneer work of Anderson (1961) and to include the corresponding exchange integrals Jdd and Jdc not yet included in Eq. (2.1). As usual, Vkm labels the hybridization term between a |dm state among the Dve 3d orbitals of a given spin and a |k substrate band state of same spin. For simplicity, in this preliminary study, we only took two “band” orbitals to describe the 2pz band of graphite in the Fermi energy region: one below (k = ) and one above (k = ∗ ) the Fermi level, representing the bonding (ligand) and antibonding (conduction) parts of the graphite -band. In our Hamiltonian, the two “band” orbitals are not coupled directly but rather through the transition-metal 3d orbitals by the hybridization Vkm . This simplest model is equivalent to considering only two carbon atoms (2sz spin orbitals) like a diatomic C2 molecule, hybridized with a vanadium atom (5 spin d orbitals) just sitting at a bridge position upon a diatomic molecule and building a VC2 cluster. Also, in this preliminary calculation, we limit ourselves to the case of a nominally 3d 3 adsorbed vanadium atom, i.e. we focus on the divalent atomic conDguration 3d 3 4s2 . Altogether we got 5 electrons (3 “d” and 2 “pz ”) which are ascribed to our considered VC2 cluster in the initial state (ground state) of the core level photoemission process, when the core orbital is still Dlled. Then we diagonalized the Hamiltonian in the subspace |d 3 ; |d 4 ; |d 2 ∗ and |d 3 ∗ , where means a hole in the ligand orbital. In the Dnal state of the photoemission process, where a 3s core hole c had just been created, we took the subspace |cd 3 ; |cd 4 ; |cd 2 ∗ and |cd 3 ∗ . The charge transfer energies, = E(d 4 ) − E(d 3 ) and ∗ = E(d 2 ∗ ) − E(d 3 ) are deDned between |d 3 and |d 4 , |d 2 ∗ , respectively. Our parameters values (Parlebas et al., 1997) appear in the caption of Fig. 4.5. Comparable values for the Coulomb integrals Udd and Udc had already been used within similar calculations on V sesquioxides V2 O3 (Section 4.1). The low spin conDguration (in Fig. 4.5, Sd = 1=2) shows essentially a two peak structure with a small charge transfer satellite at about 7 eV from the main line. In this case, the exchange splitting cannot be resolved. On the contrary, in the high spin conDguration (Fig. 4.5, Sd = 3=2) the exchange splitting is quite large and gives rise to a big exchange satellite at about 4 eV from the main line. Also this last feature of the high spin conDguration is not very sensitive neither to the charge transfer energies and ∗ , nor to the hybridization strength Vkm = Vd = Vd∗ . As a matter of fact, already within the present preliminary model, the spectrum of the low spin conDguration Dts much better the experimental data (small satellite) of Binns et al. (1992) than the high spin conDguration. Our preliminary conclusion was that the satellite in V 3s-XPS spectra of dilute V upon graphite (VC2 cluster) was most likely due to charge transfer, i.e. core hole screening, rather than to localized magnetism on the V atom. This does not exclude the presence of a small V magnetic moment. The same conclusion was already suggested in a very crude model with no exchange interaction, no d orbital degeneracy and with a Dlled d band approximation (Rakotomakevitra et al., 1995). The present model is now systematically extended, in the following subsection, to various geometrical positions of the adsorbed atom. Special emphasis is put on the study of a VC6 cluster. 4.3.2. Hollow position in VC6 (C6v symmetry) and other positions First of all, let us consider in detail the “hollow position” (Kruger et al., 1997) where the adatom sits above the center of a C6 ring at a distance h from it (C6v symmetry). The
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Fig. 4.5. Calculated 3s-XPS spectra of V upon graphite i.e. a VC2 cluster for (a) low (1=2) and (b) high (3=2) V 3d spin conDgurations, using a phenomenological model and the following parameters (eV): Udd = = ∗ = 5; Vd = Vd∗ = ' = 2; Udc = 7; Jdc = 2:5; Jdd = 0:4; ∗ − = 5; ' characterizes the spectral broadening due to Dnite lifetime of the core hole and experimental resolution (from Parlebas et al., 1997).
C-cluster Hamiltonian was diagonalized by constructing appropriate molecular orbitals (MOs) which transform as the irreducible representations of the symmetry group (Fig. 4.6). The energies of the various six carbon MOs are expressed in terms of the Slater–Koster integral pp between nearest-neighbor C atoms. Following Priester et al. (1982), the following choice was made: pp = −3:33 eV, which led to a reasonably good description of the -band structure of graphite; TM d orbitals and C MOs were then coupled through hybridization, which vanished for certain symmetry (Fig. 4.6). By the construction of symmetrical C MOs, the number of non-vanishing hybridization integrals (between the TM atom and the C cluster) was minimized. In contrast to our preliminary model (Section 4.3.1), the hybridization integrals Vkm now depend on the adatom–surface distance h, essentially through the Slater–Koster integrals pd and pd between V and C atoms, which themselves are functions of h (Kruger et al., 1997). Because of d–d correlation, the one-electron d of the TM atom has no direct physical meaning. Rather
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Fig. 4.6. Scheme of the Hamiltonian when the adatom sits at a hollow position corresponding to a VC6 cluster. Thick horizontal lines represent orbitals: transition-metal core and d orbitals on the left and C cluster (molecular) orbitals on the right. In the atomic limit (V = 0) the lower half ( ¡ p ) is double occupied, the upper half ( ¿ p ) empty. The order of the d orbitals is 3z2 –r 2 ; yz; zx; xy, x2 –y2 . Thin solid lines indicate non-zero hybridization integrals. Dotted lines symbolize electron correlation (from KrXuger et al., 1977).
than d , we therefore again specify the quantity = E(d n+1 p ) − E(d n ) = d + nUdd − p , i.e., the charge-transfer energy for moving one electron from the center of the carbon MO spectrum ( p = F ) to the TM d orbitals. Here n is the number of d electrons in the atomic limit (V = 0). Since each Cpz orbitals adds one electron, the VC6 cluster contains n + 6 electrons altogether. For clarity we give below the explicit form of the Hamiltonian which incorporates Jdd and Jdc into Eq. (2.1). Again H = HL + HTM + Hmix ; where
(4.9)
k a+ k ak ; k HTM = d a+ a+ m am + c c ac
HL =
m
+ Udd
m¡m
+ Udc
m
and Hmix =
km
(4.10)
+ a+ m am am am − Jdd
+ a+ m am ac ac − Jdc
(Vkm (h)a+ k am + h:c:) :
Sm · Sm
m¡m
Sm · Sc
(4.11)
m
(4.12)
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Fig. 4.7. Calculated V 3s-XPS spectra as a function of h for n = 3 and = 2:5; 5:0 eV. Both values of correspond to small charge transfer between V and C6 (hollow position) (from Kruger et al., 1997).
Here c, m and k refer to the 3s core orbital, the Dve 3d orbitals and the cluster molecular + and a+ are the corresponding creation operators; labels the orbitals respectively; a+ c ; am k 1 two spin directions. Sm = 2 a+ m am is the spin operator for electrons in orbital m. Here are the matrix elements of the Pauli spin matrices = (x ; y ; z ). Jdd and Jdc are the exchange integrals between two d electrons and between a d and a core electron, respectively. As in the preceding subsection the Hamiltonian was diagonalized in the conDguration– interaction approach; the values for Udd , Jdd , Udc and Jdc were kept the same as before (Parlebas et al., 1997), but now the h-dependent hybridization is fully calculated (Kruger et al., 1997). Recently Peng et al. (1996) determined the equilibrium position of a single non-magnetic Fe atom adsorbed on the graphite surface by ab initio “full potential linearized augmented plane wave” (FLAPW) total energy calculation. They actually considered an extremely dilute Fe monolayer. W above the surface was found to be the stable position. With this value The hollow site 1:6 A W (Fig. 4.7). In the as a rough estimation for vanadium, h was varied between 1.0 and 2:5 A atomic state, vanadium has, as in the preceding subsection (Section 4:2:1), a 3d 3 4s2 conDguration. We shall take n = 3 in Fig. 4.7 which leads to the correct atomic limit (V → 0 ⇔ h → ∞). For small heights, i.e. strong hybridizations, the system has a low spin behavior with a total W for = 2:5 eV, the system d-spin Sd = −7=27B = 0:6. At a critical h, which is about 1:5 A switches to a high-spin state with Sd around 3=2. The spin transition is a consequence of the competition between magnetic and chemical energies, i.e. adsorption energy. Also there is a strong correlation between Sd and the occupancies of the |xy, |x2 –y2 and |3z 2 –r 2 orbitals. The magnetic transition can be phenomenologically explained using an atomic model which includes a crystal Deld term (Kruger et al., 1997). In Fig. 4.7, the calculated XPS spectra are shown as a function of h, for = 2:5 and 5:0 eV in the case of n = 3. The low-to-high spin
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transition occurs when the vanadium adatom moves away from the C6 surface. The low-spin spectra (low h) are characterized by a main line and a small satellite which is due to Dnal-state charge transfer as a result of core hole screening; the exchange splitting cannot be resolved since it is too small. In the high-spin regime (high h) the spectral shapes depend strongly on the parameter . For = 5:0 eV they are very atomic-like with a simple exchange splitting of Jdc (Sd + 1=2) = 5 eV; Dnal-state charge-transfer features are very small. For = 2:5 eV, however, the spectra are more complex, showing essentially a three-peak structure. No clear distinction between the exchange-split main line and charge-transfer satellite can be made. Again if we compare these spectra with the experimental ones by Binns et al. (1992), we Dnd the best agreement in the low-spin regime close to the transition point, although the main-line in the satellite splitting is a little larger here than in the experiment. Quite recently Kruger et al. (1999) have extended the preceding study to the whole 3d-series selfconsistently for the three characteristic adsorption sites: hollow, on-top and bridge. In addition to the previously considered work of Peng et al. (1996), on the same subject, at least for W we should also an Fe adatom, which was shown to be stable at the hollow site with h = 1:6 A, recall two other calculations in the Deld. Montero et al. (1994) compared the stability of the di1ering adsorption sites for a V atom on graphite by means of electronic structure calculations using the semi-empirical complete neglect of di1erential overlap (CNDO) method; they found that the stable position of the V adatom is the hollow site with an adatom–surface distance h W and that bridge and on-top sites are less stable. Also Du1y et al. (1998) between 1.7 and 1:8 A, calculated the stable positions and magnetic moments of 3d-TM adatoms on graphite, using a linear combination of atomic orbitals (LCAO) molecular approach within density functional theory (DFT): they obtained that the early 3d-TM elements up to Cr and Mn (thus including W while the late 3d-TM elements Fe, Co, Ni occupy V) occupy the on-top position with h = 2:1 A, W the hollow site with h = 1:5 A. Moreover, for all the 3d elements, except Mn, the d-charge of the adatom was found closer to that of a monovalent ion, i.e. 3d n+1 4s1 rather than of a divalent ion i.e., 3d n 4s2 , as given by the free atom electronic conDguration. However the Sd spin values reported by Du1y et al. (1998) were smaller by an amount of 0.2– 0.5, as compared to the Drst Hund’s rule results applied to the monovalent 3d-ions, i.e. Sd = |no − 5|=2 with no = n + 1. Again, Mn is an exception, with Sd = 2:3, which is by 0.3 larger than the corresponding Hund’s rule for a 3d 6 ion. Now, taking the preceding equilibrium positions and related h values, Kruger et al. (1999) found that the various 3d-adatoms along the series are each in a high spin (Hund’s rule) ground state. Upon decreasing h, however, they observed in some cases, a Drst order spin transition to a low spin state. These spin transitions are accompanied by electronic transitions between di1ering d-orbitals and can be explained in an e1ective crystal Deld model. The photoemission spectra obtained by Kruger et al. (1999) show intra-atomic exchange splitting as well as charge transfer satellites from Dnal state core hole screening by graphite -electrons; the intensity of the charge transfer satellites increases with no , the d-electrons number, which reJects the increasing electron-electron correlation, from light to heavy 3d-elements. Finally, we again address the comparison with the experimental data of 3s-XPS of V clusters on graphite reported by Binns et al. (1992). In the beginning of the present subsection we showed that these experimental spectra can be interpreted by the V 3d adatom (hollow site) being in the low-spin regime; the observed satellite is then due to the charge transfer e1ect rather than to a magnetic moment on V. This seems to contradict recent results of Kruger et al. (1999),
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where a large magnetic moment of 47B was found for the (monovalent) V adatom at the on-top W It must be kept in mind, however, that our model deals with equilibrium-position with h=2:1 A. a single adatom while experiments were carried out on clusters with a size of a few nanometers. Finally, we believe that a single V adatom on graphite may be in a high spin state in agreement with Du1y et al. (1998), but that the V clusters of Binns et al. (1992, 1999) are in a low spin state (or in an antiferromagnetic state) due to later-atomic d–d electron hopping. We hope that this issue will be clariDed by new experimental and theoretical studies in near future. 5. Chromium compounds The most controversial topics in TMCs, i.e. the character of the valence and conduction electron states and the origin of band gaps, have been also studied and discussed for di1erent types of chromium compounds (mainly oxides and halides). In this section we brieJy review a few valence band XPS and BIS results for chromium compounds (Section 5.1); then we present some corresponding core level XPS spectra (Section 5.2). 5.1. Valence band XPS and BIS Metallic CrO2 , with a Curie temperature Tc ∼390 K is the only ferromagnet of this class. Schwarz (1986) used the LSDA band theory to predict that the spin moment would be the full 27B required by Hund’s rules for the Cr 4+ (3d 2 ) ion. The Fermi level lies in a partly Dlled band for the majority (up-spin) electrons, but for minority (down-spin) electrons it lies in the semiconducting gap, which separates the Dlled oxygen 2p levels from the chromium 3d levels. This behavior has been named “half-metallic” by de Groot et al. (1983). Recent LSDA band calculations (Lewis et al., 1997) agree with the Schwarz’s band structure and provide an analysis of the transport measurements using band theoretical parameters. Resistivity measurements have given an electron mean free path, which is only a few angstroms at 600 K, with the resistivity always rising as the temperature increases (Lewis et al., 1997). The electronic structure of CrO2 has been also studied by PES and by speciDc heat measurements (Tsujioka et al., 1997). The Cr 3d band shows a splitting into a upper and lower Hubbard bands, with a small but Dnite density of states at the Fermi level, consistent with the metallic behavior. Lewis et al. (1997) reported a combined plot of UPS and BIS spectra of CrO2 compounds which they compared with theoretical spectra deduced from LSDA and LSDA + U band structure calculations (Fig. 5.1). One can notice that the large splitting (∼5 eV) between the prominent Cr 3d peaks is closer to the value (∼4:5 eV) obtained from the LSDA+U calculation rather than that (∼1:6 eV) obtained from the LSDA calculation. The underestimate of the splitting in the LSDA calculation is due to the neglect of the Hubbard splitting of the 3d band by ∼U . This quantity is estimated to be ∼ 3:4 eV, close to the value used in the LSDA + U calculation, which predicts an insulating U= gap only for U ¿6 eV. In conclusion, these recent results put CrO2 into the category of the “bad metals”, together with the high-Tc superconductors, the high-T metallic phase of VO2 and the ferromagnet SrRuO3 . Either for insulating Cr oxides, such as Cr 2 O3 or for halides of chromium, such as CrX3 (X=Cl, Br), the self-consistent one-electron band-structure description breaks down, because of
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Fig. 5.1. Combined UPS and BIS spectra for CrO2 compounds: experimental data (thick line); theoretical results from LSDA (dotted line) and LSDA + U (thin line) (adapted from Lewis et al., 1997).
the strong electron correlations (Antoci and Mihich, 1978). As for the origin of the bandgaps (EG ), Ronda et al. (1987) pointed out that the most reliable methods for determining EG are a comparison of direct and inverse PES, or better, photoconductivity measurements. CrCl3 and CrBr 3 compounds are magnetic insulators with an energy gap of about 2.3 and 2:1 eV, respectively, and an ionicity parameter fiDT ≈0:80 (Pollini, 1998). This parameter was calculated within the framework of Phillips theory (Phillips and van Vechten, 1969), earlier applied to divalent TMCs with open shell conDguration (Thomas and Pollini, 1985). The low temperature structure of CrX3 is like that of layered BiI3 crystals. CrCl3 has a hexagonal unit cell with 3 (monoclinic with four forW and c = 17:47 A W below 238 K. The space group is C2h a = 5:952 A 3 mula units per unit cell) above 240 K and C2i (rhombohedral with two formula units per unit W and c = 18:20 A W cell) at lower temperatures. CrBr 3 has a hexagonal unit cell with a = 6:26 A 3 and its low temperature structure has the space group C2h . Valence XPS results are reported in Fig. 5.2 for layered CrCl3 and CrBr 3 crystals and have been compared to the experimental v-XPS (and BIS) spectra of Cr 2 O3 insulating compounds (Uozumi et al., 1997). These spectra are rather similar and can be divided into the main band (1–10 eV) and the satellite region (10 –15 eV). A weak satellite feature was Drst detected in the VB-XPS spectrum of a high quality CrCl3 crystal, but was not considered in the earlier discussion of the data so that the main emission band was simply assigned to Cr 3d 2 (poorly screened) Dnal states (Pollini, 1994). In Cr 2 O3 , the main peak and satellite are largely separated because of the strong p–d hybridization strength with an energy distance Es given by Eq. (4.2). In CrX3 , the main Cr 3d structure, partly mixed with the nearby Cl 3p and Br 4p bands, is located at binding energies around 3 eV and the Dnal state peak position for a BIS experiment has been indicated in Fig. 5.2. The BIS position has been extrapolated by inspection of vacuum ultraviolet reJectance (Fig. 5.3) and EELS data measured on CrCl3 and CrBr 3 crystals (Carricaburu et al., 1986; Pollini et al., 1989) and by taking into consideration the various
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Fig. 5.2. Valence XPS spectra for layered CrCl3 (a) and CrBr 3 (b) compounds (from Pollini, 1999).
Fig. 5.3. Vacuum ultraviolet reJectance for CrCl3 (a) and CrBr 3 (b) compounds (from Carricaburu et al., 1986).
electronic processes discussed in these experiments. The PES and BIS peaks give an indication of the Cr 3d 2 and Cr 3d 4 energy level positions (Hubbard bands), whose separation yields the Coulomb energy U (=Udd ) to zero order both in CrBr 3 and CrCl3 (U =3:6 and 4:6 eV). Energy corrections due to p–d hybridization reduce then the solid state value U and the value of atomic origin Ubare , which has been determined by estimating approximately the hybridization shifts p–d in Cr halides. In Cr 2 O3 , the satellite feature (S) has an energy separation (ES ) from the main line given by ES ≈11:5 eV, while in CrCl3 and CrBr 3 , the weak satellite structures detected
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Table 5.1 Solid state parameters (eV) of early transition metal compounds rough estimated through the analysis of 2p-XPS and v-XPS, UV reJectance and EELS spectraa Compounds
dn
U (∗ )
V
Ve1
ES
EG
References
Ti2 O3 V 2 O3
d1 d2
3.0 3.9 2.3 3.5
6.9 6.8 6.5 7.6
2.5
d3
11.1–11.5
2.6 –2.8
CrCl3
d3
1.7
3.6
7.9
2.3
Bocquet et al. (1996) Bocquet et al. (1996) Zimmermann et al. (1999) Uozumi et al. (1997) Bocquet et al. (1996) Pollini (1999)
CrBr 3 MnO
d3 d5
5.5 6.5
4.5 4.0 3.8 5.2 5.5 4.0 3.5 3.5 7.0
12.0 12.0
Cr 2 O3
6.0 4.0 5.5 5.2 5.5 6.5
1.7 1.9
3.6 4.2
7.7 10.5
2.1 4.5 – 6.0
Pollini (1999) Park et al. (1988)
a
Ue1 = U − 9: the e1ective U1 can be reduced considerably from the bare U by an hybridization shift 9, since the states near the energy gap can have a large ligand p character.
in VB-XPS have a distance from the main emission of ES ≈7:9 eV and ≈7:7 eV, respectively. The satellite origin in the spectra of CrX3 is due, in the spirit of the CI cluster model, to the e1ective hybridization Ve1 between the 3d 2 and 3d 3 L Dnal states Ve1 = [(6 − n)(Tt2g )2 + 4(Teg )2 ]1=2 ;
(5.1)
where Tt2g and Teg are the one-electron mixing matrix elements: Teg = eg |H |eg = V and Tt2g = t2g |H |t2g = 0:5 V. In the simple case, when the Hamiltonian is restricted to d n and d n+1 L
conDgurations, the main peak–satellite splitting ES in VB-XPS is given by Eq. (4.2). In many early TMCs the parameters , U and Ve1 have comparable values and the e1ect of covalency is important in the properties of the ground and Drst ionization state, suggesting that the hole introduced in the ionization state is spread over the neighbouring ligand sites. However, the e1ective hybridization parameter in CrX3 has a value of Ve1 ∼ 3:7 eV, which is slightly lower than in early TMCs, where Ve1 = 6–9 eV (Bocquet et al., 1996; Uozumi et al., 1997). Thus the satellite structure may be assigned to slightly mixed 3d 3 L and 3d 2 conDgurations. Also the main Cr 3d emission, formerly assigned to poorly screened 3d 2 Dnal hole-state energies (¿U ), is in fact weakly screened by a ligand electron transferred into the 3d Dnal state conDguration. In this case, however, the Coulomb interaction, due to the 3d valence hole, does not seem strong enough to widely separate the di1erent Dnal state conDgurations and only a CM calculation could predict their relative weights quantitatively. In Table 5.1, the cluster model parameters for Cr trihalides (Pollini, 2000) and some typical early TMOs have been listed. We remark that Ti2 O3 and V2 O3 may be classiDed as MH compounds (Bocquet et al., 1996; Saitoh et al., 1995), while Cr 2 O3 seems an intermediate compound. In the case of CrCl3 and CrBr 3 insulators, we have U ¡, i.e. the Coulomb repulsion energy U is lower than the CT energy, but larger than the dispersional 3d bandwidth w (0:5 eV); the e1ective hybridization Ve1 (3.7 eV) is of the same order of magnitude of U and . It is also worth noting that the V values here found (1.7–1:8 eV) for CrX3 almost agree with the value (V ∼1:5 eV) earlier estimated with the spectroscopic model (Pollini, 1998) and that, besides, they are consistent with the values
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Fig. 5.4. Charge transfer versus Mott–Hubbard classiDcation in terms of (; U ) (adapted from Fujimori et al., 1987).
of many early TMOs. Thus, although it is possible that Cr halides might have an intermediate character as some early TMOs, one can however see from Fig. 5.4 that the ionic Cr trihalides are inside the MH region (large values of ) and located close to MH insulators, such as TiO and V2 O3 (Fujimori et al., 1987) (Fig. 5.4). The analysis of the valence band spectrum shows that the satellite–main peak distance is governed by the e1ective hybridization, which may inJuence the classiDcation scheme of light TMCs. However, the ionicity parameter of CrX3 points out that the calculated p–d hybridization energy in CrCl3 and CrBr 3 is not as large as in early TMOs. Also the importance of the correlation energy U in determining the insulating nature of CrX3 is conDrmed. Nevertheless, the correct picture of each Dnal state (main peak or satellite) in these compounds in the valence band region is represented by a mixture of screened and unscreened conDgurations, with a lower degree of e1ective screening in the states at lower binding energy. 5.2. Core level XPS Core-level spectra obtained in crystalline CrCl3 (Pollini, 2000) seem to reproduce again the same features observed in the valence band region, with satellites which can be interpreted either within the charge transfer or, possibly, by the competing exciton model. However, in the case of core-level photoemission, one has also to check whether the observed satellites arise exclusively from intrinsic Dnal-state e1ects, or whether extrinsic losses, e.g. due to plasmon excitation, may contribute to them, especially in cases where the main line-to-satellite separation takes a value of a typical plasmon energy. Fig. 5.5 shows the core-level Cr 3p-XPS and Cr 3s-XPS results for CrCl3 for binding energies between 25 and 125 eV. The general aspect of the photoemission core-level spectrum is rather similar to that previously reported (Pollini, 1994), but the improved
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Fig. 5.5. 3s and 3p Cr XPS spectra of CrCl3 compounds (from Pollini, 2000).
quality surface of freshly grown crystals has shown new features and weak details which were not observed before. In the Cr 3p-XPS, at binding energy between 35 and 60 eV, a satellite structure (S) is well separated from the main line, but the analysis is complicated by the existence of a pronounced multiplet splitting and spin-orbit e1ects. The main line at about 45 eV ∼ 8:8 eV from the main line and a spread of is followed by the weak satellite (S) at ES = spectral features up to about 70 eV. In Cr 3s-XPS of CrCl3 the main emission, split by ex∼ 75 and 79:3 eV, accompanied by a satellite (S) at change interaction, presents peaks at EB = ∼ ES = 10 eV and a broad band at about 22 eV from the main peak. These photoemission spectra resemble somewhat the Cr 3p and Cr 3s core level spectra reported for CrCl3 (Okusawa, 1984) and Cr 2 O3 (Uozumi et al., 1997), but with di1erent energy values for satellites. In the Cr 3s-XPS of Cr 2 O3 measured by Uozumi et al. (1997), in addition to the two main peaks ∼ 10 eV and split by the 3d–3s exchange interaction, one also observes two CT satellites at EB =
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Fig. 5.6. EELS spectrum of CrBr 3 (from Pollini et al., 1989).
∼ 20 eV (see Fig. 3 in Okusawa’s paper). The Cr 3s spectra are governed a CI satellite at EB = by the exchange coupling of the spin of the 3s core hole to the total spin with the intensity ratio between the split peaks given by S=(S + 1). For late TMCs this picture seems to work already with diEculties, suggesting one to use with cautions the splitting as a rough measure for the 3d occupation and magnetic moment (Fadley et al., 1969; Hermsmeier et al., 1988; van Acker et al., 1988; Kinsinger et al., 1990). Also, for intermediary-to-early TMCs, there seems to be no direct correlation of the exchange splitting with the actual magnetic moment (see also Section 4.3.2). In Cr 3p-XPS Dnal states of CrCl3 , the Coulomb multiplet coupling between the 3p hole and the 3d electrons becomes larger and a1ects the spectral shape, as in the Cr 3p spectrum of Cr 2 O3 . The satellite structure (S) observed in Fig. 5.5 can be due to a CT process: this would be supported by the observation and analysis of a similar satellite in Cr 3p-XPS in Cr 2 O3 observed ∼ 10 eV from the main line. However, it cannot be excluded that, owing to the larger at ES = value of the main-to-satellite ES distance, the satellite could be also assigned to an excitonic process involving the 4s, 4p metal orbitals, in correspondence with the energy values observed for interband p–s transitions in the optical spectra. As for the 22 eV satellite band, which has the same energy distance as the plasmon maximum seen in EELS spectra of CrCl3 (Carricaburu et al., 1986; Pollini et al., 1989), it could be assigned to plasma oscillations (extrinsic satellite) on these grounds. Plasmon maxima have been observed in EELS spectra at 20 eV (CrBr 3 ) and 22 eV (CrCl3 ); Fig. 5.6 shows an example of an EELS spectrum measured in CrBr 3 samples. Thus, it seems that the satellite structures observed in photoelectron spectra of CrCl3 can be described within the approach of the charge transfer model as in early and late TMCs, although the electronic conDgurations for the main line and satellites may not be the same. It is however not sure that the charge transfer model can explain satellite structures of all the transition metal photoelectron spectra, for instance in the photoemission core-level spectra. In this connection, we have to recall that the exciton satellite mechanism may be dominant over the charge-transfer satellites, when is large and U is small and there are many empty 3d orbitals.
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Fig. 5.7. Cr 2p-, 3s- and 3p-XPS spectra of CrO2 and Cr 2 O3 compounds (adapted from Zimmermann et al., 1999).
Fig. 5.7 shows a summary of the experimental core level XPS spectra in CrO2 and Cr 2 O3 in relation to cluster model results (Zimmermann et al., 1999). The metallic ferromagnet CrO2 is considered as an important material widely used for its magnetic properties; it is one of the few TMOs both metallic and magnetic that belongs to the class of itinerant ferromagnets. Surprisingly, band structure calculations, transport properties, Hubbard splitting and electron correlations have been studied only very recently (Schwarz, 1986; Lewis et al., 1997; Tsujioka et al., 1997). The Cr 2p spectra are characterized by a strong spin-orbit splitting into the 2p1=2 and 2p3=2 parts. The coupling increases with the atomic number of the TM from TiO2 through V oxides (7–8 eV) and the Cr oxides (nearly 10 eV) to a value of more 12 eV for Fe oxides. The main Cr 3s photoemission line shows a well separated exchange satellite, which is split o1 by 4:1 eV in Cr 2 O3 . The Cr 3p-XPS spectra also exhibit a satellite structure well separated from the main line, but contrary to the case of the Cr 2p-XPS spectra, the spin–orbit splitting is small and is only reJected in a broadening of the main lines. However, for Cr compounds the pronounced multiplet splitting makes a precise determination of satellite position and weight very diEcult. In CrF2 compounds for a study of the photoemission of Cr 2p and 3s XPS lines
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Fig. 6.1. Theoretical charge transfer energy versus d-electron number n (from Kotani et al., 1995).
on one hand and of the CrL2; 3 absorption edges on the other hand, we refer to Okada et al. (1994) and Theil et al. (1999), respectively. 6. Systematic trends The electronic structure of early TMCs seems determined by a balance between charge-transfer energy, dd electron correlations and covalency. In the previous section a list of model parameter values for (; U; V ) as well as for the resulting ES , the main line–satellite distance, and the bandgap energy EG has been reported in a few cases in order to discuss recent results on early TMCs (Table 5.1). However, one further needs to know how , U and V are related to n, the net d-electron occupancy and whether the trends established for the late TMCs continue along the whole Drst transition metal row. Hybridization between the metal 3d orbitals and the ligand 2p O orbitals, in the case of TMOs, is included via the one-electron mixing matrix elements deDned as d7 |H |L7 = V7 ;
(6.1)
where V7 =V if 7 indicates the 3z 2 − r 2 or x2 − y2 orbitals; V7 =V if 7 indicates the xy, yz or zx orbitals; L7 and d7 are a ligand electron and a 3d electron, respectively, with the same orbital symmetry. The Slater–Koster parameters (Harrison, 1980) pd and pd, where pd=pd = −2:2, reJect the anisotropic √ hybridization strength of the ligand p orbitals with the metal eg and t2g orbitals; also V = 3 pd and V = 2 pd. The spin and orbital degeneracy is accounted for by the number of conDgurations in the transfer integrals. These points are touched upon in Figs. 6.1– 6.3, which we have adapted from Kotani et al. (1995) and Bocquet et al. (1996). The values of the , U and pd parameters are displayed graphically versus the d-electron number n. These values were carried out by means of a cluster model (CIAM) with the full multiplet structure of the TM ion: they were chosen in such a way that the calculated spectra of XPS and BIS are in good agreement with the experimental data (Uozumi et al., 1997). One can see in Fig. 6.1 that, for a series of compounds with the same ligand, the parameter falls
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Fig. 6.2. Theoretical Coulomb energy U (=Udd ) versus d-electron number n (adapted from Bocquet et al., 1996).
by about 1–2 eV for each decrease in the formal d-electron number and valence state, due to the lowering of the 3d orbitals in the energy diagrams. As one goes along the TM series from Ti to Fe, for any series of compounds, one Dnds a general linear decrease in the parameter reJecting the increasing electronegativity of the metal cation. As for the U parameter (Fig. 6.2), one expects that its value for the early TMCs could be relatively small, as the wave function of the 3d electrons should be fairly extended and the correlation e1ects rather weaker. This can also be seen from the overall decrease in U by going from Fe to Ti. For oxides with the same cation (for example VO2 and V2 O5 ), it turns out that U increases slightly with the decreasing of the d-electron number, probably due to the shrinking of the 3d orbital radius. The correlation becomes larger as the nuclear charge increases in the 3d elements. As the nuclear charge increases, the radius becomes smaller and the e1ective electron density increases. The radius then increases at the end of the 3d period. The metal-insulator transition often appears at both ends of the 3d period, that is, around (Ti, V) and (Ni, Cu) and the magnetic insulators appears at the middle of the period. Thus there seems to be boundaries between (Ti, V) and (Ni, Cu) within which the correlation is large enough to make, for example, oxides insulators. The correlation is not so large in 4d and 5d elements and for most of their oxides the band picture works rather well. However, it must be noted that this statement is an oversimpliDed view. In fact, there are many metallic oxides of the 3d elements other than Ti, V, Ni and Cu. The most remarkable variation between the early and late TMCs is in the value of pd (Fig. 6.3), which falls between 2 and 3 eV for the Ti and V oxides, indicating a larger hybridization than for the late TMOs
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Fig. 6.3. Theoretical Slater–Koster parameter pd versus d-electron number n (adapted from Bocquet et al., 1996).
(for example about 1 eV for MnO and FeO). While trends for pd are governed principally by the interatomic distances within the cluster, larger values are also found for compounds with small d-electron numbers (Bocquet et al., 1992; Saitoh et al., 1995). Trends for pd across the TM series can be estimated from the relative size of rd , the radial extent of the 3d orbital and dTM−O , the metal–oxygen (for TMOs) distance, using the relationship by Harrison (1980): pd ˙ rd1:5 =d3:5 TM−O :
(6.2)
In summary, it seems from these relatively few results obtained in early TMCs that the parameter values deduced for the charge transfer and Coulomb energy U somewhat continue the systematic trends established for the late TMCs. DeDnitively, the charge transfer mechanism cannot be neglected to analyze satellite structures of early TMCs. Those compounds are characterized by a large hybridization energy and a strong resulting covalency. Actually according to the values of ; U and pd many early TMCs should belong to a new category between the MH and CT regime (Uozumi et al., 1997). 7. Spectroscopic and elastic properties of substoichiometric TiCx , and TiNx compounds Titanium carbides and nitrides are considered as high technology materials often used in microelectronics, space technology, aeroplanes industry and biomaterials, due to their exceptional physical and chemical properties. The conductivity of these materials is of the same order
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of magnitude as pure titanium metal and their melting point, higher than 3000 K, enables to classify them within refractory metals (Toth, 1971; Storms, 1967). At room temperature these compounds are chemically very stable (for example TiN is a good solid solution but CoN is not because of clustering of Co) and present excellent corrosion resistance; thus they can be used as biocompatible layers on orthopaedic and dental implants. Lastly, their hardness, among the highest after diamond, has contributed to the industrial use of titanium carbides as coating for cutting tools and besides, has designate them, as well as titanium nitrides, as good candidates for applications needing high wear resistance (Ohring, 1992; Bunshah and Deshpandey, 1989). On the other hand, titanium nitrides are widely used in semi-conductors technology as di1usion barriers (Kim et al., 1997; Wang and Allen, 1996; Faltermeier et al., 1997). Moreover, a characteristic property of these compounds is to exist on a large substoichiometric scale. As a consequence, the lattice can show an important proportion of vacancies while keeping the same crystallographic structure. This property is very interesting to study experimentally as well as theoretically, since the physical and chemical properties of these systems depend widely upon their vacancy concentration (Williams, 1971; Goldschmidt, 1967). Nevertheless, one should notice that most papers published presently are dealing only with stoichiometric or near stoichiometric titanium carbides and nitrides, due to the usual idea that properties of these materials are then most interesting. In this section we report studies on TiC0:49 ; TiC0:78 ; TiN0:45 and TiN0:61 compounds, i.e. containing vacancies with very di1erent and signiDcant concentrations. These compounds were synthesised by C+ or N+ multiple energy ion implantation (180, 100, 50, 20 keV) in polycrystalline titanium. This method is actually very convenient in order to synthesize well-deDned titanium carbides and nitrides. Layers, about 500 nm thick, were prepared by multiple energy (180, 100, 50 and 20 keV) ion implantation of C+ or N+ in polycrystalline titanium. The dose corresponding to each energy was deDned according to Ziegler et al. (1985). The implanted ion proDles were determined by combining RBS (Rutherford backscattering spectrometry) and SIMS (secondary ion mass spectroscopy) measurements and the crystallographic structure of the layers was obtained by glancing incidence X-ray di1raction (Guemmaz et al., 1996, 1997). The chemical bonds were studied by XPS, performed with an apparatus equipped with two chambers. We refer to Guemmaz et al. (1997a) for details of the experiment. The section is organized as follows: in Section 7.1, we recall a few core-level spectra of titanium carbides and nitrides; then in Section 7.2, we connect the electronic structure obtained by tight binding-linear muEn tin orbitals (TB-LMTO) calculations to the v-XPS and discuss the inJuence of the vacancies on chemical bonds, and, Dnally, in Section 7.3, we present the variation of the bulk modulus due to the changes of the chemical bonds for the whole range of vacancy concentration. 7.1. Core level XPS Fig. 7.1 shows the two spin–orbit components Ti 2p3=2 and Ti 2p1=2 corresponding to pure titanium and to the various titanium carbides (Fig. 7.1a) and nitrides (Fig. 7.1b) Ti 2p-XPS spectra. The Ti 2p peaks are shifted toward higher binding energies when the ligand concentration increases which is due to the charge transfer from titanium to metalloid atoms during the formation of Ti–C or Ti–N bonds (Johansson et al., 1977). As several authors have indicated (Porte et al., 1983; Hofmann, 1986; Strydom and Hofmann, 1990) the line shapes for titanium nitrides are rather complicated and there are, at least, two contributions: the Drst one,
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Fig. 7.1. XPS spectra of Ti 2p3=2 and Ti 2p1=2 lines recorded on: (a) pure Ti, TiC0:49 and TiC0:78 and (b) pure Ti, TiN0:45 and TiN0:61 .
due to actual titanium nitride and the second one, associated with titanium oxide located at the surface, as a consequence of gettering of residual oxygen. In their Ti 2p-XPS studies of 9-TiNx (0:50 6 x 6 1) titanium nitrides, Porte et al. (1983) reported the existence of a new line at 2:2 eV above the main peak for compositions near stoichiometry. This is related to a decreased screening e1ect of conduction electrons. On the other hand, the Ti 2p peak shape was shown to be a function of nitrogen concentration by Vasile et al. (1990) and also oxygen contributes signiDcantly to the background formation. Moreover, in a study of titanium nitride energy losses, Strydom and Hofmann (1990) pointed out that the losses due to intraband transitions change the line shapes of the Ti 2p transitions. For C1s XPS of TiCx as well as for N1s XPS of TiNx we refer to Guemmaz et al. (1997b). 7.2. Electronic band structure and valence band XPS of Ti carbides and nitrides In Section 7.2.1, we brieJy recall the case of stoichiometric compounds; then we compare v-XPS spectra with ab initio TB-LMTO band structure calculations (Section 7.2.2) in substoichiometric compounds. 7.2.1. Stoichiometric TiC and TiN The electronic structure of stoichiometric titanium carbide and nitride has been determined by many authors since the pioneering work of Ern and Switendick (1965) and various theoretical
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Fig. 7.2. DOS curves for: (a) stoichiometric titanium carbide and (b) stoichiometric titanium nitride.
methods have been used afterwards: for example, EPM (empirical pseudo-potential method), LCAO (linear combination of atomic orbitals), APW (augmented plane wave) and MAPW (modiDed augmented plane wave) methods as well as cluster techniques (see Calais, 1977; Neckel, 1983 and refs. therein). From these works, it appears that the TiC electronic structure is composed of several bands in the following way: at the bottom, far from the Fermi level, the s band is mainly built of C 2s or N 2s states, with a small contribution of the Ti 3d states. Then, we Dnd a p band essentially formed by C 2p or N 2p states and also by Ti 3d states, just below the Fermi level. Finally, at higher energies, above the Fermi level, there are the Ti 3d bands, which are mostly mixed with C 2p or N 2p states. The electronic density of states (DOS) calculated by TB-LMTO method is shown for TiC and TiN in Fig. 7.2a and b, respectively (Guemmaz et al., 1997) and present a pseudo-gap between occupied and empty states. In the case of titanium carbide, the Fermi level is in the middle of the pseudo-gap at the DOS minimum, This allows to predict a very strong TiC bond (Ahuja et al., 1996). In the opposite, the Fermi level of titanium nitride is somewhat towards the edge of the pseudogap, due to an additional 2p electron in nitrogen. This situation gives rise to the metallic conductivity observed in TiN (as well as VN) nitrides. In stoichiometric TiC (as in TiN) we Dnd three typical
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types of covalent bonds: (i) the Ti 3d orbitals of symmetry eg (dx2 −y2 ; d3z2 −r 2 ), which interacts with C 2p orbitals of the neighboring carbon atoms and form a pd bond; (ii) the Ti 3d orbitals of symmetry t2g (dxy ; dyz ; dxz ), which creates pd bonds with the C 2p orbitals of the neighboring carbon atoms. Both pd and pd bonds appear mostly in the p band region. The particularly large stability of these compounds is due to the Dlled metal 3d-ligand 2p bonding states; (iii) the Ti t2g orbitals also interact with neighboring titanium orbitals of the same symmetry to create dd bonds. These bonds appear mainly in the higher d band part, which remains unoccupied in the stoichiometric limit. From the experimental point of view, in most papers dealing with titanium carbides (or nitrides), the occupied DOS has been compared to the v-XPS recorded through ad hoc spectroscopic techniques. For example, Ihara et al. (1975) connected the v-XPS recorded on a monocrystalline TiC0:98 with the DOS calculated by the APW method. In the same way, photoelectron spectra measured with synchrotron radiation, as well as with HeI and HeII lines, on monocrystalline stoichiometric titanium carbides have been compared by Johansson et al. (1980) with the DOS calculated by the APW method. These and other works (Weaver et al., 1980; HagstrXom et al., 1976, 1977; Soriano et al., 1997) have allowed to conclude that the peak positions and band widths, obtained by ab initio calculations on these stoichiometric compounds, are in fairly good agreement with the experimental spectra. 7.2.2. Substoichiometric TiCx and TiNx In Figs. 7.3a–7.6a v-XPS spectra, recorded on our various substoichiometric titanium compounds TiC0:49 ; TiC0:78 ; TiN0:45 and TiN0:61 present a characteristic peak at about 10 eV for carbides or 15 eV for nitrides below the Fermi level, the relative intensity of which changes in the same way as the ligand concentration. In the following, we shall limit our discussion to the case of titanium carbides since it would be nearly the same in the case of nitrides. The carbide spectra present a broad structure located between 1.5 and 8 eV below the Fermi level, the shape of which also varies with the carbon amount. In the case of TiC0:78 , the structure is composed of only one intense peak, contrary to the other concentration which presents clearly two shoulders (Fig. 7.3a). Since v-XPS spectra directly reJect the main characteristics of the DOS in the stoichiometric case (see Section 7.2.1), the same type of comparison in the substoichiometric case will also help to identify the various peaks and relative positions of the bands. The DOS calculations were performed using the TB-LMTO method within the atomic sphere approximation (ASA) approximation (Guemmaz et al., 1997a). Although the vacancies (symbolized by in the text and in the Dgures) were statistically distributed in our titanium carbide samples, the calculations were done with completely ordered compounds of NaCl type, composed of two fcc sublattices, the Drst one containing titanium atoms and the second one, carbon atoms and ordered vacancies. In Figs. 7.3b and 7.4b, DOS curves are displayed, corresponding, respectively, to TiC0:50 0:50 and TiC0:75 0:25 substoichiometric carbides. Let us notice the following points on DOS: (i) the intensity of the structure at 10 eV below the Fermi level originates from the C 2s states, and varies in the same way as carbon concentration; (ii) the DOS intensity just below the Fermi level increases with the vacancy concentration; (iii) the substoichiometric carbides exhibit new structures in the vicinity of the Fermi level, which fall within the pseudogap of the TiC1:0 compound. In Figs. 7.3–7.6, the v-XPS spectra are presented above the corresponding DOS curves. To help comparison, one should remember the
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Fig. 7.3. (a) Experimental v-XPS spectrum for TiC0:49 , (b) Theoretical DOS curve for TiC0:50 and without (dashed line) convolution.
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0:50
with (full line)
following points: (i) the synthesized titanium carbides have disordered vacancies in opposite to the calculated structures which are strictly ordered. Moreover, the theoretical stoichiometries do not correspond exactly to the experimental ones; (ii) the v-XPS spectra are more easily compared to DOS curves convoluted with the experimental apparatus resolution (1:4 eV). On one hand, this convolution help to smooth the DOS curves and, on the other hand, to broaden the peaks: see Figs. 7.3b–7.6b; (iii) a background increasing with binding energy is measured with the bare XPS signal. By taking into account all these points, we can conclude that the DOS reproduce the v-XPS and especially the structures near the Fermi level in a rather satisfactory manner. From the theoretical point of view, the e1ect of vacancies on the electronic structure of titanium carbides has been worked out by many authors. For example, Gubanov et al. (1984) used the Hartree–Fock–Slater method applied to clusters to study the changes of the TiO and TiC valence bands. With the help of the APW method, Redinger et al. (1985, 1986) calculated the modiDcations introduced by the substoichiometry in TiC0:75 0:25 carbides.
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Fig. 7.4. (a) Experimental v-XPS spectrum for TiC0:78 , (b) Theoretical DOS curve for TiC0:75 and without (dashed line) convolution.
0:25
with (full line)
Moreover, using the Korringa–Kohn–Rostoker (KKR) method, within the coherent potential approximation (CPA) and within the Green’s function (GF) method, Marksteiner et al. (1986) studied various titanium carbide compounds. The LMTO-GF method was applied by Ivanovsky et al. (1988, 1998) to analyze the e1ect of carbon and titanium vacancies on the electronic properties of titanium carbides. With the help of a LCAO method, Capkova and Skala (1992) described the rearrangement of inter-atomic interactions in TiCx and their e1ect on the DOS due to vacancies. Several authors studied titanium nitrides as well. In particular, Klima (1979, 1980) used a CPA-LCAO method to determine the DOS for TiNx (0:6¡x¡1:0) compounds. Besides, Marksteiner et al. (1986) exhibited KKR-CPA and GF methods to characterize the
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Fig. 7.5. (a) Experimental v-XPS spectrum for TiN0:45 , (b) Theoretical DOS curve for TiN0:50 and without (dashed line) convolution.
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0:50
with (full line)
inJuence of substoichiometry on TiNx (0:5¡x¡1) nitrides. Also, Herzig et al. (1987) applied APW method to study TiN0:75 assuming long range order for vacancy distribution on the nitrogen sublattice. The e1ect of vacancies (both metallic and non-metallic sites) on the electronic structure of titanium nitrides as well as the inter-atomic bond rearrangements in the vicinity of a considered vacancy were studied by Ivanovsky et al. (1988) within a LMTO-GF method. Lastly, describing titanium nitrides by clusters within Hartree–Fock approximation, Capkova and Skala (1992) determined the inJuence of the deviation from stoichiometry upon chemical bonds and lattice relaxation. In summary, all these authors outline the important e1ect of vacancies on the electronic structure and the appearance of new structures in the DOS near the Fermi level. These new structures can clearly be seen on Figs. 7.3–7.6 and are called
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Fig. 7.6. (a) Experimental v-XPS spectrum for TiN0:61 , (b) Theoretical DOS curve for TiN0:75 and without (dashed line) convolution.
0:25
with (full line)
“vacancy state associated structures”. Their origin can be explained by symmetry changes resulting from the vacancy sites in the lattice. Titanium atoms which are completely equivalent in a perfect stoichiometric rocksalt structure, are no more so in substoichiometric structures. This is related to the formation of new bonds associated with the peaks in the vicinity of the Fermi level. In addition to the three covalent bonds pd; pd and dd present in the stoichiometric carbides, two new bonds are observed in the substoichiometric carbides (Redinger et al., 1986). Both Ti[4] d neighboring levels of (3z 2 − r 2 ) symmetry interact together through a vacancy in order to establish a Ti[4] (d3z2 −r 2 )–Ti[4] (d3z2 −r 2 ) covalent bond. The peaks observed near the Fermi level on the v-XPS spectra and on the DOS curves take their origin mainly from this
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Table 7.1 Reduced modulus Er and Young’s modulus E a; b values obtained by nanoindentation: (a) with a constant Poisson coeEcient # = 0:17 for TiCx and # = 0:194 for TiNx ; (b) with a linear interpolated Poisson coeEcient, between # = 0:17 for TiC or # = 0:194 for TiN, and # = 0:34 for pure titanium. For stoichiometric titanium carbides, the values were obtained by linear extrapolation Compounds
Er (GPa)
E=(1 − #2 ) (GPa)
E (a) (GPa)
E (b) (GPa)
TiC0:26 TiC0:49 TiC0:78 TiN0:45 TiN0:61
163 218 290 214 241
192.5 274.3 399 268 311.8
187 265 386 258 300
177 257 383 248 281
type of orbital mixing. Thus, the existence of vacancies leads to the formation of new bonds between titanium atoms, absent in the stoichiometric titanium compounds. Moreover, this interpretation is in good agreement with Skala and Capkova (1990) on substoichiometric vanadium nitrides where there are also similar important V–V bonds, through vacancies. 7.3. Young’s modulus A nanoindenter apparatus was used to investigate the mechanical properties of the synthesized layers (Guemmaz et al., 1999). In this last section, we report (Section 7.3.1) our measurements of Young’s modulus by nanoindentation, then (Section 7.3.2) we discuss our data in terms of vacancy concentration (Guemmaz et al., 1999). 7.3.1. Determination by nanoindentation In order to determine the mechanical properties of our samples we carried out nanoindentation tests. From the unloading curves the reduced Young’s modulus Er is determined, by the following relation (Guemmaz et al., 1996, 1997b and refs. herein): 1 (1 − #2 ) (1 − #2i ) = + Er E Ei
(7.1)
where (E; #) and (Ei ; #i ) are the Young’s modulus and Poisson coeEcient of the sample and nanoindenter tip, respectively. Since the tip is made of diamond, its elastic constants are Ei = 1050 GPa and #i = 0:104 (Spear, 1989). Also the preceding relation allows us to calculate the E=(1 − #2 ) value for a considered sample. In Table 7.1 the Er values are given for three various carbides and two nitrides. To derive Young’s modulus we need to know the corresponding Poisson coeEcients. To our knowledge, there is no published work giving the evolution of Poisson coeEcient as a function of carbon or nitrogen concentrations for the systems we are interested in. To overcome this problem, we used two di1ering approaches: the Drst one considers the Poisson coeEcient as independent from the stoichiometry and equal to the value # = 0:17 for TiC1:0 (Chang and Graham, 1966) and # = 0:19 for TiN1:0 . In the second approach, we consider values interpolated linearly between # = 0:32 for the pure titanium and
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the preceding values for the stoichiometric compounds. Linear interpolation is justiDed by the fact that the Poisson coeEcient is an elastic property of the matter as well as Young’s modulus. 7.3.2. Evolution as a function of vacancy concentration From Table 7.1, in the case of carbides, both previously considered approaches lead to Young’s modulus values close to each other but also very dependent on the vacancy concentration. The values evolve from 187 GPa for TiC0:26 to 386 GPa for TiC0:78 in the Drst approach and from 177 to 383 GPa in the second one. The extrapolation for the stoichiometric compound gives a value equal to 468 GPa in both cases. Young’s modulus values, found in the literature, mainly concern the titanium carbides at (or near) stoichiometry. Thus, Gilman and Roberts (1961) and Williams and Schaal (1962) reported values, respectively, equal to 458 and 447 GPa, for monocrystalline TiC0:94 . Chang and Graham (1966) estimated Young’s modulus at 438 GPa, for monocrystalline TiC0:91 and for temperatures ranging from 4.2 to 298 K. Measurements performed on titanium carbides deposited on quartz crystals gave a value of 200 Gpa (Kinbara and Baba, 1983), but the carbon concentration was not speciDed in this study. For TXorXok et al. (1987) Young’s modulus is most likely equal to 460 GPa which is very close to our extrapolated result. A few years ago, a value equal to 494 GPa was found by X-ray di1raction on titanium carbide Dlms deposited by chemical vapor deposition (CVD) on steel (Sawada et al., 1992). For titanium nitrides, Young’s modulus reported in literature depends upon the preparation mode, measurement techniques, grain magnitude, sample texture and of course stoichiometry: the intensity varies from 250 to 640 GPa (TXorXok et al., 1987). For example in the case of TiN prepared by physical vapor deposition (PVD) and with the help of X-ray di1raction, Sue (1992) measured values comprised between 408 and 447 GPa, whereas by nanoindentation, Wittling et al. (1995) found a value of 400 GPa for TiN. Let us report the bulk modulus evolution as a function of carbon concentration in TiCx compounds (Guemmaz et al., 1999, 2000). In Fig. 7.7 we show the bulk modulus values (i) derived from the nanoindentation experiments and (ii) obtained by the full potential LMTO method (FPLMTO), as a function of stoichiometry. Here let us point out that the “FP” version of the LMTO method, i.e. beyond the ASA option, is necessary to treat the elastic property (Guemmaz et al., 1999). Actually, the calculated bulk modulus evolves as a function of x, in a similar way as the corresponding experimental data. Moreover, the K values obtained from the linearly interpolated Poisson coeEcients give the best agreement with the calculated ones. This can be explained by the fact that both E and # are elastic constants and, since E changes with stoichiometry, # changes automatically too. Finally the agreement obtained between the experimental and calculated bulk modulus, indicates that the substoichiometric titanium carbide structure considered in the calculations, although completely ordered, is suEciently realistic to provide an agreement with experience. The observed variation of Young’s modulus as a function of the vacancy concentration is related to the evolution of chemical bonds (Table 7.1). We have seen above, that the electronic structure is deeply modiDed after the appearance of vacancy sites: new chemical bonds are then responsible of changes in the cohesion of the considered substoichiometric compounds. Thus, if Young’s modulus decreases as a function of the vacancy concentration, it is a consequence of the formation of Ti–Ti bonds. At high vacancy concentration, the lattice behaves like
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Fig. 7.7. Bulk modulus K versus carbon atomic concentration. The triangles are the experimental values obtained from nanoindentation either with # = 0:17 (4) or with # depending linearly on carbon concentration (5). The circles are the theoretical values deduced from a FP-LMTO calculation: a FP version of the LMTO code is necessary to handle this type of problem (from Guemmaz et al., 1999).
a pure titanium rather than a carbide or nitride structure. Young’s modulus evolves to the pure titanium limit, which is equal to about 170 GPa. As a result, we can say that the elastic constants indirectly reJect the evolution of the chemical bonds due to substoichiometry and express the structure sti1ness. 8. Summary and outlook This review was intended to survey the present status of our knowledge of the electronic properties of early transition metal compounds, which depend essentially on the electron–electron correlations, i.e. many-body e1ects. The review is by no means a complete compilation. We hope, however, that we have complied with pointing out the nature of the generalization and limitations as well as the experimental and theoretical diEculties in the area of the electronic properties of these compounds. The subject is changing and still developing so rapidly at the present time that, despite the existence of excellent reviews (Adler, 1968; Brandow, 1977; Henrich, 1985; Davis, 1986; Kanamory and Kotani, 1988; Mott, 1990; Fulde, 1991; Hufner, 1994), it seemed worth trying to put some of the recent developments in a form accessible to non-specialists. Also, at least two major new developments have occurred in the last 15 years which seemed appropriate to be recalled, although the historical view (Mott compound) has not been challenged as yet. First, it was realized that NiO is not a Mott insulator in terms of the energy level diagram, but rather a more conventional substance, namely a charge transfer material in the sense of chemists (Pauling, 1960; Jorgensen, 1970). A second new event has been the discovery of the so-called high-temperature superconductors (Bednorz and Muller, 1986),
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a class of compounds where copper–oxygen planes (CuO2 planes) give rise to superconductivity up to above 100 K. Further developments, which have added new and unexpected results, have been the reDnements of the technique of photoelectron spectroscopy, which has allowed to map the electronic band structure of a few of these materials such as TiO2 , NiO (Hufner, 1995), NiI2 (Stanberg et al., 1986), CoO (Brooks et al., 1989; Shen et al., 1990) and RuCl3 (Pollini, 1996). Also a new e1ort in the local density approximation and beyond LDA for the calculation of the band structures gives at least an ad hoc basis to include the electron–electron correlation and thereby to come to a more meaningful calculation of energy gap values. Especially, there have been recent attempts to incorporate the Mott–Hubbard type correlations into the standard LSDA formalism (Svane and Gunnarsson, 1990; Anisimov et al., 1991). One of the reasons why the extended LSDA and LSDA + U calculations have been rather successful, although the Hubbard type correlations are still not always adequately taken into account, stems from the physical concept of quasi-particle developed in Landau’s Fermi-liquid theory. There it was shown that one can treat a system of interacting electrons like one of the non-interacting ones provided the electron mass is renormalized (Ziman, 1969; Madelung, 1978; Nakajima et al., 1980). A new e1ort in the past decades to explain the electronic structure and electronic excitations in TMCs has been also made with single-ion models, most notably the cluster approaches based on the impurity Anderson Hamiltonian (Fujimori and Minami, 1984; Zaanen and Sawatzky, 1990; Okada and Kotani, 1992a and Sections 2.1, 3.2 and 4.3 of the present review). Also, many TMCs have been reclassiDed as intermediate compounds between MH and CT compounds. These e1orts have been very successful and have provided a considerable insight into the electronic structure of these systems. In particular in the present paper, we have put emphasis on the fairly good connection between: (i) v-XPS spectra and standard LMTO band calculations (Sections 3.1, 7.2), even in the case of substoichiometric compounds; (ii) c-XPS and CI-CT models (Sections 2.2, 3.2), including the case of localized magnetic moments (Section 4.3); (iii) the full potential LMTO band calculations and the analysis of elastic subtoichiometric properties (Section 7.3). Moreover, one of the aims of the article was to reconsider the Hubbard model (Austin and Mott, 1969; Mott and Zinamon, 1970) with today’s standard and also recall some of its developments. In spite of its simple deDnition, the Hubbard model is still believed to exhibit various interesting phenomena, including metal–insulator transitions, antiferromagnetism, ferrimagnetism, Tomonaga–Luttinger liquid and superconductivity (Tasaki, 1998; Lieb, 1998). Nevertheless, more generally, let us recall that the e1ects of electron correlation on spectroscopic properties, like XPS, electron scattering and XAS, are still treated today in a rather incomplete way (Hedin, 1999). Although transition metals and metallic alloys involving transition metals have not been studied in this review, the experimental and theoretical techniques described here are relevant to such materials as well (Bennett et al., 1983; Parlebas et al., 1990). BIS and inverse photoemission have become very useful and standard methods of studying the states above the Fermi energy (Speier et al., 1984). Auger photoelectron coincidence spectroscopy (Haak et al., 1978) is another technique which has kept the promises of interest raised at the outset. These measurements are diEcult, however several issues regarding the local atomic conDguration of satellites in XPS and Auger spectra have been resolved both for TMs and TMCs. Electron correlations are present in varied degrees in metals as well. In the Deld of metals and TMs, the simplest system is the free-electron metal (e.g. Na), where only small electron–electron
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correlations are present (Jensen and Plummer, 1985; Whan and Plummer, 1988) and conductivity takes place because one has an only half-Dlled band. Conductivity can be described here by the creation of s2 conDgurations. In a typical TM, namely Ni, large correlations are present, however, the band-width can accomodate these correlations so that a conducting state is achieved. In the context of this review, this situation can be called a Mott–Hubbard metal. The metallic phases of TiO2 and VO2 probably fall into this category. Another class of materials, where highly correlated electrons take part in the conductivity are the mixed-valence compounds containing rare-earth or transition elements. A prototype in this class is metallic SmS, where the 4f 5 and 4f 6 conDgurations coincide at the Fermi energy. This makes a charge Juctuation of the type 5d 3 4f 5 → 5d 3 4f 6 possible and the occurrence of metallic conductivity. NiS is with respect to its electronic structure the case perhaps nearest to the high Tc superconductors (Hufner, 1994). A large number of TMCs are semiconductors, although few of them have been investigated in any detail. The great diEculties encountered in preparing single crystals of nearly all these compounds have greatly hampered the e1orts to understand their behavior in any fundamental way. The measurements of electrical properties on powders or sintered samples must always be regarded as an uncertain procedure, as these measurements are in suEcient disagreement with the single crystal measurements, because they are greatly inJuenced by interfacial e1ects between single crystal grains. Nevertheless, a general picture of the behavior of TMCs has been obtained. Many of these materials are of interest because of their magnetic properties and a great deal of information derived from studies of the latter is useful in interpreting the electrical behavior. Thus, information about structure, number of unpaired spins, the NZeel temperature for the transition to antiferromagnetic state can be studied in this way. The compounds whose electronic structure has been studied in most detail have been the 3d TMOs, although a number of studies of sulDdes and halides have been carried out and some 4d and 5d TMCs have been investigated. Rutile oxides show, in particular, an interesting range of physical properties which can be correlated with the number of free d electrons of the metal ion (Riga et al., 1977; Beatham and Orchard, 1979; Daniels et al., 1984; Xu et al., 1989). This interest derives from unresolved questions regarding the inJuence of the electron correlations and the associated magnetic structure, beside numerous technological applications of the materials. Electrically, the outstanding characteristic of the insulating TMCs is their very low carrier mobility. This has been generally interpreted as indicating a “hopping process” for the conduction mechanism, in which the carriers are localized on a particular cation within the activation energy required for the carrier transport to an adjacent cation because of a potential barrier betwen cations. Thus, for a hopping process, the mobility 7 increases with temperature, whereas in normal bands 7 decreases with increasing temperature due to increased lattice scattering. Also, some of these compounds are metals with mobilities in the range 10−2 to 1 cm2 V−1 s−1 . The mobilities for the compounds which are insulators are very low, below 10−5 cm2 V−1 s−1 . The large di1erences in the electrical properties of these compounds hinge upon the possibility that an energy band is formed by the d-type wave functions. The lower lying electronic levels are completely Dlled, but the valence d levels are either empty or partially Dlled and the possibility of electrical conduction in these levels exists. If they form a band of levels which is empty, then their behavior might be expected to resemble that of normal semiconductors, like germanium and silicon. On the other hand, if they form a band which is partially Dlled they should be metallic in character. Finally, if no band is formed, but the levels are partially Dlled, a hopping mechanism can be
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expected to give rise to a very low conductivity, the compounds being properly classiDed as insulators in this case. Experimentally, many of these situations have been conDrmed for several of these materials (Rao and Rao, 1970; Tsuda et al., 1991). Let us mention quite a new Deld of research for those materials, i.e. the study of their properties when they can be prepared as reduced dimensional systems (i.e. by epitaxy; or as in Section 4.3) or systems containing a controlled number of defects as in Section 7. From the spectroscopic and electronic point of view, conDguration crossover in TMCs between sub-bands of the d-band manifold (i.e. the t2g and eg orbitals with octahedral coordination for high-spin=low-spin e1ects) can strongly a1ect the lattice parameter of the compounds and greatly alter the conduction characteristics. For example, in the 4d 5 low-spin semiconductor -RuCl3 (t 5 t2g ), the lattice parameter is actually smaller than in the high-spin FeCl3 (t 3 t2g e2g ) 3 e2 ) is an insulator, and the low-spin IrO2 (t5 t2g ) is metallic, whereas the high-spin MnTe (t2g g 3 e2 seems to be the key insulator. Adoption of the high spin in half-Dlled shell conDguration t2g g to the Mott-insulating character for many compounds, holding even for tellurides, such as MnTe and MnTe2 . Despite the considerable p–d hybridization in these tellurides through the proximity of ligand 5p and cation 3d energies, the resulting d-bandwidth still remains insuEcient to overcome the large exchange and correlation energies operative with the high-spin 3d 5 Hund’s rule groundstate 6 A1g (Wilson, 1972). Furthermore, TMOs and TMHs are ideal subjects for investigating the insulating and metallic states, because of the wide diversity of the electrical properties observed in apparently similar materials. For example, layered CrCl3 ; CrBr 3 and Cr 2 O3 materials are probably all Mott insulators, as the layered RuCl3 compound (Pollini, 1996), while cubic RuO2 is a metal (Mattheiss, 1976; Daniels et al., 1984; Xu et al., 1989; Glassford and Chelikowsky, 1992, 1994). Besides, -RuCl3 ; -TiCl3 , and -TiCl3 , are non-magnetic compounds at room temperature through cation interaction and once more they are non-standard semiconductors (Wilson, 1972; Pollini, 1983; Maule et al., 1988). As far as 4d TM compounds are concerned and, in connection with Sections 7.2 and 7.3, the electronic and elastic properties of substoichiometric ZrNx compounds are now under investigation and will be published in near future. In conclusion, this review has been purposely limited in scope, being mainly devoted to comments concerning the early 3d transition metal compounds with special emphasis on the experimental and theoretical v- and c-XPS spectra. However, that part of the photoemission Deld and theoretical models (band or cluster) covered seems now mature enough in many respects beyond the boundaries we have imposed to ourselves. The potential impact of knowledge gained in this and connected Delds seems rather near to be ready for some applications or, in a more fundamental way, for generalization in close research Delds, such as physical-chemistry (i.e. molecules). 9. Glossary of symbols and acronyms APW ARUPS ASA BIS CB
augmented plane wave angle resolved ultraviolet photoelectron spectroscopy atomic sphere approximation bremsstrahlung isochromat spectroscopy conducting band
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CI CIAM CM CNDO CPA CT CVD c-XPS DFT DOS EELS EPM FLAPW FPLMTO GF GGA GWA KKR LB LCAO LDA LFT LMTO LSDA MAPW MH MO PAM PES PVD RBS RPES SIAM SIC SIMS SXES TB-LMTO TM TMC TMH TMO UPS VB VUV
conDguration interaction cluster impurity Anderson model cluster model complete neglect of di1erential overlap coherent potential approximation charge transfer chemical vapor deposition core level X-ray spectroscopy density functional theory density of states electron energy loss spectroscopy empirical pseudo-potential method full potential linearized augmented plane wave full potential linear muEn tin orbital Green’s function generalized gradient approximation one-electron Green function + W screened potential approximation Korringa–Kohn–Rostoker ligand band linear combination of atomic orbitals local density approximation ligand Deld theory linear muEn tin orbital local spin density approximation modiDed augmented plane wave Mott–Hubbard molecular orbital periodic Anderson model photoemission spectroscopy physical vapor deposition Rutherford backscattering spectroscopy resonant photoemission spectroscopy single impurity Anderson model self interaction correction secondary ion mass spectroscopy soft X-ray emission spectroscopy tight binding linear muEn tin orbital transition metal transition metal compound transition metal halides transition metal oxide ultraviolet photoelectron spectroscopy valence band vacuum ultraviolet
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v-XPS valence band X-ray spectroscopy WSS Wigner–Seitz sphere XAS X-ray absorption spectroscopy XES X-ray emission spectroscopy XPS X-ray photoelectron spectroscopy ZSA Zaanen–Sawatsky–Allen Acknowledgements In the past years a great number of people have been helpful in discussing the various problems mentioned in the present paper. In particular the authors gratefully thank the following professors: M. Alouani, M. Avignon, L. Falicov, P. Fulde, F. Gautier, M. Guemmaz, J. Hugel, B. Johansson, J. Kanamori, A. Kotani, T. Jo, P. Rennert, S. Shin and doctors: R. Ahujab, A. Bocquet, E. Beaurepaire, C. Demangeat, J.P. Kappler, J.-J. Grob, M.A. Khan, P. Kruger, G. Moraitis, K. Okada, D. Raiser, A. Rakotomakevitra, M. Taguchi, T. Uozumi. This work has also much beneDted from past cooperation research program between JSPS and CNRS. Finally, we are greatly indebted to Prof. D.L. Mills for having encouraged us to write this review. References Adler, D., 1968. Solid State Physics, Vol. 21. In: Ehrenreich, H., Turnbull, D. (Eds.), Academic Press, New York, p. 1. Ahuja, R., Eriksson, O., Wills, J.M., Johannsson, B., 1996. Phys. Rev. B 53, 3072. Anderson, P.W., 1961. Phys. Rev. 124, 41. ∗ ∗ ∗ Anisimov, V.I., Zaanen, J., Andersen, O.K., 1991. Phys. Rev. B 44, 943. Antoci, S., Mihich, L., 1978. Phys. Rev. B 18, 5765. Aryasetiawan, F., Gunnarsson, O., 1995. Phys. Rev. Lett. 74, 3221; 1998. Rep. Prog. Phys. 61, 237. Asada, S., Sugano, S., 1976. J. Phys. Soc. Japan 41, 1291. Austin, I.G., Mott, N.F., 1969. Adv. Phys. 18, 41. Barth, J., Gerken, F., Kuntz, C., 1985. Phys. Rev. B 31, 2022. Beatham, N., Orchard, A.F., 1979. J. Electron Spectrosc. Relat. Phenom. 16, 77. Beaurepaire, E., Lewonczuk, S., Ringeissen, J., Parlebas, J.C., Uozumi, T., Okada, K., Kotani, A., 1993. Europhys. Lett. 22, 463. ∗∗ Becke, A.D., 1988. Phys. Rev. A 38, 3098. Bednorz, J.G., Muller, K.A., 1986. Z. Phys. 64, 189. Bengone, O., Alouani, M., BlXochl, P., Hugel, J., 2000. Comp. Mater. Sci. 17, 146; Phys. Rev. B 62, 16 392. ∗ Bennett, P.A., Fuggle, J.C., Hillebrecht, F.U., Lenselink, A., Sawatzky, G.A., 1983. Phys. Rev. B 27, 2194. Bethel, E., Stockbauer, R., Madey, T.E., 1983. Phys. Rev. B 27, 1939. Binns, C., Baker, S.H., Demangeat, C., Parlebas, J.C., 1999. Surf. Sci. Rep. 105. Binns, C., Derbyshire, H.S., Bayliss, S.C., Norris, C., 1992. Phys. Rev. B 45, 460. ∗∗ Blaka, B., Dufek, P., Sliwko, V., Swartz, K., 1996. J. Magn. Magn. Matter 140 –144, 173. Blaauw, C., Leenhouts, F., van der Woude, F., Sawatzky, G.A., 1975. J. Phys. C 8, 459. Bocquet, A.E., Mikozawa, T., Morikawa, K., Fujimori, A., Barman, S.R., Maiti, K., Sarma, D.D., Tokura, Y., Onada, M., 1996. Phys. Rev. B 53, 1161. ∗ ∗ ∗ Bocquet, A.E., Mizokawa, T., Saitoh, T., Namatame, H., Fujimori, A., 1992. Phys. Rev. B 46, 3771. ∗∗ Brandow, B.H., 1977. Adv. Phys. 26, 651. Brooks, N.B., Law, D.S., Warburton, D.R., Wincott, P.L., Thornton, G., 1989. J. Phys. Condens. Matter 1, 4267.
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PERTURBATIVE QUANTUM FIELD THEORY IN THE STRING-INSPIRED FORMALISM
Christian SCHUBERT Laboratoire d+Annecy-le-Vieux de Physique TheH orique LAPTH, Chemin de Bellevue, BP 110, F-74941 Annecy-le-Vieux CEDEX, France Instituto de Fisica y MathemaH ticas, Universidad Michoacana de San NicolaH s de Hidalgo, Apdo. Postal 2-82, C.P. 58040, Morelia, MichoacaH n, MeH xico California Institute for Physics and Astrophysics 366 Cambridge Ave, Palo Alto, Califormia, USA
AMSTERDAM } LONDON } NEW YORK } OXFORD } PARIS } SHANNON } TOKYO
Physics Reports 355 (2001) 73–234
Perturbative quantum eld theory in the string-inspired formalism Christian Schubert a; b; c a
Laboratoire d’Annecy-le-Vieux de Physique Theorique LAPTH, Chemin de Bellevue, BP 110, F-74941 Annecy-le-Vieux Cedex, France b Instituto de F'sica y Matematicas, Universidad Michoacana de San Nicolas de Hidalgo, Apdo. Postal 2-82, C.P. 58040, Morelia, Michoacan, Mexico c California Institute for Physics and Astrophysics, 366 Cambridge Ave., Palo Alto, CA, USA Received December 2000; editor: A: Schwimmer
Contents 1. Introduction: strings vs. particles, rst vs. second quantization 2. The Bern–Kosower formalism 2.1. The in nite string tension limit 2.2. The Bern–Kosower rules for gluon scattering 3. Worldline path integral representations for e5ective actions 3.1. Scalar eld theory 3.2. Scalar quantum electrodynamics 3.3. Spinor quantum electrodynamics 3.4. Non-abelian gauge theory 4. Calculation of one-loop amplitudes 4.1. The N -point amplitude in scalar eld theory 4.2. Photon scattering in quantum electrodynamics 4.3. Example: QED vacuum polarization
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4.4. Scalar loop contribution to gluon scattering 4.5. Spinor loop contribution to gluon scattering 4.6. Gluon loop contribution to gluon scattering 4.7. Example: QCD vacuum polarization 4.8. N photon=N gluon amplitudes 4.9. Example: gluon–gluon scattering 4.10. Boundary terms and gauge invariance 4.11. Relation to Feynman diagrams 5. QED in a constant external eld 5.1. Modi ed worldline Green’s functions and determinants 5.2. Example: one-loop Euler–Heisenberg– Schwinger Lagrangians 5.3. The N -photon amplitude in a constant eld
E-mail address:
[email protected] (C. Schubert). c 2001 Elsevier Science B.V. All rights reserved. 0370-1573/01/$ - see front matter PII: S 0 3 7 0 - 1 5 7 3 ( 0 1 ) 0 0 0 1 3 - 8
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5.4. Explicit representations of the modi ed worldline Green’s functions 5.5. Example: the scalar=spinor QED vacuum polarization tensors in a constant eld 5.6. Example: photon splitting in a constant magnetic eld 6. Yukawa and axial couplings 6.1. Yukawa couplings from gauge theory 6.2. N scalar=N pseudoscalar amplitudes 6.3. The spinor loop in a vector and axialvector background 6.4. Master formula for the one-loop vector-axialvector amplitudes 6.5. The VVA anomaly 6.6. Inclusion of constant background elds 6.7. Example: vector–axialvector amplitude in a constant eld 7. E5ective actions and their inverse mass expansions 7.1. The inverse mass expansion for nonabelian gauge theory 7.2. Other backgrounds 8. Multiloop worldline Green’s functions 8.1. The 2-loop case 8.2. Comparison with Feynman diagrams 8.3. Higher loop orders
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8.4. Connection to string theory 8.5. Example: three-loop vacuum amplitude 9. The QED photon S-matrix 9.1. The single scalar loop 9.2. The single electron loop 9.3. The general case 9.4. Example: the two-loop QED -functions 9.5. Quantum electrodynamics in a constant external eld 9.6. Example: The two-loop Euler– Heisenberg lagrangians 9.7. Some more remarks on the two-loop QED -functions 9.8. Beyond two loops 10. Conclusions and outlook Acknowledgements Appendix A. Summary of conventions Appendix B. Worldline Green’s functions Appendix C. Symmetric partial integration Appendix D. Proof of the cycle replacement rule Appendix E. Massless one-loop 4-point tensor integrals Appendix F. Some worldloop formulas References
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Abstract We review the status and present range of applications of the “string-inspired” approach to perturbative quantum eld theory. This formalism o5ers the possibility of computing e5ective actions and S-matrix elements in a way which is similar in spirit to string perturbation theory, and bypasses much of the apparatus of standard second-quantized eld theory. Its development was initiated by Bern and Kosower, originally with the aim of simplifying the calculation of scattering amplitudes in quantum chromodynamics and quantum gravity. We give a short account of the original derivation of the Bern–Kosower rules from string theory. Strassler’s alternative approach in terms of rst-quantized particle path integrals is then used to generalize the formalism to more general eld theories, and, in the abelian case, also to higher loop orders. A considerable number of sample calculations are presented in detail, with an c 2001 Elsevier Science B.V. All rights reserved. emphasis on quantum electrodynamics. PACS: 03.70.+k; 11.10.−z
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1. Introduction: strings vs. particles, rst vs. second quantization One of the main motivations for the study of string theory is the fact that it reduces to quantum eld theory in the limit where the tension along the string becomes in nite [1– 4]. In this limit all massive modes of the string get suppressed, and one remains with the massless modes. Those can be identi ed with ordinary massless particles such as gauge bosons, gravitons, or massless spin- 12 fermions. Moreover, the interactions taking place between those massless modes turn out to be familiar from quantum eld theory. In particular, it came as a pleasant surprise that a consistent theory of fundamental strings must by necessity include quantum gravity [5]. Regardless of whether string theory is realized in nature or not, those mathematical facts already lend us a new perspective on quantum eld theory, which we now nd embedded in a theory which is not only vastly more complex, but also structurally di5erent. In particular, a major di5erence between string theory and particle theory is that, in string theory, the full perturbative S-matrix is calculable in rst quantization using the Polyakov path integral, which describes the propagation of a single string in a given background. An adequate second quantized eld theory for strings was built after considerable e5orts [6,7], allowing one to de ne o5-shell amplitudes with the correct factorization properties, and even to compute nonperturbative e5ects in string theory [8–10]. Nevertheless, so far the second quantized approach has not led to improvements over the rst quantized formalism as far as the calculation of perturbative string scattering amplitudes is concerned. In ordinary quantum eld theory, of course, perturbative calculations are usually performed using second quantization, and Feynman diagrams. First quantized alternatives have been developed already at the very inception of relativistic quantum eld theory [11], and will, in fact, be the main subject of the present review. However it appears that, until recently, they were hardly ever seriously considered as an eNcient tool for standard perturbative calculations. This discrepancy between string and particle theory becomes something of a puzzle as soon as one considers the latter as a limiting special case of the former. And we would like to convince the reader in the following that this apparent paradox owes more to historical development than to mathematical fact. If string theory reduces to eld theory in the in nite string tension limit, then clearly it should be possible to obtain S-matrix elements in certain eld theory models by analyzing the corresponding string scattering amplitudes in this limit. It goes without saying that the calculation of string amplitudes is generally much more diNcult than the calculation of the corresponding eld theory amplitudes, so that the practical value of such a procedure may appear far from obvious. Nevertheless, it turns out to be suNciently motivated by the di5erent organization of both types of amplitudes, and by the di5erent methods available for their computation. As early as 1972 Gervais and Neveu observed that string theory generates Feynman rules for Yang–Mills theory in a special gauge that has certain calculational advantages [12]. At the loop level, the rst explicit calculation along these lines was performed in 1982 by Green, Schwarz and Brink, who obtained the one-loop 4-gluon amplitude in N = 4 super Yang–Mills theory from the low energy limit of superstring theory [13]. However, serious interest in this subject commenced only in 1988, when this limit was used by several authors to obtain eld
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Fig. 1. The loop expansion in string perturbation theory.
theory -functions from various string models [14–17]. In particular, in Ref. [16] the one-loop -function for Yang–Mills theory was extracted from the genus one partition function of an open string coupled to a background gauge eld. This calculation gives also some insight into the well known fact that this -function coeNcient vanishes for D = 26, the critical dimension of the bosonic string, when calculated in dimensional regularization [280,18]. A systematic investigation of the in nite string tension limit was undertaken in the following years by Bern and Kosower [19 –21]. This was done again with a view on application to non-abelian gauge theory, however now to the computation of complete on-shell scattering amplitudes. Again the idea was to calculate, say, gauge boson scattering amplitudes in an appropriate string model containing SU (Nc ) gauge theory, up to the point where one has obtained an explicit parameter integral representation for the amplitude considered. At this stage one performs the in nite string tension limit, which should eliminate all contributions due to propagating massive modes, and lead to a parameter integral representation for the corresponding eld theory amplitude. In the present work, we will concentrate on a di5erent and more elementary approach to this formalism, which does not rely on the calculation of string amplitudes any more, and uses string theory only as a guiding principle. Only a sketchy account will therefore be given of string perturbation theory, and the reader is referred to the literature for the details [5,22]. The basic tool for the calculation of string scattering amplitudes is the Polyakov path integral. In the simplest case, the closed bosonic string propagating in Pat spacetime, this integral is of the form Dh Dx( ; )V1 · · ·VN e−S[x; h] : V1 · · ·VN ∼ (1.1) top
This path integral corresponds to rst quantization in the sense that it describes a single string propagating in a given background. The parameters ; parametrize the world sheet surface swept out by the string in its motion, and the integral Dx( ; ) has to be performed over the space of all embeddings of the string world sheet with a xed topology into spacetime. The integral Dh is over the space of all world sheet metrics, and the sum over topologies top corresponds to the loop expansion in eld theory (Fig. 1). If the closed string is assumed to be oriented, there is only one topology at any xed order of loops. In the case that the background is simply Minkowski spacetime the world sheet action is given by √ 1 S[x; h] = − d d hh 9 x 9 x ; (1.2) 4
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Fig. 2. Vertex operators inserted on the boundary of the annulus. Fig. 3. In nite string tension limit of a string diagram.
where 1=2 is the string tension. Note that in Polyakov’s formulation the action is quadratic in the coordinate eld x. The vertex operators V1 ; : : : ; VN represent the scattering string states. In the case of the open string, which is the more relevant one for our purpose, the world sheet has a boundary, and the vertex operators are inserted on this boundary. For instance, for the open oriented string at the one-loop level the world sheet is just an annulus, and a vertex operator may be integrated along either one of the two boundary components (Fig. 2). The vertex operators most relevant for us are of the form V [k] = d eik · x() ; (1.3) A ˙ eik · x() : (1.4) V [k; ; a] = d T a · x() They represent a scalar and a gauge boson particle with de nite momentum k and polarization vector . T a is a generator of the gauge group in some representation. The integration variable parametrizes the boundary in question. Since the action is Gaussian, Dx can be performed by Wick contractions, x (1 )x (2 ) = G(1 ; 2 ) ;
(1.5)
where G denotes the Green’s function for the Laplacian on the annulus, restricted to its boundary, and the Lorentz metric. In D = 26, the critical dimension of the bosonic string, the Polyakov path integral is conformally invariant. The remaining path integral over the in nite dimensional space of all world sheet metrics h can then be reduced to the space of conformal equivalence classes, which is nite dimensional. The actual integration domain, moduli space, is somewhat smaller, since a further discrete symmetry group has to be taken into account. At this stage, then, the amplitude is in a form suitable for performing the in nite string tension limit. It turns out that, in this limit, only certain corners of the whole moduli space contribute. The amplitude thus splinters into a number of pieces, which individually are parameter integrals of the same type encountered in eld theory Feynman diagram calculations.
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For illustration, consider the following two-point “Feynman diagram” for the closed string (Fig. 3). In the → 0 limit this Riemann surface gets squeezed to a Feynman graph, although not to a single one; two Feynman diagrams of di5erent topologies emerge. This proliferation becomes, of course, much worse at higher orders. Moreover, in gauge theory or gravity it is further enhanced by the existence of quartic and higher order vertices, which lead to many more possible topologies. The generating string theories thus have a much smaller number of “Feynman diagrams” then the limiting eld theories, which is another major motivation for the use of string techniques in eld theory. The uses of the Polyakov path integral are not restricted to the calculation of scattering amplitudes. As pointed out by Fradkin and Tseytlin [23], it is equally useful for the calculation of string e5ective actions. For instance, an open oriented string propagating in the background of a Yang–Mills eld A would generate an e5ective action for this background eld given by the following modi cation of the Polyakov path integral, [A]∼ Dh Dx( ; )e−S0 −SI ; top
√ 1 d d hh 9 x 9 x ; S0 = − 4 M d iex˙ A (x()) : SI = 9M
(1.6)
The sum now extends over all oriented bounded manifolds. The free term S0 is the same as above, Eq. (1.2), and the interaction term SI has to be integrated along all components of the boundary. For simplicity, we have written the interaction term for the abelian case; in the non-abelian case, a color trace and path ordering would have to be included. This will be discussed later on in the eld theory context. Metsaev and Tseytlin calculated the one-loop path integral exactly for the constant eld strength case [16], and veri ed that the → 0 limit coincides with the corresponding e5ective action in Yang–Mills theory. In particular, the correct -function coeNcient can be read o5 from 2 -term. This procedure is not completely rigorous, though, since the open bosonic string the F theory cannot be consistently truncated from 26 down to four dimensions. In their analysis of the N -gluon amplitude [21], Bern and Kosower therefore used, instead of the open string, a certain heterotic string model containing SU (Nc ) Yang–Mills theory in the in nite string tension limit. This allows for a consistent reduction to four dimensions, at the price of a more complicated representation of this amplitude. By an explicit analysis of the in nite string tension limit, they succeeded in deriving a novel type of parameter integral representation for the on-shell N -gluon amplitude in Yang–Mills theory, at the tree- and one-loop level. Moreover, they established a set of rules which allows one to construct this parameter integral, for any number of gluons and choice of helicities, without referring to string theory any more. While those rules are very di5erent from the corresponding eld theoretic Feynman rules, the precise connection and equivalence between both sets of rules were soon established [24]. Moreover, once an understanding of the rules had been reached, it emerged that the consistency requirements motivating the choice of the heterotic string were not really relevant in their
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derivation. An alternative derivation using the naive truncated open string was given [25], and even yielded a somewhat simpler set of rules. This set of rules will be discussed in detail in Section 2. For the moment, let us just mention some advantages of the “Bern–Kosower rules” as compared to the Feynman rules: 1. Superior organization of gauge invariance. 2. Absence of loop momentum, which reduces the number of kinematic invariants from the beginning, and allows for a particularly eNcient use of the spinor helicity method. 3. The method combines nicely with spacetime supersymmetry. 4. Calculations of scattering amplitudes with the same external states but particles of di5erent spin circulating in the loop are more closely related than usual. The last two points are, of course, closely related. The eNciency of these rules has been demonstrated by the rst complete calculation of the one-loop ve-gluon amplitude [26]. A similar set of rules for graviton scattering was derived from closed string theory in [27]. Those have been used for the rst calculation of the complete one-loop four-graviton amplitude in quantum gravity [28]. Since the Bern–Kosower rules do not refer to string theory any more, the question naturally arises whether it should not be possible to re-derive them completely inside eld theory. Obviously, such a re-derivation should be attempted starting from a rst-quantized formulation of ordinary eld theory, rather than from standard quantum eld theory. As we mentioned in the beginning, such formulations have been known for decades, albeit only for a very limited number of models. Already in 1950, in Appendix A of his famous paper “Mathematical Formulation of the Quantum Theory of Electromagnetic Interaction” [11], Feynman presented such a formalism for the case of scalar quantum electrodynamics, “for its own interest as an alternative to the formulation of second quantization”. What he states here is that the amplitude for a charged scalar particle to move, under the inPuence of the external potential A , from point x to x in Minkowski space is given by s d x d x 2 1 2 i s ds Dx() exp − im s exp − d −i d A (x()) 2 2 0 d d x(0) = x 0 0 s i 2 s d x d x − e (1.7) d d ! (x() − x( )) : 2 d d + 0 0
∞
x(s) = x
That is, for a xed value of the variable s (which can be identi ed with Schwinger proper time) one can construct the amplitude as a certain quantum mechanical path integral. This path integral has to be performed on the set of all open trajectories running from x to x in the xed proper time s. The action consists of the familiar kinetic term, and two interaction terms. Of those the rst represents the interaction with the external eld, to all orders in the eld, while the second one describes an arbitrary number of virtual photons emitted and re-absorbed along the trajectory of the particle (!+ denotes the photon propagator). In second quantized eld theory, this amplitude would thus correspond to the in nite sequence of Feynman diagrams shown in Fig. 4.
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Fig. 4. Sum of Feynman diagrams represented by a single path integral.
As Feynman proceeds to show, this representation extends in an obvious way to the case of an arbitrary xed number of scalar particles, moving in an external potential and exchanging internal photons, and thus to the complete S-matrix for scalar quantum electrodynamics. Every scalar line or loop is then separately described by a path integral such as the one above. The path integrals are coupled by an arbitrary number of photon insertions. The derivation of this type of path integral will be discussed in detail in Section 3. In the present work, we are mainly concerned with path integrals for closed loops. Let us therefore rewrite Feynman’s formula for the case of a single closed loop in the external eld, with no internal photon corrections. What we have at hand then is simply a representation of the one-loop e5ective action for the Maxwell eld, 1 T ∞ dT −m2 T 1 2 Dx exp − [A] = d : (1.8) e x˙ + ieA x˙ T 4 0 0 The path integral runs now over the space of closed trajectories with period T; x (T ) = x (0). The comparison of the Fradkin–Tseytlin path integral (1.6) with the path integral (1.8) shows that the former is clearly a string theoretic generalization of the latter. Conversely, at least at the one-loop level it is not diNcult to show that the Feynman path integral is precisely the in nite string tension limit of the Fradkin–Tseytlin path integral. Naively, one can think of the annulus in Fig. 2 being squeezed to its boundary. The path integral representation Eq. (1.8) generalizes in various ways to spinor quantum electrodynamics. In the fermion loop case, one has a basic choice to make in the treatment of the spin degrees of freedom. Those can be incorporated either by explicit "-matrices [29,30], or by Grassmann variables [31–34], which carry the same algebraic properties. The rst, “bosonized”, version may be preferable for certain purposes such as the evaluation of path integrals by numerical or saddle point approximation. Nevertheless, we will generally use the second alternative, since it o5ers the possibility to use worldline supersymmetry in a computationally meaningful way. Supersymmetric worldline Lagrangians were constructed soon after the advent of supersymmetry. This led to an intensive study of eld theories in 1 + 0 dimensions, and to the discovery that the worldline Lagrangian appropriate for the description of a Dirac particle is precisely the one for N = 1 supergravity [35,36]. As a consequence of that work, the generalization of our path integral Eq. (1.8) for the one-loop e5ective action to spinor QED can be written as 1
The proper time parameter s has been rescaled and Wick rotated, s → − i2T . The spacetime metric will also be taken as Euclidean, except when stated otherwise (upper and lower indices will be used purely for typographical convenience). Moreover, we anticipate dimensional regularization and thus usually continue to D Euclidean dimensions.
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a super path integral [31,37–39] 1 ∞ dT −m2 T [A] = − e 2 0 T × Dx D exp − P
A
0
T
1 2 1 d · ˙ + ieA x˙ − ie x˙ + 4 2
81
F
:
(1.9)
In addition to the integral over the periodic functions x (), we have now a second path integral over the functions (), which are Grassmann valued and antiperiodic (the periodicity properties are expressed by the subscripts P; A on the path integral). The global minus sign accounts for the Fermi statistics of the spinor loop. Comparing this formula with Eq. (1.8), it becomes immediately clear that one can think of this double path integral as breaking up the Dirac spinor into an “orbital part” and a “spin part”. The former is represented by the same coordinate path integral as the scalar particle, the latter by the additional Grassmann path integral. In writing this path integral, we have already gauge- xed the local one-dimensional supergravity. This leaves over a global supersymmetry, “worldline supersymmetry” !x = − 2 !
= x˙
; (1.10)
with a constant Grassmann parameter . As we will see later on, the existence of this symmetry has far-reaching calculational consequences. A vast amount of literature is available on this type of relativistic particle Lagrangians, and the corresponding path integrals. However, only a minor part of it is concerned with attempts to use them as a tool for actual calculations in quantum eld theory. Much of particularly the early literature emphasizes the one-dimensional over the spacetime point of view, or is concerned with the formal properties of such worldline eld theories. In particular, one-dimensional eld theories are often used for a comparative study of the various known quantization procedures (see, e.g., [40]). Of those applications which have come to this author’s notice, let us mention the work of Halpern et al. [41,42], who proposed to use rst quantized path integrals for a construction of the strong-coupling expansion in non-abelian gauge theory. More recently, various attempts have been made to apply worldline path integrals to nonperturbative calculations, using various approximation schemes for the path integral. See, e.g., [43] for scalar eld theory, [44] for heavy-meson theory, [45,46] for QCD, and [47] for meson–nucleon theory applications. Some applications to QED can be found in [48,49,38,50 –53], to statistical physics in [54]. Probably best known is, however, the application to the calculation of anomalies and index densities [55 – 62]. Here a number of special cases of the Atiyah–Singer index theorem could be reproduced in an elementary way by rewriting supertraces of heat kernels for the corresponding operators in terms of supersymmetric particle path integrals [56 –58]. Nevertheless, despite this remarkable success it seems that, until recently, the rst quantized formalism was never seriously considered as a competitor to the usual Feynman diagrammatic approach with regard to everyday life calculations of scattering amplitudes or e5ective actions.
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Fig. 5. Expanding the path integral in powers of the background eld.
The principle of how one might reproduce ordinary perturbation theory in the rst quantized formalism, simply by mimicking string perturbation theory, was already sketched in Section 9 of Polyakov’s book [63]. However it was only after the work of Bern and Kosower, when it had become clear that techniques from rst-quantized string perturbation theory do have the potential to improve on the eNciency of eld theory calculations, that such an approach was seriously investigated by Strassler [64,65]. Let us demonstrate the method for the example of the scalar loop, Eq. (1.8). The basic idea is simple: We will evaluate this path integral in precisely the same way as one calculates the Polyakov path integral in string theory, i.e. in a one-dimensional perturbation theory. If we expand the “interaction exponential”, T ∞ N (−ie)N T exp − d ieA x˙ = di [x˙ (i )A (x(i ))] ; (1.11) N ! 0 0 N =0
i=1
the individual terms correspond to Feynman diagrams describing a xed number of interactions of the scalar loop with the external eld (Fig. 5). By standard eld theory, the corresponding N -photon correlator is then obtained by specializing to a background consisting of a sum of plane waves with de nite polarizations, A (x) =
N
i eiki · x
(1.12)
i=1
and picking out the term containing every i once (this also removes the 1=N ! in Eq. (1.11)). We nd thus exactly the same photon vertex operator used in string perturbation theory, Eq. (1.4), inserted on a circle instead on the boundary of the annulus. At this stage the path integral has become Gaussian, which reduces its evaluation to the task of Wick contracting the expression x˙1 1 eik1 · x1 · · ·x˙NN eikN · xN :
(1.13)
The Green’s function to be used is now simply the one for the second-derivative operator, acting on periodic functions. To derive this Green’s function, rst observe that Dx() contains the constant functions, which we must get rid of to obtain a well-de ned inverse. Let us therefore restrict our integral over the space of all loops by xing the average or “center of mass” position x0 of the loop 1 T d x () : (1.14) x0 ≡ T 0
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For e5ective action calculations this reduces the e5ective action to the e5ective Lagrangian. In scattering amplitude calculations, the integral over x0 just gives momentum conservation. The reduced path integral Dy() over y() ≡ x() − x0 has an invertible kinetic operator. The inverse is easily seen to be, up to an irrelevant constant, −2 d 21 | |2 = GB (1 ; 2 ) (1.15) d with the “bosonic” worldline Green’s function (1 − 2 )2 (1.16) T (a “fermionic” worldline Green’s function GF will be introduced later on). For the performance of the Wick contractions, it is convenient to formally exponentiate all the x˙i ’s, writing GB (1 ; 2 ) = |1 − 2 | −
i · x˙i eiki · xi = ei · x˙i +iki · xi |lin(i ) :
(1.17)
This allows one to rewrite the product of N photon vertex operators as an exponential. Then one needs only to “complete the square” to arrive at the following closed expression for the one-loop N -photon amplitude in scalar QED, scal [k1 ; 1 ; : : : ; kN ; N ] N D ki = (−ie) (2) !
0
×
N
i=1 0
T
∞
dT 2 (4T )−D=2 e−m T T
N 1 1 di exp GBij ki · kj − iG˙ Bij i · kj + GS Bij i · j 2 2 i; j = 1
: (1.18) multi-linear
Here it is understood that only the terms linear in all the 1 ; : : : ; N have to be taken. Besides the Green’s function GB also its rst and second derivatives appear, (1 − 2 ) G˙ B (1 ; 2 ) = sign(1 − 2 ) − 2 ; T 2 GS B (1 ; 2 ) = 2!(1 − 2 ) − : (1.19) T Dots generally denote a derivative acting on the rst variable, G˙ B (1 ; 2 ) ≡ 9= 91 GB (1 ; 2 ), and we abbreviate GBij ≡ GB (i ; j ), etc. The factor [4T ]−D=2 represents the free Gaussian path integral determinant factor. Expression (1.18) which we have arrived at in this quite elementary way is identical with the “Bern–Kosower master formula” for the special case considered [21,66]. We will discuss various generalizations and applications of this formula later on. For now, the important point to note is that we have at hand here a single unifying generating functional for the one-loop photon S-matrix—something for which no known analogue exists in standard eld theory. How does this master integrand relate to the integrals appearing in an ordinary Feynman parameter calculation of this amplitude? Note that in (1.18) every photon leg is integrated
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Fig. 6. Sum of one-loop diagrams with permuted legs.
Fig. 7. Sum of two-loop diagrams with di5erent topologies.
around the loop independently. As we will see in detail later on, once one restricts the integration domain to a xed ordering i1 ¿i2 ¿· · ·¿iN , it is not diNcult to identify the integrand with the corresponding Feynman parameter integral. In particular, there is an exact correspondence between the !-function appearing in the second derivative of GB , and the seagull vertex of scalar quantum electrodynamics. However the complete integral does not represent any particular Feynman diagram, with a xed ordering of the external legs, but the sum of them (Fig. 6): This fact may not seem particularly relevant at the one-loop level. However it is important to note that the path integral representation Eq. (1.8) and the resulting integral representation Eq. (1.18) are valid o5-shell. 2 We can therefore use this formula to sew together a pair of legs, say, legs number 1 and N , and obtain a parameter integral representing the complete two-loop (N − 2)-photon amplitude (Fig. 7): This is interesting, as we have at hand a single integral formula for a sum containing many diagrams of di5erent topologies. We may think of it as a remnant of the “less fragmented” nature of string perturbation theory mentioned before (Fig. 3). Moreover, it calls certain well-known cancellations to mind which happen in gauge theory due to the fact that the Feynman diagram calculation splits a gauge invariant amplitude into gauge non-invariant pieces. For instance, to obtain the 3-loop -function coeNcient for quenched (single spinor-loop) QED, one needs to calculate the sum of diagrams shown in Fig. 8. Performing this calculation in, say, dimensional regularization, one nds that 1. All poles of order higher than 1= cancel. 2
This fact was not obvious in the original string-theoretic derivation of (1.18), since before the in nite string tension limit the requirement of conformal invariance forces the external states to be on-shell.
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Fig. 8. Sum of diagrams contributing to the 3-loop QED -function.
2. Individual diagrams give contributions to the -function proportional to *(3) which cancel in the sum, leaving a rational coeNcient. The rst property is known to be a consequence of gauge invariance, and to hold true to all orders of perturbation theory [67]. The second one, i.e. the absence of irrational numbers in the quenched QED -function, has been explicitly veri ed to four-loop order in spinor QED [68,69], and to three-loop order in scalar QED [70]. Recently arguments from knot theory have been given which link both properties [70], indicating that this property should hold to all orders, too. As every practician in quantum eld theory knows, similar extensive cancellations abound in calculations in gauge theory. It seems therefore very natural to apply the Bern–Kosower formalism to this type of calculation. However, in its original version the Bern–Kosower formalism was con ned to tree-level and one-loop amplitudes. The extension of this formalism beyond one loop is obviously desirable, and has already been attempted along quite di5erent lines: In the original approach of Bern and Kosower, going beyond one loop would imply nding the particle theory limits of higher genus string amplitudes, a formidable task considering the complicated structure of moduli space for genus higher than one. While a suitable representation of the N -gluon amplitude at arbitrary genus was already given in [71], and substantial progress was achieved in the analysis of the in nite string tension limit [72–78], so far this line of work has not yet led to the formulation of multiloop Bern–Kosower type rules. Another, and in some sense opposite route has been taken by Lam [79], who sets out with the usual Feynman parameter integral representation of multiloop diagrams, and uses the electric circuit analog [80,81] to transform those into the Koba–Nielsen type representation which one would expect from a string-type calculation. Yet another approach has been followed by McKeon [82], who proposes to perform multiloop calculations by writing a separate worldline path integral for every internal propagator of a diagram. A Hamiltonian approach was considered in [83]. In principle one could, of course, also construct Bern–Kosower type multiloop formulae using the explicit sewing procedure indicated above. However, we will describe another multiloop formalism here, proposed by M.G. Schmidt and the author [84,85], which is based on a more eNcient way of inserting propagators into one-loop amplitudes. This approach is a direct generalization of Strassler’s one-loop formalism, and preserves its main properties. Its distinguishing features are the following: 1. We will generalize Eq. (1.16) for the one-loop Green’s function to the construction of Green’s functions de ned on multiloop graphs. 2. The super eld formalism for the fermion loop will carry over to the multiloop level. 3. All eld theory vertices will be represented by worldline quantities.
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Though not explicitly referring to string theory any more, the resulting formalism may still be called “string-inspired” in the sense that it has a natural interpretation in terms of a onedimensional eld theory de ned on graphs. As one would expect from a generalization of the Bern–Kosower method, it allows one to derive well-organized parameter integral representations for dimensionally regularized o5-shell amplitudes, without the need for computing momentum integrals or Dirac traces. At the multiloop level, this formalism has been worked out comprehensively for scalar eld theories [84,86 –90] as well as for scalar and spinor QED [85,91–93]. In those models, in principle it applies to the calculation of arbitrary o5-shell amplitudes involving only spin 0 and spin 1 scattering states, or of the corresponding e5ective Lagrangians. More recently along these lines preliminary results have been obtained also for Yang–Mills Theory at the two-loop level [94,95]. Our applications center around the photon S-matrix in quantum electrodynamics as our main paradigm. Here the formalism has been developed to a point where it shows some distinct advantages over the more standard methods, in particular for problems involving constant external elds. At the one-loop level, we present also a number of calculations involving Yukawa and axial couplings, as well as non-abelian gauge elds. The material is organized as follows. In Section 2 we state the Bern–Kosower rules for the case of gluon scattering, and shortly sketch their derivation from the open bosonic string. In Section 3 we give derivations for the most basic worldline Lagrangians used in this work, describing the coupling of particles with spin 0; 12 , and 1 to external gauge elds. The starting point in this derivation is always the proper-time representation of the one-loop e5ective action in terms of the one-loop functional determinant. We discuss in particular detail the path integral representation of spin-1 particles [64,92], since here the application of the string-inspired technique requires a non-standard approach. The principle of how to calculate such path integrals within the “string-inspired formalism” is then explained in Section 4. The advantages of the technique compared to the standard Feynman diagram technique are then demonstrated using the example of the QED and QCD vacuum polarizations, and QCD gluon–gluon scattering. We investigate the systematics of the Bern–Kosower partial integration procedure for the general N -photon=N gluon amplitudes, and determine the structure of the resulting integrand. We also clarify the relation of the worldline parameter integrals to the ones arising in standard Feynman parameter calculations of the same amplitudes. Section 5 is devoted to the treatment of QED amplitudes in a constant electromagnetic background eld. This case is given special attention since it provides a particularly natural application of the string-inspired technique [96 –100,92]. In Section 6 we consider more general eld theories. While worldline path integral representations have been known and investigated for decades for the case of spin-0 and spin- 12 particles minimally coupled to gauge and gravitational backgrounds, worldline Lagrangians describing the coupling of a Dirac fermion to a general background consisting of a scalar, pseudoscalar, vector, axialvector and antisymmetric tensor eld were obtained only recently [101–105]. In the present review we restrict ourselves to two special cases which admit particularly elegant formulations, namely the scalar–pseudoscalar and the vector–axialvector amplitudes.
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Section 7 deals with the application of the formalism to the calculation of e5ective actions in the higher derivative or heat-kernel expansion [96,106,107]. While our discussion concentrates on the gauge theory case, we also shortly discuss some mathematical subtleties which arise in the generalization of the formalism to curved backgrounds, and which were clari ed only very recently. In Section 8 we generalize the worldline formalism to the calculation of multiloop amplitudes in scalar self-interacting eld theories. This generalization is based on the concept of Green’s functions de ned on graphs, and follows the string analogy closely. We derive explicit expressions for those Green’s functions for a large class of graphs, the so-called “Hamiltonian graphs”, and verify for some simple two-loop examples that their application reproduces the same amplitudes as the corresponding Feynman diagram calculations. This formalism is then generalized to quantum electrodynamics in Section 9, where in principle it applies to the calculation of the whole photon S-matrix. At the two-loop level, we present a detailed recalculation [92,108] of the Euler–Heisenberg Lagrangians in scalar and spinor QED, including the -function coeNcients. An interesting cancellation which occurs in the spinor QED case is explained by an analysis of the renormalization procedure. In the conclusion we give a short overview over the present range of applicability of the worldline technique, and point out some possible future directions. There are several technical appendices. Appendix A contains a summary of the conventions used in the present work, 3 including the rules for continuation from Euclidean to Minkowski space. In Appendix B we give detailed calculations of the various worldline propagators which are used in the main text. Appendix C contains a more detailed discussion of the partial integration procedure introduced in Section 4.8, and a summary of the resulting worldline integrands up to the six-point case. The results are used in Appendix D for a simple proof of the basic fermionic replacement rule (2.15). In Appendix E we explain a technique for the calculation of four-point massless on-shell tensor parameter integrals, following [109]. Finally, Appendix F contains a collection of formulas which we have found useful, or at least amusing. 2. The Bern–Kosower formalism This section is devoted to a statement of the Bern–Kosower rules, and to a short account of their derivation from string theory. We follow not the original derivation from the heterotic string [19 –21] but the simpler one using the open bosonic string, as given in [25,66]. 2.1. The in
Those di5er in some points from previous work by this author.
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spacetime, and the integral over moduli space. In the case of the annulus there is only one modular parameter [5] = − 12 ln(q) ;
(2.1)
where q can be interpreted as the square of the ratio of the two radii de ning the annulus. Since the two-dimensional worldsheet theory is free, the path integral can be computed by a repeated application of Wick’s theorem, using the Green’s function GBann for the Laplacian on the annulus. If we assume all of the vertex operators to be on the same boundary, one has explicitly 212 2 ann y (1 )y (2 ) = g GB (1 ; 2 ) = − g ln |2 sinh(12 )| − − 4q sinh (12 ) + O(q2 ) ; (2.2) where denotes the length of the boundary, i the location of the ith vertex operator along the boundary, N = 0, and ij ≡ i − j . One obtains a parameter integral (compare Eq. (1.18)) ∞ N
−1 (N=2−2) −D=2 d Z() di /(i − i+1 ) [k1 ; 1 ; : : : ; kN ; N ]∼( ) 0
×exp
N
i¡j = 1
i=1
0
1 ann 1 √ ˙ ann ann GBij ki · kj + G Bij (ki · j − kj · i ) − GS Bij i · j 2 4
: (2.3) multi-linear
Here we have omitted the color trace and some global factors. Z() is essentially the string vacuum partition function, and given by Z = q−1
∞
(1 − q n )−2 = q−1 + 2 + O(q) :
(2.4)
n=1
The analysis of the in nite string tension limit → 0 can be simpli ed by rst removing all ann second derivatives GS Bij by suitable partial integrations in the variables i . This is always possible [20], and will be discussed later on in the eld theory context. The integrand then becomes homogeneous in . The possible boundary terms appearing in the integration by parts can be made to vanish by a suitable analytic continuation in the external momenta. In the → 0 limit, rst one has to extract massless poles in the S-matrix, which can appear for regions where i → j . Those are of the form 1 1 → 0 di 1+ ki · kj → − (2.5) k i · kj ij and yield the so-called “tree” or “pinch” contributions. The limit itself is to be taken on the sum of the unpinched expression together with all pinch contributions. It is consumed by taking ; |ij | → ∞, which corresponds to the ratio of radii approaching 1, and thus to the annulus ann being squeezed to a eld theory loop. Analyzing GBann and G˙ B in this limit, one nds that they
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can be replaced by
exp[GBann (12 )] → const:×exp
212 − |12 | ;
89
(2.6)
12 ann G˙ B (12 ) → − sign(12 ) + 2 (2.7) + 2 sign(12 )(qe2|12 | − e−2|12 | ) : Since this limit corresponds to q → 0, and the expansion of Z as a power series in q starts with a q−1 , there are again two types of contributions. The rst type arises by picking the next-to-leading constant term in Z. Then only the leading ann order terms from the integrand can survive the limit, so that G˙ B is further truncated to the rst two terms of Eq. (2.7). The second type is obtained by combining the leading order term from Z with a next-to-leading ann term in the integrand. Then subleading terms in the G˙ Bij can potentially contribute, however it turns out that a too strong suppression of the integrand for q → 0 can be avoided only for those ann terms in the integrand which contain a closed cycle of G˙ Bij ’s, i.e. a factor ann ann ann G˙ Bi1 i2 G˙ Bi2 i3 · · ·G˙ Bin i1 :
(2.8)
Even then, the cycle can only survive the limit if the indices follow the ordering of the external legs. Of course it is also possible to combine the leading order terms from both Z and the integrand. This produces terms which diverge in the limit. Those are discarded, since they can be identi ed as being due to an unphysical tachyonic scalar circulating in the loop. In this way one arrives at the Bern–Kosower rules for the gluon loop as given in [21,25]. In the following we will give those rules in a slightly di5erent version, which is more in line with the worldline path integral approach with respect to the treatment of the color algebra. The original string-based approach naturally yields the gluon amplitudes in color-decomposed form, i.e. with the group theory factors expressed in terms of the fundamental instead of the adjoint representation [110,111]. If in the above derivation the gluon vertex operators are taken to be in the fundamental representation, as it was done in [21,25], then the resulting eld theory amplitude represents the so-called “leading color partial amplitude”, where leading refers to the large Nc limit of SU (Nc ) gauge theory. (There is an extra overall factor of Nc in this approach, coming from a second index in the fundamental representation, which is untouched by all the gluon vertex operators, leading to tr(1) = Nc .) The missing subleading amplitudes would be obtained by the inclusion of string theory amplitudes with vertex operator insertions on both boundaries of the annulus, although it turns out that they can also be constructed directly as sums of permutations of the leading color amplitude [112,113]. The form of the spin 1 rules which we give here instead does not use color decomposition; in Eqs. (2.18), (2.19) below, as well as in the remainder of this review, a color matrix T a always refers to the adjoint representation in the gluon loop case. The equivalence of these rules to the original version of [21,25] follows from recent work on the color decomposition [113]. The analogous rules for the spin 12 loop can be derived by repeating the same analysis for the open superstring. Remarkably, this leads to a rule which allows one to infer all contributions from worldsheet fermions to the nal integrand from the purely bosonic terms. This “replacement rule” will play a prominent role in many of our applications. Its validity can be shown to be
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a consequence of worldsheet supersymmetry [66]. For the spin 0 loop, one nds only the rst type of contributions above. 2.2. The Bern–Kosower rules for gluon scattering For the statement of the rules, we rewrite them in the conventions used throughout the remainder of this work. In particular, we work in the Euclidean. The Bern–Kosower rules give a prescription for the construction of integral representations for one-loop photon or gluon scattering amplitudes in non-abelian gauge theory [21]. We write them down here for the case of a non-abelian gauge theory with massless scalars and fermions. To obtain the one-loop on-shell amplitude for the scattering of N gluons, 4 with momenta ki and polarization vectors i , the following steps have to be taken: Step 1: Consider the following kinematic expression:
N
˙ S K= dui exp[GBij ki · kj + iG Bij (ki · j − kj · i ) + G Bij i · j ] ; (2.9) i¡j i=1 multi-linear where “multi-linear” means that only terms linear in each of the 1 ; : : : ; N are to be kept. Step 2: An on-shell gluon has only two physical polarizations, denoted by “+” and “−”. Consider one helicity amplitude at a time, and denote, for example, by A(+; +; −; · · ·; −) the amplitude for the process where the rst two gluons have the same helicity, and all remaining ones the opposite one. Each helicity amplitude is separately gauge invariant, i.e. insensitive to a rede nition of any of the polarization vectors by a transformation i± → i± + 4ki :
(2.10)
This freedom can be used to choose—for given external momenta {k1 ; : : : ; kN }—a set of polarization vectors which makes a maximal number of the invariants ki · j and i · j vanish. A systematic way of nding such a set of polarization vectors for a given choice of helicities is provided by the Spinor helicity method (see, e.g., [66,114]). At the end, all surviving invariants are rewritten as functions of the external momenta alone. Step 3: Expand out the kinematic expression, and perform integrations by parts, till all double derivatives of GB are removed (ignore boundary terms). We have now an expression
N
dui Kred exp[GBij ki · kj ] ; (2.11) i=1
i¡j
where Kred , the “reduced kinematic factor”, is a sum of terms that are products of G˙ Bij ’s, and of dot products ki · kj ; ki · j ; i · j . Step 4: Draw all possible labelled 3 1-loop diagrams Di with N external legs (Fig. 9): but excluding tadpoles, and diagrams where the loop is isolated on an external leg (Fig. 10): The labels follow the cyclic ordering of the trace. One also attaches a label to every internal line in the tree part of a diagram; for de niteness, this is taken to be the smallest one of the labels of the two lines into which the line splits to the outward. 4
By “gluon” we denote any non-abelian gauge boson.
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Fig. 9. Diagrams in 3 theory. Fig. 10. Diagrams to be omitted.
Fig. 11. Removal of trees.
Every diagram Di contributes a parameter integral 1 ui ui 1 m −2 D Pi (ui1 ; : : : ; uim ) Di = m − dui1 dui2 · · · duim−1 : m 2 [ − r¡s GBir is Kir · Kis ]m−(D=2) 0 0 0
(2.12)
Here D is the space-time dimension, m is the number of legs directly attached to the loop, and GBir is = GB (uir ; uis ) = |uir − uis | − (uir − uis )2 = (uir − uis ) − (uir − uis )2
(2.13)
(uim = 0): Kir denotes the sum of the external momenta Powing into the tree which enters the loop at the point carrying the label ir . Pi is a polynomial function of the loop parameters, and of the external momenta and polarization vectors; it will be determined in Steps 5 and 6. Step 5: tree replacement rules. Remove all trees, working from the outside of the diagram toward the loop. If the diagram contains a vertex as shown in Fig. 11 keep only those terms in Kred which contain exactly one G˙ Bij . In those, replace G˙ Bij , with i¡j, by 2=k 2 , and replace all remaining G˙ Bjr by G˙ Bir . Repeat this procedure, till only the naked loop is left. Step 6: loop replacement rules. It is only at this stage that one has to distinguish between the scalar, the fermion, and the gluon loop. Scalar loop: Simply write out Kred (i.e. what became of Kred in step 5) in terms of the integration variables, by substituting G˙ Bij → sign(ui − uj ) − 2(ui − uj ) :
(2.14)
Multiply by an overall factor of 2 if the scalar is complex. Spinor loop: Replace simultaneously every closed cycle G˙ Bi1 i2 G˙ Bi2 i3 · · ·G˙ Bik i1 appearing in Kred (which may or may not follow the ordering of the external legs) by G˙ Bi1 i2 G˙ Bi2 i3 · · ·G˙ Bik i1 − GFi1 i2 GFi2 i3 · · ·GFik i1 :
(2.15)
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An expression is considered a cycle already if it can be put into cycle form using the antisymmetry of G˙ B (e.g. G˙ Bij G˙ Bij = − G˙ Bij G˙ Bji ). Then replace all G˙ B ’s as in the scalar case, and all GF ’s by GFij → sign(ui − uj ) :
(2.16)
Multiply by an overall factor of −4 for a Dirac fermion −2 for a Weyl fermion Gluon loop: In this case, there are two types of contributions, which have to be summed: type 1: Replace G˙ Bij as above. type 2: For every closed cycle of G˙ Bij ’s appearing in a term, with the ordering of the indices following the ordering of the external legs, write down one additional contribution, obtained in the following way: replace G˙ Bi1 i2 G˙ Bi2 i1 → 4 ; G˙ Bi1 i2 G˙ Bi2 i3 · · ·G˙ Bik i1 → 2k−1
(k¿2)
(2.17)
and all remaining G˙ B ’s—even those belonging to other cycles—as in the scalar case. Multiply by a factor of 2 for both types. The expression obtained from Kred in this way is the polynomial Pi above. Step 7: Perform the parameter integrations. Techniques for their calculation may be found in [109,115,116]. The four-point case is treated in Appendix E, following [109]. Step 8: Finally the amplitude is given by a sum over all diagrams, with an overall normalization factor (42 )−j=2 a1 ···aN [k1 ; 1 ; : : : ; kN ; N ] = (−ig)N tr(T a1 · · ·T aN ) Di : (2.18) 322 diagrams
Here j = D − 4, and is the usual unit of mass appearing in the dimensional continuation. T ai is a color matrix in the representation of the loop particle. We have specialized to a speci c version of dimensional regularization, the so-called four-dimensional helicity scheme [21]. This is only the partial amplitude corresponding to the considered xed ordering of the external states. To obtain the complete amplitude one must still sum over all possible non-cyclic permutations of the states, so that ({ai ; ki ; i }) = a(1) ···a(N ) [k(1) ; (1) ; : : : ; k(N ) ; (N ) ] : (2.19) ∈ SN =ZN
In Section 4 we will explicitly apply these rules to the four-gluon case. See [27,28] for the analogous rules for graviton scattering in quantum gravity. 3. Worldline path integral representations for e&ective actions In this section, we derive worldline path integral representations for a number of one-loop e5ective actions involving some of the most basic interactions in quantum eld theory. Those
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derivations are based on the fact that one-loop e5ective actions can generally be expressed in terms of the determinant of the kinetic operator in eld theory. Using the ln(det) = tr(ln)-formula and the Schwinger proper-time representation, one obtains an integral over the space of all closed trajectories of a quantum mechanical particle moving in spacetime. Generally, to every such closed loop path integral one nds associated a similar open-ended path integral, representing the eld theory propagator of the loop particle in the background eld. The propagator path integral has the same worldline Lagrangian, possibly with some boundary terms added. It is to be performed over the space of trajectories connecting two xed points in spacetime, with appropriate boundary conditions. In the present review we will concentrate on the e5ective action, or closed loop case, simply because almost all explicit calculations which have been done so far pertain to this case. By this we do not mean to imply that propagator path integrals may not play an important role in future extensions of this formalism. Our derivations are mostly formal. Only in the spin-1 loop case will we touch upon the subtle issues connected with the existence of di5erent discretization prescriptions, etc. Those have recently been investigated in much detail for the case of curved backgrounds, a subject which will be shortly discussed in Section 7.2. 3.1. Scalar <eld theory Let us begin with the simplest case of a real massive scalar eld with a self-interaction potential U (). According to standard quantum eld theory (see, e.g., [117]) the Euclidean one-loop e5ective action for this eld theory can be written as 5 − + m2 + U () 1 [] = − Tr ln : (3.1) 2 − + m2 We use the formula ∞ A dT −Tr ln = Tr(e−AT − e−BT ) B T 0
(3.2)
valid for positive de nite operators A; B, delete the irrelevant -independent term, and perform the functional trace in x-space. This gives 1 ∞ dT [] = d D xx|exp{−T [ − + m2 + U ((x))]}|x : (3.3) 2 0 T Now compare this with Feynman’s path integral formula for the evolution operator in nonrelativistic quantum mechanics. For a particle with mass m˜ moving in a time-independent potential V˜ (x) this formula reads (see, e.g., [118,119]), x(t ) = x t −i(t −t )H ˜ x˙2 −V˜ (x)] x |e |x = Dx(t) ei t dt[(m=2) : (3.4) x(t ) = x
5
We work with relativistic quantum eld theory conventions, ˝ = c = 1. Functional traces are denoted by Tr, nite dimensional traces by tr.
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We can therefore interpret our kinetic operator above as the Hamilton operator H for a ctitious particle moving in D dimensions, p2 + V˜ (x) ; 2m˜ by identifying H=
(3.5)
V˜ (x) = m2 + U ((x)) ; m˜ = 12 ; i(t − t ) = T :
(3.6)
Without retracing the usual path integral discretization procedure [118,119] we can thus immediately write x(T ) = x T 2 2 x| exp{−T [ − + m2 + U ((x))]}|x = Dx()e− 0 d[1=4x˙ +m +U ((x()))] (3.7) x(0) = x
( = it). Taking into account that dD x Dx() = x(0) = x(T ) = x
x(0) = x(T )
Dx() ;
(3.8)
we obtain the desired path integral representation for the e5ective action, T 2 1 ∞ dT −m2 T [] = Dx()e− 0 d(1=4x˙ +U ((x()))) : e 2 0 T x(T ) = x(0)
(3.9)
In a completely analogous way one derives the path integral representation for the scalar propagator in the background eld , ∞ −1 2 x |[ − + m + U ((x))] |x = dT x | exp{−T [ − + m2 + U ((x))]}|x
=
0
∞
2
dT e−m T
0
x(T ) = x
x(0) = x
Dx()e−
T 0
d(1=4x˙2 +U ((x())))
:
(3.10)
3.2. Scalar quantum electrodynamics The path integral for a massive (complex) scalar eld minimally coupled to a background Maxwell eld can also be found simply by recurring to quantum mechanics. The eld theory kinetic operator now reads (9 + ieA)2 − m2
(3.11)
with a ctitious Hamiltonian H=
(p + eA)2 + m2 : 2m˜
(3.12)
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This translates into −(9 + ieA)2 + m2 1 scal [A] = − Tr ln 2 − + m2 ∞ T 2 dT −m2 T = Dx() e− 0 d(1=4x˙ +iex˙ · A(x())) e T x(T ) = x(0) 0
95
(3.13)
and analogously for the propagator. Note that the global factor of 12 has disappeared, since in taking the trace we have to take the double number of degrees of freedom of the complex scalar into account. 3.3. Spinor quantum electrodynamics For the spin 12 -particle various worldline path integral representations can be found in the literature. The basic choice is between bosonic [29,30] and Grassmann [31– 40] representations. The latter can be derived using either coherent state methods [120,121,102,104] or the “Weyl symbol” method [34,122]. Our following treatment of the spin 12 -case uses the coherent state formalism. We would like to nd a path integral representation for the Euclidean e5ective action 6 spin [A] = ln Det[ p= + e A= − im] :
(3.14)
This time we will not be able to just take over results from quantum mechanics; we have to construct our path integral by brute force. Let us start with the well-known observation that we can rewrite i ( p= + e A=)2 = − (9 + ieA )2 − e F (3.15) 2 ( = 12 [" ; " ]). Using the usual argument that Det[( p= + e A=) − im] = Det[( p= + e A=) + im] = Det1=2 [( p= + e A=)2 + m2 ] ; we can then write the e5ective action in the following form: ∞ 1 dT i 2 2 spin [A] = − Tr : exp −T −(9 + ieA) − e F + m 2 T 2 0
(3.16) (3.17)
Up to the global sign, this is formally identical with the e5ective action for a scalar loop in a background containing, besides the gauge eld A, a potential term i V ≡ − e F : (3.18) 2 We will now use the formalism developed in [120] to transform this functional trace into a quantum mechanical path integral. Our treatment closely parallels the one in [102,104], except − that they work in six dimensions. De ne matrices a+ r and ar , r = 1; 2, by 1 a± 1 = 2 ("1 ± i"3 ); 6
1 a± 2 = 2 ("2 ± i"4 ) :
Our Euclidean Dirac matrix conventions are {" ; " } = 2g 5; "† = " ; "5 = "1 "2 "3 "4 .
(3.19)
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Those satisfy Fermi–Dirac anticommutation rules − {a+ r ; as } = !rs ;
a+ r
Thus we can use vacuum de ned by
+ − − {a+ r ; as } = {ar ; as } = 0 :
and
a− r
(3.20)
as creation and annihilation operators for a Hilbert space with a
+ a− r |0 = 0|ar = 0 :
(3.21)
Next we introduce Grassmann variables r and Xr , r = 1; 2, which anticommute with one another and with the operators a± r , and commute with the vacuum |0. The coherent states are then de ned as − | ≡ i0|(1 − a− 1 )(2 − a2 )
+ | ≡ exp(−1 a+ 1 − 2 a2 )|0 ;
− X| ≡ 0| exp(−a− 1 X1 − a2 X2 )
+ | ≡ i(X1 − a+ 1 )(X2 − a2 )|0 :
(3.22)
It is easily veri ed that those satisfy the de ning equations for coherent states, |a− r = |r
a− r | = r | |X = exp(1 X1 + 2 X2 ) ;
X|a+ r = X|Xr
a+ r |X = Xr |X X| = exp(X1 1 + X2 2 ) :
Also one introduces the corresponding Grassmann integrals, de ned by i di = Xi d Xi = i :
(3.23) (3.24)
The dr ; d Xr commute with one another and with the vacuum, and anticommute with all Grassmann variables and the a± . This leads to the completeness relations r 5 = i || d 2 = − i d 2 X |XX| (3.25) (d 2 = d2 d1 ; d 2 X = d X1 d X2 ), and to the following representation for a trace in the Fock space generated by the a± , r Tr(U ) = i d 2 −|U | : (3.26) We can now apply these fermionic coherent states together with the usual complete sets of coordinate states to rewrite the functional trace in (3.17) in the following way: −T: 4 Tr e = i d x d 2 x; −|e−T: |x; =i
N
N
(d 4 xi d 2 i xi ; i |e−(T=N ): |xi+1 ; i+1 ) ;
(3.27)
i=1
where : = − (9 + ieA)2 + V . The boundary conditions on the x and integrations are (xN +1 ; N +1 ) = (x1 ; −1 ). For the evaluation of this matrix element it will be useful to look rst at the matrix elements of products of Dirac matrices. For the product of two "’s one nds i i+1 |" " | = − i d 2 Xi; i+1 i |Xi; i+1 Xi; i+1 |i+1 2 i i+1 ; = ; (3.28)
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where i+1 1; 2 ≡
1 2
i; i+1 √ (i+1 1; 2 + X1; 2 );
i+1 3; 4 ≡
i
i; i+1 √ (i+1 1; 2 − X1; 2 ) ;
2
1 i i i; i+1 i (3.29) 3; 4 ≡ √ (1; 2 − X1; 2 ) : 2 2 To verify this equation one rewrites the Dirac matrices in terms of the a± r and then inserts a complete set of coherent states |Xi; i+1 in between them. With this information it is now easy to compute that i
1; 2 ≡
√ (i1; 2 + Xi;1;i+1 2 );
xi ; i |e−(T=N ):[p; A; " " ] |xi+1 ; i+1 i i; i+1 i i+1 i; i+1 i i+1 =− d 4 pi; i+1 d 2 Xi; i+1 ei(x −x )p +( − )r Xr 4
(2)
T × 1 − :[pi; i+1 ; Ai; i+1 ; 2i N
i+1 ]
T2 +O N2
:
(3.30)
Here the superscript i; i+1 on a eld denotes the average of the corresponding elds with superscripts i and i +1. Inserting this result back into Eq. (3.27) one obtains, after symmetrizing the positions of the Grassmann variables in the exponentials, Tr e−T: =
1 (2)4N ×exp
N i=1
2 T T d 4 xi d 4 pi; i+1 d 2 i d 2 Xi; i+1 1 − :i + O N N2
N
i
i(x − x
i=1
i+1
)p
i; i+1
1 i 1 i i−1; i i+1 i; i+1 i; i+1 + (r − r )Xr − r (Xr − Xr ) : 2 2
(3.31)
Introducing an interpolating proper-time such that 1 = T , N +1 = 0, and i − i+1 = T=N , and taking the limit N → ∞ in the usual naive way, we nally obtain the following path integral representation, T 1 ˙ 1 −T: Tr e = Dp Dx DDX exp d ix˙ · p + ˙r Xr − r Xr − :[p; A; 2 ] : 2 2 A 0 (3.32) The “A” denotes the antiperiodic boundary conditions which we have for ; . X The continuum limits of Eqs. (3.29) are 1 i (3.33) 1; 2 () = √ (1; 2 () + X1; 2 ()); 3; 4 () = √ (1; 2 () − X1; 2 ()) : 2 2 This suggests a change of variables from ; X to , which we complete by rewriting the fermionic kinetic term 1 ˙ X − 1 r X˙ = − 1 ˙ : (3.34) 2 r r
2
r
2
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The boundary conditions are now (x(T ); (T )) = (x(0); − (0)). Finally, we note that the momentum path integral is Gaussian, and perform it by a naive completion of the square (for a less unscrupulous treatment of this point see again [102,104], as well as for the various normalization factors involved). This brings us to our following nal result: 7 T 1 ∞ dT −m2 T spin [A] = − Dx D e− 0 d Lspin ; (3.35) e 2 0 T P A Lspin = 1 x˙2 + 1 ˙ + iex˙ A − ie F ; (3.36) 4
2
which we already quoted in the introduction, Eq. (1.9). Although in the present review we will be exclusively concerned with four-dimensional eld theories, it should be mentioned that the obtained path integral representation is valid for all even spacetime dimensions. Note also that only the even subspace of the Cli5ord algebra came into play in the above. This is di5erent in the case of an open fermion line, and is the reason why the corresponding path integral representation for the electron propagator in a background eld is signi cantly more complicated [38,123,121,39,40,92]. In the introduction it was also mentioned that the worldline Lagrangian (3.36) has a global supersymmetry, (1.10). One consequence of this is that we can make use of a one-dimensional super eld formalism. Introducing √ X = x + 2/ ; (3.37) Y = X − x0 ; 9 9 D= −/ ; 9/ 9 d/ / = 1 ;
(3.38) (3.39) (3.40)
we can combine the x- and -path integrals into the following super path integral [63,124,125,85]: T 1 ∞ dT −m2 T 3 DX e− 0 d d/[−1=4X · D X −ieDX · A(X )] : (3.41) spin [A] = − e 2 0 T Written in this way, the spinor path integral becomes formally analogous to the scalar one, and can be considered as its “supersymmetrization”. Note, however, that the supersymmetry is broken by the di5erent periodicity conditions which we have for the coordinate and the Grassmann path integrals. For a constant Grassmann parameter those are not respected by the supersymmetry transformations (1.10). 3.4. Non-abelian gauge theory 3.4.1. Scalar loop contribution to the gluon e@ective action The simplest non-abelian generalization which one can consider is the contribution to the gluon scattering amplitude due to a scalar loop. Retracing the above derivation of the scalar 7
Our de nition of the Euclidean e5ective action di5ers by a sign from the one used in [102,104].
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path integral for photon scattering, one nds that the non-abelian nature of the background eld leads to the following changes in Eq. (1.8): (1) The trace now includes a global color trace. (2) The corresponding quantum mechanical Hamilton operators at di5erent times need not commute any more, so that the exponential must be taken path-ordered. We have thus
scal [A] = tr
∞
0
dT −m2 T e T
Dx()Pe−
T
2 0 d(1=4x˙ +igx˙ · A(x()))
;
(3.42)
where now A = Aa T a . P denotes the path ordering operator, and tr the color trace. 3.4.2. Spinor loop contribution to the gluon e@ective action In addition to these two changes, in the spinor loop case the F appearing in the worldline Lagrangian (3.36) must now be taken to be the full non-abelian eld strength tensor, including the commutator term [A ; A ] [126,127]. One may wonder how this commutator term is to be accommodated in the super eld formalism. As was shown in [125], a very convenient way of doing so is to introduce a super path ordering. The ordinary path ordering can be de ned by T 1 T N − 1 T N
−1 N T
P di ≡ N ! d1 d2 · · · dN = N ! d1 · · · dN /(i − i+1 ) : i=1 0
0
0
0
0
0
i=1
(3.43)
The super path ordering is obtained from this simply by replacing the proper-time di5erences in the arguments of the /-functions by super-di5erences, ˆij ≡ i − j + /i /j
(3.44)
so that Pˆ
N
i=1 0
T
di
d/i ≡ N !
0
T
d1
d/1 · · ·
0
T
dN
(here and in the following we use the convention that ing the /-functions one nds /(ˆi(i+1) ) = /(i − i+1 ) + /i /i+1 !(i − i+1 )
d/N
N
N
−1
/(ˆi(i+1) )
(3.45)
i=1
i = 1 d/i
≡ d/1 d/2 · · ·d/N ). Then expand-
(3.46)
and the !-function terms will generate precisely the commutator terms above. With this de nition of Pˆ , we can thus generalize Eq. (3.41) to the non-abelian case as ∞ T 1 dT −m2 T 3 DX Pˆ e− 0 d d/[−1=4X · D X −igDX · A(X )] : (3.47) spin [A] = − tr e 2 0 T This remarkable interplay between worldline supersymmetry and spacetime gauge symmetry has recently attracted some attention [128].
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3.4.3. Gluon loop contribution to the gluon e@ective action We proceed to the much more delicate case of the gluon loop, i.e. we wish now to derive a path integral describing a spin-1 particle coupled to a spin-1 background. Here one would expect to run into diNculties. It is well known how to construct free path integrals for particles of arbitrary spin (see, e.g., [129,130]). However, the quantization of those path integrals usually leads to inconsistencies as soon as one tries to couple a path integral with spin higher than 1 2 to a spin-1 background. In [64] this problem had been circumvented by the introduction of auxiliary degrees of freedom, and we will follow a similar approach here [92]. We employ the background gauge xing technique so that the e5ective action [Aa ] becomes a gauge invariant functional of Aa [131,132]. The gauge xed classical action reads, in D dimensions, 1 1 D a a S[a; A] = − d x F (A + a)F (A + a) − d D x(Dab [A]ab )2 : (3.48) 4 2 A priori, the background eld Aa is unrelated to the quantum eld aa . The kinetic operator of the gauge boson Puctuations is obtained as the second functional derivative of S[a; A] with respect to aa , at xed Aa . This leads to the inverse propagator ab ab D = − D?ac D?cb ! − 2igF
(3.49)
and the e5ective action 1 ∞ dT 1 glu [A] = − ln det(D) = (3.50) Tr(e−T D ) : 2 2 0 T In writing down Eq. (3.49) we have adopted the Feynman gauge = 1. The covariant derivative ab ≡ F c (T c )ab are matrices in the adjoint representation D ≡ 9 +igAa T a and the eld strength F 8 of the gauge group. The full e5ective action is obtained by adding the contribution of the Faddeev–Popov ghosts to Eq. (3.50). The evaluation of the ghost determinant proceeds along the same lines as scalar QED, and will be dealt with later on. In order to derive a path integral representation of the heat-kernel Tr(e−T D ) ;
(3.51)
we rst look at a slightly more general problem. We generalize the operator D to hˆ ≡ − D2 ! + M (x) ;
(3.52)
where M (x) is an arbitrary matrix in color space. In particular, we do not assume that the Lorentz trace M is zero. Given M , we construct the following one-particle Hamilton operator Hˆ = (pˆ + gA (x)) ˆ 2 − : X M (x) ˆ ˆ :: (3.53) The system under consideration has a graded phase-space coordinatized by x ; p and two sets of anti-commuting variables, and X , which obey canonical anti-commutation relations ˆ Xˆ + Xˆ ˆ = ! : (3.54)
8
Our de nition for the non-abelian covariant derivative is D ≡ 9 + igAa T a , with [T a ; T b ] = ifabc T c . The adjoint representation is given by (T a )bc = − ifabc .
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For a reason which will become obvious in a moment we have adopted the “anti-Wick” ordering in (3.53): all X ’s are on the right of all ’s, e.g. : ˆ Xˆ := ˆ Xˆ ; : Xˆ ˆ := − ˆ Xˆ :
(3.55)
We can represent the commutation relations on a space of wave functions A(x; ) depending on x and a set of classical Grassmann variables . The “position” operators xˆ = x ; ˆ = act multiplicatively on A, the conjugate momenta as derivatives pˆ = − i9 and Xˆ = 9= 9 . Thus the Hamiltonian becomes [133] 9 Hˆ = − D2 + M (x) : (3.56) 9 The wave functions A have a decomposition of the form A(x; ) =
D 1 p=0
p!
(x) (p) 1 ···p
1
···
p
:
(3.57)
This suggests the interpretation of A as an inhomogeneous di5erential form on RD with the fermions playing the role of the di5erentials d x [134,135]. The form-degree or, equivalently, the fermion number is measured by the operator 9 Fˆ = ˆ Xˆ = : (3.58) 9 We are particularly interested in one forms:
A(x; ) = ’ (x)
:
(3.59)
The Hamiltonian (3.56) acts on them according to (Hˆ A)(x; ) = (hˆ ’ )
:
(3.60)
We see that, when restricted to the one-form sector, the quantum system with the Hamiltonian (3.53) is equivalent to the one de ned by the bosonic matrix Hamiltonian hˆ [133,135]. The Euclidean proper time evolution of the wave functions A is implemented by the kernel K(x2 ;
2 ; 2 |x1 ;
1 ; 1 ) = x2 ;
−(2 −1 )Hˆ |x1 ; 2 |e
which obeys the SchrSodinger equation 9 + Hˆ K(x; ; T |x0 ; 0 ; 0) = 0 9T
1
;
(3.61) (3.62)
with the initial condition K(x; ; 0|x0 ; 0 ; 0) = !(x − x0 )!( − 0 ). It is easy to write down a path integral solution to Eq. (3.62). For the trace of K one obtains T −T Hˆ W Tr(e )= Dx() D ()D X ()tr Pe− 0 d L (3.63) P
A
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with
1 2 d X ! − M (3.64) L = x˙ + igx˙ A + 4 d (the subscript “W” for H will be explained in a moment). We have again periodic boundary conditions for x (), and anti-periodic ones for (). At this point we have to be careful. If we regard the Hamiltonian (3.53) as a function of the anti-commuting c-numbers and X it is related to the classical Lagrangian (3.64) by a standard Legendre transformation. As is well known, the information about the operator ordering is implicit in the discretization which is used for the de nition of the path-integral. Di5erent operator orderings correspond to di5erent discretization prescriptions (see, e.g., [136]). In our derivation of the worldline path integral representation for the spinor loop e5ective action in the previous section we used the so-called midpoint prescription [137] for the discretization. The reason for this choice is that, by Sato’s theorem [138], only in this case the familiar path-integral manipulations are allowed. Those will be needed to justify the naive one-dimensional perturbation expansion which we have in mind. It is known [137–141] that, at the operator level, this is equivalent to using the Weyl ordered Hamiltonian Hˆ W . This is the reason why we wrote Hˆ W rather than Hˆ on the l.h.s. of Eq. (3.63). In order to arrive at relation (3.60) we had to assume that the fermion operators in Hˆ are “anti-Wick” ordered. Weyl ordering amounts to a symmetrization in X and so that Hˆ W = (pˆ + gA (x)) ˆ 2 + 12 M (x)( ˆ ˆ X − X ˆ )
ˆ : = Hˆ − 12 M (x)
(3.65)
In the second line of (3.65) we used (3.53) and (3.54). (With respect to xˆ and pˆ Weyl ordering is used throughout.) If we employ (3.65) in (3.63) we obtain the following representation for the partition function of the anti-Wick ordered exponential: T 1 −T Hˆ X Tr(e )= Dx() D ()D ()tr P exp − d L() + M (x()) : (3.66) 2 P A 0 ˆ which is a generalization of the Let us now calculate the partition function Tr(exp(−T h)) heat-kernel needed in Eq. (3.50). By virtue of Eq. (3.60) we may write ˆ
ˆ
Tr(e−T h ) = Tr 1 (e−T H ) ;
(3.67)
where “Tr 1 ” denotes the trace in the one-form sector of the theory which contains the worldline fermions. In order to perform the projection on the one-form sector we identify M with M = C! − 2igF ; where C is a real constant. As a consequence, Hˆ = Hˆ 0 + C Fˆ
(3.68) (3.69)
with Hˆ 0 ≡ (pˆ + gA (x)) ˆ 2 − 2igF (x) ˆ ˆ X ;
(3.70)
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denoting the Hamiltonian which corresponds to the inverse propagator D. The fermion num ber operator Fˆ ≡ ˆ Xˆ is anti-Wick ordered by de nition. Its spectrum consists of the integers p = 0; 1; 2; : : : ; D. Note that M = DC, and that because of the antisymmetry of F the Hamiltonian Hˆ 0 has no ordering ambiguity in its fermionic piece. It will prove useful to apply Eq. (3.66) not to Hˆ directly, but to Hˆ − C = Hˆ 0 + C(Fˆ − 1). This leads to T D ˆ −CT (F−1) −T Hˆ 0 Tr(e e ) = exp −CT −1 Dx() D ()D X ()tr Pe− 0 d Lglu (3.71) 2 P A with
1 d Lglu = x˙2 + igx˙ A + X ! − C − 2igF 4 d
:
(3.72)
After having performed the path integration in (3.71) we shall send C to in nity. While this has no e5ect in the one-form sector, it leads to an exponential suppression factor exp[ − CT (p − 1)] in the sectors with fermion numbers p = 2; 3; : : : ; D. Hence only the zero and the one forms survive the limit C → ∞. In order to eliminate the contribution from the zero forms we insert the projector [1 − (−1)Fˆ ]=2 into the trace. It projects on the subspace of odd form degrees, and is easily implemented by combining periodic and anti-periodic boundary conditions for . In this way we arrive at the following representation of the partition function of Hˆ 0 , restricted to the one-form sector: 1 ˆ −T Hˆ 0 −T Hˆ 0 Fˆ −CT (F−1) Tr 1 [e ] = lim Tr (1 − (−1) )e e C →∞ 2 D = lim exp −CT −1 C →∞ 2 T 1 D ()D X ()tr Pe− 0 d Lglu : − (3.73) × Dx() 2 A P P Because Tr(exp(−T D)) = Tr 1 (exp(−T Hˆ 0 )), Eq. (3.73) implies for the e5ective action [64,92] ∞ 1 1 dT D glu [A] = D DX −1 Dx − lim exp −CT 2 C →∞ 0 T 2 2 A P P T 1 2 d X ×tr P exp − d ! − C − 2igF : x˙ + igx˙ A + 4 d 0 (3.74) Note that, from the point of view of the worldline fermions, C plays the role of a mass. The factor exp[ − CTD=2] in (3.74) is due to the di5erence between the Weyl and the anti-Wick-ordered Hamiltonian. It is crucial for obtaining a nite result in the limit C → ∞. In fact, for D = 4 it converts the prefactor eCT to a decaying exponential e−CT . 9 9
This reordering factor was missing in [64], where the change of the sign in D = 4 was instead erroneously attributed to a di5erence between Minkowski and Euclidean spacetime.
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A similar worldline path integral representation can also be written down for the gluon propagator in a background Yang–Mills eld [92]. This may be useful for future extensions of the string-inspired formalism. 4. Calculation of one-loop amplitudes We proceed to the evaluation of worldline path integrals at the one-loop level. As was already mentioned the method used is a very speci c one, and analogous to the techniques used in string perturbation theory. All path integrals will be manipulated into Gaussian form, which reduces their evaluation to the calculation of worldline propagators and determinants, and standard combinatorics. Of course there exist many alternatives to this procedure (see, e.g., [48,38,50,52,44,47,53,45]). Those will not be discussed here. While most of the formalism developed here applies to an arbitrary spacetime dimension, or at least to even dimensions, in this review all of our applications will be to four dimensional eld theories (for some calculations in D = 2 see [103], in D = 3 [142]). 4.1. The N -point amplitude in scalar <eld theory At the one-loop level, the worldline formalism has been used for a large variety of purposes. Let us begin with the simplest possible case, the one-loop N -point amplitude in massive 3 -theory. Choosing 4 3 ; 3! in Eq. (3.9), the path integral for the corresponding e5ective action reads T 2 1 ∞ dT −m2 T Dx()e− 0 d(1=4x˙ +4(x())) : [] = e 2 0 T x(T ) = x(0) U () =
We intend to calculate this path integral using the elementary Gauss formula −1 d x e−x · A · x+2b · x ∼(det(A))−1=2 eb · A · b :
(4.1)
(4.2)
(4.3)
As we already explained in the introduction, rst one has to deal with the zero-mode contained in the coordinate path integral Dx(), i.e. the constant loops. This is done by separating o5 the integration over the loop center of mass x0 , which reduces the coordinate path integral to an integral over the relative coordinate y: D x = d x0 D y ; x () = x0 + y () ; T d y () = 0 : 0
(4.4)
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The e5ective action thereby gets expressed in terms of an e5ective Lagrangian Le5 , = d x0 Le5 (x0 )
105
(4.5)
and Le5 (x0 ) is represented as an integral over the space of all loops with xed common center of mass x0 . In the reduced Hilbert space without the zero mode, the kinetic operator is invertible, and the inverse is easily found using the eigenfunctions of the derivative operator on the circle with circumference T , {e2in =T ; n ∈ Z \ {0}}: −2 ∞ d e2in(1 −2 )=T (1 − 2 )2 T 21 | |2 = 2T = | − | − − (4.6) 1 2 2 d (2in) T 6 n = −∞ n = 0
(1 − 2 ∈ [ − T; T ]). It will be seen later on that the constant −T=6 drops out of all physical results, so that we can delete it at the beginning. The remainder is the “bosonic” Green’s function which we already introduced in Eq. (1.16), GB (1 ; 2 ) = |1 − 2 | −
(1 − 2 )2 : T
Note that it is continuous as a function on S 1 ×S 1 . Its value depends neither on the location of the zero on the circle, nor on the choice of orientation. This Green’s function we will always use as the correlator for the coordinate “ eld”, y (1 )y (2 ) = − g GB (1 ; 2 ) :
(4.7)
The zero mode xing prescription, and consequently also the form of this worldline correlator, are not unique [106,107,143,144]. This ambiguity is of some technical importance, and will be discussed in Section 7 in connection with the calculation of the e5ective action itself. The only other information required for the execution of a Gaussian path integral is the free path integral determinant. With our conventions, the free coordinate path integral at xed proper-time T is T 1 Dy exp − d y˙ 2 = (4T )−D=2 : (4.8) 4 0 Here the T -dependence can be easily determined by, e.g., *-function regularization [121], while the factor [4]−D=2 corresponds to the usual loop-counting factor in quantum eld theory. How to continue now depends on whether we wish to compute the e5ective action itself, or the corresponding scattering amplitudes. For the calculation of the e5ective action, one way to proceed after the expansion of the interaction exponential is to Taylor-expand the external eld at the loop center of mass x0 , (x) = ey · 9 (x0 ) :
(4.9)
As we will see in Section 7, this leads to a highly eNcient algorithm [96,106,107,143] for calculating derivative expansions of e5ective actions.
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At the moment we are interested in the calculation of the N -point amplitude, which proceeds somewhat di5erently. According to standard quantum eld theory (see, e.g., [117]), the (one-particle irreducible) N -point function can be obtained from the one-loop e5ective action [] by a N -fold functional di5erentiation with respect to . In x-space, we can implement this operation simply by expanding the interaction exponential to N th order, and inserting appropriate !-functions into the path integral [63]: ∞ T
N T 2 dT −m2 T 1 N 1PI [x1 ; : : : ; xN ] = (−4) Dx di !(x(i ) − xi )e− 0 d(1=4x˙ ) : e 2 T 0 0 i=1
(4.10)
Thus only trajectories running through the prescribed points x1 ; : : : ; xN will contribute to the amplitude. The subscript “1PI” stands for one-particle-irreducible, and needs to be introduced here since, in contrast to the N -photon amplitude treated in the introduction, the N -point function in 3 -theory has also one-particle-reducible contributions. Similarly, the N -point function in momentum space can be obtained by specializing the background to a sum of plane waves, N (x) = eipi · x : (4.11) i=1
Then one picks out the term containing every pi precisely once (compare Eqs. (1.11), (1.12)). This leads to ∞ N 1 dT −m2 T T
N 1PI [p1 ; : : : ; pN ] = (−4) di d x0 Dy e 2 T 0 0 ×exp i
N
pi · x(i ) e−
i=1
T
2 0 d(1=4x˙ )
:
(4.12)
i=1
Note that now every external leg is represented by a scalar vertex operator, Eq. (1.3). 10 Since xi ≡ x(i ) = x0 + y(i ), the x0 -integral just gives momentum conservation N N D d x0 exp ix0 · pi = (2) ! pi : (4.13) i=1
i=1
The y-path integral is now Gaussian, and can be simply calculated by “completing the square”. One obtains the following parameter integral, ∞ dT 1 2 N D 1PI [p1 ; : : : ; pN ] = (−4) (2) ! pi (4T )−D=2 e−m T 2 T 0 N T N
1 × di exp GB (i ; j )pi · pj : (4.14) 2 0 i=1
10
i; j = 1
In our present conventions momenta appearing in vertex operators are ingoing.
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Fig. 12. One-loop N -point diagram.
This representation of the one-loop N -point amplitude in 3 -theory appears, as far as is known to the author, rst in [63]. It is the simplest example of a Bern–Kosower type formula. Note that a constant added to GB would drop out immediately on account of momentum conservation. Using momentum conservation this parameter integral can, for any given ordering of the external legs along the loop, be readily transformed into the corresponding standard Feynman parameter integral [21,24,145,146]. To obtain the contribution of the Feynman diagram with the standard ordering of the momenta p1 ; : : : ; pN (Fig. 12), one simply restricts the -integrations to the sector de ned by T ¿ 1 ¿ 2 ¿ · · ·N = 0, and transforms from -parameters to -parameters: 1 = T − 1 ; 2 = 1 − 2 ; .. .. . . N = N −1 :
(4.15)
Here we have made use of the freedom to choose the zero somewhere on the “worldloop” for setting N = 0. This is always possible, since our worldline Green’s function GB (1 ; 2 ) is translation invariant in . The complete parameter integral represents the sum of that particular Feynman diagram together with all the “crossed” ones. The case of 4 -theory is only marginally di5erent at the one-loop level. A eld theory potential of U () = 4=4!4 leads to a worldline interaction Lagrangian of Lint = 4=22 . The formula for the 2N -point amplitude analogous to Eq. (4.14) is ∞ 1 dT −m2 T 1PI [p1 ; : : : ; p2N ] = (−4)N e 2 T 0 N T
N T 2 × di d x0 Dy exp i (p2i−1 + p2i ) · x(i ) e− 0 d(1=4x˙ ) 0
i=1
+permuted terms :
i=1
(4.16)
Here one must explicitly sum over all possible ways of partitioning the 2N external states into N pairs.
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4.2. Photon scattering in quantum electrodynamics Since the scalar loop contribution to the one-loop N -photon scattering amplitude has already been discussed in the introduction, we immediately turn to the spinor loop case. The appropriate path integral representation for the one-loop e5ective action was given in Eq. (1.9), 1 ∞ dT −m2 T spin [A] = − Dx D e 2 0 T P A T 1 2 1 ˙ ×exp − d · + ieA · x˙ − ie · F · : (4.17) x˙ + 4 2 0 Besides the periodic coordinate functions x () we now need to also integrate over the ()’s, which are anti-periodic Grassmann functions. As was explained before, the scalar loop case is obtained from this simply by discarding all Grassmann quantities. As a consequence, in this formalism all calculations performed in fermion QED include the corresponding scalar QED results as a byproduct (as far as the calculation of the bare regularized amplitudes is concerned). Thus the calculation of the x-path integral proceeds as before. For the -path integral, rst note that there is no zero mode due to the anti-periodicity. To nd the appropriate worldline Green’s function, we now need to invert the rst derivative in the Hilbert space of anti-periodic functions. This yields −1 ∞ d e2i(n+1=2)(1 −2 )=T 21 | |2 = 2 (4.18) = sign(1 − 2 ) ≡ GF (1 ; 2 ) d 2i(n + 12 ) n = −∞ (1 − 2 ∈ [T; −T ]). Thus we have now the following two Wick contraction rules: (1 − 2 )2 y (1 )y (2 ) = − g GB (1 ; 2 ) = − g |1 − 2 | − ; T
(1 )
(2 ) = 12 g GF (1 ; 2 ) = 12 g sign(1 − 2 ) :
With our present conventions the free -path integral is normalized as 11 T 1 ˙ D exp − d · =4 : 2 0
(4.19)
(4.20)
This takes into account the fact that a Dirac spinor in four dimensions has four real degrees of freedom. One-loop scattering amplitudes are again obtained by specializing the background to a nite sum of plane waves of de nite polarization. Equivalently one introduces a photon vertex operator V A representing an external photon of de nite momentum and polarization (compare Eq. (1.4)). For the spinor loop case this vertex operator is T A Vspin [k; ] = d [ · x˙ + 2i · k · ] eik · x : (4.21) 0
11
This convention di5ers from the one used in [96,84,92].
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A [k; ] for the scalar loop is given by the same expression without The photon vertex operator Vscal the Grassmann term. We can then express the scalar and spinor QED N -photon amplitudes in terms of Wick contractions of vertex operators as follows: ∞ dT −m2 T N A A scal [k1 ; 1 ; : : : ; kN ; N ] = (−ie) (4T )−D=2 Vscal; (4.22) e 1 · · ·Vscal; N ; T 0 ∞ dT −m2 T A A spin [k1 ; 1 ; : : : ; kN ; N ] = − 2(−ie)N (4T )−D=2 Vspin; (4.23) e 1 · · ·Vspin; N : T 0 Here the zero-mode integration has already been performed, and the resulting factor (4.13) been omitted. The normalization refers to the complex scalar and Dirac fermion cases. The bosonic Wick contractions may be performed using the formal exponentiation as explained in the introduction, Eq. (1.17), leading to the Bern–Kosower master formula Eq. (1.18). Alternatively, one may follow the following simple general rules for the Wick contraction of expressions involving both elementary elds and exponentials:
1. Contract elds with each other as usual, and elds with exponentials according to y (1 ) eik · y(2 ) = iy (1 )y (2 )k eik · y(2 )
(4.24)
(the eld disappears, the exponential stays in the game). 2. Once all elementary elds have been eliminated, the contraction of the remaining exponentials yields a universal factor N 1 eik1 · y1 · · ·eikN · yN = exp − ki y (i )y (j )kj : (4.25) 2 i; j = 1
Since the photon vertex operator Eq. (4.21) contains x˙ we will also need to Wick-contract expressions involving y. ˙ It is always assumed that Wick-contractions commute with derivatives. Therefore the rst and second derivatives of GB will appear, (1 − 2 ) G˙ B (1 ; 2 ) = sign(1 − 2 ) − 2 ; T 2 GS B (1 ; 2 ) = 2!(1 − 2 ) − : (4.26) T Turning our attention to the -path integral, rst note that its explicit execution would be algebraically equivalent to the calculation of the corresponding Dirac traces in eld theory. For example, the correlator of four ’s gives
D
(1 )
4
(2 )
(3 )
(4 ) = 14 [GF12 GF34 gD4 g − GF13 GF24 gD g4 + GF14 GF23 gD g4 ] : (4.27)
Choosing a de nite ordering of the proper-time arguments, e.g. 1 ¿2 ¿3 ¿4 , we have the familiar alternating sum of products of metric tensors at hand which appears also in the trace of the product of four Dirac matrices. However, the explicit computation of the -integral can be avoided, due to the following remarkable feature of the Bern–Kosower formalism. After the evaluation of the x-path integral, one is left with an integral over the parameters T; 1 ; : : : ; N , where N is the number of
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external legs. The integrand is an expression consisting of the ubiquitous exponential factor exp[ 12 Ni;j = 1 GB (i ; j )ki · kj ] multiplied by a prefactor PN which is a polynomial function of the G˙ Bij ’s and GS Bij ’s, as well as of the kinematic invariants. As Bern and Kosower have shown in Appendix B of [20], all the GS Bij ’s can be eliminated by suitable chains of partial integrations in the parameters i , leading to an equivalent parameter integral involving only the GBij ’s and G˙ Bij ’s. According to the Bern–Kosower rules for the spinor loop case, all contributions from fermionic Wick contractions may then be taken into account simply by simultaneously replacing every closed cycle of G˙ B ’s appearing, say G˙ Bi1 i2 G˙ Bi2 i3 · · ·G˙ Bin i1 , by its “supersymmetrization”, G˙ Bi1 i2 G˙ Bi2 i3 · · ·G˙ Bin i1 → G˙ Bi1 i2 G˙ Bi2 i3 · · ·G˙ Bin i1 − GFi1 i2 GFi2 i3 · · ·GFin i1
(4.28)
(see Eq. (2.15)). The validity of this rule can be understood either in terms of worldsheet [66] or worldline [64] supersymmetry. We will give a direct combinatorial proof in Appendix D, using the worldline super eld formalism introduced in Section 3.3 and results from Section 4.8 below. The rule reduces the transition from the scalar to the spinor loop case to a mere pattern matching problem. Yet another option in the spinor loop calculation is the explicit use of the super eld formalism. If one takes the super path integral representation of the e5ective action, Eq. (3.41), as a starting point in the construction of the photon scattering amplitude, one still obtains Eq. (4.23). However the photon vertex operator then appears rewritten as T A Vspin [k; ] = d d/ · DX exp[ik · X ] : (4.29) 0
This allows one to combine the two Wick contraction rules Eqs. (4.19) into a single one, ˆ 1 ; /1 ; 2 ; /2 ) Y (1 ; /1 )Y (2 ; /2 ) = − g G(
(4.30)
with a worldline superpropagator ˆ 1 ; /1 ; 2 ; /2 ) ≡ GB (1 ; 2 ) + /1 /2 GF (1 ; 2 ) : G(
(4.31)
One can then write down a master formula for N -photon scattering [65] which is formally analogous to the one for the scalar loop, Eq. (1.18), ∞ dT 2 N D spin [k1 ; 1 ; : : : ; kN ; N ] = −2(−ie) (2) ! ki (4T )−D=2 e−m T T 0 N T N
1 ˆ × di d/i exp G ij ki · kj 2 0 i=1
i; j = 1
1 +iDi Gˆ ij i · kj + Di Dj Gˆ ij i · j 2
: 1 :::N
(4.32)
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Here, as well as in (4.29), we introduce the further convention that also the polarization vectors 1 ; : : : ; N are to be treated as Grassmann variables. Thus we have now all ’s, /’s, d/’s, and ’s anticommuting with each other. The overall sign of the master formula refers to the standard ordering of the polarization vectors 1 2 : : : N . 12 The super eld formalism thus also avoids the explicit execution of the Grassmann–Wick contractions. Those are now replaced by a number of Grassmann integrations, which have to be performed at a later stage of the calculation. Ultimately the super eld formalism leads to the same collection of parameter integrals to be performed as the component formalism, however it is often useful for keeping intermediate expressions compact. 4.3. Example: QED vacuum polarization As a rst example, let us recalculate in detail the one-loop vacuum polarization tensors in scalar and spinor QED [64]. 4.3.1. Scalar QED According to the above the one-loop two-photon amplitude in scalar QED can be written as ∞ T T dT −m2 T 2 scal [k1 ; k2 ] = (−ie) Dx d1 d2 x˙ (1 )eik1 · x(1 ) x˙ (2 ) e T 0 0 0 ×eik2 · x(2 ) e−
T
2 0 d(1=4)x˙
:
(4.33)
Separating o5 the zero mode according to Eqs. (4.4), (4.13), one obtains [k1 ; k2 ] = (2)D !(k1 + k2 )Escal (k1 ) scal Escal (k)
= −e
2
0
∞
dT −m2 T e T
−ik · y(2 ) −
×e
e
T
T
0
2 0 d(1=4)y˙
d1
0
T
d2
Dy y˙ (1 )eik · y(1 ) y˙ (2 )
:
(4.34)
The Wick contraction of the two photon vertex operators according to the above rules produces two terms 2
y˙ (1 )eik · y(1 ) y˙ (2 )e−ik · y(2 ) = {g GS B12 − k k G˙ B12 }e−k
2
GB12
:
(4.35)
Now one could just write out GB and its derivatives, and perform the parameter integrals. It turns out to be useful, though, to rst remove the GS B12 appearing in the rst term by a partial integration in the variable 1 or 2 . The integrand then turns into 2
{g k 2 − k k }G˙ B12 e−k
2
GB12
:
(4.36)
Note that this makes the transversality of the vacuum polarization tensor manifest. We rescale to the unit circle, i = Tui ; i = 1; 2, and use the translation invariance in to x the zero to be 12
With our conventions a Wick rotation ki4 → − iki0 ; T → is yields the N -photon amplitude in the conventions of [136].
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Fig. 13. Scalar QED vacuum polarization diagrams.
at the location of the second vertex operator, u2 = 0; u1 = u. We have then GB (1 ; 2 ) = Tu(1 − u) ; G˙ B (1 ; 2 ) = 1 − 2u :
(4.37)
Taking the free determinant factor Eq. (4.8) into account, and performing the global proper-time integration, one nds Escal (k)
1 dT −m2 T 2 −D=2 2 = e [k k − g k ] (4T ) T du(1 − 2u)2 e−Tu(1−u)k e T 0 0 1 e2 D [k k − g k 2 ] 2 − du(1 − 2u)2 [m2 + u(1 − u)k 2 ](D=2)−2 : = 2 (4)D=2 0 (4.38) 2
2
∞
The reader is invited to verify that this agrees with the result reached by calculating, in dimensional regularization, the sum of the corresponding two Feynman diagrams (Fig. 13). 4.3.2. Spinor QED For the fermion loop the path integral for the two-photon amplitude becomes, in the component formalism, ∞ T T 1 dT −m2 T 2 spin [k1 ; k2 ] = − (−ie) Dx D d1 d2 (x˙1 + 2i 1 1 · k1 ) e 2 T 0 0 0 ×eik1 · x1 (x˙2 + 2i
2
ik2 · x2 − e 2 · k2 )e
T
2 0 d(1=4x˙ +1=2
˙)
:
The calculation of Dx is identical with the scalar QED calculation. Only the calculation of D is new, and amounts to a single Wick contraction, (2i)2
1 1 · k1
2
2 2 · k2 = GF12 [g k1 · k2
− k2 k1 ] :
(4.39)
Adding this term to the bosonic result shows that, up to the global normalization, the parameter integral for the spinor loop is obtained from the one for the scalar loop simply by replacing, in Eq. (4.36), 2 2 2 G˙ B12 → G˙ B12 − GF12 :
(4.40)
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This is in accordance with Eq. (2.15). The complete change thus amounts to supplying Eq. (4.38) with a global factor of −2, and replacing (1 − 2u)2 by −4u(1 − u). This leads to 1 D e2 2 [k k − g k ] 2 − du u(1 − u)[m2 + u(1 − u)k 2 ](D=2)−2 Espin (k) = 8 2 (4)D=2 0 (4.41) again in agreement with the result of the standard textbook calculation. 4.4. Scalar loop contribution to gluon scattering As we discussed in Section 3.4.1, the path integral representing the e5ective action for the scalar loop in a gluon eld di5ers from the photon case, Eq. (1.8), only by the path ordering of the exponentials, and the addition of a global color trace. The gluon vertex operator Eq. (1.4) therefore di5ers from the photon vertex operator Eq. (4.21) only by the additional T a factor, which denotes a gauge group generator in the representation of the loop particle. However, the path ordering ( = proper-time ordering = color ordering) of the path integral has the e5ect that now those N vertex operators appear inserted on the worldloop in a xed ordering. Thus the global color trace factors out, and the scalar loop Bern–Kosower master formula Eq. (1.18) generalizes to the gluon scattering case as follows: a1 :::aN [k1 ; 1 ; : : : ; kN ; N ] N
a1
aN
D
= (−ig) tr(T · · ·T )(2) !
ki
∞
0 N 1 × d1 d2 : : : dN −1 exp GBij ki · kj 2 0 0 0 i; j = 1 1 −iG˙ Bij i · kj + GS Bij i · j : 2
T
1
2
dT (4T )−D=2 e−m T
N − 2
multi-linear
(4.42)
Here we have already eliminated one integration by setting N = 0. Note that it can now happen that a !(i − i+1 ), generated by the Wick contractions, appears multiplied by a /(i − i+1 ), generated by the path-ordering. Symmetry then dictates that just one half of this !-function should be allowed to contribute to the color ordering under consideration. The interpretation of this non-abelian master integral di5ers from its abelian counterpart in two important ways. Firstly, note that in the abelian case we can construct the complete amplitude in either of two ways. We can calculate the integral in the ordered sector 1 ¿2 ¿· · ·¿N = 0, and then generate the “crossed” contributions by explicit permutations of the result. Alternatively, we can generate those permuted terms by including the other ordered sectors in the integration. The second option does not exist in the non-Abelian case, since the crossed terms now generally have di5erent color traces. Secondly, in contrast to the one-loop photon scattering amplitudes the gluonic amplitudes generally have one-particle reducible contributions in addition to the irreducible ones. Since our derivation of the above master formula was based on a path integral representing the one-loop e5ective action for the gluon eld, which is the generator for the one-particle irreducible gluon
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correlators, the master formula as it stands yields precisely the contributions of all one-particle irreducible graphs to the amplitude in question. To complete the construction of the full one-loop N -gluon amplitude inside eld theory one would now have to generate the missing diagrams by an explicit Legendre transformation, amounting to sewing trees onto the one-particle irreducible diagrams. While feasible [66], this would to some extent spoil the elegance of the string-inspired approach. Fortunately, we have seen already in Section 2 that, as far as the on-shell amplitude is concerned, the full set of string-derived rules tells us how to bypass this procedure. Steps 3–5 of the rules instruct us, starting from the master formula, to remove all second derivatives GS B ’s, and then to apply the tree replacement (“pinch”) rules. In this way all the one-particle-reducible terms are included automatically. This is a remnant of the fact that the fragmentation of the amplitude into one-particle reducible and irreducible diagrams appears only in the in nite string tension limit, when parts of the string worldsheet get “pinched o5” [21,66]. 4.5. Spinor loop contribution to gluon scattering As we have already seen in the previous section, the fermion loop case is more involved. In the non-abelian case, the worldline Lagrangian Eq. (1.9) now contains a term [A ; A ] . In the component formalism this would force one to introduce, besides the one-gluon vertex operator T A a Vspin [k; ; a] = T d( · x˙ + 2i · k · ) exp[ik · x] (4.43) 0
an additional two-gluon vertex operator [64,94]. This is not necessary in the super eld formalism, where the single gluon super vertex operator T A [k; ; a] = T a d d/ · DX exp[ik · X ] (4.44) Vspin 0
remains suNcient. This is because, as explained in Section 3.4, the above commutator terms are then generated automatically by the super /-functions implicit in the path ordering. Our Eqs. (4.22), (4.23) for the N -point functions in the abelian case thus generalize to the non-abelian case simply as follows: ∞ dT −m2 T a1 :::aN N A A 1PI; [k ; ; : : : ; k ; ] = ( − ig) tr (4T )−D=2 Vscal; e N N 1 · · ·Vscal; N scal 1 1 T 0 ×! a1 :::aN 1PI; spin [k1 ; 1 ; : : : ; kN ; N ]
Here it is understood that
T
= −2(−ig) tr −1 N
N
carries a
T
/(i − i+1 ) ;
i=1 N
×!
ViA
−1 N
N
i=1
T ai .
0
∞
(4.45)
dT −m2 T A A (4T )−D=2 Vspin; e 1 · · ·Vspin; N T
/(ˆi(i+1) ) :
(4.46)
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4.6. Gluon loop contribution to gluon scattering In Section 3.4 we derived the following path integral representation of the one-loop e5ective action in pure Yang–Mills theory, using the background eld method in Feynman gauge, ∞ 1 1 dT D glu [A] = D DX −1 Dx − lim exp −CT 2 C →∞ 0 T 2 2 A P P T 1 d ×tr P exp − d x˙2 + igx˙ A + X − C ! − 2igF : 4 d 0 Recall that this path integral describes a whole multiplet of p-forms, p = 0; : : : ; D circulating in the loop; the role of the limit C → ∞ is to suppress all contributions from p ¿ 2, and the contributions from the zero form cancel out in the combination A − P . The Grassmann path integral now appears both with anti-periodic and periodic boundary conditions. The worldline Green’s function to be used for its evaluation is [92] −1 d C G (1 ; 2 ) ≡ 1 | −C |2 (4.47) d and reads for periodic and anti-periodic boundary conditions, respectively, GPC (1 ; 2 ) = − [/(2 − 1 ) + /(1 − 2 )e−CT ]
eC(1 −2 ) ; 1 − e−CT
(4.48)
GAC (1 ; 2 ) = − [/(2 − 1 ) − /(1 − 2 )e−CT ]
eC(1 −2 ) : 1 + e−CT
(4.49)
We observe that for C → ∞ there is an increasingly strong asymmetry between the forward and backward propagation in the proper time. The derivation of these Green’s functions is given in Appendix B. The representation (4.47) of the e5ective action does not coincide with the one used by Strassler [64]. While he uses the same kinetic term in the fermionic worldline Lagrangian, he modi es the interaction term according to X F
→ 12 F F F ≡ X F
+ 12 F (
+ X X );
(4.50)
where F () ≡
() + X () :
(4.51)
After this modi cation, Wick contractions involve the 2-point function of F, i.e. F (1 )F (2 ) = g G F (1 ; 2 ) ;
(4.52)
G F (1 ; 2 ) ≡ G C (1 ; 2 ) − G C (2 ; 1 ) :
(4.53)
where
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From (4.48), (4.49) we obtain explicitly GPF (1 ; 2 ) = sign(1 − 2 )
sinh[C(T=2 − |1 − 2 |)] ; sinh[CT=2]
GAF (1 ; 2 ) = sign(1 − 2 )
cosh[C(T=2 − |1 − 2 |)] : cosh[CT=2]
(4.54)
These Green’s functions still do not quite coincide with the ones given by Strassler [64]; however, they become e5ectively equivalent in the limit C → ∞. The modi ed version is more convenient for the calculation of scattering amplitudes. Although the worldline Lagrangians for the gluon and for the spinor loop are similar in structure, there is no analogue of the supersymmetry transformations Eq. (1.10) in the spin-1 case. As was noted in [94], it is useful nevertheless to introduce the super eld formalism as a book-keeping device. In complete analogy to the spinor loop case we introduce new super elds √ X˜ = x + 2/F ; Y˜ = X˜ − x0
(4.55)
with the same super conventions as before. The gluon vertex operator becomes (compare Eq. (4.44)) T A a Vglu [k; ; a] = T d d/ · DX˜ exp[ik · X˜ ] (4.56) 0
(with T a in the adjoint representation). The appropriate worldline super propagator is F F Gˆ P; A (1 ; /1 ; 2 ; /2 ) ≡ GB (1 ; 2 ) + 2/1 /2 GP; A (1 ; 2 ) :
(4.57)
The super Wick contraction rule
F
Y˜ (1 ; /1 )Y˜ (2 ; /2 )P; A = − g Gˆ P; A (1 ; /1 ; 2 ; /2 )
(4.58)
then correctly reproduces the component eld expressions. It allows us to take over all the conveniences of the super eld formalism encountered before, and to write the one-particle-irreducible o5-shell N -gluon Green’s function in a way analogous to Eq. (4.46), ∞ 1 dT −CT a1 :::aN N 1PI; glu [k1 ; 1 ; : : : ; kN ; N ] = − (−ig) lim tr (4T )−D=2 e C →∞ 4 T 0 ×
p = P;A
A A
p Zp Vglu; 1 · · ·Vglu; N p !
−1 N
N
T
/(ˆi(i+1) ) :
(4.59)
i=1
Here we have de ned P = 1; A = − 1. ZA; P are the fermionic determinant factors D d d − C ! = Det A; P −C : ZA; P ≡ Det A; P d d
(4.60)
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Those can be easily calculated using the same basis of circular eigenfunctions of the derivative operator as was used in the computation of GB ; GF above. The result is ZA = (2 cosh[CT=2])D ; ZP = (2 sinh[CT=2])D :
(4.61)
Note that we have already set D = 4 in the reordering factor, and the same will be done for ZA; P . This corresponds to the choice of a certain dimensional reduction variant of dimensional regularization, the four-dimensional helicity scheme developed by Bern and Kosower [21] (compare Section 2). The super formalism allows us not only to do without an additional two-gluon vertex operator, but also to generalize the replacement rule Eq. (2.15) to the gluon loop case. This means that one can, for nite C and a xed choice of the fermionic boundary conditions, rst perform the bosonic Wick contractions, then partially integrate away all GS Bij ’s, and nally include the terms from the fermionic sector by replacing F F F G˙ Bi1 i2 G˙ Bi2 i3 · · ·G˙ Bin i1 → G˙ Bi1 i2 G˙ Bi2 i3 · · ·G˙ Bin i1 − 2n Gpi (4.62) i Gpi i · · ·Gpi i : 1 2
2 3
n 1
The analysis of the C → ∞ limit [21,64] shows, however, that in this limit all terms containing multiple products of fermionic cycles get suppressed. Moreover, of the terms containing precisely one cycle, only those survive for which the ordering of the indices follows the ordering of the external legs. For those, the C-dependence is isolated in a factor of 1 F F F zn (C) ≡ e−CT
p Zp Gpi Gpi · · ·Gpi : (4.63) 1 i2 2 i3 n i1 2 p = P;A
In the limit this expression goes to a constant, namely 2 n=2 ; lim zn (C) = 1 n¿2 indices follow legs ; C →∞ 0 n¿2 all other orderings :
(4.64)
Here the sign in the second line refers to the descending ordering i1 ¿i2 ¿· · ·¿in :
(4.65)
If the ordering of the indices follows the ordering of the legs, this can always be arranged F for using the cyclicity and the antisymmetry of GA; P . This then leads just to the gluonic cycle rule part of the Bern–Kosower rules, Eq. (2.17). For the remaining purely bosonic terms the C-dependence is trivial, and gives a factor lim e−CT 21 (ZA − ZP ) = 4 :
C →∞
(4.66)
The bosonic part alone will therefore yield four times the contribution of a real scalar in the loop. This corresponds just to the four degrees of freedom of the gluon. But as we know from eld theory the ghost contribution to the amplitude is −2 times the contribution of a real scalar. Therefore it just subtracts the contribution of the two unphysical degrees of freedom of the gluon, and the whole ghost contribution can be taken into account simply by changing the above factor from 4 to 2. This explains the factor 2 which we had in the Bern–Kosower rules for the “type 1” contributions to the gluon loop.
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4.7. Example: QCD vacuum polarization With all this machinery in place, we can now easily generalize our results for the QED vacuum polarization tensors, Eqs. (4.38) and (4.41), to the QCD case [64,94]. For the QCD gluon vacuum polarization we have to take the scalar, spinor, gluon, and ghost loops into account. For the scalar and spinor loops, the replacement of Eqs. (4.22), (4.23) by their non-abelian counterparts Eqs. (4.45), (4.46) has the sole e5ect that the amplitude gets multiplied by a color trace tr(T a1 T a2 ). The !-function term contained in /(1 − 2 + /1 /2 ) after the /-integrations produces a term proportional to !12 GF12 e−k
2
GB12
;
however its -integral vanishes since GF12 (0) = 0. For the gluon loop, the coordinate part together with the ghost loop give the same as a complex scalar. Again the terms produced by the !-function drop out due to the antisymmetry F of GA; P (1 ; 2 ). The only nontrivial new contribution comes from the gluon spin in the loop. This one is a two-cycle, and according to the above is related to the scalar contribution by a replacement of G˙ B12 G˙ B21 → 4 : If we assume the loop scalars and fermions to be massless and in the adjoint representation, the results can be combined into the following single parameter integral (compare Eqs. (4.37), (4.38), (4.41)), 1 g2 D a 1 a2 2 Eadj (k) = tr(Tadj Tadj ) [k k − g k ] 2 − du[u(1 − u)k 2 ](D=2)−2 2 (4)D=2 0 Ns × (4.67) − Nf + 1 (1 − 2u)2 + Nf − 4 : 2 Here Ns denotes the number of (real) scalars, Nf the number of Weyl fermions. It should be remembered that, in eld theory terms, this result corresponds to a calculation using the background eld method and Feynman gauge. It is nice to verify [66,64] that the second line vanishes for Ns = 6, Nf = 4, corresponding to the case of N = 4 Super Yang–Mills theory, which is a nite theory. Note that the amplitude then vanishes already at the integrand level. 4.8. N photon=N gluon amplitudes Before proceeding to higher numbers of external legs, let us introduce some notation to keep the formulas manageable. Writing out the exponential in the master formula Eq. (1.18) for a xed number N of photons, one obtains an integrand N 1 exp {} |multi-linear = (−i)N PN (G˙ Bij ; GS Bij ) exp GBij ki · kj (4.68) 2 i; j = 1
with a certain polynomial PN depending on the various G˙ Bij ’s, GS Bij ’s, as well as on the kinematic invariants. To be able to apply the Bern–Kosower rules, we need to remove all second
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derivatives GS Bij appearing by suitable partial integrations in the variables 1 ; : : : ; N . This transforms PN into another polynomial QN depending only on the G˙ Bij ’s alone: PN (G˙ Bij ; GS Bij ) e1=2
GBij ki · kj
part:int:
→
QN (G˙ Bij ) e1=2
GBij ki · kj
:
(4.69)
As a result of the partial integration procedure certain combinations of the kinematic invariants are going to appear, the “Lorentz cycles” Zn , Z2 (ij) ≡ i · kj j · ki − i · j ki · kj ; Zn (i1 i2 : : : in ) ≡ tr
n
[kij ⊗ ij − ij ⊗ kij ]
(n ¿ 3) :
(4.70)
j=1
Those generalize the transversal projector which is familiar from the two-point case. (In the (abelian) e5ective action they would correspond to a tr(F n ).) We also introduce the notion of a “-cycle”, which is a product of G˙ Bij ’s with the indices forming a closed chain, (4.71) G˙ Bi1 i2 G˙ Bi2 i3 · · ·G˙ Bin i1 : (It should be remembered from Section 2 that an expression is considered a cycle already if it can be put into cycle form using the antisymmetry of G˙ B .) With these notations, the two-point result is Q2 = Z2 (12)G˙ B12 G˙ B21 : (4.72) In the three-point case one starts with P3 = G˙ B1i 1 · ki G˙ B2j 2 · kj G˙ B3k 3 · kk −[GS B12 1 · 2 G˙ B3i 3 · ki + (1 → 2 → 3) + (1 → 3 → 2)] :
(4.73)
Here and in the following the dummy indices i; j; k should be summed over from 1 to N , and one has G˙ Bii = 0 by antisymmetry. Removing all the GS Bij ’s by partial integrations one nds Q3 = G˙ B1i 1 · ki G˙ B2j 2 · kj G˙ B3k 3 · kk + 12 {G˙ B12 1 · 2 [G˙ B3i 3 · ki (G˙ B1j k1 · kj − G˙ B2j k2 · kj ) +(G˙ B31 3 · k1 − G˙ B32 3 · k2 )G˙ B3j k3 · kj ] + 2 permutations} = Q33 + Q32 ;
(4.74)
where Q33 = G˙ B12 G˙ B23 G˙ B31 Z3 (123) ; Q32 = G˙ B12 G˙ B21 Z2 (12)G˙ B3i 3 · ki + (1 → 2 → 3) + (1 → 3 → 2) :
(4.75)
Here we have decomposed the result of the partial integration, Q3 , according to its “cycle content”, which is indicated by the upper index. Q33 contains a 3-cycle, while the terms in Q32 have a 2-cycle. Note that a -cycle comes multiplied with the corresponding Lorentz cycle. This turns out to be true in general, and motivates the further de nition of a “bi-cycle” as the product of the two, ˙ 1 i2 : : : in ) ≡ G˙ Bi1 i2 G˙ Bi2 i3 · · ·G˙ Bin i1 Zn (i1 i2 : : : in ) : G(i (4.76)
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After all bi-cycles have been separated out in a given term, whatever remains is called the “tail” of the term, or “m-tail”, where m denotes the number of left-over G˙ Bij ’s. In the case of Q32 we have just a 1-tail G˙ B3i 3 · ki . In the abelian case the 3-point amplitude must vanish according to Furry’s theorem. In the present formalism this can be immediately seen by noting that the integrand is odd under the orientation-reversing transformation of variables i = T − i ; i = 1; 2; 3. In the three-point case, Q3 is still unique; all possible ways of performing the partial integrations lead to the same result. The same is not true any more in the four-point case, where the result of the partial integration procedure turns out to depend on the speci c chain of partial integrations chosen. In [147] it was shown that this ambiguity can be xed by requiring QN to have, like PN , the full permutation symmetry in the external legs, and a particular algorithm for the partial integration was given which manifestly preserves this symmetry. This algorithm is explained in detail in Appendix C, where we also explicitly write down the resulting polynomials QN up to the six-point case. For the four-point case the result can be written in the following form: Q4 = Q44 + Q43 + Q42 + Q422 ; Q44 = G˙ B12 G˙ B23 G˙ B34 G˙ B41 Z4 (1234) + 2 permutations ; Q43 = G˙ B12 G˙ B23 G˙ B31 Z3 (123)G˙ B4i 4 · ki + 3 perm ; 2 ˙ ˙ G˙ B3i 3 · ki G˙ B4j 4 · kj Q4 = G B12 G B21 Z2 (12) 1 ˙ ˙ ˙ + G B34 3 · 4 [G B3i k3 · ki − G B4i k4 · ki ] + 5 perm ; 2
Q422 = G˙ B12 G˙ B21 Z2 (12)G˙ B34 G˙ B43 Z2 (34) + 2 perm :
(4.77)
Here the terms in the partially integrated integrand have already been grouped according to their cycle content. The appearing in the two-tail of Q42 means that in the summation over the dummy variables i; j the term with i = 4; j = 3 must be omitted, since for these values an additional two-cycle would be present in the tail. Thus our nal representation for the four-photon amplitude in scalar QED is the following: ∞ e4 dT 4−(D=2) −m2 T scal [k1 ; 1 ; : : : ; k4 ; 4 ] = e T D=2 T (4) 0 1 4 T ˙ × du1 du2 du3 du4 Q4 (G Bij ) exp GBij ki · kj : (4.78) 2 0 i; j = 1
Here we have already rescaled to the unit circle, i = Tui ; GBij = |ui − uj | − (ui − uj )2 . Note that this is already the complete (o5-shell) amplitude, with no need to add “crossed” terms. The summation over crossed diagrams which would have to be done in a standard eld theory calculation here is implicit in the integration over the various ordered sectors.
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As an unexpected bonus of the whole procedure it turns out that this decomposition according to cycle content coincides, for arbitrary N , with a decomposition into gauge invariant partial amplitudes. Every single one of the 16 terms contained in the decomposition of Q4 is individually gauge invariant, i.e. it either vanishes or turns into a total derivative if the replacement i → ki is made for any of the external legs. For example, if in Q43 we substitute k4 for 4 (in the un-permuted term) we have the following total derivative at hand: 94 [G˙ B12 G˙ B23 G˙ B31 e1=2GBij ki · kj ] :
These total derivatives become more and more complicated with increasing lengths of the tails [147]. Note also that the partial integration has the e5ect of homogeneizing the integrand; every term in QN has N factors of G˙ Bij and N factors of external momentum. In the four-point case this has the additional advantage of making the UV niteness of the photon–photon scattering amplitude manifest. As is well known, in a Feynman diagram calculation this property would be seen only after adding up all diagrams. Similarly, in the present approach the initial parameter integral still contains spurious divergences, since P4 has terms involving products of two GS Bij ’s which lead to a logarithmically divergent T -integral. After the partial integration the integrand is nite term by term so that there is no necessity any more for an UV regulator; as far as the QED case is concerned, we can set D = 4 in (4.78). Applying the Bern–Kosower rules to the above integrand we can immediately obtain the corresponding parameter integrals for (o5-shell) photon–photon scattering in spinor QED, as well as for (on-shell) gluon–gluon scattering in QCD. As was already mentioned, in the latter case the presence of color factors forces one to restrict the integrations to the standard ordered sector 1 ¿ u1 ¿ u2 ¿ u3 ¿ u4 = 0, and to explicitly sum over all non-cyclic permutations. The same partial integration algorithm can also be used for the fermion loop in the super eld formalism. In Appendix D this will be used for a simple proof of the replacement rule (2.15). Finally, what does one gain by the partial integration procedure in terms of the diNculty of the arising parameter integrals? The partial integration increases the total number of terms in the integrand, but decreases the number of independent integrals. But then one must take into account the fact that those fewer independent integrals have, on the average, more complicated Feynman numerators (this fact turned out to be of signi cance in the calculation of the ve-gluon amplitude [148]). It is therefore diNcult to say in general, and to confuse matters more we will see in Section 9.6 that sometimes even taking a linear combination of both integrands can be useful. 4.9. Example: gluon–gluon scattering Let us now have a look at a somewhat more substantial calculation, namely the one-loop gluon–gluon scattering amplitude in massless QCD. While this amplitude still does not present a challenge by modern standards, and was obtained by Ellis and Sexton many years ago [149], the advantages of the following recalculation over a standard Feynman diagram calculation should be evident. The calculation proceeds in two parts. The rst step is concerned with the reduction of this amplitude to a collection of parameter integrals; the second one with their explicit calculation. As always one nds those integrals to be of the same type as the corresponding Feynman parameter
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Fig. 14. Gluon–gluon scattering amplitude.
integrals. At the 4-point level those integrals are still fairly easy. A convenient method for their calculation is described in Appendix E, following [109]. Gluon scattering amplitudes are usually calculated in a helicity basis. A gluon has only two di5erent physical polarizations, which are chosen to be helicity eigenstates “+” and “−”. In QCD those amplitudes are not independent, since CP invariance implies that simultaneously Pipping all helicities is equivalent to changing all momenta from ingoing to outgoing, and vice versa. Therefore in the 4-point case there are only four amplitudes to consider, A (+ + ++); A(− + ++); A(− − ++), and A(− + −+). As always in the non-abelian case we have to x the ordering of the external legs, which we choose as the standard ordering (1234). In the end one must sum over all non-cyclic permutations. Quite generally it turns out that the calculation of the N gluon or photon amplitudes is easiest for the all “+” (or all “−”) cases, and most diNcult for the completely “mixed” ones. 13 In the 4-point case the calculation of A(+ + ++) is similar to the one of A(− + ++), while A(− − ++) is similar to A(− + −+). We will therefore restrict ourselves to the computation of A(− + ++) and A(− − ++). Let us start with the easier one, A(− + ++) (Fig. 14). A given assignment of polarizations can be realized by many di5erent choices of polarization vectors, and it is desirable to make this choice in such a way that the number of non-zero kinematic invariants i · j ; i · kj is minimized. A convenient way of nding such a set of polarization vectors for a given polarization assignment is provided by the spinor helicity technique (see, e.g., [111]), which makes use of the freedom to perform independent gauge transformations on all external legs. While this technique is already very useful in the corresponding eld theory calculation, here its ef ciency is further enhanced by the fact that there is no loop momentum, which reduces the number of kinematic invariants from the very beginning. Appropriate sets of polarization vectors have been given in [21,66]. For A(− + ++) they nd that using a reference momentum k4 for the rst gluon and k1 for the other ones makes all products of polarization vectors vanish, i · j = 0; 13
i; j = 1; : : : ; 4 :
(4.79)
It should be noted that A(+ + ++) does not describe a helicity conserving process, due to the convention used here that all momenta are ingoing.
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Fig. 15. Four-point 3 -diagrams.
Moreover one has the following further relations: k4 · 1 = k1 · 2 = k1 · 3 = k1 · 4 = 0 ; k3 · 1 = − k2 · 1 ;
k4 · 2 = − k3 · 2 ;
k4 · 3 = − k2 · 3 ;
k3 · 4 = − k2 · 4 :
(4.80)
Those allow one to express all non-zero invariants in terms of 1 · k3 ; 2 · k4 ; 3 · k4 ; 4 · k3 . Using them in the representation Eq. (4.77) for Q4 we nd that they make most of the Lorentz cycles Zn vanish; the surviving ones are Z3 (234) = 22 · k4 3 · k4 4 · k3 ; Z2 (23) = 2 · k4 3 · k4 ; Z2 (24) = − 2 · k4 4 · k3 ; Z2 (34) = 4 · k3 3 · k4 :
(4.81)
This leads to the vanishing of all pure cycle terms, Q44 = Q422 = 0 :
(4.82)
The surviving structures Q43 and Q42 yield
2
Q4 = C−+++ (G˙ B13 − G˙ B12 ){2G˙ B23 G˙ B34 G˙ B42 − G˙ B23 (G˙ B43 − G˙ B42 ) 2
2
(4.83)
C−+++ ≡ 1 · k3 2 · k4 3 · k4 4 · k3 :
(4.84)
+G˙ B24 (G˙ B34 − G˙ B32 ) − G˙ B34 (G˙ B24 − G˙ B23 )} ; where
According to step 4 of the Bern–Kosower rules we should now search for possible pinch terms. The relevant 3 diagrams at this order are shown in Fig. 15.
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Considering diagram (b) we note that it is indeed a candidate for a pinch term, since Q4 contains G˙ B12 linearly. Step 5 of the Bern–Kosower rules tells us that to nd its contribution we have to replace G˙ B12 by 2=(k1 + k2 )2 , and replace the variable u2 by u1 in the remainder of the expression. This transforms Q4 into Pb = −2
C−+++ 2 2 {2G˙ B13 G˙ B34 G˙ B41 − G˙ B13 (G˙ B43 − G˙ B41 ) + G˙ B14 (G˙ B34 − G˙ B31 ) 2 (k1 + k2 ) 2
−G˙ B34 (G˙ B14 − G˙ B13 )} :
(4.85)
Diagram (c) looks like another candidate, however its pinch contribution disappears since the rst factor in Q4 vanishes if u3 is replaced by u2 . Similarly the pinch contributions of all remaining diagrams can be seen to be zero. Thus for the scalar loop we have to calculate the following two parameter integrals: 1 u1 u2 D Pa (u1 ; u2 ; u3 ; u4 ) Da = 4 − du1 du2 du3 ; 4 2 [ − i¡j = 1 GBij ki · kj ]4−D=2 0 0 0 1 u1 D Pb (u1 ; u3 ; u4 ) du1 du3 (4.86) Db = 3 − 2 [ − i¡j = 1; 3; 4 GBij ki · kj ]3−D=2 0 0 (with Pa the Q4 of Eq. (4.83)). Both integrals turn out to be nite, so that one can set D = 4. For their calculation we introduce Mandelstam variables s; t according to Eq. (E.4) of Appendix E, and write out the various GBij ’s, G˙ Bij ’s for the standard ordering of the external legs, u1 ¿u2 ¿u3 ¿u4 = 0 (the usual rescaling to the unit circle has already been done). Then the rst integral turns into 1 u1 u2 (u2 − u3 )2 u3 (1 − u2 ) Da = − 16C−+++ du1 du2 du3 : (4.87) [s(u2 − u3 )(1 − u1 ) + t(u1 − u2 )u3 ]2 0 0 0 Beginning with u1 all three integrations can be done elementarily, and one obtains 8 C−+++ : 3 st Similarly the other diagram yields a Da = −
(4.88)
8 C−+++ : (4.89) 3 s2 To nd the corresponding numerator polynomials for the spinor and gluon loops, we have to apply the cycle replacement rules Eqs. (2.15), (2.17). For example, the rst term inside the braces in Eq. (4.83) is a 3-cycle, and will give additional terms both for the fermion and the gluon cases, and both for diagrams (a) and (b). However, adding up all terms generated by the application of the replacement rule, and writing them out in the ui , one nds them to cancel out exactly. Thus for this particular helicity component there are no further integrals to calculate, and the amplitudes for the scalar, fermion and gluon loop cases di5er, apart from the group theory factor, only by the global factors counting the di5erences in statistics and degrees of Db = −
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freedom a1 ···a4 scal (− + ++) = − a1 ···a4 (− + ++) = spin
g4 a1 a4 s + t tr(Tscal · · ·Tscal ) 2 C−+++ ; 2 12 st
g4 a1 a4 s + t tr(Tspin · · ·Tspin ) 2 C−+++ ; 2 6 st
a1 ···a4 glu (− + ++) = −
g4 a1 a4 s + t tr(Tadj · · ·Tadj ) 2 C−+++ : 2 6 st
(4.90)
Here the normalizations refer to a real scalar and to a Weyl fermion, and it should be remembered that the gluon loop contribution includes the contribution of its ghost. The factor C−+++ can, using the spinor helicity method, be expressed in terms of the Mandelstam variables up to a complex phase factor [111,66]. Remember that this is not yet the complete amplitude, but must still be summed over all non-cyclic permutations according to (2.19). As in the case of the vacuum polarization, things become particularly simple if the scalars and fermions are also in the adjoint representation, which is the case in N = 4 Super–Yang–Mills theory. Here one has spin (− + ++) = − 2scal (− + ++) ; glu (− + ++) = 2scal (− + ++) :
(4.91)
These simple relations hold only for A(− + ++) and A(+ + ++), and can be derived from the spacetime supersymmetry [66]. In contrast to a standard eld theory calculation here they are visible already at the integrand level. We proceed to the more substantial calculation of A(− − ++). Again following [21,66] we choose reference momenta (k4 ; k4 ; k1 ; k1 ), which leads to the following relations: 1 · 2 = 1 · 3 = 1 · 4 = 2 · 4 = 3 · 4 = 0 ; 1 · k4 = 2 · k4 = 3 · k1 = 4 · k1 = 0 ; 1 · k3 = −1 · k2 ; 2 · 3 =
2 · k1 = − 2 · k3 ;
3 · k4 = − 3 · k2 ;
4 · k2 = − 4 · k3 ;
2 · k3 3 · k2 : k2 · k3
(4.92)
Using these relations in Q4 one nds that this time Q43 is vanishing. The others all do contribute, yielding 2
2
2
Q4 = C−−++ {G˙ B12 G˙ B34 + G˙ B12 [G˙ B23 G˙ B34 − G˙ B23 G˙ B24 − G˙ B24 G˙ B34 ] 2
+G˙ B34 [G˙ B12 G˙ B23 − G˙ B12 G˙ B13 − G˙ B13 G˙ B23 ] + G˙ B12 G˙ B13 G˙ B24 G˙ B34 +G˙ B13 G˙ B14 G˙ B23 G˙ B24 − G˙ B12 G˙ B14 G˙ B23 G˙ B34 }
(4.93)
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with C−−++ ≡ 1 · k2 2 · k1 3 · k2 4 · k2 :
(4.94)
For this helicity component all pinch terms turn out to vanish. For example, Q4 contains four terms containing G˙ B12 linearly, however they cancel in pairs once the replacement u2 → u1 is made. On the other hand, this time the replacement rules will have an e5ect. For example, for the spinor loop the rst term in Q4 has to be replaced by 2 2 2 2 2 2 G˙ B12 G˙ B34 → (G˙ B12 − G˙ F12 )(G˙ B34 − G˙ F34 ) :
(4.95)
The analogous replacement has to be made for the three 4-cycles appearing. For the gluon loop case the rst term should instead be replaced by 2 2 2 2 2 2 G˙ B12 G˙ B34 → G˙ B12 G˙ B34 − 4G˙ B12 − 4G˙ B34 :
(4.96)
Of the three 4-cycles only one has the ordering of the indices following the (standard) ordering of the external legs, and thus needs to be replaced by G˙ B12 G˙ B14 G˙ B23 G˙ B34 → G˙ B12 G˙ B14 G˙ B23 G˙ B34 − 8 :
(4.97)
Writing out the results in the variables ui , and then transforming to Feynman parameters according to Eq. (E.3) of Appendix E, one obtains the Feynman polynomials to be integrated. For the case of the gluon loop one nds, taking the global factor of 2 into account, Pa = 2C−−++ (8 − 12a3 − 20a1 a3 − 16a2 a3 + 16a1 a2 a3 + 16a22 a3 − 4a23 + 32a1 a23 +64a2 a23 − 48a1 a2 a23 − 48a22 a23 + 32a33 − 32a1 a33 − 48a2 a33 − 16a43 ) :
(4.98)
For this helicity component the parameter integrals turn out to be divergent, so that their calculation is not elementary any more. In Appendix E we explain a method [109] for the calculation of arbitrary on-shell massless four-point tensor integrals in dimensional regularization. For the numerator polynomial Pa this yields the following: 2 ln(s) + 2 ln(t) − 11 C−−++ (1 − =2)2 (1 + =2) 8 3 Da = 32 + st (1 + ) 2 11 2 32 +ln(s) ln(t) − ln(t) − (4.99) + + O() : 6 2 9 The double pole in this expansion is due to infrared divergences alone, while the simple pole comes from both infrared and ultraviolet divergences. The infrared divergences will ultimately cancel against contributions from the ve-gluon tree amplitude, however the ultraviolet divergence must be removed by renormalization. The nal result for the gluon loop contribution becomes a1 ···a4 glu (− − ++) =
g4 a1 a4 tr(Tadj · · ·Tadj )(4)−=2 Daren ; 322
(4.100)
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where Daren
2 ln(s=2 ) + 2 ln(t=2 ) − C−−++ (1 − =2)2 (1 + =2) 8 = 32 + st (1 + ) 2 s t 11 t 2 32 +ln 2 ln 2 − ln 2 − + 6 2 9
127
22 3
(4.101)
and is the renormalization scale. As usual we have worked in the Euclidean; the analytic continuation to physical momenta requires the use of the appropriate i prescription for the Mandelstam variables, s → s − i, etc.14 4.10. Boundary terms and gauge invariance So far we have completely disregarded possible boundary terms in the partial integration procedure. In the abelian case boundary terms are clearly absent, since all integrations are over the complete circle, and the integrand is written in terms of the worldline Green’s functions, which have the appropriate periodicity properties. This is di5erent in the non-abelian case, where boundary terms will generally appear. When using the string-derived rules for the calculation of the gluon scattering amplitudes those can still be ignored, since their contributions are automatically included by the application of the pinch rules. However, the validity of the original Bern–Kosower pinch rules is restricted to the on-shell case. 15 Therefore the boundary terms come into play if one wishes to apply the partial integration procedure to the calculation of the nonabelian e5ective action itself, or to the corresponding one-particle-irreducible o5-shell vertex function 1PI [k1 ; : : : ; N ]. Let us therefore investigate their structure for the simplest case of a scalar loop (in this section we closely follow [150]). Let us thus consider the standard low-energy expansion of the one-loop e5ective action in gauge theory, to be discussed at length in Section 7 below. In scalar QED, the rst non-trivial term in this expansion is proportional to the Maxwell term F F . Its coeNcient is given by the zero-momentum limit of the vacuum polarization tensor, Eq. (4.38). This term must also appear, with the same coeNcient, in the scalar loop contribution to the QCD e5ective action. However by gauge invariance it must now involve the full non-Abelian eld strength tensor F = 9 A − 9 A + ig[A ; A ]
(4.102)
and thus be of the form 0 0 0 tr F F = tr F F + 2ig tr F [A ; A ] + · · · :
(4.103)
Here we have de ned 0 F = 9 A − 9 A
(4.104)
14 Note that, due to our use of the metric (− + ++), our de nition of the Mandelstam variables di5ers by a sign from the one used in [21,66]. 15 Some steps towards an extension of those rules to the o5-shell case were taken in [64].
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Fig. 16. Structure of three-point boundary term.
as the “abelian” part of the non-Abelian eld strength tensor. Obviously, in the non-Abelian 0 F 0 -part. The additional terms involve the case the “bulk” integral will just produce the F color commutator, and thus, in eld theory terms, a quartic vertex. But Feynman diagrams involving quartic vertices have, compared to those with only cubic vertices, a smaller number of internal propagators, and thus of Feynman parameters. Since the boundary terms appearing in the worldline partial integration have fewer integrations but the same number of polarization vectors as the main term, they have the right structure to represent the missing color commutator terms, and a closer analysis of the e5ective action reveals that this is indeed their role. Here we will be satis ed with seeing how the second term in Eq. (4.103) makes its appearance. Consider the three-point integrand before partial integration, the P3 of Eq. (4.73). If we take just the term −GS B12 1 · 2 G˙ B3i 3 · ki
(4.105)
multiply by the three-point exponential, and partially integrate it in the variable 2 , we nd a boundary contribution T 1 d1 G˙ B12 1 · 2 G˙ B3i 3 · ki e1=2 GBij ki · kj |22 = = 3 0
=−
0
T
d1 G˙ B13 1 · 2 G˙ B31 3 · k1 eGB13 k1 · (k2 +k3 )
(4.106)
(3 = 0; only the lower boundary contributes, since G˙ B12 = 0 for 1 = 2 ). We note that the new integrand is, but for the Lorentz factors, identical with the one which we got in the two-point case after the partial integration, Eq. (4.36). The momenta k2 and k3 now appear only in the combination k2 + k3 , so that we may think of the vertex operators V2 and V3 as having merged to form a quartic vertex (Fig. 16). In our comparison with the e5ective action we thus should clearly identify leg number 1 with the A- eld appearing in F 0 , and indeed with this assignment the correct Lorentz structure ensues. What still needs to be seen is the color commutator. So far we have just a global color factor of tr(T a1 T a2 T a3 ), but we must remember that the full amplitude is only obtained after summing over all non-cyclic permutations of the result reached with the standard ordering. In the three-point case we have already two non-equivalent orderings, and taking the other ordering into account one nds a second boundary contribution which is identical to the one above except that it has the reverse color trace. Both traces can then be combined to a tr(T a1 [T a2 ; T a3 ]).
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We conclude that 1. Boundary terms always involve color commutators, and thus in the e5ective action picture contribute merely to the “covariantization” of the main term. 2. They lead to integrals which are already known from lower-point calculations. The rst fact may have been expected on general grounds; the second one is non-trivial, and has been veri ed only to low orders. 4.11. Relation to Feynman diagrams Finally, how do the integrand polynomials PN relate to the ones encountered in an ordinary Feynman parameter calculation of the N -photon amplitude? For the scalar QED case the connection is still very direct [24]. Consider again the two Feynman diagrams for the scalar QED vacuum polarization, Fig. 13. The rst one is a tadpole diagram involving the seagull vertex. Now the result of the worldline calculation before partial integration, Eq. (4.35), contained a GS Bij , and thus a !(1 − 2 ). This !-function also creates a quartic vertex, and comparing the parameter integrals one nds that, not surprisingly, its contribution to the amplitude matches with the tadpole diagram. This correspondence carries over to the N -point case, if one xes the ordering of the external legs, and transforms from - to -parameters according to Eq. (4.15). The partially un-integrated Bern–Kosower integrand is thus obtained from the Feynman parameter integrand by a transformation of variables, and a certain regrouping of terms. This transformation has two e5ects. First, it allows one to combine into one expression an individual Feynman diagram and all the ones related to it by a permutation of the external states. Second, by regrouping the -parameter expressions in terms of GBij , G˙ Bij , GS Bij , which are functions well adapted to the circle, the integrand is brought into a form suitable for partial integration, since now one needs, at least in the abelian case, not to worry about possible boundary terms. In the spinor-loop case, comparison with the Feynman calculation is not quite so straightforward. The resulting parameter integrals obviously include those from the scalar loop, and thus contain contributions from diagrams including the seagull vertex. Clearly they cannot correspond to the parameter integrals obtained from the standard QED Feynman rules. It turns out that they correspond to a di5erent break-up of those photon scattering amplitudes, a break-up according to a second order formalism for fermions [151–154,24,64,155] (this holds true also for the more general theories considered in [101,103,156]). The Feynman rules for (Euclidean) spinor QED in the second order formalism (see [155] and references therein) are, up to statistics and degrees of freedom, the ones for scalar QED with the addition of a third vertex (Fig. 17). The third vertex involves = 12 [" ; " ] and corresponds to the F -term in the worldline Lagrangian Lspin (compare Eq. (3.17)). For the details and for the non-abelian case see [155]. There also an algorithm is given, based on the Gordon identity, which transforms the sum of Feynman (momentum) integrals resulting from the rst order rules into the ones generated by the second order rules. This explains the close relationship between scalar and spinor QED calculations in the worldline formalism, which we already noted before, and will encounter again at the multiloop level.
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Fig. 17. Second order Feynman rules for spinor QED.
5. QED in a constant external eld An important role in quantum electrodynamics is played by processes involving constant external elds. An obvious physical reason is that in many cases a general eld can be treated as a constant one to a good approximation. In QED this is expected to be the case if the variation of the eld is small on the scale of the electron Compton wavelength. Mathematically, the constant eld is distinguished by being one of the very few known eld con gurations for which the Dirac equation can be solved exactly, allowing one to obtain results which are non-perturbative in the eld strength. For QED calculations in constant external elds it is possible and advantageous to take account of the eld already at the level of the Feynman rules, i.e. to absorb it into the free electron propagator. Suitable formalisms have been developed many years ago [157–160]. However, beyond the simplest special cases they lead to extremely tedious and cumbersome calculations. As we will see in the following section, in the string-inspired formalism the inclusion of constant external elds requires only relatively minor modi cations [96 –99,92]. For this reason it has been extensively applied to constant eld processes in QED in four [96 –100,92,108,161–164] as well as in three dimensions [142]. 5.1. Modi<ed worldline Green’s functions and determinants Similar to the absorption of a constant eld into the electron propagator in standard eld theory, in the worldline approach we would like to absorb the eld into the basic worldline correlators. Let us thus assume that we have, in addition to the background eld A (x) we started with a second one, AX (x), with constant eld strength tensor FX . Using Fock–Schwinger gauge
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centred at x0 [96] we may take AX (x) to be of the form
AX (x) = 12 y FX :
(5.1)
The constant eld contribution to the worldline Lagrangian (3.36) may then be written as ZLspin = 12 iey FX y˙ − ie
FX
;
(5.2)
in components, or as ZLspin = − 12 ieY FX DY ;
(5.3)
in the super eld formalism. Since it is still quadratic in the worldline elds, we need not consider it as part of the interaction Lagrangian; we can absorb it into the free worldline propagators. This means that we need to replace the de ning Eqs. (1.15) and (4.18) for the worldline Green’s functions by −1 2 d d 21 | − 2ieFX |2 ≡ GB (1 ; 2 ) ; (5.4) d2 d −1 d X 21 | − 2ieF |2 ≡ GF (1 ; 2 ) : (5.5) d These inverses are calculated in Appendix B, with the result (deleting the “bar”) T Z −iZG˙ B12 ˙ GB (1 ; 2 ) = + iZG B12 − 1 ; e 2(Z)2 sin(Z) GF (1 ; 2 ) = GF12
e−iZG˙ B12 ; cos(Z)
(5.6)
where Z ≡ eFT . These expressions should be understood as power series in the Lorentz matrix Z (note that Eqs. (5.6) do not require the eld strength tensor F to be invertible). Equivalent formulas have been given for the magnetic eld case in [142], and for the general case in [99]. Note also that the generalized Green’s functions are still translation invariant in , and thus functions of 1 − 2 . By writing them in terms of the ordinary Green’s functions GB GF we have avoided an explicit case distinction between 1 ¿2 and 1 ¡2 which would become necessary otherwise [99]. Note the symmetry properties GB (1 ; 2 ) = GBT (2 ; 1 );
T
G˙ B (1 ; 2 ) = − G˙ B (2 ; 1 );
GF (1 ; 2 ) = − GFT (2 ; 1 ) :
(5.7)
Those generalized Green’s functions are, in general, non-trivial Lorentz matrices, so that the Wick contraction rules Eq. (4.19) have to be replaced by y (1 )y (2 ) = − GB (1 ; 2 ) ;
(1 )
(2 ) = 12 GF (1 ; 2 ) :
(5.8)
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We also need the generalizations of G˙ B ; GS B , which are (see Appendix B) −1 i Z d −iZG˙ B12 ˙ GB (1 ; 2 ) ≡ 21 | − 2ieF |2 = −1 ; e d Z sin(Z) −1 −1 d 2 Z ˙ GS B (1 ; 2 ) ≡ 21 | 5 − 2ieF |2 = 2!12 − e−iZGB12 : d T sin(Z)
(5.9)
Let us also write down the rst few terms in the expansion in F for all four functions, T i ˙ T 2 T3 GB12 = GB12 − − G B12 GB12 TeF + (eF)2 + O(F 3 ) ; G − 6 3 3 B12 90 2 T ˙ ˙ GB12 = G B12 + 2i GB12 − eF + G˙ B12 GB12 T (eF)2 + O(F 3 ) ; 6 3 T S ˙ S GB12 = G B12 + 2iG B12 eF − 4 GB12 − (eF)2 + O(F 3 ) ; 6 GF12 = GF12 − iGF12 G˙ B12 TeF + 2GF12 GB12 T (eF)2 + O(F 3 )
(5.10)
2 (here we used the identity G˙ B12 = 1 − 4=T GB12 ). To lowest order in this expansion the eld dependent worldline Green’s functions coincide, of course, with their vacuum counterparts. Contrary to the vacuum case, in the constant eld background one nds nonvanishing coincidence limits not only for GB , but also for G˙ B and GF : T GB (; ) = (Z cot(Z) − 1) ; 2(Z)2
G˙ B (; ) = i cot(Z) −
i ; Z
GF (; ) = − i tan(Z) :
(5.11)
To correctly obtain this and other coincidence limits, one has to apply the rules G˙ B (; ) = 0;
2 G˙ B (; ) = 1
(5.12)
which follow from symmetry and continuity, respectively. Again GB and GF may be assembled into a super propagator, Gˆ (1 ; /1 ; 2 ; /2 ) ≡ GB (1 ; 2 ) + /1 /2 GF (1 ; 2 ) ;
(5.13)
allowing one to generalize (4.30) to
Y (1 ; /1 )Y (2 ; /2 ) = − Gˆ (1 ; /1 ; 2 ; /2 ) :
(5.14)
At rst sight this de nition would seem not to accommodate the non-vanishing coincidence limit of GF (which cannot be subtracted). Nevertheless, comparison with the component eld formalism shows that the correct expressions are again reproduced if one takes coincidence
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limits after superderivatives. For instance, the correlator D1 X (1 ; /1 )X (1 ; /1 ) is evaluated by calculating D1 X (1 ; /1 )X (2 ; /2 ) = /1 G˙ B12 − /2 GF12
(5.15)
and then setting 2 = 1 . This is almost all we need to know for computing one-loop photon scattering amplitudes, or the corresponding e5ective action, in a constant overall background eld. The only further information required at the one-loop level is the change in the free path integral determinants due to the external eld. As we will show in a moment, this change is ([96], see also [38,165]) −1=2 sin(Z) −D=2 −D=2 (4T ) → (4T ) det (Scalar QED) ; (5.16) Z −1=2 tan(Z) −D=2 −D=2 (4T ) → (4T ) det (Spinor QED) : (5.17) Z Since those determinants describe the vacuum amplitude in a constant eld one nds them to be, of course, just the integrands of the well-known Euler–Heisenberg–Schwinger formulas. 5.2. Example: one-loop Euler–Heisenberg–Schwinger Lagrangians Let us shortly retrace this calculation. In the scalar QED case, we have to replace T d2 1 2 −1=2 Dy exp − − 2 = (4T )−D=2 d y˙ = DetP 4 d 0 by
−1 2 d d d Det−1=2 5 − 2ieF − 2 + 2ieF = (4T )−D=2 Det −1=2 P P
d
d
d
(5.18)
(5.19)
(as usual, the prime denotes the absence of the zero mode in a determinant). Application of the ln det = tr ln identity yields 16 ∞ −1 d 1 (2ie)n d −n −1=2 n 5 − 2ieF Det P = exp tr [F ] Tr d 2 n d n=1 ∞ 1 B sin(eFT ) n −1=2 n n = exp ; (5.20) (2ieT ) tr[F ] − 2 = det n!n eFT n=2 n even
16 Note that, although the determinants considered here become formally identical with the ones appearing in (4.60) by 2ieF → C! , in the periodic case the results are not of the same form. The reason is that here the zero-mode had to be excluded from the determinant, while it needs to be included in the calculation of ZP .
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where the Bn are the Bernoulli numbers. In the second step Eq. (B.9) was used. The analogous calculation for the Grassmann path integral yields a factor −1 d (5.21) 5 − 2ieF Det+1=2 = det1=2 [cos(eFT )] : A d For spinor QED we therefore nd a total overall determinant factor of −1=2 tan(eFT ) −D=2 (4T ) det : eFT
(5.22)
Expressing these matrix determinants in terms of the two standard Lorentz invariants of the Maxwell eld (see Section 5.4 below) and continuing to Minkowski space, one obtains the well-known Schwinger proper-time representation of the (unrenormalized) Euler–Heisenberg– Schwinger Lagrangians [166 –169], ∞ 1 ds −ism2 e2 ab Lscal = − e ; (5.23) 162 0 s sin(eas) sinh(ebs) ∞ 1 ds −ism2 e2 ab Lspin = 2 e ; (5.24) 8 0 s tan(eas) tanh(ebs) where a2 − b2 ≡ B2 − E2 , ab ≡ E · B. 5.3. The N -photon amplitude in a constant <eld Retracing our above calculation of the N -photon path integral with the external eld included we arrive at the following generalization of Eq. (1.18), representing the scalar QED N -photon scattering amplitude in a constant eld [99,92]: scal [k1 ; 1 ; : : : ; kN ; N ] ki = (−ie)N (2)D !
0
×
N
i=1 0
T
∞
dT sin(Z) 2T (4T )−D=2 e−m det −1=2 T Z
N 1 1 di exp ki · GBij · kj − ii · G˙ Bij · kj + i · GS Bij · j 2 2 i; j = 1
: multi-linear
(5.25)
From this formula it is obvious that adding a constant Lorentz matrix to GB will have no e5ect due to momentum conservation. As in the vacuum case, we can use this fact to get rid of the coincidence limit of GB , (5.11). Thus instead of GB we will generally work with the equivalent Green’s function GX B , de ned by −iG˙ B12 Z − cos(Z) T e GX B (1 ; 2 ) ≡ GB (1 ; 2 ) − GB (; ) = (5.26) + iG˙ B12 : 2Z sin(Z) No such rede nition is possible for G˙ B or GF .
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The transition from scalar to spinor QED is done as in the vacuum case, again with only some minor modi cations. The spinor QED integrand for a given number of photon legs N is obtained from the scalar QED integrand by the following generalization of the Bern–Kosower algorithm: 1. Partial Integration: After expanding out the exponential in the master formula (5.25), and taking the part linear in all 1 ; : : : ; N , remove all second derivatives GS B appearing in the result by suitable partial integrations in 1 ; : : : ; N . 2. Replacement Rule: Apply to the resulting new integrand the replacement rule (2.15) with G˙ B ; GF substituted by G˙ B ; GF . Since the Green’s functions GB ; GF are, in contrast to their vacuum counterparts, non-trivial matrices in the Lorentz indices, it must be mentioned here that the cycle property is de ned solely in terms of the -indices, irrespectively of what happens to the Lorentz indices. For example, the expression 1 · G˙ B12 · k2 2 · G˙ B23 · 3 k3 · G˙ B31 · k1 would have to be replaced by 1 · G˙ B12 · k2 2 · G˙ B23 · 3 k3 · G˙ B31 · k1 − 1 · GF12 · k2 2 · GF23 · 3 k3 · GF31 · k1 : The only other di5erence to the vacuum case is due to the non-vanishing coincidence limits (5.11) of G˙ B ; GF . Those lead to an extension of the “cycle replacement rule” to include one-cycles [92]: G˙ B (i ; i ) → G˙ B (i ; i ) − GF (i ; i ) :
(5.27)
3. The scalar QED Euler–Heisenberg–Schwinger determinant factor must be replaced by its spinor QED equivalent, −1=2 sin(Z) −1=2 tan(Z) → det det : (5.28) Z Z 4. Multiply by the usual factor of −2 for statistics and degrees of freedom. 5.4. Explicit representations of the modi<ed worldline Green’s functions For the result to be practically useful it will be necessary to write GB ; GF in more explicit form. This can be done by choosing some special Lorentz system, such as the one where E and B are both pointing along the z-direction, and working with the explicit matrix form of the worldline correlators, which becomes particularly simple in such a system. This approach turns out to be quite adequate for the case of a purely magnetic (or purely electric) eld [92,100]. However, it is also possible to directly express all generalized worldline Green’s functions in terms of Lorentz invariants, without specialization of the Lorentz frame. This procedure is not only more elegant but appears also to be more eNcient computationally in the general case.
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5.4.1. Special constant <elds 1. Magnetic <eld case: With the B- eld chosen along the z-axis, introduce matrices g⊥ and g projecting on the x; y- and z; t- planes, so that 0 B 0 0 1 0 0 0 0 0 0 0 −B 0 0 0 ; g⊥ ≡ 0 1 0 0 ; g ≡ 0 0 0 0 : (5.29) F= 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 We also introduce z = eBT , and Fˆ = F=B. With these notations, we can rewrite the determinant factors Eqs. (5.16), (5.17) as z −1=2 sin(Z) det = ; Z sinh(z) z −1=2 tan(Z) det = : (5.30) Z tanh(z) The worldline correlators Eqs. (5.6), (5.9), (5.26) specialize to T (cosh(z G˙ B12 ) − cosh(z)) g⊥ 2 z sinh(z) T sinh(z G˙ B12 ) + − G˙ B12 iFˆ ; 2z sinh(z) ˙ B12 ) ˙ B12 ) 1 sinh(z G cosh(z G ˆ G˙ B (1 ; 2 ) = G˙ B12 g + − iF; g⊥ − sinh(z) sinh(z) z ˙ B12 ) z sinh(z G˙ B12 ) ˆ z cosh(z G GS B (1 ; 2 ) = GS B12 g + 2 !12 − g⊥ + 2 iF; T sinh(z) T sinh(z) GX B (1 ; 2 ) = GB12 g −
GF (1 ; 2 ) = GF12 g + GF12
cosh(z G˙ B12 ) sinh(z G˙ B12 ) ˆ g⊥ − GF12 iF : cosh(z) cosh(z)
(5.31)
Note that from GB we subtracted already its coincidence limit, indicated by the “bar”. Not removable are the coincidence limits for G˙ B and GF , 1 ˆ ˙ GB (; ) = − coth(z) − iF; z GF (; ) = − tanh(z)iFˆ :
(5.32)
2. Crossed <eld case: In a “crossed eld”, de ned by E ⊥ B; E = B, both invariants B2 − E 2 and E · B vanish. For such a eld F 3 = 0, so that the power series (5.6) break o5 after their quadratic terms. The worldline correlators thus get truncated to those terms which were
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given in (5.10). The determinant factors are trivial, sin(Z) tan(Z) = det−1=2 =1 : (5.33) det −1=2 Z Z The importance of this case lies in the fact that a general constant eld can be well-approximated by a crossed eld at suNciently high energies (see, e.g., [170]). 5.4.2. Lorentz covariant decomposition for a general <eld De ning the Maxwell invariants f ≡ 14 F F = 12 (B2 − E 2 ) ; g ≡ 14 F F˜ = iE · B ; we have the relations 2 F 2 + F˜ = − 2f5 ;
(5.34) (5.35)
F F˜ = − g5 : De ne N 2 F − N+ N− F˜ F± ≡ ± 2 ; N± − N∓2 N± ≡ n+ ± n− ; ) f±g n± ≡ : 2 Then one has
(5.36) (5.37) (5.38) (5.39)
F = F+ + F− ;
(5.40)
F 2 F± = − N±2 F± ;
(5.41)
F+ F− = 0 : With the help of these relations one easily derives the following formulas: feven (F) = feven (iN+ ) =
N+2
N+2
F−2 F+2 + f (iN ) even − (iN− )2 (iN+ )2
1 {−feven (iN+ )[N−2 5 + F 2 ] + feven (iN− )[N+2 5 + F 2 ]} ; − N−2
fodd (F) = fodd (iN+ ) =
(5.42)
F+ F− + fodd (iN− ) iN+ iN−
i {[N− fodd (iN− ) − N+ fodd (iN+ )]F − N−2
+ [N− fodd (iN+ ) − N+ fodd (iN− )]F˜ } ;
(5.43)
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where feven (fodd ) are arbitrary even (odd) functions in the eld strength matrix regular at F = 0, feven (F) =
∞
c2n F 2n ;
fodd (F) =
n=0
∞
c2n+1 F 2n+1 :
(5.44)
n=0
Decomposing GB; F into their even (odd) parts SB; F (AB; F ), GB; F = SB; F + AB; F
(5.45)
and applying the above formulas we obtain the following matrix decompositions of GB ; G˙ B ; GS B ; GF , SB12
AB12
A− T A+ B12 ˆ 2 B12 ˆ 2 = Z + Z 2 z+ + z− − 2 + 2 z− + AB12 A− z+ T − 2 B12 5 Z = A − A + − 2 − z2 ) z+ B12 z− B12 z+ z− 2(z+ − ˆ+ ˆ− Z Z iT − + = (SB12 − G˙ B12 ) + (SB12 − G˙ B12 ) 2 z+ z− z+ − iT − + ˙ B12 ) − z− (S + − G˙ B12 ) Z ˜ = [S − S ] Z + (S − G B12 B12 2 − z2 ) z− B12 z+ B12 2(z+ − 2
2
− ˆ + ˆ ˙ B12 = −SB12 S Z+ − SB12 Z− =
2 z+
1 − 2 − 2 + + {[z+ SB12 − z− SB12 ]5 + [SB12 − SB12 ]Z 2 } 2 − z−
+ ˆ ˆ A˙B12 = −i[A− B12 Z− + AB12 Z+ ]
=
2 z+
i − + + ˜ {[z− A− B12 − z+ AB12 ]Z + [z− AB12 − z+ AB12 ]Z} 2 − z−
− ˆ2 S B12 = GS B12 5 + 2 [z+ A+ ˆ2 S B12 Z+ + z− AB12 Z− ]
T
= GS B12 5 + S B12 = A =
2 T (z+
2 − − 2 2 + 2 {[z− z+ A+ B12 − z+ z− AB12 ]5 + [z+ AB12 − z− AB12 ]Z } 2) − z−
2i − ˆ + ˆ Z+ + z− SB12 Z− ] [z+ SB12 T 2i − 2 + 2 − + ˜ {[z+ SB12 − z− SB12 ]Z + z+ z− [SB12 − SB12 ]Z} 2 − z− )
2 T (z+
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2
− ˆ + ˆ SF12 = −SF12 Z+ − SF12 Z− =
139
1 − 2 − 2 + + {[z+ SF12 − z− SF12 ]5 + [SF12 − SF12 ]Z 2 } 2 − z2 z+ −
+ ˆ ˆ AF12 = −i[A− F12 Z− + AF12 Z+ ]
=
2 z+
i − + + ˜ {[z− A− F12 − z+ AF12 ]Z + [z− AF12 − z+ AF12 ]Z} : 2 − z−
(5.46)
Here we have further introduced ˜ ˜ ≡ eT F; Z
z± ≡ eN± T;
Z± ≡ eTF± =
2Z−z z Z z± + − ˜ ; 2 2 z± − z∓
ˆ±≡ Z
Z±
z±
:
(5.47)
˜ = − z+ z− 5; Z ˆ ± . The scalar, dimensionless coeNcient functions appearing ˆ 3± = − Z Note that ZZ in these formulas are given by ± SB12 =
sinh(z± G˙ B12 ) ; sinh(z± )
± SF12 = GF12
A± B12 =
cosh(z± G˙ B12 ) ; cosh(z± )
cosh(z± G˙ B12 ) 1 − ; sinh(z± ) z±
A± F12 = GF12
sinh(z± G˙ B12 ) : cosh(z± )
(5.48)
± (A± Note that SB=F12 B=F12 ) are odd (even) in 1 − 2 . Thus the non-vanishing coincidence limits ± are in AB; F ,
A± Bii = coth(z± ) −
1 ; z±
A± Fii = tanh(z± ) :
(5.49)
In the string-inspired formalism, the functions (5.48) are the basic building blocks of parameter integrals for processes involving constant elds. Let us also write down the rst few terms of the weak eld expansions of these functions, 2 2 GB12 2 2 GB12 2 GB12 4 6 1− z± + O(z± ) ; z + + 3 T ± 45 T 15 T 2 2 GB12 1 1 2 GB12 ± 3 5 −2 z± + − + z± + O(z± ); AB12 = 3 T 45 3 T 2 2 GB12 2 2 GB12 GB12 ± 4 6 SF12 = GF12 1 − 2 z + + 2 z± + O(z± ) ; T ± 3 T T 1 2 GB12 3 5 ˙ A± = G z − z + O(z ) : G + ± F12 B12 ± ± F12 3 3 T
± SB12 = G˙ B12
(5.50)
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In the same way one nds for the determinant factors (5.16), (5.17) z+ z− sin(Z) = det −1=2 ; Z sinh(z+ ) sinh(z− ) z + z− −1=2 tan(Z) det = : Z tanh(z+ ) tanh(z− )
(5.51)
Using the above formulas we can obtain explicit results in a Lorentz covariant way. Nevertheless, it will be useful to write down these formulas also for the Lorentz system where E and B are both pointing along the positive z-axis, E = (0; 0; E); B = (0; 0; B). (For this to be possible we have to assume that E · B¿0.) In this Lorentz system g = iEB, so that n± = 12 (B ± iE); where
N+ = B;
0 1 0 0 −1 0 0 0 r⊥ ≡ 0 0 0 0 0 0 0 0
N− = iE;
F+ = Br⊥ ;
0 0 0 0 0 0 : 0 0 1 0 −1 0
0 0 r ≡ 0 0
;
F− = iEr ;
(5.52)
(5.53)
Using those and the projectors g⊥ ; g introduced in (5.29) the matrix decompositions (5.46) can be rewritten as follows, SB12 =−
˙ B12 = S
T AB12 g ; 2 z
A B12 =
= ⊥;
SB12 g ;
= ⊥;
AB12 r ;
= ⊥;
S B12 = GS B12 g − S SF12 =
A˙B12 = − i
= ⊥;
−G ˙ B12 iT SB12 r ; 2 z
2 z AB12 g ; T = ⊥;
SF12 g ;
A F12 = − i
= ⊥;
S B12 = A
2i z SB12 r ; T = ⊥;
AF12 r
(5.54)
= ⊥;
− + with S=A⊥ B=F ≡ S=AB=F (z+ = eBT ≡ z⊥ ); S=AB=F ≡ S=AB=F (z− = ieET ≡ z ).
5.5. Example: the scalar=spinor QED vacuum polarization tensors in a constant <eld We now apply this formalism to a calculation of the scalar and spinor QED vacuum polarization tensors in a general constant eld. For the 2-point case the master formula (5.25) yields the following integrand, X exp {: : :} |multi-linear = [1 · GS B12 · 2 − 1 · G˙ B1i · ki 2 · G˙ B2j · kj ]ek1 · GB12 · k2 ;
(5.55)
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where summation over i; j = 1; 2 is understood. Removing the second derivative in the rst term by a partial integration in 1 this becomes X [ − 1 · G˙ B12 · 2 k1 · G˙ B1j · kj − 1 · G˙ B1i · ki 2 · G˙ B2j · kj ]ek1 · GB12 · k2 :
(5.56)
We apply the “cycle replacement rule” to this expression and use momentum conservation, k ≡ k1 = − k2 . The content of the brackets then turns into 1 I 2 , where k · GF12 · k I = G˙ B12 k · G˙ B12 · k − GF12 4 D − [(G˙ B11 − GF11 − G˙ B12 )4 (G˙ B21 − G˙ B22 + GF22 )D + GF12 GF21 ]k D k 4 :
(5.57)
Next we would like to use the fact that this integrand contains many terms which integrate to zero due to antisymmetry under the exchange 1 ↔ 2 . This we can do by decomposing GB and GF as in (5.45). First note that only the Lorentz even part of GB contributes in the exponent, k1 · GX B12 · k2 = k1 · (SB12 − SB11 ) · k2 ≡ − Tk · A12 · k ;
(5.58)
I turns, after decomposing all factors of G˙ B ; GF as above, and deleting all -odd terms, into
D4
4
D
4 D4 D ˙ B12 S ˙ B12 − S ˙ B12 S ˙ B12 ) − (SF12 ≡ {(S SF12 − SF12 SF12 ) Ispin D D 4 + (A˙B12 − A˙B11 + AF11 )4 (A˙B12 − A˙B22 + AF22 )D − A4 F12 AF12 }k k
(5.59)
(here we used (5.7)). In this way we obtain the following integral representations for the dimensionally regularized scalar=spinor QED vacuum polarization tensors [171], Escal (k) =
e2 − (4)D=2
Espin (k) = 2
e2 (4)D=2
∞
0
0
∞
1 dT 2−D=2 −m2 T −Tk · A12 · k −1=2 sin(Z) e det du1 Iscal e ; (5.60) T T Z 0
1 dT 2−D=2 −m2 T −Tk · A12 · k −1=2 tan(Z) e det du1 Ispin e : T T Z 0
(5.61)
Here Iscal is obtained simply by deleting, in Eq. (5.59), all quantities carrying a subscript “F”. As usual we have rescaled to the unit circle and set u2 = 0. Note that again the transversality of the vacuum polarization tensors is manifest at the inte grand level, k Iscal=spin = Iscal=spin k = 0. The constant eld vacuum polarization tensors contain the UV divergences of the ordinary vacuum polarization tensors (4.38), (4.41), and thus require renormalization. As is usual in this
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context we perform the renormalization on-shell, i.e. we impose the following condition on the renormalized vacuum polarization tensor EX (k) (see, e.g., [172]), lim EX (k) = 0 : lim 2
(5.62)
k →0 F →0
Counterterms appropriate to this condition are easy to nd from our above results for the ordinary vacuum polarization tensors. From the representations Eqs. (4.38), (4.41) for these tensors it is obvious that we can implement (5.62) by subtracting those same expressions with 2 the last factor e−GB12 k deleted. In this way we nd for the renormalized vacuum polarization tensors ∞ 2 dT −m2 T 1 2 X Escal (k) = Escal (k) + (g k − k k ) du1 G˙ B12 ; e 4 T 0 0 ∞ 1 dT −m2 T 2 2 (k) − (g k 2 − k k ) du1 (G˙ B12 − GF12 ): (5.63) EX spin (k) = Espin e 2 T 0 0 The remaining u1 -integral can be brought into a more standard form by a transformation of variables v = G˙ B12 = 1 − 2u1 . Writing the integrands explicitly using formulas (5.46) and continuing to Minkoswki space 17 we obtain our nal result for these amplitudes [164], ∞ ds −ism2 1 dv z+ z− EX scal (k) = − e 4 0 s 2 sinh(z + ) sinh(z− ) −1 s AB12 − AB11 2 ˆ ·k ×exp −i k ·Z 2 = +;− z
×
2
; = +;−
2
2
2
ˆ · k − (Z ˆ ) k · Z ˆ k) (Z ˆ k) ] (SB12 SB12 [(Z
ˆ k) (Z ˆ k) ) − ( k 2 − k k )v2 − (AB12 − AB11 )(AB12 − AB22 )(Z EX spin (k) = 2
∞
1
; = +;−
17
dv 2
;
(5.64)
z+ z− tanh(z+ ) tanh(z− ) 0 −1 s AB12 − AB11 2 ˆ ·k ×exp −i k ·Z 2 = +;− z
×
ds −ism2 e s
*
ˆ 2 · k − (Z ˆ 2 ) k · Z ˆ 2 k) (Z ˆ 2 k) ] ([SB12 SB12 − SF12 SF12 ][(Z
For the Maxwell invariants this means f → F; g → iG; N+ → a; N− → ib (to be able to x all signs we assume G ¿ 0). Note also that r⊥ k → k˜⊥ ; r k → − ik˜ .
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ˆ k) (Z ˆ k) ) − [(AB12 − AB11 + AF11 )(AB12 − AB22 + AF22 ) − AF12 AF12 ](Z 2 2 −( k − k k )(v − 1) ; (5.65) where now z+ = iesa ; ˆ+= Z
z− = − esb ;
aF − bF˜ ; a2 + b2
ˆ−= −i Z
F 2 − b2 5 ; a2 + b2
2
ˆ+= Z
bF + aF˜ ; a2 + b2
2
ˆ−= − Z
F 2 + a2 5 : a2 + b2
(5.66)
Here a; b denote the standard ‘secular’ invariants which we already introduced in Eqs. (5.23), (5.24). In terms of the invariants F; G those read a≡
+,
F2 + G 2 + F ;
b≡
+,
F2 + G 2 − F :
(5.67)
(F = 12 (B2 − E 2 ); G = E · B). For fermion QED, the constant eld vacuum polarization tensor was obtained before by various authors [173–176]. For the sake of comparison with their results, let us also specialize to the Lorentz system where E = (0; 0; E) and B = (0; 0; B). In this system a = B; b = E. Denoting k = (k 0 ; 0; 0; k 3 );
k⊥ = (0; k 1 ; k 2 ; 0) ;
k˜ = (k 3 ; 0; 0; k 0 );
k˜⊥ = (0; k 2 ; −k 1 ; 0) ;
(5.68)
our result can be written as follows: EX
spin scal
(k)
=− 4
−2
1
0
×e−isA0
; = ⊥; 2
∞
ds s
1
−1
dv 2
zz sin(z) sinh(z )
cos(z) cosh(z ) 1
s
( k 2 spin
scal
− e−ism ( k 2 − k k )
− k k ) + a
v2 − 1 v2
;
spin scal
k˜
˜ k
(5.69)
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where z = eBs; z = eEs, and 2 2 k⊥ cos(zv) − cos(z) k cosh(z v) − cosh(z ) − A0 = m + ; 2 z sin(z) 2 z sinh(z ) 2
⊥⊥ = sscal ⊥ ; ⊥
sscal
(5.70)
sin2 (zv) ; sin2 (z) =
sin(zv) sinh(z v) ; sin(z) sinh(z )
sinh2 (z v) ; sinh2 (z ) cos(zv) − cos(z) 2 ⊥⊥ ascal = ; sin(z)
sscal =
cos(zv) − cos(z) cosh(z v) − cosh(z ) ; sin(z) sinh(z ) cosh(z v) − cosh(z ) 2 ; ascal = sinh(z ) ⊥ ; ⊥
ascal
⊥⊥ sspin = ⊥ ; ⊥
sspin
=−
(5.71)
sin2 (zv) cos2 (zv) − ; cos2 (z) sin2 (z) =
sin(zv) sinh(z v) cos(zv) cosh(z v) − ; sin(z) sinh(z ) cos(z) cosh(z )
sinh2 (z v) cosh2 (z v) ; − sinh2 (z ) cosh2 (z ) 2 cos(zv) − cos(z) sin2 (zv) a⊥⊥ = − tan(z) − ; spin sin(z) cos2 (z) cos(zv) − cos(z) cosh(z v) − cosh(z ) ⊥ ; ⊥ aspin = − − tan(z) + tanh(z ) sin(z) sinh(z )
sspin =
− aspin
=
sin(zv) sinh(z v) ; cos(z) cosh(z )
cosh(z v) − cosh(z ) + tanh(z ) sinh(z )
2
−
sinh2 (z v) : cosh2 (z )
(5.72)
In this form it can be easily identi ed with the eld theory results of [159] (scalar QED) and [174,175] (fermion QED).
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5.6. Example: photon splitting in a constant magnetic <eld Photon splitting in a constant magnetic eld is a process of potential astrophysical interest. Its rst exact calculation, valid for an arbitrary magnetic eld strength and photon energies up to the pair creation threshold, was performed by Adler in 1971 [177]. This calculation amounts essentially to the calculation of the QED one-loop three-photon amplitude in a constant eld. This amplitude is nite, so that one can set D = 4. 5.6.1. Scalar QED To obtain the photon splitting amplitude for scalar QED, we have to use the correlators (5.31) for the Wick contraction of three photon vertex operators V0 and V1; 2 , T A Vscal; i [ki ; i ] = di i · x( ˙ i ) exp[iki · x(i )] ; 0
representing the incoming and the two outgoing photons. The calculation is greatly simpli ed by the peculiar kinematics of this process. Energymomentum conservation k0 + k1 + k2 = 0 forces collinearity of all three four–momenta, so that, writing k0 ≡ k ≡ !n, !1 !2 (5.73) k1 = − k; k2 = − k; k 2 = k12 = k22 = k · k1 = k · k2 = k1 · k2 = 0 : ! ! By a simple Lorentz invariance argument [177] one can assume k to be orthogonal to the magnetic eld direction. Moreover, in [177] it was shown, using CP invariance together with an analysis of dispersive e5ects, that there is only one non-vanishing polarization case. This is the case where the magnetic vector kˆ ׈0 of the incoming photon is parallel to the plane ˆ and those of the outgoing containing the external eld and the direction of propagation k, ones are both perpendicular to this plane. Taking the magnetic eld in the z-direction, and choosing 18 n = (1; 0; 0; −i), we can implement this case by taking 0 = (0; 1; 0; 0) and 1 = 2 = (0; 0; 1; 0). This leads to the further vanishing relations 1; 2 · 0 = 1; 2 · k = 1; 2 · F = 0 which leave us with the only a small number of nonvanishing Wick contractions: A A A Vscal; 0 Vscal; 1 Vscal; 2 - 2
T
=
i=0
-
=
i=0
=−i 18
T
0
2
i=0
.
di i · x˙i exp[iki · x(i )]
0
2
(5.74)
0
di i · x˙i exp in · T
2
. !X j xj
j=0
2 2 1 di exp !X i !X j n · GX Bij · n 1 · GS B12 · 2 !X i 0 · G˙ B0i · n : 2 i; j = 0
Note that we are still using Euclidean conventions.
i=0
(5.75)
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To keep the notation compact, we have de ned !X 0 = !; !X 1; 2 = − !1; 2 . A number of terms which vanish by the above relations have been omitted. For example, to see that the term involving 0 · x˙0 1 · x˙1 = 0 · GS B01 · 1
vanishes, remember that GS B is a power series in the matrix F . The rst term in this expansion is proportional to the Lorentz identity and gives zero since 0 · 1 = 0; all remaining ones give zero because F · 1 = 0. Performing the Lorentz contractions, and taking the determinant factor Eq. (5.30) into account, one obtains the following parameter integral for the three-point amplitude: ∞ dT −m2 T z 3 A A A scal [k0 ; k1 ; k2 ] = (−ie) (4T )−2 Vscal; e 0 Vscal; 1 Vscal; 2 T sinh(z) 0 ∞ T 2 z cosh(z G˙ B0i ) 3 −m2 T −2 = ie dT e (4T ) d1 d2 GS B12 !X i sinh(z) 0 sinh(z) 0 i=0 2 1 ˙ cosh(z G ) T Bij ×exp − !X i !X j GBij + ; (5.76) 2 2z sinh(z) i; j = 0
(z = eBT ). Translation invariance in has been used to set the position 0 of the incoming photon equal to T . Normalizing the amplitude according to Eq. (25) in [177], the nal result becomes Cscal [!; !1 ; !2 ; B] =
∞ T 2 m8 e−m T dT T d1 d2 GS B12 8!!1 !2 0 z 2 sinh2 (z) 0 2 2 1 ˙ Bij ) cosh(z G T × !X i cosh(z G˙ B0i ) exp − !X i !X j GBij + : 2 2z sinh(z) i=0
i; j = 0
(5.77) (Cscal corresponds to C2 there). 5.6.2. Spinor QED For the fermion loop case let us, for a change, use the super eld formalism rather than the cycle replacement rule. Using the super eld representation (4.29) of the photon vertex operator, A Vspin [k; ] =
0
T
d d/ · DX eik · X ;
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we can write the result of the Wick contraction for the spinor loop in complete analogy to the scalar loop result Eq. (5.75): 2 T 2
1 A A A Vspin; di d/i exp !X i !X j n · GXˆ ij · n 0 Vspin; 1 Vspin; 2 = i 2 0 i=0
×1 · D1 D2 Gˆ 12 · 2
i; j = 0
2
!X i 0 · D0 Gˆ 0i · n :
(5.78)
i=0
Here Gˆ denotes the constant eld worldline super propagator, Eq. (5.13). The Lorentz contractions are performed as before. The only di5erence is in the additional /-integrations, which are easy to do. The nal result becomes ∞ 2 m8 e−m T Cspin [!; !1 ; !2 ; B] = dT T 2 4!!1 !2 0 z sinh(z) T 2 1 T cosh(z G˙ Bij ) × d1 d2 exp − !X i !X j GBij + 2 2z sinh(z) 0
i; j = 0
× [ − cosh(z)GS B12 + !1 !2 (cosh(z) − cosh(z G˙ B12 ))]
! cosh(z G˙ B01 ) cosh(z G˙ B02 ) × − !1 − !2 sinh(z) cosh(z) sinh(z) sinh(z)
!!1 !2 GF12 ˙ ˙ + [sinh(z G B01 )(cosh(z) − cosh(z G B02 )) − (1 ↔ 2)] : cosh(z) (5.79)
A numerical analysis of this three-parameter integral has shown [100,178] it to be in complete agreement with other known integral representations of this amplitude [177,160,179,180]. See [181] for a more extensive analysis, as well as for a discussion of the relevance of photon splitting for the spectra of -ray pulsars and soft -repeaters. 6. Yukawa and axial couplings Up to now we have been concentrating almost exclusively on QED and QCD amplitudes. This rePects the present state-of-the-art, since almost all non-trivial applications of the string-inspired technique have been to these theories. In fact, until recently worldline path integral representations for more general theories were not available in the literature. It will be recalled that, with the exception of the gluon loop case, the worldline path integrals which we have used for QED and QCD have essentially been known since the 1960s. It is thus quite surprising that some relatively straightforward extensions of these formulas were apparently never considered until the recent revival of this subject triggered by the work of Bern and Kosower, and Strassler.
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Obviously, to be able to treat arbitrary fermion loop processes in the standard model one would need a worldline representation for the coupling of a spin 12 -loop to a more general background including a scalar eld , pseudoscalar eld 5 , vector eld A, and axialvector eld A5 . In (Minkowski space) eld theory we would thus be dealing with the following action: (6.1) S[; 5 ; A; A5 ] = − d x X [9= + + i"5 5 + i A= + i"5 A=5 ] : Here we absorbed the coupling constants into the background elds. The corresponding Euclidean e5ective action is given by E [; 5 ; A; A5 ] = ln Det[ p=E − i + "E5 5 + A=E + "E5 A=E5 ] :
(6.2)
In the following we work in Euclidean space-time as usual and drop the subscript E. In contrast to the QED or QCD e5ective action, which develops an imaginary part only due to threshold or nonperturbative e5ects, in the presence of axial vectors or pseudoscalars the e5ective action can become imaginary already in Euclidean perturbation theory. To be precise, in Euclidean space-time Feynman graphs with an even (odd) number of "5 -vertices contribute to Re(E ) (Im(E )). A worldline path integral representation for this e5ective action was constructed in [101,103] for the abelian case, though in a heuristic way. In [102,104] a completely rigorous treatment was given which also includes the antisymmetric tensor coupling and the non-abelian case. A detailed treatment of the general case would be lengthy. We will therefore restrict ourselves to two special cases of particular interest, the scalar-pseudoscalar and the vector–axialvector backgrounds. Moreover, we will take all elds to be abelian, and refer again to [102,104] for the non-abelian generalization. 6.1. Yukawa couplings from gauge theory For a beginning, let us restrict our attention to the case of only a scalar and a pseudoscalar eld. Here it is possible to give a simple and instructive solution of the problem [101], using a dimensional reduction procedure. As usual in this formalism we try to stay in line with string theory. Now in string theory Yukawa couplings are usually generated in the process of compactifying some of the unphysical dimensions. Our ansatz for this worldline action is therefore to take 5 and as the fth and sixth components of a Yang–Mills eld in six dimensions, with the other four components vanishing, A = (0; 0; 0; 0; 5 ; ). It was already mentioned that the path integral representation Eq. (1.9) for the coupling of the spinor loop to a background gauge eld remains valid for any even spacetime dimension. In six dimensions and for an A- eld of the form above, the worldline Lagrangian becomes L = 14 x˙2 + 14 x˙25 + 14 x˙26 + −2ig
5 9 5
1 2
− 2ig
˙+
1 2
5
6 9
˙ + 5 ;
1 2
6
˙ + igx˙5 5 + igx˙6 6 (6.3)
where g denotes the Yang–Mills coupling. We assume and 5 to depend only on the four physical dimensions, so that the index runs only from 1 to 4. The six-dimensional path
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integral is then Gaussian in the coordinate elds x5 , x6 . Integrating those out we obtain a new Lagrangian, L = 14 x˙2 +
1 2
˙+
1 2
5
˙ + 5
1 2
6
˙ + g2 2 + g2 25 − 2ig 6
5 9 5
− 2ig
6 9
:
(6.4)
So far the loop fermion was taken massless (which implies, in particular, that we cannot yet distinguish between the scalar and the pseudoscalar elds). To generate a mass term for the loop fermion we now use the scalar eld as a Higgs eld, i.e. we give it a non-vanishing vacuum expectation value by shifting m → + : (6.5) g Moreover, since we do not insist on gauge invariance, we can choose di5erent couplings 4; 45 for ; 5 . Our nal result for the worldline Lagrangian then becomes Lyuk = m2 + 14 x˙2 +
1 2
+ 452 25 + 2i45
˙+ 5
1 2
5
˙ + 5
1 2
6
˙ + 42 2 + 2m4 + 2i4 6
· 95 :
This Lagrangian contains two new worldline elds, are the same as for the other -components,
1 5; 6 (1 ) 5; 6 (2 ) = 2 GF (1 ; 2 )
:
6
· 9
(6.6) 5
and
6.
The Wick contractions for those (6.7)
Their free path integrals are normalized to unity. Note also the presence of several nonlinear terms in this worldline Lagrangian. For momentum space amplitude calculations those have to be treated in the same way as in the case of 4 theory above (see Section 4.1). This Lagrangian looks certainly less compelling than its gauge theory analogue. Nevertheless, precisely the same Lagrangian was also obtained by D’Hoker and Gagn\e in their more rigorous derivation [102,104,182,183]. For now, let us shortly explore its practical usefulness for amplitude calculations. 6.2. N scalar=N pseudoscalar amplitudes Consider the one-loop one-particle-irreducible amplitude involving either N massless scalars or N massless pseudoscalars, interacting with a fermion loop via Yukawa interactions. Using the above worldline Lagrangian in the usual procedure one nds that a master formula for this amplitude can be obtained from the one for the N -photon amplitude, Eq. (4.2), by the following modi cations: 1. Write Eq. (4.32) in D = 5, but with the same path integral determinant factor (4T )−2 as in four dimensions. 2. Take all polarization vectors ˜i in the unphysical dimension, ˜i = (0; 0; 0; 0; 1), and all momenta in the physical dimensions, k˜i = (ki ; 0). 3. Delete the constant term in the second derivative of the bosonic worldline Green’s function, 2 GS Bij = 2!ij − → 2!ij : (6.8) T
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Since point 2 implies that all ve-dimensional Lorentz products ˜i · k˜j vanish we can write N T ∞ dT
(5) N D −D=2 yuk [k1 ; : : : ; kN ] = − 2(i4(5) ) (2) ! ki di d/i (4T ) T 0 i=1 0 N 1 1 (GBij + /i /j GFij )ki · kj − (GFij + /i /j 2!ij )˜i · ˜j ×exp : (6.9) 2 2 i; j = 1
˜1 :::˜N
Here the ˜i · ˜j are all equal to unity, but before making use of this the exponential must be expanded, and the ˜i ’s anticommuted to the standard ordering ˜1 : : : ˜N . In the massive case we have to distinguish between the scalar and pseudoscalar cases. The simpler one is the pseudoscalar case, since according to Eq. (6.6) here the only di5erence between the massless and massive cases is in the usual m2 -term. Thus all that is needed to generalize the master formula Eq. (6.9) to the massive loop case is to supply it with the usual 2 proper-time exponential e−m T : N T ∞ dT
2 5 yuk [k1 ; : : : ; kN ] = − 2(i45 )N (2)D ! ki di d/i e−m T (4T )−D=2 T 0 i=1 0 N 1 1 ×exp : (6.10) (GBij + /i /j GFij )ki · kj − (GFij + /i /j 2!ij )˜i · ˜j 2 2 i; j = 1
˜1 :::˜N
In the scalar case we have the same mass term, but also the term 24m. After the usual formal exponentiation it produces an additional term in the master exponent, so that the massive master formula for the scalar case becomes somewhat more complicated than the pseudoscalar one: N T ∞ dT
N D −m2 T −D=2 yuk [k1 ; : : : ; kN ] = − 2(i4) (2) ! ki (4T ) di d/i e T 0 i=1 0 N N 1 1 ×exp ˜i /i : (GBij + /i /j GFij )ki · kj − (GFij + /i /j 2!ij )˜i · ˜j + 2im 2 2 i; j = 1
i=1
˜1 :::˜N
(6.11) Let us look explicitly at the two-point cases. For the scalar case we get from Eq. (6.11) the parameter integral ∞ T T dT −m2 T 2 D −D=2 [k1 ; k2 ] = −24 (2) !(k1 + k2 ) (4T ) d1 d2 d/1 d/2 e T 0 0 0 ×[(1 + /1 /2 GF12 k1 · k2 )(GF12 + /1 /2 2!12 ) − 4m2 /1 /2 ]eGB12 k1 · k2 :
(6.12)
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As usual we rescale to the unit circle, and set 2 = 0. Performing the /- and T -integrals, and using energy-momentum conservation, we obtain D D −D=2 2 D−2 2 2 (k) = 2(4) 4 2 1 − m − (4m + k ) 2 − 2 2 1 du[m2 + u(1 − u)k 2 ]D=2−2 : (6.13) × 0
The two-point function for the pseudoscalar case is obtained simply by deleting the term proportional to 4m2 (and replacing 4 by 45 ). In both cases the parameter integrals which we have at hand can be identi ed with the corresponding eld theory Feynman parameter integrals, albeit only after suitable integrations by parts. This generalizes, of course, to the N -point case. However, it should be noted that our parameter integrals have a certain advantage insofar as they are already of the scalar type. The master formula involves, apart from the usual factor of e1=2GBij ki · kj which in the T -integration turns into the Feynman denominator, only !(i − j ) and sign(i − j ). After restriction to an ordered sector the Feynman numerators are therefore constants. This is not the case in a straightforward Feynman parameter integral calculation of these amplitudes, where one would generally encounter non-trivial numerator polynomials. For example, the above master formula allows one to write the, say, six-point amplitudes immediately in terms of scalar triangle, box, pentagon, and hexagon integrals, without the need to perform a Passarino–Veltman type reduction. Thus it seems that, for the scalar=pseudoscalar amplitudes, one should use the master formula as it stands, without performing partial integrations. At this point it must be observed that we have been cheating a bit in the pseudoscalar case. From a Feynman diagram analysis it can be easily seen that, in Euclidean space, the amplitude with a massive fermion loop and any number of scalar and pseudoscalar legs is real (imaginary) for an even (odd) number of pseudo-scalars. Since our master formula (6.10) obviously vanishes for N odd, we have seemingly lost the imaginary part of the pseudoscalar amplitude. This was to be expected, since in our heuristic derivation we started from the gauge theory amplitude in six dimensions, which is real in Euclidean space. The missing imaginary part can also be represented on the worldline, though in a somewhat less natural way [101–104]. In [184] the resulting path integral representation was applied to the calculation of the radiative decay of the axion into two photons in a constant electromagnetic eld, and moreover generalized to the nite temperature case. 6.3. The spinor loop in a vector and axialvector background Another generalization of obvious interest is the inclusion of axialvectors. Here we will not follow the approach taken in [102,104], based on the introduction of auxiliary dimensions, but a more direct construction, which was proposed in [105] and further elaborated in [156,185]. This will also allow us to avoid the separation into the real and the imaginary part of the e5ective action which was implied in the approach of [102,104]. Thus we would now like to nd a path integral representation for (in Euclidean space) spin [A; A5 ] = ln Det[ p= + A= + "5 A=5 − im] :
(6.14)
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The method is a straightforward generalization of the one used in Section 3.3 for the pure vector case, and we will indicate only the necessary changes. First, Eq. (3.15) can be generalized to ( p= + A= + "5 A=5 )2 = − (9 + iA )2 + V ;
(6.15)
A ≡ A − "5 A5 ;
(6.16)
i V ≡ − (9 A − 9 A ) + i"5 A5; + (D − 2)A25 : 2
(6.17)
where
Here we have used the four-dimensional Dirac algebra, but dimensionally continued with an anticommuting "5 . Using Det[ p= + A= + "5 A=5 − im] = Det[ p= + A= + "5 A=5 + im] = Det1=2 [( p= + A= + "5 A=5 )2 + m2 ] ; one obtains 1 spin [A; A5 ] = − Tr 2
0
∞
dT exp{−T [ − (9 + iA )2 + V + m2 ]} : T
(6.18)
(6.19)
Up to a global factor, this is formally identical with the e5ective action for a scalar loop in a background containing a (Cli5ord algebra valued) gauge eld A and a potential V . Note that the exponent is not hermitian, which is the price we have to pay for writing down the whole e5ective action in one piece. However it is still positive for weak background elds, which is suNcient for our perturbative purposes. Applying the coherent state formalism in the same way as in Section 3.3 we arrive at the following representation, corresponding to (3.27), −T: 4 Tr e = i d x d 2 x; −|e−T: |x; =i
N
N
(d 4 xi d 2 i xi ; i |e−T=N: |xi+1 ; i+1 ) ;
(6.20)
i=1
where now : = − (9 + iA )2 + V . The only essential novelty is the presence of the "5 -matrix. To take it into account, it is crucial to observe that, expressed in terms of the a± r , it is identical to the fermion number counter or “G-parity operator” (−1)F [56,5], "5 = (−1)F = (1 − 2F1 )(1 − 2F2 ) ;
(6.21)
where F ≡ F1 + F2
− with Fi = a+ i ai :
(6.22)
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From the identity F
−|(−) = i0|
2
(−r −
a− r )(1
−
− 2a+ r ar ) = i0|
r=1
2
(−r + a− r ) = | ;
(6.23)
r=1
it is clear that the presence of (−1)F can be taken into account by switching the boundary conditions on the Grassmann path integral from antiperiodic to periodic. Thus (3.30) generalizes to xi ; i |e−T=N :[p; A; A5 ; " " ; "5 ] |xi+1 ; i+1 i i; i+1 i i+1 i; i+1 i i+1 =− d 4 pi; i+1 d 2 Xi; i+1 ei(x −x )p +( − )r Xr 4
(2) T × 1 − :[pi; i+1 ; Ai; i+1 ; Ai;5 i+1 ; 2i N
i+1 F ; (−1) ]
2 T +O : N2
(6.24)
In the continuum limit this leads to T 1 ∞ dT −m2 T Dx D e− 0 d LVA ; spin [A; A5 ] = − e 2 0 T A LVA = 14 x˙2 +
1 2
· ˙ + ix˙ A − i
F
− 2i"ˆ5 x˙
A5
+ i"ˆ5 9 A5 + (D − 2)A25 :
(6.25)
The operator (−1)F has turned into an operator "ˆ5 whose only raison d’ˆetre is to determine the boundary conditions of the Grassmann path integral; after expansion of the interaction exponential a given term will have to be evaluated using antiperiodic (periodic) boundary conditions on D , if it contains "ˆ5 at an even (odd) power. Once the boundary conditions are determined "ˆ5 can be replaced by unity. The perturbative evaluation of this double path integral can be done in the usual way. For the coordinate path integral everything proceeds as before. But for the Grassmann path integral one now has to proceed di5erently depending on the boundary conditions. In the antiperiodic case (“A”) there is again nothing new, we can compute it using the by now familiar Green’s function GF . In the periodic case (“P”) however one now encounters a fermionic zero mode. As for the coordinate path integral we must rst remove this zero mode before executing the path integral. Analogously to Eq. (4.4) we can do this by factorizing the Hilbert space of periodic Grassmann functions into the constant functions 0 and their orthogonal complement L(), D = d 0 DL ; P
0
() =
T
0
+ L () ;
d L() = 0 :
The zero mode integration then produces the expected -tensor via d 4 0 0 0 0D 04 = D4
(6.26)
(6.27)
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and the L-path integral can be performed using the correlator 1 − 2 1 L (1 ) L (2 ) = g = g 21 G˙ B12 : 2 sign(1 − 2 ) − T
(6.28)
The free L-path integral is, in four dimensions, normalized to unity. Summarizing, in the vector– axial vector case we have the Wick contraction rules y (1 ) y (2 ) = − g GB (1 ; 2 ) ;
(1 )
(2 )A = g 21 GF (1 ; 2 ) ;
L (1 ) L (2 )P = g 12 G˙ B (1 ; 2 )
(6.29)
and the free path integral determinants T 1 2 Dy e− 0 d 4 y˙ = (4T )−D=2 ; A
P
T
1 ˙ 2 ·
T
1 ˙ 2L·L = 1 =
D e− D L e−
0 d
0 d
= 4 = : NA ; : NP :
(6.30)
6.4. Master formula for the one-loop vector–axialvector amplitudes To extract the scattering amplitude from the e5ective action, as usual we must specialize the background elds to plane waves, and then keep the part of the e5ective action which is linear in all polarization vectors. As a preliminary step, it is convenient to linearize the term quadratic in A5 by introducing an auxiliary path integration, writing T 2 T √ z 2 exp −(D − 2) d A5 = Dz exp − d : (6.31) + i D − 2"ˆ5 z · A5 4 0 0 The Wick contraction rule for this auxiliary eld is simply z (1 )z (2 ) = 2g !(1 − 2 )
(6.32)
and its free path integral is normalized to unity. This allows us to de ne an axialvector vertex operator as follows: T √ A5 V [k; ] ≡ "ˆ5 d(i · k + 2 · x˙ · + D − 2 · z) eik · x : (6.33) 0
Before using this vertex operator for Wick contractions, as usual it is convenient to formally rewrite it as a linearized exponential, T √ √ √ A5 V [k; ] = "ˆ5 d d/ exp[ik · x + i/ · k + 2 · + 2/x˙ · + D − 2 / · z]|lin() : 0
(6.34)
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Here / is a Grassmann variable with d// = 1, and must now also be formally treated as Grassmann. The vectors are represented by the usual photon vertex operator (4.21) T A V [k; ] = d[ · x˙ + 2i · k · ] eik · x 0
=
T
0
d
√
d/ exp[ik · (x +
2/ ) + · (−/x˙ +
√
2 )]|lin() :
(6.35)
Those de nitions allow us to represent the one-loop amplitude with M vectors and N axial vectors in the following way: ∞ 1 dT −m2 T M +N [{ki ; i }; {k5j ; 5j }] = − NA; P (−i) (4T )−D=2 e 2 T 0 ×V A [k1 ; 1 ] : : : V A [kM ; M ]V A5 [k51 ; 51 ] : : : V A5 [k5N ; 5N ] ;
(6.36)
where the global sign refers to the ordering 1 2 : : : M 51 52 : : : 5N of the polarization vectors. It is then straightforward to perform the bosonic path integrations, 19 T x˙2 z 2 Dx Dz V A [k1 ; 1 ] : : : V A5 [k5N ; 5N ] e− 0 d( 4 + 4 ) −D=2
= (4T )
0
T
d1 · · ·
d/M
0
T
d51 · · ·
d/5N
√ 1 GBIJ KI · KJ − i/i G˙ BiJ i · KJ − i 2/5i G˙ BiJ 2 √ 1 − GS Bij /i /j i · j − 2GS Bij /i /5j i · j − GS Bij /5i /5j 2 √ √ + 2(i · i + 5j · j ) + 2i/i ki · i + i/5i 5i · k5i / − 2(D − 2)!(i − j )/5i /5j 5i · 5j |lin({i };{5j }) : ×exp
i
· KJ
i
·
j
(6.37)
Here and in the following all lower case repeated indices run over either the vector or the axial vector indices, depending on whether /i ; i or /5i ; 5i are involved, while capital repeated indices run over all of them ({KI } denotes the set of all external momenta). We omit the momentum conservation factor (4.13). The remaining -path integral is still Gaussian. For its performance we must now distinguish between even and odd numbers of axialvectors. For the antiperiodic case, N even, there is no zero-mode, and the integration can still be done in closed form. The only complication is the existence of the term −GS Bij /5i /5j i · j . It modi es the worldline propagator GF to (N ) GF12 ≡ 21 |(9 + 2B(N ) )−1 |2 ;
19
(6.38)
In [156] the factor of 2 in front of the term involving (D − 2)!(i − j ) had been missing in Eqs. (2.14), (2.19), and (2:22).
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where B(N ) denotes the operator with integral kernel B(N ) (1 ; 2 ) = !(1 − i )/5i GS Bij /5j !(j − 2 )
(6.39)
(B(N ) acts trivially on the Lorentz indices, which we suppress in the following). Expanding the right hand side of Eq. (6.38) in a geometric series and resuming one obtains a matrix (N ) representation for GF12 , GF(N ) =
GF S F + ··· : = GF − GF NG S F 5 + NG
(6.40)
Here NS ij is the antisymmetric N ×N matrix with entries /5i GS Bij /5j (no summation). Moreover, the fermionic path integral determinant changes by a factor S F )D=2 Det1=2 (5 + 2B(N ) 9−1 ) = det(5 + NG
(6.41)
as is easily seen using the ln det = tr ln-formula (note that on the left hand side we have a functional determinant, on the right hand side the determinant of a N ×N matrix). Using these results the fermionic path integral can be eliminated, yielding the following master formula for this amplitude [156], ∞ NA dT −m2 M +N even [{ki ; i }; {k5j ; 5j }] = − (−i) e T (4T )−D=2 2 T 0 T T S F )D=2 × d1 d/1 · · · d5N d/5N det(5 + NG 0
×exp
0
1 1 GBIJ KI · KJ − i/i G˙ BiJ i · KJ − GS Bij /i /j i · j 2 2
(N ) GFij (i + i/i ki + 5i − i/5i G˙ BiR KR + GS Bir /5i /r r ) 2 ×(j + i/j kj + 5j − i/5j G˙ BjS KS + GS Bjs /5j /s s ) + i/5i 5i · k5i −2(D − 2)!(i − j )/5i /5j 5i · 5j |lin ({i }; {5j }) : (6.42)
−
Let us verify the correctness of this formula for the case of the massive 2-point axialvector function in four dimensions. Expanding out the exponential as well as the determinant factor, and performing the two /-integrals, we obtain the following parameter integral, ∞ T T dT −m2 T [k1 ; 1 ; k2 ; 2 ] = 2 (4T )−D=2 d1 d2 e T 0 0 0 2 ×eGB12 k1 · k2 {2(D − 2)!(1 − 2 )1 · 2 − (D − 1)GS B12 GF12 1 · 2 2
2 −GF12 G˙ B12 (1 · 2 k1 · k2 − 1 · k1 2 · k2 ) − 1 · k1 2 · k2 } :
(6.43)
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As usual we rescale 1; 2 = Tu1; 2 , and use the translation invariance in to set u2 = 0. Setting also k = k1 = − k2 this leads to ∞ dT −m2 T (k) = 2 (4T )−D=2 {2(D − 2)Tg − 2(D − 1)Tg e T 0 1 2 + du e−Tu(1−u)k [2(D − 1)Tg + (1 − 2u)2 T 2 (g k 2 − k k ) + T 2 k k ]} : 0
(6.44)
In the massless case the rst two terms in braces do not contribute in dimensional regularization, since they are of tadpole type. For the remaining terms both integrations are elementary, and the result is, using -function identities, easily identi ed with the standard result for the massless QED vacuum polarization. A suitable integration by part veri es the agreement with eld theory also for the massive case. Here the tadpole terms do contribute, and the comparison shows that to get the precise D-dependence of the amplitude, appropriate to dimensional regularization using an anticommuting "5 , it was essential to keep the explicit D-dependence of the A25 -term in the worldline Lagrangian LVA . For an odd number of axial vectors, we need to go back to Eq. (6.37) and replace by +L. The D L-path integral is then executed in the same way as before, but with the propagator 0 GF changed to G˙ B . The nal result becomes [156] odd [{ki ; i }; {k5j ; 5j }] ∞ NP dT −m2 T M +N (4T )−D=2 =− (−i) e 2 T 0 T T × d1 d/1 · · · d5N d/5N det(5 + NS G˙ B )D=2 0
×
d4
0 exp
0
1 1 GBIJ KI · KJ − i/i G˙ BiJ i · KJ − GS Bij /i /j i · j 2 2
(N ) √ G˙ Bij − (i + i/i ki + 5i − i/5i G˙ BiR KR + GS Bir /5i /r r + 2GS Bir /5i /5r 0 ) 2 √ ×(j + i/j kj + 5j − i/5j G˙ BjS KS + GS Bjs /5j /s s + 2GS Bjs /5j /5s 0 ) √ + i/5i 5i · k5i − 2(D − 2)!(i − j )/5i /5j 5i · 5j − i 2/5i G˙ BiJ 0 · KJ * √ √ √ − 2GS Bij /i /5j i · 0 + 2 i + 5j · 0 + 2i/i ki · 0 ({i }; {5j }) : lin
(6.45)
(N ) Here G˙ B is de ned analogously to Eq. (6.40), (N ) G˙ B =
G˙ B : S G˙ B 5+N
(6.46)
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Fig. 18. Sum of triangle diagrams in eld theory.
The integrand still depends on the zero-mode Eq. (6.27).
0,
which is to be integrated according to
6.5. The VVA anomaly Any new formalism for calculations involving axialvectors must, of course, be confronted with the existence of the chiral anomaly [186,187]. Let us thus verify that our formulas above correctly reproduce the anomaly for the VVA case. Calculation of the VV 9 · A amplitude, either using the master formula Eq. (6.45) or a direct Wick contraction of Eq. (6.36), yields the following parameter integral: k3? A [k1 ]A [k2 ]A?5 [k3 ] = 2D4 k1D k24
0
∞
3
dT (4T )−2 T
i=1 0
T
di
×exp[(GB12 − GB13 − GB23 )k1 · k2 − GB13 k12 − GB23 k22 ] ×{(k1 + k2 )2 + (G˙ B12 + G˙ B23 + G˙ B31 )(G˙ B13 − G˙ B23 )k1 · k2 − (GS B13 + GS B23 )} : (6.47)
Here momentum conservation has been used to eliminate k3 . It must be emphasized that this parameter integral represents the complete three-point amplitude, and thus corresponds to the sum of the two di5erent triangle diagrams in eld theory, shown in Fig. 18. Removing the second derivatives GS B13 (GS B23 ) by a partial integration in 1 (2 ), the expression in brackets turns into 2 2 2 2 2 k1 · k2 {2 − (G˙ B12 + G˙ B23 + G˙ B31 )2 + G˙ B12 − G˙ B13 − G˙ B23 } + k12 (1 − G˙ B13 ) + k22 (1 − G˙ B23 )
=−
4 [(GB12 − GB13 − GB23 )k1 · k2 − GB13 k12 − GB23 k22 ] : T
(6.48)
In the last step we used the identities (F:25); (F:26). This is precisely the same expression which appears also in the exponential factor in (6.47). After performing the trivial T -integral we nd therefore a complete cancellation between the Feynman numerator and denominator polynomials, and obtain without further integration the desired result, k3? A A A?5 =
8 D4 D 4 k1 k2 : (4)2
(6.49)
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This is the usual expression for the divergence of the axialvector current [186,187]. Note that in the present formalism this divergence is unambiguously xed to be at the axialvector current. This can, in fact, be already seen at the path integral level, since the vectors are represented by the photon vertex operator (6.35) which, as is familiar from string theory, turns into a total derivative when contracted with its own momentum. This will lead to the vanishing of the whole amplitude, independently of the possible divergence of the global T -integration. Nothing analogous holds true for the axialvector vertex operator. The behavior of the present formalism with respect to the chiral symmetry in the dimensional cointinuation is thus somewhat unusual. It should be remembered that, in eld theory, one has essentially a choice between two evils. If one preserves the anticommutation relation between "5 and the other Dirac matrices [188] then the chiral symmetry is preserved for parity-even fermion loops, but Dirac traces with an odd number of "5 ’s are not unambiguously de ned in general, requiring additional prescriptions. The main alternative is to use the ’t Hooft– Veltman-Breitenlohner–Maison prescription [189,190]. In this case there are no ambiguities, but the chiral symmetry is explicitly broken, so that in chiral gauge theories nite renormalizations generally become necessary to avoid violations of the gauge Ward identities [191]. Since our path integral representation was derived using an anticommuting "5 , we have not broken the chiral symmetry. In particular, in the massless case the amplitude with an even number of axialvectors should coincide with the corresponding vector amplitude, and we have explicitly veri ed this fact for the two-point case. (Even though the structure of the resulting Feynman numerators is quite di5erent from the equivalent ones derived from the ordinary Bern–Kosower master formula.) Nevertheless, we did not encounter any ambiguities even in the parity-odd case, not even in the anomaly calculation. This property of the formalism may be useful for applications to chiral gauge theories. 6.6. Inclusion of constant background <elds As in the pure vector case, it is easy to take into account an additional (vector) background eld with constant eld strength tensor F [192]. The presence of the background eld modi es the Wick contraction rules (6.29) to y (1 )y (2 ) = − GB (1 ; 2 ) ;
(1 )
(2 ) = 12 GF (1 ; 2 ) ;
L (1 )L (2 ) = 12 G˙ B (1 ; 2 )
(6.50)
(compare Section 5). The Gaussian path integral determinants (6.30) become T 2 sin(Z) Dy e− 0 d(1=4y˙ +1=2ie y F y˙ ) = (4T )−D=2 det −1=2 ; Z T 1 ˙ D e− 0 d( 2 · −ie F ) = 4 det 1=2 [cos Z] ; A
P
−
DLe
T 0
1 ˙ d( 2 L · L−ie L F L )
= det
1=2
sin(Z) Z
:
(6.51)
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Note that for the case of periodic Grassmann boundary conditions the eld dependence of the determinant factors cancels out between the coordinate and Grassmann path integrals. This cancellation is a consequence of the worldline supersymmetry (1.10) [20,21,64,156]. It does not occur in the antiperiodic case since here the supersymmetry is broken by the boundary conditions (see Appendix C of [156] for more on this point). In the periodic case the external eld must, moreover, be also taken into account in the zero-mode integration, as shown in the following example. 6.7. Example: vector–axialvector amplitude in a constant <eld As an explicit example, we calculate the vector–axialvector two-point function in a constant eld. This amplitude is relevant, for example, for photon–neutrino processes at low photon energies (see, e.g., [193,194,171]). According to the above we can represent this amplitude as follows: 1 ∞ dT −m2 T Dx A (k1 )A5 (k2 ) = D e 2 0 T P T 1 2 1 i ˙ ×exp − d x˙ + · + ex · F · x˙ − ie · F · 4 2 2 0 T × d1 (x˙ (1 ) + 2i (1 )k1 · (1 ))eik1 · x1 0
×
0
T
d2 (ik2 + 2 (2 )x( ˙ 2 ) · (2 ))eik2 · x2 :
(6.52)
This amplitude is nite, so that we can set D = 4 in its evaluation. As a rst step, the zero-modes of both path integrals are separated out according to Eqs. (4.4), (6.26), and the Grassmann zero mode integrated out using Eq. (6.27). All terms which do not contain all four zero mode components precisely once give zero. To explicitly perform this integration we note that by Eq. (6.26) we can rewrite, in the exponent of Eq. (6.52), T T d () · F · () = T 0 · F · 0 + d L() · F · L() : (6.53) 0
0
Thus for the case at hand the Grassmann zero mode integral can appear in the following three forms: (eT )2 d 4 0 eieT 0 · F · 0 = − F FD4 = − (eT )2 F · F˜ ; 2 D4 d 4 0 eieT 0 · F · 0 0 0 = ieTD4 FD4 = 2ieT F˜ ;
d4
ieT 0e
0
·F ·
0
0 0 0D 04 = D4
:
(6.54)
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161
In the next step, both path integrations are performed using the eld-dependent Wick contraction rules Eqs. (6.50). This results in the following parameter integral representation for the vector– axialvector vacuum polarization tensor 20 ee5 ∞ dT −m2 T T X E5 (k) = 2 e d1 d2 J5 (1 ; 2 ) e−k · G12 · k ; 8 0 T 3 0 ˜ Z Z · ? ? ˜ − J5 (1 ; 2 ) = [GS 12 − (G˙ 21 − G˙ 22 )(G˙ 11 − G˙ 12 )k k? ] iZ G˙ 22 4 ˜ ? ˜ ? Z·Z Z Z · ? ? ? ˜ ? − + G˙ 11 + k? k (G˙ 21 − G˙ 22 ) (G˙ 11 − G˙ 12 )k? k + k k? iZ 4 4 ˙ ˜ ˙ ˜ ˙ ˜ ˙ ˜ ˜ − G˙ ˜ × + i(G˙ 22 Z 12 Z + G12 Z + G12 Z − G12 Z + G11 Z ) ˜ Z·Z − (G˙ 11 G˙ 22 − G˙ 12 G˙ 12 + G˙ 12 G˙ 12 ) ;
4
(6.55)
˜ As usual it is useful to perform a partial integration on the one ˜ ≡ eT F. where k = k1 = − k2 ; Z S term involving G12 , leading to the replacement
GS 12 → G˙ 12 k · G˙ 12 · k :
(6.56)
By this partial integration, and the removal of some terms which cancel against each other, J5 (1 ; 2 ) gets replaced by ˜ ˜ ? ? Z·Z Z Z · ? ? ˙ ˙ ? ˜ − G˙ 22 − G˙ 12 k k k k [G12 G12 + (G˙ 21 − G˙ 22 )G˙ 12 ] iZ 4 4 ˜ ? ? ? ˙ ˙ ˙ ˜ + k ? k (G˙ 21 − G˙ 22 ) + Z · Z (G˙ + ik k ? Z 12 G12 − G12 G12 ) 4
˜ − G˙ 12 Z ˜ + G˙ 12 Z ˜ + G˙ 12 Z ˜ − G˙ 12 Z ˜ : (6.57) + i G˙ 22 Z As in the vector–vector case, we decompose Gij as Gij = Sij + Aij ;
(6.58)
where S(A) are its parts even (odd) in F. We can then delete all terms odd in 1 − 2 since they vanish upon integration. After using the identity F F˜ = − g5 and some combining of terms, J5 nally turns into the following, nicely symmetric expression I5 , 0 ˜ k U12 k + [(Z ˜ k) (U12 k) + ( ↔ )] I5 (1 ; 2 ) = i Z 20
Since GF ; GF do not occur for the periodic case we delete the subscript “B” in the remainder of this section.
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* ˙ 12 k − [(Z ˙ 12 k) + ( ↔ )] ˙ 12 ) k S ˙ 12 k) (S ˜S ˜S −(Z
+
˜ 0 Z·Z −A˙12 k U12 k − [(A˙12 k) (U12 k) + ( ↔ )] 4
* ˙ 12 k + [(A˙22 S ˙ 12 k) + ( ↔ )] : ˙ 12 ) k S ˙ 12 k) (S +(A˙22 S
Here in addition to A and S we have introduced the combination U, i 1 − cos(ZG˙ 12 )cos(Z) 2 ˙ ˙ ˙ ˙ : U12 = S12 − (A12 − A22 ) A12 + = Z sin2 (Z)
(6.59)
(6.60)
De ning also i ; (6.61) Z this expression can be further compressed to ˜ Z·Z I5 (1 ; 2 ) = {−Aˆ12 k U12 k − [(Aˆ12 k) (U12 k) + ( ↔ )] 4 ˙ 12 k + [(Aˆ22 S ˙ 12 k) + ( ↔ )]} : ˙ 12 ) k S ˙ 12 k) (S +(Aˆ22 S (6.62) ˙ ; A˙, given in Eq. (5.45), to write the We can now use the matrix decompositions of S; S ˆ ±; Z ˆ 2± } or integrand in explicit form. In this we have a choice between the matrix bases {Z ˜ F 2 }. We will use the former one here since it leads to a somewhat more compact {5; F; F; expression. After the usual rescaling to the unit circle, a transformation of variables v = G˙ 12 , and continuation to Minkowski space, we obtain our nal result for the vector–axialvector amplitude in a constant eld [192], 1 ∞ Aˆ − Aˆ e3 e5 s dv 2 2 B12 B11 ˆ ·k E5 (k) = G ds s e−ism k ·Z exp −i 82 2 = +; − z 0 −1 2 Aˆ ≡ A˙ +
×
; = +;−
2 [Aˆ 12 ((Aˆ 12 − Aˆ 22 )Aˆ 12 − (S12 ) ) + Aˆ 22 S12 S12 ] 2
2
2
ˆ kZ ˆ k + (Z ˆ k) (Z ˆ k) + (Z ˆ k) (Z ˆ k) ] ; ×[Z
(6.63)
where ± S12 =
sinh(z± v) ; sinh(z± )
± cosh(z± v) Aˆ 12 = ; sinh(z± )
± Aˆ ii = coth(z± )
(6.64)
ˆ ± ; a; b are the same as in (5.66), (5.67). As in the vector–vector case, this expression and z± ; Z becomes somewhat more transparent if one specializes to the Lorentz system where E and B are both pointing along the positive z-axis, E = (0; 0; E); B = (0; 0; B). Here one obtains 1 dv −isA0 ˜ 2 e2 e5 ∞ E5 (k) = i 2 ds c [F k + (F˜ k) k + (F˜ k) k ] ; (6.65) e 8 0 2 −1 ; = ⊥;
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where z = eBs; z = eEs; k⊥ = (0; k 1 ; k 2 ; 0); k = (k 0 ; 0; 0; k 3 ), 0 0 0 B 0 0 0 0 0 0 0 0 0 − E ; (F˜ ⊥ ) ≡ (F˜ ) ≡ 0 0 E 0 0 0 0 0 0 0 −B 0 0 0 2 2 k k cos(zv) − cos(z) cosh(z v) − cosh(z ) − A0 = m2 + ⊥ 2 z sin(z) 2 z sinh(z ) cos(zv) − cos(z) ; c⊥⊥ = z sin3 (z) c⊥ =
0 0 ; 0 0
;
163
(6.66) (6.67)
z cos(zv) cosh(z )cosh(z v) − 1 z cos(z)sin(zv) sinh(z v) − ; sin(z) sinh(z ) sinh2 (z ) sin2 (z)
c ⊥ = −
z cosh(z v) cos(zv)cos(z) − 1 z cosh(z )sinh(z v) sin(zv) − ; sinh(z ) sin(z) sin2 (z) sinh2 (z )
cosh(z v) − cosh(z ) : sinh3 (z ) This result can still be slightly simpli ed using the relations F˜ k2 = (F˜ k) k − (F˜ k) k c = − z
(6.68)
(6.69)
( = ⊥; ). It agrees, even at the integrand level, with the recent eld theory result of [195]. We remark that via the axial Ward identity k2 A (k1 )A5 (k2 ) = − 2imA (k1 )5 (k2 )
(6.70)
E5
one can also immediately obtain the vector–pseudoscalar amplitude in a constant eld. from This amplitude leads, for example, to a eld-induced e5ective axion–photon interaction [196]. More generally, from the Ward identity it is clear that the Lorentz contraction V A5 [k; k] of the axialvector vertex operator (6.33) can e5ectively serve as a pseudoscalar vertex operator. 7. E&ective actions and their inverse mass expansions In the previous sections we have derived worldline path integrals representing e5ective actions, but applied them mainly to the calculation of scattering amplitudes. In this section, we calculate the e5ective action directly in x-space, in a higher derivative expansion. The method used is a pure x-space version of the one used by Strassler in [150], made manifestly gauge invariant by the use of Fock–Schwinger gauge. 7.1. The inverse mass expansion for non-abelian gauge theory The higher derivative expansion is a standard tool for the approximative calculation of one-loop e5ective actions, and considerable work has gone into the determination of its coeNcients for various theories (see [197,198] and Refs. therein).
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This expansion exists in several versions, which di5er by the grouping of terms. The one which we will consider here is the “inverse mass expansion”, which is just the expansion in powers of the proper-time parameter T . This groups together terms of equal mass dimension. Up to partial integrations in space-time, it coincides with the (diagonal part of the) “heat kernel expansion” for the second order di5erential operator in question. In particular, every coeNcient in this expansion is separately gauge invariant. Alternatively, one may calculate the same series up to a xed number of derivatives, but with an arbitrary number of elds or potentials [199 –202,142]. In QED this corresponds, to zeroeth order, to the approximation of the e5ective Lagrangian by the Euler–Heisenberg Lagrangian (see Section 5). See [97,98] for a calculation of the rst gradient correction to the Euler–Heisenberg Lagrangian. Yet another option is to keep the number of external elds xed, and sum up the derivatives to all orders. This leads to the notion of Barvinsky–Vilkovisky form factors [203–205]. For the calculation of some such form factors in the string-inspired formalism see [96]. Individual terms in this expansion are also relevant for the determination of counterterms in the corresponding eld theories, de ned at a spacetime dimension D which is related to the mass dimension of the term considered [206]. We consider a background consisting of a scalar eld and=or a gauge eld, both possibly non-abelian. In this background, the scalar loop path integral (1.8) generalizes to
scal [A; V ] =
0
∞
dT −m2 T tr e T
Dx exp −
0
T
d
1 2 x˙ + igx˙ · A + V 4
:
(7.1)
Here we have rewritten V (x) ≡ U ((x)), where U () is the eld theory interaction potential of Section 3. The path integral is path ordered except if both A and V are abelian. In most applications of the heat kernel expansion the loop spin can be taken into account by an appropriate choice of the scalar part V (see, e.g., [207,208]). In this context we will therefore restrict ourselves to a treatment of the scalar loop case, although the evaluation technique outlined below extends to the spinor and gluon path integrals in an obvious way; see [96 –98] for the case of the fermion loop in QED. For QED this approach to the calculation of the e5ective action has also been generalized to the nite temperature case [162]. As always we separate out the ordinary integral over the loop center of mass x0 , which reduces the e5ective action to the e5ective Lagrangian,
[A; V ] =
d x0 L[A; V ](x0 ) :
To obtain the higher derivative expansion, we Taylor-expand both A and V at x0 , V (x) = ey · 9 V (x0 ) ; x˙ A (x) = y˙ ey · 9 A (x0 ) :
(7.2)
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The path-ordered interaction exponential is then expanded to yield ∞ T 1 n − 2 ∞ dT −m2 T (−1)n D d x0 d1 d2 : : : dn−1 scal [A; V ] = tr e T T n 0 0 0 0 n=0 y(1 )9(1) (1) × Dy[ig y˙ 1 (1 )ey(1 )9(1) A(1) V (x0 )] : : : 1 (x0 ) + e n
y(n )9(n)
×[ig y˙ (n )e
A(n) (x0 ) n
y(n )9(n)
+e
V
(n)
(x0 )]exp −
0
T
y˙ 2 d 4
:
(7.3)
Here we have labelled the background elds, and xed n = 0. This is also the origin of the factor of 1=n. We then use the Wick contraction rules for evaluating the individual terms in this expansion, e.g., ey(1 )9(1) ey(2 )9(2) = e−GB (1 ; 2 )9(1) · 9(2) ; y˙ (1 )ey(1 )9(1) ey(2 )9(2) = − G˙ B (1 ; 2 )9(2) e−GB (1 ; 2 )9(1) · 9(2) :
(7.4)
As in our earlier constant background eld calculations, we can enforce manifest gauge invariance by choosing Fock–Schwinger gauge centred at x0 . The gauge condition is y A (x0 + y()) ≡ 0 : In this gauge, A (x0 + y) = y
?
1
0
(7.5)
d F? (x0 + y)
(7.6)
and F? and V can be covariantly Taylor-expanded as (see, e.g., [209]) F? (x0 + y) = ey · D F? (x0 ) ; V (x0 + y) = ey · D V (x0 ) :
(7.7)
This leads also to a covariant Taylor expansion for A: 1 1 1 d y? ey · D F? (x0 ) = y? F? + y y? D F? + · · · : A (x0 + y) = 2 3 0
(7.8)
Using these formulas, we obtain the following manifestly covariant version of Eq. (7.3): ∞ T 1 n − 2 ∞ dT −m2 T (−1)n scal [F; V ] = tr d D x0 d1 d2 : : : dn−1 e T T n 0 0 0 0 n=0 n T y˙ 2 1 y(j )D(j) (j) × Dy exp − d e V (x0 ) + igy˙ j (j )y?j (j ) 4 0 j=1
×
0
1
j y(j )D(j)
dj j e
F?(jj)j (x0 )
:
(7.9)
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From this master formula, the inverse mass expansion to some xed order N,
scal [F; V ] =
0
∞
2
dT e−m T tr T (4T )D=2
d x0
N (−T )n n=1
n!
On [F; V ] ;
(7.10)
is obtained in three steps: Wick contractions: Truncate the master formula to n = N , and the covariant Taylor expansion Eq. (7.8) accordingly. Perform the Wick contractions. Alternatively, one may also rst Wick contract the complete expression for n = N , and truncate the Taylor expansion afterwards. (This procedure is preferable in the pure scalar eld case.) Integrations: Perform the -integrations. The integrand is a polynomial in the worldline Green’s function GB ; G˙ B , and GS B . As usual, the -integrals can be rescaled to the unit circle, i = Tui . The !-function in GS B (ui ; uj ) only contributes if ui and uj are neighbouring points on the loop. (Note that this includes the case GS B (1; un ).) In the non-Abelian case the coeNcient 2 in front of the !-function has to be omitted, since only half of the !-function contributes to the ordered sector under consideration (in the scattering amplitude context this rule was already stated in Section 4.4). Reduction to a minimal basis: The result of this procedure is the e5ective Lagrangian at the required order, albeit in redundant form. To be maximally useful for numerical applications, it still needs to be reduced to a minimal set of invariants, using all available symmetries. Those are 1. Cyclic invariance under the trace. 2. Bianchi identities. 3. For real representations of the gauge group one has an additional symmetry under transposition (up to a sign for every factor of F ). Usually those symmetry operations would have to be combined with judiciously chosen partial integrations performed on the e5ective Lagrangian. It is a remarkable property of the present calculational scheme that the reduction of our result for the e5ective action to a minimal basis of invariants can be achieved without any such partial integrations. In particular, for the pure scalar case the reduction process amounts to nothing more than the identi cation of cyclically equivalent terms (in fact, for this special case the whole procedure can be condensed into a purely combinatorial formula [210]). In the general case, the reduction to a minimal basis of invariants is much more involved. The method adopted here follows a proposal by MSuller [211,212], which is also explained in [107]. Let us give the explicit result of this procedure up to order O(T 4 ) (absorbing the coupling constant g into the elds, and abbreviating FD4 ≡ DD D4 F , etc.): O1 = V ; O2 = V 2 + 16 FD4 F4D ; O3 = V 3 + 12 VD VD + 12 VFD4 F4D −
2 15
i FD4 F4 FD +
1 20 FD4 FD4
;
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O4 = V 4 + 2VVD VD + 15 VD4 V4D + 35 V 2 FD4 F4D + 25 VFD4 VF4D − 45 i FD4 V4 VD −
8 15
i VFD4 F4 FD + 15 VFD4 FD4 −
+ 13 FD4 F4D V + 13 FD4 V F4D + 1 − 21 FD4 F4 F FD −
8 105
11 + 420 FD4 F F4D F +
2 35 FD4 F4D F F
i FD4 F4 FD −
1 70 FD4 F4D
6 35
+
2 15 FD4 F4 VD 4 35 FD4 F4 FD F
i FD4 F4 FD
:
With the present method, a complete calculation of all coeNcients was achieved to order O(T 6 ) in the general case, 21 and to order O(T 12 ) in the case with only a scalar eld [106, 143,107]. With conventional methods this expansion was previously obtained to order O(T 5 ) in the general case [206], and only recently to order O(T 7 ) in the scalar case [214]. Detailed comparisons with other methods [215,201,206] of calculating the higher derivative expansion have been made in [106,107]. We consider here only the Onofri–Zuk method, which is the one most closely related to the worldline technique. In Onofri’s work [216], the Baker–Campbell–Hausdor5 formula was employed to represent the coeNcients for the pure scalar case by Feynman diagrams in a one-dimensional auxiliary eld theory. Those Feynman diagrams are calculated using the Green’s function G 0 (1 ; 2 ) = |1 − 2 | − (1 + 2 ) +
2 1 2 T
(7.11)
which is the kernel for the second derivative operator on an interval of length T appropriate to the boundary conditions x(0) = x(T ) = 0 :
(7.12)
This representation was then used by Zuk to calculate the e5ective Lagrangian for the pure scalar case up to the terms with four derivatives [199,200]. This author further generalized the method to the gauge eld case, and also used Fock–Schwinger gauge to enforce manifest gauge invariance [217]. The same Green’s function is used in the “quantum mechanical path integral method” [218,82,219,220], which may be considered as an extension of the Onofri–Zuk formalism. To see the connection to our formalism, rst note that the Green’s function which we used for the evaluation of the reduced path integral Dy(), Eq. (1.16), is by no means unique. Since the naive de ning equation 1 92 G(1 ; 2 ) = !(1 − 2 ) 2 921
21
(7.13)
See the recent [213] for an application of this result to the approximate calculation of the one-loop contribution by massive quarks to the QCD vacuum tunneling amplitude.
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has no periodic solutions, it needs to be modi ed by the introduction of a background charge, leading to [64] 1 92 ? G (1 ; 2 ) = !(1 − 2 ) − ?(1 ) : 2 921
(7.14)
The distribution of the background charge ? along the circle is arbitrary, except that it should integrate to unity, T d?() = 1 : (7.15) 0
That this is necessary can be seen by integrating Eq. (7.14) in the rst variable. If one further requires the Green’s function G ? to be symmetric in its both arguments, periodicity determines it up to an irrelevant constant. The solution can be expressed in terms of the standard Green’s function GB , Eq. (1.16), as follows: T T d ?( )GB ( ; 2 ) − d GB (1 ; )?( ) G ? (1 ; 2 ) = GB (1 ; 2 ) −
+
0
T
d 1
0
T
0
0
d 2 ?( 1 )GB ( 1 ; 2 )?( 2 ) :
(7.16)
Any such G ? can be used as a Green’s function for the evaluation of the path integral Eq. (1.8). Di5erent choices of ? must lead to the same e5ective action or scattering amplitude. This is easily veri ed by the following little argument well known from string perturbation theory. Let us consider the scalar eld theory case rst. For the 3 scattering amplitude, Eq. (4.14) becomes, if a general ? is used, ∞ dT 1 2 N D [p1 ; : : : ; pN ] = (−4) (2) ! pi (4T )−D=2 e−m T 2 T 0 N T N
1 × di exp G ? (i ; j )pi · pj : (7.17) 2 0 i=1
i; j = 1
Using Eq. (7.16) and momentum conservation it is immediately seen that all ?-dependence drops out in the exponent. For the e5ective action the equivalence works in almost the same way. If only V is present, Eq. (7.3) after Wick contraction turns into ∞ T 1 n − 2 ∞ dT (−1)n −D=2 −m2 T D scal [V ] = (4T ) T e tr d x0 d1 d2 : : : dn−1 T n 0 0 0 0 n=0 n 1 ? ×exp − G (i ; j )9(i) · 9( j) V (1) (x0 ) : : : V (n) (x0 ) : (7.18) 2 i; j = 1
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Using Eq. (7.16) we may rewrite the exponent as follows: n n 1 (?) 1 − G (i ; j )9(i) · 9( j) = − GB (i ; j )9(i) · 9( j) 2 2 i; j = 1
+ 1 − 2
T
0
d ?( )
i; j = 1
n i=1
0
T
d 1
0
T
9(i) ·
n
GB ( ; j )9( j)
j=1
d 2 ?( 1 )GB ( 1 ; 2 )?( 2 )
n
9(i) · 9( j) :
(7.19)
i; j = 1
This shows that all ?-dependent terms in the e5ective Lagrangian carry at least one factor of ni= 1 9(i) . They are thus total derivative terms and will disappear in the nal x0 -integration (under appropriate boundedness conditions on the background eld at in nity). The same applies to the dependence on a constant which one could always add to G ? . This argument easily carries over to the gauge theory case, if one uses the Bern–Kosower master formula Eq. (1.18) and its e5ective action analogue. Di5erent admissible Green’s functions will thus in general produce di5erent e5ective Lagrangians, but the same e5ective action and scattering amplitudes. However, from string theory it is also known that the choice of the background charge can have some technical signi cance at intermediate stages of calculations [22,221]. The Green’s function GB usually used in the string-inspired approach corresponds to the choice of a constant ?, 1 ?() = : (7.20) T This is the only choice leading to a translation-invariant Green’s function. If one chooses the function ?() = !() (7.21) instead, Eq. (7.16) yields just the one used by Onofri, Eq. (7.11). Expansion of the path integral as in Eq. (7.9) then precisely generates Zuk’s one-dimensional Feynman rules. The nal result after integrating out x0 will be the same. However, the di5erence in the choice of ? turns out to have some nontrivial technical consequences: With our choice of the worldline Green’s function partial integrations never become necessary in the reduction process. This is not true for the Onofri–Zuk approach, a fact which becomes particularly conspicuous in the pure scalar case [106]. Here the e5ective Lagrangian resulting from our method is already minimal after identi cation of cyclically equivalent terms, while large numbers of partial integrations turn out to be necessary to further minimize Zuk’s result. Moreover, due to the translational invariance of the worldline Green’s function Eq. (1.16) cyclically equivalent terms always come with the same numerical coeNcient. This considerably facilitates the cyclic identi cation process. Again, this property does not hold true if one uses the Green’s function Eq. (7.11); for example, of the three cyclically equivalent terms V V V , VV V and V VV appearing in the scalar e5ective action at O(T 4 ) the rst two then get assigned the same coeNcient, while the coeNcient of the third one is di5erent.
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Finally, let us note that the ambiguity in the choice of ? has an interpretation already at the path integral level. From Eq. (7.16) it can be read o5 that G ? ful lls T T d ?( )G ? ( ; 2 ) = d G ? (1 ; )?( ) = 0 (7.22) 0
0
(for any 1 ; 2 ). This indicates that a given background charge ? corresponds to the following generalization of the zero mode xing Eqs. (4.4): Dx = d x0 Dy ; x () = x0 + y () ; T d?()y () = 0 : 0
(7.23)
In particular, for ?() = !() one recovers the boundary conditions Eq. (7.12), y(0) = y(T ) = 0. In the “string-inspired” formalism, the e5ective Lagrangian L(x0 ) is obtained as a path integral over the space of all loops having x0 as their common center of mass; in the Onofri– Zuk formalism, as a path integral over the space of all loops intersecting in x0 . And indeed, for Onofri’s original formalism precisely this path integral representation was already provided by Fujiwara et al. [222]. Clearly the center of mass choice is more “symmetric”, so that it is intuitively reasonable that it should lead to a more compact form for the e5ective Lagrangian. 7.2. Other backgrounds The above procedure can be extended to the case of a mixed vector–axialvector background without diNculties; see [105] for the computation of some heat kernel coeNcients for this background along the above lines. To the contrary, the inclusion of gravitational backgrounds poses new and interesting conceptual problems. Here the problems connected to the existence of di5erent ordering prescriptions for the quantum mechanical Hamiltonian, which we already briePy encountered in our discussion of the gluon loop, are of a more serious nature. Classically, the worldline Lagrangian for a scalar point particle coupled to a background gravitational eld could be taken as Lcl = 12 x˙ g (x)x˙ :
(7.24)
Quantum mechanically, the ambiguity in the factor ordering of the Hamiltonian leads to the possible appearance of further terms in the path integral action, which are of order ˝2 . According to the theorem by Sato [138] mentioned earlier our naive way of evaluating Gaussian path integrals requires the use of the Weyl-ordered Hamiltonian, which in turn is equivalent to using the mid-point rule in the standard time-slicing de nition of the path integral [137–141]. This procedure leads to the following “quantum” Lagrangian [140], ˝2 1 D 4 Lqu = g x ˙ + D ); x ˙ (R + g 4 TS 2 8 D denotes the Christo5el symbol. where 4
(7.25)
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Moreover, in a curved background the path integral measure also becomes nontrivial. Since, as usual, we wish to evaluate the path integral in terms of one-dimensional Feynman diagrams, it is natural to absorb this measure into the action by the introduction of appropriate ghost terms into the action, in the spirit of Lee and Yang [223]. In the present context this can be done in two slightly di5erent ways [61,224]. To further complicate matters, a careful analysis performed over the past few years by van Nieuwenhuizen, Bastianelli, and their coworkers [60 – 62,225,144,226] has established that the coeNcients of the above ˝2 -terms have no absolute meaning; they depend on the choice of the regularization prescription which is implicit in the de nition of the path integral. For example, another plausible de nition would be to expand all worldline elds, including the ghosts, in a sine expansion about the classical trajectories, and then integrate over the Fourier coeNcients. Regulating the resulting expressions by a universal cuto5 on these Fourier mode sums one arrives at the so-called “mode regularization”. The explicit computation [225,226] shows that, if one wishes to reproduce in this scheme the known results for the heat kernel in curved space, then the above Lagrangian must be replaced by 1 ˝2 1 D4 qu ?
LMR = x˙ g x˙ + R − g g g? D 4 : (7.26) 2 8 3 This ambiguity has also a natural interpretation in terms of the one-dimensional quantum eld theory de ned by the path integral. This eld theory constitutes a super-renormalizable nonlinear sigma model, which by power counting has super cial ultraviolet divergences at the one- and two-loop levels, but not at higher loop orders. 22 Those ultraviolet divergences cancel out in the sum of terms, but, as is usual in such cases, leave a nite ambiguity embodied by the above ˝2 -terms. Their coeNcients cannot be determined inside the one-dimensional eld theory without further input. Requesting the result of the worldline perturbation series to reproduce the usual heat kernel expansion for the space-time eld theory provides such an input, and suNces to x them completely. Once this has been done, the coeNcient of the R-term turns out to be universal for the regularization schemes considered in the works quoted above. Since the heat kernel is a covariant quantity, the appearance of the other, noncovariant terms in the worldline Lagrangians (7.25), (7.26) is clearly connected to the fact that both regularizations used, time-slicing and mode regularization, break the covariance; the role of the explicit noncovariant terms in the Lagrangians is to compensate for this. This leads to the question whether some regularization can be found which would avoid this covariance breaking. Very recently, Kleinert and Chervyakov [227–229] have claimed that one-dimensional regularization provides such a scheme, and they veri ed the absence of any ˝2 -terms in a simpler model with a one-dimensional target space. In [230,231] it was then shown that, in the four-dimensional case, this scheme is indeed free of noncovariant counterterms, although the R-term is still necessary: 1 ˝2 Lqu = g x ˙ + x ˙ R: DR 2 8 22
(7.27)
Here the loop counting refers to the one-dimensional worldline eld theory; in terms of the four-dimensional eld theory our discussion is, of course, at the one-loop level.
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This scheme therefore seems to be the most promising one for future applications of the string-inspired formalism to curved-space calculations. 23 To use any of these worldline actions for the calculation of the higher derivative expansion of the gravitational e5ective action, one would now wish to choose a Riemann normal coordinate system centered at the loop center of mass x0 . This is the gravitational analogue of Fock– Schwinger gauge, and allows one to rewrite the Taylor expansion of the metric at x0 in terms of covariant derivatives of the curvature tensor [232,233], g (x) = ! + 13 y y R + 16 y" y y ∇" R + · · · :
(7.28)
Individual terms in the higher derivative expansion can then again be calculated by the application of appropriate Wick contraction rules [60,61,224,62,225,144,226,185]. However, here another rather subtle complication appears in the choice of the worldline propagator. As we remarked in the previous section for the gauge theory case, and explicitly demonstrated for scalar eld theory, there exists a large family of admissible worldline propagators, and the e5ective actions computed with di5erent such propagators di5er by total derivative terms. While this statement remains true in the curved space case, an explicit computation at the two-loop level has revealed [144] that those total derivative terms are in general not covariant. If one wishes to reproduce precisely the standard heat kernel expansion, which is manifestly covariant, then the Onofri–Zuk propagator (7.11) must be used, corresponding to the boundary conditions y(0) = y(T ) = 0 on the path integral after the zero mode xing. The use of the standard worldline Green’s function GB , on the other hand, leads to a result which di5ers from this by noncovariant total derivative terms. This poses no problems in principle, but in practice, since it invalidates the application of Riemann normal coordinates, which are useful only for the computation of covariant quantities. 24 Thus the present state of a5airs is that, in the application of the worldline technique to curved space e5ective actions, one either has to forgo the convenience of using the translation invariant worldline propagator GB or, worse, of the use of Riemann normal coordinates. 25 8. Multiloop worldline Green’s functions While one could clearly construct multiloop formulas of the Bern–Kosower type starting with formulas such as Eqs. (4.14), (1.18), (4.32), and then sewing together pairs of external legs, such a procedure turns out to be unnecessarily cumbersome. In the spirit of the Bern– Kosower formalism, we would rather like to completely avoid the appearance of internal momenta. This is not only for aesthetic reasons; it is the absence of internal momenta which, in the 23 As a necessary preliminary step to any such application in [231] it was shown how to extend dimensional regularization to a compact time interval. 24 To be precise, the application of such coordinates to the computation of a noncovariant quantity will produce an apparently covariant result which is correct in that particular coordinate system but not in others. 25 It should be noted that, a priori, the analogous problem could have appeared also in the gauge theory case in the form of non-gauge invariant total derivative terms. Our use of Fock–Schwinger coordinates in the previous section was possible only because the compatibility of the standard propagator GB with gauge invariance is known on general grounds (see Section 4).
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Fig. 19. Change of the one-loop Green’s function by a propagator insertion.
Bern–Kosower formalism, reduces the number of independent kinematic invariants from the very beginning, thereby rendering the spinor helicity formalism even more useful than usual. We will rather follow the example of string theory, where multiloop amplitudes can be represented as path integrals over Riemann surfaces of higher genus, embedded into some target space. The Green’s function of the Laplacian (or some other kinetic operator) for those surfaces [5,22] is then the basic quantity needed for their calculation. In the in nite string tension limit, the Riemann surfaces correspond to graphs. If we wish to preserve the analogy with string theory, we ought to nd out how to calculate path integrals over graphs embedded into spacetime. This is by no means a new idea. For the example of 3 -theory, it has been repeatedly pointed out that it should be possible to construct multiloop amplitudes in terms of path integrals over graphs [234,235,121,236,237]. The proper-time lengths of the propagators making up those graphs would then just correspond to the moduli parameters in string theory. However, none of those authors provided an explicit computational prescription for the evaluation of this kind of path integral. In [84], M.G. Schmidt and the present author proposed such a prescription, based on the concept of Green’s functions de ned on graphs. Let us therefore begin with explaining how to construct such “multiloop worldline Green’s functions” at the two-loop level. 8.1. The 2-loop case At rst glance, this looks like an ill-de ned problem. In contrast to the circle, a general graph is not a di5erentiable manifold, and it is a priori not obvious how to de ne the second derivative operator at the node points. Instead, we will pose the following simple question. How does the Green’s function GB (1 ; 2 ) between two xed points 1 ; 2 on the circle change, if we insert, between two other points a and b , a (scalar) propagator of xed proper-time length TX (Fig. 19)? To answer this question, let us start with the worldline-path integral representation for the one-loop two-point amplitude in 3 -theory, and sew together the two external legs. The result is, of course, the vacuum path integral with a propagator insertion ∞ T T T x˙2 dT −m2 T (2) Dx vac = da db (x(a ))(x(b )) exp − d : (8.1) e T 4 0 0 0 0 Here (x(a ))(x(b )) is the x-space scalar propagator in D dimensions, which, if we specialize to the massless case for a moment, would read (D=2 − 1) (x(a ))(x(b )) = : (8.2) 4D=2 [(xa − xb )2 ](D=2−1)
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Clearly this form of the propagator is not suitable for calculations in our auxiliary onedimensional eld theory. The approach based on worldline Green’s functions which we have in mind will work nicely only if we can manipulate all our path integrals into Gaussian form. To obtain a Gaussian path integral, we therefore make further use of the Schwinger proper-time representation to exponentiate the propagator insertion, ∞ (x(a ) − x(b ))2 2 X (x(a ))(x(b )) = d TX e−m T (4TX )−D=2 exp − ; (8.3) 4TX 0 where the propagator mass was also reinstated. We have then the following path integral representation of the two-loop vacuum amplitude: ∞ T T dT −m2 T ∞ X (2) −D=2 −m2 TX X vac = d T (4T ) e da db e T 0 0 0 0 T x˙2 (x(a ) − x(b ))2 × Dx exp − d − : (8.4) 4 4TX 0 The propagator insertion has, for xed parameters TX ; a ; b , just produced an additional contribution to the original free worldline action. Moreover, this term is quadratic in x, so that we can hope to absorb it into the free worldline Green’s function. For this purpose, it is useful to introduce an integral operator Bab with integral kernel Bab (1 ; 2 ) = [!(1 − a ) − !(1 − b )][!(a − 2 ) − !(b − 2 )] (Bab acts trivially on Lorentz indices). We may then rewrite T T 2 d1 d2 x(1 )Bab (1 ; 2 )x(2 ) : (x(a ) − x(b )) = 0
0
(8.5)
(8.6)
Obviously, the presence of the additional term corresponds to changing the de ning equation for GB , Eq. (1.15), to - −1 . d2 B ab GB(1) (1 ; 2 ) = 2 1 − (8.7) 2 : 2 X d T After eliminating the zero-mode as before, this modi ed propagator can be constructed simply as a geometric series 2 −1 −2 −2 d Bab d d Bab d −2 − = + d2 d d TX TX d −2 d Bab d −2 Bab d −2 + + ··· : (8.8) d TX d TX d Noting that −2 d Bab Bab = − GBab Bab ; d
(8.9)
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we can explicitly sum this series, and obtain [84] GB(1) (1 ; 2 ) = GB (1 ; 2 ) +
1 [GB (1 ; a ) − GB (1 ; b )][GB (a ; 2 ) − GB (b ; 2 )] : 2 TX + GB (a ; b )
(8.10)
The worldline Green’s function between points 1 and 2 is thus simply the one-loop Green’s function plus one additional piece, which takes the e5ect of the insertion into account. Observe that this piece can still be written in terms of the various one-loop Green’s functions GBij . However it is not a function of 1 − 2 any more, nor is its coincidence limit a constant (for alternative derivations of this expression see [84,91,86]). Knowledge of this Green’s function is not yet suNcient for performing two-loop calculations. We also need to know how the path integral determinant is changed by the propagator insertion. Using the ln det = tr ln-formula this can be easily calculated, and yields T Dy exp[ − 0 dy˙ 2 =4 − (y(a ) − y(b ))2 =4TX ] T Dy exp[ − 0 dy˙ 2 =4] 2 3−D=2 Det P d 2 =d2 − Bab = TX GBab −D=2 = = 1+ : (8.11) 2 3−D=2 TX DetP d 2 =d2 To summarize, the insertion of a scalar propagator into a scalar loop can, for xed values of the proper-time parameters, be completely taken into account by changing the path integral normalization, and replacing GB by GB(1) . The vertex operators remain unchanged. In this way we arrive at the following two-loop generalization of Eq. (4.14), ∞ T T dT ∞ X −m2 (T +TX ) dT e (4)−D da db [T TX + TGB (a ; b )]−D=2 T 0 0 0 0 N T N
× di exp GB(1) (k ; l )kk · kl : (8.12) i=1 0
k;l = 1
For xed N , this integral represents a certain linear combination of two-loop diagrams in 3 -theory which have N legs on the loop, and no leg on the internal line (Fig. 20). Observe that we have diagonal terms in the exponential, since GB(1) has GB(1) (; ) = 0 in general. Self-contractions of vertex operators must therefore be taken into account. As always, momentum conservation allows one to absorb the diagonal terms into the non-diagonal ones, however in contrast to our previous experiences the coincidence limit of GB(1) is not constant. To ensure that the subtracted Green’s function has a zero coincidence limit, it must now be de ned in the following way: 26 (1) GX B (1 ; 2 ) ≡ GB(1) (1 ; 2 ) − 12 GB(1) (1 ; 1 ) − 12 GB(1) (2 ; 2 ) :
26
(8.13)
We remark that an analogous ambiguity appears in the electric circuit approach to Feynman parameter integration (1) [80,81,79]. In the terminology of [79] the condition GX B (; ) = 0 corresponds to the choice of a “level zero scheme”.
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Fig. 20. Summation of diagrams with N legs on the loop.
Fig. 21. Two di5erent parametrizations of the two-loop diagram.
Explicitly this gives 1 (GB1a − GB1b − GB2a + GB2b )2 (1) : GX B (1 ; 2 ) = GB12 − 4 TX + GBab
(8.14)
We note the following properties of this “subtracted two-loop worldline Green’s function”: (1) 1. As one would expect GX B reduces to GB in the limit where the proper-time TX of the inserted propagator becomes in nite. 2. For the case that both points 1; 2 are located on the same side (say, the left one) of the (1) propagator insertion we can rewrite GX B as the one-loop Green’s function GB with a modi ed global proper-time T → T . In the parametrization of (Fig. 21b) this new proper-time is given by T = T1 + T2 T3 =(T2 + T3 ): 27 (1) 3. GX B is not translation invariant, i.e. the equation 9 9 (1) + (8.15) GX B (1 ; 2 ) = 0 91 92
27
This property also has a direct analogue in the electric circuit formalism.
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is not true in general. It does hold, however, for 1; 2 on the same side of the propagator insertion, as follows directly from the previous property. So far we are restricted to inserting vertex operators on the loop only. Due to the symmetry of the diagram, this restriction is easily removed. Obviously, for any two points on the two-loop vacuum graph we may regard those to be on the loop, and the remaining branch—or one of the remaining branches—to be the inserted line. We can thus always use our formula Eq. (8.10) up to a reparametrization. In contrast to the string-theoretic worldsheet Green’s function the worldline Green’s function is a “bi-scalar”, i.e. it transforms trivially under reparametrizations. We therefore now nd it convenient to switch to the parametrization of Fig. 21b, which is symmetric with regard to the three branches. We have then three “moduli parameters” T1 ; T2 and T3 , and the location of a vertex operator on branch i will be denoted by a parameter i running from 0 to Ti . If we now x the number of external legs on branch i to be ni , carrying momenta k1(i) ; : : : ; kn(i)i , we obtain the following obvious generalization of Eq. (8.12): 3 ∞
2 dTa e−m (T1 +T2 +T3 ) (4)−D (T1 T2 + T1 T3 + T2 T3 )−D=2 a=1 0
×
n3 n1
n2
T1
i=1j=1m=1 0
d(1) i
T2
0
d(2) j
0
T3
d(3) k
ns nr nr 3 1 sym (r) (s) (r) sym (r) (r) (r) ×exp GBrs (k ; l )kk · kl(s) + GBrr (k ; l )kk · kl(r) : r¡s k = 1 l = 1
r=1
2
k;l = 1
(8.16) sym sym sym ; GB33 ; GB13 are related to GB(1) by the mentioned reparametrization. Again it Here the GB11 is convenient to absorb the diagonal coincidence terms from the beginning via Eq. (8.13). After this subtraction, the two-loop worldline Green’s function in symmetric parametrization becomes sym (1) (1) (1) (1) (1) GX B11 ((1) 1 ; 2 ) = Q|1 − 2 |[(T1 − |1 − 2 |)(T2 + T3 ) + T2 T3 ] (1) (1) (1) 2 = |(1) 1 − 2 | − Q(T2 + T3 )(1 − 2 ) ; sym GX B12 ((1) ; (2) ) = Q[T3 ((1) + (2) )[T1 + T2 − ((1) + (2) )]
+ (2) (T2 − (2) )T1 + (1) (T1 − (1) )T2 ] = (1) + (2) − Q[((1) )2 T2 + ((2) )2 T1 + ((1) + (2) )2 T3 ] Q = (T1 T2 + T1 T3 + T2 T3 )−1 ; plus permuted GX Bij ’s.
(8.17)
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Fig. 22. Reparametrization of the two-point two-loop diagrams.
8.2. Comparison with Feynman diagrams Let us look more closely at the two-point case, and compare our approach with the corresponding Feynman diagram calculation. For N = 2, Eq. (8.12) reads ∞ T T dT ∞ X −m2 (T +TX ) −D dT e (4) da db [T TX + TGB (a ; b )]−D=2 T 0 0 0 0 T T 1 (1) 1 (1) (1) × d1 d2 exp (8.18) GB (1 ; 1 ) + GB (2 ; 2 ) − GB (1 ; 2 ) k 2 : 2 2 0 0 This should correspond to the sum of graphs (a) – (c) of Fig. 20. A straightforward Feynman parameter calculation of graph (a) results in (k = k1 = − k2 ) 28 ∞ 5 5
2 ˆ T −D −m d Tˆ (4) e di ! Tˆ − i [P (a) (i )]−D=2 exp[ − Q(a) (i )k 2 ] (8.19) 0
i=1
i=1
with P (a) = 5 (1 + 2 + 3 + 4 ) + 3 (1 + 2 + 4 ) ; P (a) Q(a) = 1 [5 (2 + 3 + 4 ) + 2 3 + 3 4 ] :
(8.20)
The analogue of the transformation Eq. (4.15) can be directly read from Fig. 22a 1 = 1 − 2 ; 2 = T − 1 + b ; 3 = a − b ; 4 = 2 − a ; 5 = TX : 28
P (a) was misprinted in [84].
(8.21)
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As expected, it leads to the identi cations 1 (a) X P = T T 1 + GB (a ; b ) ; TX (1) Q(a) = GX B (1 ; 2 ) :
(8.22)
The Feynman calculation for diagram (b) yields polynomials P (b) = 5 (1 + 2 + 3 + 4 ) + (1 + 2 )(3 + 4 ); P (b) Q(b) = 5 (2 + 3 )(1 + 4 ) + 1 2 (3 + 4 ) + 3 4 (1 + 2 ) ;
(8.23)
which are di5erent as functions of the variables i , but after the corresponding transformation 1 = 1 − a ; 2 = b + T − 1 ; 3 = 2 − b ; 4 = a − 2 ; 5 = TX ;
(8.24)
identify with the same expressions (8.22). It should be noted that this becomes apparent only after everything has been expressed in terms of GB , due to the absolute sign contained in that function. Our worldline formula Eq. (8.18) thus indeed uni es the three -parameter integrals, arising in the calculation of diagrams (a) – (c), in a single -parameter integral. This correspondence has also been checked for a number of diagrams with more external legs, as well as for diagrams with legs on all three branches. This is remarkable since in eld theory diagrams (a) and (b) have very di5erent properties. Both in the massless and in the massive cases the integrals arising from topology (b) are less elementary than the ones from (a). To understand how this comes about we need only remember (1) a property of GX B stated above, namely that it is translation invariant for (a) but not for (b). Therefore for (a) the parameter integral has a redundancy, and Eq. (8.15) can be used to reduce the number of integrations by one. Quite obviously we have found here a universality property which is not visible in ordinary Feynman parameter calculations, and constitutes a eld theory relic of the fact mentioned in the introduction, namely that string perturbation theory does not su5er from the usual proliferation of terms due to the existence of many di5erent topologies. 8.3. Higher loop orders The whole procedure generalizes without diNculty to the case of m propagator insertions, resulting in an integral representation combining into one expression all diagrams with N legs
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on the loop, and m inserted propagators: m ∞ T T m ∞ TX j ) −m2 (T +
dT −D=2 −(m+1)D=2 j = 1 (4) d TX j e daj dbj T T 0 0 0 0 ×
N
T
i=1 0
di N (m)
D=2
j=1
N 1 exp GB(m) (k ; l )kk · kl ; 2
(8.25)
k;l = 1
where N (m) = Det(A(m) ) GB(m) (1 ; 2 ) = GB (1 ; 2 ) +
m 1 (m) [GB (1 ; ak ) − GB (1 ; bk )]Akl [GB (2 ; al ) − GB (2 ; bl )] 2 k;l = 1
(8.26)
and the symmetric m × m-matrix A(m) is de ned by C −1 (m) X A = T− ; 2 TX kl = TX k !kl ; (8.27) Ckl = GB (ak ; al ) − GB (ak ; bl ) − GB (bk ; al ) + GB (bk ; bl ) : Here TX 1 ; : : : ; TX m denote the proper-time lengths of the inserted propagators. The coincidence terms can be subtracted as in the two-loop case, leading to “subtracted” (m) Green’s functions GX B related to the GB(m) via the same Eq. (8.13). Both choices lead to the same scattering amplitudes. Of course one is free to choose the masses of the inserted propagators to be di5erent from each other, and from the loop mass m. To give di5erent masses to the individual eld theory propagators making up the loop is also possible, albeit only if one xes the ordering i1 ¿ i2 ¿ · · ·
(8.28) 2
of the interaction points around the loop. In this case, instead of the global factor of e−m T one has to insert one factor of 2
e−mj (ij −ij+1 )
(8.29)
for every massive propagator, where mj denotes the mass for the propagator connecting ij and ij+1 . Note that the formula above gives the worldline correlator only between points on the loop. Beyond the two-loop level, the construction of the correlators involving points on the inserted propagators cannot be achieved by symmetry arguments any more. This extension was studied by Roland and Sato [87,88], who obtained explicit formulas similar to Eqs. (8.26), (8.27) for the Green’s function between arbitrary points on the same class of graphs. This knowledge then is suNcient to write down worldline representations for all 3 graphs which have the topology
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of a loop with insertions. According to graph theory [238], the set of such graphs is surprisingly large. For the rst few orders of perturbation theory such a loop, or “Hamiltonian circuit”, can always be found; all trivalent graphs with less than 34 vertices do have this property. 29 This would, of course, do for most practical purposes, but still poses a problem in theory. But there is a more bothersome problem, which so far we have swept under the carpet. In checking the correspondence to Feynman graph calculations, we veri ed that the correct integrands were produced, but left aside the global statistical and symmetry factors for the individual diagrams. As it turns out those do not work out in the case of 3 -theory. Even in the simple example analyzed above the complete -integral contains diagram (a) and (b) in a ratio of 2:1, while in eld theory one would have a ratio of 1:1. Both these problems can be solved at the same time by an appropriate reformulation of the theory at the eld theory level [89]. This can be done in various ways. In 43 -theory, the basic idea is to rewrite the generating functional 1 2 3 Z[] = D(x) exp d x − (x)(− + m )(x) + (x)(x) − 4 (x) (8.30) 2 in the following way: Z[] = DA(x)D(x)!(A − ) ×exp
1 2 2 d x − (x)(− + m )(x) + (x)(x) − 4A(x) (x) :
2
(8.31)
After a Fourier transformation of the functional !-function one nds that the new scalar eld theory obtained does not su5er from the above problems, since the interaction between and the auxiliary eld A is of the Yukawa type. For this theory the whole S-matrix can be exhausted by Hamiltonian graphs, and moreover letting “legs slide around loops” generates the correct statistical factors. A similar reformulation exists for Yang–Mills theory [95]. However the practical value of this procedure has not been established yet, and we will not pursue this matter further here. It is interesting to compare these diNculties in the correct generation of the set of all Feynman diagrams to the situation in string perturbation theory. There the apparent advantage of being able to write down the full amplitude “in one piece” o5ered by the Polyakov path integral approach may, at higher loop orders, become increasingly illusory due to the absence of a convenient global parametrization of the corresponding moduli spaces. The cell decomposition of moduli space provided by second quantized string eld theory [7] may then turn into an advantage. 8.4. Connection to string theory Roland and Sato [87] provided a link back to string theory by analyzing the in nite string tension limit of the Green’s function GBRS(m) of the corresponding Riemann surface, and identifying 29
This statement assumes that we disallow insertions of the trivial one-loop propagator bubble graph. (I thank D. Kreimer for pointing this out to me.)
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Fig. 23. Planar and nonplanar sectors. (m) GX B with the leading order term of GBRS(m) in the 1= -expansion →0 1 (m) GBRS(m) (z1 ; z2 ) → GX B (1 ; 2 ) + nite : (8.32) Note that their derivation automatically leads to the “subtracted” version of the multiloop Green’s functions.
8.5. Example: three-loop vacuum amplitude Finally, let us have a look at the simplest example of a three-loop parameter integral calculation in this formalism [239]. This is the one where the integrand consists just of the bosonic three-loop determinant factor (N (2) )D=2 (see Eq. (8.25)). In dimensional regularization it reads ∞ dT −m2 T 6−3=2D (3) −3=2D vac (D) = (4) T I (D) ; (8.33) e T 0 ∞ 1 2 −D=2 C I (D) = d Tˆ 1 d Tˆ 2 da db dc dd (Tˆ 1 + GBab )(Tˆ 2 + GBcd ) − : (8.34) 4 0 0 Here Tˆ 1; 2 = T1; 2 =T denote the proper-time lengths of the two inserted propagators in units of T , and C ≡ GBac − GBad − GBbc + GBbd . In the following we will show how to compute the 1= j-pole of this amplitude, which is the quantity needed for applications to the calculation of renormalization group functions. In writing Eq. (8.33) we have already rescaled to the unit circle, and separated o5 the global proper-time integral. This integral decouples, and just yields an overall factor of ∞ dT −m2 T 6−3=2D 3 2 T = 6 − D m3D−12 ∼ − (8.35) e T 2 3j 0 (j = D − 4). The nontrivial integrations are 01 da db dc dd ≡ abcd , representing the four propagator end points moving around the loop (Fig. 23). This fourfold integral decomposes into 24 ordered sectors, of which 16 constitute the planar (P) (Fig. 23a) and 8 the nonplanar (NP) sector (Fig. 23b). Due to the symmetry properties of the integrand, all sectors of the same topology give an equal contribution. The integrand has
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a trivial invariance under the operator 9= 9a + 9= 9b + 9= 9c + 9= 9d, which just shifts the location of the zero on the loop. As a rst step in the calculation of I (D), it is useful to add and subtract the same integral with C = 0 and rewrite I (D) = Ising (D) + Ireg (D) ; ∞ d Tˆ 1 d Tˆ 2 Ising (D) = 0
abcd
(8.36) [(Tˆ 1 + GBab )(Tˆ 2 + GBcd )]−D=2 ;
(8.37)
where Ising (D) factorizes into two identical three-parameter integrals, which are elementary: ∞ 2 1 2B(2 − (D=2)2 − (D=2)) 2 dT du[T + u(1 − u)]−D=2 = : (8.38) Ising (D) = D−2 0 0 The point of this split is that the remainder Ireg (D) is nite. To see this, set D = 4, expand the original integrand in C 2 =(GBab GBcd ), and note that for all terms but the rst one the zeroes of GBab (GBcd ) at a ∼ b (c ∼ d) are neutralized by zeroes of C 2 . Since we want only the 1= j-pole (3) we can set D = 4 in the calculation of Ireg (D). The integrations over Tˆ 1 ; Tˆ 2 are then of vac elementary, and we are left with 4 C2 1 Ireg (4) = − 2 ln 1 − − : (8.39) C 4GBab GBcd GBab GBcd abcd For the calculation of this integral, observe the following simple behavior of the function C under the operation Dab ≡ 9= 9a + 9= 9b : 2 Dab C = ± 2FNP ; Dab C = 2(!ac − !ad − !bc + !bd ) ; (8.40) where FNP denotes the characteristic function of the nonplanar sector (i.e. it is zero on the planar and one on the nonplanar sector). From these identities and the symmetry properties one can easily derive the following projection identities, which e5ectively integrate out the variable C: 1 a f(C; GBab ; GBcd ) = 4 da dc(a − c)f(−2c(1 − a); a − a2 ; c − c2 ) ; P 0 0 1 a −2c(1−a) f(C; GBab ; GBcd ) = − 4 da dc dCf(C; a − a2 ; c − c2 ) : (8.41) NP
0
0
0
Here f is an arbitrary function in the variables G; GBab ; GBcd , and 0C dCf denotes the integral of this function in the variable C, with the other variables xed. The integrals on the left hand side are restricted to the sectors indicated. For f the integrand of our formula Eq. (8.39), we have C 4 4 C C2 1 C √ √ dCf = − + ln 1 − + arctanh : GBab GBcd C 4GBab GBcd 2 GBab GBcd GBab GBcd 0 (8.42) Inserted in the second equation of (8.41) this leaves us with three two-parameter integrals, of which the rst one is elementary. Applying the substitution c(1 − a) y= (8.43) a(1 − c)
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Fig. 24. 3-loop 4 vertex diagrams.
to the second integral, and c(1 − a) y2 = a(1 − c)
(8.44)
to the third integral, those are transformed into known standard integrals, tabulated for instance in [240]. The result is f = 12*(3) − 8*(2) : (8.45) NP
The calculation in the planar sector is elementary, and we just give the result, f = 4*(2) − 4 : P
Putting the pieces together, we have, up to terms of order O(j0 ), 3 (3) 3D−12 −(3=2)D vac (D) = m (4) 6− D 2 2B(2 − (D=2); 2 − (D=2)) 2 + 12*(3) − 4*(2) − 4 : × D−2
(8.46)
(8.47)
This calculation method generalizes in an obvious way to the tensor integrals which appear in worldline calculations of three-loop renormalization group functions in other abelian theories such as QED or the Yukawa model. The basic scalar integral considered here appears in the calculation of the 3-loop -function for 4 -theory. This permits an easy check of the above (3) (D) calculation against a Feynman diagram calculation. In diagrammatic terms, the integral vac corresponds to a weighted sum of the two scalar 3-loop vertex diagrams depicted in Fig. 24, calculated at zero external momentum, with massive propagators along the loop, and massless propagator insertions. It is not diNcult to verify that, with an appropriate normalization, the relation (3) vac (D) = 16DP + 8DNP
indeed holds true for the singular parts of the 1= j-expansions.
(8.48)
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9. The QED photon S -matrix We generalize this “multiloop worldline formalism” to the case of photon scattering in quantum electrodynamics [85,92]. 9.1. The single scalar loop We begin with studying scalar electrodynamics at the two-loop level, i.e. a scalar loop with an internal photon correction. A photon insertion in the worldloop may, in Feynman gauge, be represented in terms of the following current-current interaction term inserted into the one-loop path integral, T e2 (4) T x( ˙ a ) · x( ˙ b) − da db (9.1) 4+1 2 4 ([x(a ) − x(b )]2 )4 0 0 (4 = D=2−1). This is essentially still Feynman’s formula Eq. (1.7), except that we have rewritten it in D dimensions, and Euclidean conventions. As in the case of the scalar propagator, we can exponentiate the o5ending “non-Gaussian” denominator, ∞ 2 (4) X (4TX )−D=2 exp − (x(a ) − x(b )) = d T (9.2) 4TX 44+1 ([x(a ) − x(b )]2 )4 0 and absorb it into the worldline Green’s function. We obtain then, of course, the same 2-loop worldline Green’s function Eq. (8.10) and determinant factor Eq. (8.11) as before. The numerator x˙a · x˙b remains, and will participate in the Wick contractions. This treatment of the photon propagator may appear somewhat unnatural, but will be seen to work quite well in practice. Moreover, only this procedure will enable us to use the same universal Green’s functions both for scalar eld theory and gauge theory calculations. As in the scalar eld theory case, the generalization from one- to two-loop calculations of photon amplitudes in scalar QED requires no changes of the formalism itself, but only of the Green’s functions used, and of the global determinant factor. The generalization to an arbitrary xed number of photon insertions is obvious. To obtain a parameter integral representation for the sum of all diagrams with one scalar loop and xed numbers of photons, N external and m internal, we have to Wick contract N photon vertex T T operators, together with m factors of 0 da 0 db x˙a · x˙b , using the (m+1)-loop Green’s function G (m) : (m+1) scal [k1 ; 1 ; : : : ; kN ; N ] 2 m ∞ e dT −m2 T −D=2 N T (4)−(m+1)D=2 = (−ie) − e 2 T 0 - m . T m T
∞
D=2 A A × d TX j daj dbj N (m) x˙aj · x˙bj Vscal; : 1 · · · Vscal; N j=1 0
0
0
(9.3)
j=1
This is our (m + 1)-loop generalization of the one-loop photon scattering formula Eq. (4.22) (to avoid a further complication of nomenclature, it should simply be understood in the following
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that this is only the “quenched” part of the amplitude). As in the one-loop case, one could immediately translate this into a master formula of the type Eq. (1.18). It is important to note that precisely the same integral representation could be obtained starting from the one-loop formula Eq. (4.22) with (N +2m) external photons, and then sewing together m pairs of them, using Feynman gauge. Of course this would be much more laborious. This also explains why the multiloop Green’s functions can be rewritten in terms of the one-loop Green’s function GB . It has the nontrivial consequence that one is still allowed to use the one-loop replacement rule; after writing out the result of the Wick-contractions in terms of the one-loop Green’s function GBij , one can generate the corresponding spinor loop integrand by the usual partial integration routine, and use of Eq. (2.15). While this multiloop construction is done most simply using Feynman gauge for the propagator insertions, other gauges can be implemented as well (the gauge freedom was also discussed in [91]). In an arbitrary covariant gauge, the photon insertion term Eq. (9.1) would read T T e2 1 1+ D x˙a · x˙b − da db −1 D=2 2 4 2 2 [(xa − xb )2 ](D=2)−1 0 0 D x˙a · (xa − xb )(xa − xb ) · x˙b + (1 − ) : (9.4) 2 [(xa − xb )2 ]D=2 Here = 1 corresponds to Feynman gauge, = 0 to Landau gauge. The integrand may also be written as D x˙a · x˙b 9 9 1− D −1 − −2 [(xa − xb )2 ]2−D=2 : (9.5) D=2−1 2 2 4 2 9a 9b [(xa − xb ) ] This shows that, on the worldline, gauge transformations correspond to the addition of total derivative terms. This form of the photon insertion is also the more practical one for actual calculations. The power of (xa − xb )2 appearing in the second term is then again to be exponentiated. Before completing this section, let us mention that sometimes it can be useful to exponentiate also the numerator of the inserted Feynman propagator, rewriting 1 9 9 9 −(xa −xb )2 x˙a · x˙b = lim lim lim e : (9.6) 2 a →a b →b →0 9 9a 9b This little point-splitting trick turns out to be surprisingly useful for the computerization of the algorithm [241]. For example, it allows one to generate the integrand for the three-loop scalar QED vacuum amplitude by di5erentiations performed on the ve-loop determinant factor N (4) , instead of Wick contractions at the three-loop level. 9.2. The single electron loop As in the one-loop case, the transition to spinor electrodynamics is most simply accomplished by supersymmetrization. According to the supersymmetrization rules, the photon insertion Eq. (9.1) generalizes to the spinor loop as follows: T e2 (4) T DXa · DXb da d/a db d/b : (9.7) 2 44+1 0 ((Xa − Xb )2 )4 0
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The simplest way to verify the correctness of this expression is to write the one-loop two-photon amplitude in the super-formalism, and then sewing together the external legs to create an internal photon, using Feynman gauge. Just as a demonstration of the usefulness of the super eld formalism, let us rewrite the double integral in components: T T (xa − xb )( b b x˙a − a a x˙b ) x˙a x˙b da db − − 44 ((xa − xb )2 )4 ((xa − xb )2 )4+1 0 0 (xa − xb )(xa − xb ) a b aD bD ( a b )2 + 84 − 164(4 + 1) : (9.8) ((xa − xb )2 )4+1 ((xa − xb )2 )4+2 The denominator of Eq. (9.7) being bosonic, we can again use the proper-time representation Eq. (9.2) to get it into the exponent, and then absorb this exponent into the worldline superpropagator. The algebra is completely identical to the scalar case, and leads to modi ed (m) superpropagators Gˆ which are given by the same formulas as in Eqs. (8.10) and (8.26), with all the one-loop Green’s functions appearing on the right-hand sides replaced by the corresponding one-loop superpropagators Eq. (4.31). The same applies to the determinant factor (N (m) )D=2 . The generalization of the (m + 1)-loop N -photon scattering formula Eq. (9.3) to the spinor loop case is equally trivial, and there is no point in writing it down here. Again what we have at hand is a parameter integral combining into one formula all Feynman diagrams with one electron loop, and xed numbers of external and internal photons. For instance, for N = m = 2 this just corresponds to the diagrams of Fig. 8 which we discussed in the introduction. Finally, note that formulas (9.4), (9.5) for an arbitrary covariant gauge also carry over to the spinor loop case mutatis mutandis. 9.3. The general case The general case of a multiple product of scalar or electron loop path integrals coupled by photon insertions requires only two new considerations. Firstly, every scalar=electron path integral has its own zero-mode integral, which must be separated o5, and yields momentum conservation for the photon momenta entering that particular loop. Total momentum conservation is obtained only after all zero mode integrals are performed. Secondly, since we now have to Wick-contract vertex operators attached to the di5erent loops, the multiloop worldline Green’s function becomes a matrix in the space of loops. For instance, in the case of just two scalar=electron loops one has a two × two matrix of Green’s functions GB . The matrix element GB11 has to be used for the Wick contraction of two photon vertex operators both on loop 1, GB12 for the contraction of one vertex operator on loop 1 and one on loop 2, etc. The explicit expressions for these Green’s functions can be found by the same procedure which we described in the previous chapter. This leads to formulas similar to Eqs. (8.26), (8.27), which we will not write down here (the simplest case of two loops connected by a single propagator was also considered in [91]). Note that in the QED case we encounter neither of the two problems discussed in the last chapter, which motivated the introduction of an auxiliary eld formalism. First, Feynman’s
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formula Eq. (1.7) and its supersymmetrization allow one to neatly exhaust the complete photon S-matrix in terms of scalar=electron path integrals connected by photon insertions. The question of nonHamiltonian graphs therefore does not arise. Moreover, from the same formula it follows that the contributions of individual Feynman diagrams are always generated with the appropriate statistical factors. Note that we do not consider here amplitudes involving external electrons; the corresponding formalism has not yet been suNciently developed. For a treatment of external scalars in scalar QED along the present lines see [91]. 9.4. Example: the two-loop QED -functions As an illustration, we will use this calculus for a re-derivation of the two-loop QED -function, both for scalar and for spinor electrodynamics [85,93]. As usual, matters considerably simplify if one is interested only in the -function contribution, as opposed to a calculation of the complete two-loop vacuum polarization amplitude. Our strategy here will be to use the e5ective action formalism with a constant background eld F , and read o5 the -function from the coeNcient of the induced F F -term. Standard dimensional regularization will be used for the treatment of the UV divergences. Just for setting the stage, let us rst redo the one-loop calculation. As always in the constant eld case we choose Fock–Schwinger gauge centered at x0 , so that A = 12 y? F? . Using this A- eld in the spinor-loop path-integral Eq. (1.9), expanding the interaction exponential to second order, and performing the Wick contractions, one obtains T 1 ∞ dT −m2 T 1 2 1 ˙ (1) spin [F] = − DxD exp − d e x˙ + 2 0 T 4 2 0 2 T T e 1 × − d1 d2 x˙1 F x1 x˙2 F x2 + 1 F 1 2 F 2 2 4 0 0 T T 2 e2 ∞ dT −D=2 −m2 T 2 ˙ = e d1 d2 G B12 − GF12 d x0 F F : (9.9) (4T ) 2 0 T 0 0 The parameter integral gives T T 2 2 d1 d2 (G˙ B (1 ; 2 ) − GF2 (1 ; 2 )) = − T 2 : 3 0 0 The singular part of the one-loop e5ective action becomes 2 (1) −2 1 2 spin [F] ∼ (4) e d x0 F F 3 j
(9.10)
(9.11)
(j = D − 4). From this one can read o5 the one-loop photon wave-function renormalization factor (Z3 − 1)(1) =
2 ; 3j
(9.12)
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Fig. 25. Diagrams contributing to the two-loop vacuum polarization.
leading to the usual value for the one-loop spinor QED -function, 2 2 (1) () = spin (9.13) 3 ( = e2 =4). The corresponding result for scalar QED is simply obtained by omitting, in Eq. (9.9), the term involving GF , and the global factor of −2. This yields 1 2 (1) () = scal : (9.14) 6 Now let us tackle the two-loop calculation. In the corresponding Feynman diagram calculation (see, e.g., [117]), one would have to separately calculate the three diagrams of Fig. 25, and then extract their 1= j-poles. Cancellation of the 1= j2 -poles would be found in the sum of the results, indicating a cancellation of subdivergences due to gauge invariance. Let us begin with the purely bosonic contributions, which correspond to the scalar QED calculation. Those are obtained by inserting the worldline current–current interaction term Eq. (9.1) into the bosonic one-loop path-integral. After exponentiation of the denominator and absorption into the worldline Green’s function, this results in (2) (2) bos [F] = −2scal [F] = −2(4)−D
T dT −m2 T −D=2 ∞ X T T dT da db [TX + GBab ]−D=2 e T 0 0 0 0 2 2 T T 1 −e × d1 d2 d x0 y˙ 1 F y1 y˙ 2 F y2 y˙ 4a y˙ b4 : (9.15) 2 4 0 0 Note the appearance of the two-loop determinant factor [TX + GB (a ; b )]−D=2 . The Wick contraction of ∞
y˙ 1 y1 y˙ 2 y2 y˙ 4a y˙ b4
(9.16)
has now to be done, using the two-loop Green’s function Eq. (8.10). Due to the symmetries of the problem there are only two nonequivalent contraction possibilities, namely (1)
(1)
(1)
(1)
(1)
y˙ 1 y2 y1 y˙ 2 y˙ 4a y˙ b4 = − Dg g 91 GB12 92 GB12 9a 9b GBab ; (1)
y˙ 1 y˙ 2 y1 y˙ 4a y2 y˙ b4 = − g g 91 92 GB12 9a GB1a 9b GB2b :
(9.17)
Those occur with multiplicities 2 and 8, respectively. Care must be taken with Wick contractions involving y˙ a ; y˙ b , as the derivatives should not act on the a ; b explicitly appearing in that Green’s function. The result is written out in terms of the bosonic one-loop Green’s function and
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Fig. 26. De nition of the six integration parameters.
its derivatives. As in the one-loop calculation, one next eliminates all factors of GS B appearing by partial integrations with respect to 1 ; 2 ; a ; b . As the next step, all fermionic contributions are included by applying the one-loop replacement rule (2.15). For example, one replaces G˙ B12 G˙ B21 G˙ Bab G˙ Bba → (G˙ B12 G˙ B21 − GF12 GF21 )(G˙ Bab G˙ Bba − GFab GFba ) ; (9.18) etc. At this stage, we have the desired contribution to the two-loop e5ective action in the form of a sixfold integral (see Fig. 26), ∞ 4 ∞ dT (2) −D e −m2 T −D=2 Lspin [F] = −2(4) T d TX e 16 0 T 0 T da db d1 d2 P(T; TX ; a ; b ; 1 ; 2 )F F : (9.19) × 0
The integrand function P is a polynomial in the various GBij ; G˙ Bij ; GFij , multiplied by powers of " ≡ [TX + GB (a ; b )]−1 . Let us just write down its purely bosonic part Pbos , which is 30 2 2 Pbos = "D=2 {D(D − 1)"G˙ Bab G˙ B12 +8D"G˙ Bab G˙ B12 G˙ B1a G˙ B2b +8"G˙ B1a G˙ Bab G˙ B12 [G˙ B2a − G˙ B2b ] −4"G˙ B1a G˙ B2b [G˙ B1a − G˙ B1b ][G˙ B2a − G˙ B2b ] 2 + (D + 2)(D − 1)"2 G˙ Bab G˙ B12 [G˙ B1a − G˙ B1b ][GB2a − GB2b ]} : (9.20) In writing this polynomial, we have used the symmetry with regard to interchange of 1 and some terms, and omitted some terms which are total derivatives with respect to 2 to combine d1 or d2 (those terms are easy to identify at an early stage of the calculation). It is convenient to begin with the integrations over 1 ; 2 . Those are polynomial, and easily performed using a set of relations of the type 1 1 du2 G˙ B12 G˙ B23 = 2GB13 − ; 3 0
0
30
1
1 1 : du2 GB12 GB23 = − GB2 13 + 6 30 .. .. . .
In [85] this integrand was given incorrectly.
(9.21)
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All those relations may be derived from the following master identities proven in Appendix B: 1 2n du2 : : : dun G˙ B12 G˙ B23 : : : G˙ Bn(n+1) = − Bn (|u1 − un+1 |) signn (u1 − un+1 ) ; n! 0 1 2n−1 du2 : : : dun GF12 GF23 : : : GFn(n+1) = En−1 (|u1 − un+1 |) signn (u1 − un+1 ) : (9.22) (n − 1)! 0 In writing these identities, we have scaled down to the unit circle again. Bn denotes the nth Bernoulli-polynomial, and En the nth Euler-polynomial. Due to the fact that those polynomials can be rewritten as Bn (x) = Pn (x2 − x)
(n even) ;
Bn (x) = Pn (x2 − x)(x − 12 )
(n odd)
(9.23)
with another set of polynomials Pn (x) (the same property holds true for En (x)), the right hand sides can always be re-expressed in terms of GB ; G˙ B and GF , so that explicit ui ’s will never appear in those relations. Those integrals needed for the present calculation are listed in Appendix F. Next, we perform the TX -integration, which is trivial: ∞ 1−(D=2)−k GBab d TX [TX + GBab ]−(D=2)−k = (k = 1; 2) : (9.24) (D=2) + k − 1 0 Collecting terms, and using (F:25), we get ∞ T T X dT d1 d2 P(T; TX ; a ; b ; 1 ; 2 ) 0
0
=
0
16 1−(D=2) 2−(D=2) {(D − 4)(D − 1)GBab T + (D − 2)(D − 7)GBab }: 3D
(9.25)
The corresponding expression for scalar QED is obtained by using only the bosonic part Pbos of the function P: ∞ T T d TX d1 d2 Pbos (T; TX ; a ; b ; 1 ; 2 ) 0
0
0
32 2 −(D=2) 2 1−(D=2) − 4 GBab T = (D − 1)GBab T + (D − 1) 3 3D 16 2−(D=2) : + (D − 2)(D − 7)GBab 3D
Setting a = 0, the integration over b produces a couple of Euler Beta-functions, T T D D k−(D=2) da db GBab =B k + 1 − ;k + 1 − T 2+k−(D=2) : 2 2 0 0
(9.26)
(9.27)
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As in the one-loop case, the remaining electron proper-time integral just gives a -function: ∞ dT −m2 T 4−D T = (4 − D)m2(D−4) : (9.28) e T 0 Combining terms and performing the j-expansions for the e5ective Lagrangians, we obtain 1 4 e (4)−4 F F + O(j0 ) ; 2j 3 (2) Lspin [F] ∼ − e4 (4)−4 F F + O(j0 ) : j (2)
Lscal [F] ∼
(9.29)
So far this is a calculation of the bare regularized e5ective action. What about renormalization? The counterdiagrams due to electron wave function and vertex renormalization need not be taken into account, since they cancel by the QED Ward identity (Z1 = Z2 ). However, we have used the electron mass as an infrared regulator for the electron proper-time integral Eq. (9.28); mass renormalization must therefore be dealt with. Since our calculation corresponds to a eld theory calculation in dimensional regularization, we need to know the corresponding one-loop mass renormalization counterterms, both for scalar and spinor QED. This is a simple textbook calculation, of which we give the result only: !m2scal 6 2 = e (4)−2 ; j m2scal
!mspin 6 2 = e (4)−2 : mspin j
(9.30)
We insert those counterterms into the one-loop path integrals, and obtain the following additional contributions to the two-loop e5ective Lagrangians, 9 (1) 1 (2) Zscal [F] = !m2scal 2 scal [F] ∼ e4 (4)−4 d x0 F F + O(j0 ) ; 9m 2j 9 (1) 4 4 (2) −4 d x0 F F + O(j0 ) : (9.31) Zspin [F] = !mspin spin [F] ∼ e (4) 9m j The extraction of the -function coeNcients proceeds in the usual way. From the total e5ective Lagrangians 1 4 e (4)−4 F F ; j 1 (2) (2) Lspin [F] + ZLspin [F] ∼ e4 (4)−4 F F ; j (2)
(2)
Lscal [F] + ZLscal [F] ∼
(9.32)
one obtains the two-loop photon wave-function renormalization factors, and from those the standard results for the two-loop -function coeNcients [242,243], (2) (2) scal () = spin () =
3 : 22
(9.33)
Observe that in the spinor-loop case, the integrand after performance of the rst three integrations, Eq. (9.25), has only one term which would be divergent for D = 4 when integrated over
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b . Moreover, the coeNcient of this term vanishes for D = 4. This suggests that this calculation can be further simpli ed by using some four-dimensional regularization scheme. And indeed, if we do the spinor-loop calculation in four dimension, then instead of Eq. (9.25) we nd simply ∞ T T d TX d1 d2 P(T; TX ; a ; b ; 1 ; 2 ) = − 8 : (9.34) 0
0
0
This time there is no dependence on a ; b left, so that one immediately gets ∞ dT −m2 T (2) −4 4 Lspin [F] = (4) e F F : (9.35) e T 0 It is only the nal electron proper-time integral that now needs to be regularized. This can be done by introducing a proper-time cuto5 T0 at the lower integration limit, which replaces Eq. (9.28) by ∞ dT −m2 T ∼ −ln(m2 T0 ) (9.36) e T T0 (Pauli–Villars regularization could be used as well, although proper-time regularization appears more natural in the worldline formalism). With this regulator, the two-loop e5ective Lagrangian becomes (2)
Lspin [F] ∼ −ln(m2 T0 )(4)−4 e4 F F + nite :
(9.37)
In spite of the manifest suppression of subdivergences, there is again a contribution from mass renormalization, which can be determined by comparison with the corresponding Feynman calculation. On-shell renormalization of spinor QED using a proper-time cuto5 has been studied in [158,172]. It leads to a one-loop mass renormalization counterterm !m (9.38) = 3 ln(m2 T0 )e2 (4)−2 + nite : m Insertion of this counterterm into the one-loop path integral gives 9 (1) (2) 2 −4 4 Zspin [F] = !m [F] ∼ 2 ln(m T0 )(4) e d x0 F F + nite ; (9.39) 9m so that mass renormalization now just amounts to a sign change for the e5ective Lagrangian: (2)
(2)
Lspin [F] + ZLspin [F] ∼ ln(m2 T0 )(4)−4 e4 F F :
(9.40)
(2) The extraction of the (still scheme-independent) -function coeNcient spin () is again standard [172], and leads back to Eq. (9.33). Let us summarize the properties of this calculation:
1. Neither momentum integrals nor Dirac traces had to be calculated. 2. The three diagrams of Fig. 25 were combined into one calculation (in fact, in this formalism it is somewhat easier to compute the sum than any single one of them). 3. In the spinor-loop case, we have managed to obtain the correct two-loop coeNcient without performing any nontrivial integrals. We will see later on that the cancellations which led to the last property are not accidental, but a direct consequence of renormalizability.
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9.5. Quantum electrodynamics in a constant external <eld As in the one-loop case, it takes only minor modi cations to extend this formalism to the case when an additional constant external eld F is present [92]. Again we begin with the simplest case, which is scalar QED at the two-loop level. Combining the contributions to the quadratic part of the worldline Lagrangian due to the external eld Eq. (5.2) and to the propagator insertion Eq. (9.2), we obtain the new total bosonic kinetic operator d2 d Bab : (9.41) − 2ieF − 2 d d TX The inverse of this operator can still be constructed as a geometric series, 2 −1 2 −1 d Bab d d d − 2ieF − = − 2ieF d2 d d2 d TX 2 −1 −1 d d d Bab d 2 + − 2ieF − 2ieF + · · · : (9.42) d2 d d TX d2 Summing this series one obtains the following Green’s function: (1)
GB (1 ; 2 ) = GB (1 ; 2 ) +
1 [GB (1 ; a ) − GB (1 ; b )][GB (a ; 2 ) − GB (b ; 2 )] ; 2 TX − 12 Cab
(9.43)
where we have de ned Cab ≡ GB (a ; a ) − GB (a ; b ) − GB (b ; a ) + GB (b ; b )
=T
cos(Z) − cos(ZG˙ Bab ) : (Z) sin(Z)
(9.44)
This is almost but not quite identical with what one would obtain from the ordinary bosonic two-loop Green’s function, Eq. (8.10), by simply replacing all GBij ’s appearing there by the corresponding GBij ’s. The more complicated structure of the denominator is due to the fact that T , the GBij ’s are not any more symmetric under the interchange i ↔ j, rather we have GBij = GBji and moreover have non-vanishing coincidence limits. The denominator is now in general a nontrivial Lorentz matrix, and must be interpreted as a matrix inverse (of course, all matrices appearing here commute with each other). The free Gaussian path integral is again easily calculated using the ln det = tr ln-identity, yielding −1 2 2 d d d d B DetP − 2 + 2ieF + ab = DetP − 2 Det P 5 − 2ieF d d d d TX −1 Bab d 2 sin(Z) 1 d D 5 − − 2ieF = (4T ) det det C ×Det P 5 − ab : (9.45) d Z 2TX TX d2
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Thus we have now a product of two Lorentz matrix determinants. The rst one is identical with the by now familiar Euler–Heisenberg integrand, Eq. (5.16), while the second one generalizes the two-loop determinant factor Eq. (8.11) to the external eld case. As in the case without a background eld, the whole procedure goes through essentially unchanged for the fermion loop, if the super eld formalism is used. As a consequence, one nds the same close relationship as before between the parameter integrals for the same amplitude calculated for the scalar and for the fermion loop: They di5er only by a replacement of all GB ’s by Gˆ ’s, and by the additional /-integrations. Of course, one must also replace the scalar QED one-loop Euler–Heisenberg factor Eq. (5.16) by its spinor QED equivalent Eq. (5.17), and as always the global factor of −2 must be taken into account. The generalization to an arbitrary xed number of photon insertions is straightforward, and leads to formulas for the generalized (super-) Green’s functions and (super-) determinants idenˆ tical with the ones given above for the vacuum case, up to a replacement of all GB ’s (G’s) by GB ’s (Gˆ ’s). The only point to be mentioned here is that care must now be taken in writing the indices of the GBij ’s appearing. For instance, the (un-subtracted) bosonic three-loop Green’s function, Eq. (8.26) with m = 2, must be written as 2 1 (2) GB (1 ; 2 ) = GB (1 ; 2 ) + [GB (1 ; ak ) − GB (1 ; bk )]A(2) kl [GB (al ; 2 ) − GB (bl ; 2 )] : 2 k;l = 1
(9.46)
The matrix A appearing here is the inverse of the matrix T1 − 12 (GBa1 a1 − GBa1 b1 − GBb1 a1 + GBb1 b1 ) − 12 (GBa1 a2 − GBa1 b2 − GBb1 a2 + GBb1 b2 ) − 12 (GBa2 a1 − GBa2 b1 − GBb2 a1 + GBb2 b1 )
T2 − 12 (GBa2 a2 − GBa2 b2 − GBb2 a2 + GBb2 b2 ) (9.47) and T1 ; T2 denote the proper-time lengths of the two inserted propagators. 9.6. Example: the two-loop Euler–Heisenberg lagrangians As an application of the constant eld formalism at the two-loop level, in this section we will calculate the rst radiative corrections to the Euler–Heisenberg–Schwinger formulas Eqs. (5.23), (5.24) [92,108]. 9.6.1. Scalar QED According to the above, for the scalar QED case we can write this e5ective Lagrangian in the form 2 ∞ T e dT −m2 T −D=2 ∞ X T (2) −D − Lscal [F] = (4) T dT da db e 2 T 0 0 0 0 1 −1=2 sin(Z) −1=2 X T − Cab y˙ a · y˙ b : ×det det (9.48) Z 2 In this fourfold parameter integral, T and TX represent the scalar and photon proper-times, and a; b the endpoints of the photon insertion moving around the scalar loop.
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A single Wick contraction is to be performed on the “left-over” numerator of the photon insertion, using the modi ed worldline Green’s function Eq. (9.43). This yields ˙ Baa − G˙ Bab )(G˙ Bab − G˙ Bbb ) 1 G ( y˙ a · y˙ b = tr GS Bab + : (9.49) 2 TX − 12 Cab After replacing the one-loop Green’s functions G˙ Bij ’s as well as Cab by the explicit expressions given in Eqs. (5.9) and Eq. (9.44), we have already a parameter integral representation for the bare dimensionally regularized e5ective Lagrangian. Alternatively one may remove GS B by a partial integration with respect to a or b . Using the formula d det(M ) = det(M )tr(dMM −1 ) T and G˙ Bab = − G˙ Bba , one obtains the equivalent parameter integral 2 ∞ T e dT −m2 T −D=2 ∞ X T (2) −D − Lscal [F] = (4) T dT da db e 2 T 0 0 0 0 1 −1=2 sin(Z) −1=2 X T − Cab ×det det Z 2 ˙ Baa − G˙ Bab )(G˙ Bab − G˙ Bbb ) ˙ Bab 1 ( G G tr G˙ Bab tr + tr : × 2 TX − 12 Cab TX − 12 Cab
(9.50)
(9.51)
To facilitate the further evaluation and renormalization of this Lagrangian, we specialize the constant eld F to a pure magnetic eld. It will be instructive to do this calculation in two di5erent regularizations, proper-time and dimensional regularization. The renormalization will be performed on-shell in both cases. Let us begin with the proper-time regularized version. This regularization keeps the integrations fairly simple, and was used in all previous calculations of two-loop Euler–Heisenberg Lagrangians [244 –247,172]. It means that in the following we set D = 4, and instead introduce proper-time UV cuto5s for the various proper-time integrals later on. As in the photon-splitting calculation of Section 5.6, we choose the eld in the z-direction. For this case the generalized worldline Green’s functions and determinants were given in (5.30), (5.31). The combination Cab becomes z Cab = − 2GBab g − 2GBab g⊥ ;
(9.52)
where z ≡ GBab
T (cosh(z) − cosh(z G˙ ab )) 1 2 2 = GBab − G z + O(z 4 ) : 2 z sinh(z) 3T Bab
We will also use the derivative of this expression, sinh(z G˙ Bab ) z G˙ Bab = : sinh(z)
(9.53)
(9.54)
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Introducing the further abbreviations " ≡ (TX + GBab )−1 ; z )−1 ; "z ≡ (TX + GBab
we can then rewrite the various Lorentz traces and determinants appearing in Eqs. (9.48), (9.51) as 1 z −1=2 sin(Z) det = TX − Cab ""z ; Z 2 sinh(z) 4 5 z cosh(z G˙ Bab ) tr GS Bab = 8!ab − 4 − 4 ; sinh(z) ˙ Bab ) ˙ Bab ) z G˙ Bab 1 ˙ sinh(z G sinh(z G = 2 G˙ Bab + G˙ Bab " + tr GBab tr " ; 2 sinh(z) sinh(z) TX − 12 Cab 2 2 ˙ ˙ 1 (G˙ aa − G˙ ab )(G˙ ab − G˙ bb ) 2 z sinh (z G Bab ) + [cosh(z G Bab ) − cosh(z)] = − " − G˙ Bab " : tr 2 1 X 2 T − 2 Cab sinh (z) (9.55)
As usual we rescale to the unit circle, a; b = Tua; b , and use translation invariance in to set b = 0, so that GB (a ; b ) = TGB (ua ; ub ) = T (ua − ua2 ) ; G˙ B (a ; b ) = G˙ B (ua ; ub ) = 1 − 2ua : After performance of the TX -integration, which is nite and elementary, Eq. (9.51) turns into 1 e2 ∞ dT −m2 T z (2) Lscal [B] = − (4)−4 e dua A(z; ua ) (9.56) 2 0 T3 sinh(z) 0 with
A=
z ) ln(GBab =GBab A2 A3 A1 + z + z z z ) 2 (GBab − GBab ) (GBab )(GBab − GBab ) (GBab )(GBab − GBab
;
z z coth(z) − GBab ] ; A1 = 4[GBab z
z A2 = 1 + 2G˙ Bab G˙ Bab − 4GBab z coth(z) ; 2
z
A3 = − G˙ Bab − 2G˙ Bab G˙ Bab :
(9.57)
All Green’s functions now refer to the unit circle, T = 1. Here and in the following we often 2 use the identity G˙ Bab = 1 − (4=T )GBab to eliminate G˙ Bab in favor of GBab .
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Renormalization must now be addressed, and will be performed in close analogy to the discussion in [172]. The integral in Eq. (9.56) su5ers from two kinds of divergences: 1. An overall divergence of the loop proper-time integral 0∞ dT at the lower integration limit (which is already familiar from the -function calculation of Section 9.4). 1 2. Divergences of the 0 dua parameter integral at points 0; 1 where the endpoints of the photon propagator become coincident, ua = ub . The rst one will be removed by one- and two-loop photon wave function renormalization, the second one by the one-loop renormalization of the scalar mass. As was already mentioned, vertex renormalization and scalar self energy renormalization need not be considered in this type of calculation, since they must cancel on account of the QED Ward identity. By power counting, an overall divergence can exist only for the terms in the e5ective Lagrangian which are of order at most quadratic in the external eld B. Expanding the integrand of Eq. (9.56), K(z; ua ) ≡ z=sinh(z)A(z; ua ), in the variable z, we nd 3 12 1 1 1 − + − + + 2 z 2 + O(z 4 ) : (9.58) K(z; ua ) = 2 2 GBab 2 GBab GBab GBab The complicated singularity appearing here at the point ua = ub indicates that this form of the parameter integral is not yet optimized for the purpose of renormalization. In particular, it shows a spurious singularity in the coeNcient of the induced Maxwell term ∼ z 2 . This comes not unexpected as the cancellation of subdivergences implied by the Ward identity has, in a general gauge, no reason to be manifest at the parameter integral level. We could improve on this either by switching to Landau gauge, or by performing a suitable partial integration on the integrand. The latter procedure is less systematic, but easy enough to implement for the simple case at hand: Inspection of the two versions which we have of this parameter integral, the original one Eq. (9.48) and the partially integrated one Eq. (9.51), shows that we can optimize the integrand by choosing a certain linear combination of both versions, namely (2)
Lscal [B] = 34 × Eq: (9:48) +
1 4
× Eq: (9:51) :
(9.59)
(taking the photon insertion in Landau gauge would yield a similar simpli cation, though the resulting parameter integrals are not identical). 31 After integration over TX , this leads to another version of Eq. (9.56), 1 2 ∞ dT z (2) −4 e −m2 T Lscal [B] = − (4) e dua A (z; ua ) (9.60) 2 0 T3 sinh(z) 0 with a di5erent integrand z ) z ln(GBab =GBab ln(GBab =GBab ) A = A0 + A 1 z ) z )2 (GBab − GBab (GBab − GBab A3 A2 + + z z ) z ) (GBab )(GBab − GBab (GBab )(GBab − GBab 31
;
We remark that they would be identical for the special case of a self dual eld.
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z A0 = 3 2z 2 GBab − 2
199
z −1 ; tan(z) z2
A1 = A1 − 32 [G˙ Bab − G˙ Bab ] ; z z2 A2 = A2 − 32 [G˙ Bab G˙ Bab + G˙ Bab ] ; 2 z A3 = A3 + 32 [G˙ Bab + G˙ Bab G˙ Bab ] :
(9.61)
We have not yet taken into account here the term involving !ab , stemming from GS Bab , which was contained in the integrand of Eq. (9.60). This term corresponds, in diagrammatic terms, to a tadpole insertion, and could therefore be safely deleted. However, it will be quite instructive to keep it and check explicitly that it is taken care of by the renormalization procedure. It leads 2 to an integral 0∞ d TX = TX which we regulate by introducing an ultraviolet cuto5 for the photon proper-time, ∞ X 1 dT = : (9.62) 2 TX 0 TX 0 TX (2) It gives then a further contribution E(TX 0 ) to Lscal [B], ∞ 1 dT −m2 T z e E(TX 0 ) = − 3(4)−4 e2 : (9.63) X sinh(z) T0 0 T2 Expanding the new integrand, K (z; ua ) ≡ z=sinh(z)A (z; ua ), in z, we nd a much simpler result than before, 1 K (z; ua ) = − 6 + 3z 2 + O(z 4 ) : (9.64) GBab In particular, the absence of a subdivergence for the Maxwell term is now manifest. We delete the irrelevant constant term, and add and subtract the Maxwell term. If we de ne 1 K02 (z; ua ) = − 6 + 3z 2 ; (9.65) GBab the Lagrangian becomes ∞ dT −m2 T 2 (2) X Lscal [B] = E(T 0 ) − e 3z 2(4)3 0 T 3 ∞ dT −m2 T 1 − e dua [K (z; ua ) − K02 (z; ua )] : (9.66) 2(4)3 0 T 3 0 The second term, which we denote by F, is divergent when integrated over the scalar proper-time T . We regulate it by introducing another proper-time cuto5 T0 for the scalar proper-time integral: ∞ dT −m2 T 2 e 3z (9.67) F(T0 ):= − 2(4)3 2T0 T 3
(compare [172]). The third term is convergent at T = 0, but still has a divergence at ua = ub , as it contains negative powers of GBab .
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Expanding the integrand in a Laurent series in GBab , one nds f(z) 0 K (z; ua ) − K02 (z; ua ) = + O(GBab ) GBab z z 2 cosh(z) f(z) = 3 2 − − : (9.68) sinh(z) sinh(z)2 Again the singular part of this expansion is added and subtracted, yielding ∞ dT −m2 T 1−T0 =T f(z) (2) X Lscal [B] = E(T 0 ) + F(T0 ) − e dua 3 3 2(4) 2T0 T GBab T0 =T ∞ dT −m2 T 1 f(z) − e dua K (z; ua ) − K02 (z; ua ) − : (9.69) 2(4)3 0 T 3 GBab 0 The last integral is now completely nite. The third term, which we call G(T0 ), is nite at T = 0, as f(z) = O(z 4 ) by construction. Here we have introduced T0 for the purpose of regulating the divergence at ua = ub . Obviously this term cannot be made nite by photon wave function renormalization, so we must try to use the scalar mass renormalization for the purpose. This will be seen to work out in a quite nontrivial way. The ua -integral for this term is readily computed and yields, in the limit T0 → 0, a contribution 1−T0 =T 1 T0 dua = − 2 ln = − 2 ln(m2 T0 ) + 2 ln(m2 T ) : (9.70) GBab T T0 =T We have rewritten this term for reasons which will become apparent in a moment. The upshot is that we can relate the function f(z) to the scalar one-loop Euler–Heisenberg Lagrangian, Eq. (5.23). If we write this Lagrangian for the pure magnetic eld case, and subtract the two divergent terms lowest order in z, we obtain ∞ 1 dT −m2 T z z2 (1) X Lscal [B] = e (9.71) + −1 : (4)2 0 T 3 sinh(z) 6 On the other hand, we can write z z 2 cosh(z) 1 z z2 3 d − = 3T : (9.72) f(z) = 3 2 − + −1 sinh(z) sinh(z)2 dT T 2 sinh(z) 6 By a partial integration over T , we can therefore re-express ∞ ∞ 1 dT −m2 T dT −m2 T z m2 z2 e f(z) = 3 e + −1 (4)2 0 T 3 (4)2 0 T 2 sinh(z) 6 = −3m2
9 X (1) L [B] 9m2 scal
(9.73)
(there are no boundary contributions since f(z) = O(z 4 )). Next note that, at the two-loop level, the e5ect of mass renormalization consists in the following shift produced by the one-loop mass displacement !m2 , 9 X (1) (2) [B] = !m2 2 L [B] : (9.74) !Lscal 9m scal
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To proceed, we need thus again the value of the one-loop mass displacement in scalar QED, but this time in proper-time regularization, and including its nite part. Again we give only the result, 3 2 1 2 2 !m = m −ln(m T0 ) − " + c + 2 ; (9.75) 4 m TX 0 where " denotes the Euler–Mascheroni constant, 32 and c is a renormalization scheme dependent constant. Using this result, we may rewrite 9 X (1) 2 2 G(T0 ) = !m − 3c m − 3 Lscal [B] 4 4TX 0 9m2 ∞ dT −m2 T − e [ln(m2 T ) + "]f(z) : (9.76) 3 (4) 0 T 3 As expected the 1= TX 0 -term introduced by the one-loop mass renormalization cancels the tadpole term E(TX 0 ), up to its constant and Maxwell parts. Moreover, the complete divergence of G(T0 ) for T0 → 0 has been absorbed by !m2 . This is precisely the mechanism which had been found already in Ritus’ analysis [247,244,246]. Putting all pieces together, we can write the complete two-loop approximation to the e5ective Lagrangian in the following way: ∞ 1 2 1 dT −m20 T z 2 9 X (1) (1) ( 6 2) X scal Lscal [B0 ] = − B0 − e [B0 ] + !m20 2 L [B0 ] +L 2 3 2 (4) T0 T 6 9m0 scal ∞ 5 0 2 9 X (1) 0 dT −m20 T 4 −3c m0 2 Lscal [B0 ] − e ln(m20 T ) + " f(z) 3 3 4 9m0 (4) 0 T ∞ 0 dT −m20 T 1 f(z) − e du K (z; u ) − K (z; u ) − a a a 02 2(4)3 0 T 3 GBab 0 ∞ 0 dT −m20 T 2 T e z 3− : (9.77) − 3 3 2(4) 2T0 T TX 0 We have rewritten this Lagrangian in bare quantities, since up to now we have been working in the bare regularized theory. Only mass and photon wave function renormalization are required to render this e5ective Lagrangian nite: m2 = m20 + !m20 ; e = e0 Z31=2 ; B = B0 Z3−1=2 :
(9.78)
(note that this leaves z = e0 B0 T una5ected). Here !m20 has already been introduced in Eq. (9.75), while Z3 is chosen so as to absorb the diverging one- and two-loop Maxwell terms in Eq. (9.77). 32
In comparing with [247,244,172,92] note that there this constant had been denoted by ln(").
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The nal answer becomes ( 6 2) Lscal [B]
z2 dT −m2 T z + −1 e T3 sinh(z) 6 0 ∞ 2 1 dT −m2 T z z2 +3c m e + −1 4 (4)2 0 T 2 sinh(z) 6 ∞ dT −m2 T 1 f(z) e dua K (z; ua ) − K02 (z; ua ) − − 2(4)3 0 T 3 GBab 0 ∞ dT −m2 T e [ln(m2 T ) + "]f(z) ; − 3 (4) 0 T 3
1 1 = − B2 + 2 (4)2
∞
(9.79)
e now denotes the physical charge. However, the result still contains the undetermined constant c, which appeared in the nite part of the one-loop scalar mass renormalization Eq. (9.75). What remains to be done is to determine the value of c for which the renormalized mass m becomes the physical mass. The worldline formalism applies, at least at the present stage of its development, only to the calculation of bare regularized amplitudes. As we have seen already in our -function calculations, the renormalization of those amplitudes has to rely on auxiliary calculations in standard eld theory. Usually worldline calculations are done in dimensional regularization, where there is no harm done in using di5erent formalisms for the calculation of graphs and countergraphs, due to the universality of the regulator and the minimal subtraction prescription. 33 The same is not true for multiloop calculations using a proper-time cuto5, where one must make sure that the precise way of applying the cuto5 is chosen consistently between graphs and countergraphs; otherwise one may have e5ectively performed unwanted nite renormalizations. This problem is well known from multiloop calculations in scalar eld theory performed with a naive momentum space cuto5 (see [248] and references therein). To ensure a correct identi cation of the physical scalar mass, we nd it easiest to retrace the same calculation in dimensional regularization. If one keeps the integrands in Eqs. (9.48), (9.51) in D dimensions, instead of Eq. (9.55) one obtains z 1 −1=2 sin(Z) X det T − Cab = "(D=2)−1 "z ; Z 2 sinh(z) 1 4 z cosh(z G˙ Bab ) tr[GS Bab ] = 2D!(a − b ) − 2(D − 2) − ; T T sinh(z) 1 G˙ Bab 1 ˙ z z = [(D − 2)G˙ Bab + 2G˙ Bab ][(D − 2)G˙ Bab " + 2G˙ Bab "z ] ; tr GBab tr 1 2 2 TX − 2 Cab 4 (G˙ aa − G˙ ab )(G˙ ab − G˙ bb ) 1 1 2 z2 2 z2 = − (D − 2)G˙ Bab " − G˙ Bab + 2 z GBab "z : (9.80) tr 2 2 T TX − 12 Cab 33
This may not hold in certain cases involving "5 or spacetime supersymmetry.
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The term involving !(a − b ) can now be omitted, since in dimensional regularization it will −D=2 not even contribute to the unrenormalized e5ective action (it leads to an integral 0∞ d TX TX which vanishes according to the rules of dimensional regularization). The linear combination Eq. (9.59) generalizes to D−1 1 (2) Lscal [B] = × Eq: (9:48) + × Eq: (9:51) : (9.81) D D We rescale to the unit circle, a; b = Tua; b , set b = 0 as usual, and also rescale TX = T Tˆ . This yields an integral ∞ 1 2 ∞ dT (2) −D e −m2 T 2−D ˆ Lscal [B] = − (4) T dT dua I (z; ua ; Tˆ ; D) ; (9.82) e 2 0 T 0 0 which is the D-dimensional version of Eq. (9.60). Note that the rescaled integrand I (z; ua ; Tˆ ; D) depends on T only through z. In contrast to the calculation in proper-time regularization, the Tˆ -integration is nontrivial in dimensional regularization. It will therefore now be easier to extract all subdivergences before performing this integral. The analysis of the divergence structure shows that Eqs. (9.64), (9.68) generalize to the dimensional case as follows: ∞ 1−(D=2) 2−(D=2) K (z; ua ; D) ≡ d Tˆ I (z; ua ; Tˆ ; D) = K02 (z; ua ; D) + f(z; D)GBab + O(z 4 ; GBab ) 0
(9.83)
with K02 (z; ua ; D) = −4
D − 1 1−(D=2) 2 1−(D=2) + GBab [(D − 1)(D − 4)GBab D−2 3D(D − 2)
2−(D=2) 2 +(−2D2 + 18D − 4)GBab ]z ; z z 2 cosh(z) D−1 2 2 4D − (D − 4)z + (8 − 4D) −8 = O(z 4 ) : f(z; D) = D(D − 2) 3 sinh(z) sinh2 (z) (9.84)
After splitting o5 these two terms, the integral over the remainder is already nite, so that one can set D = 4 in its computation. The Tˆ -integral then becomes elementary, and one is led back to Eq. (9.61), since K (z; ua ; 4) = K (z; ua ). Turning our attention to the second term on the right hand side of Eq. (9.83), let us denote its contribution to the e5ective Lagrangian by Gscal (z; D), 1 2 ∞ dT 1−(D=2) −D e −m2 T 2−D Gscal (z; D) = − (4) T dua f(z; D)GBab : (9.85) e 2 0 T 0 The equivalent of Eq. (9.70) now reads 1 D D 4 1−D=2 dua GBab = B 2 − ; 2 − = − + 0 + O() ; 2 2 0 where B denotes the Euler Beta-function.
(9.86)
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The identity Eq. (9.72) generalizes to D dimensions as follows: d z D−1 z2 T −D=2 : f(z; D) = 8 T D=2+1 + −1 D(D − 2) dT sinh(z) 6
(9.87)
The partial integration over T now produces two terms, ∞ ∞ dT −m2 T 2−D dT −m2 T 3−D z D−1 z2 2 T f(z; D) = 8 m T e e + −1 T D(D − 2) T sinh(z) 6 0 0 ∞ dT −m2 T 2−D z D−4 z2 + T : (9.88) e + −1 2 T sinh(z) 6 0 We now need the complete one-loop mass displacement calculated in dimensional regularization, which is 34 6 2 2 0 2 − + 7 − 3[" − ln(4)] − 3 ln(m ) + O() : !m = m (9.89) 4 Expanding Eqs. (9.86), (9.88), and (9.74) in one nds that, up to terms of order O(), ∞ 2 dT z z 2 (1) 0 2 9 2 −m T X [B0 ] + m L e Gscal (z; D) = !m + −1 9m2 scal (4)3 0 T 2 sinh(z) 6 3 9 : (9.90) × −3" − 3 ln(m2 T ) + 2 + mT 2 Note that again the whole divergence of Gscal (z; D) for D → 4 has been absorbed by !m2 . Our nal answer for the two-loop contribution to the nite renormalized scalar QED Euler– Heisenberg thus becomes ∞ dT −m2 T 1 f(z) (2) Lscal [B] = − e dua K (z; ua ) − K02 (z; ua ) − 2(4)3 0 T 3 GBab 0 ∞ 3 dT −m2 T z z2 9 2 2 + m e : + − 1 −3" − 3 ln(m T ) + 2 + (4)3 T2 sinh(z) 6 mT 2 0 (9.91) This parameter integral representation is of a similar but simpler structure than the one obtained in [244,245], and we have not succeeded at a direct identi cation of both formulas. However, we have used MAPLE to expand both formulas in a Taylor expansion in B up to order O(B20 ), and found exact agreement for the coeNcients. Let us give the rst few terms in this expansion, m4 1 275 B 4 5159 B 6 (2) Lscal [B] = − (4)3 81 8 Bcr 200 Bcr 34
Note that this di5ers by a sign from !m2 as used in Eq. (9.30)—here !m2 denotes the mass displacement itself, while there it denoted the corresponding counterterm.
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2255019 + 39200
B Bcr
8
931061 − 3600
B Bcr
10
205
+ ···
:
(9.92)
The expansion parameter has been rewritten in terms of Bcr ≡ m2 =e ≈ 4:4 × 1013 G. 9.6.2. Spinor QED The corresponding calculation for fermion QED is completely analogous, and we will present only the version in dimensional regularization. In the super eld formalism, formulas (9.48), (9.49) immediately generalize to the following integral representation for the two-loop e5ective action due to the spinor loop, 2 ∞ e dT −m2 T −D=2 ∞ X T (2) −D − Lspin [F] = (−2)(4) T dT da db d/a d/b e 2 T 0 0 0 1ˆ − 12 tan(Z) −1=2 X ×det det T − Cab −Da ya · Db yb ; Z 2
1 Da (Gˆ aa − Gˆ ab )Db (Gˆ ab − Gˆ bb ) −Da ya · Db yb = tr Da Db Gˆ ab + 2 TX − 12 Cˆ ab
:
(9.93)
Performing the Grassmann integrations, and removing GS Bab by partial integration over a , we obtain the equivalent of Eq. (9.51), ∞ T dT −m2 T −D=2 ∞ X T (2) Lspin [F] = (4)−D e2 T dT da db e T 0 0 0 0 ˙ G tan Z ) 1 1 G Bab Fab − tr GFab tr TX − Cab × det −1=2 tr G˙ Bab tr Z 2 2 TX − 12 Cab TX − 12 Cab (G˙ Baa − G˙ Bab )(G˙ Bab − G˙ Bbb + 2GFaa ) + GFab GFab − GFaa GFbb : (9.94) + tr TX − 12 Cab As we will discuss in more detail in the next section, in the spinor loop case this partially integrated integral is already a suitable starting point for renormalization. Specializing to the magnetic eld case, it is again easy to calculate the Lorentz determinants and traces. In addition to the bosonic ones calculated in Eq. (9.80) we now need also cosh(z G˙ Bab ) cosh(z G˙ Bab ) z GFab tr GFab tr = GFab (D − 2) + 2 GFab (D − 2)" + 2 " ; cosh(z) cosh(z) TX − 12 Cab
(G˙ Baa − G˙ Bab )GFaa tr TX − 12 Cab
= 2"z
cosh(z G˙ Bab ) 1− cosh(z)
;
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2 2 ˙ ˙ GFab GFab z cosh (z G Bab ) + sinh (z G Bab ) ; tr = (D − 2)" + 2" TX − 12 Cab cosh2 (z) GFaa GFbb tr = 2 tanh2 (z)"z : 1 X T − 2 Cab
After rescaling to the unit circle, one obtains a parameter integral ∞ dT −m2 T 2−D ∞ ˆ 1 (2) −D 2 Lspin [B] = (4) e T dT dua J (z; ua ; Tˆ ; D) : e T 0 0 0 The extraction of the subdivergences yields ∞ 1−D=2 2−D=2 L(z; ua ; D) ≡ d Tˆ J (z; ua ; Tˆ ; D) = L02 (z; ua ; D) + g(z; D)GBab + O(z 4 ; GBab ) 0
(9.95)
(9.96)
(9.97)
with 1 − D=2 L02 (z; ua ; D) = − 4(D − 1)GBab −
4 1 − D=2 2 − D=2 2 +(D − 2)(D − 7)GBab ]z [(D − 1)(D − 4)GBab 3D (9.98)
(compare Eq. (9.25)), and g(z; D) = −
z2 4D−1 2 + 3(D 6 − 2)z coth(z) − (D − 4)z − 3D = O(z 4 ) : 2 3 D sinh (z)
(9.99)
L02 is again removed by photon wave function renormalization. Denoting the contribution of the second term by Gspin (z; D), we note that the ua -integral is the same as in the scalar QED case, Eq. (9.86). Using the following identity analogous to Eq. (9.72): z z2 D − 1 (D=2)+1 d −(D=2) g(z; D) = 8 T − −1 ; (9.100) T D dT tanh(z) 3 we partially integrate the remaining integral over T . The 1=-part of Gspin is then again found to be just right for absorbing the shift induced by the one-loop mass displacement, 0 6 2 !m0 = m0 − + 4 − 3[" − ln(4)] − 3 ln(m0 ) + O() : (9.101) 4 Up to terms of order one obtains
∞ dT −m20 T z z2 (1) 2 0 X L [B0 ] + m0 e − −1 Gspin (z; D) = !m0 9m0 spin (4)3 0 T 2 tanh(z) 3 12 × 12" + 12 ln(m20 T ) − 2 − 18 : m0 T 9
(9.102)
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Our nal result for the on-shell renormalized two-loop spinor QED Euler–Heisenberg Lagrangian is ∞ dT −m2 T 1 g(z; 4) (2) Lspin [B] = e dua L(z; ua ; 4) − L02 (z; ua ; 4) − (4)3 0 T 3 GBab 0 ∞ 12 dT −m2 T z z2 2 2 m e − − 1 18 − 12" − 12 ln(m T ) + 2 − (4)3 T2 tanh(z) 3 mT 0 (9.103) with
z ) ln(GBab =GBab z B2 B3 L(z; ua ; 4) = B1 ; + z )2 + G z (G z z ) tanh(z) (GBab − GBab GBab (GBab − GBab Bab − GBab ) Bab z B1 = 4z(coth(z) − tanh(z))GBab − 4GBab ; z
z B2 = 2G˙ Bab G˙ Bab + z(8 tanh(z) − 4 coth(z))GBab −2 ; z z B3 = 4GBab − 2G˙ Bab G˙ Bab − 4z tanh(z)GBab +2 ;
L02 (z; ua ; 4) = −
12 + 2z 2 ; GBab
z2 + z coth(z) − 2 : g(z; 4) = − 6 sinh(z)2
(9.104)
Comparing with the previous results by Ritus [247,246] and Dittrich-Reuter [172], we have again not succeeded at a direct identi cation with the more complicated parameter integral given by Ritus. However, as in the scalar QED case we have veri ed agreement between both formulas up to the order of O(B20 ) in the weak- eld expansion in B. The rst few coeNcients are m4 1 B 4 (2) Lspin [B] = 64 (4)3 81 Bcr 1219 B 6 135308 B 8 791384 B 10 − + − + ::: : (9.105) 25 Bcr 1225 Bcr 1575 Bcr On the other hand, our formula almost allows for a term by term identi cation with the result of Dittrich-Reuter [172], as given in Eqs. (7.21), (7.22), (7:37) there. This requires a rotation to Minkowskian proper-time, T → is, a transformation of variables from ua to v:=G˙ Bab , the use of trigonometric identities, and another partial integration over T for the second term in Eq. (9.103). The only discrepancy arises in the constant 18, which reads 10 in the Dittrich-Reuter formula. One concludes that the results reached by Ritus and Dittrich-Reuter for the two-loop Euler–Heisenberg Lagrangian are incompatible, and di5er precisely by a nite electron mass
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renormalization. 35 Moreover, it is clearly Ritus’ formula which correctly identi es the physical electron mass. The corresponding result for the case of a pure electric eld is obtained from this by the substitution B2 → −E 2 . This sign chance makes an important di5erence, since it creates an imaginary part for the e5ective action, indicative of the possibility of pair creation in an electric eld [169]. In [249] an approximation for this imaginary part was obtained from the weak eld expansion for the magnetic case, using Borel summation and a dispersion relation. The generalization of this calculation to the case of a general constant background eld is straightforward [161]. Concerning the physical relevance of this type of calculation, let us mention the experiment PVLAS in preparation at Legnaro, Italy, which is an optical experiment designed to yield the rst experimental measurement of the Euler–Heisenberg Lagrangian [250,251]. It is conceivable that the technology used there may even allow for the measurement of the two-loop correction in the near future [252,253]. 9.7. Some more remarks on the two-loop QED -functions The calculation of the two-loop Euler–Heisenberg Lagrangian has to teach us also something about our previous -function calculation. Of course, the -function coeNcients can be simply retrieved from the Euler–Heisenberg Lagrangians. Up to the contributions from mass renormalization, they can be read o5 from the expansions Eqs. (9.64), and Eq. (9.104) 6 K (z; ua ; 4) = − + 3z 2 + O(z 4 ) ; GBab 12 + 2z 2 + O(z 4 ) : (9.106) L(z; ua ; 4) = − GBab For example, coeNcient 2 appearing in the second line is nothing else but the −8 which we found in Eq. (9.34) (up to the global factor of −2, and another factor of 2 which is due to the di5erent choice of eld strength tensors). Comparing with that calculation we see that the use of the generalized Green’s functions GB ; GF has saved us two integrations: The same formulas Eq. (9.22) which there had been employed for executing the integrations over the points of interaction 1 ; 2 with the external eld, have now entered already at the level of the construction of those Green’s functions. Of course, for the -function calculation all terms of order higher than O(F 2 ) are irrelevant, so that one could then as well use the truncations of those Green’s functions given in Eq. (5.10). Moreover, the choice of an external eld with the property F 2 ∼ 5 is then more convenient than a magnetic eld (this variant of the two-loop -function calculation was presented in [93]). More interestingly, we had noted before that, if the spinor QED -function is calculated in a four-dimensional scheme, a subdivergence-free integrand is obtained proceeding directly from the partially integrated version Eq. (9.94). We are now in a position to see that this fact is not accidental, but a consequence of renormalizability itself. 35
The two formulas had been compared in [172] only in the strong eld limit, which is not sensitive to this discrepancy.
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Let us retrace our two-loop Euler–Heisenberg calculations, and analyze how the removal of all divergences worked for the Maxwell term. In the scalar QED case, there are three possible sources of quadratic divergences for the induced z 2 -term: 1. The contact term containing !ab . 2 )z 2 in the (1=G 2. The leading order term ∼ (1=GBab Bab )-expansion of the main term (see, e.g., Eq. (9.58)). 3. The explicit 1= TX 0 appearing in the one-loop mass displacement Eq. (9.75). The last one should cancel the other two in the renormalization procedure, if those are regulated by the same UV cuto5 TX 0 for the photon proper-time, and this is indeed the case, as we veri ed in various versions of this calculation. In the spinor QED case the fermion propagator has no quadratic divergence (this is, of course, manifest in the standard rst order formulation, while in the second order formulation discussed in Section 4.11 there are several diagrams contributing to the electron self energy, and the absence of a quadratic divergence is due to a cancellation among them). The third term is thus missing, and the other two have to cancel among themselves. In particular, the completely partially integrated version of the integrand has no !ab -term any more, and consequently the second term must also be absent. But the structure of the integrals is such that, if one does this calculation in D = 4, the (1=GBab )-expansion of the main contribution to the Maxwell term is always of the form shown in Eq. (9.58), B A + + C tr(F 2 ) (9.107) 2 GBab GBab with coeNcients A; B; C. In the partially integrated version rst the absence of a quadratic subdivergence allows one to conclude that A = 0, and then the absence of a logarithmic subdivergence that B = 0. Note that this argument does not apply to the scalar QED case, nor does it to spinor QED in dimensional renormalization, due to the principal suppression of quadratic divergences by that scheme. In both cases one would have only one constraint equation for the two coeNcients A and B appearing in the partially integrated integrand, and indeed they turn out to be nonzero in both cases. In the present formalism, the fermion QED two-loop -function calculation thus becomes simpler when performed not in dimensional regularization, but in some four-dimensional scheme such as proper-time or Pauli–Villars regularization. 9.8. Beyond two loops This cancellation mechanism is interesting in view of some facts known about the three-loop fermion QED -function [68,254,70]. Apart from the well-known cancellation of transcendental numbers occurring between diagrams in the calculation of the quenched (one fermion loop) contribution to this -function [68,70], which takes place in any scheme and gauge, even more spectacular cancellations were found in [254] where this calculation was performed in four dimensions, Pauli–Villars regularization, and Feynman gauge. In that calculation all contributions from non-planar diagrams happened to cancel out exactly. It appears that previously gauge invariance was considered as the only source of cancellations in this type of calculation. The cancellation mechanism which we exhibited in the previous
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section is clearly of a di5erent type, and speci c to spinor QED. Whether this mechanism has a generalization to the three-loop level, or perhaps even relates to the cancellation found by Brandt, remains to be seen. In a preliminary study [255] we have computerized the generation of the partially integrated integrand for the three-loop spinor QED vacuum amplitude in a constant eld, using MAPLE and M [256]. In the three-loop case the integrations over the proper-time parameters for the two inserted photons are still elementary. Therefore it turns out to be relatively easy to generate the unrenormalized three-loop Euler–Heisenberg Lagrangians, which are now four-parameter integrals. However the analysis of the divergence structure along the above lines is a rather formidable problem, and so far no conclusive results have been reached. This is due not only to the large number of terms generated at the three-loop level, but also to the existence of several di5erent subdivergences (in the notation of the example in Section 8.5 those are at a ∼ b, c ∼ d, (a; b) ∼ (c; d), and (a; b) ∼ (d; c)). After discovering the rationality of the three-loop quenched QED -function many years ago, Rosner [68] conjectured that this -function may perhaps provide a window to high orders in perturbation theory. Whether or not the worldline formalism will eventually allow one to go beyond the four orders presently accessible to other methods in this calculation, is impossible to say at present. Still we believe that this is a question very much worth pursuing, and that the answer will be a good indicator for the ultimate usefulness of this formalism. 10. Conclusions and outlook In this work we have reviewed the present status of the “string-inspired” technique, and its range of applications in perturbative quantum eld theory. Although we have given a sketch of the original derivation of the Bern–Kosower rules from string theory, based on an analysis of the eld theory limit of the string path integral, our overall emphasis has been on Strassler’s more elementary “worldline” approach, using rst-quantized particle path integral representations for one-loop e5ective actions. From our discussion of QCD amplitudes in Section 4 it should have become clear that these two approaches complement each other nicely. The worldline approach provides a simple and eNcient method for computing either the QCD e5ective action itself, or the one-particle irreducible N -gluon vertex function. The original string-based approach appears to be more powerful when it comes to the calculation of the N -gluon scattering amplitude. Here the worldline approach can still be used to correctly generate the input integrand for the Bern–Kosower rules, and also for a rederivation of the “loop replacement” part of those rules. However it presently still falls short of yielding a complete rederivation of the “pinch” part of the Bern–Kosower rules. On the other hand, the string-based approach is a priori restricted to the on-shell case, due to the requirement of conformal invariance. The o5-shell continuation of string amplitudes in general leads to ambiguities, although at the one-loop level those seem now to be under control [72,73,75]. In particular, in [73] a method of continuation was given which in the eld theory limit yields the gluon amplitudes in background eld Feynman gauge. This line of work has recently led to the formulation of a general algorithm for computing the o5-shell, one-loop multigluon amplitudes from the bosonic string [257]. Some complementarity holds also concerning the present range of applications of both methods. The string-based method has so far, apart from scalar amplitudes [72,76,77], essentially
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been applied only to one-loop gluon [26] and graviton [27,28,258] scattering amplitudes. The worldline path integral approach has provided a simple means to generate Bern–Kosower type master formulas also for eld theories involving Yukawa [101,102] and axial [103–105,156] couplings. On the other hand, its application to gravitational backgrounds requires the mastery of some technical subtleties [60 – 62,144,225 –231], as we discussed in Section 7.2. In this context most authors have concentrated on the computation of anomalies [56 – 62] or of the e5ective action [224,226,230,231], although recently an interesting application was also given to the calculation of scattering amplitudes in eleven-dimensional supergravity [259]. Beyond the one-loop level, our presentation has been con ned to the cases of scalar eld theory and QED. As we already mentioned in the introduction, the construction of multiloop QCD amplitudes in the string-based formalism has turned out to be a formidable technical challenge [260]. Nevertheless, the recent progress in this line of work [72–78] seems to indicate that at least a computation of the two-loop QCD -function may now be in reach. A parallel development using the worldline approach [89,95] has led to the derivation of a two-loop Euler– Heisenberg type action for pure Yang–Mills theory [261], albeit not yet in a form which would allow one to extract the two-loop Yang–Mills -function. As we have shown here, things are much easier in the abelian case. Here the worldline path integral approach provides an easy route to the construction of multiloop Bern–Kosower type formulas, based on the concept of multi-loop worldline Green’s functions. Those Green’s functions were explicitly constructed for Hamiltonian graphs, and carry the full information on the internal propagators inserted into the Hamiltonian loop. Concerning the signi cance of this concept, from Roland and Sato’s result Eq. (8.32), one can see that this treatment of scalar propagator insertions is natural and “stringy”. Whatever the ultimate form of the multiloop Bern–Kosower rules may turn out to be, it seems likely that those functions will gure in them prominently. The same cannot be said for our treatment of photon insertions. Here it is rather clear that a truly string-based approach will incorporate internal photons in a more organic way. Nevertheless, our experience with the formalism at the two- and three-loop level clearly shows that it has many of the properties which one expects from a string-derived approach. It also seems to indicate that, at the multiloop level, quantum electrodynamics is a eld theory which should be particularly suited to the application of string-derived techniques. This impression is based on the following properties: • Particularly extensive cancellations are known to occur in multiloop QED calculations, sug-
gesting that the standard eld theory methods are far from optimized for this task. • In contrast to scalar and nonabelian gauge theories, for QED amplitudes there exists a natural worldline parametrization exhausting the complete S-matrix. • Sums of Feynman diagrams are always generated with the correct statistical weights. • No color factors exist which may prevent us from combining diagrams by “letting legs slide along lines”. In the presentation of this formalism given here we have tried to make maximal use of the worldline supersymmetry (which comes as a free gift from heaven in the worldline formalism). There are several aspects to this. The existence of this supersymmetry alone already leads to functional relationships between the parameter integrals for processes with the same external
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states, but di5erent types of virtual particles involved. Moreover, the worldline super eld formalism allows one to treat scalar, spinor and gluon loops in a uniform manner, as well as to avoid the introduction of two di5erent types of vertex operators. The introduction of multiloop worldline Green’s functions and of worldline super elds have a principle in common, which is that one should always try to absorb a maximum of information into the worldline propagators themselves. A third example of this principle was our introduction of worldline Green’s functions incorporating constant external electromagnetic elds. As we have seen in Section 9.5 those three instances of the principle can be freely combined with each other. We have discussed a large number of applications, often in considerable detail. Some of those calculations have been of an illustrative nature, or mere consistency checks. Others were state-of-the-art calculations, such as our recalculations of the QED photon splitting amplitudes and of the two-loop Euler–Heisenberg Lagrangians. Those examples clearly display the technical advantages over standard eld theory techniques which one can hope to achieve in this formalism, at least for certain types of calculations. However, they also show that the string-inspired technique is presently still a rather specialized tool. All of our applications have been to processes involving a loop with no change of the identity of the particle inside the loop. This class of amplitudes seems to be the most natural one to consider in this context, although it is by no means the only one to which string-inspired techniques can be applied. Notably a generalization of the Bern–Kosower formalism to amplitudes involving external quarks was constructed by Dixon [262], but in preliminary studies has not led to as signi cant an improvement over eld theory methods as was found in the calculation of the four and ve gluon amplitudes [263]. 36 Similarly also the worldline formalism can be easily extended to the computation of fermion self energies [219,266,267]. However the resulting formalism seems somewhat less elegant than in the photon self energy case, and has been too little explored yet to be presented here. Another omission in the present review is the extension to the nite temperature case, which has been considered by various authors [268–271,184,272]. The most general result which has been reached in this line of work is a generalization of the QED Bern–Kosower master formula to the N -photon amplitude at nite temperature and chemical potential [272]. However, this result holds for the Euclidean amplitude, and for most physical applications would still have to be analytically continued. Here one encounters the same type of ambiguities as in Feynman parameter calculations at nite temperature [273,274], which makes it presently diNcult to judge the practical usefulness of this generalization. Finally, it is quite possible that the use made of worldline path integrals in the present work may appear overly modest to the enterprising reader. Our whole aim here was to reproduce the S-matrices of known renormalizable eld theories in a way which avoids some of the shortcomings of conventional Feynman diagram calculations. Clearly it is always possible to rewrite a given eld theory amplitude in terms of worldline path integrals, although not in all cases this will lead to calculational improvements. Much less obvious is the converse question, which is whether a “sensible” worldline Lagrangian must always be induced by a spacetime Lagrangian, or whether worldline path integrals can perhaps be used to de ne physically relevant 36
This evaluation could possibly change due to recent progress [264,265] with regard to the computation of heterotic string amplitudes involving external fermions.
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S-matrices which do not correspond to Lagrangian eld theories. The use of an additional worldline curvature term [275 –277] could be seen as a step in this direction. Acknowledgements I would like to thank M.G. Schmidt for a fruitful collaboration on many of the topics dealt with in this work, as well as S.L. Adler, F.A. Dilkes, G. Dunne, D. Fliegner, P. Haberl, D.G.C. McKeon, M. Mondrag\on, L. Nellen and M. Reuter. Of the many other colleagues who supported this work in one way or the other I wish to particularly thank: Z. Bern for kind hospitality during a visit at UCLA, as well as for many explanations on the Bern–Kosower formalism; D. Broadhurst for encouragement, and for telling me many things about quantum electrodynamics which I would have never learned otherwise; L. Dixon for detailed informations on his unpublished extension of the Bern–Kosower formalism to include external quarks, as well as for explanations on [113]; A. Laser for performing a recalculation of the two-loop spinor QED -function in the second order formalism, as a check on the worldline calculation presented in Section 9.4; F. Bastianelli, T. Binoth, L. Dixon, R. Russo, H.-T. Sato, A.A. Tseytlin, and B. Zwiebach for useful comments on the preprint version of this work. Discussions and correspondence with V.N. Baier, F. Bastianelli, J. Biebl, A. Davydychev, E. D’Hoker, W. Dittrich, H. Dorn, D. Dunbar, H. Gies, H. Kleinert, D. Kreimer, O. Lechtenfeld, N.E. Ligterink, D. LSust, A. Morgan, U. MSuller, C. Preitschopf, M. Rausch, V.I. Ritus, J.L. Rosner, R. Stora, M.J. Strassler, O. Tarasov, B. Tausk, and P. van Nieuwenhuizen are gratefully acknowledged. 37 I also thank the Institute for Advanced Study, Princeton, and the Theory Group of Argonne National Laboratory for hospitality. Appendix A. Summary of conventions At the path integral level, we work in the Euclidean throughout with a positive de nite metric (g ) = diag(+ + · · · +). Our Euclidean Dirac matrix conventions are {" ; " } = 2g 5;
"† = " ;
"5 = "1 "2 "3 "4 ;
= 12 [" ; " ] :
(A.1)
The Euclidean eld strength tensor is de ned by F ij = ijk Bk ; i; j = 1; 2; 3; F 4i = − iEi , its dual by F˜ = 12 F with 1234 = 1. The corresponding Minkowski space amplitudes are obtained by replacing g → ; k 4 → −ik 0 ; T → is ; 4i 0i ˜ ˜ F → F = Ei ; F → −iF ;
1234 → i1230 ;
0123 = 1.
(A.2)
where ( ) = diag(− + ++); The nonabelian covariant derivative is D ≡ 9 + igAa T a , with [T a ; T b ] = ifabc T c . The adjoint representation is given by (T a )bc = − ifabc , and the generators in the fundamental representation of SU (Nc ) are normalized as tr(T a T b ) = 12 !ab . Momenta appearing in vertex operators are ingoing. 37
Special thanks to the referee for a large number of useful suggestions and criticisms!
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Appendix B. Worldline Green’s functions C In this appendix we derive the generalized worldline Green’s functions GP; A needed for the evaluation of the gluon-path integral (Section 4:6) and GB; F for the scalar=spinor path integral in a constant external eld (Section 5). All those Green’s functions are kernels of certain integral operators, acting in the real Hilbert space of periodic or antiperiodic functions de ned on an interval of length T . We denote by HX P the full space of periodic functions, by HP the same space with the constant mode exempted, and by HA the space of antiperiodic functions. The ordinary derivative acting on those functions is correspondingly denoted by 9P ; 9XP or 9A . With those de nitions, we can write our Green’s functions as
GB (1 ; 2 ) = 21 |(92P − 2iF 9P )−1 |2 ; GF (1 ; 2 ) = 21 |(9A − 2iF)−1 |2 ;
GPC (1 ; 2 ) = 1 |(9XP − C)−1 |2 ; GAC (1 ; 2 ) = 1 |(9A − C)−1 |2
(B.1)
(in this appendix we absorb the coupling constant e into the external eld F). Note that GAC is, up to a conventional factor of 2, formally identical with GF under the replacement C → 2iF. GB and GF are easy to construct using the following representation of the integral kernels for inverse derivatives [85] 1 u|9−n (B.2) Bn (|u − u |) signn (u − u ) ; P |u = − n! 1 u|9−n (B.3) En−1 (|u − u |) signn (u − u ) : A |u = 2(n − 1)! Here Bn (En ) denotes the nth Bernoulli (Euler) polynomial, and we have set T = 1. Those formulas are valid for |u − u | 6 1. Let us shortly prove the rst identity; the proof of the second one is completely analogous. First observe that, by construction, 12 G˙ B is the integral kernel inverting the rst derivative 9P acting on periodic functions. We may therefore write 1 Kn (u1 − un+1 ) := du2 : : : dun G˙ B12 G˙ B23 : : : G˙ Bn(n+1) 0
= 2 u1 |9−n P |un+1 : n
This leads to the recursion relation 9 Kn (u − u ) = 2n u|9P−(n−1) |u = 2Kn−1 (u − u ) : 9u We want to show that the same recursion relation is ful lled by the polynomial K˜ n , 2n K˜ n (u − u ):= − Bn (|u − u |)signn (u − u ) : n!
(B.4) (B.5)
(B.6)
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Explicit di5erentiation yields 9 ˜ 2n K n (u − u ) = − Bn (|u − u |)sign(u − u )signn (u − u ) 9u n!
=−
2n Bn−1 (|u − u |)signn+1 (u − u ) = 2K˜ n−1 (u − u ) : (n − 1)!
(B.7)
Here the recursion relation for the Bernoulli polynomials was used, d=d x Bn (x) = nBn−1 (x). An additional term arising by di5erentiation of the signum function for n odd can be deleted due to the fact that !(x)Bn (|x|) = !(x)Bn (0) = 0
(B.8)
for n odd, n ¿ 1. The proof is completed by checking that the master identity works for n = 1 (B1 (x) = x − 12 ), and on the diagonal u1 = un+1 for any n. The second statement is trivial for odd n, since here both sides vanish by antisymmetry. For even n it becomes Bn Tr(9−n (B.9) P )= − n! with Bn = Bn (0) the nth Bernoulli number. This identity is easily shown by writing the trace in the eigenbasis {e2iku ; k ∈ Z \ {0}}, and using the well-known relation between the Bernoulli numbers and the values of the Riemann *-function at positive even numbers, *(2n) = (2)2n = (2(2n)!|B2n |). Using (B.2), we can compute GB for the unit circle as follows: ∞ GB (u1 ; u2 ) = 2u1 |(92P − 2iF 9P )−1 |u2 = 2 (2iF)n u1 |9−(n+2) |u2 P n=0
= −2
∞ (2iF)n−2 signn (u1 − u2 ) n=2
n!
Bn (|u1 − u2 |)
1 sign(u1 − u2 )e2iF(u1 −u2 ) sign(u1 − u2 ) 1 + B1 (|u1 − u2 |) − iF iF 2F 2 e2iFsign(u1 −u2 ) − 1 1 F −iF G˙ B12 ˙ = + iF G B12 − 1 : e 2F 2 sin F =−
In the next-to-last step we used the generating identity for the Bernoulli polynomials ∞ tn text B (x) = : n et − 1 n!
(B.10)
(B.11)
n=0
This is GB as given in Eq. (5.6) up to a simple rescaling. The computation of GF proceeds in a completely analogous way. This method does not work for the determination of GPC , since negative powers of 9XP are not C even well de ned in the presence of the zero mode. In the following, we will calculate GP; A in a di5erent way, which corresponds to the usual construction of the Feynman propagator in
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eld theory. In order to determine GAC (), say, we employ the following set of basis functions over the circle with circumference T : 2 1 −1=2 fn () = T exp i n+ ; n∈Z : (B.12) T 2 They satisfy T dfn∗ ()fm () = !nm ; 0 ∞
n = −∞
fn (2 )fn∗ (1 ) =
∞
!(2 − 1 − mT )
(B.13)
m = −∞
and fn ( + T ) = − fn (). In this basis, the Green’s function (4.49) becomes ∞ 1 exp[i (2=T )(n + 12 )(1 − 2 )] C C : GA (1 ; 2 ) = GA (1 − 2 ) = T n = −∞ i(2=T )(n + 12 ) − C By introducing an auxiliary integration in the form ( ≡ 1 − 2 ) i! ∞ ∞ 2 e 1 1 C d! ! !− n+ GA () = T n = −∞ T 2 i! − C −∞
(B.14)
(B.15)
and using Poisson’s resummation formula, the Green’s function assumes the suggestive form [119] ∞ C (−1)n G∞ ( + nT ) (B.16) GAC () = n = −∞
with C G∞ () ≡
∞
−∞
d! ei! : 2 i! − C
We verify that d C − C G∞ () = !() ; d ∞ d − C GAC () = (−1)n !( + nT ) ; d n = −∞
(B.17)
(B.18) (B.19)
C is a Green’s function on the in nitely extended real line, while G C is which shows that G∞ A de ned on the circle. The integral (B.17) yields for C ¿ 0 C G∞ () = − /(−)eC :
(B.20)
Hence, from (B.16) GAC () =
C
−e
∞
(−1)n /(− − nT )enCT :
n = −∞
(B.21)
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For ∈ (0; T ) only the terms n = − ∞; : : : ; −1 contribute to the sum in (B.21), while for ∈ (−T; 0) a nonzero contribution is obtained for n = − ∞; : : : ; 0. Summing up the geometric series in either case and combining the results we obtain the expression given in Eq. (4.49). It is valid for −T ¡ ¡ + T . Using a basis of periodic functions the same arguments lead to GPC as stated in (4.48). Note that in the limit of a large period T C C lim GA; P () = G∞ ()
(B.22)
T →∞
C and G C have a well-de ned limit: as it should be. For C → 0, both G∞ A 0 G∞ () = − /(−) ;
GA0 () = 12 sign() :
(B.23) −1
The periodic Green’s function GPC blows up in this limit because 9XP does not exist in presence of the constant mode. It is important to keep in mind that GPC is de ned in such a way that it includes the zero mode of d=d. In the perturbative evaluation of the spin-1 path integral one has to deal with traces over chains of propagators of the form −n d n
A; P (C) ≡ Tr A; P −C : (B.24) d Because n
A; P (C) =
1 (n − 1)!
d dC
n−1
1
A; P (C) ;
(B.25)
1 (C). The subtle point which we would like to mention here is that it is suNcient to know A; P strictly speaking the sum de ning A1 , say, ∞ 1
A1 (C) = (B.26) i(2=T )(n + 12 ) − C n = −∞
does not converge as it stands, and is meaningless without a prescription of how to regularize it. The usual strategy is to combine terms for positive and negative values of n, and to replace (B.26) by the convergent series −1 2 ∞ 2 1 2 T CT 1 2
A (C) = − 2C n+ +C = − tanh : (B.27) T 2 2 2 n=0
It is important to realize that this de nition implies a well-de ned prescription for the treatment C of the / functions in GA; P at = 0. In fact, T
A1 (C) = d GAC ( − ) = TGAC (0) (B.28) 0
and by combining Eqs. (B.27) and (B.28) we deduce that we must set 1 lim /() = lim /(−) = : 0 0 2
(B.29)
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With (B.27) we obtain 1
An (C) = − (n − 1)!
n
T 2
d dx
n−1
tanh(x)
The analogous relation in the periodic case is n n−1 1 T d coth(x)
Pn (C) = − (n − 1)! 2 dx
:
(B.30)
x = CT=2
(B.31) x = CT=2
if the zero mode of d=d is included in the trace (B.24), and n n−1 1 T d n −1 {coth(x) − x }
P (C) = − (n − 1)! 2 dx
(B.32)
x = CT=2
if the zero mode is omitted. For C suNciently small one nds the power series expansions ∞ 1 (22k − 1)B2k 2k 2k−n n
A (C) = − ; T C (n − 1)! 2k(2k − n)!
Pn (C) = −
1 (n − 1)!
k = n=2 ∞
k = n=2
B2k T 2k C 2k−n ; 2k(2k − n)!
(B.33)
An and Pn have well-de ned limits for C → 0: (2n − 1)Bn n 1 En−1 (0) n
An (0) = − (n even) ; T = T n! 2 (n − 1)! Bn (B.34)
Pn (0) = − T n (n even) n! (those limits vanish for n odd). This brings us, of course, back to Eqs. (B.2) and (B.3). Appendix C. Symmetric partial integration In this appendix we explain a partial integration algorithm which allows one to remove all GS Bij ’s contained in the original numerator polynomial PN (4.68) of the N -photon amplitude, and which preserves the full permutation symmetry in the N photons. Such an “impartial” partial integration algorithm can be de ned in the following way: 1. In every step, partially integrate away all GS Bij ’s appearing in the term under inspection simultaneously. This is possible since di5erent GS Bij ’s do not share variables to being with, and this property is preserved by all partial integrations. New GS Bij ’s may be created. 2. In the rst step, for every GS Bij partially integrate both over i and j , and take the mean of the results. 3. At every following step, any GS Bij appearing must have been created in the previous step. Therefore either both i and j were used in the previous step, or just one of them. If both, the rule is to again use both variables in the actual step for partial integration, and take the
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219
mean of the results. If only one of them was used in the previous step, then the other one should be used in the actual step. For example, the term GS B12 GS B34 appearing in P4 in the rst step transforms as follows: GS B12 GS B34 → 1 G˙ B12 G˙ B34 {[G˙ B1i k1 · ki − G˙ B2i k2 · ki ][G˙ B3j k3 · kj − G˙ B4j k4 · kj ] 4
−GS B13 k1 · k3 + GS B14 k1 · k4 + GS B23 k2 · k3 − GS B24 k2 · k4 } :
(C.1)
The terms in the second line have to be further processed. Considering just the rst one of them, since both variables appearing in GS B13 were active in the rst step, both must also be used in the second one. This yields − 1 G˙ B12 G˙ B34 GS B13 → 1 G˙ B12 G˙ B34 G˙ B13 [G˙ B1i k1 · ki − G˙ B3i k3 · ki ] 4
8
+ 18 G˙ B13 [GS B12 G˙ B34 − G˙ B12 GS B34 ] :
(C.2)
Considering again the rst term in the second line, only 1 was active in the previous step. Therefore only 2 must be used now, and the third step is the nal one, 1 ˙ 1 ˙ S ˙ ˙ ˙ ˙ (C.3) 8 G B13 G B12 G B34 → 8 G B13 G B12 G B34 G B2i k2 · ki : This prescription treats all variables on the same footing, and therefore must lead to a permutation symmetric result. The nontrivial fact is that the process terminates after a nite number of steps, and does not become cyclic (as would be the case if, for example, one would always treat the indices in a GS Bij symmetrically). This is not diNcult to derive from the fact that, for any term in PN , the indices appearing in the GS Bij ’s and the rst indices of the G˙ Bij ’s are associated to the polarization vectors, and thus must all take di5erent values. This algorithm transforms P4 into Q4 = G˙ B1i 1 · ki G˙ B2j 2 · kj G˙ B3k 3 · kk G˙ B4l 4 · kl + { 12 G˙ B12 1 · 2 {G˙ B3i 3 · ki G˙ B4j 4 · kj [G˙ B1k k1 · kk − G˙ B2k k2 · kk ] + [G˙ B3i 3 · ki (G˙ B41 4 · k1 − G˙ B42 4 · k2 )G˙ B4k k4 · kk + (3 ↔ 4)] + [(G˙ B31 3 · k1 − G˙ B32 3 · k2 )G˙ B43 4 · k3 G˙ B4k k4 · kk + (3 ↔ 4)]} + 5 permutations} + { 14 G˙ B12 G˙ B34 1 · 2 3 · 4 {[G˙ B1i k1 · ki − G˙ B2i k2 · ki ][G˙ B3j k3 · kj − G˙ B4j k4 · kj ] + 12 [G˙ B13 k1 · k3 − G˙ B23 k2 · k3 − G˙ B14 k1 · k4 + G˙ B24 k2 · k4 ] ×[G˙ B1i k1 · ki + G˙ B2i k2 · ki − G˙ B3i k3 · ki − G˙ B4i k4 · ki ]} + 2 perm:} :
(C.4)
This expression can be rewritten more compactly as follows: Q4 = q44 + q43 + q42 − q422 where q44 = G˙ B12 G˙ B23 G˙ B34 G˙ B41 Z4 (1234) + 2 permutations ; q43 = G˙ B12 G˙ B23 G˙ B31 Z3 (123)G˙ B4i 4 · ki + 3 perm: ;
(C.5)
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q42 = G˙ B12 G˙ B21 Z2 (12){G˙ B3i 3 · ki G˙ B4j 4 · kj + 12 G˙ B34 3 · 4 [G˙ B3i k3 · ki − G˙ B4i k4 · ki ]} + 5 perm: ; q422 = G˙ B12 G˙ B21 Z2 (12)G˙ B34 G˙ B43 Z2 (34) + 2 perm:
(C.6)
and the “Lorentz cycles” Zn , Z2 (ij) ≡ i · kj j · ki − i · j ki · kj ; Zn (i1 i2 : : : in ) ≡ tr
n
[kij ⊗ ij − ij ⊗ kij ]
(n ¿ 3)
j=1
have already been introduced in (4.70). Note that the product of two-cycles q422 appears with a minus sign in Eq. (C.5). The reason is that we corrected for an over-counting here; q422 is also contained twice in q42 , and separating it out from there will change the “−” to a “+”. Before proceeding to higher point amplitudes, it will be prudent to further condense the notation. We thus abbreviate G˙ ij ≡ G˙ Bij i · kj ; G˙ ij ≡ G˙ Bij i · j ; G˙= ij ≡ G˙ Bij ki · kj ; ˙ 1 i2 : : : in ) ≡ G˙ Bi1 i2 G˙ Bi2 i3 · · · G˙ Bin i1 Zn (i1 i2 : : : in ) : G(i
(C.7)
As was already mentioned in the main text, it is known from the previous work [20,21,150] that a closed “-cycle” G˙ Bi1 i2 G˙ Bi2 i3 · · · G˙ Bin i1 after the partial integration will always appear multiplied by a factor of Zn (i1 i2 : : : in ). This motivates the last one of the abbreviations above, and also explains why the formulation of the “cycle substitution” part of the Bern–Kosower rules did not require the speci cation of a particular partial integration algorithm. ˙ · ), multiplied by a remainder. A given term in QN thus will be a product of “bi-cycles” G( Following [150] we call this remainder “tail”, or “m-tail”, where m denotes the number of indices not appearing in any of the cycles. For example, q42 is the product of a 2-bi-cycle and a 2-tail. Only the tails depend on the choice of the partial integration algorithm. The tail generated by our speci c symmetric algorithm will be denoted by Tm (i1 : : : im ). The 1-tail is (unambiguously) given by T1 (i) = G˙ ij (i being xed and j summed over). With the above abbreviations, the result for Q5 obtained by an application of the symmetric algorithm can be written as follows: Q5 = q55 + q54 + q53 + q52 − q532 − q522 ; where ˙ q55 = G(12345) + 11 permutations ; ˙ G˙ 5i + 14 perm: ; q54 = G(1234)
(C.8)
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˙ q53 = G(123) {G˙ 4i G˙ 5j + 12 G˙ 45 [G˙= 4i − G˙= 5i ]} + 9 perm: ; ˙ q52 = G(12) {G˙ 3i G˙ 4j G˙ 5k + 12 G˙ 34 [G˙ 5k [G˙= 3i − G˙= 4i ] + G˙= 5i [G˙ 53 − G˙ 54 ]] + 12 G˙ 35 [G˙ 4k [G˙= 3i − G˙= 5i ] + G˙= 4i [G˙ 43 − G˙ 45 ]] + 12 G˙ 45 [G˙ 3k [G˙= 4i − G˙= 5i ] + G˙= 3i [G˙ 34 − G˙ 35 ]]} + 9 perm: ; ˙ ˙ q532 = G(123) G(45) + 9 perm: ; ˙ ˙ q522 = G(12) G(34) G˙ 5i + 14 perm:
(C.9)
Again we have an over-counting; q532 is contained once in both q53 and q52 , and q522 is contained twice in q52 . Comparing the 2- and 3-tails appearing in (C.9) with our earlier results (4.72), (4.74) for Q2 and Q3 , we note that there is a simple relation: The tail Ti can be obtained from Qi , in its un-decomposed form, by rewriting Qi in the tail variables, and then extending the range of all dummy indices to run over the complete set of variables 1 ; : : : ; 5 . It is not diNcult to see that this relation generalizes to an arbitrary Qm ; Tm . Consider (the unpermuted term of) qN2 , which ˙ has a 2-cycle G(12) and a tail TN −2 (3 : : : N ). It suNces to consider those terms in qN2 having a 1 · k2 2 · k1 as their Z2 (12)-component. From the master formula Eq. (1.18) one infers that for this part of qN2 the partial integration procedure have involved only partial integrations over the tail variables 3 ; : : : ; N . Thus the calculation of TN −2 and the lower order calculation of QN −2 are identical as far as the tail indices are concerned. The presence of the cycle variables for the tail makes itself felt only through an extension of the momentum sums in the master formula, leading to the stated extension rule for dummy indices. The same type of argument shows that the structure of Tm does not depend on the number and lengths of the cycles it multiplies. At this point it should be noted that every term in QN must have at least one cycle factor (this is a combinatorial consequence of the fact that each such term contains a total of 2N indices, of which only N are di5erent). Thus the maximal tail occurring in QN has length N − 2. The above connection between TN and QN thus allows us to write down, without going through the partial integration procedure again, Q6 as follows, Q6 = q66 + q65 + q64 + q63 + q62 − q642 − q633 − q632 − q622 + q6222 ; where
5! + permutations = 60 in total ; 2 6 4! ˙ q65 = G(12345)T = 72 in total ; 1 (6) + perm: 2 1
˙ q66 = G(123456)
˙ q64 = G(1234)T 2 (56) + perm:
(45 in total) ;
˙ q63 = G(123)T 3 (456) + perm: (20 in total) ;
(C.10)
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˙ q62 = G(12)T 4 (3456) + perm: (15 in total) ; ˙ ˙ q642 = G(1234) G(56) + perm:
(45 in total) ;
˙ ˙ q633 = G(123) G(456) + perm:
(10 in total) ;
˙ ˙ q632 = G(123) G(45)T 1 (6) + perm:
(60 in total) ;
˙ ˙ q622 = G(12) G(34)T 2 (56) + perm:
(45 in total) ;
˙ ˙ ˙ q6222 = G(12) G(34) G(56) + perm:
(15 in total) :
(C.11)
Here the only new ingredient, T4 , according to the above is related to the un-decomposed Q4 of Eq. (4.77) simply by a relabelling, and an extension of the range of all dummy indices to run from 1 to 6: T4 (abcd) = G˙ ai G˙ bj G˙ ck G˙ dl + { 1 G˙ {G˙ ci G˙ dj (G˙= − G˙= ) 2
ab
ak
bk
+[G˙ ci (G˙ da − G˙ db )G˙= dk + (c ↔ d)] +[(G˙ ca − G˙ cb )G˙ dc G˙= dk + (c ↔ d)]} + 5 perm:} + { 14 G˙ ab G˙ cd {[G˙= ai − G˙= bi ][G˙= cj − G˙= dk ] + 12 [G˙= ac − G˙= bc − G˙= ad + G˙= bd ] ×[G˙= ai + G˙= bi − G˙= ci − G˙= di ]} + 2 perm:} :
(C.12)
Note that the integrand is not yet quite suitable for the application of the cycle substitution rules, since the tails still contain cycles. For this purpose, one should further rewrite Q6 as 38 Q6 = Q66 + Q65 + Q64 + Q63 + Q62 + Q642 + Q633 + Q632 + Q622 + Q6222 ;
(C.13)
where, by de nition, Q6(···) is obtained from the corresponding q6(···) by restricting the range of the dummy indices appearing in its tail so as to precisely eliminate all additional cycles. This also removes the over-counting, so that now all coeNcients are unity. It is now obvious that in this way one arrives at a canonical permutation symmetric version of the Bern–Kosower integrand for the one-loop N -photon=gluon amplitude. Moreover, in [147] it was shown that the cycle decomposition has the additional advantage of constituting a maximal gauge invariant decomposition of this amplitude. Appendix D. Proof of the cycle replacement rule Combining the results of the previous appendix with the worldline super eld formalism we are now in a position to give a direct and simple proof of the basic cycle replacement rule (2.15) connecting the scalar and fermion loop integrands for the N -photon=gluon amplitudes. 38
(·)
When comparing with [147] note that our present de nition of qN( · ) (QN( · ) ) there corresponds to QN( · ) (Qˆ N ).
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Expanding out the super eld master formula (4.32) for the fermion loop we obtain a formula isomorphic to the scalar case, N 1 ˆ 1 ˆ ˆ exp [ 2 G ij ki · kj + iDi G ij i · kj + 2 Di Dj G ij i · j ] i; j = 1
1 :::N
N 1 ˆ = (−i)n PN (−Di Gˆ ij ; Di Dj Gˆ ij ) exp G ij ki · kj : 2
(D.1)
i; j = 1
Here PN is the same polynomial which appears in the expansion (4.68) of the ordinary master formula. We now apply to this integrand the same partial algorithm as in the previous appendix, just with ordinary derivatives replaced by super derivatives. In this way we can remove all second derivatives Di Dj Gˆ ij . A recursive analysis analogous to the one performed in Appendix C shows, that the nal result of the partial integrations is almost isomophic to the ordinary QN , except that we have to take into account that Di Gˆ ij = − Dj Gˆ ij . This means that, for example, the formula for the ordinary 2-tail T2 (ab) = G˙ ai G˙ bj + 1 G˙ (G˙= − G˙= ) ; (D.2) 2
ab
ai
bi
in the super case has to be written di5erently, Tˆ 2 (ab) = Da Gˆ ai a · ki Db Gˆ bj b · kj + 12 [Da Gˆ ab a · b Db Gˆ bi kb · ki + (a ↔ b)] :
(D.3)
The general structure found above remains the same, i.e. the nal integrand QN can be decomposed into a sum of terms QN(···) which individually are products of bi-cycles and tails, where the tails do not contain closed cycles of indices. Each term contains precisely N factors of Di Gˆ ij ’s. Now, observe that, since in the master formula (4.32) Di appears only together with i , all terms in the original integrand PN contain every Di precisely once. Since this property is preserved by the partial integrations, and the cycles contain only Di ’s in the cycle variables, it follows that the tails can contain only Di ’s in the tail variables. From Di Gˆ ij = − /i G˙ Bij + /j GFij ; (D.4) it is then clear that a GFij produced by a Di Gˆ ij in a tail cannot survive the Grassmann integrations if one of the indices i; j is a cycle index; terms of this kind have too many cycle-/’s and too few tail-/’s. Therefore after the /-integrations for all surviving GFij ’s the indices i; j are either both cycle variables, or both tail variables. Since from the structure of the component eld integral D it is clear that GFij ’s can generally only appear in cycles, GFij ’s from tails would therefore have to form cycles among themselves; but this is not possible, since the tails do not contain closed chains of indices. We conclude that, in fact, all GFij ’s coming from tails must drop out in the /-integrations. This leaves us with the basic super cycle integrals, for which we can directly verify that d/i1 · · · d/in Di1 Gˆ i1 i2 Di2 Gˆ i2 i3 · · · Din Gˆ in i1 = (−1)(n(n+1)=2) (G˙ Bi1 i2 G˙ Bi2 i3 · · · G˙ Bin i1 − GFi1 i2 GFi2 i3 · · · GFin i1 ) :
(D.5)
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Appendix E. Massless one-loop 4-point tensor integrals In the worldline parametrization the massless scalar box becomes ∞ 1 u1 u2 dT 4−D=2 du1 du2 du3 exp T GBij ki · kj B[k1 ; k2 ; k3 ; k4 ] = T T 0 0 0 0 i¡j
= 2−
j
2
1
0
du1
u1
0
du2
u2
0
du3
[−
1
i¡j
GBij ki · kj ]2−(j=2)
:
(E.1)
This is essentially Eq. (4.14) specialized to N = 4. We have already rescaled to the unit circle and put u4 = 0. We return to Feynman parameters ai via Eq. (4.15), a1 = 1 − u1 ;
a2 = u1 − u2 ;
(i = Tai ) so that 1 u1 du1 du2 0
0
0
u2
a3 = u2 − u3 ;
du3 =
1
0
a4 = u3
da1 da2 da3 da4 ! 1 −
(E.2) 4
ai
:
(E.3)
i=1
Also we introduce the Mandelstam variables s = (k1 + k2 )2 ;
t = (k2 + k3 )2 :
With all external legs massless and on-shell, GBij ki · kj = − a1 a3 s − a2 a4 t :
(E.4) ki2 = 0,
the Feynman denominator simpli es to (E.5)
i¡j
Generally we will also have a numerator polynomial P({ai }). Let us thus de ne 1
4 4 P({ai }) j dai ! 1 − ai : I [P({ai })] ≡ 2 − 2 0 [a1 a3 s + a2 a4 t]2−j=2 i=1
(E.6)
i=1
We will now perform some manipulations which allow us to generate the numerator polynomial by di5erentiations performed on the denominator polynomial. First it is useful to homogeneize the denominator by the following well-known transformation due to ’t Hooft and Veltman [278], i xi ai = 4 ; i = 1; 2; 3 ; j = 1 j xj a4 =
4 (1 − x1 − x2 − x3 ) 4 j = 1 j xj
with the constraints 1 1 1 3 = ; 2 4 = : s t Note that this implies 4 aj 1 = 4 : j j = 1 j xj j=1
(E.7)
(E.8)
(E.9)
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This transformation leads to 1
4 4 ( 4i = 1 i )( 4j = 1 j xj )−j P({ai }) j I [P({ai })] = 2 − d xi ! 1 − xj : 2 0 [x1 x3 + x2 x4 ]2−j=2 i=1
j=1
(E.10) If we now specialize P to be a polynomial Pm of de nite degree m, this can be further rewritten as 4 1
4
Pm ({i xi })( 4j = 1 j xj )−m−j j I [P] = i 2 − d xi ! 1 − xj 2 0 [x1 x3 + x2 x4 ]2−j=2 i=1
=
4
i=1
i 2 −
i=1
×
j
2
(1 − j − m) (1 − j)
Pm ({i (9= 9i )})( 4j = 1 j xj )−j
1
4
0 i=1
d xi ! 1 − xj
[x1 x3 + x2 x4 ]2−j=2 4 9 I [1] (1 − j − m)
= i Pm i : 4 (1 − j) 9i i i=1
(E.11)
i=1
Here it is understood that 9 n 9 n n i ≡ i : 9i 9i
(E.12)
It remains to calculate I [1], 4 1
( 4j = 1 j xj )−m−j j 4 I [1] = i 2 − d x! 1− xj 2 0 [x1 x3 + x2 x4 ]2−j=2 i=1
= 2−
j
2
0
1
d4 a ! 1 − ai
1 : [a1 a3 s + a2 a4 t]2−j=2
(E.13)
This can be easily done using the following transformation of variables going back to Karplus and Neuman [279], a1 = (1 − y)(1 − z);
a2 = z(1 − y);
which has a Jacobian 9(a1 ; a2 ; a3 ) 9(x; y; z) = y(1 − y) :
a3 = y(1 − x);
a4 = xy
(E.14) (E.15)
This transformation leads to 1 1 1 j 1 dx dy dz : (E.16) I [1] = 2 − 1−j=2 2 0 [y(1 − y)] [(1 − x)(1 − z)s + xzt]2−j=2 0 0
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The y-integral factors out and gives an Euler Beta-function, 1 j j −1+j=2 dy[y(1 − y)] =B : ; 2 2 0
(E.17)
For the remaining double integral the x-integration is elementary, and the resulting z-integral can be expressed in terms of hypergeometric functions using the formula 1 (a)(b − a) dt t a−1 (1 − t)b−a−1 (1 − 4t)−c = (E.18) 2 F1 (a; c; b; 4) : (b) 0 In this way one arrives at j j j 8 t t −1 (j=2)−1 (j=2)−1 I [1] = 2 r s −s t ; ; 1; 1 + ; 1 + 2 F1 2 F1 1; 1; 1 + ; 1 + j 2 2 s 2 s (E.19) where r ≡
(1 − j=2)2 (1 + j=2) : (1 + j)
(E.20)
The j-expansion of this expression is 8 2 1 3 −j=2 −j=2 2 I [1] = r 1 2 3 4 2 ((1 3 ) − + O(j) : + (2 4 ) ) − ln j 2 4
(E.21)
This is suNcient as far as the naked box integral is concerned, but not if used in formula (E.11), since for nontrivial P the factor (1 − j − m) in front has a pole, so that I [1] here would be needed to order O(j). Rather than using this formula as it stands, it is simpler to start the recursion with the four polynomials of degree m = 1. In this case there is no singular prefactor, and it suNces to give those polynomials to order j0 : 4 1 3 2 1 3 −j=2 2 I [a1 ] = I [a3 ] = r 1 2 3 4 (2 4 ) − ln + + O(j) ; j2 2(1 3 + 2 4 ) 2 4 4 2 4 1 3 (1 3 )−j=2 − ln2 + 2 + O(j) : I [a2 ] = I [a4 ] = r 1 2 3 4 2 j 2(1 3 + 2 4 ) 2 4 (E.22) Appendix F. Some worldloop formulas In this appendix we abbreviate GB = G, GB = G. All formulas are written for the unit circle, T = 1. ˙ GF Chain integrals involving G; 1 du2 : : : dun G˙ 12 G˙ 23 : : : G˙ n(n+1) = 2n u1 |9−n P |un+1 0
=−
2n Bn (|u1 − un+1 |)signn (u1 − un+1 ) ; n!
(F.1)
C. Schubert / Physics Reports 355 (2001) 73–234
0
1
227
du2 : : : dun GF12 GF23 : : : GFn(n+1) = 2n u1 |9−n A |un+1
2n−1 (F.2) En−1 (|u1 − un+1 |)signn (u1 − un+1 ) : (n − 1)! Here Bn (x) (En (x)) denotes the nth Bernoulli (Euler) polynomial. The right hand sides can be ˙ GF : rewritten in term of G; G; 1|9−1 |2 = 1 G˙ 12 ; (F.3) =
P
2
1 1 1|9−2 P |2 = 2 (G12 − 6 ) ;
(F.4)
1|9−3 P |2 = −
1 ˙ 12 G 12 G12
;
1|9−4 P |2 = −
2 1 24 (G12
1 30 )
−
(F.5) ;
1 ˙ 1 2 G 12 ( 2 G12 + 16 G12 ) ; 5! 1 3 1|9−6 (G + 1 G 2 − 1 ) ; P |2 = 6! 12 2 12 42 1|9−5 P |2 =
1 1|9−1 A |2 = 2 GF12 ;
(F.6) (F.7) (F.8) (F.9)
1 ˙ 1|9−2 A |2 = − 4 GF12 G 12 ;
(F.10)
1 1|9−3 A |2 = − 4 GF12 G12 ;
(F.11)
1 1 ˙ 1|9−4 A |2 = 24 GF12 G 12 (G12 + 2 ) ;
(F.12)
etc. Chain integrals involving G: 1 2 1 du2 G12 G23 = − 16 G13 + 30 0 1 1 1 3 1 6 2 du2 du3 G12 G23 G34 = (G14 + 12 G14 )+ − : 90 180 7! 0 0 Chain integrals involving G˙ : 1 1 9 ˙ e−iZG˙ 12 [cos(Z) + iG˙ 12 sin(Z)] + 2 : du G˙ 1u G˙ u2 = − i G12 = − 2 9Z Z sin (Z) 0 Integrals appearing in the Calculation of the two-loop spinor QED -function 1 1 2 2 du1 du2 (G˙ 12 − GF12 ) = − 23 ; 0 0 1 2 du1 (G13 − G14 )2 = 13 G34 ; 0
(F.13) (F.14)
(F.15)
(F.16) (F.17)
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1 2 du1 du2 (G˙ 12 G˙ 23 G˙ 34 G˙ 41 − GF12 GF23 GF34 GF41 ) = − 83 G34 − 43 G34 ; 0 0 1 1 2 du1 du2 (G˙ 13 G˙ 32 G˙ 24 G˙ 41 − GF13 GF32 GF24 GF41 ) = 4G34 + 83 G34 − 89 : 1
0
0
(F.18) (F.19)
Two-point integrals: 1 G(u1 ; u2 )2n du[G(u; u1 ) − G(u; u2 )]2n = (F.20) (n ∈ N) ; 2n + 1 0 1 sinh[cG(u1 ; u2 )] du exp{c[G(u; u1 ) − G(u; u2 )]} = ; (F.21) cG(u1 ; u2 ) 0 1 i k+l−i+1 l k!l! (1 − (−1)k+l−i+1 )G˙ 12 − (1 − (−1)i )G˙ 12 k l ˙ ˙ du G (u1 ; u)G (u; u2 ) = : 2 i!(k + l − i + 1)! 0 i=0
(F.22)
Three-point integrals 1 ˙ u1 )G(u; ˙ u2 )G(u; ˙ u3 ) = − 2 {G˙ 23 [G12 − G13 ]+ G˙ 31 [G23 − G21 ]+ G˙ 12 [G31 − G32 ]} du G(u; 3 0
= − 16 (G˙ 12 − G˙ 23 )(G˙ 23 − G˙ 31 )(G˙ 31 − G˙ 12 ) : n-point integrals: n n 1 ˙ n j = 1 cj G ij sinh(c ) e ˙ i G(u; u c ) i = 1 i i i = 1 du e = : n 0 i = 1 ci
(F.23)
(F.24)
Miscellaneous identities: 2 G˙ ij = 1 − 4Gij ;
(F.25)
G˙ 12 + G˙ 23 + G˙ 31 = − GF12 GF23 GF31 ;
(F.26)
dn (G n ) = n!Pn (G˙ ij ) ; duin ij
(F.27)
where Pn denotes the nth Legendre polynomial. References [1] [2] [3] [4] [5] [6] [7] [8]
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Physics Reports 355 (2001) 235–334
Quantum phase transitions and vortex dynamics in superconducting networks Rosario Fazioa; b; ∗ , Herre van der Zantc a
Dipartimento di Metodologie Fisiche e Chimiche (DMFCI), Universita di Catania, viale A. Doria 6, 95125 Catania, Italy b Istituto Nazionale per la Fisica della Materia (INFM), Unita di Catania, Italy c Department of Applied Sciences and DIMES, Delft University of Technology, Lorentzweg 1, CJ 2628, Delft, Netherlands Received February 2001; editor : C:W:J: Beenakker Contents 1.
2.
3.
Introduction 1.1. Josephson-junction arrays 1.2. Phase-number relation 1.3. Structure of the review Quantum phase transitions 2.1. The model of a Josephson-junction array 2.2. The zero-8eld phase diagram 2.3. Magnetic frustration 2.4. Charge frustration and the supersolid 2.5. Dissipation induced S–I transition 2.6. Transport properties 2.7. One-dimensional arrays 2.8. Field-tuned transitions Quantum vortex dynamics 3.1. Classical equation of motion 3.2. Ballistic vortex motion 3.3. E?ective single vortex action
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3.4. Quantum vortices 3.5. One-dimensional vortex localization 4. Future directions 4.1. Persistent vortex currents 4.2. The quantum Hall e?ect 4.3. Quantum computation with Josephson junctions Acknowledgements Appendix A. Array fabrication and experimental details Appendix B. Triangular arrays and geometrical factors Appendix C. Phase correlator Appendix D. Derivation of the coupled Coulomb gas action Appendix E. E?ective single vortex action Appendix F. List of symbols References
300 307 313 313 316 317 321 321 322 323 324 325 326 327
∗ Corresponding author. Dipartimento di Metodologie Fisiche e Chimiche (DMFCI), UniversitCa di Catania, viale A. Doria 6, 95125 Catania, Italy. Fax: +39-95-333231. E-mail address:
[email protected] (R. Fazio).
c 2001 Elsevier Science B.V. All rights reserved. 0370-1573/01/$ - see front matter PII: S 0 3 7 0 - 1 5 7 3 ( 0 1 ) 0 0 0 2 2 - 9
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Abstract Josephson-junction arrays are ideal model systems to study a variety of phenomena such as phase transitions, frustration e?ects, vortex dynamics and chaos. In this review, we focus on the quantum dynamical properties of low-capacitance Josephson-junction arrays. The two characteristic energy scales in these systems are the Josephson energy, associated with the tunneling of Cooper pairs between neighboring islands, and the charging energy, which is the energy needed to add an extra electron charge to a neutral island. The phenomena described in this review stem from the competition between single-electron e?ects with the Josephson e?ect. They give rise to (quantum) superconductor–insulator phase transitions that occur when the ratio between the coupling constants is varied or when the external 8elds are varied. We describe the dependence of the various control parameters on the phase diagram and the transport properties close to the quantum critical points. On the superconducting side of the transition, vortices are the topological excitations. In low-capacitance junction arrays these vortices behave as massive particles that exhibit quantum behavior. We review the various quantum–vortex experiments and theoretical c 2001 Elsevier Science B.V. All rights reserved. treatments of their quantum dynamics. PACS: 74.20.Fg; 73.23.Hk; 74.80.Bj
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1. Introduction 1.1. Josephson-junction arrays The 8rst arti8cially fabricated Josephson-junctions arrays (JJAs) were realized 20 years ago at IBM [1] as part of their e?ort to develop an electronics based on superconducting devices. In the 8rst 10 years of their existence, Josephson arrays were intensively studied to explore a wealth of classical phenomena [2– 6]. JJAs proved to be an ideal model system in which classical phase transitions, frustration e?ects, classical vortex dynamics, non-linear dynamics and chaos could be studied in a controlled way. The observation of the Berezinskii–Kosterlitz– Thouless (BKT) transition [7,8] in Josephson arrays is probably one of the most spectacular experiments [9] in this respect. All classical phenomena can be successfully explained by studying the classical (thermo) dynamics of the phases of the superconducting order parameter on each island. This approach is justi8ed because experiments are usually carried out at temperatures well below the BCS transition temperature. Each island is then superconducting with a well de8ned gap, but phase Luctuations are still allowed. Under these conditions, classical JJAs are a physical realization of the two-dimensional XY model and above the BKT transition temperature, phase Luctuations destroy global phase coherence preventing the system to reach the superconducting state. Global phase coherence is only restored below temperatures corresponding to the Josephson coupling energy EJ , which is the energy scale associated with Cooper pair tunneling between neighboring islands. As we now understand, the relatively large junctions at that time had resistances too low 1 to observe clear quantum e?ects. By the end of the 1980s semiconductor technology had pushed device dimensions well below the micron size. It became possible to fabricate arrays with Josephson tunnel junctions of sizes 100 × 100 nm2 . Circuits with such small junctions showed single-electron e?ects when cooled down to temperatures corresponding to the charging energy EC , the energy needed to add an extra electron charge to a neutral island. It was soon realized that the competition between single-electron e?ects [10,11] and the Josephson e?ect would lead to new, exciting physics. An appealing feature of JJAs already emerges at this stage as they can be visualized as model systems to investigate quantum (zero-temperature) phase transitions [12,13]. In recent years, this 8eld of research has attracted the attention of many physicists. Experiments on thin, superconducting 8lms, high-temperature superconductors, spin systems and two-dimensional electron gases have all shown the existence of quantum critical points. In arrays made of submicron junctions, the quantum Luctuations drive the system through a variety of quantum phase transitions. A quantum JJA may be insulating at zero temperature even though each island is still superconducting. In the classical limit EJ EC , the system turns superconducting at low temperatures since the Luctuations of the phases are weak and the system is globally phase coherent. 1
A simple estimate for value of the junction resistance above which clear quantum e?ects become visible can be obtained by using the Heisenberg relation PE P¿˜. By taking the charging energy (e2 =2C) for PE and of the order of the junction RC-time one 8nds that the junction resistance RN should satisfy the inequality RN ¿ RQ =h=4e2 for quantum e?ects to be observable.
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In the opposite limit, EJ EC , the array becomes a Mott insulator since the charges on each islands are localized and an activation energy of the order of EC is required to transport charges through the system (Coulomb blockade of Cooper pairs). Strong quantum Luctuations of the phases prevent the system from reaching long-range phase coherence in this regime. Granular superconducting thin 8lms are closely related to arrays. In granular 8lms, superconducting islands of various sizes and with various coupling energies are connected together. Therefore, disorder plays a crucial role in these granular materials while it is virtually absent in JJAs (or it can be introduced in a controlled way). Models based on the behavior of Josephson arrays also form the starting point to describe the physics of ultra-thin, amorphous superconducting 8lms in which superconductivity is quenched by disorder or by an applied magnetic 8eld. In these two-dimensional homogeneous 8lms it is believed that, although the order parameter is suppressed, phase Luctuations are still responsible for driving the system through the superconductor–insulator (S–I) transition [14]. Another important 8eld of investigation addressed with JJAs, is the study of the quantum dynamics of macroscopic objects. In the classical limit vortices are the topological excitations that determine the (thermo)dynamic properties of JJAs. In the opposite situation (EJ EC ) the charges on each island are the relevant degrees of freedom. Vortices and charges play a dual role and many features of JJAs can be observed in the two limits if the role of charges and vortices are interchanged. By fabricating arrays with di?erent geometries, vortices can be manipulated to a great extent. Quantum dynamics of macroscopic objects requires knowledge of the coupling to the surrounding environment [15]. To a certain degree, the dissipative environment can be modeled and therefore JJAs are prototype systems to study macroscopic quantum mechanics as well. Born as a problem related to the foundations of quantum mechanics, macroscopic coherence in superconducting nanocircuits is acquiring increasing attention since the advent of quantum computation. 1.2. Phase-number relation Throughout this review, the interplay between the phase of an island and number of charge carriers Q on it plays a crucial role. Together they determine the properties of quantum Josephson networks. The competition between these two canonically conjugated variables is captured by the following Heisenberg relation [16]: [ i ; Qj ] = 2eiij ; where the subscripts i and j label the island positions. An elegant illustration of this competition is presented by what became known as the Heisenberg transistor [17,18]. The aim of the experiment was to control and measure the quantum phase Luctuations through a modulation of the critical current of the system. In Fig. 1 the layout of the device is shown. Two junctions in series (indicated by crosses) are connected to a current source. The junction parameters are such that EJ ∼ EC , i.e., quantum mechanical Luctuations of the number of Cooper pairs and of the phase of the central island are comparable. A large superconducting reservoir is coupled to the island through a superconducting quantum interference device (SQUID).
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Fig. 1. Left hand side: a scanning-electron-microscope (SEM) photograph of the Heisenberg transistor. The leads in upper, left corner are used to perform a four-terminal measurement on the two junctions in series. O?set-charges on the central island are nulled out by the gate capacitor situated below the central island. (Picture taken by W.J. Elion.) Right hand side: a schematic drawing of the Heisenberg transistor [17]. The phase and the charge on the island are quantum mechanical conjugated variables. By varying the Lux through the SQUID ring the e?ective Josephson energy is tuned and, as a result, phase Luctuations on the central island can be varied. (Reprinted by permission from Nature 371 (1994) 594 copyright 1994 Macmillan Magazines Ltd.)
In the experiment the critical current was measured as a function of the applied Lux through the SQUID ring. It shows a periodic modulation with a period equal to the superconducting Lux quantum (0 = h=2e). The role of the SQUID is to provide a tunable coupling to the reservoir of Cooper pairs. When the Lux equals an integer times half a Lux quantum (n0 =2) the coupling is turned o? and Luctuations in the number of Cooper pairs are suppressed. At the same time, phase Luctuations reach their maximum as indicated by the Heisenberg relation. At 8elds equal to zero or an integer number of Lux quanta, the coupling is maximum so that the amount of charge Luctuations reaches a maximum as well. In the experiment, the critical current probes the scale of charge Luctuations: The situation with large charge Luctuations corresponds to favorable Cooper-pair tunneling and a high critical current. Thus, for zero applied 8eld a high critical current is measured because charge Luctuations are at their maximum. At half a Lux quantum applied, the critical current reaches its minimum value because phase Luctuations are at their maximum. 1.3. Structure of the review This review is organized as follows. In Section 2 the basic physical properties and models are introduced. Some theoretical tools to study the phase diagram including the boundary of the S–I transition are brieLy discussed: the mean-8eld approximation, the coarse-graining approach to derive a Ginzburg–Landau e?ective free energy, and the Villain transformation that leads to a description in terms of charges and vortices. These approaches capture most of the essential physics. Sections 2.3 and 2.4 are devoted to a description of the phase diagram when charge and=or magnetic frustration are included. Since the number of control parameters that can be
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varied is large, the phase diagram is discussed in some limiting cases only. Section 2.5 describes the various sources of dissipation in JJAs and their e?ect on the phase diagram. The 8nal three sections of Section 2 report on the transport properties close to the S–I transition, the S–I transition in one-dimensional arrays and the physics of the 8eld-tuned S–I transitions. In all sections, comparison of the theoretical phase diagrams with the experimental results is discussed as well. Section 3 deals with quantum vortex dynamics. After introducing the important vortex properties (vortex mass, pinning potential, etc.) and its classical equation of motion, a theoretical description of quantum corrections to the classical equation of motion is presented. The remainder of the section concerns the description of quantum vortex-experiments. We start with the single vortex properties (tunneling, interference, and Bloch oscillations) and then proceed with collective vortex motion in quasi-one-dimensional samples (Mott insulation of vortices and Anderson vortex localization). In the last section, some future directions are explored. The theoretical description of two new experiments is outlined: persistent vortex currents in Corbino geometries and the quantum Hall e?ect for vortices=charges. The experimental technicalities for the observation of these phenomena are described as well. We end this review with a brief discussion on Josephson qubits in which fundamental aspects of quantum mechanics and quantum information theory can be studied. We tried to keep this review self-contained and, at the same time, to give a comprehensive overview of the quantum properties of Josephson networks. For each of the sections, we present the main ideas without going into a detailed enumeration of all the results obtained in the 8eld. There are some topics which are not discussed here. Probably the most important, which would require a review by itself, is the e?ect of disorder which seems to be more important for granular materials and ultra-thin 8lms. Basics in superconductivity and Josephson physics can be found in the books by Tinkham [19] and by Barone and PaternCo [20]. Since many ideas (e.g. persistent current, localization) were born in the 8eld of more traditional mesoscopic physics we refer for these topics to the books by Beenakker [21] and Imry [22] and to the conference proceedings [23,24]. Various other aspects of JJAs have already been discussed in previous reviews devoted to this topic [25 –27]. Throughout the review we put ˜ = kB = c = 1. Distances are expressed in units of the lattice constant a. We restore S.I. units in the formulas expressing measurable quantities. 2. Quantum phase transitions A quantum Josephson array consists of a regular network of superconducting islands weakly coupled by tunnel junctions. Owing to submicron lithography, array’s parameters (associated to the shape of the islands, the thickness of the oxide barrier, etc.) can be made uniform (virtually identical) across the whole array. With present-day technology variations in junction parameters are below 20% across the array. The dimensions of the unit cell are of the other of few m2 while the superconducting islands have an area of about 1 m2 . The largest samples consist of about 10 000 junctions (e.g. 100 × 100 or 1000 × 7).
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Fig. 2. A scanning-electron-microscope (SEM) photograph of a two-dimensional, square array of small tunnel junctions produced by shadow evaporation. The white bar is 1 m long. (Picture taken from the Ph.D. Thesis of L.J. Geerligs, Delft 1990 (unpublished).)
Quantum arrays are fabricated of all-aluminum high-quality Josephson tunnel junctions with a shadow-evaporation technique. The evaporation mask is made of electron sensitive resists in which the bottom resist layer has an undercut to ensure a proper lift-o? after evaporation. Junctions are formed by evaporating a thin aluminum layer (25 nm) as the bottom electrode followed by in situ oxygen oxidation and evaporation of the counter electrode of about 50 nm from an opposite angle. A more detailed discussion of the fabrication techniques is presented in Appendix A. 2.1. The model of a Josephson-junction array 2.1.1. Quantum phase model In Fig. 2, we show a scanning-electron-microscope (SEM) picture of a Josephson junction array. Its schematic representation is given in Fig. 3. In this square geometry the coordination number, z = 4. The coupling strength between adjacent islands is determined by the Josephson energy EJ = 0 Ic =(2). This coupling energy is inversely proportional to the normal-state junction resistance RN . Experimentally, RN is determined from the normal-state array resistance rN measured at 4:2 K; RN = rN (My + 1)=Mx , where My is the number of cells across the array and Mx is the number of cells along its length. The maximum junction critical current Ic , in the absence of charging e?ects and thermal Luctuations, is assumed to be given by the Ambegaokar–Barato? value [28] Ic RN =
2e
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Fig. 3. A Josephson array consists of a regular network of superconducting islands weakly coupled by tunnel junctions. Each junction is characterized by the Josephson coupling EJ and the junction capacitance C; each island by the capacitance to the ground C0 .
with the measured critical temperature Tc . For a BCS critical temperature Tc = 1:35 K one gets Ic RN = 0:32 mV at low temperatures. Quantum e?ects in Josephson arrays come into play when the charging energy (associated with non-neutral charge con8gurations of the islands) is comparable with the Josephson coupling (the physics associated with charging e?ects in single normal and superconducting junctions has been reviewed in Refs. [10,11]). In addition, as explained in the introduction, the junction resistance should be of the order of (or larger than) RQ [29]. Arrays are made in a planar geometry, in which each island is coupled to each of the other islands and to a far-away ground by its self-capacitance C0 . The junctions are made of two overlapping superconducting layers separated by a thin oxide layer and the main contribution to the capacitance therefore comes from the junction capacitance C. An estimate of the total island capacitance C is obtained from measuring the voltage o?set (Vo?set ) in the I –V characteristics at high bias currents at T = 10 mK in a magnetic 8eld of 2 T. Using the so-called local rule [10,30], C = Me2 =2Vo?set . For junctions of 0:01 m2 , C is found to be 1:1 fF. If one identi8es C with C , the speci8c capacitance is 110 fF= m2 . Measurements on large-area junctions have yielded a speci8c capacitance that is about a factor of two lower. This discrepancy shows that stray capacitance (capacitance between next-nearest and further neighbors) may play a role in Josephson circuits as pointed out by Lu et al. [31]. However, for simplicity one often identi8es the measured C as the junction capacitance C. Reliable estimates of this self-capacitance (C0 ) are obtained from separate measurements on small series arrays with high EC =EJ ratio. A magnetic 8eld of 2 T is applied so that the series arrays are in the normal state; C0 is then measured by varying the potential of the circuit with respect to the ground potential. Recording the current through the circuit yields a periodic signal with period e=C0 . For islands of 1 m × 1 m; C0 ≈ 1:2 × 10−17 F which is much smaller than C [32].
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As already mentioned, the electrostatic energy can be determined once the capacitance matrix Cij and the gate voltages (if present) are known [30,33]. Generally one only considers the junction capacitance C and the capacitance to the ground C0 . In this case the capacitance matrix has the form Cii = C0 + zC, Cij = −C (if i; j nearest neighbors) and zero in all other cases. Consequently the charging energy (for two charges placed in islands i and j of coordinates ri and rj , respectively) is given by dk eik·(ri −rj ) e2 e2 Eij(ch) = Cij−1 = ; (1) 2 2 42 C0 + 2C(1 − cos kx ) + 2C(1 − cos ky ) which is well approximated by the expression |ri − rj | e2 (ch) : K0 Eij ∼ 4C
(2)
Here, K0 is the modi8ed Bessel function. The charging interaction increases logarithmically up to distances of the order of the screening length and then dies out exponentially. The characteristic energy scale is e2 : 2C Eq. (2) assumes three-dimensional screening and the range of the electrostatic interaction between Cooper pairs is given by (in units of the lattice spacing): = C=C0 : EC =
If the two-dimensional limit is considered (if e.g. the array is sandwiched between two media with large dielectric coeUcients) the screening length scales linearly with C=C0 yielding a longer ranged interaction. From all these considerations, we arrive at the following Hamiltonian which describes Cooper pair tunneling in superconducting quantum networks (quasi-particle tunneling is ignored at this stage [34,35]). This model is frequently called the quantum phase model (QPM) and in its most general form it is given by H = Hch + HJ 1 = (Qi − Qx; j )Cij−1 (Qj − Qx; j ) − EJ cos( i − j − Aij ) : 2 i; j
(3)
i; j
The 8rst term in the Hamiltonian is the charging energy in which the Cij−1 is the capacitance matrix; the second is due to the Josephson tunneling. An external gate voltage Vx; i gives the contribution to the energy via the induced charge Qx; i = 2eqx = j Cij Vx; j . This external voltage can be either applied to the ground plane or it may be (unintentionally) induced by trapped charges in the substrate. In this latter case Qx; i will be a random variable. A perpendicular j magnetic 8eld with vector potential A enters the Hamiltonian of Eq. (3) through Aij = 2e i A · dl. The relevant parameter that describes the magnetic frustration is f = (1=2) Aij ; P
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where the summation runs over an elementary plaquette. In quantum arrays, the 2D Lux penetration depth ⊥ (T ) = 0 =2&0 Ic (T ) is much larger than the array size so that the magnetic 8eld is essentially uniform over the whole array, i.e., f is position independent. A similar conclusion can be drawn by considering the ratio of the geometric inductance (we estimate it to be of the order of 1 pH) to the Josephson inductance (larger than 1 nH). Throughout this review, two limits of the QPM are frequently discussed: C C0 and C C0 . The former limit has already been discussed in detail as it is the appropriate regime of Josephson arrays. The latter limit is more appropriate for granular 8lms that have a short-range Coulomb interaction. To describe these systems we use the following notation. When the on-site contribution is dominant, the characteristic energy is e2 −1 C : 2 00 However, some properties (see e.g. discussion in Section 2.4) are crucially dependent on the details of the electrostatic energy at small distances, i.e., on whether the nearest-neighbor interaction is also included or not. E1 represents the variable denoting this nearest-neighbor interaction; E2 the interaction between next nearest-neighbors and so on. The two contributions in the Hamiltonian of Eq. (3) favor di?erent types of ground states. The Josephson energy tends to establish phase coherence which can be achieved if supercurrents Low through the array. On the other hand, the charging energy favors charge localization on each island and therefore tends to suppress superconducting coherence. This interplay becomes evident if one recalls the Josephson relation (which here can be obtained at the operator level by calculating the Heisenberg equation of motion for the phase) E0 =
d i 2e 2e (4) = Vi = Cij−1 Qj : dt ˜ ˜ A constant (in time) charge on the islands implies strong Luctuations in the phases. On the other hand, phase coherence leads to strong Luctuations in the charge. The low-lying excitations of the model de8ned in Eq. (3), are long wavelength phase waves whose dispersion relation can be obtained by considering the QPM in the harmonic approximation 1 EJ H∼ Qi Cij−1 Qj − ( i − j )2 : (5) 2 i; j 2 i; j
The dispersion relation of these modes (usually named spin-waves from the magnetic analogy of the Josephson coupling with the XY model) depends on the form of the capacitance matrix (see Section 3.2). The QPM possesses topological excitations as well, charges and vortices, that will be discussed in Section 2.2.3. A qualitative understanding of the phase diagram can be obtained by considering the two limiting cases in which one of the two coupling energies is largest. For simplicity, we look at the ground state of the system ignoring external voltages and magnetic 8eld. If the Josephson term is dominant, the array minimizes its energy by aligning all the phases, i.e. it is in the superconducting state. If instead the charging energy is dominant, each island has a zero charge in the ground state. In order to put an extra charge on the island one has to overcome a
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Coulomb gap of the order of the charging energy (max {EC ; E0 }). The array behaves as an insulator although each island is still in the superconducting state. 2.1.2. Dissipative models Since the seminal paper by Caldeira and Leggett [36,37] it became clear that dissipation changes the quantum dynamics of macroscopic systems. One can therefore ask the question to what extent dissipation plays a role in Josephson-junction arrays and what its role is on the S–I transition in quantum arrays. At low temperatures one expects quasi-particle tunneling not to be present since the charging energy is smaller than the superconducting gap . Experiments on small arrays indicate that even at mK-temperatures a small but 8nite amount of quasi-particles is always present, although it has not been possible to discriminate the exact details of dissipation. Therefore, we treat the various models that have been proposed to describe dissipation in superconducting networks. The QPM de8ned in Eq. (3) only accounts for Cooper pair tunneling between neighboring islands and needs to be generalized. The appropriate description is formulated in terms of an e?ective action by the authors of Refs. [10,38]: Z= D i () exp[ − S { }] : (6) i
The Euclidean e?ective action S { }, corresponding to the Hamiltonian of Eq. (3) has the form (for simplicity we ignore charge and magnetic frustration for the time being) + C0 ˙ )2 + C ˙ − ˙ )2 − EJ S[ ] = d ( ( cos( − ) (7) i j i i j 8e2 8e2 0 i ij
ij
with + being the inverse temperature. The 8rst two terms are easily recognized as charging energies expressed in terms of voltages (see Eq. (4)). In the presence of dissipative tunneling, the e?ective action has a Caldeira–Leggett form and acquires an additional term 1 + SD [ ] = d d ,( − )F( ij () − ij ( )) ; (8) 2 0 ij
where ij = i − j . Both the dissipative kernel ,(), related to the I –V characteristic of tunnel junctions [10], and the function F({ }) depend on the nature of the dissipation. Various mechanisms: tunneling of quasi-particles and=or the Low of ohmic currents through the substrate or between the junctions themselves [20,35]. For an ohmic junction, as is the case when the bath is formed by quasi-particle excitations in normal metals (or gapless superconductors), the kernel is 1 1 ,() = 2 : 2 2 2e RN + sin (=+) Here, the normal-state resistance RN controls the coupling to the environment [39]. For ideal superconducting islands on the other hand, the BCS gap inhibits leakage currents at small voltages and as a consequence the dissipative kernel is short range in imaginary time. Therefore,
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quasi-particle tunneling results in a renormalization of the junction capacitance [38] 3 C→C+ : (9) 32PRN The dissipation mechanism also a?ects the form of the function F( ) in Eq. (8). If normalelectron tunneling occurs via discrete charge transfer, as it is for a quasi-particle current, F( ) is a periodic function of the phase ij () − ij ( ) FQP ( ij () − ij ( )) = 1 − cos : (10) 2 If, on the contrary, dissipation is due to normal shunts or more generally to the interaction with the electromagnetic environment, F is quadratic in 1 ij () − ij ( ) 2 FN ( ij () − ij ( )) = (11) 2 2 indicating that the charge at a junction can assume continuous values. One can also consider dissipation due to currents Lowing to the substrate. This local-damping model plays an important role in classical, proximity coupled Josephson arrays. Voltage Luctuations compared to the ground instead of voltage di?erences between junctions are the crucial variables: the dissipative part of the action now depends on the phase i of each island and not on the phase di?erence ij 1 i () − i ( ) 2 FLD ( i () − i ( )) = : (12) 2 2 The path integration in Eq. (6) depends on the nature of dissipation mechanism and it is related to the charge on the islands being a continuous or discrete variable [10]. In the 8rst case, the phase is considered an extended variable and in the path integral i (0) = i (+). If the charge is a discrete variable, a summation over winding number is implied in Eq. (6) i0 +2mi 2 D → d i0 D i () : (13) i
0
{mi =0;±1;:::} i0
These non-trivial boundary conditions express the fact that the charges of the grains are integer multiples of 2e. The dissipative coupling strength is usually expressed in the form , = Re =RQ . The exact value of the e?ective resistance Re is not a priori clear. Consider an array of unshunted tunnel junctions. If thermally excited quasi-particles were the only source of damping, a measure of Re would be the subgap resistance which is many orders of magnitude larger than the normal-state resistance RN . However, measurements hint at a much smaller Re that is closer to RN . The exact mechanism is not clear but one should always keep in mind that Josephson junctions are highly non-linear elements. Some coupling to higher energy scales may therefore occur and Re is smaller than the subgap resistance but not lower than RN . This coupling may for instance occur when a vortex crosses a single junction thereby producing voltage spikes. Throughout this review, we keep the notation simple and use , to denote the dissipation strength regardless of the underlying dissipation mechanism. Its origin will be speci8ed from case to case.
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2.1.3. Related models The S–I transition has been investigated by studying model Hamiltonians, the so-called Bose– Hubbard and XXZ models, closely related to the QPM of Eq. (3). We follow the notation used currently in the literature and point out the connection with the couplings used to de8ne the QPM. The Bose–Hubbard (BH) model [40] is de8ned as H=
1 t −iAij † ni Uij ni − & ni − (e bi bj + h:c:) : 2 i 2 i
(14)
ij
Here, b† ; b are the creation and annihilation operators for bosons and ni = b†i bi is the number operator. Uij describes the Coulomb interaction between bosons (Uij → Eij(ch) ), & is the chemical potential, and t the hopping matrix element. The connection between the Bose–Hubbard model and the QPM is easily seen by writing the 8eld bi in terms of its amplitude and phase and by subsequently approximating the amplitude by its average. This procedure leads to the identi8cation bi ∼ ei i . The hopping term is then associated with the Josephson tunneling ( n t → EJ ) while the chemical potential plays the same role as the external charge in the QPM (& → Qx ). The mapping becomes more accurate as the average number of bosons per sites increases. In the case of strong on-site Coulomb interaction Uii → ∞ and very low temperatures only few charge states are important. If the gate voltage is tuned close to a degeneracy point, the relevant physics is captured by considering only two charge states for each island, and the QPM is equivalent to an anisotropic XXZ spin-1=2 Heisenberg model [41,42] HS = −h Siz + Siz Uij Sjz − EJ (eiAij Si+ Sj− + e−iAij Sj+ Si− ) : i
i =j
(15)
i; j
The operators Siz ; Si+ ; Sj− are the spin-1=2 operators, Siz being related to the charge on each island (qi = Siz + 1=2), and the raising and lowering Si± operators corresponding to the “creation” ±i j of the QPM. The “external” 8eld h is related to the external and “annihilation” operators e charge by h = (qx − 1=2) j Uij . Di?erent magnetic ordered phases of the XXZ Hamiltonian correspond to the di?erent phases in the QPM. Long-range order in S + indicates superconductivity in the QPM while long-range order in S z describes order in the charge con8guration (Mott insulator). The three models are equivalent in the sense that they belong to the same universality class (they lead to the same Ginzburg–Landau e?ective free energy). The non-universal features like the location of the phase transitions, the shape of the phase diagram (and sometimes the very existence of some intermediate phase) depend quantitatively on the speci8c choice of the model. A more rigorous discussion of the di?erent dynamical algebras realized by the three models can be found in Refs. [43– 47]. A generalization of these models to the case in which amplitude Luctuations are coupled to phase Luctuations was discussed in Ref. [48]. Very recently Yurkevich and Lerner [49] developed a nonlinear sigma model description of granular superconductors that reduces to the BH model in the limit in which amplitude Luctuations are ignored.
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Fig. 4. The zero-8eld linear resistance per square measured as a function of the temperature for six di?erent square arrays. The curved dashed lines are 8ts to the vortex-BKT square root cusp formula. The dashed horizontal line indicates the zero-temperature universal resistance at the S–I transition calculated in Ref. [150]. (From Ref. [51].)
2.2. The zero-<eld phase diagram Superconductor–insulator transition has been investigated in great detail in JJAs [50 –53] as well as in granular systems [54 – 60] and uniform ultra-thin 8lms [61– 63]. The 8rst controlled measurements on the S–I transition in junction arrays have been carried out by Geerligs et al. [50]. Part of the data of Ref. [50] together with the new data of Ref. [51] are presented in Fig. 4. It shows the resistive behavior of six di?erent square arrays in zero magnetic 8eld. The zero-bias resistance per junction R0 (T ) has been measured with a very small transport bias (current per junction smaller than ¡ 10−3 Ic ) in the linear part of the current–voltage characteristics. Three arrays become superconducting; two arrays insulating and one array that lies very close to the S–I transition shows a doubly reentrant dependence. The horizontal dashed line in Fig. 4 is the critical resistance value of 8RQ = (see Eq. (44)). For the three arrays that become superconducting, the data are 8tted to the predicted BKT square-root cusp dependence on temperature, R0 (T )=RN = c exp(−b[EJ =(T − TJ )]1=2 ) with b and c constants of order one. In order to compensate for the temperature dependence of EJ , it is convenient to de8ne a normalized temperature is de8ned as T=EJ (T ). From the 8ts the normalized BKT transition temperature TJ is determined. Near the S–I transition TJ is substantially smaller than the classical value of 0:90EJ . Note that at low resistance levels (R0 (T ) ¡ 10−3 RN ), deviations from the square-root cusp dependence are found and that the
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resistance decreases exponentially. This is indicative of thermal activation of single vortices across the whole array width [64]. The two arrays with a ratio of EJ =EC 60:55 show a continuous increase of the resistance as the temperature is lowered, i.e., the arrays become insulating at zero temperature despite the fact that each island is still superconducting with a well developed BCS gap. It has been proposed that, due to the long range interaction between the charges, the conductance will follow a square-root cusp dependence on temperature in a similar way as the resistance for the superconducting samples. This square-root cusp dependence characteristic for a charge– BKT transition is generally not observed. Instead the conductance decreases exponentially as temperature is lowered. This issue will be discussed in some detail in the section devoted to the BKT transition (Section 2.2.4). The resistance of sample with EC =EJ = 1:7 has a very remarkable dependence on temperature. Starting at high temperatures, R0 (T ) 8rst decreases when the temperature is lowered. Below T = 150 mK, however, R(T ) increases by more than three orders of magnitude and at the same time a charging gap develops in the I –V curve. Finally at 40 mK; R0 (T ) starts to decrease again. The second reentrant transition at 40 mK seems to be a more general feature of arrays near the S–I transition which is also present in a magnetic 8eld. Reentrant behavior in the resistance has also been observed in granular superconductors. Already in the early theoretical works on the QPM various explanations have been proposed. We will summarize some of the ideas. Efetov [65] suggested that the thermally excited quasi-particles could screen the Coulomb energy thereby lowering the threshold for the onset of phase coherence. Stimulated by Efetov’s work a number of theoretical papers showed that a reentrant phase boundary can be obtained in the QPM however it turned out that the very existence of the re-entrance was sensitive to the approximation scheme used. Moreover it was shown that even if there is no re-entrance in the phase diagram, the QPM leads to a Luctuation dominated region [66 – 68] which may account for the observed re-entrance. Many physical ingredients not contained in the QPM (random o?set, dissipation, etc.) may play a role as well. Recently Feigelman et al. [69] proposed that the parity e?ects may be responsible. At intermediate temperatures the screening of quasi-particles would decrease the e?ective Coulomb interaction (and therefore the resistance). At lower temperatures screening disappears due to the excess free energy associated with odd grains, leading to an upturn of the resistance curve. The data of Fig. 4 can be used to construct a phase diagram for phase transition of Josephson arrays in zero-magnetic 8eld [51] as shown in Fig. 5. In this 8gure the superconducting-to-normal phase boundary is the vortex–BKT phase transition. Temperature on the vertical axis in this 8gure is given in units of EJ and scaled to the classical (in absence of charging e?ects) BKT transition TJ(0) . The experimental value TJ(0) = 0:95EJ is close to the theoretical value of 0.90 determined from Monte Carlo simulations. On the insulating side of the 8gure no strict phase transition was observed. The dashed line therefore is somewhat arbitrary. It represents the crossover to the low-temperature region with R0 ¿ 103 RN . Fig. 5 indicates that at zero temperature the S–I transition takes place at EC =EJ ≈ 1:7. The existence of a zero temperature (quantum) phase transition can be understood by simple arguments as already discussed in the previous section. Mean-8eld [70], variational approaches [71–73], 1=z expansion [74], Monte Carlo simulations [75,76] and cluster expansions [77] were also applied to the QPM model of Eq. (3). We refer the reader to the various articles for a
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Fig. 5. Measured phase diagram in zero magnetic 8eld for square (solid squares) and triangular (solid triangles) arrays showing the S–I transition at EC =EJ ∼ 1:7. The solid line is a guide to the eye connecting the data points and the dotted line on the superconducting side represents the result of the calculation of Ref. [76]. (From Ref. [51].)
discussion and comparison. Here, we present a few approaches that give a self-contained description of the phase diagram. We start with the mean-8eld calculation. Inaccurate in determining the critical behavior, this approach is yet capable to capture most of the features measured in the experiments. We 8rst consider the case in which both charge and magnetic frustration are absent. Note that for the time being, dissipation is not included. It will be considered in detail in Section 2.5. 2.2.1. Mean-<eld approach The mean-8eld decoupling consists in approximating the Hamiltonian of Eq. (3) by [65,70] 1 HMF = Qi Cij−1 Qj − zEJ cos MF cos j : 2 i; j j The average cos( ) MF plays the role of the order parameter and it should be calculated self-consistently cos( ) MF = Tr {cos( i )exp(−+HMF )}=Tr {exp(−+HMF )} :
Close to the transition point the thermal average on the r.h.s. can be evaluated by expanding it in powers of cos( ) MF . (This can be done because the transition is continuous.) The transition line is determined by the equation + 1 − zEJ d cos i ()cos i (0) ch = 0 ; (16) 0
where the average · · · ch is calculated using the eigenstates of the charging part of the Hamiltonian only. Note that the (imaginary) time evolution of the phase is due to charging as well.
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Fig. 6. Sketch of the phase diagram for short-range interaction between charges. The phase incoherent state is resistive with an activated behavior of the resistance. At T = 0 the array is an insulator. The dependence of the quantum critical point on the capacitance matrix is all contained in E0 .
In the classical limit the phase correlator is unity (there are no quantum Luctuations) and the mean-8eld transition temperature is zEJ =2. Charging e?ects inhibit phase Luctuations and the critical temperature decreases. Explicit formulas for the phase–phase correlator are given in Appendix C. At T = 0 in the self-charging limit (Eij(ch) = E0 ij ) the correlator reads cos i ()cos i (0) ch = 12 exp{−4E0 (1 − =+)} :
By substituting this expression in Eq. (16) one gets a value for the S–I transition which coincides with the simple estimate based on energy considerations. The detailed structure of the phase diagram depends, even in the absence of (magnetic or electric) external frustration, on the range of the electrostatic energy. The phase diagram in the short-range case is sketched in Fig. 6. Very recently a detailed analysis of the dependence of the phase boundary on the form of the capacitance matrix has been performed in Ref. [78] using perturbation theory and numerical simulations. For two-dimensional arrays the transition to the superconducting state is of the BKT type with no spontaneous symmetry breaking. Quantum Luctuations renormalize the value of the transition temperature but do not change the universality class of the transition. The corrections to the BKT transition due to quantum Luctuations have been evaluated in a semi-classical approximation in Ref. [79]. For EJ EC , the JJA behaves as a classical XY model but with a renormalized EJ . This approach breaks down when quantum Luctuation drive the transition to zero. At zero temperature there is a dimensional crossover and the S–I transition belongs to the (d + 1)-XY universality class. 2.2.2. Coarse-graining approach Although it is very useful in determining the structure of the phase diagram, the mean-8eld approach has various shortcomings. For a more accurate description of the quantum critical regime one has to resort to di?erent approaches. Universality implies that the critical behavior of the system depends only on its dimensionality and on the symmetry which is broken in
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the ordered phase. Many properties of JJAs can be extended to other systems that show a S–I transition. By using the coarse-graining approximation it is possible to go from the QPM model to a Ginzburg–Landau model with an e?ective free energy which is a function of the order parameter [80,81] only. Since the transition is governed by quantum Luctuations, the order parameter depends both on space and (imaginary)-time [13,82]. The coarse-graining proceeds in two steps: • An auxiliary 8eld
(x; ) (which has the role of the order parameter) is introduced through a Hubbard–Stratonovich transformation. The partition function is then expressed as a path integral over . • The assumption that the order parameter is small close to the transition allows for a cumulant expansion to obtain the appropriate Ginzburg–Landau free energy. The coeUcients depend on the details of the microscopic model. The partition function of the QPM is given by Z = Tr {e−+HQPM } = Zch T e−
+ 0
dHJ ()
ch ;
(17)
where the subscripts ch and J refer to the charging and Josephson part of the Hamiltonian in Eq. (3). By applying the Hubbard–Stratonovich transformation to the Josephson term one gets (in absence of magnetic and charge frustration) E + J i i −i j exp d e e + h:c: 2 0 i; j
∼
+ ∗ D D exp − d [EJ ]−1 ij 0 i; j
∗ i () j ()
+
0
+
d
[
i
∗ i i () i ()e
+ h:c:] : (18)
Here we introduced a matrix [EJ ]ij which is equal to EJ if i and j are nearest neighbors and zero otherwise. The partition function can be written as Z = Zch D ∗ D exp{−F[ ]} : (19) Close to the phase transition one can perform a gradient expansion −1 ∗ F= d d i ()[[EJ ]ij ( − ) − g( − )ij ] j ( ) + 6 d| i ()|4 : The dynamics of the 8eld
(20)
i
i; j
is governed by the phase–phase correlator
g( − ) = exp[ i () − i ( ) ch that was already encountered in Eq. (16). The coeUcient 6 is related to the four-point phase correlator. The partition function in Eq. (19) can be calculated using a mean-8eld approximation for the phase by evaluating it in the saddle point approximation. The results coincide with that of
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the previous subsection. In the coarse-graining approach, however, a systematic treatment of the Luctuations is possible and it allows to study transport as well. In the case of zero o?set charges and zero external magnetic 8eld, by expanding the phase correlator around the zero-frequency and zero-momentum value, the quadratic part of Eq. (20) can then be rewritten as F (0) [ ] = T [8 + 962 + :!n2 ]| (k; !n )|2 ; k;!n
where using the expression given in Appendix C, the coeUcients can be expressed as 8=
g−1 (!n = 0) EJ − ; 2E0 E0
9=
g−1 (!n = 0) ; 8E0
:=
@2!n g(!n )|!n =0 : 4E0
(21)
At T = 0, this system belongs to the same universality class as the (d + 1) XY model. One can readily obtain all the critical exponents from what is known from the XY model [83]. The dynamical critical exponent is z = 1 due to the symmetry between space and time. At 8nite temperatures the transition belongs to the Berezinskii–Kosterlitz–Thouless universality class and there is no spontaneous breaking of the symmetry. The dynamical critical exponent and the dimensional crossover is modi8ed in the case of 1=r-Coulomb interaction between charges [84]. 2.2.3. Duality transformations Duality transformations have proven to be a powerful tool in 8eld theory and statistical mechanics [85]. The idea behind it is that the weak coupling region of a particular system can be mapped onto the strong coupling range (and vice versa). The symmetries of the system under this transformation lead to important insight into the structure of the model, especially in the region of intermediate couplings which is usually elusive to standard treatments. Dual transformations constitute a powerful approach since it is possible to recast the partition function solely in terms of the topological excitations of the system [86 –89]. In this section we derive some properties of quantum JJAs derived from duality. There is a dual transformation [90 –92] relating the classical vortex limit, EJ EC , to the opposite charge limit, EJ EC . The situation is most transparent in the case C0 C, which might be more relevant for arrays. The interaction between charges on islands is then logarithmic, analogous to vortex interactions in classical, superconducting arrays. The charges form a 2D Coulomb gas and are (0) expected to undergo a BKT transition at Tch ∼ EC [93] (see also next subsection). Using the results discussed in Appendix D, the partition function of a JJA can be expressed as a sum over charge q and vortex v con8gurations Z= e−S{q; v} : (22) [q;v]
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The e?ective action S {q; v} reads + d 2e2 qi ()Cij−1 qj () + EJ vi ()Gij vj () S {q; v} = 0
ij
1 q˙ ()Gij q˙j () : + iqi ()>ij v˙j () + 4EJ i
(23)
This action describes two coupled Coulomb gases. We have used a continuous time notation for clarity. Since q and v are integer valued 8elds, the path integral is well de8ned on a discretized time expression. The charges interact via the inverse capacitance matrix (8rst term). The interaction among the vortices (second term) is described by the kernel Gij , which is the Fourier transform of k −2 . At large distances rij a between the sites i and j it depends logarithmically on the distance: Gij ∼ − 12 ln rij : The third term describes the coupling between the topological excitations in the two limits, i.e., it describes the coupling between charges and vortices. The function yi − yj >ij = arctan xi − xj represents the vortex-phase con8guration at site i when its center is placed at the site j. The coupling has a simple physical interpretation: a change of vorticity at site j produces a voltage at site i which is felt by the charge at this location. The last term qG ˙ q˙ stems from the spin-wave contribution to the charge-correlation function. In the limit in which EJ → 0 or EC → 0 the action for a classical system of Cooper pair charges or vortices is recovered. The e?ective action in Eq. (23) shows a high degree of symmetry between the vortex and charge degrees of freedom. In particular, in the limit C0 C the inverse capacitance matrix has the same functional form as the kernel describing vortex interactions: EC e2 Cij−1 = Gij : Hence charges and vortices are dual. There is a critical point for which the system is self-dual with respect to interchanging them: EJ 2 = 2 : EC The duality is strict for vanishing self-capacitance and in the absence of the spin-wave duality breaking term (qG ˙ q) ˙ in Eq. (23). This latter term is irrelevant at the critical point, i.e., it merely shifts the transition point. However, it has important implications for the dynamical behavior. Duality transformations have also been applied to a three-dimensional JJA consisting of two arrays placed on top of each other [94 –96]. The authors of these papers assume that there is only capacitive coupling between them (no Josephson coupling). The most interesting situation arises when one array is in the quasi-classical (vortex) regime while the other is in the quantum, charge regime. Then, vortices in one layer and charges in the other one are well de8ned.
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The Euclidean e?ective action, in term of phases, S { 1 ; 2 } has the form + C0& ˙ 2 Cx ˙ 2 ˙ S { 1 ; 2 } = d ( ) + 2 ( i1 − i2 ) 8e2 i& 8e 0 &=1;2
i
C&
+
ij
8e2
( ˙ i& − ˙ j& )2 − EJ& cos( i& − j& )
;
(24)
where C0& are the island capacitances in array & relative to ground, C& are the junction capacitances in the array &, and Cx are the interlayer capacitances between islands on top of each other, while EJ& are the Josephson coupling constants in the layers. Similar as in a single array, we move from a description in terms of phases to one in terms of charges and vortices, and use the duality of the resulting action to investigate the transition. Before we proceed with the calculation it is necessary to stress that in the regime of interest the interlayer capacitances Cx not only couple the layers, but also renormalize the island capacitances C01 and C02 to ground. The physical reason for this is that due to the strong Luctuations of charges in layer 2 and vorticities in layer 1 these variables are e?ectively continuous, and hence a coupling to the other array plays the same role as a coupling to ground. Due to screening, the interaction between charges in each layer has a 8nite range for any non-zero Cx , and the BKT transition is replaced by a crossover. However, in the limit C01 Cx C1 the screening length ?1 ∼ (C1 =Cx )1=2 can be large enough to make it meaningful to speak about the charge–unbinding transition (the transition is exponentially sharp). Below we consider this case. For not so weak coupling, on the other hand, this description becomes meaningless, since the crossover is strongly smeared, and the insulating phase is absent. It is possible to introduce vortex degrees of freedom in the same way as for one array. We obtain the e?ective action for charges qi1 () in layer 1 and vorticities vi2 () in layer 2 (to be referred below as qi and vi ) + 1 2EC1 S {q; v} = d qi ()Gij qj () + q˙ ()Gij q˙j () 4EJ1 ij i 0 ij + EJ2
ij
× Gij −
vi ()Gij vj () +
v˙i () 8EC2 ij
Cx2 >ik Gkl >lj 42 C1 C2 kl
v˙j () +
iCx
2C1
ijk
v˙i ()>ik Gkj qj () :
(25)
This equation looks rather similar to the e?ective charge–vortex action in one Josephson junction array (see Eq. (23)). The most important di?erence is that while in one layer either charges or vortices are well-de8ned degrees of freedom, Eq. (25) describes the system of two well-de
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Fig. 7. The phase diagram for a quantum JJA in the limit of long-range (logarithmic) interaction between charges. Similarly to vortices the charges undergo a BKT transition leading to insulating behavior at low temperature and small EJ =EC ratio.
numerical coeUcients. However, close enough to the transitions these terms only produce a small renormalization of the transition temperature, and are therefore irrelevant. Another interesting feature of this action is that the last term, describing the interaction between charges and vortices, is also small, while in a single-layer array the interaction is always of the same order of magnitude as the other terms. 2.2.4. Berezinskii–Kosterlitz–Thouless transitions In classical arrays, it is well established that JJAs undergo a BKT phase transition to the superconducting state. In the opposite limit where charging dominates, quantum Luctuations of the phases are strong, and vortices are ill-de8ned objects. In this regime the duality transformations discussed in the previous section show that charges on the islands are the relevant variables. Similarly to vortices in classical arrays, they interact logarithmically with each other and are expected to undergo a charge–BKT transition leading to insulating behavior [93]. A critical point separates the superconducting and insulating regime at T = 0. As discussed before, the various models give di?erent estimates for the value of the critical point, but in all cases it lies close to EJ =EC ∼ 1. The theoretical phase diagram in the limit of logarithmically interacting charges is shown in Fig. 7. For any T = 0 the array has three phases. Next to the superconducting and insulating phase, there is a region with normal conduction. Here, there are always some free vortices=charges present as they are generated by thermal or quantum Luctuations. Experimental veri8cation of this diagram has been reported in Refs. [97,98] In the remainder of this subsection, we discuss the experimental aspects of BKT transitions in arrays. The BKT transition in the classical case has been studied in great detail (see e.g. Refs. [2,3]). On approaching the critical point of the S–I transition, the vortex–BKT transition temperature
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is lowered by quantum Luctuations. For C0 = 0 the shift of the transition temperature is [76,99] 4 E0 EJ 1− : (26) TJ = 2 3 EJ A reduction of the transition temperature is generally observed in quantum arrays. The Delft data (shown in Fig. 4), however, show that with increasing EC =EJ ratio the reduction of the transition temperature goes faster than predicted above (Eq. (26)). On the charging side (EJ EC ) of the phase diagram, the charge–BKT transition occurs at a temperature [91] E2 EC − 0:31 J : EC Note that the charge–BKT transition exists for both arrays with superconducting and normal islands [100]. The existence of a charge–BKT transition implies that arrays are insulating below the transition temperature. The conductance should vanish with a characteristic square-root dependence according to Tch =
−1=2 G ∼ R−1 }; N exp{−2b[T=Tch − 1]
(27)
where the constant b ∼ 1. The temperature dependence of the conductance in the charge regime has been investigated by several groups [101–103]. Instead of the predicted square-root cusp behavior, an exponential (activated) temperature dependence G ∼ R−1 N exp{−Ea =T } ; has been observed with an activation energy Ea ∼ + 0:24EC : In arrays the screening length is about 102 lattice constants and therefore there is no a priori reason for observing such dramatic deviations from the BKT theory. Recently, Feigelman et al. [69] re-examined the problem and found that parity e?ects together with the screening of the Coulomb interaction due to thermally activated quasi-particles is responsible for masking the charge–BKT transition. At temperatures above a crossover temperature T ∗ where parity e?ects [104] set in, the presence of quasi-particles rules out the possibility to observe the charge–BKT transition. If the BKT transition temperature is larger than the crossover temperature, the array behaves as a normal one and a charge BKT transition occurs associated with the unbinding of quasi-particles at temperatures close to EC =4. The presence of free charges screens the interaction between Cooper pairs resulting in the unbinding of pairs. The resistance as calculated by Feigelman et al. [69] is expected to be 1=2 R(T ) b FP (T ) EC ln ∼ min ; ;√ R(EC =4) T − EC =4 4 T − EC =4 where FP (T ) is the free energy di?erence between islands with an even and an odd number of electrons [104]. From analogous considerations one may conclude [69] that normal arrays are better suited for studying the charge unbinding transition.
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2.3. Magnetic frustration When in the classical limit (EC EJ ) a perpendicular magnetic 8eld is applied, vortices enter the array above a certain threshold 8eld [64]. Just as in 8lms, the vortex density increases with increasing magnetic 8eld. In junction arrays the periodic lattice potential, however, prevents vortices to move at low temperatures. Only above the depinning current, vortices move (Lux-Low branch). The resistance in this branch increases approximately linearly with f = =0 up to f ∼ 0:2 ( is the magnetic Lux piercing through an elementary plaquette). A phenomenological model, analogous to the Bardeen–Stephen model used to describe Lux-Low in 8lms, is in good agreement with experiments providing that coupling between spin waves and vortices is taken into account (see Section 3.3). The properties of arrays at low frustration (low vortex densities) are dominated by single-vortex properties and are the subject of the next chapter. In larger magnetic 8elds commensurability e?ects come into play and the behavior of junction arrays is richer than that of 8lms. A magnetic 8eld applied perpendicularly to the array leads to frustration [106]. The presence of the magnetic 8eld induces vortices in the system and if the frustration is a rational number, f = p=q, the ground state consists of a checkerboard con8guration of vortices with a q × q elementary cell. The stability of the vortex lattice against a bias current leads to a decrease in the small-bias resistance at 8nite fractional 8llings. In order of their relative strength, one expects dips at f = 1=2; 1=3; 1=4; 2=5; : : : in square arrays and at f=1=2; 1=4; 1=3; 3=8; : : : in triangular arrays as is illustrated in Fig. 8. Near these fractional values of f, defects from the ordered lattice (excess single vortices or domain walls) are believed to determine the array dynamics in a similar way as the 8eld induced vortices determine array dynamics near f = 0. Therefore, arrays near commensurate values with high stability such as f = 1=2 may qualitatively behave in a similar way as near zero magnetic 8eld. Because all properties are periodic in f with period f = 1 an increase beyond f = 1=2 does not lead to new physics. A particularly interesting case is the fully frustrated situation (f = 1=2) in square arrays. The two degenerate ground states consist of a vortex lattice with a 2 × 2 elementary cell. The current corresponding to this vortex arrangement Lows either clockwise or anti-clockwise in each plaquette (chiral ground state). Interaction between domain wall excitations with 1=4 fractionally charged vortices (at the corners of a domain wall) and excess single integer vortices are believed to trigger a combined BKT–Ising transition. A fully frustrated array has two critical temperatures related to the Z2 and U (1) symmetries of the problem. Their existence has been investigated both by analytical methods and Monte Carlo simulations. Even at the classical level, the complete scenario is not fully understood yet. There is numerical evidence either supporting the existence of two very close critical temperatures with critical behavior typical of Ising and BKT transitions, respectively, or the existence of a single transition with novel critical behavior. Further reference to classical frustrated arrays can be found in Ref. [107]. At the mean-8eld level the full phase-diagram including charging e?ects and magnetic 8elds is obtained by solving an eigenvalue problem equivalent to the Hofstadter problem [108]. The resulting phase boundary as a function of vorticity shows commensurability e?ects. Although the superconducting transition temperature is reduced, the average con8guration of the phases and the supercurrent Low patterns are unchanged. The ground state is still chiral [109]. More detailed calculations based on expansion in EJ =E0 of the QPM [110] and on the BH model
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Fig. 8. Zero-bias resistance vs. magnetic frustration for a square (a) and a triangular (b) array. The dip at f = 1=2 is the most pronounced one in both 8gures. In the triangular array the dip at f = 1=4 is more pronounced than the one at f = 1=3. In the square array the opposite occurs. (From Ref. [51].)
[111] con8rm the butterLy-like behavior of the S–I transition. In Fig. 9 the theoretical results obtained by Kim et al. [110] are shown. As in the unfrustrated case, measurements indicate a S–I transition at T = 0. For the same set of samples as presented in Fig. 4 the S–I transition for f = 1=2 has been studied. The phase diagram is shown in Fig. 10. The transition takes place very close to a normal-state resistance of 11 kZ. The critical EC =EJ ratio is about 1.2, a factor 0.7 lower than the zero-8eld value. This decrease of is consistent with the simple model that involves a reduction of e?ective Josephson √ coupling energy at f = 1=2: the interaction energy of a vortex pair is a factor 2 smaller than in zero 8eld. With this √ lower e?ective coupling the critical value of EJ =EC of the S–I transition is reduced by a factor 2, which is close to the observed reduction of 0.7. The experimental data agree rather well with the quantum Monte Carlo calculations [112] in the classical limit. The experimental points of the transition temperatures are, however, lower than the calculated ones by entering in the quantum regime. At present, there is no explanation for this discrepancy. It would require a more detailed study and better understanding of the phase transition at f = 1=2. The calculations do indicate, on the other hand, a S–I transition at EJ =EC ≈ 1, in agreement with the experiment. In Fig. 11, the critical EC =EJ ratio as a function of applied magnetic 8eld for square arrays is plotted. The three points at f = 0; 1=2, and 1=3 are combined with two data points of the
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Fig. 9. Phase boundaries between the Mott insulating phase (below each curve) and the superconducting phase (above). Boundaries for various ratios of the junction capacitance C to the self-capacitance C0 are shown: C=C0 = 0:0001 ( ); 0:1 (◦); 0:2 (), and 1:0 (). (From Ref. [110].) Fig. 10. Measured phase diagram for square arrays at f = 1=2, showing the S–I transition at EC =EJ ∼ 1:2. In the 8gure the temperature axis is normalized to the Josephson coupling, i.e., = T=EJ . The solid and dashed lines are guides to the eye connecting the data points. (From Ref. [51].)
Fig. 11. Measured phase diagram for square arrays in a magnetic 8eld. Below the dotted line samples become superconducting at low temperatures; above this line samples become insulating. At non-commensurate magnetic 8elds, the S–I transition is not sharp and there is an additional (intermediate) metallic region not shown in the 8gure. (From Ref. [51].)
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8eld-tuned S–I transition (see Section 2.8). After a rapid decrease the critical ratio is almost constant for f ¿ 0:1. The critical EC =EJ ratio at f=1=2 is larger than at other nearby values of f, indicating once again the stability of the phase con8guration at f=1=2. Fig. 11 also indicates that arrays in the range 1:2 ¡ EC =EJ ¡ 1:7 do not show special behavior at commensurate f-values (e.g. dips in the magnetoresistance); arrays are superconducting in zero 8eld but insulating at f = 1=2; 1=3; 1=4; : : : : It is possible to derive a Ginzburg–Landau free energy also in the presence of rational frustration [113]. The calculation proceeds along the lines outlined for the f = 0 case in Section 2.2.2. The important di?erence is that one should expand the free energy about the most Luctuating modes. In the unfrustrated case this means an expansion about k =(0; 0). For the fully frustrated case (f = 1=2) the expansion is carried around the two points: k = (0; 0) and k = (0; ) thereby reLecting the superlattice structure of the ground state. The resulting free energy depends on a multicomponent (complex) order parameter (e.g. two coupled complex 8elds in the fully frustrated case). The magnetic-8eld dependence of the critical exponent zB (B governs the divergence of the correlation length and z is the dynamical critical exponent) was considered by Niemeyer et al. [111]. In zero 8eld the mapping onto a three-dimensional XY model implies that zB=0:67. Their analysis hints to a dynamical exponent that increases with the magnetic 8eld. It is, however, diUcult to draw conclusions on the values of zB. It could smoothly increase with magnetic 8eld, or immediately jump to one on once the magnetic 8eld has switched on. Combining the fact that zB ¡ 1 for f = 1=2 and that a higher-order expansion predicts zB = 1 there, the authors conclude that the answer is zB = 1 for all non-zero magnetic 8elds. Monte Carlo simulations by Cha and Girvin [114] obtain for the f = 1=2 and 1=3 cases the values z = 1 and 1=B = 1:5, consistent with the analysis outlined above. Finally, we mention that, in addition to two-dimensional arrays, frustration e?ects can be studied in quantum ladders as proposed in Ref. [115]. 2.4. Charge frustration and the supersolid A uniform charge can be introduced in a quantum JJA by applying a gate voltage Vx with respect to the ground plane. This e?ect is known as charge frustration. Although from a theoretical point of view charge and magnetic frustration are dual to each other, experimentally it is only possible to tune the magnetic frustration in a controlled way. In all arrays random o?set charges, presumably caused by defects in the junctions or in the substrate [116], are present. Electron or quasi-particle tunneling will partly compensate these o?set charges so that their value lies between −e=2 and +e=2. These charges can, in principle, be nulled out by the use of a gate for each island; this procedure, however, works only for small networks. In large arrays they cannot be compensated because too many gate electrodes would be necessary, requiring too complicated fabrication procedures. A uniform charge frustration has therefore not been realized yet in two-dimensional Josephson arrays. Lafarge et al. [117] have investigated charge frustration by placing a gate underneath a Josephson array. They managed to obtain a 40% variation of the resistance between the unfrustrated and the (nominally) fully frustrated array. But most importantly, in studying the current–voltage characteristics, it was impossible to quench the Coulomb blockade as it can be done in circuits with few junctions. In future
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Fig. 12. The T = 0 phase diagram in the limit of on-site interaction as a function of the charge frustration. At the values of qx for which two charge states are degenerate, the superconducting phase extends to arbitrary small Josephson coupling.
arrays, charge frustration may be applied more uniformly if the inLuence of o?set charges can be drastically lowered. 2.4.1. Phase diagram The energy di?erence of two charge states in each island with q and q + 1 extra Cooper pairs may be reduced by changing Vx (which means to change the external charge qx ). Consequently the e?ects of a 8nite charging energy are weakened and the superconducting region in the phase diagram is enlarged. It turns out that for certain values of the gate voltages the energy di?erence vanishes implying that the Mott gap, and therefore the insulating behavior, is completely frustrated. At the degeneracy point even a small Josephson coupling makes the system superconducting since there is no pay in energy for moving a charge through the whole array. In general it is intuitive to expect that the extension of the insulating lobe will be maximum at integer values of the external charge since in this case the excitation energy is highest. A quantitative analysis of this phenomenon has only been obtained in models with a shortrange electrostatic potential. Uniform charge frustration gives rise to two new e?ects: • lobe structures appear in the phase diagrams, • new states in the phase diagram may occur (Wigner-like crystals and the supersolid).
The remainder of this subsection is devoted to the lobe structures and the Wigner-like charge ordering; the next subsection treats the supersolid. The lobe-structure already follows from a mean-8eld analysis with on-site interaction only. The corresponding phase diagram can be obtained by evaluation of the correlator given in Eq. (16) in the presence of an uniform charge. In Fig. 12, the mean-8eld phase boundary in the presence of a uniform background charge is shown. The detailed structure of the lobes is very
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sensitive to the model used (QPM, BH, XXZ) and on the approximation made [40,118–123]. The lobes in the Monte Carlo calculation are sharper than predicted by mean-8eld, but smoother than expected from the strong coupling expansion. In the case of a 8nite-range interaction a number of new classical ground states exists characterized by a crystal like ordering of charges. The phase diagram contains extra lobes where the charge density is pinned to a given fractional 8lling. Their range of existence, in the limit of vanishing Josephson coupling, is found by minimizing the charging part of the Hamiltonian for a given charge 8lling. We illustrate this by considering the simple case of short range charging energy (only on-site and next-neighbors). In the case of 0¿qx ¿1=2 (and for square lattices) only three di?erent charge con8guration should be considered. • All the islands are neutral. • A checkerboard state can be formed in which a sublattice is neutral and the other is charged
with one extra Cooper pair.
• All the islands can be uniformly charged with charge 2e.
The corresponding energies of the di?erent ground states are, respectively, z (28) Ech; 00 =4N = E0 qx2 + E1 qx2 ; 2 1 1 z Ech; 01 =4N = E0 qx2 + E0 (1 − qx )2 − E1 qx (1 − qx ) ; (29) 2 2 2 z Ech; 11 =4N = E0 (1 − qx )2 + E1 (1 − qx )2 ; (30) 2 where N is the number of islands in the array. The ground state energy is given by Ech; 00 for 0¿qx ¿qx; 1 , Ech; 01 for qx; 1 ¿qx ¿qx; 2 and Ech; 00 for larger qx . The critical values at which the ground state changes are 1 1 qx; 1 = ; 2 1 + (z=2)E1 =E0 qx; 2 =
1 1 + zE1 =E0 : 2 1 + (z=2)E1 =E0
The S–I boundary (for 8nite EJ ) can be determined, for example, using a mean-8eld approach. The result is presented in Fig. 13. The checkerboard state for qx; 1 ¿qx ¿qx; 2 can be thought of as a Wigner crystal of Cooper pairs. The role of a longer-range charging energy (next-nearest neighbors, etc.) is to stabilize the crystalline phases with lower 8llings (1=4; 1=8; : : :). Not all the lobes extend down to the EJ =0 axis. First-order phase transitions between di?erent checkerboard states are then possible. Note, that the presence of charge ordering, characterized by a periodicity 2=kx ; 2=ky , is detected by studying the structure factor S at a given wave-vector 1 ik·ri S(kx ; ky ) = 4 e qi q0 : (31) L ij A uniform charge frustration changes the symmetry properties of the system. At qx = 0 the energy cost to create (or remove) a Cooper pair in a given island is the same. The system possesses particle–hole symmetry. For generic values of the external charge this symmetry is
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Fig. 13. The T = 0 phase diagram calculated with on-site interaction and a small nearest neighbor charging term E1 . Around qx = 1=2 the half-integer lobe appears. Inside the lobes, each number represents the number of Cooper pairs on a particular island. For example, the intermediate lobe, centered around qx = 1=2 has a checkerboard structure.
broken (i.e., Ech; 00 = Ech; 11 ). In the phase diagram shown in Fig. 13 the tips of the lobes correspond to a particle–hole symmetric case while away from the tips the symmetry is broken. This change of symmetry is reLected as a new term in the quadratic part of the Ginzburg– Landau free energy. This new contribution is d i∗ ()@ i () ; (32) i
where =i
@!n g(!n ; qx )|!n =0 : 2E0
(33)
The coeUcient vanishes in the particle–hole symmetric case (see Appendix C). The particle– hole symmetry has important consequences for the critical behavior of the system [40]. The dynamical critical exponent z changes from z = 1 at the tip of the lobes to z = 2 in the generic case. 2.4.2. Supersolid A solid phase is characterized by charges being pinned on the islands whereas a superLuid phase is characterized by phase coherence over the whole system (i.e., charges are delocalized). At the end of 1960s [124 –126,41] it was suggested that, in addition to the solid and superLuid phases, a new state should appear, characterized by the coexistence of o?-diagonal (superLuid) and diagonal (charge-crystalline) long-range order. This phase is known as the supersolid. If vacancies in a quantum crystal such as solid 4 He Bose–Einstein condense, they do not necessarily destroy the crystal structure and they may form a superLuid solid (or supersolid). Experiments have been performed on 4 He, but no positive identi8cation of this coexistence phase has yet been made. There are, however, hints that such a phase exists [127,128].
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Fig. 14. Mean-8eld phase diagram in the presence of charge frustration obtained in the hard-core limit. As discussed in the text the 8ctitious 8eld h is related to qx − 1=2 and the coupling J corresponds to EJ . In the 8gure nn and nnn charging terms are di?erent from zero E2 = 0:1E1 . The central lobe corresponds to the half-8lling case and the two small ones on the side are the quarter-8lling lobes. Finally SS1 and SS2 are di?erent types of supersolid. (From Ref. [42].)
An exciting possibility that attracted a lot of attention was the idea to observe supersolids in Josephson arrays [42,129 –133]. The supersolid phase is located in an intermediate region around the half-8lling lobe. A simple way to understand its existence is to focus on a region close to the phase boundary at qx ∼ 1=2. At densities corresponding to half-8lling the particles form an incompressible solid. Away from half-8lling vacancies in the charge-solid appear. As they have a bosonic character, they can Bose condense, and therefore they are able to move freely through the system. For a small enough density of vacancies one expects that the crystal order is not destroyed. In the limit of very large on-site charging (hard-core limit in which the island charge can only be zero or one) the existence of the supersolid is related to the 8nite next-nearest-neighbor interaction as it does not exist for nearest-neighbor interaction only. Furthermore, there is no supersolid phase at exactly half-8lling. In Fig. 14 the mean-8eld phase diagram in the hard-core limit is shown. The supersolid region appears in a tiny region away from half 8lling between the superconducting and Mott insulating phases. If higher values of charge are allowed, the supersolid phase already exists for nearest-neighbor interaction and also at half-8lling on the tip of the checkerboard lobe. This is related to excitations which are forbidden in the hard-core limit. A large nearest-neighbor interaction or small on-site interaction favors the supersolid, whereas in the hard-core limit the supersolid is suppressed. Thus, it seems that the system itself generates the defects (particle–hole excitations) that Bose condense, thereby turning the solid into the supersolid. The phase diagram for soft-core bosons is shown in Fig. 15.
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Fig. 15. Top: Mean-8eld phase diagram for soft-core bosons, as obtained from the analysis of the QPM with on-site and nearest-neighbor (E1 =E0 = 1=5) interaction. The symbols are the Monte Carlo data (the region between the dotted circles and the crossed circles is the supersolid). The checkerboard charge-density wave is denoted by “Sol”, the supersolid phase by “SSol”, the superLuid phase is denoted by “SF” and the Mott-insulating phase by “MI”. Bottom: Supersolid region “SSol” at qx = 0:5 as a function of U1 =U0 in the mean-8eld approximation of. Inset: Occupation-number probability |c2 |2 at qx =0:5 for the two sub-lattices A and B at the particular value of E1 =E0 =0:2. (The notation is slightly di?erent from that used in this review, U0 → E0 , U1 → E1 , J → EJ , n0 → qx .) (From Ref. [131].)
Since the supersolid phase is very sensitive to Luctuations, it was important to obtain independent checks of its existence. Monte Carlo simulations on the QPM [130,142] and the BH model [132] have con8rmed the qualitative picture discussed above. In Fig. 15 the symbols represent the phase diagram as obtained from Monte Carlo simulations by van Otterlo and Wagenblast [130,131]. Note that the supersolid region is considerably reduced as compared to the mean-8eld estimates.
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By changing the electrostatic interaction new phases named collinear supersolid were found by Scalettar et al. [132] and by Frey and Balents [133]. A detailed analysis of various supersolids including striped phases has recently been performed by Pich and Frey [134]. Supersolid phases in frustrated systems have been studied as well either by combining the e?ect of charge and magnetic frustration [135] or by considering arrays on KagomCe lattices [136]. Several other kinds of coexistence phases were studied. The possibility of a spontaneous vortex anti-vortex lattice in superLuid 8lms was explored in Refs. [137–139] and a coexistence phase of superLuidity and hexatic orientational order was proposed in Ref. [140]. Orientational order in incompressible quantum Hall Luids is discussed in Ref. [141]. Finally, we mention the relation between two-dimensional bosons and three-dimensional Lux-lines in type II superconductors (high-Tc materials) in a magnetic 8eld [143,144]. Also in these systems di?erent kinds of long-range order may coexist and the equivalent of the supersolid is discussed in Refs. [145,146]. Related is the question whether or not vortices may form a disentangled liquid, which would imply a normal ground state for bosons with long-range Coulomb interaction. 2.5. Dissipation induced S–I transition The behavior of a single Josephson junction with Ohmic dissipation has been discussed in a pioneering work by Schmid [147] who found that there is a zero-temperature phase transition governed by the dissipation strength ,. Above the critical value , = 1, dissipation suppresses quantum Luctuations thereby restoring the classical behavior with a 8nite critical current. For weak damping, on the other hand, quantum Luctuations destroy global phase coherence. The supercurrent is suppressed to zero and the junction is in the insulating state. Experimentally, this transition has only very recently been detected by Penttila et al. [148]. The reason is that for a single junction the high-frequency coupling to the environment determines the e?ective damping. Consequently, the e?ective impedance is of the order of 100 Z [149], which is about two order of magnitude smaller than the quantum resistance. Penttila et al. increased this impedance, i.e., they decoupled their single junction from its environment by placing high-ohmic, chromium resistors in the leads close by. Dissipation plays an important role in quantum phase transitions of JJAs as well. Originally the interest was stimulated by the idea that dissipation could be responsible for the observed critical resistance at the S–I transition in arrays and granular 8lms. Later Fisher [150] and Wen and Zee [151] pointed out that the observed critical resistance is a zero temperature property associated with the existence of a quantum phase transition and it is not related to the presence of an “extrinsic” source of dissipation. The next section discusses the critical behavior on transport properties in more details. Here, we discuss the inLuence of dissipation on the phase diagram. The coupling to a dissipative bath has the e?ect to suppress the quantum Luctuations of the phase, i.e. to quench the insulating region. The properties of the environment and the type of dissipation are important ingredients in Eqs. (10) and (11). As stated before, various sources of dissipation should be considered for JJAs. Quasi-particles may still play a role at mK as they may be generated by the environment or the motion of vortices themselves. From a theoretical point of view and in view of the recent experimental advances, it is also possible to realize arrays in which Ohmic shunts are important. These shunts can be normal wires placed parallel
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to the junctions in a similar way as was done for a single junction. Ohmic shunts may also arise because of coupling to the substrate (local damping). Very recently the group of Kobayashi succeeded in fabricating a JJA in which each junction is shunted by a Cr resistor [152]. A di?erent and controlled environment was investigated in the experiments of Rimberg et al. [153] by placing a two-dimensional electron gas underneath the Josephson array. We brieLy discuss all these sources of dissipation. 2.5.1. Quasi-particle dissipation When the mechanism responsible for dissipation is quasi-particle Tunneling, the e?ective action is that given in Eq. (10). Theoretically this model was studied by means of mean-8eld calculation [154,155], variational approaches [156 –159], and Monte Carlo simulations [160]. The dominant e?ect coming from quasi-particle tunneling enters in the renormalization of the e?ective junction capacitance given in Eq. (9). In the mean-8eld calculation, this amounts to a modi8cation of the capacitance matrix in the evaluation of the phase correlator. The zero-temperature phase boundary (for short range Coulomb interaction) obtained by Chakravarty et al. [154], is given by the expression 1=
3 2 1 , : 4 ln(1 + 3=4(E0 =EJ ),2 )
It is important to stress that in the case of quasi-particle tunneling the array is either insulating or superconducting. The interplay between the long-range Coulomb interaction and quasi-particle dissipation has been discussed in Ref. [91]. In Section 2.2, we have interpreted the S–I transition as being driven solely by Coulomb interactions. However, given the uncertainty in the damping resistance (e.g. sub-gap resistance of normal-state resistance) of the junctions the possibility that the transition is driven by quasi-particle dissipation cannot be ruled out. The data do not exclude the possibility that the S–I transition is inLuenced by the normal-state resistance. In fact, the Chalmers group [161] and the group of Kobayashi and collaborators [60] have interpreted their data in terms of a Schmid-like diagram. In Fig. 16 we show the results from the Chalmers group. The normal-state resistance RN is used as the resistance determining the dissipation parameter. If this resistance is used, a reasonable agreement with the theoretical models is obtained. 2.5.2. Ohmic dissipation The inLuence of Ohmic shunts on the phase diagram has been intensively investigated as well. A new phase with local phase coherence is possible, i.e., phase coherence only exists as a function of time. Various theoretical methods have been applied in this case such as coarse graining [162–164], variational [163,165 –168] and renormalization group [163,166,169] approaches. The general trend is, as expected, that the critical value of EJ =E0 for the onset of phase coherence is lowered. The dependence on the dissipation strength is stronger as compared with the case of quasi-particle damping. As for a single junction, a true dissipative transition occurs. A rigorous analysis can be performed in various limits in the EJ − , phase diagram [163,166]. For simplicity we discuss only the T = 0 case and follow the discussion presented in Ref. [170].
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Fig. 16. Measured low-temperature phase diagram for 15 square arrays plotted as the EJ =EC -ratio vs. RQ =RN . Samples ] 1–8 show a decreasing resistance as temperature is lowered (superconducting arrays). Samples ] 9–15 show an increasing resistance as temperature is lowered (insulating arrays). This classi8cation roughly agrees with the theoretical prediction [91] which states that insulating arrays are to be found in the region bounded by the dotted line. Only the insulating samples ] 9–11 fall slightly outside this area. The diagonal line represents 0 = 2EC and the dashed line corresponds to a Stewart McCumber parameter of +c = 1. (From Ref. [161].)
• In the large , limit time-like Luctuations of the phase are strongly suppressed and they only
contribute to the renormalization of the e?ective Josephson coupling. The system behaves like a classical JJA with an e?ective Josephson coupling , 1 EJe? = EJ 1 − : ln 1 + ,z 2
At zero temperature the array is in the superconducting phase independent on the ratio EJ =E0 .
• In the case EJ =E0 1 there is a phase transition at
1 ; z which separates two phases that both exhibit long-range coherence. Evidence of such a phase transition could be detected by measuring the voltage noise power spectrum. • In the limit of small damping, the critical ratio of EJ =E0 is renormalized to smaller values indicating that dissipation enhances the superLuid phase. • If the ratio EJ =E0 is very small, a dissipative transition to a phase with local order can take place at a critical value of dissipation given by 2 ,= : z ,=
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Very recently, Takahide et al. [152] fabricated a JJA in which each junction was shunted by a Cr resistor. By varying the resistance of the shunts and the ratio EJ =E0 they were able to map out the phase diagram. The results are in good agreement with the theories of dissipation induced quantum phase transition discussed above. 2.5.3. Local damping Dissipation may also arise from coupling with the substrate by means of what is named as the ‘local damping’ model. Local damping changes the universality class of the S–I transition [174], and inLuences the low-frequency dispersion of the vortex response in classical arrays [172,173]. As shown in Eq. (12), the local damping model correlates the island phase in time. In proximity-coupled arrays, which consist of superconducting islands on top of a metallic 8lm, the model with local damping is appropriate to describe the Low of normal electrons into the substrate. Aluminum tunnel junction arrays are always placed on insulating substrates so that it is not appropriate. Dissipation due to local damping is associated with the phase i , rather with the phase di?erence i − j as in the resistively shunted junctions (RSJ) model. The number of Cooper pairs in each island is allowed to decay, whereas the RSJ model describes only charge transfer between neighboring islands. By going through the same steps outlined in the section on the coarse-graining method, it is possible to obtain also in this case an e?ective Ginzburg–Landau free energy. The only di?erence is that now the phase–phase correlator g() has to be evaluated including the local damping term. For small frequencies the Fourier transform reads (for more details see Ref. [174]) g(!& ) = g(0) − H|!& |s − :!&2
with s =
2 −1 : ,
(34)
The coeUcients H and : can be determined from the phase correlator (their value is not important for our purposes). Using this expression for g(!& ), the free energy in Eq. (20) contains a non-ohmic dissipative term (˙ |!& |s ) (reducing to ohmic, or “velocity proportional” damping only in the special case s = 1). This means that ohmic damping in the quantum phase model yields non-ohmic dynamics for the coarse-grained order-parameter. The phase boundary in the saddle point approximation is shown in the inset of Fig. 17. By increasing damping strength, the superconducting region is enlarged. At T = 0 a quantum phase transition is ruled out beyond the critical value , = 2. 2.5.4. Tunable dissipative environment A controlled study of the dissipative S–I transition has been performed by Rimberg et al. [153]. They placed a Josephson array on top of a two-dimensional electron gas (2DEG). Junction parameters are chosen such that in the absence of the 2DEG the array is insulating. The array is capacitively coupled to the electron gas and its screening currents provide a source for dissipation. By tuning the back-gate voltage, the electron density and the sheet resistance Rg of the 2DEG are varied without changing the array parameters. As the resistance of the 2DEG increases the current–voltage characteristics of the array change from superconducting to insulating with a Coulomb gap as illustrated in Fig. 18. Moreover the resistance of the array is
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Fig. 17. The phase diagram at T = 0 in the local damping model. Along the solid line the conductivity is universal, whereas it is a function of the dissipation strength along the dotted line. (From Ref. [171].)
Fig. 18. Measured array I –V characteristics for eight values of the back-gate voltage corresponding to di?erent ground plane resistances Rg . The I –V ’s change from superconducting-like to insulator-like as a function of Rg . (From Ref. [153].) Fig. 19. Measured zero-bias array resistance R0 as a function of the ground plane resistance Rg . The inset shows the temperature dependence of the resistance. (From Ref. [153].)
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Fig. 20. Calculated phase diagram of an array coupled capacitively to a 2DEG. The insets show the resistance as a function of the temperature in the di?erent regions of the phase diagram. (From Ref. [174].)
very sensitive on Rg as shown in Fig. 19. Note that in the experiment the island capacitance (to the 2DEG) exceeds the estimated junction capacitance of 0.5 fF by a factor of 6. Wagenblast et al. [171] analyzed these measurement and modeled the experimental setup by capacitively coupling the array to the 2DEG. Assuming ohmic dynamics of the 2DEG they obtained the following (Caldeira–Leggett like) e?ective action for the array: 1 Se? [’] = dk D0−1 (k; !& )|’k; !& |2 + SJ (35) 2! &
(SJ is the action related to the Josephson coupling) with the propagator D0−1 (k; !& ) =
k 2 !&2 C 2 2 C0 k ! + ; & 4e2 4e2 k 2 + |!& |=K0
(36)
where C0 now represents the capacitive coupling to the 2DEG and where 1=K0 = Rg C0 . The e?ective action for the array is Ohmic only in an intermediate frequency range. At the lowest and highest frequencies the dynamics is capacitive. The two energy scales are well separated in the case C0 C and a quantum phase transition is driven by the action at the lowest frequencies. As the dissipative action is cut o? at the lowest frequencies, a dissipation driven transition cannot occur in the strict sense. However, quasi-critical behavior can be observed at temperatures and voltages exceeding the low energy scale K0 . In the limit K0 → 0 (C0 → ∞) this behavior converges to a true dissipation-tuned transition. An analysis of the conductivity as a function of EJ and , suggests a phase diagram of the type represented in Fig. 20. The insets show R0 (T ) as a function of the temperature in di?erent regions. The experiments of Rimberg et al. belong to the right-lower sector of the Schmid diagram. The theoretical temperature dependence of the resistivity R0 (T ) as well as the exponential relation between R0 and Rg are in good agreement with the experiments [153].
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2.6. Transport properties The unique feature of quantum critical points is that quantum Luctuations, which drive the system through the transition, govern its dynamical behavior. The interest in understanding charge transport near a quantum phase transition goes beyond the study of JJAs. Important examples are the transition in quantized Hall systems [12], localization in Si-MOSFETs [175] and the quantum critical point in cuprates [176]. In two dimensions right at the S–I transition the zero-temperature conductance [177] has been predicted to be 8nite and universal. This metallic behavior is present even in the absence of extrinsic dissipation and it is entirely due to the presence of collective modes which become soft at the zero-temperature transition point. Universality at a quantum phase transition then implies that the properties of the system are governed by a set of critical exponents. Wen [178] employed a scaling theory of conserved currents at anisotropic critical points identifying universal amplitudes. One of these amplitudes in two dimensions reduces to the universal conductance. A very simple argument [179] leading to a 8nite and universal conductance at zero temperature can be discussed using the duality between charges and vortices formulated in Section 2.2. Strictly speaking it applies to the case C0 = 0, i.e. for logarithmic interacting charges. From the Josephson relation the voltage drop across an array is given by the rate of vortices crossing the sample boundary. The current is given by the number of Cooper pairs which Low through the system per unit time, i.e., V=
h v˙ ; 2e
I = 2e q˙ :
At the self-dual point v˙ = q˙ and therefore the conductance at the transition (denoted with L∗ ) is 8nite, universal and corresponds to the quantum of resistance for Cooper pairs: L∗ =
4e2 : h
The value of L∗ changes in the case of short-range charging and=or in the presence of disorder. Nevertheless it remains universal (independent on the sample parameters). A large amount of theoretical work has been devoted to the determination of the critical value ∗ L and the scaling behavior of the conductance. The universal conductance in a model with no disorder was considered in Ref. [180] by means of a 1=N expansion and Monte Carlo simulations and in Ref. [181] by means of an 8-expansion. The dirty boson system and the transition to the Bose glass phase (including the case of long-range Coulomb interaction) was extensively studied by Monte Carlo simulations [182–185] and Lanczos diagonalization [186] as well as by using analytic calculations [187]. The 8nite-frequency properties close to the transition point were analyzed in Refs. [188–190]. More recently, in a series of papers, Sachdev and coworkers [191–193] studied non-zero temperature transport properties by means of a Boltzmann equation. A general analysis of the conductivity close to the transition can be performed based on scaling arguments. The frequency dependence of the conductivity L(!) has been obtained from its relation with the frequency dependent sti?ness Ms (!) (related to the increase of the free energy due to a time dependent twist) L(!) = 4e2 Ms (−i!)=i!. Close to the transition it can be
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shown that the conductance obeys the scaling form [150,180,194] h ! L(; T; !) = 2 f ; (37) ; 4e T T zB where measures the distance from the critical point, and where f(x; y) is a dimensionless scaling function. In the limit of very low temperature (compared to the frequency), the temperature drops out of the previous expression and the two scaling variables enter in the form h ˜ L(; T = 0; !) = 2 f : (38) 4e !zB In view of the scaling behavior of the conductance, one should consider two limits !T and !T . This point, emphasized in Refs. [191–193], is important both from a conceptual point of view and for a detailed comparison with experiments. The two situations correspond to two di?erent experimental setups for transport measurements. In the !T case, charge transport is governed by inelastic scattering between thermally excited carriers. In the opposite !T situation, collision between carriers can be neglected. In Eq. (37) the two limiting cases correspond to f(0; 0) and f(∞; 0) respectively. It turns out that both values are universal but di?erent. Here, we discuss the most prominent features of the conductivity close to the S–I transition by means of the Ginzburg–Landau free energy of Eq. (20). Connections to other models will be given. In the linear response regime the conductivity follows from the functional derivatives of the partition function. In the presence of an electromagnetic potential A, the gradient term in the Ginzburg–Landau free energy enter in a gauge-invariant form 2 ∇→∇− A: 0 By noticing that the current is the derivative of the free energy with respect to the vector potential and that the electric 8eld is the time derivative of the vector potential (with a negative sign), the conductivity, in imaginary time, is expressed as (a(b) = x; y) 1 2 ln Z 2 Lab (!& ) = d r d ei!& : (39) !& Aa (r; )Ab (0) A=0 Using Eq. (20), the longitudinal conductivity Laa = L can be expressed in terms of two- and four-point Green’s functions. In the absence of charge and magnetic frustration and by evaluating the correlators in the Gaussian approximation, one obtains [180] 2 1 L(!& ) = d k k 3 G(k; !B )[G(k; !B ) − G(k; !B + !& )] ; (40) RQ !& + B where G(k; !& ) =
8+
1 : + :!&2
k2
This turns out to be the 8rst term in a 1=N expansion [180]. The sum over the Matsubara frequencies can be performed by contour integration. The result is ∞ ∞ 1 dz L(!) = d kk 3 IG R (k; z)[RG R (k; z) − RG R (k; z + !)] ; (41) 4RQ ! −∞ 1 − e−+z 0
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where the retarded G R and advanced G A Green’s functions are given by G R=A (k; !) = G(k; !B → ! ± iH) : The previous expression can be evaluated in various important limits. 2.6.1. Zero-temperature conductivity Performing the k-integral the real and imaginary parts of the conductivity are !c2 RL(!) = 1 − 2 >(!2 − !c2 ) ; 8RQ ! ! − !c 1 2!c !c2 ; − IL(!) = + 1 − 2 ln 8RQ ! ! ! + !c where
!c = 2
(42) (43)
8 : :
At low frequencies the real part of the conductivity exhibits an excitation gap equal to !c . In the insulating region the system, as can be deduced from the behavior of the imaginary part of the conductivity, behaves as an e?ective capacitor with 1 Ce? = : 6RQ !c The previous expressions can be calculated in the lowest order in 1=N and obey the scaling law with zB = 1. The threshold frequency vanishes at the S–I transition leading to a 8nite d.c. (! → 0) conductivity, 4e2 : (44) 8 h As explained in the 8rst part of this section, this corresponds to the evaluation of the scaling function for !=T → ∞ (the collision-free regime). Corrections to the next order in the 1=N expansion correct this Gaussian result by roughly 30%. Another powerful method for evaluating critical quantities is the 8-expansion. In order to set up the 8-expansion one should move away from two dimensions and consider systems with d − 1 spatial dimensions. This approach allows one to obtain the scaling form of the frequency dependent conductance [181] in d dimensions. In the fully frustrated case (f=1=2) the conductance at the S–I transition is still 8nite but with a value which is di?erent from the f = 0 case. It is possible to evaluate it in a 1=N -expansion [195] and to the lowest order the conductance is twice the value of the critical conductance in zero 8eld. L? =
L? (f = 1=2) = 2L? (f = 0) : This factor of two is reminiscent of the superlattice structure at full frustration. There are, however, no fundamental reasons why this ratio should hold in general.
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2.6.2. Finite-temperature conductivity At low temperatures (T !c ), the real part of the conductivity is given by [188,191] 2 −+!c !c2 RL(!) = Te (!) + 1 − 2 >(!2 − !c2 ) 1 + 2e−+|!|=2 : (45) RQ 8RQ ! The imaginary part is obtained by means of Kramers–Kronig dispersion relations 2 IL(!) = (46) T e−+!c + !Ce? ; RQ ! where 1 T 2 −+!c Ce? = 1 + 24 2 e : (47) 6!c RQ !c The Drude peak in the real part of the conductivity arises due to a lack of dissipation or disorder in this model. Once electron or hole like excitations are created, they will propagate without damping thus leading to perfect conductivity. Although the system is a perfect conductor it is not a superconductor since it shows no Meissner e?ect. The response of the system to a static k dependent magnetic 8eld, is proportional to k 2 , i.e. it vanishes at long wavelengths. The scale for the crossover to the classical behavior is set by T ∼ !c . In the high-temperature limit (T !c ), the real and imaginary part of the conductivity read !c2 MD T RL(!) = (!) + 1 − 2 >(!c2 − !2 ) ; (48) RQ 2RQ |!| ! MD T IL(!) = !; (49) + RQ ! 4RQ !c2 where MD ∼ T and the expression for the imaginary part is valid at frequencies much smaller than !c . Damle and Sachdev [191,192] pointed out that since the conductance is a universal function of !=T , it makes a di?erence which limit is taken 8rst (either ! → 0 or T → 0). In order to study the collision-dominated regime they used a Boltzmann like approach in which the current is expressed in terms of distribution functions for the particle and hole-like excitations. By solving the appropriate Boltzmann equation (the collision term can be obtained by Fermi golden rule) they showed that also in the collision dominated regime the conductivity is a universal function at the critical point. We refer to the book by Sachdev for a clear and comprehensive presentation of these aspects of transport close to quantum critical points [13]. 2.6.3. Non-universal behavior Despite the conceptual elegance of the theories predicting a universal conductance at the transition, the experiments on JJAs and two-dimensional superconducting 8lms show critical resistivities that are di?erent (by a factor up to 10) as compared to the predicted universal values. Wagenblast et al. [174] developed a theory of this non-universal behavior using the local-damping model. The evaluation of the dynamical conductivity proceeds along the same lines discussed before (see Eq. (34)). The advanced and retarded Greens functions are given by s s [G A=R (k; !)]−1 = 8 + k 2 − :!2 + H|!|s cos ± isign(!) sin ; (50) 2 2
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Fig. 21. A scanning-electron-microscope (SEM) image of a SQUID chain. (From Ref. [197].)
With increasing damping the Mott gap is smeared out. For s ¡ 2 and low frequencies !!c one 8nds 1 H2 sin2 ((=2)s) [O(1 + s)]2 2s |!| : Re L(!) = RQ 682 O(2 + 2s)
(51)
The conductivity shows a power-law behavior at low frequency, where the power depends on the dissipation strength for s62. Of particular interest is the DC conductivity at the transition, which becomes a function of the strength of Ohmic damping for , ¿ 2=3. This model with local damping was further explored by Dalidovich and Phillips [196] in the case s = 1. In the limit of weak damping they 8nd that dissipation leads to a leveling o? of the DC conductivity at intermediate temperatures. Their estimates indicate resistance plateaus of the order of 10 kZ in the mK range, compatible with the experiments. These results seem more applicable to uniform 8lms rather than to JJAs. In any case they o?er an interesting explanation for the experimental observation that the critical resistance is not universal. 2.7. One-dimensional arrays Josephson-junction chains have been much less investigated (both theoretically and experimentally) as compared to two-dimensional systems and only recently the S–I transition in one-dimensional samples has been measured [197]. In addition to the possibility to fabricate arrays with controlled couplings, in Josephson chains the ratio of the Josephson to the charging energy can be varied in situ by connecting mesoscopic SQUIDS in series (as illustrated in Fig. 21). In this setup, the sample behaves as a chain of junctions with a tunable Josephson coupling EJ () = 2EJ cos(=0 ) depending on the magnetic Lux piercing the SQUID. By
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varying , it is possible to sweep through the S–I transition while measuring on the same sample. At zero temperature and for short range Coulomb interaction the S–I transition of a d-dimensional array is of the same universality class as a classical XY model in d + 1 dimensions. Therefore, Josephson chains should exhibit a BKT-like transition. By means of duality transformation it is possible to map the XY model onto a gas of logarithmic interacting vortices [198]. Vortices are bound in pairs (of opposite vorticity) below the transition temperature and are in a plasma phase in the disordered (high-temperature phase). In a quantum chain the relevant topological excitations, which correspond to vortices in space-time, are (quantum) phase slips. The mapping of a Josephson chain onto a gas of interacting phase slips has been performed by Bradley and Doniach [199]. Consider, for simplicity, only the charging part of the Hamiltonian and neglect the contribution due to the junction capacitance. + + Zch = D i ()Dqi () exp − d 4E0 qi2 + i d qi ˙ i : (52) 0
i;
i
i
0
The summation over the winding numbers 8xes the charges to be integers in units of 2e. By discretizing the path integral (with a time slice 8 and performing the summation over the integers qi the charging contribution to the partition function can be recasted into the form Zch = d i; exp[ − (1=88 E0 ) ( i; − i; +8 − 2ni; )2 ] : (53) i;
i;
[n]
Eq. (53) is the Villain approximation of the XY potential [86] if one identi8es i; − i; +8 as the dynamical variable and 1=8 E0 as the e?ective coupling. The time √ slice 8 can then be chosen such that the coupling in space and time is isotropic (8 ∼ 1= 8EJ E0 ) [200]. The XY model in space–time has a reduced coupling proportional to the ratio EJ =8E0 . Therefore, all the known results for the classical XY model directly apply with the following replacement EJ EJ → : (54) T 8E0 The identi8cation of the charging energy with the e?ective temperature shows the analogy between classical (thermal) and quantum (induced by charging e?ects) Luctuations. The partition function can now be expressed in terms of interacting phase slips (in the same fashion as in the classical where it is expressed in terms of vortices): 2 2 EJ Z= exp − p(k; !)G0 (k; !)p(−k; −!) ; (55) N N E x 0 p k;!
where p ± 1 are the “charges” associated with the occurrence of a phase slip. The function G0 (k; !) ∼ (k 2 + !2 )−1 ;
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Fig. 22. Calculated phase diagram of a Josephson chain shown as a function of the Josephson coupling and the dissipation strength. The dissipative part of the action leads to two new phases. For , ¿ 1=2 the dipoles are bound in a gas of quadrupoles and moreover for strong dissipation there is an additional phase transition which separate the quadrupole phase from a phase in which the system shows local order. (From Ref. [204].)
implies that phase slips interact logarithmically in space–time. The chain undergoes a BKT phase transition at a critical value EJ =E0 ∼ 2= : When a Josephson coupling is larger than the critical value the phase correlator decays algebraically (quasi-long-range order) and the chain is superconducting. Phase slips are bound in pairs of opposite sign and therefore they do not lead to any dissipation over a macroscopic region (the Josephson relation implies that the occurrence of phase slip leads to a voltage drop). In the opposite regime the chain is in the insulating phase. Phase slips are not paired and any current leads to a voltage. The correlation length is then given by b ? ∼ exp − " 1 − [2 EJ =16E0 ]1=2 (b ∼ 1). Due to the isotropy in the space–time direction, one can now de8ne an e?ective Coulomb gap ∼E0 ?−1 . As long as there is particle–hole symmetry a 8nite range Coulomb interaction does not change the universality class of the transition. A detailed analysis of the phase diagram, for realistic Coulomb interactions, as a function of the charge frustration has been performed by Odintsov [201]. The presence of dissipation (see also Section 2.5) modi8es the critical behavior of the chain. The case of a Josephson chain with ohmic dissipation has been considered by several authors [202–205] by means of dual transformations and Monte Carlo simulations. The main conclusions of this series of works is the zero-temperature-phase diagram as a function of dissipation strength and Josephson coupling as shown in Fig. 22. In addition to the S–I phase boundary there are two new phases induced entirely by dissipation:
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Fig. 23. Resistance vs. temperature for eight values of the magnetic 8eld in the range 0 –64 G. The two sets of curves correspond to two di?erent one-dimensional arrays with N = 255 (solid lines) and N = 63 (dashed lines) junctions. The longer array shows a sharper S–I transition. At the point J ? (J = EJ =EC ) the resistance is length independent. (From Ref. [197].)
• for small Josephson coupling and large dissipation the chain is in a phase with local order.
The phase di?erence at each junction is locked in time but the chain has no quasi long-range order, • for large dissipation and large Josephson coupling there is a new type of superconducting phase characterized by the phase slips bound in quadrupoles. The four di?erent phases can be measured by considering di?erent setups as discussed in Ref. [205]. We conclude this section by reviewing the experiments of Chow et al. [197]. The dependence of the resistance on the temperature, shown in Fig. 23, shows a non-trivial scaling behavior. The two set of curves (solid and dashed) refer to two chains of di?erent length. While in the insulating phase the resistance increases with the number of junctions, in the superconducting phase the opposite trend is visible. By identifying the scale independent value of R0 (T ), Chow et al. were able to trace out the zero-temperature critical point (indicated with J ∗ in the 8gure).
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Fig. 24. The magnetic 8eld dependence of the threshold voltage for one-dimensional arrays of di?erent length. The inset shows the corresponding I –V curves. (From Ref. [197].)
The resistance in the superconducting chains can be explained in terms of formation of phase slips. The Lat tails in the curves are due to a 8nite-size e?ect and occur for temperatures of the order of the e?ective Coulomb gap. In this region the probability of a phase slip event obtained √ in the Coulomb gas picture presented above scales with the number of junctions as 2−
EJ =8E0
Nx . Quantum phase Luctuations are suppressed by increasing the system size. In the insulating regime the I –V curves show Coulomb blockade with a threshold voltage which depends on the magnetic Lux piercing the SQUID [206] as shown in Fig. 24. Thus, there is reasonable agreement with the theory, but 8nite-size e?ects make a quantitative analysis diUcult because of the rapidly diverging correlation lengths.
2.7.1. 1D arrays as Luttinger liquids The interest in one-dimensional arrays goes further as they can be described in terms of the Luttinger liquid (LL) model [207]. The low energy excitations of the interacting electron gas in one dimension are long-wavelength spin and charge oscillations, rather than fermionic quasi-particle excitations. Accordingly, the transport properties cannot be described in terms of the conventional Fermi-liquid approach. The density of states shows asymptotic power-law behavior at low energies. Depending on the sign of the interaction an arbitrarily weak barrier in a quantum wire leads to perfectly reLecting (for repulsive interactions) or transmitting behavior at low voltages [208]. It is customary to characterize this interaction by a parameter g such that g = 1 in the non-interacting situation while g ¿ (¡)1 in the attractive (repulsive) case. A Josephson chain seems an ideal system to explore LL correlations [209]. In the limit of large Josephson coupling g = EJ =8E0 1; i.e., the chain behaves as an attractive LL. Glazman
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and Larkin [210] showed that in a certain region of parameters (close to qx = 1=2) between the Mott lobe and the superconducting region, there is a new intermediate phase which is equivalent to the chain behaving as a repulsive LL. In order to characterize this repulsive behavior one should consider a Josephson chain with a defect. One of the junctions could for example be made with a Josephson coupling much smaller than the charging energy [210,209]. The di?erent phases in the phase diagram can be characterized by the dependence of the Josephson current on the chain length. In the superconducting phase, the defect has no e?ect as the number of junctions increases. On the contrary, in the repulsive LL phase there is a strong dependence on the number of junctions. The LL phase can be studied also by means of Andreev tunneling spectroscopy along the lines discussed in Ref. [211]. Repulsive LL behavior is also present in a Josephson ladder as discussed in Ref. [212]. The possibility of repulsive LL behavior is related to a normal phase of interacting bosons at zero temperature. In one-dimensional systems such a possibility cannot be excluded and Monte Carlo simulations on a Josephson chain [213] show a phase in which there is neither crystalline nor superLuid order. The existence of a normal phase has been questioned in Ref. [214] through density matrix renormalization group of the BH model. One should note however that phase boundaries are non-universal and therefore the QPM and BH system can lead to di?erent results. 2.8. Field-tuned transitions In arrays which are in the superconducting state at f = 0 but have an EJ =EC ratio close to the critical value, a magnetic 8eld can be used to drive the array into the insulating state. This 8eld-tuned transition has been considered theoretically by Fisher [215] in disordered systems and has 8rst been observed by Hebard and Palaanen [216,217] in thin InOx 8lms. The interplay between disorder and vortex–vortex interactions plays an essential role. At low magnetic 8elds vortices at T =0 are pinned (by disorder) but for higher 8elds, the vortex density increases and at some critical density, vortices Bose-condense (a vortex superLuid leads to an in8nite resistance). By employing duality arguments this transition can also be thought as a Bose condensation of vortices that occurs by changing the applied magnetic 8eld. The general characteristic of the 8eld-tuned S–I transition is that when f is increased from zero, the temperature derivative of the resistance changes sign at critical values ±fc . Fisher’s analysis [215] leads to the following scaling for the resistivity tensor close to the 8eld tuned transition h f − fc M,; + = 2 M˜ ,; + ; (56) 4e T 1=zB where B is the exponent which controls the divergence of the correlation length at the transition and z is the dynamical critical exponent (with zB¿1). The resistivities are predicted to be universal at the transition and should satisfy the relation "
M2x; x + M2x; y =
h : 4e2
(57)
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283
Fig. 25. Temperature dependence of the resistance for a square (a) and triangular (b) two-dimensional Josephson array measured at di?erent applied magnetic 8elds. The 8eld tuned S–I transition occurs at that frustration where the temperature dependence of the resistance changes sign. In both cases this change occurs at f = 0:10– 0.15. (From Ref. [51].)
These predictions were tested in Josephson arrays by the Delft [51,218] and the Chalmers groups [52]. For several values of the frustration, the resistance as a function of temperature is shown in Fig. 25. Below a critical value fc , the resistance decreases upon cooling down (dR0 =dT ¿ 0). Above fc the resistance increases (dR0 =dT ¡ 0) and for low temperatures reaches a value that might be orders of magnitudes higher than the normal-state resistance. This sign change in the temperature dependence corresponds to a change in the I –V characteristics shown in Fig. 26. For f ¡ fc , a critical current is observed in the I –V characteristics, whereas above fc a charging gap develops. Note that at low temperatures, the resistance Lattens o?. This is most likely a 8nite size e?ect involving quantum tunneling of vortices. Finite-size e?ects are expected to play a more prominent role in JJAs as compared to 8lms because arrays are typically 100 cells wide. In units of the coherence length, 8lms are much larger.
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Fig. 26. I –V characteristics measured at low temperature (10 mK) for three values of the applied 8eld. The square two dimensional array has an EC =EJ ratio of 1.25. The crossover from the superconducting to the insulating behavior is related to the 8eld tuned S–I transition. (From Ref. [51].) Fig. 27. Field dependence of the resistance of a triangular two-dimensional array measured at di?erent temperatures, T = 50 mK (solid line), T = 120 mK (dotted line), and T = 160 mK (dashed line). The 8eld-tuned transition is observed around di?erent fractional values of the frustration indicated by the open circles. (From Ref. [51].)
A more detailed way of observing the 8eld-tuned S–I transition is obtained by measuring the resistance versus magnetic 8eld for di?erent temperatures (see Fig. 27). In the range 0 ¡ f ¡ 1=3, the R(f) curves are very similar to the ones measured in thin 8lms. Below the critical 8eld fc = 0:14 the resistance becomes smaller when the temperature is lowered and above fc the resistance increases. From the scaling analysis [215], it follows that the slopes of the R(f) curves at fc should follow a power-law dependence on T with power −1=(zB). When on a double logarithmic plot the slopes of the R(f) curves at fc are plotted versus 1=T , one 8nds straight lines in the temperature range 50 ¡ T ¡ 500 mK. From the reciprocal of the slope, the product zB can be determined. Values in JJAs range from 1.2 to 2 for the Delft data and from 1.5 to 8.2 for the Chalmers data, in agreement with the theoretical expectations z = 1 and B¿1. The scaling resulting from Eq. (56) is best seen by plotting the resistance as a function (f − fc )=T 1=zB as illustrated in Fig. 28: A universal function is obtained by plotting the resistance as a function of EJ (f − fc )=(EC T 1=zB ). The tails on the superconducting side (bottom curve) correspond to the 8nite-size e?ect mentioned above. The exponent z can also be obtained from the measurements by plotting fc as a function of the zero-8eld BKT transition temperature: fc ˙ TJ2=z :
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285
Fig. 28. Resistance as a function of the scaling parameter |f − fc |T −1=zB BB for two di?erent arrays (data taken in the range 0 ¡ f ¡ 0:2). The data collapse onto a single curve: the upper part for the insulating transition and the lower part for the superconducting transition. The inset in the 8gure on the left shows the scaling for the array close to full frustration (data taken in the range 0:5 ¡ f ¡ 0:6). The inset in the 8gure on the right shows a log–log plot of the critical frustrations fc as a function of the BKT transition temperature for the four measured samples. The line through the data yields a critical exponent z = 1:05. (From Ref. [52].)
The Delft-data points on the triangular arrays yield a rough estimate of z ≈ 0:34 and their two data points on the square arrays of z ≈ 1:4. The Chalmers data provide a more accurate 8t yielding z = 1:05, in good agreement with the theoretical expectation. Measurements on di?erent thin 8lms show that the resistance right at the transition is of the order of RQ but measurements are not conclusive regarding the universality. In arrays, this resistance is again of order RQ , yet in di?erent arrays it varies between 1.6 and 12:5 kZ. The Chalmers group has also measured the Hall resistance in order to check the validity of Eq. (57). For two arrays, the Hall resistance at the critical point is of the order of 30 Z, but again " M2x; x + M2x; y is not a universal quantity. A new feature introduced by JJAs is the existence of 8eld-tuned transitions near commensurate values of the applied 8eld, i.e., at fcomm ± fc [53,218]. Studying the R(f) curves of JJAs in more detail, critical behavior is not only seen around f = 0, but also around f = ±1=4, ±1=3, ±1=2, ±2=3, and ±3=4. For each commensurate f-value zB can be determined as described above and the values of zB are close to one. The sample dependent critical resistances are of the order of a few kZ. Calculations on the Bose–Hubbard model in a magnetic 8eld [114] show that the product zB at f = 1=2 is close to 1 in agreement with the measurement.
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Fig. 29. Phase con8guration of a vortex con8guration. The arrows indicate the phase of each island with respect to a given reference direction.
3. Quantum vortex dynamics A vortex (antivortex) is a topological excitation. When going around its center in a closed loop, the phases of the islands wind up to 2 (−2). Vortices have extensively been studied in classical arrays where they both determine both the phase diagram and the dynamical properties. To a large extent, this still holds for quantum arrays where the interplay between vortex and charge dynamics plays the central role. The 8eld tuned transition for instance can be understood as a Bose condensation of vortices or charges and the S–I transition in zero 8eld can be analyzed using the duality between charge and vortex excitations. In this section we show that vortices characterize the quantum dynamics of arrays as well. We concentrate on the superconducting side of the S–I transition where vortices are well de8ned excitations. As discussed before, quantum arrays have Lux penetration depths that are larger than the array sizes. The magnetic 8eld is almost uniform over the whole array area indicating that there is not one Lux quantum in particular cell. 2 The essential aspect of vortices in junction arrays is therefore not the Lux, but the distribution of phases. The phase con8guration of a vortex (shown in Fig. 29) in a large two-dimensional 2 Self-8eld e?ects may play a role in classical arrays since critical currents are substantially larger. Generally speaking, self-8eld e?ects manifest themselves in two ways [219]. First, there are self-inductance e?ects which are short-ranged and caused by the self-inductance of the cell loop. It turns out that the cell-to-cell energy barrier is dominated by these short-range interactions. Second, there are mutual-inductive e?ects which have a longer range. For example, the current distribution around a vortex changes from an exponential fall-o? for self-inductances to an algebraic fall-o? when the mutual inductances between all cell pairs are included.
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array can be approximated by the following analytical expression: yi − y i = ±arctan ; xi − x
287
(58)
where the site i has coordinates ri = [xi ; yi ] and the vortex center is placed at (r = [x; y]). The ± sign refers to the vortex (antivortex) con8guration. For most purposes Eq. (58) is accurate even very close to the vortex center. As we will see, the approximate arctan solution is very useful since it allows to express the action in terms of the coordinates of the vortex center r only (instead of in term of all the phases). This appears to be a reliable description as long as the vortex can be considered as a rigid body. In most of the cases we discuss in this review, this turns out to be a good approximation. An important property of vortices is that supercurrents around them fall o? inversely proportional to the distance r from their core. Vortex–vortex interactions therefore have a long-distance character as they are proportional to ln r. Eq. (58) is the solution for a single vortex in an in8nite system. In 8nite systems, vortices interact with boundaries. The interaction of vortices with the open edges can be viewed as the attraction of a vortex with an image antivortex outside the array. Superconducting banks repel vortices; the interaction with these edges can be viewed as the one with an image vortex (of the same sign) outside the array. The interaction between boundaries indicates that especially in small arrays the approximation given in Eq. (58) is no longer valid. Numerical calculations are then used to extract the quasi-static phase con8guration around a vortex. Experimentally, single-vortex dynamics is studied by applying a small magnetic 8eld (low vortex densities) and performing transport measurements. On the theoretical side, both numerical simulations and phenomenological models which lump the collective dynamics of the phases into the description of the motion of the vortex center, have been investigated. In this chapter, we 8rst derive the classical equation of motion and show that vortices in underdamped arrays can be viewed as massive, point-like particles. We then continue with the quantum corrections to the equation of motion and discuss their consequences. First, there is a renormalization of the vortex parameters (like the mass, damping, etc.) due to quantum Luctuations. In addition there is a class of new phenomena which arise from the quantum dynamics of vortices which are treated in Sections 3.4 and 3.5. They include macroscopic quantum tunneling of vortices in arrays, quantum interference in a hexagon-shaped array, Bloch oscillations of vortices in the periodic lattice potential and vortex localization in quasi-one-dimensional arrays. In some experiments, evidence has been found that the (independent) single-vortex picture breaks down. In these cases, we brieLy comment on the inLuence of vortex–vortex interactions. 3.1. Classical equation of motion In this section, we review the steps to derive the equation of motion for a vortex. We analyze all contributions, i.e. its inertia, its dissipation, the external potential and the applied forces. To keep the notation simple we suppose, for the moment that the vortex moves along a given (say x) ˆ direction with an average vortex velocity vx . Vortex mass: Moving vortices lead to phase changes across junctions and they therefore contribute to an electric energy. In principle all capacitances contribute, but as discussed before the
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main e?ect comes from the junction capacitance so that Ech = 1=2 ij CVij . (The contribution due to C0 will be discussed in Section 3.3.) In a quasi-static approach this sum is calculated by comparing the phase di?erences across each junction at times t and t + 1=vx (in units of the lattice constant): P ij = ij (t + 1=vx ) − ij (t). The electric energy then acts like a kinetic energy term and the proportionality factor de8nes the vortex mass [220 –223]:
Ech = 12 Mv vx2 with Mv =
1 (P ij )2 : 8EC
(59) (60)
ij
The problem of calculating the vortex mass is now reduced to 8nding the phase di?erences across junctions at times t and t + 1=vx . Note, that Eq. (60) can be applied to various array geometries if the phase con8guration around a vortex is known. In large two-dimensional Josephson arrays the arctan form given in Eq. (58) [224] is used to evaluate the sum in Eq. (60). With the assumption that this arctan-phase con8guration remains the same when the vortex moves through the array, numerical evaluation of the phase di?erences in a large two-dimensional square array yields the Eckern–Schmid value of the vortex mass 2 −1 (61) E : 4 C In this calculation roughly half of the vortex mass is due to the junction the vortex crosses; the other half comes from all the other junctions in the array. For a triangular two-dimensional array, a similar calculation can be done and the vortex mass is twice the mass of a square array. For typical arrays with C = 1 fF and a2 = 10 m2 , the vortex mass is 500 times smaller than the electron mass. This small value already indicates that quantum e?ects are likely to occur. It has been shown [225] that near array edges the vortex mass vanishes when it approaches a free boundary of the array. These boundary e?ects are, however, negligible if the vortex is a few lattice spacings away from the edge. One can also include self-8eld e?ects. Currents now extend over a distance of the penetration depth ⊥ from the vortex center so that the arctan approximation can no longer be used. As the vortex is e?ectively reduced in size, the sum of the Vi ’s can be restricted to those junctions which are ⊥ from the vortex center. The result is a smaller vortex mass and its decrease with decreasing ⊥ is given in Ref. [226]. The vortex mass is dependent on the EJ =EC ratio as well. Quantum corrections to the mass, on approaching the S–I transition [227] are discussed in Section 3.3. To a good approximation it has the value given in Eq. (61) for arrays that are not in the critical region EJ ∼ EC . The vortex mass can also be calculated in geometries other than two-dimensional arrays. In a purely one-dimensional array (N junctions in parallel connected by two superconducting leads), the vortex phase con8guration is given by Mv =
i = 4 arctan[exp((xi − x)=PJ )] ; where x denotes the position of the vortex center. For PJ ¡ N , the vortex has a kink-like shape, which extends over a distance of the order of PJ ; for PJ ¿ N the vortex is spread out equally over the whole system with x+1 − x ≈ 2=N . In this latter regime, P t+1=vx (x +
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1; x) − P x+1; x (t) = 2=N and Mv = h2 =(8EC N 2 ). For 1 ¡ PJ ¡ N the sum can be computed numerically or in a continuum approximation. The sum over the phase di?erences squared is equal to 8=PJ [228] and hence 1 (62) Mv1D = EC−1 for 1 ¡ PJ ¡ N : PJ In other geometries, the phase con8guration of a vortex is not exactly known. It can in principle be calculated. However, since a substantial contribution comes from the junction that the vortex crosses, Eq. (61) can be used as an estimate for the quasi-static vortex mass in these cases. The notion that the concept of the vortex mass is not new in the theory of superLuids. It has been discussed extensively for type-II superconductors and superLuid He [229 –231]. Dissipation: As the vortex is a macroscopic object, it couples to the environment and experiences dissipation. Quantum arrays generally have junctions that are underdamped, i.e., have a McCumber parameter +c ¿ 1. One might therefore expect that vortex motion is underdamped as well. In the simplest approximation one can assume that a moving vortex experience a viscous drag force characterized by a viscous coeUcient H. In a Bardeen–Stephen like model H is calculated using the following argument. The total power loss is the sum of all the resistive losses in the junctions. Assuming Re to be identical for all junctions, the sum in the total power is the same as in the calculation of the vortex mass. For example, 2 H= 0 2Re for a large square two-dimensional array. Here, Re is the e?ective voltage-bias resistance, i.e., the e?ective shunt resistance of each junction. At low temperatures Re is the subgap resistance which is many orders of magnitude larger than the normal state resistance, indicating the vortices can move through the medium with negligible damping. However, the simple model presented here does not take into account other sources of dissipation like the coupling to the low lying modes of the array (spin-waves) or quasi-particle tunneling (see next sections). Lattice potential: The total Josephson energy associated to a vortex con8guration (calculated for example by means of Eq. (58)) depends on the vortex position. The energy has a minimum value when the vortex is in the middle of a cell; the maximum value is reached when the vortex right on top of a junction. Neglecting vortex–vortex interaction and the inLuence of the array edges, vortices are only subject to a periodic lattice potential: Uv (x) = 12 9EJ sin(2x) : Here, 9 is the energy barrier in units of EJ a vortex has to overcome when moving from one cell to the next. In large two-dimensional arrays with no self 8elds, 9 = 0:2 in a square geometry. In a triangular array the barrier is about a factor 8ve lower, 9 = 0:043 [224]. Inclusion of self-8eld e?ects can be done and 9 increases dramatically for ⊥ ¡ 1 [219]. In contrast, there is no energy barrier in one-dimensional arrays if PJ ¿ 1 [232]. Equation of motion for a single vortex: A vortex in a Josephson array moves under inLuence of a Lorentz force 0 I in a direction perpendicular to the current Low. The phenomenological damping term and the periodic lattice potential U (x) provide additional forces.
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Gathering all the ingredients discussed so the equation of motion can be written as Mv r^ + Hr˙ = −∇r Uv − 0 zˆ × I ;
(63)
where r is the vortex position, I is the applied current per junction and zˆ is the unit vector perpendicular to the array [233]. The dynamics can be visualized as that of a massive particle moving in a washboard potential analogous to the dynamics of a single junction in the RCSJ model. For the junction problem, the motion is in arti8cial -space; for the vortex problem the motion is in real space. The mapping is exact for 2x → indicating that vortices in arrays produce the same dynamics as a single junction with a critical current of 9Ic =2 per junction, a McCumber parameter +c; v = 9+c √ and a plasma frequency !p; v = 9!p . Numerical evidence for massive vortices has been found by Hagenaars et al. [234]. Their data show that a vortex may be reLected at an array edge, thereby changing its sign (i.e., it becomes an antivortex). This behavior can be qualitatively understood within the model of a massive vortex interacting logarithmically with the image vortices outside the array. In the same paper the authors also note that the way in which vortex inertia manifests itself depends on the dynamical situation considered. In the next three subsections, we 8rst summarize the experimental details on classical arrays with +c; v ¿ 1 and then discuss two phenomena that are not included in Eq. (63). First, experiments show that vortices in highly underdamped arrays experience more damping than can be expected from the simple approach we followed above. As it turns out coupling to spin waves becomes dominant in these arrays. Second, we review the theoretical and experimental results on the Hall e?ect. 3.1.1. Experiments on classical, underdamped arrays In Fig. 30 a typical example [235,236] of a current–voltage I –V characteristic of a classical two-dimensional array is shown. The applied magnetic 8eld corresponds to f = 0:1. At low temperatures, hysteresis near the depinning current indicates that +c; v ¿ 1, consistent with the existence of a mass term in the equation of motion. The depinning current itself is close to the expected value of (9=2)Ic = 0:1Ic per junction. From the analogy with the single junction problem, a RCSJ-like I –V characteristic is expected. This is generally not observed. The I –V curves, instead, show a slight bending in the direction of the voltage axis opposite to what is expected from the single-junction analogy. This is also seen in simulations on the properties of a single vortex in a 2D array with periodic boundaries [237]. Their numerical data points at a nonlinear viscous damping of the form A H= ; 1 + Bvx where the A and B depend only on the McCumber parameter of the junctions. For currents well above depinning (above 50 A and not visible in Fig. 30), the Lux-Low state becomes unstable. The I –V enters a row-switched state [105] where rows of junctions across the whole array width start to oscillate coherently [238]: all phases rotate continuously in time with a phase shift between them. In this regime, a description of the array in terms of vortex motion is no longer appropriate.
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291
Fig. 30. A current–voltage characteristic of an underdamped two-dimensional square, aluminum array measured at low temperature (10 mK) in a magnetic 8eld of 0:10 applied per cell (f = 0:1). The arrow at the left indicates the expected depinning current of 0:1NIc with N the number of junctions perpendicular to the direction of the current Low. For small voltages hysteresis is seen. The Lux-Low region is found above the depinning current but below the current at which row switching sets in (arrow at the left). (From Ref. [236].)
One should realize that the I –V curve was recorded at an applied magnetic Lux of 0:10 per cell (f = 0:1). Thus, on average there is approximately one vortex per 1=f cells so that the distance between vortices is only three cells. At such a short distance, vortices will interact with each other. The inLuence of these vortex–vortex interactions on the measured I –V characteristics is not known in detail. 3.1.2. Spin-wave damping As shown in Fig. 30 experimental I –V characteristics in the Lux-Low regime are generally more or less straight lines. Neglecting the inLuence of the pinning potential Uv , Eq. (63) indicates that for high bias the slope of this line should approach a conductance value corresponding to 1=(2fRe ) per junction. The I –V curves therefore provide a way to estimate Re in the regime where vortices are driven with relatively large currents. A systematic study on highly underdamped arrays has been performed by the Delft group [239]. The surprising result is that for the most underdamped arrays Re is much lower than the normal-state resistance. Such a low resistance cannot be explained by the Bardeen–Stephen model. Apparently, vortices, when driven with a large current, experience more damping than can be explained by ohmic dissipation alone. A similar conclusion was drawn by Tighe et al. [240], who concluded that in their underdamped arrays vortices moved in an overdamped manner. Several authors have suggested the possibility that energy can be lost in the wake of the moving vortex [235,239,240]. The e?ective viscosity due to coupling to spin-waves can be calculated in a semi-quantitative model [239] using the following argument: The oscillating part of the junction is modeled by an LJ − C circuit and the voltage drop across it is V =0 vP =(2)
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which occurs for a time interval of the order of v−1 . As a response to this voltage step the phase di?erence starts to oscillate (note that at the same time the average phase advances when the vortex moves across the junction). Equating the total power dissipated in the LJ − C circuit to Hsw v2 , the result for the e?ective viscosity due to the plasma oscillations (in square two-dimensional arrays) is 1 1 o2 Hsw = : 2N LJ =C When comparing this viscosity coeUcient to the Bardeen–Stephen viscosity coeUcient, Re Hsw ∼ Hv LJ =C one sees that the more underdamped the arrays are, the more dominant the damping due to energy lost in the wake of the vortex becomes. These observations have been con8rmed by more systematic calculations discussed in Section 3.3. In particular it is possible to obtain a self-consistent picture of vortex dynamics which includes the interaction with its environment. There are two main advantages for discussing the dynamics from this perspective. Firstly, it is possible to evaluate quantum corrections to the classical equation of motion and secondly, it is possible to analyze the quantum dynamics in detail. 3.1.3. The Hall eEect In addition to the Lorentz force which is due to the external current, a vortex is subject to a Magnus force which is transverse to vortex velocity. The study of the Magnus force in superLuids has a very long history. A detailed discussion is outside the scope of this review (see Ref. [241] and references therein). In Josephson arrays, in presence of a gate to the ground plane, particle–hole symmetry is broken. A vortex feels a Magnus force [242,243] given by F = Qx 0 zˆ × r˙ :
(64)
Here we assumed for simplicity a homogeneous gate charge. As a result of the combined e?ect of the Magnus force and the Lorentz force, the vortices move at a certain angle, the Hall angle, with respect to the current. Its measurement yields informations on the di?erent dissipation sources in the system. Combining all the terms the equation of motion in the stationary limit the vortex moves at a constant velocity v = [vx ; vy ] = [v cos QH ; v sin QH ] obeying the following equations: Hv cos QH = Iy 0 − Qx v0 sin QH ; Hv sin QH = Qx v0 cos QH ;
(65)
lead to the resistance tensor 02 =H ; Rxx = Rxx = 1 + (Qx 0 =H)2 Rxy = −Ryx =
Qx 03 =H2 1 + (Qx 0 =H)2
(66)
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293
and to the Hall angle QH Qx 0 tan QH = : (67) H The main consequence of the previous results is that the Hall e?ect should be larger in low resistance samples. In experiments on classical Josephson arrays the Hall angle is usually found to be very small (see e.g. Ref. [244]). Samples are usually characterized by random o?set charges and as a result the Magnus force averages to approximately zero. However, up to now there is no general agreement on this explanation. From a theoretical point of view there are questions related to the derivation of Eq. (64) from 8rst principles. In Ref. [242] the Magnus force was obtained from the QPM, implying that only the external charge enters in determining the Hall angle. A reexamination of the problem by Makhlin and Volovik [245] related the (apparent) absence of the Hall angle to the near exact cancellation of the Magnus force with the spectral-Low force. On deriving the e?ective action from the BCS Hamiltonian Volovik [246] shows that the o?set charges, contributing to the Hall angle, have two di?erent physical origins. In addition to the one stemming from the coupling to the ground plane, there is an additional contribution which depends on the particle–hole asymmetry of the spectrum. This latter term is of the order of the small factor (P=EF )2 , with EF being the Fermi energy. This con8rms the expectation that the Hall angle should be small in Josephson arrays [247]. In quantum Josephson arrays Hall measurements have been performed by the Chalmers group [161,248]. Their results are shown in Fig. 31. The transverse resistance is odd in the magnetic 8eld. Combining the results for the longitudinal and transverse part the Hall angle can be extracted. Comparing the results with the classical results [244], the Chalmers experiments indicate a larger Hall angle. The only apparent di?erence is the smaller ratio EJ =EC and this is consistent with the theoretical expectation that the o?set charges are responsible for the Hall e?ect. It is reasonable to expect that these have a negligible e?ect on approaching the classical limit. Finally, it is worth mentioning that the 8eld-tuned transition discussed previously also manifests itself in Rxy . An interesting feature which still remains unexplained is that Rxx and Rxy are related by the following empirical relation @Rxx Rxy ∼ (68) @f similarly to what happens in the quantum Hall e?ect. 3.2. Ballistic vortex motion Besides the experimental veri8cation of the mass term in the equation of motion [235,240] a considerable interest was focused on the direct observation of the ballistic motion. Ballistic vortex motion has not only been observed in long continuous junctions [249], where energy barriers for cell-to-cell motion and spin-wave coupling are absent, but also in discrete one-dimensional arrays [250 –252] and in two-dimensional aluminum arrays [253]. The idea goes as follows: If vortices are massive particles, they should keep on moving if the current is turned o?. In
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Fig. 31. Longitudinal resistance R0xx (a) and the Hall resistance R0xy (b), and the Hall angle > (c) as a function of frustration. R0xx , and R0xy are shown for various temperatures ranging from T = 20 (top), 75, 100, 125, 150, 175 mK. R0xx is symmetric around f = 0 and ±1=2 whereas R0xy changes sign upon passing through these frustrations. (From Ref. [161].)
an experiment, this concept can be realized by accelerating vortices up to a high velocity v0 so that their kinetic energy is much larger than the lattice potential. With Eq. (63) one 8nds that v0 ≈ 0 I=H if one neglects the lattice potential. Then, these fast-moving vortices can be launched into a force-free environment where voltages probes can be used to detect their path through this region.
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The observation of ballistic motion is only possible in a velocity window (vmin ¡ v ¡ vmax ) bound from below by the presence of the pinning potential and from above by the various damping mechanisms which set in at high velocities. The criterion Ekin ¿Epot translates into a lower bound for the vortex velocity required to observe ballistic motion √ 9!p vmin = : (69) Note, that for a one-dimensional system with PJ ¿ 1, 9 ≈ 0 so that the minimum vortex velocity is small. The vortex velocity cannot be chosen arbitrarily large. Fast moving vortices can trigger row switching in the array [105,254]. Simulations [255] indicate that in two-dimensional arrays the vortex velocity must be limited to v ¡ !p . Another limitation comes from coupling to spin-waves. In two-dimensional arrays, there is a threshold vortex velocity below which this coupling is weak. It has been shown [256,257] that a moving vortex only couples to spin-waves above vmax ≈ 0:1!p . The requirement that √ 9!p ¡ v ¡ 0:1!p ; indicates that ballistic motion is possible in triangular arrays just above depinning. (Quantum Luctuations make the window larger; details are discussed in the next section.) Let us analyze the ballistic motion using Eq. (63). With no current applied and for vortices launched at high enough velocity (to ignore pinning e?ects), the equation of motion reduces to Mv v˙x + Hvx = 0 ; indicating that the vortex velocity decreases exponentially in time as vx = v(0) exp[ − Mv t=H]. A mean free vortex path can then be de8ned as v(0)Mv I (70) = −1 +c ; free = H Ic for a square two-dimensional array. The factor I=Ic is typically of order 0.1 so that at high temperatures with Re = RN ; free ≈ 1. At low temperatures with Re RN (the corresponding +c can be high as 107 ), free 1. The experiment with two-dimensional arrays was performed by van der Zant et al. [97] using the sample con8guration shown in Fig. 32. It consists of two two-dimensional arrays which are connected by a narrow channel of 20 cells long and 7 cells wide. Superconducting banks on both side of the channel con8ne the vortices in the channel and, in order to reduce the inLuence of the lattice potential, the arrays and the channel were made in a triangular geometry. In the array on the left hand side, vortices generated by a small magnetic 8eld, are accelerated up to a high velocity. Some of these high energetic vortices will enter the channel and will then be launched into the detector array (array on the right hand side). There is no driving current applied to this array. A set of voltage probes around the detector array is used to detect the places where vortices leave the force-free environment. (The voltage measured across two probes is proportional to the number of vortices passing the probes per unit time.) The results of the local voltage drops as a function of temperature it is shown in Fig. 33. At high temperatures, vortices move di?usively and voltages are observed between all voltage probes.
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Fig. 32. Sample lay-out used to measure ballistic vortices in two-dimensional Josephson arrays. (From Ref. [97].)
As a consequence, the voltage across the probes V3 and V10 is much smaller than the channel voltage. At low temperatures, the voltage measured between the two probes situated just opposite to the channel is almost equal to the channel voltage. Vortices cross the second array in a narrow beam (see Fig. 32). This ballistic vortex motion is observed for T ¡ 500 mK, for small applied magnetic 8elds (0:01¡f¡0:025) and for currents just above depinning. For high magnetic 8elds, vortex–vortex interactions start to play a role when more than one vortex is in the channel at the same time and for too high currents coupling to spin-waves probably starts to play a role. 3.3. EEective single vortex action In this section we derive a more general approach to describe the single vortex dynamics which incorporates the coupling to spin-waves and which is also valid in the quantum regime where EJ ≈ EC [130,227,258]. After introducing the e?ective vortex action, we 8rst consider the classical limit and show that all known results can be recovered. We then proceed to the quantum regime. The vortex mass is calculated as well as the velocity above which spin-wave damping starts to become e?ective.
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Fig. 33. Voltage vs. Temperature measured across the channel (a), between probes V3 and V10 (b), between probes V5 and V8 (c), and between probes V3 and V4 (d). In the inset voltages are plotted in units of the voltage across the channel. The dashed line corresponds to the voltage across V3–V10 when the current direction is reversed. At low temperatures, the voltage across the two probes opposite from the channel is almost equal to the voltage across the channel: all vortices that go through the channel leave the array between V7 and V8. With reversed current direction vortices are accelerated in the opposite direction and no ballistic motion is observed. (From Ref. [97].)
As discussed in Appendix E, the e?ective action for a single vortex is given by 1 d d r˙a ()Mab [r() − r( ); − ]r˙b ( ) ; Se? = 2 a;b=x;y
Mab =
∇a >(r() − rj ) qj ()qk ( ) ∇b >(rk − r( ))
(71)
jk
(>(r) = arctan(y=x)). Thus, vortex dynamics is governed by the charge–charge correlation, which depends on the full coupled charge–vortex gas. The e?ective action Eq. (71) describes dynamical vortex properties in the whole superconducting region and is therefore a good starting point for the investigation of vortex properties down to the S–I transition. The cumulant expansion that leads to Eq. (71) is correct in the EJ EC limit where the charges can be considered as continuous variables and where the vortex Luctuations can be disregarded. In general the average de8ned in Eq. (E.1) is far from being gaussian so that one may argue that higher order cumulants should be considered. Nevertheless nothing prevents us to analyze Eq. (71) keeping in mind that a full description of vortex motion may require the analysis of a dynamical equation that contains also terms proportional to higher powers of the vortex velocity. The expression given in Eq. (71) reproduce the known results in the classical limit where EJ EC . In this region of the phase diagram the charges may be considered to be continuous variables and vortex Luctuations may be neglected so that the charge–charge correlation reads 1 qq k; !& = EJ k 2 2 (!& + !k2 )
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and k2 4e2 EJ : C0 1 + 2 k 2 The spin-wave dispersion is described by !k . It is optical, i.e. !k = !p , for long range Coulomb interactions, whereas for on-site interactions we have !k = !_ p k. Here !_ p = 4e2 EJ =C0 is the plasma frequency for the case of on-site Coulomb interactions. Action (71) reduces to that of a free particle in the limit of small velocities r(). ˙ The corresponding adiabatic vortex mass Mv is + d Mxx (0; ) ; Mv = !k2 =
0
which reduces in the classical limit to 2 C0 Mv = + 2 ln(L) 4EC 4e with L the array dimension. Thus both C0 and C yield a contribution to the mass. The self-capacitance contribution depends on the system size L. For generic sample sizes and capacitance ratio’s the size dependent contribution is smaller than the Eckern–Schmid mass (the value in Eq. (61)). The e?ect of a uniform background charge on the vortex mass was considered by Luciano et al. [259,260]. The frustration of charging leads to a renormalization of the mass towards the classical value. The spin-wave damping that a moving vortex experiences may also be calculated from Eq. (71). Varying the vortex coordinate r a () in Eq. (71) yields the equation of motion d Mab (r() − r( ); − )r˙b ( ) (72) 28ab Ib =Ic = @ (8xx = 8yy = 0; 8xy = −8yx = 1). Its constant velocity solutions in the presence of an external current determine the non-linear relation between driving current and vortex velocity (i.e. the current–voltage characteristics), once the charge–charge correlation is analytically continued to real frequencies (i.e., if i!B → ! + i) [256]. The relevant information is contained in the real part of Eq. (72), which reads in Fourier components and for a constant vortex velocity ky2 v y I =Icr = d! d 2 k 2 (! − vkx )[(! − !k ) + (! − !k )] : (73) 4 k The delta functions express the spin-wave dispersion (from the analytic continuation of the charge–charge correlation) and the vortex dispersion, respectively. The overlap integral determines the amount of dissipation a moving vortex su?ers from coupling to spin-waves. Adopting the smooth momentum integration cut-o?, introduced in Ref. [221], one recovers in the classical limit the results of Refs. [256,257]. While the static damping is zero for vortex velocities below threshold (which implies the possibility of ballistic motion), a dynamical friction due to the coupling to the plasma oscillations is always present for frequencies higher than a given frequency threshold [261]. The latter contribution approaches to zero when the velocity increases to the threshold velocity. However, radiative dissipation of the vortex a?ects the threshold for ballistic motion. What is important
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in this analysis is that by changing the frequency of the applied current, one is able to extract the domain of validity where a vortex can be de8ned as a macroscopic object. Inclusion of quantum e?ects contributes to the opening of a more robust velocity interval where ballistic motion can be observed. When the ratio EJ =EC decreases the charge–charge correlation must be calculated beyond the classical approximation: The discreteness of charge transfer has to be taken into account, which is in particular important at short distances. For long range Coulomb interactions and in the absence of vortex Luctuations the charge–charge correlation function may be rewritten as k 2 EJ qq k; !& = ; !_ 2k = !k2 + 42 EJ ?2 k 2 ; (74) 2 2 !& + !_ k where the correlation length is EC −√EC =EJ c= 1=(1−) e ; ?2 ∼ EJ
=
EC EJ 2
and the constant c is of order one. The S–I phase transition takes place at EJ =EC = 1=2 . Thus, without vortex Luctuations the phase transition is at a smaller EJ =EC value than the 2=2 that follows from a duality argument [91]. The spin-wave dispersion is a?ected at small distances (large k) and the mass is now given by M= ln[1 + 4?2 ] : 16EC ?2 In the limit of small ? the Eckern–Schmid mass is recovered. An extrapolation to the S–I transition where ? → ∞ yields a mass that vanishes at the transition. With the charge–charge correlation given in Eq. (74) we may calculate the spin-wave damping of vortex motion due to the coupling to spin-waves beyond the classical limit. Replacing !k by !_ k in Eq. (73), the overlap integral over the delta functions only contributes for vortex velocities that are higher than a threshold velocity vmax ∼ ? 8EJ EC : By taking into account quantum e?ects the spin-wave spectrum enlarges the velocity window in which vortices move over the lattice potential without emitting spin-wave. An extrapolation to the S–I transition yields a diverging threshold velocity so that vortices and spin-waves decouple. Note, that the outcome of this calculation also has consequences for the classical equation of motion. For instance, it shows that coupling to spin-wave dissipation is reduced for velocities ¡ 0:5!p . This value is a factor 8ve higher than expected from classical considerations [256,257]. Spin-wave damping is not the only source of dissipation. Vortices, in their motion, can excite quasi-particles as well if the local voltage drop (due to the 8nite velocity) exceeds the quasi-particle gap. The e?ect of quasi-particles damping on vortex motion was considered in Ref. [262]. In this case the equation of motion takes the form t ˙ ) r(t r^ + Hr˙ − H dt sinc[2(t − t )] = −0 zˆ × I ; (75) |r(t) − r(t )|2 where sinc x ≡ (2=x) sin x. Despite the non-linear form of the damping kernel, Choi et al. [262] showed that the frictional force on a vortex is linear in the vortex velocity for any practical
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Fig. 34. Measured zero-bias resistance per junction versus the inverse normalized temperature measured for two di?erent square arrays. At low temperatures the resistance of the sample with the smaller EJ =EC ratio (b) is temperature independent indicative for quantum tunneling of vortices. (From Ref. [236].)
purpose (in particular in the long-time and long-wavelength limit, where the semiclassical equation of motion is mostly concerned). 3.4. Quantum vortices If vortices are massive particles that move ballistically, one can think of them as quantum mechanical objects. Like an electron, a vortex in a periodic potential will have a Bloch wave function with momentum p = ˜k and thus a wavelength of h=vMv . At present, many experiments have veri8ed the concept of a quantum vortex [51,53,235,240,263–265]. In this section, we discuss three examples: macroscopic quantum tunneling of vortices [51,53,235,240], the observation of vortex interference in a hexagon-shaped array [266] and Bloch oscillations of vortices in the periodic lattice potential [267]. 3.4.1. Macroscopic quantum tunneling of vortices In a classical description vortices oscillate in the minima of the washboard potential with frequency !p; v . In quantum arrays, these oscillations are quantized. To estimate when quantum 1 Luctuations in the vortex position become important, we compare the zero-point energy 2 ˜!p; v = 1 89EJ EC to the energy barrier Ubar = 9EJ . The two energies are equal if EJ =EC = 2=9. In 2 this quantum vortex regime, the zero-point Luctuations are large enough to allow for quantum tunneling of vortices. In Fig. 34, the resistance per junction (linear response) as a function of temperature is given for two square arrays [51]; in the classical (a) and in the quantum regime (b). The resistance
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of the classical array decreases exponentially all the way down to the lowest temperatures. The slopes de8ne the barrier for this thermally activated process: R ˙ exp[ − Ubar =kB T ]=exp[ − +9EJ ]. In contrast, the resistance of the quantum array levels o? at a temperature Tcr below which it remains constant. We denote this constant value with Rc . Above Tcr , again thermally activated behavior is observed. Similar resistance curves have been reported by the Chalmers group [53]. A 8rst estimate of the tunnel rates and of Rc can be obtained from the analogy with the single-junction problem. In the moderate damping regime [268]: √ Rc ≈ 140RQ f Se−S (76) where the action S is given by # $ 9EJ 0:87 S = 0:95 1+ : EC +c; v
(77)
The EJ =EC dependence of the critical resistance, implied by the previous equation, has indeed been reported by the Chalmers group [53]. An estimate for Tcr can be obtained by equating S to 9EJ =Tcr . Neglecting the term with +c; v , the result is 2:5 EJ EJ =Tcr = √ : (78) 9 EC
With EJ of the order of EC , the inverse normalized critical temperature (EJ =Tcr ) is typically somewhat larger than 2.5 in agreement with the data. Comparing the measured values of Rc with the estimates given in Eqs. (76) and (77), the tunnel rates in the measurements are lower than expected even when taking RN as the resistance determining +c; v . A smaller Rc is consistent with a single vortex model in which the vortex mass is an order of magnitude larger than the one calculated in the quasi-static approximation. It is likely that vortices do not move as rigid objects and calculations have shown that the dynamic band mass of a vortex can be an order of magnitude larger [269]. However, considering this uncertainty, no de8nite conclusion about the validity of the single-junction model can be drawn for the observed Lattening of the resistance. Other models like collective tunneling cannot be excluded. A surprising result is that the array (a) in Fig. 34 does not show any signature of quantum tunneling. Our simple argument given above indicates that for this array the zero-point energy is of order Ubar . The absence of quantum tunneling is explained by the fact that in Fig. 34 the measured energy barriers are of order EJ , instead of 0:2EJ . The Delft group has reported [51] a systematic increase of the measured energy barrier in the range 2 ¡ EJ =EC ¡ 20. This increase is not yet understood. The theoretical analysis of inter-site vortex tunneling in Josephson arrays was 8rst formulated by Korshunov [270,271] who evaluated the instanton action Sinst associated to vortex tunneling between adjacent plaquettes. Sinst , related to a hop from one plaquette to a neighboring one,
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determines tunnel rates and the depinning current. It can be obtained in the language of the Coupled Coulomb gas approach [92] by evaluating the action associated with the trajectory v˙i; t = vi; +8 − vi; = ; t [i; x+1 − i; x ] for a hop from (x; ) → (x + 1; + 8 ), with the result (see Eq. (71)) Sinst = 12 Mxx (0; 0) : In the limit of large Josephson coupling one recovers all known results, i.e. for general capacitance matrix # √ $ 2 2 EJ √ 2 4 Sinst = + 4 + ln : (79) + 1+ 2 4!p 2 It reduces to Sinst = 3=2 EJ =4!_ p and Sinst = 2 EJ =2!p for C = 0 and C0 = 0, respectively. Korshunov pointed out that instanton–instanton interaction cannot be neglected. Vortex tunnel√ ing is incoherent in the temperature range EJ T EJ E0 . In this case the tunneling probability W is given by EJ1=2 e−Sinst 2 ln 2T=EJ W ∼ e T 3=2 √ up to the crossover temperature T ∼ EJ E0 where the activated behavior takes place. Dissipation associated to vortex tunneling was discussed by Io?e and Narozhny [272]. Since the time associated to vortex tunneling is slow compared to −1 , the dissipation which accompanies this process arises from rare processes when a vortex excites a quasi-particle above the gap. These authors 8nd that this source of dissipation can be signi8cant even in the adiabatic limit. 3.4.2. Vortex interference: the Aharonov–Casher eEect In 1984 Aharonov and Casher [273] studied the interference of particles with a magnetic moment moving around a line charge. This AC e?ect is the dual of the Aharonov–Bohm e?ect [274] that describes quantum interference of charged particles moving around magnetic Lux. The AC e?ect has 8rst been observed using neutron beams in Ref. [275]. The concept of vortex-interference in superconductors has been introduced by Reznik and Aharonov [276] and their ideas have been adapted to ring-shaped Josephson arrays by van Wees [277] and by Orlando and Delin [278]. Although there are similarities with the conventional AC e?ect, there are important di?erences. In JJAs vortices do not carry a Lux unlike the Abrikosov vortices in a bulk superconductor. Moreover, as already stressed, JJAs form an arti8cial 2D space. The observation of the AC e?ect for vortices has been reported by Elion et al. [266], only two years after the 8rst theory papers [277,278] appeared. In the array of Elion et al., vortices follow trajectories indicated as dotted lines in the Fig. 35. The sample consists of a hexagon-shaped array with six triangular cells. Large-area junctions couple the hexagon to superconducting banks so that only two paths for vortex motion are possible. The large-area junctions con8ne vortices to the hexagon, but the coupling to the superconducting banks is not so strong that phases of the islands are set by the banks. A gate controls the charge on the superconducting island in the middle of the hexagon.
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Fig. 35. Schematic drawing of the hexagon-shaped Josephson array to measure vortex interference. (From Ref. [266].)
In the experiment the di?erential resistance in the Lux-Low regime has been measured as a function of the gate voltage. When 8xing the current through the array, clear oscillations in the di?erential resistance are observed. The measured period, however, corresponds to e, half the value expected from theory. The factor of two arises from tunneling of quasi-particles which e?ectively limit the quantum phase di?erence to values in the range −=2 to =2. (Quasi-particle tunneling changes the vortex phase di?erence by and becomes favorable as soon as the induced charge equals e=2, i.e. when the phase di?erence equals =2.) It is possible to derive the AC e?ect from the quantum phase model. In a more transparent way, although less rigorous, one starts from the representation of the partition function as a path integral over the phases and charges (see Eq. (D.2)) with the inclusion of a uniform background charge, i.e. q → q − qx . The term in the action of the QPM which is relevant for the AC e?ect is the one which is linear in ˙ i in Eq. (D.2). If a vortex is present, the con8guration of the phases will be related to its position r() and, by going over the same steps that lead to the single vortex action in Eq. (71), there is an additional term to the e?ective action equal to qx; i + SAC = −i d r˙ · ∇>[ri − r()] : (80) 2e 0 i Eq. (80) de8nes a pseudo-charge gauge 8eld for the vortex 2eAQ (r) = qx zˆ × r=r 2 ;
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Fig. 36. A sketch of the sample layout of a quasi one-dimensional Josephson array. The current is injected in the middle while the voltage probes are situated at the end of the busbars. (From Ref. [263].)
which is singular at the origin of the vortex. Thus the external charge act like a vector potential for the vortex. The phase factor S implied by Eq. (80) is qx; i S = 2 ; (81) 2e i⊂O
where the sum extends to all islands enclosed by the trajectory O. 3.4.3. Bloch oscillations Electrons in metals move in the periodic potential created by the positively charged ions. The electron wave functions overlap and energy bands are formed. A constant electric 8eld accelerates electrons, but in the absence of scattering, electrons would be Bragg reLected at the Brillouin zone edges. Electrons then undergo an oscillatory motion in space (Bloch oscillations). No charge would be transported. In metals scattering takes place before the electrons can reach the zone edge so that Bloch oscillations do not appear. In semiconductor superlattices [279] Bloch oscillations have been observed because of the larger superlattice period and because of less scattering in the controlled fabricated structures. Coherent Cooper pair tunneling in current bias Josephson junctions leads to a phenomenon analogous to Bloch oscillations [280]. Vortices in a periodic potential should also form energy bands. It is possible to study Bloch oscillation for vortices [267] in a quasi-one-dimensional Josephson array that is a few cells wide and 1000 cells long. A sketch of the sample layout is shown in Fig. 36. For low densities, the bus-bars force the vortices to move in the middle row so that they experience a purely one-dimensional sine potential. For a free vortex the energy depends quadratic on the wave vector k: E(k) = k 2 =2Mv , which equals E(k) = 2EC at the Brillouin zone edge. In a periodic potential, energy gaps open up at the zone edges. The gap is equal to the Fourier coeUcient of the lattice potential [281]. For a sine potential 12 9EJ sin(2x), the gap is then 9EJ . Thus, vortices in arrays form energy bands with a bandwidth of the order EC and an energy gap of 12 9EJ as illustrated in Fig. 37. Assuming the lowest band to be cosine-shaped (E(k) = 12 EC (1 − cos(k))), the equation of motion F = ˜ d k=dt is dk 1 dE(k) EC ˜ sin(k) : (82) = 0 I − Hv(k) where Hu(k) = = dt ˜ dk 2˜
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Fig. 37. Schematic drawing of the energy bands for a vortex moving in a quasi-one-dimensional Josephson array. Dots: numerical calculated energy bands starting from Schr^odingers equation with a cosine potential. The dashed line shows the 8rst band of a cosinusoidal dispersion relation with the same band width. (From Ref. [236].)
As de8ned before, I denotes the applied current per junction, H the phenomenological viscosity and v(k) the average vortex velocity. In the absence of damping, with a small current applied, the wave vector changes linearly in time: the vortex thus reaches the Brillouin zone edge where it will be Bragg reLected. This Bragg reLection results in an oscillatory motion in k-space. On average the vortex velocity is zero and the time it takes the vortex to complete one oscillation follows from Pt = Pk= d k=dt with Pk = 2. The corresponding Bloch oscillation frequency (BB ) is: I BB = (83) 2e and the amplitude of the oscillation is 1 E C Ic = : (84) x= 0 I EJ I When biasing an array with 1000 junctions with currents of the order of A, Bloch frequencies are in the range 1–10 GHz. Since EC ≈ EJ and Ic =I is typically 100, the Bloch oscillations extend over 10 cells. A characteristic feature of Bloch oscillating vortices is a nose-shaped form of the DC current– voltage characteristic (see Fig. 38). For very small bias, there is a small supercurrent because vortices need to overcome the energy barriers near the array edges (8nite-size e?ect). Just above the depinning current, any amount of dissipation prevents vortices from reaching the zone edges. Bloch oscillations do not exist in this regime and an increase of the current yields an increase of the measured voltage across the array. When increasing the current beyond some point, dissipation is not strong enough to prevent the vortices from reaching the zone edges. Bloch oscillations are now possible. In the I –V characteristic a sudden decrease of the voltage is then expected with a negative di?erential
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Fig. 38. A nose-shaped I –V characteristic indicative for Bloch oscillating vortices. Data (circles) has been obtained in a quasi-one-dimensional array with size 7 by 1000 and has been measured at 10 mK for a one-dimensional vortex density (n) of 0.04. The solid line is the analytical result discussed in the text. (From the Ph.D. Thesis of A. van Oudenaarden, Delft, 1998, unpublished.)
resistance: the oscillating vortices do not contribute to the net transport of vortices through the array. Eq. (82) can be solved with the result V (I ) = n
EC I e I0
if I ¡ I0 ;
(85)
% & 2 & EC I & I0 ' 1− 1− V (I ) = n
e I0
I
if I ¿ I0 ;
(86)
where I0 = HEC W=(20 ) and n is the one-dimensional vortex density (= number of vortices divided by the number of cells in the direction of motion). The solid line in Fig. 38 is a 8t to these equations. (The line is o?set by a small positive current to correct for the depinning current.) Although the overall shape of the experimental curve resembles that of the theoretical prediction, the experimental value of the band width (EC is the previous discussion) is one order of magnitude smaller than expected. This discrepancy is not understood, but is consistent with a vortex mass that is larger than the calculated, quasi-static vortex mass. Note that the data extracted from the study of the macroscopic quantum tunneling also indicated the same trend for the experimental vortex mass. Additional information on Bloch vortices can be obtained by irradiating the sample with a microwave signal. Steps occur in the I –V characteristics when the external frequency locks to the Bloch frequency. Surprisingly, the experiments show that the Bloch frequency depends on the vortex density. This dependence is not accounted for by the independent vortex model presented above (see Eq. (83)) and suggests a collective oscillation of the one-dimensional vortex chain (see next section). We will come back to this issue when discussing the formation of a Mott insulator in these quasi-one-dimensional arrays.
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3.5. One-dimensional vortex localization In Section 2.7, we already introduced quasi-one-dimensional arrays as model systems for the study of interacting bosons in one dimensions. For an ideal periodic potential and in the absence of interactions between the particles, the solution of the Schr^odinger equation consists of Bloch waves that extend throughout the whole chain. In quasi-one-dimensional Josephson arrays, however, vortex localization can occur in several ways. We already discussed the existence of Bloch oscillations when an external force is exerted on a vortex. One can view this oscillatory motion in space as localization of individual bosons analogous to Wannier–Stark localization of electrons [282]: the extend of the wave function is decreased when the external force on the quantum particle is increased. In the next two subsections, we treat two other cases where boson localization occurs. Commensurability e?ects [283] may lead to localization of vortices in quasi-one-dimensional arrays. The repelling interaction between vortices plays a crucial role in this so-called Mott-localization [40]. Another mechanism to localize Bloch waves is disorder and this phenomenon is known as Anderson localization [284]. Two important remarks should be made at this point. First, vortex localization, as discussed in the next subsection, has been studied by measuring the zero-bias resistance. In this case only a very small current is applied in contrast to the experiment showing the Bloch oscillations. Second, one should realize that in contrast to localization in electronic systems, vortex localization leads to a zero resistive state. In superconductors motion of vortices is the cause of dissipation. If they are localized, the sample is superconducting. 3.5.1. Mott insulator of vortices More than 10 years ago, localization of bosonic particles with a short-range repulsive interaction has been studied theoretically by Fisher et al. [40]. They found a Mott insulating phase for commensurate 8lling and a superLuid phase for incommensurate 8lling. Strong disorder destroys the Mott phase and there is the possibility to have a Bose glass (for theoretical studies of the phase-diagram of bosons on a chain see Refs. [118,119,285] and references therein). Experimentally, Mott localization of vortices has been studied by van Oudenaarden et al. [263,265]. They explored the inLuence of the interaction strength, the bandwidth, the sample geometry and temperature on the stability of the Mott states. First, we will summarize the experimental results. Then, we discuss their experiment in the context of a recent theory of Bruder et al. [286]. The experiment consists in measuring the zero-bias resistance vs. magnetic 8eld for quasi-onedimensional arrays of di?erent lengths and EJ =EC ratios. It is convenient to de8ne a one-dimensional frustration n, which is associated with magnetic 8eld piercing through a cell of area W (in units of the lattice constant): n = WB=0 = Wf. In Fig. 39 the results are shown for arrays of three and seven cells wide, i.e., W = 3 and 7, respectively. The plot for the W = 7-sample is mirrored with respect to the x-axis for clarity. For both samples, clear dips occur for certain values of the two-dimensional frustration index f, i.e., for certain values of the vortex density. When plotted vs. this one-dimensional frustration index, the nature of the sharp dips becomes visible. Dips are found at the same rational values of n = 1=3; 1=2; 1; 2.
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Fig. 39. Zero-bias resistance R0 as a function of the one-dimensional vortex density (n = Wf) for two samples with di?erent widths W . The sample length is 1000 cells and data has been obtained at 30 mK. The bottom curve (W = 7) is mirrored with respect to the x-axis for clarity. (From Ref. [265].)
More detailed measurements of the resistance dips show that they are not in8nitely sharp. There is a certain window of n in which the resistance vanishes and the vortex chain is pinned. The interaction energy proportional to EJ (see below) dominates the bandwidth proportional to EC in this regime. Beyond this window the resistance increases sharply, indicating that the vortex chain is depinned and that the bandwidth dominates the interaction energy. From this consideration, one expects the window to be larger for samples with a larger EJ =EC ratios. In the experiment, this dependence has indeed been observed. Around commensurate 8lling, the system is incompressible: small changes of the magnetic 8eld (i.e. n) do not lead to a change in the number of vortices in the chain. This process costs a certain energy, called the Mott gap. By analyzing thermally activated transport in the Mott states, the value of this gap can be deduced from the experiments and values in the order of Kelvins are reported. The inLuence of the array length was also studied. No signi8cant di?erences were observed in arrays with lengths larger than 200 cells. This observation demonstrates that edge e?ects do not play an important role and that the long arrays are indeed one-dimensional systems. A quantitative theory of the commensurate-incommensurate transition of a vortex chain in quasi-one-dimensional Josephson arrays has been formulated by Bruder et al. [286]. They showed that the transition to the incommensurate state is due to the proliferation of soliton excitations of the vortex chain. Since the range of the interaction between the vortices is much longer than the inter-vortex distance, solitons consist of many vortices, and possess a large e?ective mass. The transfer of one Lux quantum between the array edges is then due to soliton propagation through the sample. The number of solitons necessary to transfer one vortex is equal to the ratio of the periods of the vortex lattice and the junction array.
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Fig. 40. Activation energy as a function of n − n0 (n0 is the commensurate density) in the case that the boundary pinning Eb dominates over the soliton formation energy Es . On the incommensurate side of the transition, n ¿ nC , solitons form spontaneously and the physics is determined by boundary pinning and the elastic energy. (From Ref. [286].)
The analysis of Bruder et al. [286] focuses on determining the energy barrier ER of the observed thermal activation in the resistance vs. temperature curves. Although this approach is of semi-classical nature, quantum e?ects are crucial in relating the parameters of the e?ective theory (expressed in terms of soliton excitations) to the microscopic couplings (EC and EJ ) of the Josephson array. In the commensurate phase, there are two contributions to ER . The 8rst contribution comes from the activation energy of a soliton and the second term summarizes the boundary pinning energies. This boundary e?ect can be understood as follows: Because of commensurability, the process of vortex Low through the array can be viewed as the motion of a rigid vortex chain. Therefore, the vortex chain cannot adjust itself to the boundary pinning potential. The potentials produced by the two array ends both contribute to ER and the relative phase of these two contributions depends on whether the total Lux piercing the junction array equals an integer number of Lux quanta or not. Consequently, the second boundary pinning term to ER oscillates with the magnetic Lux piercing the array. In Fig. 40 ER is plotted for the case of small boundary pinning while the opposite situation is shown in Fig. 41. The short period oscillations are determined by boundary pinning term while the vanishing of ER at the edge of Mott lobe is driven by the soliton energy. In the incommensurate state, the vortex chain is compressible, and can adjust itself to the boundaries of the array. As a result, the main contribution to the activation energy is due to the boundary pinning potential and the elastic energy. The comparison between theory and experiment is shown in Fig. 42. The theoretical results lead to an estimate for the soliton energy much larger than the boundary pinning which is of
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Fig. 41. Activation energy as a function of n − n0 when the soliton formation energy Es dominates over the boundary pinning Eb (opposite limit as considered in Fig. 40). (From Ref. [286].)
Fig. 42. Measured activation energy of an array of 1000 × 7 cells with EJ = 0:9 K and EC = 0:7 K. The dashed line is a 8t to the data yielding the width of the Mott region. The inset shows ER inside the Mott phase. (From Ref. [286].)
the order of ∼ 0:5 K. The theoretical value of the activation energy ER ∼ 8 K ; is in very good agreement with the experiments.
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Two observations provide additional support for the theory of Bruder et al. First, the regions of n corresponding to the Mott phase are extremely narrow suggesting, at least in the conventional non-interacting picture, a weak interaction between the particles. Consequently, within the Mott phase the activation energies for particle transport are expected to be small as well. However, the observed value of ER is one order of magnitude larger than the energies EC and EJ , which determine the single-vortex band spectrum. A second feature in favor is the strong oscillating behavior of the resistance (with a period proportional to the inverse length of the chain) outside the Mott region. These oscillations would not be expected in a model of almost-free quasi-particles within the delocalized phase. 3.5.2. Anderson localization of vortices In 1958, Anderson [284] showed that disorder has a dramatic inLuence on transport properties. Disorder reduces the spatial extent of wave functions to such an extent that transport can completely be blocked. Three years later Mott and Twose [287] showed that one-dimensional systems are particularly susceptible for disorder: even weak disorder leads to strong localization. Many studies on Anderson localization have been performed on three- and two-dimensional samples. One-dimensional model systems are harder to 8nd. Josephson-junction arrays have the great advantage that disorder can be introduced in a controlled way. Experimentally, Anderson localization of vortices has been studied by the Delft group [264]. Their results will be outlined in this subsection. In the experiment by the Delft group disorder has been introduced by constructing superlattice structures. The superlattice is formed by replacing all the junctions of a column by junctions that are twice as large. Consequently, these barrier junctions have a Josephson energy that is two times larger than that of the adjacent junctions. The barrier junctions yield a peak in the potential landscape for vortices traveling through the array. Numerical calculations show that this barrier is 1:7EJ , which is about one order of magnitude larger than the energy barrier for cell-to-cell motion. A perfect superlattice structure is made by introducing columns of barrier junctions on a distance of exactly 10 lattice cells. For an array of length 1000, this means that there are 100 columns that have been changed. Disorder is now introduced by changing the distance between two barriers. Samples with di?erent amounts of disorder have been fabricated. In the least disordered samples (labeled with = 1) the barriers were separated by 9; 10 or 11 lattice cells with equal probabilities. In other disordered samples (labeled with = 2) barriers placed at distances 8; 9; 10; 11, or 12 lattice constants again with equal probability. All samples contained 100 barriers. The vortex quantum properties are probed by measuring the zero-bias resistance as a function of temperature for the perfect periodic array as well as for the disordered arrays. Since the topic of interest is the study of quantum transport, the vortex density needs to have a non-commensurate value to avoid the Mott state as discussed in the previous subsection. The result for n = 0:44 is shown in Fig. 43. At high temperatures all three arrays show the same behavior: transport is thermally activated with an energy barrier of 3EJ , a factor of two larger than the expected value. When the temperature is lowered, however, a signi8cant di?erence is observed between the periodic sample and the two disordered samples. For the perfect periodic
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Fig. 43. Arrenius plot of the linear resistance for the ordered (squares) and for the disordered (triangles, circles) arrays. At low temperatures (right hand side of the 8gure), the resistance of the disordered arrays (triangles and circles) has dropped below measuring accuracy (dashed line) indicating vortex localization. (From Ref. [264].)
sample a 8nite resistance is measured at the lowest temperatures. In this regime the resistance is independent on temperature indicating vortex transport by quantum tunneling. In contrast, the resistances of the disordered samples have dropped below the measuring accuracy of the set-up. Thus, in the perfect array vortices are mobile whereas they are localized in the disordered arrays. The zero-bias resistance has been studied for several values of the vortex density. For n¡0:3, the zero-bias resistance is too small to be resolved for the periodic array. In the range 0:3¡n¡0:8, the resistance of the periodic array is signi8cantly larger than that of the disordered arrays and this is the region where vortex localization occurs. But for even larger vortex densities the behavior of all three arrays is almost the same showing a Lattening of the resistance at the lowest temperatures. In all three arrays vortices are now mobile. At these high vortex densities the distance between them is small and their repulsive interaction can no longer be neglected. The experiment shows that in this case delocalization occurs. Above we have discussed the experiment in terms of Anderson localization which strictly speaking occurs when the interaction between the bosons is very weak; i.e., when the bosons act as independent particles that are localized for arbitrarily weak disorder. In the experiment, vortex densities are large. Disorder now competes with the interaction strength. A suUciently strong interaction can delocalize the particles, whereas strong disorder will localize them again in a Bose-glass phase. To distinguish between Anderson localization and localization in a Bose
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glass, more measurements are needed. Experiments should be performed on arrays with 8xed disorder but di?erent EJ to clarify the role of the interaction strength. 4. Future directions In this last section of the review, we discuss some future directions for research on Josephson networks. Of course, additional information on the quantum nature of Josephson networks can be obtained by using new measuring techniques (e.g. the AC-measuring method that has successfully been applied to classical arrays) and better samples (e.g. arrays with a well-controlled environment). Here, we outline the concepts behind two experiments—persistent vortex currents and vortex quantum Hall e?ect—of which some theoretical calculations are available. We summarize the main ideas as well as the experimental requirements=improvements for observation of these e?ects. Future lines of research may also involve the use of Josephson networks as model systems in unexplored areas of physics. Biophysics may be such a 8eld and an interesting example in this respect is vortex transport in ratchet arrays. Undoubtedly, the most exciting new line of future research is quantum computation. In this paper, we do not have space to treat quantum computation with Josephson circuits in great depth. We therefore only present the concept and summarize the current status of art. 4.1. Persistent vortex currents Persistent currents in rings made of normal metals are a manifestation of the Aharonov– Bohm e?ect for quantum coherent electrons in a multiply connected geometry. In the absence of any driving current, the Lux through the ring induces the motion of charge carriers that can be detected at low temperatures [288]. Charge–Lux duality indicates that in Josephson rings a persistent current of vortices may be expected generated by the charge induced on the inner island. A persistent vortex current leads to a persistent voltage across the ring and as such would be manifestation of the Aharonov–Casher e?ect discussed in Section 3.4. Although vortex interference in an open Josephson circuit has been reported by Elion et al. [266], there are no experiments reporting the existence of persistent vortex currents in Josephson Corbino circuits. The proposed setup to measure persistent vortex current is shown in Fig. 44. When the mean free vortex path is long enough, we can neglect the dissipation in the dynamics and the Hamiltonian of a vortex in a discrete Josephson ring (with N junctions) can be written as 2 1 0 H= P− (87) Qx ; 2Mv N where P is the vortex momentum and Qx the charge on the inner island of the ring. The vortex dynamics is quantized and the set of discrete energy values is given by 2 (2eB − Qx )2 N EB = with Ce? = Mv ; (88) 2Ce? 0 where B is an integer and where Ce? agrees with the continuum result of Ref. [289] and with Ckin in Ref. [278]. Thus, a Josephson ring with a vortex trapped inside acts like a perfect
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Fig. 44. The experimental setup to detect interference e?ects of vortices: The Corbino disk (From Ref. [277].)
capacitor. A similar situation arises in a single classical junction in the absence of screening. With exactly one vortex trapped, its critical current is zero. The di?erence lies in the value of the e?ective capacitance, which in quantum rings di?ers from the geometrical one. For example, for a ring consisting of a square two-dimensional array Ce? =(N 2 =2)C and for a one-dimensional ring with PJ ¡ N; Ce? = (2N=2 PJ )NC. Although this change in capacitance can in principle be measured, quantum rings should exhibit a more interesting phenomenon: persistent vortex currents. This can easily been seen from the Hamiltonian of Eq. (87) since it has the same from as the Hamiltonian for electrons in a metal ring. Duality arguments then indicate the existence of a persistent voltage due to the persistent motion [225,277,290] of a vortex induced by Qx . The basic reasoning is as follows: With some disorder, small gaps open up in the energy spectrum of Eq. (88) and energy bands form. With no current applied, the charge Qx dictates the quantum dynamics of the ring. It determines the vortex velocity and therefore the voltage across the ring because the vortex speed is proportional to @En =@Qx . As a result, a persistent voltage appears across the ring which is periodic in Qx with period 2e. Within the free-particle model, the maximum voltage can be estimated. The voltage V due to a circling vortex with velocity v is equal to 0 v=N . The velocity is equal to (N=0 )@EB =@Qx . Then for B = 0 and Qx = e, the maximum voltage equals e=Ce? = e=Mv (0 =N )2 . For a ring consisting of a square two-dimensional array Vmax = (2e=CN 2 ). With N = 10; C = 1 fF; Vmax ∼ 3 V. In the picture presented above, the vortex is treated as a free, non-interacting particle. A detailed description of vortex dynamics, however, is only possible if one considers its dissipative environment. As discussed before, an important dephasing mechanism is the existence of linear spin-waves. This coupling leads to damping and hence to a 8nite phase-coherence length for vortices. This problem was studied in Ref. [291]. For the persistent voltage one obtains ∞ n sin(2BQx =2e) exp(−SB ) 4 B=1 ∞ ; 2eV = + 1 + 2 B=1 cos(2BQx =2e) exp(−SB )
(89)
where SB corresponds to the action in the sector of B winding numbers. In the adiabatic mass approximation in the limits EJ EC and RN . RK , the action reduces to that of a free particle
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with mass Mv = 2 =4EC . The saddle point action is SB0 =
Mv (BN )2 2+
(90)
and the persistent voltage V is a sawtooth function of Qx at zero temperature and a sine-like function at higher temperatures. It depends on the system size through the ratio of the radius N= to the thermal wavelength of the vortex T = 2=Mv T = 8+EC =. To go beyond the adiabatic mass approximation, the actions SB can be calculated with the full non-local kernel (see Eq. (71)) that incorporates the e?ect of inelastic processes due to the interaction with spin-waves. An important consequence of the quantum corrections is that the vortex dephasing length L’ = !p ?=T
(91)
increasing with the ratio EC ∼ EJ (? gets larger on approaching the S–I transition). For EC ∼ EJ ∼ 1 K; L’ may be as large as 100 lattice constants at T = 10 mK, which is much larger than the thermal wavelength T for the vortex in the adiabatic mass approximation. In the limit ?N and !B !p the adiabatic result is recovered. If ? ∼ 1 a larger persistent voltage is found. If the ratio EJ =EC is reduced, the coherence length ? grows beyond the radius of the system, and the persistent voltage grows even more. As compared to the adiabatic mass limit, the persistent voltage including the vortex–spin-wave coupling is always larger. For EJ =EC =1 K; T =10 mK and N = 10, a persistent voltage in the microvolt range is expected, which is observable. Other interesting issues involve the e?ect of the underlying lattice and of disorder on the persistent voltage. They both induce backscattering and the opening of gaps in the band structure E(Qx ). One expects Zener tunneling across the band gaps to occur if the voltage Vx is switched on fast, yielding a higher transient current that relaxes to the persistent current after some relaxation time. Experimentally, fabrication of quantum rings is diUcult, but does not seem to be impossible. First of all vortices should have long mean free paths. As we have seen, two-dimensional arrays exhibit only a small window for which free propagation of vortices is possible. Purely one-dimensional discrete Josephson rings with one vortex trapped [228] seem to be more promising in this respect since for PJ ¿1 there is no energy barrier for vortex motion. Coupling to spin-waves can be small and long mean free vortex paths are expected [292]. Secondly, Josephson rings should be decoupled from their environment. This can be achieved by placing high-ohmic resistors or alternatively arrays of junctions in the leads close by. A third restriction comes from the charging energy. Calculations on the continuous Josephson system [293] show that the temperature has to be smaller than e2 =(22 Ce? ) in order to observe quantum vortex dynamics. Temperatures of the order of 100 mK therefore require Ce? to be 1 fF. This requirement indicates that aluminum junctions have to be smaller than 0:01 m2 and that the whole ring structure cannot be made too large. The capacitances to ground would otherwise be too dominant. The fourth restriction comes from the shadow-evaporation method itself. Both the wire connecting the central island in the middle of the ring and the gate capacitor—preferable situated underneath or on top of the central island—have to be made in separate fabrication steps. Alignment and good electrical isolation between the di?erent layers have to be established.
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4.2. The quantum Hall eEect Quantum electron transport in two-dimensional systems in the presence of an applied magnetic 8eld is one of the most intensively investigated areas in condensed matter. When the 8lling factor (ratio between the electron density and the density of Lux quanta) is of the order of one, the quantum Hall e?ect (integer and fractional) occurs [294]. In the previous sections we showed that charges and vortices behave as quantum particles hopping coherently in the arti8cial two-dimensional space created by the Josephson array. The duality between charges and vortices can be shown also in the presence of external frustration; magnetic 8eld for charges and o?set charges for vortices. In a series of papers [295 –297] it was suggested that a quantum Hall e?ect could be observed in Josephson arrays both for charges and vortices. The three proposals address di?erent regimes: Nazarov and Odintsov [295] describe the possibility of Hall states for Cooper pairs while the authors in Refs. [296,297] consider the Hall Luid of vortices. In the limit EC EJ and in the case of very low density, charges behaves as a dilute Bose gas with strong repulsion. Under these conditions, analytic and numerical calculations [295] support the idea that in a magnetic 8eld Cooper pairs form Laughlin-type incompressible states (a Cooper pair Luid): the charge density changes in a stepwise function by changing the external parameters. The incompressible states give rise to the quantization of the Hall conductance Lxy =
4e2 B ; h
where the 8lling factor B = q=f is given by the ratio between the charge density q and the magnetic frustration f. According to Odintsov and Nazarov two sets of Hall plateaus exist. One corresponds to the fractional quantum Hall e?ect with B = 2m (m integer). The other corresponds to the integer quantum Hall e?ect with B = l=2 (l integer). The opposite limit in which the Josephson energy dominates, has been considered in Refs. [296,297]. In this case vortices condense to form a quantum Hall Luid and the transverse conductivity is now given by Lxy = 2m
4e2 : h
Despite the similarities highlighted here, there are important di?erences as compared to the “electronic” case. Both charges and vortices are bosons, moreover they are interacting particles. Interactions modify the results. For instance, in the vortex case the logarithmic interaction changes the longitudinal response [297] as well. Up to now there is no experimental evidence for the quantum Hall e?ect in Josephson arrays. For the Cooper pair Luid, the random o?set charges form a serious obstacle for observation of the quantum Hall states. For the vortex Luid the situation is less clear. Additional theoretical work is required to locate the region in parameter space where the Hall Luid is the ground state (as opposed to the Abrikosov lattice for example). E?ects related to disorder, quantum correction of the mass, and dissipation should all be taken into account.
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4.3. Quantum computation with Josephson junctions Quantum computation (QC) has recently excited many scientists from various di?erent areas of physics, mathematics and computer science. In contrast to its classical counterpart, quantum information processing is based on the controlled unitary evolution of quantum mechanical systems. The great interest in this 8eld is certainly related to the fact that some problems which are intractable with classical algorithms can be solved much faster with QC. Factorization of large numbers as proposed by Shor is probably the best known example in this respect. This section brieLy reviews the recent work in this 8eld using Josephson nano-circuits. For excellent reviews devoted to QC we refer to Refs. [298,299]. Furthermore, many elementary books on quantum mechanics treat the physics of two-level systems. A review devoted to the implementation of quantum computation by means of Josephson nano-circuits just appeared [300]. The elementary unit of any quantum information process is the qubit. The two values of the classical bit are replaced by the ground state (|0 ) and excited (|1 ) state of a two-level system. (Note that it is common to adopt the spin-1=2 language as we will do here.) Already at this stage a fundamental di?erence between classical and quantum bits emerges. While information is stored either in 0 or in 1 in a classical bit, any state | (t) = a(t)|0 + b(t)|1 can be used as a qubit. Manipulations of spin systems have been widely studied and nowadays NMR physicists can prepare the spin system in any state and let it evolve to any other state. Controlled evolution between the two degenerate states |0 and |1 is obtained by applying resonant microwaves to the system but state control can also be achieved with a fast DC pulse of high amplitude. By choosing the appropriate pulse widths, the NOT operation can be established |0 → |1 ;
(92)
|1 → |0
or the Hadamard transformation 1 2 1 |1 → √ (|0 − |1 ) : 2
|0 → √ (|0 + |1 ) ;
(93)
These unitary operations alone do not make a quantum computer yet. Together with one-bit operations it is of fundamental importance to perform two-bit quantum operations; i.e., to control the unitary evolution of entangled states. Thus, a universal quantum computer needs both one and two-qubit gates (it has been shown that most of two-qubit gates are universal [301]). One example of a two-qubit gate is the Control-NOT operation: |00 → |00 ; |01 → |01 ;
(94)
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Fig. 45. (a) A charge qubit. (b) Improved qubit design as proposed by the Karlsruhe group. The island is coupled to the circuit via two Josephson junctions with parameters CJ and EJ . This DC-SQUID is tuned by the external Lux which is controlled by the current through the inductor loop (dashed line). The setup allows switching the e?ective Josephson coupling to zero. (Reprinted by permission from Nature 398 (1999) 305 (copyright 1999 Macmillan Magazines Ltd.))
|10 → |11 ; |11 → |10 :
The unitary single-bit operations and this Control-NOT operation are suUcient for performing all tasks of a quantum computer. Therefore, quantum computers can be viewed as programmable quantum interferometers. Initially prepared in a superposition of all the possible input states, the computation evolves in parallel along all its possible paths, which interfere constructively towards the desired output state. It is this intrinsic parallelism in the evolution of quantum systems that allows for exponentially more eUcient ways of performing computation. It is of crucial importance that qubits are protected from the environment, i.e., from any source that could cause decoherence [302]. This is a very diUcult task because at the same time one also has to control the evolution of the qubits, which inevitably means that the qubit is coupled to the environment. In quantum optics experiments, single atoms are manipulated which are almost decoupled from the outside world. Large-scale integration (needed to make a quantum computer useful) seems to be on the other hand impossible. Qubits made out of solid-state devices (spins in quantum dots or superconducting nano-devices), may o?er a great advantage in this respect because fabrication techniques allow for scalability to a large number of coupled qubits. At present di?erent proposals have been put forward to use superconducting nano-circuits [303–307] for the implementation of quantum algorithms. Depending on the operating regime, they are commonly referred to as charge [303,304,307,308] and Lux [305,306] qubits. We brieLy discuss both approaches and summarize the experimental advances made so far. Charge qubits [303,304]: In this case the qubit is realized by the two nearly degenerate charge states of a single-electron box as shown in Fig. 45. They represent the states |0 ; |1 of
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the qubit. In the computational Hilbert space the ideal evolution of the system is governed by the Hamiltonian EJ H = −PEch; i ((|0 0| − |1 1|) − (|0 1| + |1 0|) (95) 2 where PEch; i =Ech (nx −1=2). Any one bit operation can be realized by varying the external charge nx and, in the proposal of Ref. [303] by varying the Josephson coupling as well. Modulation of EJ is achieved by placing the Cooper pair box in SQUID geometry. The advantage of this choice is that during idle times the Hamiltonian can be “switched o?” completely eliminating any trivial phase accumulation which should be subtracted for computational purposes. As discussed before, a quantum computer can be realized once two bit gates are implemented. The Karlsruhe group has proposed an inductive coupling between qubits which lead to a coupling of the type HC = −EL Ly(1) Ly(2) :
(96)
This type of coupling is very close in spirit to the coupling used in the ion-trap implementation of QC. The main advantage of this choice is that qubits are coupled via an in8nite range coupling and that the two bits can easily be isolated. A di?erent scheme has been proposed in Ref. [304]. They emphasize the adiabatic aspect of conditional dynamics and suggest to use capacitive coupling between gates as to reduce unwanted transitions to higher charge states. The coupling reads HC = −EC Lz(1) Lz(2) and the qubit is now de8ned as a 8nite one-dimensional array of junctions. By means of gate voltages applied at di?erent places in the array the bit–bit coupling can be modulated in time and a Control-NOT can be realized. The experiments on the superposition of charge states in Josephson junctions [309,310] and the recent achievements in controlling the coherent evolution of quantum states in a Cooper pair box [311] render superconducting nanocircuits interesting candidates to implement solid state quantum computers. The experiment by Nakamura et al. [311] goes as follows. Initially, the system is prepared in the ground state. Appropriate voltage pulses bring the system in resonance so that the two charge states are in a coherent superposition a(t)|0 + b(t)|1 . The 8nal state is measured by detecting a tunneling current through an additional probe-junction. For example, zero tunneling current implies that the system ended up in the |0 state, whereas a maximum current indicates that the 8nal state corresponds to the excited one. In the experiment the tunneling current shows an oscillating behavior as a function of pulse length, thereby demonstrating the evolution of a coherent quantum state in the time domain. Nakamura et al. also estimate the dephasing time and report it to be of the order of few nanoseconds. The probe junction and 1=f noise presumably due the motion to trapped charges are the main source of decoherence. In their absence, the main dephasing mechanism is thought to be spontaneous photon emission to the electromagnetic environment. Decoherence times of the order of 1 s should then be possible. Phase qubits [305,306]: A qubit can also be realized with superconducting nano-circuits in the opposite limit EJ EC . An RF SQUID (a superconducting loop interrupted by a Josephson junction) provides the prototype of such a device. The Hamiltonian of this system reads ( − x )2 Q2 H = −EJ cos 2 + + : (97) 0 2L 2C
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Fig. 46. The three-junction Lux qubit. Josephson junctions 1 and 2 have the same Josephson energies EJ and capacitance C; Josephson junction 3 has a Josephson energy and capacitance that are , times larger. The islands are coupled by gate capacitors Cg = 9C to gate voltages VA and VB . The arrows de8ne the direction of the currents. (From Ref. [305].)
Here, L is the self-inductance of the loop and the phase di?erence across the junction (2=0 ) is related to the Lux in the loop. The externally applied Lux is denoted by x . The charge Q is canonically conjugated to the Lux . In the limit in which the self-inductance is large, the two 8rst terms in the Hamiltonian form a double-well potential near = 0 =2. Also in this case the Hamiltonian can be reduced to that of a two-state system. The term proportional to Lz measure the asymmetry of the double well potential and the o?-diagonal matrix elements depend on the tunneling amplitude between the wells. By controlling the applied magnetic 8eld, all elementary unitary operations can be performed. In order to ful8ll various operational requirements more re8ned designs should be used. In the proposal of Mooij et al. [305], qubits are formed by three junctions (see Fig. 46). Flux qubits are coupled by means of Lux transformers which provide inductive coupling between them. Any loop of one qubit can be coupled to any loop of the other, but to turn o? this coupling, one would need to have an ideal switch in the Lux transformer. This switch is to be controlled by high-frequency pulses and the related external circuit can lead to decoherence e?ects. At present, both the Stony Brook group (Friedman et al. [312]) and the Delft group (Van der Wal et al. [313]) have demonstrated superposition of two magnetic Lux states in superconducting loops. One state corresponds to the magnetic moment of A-currents Lowing clockwise whereas the other corresponds the same moment but of opposite sign due to the current Lowing anti-clockwise. Coherent quantum oscillations have not yet been detected. To probe the time evolution, pulsed microwaves instead of continuous ones have to be applied. Observation of such oscillations would imply the demonstration of macroscopic quantum coherence (MQC). It is called macroscopic because the currents are built of billions of electrons coherently circulating within the superconducting ring. There are two main di?erences between the approaches of the two groups. The Stony Brook group uses the excited states of an RF SQUID. The Delft circuit consists of a three-junction
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system and the continuous microwaves induce transitions between the ground state and the 8rst excited state only. The three-junction geometry has the advantage that it can be made much smaller so that it is less sensitive to noise introduced by inductive coupling to the environment. Nevertheless, recent insights indicate that in all designs put forward so far the measuring equipment destroys quantum coherence. The meter is believed to be main obstacle to study the “intrinsic” decoherence times. Future work must evaluate the role of the measuring equipment and new measuring schemes should be developed in order to study MQC in Josephson loops.
Acknowledgements Special thanks go to J.E. Mooij and G. Sch^on for many years of fruitful collaboration, guidance and for sharing their insight and knowledge with us. With C. Bruder, G. Falci, B. Geerligs, G. Giaquinta, T.P. Orlando, A. van Otterlo we had a long-standing fruitful and pleasant collaboration, we would like to thank them for this. We also thank C. Bruder for a careful reading of the manuscript. We acknowledge L. Amico, R. Baltin, P.A. Bobbert, Ya. Blanter, A. Kampf, W. Elion, A. Tagliacozzo, K.-H. Wagenblast, G. T. Zaikin, D. ZappalCa, and G.T. Zimanyi for valuable collaboration on these topics. Finally, we thank O. Buisson, J. Clarke, P. Delsing, A. Fubini, L. Glazman, D.B. Haviland, P. Martinoli, Yu. Nazarov, A. Odintsov, A. van Oudenaarden, B. Pannetier, M. Rasetti, P. Sodano, L. Sohn, M. Tinkham, V. Tognetti for many useful conversations. R.F is supported by the European Community under TMR and IST programmes and by INFM-PRA-SSQI, H.v.d.Z. is supported by the Dutch Royal Academy of Arts and Sciences (KNAW).
Appendix A. Array fabrication and experimental details There are two types of junctions arrays: proximity coupled arrays and arrays made of Josephson tunnel junctions. The proximity coupled arrays consist of superconducting islands (Nb or Pb) on top of a normal metal 8lm (Cu) and are solely used for the study of classical phenomena. The main reason for this is the low (less than 1 Z) normal-state resistance of the junctions. At present there are two technologies to fabricate arrays with Josephson tunnel junctions. Commercial niobium junctions are fabricated using a trilayer with aluminum oxide as insulating barrier. Reliable niobium junctions, however, are too large to observe the quantum e?ects discussed in this review. Quantum arrays are built up of all aluminum, tunnel junctions. These Josephson junctions are fabricated with a shadow-evaporation technique [314]. We will now outline the most recent fabrication technique as used in the Delft group.
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Samples are fabricated on silicon substrates with an insulating SiO2 top layer. In the 8rst step two resist layers are spun on the substrate. The lower layer is a solution of PMMA=MAA copolymer in acetic acid and the upper layer is a solution of PMMA in chlorobenzene. The ◦ resist sandwich is baked at 180 C for 1 h. Then the sample mask is written by high-resolution electron-beam lithography at 100 kV. After writing, the exposed resist is developed in a 1 : 3 mixture of MIBK and 2-propanol for 1 min. The solubility of the lower resist is larger than the upper layer, which leads to an undercut in the lower resist layer. This undercut is necessary for the formation of free-hanging bridges below which the junctions are formed in the shadow evaporation technique. After mask de8nition, a 24 nm thick aluminum layer is evaporated under a given angle. Then the aluminum layer is exposed to pure oxygen at a controlled pressure. By changing the pressure the thickness of the aluminum oxide barrier is varied. In the second evaporation step (the sample is not taken out of the vacuum) a 40 nm thick aluminum layer is evaporated under the opposite angle. After this step, the tunnel junction is formed. The remaining resist layers with the unwanted aluminum on top are removed by rinsing the sample in acetone. We end this appendix with some remarks on the measuring set-up. Arrays are measured in a dilution refrigerator inside &-metal and lead magnetic shields at temperatures down to 10 mK. To protect the arrays from high energy photons generated by room-temperature noise and radiation, extensive 8ltering and placing the arrays in a closed copper box are minimum requirements. Therefore, a typical set-up for the measurements of quantum arrays has the following characteristics. At the entrance of the cryostat, electrical leads are 8ltered with radio-frequency interference (RFI) feedthrough 8lters. Arrays are placed inside a closed, grounded copper box (microwave-tight). All leads leaving this box are 8ltered with RC 8lters for low-frequency 8ltering (R = 1 kZ and C = 470 pF) and with microwave 8lters. A microwave 8lter consists of a coiled manganin wire (length ∼ 5 m), put inside a grounded copper tube that is 8lled with copper powder (grains ¡30 m). The resistance of the wire in combination with the capacitance to ground via the copper grains provide an attenuation over 150 dB at frequencies higher than 1 MHz. The copper box with the RC and microwave 8lters is situated in the inner vacuum chamber and is mounted on the mixing chamber in good thermal contact.
Appendix B. Triangular arrays and geometrical factors In comparing properties of square and triangular arrays some care is necessary. The energy required to store an additional electron on an island is e2 =2C , where C is the sum of the capacitances to other islands and to ground. As in triangular arrays all islands are coupled with z = 6 instead of z = 4 junctions, the required energy is 2=3 times smaller than that of an island in a square lattice. Similarly, the freedom of the phase on a particular island is determined by the Josephson coupling energy of all junctions connected to the island and therefore it seems reasonable to assume that in a triangular array the e?ective Josephson coupling energy is 3=2 times that of a square array. Summarizing, we come to the conclusion that the e?ective EC =EJ -ratio for a triangular array is a factor 4=9 lower than that of a square array.
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Appendix C. Phase correlator In this appendix we show how to evaluate the phase-phase correlator gij introduced in Section 2.2 (see also Ref. [121]). The starting point is gi0 () = exp[ i () − 0 (0) ch j0 +2nj 1 = d j0 D j e−Sch e2i j qxj nj +i[ i ()− 0 (0)] : Zch j0 j
(C.1)
{nj }
By making use of the parametrization i () = i0 + 2ini + Qi () ; + it is possible to verify that all the o?-diagonal elements of the correlation function, viz. gi0 () for i = 0 vanish because of the integrations over j0 . The reason is that Sch does not depend on the phase 0 () itself but only on its time derivative. It is therefore suUcient to calculate the on-site correlation function at site 2 4 1 g() ≡ g00 () = exp2i qxj nj − T n C n exp(−2iTn0 )gc () ; 2 i ij j Zch 8e j ij {nj }
(C.2) where gc (), the correlation function for the case of continuous charges, results from the remaining integral over Q(). It is given by −1 gc () = exp[ − 2e2 C00 (1 − T )] :
(C.3)
By using the Poisson resummation formula N exp− ni Aij nj + 2 zi ni = exp − (qi + zi )A−1 ij (qj + zj ) det A ij i ij {ni }
{qi }
(C.4) and performing the Fourier transform + d exp(i!B )g() g(!B ) = 0
one obtains g(!B ) =
−1 4e2 C00 1 −(2e2 =T ) ij (qi −qx )Cij−1 (qj −qx ) e : −1 2 −1 Zch ] − [4e2 j C0j (qj − qx ) − i!B ]2 [2e2 C00 {qi }
(C.5) The expressions for the coeUcients 8; 9, : and an expansion at small frequencies of Eq. (C.5).
in the coarse-graining approach follow from
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Appendix D. Derivation of the coupled Coulomb gas action In this appendix we brieLy discuss the steps leading to Eq. (23) following the calculation in Refs. [91,92]. First a path integral representation is introduced for the island charges. In terms of phase trajectories i () and charges qi () = Qi ()=2e the partition function takes the form Z= dq j0 Dqj D i () exp(−S { ; q}) ; (D.1) j
i {mi }
where the phases obey the boundary conditions j (0) = j (+) + 2nj , while the charge paths are periodic, qj (0) = qj (+) = qj0 . The e?ective action in the mixed representation is + −1 2 ˙ d 2e qi ()Cij qj () + i qi () i () − EJ cos( i − j ) : (D.2) S { ; q} = 0 i; j
i
ij
Summation over winding numbers {ni } 8xes the charges qi to be integer valued. Starting from the partition function (D.1), we 8rst introduce the vortex degrees of freedom. This can be done by means of the Villain transformation [86] (see also [87]); the time dependent quantum problem requires some additional steps [91,92]. We introduce the lattice with spacing 8 in time direction; this spacing is of order of inverse Josephson frequency: 8 ∼ (8EJ EC )1=2 . In the Villain approximation one replaces , 8EJ F(8EJ ) exp −8EJ [1 − cos( i; − j; )] → exp − |∇ i − 2mi |2 : 2 i; ij;
{mi }
(D.3) Here, we have introduced a two-dimensional vector 8eld mi , de8ned on the dual lattice (alternatively, it can be considered as a scalar 8eld de8ned on bonds). The function 1 1 F(x) = → ; x 1 ; 2x ln(J0 (x)=J1 (x)) 2x ln(4=x) has been introduced to “correct” the Villain transformation for small EJ . As we see, its entire e?ect is to renormalize (increase) the Josephson coupling EJ → EJ F(8EJ ), but it does not a?ect the physics. In the following we will use only the renormalized constant. The rhs. of Eq. (D.3) can be rewritten as , 1 exp − |Ji |2 − iJi ∇ i : 28EJ i; {Ji }
Now the Gaussian integration over the phases can be easily performed, yielding 1 Z= exp −2e2 8 qi Cij−1 qj − |Ji |2 28EJ q i
Ji
i; j;
i;
and the summation is constrained by the continuity equation, ∇ · Ji − q˙i = 0 :
(D.4)
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˙ = 8&−1 [f( + 8& ) − f()]. The constraint The time derivative stands for a discrete derivative f() is satis8ed by the parameterization [88] Ji(&) = n(&) (n∇)−1 q˙i + 8(&B) ∇B Ai :
Here the operator (n∇)−1 is the line integral on the lattice (in Fourier space it has the form i(kx + ky )−1 ), 8(&B) is the antisymmetric tensor, while Ai is an unconstrained integer-valued scalar 8eld. It is important to mention that the continuity equation can be solved in di?erent ways which can be mapped onto each other by gauge transformations (see Ref. [90]). With the use of the Poisson resummation (which requires introducing a new integer scalar 8eld vi ) the partition function can be rewritten as Z= exp − S {q; v} : [qi ; vi ]
The e?ective action for the integer charges qi and vorticities vi is 1 (&) [n (n∇)−1 q˙i ]2 S[q; v] = 2e2 8 qi Cij−1 qj − 28E J i ij 8EJ i (&B) (&) −1 − 2vi − 8 ∇B n (n∇) q˙i 4 ij 8EJ
×Gij
i (&B) 2vi − 8 ∇B n(&) (n∇)−1 q˙j 8EJ
:
The kernel Gij is the lattice Green’s function, i.e., the Fourier transform of k −2 . Finally, after some algebra [92] one arrives to the e?ective action of Eq. (23), which we rewrote, for simplicity, in the continuous notations. Appendix E. E)ective single vortex action It is possible to derive from the coupled Coulomb gas action an e?ective action for a single vortex of vorticity v = ± 1. The single vortex e?ective action includes the e?ect of the interaction with Luctuating charges and the other vortices which are present in the system due to quantum Luctuations. The desired e?ective action is formally obtained by performing the sum in the partition function over all charge and vortex con8gurations excluding the vortex whose dynamics is to be studied. We introduce the vortex trajectory vi; = v(ri − r()) : The single vortex e?ective action can then be written as Se? = −ln 82EJ v vi; Gij (rj − r()) ij;
+ iv
ij;
q˙i; >ij (rj − r()) + iv8
ij;
Ii; · ∇>ij (rj − r())
;
(E.1)
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where the average is to be taken with the full coupled Coulomb gas action. The 8rst term describes the static interaction with other vortices, whereas the second describes the dynamical interaction with charges. This expression, although exact, cannot be evaluated explicitly because of the non-linearity of the action. To proceed we expand the dynamical part of the average in Eq. (E.1) in cumulants up to second order in the vortex velocity r(). ˙ For a uniform external current distribution the result is Se? =
1 a r˙ ()Mab (r() − r( ); − )r˙b ( ) + 2iv8 8ab Ia rb () ; 2
Mab =
∇a >(r() − rj ) qj qk ∇b >(rk − r( )) ;
(E.2)
jk
where a; b = x; y and 8ab is the anti-symmetric tensor [315].
Appendix F. List of symbols Josephson energy Junction capacitance Ground capacitance Capacitance matrix Charging energy (junction) Charging energy (ground) Index labels for the islands in the array Superconducting order parameter Charge on the island External charge Vector potential Flux quantum Magnetic frustration per plaquette External current Junction critical current Quantum of resistance Dissipation strength Josephson plasma frequency BCS transition temperature Vortex unbinding transition Charge unbinding transition Universal conductance at the S–I transition Vortex mass McCumber parameter
EJ C C0 Cij EC = e2 =(2C) −1 E0 = e2 C00 =2 i; j Pei Q Qx Aij = ij A · dl 0 = h=2e f I Ic RQ = h=(4e2 ) , √ !p = 8EJ EC Tc TJ Tch L∗ Mv +c
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Josephson-Junction Array Superconductor–insulator Berezinskii–Kosterlitz–Thouless Aharonov–Bohm Aharonov–Casher Bose–Hubbard
327
JJA S–I BKT AB AC BH
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COLD ATOMS IN DISSIPATIVE OPTICAL LATTICES
G. GRYNBERG, C. ROBILLIARD ElectriciteH de France, Div. R&D, MFTT, 6 Quai Watier, 78400 Chatou, France Energy Conversion Department, Chalmes University of Technology, S-41296 GoK teborg, Sweden
AMSTERDAM } LONDON } NEW YORK } OXFORD } PARIS } SHANNON } TOKYO
Physics Reports 355 (2001) 335–451
Cold atoms in dissipative optical lattices G. Grynberga; ∗ , C. Robilliardb a
Laboratoire Kastler Brossel, Departement de Physique de l’Ecole Normale Superieure, 24 rue Lhomond, F-75231 Paris Cedex 05, France b Laboratoire Collisions, Agregats, Reactivite, IRSAMC, Universite Paul Sabatier, 118 Route de Narbonne, F-31062 Toulouse Cedex 4, France Received October 2000; editor: J: Eichler
Contents 1. Introduction 2. A classical approach to optical lattices 2.1. Atom with an elastically bound electron in a standing wave 2.2. One-dimensional lattices 2.3. Two-dimensional periodic lattices 2.4. Quasiperiodic lattices 2.5. Three-dimensional lattices 3. Sisyphus cooling in optical lattices 3.1. The one-dimensional model 3.2. Kinetic temperature 3.3. Generalization to higher dimensions 3.4. Bright lattices and grey molasses 3.5. Universality of Sisyphus cooling 4. Theoretical methods 4.1. Generalized optical Bloch equations 4.2. The band method 4.3. Semi-classical Monte-Carlo simulation 4.4. The Monte-Carlo wavefunction approach 5. Probe transmission spectroscopy: vibration, propagation and relaxation
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5.1. General considerations on probe transmission 5.2. Raman transitions between eigenstates of the light-shift Hamiltonian 5.3. Raman transitions associated with the vibrational motion 5.4. Propagating excitation. The Brillouinlike resonance 5.5. Stimulated Rayleigh resonances— relaxation of nonpropagative modes 5.6. Coherent transients—another method to study the elementary excitations 5.7. Intense probe beam 6. Temperature, >uorescence and imaging methods 6.1. Temperature 6.2. Fluorescence 6.3. Spatial di@usion 7. Bragg scattering and four-wave mixing in optical lattices 7.1. Basics of Bragg scattering 7.2. Bragg scattering and four-wave mixing
∗
Corresponding author. Fax: +3333-45-350076. E-mail address:
[email protected] (G. Grynberg).
c 2001 Elsevier Science B.V. All rights reserved. 0370-1573/01/$ - see front matter PII: S 0 3 7 0 - 1 5 7 3 ( 0 1 ) 0 0 0 1 7 - 5
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7.3. The Debye–Waller factor—sensitivity to atomic localization 7.4. Backaction of the localized atoms on the lattice beams 8. Atomic interactions in optical lattices 8.1. Light-induced interactions in optical lattices 8.2. Study of collisions 9. E@ect of a magnetic Ield 9.1. Paramagnetism 9.2. Antidot lattices 9.3. Lattice in momentum space
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9.4. The asymmetric optical lattice: an analogue to molecular motors 10. Nanolithography 10.1. The principles of atomic nanolithography 10.2. Experimental achievements 10.3. Latest research directions 11. Conclusion Acknowledgements Appendix A. Index of notations References
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Abstract We present a review of the work done with dissipative optical lattices so far. A dissipative optical lattice is achieved when a light Ield provides both velocity damping and spatial periodicity of the atomic density. We introduce the geometric properties of optical lattices using a classical model for the atoms. We discuss the Sisyphus cooling mechanism and its extension to di@erent optical lattices, and we present the main theoretical approaches used to describe the atomic dynamics in optical lattices. The major experimental tools and studies are then discussed. This includes pump–probe spectroscopy, experiments based on >uorescence, Bragg scattering and studies of atomic interactions. We also present di@erent phenomena occurring in the presence of an additional static magnetic Ield. Atomic nanolithography is Inally brie>y c 2001 Elsevier Science B.V. All rights reserved. discussed as an application of optical lattices. PACS: 32.80.Lg; 32.80.Pj; 32.60.+i Keywords: Laser cooling; Optical lattices
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1. Introduction Optical lattices consist of arrays of atoms bound by light. They oLcially came to the world in 1991–1992, when two groups observed signals originating from atoms spatially ordered in a standing wave [1–3]. In those experiments, light has two e@ects, Irstly to attract the atoms around points located on a periodic lattice having a spatial period on the order of the optical wavelength, and secondly to cool down the atoms. In fact, optical lattices were conceived a few years earlier by Dalibard and Cohen-Tannoudji [4] and independently by Chu and his group [5] when they proposed their famous model to interpret the sub-Doppler cooling recently discovered by Phillips and his group [6]. They showed indeed that two counter-propagating laser beams having crossed linear polarizations can be used to damp the atomic velocity along their axis of propagation and to achieve very low temperatures. Although the fact that atoms should be regularly distributed along the standing wave is a byproduct of their model, this point was not much considered before 1991. In fact, the initial e@ort was mainly focused on the study of the distribution in momentum space rather than in real space. The atomic dynamics in such a light Ield was thus rather neglected. However, in 1990, Westbrook et al. [7] observed that the emission peak of laser cooled atoms was much narrower than the Doppler width, and this was a very serious indication that the atoms are bound around some points where they oscillate (Dicke narrowing [8]). The light Ield conIguration used by Westbrook et al. consists of three pairs of cross-polarized counter-propagating beams, one along each axis, which allows velocity damping along any direction. Such a Ield conIguration was called molasses by Chu et al. [9] because of the friction force that slows down the atoms. In fact, optical molasses and optical lattices are very often two sides of the same object. This object is named molasses when one considers velocity damping and momentum distribution, and lattice when the distribution in real space is investigated. To be more precise, those objects that provide both velocity damping and spatial periodicity can be called “dissipative optical lattices” and their description is the main topic of this review. These dissipative optical lattices are obtained by tuning the incident beams in the neighbourhood of an atomic resonance. By contrast, when the beams frequency is far from any resonance, the velocity damping is no longer eLcient but it is still possible to obtain a periodic atomic structure provided that the atoms are cooled by an independent method. These other lattices are usually called “far o@-resonant optical lattices” and their properties will be reviewed in a forthcoming paper. The light Ield conIguration used to obtain atomic optical lattices can also be adapted to achieve periodic patterns of mesoscopic objects. Many properties of these “mesoscopic optical lattices” are similar to those of “far o@-resonant optical lattices” because in both cases, the main e@ect of the interaction between light and matter is to create a periodic dipole force originating from the dynamic Stark e@ect (see [10], Section II-E). These mesoscopic optical lattices provide a simple and elegant visual demonstration of the lattice structure [11]. They also provide a link with advanced technology [12]. All these lattices share a common feature: the lattice structure and the translation symmetries only depend on the directions of the laser beams [13,14]. Another general result concerns the topography of the electromagnetic forces Ield. This topography is independent of the phases of the laser beams when the minimum number of beams is used, and this number is equal to the dimensionality of the lattice plus 1 [13]. For instance, a phase-independent three-dimensional
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lattice is obtained with four beams. These results are introduced in Section 2 using the classical description of atoms with an elastically bound electron. This model system is used here to describe the general geometric properties of optical lattices. It can often be applied to the far o@-resonant and mesoscopic lattices with only minor modiIcations. The cooling in dissipative optical lattices originates from the Sisyphus e;ect [15]. This process relying on the polarization gradient of the laser Ield is described in Section 3. The starting point is the famous one-dimensional “lin ⊥ lin” conIguration for a transition connecting levels having angular momenta Jg = 1=2 and Je = 3=2 [16]. Several extensions of this model system are presented: to higher dimensions, to other beam conIgurations, to other transitions. In particular, transitions accommodating an internal dark state [17] provide a Sisyphus mechanism with signiIcantly reduced >uorescence. The main theoretical methods used to predict the behaviour of the atoms in an optical lattice are presented in Section 4. They range from approaches where the external degrees of freedom are described in the framework of classical mechanics to more sophisticated methods where a full quantum mechanical treatment is used. Several experimental methods were used to study the properties of atoms in an optical lattice. We start by describing probe transmission spectroscopy (Section 5). This method was applied successfully to several problems, and in particular the study of the vibration of atoms around their equilibrium position and the demonstration of atomic propagating modes. In Section 6, we describe methods involving >uorescence or imaging. These methods were used in particular to measure atomic kinetic temperature and atomic spatial di@usion. Because of the periodic variation of the atomic density in an optical lattice, Bragg scattering can also be used. This technique and its applications are presented in Section 7. Although the separation between Sections 5 –7 is perfectly consistent with respect to the experimental approach, it should be emphasized that the same physical phenomenon (atomic vibration for instance) can often be studied by di@erent methods. Therefore, these sections are tightly connected. The next two sections deal with perturbative e@ects. First, we discuss atom-atom interactions and collision processes in an optical lattice (Section 8). Second, we describe several phenomena occurring when an optical lattice is submitted to a static magnetic Ield (Section 9). This section covers a broad domain which includes paramagnetism, antidot lattices and atomic motors. The last section is devoted to the applications of the optical lattices concepts to nanolithography (Section 10). Experiments showing regular patterns of atoms deposited on a surface are presented. Under many circumstances, we have inserted comments in the core of the text. These comments can be skipped in a Irst reading. They generally concern more elaborate subjects that should be of interest mainly for those working in laser cooling and for those outside this Ield who want to deepen their knowledge on a particular point. An index of the notations used in the paper is presented in Appendix A. 2. A classical approach to optical lattices Although optical lattices were mostly developed and studied in the context of sub-Doppler cooling of atoms which requires a quantum description of the internal degrees of freedom,
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a classical approach performed in the spirit of the dipole force caused by Ield amplitude variations can be useful as a Irst step. In particular, the basis of optical lattices crystallography can be derived from such an approach. The conclusions of this section can also be readily applied to the case of the far-detuned traps for many atoms. 2.1. Atom with an elastically bound electron in a standing wave In this section, we consider the classical model of an atom with an elastically bound electron and we show that the force acting on the atom has a component that derives from a potential. We will show that this dipole potential is proportional to the Ield intensity. The position of the electron in the atomic frame is re and its resonance frequency is !0 . We denote by R the position of the atomic centre of mass and we assume that the atom is immersed in a standing wave E(r; t) = E0 (r) Re[” exp − i(!t − )] :
(1)
In this section we assume that the polarization ” can be space dependent, but not the phase . The electric dipole interaction between the atom and the Ield is VAL = − qe re · E(R; t) ;
(2)
where qe is the electron charge. By solving the dynamics equation for the electron, we Ind for the steady state value of the electric dipole moment d = qe re d = Re[ 0 0 E0 (R)” exp − i(!t − )] ;
(3)
with the following value for the polarizability 0 : qe2 1 0 = − : (4) 2me ! 0 + i=2 In this expression me is the electron mass, = ! − !0 the detuning from resonance and the classicalradiative width [19]. The force F acting on the atom is obtained by averaging −∇R VAL = i = x; y; z di ∇R Ei over a time long compared to 2=!. 0 F = 0 E0 (R)∇R E0 (R) ; (5) 2 where 0 is the real part of the polarizability ( 0 = 0 + i 0 ). Eq. (5) shows that this force derives from a potential U (R), 0 U (R) = − 0 E02 (R) : (6) 4 This is the classical prediction for the dynamic Stark e@ect. For a red detuning (¡0) of the incident Ield, the minima of the potential are found at points where the intensity is maximum. The atom is thus attracted towards the points of maximum intensity. By contrast, for a blue detuning (¿0) 0 is negative and the points of highest intensity correspond to maxima of the potential. The atom is then expelled from the high Ield intensity domains. Comments: (i) When the saturation of the upper level is negligible, this model can be readily applied to a real atom on a transition connecting a ground state of angular momentum Jg = 0 to an excited state for which Je = 1. More generally, it can be applied whenever the light-shift
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e@ective Hamiltonian [18,20] is scalar. This is for example the case for a ground state of angular momentum Jg = 1=2 when the detunings from the Jg = 1=2 → Je = 1=2 and Jg = 1=2 → Je = 3=2 resonances are much larger than the Ine structure of the excited state. If this condition is not fulIlled (i.e. the atom is excited either near a Jg = 1=2 → Je = 1=2 or near a Jg = 1=2 → Je = 3=2 transition) the light-shift Hamiltonian remains scalar if the light Ield polarization is linear (not necessarily in the same direction everywhere). (ii) In the case where the phase is also space dependent, there is an additional contribution F to the force which corresponds to the radiation pressure 0 (7) F = 0 E02 (R)∇R (R) : 2 This component of the force is associated to a transfer of momentum between the Ield and the atoms in a scattering process [21] (see also [10, Section V.C.2]). In the general case, the total force acting on the atom is F = F + F . In the situations where E0 (R) and (R) exhibit variations having similar spatial scales, the ratio F =F is on the order of 0 = 0 ∼||=. If ||, the radiation pressure is then negligible compared to the dipole force. (iii) In fact, the dipole force (Eq. (5)) was derived for a polarization ” = i = x; y; z fi eii ei 2 that i fi ∇R i = 0. If this condition is not fulIlled, ∇R should be replaced by veriIes 2 i fi ∇R i in the expression of the dissipative force (Eq. (7)). (iv) If the Ield arises from the superposition of several beams of (real) amplitude Ei , polarization ”i , wavevector ki and phase i , the dipole force and the dissipative force are, respectively, equal to i(kl − kj )El Ej (”l · ”∗j )ei[(kl −kj ) · R+l −j ] ; (8) F = 0 0 4 l = j
F = 0
0 2
t
El2 kl + 0
0 (kl + kj )El Ej (”l · ”∗j )ei[(kl −kj ) · R+1 −j ] : 4
(9)
l = j
(v) The dipole potential U (R) depends on the Ield intensity but not on its polarization because the atomic polarizability 0 was assumed to be scalar. 2.2. One-dimensional lattices We study in this section the dipole potential generated by two counter-propagating plane waves E1 and E2 having the same polarization ” E1 (r; t) = E0 Re[” exp − i(!t − kz − 1 )] ;
(10)
E2 (r; t) = E0 Re[” exp − i(!t + kz − 2 )] :
(11)
The total Ield is then
1 − 2 E(r; t) = 2E0 cos kz + 2
1 + 2 Re ” exp − i !t − :
2
(12)
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Fig. 1. Dipole potential for two counter-propagating beams of same polarization and frequency. The potential exhibits wells located periodically every =2.
Using Eq. (6), we Ind the following value for the dipole potential: 1 − 2 : U (R) = − 0 0 E02 cos2 kZ + 2
(13)
The potential is periodic (Fig. 1) with a spatial periodicity =2 ( being the wavelength of the incident Ield = 2=k). It consists of a succession of potential wells, each having a depth 0 | 0 |E02 . The possibility to trap atoms in such potential wells was considered by Lethokov et al. [22]. The typical range of potential depths available with usual lasers lies in the mK and sub-mK domain. To have an eLcient trapping, it is thus necessary to get very cold atoms. In principle, the Ields (10) and (11) that create the potential can also be used to cool the atoms. However the cooling mechanism (Doppler cooling [23]) which arises from the radiation pressure is generally not suLcient to achieve the very low temperatures required to trap the atoms. An external cooling mechanism is thus necessary. This can be obtained with additional laser beams using the polarization gradient cooling method that we describe in Section 3. One of the interesting features of Eq. (13) is that the topography of the potential does not change under a variation of the phases 1 and 2 of the incident beams: a phase variation just induces a translation of the potential. More precisely, Eq. (12) shows that a phase shift corresponds to a space translation and a time translation. It is always possible, using the appropriate translations, to write the total electric Ield as 2E0 cos kz Re[” exp − i!t]. In this Ield conIguration the number of independent phases (two) is equal to the number of possible translations. Comments: (i) The topography of the potential is the same for red- and blue-detuned Ields. However atoms are attracted to high intensity domains in the Irst case and to low intensity domains in the second case. (ii) The channeling of atoms along the nodal lines of a blue-detuned standing wave was experimentally observed by Salomon et al. [24]. (iii) The dipole potential changes with the polarization of the counter-propagating beams. For instance, if these beams have crossed polarizations either linear (lin ⊥ lin conIguration) or circular ( + – − or corkscrew conIguration), potential (6) is >at because the Ield intensity is space independent. We will see in Section 3 that this result does not hold when the ground
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state is degenerate and the light-shift Hamiltonian is not scalar. In this case the polarization gradient creates a spatially modulated potential. 2.3. Two-dimensional periodic lattices Several beam conIgurations can be used to generate a periodic 2D lattice. We Irst present (Section 2.3.1) a conIguration with three travelling waves and we show that the corresponding dipole potential has a topography which is invariant under phase variations of the incident beams [13]. We then use crystallographic results to determine the symmetries of the lattice and we show how the primitive translations of the reciprocal lattice are deduced from the beam wavevectors [14] (Section 2.3.2). In the following section (Section 2.3.3) we study the conIgurations with two standing waves i.e. four travelling waves. Although more intuitive than the three-beam conIguration, these conIgurations lead to dipole potentials the topography of which can depend on the relative phases of the beams [25]. Generally, the potentials appear as a periodic array of potential wells; however, it is also possible to generate a periodic array of antidots [26] (Section 2.3.4). Finally we show that the four-beam conIguration allows to generate a superlattice [27], i.e. a periodic potential with two di@erent spatial scales, microscopic potential wells being embedded in potential wells of much larger size (Section 2.3.5). 2.3.1. Phase-independent lattices: three-beam con=guration Consider the case of three waves E1 , E2 , E3 , of same polarization ez and wavevectors k1 , k2 , k3 , located in the xOy plane (Fig. 2a): Ei (r; t) = E0 ez Re[exp − i(!t − ki · r − i )];
i = 1; 2; 3 :
The dipole potential (6) associated with this Ield conIguration is 0 2 U (R) = − 0 E0 3 + 2 cos[(ki − kj ) · R + i − j ] : 4
(14)
(15)
i¿j
The potential consists in a periodic array of two-dimensional wells, as shown by the plot of Fig. 2b obtained for a red detuning ( 0 ¿0). Here again phase variations do not a@ect the topography of the potential but just lead to a global translation. This is because the potential U in Eq. (15) depends on two independent parameters, (1 − 2 ) and (2 − 3 ), which correspond to the two coordinates X0 and Y0 of a space translation. More precisely, the three phases 1 , 2 and 3 of the three-beam conIguration are associated with three free parameters, namely the origin in time and space: this is why the topography of the potential is invariant under phase variations [13]. When the wavevectors k2 and k3 propagate in symmetric directions with respect to k1 : k1 = kex ; k2 = k[ex cos # + ey sin #] ; k3 = k[ex cos # − ey sin #] ;
(16)
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(a) θ
θ
y x
z
θ/
0
0.5 ) /λ
1
Ys
X(
-0.
0 1-co sθ
5
in
-0.5
-1
-1
λ
0.
5
1
(b)
Fig. 2. (a) Three-beam conIguration, (b) Dipole potential in the case of red detuned laser beams. This potential shows a periodic array of potential wells and its topography is not changed by phase variations of the incident beams.
the dipole potential deduced from (15) is 0 2 U (R) = − 0 E0 3 + 2 cos(2K⊥ Y + 2 − 3 ) 4 2 − 3 2 + 3 + 4 cos KX + 1 − cos K⊥ Y + ; (17) 2 2 with K⊥ = k sin # and K = k(1 − cos #). The potential is periodic along the Oy and Ox directions with spatial periodicities ⊥ = =sin # and % = =(1 − cos #), respectively. The symmetry of this potential generally corresponds to a centred rectangular lattice [14]. It becomes a hexagonal lattice for # = =3; 2=3 and a square lattice for # = =2. The fact that the potential depends on two phase parameters (2 − 3 ) and (1 − (2 + 3 )=2) corresponding to a translation along Oy and Ox, respectively, is particularly clear on Eq. (17). Comments: (i) The three-beam conIguration gives both a dipole force and a radiation pressure force. (ii) In the general case where the beams have amplitudes Ei and polarizations ”i which can be di@erent, the dipole potential is equal to 0 2 ∗ U (R) = − 0 Ei + Ei Ej (”i · ”j ) exp i[(ki − kj ) · R + i − j ] : (18) 4 i
i = j
It can be easily checked that the dipole force given by Eq. (8) is equal to −∇R U (R).
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2.3.2. Spatial periodicity and reciprocal lattice Eq. (15) shows that the potential U (R) is invariant under a space translation Rmn = ma1 +na2 (m; n ∈ Z) with (k1 − k2 ) · a1 = 2
(k1 − k3 ) · a1 = 0 ;
(19)
(k1 − k2 ) · a2 = 0
(k1 − k3 ) · a2 = 2 :
(20)
Spatial periodicity is generated through the basis vectors a1 and a2 . Eqs. (19) and (20) also show that the vectors a1? = (k1 − k2 ) and a2? = (k1 − k3 ) are primitive translations of the reciprocal lattice (they form a basis of the reciprocal lattice) [13]. In practice, it is straightforward to characterize the reciprocal lattice from the beam wavevectors, and the primitive translations of the lattice in real space are found from Eqs. (19) and (20). The notion of reciprocal lattice is particularly useful in the context of Bragg scattering [28] (see also Section 7.1). Let us consider an ensemble of atoms immersed in the potential of Fig. 2 and assume that a probe beam with wavevector kp (not necessarily of frequency !) is sent into the atomic cloud. Bragg di@raction occurs along directions kB such as kB = kp + G ;
(21)
where G = pa1? + qa2? (p; q ∈ Z) is a vector of the reciprocal lattice. In fact, in nonlinear optics the origin of the coherent emission along kB will be credited to a multiwave-mixing process involving exchange of photons between the lattice beams. In this approach, Eq. (21) appears as a phase-matching condition [13] for the multiwave-mixing process: kB = kp + (p + q)k1 − pk2 − qk3 :
(22)
The scattering process has involved the absorption of (p + q) photons in the beam E1 and the stimulated emission of p photons in the beam E2 and q photons in the beam E3 . Comments: (i) From the comparison between the Bragg condition and the phase-matching condition, one can directly infer that the translation symmetries of the potential are imposed solely by the beam directions. For a three-beam conIguration, a1? = (k1 − k2 ) and a2? = (k1 − k3 ) are primitive vectors of the reciprocal lattice whatever the Ields amplitudes and polarizations. The determination of the lattice structure (oblique, rectangular, square, hexagonal) is thus determined by k1 , k2 , k3 . Of course, the shape of the potential inside a unit cell depends also on amplitudes and polarizations. (ii) If the three wavevectors do not belong to the same plane, the dipole potential is still modulated only in two dimensions. This is because all the vectors of the reciprocal lattice belong to the plane P containing a1? = (k1 − k2 ) and a2? = (k1 − k3 ) (for example k2 − k3 = a2? − a1? ). This implies that in real space, the potential does not exhibit any spatial modulation along the direction orthogonal to P. To achieve a real 3D potential, four beams at least are necessary. (iii) Several choices for the primitive vectors in the reciprocal lattice are possible. Instead of a1? and a2? , one can for example choose b1? = (k1 − k2 ) and b2? = (k2 − k3 ).
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(a)
y x
z
0
-1 5
Y/
-0.5
λ
0. 5
1
(b)
1
-1
0.5
1
-1
λ
-0.
0 X/
0
-1 5
Y/
-0.5
λ
0. 5
1
(c)
λ
-0.
0 X/
0.5
Fig. 3. (a) Beam conIguration generating a phase-dependent 2D periodic lattice. The Igure is drawn with k1 ⊥ k2 . (b) Dipole potential for red detuned laser beams and 1 = 2 = 3 = 4 = 0. (c) Dipole potential for red detuned laser beams and 1 = 3 = =2; 2 = 4 = 0.
2.3.3. Phase-dependent lattices Consider now the case of four beams E1 , E2 , E3 , E4 of same polarization ez and wavevectors k1 , k2 , k3 = − k1 ; k4 = − k2 (Fig. 3). The phases of the beams are, respectively, 1 , 2 , 3 , 4 . This situation corresponds to the interference of two 1D standing waves (studied in Section 2.2) but propagating along di@erent axis. The dipole potential (6) for this
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situation is
1 − 3 2 − 4 2 U (R) = cos k1 · R + + cos k2 · R + 2 2 1 − 3 2 − 4 1 + 3 − 2 − 4 + 2cos cos k1 · R + cos k2 · R + : 2 2 2 (23) − 0 0 E02
2
This potential is still periodic but it is no longer invariant under phase variations. This is because U depends on three independent parameters (1 − 3 ), (2 − 4 ) and (1 +3 − 2 − 4 ) whereas a space translation can only cope with two. This phase dependence is clearly seen in Figs. 3b and c where we have plotted the potentials for red detuned beams, 1 − 3 = 2 − 4 = 0 and (1 + 3 − 2 − 4 ) = 0 in Fig. 3b and (1 + 3 − 2 − 4 ) = in Fig. 3c. In the four-beam situation, it is necessary to control the phases of the beams to keep the topography of the potential unchanged. Comments: (i) With the three beams E1 , E2 , E3 one achieves a periodic potential and a set of primitive translations for the reciprocal lattice are a1? = k1 − k2 and a2? = k1 − k3 = 2k1 . The addition of the beam E4 introduces a third possible primitive translation a3? = k1 − k4 = k1 + k2 . However a3? belongs to the lattice generated by a1? and a2? because a3? = a2? − a1? . This is why this four-beam lattice remains periodic. The conIgurations where a3? cannot be expanded as a superposition of a1? and a2? with rational coeLcients will be studied in Section 2.4. (ii) Consider the three-beam conIguration with k1 = − k3 = kex and k2 = key . This beam conIguration generates a lattice with a square unit cell (see comment in Section 2.3.1). The same translational symmetry is found with the four-beam conIguration, where a fourth beam is added along k4 = − key . If all the beams have the same phase and the same amplitude, a re>ection symmetry is obtained in the three-beam case, but the lattice is invariant under a =2 rotation only for the four-beam conIguration. This example shows that the translation group of the lattice is determined by the incident beams wavevectors, but the potential may have additional symmetries. A similar distinction is encountered in crystallography where one can associate to each point of a Bravais lattice a basis of atoms with its own point-group symmetry [28,29]. (iii) The topography of 2D four-beam lattices is generally phase-dependent. However there are some exceptions to this rule. Consider the beam conIguration where E1 and E3 are counterpropagating beams travelling along Ox and polarized along Oy, and E2 and E4 are counterpropagating beams travelling along Oy and polarized along Ox. The dipole potential for this situation is 1 − 3 2 − 4 2 2 2 U (R) = − 0 0 E0 cos kX + + cos kY + : (24) 2 2 There are only two independent phase parameters in Eq. (24) and a phase variation just induces a translation of the potential without changing its topography. Such a result is obtained because the two standing waves have orthogonal polarizations and the dipole potential only depends on the Ield intensity. The interference between the two standing waves is thus not crucial for the dipole potential. This property does not hold in the
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0
-1 0 λ
0.5 1
-1
X/
Y/
-0. 5
λ
-0.5
Fig. 4. Dipole potential for the four-beam conIguration of Fig. 3 with 1 = 2 = 3 = 4 = 0. The beams are tuned on the blue side of the resonance. The shape of the potential corresponds to an antidot lattice. Fig. 5. (a) Beam conIguration. The angle # is assume to be small (#¡=2) and cos # = 1 − 1=p with p ∈ N. (b) Section of the dipole potential along the x-axis for p = 3. The potential has a periodicity p but a smaller structure with a dimension =2 is superimposed on the long range variation.
general case of a nonscalar light-shift Hamiltonian because the light-shifts then depend both on the intensity and on the polarization. 2.3.4. Antidot lattices The potential shown in Fig. 3b for red-detuned beams ( 0 ¿0) consists of a periodic array of potential wells. In the case of blue-detuned beams ( 0 ¡0), and for the same values of the phases, one obtains a potential exhibiting antidots (Fig. 4). Instead of wells associated with localized minima of the potential, one Inds here that the minima correspond to continuous lines. Because the atoms can move along these lines, the atomic dynamics is expected to be markedly di@erent from that in a lattice exhibiting wells where the atoms can be trapped. Comments: (i) In the general case, a 2D periodic potential exhibits both localized minima and maxima. There are thus both wells and antidots. If the potential wells are suLciently deep, the trapping in the wells is however dominant in the atomic dynamics. (ii) The notion of antidot lattices is obviously connected to the lattice dimensionality. There exists antidot lattices in 2D and 3D, but not in 1D where a description in terms of wells is always appropriate. 2.3.5. Superlattices We consider again the case of four waves E1 ; E2 ; E3 ; E4 of same frequency and polarization ez , but the beam conIguration (Fig. 5a) is obtained by adding a beam of wavevector k4 = − kex to the three-beam conIguration of Eq. (16).
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Assuming 1 = 2 = 3 = 0 for the sake of simplicity, the dipole potential in this case is U (R) = − 0 0 E02 {1 + 12 cos(2kX − 4 ) + 12 cos 2K⊥ Y + cos K⊥ Y (cos KX + cos[(2k − K)X − 4 ])} ;
(25)
with K⊥ = k sin # and K = k(1 − cos #). This potential is periodic along Oy with a periodicity ⊥ = =sin #, but along Ox, U is periodic only if (1 − cos #) is rational. This is because U is a combination of functions having di@erent spatial periods along Ox. One spatial period is =2 and the other one is % = =(1 − cos #). We assume in this section that the potential is periodic. For instance, in the case where (1 − cos #) = 1=p (p ∈ N), the periodicity of the potential along Ox is p . However, potential minima separated by a distance on the order of =2 are found along a x-section of the potential (see Fig. 5b). The coexistence of this small scale =2 with the large scale % is characteristic of a superlattice. Comments: (i) If (1 − cos #) = q=p, the periodicity along the x-axis is p for q odd and p =2 for q even. (ii) In principle (i.e. mathematically), a periodic lattice is found for any rational value q=p of (1 − cos #). However, from a physical point of view, there are some limitations originating for example from the size of the sample interacting with the beams. If the sample has a dimension L, the periodicity can be observed only if p L. (iii) If (1 − cos #) is not rational, the potential is called quasi-periodic. This type of potential will be considered in Section 2.4. (iv) The connection between the three possible translation vectors of the reciprocal lattice a1? = k1 − k2 , a2? = k1 − k3 and a3? = k1 − k4 is (1 − cos #)a3? = a1? + a2? :
(26)
When (1 − cos #) = q=p (and for instance p and q odd), a possible choice of primitive vectors ? in the reciprocal lattice is ci? = ((a? i )X =q)ex + (ai )Y ey (i = 1; 2). It can be easily checked that ? ? ? ? ? a1 ; a2 and a3 can be expanded on c1 and c2 with integer coeLcients. 2.4. Quasiperiodic lattices As illustrated by Eq. (25), the dipole potential generated by a 2D four-beam conIguration often appears as a superposition of periodic functions with di@erent periodicities. If the ratio of the periods is rational, the potential remains periodic (superlattices case). If the ratio is irrational, the potential is called quasiperiodic. Nevertheless, the quasiperiodic potential exhibits some long range order, as proved by the occurrence of discrete peaks in the Fourier transform of the potential. The concept of quasiperiodic lattices has been investigated in great details [30 –32] after the discovery of incommensurate phases [33] and quasicrystals [34] in solid-state physics. However the mathematical description has started earlier with the work of Bohr [35] on quasiperiodic functions, and it is this approach, in terms of continuous functions, that is more appropriate in the case of optical lattices.
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Although nonperiodic, a quasiperiodic potential is for a physicist very similar to a periodic structure. In particular, there is a discrete but inInite set of translations T for which U (X + T ) and U (X ) are almost equal. This means that a sequence of potential wells found around some point in X0 will be reproduced in an almost identical way in a large number of locations. Under many circumstances, the quasiperiodic dipole potential U is also weakly dependent on the phase of the beams. When the phases i change, a new potential U is achieved which cannot be superimposed to the Irst one U through a translation. However, here again, any sequence of potential wells found in U will be also found almost identically around some other point in U . This property (called “local isomorphism” [36,37]) explains why most physical results do not depend on the phases i in a quasiperiodic potential [38]. Comments: (i) A 2D quasiperiodic potential is the section of a periodic potential in a higher dimension space [39]. For instance, the potential given by Eq. (25) is the section in the plane Z = X − (4 =2k) of the 3D periodic potential U (3) U (3) (X; Y; Z) = − 0 0 E02 {1 + 12 cos(2kZ) + 12 cos 2K⊥ Y + cos K⊥ Y (cos KX + cos[2kZ − KX ])} :
(27)
Indeed, changing the phase 4 results in a translation of the plane used for the cut. The local isomorphism means that all the 2D sections performed with a plane parallel to Z = X are equivalent. (ii) Because three beams are necessary to obtain a periodic potential in 2D and four beams in 3D, we may infer that the most general distribution of four beams in a plane corresponds to a quasiperiodic potential originating from the cut of a 3D periodic potential. In the case of Ive beams in a plane, the periodic potential should generally be found in a 4D space. (iii) A particularly interesting example of a Ive-beam quasi-periodic lattice is found using Ive travelling waves (Fig. 6a) having wavevectors 2n 2n kn = k ex cos (n = 0; : : : ; 4) : + ey sin 5 5 If all the beams have the same polarization ez and the same phase, the potential is invariant in a rotation of 2=5 around the origin. This potential is a 2D section of a 4D periodic potential and all the potentials obtained through phase variations are locally isomorphic. This potential has close connections with the Penrose tiling of the plane [40]. A potential having the same symmetry can be achieved using Ive standing waves instead (Fig. 6b). In this case there are 10 independent phases and the topography of the quasiperiodic potential depends on the relative phases. 2.5. Three-dimensional lattices Most properties of 3D lattices are simple extensions of the results obtained in 2D. The presentation will thus be short. We Irst present the four-beam conIgurations (Section 2.5.1) which always lead to a periodic lattice with a phase-independent topography [13]. We then consider the lattices generated by a number of beams larger than 4 (Section 2.5.2) and we show that they can lead to periodic [41] or quasiperiodic [38] lattices depending on the beam
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Fig. 6. (a) Five-beam conIguration generating a quasi-periodic potential having a Ive-fold symmetry. (b) If standing waves are used instead of travelling waves, the potentials obtained through phase variations are no longer locally isomorphic.
wavevectors. In the last subsection (Section 2.5.3) we show that di@raction of the light by a periodic mask (Talbot e@ect [43– 45]) can also be used to create a 3D lattice [46]. 2.5.1. Phase-independent lattices: four-beam con=gurations We consider an ensemble of four beams having wavevectors ki , amplitudes Ei and polarizations ”i . The dipole potential for this conIguration is given by Eq. (18) where the sum runs on i = 1; : : : ; 4. This expression of U shows that the potential U (R) is invariant under a space translation Rmnp = ma1 + na2 + pa3 (m; n; p ∈ Z) with (k1 − k2 ) · a1 = 2;
(k1 − k3 ) · a1 = 0;
(k1 − k4 ) · a1 = 0 ;
(k1 − k2 ) · a2 = 0;
(k1 − k3 ) · a2 = 2;
(k1 − k4 ) · a2 = 0 ;
(k1 − k2 ) · a3 = 0;
(k1 − k3 ) · a2 = 0;
(k1 − k4 ) · a3 = 2 :
a1? = (k1 − k2 );
a2? = (k1 − k3 )
(28) a3? = (k1 − k4 )
This set of equations shows that and are primitive translations of the reciprocal lattice [13]. As in the 2D case, the reciprocal lattice is currently used to determine the lattice translation symmetries. The primitive translation vectors a1 ; a2 ; a3 in real space are then determined using (28) or equivalently [28,29]. a? ×a? a? ×a? a? ×a? a1 = 2 ? 2 ? 3 ? ; a2 = 2 ? 3 ? 1 ? ; a3 = 2 ? 1 ? 2 ? : (29) a1 · (a2 ×a3 ) a1 · (a2 ×a3 ) a1 · (a2 ×a3 )
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351
Fig. 7. Example of four-beam conIguration. Two pairs of beams symmetrically located with respect of Oz propagate in the xOz and yOz planes, respectively.
The topography of these four-beam 3D lattices does not depend on the beam phases; a phase variation just leads to a translation of the potential (see comment (i)). There are many possible four-beam conIgurations [14]. We describe here a conIguration [42] that is often used. It consists of two beams propagating in the xOz plane, having wavevectors symmetrical with respect to Oz, and two other beams propagating in the yOz plane, with wavevectors symmetrical with respect to Oz (Fig. 7). k1 = k[ex sin #x + ez cos #x ];
k2 = k[ − ex sin #x + ez cos #x ] ;
k3 = k[ey sin #y − ez cos #y ];
k4 = k[ − ey sin #y − ez cos #y ] :
(30)
The knowledge of a1? ; a2? ; a3? (or any other set of primitive translation vectors) allows to determine the lattice structure in the reciprocal space. Using the relationship between the structures in the reciprocal and direct spaces [28], the lattice in direct space can be determined. In the general case, the Ield conIguration of Fig. 7 gives a face-centred orthorhombic lattice [14]. When #√x = #y = # the lattice is tetrogonal. One Inds a face-centred cubic lattice when √ # = arccos(1= 5) and a body-centred cubic lattice when # = arccos(1= 3). More precisely, if we take c1? = k1 − k2 ; c2? = k3 − k4 and c3? = k1 − k3 as a set of primitive vectors in the reciprocal space, the primitive translations c1 ; c2 ; c3 obtained using Eqs. (29) are equal to x + ex − ez ; 2 4 y + ez ; c2 = ey + 2 4 + c3 = e z ; 4 with x = =sin #x ; y = =sin #y and + = 2 =(cos #x + cos #y ). c1 =
(31)
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Fig. 8. Umbrella-like four-beam conIguration.
Comments: (i) In the case of a four-beam 3D lattice, it is possible to cancel all the phases i in expression (18) of the dipole potential through space translations. In fact, one can easily check that a translation T = i = 1; 2; 3 Ti ai where ai are the primitive translations deIned by Eqs. (28) and T1 = − (1 − 2 )=2; T2 = − (1 − 3 )=2; T3 = − (1 − 4 )=2 gives such an expression of the potential. (ii) The four-beam conIgurations usually give rise both to dipole and radiation pressure forces. (iii) In the case where the set of primitive translations a1? ; a2? and a3? lie on the same plane, the potential is modulated along two directions only and the resulting potential may have a phase-dependent topography. (iv) Another four-beam conIguration has been used [13]. This umbrella-like conIguration (Fig. 8) consists of one beam propagating along ez and three beams making the same angle # with Oz and symmetrically located with respect to Oz (i.e. the beam conIguration is invariant in a rotation of 2=3 around Oz). In the general case, the spatial periodicity of the potential corresponds to a trigonal lattice [14]. If the beams are propagating along the symmetry axis of a regular tetrahedron (i.e. when cos # = 1=3), the lattice is cubic body-centred. 2.5.2. Con=gurations with more than four beams Whereas all the four-beam conIgurations lead to the same class of potentials (periodic with a phase-independent topography), the situation is more complex when the conIguration contains more than four beams. In the case of random beam directions, the potential will probably be quasiperiodic but it can be periodic (with possibly a superlattice structure) for some particular sets of wavevectors. Here again, an inspection of the lattice in the reciprocal space is generally useful to predict the lattice structure. Consider a set of n beams (n¿4) with wavevectors ki . All the sites of the reciprocal lattice ? = k − k . If these are obtained through a combination of n − 1 vectors a1? = k1 − k2 ; : : : ; an−1 n 1 vectors can be expanded along three of them with only rational coeLcients, the lattice will be periodic and its topography will under most circumstances depend on the relative phases between the beams [41]. If this is not possible, the lattice is quasiperiodic [38]. We now describe the most widely used conIguration with a number of beams larger than four. This is a six-beam conIguration with three standing waves along the axis Ox; Oy and Oz (Fig. 9). The wavevectors of the six travelling waves are k1 = kex ; k2 = key ; k3 = kez ; k4 = − k1 ; k5 = − k2 and k6 = − k3 . The vectors a4? and a5? of the reciprocal lattice are respectively equal to a3? − a1? and a3? − a2? . The potential generated by this beam conIguration is thus periodic. A set of primitive
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353
Fig. 9. The usual six-beam conIguration consists of three standing waves along the three axis Ox; Oy; Oz.
translations is ey ; ez and ( =2)(ey + ez − ex ), which shows that this is a cubic body-centred lattice. Comment: Although the six-beam conIguration of Fig. 9 generally leads to a potential with a phase-dependent topography, there are a few examples where the topography is phaseindependent (for a scalar light-shift Hamiltonian). This is for instance the case if the beams E1 and E4 are y-polarized, the beams E2 and E5 z-polarized and the beams E3 and E6 x-polarized. 2.5.3. Talbot lattices When a plane wave is sent through a mask displaying a periodic pattern, in many cases the transmitted Ield at a Inite distance from the mask reproduces periodically the pattern of the mask. This is the Talbot e@ect [43– 45] which can be used to achieve a periodic 3D lattice [46,47]. Consider a periodic mask in the z = 0 plane with two independent primitive translations a1 and a2 . Because of the periodicity, the transmitted Ield just behind the mask can be expanded in Fourier series E(x; y; 0; t) = Re Emn ” exp i[(ma1? + na2? ) · r⊥ − !t] ; (32) m; n ∈ Z
where ” is the beam polarization, a1? and a2? are primitive translations in the reciprocal space and r⊥ = xex + yey . After propagation the Ield becomes E(x; y; z; t) = Re Emn ” exp i[(ma1? + na2? ) · r⊥ + kmn z − !t] ; (33) m; n ∈ Z
? with (in the limit ma? 1 ; na2 k) 1 2 2 ? 2 ? ? = k − [m2 (a? kmn 1 ) + n (a2 ) + 2mna1 · a2 ] : 2k
(34)
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If it is possible to deIne kT such as 2 (a? 1) = p1 kT ; 2k
2 (a? 2) = p2 kT ; 2k
(a1? · a2? )2 = pkT k
with p1 ; p2 ; p ∈ Z
(35)
then the distribution of the Ield intensity is identical in the planes z = 0 and z = q(2=kT ); q being an integer. The dipole potential generated by this Ield is thus periodic with spatial periods a1 and a2 in the xOy plane and zT = (2=kT ) along Oz. Note that zT corresponds to the largest value of kT for which Eqs. (35) are fulIlled. ? Comments: (i) In the case of a square lattice (a1? · a2? = 0 and a? 1 = a2 = 2=aM where aM is 2 the spatial period of the mask), the Talbot period zT is zT = 2aM = . In this case, a supplementary translational symmetry of the dipole potential U is found using Eqs. (33) and (34): a1 + a2 zT U (R⊥ ; Z) = U R⊥ + : (36) ;Z + 2 2 √ 2 ? ? (ii) In the case of a hexagonal lattice (a1? · a2? = (a? 1 ) =2 and a1 = a2 = 4=aM 3 where aM is the spatial period of the mask), the Talbot period is zT = 3a2M =2 . (iii) The Talbot lattice is closely connected to the lattices generated by several beams. Indeed the Ield given by Eq. (33) can be considered as the superposition of several beams e + ma? + na? and (complex) amplitudes E . However having wavevectors kmn equal to kmn z mn 1 2 the relationship between the phases of these beams is precisely determined by the transmission function of the mask and thus the topography of the potential is not sensitive to the phases. (iv) In the case where the mask has a random transmission (i.e. is a di@user), the transmitted light is a speckle Ield and one achieves a disordered dipole potential [47,48].
3. Sisyphus cooling in optical lattices The potential depths that can be achieved in an optical lattice are so small that the trapping of atoms generally requires to cool them at very low temperatures. Fortunately, the same laser beams can often be used to create the lattice and to cool the atoms. This result is not surprising because one of the major cooling mechanism makes use of the topography of the optical potential to dissipate energy. In the Sisyphus e@ect [4,5,15], the atoms lose their kinetic energy because they continuously climb potential hills. In the Irst subsection (Section 3.1) we present the 1D lin ⊥ lin conIguration, the famous conIguration used for the Irst analysis of the Sisyphus e@ect. We then present a rough estimate of the temperature in the case of a Jg = 1=2 → Je = 3=2 transition (Section 3.2) and we show that a signiIcant fraction of the atoms are trapped in potential wells. Extensions of this conIguration to 2D and 3D are presented in Section 3.3. Sisyphus cooling in dark or grey lattices is described in Section 3.4 and several conIgurations that di@er from the lin ⊥ lin case are Inally presented in Section 3.5.
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Fig. 10. (a) One-dimensional lin ⊥ lin conIguration. Two beams with cross-polarized linear polarizations are counter-propagating along the Oz axis. (b) The total Ield exhibits a polarization gradient with a =2 periodicity. (c) Light-shifts for a Jg = 1=2 → Je = 3=2 transition. The spatial modulation of the light-shifts originates from the polarization gradient. The extrema of the light-shift correspond to points where the polarization is circular. The Igure drawn for ¡0 shows that, because of optical pumping, the lowest sublevel (i.e. having the largest negative light-shift) also has the largest population.
3.1. The one-dimensional model 3.1.1. Field con=guration: linearly cross-polarized counter-propagating beams We describe here the traditional lin ⊥ lin conIguration used to explain the Sisyphus cooling [16]. Because of its simplicity it is also the starting point for the study of higher dimensional conIgurations. The 1D conIguration consists of two counter-propagating beams having the same frequency ! and the same amplitude E0 and crossed linear polarizations ex and ey (Fig. 10a). By an appropriate translation in phase and in time, it is possible to eliminate the phases of the beams (see Section 2.2) and the total electric Ield can be written as E(z; t) = Re{[E + (z)e+ + E − (z)e− ]exp − i!t } ;
(37)
where e+ and e− are the unit vectors associated with the circular polarizations + and − : e± = ∓
ex ± iey √ ; 2
(38)
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Fig. 11. Square of the Clebsch–Gordan coeLcients for a Jg = 1=2 → Je = 3=2 transition.
and E + and E − are the circular components of E: √ E + (z) = −i 2E0 sin kz ; √ E − (z) = 2E0 cos kz :
(39)
The sites where the light polarization is purely circular are then z = 0; =2; ; : : : for the − polarization and z = =4; 3 =4; : : : for the + polarization (Fig. 10b). In general the polarization is elliptical. Two fundamental and closely connected processes are at the basis of Sisyphus cooling: optical pumping [49] and light-shift [50]. Consider an atom having an angular momentum Jg in its ground state and assume that the Ield given by Eq. (37) is quasiresonant on a Jg → Je transition. Because of optical pumping [49], the atoms are transferred to a Zeeman substate mg = + Jg at points where the light is + polarized and to a substate mg = − Jg at points where the light polarization is − . 3.1.2. The situation of a Jg = 1=2 → Je = 3=2 transition Physical processes are usually studied in the simple case of a Jg = 1=2 → Je = 3=2 transition (Fig. 11). The populations -+ and -− of the two Zeeman substates mg = +1=2 and mg = − 1=2 are solutions of the optical pumping equations + d I I− -+ = p - − − -+ ; dt I I − I I+ d (40) -− = p -+ − - − ; dt I I where I ± = |E ± |2 ; I = I + + I − and p is the optical pumping rate. Its value is p = 29 s0 ; being the radiative width of the upper level and s0 the saturation parameter s0 =
012 =2 : 2 + 2 =4
(41)
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In this expression, = ! − !0 is the detuning from the atomic resonance and 01 is the resonant Rabi frequency for one travelling wave of amplitude E0 and for a transition having a Clebsch– Gordan coeLcient equal to 1. Using -+ + -− = 1; Eqs. (40) can be written as d st ); -+ = −p (-+ − -+ dt d st ); -− = −p (-− − -− dt
(42)
st = sin 2 kZ and -st = cos2 kZ are the steady-state populations for a stationary atom where -+ − located in Z. The coupling between atom and light also produces level shifts [50]. These light-shifts are particularly simple to evaluate in the case of the Jg = 1=2 → Je = 3=2 transition because there is no Raman coupling between mg = − 1=2 and mg = + 1=2 when the Ield has only + and − components. The light-shift Hamiltonian is thus diagonal in the basis (|mg = −1=2; |mg = +1=2) with eigenvalues + I I− U+ = 2˝0 ; + I 3I − I I+ U− = 2˝0 ; (43) + I 3I
where 0 = Ss0 =2 is the light-shift per beam for a closed transition having a Clebsch–Gordan coeLcient equal to 1. Because I + and I − depend on the position R of the atom, U+ and U− are also space dependent. They act as a potential in the same way as the dipole potential considered in Section 2, but this is now a bipotential which depends on the internal state of the atom. When ¡0, the largest population is found in the lowest potential curve. Instead of 0 , one can express the light-shifts in terms of the depth U0 = − 43 ˝0 of the optical potential (Fig. 10c): U0 U± = [ − 2 ± cos 2kZ] : (44) 2 As mentioned before, Sisyphus cooling originates from a combination of optical pumping and light-shift. Consider an atom initially located in the U− potential curve in Z = 0 and moving in the +Oz direction. If its velocity is such that it travels over a distance on the order of =4 in an optical pumping time p−1 , the atom climbs a potential hill, reaches the domain where the light is mostly + polarized and undergoes an optical pumping process which brings it in the potential valley corresponding to U+ . From there, a similar sequence can be repeated. The atom thus almost always climbs potential hills, and therefore its kinetic energy decreases. In fact the kinetic energy is Irst transformed into potential energy and Inally dissipated in the spontaneous emission that accompanies the optical pumping process. An important feature of this mechanism is its nonadiabaticity. Because of the time-lag ∼p−1 associated with optical st evaluated for pumping, the relative populations of moving atoms di@er from the values -±
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atoms at rest. We show in Section 3.2 that the friction force associated with Sisyphus cooling can be calculated from the actual values of -± . Comments: (i) The coeLcient 2=9 that appears in the formula giving the optical pumping rate originates from the Clebsch–Gordan coeLcients of the Jg = 1=2 → Je = 3=2 transition (Fig. 11). To have a less speciIc reference, the photon scattering rate per beam 0 = s0 =2 for a transition having a Clebsch–Gordan coeLcient equal to 1 is often introduced. (ii) The damping of the atomic velocity in the Sisyphus mechanism can also be interpreted as arising from a redistribution of photons between the two incident travelling waves. In a Raman process where the atom absorbs a photon propagating along −Oz and emits a photon propagating in the opposite direction, the atomic momentum is changed by −2˝k. The succession of these Raman processes permits to reduce the atomic velocity. This photon redistribution in the lin ⊥ lin conIguration was studied in [51]. 3.1.3. Case of a Jg → Je = Jg + 1 transition with Jg ¿1=2 In the case where the ground state has an angular momentum Jg ¿1=2, the light-shift Hamiltonian Uˆ is not diagonal because two sublevels m and m + 2 of the ground state are coupled by a Raman process involving the absorption of a photon + and the stimulated emission of a photon − . This implies that the eigenstates |1n of Uˆ do not generally coincide with the Zeeman substates |m. The operator Uˆ can be written as [16] (see Section 4.1) 2
E (R) − + Uˆ (R) = ˝0 0 2 [”∗ (R) · dˆ ][”(R) · dˆ ] ; E0
(45)
where E0 (R) is the (real) Ield amplitude in R; ”(R) is the (generally complex) local polarization + and dˆ is the reduced dipole operator, the matrix elements of eq (R) · dˆ (with q = − 1; 0; 1) being the Clebsch–Gordan coeLcients for the Jg → Je transition (see also Section 4.1). The eigenstates |1n and the eigenvalues Un of Uˆ are generally space dependent: Uˆ (R) = Un (R)|1n (R)1n (R)| : (46) n
The |1n correspond to the adiabatic basis and the Un are the adiabatic energies. We have plotted them in Fig. 12a in the case of the Jg = 4 → Je = 5 transition. Comments: (i) For several problems, the detailed knowledge of |1n (R) and Un (R) everywhere is not necessary. For example, one can be interested only in the dynamics of the atoms close to the minima of the potential curves. Because the minima are located at points where the light has a pure circular polarization + or − , the Raman couplings are small and the light-shift eigenstates are nearly equal to the Zeeman substates |m. Near those points, a reasonable approximation for the light-shift operator is Uˆ (R) Vm (R)|mm| ; (47) m
where the Vm (R) = m|Uˆ (R)|m are the diagonal elements of Uˆ (R). The Vm are often called the diabatic energies (and the Zeeman substates |m represent then the diabatic basis). These diabatic potentials are compared to the adiabatic potentials in Fig. 12.
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Fig. 12. Optical potentials for a Jg = 4 → Je = 5 transition in the lin ⊥ lin conIguration for ¡0. (a) Adiabatic potentials. (b) Diabatic potentials. The two sets of curves are similar away from the level anti-crossings of the adiabatic potentials. In particular, the potentials have a nearly identical curvature at the minima, located at points where the light polarization is circular.
(ii) The picture of velocity damping given in Section 3.1.2 in the case of a Jg = 1=2 → Je = 3=2 transition implies that the atom travels over several potential wells and hills to dissipate its kinetic energy. Another picture involving local cooling is also possible for levels of higher angular momentum. The light-shifts presented in Fig. 12 exhibit several wells around the same site of circular polarization, the deepest well being associated with a level for which |m| = Jg . Because optical pumping tends to accumulate the atoms in this level near the minima while it redistributes the populations among the various levels away from these points, a local cooling scheme where the atom travels over a distance smaller than =4 is possible, as shown in Fig. 13. In this process, the atom climbs the deepest potential well from A to B, is then optically pumped into another well having a smaller curvature where it comes down from C to D, and Inally returns to the deepest well in another optical pumping process [52]. 3.1.4. Motional coupling and topological potential The description of the Hamiltonian motion in the adiabatic basis is not always straightforward, because the |1n (R) are space dependent and nonadiabatic coupling terms proportional to 1m |∇Z 1n thus connect di@erent potential curves (|∇Z 1n is a short notation for ∇Z (|1n )). The order of magnitude of these motional terms is ˝kv with v the atomic velocity (see Section 3.4.2 for an example). As long as ˝kv|Um − Un |, the adiabatic approximation is valid and the motional terms can be neglected.
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Fig. 13. Local cooling for a transition starting from a fundamental level having an angular momentum Jg larger than 1=2 (typically, Jg ¿ 2).
When necessary, it is possible to have a more convenient image in the diabatic basis. This results in the addition of a topological potential [53] to the light-shift Hamiltonian. To understand the origin of this term we start from the Hamiltonian He@ of the atom in the light Ield P2 He@ = Z + Un |1n 1n | ; (48) 2M n and we apply to He@ a unitary transformation T which transforms the adiabatic basis into the diabatic basis: T= |m1m | : (49) m
The transform of the light-shift Hamiltonian gives T Uˆ T † = Un |nn| :
(50)
n
To calculate the kinetic term, we note that 9T [T; PZ ] = i˝ = i˝ |m∇Z 1m | : 9Z m This leads to TPZ T † = PZ + i˝
∇Z 1m |1n |mn| :
(51)
(52)
m; n
After some straightforward algebra, the Hamiltonian in the new representation can be written as 1 THe@ T † = Un |nn| + Ut + Ct ; (53) (PZ + At )2 + 2M n
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Fig. 14. Adiabatic potentials for a Jg = 2 → Je = 2 transition in the lin ⊥ lin conIguration for ¿0. The lowest potential curve is >at because the corresponding eigenstate is not coupled to the excited state. Sisyphus cooling originates from nonadiabatic transitions near the anticrossing points. The two upper curves have a two-fold degeneracy.
with At = i˝
∇Z 1m |1m ;
(54)
m
Ut =
˝2
2M
[∇Z 1m |∇Z 1m − |∇Z 1m |1m |2 ]|mm| ;
(55)
m
and where Ct is an operator that couples di@erent states |m and |n. The matrix elements of At ; Ut and Ct are very small. Typical orders of magnitude for Ut and Ct are, respectively, the recoil energy ER = ˝2 k 2 =(2M ) and k vU (where vU is the mean quadratic velocity). Therefore, when the |Um − Un | are very large compared to k v, U it is legitimate to neglect the e@ect of Ct and to use 1 Ht = Un |nn| + Ut (56) (PZ + At )2 + 2M n as an approximate Hamiltonian. Because of their analogies with the vector and scalar potentials in electromagnetism, At and Ut are called topological potentials. Comment: The e@ect of these topological potentials on the atomic dynamics has been recently observed in a 1D lin ⊥ lin optical lattice Illed with 87 Rb atoms [218]. Indeed, the dependence of the lowest-band tunnelling period on the depth of the adiabatic potential can only be explained by additional gauge potentials. 3.1.5. Transitions accommodating an internal dark state For transitions connecting two levels of same angular momentum (Jg = Je ) with Jg integer and for transitions Jg → Je = Jg − 1, there exists internal dark states [17,54 –56]. These states correspond to a linear superposition of Zeeman substates for which the matrix elements of ”(R) · d (d electric dipole moment) with any substate of the excited level is equal to 0. As a result, the light-shift of these internal dark states |1NC is 0. The adiabatic potentials for a Jg = 2 → Je = 2 transition, displayed in Fig. 14, show the occurrence of such a state (there is one internal dark state for Jg = Je integer and two for Je = Jg − 1). In steady state and for atoms at rest, optical pumping collects all the population in these internal dark states from which they cannot escape by absorbing radiation.
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Sisyphus cooling is also possible in this situation, but it occurs for a blue detuning (¿0) and the mechanism is slightly di@erent [57–59]. Consider an atom moving at velocity v on the >at potential of Fig. 14. Because of the motional coupling a nonadiabatic transition is possible from |1NC to another eigenstate |1C of the light-shift operator (when 1C |∇Z 1NC = 0) and such a transition has a maximum probability to occur near the anticrossing between these two potential curves. The atom then climbs a potential hill in the state |1C and when it returns into |1NC through an optical pumping process, it has lost a fraction of its kinetic energy. Comment: We use the term dark state only in the situations where the internal dark state is also an eigenstate of the kinetic energy operator. In this case indeed, an additional cooling process using velocity selective coherent population trapping (VSCPT) occurs [60]. 3.2. Kinetic temperature The Sisyphus mechanism allows to damp the atomic velocity. To Ind out the temperature, it is necessary to know more precisely the friction force and the >uctuation processes that prevent a complete freezing of the motion. 3.2.1. Friction force for a Jg = 1=2 → Je = 3=2 transition Consider an atom moving at velocity v along Oz in the lin ⊥ lin conIguration. To evaluate the friction force, we calculate the populations -+ (t) = 12 + p(t) and -− (t) = 12 − p(t) for an atom following a trajectory Z = vt: Using Eq. (42), we Ind p2 p kv 1 p(t) = − cos 2kvt − 2 sin 2kvt : (57) 2 p2 + 4k 2 v2 p + 4k 2 v2 If there was no time-lag, the second term in the right-hand side of Eq. (57) would vanish. We assume that the atom evolves in the jumping regime, i.e. that it makes frequent jumps between the two potential curves of Fig. 10c when it travels over one spatial period. In this case, the force acting on the atom is the average value of the dipole force: FU = − -+ ∇Z U+ − -− ∇Z U− : (58) Using Eqs. (44) and (57) and averaging over a time period long compared to (kv)−1 , one Inds [4] p U0 FU = − k 2 v 2 : (59) p + 4k 2 v2 In the limit of low velocity (kvp ), Eq. (59) describes a friction force − v with a friction coeLcient = − 3˝k 2 (=). As expected, damping is found for red detuned beams (¡0). Comments: (i) Eq. (59) shows that the friction is eLcient for atoms having a velocity 2kv 6 p . The velocity vc = p =2k is known as the capture velocity for the Sisyphus mechanism. For atoms faster than vc , Doppler cooling [23,61] can damp the velocities. (ii) An atom with a very weak velocity is submitted to an average dipole force equal to st st FU = − -+ ∇Z U+ − -− ∇Z U− : (60) In this 1D situation, an average dipole potential can be deIned through the relation FU = −∇Z UU with UU = (U0 =4) sin2 2kZ.
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3.2.2. Momentum di;usion coeBcient The most important heating process in the Sisyphus mechanism arises from the >uctuations of the dipole force that occur when the atom jumps from one level to the other. To estimate the di@usion coeLcient associated with this process, we note that when the atom remains in the potential curve U+ for a time p−1 , its momentum changes by an amount (∇Z U+ )p−1 ∼kU0 p−1 . When the atom jumps back and forth between U+ and U− , it thus performs a random walk in the momentum space with elementary steps on the order of kU0 p−1 . Because the time interval between two steps is on the order of p−1 , the momentum di@usion coeLcient D can be estimated to be on the order of k 2 U02 =p . In fact, a precise calculation gives [4] 3 D = ˝2 k 2 0 : (61) 2 The equilibrium temperature T , found from the Einstein relation kB T = D= , is thus kB T = − 12 ˝0 :
(62)
The ratio kB T=U0 = 3=8 is relatively small and a good localization in the potential wells is thus expected. Comments: (i) There are some other contributions to the momentum di@usion coeLcient. As known from the theory of Doppler cooling [16,61], one should also consider the >uctuations of the momentum carried away by >uorescence photons and the >uctuations in the di@erence between the number of photons absorbed in each travelling wave. These two processes lead to a contribution to D which is on the order of ˝2 k 2 0 . However, in the limit ||, the contribution given by Eq. (61) is dominant and the temperature is only a function of 0 . In this limit, T is thus proportional to I=. Such a dependence is no longer expected for very small detunings ||∼ because the additional terms coming from the di@usion processes described above should then be included in the model. (ii) Instead of 0 or U0 , one uses sometimes the light-shift at a point where the light polarization is circular. For the 1D lin ⊥ lin conIguration, = 20 . (iii) In the theory of brownian motion, the spatial di@usion coeLcient Dsp is related to the temperature T and the friction coeLcient through the relation [62] kB T Dsp = : (63) Using Eq. (62) and the value of found in Section 3.2.1, this yields for Dsp M= ˝ (which is a dimensionless quantity) a value equal to 0 =12!R (where !R = ER = ˝ is the recoil frequency). Note however that 0 cannot be inInitely small because we assumed that the atom jumps several times between the two potential curves when it travels over . In fact, the jumping regime condition for delocalized atoms is kvp . Because Mv2 ∼U0 and p ∼0 , this condition can be written 0 =!R ||=. Therefore we expect Dsp M= ˝ ¿ ||=. However, this model has a restricted range of validity. If |0 |=!R is not large enough, there are many atoms for which the friction force is not linear in v (see Eq. (59)) and this leads to a divergence of Dsp . Using the complete friction force (Eq. (59)) and an analogous equation for D, Hodapp et al. [62] were able to derive a formula for Dsp which coincides with the value 0 =12!R for |0 |=!R 1 and shows a divergence for |0 |=!R = 135: Furthermore, from a more general point of view, these models do not describe accurately the contribution of localized
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Fig. 15. Kinetic temperature versus depth of the optical potential in the lin ⊥ lin conIguration, for a Jg = 1=2 → Je = 3=2 transition. The calculation was performed using a semi-classical Monte-Carlo simulation (see Section 4.3).
atoms and more sophisticated methods are often necessary to predict quantitatively the value of Dsp [63]. (iv) Because an atom moving at velocity v interacts in its own frame with two waves having Doppler-shifted frequencies !1 = ! − kv and !2 = ! + kv, the dependence on kv of the friction force (Eq. (59)) implies that for an atom at rest, the redistribution of photons between two waves of frequency !1 and !2 = !1 + : has a dispersive shape when : is swept, with a peak to peak distance equal to 2p . This stimulated Rayleigh resonance has indeed been studied in the context of nonlinear optics [51,64]. 3.2.3. Temperature versus potential depth In Sections 3.2.1 and 3.2.2, it was assumed that an atom makes frequent jumps between the two potential curves of Fig. 10c when it travels over one wavelength. This jumping regime corresponds however to a restricted range of parameters. For a Jg = 1=2 → Je = 3=2 transition, a more general derivation of the kinetic temperature can be obtained by solving the Fokker– Planck equation in the bipotential using a Monte-Carlo simulation [65,66] (see also Section 4.3) or with the band model [67] (see also Section 4.2). These two approaches give a similar dependence for the temperature which varies linearly with U0 for large values of U0 and exhibits a sharp increase at small U0 (see Fig. 15). In the asymptotic regime, the slope kB T=U0 is equal to 0.28, which is slightly smaller than the one predicted in Section 3.2.2. The sharp increase at low U0 is often called “decrochage”. It occurs because in this case the energy dissipated in a Sisyphus process, which is always less than U0 , cannot eLciently compensate the recoil energy transferred to the atom in an optical pumping process. The minimum temperature is on the order of 60ER (which corresponds to an average value of 5:5˝k for the momentum) and is found for U0 95ER . The orders of magnitude rather than the exact numbers are important here because similar values are found in most Sisyphus processes and for most transitions. Comments: (i) Quantitative calculations of the variations of T versus 0 were performed for various transitions with Jg ¿ 1 [52,68,69]. For all of the Jg → Je = Jg + 1 transitions (with ¡0) and for most of the transitions accommodating an internal dark state (with ¿0), a variation similar to the one of Fig. 15 is observed. The only exceptions correspond to situations
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where none of the potential curves is spatially modulated and Sisyphus cooling vanishes. This is the case for the Jg = 1 → Je = 1 transition in the lin ⊥ lin conIguration, but another cooling mechanism (VSCPT) occurs for this transition [60]. (ii) In practical situations, when there are several hyperIne sublevels, it is not always possible to describe quantitatively the temperature with a model using a well-isolated transition. This is in particular the case in the neighbourhood of a transition accommodating an internal dark state because the cooling arising from nonadiabatic couplings has a weak eLciency (see Section 3.4.2). The e@ect of distant transitions can therefore be signiIcant in this case [70,71]. 3.2.4. Oscillating and jumping regimes Because kB T ¡U0 , many atoms are found inside a potential well where they √ oscillate. If we call 0v the oscillation frequency in a well associated to U+ or U− (0v = 2 ER U0 = ˝), the situations 0v p and 0v p are naturally separated. In the Irst case, an atom makes several jumps during an oscillation period and this corresponds to the jumping regime. Note that because U0 ∼Mv2 , the condition for the jumping regime can be written kvp . The opposite condition describes the case where the atom performs several oscillations between two optical pumping processes, and it corresponds to the oscillating regime. In fact, a further inspection of the border between these domains should be performed when considering localized atoms [72]. Consider for example an atom in the − well centred in z = 0. The probability to jump in the U+ potential curve is p I + =I (see Eq. (40)) i.e. p sin2 kz. If the amplitude of the oscillation motion is a (Lamb–Dicke regime), the border between the oscillating and jumping regimes is found for 0v ∼p (ka)2 . The oscillating regime domain has increased to the detriment of the jumping regime. 3.2.5. Distribution of population in the quantum approach In the quantum approach, the description in terms of band structure is appropriate because of the periodicity of the potential (Fig. 16a, see also Section 4.2). If we consider a single potential well, we can Ind a set of eigenenergies En and eigenstates | n . Due to the tunnelling e@ect, the eigenstates of two di@erent wells having the same energy are coupled: this process gives rise to the band structure. However, for U0 ER and for the deepest bound states, the tunnelling rate is generally very weak compared to the photon scattering rate and can be neglected. (The situation can however be di@erent in far-detuned optical lattices [73–79] where dissipation is very small.) One can thus use a description either in terms of band structure or in terms of well localized eigenstates, according to one’s convenience. In the case of the Jg = 1=2 → Je = 3=2 transition, the distribution of population in the various bands (Fig. 16a) was calculated by Castin and Dalibard [67]. Fig. 16b shows the variation of the populations n of the lowest energy bands versus the depth of the optical potential. For example, the population of the lowest band 0 reaches a maximum on the order of 33% for U0 60ER . As expected, for large U0 most of the population is found in the bands associated with the bound levels of a well. The population in each band results from rate equations. In the feeding and emptying processes, two kinds of terms can be distinguished: the Irst terms are associated to optical pumping
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Fig. 16. (a) Position and population of the bands for the Jg = 1=2 → Je = 3=2 transition and for U0 = 100ER . Only the potential curve U− (Z) has been plotted. (b) Population of the lowest bands versus U0 =ER . From Castin and Dalibard [67].
processes inducing a transfer between potential curves and the second terms correspond to the redistribution of population between the various bands inside a given potential. Following Courtois and Grynberg [72], we now study the >ux of population from the level (−) | n in the potential U− , starting with the transfer towards the levels of U+ . From Eq. (40), we estimate this transition rate to be I+ (n− → l+ ) = p n(−) | | n(−) : (64) I l
When | n(−) corresponds to a deeply bound level, it is possible to replace U− (Z) with its harmonic approximation. For example, near Z = 0 we Ind 1 3 3 U− (Z) − U0 + U0 k 2 Z 2 − U0 + M0v2 Z 2 ; (65) 2 2 2 √ with ˝0v = 2 ER U0 . In this limit, | n(−) is nearly equal to the eigenstate |nho of the harmonic oscillator. From Eq. (39) we Ind I + (Z) 2E02 k 2 Z 2 and therefore ER (n− → l+ ) = p (2n + 1) : (66) ˝0v l
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Optical pumping from the deepest bound states is thus much smaller than p because of the Lamb–Dicke factor ER = ˝0v . Consider now the transition rate between two levels of the same potential. An atom in the (−) | n level can absorb a − photon from the lattice beams and spontaneously emit a − photon of wavevector k to end in the | l(−) . The transition amplitude for this process is proportional to l(−) |E − (Z)e−ik · R | n(−) . In the case of deeply bound levels (Lamb–Dicke regime), we can use the eigenstates of the harmonic oscillator and expand E − (Z)e−ik · R in the vicinity of the potential minimum. For example, near Z = 0 E − (Z)e−ik · R = 1 − ik · R + · · ·. The zeroth order transition amplitude di@ers from 0 only if l = n. Therefore, it does not contribute to a redistribution of population. The Irst order term allows transitions from |nho to the neighbour levels |n − 1ho and |n + 1ho . Because of the matrix element ho l|k · R|nho , the transition rate includes the Lamb–Dicke factor. A detailed calculation also including the possibility of absorbing and emitting a + photon gives 11 1 ER (n− → l− ) = 0 n + : (67) 9 2 ˝0v l = n
Because p = (4=9)0 , the two rates given by Eqs. (66) and (67) are on the same order. The lifetime of the deepest levels is thus considerably longer than the average time between two scattering events because of the Lamb–Dicke factor. Note however that the photon scattering rate from a deeply bound level is not reduced with respect to that of a free atom, but most of the scattering events correspond to elastic processes where the Inal state coincides with the initial one. (In the preceding calculation, these processes are associated to the zeroth order term in the expansion versus k · R.) For atoms in these deeply bound states, the Rayleigh scattering should be signiIcantly larger than the inelastic scattering [72]. This result, which has been conIrmed experimentally [3], is strongly connected with the Mossbauer and Lamb–Dicke e@ects [8,80 –82]. Comments: (i) For atoms in the deeply bound levels, the condition for the oscillating regime is 0 (ER = ˝0v )0v which is equivalent to ||. Note that it is possible to have an oscillating regime for the lowest levels and a jumping regime for levels of higher energy. (ii) In the case of the Jg = 1=2 → Je = 3=2 transition, the redistribution of population inside one well (Eq. (67)) and the transfer to the other potential curves (Eq. (66)) have the same order of magnitude. In the case of transitions Jg = J → Je = J + 1 with higher Jg value, the transfer to other potential curves is reduced for tightly bound levels because the Clebsch–Gordan coeLcient connecting mg = J to me = J − 1 is a decreasing function of J . (iii) The populations of the bands show a very smooth variation with the potential depth in Fig. 16b. Such a behaviour is not found for transitions starting from a level of higher angular momentum (typically Jg ¿ 2). Sharp resonances are then superimposed on broader curves similar to those shown in Fig. 16b. These resonances appear when a high energy band of the lowest potential curve is nearly degenerate with a band of another potential curve. Because of their coupling, the feeding of the lowest bands can be highly modiIed [83].
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(iv) In the case of a Jg = J → Je = J + 1 transition with J ¿ 1, there is in addition to Eqs. (66) and (67) a supplementary term that empties bound levels. This term originates from the fact that the adiabatic states do not coincide with the Zeeman states. For instance, the eigenstate |1−J of lowest energy U−J coincides with |m = − J in z = 0 where the polarization of light is − but has a |m = − J + 2 component proportional to E+ (z)=E− (z) near z = 0. Because of this component, an atom can leave a tightly bound level in U−J by absorbing a − photon to reach the U−J +2 ; U−J +1 or U−J potential at the end of the scattering process. Here again the harmonic approximation permits to evaluate this transition rate, and its order of magnitude is similar to the one in Eq. (66). 3.3. Generalization to higher dimensions 3.3.1. Circular components of the =eld—spatial periodicity To summarize, the essential features of the 1D lin ⊥ lin conIgurations are: (i) the + and − components of the total Ield exhibit a periodic variation with the same spatial period, (ii) the maxima of I + are located at points where I − = 0 and vice versa. These features of the Ield are to be found in the generalizations to 2D and 3D of the lin ⊥ lin conIguration. We assume here that the Ield polarization lies in the xOy plane. The Irst step to characterize ∗ · E(r)|2 the lattice consists in the study of the intensity of the two circular components I + (r) = |e+ ∗ · E(r)|2 of the Ield E(r; t) = Re[E(r)e−i!t ]. For example, if the Ield results from and I − (r) = |e− the superposition of several plane waves of real amplitudes Ej , wavevectors kj , polarizations ”j and phases j , we Ind ∗ I + (r) = Ej El (e+ · ”j )(e+ · ”∗l ) exp i[(kj − kl ) · r + j − l ] (68) j;l
and a similar expression for I − (r) obtained by changing e+ into e− . The bipotential for the Jg = 1=2 → Je = 3=2 transition is then readily obtained using Eqs. (43). The results presented in Section 2 can be applied to determine the lattice structure. For example, Eq. (68) shows that the vectors (kj − kl ) belong to the reciprocal lattice. We can take a1? = k1 − k2 ; a2? = k1 − k3 , etc. as primitive translations of the reciprocal lattice and Ind the primitive translations in real space using Eqs. (19) and (20) or (29). The discussion about periodic phase-independent and phase-dependent lattices, superlattices, quasiperiodic lattices presented in Section 2 can be simply adapted here. In particular, a periodic phase-independent lattice is obtained in 2D with three beams and in 3D with four beams [13]. Note that for a lin ⊥ lin conIguration, the primitive cell should at least contain one + well and one − well. Usually, the graphic description of potentials in the case of a Jg = 1=2 → Je = 3=2 transition consists of the map of Uinf = inf {U+ ; U− } together with the points of circular polarizations. Similarly, for transitions Jg = J → Je = J + 1 with J ¿ 1 integer, the map of the lowest potential surface together with the points of circular polarization is usually presented. When it is important to have a scheme involving several potential surfaces, sections of the surfaces are generally shown.
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Comments: (i) In the jumping regime the reactive and dissipative forces are given by the following expressions which generalize Eqs. (8) and (9): F = −˝0
El Ej
E02 l = j
− + i(kl − kj )Tr {(dˆ · ”∗j )(dˆ · ”l ) }
×exp i[(kl − kj ) · R + l − j ] ;
F = ˝0
E2 l
l
E02
−
(69)
+
kl Tr {(dˆ · ”∗l ) − (dˆ · ”l ) }
˝ E1 Ej + 0 (kl 2 2 E 0 l = j
−
+
+ kj ) Tr {(dˆ · ”∗j )(dˆ · ”∗l ) }
×exp i[(kl − kj ) · R + l − j ] ;
(70)
where is the restriction of the density matrix to the ground state (see Section 4.1). A rapid inspection shows that F is equal to −Tr { ∇Uˆ } where Uˆ is the light-shift Hamiltonian deIned in Eq. (45). This equation reminds of Eq. (5); however, contrary to the classical situation, in the multidimensional case F is not necessarily the gradient of a potential. It should also be noticed that F generally di@ers from − n -n ∇Un (where -n is the population of the state |1n ) because of nonadiabatic terms [16]. The equality occurs when the |1n are space independent (Eq. (60) was derived in this situation). (ii) If the optical potential is spatially periodic, the topological potentials are also spatially periodic. Consider the situation where a1 ; a2 ; a3 are three independent primitive translations of the potential. The basis vectors a1? ; a2? ; a3? in the reciprocal space are related to a1 ; a2 ; a3 through Eqs. (29). Because of the periodicity, the eigenstates |1n of Uˆ (R) have a Fourier expansion |1n = Cpnmj exp i pj aj? · R |m : (71) n pj ∈ Z
i = 1;2;3
All quantities are the building blocks of n|At |n = i˝∇1n |1n and n|Ut |n 1n |∇1n which 2 2 = (˝ =2M ) n = n |1n |∇1n | (see Eqs. (54) and (55)) are thus periodic functions with primitive translations a1 ; a2 ; a3 . 3.3.2. Three-beam two-dimensional con=gurations The topography of the potential surfaces is invariant under phase translations in a three-beam conIguration for reasons identical to those presented in Section 2. We start by the description of the conIguration used by Grynberg et al. [13]. Three coplanar ◦ beams of equal amplitudes E0 , making with each other an angle of 120 (Fig. 17a) and linearly polarized in the xOy plane of the beams, create a total Ield with a space dependent polarization.
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G. Grynberg, C. Robilliard / Physics Reports 355 (2001) 335–451 (a)
ey ex
(b) 1.0
+ 0.5
Y/ λ
+
+
-
-
0.0
+ -0.5
-1.0 -1.0
-0.5
0.0
0.5
1.0
X/ λ ◦
Fig. 17. (a) Beam conIguration. The three coplanar beams make an angle of 120 with each other. Their linear polarizations lie in the plane of the Igure. (b) Map of the lowest potential surface. The potential minima, corresponding to bright zones, are found in points where the light is circularly polarized, alternatively + and − .
Indeed, with an appropriate choice of phase, the total Ield is given by Eq. (37) with E0 E − (r) = √ (exp ik1 · r + j exp ik2 · r + j 2 exp ik3 · r) ; 2 E0 (72) E + (r) = − √ (exp ik1 · r + j 2 exp ik2 · r + j exp ik3 · r) ; 2 where j = exp 2i=3. The translation symmetries of I − (r) and I + (r) are those of a hexagonal lattice. Indeed, from the knowledge of the primitive vectors a1? = k1 − k2 and a2? = k1 − k3 of the reciprocal lattice, we deduce from Eqs. (19) and (20) that a1 = (4=3k 2 )k2 and a2 = −(4=3k 2 )k3 are primitive translations for I − (r) and I + (r). They are also primitive translations of the optical potential (Fig. 17b) which is given by Eqs. (43) in the case of a Jg = 1=2 → Je = 3=2 transition.
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371
It should be noticed that the maxima of I + are found at points where I − = 0 and vice versa. More precisely, inside the unit cell a pure + site is found in r = (2a1 + a2 )=3, a pure − site in r = (a1 + 2a2 )=3 and the Ield vanishes in r = 0. The alternation of + and − sites permits a Sisyphus cooling mechanism identical to the one described in Section 3.1.2 to occur. Comments: (i) The lattices exhibiting alternated + and − potential wells are often described as antiferromagnetic. This is because in steady state the atoms are optically pumped into the states |m = Jg and |m = − Jg , respectively. The resulting pattern of atoms has thus the same order as an antiferromagnetic medium. However, in the case of an optical lattice, the order results from optical pumping and not from atom–atom interactions. (ii) Another possible 2D three-beam conIguration that gives rise to a potential with alternated + and − wells is obtained with beams having wavevectors given by Eq. (16). The beam propagating along the x-axis has an amplitude E0 and a polarization ey , while the two other beams have an amplitude E0 =2 and a polarization ez . The lattice is generally centred rectangular with a basis consisting of a + and a − well [14]. 3.3.3. Four-beam two-dimensional con=gurations A 2D lin ⊥ lin lattice can also be generated with a y-polarized standing wave along Ox and x-polarized standing wave along Oy [25]. With an appropriate choice of phases (i.e. of the origin of the space coordinates) the circular components of the Ield are √ E − (r) = 2E0 [iei cos kx + cos ky] ; √ (73) E + (r) = 2E0 [iei cos kx − cos ky] ; where is the phase di@erence between the two standing waves. (If we consider the standing wave as the sum of two counter-propagating travelling waves as in Section 2.3.3, 2 = 1 + 3 − 2 − 4 .) The optical potentials given by Eq. (43) in the case of a Jg = 1=2 → Je = 3=2 transition are thus phase-dependent as expected for a 2D conIguration with more than three beams. It can be easily checked that a pattern with alternated + and − wells is obtained for = ± =2. The potential has the symmetry of a square lattice with a basis consisting of one + and one − potential wells (Fig. 18). In practice, to achieve this type of pattern it is necessary to lock the relative phase of the beams. 3.3.4. The four-beam three-dimensional con=guration The four-beam 3D lin ⊥ lin conIguration is a simple extension of the 1D lin ⊥ lin conIguration studied earlier. The beam wavevectors are arranged according to Eq. (30) and Fig. 7, the polarizations being ey for the two beams propagating in the xOz plane and ex for the two beams propagating in the yOz plane [14,42]. The beams have equal amplitude E0 . The circular components of the Ield are √ E − (r) = 2E0 exp iK− z[cos(Kx x)eiK+ z + cos(Ky y)e−iK+ z ] ; √ (74) E + (r) = 2E0 exp iK− z[cos(Kx x)eiK+ z − cos(Ky y)e−iK+ z ] ; where Kx = k sin #x ; Ky = k sin #y and K± = k(cos #x ± cos #y )=2. The relationship to the 1D lin ⊥ lin conIguration is obvious when the 3D Ield is viewed along a line parallel to Oz
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+
-
+
0.0
-
+
-
-0.5
+
-
+
-0.5
0.0
0.5
Y/ λ
0.5
-1.0 -1.0
1.0
X/λ
Fig. 18. Map of the lowest potential surface in the case of two coplanar standing waves with a linear polarization in the xOy plane and for a phase di@erence = =2.
such as x = y = O. Apart from the distance between two consecutive circular sites which is + =4 (with + = 2=K+ ) instead of =4, one Inds the same Ield conIguration. More generally, the 3D lattice exhibits alternated + and − wells, the spatial periodicity being given by the primitive translation vectors derived in Eq. (31). Maps of the potential in the xOy and xOz planes are shown in Fig. 19. It can be noticed that a potential well is generally not isotropic. Therefore the atomic vibrational frequencies along the three principal axes are generally di@erent (see Eqs. (102) in Section 5.3). Comments: (i) The data for these lattices are sometimes given as a function of the light-shift per beam 0 and sometimes as a function of the light-shift at a point of circular polarization for a transition having a Clebsch–Gordan coeLcient equal to 1. The relation between these quantities is = 80 . (ii) From the knowledge of the intensity per beam I and the detuning , one deduces = (I=Isat )(2 =) where Isat is the saturation intensity. Because Isat and are generally well known parameters, this formula is often used in practice. (iii) Another simple 3D generalization of the 1D lin ⊥ lin situation is obtained by inserting a 2D periodic mask on the trajectory of one of the beams of the 1D lin ⊥ lin conIguration. Because of the Talbot e@ect [43– 45], a periodic 3D conIguration is achieved with alternated + and − sites (Section 2.5.3). 3.4. Bright lattices and grey molasses 3.4.1. Photon scattering rate In the case of a Jg = J → Je = J +1 transition with Jg ¿ 1=2, the majority of atoms are optically pumped into + and − wells where they are trapped. Once in a well, they scatter photons
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373
Fig. 19. Section of the lowest potential surface along the xOy (a) and xOz (b) planes for the 3D four-beam lin ⊥ lin ◦ ◦ conIguration. The maps correspond to #x = 30 and #y = 50 . The spatial periods along Ox; Oy and Oz are x ; y and + =2, respectively.
at the maximum rate because the Clebsch–Gordan coeLcient connecting mg = J to me = J + 1 is equal to 1. The photon scattering from these lattices is high, hence the name bright lattice often given to them. By contrast, in the case of Jg = J → Je = J (with J integer) and Jg = J → Je = J − 1 transitions, atoms are optically pumped into internal dark states which radiate very few photons. Only motional coupling induces absorption from the internal dark states. This is why they are called grey molasses or grey lattices. Molasses here refers to the fact that the majority of atoms are in a >at potential surface because the light-shift of an internal dark state is equal to 0. Hence no localization is expected. In fact, this is not exactly true. First, the topological potential (Section 3.1.4) generally di@ers from 0 so that the potential surface associated with an internal dark state is not perfectly >at [84]. Second, the other potential surfaces are spatially modulated (and this is indeed necessary for the Sisyphus e@ect to occur, see Fig. 14). Therefore the atomic population may be modulated and the term grey lattice can also be used for these systems.
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Fig. 20. The lin # lin conIguration consists of two counter-propagating beams with linear polarizations making an angle # with each other.
Dark lattices are a special case where an eigenstate of kinetic energy can be found from the internal dark state. Because very low temperatures can be achieved in this case using VSCPT [60], the topological potentials are particularly signiIcant in this situation [53]. Comment: The addition of a small magnetic Ield to a grey molasses leads to a potential which is spatially modulated even for the internal dark state [85]. In 2D and 3D, this potential consists of antidots rather than wells in most conIgurations (see Section 9.2). 3.4.2. Blue Sisyphus cooling—a one-dimensional model The lin # lin conIguration consists of two linearly polarized beams counterpropagating along Oz, the angle between the polarizations being # (Fig. 20). To Ix the notations we take e1 = cos #=2 ey + sin #=2 ex and e2 = cos #=2 ey − sin #=2 ex as the polarization vectors. If the beams have equal amplitude E0 , the circular components are √ # ± E (z) = E0 2 cos kz ± : (75) 2 Although there exists sites of pure circular polarization, they generally do not coincide with the minima of the potentials as it can be checked in the case of the Jg = 1=2 → Je = 3=2 transition using Eqs. (43) and (75). This Ield conIguration is in fact well suited to understand the Sisyphus cooling of grey and dark molasses. For the sake of simplicity, we consider the Jg = 1 → Je = 1 transition. It can be noticed that for a Ield with only + and − components, the population of the |Jg ; mg = 0 sublevel vanishes in steady state because there is no >uorescence from |Je ; me = 0 towards |Jg ; mg = 0 (the Clebsch–Gordan coeLcient is 0). We can thus describe the cooling process within the subspace of the Zeeman components | + 1 and | − 1 of the ground state. The eigenstates of the light-shift operator are 1 # # | + 1 + cos kz − | − 1 ; |1NC = cos kz + 2 2 D(z) 1 # # | + 1 − cos kz + | − 1 ; |1C = cos kz − (76) 2 2 D(z) where D(z) = 1 + cos # cos 2kz. The corresponding adiabatic energies are UNC = 0 ; UC = ˝0 D(z) :
(77)
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375
Because UNC = 0, |1NC is the internal dark state. The motional coupling between |1NC and |1C is equal to d |1NC : (78) WMC = − i˝1C | dt For an atom moving at velocity v we have, using t = Z=v and Eqs. (76) kv sin # WMC = − i˝ : (79) D(Z) Because D(Z) is minimum near the anticrossings between UNC and UC , the transition probability from |1NC to |1C is maximum in these points (see Fig. 14). In the case of a blue detuning (0 ¿0), an atom transferred into |1C then climbs a potential hill. Because the photon absorption rate C = 0 D(Z) is maximum at the top of the potential hill, the probability to return into |1NC is maximum near these points and on the average, the kinetic energy of the atom decreases. −1 Comments: (i) Because of motional coupling, the atoms in |1NC have a Inite lifetime NC given by |WNC |2 NC : (80) = C 2 UC + (˝2 C2 =4) This formula can be used to calculate the friction force. For example, in the range 0 kv0 , ˝ , which leads to a friction the kinetic energy loss per unit time is on the order of NC 0 coeLcient on the order of −˝k 2 (=). This friction is weaker than the one found in the case of a bright lattice (Eq. (59)) by a factor (=)2 . However the momentum di@usion is also reduced and the temperature is still predicted to be on the order of ˝0 [86]. (ii) The topological potentials can be calculated for the eigenstates of Eq. (76) using Eqs. (54) and (55). In particular, for the internal dark state |1NC , one Inds At = 0 and sin2 # : (81) [1 + cos # cos 2kZ]2 As expected the order of magnitude of Ut is the recoil energy. The maxima Ut are found for 2kZ = (2p + 1) (p integer), which corresponds to the minima of the Ield intensity. Ut = ER
3.5. Universality of Sisyphus cooling Sisyphus cooling is a very general cooling process which can be accomplished in many Ield conIgurations. Although the basic ideas were already given in the case of the lin ⊥ lin model system (existence of several potential surfaces, of a dissipation process transferring the atoms to the lowest potential surface, nonadiabaticity), it is however interesting to describe a few other conIgurations either because they are often used or because they exhibit particular characteristics. 3.5.1. Six-beam molasses In many experiments, cooling is achieved using a molasses consisting of two counterpropagating beams along each of the three axis [9]. For most situations (i.e. choices of
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polarization and transition), the optical potentials are spatially modulated. Because of the combined e@ects of di@erent light-shifts and optical pumping rates, Sisyphus cooling occurs for a red detuning in the Jg = J → Je = J + 1 case and for a blue detuning in the Jg = J → Je = J and Jg = J → Je = J − 1 situations. Because of its translational symmetries (see Section 2.5.2), this structure can be described as a lattice. However, in most experiments the phases of the beams vary randomly. Hence the topography of potentials, the Ield polarization and the optical pumping rates are not Ixed and the experimental results generally correspond to an average over the phases. In that sense, the term molasses is appropriate because the relevant properties do not depend on the precise shape of the potentials. Comments: (i) Although this Ield conIguration does not generally yield alternated + and − sites, a very eLcient Sisyphus cooling is however obtained [87]. (ii) The properties of a magneto-optical trap were studied experimentally by Schadwinkel et al. [88] for several values of the relative phases between the six incident beams. It appears that in most cases, the magneto-optical trap behaves as an optical lattice. In particular, there are potential wells in which the atoms are localized. 3.5.2. Potential wells: from circular to linear polarization The 3D Rot [lin ⊥ lin] four-beam conIguration described in this paragraph is interesting because di@erent patterns of potential, leading to di@erent cooling schemes, are obtained according to the angles between the beams and the choice of polarizations. In this conIguration, the deepest potential wells can have a linear polarization. Therefore the optical pumping rates near the bottom of the wells are not reduced by a Lamb–Dicke factor as in Eq. (66) and the parameter range in which a jumping regime is found is much broader than in the lin ⊥ lin case [89]. This Ield conIguration is obtained from the 3D four-beam lin ⊥ lin conIguration (Section ◦ 3.3.4) by rotating all the polarizations by 90 . The beam wavevectors are thus still given by Eq. (30) but the polarizations of the two beams propagating in the xOz plane lie in the xOz plane and similarly the beams propagating in the yOz plane have their polarizations in this yOz plane. The main di@erence with the lin ⊥ lin case is that the Ield now has a component. In the case where #x = #y = #, the components of the Ield are √ E ± (r) = 2E0 cos #[ ± i sin(K⊥ x) exp iK+ z − sin(K⊥ y) exp − i K+ z] ; E (r) = 2E0 sin #[cos(K⊥ x) exp iK+ z + cos(K⊥ y) exp − iK+ z] ;
(82)
with K⊥ = k sin # and K+ = k cos #. In the case of a small angle #, the component remains small and the potentials have a pattern similar to those of the lin ⊥ lin conIguration with ◦ alternated + and − wells. By contrast, when # ¿ 45 , these wells are no longer attractive in the three dimensions and the atomic localization occurs near -polarized sites [14]. Contrary to the case of + and − sites where all the atoms are optically pumped in a single sublevel (m = J or −J ), the atoms are distributed over several sublevels in a site where the polarization is linear. The population increases when |m| decreases but a signiIcant fraction of the population is found outside the lowest potential surface. For example, in the case of the Jg = 1 → Je = 2 transition, the fraction of atoms in m = 0 is 9=17.
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377
Comments: (i) A local cooling mechanism similar to the one shown in Fig. 13 is certainly important in this conIguration. (ii) Eq. (82) for the Ield is relative to the (Ixed) z axis. It should however be noticed that in any point r, the Ield actually oscillates in a plane and an expansion in terms of two local circular polarizations is also possible. This is because the Ield can always be written as E(r; t) = Re[(EP (r) + iEQ (r))e−i!t ] ;
(83)
where EP (r) and EQ (r) are real vectors. In r, the Ield oscillates in the plane generated by EP (r) and EQ (r). 3.5.3. Magnetically assisted Sisyphus e;ect In the Sisyphus cooling of transitions with an internal dark state (Sections 3.1.5 and 3.4.2), the atoms are optically pumped towards the lowest potential curve while motional coupling transfers the atoms into the other potential curves, allowing the Sisyphus process to occur. Similar e@ects are found in magnetically assisted Sisyphus e@ect (MASE, also known as MILC for magnetically induced laser cooling) [90]. However this scheme also works with Jg = J → Je = J +1 transitions and red detuned beams. Consider for example a 1D + polarized standing wave and a Jg = 1=2 → Je = 3=2 transition. The atoms are optically pumped in the mg = +1=2 sublevel which has the largest light-shift (see Fig. 11). However, at the nodes of the standing wave the | + 1=2 and | − 1=2 substates have the same energy so that a small transverse magnetic Ield can mix the two states eLciently. The adiabatic eigenstates are then space dependent and the transfer from the lowest potential curve to the higher one due to motional coupling can be signiIcant near the nodes of the Ield. The Sisyphus e@ect proceeds as follows: the atoms climb a potential hill of the lower potential curve, undergo a motional-induced transition near the nodes of the Ield, fall down in a shallower potential well of the higher potential curve before being optically pumped into the lower potential curve by the + light near an antinode of the Ield. On the average, the kinetic energy has decreased. Comments: (i) In steady state most atoms are trapped in a well associated with an eigenstate which is nearly | + 1=2 near the bottom of the well. Because the magnetic moments of all the trapped atoms are nearly parallel, this type of lattice is often called ferromagnetic. (ii) Ferromagnetic optical lattices do not necessarily require an external static Ield. An example of 3D four-beam lattice where the atoms are trapped in + sites only was reported by Grynberg et al. [13]. 4. Theoretical methods In this section, we describe several methods used to study theoretically the dynamics of an atom with a degenerate ground state in an electro-magnetic Ield resulting from the interference of several waves of same frequency !=2. In the Irst paragraph, we present the basic approximations leading to the master equation (Eq. (92)) for the ground state density matrix. The two following paragraphs are devoted to two methods often used to solve this optical pumping equation. The Irst one, the band method,
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consists of a quantum treatment of all atomic degrees of freedom while the second one considers the atomic centre-of-mass motion classically. We Inally present an alternative approach to the resolution of the master equation: the Monte-Carlo wavefunction approach. 4.1. Generalized optical Bloch equations Let us consider atoms having a closed transition of frequency !0 between a fundamental state g with an angular momentum Jg and an excited state e with an angular momentum Je . The external degrees of freedom of the atom are quantized, so that the position and momentum operators R and P do not commute. The atoms evolve in the presence of a laser Ield of frequency ! = !0 + . The interaction between the atom and the laser Ield is treated classically using the electric dipole Hamiltonian in the rotating wave approximation: + − Vˆ AL (t) = Vˆ AL (t) + Vˆ AL (t) = − d+ · E(+) (R; t) − d− · E(−) (R; t) :
(84)
In this equation, E(+) (resp. E(−) ) represents the positive (resp. negative) frequency component of the complex laser Ield and d+ (resp. d− ) the upwards (resp. downwards) component of the atomic dipole. One can also write the atomic dipole d as ++d −) ; d = D(d (85) + are deIned where the reduced matrix element of the dipole D and the reduced atomic dipole d through the Wigner–Eckart theorem: Je d+ Jg ; (86) D= √ 2Je + 1 + · e |J ; m = J ; m |J ; 1; m ; q with q = 0; ±1 : Je ; me |d (87) q g g e e g g
The vacuum electro-magnetic Ield is quantized and is considered as a reservoir inducing >uctuations, hence dissipation, in the evolution of the atomic system. In particular, spontaneous emission originates from the atom–vacuum interaction. The equations describing the interaction between atoms and light are called generalized optical Bloch equations [10], and they determine the evolution of the density matrix @. This density matrix can be written as @ee @eg @= ; (88) @ge @gg where @gg and @ee are the matrices containing the populations and the Zeeman coherences of the fundamental and excited states, respectively, and @ge = @†eg contains the optical coherences. Using a unitary transformation (@gg → @gg ; @ee → @ee and @eg → @eg ei!t ) which is equivalent to using a frame rotating at frequency !, one can obtain a time-independent expression for the optical Bloch equations: d@ee 1 (+) 1 P2 (−) (89) = ; @ee + [Vˆ AL @ge − @eg Vˆ AL ] − @ee ; dt i˝ 2M i˝ d@eg 1 ˆ (+) 1 P2 (+) ˆ @eg ; (90) = ; @eg + [V AL @gg − @ee V AL ] + i − dt i˝ 2M i˝ 2
G. Grynberg, C. Robilliard / Physics Reports 355 (2001) 335–451
d@gg 1 (−) 1 P2 (+) = ; @gg + [Vˆ AL @eg − @ge Vˆ AL ] dt i˝ 2M i˝ 3 − · e∗ )e−i · R @ ei · R (d + · e) : + d 2 0 (d ee 8
379
(91)
e⊥
The last term of Eq. (91) describing the e@ect of spontaneous emission is integrated over the solid angle 0 in which a spontaneous photon of wavevector and polarization e is emitted. Before going to the next step, it is important to note that Eqs. (89) – (91) take into account all the mechanisms concerning the evolution of atomic observables and do not contain any radical approximation on laser parameters or atomic velocities. In most cases, however, the saturation parameter is small (s(R) = (0(R)2 =2)=(2 + 2 =4)1 where 0(R) is the total resonant Rabi frequency) so that the atoms spend most of their time in the ground state. Furthermore, atoms in optical lattices are generally much colder than the Doppler limit, so that their Doppler-shift kv is negligible compared to the natural width . When these two conditions are fulIlled, it is possible to eliminate adiabatically the excited state and the optical coherences: Eq. (91) indicates that the evolution time of @gg is on the order of the optical pumping rate AP −1 while @ee and @ge evolve much faster, with rates on the order of (see Eqs. (89) and (90)). After the adiabatic elimination, one obtains an equation related only to the restriction of the density matrix @ to the fundamental state, @gg [16,91]. From now on, we use the simpliIed notation = @gg . Let us remark that is a square matrix of dimension 2Jg +1 whose diagonal elements are the populations of the Zeeman sub-levels of the fundamental state, the o@-diagonal elements corresponding to Zeeman coherences (light-induced couplings between the Zeeman sub-levels). The master equation for the ground state density matrix is d 1 d : (92) = [H ; ] + dt i˝ e@ dt relax Eq. (92) has the same structure as the traditional optical pumping equation [18,16] even though the centre-of-mass motion is here quantized. We now analyse the di@erent terms of Eq. (92). • The Irst term corresponds to the Hamiltonian evolution due to the e@ective Hamiltonian
He@ = P2 =2M + Uˆ (R). The light-shift operator Uˆ (R) may also be written as Uˆ (R) = ˝ (R)u(R), ˆ where (R) = s(R)=2 and u(R) ˆ is the dimensionless light-shift operator: − · ”∗ (R)][d + · ”(R)] : u(R) ˆ = [d
(93)
Note that because the atomic position R and momentum P are quantized, the light-shift operator and the kinetic energy operator do not commute. Consequently, one cannot write the eigenstates of the total Hamiltonian as a product of the eigenstates of the light-shift and of the kinetic energy. The vector noted ”(R) represents the polarization of the laser Ield at point R.
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• The second term of Eq. (92) is a dissipative term and can be written as d (R) =− {u(R); ˆ }
dt
relax
2
3 (R) + 8
d 2 0
† Bˆ e (R)e−i · R ei · R Bˆ e (R) ;
(94)
e⊥
where (R) = s(R)=2 and {A; B} = AB + BA. The Irst term on the right-hand side in Eq. (94) describes the depopulation of the fundamental state due to the absorption of photons in the laser Ield, while the second term corresponds to the repopulation of the fundamental state by spontaneous emission. The nonhermitian operator Bˆ e (R) is − · ”∗ (R)][d + · e] : (95) Bˆ e (R) = [d There is generally no analytic solution to this equation, even in simple cases. Some supplementary approximations as well as the use of numerical simulations are thus necessary. We do not describe the technique consisting of a direct integration of the equation after discretization of the momentum space [92,93]. This method is indeed extremely power demanding and its 3D generalization cannot be implemented on usual computers. 4.2. The band method This fully quantum method, developed in the secular limit, takes advantage of the strong analogy between optical lattices and periodic solid state materials. It has been used for the Irst time in the frame of laser-cooled atoms by Castin and Dalibard [67,92] in a one-dimensional conIguration. The basic idea is that an atom experiencing a periodic optical potential satisIes to the Bloch theorem. One can thus develop the wavefunction for the atomic position on the basis of the Bloch states, which leads to the existence of allowed and forbidden energy bands for the atoms, as for the electrons in solid-state physics. The resolution of the master equation is made perturbatively in three steps. First, one calculates the eigenstates |n; q; m (n ¿ 0 being the band number, q the Bloch index in the Irst Brillouin zone and m the internal state) and the energy spectrum En; q; m of the Hamiltonian He@ in Eq. (92). One then takes into account the relaxation part of the master equation by calculating the rates of transfer Cn; q; m → n ; q ; m from |n; q; m to |n ; q ; m . One Inally obtains the steady-state populations (n; q; m) by solving the rate equations 0 = (n; ˙ q; m) = − Cn; q; m → n ; q ; m (n; q; m) + Cn ; q ; m → n; q; m (n ; q ; m ) : (96) n ;q ; m
n ;q ; m
All steady-state quantities, such as population and momentum distributions, can then be deduced from this set of eigenstates and steady-state populations. An inspection of the band method shows that this approach is valid if the Hamiltonian evolution dominates the dynamics, i.e. if the Bohr frequencies of the Hamiltonian are much larger than the damping rate . In this case, one can neglect the nondiagonal elements of the density matrix n; q; m| |n ; q ; m with (n; q; m) = (n ; q ; m ). This approximation is called
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the secular approximation [10] and is generally valid for large detunings. More precisely, in one-dimensional situations √ the energy splitting between two states is on the order of the oscillation frequency 0 ˙ U0 ER of the atoms in a potential well, which leads to the condition v U0 =ER =. The increase of the band degeneracy in higher dimensions makes the extension of the band method to two and three dimensions problematic: in 2D situations, for example, the average energy splitting between two bound states is on the order of the recoil energy ER , leading to a condition U0 =ER |=| [93]. The condition becomes even more restrictive in 3D. On the contrary, the secular approach can be applied to transitions with angular momenta higher than 1=2 without any particular diLculty. It has been often used, in particular to predict pump–probe spectra [72,83] and to study the temperature and the magnetism of atoms in optical lattices [69,94,95], and led to results in good agreement with the experiments. 4.3. Semi-classical Monte-Carlo simulation Here, the solution of the master equation for the ground state density matrix is evaluated by calculating the temporal evolution of a given number of atoms. In the semi-classical approach, only the internal degrees of freedom of the atom are treated using quantum mechanics, the external degrees of freedom being treated semi-classically. The validity of this approximation requires that spatial coherence length of the wavefunction associated with the atom position to be small compared to the optical wavelength , which, through the Heisenberg inequality, implies the SP ˝k. This condition means that the momentum change due to the absorption or the emission of a photon has to be small compared to the momentum distribution width. In this regime, it is convenient to use the Wigner representation, which is particularly well adapted to the semi-classical approximation u 1 3 u iP · u W (R; P; t) = d 3 u R + (t) R − exp − : (97) 2˝ 2 2 ˝ W (R; P; t) is thus a matrix representing a quasiprobability distribution in phase space. Applying the Wigner transformation to the master equation, one obtains an equation for the evolution of the Wigner transform W (R; P; t) of . This equation is nonlocal in momentum P, but in the regime where a semi-classical approximation is justiIed, one can expand the equation up to second order in the small parameter ˝k=SP. One more approximation is necessary to get Fokker–Planck-like equations for the quasipopulations -i (R; P; t) = 1i |W (R; P; t) |1i of the eigenstates |1i of the light-shift operator: the adiabatic approximation, meaning that one neglects the motion-induced couplings between two adiabatic states |1i and |1j . This approximation amounts to neglect nondiagonal elements in the Wigner distribution in the adiabatic basis, provided one adds to the potential topological terms (see Section 3.1.4) [52]. In this model, the atom is thus submitted to an optical potential, to radiation pressure forces and to optical pumping that can induce jumps between di@erent sublevels. In addition, a momentum di@usion coeLcient takes into account the recoil of the atom during elementary absorption
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or emission processes. This momentum di@usion coeLcient is simulated in the calculations by random forces with a zero average. The semi-classical Monte-Carlo simulation has been frequently used [26,52,93,96 –99]. The advantage of this method is that one can easily control each term and thus determine the precise origin of an observed e@ect. Note however that the validity regime of the adiabatic approximation becomes narrower for transitions with high angular momenta [52] because the energy separation between the levels is smaller: the probability for the atom to undergo a nonadiabatic transition is then all the more important. Nevertheless, in situations similar to the lin ⊥ lin conIguration, the localized atoms hardly attain the anti-crossing points, so that the semi-classical Monte-Carlo simulations remain valid to study the e@ects originating from the majority of atoms which are localized (temperature for instance). Comment: For Jg ¿ 1, the full set of potential curves can often be replaced by an e@ective bipotential [52]. The lowest curve of the bipotential corresponds to the lowest adiabatic potential, the upper curve being the average of the other potentials with a weight proportional to their populations.
4.4. The Monte-Carlo wavefunction approach Introduced in 1992 [100], this method allows to handle a wide variety of dissipative problems in quantum optics without integrating the master equation Eq. (92). It consists of replacing the calculation of the atomic density matrix by the calculation of the temporal evolution of a statistical ensemble of wavefunctions. The required information for a quantity A is then obtained by averaging the corresponding observable Aˆ over this statistical ensemble, instead of calculating ˆ the trace of A. Although a wavefunction approach is apparently incompatible with the existence of dissipation, spontaneous, emission is taken into account here by the addition of a stochastic element to the Hamiltonian evolution of the “atom+quantized Ield” system: after a short phase dt during which the system has evolved under the e@ect of the atom-laser Hamiltonian, one calculates the probability dp that a spontaneous photon was emitted during dt (dt is chosen small enough to guarantee dp1). Comparing dp and a pseudo-random number uniformly distributed in [0,1], one then makes a “gedanken measurement” which projects the wavefunction onto either of its components corresponding to 0 or 1 photon in the quantized Ield. The possibly emitted photon is destroyed immediately after being “detected”: one derives here the evolution of only the atomic part of the wavefunction. This method is equivalent to the optical Bloch equation approach [100]. Its main advantage is that the size of the wavefunction scales as the number N of atomic states, while the density matrix scales as N 2 . This method is thus very powerful, in particular to handle the case of atomic transitions with high angular momenta. It has been used to calculate temperatures and >uorescence spectra in 1D and 3D molasses [68,101,102], and also to study anomalous di@usion in optical lattices [63]. The drawback of this method lies in a relative lack of transparency: it is indeed sometimes diLcult to get a precise idea of the elementary phenomenons leading to a given macroscopical e@ect.
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5. Probe transmission spectroscopy: vibration, propagation and relaxation Probe transmission is obviously a widespread and eLcient method to study the elementary excitations of an optically active medium. A Irst point that should be stressed is that this detection method is sensitive to the excitation modes of the free system (i.e. in the absence of probe) as long as signals linear into the probe are concerned. The only in>uence of the probe is to excite some particular dynamical modes of the system because of its polarization and its direction. In that respect, the information provided by this method is not qualitatively di@erent from those obtained by >uorescence spectroscopy (Section 6). The use of one method or the other is thus not a question of principle but merely a question of convenience. After a brief introduction (Section 5.1), we describe di@erent types of processes that were studied with this method, among which we made a classiIcation according to the detuning : between the probe beam and the lattice beams. Starting from large values of |:|, we Irst describe Raman transitions between eigenstates of the light-shift Hamiltonian that are di@erently populated by the Sisyphus mechanism (Section 5.2). We then present Raman transitions between di@erent vibrational levels inside the same potential well (Section 5.3). These Raman resonances being associated with transitions between atomic eigenstates, their position is independent of the probe direction. The next subsection concerns the observation of propagation modes associated with a single atom that give rise to resonances analogous to those found in Brillouin scattering (Section 5.4). In particular, the position of these resonances vary with the probe direction. The relevant parameter here is the velocity of the propagation mode. We continue this description with the central part of the probe transmission spectrum, i.e. the stimulated Rayleigh line which originates from light scattering on non-propagating modulation of atomic observables (Section 5.5). All these subsections deal with the steady-state regime but the study of the transmission in the transient regime (Section 5.6) provides the same information in a form that is sometimes more convenient. Finally, we describe multiphotonic e@ects that appear with a stronger probe beam (Section 5.7). 5.1. General considerations on probe transmission The spatial evolution of a probe beam inside a material medium is found from the Maxwell equations. In the case of a dilute medium of length L and in the slowly varying envelop approximation, the variation of the complex amplitude Ep of the probe beam at the exit of the medium is ikL Ep (L) = Ep + P; (98) 2 0 where P is the complex amplitude of the Fourier component of the atomic polarization along the probe direction. For a weak monochromatic probe beam of frequency !p = ! + :(|:|!); P = 0 [F (!p ) + iF (!p )]Ep ;
where F = F + iF is the probe linear susceptibility. Eq. (98) thus becomes kL Ep (L) = Ep 1 + (−F (!p ) + iF (!p )) : 2
(99) (100)
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Usually, one measures the intensity of the transmitted probe beam |Ep |2 (1 − F kL) and Inds therefore the spectral dependence of F (!p ) when the probe beam frequency is scanned. Comments: (i) From the knowledge of F , one can deduce F using the Kramers–KrEonig relation i +∞ F(!) F(0) = d! : (101) −∞ 0 − ! (ii) In most situations, the linear susceptibility is a tensor Fij (!p ) rather than a scalar. In optical lattices, the directions along which this tensor is diagonal can often be deduced from symmetry arguments. 5.2. Raman transitions between eigenstates of the light-shift Hamiltonian In a probe transmission spectrum, broad resonances are found for probe detunings on the order of the light-shift . These resonances correspond to Raman transitions between potential surfaces associated with eigenstates of the light-shift Hamiltonian which are di@erently populated by the sub-Doppler cooling mechanism. In principle, a resonant enhancement of the absorption (F ¿0) is found for :¿0, which corresponds to a Raman process where a probe photon is absorbed and a photon is emitted in the lattice beams; conversely, a resonant enhancement of the probe ampliIcation (F ¡0) is found for :¡0 with a permutation between the probe and the lattice beams in the Raman process. The width of these resonances is on the order of or larger than the optical pumping rate p (the spatial dependence of the light-shifts can contribute to an inhomogeneous width). This type of resonances was Irst observed by Grison et al. [103] and Tabosa et al. [104]. Examples of resonances observed in 3D four-beam optical lattices (lin ⊥ lin lattice, see ◦ Section 3.3.4, and Rot[lin ⊥ lin] lattice, see Section 3.5.2, with # = 55 ) are shown in Fig. 21. The experiments [89] are performed with cesium atoms. The lattice beams are tuned to the red side of the 6S1=2 (F = 4) → 6P3=2 (F = 5) transition ( = − 13), the light-shift per beam being 0 −65ER . The transverse probe propagates along the Ox axis. In the lin ⊥ lin case (Fig. 21a), the broad resonance is observed with a -polarized probe and the absorption peak 0S is signiIcantly larger than the ampliIcation peak. The asymmetry between absorption and ampliIcation Irst arises from the di@erent Clebsch–Gordan coeLcients for the probe absorption (F; m p F +1; m F; m ± 1) and the probe ampliIcation (F; m F +1; m ± 1 p F; m ± 1). However the main reason is that the atoms are localized around sites where the lattice Ield has a pure circular polarization. (Note that m is a good quantum number at points where the light is + ; − or polarized.) Because of the weak intensity of the minority circular polarization ( + for example), the Raman process F; m = − F F + 1; m = − F + 1 p F; m = − F + 1 is reduced and therefore the probe ampliIcation also. Furthermore, the Clebsch–Gordan coeLcient connecting F; m = − F to F + 1; m = − F + 1 is also small for large F. For a -polarized probe, the analogous resonances are weaker and this is due to the combined e@ect of strong localization and smaller Clebsch–Gordan coeLcients. The spectra obtained with Rot[lin ⊥ lin] lattice are markedly di@erent. First, with a -polarized ◦ probe the 0S resonance is almost absent and this is explained by the fact that for # = 55 , the atoms are localized around sites where the lattice Ield is -polarized (see Section 3.5.2).
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Fig. 21. Probe transmission spectrum in a Cs four-beam optical lattice on the red side of the 6S1=2 (F = 4) → 6P3=2 (F = 5) transition. The probe frequency is scanned here on a relatively large range to observe ◦ transitions between di@erent eigenstates of the light-shift Hamiltonian. (a) Lin ⊥ lin lattice with #x = #y = # = 55 . ◦ The probe is -polarized. (b) Rot[lin ⊥ lin] lattice with # = 55 . The probe is -polarized. From Mennerat-Robilliard et al. [89].
Therefore, a Raman process between two Zeeman substates of di@erent magnetic quantum number m cannot be induced by the two -polarized Ields. By contrast, the 0S resonance is observed with a -polarized probe (Fig. 21b). These examples show that the relative magnitude of these broad resonances for di@erent probe polarizations gives valuable information on the atomic localization inside the lattice. Comment: In the case of grey molasses, the asymmetry between absorption and ampliIcation is even more important, the ampliIcation being vanishingly small [105]. This is because most atoms are optically pumped into the internal dark state and because the molasses Ield does not connect the internal dark state to the excited state. 5.3. Raman transitions associated with the vibrational motion When an atom is trapped inside a potential well, it oscillates around the potential minimum. The probe transmission spectrum exhibits resonances located at the frequency of this oscillation. We present now the characteristics of these resonances in the oscillating (Section 5.3.1) and jumping (Section 5.3.3) regimes. Although a quantum approach is often used to describe these resonances, a classical point of view (Section 5.3.2) also provides some valuable information with a simple formalism.
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5.3.1. Oscillating regime √ In a potential well of depth U0 ; the oscillation frequency 0v is on the order of ER U0 (see Section 3.2.5). Because U0 ∼| | and (generally) ER | |, one Inds 0v | |. Therefore the resonances associated with the vibration motion occur at a probe detuning (:∼0v ) smaller than the one (:∼| |) associated with transitions between di@erent potential surfaces. These vibrational resonances were Irst reported in [1,2] and their main characteristics were described by Courtois and Grynberg [72]. The origin of these resonances can be easily understood in the quantum approach. Because of the population di@erence between the vibrational levels (see Section 3.2.5), stimulated Raman processes occur with probe ampliIcation when : = − 0v (Fig. 22a) and with probe absorption when : = 0v (Fig. 22b). Examples of probe transmission spectra in a 3D four-beam lin ⊥ lin lattice (Section 3.3.4) Illed with cesium atoms are shown in Figs. 22c and d. The experiment is performed on the red side of the 6S1=2 (F = 4) → 6P3=2 (F = 5) transition with = − 15; 0 = − 190ER and ◦ #x = #y = 30 [106]. Fig. 22c obtained with a probe propagating along Oz (and polarized along a direction orthogonal to that of the copropagating lattice beams) shows evidence of vibrational resonances located in : = 0z and −0z . Fig. 22d obtained with a probe propagating along Ox (and polarized along Oy) shows other vibrational resonances located in : = 0x and −0x . The occurrence of di@erent vibrational frequencies is not surprising because the potential wells do not have a spherical symmetry in the general case. More precisely, the bipotential for a Jg = 1=2 → Je = 3=2 transition can be found from Eqs. (43) and (74) and an expansion near the potential minima gives ˝0x; y = 2 sin #x; y 3U0 ER ; ˝0z = 2(cos #x + cos #y )
U0 ER ;
(102)
where U0 = − 4˝0 =3 is the depth of the 1D optical potential. The dependence of the vibrational frequencies with the angles #xand #y has been studied by Verkerk et al. [42] and Morsch et al. [107]. The proportionality to |0 | has also been checked by varying both the beams intensity and the detuning from resonance [89,107]. Comments: (i) Eqs. (102) for the vibration frequencies are adapted to the particular case of a Jg = 1=2 → Je = 3=2 transition. In a more general situation, it is necessary to expand the adiabatic potential near the potential minima. For a suLciently large value of Jg (typically Jg ¿ 3), the following formulas are a better approximation of the vibration frequency: ˝0x; y = sin #x; y 2˝| |ER ; ˝0z = (cos #x + cos #y ) ˝| |ER :
(103)
(ii) The relative intensities of the vibrational transitions between the levels {n} = {nx ; ny ; nz } and {n } = {nx ; ny ; nz } of a lin ⊥ lin lattice are given in the case of a Jg = 1=2 → Je = 3=2 transition by ˆ m; {n }|2 ; I (m; {n} → m; {n }) ˙ [({n}) − ({n })]|m; {n}|P|
(104)
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Fig. 22. Raman transitions between vibrational levels. (a) Absorption of a photon of a lattice beam and stimulated emission of a probe photon. (b) Reverse process with probe absorption. (c) and (d) Probe transmission spectra in a 3D four-beam lin ⊥ lin lattice Illed with cesium atoms. The probe propagates along Oz in (c) and along Ox in (d). Spectrum (c) is obtained with a x-polarized probe (i.e. its polarization is orthogonal to that of the copropagating lattice beams). From Grynberg and TrichYe [106].
where m is an eigenvalue of Jz = ˝ and ({n}) is the population of the vibrational level nx ; ny ; nz . The operator Pˆ is equal to Ep∗− (R)E− (R) Ep∗+ (R)E+ (R) Jz Jz ˆ P= 1− + 1+ ; (105) E0 Ep ˝ E0 Ep ˝ where E− (R) and E+ (R) are the circular components of the lattice Ield given in Eq. (74) and Ep− (R) and Ep+ (R) the circular components of the probe beam. The extension of Eqs. (104) and (105) to higher angular momenta gives di@erent coeLcients for the magnetization term and requires the introduction of tensors of higher rank [72].
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Note that Eq. (105) shows that a probe propagating along Oz cannot excite the 0x resonance (nx − nx = ± 1; ny = ny ; nz = nz ) because there is no term linear in X in the expansion of the operator Pˆ whatever the probe polarization. (iii) Experimental spectra often show the occurrence of overtones located in the vicinity of 20x ; 20y or 20z , depending on the probe direction. These overtones have weaker intensities because they appear at a higher order in the expansion in R= of the Ields in Eq. (105) (see also Section 3.2.5). (iv) Soon after the observation of the vibrational resonances, it was realized that their position slightly di@ers from 0v [72]. Indeed, accurate measurements [107,108] show that the center of the resonance is found at a frequency smaller than 0v . This is due to the anharmonicity of the potential that we discuss now in the 1D lin ⊥ lin case and for a Jg = 1=2 → Je = 3=2 transition. The expansion of U− (Z) (Eq. (44)) to order 4 in kZ gives an additional term −U0 (kZ)4 =3 to the harmonic expansion in Eq. (65). Considering this supplementary term as a small correction, one obtains using perturbation theory to Irst order: En+1 − En = ˝0v − (n + 1)ER :
(106)
The resonance frequency is thus shifted to a lower value and the distance between two consecutive vibrational transitions is equal to the recoil frequency. This splitting between the vibrational lines has not been observed yet but an indirect evidence for it was obtained through the observation of revivals in transients [109]. Because all these vibrational resonances overlap, one can estimate the location of the centre of their superposition. The weight pn of the n → n+1 line, deduced from Eq. (104), is proportional to [(n + 1) − (n)](n + 1) (the factor (n + 1) originates from the matrix element of Z that appears in the Ield expansion). Assuming a Boltzmann distribution with a temperature T for (n); one Inds an average shift Hv −(kB T=2U0 ) 0v . Because kB T=U0 0:28 (see Section 3.2.3), this model predicts a relative shift Hv =0v on the order of 14%, in reasonable agreement with experimental observations [108]. This is also in agreement with the result of a full quantum calculation [72]. (v) Since the Irst observation of vibrational transitions [1,2], their width was a subject of considerable discussion. First, it was realized that their width was much smaller than the photon scattering rate because the strong elastic component (Section 3.2.5) does not contribute to the width [72]. If the vibrational lines were well isolated, the width of each vibrational line would be given by half the sum of their lifetimes which are given by Eqs. (66) and (67). The width of the n → n + 1 transition would thus be on the order of n (ER = ˝0v ) [72]. Although this is smaller than because of the Lamb–Dicke factor, the observed width is generally even smaller: this is because in a harmonic potential the transfer of coherence gives much narrower lines [10]. In fact, in the framework of this model one predicts a linewidth on the order of 2n U p (ER = ˝0v )∼p (kB T=2U0 ) [106]. However, anharmonicity prevents a full transfer of coherence and accounts for a signiIcant fraction of the width. In fact, it was shown experimentally by Morsch et al. [107] that there is a relatively large range of parameters where the width is proportional to 0v . Such a relation appears naturally if the width corresponds to the mean quadratic value of the anharmonic shifts (see also comment (i) in Section 5.3.2). (vi) A more correct method to predict the lineshape consists of calculating the density matrix of the system (Section 4) and to deduce from it the probe absorption [72]. In the presence
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of the probe beam, the steady-state density matrix can be written as a Fourier expansion = 0 + ( 1 exp − i:t + 1† exp i:t) + · · · : The absorption
F
F = C0 Im
in a lin ⊥ lin lattice is then given by [72] 0 0 − i Tr Pˆ 1 ; 2
(107)
(108)
where C0 is a constant proportional to the atomic density. In the case of the 1=2 → 3=2 transition, the transition operator Pˆ writes J z Pˆ = ei(kj −kp ) · R − ei(kj −kp ) · R : (109) ˝ j⊥p j p In this expression, j p (resp. j⊥p ) means that the summation runs over the lattice beams having the same polarization as the probe beam (resp. a polarization orthogonal to that of a probe beam). This formalism was applied to 1D lattices in [72]. 5.3.2. Semi-classical approach to atom dynamics In classical terms, the action of the probe is to shake the potentials and thus to induce the oscillation motion. For the Jg = 1=2 → Je = 3=2 transition and a 3D lin ⊥ lin lattice, the potential U− (t) associated with |mg = − 1=2 becomes U− + U−(p) where U− is given by Eq. (43) and ˝0 1 ∗ (p) ∗ i:t U− = 2 Ep− (R)E− (R) + Ep+ (R)E+ (R) e + c:c: : (110) I 3 The force acting on the atom F = − ∇U− (t) can be expanded in the vicinity of a minimum of U− (for example R = 0). The lowest order term gives a driving force oscillating at frequency :. Among the higher order terms, some lead to a parametric excitation, resonant when : = 20j ( j = x; y or z). These resonances correspond to a transition nj − nj = 2 in the quantum description (see Section 5.3.1, comment (iii)). In the case of the 1D lin ⊥ lin lattice, U−(p) and U+(p) can be easily calculated and this is interesting to illustrate the in>uence of the probe polarization. We thus assume that the probe beam Ep sin (kp z − !p t) propagates in the +z direction and that its polarization is either ey or ex . In the Irst case (labelled ⊥), the probe has a polarization perpendicular to the copropagating lattice beam. In the second case (labelled ), both have the same polarization. The shifted potentials are [72] Ep 1 U±(p) = U0 −cos(2kZ − :t) ± cos :t (111) E0 2 for a ⊥ probe and Ep 1 U±(p) = U0 ± sin(2kZ − :t) + sin :t E0 2
(112)
for a probe (similar studies were also performed in 2D four-beam lattices [110]). The total potentials U+ (t) and U− (t) at di@erent time intervals are shown in Figs. (23) and (24). It can be seen that the potentials oscillate back and forth around their mean value and this is the source
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Fig. 23. Optical bipotential for a probe having a polarization orthogonal to the one of the copropagating lattice beam (⊥ conIguration). The shape of the potentials is represented at four di@erent times during a period T = 2= |:|. Fig. 24. Optical bipotential for a probe having a polarization parallel to the one of the copropagating lattice beam ( conIguration). The shape of the potentials is represented at four di@erent times during a period T = 2= |:|.
of the Raman resonance occurring at : = 0v . It can also be seen that the potential curvature oscillates periodically and this induces the parametric excitation. Finally, it can be noticed in Figs. 23 and 24 that the depths of the two potentials oscillate in phase in the case and that they are phase-shifted in the ⊥ case. For the evaluation of the additional force −∇Up(±) , the only relevant term is the Irst term in the brackets of Eqs. (111) and (112) because it is space dependent. This term is twice as large for the ⊥ probe, which thus excites more eLciently the vibrational motion. To be more precise, consider as an example the motion of an atom in the − well located in z = 0 of a 1D lattice, and submitted to a ⊥ probe. The atom’s dynamics is described by the equation dZ 0v2 d2 Z 0v2 Ep + sin(:t − 2kZ) : + sin 2kZ = dt 2 M dt 2k k E0
(113)
As a Irst approximation, we can assume that the atom remains in the neighbourhood of Z = 0. Using sin 2kZ ∼2kZ and sin(:t −2kZ)∼sin :t, we Ind the equation of a driven harmonic oscillator which exhibits a resonance when : = 0v . A more precise approximation is to expand sin(:t − 2kZ) to Irst order in kZ. We then Ind an additional driving term −2kZ cos :t which yields a parametric resonance when : = 20v . If we expand the static dipole force up to terms of order (kZ)3 , we Ind an anharmonic contribution −2k 2 0v2 Z 3 =3. If :∼0v , the equation of motion can have more than one stable
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steady state. In fact, in a certain range of parameters, two stable oscillations with di@erent amplitudes are found [111]. An experimental evidence of this mechanical bistability in an optical lattice is presented in Section 5.7.1. If the driving force has a frequency :∼0v =3, the unperturbed oscillation Z0 = Re[aei:t ] yields, due to the anharmonic term, a driving force oscillating at frequency 3:. Because 3:∼0v , the mechanical resonance can be excited [111]. Such a nonlinear resonance has been observed experimentally (see Section 5.7.2) and can be interpreted as a hyper-Raman transition in the lattice. Comments: (i) The shift of the vibrational resonance due to anharmonicity and temperature can also be easily found from Eq. (113). Instead of expanding sin 2kZ around Z = 0, the expansion is performed around Z0 which di@ers from 0 because of >uctuations. The new frequency 0v [cos (2kZ0 )]1=2 is shifted from 0v by a quantity H(Z0 )v −k 2 Z02 0v . Using M0v2 Z02 = kB T and √ 0v = 2 ER U0 yields an average shift HUv = − (kB T=2U0 )0v , in perfect agreement with the result of the quantum analysis (Section 5.3.1, comment (iv)). If the width of the resonance is to be associated with the dispersion of anharmonicity (see U comment (v) in Section √ 5.3.1), it should vary as the mean quadratic value of [H(Z0 )v − Hv ], U a quantity equal to 2Hv . It turns out that this analysis is in reasonable agreement with the experiments [107]. (ii) The probe beam is also the source of a time-dependent dissipation [72]. The fact that optical pumping exhibits a periodic variation at frequency : has important consequences on the relative occupation of + and − wells (see Section 5.5.1). (iii) Once the atomic motion R(t) is known, the transmission of the probe can be evaluated by a method similar to the one described in Section 5.3.1 (comment (vi)) for the quantum case. The formula for F is identical to Eq. (108) but for the replacement of Tr (Pˆ 1 ) by Pˆ exp i:t, where Pˆ is given by Eq. (109) with a time-dependent R and the average is taken on the internal state and on the classical probability distribution of amplitude and phase. In the case of a 3D four-beam lin ⊥ lin lattice and √ a y-polarized probe (called ⊥ conIguration in the limit #x = #y = 0), using the notation Z = 2A sin(:t + I) for the position relative to the potential minima along the z axis (and X = Y = 0) one Inds √ ˆ P exp i:t = 2 dA dI J1 [k(1 + cos #y ) 2A]e−iI {-+ (A; I) + -− (A; I)} −
√
dA dI J1 [k(1 − cos #x ) 2A]e
−iI
{-+ (A; I) − -− (A; I)}
;
(114)
where -± (A; I) is the quasipopulation in phase space expressed using the {A; I} coordinates and in the mg = ± 1=2 sublevel [112]. Note that in the case of large detunings (||), the absorption is proportional to Im Pˆ ei:t and therefore to the average value of sin I, which is reasonable because I corresponds to the phase-shift between the force (see Eq. (111)) and the position. 5.3.3. Jumping regime In the jumping regime, the atom undergoes many jumps between the potential surfaces during a single oscillation period. If nonadiabatic terms can be neglected, the relevant force acting on
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Fig. 25. Probe transmission spectrum in a Rot[lin ⊥ lin] lattice Illed with cesium atoms. The probe is polarized along Oz. Narrow vibrational resonances are observed. From Mennerat-Robilliard et al. [89].
the atom is − n -n ∇Un , where -n is the average population in the nth potential surface (see Section 3.3.1). In the case of a restoring force linear in the amplitude of the displacement, a harmonic motion is found. This motion also leads to sharp vibrational resonances, as shown by Mennerat-Robilliard et al. [89,113]. To achieve the jumping regime, they used a Rot[lin ⊥ lin] ◦ conIguration with # = 55 (see Section 3.5.2). In this case, the atoms are mostly located near sites where the light is linearly polarized along Oz and they jump from one potential to the other at the optical pumping rate. An example of vibrational spectrum obtained on the red side of the 6S1=2 (F = 4) → 6P3=2 (F = 5) cesium transition ( = − 13) and for a light-shift per beam 0 = − 90ER is shown in Fig. 25. This recording was obtained with a -polarized transverse probe, i.e. propagating along Ox. Here again the variation of the position 0x of the resonance with |0 | is in reasonable agreement with a square root law [89]. Although the atoms remain in the same potential surface for a time shorter than p−1 , the width of the vibrational resonance is much narrower than p . As shown by Mennerat-Robilliard et al. [113], this is due to a motional narrowing e@ect [114]. Let us denote 0n the vibration frequency associated with the nth potential curve. One can deIne an average frequency 0U = n -n 0n U 2 ]1=2 . The vibrational resonance is centred in 0U and the and a dispersion = [ n -n (0n − 0) frequency dispersion contributes to the width by a term on the order of 2 p−1 only. This is smaller than by a factor =p which decreases when p increases. The random and fast sampling of di@erent potential curves transforms here a broad inhomogeneous width into a narrow homogeneous width. Comment: Probe transmission was used by Schadwinkel et al. [88] to show the localization and the oscillation of atoms at the bottom of potential wells in a magneto-optical trap. This experiment gives actual evidence that a magneto-optical trap with well stabilized phases is closely related to a lattice.
5.4. Propagating excitation. The Brillouin-like resonance In >uids, light scattering on propagative excitation modes such as sound waves gives rise to Brillouin resonances [115,116]. Identical processes have not been observed up to now in optical lattices because the density is generally too low to have suLcient particle interactions. However,
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Fig. 26. (a) Section of the optical bipotential of a Jg = 1=2 → Je = 3=2 four-beam lin ⊥ lin lattice along the direction Y = Z = 0. An atom can propagate along this periodic structure through a sequence of half-oscillations in a well followed by optical pumping transitions. (b) Example of atomic trajectory: the empty (resp. black) dots correspond to a m = + 1=2 (resp. m = − 1=2) state for the atom. From Courtois et al. [96].
a di@erent propagative excitation mode is found in optical lattices. It consists of repeated cycles of half oscillations in a potential well followed by optical pumping towards an adjacent potential well [96]. Consider atoms in a four-beam lin ⊥ lin lattice (Section 3.3.4). The variation of the optical potential along an x-axis connecting the potential minima is shown in Fig. 26a for the Jg = 1=2 → Je = 3=2 transition. The light polarization at the bottom of the wells is alternatively + and − and the distance between two adjacent wells is =2 sin #x . Semiclassical Monte-Carlo simulations (Section 4.3) of the atomic motion [96] show that the dominant propagation mode along the x direction for delocalized atoms consists of the following steps: (i) half oscillation in a − potential well (the atom moves from A to B); (ii) optical pumping from a − to a + potential surface; (iii) half oscillation in a + potential well (the atom moves from B to C); (iv) optical pumping from a + to a − potential surface; and so on. A typical example of computed trajectory is shown in Fig. 26b; an initially localized atom travels over eight potential wells before being trapped again. To estimate the velocity vU associated with this propagation mode, we remark that the atom travels from A to B, B to C in a typical time A equal to half the oscillation period, i.e. A = =0x . As AB =2 sin #x , we Ind vU 0x =k sin #x .
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Fig. 27. Probe transmission spectrum in a 3D lin ⊥ lin lattice Illed with Cs atoms. The probe propagates along Oz and is y-polarized (its polarization is parallel to that of the copropagating lattice beams). The spectrum shows vibrational resonances located in ±0z and resonances associated with the propagating excitation in ±0B . From Grynberg and TrichYe [106].
Such a density wave can be driven by the modulation of the optical potential and of the optical pumping rate induced by the lattice–probe interference pattern provided that its phase velocity along the x direction be equal to ±v. U Considering a lattice beam having the same polariztion as the probe beam, the interference pattern moves along Ox with the velocity v = :=(kp − k) · ex where kp and k are the wavevectors of the probe and lattice beams respectively. The resonant driving condition |v| = vU yields sin #x − sin #p ; : = 0x (115) sin #x where #p is the angle between kp and ez (note that #p and #x can have opposite signs). The occurrence of a resonance when Eq. (115) is fulIlled has been checked experimentally and numerically by Courtois et al. in the limit of small #x and #p [96]. When the probe beam propagates along Oz, this Brillouin-like resonance is expected to be found near : = 0x . This ◦ ◦ is what is observed in Fig. 27 where a spectrum obtained for #x = 30 , #y = 15 , = − 11 (detuning from the 6S1=2 (F = 4) → 6P3=2 (F = 5) Cs transition), 0 = − 250ER is shown. In spite of the fact that the 0B resonance is located at the expected position for the 0x vibration frequency, it does not correspond to a Raman vibrational transition, for several reasons. First, as explained in Section 5.3.1, this probe geometry does not permit the x vibrational mode to be excited. Second, when the probe polarization is rotated by =2, the resonance disappears (see Fig. 22c). Third, when the probe direction changes the resonance 0B is shifted whereas a vibrational transition remains always located at the same frequency [96]. Comments: (i) Actually, the propagating excitation mode can be driven by the interference pattern created by the probe and any of the two copropagating lattice beams. It follows that when the probe is not aligned along Oz, the 0B resonance splits into two resonances corresponding to these two patterns. One resonance frequency is smaller than 0x and the second one higher [96]. (ii) The polarization pattern of the four-beam lin ⊥ lin lattice promotes these propagation modes. In particular, it can be noticed that transitions from m = − 1=2 to +1=2 are strongly
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suppressed near the bottom of a − potential well because the + intensity varies as x4 (x being the distance to the potential minimum). (iii) The mechanism of these Brillouin-like resonances, which requires spontaneous emission (i.e. a dissipative process) reminds the stochastic resonances [117]. 5.5. Stimulated Rayleigh resonances—relaxation of nonpropagative modes Rayleigh scattering results from the scattering of light on any nonpropagative modulation of atomic observables [64]. In the case of optical lattices and molasses, the observables that were considered up to now are the magnetization (Section 5.5.1), the density (Sections 5.5.2 and 5.5.3) and the velocity (Section 5.5.4). The width of the Rayleigh resonance gives information on the damping time of these observables. 5.5.1. Magnetization In a lin ⊥ lin lattice, the population is on average the same in the + and − wells. When a probe beam is added, the balance between atoms in the m = + Jg and in the m = − Jg states can be broken, leading to a global magnetization of the lattice. If the probe–lattice detuning is :, the magnetization oscillates at frequency : but with a phase-shift with respect to the applied Ield because of the time delay AD associated with the atomic response time. The atomic polarization thus exhibits a nonzero component being =2 phase-shifted with respect to the probe beam. The work of the probe Ield onto this component leads to the modiIcation of its intensity as displayed by a resonance on the probe transmission spectrum. This resonance is centred around : 0 because it is associated to a nonpropagative observable and its width is on the order of A−1 D . In the general case, there is no phase-shift when : = 0 and the transmitted probe is neither ampliIed nor absorbed. The generic shape of the stimulated Rayleigh resonance is thus dispersive with a peak-to-peak distance on the order of A−1 D [64]. To be more precise, we now discuss the particular case of the 1D lin ⊥ lin lattice with a probe beam cross-polarized with the copropagative lattice beam [72]. In this situation labelled ⊥ in Section 5.3.2, the di@erence (I + − I − ) between the two circular components of the total Ield is I + − I − = − E0 Ep cos :t :
(116)
The optical pumping is thus time-modulated. However there is not a single time constant AD for the magnetization here. The optical pumping rate depends on the vibrational level (see Eq. (66)) and di@erent relaxation modes with di@erent rates ranging from p ER = ˝0v to p are expected to be involved in the Rayleigh scattering. The resonance thus appears as the sum of several dispersive curves with di@erent peak-to-peak distances. This gives rise to an uncommon shape with a steep slope at the center and broad wings. This is indeed what was observed by Verkerk et al. [2]. Note that there is a persistent memory of this lineshape in the 3D lin ⊥ lin lattice when the probe beam has a polarization orthogonal to that of the copropagating lattice beams (see Fig. 22c). Comments: (i) In fact, the preceding analysis is oversimpliIed because it does not include the reactive e@ects and the density modulation [72]. As shown in Fig. 23, the probe also changes the relative depth of the + and − wells. Optical pumping by the lattice beams tends to adapt the population in each state to the time-dependent modiIcation of the potentials. The >ow
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of population back and forth between the + and − wells creates a time-dependent density modulation having a spatial period =2. This density modulation gives a contribution to the Rayleigh resonance having an importance nearly equal to that of the magnetization [72]. (ii) In the conIguration for the probe (Section 5.3.2), (I + − I − ) is equal to E0 Ep sin(:t − 2kz) in the 1D case. Because this Ield conIguration induces the same time dependence in the evolution of U− (t) and U+ (t) (see Fig. 24), the probe only induces a redistribution of population inside each well without any net transfer from + to − wells or vice versa. The Rayleigh resonance originates from the time evolution of this antiferromagnetic magnetization grating [72]. The amplitude of this resonance is predicted to be very small and in most experiments it is hidden by noise or parasitic e@ects. An example of this kind of resonance was however reported for a 3D lin ⊥ lin lattice (Fig. 13a of [106]). 5.5.2. Density—connection with spatial di;usion The interference pattern created through the superposition of the probe and the lattice beams induces a variation of the optical potential and of the optical pumping which generally leads to a density modulation. Consider for example the interference created by a lattice beam (k) and a probe beam (kp ) of same polarization. A spatial variation of the form N = Re[N0 exp i (k−kp ) · r] can be expected for the atomic density, N0 being proportional to the probe amplitude. In the case where the density modulation relaxes by spatial di@usion with an isotropic spatial di@usion coeLcient Dsp , the equation 9N (117) = Dsp SN ; 9t yields a relaxation rate Csp Csp = Dsp |k − kp |2 :
(118)
Thus, if the probe and the copropagating lattice beams have the same polarization, it is in principle possible to measure Dsp from the width of the Rayleigh line and this method has been used by Jurczak et al. [97]. However it must be emphasized that this requires a careful examination of the experimental conditions [118]. First, spatial di@usion is often anisotropic. In this case one should replace the right-hand side of Eq. (118) by j Dj (k − kp )2j where Dj are the eigenvalues of the spatial di@usion tensor. Second, a di@usion model cannot always be applied to the spreading of the atomic density. This requires that the mean free path is much smaller than the typical length scale 2= |k − kp | of the density modulation. In particular, when decreasing the optical potential depth, there can be a transition from Gaussian spatial di@usion to anomalous di@usion with LYevy walks [63,119] (see Section 6.3.2). It should also be noticed that escape channels along which atoms are accelerated can be found in 2D and 3D lattices [63]. Comment: In the case of free or almost free atoms the density modulation is washed out through ballistic atomic motion. The associated resonance corresponds to the derivative of thevelocity distribution (generally a Gaussian with a peak-to-peak distance equal to 2|k − kp | kB T=M where T is the atomic kinetic temperature) [64,120 –123]. This “recoil-induced” resonance can be interpreted either in terms of stimulated Rayleigh scattering from the atomic density modulation in the probe–lattice interference pattern [122] or in terms of stimulated
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Fig. 28. Probe transmission spectrum in a 3D lin ⊥ lin lattice. The probe beam propagating along Oz has the same polarization as the copropagating lattice beams. The narrow central resonance (see inset) is very close to a Lorentzian. Form Guibal et al. [99].
Raman (or Compton) scattering between di@erently populated atomic velocity groups [120,123] (see Section 6.1.1, comment (iii)). 5.5.3. Phase-shift between the density modulation and the interference pattern The generic stimulated Rayleigh lineshape is dispersive with no probe gain when : = 0 [64]. However, probe transmission spectra recorded in optical lattices often show the occurrence of an almost lorentzian central structure, as shown in Fig. 28 [99]. This spectrum was obtained in a ◦ 3D lin ⊥ lin lattice Illed with Rb atoms for #x = #y = 20 ; 0 = − 200ER ; = − 5. The probe is aligned along Oz and has the same polarization as the copropagating lattice beams. Apart from vibrational (0z ) and Brillouin-like (0B ) structures, one clearly distinguishes a central resonance with a maximum probe ampliIcation for : = 0. The It using an adjustable superposition of a lorentzian and a dispersion shown in the inset proves that this resonance is very close to a lorentzian. The occurrence of gain for : = 0 shows that the density modulation induced by the probe has a component which is =2 phase-shifted with respect to the probe-lattice interference pattern. In other words, the density modulation for : = 0 exhibits a shift with respect to the optical potential in the presence of the probe. As shown by Guibal et al., this shift originates from the radiation pressure [99]. Schematically, the probe and the copropagating lattice beams are in phase at the optical potential minima. The radiation pressure from the counter-propagating lattice beams is no longer balanced and the atoms are displaced towards another position where an equilibrium between radiation pressure and the dipole force is found. Because the ratio between radiation pressure and the dipole force originating from light-shift varies as = || (Section 2.1), the resonance tends to distort from a lorentzian to a dispersion as || increases [99]. Comment: A similar beam coupling process is found in some photorefractive materials [124,125].
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5.5.4. Velocity Radiation pressure acting on an unbound atom in a molasses or a lattice oscillates at frequency : because of the time variation of the relative phase between the probe and the lattice beams. Because of this driving force, the average velocity of these atoms also oscillates but with a phase shift depending on the relative value of the friction coeLcient and the oscillation frequency. If the atomic polarization depends on the velocity, a Rayleigh resonance having a width on the order of the friction coeLcient can be observed, as shown by Lounis et al. in the case of a 1D + − − molasses [126]. A detailed theoretical description of this resonance is presented by Courtois and Grynberg [127]. 5.6. Coherent transients—another method to study the elementary excitations In probe transmission spectroscopy, one generally studies the steady-state value of the probe transmitted intensity. However, it is also possible to interrupt the probe or to shift its frequency and study the transient atomic emission. If the probe is weak, the signals are proportional to the probe amplitude and the information contained in the transient spectrum is identical to that in the steady state spectrum. However, the transient signal can be more convenient for some measurements. In fact, this duality has been considered a few years ago in laser spectroscopy [128]. In optical lattices, the transient response of the probe has been used by Hemmerich et al. [41], TrichYe et al. [118,129] and Morsch et al. [107]. We Irst present this technique in Section 5.6.1 and then discuss a similar technique in Section 5.6.2, where no additional probe beam is needed as one monitors the lattice beams themselves to study the redistribution of photons among them. 5.6.1. Transient response of the probe beam In the coherent transient method, the probe frequency is kept Ixed until the atoms reach a steady state; then the probe frequency is suddenly switched from !p = ! + : to !p at time t = 0 (the intensity remaining constant). It is generally convenient to choose !p such as the atoms do not interact with the probe after the switch. Thus the signal for t¿0 corresponds to the beat note between the probe at its new frequency and the light scattered by the atoms in the probe direction during their relaxation to their new steady state. Examples of signals obtained in a 3D lin ⊥ lin lattice Illed with Cs atoms are shown in Figs. 29a and c [129]. In this experiment, the probe is on the Oz symmetry axis and has the same polarization as the copropagating ◦ beams. The other conditions are #x = #y = 55 ; = − 14; 0 = − 630ER ; !p − !p = 1:5 MHz. Fig. 29a is obtained for a resonant Rayleigh excitation and Fig. 29c for a resonant Raman excitation (: 125 kHz). These Igures permit to measure the decay of non-propagative modes (Fig. 29a) and of the vibrational coherence (Fig. 29c). The square of the Fourier transforms of Figs. 29a and c are shown in Figs. 29b and d, respectively. By comparison with the probe transmission spectra shown earlier (Figs. 22 and 28), the interest of Figs. 29b and d is that there is almost no overlap between the Rayleigh and Raman resonances and it is possible to magnify one or the other by an appropriate choice of :. However, there is no additional information in the transient experiment. In fact, it can be readily shown that the amplitude of the 0 component of the Fourier transform of the transient
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Fig. 29. (a) and (c) Transient spectra in the 3D lin ⊥ lin lattice for a probe propagating along Oz, having the same polarization as the copropagating lattice beams and producing (a) a resonant Rayleigh excitation and (c) a resonant Raman excitation. (b) and (d): Square of the Fourier transform of spectra (a) and (c), respectively. From TrichYe et al. [129].
is equal to [129] kL F(0) − F(!p ) S(0) = Ep2 ; 8 0 − !p
(119)
where L is the length along Oz of the atomic distribution. This equation shows that the same physical quantity F(!) can be deduced from the transient and from the steady-state experiments. Furthermore, in the usual case where F(!) ˙ [(! − !a ) + ia ]−1 , it can be readily shown that S(0) ˙ [(0 − !a ) + ia ]−1 . The same spectral dependence is therefore found for F and S in this case. 5.6.2. Photon redistribution: a particular coherent transient Instead of monitoring the transient signal on an additional probe beam, one can also use the lattice beams themselves and record their transient intensity after changing the optical potential in a coherent way. In this case, the intensity modulation of a lattice beam with wavevector k originates from its interference with the light scattered from another lattice beam into the same direction k. This technique was demonstrated by Kozuma et al. [130] in a rubidium 1D optical lattice consisting of two beams with linear parallel polarizations, intersecting at an angle #. These beams are tuned to the red side of the resonance (¡0) and their frequencies are, respectively, ! and ! + :L (with |:L |||). At time t = 0; :L is suddenly switched from 0 to more than 100 kHz, a value much larger than the atomic oscillation frequency in the potential wells. The atoms do not follow the potential moving at v0 = :L =(2k sin #=2) and experience an alternating dipole force as they cross the potential hills and wells. The corresponding photon
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Fig. 30. Experimental (solid line) and theoretical (dotted line) power transfer ratio induced by wave packet oscillations initiated by a sudden shift of the optical potential :z = 0:14 : U0 = 1000ER and = − 10. The theoretical line is obtained with quantum Monte-Carlo wavefunction simulations and is displaced from the experimental line for clarity. From Raithel et al. [109], reproduced by permission of the authors.
redistribution signal exhibits damped oscillations with a period :L . On the other hand, if one sets :L so that the kinetic energy of the atoms in the moving frame is much smaller than the optical potential depth, the atomic wave packets will remain localized in the moving potential. In this case, the period of the damped oscillations is the atomic vibration period, independently of the potential velocity [131]. In both experiments, the oscillations decay in a time on the order of 10 microseconds, which is much longer than the optical pumping time p−1 . Such a slow decay in a linearly polarized Ield can originate from a motional narrowing e@ect [114] (see Section 5.3.3) that prevents inelastic photon scattering from immediately washing out the coherences. As these experiments are performed in the jumping regime, the signal decays after a few oscillations in the potential and the observation of e@ects associated with the quantization of the atomic motion is not expected. Making a similar experiment in the oscillating regime should allow a deeper insight in the nature—dispersive or dissipative—of the decay: dispersion through potential anharmonicity should give birth to a decay followed by quantum revivals provided that the vibrational motion is quantized, while dissipation through spontaneous emission should lead to a damping of the signal. Such an experiment was performed by Raithel et al. [109] with a 1D lin ⊥ lin lattice, which means that the light polarization is circular at the bottom of the wells and consequently that dissipation is reduced by the Lamb–Dicke e@ect [8]. If one of the beams is suddenly phase-shifted after the atoms thermalize in the potential wells, the optical potential is spatially shifted. In a semi-classical picture, the atoms are on the edge of the potential wells and start oscillating in phase under the action of the dipole force. We show in Fig. 30 the experimental power transfer induced by coherent wave packet oscillations after a sudden shift of the lattice potential [109]. In agreement with the previous observations, the oscillations Irst decay with a time constant on the order of a few tens of s. This time constant is not related to dissipation, but rather to the anharmonicity of the potential wells which induces a dephasing of the oscillations. This is conIrmed by the existence in Fig. 30 of revivals corresponding to a subsequent rephasing of the oscillation. Note that a full analysis of the signal requires a quantum treatment to take
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into account tunneling e@ects, the most appropriate description being the band model [109] (see Section 4.2). Comments: (i) Such a transient experiment can also be performed by looking at the >uorescence of the atoms [132]. This is described in Section 6.2.4. (ii) It is possible to stimulate a revival using two spatial shifts of the potential separated in time. This method, analogous to photon echoes [133], permits to measure the coherence relaxation time [134]. 5.7. Intense probe beam The situations where the driving of the atomic motion by the probe is no longer linear is interesting although it has not been studied very often in dissipative optical lattices. The e@ects already considered are mechanical bistability (Section 5.7.1) and hyper-Raman transitions involving several probe photons (Section 5.7.2). 5.7.1. Mechanical bistability A forced nonlinear oscillator has two stable states in a large range of parameters [111]. One state corresponds to a large amplitude motion and the other to a small amplitude motion. A beautiful microscopic example of this mechanical bistability was provided by the cyclotron resonance of a relativistic electron revolving in a static magnetic Ield under the action of a nearly resonant electromagnetic wave [135 –137]. Although experiments in optical lattices were mostly focused on the harmonic motion, mechanical bistability originating from the anharmonic terms in the potential (see Section 4:3:2) was also observed with atoms. The dynamics of an atom driven by a probe propagating along Oz in a − well of a 1D lin ⊥ lin lattice can be described by Eq. (113). In the case of a weak probe the atom remains close to z = 0 and the harmonic expansion of the potential gives one steady-state solution Z = Im[a exp i(:t + I)]. However two solutions having di@erent amplitudes of oscillation a1 and a2 can be found with an intense probe because of the potential anharmonicity [111]. This is because the detuning from the vibrational resonance is then a function of the amplitude itself and the equation that yields the amplitude is therefore nonlinear. The experiment [118,138] is performed in two steps. In the Irst step, the atoms reach a steady state under the combined action of the lattice and probe beams and they attain an amplitude in the vicinity of a1 or a2 . In the second step, starting at time t = 0, one records the coherent transient (see Section 5.6) that follows the abrupt switching of the driving probe to a new frequency. The emission from the lattice shows beats between transient oscillations of di@erent frequencies. In fact, because of potential anharmonicity, the free oscillation frequency 0(a) depends on the amplitude a. Therefore, two emission frequencies located around the values 01 = 0(a1 ) and 02 = 0(a2 ) associated with the two stable amplitudes a1 and a2 of oscillation in steady state are expected. The experiment is performed in a four-beam lin ⊥ lin conIguration. The lattice beams are tuned to the red side of the cesium 6S1=2 (F = 4) → 6P3=2 (F = 5) transition and the probe beam of frequency !p = ! + : is sent along the Oz symmetry axis of the lattice. ˜ The amplitude S(t) of the beat note signal between the transmitted probe and the light scattered by the atoms in the probe direction for t¿0 is shown in Fig. 31a. This curve, obtained with
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Fig. 31. (a) Transient emission of Cesium atoms in a lin ⊥ lin lattice after an interrupted probe excitation. The experimental conditions are = − 29; I = 30 mW=cm2 ; Ip = 0:7 mW=cm2 ; : = 92 kHz. The fast oscillation corresponds to the beat frequency between the transmitted probe frequency !p and the scattered light, having a frequency in the neighborhood of ! + 0v . The slow oscillation shows that the distribution of frequency of the scattered light has two peaks located in ! + 01 and ! + 02 . Note that the signal has been Iltered to eliminate the weak contribution of the Rayleigh component and of the anti-resonant Raman component. (b) Square of the Fourier transform of the transient emission. The horizontal axis has been shifted so that 0 = ! corresponds to the static response. Two peaks located in 01 and 02 are clearly visible in the Igure. From Grynberg et al. [138].
Ip =I ≈ 2×10−2 , can be interpreted as arising from the beat between two damped oscillators having di@erent free oscillation frequencies. Such an image is conIrmed by the observation of ˜ |S(0)|2 (Fig. 31b), where S(0) is the Fourier transform of S(t). Two peaks located in 01 and 02 are clearly observed on this Igure. These peaks are located on both sides of the excitation frequency : and the values of 01 and 02 are smaller than the value 0v predicted for the harmonic frequency. The relative intensities of the peaks are related to the atomic population in the neighbourhood of each steady state [138]. When : is swept, the positions of the maxima of emission follow a bistable curve similar to those observed in optics [139]. 5.7.2. Hyper-Raman transitions In usual probe transmission spectra, the vibrational resonances are located near : 0v . However, with an intense probe other resonances originating from nonlinear resonances [111] and located near 0v =n with n = 2; 3; : : : can also be found. A probe transmission spectrum obtained with a probe beam having an intensity comparable to that of the lattice beams is shown
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Fig. 32. (a) Hyper Raman transition involving two probe (gray arrows) and two lattice photons (black arrows). (b) Probe transmission in a 3D Rubidium lattice with a probe having an intensity comparable to that of the lattice beam. The Raman vibrational resonance is found for |:| 165 kHz, and additional resonances occur for |:| 82 kHz. (c) The dotted line is the second derivative of the probe transmission spectrum shown in solid line (this corresponds to the fraction of the spectrum shown in (b) between : = − 100 and −25 kHz). From Hemmerich et al. [140], reproduced by permission of the authors.
in Fig. 32b [140]. A resonance located around : 0v =2 is observed. Although this subharmonic resonances can be interpreted classically (see Section 4:3:2), a quantum picture (Fig. 32a) showing how photons are exchanged between the beams and the atom gives a particularly clear physical picture of the process. Note that a careful examination of the shape of the transmission spectrum shows also the occurrence of other subharmonics 0v =n with n = 3; 4; 5. (This is done by calculating the second derivative of the spectrum, see Fig. 32c.)
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6. Temperature, *uorescence and imaging methods This section presents the information that can be deduced from >uorescence or imaging techniques. We start with the description of temperature measurements (Section 6.1) because the time-of->ight method which is commonly used involves the study of the >uorescence of atoms released from the lattice and crossing a probe beam. We then proceed with the description of the emission spectrum in the lattice itself (Section 6.2) and we Inish with spatial di@usion which can be reliably measured using >uorescent light by taking pictures of the expanding cloud of atoms (Section 6.3). 6.1. Temperature The linear dependence of temperature with light-shift is one of the most robust results concerning optical lattices. It was demonstrated both in the jumping regime and in the oscillating regime (Section 3.2). In fact, this variation was Irst observed experimentally by Lett et al. [6,141] using a time-of->ight method, a technique which is now widely used to study kinetic temperature of cold atoms. 6.1.1. Time-of-Iight In the time-of->ight method (see Fig. 33a), the atoms cooled in the optical lattice are released when the lattice beams are suddenly switched o@. Because of gravity, the atoms fall towards a probe beam located a few centimeters below the intersection of the lattice beams. As the atoms pass through the nearly resonant probe laser beam, they absorb and >uoresce light. The time variation of absorption and >uorescence is related to the initial velocity distribution of the atoms in the lattice and therefore to the kinetic temperature. If the spatial extension of the lattice and of the probe beam can be neglected (this is the case when the distance h between the lattice and the probe beam is suLciently large), the width :t of the time-dependent absorption is related to the width :v of the velocity distribution by :t = :v=g. An example of time-of->ight signal is shown in Fig. 33b. Comments: (i) The scheme described above permits to determine the velocity distribution only along the vertical direction. However, the velocity distributions along both horizontal axis can also be measured if the whole pattern of the >uorescence emitted by the atoms as they cross the probe beam is recorded by a CCD camera. Such an experiment requires a probe beam signiIcantly broader than the atomic cloud in the horizontal plane. (ii) Corrections due to the Inite size of the atomic cloud and of the probe beam’s vertical waist are usually determined independently. It is also possible to have two probe beams located at di@erent heights h1 and h2 . An alternate recording of the corresponding time-of->ight signals permits to deduce these corrections [70]. (iii) The “recoil-induced resonances” [64,120 –122] can also be used to measure the velocity distribution of cold atoms [123]. In this case, two beams of frequencies ! and ! + : with wavevectors k and k + q are sent onto the cloud of cold atoms once the lattice beams are extinguished. Stimulated Compton transitions with the atomic momentum changing from P to
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Fig. 33. Time-of->ight temperature measurement. (a) Scheme of the method. Atoms released from the lattice fall ballistically towards the probe beam. The distribution of arrival times, as measured on absorption (represented on the Igure) or >uorescence, allows for the determination of atomic kinetic temperature. (b) Example of time-of->ight distribution. This recording, obtained with Cs atoms and a distance h = 10 cm between the lattice and the probe beam, corresponds to a temperature T = 1:5 K. From TrichYe [118].
P − ˝q are possible provided that energy is conserved, i.e. for q · P ˝q 2 − : (120) M 2M The transition amplitude scales as [(P) − (P − ˝q)] where (P) is the atomic momentum distribution. In the limit where qP= ˝, this term is equal to ˝q · ∇P (P). The transition amplitude is thus proportional to the derivative of the momentum distribution along q [120]. The total distribution along this direction is found by sweeping :. With several directions for q, the distribution in the whole momentum space can be recovered. The identity between predictions of the recoil-induced resonances and the time-of->ight method was experimentally demonstrated by Meacher et al. [123]. (iv) The relative intensities of the vibrational lines of the >uorescence spectrum are also connected to temperature. This will be discussed in Section 6.2. :=
6.1.2. Experimental results All measurements performed showed that temperature increases linearly with the lattice beams intensity, apart from the range of very small intensities where a “decrochage” is observed. This includes experiments on 1D lattices [108], on 3D four-beam lin ⊥ lin lattices [89,108], on
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Fig. 34. Temperature as a function of intensity in a 3D six-beam molasses for various detunings. From Salomon et al. [87].
3D four-beam Rot[lin ⊥ lin] lattices [89], on six-beam molasses and lattices [141,142,87,143], on quasiperiodic lattices [38], on Talbot lattices [46], on grey molasses and antidot lattices [70,71]. We show in Fig. 34 an example of results obtained with a 3D cesium molasses. The linear dependence with 1= || is also generally observed, although the range of variation of is often reduced by the presence of neighboring transitions. For example, in the case of Cs, the 6S1=2 (F = 4) → 6P3=2 (F = 5) transition is located at a distance on the order of 50 (=2 = 5:2 MHz) from the 6S1=2 (F = 4) → 6P3=2 (F = 4) transition. The approximation of a well-isolated transition thus breaks down when || becomes larger than 25. In these experiments the lowest temperature is on the order of a few recoil energies, as expected (see Section 3.2.3). For instance, cesium atoms were cooled to about 1:2 K (i.e. 6TR ) in a four-beam lin ⊥ lin bright lattice. Furthermore, by turning down the intensity of the lattice beams the adiabatic expansion permits a further cooling to 0:7 K [144]. In the case of a four-beam grey molasses, cesium atoms were cooled to about 0:8 K [70]. Comments: (i) For the lowest temperature values, the width of the velocity distribution slightly overestimates the temperature inside the lattice. This is because the lowest eigenstate of the lattice Hamiltonian has a non-zero spread in momentum space. (ii) In some circumstances, it is possible to deIne a spin temperature in the lattice [145]. This temperature generally di@ers from the kinetic temperature (see Section 9.1). (iii) In a 3D lattice, the temperature can be anisotropic. For instance, in the Talbot lattice the temperatures in the longitudinal and in the transverse directions are signiIcantly di@erent [46]. (iv) In a quasi-periodic lattice (see Section 2.4), the optical potentials depend on the relative phases of the laser beams. However Guidoni et al. [38] did not observe any signiIcant variation of the temperature with these phases. The temperature was found to be a function of I and only. This is because all the potentials generated by phase variations in their experiment were topologically equivalent.
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Fig. 35. Experimental scheme to observe the spectrum of light emitted in optical lattices and molasses. AOM denotes acousto-optic modulator.
6.2. Fluorescence Many experiments on laser-cooled atoms use a scheme that either destroys the atomic cloud (time-of->ight for instance) or at least induces some perturbation (probe transmission for instance). It is however possible to collect a signiIcant amount of information by simply observing the >uorescence spectrum of the light emitted by the atoms in the lattice. This light originates from the scattering of the lattice beams by the cooled atoms. In fact, the same information can often be found by probe transmission spectroscopy and by >uorescence spectra. Vibration and propagating excitations were for example studied by these two techniques and the use of one method or the other is often merely a question of convenience. The starting point of these studies was the observation by Westbrook et al. [7] of a narrow resonance in the >uorescence spectrum of atoms cooled in 3D six-beam molasses. This resonance was interpreted by the Dicke narrowing of light emitted by atoms conIned in a well having a dimension smaller than the wavelength [8]. This experiment used a heterodyne detection method presented in Section 6.2.1 and also employed in the following experiments on 1D and 3D lattices. Examples of information deduced from these spectra are presented in Section 6.2.2. We describe in the following section (Section 6.2.3) photon correlation spectroscopy and in Section 6.2.4 experiments where information on atomic dynamics can be deduced from time-resolved >uorescence when the optical potential is changed. 6.2.1. Heterodyne detection of the Iuorescence spectrum In this method, light emitted by atoms in an optical lattice is combined on a beam splitter with a strong local oscillator beam derived (with a frequency o@set) from the same laser as the one creating the lattice (see Fig. 35). Any frequency >uctuations due to technical noise in the laser thus cancel in the beat note and a frequency resolution in the kHz range can be achieved [7]. It should be emphasized that the natural width of the excited state does not play any role in
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Fig. 36. Heterodyne detection of the emission spectrum of rubidium atoms in a 1D lin ⊥lin lattice. From Jessen et al. [3], reproduced by permission of the authors. Fig. 37. Quantum description of the physical processes leading to the di@erent components of an emission spectrum: (a) Rayleigh line, (b) Raman Stokes transition, (c) Raman anti-Stokes transition.
the linewidth because these measurements concern Rayleigh scattering which is perfectly elastic for a stationary two-level atom [10]. The frequency shift of the local oscillator with respect to the lattice beams should be large enough so that the Rayleigh line and its possible sidebands are located far away from the zero frequency. This is useful both for improving the signal-to-noise ratio and for discriminating the Stokes and anti-Stokes sidebands of the Rayleigh line which can have di@erent magnitudes. 6.2.2. The Rayleigh line and its vibrational sidebands An example of emission spectrum obtained from rubidium atoms (85 Rb) trapped in a 1D optical lattice is shown in Fig. 36 [3]. The spectrum mainly consists of a central peak at the frequency of the lattice beams (elastic scattering) and two much smaller sidebands at about 85 kHz, which roughly corresponds to the vibration frequency 0v of atoms trapped in a potential well. Quantum mechanically, the central Rayleigh line is associated with a scattering process where the initial and Inal vibrational levels are identical (Fig. 37a). The Stokes sideband corresponds to a process beginning on a given vibrational level and returning to the next higher level (Fig. 37b). The anti-Stokes sideband originates from processes returning to the next lower level (Fig. 37c). A careful examination of Fig. 36 shows the occurrence of still smaller sidebands where the vibrational quantum number changes by two units. The smallness of the sidebands re>ects the smallness of the matrix element of the transition operator connecting two di@erent vibrational levels (see Section 3.2.5). More precisely, the intensity of the Irst sideband is smaller than the intensity of the Rayleigh peak by a factor on the order of (2nU + 1)ER = ˝0v where nU is the average vibrational number and ER = ˝0v is the Lamb–Dicke factor [72]. Thus the sidebands are all the smaller (with respect to the Rayleigh line) as the atoms are well localized in the potential well [8,80 –82].
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Fig. 38. Self-beating spectrum of rubidium atoms in a 3D lin ⊥ lin lattice, exhibiting the vibrational resonance 0z in the z direction and the Brillouin resonance 0B (see Section 5.4). Detection is performed along Oz . From Jurczak et al. [147], reproduced by permission of the authors.
An important feature that distinguishes the emission spectrum (Fig. 36) from a probe transmission spectrum (Fig. 22) is the di@erence of magnitude between the Stokes and anti-Stokes sidebands in emission spectra. This originates from the fact that the red sideband comes from lower vibrational levels which are more populated than the higher ones. In the harmonic approximation, the intensity of the Stokes sideband is proportional to (0)|S01 |2 +(1)|S12 |2 + · · · where (n) is the population of the vibrational level n and Spn is the transition amplitude between levels p and n. Conversely, the intensity of the anti-Stokes line is proportional to (1)|S10 |2 + (2)|S21 |2 + · · · . For a thermal distribution, (n)=(n − 1) = exp − (˝0v =kB T ) and therefore, the ratio of the sideband strengths is given by the Boltzmann factor exp − (˝0v =kB T ). This provides an independent estimation of the temperature of the atoms in the lattice. This value is in reasonable agreement with theoretical predictions (Section 3.2) and with time-of->ight experiments (Section 6.1). Comments: (i) Many e@ects described in Section 5 for probe transmission are also found on the >uorescence spectrum. For example, di@erent emission spectra are found along the x- and z-axis of a 3D lin ⊥ lin lattice because the sti@ness of the potential is di@erent in these two directions [108]. The sideband is also located at a frequency smaller than the value predicted by the harmonic approximation because of the anharmonicity of the optical potential. (ii) The Rayleigh peak of the emission spectrum is theoretically the sum of an elastic contribution (a Dirac : function) and of an inelastic contribution having the same width as the stimulated Rayleigh resonance (Sections 5.5 and 5.6). Note that this inelastic contribution generally originates from several relaxation modes having di@erent damping times. (iii) If the local oscillator is removed from the experimental scheme (Fig. 35), a self-beating signal is observed [146]. Vibrational resonances can then be detected as shown in Fig. 38 [147]. However, the asymmetry between the sidebands of Fig. 36 is lost in Fig. 38. A careful comparison of the relative advantages of heterodyne and self-beating spectroscopy is presented by Westbrook [148].
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6.2.3. Photon correlation spectroscopy Instead of using a single detector for >uorescence, it is possible to study correlations between two spatially separated detectors observing the same optical lattice. It is also possible to study correlations between two detectors looking in the same direction but detecting cross-polarized photons. To illustrate the possibilities of this approach, let us consider a single atom in a lin ⊥ lin lattice. When the atom is bounded in a + well, it scatters mainly + photons. When it hops into a − well, the scattered light changes polarization. Thus the time scale over which polarization remains correlated should be related to the time spent in a given well. The photon correlation method was used to study spatial di@usion [97] and spontaneous Brillouin scattering associated with the propagating excitation presented in Section 5.4 [147]. 6.2.4. Time resolved Iuorescence In steady state, atomic density and magnetization match the local polarization and intensity of the lattice Ield. When this Ield is changed in a time scale short compared to the atomic response time, there is a shift between the atomic and the light patterns which yields a modiIcation of the total >uorescence. This phenomenon is the spontaneous analog to the photon redistribution between the lattice beams (see Section 5.6.2). Consider for instance the 1D lin ⊥ lin lattice (Fig. 10) and assume that a sudden phase-shift of the incident beam translates the bipotential by =4. Because atoms located at a + site are in the mg = − 1=2 state immediately after the translation, the >uorescence suddenly decreases because of smaller Clebsch– Gordan coeLcients (see Fig. 11 for the case of the Jg = 1=2 → Je = 3=2 transition). Then the >uorescence returns to its steady-state value with a temporal behaviour depending on the eigenvalues of the dynamical modes of the optical lattice. If the translation is smaller than =8, >uorescence should exhibit a variation associated with the damped oscillation of atoms in a potential well. Instead of translating the potential, one can modify its depth by changing the beams intensity or detuning [132]. If the potential depth is suddenly changed from U0 to a new value U0 , the initial wave packet is projected unaltered into the new potential and undergoes damped oscillations in a breathing mode around the potential minima. Calling 0v the oscillation frequency in the harmonic approximation, the period is only =0v because of the symmetry of the wave packet. Indeed, after this time interval the atoms initially located on the right-hand side of the potential minimum occupy the symmetric position on the left-hand side of the potential minimum. The atoms initially on the left-hand side being conversely on the right-hand side, the initial wave packet is recovered, provided that damping processes can be neglected during =0v . This e@ect was observed by Rudy et al. [132] in a 3D optical lattice Illed with sodium atoms (see Fig. 39). 6.3. Spatial di;usion The problem of atomic transport in optical lattices is both very diLcult and particularly important to understand the atomic dynamics in this kind of structure. Depending on the depth of the optical potential and thus on the cooling eLciency, one can distinguish three regimes: if the potential is deep enough, the motion of the atoms is di@usive, which corresponds to d X 2 =dt = 2Dx ; Dx being the spatial di@usion coeLcient along the x direction. At the opposite
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Fig. 39. Normalized >uorescence of sodium atoms in a 3D lattice after a sudden change of the potential depth. Fluorescence is detected at a Ixed time (2:6 s) after the potential change and the data are plotted versus the expected period =0v . Each experimental result (triangles) thus corresponds to a di@erent potential depth. The solid curve is the expected result. From Rudy et al. [132], reproduced by permission of the authors.
limit, the atoms are nearly free and their motion is ballistic, X 2 increasing quadratically with t. Between these two situations, one Inds a regime corresponding to anomalous di@usion, where few atoms undergoing LYevy walks provide the main contribution to the evolution of the spatial distribution. The simplest theoretical model describing spatial di@usion uses standard results of Brownian motion in the linear regime, i.e. with a friction force linear in velocity and a uniform momentum di@usion coeLcient (see Section 3.2.2, comment (iii)). The validity condition for such a simple model is that the average displacement of an atom during one optical pumping time is small compared to the characteristic length of the potential (i.e. v= U p ); using Eq. (62), one gets I ˝0 ; (121) ER where I is a numerical coeLcient depending on the geometry and the atomic transition. In fact, as soon as 0 ER , Eq. (121) corresponds to the condition for the jumping regime. Outside this regime, an extension of the Brownian model can still be used, starting with more realistic expressions for the friction force and the momentum coeLcient [62,65]. This leads to predictions in reasonable agreement with the experimental results for deep enough potential wells. However, quantitative predictions require a complete treatment taking into account both the contribution of atoms with energies far above the potential wells and the contribution of localized atoms [63]. We describe in Section 6.3.1 the experiments studying Gaussian spatial di@usion and in Section 6.3.2 the experimental results obtained about anomalous di@usion. 6.3.1. Gaussian di;usion Historically, the Irst estimates of spatial di@usion coeLcients Dsp of atoms in optical molasses were deduced from the molasses lifetime [9,141]. Unfortunately, it appeared that the values obtained were up to 2 orders of magnitude larger than those predicted by simple theoretical models. The reason for this discrepancy is probably that the main loss mechanism is background collisions rather than atomic di@usion.
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A more relevant experimental method was thus developed to study spatial di@usion of atoms in optical molasses [62]. It consists of taking a series of pictures of an atomic cloud expanding inside the spatial domain delimited by the molasses beams. A careful analysis of these images then allows to measure the size X 2 of the cloud as a function of time and, after checking that the square X 2 of this size evolved linearly with time, to calculate the spatial di@usion coeLcient Dx =
1 d X 2 : 2 dt
(122)
The most extensive study of this kind was performed on rubidium atoms with a 3D beam conIguration consisting of three pairs of cross-polarized counter-propagating beams along the three directions Ox; Oy and Oz [62]. Strictly speaking, this conIguration leads to a molasses rather than a lattice, because the potential topography changes with the relative phases of the beams (see Section 2.5.2). It is however interesting to describe the results obtained as they are similar to what is expected in an optical lattice. First, the di@usion coeLcient Dx is found to be on the order of a few 100˝=M , which is in good agreement with theoretical predictions. Second, in the limit of deep potential wells Dx increases linearly with the Ield intensity (the points of largest light-shift in Fig. 40 correspond to a small ||). Third, Dx reaches a minimum for a given value of the light-shift min ; then it increases and Inally diverges for smaller values of . Fig. 40 shows the measurements performed by Hodapp et al. for di@erent values of the detuning and of the intensity I of the laser beams, as a function of the light-shift per beam 0 . Comments: (i) In principle, one can also measure spatial di@usion coeLcients indirectly, either through photon correlation spectroscopy [97] (see Section 6.2.3) or through pump–probe spectroscopy [118] (see Section 5.5). Such a method seems much simpler and faster than the direct technique described above. Nevertheless, quantitative relations such as Eq. (118) between the width of the zero-frequency resonance on experimental spectra and the di@usion coeLcient are often not veriIed because the validity conditions are not met [118]. (ii) With the same experimental technique, spatial di@usion was also studied in a Talbot lattice (see Section 2.5.3) Illed with cesium atoms. We show in Fig. 41 the shape of an initially spherical cloud after a lattice phase of 60 ms. Spatial di@usion is much faster in the transverse directions where the spatial period of the potential is much larger than in the longitudinal direction [46]. (iii) Di@usion measurements were also performed with nonperiodic structures, namely atoms cooled in a quasiperiodic optical lattice [149] and in a speckle Ield [150]. These two experiments show that investigation of spatial di@usion gives a lot of information on atomic dynamics, and especially on the role of periodicity. 6.3.2. Anomalous di;usion Both experimental results showed in Fig. 40 and theoretical predictions show a divergence of the spatial di@usion coeLcient for low values of the light-shift. In the case of the 1D lin ⊥ lin conIguration and for a Jg = 1=2 → Je = 3=2 transition, this divergence occurs for U0 ¡61:5ER [63,119].
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Fig. 40. Spatial di@usion coeLcients Dx obtained by the measurement of the expansion of an atomic cloud in a 3D optical molasses generated by three pairs of cross-polarized counter-propagating beams. The data are plotted as a function of the light-shift per beam |0 |. For the largest values of |0 |; Dx increases linearly while it reaches a minimum for |0 | 0:06 and increases for lower values. From Hodapp et al. [62], reproduced by permission of the authors. Fig. 41. Pattern of the >uorescence emitted by cesium atoms in a Talbot lattice. Because spatial di@usion is slower along Oz, the shape of the cloud becomes an ellipsoid. From Mennerat-Robilliard [46].
This regime of shallow potential wells has been investigated by Katori et al. [119] with a single ion radially conIned in a 2D radio-frequency quadrupole trap and cooled in a 1D lin ⊥ lin lattice along the longitudinal direction. The position of the ion is traced through >uorescence photons. Additionally, a weak harmonic electric potential is superimposed on the optical potential, allowing for an easy relationship between the position and the potential energy of the ion. A careful statistical analysis of the recorded ion position as a function of time shows that for shallow potential wells, some anomalous >uctuations of the ion’s kinetic energy occur with long range correlations. These correlations originate from the existence of LYevy >ights, i.e. periods during which the ion has enough energy to travel over many wavelengths before being trapped again. Although quantitative comparison of the experimental results with theoretical predictions [63] is not straightforward because of the weak electric potential and of the residual micromotion of the ion in the RF trap, this experiment is a nice evidence for signiIcant deviations from the Gaussian di@usion law. 7. Bragg scattering and four-wave mixing in optical lattices In this section, we present additional diagnostic techniques related to Bragg scattering. We Irst study the basics of Bragg di@raction in optical lattices (Section 7.1). In the second paragraph (Section 7.2), we discuss the relationship between Bragg scattering and four-wave mixing in optical lattices and present some four-wave mixing experiments. The third paragraph (Section 7.3) is devoted to experiments using the dependence of the Debye–Waller factor on atomic localization to get some dynamical information on cooling and localization. In the last paragraph
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Fig. 42. Ewald construction yielding the set of Bragg di@racted wavevectors kB for a given incident probe beam kp . O being the origin of the reciprocal lattice, a sphere of radius kp is drawn around the point A such that OA = kp . A di@raction peak can be observed if the surface of the sphere contains a point B that belongs to the reciprocal lattice. The di@racted wave is emitted in the direction BA = kB .
(Section 7.4), we discuss the modiIcations of light propagation in the optical lattice due to localized atoms. 7.1. Basics of Bragg scattering In solid-state physics, X-ray Bragg scattering is commonly used to test long-range order in materials. Because of the structural analogy between crystals and optical lattices, it is natural to try and apply the same technique to optical lattices. The basic idea is that a beam with an incident wavevector kp can be Bragg di@racted by a periodic structure into a beam having a wavevector kB , provided that the di@erence kB − kp be a vector G of the reciprocal lattice [29]. Considering that the reciprocal lattice is discrete and that Bragg scattering is essentially an elastic process (|kp | = |kB |), there is generally no wavevector matching the Bragg condition for a given incident monochromatic beam. The Ewald construction presented in Fig. 42 can be helpful to determine the set of Bragg di@racted beams for a given incident beam. In the case of an ideal crystal where the atoms are rigidly placed at the lattice sites, the width of the Bragg peaks is di@raction limited. Taking into account the vibration of the ions around their equilibrium position, two changes occur: (i) the intensity of the Bragg peaks decreases because of the Debye–Waller factor which is associated with atomic localization (see Section 7.3), (ii) a di@use background due to multiphonon processes appears. This discussion can be readily transposed to optical lattices, taking into account that the typical distance between two sites is now on the order of the optical wavelength instead of one \ Angstr Hm for solid crystals. Bragg di@raction of electromagnetic waves can therefore occur in the optical range instead of X-rays. We have seen in Section 2.3.2 that for optical lattices, the reciprocal lattice is generated by the di@erences between the lattice beams wavevectors. This implies in particular that the Bragg condition is automatically matched for a di@raction process between two lattice beams (because kj = ki + G with G = kj − ki ).
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Fig. 43. Phase conjugation in an optical lattice. (a) Photon exchange in the energy diagram and (b) phase-matching condition in a 1D lattice. The lattice beams are represented by double arrows, the probe beam with a simple arrow and the conjugate beam with a dashed arrow.
Experimentally, Bragg scattering in optical lattices was demonstrated by the groups of Phillips [151] and of H]ansch [152]. Both found out that the Bragg peaks have an intensity several orders of magnitude larger than the background due to di@use scattering, and that their angular breadth, on the order of 1 mrad, is limited by the collimation of the incident probe beam and not by the di@raction due to the Inite size of the optical lattice. Because of the frequency dependence of the scattering cross-section by the atoms, the probe beam frequency should be close to an atomic resonance to achieve a signiIcant Bragg re>ectivity. By sweeping the probe frequency around an atomic resonance, one can record a frequency spectrum of Bragg re>ectivity. As expected, the spectrum exhibits a peak having a width on the order of the natural linewidth of the transition [152]. For increasing atomic densities however, the peak broadens and a dip even appears at reasonance [151], accounting for the huge absorption cross-section in these conditions. Comments: (i) A question arising when considering Bragg scattering in optical lattices concerns the low Illing factor, which is generally on the order of a few percents or less in 3D lattices. One might fear that the random distribution of atoms among the lattice sites destroys the coherence of the scattering process. This is however not the case, because the empty lattice sites do not distort the potential which is imposed by the lattice beams. Contrary to the situation in solid-state physics, empty sites are not defects. (ii) In principle, the angular width of the Bragg peak is related to the spatial extension of the illuminated sample, provided the incident probe beam is well collimated enough [153]. (iii) Bragg di@raction was also used in a quasiperiodic optical lattice (see Section 2.4) to prove the long-range quasiperiodic order in such a structure [38]. 7.2. Bragg scattering and four-wave mixing We have already mentioned in Section 2.3.2 that Bragg di@raction in optical lattices is connected to four-wave mixing processes including two lattice photons, a photon from the incident probe beam and one in the di@racted beam (see Section 2.3.2). For a deeper analysis, one can distinguish two types of four-wave mixing, namely phase conjugation-like processes and nonlinear elastic scattering (also called distributed feedback processes). We show in Fig. 43a the energy diagram for phase conjugation and in Fig. 43b the corresponding phase matching geometry in the 1D case. More generally, in these processes two photons from the lattice beams are absorbed, a stimulated emission occurs in the probe beam and a photon is emitted in the direction kpc that satisIes the phase-matching condition for this
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four-wave mixing process kpc + kp − k1 − k2 = 0 ;
(123)
where k1 and k2 are the wavevectors of two lattice beams. Eq. (123) di@ers from a Bragg condition because k1 + k2 does not generally belong to the reciprocal lattice. Phase conjugation in an optical lattice can also be interpreted as the scattering of a lattice beam on the atomic polarizability grating induced by the probe and a lattice beam interference pattern. It is thus clear that such a process does not constitute any evidence for the long-range order in the optical lattice [13]. Four-wave mixing spectra were recorded in a + − + one-dimensional lattice with a small transverse magnetic Ield (see Section 3.5.3). The intensity of the phase-conjugated emission exhibits a resonant enhancement when the probe beam frequency !p satisIes !p − ! = 0; ±n0v where n is an integer and 0v is the vibrational frequency [154]. These resonance conditions correspond, respectively, to the situations where the intermediate energy level in the four-wave mixing process is the initial one (degenerate four-wave mixing: !pc = !p ) or that of a neighbouring vibrational level (nondegenerate four-wave mixing: !pc = 2! − !p ). This type of four-wave mixing spectroscopy was also applied to a 3D six-beam optical lattice [155]. The spectra obtained (see Fig. 44) have a general shape similar to the ones in 1D lattices, except that the Rayleigh resonance is much narrower than the Raman vibrational sidebands. Indeed, the process which is responsible for the decay of the long-range grating induced by the pump–probe interference pattern is in this case atomic di@usion, which is very slow with respect to other relaxation processes including optical pumping. The information that can be deduced from a phase-conjugation spectrum is in fact equivalent to that found by probe transmission spectroscopy [72]. The second type of four-wave mixing process includes the absorption of only one photon in a lattice beam. A probe photon is also absorbed and there is a stimulated emission of one photon into another lattice beam. Finally, a photon is emitted into the di@racted beam. The corresponding phase-matching condition reads kd − kp + k1 − k2 = 0 :
(124)
For a given probe wavevector, the condition imposed by Eq. (124) generally corresponds to scattering directions di@erent from those satisfying Eq. (123). By contrast, Eq. (124) coincides with a Bragg condition because k1 − k2 belongs to the reciprocal lattice. This four-wave mixing process was studied in the case of the 2D three-beam optical lattice [13,156] (see Section 2.3.1). In the experiment, a probe beam counter-propagating with respect to one of the lattice beams (kp = − k1 ) is scattered into two beams counter-propagating with respect to each of the two other lattice beams (kd = − k2 and kd = − k3 ). The four-wave mixing spectra obtained by sweeping !p yield some information similar to those obtained by probe transmission spectroscopy. Fig. 45 shows how photons are exchanged in the energy diagram in this nonlinear elastic scattering. Two di@erent processes can occur. In Fig. 45a, the sequential order of elementary steps is the following: absorption of a probe photon, stimulated emission in beam 1, absorption from beam 2, emission of a photon of wavevector kd . This four-wave mixing process is resonant when : = 0; ±n0v but the emitted photon has the same frequency as the probe (!d = !p ). Another equivalent interpretation of this process is that the beam 2 is scattered by the atomic
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Fig. 44. Phase-conjugation spectrum for a weak probe beam directed through a cubic 3D six-beam optical lattice Illed with rubidium atoms. The phase-conjugated re>ectivity exhibits resonances when : = ± n0v and : = 0. The inset shows the central part of the spectrum with a better resolution. From Hemmerich et al. [155], reproduced by permission of the authors. Fig. 45. Scheme of two possible four-wave mixing processes that yield nonlinear elastic scattering. The lattice (resp. probe) beam photons are represented by a double (resp. simple) arrow. The dashed arrow corresponds to the scattered wave. Only the process of (b) can be considered as Bragg di@raction in the optical lattice.
grating originating from the interference pattern created by the beam 1 and the probe beam. The emission is therefore not associated with the density modulation in the optical lattice in the absence of the probe beam. In the diagram shown in Fig. 45b, the sequential order is di@erent. Here, the absorption in beam 2 followed by the emission in beam 1 occurs before the absorption of a probe photon and the emission of a photon with wavevector kd . This four-wave mixing process does not exhibit any resonance when : coincides with the vibration frequency. In the classical picture, the emission results from the scattering of the probe beam o@ the atomic grating created by the lattice beams 1 and 2. This is therefore a Bragg di@raction process. True Bragg di@raction can be observed if the experiment is not sensitive to four-wave mixing processes described by the diagrams of Fig. 45a. This can be done either by switching o@ abruptly the lattice beams before the Bragg scattering observation [151], or by resonantly enhancing the diagram of Fig. 45b with respect to the one of Fig. 45a. For instance, the choice of a probe beam frequency far enough from that of the lattice beams creates a lattice-probe interference pattern that moves too fast to imprint any grating on the atomic observables [152]. Comments: (i) Note that neither of the two experimental methods described above is able to prove unambiguously a periodic modulation of the atomic density. Other atomic observables can be modulated and give rise to di@raction in the same way as a density modulation. In particular, an excited state population grating or a magnetization grating can both produce Bragg scattering, even with an uniform density. Switching o@ the lattice beams rapidly eliminates the excited state population grating, but not the magnetization grating. This question was addressed experimentally by comparing Bragg re>ectivities for di@erent polarizations of the Bragg probe [153].
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(ii) In Figs. 43 and 45, additional diagrams yielding the same physical conclusions can be drawn. For example, in Fig. 45a the following order is also possible: absorption of 2, emission of d, absorption of p, emission of 1. 7.3. The Debye–Waller factor—sensitivity to atomic localization Let us now consider Bragg di@raction in a more quantitative way. We assume that each atom is localized at some lattice site R, bound in a harmonic potential well considered as isotropic for the sake of simplicity. Note that for deeply bound vibrational levels, the harmonic approximation is excellent. A linearly polarized plane travelling wave of intensity Ip and wavelength p is incident on the lattice. The amplitude A1 scattered by a single atom is proportional to (= p2 ) Ip sin H| 0 |, where H is the angle between the incident polarization and the di@racted wavevector and 0 is the projection of the atomic polarizability tensor on the incident polarization vector. Then the average power di@racted into the solid angle d by N atoms randomly distributed among Nl lattice sites is proportional to [157] 2 2 N‘ (N‘ − N ) dP N |A1 |2 L2 |S |2 eiSk · R + N‘ (1 − L2 ) + : (125) = d N‘ N R
In this equation, the last two terms correspond to incoherent contributions coming, respectively, from the Inite temperature of the lattice and from the random distribution of atoms among the lattice sites. In the following, we shall consider only the Irst term in the brackets of Eq. (125) associated with coherent scattering and predominant in experiments with optical lattices. In this coherent term, the sum has to be taken over all lattice sites R. It averages to zero as soon as Sk = kB − kp does not satisfy the Bragg condition. The quantity S = j exp(iSk · rj ) is the structure factor of the unit cell (rj is the position of the jth atom of the basis) and L = exp(iSk · :R) is the Debye–Waller factor, where the bar represents the average over the distribution of position deviation :R with respect to the center of the potential well. If we assume a thermal energy distribution of the atoms in the potential wells, then the position distribution is Gaussian and the Debye–Waller factor takes the well known exponential form 1 L = e−W with W = [(Skx )2 :X 2 + (Sky )2 :Y 2 + (Skz )2 :Z 2 ] : (126) 2 In view of Eq. (125), the scattered intensity appears as an accurate probe for atomic localization. Unfortunately, a direct relation between the absolute di@racted intensity and localization requires a good knowledge of the number of trapped atoms, which is diLcult to measure precisely. However, some relative measurements provide some information on the evolution of localization. Sensitivity to atomic localization was Irst shown independently in two experiments: in the Irst one, the scattered intensity was measured as a function of time after the lattice beams were switched o@ [151]. Because of the initial momentum distribution, the position distribution spreads as :X (t)2 = :X (0)2 + Px2 t 2 =M 2 , leading to a temporal decrease of the di@racted intensity according to a Gaussian law. The time constant of this decay is on the order of a few microseconds, in agreement with independent temperature measurements [151]. In the second
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experiment [152], the lattice was created with infrared beams ( = 780 nm) tuned on the red side of the 5S1=2 → 5P3=2 rubidium transition whereas the Bragg beam is nearly resonant on the 5S1=2 → 5P1=2 transition and has therefore a completely di@erent wavelength ( = 422 nm). A second probe beam at p 780 nm can induce stimulated Raman excitations of the vibrational motion when !p − ! = ± 0v (see Section 5.3). When this condition is fulIlled the oscillation motion of the atom is resonantly excited and :X 2 increases. This yields a drop in the intensity of the Bragg di@racted beam (by an amount on the order of 20% [152]) because of the weaker localization. One fundamental use of Bragg scattering in optical lattices is the dynamical study of cooling and localization [158]. Bragg scattering is indeed both extremely sensitive to localization because of the exponential form of the Debye–Waller factor, and instantaneous, contrary to pump– probe experiments for instance, which require a scanning of the probe beam frequency. Using 2 of the atoms along Sk can be deduced from Eq. (126), the mean-square position spread :XSk 2 = − ln[I (t)]=(SK)2 + C, C being an additive the Bragg scattered intensity IB (t) through :XSk B constant which needs some further assumptions to be determined. In the harmonic approximation 2 = K T=02 in the case of thermal equilibrium. and for an isotropic potential, one Inds :XSk B v 2 The time evolution of :XSk therefore measures the evolution of both the localization and the cooling. We show in Fig. 46 the results obtained by Phillips and coworkers [158] with cesium atoms in 1D and 3D lin ⊥ lin optical lattices. The curves have several striking features. First, for potential wells suLciently deep, the steady-state value reached by the mean-square position spread is independent of the lattice beams intensity and detuning, which is expected because both the kinetic temperature kB T and 0v2 = 4ER U0 = ˝2 are proportional to the potential depth U0 (see Section 3.2.3). Second, the cooling rates are on the order of tens of microseconds, and for a given potential depth the evolution in 1D is around 6 times faster than in 3D. Third, an 2 as a function of both intensity and detuning showed that the cooling extensive study of :XSk rate is proportional only to the photon scattering rate , i.e. that whatever the conditions, an atom needs to scatter about 30 photons to be cooled in 1D and about 200 in 3D [158]. This result, which was conIrmed by both semiclassical and wavefunction Monte-Carlo simulations, is unexpected in view of the theory of Sisyphus cooling, which predicts a cooling rate independent of the lattice beams intensity [4]. This suggests that the actual cooling mechanism in a realistic optical lattice is partly di@erent from the usual theoretical description (Section 3.2). In particular, e@ects associated with localization could be signiIcant. In the same spirit, one can also use Bragg di@raction in the transient regime to study the reaction of trapped atoms to a sudden change in the potential. If one modiIes the depth of the potential by a sudden change in the lattice beams intensity, the Bragg re>ectivity exhibits oscillations corresponding to “breathing modes” of the atomic wave packets [159,160], 2 at twice the oscillation frequency in the potential well. We show in i.e. oscillations of :XSk Fig. 47 an example of such oscillations, on which di@erent time constants appear: the fastest one corresponds to twice the atomic oscillation period, as is expected for a breathing mode (see Section 6:2:4). These oscillations decay after a few periods, both because of the anharmonicity of the potential wells and of the damping of the vibrational coherences which are excited when the potential depth changes. But one also sees a slower evolution time, on the order of 40 s,
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Fig. 46. Mean-square position spread deduced from Bragg scattering experiments in cesium 1D and 3D lin ⊥ lin optical lattices. The solid lines represent exponential Its of the experimental data with a free additive parameter. 2 reaches a steady-state value independent of the parameters. In both lattices, for deep enough potential wells, :XSk From Raithel et al. [158], reproduced by permission of the authors. Fig. 47. Wave packet oscillations of the mean-square position spread in a 3D lin ⊥ lin lattice induced by a sudden increase of the potential depth from U0 = 850ER to U0 = 3200ER at t = 0 ( = − 5). The long-term heating is best Itted with a time constant of 44 s. From Raithel et al. [160], reproduced by permission of the authors.
which corresponds to a heating of the atoms towards the steady-state temperature corresponding to the new potential. Additionally, both groups (NIST and Munich) report the appearance of some structure above the noise after the oscillations decay [159,160]. Although this feature is not clear enough to be analysed quantitatively, it probably corresponds to quantum revivals, which occur if the anharmonicity-induced dephasing is faster than the damping of the coherences (a similar example of competition between dispersion and dissipation has been discussed in Section 5.6.2). Comments: (i) To get an absolute measurement of atomic localization, one can compare the intensity scattered in two di@erent Bragg spots [159]. (ii) Contrary to pump–probe spectroscopy which addresses all the populated vibrational levels in a potential well, the Bragg scattering technique is mainly sensitive to the lowest lying levels which correspond to the best localization. The Bragg re>ectivity decreases indeed exponentially with localization. This selectivity on the low lying levels was experimentally observed in [152], where the dips in Bragg re>ectivity recordings were signiIcantly narrower than the width of vibrational resonances on the corresponding pump–probe spectra. 7.4. Backaction of the localized atoms on the lattice beams Until now, we have considered the action of the lattice beams on the atomic dynamics. However, the Bragg condition being automatically fulIlled between two lattice beams, the lattice
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waves are Bragg scattered into one another. This redistribution of the Ield energy induced by the atoms modiIes the beams propagation and this reacts on the position of the atoms. A rigorous treatment of optical lattices thus requires a self-consistent approach, which is theoretically extremely complicated. However, one can have a simple insight in the physical mechanisms involved by modeling a 1D optical lattice as planes of polarizable atoms of polarizability 0 and with a planar density M, perpendicular to a standing wave (i.e. two counter-propagating travelling waves E0 ei(±kz−!t) with identical polarizations) [161]. Because of the small atomic densities in optical lattices, we neglect multiple scattering and other losses. Let us now consider one atomic plane located at z = z0 and calculate the amplitude to the total light Ield on each side of the plane [161]. The one-dimensional propagation equation in the vicinity of z = z0 is !2 !2 !2 2 9z + 2 E = − P = − M 0 :(z − z0 )E : (127) c 0 c2 c2 The tangential electric Ield is continuous at the boundary: E(z = z0− ) = E(z = z0+ ) :
(128)
By contrast, the derivative of the Ield exhibits a jump that can be calculated from Eq. (127): !2 M 0 E(z = z0 ) : (129) c2 For a bright lattice, i.e. for red-detuned laser beams, the atoms are localized in planes of maximum intensity. Let us assume that such a plane is found in z0 = 0 and that the solution of the Maxwell equations is 9z E(z = z0− ) − 9z E(z = z0+ ) =
Eleft = 2E0 cos(kz − ) cos !t ;
(130)
Eright = 2E0 cos(kz + ) cos !t ;
(131)
where Eleft and Eright , respectively, correspond to the Ield in the − latt ¡z¡0 and 0¡z¡ latt regions ( latt is the spatial periodicity of the optical lattice). This Ield is obviously a solution of Eq. (127) outside z = 0 with k = !=c. It is also continuous in z = 0. Finally, we deduce from Eq. (129) that tan = (k=2M) 0 . This procedure can be repeated at each antinode where the atoms are localized. From Eqs. (130) and (131), it obviously appears that the distance between nodes and antinodes is reduced by comparison to the vacuum case. More precisely, the Irst node for z¿0, corresponding to z = latt =2, is found when kz + = =2. This equation directly yields 2 latt = 1− : (132) 2 This spatial periodicity can be compared with that found in a dilute disordered gas with the same mean density N = 2M= latt kM=. The refractive index is nrandom = 1 + n 0 =2 1 + . The wavelength random inside this gas is therefore random = (1 − =). The distance between two antinodes is random =2. The comparison with Eq. (132) shows that the periodic arrangement of the atoms induces an e@ect twice larger through coherent interference e@ects.
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Fig. 48. Shift of the Bragg angle as a function of density. The shaded area represents the experimental conIdence interval. The dotted line corresponds to the expected value without localization, and the dashed line represents the calculated value with the actual atomic localization. From Weidem]uller et al. [153], reproduced by permission of the authors.
The situation for a blue-detuned lattice is quite di@erent. The atoms are indeed localized at points of minimum intensity. A solution of the Maxwell equations of the form Eleft = Eright = 2E0 cos kz fulIls the boundary conditions (128) and (129) in a plane such as z = =4 where the atoms should be found. The lattice spatial period thus remains equal to =2. One can understand the di@erence between red and blue detuned lattices with simple arguments: compared to a randomly distributed medium, the interaction of the atoms with light is suppressed for blue detuned lattices because the atoms are localized at nodes of the Ield, while it is enhanced for bright lattices where the atoms are localized at the antinodes of the Ield. For a realistic 3D optical lattice, more complicated models have to be developed, taking into account Bragg scattering of all lattice beams into one another [153]. Experimentally, the change of lattice constant can be detected through the change in the Bragg angle # between the incident and di@racted beams, because the Bragg condition can be written 2d cos # = B where d is a spatial period of the lattice [29]. A contraction Sd∼10−4 d was indeed observed in red-detuned optical lattices for high enough atomic densities [151,153]. We show in Fig. 48 the shift of Bragg angle as a function of the atomic density, for constant lattice parameters [153]. The shaded area represents the experimental uncertainties due to the error in determining the density and also to the error in the calibration of the Bragg angle. The dotted line corresponds to the expected value for a disordered atomic distribution and the dashed line, to the value calculated with atomic localization. One sees clearly the e@ect of localization on the lattice contraction. Coherent scattering of light by the atoms localized in an optical potential can also lead to photonic bandgaps, similarly to the electronic bandgaps that are studied in solid crystals as a result of spatial periodicity [161]. From an optical point of view, photonic bandgaps arise from interferences between di@erent Bragg di@racted beams that strongly enhance re>ection and suppress propagation of light inside the optical lattice. The engineering of materials with three-dimensional bandgaps (which means that in a given frequency range no light propagates in the material whatever the direction) is a very active
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Ield of research. Such an e@ect requires that there is a strong Bragg scattering in all directions. As far as optical lattices are concerned, their Illing factor is very small, and the atomic scattering cross-section is reasonably large only at resonance, so that optical lattices cannot contain multi-dimensional bandgaps. Nevertheless, a method using an interference pattern to engineer a 3D photonic crystal has been demonstrated recently [12]. This technique consists of illuminating a photoresist with the 3D interference pattern originating from 4 noncoplanar laser beams, and then dissolving the nonilluminated parts. The obtained structure is periodically modulated in three dimensions and can be used as a template for materials with a higher index of refraction.
8. Atomic interactions in optical lattices This section deals with the atom–atom interactions in a dissipative optical lattice. We Irst discuss the di@erent types of light-induced interactions that may occur (Section 8.1), and then we study the consequences of atomic localization on collisional rates (Section 8.2). 8.1. Light-induced interactions in optical lattices Most of the time, the atomic dynamics in dissipative optical lattices can be described without taking into account any interaction between the atoms. The Illing factor in these structures is indeed very low (on the order of the percent in 3D lattices), which means that the mean distance between two trapped atoms is on the order of a few microns. However, a theoretical study of long range molecular interactions shows that the interaction energy of two atoms in nearby potential wells of a dissipative optical lattice is not negligible with respect to the potential depth for one atom [162]. For example, for a 1D optical lattice generated by two counter-propagating waves with parallel linear polarizations, the interaction is attractive and for two atoms in adjacent wells, the interaction energy can be as large as 1=6 of the potential depth. The molecular interaction energy of two atoms in the same potential well of a dissipative optical lattice is about twice the depth of the potential well, which forbids to put more than a single atom in one well [162]. Other kinds of light-induced interactions, such as multiple photon scattering, are possible in principle, but cannot be identiIed in practice because of the low densities achieved in near-resonant lattices. Comments: (i) Note however that the situation is very di@erent in far-o@ resonant lattices, where no spontaneous emission occurs and where molecular interactions are much smaller (they scale as 1= [162]): the densities achieved in these systems can reach one atom per well [79,163] and Bose–Einstein condensation might even be possible. (ii) In optical molasses, the phase-space density is intrinsically limited by light reabsorption [164 –167]. This is probably the case also for near-resonant bright optical lattices, although the localization of atoms in potential wells makes the situation di@erent. (iii) It is possible to have a spatial period much larger than in an optical lattice. This is for instance the case in a four-beam lattice with small angles #x and #y (Section 3.3.4). In
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Fig. 49. (a) Penning ionization detected during the lattice phase and afterwards, averaged over 50 cycles. (b) Comparison of collision rate in the lattice to that for free atoms as a function of time. The detuning is = − 4800 and the intensity is I = 5 W=cm2 per beam. From Lawall et al. [169], reproduced by permission of the authors.
this situation, two atoms in the same well can be suLciently far away so that their interaction remains smaller than the potential depth. 8.2. Study of collisions In optical lattices, the atoms are strongly localized (:X ∼0:1 ) around the bottom of the potential wells. The question of collisions then arises: does atomic localization reduce the collisional rate by isolating each atom in its own potential well, or does the optical potential channel the atoms towards the wells, making them more probable to collide with one another? This question was addressed in two experiments performed at the University of Tokyo [168] and at NIST [169]. Both use metastable atoms and detect ionizing collisions, which are dominant [170,171]. The metastable atoms (argon, krypton or xenon) are Irst cooled in a magneto-optical trap and then transferred into a 3D four-beam optical lattice. After the lattice phase during which the collisional rate is recorded continuously, the lattice beams are shut o@. We show in Fig. 49a a typical signal obtained with 132 Xe [169]: the collisional rate drops slowly because of the reduction in density due to atomic di@usion and collisions, and also because of localization inside the potential wells. Then the lattice beams are switched o@ at t = 100 ms, and the collisional rate increases again, showing that a suppression of collisions indeed occurs in the lattice. The rapid decrease afterwards originates from the ballistic motion of the atoms once in the dark.
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Fig. 50. Open squares: Rlatt =Rfree versus . Solid circles: Rlatt =Rtrav , where Rtrav is the measured collision rate for atoms submitted to a travelling wave (i.e. experiencing no optical potential) having an intensity 8I equal to that at the bottom of the lattice wells. I = 5 W=cm2 per beam. From Lawall et al. [169], reproduced by permission of the authors.
Fig. 49b shows the ratio Rlatt =Rfree of the collisional rate in the lattice compared to that for free atoms for a similar density (i.e. measured immediately after the lattice beams are shut o@), for di@erent lattice durations. As the lattice beams are turned on, the atoms are rapidly channelled into the potential valleys, and thus the peak density is higher than for randomly distributed atoms. Because the binary collision rate is proportional to the square of the density, a modulated density leads to a larger rate than a uniform one. Consequently, it seems that the atomic dynamics in the lattice Irst enhances collisions, which corresponds to values of Rlatt =Rfree initially larger than 1 in Fig. 49b. Then Rlatt =Rfree decreases towards an asymptotic value Rlatt =Rfree 0:5. This re>ects a suppression of collisions when the thermalized atoms are localized inside potential wells [168,169]. This interpretation is supported by the fact that the temperature has the same temporal dependence as Rlatt =Rfree [169]. Note that the ratio discussed above compares collisional rates in an optical lattice and in the darkness. One should however take into account the e@ect of the excited state population on the mean scattering cross-section, as studied for example in [172], which is not related to the lattice dynamics itself. It can thus be interesting to measure also the collisional rate for atoms in a travelling wave of intensity 8I , which corresponds to the intensity at the bottom of the wells induced by a 3D four-beam lattice. We show in Fig. 50 both the previous ratio Rlatt =Rfree and the ratio Rlatt =Rtrav of collisional rates in the lattice and in the corresponding travelling wave, in steady state, as a function of lattice laser detuning [169]. As expected, for large detunings both ratios correspond, because then the excited state population is negligible. As the detuning decreases however, Rlatt =Rtrav decreases while Rlatt =Rfree increases and becomes larger than 1. This shows that the dominant e@ect for || 6 2000 is due to resonant enhancement [172] rather than to intrinsic lattice dynamics. Therefore, the detailed analysis of the dependence of Rlatt =Rfree on the lattice beams intensity and detuning
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[168] requires some precautions and a complete theoretical treatment accounting for resonant excitation. 9. E4ect of a magnetic 5eld 9.1. Paramagnetism We have seen in Section 3.1.3 that the optical potential resulting from the lin ⊥ lin conIguration has an antiferromagnetic structure, i.e. it exhibits potential wells with alternating + and − light polarization. In the presence of a small static magnetic Ield B0 along Oz (small meaning here that the maximum Zeeman shift is small compared to the maximum light-shift), this property leads to a paramagnetic behaviour of the atoms in the lattice [145]. Let us consider an atom having an F → F = F + 1 transition with F ¿ 1 in a 1D lin ⊥ lin optical lattice. The eigenstates of light-shift are generally a superposition of several Zeeman sublevels, but at the bottom of ± potential wells they correspond to pure |F; mF = ± F states. In the absence of any magnetic Ield, the + and − potential wells have identical depths and pumping rates. The atoms are thus randomly distributed in the lattice sites, with equal probabilities of occupying a + or a − well. On the contrary, if a small static longitudinal magnetic Ield B0 is applied, the Zeeman shift is opposite for |F; mF = + F and |F; mF = − F sublevels. A typical potential obtained in this conIguration is showed in Fig. 51. From statistical physics intuition, one expects the atoms to be transferred to the potential wells corresponding to the lowest energy, leading to a macroscopic magnetization of the lattice. However, atoms in an optical lattice are not coupled to a thermal bath and the occurrence of a magnetization is not obvious. Most atoms being localized in + and − wells, they mainly populate the extreme Zeeman sublevels |F; mF = + F and |F; mF = − F , respectively, which have very di@erent absorbing cross-sections for + and − light: in the case of cesium, for example, the probability for an atom in the |F = 4; mF = + 4 sublevel of absorbing a − photon is 45 times smaller than that of absorbing a + photon for equal incident intensities. Consequently, for atomic transitions with large angular momenta one can consider that a weak + -polarized beam probes the atoms in + wells while a − -polarized beam probes those in − wells. The magnetization of the atoms can thus be deduced from the transmission of a circularly polarized probe beam through the atomic cloud [145,173]. Instead of measuring the transmission, it is possible to measure the stimulated Raman gain (or absorption) with a circularly polarized probe in the vicinity of the vibrational resonance (see Section 5.3). The gain for a ± -polarized probe is proportional to the atomic population -± in ± potential wells (provided one neglects the very weak absorption of ± light by atoms localized in ∓ wells). The ratio of the gains found with − and + probes is then simply equal to -− =-+ . We show in Fig. 52 this population ratio as a function of the applied magnetic Ield. The exponential dependence on B0 of -− =-+ reminds a thermal dependence and suggests the introduction of a phenomenological spin temperature. This temperature was found to be twice the zero-Ield kinetic temperature, in good agreement with numerical simulations [145,173]. This di@erence re>ects the well known fact that laser-cooled atoms are not in thermodynamics
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Fig. 51. Optical potential for an atom with a F = 4 → F = 5 transition in the 1D lin ⊥ lin conIguration. (a) In the absence of magnetic Ield, the + and − potential wells are identical. (b) With a small static longitudinal magnetic Ield B0 , these potential wells experience opposite Zeeman shifts.
Fig. 52. Ratio -− =-+ of the population in the − wells to that in the + wells, versus magnetic Ield B0 . The exponential variation permits to deIne a phenomenological spin temperature. From Meacher et al. [145].
equilibrium. Therefore, it is not surprising that the steady state of the system cannot be described by an unique temperature. The global magnetization of the atoms increases with the magnetic Ield until the Zeeman shift becomes on the same order of magnitude as the light-shift. The magnetization then decreases if B0 further increases, because then the potential curves are separated and associated to almost pure Zeeman sublevels, so that the situation resembles the Jg = 1=2 → Je = 3=2 transition (see comment below).
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The dynamics of magnetization, which is strongly correlated to localization, can also yield some information on the local character of the cooling process. We have seen in Section 3.1.2 that for a Jg = 1=2 → Je = 3=2 transition, Sisyphus cooling can occur only when the atoms jump between di@erently polarized potential wells. On the contrary, for atomic transitions with higher angular momenta a local cooling mechanism is also possible (see Fig. 13). Measuring both the cooling time constant and the magnetization (i.e. localization) time constant can thus help determining whether local processes play an important role in the cooling dynamics of “realistic” atoms. Such a dynamical study was performed with cesium atoms by Phillips and coworkers [173]. They Irst localize the atoms in an optical lattice with a small magnetic Ield and then switch o@ the magnetic Ield and measure the transmission of a weak circularly polarized probe beam. The magnetization time constant was found to be on the order of 100 s, around 3 times longer than the cooling time constant which was measured independently by Bragg re>ection (see Section 7). This shows that it is not necessary for an atom to hop from one well to a distant one to dissipate energy. It indicates that for atoms with high angular momenta, local cooling plays a signiIcant role in the Sisyphus process eLciency. Comments: (i) For an atom having a Jg = 1=2 → Je = 3=2 transition, no paramagnetic behaviour is expected in the lin ⊥ lin conIguration. The eigenstates of the light-shift operator are indeed the Zeeman sublevels |J; mJ = ± J , which are also eigenstates of Jz . Thus neither the eigenstates nor the pumping rates are modiIed by the addition of a magnetic Ield: the only e@ect of B0 consists of shifting globally both potential curves in opposite directions, without a@ecting the atomic dynamics. (ii) For J → J and J → J − 1 transitions (see Section 9.2), a paramagnetic behaviour is also expected and observed [69,70]. The magnetization increases linearly with magnetic Ield for small B0 , and then decreases towards zero for large B0 . 9.2. Antidot lattices In Section 2.3.4, we found that under some circumstances, an antidot lattice can be achieved with a far blue-detuned lattice beam. We show here that the optical potential of a grey molasses (see Section 3.4) in the presence of a static magnetic Ield generally consists of antidots rather than wells [105]. In an antidot lattice, there is no bound state and all the atoms are free. In fact, atoms undergo an erratic motion in the lattice by occasionally bouncing on the antidots in a manner similar to a ball in a pinball game [26]. Consider the situation of a 1D lin ⊥ lin lattice (Section 3.1.1) in the presence of a static magnetic Ield B0 applied along the Oz axis. If the lattice beams are tuned on the blue side of a transition connecting a ground state of angular momentum F to an excited state of angular momentum F or F − 1, Sisyphus cooling can be achieved [85]. In fact, the cooling is most eLcient for low B0 and for high B0 [105]. We consider this last case in the remaining of the section. The Zeeman splitting 2F00 between the extreme Zeeman sublevels of the ground state is assumed to be larger than the light shift . Therefore, the light-shift can be treated as a perturbation compared to the Zeeman e@ect and the magnetic quantum numbers can be used to label the potential surfaces. In this situation, a majority of atoms are found in the extreme Zeeman sublevels (mF = ± F for transitions F → F and mF = ± F, ±F − 1 for transitions F → F − 1) and the cooling occurs through the mechanism shown in Fig. 53. The dependence
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Fig. 53. Sisyphus cooling in a 1D lin ⊥ lin lattice with a magnetic Ield B0 . The potentials correspond to a blue detuning from a Fg = 1 → Fe = 1 transition. The atom climbs a potential hill before being optically pumped into another Zeeman sublevel where it also climbs a potential hill.
of the temperature T with and 00 was studied theoretically [69] and experimentally [70]. In particular, for a Ixed value of 00 , T is a linear function of . In the 1D case, the potential curves resulting from the combined e@ect of the Zeeman shifts and of the light-shifts exhibit potential wells (Fig. 53). Note that the light-shift vanishes at the bottom of the wells associated with the extreme Zeeman sublevels. An atom localized around these points is very weakly coupled to the incident beams. In higher dimensions, the potential surfaces generally exhibit antidots rather than wells, as shown in Fig. 54a where a section in the xOy plane of the mF = − 1 potential surface for a Fg = 1 → Fe = 1 transition is shown. This surface was obtained for a four-beam lin ⊥ lin lattice (Section 3.3.4) and a Ield B0 along Oz. The shape of the antidot potential suggests that most of the atoms should be located near the bottom of the valleys between the antidots, where the potential is minimum and where the optical pumping rate vanishes (these lines are associated with a pure circular polarization of the lattice Ield). The steady-state atomic density in the xOy plane is shown in Fig. 54b. This density is calculated using a semi-classical Monte-Carlo simulation (Section 4.3) for the Fg = 1 → Fe = 1 transition. As expected, the density is maximum in the valleys between the antidots. A precise examination of a single semi-classical trajectory shows that a ballistic motion over several wavelengths is a relatively seldom event. In most circumstances, the atoms di@use in the lattice by repeatedly bouncing on the antidots as shown in Fig. 54c. Hence, the antidots act as pinball bumpers for the atoms. Because the height of the antidots is on the order of , one expects that the typical frequency for an atom bouncing back and forth between two √ antidots should be on the order of (as for the vibration motion considered in Section 5.3). To excite these transient oscillations between antidots, a probe beam of frequency !p = ! + :
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X Y
Y/λ
(b) 2
1
0 0
(c)
1
2
X /λ
2
Y/λ
1.5
1
0.5
0
0.5
1 X /λ
1.5
2
Fig. 54. Antidot optical lattice. The calculations are made in the case of a four beam lin ⊥ lin lattice with ◦ #x = #y = # = 30 in the presence of a large static magnetic Ield applied along the Oz axis. The data correspond to the mF = − 1 potential surface for a Fg = 1 → Fe = 1 transition. (a) Section of the potential in the xOy plane. (b) Distribution of the atomic density. The dark zones which correspond to increased density are found near the valleys of the potential. (c) Typical semi-classical trajectory showing the bouncing of an atom between antidots. From Petsas et al. [26].
can be added to shake the optical potential at frequency :. Raman resonances characteristic of this bouncing motion are found in the probe transmission spectrum both in the experiment and in the numerical simulation. These resonances were extensively studied in [26]. They are
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Fig. 55. Raman resonances between Zeeman sublevels in the case where the light beams have only + and − components with respect to B0 . The resonance occurs for !1 − !2 ±200 .
broader than the vibrational resonances shown in Section 5.3 because of the short duration of these transient oscillations. Comments: (i) A similar dynamical behaviour has been previously reported for electrons in 2D antidot semiconductor superlattices [174 –176]. However, one main di@erence with the solid-state experiments is that there is not a simple parameter (such as the magnetic Ield for electrons) to act on the atomic motion. Some interesting developments inspired by the solid-state physics experiments can nonetheless be considered. For example, it was shown that the dynamics of electrons is di@erent in square and in hexagonal antidot superlattices [177] because of the possible weak localization of electrons in the latter case [178,179]. An analogous study in an optical lattice has not yet been performed. (ii) Although antidot lattices are generally encountered in the case of a 3D grey molasses with magnetic Ield, it is also possible to achieve a lattice exhibiting wells in some Ield conIgurations [180]. 9.3. Lattice in momentum space The localization of atoms on a lattice in the momentum space can also be found when a static magnetic Ield is added to a grey molasses or to a bright lattice [181,182]. The origin of this e@ect and the construction of the lattice in the momentum space are described in this section. Sisyphus cooling originates from a redistribution of photons between the cooling beams (a process known as two-beam coupling or stimulated Rayleigh scattering, see Section 3.2.2). In the case of free atoms, two-beam coupling is resonant around frequencies !1 and !2 for the laser beams such as !1 − !2 0 (Rayleigh resonance) and !1 − !2 !gg where !gg coincides with an atomic Bohr frequency (Raman resonance) [51,183]. If we consider a ground state having an angular momentum F and a LandYe factor gF , the Raman resonance (Fig. 55) occurs for !1 − !2 ±200 (with 00 = gF OB B0 ) if all the Ield polarizations are perpendicular to B0 . (In the case where the light Ield has + , − and components with respect to B0 , the resonance condition becomes !1 − !2 ±00 .). In most cases, experiments are performed with Ields having the same frequency !. However in the atomic frame, for an atom with a velocity v, these Ields are Doppler-shifted (!i = ! − ki · v where ki is the wavevector of the beam i).
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As a result, the Rayleigh and Raman two-beam couplings exhibit resonances for the velocity groups that satisfy: (ki − kj ) · v = p200
(p = − 1; 0; 1) :
(133)
These conditions determine the velocity groups around which the velocity damping occurs. In the 1D case with two counter-propagating beams (k1 = kez ; k2 = − kez ), the bunching occurs around three velocities vz = − 00 =k; 0; 00 =k proportional to B0 . This was observed by Van der Straten et al. [184,185] and Valentin et al. [186]. If the beam geometry makes use of the minimum number of beams (i.e. 4 in 3D), a periodic optical lattice with a basis {a1 ; a2 ; a3 } in the direct space is found. This basis is related to the ? = k − k of the reciprocal lattice by the relations a? · a = 2: vectors aj−1 j 1 l l; j (see Eqs. (28)). j The comparison between these equations and Eq. (133) shows that the velocities are located on a lattice with a basis vl = al 00 = with (l = 1; 2; 3) [181]. To have a better insight, we present the localization found in the 2D case when the cooling originates from three beams k1 ; k2 ; k3 (Fig. 56a). The basis of the reciprocal lattice (a1? and a2? ) and of the direct lattice (a1 and a2 ) are shown in Fig. 56b. Consider for example the bunching of velocities around v1 = a1 00 =. It results from a Rayleigh two-beam coupling originating from beams 1 and 3 (a2? · v1 = 0) and from Raman two-beam couplings originating, respectively, from beams 1 and 2 (a1? · v1 = 200 ) and from beams 2 and 3 ((a2? − a1? ) · v1 = − 200 ). The three pairs of beams thus create a 2D restoring force that pulls the nearby velocity towards v1 . There are seven velocity groups (0; ±v1 ; ±v2 ; ±(v1 + v2 )) that share this property (see Fig. 56c). Actually we have plotted in Fig. 56d all the lines for which two pairs of beams create a damping force that attracts the atoms along these lines. The seven velocity groups mentioned above correspond to the intersection of these lines. The experimental observation of this bunching in momentum space was performed by TrichYe et al. [181] using a 3D four-beam lin ⊥ lin lattice and by Rauner et al. [182] in the case of a 2D lattice. We give some details about this last experiment which is performed starting from a beam of cold metastable neon atoms prepared in a tilted magneto-optical funnel [187]. This beam is normal to the lattice plane xOy deIned by two orthogonal standing waves of same frequency, linearly polarized in the plane (see Sections 2.3.3 and 3.3.3). The beams are tuned to the red side of the 3S3=2 (J = 2) → 3P5=2 (J = 3) transition and the magnetic Ield B0 has components along the three ex ; ey and ez axis. After travelling a distance of 24 cm from the molasses, the metastable neon atoms are detected using a microchannel plate with an adjacent phosphor screen and a CCD camera. A direct 2D image of the transverse atomic distribution is thus obtained as shown in Fig. 57a. The bunching of atoms along a few directions is clearly observed. These directions correspond to transverse velocities located on a lattice with a basis vl = al 00 =2 (the factor 12 with respect to the previous formula arises from the additional e@ect of photons in the Raman resonance condition). Indeed, the expected pattern shown in Fig. 57b is in perfect agreement with the experiment. In both experiments [181,182], it was also shown that the size of the basis vectors of the lattice in momentum space increases proportionally to B0 , as expected. Note Inally that this lattice can be found both in the case of a bright lattice [182] and in the case of a grey molasses (the experiment of TrichYe et al. was performed on the blue side of the 6S1=2 (F = 3) → 6P3=2 (F = 2) transition of cesium) [181].
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Fig. 56. Bunching of the velocities in a three-beam 2D lattice. (a) Wavevectors of the incident Ield. (b) Basis vectors in the reciprocal space and in the direct space. (c) Velocity groups around which bunching occurs. Note that v1 and v2 are parallel to a1 and a2 . (d) The velocity groups around which bunching occurs are found at the intersection of three lines corresponding to a Rayleigh or a Raman resonance. The splitting between the lines is proportional to B0 .
9.4. The asymmetric optical lattice: an analogue to molecular motors Combining light-shift and Zeeman shift, one can tailor nearly any potential shape, thus modelling very di@erent physical situations. For example, the addition of a small static magnetic Ield B0 along the direction of propagation of two laser beams in the lin # lin conIguration (see Section 3.4.2 and Fig. 58a) gives rise to a potential curve which is periodic but asymmetric (Fig. 58b). The light frequency is set a few above atomic resonance, so that the internal dark state (see Section 3.4) existing for the F = 1 → F = 1 transition considered here has the lowest potential energy. This potential energy is modulated by the small magnetic Ield, which gives rise to the sawtooth potential of Fig. 58b. In fact, the physical situation realized here models that of molecular motors in biological cells, which are proteins moving along protein Ibers [188,189]. Proteins being polar molecules,
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Fig. 57. Bunching of the velocities in a four-beam 2D lattice [182]. The experiment is performed with a cold and slow beam of metastable neon (frequency detuning from the J = 2 → J = 3 transition = − 9, saturation parameter s = 2:6×10−3 , static Ield B0 = 0:21 G). (a) Experimental observation of the momentum distribution. Dark regions correspond to increased atomic density. The extension of the image is ∼44˝k. (b) Expected pattern. From Rauner et al. [182], reproduced by permission of the authors.
Fig. 58. (a) Beam conIguration used to create an asymmetric potential and consisting of two counter-propagating beams with linear polarizations making an angle #. A small magnetic Ield B0 is added along the propagation axis. (b) Resulting potential for an atom having a F = 1 → F = 1 transition and for blue-detuned light. The arrows indicate the jumping process inducing a net movement of the atoms. The potential curve corresponding to the mg = 0 state is not represented because this state is not populated by optical pumping.
the potential experienced by a motor due to the Iber has a sawtooth shape, i.e. it is periodic but asymmetric. Although the periodicity of the potential ensures that no macroscopic force acts on the motors, a deterministic >ux of particles is obtained in the presence of dissipation, which arises from ATP burning. More generally, a net movement requires the breaking of spatial and temporal symmetries [190]. In optical lattices, the spatial symmetry breaking comes from the
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◦
Fig. 59. Experimental images obtained for # = 45 , = 2 and 0 100!R , after 6 ms of asymmetric optical lattice, for B0 = 0:5 G (a), B0 = 0 (b) and B0 = − 0:5 G (c). Because of gravity g, the cloud has moved less in (c) than in (a). From Mennerat-Robilliard et al. [191].
simultaneous use of light and magnetic Ields, and dissipation originates from optical pumping that transfers the atoms from one potential curve to the other. The process leading to a net atomic >ux is represented in Fig. 58b: in the Irst step, the atom is localized in a potential well of the internal dark state where it oscillates (A–B in Fig. 58b). It is then transferred to the coupled state by an optical pumping process (M–N) where it starts oscillating (N–P) before being optically pumped back to the lowest potential surface (P–Q). As the potential minima of the two curves are shifted with respect to one another, the probability that the atom has moved towards the well located on the right of the initial well is larger than the probability that it jumped into the well situated on the left. This simple theoretical model is conIrmed by semi-classical numerical simulations [191]. Although the occurrence of a directed motion looks obvious in this scheme, a detailed inspection shows that it is not the case. In fact, the directed motion is closely related to the optical pumping rates between the internal dark state and the coupled state. When the transition rates are forced to obey the detailed balance, the directed motion disappears. The motion exists because the atoms are not at thermodynamic equilibrium. The experimental study was performed with rubidium atoms (87 Rb) [191]. We show in Fig. 59 images of the atomic cloud for di@erent values of B0 : the image (b) corresponds to a zero magnetic Ield, and thus to a symmetric potential. In this case the cloud does not move. In images (a) and (c), the atoms have moved in opposite directions because they correspond to opposite magnetic Ields, i.e. to opposite asymmetries. The mean velocity of the atoms in the optical lattice is on the order of 10 vR (vR being the recoil velocity). Provided that the axis of the lattice is chosen vertical, one can also measure the momentum distribution of the atoms, using a time-of->ight technique (see Section 6.1). We show in Fig. 60 the signals obtained for B0 = 0 (Fig. 60b) and for B0 = ± 0:5 G (Fig. 60a and c) [191]. As expected, the momentum distribution corresponding to a symmetric potential is symmetric while the distributions (Fig. 60a and c), which are obtained for opposite magnetic Ields, exhibit opposite asymmetries. From the shape of the distributions one can infer that the atoms are most of the time localized in a potential well where their average velocity is zero, and they hop only from time to time, then contributing to the asymmetric wing of the distribution. Finally, the need of spatial symmetry breaking appears in the fact that for all parameters leading to a symmetrical potential (B0 = 0 or # = 0 or =2 for example), no net >ux is observed. The role of dissipation can also be studied by varying both the light intensity and the detuning
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◦
Fig. 60. Experimental momentum distribution measured with a time-of->ight technique for # = 45 , = 2 and 0 200!R , after 6 ms of asymmetric optical lattice, for B0 = 0:5 G (a), B0 = 0 (b) and B0 = − 0:5 G (c). From Mennerat-Robilliard et al. [191].
to keep 0 constant while varying the optical pumping rate: as expected, the mean velocity increases with dissipation [191]. 10. Nanolithography Nanolithography is the most developed application of optical lattices to date. The ability of designing and fabricating devices on the nanometer scale is indeed one of the very important technological and industrial challenges of the time. Various technologies are presently under development, but none o@ers yet both a resolution in the tens of nanometers and the possibility of a parallel fabrication. For example, the resolution of electron or ion beam lithography goes down to the 10-nm range [192], but massive fabrication is still diLcult to implement. The STM (scanning tunneling microscopy) technology has also been adapted several years ago for very high resolution lithography [193], and it becomes now possible to design very small active
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devices by positioning the atoms one by one. However, this kind of technique is not suited for industrial fabrication yet. The idea thus arises to use the natural periodicity of light interference patterns to make light masks for atoms. Compared with traditional material masks [194], these light masks o@er an almost perfect periodicity and an appreciable robustness. We Irst present the principle of an atom nanolithography experiment (Section 10.1). In a second paragraph, we discuss the main experimental achievements (Section 10.2) and we Inally sketch the present research directions concerning laser-controlled nanolithography (Section 10.3). 10.1. The principles of atomic nanolithography The Irst idea of focusing atoms on a nanometer-scale with near-resonant light was proposed by Balykin and Letokhov [195]. More precisely, they suggested to use a focused doughnut mode (TEM? 01 ) laser beam as a lens for neutral atoms, similarly to magnetic lenses for charged particles. Experimentally, the atoms are rather focused in a laser standing wave, which allows for an intrinsic massive parallelism. Fig. 61 shows the schematic of an experiment. An atomic beam is collimated by transverse laser cooling and is then focused in a standing wave just before depositing on a substrate. Depending on the sign of the light mask detuning with respect to atomic resonance, the atoms are channelled along the maximum or minimum intensity lines. Using an atom optics approach, one can describe each node of the standing wave as a lens and predict the Irst order properties and aberrations of this lens, either with a time-dependent trajectory analysis [196] or with a particle optics approach [197]. Numerical simulations can also be useful to go beyond the Irst order approximation. We show in Fig. 62 the trajectories resulting from an exact numerical solution of the equation of motion for an atom in a Gaussian standing wave [197]. The >ux appears to be very well focused. However, such results are not obtained in practice, partly because the initial velocity distribution is usually thermal and mainly because the incoming beam is not perfectly collimated. Other technical reasons also contribute to broaden the atomic >ux, such as vibrations of the apparatus or atomic mobility on the substrate. 10.2. Experimental achievements The Irst focusing experiment with a laser light Ield consisted of focusing a beam of metastable He atoms on a single period of a large period standing light wave. This allowed the imaging of an object with this neutral atoms lens [198]. Shortly afterwards, a one-dimensional standing light wave was used to focus an atomic beam before deposition on a substrate, Irst with sodium atoms [199,200], and then also with chromium [201,202] and aluminium [203] atoms. A two-step laser-controlled atom nanolithography process is also possible, by using either metastable [204 –206] or cesium atoms [207] that react on a self-assembled monolayer (SAM). A subsequent chemical etching engraves the substrate where the SAM was exposed. In all these experiments, the period of the deposited line-pattern is =2, thus on the order of a few hundreds of nanometers. However, one can overcome this limitation by taking into account the polarizations of the laser beams: for a lin ⊥ lin conIguration and an atomic transition
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Fig. 61. Schematic of a laser-controlled nanolithography experiment. An atomic beam is collimated by laser techniques before being focused by a standing wave and deposited on a substrate. Note that the mirror retro-re>ecting the beam is usually glued on the substrate itself, to prevent the deposited pattern from being washed out by vibrations. Fig. 62. Exact trajectory calculation of laser focusing of chromium atoms in a standing wave with a Gaussian envelope. A series of trajectories are shown for varying initial x values. All trajectories are given for the same initial velocity of 926 m=s and a zero initial angle relative to the z axis. Also shown is a plot of the atomic >ux at the focal plane, assuming a uniform >ux entering the lens, and laser beam proIles I (x; z) along x (bottom) and z (left). The laser intensity and detuning are chosen to satisfy the condition required for focusing at the center of the beam [197]. From McClelland [197], reproduced by permission of the author.
J = 1=2 → J = 3=2, the distance between two adjacent potential wells is =4 (see Section 3.1.2). For J → J + 1 atomic transitions with J ¿ 1, the anticrossings of adiabatic potentials (see Fig. 12a) give rise to supplementary potential minima for the upper potential curves, where the atoms can also be focused. A =8 period was achieved this way by the group of McClelland [208]. In order to populate more eLciently the potential wells of the upper potential curves, one can also use a static homogeneous magnetic Ield perpendicular to the light mask plane: the induced mixing of levels splits the potential curves and makes the adiabatic approximation easier to fulIl, even at anticrossing points. As a result, the atoms follow their own adiabatic curve without undergoing any nonadiabatic transition and can populate the secondary potential wells more eLciently [209]. Another extension of this technique consists of using two-dimensional standing light waves. One then obtains square lattices [210] or honey-comb [211] structures, depending on the number and the geometry of the laser beams. We show in Fig. 63 atomic force microscope pictures of two-dimensional chromium nanostructures. The atoms are focused by a standing wave resulting ◦ from the interference of three coplanar beams with wavevectors making 120 angles. Fig. 63a was obtained for red-detuned laser beams (thus the atoms are focused on the antinodes of the light Ield) while Fig. 63b corresponds to the same geometry, with a blue detuning (the atoms are focused on the nodes of the Ield).
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Fig. 63. AFM pictures of chromium nanostructures obtained by focusing on a 2D standing light wave resulting ◦ from the interference of three coplanar beams propagating with angle 120 with respect to each other. (a) The laser is red-detuned, so that the atoms are attracted into high intensity regions. (b) The laser is blue-detuned. The atoms are expelled from the points of highest intensity. From Drodofsky et al. [211], reproduced by permission of the authors.
10.3. Latest research directions The ability of depositing nanoscale patterns of neutral atoms with light masks is now well established. Three main research axes develop now. First, one must reduce further the width of the deposited features, in order to improve the resolution and to be competitive with other techniques [212]. This requires a good understanding of the growth process and a precise control of the deposition conditions. The second axis aims at depositing arbitrary Igures, which is a necessary step before considering an industrial development. One can think of using holographic techniques or photorefractive materials to generate an arbitrary light pattern onto which the atoms are focused. A third research direction consists of fabricating with the existing technique “useful” nanostructures: for instance, one can think of depositing isolated dots of a magnetic atom to make magnetic memories. Another idea would be to take advantage of the material selectivity of laser-controlled nanolithography to deposit complex three-dimensional structures, for example one atomic species homogeneously and another according to a light mask [213]. This would result in a material with a nanostructured doping, i.e. a candidate for a photonic crystal. 11. Conclusion We have presented in this review what we believe to be the most interesting results concerning dissipative optical lattices. In fact, this is a timely period to write a review. This subject is now mature and the probability to Ind a totally unexpected and interesting result is rather low. Although the subject is still active, most of the present researches tend to improve our knowledge about phenomena that have not been studied yet with enough precision. This is for example the case of spatial di@usion and of nonlinear atom dynamics. However, it should also be kept in mind that optical lattices are a fantastic scale model to study processes found in solid-state materials and in statistical physics, but with totally di@erent scales. Furthermore, there is a relatively large >exibility in the design of the optical potentials. This >exibility was
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used to produce for example an atomic motor and can be adapted tomorrow to many other problems. Nevertheless, those who wish to Ind a major breakthrough have probably more chances to reach their goal by studying far o@ resonant optical lattices or perhaps mesoscopic optical lattices. Because of their vanishingly small dissipation, far o@ resonant lattices are particularly well adapted to quantum studies about the external degrees of freedom. A few spectacular e@ects, such as Bloch oscillations [75], quantum chaos [214,215] and mesoscopic quantum coherence [219], are examples of what can be expected in this direction. Still more interesting are the e@ects that could be observed by Illing an optical lattice with a Bose condensate [216,217] or a degenerate Fermi gas. But this is another world, beyond the scope of this paper. Acknowledgements The authors are indebted to all the physicists that share their interest on optical lattices and had a discussion with them during the last years. In particular, they would like to acknowledge daily enlightening conversations and meetings with their colleagues from the ENS and most particularly with Claude Cohen-Tannoudji, Yvan Castin, Jean-Yves Courtois, Jean Dalibard, Christophe Salomon, Philippe Verkerk, Peter Horak, David M. Lucas, David R. Meacher, Samuel Guibal, Luca Guidoni, Brahim Lounis, Konstantinos I. Petsas and Christine TrichYe. The friendly competition with the groups of William D. Phillips and Theodor W. H]ansch was another source of inspiration. Finally, fruitful discussions with distinguished colleagues during visits or conferences were also extremely helpful. Among them the authors wish especially to thank William D. Phillips, Alain Aspect, Paul R. Berman, Harold J. Metcalf, Jacques ViguYe, Herbert Walther and all their colleagues from the TMR network “Quantum Structures” (European Commission contract FRMX-CT96-0077). Laboratoire Kastler Brossel is an unitYe de recherche de l’Ecole Normale SupYerieure, de l’UniversitYe Pierre et Marie Curie et du CNRS (UMR 8552). Laboratoire Collisions, AgrYegats, RYeactivitYe is an unitYe de recherche de l’UniversitYe Paul Sabatier et du CNRS (UMR 5589). Appendix A. Index of notations Notation
De=nition
a ai ai? aM At d dˆ d± D
amplitude of the vibration motion primitive translation of the optical lattice primitive translation of the reciprocal lattice transverse spatial period (Talbot lattice) topological vector potential electric dipole moment reduced dipole operator upwards (downwards) component of the atomic dipole momentum di@usion coeLcient
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D
Dsp e± E0 E0 (R) E± E E (±) Ep ER = ˝2 k 2 =2M En F (R) F (R) FU G He@ = P 2 =2M + Uˆ I ± = |E ± |2 I ki k = !=c kp kB K⊥ = k sin # K = k(1 − cos #) Kx;y = k sin #x;y K± = k(cos #x ± cos #y )=2 L me |m |nho M P P Pˆ qe re R s0 s(R) S ˜ S(t); S(0) T u(R) ˆ
reduced matrix element of the atomic dipole spatial di@usion coeLcient unit vectors for the circular polarizations Ield amplitude of a single lattice beam lattice Ield amplitude in R circular components of the Ield lattice Ield component along Oz positive (negative) frequency component of the Ield probe beam amplitude recoil energy energy of the vibrational state | n reactive (dipole) force dissipative force (radiation pressure) average value of the dipole force (multilevel atom) vector of the reciprocal lattice Hamiltonian (external degrees of freedom) intensity of the circular component of the Ield Ield intensity wavevector of a lattice beam modulus of the wavevector of a lattice beam probe beam wavevector Boltzmann constant transverse spatial frequency longitudinal spatial frequency (3-beam lattice) transverse spatial frequency (3D 4-beam lattice) longitudinal spatial frequency (3D 4-beam lattice) length of the atomic cloud electron mass Zeeman substate eigenstate of the harmonic oscillator atomic mass atom momentum atomic polarization e@ective coupling operator (external state variables) electron charge electron position in the atomic rest frame atom position saturation parameter for a single beam saturation parameter in R structure factor transient probe emission (time and frequency domain) kinetic temperature dimensionless light-shift Hamiltonian
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U (R) Uˆ U± ; Un U0 = −4˝0 =3 Ut U±(p) v vU vR = ˝k=M Vm = m|Uˆ |m VˆAL W (R; P; t) zT 0 L p 0 = s0 =2 (R) = s(R)=2 : = !p − ! :L 0 = Ss0 =2 (R) = Ss(R)=2 ”i ”(R) M #; #x;y = 2=k x;y = = sin #x;y + =2 = =(cos #x + cos #y ) ⊥ = = sin # % = =(1 − cos #) Hv (n) -± -n @ ± i
dipole potential light-shift Hamiltonian eigenvalues of the light-shift (adiabatic energies) depth of the 1D lin ⊥ lin lattice scalar topological potential shift of the optical potential induced by the probe atomic velocity average atomic velocity recoil velocity diabatic potentials electric dipole interaction Wigner representation of the atomic density matrix Talbot length friction coeLcient atomic polarizability Debye–Waller factor radiative width of the upper level optical pumping rate photon scattering rate per beam photon scattering rate in R photon scattering rate at a point where the lattice Ield is ± probe–lattice frequency detuning frequency di@erence between lattice beams light-shift per beam light-shift in R light-shift at a point where the lattice Ield is ± polarization of a lattice beam local polarization of the Ield planar density of atoms angles between lattice beams wavelength of a lattice beam transverse spatial period (3D 4-beam lattice) longitudinal spatial period (3D 4-beam lattice) transverse spatial period longitudinal spatial period shift of the vibrational resonance caused by anharmonicity population of the vibrational level (band) n population of the Zeeman substates (J = 1=2) population of the adiabatic state |1n atomic density matrix reduced atomic density matrix (ground state) circular polarizations phase of a lattice beam
G. Grynberg, C. Robilliard / Physics Reports 355 (2001) 335–451
|1n |1NC
F
|
n ! !0 !p = ! + : !R = ER + ˝ 01 0v 0x;y;z 0B 00
443
adiabatic states internal dark state atomic susceptibility vibrational eigenstate frequency of a lattice beam atomic resonance frequency probe beam frequency recoil frequency resonant Rabi frequency for a single beam vibration frequency vibrational frequency along x; y; z Brillouin resonance Zeeman splitting
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CONTENTS VOLUME 355 I. Pollini, A. Mosser, J.C. Parlebas. Electronic, spectroscopic and elastic properties of early transition metal compounds C. Schubert. Perturbative quantum "eld theory in the string-inspired formalism
1 73
R. Fazio, H. van der Zant. Quantum phase transitions and vortex dynamics in superconducting networks
235
G. Grynberg, C. Robilliard. Cold atoms in dissipative optical lattices
335
Contents volume 355
453
Forthcoming issues
454
PII: S0370-1573(01)00078-3
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FORTHCOMING ISSUES V.M. Shabaev. Two-time Green's function method in quantum electrodynamics of high-Z few-electron atoms G. Bo!etta, M. Cencini, M. Falcioni, A. Vulpiani. Predictability: a way to characterize complexity A. Wacker. Semiconductor superlattices: a model system for nonlinear transport I.L. Shapiro. Physical aspects of the space}time torsion Ya. Kraftmakher. Modulation calorimetry and related techniques M.J. Brunger, S.J. Buckman. Electron}molecule scattering cross sections. I. Experimental techniques and data for diatomic molecules S.Y. Wu, C.S. Jayanthi. Order-N methodologies and their applications V. Barone, A. Drago, P. Ratcli!e. Transverse polarisation of quarks in hadrons M. Baer. Introduction to the theory of electronic non-adiabatic coupling terms in molecular systems S.-T. Hong, Y.-J. Park. Static properties of chiral models with SU(3) group structure W.M. Alberico, S.M. Bilenky, C. Maieron. Strangeness in the nucleon: neutrino}nucleon and polarized electron}nucleon scattering M. Bianchetti, P.F. Buonsante, F. Ginelli, H.E. Roman, R.A. Broglia, F. Alasia. Ab-initio study of the electronic response and polarizability of carbon chains C.M. Varma, Z. Nussinov, W. van Saarloos. Singular Fermi liquids J.-P. Blaizot, E. Iancu. The quark}gluon plasma: collective dynamics and hard thermal loops A. Sopczak. Higgs physics at LEP-1 A. Altland, B.D. Simons, M. Zirnbauer. Theories of low-energy quasi-particle states in disordered d-wave superconductors
PII: S0370-1573(01)00079-5
