Advances in Atomic, Molecular, and Optical Physics, Volume 59 (Advances in Atomic, Molecular and Optical Physics)

Editors ENNIO ARIMONDO University of Pisa Pisa, Italy PAUL R. BERMAN University of Michigan Ann Arbor, Michigan CHUN C. LIN University of Wisconsin Madison, Wisconsin

EDITORIAL BOARD P.H. BUCKSBAUM SLAC Menlo Park, California M.R. FLANNERY Georgia Tech Atlanta, Georgia C. JOACHAIN Universit�e Libre de Bruxelles Brussels, Belgium J.T.M. WALRAVEN University of Amsterdam Amsterdam, The Netherlands

ADVANCES IN

ATOMIC, MOLECULAR, AND OPTICAL PHYSICS Edited by

E. Arimondo PHYSICS DEPARTMENT UNIVERSITY OF PISA PISA, ITALY

P. R. Berman PHYSICS DEPARTMENT, UNIVERSITY OF MICHIGAN, ANN ARBOR, MI, USA

C. C. Lin DEPARTMENT OF PHYSICS, UNIVERSITY OF WISCONSIN, MADISON, WI, USA

Volume 59

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CONTRIBUTORS Numbers in parentheses indicate the pages on which the author’s contributions begin.

JAMES F. BABB (1), ITAMP, Harvard-Smithsonian Center for Astrophysics, MS 14, 60 Garden St., Cambridge, MA 02138, USA VISHAL SHAH (21), Symmetricom Technology Realization Center, 34 Tozer Road, Beverly, MA 01915, USA JOHN KITCHING (21), Time and Frequency Division, NIST, 325 Broadway, Boulder, CO 80305, USA RAINER JOHNSEN (75), Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 16260, USA STEVEN L. GUBERMAN (75), Institute for Scientific Research, 22 Bonad Road, Winchester, MA 01890, USA TIM CHUPP (129), FOCUS and MCTP, Physics Department, University of Michigan, Ann Arbor, MI 48109, USA PAUL R. BERMAN (175), Michigan Center for Theoretical Physics, Physics Department, University of Michigan, 450 Church Street, Ann Arbor, MI 48109-1040, USA GEORGE W. FORD (175), Michigan Center for Theoretical Physics, Physics Department, University of Michigan, 450 Church Street, Ann Arbor, MI 48109-1040, USA SHAUL MUKAMEL (223), Department of Chemistry, University of California, Irvine, CA 92697, USA SAAR RAHAV (223), Department of Chemistry, University of California, Irvine, CA 92697, USA

ix

PREFACE Volume 59 of the Advances Series contains six contributions, covering a diversity of subject areas in atomic, molecular, and optical physics. James Babb presents an interesting discussion of the Casimir effect in atomic, molecular, and optical physics. Casimir effects are quantum in origin and are related to vacuum fluctuations that give rise to forces between atoms, molecules, and surfaces. The Casimir effect has received a great deal of attention over the last several years. Babb reviews both neutral atom— neutral atom and ion—neutral atom interactions and looks at how these effects vary with distance. He compares results obtained via traditional methods and those involve “dressing” of the atoms by the vacuum field. In doing so, he provides new insight into the numerical factors that arise in these theories. The chapter by Rainer Johnsen and Steven Guberman focuses on the dissociative recombination of H+3 ions with electrons. For several decades, this seemingly simple process had been a great puzzle to researchers in this field, with a strong disparity between the results of theoretical and experimental studies. Johnsen and Guberman discuss recent progress, which has reduced many of the “contradictions” and reconciled the remaining discrepancies. In particular, they discuss and compare disso­ ciative combination that is produced in beam experiments with those employing plasma afterglow techniques. In their contribution, Vishal Shah and John Kitching review recent advances in the field of coherent population trapping as applied to atomic frequency standards and atomic clocks. The very narrow absorp­ tion lines associated with atomic coherence quantum interference has stimulated a large interest within the atomic clock community, leading to the development of new atomic clocks employing both standard and “chip-scale” atomic vapor cells. The authors review the progress that has been made in improving the resonance contrast, decreasing the clock line width, and the reducing light shifts that affect the long-term stability of these devices. The expected impact of these new approaches on future generations of laboratory and commercial instruments is examined. Timothy Chupp provides an overview of the search for permanent electric dipole moments of atoms, molecules, and elementary particles. Attempts to measure an electric dipole moment of the electron or the neutron have been underway for decades. In recent years, these and other xi

xii

Preface

searches have been connected with predictions of theories that go beyond the Standard Model of particle physics. Chupp reviews several experi­ mental techniques that have been used to date, along with the current experimental limits on electric dipole moments of atoms, molecules, and elementary particles. He then gives a critical discussion of proposed experimental techniques that may lead to improved precision and impor­ tant tests of physics beyond the Standard Model. Spontaneous emission from an isolated atom is the subject of the contribution of Paul Berman and George W. Ford. Although this is an old subject, it is one that has been plagued by mathematical anomalies. Berman and Ford present a detailed calculation of both the excited state decay dynamics and the spectrum of the emitted radiation. Using differ­ ent models for the atom—vacuum field interaction, they show that, while exponential decay and the Lorentzian spectrum originally predicted by Weisskopf and Wigner are good approximations to the actual decay and spectral density associated with spontaneous emission, the actual decay and spectrum must differ from the Weisskopf—Wigner result. An integral expression is obtained for the excited state probability amplitude and an analytic expression for the spectrum. Shaul Mukamel and Saar Rahav present a diagrammatic approach to calculating the response of molecules to a number of applied optical fields. Their approach provides a consistent treatment of multi-wave mixing in which both the fields and the atoms are quantized. In effect, they are able to use an amplitude approach to keep track of the various multi-photon processes that contribute to the observed signals. A closedtime-path-loop diagrammatic method plays a critical role in their analy­ sis. They apply the formalism to pump—probe and coherent anti-Stokes Raman spectroscopy to elucidate the role played by two-photon absorp­ tion and stimulated Raman scattering. The editors would like to thank all the contributing authors for their contributions and for their cooperation in assembling this volume. They would also like to express their appreciation to Ms. Gayathri Venkata­ samy at Elsevier for her invaluable assistance. Ennio Arimondo Paul Berman Chun Lin

CHAPTER

1

Casimir Effects in Atomic, Molecular, and Optical Physics James F. Babb ITAMP, Harvard-Smithsonian Center for Astrophysics, MS 14, 60 Garden St., Cambridge, MA 02138, USA

Contents

Introduction What’s a Micro Effect; What’s a Macro Effect? Relativistic Terms Yet Another Repulsive Interaction Nonrelativistic Molecules and Dressed Atoms Not a Trivial Number Reconciling Multipoles 7.1 Two Atoms 7.2 An Electron and an Ion 8. Conclusion Acknowledgments References

Abstract

The long-range interaction between two atoms and the longrange interaction between an ion and an electron are compared at small and large intersystem separations. The vacuum dressed atom formalism is applied and found to provide a framework for interpretation of the similarities between the two cases. The van der Waals forces or Casimir–Polder potentials are used to obtain insight into relativistic and higher multipolar terms.

1. 2. 3. 4. 5. 6. 7.

1 2 4 5 9 12 13 13 15 16 16 17

1. INTRODUCTION Distance changes everything. The same is the case for electromagnetic inter­ action potential energies between polarizable systems. In atomic, molecular, and optical physics, the small retarded van der Waals (or Casimir—Polder) Advances in Atomic, Molecular, and Optical Physics, Volume 59
2010 Elsevier Inc. ISSN 1049-250X, DOI: 10.1016/S1049-250X(10)59001-3 All rights reserved.

1

2

James F. Babb

potentials between pairs of polarizable systems (either of which is an atom, molecule, surface, electron, or ion) for separations at long ranges where exchange forces are negligible have been well studied theoretically. There are also three- and higher-body potentials (cf. Salam, 2010), antimatter appli­ cations (Voronin et al., 2005), and more. Much of the current interest in the Casimir interactions between atoms and walls is due to interests related to nanotechnologies (cf. Capasso et al., 2007) and related attempts to engineer repulsive forces at nanoscales (Marcus, 2009). Numerous topical surveys and reviews, monographs, book chapters, conference proceedings, and popular texts touching on particular pairwise potentials are in print–literally a “mountain of available information” (Bonin & Kresin, 1997, p. 185)–and many sources contain extensive bibliographies. It is not uncommon to run across statements indicating that there has been a rapid increase in the number of available papers related to the Casimir effect. Even a recent book, Advances in the Casimir Effect (Bordag et al., 2009), focusing mainly on recent results, comes to over 700 pages. This chapter is concerned with bringing to light some connections between theoretical results from various formulations for the zerotemperature limits of interactions between ground state atoms, ions, or molecules. The case has been advanced that the practical relevance of zero-temperature results is questionable (see Wennerstrom et al., 1999), and although the case is reasonable, you have to start some­ where. This chapter is therefore more selective than comprehensive and it is organized as follows. In Section 2 the microscopic and macroscopic natures of Casimir effects are very briefly surveyed, and the interaction between two atoms is reviewed in Section 3 including discussion of the terms of relativistic origin arising for small atomic separations. In Sec­ tion 4 the change in the form of the interaction, when one of the two polarizable systems is charged, is studied. The vacuum dressed atom approach is introduced and applied to the case of an electron and an ion in Section 5, and it is used in Section 6 to gain insight into the origin of the numerical factor “23” in expressions for potentials related to the Casimir effect. Finally in Section 7 the treatment of multipoles beyond the electric dipole is discussed for two atoms and for an electron and an ion.

2. WHAT’S A MICRO EFFECT; WHAT’S A MACRO EFFECT? The picture of two well-spaced systems interacting through fluctuating electromagnetic fields can describe many phenomena. The usual definitions are that the Casimir–Polder potential (Casimir & Polder, 1948) is the retarded interaction between two atoms or an atom and a wall and a Casimir effect (Casimir, 1948) is the “observable non-classical electromagnetic force of

Casimir Effects in Atomic, Molecular, and Optical Physics

3

attraction between two parallel conducting plates” (Schwinger, 1975). Milton (2001, p. 3) traced the change in Casimir’s perspective from action at a distance (Casimir & Polder, 1948) to the local action of fields (Casimir, 1948) or an equivalence in physical pictures of fluctuating electric dipoles or fluctuating electric fields. The conceptual realizations of the Casimir effect and of the Casimir—Polder potential have been extended well beyond their original theoretical models; an extensive tabulation can be found in Buhmann and Welsch (2007). The term Casimir effect will be used rather more loosely in the present work recognizing in advance the connection already established in the literature with the more general pictures of “dispersion forces” (Mahanty & Ninham, 1976) or “van der Waals forces” (Barash & Ginzburg, 1984; Parsegian, 2006). Also as noted by Barton (1999), “By tradition, ‘Casimir effects’ denote macroscopic forces and energy shifts; yet for connected bodies the macroscopic must be matched to microscopic physics, and no purely macroscopic model can be guaranteed in advance to reproduce the results of this matching adequately for whatever purpose is in hand.” And as Barash and Ginzburg (1984) write, “The fluctuation nature of van der Waals forces for macroscopic objects is largely the same as for individual atoms and molecules. The macroscopic and microscopic aspects of the theory of van der Waals forces are therefore intimately related.” Moreover, there are macroscopic formulations that can yield results for microscopic systems by taking various limits (Buhmann & Welsch, 2007; Milonni & Lerner, 1992; Spagnolo et al., 2007), but local field corrections require close study (Henkel et al., 2008). For the present purposes, the concern is largely with pair-wise poten­ tials and their comparison with results from various approaches. Atomic units with h
= e = me = 1 are used throughout, wherein the fine structure constant is
= 1/c, though for some formulae h
and c are restored. It is useful to define the reduced Compton wavelength of the electron h/mec. The notational convention of Spruch and Tikochinsky l
C

(1993) is followed where the subscripts At, Ion, and El denote, respec­ tively, an atom, ion, and electron. The Casimir—Polder potential for the interaction between two identical atoms is written as (Casimir & Polder, 1948) 1 VAtAt ðrÞ ¼  6 r

1 ð

d! expð2
!rÞ½
e ði!Þ 2 Pð
!RÞ;

ð1Þ

0

with PðxÞ ¼ x4 þ 2x3 þ 5x2 þ 6x þ 3;

ð2Þ

and where the dynamic (frequency dependent) electric dipole polariz­ ability is

4

James F. Babb


e ð!Þ ¼

X u

fu ; E2u0  !2

ð3Þ

fu is the electric dipole oscillator strength from the ground state 0 to the excited state u, Eu0 = Eu—E0 is the energy difference, the summation includes an integration over continuum states, and ! is the frequency. An alternative form of Equation (1) (Boyer, 1969; Spruch & Kelsey, 1978) is hc
VAtAt ðrÞ ¼  lim  !0

1 ð

dk k6 e  k ½
e ð! Þ 2 IðkrÞ;

ð4Þ

0

with ! = kc and where IðxÞ ¼ sinð2xÞðx  2  5x  4 þ 3x  6 Þ þ cosð2xÞð2x  3  6x  5 Þ:

ð5Þ

The interaction potentials given in Equations (1) and (4) are valid for all separations larger than some tens of a0, and do not take into account electron charge cloud overlap, spin, and magnetic susceptibilities, for example, though these have all been studied. The interaction potential VAtAt(r), given by either Equation (1) or (4), does contain the van der Waals interaction, certain relativistic effects, and higher order effects, as well as the asymptotic form first obtained by Casimir and Polder (1948),
hc ½
e ð0Þ2 ; 4r7 For the hydrogen atom,
e(0) = 9/2. VAtAt ðrÞ ! 23

r ! 1:

ð6Þ

3. RELATIVISTIC TERMS Before studying the long-range Casimir—Polder interaction potential in detail, it is useful to look at the “small r” expansion1 of the full potential Equation (1) for distances, say, of the order 20 a0. Expanding Equation (1) for small r, the potential is VAtAt ðrÞ  

C6 W4
3 þ
2 4 þ O 3 ; 6 r r r

r  20 a0 :

ð7Þ

1 The term “short-range” is avoided and reserved for exchange, overlap, and forces that are, for example, exponentially decaying (Barash & Ginzburg, 1984). Thus, the term “long-range” interaction potential here will indicate the form valid for intersystem separations, typically from several to tens of a0 to infinity, such as those in Equations (1) and (4), which have a “small r” expansion [Equation (7)] and a “large r” expansion [Equation (6)].

Casimir Effects in Atomic, Molecular, and Optical Physics

5

The first term in the expansion is the van der Waals potential with van der Waals constant, 1 ð 3 C6 ¼ d! ½
e ði! Þ  2 ; ð8Þ  0

and for two H atoms, C6 = 6.499 026 705 405 84 (Watson, 1991). The term of order
2 relative to the van der Waals potential is a relativistic correction 1 ð 1 W4 ¼ d! !2 ½
e ði!Þ2 ; ð9Þ  0

which can be traced back (Power & Zienau, 1957) to the “orbit—orbit” effective potential appearing in the Breit—Pauli reduction of the Dirac equation (Meath & Hirschfelder, 1966). The numerical value of W4 for two H atoms is 0.462 806 538 843 273 according to Watson (1991), who used a momentum space approach and expansion in Pollaczek polynomials; he also obtained the highly accurate value of C6 quoted above. Certain exact representations of the dynamic polarizability function of H also facilitate evaluations of W4 (Deal & Young, 1971) and of C6 (O’Carroll & Sucher, 1968). Small relativistic terms were applied in a few cases to potential energy functions of light diatomic molecules, see for example Przybytek et al. (2010), and where improved accuracies were sought for precision calcu­ lations, for example, such as those of low-energy ultra-cold atomic colli­ sions (Zygelman et al., 2003) or of the ionization potential of the hydrogen molecule (Piszczatowski et al., 2009). Recently, Pachucki (2005) reanalyzed the Casimir—Polder potential complete to terms of Oð
2 Þ, but expressed it in such a way that its form is valid over all distances sufficiently large that the atomic wave functions do not overlap, not just in the large r limit.

4. YET ANOTHER REPULSIVE INTERACTION In the previous section, the original Casimir—Polder potential was intro­ duced and seen to be attractive, but there are several known cases where repulsive potentials have been predicted theoretically.2 Thus, Feinberg and Sucher (1968, 1970) used a general dispersion-theoretic scattering approach to show that the potential given in Equation (6) can be general­ ized for two systems A and B to an expression bilinear in the electric and the magnetic polarizabilities of each system, 2 V. Hushwater, Survey of Repulsive Casimir Forces, unpublished talk, ITAMP Casimir workshop, Cambridge, MA, November 16, 2002.

6

James F. Babb


c   A B h A B 23
e
e þ
m
m 7 4r     B A 7
eA
m þ
eB
m  þ O r9 ;

VAtAt ðrÞ  

r  1;

ð10Þ

B where
A e and
e are the static polarizabilities
e(0), respectively, of A and A B , with
m

m(0), are, respectively, the static magnetic B, and
m and
m polarizabilities of A and B. Note that the cross term in the potential contain­ ing the product of
e and
m leads to a repulsive force.3 More detailed discussions concerning the treatment of the magnetic terms for the interac­ tion between two atoms can be found in Salam (2000, 2010). There is another repulsive Casimir—Polder potential, perhaps not as well known. For the scattering interaction between a charged, structure­ less particle B and a neutral polarizable particle A (Bernabeu & Tarrach, 1976; Spruch & Kelsey, 1978) or for the interaction between a charged, structureless particle B and an ion A (Kelsey & Spruch, 1978b; Spruch & Kelsey, 1978), there is an interaction (for B an electron) given by  l–C e2  A r  1: ð11Þ 11
A VAtEl ðrÞ  VIonEl ðrÞ  e þ 5
m ; 5 4r

For either of the two cases (the target is neutral or it is charged), the asymptotic result [i.e., Equation (11)] applies, but the complete poten­ tials including other corrections are not identical, due to the remnant 1/r Coulomb interaction in the charged particle and ion case higher order corrections at large r differ, as emphasized by Au (1986, 1989). The long-range Casimir potential VIonEl(r) is the present object of interest, but it is useful sometimes to write the full potential with “instantaneous” Coulomb interactions as well. Therefore, the full poten­ tial U(r), including the charged particle and ion electric interactions (but neglecting the dominant 1/r Coulomb potential), is at large r 1 11 ð12Þ UðrÞ ¼  e2
e r  4 þ l–C e2
e r  5 . . . r  1; 2 4 where
e is the polarizability of the ion. The first term in Equation (12) is the polarization potential. The second term is the asymptotic Casi­ mir—Polder-type interaction, which was confirmed theoretically (Au, 1986; Feinberg & Sucher, 1983). The Casimir—Polder potential for the interaction between an electron and an ion was expressed in a fashion similar to the result for two atoms given in Equation (1) by Au et al. (1984) using a dispersion relation analysis, and later using Coulomb

B 3 The
A e
m term supports the result that the interaction between two plates is repulsive, if one plate (A) has only infinite permittivity and one plate (B) is only infinitely permeable (Boyer, 1974); an alternative argument not making use of Equation (10) is given by Schaden and Spruch (1998).

Casimir Effects in Atomic, Molecular, and Optical Physics

7

gauge, old-fashioned perturbation theory by Babb and Spruch (1987) and Au (1988). It can be expressed (Babb & Spruch, 1987) in the compact form similar to Equation (4), e2 VIonEl ðrÞ ¼ l–C lim !0  where FðkÞ ¼

1 ð

dk e  k k4 FðkÞI ðkrÞ;

ð13Þ

0

X

fu

u

½Eu0 ðEu0 þ Ek Þ

:

ð14Þ

Taking account of the Coulomb interactions, the potential has the expansion for small r,  2 1
UðrÞ þ VIonEl ðrÞ   e2
e r  4 þ 3r  6 þ r  4 . . . ; r  a few a0 ; 2 Z2 ð15Þ where Z is the charge of the ion, and  = 43/(8Z6) (Dalgarno et al., 1968; Kleinman et al., 1968). Note the disappearance for small r of the r5 term that was present in the large r potential, Equation (12). It was emphasized by Kelsey and Spruch (1978a) and Feinberg et al. (1989) that the 3r—6 term disappears at large r. Thus, the potential can be written in the form (Babb & Spruch, 1987) ð1 hc
dk k6 e  k IðkrÞ VIonEl ðrÞ þ 3r  6 ¼  lim  !0 0  hc

e ðkÞ
out ðkÞ  2 2
out ðkÞ½ðkÞ   ; ð16Þ e using e2 ; m!2 k FðkÞ ¼
e ðkÞ  2 ðkÞ;

out ð!Þ ¼ 

and

ðkÞ ¼

1X f   u  : 2 u Eu0 E2u0 þ E2k

ð17Þ ð18Þ

ð19Þ

The effective potential Equation (13) was evaluated numerically and used to study theoretically the energy shift arising from the interaction between an electron bound in a high Rydberg state |lni with principal quantum number n, and with angular momentum l  n. Experiments on highly excited n = 10 Rydberg states of He were carried out and several

8

James F. Babb

theoretical formulations were developed. The details about experiments and theories, and their comparisons, are very completely presented in the book edited by Levin and Micha (1992) (see also Hessels, 1992; Stevens & Lundeen, 2000; Drake, 1993; Lundeen, 2005). Many other terms must be considered carefully in the theoretical calculations; the details do not directly relate further to this chapter. Recently two groups have reproduced the electric (
e) part of Equation (11) using different arguments, but both approaching the interaction between a charged particle and a neutral particle as oneloop quantum field theoretic calculations. Panella et al. (1990) obtained Equation (11) and traced the r5 result back to a change in the mass induced by “condensed-matter renormalization” of the elec­ tromagnetic fluctuations (Panella & Widom, 1994). The repulsive potential is attributed to soft-photon infrared renormalization. In contrast, Holstein and Donoghue (2004) were seeking to identify cases where classical effects are found within one-loop diagrams. Using an effective field theory approach, they find quantum correc­ tions to the classical polarization potential; their result is identical to the
e part of Equation (11). They identify the r5 term as a quantum correction to the polarization potential, which arises from the infrared behavior of the Feynman diagrams, when at least two massless pro­ pagators occur in a loop contribution. In a subsequent study, Holstein (2008) obtained Equation (11) and observed that the r5 term in Equation (11) might be associated with zitterbe­ wegung and he noted that, under the influence of this effect, “in the quantum mechanical case the distance between two objects is uncertain by an amount of order the Compton wavelength due to zero point motion,” r  l
C ; hence VðrÞ 

1 1 1 1 ! » +4l–C 5 : r4 ðr – r Þ 4 r4 r

ð20Þ

It is an intriguing argument, though the ambiguity of the sign is unresolved. Zitterbewegung would normally be attributed to virtual electron­ positron transitions (Milonni, 1994, p. 322) at the length scale less than l
C, which would seemingly place the effect outside of the realm of the lowenergy fluctuation arguments used by, for example, Spruch and Kelsey (1978) in deriving Equation (11). Nevertheless, the calculations of Holstein and Donoghue (2004) are concerned with large r, so there must be a connec­ tion to the scale of l
C and this will be addressed in the next section. The effective field theory of Holstein (2008) allows the longest-range parts of electromagnetic scattering processes to be isolated, and he extended the asymptotic (large r) results for the interactions between two systems, with and without spin, to the case where one or both systems are electrically neutral; see also Sucher and Feinberg (1992).

Casimir Effects in Atomic, Molecular, and Optical Physics

9

5. NONRELATIVISTIC MOLECULES AND DRESSED ATOMS In the theoretical “vacuum dressed atom” approach, a ground state “bare” atom interacts with the vacuum electromagnetic field. The combi­ nation system of the atom and the field is taken to be in the lowest possible energy state of the noninteracting atom—field system, and the zero-point fluctuations of the field are seen as inducing virtual absorption and re-emission of photons in the atom–the “vacuum dressed atom” is then the system comprised of the atom and the associated cloud of virtual photons (Compagno et al., 1995a,b). A good account of the development of the concept of atoms dressed by the vacuum electromagnetic field is given by Compagno et al. (2006). The energy density can be calculated and used to obtain expressions for long-range potentials and other phy­ sical quantities, as was shown in quantum optics (Cirone & Passante, 1996; Compagno et al., 1995b; Salam, 2010) for the nonrelativistic free electron interacting with the vacuum electromagnetic field and the non­ relativistic hydrogen atom. Compagno and Salamone (1991) (see also Compagno et al., 1995b) considered a slow electron interacting with the vacuum field. The key observation is that the cloud around the electron is due to emission and reabsorption of virtual photons in the course of recoil events. They point out that the zitterbewegung due to relativistic fluctuations would enter and contribute a cloud of size of order l
C. This effectively limits their nonrelativistic model to distances  l
C, so that the positron cloud can be neglected and the electron has the physical charge e; accordingly, only low-frequency photons enter. In this picture the virtual cloud affects the field surrounding the charge and changes the average values of the squares of the electric and magnetic fields. They calculate the classical and quantum contributions to the energy density around the electron both moving and at rest, and for the electron at rest, they find (for r  l
C)
X hc e2 5 e2 l–C þ þ k ð21Þ hEe ðrÞi ¼ 4 2 4 16 r r 4V k 8r and hEm ðrÞi ¼ 


X hc 5 e2 l–C þ k; 2 4 4V k 16 r r

ð22Þ

where Ee(r) and Em(r) are, respectively, the electric and magnetic energy densities at a distance r from the electron, V is the quantization volume, and the sum is over the vacuum field modes. For the present purpose, a key observation is the appearance of the r5 contribution which Compagno and Salamone (1991) attribute to the vir­ tual photon cloud surrounding the electron fluctuations that arise due to

10

James F. Babb

interference between the virtual photons emitted and absorbed by the electron and zero-point field fluctuations. The r5 term is deemed purely quantum in nature (Compagno et al., 1995b). The energy densities can be directly related to the Casimir—Polder potential, as noted by Passante and Power (1987). What is striking is the similarity in form between Equations (21) and (22) and the large r potential for the interaction of an electron and an ion, Equation (12). Evidently, both the classical polarization potential and the retarded asymptotic Casimir—Polder potential are present. As discussed above, Holstein and Donoghue (2004) showed that, within a diagrammatic, effective field theory approach, classical effects can arise. In particular, the energy density of a particle in a plane wave calculated by Holstein and Donoghue (2004) agrees in form with the dressed electron result containing both a r4 polariza­ tion potential and the “purely quantum” r5 asymptotic retarded potential. For the nonrelativistic hydrogen atom, the analysis was carried out again within the vacuum dressed atom formalism; the extensive calcula­ tions can be found in Passante and Power (1987) and Compagno et al. (1995b, 2006). The analysis is complicated, but it is similar to that carried out by Babb and Spruch (1987) and Au (1989). For example, Equation (7.148) of Compagno et al. (1995b) describes the longitudinal electric field and transverse electric field contributions to the energy density around a hydrogen atom, 1 0 1 hjE jj ðxÞ  E\ ðxÞji0  3 4 r

ð

k2 j2 ðkrÞ dk ! N þ !k

ð23Þ

and it is almost identical to VIT, Equation (4.16) found by Babb and Spruch (1987) for the contribution of one instantaneous Coulomb photon and one transverse photon to the effective potential in the case of an electron and an ion. In an earlier study using the virtual photon cloud picture, Passante and Power (1987) note that the r6 term in the description of the energy density around a ground state hydrogen atom disappears at large r similarly to the way the van der Waals form r6 form is replaced at asymptotic distances by the r7 form. They find that nonretarded effects of order r6 in the energy density are absent in the far zone of the hydrogen atom, and they obtain the simple form for the energy density with an Oðr  7 Þ term related to the virtual charge cloud, 1 23
hc
e ; hjF2 jiEzp ¼ 8 162 r7 where F is the electric field.

ð24Þ

Casimir Effects in Atomic, Molecular, and Optical Physics

11

_ Radozycki (1990) carried out a relativistic calculation of the electro­ magnetic virtual cloud of the ground-state hydrogen atom using a Dirac formalism. According to Compagno et al. (2006), his work was supposed to be an independent calculation of the energy density of the vacuum dressed hydrogen atom. For the energy density due to the electric field, his result in the large r limit is 1 1 2 13
hce2 X 1 h1jxjnihnjxj1i 7 hF ðrÞi¼ 2 r 2 16 n En1 þ

^ri^rj 7
hce2 X 1 h1jxi jnihnjxj j1i 7 : 2 16 n En1 r

ð25Þ

Identifying the tensor electric dipole polarizability in Equation (25)

2

X 1 h1jxi jnihnjxj j1i¼
e; ij ; En1 n

ð26Þ

Equation (25) agrees with the two-level “Craig—Power” model (Compagno et al., 1995b) result for the energy density in the large r limit, hEe ðrÞi¼

1  1

ij 13ij þ 7^ri^rj 7 : hc 2 r 32

ð27Þ

In an unrelated study of the Casimir—Polder potential for an electron interacting with a hydrogen molecular ion core, Babb and Spruch (1994) obtained an expression almost identical to Equations (25) and (27). The tensor polarizability arises from the anisotropic interaction arising from the cylindrical symmetry of the diatomic molecule core. Compagno et al. (1995b) interpret the large r Casimir—Polder potential as the interaction between the vacuum dressed “source” atom with polarizability
S and the “test” atom with polarizability
T VðrÞ ¼

23 1
S
T 7 ¼ 4
T hESe ðrÞi; hc
4 r

ð28Þ

where the energy density is generated by the source at point r in the absence of the test atom. Another interesting point emphasized by Compagno et al. (1995b) is that VðrÞ ¼ 4
T hESe ðrÞi;

ð29Þ

12

James F. Babb

“thus the van der Waals force provides a means of measuring directly the electric energy density of a source both in the near and in the far regions.”

6. NOT A TRIVIAL NUMBER In his contribution to the proceedings of a conference held in Maratea, Italy,4 Casimir (1987) wrote, “In the theory of the so-called Casimir effect two lines of approach are coming together. The first one is concerned with Van der Waals forces, the second one with zero-point energy.” Today, that connection is well established, though the “reality” of zeropoint energy is still debatable; see the very accessible article by Rugh et al. (1999) and also Jaffe (2005). In Equation (6), it was shown that the asymptotic potential for the interaction between two electrically polarizable particles contains the factor 23, as does the asymptotic potential for two magnetically polariz­ able particles, see Equation (10). The factor 23 has reappeared in other situations. In the asymptotic interaction between an electron and an ion, expanding Equation (16) for large r and keeping one more term past that given in Equation (12) (Feinberg & Sucher, 1983), the potential is 1 UðrÞ  e2
e r  4 þ VIonEl ðrÞ 2 1 11
e 23 e2 l–C ð0Þ þ . . .; r  1: ð30Þ   e2
e r  4 þ l–C e2 5 þ 2 4 r 4 a20 r7 According to Feinberg et al. (1989), when told of this result [i.e., Equation (30)], at the Maratea conference, Casimir replied, “23 is not a trivial number. I am happy to see that.” However, the appearance of the 23 in a way nearly identical to the result for the asymptotic atom—atom poten­ tial was not explained completely (Feinberg & Sucher, 1983). In addition, the sign of the term containing 23 is opposite to that for the case of two atoms. Evidently, while the complete potential for the electron—ion case can be expressed as Equation (13), expansion for large r yields the two terms of Equation (11). I conjecture that the Oðr  5 Þ term can be interpreted as the effective potential arising from the energy density of the weakly bound, vacuum dressed, electron “source” interacting with the ion core “test”
e,   1 1 ð31Þ
e hE2out i
e 4 þ l–C 5 ; r r

4

June 1—14, 1986.

Casimir Effects in Atomic, Molecular, and Optical Physics

13

in accord with the ideas of Compagno and Salamone (1991), supported by the large r one-loop calculations of Holstein and Donoghue (2004) and Holstein (2008). The other term r7 can be interpreted as the source term of the fluctuating vacuum dressed polarizable ion core “source” acting on the electron “test” particle, 23
 7; hcr ð32Þ 4 where E Se indicates that the electric field energy density is modified due to the Coulomb binding and ! is a characteristic energy. This is to be expected based on arguments given by Au (1986, 1989) for the Rydberg helium case and by Compagno and Salamone (1991) for the vacuum dressed slow electron and vacuum dressed hydrogen atom. Using Equa­ tion (17) for
out(k) = l
Cak2, where kc = !, evaluated at ! ¼
c=a0 (Feinberg & Sucher, 1983). Equation (32) yields a term in general agree­ ment with Equation (30), 4
out ð!ÞhE Se ðrÞi
out ð!Þ

23 4
out ð! ÞhE Se ðrÞiþl–C r  7 : 4

ð33Þ

The approach of adding the two interactions is consistent with the inter­ pretation of the fluctuating field approach to Casimir—Polder interactions proposed by Power and Thirunamachandran (1993). Namely, that the dipole in each particle is induced by the vacuum fluctuations of the electromagnetic field.

7. RECONCILING MULTIPOLES 7.1 Two Atoms The extension beyond electric and magnetic dipoles for the retarded van der Waals (or Casimir—Polder) potential between two neutral spinless systems was achieved by Au and Feinberg (1972). Using scattering ana­ lysis, they were able to obtain integral forms for the complete potential for each multipole, valid for all separations greater than some tens of a0, and they gave the first several terms in each of the small r and large r expansions of the potentials. Their result, as noted by Feinberg (1974), and as emphasized by Power and Thirunamachandran (1996), included the property “that the expansions of the electric (magnetic) form factors include high-order magnetic (electric) susceptibilities in addition to elec­ tric (magnetic) polarizabilities.” The first several electric multipole results of Au and Feinberg (1972) were used for applications to calculations of the binding energy of the helium dimer by Luo et al. (1993) and by Chen and Chung (1996) and for applications to ultra-cold atom scattering by Marinescu et al. (1994), who

14

James F. Babb

evaluated the expressions for a pair of hydrogen atoms and for a pair of like alkali-metal atoms. A few years later, in a thorough analysis, Salam and Thirunamachandran (1996) and Power and Thirunamachandran (1996) pointed out that the results of the Au and Feinberg (1972) analysis did not concur with other results obtained evidently independently by Jenkins et al. (1994), who used a different approach. Power and Thirunamachandran (1996) argued that the correct form for the next order Casimir—Polder interaction, arising from the interaction of an electric dipole and an electric quadrupole, is 1 V12 ðrÞ ¼  3

1 ð

d!expð2
!rÞ
e ði!Þ
e ; 2 ði!ÞP12 ð!
rÞ;

ð34Þ

0

where
e;2 is the electric quadrupole polarizability and P12 ðxÞ ¼

1 6 27 x þ 3x5 þ x4 þ 42x3 þ 81x2 þ 90x þ 45: 2 2

ð35Þ

C8 W6 þ
2 6 þ . . .; r8 r

ð36Þ

d!!2
e ði!Þ
e ; 2 ði!Þ;

ð37Þ

For small r, V12 ðrÞ   where 15 C8 ¼ 

1 ð

0

and the coefficient of the relativistic correction is 3 W6 ¼ 

1 ð

d!!2
e ði!Þ
e ; 2 ði!Þ:

ð38Þ

0

Meath and Hirschfelder (1966) obtained the relativistic corrections of relative order
2 for two hydrogen atoms. For the orbit—orbit interaction HLL;0, the corresponding effective potential contributions arise as powers of r4 and r6. Their approach is not valid in the large r limit, but it should agree with the small r form of VAtAt(r) and V12(r). In turn Marinescu et al. (1994) noticed a discrepancy between the result of Meath and Hirschfelder and Johnson et al. (1967) for the r6 relativistic term for small distances and the result obtained by expanding the potential of Au and Feinberg for small r. In contrast, expansion of the revised dipole—quadrupole potential V12(r) of Jenkins et al. (1994) for small r provides a value for W6, see Equation (38), in agreement with

Casimir Effects in Atomic, Molecular, and Optical Physics

15

the expression of Meath and Hirschfelder (1966) and Johnson et al. (1967) for the r6 term in the expansion of the Breit—Pauli equation. Accordingly, earlier results, such as the results of Chen and Chung (1996), for W6 should be multiplied by 3/2. Later, Marinescu and You (1999) rederived the atom—atom potential accounting for magnetic and other terms to higher order (see also Salam, 2000). Marinescu and You note that numerically, at least, for their evalua­ tions of the like alkali-metal atom pairs, the relative error between results from the two approaches is smaller than 105. In any case, other terms, such as mass polarization, Darwin interaction terms, and Lamb shifts, would have to be included at the correct order for a complete description. Asymptotically, for large r (Jenkins et al., 1994; Marinescu & You, 1999; Thirunamachandran, 1988; Yan et al., 1997) V12 ðrÞ  

531
hc
e ð0Þ
e ; 2 ð0Þ: 16r9

ð39Þ

Feinberg (1974) expected that for atoms–with the exception of possibly the magnetic—magnetic case for two hydrogen atoms (see Feinberg and Sucher, 1968)–magnetic and higher order multipole effects would be negligible for domains where retardation was important. Thus, Feinberg was motivated to use the scattering approach to study the case of two superconducting spheres, and he obtained a series in powers of the sphere—sphere separation distance. Using a new formalism based on a scattering approach, Emig (2008) and Emig and Jaffe (2008) investigated the Casimir energy between two spheres. For large separations, they obtain an expansion in the separation distance r. The lead term is of order r7 and is given by Equation (10), where the polarizabilities correspond to those of the spheres. Moreover, the next term of order r9 is given by Equation (39). In another calculation using the scattering approach, Emig (2010) obtains the large r interaction potential for two anisotropic objects; his result is in agreement with the earlier results for two anisotropic particles given by Craig and Power (1969). 7.2 An Electron and an Ion Expanding Equation (13) for small r, it was shown above in Equation (15) that there is a term (a2/Z2)r4. This relativistic term is identical to the atom—atom case, which is known to result from perturbation treatment of the Breit interaction with the Coulomb interaction (Au, 1989; Power & Zienau, 1957). Some time after Equation (4) was obtained, the complete long-range potential including multipoles for an ion and a neutral spinless system

16

James F. Babb

was obtained by Feinberg and Sucher (1983) and by Au (1985). According to Hessels (1992), the result of Au (1985) for the next term is VE ; 1 ðrÞ ¼

9 2 6
r : 16

ð40Þ

Hessels (1992) carried out a perturbation theoretic calculation of the relativistic corrections for the ion—electron system, analogously to the calculation of Meath and Hirschfelder (1966) for the atom—atom interac­ tion. His analysis is in disagreement with Au (1985), but in agreement with comprehensive calculations by Drake (1992), indicating an unresolved discrepancy between the dispersion theoretic result and perturbation theoretic results at Oð
2 r  6 Þ for small r limit of the ion—electron system.

8. CONCLUSION The Casimir effects for the interaction between two atoms and for the inter­ action between an ion and an electron were investigated and, respectively, their expansions lead to asymptotic terms of order r7 and r5. The second correction at large r for the ion and electron case is similar to the leading term at large r for the case of two atoms. It was shown that the vacuum dressed atom picture provides a framework for interpretation of this similarity. Reconciliation of interaction potentials for electric dipole and electric quadrupole multipoles between atom—atom and ion—electron cases led to insight concerning a discrepancy between a scattering dispersion theore­ tic calculation and a perturbation theoretic calculation of the ion—electron interaction for the electric quadrupole relativistic term. As interest in the potential applications of Casimir effects in atomic, molecular, and optical physics increases, limiting results for interaction potentials at zero temperature–such as those presented here–may pro­ vide useful insights and checks on calculations for more complicated geometries. Hopefully, it will be a long time until it is true that nothing can be added to vacuum studies.

ACKNOWLEDGMENTS I am indebted to several colleagues who have shared their knowledge with me over the years on topics related to this chapter. In particular, Larry Spruch Alex Dalgarno, Joe Sucher, Akbar Salam, and Peter Milonni provided helpful insights. ITAMP is partially supported by a grant from the NSF to Harvard University and the Smithsonian Astrophysical Observatory.

Casimir Effects in Atomic, Molecular, and Optical Physics

17

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Compagno, G., & Salamone, G. M. (1991). Structure of the electromagnetic field around the free electron in nonrelativistic QED. Physical Review A, 44, 5390—5400. Craig, D. P., & Power, E. A. (1969). The asymptotic Casimir-Polder potential from secondorder perturbation theory and its generalization for anisotropic polarizabilities. Interna­ tional Journal of Quantum Chemistry, 3, 903—911. Dalgarno, A., Drake, G. W., & Victor, G. A. (1968). Nonadiabatic long-range forces. Physical Review, 176, 194—197. Deal, W., & Young, R. (1971). The long-range retarded interaction between two hydrogen atoms. Chemical Physics Letters, 11, 385—386. Drake, G. W. F. (1992). High-precision calculations for the Rydberg state of helium. In F. S. Levin & D. Micha (Eds.) Long range Casimir forces: Theory and recent experiments in atomic systems (pp. 107—217). New York: Plenum Press. Drake, G. W. F. (1993). Energies and asymptotic analysis for helium Rydberg states. In Advances in atomic, molecular, and optical physics (Vol. 31, pp. 1—62). San Diego: Academic Press. Emig, T. (2008). Fluctuation-induced quantum interactions between compact objects and a plane mirror. Journal of Statistical Mechanics, 2008, P04007. Emig, T. (2010). Casimir physics: Geometry, shape and material. arxiv.org/abs/1003.0192. Emig, T., & Jaffe, R. L. (2008). Casimir forces between arbitrary compact objects. Journal of Physics A, 41, 164001. Feinberg, G. (1974). Retarded dispersion forces between conducting spheres. Physical Review B, 9, 2490—2496. Feinberg, G., & Sucher, J. (1968). General form of the retarded van der Waals potential. Journal of Chemical Physics, 48, 3333—3334. Feinberg, G., & Sucher, J. (1970). General theory of the van der Waals interaction: A modelindependent approach. Physical Review A, 2, 2395—2415. Feinberg, G., & Sucher, J. (1983). Long-range forces between a charged and neutral system. Physical Review A, 27, 1958—1967. Feinberg, G., Sucher, J., & Au, C. K. (1989). The dispersion theory of dispersion forces. Physics Reports, 180, 83—157. Henkel, C., Boedecker, G., & Wilkens, M. (2008). Local fields in a soft matter bubble. Applied Physics B, 93, 217—221. Hessels, E. A. (1992). Higher-order relativistic corrections to the polarization energies of helium Rydberg states. Physical Review A, 46, 5389—5396. Holstein, B. R. (2008). Long range electromagnetic effects involving neutral systems and effective field theory. Physical Review D, 78, 013001. Holstein, B. R., & Donoghue, J. F. (2004). Classical physics and quantum loops. Physical Review Letters, 93, 201602. Jaffe, R. L. (2005). Casimir effect and the quantum vacuum. Physical Review D, 72, 021301. Jenkins, J. K., Salam, A., & Thirunamachandran, T. (1994). Retarded dispersion interaction energies between chiral molecules. Physical Review A, 50, 4767—4777. Johnson, R. E., Epstein, S. T., & Meath, W. J. (1967). Evaluation of long-range retarded interaction energies. Journal of Chemical Physics, 47, 1271—1274. Kelsey, E. J., & Spruch, L. (1978a). Retardation effects and the vanishing as r  1 of the nonadiabatic r6 interaction of the core and a high-Rydberg electron. Physical Review A, 18, 1055—1056. Kelsey, E. J., & Spruch, L. (1978b). Retardation effects on high Rydberg states–retarded r5 polarization potential. Physical Review A, 18, 15—25. Kleinman, C. J., Hahn, Y., & Spruch, L. (1968). Dominant nonadiabatic contribution to the long-range electron-atom interaction. Physical Review, 165(1), 53—62. Levin, F. S., & Micha, D. (Eds.) (1992). Long range Casimir forces: Theory and recent experiments in atomic systems. New York: Plenum Press. Lundeen, S. R. (2005). Fine structure in high-L Rydberg states: A path to properties of positive ions. In Advances in atomic, molecular, and optical physics (Vol. 52, pp. 161—208). San Diego: Elsevier Academic. Luo, F., Kim, G., Giese, C. F., & Gentry, W. R. (1993). Influence of retardation on the higherorder multipole dispersion contributions to the helium dimer potential. Journal of Chemical Physics, 99, 10084—10085.

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Spruch, L., & Tikochinsky, Y. (1993). Elementary approximate derivations of some retarded Casimir interactions involving one or two dielectric walls. Physical Review A, 48, 4213—4222. Stevens, G. D., & Lundeen, S. R. (2000). Experimental studies of helium Rydberg fine structure. Comments on Atomic and Molecular Physics, Comments on Modern Physics, Part D, 1, 207—219. Sucher, J., & Feinberg, G. (1992). Long-range electromagnetic forces in quantum theory: Theortical formulations. In F. S. Levin & D. Micha (Eds.) Long range Casimir forces: Theory and recent experiments in atomic systems (pp. 273—348). New York: Plenum Press. Thirunamachandran, T. (1988). Vacuum fluctuations and intermolecular interactions. Phy­ sica Scripta, T21, 123—128. Voronin, A. Y., Froelich, P., & Zygelman, B. (2005). Interaction of ultracold antihydrogen with a conducting wall. Physical Review A, 72, 062903. Watson, G. I. (1991). Two-electron perturbation problems and Pollaczek polynomials. Journal of Physics A, 24, 4989—4998. Wennerstrom, H., Daicic, J., & Ninham, B. W. (1999). Temperature dependence of atom— atom interactions. Physical Review A, 60, 2581—2584. Yan, Z.-C., Dalgarno, A., Babb, J. F. (1997). Long-range interactions of lithium atoms. Physical Review A, 55, 2882—7. Zygelman, B., Dalgarno, A., Jamieson, M. J., & Stancil, P. C. (2003). Multichannel study of spin-exchange and hyperfine-induced frequency shift and line broadening in cold colli­ sions of hydrogen atoms. Physical Review A, 67, 042715.

CHAPTER

2

Advances in Coherent Population Trapping for Atomic Clocks Vishal Shaha and John Kitchingb a

Symmetricom Technology Realization Center, 34 Tozer Road,

Beverly, MA 01915, USA

b Time and Frequency Division, NIST, 325 Broadway, Boulder,

CO 80305, USA

Contents

1. 2.

3.

4.

Coherent Population Trapping 1.1 Introduction 1.2 Basic Principles Atomic Clocks 2.1 Introduction 2.2 Vapor Cell Atomic Clocks 2.3 Coherent Population Trapping in Atomic

Clocks 2.4 Stability of Vapor Cell Atomic Clocks 2.5 Light Shifts Advanced CPT Techniques 3.1 Contrast Limitations due to Excited-State

Hyperfine Structure 3.2 Contrast Limitations due to Zeeman

Substructure 3.3 High-Contrast Resonances Using

Four-Wave Mixing 3.4 Push–Pull Laser Atomic Oscillator 3.5 The CPT Maser 3.6 N-Resonance 3.7 Raman–Ramsey Pulsed CPT 3.8 CPT in Optical Clocks Additional Considerations 4.1 Light-Shift Suppression 4.2 Laser Noise Cancellation 4.3 Light Sources for Coherent Population

Trapping

22

22

23

27

27

29

30

35

38

39

40

42

51

52

54

55

56

59

60

60

61

63

Advances in Atomic, Molecular, and Optical Physics, Volume 59
2010 Elsevier Inc. ISSN 1049-250X, DOI: 10.1016/S1049-250X(10)59002-5 All rights reserved.

21

22

Vishal Shah and John Kitching

4.4 Dark Resonances in Thin Cells 4.5 The Lineshape of CPT Resonances: Narrowing Effects 5. Conclusions and Outlook Acknowledgments References

Abstract

65 65 66 67 67

We review advances in the field of coherent population trapping (CPT) over the last decade with respect to the application of this physical phenomenon to atomic frequency references. We provide an overview of both the basic phenomenon of CPT and how it has traditionally been used in atomic clocks. We then describe a number of advances made with the goal of improving the resonance contrast, decreasing its line width, and reducing light shifts that affect the long-term stability. We conclude with a discussion of how these new approaches can impact future generations of laboratory and commercial instruments.

1. COHERENT POPULATION TRAPPING 1.1 Introduction Coherent population trapping (CPT) (Arimondo, 1996) refers to the pre­ paration of atoms in coherent superposition states by use of multimode optical fields. This phenomenon, as investigated using hyperfine (Alzetta et al., 1976; Arimondo & Orriols, 1976) and optical (Whitley & Stroud, 1976) transitions in 1976, has led to significant advances in a variety of areas of optical and atomic physics including laser cooling (Aspect et al., 1988), nonlinear optics (Hemmer et al., 1995), precision spectroscopy (Wynands & Nagel, 1999), slow light (Schmidt et al., 1996), atomic clocks (Kitching et al., 2000; Thomas et al., 1981, 1982; Vanier et al., 1998; Zanon et al., 2005; Zanon-Willette et al., 2006), and other precision spectroscopic instrumentation (Nagel et al., 1998; Schwindt et al., 2004). The central principle that underlies the value of CPT in this diverse set of applica­ tions is the idea that certain coherent superposition states do not absorb light from the excitation field. This reduced absorption leads both to a spectroscopic signal on the light field and to a modified atom—light interaction. The use of CPT in atomic clocks is a particularly important application that has sustained interest over three decades. Early work to use micro­ wave CPT (Arimondo & Orriols, 1976; Orriols, 1979) in atomic beam clocks (Thomas et al., 1981, 1982) has been adapted for application to

Advances in CPT for Atomic Clocks

23

vapor cell clocks (Cyr et al., 1993; Vanier et al., 1998) and led most recently to microfabricated atomic clocks (Knappe et al., 2004). A previous review of atomic microwave clocks based on CPT is given in the work of Vanier (2005). Possible future application to optical clocks has also been consid­ ered (Hong et al., 2005; Santra et al., 2005). In these clock designs, the CPT resonance is used to directly measure the atomic transition frequency. The performance of the clock therefore depends intimately on the quality of the CPT resonance, and most specifically on its line width and contrast. This chapter reviews research over the last decade to understand and extend the phenomenon of CPT with respect to its application to atomic frequency standards. Special emphasis is placed on novel excitation and detection schemes, and other new phenomena that improve the reso­ nance contrast, reduce its line width, or minimize the effect of the light fields on the resonance frequency. In Section 1.2, we review the basic phenomenon of CPT and describe how CPT resonances have traditionally been excited and detected. In Section 2, we provide an introduction to atomic clocks and discuss the differences between conventional atomic clocks and those based on CPT. In Section 3, we consider the limitations of the simplest CPT excitation schemes and describe several schemes that have been recently developed to enhance the quality of the CPT reso­ nance for use in atomic clocks. In Section 4, we address a number of outstanding issues including light shifts, light sources for CPT, and unique experimental environments in which CPT is observed. Finally in Section 5, we offer some conclusions and discuss how the new ideas connected with CPT may ultimately impact the development of future atomic frequency references. 1.2 Basic Principles A two-level atom illuminated by a monochromatic electromagnetic field is perhaps the simplest spectroscopic system. When the frequency of the illumination field is tuned into resonance with the transition between the atomic levels, radiation is scattered by the atom via spontaneous emis­ sion, causing fluorescence, and a corresponding change in the intensity and phase of the transmitted radiation (see Figure 1a). The fluorescence signal, combined with a measurement of the radiation wavelength or frequency, often gives highly precise information about the internal structure of the atom. In atoms with more than two energy levels, a variety of more complex phenomena can occur. One example of this is optical pumping, in which light resonant with one optical transition causes atomic population to accumulate in a state not excited by the light field (see Figure 1b). Once pumped into this third state, the atoms stop scattering light to the extent that the third state is stable and not resonant with the light field. Such a state is referred to as a dark state.

24

Vishal Shah and John Kitching

(a)

|3>

|1>

(b)

|3>

(c)

|3>

|1>

|1>

|2>

|2>

Figure 1 Optically excited transitions in atoms. (a) Simple excitation of a two-level atom. (b) Optical pumping in a three-level atom. Level |2> is an incoherent dark state. (c) Bichromatic excitation of a three-level atom. A superposition of levels |1> and |2> is a coherent dark state

CPT is a phenomenon that occurs in atoms with more than two energy levels excited by coherent, multimode optical fields. Under the right conditions, atoms are optically pumped into a superposition of two of the levels that does not scatter light from the multimode field (see Figure 1c). This coherent dark state has an electromagnetic moment that oscillates at one of the beat frequencies of the multimode field. Its excita­ tion can be thought of as a nonlinear process: a nonlinear resonator (the atom) is driven with a force with two spectral components (the light), and through the nonlinearity, an oscillation at the sum or difference of the two driving frequencies is established. The phase of this oscillation, with respect to the phase of the driving fields, is such that no energy is absorbed. CPT between hyperfine atomic levels was first observed experimen­ tally in a seminal paper by Alzetta et al. (1976), in which a light field from a multi-longitudinal-mode dye laser was sent into a vapor cell containing saturated Na and a buffer gas. The laser mode spacing had a harmonic near the frequency of the ground-state hyperfine splitting of Na. A long­ itudinal magnetic field gradient was applied to the cell, and the fluores­ cence from the cell was measured as a function of longitudinal position. Dark lines were observed in the fluorescence at locations where the magnetic field had shifted magnetically sensitive hyperfine levels into resonance with a mode spacing harmonic. Data from Alzetta et al. (1976) are shown in Figure 2. The observations were explained theoretically by use of a density matrix analysis, in which the excited-state population, and hence the fluorescence rate, in the three-level system was calculated as a function of the relative detuning of a pair of resonant optical fields (Arimondo & Orriols, 1976; Gray et al., 1978; Whitley & Stroud, 1976;). The dark line in the fluorescence was identified as resulting from (destructive) interfer­ ence of absorption pathways in the coherently excited atomic system.

Advances in CPT for Atomic Clocks

25

Figure 2 Data showing the fluorescence from a Na vapor cell under illumination by a multimode optical field. A magnetic field is applied with a gradient along the axis of the light propagation. Three dark lines are observed in the fluorescence spectrum.
Reprinted figure with permission from Alzetta et al. (1976);
1976 of the Societa Italiana de Fisica

A similar phenomenon on the Zeeman rather than hyperfine coherences was reported earlier by Bell and Bloom (1961), and a review of CPT is given by Arimondo (1996). A key aspect of this phenomenon is that coherences in atoms at microwave frequencies can be excited even with no microwave fields being present at the location of the atoms. The hyperfine coherences are excited entirely by two-photon processes involving only optical fields and have a line width determined by the hyperfine relaxation rate. In addition, the presence of the coherence can be detected easily by mon­ itoring the fluorescence (or absorption) by the atomic sample. CPT can also be understood as a combination of quantum interference and optical pumping. In the three-level model, a bichromatic optical field couples two long-lived states (denoted |1> and |2> in the Figure 1c) to a single upper level (denoted |3>). The energy of the ith level is denoted Ei and the optical field is denoted EðtÞ ¼ "1 e  ið!1 tþ1 Þ þ "2 e  ið!2 tþ2 Þ ;

ð1Þ

where "i, !i, and i are the (complex) amplitude, frequency, and phase of the ith field component, respectively. We may write orthogonal super­ positions of states |1> and |2> that interact with the optical field in distinct ways: jNCi ¼ c2 j1i  c1 j2i jCi ¼ c1 j1i þ c2 j2i:

ð 2Þ

i "i ffiffiffiffiffiffiffi ei½ðEi =hÞtþi  ; ci ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 j1 "1 j þ j2 "2 j2

ð3Þ

Here

26

Vishal Shah and John Kitching

and i is the electric dipole moment between state |i> and state |3>. It can be shown that when the two-photon resonance condition, !1–!2 = (E1–E2)/h, is fulfilled, the transition amplitude from the state |NC> to the excited state |3> is zero. The |NC> state is therefore a dark (or “non­ coupled”) state, because no light is scattered when the resonance condi­ tion is fulfilled. The suppression of the transition amplitude can be interpreted as a result of quantum interference between transitions from states |1> and |2> to state |3> under the influence of the exciting optical field (Alzetta et al., 1976; Arimondo, 1996; Lounis & Cohen-Tan­ noudji, 1992). An atomic sample initially in a thermal state will develop coherences when illuminated with a bichromatic field satisfying the resonance con­ dition. This can be thought of as an optical pumping effect: atoms in the “bright” (or “coupled”) state |C> will be excited to level |3> and will eventually fall into the dark state, where they no longer interact with the optical field. Population therefore builds up in the dark state and the absorption of the optical field (and fluorescence from the atomic sample) is reduced. As a function of two-photon detuning, we therefore observe a resonance in the absorption/fluorescence signal. In the data shown in Figure 2, CPT resonances are observed in the fluorescence spectrum of a cell subjected to a magnetic field gradient. CPT resonances are also frequently observed in the transmission spec­ trum of light passing through an atomic sample in the presence of a uniform magnetic field, as shown in Figure 3. In this case the frequency difference between the two excitation fields is scanned over the hyperfine transition frequency and a spectrum containing a number of absorption resonances can be observed as a function of difference frequency, corre­ sponding to transitions between different pairs of Zeeman-split hyperfine levels. In the case of circularly polarized light and a longitudinal mag­ netic field, only transitions between Zeeman levels with DmF = 0 are observed, resulting in a spectrum consisting of 2I lines for an atom with nuclear spin I. For atoms with half-integer nuclear spin, the central line (corresponding to mF = 0 ! mF = 0) occurs at a frequency close to the f1

f2

λ /4 Bichromatic light source

Full transmission

Magnetic field 87Rb

vapor cell

Photo detector f1-f2

Figure 3 Coherent population trapping resonance spectrum observed in the transmitted light through a vapor cell subject to a uniform magnetic field

Advances in CPT for Atomic Clocks

27

zero-field hyperfine splitting of the atom. The other lines are separated from the central line by even multiples of the Larmor frequency fL = B0, where  is the gyromagnetic ratio of the atoms. Atoms with higher nuclear spin have more Zeeman levels, and the CPT spectrum consists of correspondingly more lines. If the magnetic field is rotated so it has a transverse component, the CPT spectrum displays additional lines half­ way between the DmF = 0 lines corresponding to DmF = 1 transitions. Transitions with DmF = 2 can also be excited. For a more complete description of the CPT spectrum, see Wynands and Nagel (1999) and Knappe (2001). The bichromatic light needed to excite the CPT resonance can be generated in a number of ways. The two most common methods are direct modulation of the injection current of a laser diode (Cyr et al., 1993) and phase-locking of two lasers (Nagel et al., 1998). Direct modula­ tion is simpler to implement but results in an optical spectrum typically consisting of more than two frequencies with spectral amplitudes that are difficult to control. The use of injection-locked lasers requires more com­ plicated locking electronics but results in only two optical frequencies and allows a high degree of freedom in controlling the relative polariza­ tion and amplitude of these components.

2. ATOMIC CLOCKS 2.1 Introduction Most atomic clocks are based on alkali atoms (in particular H, Rb, Cs), which have a single valence electron. In these atoms, the energy spectrum is relatively simple, and long-lived ground states result in slow relaxa­ tion, narrow transition line widths, and correspondingly high precision. In addition to charge, both the atomic nucleus and the valence electron of all alkali atoms have spin angular momentum, and therefore a magnetic moment. Microwave frequency references are based on hyperfine transi­ tions between atomic states that differ in the relative orientation of the nuclear and electron magnetic moments. This difference in energies is on the order of 1—10 GHz, when translated into frequency units by dividing by Planck’s constant h. With some notable exceptions (Post et al., 2003; Taichenachev et al., 2005b), microwave atomic clocks are based on transitions between the magnetically insensitive mF = 0 substates of different hyperfine mani­ 6 0 (but DmF = 0) require a folds. Clocks based on states for which mF ¼ careful simultaneous measurement of the local magnetic field in order to prevent variations in this parameter from resulting in variations of the clock frequency. While this latter approach is not altogether prohibitive, it

28

Vishal Shah and John Kitching

adds significant complications to the operation of the device and has not found widespread popularity. On the other hand, optical clocks based on fermionic alkaline earth atoms have a small linear Zeeman shift that is effectively and routinely removed with appropriate interrogation techni­ ques (Akatsuka et al., 2008). The measurement of transitions between these atomic states can be accomplished in several ways. Perhaps the simplest measurement method is the passive excitation method, where an oscillating magnetic field is applied to the atoms at a frequency corresponding to the energy difference. When the frequency of the applied field is close to the fre­ quency of the atomic transition, an oscillating moment (coherence) is excited in the atom. Most frequently, magnetic dipole moments are excited. This coherence allows energy to be transferred from the field (atom) to the atom (field), and changes the internal state of the atom. Because this energy transfer is a resonant effect, the internal state of the atom can be monitored to determine when the frequency of the applied field corresponds to the energy splitting of the atomic states being coupled. A typical atomic frequency reference can be thought of as a series of steps. The atoms are first prepared in one specific atomic state (by magnetic state selection, optical pumping, or some other means). The oscillating magnetic field, generated by a “local oscillator” (LO), is then applied, causing some fraction of the atoms to change their state; this fraction depends on whether the frequency of the oscillating field is onresonance with the atoms. Finally the number of atoms in the final state (or the initial state) is detected, again by optical or magnetic means. Because of the resonant nature of the interaction, the number of atoms in the final state depends on the difference between the frequencies of the oscillating field and the atomic transition, and a measurement of this quantity can therefore be used in a feedback loop to lock the frequency of the oscillating field to the atomic transition frequency. The output of the clock is simply the frequency of the locked LO. The operation of a basic passive atomic frequency reference is shown in Figure 4. Atomic clocks based on alkali atoms can be divided into four main categories. Fountain clocks (Clairon et al., 1991; Kasevich et al., 1989; Zacharias, 1953), the most accurate atomic clocks at present, are large devices that often take up the better part of an entire room and require several hundred watts of power. There exist perhaps 10 such instruments worldwide and each typically takes several person-years to construct and evaluate. Hydrogen masers (Gordon et al., 1954), highly stable over long time periods, are about the size of a large filing cabinet. Cs beam clocks (Essen & Parry, 1955; Ramsey, 1950), based on beams of alkali atoms in a vacuum, are also highly accurate and are manufactured in rack-mounted enclosures. Vapor cell atomic clocks are based on atoms confined in a cell with a buffer gas (Arditi, 1958; Carver, 1957; Dicke, 1953) or wall coating

Advances in CPT for Atomic Clocks

(a)

29

(b)

LO

Number of atoms in final state

E2

E2 E1 State preparation

E1

State

detection

Transition excitation

LO frequency

Figure 4 (a) The operation of a passive atomic frequency reference typically proceeds in three steps. First the atom is prepared in some energy state, E1. The frequency from the local oscillator is then applied, causing transitions to another state with energy E2. The number of atoms in the final state is detected. (b) With this method, the frequency of the local oscillator can be determined with respect to the atomic transition

(Goldenberg et al., 1961; Robinson et al., 1958). The highest-performance vapor cell clocks are manufactured for installation on GPS satellites. These clocks are typically stable to 1013 or better over long periods but are intrinsically accurate (without calibration) only to about 109. Com­ pact vapor cell atomic clocks, developed for the telecommunications industry, can be held in the palm of one hand and are stable to about 1011. 2.2 Vapor Cell Atomic Clocks A schematic of a traditional Rb vapor cell frequency reference (Vanier & Audoin, 1992) is shown in Figure 5. The heart of the frequency reference is a vapor cell that contains the Rb vapor, along with an appropriate density of buffer gas. The vapor cell is contained within a microwave cavity into which a microwave field is injected. The microwave field is Physics package 87Rb lamp

Microwave cavity

85Rb

filter cell

Local oscillator

87Rb

vapor cell

Photo detector

Control system

Figure 5 Schematic of the major components of a traditional vapor cell frequency reference

30

Vishal Shah and John Kitching

generated by a LO, which is usually based on an electromechanical resonator (such as a quartz crystal resonator). Because the LO is an electromechanical device, it is typically rather unstable over long periods; the atoms therefore provide a stable reference frequency to which the LO can be locked. The atoms in the cell are illuminated by light generated by a Rb lamp. The Rb lamp is a second glass cell, containing Rb, through which an RF discharge is excited. Light from the lamp, with appropriate filtering, serves to prepare the atoms in the reference cell in one of the two hyperfine ground states via optical pumping. Because of this optical pumping, the atoms in the reference cell become less absorbing than they would in a thermal dis­ tribution. As the frequency of the RF field is tuned to near the atomic resonance, the population distribution in the reference cell changes again as the hyperfine populations are returned closer to their thermal distribu­ tion. This in turn increases the atomic absorption, and the increased absorption can be detected by monitoring the optical power transmitted through the cell with a photodiode. The transmitted optical power, as a function of microwave frequency, therefore becomes the “signal” to which the LO is locked. 2.3 Coherent Population Trapping in Atomic Clocks CPT was first used in atomic clocks in the early 1980s (Ezekiel et al., 1983; Hemmer et al., 1983a, 1984; Thomas et al., 1981, 1982). In these experi­ ments, modulated dye lasers were used to excite microwave transitions in a Na beam atomic clock. In effect, CPT zones replaced the microwave cavities employed in a conventional beam clock based on Ramsey’s method of separated oscillatory fields: in the first CPT zone the atomic coherence was created, and in the second, its phase was compared to that of the drive signal. Although initial investigations focused on the hyperfine transition in Na at 1.77 GHz, the use of optical fields to excite the coherence opened the door to the possibility of exciting atomic coherences in fre­ quency bands far beyond the gigahertz range. A Ramsey zone separation of 15 cm led to a resonance line width of 2.6 kHz (see Figure 6), and a corresponding frequency instability of 8  1010 at 1 second was measured, as shown in Figure 7. A subsequent experiment using a Cs atomic beam, excited by a modulated diode laser, demonstrated resonance widths of 1 kHz and a projected instability of 6  1011 at 1 second (Hemmer et al., 1993). The signal-to-noise ratio was about 10 times worse than that pre­ dicted by photon shot noise and was limited by frequency noise on the diode laser being translated into intensity noise on the measured atomic fluorescence. The conversion of FM to AM noise continues to be an important source of instability in the current generation of laser-pumped atomic clocks (Camparo & Coffer, 1999; Kitching et al., 2000).

Advances in CPT for Atomic Clocks

(a)

31

520 kHz

(b) 43 kHz (c) 1.3 kHz

ω2 Figure 6 (a) and (b) Rabi fringes from a Na atomic beam excited by a modulated dye laser. (c) Raman–Ramsey fringes with a width of 2.6 kHz. Reprinted figure with permission from Thomas et al. (1982);
1982 of the American Physical Society

10−6 10−7 10−8 σy (τ) 10−9 10−10 10−11 10−12 −3 10 10−2 0.1

1 10 102 103 104 τ (seconds)

Figure 7 Allan deviation of a frequency reference based on Raman–Ramsey excitation of CPT resonances in a Na atomic beam. Reprinted figure with permission from Hemmer et al. (1983b);
1983 of the Optical Society of America

These early experiments not only demonstrated the viability of the use of CPT in atomic frequency references but also identified the major limitations to both the short-term frequency stability and the accuracy. Sources of frequency instability common to many types of CPT frequency references include the FM—AM noise conversion mentioned above, atom and photon shot noise, and instability arising from the light shift (Hemmer et al., 1989). Sources of frequency instability associated specifically with the

32

Vishal Shah and John Kitching

Raman—Ramsey scheme include misalignment of the beams from a copro­ pagating configuration, birefringence, and polarization differences between the beams and changes in the optical path length (Hemmer et al., 1986). As mentioned above, semiconductor lasers have been used for CPT excitation in atomic beam clocks (Hemmer et al., 1993). The advantages of semiconductor lasers over dye lasers in this type of experiment are clear: smaller size and simpler operation. In addition, it is possible to modulate the optical field output of the laser by directly modulating the injection current. Resonances of width 1 kHz were obtained in a 133Cs beam by use of an edge-emitting AlGaAs diode laser modulated at 4.6 GHz. The two coherent first-order sidebands created the 9.2-GHz frequency difference needed to excite the Raman—Ramsey fringes. In 1993, Cyr, Tetu, and Breton described a method for exciting and detecting CPT resonances in an alkali vapor cell by use of a single diode laser (Cyr et al., 1993). The details of the experiment are shown in Figure 8. The injection current of the laser was modulated near the sixth subharmo­ nic (1.139 GHz) of the 87Rb hyperfine frequency (6.835 GHz), creating sidebands on the optical carrier, several of which are separated by approxi­ mately the atomic resonance frequency. When one of these sideband pairs was tuned to be in optical resonance with the atomic transitions, and their (a)

(c)

νeg

νef

e

1.0 m=0

f

νo νo

g

(b)

fm

fm

νI

S

P LD

0.8 Intensity Id [a.u.]

νef νeg

Probe p Pum

θ

AN PD

m = −1

m=1

0.6 0.4 0.2 0 −60 −40 −20

0

20

40

60

Detuning of f m [kHz]

λ /4

Synthesizer

Figure 8 Excitation of CPT resonances in an alkali vapor cell with a modulated diode laser. (a) The atomic energy level spectrum and optical frequency spectrum of the modulated diode laser. The CPT resonance is excited when the frequency splitting,  0 = nfm, between two components of the diode laser optical spectrum is equal to the atomic ground-state hyperfine splitting,  eg- ef. (b) Experimental setup. LD, laser diode; P, polarizer; S, solenoid; AN, polarization analyzer; PD, photodetector; /4, quarterwave plate. (c) The photodetector signal as a function of the detuning of the laser modulation frequency from the sixth subharmonic of the atomic resonance frequency. Reprinted figure with permission from Cyr et al. (1993);
1993 IEEE

Advances in CPT for Atomic Clocks

33

difference frequency (determined by the modulation frequency) made equal to the microwave transition, a microwave coherence was excited in the atoms. This coherence was detected by monitoring the polarization rotation of a probe beam derived from the same laser (Figure 8c). This idea is particularly noteworthy in the context of miniaturized frequency references, because all components used in the experiment are small. The use of a vapor cell lends itself well to miniaturization compared to previous CPT experiments based on atomic beams because much smal­ ler cell sizes can be used to achieve a given short-term frequency stability, due to the presence of the buffer gas. Some time later it was demonstrated that a single laser beam could also be used, allowing for even further simplification of the experimental setup (Levi et al., 1997). Since then, there has been considerable study of CPT-excited vapor cell frequency references. The noise processes in these instruments have been identified (Kitching et al., 2000) and short-term instabilities as low as 1.3  1012/H have been demonstrated in large-scale systems (Zhu & Cutler, 2000). Vapor cell CPT atomic clocks have been investi­ gated in some detail theoretically (Vanier et al., 1998, 2003a,c,d) and experimentally (Godone et al., 2002d; Knappe et al., 2001, 2002; Levi et al., 2000; Merimaa et al., 2003; Stahler et al., 2002). They have also been compared both theoretically (Vanier, 2001b) and experimentally (Lutwak et al., 2002) to conventional optically pumped vapor cell refer­ ences with the conclusion that the short-term frequency stability of CPTbased instruments should be comparable to or better than conventional ones. A review of atomic clocks based on CPT is given in the work of Vanier (2005). In 2001, a compact physics package for CPT frequency reference was demonstrated by Kitching et al. (2001a,b). This device had a volume of about 14 cm3, and the short-term stability of this device was 1.31010/ H. A photograph of the device is shown in Figure 9a, and the CPT resonance and Allan deviation are shown in Figure 9b and c, respec­ tively. Similar work was being explored simultaneously (Delany et al., 2001; Vanier, 2001a), connected with the ultimate development of com­ pact, commercial CPT clocks (Deng, 2008; Vanier et al., 2005). These early efforts to use CPT to miniaturize atomic frequency standards used glass-blown vapor cells with a diameter of several millimeters or more. The cells were assembled as discrete components with a laser, optics, and photodetector to form the functioning physics package. Complete miniaturized CPT frequency references for use in a com­ mercial setting have also been demonstrated (Deng, 2008; Vanier et al., 2004, 2005). This work demonstrated integration of a compact CPT phy­ sics package with a low-power LO and compact control electronics. The opportunities for atomic clock miniaturization afforded by the use of CPT led to some significant developments related to use of

34

Vishal Shah and John Kitching

(a) Laser Lens

Attenuator

Waveplate

Magnetic shielding

Photodetector

(b)

(c) 1.003 1.002

620 Hz at 4.6 GHz

σ y (τ)

Change in absorption

Cell

1.001

10−10

10−11 1.000 −1000

−500

0

500

1000

Frequency offset from 4.6 GHz (Hz)

100 101

102

103

104

105

τ (seconds)

Figure 9 A miniaturized physics package for a CPT frequency reference. (a) Photograph of the instrument with major components identified. (b) CPT resonance and (c) fractional frequency instability (Allan deviation) as a function of integration period. Reprinted figure with permission from Kitching et al. (2001b)

micromachining processes in atomic clocks. A preliminary analysis suggested that CPT clocks based on millimeter-scale vapor cells could achieve short-term frequency instabilities of a few parts in 1011 at 1 second of integration (Kitching et al., 2002). While the stability was expected to be worse than that of their larger counterparts, it was recognized that these micromachined or “chip-scale” atomic clocks could serve an important role in providing precise timing for portable, battery-operated instruments. Because of the small size, the power required to maintain the cell at its operating temperature could be drastically reduced. When combined with similar improvements in power resulting from the use of a laser, rather than a lamp, as the light source, operation with small batteries could be envisioned. A review of chip-scale atomic frequency references can be found in the work of Knappe (2007). Because CPT played an important role in many of these new micromachined clock designs, considerable research was carried out to improve and optimize CPT techniques specifically for this new development.

Advances in CPT for Atomic Clocks

35

2.4 Stability of Vapor Cell Atomic Clocks The stability of an atomic clock is most often characterized by its Allan deviation. The Allan deviation (Allan, 1966; Barnes et al., 1971), devel­ oped to quantify the fluctuations of nonstationary random variables, is a measure of the frequency instability of the clock obtained after integrat­ ing a measurement of its frequency for a period  and is denoted y(). For passive atomic clocks, in which white frequency noise is the domi­ nant noise source, the Allan deviation is given by y ðÞ ¼

pffiffiffi ; QðS=NÞ 

ð4Þ

where Q is the Q-factor of the atomic resonance,  is the integration period, S/N is the measurement signal-to-noise ratio (in units of HHz), and is a constant of order unity related to how the resonance is mea­ sured. The frequency instability is hence proportional to the resonance line width, or relaxation rate, of the atoms and inversely proportional to the signal strength. In a conventional optical-microwave double-resonance (OMDR) fre­ quency reference, the optical transmission is high when the microwave field is tuned away from resonance and decreases on resonance as the atoms are repumped into an equilibrium state by the microwave field (see Figure 10a). The OMDR resonance is characterized by its width W and its transmission contrast A/B, according to Figure 10a. The CPT resonance, shown in Figure 10b, has low transmission when the frequency difference between the optical fields is off resonance; the transmission increases on resonance when the dark state is populated. The resonance can be characterized by its width and its absorption con­ trast, denoted by W and CA = A/B in Figure 10b in the limit that the quantity B << 1. This last condition occurs when the cell temperature and length are such that the cell is optically thin for a weak optical field tuned to the center of the optical resonance. It implies that the quantity CA is a characteristic of a single atom and the way it is excited and does not (b) Full transmission

W

A B

Raman detuning, δr

1

Transmission

Transmission

(a)

Full transmission

W

A B 1

Raman detuning, δr

Figure 10 (a) Parameterization of conventional OMDR resonance. (b) Parameterization of CPT resonance. See text for explanation

36

Vishal Shah and John Kitching

depend on the atom density, the longitudinal location within the cell, or the effects of propagation of light through the (possibly optically thick) atomic sample. The width of the resonance, W, is determined by relaxation processes associated with the hyperfine coherence. There are many such processes, but the ones that dominate in vapor cell atomic clocks are collisions of alkali atoms with the cell walls, collisions with buffer gas atoms, colli­ sions with other alkali atoms, optical power broadening due to the pre­ sence of the optical field, and the relative phase stability of the excitation optical fields. This width can be minimized through the correct choice of buffer gas pressure and optical intensity. An analysis of these effects under some assumptions can be found in the work of Kitching et al. (2002). Considering still optically thin vapor cells, we may associate the signal S in Equation (4) with the change in optical power as the modulation frequency is moved from off the CPT resonance to on resonance. The fractional (off-resonance) absorption B can be written as   B ¼ 1  e  n0 L » n0 L; ð5Þ where n is the density of atoms, 0 is the (unpolarized) optical absorption cross section, and L is the length of the cell. The signal S can therefore be written S ¼ APin ¼ CA n0 LPin ;

ð6Þ

where Pin is the optical power incident on the cell. We therefore see that the signal is proportional to the column density of atoms, the incident optical power, and the absorption contrast. The typical absorption con­ trast for a CPT resonance is in the range of 0.1—10%; the physical effects that limit this contrast are discussed below. As the cell temperature is increased and the optically thick regime is approached, the assumption B << 1 above breaks down. In this regime, propagation effects must be considered, and propagation-induced nar­ rowing of the resonance and modification of the contrast occurs (Godone et al., 2002d). Experimentally, it has been found that for centimeter-scale cells, the optimal cell temperature is that for which the optical absorption (away from the CPT resonance) is approximately 0.5 (Knappe et al., 2002). At lower atomic densities, the signal is reduced due to the dependence described in Equation (6) (because very few atoms contribute toward the signal), while at higher densities, the strong optical absorption by the atomic sample attenuates the signal. While a more complete treatment of propagation effects may uncover ways to improve the clock performance at high optical densities, these remain unclear at present. Ultimately, increasing the atomic density will lead to increased hyperfine relaxation due to alkali—alkali spin-exchange or spin-destruction collisions, and any

Advances in CPT for Atomic Clocks

37

improvements gained by the higher atom number may be offset by collision-induced broadening of the resonance line width. The noise on the signal comes from a number of sources. Of these, photon shot noise and atom shot noise (spin projection noise) are the most funda­ mental. In most cases the photon shot noise is larger than the atom shot noise for measurements of hyperfine resonances; the power spectral density function of this (optical power) noise is given by (Yariv, 1997) SDP  N2 ¼ 2hvð1  BÞPin » 2hvð1  n0 LÞPin ;

ð7Þ

with the last approximation again relying on the assumption that B << 1. Additional noise sources include (a) AM noise on the laser, (b) FM laser noise, converted to AM noise by the atomic absorption profile (Camparo & Coffer, 1999), and (c) noise in the detection electronics. Often it is possible to reduce these additional sources of noise to a level close to the photon shot noise through the use of feedback techniques or intrinsically low-noise lasers. The signal-to-noise ratio can be expressed in terms of the CPT resonance parameters above as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S ð1  n0 LÞPin ¼ CA n0 L : ð8Þ N 2h Expressed in this way, the signal-to-noise ratio is proportional to the absorption contrast and the square root of the input power, but has a somewhat complicated dependence on the alkali atom density and cell length. To simplify this problem, the signal-to-noise ratio can be expressed instead in terms of the transmission contrast CT = A/(1 – B) and the output power Pout = (1 – B) Pin: rffiffiffiffiffiffiffiffiffiffi S Pout ¼ CT : ð9Þ N 2h In this expression, all of the effects of the alkali density are contained within the parameters Pout and CT, which can be measured experimen­ tally in a direct manner. We see therefore that the characterization of the CPT resonance in terms of the absorption contrast CA allows for a density-independent measure of the CPT resonance parameters, while the characterization in terms of the transmission contrast CT allows for a simple evaluation of the signal-to-noise ratio and clock stability. Both measures of the CPT resonance contrast are used in the literature, depending on the context of the discussion. We therefore find that there are three major routes to improving the short-term stability of the atomic clock: narrowing the resonance line width, increasing the resonance contrast, and reducing the noise. In the sections below, we describe a number of improvements and

38

Vishal Shah and John Kitching

modifications to the conventional CPT excitation scheme that result in some combination of reduced line width and increased signal contrast. These methods have been developed mostly within the last 10 years and represent a significant enhancement of the understanding of CPT, as applied to atomic frequency references. 2.5 Light Shifts Shifts in the atomic energy levels due to the presence of optical fields, known as AC Stark shifts or light shifts, are a major source of long-term instability for atomic clocks (Arditi & Carver, 1961; Kastler, 1963; Vanier & Audoin, 1992). The use of CPT provides some additional complications, but also opportunities with respect to the light shift, compared to con­ ventional clocks based on OMDR. The light shift of a two-level atom illuminated by a monochromatic optical field is (Cohen-Tannoudji et al., 1992)

L ¼

1 O2 D ; 4 ðG=2Þ2 þ D2

ð10Þ

where  is the Rabi frequency of the optical field—atom interaction, D is the detuning of the optical field from the atomic resonance, and G is the radiative decay rate of the atom. The shift is therefore proportional to the intensity of the optical field and traces a dispersive profile as a function of detuning from the atomic resonance. For the light intensities typically used in conventional OMDR clocks, the magnitude of the shift is on the order of a few parts in 109 (Arditi & Carver, 1961), making it a significant contribution to the instability of this type of clock at the level of 1011 for 1% changes in the light intensity. In a system exhibiting CPT, the interplay between the coherence of the atoms and the bichromatic nature of the light field leads to changes in the properties of the light shift. The most dramatic of these changes is the complete elimination of the light shift on Raman resonance for a perfect three-level system illuminated by a light field in which the two spectral components of the CPT field have equal Rabi frequencies. This elimination is a consequence of the fact that a perfect coherent dark state does not interact dissipatively with the light field and hence does not experience a Stark shift. For unequal intensities, the Stark shift has been calculated to be equal to (Arimondo, 1996; Kelley et al., 1994; Vanier et al., 1998)

L ¼

  2 1 D 1  22 ; 2 2 4 ðG=2Þ þ D

ð11Þ

where 1 and 2 are the Rabi frequencies associated with the two spectral components. The fact that the light shift can in principle be eliminated is

Advances in CPT for Atomic Clocks

39

of high interest to the application of CPT to atomic clocks, as this major source of long-term instability can potentially be eliminated. However, as we will see below, the complications associated with the multiplicity of levels, Doppler broadening, and other effects present in real atomic systems limit the extent to which the cancellation of the light shift can be accomplished in practice. Measurements of the light shift in CPT systems were described by Nagel et al. (1999) and Zhu and Cutler (2000), and a magnitude of the light shift of approximately 107/(mW/cm2) was obtained for cells with a roughly optimal buffer gas pressure. This shift is comparable to that obtained in conventional OMDR systems. The shift was found to be roughly linear in the CPT intensity and was smaller for higher buffer gas pressures.

3. ADVANCED CPT TECHNIQUES In its simplest form, CPT can be excited in atoms with only three energy levels by use of a single pair of coherent light fields. In principle under these assumptions, and with sufficient light intensity, atoms can be optically pumped into a coherent dark state with an efficiency of nearly 100%. Optically pumping a very large fraction of the atoms into a coher­ ent dark state even under ideal conditions requires the optical pumping rate to be much greater than the ground-state relaxation rate. This results in significant power broadening and is not ideal for clock operation, as it reduces the resonance Q-factor. Typically, the optical intensity for normal clock operation is chosen such that the power broadening rate is roughly equal to the ground-state relaxation rate; under these circumstances, about 50% of the atoms are optically pumped into the coherent state. Laboratory CPT experiments are usually carried out with alkali atoms, especially when the application is microwave atomic clocks. The presence of additional energy levels in these atoms can interfere with the efficient excitation of the coherent dark states and thereby reduce the contrast of CPT resonances. As discussed above, the reduced contrast results in higher instability for the clock. In addition, unwanted energy levels introduce asymmetries in the CPT resonance lineshape (Post, 2003) and produce AC Stark shifts, both of which adversely affect the atomic clock performance by producing timedependent frequency shifts of the clock output. Some of these effects are further amplified if a light field with more than two modes is used for CPT excitation. Over the last several years, a number of advanced interrogation techniques have been developed with the goal of mitigat­ ing some of these effects and thereby improving various aspects of the atomic clock performance.

40

Vishal Shah and John Kitching

The following section is not an exhaustive review of the literature but seeks instead to provide an overview and the general direction of many of the approaches. 3.1 Contrast Limitations due to Excited-State Hyperfine Structure The existence of a magnetic moment in the nucleus of all stable alkali atoms creates hyperfine structure not only the in the ground S1/2 state but also in the P states to which the S state is coupled by the optical fields. The frequency separation of these excited-state hyperfine states (500 MHz) is often smaller than the homogeneous broadening of the optical transition resulting from the buffer gas (2 GHz for a buffer gas pressure of 10 kPa). As a result, the optical fields used to excite the CPT resonance typically couple to several excited-state hyperfine levels simultaneously. While some of the transitions help optically prepare atoms in the coherent dark state, other transitions can simultaneously depopulate the dark state through incoher­ ent optical pumping of atoms out of the dark state. For example, in 87Rb, the F’ = 0 and F’ = 3 excited-state hyperfine levels in the 5P3/2 state are coupled to only one of the two ground states, due to single-photon selection rules. As a result, a coherence between the two ground states cannot be generated by these levels via CPT, but single-photon transitions out of the CPT state to these levels are allowed (Nagel et al., 2000). In addition to incoherent optical pumping out of the dark state, another important destructive mechanism occurs when multiple Lambda systems are formed between a common pair of ground states and different excited states. As described in the Equation (2), the phase of the dark state is governed by the complex optical Rabi frequencies that depend on both the phase of the optical fields and the coupling coefficients between the light and atomic energy levels. When multiple Lambda systems are excited between a common pair of ground states, the existence of an overall dark state in the system is not guaranteed and depends on the relative phase between the dark states of the individual Lambda systems. If the individual dark states are out of phase, then the strength of the overall CPT resonance can be reduced or even eliminated completely in the case of perfect destruc­ tive interference (Nagel et al., 2000; Stahler et al., 2002). Consider, for example, transitions that are excited on the D1 and the D2 line of 87Rb between mF = 0 ground states and the excited states by use of circularly polarized light fields exciting þ transitions (see level diagram in Figure 11a). In addition, we assume that the buffer gas pressure in the cell is high enough that the broadening of the optical transitions is larger than the excited hyperfine splitting. On the D1 line, there are two Lambda systems and no single-photon transitions. The two Lambda systems are and

!

jF ¼ 1; mF ¼ 0 > ! jF 0 ¼ 1; mF ¼ 1 >

jF ¼ 2; mF ¼ 0 > ;

Advances in CPT for Atomic Clocks

F′=3

(a)

41

(b)

F′=2 F′=1 F′=2

νopt

F′=1 νopt F=2

F=2

δr

Δhfs

F=1

δr

Δhfs

Transmission

Transmission

F=1

Raman detuning, δr

Raman detuning, δr

Figure 11 Simplified energy level diagram of 87Rb, showing the (a) D2 and (b) D1 transitions. The gray lines show the original three-level CPT model, while the dark lines show the additional transitions due to the presence of the excited-state hyperfine structure. The solid lines indicate CPT transitions, while the dashed line indicates a single-photon transition

!

jF ¼ 1; mF ¼ 0 > ! jF 0 ¼ 2; mF ¼ 1 >

jF ¼ 2; mF ¼ 0 > :

The transition amplitudes are such that the individual dark states are in phase and therefore add constructively (Stahler et al., 2002). On the D2 transition, the same two Lambda systems are again excited. However, due to the different Clebsch—Gordon coefficients, the phases of the two dark states are quite misaligned, which results in a partial destruction of the CPT resonance. For example, apart from a constant phase factor on the D2 87Rb line on the mF = 0 ground states, the dark state for F = 1 ! F’ = 2 F = 2 is jNCi ¼ p1ffiffi2 ðj1i þ j2iÞ, and for the F = 1 ffi ð5j1i þ j2iÞ. On the other ! F’ = 1 F = 2 Lambda system, jNCi ¼ p1ffiffiffi 26 hand, on the D1 line, the dark state is given by jNCi ¼ p1ffiffi2 ðj1i þ j2iÞ for both Lambda systems. In addition, on the D2 line there is one single-photon transition that depopulates the dark state: !

!

jF ¼ 2; mF ¼ 0 > !jF 0 ¼ 3; mF ¼ 1 > : The combined influence of these two effects is that the strength of the CPT resonance on the D2 transition is significantly smaller than that on the D1 transition.

42

Vishal Shah and John Kitching

CPT resonance contrast

0.10 A

0.08 0.06 0.04

B

0.02 0.00 −8

−6

−4

−2

0

2

4

6

8

10

Detuning (kHz) Figure 12 Comparison of CPT resonances excited using light resonant with the D1 (Trace A) and D2 (Trace B) optical transitions in 85Rb. An improvement in the absorption contrast by a factor of 10 is obtained by use of the D1 resonance. The line width of the D1-excited resonance is also narrower. Reprinted figure with permission from Stahler et al. (2002);
2002 of the Optical Society of America

The influence of additional energy levels on CPT resonances becomes clearly evident when CPT resonances using D1 and D2 transitions are compared by use of the atomic vapor cell (Zhu, 2002). It is found experimentally that CPT resonances excited by use of the D1 transition are almost an order of magnitude stronger than those seen by use of the D2 transition (Lutwak et al., 2003; Stahler et al., 2002). A comparison of CPT resonances obtained using D1 and D2 excitation, but otherwise under similar conditions, is shown in Figure 12. The discrepancy between the strengths of the CPT resonances on the D1 and D2 transi­ tions is in contrast with microwave resonances seen in conventional optically pumped clocks, in which efficient optical pumping can be achieved by use of both D1 and D2 transitions. The reason CPT resonances are so sensitive to excitation pathways is that CPT reso­ nances rely equally on both optical pumping and quantum interference. This is contrasted with conventional microwave resonances that rely only on optical pumping. 3.2 Contrast Limitations due to Zeeman Substructure The multiplicity of Zeeman levels present in the ground states of alkali atoms is another primary limitation to the signal strength. In the pre­ sence of a small magnetic field, the ground-state hyperfine levels are Zeeman split into (2F þ 1) magnetic ground states. In atomic clocks, a small magnetic field is typically applied to separate the various ground

43

Advances in CPT for Atomic Clocks

states so that the magnetically insensitive mF = 0 ground states can be uniquely interrogated without interference from the magnetically sen­ sitive transitions. However, the presence of the unwanted but closely spaced mF 6¼ 0 states also influences the strength of the CPT resonance. In thermal equilibrium, atoms populate all of the ground-state mag­ netic sublevels with roughly equal probability, as shown in Figure 13a. In atomic clocks, the optimized light intensity used for CPT excitation is generally weak enough that it causes some, but not significant, redistribution of atomic population between the various magnetic sub­ levels. The useable CPT resonance signal (quantity A in Figure 10b) is generated by atoms that are in the mF = 0 states, but many atoms in states with mF 6¼ 0 contribute to absorption of the incident light (quan­ tity B in Figure 10b). The absorption contrast is therefore reduced by roughly the ratio of the number of mF = 0 ground states to the total number of ground states, under the assumption that the light intensity is weak enough that it does not significantly redistribute the level populations. Another issue that is frequently discussed is optical pumping loss to the “trap state” (Renzoni & Arimondo, 1998; Vanier et al., 2003b). When

(a)

(b)

νopt

νopt

F=2

δr

F=2

Δhfs

δr

F=1 Transmission

Transmission

F=1

Raman detuning, δr

Δhfs

or

Raman detuning, δr

Figure 13 (a) Reduction of the CPT resonance contrast due to the presence of groundstate Zeeman structure. An even distribution of atoms among the Zeeman groundstate sublevels implies that only a small fraction of atoms contribute to the magnetically-insensitive mF = 0 ! mF = 0 transition. (b) Effects of the “trap” state (population indicated by the dark bar), which does not form a CPT resonance and into which atoms are pumped by the optical fields

44

Vishal Shah and John Kitching

circularly polarized light resonant with the D1 transition is used to excite CPT resonance, for example, a significant fraction of the population can be pumped to the magnetic end states (F = I þ 1/2, mF = + F). As seen in Figure 13b, the magnetic end state is an incoherent state, which is dark due to selection rules which prohibit excitation by circularly polarized light fields (þ transitions in this case). Atoms can therefore be “trapped” in this end state (or even in other states, depending on the laser polariza­ tion) and stop interacting with the light fields. Because the optical con­ figuration described above is used in current generation of microwave CPT clocks, the problem of the trap state has received significant attention from the research community. This configuration in which a Lambda system is excited in the presence of an incoherent dark state is often referred to as an open Lambda system. The atoms that are trapped or lost to the end state contribute neither to the CPT resonance signal nor to the optical absorption. They are simply invisible to the incident light fields. To compensate for this loss of atomic population, alkali density can be increased, but this also increases relaxa­ tion through alkali density dependent decoherence mechanisms such as spin exchange (Happer, 1972), which increases the line width of the CPT resonance. There have been a number of solutions that have been pro­ posed to depopulate the end state. Some of the approaches are outlined below. 3.2.1 Depopulation Pumping Using -Polarized Light This idea involves use of an additional linearly polarized laser light resonant with the F = I þ 1/2 ground state and the F’ = I þ 1/2 excited state (Kazakov et al., 2005b). The linearly-polarized light travels in a direction perpendicular to the applied magnetic field such that it excites -transitions (DmF = 0). This secondary light field is used to depopulate the end state but does not depopulate the atomic population in the F = I þ 1/2, mF = 0 ground because of selection rules that prohibit such a transition. While this technique works well in theory, there are several difficulties associated with using it in a practical device. Besides the technical diffi­ culties arising from the use of separate laser beams traveling in perpen­ dicular directions, this technique requires selective excitation of the F = I þ 1/2 ground state by use of the F’ = I þ 1/2 excited state. This limits the amount of buffer gas that can be used in the vapor cell to approximately below 1 kPa in Rb and Cs such that the levels in the excited state can be clearly resolved. This technique therefore presents difficulties for use with smaller cells, which typically use higher buffer gas pressures to avoid broadening of the hyperfine transition due to wall

Advances in CPT for Atomic Clocks

45

collisions. In addition, the narrow optical line width resulting from the low buffer gas pressure requires tighter restrictions on the amount of laser drift and laser frequency noise that can be tolerated. 3.2.2 Excitation with Orthogonal s-Polarized Light Fields Another approach that has been proposed to reduce the atomic population in the unused Zeeman levels is simultaneous excitation of the CPT resonance by use of a combination of þ and — light fields. This technique addresses both the thermal population in all Zeeman levels and the population build-up in the trap state due to optical pumping. A linear light field traveling along the magnetic field is the simplest example of þ and — light fields. The transitions that are excited by a combination of þ and — light fields are shown in Figure 14. The þ and — light fields independently excite Lambda systems on the mF = 0 states. As can be seen from the figure, there are no end states that are dark in this optical configuration. Unfortunately, simply using linearly polarized light to excite a closed Lambda system between mF = 0 levels does not work. The reason for this is that the individual dark states excited by the þ and the — light are out of

νopt

F=2

δr

Δhfs

Transmission

F=1

Raman detuning,δr Figure 14 Excitation of CPT resonances by use of a combination of sþ and s– light fields. Destructive interference can be avoided by independently adjusting the phase of the modulation in each polarization component. Solid lines indicate optical fields with sþ polarization that excite CPT transitions, while dashed lines indicate optical fields with s– polarization. Dotted lines indicate depopulation pumping of “trap” states

46

Vishal Shah and John Kitching

phase. In other words, the atoms that appear dark to the þ polariza­ tion appear bright to the — polarization, and vice versa. The result of this destructive interference is that the overall CPT resonance is not observed. One solution to this problem is to introduce time delay between þ and the — light fields such that the individual dark states constructively add together in phase. This principle has been successfully demonstrated in two ways. Jau et al. (2004a), Taichenachev et al. (2004b), and Karga­ poltsev et al. (2004) have proposed splitting the light fields into two parts with orthogonal polarization and recombining the fields after introdu­ cing a relative path delay between the þ and — components. The relative path difference of half a microwave wavelength (ground-state hyperfine splitting) shifts the phase of the dark state such that the reso­ nances constructively interfere. This path length difference can be intro­ duced by use of polarization filtering in a copropagating geometry or by reflecting the light back through the cell. A significant gain in the CPT contrast was reported by Jau et al. (2004a). The difficulty in using additional optics in splitting and recombining the light field after introducing the path delay led to development of another similar approach (Shah et al., 2006b). In this approach two separate lasers were used to avoid the difficulty in splitting and recom­ bining the light fields. Each of the two independent lasers had opposite circular polarizations, and they independently excited CPT resonance on the mF = 0 ground states. The lasers were modulated by use of the same microwave source, and an electronic microwave phase shifter was inserted into the RF path to one laser to shift the relative phase of the microwave modulation on the two lasers by . This technique works well and can also be implemented in a miniature package; however, the use of two separate lasers adds some complexity to the overall implementation and to the control system in particular. Another strategy is to generate a coherence on the mF = 0 ! mF = 0 transition using one polarization and measure the resonant change in birefringence with a weak optical field with an orthogonal polarization (Zhu, 2003). 3.2.3 CPT Excitation on DmF = 2 Transitions A third approach that has been proposed involves the direct use of linearly polarized light and excitation of CPT resonance between mF = þ1 and mF = 1 (Taichenachev et al., 2005b). This scheme, often referred to as "lin // lin" since two optical fields with parallel linear polarization are used, is by far the simplest way of exciting a closed Lambda transition by use of a combination of þ and — light fields. The transitions that are excited are shown in Figure 15. The relative phase between the dark states excited between mF = þ1 ! mF = –1 and mF = –1 ! mF = þ1 is no

Advances in CPT for Atomic Clocks

47

νopt

F=2 Δhfs Δ hfs

δr

Δhfs

Transmission

F=1

Raman detuning, δr Figure 15 levels

Excitation of CPT resonance on DmF = 2 transitions between hyperfine

longer of concern, because the dark states are excited between two independent sets of ground states. In this scheme, the transitions have an interesting dependence on magnetic field. Because of the slightly different g-factors for states in each of the two hyperfine levels, there is a small first-order magnetic field sensitivity for each of the mF = þ1 ! mF = –1 and mF = –1 ! mF = þ1 transitions. In 87Rb for example, individual mF = +1 states are shifted by over 300 kHz in 50 mT magnetic field. However, the difference frequency between the states jF ¼ 2; mF ¼ þ1Þ and jF ¼ 2; mF ¼ 1Þ shifts by only a few hundred hertz because of their much smaller linear magnetic field dependence. In addition, the sign of this residual linear sensitivity is negative for one transition (jF ¼ 2; mF ¼ þ1i $ jF ¼ 2; mF ¼ 1i) and positive for the other (jF ¼ 2; mF ¼ 1i $ jF ¼ 2; mF ¼ þ1i). As a result, when the resonance is excited using both pairs of ground states simulta­ neously, the linear dependence on magnetic field of the center point of the resonance vanishes. The presence of a small magnetic field therefore produces only a broadening of the overall resonance. Kazakov et al. (2005a) have proposed using this mechanism to produce a pseudoreso­ nance by applying a magnetic field large enough that the resonances on the individual pairs of ground states are shifted such that there is a dip in the center to which an LO can be stabilized. The main perceived drawback of this technique is that it was predicted to be effective only in 87Rb (or other atoms that have a nuclear spin of 3/2) atoms but not in 133Cs (which has nuclear spin 7/2). Atoms with

48

Vishal Shah and John Kitching

larger nuclear spin have a greater number of levels in the excited state, which cause additional Lambda systems to be excited that destroy the CPT resonance through destructive interference. This is due to the fact that it requires selective excitation of the Lambda system by use of the F’ = 1 excited state. This requires the use of very low buffer gas pressures, possibly in combination with a wall coating to reduce the wall-induced relaxation. Interestingly, high-contrast resonances have been observed using a similar scheme in a cell containing 133Cs and a low buffer gas pressure (Watabe et al., 2009). This experimental result suggests that the excitation of multiple Lambda systems affects the resonance contrast only modestly. 3.2.4 The Use of End Resonances The diluted atomic population participating in the clock transition can be improved by use of optical pumping, for example, with -polarized light, as mentioned in Section 3.2.1 above. Ideally, the population accumulates in the mF = 0 states, which are first-order magnetically insensitive. How­ ever, population can also be pumped into the “end” mF = F states, which can then be used to measure the hyperfine frequency. Because transitions between end states are first-order sensitive to magnetic fields, the local magnetic field must be measured simultaneously in a precise manner in order to reduce the field dependence of the clock output frequency. This can be done by measuring the Zeeman resonance frequency simulta­ neously with the hyperfine end-resonance frequency. An additional advantage gained by the use of end resonances is the suppression of spin-exchange broadening. At high alkali atom densities, spin-exchange collisions can be the dominant source of hyperfine relaxa­ tion (Walter & Happer, 2002). An atomic sample perfectly polarized in the end state does not undergo spin-exchange relaxation because all atoms are oriented in the same direction, and therefore no angular momentum can be exchanged between any two colliding atoms. However, a small population in other states creates some relaxation, and hence only a partial suppression of the spin-exchange relaxation can be achieved in real atomic systems. A final advantage of this scheme is that atoms can be optically pumped even at very high buffer gas pressures where the optical transi­ tions from the ground-state hyperfine levels are broadened far beyond the state frequency splitting. The traditional OMDR configuration has considerably degraded performance in the presence of high buffer gas pressure, because the hyperfine optical pumping is very inefficient. This proposal and accompanying experiments in 87Rb are described by Jau et al. (2004b) and shown schematically in Figure 16. Microwave excitation of the hyperfine transitions was used in the experiment, as opposed to CPT transitions, although the enhanced contrast and narrow

Advances in CPT for Atomic Clocks

49

(a) νopt

F=2

ω RF ω mw

Δhfs

ωL ωL

F=1 (c)

3ω L

Δhfs Microwave frequency, ω mw

Transmission

Transmission

(b)

ωL

RF frequency, ω RF

Figure 16 “End-resonance” method for increasing the absorption contrast and decreasing the line width of CPT resonances. (a) Atoms are optically pumped into the “end” state with maximal angular momentum, resulting in high transmission of the pumping light through the cell. A microwave field w mw and RF field w RF are applied simultaneously and the pump light transmission monitored as a function of (b) w mw and (c) w RF, providing a simultaneous measurement of w L and Dhfs –3 w L (for 87Rb)

line width should be equally present in CPT resonances. Line width suppression by a factor of about three was measured, as was considerably enhanced signal contrast (Post et al., 2003). Simultaneous measurement of a magnetically sensitive hyperfine transition frequency and Larmor pre­ cession frequency by use of a “tilted 0-0 state” was demonstrated by Jau and Happer (2005). When the system was operated as an atomic clock, a short-term instability of 6  1011/H was obtained in a compact physics package (Braun et al., 2007). 3.2.5 Amplitude-Modulated Versus Frequency-Modulated CPT Excitation Sources CPT resonances are often excited in alkali vapor cells with dimensions on the order of 1 cm. In order to prevent rapid relaxation of the hyperfine coherence due to wall collisions, a buffer gas is usually added to the cell. The buffer gas species can be chosen so that its effect on the hyperfine relaxation rate is rather small. The optimal buffer gas pressure varies roughly as the inverse of the smallest cell dimension and balances colli­ sion-induced relaxation with relaxation caused by residual diffusion

50

Vishal Shah and John Kitching

through the buffer gas to the walls of the cell (Kitching et al., 2002; Knappe, 2007). The buffer gas also significantly broadens the optical transitions involved in the CPT resonance. As the buffer gas pressure is increased, the optical transitions from the ground-state hyperfine levels can go from being completely resolved to being completely unresolved. When a single modulated laser is used to excite the resonances, the number of modula­ tion sidebands that interact with the atoms can vary from two to many. In Figure 17a, the spectrum from the optical transitions is plotted in these

Transmission

Optical Spectrum

Δhfs

(a)

A B

Optical frequency

Saturation S

(b) 100 Ideal saturation AM Ideal saturation FM AM data FM data Computer model fit

10−1

10−2 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Buffer gas Pressure p (atm at 21°C) Figure 17 (a) Comparison of atomic optical absorption spectrum with laser modulation spectrum for two qualitatively different buffer gas pressures. Trace A is for a low buffer gas pressure, for which the optical transitions from each hyperfine level are well resolved, while Trace B is for a higher buffer gas pressure, for which the transitions are not resolved. For the case of Trace A, only two of the frequencies in the optical spectrum interact with the atoms, while for the case of Trace B, almost all do. (b) Experimental data comparing the strength of the CPT resonance (identified with the saturation parameter S) as a function of buffer gas pressure for FM- and AMmodulated light fields. Reprinted figure with permission from Post et al. (2005);
2005 of the American Physical Society

Advances in CPT for Atomic Clocks

51

two limits and compared with the spectrum of a light field modulated at one-half the hyperfine splitting. When the transitions are resolved and only two optical field frequencies interact with the atoms (Trace A in Figure 17a), a dark state is created with a phase defined by the relative phase of the two relevant optical fields, and the absorption contrast can be quite large. However, when the transitions are not resolved, many optical frequencies interact with the atoms (Trace B in Figure 17), and the dark state tries to adjust its phase to correspond to the phase of each pair of sidebands separated by the hyperfine splitting. If the light source is amplitude modulated, the beatnote between each pair of sidebands has the same phase and the dark states created by each pair independently add constructively to form a single dark state with high contrast. If, on the other hand, the light source is frequency modulated, some pairs of sidebands are out of phase with the other pairs, and the dark states add destructively. For the high buffer gas pressures required for small vapor cells, therefore, the modulation properties of the light source are critically important. A careful study of this phenomenon is presented by Post et al. (2005), largely supporting the intuitive reasoning presented here. Data from Post et al. (2005) are shown in Figure 17b. 3.3 High-Contrast Resonances Using Four-Wave Mixing One of the issues associated with microwave CPT atomic clocks is that the transmission contrast of the CPT resonance is typically small (in the range of a few percent) when the atomic clock is fully optimized. This is due to several reasons, including the presence of modes generated by micro­ wave laser modulation that do not participate in the formation of CPT resonances. A very large fraction of the laser noise that affects the CPT clock performance can be eliminated by removing the background light. Shah et al. (2007) demonstrated a novel technique based on four-wave mixing in a double-Lambda system, shown in Figure 18a, to eliminate most of the background light falling on the photodetector. The experimental setup from Shah et al. (2007) is shown in Figure 18b. In this experiment, the þ light is used to create a dark state (coherence) in atoms by use of conventional CPT laser modulation. By use of a second single-frequency laser with opposite circular polarization, the coherence generated in the atoms is gently probed to stimulate emission of a con­ jugate light field whose frequency is separated from that of the original probe beam by the ground-state splitting. Through a combination of spectral and polarization filtering, the power in all of the incident light fields other than the conjugate field is then largely eliminated. When the pump beams satisfy the two-photon resonance condition, there is brightness observed on the photodetector from the incident conjugate field. When the two-photon condition is not satisfied, the conjugate field

52

Vishal Shah and John Kitching

(a)

⏐F ′ = 2, m f +1〉

⏐F ′ = 2, m f −1〉

52P1/2

Γ

σ−

Γ

Pump σ+ Beam-2

ΩP

D1

795 nm

Generated conjugate field ⏐F = 2, m f 〉 52S1/2

Ω

Ω

Pump Beam-1

γ

⏐F = 1, m f 〉 6.834 GHz (b) Beam-1 +

+

+

+

Beam-2 87Rb

λ/4

85

Rb

λ/4 Polarizar Photodetector Conjugate field power (nW)

Laser-2

Laser-1

20

Contrast: 100 × h /H = 90.3%

h H

A 0 −10 10 −5 5 0 Two photon detuning (KHz)

Figure 18 (a) Level diagram showing the polarizations and tunings of the optical fields for CPT resonance contrast enhancement using four-wave mixing and a filter cell. (b) Experimental setup and contrast measurement. From Shah et al. (2007); reproduced with permission from the Optical Society of America

is not generated, and therefore there is no light incident on the photo­ detector. Experimentally the transmission contrast seen in this way approaches 95% and is limited only by the efficiency of the optical filter­ ing in the setup. 3.4 Push–Pull Laser Atomic Oscillator Jau and Happer (2007) have demonstrated a novel technique in which they show a “mode-locking” type behavior in a laser cavity in the pre­ sence of alkali atoms. In this self-oscillating system, the frequency

Advances in CPT for Atomic Clocks

λ /4 (2) Vapor cell

(a)

Semitransparent mirror

Grain medium

λ/4 (1)

Alkali-metal atoms

53

Mirror

Photon spin of the laser light Time

Electron spin of alkali-metal atoms

Amplitude (a.u.)

(b) Fabry−Perot signal:

12

Without SPPP Single optical peak:

8

With SPPP An optical comb:

Generation of optical comb

4 0 –10

–5

0

5

Relative frequency (GHz) Figure 19 (a) Operation of the “push–pull” laser-atomic oscillator. (b) The comb of output frequencies spaced by the hyperfine frequency of the alkali atoms in the cavity. Reprinted figure with permission from Jau and Happer (2007);
2007 of the American Physical Society

separation of the spontaneously generated modes is given by the ground state mF = 0 ! mF = 0 transition frequency, introducing the prospects of operating the system as an atomic clock. The schematic of their experi­ mental setup is shown in Figure 19a. The basic idea is the following: imagine that all the atoms are initially prepared in a dark state between the mF = 0 hyperfine ground states. The atoms in the dark state precess at the hyperfine frequency and appear continuously transparent only to light fields that excite a Lambda system. Two or more optical modes that are separated by the hyperfine frequency and have a constant phase relation between them suitable for exciting a Lambda system are thus the “allowed” cavity modes in the system. Because of the fixed phase relationship between the optical modes, the cavity operation is analogous to that in a mode-locked laser. The result is that the light coupled out of the cavity has a frequency spectrum similar to that of a frequency comb (see Figure 19b).

54

Vishal Shah and John Kitching

Through the use of two /4 wave plates, the optical arrangement in the cavity was such that the light excites þ transitions traveling along one direction and — transitions when traveling in the opposite direction. The purpose of this was to prepare atoms in a pure superposition of mF = 0 states without the usual fraction in the end state. The length of the cavity was chosen to be an odd integral multiple of half the hyperfine wave­ length, such that the dark state due to oppositely traveling light fields remained in phase. This system has the important feature that no LO is needed to excite the resonance; the system here is an active system and oscillates at the hyperfine frequency. A related experiment was described by Akulshin and Ohtsu (1994), in which an alkali cell was placed in an external cavity providing optical feedback to a distributed feedback (DBF) laser. A second laser beam, separated in frequency by approximately the alkali ground-state hyper­ fine frequency, was sent through the cell parallel to the first laser beam. It was found that the laser with feedback optically locked to the second laser with a frequency difference exactly equal to the ground-state hyper­ fine splitting. CPT-induced polarization rotation has also been used in a similar context (Liu et al., 1996). A number of other self-oscillating systems based on CPT have been developed (Strekalov et al., 2003, 2004; Vukicevic et al., 2000), in which RF rather than optical feedback was used to sustain the oscillation. 3.5 The CPT Maser A CPT maser (Godone et al., 1999; Vukicevic et al., 2000) is an active frequency standard in which a coherent microwave signal is directly recovered from the atoms instead of an indirect signal in the form of change in optical transmission. This approach can also eliminate the need for an external microwave oscillator for laser modulation. Once the microwave oscillation is started, it can be sustained indefinitely by use of the microwaves obtained directly from the atoms in a feedback configuration. In a CPT maser, atoms are enclosed in a microwave cavity whose frequency is tuned close to the difference frequency between the mF = 0 hyperfine ground states (see Figure 20). The coherence generated in the atoms through dark-state excitation couples with one of the cavity modes to stimulate emission of microwaves by the atoms at the hyperfine frequency. The microwaves emitted by the atoms can, in turn, be used to modulate the laser to sustain the maser operation. A complete CPT maser prototype was demonstrated and evaluated and an instability of 31012/H was measured, integrating down to below 1013 at 1 hour, once the linear drift had been removed (Levi et al., 2004). A variety of noise contributions were also measured, with thermal

Advances in CPT for Atomic Clocks

55

B2 Pout Ptr

Lasers

rf Field Cesium cell & Buffer gas

P11

Microwave cavity

Figure 20 The CPT maser, in which a ground-state atomic coherence is generated using CPT and the power radiated by the corresponding magnetic moment, is captured in a microwave cavity. Experimental setup. Reprinted figure with permission from Vanier et al. (1998);
1998 of the American Physical Society

noise being the most important at short integration times and temperature-related effects dominating at long integration times. Considerable theoretical work was also carried out to understand the operation of the CPT maser in detail (Godone et al., 2000; Vanier et al., 1998), as well as a number of interesting independent phenomena related to its operation and underlying physics (Godone et al., 2002a,b,c,d). 3.6 N-Resonance A novel alternative to CPT, the N-resonance, was proposed (Zibrov et al., 2005) and studied subsequently (Novikova et al., 2006a,b). This scheme, which has its origins in earlier work on multiphoton resonances in alkali atoms (Zibrov et al., 2002), can be thought of as a modification of the conventional OMDR technique. Just as in the OMDR technique, atoms are optically pumped into one hyperfine level with an optical “probe” field resonant with a transition from the other hyperfine level. However, instead of exciting the microwave transition by use of a microwave field, a bichromatic optical field is used, close to Raman resonance with the microwave transition, but detuned from the optical transition. When the Raman resonance condition is achieved, atoms are optically pumped back into the depleted hyperfine level, leading to increased optical absorption of the pump field. This absorptive resonance is in contrast with the conventional CPT resonance, in which reduced absorption is seen when the two-photon condition is satisfied. Among the advantages of the N-resonance scheme is that unlike CPT resonance, this scheme produces signals of high contrast (as high as 30% transmission contrast) on both the D1 and the D2 transitions. Clock in stabilities of 1.5  1010/H have been obtained in 87Rb confined in a cell

56

Vishal Shah and John Kitching

of diameter 2.5 cm. It has also been shown that despite the inherently offresonant operation of N-resonance-based atomic clocks, the light shifts can be canceled to first order by appropriately choosing the intensity in the pump and the probe beams, allowing the possibility of building an atomic clock with good long-term stability based on N-resonances. This scheme still requires that the excited-state hyperfine resonances be resolved, and hence has the same limitations with respect to buffer gas pressure as some of the techniques discussed above. Although in its most general form requiring optical fields at three unique frequencies, this scheme can be implemented by use of only two optical fields by allowing a single field to do double duty, both as the probe field and as one of the legs of the Raman field. An energy-level diagram of the N-resonance excitation mechanism is shown in Figure 21a, and the experimental implementation using a single modu­ lated diode laser is shown in Figure 21b. One of the weak sidebands generated by the modulator serves as the probe and one of the Raman fields, while the strong optical carrier provides the second Raman field. An etalon is placed after the cell to attenuate the strong Raman field and therefore increase the resonance contrast. 3.7 Raman–Ramsey Pulsed CPT As described above, light shifts play a major role in determining the longterm stability of vapor cell atomic clocks. In CPT clocks, the presence of the off-resonant optical modes and an imbalance between the intensities in the two arms of the Lambda system can cause significant light shifts. A com­ monly used technique to avoid light shifts is to pulse the light fields and allow the atoms to evolve in the dark. An additional advantage of pulsing the light fields is that atoms can be prepared in a coherent superposition state with higher efficiency by use of strong light fields while avoiding power broadening to a large extent. A pulsed technique for CPTclocks has also been recently proposed (Zanon et al., 2004b, 2005) and demonstrated (Guerandel et al., 2007), and it has shown excellent both stability (Boudot et al., 2009) and a high degree of insensitivity to light shifts (Castagna et al., 2009). In this technique, the light fields are switched on and off at regular intervals. The operation of the clock can be understood as follows: during the first pulse, CPT light fields prepare atoms in a coherent dark state. After the state preparation is nearly complete, the light fields are turned off for a period roughly equal to the ground-state relaxation time. During this period, the atoms freely evolve at the ground-state hyperfine fre­ quency without being perturbed by the light fields. When the second light pulse is turned on, the hyperfine frequency of the atoms in the dark is inferred from the initial absorption signal of the light by the atoms. If the frequency of the microwave oscillator used to modulate the light fields is the same as that of the atomic hyperfine frequency (in the

57

Advances in CPT for Atomic Clocks

(a)

(b)

(a)

v0

Ωp

Ωp Ωp

Ωp

|b〉

lbg

Δv

A

Probe frequency

|b〉

v0

|c〉 Probe transmission

Probe transmission

|c〉

Ωp

Ωp

|b〉

v0

(c)

|a〉

|a〉

Δv A lbg

Ωp

v0

|c〉 Probe transmission

|a〉

Probe frequency

Δv

lbg A

Microwave frequency

(b) (a) Rb cell inside magnetic shielding Laser

EOM

Fabry−Perot etalon

PD

λ /4 Solenoid 6.835 GHz frequency synthesizer From H-maser

Slow frequency modulation (f m = 330 Hz)

Lock-In amplifier Output

Figure 21 (a) Atomic level diagram for the N-resonance. (b) Experimental implementation. Reprinted figure with permission from Zibrov et al. (2005);
2005 of the American Physical Society

dark), then the atoms appear transparent to the light fields the instant they are turned on. This is because the microwave modulation on the laser and the atomic evolution in the dark remain in phase; as a result, the atoms continue to appear transparent to the light fields. The later part of the second pulse repumps the fraction of the atoms that relaxed during the period when the light fields were turned off. The pulsed excitation scheme and atomic energy level diagram are shown in Figure 22a. Figure 22b shows experimentally observed Ramsey fringes when the CPTclock is operated in

58

Vishal Shah and John Kitching

Preparation pulse

(a)

Detection pulse

Free evaluation

τ1 |3〉

Δ0

Γ

|3〉

Ω32 Ω31

δR

Γ

Ω42

|2〉

γc

|4〉

Δ0 Ω32

δR

Ω91

τ2

T

Ω41 |2〉

γc

|1〉

|1〉

Normalized transmission

(b) 1024

1016

1008

1000

992 –30 –25 –20 –15 –10

–5

0

5

10

15

20

Raman detuning δR (kHz)

Figure 22 Raman–Ramsey excitation of hyperfine clock transitions. (a) Pulsed excitation scheme and atomic energy spectrum. (b) Raman–Ramsey pulses. Reprinted figure with permission from Zanon et al. (2005);
2005 of the American Physical Society

the pulsed mode. The individual fringe width can be narrower than the width of a zero-intensity continuously-excited CPT resonance if the delay between successive pulses is greater than the ground-state relaxation time. The pulsed CPT interrogation scheme has been operated as an atomic frequency reference and studied in some detail. It was shown that both the short-term frequency stability and the light shift could be improved considerably compared to continuous interrogation in the pulsed config­ uration (Castagna et al., 2007). A short-term instability of 71013/H was obtained, integrating down to 21014 at 1000 seconds when the linear frequency drift was removed. Dominant contributions to the

Advances in CPT for Atomic Clocks

59

short-term instability were amplitude noise on the laser and electronic noise in the photodetection system. Recently, “pulsed” vapor cell atomic clocks, conventional OMDR clocks in which the optical pumping field and microwave field are applied at different times, have also been the subject of some research (Godone et al., 2004, 2006a,b). In addition, there has been some considera­ tion of pulsed CPT excitation of cold atom systems in order to eliminate the buffer gas shifts present in vapor cell clocks (Farkas et al., 2009; Zanon et al., 2003, 2004a). A novel system based on transient excitation of a hyperfine coherence and feedforward to correct the LO frequency has also been investigated (Guo et al., 2009). 3.8 CPT in Optical Clocks So far we have focused on the role of CPT resonances in microwave clocks. In recent years, CPT-based atomic clocks operating in the optical or the terahertz regime have been proposed (Hong et al., 2005; Santra et al., 2005; Yoon, 2007). Optical clocks have the great advantage over microwave clocks that the transition frequencies, and hence the Q-factors of the resonances, are orders of magnitude higher. In order to obtain narrow line widths, forbidden transitions such as the intercombination lines in alkaline earth atoms are often used. Some of these transitions, such as the 1S0 $ 3P0 line at 698 nm in bosonic 188Sr, are forbidden to any order and cannot be accessed by use of single-photon excitation. It is, however, possible to observe the transition indirectly by use of twophoton (CPT) excitation as shown in Figure 23. The line width of the (b)

(a)

Γ31

Γ32 Δ2

Δ1

γc 2

Ω2 Ω1

3P = |2〉 0

γc1

γc 1S = |1〉 0

Percentage in ground-state population transfer

1P = |3〉 1

1.00

Δ0 / 2π = 80 MHz

0.95 0.90 0.85 0.80

Ω2 = Ω Ω1 = 15.16 Ω −6 −4 −2 0 2 4 6 Raman detuning δr /2π (Hz)

Figure 23 CPT excitation of intercombination transitions in alkaline earth atoms. (a) Atomic level structure and optical fields. (b) Predicted Raman lineshape. Reprinted figure with permission from Zanon-Willette et al. (2006);
2006 of the American Physical Society

60

Vishal Shah and John Kitching

transition can be conveniently tuned from megahertz to below 1 Hz by controlling the power broadening of this transition. It has been predicted that the accuracy of such a clock can be better than 1017. To eliminate light shifts, a pulsed CPT scheme similar to the scheme proposed for microwave clocks (see pulsed CPT section above) has been proposed and demonstrated (Zanon-Willette et al., 2006).

4. ADDITIONAL CONSIDERATIONS 4.1 Light-Shift Suppression Several interesting techniques have been developed to reduce the effects of light shift on the long-term instability of vapor cell atomic clocks. One such scheme involves the use of the multiplicity of sidebands created when the injection current of a diode laser is modulated (Zhu & Cutler, 2000). Current modulation of a diode laser results in both AM and FM modulation of the output optical field. As the modulation index is increased, a comb of optical frequencies is therefore produced, separated from the carrier by multiples of the modulation frequency. Each of these optical frequencies produces a light shift for each of the ground-state hyperfine frequencies, and each shift can be either positive or negative, depending on the detuning of the specific frequency from the relevant transition. By adjusting the modulation index, it is therefore possible to modify the light shift and reduce it to near zero. A calculation from Zhu and Cutler (2000) is shown in Figure 24a, indicating that for two different operating conditions, the first-order light shift is reduced to zero at a modulation index of about 2.5. Measurements shown in Figure 24b con­ firm the effect. A frequency instability of about 1013 was obtained at an integration period of 1000 seconds by use of this technique (Zhu & Cutler, 2000). It should be noted that, in principle, this same technique can be used to reduce or eliminate the light shift in OMDR clocks (Affolderbach et al., 2003). However, the required modulation of the laser injection current would have to be added to this configuration, while this is present quite naturally in the CPT configuration. The technique above allows for the modulation index to be set such that small changes in the light intensity do not (to first order) affect the clock frequency. Such changes in light intensity can occur, for example, if the laser generating the optical fields ages in some way. However, this aging can also result in a change in the electrical impedance of the laser. If the laser impedance changes, the coupling of the RF modulation field to modulation on the optical field in general changes. As shown in Figure 24b, a change in the coupled RF power by 1 dB can result in a frequency shift on the order of 1010.

61

Advances in CPT for Atomic Clocks

(b)

(a) 60

40

for CPT-generation

30 20 10

4

2 Δf/f (10−10)

Light shift (Hz)

50

Constant input intensity Constant Rabi frequencies

Microwave power 95 dBm (β-208) 105 dBm (β-208) 115 dBm (β-208)

0

−2

0 −10 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Frequency modulation index

−4 0

20

40

60

80

Laser beam intensity (μW/cm2)

Figure 24 Suppression of light shifts using optimized modulation of a diode laser. (a) Theoretical predictions of the light shift as a function of madulation index. (b) Measurements of the frequency shift as a function of laser beam intensity for several values of the modulation index. A local minimum is observed for  2.33 and I  40 m}go:goW/cm2. Operation of the clock at this setpoint should result in improved long-term frequency stability when light shifts dominate. Reprinted figure with permission from Zhu and Cutler (2000)

In order to address the effects of such changes in laser impedance, a modification of the RF modulation technique was suggested (Shah et al., 2006a). After adjusting the RF power to the zero-light-shift point, the power of the light field is modulated at a low frequency (17 Hz in this experiment) by use of a variable attenuator. This modulation in intensity will cause a corresponding change in the frequency of the CPT resonance if the zero-light-shift condition is not satisfied. Assuming that the synthesizer frequency is more stable than the CPT resonance at the modulation fre­ quency, the modulated frequency shift can be detected through the normal comparison of synthesizer frequency to CPT resonance frequency. The RF power can then be corrected to maintain the zero-light-shift condition. A schematic of the optical/electronic arrangement is shown in Figure 25. Under exaggerated conditions, an improvement in the insensitivity of the output frequency to RF impedance (as adjusted by modulating the laser temperature) by a factor of 10 is obtained (Shah et al., 2006a). 4.2 Laser Noise Cancellation One of the problems frequently encountered in laser-based atomic inter­ rogation is the conversion of laser frequency noise to current noise in the detector output by the optical resonance. The laser frequency noise, which otherwise cannot be seen by the photodetector, appears because the atoms act as a sharp discriminator of the optical frequency in the vicinity of the optical resonance. The amount of noise that appears on

62

Vishal Shah and John Kitching

Physics package LCD attenuator

B

Photo detector

VCSEL

Polarizer λ/4 Vapor cell

17 Hz Signal generator

RF-synthesizer

Frequency error signal

Lock-in 2

Amplifier

RF power error signal

Servo Experimental setup

Figure 25 Schematic of a technique to maintain the RF power at the zero-light-shift point as the laser impedance ages. Reprinted figure with permission from Shah et al. (2006a);
2006 of the Optical Society of America

the photodetector signal depends on the amount of frequency noise present on the light (the laser line width), the width of the optical resonance, and the tuning of the laser frequency. There are several solutions with varying complexity that can be used to reduce this excess laser frequency to photodetector current noise. Among the simplest is to choose a laser with a narrow line width. However, this is not always feasible for commercial or other technical reasons. Another simple alternative is to broaden the optical resonance using higher buffer gas pressure which reduces the slope of frequency discrimination by the optical resonance. Here again, there are limitations since buffer gas pressure cannot be arbitrarily increased without affecting the clock performance. For example, in the regime in which most CPT/OMDR clocks operate, one of the immediate consequences of using higher buffer gas pressure is that the optical depth of the atomic medium at a given temperature is correspond­ ingly reduced. To compensate for the loss in the optical depth, the atomic vapor pressure can be increased by increasing the cell temperature. But this increases the width of the ground-state resonance by increasing alka­ li—alkali spin-exchange contribution to the ground-state relaxation. Another alternative to reducing the laser noise is to use external means such as real-time laser noise cancellation using differential detection. This was accomplished, for example, by Gerginov et al. (2008). Here, a split wave plate was used such that light in one half of the vapor cell was circularly polarized and light in the other half was linearly polarized (see Figure 26). The circular and the linear components of the light beam were collected on two spatially separated photodetectors. While the CPT

Advances in CPT for Atomic Clocks

VCSEL

λ/4 σ +

L P ND G

B

PD1

lin PD2 Rb cell + shield

–

RF

63

To exp.

+

Figure 26 Experimental setup used to reduce the laser noise. VCSEL: L, lens; P, polarizer; ND, neutral density filter; /4, quarter-wave plate; G, glass plate sþ; PD, photodiode. Reprinted figure with permission from Gerginov et al. (2008);
2008 of IEEE

resonance is excited in only the first part of the vapor cell and thus seen only by PD1, the laser noise appears in both the channels. Subtracting the signals from the two photodetectors thus removes the laser noise without affecting the CPT resonance seen using PD1. Similar techniques have also been developed for OMDR clocks (Deng, 2001; Mileti et al., 1998; Rosen­ bluh et al., 2006). Yet another technique for laser noise cancellation was proposed and demonstrated by Rosenbluh et al. (2006). In this technique, the noise originating with the laser was reduced and the effects of optical pumping to the dark end state were simultaneously eliminated. A CPT resonance was excited using a combination of copropagating left and right circu­ larly polarized light obtained from a common laser. A relative path delay, equal to one quarter of the microwave wavelength, was introduced between the right and the left circular components of the light beam. After propagating the light through the vapor cell, the two components were separated using an arrangement of a quarter-wave plate (/4) and a polarizing beam splitter; each of the components was separately moni­ tored using photodetectors. Due to the /4 path delay between the two polarization components of the light beam that excite the CPT resonance, the phase of the coherent dark state that is excited by the two beams combined is partially shifted in phase with respect to the individual polarization components of the beam. This phase shift, which has equal but opposite signs for the two light components, introduces an asymme­ try into the CPT resonance lineshape by adding a dispersive component to an otherwise Lorenztian profile. While the CPT resonance seen by adding the signals from both photodetectors still has a purely Lorenztian lineshape (Figure 27a), the difference signal is purely dispersive (Figure 27b). Because differential detection is employed in the latter case, the common mode laser noise is removed, without any effect on the overall strength of the CPT resonance signal. It can be seen even from the oscilloscope traces of the CPT resonances that the trace in Figure 27b has lower noise compared to Figure 27a, in which differential detection is not employed.

Vishal Shah and John Kitching

0.665 (a) 0.660 0.655 0.650 0.645 0.640 0.635 0.630

–2 2 4 –4 0 Frequency detuning, δr [kHz]

0.015 Photodetector signal [V]

Photodetector signal [V]

64

(b)

0.010 0.005 0.000 –0.005 –0.010 –0.015

–2 2 4 –4 0 Frequency detuning, δr [kHz]

Figure 27 Lineshapes of atomic-coherence-induced resonances for phase-shifted, two-beam excitation. (a) One beam blocked, conventional CPT resonance, (b) both beams present, signals from balanced photodetectors subtracted. Reprinted figure with permission from Rosenbluh et al. (2006);
2006 of the Optical Society of America

4.3 Light Sources for Coherent Population Trapping A number of types of light sources have been used to generate the bichromatic optical field needed to excite hyperfine CPT resonances. Although the coherence requirements of the light source are not nearly as stringent as in optical spectroscopy, the lamps currently used in con­ ventional atomic clocks appear to be too incoherent to generate CPT resonances of any reasonable contrast. The requirement on the coherence is nominally that the line width of the light source be smaller than the buffer gas broadened optical transitions in the alkali atoms, typically ranging from a few hundred megahertz to several tens of gigahertz. Most lasers satisfy this requirement, which allows great latitude in laser choice to optimize the system with respect to other criteria. The earliest experiments on CPT were carried out using multimode dye lasers (Alzetta et al., 1976). Since then, a variety of more sophisticated light sources have been used, including acousto-optically modulated dye lasers (Thomas et al., 1982); injection-current-modulated edge-emitting diode lasers (Hemmer et al., 1993; Levi et al., 1997); phase-locked external cavity diode lasers (ECDLs) (Brandt et al., 1997; Zanon et al., 2005) or edge-emitting lasers (Zhu & Cutler, 2000); injection-current-modulated vertical-cavity surface-emitting lasers (VCSELs) (Affolderbach et al., 2000; Braun et al., 2007; DeNatale et al., 2008; Kitching et al., 2000; Lutwak et al., 2003; Serkland et al., 2007; Youngner et al., 2007); and electro­ optically modulated ECDLs (Jau et al., 2004a). Each of these light sources has relative merits and detriments. For example, VCSELs have very low threshold currents, making them ideal for low-power instruments based on CPT, but suffer from inflexibility with respect to the modulation side­ band spectrum. The spectrum of phase-locked ECDLs can be controlled

Advances in CPT for Atomic Clocks

65

very precisely, but the locking is difficult to implement experimentally. Acousto- and electro-optically modulated sources can have nearly perfect amplitude modulation with no associated phase modulation, but are large, cumbersome, and expensive. Novel lasers, developed for CPT experiments, include VCSELs in an extended cavity (Gavra et al., 2008), and very short edge emitting lasers with a low-reflectivity coating on one facet (Kargapoltsev et al., 2009). These lasers were developed to have high modulation bandwidths without the need for even a modest amount of output power (in many experiments 10 mW of optical power is sufficient to excite high-contrast resonances). 4.4 Dark Resonances in Thin Cells Considerable recent work has focused on the optical properties of alkali atoms confined in cells for which the longitudinal dimension is such that the transit time across the cell is shorter than the optical relaxation period (Briaudeau et al., 1996). In these cells, atoms in velocity classes perpendi­ cular to the cell walls do not build up appreciable optical coherence before the wall-induced relaxation and therefore do not contribute to the optical absorption. By contrast, atoms in velocity classes parallel to the cell walls do not collide with the walls as frequently and hence do build up coherence and exhibit corresponding absorption. Some recent work in this area has focused on the understanding of CPT resonances that are observed in these media (Failache et al., 2007; Fukuda et al., 2005; Petrosyan & Malakyan, 2000; Sargsyan et al., 2006). While the CPT line widths are typically very large (greater than 1 MHz), some work is proceeding to evaluate how these systems might be used in compact atomic clocks (Lenci et al., 2009). 4.5 The Lineshape of CPT Resonances: Narrowing Effects The simplest theories of the CPT resonance lineshape typically involve collisional or diffusion-induced relaxation processes or radiative relaxa­ tion. These relaxation mechanisms result in the typical Lorentzian line width found for most spectroscopic signals. However, there are a number of physical effects that occur in real experiments that can distort the CPT resonance lineshape. It has been found, for example, that if the transverse intensity distribution of the excitation light field is non-uniform, or if the beam diameter is smaller than the diffusion length of the atoms over the ground-state relaxation time scales, then a narrowing of the resonance near its center can result (Levi et al., 2000; Taichenachev et al., 2004a, 2005a) to form a pointed resonance lineshape. This lineshape can also be explained by diffusion-induced narrowing (Xiao et al., 2006, 2008) based on the Ramsey effect, in which atoms diffuse out and then re-enter the

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Vishal Shah and John Kitching

excitation beam before relaxing. Finally, propagation effects are known to cause modifications of the resonance line width compared to that observed in an optically thin medium (Godone et al., 2002d). The optical absorption coefficient is smaller when the CPT resonance condition is satisfied, and because the absorption in an optically dense medium is nonlinear as a function of propagation length, significantly more light may be transmitted when on resonance compared to when away from resonance, producing an artificial narrowing effect. However, it remains unclear whether these unusual lineshapes can be effectively used to improve the performance of a CPT atomic clock in some way.

5. CONCLUSIONS AND OUTLOOK After considering the many possibilities for improving CPT resonances for use in atomic clocks, it is perhaps important to ask to what extent these techniques have impacted the design and performance of actual devices. To some extent this question is premature, as it often takes considerable time for new knowledge, even if it allows clear performance improvements, to find its way into realized systems implemented in the laboratory or in a commercial setting. The cost, time, and risk associated with implementing a new technique to replace, for example, an already proven commercial instrument or an already-operating laboratory instru­ ment are often considered too high. In addition, until experiments are engineered at a level that their importance emerges, issues believed to be ultimately important, such as the effects of the light shift on the long-term stability of the clock, are often masked by other more technical effects, such as temperature-induced shifts. It is therefore often difficult to estab­ lish a clear measure of the improvements certain techniques will allow [see, for example, the work by Shah et al. (2006a)]. However, CPT as a whole has now been not just successful for labora­ tory instruments, but appears to be on the verge of commercial success (DeNatale et al., 2008; Deng, 2008; Lutwak et al., 2007; Vanier et al., 2005; Youngner et al., 2007). In particular, CPT has been shown to be the method of choice for microfabricated vapor cell frequency standards. This is interesting because comparisons of CPT techniques to conven­ tional OMDR techniques for vapor cell frequency references (Kitching et al., 2002; Lutwak et al., 2002; Vanier, 2001b, 2005) have suggested that there is no clear advantage to be gained through the use of CPT with respect to short-term stability. From a technical viewpoint, the biggest strengths of the CPT approach at present therefore appear to be that (a) the physics package design is simple to implement, and (b) that the considerable work over the last 10 years on miniaturized CPT frequency references has clearly established the technical viability of this approach.

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While it remains possible that highly miniaturized frequency references based on OMDR will ultimately be competitive with their CPT counter­ parts, considerable work is needed to develop the OMDR approach further for miniaturized devices. One additional, and perhaps over­ looked, advantage of CPT is that the resonances can be excited by use of modulation at a subharmonic of the hyperfine frequency. This allows the LO to be placed in close proximity to the physics package without having radiated RF power interfere with the atomic transition. Probably the most important improvement among the techniques described above has been the use of the D1 line rather than the D2 line to excite the resonances. The D2 line was used initially because of the availability of commercial diode lasers at the 852 nm D2 transition of Cs. The improvements gained by using the D1 line have, to some extent, motivated the development of new lasers, and it now appears that (a) the use of the D1 line is clearly superior with respect to the short-term stability and (b) that there is no significant disadvantage to this approach. Certain other design improvements focused on improving the resonance contrast, such as the end-resonance technique, push—pull optical pump­ ing, and the linear polarization techniques, continue to appear promising in principle, but work is needed to quantify the level of improvement in a real clock experiment. The additional system complexity and correspond­ ing impact on reliability will also be a factor in determining the extent to which these techniques will be used in real-world instruments. Techni­ ques to reduce the light shift, such as pulsed CPT and sideband spectrum engineering, are expected to be important for future generations of CPT clocks, for which the engineering has progressed to the point where technical limitations to the long-term instability have been suppressed. Whatever the outcome from an instrumentation perspective, it is clear that the understanding of CPT as it applies to atomic clocks has advanced considerably over the last decade.

ACKNOWLEDGMENTS We gratefully acknowledge valuable comments from S. Knappe and A. Post. This work is a partial contribution of NIST, an agency of the US Government, and is not subject to copyright.

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Strekalov, D., Savchenkov, A., Matsko, A., Yu, N., & Maleki, L. (2004, August 23—27). Alloptical atomic clock on a chip: progress report. Proceedings of the 2004 IEEE Interna­ tional Ultrasonics, Ferroelectrics, and Frequency Control 50th Anniversary Joint Con­ ference, Montreal, PQ. Piscataway, NJ: IEEE, 104—108. Taichenachev, A. V., Tumaikin, A. M., Yudin, V. I., Stahler, M., Wynands, R., Kitching, J., et al. (2004a). Nonlinear-resonance line shapes: Dependence on the transverse intensity dis­ tribution of a light beam. Physical Review A, 69(2), 024501. Taichenachev, A. V., Tumaikin, A. M., Yudin, V. I., Stahler, M., Wynands, R., Kitching, J., et al. (2005a). Nonlinear-resonance line shapes: Dependence on the transverse intensity distribution of a light beam (vol A 69, art non 024501, 2004). Physical Review A, 71(2), 029903. Taichenachev, A. V., Yudin, V. I., Velichansky, V. L., Kargapoltsev, S. V., Wynands, R., Kitching, J., et al. (2004b). High-contrast dark resonances on the D-1 line of alkali metals in the field of counterpropagating waves. JETP Letters, 80(4), 236—240. Taichenachev, A. V., Yudin, V. I., Velichansky, V. L., & Zibrov, S. A. (2005b). On the unique possibility of significantly increasing the contrast of dark resonances on the D1 line of Rb-87. JETP Letters, 82(7), 398—403. Thomas, J. E., Ezekiel, S., Leiby, C. C., Picard, R. H., & Willis, C. R. (1981). Ultrahighresolution spectroscopy and frequency standards in the microwave and far-infrared regions using optical lasers. Optics Letters, 6(6), 298—300. Thomas, J. E., Hemmer, P. R., Ezekiel, S., Leiby, C. C., Picard, R. H., & Willis, C. R. (1982). Observation of Ramsey fringes using a stimulated, resonance Raman transition in a sodium atomic-beam. Physical Review Letters, 48(13), 867—870. Vanier, J. (2001a). US Patent # 6,320,472. US Patent and Trademark Office. Vanier, J. (2001b, September 9—14). Coherent population trapping and intensity optical pumping: On their use in atomic frequency standards. Proceedings of the 6th Sympo­ sium on Frequency Standards and Metrology, St. Andrews, Scotland. Singapore: World Scientific, 155—166., September 9—14). Vanier, J. (2005). Atomic clocks based on coherent population trapping: A review. Applied Physics B, 81(4), 421—442. Vanier, J., & Audoin, C. (1992). The quantum physics of atomic frequency standards. Bristol: Adam Hilger. Vanier, J., Godone, A., & Levi, F. (1998). Coherent population trapping in cesium: Dark lines and coherent microwave emission. Physical Review A, 58(3), 2345—2358. Vanier, J., Godone, A., Levi, F., & Micalizio, S. (2003a, May 4—8). Atomic clocks based on coherent population trapping: Basic theoretical models and frequency stability. Proceed­ ings of the Joint Meeting 2003 IEEE International Frequency Control Symposium & PDA Exhibition and 17th European Frequency and Time Forum, Tampa, FL, 2—15. Vanier, J., Levine, M., Kendig, S., Janssen, D., Everson, C., & Delany, M. (2004, August 23—27).Practical realization of a passive coherent population trapping frequency standard. Proceedings of the 2004 IEEE International Ultrasonics, Ferroelectrics, and Frequency Control 50th Anniversary Joint Conference, IEEE, Piscataway, NJ, 92—99. Vanier, J., Levine, M. W., Janssen, D., & Delaney, M. (2003b). Contrast and linewidth of the coherent population trapping transmission hyperfine resonance line in Rb-87: Effect of optical pumping. Physical Review A, 67(6), 065801. Vanier, J., Levine, M. W., Janssen, D., & Delaney, M. J. (2003c). The coherent population trapping passive frequency standard. IEEE Transactions on Instrumentation and Measure­ ment, 52(2), 258—262. Vanier, J., Levine, M. W., Janssen, D., & Delaney, M. J. (2003d). On the use of intensity optical pumping and coherent population trapping techniques in the implementation of atomic frequency standards. IEEE Transactions on Instrumentation and Measurement, 52(3), 822—831. Vanier, J., Levine, M. W., Kendig, S., Janssen, D., Everson, C., & Delaney, M. J. (2005). Practical realization of a passive coherent population trapping frequency standard. IEEE Transactions on Instrumentation and Measurement, 54(6), 2531—2539. Vukicevic, N., Zibrov, A. S., Hollberg, L., Walls, F. L., Kitching, J., & Robinson, H. G. (2000). Compact diode-laser based rubidium frequency reference. IEEE Transactions on Ultra­ sonics Ferroelectrics and Frequency Control, 47(5), 1122—1126.

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Walter, D. K., & Happer, W. (2002). Spin-exchange broadening of atomic clock resonances. Laser Physics, 12(8), 1182—1187. Watabe, K., Ikegami, T., Takamizawa, A., Yanagimachi, S., Ohshima, S., & Knappe, S. (2009). High-contrast dark resonances with linearly polarized light on the D-1 line of alkali atoms with large nuclear spin. Applied Optics, 48(6), 1098—1103. Whitley, R. M., & Stroud, C. R. (1976). Double optical resonance. Physical Review A, 14(4), 1498—1513. Wynands, R., & Nagel, A. (1999). Precision spectroscopy with coherent dark states. Applied Physics B, 68(1), 1—25. Xiao, Y., Novikova, I., Phillips, D. F., & Walsworth, R. L. (2006). Diffusion-induced Ramsey narrowing. Physical Review Letters, 96, 043601. Xiao, Y., Novikova, I., Phillips, D. F., & Walsworth, R. L. (2008). Repeated interaction model for diffusion-induced Ramsey narrowing. Optics Express, 16(18), 14128—14141. Yariv, A. (1997). Optical electronics in modern communications. Oxford, UK: Oxford University Press. Yoon, T. H. (2007). Wave-function analysis of dynamic cancellation of ac Stark shifts in optical lattice clocks by use of pulsed Raman and electromagnetically-induced-transpar­ ency techniques. Physical Review A, 76(1), 013422. Youngner, D. W., Lust, L. M., Carlson, D. R., Lu, S. T., Forner, L. J., Chanhvongsak, H. M., et al. (2007, June 10—14). A manufacturable chip-scale atomic clock. Proceedings of the IEEE Transducers ’07 & Eurosensors XXI Conference, Lyon, France, 39—44. Zacharias, J.R. (1953). Unpublished. Described in N. F. Ramsey, Molecular beams (pp. 138). Oxford, UK: Oxford University Press, 1956. Zanon, T., Guerandel, S., De Clercq, E., Dimarcq, N., & Clairon, A. (2003, May 4—8). Coherent population trapping with cold atoms. Proceedings of the Joint Meeting 2003 IEEE Inter­ national Frequency Control Symposium & PDA Exhibition and 17th European Fre­ quency and Time Forum, Tampa, FL, 49—54. Zanon, T., Guerandel, S., De Clercq, E., Holleville, D., Dimarcq, N., & Clairon, A. (2004a). Coherent population trapping on cold atoms. Journal de Physique IV, 119, 291—292. Zanon, T., Guerandel, S., De Clercq, E., Holleville, D., Dimarcq, N., & Clairon, A. (2005). High contrast Ramsey fringes with coherent-population-trapping pulses in a double lambda atomic system. Physical Review Letters, 94(19), 193002. Zanon, T., Guerandel, S., De Clercq, E., Hollville, D., & Dimarcq, N. (2004b, April 5—7). Observation of Ramsey fringes with optical CPT pulses. Proceedings of the 18th Eur­ opean Frequency and Time Forum, Guildford, UK. Session 1B. Zanon-Willette, T., Ludlow, A. D., Blatt, S., Boyd, M. M., Arimondo, E., & Ye, J. (2006). Cancellation of stark shifts in optical lattice clocks by use of pulsed Raman and electro­ magnetically induced transparency techniques. Physical Review Letters, 97(23), 233001. Zhu, M. (2002). US Patent # 6,359,916. US Patent and Trademark Office. Zhu, M. (2003, May 4—8). High contrast signal in a coherent population trapping based atomic frequency standard application. Proceedings of the Joint Meeting 2003 IEEE International Frequency Control Symposium & PDA Exhibition and 17th European Frequency and Time Forum, Tampa, FL, 16—21. Zhu, M., & Cutler, L. S. (2000, November 28—30). Theoretical and experimental study of light shift in a CPT-based Rb vapor cell frequency standard. Proceedings of the 32nd Annual Precise Time and Time Interval (PTTI) Meeting, Reston, VA, 311—324. Zibrov, A. S., Ye, C. Y., Rostovtsev, Y. V., Matsko, A. B., & Scully, M. O. (2002). Observation of a three-photon electromagnetically induced transparency in hot atomic vapor. Physical Review A, 65(4), 043817. Zibrov, S., Novikova, I., Phillips, D. F., Taichenachev, A. V., Yudin, V. I., Walsworth, R. L., et al. (2005). Three-photon-absorption resonance for all-optical atomic clocks. Physical Review A, 72(1), 011801.

CHAPTER

3

Dissociative Recombination of H3þ Ions with Electrons: Theory and Experiment Rainer Johnsena and Steven L. Gubermanb a

Department of Physics and Astronomy, University of Pittsburgh,

Pittsburgh, PA 16260, USA

b Institute for Scientific Research, 22 Bonad Road, Winchester,

MA 01890, USA

Contents

1. 2. 3.

Introduction Basic Definitions Experimental Techniques 3.1 Afterglow Techniques 3.2 Single-Pass Merged-Beam and Ion-Storage

Ring Experiments 4. Theory 4.1 DR Mechanisms 4.2 H3þ Potential Curves and Surface 4.3 Vibrational and Rotational Considerations 4.4 One- and Two-Dimensional Theory 4.5 Three-Dimensional Treatments of H3þ DR 5. History of Experimental H3þ Recombination

Studies 6. Reconciling Afterglow and Storage Ring Results 6.1 Afterglow Measurements That Yielded

Very Low Recombination Coefficients 6.2 Afterglow Measurements That Yielded

High Recombination Coefficients 6.3 Third-Body Stabilized Recombination of H3þ 7. Comparison of Storage Ring Data 8. H3þ Product Branching 9. Isotope Effects 10. Conclusions Acknowledgments References

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Advances in Atomic, Molecular, and Optical Physics, Volume 59
2010 Elsevier Inc. ISSN 1049-250X, DOI: 10.1016/S1049-250X(10)59003-7 All rights reserved.

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76 Abstract

Rainer Johnsen and Steven L. Guberman

Four decades of experimental and theoretical studies of the dissociative recombination of the seemingly “simple” H3þ ions with electrons have often given strongly disagreeing results. The literature on the subject abounds in terms like enigma and puzzles, and several authors have asked if the “saga” is finally approaching a satisfactory ending. Fortunately, recent progress in theory and experiment has greatly reduced many of the apparent contradictions. In this review, we attempt to reconcile the remaining discrepancies, in particular those between beam experiments and those employing plasma afterglow techniques. We conclude that there are no true contradictions between those results if one examines the conditions under which the data were taken and includes effects arising from third-body-assisted recombination. The best available theoretical treatments of purely binary recombination now agree rather well with state­ of-the-art ion-storage ring results, but we think that further refinements in the complex theoretical calculations are required before it can be said that the mechanism of the recombination is understood in all details and that the “saga” has truly come to an end.

1. INTRODUCTION H3þ, the simplest of all polyatomic molecular ions, consists of three protons arranged in an equilateral triangle, held together by two elec­ trons. The physics and chemistry of this ion has occupied a special niche in the molecular physics community for many years, and it is a fair question to ask why it continues to be of interest today and what pro­ gress has been made in understanding its basic properties. The apparent simplicity of this ion makes it attractive as a test case for ab initio quantum-chemical calculations and that certainly has stimulated much theoretical work. A second important motivation comes from astrophy­ sics: H3þ is perhaps the second most abundant molecular species (after H2) in interstellar clouds, in the ionospheres of the outer planets, and plays a central role in determining the ionization balance and in building more complex ions that determine the physical properties in these starforming regions (Herbst, 2007; McCall, 2006; McCall et al., 2002). While the ion is quite stable, the relatively small proton affinity of H2 (4.2 eV) enables efficient proton transfer to other molecules. However, if H3þ ions recombine efficiently with electrons and dissociate into H2 and/or H atoms in the process, the same species from which they were formed by several slow steps, the reaction chain is essentially terminated, and recombination limits the rate of molecule formation. The effect of H3þ on

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the interstellar chemistry can be quite complicated and lead to bistable chemical evolutions, as has been discussed in detail by Pineau Des Foreˆ ts and Roueff (2000). In this review, we focus on the dissociative recombination (DR) of H3þ ions with electrons, a process that can be symbolically represented as  Hþ 3 þe !HþHþH ! H2 þ H

ð1Þ

Anticipating later discussions, we note that Equation (1) may be read either as representing an ion—electron binary collision or as a reaction equation that describes a more complex process in an ionized gas. We adopt the first interpretation but note that other electron—ion recombina­ tion mechanisms exist in which part of the energy released by recombi­ nation is transferred to third bodies (atoms, molecules, or other electrons) or is removed by emission of radiation. We will discuss such third-body­ assisted recombination only to the extent that it affects the interpretation of experimental data. All experimental studies of DR face the problem that two charged species, ions and electrons, must be brought together in a controlled manner with a small relative velocity. Theorists have an equally and perhaps even more difficult task. A slow electron that is captured by a molecular ion can give rise to numerous excited states of the molecule, and it requires extensive quantum mechanical calculations to decide which of those states eventually lead to dissociation. The task is further complicated by the fact that recombination is sensitive to the rotational and vibrational states of the ion and that the ion exists in two nuclear spin modifications, denoted as para-H3þ (two of the three proton spins aligned) and orho-H3þ (three proton spins aligned). As in other fields of physics and chemistry, experiment and theory sometimes have often given conflicting answers to some of the basic questions. For many years there was considerable doubt that efficient recombination of H3þ actually occurred! Many open and once difficult questions have been clarified in recent years by advances in theory and by new and powerful experimental techniques, especially ion-storage rings that supply more detailed information than the plasma-based experimental methods. Progress in theory has been commensurate with that in experiment: what Bates (1993), the “founding father” of DR, once described as an “enigma” has largely been solved, but some finer details may still need to be worked out. This review is intended to present a critical but not necessarily complete analysis of all experiments and theories. We seek to reconcile experiment and theory as far as possible given the current state of knowledge, and to see if remaining discrepancies are “real” in

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Rainer Johnsen and Steven L. Guberman

the sense that they indicate deficiencies in our understanding as opposed to incomplete or erroneous interpretations of experimental observations. The literature on DR is extremely large and H3þ is certainly not the only ion of interest. Several excellent reviews on DR in general have appeared in the last few years that include extensive lists of measured rate coefficients and other data (Florescu-Mitchell & Mitchell, 2006; Larsson & Orel, 2008). A previous review of H3þ recombination mea­ surements by Johnsen (2005) contains much additional material that we will not repeat here, and some proposed solutions of apparent contra­ dictions have now been ruled out by new experimental and theoretical work.

2. BASIC DEFINITIONS We begin by reviewing some basic definitions, most of which are com­ mon in the physics of atomic collisions, but others are specific to parti­ cular experiments and require a few words of explanation. Consider an ion that moves in a region containing uniformly distrib­ uted free electrons at density ne (cm3). The recombination coefficient
is defined by the probability dP that the ion captures an electron during time dt and dissociates before releasing it by autoionization, i.e., dP ¼
ne dt:

ð2Þ

This defines a “raw” or “effective” recombination coefficient that still depends on the distribution of the relative ion—electron speeds f(vrel). If the recombination is purely binary, one can define a recombi­ nation cross section (vrel), which is related to the rate coefficient by the average
¼ hðvrel Þvrel f ðvrel Þi;

ð3Þ

where the brackets indicate averaging over all vrel. If f(vrel) is sufficiently narrow to be reasonably approximated by a delta function centered at , the cross section is closely given by the ratio
: ð4Þ ðvrel Þ ffi hvrel i This approximation is fairly good in merged-beam experiments, but fails at very low vrel. For that reason merged-beam experimenters often report their raw results not as cross section but as a nonthermal recombi­ nation coefficient as a function of the “detuning energy.” However, they usually deconvolute the recombination coefficient to obtain the cross section, and then compute the thermal recombination coefficient by

Dissociative Recombination of H3þ Ions with Electrons

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convolving the cross section again with a Maxwell distribution. The deconvolution may require extrapolation to very low energies. Plasma afterglow experiments directly yield the thermal recombina­ tion coefficient, although often only over a narrow range of temperatures. Those results are typically given in the form of a power-law dependence
 Te  x
ðTe Þ ¼
ð300KÞ ; ð5Þ 300 as a function of the electron temperature Te. In such experiments, the ion translational ion temperature Ti is almost always the same as the gas temperature Tg, but Te can be greater than Ti. It can hardly ever be assumed that the internal degrees of freedom of the ions, especially their vibrations, are in thermal equilibrium at the translational temperature. Theoretical calculations usually generate cross sections for a set of discrete collision energies. To facilitate comparison to experiment, theor­ ists often calculate (a) the thermal rate coefficient and (b) an “effective” rate coefficient that should be measured in beam experiments with a finite energy resolution. The procedure “washes out” some of the finer structure in the theoretical cross section but, unlike the thermal rate coefficient, retains some of its structure.

3. EXPERIMENTAL TECHNIQUES The experimental techniques used to study DR can be divided into two broad categories, plasma afterglow experiments and merged-beam experiments. In afterglow experiments, electron—ion recombination rate coefficients and product yields are derived from observations of ion and electron densities, optical emissions, and neutral products during the afterglow phase of a plasma. The analysis of afterglow plasmas can be complicated by reaction processes that occur in addition to electron—ion recombina­ tion, and it also is not always obvious that recombination in a plasma involves only simple binary recombination. However, what is regarded as a “complication” in the context of recombination may be of great interest to the physics of ionized gases in general and this should be kept in mind. Merged-beam and ion-storage ring methods, while requiring far greater experimental effort, are closer to the theorists’ ideal experiment and can provide more detailed information. The outstanding progress that has been made in refining these techniques now permits studies with very high energy resolution as well as determinations of the chemical

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identity of neutral reaction products and their kinetic energy. Such data, of course, provide far more sensitive tests of theoretical calculations of recombination than the thermally averaged rate coefficients obtained by afterglow techniques. 3.1 Afterglow Techniques The two principal subcategories, “stationary” or “flowing” afterglows, have much in common, but they differ in the way that the plasma is produced and observed. We will discuss them together while pointing out relative strengths and weaknesses. In the stationary afterglow, more appropriately called a “pulsed” after­ glow, a plasma in a pure gas or gas mixture is created by repeated pulses of microwaves, high voltages applied to discharge electrodes, ultraviolet light, or other ionizing radiation (see Figure 1). All afterglow observations are carried out in the same volume as a function of time. In the flowing afterglow method (see Figure 2), a pure gas (helium most often) is first ionized, usually in a microwave discharge, and then flows at high speed down the flow tube and is eventually discharged into a fast pump. At some point, reagent gases are added that convert the primary ions and metastable atoms to the desired molecular ion species. Recombination occurs in the region downstream from the reagent inlet, and observations are carried out as a function of distance from the gas inlet. The flow tube method has the advantage of greater chemical flex­ ibility and it avoids exposing the molecular gases directly to an intense discharge, which can lead to undesired excitation or dissociation. It also has some disadvantages: There is only an approximate correspondence between time and distance since the gas flows faster at the center of the tube than it does near the wall and the spatial distribution of particles in the plasma is not necessarily uniform. Also, the mixing of gases at the reagent inlet is not instantaneous and this can complicate the data analysis. A frequently employed method to convert the active species flowing out of the discharge to ions consists of adding argon at a point upstream from the reagent inlet. This converts metastable helium to argon ions, which are subsequently used as precursors for the ion—molecule reactions that generate the desired ion species. What is often ignored is that along with the argon ions some undesired energetic particles and ultraviolet photons also enter the region downstream from the reagent inlet, for instance metastable argon atoms (see, e.g., Skrzypkowski et al., 2004) that are produced by collisional radiative recombination of argon ions. Ultraviolet photons, in particular “trapped” helium resonance radiation, can enter the reaction zone unless one adds a sufficient amount of argon to destroy them by photoionization of argon. Fortunately, such effects do

Dissociative Recombination of H3þ Ions with Electrons

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Pulsed microwave for plasma generation

Vacuum

Mass spectrometer Gas inlet

Langmuir probe

Figure 1 Schematic diagram of a stationary or pulsed afterglow apparatus. A Langmuir probe or a microwave frequency method is used to record the decay of the electron density subsequent to an ionizing pulse. Typical linear dimensions of the plasma chamber are 10–40 cm

Microwave discharge

Mass spectrometer

Reagent gas inlet

z

Langmuir probe

Figure 2 Schematic diagram of a flowing afterglow Langmuir probe apparatus (FALP). A movable Langmuir probe records the electron density as a function of distance from the reagent inlet

not interfere much with measurements of recombination coefficients, but they can be important in spectroscopic studies of reaction products. Different afterglow experiments employ different reaction sequences to produce H3þ ions. One frequently used scheme makes use of the fast two-step reaction sequence

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Rainer Johnsen and Steven L. Guberman

Arþ þ H2 ! ArHþ þ H þ 1:53eV

ð6Þ

ArHþ þ H2 ! Ar þ Hþ 3 þ 0:57eV

ð7Þ

followed by þ

which releases sufficient energy to produce H3 in vibrational states up to v = 5. If argon is present in sufficient concentration, subsequent proton transfer to Ar, þ Hþ 3 ðvÞ þ Ar ! ArH þ H2 ;

ð8Þ

þ

destroys all H3 ions with internal energies above 0.57 eV, leaving only those in the ground state [A1 (0, 00)], in the v2 = 1 bending-mode vibration [E (0, 11) at 0.3126 eV], and in the v1 = 1 breathing-mode vibra­ tion [A1 (1, 00) at 0.394 eV]. The radiative lifetime of the v1 = 1 state is very long (1.2 s). Radiative decay of ions in the v2 = 1 level is faster (4 ms), but does not necessarily occur at the time scale of recombination measurements. The electron density can be measured by several methods: Langmuir probes return local values of ne, while microwave methods have low spatial resolution and yield a “microwave-averaged electron density.” The flow tube has the significant practical advantage that the gas is exchanged rapidly, on a time scale of milliseconds. In stationary after­ glows, outflow of gases occurs only through the small sampling orifice used for mass spectrometric sampling of ions, but the gas exchange time is usually on the order of many minutes. For this reason, impurity problems tend to be less serious in flow tubes than in stationary afterglows. The methods to measure recombination coefficients are essentially the same in both types of afterglows. In the simplest case, when only a single ion species is present and the plasma is quasi-neutral, e.g., ne = ni, the electron continuity equation is given by @ne ðt; ! rÞ r Þ; ð9Þ ¼ 
n2e ðt; ! r Þ þ Da r2 ne ðt; ! @t where Da is the ambipolar diffusion coefficient of the ion. If diffusion is sufficiently slow that it can be ignored, the reciprocal electron density varies with time as 1 1 Þ ¼ Þ þ
t; r ne ð0; ! r ne ðt; !

ð10Þ

and hence the recombination coefficient can be obtained directly from the slope of a graph of the measured reciprocal electron densities as a func­ tion of time. This simple form of analysis yields reasonably accurate

Dissociative Recombination of H3þ Ions with Electrons

83

recombination coefficients only if the diffusion current of ions into or out of the volume in which ne is measured is very small compared to the volume loss rate of electrons due to recombination. A frequently used, but not entirely satisfactory, approximation “corrects” for the diffusion loss of electrons by fitting the observed electron density decays to an equation of the form Da ne dne ðtÞ ¼ 
n2e ðtÞ  ; dt 2

ð11Þ

in which L2 is the fundamental diffusion length of the plasma container, and the electron density is measured at the center of the container (or points on the axis of a flow tube). The equation is correct only in the limits when either of the two loss terms greatly outweighs the other since it ignores the fact that quadratic recombination loss tends to “flatten” the spatial distribution of electrons and ions. As a conse­ quence, the diffusion current away from the center is reduced, and Equation (11) overestimates the diffusion loss, but underestimates the recombination loss. The pulsed microwave afterglow measurements often employed numerical solutions of the continuity equations to ana­ lyze the data while the analysis of flow tube data is usually carried out using Equation (11). The time scale of recombination experiment is of practical interest. From Equation (10), it follows that the electron density during the after­ glow decays by a factor of 2 from its value at time t whenever the time increases by the “half-time”  1/2, given by 1 : ð12Þ 1=2 ¼
ne ðtÞ Accurate determinations of recombination coefficients require obser­ vation of ne over a significant range, a factor of 4 or preferably more. Hence, for an initial electron density of ne(t = 0) = 1010 cm3 and a typical recombination coefficient
= 107 cm3/s, one must measure ne(t) over a time of at least (1 þ 2) = 3 ms, longer if the initial electron density is only 109 cm3. Obviously, the ion—molecule reactions that form the desired ions should go essentially to completion in a time short compared to the time scale of recombination, and the ions under study must not convert to a different type during this time by reacting with any of the gases in the afterglow plasma or impurity gases. We will show later (see Section 6.1) that serious errors ensue when these requirements are not fulfilled. The gas temperature in afterglows can be adjusted fairly easily over a limited range from liquid-nitrogen temperature (77 K) to roughly 600 K by heating the entire apparatus. This is more useful as a means to control equilibrium concentrations of weakly bound ions (for instance shifting the chemical equilibrium from H3þ to H5þ ions) than as a means to measure the

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temperature variation of recombination coefficients. Much higher electron temperatures (up to 10,000 K) can be reached by microwave heating of the electron gas, a technique that was used extensively in stationary afterglows and that has provided data on many important ion species, including H3þ. The technique is subject to complications in the presence of molecular additives (Johnsen, 1987). In the afterglow measurements on H3þ, such effects are not important and we will not discuss this subject further. Higher gas temperatures (up to nearly 10,000 K) can be reached by employing shock heating of the afterglow plasma (Cunningham et al., 1981), a technique that has been applied to several recombination processes of atmospheric interest but not to H3þ. 3.2 Single-Pass Merged-Beam and Ion-Storage Ring Experiments With the development of ion-storage rings, experiments on DR were transformed from small-scale “table-top” experiments to large-scale mul­ tiuser type operations that made use of technologies from nuclear and high-energy physics. The impact of these new machines cannot be over­ stated: the considerable investment in the experimental facilities revita­ lized and revolutionized experimental studies of DR. We will only summarize the basic principles and current capabilities since extensive reviews have been written by authors who are more familiar with experi­ mental details (Larsson & Orel, 2008). The predecessor of the storage rings, the single-pass merged-beam method was developed at the University of Western Ontario (see, e.g., Auerbach et al., 1977). While it was an important step forward and resulted in many important results, the single-pass merged beam has been superseded by the more powerful ion-storage ring technique. Both have in common that recombination of ions and electrons takes place between parallel ion and electron beams of nearly the same velocity. In a single-pass merged beam, the ion beam passes through the electron target beam once and is then discarded; in a storage ring the ions circulate in the ring and pass through the interaction region (see Figure 3) many times. It is not the more “efficient” use of ions in storage rings that makes them preferable but the fact that the longer storage time (up to 10 seconds) in a ring removes all excited ions that radiate on that time scale, for instance infrared active vibrationally excited ions. In merged beams, the relative velocity between the two beams can be made very small. More importantly, the velocity spread in the electron target gas can be greatly reduced by accelerating the (initially “hot”) electrons to a high velocity that closely matches that of the ions. The narrowing of the electron velocity distribution in the direction of the beam (but not transverse to it) is a purely kinematic effect that follows from the classical equations of motion. However, at finite electron

Dissociative Recombination of H3þ Ions with Electrons

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Electron collector

Electron gun

Electron beam

Circulating ion beam

Beam deflector Merging region

Circulating ion beam

Demerging region Neutral products

Interaction length

Detector

Figure 3 Schematic diagram of the electron cooler and interaction region of an ion-storage ring. The length of the interaction region is typically on the order of 1 m

densities Coulomb interactions between electrons occur and the actual velocity spread in the beam direction is somewhat larger than that calcu­ lated from the kinematic equations. In addition, the effective energy resolution for ion—electron collisions depends also on the velocity com­ ponents transverse to the beam. It is common practice to model the electron velocity distribution by a two-temperature Maxwellian function with temperatures Ti for the parallel velocity component and T\ for the two transverse components. Several methods are available to reduce the transverse velocity spread and thereby improve the energy resolution: In the single-pass merged-beam apparatus (Auerbach, 1977), improvements in the energy resolution were made by using trochoidal analyzers to merge electron and ion beams, while storage rings employ “electron coolers” in which the electrons are cooled by expansion in a magnetic guiding field. Cooling and recombination can be accomplished either in the same section of the ring or in the two separate sections. In addition, the coolers also cool the ion beam by a “friction” effect and reduce the diameter of the ion beam. In all merged-beam techniques, recombination events are detected by counting recombination products using an energy-sensitive barrier detec­ tor. The detector ideally registers one count of full pulse height when all products from a single event strike the detector simultaneously. In that case the number of counts received for a single traversal of a single ion through the electron target is N ¼
ne Dt:

ð13Þ

Here
is the recombination rate coefficient appropriate to the experi­ mental velocity distribution, ne the electron density, and Dt the time of traversal of an ion through the interaction region. If the ion beam is much narrower than the electron beam, which is the case in storage rings, there

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is no need to consider overlap factors. To obtain absolute values of
, one needs to know the ion beam current, which can be measured either by collecting ions or using a current transformer. While ion-storage rings come very close to the theorist’s perception of an ideal experiment, there are some, fortunately minor, imperfections that should be mentioned. Ion—electron collisions also occur in the mer­ ging and demerging regions (magnets in storage rings) where the ion and electron beams are obviously not parallel and their relative velocities are larger than those in the straight part of the interaction region; however, this “toroidal correction” is not large and can be taken into account. In addition to providing high-resolution recombination cross sections, storage rings have an outstanding ability to determine the relative abun­ dance of recombination products by placing grids in front of the detector and analyzing the pulse-height spectra (for details, see Larsson & Orel, 2008). The single-pass merged beam also has been employed for such studies but the small event rate made quantitative product determina­ tions tedious and time consuming.

4. THEORY 4.1 DR Mechanisms If all internuclear distances in a polyatomic molecule are held constant except for the dissociation coordinate, a potential curve similar to that for a diatomic molecule can be used to illustrate the fundamental features of DR. Figure 4 shows such a slice through the potential surfaces of the ion, Rydberg, and dissociative states.  denotes the electron energy at which capture takes place into a repulsive state of the neutral molecule from an ion in some vibrational level. Note that any electron energy will do, even zero, since varying the electron energy only varies the point of capture. Once in the repulsive state, the neutral molecule can emit the captured electron or dissociate. If dissociation takes the internuclear distance beyond the crossing point of the neutral and ion curves, electron emission (autoionization) is no longer possible and dissociation is completed. This is the direct mechanism for DR originally proposed by Bates (1950). Superexcited states of the neutral molecule are generally found at the same total energies as that for the ion ground state. Electron capture also occurs into these superexcited states and competes with capture into the dissociative state. Among these states are the vibrationally excited Rydberg states that have the ground state of the ion as core. The v = 0 ground core Rydberg levels all lie below the ion, but the v = 1 ion level is the energetic limit of an infinite number of Rydberg levels as are the other excited ion vibrational levels. Capture into one of these levels at electron

Dissociative Recombination of H3þ Ions with Electrons

87

Energy (Hartrees)

−112.60

−112.61

ε� ε

Ion

−112.62 Rydberg

Dissociative state

−112.63

−112.64 1.8

2.0

2.2

2.4

Internuclear distance (Bohr) Figure 4 The direct (at electron energy e) and indirect (at electron energy e 0 ) mechanisms of dissociative recombination

energy "0 is shown in Figure 4. After capture, the electron can be emitted or the Rydberg level can be predissociated by the dissociative state of the direct mechanism. This is the indirect DR mechanism, first introduced by Bardsley (1968). Both the direct and indirect mechanisms are paths to the same dissociation products and can interfere with each other. Any tech­ nique for calculating the DR cross section must account for this inter­ ference. A recent addition to the indirect DR mechanism, Rydberg states having an excited ion core (Guberman, 2007), will not play a role in H3þ DR at low electron energies since the first excited ion states lie too high above the ground state (Shaad & Hicks, 1974). A second-order mechan­ ism (Guberman & Giusti-Suzor, 1991; Hickman, 1987; O’Malley, 1981) also can take place in which the neutral repulsive state acts as an inter­ mediate between the electron—ion and a bound Rydberg state. In this manner, an electron can be captured by an electron—electron interaction into a Rydberg state. 4.2 H3þ Potential Curves and Surface Figure 6 has potential curves for H3þ and for several H3 states that are important for DR. These states have been calculated in C2v symmetry with the nuclear configuration shown in Figure 5, i.e., R1, the distance between two H atoms has been kept constant at the equilibrium separa­ tion, 1.63 a0. The remaining atom moves along R2, which is perpendicular

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Rainer Johnsen and Steven L. Guberman

H

R1

Θ

H R2

H Figure 5 Jacobi coordinates for H3

to R1 and intersects R1 at its midpoint. The potential curves are calculated with [4s, 3p, 2d, 1f] Gaussian basis sets centered on each H atom. For the description of Rydberg surfaces, this basis set is supplemented with six diffuse s and six diffuse p basis functions placed at the center of mass. Orbitals are determined in Hartree-Fock (HF) calculations on H3þ, and the final energies are obtained from CI wave functions calculated by taking all single and double excitations to the virtual orbitals from a large reference set of configurations. The potential curves are identified by the symmetries in C2v as well as the symmetries at the equilateral triangle configuration in D3h. It is clear from Figure 6 that no neutral state potential curves cross the X1A1 ground-state ion curve (X1A0 1 at the equilateral triangle configuration), the highest potential curve in the figure. The two possible dissociative routes are the lowest curves, 12A1 and 12B2. These curves are degenerate at the equilateral triangle position where they have 12E0 symmetry, and they have asymptotes that lie below

−1.0

H3+ X1A1(1A1�)

Energy (Hartrees)

−1.1 −1.2

22B2(22E�)

22B1(B2A2′′)

−1.3 −1.4

22A1(12A1�)

−1.5

12B1(12A2′′)

−1.6

12B2(12E�)

−1.7

32A1(22E�) 12A1(12E�)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Internuclear distance (Bohr) Figure 6 Potential curves for the approach of one H in C2v geometry along R2 (as in Figure 5) to the midpoint of the other two H atoms held at R1 = 1.63 ao

Dissociative Recombination of H3þ Ions with Electrons

89

the ground vibrational state of H3þ. No other states can provide DR routes at low electron energies. These states consist of a 2px or 2py orbital bound to the ion ground state where the xy plane is the plane of the molecule. Because these states do not cross the ion, DR was initially thought to be slow for H3þ. This is discussed further below. The neutral curves shown in Figure 6 are qualitatively similar to those shown in Figure 3 of Petsalakis et al. (1988). A precise comparison is not possible due to the different geometries used in their figure. However, at the equilateral triangle geometry, the curves in Figure 6 are about 0.12 eV lower than those of Petsalakis et al. (1988). Also shown in the figure are the 32A1 and 22B2 states, which are the components of the 22E0 doubly degenerate state at the equilateral triangle configuration. These states are too high in energy to be dissociative channels at low electron energies. Figure 7 shows a two-dimensional surface for 12A1, 12B2 and the ion ground state. In the plot, both R1 and R2 are varied and , as shown in Figure 5, is fixed at 90. Both neutral surfaces intersect at the equilateral triangle configuration. As shown in the figure, the 12A1 surface leads to H2 þ H and both 12A1 and 12B2 can generate H þ H þ H.

−1.26 −1.62

−1.44

Energy (Har trees)

−1.08

H3 + X1A1

1.0

H+H+H

4.0 3.0 ) r 2.0 (Boh R2

H312B2 H2 + H

H312A1

5.0

.00 .00 5 .00 4 3 0 2.0 hr) 1.00 R 1 (Bo

Figure 7 Potential surfaces for H3 and H3þ using the coordinates of Figure 5 with  = 90

90

Rainer Johnsen and Steven L. Guberman

4.3 Vibrational and Rotational Considerations The nuclear configuration of the ground state of H3þ is an equilateral triangle and belongs to symmetry group D3h. The normal modes are shown in Figure 8, labeled using the notation of Herzberg (1945). The first normal mode, labeled  1, is the symmetric stretch or breath­ ing mode. The remaining normal modes,  2a and  2b, are degenerate, i.e., they have the same frequency. Indeed,  2a and  2b, as shown in Figure 8, are not unique. An infinite number of pairs of modes can be obtained by taking orthogonal linear combinations of  2a and  2b, and they are all equally valid. If one degenerate mode is superposed upon another with different phases for the vibrational motion, the H atoms will move in ellipses (Herzberg, 1945). If the motion in the two modes is out of phase by 90 (i.e., when the atoms in one mode are passing through the equilibrium position, the atoms in the other mode are at the maximum displacement), the H atoms will move on circles and the motion can be described with a vibrational angular momentum quantum number, ‘. Instead of describing the vibrational state of the molecule with quantum numbers, v1, v2, and v3, it is now common practice to use (v1, v2‘). For H3þ (Watson, 2000), ‘ = —v2, —v2 þ 2, . . . , v2 — 2, v2. Since H3þ is a symmetric top (i.e., two of its moments of inertia are equal), the quantum numbers specifying the rotational energy levels are Nþ, the total angular momentum, and Kþ, the projection of Nþ, upon the molecular symmetry axis. Each proton has a spin of 1/2 and the total nuclear spin, I, can be 3/2 (ortho) or 1/2 (para). For the ortho states, Kþ = 3n, where n is an integer (Pan & Oka, 1986). For the para states, Kþ = 3n + 1 (Pan & Oka, 1986). It can be shown that the state with (Nþ, Kþ) = (0, 0) does not exist. The lowest energy rota­ tional state is for (1, 1) and is para. The second level, at 23 cm1 above (1, 1), is (1, 0) and is ortho. The (1, 0) level is highly metastable since an ortho—para transition is forbidden. The lowest ortho levels are (1, 0), (3, 3), (3, 0), and (4, 3). The lowest para levels are (1, 1), (2, 2), (2, 1), and (3, 2). It is interesting to note, especially for the

H1

H1

ν1 H2

H1

ν2a H3

H2

ν2b H3

H2

Figure 8 The three normal mode vibrations of the ground state of H3þ

H3

Dissociative Recombination of H3þ Ions with Electrons

91

interpretation of DR experiments, that all of these levels have very long lifetimes (Pan & Oka, 1986). The radiative lifetimes are 1.2  106 seconds for (2, 2), 15  106 seconds for (2, 1), 3.3  104 seconds for (3, 2), 2.2  104 seconds for (3, 0), and 2.2  104 seconds for (4, 3) (Pan & Oka, 1986). Once generated, these ions will not decay by photoemission during DR experiments. 4.4 One- and Two-Dimensional Theory 4.4.1 Direct Recombination The direct recombination cross section for vibrational level v0 , v0 , is given by (Bardsley, 1968; Flannery, 1995; Giusti, 1980) v 0 ¼

2 Gv 0 r 2 k ð1 þ Sv Gv Þ2

ð14Þ

where Gv 0 ¼ 2 jðYd Xd jHjYi Xv 0 Þj2 , r is the ratio of the statistical weights of the neutral and ion states, k is the wave number of the incident electron, v runs over the open ion vibrational levels, Xd and Xv0 are dissociative and bound vibrational wave functions, respectively, Yd and Yi are electronic wave functions of the dissociative and the ion states, respectively, and H is the electronic Hamiltonian. Equation (14) does not account for the inter­ mediate Rydberg levels. In the expression for Gv0, the integration is over the electronic and nuclear coordinates. If the dissociative potential curve does not cross within the turning points of the ion vibrational level, the small vibrational overlap will lead to a small v0 . Figure 6 shows that the dissociative potential curves, 12A1 and 12B2, in a one-dimensional view, do not cross the ion. This feature alone led theorists (Kulander & Guest, 1979; Michels & Hobbs 1984) to predict that the DR rate constant for H3þ is small. At the time, the direct recombination process was thought to be much more important than the indirect process. 4.4.2 Multichannel Quantum Defect Theory Because of the large literature on Multichannel Quantum Defect Theory (MQDT), a full description of the technique is not given here. Instead we guide the reader to the most relevant literature. The primary advantages of MQDT for the study of DR is that one can account for interference between direct DR and indirect DR with both being treated equally and one can treat entire Rydberg series rather than concentrating upon individual states as would be the case with a scattering theory approach. The pioneering studies which introduced MQDT to the study of DR were those of Lee (1977) and Giusti (1980). The approach of Giusti (1980) modified by Nakashima et al. (1987) to incorporate Seaton’s (1983) closed-channel

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Rainer Johnsen and Steven L. Guberman

elimination procedure for the S matrix is the approach used today by most theorists. The theory involves a K or reaction matrix which contains the interaction matrix elements between all channels. The K matrix is calculated perturbatively from the Lippmann—Schwinger equation. The first papers used a K matrix limited to first order. The usage of a second-order K matrix was introduced by Guberman and Giusti-Suzor (1991). The original approach has been revised to include rotation (Schneider et al., 1997; Takagi, 1993; Takagi et al., 1991), derivative couplings (Guberman, 1994), Rydberg states with excited cores (Guberman, 2007), and spin—orbit coupling (Guberman, 1997). An excellent reference on MQDT is the volume by Jungen (1996) and the papers contained therein. 4.4.3 Dissociative Recombination of HeHþ and One-Dimensional H3þ A clue that the theoretical view of H3þ could be wrong came in calcula­ tions on a diatomic molecule that shares the noncrossing features of H3þ. Because we can think of HeHþ as H3þ with two of the protons super­ posed, they are expected to have similar recombination mechanisms. Figure 9 shows the ground-state potential curve for HeHþ and curves for seven HeH states (Guberman, 1994, 1995). All the HeH states in Figure 9 are Rydberg with the exception of the ground state. None of the states cross the ion curve. For this case, it was shown that electron capture could occur by breakdown of the Born—Oppenheimer principle, which also drives indirect DR. Because all the states found to be involved in DR are adiabatic Rydberg states, there are no electronic couplings between these states. Instead, derivative couplings were introduced to drive DR between the adiabatic states. The cross section was calculated for 3HeH up to 0.3 eV, using the MQDT approach (Giusti, 1980; Guber­ man & Giusti-Suzor, 1991), and over most of this region, the indirect process was much more important than direct recombination. Indeed, inclusion of the indirect mechanism increased the cross section by a factor of 49 (Guberman, 1995). For 3HeH, it was also found that He þ H(2s) are the main dissociation products at low electron energies. The total rate coefficient at 300 K was 2.6  108 cm3/s, giving a clear example of how DR, dominated by the indirect mechanism, can have a high rate coeffi­ cient. Indeed, the rate would have been higher if it had been calculated for the true analog of H3þ, the unphysical 2HeH. The potential curves shown in Figure 8 apply also to 2HeH, but the lower mass, compared to 3 HeH, raisesPthe vibrational levels in the well leading to higher overlap with the C2 þ dissociative state. Other calculations (Sarpal et al., 1994) for 4HeH using an R-matrix approach did not report a rate coefficient but also found that indirect recombination dominated the cross section. The main dissociation products were He þ H (1s) at low electron energies. This was surprising since the identity of the dissociation products found

93

HeH+ X1Σ+

−3.0

He+H (n = 3)

A2Σ+,C2Σ+,D2Σ+,3p2Σ+

−3.3

−3.2

−3.1

He+H (n = 2)

HeH X2Σ+

−3.5

−3.4

Energy (Hartrees)

−2.9

−2.8

Dissociative Recombination of H3þ Ions with Electrons

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Internuclear distance (Bohr) Figure 9 Potential curves for HeH and HeHþ from Guberman (1995). Reprinted by permission from the AIP Press

with the MQDT approach was a qualitative and not a quantitative result. Experiments (Stro¨ mholm et al., 1996) have since verified that the main products are He þ H (2s). Takagi (2003) reported a one-dimensional MQDT treatment of DR using the potential curves of Michels and Hobbs (1984). He found that the rate coefficient of the recombining ion was highly sensitive to the initial rotational level with the N  4 levels of the vibrational ground state having large rate coefficients. 4.4.4 Derivative Couplings for H3þ In a study of the predissociation of H3 (Schneider & Orel, 1999), d/dR1 and d/dR2 (see Figure 5 in their paper) derivative couplings connecting the lowest 2A1 dissociative state with 2s2A1 and 3s2A1 were reported. For the 2s state, the d/dR1 coupling at the ion equilibrium separation (R1 = 1.65 ao and R2 = 1.43 ao) is 0.15 ao1 and that for d/dR2 is —0.20 ao1. (The phase of the coupling is arbitrary since it depends upon the phases of the orbitals and the total wave function.) The largest d/dR1 coupling is 0.75 ao1 at R1 = 1.15 ao1 and R2 = 0.92 ao1 and for |d/dR2| it is >0.95 near R1 = 1.15—1.85 and R2 = 0.93. The largest coupling in this case is for R1 near the equilibrium separation but for R2 smaller than the equilibrium separation. The couplings with the 3s Rydberg state, as

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Rainer Johnsen and Steven L. Guberman

expected, are much smaller. Tashiro and Kato (2002) have reported derivative couplings calculated in hyperspherical coordinates between the two 2pE 0 (12A1 and 12B2 states in C2v) dissociative states and the 2s2A1. They found a large coupling that peaks at 5 ao1 for a hyperradius of 1.5 ao and hyperangles of  = 1/2 and = /6 radians and for the upper 2pE 0 state (12B2). Couplings with the lower 2pE 0 state (12A1) were found to be much smaller in agreement with the results of Schneider and Orel (1999). 4.4.5 Two-Dimensional Cross Sections Using a combined wave packet MQDT approach and derivative cou­ plings, a two-dimensional calculation (varying R1 and R2 as in Figure 5) was performed for DR along the 2B2 surface (Schneider et al., 2000). For direct recombination, they found that the calculated cross section is 4—5 orders of magnitude below the experimental cross section (Larsson et al., 1997). However, the inclusion of Rydberg states coupled together by the R1 and R2 dependence of the quantum defect led to a dramatic increase in the cross section although the theory was still two orders of magni­ tude less than the experimental cross section. The authors concluded that the Rydberg channels, via the indirect mechanism, played a crucial role in the DR of H3þ. They attributed the difference between theory and experiment to the lack of a full three-dimensional treatment and to the absence of the 2A1 dissociative state in the theoretical treatment. They also tested the proposal of Bates that DR in H3þ may occur via inter­ connected Rydberg states in which the connection is mainly between states differing by Dv = 1. They found that Dv  1 connections are also very important. 4.5 Three-Dimensional Treatments of H3þ DR The first three-dimensional theoretical treatment of the DR of a polyatomic molecule (Kokoouline et al., 2001) combined several new theore­ tical methods for the study of DR with aspects of the MQDT approach. In these pioneering calculations, a new driving mechanism, not present in diatomic molecules, was introduced. The next section contains a brief description of the adiabatic hyperspherical approach. Section 4.5.2 describes the role of Jahn—Teller (JT) coupling in the DR of H3þ. Section 4.5.3 sum­ marizes the role of the nuclear spin. The approach to calculating the cross sections used in the first paper (Kokoouline et al., 2001) is given in Section 4.5.4. The revised approach used in later papers is discussed throughout and described further in Section 4.5.5. The last section contains suggestions for future theoretical research.

Dissociative Recombination of H3þ Ions with Electrons

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4.5.1 Hyperspherical Coordinates and the Adiabatic Approximation The calculations describe the nuclear motion with hyperspherical coordi­ nates consisting of a hyperspherical radius, R, and two hyperangles,  and . The coordinates can be defined in terms of the distances between the H atoms. Taking ri to be the distance between atom i and the center of mass, the hyperradius is given by R2 = H3 (r12 þ r22 þ r32) (Kokoouline et al., 2001). In later papers (Kokoouline & Greene, 2003a,b), the expres­ sion for R remains the same but ri is taken to be the distance between atoms j and k and the coordinates are given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi 2 r1 ¼ 3  1 = 4 R 1 þ sin sin þ ; 3

r2 ¼ 3

1=4

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi 2 R 1 þ sin sin  ; 3

and r3 ¼ 3  1 = 4 R

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ sin  sin :

ð15Þ

ð16Þ

ð17Þ

From Equations (15)—(17) one can derive expressions for  and , which become intuitively meaningful by consulting Figure 6 in the work of Kokoouline and Greene (2003a) for a valuable demonstration of the mean­ ings of these angles. [For further discussion of hyperspherical coordinates, the reader is referred to the review by Lin (1995)]. The general idea is that the hyperradius describes the overall size of the molecule, whereas the hyperangles, which are not explicitly defined in the first paper, describe the shape of the molecule. These considerations lead to the adiabatic hyperspherical approximation in which motion in R is considered to be much slower than the motion in the hyperangles, i.e., as the atoms traverse the potential surface, the shape of H3þ changes more rapidly than the overall size of the molecule. With the motion in the hyperangles separated from the motion in R, a Schro¨dinger equation at a single value of R can be written in which the eigenvalue is a point on the potential curve. The hyperradius, R, is identified as the polyatomic analog to the familiar diatomic internuclear distance. But is this analogy appropriate? The famil­ iar Born—Oppenheimer approximation is an adiabatic treatment of the nuclear motion and is justified by the great difference in the electron and nuclear masses. However, in the adiabatic hyperspherical approach for H3þ, the particles are all of equal mass. This approach is tested (Kokoouline & Greene, 2003a and 2003b) by solving for the nuclear vibrational energies within the generated potential curves. The eigenvalues for several lowlying levels differ by less than 23 cm1 from a full three-dimensional

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Rainer Johnsen and Steven L. Guberman

diagonalization (Jaquet et al., 1998), and the results support the use of this approximation. In later work (Fonseca dos Santos et al., 2007), this approx­ imation is partially dropped (see below). While the adiabatic hyperspherical approximation for H3þ appears to be successful for the vibrational energies, there have been no reported tests of the accuracy of the amplitudes of the vibrational wave functions. Inaccuracies in the amplitudes may significantly affect the values of important matrix elements between Rydberg states. It is interesting to note that the adiabatic hyperspherical approach fails for H2Dþ and D2Hþ (Kokoouline & Greene, 2005). 4.5.2 Potential Surfaces and Jahn–Teller Coupling The potential surface of H3þ used by Kokoouline et al. (2003) is from Cencek et al. (1998) and Jaquet et al. (1998), and the H3 surfaces are from the work of Siegbahn and Liu (1978), Truhlar and Horowitz (1978), and Varandas et al. (1987). These surfaces need to be interpolated to be converted to a grid in hyperspherical coordinates, but this is not covered in the published papers. The main driving force for DR, introduced for the first time in these calculations, is the JT coupling (Jahn & Teller, 1937), a coupling which does not occur in diatomic molecules. Figure 6 shows that the two lowest states of H3 intersect. The intersection point is at the equilateral triangle configuration where the molecule has D3h symmetry. The two lowest states have two electrons distributed in the three H 1s orbitals and one electron that is either in a 2px or 2py orbital where the molecule is in the xy plane. As R2 (see Figure 5) moves away from the equilateral triangle value (1.43 a0) but with  fixed at 90, the molecular symmetry is lowered to C2v and the degeneracy is split. The splitting is known as the static JT effect. The degeneracy at the D3h configuration appears as a point of conical intersection when the potential surfaces are plotted in normal coordinate space, not Cartesian coordinate space. A further splitting of the energies of the original vibrational levels (in the wells of the 2px or 2py electronic states) occurs when the levels are determined in the mixed state. This splitting arises from the dynamic JT effect. An excellent description of the JT effects can be found in the work of Herzberg (1966). For H3 (Greene et al., 2003), the JT mixing has used the K-matrix form of Staib and Domcke (1990) and the JT mixing parameter and quantum defects, , of Mistrık et al. (2000), which was obtained from a fit to ab initio surfaces. The nature of the conical intersection allows one to represent the coupling with two parameters [see Equation (4.7) of Mistrık et al. (2000)]. This is an enormous simplification compared to other situations where a non-JT coupling may need to be represented by a surface of derivative couplings. Of further importance, Staib and Domcke (1990) reported that

Dissociative Recombination of H3þ Ions with Electrons

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the fit to the ab initio results of Nager and Jungen (1982) shows that the conical shape of the potential surfaces is close to a true cone, although data to support this observation were not reported. This observation means that higher JT interaction terms beyond linear may not be needed and that the interaction of the np series with the ns or nd series is not important since if they were important a distorted conical shape would occur. Here, n is the principal quantum number. These observations provide some justification for the use of only an ‘ = 1 partial wave for the incoming electron in the calculations of Kokoouline and Greene (2003). On the other hand, Mistrık et al. (2000, 2001) found evidence for strong mixing of the ns and nd Rydberg states with 3p, 4p, and 5p states built on H3þ cores having the degenerate asymmetric vibrational motion. A point on the 5pE0 surface was found to have only 80% p character. The JT coupling parameter and the quantum defects used (Kokoouline et al., 2003a, b) are for the 4p state of H3 (Mistrık et al., 2000). Ideally, the best coupling parameters and the best quantum defects would vary with the Rydberg or continuum orbital energy. However, these are not available. In addition, for n  3 the quantum defect varies only slightly with n but that for n = 2 differs considerably from those for n  3. Because the MQDT approach requires a single coupling parameter and a single quantum defect surface, one must choose a compromise value. Usage of the n = 4 quantum defects (Kokoouline et al., 2003) should produce only very small errors in the positions of resonances, but the n = 2 states will suffer the largest shift in energy from the true positions. The JT interaction is generally thought of as that between states having the same n. Here n is the effective principal quantum number, i.e., n = n — , and is the quantum defect. In the MQDT approach used by Greene and coworkers, the JT interaction not only describes the interac­ tion between the two E0 states having outer orbitals 2px and 2py but also accounts for the mixing of Rydberg states with different n, the mixing of Rydberg states with continuum states, and the mixing of the Rydberg and continuum states with the 2pE 0 dissociative states. This assumes that these mixings are symmetry allowed. Rydberg orbitals of differing n, although orthogonal to each other, are quite similar near the nuclei except for a normalization factor of 1/n3/2. Since it is the Rydberg amplitude near the nuclei that is most important, the JT effect will occur between Rydberg states and the two dissociative states, scaled by the 1/n3 factor. The normalization constant is squared since the width [see the expression below Equation (14)] has the square of the interaction matrix element. If the incoming electron is in a px continuum orbital, it can be captured into an npy orbital also scaled by the 1/n3 factor. For two different Rydberg states having effective principal quantum numbers of n1 and n2, the connecting width would scale as 1/(n1n2)3.

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The couplings mix the Rydberg and continuum states with the lowest n = 2 dissociative states along which DR is finalized. The mixing of all the Rydberg states with each other and that of the continuum state with all the Rydberg states means that many possibilities for DR can occur. The continuum electron can be directly captured into the 2px,y states followed by dissociation, or it can be captured into a higher state which, via couplings to other intermediate levels, can even­ tually lead to the dissociative levels. The mechanism is remarkably simi­ lar to one originally proposed by Bates et al. (1993). 4.5.3 Nuclear Spin The nuclear spin has been included in prior theoretical studies of homo­ nuclear diatomic DR whenever molecular rotation is considered. The calculations reported by Greene and coworkers also include nuclear spin in the theory. Nuclear spin cannot be ignored because in H3, the nuclei are fermions and the total wave function must change sign for an interchange of any two protons. (The exchange is equivalent to a rotation by 180 around an axis perpendicular to the main symmetry axis.) This requirement places restrictions upon the allowed values for the rotational quantum numbers and requires that the total symmetry (i.e., the product of the symmetries of the vibrational, rotational, nuclear spin and electro­ nic wave functions) be that of the A02 or A2† representations of D3h. The ortho and para states have total nuclear spin of 3/2 and 1/2, respectively. 4.5.4 Calculation of the DR Cross Section and Rate Coefficient The first paper (Kokoouline et al., 2001) reported preliminary calculations which made use of the hyperspherical adiabatic approach and an expres­ sion derived by O’Malley (1966) for the direct DR cross section, , of diatomic molecules: ¼

X 2 G R     YR 2 0   Eel U  R 

ð18Þ

In Equation (18)  is an index that runs over the dissociative routes, Eel is the electron energy, R is the value of the hyperradius for the  th dissociative route at an energy, Eel, above the ion rovibrational level undergoing DR, G is the width for capture into the  th dissociative 0 route, Ub is the slope of the th dissociative route, and Y(R) is the dissociative nuclear wave function. The use of this expression follows from the observation that when the potential curves are plotted as a function of the hyperradius, all the Rydberg states cross the ion ground state.

Dissociative Recombination of H3þ Ions with Electrons

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Equation (18) omits the survival factor [the denominator within parentheses in Equation (14)] and thereby does not account for autoioni­ zation. There are several other caveats to consider. This expression was derived for diatomic molecules and its use for H3þ entails replacing the internuclear distance by the hyperradius. This replacement is not likely to lead to quantitative results. In diatomics, the nuclear configuration depends solely upon the internuclear distance, and the Franck—Condon factor in G has a rigorous dependence upon this distance. In a triatomic, in hyperspherical coordinates, the Franck—Condon factors will depend upon both the hyperradius and the hyperangles. Since the hyperradius is often viewed as a measure of the size of the molecule, taking G to depend only upon R is making the approximation that the Franck— Condon factors depend more upon molecular size than upon the details of molecular shape. This approximation is not expected to be reliable. It is probably for these reasons that the results (Kokoouline et al., 2001) are referred to as preliminary and approximate. Both upper and lower bound cross sections were reported. The lower bound cross sections included only the 2p states. Both the 2p and higher np states are included in the upper bound cross section. In a later paper (Kokoouline & Greene, 2003b), it is noted that the cross sections reported in the first paper (Kokoouline et al., 2001) need to be multiplied by a factor of 2 due to inconsistencies in the literature concerning the definition of the K matrix. Surprisingly, if one multiplies the 2001 results by 2, the upper bound cross section is in quite good agreement with the storage ring results (Jensen et al., 2001). The calculated cross sections are structureless as are the experimental results to which they were compared. Using only the 2p states, it is estimated that 70% of the DR events lead to H þ H þ H compared to the experimental result (Datz et al., 1995a, b) of 75% + 8%. For the H þ H2 channel, the peak H2 vibrational distribution occurs at v = 5—6 compared to the broad distribution found experimentally, which peaks at v = 5 (Strasser et al., 2001). The upper bound thermal rate coefficient at 300 K is 1.2  107cm3/s after correction by the p2 factor and compares well to the storage ring results of 1.0  107cm3/s (Jensen et al., 2001) and 1.15  107cm3/s (Sundstro¨m et al., 1994). The usage of Equation (18) to calculate these results would lead one to conclude that this agreement must be fortuitous. However, the agreement reported not only for the cross section and rate constant but also for the branching fraction and vibrational distribution argues otherwise. 4.5.5 Improved Cross Sections The lack of structure in the calculated cross section was corrected in a later detailed paper (Kokoouline & Greene, 2003b), which used an MQDT approach instead of Equation (18). The use of the adiabatic

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hyperspherical approximation has been described above as has the K matrix having the JT coupling. The calculated rate coefficients are reported (Kokoouline & Greene, 2003a and 2003b) to be accurate to better than 20% due to the incomplete set of states that are included in the calculations. The states are character­ ized by the quantum numbers [I, G, Nþ, N], where I represents the two values for the total nuclear spin, 3/2 (ortho) and 1/2 (para), G denotes the 0 total molecular symmetry (A2 or A2†), and Nþ and N denote the rotational quantum number for H3þ and H3, respectively. (The total molecular symmetry is determined by the need to have the total wave function change sign upon a swap of any two nuclei.) In the first detailed report of the calculations (Kokoouline & Greene, 2003b), 17 sets of these quantum numbers were used, each with 8—12 vibrational wave functions (includ­ ing the continuum) and 50—100 hyperspherical potential curves. A tabu­ lation of the levels is not included. In the most recently reported calculations (Fonseca dos Santos et al., 2007), several improvements were incorporated into the cross section and rate constant calculations. The adiabatic hyperspherical approximation was relaxed by including couplings between the adiabatic channels. The slow variable discretization approach was used to incorporate these couplings, but the details of these new calculations are not reported. A comparison of the calculated vibrational energies for 26 low-lying vibrational states with a full three-dimensional diagonalization (Jaquet et al., 1998) shows a clear improvement over the earlier full adiabatic approach (Kokoouline & Greene, 2003b). The positions of the Rydberg resonances are improved with this revision. However, the physical interpretability of these calculations is some­ what problematic. Potential curves plotted as a function of the hyperra­ dius are much more difficult to interpret than the more familiar surfaces plotted as a function of Cartesian coordinates. Furthermore, if one improves upon the adiabatic hyperspherical approach by including more couplings between the curves, the concept of a potential curve as a function of the hyperradius becomes weak. In the limit of completely dropping the adiabatic hyperspherical approximation, potential curves are no longer meaningful. These considerations must be balanced against the reasonable agreement that has been obtained to date between these calculations and experiment. This is discussed further below. An important additional improvement in the most recent calculations (Fonseca dos Santos et al., 2007) is the addition of more resonance states. Rotational states up to Nþ = 5 are included compared to the prior calcula­ tions which included levels up to Nþ = 3 (but not Kþ = 1) and (4, 3) for the ground vibrational level. A detailed accounting of the included vibra­ tional levels is not presented, which makes it difficult to assess whether or not the theoretical treatment is adequate at particular electron energies.

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4.5.6 Toroidal Correction In the storage ring experiments, a beam of molecular ions circulates in a large ring (51.6-m circumference) (Stro¨ mholm et al., 1996) and merges with a beam of electrons in only a small section (0.85 m) (Stro¨ mholm et al. 1996) of the ring known as the electron cooler (the region between the merging and demerging regions in Figure 3). The electron beam is bent by a toroidal magnetic coil at the beginning and end of the overlap region. Collisions between the continuously renewed electron beam and the ions serve to reduce the random motions of the ions leading to a high energy resolution. The ion beam is generally a few mm in diameter compared to the electron beam which is a few cm in diameter. The cooler is also the location where DR takes place. For measure­ ments at “zero” center of mass energy, the electron beam is velocity matched with the ion beam. For other center of mass energies, the elec­ tron beam energy is shifted up or down from the “zero” energy measure­ ment. For most of the length of the cooler, the electron beam is very closely collinear with the ion beam and the intended center of mass energy is appropriate. However, in the merging and separating regions at both ends of the cooler, the ion and electron beams are not parallel and the center of mass energy changes with the angle between the two beams. The result is that a measurement of the DR rate constant at a single center of mass energy (appropriate in the straight section of the electron beam cooler) is actually an average of rate constants for different center of mass energies over the length of the cooler from the beginning of the merging region to the end of the separating region. The bending region comprises only about 15% (Amitay et al., 1996) of the full length of the overlap of electron and ion beams and was thought to not play a significant role in deriving the value of the rate constants. However, an important recent study by Kokoouline and Greene (2005) on H3þ indi­ cates that the experimental data deviate considerably from the theoretical values near 0.03 eV, 0.1 eV, and above 0.8 eV. In the latter region the difference between experiment and theory is over an order of magnitude. If the theoretical results are averaged over the full cooler length, account­ ing for the higher relative center of mass energies at the ends of the cooler, the theory agrees with experiment above 0.8 eV and shows improved agreement at 0.03 and 0.1 eV. The results indicate that raw storage ring data must be corrected to remove the effect of the electron bending regions. The deconvolution procedure for accomplishing the correction (Lampert et al., 1996) introduces considerable uncertainty because the rate constants needed at higher energies have often not been measured, and in the case of those that have been measured, they too must be corrected. The result is an iterative procedure which is usually carried out to first order (i.e., a single iteration) (Stro¨ mholm et al., 1996).

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4.5.7 Breit–Wigner Cross Sections If one assumes that all electron captures into Rydberg states lead to dissociation in one way or another and that there is no direct dissociative channel that would interfere with the dissociation through the Rydberg states (as is the case for H3), the Breit—Wigner expression can be used for calculating DR cross sections. An important innovative approach along these lines has been reported by Jungen and Pratt (2009). They treated the linear JT effect, restricting capture into v2 = 1 Rydberg levels (from the ion ground state). Capture into v1 = 1 Rydberg levels was not considered. Using spectroscopic data for the 3pE 0 state and previously determined JT coupling parameters, they show that after averaging over the closely spaced v2 = 1 resonances, a simple cross section expression results which is independent of n and structureless. 4.5.8 Comparison of Theory and Experiment For the four isotopomers, Jungen and Pratt (2009) show that there is a factor of two disagreement with the experimental rate coefficient at some energies for D3þ and a factor of 2—3 disagreement for D2Hþ. For H2Dþ and H3þ the agreement is even better except near 0.006 eV for H3þ. The resulting rate coefficients show remarkable agreement with experimental results for the four isotopomers considering the simplicity of the cross section expression. Figure 14 has the latest results of Greene and coworkers (Fonseca dos Santos et al., 2007), Jungen and Pratt (2009), and the CRYRING (McCall, 2004) data for H3þ. The theoretical results of Fonseca dos Santos et al. (2007) show much more structure than the CRYRING data. Although the theory and experiment are in generally good agreement, there is clearly room for improvement. 4.5.9 Suggestions for Future Theory The pioneering research of Greene, Kokoouline, and coworkers has made an enormous contribution to our understanding of the DR of H3þ. Never­ theless, many of the details remain to be uncovered. We still do not know which Rydberg states drive DR. The identities of the important states will change with electron energy as will the details of the mechanism. An important contribution in this regard has been the theoretical work of Tashiro and Kato (2002, 2003) on the predissociation lifetimes of H3 0 Rydberg states. They found that the 2s2A1 state has a large coupling with the upper 2pE 0 state (see Section 4.4.4) and may be a feeder state for DR from higher Rydberg states. They propose that in DR, initial electron capture occurs into high n (n = 6 or 7) states with low vibrational excitation followed by coupling to lower n states with higher vibrational

Dissociative Recombination of H3þ Ions with Electrons

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0

excitation. The coupling eventually leads to the 2s2A1 state, which is predissociated mostly by the upper 2pE 0 state. They propose that if electron capture involving a single vibrational quantum is most impor­ 0 tant, the 6s2A1 (1, 00) and 7p2E 0 (0, 11) are important for DR at electron energies just above the lowest vibrational level of H3þ. By propagating a wave packet from 7pE 0 (0, 11), they find that the predissociation involves the intermediate states 5p2E 0 , 4s2A10 , 3s2A10 , 3p2E 0 , 2s2A10 , and finally DR via 2pE 0 . However, the precise identification of these states requires greater accuracy in the quantum chemical determinations of their posi­ tions and widths and would be a valuable contribution. Note that the 0 2s2A1 state and those for n > 2 are not included in the calculations of Greene, Kokoouline, and coworkers or those of Jungen and Pratt (2009) and should be considered for future work. Future theoretical studies should explore the role of the ‘ = 0 and 2 partial waves. The work of Tashiro and Kato (2002) indicates that the ‘ = 0 wave may be more important than ‘ = 2. The calculations of Greene, Kokoouline, and coworkers and Jungen and Pratt (2009) treated only ‘ = 1. The inclusion of the ‘ = 0, 2 partial waves may account for some of the differences between theory and experiment. The JT coupling explored by Greene and coworkers is probably the dominant coupling that drives DR. But other derivative couplings that have been identified in prior calculations (Schneider & Orel, 1999; Schneider et al., 2000; Tashiro & Kato, 2002) need to be included in future three-dimensional calculations. The failure of the adiabatic hyperspherical approach for H2Dþ and D2Hþ leads one to ask if it is entirely adequate for H3þ. Instead of calculating vibrational energies to determine the accuracy of this approach, it may be more meaningful to compare the values of S matrix elements resulting from the adiabatic hyperspherical approach to elements calculated by relaxing this approach.

5. HISTORY OF EXPERIMENTAL H 3 þ RECOMBINATION STUDIES As may be seen in Figure 10, the measured recombination coefficients have varied considerably over the years. While all afterglow measurements carried out before 1973 probably refer to mixtures of H3þ and H5þ ions (and impurity ions), the recombining H3þ ions were clearly identi­ fied by mass analysis in the microwave afterglow studies by Leu et al. (1973). The measured recombination rates were very similar to those found for many other ions and nothing unusual was noted. Subsequent studies used either an inclined-beam (Peart & Dolder, 1974) or single-pass merged-beam (Auerbach et al., 1977) measured recombination cross

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α (H3+) [10–7 cm3/s] at 300 K Leu et al., 1973 2.5 SA 2

1.5

1

Macdonald et al., 1984 Adams et al., SA 1984 FALP

Greene, Kokoouline

Theory

Amano, 1988 IR absorption

Gougousi et al., 1995 FALP Larsson 1993 Canosa et Jensen 2001 Smith& al., 1992 McCall 2004 Španel FALP Storagering 1993 FALP Hus et al., 1988 Laube et MB

al., 1998

FALP

0.5

Glosik 2009 AISA, FALP

High H2

Low

H2

0 Figure 10 H3þ recombination coefficients inferred from different types of experiments, at electron temperatures near 300 K

sections over a wider range of energies, confirming the afterglow data within about a factor of two. Macdonald et al. (1984) extended the microwave measurements to higher electron temperatures up to 5000 K by microwave heating of the plasma electrons. While the measured 300 K rate coefficients were somewhat smaller than those of Leu et al. (1973), the temperature dependence was quite close to that expected from the merged-beam results. Not much attention was paid at that time to a theoretical argument by Kulander and Guest (1979) that the usual curve-crossing DR mechanism would not be applicable in the case of H3þ. The situation changed when Michels and Hobbs (1984) again calculated one-dimensional potentialenergy curves of H3þ and showed that the ionic ground-state curve of H3þ in the lowest vibrational states does not intersect a repulsive curve leading to neutral products. However, suitable curve crossings, were found for H3þ ions in the third or higher vibrational states. Hence, Michels and Hobbs suggested that the experimental data referred to vibrationally excited H3þ ions. Their argument was seemingly strength­ ened by new experimental data of Adams et al. (1984), who used their new “Flowing Afterglow Langmuir Probe” (FALP) technique to study the recombination of H3þ. They noticed that the initial electron-density decay was quite fast, compatible with a recombination coefficient near 107 cm3/s, but also that it changed in the later afterglow to a slower decay indicating a much smaller (<2  108 cm3/s) recombination rate coeffi­ cient. Michels’ and Hobbs’ prediction offered a ready explanation for this

Dissociative Recombination of H3þ Ions with Electrons

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observation, namely that the initial fast decay was due to recombination of vibrationally excited ions (v  3) and that the slow decay should be ascribed to H3þ in v = 0. Later, Adams and Smith reported that the recombination rate for v = 0 ions might be even smaller (1011 cm3/s). Larsson and Orel (2008) provide a brief account of these experiments, which were not published in detail. While this very low value may have been due to an experimental problem (presence of nonrecombining Heþ ions) it became generally (with some exceptions) accepted that H3þ in the vibrational ground-state recombined only slowly. The interpretation of Adams et al. experiment became highly question­ able when Amano (1988, 1990) used infrared absorption to monitor the decay of the H3þ (v = 0) density in the afterglow of a radio-frequency discharge in pure hydrogen. At a gas temperature of T = 210 K Amano measured a recombination coefficient for v = 0 ions of 1.8  107 cm3/s, fairly close to the data of Leu et al. and Macdonald et al. The ensuing debate at times became rather contentious. Responding to some criticism, Smith and Sˇ panˇel (1993) repeated the original studies by Adams et al. (1984) in much greater detail and arrived at the conclusion that v = 0 ions recombined with
= (3.6 + 1)  107 cm3/s, somewhat faster than their previous result. The measurements by Amano suggested that curve crossing might not be as essential for recombination as had been thought. Bates et al. (1993), in a paper entitled “Enigma of H3þ dissociative recombination,” pro­ posed a multistep mechanism to rationalize experimental results, but the theory was not sufficiently quantitative to dispose of the “enigma.” Additional flowing-afterglow measurements were carried in attempts to settle the question. One study (Canosa et al., 1992) gave rate coefficients of 1.1  107 cm3/s at T = 650 K for H3þ, thought to be in v = 0, and 1.5  107 cm3/s at 300 K for ions believed to be of low vibrational excitation (v £ 2). A further study in the same laboratory by Laube et al. (1998) resulted in a factor-of-two lower value of 7.8  108 cm3/s at 300 K. This value is often quoted as the afterglow measurement that agrees best with the storage ring results. However, the authors also carried out an identical experiment for D3þ and found essentially the same recombination coefficient as for H3þ, and this does not agree with the storage ring data. Gougousi et al. (1995), using the flowing afterglow techniques, observed a decline of the recombination rate at late times and found that the apparent recombination rate (inferred from the early afterglow decay) increased from 1.5  107 cm3/s to nearly 2  107 cm3/s when the experimental H2 concentration was raised from 1  1014 cm3 to 15  1014 cm3. They attempted to explain their data by a model in which H3þ recombination occurred by a three-body mechanism in which both electrons and neutral hydrogen play a role. Later studies using the single-pass merged-beam method did not lead to consistent results. For instance, Hus et al. (1988) found a nearly

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20 times larger cross DR cross section when the “rf source” was used rather than the “trap source” to produce ions. In single-pass measure­ ments, the ions do not have sufficient time to relax vibrationally or rotationally before merging with the electron beam, and hence it seemed possible that the observed effects were due to vibrational excitation of the ions, as was suggested by the authors. Mitchell [unpublished, brief accounts are contained in a paper by Johnsen and Mitchell (1998) and by Mitchell and Rogelstad (1996)] also carried out a merged-beam experi­ ment in which the deflection field used to separate product neutrals from the ions in the post-collision region was varied. The measured DR pro­ duct signal increased by a factor of 5 (approaching that obtained in storage-ring experiments) when the field strength in that region was reduced from 3000 V/cm to 200 V/cm. Mitchell ascribed this effect to field ionization of H3 Rydberg molecules that are produced by DR of H3þ. At the high field strength, but not at the lower, a substantial fraction of the recombination H3 Rydberg products would be re-ionized in the demerging region of the apparatus and would not be counted as DR products. The problem with this suggestion is that it does not really remove the discrepancy between the merged-beam and storage ring data since field ionization should also occur in the bending magnet of storage ring experiments. A convincing explanation of these observations has not yet been given. Since then, afterglow measurements on H3þ and D3þ have been carried out almost exclusively by Glosık and coworkers in Prague, who use both a flow tube and an advanced stationary afterglow apparatus (AISA). The Prague group systematically studied the dependence of the apparent rate coefficients as a function of gas densities and temperature. A nearly complete set of their data has been presented in a recent paper (Glosik et al., 2009a). The Prague group (Macko et al., 2004) also carried out a series of afterglow measurements in which the H3þ(v = 0) ion density during the afterglow was measured by optical absorption using a cavity ring-down technique. These results confirmed the spectroscopic measurements by Amano (1988 and 1990) and show that vibrationally cold ions recombine with coefficient of about 1.5  107 cm3/s. The most interesting and startling observations made by the Prague group (see, e.g., Plasˇ il et al., 2002) were that the DR rate coefficients seem to fall off rapidly (down to 1  109 cm3/s) when the H2 concentration is reduced to below 1012 cm3. It is this observation that challenges the now generally accepted rate coefficient of 1  107 cm3/s and seemingly poses a serious problem. We will show later (see Section 6.1) that those afterglow data do not really support the very low inferred recombination rates. The modern era of DR studies of H3þ and D3þ (and many other ion species) began with the extensive work using ion-storage rings, the CRYRING in Stockholm, the ASTRID ring in Aarhus, and the TSR in

Dissociative Recombination of H3þ Ions with Electrons

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Heidelberg. Unlike the afterglow and single-pass merged-beam work, this technique produced remarkably consistent results. Steady improve­ ments in the energy resolution, control and characterization of vibrational and rotational states, and beam quality were made over the years, but the conclusions never changed significantly. The latest results of the CRYRING and TSR rings show a nearly identical dependence of the rate coefficient (or cross section) on energy, including the finer structures that will be discussed later. The thermally averaged (Maxwellian) rate coefficient as a function of electron temperature, derived from the CRYRING results (McCall et al., 2004), can be expressed by an analytical fit of the form aðTe Þ½cm3 =s¼1:310  8 þ1:2710  6 Te0:48 ; 8

ð19Þ

which gives a recombination coefficient of 6.9  10 cm /s at Te = 300 K. This value refers to ions in the lowest vibrational state, and at a rotational temperature of about 30 K. 3

6. RECONCILING AFTERGLOW AND STORAGE RING RESULTS The history of afterglow measurements of H3þ recombination rates pre­ sents a rather confusing picture. If one accepts the agreeing storage ring and theoretical value of
(300 K) = 7  108 cm3/s as a “benchmark,” then some afterglow measurements yielded values that were “too small” by factors of 10 and more, while others are “too large” by factors of 2—3. The question then arises which of the afterglow observations reflect a real difference in recombination mechanisms and which ones are due to experimental errors. 6.1 Afterglow Measurements That Yielded Very Low Recombination Coefficients We will discuss the unusually low values first. There are good reasons to believe that plasma recombination can be enhanced by third-body­ assisted recombination, but it is difficult to envision a mechanism that suppresses binary H3þ recombination in the plasma environment. One might surmise that the H3þ ions in the experimental plasmas were of a particular type, for instance in a nonrecombining vibrational state or perhaps in different spin modification, e.g., ortho or para H3þ. At this time, it appears very unlikely that vibrationally excited H3þ ions recom­ bine more slowly than those in the ground state, as was suggested in a previous review by one of the present authors (Johnsen, 2005).

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New theoretical calculations by Fonseca Dos Santos et al. (2007) for the E (0, 11)] and A1 (1, 00)] vibrationally excited states actually indicate the opposite so that this “loop hole” has essentially been closed. Like­ wise, recent storage ring experiments (Tom et al., 2009) and theory (Fonseca dos Santos et al., 2007) indicate that both the ortho and para spin modifications of H3þ recombine at nearly the same rate, at least at 300 K. The storage ring results show that the ortho form recombines somewhat more slowly, but only by about a factor of 1.5. The most puzzling findings that need to be examined in some detail are those made in the extensive series of stationary- and flowing-after­ glow measurements by the Prague group (Glosik et al., 2009a; Plasˇ il et al., 2002). Their experiments seemed to indicate that the H3þ recombination rate dropped to values far below the binary value (by a factor of 10 and more) when the hydrogen concentration in the experiments was reduced from about 1  1012 cm3 to 1  1011 cm3. Very similar results were consistently obtained by the two different afterglow methods, at different temperatures, and also for D3þ ions (Glosik et al., 2009b). This finding is often mentioned as a serious problem since it is in conflict with both recent theory and experiments. It is always difficult to reanalyze experimental data that were taken by others. However, if one examines the experimental conditions, one rea­ lizes that the expected recombination rate of about 107 cm3/s could not have been observed at low H2 concentrations. While the two reactions in the sequence Arþ þ H2 ! ArHþ þ H and ArHþ þ H2 ! Ar þ H3þ are fast (rate coefficients near 109 cm3/s), it will still take roughly 10 ms at [H2] = 1011 cm3 to produce H3þ ions, but recombination of an ion with
= 107 cm3/s (at ne = 1010 cm3) proceeds at a time scale of 1/(
ne) = 1 ms. This means that the loss rate of electrons in the plasma is not limited by recombination, but by the rate at which the ion is formed. Since our criticism affects a large set of published data, we constructed a simple numerical model that simulates the afterglow pro­ cesses and the methods of analysis that were employed by the Prague group. The authors determine recombination coefficients using a form of data analysis in which one constructs a graph of the measured values of the quantity 
1 dne  D ð20Þ  2 þ ne dt ne as a function of the reciprocal electron density. Here  D describes the loss of ions and electrons due to diffusion in the fundamental diffusion mode. If the plasma contains only one recombining ion species from the very beginning, or if this condition is approached rapidly, then a graph of this kind indeed approaches the value of the recombination coefficient in the limit of ne ! 0 (i.e., the late afterglow). In practice, the asymptotic

Dissociative Recombination of H3þ Ions with Electrons

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value is attained in a short time if the reactions forming H3þ go to completion rapidly, which is the case in most afterglow measurements when the H2 density is sufficiently high. However, this is not true when the H2 density is very low. Our numerical model returns the values in the expression above as a function of the reciprocal electron density. To keep the model simple, we ignore depletion of the neutral H2 due to the ion—molecule reactions, even though it is not negligible, but including it would make the situation only worse. The model shows that an input value of
= 107 cm3/s and initial electron densities of 1  1010 cm3 leads to same inferred recombination coefficient only if [H2] > 1012 cm3. At [H2] = 1  1011 cm3 (see Figure 11) graphs of the same kind show that the asymptotic value is never approached on the time scale of the experiment (about 40 ms). The authors’ method of recovering the recombination coefficients employed a linear extrapolation (sometimes done approximately on a logarithmic graph) toward 1/ne ! 0. The procedure returns a much smaller and incorrect value of the recombination coefficient. The asymptotic value approached in the limit ne ! 0 should have been used, but in practice this value cannot be obtained at low [H2], even by curve-fitting, with any reasonable degree of precision, since diffusion becomes the dominant loss in the late afterglow. Another way of illustrating the cause of the problem is to examine the evolution of the ion composition during the afterglow, an example of which

−(ne′/ne2+VD /ne) (cm3/s)

1.E−07

1.E−08

1.E−09 0.0E+00

5.0E−10

1.0E−09

1.5E−09

2.0E−09

1/ne (cm3) Figure 11 Numerical simulation of an afterglow in an helium/argon/hydrogen mixture at a hydrogen concentration of 1  1011 cm3 for an assumed H3þ recombination coefficient of 1  107 cm3/s. The arrow indicates the extrapolation to 1/ne = 0, from which a far smaller recombination coefficient is obtained

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Density (cm−3)

1.E+10

1.E+09

1.E+08 0.00

0.02

0.04

0.06

Afterglow time (seconds) Figure 12 Numerical simulation of an afterglow in a helium/argon/hydrogen mixture at a hydrogen concentration of 1  1011 cm3 for an assumed H3þ recombination coefficient of 1  107 cm3/s. The lines indicate the evolution of density of electrons (thick line), Arþ ions (dotted line), ArHþ ions (dashed line), and H3þ ions (dash-dotted line)

is given in Figure 12. Even at an afterglow time of 40 ms, H3þ accounts for only 1/3 of all ions, which makes it impossible to obtain accurate H3þ recombination coefficients. The authors did carry out simultaneous mass spectrometric observations that seemed to indicate that the plasma was dominated by H3þ ions. However, the mass spectrometer samples ions from a region near the wall of the plasma container where the electron density and recombination loss of H3þ is lower, and hence the relative abundance of this ion is higher than it is in the center of the plasma. We conclude that the observations of very low recombination rates at low [H2] are probably in error and that consequently there is no need to search for explanations in terms of H3þ recombination mechanisms. In reality, the situation may be more complicated. A slower increase of the recombination coefficient with H2 concentration is consistently observed at much higher [H2] and this effect must have a different origin (see Section 6.3). While attempting to fit some of the published data samples, we also noticed that better fits were obtained when the model H2 con­ centration was reduced to values below the stated concentrations. This may indicate that a fraction of the H2 was dissociated during the dis­ charge phase of the experiment in the stationary afterglow experiments. Some dissociation of H2 can also occur in flowing afterglow measure­ ments due to metastable argon atoms that enter the recombination region. These remarks are speculative. It may be worthwhile to conduct some experiments to clear up such questions.

Dissociative Recombination of H3þ Ions with Electrons

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A further observation of very low recombination rates was made by Adams et al. (1984) in an afterglow experiment. This observation had a great impact and for a while since it was believed to provide evidence that H3þ ions in their vibrational ground state recombined only slowly. Their experiments showed that the electron density in an H3þ afterglow initially decayed quite fast (indicating a recombination rate of about 1.3  107 cm3/s) but then decayed much more slowly. No such effects were found when the plasma contained O2þ ions. At the time when the experiments were done, it was believed that H3þ in the vibrational ground state recombined very slowly. Hence the experimenters drew the natural conclusion that the initial decay was due to vibrationally excited H3þ and that the later slower decay was due to ground state ions. The lowest 300 K recombination coefficient derived in a later repeti­ tion of this experiment by Smith and Sˇ panˇel (1993) was 3  108 cm3/s, lower by a factor of 2.3 than the storage ring value. The authors believed that this value referred to a mixture of v = 0 and v = 1 ions. However, the accuracy of this value must be regarded as questionable. It was obtained by fitting the observed decay to a model that has too many adjustable parameters, the relative abundance of the two (or possibly three) states, two recombination coefficients, the quenching coefficient from the higher to the lower state, an estimated impurity concentration, and a diffusion rate. Also, the deviation of the decay curve from that corresponding to a simple (single-ion) decay is actually very small (only a few %), which makes it difficult to determine several coefficients by curve-fitting. While a good fit to the data was obtained, it does not necessarily result in a unique value of the recombination coefficient in the late afterglow. We constructed a simple numerical model similar to the one used by the authors and found that equally good fits could be obtained for higher recombination rate coefficients (up to about 6  108 cm3/s) in the late afterglow. If one simply fits the 1/ne(t) graph in the paper by a straight line, one obtains an upper limit of the recombination coefficient in the later afterglow of about 8  108 cm3/s. While the data show that there is indeed something “unusual” about the decay curve, the low inferred value of
(v = 0,1) = 3  107 cm3/s is not sufficiently accurate to be considered a challenge to the storage ring data. In an attempt to reduce the vibrational state to v = 0, Smith and Sˇ panˇ el carried out a second set of measurements in which they used Krþ ions to produce H3þ and, using a different fitting procedure, arrived at an even lower estimated value
(H3þ,v = 0)  (1—2)  107 cm3/s. However, the authors also found evi­ dence that the plasma contained both H3þ and KrHþ in apparent chemi­ cal equilibrium, and it is not at all obvious which of the two ions was responsible for the observed recombination loss. Similar observations of a reduced recombination rate in the later after­ glow were later made in flowing-afterglow measurements by Gougousi

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et al. (1995). Given the considerable uncertainty of the data analysis, the lowest values are not in conflict with the storage ring value at 300 K. Those authors attempted to explain their observations by a three-body mechanism in which ambient electrons induce l-mixing in the autoioniz­ ing states. The model may have contained a kernel of truth, but it relied on unrealistically long lifetimes, taken from a merged-beam experiment, that are not supported by either theory or other measurements. The explanation for the observed faster decay at early afterglow times may actually be that proposed by Smith and Sˇ panˇ el, but in somewhat modified form. We now know from theory (Fonseca dos Santos et al., 2007) that vibrational excitation enhances recombination, even for low vibrational states. Unfortunately, there are no direct measurements of such rates that would help to put this conjecture on a firmer basis. We conclude in this section that there are no afterglow measurements that give strong support for H3þ (v = 0) recombination coefficients sig­ nificantly smaller than those found in storage rings. Those afterglow measurements, in which the state of the ion was identified by spectro­ scopy, consistently yielded higher values. We now turn our attention to the question why many afterglow measurements have yielded higher recombination rates. 6.2 Afterglow Measurements That Yielded High Recombination Coefficients Most afterglow measurements, provided sufficient H2 was present in the gas mixture, yielded recombination coefficients that were higher by factors of 2—3 than those found in the storage ring experiments. The extensive compilation of data presented in the recent paper by Glosik et al. (2009a) shows quite clearly that the observed rate coefficients tend to increase with increasing neutral density (largely helium), which suggests that the recombination is enhanced in the presence of third bodies. The problem is that the conventional three-body collisionalradiative recombination mechanisms for atomic ions, in which either neutrals or electrons act as stabilizing agents, are far too slow to explain the observed three-body rate coefficients. In the next section we will explore more efficient third-body-assisted recombination mechanisms. 6.3 Third-Body Stabilized Recombination of H3þ There are several possible mechanisms that could make third-body effects on recombination more efficient in the case of molecular ions that recombine indirectly via intermediate resonant states that involve capture into high Rydberg orbitals. High Rydberg states are easily

Dissociative Recombination of H3þ Ions with Electrons

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perturbed by neighboring particles, in particular the electronic angular momentum can be altered by l-mixing collisions and the decay by dis­ sociation strongly depends on the electronic angular momentum. The resonant states that play a role in the binary recombination are primarily those in which the ion core is vibrationally excited by the JT interaction. Here the relevant electronic states have relatively low principal quantum numbers (around 6—8), and low angular momentum which makes predissociation fairly efficient compared to autoionization. If one assumes that all captured electrons predissociate, as is done in some simplified treatments (Jungen & Pratt, 2009), then electron capture is the rate limit­ ing step and any additional third-body stabilization mechanism will have no effect. On the other hand, if autoionization is not negligible, then l-mixing by third-body interactions may lead to states that are no longer capable of autoionization, but can be stabilized by further collisions. That would enhance recombination. A mechanism of this kind was once proposed (Gougousi et al., 1995) to explain H3þ recombination at a time when the binary recombination mechanism was not as well established as it is now. The l-mixing due to electrons was thought to be the most important ingredient. In hindsight, the proposed mechanism employed unrealistically long resonance lifetimes, which were based on experimen­ tal observations in merged-beam experiments. A different mechanism for a more efficient third-body-assisted recom­ bination process has recently been proposed by Glosik et al. (2009a,b). It shares some features (like l-mixing) with the model of Gougousi et al., but it focuses on resonant states formed by capture into rotationally excited core states, which form Rydberg states with higher principal quantum number (n = 40—80) and invokes l-mixing due to ambient neu­ tral atoms (helium in particular). These states do not usually contribute much to recombination since they tend to decay quickly by autoioniza­ tion, but they can have fairly long lifetimes (e.g.>10ps) and are thus good candidates for l-mixing. If one now had a further mechanism that stabilizes the population of these Rydberg molecules, i.e., renders them incapable of reverting to an autoionizing state, the overall recombination rate would be enhanced and the neutral density would be one controlling factor in the recombination in the afterglow plasma. Using theoretically calculated life­ times of the initially formed autoionizing states and estimates of the l-mixing efficiency due to helium atoms, and assuming that a large number of Rydberg states (principal quantum numbers from 40 to 100) contribute, the authors succeeded in deriving a three-body rate coefficient that comes close to the experimental value. However, the assumptions underlying his model are not realistic: Firstly, the authors’ estimate assumes a very high l-mixing efficiency of the helium atoms, that is appropriate only for small principal quantum numbers, while theoretical calculations (Hickman, 1978, 1979) show that the efficiency of l-mixing due to helium falls off

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rapidly with principal quantum number as n2.7 for n > 15. Secondly, if one invokes l-mixing due to helium atoms as the rate limiting step, one should also consider l-mixing by electrons, which is known to be faster by many orders of magnitude than that due to helium atoms, and its effi­ ciency rises with the fifth power of the principal quantum number (Dutta et al., 2001). In the range n = 40—80, in typical afterglows with ionization fractions of about 3  107, l-mixing by electrons would be more efficient by factors from 10 to 104 than by helium atoms. Thus, one would also expect a very efficient electron-assisted recombination process, for which, however, there is little experimental evidence. The relevant rate coefficients will be discussed later. The third problem is that this model leaves unan­ swered the question how the l-mixed states are eventually stabilized. Collisional stabilization by stepwise n-reducing collision with either elec­ trons or atoms may occur, but the efficiency of such collisions is not expected to be higher than for atomic systems such that the overall process is not likely to be faster than collisional radiative recombination of atomic systems. The model also has a more basic deficiency. It focuses on the lifetime of the initially formed rotational autoionizing resonances in low l-states and then assumes that higher l-states are exclusively populated by l-mixing. A more complete model should include three-body capture of electrons into all l-states by rotationally excited H3þ ions and its inverse, collisional ionization. For high n-states (with binding energies below 4 KT) colli­ sional ionization occurs on a time scale that is much shorter than the time scale of recombination in an afterglow plasma such that an equilibrium population of l-mixed states is always present. Any additional l-mixing mechanism hence is of no consequence. We will now consider a third mechanism for an efficient three-body mechanism that is an extension of the collisional dissociative process of Collins (1965), who realized that three-body capture of electrons into high Rydberg states of molecules can sometimes lead to predissociating states. Hence, the slow collisional and radiative descent from high-n to low-n states, the only stabilization route open to atomic systems, can be bypassed thus enhancing the overall recombination rate. Collins only treated a hypothetical model system with a single dissociative state, and did not consider effects of orbital angular momentum on the rate of predissociation, which should be included in a fuller treatment. In our model, we also invoke l-mixing but in the direction from high to low angular momenta and stabilization by predissociation of low l-states. We assume that in the plasma an equilibrium population of high Ryd­ berg states, denoted by H3, is maintained by three-body capture and its inverse, collisional ionization, i.e.,  e þ Hþ 3 þ M $ H3 ðnÞ þ M

ð21Þ

Dissociative Recombination of H3þ Ions with Electrons

115

and that the equilibrium constant K(n) of this reaction is approximately given by the Saha equilibrium ½H3 ðnÞ ¼ KðnÞ ¼ n2 3th eEn = kT ; ½Hþ 3 ne

ð22Þ

where n is the principal quantum number and th is the thermal de Broglie wavelength of the electrons at temperature T, i.e., th ¼

h2 ð2me kT Þ 1 = 2

ð23Þ

and En is the ionization potential of the Rydberg state. The assumption is made that three-body capture populates all l and magnetic substates ml evenly. This seems justified since the inverse process, collisional ioniza­ tion, depends only weakly on angular momentum of the Rydberg state. In the traditional theory of collisional radiative recombination of atomic ions, one now considers the departures from the thermal equilibrium due to the downwards collisional and radiative cascading transitions (Stevefelt et al., 1975). Under the conditions of the afterglow experiments discussed here (electron densities <4  1010 cm3, helium densities <3  1017 cm3), the effective binary rate coefficients due to either electron or helium stabilized recombination at 300 K are on the order of 109 cm3/ s, and make a negligible contribution to the binary recombination coeffi­ cient of 107 cm3/s. Since we are seeking a three-body mechanism that is far more efficient, we ignore all collisionally induced and radiative transitions. Instead, we focus on other mechanisms that stabilize H3(n). Any process, that stabilizes H3(n) with frequency vs, will enhance the overall recombination by the amount D
ðnÞ ¼ KðnÞ s ðnÞ:

ð24Þ

If many such states exist, the overall recombination rate will exceed the binary rate by the sum of D
over a range of n D
¼

nX max

KðnÞ s ðnÞ;

ð25Þ

nmin

and the effect may become comparable to the binary rate coefficient. The range of n will be discussed further below. For exploratory purposes, we assume that only s-states within a range of n predissociate on a time scale faster than l-mixing so that l-mixing becomes the rate limiting step. Let us first consider l-mixing due to electrons. It is known that electrons are very efficient in inducing l-mixing. The cross section for the process at an electron energy of 5 meV is approximately (Dutta et al., 2001)  0e ; mix ¼ 4:4  a02 n5 ;

ð26Þ

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slightly smaller (by a factor of 0.65) for electrons with thermal energy at 300 K. The corresponding rate coefficient is taken as k 0e ; mix ¼ ve  0e ; mix :

ð27Þ

This rate coefficient describes the transfer from a given l,ml state into any one of the other n2 — 1 states. What we need for our purpose is the rate of transfer from any of the n2 — 1 states to a particular state, namely l = 0, which will be smaller by the factor 1/(n2 — 1). For simplicity, we take the factor as 1/n2. Hence, the needed l-mixing rate ke,mix is taken as k0e,mix/n2 and rises with n as n3. Inserting the relevant numbers leads to an estimate of ke,mix =n3  2.7  109 [cm3/s]. The actual mixing rate vmix, i.e., the product ne ke,mix [1/s], is not necessarily equal to the rate limiting stabilization frequency in Equation (24), at least not for all values of n. If one made that assumption, the summation in Equation (25) would diverge strongly! One needs to take into account that the rate of predissociation of the s-states will be a declining function of n. A crude estimate may be based on the classical expectation that the rate at which an electron in an s-state will “collide” with the ion core is proportional to the classical orbiting frequency, which scales as 1/n3. This means that the l-mixing frequency increases as n3, while the predissociation frequency declines as 1/n3. If one views l-mixing and predissociation as two “conductances” in series, the combined inductance would rise as n3 at low n, but fall off as 1/n3 at high n, but we do not know a priori where the “crossover ” might be. Experimental measurements of H3(n) predissociation spectra (Mistrık et al., 2001) show that predissociation rates of s-states fall below 106 [1/s] around n = 40, which is to be compared to the expected l-mixing rate at an electron density of 1010 [cm3] of about 1.7  106 [1/s]. The exact numbers are not critical but it seems plausible that the summation in Equation (25) should be cut off somewhere around n = 40. If one now performs the summation from nmin = 12 to nmax = 40, one finds that electron stabilized recombination makes only a fairly small contribution to the overall recom­ bination. At the highest electron densities that are commonly used in afterglow experiments, ne = 4  1010 cm3, the effect would be to increase the binary rate by only about 1  108 cm3/s, larger than the enhancement by purely collisional radiative recombination, but still only a small part of the binary rate coefficient. Our estimates indicate that the electron density will play only a minor role in typical afterglow experiments of H3þ recom­ bination. This agrees with experimental observations. The situation is quite different when one considers helium atoms as third bodies. The estimates follow very much the same scheme as for electrons, but the l-mixing rate is now taken as kmix ; He ¼ 3:1  10  5

1 1 ½cm3 =s: n2 n2:7

ð28Þ

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This value is obtained from the cross sections given by Hickman (1978, 1979, and 1981), multiplied with the average velocity for H3þ/He collisions at thermal energy (300 K). The fast fall-off with increasing n makes the summation in Equation (25) convergent at high n so that the high-n cutoff does not matter much. However, it is not obvious where to cut off the summation at low n. We chose a low-n cutoff at the value of n for which the ionization potential of that state exceeds thermal energy by a factor of 4 or greater. The summation then leads to the estimate that the recombi­ nation rate at n(He) = 3  1017 [cm3] (10 Torr at 300K) would exceed the binary rate by 8  108 [cm3/s] (see Figure 13). The corresponding threebody rate coefficient with helium at 300 K would be 2.6  1025 [cm6/s], which agrees with the experimental value of (2.5 + 1.2)  1025 [cm6/s] (Glosik et al., 2009a), far better than one has a right to expect. The corresponding rate coefficient for D3þ would be (2.2 + 1.2)  1025 [cm6/ s], very similar to the measured values of (1.8 + 0.6)  1025 [cm6/s] (Glosik et al., 2009b). Our model would predict only a small increase of the three-body coefficient at reduced temperatures, by about 50% at 100 K. We note that helium is particularly effective in inducing l-mixing because the electron—helium momentum transfer cross section is large and nearly independent of energy. By comparison, neon should be far less effective (see Hickman, 1978). There is only a single measurement (Macdonald et al., 1984) of H3þ recombination in neon buffer gas (at 20 Torr) which actually yielded a significantly lower recombination rate than a very similar measurement in helium. We also consider a third stabilization mechanism that involves the hydrogen gas that is usually present in afterglow experiments of H3þ recombination. It is known that H5þ ions recombine much more rapidly 3

αeff [10−7cm3/s]

2.5 2 1.5 1 0.5 0

0

1

2

3

4

5

6

7

8

[He] 1017cm−3 Figure 13 Observed dependence of the H3þ recombination coefficient at T = 300 K on the experimental helium density. Squares and triangles: data from Glosik (2009a). Cross: data from Leu et al. (1973). The line indicates the density dependence expected from the model described in the text

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with electrons with a 300 K rate of a(H5þ) = 1.8  106 [cm3/s] (Macdonald et al., 1984). While the equilibrium concentration of H5þ is small compared to that of H3þ, transient H5 Rydberg molecules could be formed in collisions of H3 with H2, i.e., a reaction of the type H3 þ H2 $ H5

ð29Þ

The forward rate coefficient should be comparable to that of fast ion—molecule reactions, i.e., k(H2) 2  109 cm3/s. Once formed, these H5 probably predissociate very rapidly, since the recombina­ tion of H5þ ions is extremely fast (Macdonald et al., 1984). If one adds this as an additional stabilization mechanism to H3þ recombination, the effective stabilization frequency that enters Equation (24) would be given by [H2]k (H2). The proper choice of nmax in the summation over n is not obvious in this case. To reproduce the large experimental value (Gougousi et al., 1995) of the three-body rate of about 2.5  1023 cm6/s, the summation would have to include n values up to about 70 and this does not seem to be unreasonable. This may also explain the rather high value obtained by Amano (1990), how­ ever, that experiment also employed very high electron densities (5  1011 cm3), and it is difficult to separate possible effects of elec­ trons from those of H2. The experiment did not show a measurable effect of H2 on the observed recombination rate. The foregoing estimates indicate that H3þ recombination in an after­ glow plasma, at least in part, involves a mechanism in which the neutral gas density plays a role. This agrees with experimental findings. Helium seems to particularly effective in promoting recombination. One would not expect a significant effect of the electron density if it is below 4  1010 cm3. Again, this agrees with experiments in the range of electron densities commonly present in afterglows. The three-body model that we propose is by no means complete: The assumption that the equilibrium concentrations of the Rydberg states are only slightly perturbed is a serious simplification and ignores competition between different stabilization mechanisms. Constructing a more rigorous model, however, looks like an exceedingly complicated task.

7. COMPARISON OF STORAGE RING DATA The storage ring measurements have yielded remarkably consistent data over the years. While some of the early measurements (Jensen, 2001; Larsson et al., 1993) gave slightly higher cross sections than was found in later work, those were shown to be due to rotational excitation of the

Dissociative Recombination of H3þ Ions with Electrons

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ions. Nearly complete control of the rotational population was eventually achieved by using either a supersonic jet expansion ion source (CRYR­ ING) or a cryogenically cooled radio-frequency multipole trap (TSR). The actual rotational populations were verified by optical absorption mea­ surements, and it was shown that only the two lowest rotational states were populated. As far as is known from experiment, the rotational populations in the circulating beam do not differ significantly from those injected from the ion source. If one compares the two results obtained in the CRYRING (McCall et al., 2004) and in the TSR (Kreckel et al., 2005), one is immediately struck by the fact that the observed energy dependence is nearly identical in both experiments. It should be noted, however, that the TSR data were normalized to the CRYRING data at an energy of 10eV. The absolute magnitude of the cross sections and the overall dependence on energy are well reproduced by theory which seems to say that the binary recom­ bination cross section has been firmly established. However, if one com­ pares the finer structure of the measured and calculated cross sections (see Figure 14), one notices that the theoretical results show several narrow peaks that are not present in the experimental data. It is not clear at this time if this discrepancy is due to approximations made in the theory or if it indicates a possible systematic problem in the

Rate coefficient (cm3/s)

1.00E−06

1.00E−07

1.00E−08

1.00E−09 1.00E−04

1.00E−03

1.00E−02

1.00E−01

1.00E+00

Energy (eV) Figure 14 Comparison of the experimental CRYRING data of McCall et al. (2004) (dashed line) to the theoretical results of Fonseca dos Santos (2007) (solid line), for a rotational temperature of 13 K. The theoretical data have been convoluted with the experimental energy resolution and have been corrected for the “toroidal effect.” Drawn from data supplied by M. Larsson and V. Kokoouline. The dotted line represents results of the approximate Jahn–Teller theory of Jungen and Pratt (2009)

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measurements. Third-body-assisted recombination of the kind that seem to occur in afterglows can almost certainly be excluded since the relevant particle densities are lower by many orders of magnitude. Such effects have been considered in storage ring experiments of dielectronic recom­ bination to explain the higher-than-expected recombination cross sec­ tions in the limit of very energies (Pajek and Schuch, 1997, 1999). Since DR is much faster than dielectronic recombination, it should be much less affected by third-body effects and such effects are probably negligible. One may also ask if the magnetic field in the interaction region of storage rings (570 mT in the TSR, 300 mT in the CRYRING) and small electric stray fields (estimated to be on the order of 1 V/cm) have an effect on the recombination. Wolf et al. (2006) mention such effects without reaching a conclusion as to their importance. Several possibilities exist: If for some reason l-mixing due to small fields were to occur on a time scale comparable to that of autoionization or predissociation, this could affect the relative importance of the two decay channels and alter the recombination rate. If l-mixing should produce long-lived high-l states of H3, those presumably would be field-ionized in the demerging magnets and the net effect would be that the observed recombination rate would be too small. If, on the other hand, l-mixing enhances the rate of predissociation, then the observed rates would come out larger than they would be in the absence of fields. One experiment (Larsson et al., 1997) et al. has been performed (for D3þ) in which small electric fields (on the order of 30 V/cm) were deliberately added and those gave negative results. The negative finding is not totally conclusive as was mentioned by the authors of that study. Ideally one should remove all stray fields, rather than adding to them, which in practice, of course, is impossible to achieve. Some theoretical calculations have been made to assess l-mixing due to small static fields (Chao et al., 1998) in the context of ZEKE spectroscopy. It appears that small fields on the order of a few V/ cm can induce l-mixing in Rydberg states, but the time constants are found to be on the order of 1—10 ns, which is much longer than likely lifetimes of H3 autoionizing states. Hence, in the absence of evidence to the contrary, we do not believe that stray fields will have significant effects but the question may deserve further scrutiny. The same conclusion has been reached by the storage ring experimenters (Wolf, private communication).

8. H 3þ PRODUCT BRANCHING The first measurements of the product branching ratios  ! Hþ 3 þe ! !

H þ H þ H ð
Þ ðÞ H2 þ H ð Þ H3

ð30Þ

Dissociative Recombination of H3þ Ions with Electrons

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were carried out by an extension the MEIBE merged-beam apparatus in which a grid was placed in front of the energy-sensitive detector. These experiments (Mitchell et al. 1983) were extremely challenging, suffered from fairly poor signal-to-noise ratio, and the vibrational state of the recombining ion was not known very well. Florescu-Mitchell and Mitchell (2006) provide a synopsis of the results of this work: channel
accounted for typically 52%, channel  for 40%, and surprisingly channel for the remaining 8%. In later work it was found that the third channel appeared to increase when the field strength in the demer­ ging region of the experiment was reduced which seems to indicate that a fraction of the H3 were field-ionized in that region. If that interpreta­ tion is correct, which is still not clear at this time, then the lifetime of these metastable particles must have been on the order of or larger than about 100 ns, the flight time from the interaction region to the detector. It was this observation that motivated Gougousi et al. (1995) to propose that the recombination of H3þ in afterglow plasmas involved stabiliza­ tion of metastable H3 by subsequent reactions with hydrogen molecules. Later work by Datz et al. (1995) using the CRYRING provided more detailed and presumably more accurate energy-resolved branching fractions for H3þ recombination. At energies below 0.3 eV, decay channel
into three H-atoms was found to account for 75% of the total, channel  for 25 %, and channel was not observed at all. Datz et al. state that metastable H3 Rydberg molecules with principal quantum numbers below about 7 should have survived without being field-ionized while passing through the demerging magnet and hence should have been detectable. The authors concluded that formation of high Rydberg states probably does not contribute much to the recombination. The branching fractions observed by Datz et al. are very well reproduced by the statistical model of Strasser et al. (2003). Very detailed two-dimensional investigations of the kinematics of the dissociation into three H atoms and vibrational distributions of the H2 product were also reported by Strasser et al. (2001, 2002a, and 2002b). It is difficult to determine precise branching fractions for H3þ recombination afterglow measurements since there are numerous extraneous sources of H atoms. In the only such experiment (Johnsen et al., 2000) that has been performed, the H atom yield was mea­ sured by converting the H atoms to OH by reacting H with NO2 and then measuring the OH concentration using laser-induced fluores­ cence. The results indicated that branch
accounts for 63% of the total, roughly compatible with the storage ring data. It was not possible to measure the H2 yield directly or to find evidence of long-lived H3.

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9. ISOTOPE EFFECTS It is difficult to perform afterglow studies of the recombination of H2Dþ and HD2þ, but several storage ring measurements have been published. A detailed discussion and references to theoretical papers can be found in the book by Larsson and Orel (2008). It appears that the agreement between theory and experiment is less satisfactory, especially for HD2þ, and that more theoretical work is needed. D3þ has been studied by both afterglow (Glosik et al., 2009b; Gougousi et al., 1995; Laube et al., 1998) and storage ring methods (Larsson et al., 1997). Most of the experimental data indicate that D3þ recombines more slowly than H3þ by a factor of 2—3, although one afterglow experiment (Laube et al., 1998) gave nearly the same value for both ions. Arguing from a simplified theoretical treatment, Jungen and Pratt (2009) suggest that the recombination coefficient of D3þ should be smaller than that of H3þ by a factor of H2, a weaker isotope effects than predicted by the theory of Kokoouline and Greene (2003).

10. CONCLUSIONS Largely motivated by applications to ionospheric physics, astrophysics, and technical applications of cold, nonequilibrium plasmas, studies of DR have been an ongoing communal effort for many years. In the case of H3þ, results have often been difficult to reconcile, and while the discus­ sions have been contentious at times, most often they were conducted in the spirit of collegiality, and we have now reached a state that is close to a consensus. Larsson et al. (2008), in their concise and well-presented review of the status of H3þ recombination studies, ask the question if “the saga has come to an end.” They conclude that the “saga” is not quite finished but that it is approaching a satisfactory finale in which experiment and theory converge to a common picture of this important process. We largely concur with their assessment. As far as binary recombination is concerned, the consistency of storage ring data obtained in different experiments strongly suggests that the data are reliable and are applicable to conditions in the interstellar medium. The absolute magnitude of the coefficients is well reproduced by theory, but it would be highly desirable to refine the theoretical treatments to the point where they accurately reproduce the finer structure of the observed energy dependence of the cross section. Larsson et al. in their review point to the outstanding problem of reconciling the afterglow and storage ring measurements, in particular those that yielded coefficients far below the binary value. We have examined the data in some detail and find that those measurements

Dissociative Recombination of H3þ Ions with Electrons

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were not performed under conditions that permit an unambiguous inference of substantially lower recombination coefficients. We also conclude that recombination in a plasma containing substantial densi­ ties of neutral and charged particles involves additional third-body­ assisted mechanisms that should be taken into account in applications other than those to highly dilute interstellar media. In this we agree with the conclusion of Glosik et al. (2009a and 2009b) that neutral particles play an important part in such processes, but we propose an alternate, although still very approximate, model that seems to fit observations fairly well.

ACKNOWLEDGMENTS The authors are indebted to several colleagues for helpful discussions, providing alternate viewpoints, and providing data files, in particular to M. Larsson, S. Kokoouline, J. Glosik, A. Wolfe, and C. Greene. SLG gratefully acknowledges support from the US National Science Founda­ tion under grant ATM-0838061. This research is also supported by NASA Grants NNX08AE67G and NNX09AQ73G. SLG dedicates this chapter to the memory of his beloved wife and precious companion, Susan L. Greenblatt, who in spite of her serious illness continued to encourage this work but unfortunately did not live to see its completion.

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CHAPTER

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Permanent Electric Dipole Moments of Atoms and Molecules Tim Chupp FOCUS and MCTP, Physics Department, University of Michigan, Ann Arbor, MI 48109, USA

Contents

1. 2.

Introduction Historical Perspectives 2.1 The Neutron EDM 2.2 Early Atomic Beam Measurements 2.3 Paramagnetic Atoms: Cesium, Thallium,

and Metastable Xenon 2.4 Early Molecular Beam Experiments
TlF 2.5 Diamagnetic Atoms
Xenon and Mercury 2.6 EDM Measurements in Other Systems 2.7 Summary of EDM Measurements to Date 3. Contemporary Theoretical Motivations
The

Standard Model and Beyond 4. EDM Measurements 4.1 Precision 4.2 Systematic Effects and False EDM Signals 5. Contemporary Experiments 5.1 Liquid Noble Gases 5.2 Cold Alkali-metal Atoms: Fountains and

Lattices 5.3 Paramagnetic Molecules 5.4 Rare Atoms 5.5 Other Directions 6. Conclusion Acknowledgments References

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130 Abstract

Tim Chupp

The permanent electric dipole moment (EDM) of a particle or system is a separation of charge along the direction of the total angular momentum of the system and arises from elementary particle interactions that directly violate parity and time-reversal symmetry. Assuming symmetry under combined charge-conjugation, parity, and time-reversal transformations, an EDM is a signal of simultaneous violation of charge-conjugation and parity symmetry (CP violation), and thus probes very weak interactions in the background of much stronger interactions that bind the atom and its nucleus. CP violation arises naturally in the Standard Model of strong, weak, and electromagnetic interactions through two mechanisms: the phase of a complex amplitude in the weak interactions, the CabibboKobayashiMaskawa phase, which is constrained by measurements with neutral K and B mesons, and the vacuum phase of the strong interaction,
QCD, which is constrained by EDM measurements. The Standard Model does not, however, appear complete. For example, the predominance of matter over antimatter in the universe is not compatible with Standard Model mechanisms of baryogenesis, and CP-violating physics beyond the Standard Model’s interactions could produce the observed baryon asymmetry as well as detectable EDMs. For over five decades, experimenters have measured steadily smaller upper limits on the EDMs of atoms, molecules, and elementary particles, and the searches continue with new motivations and new techniques that promise improved sensitivity to CP violation. In this article, we set the stage for contemporary efforts and discuss current and near-term experimental endeavors to measure EDMs in a variety of systems.

1. INTRODUCTION The experimental study of charge-conjugation/parity (CP) and timereversal (T) invariance violation is over 45 years old, born with the dis­ covery that the neutral kaon system is a mixture of CP eigenstates. Remarkably, CP violation remains one of the most important open issues in elementary particle physics and motivates the searches for permanent electric dipole moments (EDMs). An EDM in a quantum mechanical system can be defined as the projection of the charge distribution along the total angular momentum ~ J in the state having magnetic quantum number mJ = J, that is

Permanent EDMs of Atoms and Molecules

ð rÞd3 r: dz ¼ zjj ð~

131

ð1Þ

Thus, ~ d ¼ gd~ J, where gd is analogous to the magnetic moment g-factor. ~ Under P, d changes sign, but ~ J does not, and under T, ~ J changes sign, ~ but d does not. Thus, gd must be odd under both P and T transforma­ tions, and an EDM is both T-odd and P-odd. We can think of the EDM as arising from the electrical polarization of the system along ~ J induced by elementary particle interactions that violate P and T, and assuming CPT invariance, violate CP. The contemporary view of EDM measurements is that they probe new physics, that is, physics beyond the Standard Model of elementary particle interactions. The CP viola­ tion that arises naturally with any new physics may give rise to EDMs of fundamental particles, that is, leptons and quarks, as well as to composite systems (neutron, proton, nuclei, atoms, and molecules) due to the constituent EDMs and due to CP violating interactions among the constituents. For more than five decades, experimenters have been searching for EDMs, first in the neutron beam experiments of Ramsey and Purcell and subsequently in experiments on atomic and molecular systems as well as elementary particles. Over this time, the experiments have become increasingly more sensitive for many reasons including better statistical precision, improved understanding and control of systematic errors and their uncertainties, new technologies and techniques such as ultracold neutrons (UCN) and lasers, and new theoretical insights that have uncov­ ered systems or observables that are potentially much more sensitive to CP violation. The chase has been a long one, but the enthusiasm has been sustained and has even grown, and the motivations for such studies provided by new developments in elementary particle physics have become stronger. With the most recent measurement of the atomic EDM of 199Hg (Griffith et al., 2009) the bar is set high. The result,1 d199Hg = (0.49 + 1.29 + 0.76)  1029 e-cm, is two orders of magnitude more sensi­ tive than for any other system and quantitatively constrains a combi­ nation of parameters that characterize T violation in an atom and its nucleus. The 199Hg result, which is consistent with no EDM and no T violation, cannot by itself constrain each parameter of the minimal set of CP violating parameters. This set of parameters includes the vacuum phase of the strong interaction,
QCD, a Standard Model parameter that is not constrained by the theory or by other 1 Results are presented with errors provided by the authors in the original papers. In the cases where both statistical and systematic errors are presented, the statistical error will come first, in this case, d199Hg = (0.49 + 1.29(stat) + 0.76(sys))  1029 e-cm.

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experiments. Significant model dependence is necessary to constrain the physics that generates EDMs, even when the 199Hg result is combined with the neutron and paramagnetic thallium atom EDM measurements. The conclusion is that a set of new experimental approaches is required that will improve sensitivities to the sources of CP violation in many systems. This requires exploiting enhance­ ments that arise, for example, in paramagnetic molecules and octu­ pole deformed nuclei, and exploring new techniques and technologies including cold atoms and molecules at the same time that the neutron EDM is pushed to greater sensitivity. There have been many fine summaries, reviews, and monographs on EDMs of specific systems as well as more general overviews
both theoretical and experimental (e.g., Ramsey, 1982; Khriplovich & Lamoreaux, 1997; Sandars, 2001; Pospelov & Ritz, 2005); and there is no substitute for the information provided in the authors’ original papers. The goal here is therefore to drop a pin between historical and contemporary efforts in the landscape of EDM measurements that span atomic, molecular, nuclear, and even high-energy physics. From this perspective, we aim to unify past with present across the sub­ fields of physics and to emphasize the importance of pushing ahead on several fronts.

2. HISTORICAL PERSPECTIVES 2.1 The Neutron EDM The first direct EDM measurements were motivated by the suggestion (Purcell & Ramsey, 1950) that parity symmetry should be tested by searching for the neutron EDM. They undertook the experiment at Oak Ridge using a beam of neutrons. The first result was astonishingly sensitive, probing at the level of 1020 e-cm, less than 0.1 ppm of the size of the neutron. The neutron EDM was apparently consistent with zero, and it was recognized that this did not imply symmetry under parity transformations, which by then had been shown to be violated (Wu, 1957), rather the apparent symmetry was time-reversal or CP symmetry (Landau, 1957). In the modern theoretical context, the neu­ tron EDM arises due to a combination of quark EDMs and CP violation in the couplings of the quarks (Pospelov & Ritz, 2005). These in turn relate to fundamental interactions that arise either due to Standard Model physics or many possible extensions of physics beyond the Standard Model. As shown in Figure 1, the neutron-EDM sensitivity has been steadily improved
by six orders of magnitude over nearly six decades
and

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10−19

dn upper limit (e -cm)

10−20 10−21 10−22 10−23 10−24 10−25 10−26 1960

1970

1980

1990

2000

2010

Publication date Figure 1 History of the neutron EDM over six decades in time and sensitivity. Upper limits, determined as described in Note 2, are shown. Solid squares are neutron beam experiments at Oak Ridge (Smith et al., 1957; Miller, 1967; Baird et al., 1969; Dress et al., 1977; respectively), the hashed square is a neutron scattering experiment by Shull (Shull & Nathans, 1967), and the open squares are UCN bottle experiments at Leningrad Nuclear Physics Institute (Altarev et al., 1980, 1981) and ILL (Smith et al., 1990; Harris et al., 1999; Baker et al., 2006)

the search continues.2 In the 1980s, the cold-neutron beam techniques reached their statistical and systematic limits. This was followed by the development of UCN techniques (Golub & Pendlebury, 1979; Sha­ piro, 1970), which allowed neutron-EDM experiments to be carried out using storage cells or bottles. UCN provide two significant advantages: much longer storage times with correspondingly narrower line-widths and suppression of the motional ~ ~ E effect, which couples to the (parity-allowed) magnetic moment potentially producing a false EDM signal. Another notable improvement came with the introduction of a second species or comagnetometer in the neutron storage volume. The comagnetometer was made practical by the advances of atomic-EDM techniques, specifically with 199Hg, which is discussed in Section 2.5, 2 Note on upper limits: many authors provide upper limits on |d| with 90% or 95% confidence level using different approaches. In this chapter, we have chosen to compile the results as the magnitude of the value furthest from zero within a 90% confidence interval around a central value. For example, for the most recent neutron-EDM measurement (Baker et al., 2006), dn = (0.2 + 1.5 + 0.7)  1026 e-cm. Combin­ ing the statistical and systematic errors in quadrature, assuming a Gaussian error distribution, and using 1.64, the 90% confidence interval spans (2.5 £ dn £ 2.9)  1026 e-cm. We report this as an upper limit of 2.9  1026 e-cm. This is not strictly a limit on |dn|, which would imply (2.9 £ dn £ 2.9)  1026 e-cm, that is, a 92% confidence interval; however, assigning an upper limit to |dn| is not truly equivalent to the restriction on a measured quantity that must be positive, that is, a rate or branching ratio as discussed, for example by Cowan, 2008; Feldman & Cousins, 1998.

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though in the original idea (Ramsey, 1984) 3He was suggested as the comagnetometer. Neutron measurements with UCN bottles continue, and several major efforts are underway worldwide. The “CryoEDM” experiment (van der Grinten, 2009) at Institut Laue-Langevin (ILL) in Grenoble, France, has been constructed to take advantage of several low-tempera­ ture phenomena: (1) the UCN are produced by down scattering a fixed­ energy-cold-neutron beam in superfluid 4He to provide much higher numbers of UCN (Golub & Pendlebury, 1975; Golub et al., 1983); (2) the dielectric strength of the superfluid 4He allows a significantly higher electric field than a vacuum bottle (Gerhold, 1998); (3) superconducting magnetic shielding will be used to mitigate magnetic noise from sources outside the shields. In addition, low-temperature-SQUID magnet­ ometers can measure and control the distribution of magnetic fields around the storage cell. The CryoEDM experiment’s aim is to reach a sensitivity of 1028 e-cm (van der Grinten, 2009). At the Paul Scherrer Institute in Zurich, a new UCN source based on spallation neutron production, followed by thermalization in D2O and further cooling in solid deuterium is anticipated to provide 10—100 times greater UCN density than the completed ILL neutron bottle measurements (Atchison et al., 2005). This statistical improvement combined with higher electric field and improvements to the room-temperature approach are also expected to provide sensitivity in the 1028 e-cm range. The plans include an improved comagnetometer system possibly based on xenon. Similar efforts are planned at Petersburg Nuclear Physics Insti­ tute (Serebrov et al., 2009) and KEK/TRIUMF. A new United States effort at the Oak Ridge Spallation Neutron Source (SNS) also projects one to two orders of magnitude greater sensitivity and measurement of dn at the 1028 e-cm level. In addition to production of UCN by down scatter­ ing in superfluid 4He, the experiment plan envisions using 3He both as a comagnetometer and detector of neutron spin-precession (Golub & Lamoreaux, 1994; Ito, 2007). 2.2 Early Atomic Beam Measurements There are several reasons that the neutron was the focus of the early EDM measurements
most importantly it is neutral and would not be acceler­ ated from the measurement region by a large static electric field. Neutral atoms can also be contained, but the charged constituents, the electron and nucleus, are significantly shielded from the large external field by the polarization of the atom
an effect embodied in Schiff’s theorem (Commins et al., 2007; Schiff, 1963). As discussed by Schiff, the shielding is not perfect because the nucleus has finite size and structure, which is probed by the atomic electrons so that sensitivity to T and P violation in

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the nucleus scales as roughly Z2. In the case of paramagnetic atoms, the spin-orbit force leads to an effective enhancement of the sensitivity to an electron EDM that is approximately proportional to Z3a2. Moreover, an atomic EDM can arise due to T and P violation in the electron—nucleus interaction
a neutral current that may have a scalar or tensor nature, and these effects also increase with Z. For these reasons, atomic and molecular beam apparatus were adapted to search for EDMs. Limits on the proton and electron EDMs were established by studies of corrections to the Lamb shift (Sternheimer, 1959) and scattering of elec­ trons from helium (Goldemberg, 1963); however, the first direct atomic EDM experiment
measurement of the frequency shift of the cesium atom EDM in an atomic beam with a modulated electric field
were undertaken by Sandars and collaborators (Sandars & Lipworth, 1964). There are many challenges to such a measurement that have led to the techniques applied to contemporary undertakings. First, the atomic beam, traveling at several hundred meters per second, transited an appa­ ratus of length less than a meter so that line-widths of several kilohertz were observed (Ramsey’s separated oscillatory field technique was used). Second, due to the unpaired electron, the cesium atom has a large magnetic moment that couples both to external magnetic fields and to ~ E. The magnetic field produced by the motional magnetic field ~ Bm ¼ ~ any leakage currents that arise due to the high voltage would change with the modulation of the electric field and could provide a false­ EDM signal. Misalignment of the applied magnetic field and ~ E also produces a false signal. By splitting the line by more that 10,000, the frequency shift sensitivity was of order 0.1 Hz, and they obtained the result dCs = (2.2 + 0.1)  1019 e-cm with an electric field up to 60 kV/cm. The error is statistical only, and the finite EDM signal is attributed to the motional effect due to a misalignment of 10 mrad (Sandars & Lipworth, 1964). Subsequent work using other alkali-metal species with lower Z and less sensitivity to T-odd/P-odd interactions in order to monitor magnetic field effects
now called a comagnetometer
led to the result dCs = (5.1 + 4.4)  1020 e-cm (Carrico et al., 1968). Shortly after that publication Weisskopf et al. (1968) presented a significantly improved result based on a longer interaction region, correspondingly narrower resonance lines and a sodium comagnetometer: dCs = (0.8 + 1.8)  1022 e-cm (Weisskopf et al., 1968). As noted above, the cesium EDM could arise from a combination of sources including an electron EDM, T and P violation in the nucleus, and a T-violating mixing of opposite parity states of a scalar, pseudoscalar, or tensor nature. For paramagnetic systems, the scalar contributions are likely to be several orders of magnitude stronger than tensor and pseudoscalar contributions, given comparable strength of the intrinsic couplings (Ginges & Flambaum, 2004). As discussed in Section 2.5, the nuclear

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contributions are strongly constrained by other measurements, thus for the cesium atom, one can write (Bouchiat, 1975; Carrico et al., 1968; Ginges & Flambaum, 2004) dCs » e dd þ kCS

ð2Þ

where CS is a combination of scalar couplings of the valence electron to the proton and the neutron; for cesium, CS = 0.41csp þ 0.59csn (Bouchiat, 1975). A variety of calculations place e in the range þ100 to þ140, and kCS » 7  10  19 e-cm for cesium (Ginges & Flambaum, 2004; Nataraj, 2008). Most authors use the single EDM result to set limits on the individual contributions assuming all other contributions are negligible, for exam­ ple, if CS = 0, then the result of Weisskopf et al. (1968) could be interpreted as a 90% limit on the electron EDM of de < 3.3  1024 e-cm (see Note 2 for a discussion of upper limits). There is, however, no substantial reason to assume that the contribution to the cesium atomic EDM due to scalar interactions is much smaller than the electron EDM contribution. Thus, measurements in more than one system sensitive to both de and CS would be required to truly constrain possible sources of an atomic EDM. We also note that an atomic EDM can arise due to higher-order T-odd and P-odd nuclear moments when the nucleus has spin  1. The magnetic quadru­ pole moment, a P-odd, and T-odd distribution of currents in the nucleus would induce an atomic EDM by coupling to an unpaired electron (Derevianko, 2005). A nuclear magnetic quadrupole moment effect could induce an atomic EDM in 133Cs, which has nuclear spin 7/2 (Dmitriev et al., 1994). 2.3 Paramagnetic Atoms: Cesium, Thallium, and Metastable Xenon The cesium-atomic-beam EDM experiments were ultimately limited by the line widths, count-rate limitations, and by systematic errors due to motional magnetic field effects, though the atomic beams machines pro­ vided the capability to use other, lighter alkali-metal species, that is, a comagnetometer, significantly reducing the motional-field systematic errors. One approach was to develop and refine atomic or molecular beam experiments in systems with significantly enhanced sensitivity to T-odd/P-odd interactions relative to cesium, that is, a metastable atomic state in xenon, a heavier atom, thallium, used by Commin’s group and a molecular beam of TlF used by Sandars and independently by Ramsey. Sandar’s group performed an atomic beam experiment to search for an EDM in the 3P2 metastable state of xenon with a comagnetometer beam of krypton (Player & Sandars, 1970). In the strong applied electric field, the parity-allowed splittings are proportional to m2J E2 and to the magnetic field component parallel to ~ E. The EDM signal would be a splitting

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linear in ~ E; however, transitions that change the magnetic quantum number mJ are not practical due to the sensitivity of the E2 term to a change of the magnitude of the electric field. Thus, the D|mJ|= 0 transi­ tion mJ = 1 ! mJ = þ1 was measured. With the xenon—krypton compar­ ison, the difference of EDMs was found to be jdXe  dKr j¼ð0:7 – 2:2Þ  10  22 e-cm:

ð3Þ

For xenon and krypton, e  130 and e  20, respectively. Another approach was the vapor cell experiment developed by Hunter and collaborators (Murthy et al., 1989). The confined atoms provided much narrower resonance line widths, approximately 50 Hz, and also greatly mitigated motional field effects. Though a comagnetometer was not practical in the vapor cell, the leakage currents were directly measured and set the systematic-error uncertainty in the final result dCs ¼ ð1:8 – 6:7 – 1:8Þ  10  24 e-cm:

ð4Þ

Commins and colleagues (1994) developed a vertical counter propagating thallium atomic beam and subsequently added sodium beams as a comagnetometer (Regan et al., 2002). The two beams moving in opposite directions through the same volume would mitigate the ~ ~ E effects as well as a geometric phase effect (Barabanov, 2006; Commins, 1991; Pen­ dlebury et al., 2004). The most recent result can be interpreted as dTl ¼ ð4:0 – 4:3Þ  10  25 e-cm:

ð5Þ

The results for dCs and dTl can be combined using Equation (2) to con­ strain de and CS. We use e = 133 and kCs = 7  1019 e-cm for cesium and e = 585 and kCs = 5  1018 e-cm for thallium (Ginges & Flambaum, 2004). The result is de = (0.4 + 1.4)  1025 e-cm, and 5 CS = (0.4 + 1.6)  10 , with an upper limit on de of 2.7  1025 e-cm. The result reported in Regan et al. (2002), |de| < 1.6  1027 e-cm, assumes CS = 0, which can be considered model dependent in the sense that this assumption would only be valid in the context of specific models of T- and P-violating interactions. The tensor interaction of electron and nucleus and the nuclear Schiff moment also contribute to atomic EDMs; however, these are suppressed in paramagnetic atoms compared to dia­ magnetic atoms and molecules and are therefore discussed in more detail in Sections 2.4 and 2.5. 2.4 Early Molecular Beam Experiments
TlF Molecular beam experiments using TlF were pursued by Sandars (Harri­ son et al., 1969; Hinds & Sandars, 1980), Ramsey (Wilkening et al., 1984), and Hinds (Cho et al., 1991; Schropp et al., 1987). For molecular beams, the

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systematic errors associated with the ~ ~ E and leakage current effects are mitigated by using a relatively small applied electric field to align the intermolecular axis. This results in a large internal electric field at the thallium nucleus (Coveney & Sandars, 1983). The experiment is set up to detect an alignment of a spin or angular momentum along the electric field by detecting precession around the internuclear axis, that is, the frequency shift when the relative orientation of applied electric field and magnetic fields are reversed. When the averaged projection of the thallium nuclear spin on the internuclear axis is taken into account (hcos
i = 0.524), a frequency shift for full polarization is determined to be d = (0.13 + 0.22)  103 Hz. With the applied elec­ tric field of 29.5 kV/cm, this is interpreted as a permanent dipole moment of the thallium molecule of dT1F ¼ ð1:7 – 2:9Þ  10  23 e-cm:

ð6Þ

For TlF, the electron spins form a singlet, but both stable isotopes of thallium (203Tl and 205Tl) have J = 1/2þ, and the dipole distribution in the nucleus would be aligned with the spin through T and P violation. This gives rise to the Schiff moment, ~ S, which is probed by the molecular electrons through the interaction ! H ¼ 4r ð0Þ  ~ S; ð7Þ ! where r ð0Þ is the gradient of the electron density at the nucleus. The Schiff moment is the mean-square radius of the difference between the EDM and charge distributions given by 1 1 ~ ri hr2 ihe~ ri: S ¼ her2~ 10 6

ð8Þ

This can be viewed as an effective electric field in the nucleus that has a T- and P-violating permanent projection along ~ J, the total angular momentum of the nucleus (Flambaum, 2002) and thus produces a force on the electrons that is canceled by the long-range Coulomb interaction with the nucleus. (Schiff’s theorem has been reformulated in work show­ ing that these formulas are approximations that may not be justified (Liu et al., 2007; Sen’kov, 2008).) Thus, T violation induces the molecular EDM through a series of mechanisms originating with CP-violating interaction among the nucleons, or more fundamentally the nucleon constituents (quarks and gluons). For example, CP violation evolves from a fundamental level (e.g.,
QCD) to the phenomenological quark—quark or quark—gluon level to the nucleon—nucleon (NN) level, and so on: Fundamental ! qq or qg ! N or NN ! ~ S ! atomic=molecular EDM

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The Schiff moment that would result from a CP violating NN interaction in 203/205Tl of course depends on nuclear structure considerations, but has been schematically parameterized as STl = (1.2pp  1.4nn)  108 e-fm3 (Flambaum & Khriplovich, 1985). An alternative (and the original) interpretation is based on the obser­ vation that in the odd-A thallium isotopes, one proton remains unpaired and can induce the molecular EDM through both the resulting Schiff moment and through magnetic interactions (Coveney & Sandars, 1983). Separating these, the proton EDM would produce a magnetic contribu­ p-mag tion to a molecular EDM of dT1F ¼ 0:13dp , and a contribution to the Schiff moment that would produce a molecular EDM estimated to be p-vol dT1F ¼ 0:46dp . The TlF molecular EDM can also arise from the electron EDM and from P- and T-violating electron—hadron interactions, that is, scalar (CS) and tensor (CT). As discussed in Section 2.3, paramagnetic systems are more sensitive to CS and diamagnetic systems such as TlF are more sensitive to CT. Combining these sources dTIF ¼ 0:13dp þ 81de þ ð1:1CT þ 0:03CS Þe-cm þ7:410  14 STl cm=fm3 :

ð9Þ

The proton-EDM contribution to the Schiff moment is not explicitly written in Equation (9), rather it is part of STl. From Section 2.3, de is less than 2.7  1025 e-cm, and the 199Hg result constrains the Schiff moment to less than 1012 e-fm3. If CT and CS are comparable in magni­ tude (a model-dependent assumption), the sensitivity to tensor coupling is dominant and also tightly constrained by 199Hg. Thus, this measure­ ment could be interpreted as a (model dependent) measurement of the proton EDM: dp = (3.7 + 6.3)  1023 e-cm. 2.5 Diamagnetic Atoms
Xenon and Mercury Diamagnetic atoms have the experimentally attractive feature that they can be contained in room-temperature bottles or cells because the angular momentum of the atom, which resides in the nucleus, is well shielded by the closed electron shell, even as the atom sticks to the wall for short times. Diamagnetic atoms can also be spin polarized using optical-pump­ ing techniques, providing the largest possible signal-to-noise ratios and optimal statistical precision. Combined with techniques to carefully monitor and control systematic effects, measurements with 129Xe (Vold et al., 1984), with 129Xe/3He (Rosenberry & Chupp, 2001) and the series of measurements with 199Hg are the most sensitive EDM measurements to date. The most recent 199Hg result stands alone in its sensitivity to various sources of CP violation.

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Diamagnetic atoms acquire EDMs from several possible sources. Sen­ sitivity to the Schiff moment of the nucleus increases approximately as Z2 due to the electron momenta, relativistic effects, and the size of the nucleus (Dzuba et al., 2002). Diamagnetic atoms are also sensitive to T-odd/P-odd neutral current interactions between the electrons and the nucleus (tensor, scalar, or pseudoscalar) (Ginges & Flambaum, 2004; Ma˚ rtensson-Pendrill, 1985). The EDM of a diamagnetic atom can also be induced, at higher order, by the electron’s EDM (Flambaum & Khriplovich, 1985; Ma˚ rtensson-Pendrill & Oster, 1987). In general, one can write dA ¼ kS S þ e de þ ðkT CT þ kS CS þ kP CP Þ:

ð10Þ

Here CT, CS, and CP refer to P-odd and T-odd couplings of the electron to a current-density of the nucleus that transforms as a tensor, scalar, and pseudoscalar, respectively. For an infinitely heavy nucleus, the recoil velocity is zero and the pseudoscalar contribution would vanish. The Schiff moment itself can arise from T-odd/P-odd NN interactions and from the EDMs of the individual nucleons (both 129Xe and 199Hg have an unpaired neutron); however, these sources can be related depending on the nature of the T-odd/P-odd interactions. The T-odd/ P-odd contributions to the Schiff moment are generally separated into isospin contributions (deJesus & Engel, 2005; Dmitriev, 2005):   S ¼ gNN a0 g 0CP þa1 g 1CP þa2 g 2CP : ð11Þ Here gpNN = 13.5 is the strong pNN coupling constant and g 0;1;2 are CP isoscalar, isovector, and isotensor contributions to CP-violating pNN couplings, and the coefficients a0,1,2 depend on the details of the assumed NN interaction. Each isospin contribution may isolate specific physics; for example, the QCD vacuum phase contributes to the isoscalar piece: g 0CP » 0:027
QCD (Crewther et al., 1979, 1980), and quark EDMs contribute to the isovector component. The Schiff moment could also arise from the proton or neutron EDM (Dmitriev & Sen’kov, 2003). 2.5.1 Xenon Xenon is the heaviest stable noble gas, and 129Xe is a spin-1/2 isotope. Spin 1/2 has advantages because only dipole interactions with external fields, other atoms, and cell walls are allowed. This leads to longer spincoherence times and narrow linewidths
in fact spin relaxation times of several hundreds of seconds and longer are observed for free-induction decay. The stable spin-3/2 isotope, 131Xe, has a quadrupole moment, which leads to shorter spin-coherence times and cell-geometry depen­ dent effects (Chupp & Hoare, 1990; Wu et al., 1990). In natural xenon, the abundance of 129Xe is 26%; however, isotopically enriched gas is

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available. Polarization of 129Xe greater than 10% is possible using spinexchange with laser-optically-pumped alkali-metal vapor (Zeng et al., 1985). Spin exchange is mediated by the contact hyperfine interaction that occurs during a binary collision or during the lifetime of weakly bound van der Waals molecules. This also makes it possible to monitor the free-precession of 129Xe polarization, which weakly polarizes the alkali-metal vapor. The first EDM measurement in 129Xe (Vold et al., 1984) used spin exchange with laser-optically-pumped rubidium to polarize 129Xe in a stack of three cylindrical cells with alternating electric fields of magni­ tude 3.2—4.9 kV/cm applied parallel and antiparallel to a uniform and well shielded 0.1 mG magnetic field. The stack of cells, treated as mag­ netometers, allows sums and differences of the free-precession frequen­ cies to be used to determine the average magnetic field, and the average magnetic field gradient. A third combination of the three frequencies is the EDM signal. The magnitude of the applied electric field is notably much lower than in the atomic/molecular beam and neutron EDM experiments due to the compactness of the cells, the materials and the buffer gas. The cells contained 0.2 torr (at room temperature) of natural xenon and 220 torr of N2 along with a small amount of rubidium metal. The relatively low xenon pressure points out a feature of spin-exchange optical pumping: the noble gas is a strong relaxation mechanism for the rubidium spin, which was produced by optical pumping. The experi­ ment could be optimized, or at least a favorable trade-off established, by adjusting the noble gas density, laser power and the rubidium vapor pressure. The rubidium vapor pressure was maintained by controlling the cells’ temperature at 65C. One potential systematic error for such a system would be the effective magnetic field, due to the hyperfine inter­ action, caused by any rubidium polarization projection along the electric field axis that somehow changed when the electric fields were changed. Another concern was any change in the leakage currents that would flow across the cells due to the applied voltages that was different for different cells. Both of the effects were studied and found to be small compared to the statistical error of the measurement. The EDM of 129Xe was measured to be dXe = (0.3 + 1.1)  1026 e-cm, where the error is statistical only. Another approach to measure the 129Xe EDM used spin-exchange pumped noble-gas masers (Bear et al., 1998; Chupp et al., 1994; Stoner et al., 1996). Spin-exchange optical pumping is practical, in principle, for any odd-A noble gas, and a population inversion can be pumped in multiple species with the same sign of the magnetic moment. Two spe­ cies, 129Xe and 3He, were used. Due to the Z2 dependence, the two species have very different sensitivity to the Schiff moment and other sources of T and P violation in the atom, but similar sensitivity to magnetic field effects, particularly those produced by leakage currents that can change

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when the electric field is changed. Thus, the 3He served as a comagnet­ ometer occupying nearly the same volume as the 129Xe in a single mea­ surement cell. The spin-exchange cell, which required a large rubidium density to allow both the 3He and the 129Xe masers to operate effectively, was separated from the EDM measurement cell by a transfer tube through which the spin-polarized atoms diffuse. In the measurement cell, radiation damping, due to the coupling of the precessing spins to a coil, would increase the projection of the magnetization vector onto the plane transverse to the applied magnetic field at rate GRD; while the spin diffusion would increase the projection along the applied magnetic field (z-axis) at a rate z; and relaxation (at the rate 1/T1) would reduce the longitudinal component. (The magnetic moments of 3He and 129Xe are both negative so that high-energy Zeeman state has spin parallel to the applied magnetic field.) Decoherence of the ensemble at the rate 1/T2, mostly due to magnetic field inhomogeneity, would tend to reduce the magnitude of the transverse component of magnetization along the x-axis. The combination of these effects, illustrated in the rotating frame depiction in Figure 2, leads to a steady state equilibrium mag­ netization for each species in the frame rotating at the respective Larmor frequency. For the EDM measurement, the 129Xe precession frequency was phase locked to a local oscillator that controlled the magnetic Mz

γz

T2−1 T1−1 → M

ΓRD

Mx ! Figure 2 Principle torques on the magnetization vector, M , in the noble-gas maser depicted in the rotating frame. The magnetic field is along z and the gain and signal are proportional to the transverse projection along x

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field. Thus, any change in the 129Xe frequency when the electric field was reversed would lead to a change of the magnetic field and a resulting change of the 3He by about three times more than the 129Xe frequency shift. The result was dð129 XeÞ ¼ ð0:7 – 3:3 – 0:1Þ  10  27 e-cm:

ð12Þ

The statistical sensitivity was limited by instability of the masers that would arise due to a change of one of the torques on the magnetization vector illustrated in Figure 2. Specifically, any change of the rate at which spin diffuses into the measurement cell due, for example, to fluctuations of laser power, temperature, and magnetic field would lead to small oscillations around the equilibrium magnetization and thus a change in the magnetization projection along the z-axis. The resulting change in magnetization effectively changed the magnetic field around which the spins precess and introduced frequency noise and fluctuations. This would vanish in perfectly spherical cells, but our cells are far from spherical. The effects are also mitigated as the projection of the magneti­ zation onto the transverse plane approaches a maximum (
= p/2). We believe that significant improvement of this passive maser system is possible. An active maser based on amplified feedback has also been developed (Yoshimi et al., 2002). Contemporary efforts using polarized liquid 129Xe are discussed in Section 5.1. 2.5.2 Mercury The 199Hg experiments undertaken by Fortson and collaborators have built on the ideas used in their 129Xe buffer-gas cell experiment (Vold et al., 1984); however, there are two crucial differences with mercury: it is more chemically reactive resulting in shorter coherence times, and it is heavier and thus generally more sensitive to sources of T and P violation. The most recent experiment (Griffith et al., 2009) used a stack of four cells and directly pumped and probed the 199Hg with a 254 nm laser (Harber & Romalis, 2000). The outer two of the four cells have no electric field and the inner two have electric fields in opposite directions so that a differ­ ence of the free-precession frequencies for the two inner cells is an EDM signal. An EDM-like difference of the outer cell frequencies would be attributed to spurious effects such as non-uniform leakage currents that would be correlated with the electric field reversals and are therefore scaled and subtracted from the inner-cell frequency difference to deter­ mine the EDM frequency shift. The magnitudes of the leakage currents were also monitored directly and used to set a maximum E-field corre­ lated frequency shift that contributed to the systematic-error estimate.

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Other systematic-error sources explored are effects of high-voltage sparks on the EDM signals and a number of possible correlations of experimen­ tally monitored parameters (e.g., laser power and magnetic field fluctua­ tions outside the magnetic shields). There are no apparent correlations, and the leakage current (+ 0.5 pA) was so small that only upper limits on the systematic errors could be estimated. The final result is dð199 HgÞ ¼ ð0:49 – 1:29 – 0:76Þ  10  29 e-cm:

ð13Þ

To further explore some of the implications of this null result, we rewrite Equation (10) for 199Hg: dHg ¼ ð2  10  20 CT þ 6  10  22 CS þ 6  10  23 CP Þe-cm 2:8  10  17 SHg cm=fm3 þ 0:01de :

ð14Þ

The contributions to SHg include T-odd and P-odd NN interactions and intrinsic EDMs of the proton or neutron. Comparing these two contribu­ tions:


QCD SHg ¼ a0 gNN g 0CP ¼ 3:6  10  3 e-fm3
QCD




QCD SHg ¼ sn dn ¼ 6:8  10  3 e-fm3
QCD

ðNN interactionsÞ; ðdn from SHg Þ:

ð15Þ ð16Þ

For the NN interaction contribution, we use g 0CP ¼ 0:027
QCD (Crewther et al., 1980) and a0 = 0.01 e-fm3, from the (SkO) potential in deJesus and Engel (2005). For the neutron-EDM contribution, we take SA = spdp þ sndn, with sp = 0.2 fm2 and sn = 1.9 fm2; the uncertainties could be as high as 30% (Dmitriev & Sen’kov, 2003). Note that the T-odd and P-odd NN interaction and dn may be related to each other through more fundamental interac­ tions. For example,
QCD contributes to the neutron EDM (dn = 3.6  10 16
QCD e-cm) as well as to g 0CP . Interestingly, if we assume all other contributions in Equation (14) are zero, this result provides an indirect mea­ surement of the neutron EDM: dn(199Hg) = (0.9 + 2.8)  1026e-cm. This is comparable to the current sensitivity of direct neutron-EDM measure­ ments. Similarly, the model-dependent result for the electron EDM would be de(199Hg) = (0.49 + 1.5)  1027e-cm. Intrinsic and induced EDMs of the quarks would also show up in both the neutron EDM and Schiff moment (Pospelov, 2001; Pospelov & Ritz, 2005), and a comparison is provided in Griffith et al. (2009). 2.6 EDM Measurements in Other Systems A measurement of the muon EDM was made possible in conjunction with the measurement of the magnetic moment anomaly a = (g  2)/2. Muons were collected in a horizontal storage ring with a vertical

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confining magnetic field, ~ B, and electric-quadrupole focusing fields, ~ E (Hertzog & Morse, 2004). The magnetic moment anomaly appears as the muon-spin and muon-momentum vectors precess at different rates due to the anomalous magnetic moment. The rate difference is ! s  !c ¼

eam Bavg ; mm c

ð17Þ

where !s is the spin-precession frequency, !c is the cyclotron frequency, and Bavg the magnetic field averaged over the muon orbits. This averaged magnetic field is measured by an array of proton NMR probes. The muon spin direction with respect to its momentum is measured by the angular distribution of muon-decay electrons in the horizontal plane. A modula­ tion in the vertical direction would arise due to the muon EDM: ~ B þ~ EÞ. Detectors above and below the plane of the !EDM ¼ d ð~ muon orbits can detect this modulation. The result is (Bennett et al., 2009) dm ¼ ð0:0 – 0:9Þ  10  19 e-cm:

ð18Þ

The L0 hyperon EDM was also measured in conjunction with a magnetic moment measurement by looking for an out-of-plane component of the ~ B. The result was spin due to the motional electric field ~ Emot ¼ ~ (Pondrom et al., 1981) dL ¼ ð3:0 – 7:4Þ  10  17 e-cm:

ð19Þ

The EDMs of the neutrino (Rosendorff, 1960) and the tau lepton (Escri­ bano & Masso, 1997) have also been indirectly determined. The limits set for these particles are all significantly larger than the directly measured EDMs discussed here. 2.7 Summary of EDM Measurements to Date In Table 1, we summarize the most precise experimental results for the systems discussed in this section. We can summarize the situation as follows: • Results of direct measurements published over the past 50 years for the neutron, atoms, and molecules are all consistent with no EDM for any of these systems. • There are several possible sources of EDM in each composite system. These include
QCD, an intrinsic EDM of the quarks or electron and Podd and T-odd interactions among the constituents. No single measurement has isolated or set a limit on just one phenomenological or fundamental source of an EDM. • Results for the neutron, TlF, 199Hg, and the paramagnetic atoms cesium and thallium suggest that most sources of CP violation are either small

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Table 1 Summary of most precise direct EDM results in systems discussed in this section. System Cesium 129 Xe Thallium TlF Neutron Muon 199 Hg

Primary sensitivity de, CS
QCD, gpNN, CT de, CS
QCD, gpNN, dp, CT
QCD, quark-EDMs dm
QCD, gpNN, CT

d (e-cm)

References 24

(1.8 + 6.9)  10 (0.7 + 3.3)  1027 (4.0 + 4.3)  1025 (1.7 + 2.9)  1023 (0.2 + 1.7)  1026 (0.0 + 0.9)  1019 (0.49 + 1.50)  1029

Murthy et al. (1989) Rosenberry et al. (2001) Regan et al. (2002) Cho et al. (1991) Baker et al. (2006) Bennett et al. (2009) Griffith et al. (2009)

In this table, we have combined statistical and systematic errors in quadrature; however in the text, we provide the errors as presented in the original papers.

or that fortuitous cancellations have conspired to make all these EDMs smaller than current experimental sensitivities. • The next-generation neutron EDM experiments are likely to achieve greater sensitivity
up to two orders of magnitude improvement is predicted for the coming decade. • Improvements to the vapor-cell and molecular-beam measurements
even in199Hg
are not likely to gain much more than an order of magnitude in sensitivity. (Paramagnetic molecules, discussed in Section 5 do promise significant advances.) For atomic and molecular EDM measurements, significant advances in sensitivity will require new systems and new techniques. New systems that may be practical for EDM experiments gain sensitivity through enhance­ ments, that is because they are significantly more polarizable by CP-violat­ ing elementary particle forces. New techniques include cold atoms in fountains or lattices, cold molecular beams, and condensed systems
polarized liquids and solid state systems.

3. CONTEMPORARY THEORETICAL MOTIVATIONS
THE STANDARD MODEL AND BEYOND Interactions that induce an EDM must violate P and T and, as a conse­ quence of the CPT theorem, must also violate CP. One way to view this is that a particle or system is electrically polarized along ~ J by elementary particle forces through amplitudes that are complex, that is, nonzero phases must arise. CP violation can also be used to reveal a weaker interaction in the presence of the dominant strong and electroweak inter­ actions of the Standard Model.

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In the Standard Model, complex phases that induce EDMs have two sources:
QCD, discussed in the previous section, and complex weak-inter­ action amplitudes that mix heavy and light quark flavors as described by the Cabibbo—Kobayashi—Maskawa (CKM) matrix. The dimensionless para­ meter
QCD is apparently very small, or its effects are canceled by other non-Standard Model interactions in multiple systems (i.e., the neutron and 199 Hg). Thus,
QCD is either identically zero or in effect suppressed. If
QCD is zero, there must be a symmetry that is not inherent in the Standard Model; if
QCD is suppressed by another interaction or symmetry other effects are predicted, specifically the existence of an axion (Wilczek, 1978). A number of direct axion searches and indirect studies have not provided any evidence of an axion, and the search continues (Hagmann et al., 2008). The third possibility is that
QCD is just accidentally small. EDM measure­ ments play the fundamental role in clarifying this puzzle, which is now called the Strong CP problem. The weak interaction mixes the generations of quarks, that is, the u-quark can transform into a d-quark, s-quark, and b-quark. The ampli­ tudes for such “flavor-changing” interactions are in general complex (i.e., the amplitudes include a magnitude and a phase); however, with three generations of quarks (up-down, charm-strange, and top-bottom) and the constraints of unitarity of the CKM matrix, there is a single independent phase along with five independent magnitudes. Weak interactions allow mixing of the heavier generations in higher-order or multi-loop radiative corrections that can induce an EDM of the quarks and electron through the CP-violating phases. A bound system of nucleons or atoms could have an EDM induced by similar radiative corrections. Experiments in the neutral kaon and b-meson systems that confirm CKM unitarity do not constrain the CKM phase, that is, CP violation is maximal, but it is small due to the small amplitudes for mixing the generations. Moreover, additional cancellations of leading order contributions mean that EDMs arise at the three- and four-loop level leading to CKM EDMs smaller, by many orders of magnitude, than the current experimental limits in any  38 e-cm; for the neutron dnSM »10  32 system
for the electron, dSM e »10 e-cm. In addition to the role of EDM measurements in addressing the Strong CP problem, it has become appreciated that our universe, made up of more matter than antimatter, may have its origin in baryogenesis. Sakharov recognized that baryogenesis has three required elements: (1) baryon-number violation; (2) CP violation; (3) non-equilibrium expan­ sion (Sakharov, 1967). For example, proton or quark couplings to leptons (e.g., proton decay) are predicted by grand-unified theories. Due to CP violation, quark and antiquark couplings would be different, and thus asymmetric densities of matter and antimatter could freeze out of the non-equilibrium expansion. CP violation from the CKM matrix could

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qualitatively drive baryogenesis if baryon-number violation takes place at a similar energy scale, the electroweak scale; however, this does not work quantitatively: far too small a baryon asymmetry would result. It is therefore necessary to find CP violation in physics beyond the Standard Model. This would, in turn, produce EDMs much larger than Standard Model EDMs (Cohen et al., 1993; Dolgov, 2009; Trodden, 1999). There is a caveat: baryogenesis could arise via the lepton sector through the neu­ trino equivalent to the CKM matrix, which also has a CP violating phase. In fact, a tighter constraint on the electron EDM could rule out electro­ weak baryogenesis (Cirigliano et al., 2006). Mechanisms that produce the observed baryon asymmetry without new physics have been hypothe­ sized; however, the phase has yet to be measured. Most significant extensions of the Standard Model introduce additional phases that could produce the baryon asymmetry and could lead to an EDM many orders of magnitude larger that the CKM values (Pospelov, 2001). For example, supersymmetric models introduce phases that could produce the baryon asymmetry at the electroweak scale and produce EDMs of atoms or the neutron close to the current limits of sensitivity (Abel et al., 2002). The number of parameters that enter with new physics can be unwieldy; for example, a study of constraints on certain supersymmetric CP-violating phases provided by the combination of neutron, thallium, and 199Hg measurements can be reduced to five parameters. Constraining three masses isolates a part of the parameter space and leaves just two phases to be determined by experiments. The values of these phases that are compatible with the three measurements are tightly con­ strained near zero as shown in Figure 3 (Olive et al., 2005). A final note: the Standard Model and extensions that include amplitudes and phases do not predict the values of the phases, though they may invoke symmetries that require a set of phases to vanish. Physics has long sought a fundamental theory, which might predict the strengths of interactions and therefore ultimately predict the scale of an EDM. The framework for such a fundamental theory may be found in forms of string theory.

4. EDM MEASUREMENTS An EDM measurement is conceptually simple: the Zeeman splitting, h! ¼ 2 B – 2dE;

ð20Þ

is measured in the presence of parallel or antiparallel electric and mag­ netic fields. Thus, ! is measured as a function of E with a nominally constant B-field. The general approach includes (1) polarization or state selection; (2) a free precession interval T; (3) phase detection, which may be continuous during free precession or separated in time or space

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0.5

0.5 B

50

10

0

D

10

10

199Hg

θA/π

θA/π

5

Neutron

Thallium

0

Neutron 10

−0.5 −0.01

10

5

10

50 0 θμ/π

−0.5 0.01 0.99

1

θμ/π

Thallium

0.5

0.5 L Thallium 5 10

Thallium 1.01

M Neutron

5

θA/π

θA/π

199Hg 0

0

10 −0.5 −0.005 Neutron 0 θμ/π

−0.5 0.005 −0.02 −0.01

5 0

0.01

0.02

θμ/π

Figure 3 An analysis of the constraints on CP violating phases due to supersymmetric phases. The horizontal and vertical axes are the two parameters, that is, phases that remain free when the masses of supersymmetric parameters are fixed. Each panel (labeled B, D, L, M) represents a different set of masses of supersymmetric partners of customary particles (see Olive et al., 2005)

(e.g., Ramsey’s method). The frequency ! measured over the interval is thus defined as D ; ð21Þ T where D is the accumulated phase. Magnetic field variations, specifically those that might be correlated with ~ E, must be minimal, or even better measured directly. Basically, it is necessary to measure both terms in Equation (20), and thus the EDM measurement must incorporate a comagnetometer or other means of monitoring the magnetic field. In many cases, this is accomplished with a second or multiple species with intrinsically different sensitivity to P and T violation. !¼

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4.1 Precision The precision of an EDM measurement is determined by the precision with which D in Equation (21) can be measured and thus by all the sources of noise for the phase measurement. Consider the signal for a single frequency measurement accumulated over a time T due to parti­ cles emitted from or photons transmitted by the sample. The signal can be represented as SðTÞ ¼ S0 þ S1 e  T = T2 sinð  0 Þ;

ð22Þ

where S0 is the signal that would be measured in the absence of polariza­ tion, S1 is the amplitude of the modulation due to the polarization and the accumulated phase, and T2 is the coherence time of the ensemble of spins. The precision with which D can be measured is    d    S ; ð23Þ D ¼  dS  where the average is taken over the measurement interval T so that the precision of the frequency measurement is ! ¼

1 S : T e  T = T2 S1

ð24Þ

This is optimal for T = T2 and sin(  0) = 0. For a counting experiment, we can define an analyzing power for the measurement, A ¼ S1 =<S>. If A = 1, the signal has maximum sensitivity to the phase-sensitive effect. Thus, the precision of the EDM measurement is d ¼

2:7h S : 2AET2 <S>

ð25Þ

The factor of 2.7, which arises for decaying signals, is not usually presented in discussions of EDM sensitivity; it would not be present when T2 >> T or when the signal is constant. Assuming detector noise, backgrounds and similar pffiffiffiffi effects are small in the counting experiment, we expect 1=<S> ¼ 1= N, and d »

h pffiffiffiffi ðcount-rate limitedÞ: AET2 N

ð26Þ

For the neutron EDM measurement described in Harris et al. (1999), a single run N = 13,000 neutrons, A = 0.5, and T2 = 130 s. By eliminating the effects of magnetic field noise with the 199Hg comagnetometer, the expected statistical precision was achieved. For the 199Hg EDM experi­ ment, frequency precision of !  2p  109 Hz was reported for T2  100 s and nearly 100% modulation of the intensity of 254 nm light due to Fara­ day rotation. The measurement of the vapor polarization using Faraday

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rotation with light detuned from both the F = 1/2 and F = 3/2 absorption lines has minimal effect on T2 and thus < S > is effectively the total number of photons counted during the measurement, which would be estimated to be 3  1013 or an average detected power of about 0.2 mW. For a non-counting experiment, for example, an NMR-like measure­ ment for which S0 is nominally 0, S1 is proportional to the polarization or magnetization of the sample, and S is not dominated by counting noise. For example, in the noble-gas maser (Section 2.5), Johnson noise in the room-temperature pick-up coils was the dominant source of white-phase noise for the magnetometer species, while the free-running component of the two-species maser was subject to white-frequency noise due to the fluctuating magnetization associated withpffiffiffi maser instabilities. In the case of white-phase noise, S is proportional to B, where B is the bandwidth of the measurement, that is, B/1/T (T2 is effectively infinite) and thus ! / T  3 = 2

ðwhite phase noiseÞ: ð27Þ pffiffiffiffi For white frequency noise, S grows with T as the phase makes a random walk away from the frequency-noise-free phase. In this case ! / T  1 = 2

ðwhite frequency noiseÞ:

ð28Þ

Studies of ! and related quantities such as the Allan deviation are used to isolate the noise sources and to guide improvements to the experi­ ments as shown in Figure 4 in Section 5.1. Also, see discussions in Stoner et al. (1996) and Bear et al. (1998). 4.2 Systematic Effects and False EDM Signals There is no single set of systematic effects that dominate all experiments; however, most experiments completed or anticipated must address the following: • leakage currents that change when the electric field is changed • geometric phases that arise as the polarized spins evolve in space and time • ~ ~ E effects
even in storage cells and bottles where h~ i » 0 but h~   ~ Ei¼0: 6 One general approach is to make use of two or multiple species or transitions insensitive or less sensitive to the P-odd and T-odd EDM effects. This is generally called a comagnetometer. An ideal comagnetometer would average only the magnetic field, in exactly the same space, with exactly the same trajectories, over exactly the same time, with exactly the same T2, as the species considered most sensitive to P-odd and T-odd effects. This is diffi­ cult. For example, in the neutron-EDM experiment with 199Hg comagnet­ ometer, the UCN velocity distribution is quite different from that of the

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room temperature 199Hg used as the comagnetometer. Consequently, the average position of the two ensembles due to gravity was slightly different, and the two species averaged the storage cell magnetic field and ~ ~ E effects differently (Barabanov, 2006; Commins, 1991; Pendlebury et al., 2004). In the molecular EDM approaches described in Section 5.3, the comagnetometer is effected by using sums and differences of frequencies that separate sensitiv­ ity to dE and B. This has the advantage that the accumulated signal averages the same space for the EDM and comagnetometer measurements and could be ideal provided the effective T2’s are the same. We also mention the adoption of “blind analysis” techniques (Klein & Roodman, 2005). For example, in the most recent 199Hg measurement, much of the data were analyzed with a hidden offset to the EDM signal. The motivation for blind analyses in experiments with systematic errors comparable or larger than the statistical error is to eliminate influences that could enter the analyst’s approach because of bias for a specific outcome, for example, an EDM consistent with zero. In EDM measure­ ments, the frequency or EDM signal is determined under varying condi­ tions, for example, a magnetic field reversal motivated by the expectation that an EDM signal will not change; however, the manner in which the two experimental results are combined may be determined by foreseeing the result. There is nothing intrinsically wrong with this in general, and many important systematic effects have been discovered by understand­ ing the results of such correlations; however, we can expect blind ana­ lyses to become more common in future measurements.

5. CONTEMPORARY EXPERIMENTS In this section, we discuss current and planned experiments that are not direct extensions of the earlier efforts discussed in Section 2. The one effort that, in a way, spans the gap between these new endeavors and the ones described previously involves the paramagnetic molecule YbF (Hudson et al., 2002) and is described in Section 5.3. Contemporary experiments are generally focused on very large enhancements of T-odd and P-odd effects in specially selected systems and on applications of recently developed technologies, specifically cold atoms/molecules and condensed systems, or a combination of enhancements and new technologies. Experiments in systems investigated in earlier times, that is, cesium and xenon, are underway using cold-atom techniques and polarized liquid xenon, respectively. Molecular beam experiments with paramagnetic molecules continue to generate significant enthusiasm due to the very large effective enhancement of T-odd and P-odd effects (de and CS). Enhancements of the Schiff moment of diamagnetic atoms with nuclei that have octupole deformation have motivated experiments

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with radioactive atoms. We also describe efforts to measure the electron EDM in solid systems where electric effects would induce detectable magnetization or magnetic effects would induce detectable electric fields. Storage ring techniques similar to those applied for measurement of the muon EDM are contemplated for measurement of the proton and deuteron EDM. 5.1 Liquid Noble Gases The EDM of 129Xe is primarily sensitive to the same phenomenological P-odd and T-odd effects as 199Hg but with about an order of magnitude less sensitivity; moreover, the most recently published results for the two species differ in sensitivity to the atomic EDM by two orders of magni­ tude. Nevertheless, there is significant interest in improving the sensitiv­ ity to the EDM of 129Xe for two reasons: techniques using laser polarized xenon to produce a polarized liquid have the potential for many orders of magnitude improvement in signal-to-noise, and 129Xe appears to be an attractive magnetometer for next generation neutron-EDM efforts (Atchison et al., 2005). The current result, dXe < 6  1027, should be improved by one or two orders of magnitude in order to be useful as a neutron-EDM comagnetometer. Polarization by spin-exchange can be used to produce significant volumes of polarized 129Xe (Zeng et al., 1985) which can be condensed to a liquid and used over a broad temperature range down to the triple point of 161 K (Sauer et al., 1997). In the liquid state, the polarization lifetime can be T1 > 10 min at lower temperatures, and T2 is comparable to T1 in uni­ form magnetic fields and spherical containers. With 129Xe polarization of 10%, the magnetization of the liquid can be 10—100 times larger than in the gas-phase measurements, and a number of interesting but potentially confounding effects can arise due to the self-interaction of the precessing ensemble. For example, non-exponential decay (Romalis & Ledbetter, 2001) and amplification (Ledbetter et al., 2005) of the transverse magneti­ zation have been observed in development work for a liquid-129Xe EDM experiment. Cross-relaxation with the spin-3/2 isotope has been observed and depends on the 129Xe enrichment of the sample (Gatzke et al., 1993). Another potential advantages of the liquid-xenon system is the high dielectric strength that would allow higher electric fields; however, the EDM of an atom of liquid Xe has been shown to be suppressed by about 40% due to screening effects (Ravaine & Derevianko, 2004). One experiment underway envisions the use of liquid-helium-cooled SQUIDs to monitor the precessing magnetic moment of a sample of liquid xenon contained within a sapphire cell with E =75 kV/cm (Romalis & Ledbetter, 2001). The applied magnetic field would also be monitored or feedback-stabilized by one or more SQUIDs. In this case the signal

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Access tube C -axis sapphire Conductive coating +HV (50 kV)

Mx

Bz

Mz

18 mm

Machined hemisphere (φ 10 mm) −HV 18 mm 10−3 Slope = −1.5

σy

10−4 10−5 10−6 10−7 10−1

Data Simulation w/ white noise

100

101 sec

E-field reversal half-period, t Figure 4 Top: Schematic diagram of proposed liquid-xenon EDM experiment using low temperature SQUIDs and the three-SQUID coil configurations designed to be sensitive to transverse magnetization, Bz field and longitudinal magnetization. Bottom: The frequency precision as a function of T (see Equation (24)); for measurement time T > 2 s, frequency noise dominates and breaks the slope of T3/2. Courtesy of M. Romalis

S (Equation (22)) is the magnetic field due to the precessing-polarized 129 Xe liquid, and S is the SQUID noise for the measurement averaged pffiffiffi over ffi time T. At a frequency of 1 Hz it is anticipated that S »0:5 fT= T, while the field due to the 129Xe may provide S1 = 1 nT. For a pair of measurements with T = 10 s and |E| = 75 kV/cm, d  1027e-cm. Fluctuations of the applied magnetic field at the position of the liquid xenon would need to be monitored or stabilized at the 2 fT level, which is well within the noise capability of the SQUIDs. This approach does rely on the external SQUIDs to monitor all magnetic field variations including leakage currents, which is not the same as a comagnetometer occupying exactly the same volume; however, the approach is similar to the 199Hg measurement in the sense that magnetometers (in this case SQUIDs) monitor magnetic fields in a geometry that surrounds the measurement region. A schematic is shown in Figure 4.

Permanent EDMs of Atoms and Molecules

Top

155

Side Plug 2 mm

Electrodes

B0 Liquid Xe droplet

Xe B0 supply tube

MicroPick-up structured coils plate at −115 °C

Figure 5 Schematic diagram of proposed liquid-xenon EDM experiment using rotating electric fields. The filled circles indicate the liquid-xenon spheres, and the open circles are electrodes. Courtesy of P. Fierlinger

Another approach using polarized liquid xenon is under development (Fierlinger, 2010). In this scenario, illustrated in Figure 5, the liquid xenon is contained in an array of small spheres of 600 mm diameter. The xenon condenses into wells etched into a glass plate from a highly polarized gas injected above the plate. The central row of three spheres is surrounded by a set of electrodes (open circles) that apply the electric field differently to the different xenon samples. An EDM would lead to precession into or out of the plane. The resulting magnetization misalignment for each sphere would be measured by a nearby superconducting pick-up coil coupled to a low-temperature SQUID. Low noise in the superconducting coil and SQUID-based detection system, high 129Xe polarization and long T2 are anticipated. With somewhat conservative estimates of T2 = 100 s, E = 10 kV/cm, 10% polarization of the xenon and 100 simultaneous mea­ surements in an array of 10  10 spheres, d  2  1027 e-cm for a 100 s measurement. Leakage currents that flow around a sphere when the electric field is applied could add or subtract to ~ B0 and thus generate a false EDM signal. The spheres with no applied electric field are within a few millimeters of the spheres with applied electric field and are also sensitive to such an effect so that correlations could allow a false effect to be separated. The high sensitivity predicted for the polarized-liquid-xenon EDM experiments arises from the potentially very large signal-to-noise ratio of SQUID magnetometers and superconducting pick-up coils, large elec­ tric fields and narrow lines (corresponding to a long T2). It will require correspondingly low frequency noise to realize this sensitivity, and the liquid itself is potentially the greatest source of frequency noise due to the large magnetization. Such effects are mitigated in a spherical sample, and the manufacturing specifications of the sapphire sphere or the actual shape of the liquid drop wetted on the glass surface may ultimately limit these experiments. Systematic effects due to motion of the spins

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(~ ~ E and geometric-phase effects) should be significantly suppressed compared to gas-phase 199Hg. Leakage-current-false-EDM effects would be monitored by external magnetometers. While difficult, these experi­ ments show great promise. 5.2 Cold Alkali-metal Atoms: Fountains and Lattices The EDM measurements in cesium and thallium are motivated by the Z3 enhancement of sensitivity to the electron EDM and to the scalar coupling of the electron to the nucleus. The experimental sensitivity was limited most importantly by the short coherence time (T2) of cesium in a cell and short or observation time (T) for the thallium beam. For thallium, effects due mainly to the geometric phase and related ~ ~ E effects were the dominant systematic limitations. Laser cooling and trapping of the hea­ viest alkali-metal species, cesium and francium, offer promising new directions and several approaches are being pursued. Cesium atomic fountain clocks based on launching atoms from a laser cooled or trapped sample have moved to the forefront of time keeping. Narrow line widths (T2  1 s) are attained as the atoms move up and then down through a resonance region. While the 133Cs atomic frequency stan­ dard uses the DmF = 0 transition, which is insensitive to magnetic fields in first order, an EDM measurement must use DmF  1. From Equation (24), with T = 1 s, T2 >> T and N = 106, the expected uncertainty on ! is expected to be about !  103 Hz, which is consistent with observations of the Allan variance representing the short-term instability of cesium fountain clocks (!/!  1013 for ! = 2p  9.2 GHz (Weyers et al., 2009)). For an EDM measurement with an electric field of 100 kV/cm, which may be feasible, each 1 s shot would have a sensitivity of 6  1024 e-cm, comparable to the sensitivities of both the cesium and thallium measure­ ments presented in Table 1. Thus, significant improvement is possible, and a demonstration experiment with about 1000 atoms per shot and E = 60 kV/cm was reported as a measurement of de by Amini et al. (2006). The result can be interpreted as dCs = (0.57 + 1.6)  1020 e-cm (the authors use e = 114 for cesium). The major limitation in this demonstration was the necessity to map out the entire resonance-line shape spectrum, which is the combination of transitions among the nine F = 4 hyperfine sub-levels complicated by inhomogeneities of the applied magnetic field in the reso­ nance region. This subjected the measurement to slow magnetic field drifts that would be monitored or compensated in the final experiment. If these problems are taken into hand, the statistical sensitivity could be significantly improved with several orders of magnitude more atoms, higher electric field, and duty-factor improvements; however, the major systematic effect due to ~ ~ E was about 2  1022 e-cm (Amini et al., 2006). This could ultimately limit the sensitivity of a single species fountain measurement.

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As discussed in Section 2.3, the EDM of a paramagnetic atom is most sensitive to the electron EDM and the T-odd and P-odd scalar coupling of electrons to the nucleus, which scale approximately as Z3. For francium, e (from Equation (2)) is in the range 900—1200 (Ginges & Flambaum, 2004), and kCs should be similarly enhanced. Francium can be produced in significant quantities in isotope-separator rare-isotope production facilities, and 210Fr has been produced, laser cooled, and trapped in a magneto-optical trap (MOT) at Stony Brook, New York (Gomez et al., 2006). The experiment has been moved to the isotope-separator facility (ISAC) at TRIUMF in Vancouver, Canada. Francium isotopes have halflives of 20 min (for 212Fr) or less, and any experiment would need to be “online,” that is, the EDM apparatus would be at the site of the rareisotope production facility. The development of the cesium-fountain EDM measurement may lead the way to a future program with francium at a future facility such as the Facility for Rare Isotope Beams (FRIB) at Michigan State University. The fountain concept allows line widths on the order of 1 Hz, limited by the time for the cold atoms with vertical velocity of a few m/s to rise and fall about 1 m. Another idea being pursued by D. Weiss and collaborators is to stop and cool alkali-metal atoms in optical molasses near the apogee of their trajectory and trap them in an optical lattice formed in a build-up cavity (Fang & Weiss, 2009). Storage times in the lattice could be many seconds. The lattice would be loaded with multiple launches, filling sites that extend over 5—10 cm, and 108 or more atoms could be used for the EDM measure­ ment. After loading the lattice, the atoms would be optically pumped to maximum polarization and then the population transferred to the mF = 0 state by a series of microwave pulses. A large electric field (e.g., 150 kV/cm) would define the quantization axis in nominally zero magnetic field, and the energies would be proportional to m2F due to the parity-allowed inter­ action. In another planned innovation, a Ramsey separated-field approach would be used with the free-precession interval initiated by pulses that transfer atoms to a superposition of mF = F and mF = F states and termi­ nated by a set of pulses coherent with the initial pulses. The relative populations transferred back to the mF = 0 state would be probed by optical fluorescence that could be imaged with about 1 mm spatial resolution. With the large size of the lattice, the superposition of stretched levels (mF = + F) would amplify the sensitivity by a factor of F relative to experiments that monitor DmF = 1 transitions (Xu & Heinzen, 1999). In a measurement time T = 3 s, and N = 2  108, an EDM sensitivity of 6.5  1026 e-cm for the cesium atom is expected. The optical lattice can also trap rubidium, which would be used as a comagnetometer. Cold atom techniques have been developed over the past 25 years by a large number of groups, and appear to be able to provide statistical power and ways to monitor systematic effects. The continuing advances

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in technology of fountain clocks provides encouragement, though an EDM measurement provides a set of distinctly separate systematic effects. An EDM measurement in a lattice would allow measurements with two or more species, thus providing a comagnetometer to monitor leakage-current and other effects. 5.3 Paramagnetic Molecules Molecular beam EDM experiments with TIF are discussed in Section 2, where it is pointed out that the electric field along the interatomic axis is generally much larger than the external field required to align that axis. For a heavy polar paramagnetic molecule the effective internal electric field interacting with the unpaired electron can be volts per angstrom, for example, for YbF, Eint  2.6  1010 V/cm (Mosyagin et al., 1998) when the molecule is fully polarized. Thus, e  25,000 in the case of YbF (Hudson et al., 2002). Alternatively, this can be viewed as a strong polarizability due to the mixing of rotational states of opposite parity; T-odd and P-odd spin-axis interactions would result in a component of ~ J along ~ E and a ~ resonance-frequency shift proportional to E. For -doublet systems, the electric polarizability of the molecule is so large that relatively small laboratory electric fields can be used, greatly mitigating the leakage-cur­ rent and motional ð~ ~ EÞ effects. For a molecular beam measurement, the observation time T (from Equation (24)) is limited to the flight-time through the apparatus. This can be significantly increased by slowing the beam along its direction of motion (e.g., longitudinal cooling). The observed number of molecules can also be significantly increased and the flight-path made longer with transverse cooling. In general, the large enhancements due to the internal electric field are balanced by the rela­ tively low numbers of molecules synthesized and observed; however, significant improvements over the thallium-EDM measurement are expected in several systems including YbF (Hudson et al., 2002), metastable PbO (Bickman et al., 2009), WC (Lee et al., 2009), and ThO (Vutha et al., 2009) as well as a trapped molecular ion such as HfFþ (Meyer et al., 2006). Hinds and collaborators have developed an experiment using 174YbF, which is illustrated in Figure 6. The molecular beam is cooled to the electronic, vibrational, and rotational ground state. The hyperfine split­ ting of the F = 0 and F = 1 levels formed by the combination of 19F nuclear spin (1/2) and the unpaired electron spin is 170 MHz. The (F = 1, mF = 0) state is lowered in energy relative to the mF = +1 states due to the electric field. The EDM signal is the splitting of the mF = 1 and mF = þ1 states. A small magnetic field is applied so that both ~ E and ~ B can be reversed with respect to the laboratory coordinate system. The magnetic field must be well aligned with the electric field to avoid a false-EDM signal due to a motional field
the transverse components are estimated to be 10 mG.

159

Probe fluorescence (V)

Permanent EDMs of Atoms and Molecules

Probe PMT 1.0 0.8

Lens

0.6 Probe beam

0.4 0.2

Lens

B

0.0 1.8

2.0

2.2

2.4

RF out Mirror

2.6 E-field plates

Time after Q-switch (ms)

Pump beam

RF in

Yb target mF = −1 0

Skimmer

F=1

170 MHz

Ablation laser

Valve

+1

F=0

Normalized signal (arb. units)

2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 − 200

−100

0

100

200

Applied B-field (nT) Figure 6 Schematic diagram of YbF molecular beam experiment. The fluorescence signal, shown in the upper left, measured by the probe photomultiplier tube (PMT) represents the time-of-flight of molecules after the Q-switched ablation-laser pulse. The bottom graph shows the phase of the superposition of the mF = 1 and mF = 1 levels as a function of B, which is probed by the population of the F = 0 state after the second p-pulse. Courtesy of E. Hinds and J. Hudson

The experiment uses lasers to prepare the molecules in the F = 0 state followed by a p-pulse in the region of combined DC electric field (3.3 kV/cm) and 170 MHz RF magnetic field along the x-direction,

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Tim Chupp

which produces the superposition of mF = 1 and mF = þ1 levels. The superposition evolves as the molecules travel through the region of combined electric and magnetic field and then the superposition is probed by a second p-pulse and laser fluorescence to detect the F = 0 population. The pump and probe RF pulses are not in phase, thus the measurement is not a Ramsey separated-field experiment. A combination of reversals of electric and magnetic fields as well as a calibration mag­ netic field are used to determine the EDM signal. The most recent result is (Hudson et al., 2002) de ¼ ð0:2 – 3:2Þ  10  26 e-cm:

ð29Þ

False-EDM signals arising from leakage currents and the magnetic field misalignment are estimated to be two or more orders of magnitude smaller than the statistical error. Refinements to the apparatus include a pulsed supersonic beam, rather than an effusive thermal beam; more efficient cou­ pling of the RF allowing shorter p-pulses, which improves polarization con­ trol and reduces some systematic effects; and a more uniform electric field that reduces geometric phase effects and problems from transverse electric fields. The signal-to-noise ratio for the fluorescence probe and the super­ position signal as a function of magnetic field are shown in Figure 6, and demonstrate the prospect for significantly improving the EDM sensitivity. Experiments in -doublets, first suggested by Kozlov and DeMille (2002) for metastable PbO and by Cornell and coworkers for HfFþ and ThFþ (Meyer et al., 2006), are also being pursued using tungsten-carbide (WC) by Leanhardt and collaborators (Lee et al., 2009) and using thorium­ oxide (ThO) by DeMille, Doyle, Gabrielse, and collaborators (Vutha et al., 2009). The relevant levels of WC are shown in Figure 7. The ΔEup C

μ elE lab

− μB

+deE int

+μB

−deE int E eff

~ ~

~ ~

E lab W

m = −1

B

m=0

m = +1

W ~ ~

E eff C

−μ elE lab −μB

~ ~ −deE int +μB +deE int ΔEdown

Figure 7 Molecular levels of tungsten carbide (WC) in the presence of applied magnetic and electric fields

Permanent EDMs of Atoms and Molecules

161

electronic ground state has structure 3D1 with triplet spin  = 1, orbital angular momentum L = 2, and total angular momentum  = 1. In the absence of an external electric field, the separate m levels are paritydoublets, and in the presence of an applied external electric field, the molecule is polarized and each m level splits into two states with the splitting proportional to |m|, as shown. Any magnetic field, including that due to a leakage current or motional field would split the m = 1 and m = þ1 levels in opposite directions for both molecular orientations while an EDM couples to the internal field. Thus, the energy shifts would also depend on the molecular orientation with respect to the external field. The two energy splittings of the m = 1 and m = þ1 states are DEup ¼ 2dEint þ 2 B

and DEdown ¼ 2dEint þ 2 B

ð30Þ

The difference DEupDEdown = 4deEint measures the EDM and the sum DEup + DEdown = 4 B is the comagnetometer. This built-in comagnet­ ometer is an extremely important feature of these systems, and provides additional motivation as the experimenters develop the techniques and improve the rates. An additional advantage of 21/2 and 3D1 molecular states may be the near cancellation of the orbital L = 2 and spin-triplet  = 1 magnetic moments resulting in a very small that further mitigates leakage current and motional field effects. 5.4 Rare Atoms Rare or radioactive atoms provide attractive systems due to enhancements of atomic or nuclear polarizability. For atomic radium, enhanced atomic polarizability arises due to the near degeneracy of the two electrons in the spin-triplet 7s7p and 7s6d states, which are separated by about 5 cm1 or 103 eV. Thus, a laboratory electric field polarizes the atoms and results in a large internal electric field that interacts with the triplet electrons. The resulting enhanced sensitivity to the election EDM is estimated to be e  5000 (Dzuba et al., 2000). All radium isotopes are radioactive and can be extracted only from sources or isotope production facilities. The isotope 226 Ra (t1/2 = 1600 years) is a daughter in the decay chain of 238U, and 225Ra (t1/2=14.9 days) and can be extracted from a source of 229Th. Sources of 1

Ci and a few millicuries, respectively, that provide samples of about 1014—15 atoms are quite practical. Alternatively, isotopes can be produced at an accelerator facility, for example, at the Kernfysisch Versneller Insti­ tuut in Groningen, The Netherlands, where a program is under develop­ ment to cool and trap radium in a MOT for an EDM measurement (Jungmann, 2007). For diamagnetic atoms, including ground-state radium, the nuclear Schiff moment may be enhanced by near degeneracy of nuclear levels in a manner very similar to the molecular enhancements. For nuclei, the

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analogue to the polar molecules is characterized by the combination of quadrupole and octupole deformation that arises due to the nature of nuclear forces. Octupole deformed nuclei are reflection symmetric so that the states of definite parity are even and odd combinations of the two mirror-image configurations with very small splitting and a resulting large polarizability. Thus, the large intrinsic dipole moment and Schiff moment in the body-fixed frame of the nucleus is polarized along the total angular momentum ~ J by T-odd and P-odd interactions (Auerbach et al., 1996; Engel et al., 2000; Haxton & Henley, 1983; Spevak et al., 1997). The Schiff moment that would result can be expressed in terms of quadrupole and octupole deformation parameters, b2 and b3 respectively, as (Spevak et al., 1997) S » 0:05e

2 23 ZA2 = 3 r30  ; Eþ  E

ð31Þ

where Eþ and E— are the energies of opposite parity states, and  is the strength of the effective T-odd and P-odd interaction of the nucleons (nn, np, or pp are components of the different isospin combinations). For a core density  the potential can be characterized as (Auerbach et al., 1996) G X! ! pffiffiffi i  r: ð32Þ V TP ¼  2mp 2 i In Table 2, we list several cases with potentially large octupole enhance­ ments, which were selected because of work underway to develop techni­ ques for EDM measurements. There are several interesting cases not shown, most notably, 229Pa (protactinium), which has an exceptionally small splitting compared to the scale of nuclear binding energies, that is, dE  0.22 keV, and a Schiff moment that may be 10,000—30,000 times as large as that of 199Hg.

Table 2 Predicted Schiff moments (S) and atomic EDMs (dA) for a generic CP-violating coupling h based on the work of Spevak et al. (1997) and Dzuba et al. (2002). 223

223

225

23.2m 7/2 37 — 1000 3300

11.4d 3/2 170 50.2 400 —3400

14.9d 1/2 47 55.2 300 —2550

Rn

t1/2 I dE th (keV) dE exp (keV) 108S/ (e-fm3) 1025dA/ (e-cm)

Ra

Ra

223

Fr

22m 3/2 75 160.5 500 2800

129

Xe

199

Hg

1/2

1/2

1.75 0.66

—1.4 3.9

 dE = Eþ—E— is the splitting of low-lying-opposite-parity levels measured or predicted by Spevak et al. (1997) using a Woods-Saxon potential.

Permanent EDMs of Atoms and Molecules

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An experiment with 225Ra is underway in the group of Z.T. Lu and collaborators at Argonne National Lab in the United States (Guest et al., 2007). Radium from a source is heated to form a beam and then Zeeman slowed and trapped in a MOT operating on the 1S0—3P0 transition. One of the interesting discoveries of this work was that the MOT repumping, which is required to pump atoms out of the 3D1 to the state, could be provided by blackbody radiation, which transferred atoms from the 3D1 to the 3P0 state. Cold atoms would then be transferred from the MOT to a red-detuned far-off-resonance trap (FORT) at the focus of a laser beam provided by an erbium-fiber laser. The FORT trapped atoms will remain at the laser focus as the focusing optics are physically translated by about 1 m to place the 225Ra sample in the measurement apparatus within a multilayer mu-metal magnetic shield. An EDM measurement would take place in the FORT trap. Nuclear polarization would be optically pumped with circularly polarized light from the 7s2 ground state to the 7s7p 1P1 level with F = 1/2, followed by a Ramsey-type measurement. It is antici­ pated that N = 104 atoms can be trapped in high vacuum so that the atomstorage lifetime will allow a measurement interval of T = 100 s. The high vacuum can allow an electric field as large as 100 kV/cm. The near-term goal is a measurement of the atomic EDM with sensitivity of 1026 e-cm. From Table 2 it appears that 225Ra is about 600 times more sensitive to T-odd and P-odd nuclear interactions than 199Hg. More specifically, the isoscalar, isovector, and isotensor contributions given in Equation (11) for 225 Rn (Dobaczewski & Engel, 2005) and 199Hg (deJesus & Engel, 2005) using the SkO model are e-fm3 1CP þ 0:018g2CP Þ; S199Hg ¼ gNN ð0:01 g0CP þ 0:074g S225Ra ¼

gNN ð1:5 g0CP

þ

6:0 g1CP



4:0g2CP Þ:

ð33Þ ð34Þ

The uncertainty of the coefficients due to the nuclear-force models may be a factor of two; however, a measurement of the EDM of 225Rn at the level of 1026 e-cm would be comparable to the current 199Hg result. The red-detuned FORT is not species specific, in contrast to the Zeeman slower and MOT, so it will be possible to trap more than one species in the FORT to use as a comagnetometer. This would require an additional Zeeman slower and additional MOT lasers, but it will likely be essential in the long term. Alternatively, a different isotope of radium with no octupole deformation could be loaded into the FORT. Frequency shifts that arise in the FORT due to light shifts, particularly residual circular polarization, and due to parity mixing induced by the static electric field, may present challenges. The static field effects are linear in the magnitude of ~ E and could produce a false EDM. These have been worked out for mercury by Romalis and Fortson (1999), and certain orientations of the fields can mitigate the effects. Longer-term

164

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improvements will require more atoms, which may be produced by a more active source or by producing isotopes at an online facility accel­ erator facility such as TRIUMF or FRIB. Table 2 also shows significant potential enhancement of the 223Rn Schiff moment and atomic EDM. This enhancement is somewhat less certain than for 225Ra due to the lack of data on the nuclear level structure and complications in calculating the enhancements (Engel, 2010); how­ ever, the possibility of adapting techniques used for the xenon—helium comparison has led us to develop a program of nuclear structure mea­ surements and the development of techniques for a radon EDM experi­ ment. Due to the short half-life of 23 min, the experiment must be “online,” that is, interfaced to the production-facility beam line. In an experiment being set up at TRIUMF in Vancouver, Canada, the radon is produced by spallation from 500 MeV protons incident on a target of thorium or uranium. Radon evaporates from the target, is ionized, accel­ erated to 30 keV, and transported using magnetic and electrostatic optics to be imbedded in a foil target. After a collection interval of about two half-lives, the foil can be heated and the radon transferred to a measure­ ment cell by freezing to an intermediate cold-finger at liquid-nitrogen temperature and then carried into the cell with nitrogen transfer gas (Nuss-Warren et al., 2004). Once the radon is in the cell, the cell is closed off isolating the nitrogen-radon mixture. The cell also contains rubidium, which is optically pumped, and the radon is polarized by spin exchange. The nitrogen serves several purposes: it is the transfer gas, it assists optical pumping by suppressing radiation trapping (Chupp & Coulter, 1985), and allows higher electric fields for the EDM measurement. Polarization studies with 209Rn (Kitano et al., 1988; Tardiff et al., 2008) and 223Rn (Kitano et al., 1988) have shown that gamma-ray anisotropies can be used to monitor the nuclear polarization. Figure 8 shows data on the temperature dependence of the polarization measured by monitoring the angular dependence of the 337 keV gamma-ray emitted when 209Rn (t1/2= 28.5 m) decays to 209At. The data were taken at the Stony Brook Nuclear Structure Lab with the radon collection and optical pumping apparatus installed on the beam line developed for the original francium trapping studies (Gomez et al., 2006). The isotope 209Fr (t1/2=50 s) was produced by the reaction 16O þ 197Au ! 209Fr þ 4n with an 16O beam energy of about 91 MeV. The francium in the gold target was evaporated and surface ionized, accelerated to 5 keV and transported to the collection foil. Collection times of about 1 h allowed the 209Rn from 209Fr decay to build up to near the maximum. The foil was isolated and heated to drive off the radon (Warner et al., 2005), which was then transferred to the optical pumping cell. The temperature dependence of the radon nuclear polarization shown in Figure 8 illustrates the combination of several competing processes compiled in the rate equation:

Permanent EDMs of Atoms and Molecules

165

Uncoated cell 1.05

R

1 0.95 0.9

Coated cell 1.05

R

1 0.95 0.9 140

160 180 200 Cell temperature (°C)

220

Figure 8 Temperature dependence of the polarization of 209Rn produced by spin exchange with optically pumped rubidium. R is the laser-on/laser-off ratio of 0/90 ratio of count rates for the 337 keV gamma ray from decay of the an excited state in 209 At populated by decay of polarized 209Rn

dPRn PRn ¼ kSE ½RbðPRb  PRn Þ  ; dt T1

ð35Þ

where PRb is the rubidium polarization, [Rb] is the rubidium number density, T1 is the polarization relaxation time, and kSE = hSEvi is the rate constant for spin-exchange. The cross section has been calculated by Walker (1989) with the result SE = 2.5  1021 cm2 so that kSE  9  1017 cm2/s for binary collisions. The equilibrium radon polarization is PRn ¼

kSE ½Rb PRb : kSE ½Rb þ 1=T1

ð36Þ

The factors PRb, [Rb], and T1 are all strongly temperature dependent. The rubidium density increases exponentially with temperature by a factor of about 1.6 for every 10C, and [Rb]  1015 cm3 at 200C; the rubidium polarization produced by laser optical pumping decreases at high tem­ perature due to electron-spin relaxation processes for Rb—Rb and Rb—N2 collisions; and the relaxation rate depends on the residence time of atoms at the wall, which is expected to decrease with increasing temperature: 1 ¼ Gð1ÞeT0 = T ; T1

ð37Þ

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where G(1) is the infinite temperature rate and kT0 is a typical binding energy of atoms at the wall. These depend on the wall material and coating. Cells without and with coating were investigated. The coated cell used OTS or octadecyltrichlorosilane, a coating developed for our xenon—helium EDM experiment (Rosenberry & Chupp, 2001). By using Walker’s spin-exchange cross section, the data could be interpreted to estimate T1  30 s (Tardiff et al., 2008), which is also the limit on T2. With N = 1010 atoms, 108 decays could be observed by monitoring the betaasymmetry in each 30 s measurement; with E = 10 kV/cm, an EDM sensitivity of d  2  1024 e-cm is possible with each measurement, and 1026 e-cm could be achieved in about 2 weeks. With the Schiff moment enhanced by 800 or more, this would also be comparable to the sensitivity of 199Hg. The spin-exchange optical pumping technique offers the ability to polarize any odd-A noble gas species, and thus one or more comagnetometer species could be monitored in the measurement cell and discriminated by precession frequency or nuclear transition. More than one species may be useful to monitor effects of higher order than magnetic dipole. It is also attractive to consider the possibility of a laser experiment that would directly optically pump and probe the radon and take advantage of higher production rates that may be avail­ able at future facilities such as FRIB. A single-photon transition from the ground state would require 178 nm light, which is currently not practi­ cal. Two-photon techniques are also possible and currently being explored. Finally we note that with nuclear spin  1, nuclear moments of higher order, including a T-odd, P-odd magnetic quadrupole moment, could induce an atomic EDM. An interesting hybrid of rare-atom and molecular beam techniques has been suggested using, for example, RaO. In such a molecule, the large internal field along the interatomic axis combined with the enhanced Schiff moment, for example 225Ra, leads to an estimated sensi­ tivity to T-odd/P-odd interactions 500 times greater than TIF (Flambaum, 2008). From Equations (9) and (14), the Schiff-moment contributions to the EDMs are dSTlF ¼ 7:4  10  14 STl cm=fm3 and dSHg ¼ 2:8  10  17 STl cm=fm3 . The sensitivity of the most recent TlF measurements was d = 3  1023 e-cm (Cho et al., 1991). A molecular EDM measurement in RaO at this level would be 10 times more sensitive to T-odd and P-odd NN interactions compared to the 199Hg measurement. Experiments with rare-radioactive atoms, francium and radium, and isotopes, 223Ra and 225Rn, provide significant new technical challenges and significantly lower statistical sensitivity than stable-atom experi­ ments. Several approaches are motivated by enhancements of two or more orders of magnitude, which would balance the loss of four or more orders of magnitude in detection rate. These experiments represent

Permanent EDMs of Atoms and Molecules

167

new applications of cold-atom techniques and spin-exchange optical pumping that may have broader applications. 5.5 Other Directions We finally mention two other directions that are not strictly atomic or molecular EDM measurements, but are important to the landscape that lies ahead: solid-state systems, in which the EDMs of a sea of electrons is measured through the induced electric or magnetic effects, and ion sto­ rage rings. In solid-state systems, the EDM of the unpaired electrons is detectable either through the magnetic field produced when the electron EDMs are EÞ or through the electric aligned by the strong internal electric field ð~ Bind ~ field induced when the electron magnetic moments are polarized by a ! BÞ (Buhmann, 2002; Liu & Lamoreaux, 2004; strong magnetic field ð E ind ~ Shapiro, 1968; Sushkov et al., 2009, 2010). For example, in PbTiO3, a ferro­ electric crystal, sensitivity to the electron EDM is enhanced due to the large number of electrons in the solid and due to the strong internal electric field in a cooled crystal (Mukhamedjanov & Sushkov, 2005). A similar measure­ ment in gadolinium—gallium garnet is underway (Liu & Lamoreaux, 2004). Another approach using ferromagnetic gadolinium—iron garnet would detect the electric field produced by the electron EDMs aligned with the magnetically polarized spins (Heidenreich et al., 2005). A charged particle EDM is defined as the displacement of the center of charge with respect to the center of mass, and can be detected for ions contained by electric and magnetic fields, for example, in a storage ring similar to that used for the muon magnetic-moment and EDM measurements (Bennett et al., 2009). A charged particle in a storage ring is guided by the magnetic field normal to the plane of the ring and addi­ tional electromagnetic fields to constrain the particle. For a particle with a magnetic moment aligned with the momentum at some time, the spin will precess with respect to the momentum in the plane of the ring at a rate that depends on the anomalous magnetic moment and the velocity. An EDM will also lead to a torque that causes spin precession that is out-of-phase with the magnetic moment precession and leads to a spin component that is perpendicular to the plane of the ring. Thus, if the spin direction is measured by detectors that distinguish alignment above the plane of the ring from alignment below the plane of the ring, the anomalous magnetic moment would be measured by the average of the up-down detectors, and the EDM would be measured by the difference of the up-down detectors. A similar approach is proposed for a deuteron EDM measurement (Orlov et al., 2006). The deuteron measurement would measure the EDM of the bare nucleus and thus the Schiff shielding of external electric fields by atomic electrons is not a factor.

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6. CONCLUSION The search for EDMs has been underway since the first neutron measure­ ment over 50 years ago. The potential for discovery of an EDM in any system is strong motivation to continue, but in order to fully parameter­ ize P-odd and T-odd physics and thus explore CP violation, experiments in several different systems will be necessary. Unfortunately, high-energy theory is not a guide to how sensitive future experiments need to be, but theory does provide strong motivation to expect new sources of CP violation along with the Standard Model parameter
QCD. Several new experimental approaches in several new systems are under­ way that may push the limits currently set by the neutron, 199Hg, and thallium. We like to say that there are many orders of magnitude to explore before we reach the Standard Model predictions for EDMs, and thus there are many orders of magnitude of opportunity to discover new physics; however, we have learned a great deal by constraining new physics with the limits already set. It is tempting but inappropriate to anticipate where the break­ throughs will be
but it is safe to expect that the next 50 years might bring the discovery of the first EDM and an era of precision measurements that will set the parameters of CP violation in the Standard Model and beyond.

ACKNOWLEDGMENTS Many have helped in developing the perspectives, the ideas, and the details presented here. The many collaborators as well as colleagues actively working on EDM experiments and related theory who have provided input and updates on their efforts for this chapter and recent review talks include Naftali Auerbach, John Behr, Dave deMille, Jon Engel, Peter Fierlinger, Norval Fortson, Clark Griffith, Ed Hinds, Klaus Jungmann, Gordon Kane, Aaron Leanhardt, Wolfgang Lorenzon, Z.T. Lu, Matt Pearson, Aaron Pierce, Michael Ramsey-Musolf, Michael Romalis, Gene Sprouse, Carl Svensson, and Eric Tardiff.

REFERENCES Abel, S., Khalil, S., Lebedev, O. (2002). Additional stringy sources for electric dipole moments. Physical Review Letters, 89, 121601. Altarev, I. S., Borisov, Y. V., Borovikova, N. V., Brandin, A. B., Egorov, A. I., Ezhov, V. F., et al. (1981). A new upper limit on the electric dipole moment of the neutron. Physics Letters B, 102, 13—16. Altarev, I. S., Borisov, Y. V., Brandin, A. B., Egorov, A. I., Ezhov, V. F., Ivanov, S. N., et al. (1980). A search for the electric dipole moment of the neutron using ultracold neutrons. Nuclear Physics A, 341, 269—283. Amini, J.M., Munger Jr, C.T., & Gould, H. (2006). Demonstration of a cold atom fountain electron electric dipole moment experiment. arXiv/physics, 0602011.

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CHAPTER

5

Spontaneous Decay, Unitarity, and the Weisskopf-Wigner Approximation Paul R. Berman and George W. Ford Michigan Center for Theoretical Physics, Physics Department, University of Michigan, 450 Church Street, Ann Arbor, MI 48109-1040, USA

Contents

Introduction Excited State Time Evolution 2.1 General Solution 2.2 Spectral Weight Function F1(!k) 2.3 Spectral Weight Function F2(!k) 2.4 Spectral Weight Function F3(!k) 2.5 Spectral Weight Function F4(!k) 2.6 Spectral Weight Function F5(!k) 2.7 Spectral Weight Function F6(!k) 3. Spectrum and Unitarity 3.1 Spectral Weight Functions F1(!k) and F2(!k) 3.2 Spectral Weight Functions F3(!k) and F4(!k) 3.3 Spectral Weight Functions F5(!k) and F6(!k) 4. Discussion Acknowledgments References

Abstract

The theory of spontaneous emission presented by Weisskopf

and Wigner [Weisskopf, V., & Wigner, E. (1930) Z. Phyz. 63,

54-73] provides an excellent approximation to the actual decay

atoms undergo on optically allowed transitions. However, the

theory cannot be rigorously correct since, when applied in a

consistent fashion that maintains unitarity, it requires that

negative frequencies be permitted in the emitted spectrum. To

avoid such problems, a better treatment is needed. Starting

1. 2.

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with a model Hamiltonian that approximates the decay of a discrete state into the continuum within the rotating-wave approximation, we show explicitly that unitarity is respected for different classes of spectral densities of the vacuum field, physical or not. When the spectral density is bounded from below or above, there is always deviation from exponential decay. In this limit, it is possible for there to be some nonzero probability for the discrete state amplitude to be nonvanishing as the time goes to infinity, even if the spectral density overlaps the transition frequency. The discrete state amplitude is expressed in terms of a finite integral that can be evaluated numerically plus contributions from at most two poles on the imaginary axis that result in terms that do not decay with time. Although the time evolution of the excited state amplitude must be calculated numerically, analytic expressions can be obtained for the spectrum of the spontaneously emitted radiation.

1. INTRODUCTION The problem of spontaneous emission plays a central role in atomic phy­ sics. To a very good approximation, the probability for an isolated atom to remain in an excited state decays exponentially in time. Moreover, the spectrum emitted by an atom in decaying from its first excited state to its ground state is approximately a Lorentzian centered at the atomic transi­ tion frequency. In fact, this characterization of atomic decay is exactly that predicted in the landmark paper by Weisskopf and Wigner (1930). To arrive at their results, Weisskopf and Wigner made two critical assumptions. They assumed that the density of states for the vacuum radiation field could be evaluated at the atomic transition frequency and they extended the fre­ quency integration over vacuum field modes to minus infinity in calculat­ ing the excited state transition amplitude. The Weisskopf-Wigner approximation works exceptionally well. As far as we know, there are no experimental results on isolated atoms in free space for which deviation from exponential decay has been detected. Despite its success, however, a theory based on the Weisskopf-Wigner approximation cannot be rigor­ ously correct since it admits negative frequencies in the spontaneous emis­ sion spectrum. In this chapter we examine the problem of spontaneous emission without invoking the Weisskopf-Wigner approximation, provid­ ing a unified treatment of both the decay of the initial state amplitude and the spectrum of the emitted radiation. An atom in an excited state decays as a result of its interaction with the vacuum field. To describe this interaction, we must solve coupled ampli­ tude equations of the form

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rffiffiffiffiffi Z1  b_ 2 ¼ i d!k f ð!k Þe  i ð!k  !0 Þ t b1k ; 

ð1aÞ

rffiffiffiffiffi  b_ 1k ¼ i f ð!k Þei ð!k  !0 Þ t b2 ; 

ð1bÞ

1

where b2 is the probability amplitude for the atom to be in excited state |2i with no photons in the radiation field, b1k is the probability amplitude for the atom to be in its ground state |1i with a photon having frequency !k = kc in the field, 2 is the excited state decay rate in a Markovian limit, !0 is the atomic transition frequency, and f(!k) is a dimensionless real function that reflects the frequency dependence of the density of states of the vacuum radiation field. The rotating-wave approximation (RWA) is implicit in Equa­ tions (1a) and (1b), since terms involving the simultaneous emission of a photon and excitation of the atom have been neglected. Equations (1a) and (1b) are, in some sense, generic equations that correspond to the decay of a discrete state into a continuum. All angular averages have been absorbed into these equations and the transition from a sum over discrete vacuum field modes has been converted to an integral over these modes. For example, if the atom-field interaction in dipole approximation is taken to be ep  A/m (e is the magnitude of the electron charge, p is the momentum operator for the electron of a one-electron atom, m is the electron mass, and A is the vector potential of the vacuum radiation field), then f ð!k Þ ¼

 pffiffiffiffiffiffiffiffiffiffiffiffiffi !k =!0 0

!k  0 : !k < 0

ð2Þ

If the atom-field interaction is taken to be er  E (r is the position operator of the electron in the atom and E is the vacuum electric field), then 

f ð!k Þ ¼

ð!k = !0 Þ 3 = 2 0

!k  0 !k < 0

ð3Þ

Equations (1a) and (1b) are written in an interaction representation, with the state vector given by jcðtÞi ¼ b2 ðtÞe  i!0 t j2; 0iþ

Z1

d!k b1k ðtÞe  i!k t j1; !k i;

ð4Þ

1

where the ket |2; 0i corresponds to the atom in its excited state with no photons in the field and ket |1; !ki to the atom in its ground state and a photon of frequency !k in the field. The ket normalization is such that h1; ! k0 j1; !k i¼ð!k  ! k0 Þ while the initial condition is b2 ð0Þ ¼ 1; bflkgðtÞ ¼ 0;

ð5Þ

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that is, the atom is prepared in its excited state. The goal is to calculate the subsequent time evolution of the excited state, as well as the state ampli­ tude b1k(t). The quantity Sð!k Þ ¼ lim jb1k ðtÞj2

ð6Þ

t!1

represents the spectrum of the spontaneously emitted radiation. Note that |b1k(t)|2 is actually a probability density, having units of !1 k . It is easy to show from Equations (1a) and (1b) that 0 d@jb2 ðtÞj þ

Z1

2

1 jb1k ðtÞj d!k A 2

1

¼ 0;

dt

ð7Þ

which is a statement of conservation of probability. Thus, the equations are unitary, since, in fact, they can be derived from a Hermitian Hamiltonian, Z1 H ¼ h!0 j2; 0ih2; 0jþ h

d!k !k j1; !k ih1; !k j 1

rffiffiffiffiffi Z1  þ h d!k f ð!k Þ½j2; 0ih1; !k jþj1; !k ih2; 0j: 

ð8Þ

1

However, if one is not careful, one can run into inconsistencies in dealing with Equations (1a) and (1b). For example, in the Weisskopf-Wigner approximation, the excited state amplitude decays exponentially, b2 ðtÞ ¼ e  t :

ð9Þ

If this result is substituted into Equation (1b), we find rffiffiffiffiffi Z1  b1k ð1Þ ¼ i f ð!k Þ dt ei ð!k  !0 Þ t b2 ðtÞ  0

rffiffiffiffiffi  f ð!k Þ ¼ i :    ið!k  !0 Þ

ð10Þ

As a consequence, Sð!k Þ ¼ jb1k ð1Þj2 ¼

 Fð!k Þ ;   2 þ ð!k  !0 Þ 2

ð11Þ

where Fð!k Þ ¼ ½f ð!k Þ 2

ð12Þ

Spontaneous Decay, Unitarity, and the Weisskopf-Wigner Approximation

179

will be referred to as the spectral weight function associated with the atom-field interaction. For f(!k) given by Equation (2) or (3), the integral Z1 S¼

Z1 d!k jb1k ðtÞj2

d!k Sð!k Þ ¼ lim 1

t!1

ð13Þ

1

diverges, a result that clearly violates unitarity. Even if a cutoff is intro­ duced into Equation (13), unitarity is still violated since S is not equal to unity. This result implies that, for the density of states associated with Equa­ tions (2) and (3), the decay cannot be purely exponential. In other words, if the correct expression for b2(t) is used in Equation (10), Equation (7) must hold. The fact that the decay is not exponential is well known (Knight & Milonni, 1976; Seke & Herfort, 1989). Approximate expressions for the excited state amplitude that have been derived indicate that there are deviations from exponential decay around t=0 and for t >> 1. In general, however, these treatments do not give an expression that is valid for all time, nor do they examine the final spectrum to see if unitarity is respected. Thus, it may prove instructive to examine Equations (1a) and (1b) for different classes of spectral weight functions F(!k) to see how they affect both the excited state decay and the emitted spectrum. We would like to stress that Equations (1a) and (1b) may not correspond to physical reality for specific choices of F(!k). For example, in the RWA, it is known (Ber­ man, 2004; Milonni et al., 2004) that Equations (1a) and (1b) lead to acausal emission. That is, the signal is nonvanishing at any distance R from the atom for t > 0. Generalizing the results to include non-RWA terms (Berman, 2004; Milonni et al., 2004) removes this embarrassment and leads to a nonvanishing signal only for R £ ct. Spectral densities of the form (2) and (3) are also know to lead to divergences in level shifts that depend at least linearly on an imposed cutoff frequency. The cutoff appears naturally if the dipole approximation is not made and retarda­ tion effects are included. In addition, there are problems associated with the assumption that the atom is excited instantaneously at t=0. A more realistic excitation protocol leads to a natural cutoff in F(!k) that is of the order of inverse of the rise time of the excitation pulse (Berman, 2005). All of this is very interesting, but is not of concern here. What we want to show explicitly is that Equations (1a) and (1b) lead to unitary results for different classes of spectral weight functions F(!k) and to examine the nature of the excited state evolution and the emitted spectrum for different F(!k). In doing so, we explain why the Weisskopf-Wigner approximation leads to unitary results when applied in a consistent fashion. We are also able to follow the transition from near-Markovian behavior when the spectral range F(!k) is a broad function that overlaps

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Paul R. Berman and George W. Ford

the transition frequency !0 to no decay at all when the spectral function does not overlap !0. It turns out that, even when F(!k) completely encom­ pass !0, there can be a small, but finite, probability to remain in state 2 as t  1 for specific choices of F(!k). Moreover, for very long times, the asymptotic form for the excited state amplitude can exhibit somewhat unusual behavior, in that it does not vary solely as an inverse power of the time (Knight & Milonni, 1976; Seke & Herfort, 1989). While the time evolution of the excited state amplitude must be calculated numerically in most cases, it turns out, rather remarkably, that it is possible to obtain a very general analytic expression for the emitted spectrum. It should be noted that the transition from exponential decay to Rabi oscillations (no decay) was studied by Cohen-Tannoudji and Avan (1977) as a function of coupling strength (vacuum-induced level shifts) using graphical methods (Cohen-Tannoudji et al., 1994). To illustrate the physical concepts, we consider the following spectral weight functions: F1 ð!k Þ ¼ 1 F2 ð!k Þ ¼

!2w ð!k  !0 Þ 2 þ !2w 

F3 ð!k Þ ¼ 

ð14bÞ

1 0  !k  !c 0 otherwise

exp½ð!k  !0 Þ=!c  !k 0 !k  !k =!k 0  !k  0 F5 ð!k Þ ¼ otherwise 0  ð!k =!0 Þexp½ð!k  !0 Þ=!c  F6 ð!k Þ ¼ 0 F4 ð!k Þ ¼

ð14aÞ

ð14cÞ 0 <0

ð14dÞ ð14eÞ

!k  0 !k < 0

ð14fÞ

where !c > 0 is a cutoff frequency and !w > 0 is the frequency width of the Lorentzian spectral weight function. The spectral weight function F1(!k) corresponds to the Weiss­ kopf-Wigner (1930) approximation and leads to pure, simple exponen­ tial decay. Spectral weight function F2(!k) is chosen to illustrate that the early time dependence of the excited state amplitude has quadratic time Z 1 dependence whenever Fð!k Þd!k exists. Moreover, it allows us to see 1

how the emitted spectrum can split into a doublet for  < !0. With F3(!k),  we move to spectral weight functions that are identically zero for !k < 0, a necessary condition when one considers spontaneous decay. For such spectral weight functions, one must include a cutoff at large !k to avoid

Spontaneous Decay, Unitarity, and the Weisskopf-Wigner Approximation

181

infinite level shifts. We shall see that the sharp cutoffs at !k=0 and !k=!c in F3(!k) lead to small, but finite, contributions to b2(1) even if !c >> !0 >> . The contribution to b2(1) from the cutoff at !k=!c is removed when the high frequency cutoff is made smoothly, as with the spectral weight function F4(!k). The function F5(!k) is chosen to simulate the ep  A/m interaction. The linear dependence on !k in F5(!k) eliminates the contribu­ tion to b2(1) from the region about !k=0, provided the line shift is less than !0. The line shift in this case has a component that depends linearly on !c and allows us to examine a type of phase transition that occurs when the line shift is equal to !0. The spectral weight function F6(!k) corresponds to the ep  A/m interaction with a smooth cutoff at (!k - !0  !c) and removes all contributions to b2(1), provided the line shift is less than !0. For F4(!k) and F6(!k), it is assumed that !c >> !0. All the spectral weight functions are chosen such that F(!0) = 1. Any contributions to b2(1) are extremely small when !c >> !0 >> , provided the shifts are less than !0. When such contributions are neglected, one can obtain the excited state amplitude in the form of a single integral that can be evaluated numerically. The integral form for the excited state amplitude enables one to obtain asymptotic expressions valid for large t. In Section 2, we derive expressions for the excited state amplitude for each of the spectral densities (14). In Section 3, we calculate jb1k ðtÞj2 , as well as the spectrum S(!k) of the emitted radiation. It is shown explicitly that the conservation of probability, Equation (7), holds for each of the spectral densities given in Equations (14). A summary and discus­ sion of the results is given in Section 4.

2. EXCITED STATE TIME EVOLUTION Integrating Equation (1a) formally and substituting the result back into Equation (1b), we find  b_ 2 ¼  

Zt 0

dt 0

Z1

0

d!k Fð!k Þe  i ð!k  !0 Þ ðt  t Þ b2 ðt 0Þ:

ð15Þ

1

Before examining the specific forms of F(!k), we derive some general results. If we assume that Z1

F¼

d!k Fð!k Þ

ð16Þ

1

and 1 !¼ F

Z1 d!k ð!k  !0 ÞFð!k Þ 1

ð17Þ

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Paul R. Berman and George W. Ford

exist, and that D >  is some characteristic maximum value of |!k - !0| that contributes in Equation (15), then for t << D-1, one can expand the exponential factor in Equation (15) and obtain  b_ 2 ðtÞ   

¼

Zt

F 

or

dt 0

0

Zt

Z1

d!k Fð!k Þ½1  ið!k  !0 Þðt  t 0Þb2 ðt 0Þ

1

dt 0b2 ðt 0Þ þ

0

i!F 

Zt

dt 0ðt  t 0Þb2 ðt 0Þ;

ð18Þ

0

€b 2   F b2  i!b_ 2 : 

ð19Þ

For early times, Dt << 1, the approximate solution of this equation is     F i! b2 ðtÞ  b2 ð0Þ 1  t2 þ b02 ð0Þ t  t2 ; 2 2

ð20Þ

which, for the initial condition (5), reduces to b2 ðtÞ  1 

F 2 t : 2

ð21Þ

0

Thus, with b2(0) = 1 and b2(0) = 0, the excited state amplitude varies Z 1 quadratically with time as t  0, provided d!k Fð!k Þ exists (Cohen­ 1

Tannoudji & Avan, 1977; Cohen-Tannoudji et al., 1994). We defer a discussion of the asymptotic expression for long times, but note that a sharp cutoff in the spectral function or its derivatives at !k = 0 or !k = !c gives rises to an inverse power law dependence on t for t >> 1; for even longer times, this dependence can be modified by a factor that is a logarithmic function of !ct. It is possible to get an approximate solution that is valid when the spectral function does not overlap the transition frequency. If we assume 6 0 for 0 £ !k £ !c < !0 and that both !0/ >> 1 and that F(!k) ¼ ð!0  !c Þ= >> 1, we can solve for Z!c jb2 ðtÞj2 ¼ 1 

jb1k ðtÞj2 d!k

ð22Þ

0

using a perturbation theory result for b1k(t), since jb1k ðtÞj << 1 in this limit. Explicitly from Equation (1b) with b2k(t) replaced by unity, we find jb1k ðtÞj2 ¼

4 sin 2 ½ð!0  !k Þt=2 Fð!k Þ ;  ð!0  !k Þ 2

ð23Þ

which when substituted into Equation (22) yields 4 jb2 ðtÞj ¼ 1  

Z!c

2

d!k Fð!k Þ 0

sin 2 ½ð!0  !k Þt=2 : ð!0  !k Þ 2

ð24Þ

Spontaneous Decay, Unitarity, and the Weisskopf-Wigner Approximation

183

The upper state probability is equal to unity at t = 0 and then oscillates as it approaches its steady state value of 2 jb2 j  1  

Z!c

2

d!k 0

Fð!k Þ ð!0  !k Þ 2

ð25Þ

for !ct >> 1. The behavior corresponds to inhomogeneously broadened, off-resonant driving of an atom-field system. A graph of the solution is presented later in this section. 2.1 General Solution Equation (15) can be solved by Laplace transform techniques. With Z1 BðsÞ ¼

e  st b2 ðtÞdt

ð26Þ

1 ; s þ GðsÞ

ð27Þ

0

and b2(0) = 1, one finds BðsÞ ¼

where GðsÞ ¼

¼

   

Z1

Z1 d

d!k Fð!k Þe  i ð!k  !0 Þ  e  s

1

0

Z1 d!k 1

Fð!k Þ : s þ ið!k  !0 Þ

ð28Þ

On taking the inverse transform of Equation (27), one arrives at the formal solution b2 ðtÞ ¼

1 2i

þ Z i1

  i1

est ds ; s þ GðsÞ

ð29Þ

where  is a small positive frequency. It is possible to derive some very general results without specifying the exact form of the spectral weight function. Let us suppose that Fð!k Þ ¼ 0;

!k < !a and !k > !b ;

ð30Þ

and that F(!) is an analytic function for !a < ! < !b. If need be, the limits !a  -1, !b  1 can be taken. With a simple change of variable, Equation (29) can be written as b2 ðtÞ ¼

ei!0 t 2i

þ Z i1

  i1

est ds ; s þ i!0 þ GðsÞ

ð31Þ

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Paul R. Berman and George W. Ford

where  GðsÞ ¼ Gðs þ i!0 Þ ¼ 

Z!b d! !a

Fð!Þ : s þ i!

ð32Þ

In this form, it is apparent that G(s) has a branch cut along the imaginary axis between s = -i!b and s = -i!a. To prove this, consider values of s equal to +x þ iy in the limit that x  0 and evaluate Gd ðyÞ ¼ lim ½Gðx þ iyÞ  Gðx þ iyÞ x!0

¼

 lim  x!0

Z!b d! !a

2xFð!Þ ð! þ y Þ 2 þ x2

Z!b ¼ 2

d!Fð!Þð! þ yÞ;

ð33Þ

!a

where (!) is a Dirac delta function. If Gd(y)  0 the point s = iy is not on the branch cut, but if Gd(y) is finite, the point s = iy is on the branch cut since the value of the function changes as we move across the imaginary axis. Clearly, it follows from Equation (33) that the branch cut exists for values of y between -!b and -!a or, equivalently, between s = -i!b and s = -i!a. To evaluate the inverse Laplace transform (31), one can choose a contour that excludes the branch cut such as the one shown in Figure 1. Using the residue theorem and the fact that the contributions along all the curved segments of the contour go to zero, one then evaluates

– ωa

– ωb

Figure 1 Bromwich contour for the inverse Laplace transform. The full vertical line is displaced by a positive frequency s from the y-axis and the vertical lines around the branch cut are displaced by –", where " is a positive frequency. Ultimately, the limit is taken in which " and s tend to zero. There are at most two poles in the contour, above and below the branch cut, on the imaginary axis. For physically acceptable spectral weight functions such as F3(w k)-F6(w k), w a = 0 since emission must occur at positive frequencies

Spontaneous Decay, Unitarity, and the Weisskopf-Wigner Approximation  Z!a

ei!0 t 2

b2 ðtÞ þ

 !b

ei!0 t þ 2 ¼

X

185

eiyt dy iy þ i!0 þ Gð" þ iyÞ

 Z!b

 !a

eiyt dy iy þ i!0 þ Gð" þ iyÞ ð34Þ

Rðp; tÞ;

p

where the sum is over all the residues R(p,t) of the function Wðs; tÞ ¼

ei!0 t est s þ i!0 þ GðsÞ

ð35Þ

at the poles p contained in the contour, and " is a small positive frequency. Using Equation (33), we transform this equation into  Z!a ei!0 t 2FðyÞeiyt dy lim b2 ðtÞ ¼ ½y þ !0  iGð" þ iyÞ½y þ !0  iGð" þ iyÞ 2 "!0  !b X þ Rðp; tÞ:

ð36Þ

p

It remains only to find the poles in the contour shown in Figure 1. It is not too difficult to prove that there are at most two poles and that these poles are located on the imaginary axis. The poles are found as solutions of the equation  s þ i!0 þ GðsÞ ¼ s þ i!0 þ 

Z!b d! !a

Fð!Þ ¼ 0; s þ i!

ð37Þ

with s = x þ iy. Setting the real part of this equation equal to zero, we find  xþ 

Z!b d! !a

xFð!Þ ¼ 0: x2 þ ð! þ y Þ 2

ð38Þ

Since F(!) is a positive analytic function in the range of integration, the only solution of this equation can be x=0; any poles must be on the imaginary axis. Setting x = 0 and taking the imaginary part of Equation (37), we find that the poles exist inside the contour provided y þ !0 ¼

 

Z!b d! !a

Fð!Þ : !þy

ð39Þ

Solutions of Equation (39) must be restricted to values of y >!a and/or y > -!b since points on the branch cut -!b £ y £ -!a are not contained in the

186

Paul R. Berman and George W. Ford

contour. Thus, there can be poles on the imaginary axis that lie outside the branch cut and inside the integration contour shown in Figure 1. Whether the poles exist depend on the specific form of F(!); however, there is a maximum of one pole for y>!a and one pole for y > -!b as can be verified by a simple graphical analysis, given the fact that F(!) > 0. As a consequence, Equation (36) can be rewritten as ei!0 t lim b2 ðtÞ ¼ 2 "!0

 Z!a

 !b

2FðyÞeiyt dy ½y þ !0  iGð" þ iyÞ½y þ !0  iGð" þ iyÞ

2 X þ Rðyj ; tÞ;

ð40Þ

j¼1

where Rðyj ; tÞ ¼ rj ei!0 t eiyj t ;

ð41Þ

[a result that follows from Equation (35)], yj is a solution of  yj þ !0  

and

Z!b d! !a

Fð!Þ ¼ 0; ! þ yj

2 31 Z!b h i1  Fð!Þ 5 >0 ¼ 41 þ d! rj ¼ 1 þ dGðsÞ=dsjs ¼ iyj  ð! þ yj Þ 2

ð42Þ

ð43Þ

!a

is the magnitude of the residue associated with the pole at yj. Equation (40) is pretty remarkable. From the Riemann-Lebesgue lemma (Whittaker & Watson, 1927), the integral term vanishes as t  1. However, the residue terms always give rise to a finite contribution to b2(1), independent of the value of !0, lim b2 ðtÞ ¼

t!1

2 X

rj ei!0 t eiyj t :

ð44Þ

j¼1

As long as the spectral weight function is bounded from above or below, b2(1) does not necessarily go to zero. Similar conclusions have been reached in looking at the decay of a discrete state into a bounded con­ tinuum (Longhi, 2007; Miyamoto, 2005). If !a  -1, !b  1 (as in the Weisskopf-Wigner approximation), the residues from the poles asympto­ tically go to zero and b2(1) is exactly equal to zero. Even for finite !a and infinite !b it is possible that b2(1) = 0, provided that the spectral function does not admit poles for y > -!a, as is the case for the spectral function given in Equation (14f). Equation (40) is a central result of this chapter. Although the residues contribute to b2(1), the contributions are negli­ gibly small provided that (1) !0/ >> 1, (2) the spectral function has a range

Spontaneous Decay, Unitarity, and the Weisskopf-Wigner Approximation

187

D!k >> , and (3) the transition frequency, including any level shifts, is somewhere in the central region of D!k; in other words, the overlap of the spectral function and the transition frequency is significant. On the other hand, at least one of the poles can contribute dominantly if the spectral function does not overlap the transition frequency. We now proceed to examine the specific spectral weight functions given in Equation (14). 2.2 Spectral Weight Function F1(w k) In this and Section 2.3, it is possible to use Equation (40) to obtain b2(t) by setting !a = -!c, !b = !c, and taking the limit that !c  1. However, for the spectral weight functions (14a) and (14b) it is much simpler to solve Equation (15) directly. This is the method we use. If F1(!k) = 1, it follows from Equation (15) that  b_ 2 ðtÞ ¼  

Zt

dt 0

¼2

0

d!k e  i ð!k  !0 Þ ðt  t Þ b2 ðt 0Þ

1

0

Zt

Z1

dt 0ðt  t 0Þb2 ðt 0Þ ¼ b2 ðtÞ

ð45Þ

0

and b2 ðtÞ ¼ e  t :

ð46Þ

The decay is purely exponential. There is no shift of the transition fre­ quency, owing to the fact that the (nonphysical) spectral weight function F1(!k) = 1(1 < !k < 1) is, in effect, symmetric about !k = !0. Since F1(!k) = 1 corresponds to the Weisskopf-Wigner spectral weight function, we see that the Weisskopf-Wigner approximation leads to purely expo­ nential decay. Moreover, we will see below that, when applied in a consistent fashion, the Weisskopf-Wigner approximation maintains uni­ on t for t0, but tarity. The amplitude b2(t) does not depend quadratically Z a quadratic dependence is guaranteed only if is not the case for F1(!k).

1

1

d!k F1 ð!k Þ exists, which

2.3 Spectral Weight Function F2(w k) Again, it is simplest and most instructive to use Equation (15) to evaluate b2(t). Combining Equations (14b) and (15), we find !w b_ 2 ðtÞ ¼   2

Zt 0

dt 0

Z1

0

d!k 1

e  i ð!k  !0 Þ ðt  t Þ b2 ðt 0Þ ð!k  !0 Þ 2 þ !2w

188

Paul R. Berman and George W. Ford

Zt

0

dt 0e  !w ðt  t Þ b2 ðt 0Þ;

ð47Þ

€b2 ðtÞ ¼ !w b_ 2 ðtÞ  !w b2 ðtÞ:

ð48Þ

¼!w 0

from which it follows that



The initial conditions are b2(0)=1; b2(0)=0. Equation (48) is recognized as the differential equation of a damped harmonic oscillator. As a consequence the decay can be classified as overdamped if !w > 4, underdamped if !w < 4 and critically damped if !w ¼ 4. In all cases, the solution varies as b2 ðtÞ  1 

!w 2 t 2

near t=0, in agreement with Equation (21), since F2 ¼

ð49Þ Z 1 1

d!k F2 ð!k Þ ¼ !w .

The solution of Equation (48) subject to the initial conditions b2(0) = 1; b_ 2 ð0Þ ¼ 0 0

1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!# vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " u B 1B 1 4 u 1 C C b2 ðtÞ ¼ B1 þ u Cexp  !w t 1  1  t 4 @ A 2 2 !w 1 !w 0 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!# vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " u C 1 4 1B u 1 C B : þ B1  u Cexp  !w t 1 þ 1  t 4 A 2 !w 2@ 1 !w

ð50Þ

For !w >> 4; b2 ðtÞ&e  t, for !w ¼ 4; b2 ðtÞ ¼ e  2t ð1 þ 2tÞ, and for !w << 4, b2 ðtÞ&e  !w t = 2 cos

pffiffiffiffiffiffiffiffiffi  !w t :

ð51Þ

If the spectral weight function (14b) is replaced by one that is centered at zero frequency, Fð!k Þ ¼

!2k

!2w ; þ !2w

ð52Þ

the excited state evolution is still described by a bi-exponential decay (except in the case of critical damping). Even for F(!k) given by Equation (52) and !w
Spontaneous Decay, Unitarity, and the Weisskopf-Wigner Approximation

189

with the atomic transition frequency, the excited state amplitude decays to zero for sufficiently long times. It is the overlap of the tail of the distribution with the atomic transition frequency in this limit that allows population to “leak” back to the ground state. 2.4 Spectral Weight Function F3(w k) We now consider the spectral weight function, 

F3 ð!k Þ ¼

1 0  !k  !c ; 0 otherwise

ð53Þ

that vanishes identically for !k < 0. Any physically acceptable spectral function must vanish for !k < 0. Since the spectral weight function given in Equation (53) leads to many features that are characteristic of decay of a discrete state into a bounded continuum, we analyze it in great detail. The early time behavior is given by Equation (21), b2 ðtÞ  1 

since

F3 ¼

Z ! c 0

!c 2 t ; 2

ð54Þ

d!k F3 ð!k Þ ¼ !c . Equation (54) is valid for times

j!c  !0 jt; !0 t << 1. The solution for arbitrary times is obtained using Equations (40), (32), and (53) with !a ¼ 0 and !b ¼ !c . From Equation (32), we find GðsÞ ¼

 

Z!c d! 0

 1 i i!c ¼  ln 1 þ : s þ i!  s

ð55Þ

The branch cut of ln z is along the negative real axis, which implies that G(s) has a branch cut on the imaginary axis between s ¼ i!c and s=0 with  lim Gð– " þ iyÞ ¼  lim i ln 1 þ i!c "!0 "!0  –" þ iy   i !c ¼ ln 1  – ;  y

ð56Þ

provided !c  y  0 and " is a positive frequency. When this result is substituted into Equation (40), we obtain b2 ðtÞ ¼

ei!0 t 

þ

Z0

eiyt dy     !c 2  !c y þ !0  ln 1  þ 2  y

2 X Rðyj ; tÞ: j¼1

ð57Þ

190

Paul R. Berman and George W. Ford

This result is valid equally for !0 < !c and !0 > !c. In other words, the results presented allow one to pass continually from the limit where the spectral function completely encompasses the atomic transition fre­ quency to the limit where the atomic transition frequency lies outside the spectral weight function. The excited state amplitude is written as the sum of a finite integral that can be evaluated numerically, plus contributions from the poles in the contour. The integrand is perfectly well defined over the integra­ tion range, with ln ð1  !yc Þ varying from 1 to 1 as y varies from !c to 0. The integral term in Equation (57) reduces to purely exponential decay if the ln term is dropped and the integral is extended from 1 to 1. However, as we shall see, Equation (57) exhibits a much more com­ plex behavior than simple exponential decay. The first step in analyzing Equation (57) is to find the poles that contribute to b2(t). 2.4.1 Poles The poles occur at values of yj which are solutions of    !c yj þ !0  ln 1 þ ¼ 0;  yj

ð58Þ

a result that follows from Equation (39) with F(!)=1 and ð!a ¼ 0; !b ¼ !c Þ. It is easy to show graphically that there is a pole both for y1 > 0 and one with y2<!c. The residues at these poles obtained from Equations (41) and (43) are given by Rðs ¼ iyj ; tÞ ¼

ei!0 t eiyj t : !c 1þ yj ðyj þ !c Þ

ð59Þ

It then follows from Equations (57) and (59) that the excited state amplitude evolves as b2 ðtÞ ¼

ei!0 t 

Z0

eiyt dy     !c 2  !c y þ !0  ln 1  þ 2  y

þei!0 t

2 X

eiyj t ; !c j¼1 1 þ yj ðyj þ !c Þ

ð60Þ

where the sum is over the two poles occurring at s ¼ iy1 ; iy2 . This represents an exact solution to the problem, satisfying the initial condi­ tion b2(0) = 1.

Spontaneous Decay, Unitarity, and the Weisskopf-Wigner Approximation

191

Equation (58) cannot be solved analytically; however, it is possible to obtain approximate solutions for specific cases of interest. To restrict that range of parameters somewhat, we assume that !0  10;

ð61Þ

but we want to stress that Equation (60) is valid for arbitrary values of the parameters. With this restriction, we consider two additional limits, ðiÞ 100!0  ð!c  !0 Þ >> 10 ðiiÞ

0  !c << ð!0  10Þ:

ð62aÞ ð62bÞ

The first case corresponds to a spectral weight function that completely encompasses the atomic transition frequency, while the second to a spectral weight function that does not overlap the atomic transition frequency (see Figure 2). In limit (62a), the solutions of Equation (58) are given approximately by y1  !c e  !0 =  ;

ð63aÞ

y2   !c  !c e   ð!c  !0 Þ = 

ð63bÞ

and the residues Rðyj ; tÞ (Equation 59) by Rðy1 ; tÞ&

!c i!0 t  !0 =  ; e e 

(a) Fωk

ω0 ωc

ωk

(b) Fωk

ωc

ω0

ωk

Figure 2 Spectral weight functions that overlap (a) and do not overlap (b) the transition frequency

ð64Þ

192

Paul R. Berman and George W. Ford

Rðy2 ; tÞ&

!c i!0 t  i!c t   ð!c  !0 Þ =  e e e : 

ð65Þ

Thus, although the contributions are nonvanishing, they are exponen­ tially small in the parameters !0 = and ð!c  !0 Þ=, with b2 ðtÞ 

ei!0 t 

Z0

eiyt dy     !c 2  !c y þ !0  ln 1  þ 2  y

!c i!0 t  !0 =  e ðe þ e  i!c t e   ð!c  !0 Þ =  Þ  Z 0 ei!0 t eiyt dy : &      !c 2  !c þ 2 y þ !0  ln 1   y

þ

ð66Þ

As the value of !c diminishes, the contribution to b2(t) from the pole near y=0 remains small in the limit (61), but the contribution from the pole having y < !c begins to grow. As !c passes through !0 to values where 0  !c << ð!0  10Þ, limit (62b), one can estimate the pole locations as y1  !c e  !0 =  ;    !c y2  !0 þ ln 1   !0

ð67aÞ ð67bÞ

and the residues at these poles as !c i!0 t  !0 =  e e     it !c exp ln 1   !0 : Rðy2 ; tÞ& !c    1þ  !c !0 !0  !c  ln 1   !0 Rðy1 ; tÞ&

If

!c   << 1  !c !0 !0  !c  ln 1   !0 

ð68Þ

ð69Þ

ð70Þ

which is satisfied for !c =!0 < 0:75, the residue R(y2) provides the domi­  nant contribution to b2(t) in Equation (60); in other words,    it !c b2 ðtÞ  exp ln 1   !0

ð71Þ

Spontaneous Decay, Unitarity, and the Weisskopf-Wigner Approximation

193

in this limit. There is no decay (approximately) if the spectral function does not overlap the transition frequency. Equation (71) does not agree 2 with Equation (57) for t < !1 0 , since terms of order !c =!0 have been neglected in Equation (71). 2.4.2 Integral Term We now turn our attention to the integral

IðtÞ ¼

ei!0 t 

Z0  !c

eiyt dy     !c 2 þ 2 y þ !0 þ ln 1   y

ð72Þ

appearing in Equation (60). Although this integral can be evaluated numerically, the presence of the ln term complicates attempts to obtain analytic solutions. Before presenting some numerical solutions below, we obtain approximate expressions for the integral when condition (61) is satisfied in the limits defined in Equation (62). Under these conditions, the ln term contributes appreciably only near the endpoints of the inte­ gral. We return to the endpoint contributions shortly, since they give rise to the asymptotic value of the integral for very long times, but neglect such contributions for the moment and approximate Equation (72) as ei!0 t I1 ðtÞ ¼ 

Z0  !c

eiyt dy : ½y þ !0  2 þ  2

ð73Þ

We consider each of the limits in Equation (62) separately. 100!0  ð!c  !0 Þ >> 10. In this limit, the contribution from the poles is negligible and the integral (73) provides a good approximation to b2(t). The integral can be evaluated analytically and one obtains I1 ðtÞ ¼

e  t

2e2t þ i e2t Ei½ð  i!0 Þt  i Ei½ð þ i!0 Þt 2

þ i Ei½ð  ið!c  !0 ÞÞt  i e2t Ei½ð þ ið!c  !0 ÞÞtg;

where

Z1 EiðzÞ ¼  z

et dt t

ð74Þ

ð75Þ

is an exponential integral (the principal value of the integral must be taken). To a good approximation, I1 ðtÞ&e  t , a result that follows if the integration limits in Equation (73) are extended to +1. For early times, !0 t << 1; ð!c  !0 Þt << 1, Equation (74) reduces to

194

Paul R. Berman and George W. Ford

I1 ðtÞ  1 

!c t þ i ln !0 ð!c  !0 Þ 



 !c !c t2 1  !0 2

ð76Þ

and, for t >> 1, Equation (74) reduces to I1 ðtÞ  e

 t

 ei!0 t 1 e  i!c t þ  ; ð!c  !0 Þ 2 it !02

ð77Þ

giving the asymptotic behavior for long times. Owing to the neglect of the ln term, b2(0) 6¼ 1 and b02 ð0Þ ¼ 6 0; that is, the approximate solution given by Equation (76) does not satisfy the correct initial conditions. Explicitly one finds from Equation (76) that !c ; !0 ð!c  !0 Þ    !c  !0 b02 ð0Þ&I10 ð0Þ ¼ i ln :  !0

b2 ð0Þ&I1 ð0Þ ¼ 1 

ð78Þ ð79Þ

c Thus, corrections to the exact results are of order !0 ð!!c ! ; moreover, 0Þ given that b2(0) ¼ 6 1 and b02 ð0Þ 6¼ 0, Equation (76) does not correctly predict the early time behavior predicted by Equation (21). 0  !c << ð!0  10Þ: In this limit, most of the contribution to b2(t) originates from the pole near s ¼ i!0, although the integral provides a c . Explicitly, one finds for the integral small contribution, of order !0 ð!!c ! 0Þ term

I1 ðtÞ ¼

i e  t 2t e Ei½ð  i!0 Þt  Ei½ð þ i!0 Þt 2

 þ Ei½ð  ið!c  !0 ÞÞt  e2t Ei½ð þ ið!c  !0 ÞÞt ;

ð80Þ

which is plotted in Figure 3 for !0 = ¼ 50 and !c = ¼ 10. The function jI1 ðtÞj << 1 at t=0 and decays to even smaller values in a time of order  1. For the parameters shown, the exact integral expression given by Equa­ tion (72) and the approximate integral expression given by Equation (80) virtually coincide. The dominant contribution to b2(t) originates from the pole near s ¼ i!0 and not from the integral. 2.4.3 Long-Time Behavior We consider only the limit (62a) in which the spectral range overlaps the transition frequency. The pole contributions dominate the long-term behavior if this is not the case. In the limit of large t it is easy to estimate the endpoint contributions from Equation (73) using integration by parts. One finds  ei!0 t 1 e  i!c t  I1 ðtÞ  ð!c  !0 Þ 2 it !02

ð81Þ

195

Spontaneous Decay, Unitarity, and the Weisskopf-Wigner Approximation

| / 1(t )| x 1000 1.5

1.0

0.5

0

0

1

2

3

4

γt

5

Figure 3 The approximation of the integral contribution to the excited state amplitude when the spectral weight function F3(w k) does not overlap the transition frequency: w 0/g = 50 and w c/g = 10

consistent with Equation (77). The asymptotic result (81) becomes domi­ nant for times satisfying e  t <<

  ; : t!20 t ð!c  !0 Þ 2

ð82Þ

It is not easy to estimate the next-order corrections to Equation (81) resulting from the use of the correct integral h expression given by

iEqua­ tion (72). The reason for this is that the

y þ !0 

 

ln 1

!c y

term

goes to zero near the endpoints and the logarithm term diverges at the endpoints. From the numerical solutions, it appears that endpoint correc­ tions to Equation (81) are of order 2 ln ð!c tÞ 2ln ð!c tÞ ; ; !0 ð!c  !0 Þ

which is consistent with replacing y in the logarithm term in Equation (72) by 1/t. Therefore, for sufficiently small =!0 ; =ð!c  !0 Þ, Equation (81) represents a good asymptotic expansion provided that Equation (82) holds and that !c t << e!0 = 2 ;

!c t << e ð!c  !0 Þ = 2 :

ð83Þ

The oscillatory behavior of jb2 ðtÞj&jI1 ðtÞj, implicit in Equation (81), is seen clearly in Figure 4 for t&30 and !0 = ¼ ð!c  !0 Þ= ¼ 500. The exact (Equation 60) and asymptotic (Equation 81) solutions differ by no more than 2% in this region, except near values of I1(t)=0. For these parameters, the crossover from exponential to power law dependence in

196

Paul R. Berman and George W. Ford

| b2(t ) | x 108 7.5

5

2.5

0

29.995

30

30.005

γt

Figure 4 Oscillatory behavior of |b2(t)| for the spectral weight function F3(w k) with g t  30 and w 0 /g = (w cw 0)/g = 500

the total solution for the excited state amplitude occurs for t&8. For smaller values of !0 = and ð!c  !0 Þ=, such as !0 = ¼ ð!c  !0 Þ= ¼ 10, the qualitative behavior of the exact and asymptotic solutions is similar, but the asymptotic amplitude is roughly 60% of the exact ampli­ ln ð!c tÞ tude, reflecting corrections of order 2 ! &0:4 that were discussed 0 above. The power law dependence cannot remain valid for extremely long times, when conditions (83) are violated. To see this, one need only return to the exact equation, Equation (72). Since the integrand vanishes at the endpoints, a simple asymptotic expansion fails. How­ ever, asymptotic expansions of integrals of the type (72) containing logarithmic singularities at the endpoints are known (Wong, 1989). Near the endpoints, the  2 term in the denominator of Equation (72) can be neglected and, with a few changes of variables, Equation (72) can be recast as !c ei!0 t IðtÞ  

Z1 0

e  iy!c t dy    2 ;  !c 1 y!c þ !0  ln  y

ð84Þ

where only the endpoint contributions are to be calculated. The calcula­ tion from the lower endpoint is obtained by neglecting the 1 in the logarithm and the y!c term in the denominator, and by making the change of variable y ! y expð!0 =Þ;

Spontaneous Decay, Unitarity, and the Weisskopf-Wigner Approximation

197

which transforms Equation (84) into IðtÞ 

!c ei!0 t expð!0 =Þ 

Z 0

exp ½iy!c t exp ð!0 =Þdy    2 y ln !c

ð85Þ

where only the lower endpoint is considered. This is exactly in the form needed to apply Equation (2.46) of Wong (1989), and one finds I1 ðt; y ¼ 0Þ 

ei!0 t it½ln ð!c tÞ2

;

ð86Þ

provided !c t expð!0 =Þ >> 1:

ð87Þ

A similar treatment for the upper endpoint of Equation (84) yields I1 ðt; y ¼ 1Þ  

ei!0 t e  i!c t it½ln ð!c tÞ2

;

ð88Þ

provided !c t exp½ð!c  !0 Þ= >> 1:

ð89Þ

Thus in the very long time limit defined by both Equations (87) and (89), the asymptotic form is I1 ðtÞ 

 ei!0 t it½ln ð!c tÞ

2

  1  e  i!c t :

ð90Þ

It is clear from Equations (83), (87), and (89) that the asymptotic results given by Equations (81) and (90) occur in nonoverlapping time regions. 2.4.4 Numerical Results The exact equation for the upper state amplitude as a function of time is given by Equation (60). Numerical evaluations of the integral and residue contributions appearing in that equation can be obtained easily, provided !0 = is not too large. We present a few exact results for jb2 ðtÞj in the limit that t <  10, avoiding the long-time asymptotic contributions. The first case shown (Figure 5a) is for !c = ¼ 2!0 = ¼ 200 and approximates exponential decay. The second case shown (Figure 5b) is for !c = ¼ 2!0 = ¼ 20 and departs from exponential decay with this reduced value of !0 =. In these examples, the poles contribute negligibly.

198

Paul R. Berman and George W. Ford

(a)

(b) | b2(t ) |

| b2(t ) |

1.0

1.0 0.75

0.75

0.5

0.5

0.25

0.25

0

0

1

2

3

4

5

γt

(c)

0

1

2

3

4

5

γt

(d)

| b2(t ) |

| b2(t ) |

1.0

1.0

0.99

0.75

0.5

0

0

2.5

5

7.5

10

γt

0.98

0

2.5

5

7.5

γt 10

Figure 5 The excited state amplitude |b2(t)| for the spectral weight function F3(w k) and (a) w c / g = 2w 0/g = 200, (b) w c / g = 2w 0/g = 20 (the dashed curve is the approximate solution given in Equation 74), (c) w c / g = w c / g = 10, and (d) w c / g = w 0 / 5g = 2

The approximate solution given by Equation (74) is also shown in Figure 5b and agrees well with the exact solution except near t=0 (the approximate and exact solutions overlap in Figure 5a). As we reduce !c =!0 keeping !0 =  10, the pole y1 in Equation (60) still contributes negligibly, but the pole at y2 begins to contribute. For !c = ¼ !0 = ¼ 10 (Figure 5c) one sees that a steady state is reached in which jb2 ðtÞj ¼ 0:736. Finally, for !c = ¼ !0 =5 ¼ 2 (Figure 5d), the transition frequency is well outside the range of the spectral weight function. In this limit, a perturba­ tion theory approach is valid and the solution shown in Figure 5d is approximated very well by Equation (24) with F(!k)=1. The magnitude of the upper state amplitude quickly settles to the steady state value 2 2 jb2 ð1Þj41  

Z!c 0

31 = 2 1 !c 5 &1  d!k ¼ 0:992 !0 ð!0  !c Þ ð!0  !k Þ 2

given by Equation (25).

ð91Þ

199

Spontaneous Decay, Unitarity, and the Weisskopf-Wigner Approximation

2.5 Spectral Weight Function F4(w k) We now remove the sharp cutoff at !c by choosing the spectral weight function 

F4 ð!k Þ ¼

exp½ð!k  !0 Þ=!c  0

!k  0 !k < 0

ð92Þ

and consider only the case !c >> !0. There are two effects of the smooth cutoff. The second pole y2 is eliminated and there is no asymptotic contribution from the lower endpoint of the integral. The early time behavior is given by Equation (21), F4 ¼

Z 1 0

b2 ðtÞ  1 

d!k F4 ð!k Þ ¼ !c e!0 = !c .

!c e!0 = !c 2 t , 2

since

From Equation (32), one finds GðsÞ ¼

e!0 = !c 

Z

1

d! 0

  e  ! = !c i is ¼ e!0 = !c e  is = !c G 0;  ; s þ i!  !c

ð93Þ

where   Z 1 is G 0;  ¼ t  1 e  t dt !c z

is the incomplete gamma function. The branch cut of G 0;  the negative imaginary axis and

ð94Þ is !c

is along

 !0 = !c ið–" þ iyÞ  is = !c lim Gð–" þ iyÞ ¼  lim ie e G 0; "!0 "!0  !c ¼

   ie!0 = !c y = !c y G 0; þ i – e ðy þ !0 Þ = !c ; e  !c

ð95Þ

provided 1 < y  0. As a consequence, to calculate b2 using Equation (31) we can choose the contour shown in Figure 6 to arrive at ei!0 t b2 ðtÞ ¼ 

þ

Z0

e ðy þ !0 Þ = !c eiyt dy i2 h i2 h  ðy þ !0 Þ = !c e ½Gð0; y=!c Þ þ i þ e ðy þ !0 Þ = !c  1 y þ !0   ei!0 t eiy1 t !0 = !c

e 1þ y1

y1 þ !0  !c

:

ð96Þ

since there is now only one pole at s ¼ iy1 ðy1 > 0Þ inside the contour. The pole location is found as a solution of

200

Paul R. Berman and George W. Ford

Figure 6 Bromwich contour for the weight functions F4(w k) and F6(w k)

  y1  y1 þ !0  e ðy1 þ !0 Þ = !c G 0; ¼ 0:  !c

ð97Þ

The residue was calculated using Equations (41), (43), and (97). For !0 =  10, the contribution from the pole is negligible and the major difference between Equations (57) and (96) (when !c/!0  1) is in their asymptotic expansions. Since there is no endpoint contribution from y = 1, the only endpoint contribution is from y = 0 and the result analogous to Equation (81) is I1 ðtÞ 

ei!0 t e!0 = !c : it!20

ð98Þ

In contrast to Equation (81), the behavior of jI1 ðtÞj is no longer oscillatory. 2.6 Spectral Weight Function F5(w k) The spectral weight function  F5 ð!k Þ ¼

!k =!0 0

!c  !k  0 otherwise

ð99Þ

is the one that is encountered if the atom-vacuum field interaction is taken to be epA/m. From Equations (32), (56), and (40), one finds

201

Spontaneous Decay, Unitarity, and the Weisskopf-Wigner Approximation

GðsÞ ¼

 !0

Z

1

d! 0

 ! i!c s i!c ¼ þ ln 1 þ s þ i! !0 !0 s

ð100Þ

and b2 ðtÞ ¼

ei!0 t 

Z

þ ei!0 t

0

!c

ðy=!0 Þeiyt dy     2 !c y !c 2 y þ ln 1  þ y þ !0  !0 y !0 !0

X2

eiyj t

j¼1

!; !2c 1 !20 ð!c þ yj Þ

!0 yj

ð101Þ

where the sum is over the contributions from the two poles on the imaginary axis within the contour shown in Figure 1 with !a=0 and !b=!c. The pole locations are found as solutions of yj þ !0 

 yj  !c !c þ ln 1 þ ¼ 0: yj !0 !0

ð102Þ

The magnitude of the residues, rj ¼

!0 yj

1 !; !2c 1 !20 ð!c þ yj Þ

ð103Þ

appearing in Equation (101) were calculated using Equations (41), (43), and (102). In the discussion below, we assume always that !0 =  10. The early time behavior is given by Equation (21), b2 ðtÞ  1  F5 ¼

Z ! c 0

d!k F5 ð!k Þ ¼

!2c =2!0 .

!2c 2 4!0 t ,

since

If the logarithm term is neglected in the

integrand in Equation (101), the integral can be evaluated in terms of exponential integrals. Qualitatively, the results are similar to that for F3(!k). However, there are a few differences. First, there is now a linear shift in the transi­ tion frequency !c =!0 , as well as the logarithmic shift. If so desired, the linear shift can be “renormalized” away,1 but we do not do so and allow

One way to “renormalize” the result is to add a term h

1

 

Z

1 1

d!½Fð!k =!k Þj2ih2j to the Hamiltonian

given in Equation (8). This term leads to an energy shift of level 2 that exactly cancels the !c/p!0 term in Equation (101). In other words, it guarantees that level 2 cannot be shifted below level 1 in energy.

202

Paul R. Berman and George W. Ford

!00 ¼ !0 

!c !0

ð104Þ 0

to take on both positive and negative values. If !0 > 0, there is no pole for y1 > 0 owing to the fact that the ln term in Equation (101) is multiplied by y. For !00 <0, there is a pole for y1 > 0 that contributes to b2(1). There is always a pole for y2 < !c The second difference occurs in the asymptotic expansions. Owing to the linear dependence of the integrand on y, a second integration by parts is needed to obtain the contribution from the endpoint at y = 0. As a result, Equation (81) must be replaced by "  # ei!0 t 1 e  i!c t !c 1 þ  : I1 ðtÞ   t !0 t !002 t ð!c  !00 Þ2 i

ð105Þ

Moreover, if we are interested in the very long-time asymptotic behavior corresponding to Equation (90), there is no logarithmic contribution from the endpoint near y=0 owing to the factor of y that multiplies the logarithm. In summary, the decay is qualitatively similar to that for the spectral < 10. That is, b2(1)  0 if !c/!00  > 1.3, weight function F3(!k) if 0< !00 /  0 6 0 if !c =!0 < 1:3 (when the upper bound of the spectral weight and b2(1) ¼  function approaches the transition frequency). On the other hand, for !00 < 0 we enter a qualitatively new regime in which the effective transition frequency is negative. In this case, the spectral weight function can never overlap the transition frequency and the excited state amplitude b2(1)6¼0. The excited state amplitude is shown in Figure 7 for !0/=10 and |b2(t )| 1.0

0.9

0.8

0

0.25

0.5

0.75

1.0

γt

Figure 7 Excited state amplitude |b2(t)| for the spectral weight function F5(w k) and w 0 / g =10,w c / g =400(w 00 =2.73)

Spontaneous Decay, Unitarity, and the Weisskopf-Wigner Approximation

203

!c = ¼ 400ð!00 = ¼ 2:73Þ, exhibiting behavior similar to that shown in Figure 5d. The dominant pole is at y1=2.35 and the magnitude of the residue at this pole is 0.883. There is something like a phase transition that occurs for fixed != < 10 as !c is varied around !c=!p, where  !p ¼ !20 =:

ð106Þ

The value !c=!p corresponds to !00 ¼0. For 2!0 < !c < ! , there is no pole  p for y1 > 0 and the pole with y2 < !c contributes negligibly, b2 ð1Þ& 0. On the other hand, for !c > !p, there is a pole for y1 > 0 that contributes significantly to the excited state population as t  1. When !c > !p one can use Equation (102) to estimate the pole position as y1&

where

!00 ; 1 þ ~ ln ð1  !c =!00 Þ

ð107Þ

 : !0

ð108Þ

~ ¼

  Equation (107) is valid provided ~ ln 1  !c =!00 << 1 . The magnitude of the residue associated with this pole, obtained from Equation (103), can be approximated as r1 ¼ jb2 ð1Þj&

1 ; 1 þ ~ ln ð1  !c =!00 Þ

ð109Þ

where the assumption y1 < j!00 j << !c was used. The amplitude |b2(1)| rises from a value of zero for !c < !p to a value of order unity for !c>!p, in a frequency window of order !ce1/~ centered at  !c, and r1 approaches the asymptotic value !c ¼ !p . For !c >> !p ; !00 &~ r1 ¼ jb2 ð1Þj

1  1 þ ~ ln 1 þ ~

1

;

ð110Þ

Note that this asymptotic result depends only on ~ and not !c. For !0/ =10 and !c << !p , |b2(1)| approaches an asymptotic value of 0.923.2 The value of |b2(1)| obtained from a numerical solution for the residues is plotted in Figure 8 as a function of !c = for !0/ = 10. The approximate solution given by Equation (109) is also shown in the figure. As can be seen in the figure, although Equation (109) was derived assuming that ~ ln ð1  !c =!00 Þ << 1 it provides a good approximation to |b2(1)| 2 Equation (110) leads to a value |b2(1)|=0.900, but the position of the pole given by Equation (107) and the residue given by Equation (109) are not exact–the value 0.923 is found from a numerical evaluation of the residue. A somewhat better estimate for the asymptotic value of |b2(1)| is ð1 þ ~ Þ where  ¼ ½1 þ ~ Inð1 þ ~  1 Þ   1 is the value given in Equation (110); this results in a value equal to 0.926. Equations (107) and (109) give better approximations to the exact results with decreasing ~ .

204

Paul R. Berman and George W. Ford

|b 2(∞)| 1.0 0.75 0.5 0.25 0 100

300

500

700

ωc /γ

Figure 8 Excited state amplitude |b2(1)| as a function of w c / g for the spectral weight function F5(w k) and w 0/g =10. The solid curve is obtained using the residues calculated numerically, while the dashed curve is the approximate solution given by Equation (109)

for all frequencies !c =!p > 1. The “phase transition” occurs at !c = ¼ !p = ¼ 100. The point !c ¼ !p is actually problematic since the pole now lies on the branch point at y = 0. In this case the pole lies inside the contour shown in Figure 1 so it would appear that the pole does not contribute; however, the contribution from the semicircle around the pole should probably be examined in more detail. 2.7 Spectral Weight Function F6(w k) Finally, for the spectral weight function  F6 ð!k Þ ¼

ð!k =!0 Þexp½ð!k  !0 Þ=!c  0

!k  0 ; !k < 0

ð111Þ

one finds GðsÞ ¼

e!0 =!c !0

Z1 d! 0

  !e !=!c i!c e!0 =!c s !0 =!c is=!c is ¼ þ e e G 0;  ; s þ i! !0 !0 !c

ð112Þ

and b2 ðtÞ ¼

ei!0 t 

þ

Z0  1

ðy=!c Þe ðyþ!0 Þ =!c eiyt dy 2  y ðyþ!0 Þ =!c y ðyþ!0 Þ =!c 2 e ½Gð0;y=!c Þþi þ e yþ!000 þ !0 !0

!c e!0 =!c !0 y1

ei!0 t eiy1 t Yð!000 Þ y21 þ!0 ð!c þy1 Þ  !c y1

ð113Þ

Spontaneous Decay, Unitarity, and the Weisskopf-Wigner Approximation

205

where !000 ¼ !0 

!c !0 = !c e !0

ð114Þ

and Y is the Heaviside function. The pole location is found as a solution of   !c !0 = !c y1  ðy1 þ !0 Þ = !c y1 y1 þ !0  e þ e G 0; ¼ 0: !c !0 !0

ð115Þ

The residue was calculated using Equations (41), (43), and (115). There is a pole inside the contour only if !000 < 0; otherwise the excited state amplitude is given solely by the integral term in Equation (113) and b2 ð1Þ  0. For !000 < 0, the analysis follows that give for F5 ð!k Þ. The early time behavior is given by Equation (21), since F6 ¼

Z ! c 0

b2 ðtÞ  1 

!c2 e!0 =!c 2 2!0 t ,

d!k F6 ð!k Þ ¼ !2c e!0 = !c =!0 .

For !c >> !0, the major difference of Equations (101) and (113) is in their asymptotic expansions. Since there is no endpoint contribution from y ¼ 1, the only endpoint contribution is from y=0 and the result analo­ gous to Equation (105) is (Knight & Milonni, 1976; Seke & Herfort, 1989) I1 ðtÞ  

ei!0 t e!0 = !c : !0 !0002 t2

ð116Þ

There is no logarithmic time dependence in this case, even in the limit of very long times.

3. SPECTRUM AND UNITARITY We now turn our attention to the spectrum of the emitted radiation. The probability to find the atom in state 1 with a photon having frequency !k in the field at time t is obtained from Equation (1b) as Sð!k ; tÞ ¼ jb1k ðtÞj2 ¼

t  Z 2  0 Fð!k Þ dt 0ei ð!k  !0 Þ t b2 ðt 0Þ : 

ð117Þ

0

At any time t, conservation of probability requires that Z1 jb2 ðtÞj2 þ

Sð!k ; tÞd!k ¼ jb2 ð0Þj2 ¼ 1:

ð118Þ

1

To obtain an analogous expression related to conservation of energy, we evaluate the expectation value of the Hamiltonian (8) for the state vector (4) and find

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Paul R. Berman and George W. Ford

Z1 h!0 jb2 ðtÞj2 þ

h!k Sð!k ; tÞd!k þ 2 h 1

rffiffiffiffiffi Z1  Re f ð!k Þei ð!k  !0 Þ t b2 ðtÞb1k ðtÞd!k  1

2

¼ h!0 jb2 ð0Þj ¼ h!0 :

ð119Þ

The third term on the left-hand side of the equation is the expectation value of the atom-field interaction energy and is nonvanishing as t1 if b2(1)6¼0. We have been able to verify that Equations (118) and (119) are respected for each of the spectral weight functions considered in this work. In the case of the spectral weight functions F3 - F6, the calculations must be done numerically. Thus, there is no problem with either con­ servation of probability (unitarity) or conservation of energy if Equations (1a) and (1b) are solved without approximation. To examine the spectral distribution of the emitted radiation, it is necessary to evaluate Sð!k Þ ¼ lim Sð!k ; tÞ: t!1

ð120Þ

Only if b2 ð1Þ  0 does Z1 S¼

Sð!k Þd!k ¼ 1;

ð121Þ

1

otherwise S ¼ 1  jb2 ð1Þj2 : It is possible to obtain a simple analytic expression for S(!k). At first glance, it might seem that such an expression can be derived using  2 Z1     i ð!  ! Þ t Sð!k Þ ¼ Sð!k ; 1Þ ¼ Fð!k Þ dt e k 0 b2 ðtÞ ;   

ð122Þ

which, when combined with Equations (26), (27), and (32) yields  Fð!k ÞjB½ið!k  !0 Þj2     2  1   ; ¼ Fð!k Þ   !k  !0 þ iGði!k Þ 

Sð!k Þ ¼

ð123Þ

where G(s) is given by Equation (32). Equation (123) is valid only if there are no poles inside the contour shown in Figure 1. The problem in its derivation is that the limit t ! 1 was taken directly in the integral, rather than squaring the integral and then taking the limit. If the proper limiting process is carried out, the poles give rise to an additional contribution to Sð!k Þ. A simple example illustrates this point. Consider the function b2 ðtÞ ¼ ae  t þ

N X j¼1

rj ei ðyj þ !0 Þ t ;

ð124Þ

Spontaneous Decay, Unitarity, and the Weisskopf-Wigner Approximation

207

where a is a complex constant and yj and rj are real constants with rj > 0. The Laplace transform of this function is BðsÞ ¼

N X rj a þ ;  þ s j ¼ 1 s  iðyj þ !0 Þ

ð125Þ

such that B½ið!k  !0 Þ ¼

N X rj a þ :   ið!k  !0 Þ j ¼ 1 ið!k þ yj Þ

ð126Þ

To calculate the spectrum correctly, we start from Equation (1b), 0 1 rffiffiffiffiffi Zt N X  0 0 0 b1k ðtÞ ¼ i f ð!k Þ dt 0 ei ð!k  !0 Þ t @ae  t þ rj ei ðyj þ !0 Þ t A  j¼1 20  3  rffiffiffiffiffi N  t þ i ð!k  !0 Þ t i ð!k þ yj Þ t X a 1  e r ðe  1Þ  j 5 ¼i f ð!k Þ4 þ    ið!k  !0 Þ ið!k þ yj Þ j¼1

ð127Þ

and evaluate

 2  N N ið!k þyj Þt  X X  r e r  a j j  ; Sð!k Þ¼lim jb1k ðtÞj2 ¼ Fð!k Þlim  þ þ t!1 t!1ið!k !0 Þ ið!k þyj Þ j ¼ 1 ið!k þyj Þ   j¼1 ð128Þ

where have set lim e  t ¼ 0. The cross term involving the oscillating t!1 exponential does not contribute as t ! 1 and we are left with 2 3 2   2 N N X X r   r  a j j 7 6  þ Sð!k Þ ¼ Fð!k Þ4 þ 5  ð!k þ yj Þ 2     ið!k  !0 Þ j ¼ 1 ið!k þ yj Þ  j¼1 0 1 N X r2j  2 A ¼ Fð!k Þ@jB½ið!k  !0 Þj þ  ð!k þ yj Þ 2 j¼1 0 1  2 N X r2j    1  þ A; ¼ Fð!k Þ@ ð!k þ yj Þ 2  !k  !0 þ iGði!k Þ  j¼1

ð129Þ

which is the correct expression for the spectrum when there are poles on the imaginary axis that lead to a finite value for b2(1). Equation (123) agrees with this expression only if all the rj vanish, that is, if b2(1)=0. In the limit that t  1, the equation for conservation of probability, Equation (118), reduces to

208

Paul R. Berman and George W. Ford

2 X j¼1

Z1 rj2 þ

Sð!k Þd!k ¼ 1;

ð130Þ

1

2

where jb2 ð1Þj has been evaluated using Equation (44) (recall that rj is the magnitude of the residue from pole yj) and S(!k) is given by Equation (129). In a similar manner, one can show that the analogous equation for conservation of energy, Equation (119) evaluated in the limit that t  1, is !0

2 X j¼1

Z r2 j þ

1

1

!k Sð!k Þd!k  2

 

Z

1

1

Fð!k Þ

2 X

r2 j

j¼1

ð!k þ yj Þ

d!k ¼ !0 :

ð131Þ

If Equation (42) is used, this can be transformed into !0

2 X j¼1

Z1 rj2 þ 1

2 X !k Sð!k Þd!k  2 ð!0 þ yj Þrj2 ¼ !0 :

ð132Þ

j¼1

which applies equally for all spectral weight functions.3 Equation (129) is truly remarkable in that it gives the spectrum of sponta­ neous emission in terms of the Laplace transform of the excited state ampli­ tude and the position of any poles and their residues in the contour integration used to obtain the inverse Laplace transform of B(s). Since we have relatively simple, analytic forms for both G(s) and the magnitudes of the residues rj for each of the spectral weight functions F(!k), the only quantities that must be calculated numerically in Equation (129) are the positions, yj, of the poles. Moreover, finding the positions of the poles is a simple task since it involves only a numerical solution of Equation (39). Thus, in effect, we have derived an analytic expression for the spectrum. We use Equation (129) to evaluate the spectrum S(!k) for the spectral weight functions F3(!k)F6(!k). The spectrum for the spectral weight functions F1(!k) and F2(!k) can be calculated directly using Equation (122) and the analytic expressions we found previously for b2(t); Equation (122) can be used since b2(1)  0. For reference purposes, the probabilities Sð!k ; tÞ ¼ jb1k ðtÞj2 are given for the spectral weight functions F3(!k)F6(!k); the expressions in these equations must be calculated numerically. A word of caution is in order here. Although we calculate and plot the spectrum given by Equation (129), it is not obvious that this spectrum corresponds to an experimentally measurable quantity when 3 The interpretation of the interaction term (third term in Equation 132) remains somewhat of a mystery to us, although an analogous term appears in the off-resonant Jaynes-Cummings model. The interaction term can be substantial. For example, for the spectral weight function F5(!k) with !0/=10 and !c/=400 (yp=2.35, rp=0.883), the first term in Equation (132) equals 7.808, the second term 21.467, and the interaction term–19.269, which sum to 10 as required.

Spontaneous Decay, Unitarity, and the Weisskopf-Wigner Approximation

209

b2(1) 6¼ 0. It does appear that the second term in Equation (129) gives rise to a far field that would contribute to an experimentally measured spec­ trum, but this feature needs to be analyzed in more detail. Since the expectation value of atom-field interaction energy is nonvanishing for b2(1) 6¼ 0, the field continues to interact with the atom for all times in this limit. 3.1 Spectral Weight Functions F1(w k) and F2(w k) For the spectral weight function F1(!k) given by Equation (14a), it follows from Equations (122) and (46) that Sð!k Þ ¼

1  :  ð!k  !0 Þ 2 þ  2

ð133Þ

The spectrum integrated over !k is equal to unity, S=1. The spectral weight function F1(!k) corresponds to the Weisskopf-Wigner approxima­ tion. We see that the Weisskopf-Wigner approximation leads to a Lor­ entzian spectrum that is consistent with unitarity. Of course, this spectrum is not physical since it extends to negative frequencies. For the spectral weight function F2(!k) given by Equation (14b), the decay is bi-exponential, except in the case of critical damping. It follows from Equations (122), (50), and (121) that Sð!k Þ ¼

 

1  2 ð!k  !0 Þ 2 ð!k  !0 Þ 2 þ   !w

ð134Þ

and S=1. If !w >> 4 the spectrum is approximately the single Lorentzian given by Equation (133). In the opposite limit when !w << 4 the spectrum consists approximately of a Lorentzian doublet with maxima located at pffiffiffiffiffiffiffiffiffi !k ¼ !0 – !w . The width (FWHM) of each component is equal to !w in this limit. These results follow directly from Equation (134). The spectrum is shown in Figure 9 for !0/=10 and !w/=0.1. Since it is overdamped oscillations produced by the vacuum field that leads to the splitting, the separation of the peaks can be interpreted as a vacuum Rabi splitting. As was the case for F1(!k), this spectrum is not physical since it extends to negative frequencies. 3.2 Spectral Weight Functions F3(w k) and F4(w k) For the spectral weight function F3(!k) given by Equation (14c), it follows from Equations (6), (1b), and (60) that

210

Paul R. Berman and George W. Ford

S(ωk ) 3

2

1

0

9

10

ωk /γ

11

Figure 9 Spectrum for the spectral weight function F2(w k) and w 0/g = 10; w w/g = 0.1

   Z0   ð1  eið!k þ yÞ t Þ=ð!k þ yÞ  dy  Sð!k ; tÞ ¼ ½Yð!k ÞYð!c  !k Þ      !c 2   !c þ 2 y þ !0  ln 1    y

þ

2 X j¼1

j

2

ð1  eið!k þyj Þ t Þ=ð!k þ yj Þ !c 1þ : yj ðyj þ !c Þ

ð135Þ

The spectrum, obtained from Equations (129), (14c), (27), (55), and (59) is given by

" Sð!k Þ ¼

 ½Yð!k ÞYð!c  !k Þ 

þ

X2 j¼1



1    2  !c !k  !0 þ ln 1 þ 2  !k

ð!k þ yj Þ  2 1 þ

!c yj ðyj þ !c Þ

2

#

;

ð136Þ

where the pole locations are determined by Equation (58). The spectrum given by Equation (136) is shown in Figure 10a for !0/ = 10 and !c/ = 20. In this case the poles contribute negligibly and the spectrum is approximately Lorentzian, although it cuts off abruptly for !k/ < 0 and !k/ > 20. Numerically, one can integrate Equation (136) to show that S=1.

211

Spontaneous Decay, Unitarity, and the Weisskopf-Wigner Approximation

(a)

(b)

S(ωk )

S(ωk )

3

0.3

2

0.2

1

0.1 0

0

5

10

15

20

ωk /γ

(c)

0

0

4

8

12

ωk /γ

(d)

S(ωk )

S(ωk )

0.3

0.008

0.2 0.004

0.1 0

0

2.5

5

7.5

10

ωk /γ

0 0

0.5

1

1.5

2

ωk /γ

Figure 10 Spectra for the spectral weight function F3(w k) and (a) w 0/ = 10; w c/ = 20; (b) w 0/=10; w c/=12, (c) w 0/=10; w c/=10, and (d) w 0/=10; w c/ = 2

If the logarithm term in the integrand is neglected, the spectrum changes only slightly. In other words in the limits given by Equations (61) and (62a), the neglect of the logarithm term introduces small h i2 !c corrections, of order !0 ð!c !0 Þ , in the spectrum. However, in the integrated spectrum, these small errors add; for example, with the neglect of the logarithm term for the parameters of Figure 10a, S=0.924. Had we taken !0/=100 and !c/=200 and neglected the logarithm term, S=0.994. Thus, unitarity is maintained strictly only if the logarithm term is included. With the neglect of the logarithm term, Equation (135) can be evaluated analytically in terms of exponential, sine and cosine integrals. There is some additional structure hidden in Figure 10a. In the limit !0 = ¼ 0 and !c = > 13, the spectrum is given approximately by the first term in Equation (136), since the poles contribute negligibly. In this limit (and assuming that !c =!0 < 100), the ln term shifts the resonance posi­ tion from !k=!0 to    !c !k & !0  ln 1 þ :  !0

ð137Þ

212

Paul R. Berman and George W. Ford

The shift, which can be positive, negative, or zero (it is zero if !c/!0 = 2, as for the parameters of Figure 10a), can be interpreted as arising from the light shifts produced by the off-resonant vacuum field modes. The light shifts of off-resonant modes give rise to additional sharp resonances centered at !k & !c e  !0 =  ;

ð138aÞ

!k & !c ð1  e   ð!c  !0 Þ = Þ;

ð138bÞ

h corresponding to values for which the

!k  !0 þ

 



i2 ln 1 !!lc term in

Equation (136) vanishes. The amplitudes of theses additional resonances, which are not resolved in Figure 10a, are identical to that of the central resonance. To see these resonant structures, it is necessary to go to times given by !c t > e!0 = 2 ; e ð!c  !0 Þ = 2 , for which the resonance widths, of 

 !0 = 

order !c e , !c e   ð!c  !0 Þ = , would be impossible to resolve. At finite times, the resonances are broadened. The existence of the extra resonances (Equation 138) is linked to the discontinuity in the spectral weight functions at !k = 0 and !k= !c. Equa­ tions (138) are identical to Equations (63). More generally, the resonances near !k = 0 and !k = !c occur whenever there are poles that contribute to nonvanishing values of b2(1). Thus the existence of this structure is a signature of the fact that b2(1) 6¼ 0, even if the value of b2(1) is exponen­ tially small in the parameters p!0/ and p(!c!0)/. An equivalent condition for b2(1) 6¼ 0 was given by Miyamoto (2005). As !c/!0 is reduced to values of order unity, the contribution from the residue associated with the pole y2 begins to play a role. This can be seen in Figure 10b, drawn for !0/=10 and !c/=12. Although the residue r2=0.062 associated with the pole at y2=12.021 is small, the fact that the amplitude of the sharp peak in S(!k) near !k=12 is larger that the amplitude of the peak near !k=10 is a signature of the fact that the residue term is contributing. If the residue term had been neglected, both peaks would have had the same amplitude. The spectrum given by Equation (136) is shown in Figure 10c for !0/=10 and !c = ¼ 10. For these parameters, the spectral weight function cuts off exactly at the transition frequency. In this case the pole at y2 = ¼ 10:821, obtained as a solution of Equation (58) makes a significant contribution; the magnitude of the residue equal to 0.736. This term would lead to a delta function contribution to the spectrum at !k = ¼ 10:821 but, since the spectrum equals zero for !k = > !c = ¼ 10 the only the “tail” of the residue term, along with the “half-Lorentzian” of the first term in Equation (136), contributes to the spectrum. One finds numerically that jb2 ð1Þj¼0:736 and

213

Spontaneous Decay, Unitarity, and the Weisskopf-Wigner Approximation

S ¼ 0:458 ¼ 1  jb2 ð1Þj2 :

ð139Þ

Note that the sharp resonance remains near !k = ¼ 0, owing to the discontinuity in the spectral weight function at !k = 0. The spectrum given by Equation (136) is shown in Figure 10d for !0/=10 and !c = ¼ 2. In this case the pole at y2 = ¼ 10:0705, obtained as a solution of Equation (58) makes an important contribution to the spectrum; the magnitude of the residue equal to 0.9922. This term would lead to a delta function contribution to the spectrum at !k = ¼ 10:0705 but, since the spectrum equals zero for !k = > !c = ¼ 2 it is the “tail” of the residue term that contributes. One finds numerically that jb2 ð1Þj ¼ 0:9922 and S ¼ 0:0155 ¼ 1  jb2 ð1Þj2 :

ð140Þ

The contribution from the two terms in Equation (136) are approximately equal. More generally, if !c =!0 << 1, the pole position is given by y2 &!0 and the residue from the pole is approximately unity, such that each of the terms in Equation (136) vary as ð!k  !0 Þ  2 and contribute an equal amount  

Z!c 0

d!k ð!k  !0 Þ  2 &

 !c << 1 !0 !0  !c

ð141Þ

to the integrated spectrum S. There is still a sharp resonance near !k = 0. For the spectral weight function F4 ð!k Þ given by Equation (14d) in the limit that !0 =  10, the pole near y1=0 makes a negligible contribution and the upper state amplitude can be approximated by the integral term in Equation (96). It then follows from Equations (6), (1b), and (96) that  Sð!k ; tÞ ¼ Yð!k Þe  ð!k  !0 Þ = !c    Z 0 e ðy þ !0 Þ = !c ð1  ei ð!k þ y Þ t Þ=ð!k þ yÞ 

 dy h i2 h i2    1  y þ !0  e ðy þ !0 Þ = !c ½Gð0; y=!c Þ þ i þ e ðy þ !0 Þ = !c 

 2    :  

ð142Þ

The spectrum, obtained from Equations (129), (14d), (27), and (93) is given by

Sð!k Þ ¼

 ð!k Þe  ð!k !0 Þ = !c i2 : ð143Þ h i2 h   !k  !0 þ e  ð!k  !0 Þ =!c ½Gð0; !k =!c Þ þ i þ e  ð!k !0 Þ = !c 

214

Paul R. Berman and George W. Ford

For !c >> !0, the spectrum is essentially identical to that for F3(!k), except that there is now a smooth cutoff as !k  1 instead of the sharp cutoff at !k ¼ !c . Numerically, one can show that S=1. 3.3 Spectral Weight Functions F5(w k) and F6(w k) For the spectral weight function F5(!k) given by Equation (14e), it follows from Equations (6), (1b), and (101) that Sð!k ; tÞ ¼

 !k ½Yð!k ÞYð!c  !k Þ  !0    Z  0 ðy=!0 Þð1  ei ð!k þ y Þ t Þ=ð!k þ yÞ 

 dy   2     !c !c y !c 2 y  þ ln 1  þ y þ !0   !0 y !0 !0 2    2 i ð! þ y Þ t X ð1  e k j Þ=ð!k þ yj Þ  !  : þ  !2c j ¼ 1 !0  1  2 yj !0 ð!c þ yj Þ

ð144Þ

The spectrum, obtained from Equations (129), (14e), (27), (100), and (101) is given by

"

 !k Sð!k Þ ¼ ½Yð!k ÞYð!c  !k Þ  !0

þ

X2 j¼1

1     2  !c !k  !c !k 2 !k  !0 þ þ ln 1 þ !0 !0 !k !0

ð!k þ yj Þ 2



yj !0

2

!2c 1 þ yj Þ

!20 ð!c

!2

# :

ð145Þ

If !00 = ¼

  !0 !c 1  2 >> 1;  !0

ð146Þ

the spectrum is essentially the same as for the spectral weight function F3(!k), except that there is a linear dependence on !k near !k=0 rather than a sharp resonance near !k=0. The sharp resonance near !k ¼ !c is broadened by a factor of !c =!0 . For !00 = < 0, there is a qualitative change in the spectrum, since the transition frequency is shifted to negative values and does not overlap

Spontaneous Decay, Unitarity, and the Weisskopf-Wigner Approximation

215

S(ωk)

0.002

0.001

0

25

50

75

100

ωk /γ

Figure 11 Spectrum for the spectral weight function F5(w k) and w 0 / g = 10; w c / g = 400, when w 00 /g < 0

the spectral weight function. The spectrum given by Equation (145) is shown in Figure 11 for !0/=10 and !c/=400. In this case !00 = < 0 and the pole at y1 = ¼ 2:348, obtained as a solution of Equation (102), makes an important contribution to the spectrum,the magnitude of the residue equal to 0.8833. This term would lead to a delta function contribution to the spectrum at !k = ¼ 2:348, but, since the spectrum equals zero for !k = < 0, only the “tail” of the residue term contributes to the spectrum. One finds numerically that jb2 ð1Þj¼0:780 and S ¼ 0:220 ¼ 1  jb2 ð1Þj2 :

ð147Þ

Thus, even though the upper state is shifted below the ground state in energy, unitarity is maintained, as required. For !0/=10 the value of S as a function of !c = is given by 1 the square of the curve in Figure 8. For large !c =; S  1  0:9232 ¼ 0:148. The contributions from the two terms in Equation (145) are approxi­ mately equal. More generally, for !c << !p ¼ !20 = and ~ ¼ =!0 << 1, the residue from the pole is approximately unity and the pole position is given by y1& ~ !c ; as a consequence each of the terms ~ c Þ 2 and contribute an equal in Equation (145) is equal to ~ ð!k þ ! amount Z!c ~

~ c Þ  2& ½ln ~ ð1 þ ~ d!k !k ð!k þ !

1

Þ  1 << 1

ð148Þ

0

to the integrated spectrum S. For the spectral weight function F6(!k) given by Equation (14f), there is no pole near y1=0 if !000 > 0. It follows from Equations (6), (1b), and (113) that

216

Paul R. Berman and George W. Ford

 !k Sð!k ;tÞ ¼ Yð!k Þ e  ð!k !0 Þ = ð!c Þ  !0   y ðyþ!0 Þ=!c Z e ð1  eið!k þyÞt Þ=ð!k þ yÞ  0  !0

 dy 2   1  y ðyþ!0 Þ = ð!c Þ y ðyþ!0 Þ = ð!c Þ 2 00  e ½Gð0; y=!c Þ þ i þ e y þ !0 þ  !0 !0

þ

j

2

ð1  eið!k þyj Þt Þ=ð!k þ y1 Þ Yð!000 Þ !c e!0 =!c y21 þ !0 ð!c þ y1 Þ ;  !c y1 !0 y1

ð149Þ

where !000 is defined by Equation (114) and the pole y1 is obtained as a solution of Equation (115). The spectrum, obtained from Equations (129), (14e), (27), (100), and (101) is given by  !k Sð!k Þ¼ Yð!k Þ eð!k !0 Þ=ð!c Þ  !0

"



1  2  !c !0 =!c !k ð!k !0 Þ=!c !k ð!k !0 Þ=!c 2 e þ e ½Gð0;!k =!c Þþi þ e !k !0 þ !0 !0 !0

þð!k þyj Þ 2



!c e!0 =!c y21 þ!0 ð!c þy1 Þ  !c y1 !0 y1

2

#

Yð!000 Þ

ð150Þ

For !c >> !0, the spectrum is essentially identical to that for F5 ð!k Þ, except that there is now a smooth cutoff as !k  1 instead of the sharp cutoff at !k ¼ !c . The spectrum is shown in Figure 12 for !0 = ¼ 1000 and !k = ¼ 50; 000. Although only a part of the spectrum is shown, there are no longer any sharp cutoffs in the spectrum as a function of !k ; that is, Sð!k Þ is zero for !k<0, depends linearly on !k for 0  !k << !0 and varies as ð!k =!0 Þe  ð!k  !0 Þ = !c for ð!k  !0 Þ=!c >> 1. The shift of the maximum of the spectrum from !k=!0 can be calculated from the 

!c !0 = !c !k  ð!k !0 Þ = !c e  e ½Gð0; !k =!c Þ þ i !0 !0

terms in the denominator of the first term in Equation (150), evaluated at !k=!0; for the parameters of Figure 12, this leads to a shift of !k =&17:3. Numerically, one can show that S=1.

Spontaneous Decay, Unitarity, and the Weisskopf-Wigner Approximation

217

S(ωk)

0.3

0.2

0.1

0 960

980

1000

ωk /γ

Figure 12 Spectrum for the spectral weight function F6(w k) and w 0 / g = 1000;

w c / g = 50,000. Although the entire spectrum is not shown, there are no discontinuities

in S(w k)

4. DISCUSSION We have considered a model Hamiltonian to explore features of the decay of a discrete state into a continuum, specifically, spontaneous emission. The Hamiltonian represents the interaction of an atom with the vacuum field in the RWA, allowing for an arbitrary spectral density of the vacuum field. As was stressed in Section 1, both the Hamiltonian and the initial condition ½b2 ð0Þ ¼ 1 can, at best, represent an approx­ imate description of the atom-field interaction. However, the validity of the Hamiltonian and the initial condition was not of primary concern. Instead, the major objective of this paper was to develop a relatively simple formula for the time evolution of the excited state amplitude and to show that unitarity is maintained, provided one uses the correct excited state amplitude in the equation that determines the spectral weight function distribution of the spontaneously emitted radiation. With this goal in mind, we were able to show why the Weiss­ kopf-Wigner approximation leads to pure exponential decay and a Lorentzian spectrum, albeit for an unphysical spectral weight function that admits negative frequencies. For spectral weight functions that are identically zero for !k < 0, many interesting features were found. If the spectral weight functions have discontinuities, there is always some excited state probability that remains as t  1. This can be traced to the existence of eigenstates of the atom-field system that contain an admixture of the excited state. More formal analyses of the conditions for the existence of such eigenstates is given in Miyamoto (2005) and Longhi (2007). Once such discontinuities are removed, the final state

218

Paul R. Berman and George W. Ford

amplitude goes to zero as t  1, provided the spectral weight function overlaps the transition frequency. On the other hand, if the spectral weight function does not overlap the transition frequency (including any level shifts), there is bound to be a considerable excited state amplitude as t  1. The models considered in this work let one pass continuously from a Markovian limit in which there is (approximately) exponential decay to a limit in which considerable population remain trapped in the excited state. Since any Hermitian Hamiltonian leads to unitarity, the results pre­ sented in Section 3 allow us to generate new classes of integral iden­ tities. We want a situation in which no poles contribute to the integral. To assure that this is the case, we assume that the spectral weight function F ð!k Þ vanishes for !k  0 and as !k  1, and that Fð!k Þ is analytic for 0  !k < 1. This alone does not guarantee that no poles exist, since the resonance frequency can be shifted below zero, as we found for F5 ð!k Þ and F6(!k). One way to guarantee that that no poles exist is to add a term  h!r j2ih2j to the Hamiltonian given in Equation (8), with  !r ¼ 

Z1 d!k ½Fð!k Þ=!k ;

ð151Þ

1

since such a “renormalization” procedure leads to an average value for the effective transition frequency that is always positive. Let us assume, however, that such a renormalization is not necessary; that is, we assume that !0>!r. For spectral weight functions satisfying these criteria, the inverse Laplace transform can be carried out using the contour shown in Figure 6 with no poles in the contour. As a result the spectrum is given by Equation (122) and, since b2 ð1Þ ¼ 0 owing to the fact that there are no poles in the contour, it follows that  

Z1 0

  d!k Fð!k Þ

2  1  ¼ 1; !k  !0 þ iGði!k Þ 

ð152Þ

where G(s) is given by Equation (32). Since GðsÞ ¼ Gðs þ i!0 Þ ¼

 

Z!b d! !a

Fð!Þ s þ i!

ð153Þ

can be evaluated analytically for an infinity of choices for F(!), entirely new classes of integral identities can be generated using this procedure. For example, for the spectral weight function F6(!k), we find

Spontaneous Decay, Unitarity, and the Weisskopf-Wigner Approximation

 

Z1 d!k  0

219

ð!k =!0 Þeð!k !0 Þ=!c

!k !0 þ

2  ¼1 !c !k ð!k !0 Þ=!c y ð!k !0 Þ=!c 2 ! =! 0 c þ e ½Gð0;!k =!c Þþi þ e e !0 !0 !0 ð154Þ

provided ð!c =!20 Þe!0 = !c < 1. If ð!c =!20 Þe!o = !c > 1, one can add a term h!r j2ih2j, with !r ¼ ð!c =!0 Þe!o = !c > 1 given by Equation (151), to the Hamiltonian and replace !0 by !0þ!r in Equation (154). The net effect would be to eliminate the shift term ð!c =!20 Þe!0 = !c in the denominator of Equation (154). A value of !0/=10 was chosen for many of the examples to illustrate deviations from exponential decay. For !0  ¼ 106 ; !c =!0 ¼ 102  105, typical of optical transitions, the corrections to exponential decay and a Lorentzian spectrum are prohibitively small. As far as we know, there has never been any experimental evidence for other than pure exponential decay on isolated atomic transitions. Deviations from exponential decay have been observed in systems in which the vacuum field that interacts with the atoms of interest has been modified in some manner. For exam­ ple, the atoms undergoing decay can be placed in a cavity or could be located in a photonic crystal. For atoms in “free space,” it is unlikely that such deviations could be seen. The short time deviations occur only for times satisfying !0t<1; as a consequence, the excitation of the atoms must occur on a time scale faster than !1 0 to observe such effects. Aside from the practical problems associated with such a fast excitation time, there is the plethora of complications that arises from additional atomic excita­ tions that will undoubtedly be produced by ultra-short excitation pulses. It may also be all but impossible to observe long-time deviations from exponential decay for atoms undergoing spontaneous emission in free space. It the theory presented in this paper were correct, deviations would occur for times such that e  t &

 ; 2 !0 !00 t2

ð155Þ

a result that follows from Equation (116). For optical transitions, this condition implies that t is of order 64, requiring an extremely large number of atoms to have any observable signal (moreover, a transition must be used in which the lower level is not the ground state to avoid reabsorption of the emitted radiation). The applicability of the theory presented in this paper to the long-time behavior of real atom-vacuum field systems is questionable. From the asymptotic form of the excited state amplitude given by Equation (116), it follows that b2(t) varies as exp ði!0 tÞ in the interaction representation. As a consequence, b2(t) does not

220

Paul R. Berman and George W. Ford

oscillate at all in the Schro¨dinger representation–the long-time emission is essentially that of a zero-frequency field. As a consequence, the use of the RWA is certainly suspect in modeling the true atom-field Hamilto­ nian. A complete and correct solution to the spontaneous emission pro­ blem would require a calculation of the modification of the signal measured at a detector following the excitation of atoms from their ground states (dressed by the vacuum field). Moreover, it may be nontrivial to specify the initial state of the atom-field system if the rotating were approximation is not valid, since the ground state of the atom-field system contains an admixture of the excited state of the atom; there is no guarantee that the atom-field system is in this dressed ground state. The actual excitation process must be included in the calculation as well, but the long-time behavior should not depend critically on the excitation pulse provided it has a duration that is much larger than !1 0 , but much shorter than  1. In other words, for a given Hermitian Hamiltonian whose energy levels are bounded from below, the methods outlined in this paper can be used to examine the decay dynamics. However, the Hamiltonian considered in this paper cannot correspond exactly to the true atomfield Hamiltonian. It does provides a good approximate solution for exponential decay, but the very small corrections to exponential decay that were derived, while correct for the model Hamiltonian, must be re­ evaluated for the correct atom-field Hamiltonian.

ACKNOWLEDGMENTS P.R.B. is pleased to acknowledge very helpful discussions with P. Miller of the University of Michigan and with P. Milonni. In particular, P. Miller steered him to Wong (1989) for asymptotic solutions involving logarith­ mic singularities, and P. Milonni provided him with insightful comments after reading a draft of this manuscript.

REFERENCES Berman, P. R. (2004). Causality in field emission from an atomic dipole: Schro¨dinger picture approach. Physics Review A, 69, 022101. Berman, P. R. (2005). Wigner-Weisskopf approximation under typical experimental condi­ tions. Physical Review A, 69, 025804. Cohen-Tannoudji, C., & Avan, P. (1977). Discrete state coupled to a continuum: Continuous transition between the Weisskopf-Wigner exponential decay and the Rabi oscillation. In Feneuille, S. & Lehmann, J. C. (Eds.), Etats Atomiques et Moleculaires Couples a un Continuum. Atomes et Molecules Hautement Excites (Atomic and molecules states coupled to a continuum. Highly excited atomic and molecules). No. 273 in Colloques Internationaux du C.N.R.S. Editions du CNRS, Paris, pp. 93-106. Cohen-Tannoudji, C., Dupont-Roc, J., & Grynberg, G. (1994). Atom-photon interactions (pp. 239-255). New York: Wiley Interscience.

Spontaneous Decay, Unitarity, and the Weisskopf-Wigner Approximation

221

Knight, P.L., & Milonni, P.W. (1976). Long-time deviations from exponential decay in atomic spontaneous emission theory. Physics Letters A, 56, 275-278, and references therein. Longhi, S. (2007). Bound states in the continuum in a single-level Fano-Anderson model. The European Physics Journal B, 57, 45-71. Milonni, P., James, D.V.P., & Fearn, H. (2004). Photodetection and causality in quantum optics. Physical Review A, 52, 1525-1537, and references therein. Miyamoto, M. (2005). Bound state eigenenergy outside and inside the continuum for unstable multilevel systems. Physical Review A, 72, 063405. Seke, J., & Herfort, W. (1989). Finite-time deviations from exponential decay in the Weiss­ kopf-Wigner model of spontaneous emission. Letters in Mathematical Physics, 18, 185-191, Weisskopf, V., & Wigner, E. (1930). Berechnung der natu¨auf grund der diracschen lichtthe­ orie (Calculation of the natural line width on the basis of Dirac’s theory of light). Zeitschrift fu¨ r Physik, 92, 54-73. This article is translated by J. B. Sykes and reprinted in Hindmarsh, W. (1967). Atomic spectra (pp. 304-327). Oxford: Pergamon Press. Whittaker, E. T., & Watson, G. N. (1927). A course of modern analysis (4th ed., pp. 172-176). New York: Cambridge University Press. Wong, R. (1989). Asymptotic approximations of integrals (pp. 66-76). Boston: Academic Press

CHAPTER

6

Ultrafast Nonlinear Optical Signals Viewed from the Molecule’s Perspective: Kramers-Heisenberg Transition-Amplitudes versus Susceptibilities Shaul Mukamel and Saar Rahav Department of Chemistry, University of California, Irvine, CA 92697, USA

Contents

1. 2. 3. 4.

5. 6.

7. 8.

Introduction Quantum-Field Description of Heterodyne

Signals Transition-Amplitudes and the Optical

Theorem for Time-Domain Measurements 3.1 Purely Dissipative Signals CTPL Representation of Optical Signals 4.1 Rules for the CTPL Diagrams in the Time

Domain 4.2 Rules for the CTPL Diagrams in the

Frequency Domain The Pump-Probe Signal The Pump–Probe signal Revisited: Transition

Amplitudes 6.1 Unrestricted Loop Diagrams 6.2 The Two-Photon-Absorption and

Stimulated-Raman Components of the

Pump-Probe Signal Coherent Anti-Stokes Raman Spectroscopy Cars Signals Recast in Terms of Transition

Amplitudes

224

228

231

236

236

237

238

239

243

243

244

249

252

Advances in Atomic, Molecular, and Optical Physics, Volume 59  2010 Elsevier Inc. ISSN 1049-250X, DOI: 10.1016/S1049-250X(10)59006-2 All rights reserved.

223

224

Shaul Mukamel and Saar Rahav

9.

CARS Resonances Can be Viewed as a Double-Slit Interference of Two Two-Photon Pathways 10. Purely-Dissipative Spectroscopic Signals 11. Summary Acknowledgments References

Abstract

256 258 260 261 261

Coherent nonlinear optical signals are commonly calculated using a semiclassical approach that assumes a quantum system interacting with classical fields. Compact expressions for the signals are then derived in terms of nonlinear susceptibilities. We present an alternative approach based on a quantum description of both matter and field. The signals are further recast in terms of transition amplitudes, which provide a clearer picture for the underlying molecular processes and may be intuitively represented by closed-time-path-loop diagrams. Unlike the semiclassical approach that treats the signal mode macroscopically using Maxwell’s equations, the present formalism allows for a fully microscopic calculation of the entire process. For example, n þ 1 wave mixing appears as a concerted n þ 1 photon event and all n þ 1 field modes (including the signal) are treated on the same footing. Resonant contributions to nonlinear optical signals that carry useful molecule-specific information are recast as the modulus square of transition amplitudes and are clearly separated from the parametric background. Purely dissipative signals that can be measured using a collinear beam geometry and manipulated by pulse shaping techniques are proposed. The approach is demonstrated by applications to the stimulated Raman and the two-photon absorption components of pump-probe, and to coherent anti-Stokes Raman spectroscopy.

1. INTRODUCTION Nonlinear spectroscopy provides detailed information on molecular struc­ ture and dynamical processes through specific electronic or vibrational resonances. Spectroscopic techniques may be broadly classified as fre­ quency- or time-domain type and are conveniently and systematically analyzed order by order in the incoming fields. In the semiclassical (quan­ tum matter coupled to classical fields) description of an n þ 1 wave mixing process, the system is subjected to n incoming pulses that generate an n’th order polarization P(n) (Mukamel, 1995; Scully & Zubairy, 1997; Shen, 2002)

Ultrafast Nonlinear Optical Signals

P

ðn Þ

Z1 ðtÞ ¼

Z1 dt1   

0

225

dtn S ðn Þ ðtn ; . . . ; t1 ÞEðt  tn Þ    Eðt  tn     t1 Þ: ð1Þ

0

This can be alternatively recast in the frequency domain Z Z ðn Þ P ð!Þ ¼ d!1    d!n � ðn Þ ð!; !1 ; . . . ; !n ÞEð!1 Þ    Eð!n Þ;

ð2Þ

R where Eð!Þ ¼ dtEðtÞei!t . The response functions S(n) or their frequency domain counterparts, the susceptibilities �(n), contain all the material informa­ tion necessary for calculating and analyzing n’th order processes. The signal field is calculated by substituting P(n) as a source in Maxwell’s equations. The homodyne detected signal is quadratic in the polarization Z  2 ðnÞ SHOM  P ðn Þ ðtÞ dt; ð3Þ while heterodyne signals (Figure 1) depend linearly on the polarization and provide both its amplitude and its phase (Mukamel, 1995) Z ðnÞ SHET  JEðtÞP ðn Þ ðtÞdt: ð4Þ [For the precise expression see Equation (21).] One problem with the molecular-level interpretation of optical signals is that the polarization (like any other quantum observable) is determined by interactions occurring on both the bra and the ket of the matrix elements of the dipole operator � P ðn ÞðtÞ ¼

n X

h

ðm Þ

^j ðtÞj�

ðn  m Þ

ðtÞi:

ð5Þ

m¼0

 Here  ðn Þ ðtÞi is the wave function calculated to n’th order in the field-matter interaction Hint [Equation (13)]. The susceptibilities depend on various Liouville space pathways which count the various orders in Equation (5) as well as the relative time ordering of the interactions. Different pathways interfere, and this interference complicates the simple intuitive interpretation of signals. In an alternative approach, some optical signals are traditionally inter­ preted in terms of transition amplitudes rather than susceptibilities. The description of four-wave mixing signal from an ensemble of two-level atoms in terms of transition amplitudes was discussed by Dubetsky and Berman (1993). A notable example is the Kramers- Heisenberg formula for spontaneous light emission (Cohen-Tannoudji et al., 1997) where a photon !1 is absorbed and !2 is emitted

226

Shaul Mukamel and Saar Rahav

k4 Sample

k3 k2 k1

Detector t1

t3

t2

Time k2

k1

k3

k4

Figure 1 Schematic depiction of a heterodyne detected four-wave mixing process. The signal is generated in the direction k4 = –k1 – k2 – k3. Here the incoming k4 beam passes through the sample and stimulates the signal. In ordinary heterodyne detection the beam mixes only with the signal and does not pass through the sample. The two configurations yield identical signals to the first order in the k4 beam amplitude

Sð!1 ; !2 Þ ¼

X

 2 PðaÞjE 1 j2 ~ Tca  �ð!1  !2  !ca Þ:

ð6Þ

a;c

Here ~ca ¼ T

X b

�cb �ba ; !1  !ba þ i�

ð7Þ

is the transition amplitude and P(a) the equilibrium population of state a. [Two-photon absorption is also given by Equation (6), by replacing !1  !2 with !1 þ !2.] There is no ambiguity as to what is going on in the molecule in this formulation. The signal is given by the modulus square of a transition ~ca from the initial ðjaiÞ to the final ðjciÞ state. That transition amplitude T amplitude is in turn given by the sum over all possible quantum pathways which may contain interferences. The interference of single- and two-photon pathways in photoelectron detection (Figure 2) was pointed out by Glauber (2007). In this case the transition amplitude has the form ~ca ¼ �ca Eð2!Þ þ T

�cb �ba E2 ð!Þ: !  !ba þ i�

ð8Þ

This interference may be controlled by varying the relative phase of the two fields E(!) and E(2!). A simpler interference between one and two photon processes was used to control the photocurrents in semiconductors by varying the relative phase of the two beams (Hache et al., 1997). In this review we address the following question: under what condi­ tions is it possible to represent heterodyne detected nonlinear optical

Ultrafast Nonlinear Optical Signals

227

c

ω

2ω

b

ω a Figure 2 The transition pathways of Equation (8). A direct transition where a single 2w photon is absorbed interferes with the absorption of two w photon

signals in terms of transition amplitudes rather than susceptibilities? Apart from the obvious advantage for the interpretation, amplitudes are simpler to calculate, since they are lower order and contain fewer terms than susceptibilities. The results presented here have been devel­ oped in a series of articles (Marx et al., 2008; Rahav & Mukamel, 2010; Rahav et al., 2009; Roslyak et al., 2009). Here we provide an overview and discuss possible generalizations. By using a quantum description of the field we show that examina­ tion of the relevant processes from the viewpoint of the material natu­ rally leads to a description in terms of transition amplitudes rather than susceptibilities. Once the optical signals have been recast in terms of these transition amplitudes, the material processes involved become evident. We further show that nonlinear signals generally contain two types of contributions: resonant dissipative processes where the matter participates actively and changes its state at the end, and parametric processes where the matter only serves as a passive “catalyst” for exchange of energy among field modes and it returns to its initial state at the end of the process. Only the former, which are most interesting for spectroscopic applications, can be generally recast in a generalized Kramers-Heisenberg form, whereas the latter merely provide an offresonant background. Intuitive closed-time-path-loop (CTPL) diagrams will be introduced and used to dissect the signal into the two compo­ nents. We further discuss signals that eliminate the parametric process and solely provide the desired resonant contributions. These ideas will be illustrated by applications to pump-probe spectroscopy and to coherent anti-Stokes Raman spectroscopy (CARS). The structure of this review is as follows. Sections 2-4 contain the necessary background material for the fully quantum calculation and

228

Shaul Mukamel and Saar Rahav

analysis of optical signals. In Section 2 we present a quantum field approach for the calculation of heterodyne detected signals. Here all n þ 1 active field modes are considered on the same footing. In Section 3 we examine optical processes from the viewpoint of the material degrees of freedom and introduce the transition amplitudes which represent the material processes. CTPL diagrams provide a convenient bookkeeping tool for nonlinear optical signals. These are introduced in Section 4. In Sections 5-8 we calculate the optical signals and dissect them into various contributions from material processes. Pump-probe (two-photon absorption and stimulated Raman) signals are presented in Sections 5 and 6, whereas CARS signals are described in Sections 7 and 8. Using the results of Section 8 we show in Section 9 that the resonant part of the CARS signal can be interpreted as originating from double-slit interfer­ ence. In Section 10 we show that the purely dissipative signals, defined in Section 3.1, can be used to distinguish between different resonant transi­ tions in matter. We conclude in Section 11 with some remarks on the generality of the present approach.

2. QUANTUM-FIELD DESCRIPTION OF HETERODYNE SIGNALS Traditionally, nonlinear optical signals are calculated in a semiclassical framework whereby a classical field interacts with quantum matter (Mukamel, 1995; Scully & Zubairy, 1997; Shen, 2002). This assigns differ­ ent roles to the n fields interacting with the system and to the (n þ 1)’th “local oscillator” field used for heterodyne detection. In the following we present a fully quantum description of both matter and field. In this approach, which can describe both spontaneous and stimulated pro­ cesses, the system is allowed to interact with the “local oscillator,” and the signal measures the change of the number of photons in the detected modes. n þ 1 wave mixing naturally appears as a single event involving all n þ 1 field modes which are treated on an equal footing. A molecule interacting with an optical field is described by the Hamiltonian ^ int ; ^0 þ H ^F þ H ^ ¼H H ^ 0 represents the free molecule, and where H X ^F ¼ h!s^a †s ^a s ; H

ð9Þ

ð10Þ

s

is the Hamiltonian of the field degrees of freedom. The optical electric field operator is ^ r; tÞ ¼ E^ ðr; tÞ þ E^† ðr; tÞ; Eð

ð11Þ

Ultrafast Nonlinear Optical Signals

with the positive-frequency component  X  2p h!s 1 = 2 ^ ^a s eiks  r  i!s t : E ðr; tÞ ¼ W s

229

ð12Þ

The quantities ^a †s ð^a s Þ are boson creation (annihilation) operators, W is the quantization volume, and cgs units are used. The molecule-field interaction in the rotating wave approximation (RWA), which neglects off-resonant terms, is given by ^ int ðtÞ ¼ E^ ðr; tÞV ^ † þ E^ † ðr; tÞV ^; H

ð13Þ

^ ¼ a b > a � jaihbj is the part of the dipole operator describing where V ab transitions down in energy. The entire moleculeþfield system is represented by the density matrix ^ ðtÞ. We denote expectation values with respect to this density matrix by � (  )�. Using perturbation theory these will be expanded in terms of averages h  i over the initial non-interacting density matrix at t!1. In a quantum description of time-domain optical signals where the system interacts with the field only during finite pulses, the signal S is defined as the net change of the photon number between the initial (i) and final (f) states, that is Z S

dt

d ^ ^ i  hN ^i; ðN Þ � ¼ hN f i dt

where ^  N

X

^a †s ^a s ;

ð14Þ

ð15Þ

s

and the sum runs over the detected modes. Taking the frequency domain (FD) limit should be done with care, since Equation (14) may turn out to be infinite. It is then natural to drop the t integration in the definition of the signal and redefine it as the rate of change in the number of photons, S

d ^ ðN Þ � dt

ð16Þ

We will use both definitions of the signal in the following. Equation (16) is adequate for the pump-probe application studied in Section 5. Equation (14) will be used in Section 7 for the CARS signal. The reasons behind this will be discussed in the relevant sections.

230

Shaul Mukamel and Saar Rahav

The time derivative in Equation (14) or (16) will be calculated using the Heisenberg equations of motion and the Hamiltonian Equation (9) * + * + i X ih d ^ d ^ † ^ NH ¼ ð17Þ H int ðtÞ; ^a s;H ^a s ; H ; ðN Þ �  dt dt h s The commutator is easily calculated, leading to    d ^ 2 † ^ ^ ðN Þ � ¼  Im E ðr; tÞV : � dt h

ð18Þ

The density operator at time t can be expressed starting with the initial (t!1) density operator whose matter and field degrees of freedom are uncoupled, which is then propagated by the Hamiltonian Equation (9). This propagation is most compactly described in terms of Liouville space “left” and “right” superoperators (Harbola & Mukamel, 2008; Mukamel, 2003; Cohen & Mukamel, 2003) which provide a clean bookkeeping device for all interactions. These are defined as ^ X; ^ A ^ ^L X A ^: ^ RX ^A ^ X A

ð19Þ

^ L ðA ^ R Þ corresponds to an A ^ appearing to the left (right) of X ^ in Hilbert A space. We further introduce linear combinations of L/R operations, which will be referred to as þ/ operations 1 ^ ^ –  pffiffi ^ R : ffi ½A L – A A 2

ð20Þ

^ L, A ^ R , or equivalently A ^ þ, A ^  , form complete sets of superoperators A which are connected by a unitary transformation. h i ^ d� ^ � ^ ^ ðtÞ by solving the Liouville equation dt Propagating � ¼ hi H; gives (Marx et al., 2008; Roslyak et al., 2009) 2* ( )+# Zt pffiffiffi d ^ 2 4 i † ^ ^ ðtÞexp  T E L ðr; tÞV ðN Þ � ¼  = d� 2Hint  ð�Þ : L dt h h

ð21Þ

1

This together with Equation (14) or (16) provides an exact compact formal expression for the signals. A key ingredient in Equation (21) is the time ordering operator in Liouville space T which reorders superoperators so that ones with earlier times appear to the right of those with later time variables. Thanks to this operator we can use an ordinary exponent in Equation (21) without worrying about time ordering.

Ultrafast Nonlinear Optical Signals

231

In the following we ignore spontaneous signals and focus solely on stimulated processes. We assume that the field is initially in a coherent state, ( ) X X 2 † ^a s �s j0i: jYF i ¼ exp ð j�s j Þexp ð22Þ s

s

In Equation (22) �s is the eigenvalue of the photon annihilation operator ^a s , ^a s jYF i¼�s jYF i, and j0i is the vacuum state of the field. The expecta­ tion value of the field is then ^ tÞjYF i¼Eðr; tÞ þ c:c:; hYF jEðr;

ð23Þ

where Eðr; tÞ ¼

X  2ph!s 1 = 2 s

W

eiks  r  i!s t �s ;

ð24Þ

is the field amplitude at space-point r. Using Equation (22) the field expectation values can be calculated by simply replacing E^ everywhere with its classical expectation value E(r,t). Equation (21) contains all orders in the fields and can serve as a starting point for a perturbative calculation of specific signals. These are generally given by products of correlation functions of field and matter degrees of freedom. The different terms in the perturbative expansion of Equation (21) represent the various possible optical signals. These are conveniently described in terms of the loop diagrams which will be presented in Section 4.

3. TRANSITION-AMPLITUDES AND THE OPTICAL THEOREM FOR TIME-DOMAIN MEASUREMENTS We wish to study optical signals from the viewpoint of the molecule, rather than the field, with the ultimate goal of relating the signals to material processes. This will be done in this section by recasting the signals in terms of transition amplitudes which originate from the per­ turbation theory of the molecular wave functions. We start by considering a time-domain setup where the molecule interacts with a finite optical pulse. Namely, E(�) 6¼ 0 only for t0<�
232

Shaul Mukamel and Saar Rahav

E(τ)

τ

t0

t

Figure 3 A time-domain experiment where a molecule interacts with a pulse (or a series of pulses) for a finite time, between an initial time, t0, and a final time t

  ^ ðt; t0 Þjaðt0 Þi2 ; Pa ! c ¼ hcðtÞjU ð25Þ h i R t where U^ ðt; t0 Þ ¼ exp þ  hi t d�H0I ð�Þ is the time evolution operator in 0

the interaction picture with respect to H0, H0I ð�Þ ¼ U0† ð�; t0 ÞH 0 ð�ÞU0 ð�; t0 Þ: U0 is the evolution operator of the non-interacting field and matter, while jaðtÞiU0† ðt; t0 Þjai. The time-ordered exponential is defined as

 Z i t d�H0I ð�Þ ¼ exp þ  h t0

  Z Z �n Z� 2

1 X i n t 1þ  d� n d� n  1    d� 1 H0I ð� n ÞH0I ð� n  1 Þ    HI0 ð� 1 Þ:  h t t 0 0 n¼1 t0

ð26Þ ^ satisfies the integral equation U ^ ðt; t0 Þ ¼ 1  i U h

Zt

^ ð�; t0 Þ; d�H0I ð�ÞU

ð27Þ

t0

which allows to recast its matrix elements in the form ^ ðt; t0 Þjaðt0 Þi ¼ �ca e hcðtÞjU



i h

"a ð t  t0 Þ

with  h!ca ¼ "c  "a ,

i

i  ð "c t  "a t0 Þ  e h Tca ð!ca Þ; h

ð28Þ

Z Tca ð!Þ ¼

dt ei!� Tca ð�Þ;

ð29Þ

Ultrafast Nonlinear Optical Signals

and

2 Tca ð�Þ 

6 i hcð�ÞjH0I ð�Þexpþ 4

h

3

Z� d�

0

i

7 H0I ð� 0 Þ5jaðt0 Þie h

"a ð�  t0 Þ

233

ð30Þ

t0

are the transition amplitudes. Conservation of probability, or equivalently, unitarity of U^ , implies P  2 ^ that ¼ 1. Substitution of Equation (28) in this relation leads to c U ca the optical theorem 1 X jTca ð!ca Þj2 ; ð31Þ JTaa ð!aa ¼ 0Þ ¼  2 h c where Taa is given by Equation (30) with c(�) replaced by a(�). The c summation runs over all states including c=a. This is analogous but different from the optical theorem of stationary (steady state) scattering theory (Newton, 1982), since here we consider pulsed excitation and the T matrix Equation (30) carries the full-time dependence of the fields. Expanding Equation (31) in powers of the field gives Z ~ ð1Þ ð!Þ�ð!ca  !Þ Tca ð!ca Þ ¼  d!Eð!ÞT ca þ

1 2p h 1

Z

~ ð2Þ ð!2 ; !1 Þ�ð!ca  !1  !2 Þ d!1 d!2 Eð!1 ÞEð!2 ÞT ca Z

 h2 4p2 

d!1 d!2 d!3 Eð!1 ÞEð!2 ÞEð!3 Þ

~ ð3Þ ð!3 ; !2 ; !1 Þ�ð!ca  !1  !2  !3 Þ þ    T ca

ð32Þ

where ~ ð1Þ ð!1 Þ  �ca ; T ca ~ ð2Þ ð!2 ; !1 Þ  T ca ~ ð3Þ ð!3 ; !2 ; !1 Þ  T ca

X � 1 ;� 2

X �

�c� ��a ; !1  !�a þ i�

�c�2 �� 2 � 1 ��1 a ð!1 þ !2  !� 2 a þ i�Þð!1  !�1 a þ i�Þ

ð33Þ ð34Þ

ð35Þ

and so forth. ~ introduced above are defined as The partial transition amplitudes T follows: (i) each transition between states contributes a dipole operator �

234

Shaul Mukamel and Saar Rahav

factor, (ii) propagation between transitions is given by a Green’s function whose argument is the cumulative frequencies of field modes, minus the transition frequency between the current and initial state. These quantities can also be used in a frequency-domain setup. As an example, a second-order transition from a to c through �, which involve absorption of !1 and then !2 would be described by ~ ð2Þ ð!2 ; !1 Þ ¼ T ca

�c� ��a ; !1  !�a þ i�

where � is a positive infinitesimal. Each transition amplitude describes a partial contribution of a specific molecular process. The partial ampli­ tudes are multiplied by the field amplitudes and summed over to give the full transition amplitude of the process, see Equation (32). ~ ð2Þ We shall denote quantities such as T ca ð!2 ; !1 Þ, that do not include the field amplitudes, as bare transition amplitudes as opposed to the dressed (partial) transition amplitudes which include the fields as well. The two ð2Þ ~ ð2Þ are related by Tca ð!2 ; !1 Þ ¼ E 1 E 2 T ca ð!2 ; !1 Þ etc. We will mostly use bare amplitudes in what follows. To distinguish these partial amplitudes from the full transition amplitudes of Equation (30), they contain a superscript that denotes their order in the field-matter interaction. The optical theorem (31) can be represented diagrammatically. The expansion Equation (32) of the transition amplitude is depicted schema­ tically in Figure 4. The summation in Figure 4 corresponds to that of Equation (32), namely a sum over all intermediate states and all fre­ quency combinations which sum to ! (with negative signs for emission). With the help of the diagrammatic representation of Tca(!), the optical theorem (31) is depicted in Figure 5. The rates of various material processes can be expressed using the transition amplitudes. In a frequency-domain measurement the rate of a k-photon process assumes the generalized Kramers-Heisenberg form

c

c

ω

+

–

a

ω − ω1

c

ω

1 2π

Σ

+ ......

ω1

a

Figure 4 Diagrammatic representation of the transition amplitude Tca(w) [Equation (32)]

a

Ultrafast Nonlinear Optical Signals

c

a

c

ωca ωca

0

=−

Im

235

1 Σ 2ħ c a

a

a

Figure 5 Diagrammatic representation of Equation (31). The two strands on the righthand side correspond to the evolution of the ket and the bra, and are complex conjugates of each other

Ra ! c

! k  ðkÞ 2 X   / Tca ð!1 ; . . . ; !k Þ � !k  !ca : i¼1

This process makes the following contribution to an optical signal where photon i is detected Si  ARa ! c : Here A = þ1 if the photon !i is emitted, 1 if it is absorbed, and 0 if it is neither absorbed or emitted. The generalization to processes where more than one photon is emitted or absorbed is obvious. The discussion in the previous section assumed that a single-field pathway contributes to the molecular transition. When several pathways are possible they must be added at the amplitude level (as in Figure 2) and will interfere  ! k1  X  ðk1 Þ DNa ! c / Tca ð!1 ; . . . ; !k1 Þ� !i  !ca  i¼1 ðk2 Þ ð!k1 þ 1 ; . . . ; !k1 þ k2 Þ� þ Tca

!2   !i  !ca  :  þ1

kX 1 þ k2 i ¼ k1

By expandingi the brackets we see that  terms of the form h ðk1 Þ ðk2 Þ Tca Tca þ c:c � ki¼1 1 !i  !ca � ki¼1 þk1kþ2 1 !i  !ca can be interpreted as an interference correction for the number of molecular transitions.1 A similar interpretation holds also for optical signals. By recasting the optical signal in one of the above forms it can be interpreted in terms of the underlying molecular transitions. This is 1 The appearance of factor of �2 once the bracket is squared reflects the fact that the diagonal terms are naturally described in terms of the rate of a process, while non-diagonal terms are described in terms of the overall number of transitions. We will clarify this for CARS signals in Section 9.

236

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straightforward for pump-probe processes but is less obvious for CARS, due to the existence of parametric processes, which contribute to optical signals but do not represent a molecular process since they eventually leave the molecule at its initial state. 3.1 Purely Dissipative Signals Optical signals generally include contributions from two types of processes: resonant, where the matter makes a transition from one state to another, and parametric, where photons are exchanged between different field modes, but the molecule only serves as a “catalyst” and ultimately returns to its initial state. The latter typically give a broad, featureless, background to optical signals and the resonant signal from the molecule of interest may be masked by a much stronger parametric background from, for example, solvent molecules (Kirkwood et al., 2000). Removing the parametric background is of great interest for spectroscopic and imaging applications (Li et al., 2008; Pestov et al., 2008; Potma et al., 2006). Based on our analysis, this can generally be accomplished by measuring the total energy exchanged between the field and matter. This dissipative signal requires detecting all the field modes as is given by i !i Si , where Si is the signal in the ith mode and may be easily calculated from the material perspective as Z D¼ Dð!Þ  h  1

X

d!Dð!Þ;

PðgÞ!jTf g ð!Þj2 �ð!  !f g Þ:

ð36Þ ð37Þ

fg

D(!) is the energy gained by the material through transitions between states separated by energy  h!. Note that ! can be any combination of field frequencies. D may be measured by subtracting the transmitted and incoming pulse energies. Parametric processes do not affect the total field energy and thus do not contribute to D. Obviously, D will be useful as a spectro­ scopic tool provided that specific resonances can be separated out by varying pulse parameters. We will return to this point in Section 10.

4. CTPL REPRESENTATION OF OPTICAL SIGNALS In the following sections we apply the transition amplitude approach to calculate the pump-probe and CARS signals. These signals, which are fourth order in the field, will be calculated diagrammatically by expanding Equa­ tion (21), assuming that the field is initially in a coherent state [Equation (22)].

Ultrafast Nonlinear Optical Signals

237

The contribution of each diagram could be read out following the rules given below. The diagrams represent the expansion of the ordered expo­ nential in Equation (21). Note that only the imaginary part of the dia­ grams contributes to the signals. There are several types of diagrams, which differ by the bookkeeping of matter-field interactions. Time-domain measurements are commonly represented by double-sided Feynman diagrams for the density matrix (Mukamel, 1995, 2008). Only forward time evolution is required in that case. These diagrams are read from bottom to top following the evolution of both the ket and the bra in the physical time. They are well documented and we will not repeat their description here. Suffice it to note that these diagrams maintain full bookkeeping of the time ordering, namely that all interactions are ordered in time, whether they are with the ket or with the bra, this makes them particularly suitable for time-domain measurements. The loop diagrams presented below, in contrast, are not read in real (physical) time, but rather clockwise along the loop: time first runs forward on the left branch (ket) and then backwards on the right branch (bra). The interactions are ordered along the loop. Loop diagrams are therefore partially ordered in real time. (Only interactions in each branch are time ordered.) This turns out to be most convenient for frequency-domain techniques, where no specific order of interactions is enforced by the field envelopes. Fewer loop diagrams are required since each loop diagram represents a sum of several double-sided Feynman diagrams which reflect all possible time orderings of interactions on the ket and bra following (Marx et al., 2008). We now present the rules used to read these diagrams. These will then be used to calculate the pump-probe [Equation (39)] and the CARS [Equation (59)] signals. Hereafter we only use the FD rules but for completeness we also give the rules in the time domain. Example of an application of the time domain rules can be found in Marx et al. (2008).

4.1 Rules for the CTPL Diagrams in the Time Domain TD1 The loop represents the density operator. Its left branch stands for the ket, the right corresponds to the bra. TD2 Each interaction with a field mode is represented by a wavy line on either the right (R-superoperators) or the left (L-superoperators) branch. TD3 The field is indicated by dressing the wavy lines with arrows, where an arrow pointing to the right represents the field annihilation operator E(r,t), which involves the term eiðkj r!j tÞ (see Equation (12)). Conversely, an arrow pointing to the left corresponds to the field creation operator E †(r,t), associated with a e  iðkj r!j tÞ factor. This is made explicit by adding the wave vectors +kj to the arrows.

238

Shaul Mukamel and Saar Rahav

TD4 Within the RWA, each interaction with E(r,t) is accompanied by applying the operator V†, which leads to excitation of the state represented by the ket and de-excitation of the state represented by the bra, respectively. Arrows pointing “inwards” (i.e., pointing to the right on the ket and to the left on the bra) consequently cause absorption of a photon by exciting the system, whereas arrows pointing “outwards” (i.e., pointing to the left on the bra and to the right on the ket) represent de-exciting the system by photon emission. TD5 The interaction at the observation time t is always the last. As a convention, it is chosen to occur from the left. This choice is arbitrary and does not affect the result. TD6 Interactions within each branch are time ordered, but interactions on different branches are not. Each loop can be further decomposed into several fully-time-ordered diagrams (double-sided Feynman diagrams). These can be generated from the loop by simply shifting the arrows along each branch, thus changing their position relative to the interactions on the other branch. Each of these relative positions then gives rise to a particular fully-time-ordered diagram. TD7 The overall sign of the correlation function is given by (1)NR, where NR stands for the number of interactions from the right. TD8 Diagrams representing (nþ1)-wave mixing acquire a common prefactor in.

4.2 Rules for the CTPL Diagrams in the Frequency Domain FD1 Time runs along the loop clockwise from bottom left to bottom right. FD2 Each interaction with a field mode is represented by a wavy line. FD3 The field is indicated by dressing the wavy lines with arrows, where an arrow pointing to the right represents the field annihilation operator E(r,t), which involves the factor eiðks r!s tÞ. Conversely, an arrow pointing to the left corresponds to the field creation operator E †(r,t), being associated with e  iðks r!s tÞ . This is made explicit by adding the wave vectors +ks to the arrows. FD4 Within the RWA each interaction with E(r,t) is accompanied by applying the operator V†, which leads to excitation of the material system. Arrows pointing to the right cause absorption of a photon by exciting the molecule, whereas arrows pointing to the left represent de-exciting the system by photon emission. FD5 The interaction at the observation time t is fixed to be with the detected mode and is always the last. It is chosen to occur on the left branch of the loop. This choice is arbitrary and does not affect the result. FD6 The loop translates into an alternating product of interactions (arrows) and periods of free evolutions (vertical solid lines) along the loop.

Ultrafast Nonlinear Optical Signals

239

FD7

Since the loop time goes clockwise along the loop, periods of free evolution on the left branch amount to propagating forward in real time (iG(!)), whereas evolution on the right branch corresponds to backward propagation (iG†(!)). FD8 The frequency arguments of the various propagators are cumulative, i.e. they are given by the sum of all “earlier” interactions along the loop. Additionally, the ground state frequency !g is added to all arguments of the propagators. FD9 A diagram representing nþ1 mixing caries the prefactor in ð1NR Þ(NR is the number of interactions from the right).

5. THE PUMP-PROBE SIGNAL Pump-probe is the simplest nonlinear technique: the system interacts with two fields, a pump k1, and a probe k2 (which is detected). The signal is defined as the difference in the probe transmitted intensity between mea­ surements where the pump is present or absent. This difference between two large quantities amounts to “determining the weight of the captain by weighting the ship with and without the captain”. It limits the sensitivity compared to homodyne four-wave mixing signals. However, this techni­ que is simpler to implement and does not require phase control of the pulses. Stimulated Raman spectroscopy (Alfano & Shapiro, 1971; Jones & Stoicheff, 1964) carried out with a combination of broadband (femtose­ cond) and narrowband (picosecond) pulses is widely used for improving the sensitivity of spontaneous Raman signals (Laimgruber et al., 2006; Lakshmanna, 2009; Mallick et al., 2008; Wilson et al., 2009). This technique has also been used for bioimaging applications (Min et al., 2009). We shall calculate the frequency-domain pump-probe signal starting from Equation (16). We assume that the field intensities are high enough so that spontaneous emission can be safely neglected and all matter/field interactions are stimulated. The derivation starts by using Equation (16) and expanding the exponent in Equation (21) to third order, Z Z Zt 1 d� 1 d� 2 d� 3 E 2 ðtÞ SPP ¼  4 Re 3 h 1

^ † ðtÞHint  ð� 1 ÞHint  ð� 2 ÞHint  ð� 3 Þi: hT V L

ð38Þ

Only contributions proportional to jE 1 j2 jE 2 j2 where two of the interactions are with the probe, and two are with the pump, will be kept. For these contributions the expectation value in Equation (38) turns out to be independent of t. This is the reason for using Equation (16) to define the signal. An additional integration over t would result in an infinite signal.

240

Shaul Mukamel and Saar Rahav

{ | f 〉}

μef { | e 〉}

μge { | g 〉} Figure 6 The three-band (ladder) model system and transition dipoles used in the derivation of Equation (39)

We will consider the three-band model, as depicted in Figure 6. Within the RWA we neglect off-resonant contributions and only retain the resonant ones where photon absorption is accompanied by a molecular-up transi­ tion and vice versa. This excellent approximation for resonant signals limits the number of possible processes and simplifies the analysis. For instance, it excludes emission from the ground state band, as well as three consecu­ tive absorptions. Substitution of Hint in Equation (38) results in the eight terms (Roslyak et al., 2009), which are depicted diagrammatically in Figure 7. The frequency-domain signal (absorption of !2), which can be read from the diagrams with the help of the rules in Section 4.2, is given by 4 j E1 j 2 j E2 j 2 h4 n



†

† ^ † ! g þ !1 V ^ † !g þ !1 þ !2 V ^ !g þ ! 1 V ^G ^G ^ G ^ i Im hV

SPP ð!2 ; !1 Þ ¼ 





†

† ^ † ! g þ !2 V ^ † ! g þ !1 þ !2 V ^ !g þ ! 1 V ^G ^G ^ G ^ i þhV



† †

† ^ † ! g þ !1 V ^ † ! g þ !1 þ !2 V ^ !g þ ! 2 V ^G ^G ^ G ^ i þhV ^ ^G þhV

†



† †

† ^ † ! g þ !1 þ !2 V ^ !g þ ! 2 V ^G ^ G ^ i ! g þ !2 V



†



† ^ † ! g þ !2 V ^ !g þ !1  !1 V ^ !g þ ! 1 V ^G ^ G ^G ^ i þhV

† †



† ^ † ! g þ !1 V ^ !g þ !2  ! 2 V ^ † !g þ ! 2 V ^ i ^G ^ G ^G þhV

† †



† ^ † ! g þ !2 V ^ !g þ !2  ! 1 V ^ † !g þ ! 2 V ^G ^ G ^G ^ i þhV D

† †



† Eo ^ † !g þ !1 V ^ ! g þ !1  !2 V ^ !g þ !1 V ^G ^ G ^G ^  V : ð39Þ

241

Ultrafast Nonlinear Optical Signals

ω2

ω1

f

ω2

ω 2

f

e

ω2

g

g

ω2

f

ω1

e

ω2

e

e

ω1

e

ω2

g�

ω1

g e

ω1

e g

g

g (d)

e

ω1

g�

ω1

g e g (g)

(f) ω2

e

ω2

ω1

g (e)

ω2

ω2

e

ω2

g�

ω1

f

(c)

e

ω1

e

ω1

g

(b) ω 2

e

g

g

e

g

e

e

ω1

(a) ω2

ω2

ω1

f

e

ω1

e g

f

g�

e ω1 g

g�

ω2

e

ω1

g (h)

Figure 7 CTPL diagrams for the eight contributions to the pump-probe signal, respectively [Equation (39)]

^ Here Gð!Þ ¼ ð!  H0 þ i� Þ  1 is the retarded Green’s function and † ^ G ð!Þ ¼ ð!  H0  i� Þ  1 is the advanced Green’s function. The terms in Equation (39) naturally separate into two groups depending on the order of absorption and emission events along the loop. The first four terms have two consecutive absorptions followed by two emissions (VVV†V†), and the matter goes through the doubly excited f band. These contributions will therefore be termed two-photo absorption (TPA). Terms 5-8 have the form of absorption, emission, absorption, emission (VV†VV†) and only involve the g and e bands. These will be termed stimulated Raman scattering (SRS). Expanding Equation (39) in the eigenvalues results in the final expres­ sion for the pump-probe signal

242

Shaul Mukamel and Saar Rahav

SPP ð!2 ; !1 Þ ¼ STPA ð!2 ; !1 Þ þ SSRS ð!1 ; !1 Þ; with the two photon absorption component STPA ð!2 ; !1 Þ ¼4pNh   4 jE 1 j2 jE 2 j2 j�eg j2 j�f e j2 J

ð40Þ X

g;g 0 ;e;f

(

1





 !1  !eg  i� !2 þ !1  !f g  i� !1  !eg þ i� þ

1





!2  !eg  i� !2 þ !1  !f g  i� !1  !eg þ i�

þ

þ

1



!2  !eg  i� !2 þ !1  !f g  i� !2  !eg  i�



g

1



; !2  !eg  i� !1 þ !2  !f g  i� !1  !eg  i�



ð41Þ

and the stimulated Raman component h  4 jE 1 j2 jE 2 j2 J SSRS ð!2 ; !1 Þ ¼ 4pN

X

j�eg j2 j�g 0 e j2

g;g 0 ;e;f

( 



1





!2  !eg  i� !1  !1  !g 0 g  i� !1  !eg  i�





1



!1  !eg  i� !1  !2  !g 0 g  i� !1  !eg þ i�



1





!2  !eg  i� !2  !1  !g 0 g  i� !2  !eg  i�

g

1



; !2  !eg  i� !1  !1  !g 0 g þ i� !1  !eg þ i�



ð42Þ

Despite the straightforward derivation, it is not evident by a simple inspection of Equations (41) and (42) what are the material processes underlying the signal in real time since the calculation is done on a

Ultrafast Nonlinear Optical Signals

243

loop that involves both forward and backward time evolutions. We emphasize that this bookkeeping along the loop merely gives the reso­ nances that contribute to a particular signal, but since we are going forward and backward in time we cannot simply attribute a given loop diagram to a transition between an initial and a final state. This can only be done by breaking the loop into transition amplitudes and bringing them to the Kramers-Heisenberg form. This will be done next.

6. THE PUMP–PROBE SIGNAL REVISITED: TRANSITION AMPLITUDES In this section the pump-probe signal will be dissected to reveal the underlying material processes. The dissection of signals into contri­ butions corresponding to material processes is done by recasting the signals in terms of partial transition amplitudes. To that end we define slightly modified (frequency-domain) loop diagrams, termed unrest­ ricted loop diagrams, that naturally represent material processes. The diagrams of Section 4 were aimed at the calculation of optical signals. The new diagrams, in contrast, correspond to generalized Kramers-Heisenberg terms, and therefore naturally represent the material processes. 6.1 Unrestricted Loop Diagrams The unrestricted diagrams are closely related to those of Section 4.2, but with one difference: we drop the restriction that the last interaction from the left is at the latest time t, allowing for any relative time ordering between the last interactions on the ket and the bra. These will therefore be denoted unrestricted loop diagrams. We will illustrate why the new diagrams are useful, and how to read them, using a simple example. The unrestricted diagram depicted in Figure 8 is given by the sum of two restricted loop diagrams where the last interaction is on either of the two branches of the loop. To distinguish between the two types of diagrams, we represent the unrestricted part of the loop by a solid line, as opposed to the dashed line used in the previous restricted diagrams. The two loop diagrams are read according to the rules of Section 4.2, omitting rule FD5 regarding the last interaction. From these rules, as well as our definition of partial transition ampli­ tudes, the first diagram on the right-hand side of Figure 8 is proportional ð1Þ ð2Þ to Tca ð!1 ÞTca ð!3 ; !2 Þð!1  !ca  i� Þ  1 . The second diagram on the right-hand side is similar, but with an opposite sign, and the advanced Green function (!1!cai�)1 is replaced by a retarded one (!1  !ca þ i�)1. (See rule FD7.) As a result, the contribution of the unrestricted loop diagram is proportional to

244

Shaul Mukamel and Saar Rahav

ω1

c

c

ω3

ω1 b a

ω2

b

a

a

ω3

c

ω3 ω2

ω1

b

+

a

a

ω2

a

Figure 8 Example of an unrestricted loop diagram (solid line). The diagram is defined as the sum of two restricted diagrams, denoted by a dashed line along the top of the loop, where the last interaction is located either on the ket or on the bra. Phase matching requires !1  !2  !3 ¼ 0

ð1Þ ð2Þ ð!3 ; !2 Þ JTca ð!1 ÞTca



1 1  !1  !ca  i� !1  !ca þ i�



h i ð1Þ ð2Þ ¼ 2pR Tca ð!3 ; !2 Þ �ð!1  !ca Þ: ð!1 ÞTca The reason for introducing the unrestricted diagrams now becomes clear: The contribution of such diagrams to the signal takes a generalized Kramers-Heisenberg form with the branches of the loop corresponding to partial transition amplitudes and the top of the loop to the resonant �­ function. These diagrams naturally connect optical signals with the underlying material processes. While we used a simple example to demonstrate the definition and calculation of unrestricted diagrams, the generalization to any diagram is straightforward. There is only one class of special diagrams, namely ones where all interactions are either on the left branch or on the right branch, that needs to be treated separately. In this case there is no meaning to the relative ordering between branches, and we define the unrestricted diagrams to be equal to the restricted one. The contribution ðnÞ of such diagrams to the signal always takes the form  JTaa , namely the imaginary part of a diagonal (partial) transition matrix. [See Figure 13(i) for an example.] 6.2 The Two-Photon-Absorption and Stimulated-Raman Components of the Pump-Probe Signal We now dissect the pump-probe signal into contributions from various material processes. Equations (41) and (42) can be partially recast in terms of transition amplitudes by noting that the loop diagrams correspond to

Ultrafast Nonlinear Optical Signals

ψ (t)

245

G (Δω + ωg) t

τ

ψ (– ∞)

ψ (t)

ψ (– ∞)

Figure 9 By dissecting the loop along its centerline it factorizes into two single-sided Feynman diagrams. This is possible since the system remains in the same state hj ðt Þi between the topmost interaction on the two branches which occur, respectively, at times t and t. The advanced propagator G † ðD! þ !g Þ, representing backward propagation from t and t, connects the two

the product of two such amplitudes times an additions propagator, as is explained in Figure 9. This allows to rewrite the signals as STPA ð!2 ; !1 Þ ¼ 4pN h  4 jE 1 j2 jE 2 j2  X  ð2Þ ~ ð!2 ; !1 Þj2 þ T ~ ð2Þ ð!2 ; !1 ÞT ~ ð2Þ ð!1 ; !2 Þ J jT fg fg fg

1 ! þ !  !f g  i� 2 1 g;g 0 ;e;f   1 ~ ð3Þ ~ ð1Þ ~ ð3Þ ~ ð1Þ : þ T eg ð!2 ÞT eg ð!1 ; !1 ; !2 Þ þ T eg ð!2 ÞT eg ð!1 ; !2 ; !1 Þ !2  !eg  i� ð43Þ

SSRS ð!2 ; !1 Þ ¼ 4pN h  4 jE 1 j2 jE 2 j2 J

2 X  ð2Þ T ~ 0 ð! ; ! Þ  g ;g 2 1 

g;g 0 ;e;f

1 !1  !2  !g 0 g  i�

~ ð1Þ ð!2 ÞT ~ ð3Þ ð!1 ; !1 ; !2 Þ ~ ð3Þ ð!2 ; !1 ; !1 Þ þ T ~ ð1Þ ð!2 ÞT ðT eg eg eg eg ~ ð1Þ ð!2 ÞÞ ~ ð3Þ ð!2 ; !1 ; !1 ÞT þT eg eg

1 : !2  !eg  i�

ð44Þ

The first term corresponds to diagram (h) in Figure 7 whereas the second term is related to diagrams (e)-(g).

246

Shaul Mukamel and Saar Rahav

We start with the SRS signal (44). Taking the imaginary part of the first term results in a d-function. The same is true for the sum of the second and fourth terms in Equation (44). Subtracting

1 ~ ð3Þ ~ ð1Þ J T eq ð!1 ; !1 ; !2 ÞT eg ð!2 Þ !2  !eg  i�  2  ~ ð2Þ  þT g 0 g ð!1 ; !2 Þ



1 ¼0 !2  !1  !g 0 g  i�

from the terms in the sum in Equation (44) allows to bring all terms in Equation (44) to a form where the imaginary part can be taken, leading to various d-functions. This gives  2 X  ~ ð2Þ  2 2 4 2 SSRS ð!2 ; !1 Þ ¼ 4p Nh jE 1 j jE 2 j PðgÞ T ð! ; ! Þ �ð!1  !2  !g 0 g Þ 0 2 1 gg gg 0 e

 2  ~ ð2Þ  T ð! ; ! Þ �ð!2  !1  !g 0 g Þ 0 1 2 gg h i ~ ð1Þ ~ ð3Þ  T eg ð!2 ÞT eg ð!1 ; !1 ; !2 Þ þ c:c �ð!2  !eg Þ

g

h i ð1Þ ð3Þ ~ eg ~ eg  T ð!2 ÞT ð!2 ; !1 ; !1 Þ þ c:c �ð!2  !eg Þ :

ð45Þ

This has the desired generalized Kramers-Heisenberg form, allowing to identify the underlying molecular processes. The four terms in Equation (45) are represented diagrammatically by the unrestricted loop diagrams of Figure 10. For brevity we have omitted the diagrams corresponding to the two complex conjugate terms in Equation (45), but these can be easily obtained by reflecting diagrams (iii) and (iv) along a vertical line crossing the top of the loops. Interestingly, all the terms proportional to d(!2!eg) could be combined into a single term whose amplitude is the sum of three processes, with corrections which have different scaling in the field amplitude. This gives ~ SRS ð!2 ; !1 Þ ¼ 4p2 N h4 S

X g;g 0 ;e;f

~ ð2Þ0 ð!2 ; !1 Þj2 �ð!1  !2  !g 0 g Þ jE 1 j2 jE 2 j2 jT gg

~ ð2Þ0 ð!1 ; !2 Þj2 �ð!2  !1  !g 0 g Þ jE 1 j2 jE 2 j2 jT gg ~ ð1Þ ð!2 Þj2 �ð!2  !eg Þ þjE 2 j2 jT eg

247

Ultrafast Nonlinear Optical Signals

  ~ ð1Þ 2 ~ ð3Þ  E 2 T eg ð!2 Þ þ jE 1 j E 2 T eg ð!1 ; !1 ; !2 Þ

j

2

~ ð3Þ ð!2 ; !1 ; !1 Þ �ð!2  !eg Þ: þ jE 1 j2 E 2 T eg

ð46Þ

where we have introduced an additional term to eliminate the  jE 2 j2 part. The SRS signal is obtained from Equation (46) by neglecting terms of order jE 2 j2 jE 1 j4. Obviously, to obtain a Kramers-Heisenberg form we must give up the strict bookkeeping in orders of the field since that form naturally mixes the different orders. We next turn to the TPA term. It can be brought to a form in which one can take the imaginary part, leading to d-functions by adding to all terms in the sum of Equation (43)    2 1  ~ ð2Þ  ð2Þ ð2Þ ~ ~ J T f g ð!1 ; !2 Þ þ T f g ð!1 ;!2 ÞT f g ð!2 ; !1 Þ !2 þ !1  !f g  i� þ

h

i

~ ð3Þ ð!1 ;!1 ; !2 ÞT ~ ð1Þ ð!2 Þ þ T ~ ð3Þ ð!1 ;!2 ;!1 ÞT ~ ð1Þ ð!2 Þ T eg eg eg eg g�

ω2

ω1

ω1

ω2

e

ω1 g

g

e

e

g

g

e

ω1 g�

ω1

e

ω2

g (iii)

ω2

(ii)

e

g

¼ 0:

ω1

(i)

ω2

)

g�

ω2 e

1 !2  !eg  i�

ω2 ω2

g� e g

ω1 ω1

g (iv)

Figure 10 Unrestricted loop diagrams corresponding to the four terms in Equation (45), respectively

248

Shaul Mukamel and Saar Rahav

This leads to h  4 jE 1 j2 jE 2 j2 STPA ð!2 ; !1 Þ ¼ 4p2 N  2 X  ~ ð2Þ ~ ð2Þ ð!1 ; !2 Þ �ð!1 þ !2  !f g Þ  PðgÞ T ð!2 ; !1 Þ þ T fg

gg 0 ef

fg

h i ~ ð1Þ ð!2 ÞT ~ ð3Þ ð!1 ; !1 ; !2 Þ þ c:c: �ð!2  !eg Þ þ T eg eg

g

h i ~ ð1Þ ð!2 ÞT ~ ð3Þ ð!1 ; !2 ; !1 Þ þ c:c: �ð!2  !eg Þ : þ T eg eg

ð47Þ

The terms of Equation (47) are depicted diagrammatically in Figure 11. The complex conjugates of the last two terms are omitted for brevity, as they are the mirror images of diagrams 11(ii) and 11(iii). Combining the last two terms into one amplitude, as was done for the SRS signal, would give X ~ ð2Þ ð!2 ; !1 Þ ~ TPA ð!2 ; !1 Þ ¼ 4p2 N h4 jE 1 j2 jE 2 j2 jT S fg g;g 0 ;e;f

~ ð!1 ; !2 Þj2 �ð!2 þ !1  !f g Þ þT fg ð2Þ

  ~ ð1Þ 2 ~ ð3Þ þ E 2 T eg ð!2 Þ þ jE 1 j E 2 T eg ð!1 ; !1 ; !2 Þ

j

2

~ ð3Þ ð!1 ; !2 ; !1 Þ �ð!2  !eg Þ þ jE 1 j2 E 2 T eg ~ ð1Þ ð!2 Þj2 �ð!2  !eg Þ: jE 2 j2 jT eg f

ω2 ω1

e

e

ω2

ω1

ω1 +

ω2

f e e

g g

ω1

ω2

ω2 +

ω1

ð48Þ f e e

ω1

ω1

ω2 +

ω2

g g

g g

f

ω2

e

e

g

g

ω1

(i) e f

ω1

e g

ω2

ω2 g

e

ω1

(ii)

ω2 g

ω1 f

ω2

e g

ω1

(iii)

Figure 11 Unrestricted loop diagrams corresponding to the three terms in Equation (47), respectively

Ultrafast Nonlinear Optical Signals

249

Again, Equation (47) can be obtained from Equation (48) by neglecting terms of order jE 2 j2 jE 1 j4. We thus accomplished our goal of expressing the signal in a generalized Kramers-Heisenberg form.

7. COHERENT ANTI-STOKES RAMAN SPECTROSCOPY Heterodyne CARS is a four-wave mixing technique where the system interacts with four field modes (Begley et al., 1974; Evans & Xie, 2008; Lotem et al., 1976; Penzkofer et al., 1979; Silberberg, 2009). The technique provides a powerful spectroscopic tool for probing molecular vibrations and for imaging applications (Nan et al., 2006; Potma & Xie, 2008; Potma et al., 2006). Time domain femtosecond techniques with pulse shaping have been employed to enhance the degree of control over the signals (Kukura et al., 2007; Laimgruber et al., 2006; Mallick et al., 2008; Mukamel, 2009; Oron et al., 2002; Pestov et al., 2007). We will start with the time-integrated signal (14). This is convenient since the CARS contributions, which interact once with each field, will depend on t through a factor of exp[i(!1  !2 þ 3  !4)t]. (In the frequency-domain technique the field amplitudes are time independent.) The t integral then results in a factor of 2pd(!1  !2 þ !3  !4), giving a singular signal in the continuous wave (CW) limit. This further shows that only frequency combinations satisfying !1  !2 þ !3  !4 ¼ 0;

ð49Þ

can contribute to the signal. There are four possible CARS signals (Mukamel, 1995) which only differ by the choice of the detected mode. Denoting the signal obtained by measuring mode !i by Si, we have h i 4p S1 ¼  �ð!1  !2 þ !3  !4 ÞJ E 1 E 2 E 3 E 4 � ð3 Þ ð!1 ; !4 ; !3 ; !2 Þ ; ð50Þ h h i 4p S2 ¼  �ð!1  !2 þ !3  !4 ÞJ E 1 E 2 E 3 E 4 � ð3 Þ ð!2 ; !3 ; !4 ; !1 Þ ; ð51Þ h h i 4p S3 ¼  �ð!1  !2 þ !3  !4 ÞJ E 1 E 2 E 3 E 4 � ð3 Þ ð!3 ; !2 ; !1 ; !4 Þ ; ð52Þ h h i 4p S4 ¼  �ð!1  !2 þ !3  !4 ÞJ E 1 E 2 E 3 E 4 � ð3 Þ ð!4 ; !1 ; !2 ; !3 Þ : ð53Þ h The pump-probe technique only involves two field modes. The four field modes in CARS generate a larger number of terms. To keep the problem manageable, we employ the model of Figure 12 which limits the number of optical transitions. a and c are vibrational states belonging to the ground electronic state whereas b is an electronically excited state. Levels

250

Shaul Mukamel and Saar Rahav

|b〉

ω1

ω3

ω2

ω4

|c〉 |a〉 Figure 12 Level scheme and optical transitions for the CARS process

a and c are resonantly coupled by two possible Raman processes, with !1  !2 ¼ !4  !3 ’ !ca . The system is described by the Hamiltonian (Mukamel, 1995; Scully & Zubairy, 1997; Shen, 2002) H ¼ Hs þ Hf þ Hint ;

ð54Þ

Hs ¼  h!a jaihaj þ h!b jbihbj þ h!c jcihcj;

ð55Þ

with the molecular part and the field part Hf ¼

4 X i¼1

h!  i^a †i ^a i :

ð56Þ

Within the RWA, the dipole coupling between the laser field and the molecule is given by  Hint ¼

2p!1 W

1 = 2

^a 1 e  i!1 t �ba jbihajþ



2p!2 W

1 = 2

^a 2 e  i!2 t �bc jbihcj

    2p!3 1 = 2 2p!4 1 = 2 ^a 4 e  i!4 t �ba jbihaj þ h:c: ^a 3 e  i!3 t �bc jbihcjþ þ W W ð57Þ The CTPL diagrams, which correspond to the two processes contributing to the signal (53) are depicted in Figure 13. In (i) the system is initially in the lower state a, while in (ii) it starts in the vibrationally excited state c. The loop diagrams can be read according to the rules given in Section 4, leading to

251

Ultrafast Nonlinear Optical Signals

ω4 ω3 ω2 ω1

ω4

|a〉

|a〉 〈a |

|b〉

〈b |

|b〉

|c〉

ω2

ω3 |b〉

|c〉

〈a|

|a〉 (i)

ω1

〈c |

(ii)

Figure 13 CTPL representation of �(3)(w 4; w 1,w 2,w 3) which is related to the S4 signal. (i) and (ii) represent the two terms in Equation (58) respectively. In both the interaction with the detected mode (w 4) is chronologically the last

� ð3 Þ ð!4 ; !1 ; !2 ; !3 Þ ¼ 

j�ba j2 j�cb j2 h3 



PðaÞ ð!1  !2 þ !3  !ba þ i�Þð!1  !2  !ca þ i�Þð!1  !ba þ i�Þ

þ

 PðcÞ : ð!3  !4 þ !1  !bc  i�Þð!3  !4  !ac  i�Þð!3  !bc þ i�Þ ð58Þ

Substitution in Equation (53) gives

½

4p S4 ¼  4 �ð!1  !2 þ !3  !4 ÞJ E 1 E 2 E 3 E 4 j�ab j2 j�bc j2 h  

PðaÞ ð!4  !ba þ i�Þð!1  !2  !ca þ i�Þð!1  !ba þ i�Þ

PðcÞ þ ð!3  !bc þ i�Þð!2  !bc  i�Þð!2  !1  !ac  i�Þ





ð59Þ

where we have further made use of Equation (49) to rearrange some frequency combinations. P(a) is the equilibrium probability to be in state a.

252

Shaul Mukamel and Saar Rahav

These results will be used in the next section to recast the signal in terms of transition amplitudes revealing the underlying molecular processes.

8. CARS SIGNALS RECAST IN TERMS OF TRANSITION AMPLITUDES We now dissect the CARS signal (59) into components corresponding to various molecular processes. Equation (59) has two contributions, one proportional to P(a), and the other to P(c). For clarity, we only consider the P(a) contribution in detail, and then point out how to do the same for the P(c) part. The P(a) contribution to Equation (59) exhibits a different structure than the pump-probe signal. This stems from parametric processes. We now demonstrate how to separate these from the contribution of the resonant processes (which assume the generalized Kramers-Heisenberg form). The P(a) contribution is proportional to (the imaginary part of) ð4Þ ~ ð4Þ Taa ð!4 ; !3 ; !2 ; !1 Þ ¼ E 1 E 2 E 3 E 4 T aa ð!4 ; !3 ; !2 ; !1 Þ, corresponding to a fourth-order process leaving the molecule in its initial state. The model of Figure 12, allows for yet another fourth-order process with the order of interactions reversed. The contribution of said process would be ð4Þ ~ ð4Þ proportional to Taa ð!1 ; !2 ; !3 ; !4 Þ ¼ E 1 E 2 E 3 E 4 T aa ð!1 ; !2 ; !3 ; !4 Þ. Both processes are depicted diagrammatically in Figure 14. While according to Equation (59), the signal is proportional to the contribution from the first process, it is clear that both processes

ω4 ω3 ω2

a

ω1

b

ω2

c b

ω1

a b c

ω3 ω4

b

a

a

Taa (− ω 4, ω 3, − ω 2, ω 1)

Taa (− ω 1, ω 2, − ω 3, ω 4)

Figure 14 The two fourth-order sequences of interactions contributing to the CARS signal

253

Ultrafast Nonlinear Optical Signals

contribute to the frequency-domain signal. For reasons that will become clear shortly, we rewrite the signal as a sum a symmetric and an asym­ metric contribution with respect to the two processes, ~ ð4Þ ð!4 ; !3 ; !2 ; !1 Þ ¼ Tsym þ Tas ; E 1 E 2 E 3 E 4 T aa where Tsym 

1h ~ ð4Þ ð!4 ; !3 ; !2 ; !1 Þ E 1 E 2 E 3 E 4 T aa 2 i ~ ð4Þ ð!1 ; !2 ; !3 ; !4 Þ þE  E 2 E  E 4 T 1

Tas 

ð60Þ

3

ð61Þ

aa

1h ~ ð4Þ ð!4 ; !3 ; !2 ; !1 Þ E 1 E 2 E 3 E 4 T aa 2 i ~ ð4Þ ð!1 ; !2 ; !3 ; !4 Þ E  E 2 E  E 4 T 1

3

ð62Þ

aa

This is shown diagrammatically in Figure 15. Tsym and Tas turn out to correspond to resonant and parametric contributions to the signal. ~ ð4Þ It follows from equation (49) that T aa ð!4 ; !3 ; !2 ; !1 Þ ¼ ð4Þ ~ aa ð!1 ; !2 ; !3 ; !4 Þ. This allows us to write Tas as T ~ ð4Þ ð!4 ; !3 ; !2 ; !1 Þ; Tas ¼ iJðE 1 E 2 E 3 E 4 ÞT aa

ð63Þ

We associate the contribution of the asymmetric part with the parametric process for the following reason. The model allows for two time-reversed fourth-order processes: In one a photon is emitted into mode 4, while in the other a photon is absorbed. The signal is proportional to the differ­ ence between the two. Note that these contributions are linear rather than

ω4

|a〉

ω4

|a〉

ω1

|a〉

ω4

|a〉

ω1

|a〉

ω3

|b〉

ω3

|b〉

ω2

|b〉

ω3

|b〉

ω2

|b〉

ω3

|c〉

ω4

|b〉

ω2

=1 |c〉 2

ω2

−1 |c〉 2

ω1

|b〉

ω1

|b〉

|a〉

ω3

+1 2 |c〉

ω4

|b〉

|a〉

Antisymmetric (Tas)

|a〉

ω2

+1 |c〉 2

ω1

|b〉 |a〉

|a〉

Symmetric (Tsym)

Figure 15 The decomposition of the fourth-order dressed transition amplitude into its symmetric and antisymmetric parts. Time runs from bottom to top. The P(a) part of the CARS signal [Equation (59)] is proportional to the imaginary part of the diagram on the left-hand side

254

Shaul Mukamel and Saar Rahav

quadratic in the transition amplitudes. The reason is that they come from interference between the fourth-order processes and the zero-order pro­ cess in which the molecule remains in its initial state. Tsym is the sum of the two possible pathways to make a fourth-order transition starting and ending at a. It can be recast as a contribution from resonant terms using the optical theorem h i ~ ð1Þ ð!1 ÞT ~ ð3Þ ð!2 ; !3 ; !4 Þ þ c:c: �ð!1  !ba Þ 2JTsym ¼ p E 1 E 2 E 3 E 4 T ba ba h i   ~ ð3Þ ~ ð1Þ ð!4 Þ þ c:c: �ð!4  !ba Þ p E 1 E 2 E 3 E 4 T ba ð!3 ; !2 ; !1 ÞT ba h i ð2Þ   ~ ð2Þ ~ p E 1 E 2 E 3 E 4 T ca ð!2 ; !1 ÞT ca ð!3 ; !4 Þ þ c:c: �ð!1  !2  !ca Þ: ð64Þ For our model this theorem is represented in Figure 16 using the unrest­ ricted loop diagrams. Having rewritten both Tas and Tsym in a form with a clear physical interpretation, the P(a) signal can be obtained simply by substituting Equations (63) and (64) in (60), and then (60) in the first term of Equation (59). To complete the dissection of the signal we need to add the P(c) part. This is not proportional to a single transition amplitude, but it can be brought to this form by first writing ð!3  !bc þ i� Þ  1 ¼ ð!3  !bc  i� Þ  1  2�i�ð!3  !bc Þ in Equation (59), and then taking the complex conjugate of the term with three advanced Green’s functions. Notably, the resonant term, which has been split off, already has the desired generalized Kramers-Heisenberg form. ω4 ω3

ω1 ω2

a b

ω2

c

ω1

b a

ω3 ω4

+

ω3

a c

=

b a

ω2

c

ω1

b

ω3

ω4

b

b

+

b

ω4

a

a

a

c

ω2

b

ω1

a

dis 1

ω1 + a

b

ω2

c

ω3

b a

ω2 +

ω4

ω3 ω4

ω1

c b a

dis 2

ω2

b

a

ω3

c

+

+

b

ω1

b a

ω3

ω4

ω2

b

ω1

b

ω4 a

a

c

a

dis 3

Figure 16 Diagrammatic representation of the optical theorem for our model (64). The six loop diagrams represent the six terms in Equation (64) respectively. Twice the imaginary part of the diagrams on the left is equal to the imaginary part of the diagrams on the right

Ultrafast Nonlinear Optical Signals

255

The non-resonant term can now be dissected following the steps used for the P(a) terms. Collecting all terms, the signal finally takes the form, S4 ¼ S4 ; par þ S4 ; dis ;

ð65Þ

with the parametric part S4 ; par ¼

4p h4 

~ ð4Þ ð!4 ; !3 ; !2 ; !1 Þ �ð!1  !2 þ !3  !4 ÞJðE 1 E 2 E 3 E 4 Þ½PðaÞRT aa ð4Þ

~ ð!2 ; !1 ; !4 ; !3 Þ; þPðcÞRT cc

ð66Þ

and the dissipative part S4 ; dis ¼ 

2p2

� ð ! 1  ! 2 þ !3  !4 Þ 4 nh h i

ð1Þ ~ ð3Þ !2 ;  !3 ; !4 T ~ ð!1 Þ þ c:c:  PðaÞ E 1 E 2 E 3 E 4 T ba ba h i ð3Þ ~ ð!3 ;  !2 ; !1 ÞT ~ ð1Þ ð!4 Þ þ c:c:  �ð!1  !ba ÞþPðaÞ E 1 E 2 E 3 E 4 T ba ba h i ~ ð2Þ ð!2 ; !1 ÞT ~ ð2Þ ð!3 ; !4 Þ þ c:c:  �ð!4  !ba ÞþPðaÞ E 1 E 2 E 3 E 4 T ca ca h i ð3Þ   ~ ð!4 ;  !1 ; !2 ÞT ~ ð1Þ ð!3 Þ þ c:c:  �ð!1  !2  !ca ÞþPðcÞ E 1 E 2 E 3 E 4 T bc ba h i ~ ð3Þ ð!1 ; !4 ; !3 ÞT ~ ð1Þ ð!2 Þ þ c:c:  �ð!3  !bc ÞPðcÞ E 1 E 2 E 3 E 4 T bc ba h i   ð2Þ ð2Þ ~ ð!1 ; !2 ÞT ~ ð!4 ; !3 Þ þ c:c:  �ð!2  !bc ÞPðcÞ E 1 E 2 E 3 E 4 T ac ac  �ð!2  !1  !ac Þ

g ð67Þ

The physical interpretation of the resonant terms in Equation (67) is obvious: They all represent interferences of different possible molecular transitions. Note that the overall sign of each term signifies whether the !4 photon is emitted or absorbed. So far, we have focused on one of the possible CARS signals, namely S4. The other signals can be similarly dissected into their components. Once this is done, we can combine different signals in order to either enhance or suppress specific molecular pathways. As an example, the signal S1 can be obtained from S4 by changing the roles of the field modes 1, 2 and 4, 3 respectively. A simple inspection shows that the parametric part changes its sign under this operation, S1 ; par ¼ S4 ; par . The combination S1 þ S4 ¼ S1; dis þ S4; dis

ð68Þ

256

Shaul Mukamel and Saar Rahav

is thus purely dissipative. Further discussion can be found in Rahav et al. (2009). This result will be generalized to arbitrary nonlinear processes in section 10.

9. CARS RESONANCES CAN BE VIEWED AS A DOUBLE-SLIT INTERFERENCE OF TWO TWO-PHOTON PATHWAYS In the previous section we had dissected the CARS signal into para­ metric and resonant processes. We now show that the resonant part of the signal originates from an interference of two transition pathways. We assume that the molecule is initially in its ground state a, and that all frequencies are tuned off electronic resonances so that only Raman resonances are possible. The leading order of the transition amplitude can be found from Equations (25), (28), and (32), Z 2  Eð!ÞEð!ca  !Þ  1 2 2 Pa ! c ¼ j�cb j j�ba j  d! : ð69Þ !  !ba þ i�  h4 4p2  In stimulated CARS the field is made of four narrow-band pulses, cen­ tered around frequencies !i, i = 1, 2, 3, 4, 4 X Eð!Þ ¼ 2p ½E i �D ð!  !i Þ þ E i �D ð! þ !i Þ:

ð70Þ

i¼1

dD is a slightly broadened delta function, of width D, describing the (normalized) narrowband shape of the pulses. By substituting Equation (70) and using the dipole transitions of Figure 12 we find that the integral has only two contributions coming from ! ’ !1, !4.   E 1 E 2 4p2 2 2 Pa ! c ’ 4 j�cb j j�ba j  �D 0 ð!1  !2  !ca Þ !1  !ba þ i� h 2 ð71Þ  E 4 E 3  0 þ �D ð!4  !3  !ca Þ !4  !ba þ i� The functions dD0 in Equation (71) result from an integrated product of two of the band shapes dD. While the width and shape of the dD0 in Equation (71) are different from those appearing in Equation (70), these are still narrow d-like shapes. Equation (71) has a typical form of a double-slit measurement: Two interfering pathways contribute to the resonant Stokes Raman a ! c amplitude. By opening the brackets we find

Ultrafast Nonlinear Optical Signals

12 34 1234 Pa ! c ¼ Pa!c þ Pa!c þ Pa!c ’

4p2 4

257

j�cb j2 j�ba j2

h 2     2 E1E2  � 0 ð!1  !2  !ca Þ   !1  !ba þ i�  D  2   2 E 4 E 3  � 0 ð!4  !3  !ca Þ þ  !4  !ba þ i� D

 E 1 E 2 E 3 E 4 þ 2R ð!1  !ba þ i�Þð!4  !ba  i�Þ  �D 0 ð!1  !2  !ca Þ�D 0 ð!4  !3  !ca Þ :

ð72Þ

34 Here P12 a!c ðPa!c Þ represents a pump-probe process involving modes 1 and 2 (3 and 4)2. P1234 a!c describes the interference of these two pump-probe pathways. The double-slit picture has long been established for two-photon absorption and photo electron detection (Glauber, 2007). Equation (72) extends it to Raman processes. The resonant component of the stimulated CARS signal is given by P1234 a!c . For P(c) = 0 and when all frequencies are tuned off electronic reso­ nances, only one contribution remains in Equation (67). Furthermore, comparison of Equations (67) and (72) gives

1 S4; dis ¼  P1234 : 2 a!c

ð73Þ

Equation (73) relates the rate of resonant a ! c transitions to the resonant part of the CARS signal. The 1/2 factor can be easily rationalized: The sign comes from the fact that an !4 photon is absorbed, while the factor of a 1/2 signifies that only one of the interfering processes affects the number of photons in mode 4. It is amusing to note that the two processes contributing to the resonant coherent anti-Stokes Raman spectroscopy (CARS) signal are in fact a ! c Stokes processes!

2 One may be worried by the appearance of the factor of d 2D0 in those terms as the limit of narrowband shape is taken, but this is just an artifact resulting from the fact that these pump-probe processes are naturally described in terms of the rate of transitions while here we are studying the overall probability. Indeed, �2D 0 ðxÞ  �D 0 ðxÞ=D 0 and 1/D0 is proportional to the overall time where the pulses are turned on.

258

Shaul Mukamel and Saar Rahav

10. PURELY-DISSIPATIVE SPECTROSCOPIC SIGNALS At the end of Section 8 we had pointed out that it is possible to identify a linear combination of signals such that the parametric contribution is canceled out. The purely dissipative signals, which were defined in section 3.1, accomplish that goal. These signals are obtained by calculat­ ing the exchange of energy between the field and the material. Purely dissipative signals are generally given by Equation (37), which includes contributions from all possible material transitions. Such signals would be useful for spectroscopic applications once some pulse para­ meters are scanned. This can be done using pulse shaping techniques (Shim & Zanni, 2009; Tian et al., 2003; Weiner, 2009). We represent the field as Eð!Þ ¼ Að!Þei�ð!Þ , where A is the amplitude of the field while � denotes its phase. Both functions are real. Different transitions may be separated by comparing the response of D to variation of A(!) at different frequencies. We first consider the linear siganl D’ h

1

Z d!

X

!j�ba j2 A2 ð!Þ�ð!  !ba Þ:

ð74Þ

b

Variation of the field amplitude gives �ð!Þ ¼

X �D ¼ !ba j�ba j2 �ð!  !ba Þ; �A2 ð!Þ b

ð75Þ

which is the linear absorption. We now turn to Raman processes. We assume that the field is tuned off electronic resonances. The dissipative signal is then

DCARS ¼

1

X

4p2 h3

c

Z   Eð!ÞEð!ca  !Þ 2 : !ca j�cb j j�ba j  d! !  !ba þ i�  2

2

ð76Þ

(The optical pulse band shape covers the frequency regime j!j >> !ca , since !ca is a vibrational transition frequency.) Raman resonances may be obtained by taking a second-order var­ iation �2 D=�Að!1 Þ�Að!2 Þ. However, these lie on the top of a smooth background resulting from the term where each of the integrals in Equation (76) is varied once. A different approach, which only

Ultrafast Nonlinear Optical Signals

259

requires one variation is to consider a combination of a narrow band and a broad band pulse ~ Eð!Þ ’ 2pE 0 �ð!  !0 Þ þ 2pE 0 �ð! þ !0 Þ þ Eð!Þ:

ð77Þ

Below we show that variation of the E 20 part of D at frequencies near (but different from) !0 allows to separate out the different Raman resonances. The integral in Equation (76) in the RWA can now be approximated by Z d!

1 Eð!ÞEð!ca  !Þ ’ 2pE 0 Eð!ca  !0 Þ !0  !ba þ i� !  !ba þ i� 1  ; þ2pE 0 Eð!ca þ !0 Þ !0  !bc þ i�

ð78Þ

Variation with respect to the amplitude of the broadband pulse, A(!), gives    �DCARS 1 E 0 Eð!0  !ca Þ E 0 E  ð!0 þ !ca Þ 2 2 ’ 3 !ca j�cb j j�ba j R þ �Að!Þ !0  !ba  i� !0  !bc  i� ph   E 0 e  i� ð!0  !ca Þ  ½�ð!  !0 þ !ca Þ þ �ð! þ !0  !ca Þ !0  !ba þ i�  E  ei� ð!0 þ !ca Þ ð79Þ þ 0 ½�ð!  !0  !ca Þ þ �ð! þ !0 þ !ca Þ : !0  !bc þ i� Equation (79) shows sharp Raman peaks at –!0 –!ca . The above considerations illustrate that the variations of dissipative signals with incoming pulse parameters allow to distinguish different material processes. Spectroscopy with dissipative signals should be very convenient. Dissipative signals eliminate the parametric background but are not background-free since the signal is a difference of two large quantities (transmitted minus incoming field intensity). Chirped pulses (Malinovshy, 2009; Onorato et al., 2007; Pergoraro et al., 2009) as well as coherently shaped pulses in a collinear geometry (Caster et al., 2009; Li et al., 2008; Mu¨ller et al., 2009; Ogilvie et al., 2006; Roy et al., 2009; Silberberg, 2009; von Vacano and Motzkus, 2008) were found to be effective in suppressing the parametric signal. The dissipa­ tive signals, which can be easily implemented in a collinear geometry and make use of pulse shaping, present a different solution to the same problem.

260

Shaul Mukamel and Saar Rahav

11. SUMMARY In this review we have demonstrated that by examining the optical processes from the viewpoint of the matter we can recast the signals in terms of transition amplitudes which represent the molecular wave func­ tion and gain new insights. These ideas were demonstrated for two examples: pump-probe (Sections 5 and 6) and CARS (Sections 7 and 8) signals. However, the methods used here are quite general, and apply to other optical measurements, as explained in the following. Resonant and parametric processes can be clearly separated by recast­ ing the signal in terms of the transition amplitudes. The former take a generalized Kramers-Heisenberg form, namely a product of complex conjugated transition amplitudes, times a resonant d-function. When considering the rate of transitions between states of matter, one expects to find the amplitudes appearing as complete squares, such as in Equa­ tion (71), or sums over complete squares. However, the translation to contributions for optical signals breaks this form. Some of the terms may be omitted since they are not part of the signal under consideration, such as the first two terms in Equation (72), when considering the CARS signal. The terms which are kept should be weighted according to the overall change in the number of photons in the measured mode accord­ ing to both the ket and the bra. All of these considerations are highly intuitive and easy to follow, but imply that optical signals are not in general of a form of a modulus square of an amplitude. Parametric processes are linear in the transition amplitude, due to an interference between a high-order process and the zero-order process where the material does not interact with the field. For each n’th order process starting and ending at the same state there is also a time reversed process, where the transitions are “traversed backward.” The parametric contribution is proportional to the difference between the direct and timereversed processes, since a photon which is absorbed in one will be emitted in the other. Such processes were studied in Section 8 for the CARS signal but similar consideration should apply to any type of parametric process. The contributions of material processes to the optical signal can be understood by recasting it as a sum over terms which correspond to either resonant transitions or parametric processes. We argue that this may be done quite generally, using the following prescription: any loop diagram representing a signal would have retarded Green’s functions along the left branch and the advanced ones along the right branch, including the top of the loop. (See rules FD5 and FD7.) One can replace the loop diagram with a different one, where the interactions are shifted along the loop (keeping their relative ordering). This is done either by replacing the last retarded Green’s function by an advanced one, or by replacing the first advanced Green’s function by a retarded one. This

261

Ultrafast Nonlinear Optical Signals

ω3

ω1 c b a

ω3 ω1 ω2

c

=

a

a

c

ω3

b

ω2

a

+

c

b

a

a

ω1

ω2

Figure 17 Moving interactions along the loop by adding Kramers-Heisenberg terms. The diagrams are obtained from Figure 8 by (i) moving the second diagram from the right-hand side to the left hand side, and (ii) moving the last interaction from the right branch to the left branch. The second step must be compensated by a sign change due to rule FD9

replacement (Figure 17) must be compensated by an additional term, which is given by an unrestricted loop and has the generalized Kramers-Heisenberg form. Using this operation, any restricted loop diagram can be written as a linear combination of unrestricted ones plus a diagram where all the interactions are on the ket. The diagrams with interactions on the ket should then be divided into symmetric and antisymmetric parts, as was done in Figure 15. The antisymmetric parts are identified with various parametric contributions. The symmetric part can be rewrit­ ten as resonant terms using an optical theorem analogous to Equation (64). This last step is non-trivial, and needs to be studied further. Dissecting nonlinear optical signals into a sum of contributions from resonant and parametric processes enhances our understanding of opti­ cal signals and reveals which material processes contribute to each opti­ cal signal. In addition, it allows to identify combinations of signals where some material pathways are canceled out and others are enhanced, thus helping the design of new types of measurements.

ACKNOWLEDGMENTS The support of the National Science Foundation (Grant No. CHE­ 0745892) and the Chemical Sciences, Geosciences and Biosciences Divi­ sion, Office of Basic Energy Sciences, Office of Science, U.S. Department of Energy, is gratefully acknowledged.

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Index Absorption coefficient, 66

Absorption contrast, 35–37, 42, 43,

49, 51

Acousto-optically modulated laser, 64, 65

Adiabatic approximation, 95–96

Adiabatic hyperspherical approach,

94–96, 100, 103

Advanced stationary afterglow

apparatus (AISA), 106

Afterglow measurements, 83, 84, 103,

105–109, 111–113, 122

Afterglow plasma, 79, 83, 84, 114,

118, 121

Afterglow technique, 80–84, 106

Alignment, 138, 167

Alkali atoms, 27, 28, 36, 39, 40, 42, 52,

53, 55, 64, 65

Alkali density, 37, 44

Allan deviation, 31, 33–35, 151

Ambipolar diffusion, 82

Angular momentum, 7, 27, 48, 49, 90,

113, 115, 130, 138, 139, 161, 162

Annihilation operator, 231, 237, 238

Argon (Ar), 80, 82, 110, 111

ASTRID ring, 107

Asymptotic interaction, 12

Atomic absorption, 30, 37

Atomic beam measurements,

134–136

Atomic clock, 22, 23, 27–39, 42, 43,

51, 56, 59, 60, 64–67

Autoionization, 78, 86, 113, 120

Baryogenesis, 147, 148

Baryon asymmetry, 148

Baryon-number violation, 147, 148

Bichromatic light, 27

Bichromatic optical field, 25, 55, 64

Binding energy, 13, 166

Blind analysis, 152

Born–Oppenheimer principle, 92, 95

Breit–Pauli equation, 5, 15

Breit–Wigner cross sections, 102

Bromwich contour, 184, 200

Buffer gas, 24, 28, 29, 33, 36, 39, 40,

44, 45, 48–51, 56, 59, 62, 64,

118, 141

Cabibbo–Kobayashi–Maskawa (CKM)

matrix, 147, 148

Casimir effect, 1–16

Cesium, 135–137, 145, 153, 156, 157

Charge-conjugation/parity (CP), 130

Chemical flexibility, 80

Chip-scale atomic frequency, 34

Chirped pulses, 259

Circular polarization, 46, 51, 163

Clebsch–Gordon coefficients, 41

Closed-channel elimination, 91–92

Closed-time-path-loop (CTPL), 227

Coherent anti-Stokes Raman

spectroscopy (CARS),

227–229, 235–237, 249–257,

259–260

Coherent population trapping (CPT),

22–27, 64–65

Cold alkali atoms, 156–158

Cold-atom technique, 153, 157, 167

Collinear geometry, 259

Collisional ionization, 114, 115

Collisional radiative recombination,

80, 114, 115, 117

Collisional stabilization, 114

Collision-induced broadening, 37

Comagnetometer, 133–137, 149–152,

154, 157, 158, 161, 163,

166

Compton wavelength, 3, 8

Condensed-matter renormalization, 8

Correlation, 231, 238

Coulomb interaction, 6, 7, 15,

85, 138

Coulomb photon, 10

Coupling coefficient, 40

265

266

Index

Couplings, 93–94, 98, 100, 132,

140, 147

Coupling strength, function, 180

CPT excitation, 23, 32, 38, 39, 43,

46–49, 59

CPT maser, 54–55

CPT theorem, 146

Cross-relaxation, 154

CRYRING, 102, 107, 119–121

CTPL diagram, 228, 237, 238,

241, 250

Cyclotron frequency, 145

Damping, 142, 188, 209

Dark resonance, 65

Darwin interaction, 15

Decay chain, 161

Decoherence, 44, 142

Demerging magnet, 120, 121

Density matrix, 24, 229, 237

Depopulation pumping, 44–45

Derivative coupling, 92–94, 96, 103

Detuning, 24, 26, 32, 38, 60, 78

Deuteron (D), 153, 167

Diagonalization, 96, 100

Diamagnetic atom, 139–145, 153, 161

Differential detection, 62, 63

Diffusion-induced relaxation, 65

Dirac function, 5, 11, 184

Direct modulation, 27

Dispersion force, 3

Dispersion-theoretic scattering, 5

Dissipative signal, 227, 228, 236, 255,

257–259

Dissociation, 77, 80, 86, 87, 93, 98,

102, 111, 113, 122

Dissociative recombination (DR),

86–87, 77, 92–93, 104

Distributed feedback (DBF) laser, 54

Doppler broadening, 39

Double-slit interference, 228,

256–257

Edge-emitting lasers, 64

EDM measurement, 131, 132, 134,

144–153, 156, 162, 167

Effective field theory, 8, 10

Eigenvalue, 95, 231, 242

Electric dipole, 2, 3, 4, 14, 16,

26, 130

Electric (magnetic) polarizabilities, 13

Electromagnetic moment, 24

Electromagnetic scattering, 8

Electron bending, 101

Electron cooler, 85, 101

Electron density, 81–83, 85, 109, 111,

116, 117, 119, 138

Electronic noise, 59

Electro-optically modulated

source, 65

End-resonance, 48, 49, 67

Energy resolution, 79, 85, 101,

107, 120

Energy-sensitive barrier detector, 85

Energy transfer, 28

Exchange, 2, 4, 36, 44, 48, 62, 82,

98, 141, 142, 154, 164–167,

227, 257

Excited-state hyperfine structure,

40–42, 56

Exponential decay, 154, 179, 180, 187,

188, 190, 197, 217–220

Facility for rare isotope beams (FRIB),

157, 164, 166

False EDM signals, 151–153, 160

Faraday rotation, 150, 151

Far-off-resonance trap (FORT), 163

Feedback, 28, 37, 54, 143, 154

Feynman diagram, 8, 237, 238, 245

Filtering, 30, 46, 51, 52

Flavor-changing, 147

Flowing afterglow Langmuir probe,

81, 105

Fluorescence, 23–26, 30, 122, 157,

159, 160

FM–AM noise conversion, 31

Fountain clock, 28, 156, 158

Index

Four-wave mixing, 51, 52, 225,

226, 249

Franck–Condon factor, 99

Frequency domain (FD), 229, 237

Frequency splitting, 32, 48

Friction, 85

Gallium (Ga), 167

Gamma function, 199

Gamma-ray, 164

Gas-phase measurements, 154

Gradient, 24–26, 138, 141

Green’s function, 234, 241, 244,

254, 260

Ground-state relaxation, 39, 56, 58,

62, 65

267

Jacobi coordinates, 88

Jahn–Teller (JT) coupling, 94, 96–98,

100, 102, 103

Jahn–Teller theory, 120

Johnson noise, 151

Kinematic effect, 84

Kinematic equations, 85

Kramers–Heisenberg form, 225, 227,

234, 243, 244, 246, 252, 254,

259

Krypton (Kr), 136, 137

Lambda system, 40, 41, 44, 45, 48, 51,

53, 56

Lamb shift, 15, 135

Langmuir probe, 81, 82

Laplace transform, 183, 184, 207,

Harmonic oscillator, 188

208, 218

Helium (He), 8, 13, 80, 110, 113–115,

Larmor frequency, 27, 49, 142

117, 118, 135, 154, 164, 166

Laser cooling, 22, 156

Heterodyne detection, 226, 228

Laser noise, 37, 51, 61–63

Heterodyne four-wave mixing, 226

Laser polarization, 44

Heterodyne signal, 225, 228–231

Lattices, 146, 156

High-contrast resonance, 48, 51–52,

Lifetime, 82, 114, 121, 141,

65

154, 163

Homodyne four-wave mixing, 239

Light shifts, 23, 38–39, 56, 60, 61,

Hydrogen masers, 28

163, 212

Hyperfine coherence, 25

Light-shift suppression, 60–61 Hyperfine relaxation, 25, 36, 48, 49

Hyperfine splitting, 24, 27, 32, 40, 46, Linear polarization, 67

Liouville space, 225, 230

51, 54, 158

Lippmann–Schwinger equation, 92

Hyperspherical approximation, 95,

Liquid noble gases, 153–156 96, 100

l-mixing, 112–118, 120, 121

Longitudinal cooling, 158

Impedance, 60–62

Loop diagram, 8, 231, 237, 242–245,

Injection, 27, 32, 60, 64

250, 254, 260

Interference, 10, 24–26, 40, 42, 43, 45,

46, 48, 87, 91, 225, 226, 228,

Magnetic dipole, 13, 28, 166

235, 254, 256, 257, 260

Magnetic moment, 27, 40, 55, 131, 133,

Intrinsic coupling, 135

135, 141, 142, 144, 145,

Ion–electron collision, 77, 85, 86

154, 161, 167

Ion–electron system, 16

Magnetization vector, 142, 143

Ionization, 5, 76, 106, 114, 115, 117

Magnetometer, 134, 141, 151, 154–156

Ion-storage ring, 77, 79, 84–86, 107

268

Index

Magneto-optical trap (MOT), 157

Markovian limit, 177, 218

Maxwell distribution, 79

Maxwellian function, 85

Maxwell’s equations, 225

Mercury (Hg), 139, 143–144, 163

Merged-beam experiment, 78, 79, 106,

112, 113

Micro effect, 2–4 Microwave-averaged electron

density, 82

Misalignment, 32, 135, 155, 160

Modulation, 27, 32, 36, 45, 46, 50, 51,

54, 57, 60, 61, 64, 65, 67, 135,

145, 150

Modulation index, 60, 61

Molecular beam experiment, 136–139,

153

Monochromatic electromagnetic

field, 23

Multichannel Quantum Defect Theory

(MQDT), 91–94, 97, 99

Multiphoton resonances, 55

Muon orbit, 145

Octadecyltrichlorosilane (OTS), 166

Octupole deformation, 153, 162, 163

One- and two-dimensional theory, 91–94 Optical absorption, 36, 44, 50, 55, 65,

66, 107, 119

Optical-microwave double-resonance

(OMDR), 35, 38, 39, 48, 55, 59,

60, 62, 63, 66, 67

Optical pumping, 23–26, 28, 30, 39,

40, 42, 43, 45, 48, 59, 63, 67,

139, 141, 164–167

Optical signals, CTPL representation, 236–239 Optical theorem, 231–236, 254, 260

Optical transition, 23, 40, 42, 48, 50,

55, 64, 219, 249, 250

Orbit–orbit interaction, 14

Orientation, 27, 138, 161

Ortho–para transition, 90

Oscillating moment, 28

Oscillation, 24, 54

Paramagnetic atom, 135–137,

145, 157

Narrowing effect, 65–66 Paramagnetic molecule, 132, 146, 153,

Neon (Ne), 118

158–161

Neutral Kaon system, 130

Partial transition amplitude, 233, 243,

Neutron EDM, 132–134, 140, 141, 144,

244

146, 150

Perturbation, 7, 15, 16, 182, 198,

Neutron measurement, 134, 168

229, 231

Neutron spin-precession, 134

Phase effect, 137, 156, 160

Nitrogen (N), 83, 164

Phase-locked external cavity diode

Noise, 30, 31, 33, 35, 37, 45, 51, 54, 55,

laser, 64

59, 61–63, 121, 134, 139, 143,

Phase-locking, 27

150–152, 154, 155, 160

Phase matching, 244, 253

Noise cancellation, 61–64 Phase transition, 181, 203, 204

Nonrelativistic hydrogen atom, 9, 10

Photodetection, 59

Nonrelativistic molecule, 9–12 Photodetector, 32, 33, 51,

N-resonance, 55–57 52, 61, 62

Nuclear polarization, 163, 164

Photoelectron detection, 226, 257

Nuclear spin, 26, 27, 47, 48, 77, 90, 94, Photoemission, 91

98, 100, 136, 138, 158, 166

Photoionization, 80

Nuclear wave function, 98

Photomultiplier tube (PMT), 159

Index

Photon absorption, 226, 228,

240, 257

Photonic crystal, 219

Photon number, 229

Plasma afterglow experiment, 79

Plasma recombination, 108

Plasmas, 79, 108, 121, 122

P-odd and T-odd, 140, 145, 151, 157,

162, 166, 168

Polarizabilities, 5, 6, 15

Polarization potential, 6, 8, 10

Pollaczek polynomial, 5

Population trapping, 21, 22, 26,

30–34, 64

Potential curve, 86–88, 91, 92, 93, 95,

98, 100

Precision, 5, 22, 27, 109, 131, 139,

150–152, 168

Predissociation, 93, 102, 103, 115, 116,

120

Product branching, 121–122

Protactinium (Pa), 162

Pseudoscalar contribution, 135

Pulse shaping, 249, 257, 259

Pump–probe signal, 228, 229,

239–249, 252

Push–pull optical pumping, 67

Push–pull oscillator, 52–54

Q-factor, 35, 39, 59

Quadrupole, 14, 16, 136, 140, 145,

162, 166

Quantum defect, 91, 94, 96, 97

Quantum field, 8, 228–231

Quantum interference, 25, 26, 42

Quantum pathway, 226

Quark and antiquark

coupling, 147

Rabi frequency, 38, 40

Rabi oscillations, 180

Rabi splitting, 209

Radiative decay, 38, 82

Radium (Ra), 161, 163, 166

269

Raman–Ramsey pulsed CPT,

56–59

Raman–Ramsey scheme,

32, 58

Raman resonance, 38, 55,

256, 258

Raman spectroscopy, 239, 249

Ramsey effect, 65

Ramsey fringes, 31, 32, 57

Ramsey’s method, 30, 149,

160, 163

Rare atoms, 161–167

Rare-radioactive atom, 166

Reciprocal electron density,

82, 109

Recombination coefficient, 78, 79,

81–84, 103–112, 115, 117,

122, 123

Relativistic correction, 5, 14, 16

Relaxation processes, 36, 165

Renormalization, 8, 218

Repulsive interaction, 5–8

Resonance, 48–49, 50–52, 55–56,

65–66, 256–257

Resonator, 24, 30

Riemann–Lebesgue lemma, 186

Rotating frame, 142

Rotating wave approximation (RWA),

177, 179, 217, 220, 229, 237, 238,

240, 250, 258

Rubidium (Rb), 27, 29, 30, 41,

44, 49, 141, 142, 157, 164,

165

Rydberg levels, 86, 91, 102

Saha equilibrium, 115

Saturation, 50

Scalar coupling, 136, 156, 157

Scattering, 6, 13, 15, 16, 23, 92,

133–135

Schiff moment, 137–141, 144, 153, 161,

162, 164, 166, 167

Schrödinger representation, 220

Screening, 154

270

Index

Semiconductor, 32

Separated oscillatory

technique, 135

Signal-to-noise ratio, 30, 35, 37, 121,

139, 155, 160

Single-pass merged-beam, 84–86, 103,

106, 107

Spatial distribution, 80, 83

Spectral density, 37, 217

Spectral function, 180, 182,

186–190, 193

Spectral weight function, 179–181, 183,

184, 186–193, 195, 196,

198–206, 208–218

Spectrometric sampling, 82

Spectroscopic measurement, 107

Spectroscopic technique, 224

Spectroscopy, 22, 64, 112, 121, 224,

227, 259

Spectrum, 205–217

Spin, 4, 8, 26, 27, 36, 37, 44, 47, 48,

62, 77, 91, 92, 94, 98, 100, 108,

134–136, 138–143, 145, 154,

158, 161, 164–167

Spin-destruction, 36

Spin-exchange, 36, 44, 48, 62, 141, 142,

154, 164–167

Spin–orbit coupling, 92

Spin-precession frequency, 145

Spin projection noise, 37

SQUID, 134, 152, 154, 155

Stability, 30, 31, 33–38, 56, 58, 61,

66, 67

Stark shift, 38, 39

Stationary scattering

theory, 232

Stimulated Raman scattering (SRS),

242, 246–248

Storage ring, 84–86, 99, 101,

106–108, 111, 112, 119–123,

153, 167

Symmetry, 11, 87, 88, 90, 96–98, 100,

132, 147

Synthesizer frequency, 61

Tensor electric dipole polarizability, 11

Thallium, 132, 136–139, 145, 148, 156,

158, 168

Thermal de Broglie wavelength, 115

Thermal noise, 54–55

Third-body stabilized recombination,

113–119

Thorium-oxide (ThO), 160

Threshold, 64

Time domain femtosecond, 249

Toroidal correction, 86, 101

Toroidal effect, 120

Transition amplitude, 26, 41, 176,

225–228, 231–236, 242–249,

252–254, 256, 259, 260

Transition frequency, 23, 26, 28,

49, 53, 176, 177, 180, 182,

187, 189–191, 193–195,

198, 201, 202, 212, 214,

218, 234, 258

Transverse cooling, 158

Trapping, 26, 156, 164

Trap state, 43–45

Tungsten-carbide (WC), 160

Two-dimensional cross section, 94

Two-photon-absorption, 241,

244–249 Ultra-cold atom scattering, 13

Ultracold neutrons (UCN), 131, 133

Ultra-short excitation, 219

Ultraviolet photons, 80

Unitarity, 147, 175, 179,

205–217, 233

Unrestricted loop diagram, 243, 244,

246–248, 254

Vacuum dressed atom, 2, 9, 10, 16

Vacuum dressed slow electron, 13

van der Waals force, 3, 12

Vapor polarization, measurement, 150

Vector, 142, 143, 177, 205

Vertical-cavity surface-emitting laser

(VCSEL), 63–64

Index

Vibrational energies, 95, 96, 100, 103

Vibrational excitation, 102, 105,

106, 112

Wall collision, 45, 49

Weisskopf–Wigner approximation,

175, 176, 178–180, 186, 187,

209, 217

271

White-phase noise, 151

Xenon, 134, 136, 137, 139, 140–143,

152–155, 164, 166

Zero-light-shift, 61, 62

Zero-point fluctuation, 9

Zitterbewegung, 8, 9

CONTENTS OF VOLUMES IN

THIS SERIAL

Volume 1 Molecular Orbital Theory of the Spin Properties of Conjugated Molecules, G.G. Hall and A.T. Amos Electron Affinities of Atoms and

Molecules, B.L. Moiseiwitsch

Atomic Rearrangement Collisions,

B.H. Bransden The Production of Rotational and Vibrational Transitions in Encounters between Molecules, K. Takayanagi The Study of Intermolecular Potentials with Molecular Beams at Thermal Energies, H. Pauly and J.P. Toennies High-Intensity and High-Energy Molecular Beams, J.B. Anderson, R.P. Anders and J.B. Fen Volume 2 The Calculation of van der Waals Interactions, A. Dalgarno and W.D. Davison Thermal Diffusion in Gases, E.A. Mason, R.J. Munn and Francis J. Smith Spectroscopy in the Vacuum Ultraviolet, W.R.S. Garton The Measurement of the Photoionization Cross Sections of the Atomic Gases, James A.R. Samson The Theory of Electron–Atom Collisions, R. Peterkop and V. Veldre Experimental Studies of Excitation in Collisions between Atomic and Ionic Systems, F.J. de Heer Mass Spectrometry of Free Radicals, S.N. Foner

Volume 3 The Quantal Calculation of Photoionization Cross Sections, A.L. Stewart Radiofrequency Spectroscopy of Stored Ions I: Storage, H.G. Dehmelt Optical Pumping Methods in Atomic Spectroscopy, B. Budick Energy Transfer in Organic Molecular Crystals: A Survey of Experiments, H.C. Wolf Atomic and Molecular Scattering from Solid Surfaces, Robert E. Stickney Quantum, Mechanics in Gas CrystalSurface van der Waals Scattering, E. Chanoch Beder Reactive Collisions between Gas and Surface Atoms, Henry Wise and Bernard J. Wood Volume 4 H.S.W. Massey—A Sixtieth Birthday Tribute, E.H.S. Burhop Electronic Eigenenergies of the Hydrogen Molecular Ion, D.R. Bates and R.H. G. Reid Applications of Quantum Theory to the Viscosity of Dilute Gases, R.A. Buckingham and E. Gal Positrons and Positronium in Gases, P.A. Fraser Classical Theory of Atomic Scattering, A. Burgess and I.C. Percival Born Expansions, A.R. Holt and B.L. Moiseiwitsch Resonances in Electron Scattering by Atoms and Molecules, P.G. Burke

273

274

Contents of Volumes in this Serial

Relativistic Inner Shell Ionizations, C.B.O. Mohr Recent Measurements on Charge Transfer, J.B. Hasted Measurements of Electron Excitation Functions, D.W.O. Heddle and R.G. W. Keesing Some New Experimental Methods in Collision Physics, R.F. Stebbings Atomic Collision Processes in Gaseous Nebulae, M.J. Seaton Collisions in the Ionosphere, A. Dalgarno The Direct Study of Ionization in Space, R.L.F. Boyd Volume 5 Flowing Afterglow Measurements of IonNeutral Reactions, E.E. Ferguson, F.C. Fehsenfeld and A.L. Schmeltekopf Experiments with Merging Beams, Roy H. Neynaber Radiofrequency Spectroscopy of Stored Ions II: Spectroscopy, H.G. Dehmelt The Spectra of Molecular Solids, O. Schnepp The Meaning of Collision Broadening of Spectral Lines: The Classical Oscillator Analog, A. Ben-Reuven The Calculation of Atomic Transition Probabilities, R.J.S. Crossley Tables of One- and Two-Particle Coefficients of Fractional Parentage for Configurations sλstu pq, C.D. H. Chisholm, A. Dalgarno and F.R. Innes Relativistic Z-Dependent Corrections to Atomic Energy Levels, Holly Thomis Doyle Volume 6 Dissociative Recombination, J.N. Bardsley and M.A. Biondi Analysis of the Velocity Field in Plasmas from the Doppler Broadening of Spectral Emission Lines, A.S. Kaufman The Rotational Excitation of Molecules by Slow Electrons, Kazuo Takayanagi and Yukikazu Itikawa The Diffusion of Atoms and Molecules, E.A. Mason and T.R. Marrero Theory and Application of Sturmian Functions, Manuel Rotenberg

Use of Classical Mechanics in the Treatment of Collisions between Massive Systems, D.R. Bates and A.E. Kingston Volume 7 Physics of the Hydrogen Maser, C. Audoin, J.P. Schermann and P. Grivet Molecular Wave Functions: Calculations and Use in Atomic and Molecular Process, J.C. Browne Localized Molecular Orbitals, Harel Weinstein, Ruben Pauncz and Maurice Cohen General Theory of Spin-Coupled Wave Functions for Atoms and Molecules, J. Gerratt Diabatic States of Molecules—Quasi Stationary Electronic States, Thomas F. O’Malley Selection Rules within Atomic Shells, B.R. Judd Green’s Function Technique in Atomic and Molecular Physics, Gy. Csanak, H.S. Taylor and Robert Yaris A Review of Pseudo-Potentials with Emphasis on Their Application to Liquid Metals, Nathan Wiser and A.J. Greenfield Volume 8 Interstellar Molecules: Their Formation and Destruction, D. McNally Monte Carlo Trajectory Calculations of Atomic and Molecular Excitation in Thermal Systems, James C. Keck Nonrelativistic Off-Shell Two-Body Coulomb Amplitudes, Joseph C.Y. Chen and Augustine C. Chen Photoionization with Molecular Beams, R.B. Cairns, Halstead Harrison and R.I. Schoen The Auger Effect, E.H.S. Burhop and W.N. Asaad Volume 9 Correlation in Excited States of Atoms, A.W. Weiss The Calculation of Electron–Atom Excitation Cross Section, M.R.H. Rudge

Contents of Volumes in this Serial

Collision-Induced Transitions between Rotational Levels, Takeshi Oka The Differential Cross Section of Low-Energy Electron–Atom Collisions, D. Andrick Molecular Beam Electric Resonance Spectroscopy, Jens C. Zorn and Thomas C. English Atomic and Molecular Processes in the Martian Atmosphere, Michael B. McElroy Volume 10 Relativistic Effects in the Many-Electron Atom, Lloyd Armstrong Jr. and Serge Feneuille The First Born Approximation, K.L. Bell and A.E. Kingston Photoelectron Spectroscopy, W.C. Price Dye Lasers in Atomic Spectroscopy, W. Lange, J. Luther and A. Steudel Recent Progress in the Classification of the Spectra of Highly Ionized Atoms, B.C. Fawcett A Review of Jovian Ionospheric Chemistry, Wesley T. Huntress Jr. Volume 11 The Theory of Collisions between Charged Particles and Highly Excited Atoms, I.C. Percival and D. Richards Electron Impact Excitation of Positive Ions, M.J. Seaton The R-Matrix Theory of Atomic Process, P.G. Burke and W.D. Robb Role of Energy in Reactive Molecular Scattering: An Information-Theoretic Approach, R.B. Bernstein and R.D. Levine Inner Shell Ionization by Incident Nuclei, Johannes M. Hansteen Stark Broadening, Hans R. Griem Chemiluminescence in Gases, M.F. Golde and B.A. Thrush Volume 12 Nonadiabatic Transitions between Ionic and Covalent States, R.K. Janev

275

Recent Progress in the Theory of Atomic Isotope Shift, J. Bauche and R.-J. Champeau Topics on Multiphoton Processes in Atoms, P. Lambropoulos Optical Pumping of Molecules, M. Broyer, G. Goudedard, J.C. Lehmann and J. Vigué Highly Ionized Ions, Ivan A. Sellin Time-of-Flight Scattering Spectroscopy, Wilhelm Raith Ion Chemistry in the D Region, George C. Reid Volume 13 Atomic and Molecular Polarizabilities— Review of Recent Advances, Thomas M. Miller and Benjamin Bederson Study of Collisions by Laser Spectroscopy, Paul R. Berman Collision Experiments with Laser-Excited Atoms in Crossed Beams, I.V. Hertel and W. Stoll Scattering Studies of Rotational and Vibrational Excitation of Molecules, Manfred Faubel and J. Peter Toennies Low-Energy Electron Scattering by Complex Atoms: Theory and Calculations, R.K. Nesbet Microwave Transitions of Interstellar Atoms and Molecules, W.B. Somerville Volume 14 Resonances in Electron Atom and Molecule Scattering, D.E. Golden The Accurate Calculation of Atomic Properties by Numerical Methods, Brain C. Webster, Michael J. Jamieson and Ronald F. Stewart (e, 2e) Collisions, Erich Weigold and Ian E. McCarthy Forbidden Transitions in One- and TwoElectron Atoms, Richard Marrus and Peter J. Mohr Semiclassical Effects in Heavy-Particle Collisions, M.S. Child Atomic Physics Tests of the Basic Concepts in Quantum Mechanics, Francies M. Pipkin

276

Contents of Volumes in this Serial

Quasi-Molecular Interference Effects in Ion–Atom Collisions, S.V. Bobashev Rydberg Atoms, S.A. Edelstein and T.F. Gallagher UV and X-Ray Spectroscopy in Astrophysics, A.K. Dupree

Volume 15 Negative Ions, H.S.W. Massey Atomic Physics from Atmospheric and Astrophysical, A. Dalgarno Collisions of Highly Excited Atoms, R.F. Stebbings Theoretical Aspects of Positron Collisions in Gases, J.W. Humberston Experimental Aspects of Positron Collisions in Gases, T.C. Griffith Reactive Scattering: Recent Advances in Theory and Experiment, Richard B. Bernstein Ion–Atom Charge Transfer Collisions at Low Energies, J.B. Hasted Aspects of Recombination, D.R. Bates The Theory of Fast Heavy Particle Collisions, B.H. Bransden Atomic Collision Processes in Controlled Thermonuclear Fusion Research, H.B. Gilbody Inner-Shell Ionization, E.H.S. Burhop Excitation of Atoms by Electron Impact, D.W.O. Heddle Coherence and Correlation in Atomic Collisions, H. Kleinpoppen Theory of Low Energy Electron–Molecule Collisions, P.O. Burke

Volume 16 Atomic Hartree–Fock Theory, M. Cohen and R.P. McEachran Experiments and Model Calculations to Determine Interatomic Potentials, R. Du ¨ ren Sources of Polarized Electrons, R.J. Celotta and D.T. Pierce Theory of Atomic Processes in Strong Resonant Electromagnetic Fields, S. Swain

Spectroscopy of Laser-Produced Plasmas, M.H. Key and R.J. Hutcheon Relativistic Effects in Atomic Collisions Theory, B.L. Moiseiwitsch Parity Nonconservation in Atoms: Status of Theory and Experiment, E.N. Fortson and L. Wilets

Volume 17 Collective Effects in Photoionization of Atoms, M.Ya. Amusia Nonadiabatic Charge Transfer, D.S. F. Crothers Atomic Rydberg States, Serge Feneuille and Pierre Jacquinot Superfluorescence, M.F.H. Schuurmans, Q.H.F. Vrehen, D. Polder and H.M. Gibbs Applications of Resonance Ionization Spectroscopy in Atomic and Molecular Physics, M.G. Payne, C.H. Chen, G.S. Hurst and G.W. Foltz Inner-Shell Vacancy Production in Ion– Atom Collisions, C.D. Lin and Patrick Richard Atomic Processes in the Sun, P.L. Dufton and A.E. Kingston

Volume 18 Theory of Electron–Atom Scattering in a Radiation Field, Leonard Rosenberg Positron–Gas Scattering Experiments, Talbert S. Stein and Walter E. Kaupplia Nonresonant Multiphoton Ionization of Atoms, J. Morellec, D. Normand and G. Petite Classical and Semiclassical Methods in Inelastic Heavy-Particle Collisions, A.S. Dickinson and D. Richards Recent Computational Developments in the Use of Complex Scaling in Resonance Phenomena, B.R. Junker Direct Excitation in Atomic Collisions: Studies of Quasi-One-Electron Systems, N. Andersen and S.E. Nielsen Model Potentials in Atomic Structure, A. Hibbert

Contents of Volumes in this Serial

Recent Developments in the Theory of Electron Scattering by Highly Polar Molecules, D.W. Norcross and L.A. Collins Quantum Electrodynamic Effects in FewElectron Atomic Systems, G.W.F. Drake Volume 19 Electron Capture in Collisions of Hydrogen Atoms with Fully Stripped Ions, B.H. Bransden and R.K. Janev Interactions of Simple Ion Atom Systems, J.T. Park High-Resolution Spectroscopy of Stored Ions, D.J. Wineland, Wayne M. Itano and R.S. Van Dyck Jr. Spin-Dependent Phenomena in Inelastic Electron–Atom Collisions, K. Blum and H. Kleinpoppen The Reduced Potential Curve Method for Diatomic Molecules and Its Applications, F. Jenč The Vibrational Excitation of Molecules by Electron Impact, D.G. Thompson Vibrational and Rotational Excitation in Molecular Collisions, Manfred Faubel Spin Polarization of Atomic and Molecular Photoelectrons, N.A. Cherepkov Volume 20 Ion–Ion Recombination in an Ambient Gas, D.R. Bates Atomic Charges within Molecules, G.G. Hall Experimental Studies on Cluster Ions, T.D. Mark and A.W. Castleman Jr. Nuclear Reaction Effects on Atomic Inner-Shell Ionization, W.E. Meyerhof and J.-F. Chemin Numerical Calculations on Electron-Impact Ionization, Christopher Bottcher Electron and Ion Mobilities, Gordon R. Freeman and David A. Armstrong On the Problem of Extreme UV and X-Ray Lasers, I.I. Sobel’man and A.V. Vinogradov Radiative Properties of Rydberg States in Resonant Cavities, S. Haroche and J.M. Raimond

277

Rydberg Atoms: High-Resolution Spectroscopy and Radiation Interaction— Rydberg Molecules, J.A.C. Gallas, G. Leuchs, H. Walther, and H. Figger Volume 21 Subnatural Linewidths in Atomic Spectroscopy, Dennis P. O’Brien, Pierre Meystre and Herbert Walther Molecular Applications of Quantum Defect Theory, Chris H. Greene and Ch. Jungen Theory of Dielectronic Recombination, Yukap Hahn Recent Developments in Semiclassical Floquet Theories for Intense-Field Multiphoton Processes, Shih-I Chu Scattering in Strong Magnetic Fields, M.R. C. McDowell and M. Zarcone Pressure Ionization, Resonances and the Continuity of Bound and Free States, R.M. More Volume 22 Positronium—Its Formation and Interaction with Simple Systems, J.W. Humberston Experimental Aspects of Positron and Positronium Physics, T.C. Griffith Doubly Excited States, Including New Classification Schemes, C.D. Lin Measurements of Charge Transfer and Ionization in Collisions Involving Hydrogen Atoms, H.B. Gilbody Electron Ion and Ion–Ion Collisions with Intersecting Beams, K. Dolder and B. Peart Electron Capture by Simple Ions, Edward Pollack and Yukap Hahn Relativistic Heavy-Ion–Atom Collisions, R. Anholt and Harvey Gould Continued-Fraction Methods in Atomic Physics, S. Swain Volume 23 Vacuum Ultraviolet Laser Spectroscopy of Small Molecules, C.R. Vidal Foundations of the Relativistic Theory of Atomic and Molecular Structure, Ian P. Grant and Harry M. Quiney

278

Contents of Volumes in this Serial

Point-Charge Models for Molecules Derived from Least-Squares Fitting of the Electric Potential, D.E. Williams and Ji-Min Yan Transition Arrays in the Spectra of Ionized Atoms, J. Bauche, C. Bauche-Arnoult and M. Klapisch Photoionization and Collisional Ionization of Excited Atoms Using Synchrotron and Laser Radiation, F.J. Wuilleumier, D.L. Ederer and J.L. Picqué Volume 24 The Selected Ion Flow Tube (SIDT): Studies of Ion-Neutral Reactions, D. Smith and N.G. Adams Near-Threshold Electron–Molecule Scattering, Michael A. Morrison Angular Correlation in Multiphoton Ionization of Atoms, S.J. Smith and G. Leuchs Optical Pumping and Spin Exchange in Gas Cells, R.J. Knize, Z. Wu and W. Happer Correlations in Electron–Atom Scattering, A. Crowe Volume 25 Alexander Dalgarno: Life and Personality, David R. Bates and George A. Victor Alexander Dalgarno: Contributions to Atomic and Molecular Physics, Neal Lane Alexander Dalgarno: Contributions to Aeronomy, Michael B. McElroy Alexander Dalgarno: Contributions to Astrophysics, David A. Williams Dipole Polarizability Measurements, Thomas M. Miller and Benjamin

Bederson

Flow Tube Studies of Ion–Molecule Reactions, Eldon Ferguson Differential Scattering in He–He and Heþ–He Collisions at keV Energies, R.F. Stebbings Atomic Excitation in Dense Plasmas, Jon C. Weisheit Pressure Broadening and Laser-Induced Spectral Line Shapes, Kenneth M. Sando and Shih-I. Chu

Model-Potential Methods, C. Laughlin and G.A. Victor Z-Expansion Methods, M. Cohen Schwinger Variational Methods, Deborah Kay Watson Fine-Structure Transitions in Proton–Ion Collisions, R.H.G. Reid Electron Impact Excitation, R.J.W. Henry and A.E. Kingston Recent Advances in the Numerical Calculation of Ionization Amplitudes, Christopher Bottcher The Numerical Solution of the Equations of Molecular Scattering, A.C. Allison High Energy Charge Transfer, B.H. Bransden and D.P. Dewangan Relativistic Random-Phase Approximation, W.R. Johnson Relativistic Sturmian and Finite Basis Set Methods in Atomic Physics, G.W. F. Drake and S.P. Goldman Dissociation Dynamics of Polyatomic Molecules, T. Uzer Photodissociation Processes in Diatomic Molecules of Astrophysical Interest, Kate P. Kirby and Ewine F. van Dishoeck The Abundances and Excitation of Interstellar Molecules, John H. Black Volume 26 Comparisons of Positrons and Electron Scattering by Gases, Walter E. Kauppila and Talbert S. Stein Electron Capture at Relativistic Energies, B.L. Moiseiwitsch The Low-Energy, Heavy Particle Collisions—A Close-Coupling Treatment, Mineo Kimura and Neal F. Lane Vibronic Phenomena in Collisions of Atomic and Molecular Species, V. Sidis Associative Ionization: Experiments, Potentials and Dynamics, John Weiner Françoise Masnou-Seeuws and Annick Giusti-Suzor On the β Decay of [loc=pre] 187Re: An Interface of Atomic and Nuclear Physics and Cosmochronology, Zonghau Chen, Leonard Rosenberg and Larry Spruch

Contents of Volumes in this Serial

279

Progress in Low Pressure Mercury-Rare Gas Discharge Research, J. Maya and R. Lagushenko

Multiple Electron Excitation, Ionization, and Transfer in High-Velocity Atomic and Molecular Collisions, J.H. McGuire

Volume 27 Negative Ions: Structure and Spectra, David R. Bates Electron Polarization Phenomena in Electron–Atom Collisions, Joachim Kessler Electron–Atom Scattering, I.E. McCarthy and E. Weigold Electron–Atom Ionization, I.E. McCarthy and E. Weigold Role of Autoionizing States in Multiphoton Ionization of Complex Atoms, V.I. Lengyel and M.I. Haysak Multiphoton Ionization of Atomic Hydrogen Using Perturbation Theory, E. Karule

Volume 30 Differential Cross Sections for Excitation of Helium Atoms and Helium-Like Ions by Electron Impact, Shinobu Nakazaki Cross-Section Measurements for Electron Impact on Excited Atomic Species, S. Trajmar and J.C. Nickel The Dissociative Ionization of Simple Molecules by Fast Ions, Colin J. Latimer Theory of Collisions between Laser Cooled Atoms, P.S. Julienne, A.M. Smith and K. Burnett Light-Induced Drift, E.R. Eliel Continuum Distorted Wave Methods in Ion–Atom Collisions, Derrick S.F. Crothers and Louis J. Dube

Volume 28 The Theory of Fast Ion–Atom Collisions, J.S. Briggs and J.H. Macek Some Recent Developments in the Fundamental Theory of Light, Peter W. Milonni and Surendra Singh Squeezed States of the Radiation Field, Khalid Zaheer and M. Suhail Zubairy Cavity Quantum Electrodynamics, E.A. Hinds

Volume 31 Energies and Asymptotic Analysis for Helium Rydberg States, G.W.F. Drake Spectroscopy of Trapped Ions, R.C. Thompson Phase Transitions of Stored Laser-Cooled Ions, H. Walther Selection of Electronic States in Atomic Beams with Lasers, Jacques Baudon, Rudalf Du ¨ lren and Jacques Robert Atomic Physics and Non-Maxwellian Plasmas, Mich
ele Lamoureux

Volume 29 Studies of Electron Excitation of Rare-Gas Atoms into and out of Metastable Levels Using Optical and Laser Techniques, Chun C. Lin and L.W. Anderson Cross Sections for Direct Multiphoton Ionization of Atoms, M.V. Ammosov, N.B. Delone, M.Ya. Ivanov, I.I. Bandar and A.V. Masalov Collision-Induced Coherences in Optical Physics, G.S. Agarwal Muon-Catalyzed Fusion, Johann Rafelski and Helga E. Rafelski Cooperative Effects in Atomic Physics, J.P. Connerade

Volume 32 Photoionization of Atomic Oxygen and Atomic Nitrogen, K.L. Bell and A.E. Kingston Positronium Formation by Positron Impact on Atoms at Intermediate Energies, B.H. Bransden and C.J. Noble Electron–Atom Scattering Theory and Calculations, P.G. Burke Terrestrial and Extraterrestrial Hþ 3;

Alexander Dalgarno

Indirect Ionization of Positive Atomic Ions, K. Dolder

280

Contents of Volumes in this Serial

Quantum Defect Theory and Analysis of High-Precision Helium Term Energies, G.W.F. Drake Electron–Ion and Ion–Ion Recombination Processes, M.R. Flannery Studies of State-Selective Electron Capture in Atomic Hydrogen by Translational Energy Spectroscopy, H.B. Gilbody Relativistic Electronic Structure of Atoms and Molecules, I.P. Grant The Chemistry of Stellar Environments, D.A. Howe, J.M.C. Rawlings and D.A. Williams Positron and Positronium Scattering at Low Energies, J.W. Humberston How Perfect are Complete Atomic Collision Experiments?, H. Kleinpoppen and H. Handy Adiabatic Expansions and Nonadiabatic Effects, R. McCarroll and D.S.F. Crothers Electron Capture to the Continuum, B.L. Moiseiwitsch How Opaque Is a Star?, M.T. Seaton Studies of Electron Attachment at Thermal Energies Using the Flowing Afterglow– Langmuir Technique, David Smith and Patrik Španěl Exact and Approximate Rate Equations in Atom–Field Interactions, S. Swain Atoms in Cavities and Traps, H. Walther Some Recent Advances in Electron-Impact Excitation of n = 3 States of Atomic Hydrogen and Helium, J.F. Williams and J.B. Wang Volume 33 Principles and Methods for Measurement of Electron Impact Excitation Cross Sections for Atoms and Molecules by Optical Techniques, A.R. Filippelli, Chun C. Lin, L.W. Andersen and J.W. McConkey Benchmark Measurements of Cross Sections for Electron Collisions: Analysis of Scattered Electrons, S. Trajmar and J.W. McConkey Benchmark Measurements of Cross Sections for Electron Collisions: Electron Swarm Methods, R.W. Crompton

Some Benchmark Measurements of Cross Sections for Collisions of Simple Heavy Particles, H.B. Gilbody The Role of Theory in the Evaluation and Interpretation of Cross-Section Data, Barry I. Schneider Analytic Representation of Cross-Section Data, Mitio Inokuti, Mineo Kimura, M.A. Dillon, Isao Shimamura Electron Collisions with N2, O2 and O: What We Do and Do Not Know, Yukikazu Itikawa Need for Cross Sections in Fusion Plasma Research, Hugh P. Summers Need for Cross Sections in Plasma Chemistry, M. Capitelli, R. Celiberto and M. Cacciatore Guide for Users of Data Resources, Jean W. Gallagher Guide to Bibliographies, Books, Reviews and Compendia of Data on Atomic Collisions, E.W. McDaniel and E.J. Mansky Volume 34 Atom Interferometry, C.S. Adams, O. Carnal and J. Mlynek Optical Tests of Quantum Mechanics, R.Y. Chiao, P.G. Kwiat and A.M. Steinberg Classical and Quantum Chaos in Atomic Systems, Dominique Delande and Andreas Buchleitner Measurements of Collisions between LaserCooled Atoms, Thad Walker and Paul Feng The Measurement and Analysis of Electric Fields in Glow Discharge Plasmas, J.E. Lawler and D.A. Doughty Polarization and Orientation Phenomena in Photoionization of Molecules, N.A. Cherepkov Role of Two-Center Electron–Electron Interaction in Projectile Electron Excitation and Loss, E.C. Montenegro, W.E. Meyerhof and J.H. McGuire Indirect Processes in Electron Impact Ionization of Positive Ions, D.L. Moores and K.J. Reed Dissociative Recombination: Crossing and Tunneling Modes, David R. Bates

Contents of Volumes in this Serial

Volume 35 Laser Manipulation of Atoms, K. Sengstock and W. Ertmer Advances in Ultracold Collisions: Experiment and Theory, J. Weiner Ionization Dynamics in Strong Laser Fields, L.F. DiMauro and P. Agostini Infrared Spectroscopy of Size Selected Molecular Clusters, U. Buck Fermosecond Spectroscopy of Molecules and Clusters, T. Baumer and G. Gerber Calculation of Electron Scattering on Hydrogenic Targets, I. Bray and A.T. Stelbovics Relativistic Calculations of Transition Amplitudes in the Helium Isoelectronic Sequence, W.R. Johnson, D.R. Plante and J. Sapirstein Rotational Energy Transfer in Small Polyatomic Molecules, H.O. Everitt and F.C. De Lucia Volume 36 Complete Experiments in Electron–Atom Collisions, Nils Overgaard Andersen and Klaus Bartschat Stimulated Rayleigh Resonances and Recoil-Induced Effects, J.-Y. Courtois and G. Grynberg Precision Laser Spectroscopy Using Acousto-Optic Modulators, W.A. van Wijngaarden Highly Parallel Computational Techniques for Electron–Molecule Collisions, Carl Winstead and Vincent McKoy Quantum Field Theory of Atoms and Photons, Maciej Lewenstein and Li You Volume 37 Evanescent Light-Wave Atom Mirrors, Resonators, Waveguides, and Traps, Jonathan P. Dowling and Julio Gea-Banacloche Optical Lattices, P.S. Jessen and I.H. Deutsch Channeling Heavy Ions through Crystalline Lattices, Herbert F. Krause and Sheldon Datz

281

Evaporative Cooling of Trapped Atoms, Wolfgang Ketterle and N.J. van Druten Nonclassical States of Motion in Ion Traps, J.I. Cirac, A.S. Parkins, R. Blatt and P. Zoller The Physics of Highly-Charged Heavy Ions Revealed by Storage/Cooler Rings, P.H. Mokler and Th. Stöhlker Volume 38 Electronic Wavepackets, Robert R. Jones and L.D. Noordam Chiral Effects in Electron Scattering by Molecules, K. Blum and D.G. Thompson Optical and Magneto-Optical Spectroscopy of Point Defects in Condensed Helium, Serguei I. Kanorsky and Antoine Weis Rydberg Ionization: From Field to Photon, G.M. Lankhuijzen and L.D. Noordam Studies of Negative Ions in Storage Rings, L.H. Andersen, T. Andersen and P. Hvelplund Single-Molecule Spectroscopy and Quantum Optics in Solids, W.E. Moerner, R.M. Dickson and D.J. Norris Volume 39 Author and Subject Cumulative Index Volumes 1–38 Author Index Subject Index Appendix: Tables of Contents of Volumes 1–38 and Supplements Volume 40 Electric Dipole Moments of Leptons, Eugene D. Commins High-Precision Calculations for the Ground and Excited States of the Lithium Atom, Frederick W. King Storage Ring Laser Spectroscopy, Thomas U. Ku ¨ hl Laser Cooling of Solids, Carl E. Mangan and Timothy R. Gosnell Optical Pattern Formation, L.A. Lugiato, M. Brambilla and A. Gatti

282

Contents of Volumes in this Serial

Volume 41 Two-Photon Entanglement and Quantum Reality, Yanhua Shih Quantum Chaos with Cold Atoms, Mark G. Raizen Study of the Spatial and Temporal Coherence of High-Order Harmonics, Pascal Salières, Anne L’Huillier, Philippe Antoine and Maciej Lewenstein Atom Optics in Quantized Light Fields, Matthias Freyburger, Alois M. Herkommer, Daniel S. Kra ¨ hmer, Erwin Mayr and Wolfgang P. Schleich Atom Waveguides, Victor I. Balykin Atomic Matter Wave Amplification by Optical Pumping, Ulf Janicke and Martin Wikens Volume 42 Fundamental Tests of Quantum Mechanics, Edward S. Fry and Thomas Walther Wave-Particle Duality in an Atom Interferometer, Stephan Du ¨ rr and Gerhard Rempe Atom Holography, Fujio Shimizu Optical Dipole Traps for Neutral Atoms, Rudolf Grimm, Matthias Weidemu¨ller and Yurii B. Ovchinnikov Formation of Cold (T £ 1 K) Molecules, J.T. Bahns, P.L. Gould and W.C. Stwalley High-Intensity Laser-Atom Physics, C.J. Joachain, M. Dorr and N.J. Kylstra Coherent Control of Atomic, Molecular and Electronic Processes, Moshe Shapiro and Paul Brumer Resonant Nonlinear Optics in Phase Coherent Media, M.D. Lukin, P. Hemmer and M.O. Scully The Characterization of Liquid and Solid Surfaces with Metastable Helium Atoms, H. Morgner Quantum Communication with Entangled Photons, Herald Weinfurter Volume 43 Plasma Processing of Materials and Atomic, Molecular, and Optical Physics: An Introduction, Hiroshi Tanaka and Mitio Inokuti

The Boltzmann Equation and Transport Coefficients of Electrons in Weakly Ionized Plasmas, R. Winkler Electron Collision Data for Plasma Chemistry Modeling, W.L. Morgan Electron–Molecule Collisions in Low-Temperature Plasmas: The Role of Theory, Carl Winstead and Vincent McKoy Electron Impact Ionization of Organic Silicon Compounds, Ralf Basner, Kurt Becker, Hans Deutsch and Martin Schmidt Kinetic Energy Dependence of Ion–Molecule Reactions Related to Plasma Chemistry, P.B. Armentrout Physicochemical Aspects of Atomic and Molecular Processes in Reactive Plasmas, Yoshihiko Hatano Ion–Molecule Reactions, Werner Lindinger, Armin Hansel and Zdenek Herman Uses of High-Sensitivity White-Light Absorption Spectroscopy in Chemical Vapor Deposition and Plasma Processing, L.W. Anderson, A.N. Goyette and J.E. Lawler Fundamental Processes of Plasma–Surface Interactions, Rainer Hippler Recent Applications of Gaseous Discharges: Dusty Plasmas and UpwardDirected Lightning, Ara Chutjian Opportunities and Challenges for Atomic, Molecular and Optical Physics in Plasma Chemistry, Kurl Becker Hans Deutsch and Mitio Inokuti Volume 44 Mechanisms of Electron Transport in Electrical Discharges and Electron Collision Cross Sections, Hiroshi Tanaka and Osamu Sueoka Theoretical Consideration of PlasmaProcessing Processes, Mineo Kimura Electron Collision Data for PlasmaProcessing Gases, Loucas G. Christophorou and James K. Olthoff Radical Measurements in Plasma Processing, Toshio Goto

Contents of Volumes in this Serial

Radio-Frequency Plasma Modeling for Low-Temperature Processing, Toshiaki Makabe Electron Interactions with Excited Atoms and Molecules, Loucas G. Christophorou and James K. Olthoff Volume 45 Comparing the Antiproton and Proton, and Opening the Way to Cold Antihydrogen, G. Gabrielse Medical Imaging with Laser-Polarized Noble Gases, Timothy Chupp and Scott Swanson Polarization and Coherence Analysis of the Optical Two-Photon Radiation from the Metastable 22Si1/2 State of Atomic Hydrogen, Alan J. Duncan, Hans Kleinpoppen and Marian O. Scully Laser Spectroscopy of Small Molecules, W. Demtröder, M. Keil and H. Wenz Coulomb Explosion Imaging of Molecules, Z. Vager Volume 46 Femtosecond Quantum Control, T. Brixner, N.H. Damrauer and G. Gerber Coherent Manipulation of Atoms and Molecules by Sequential Laser Pulses, N.V. Vitanov, M. Fleischhauer, B.W. Shore and K. Bergmann Slow, Ultraslow, Stored, and Frozen Light, Andrey B. Matsko, Olga Kocharovskaya, Yuri Rostovtsev, George R. Welch, Alexander S. Zibrov and Marlan O. Scully Longitudinal Interferometry with Atomic Beams, S. Gupta, D.A. Kokorowski, R.A. Rubenstein, and W.W. Smith Volume 47 Nonlinear Optics of de Broglie Waves, P. Meystre Formation of Ultracold Molecules (T £ 200 K) via Photoassociation in a Gas of Laser-Cooled Atoms, Françoise Masnou-Seeuws and Pierre Pillet

283

Molecular Emissions from the Atmospheres of Giant Planets and Comets: Needs for Spectroscopic and Collision Data, Yukikazu Itikawa, Sang Joon Kim, Yong Ha Kim and Y.C. Minh Studies of Electron-Excited Targets Using Recoil Momentum Spectroscopy with Laser Probing of the Excited State, Andrew James Murray and Peter Hammond Quantum Noise of Small Lasers, J.P. Woerdman, N.J. van Druten and M.P. van Exter Volume 48 Multiple Ionization in Strong Laser Fields, R. Dörner Th. Weber, M. Weckenbrock, A. Staudte, M. Hattass, R. Moshammer, J. Ullrich and H. Schmidt-Böcking Above-Threshold Ionization: From Classical Features to Quantum Effects, W. Becker, F. Grasbon, R. Kapold, D.B. Milošević, G.G. Paulus and H. Walther Dark Optical Traps for Cold Atoms, Nir Friedman, Ariel Kaplan and Nir Davidson Manipulation of Cold Atoms in Hollow Laser Beams, Heung-Ryoul Noh, Xenye Xu and Wonho Jhe Continuous Stern–Gerlach Effect on Atomic Ions, Gu ¨ nther Werth, Hartmut Haffner and Wolfgang Quint The Chirality of Biomolecules, Robert N. Compton and Richard M. Pagni Microscopic Atom Optics: From Wires to an Atom Chip, Ron Folman, Peter Kru¨ger, Jörg Schmiedmayer, Johannes Denschlag and Carsten Henkel Methods of Measuring Electron–Atom Collision Cross Sections with an Atom Trap, R.S. Schappe, M.L. Keeler, T.A. Zimmerman, M. Larsen, P. Feng, R.C. Nesnidal, J.B. Boffard, T.G. Walker, L.W. Anderson and C.C. Lin

284

Contents of Volumes in this Serial

Volume 49 Applications of Optical Cavities in Modern Atomic, Molecular, and Optical Physics, Jun Ye and Theresa W. Lynn Resonance and Threshold Phenomena in Low-Energy Electron Collisions with Molecules and Clusters, H. Hotop, M.-W. Ruf, M. Allan and I.I. Fabrikant Coherence Analysis and Tensor Polarization Parameters of (g, eg) Photoionization Processes in Atomic Coincidence Measurements, B. Lohmann, B. Zimmermann, H. Kleinpoppen and U. Becker Quantum Measurements and New Concepts for Experiments with Trapped Ions, Ch. Wunderlich and Ch. Balzer Scattering and Reaction Processes in Powerful Laser Fields, Dejan B. Milošević and Fritz Ehlotzky Hot Atoms in the Terrestrial Atmosphere, Vijay Kumar and E. Krishnakumar Volume 50 Assessment of the Ozone Isotope Effect, K. Mauersberger, D. Krankowsky, C. Janssen and R. Schinke Atom Optics, Guided Atoms, and Atom Interferometry, J. Arlt, G. Birkl, E. Rasel and W. Ertmet Atom–Wall Interaction, D. Bloch and M. Ducloy Atoms Made Entirely of Antimatter: Two Methods Produce Slow Antihydrogen, G. Gabrielse Ultrafast Excitation, Ionization, and Fragmentation of C60, I.V. Hertel, T. Laarmann and C.P. Schulz Volume 51 Introduction, Henry H. Stroke Appreciation of Ben Bederson as Editor of Advances in Atomic, Molecular, and Optical Physics Benjamin Bederson Curriculum Vitae Research Publications of Benjamin Bederson A Proper Homage to Our Ben, H. Lustig

Benjamin Bederson in the Army, World War II, Val L. Fitch Physics Needs Heroes Too, C. Duncan Rice Two Civic Scientists—Benjamin Bederson and the other Benjamin, Neal Lane An Editor Par Excellence, Eugen Merzbacher Ben as APS Editor, Bernd Crasemann Ben Bederson: Physicist–Historian, Roger H. Stuewer Pedagogical Notes on Classical Casimir Effects, Larry Spruch Polarizabilities of 3P Atoms and van der Waals Coefficients for Their Interaction with Helium Atoms, X. Chu and A. Dalgarno The Two Electron Molecular Bonds Revisited: From Bohr Orbits to Two-Center Orbitals, Goong Chen, Siu A. Chin, Yusheng Dou, Kishore T. Kapale, Moochan Kim, Anatoly A. Svidzinsky, Kerim Urtekin, Han Xiong and Marlan O. Scully Resonance Fluorescence of Two-Level Atoms, H. Walther Atomic Physics with Radioactive Atoms, Jacques Pinard and H. Henry Stroke Thermal Electron Attachment and Detachment in Gases, Thomas M. Miller Recent Developments in the Measurement of Static Electric Dipole Polarizabilities, Harvey Gould and Thomas M. Miller Trapping and Moving Atoms on Surfaces, Robert J. Celotta and Joseph A. Stroscio Electron-Impact Excitation Cross Sections of Sodium, Chun C. Lin and John B. Boffard Atomic and Ionic Collisions, Edward Pollack Atomic Interactions in Weakly Ionized Gas: Ionizing Shock Waves in Neon, Leposava Vušković and Svetozar

Popović

Approaches to Perfect/Complete Scattering Experiments in Atomic and Molecular Physics, H. Kleinpoppen, B. Lohmann, A. Grum-Grzhimailo and U. Becker Reflections on Teaching, Richard E. Collins

Contents of Volumes in this Serial

Volume 52 Exploring Quantum Matter with Ultracold Atoms in Optical Lattices, Immanuel Bloch and Markus Greiner The Kicked Rydberg Atom, F.B. Dunning, J.C. Lancaster, C.O. Reinhold, S. Yoshida and J. Burgdörfer Photonic State Tomography, J.B. Altepeter, E.R. Jeffrey and P.G. Kwiat Fine Structure in High-L Rydberg States: A Path to Properties of Positive Ions, Stephen R. Lundeen A Storage Ring for Neutral Molecules, Floris M.H. Crompvoets,

Hendrick L. Bethlem and Gerard Meijer

Nonadiabatic Alignment by Intense Pulses. Concepts, Theory, and Directions, Tamar Seideman and Edward Hamilton Relativistic Nonlinear Optics, Donald Umstadter, Scott Sepke and

Shouyuan Chen

Coupled-State Treatment of Charge Transfer, Thomas G. Winter Volume 53 Non-Classical Light from Artificial Atoms, Thomas Aichele, Matthias Scholz,

Sven Ramelow and Oliver Benson

Quantum Chaos, Transport, and Control— in Quantum Optics, Javier Madroñero, Alexey Ponomarev, Andrí R.R. Carvalho, Sandro Wimberger, Carlos Viviescas, Andrey Kolovsky, Klaus Hornberger, Peter Schlagheck, Andreas Krug and Andreas Buchleitner Manipulating Single Atoms, Dieter Meschede and Arno

Rauschenbeutel

Spatial Imaging with Wavefront Coding and Optical Coherence Tomography, Thomas Hellmuth The Quantum Properties of Multimode Optical Amplifiers Revisited, G. Leuchs, U.L. Andersen and C. Fabre Quantum Optics of Ultra-Cold Molecules, D. Meiser, T. Miyakawa, H. Uys and P. Meystre Atom Manipulation in Optical Lattices, Georg Raithel and Natalya Morrow

285

Femtosecond Laser Interaction with Solid Surfaces: Explosive Ablation and Self-Assembly of Ordered Nanostructures, Juergen Reif and Florenta Costache Characterization of Single Photons Using Two-Photon Interference, T. Legero, T. Wilk, A. Kuhn and G. Rempe Fluctuations in Ideal and Interacting Bose–Einstein Condensates: From the Laser Phase Transition Analogy to Squeezed States and Bogoliubov Quasiparticles, Vitaly V. Kocharovsky, Vladimir V. Kocharovsky, Martin Holthaus, C.H. Raymond Ooi, Anatoly Svidzinsky, Wolfgang Ketterle and Marlan O. Scully LIDAR-Monitoring of the Air with Femtosecond Plasma Channels, Ludger Wöste, Steffen Frey and

Jean-Pierre Wolf

Volume 54 Experimental Realization of the BCS-BEC Crossover with a Fermi Gas of Atoms, C.A. Regal and D.S. Jin Deterministic Atom–Light Quantum Interface, Jacob Sherson, Brian Julsgaard and Eugene S. Polzik Cold Rydberg Atoms, J.-H. Choi, B. Knuffman, T. Cubel Liebisch, A. Reinhard and G. Raithel Non-Perturbative Quantal Methods for Electron–Atom Scattering Processes, D.C. Griffin and M.S. Pindzola R-Matrix Theory of Atomic, Molecular and Optical Processes, P.G. Burke, C.J. Noble and V.M. Burke Electron-Impact Excitation of Rare-Gas Atoms from the Ground Level and Metastable Levels, John B. Boffard, R.O. Jung, L. W. Anderson and C. C. Lin Internal Rotation in Symmetric Tops, I. Ozier and N. Moazzen-Ahmadi Attosecond and Angstrom Science, Hiromichi Niikura and P.B. Corkum Atomic Processing of Optically Carried RF Signals, Jean-Louis Le Gouët, Fabien Bretenaker and Ivan Lorgeré

286

Contents of Volumes in this Serial

Controlling Optical Chaos, SpatioTemporal Dynamics, and Patterns, Lucas Illing, Daniel J. Gauthier and Rajarshi Roy Volume 55 Direct Frequency Comb Spectroscopy, Matthew C. Stowe, Michael J. Thorpe,

Avi Pe'er, Jun Ye, Jason E. Stalnaker,

Vladislav Gerginov and

Scott A. Diddams

Collisions, Correlations, and Integrability in Atom Waveguides, Vladimir A. Yurovsky, Maxim Olshanii and

David S. Weiss

MOTRIMS: Magneto–Optical Trap Recoil Ion Momentum Spectroscopy, Brett D. DePaola, Reinhard Morgenstern and Nils Andersen All-Order Methods for Relativistic Atomic Structure Calculations, Marianna S. Safronova and

Walter R. Johnson

B-Splines in Variational Atomic Structure Calculations, Charlotte Froese Fischer Electron–Ion Collisions: Fundamental Processes in the Focus of Applied Research, Alfred Mu¨ller Robust Probabilistic Quantum Information Processing with Atoms, Photons, and Atomic Ensembles, Luming Duan and Christopher R. Monroe Volume 56 Ionizing Collisions by Positrons and Positronium Impact on the Inert Atoms, G. Laricchia, S. Armitage, Á. Kövér and D.J. Murtagh Interactions Between Thermal Ground or Excited Atoms in the Vapor Phase: Many-Body Dipole–Dipole Effects, Molecular Dissociation, and Photoassociation Probed By Laser Spectroscopy, J.G. Eden, B.J. Ricconi, Y. Xiao1, F. Shen and A.A. Senin Bose–Einstein Condensates in Disordered Potentials, Leonardo Fallani, Chiara Fort and Massimo Inguscio

Dipole–Dipole Interactions of Rydberg Atoms, Thomas F. Gallagher and Pierre Pillet Strong-Field Control of X-Ray Processes, Robin Santra, Robert W. Dunford, Elliot P. Kanter, Bertold Kra¨ssig, Stephen H. Southworth and Linda Young Optical Trapping Takes Shape: The Use of Structured Light Fields, K. Dholakia and W.M. Lee Volume 57 Driven Ratchets for Cold Atoms, Ferruccio Renzoni Quantum Effects in Optomechanical Systems, C. Genes, A. Mari, D. Vitali and P. Tombesi The Semiempirical Deutsch-Märk Formalism: A Versatile Approach for the Calculation of Electron-Impact Ionization Cross Sections of Atoms, Molecules, Ions, and Clusters, Hans Deutsch, Kurt Becker, Michael Probst and Tilmann D. Märk Physics and Technology of Polarized Electron Scattering from Atoms and Molecules, T. J. Gay Multidimensional Electronic and Vibrational Spectroscopy: An Ultrafast Probe of Molecular Relaxation and Reaction Dynamics, Jennifer P. Ogilvie and Kevin J. Kubarych Fundamentals and Applications of Spatial Dissipative Solitons in Photonic Devices, Thorsten Ackemann, William J. Firth and Gian-Luca Oppo

Supplements Atoms in Intense Laser Fields, edited by Mihai Gavrila (1992) Multiphoton Ionization, H.G. Muller, P. Agostini and G. Petite Photoionization with Ultra-Short Laser Pulses, R.R. Freeman, P.H. Bucksbaum, W.E. Cooke, G.

Gibson, T.J. McIlrath and L.D. van

Woerkom

Contents of Volumes in this Serial

Rydberg Atoms in Strong Microwave Fields, T.F. Gallagher Muiltiphoton Ionization in Large Ponderomotive Potentials, P.B. Corkum, N.H. Burnett and F. Brunel High Order Harmonic Generation in Rare Gases, Anne L’Huillier, Louis-André Lompré, Gerard Manfrey and Claude Manus Mechanisms of Short-Wavelength Generation, T.S. Luk, A. McPherson, K. Boyer and C.K. Rhodes Time-Dependent Studies of Multiphoton Processes, Kenneth C. Kulander, Kenneth J. Schafer and Jeffrey L. Krause Numerical Experiments in Strong and Super-Strong Fields, J.H. Eberly, R. Grobe, C.K. Law and Q. Su Resonances in Multiphoton Ionization, P. Lambropoulos and X. Tang Nonperturbative Treatment of Multiphoton Ionization within the Floquet Framework, R.M. Potvliege and Robin Shakeshaft Atomic Structure and Decay in High Frequency Fields, Mihai Gavrila Cavity Quantum Electrodynamics, edited by Paul R. Berman (1994) Perturbative Cavity Quantum Electrodynamics, E.A. Hinds The Micromaser: A Proving Ground for Quantum Physics, Georg Raithel, Christian Wagner, Herbert Walther, Lorenzo M. Narducci and Marlan O. Scully Manipulation of Nonclassical Field States in a Cavity by Atom Interferometry, S. Haroche and J.M. Raimond

287

Quantum Optics of Driven Atoms in Colored Vacua, Thomas W. Mossberg and Maciej Lewenstein Structure and Dynamics in Cavity Quantum Electrodynamics, H.J. Kimble One Electron in a Cavity, G. Gabrielse and J. Tan Spontaneous Emission by Moving Atoms, Pierre Meystre and Martin Wilkens Single Atom Emission in an Optical Resonator, James J. Childs, Kyungwon An, Ramanchandra R. Dasari and Michael S. Feld Nonperturbative Atom–Photon Interactions in an Optical Cavity, H.J. Carmichael, L. Tian, W. Ren and P. Alsing New Aspects of the Casimir Effect: Fluctuations and Radiative Reaction, G. Barton Volume 58 Simultaneous Emission of Multiple Electrons from Atoms and Molecules Using Synchrotron Radiation, Ralf Wehlitz CP-violating Magnetic Moments of Atoms and Molecules, Andrei Derevianko and M.G. Kozlov Superpositions of Degenerate Quantum States: Preparation and Detection in Atomic Beams, Frank Vewinger, Bruce W. Shore and Klaas Bergmann Atom Trap Trace Analysis of Rare Noble Gas Isotopes, Zheng-Tian Lu and Peter Mueller Cavity Optomechanics with WhisperingGallery Mode Optical Micro-Resonators, Albert Schliesser and Tobias J. Kippenberg