Physics Reports vol.298

DENSITY FUNCTIONAL. THEORY AND APPLICATION TO ATOMS AND MOLECULES

A¨ . NAGY Institute of Theoretical Physics, Kossuth Lajos University, H-4010 Debrecen, Hungary

AMSTERDAM — LAUSANNE — NEW YORK — OXFORD — SHANNON — TOKYO

Physics Reports 298 (1998) 1—79

Density functional. Theory and application to atoms and molecules A¨. Nagy Institute of Theoretical Physics, Kossuth Lajos University, H-4010 Debrecen, Hungary Received August 1997; editor: J. Eichler

Contents 1. 2. 3. 4.

Introduction Hohenberg—Kohn theorems The method of constrained search The Kohn—Sham scheme 4.1. Non-interacting system 4.2. Spin density functional theory 5. Exact theorems, relations, inequalities 5.1. Long-range asymptotic form of the density and the potentials 5.2. Exchange-correlation hole 5.3. Virial theorems 5.4. Coordinate scaling 5.5. Hierarchy of equations for the energy functional 5.6. Functional expansions 5.7. Adiabatic connection and perturbation theory 6. Fundamental concepts based on density functional theory 6.1. Chemical potential and electronegativity 6.2. Hardness and softness 6.3. Fukui function and local softness

4 5 6 7 7 10 12 12 13 15 20 24 26 28 29 29 30 31

6.4. Density functional theory as thermodynamics 6.5. Work formalism 7. Optimized potential method 8. Potentials from electron density 9. Functionals 9.1. Local density approximation (LDA) 9.2. Approximations containing the gradient of the density 10. Applications 10.1. Atoms 10.2. Molecules 11. Extensions of the density functional theory 11.1. Finite-temperature density functional theory 11.2. Density functional theory for excited states 11.3. Current density functional theory 11.4. Time-dependent density functional theory 11.5. Relativistic density functional theory 12. Concluding remarks References

33 35 37 39 41 41 43 45 45 48 50 50 52 60 62 68 71 73

Abstract The density functional theory is one of the most efficient and promising methods of quantum physics and chemistry. It is a theory of electronic structure formulated in terms of the electron density as the basic unknown function instead of the electron wave function. According to the fundamental theorems of Hohenberg and Kohn, the electron density carries all the information one might need to determine any property of the electron system. However, the way of obtaining it, is not at all trivial. In this report, the recent advances are summarized. After a review of the Hohenberg—Kohn theorems, the 0370-1573/98/$19.00 Copyright ( 1998 Elsevier Science B.V. All rights reserved PII S 0 3 7 0 - 1 5 7 3 ( 9 7 ) 0 0 0 8 3 - 5

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method of constrained search and the Kohn—Sham scheme, exact theorems, relations and inequalities are discussed. There are several important concepts of chemistry (e.g. electronegativity, hardness, softness) that have recently obtained a firm foundation in the density functional theory. The optimized potential method and the methods that generate the potential from the electron density are reviewed. The local and nonlocal approximate functionals are compared. Extensions of the ground-state density functional theory (excited states, time-dependent, relativistic and finite temperature) are summarized. A review of the applications to atoms and molecules is presented. ( 1998 Elsevier Science B.V. All rights reserved. PACS: 31.15.Ew Keywords: Density functional theory; Hohenberg—Kohn theorems; Constrained search; Kohn—Sham theory

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1. Introduction The story of the density functional theory dates back to the pioneering work of Thomas and Fermi [1]. After the fundamental steps taken by Gomba´s [2], Dirac [3], Slater [4] and Ga´spa´r [5] the theory has been given a firm foundation by Hohenberg and Kohn [6] and Kohn and Sham [7]. Since then an enormous amount of work has been done in this field and this approach to the solution of the many-electron problem has become competitive in accuracy with modern quantum chemical methods. An impressive development has taken place in the formalism, the basic principles and, due to the construction of more and more reliable functionals, in the applications. Several monographs, textbooks, reviews [8—39] have been devoted to this novel approach. The growing number of reviews reflects the great interest of the community of physicists and chemists. Because of the vast number of papers in the subject, even a review could focus on only selected key points. In this survey the basic theorems, relations and fundamental concepts are pinpointed, applications are outlined only from the point of view of judging the progress in the theory. We begin with a short summary of the ground state density functional theory. The essence of the ground state density functional theory is that a knowledge of the ground state electron density is sufficient in principle to determine all molecular properties. This statement can be simply understood following Bright Wilson’s [38] argument: A well-known theorem of quantum mechanics, Kato’s theorem [40] states that

K

1 ­n(r) Z "! b 2n(r) ­r

,

(1)

r/Rb where the partial derivatives are taken at the nuclei b. So the cusps of the density tell us where the nuclei (R ) are and what the atomic numbers Z are. On the other hand, the integral of the density b b gives us the number of electrons:

P

N" n(r) dr .

(2)

Thus, from the density, the Hamiltonian can be readily obtained from which every property can be determined. Certainly, we do not follow this way leading to the traditional quantum mechanics. (We have to mention, however, that this argument can only be applied to the Coulomb potential, while the density functional theory is valid for any local external potential.) The basic theorem of the novel treatment of the many-body problem, the Hohenberg—Kohn theorem is discussed in Section 2. Section 3 summarizes the constrained search method, one of the most powerful techniques of the density functional theory. Section 4 presents the Kohn—Sham scheme. Section 5 covers the exact theorems, relations of the density functional theory, such as relations for the exchange-correlation hole, different forms of the virial theorem, coordinate scaling, hierarchy equations and functional expansions. Fundamental new concepts, based on the density functional theory, including the softness, hardness and the local temparature are reviewed in Section 6. Exchange can be treated exactly via the optimized potential method which is outlined in Section 7. Section 8 describes how the Kohn—Sham and the exchange-correlation potentials can be determined exactly if the electron density is known. Section 9 outlines model functionals, including the local density approximation and approximations containing the gradient of the electron

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density. Some key points of applications to atoms and molecules are presented in Section 9. Section 10 summarizes extensions of the density functional theory to finite temperature, excited states, magnetic fields, time-dependent phenomena and the relativistic formalism.

2. Hohenberg—Kohn theorems The density functional theory is based on the theorems of Hohenberg and Kohn [6]. Consider a system of N electrons enclosed in a large box and moving under the influence of some time-independent local external potential v(r). In this section only nondegenerate ground-states are considered. The first Hohenberg—Kohn theorem states that v(r) is determined within a trivial additive constant by the knowledge of the electron density n(r). The proof proceeds by reductio ad absurdum. Suppose that there exists another potential v@(r) leading to the same density n(r) and vOv@#const. That means that we have two different ground-state wave functions W and W@ corresponding to the two external potentials v(r) and v@(r) and consequently two different Hamiltonians HK and HK @ with ground-state energies E and E@ . The Hamiltonian is 0 0 N HK "¹K #»K # + v(r ) , %% i i/1 where ¹K and »K are the kinetic and electron—electron repulsion operators: %% 1 N ¹K "! + + 2 , i 2 i/1 N 1 , »K " + %% r ij i:j N »K " + v(r ) . i i/1 Atomic units are used everywhere. From the Rayleigh—Ritz variational principle it follows that

(3)

(4)

(5)

(6)

E "SWDHDWT(SW@DHDW@T"SW@DH@DW@T#SW@DH!H@DW@T 0

P

"E@ # n(r)[v(r)!v@(r)] dr . 0

(7)

Similarly, using the variational principle for the Hamiltonian H@ with the trial function W, we have E@ "SW@DH@DW@T(SWDH@DWT"SWDHDWT#SWDH@!HDWT 0

P

"E # n(r)[v@(r)!v(r)] dr . 0

(8)

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Addition of Eqs. (7) and (8) leads to contradiction and one concludes that the density determines the external potential, consequently the Hamiltonian and thus all electronic properties of the system. If we write the total energy as

P

E [n]" n(r)v(r) dr#F [n] , v HK

(9)

the functional F [n] is the sum of the kinetic and electron—electron repulsion energies. HK The second Hohenberg—Kohn theorem states that for any trial density nJ E 4E[nJ ] 0 if nJ (r)50 and N":nJ (r) dr. The proof is based on the variational principle as for any trial wave function WI

P

SWI DHK DWI T" nJ (r)v(r) dr#F[nJ ]"E [nJ ]5E [n] . v v

(10)

(11)

Equality stands only in the true ground-state. The variation of the total energy at constant number of electrons

G

d E [n]!k v

CP

DH

n(r) dr!N

"0

(12)

leads to the Euler equation dF [n] dE [n] , k" v "v(r)# HK dn dn

(13)

where the Lagrange multiplicator k is the chemical potential (or the negative of the electronegativity). The functional F [n] is defined only for those trial n(r) that are v-representable. A v-representaHK ble density is one that is associated with a ground-state wave function of some Hamiltonian with a local external potential. The conditions for a density to be v-representable are yet unknown. It was demonstrated [41—43], however, that there exists a proper universal variational functional, which delivers the sum of the kinetic and repulsion energies and which does not require the density to be v-representable. This theory of constrained search is discussed in the next section.

3. The method of constrained search The v-representability problem was solved by the constrained search method by Levy and Lieb [41—43]. A universal functional F[n] is defined as a sum of kinetic and repulsion energies: F[n]"Min SWD¹K #»K DWT . %% W ?n

(14)

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F[n] searches all wave functions W which yield the fixed trial density n. n need not be vrepresentable. The ground-state energy is searched in two steps:

T G T G C

U

N E "Min WD¹K #»K # + v(r )DW %% i 0 W i/1 N "Min Min WD¹K #»K # + v(r )DW %% i W n i/1 ?n

P

UH

DH

"Min Min SWD¹K #»K DWT# v(r)n(r) dr . %% W ?n n Using the definition of F[n], Eq. (15) can also be written as

G

P

(15)

H

E "Min F[n]# v(r)n(r) dr 0 n "Min E[n] , n

(16)

where

P

E[n]"F[n]# v(r)n(r) dr .

(17)

For v-representable densities F[n]"F [n] . (18) HK The functional F[n] is universal because it is independent of the external potential v. The constrained search formulization eliminates the limitation of the Hohenberg—Kohn theorems that there be no degeneracy in the ground state. In the constrained search only one of a set of degenerate wave functions is selected, the one corresponding to the density n. The method of constrained search is frequently applied in the density functional theory (see Sections 4, 5.2, 8 and 11.2).

4. The Kohn—Sham scheme 4.1. Non-interacting system The ground-state electron density can be in principle determined by solving the Euler Eq. (13): dF[n] #v(r)"k . dn

(19)

However, we do not know the exact form of the functional F[n]: F[n]"¹[n]#» [n] . %% Slater [4], Ga´spa´r [5] and Kohn and Sham [7] proposed the following scheme:

(20)

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A non-interacting system, in which the electrons move independently in a common local potential v is constructed. The Hamiltonian is 4 N 1 N HK " + ! + 2 # + v (r ) . (21) 4 4 i 2 i i/1 i/1 Substituting the non-interacting wave function

A

B

1 W" det Mu ,u ,2,u N , 4 JN! 1 2 N

(22)

into the Schro¨dinger equation of the non-interacting system HK W "E W 4 4 4 4 we obtain the one-electron equations

(23)

hK u "M!1+2#v (r )Nu . 4 i i 4 i 2 i The kinetic energy of the non-interacting system is

(24)

T K A

BK U

TK

KU

1 N 1 N ¹ " W + ! +2 W " + u ! +2u , 4 4 4 i 2 i 2 i i i/1 i/1 while the density of the non-interacting system

(25)

N n(r)" + Du (r)D2 (26) i i/1 is equal to that of the interacting one. The exact kinetic energy functional ¹ is unknown, so we simply take the kinetic energy functional ¹ of the non-interacting system instead of ¹. Substituting ¹ into Eq. (20) of F[n] an 4 4 extra term, the difference ¹ "¹!¹ appears: # 4 F[n]"¹ [n]#» [n]#¹ [n] , (27) 4 %% # » [n]#¹ [n]"J[n]#E [n] , (28) %% # 9# i.e. F[n]"¹ [n]#J[n]#E [n] . (29) 4 9# With the help of the exchange-correlation energy functional E [n] the total energy E[n] has the 9# form

P

E[n]" n(r)v(r) dr#¹ [n]#J[n]#E [n] . 4 9#

(30)

The variation of Eq. (30) leads to the Euler equation dE[n] d¹ [n] dJ[n] dE [n] k" "v(r)# 4 # # 9# . dn dn dn dn

(31)

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It can also be written as d¹ [n] k"v (r)# 4 , KS dn

(32)

where v (r)"v(r)#vJ(r)#v (r) KS 9# is the Kohn—Sham potential consisting of the external v, the classical Coulomb

P

dJ[n] n(r@) v (r)" " dr@ J dn Dr!r@D

(33)

(34)

and the exchange-correlation dE [n] v (r)" 9# , 9# dn

(35)

potentials. Taking the variation of the total energy expressed with the one-electron orbitals u i N N 1 E[u ]" + v(r)Du (r)D2 dr# + u*(r) ! + 2 u (r) dr i i i i 2 i/1 i/1 N 1 N N Du (r)D2 Du (r@)D2 i j dr dr@#E n" + Du (r)D2 . # + + 9# i 2 Dr!r@D i/1 i/1 j/1 with the constraint of orthonormality of the orbitals u i

P

P C

P

P

u*(x)u (x) dx"d , i j ij

D

C

D

(36)

(37)

we obtain

C

D

1 N (38) hK KSu " ! + 2#vKS(r) u " + e u . i i ij j 2 i/1 Making use of the fact that the operator hK is Hermitian, a unitary transformation of the orbitals KS leads to Kohn—Sham equations [!1+ 2#v (r)]u "e u . (39) 2 KS i i i This is probably the most important equation of the density functional theory. It tells us that the motion of the interacting electrons can be treated exactly as a system of independent particles. The electrons can be considered as if they move in a common effective local potential v . All the KS interaction between the electrons can be merged exactly into a single local potential v . KS The Kohn—Sham equations can also be derived with constrained search. In the following subsection this technique will be applied to obtain the Kohn—Sham equations of the spin density functional theory.

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4.2. Spin density functional theory Up to this point we studied systems with scalar external potential only. The density functional theory was extended to systems in external magnetic fields by von Barth and Hedin [44] and Pant and Rajagopal [45]. To characterize a system in the presence of magnetic field B(r) we need more information beyond the electron density. The new quantity is the magnetization or electron spin density Q(r)"n (r)!n (r) , (40) ­ ¬ which is defined as the difference of the electron densities of electrons with spin up n and down n . ­ ¬ The Hamiltonian has the form 1 N N 1 N N HK "! + + 2# + # + v(r )#2b + B(r) ) s , i i % i 2 r i:j ij i/1 i/1 i/1 where

(41)

e+ b" % 2mc

(42)

is the Bohr magneton and s is the spin vector of the ith electron. Here, the interaction of the i magnetic field with the electronic current is neglected. (The current density functional theory is outlined in Section 11.3.) Let us consider the interaction with the external field N N »K " + v(r )#2b + B(r) ) s (43) i % i i/1 i/1 in the case of z-direction magnetic field b(r). Then the expectation value of »K has the form

P

P

SWD»K DWT" v(r)n(r) dr! b(r) ) m(r) dr ,

(44)

where m(r)"b (n (r)!n (r)) % ­ ¬ is the magnetization density. The constrained search leads to

T G G

U

N N E "Min WD¹K #»K # + v(r )#2b + b(r )s DW 0 %% i % i zi W i/1 i~1 "Min n­,n¬

(45)

P

Min SWD¹K #»K DWT# [v(r)n(r)!b(r)m(r)] dr %% ?n­,n¬

W

P

H H

"Min F[n ,n ]# dr [(v(r)#b b(r))n (r)#(v(r)!b b(r))n (r)] , ­ ¬ % ­ % ¬ n­,n¬

(46)

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where F[n ,n ]" Min SWD¹K #»K DWT %% ­ ¬ W ?n­,n¬ is a universal functional that can be written as

(47)

F[n ,n ]"¹ [n ,n ]#J[n #n ]#E [n ,n ] ­ ¬ 4 ­ ¬ ­ ¬ 9# ­ ¬ to obtain the Kohn—Sham equations. The non-interacting kinetic energy has the form

(48)

C

P

D

1 ¹ [n ,n ]"Min ! + f dr u* (r)+ 2u (r) ip ip ip 4 ­ ¬ 2 ip where

(49)

n (r)"+ f Du (r)D2 . (50) p ip ip ip p denotes C or B. The occupation numbers f are chosen so that the lowest eigenstates are occupied ip ( f "1) and the rest are unoccupied ( f "0). The minimization of the total energy ip ip 1 (51) E[n ,n ]"+ f dr u*(r) ! + 2#v (r) u (r)#J[n #n ]#E [n ,n ] i ip i ­ ¬ 9# ­ ¬ ­ ¬ ip 2 ip subject to normalization constraint

P

A

B

P

Du (r)D2 dr"1 ip

(52)

leads to the spin-polarized Kohn—Sham equations: [!1+ 2#vKSr(r)]u (r)"e u (r) , ip ip ip 2 where

P

n(r@) dr@#v (r) , v (r)"v (r)# 9#p KSp p Dr!r@D

(53)

(54)

v (r)"v(r)#b b(r) , (55) ­ % v (r)"v(r)!b b(r) , (56) ¬ % dE [n , n ] v (r)" 9# ­ ¬ . (57) 9#p dn p The spin density functional theory can be applied even in the absence of magnetic fields. In this case the results should reduce to the spin-compensated results provided that the exact Kohn—Sham potential is applied. However, because the exact form of the exchange-correlation functional is unknown and one should use approximate potentials the results are different and the spin density functional theory generally provides a better description of a system. This is surely the case for open-shell atoms or molecules.

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5. Exact theorems, relations, inequalities 5.1. Long-range asymptotic form of the density and the potentials The asymptotic expression for the Kohn—Sham orbitals [46] u (r)"C / (r)[1#O(r~1)] , i i i

(58)

where the functions / (r)"rbi exp(!i r) i i

(59)

decay exponentially. i "(!2e )1@2 , i i

(60)

b "Z /i !1 , i %&& i

(61)

Z "Z!N#1 . %&&

(62)

Z"+Z and N are the total nuclear charge of the atom or molecule and the number of electrons, b respectively. It can also be proven [46] that the uppermost occupied Kohn—Sham orbital energy is equal to the negative of the ionization potential e "!I. Consequently, the density has the m asymptotic behaviour: n(r)"r2bm exp(!2i r)[1#O(r~1)] . m

(63)

Studying the large-r expansion of the external and the classical electrostatic potentials the asymptotic form of the exchange-correlation potential is given by [46]:

AB

1 1 , v (r)"! #O 9# rp r

(64)

where p"4 for a nondegenerate spin-unpolarized or for the spin-up potential of a spin-polarized ground-state, otherwise p"3. The long range behavior of the exchange potential can be written as [47—50]

AB

1 1 . v (r)"! #O 9 r2 r

(65)

As the leading term in both the exchange-correlation and the exchange potentials is the same the correlation potential does not play any role in the asymptotic region. Another important exact expression for the density is the cusp condition [40] that has already been mentioned in the introduction (Eq. (1)). The cusp of the exchange and exchange-correlation potentials [51] have also been studied. Another set of properties of considerable interest is obtained by imposing rigorous bounds on the density and expectation values [52].

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5.2. Exchange-correlation hole The electron—electron interaction energy can be written [53] as

P

1 SUD»K DUT" C»K dr dr . %% %% 1 2 2

(66)

The two-particle density matrix C can be split into four terms: C 1 2, where p corresponds to spin i p ,p up or down [53]. The pair-correlation function h 1 2 is defined as p ,p C 1 2(r ,r )"(1#h 1 2(r ,r ))n 1(r )n 2(r ) . p ,p 1 2 p 1 p 2 p ,p 1 2

(67)

To obtain the exchange-correlation energy of the density functional theory the coupling constant [54—56] integrated pair-correlation function [54] (see Section 5.7) should be determined: hM

P

(r ,r )" p1, p2 1 2

1

0

hj1 2(r ,r ) dj . p ,p 1 2

(68)

hj1 2(r ,r ) is the pair-correlation function corresponding to the Hamiltonian p ,p 1 2 HK "¹K #j»K #»K . j %% j

(69)

The coupling constant integration is done in such a way that the density remains constant at any value of the coupling strength j. j"1 and j"0 give the fully interacting and the non-interacting cases, respectively. The exchange-correlation energy has the form 1 E [n ]" + 9# p 2 p1, p2

P

n 1(r )n 2(r ) p 1 p 2 hM (r , r ) dr dr . Dr !r D p1, p2 1 2 1 2 1 2

(70)

It can also be expressed with the exchange-correlation hole oN 1 2 [4,57,58]: 9# p ,p 1 E [n ]" + 9# p 2 p1, p2

P

n 1(r )oN 1 2(r , r ) p 1 9#p ,p 1 2 dr dr . 1 2 Dr !r D 1 2

(71)

Using the definition of the exchange and correlation energies one arrives at 1 E [n ]" + 9 p 2 p

P

1 E [n ]" + # p 2 p1, p2

n (r )oN (r , r ) p 1 9r 1 2 dr dr , 1 2 Dr !r D 1 2

(72)

P

(73)

n 1(r )oN 1 2(r , r ) p 1 #p ,p 1 2 dr dr . 1 2 Dr !r D 1 2

The exchange or Fermi hole can be expressed with the help of the one-particle density matrix c: oN (r , r )"!1Dc (r , r )D2/n (r ) . p 1 9p 1 2 2 p 1 2

(74)

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The exchange-correlation, the exchange and the correlation holes satisfy the following sum rules:

P

+ oN 1 2(r , r ) dr "!1 , 9#p , p 1 2 2 p2

P

(75)

oN 1(r , r ) dr "!1 , 9p 1 2 2

(76)

P

(77) + oN 1 2(r , r ) dr "0 . #p , p 1 2 2 p2 These sum rules can serve important tests of approximations. Recently, there is considerable interest in the properties of the so-called on-top electron pair density [59] defined as the diagonal of the two-particle density matrix C(r , r )"n(r )[n(r )#o (r , r )] , (78) 1 2 1 2 9# 1 2 where o (r , r ) is the exchange-correlation hole surrounding an electron at r (without the 9# 1 2 1 coupling constant integration). [n(r )#o (r , r )] is the conditional probability to find an electron 2 9# 1 2 in dr at r , given that there is one at r . For a uniform electron gas Eq. (78) becomes 2 2 1 C6/*&(n ,n ;w)"n[n#o6/*&(n ,n ;w)] (79) ­ ¬ 9# ­ ¬ with r "r #w. Then the on-top pair density for an electron gas with uniform spin densities 2 1 n and n is given by ­ ¬ CI (r, r)"C6/*&(n ,n ;w"0) . (80) ­ ¬ Numerical tests [59] show that there is a satisfactory agreement between the exact on-top exchange-correlation hole density and its LSD and GGA approximations. Based on this finding an alternative interpretation of the spin-density-functional theory (LSD) has been put forward. It is possible to set up a formal density functional theory [59] that yields the exact energy, density n and on-top pair density C(r,r). Using the constrained search approach a new functional SWD¹K #»K DWT (81) Min %% ?n/n­`n¬ ? (n­,n¬§w/0) is introduced. This means that the search is over all antisymmetric wave functions that yield the density n and the on-top pair density CI (r, r) until the minimum expectation value is reached. The total energy is defined FI [n ,n ]" ­ ¬

W CI C6/*&

A

P

B

(82) EI "Min FI [n ,n ]# dr v(r)[n (r)#n (r)] . ­ ¬ ­ ¬ ­ ¬ n ,n From the Rayleigh—Ritz variational principle EI is larger or equal to the true ground-state energy. The standard spin-density-functional theory allows to predict the spin magnetization density n !n , but the reliability of this prediction is more questionable than that of CI (r,r). This ­ ¬ alternative theory is exact in the fully spin-polarized and low density limits and more accurate than LSD or GGA and does not lead to a symmetry dilemma [59].

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5.3. Virial theorems The virial theorem of quantum mechanics in the case of a Coulomb potential in equilibrium molecular geometry has the form ¹"!E ,

(83)

where ¹ and E are the kinetic and total energies, respectively. This theorem is often used to judge the quality of approximate wave functions. Naturally, this theorem holds also in the density functional theory. However, in the Kohn—Sham scheme of the density functional theory, it takes a somewhat different form [60]: ¹ #¹ "!E . (84) 4 # The non-interacting kinetic energy ¹ differs from the interacting kinetic energy ¹. As the 4 difference ¹ is positive # ¹ '0 (85) # we arrive at the inequality ¹ (!E . 4 Using the Kohn—Sham equations one can easily derive the Levy—Perdew relation [60]

P

¹ #E "! nr ) +v (r) dr . 9# # 9#

(86)

(87)

This equation combined with Eq. (26) leads to another form of the Levy—Perdew relation [60]:

P

F#¹"! nr ) +

dF dr . dn

(88)

In the exchange-only case the Levy—Perdew relation takes the form

P

E "! n(r)r ) +v (r) , 9 9 where the exchange potential v is the functional derivative of the exchange energy E : 9 9 dE v (r)" 9 . 9 dn

(89)

(90)

In this case the virial theorem (Eq. (84)) reduces to ¹ "!E , 4 because

(91)

¹ "0 . (92) # In the following subsections the differential, the local and the spin virial theorems are summarized. For a recent review of other forms of the virial theorem, e.g., the integral and the regional virial theorems, see Ref. [61].

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5.3.1. Differential virial theorem The differential virial theorem has been derived by Holas and March [62]. The starting point of the derivation is the many-body Schro¨dinger equation: HK W"EW ,

(93)

that can be written as »K #»K !E"!(¹K WR%)/WR%"!(¹K WI.)WI. , (94) %% Differentiating Eq. (94) with respect to r , then multiplying with (WR%)2, then repeating the 1b procedure after replacing WR% by WI. and adding the two final equations and integrating, we get the differential virial theorem of Holas and March:

P

n(r)+v# dr@C(r, r@)+ w(r,r@)"1+ 2+n(r)!2 div pL HM , 3 4

(95)

where C(r, r@) is the diagonal of the two-particle density matrix and pL HM is the kinetic energy density tensor defined by pL HM(r)"1(+"+ @#+ @"+ )o(r, r@)]D . 4 r/r{ The differential virial theorem of Holas and March can be written in the following form:

+v"!f (r; [w, n, o, C]) ,

(96)

(97)

where

CP

f (r; [w, n, o, C])"

D

1 dr@C(r, r@)+ w(r, r@)! + 2+n(r)#2 div pL HM /n(r) . r 4

o(r, r@) is the one-electron density matrix and w(r, r@)"1/Dr!r@D .

(98)

(99)

Eq. (97) presents an exact expression from which the external potential can be readily determined by a line integral or from a knowledge of the external potential the exchange-correlation potential can in principle be obtained. In the Kohn—Sham scheme of the density functional theory, Eq. (97) has the form:

+v "!f (r;[n,o ]) , 4 4 4 where

(100)

(101) f (r; [n, o ])"[!1+ 2+n(r)#2 divpL HM]/n(r) . 4 4 4 4 (The subscript s refers everywhere to the non-interacting system.) For spherically symmetric systems the differential virial theorem of Holas and March Eq. (95) reduces to the differential virial theorem of Nagy and March [63] 1 1 q@ q q@"! . @@@! . v@ # ! , 8 2 KS r2 r3

(102)

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17

where the radial kinetic energy density has the form

K

K

1 ­2. (r@,r) q(r)"! 2 ­r2

. (r) 1 # + l (l #1) k , r2 2 k k r{/r k

(103)

1 q" + l (l #1). . (104) k 2 k k k . , . and v are the radial electron density, the radial electron density of orbital k and the k KS Kohn—Sham potential, respectively. For particles having zero angular momentum i.e. for s electrons the differential virial theorem reduces to the special form of March and Young [64] (105) q@"!1. @@@!1. v@ . 2 KS 8 Recently, using the differential virial theorem for spherically symmetric system [63] (Eq. (102)) an exact relation for the kinetic energy of atoms (or ions) with one p and one or more s shells has been derived in terms of the total and the s electron densities [65]. 5.3.2. Local virial theorem To derive the local virial theorem [66] we take the gradient of the Euler equation Eq. (13) and multiply by the density: n+

d¹ 4#n+v #n+v*"0 , 9# dn

(106)

where v*"v#v

(107)

div pL !n+v*"0 ,

(108)

J is the total classical electrostatic potential. The force equation can be written as

which gives the condition of static equilibrium for the system. The stress tensor pL was introduced by Bartolotti and Parr [67]. Following Ghosh and Berkowitz [68] the stress tensor pL connected with 4 the non-interacting kinetic energy is defined by d¹ 4. div pL "!n+ 4 dn

(109)

Deb and Ghosh [69] determined the form of pL . As can be seen from Eqs. (108) and (109) the 4 definition of the stress tensor is not unique and the corresponding non-interacting kinetic energy can also have several forms. The pressure is defined as p"!1tr pL . 3 The pressure connected with the non-interacting kinetic energy takes the form

(110)

p "2 t . 4 3 4 With the definitions of the exchange-correlation stress tensor pL

(111)

div pL "!n+v 9# 9#

9# (112)

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and the exchange-correlation pressure p "!1tr pL , 9# 3 9# using the total stress tensor

(113)

pL "pL #pL , 4 9# Eqs. (111) and (113) and Eq. (114) lead to the local virial theorem

(114)

2t "3(p!p ) . 4 9# If the exchange—correlation stress tensor has the form

(115)

pL "!p IK 9# 9# the local virial theorem takes the form

(116)

P

=

n(r@)+r v (r@) ) dr@ . { 9# r In the local density approximation the local virial theorem reads 2t "3p#3 4

de 2 p" t #n2 9# , dn 3 4

(117)

(118)

where e (n)"ne (n) is the exchange-correlation energy density. 9# 9# Ghosh et al. [70] introduced the concept of local temperature into the density functional theory. The local temperature ¹(r) was defined by the ideal gas expression for the kinetic energy (119) t "3nk¹ , 4 2 where k is the Boltzmann constant. With this definition the local virial theorem provides the virial equation of state p"nk¹#p

9# and in the local density scheme p"nk¹#n2

de 9# . dn

(120)

(121)

This equation shows that the deviation of the real system from the ideal system (for which p"nk¹) is due to the exchange-correlation effects which contain a kinetic contribution. It is interesting to note that for a microscopic system the virial equation of state has a closed form and unlike for the macroscopic system a virial expansion is not needed. Local virial relations have also been derived by Bader et al. [71] and Zietsche et al. [72]. (For a discussion see Ref. [73].) Several forms of the integral virial theorem [74] can be obtained by integrating the local virial theorem Eq. (115) on the whole space. The Levy—Perdew relation (Eq. (109)) leads to an integral form of the virial theorem which is the generalization of the theorem derived by Bartolotti and Parr [67]. (Another form of the virial theorem was given by Ghosh and Parr [75].) It was shown [74] that the orbital energy sum can be approximated by the integral of the pressure — :p dr. In recent years there has been considerable interest in the general properties of subdomains of systems.

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Regional virial theorem has been derived within the wave function theory [76] and in the density functional theory [77,74]. The regional analogue of the Levy—Perdew relation has also been derived [74,61]. 5.3.3. Spin virial theorem The spin virial theorem connects the difference of the spin-up and -down kinetic and potential energies [78,79]. The starting point of the derivation is given by the spin-polarized Kohn—Sham equations (Eq. (53)). After some manipulation we get the spin virial theorem

P

2(¹­!¹¬)" dr(n r ) +v !n r ) +v ) . 4 4 ­ KS­ ¬ KS¬

(122)

¹p and 4 n "+ u* u (123) p ip ip i (p) are the electron density and the non-interacting kinetic energy for spin p, respectively. The spin virial theorem is independent from the virial theorem. Contrary to the virial theorem the spin virial theorem has no classical counterpart; it is a completely quantum mechanical theorem. The spin virial theorem for free atoms, ions and molecules has the form 2D¹ "!D» , 4 where

(124)

D¹ "¹­!¹¬ , 4 4 4 D»"» !» "w !w #y !y #q #x !x . ­ ¬ ­ ¬ ­­ ¬¬ ­¬ ­ ¬ The first and second terms in Eq. (126)

(125)

P C

D

Z Z p #R ) + p dr w !w "Dw"! Q+ p pDr!R D ­ ¬ Dr!R D p p p come from the electron—nuclear attraction, where

(126)

(127)

Q(r)"n (r)!n (r) (128) ­ ¬ is the (spin) magnetization density. The third and fourth terms in Eq. (126) arise from the electron—electron repulsion

P P P

1 n (r )n (r ) ­ 1 ­ 2 dr dr , y " 1 2 ­­ 2 Dr !r D 1 2 1 n (r )n (r ) ¬ 1 ¬ 2 dr dr , y " ¬¬ 2 Dr !r D 1 2 1 2 n (r ) n (r ) q " ­ 1 ¬ 2 dr dr . ­¬ 1 2 Dr !r D 1 2

(129) (130) (131)

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The difference between the exchange-correlation virials gives the last two terms of Eq. (126):

P

x "! n r ) +v dr . p 9#p p

(132)

Another form of the spin virial theorem can be given as

P

Z p dr , DE#D¹ "DE !q !Dx# Q+ R ) + 4 9# ­¬ p pDr!R D p p where

(133)

DE"E !E , (134) ­ ¬ DE "E !E . (135) 9 9#­ 9#¬ This form of the theorem connects the total and kinetic energy differences. The spin-up and spin-down total energies are defined as

P

P

Z 1 p dr# n (r)/(r) dr#E . E "¹p! n (r)+ p 4 p 9#p Dr!R D 2 p p p The exchange-correlation energy

P

1 n (r ) p 1 . (r ,r ) dr dr E " 9#p 2 r 9#p 1 2 1 2 12 is expressed with average exchange-correlation hole

(136)

(137)

.

(r ,r )"+ n (r )hM (r ,r ) , (138) 9#p 1 2 p{ 2 9#pp{ 1 2 p{ where hM (r ,r ) is the coupling-constant averaged pair-correlation function. The spin virial 9#pp{ 1 2 theorem can be derived in the presence of magnetic field, too [79]. The spin virial theorem has conceptual and practical significance. It can be used to check the accuracy of the approximate spin orbitals. As the spin virial theorem is independent of the virial theorem, this is a new way of checking the accuracy of spin orbitals. 5.4. Coordinate scaling The density functional theory constitutes an enormous simplification of the many-electron problem. However, the exact exchange-correlation functional must be approximated. Levy and coworkers [60] have shown that coordinate scaling provides a powerful tool for constructing approximate potentials. (For reviews see Refs. [80,81].) Let U(r ,2,r ) be an eigenfunction of the 1 N Hamiltonian HK . The (uniform) coordinate scaling means that the coordinates r are changed into i ar , where a is any real constant. The scaled wave function has the form: i U (r ,2,r )"a3N@2U(ar ,2,ar ) . (139) a 1 N 1 N

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The factor a3N@2 is present to preserve normalization to N electrons. From the Rayleigh—Ritz variational principle d SU DHK DU TD "0 . a a/1 da a

(140)

One can easily check the scaling of the kinetic and potential energies: SU D¹K DU T"a2SUD¹K DUT , a a SU D»K DU T"aSUD»K DUT , a %% a %% SU D»K DU T"aSUD»K DUT , a a where the scaled density can be written as

(141) (142) (143)

n (r)"a3n(ar) . (144) a The Hohenberg—Kohn theorem implies that the ground-state wave function, the kinetic and the electron—electron potential energies are functionals of the electron density n ¹[n]"SU[n]D¹K DU[n]T ,

(145)

» [n]"SU[n]D»K DU[n]T . (146) %% %% To derive scaling relations for the functionals ¹[n] and » [n] we use the method of constrained %% search ¹[n ]#» [n ]"F[n ]"Min SWD¹K #»K DWT a %% a a %% W ?na "SW aD¹K #»K DW aT(SU aD¹K #»K DU aT , aO1 , n %% n n %% n where the wave function W a is a solution of the Schro¨dinger equation n (¹K #»K #»K )W a"E aW a . %% n n n From Eqs. (141)—(143) the inequalities

(147)

(148)

¹[n ]#» [n ](a2¹[n]#a» [n] , aO1 (149) a %% a %% can be obtained. The wave function U differs from the wave function W (if aO1) because U is a a a a solution of the following Schro¨dinger equation: EU (r ,2,r )"HK (ar ,2,ar ) [a3N@2U(ar ,2,ar )] a 1 N 1 N 1 N 1 1 1 " ¹K # »K # »K U (r ,2,r ) . %% a 1 N a a a2

A

B

(150)

Or (¹K #a»K #a»K )U "(a2E)U . %% a a

(151)

22

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It means that the wave function U minimizes the functional SWD¹K #a»K DWT. It leads to the a %% inequality SU D¹K #a»K DU T(SW aD¹K #a»K DW aT , aO1 . a %% a n %% n Making use of this inequality and Eqs. (141)—(143) one obtains

(152)

a2(¹[n]#» [n])(¹[n ]#a» [n ] , aO1 . %% a %% a Eqs. (149)—(153) lead to the inequalities:

(153)

¹[n ](a2¹[n] , a'1 , (154) a ¹[n ]'a2¹[n] , a(1 , (155) a » [n ](a» [n] , a(1 , (156) %% a %% » [n ]'a» [n] , a'1 . (157) %% a %% On the other hand, the scaling relation for the non-interacting kinetic energy ¹ takes the form 4 ¹ [n ]"a2¹ [n] (158) 4 a 4 as for the case of » "0, U "W a. %% a n From the definition of the exchange and correlation energies E [n]"SU.*/D»K DU.*/T!J[n] , (159) 9 n %% n E [n]"SW.*/D¹K #»K DW.*/T!SU.*/D¹K #»K DU.*/T , (160) # n %% n n %% n where U.*/ is the wave function that yields the given density n and minimizes the kinetic energy n ¹ [n]"SU.*/D¹K DU.*/T , (161) 4 n n we arrive at E [n ]"aE [n] , 9 a 9 E [n ]OaE [n] . # a # The correlation energy satisfies the following inequalities: E [n ](aE [n] , a(1 , # a # E [n ]'aE [n] , a'1 # a # coming from Eqs. (156), (157) and (159). Eq. (160) leads to the correlation energy E [n ]"SW.*/ D¹K #»K DW.*/ T!SU.*/ D¹K #»K DU.*/ T # a na %% na na %% na "a2[SW.*/,jD¹K DW.*/,jT!SU.*/D¹K DU.*/T] n n n n # a[SW.*/,jD»K DW.*/,jT!SU.*/D»K DU.*/T] , n %% n n %% n

(162) (163)

(164) (165)

(166)

as W.*/ "a3N@2W.*/,j(ar ,2,ar ) , j"a~1 . na n 1 N

(167)

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The wave function W.*/,j yields the density n and minimizes S¹K #j»K T. It follows that the wave na %% function a3N@2W.*/, j(ar ,2,ar ) yields the density n and minimizes S¹K #ja»K T (i.e. S¹#»K T if n 1 N j %% %% j"a~1). For small values of j substituting the perturbation expansion of the wave function W.*/,j n = W.*/,j(r ,2,r )"U.*/(r ,2,r )# + jkg (r ,2,r ) (168) n 1 N n 1 N k 1 k k/1 into Eq. (166) one obtains E [n ]"a[n]#b[n]a~1#c[n]a~2#2 . # a It leads to the inequality

(169)

lim E [n ]'!R . # a a?= For small values of a (aP0 ill. jPR) from Eq. (166) it follows

(170)

E [n ] lim # a "!g[n] , (171) j a?0 where g[n]'0, as SW.*/,jD»K DW.*/,jT!SU.*/D»K DU.*/T tends to a negative constant. n %% n n %% n From the Levy—Perdew relation (87) [60] and Eqs. (159) and (160) we arrive at other important expressions derived by Levy and Perdew [60]:

P

!jEj[n]!j dr n(r)r ) + #

P

Ej[n]# dr n(r)r ) + #

A

A

B

dEj[n] # "¹j[n] , # dn(r)

B

dEj[n] dEj[n] # "j # . dj dn(r)

(172) (173)

Eliminating the integral term between these equations, results in dEj[n] ¹j[n]"!j2 # . # dj

(174)

Making use of the definition of ¹j[n]: # ¹j[n]"SWjD¹K DWjT!SUj/0D¹K DUj/0T , # and Eqs. (172)—(175) we have

P

2¹j[n]# dr n(r)r ) + #

A

B

d¹j[n] d¹j[n] # "j # . dn(r) dj

(175)

(176)

The exact identities Eqs. (173) and (176), respectively, involve Ej[n] and ¹j[n]. # # A set of other exact relations can be derived [82]. These conditions are very important to help approximate the exchange-correlation energy functional. It is often quite easy to ascertain how a functional behaves upon coordinate scaling. E.g. GGA [83](PW92) obeys many of the known exact relationships including those obeyed by the LDA and several others that are violated by

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LDA. Anologous coordinate scaling relations hold for the second-order density matrix, the pair correlation function and the exchange-correlation hole [84]. Non-uniform coordinate scaling imposes further conditions upon the functionals [85]. 5.5. Hierarchy of equations for the energy functional For development of approximations, exact relations and criteria that are fulfilled by the exact functionals are of great help in constructing new approximate functionals. Hierarchies of equations for the energy functional [86—89] and their Legendre transformations [90] provide such relations. Here, hierarchies for the sum of the kinetic and the electron—electron energies and for the exchange and exchange-correlation are reviewed. Hierarchies of equations have been derived in several fields of physics (e.g. Bogoliubov—Born— Green—Kirkwood—Yvon (BBGKY) hierarchy [91]) or hierarchy equations for reduced density matrices [92]. 5.5.1. Hierarchy of equations for the sum of the kinetic and electron—electron energies Taking the gradient of the Euler—Lagrange equation (Eq. (13)) we obtain

+u(r)"!+

dF[n] , dn(r)

(177)

where dF u(r)"v(r)!k"! . dn

(178)

The zeroth equation of the hierarchy is the universal virial relation of Levy and Perdew (Eq. (88)) [60]. Its functional differentiation with respect to n(r) leads to the first equation of the hierarchy

P

dF[n] d¹[n] dF[n] # "!r ) + ! dr n(r )r ) + g(r, r ; n) , 1 1 1 1 dn(r) dn(r) dn(r)

(179)

where d2 F[n] du(r) g(r, r ; n)" "! (180) 1 dn(r)dn(r ) dn(r ) 1 1 is the hardness kernel [93]. With another functional differentiation we arrive at the second equation of the hierarchy:

P

dg(r, r ; n) d2 ¹[n] 1 . g(r, r ; n)# "!(r ) +#r ) + )g(r, r ; n)! dr n(r )r ) + (181) 1 1 1 2 2 2 2 dn(r ) 1 dn(r)dn(r@) 2 Further differentiations will lead to higher-order equations. The hierarchy of equations links the nth functional derivatives to the (n#1)-th functional derivatives and the electron density.

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5.5.2. Hierarchy of equations for the exchange-correlation and the exchange energies The starting point of the derivation is the Levy—Perdew relation (Eq. (87)). The first

P

dv (r; n) d¹ [n] v (r; n)# # "!r ) +v (r; n)! dr n(r )r ) + 9# , 9# 9# 1 1 1 1 dn(r ) dn(r) 1 and the second equations of the hierarchy

(182)

P

dv (r; n) 1 d2v (r; n) d2 ¹ [n] dv (r; n) 9# # 9# # "!(r ) +#r ) + ) 9# ! dr n(r )r ) + (183) 1 1 dn(r ) 2 2 2 2dn(r )dn(r ) dn(r)dn(r ) 2 dn(r ) 1 1 2 1 1 can be obtained by subsequent functional differentiation of the Levy—Perdew relation. The functional differentiation of the Levy—Perdew-relation (Eq. (89)) for exchange leads to the first

P

dv (r; n) v (r; n)"!r ) +v (r; n)! dr n(r )r ) + 9 , 9 9 1 1 1 1 dn(r ) 1 the second

P

(184)

dv (r; n) dv (r; n) 1 d2v (r; n) 9 9 "!(r ) +#r ) + ) 9 ! dr n(r )r ) + , (185) 1 1 dn(r ) 2 2 2 2 dn(r )dn(r ) dn(r ) 2 1 1 1 2 and higher order equations of the hierarchy. This hierarchy is self-contained. It means that it contains only the exchange energy functional and its functional derivatives. It is the consequence of the fact that the exchange energy functional scales homogeneously as it has been shown by Ou—Yang and Levy [94]. The hierarchies of equations can be used to check the quality of approximate functionals. One can use first the zeroth order equations of the hierarchies. Inserting the approximate expressions into both sides of the zeroth order equation, the values of two integrals, i.e., the two numbers can be compared and used for the checking. On the other hand, if the first order equations of the hierarchies are applied as criteria for the approximate functionals, the comparison of the two sides of these equations would involve a comparison of two functions of r instead of two numbers. So the first order equations of hierarchy give more information. Similarly, higher order equations of hierarchy provide even more information. Certainly, to apply these equations, higher order derivatives of the functionals are needed. It is interesting to note that the Legendre transforms of energy functionals as functionals of potentials are available. The electron density appears as a functional derivative of Legendre transforms and the first order equations of the hierarchies provide equations for the electron density. The Legendre transform of the exchange energy is studied in the weighted density and gradient approximations. The first order equation of the hierarchy gives useful differential equation for the electron density. This equation can be used in constructing or improving approximating functionals. The hierarchies of equations are exact. The exact functionals satisfy these equations. On the other hand, these equations do not generally hold for approximate functionals. Occasionally, even

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approximate functionals are exact solutions of the hierarchy equations. For instance the Thomas—Fermi [1] and the Thomas—Fermi—Dirac [3,2] models can be derived from the hierarchy equations for the non-interacting kinetic and exchange energies. The derivation is based on the locality assumption. With the assumption that the functionals are local the exact solutions are the Thomas—Fermi and Thomas—Fermi—Dirac solutions. The truncation of the hierarchies of the kinetic and exchange energies results in rigorous lower bounds to the kinetic energy and upper bounds to the exchange energy in the plane-wave approximation. In both cases an additional assumption was done (locality or truncation). This assumption includes an approximation. (So we cannot obtain the exact solution.) In these cases the hierarchy of equations cannot be used to judge the quality of the approximation and the validity of the approximation can be studied by other methods. 5.6. Functional expansions There is a growing interest in the properties of energy functionals. A recent approach in this field includes functional expansions. Functional expansions have been derived while studying hierarchies of equations for the noninteracting kinetic [86], exchange and exchange-correlation [87] functionals and the Legendre transforms of different energy functionals [90]. Important identities have been obtained making use of functional expansions [86, 87, 90, 95, 96]. For any well-behaved functional F[n] [96] up to a constant

P

PP

1 F[n]" dr n(r)F@(r; n)# 2

dr dr n(r )n(r )F@@(r , r ; n)#2 , 1 2 1 2 1 2

(186)

where it is supposed that the successive functional derivatives (denoted by primes) exist and the series converges at least in a region near n. The functional expansions have recently been applied to generate a transition functional method [97], which can be considered the functional generalization of Slater’s transition state method [98]. This transition functional method can be used to calculate energy differences if the functional derivative is known at the transition density:

P

F[n ]!F[n ]" dr [n (r)!n (r)]F@ (r; n)# third and higher order terms . 2 1 0 2 1

(187)

The second order terms disappeared. The functional derivatives are evaluated at the transition density n (r)"1(n (r)#n (r)) . (188) 0 2 1 2 For example for iso-electronic ions the kinetic and exchange energy differences have the forms

P P

¹ [n ]!¹ [n ]+ dr [n (r)!n (r)]v0 (r) , 4 2 4 1 1 2 KS

(189)

E [n ]!E [n ]+ dr [n (r)!n (r)]v0(r) , 9 2 9 1 2 1 9

(190)

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respectively, where both the Kohn—Sham v0 and the exchange v0(r) potentials are taken at the KS 9 transition density. The transition functional method is especially useful if the functional itself is unknown, and only the first functional derivative is available. A very important example is as follows: Nowadays there are several methods [99—105] to determine the exchange-correlation potential if the electron density is known (see Section 8). One can obtain, however, the exchangecorrelation potential only as a function of the radial distance and not as a functional of the electron density. So, the exchange-correlation energy cannot be calculated. Using the transition functional method one can determine exchange-correlation energy differences even if the exchange-correlation energies are not known. Several other identities have been derived for local functionals and functionals containing gradient terms. Combining the constrained-search formalism with Taylor series expansions general expansions of E [n] and ¹ [n] in terms of homogeneous functionals can be derived [106—108]: Making the # # general postulate that there exists an expansion of Ej in powers of j # = jk dkEj[n] = jk # Ej[n]" + " + A [n] (191) # k! k! k djk j/0 k/1 k/1 and inserting it into Eq. (173) one obtains

A

P

! dr n(r)r ) +

B

A

B

dA [n] k "(1!k)A [n] . k dn(r)

(192)

It means that the kth component of E [n] and consequently ¹ [n] is a homogeneous functional of # # degree (1!k) in coordinate scaling, i.e., A [n ]"a1~kA [n] , k a k where

k"1,2,2 ,

n "a3n(ar) . a With j"1, Eq. (191) becomes

(193)

(194)

= A [n] E [n]" + k . # k! k/1 Similarly, one can obtain that

(195)

= A [n] k ¹j[n]"! + . (196) # (k!1)! k/1 If one assumes that A [n] is a local functional then up to third order the correlation energy can be k expressed [106] as

P

P

E "C N#C dr n2@3# dr n1@3#C , # 1 2 4

(197)

where N is the total number of electrons and the constants C —C can be determined by fitting. 1 4

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5.7. Adiabatic connection and perturbation theory The adiabatic connection [54—56] is a key concept in the density functional theory. It is not only supposed that the electron density is the same for both the interacting and non-interacting systems, but there exists a continuous path between them. The coupling constant path is defined by the Schro¨dinger equation [¹K #a»K #»K ]DW T"E DW T . (198) %% a a a a The density is supposed to be the same for any value of the coupling constant a. a"1 corresponds to the fully interacting case, while a"0 gives the Kohn—Sham system. Consider the functional F [n] j F [n]"min S¹K #j»K T . j %% W ?n Noticing that for the interacting system

(199)

F [n]"¹[n]#» [n] , 1 %% while for the non-interacting case

(200)

F [n]"¹ [n] , 0 4 the exchange-correlation energy has the form

(201)

E [n]"F [n]!F [n]!J[n] 9# 1 0 1 ­F [n] " dj j !J[n] . ­j 0 Using the Hellmann—Feynmann theorem

P

­F [n] j "S»K T , %% j ­j

(202)

(203)

we obtain the adiabatic connection formula for the exchange-correlation energy:

P

1 dj (S»K T !J) . %% j 0 From the scaling relation for the Coulomb repulsion energy E [n]" 9#

J[n ]"a~1J[n] , 1@a another form of the adiabatic connection formula follows:

P

(204)

(205)

1 a(» [n ]!J[n ]) da . (206) %% 1@a 1@a 0 With the coupling-constant integration important expressions can be derived for the correlation energy [109]. E [n]" 9#

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It has been shown [110] that the Kohn—Sham potential has the form

C

D

dE [n ] v (r)"v (r)!a v (r)#v (r)#a # j , j"1/a , a 0 J 9 dn(r)

(207)

where v (r) is the non-interacting Kohn—Sham potential, v (r) and v (r) are the classical Coulomb 0 J 9 and exchange potentials, respectively, while E is the correlation energy. The density functional # perturbation theory is based on the Taylor series of v (r) a = v (r)" + akv (r) a k k/0 and E a

(208)

= E " + akE . (209) a k k/0 The zeroth order term is equal to the Kohn—Sham potential, while the higher order terms are related to the Coulomb, exchange and correlation potentials. It was shown that potentials v (r) are k functionals of the Kohn—Sham orbitals, eigenvalues and the potentials v (r),2,v (r). Thus all the 1 k~1 potentials v (r) can be calculated exactly for any given Kohn—Sham potential. In practice, however, k the potentials only up to some finite order can be determined. Similarly, the terms in the series of E can be obtained a

P

E "F [Mu N,Me N,v (r)2v (r)]# n(r)v (r) . k k i i 1 k~1 k

(210)

Comparing the density functional perturbation theory with the conventional quantum chemical perturbation theory (e.g. Møller—Plesset perturbation theory) the important difference is in the fact that the self-consistent procedure leading to the Kohn—Sham wave function already depends on the perturbation theory, so the perturbation comes into play at an earlier stage. With the help of the density functional perturbation theory formally exact expressions can be gained for the exchangecorrelation energy and potential. It seems reasonable that approximate functionals can be directly constructed from the Kohn—Sham orbitals and eigenvalues. It is not necessary to know how these can be explicitly formulated as functionals of the density. It is believed that functionals based directly on the Kohn—Sham orbitals and eigenvalues provide much more freedom in the construction of approximate expressions.

6. Fundamental concepts based on density functional theory 6.1. Chemical potential and electronegativity In the density functional theory the total energy E[n] is a unique functional of the density n. The variational principle leads to the Euler equation (19). The Lagrange multiplier k is the chemical

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potential that plays an important role in the density functional theory. Another expression for the chemical potential written as

A B

k"

­E ­N

(211) 7 can be easily seen by an infinitesimal change of the given ground-state energy into another ground-state energy:

P

dE"k dN# n(r) dv(r) dr ,

(212)

where v is the external potential. As the chemical potential is constant, i.e., takes the same value in every point of the molecule or solid considered, its analogy with the macroscopic chemical potential is transparent. It is even more important that one can define the electronegativity [111] s with the chemical potential: s"!k .

(213)

Mulliken [112] long ago showed that the absolute electronegativity of a species can be sensibly defined as the average of its ionization potential I and electron affinity A s "1(I#A) . (214) M 2 This is just the finite difference approximation to Eq. (211). From the definition of the electronegativity, Sanderson’s principle [113] immediately follows: the electronegativity equalizes when two species unite to form a new species leading to a single electronegativity or chemical potential (the same way as in ordinary thermodynamics). 6.2. Hardness and softness The hardness g of an electronic system is defined [114] as

A B A B

1 ­2E 1 ­k " . (215) g" 2 ­N2 2 ­N 7 7 This is a global quantity, often called absolute hardness to emphasize the fact that its value for an atom in some environment can be different from its value in isolation. The global softness [115] is the inverse of global hardness:

A B

­N 1 S" " ­k 2g

. (216) 7 From the convexity of the total energy E(N) follows that g50. It can be easily shown [22] that the global hardness can be approximated by g+1(I!A)+1(e !e ), (217) 2 2 LUMO HOMO where HOMO and LUMO stand for the highest occupied molecular orbital and lowest unoccupied molecular orbital, respectively.

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The global softness, on the other hand, can be connected with the fluctuation of the particle number [115]. Considering a grand canonical ensemble with temperature ¹, volume » and chemical potential k, the equilibrium fluctuation of the particle number is given by a well-known expression of statistical physics:

A B

1 ­SNT [SN2T!SNT2]" k¹ ­k

.

(218)

T,V

Noticing the analogy between Eqs. (216) and (218) the softness is 1 [SN2T!SNT2]"S . k¹

(219)

The concepts of chemical hardness and softness are useful in studying acid—base reactions. It turned out that [115] hard (soft) acids generally have high (low) positive charge, low (high) polarizability and small (large) size. On the other hand, hard (soft) bases possess the property of high (low) electronegativity, low (high) polarizability and it is difficult (easy) to oxidize them. Studying the concepts of chemical hardness and softness, Pearson [116] concluded that there is a principle of maximum hardness, i.e. molecules arrange themselves so as to be as hard as possible. This principle is equivalent to the principle of minimum softness. It was formally proved [117] making use of the fluctuation—dissipation theory of statistical mechanics. Another very important theorem is the so-called HSAB principle. In acid—base reactions hard (soft) acids prefer to coordinate with hard (soft) bases. Two formal proofs [118] are available for this principle. Both the principle of maximum hardness and the HSAB principle proved to be very powerful in the theory of chemical reactions. 6.3. Fukui function and local softness To understand the local properties of molecules or solids, local quantities which vary from place to place are introduced. In order to measure the chemical reactivity of a particular site in a molecule different local variables are defined. The Fukui function [119] f (r) is defined as

C D

f (r)"

dk dv(r)

.

(220)

N

It measures how sensitively the chemical potential reacts to an external perturbation at a particular point [22]. With the invertion of the order of variation and differentiation

A B

A B

­ dE d ­E " ­N dv(r) dv(r) ­N

(221)

the Fukui function f (r) has the form

A B

f (r)"

­. (r) ­N

v

.

(222)

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f (r) is established as an index of considerable importance for understanding molecular behaviour. The Fukui function measures reactivity:

A B . A B . A B

­. (r) ~ , electrophilic , (223) ­N v ­ (r) ` f `(r)" , nucleophilic , (224) ­N v ­ (r) 0 , radical (225) f 0(r)" ­N v reagents. This reactivity index can be easily approximated using the HOMO and LUMO orbital densities l and l , respectively: HOMO LUMO f ~(r)+l (r) , (226) HOMO f `(r)+l (r) (227) LUMO (r)#l (r)) . (228) f 0(r)+1(l LUMO 2 HOMO The density functional definition of the Fukui function provides a firm foundation of the frontierelectron theory [120]. The local softness [115] is defined as f ~(r)"

C D ­n(r) ­k

(229)

S" s(r) dr .

(230)

s(r)"

v(r) and it yields the global softness upon integration:

P

The relationship between local softness and the Fukui function, s(r)"Sf (r) ,

(231)

reflects that these quantities contain the same information about the relative site reactivity in a molecule. For metals, the local and the global softness [115] have the forms s(r)"g(e , r) , (232) F S"g(e ) , (233) F respectively, where g(e , r) and g(e ) are the local and global density of states at the Fermi level. F F Higher functional derivatives have also been derived [93]. The most important ones are the second functional derivatives, the hardness and softness kernels; they have a fundamental role in the hierarchy equations for energy functionals (see Section 5.4).

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6.4. Density functional theory as thermodynamics 6.4.1. Transcription of ground-state density functional theory into a local thermodynamics Ghosh et al. [70] developed a local thermodynamical picture within the framework of groundstate density functional theory. They introduced concepts like local temperature, local entropy and local free energy density. Recently, this theory has been enlarged into an exact local thermodynamics [121]. The theory of Ghosh et al. provides a phase-space approach to the density functional theory. The density functionals are considered as averages in the phase-space. A distribution function f (r, p) in the phase-space is introduced with the following properties:

P P P

dp f (r, p)"n(r) ,

(234)

dr n(r)"N ,

(235)

dp 1p2f (r, p)"t (r) . 2 4

(236)

The entropy density s and the entropy S are defined as

P

s(r)"!k dp f (ln f!1) ,

P

S" dr s(r) ,

(237) (238)

where k is the Boltzmann constant. The most probable distribution function is obtained by imposing the criterion of maximum entropy subject to the constraints of correct density (Eq. (234)) and correct kinetic energy (Eq. (236)). Thus, f (r, p)"e~a(r)e~b(r)p2@2 ,

(239)

where a(r) and b(r) are r-dependent Lagrange multipliers. The local temperature ¹(r) is defined in terms of the kinetic energy density t (r)"3n(r)k¹(r) , 4 2 i.e. by the ideal gas expression. Eqs. (236), (239) and (240) lead to b(r)"1/k¹(r) .

(240)

(241)

For the entropy density we obtain (242) s(r)"!kn ln n#3 kn ln ¹#1 kn[5#3 ln(2pk)] , 2 2 the Sackur—Tetrode equation. The exchange energy can also be derived [122]. Assuming ¹ to be a function of the density, the Thomas—Fermi—Dirac model can be obtained with numerical values of the coefficients C and C which provide improvements of the original values. Predicted F 9 exchange energies [122] and predicted Compton profiles [123] are very good.

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6.4.2. The exact thermodynamics This phase-space approach of Ghosh et al. is, however, not exact. It is possible to go beyond this theory by postulating a thermodynamic description without using any statistical method [121]. The electron system under consideration has the ground-state electronic energy functional E,

P

E[n]" e[n; r] dr ,

(243)

where the energy density e is a functional of n. According to the Hohenberg—Kohn theorem the functional E takes its minimum value E[n ]"E (244) 0 0 at the exact ground-state electron density n . The variation of Eq. (244) leads to the Euler— 0 Lagrange equation

K

dE dn

(245)

P

(246)

"k n/n0 using the constraint n dr"N .

The Lagrange multiplier k is the chemical potential of the system. Now, we postulate thermodynamics by requiring the existence of a new functional

P

E[n, s]" e[n, s; r] dr ,

(247)

where the energy density e is a functional of both the density n and the entropy density s. The extremum principle for the ground-state takes the form d

GP

H

(e[n, s; r]!¹(r)s!kJ (r)n) dr "0 .

(248)

The Lagrange multipliers, the local temperature ¹ and the local chemical potential kJ are functions of r. The Euler—Lagrange equations of thermodynamics are

K K

dEI "¹(r) , (249) ds n dEI "kJ (r) . (250) dn s To build up a thermodynamical formulation we require that the local form of the fundamental equation EI "¹S!p»#kJ N

(251)

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holds, i.e., e"¹s!p#kJ n ,

(252)

where p is the local pressure. The Kohn—Sham kinetic energy density is written as t "3nk¹ . 4 2 Certainly, we have

(253)

¹"¹[n] ,

(254)

e[¹,n]"e[n] .

(255)

We emphasize that generally ¹ is a functional of the density n, not simply a function of n. In the local density approximation, all quantities are functions of n. E.g. ¹(r)"c(n(r)). The entropy contains a correction to the Sackur—Tetrode equation (Eq. (242)) resulting from the nonideality. The locality assumption causes ambiguity in the description as it was shown in Ref. [121]. Taking advantage of this ambiguity in the limiting case of the original density functional theory the model of Ghosh et al. is recovered. There is another way of using the ambiguity: the local chemical potential is taken to be a constant, i.e., equal to the chemical potential of the original density functional theory. In this case, using the scaling argument, the well-known Thomas—Fermi and Thomas—Fermi—Dirac models are recovered. 6.5. Work formalism Harbola and Sahni [124] proposed an interesting interpretation of the exchange-correlation potential. The electron-interaction potential that contains a kinetic energy component, can also be interpreted as a work done to bring an electron from infinity to its position at r against a field E(r):

P

d(E%%[n] ) r v%%(r)" "! E(r@) ) dr@ . dn = The field E(r) can be separated into

(256)

(257) E(r)"E (r)#E #(r) . %% t The field E (r) arises from the classical Coulomb and exchange-correlation terms, while the field %% E #(r) comes from the difference of the interacting and non-interacting kinetic energy tensors. The t potential v%%(r) can be written as a sum of the works: v%%(r)"¼ (r)#¼ #(r) , %% t where

P P

r

E (r@) ) dr@ , %% = r ¼ #(r)"! E #(r@) ) dr@ . t t =

¼ (r)"! %%

(258)

(259) (260)

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This interpretation is based on the fact that

+v%%(r)"!E(r) .

(261)

The condition on the path-independence of the work is

+]E(r)"0 .

(262)

The sum of the works ¼ (r) and ¼ #(r) is path-independent and for systems of a certain symmetry %% t such as closed shell atoms, jellium metal clusters, etc., the works ¼ (r) and ¼ #(r) are separately %% t path-independent. Following Harbola and Sahni [124] we write the electric field E (r) as a sum of the classical %% Coulomb E (r) and the exchange correlation E (r) terms: J 9# E (r)"E (r)#E (r) , (263) %% J 9# where

P P

E (r)" J

E (r)" 9#

n(r@)(r!r@) dr@ , Dr!r@D3

(264)

o (r, r@)(r!r@) 9# dr@ . Dr!r@D3

(265)

They supposed that

+v (r)"!E (r) . (266) 9# 9# Making use of the exchange-correlation hole density o (r, r@) and the (coupling-constant averaged) 9# pair-correlation function hM (r, r@) (Eq. (68)) the exchange-correlation energy has the form (Eq. (70))

P

1 1 E " n(r)n(r@)hM (r, r@) dr dr@ . 9# 2 Dr!r@D

(267)

It can be rewritten as

P

P

P

o (r, r@)(r!r@) 9# dr@" dr n(r)r ) E (r) . 9# Dr!r@D3

E " dr rn(r) 9#

(268)

Using Eq. (268) with the Levy—Perdew relation (Eq. (87)) we obtain

P

¹ "! n(r)r ) [E (r)#+v (r)] dr . 9# 9# #

(269)

Comparing this relation with Eq. (266) we notice that the Harbola—Sahni conjecture on the exchange-correlation field (Eq. (265)) does not satisfy [125,126] the Levy—Perdew relation. The Harbola—Sahni expression for the exchange can be derived from the differential virial theorem of Holas and March (Eq. (98)) which can be rewritten as a path-integral expression for the exchange-correlation potential [62]:

P

v (r )"! 9# 0

r

0

=

f (r) ) dr . 9#

(270)

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Neglecting the difference between the interacting and non-interacting density matrices in f , it has 9 the form

P

1 . f (r)"! dr@[C(r, r@)!n(r)n(r@)]/n(r)+ 9 rDr!r@D

(271)

Expressing the diagonal of the two-particle density matrix C(r,r@) with the pair-correlation function h(r, r@) (Eq. (67)) one obtains

P

f (r)" 9

o (r, r@)(r!r@) 9 dr@ . Dr!r@D3

(272)

We recognize f (r) to be identical with the force field E (r) (Eq. (265)) of Harbola and Sahni. 9 9# Finally, we note here that, following the work of Holas and March [62] their expression for v has been generalized by Levy and March [127] for electron—electron interaction je2/r , where 9# ij 0(j41. They thereby have exhibited a kinetic correction to the Harbola—Sahni exchange-only potential. The path dependence of this latter contribution is thereby annulled. The Harbola—Sahni formalism has the advantage, besides the appealing interpretation character, that the exchange-correlation potential in this model shows the exact asymptotic behaviour. Several calculations have been performed using the work formalism [128]. For instance, electron removal energies and electron affinities have been determined for atoms and ions. Even excited state calculations have been done with this method [129].

7. Optimized potential method Though the Kohn—Sham approach to the density functional theory is an exact scheme to treat the ground state properties of many-particle electronic systems, unfortunately, the exchangecorrelation part of this Kohn—Sham potential is not known exactly. The exchange potential, however, can be exactly determined by finding the optimized effective potential. The question of how to obtain the local potential whose eigenfunctions would minimize a given energy functional was first investigated by Sharp and Horton [130]. Their integral equation for the local potential was independently derived by Talman and Shadwick [47]. From calculations for atoms, they and Aashamar et al. [131] found that the calculated one-electron and total energies were very close to that of the Hartree—Fock method. Norman and Koelling [132] combined the optimized effective potential method with the technique of including self-interaction correction proposed by Perdew and Zunger [133]. The optimized effective potential method was generalized for the spin polarized case by Krieger et al. [49]. Krieger et al. [49] introduced an approximate OPM method. Recently, an alternative derivation to the KLI method was proposed [134]. The optimized potential method can be applied when the total energy is given as a functional of the one-electron orbitals u : i

P

E[u ]"¹ [u ]#J[u ]#E [u ]# dr v(r)n(r) , i 4 i i 9# i

(273)

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where ¹ [u ], J[u ] and E [u ] are the non-interacting kinetic, the Coulomb and the exchange4 i i 9# i correlation energies, respectively. The one-electron orbitals u are eigenfunctions of a local effective i potential » hK u "(!1+ 2#»)u "e u , (274) i 2 i i i with » being determined by requiring that E[u ] is minimized for all u obtained from Eq. (274). i i This results in

P

dE dE du*(r@) i "+ dr@#c.c. "0 . (275) d» du*(r@) d»(r) i i The functional derivative of the one-electron orbitals u with respect to the local effective potential i » can be calculated with the help of Green’s function: du*(r@) i "!G (r@, r)u (r) , i i d»(r)

(276)

(hK !e )G (r@, r)"d(r!r@)!u (r)u*(r@) , (277) i i i i Using Eqs. (274)—(277) an integral equation for the effective exchange-correlation potential » follows: 9#

P

H(r, r@)» (r@) dr@"Q(r) , 9#

H(r, r@)"+ u*(r)G (r, r@)u (r@) , i i i i

P

(278) (279)

(280) Q(r)"+ dr@ u*(r)G (r, r@)vi (r@)u (r@) . i i 9# i i The orbital dependent potential vi is given by 9# dE [u ] vi (r)" 9# i . (281) 9# u du* i i The effective exchange-correlation potential » can be determined from the effective potential »: 9# » (r)"»!v!v . (282) 9# J If the total energy were known as a functional of the one-electron orbitals u Eq. (282) would i result in the exact exchange-correlation potential. If the method is applied to the Hartree—Fock energy functional the exact exchange-only Kohn—Sham potential can be obtained. It was found [131,49,104,50] that the results of OPM and the HF calculations are almost identical, the total energies obtained from OPM are slightly higher than that of the HF. It is very difficult to calculate the effective potential » because of various numerical problems. It has already been done only for spherically symmetric systems. Krieger et al. [49] proposed an

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accurate approximate approach to OPM. They found that the analytic expression n (r) (283) »KLI(r)"»S(r)#+ i (»M KLI!vN ) , 9i 9 9 n(r) 9i i where n (r) v 9i »S(r)"+ i (284) 9 n(r) i is the Slater potential, is an extremely accurate approximation. »M and vN are the expectation 9i 9i values of the exchange potential »KLI and the Hartree—Fock exchange potentials v with respect to 9 9i orbital u . n (r) is ith one-electron density. i i 8. Potentials from electron density The exact functional form of the Kohn—Sham potential of the density functional theory is still unknown. However, if the density is known it is possible to calculate the Kohn—Sham, the exchange or exchange-correlation potentials. Several methods have been developed to obtain the potentials from the electron density. In this section, the most important methods are outlined [136]. In a density functional calculation, usually the Kohn—Sham equations (Eq. (39)) are solved self-consistently. In the so-called inverse problem, the density is known and from Eqs. (26), (33) and (39) e ,u and v are determined. If we have only two electrons the inverse problem can be trivially i i KS solved: n"2DuD2 ,

(285)

1 1 v "e# + 2Jn . KS 2Jn

(286)

This question has been studied by Almbladh and Pedroza [102], Davidson [135] and Umrigar and Gonze [137]. In this case the exchange potential

P

1 n(r@) v "! dr@ 9 2 Dr!r@D

(287)

corresponds to the self-exchange. The correlation potential takes the form

P

1 n(r@) 1 1 + 2Jn!v! dr . v "v !v "e# # 9# 9 2 Dr!r@D 2Jn

(288)

The nontrivial cases of two- and three-level spherically symmetric systems are detailed in Refs. [61,136]. Several methods have been worked out for systems with arbitrary number of electrons. Stott et al. [103] proposed the following method. The one-electron orbital with the lowest energy can be expressed as

A

B

1@2 N u " n! + u2 . 1 k k/2

(289)

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The Kohn—Sham potential is given by 1 v " + 2u #e KS 2u 1 1 1 1 N 1@2 " + 2 n! + u2 #e . (290) k 1 2(n!+N u2)1@2 k/2 k k/2 Combining Eqs. (39) and (290) a set of coupled nonlinear second-order differential equations

A

C

B

A

B D

N 1 1@2 1 + 2 n! + u2 u "eJ u , k"2,3,2,N ! + 2u # k k k k k 2(n!+N u2)1@2 2 k/2 k k/2 are obtained, where

(291)

eJ "e !e . (292) k k 1 If the density is known the single-particle wave functions can be determined from Eq. (291). Then Eq. (290) gives the Kohn—Sham potential. Parr and collaborators [101] used the constraint 1 2

PP

[n(r)!n (r)][n(r@)!n (r@)] 0 0 dr dr@"0 Dr!r@D

(293)

to determine the orbitals corresponding to the input density n . The minimization of the kinetic 0 energy min SDD¹K DDT D?n0 using the constraint (293) leads to the one-electron equations

(294)

[!1+2#v(r)#vj(r)]uj(r)"ejuj(r) , 2 # i i i where

(295)

P

vj (r)"j 9#

n(r)!n (r) 0 dr . Dr!r@D

(296)

The Kohn—Sham orbitals are obtained by extrapolating the Lagrange multiplier j to R. To speedup the convergence the one-electron equations

C

A

B

D

1 1 ! + 2#v(r)# 1! vj(r)#vj (r) uj(r)"ejuj(r) 9# i i i 2 N J

(297)

are solved instead of Eq. (295), where the factor (1!(1/N)) in the Coulomb potential

P

n(r@) dr vj" J Dr!r@D

(298)

can be thought of as the Fermi—Amaldi self-interaction correction. The exchange-correlation potential is given by

A

B

1 vj! vj . v "lim 9# j?= # N J

(299)

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Another method of calculating the Kohn—Sham potential has been proposed by Go¨rling [104]. The linear response

P

dn(r)" dr@ G(r, r@)dv (r@) KS

(300)

of the electron density to the changes of the effective potential v can be obtained from KS perturbation theory. The Green’s function is given by 0## 6/0## u*(r) u (r)u*(r@)u (r@) s s i # c.c. G(r,r@)"2 + + i e !e i 4 i s To determine the change in the effective potential

P

dv (r)" dr@ G~1(r, r@)dn(r@) KS

(301)

(302)

the inverse of the Green’s function should be calculated. Go¨rling proposed an iterative method to obtain the Kohn—Sham potential and one-electron orbitals and energies. The present author [99] has worked out a numerical, iterative method. Starting from an appropriate potential, the Kohn—Sham equations are solved self-consistently then using the Kohn—Sham potential v(1) and the electron-density n(1) obtained, another Kohn—Sham potential KS constructed and the Kohn—Sham equations are solved again. This process goes on until the density equal to the input density is reached. It has turned out that the potential of the (i#1)th iteration can be constructed as

C

D

n »(i`1)"»(i) 0 c#(1!c) , KS KS n(i)

(303)

where c is a damping factor applied to speedup the convergence. From the Kohn—Sham potential the exhange-correlation potential can be readily obtained. If the input density is the exact one, the exact exchange-correlation potential can be determined. However, if only the Hartree—Fock density is available as an input, the exchange-correlation potential obtained by the abovementioned process is very close to the exact exchange potential [104,50]. A method very similar to this has been worked out by Baerends and coauthors [138,105]. As we have seen if the exact density is known, the exchange-correlation potential can be determined exactly. However, it should be emphasized that the exchange-correlation potential can only be obtained as a function of the radial distance r. The form of the exchange-correlation potential as a functional of the density still remained unknown.

9. Functionals 9.1. Local density approximation (¸DA) The density functional theory is exact in principle, but the exact form of the energy functional is unknown, so approximations have to be used in calculations. Numerous approximate methods

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have already been proposed. The simplest forms are the local expressions. We have to emphasize that the term “local” is used in several meanings: First, a potential may be local in multiplicative sense. E.g. the Kohn—Sham potential (Eq. (39)) is local, whereas the Hartree—Fock exchange potential is nonlocal. Secondly, a functional F is local, if

P

F[n]" dr f (n(r)) ,

(304)

i.e. the density f corresponding to F depends on r only through the electron density n. In this sense the term “local density approximation (LDA)” is used. Third, a potential is local if its value at a given point depends on the value of the density (and finite number of its derivatives) at that point only. In this sense the gradient functionals

P

F[n]" dr f (n(r), ­ n(r), ­2 n(r),2) k kl

(305)

are also local. In a great majority of the papers on density functional theory the term “local” if it refers to a functional is used in the second sense and functionals in Eq. (305) are called nonlocal. In this subsection local functionals (in the second sense) are reviewed. These are the simplest and most widely used approximations. Starting with the exchange we can easily arrive at this approximation by studying the homogeneous electron gas. In the lowest order perturbation theory for a homogeneous system one obtains the contribution to the exchange energy density e 9 3 3 1@3 (e (n)) "! e2n(r)4@3 . (306) 9 LDA 4p p

AB

It is worth mentioning that an exchange energy density proportional to the 4 power of the density 3 can be obtained in several other ways, too. It was first obtained by Dirac [3], Slater [4], Ga´spa´r [5], Kohn and Sham [7]. It can be derived, e.g., from dimensional considerations [23]. The first equation of the hierarchy of the exchange energy [87,90] with the locality assumption leads also to this expression. It can also be derived from the thermodynamical picture of the density functional theory [121]. The value of the constant in front of the 4 power of the density, was a crucial question 3 in the so-called Xa method [98] which is still in use in special calculations [139]. Considering higher order perturbations for a homogeneous system, one notices that all higher order terms contribute to the correlation energy density e . Recent local density functional # calculations are based on the results of accurate Green’s function Monte Carlo calculations of Ceperley and Alder for the homogeneous electron gas [140]. The most widely used parametrisations are due to Vosko, Wilk and Nusair [141] and to Perdew and Zunger [133]. The most recent one is [142]. From the genesis of the local density approximation, one would expect that this approximation is reasonably accurate, if the density varies slowly. The approximation gives, however, acceptable results for real electronic systems, where this condition is (strongly) violated. One reason for this is that the local density exchange-correlation-hole satisfies the exact sum rule. Nevertheless, the local density approximation leads to a number of deficiencies, the most serious being a rather imperfect cancellation of self-interaction effects, which leads to the incorrect asymptotic limit of the local density exchange-correlation potential. This leads, e.g., to the result that negative ions are poorly

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described by the local density approximation. One of the most frequently used method that incorporates self-interaction correction was proposed by Perdew and Zunger [133]. Definitely improved results can be obtained by this method. For a review see [143]. Another method, that goes beyond the local density approximation is the weighted density approximation (WDA) [144]. The gist of this approximation, which leads to nonlocal energy functional, is a more careful transcription of the exchange-correlation hole of the homogeneous electron gas to the case of an inhomogeneous system. o%9!#5(r, r@)PoWDA(r, r@)"n(r@)[h)0.(n ,Dr!r@D)!1] 0 N r . (307) 9# 9# 0 n ?n( ) The weighted density exchange-correlation hole features the correct density prefactor and the density argument of the homogeneous pair correlation function is replaced by a weighted density nN (r), which is determined by requiring that the basic sum rule (Eq. (75)) be satisfied for each value of r. This leads to an improved but not perfect cancellation of self-interaction effects. From among the several other local approximations we note in passing the Gomba´s—Lie— Clementi [145] or local Wigner functional for correlation parametrized also by Wilson and Levy [146] and Su¨le and Nagy [147]. The same expression was proposed by Lee and Parr [148] for exchange-correlation energy. 9.2. Approximations containing the gradient of the density There are several approximations that go beyond the local density approximation. Second order gradient corrections to the exchange-correlation energy functional ELDA can be obtained by 9# consideration of the lowest order energy shift due to linear response of a homogeneous system. Linear response also yields some of the higher order gradient corrections, but for a complete calculation of all contributions higher order response functions have to be considered. The static linear response argument (see Ref. [26]) leads to the second order gradient correction to the exchange-correlation energy having the form

P

E*2+[n]" dr B (n(r))(+n(r))2 . 9# 2

(308)

The coefficient B (n) obtained at the RPA level using a high density expansion is found to be 2 proportional to 1/n4@3. A more careful evaluation of the gradient contribution in the ring approximation was carried out by Geldart and Rasolt [149], Langreth and Perdew [150], and Hu and Langreth [151]. Unforturnately, it turned out that the addition of second order gradient corrections to local exchange-correlation energy functional ELDA did not lead to a systematic improve9# ment. This failure was the starting point for the development of generalised gradient approximations (GGA). The first ones were introduced starting with an ansatz of the form

P

DEGGA" dr n(r)4@3f (n,s) , 9# 9#

(309)

where s is the dimensionless variable s"((+n(r))2/n(r)8@3)1@2 .

(310)

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The function f is chosen so as to reproduce the limiting cases of high as well as low density 9# gradients correctly. For the exchange functional Becke (1992-version [152]) proposed the following form: s2 f (n, s)"!b . 9 1#6bs arcsinh(s)

(311)

For small s (i.e., low gradients) it reproduces the gradient expansion result

P

(+n)2 . DEBP !b d3r 9 n4@3 s?0 For the case of an exponential tail density one has s"ce1@3crJn~1@3

(312)

(313)

which diverges as rPR. In this limit the exchange energy density 1 n(r) , eB(r)"! 9 2 r

(314)

falls off correctly. The parameter b is fitted to reproduce the Hartree—Fock exchange energies of the noble gas atoms from He to Rn giving b"0.0042 a.u. We note in passing that it was shown by Engel et al. [48] that generalized gradient approximations for exchange cannot simultaneously reproduce the correct asymptotic forms of both the energy and the potential. They also pointed out that the correct asymptotic form of the exchange energy does not lead to significant improvement. On the other hand, Baerends [153] and coworkers argued that the correct asymptotic behaviour of the exchange potential has a crucial effect on the results. To investigate more closely the reasons for the failure of the gradient expansion, following Perdew and Wang [154], we turn to study the exchange-correlation hole. The exchange hole should be strictly negative and should satisfy the basic sum rule (Eq. (76)). On the other hand, the correlation hole satisfies the sum rule (Eq. (77)), being negative close to the electron and positive further away. It can be shown that while the holes obtained from the local density approximation satisfy these conditions, the holes calculated from the gradient expansion do not. In order to construct a generalized gradient approximation the following cut-off procedure was proposed: 1.

G

oGE oGGA" 9 9 0

if oGE40, 9 if oGE50; 9

2. oGGA"0 for Dr!r@D5R , 9 9 where R is chosen so that the sum rule is satisfied. Then the resulting numerical exchange energy 9 density is fitted to an appropriate form. A similar analysis can be applied to the correlation part. The cut-off is given by oGGA"0 for Dr!r@D5R , # #

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where R is chosen so that the sum rule for correlation (Eq. (77)) is satisfied. Regarding the rather # involved parametrisation of the numerical results we refer to [155]. It was pointed out by Engel and Vosko [156] that the Perdew—Wang form for exchange does not reproduce the exact OPM exchange potential correctly. They then constructed a new exchange energy EGGA, which reproduces the OPM exchange potential much better. (On the other hand, the 9 new form reproduces the exchange energy worse than the Perdew—Wang functional). We note in passing the Wigner-like Wilson—Levy [146](WL) functional, which was constructed to satisfy certain coordinate scaling relations:

P

an#bD+nD/n1@3 EWL[n]" dr , (315) # c#dD+nD/(n/2)4@3#r 4 a, b, c, and d are parameters [146], and r is the Wigner—Seitz radius: 4 r "(3/4pn)1@3 . (316) 4 Another frequently used approximation for the correlation energy is the Lee—Yang—Parr (LYP) [157] formula which involves the Weizsacker kinetic energy term and is based on the Colle—Salvetti expression [158]. Correlational functional involving the Laplacian of the density has recently been proposed by Proynov et al. [159]. They supposed that the local temperature (see Section 6.2) plays the role of a Fermi-hole correlation length. They found that combined with the exchange functional of Becke [160] and Perdew [161] their correlation functional gives improved results for the binding energies and the geometries of molecules.

10. Applications 10.1. Atoms First, we have to emphasize that exchange can be treated exactly within the optimized potential method (see Section 7). For atoms and atomic ions this method can be applied. To simplify the numerically rather involved method Krieger et al. [49] worked out a very accurate approximation. Table 1 presents exchange energies for several closed shell atoms. The gradient exchange functional of Becke [160] provides dramatic improvement over the local density approximation. Table 2 shows total energies. As it is expected the Hartree—Fock total energy is somewhat lower than the one obtained from the exchange-only OPM [48,131], though the difference is extremely small. One finds that the local density approximation does not perform too badly. Its error is of the same order as the error in Hartree—Fock, but in the opposite direction. In the GGA (with results above the experimental values) the accuracy is improved considerably. Turning to the correlation, following Gross and coworkers [162], the conventional quantum chemical and the density functional correlation energies are compared. In quantum chemistry the correlation energy is traditionally defined as the difference between the exact (non-relativistic) energy and the total Hartree—Fock energy: EDC"E !E . # %9!#5 HF

(317)

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Table 1 Exchange energies of atoms calculated by the Hartree—Fock [48], the OPM [48,131], the LDA [160] and the gradient-corrected functional of Becke [160](in a.u.) Atom

HF

OPM

LDA

B

He Be Ne Mg Ar

!1.026 !2.667 !12.108 !15.994 !30.185

!1.026 !2.666 !12.105 !15.988 !30.175

!0.884 !2.312 !11.03 !14.61 !27.86

!1.025 !2.658 !12.14 !16.00 !30.15

Table 2 Total energies of atoms calculated by the Hartree—Fock, the OPM, the LDA and the GGA methods (in a.u.) Atom

Exp.

HF

OPM

LDA

GGA

He C Ne Si Cl

!2.9037 !37.8450 !128.939 !289.383 !460.217

!2.8617 !37.6886 !128.547 !288.854 !459.482

!37.6865 !128.546 !288.850 !459.477

!2.975 !38.0522 !129.317 !289.912 !460.838

!2.8989 !37.8243 !128.945 !289.368 !460.162

On the other hand, in the density functional theory the correlation energy is obtained by inserting the exact ground-state density into the correlation functional EDFT # EDFT"E [n] , (318) # # which is given by the difference of the exact exchange-correlation and the exchange energies: EDFT"EDFT[n]!EDFT[n] . (319) # 9# 9 Keeping in mind that the density functional exchange energy can be given by the Hartree—Fock expression for exchange energy providing that the exact Kohn—Sham orbitals are inserted into it: EDFT"EHF[uKS] . 9 9 i The density functional correlation energy can be expressed as

(320)

EDFT"E[n]!EHF[uKS] , (321) # i because E[n]"E , while the quantum chemical correlation energy has the form %9!#5 EQC"E[n]!EHF[uHF] . (322) # i Since the Hartree—Fock orbitals are the ones that minimize the Hartree—Fock total energy, the following inequality holds EDFT4EQC . # #

(323)

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However, the difference between the two correlation energies is very small [162], as can be seen from Table 3. D in the last column denotes the value of DEQC !EDFT D/EDFT in percent. #,%9!#5 #,%9!#5 #,%9!#5 Table 4 presents correlation energies calculated by various approximate functionals: the Wilson—Levy (WL) [146], the Lee—Yang—Parr (LYP) [157], the Perdew and Wang (GGA) [83] the local Wigner functionals (LW) [163,147] and the local correlation functional of Perdew and Wang (LDA) [164]. For purposes of comparison experimental results [165] are also shown. The correlation energy is overestimated by roughly a factor of 2 in the local density approximation, while the GGA yields an accuracy below 5% (with the exception of He). The fact that total energy in the local density agrees much better with empirical data than either exchange or correlation demonstrates the often quoted cancellation of errors. All the models considered except the local density approximation yield rather close correlation energies, which agree satisfactorily with the available empirical data. There is only some relative overestimation of correlation for heavier atoms in the Lee—Yang—Parr model. The least deviation is achieved in the Wilson—Levy model. For ionic systems the picture is less consistent. All functionals fail to reproduce the correlation energy for the F~ anion. Table 3 Density functional, conventional quantum chemical correlation energies (QC) and their difference [162] (in a.u.) DFT H~ He Be`2 Ne`8 Be Ne

!0.041 !0.042 !0.044 !0.045 !0.096 !0.394

D

QC 995 107 274 694 2

!0.039 !0.042 !0.044 !0.045 !0.094 !0.390

821 044 267 693 3

#0.002 #0.000 #0.000 #0.000 #0.001 #0.004

D% 174 063 007 001 9

5.2 0.2 0.02 0.002 2.0 1.0

Table 4 Correlation energies of atoms obtained by various approximate correlation energy functionals (in a.u.) [166]

He Be Ne Mg Ar Kr Xe Li` Be2` Ne6` B` Li~ F~

WL

LYP

GGA

LW

LDA

Exp

0.042 0.094 0.383 0.444 0.788 1.909 3.156 0.044 0.045 0.109 0.101 0.0805 0.368

0.043 0.094 0.383 0.459 0.750 1.748 2.742 0.047 0.049 0.129 0.106 0.0732 0.362

0.046 0.094 0.383 0.451 0.771 1.916 3.150 0.051 0.053 0.123 0.103 0.078 0.362

0.042 0.094 0.374 0.462 0.771 1.948 3.174 0.060 0.075 0.187 0.114 0.069 0.332

0.112 0.223 0.743 0.888 1.426 3.267 5.173 0.134 0.150 0.334 0.252 0.182 0.696

0.042 0.094 0.392 0.444 0.787

0.044 0.044 0.187 0.111 0.073 0.400

48

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Recently, the exact correlation energy density has been calculated for the He atom and compared with the model functionals [166]. The correlation energy density obtained from the abovementioned model functionals have quite different local behaviour. The form of exchange, correlation and exchange-correlation potentials is of considerable importance. Using the methods described in Section 8 these potentials can be exactly determined provided that the electron density is known. They can be readily compared with model potentials. Neither the local nor the gradient-corrected approximations provide potentials having the correct asymptotic behaviour. Baerends and coworkers [153] pointed out that the correct asymptotic behaviour of the potentials is much more important than that of the corresponding energy density. Recently, there has been considerable progress in determining the difference between the interacting and non-interacting kinetic energy [60,109,167]. Several approximate expressions have been compared in Ref. [168]. 10.2. Molecules The density functional theory has developed into a cost-effective general method of calculating molecular properties. There is a vast literature on the application of the Kohn—Sham theory to chemistry. It seems impossible even to give a complete list of the reviews on molecular computations. So, only two recent reviews are referred [169,170] and the present survey deals only with a brief comparison of some frequently used functionals based on systematic studies of Pople and coworkers [171] and Handy and collaborators [172]. Table 5 [166] presents correlation energies for 21 closed shell molecules calculated with the Wilson—Levy [146], the Lee—Yang—Parr [157], the local Wigner [163,147] and the Perdew—Wang [83] functionals. For these molecules all these functionals yield a reasonable estimate of the experimental correlation energy [173]. It was, however, found [166] that there are considerable differences in the correlation energy density. Especially the Wilson—Levy model leads to a local behaviour quite different from the others. It is well known that the Hartree—Fock method predicts bond lengths to be too short, vibrational frequencies to be systematically large and binding energies to be too low. On the other hand, MP2 predictions of the bond length for single bonds are also too short, but multiple bonds can be either too short or too long. MP2 gives generally too large vibrational frequencies and too low binding energies. The local density approximation (see Section 9.1) leads to severe overbinding (see atomization energies in Table 6), while other quantities such as bond length, bond angles, vibrational frequencies tend to agree quite well with experiment. The bond length, e.g., presented in Table 7 for some first-row diatomic molecules, is overestimated (the mean deviation is about 0.01 A_ ). The Lee—Yang—Parr correlation functional [157] combined with Becke’s exchange functional [160] gives bond lengths which are somewhat long, vibrational frequencies that are often better than MP2 and quite satisfactory atomization energies. Simple hydrides with lone-pair electrons (e.g. NH ) tend to bind less, while molecules with multiple bonds, such as F , overbind at this level 3 2 of approximation. The Becke—Perdew approximation [160,161] is an improvement over the local density approximation, especially for single bonds. For certain XH bonds, however, the bond length is too long and consequently vibrational frequencies are too low. For atomization energies this functional gives high accuracy predictions.

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Table 5 Correlation energies of molecules obtained by various model correlation energy functionals Molecule

WL

LYP

LW

PW

Exp

H 2 Li 2 Be 2 B 2 C 2 N 2 O 2 F 2 H O 2 NH 3 CH 4 HF LiH LiF HCN CO H O 2 2 C H 2 2 C H 2 6 C H 2 4 CO 2

0.049 0.136 0.231 0.336 0.446 0.532 0.621 0.683 0.386 0.376 0.369 0.377 0.088 0.417 0.525 0.516 0.690 0.504 0.678 0.593 0.865

0.038 0.133 0.200 0.289 0.384 0.483 0.583 0.675 0.340 0.318 0.294 0.363 0.089 0.418 0.464 0.484 0.638 0.443 0.551 0.497 0.791

0.029 0.134 0.193 0.265 0.344 0.435 0.533 0.633 0.314 0.268 0.241 0.335 0.083 0.343 0.410 0.440 0.569 0.386 0.426 0.417 0.720

0.046 0.137 0.205 0.296 0.391 0.490 0.588 0.671 0.347 0.338 0.320 0.367 0.092 0.415 0.478 0.488 0.652 0.466 0.577 0.529 0.807

0.041 0.122 0.205 0.330 0.514 0.546 0.657 0.746 0.367 0.338 0.293 0.387 0.083 0.447 0.527 0.550 0.691 0.476 0.553 0.528 0.829

Note: The notations of model functionals are the same as in Table 4. Exp denotes the experimental correlation energies [165]. For CO and C H the experimental correlation energies were estimated by using experimental atomization energies 2 4 on the basis of Ref. [173]. All the energies are in a.u. The calculations were performed using the large TZV#3D basis Table 6 Atomization energies (in kcal mol~1) of some molecules calculated with HF, MP2, LDA, BLYP (using 6-31G* basis [171]) and BP, BRP (using 6-31G basis [172])

H 2 LiH NH 3 C H 2 2 H CO 2 F 2

HF

MP2

LDA

BLYP

BP

BRP

Exp

75.9 30.4 170.2 271.9 237.8 !34.3

86.6 39.8 232.4 365.6 335.5 36.8

100.2 57.5 306.0 438.6 417.6 83.6

103.2 54.9 270.1 383.4 361.8 54.4

107.8 55.8 289.5 398.6 371.5 49.6

106.9 58.9 286.7 404.0 372.5 47.1

103.3 56.0 276.7 388.9 357.2 36.9

The Becke—Roussel [174] exchange functional combined with the correlation functional of Perdew [161] leads to improved geometries for a lot of molecules. The predicted atomization energies are as good as the Becke—Perdew approximation and in many cases better. It was found [171] that all the model functionals predict dipole moments that are often significantly in error. Inclusion of gradient terms does not lead to improvement.

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Table 7 Bond length (in A_ ) for some first-row diatomic molecules calculated with HF, MP2, LDA, BLYP (using 6-31G* basis [171]) and BP, BRP (using 6-31G basis [172])

H 2 BeH LiF CO N 2 NO

HF

MP2

LDA

BLYP

BP

BRP

Exp

0.730 1.348 1.555 1.114 1.078 1.127

0.738 1.348 1.567 1.150 1.130 1.143

0.765 1.370 1.544 1.142 1.111 1.161

0.748 1.355 1.561 1.150 1.118 1.176

0.747 1.356 1.580 1.135 1.103 1.160

0.741 1.353 1.582 1.130 1.101 1.158

0.741 1.343 1.564 1.128 1.098 1.151

Recently, a new class of hybrid (Hartree—Fock and density functional) methods have been proposed; the simplest form of that contains an exchange-correlation energy: E "EDFT#a(E%9!#5!EDFT) . (324) 9# 9# 9 9 The paremeter a is determined by fitting to experimental thermochemical data. This expression has been rigorously founded by Go¨rling and Levy [175]. They showed that these hybrid schemes are based on model systems which are defined as the Slater determinant which yields the exact ground-state density and minimizes the expectation value of the operator ¹K #a»K with a being %% a parameter having a value between zero and one. (These hybrid schemes belong to the class of generalized Kohn—Sham schemes derived recently [176].) This kind of exchange mixing reduces average bond energy errors considerably (to about 2 kcal/mol). Reaction barrier heights are also improved. Much progress has been made over the years in solving the molecular Kohn—Sham equations. Very large molecules (e.g. of biological interest) can be treated in the density functional theory. While in the Hartree—Fock method the computational costs increase as N4 or N3, those of the Kohn—Sham scheme as N3. But, in principle, within the density functional theory, it is possible to work out methods that scale linearly [177]. The reason lies in the fact that the whole electronic structure is determined solely by the electron density. Such methods will soon become a very important tool for molecular modelling. We mention in passing that combining the density functional theory with molecular dynamics has led to an extremely fruitful method of Car and Parinello [178] making possible large-scale simulations of molecules and solids.

11. Extensions of the density functional theory 11.1. Finite-temperature density functional theory An extension of the density functional theory can be used to treat systems at finite temperatures [7,179,180]. (For reviews see [8,10].) The forerunner of the theory is the temperature-dependent Thomas—Fermi model [181,182]. Mermin’s generalization of the Hohenherg—Kohn theorem [179]

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states that in a grand canonical ensemble at a given temperature h and chemical potential k, the density n uniquely determines the difference v(r)!k. For a given v and k there exists a functional of nJ (r), the grand potential,

P

X[nJ ]" nJ (r)(v(r)!k) dr#G[nJ ]

(325)

such that it is an absolute minimum for the correct density n(r). The functional G is a universal temperature dependent functional of the density only. The proof proceeds by reductio ad absurdum like in the original Hohenberg—Kohn theorem. The equilibrium grand potential can also be written as

C

P

D

X0"Min X[n]"Min ¹[n]#» [n]!hS[n]# n(r)(v(r)!k) dr , (326) %% n where S[n] is the entropy. In the finite temperature Kohn—Sham theory [7,180] a system of non-interacting electrons with density n at temperature h is considered, where n equals the density of the interacting system. A free kinetic energy is defined as A [n]"¹ [n]!hS [n] , (327) 4 4 4 where the subscript s refers to the non-interacting system. Thus the grand potential takes the form

P

X[n]"A [n]# n(r)(v(r)!k) dr#J[n]#A [n] , 9# 4 the exchange-correlation contribution to the free energy A"¹#» !hS is given by %% A [n]"» [n]!J[n]#¹[n]!hS[n]!¹ [n]#hS [n] . 9# %% 4 4 The grand potential of the non-interacting system reads as

P

X [n]"A [n]# n(r)(v(r)!k) dr . 4 4

(328)

(329)

(330)

The Kohn—Sham equations have the form [!1+ 2#v (r,¹)]u (r)"e u (r) , 2 KS i i i where the effective potential

P

dA [n] n(r@, h) v (r, h)"v(r)# dr@# 9# KS dn(r, h) Dr!r@D

(331)

(332)

includes the external potential v(r), the Coulomb and exchange-correlation potentials. The electron density reads n(r, h)"+ Du (r)D2f (e !k) , i i i

(333)

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where f (e !k)"Mexp[b(e !k)]#1N~1 (334) i i is the Fermi function. b"1/kh and k is the Boltzmann constant. The non-interacting electrons occupy the single-particle eigenstates according to the Fermi—Dirac statistics. Eqs. (331)—(334) have to be solved self-consistently. To do this one needs approximate expressions for the exchangecorrelation potential. For local density approximations and calculations see [183—186] The adiabatic-connection expression for the exchange-correlation expression was given by Perdew [161]. 11.2. Density functional theory for excited states The density functional theory was originally formalized for the ground-state [6]. It was soon noticed [187] that the original theory can also be applied to the lowest excited states with different symmetries. To calculate excitation energies Slater [188,98] introduced the so-called transition state method. It proved to be a very efficient and simple approach and was used to solve a large variety of problems. The density functional theory was first rigorously generalized for excited states by Theophilou [189]. Formalisms for excited states have also been provided by Fritsche [190] and English et al. [191]. A more general treatment was given by Gross, Oliveira and Kohn [192]. The relativistic generalization of this formalism has also been done [193]. 11.2.1. Density functional theory for ensembles Here, only the most general treatment of Gross et al. [192] is reviewed. (The subspace theory of Theophilou [189] can be considered as a special case of the former.) The density functional theory for ensembles is based on the generalized Rayleigh—Ritz variational principle [192]. The eigenvalue problem of the Hamiltonian HK is given by HK W "E W (k"1,2,M) , k k k where

(335)

E 4E 42 (336) 1 2 are the energy eigenvalues. The generalized Rayleigh—Ritz variational principle [192] can be applied to the ensemble energy M E" + w E , k k k/1 where w 5w 525w 50. The weighting factors w are chosen as 1 2 M i w "w "2"w "(1!wg)/(M!g) , 1 2 M~g w "w "2"w "w , M~g`1 M~g`2 M 04w41/M , 14g4M!1 .

(337)

(338) (339) (340) (341)

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The limit w"0 corresponds to the eigenensemble of M!g states (w "2"w "1/(M!g) 1 M~g and w "2"w "0). The case w"1/M leads to the eigenensemble of M states M~g`1 M (w "w "2"w "1/M). 1 2 M The generalized Hohenberg—Kohn-theorems read as follows: (i) The external potential v(r) is determined within a trivial additive constant, by the ensemble density n defined as M n" + w n . k k k/1 (ii) For a trial ensemble density n@(r) such that

(342)

n@(r)50 ,

(343)

P

n@(r) dr"N

(344)

E[n]4E[n@] .

(345)

The ensemble functional E takes its minimum at the correct ensemble density n. The proof of the theorem goes exactly in the same way as for the ground-state. Using the variation principle the Euler-equation can be obtained: dE "k . dn

(346)

Kohn—Sham equations for the ensemble can also be derived: [!1+ 2#v ]u (r)"e u (r) . 2 KS i i i The ensemble Kohn—Sham potential

P

n (r) v (r; n )"v(r)# w dr#v (r; w, n ) KS w 9# w Dr!r@D

(347)

(348)

is a functional of the ensemble density MI 1!wg MI~gI I + + j Du (r)D2#w + j Du (r)D2 , + nI (r)" mj j mj j w M I~1 m/1 j m/MI~gI`1 j g is the degeneracy of the Ith multiplet. I I M "+ g I i i/1 is the multiplicity of the ensemble and j mj

04w41/M . I are the occupation numbers. The density matrix is defined as M DK M, g(w)" + w DW TSW D . m m m m/1

(349)

(350)

(351)

(352)

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The ensemble exchange-correlation potential v is the functional derivative of the ensemble 9# exchange-correlation energy functional E 9# dE [n, w] v (r; w, n)" 9# . (353) 9# dn(r) The excitation energies can be expressed with the one-electron energies e as j 1 dEI(w) I~1 1 dEI(w) EM I" #+ , g dw M dw I w/wI i/2 I w/wi where

K

K

(354)

K

N~1`MI g N~1`MI~1 ­EI dEI(w) " + e! I + e # 9# , (355) j j M ­w w dw I~1 j/N n j/N`MI~1 04w 41/M . (356) i I It is emphasized that the excitation energy cannot generally be calculated as a difference of the one-electron energies. There is an extra term ­E /­wD w to be determined. 9# n The two-particle density matrix of the ensemble is the weighted sum of the two-particle density matrices of the ground and excited states: M CM,g,w(r , r ; r@ , r@ )" + w Cm(r , r ; r@ , r@ ) . 1 2 1 2 m 1 2 1 2 m/1 The total ensemble energy has the form

(357)

P

EM,g"trMDK M,g(w)HK N"trMDK M,g(¹K #»K )N#trMDK M,g(w)»K N"FM,g(w)# n(r)v(r) dr , w %%

(358)

where n(r) is the ensemble density. The ensemble exchange energy is given by EM,g[w,n]"FM,g[w,n]!¹M,g[w,n]!J[n] , 9# 4 where

P

1 n(r)n(r@) J[n]" dr dr@ 2 Dr!r@D

(359)

(360)

is the ensemble Coulomb energy and ¹M, g[w, n] is the noninteracting ensemble kinetic energy. 4 11.2.2. Coordinate scaling and adiabatic connection formula The constrained-search formulation has also been applied for the ensemble theory [192]. The minimum is searched over the density matrices that yield the ensemble density n (361) FM,g[w; n], min trMDK M,g(¹K #»K )N . %% M,g D (w)?n Define W.*/,a as the wavefunction which yields n and minimizes S¹#a»K T. As Levy and n %% Perdew [84] and later Levy et al. [194] and Levy [195] showed the scaled wavefunction

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j3N@2W.*/,a(jr ,2,jr ) yields n and minimizes S¹K #ja»K T. Hence, j3N@2W.*/,a(jr ,2,jr ) yields n 1 N a %% n 1 N n and minimizes S¹K #»K T if j"1/a. So a %% W.*/ (r , ,r )"j3N@2W.*/,a (jr ,2,jr ) , j"1/a . (362) nj 1 2 N n 1 N To obtain scaling relations for the ensemble theory the density matrix has to be studied instead of the wavefunction. Consider the density matrix DM,g[w, n] which yields the ensemble density n and a minimizes S¹K #a»K T. DM,g[w, n] is the density matrix considered before, while DM,g[w, n] is the %% 1 0 non-interacting density matrix, i.e. DM,g[w, n] yields n and minimizes just S¹K T. The coordinate 0 scaling for the density matrix DM,g[w; n ](r ,2, r ; r@ ,2, r@ )"j3NDM,g[w; n](jr ,2,jr ; jr@ ,2, jr@ ) , j"a~1 (363) j 1 N 1 N a 1 N 1 N can be readily obtained from Eq. (352) and the scaling law for the wavefunction (Eq. (167)). From Eqs. (352) and (167) we see that j3NDM,g[w, n](jr ,2, jr ; jr@ ,2, jr@ ) yields n and minimizes a 1 N 1 N j S¹K #»K T if j"1/a. %% From Eqs. (357) and (167) we get scaling properties of the two-particle ensemble matrix: CM,g,w (r , r ; r@ , r@ )"a6CM,g,w(ar , ar ; ar@ , ar@ ) . (364) na 1 2 1 2 n 1 2 1 2 The well-known ground-state adiabatic connection formula of the exchange-correlation energy (see Section 5.6)

P

1 (»a [n]!J[. ]) da (365) %% 0 is valid for the ensemble exchange energy, too, provided that n corresponds to the ensemble density. To prove this let us consider the functional FM,g[w, n]: j E [n]" 9#

FM,g[w, n]" min trMDK M, g(w)(¹K #j»K )N . %% j DM,g(w)?n j"1 gives the interacting system with

(366)

FM,g[w; n]"FM,g[w; n]"¹M,g[w; n]#»M,g[w; n] , 1 %% while j"0 corresponds to the noninteracting case

(367)

FM,g[w; n]"¹M,g[w; n] . 0 4 The ensemble exchange-correlation energy is given by

(368)

EM,g[w; n]"»M,g[w; n]!JM,g[w; n]#¹M,g[w; n]!¹M,g[w; n] 9# %% 4 "FM,g[w; n]!FM,g[w; n]!JM,g[w, n] 1 0 1 ­FM,g[w; n] " dj j !JM,g[w, n] . (369) ­j 0 It can be shown that the Hellmann—Feynmann theorem is valid for an ensemble [196]:

P

­FM,g[w; n] j "trMDK M,g(w)»K N"S»K T . j %% %% j ­j

(370)

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Using this theorem the adiabatic connection formula for the ensemble exchange-correlation energy:

P

EM,g[w; n]" 9#

1

0

dj(S»K T !J) . %% j

(371)

can be derived. From the scaling properties of the electron—electron energy »a [n]"a»1 [n ] and the %% %% 1@a Coulomb repulsion energy J[n ]"a~1J[n], another form of the adiabatic connection formula 1@a for the ensemble exchange-correlation energy

P

EM,g[w; n]" 9#

1

0

a(» [n ]!J[n ]) da %% 1@a 1@a

(372)

follows. This is the same form as the ground-state expression. (Of course, it contains the ensemble quantities.) 11.2.3. Excitation energies To solve the Kohn—Sham equations (Eq. (347)) one needs the ensemble exchange-correlation potential. Several approximations have been proposed for the ensemble exchange-correlation potential. Gross et al. [192] calculated the excitation energies of He atom using the quasi-localdensity approximation of Kohn [197]. The first excitation energies of several atoms [198] have been calculated with parameter-free exchange potential of Ga´spa´r [199]. As this potential depends explicitly on the spin orbitals it is very flexible and can be successfully applied not only in ground-state but also in ensemble-state calculations. Higher excitation energies have also been studied [200]. Ga´spa´r’s parameter-free potential proved to be remarkably good in these calculations. Several ground-state local density functional approximations have been tested [204]. The Gunnarsson—Lundqvist—Wilkins [202], the von Barth—Hedin [203] and Ceperley—Alder [140] local density approximations parametrized by Perdew and Zunger [133] and Vosko et al. [141] are applied to calculate the first excitation energies of atoms. As ground-state exchange-correlation potentials were used the extra term in Eq. (355) does not appear. Spin-polarized calculations [204] lead to a definite improvement compared with the non-spin-polarized results, still, in most cases the calculated excitation energies are highly overestimated. The results provided by the different local density approximations are quite close to each other (Table 8). The best one seems to be the Gunnarson—Lundqvist—Wilkins approximation. (In the non-spin-polarized case the Perdew—Zunger parametrization gives results closest to the experimental data.) In these calculations the minimum (w"0) and the maximum possible values of the weighting factor were applied. The results obtained with different weighting factors w are different. However, any value of w satisfying inequality (Eq. (340)) is appropriate. If we knew the exact exchangecorrelation energy functional, any value of w satisfying condition (Eq. (340)) would lead to the same result. As the exact form of the exchange-correlation energy is, however, unknown and we have to use approximate functionals, different values of w provide different excitation energies. The effect of w on the excitation energies has also been studied [201,204]. In certain atoms it causes only a small change while in other cases there is a considerable change in the excitation energy. The change is

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Table 8 Electron configurations of ensemble states and calculated (spin-polarized) density functional, Hartree—Fock and experimental first excitation energies of atoms (in Ry). The upper rows contain the results belonging to w"0, the lower rows contain the results belonging to w"1/(g #g ) 1 2 Atom B

Na

Mg

Al

P

Ensemble state

Xa

GLW

VBH

VWN

2s2 2p ­ 2s 2s 13 2p 123 ­ ¬ ­

0.285

0.326

0.340

0.335

0.034

0.184

0.202

0.191

3s 3p 0 ­ ­ 3s­ 14 3p 34 ­

0.152

0.166

0.162

0.159

0.156

0.167

0.164

0.162

3s2 3p 0 ­ 3s 3s 101 3p 109 ­ ­ ­

0.244

0.254

0.253

0.249

0.045

0.143

0.152

0.149

3p 4s 0 ­ ­ 3p 34 4s 14 ­ ­

0.174

0.216

0.210

0.196

0.183

0.216

0.209

0.202

3p 3 4s 0 ­ ­ 3p 214 4s 34 ­ ­

0.427

0.459

0.446

0.434

0.538

0.554

0.544

0.541

HF

Exp

0.157

0.262

0.145

0.155

0.136

0.199

0.291

0.230

0.605

0.512

monotonic in all the approximations studied. We mention that a special choice of w, in certain cases, corresponds to Slater’s transition-state method. Relativistic calculations have also been performed [193]. 11.2.4. Local ensemble exchange potential Unfortunately, the currently existing ground-state exchange-correlation potentials do not always perform well for ensemble-states. Recently, a simple local ensemble potential has been proposed [205] in the form:

A B

3 1@3 v (n ,w)"!3a(w) n . 9 w 8p w

(373)

The w-dependence is incorporated in the parameter a. The corresponding exchange energy has the form

A B

P

9 3 1@3 E [n ,w]"! a(w) n4@3 dr . 9 w w 4 8p

(374)

Using the experimental energies a(w) is determined so that the calculated ensemble energy be equal to the ensemble energy obtained from the experimental energies. Two typical cases were found for light atoms. The functions a for the atoms B, C, O, Mg, Si and P have almost the same form, i.e. slight almost monotonic dependence on w. For the atom S a is almost constant. On the other hand, the function a has a very shallow minimum for the atoms F, Cl and Na.

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As we have already seen the excitation energy is not simply the difference of the one-electron energies. The extra term ­E /­wD w is generally not zero. At certain values of w it gives a significant 9# n contribution. The function ­E /­wD w for atoms B, C, O, Mg, Si, S, and P has an almost linear 9# n dependence on w. On the other hand, for the atoms F, Cl and Na there is a value w"w , where it 0 disappears. It means that at w the excitation energy can be simply given by the difference 0 e !e . The importance of the existence of this w lies in the fact that it is possible to determine N`1 N 0 the excitation energy simply as a difference of one-electron energies at a certain value of w. The value of w is 0.0144 for F, 0.113 for Na and 0.0178 for Cl. The corresponding values of a are 0 0.76100, 0.75198, 0.74256 for the atoms F, Na and Cl, respectively. Naturally, calculations can be performed at any value of w (satisfying the condition (Eq. (356))). One can select, e.g., the maximum possible value of w, (i.e. the one corresponding to the subspace theory of Theophilou [189]). Table 9 contains the values of a corresponding to w for selected .!9 atoms. 11.2.5. Ensemble exchange potential and energy for multiplets The multiplet structure has already been treated using the density functional theory. The most important approaches have been proposed by Bagus and Bennett [206], Ziegler et al. [207] and von Barth [208]. All these methods have the same feature of not being completely within the frame of the density functional theory. The method of fractionally occupied states can be used to treat the multiplet problem, too. The method of obtaining the potential from the density (see Section 8) [99] can be applied to ensemble states without any difficulty [209]. Starting out from the Hartree—Fock densities [210] the ensemble exchange potentials for multiplets have been calculated. Writing the ensemble exchange potentials in the form

A B

3 1@3 vM,g(w, n; r)"!3aM,g(w) , n 9# w 8p

(375)

the factors aM,g(w) as functions of the radial distance r show a shell structure. (For the ground state the shell structure has also been demonstrated [99].) Though the ensemble exchange potentials are Table 9

Ground- and excited state configurations and the value of a corresponding to the maximal weighting factor w Atoms

B C O F Na Mg Al Si P S Cl

a

Configurations Ground-state

Excited-state

2P1@2(2p) 3P (2p2) 0 3P (2p4) 2 3P3@2(2p5) 2S1@2(3s) 1S0(3s2) 2P1@2(3p) 3P (3p2) 0 4S (3p3) 3@2 3P2(3p4) 3P3@2(3p5)

4P1@2(2s2p2) 5S (2p3s) 0 5S (2p33s) 0 4P3@2(2p43s) 2P1@2(3p) 3P0(3s3p) 2S1@2(4s) 5S(3s3p3) 4P(3p24s) 5S(3p34s) 4P5@2(3p44s)

0.80210 0.79115 0.77350 0.76390 0.75198 0.75105 0.74810 0.74832 0.74623 0.74472 0.74295

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different from the ground state one, the difference is not too much and the factors a show a very similar behaviour. The fact that the exact exchange potential has similar behaviour for the ensemble of multiplets suggests that approximations might also be similar. Probably, a small change in the ground-state exchange functionals might lead to good approximation for ensembles of multiplets. It was emphasized in the theory of Gross et al. [192] that the ensemble exchange potential depends on w. It has been explicitly demonstrated [209] that the ensemble exchange factor for multiplets is different for different values of w. The method described in Section 8 makes it possible to calculate the ensemble energy [209]. The ensemble exchange energies are very close to the Hartree—Fock ones, the latter being somewhat lower as it is expected. It has been recently shown that the ensemble exchange energy for multiplets is linear in w [211]. Excitation energies can be calculated within the time-dependent theory that is detailed in Section 11.4. A new way of treating excited states within the density functional theory recently proposed by Go¨rling [212] is outlined in the next subsection. 11.2.6. Density functional theory for excited states via adiabatic connection and perturbation theory The density functional theory can be extended to excited states via the adiabatic connection and making use of perturbation theory [212] (see Section 5.7). The adiabatic connection characterized by the Schro¨dinger equation HK aDWaT"EaDWaT , (376) k k k HK a"¹K #a»K #»K (377) %% a represents a continuous connection between a non-interacting system and the real system. Here not only the ground-state but also the kth eigenstate Wa of the coupling constant Hamiltonian is k considered. The additional assumption here is that the energetic order of eigenstates Wa of HK a of the k same symmetry is preserved along the adiabatic connection. So the coupling constant path establishes a continuous connection between the kth eigenstate of non-interacting and the interacting Hamiltonian. The energy of the kth eigenstate

P

Ea"SU [n ]D¹K DU [n ]T#J [n ]#E [n ]#Ea [n ]# va(r)n0(r) k k 0 k 0 k 0 9, k 0 #, k 0 k

(378)

is a functional of the ground-state density n which is kept fixed in the coupling constant path. The 0 exchange and correlation energy functionals are defined as E [n ]"SU [n ]D»K DU [n ]T!J [n ] , (379) 9, k 0 k 0 %% k 0 k 0 Ea [n ]"SWa[n ]DHK aDWa[n ]T!SU [n ]DHK aDU [n ]T , (380) #, k 0 k 0 k 0 k 0 k 0 where J [n ] is the classical Coulomb energy of the kth eigenstate. k 0 In order to treat excited states in the Kohn—Sham formalism, first, the ground-state Kohn—Sham equations have to be solved. I.e. the ground-state one-electron energies and orbitals have to be determined. To obtain the excited-state exchange and correlation energy functionals E [n ] and 9,k 0 Ea [n ], the density functional perturbation theory (see Section 5.7) can be applied. Go¨rling [212] #,k 0

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has shown that the ground-state one-electron energies are not just auxiliary quantities without physical meaning; their difference provides zeroth-order approximation to excitation energies. 11.3. Current density functional theory In the spin density functional theory (Section 4.2) the interaction of the magnetic field with the electronic current was neglected. Inclusion of currents is required whenever a large magnetic field coexists with a strongly inhomogeneous electronic structure, e.g. charge-density waves, spindensity waves, atoms and molecules in strong magnetic fields. (For a review see [214].) Recently, Vignale and Rasolt constructed the non-relativistic current density functional theory [213]. The most delicate point in the theory is the following. The non-relativistic expression for the orbital current density 1 j(r)"j (r)# n(r)A(r) , 1 c

(381)

includes the paramagnetic current density j (r) and the diamagnetic current density 1n(r)A(r). Since 1 c in the variational process the vector potential A(r) is kept constant, the variation with respect to j is the same as the variation with respect to j , consequently j should be used as a basic variational 1 1 object. The paramagnetic current density, however, is not gauge-invariant. Vignale and Rasolt managed to construct a gauge-invariant theory in terms of non-gauge-invariant variables. To write the Hamiltonian the kinetic energy operator has to be replaced by

A

B

1 N 2 1 ¹K " + !i+# A(r) . 2 c i/1 Then the total energy has the form

P C

(382)

D P

1 1 E"E # n(r) v(r)# A2(r) # j (r)A(r) dr . 0 2c2 c 1

(383)

The Hohenberg—Kohn theorem states that the external scalar and vector potentials v(r) and A(r) are determined within a trivial additive constant by the knowledge of the ground-state electron and current densities n(r) and j (r) and there is a variational principle for the total energy. The proof 1 again proceeds by reductio ad absurdum and is omitted here. The variational principle should be applied by imposing the constraints

P

N" n(r) dr ,

(384)

1 + j (r)# + [n(r)A(r)]"0 1 c

(385)

which corresponds to the constancy of the total number of electrons and the continuity equation for the total current.

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The Kohn—Sham equations can also be readily obtained in the usual way (i.e. by introducing the non-interacting kinetic energy functional and performing the variation):

CA

B

D

1 2 1 !i+# A (r) #v (r) u (r)"e u (r) . %&& %&& i i i 2 c

The densities n(r) and j (r) are related to the one-electron orbitals by 1 N n(r)" + Du (r)D2 , i i/1 1 N j (r)" + (u*(r)+u (r)!u (r)+u*(r)) . 1 i i i i 2i i/1 The effective potentials in Eq. (386) have the forms: 1 v (r)"v(r)#v (r)#v (r)# [A2(r)!A2 (r)] %&& J 9# %&& 2c2 A (r)"A(r)#A (r) , %&& 9# where v (r) is the classical Coulomb potential and J dE [n, j ] 1 , v (r)" 9# 9# dn dE [n, j ] 1 . A (r)" 9# 9# dj 1 The gauge-invariancy of the theory can be expressed in the following compact form: E [n, j ]"EM [n, m] , 9# 1 9# i.e., the exchange-correlation energy is a functional of the gauge-invariant combination

(386)

(387) (388)

(389) (390)

(391) (392)

(393)

m(r)"+][ j (r)/n(r)] . (394) 1 From Eq. (393) one can easily check that the effective potentials v (r) and A (r) can be expressed %&& %&& with n(r) and m(r) and the gauge-invariancy of the effective potentials and the Kohn—Sham equations can be readily proved. Certainly, one should approximate the exchange-correlation energy. The local density approximation can be extended to the current density functional theory. In the local approximation the exchange-correlation energy density e which is a function of the n(r) and m(r) is taken to be the 9# exchange-correlation energy density of an electron gas with uniform n(r) and m(r).

P

ELDA[n,m]" n(r)e(n(r),Dm(r)D) dr . 9#

(395)

In case of a uniform magnetic field if the electron gas is at rest in the laboratory frame the current density j(r) vanishes everywhere, consequently m(r) is uniform and has the form: m(r)"m"!B/c .

(396)

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Different expressions of the exchange-correlation energy density have been proposed for weak [213,215—217], intermediate and strong magnetic fields [218]. Handy and coworkers [219] calculated magnetisabilities, polarizabilities and nuclear shielding constants using current density functional theory. It is possible to include the spin in the current density functional theory. Vignale and Rasolt [213] proposed two alternative ways, the total current and the spin-current formulations. Recently, a time-dependent generalization of the current density functional theory has been developed by Ng [220] for electronic systems in weak electromagnetic fields. An alternative of the current density functional theory has been proposed by Grayce and Harris [221]. In this magnetic field density functional theory the density is the only fundamental variable. The magnetic field B instead of the current j appears explicitly in the energy functional: 1 1 n(r, B)n(r@, B) E"G[n, B]# n(r, B)v(r)# dr dr@ . (397) Dr!r@D 2

P

P

The disadvantage of this theory is the non-universality, i.e., the dependence on the magnetic field. While G has the same form for all v, it depends explicitly on the magnetic field B. It is universal only with respect to the scalar potential, but not with respect to the vector potential. Several calculations have been done with this method, i.e. [222, 223]. 11.4. Time-dependent density functional theory The roots of the time-dependent density functional theory date back to the time-dependent Thomas—Fermi model proposed by Bloch [224]. The first time-dependent Kohn—Sham equations were obtained by Peuckert [225] and Zangwill and Soven [226]. The rigorous foundation of the time-dependent density functional theory was started by the work of Deb and Ghosh [227] and Bartolotti [228]. The general proofs of the fundamental theorems of the time-dependent density functional theory were given by Runge and Gross [229]. For a recent review of time-dependent density functional theory see [230]. 11.4.1. Runge—Gross theorem The ground-state density functional theory is based on the Rayleigh—Ritz variation principle. In the case of a time-dependent external potential, however, no minimum principle exists. Instead, there is a stationary-action principle. The starting point of studying time-dependent systems is the Schro¨dinger equation ­W(t) i "HK W(t) , ­t

(398)

where the Hamiltonian HK (t)"¹K #»K #»K (t) %% includes the kinetic ¹K , the electron—electron repulsion »K and the external potential %% N »K " + v(r ,t) i i/1

(399)

(400)

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operators. The densities of the system evolving from a fixed initial state W(t )"W (401) 0 0 are considered. The initial state W is not supposed to be an eigenstate of the initial potential v(r,t ), 0 0 i.e. the case of sudden switching is assumed. The potentials are required to be expandable in a Taylor series about t . Then the following theorem holds: The densities n(r,t) and n@(r,t) evolving 0 from a common initial state W under the influence of two potentials v(r,t) and v@(r,t) are always 0 different provided that the potentials differ by more than a purely time-dependent function: v(r, t)Ov@(r, t)#c(t) .

(402)

The reader should consult Ref. [229] concerning the proof. As a consequence of this theorem the time-dependent wave function is a functional of the time-dependent density W(t)"e~*s(t)W[n](t) .

(403)

This functional is unique up to a time-dependent phase s(t). Concerning the expectation value of any quantum mechanical operator, the ambiguity of the phase cancels out and it is a unique functional of the density. E.g. the current density j(r,t)"SW(t)D jK (r)DW(t)T 1 is also a functional of the density. Here,

(404)

1 N (405) jK (r)" + (+rjd(r!r )#d(r!r )+rj) j j 1 2i j/1 is the paramagnetic current density operator. The time-dependent particle and current densities can also be calculated from the hydrodynamical equations: ­ n(r, t)"!+j(r, t) , ­t

(406)

­ j(r, t)"P(r, t) , ­t

(407)

where P(r, t)"!iSW(t)D[ jK (r),HK (t)]DW(t)T . 1 According to the stationary-action principle the quantum mechanical action integral

P T K

(408)

K U

­ t1 dt W(t) i !HK (t) W(t) (409) ­t 0 t has a stationary point at the correct time-dependent density. Consequently, the solution of the Euler equation A"

dA[n] "0 dn(r, t)

(410)

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leads to the correct density. The functional A[n] can be separated as

P P

t1 dt n(r, t)v(r, t) dr , t0 where the time-dependent functional B[n] A[n]"B[n]!

P T K

(411)

K U

­ dt W(t) i !¹K (t)!»K W(t) %% ­t 0

t1

(412) t is universal as it is independent of the external potential v(r,t). The approach presented above is valid only for v-representable densities. There are proposals [231,232] for a Levy—Lieb type extension of the theory. In the following v-representability is assumed. B"

11.4.2. Time-dependent Kohn—Sham scheme Just like in the time-independent case a non-interacting system, in which the electrons move independently in a common local potential, is constructed. The time-dependent Kohn—Sham equations have the form

C

D

1 ­ ! + 2#v (r, t) u (r, t)"i u (r, t) . KS i 2 ­t i

(413)

The density of the non-interacting system N n(r, t)" + Du (r, t)D2 (414) i i/1 is equal to that of the interacting one. The current density built up from the non-interacting orbitals 1 N (415) j(r,t)" + (u*(r, t)+u (r, t)!u (r, t)+u*(r, t)) i i i i 2i i/1 is also identical with the true current density of the interacting system. The time-dependent Kohn—Sham potential v (r, t)"v(r, t)#v (r, t)#v (r, t) KS J 9# includes the external v, the classical Coulomb

P

n(r@,t) dr@ v (r, t)" J Dr!r@D

(416)

(417)

and the exchange-correlation dA [n] v (r, t)" 9# , 9# dn

(418)

potentials. A [n] is the exchange-correlation part of the action functional. There is an important 9# difference between the time-independent and the time-dependent schemes. Here, the functionals,

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such as A or B implicitly depend on the initial state W . Consequently, the Kohn—Sham potential 0 also depends on the initial orbitals. In principle, infinitely many Slater determinants reproduce the same electron density [233]. Any of them can, of course, be used in the formalism. If, however, both the interacting and the non-interacting initial states are non-degenerate ground-states, there is no dependence on the initial state. The formalism presented above is valid if the motion of the atomic nuclei is neglected. It can be, however, extended to treat the nuclear motion either quantum mechanically or classically (for discussion and references see [230]). 11.4.3. Time-dependent spin and current density functional theory The theory can be extended to time-dependent electromagnetic fields. First the time-dependent spin density functional theory is reviewed. The time-independent formalism was outlined in Section 4.2. Following the notations of Section 4.2, the interaction with the time-dependent external field is given by N N (419) »K " + v(r )#2b + B(r) ) s . i i % i/1 i/1 Using this operator in the action functional Eq. (409) one obtains the time-dependent Kohn—Sham equations

C

D

1 ­ ! + 2#v (r, t) u (r, t)"i u (r, t) . KSp ip 2 ­t ip

(420)

The spin-dependent Kohn—Sham potential and the density of electrons with spin p have the forms v (r, t)"v (r, t)#v (r, t)#v (r, t) , (421) KSp p J 9#p Np n (r, t)" + Du (r, t)D2 , (422) p ip i/1 respectively. The spin-dependent exchange-correlation potential v (r,t) is defined as functional 9#p derivative of the exchange-correlation action functional A [n ,n ]. 9# ­ ¬ Turning to the problem of coupling to orbital currents, the kinetic energy ¹K in Eq. (399) has to be replaced with

A

B

N 1 1 2 ¹K (t)" + !i+ # A(r , t) , (423) i c i 2 i/1 where A(r, t) is the time-dependent vector potential. The gauge-invariant current density is given by 1 j(r, t)"SW(t)D jK (r)DW(t)T# n(r, t)A(r, t) , 1 c

(424)

where jK (r) is the paramagnetic current density operator defined by Eq. (405). Then the basic 1 theorem of the time-dependent current density functional theory reads: The current densities j(r, t) and j@(r, t) evolving from a common initial state W under the influence of the four-potentials 0 (v(r, t), A(r, t)) and (v@(r, t), A@(r, t)) which differ by more than a gauge transformation are always different provided that the potentials can be expanded in Taylor series around the initial time t . 0

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The Kohn—Sham scheme can be derived in the usual way and the Kohn—Sham equations have the form:

CA

B

D

1 2 1 ­ !i+# A(r, t) #v (r, t) u (r, t)"i u (r, t) , KS i 2 c ­t i

(425)

where

A

B

1 N 1 N (426) j(r, t)" + (u*(r, t)+u (r, t)!u (r, t)+u*(r, t))# + Du (r, t)D2 A(r, t) . i i i i i c 2i i/1 i/1 These equations are rather complicated arising from the fact that here n(r, t) and j(r, t) are the basic variables. For electrons in static electromagnetic fields, Vignale and Rasolt [213] have formulated a current density functional theory in terms of the density and the paramagnetic current density. The time-dependent formalism in terms of the density and the paramagnetic current density is not available yet. Several extensions of the time-dependent formalism have been presented, e.g. superconductors in time-dependent electromagnetic fields. (For references see [230].) 11.4.4. Time-dependent linear density response Consider an electron system subject to external potentials v(r,t)"v (r)#v (r,t), where v (r) is the 0 1 0 external potential of the unperturbed system and v (r,t) is the time-dependent perturbation. The 1 unperturbed state is supposed to be the ground-state corresponding to v (r). The external potential 0 is a functional of the time-dependent density and the density—density response function can be expressed as

K

dn[v](r, t) s(r, t; r@, t@)" dv(r@, t@)

, (427) v*n0+ where the functional derivative is to be evaluated at the external potential corresponding to the unperturbed ground-state density n . The linear density response to the perturbation v (r,t) reads 0 1

P P

n (r, t)" dt@ dr@s(r, t; r@, t@)v (r@, t@) . 1 1

(428)

Expression (427) is valid in the Kohn—Sham scheme, too, writing s (r, t; r@, t@) instead of s(r, t; r@, t@) KS and v (r, t) instead of v(r, t). The response function (427) can also be given by a Dyson-type KS equation

P PP P

C

s(r, t; r@, t@)"s (r, t; r@, t@)# dr@@ dq dr@@@ dq@s (r, t; r@@, t) KS KS

D

d(q!q@) Dr@@!r@@@D

#[f (r@@, q; r@@@, q@)]s(r@@@, q@; r@, t@) , (429) 9# that expresses the relation between the interacting and non-interacting response functions. The time-dependent exchange-correlation kernel f (r, t; r@, t@)"dv (r, t)/dn(r, t) comprises all dynamic 9# 9# exchange and correlation effects to linear order in the perturbing potential. Taking the Fourier

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transform with respect to time the linear density response has the form

P

C

P A

B

D

1 n (r, u)" dr@s (r, r@; u) v (r@, u)# dr@@ #f (r@, r@@; u) n (r@@, u) . 1 KS 1 9# 1 Dr@!r@@D

(430)

The Kohn—Sham response function that can be expressed in terms of the unperturbed Kohn—Sham orbitals: u (r)u*(r)u*(r@)u (r@) j k s (r, r@; u)"+ (j !j ) j k (431) KS k j u!(e !e )#ig j k j, k has poles at the Kohn—Sham orbital energy differences. (j stands for the occupation number of the j orbital u .) The linear response formalism can be extended to systems at finite temperature [234] j and current density response theory for arbitrary time dependent electromagnetic fields has also been worked out [220]. Higher-order response theory has also been developed [230] since recently there is a growing interest in nonlinear phenomena. A hierarchy of equations for the time-dependent density response has been derived [89]. 11.4.5. Excitation energies Just like in the time-independent case one has to use approximations to the time-dependent exchange-correlation potential. The simplest possible approximation is the time-dependent or “adiabatic” local density approximation (ALDA). The functional form of the static LDA is used with the time-dependent density: vALDA(r, t)"v)0.(n(r, t)) , (432) 9# 9# where v)0. is the exchange-correlation potential of the homogeneous electron gas. This approxima9# tion leads to an exchange-correlation kernel f ALDA having no frequency-dependence at all. To 9# incorporate the frequency-dependence into f Gross and Kohn [235] proposed that 9# f LDA(r, r@; u)"f )0.(n (r), Dr!r@D; u) , (433) 9# 9# 0 i.e. the frequency-dependent exchange-correlation kernel of the homogeneous electron gas. (For a detailed discussion see Ref. [230].) The optimized potential method reviewed in Section 7 has been extended to the time-dependent case. To construct an optimised effective potential the action functional is written as in Eq. (411), where the exchange-correlation part of the action functional has the form:

P P

1N t1 u*(r@, t)u (r@, t)u (r,t)u*(r, t) j i j A [n]"! + d i j dr dr@ . (434) dt i 9# pp 2 Dr!r@D ~= i, j A procedure similar to the one described in Section 7 leads to the time-dependent optimised potential method [236]. From the several kinds of applications (such as photoresponse of finite and infinite systems, van der Waals interactions) (see Ref. [230]) we mention only that the time-dependent density functional theory can be efficiently applied to calculate excitation energies. Table 10 presents excitation energies of several atoms calculated by Gross et al. [230, 237] with local density approximation,

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Table 10 The lowest 1SP1P excitation energies of several atoms calculated by OPM, LDA, OPM#ALDA methods (in a.u.) Atoms

OPM

LDA

OPM#ALDA

Exp

Be Mg Ca Zn Sr Cd

0.196 0.164 0.117 0.211 0.105 0.188

0.200 0.176 0.132 0.239 0.121 0.214

0.199 0.165 0.118 0.209 0.106 0.185

0.194 0.160 0.108 0.213 0.099 0.199

optimised potential method and a method, in which the optimised potential method was used for v and local density approximation (ALDA) for the exchange-correlation kernel f . The values 9# 9# obtained by the optimised potential method are clearly superior to the local density results. A similar method based on the one-particle density matrix has been recently proposed by Casida [238]. Bauernschmitt and Ahlrichs [239] calculated excitation energies of several molecules by various local, gradient-corrected and hybrid functionals and found considerable improvement over Hartree—Fock-based approaches requiring comparable numerical work. 11.5. Relativistic density functional theory The relativistic extension of the ground-state density functional theory has also been done. The forerunner was the relativistic Thomas—Fermi model [240]. The relativistic counterparts of the Hohenberg and Kohn theorem were formalized by Rajagopal and Callaway [241]. The Kohn—Sham equations were derived by Rajagopal [242] and independently by MacDonald and Vosko [243]. The retardation corrections to the Coulomb interaction were taken into consideration, however, the radiative corrections were essentially neglected. A field theoretical background was addressed by Engel and Dreizler [244] and is not covered here. (For a recent review of relativistic density functional theory see [245].) 11.5.1. Relativistic Hohenherg—Kohn theorem and Kohn—Sham equations The relativistic Hohenherg—Kohn theorem is proved by reductio ad absurdum. (Renormalised quantities should be used to obtain the proof.) It states that there exists a one-to-one correspondence between the classes of external potentials » just differing by gauge transformations and the l ground-state four current jl. Fixing the gauge once and for all one arrives at the statement that all ground-state observables are unique functionals of the four current. The energy functional E[ jl] contains all relativistic kinetic effects for both electrons and photons and all radiative effects. The variational principle takes the form

P

d E[ jl]!k j0"0 djl

(435)

with the constraint of charge conservation. (All quantities are supposed to be fully renormalised.)

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In the presence of only a purely electrostatic external potential: »l"(»0, 0) the ground-state energy and all the observables are unique functionals of the zeroth component j0, i.e. the density n only. To derive the Kohn—Sham equations one has to introduce single-particle four spinors u . Then k the exact ground-state four current has the form: jl"jl #jl , V D where

C

1 jl " V 2

(436)

D

+ uN clu ! + uN clu #Djl,(0) k k k k ~m:ek eky~m

(437)

and jl " + uN clu , (438) D k k ~m:ekyeF where Djl,(0) is the counterterm and e represents the Fermi level below which all orbitals are F occupied. (For the derivation see Refs. [244,245].) The total energy can be decomposed as E[ jl]"¹ [ jl]#E [ jl]#J[ jl]#E [ jl] , (439) s %95 9# where ¹ , E , J and E are the non-interacting kinetic, external, Coulomb and exchange-correla4 %95 9# tion energy parts, respectively. The relativistic Kohn—Sham equations have the form: c0(!ic+#m#e». #v. #v. )u "e u , J 9# k k k where

P

(440)

jl(y) v (x)"e2 dy , J Dx!yD

(441)

d E [ jl] 9# vl " 9# djl

(442)

and

are the Coulomb and exchange potentials, respectively. The Kohn—Sham equations should be solved self-consistently, which is a very complicated problem, as it includes summation over all negative and positive energy solutions and renormalisation in each iterative step. The most important simplification of this tedious problem is the no-sea approximation, in which all radiative contributions to the four current and ¹ and the vacuum contribution in E are 4 9# neglected. Another simplification may be applied if the external potential is purely electrostatic. In this case, only the density n"j0 is the variational object and the spatial current j is a functional of the density. If the electron—electron interaction decomposed into longitudinal and transverse parts one arrives at another approximation. It is straightforward for the classical part: E [n]"EL[n]#ET[ j[n]] , J J J

(443)

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where

P

e2 n(r)n(r@) EL[n]" dr dr@ , J 2 Dr!r@D

P

j(r) j(r@) e2 , ET[ j[n]]"! dr dr@ J Dr!r@D 2

(444) (445)

while EL is defined by neglecting the transverse interaction to all orders in the free-electron 9# propagator and the remainder is called ET . In the longitudinal approximation the transverse 9# contributions ET and ET are neglected in the self-consistent calculations and added perturbatively J 9# to the total energy. Then the Kohn—Sham equations have the form: [!ia ) +#bm#vL]u "e u , k k k where a and b are the Dirac matrices, while vL is the total relativistic potential

(446)

vL"v#vL#vL J 9# containing the external v, the longitudinal Coulomb

(447)

P

n(r@) vL[n(r)]"e2 dr@ J Dr!r@D

(448)

and the longitudinal exchange dEL [n] vL " 9# 9# dn

(449)

potentials. The electron density is given by n" + u`u , k k ~m:ekyeF

(450)

11.5.2. Relativistic exchange-correlation functionals Just like in the non-relativistic case the exact form of the relativistic exchange-correlation functional is unknown. On the other hand, if the orbital representation of the exchange energy is known the optimised potential method can be applied (see Section 7). Relativistic extension of the optimised potential method was put forward by Talman and coworkers [246] in the longitudinal no-sea level and was recently applied to atoms by Engel et al. [247]. In the longitudinal no-pair approximation for a purely electrostatic potential the method is completely analogous to the non-relativistic case. Calculations obtained with the exchange-only optimised potential method result in ground-state energies being extremely close to the relativistic Hartree—Fock ones. In complete analogy to the non-relativistic case the relativistic local density approximation is based on the relativistic homogeneous electron gas:

P

ERLDA[n]" dr eRLDA(n(r)) . 9# 9#

(451)

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The lowest order contribution gives the exchange energy that can be written as the non-relativistic exchange energy multiplied by the relativistic correction factor (for references see [245]). The longitudinal part has the form eRLDA,L(n)"eNRLDA(n)UL(b) , 9 9 x where (3p2n)1@3 b" . mc

(452)

(453)

There is a similar expression for the transverse part. It turned out that the longitudinal contribution dominates in the low density limit and depends only weekly on b. On the other hand, the transverse part shows a stronger dependence on b and dominates in the high density limit. Relativistic correlation contribution to the local density approximation has only been studied as a partial resummation of those terms in the perturbation expansion in e2 which are the most relevant in the high density limit. The relativistic analogue of the non-relativistic weighted density approximation has also been worked out. It has the advantage of insuring the satisfactory cancellation of self-interaction effects and consequently reproducing the asymptotic r~1 proportionality of the exchange potential though with an incorrect prefactor. The local density approximation is the most frequently applied approach in the non-relativistic calculations. Though it is far from being exact, because of the partial cancellation of errors between the exchange and correlation contributions, it often leads to acceptable results. In the relativistic case, however, the situation is somewhat different. The error in the total longitudinal exchange energy is about 10% for light atoms and reduces to about 5% for heavier atoms. Considering the relativistic correction to the exchange energy, one can conclude that longitudinal correction is underestimated by about 20%, while the transverse one is overestimated by about 50%. These errors do not cancel, therefore the local density approximation reproduces rather poorly the exchange energies. Studying the exchange potential Engel and Dreizler [245] found that the local errors are quite significant. Table 11 presents the relativistic exchange and correlation energies for Ne, Xe and Hg atoms [244]. For comparison Hartree—Fock and Møller—Plesset data are also included. Correlation energy was studied using the RPA for relativistic corrections. The results are not satisfactory.

12. Concluding remarks I have made no attempt here to survay all contributions to the density functional theory. My apology to the authors of the many important papers that are not referenced in the present review. I have not covered topics of considerable importance such as Thomas—Fermi type density functional theory, momentum-space density functional theory [248], formulation of the density functional theory using local scaling transformation [24,249], numerical methods and applications. My objective has been to review several relevant aspects encountered in the development of the theory.

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Table 11 Relativistic exchange and correlation energies [244] in atomic units NROPM, ROPM, RHF, RLDA, LDA and MBPT2 denote the non-relativistic optimized potential method, the relativistic optimized potential method, the relativistic Hartree—Fock method, the relativistic local density approximation, the non-relativistic local density functional approximation and second-order Møller—Plesset method, respectively Longitudinal exchange energy (!EL) 9 NROPM ROPM RHF RLDA

RHF RLDA

MBPT2 LDA MBPT2 LDA

Transverse exchange energy (!ET) 9

Correlation energy !DEL # (longitudinal) !ET # (transverse)

Ne

Xe

Hg

12.105 12.120 12.123 10.944

179.062 184.083 184.120 174.102

345.240 365.203 365.277 347.612

0.017 0.035

5.711 9.089

22.168 34.201

0.20 0.38 1.87 0.32

37.57 64.73 108.75 39.27

203.23 200.87 282.74 113.08

I think that the future prospects of density functional theory are bright. It is already competitive with the conventional methods and prospective applications to problems in several fields, such as molecular biology are particularly promising. Still, there remain a lot of problems to solve, and there are several open questions. Though, very efficient exchange-correlation functionals are nowadays available, the search for better and better functionals will go on. It is generally believed that exact relations and theorems, quite exhaustively detailed here, are especially useful in constructing approximate functionals. The number of these constraints, however, is still growing and for the time being it is not quite clear which ones are the most important. It is expected that orbital-dependent potentials are possible candidates. The optimised potential method solves the exchange-only problem exactly. Inclusion of an appropriate orbital-dependent correlation term might lead to an even more accurate solution. In spite of several efforts and considerable developments the form of the kinetic energy fuctional is still unknown. A sufficiently accurate approximation to this functional would lead to a breakthrough in the density functional theory, because the solution of the Euler-equation would generally need considerably less numerical effort than would the Kohn—Sham equations. There are a multitude of problems to be solved in the extension of the density functional theory. Further important developments are expected in these fields. Undoubtedly, there is room for further extentions of the density functional theory.

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Acknowledgements The author is most grateful to Professor G. Parr for valuble discussions, encouragement and generous hospitality. The grant “Sze´chenyi” from the Hungarian Ministry of Culture and Education is gratefully acknowledged. This work was supported by the grant MTA-NSF No. 93, the grants OTKA No. T 16623 and F 16621 and FKFP 0314/1997.

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DARK OPTICAL SOLITONS: PHYSICS AND APPLICATIONS

Yuri S. KIVSHAR!, Barry LUTHER-DAVIES" ! Australian Photonics Co-operative Research Centre, Optical Sciences Centre, Research School of Physical Sciences and Engineering, Australian National University, ACT 0200, Canberra, Australia " Australian Photonics Co-operative Research Centre, Laser Physics Centre, Research School of Physical Sciences and Engineering, Australian National University, ACT 0200, Canberra, Australia

AMSTERDAM — LAUSANNE — NEW YORK — OXFORD — SHANNON — TOKYO

Physics Reports 298 (1998) 81—197

Dark optical solitons: physics and applications Yuri S. Kivshar!, Barry Luther-Davies" ! Australian Photonics Co-operative Research Centre, Optical Sciences Centre, Research School of Physical Sciences and Engineering, Australian National University, ACT 0200, Canberra, Australia " Australian Photonics Co-operative Research Centre, Laser Physics Centre, Research School of Physical Sciences and Engineering, Australian National University, ACT 0200, Canberra, Australia Received April 1997; editor: A.A. Maradudin

Contents 1. Introduction 2. Dark vs. bright solitons 2.1. Optical fibers 2.2. Nonlinear waveguides 2.3. When the NLS equation fails 2.4. Modulational instability and solitons 2.5. Dark solitons: mathematical tools 2.6. Integrals of motion 2.7. Physical interpretation of dark solitons 3. Perturbation theory and applications 3.1. Equations for soliton parameters 3.2. Physical applications 3.3. Dark-soliton jitter 3.4. Effect of third-order dispersion 3.5. Background of finite extent 4. Instability-induced soliton dynamics 4.1. Stability of dark solitons 4.2. Asymptotic approach 4.3. Examples of non-Kerr dark solitons

84 87 87 88 92 93 96 100 103 105 105 108 114 116 119 121 121 123 128

5. Multi-component dark solitons 5.1. Mode interaction: general overview 5.2. Dark—bright solitons 5.3. Modes with opposite dispersions 5.4. Polarization instability and domain walls 5.5. Parametric dark solitons in s(2) media 6. Experimental verifications 6.1. Dark solitons in fibers 6.2. Spatial dark solitons 6.3. Coupled dark—bright solitons 7. Dark solitons in higher dimensions 7.1. Introductory remarks 7.2. Transverse instability of plane solitons 7.3. Vortex solitons: theory 7.4. Vortex solitons: experiments 7.5. Ring dark solitons 8. Conclusion and open problems References

137 137 139 142 145 148 153 153 160 167 168 168 170 172 179 185 188 190

Abstract We present a detailed overview of the physics and applications of optical dark solitons: localized nonlinear waves (or ‘holes’) existing on a stable continuous wave (or extended finite-width) background. Together with the traditional problems involving properties of dark solitons of the defocusing cubic nonlinear Schro¨dinger equation, we also describe recent theoretical results on optical vortex solitons; ring dark solitons; polarization domain walls; parametric dark solitons in a dispersive s(2) medium; vector dark solitons; coupled dark—bright soliton pairs, and we discuss the 0370-1573/98/$19.00 Copyright ( 1998 Elsevier Science B.V. All rights reserved PII S 0 3 7 0 - 1 5 7 3 ( 9 7 ) 0 0 0 7 3 - 2

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83

instability-induced dynamics of dark solitons in the models of generalized (i.e., non-Kerr) optical nonlinearities. Special attention is paid to the experimental demonstrations of temporal dark solitons in optical fibres and spatial dark solitons, especially dark-soliton stripes and vortex solitons, in a defocusing bulk medium. ( 1998 Elsevier Science B.V. All rights reserved. PACS: 42.65.!k

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1. Introduction Optical solitary waves, temporal and spatial solitons, have been the subject of intense theoretical and experimental studies in recent years. Solitons — localized pulses in time or bounded self-guided beams in space — evolve from a nonlinear change in the refractive index of a material induced by the light intensity distribution. When the combined effects of the refractive nonlinearity and the pulse dispersion (in the case of temporal solitons) or beam diffraction (in the case of spatial solitons) exactly compensate each other, the pulse or beam propagates without change in shape and is said to be self-trapped. Nonlinear effects responsible for soliton formation in optical fibers are, in general, weak and Kerr-like, i.e. they induce a local index change directly proportional to the light intensity. In this case the main nonlinear equation governing the pulse evolution is the famous cubic nonlinear Schro¨dinger (NLS) equation for the complex amplitude envelope of the electric field which, depending on the sign of the group-velocity dispersion, has two distinct types of localized solutions, bright or dark solitons. These two types of waves look like two members of a general family of localized solutions, and this idea manifests itself in the drawing of Marc Haelterman, see Fig. 1. However, as will be seen from the results presented below, these two types of solitary waves are in fact very different, they have completely different nature and result from quite different physics. In the case of temporal solitons observed in optical fibers [Hasegawa (1989); an extended overview and history can be also found in the book by Hasegawa and Kodama (1995)], the group velocity dispersion is known to vanish at a wavelength of about 1.3 lm and is positive at larger wavelengths and negative at shorter ones. As a result, since silica optical fibers have always a positive Kerr coefficient, the two different signs of group-velocity dispersion support two different types of solitons, dark, in the former case, and bright, in the latter case. A similar situation occurs for self-guided beams or spatial optical solitons [Chiao et al. (1964); see also Chiao et al. (1993)] observed in planar waveguides or in a bulk medium. Here diffraction plays a role analogous to dispersion in the temporal domain, but now the nonlinearity may be either positive, for the so-called self-focusing nonlinear medium, or negative, for self-defocusing medium. This again gives rise to two distinct types of solitons, bright and dark, respectively.

Fig. 1. Do these ‘animals’ belong to the same soliton family? (the drawing made by Marc Haelterman in 1989).

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When the group-velocity dispersion in an optical fiber is anomalous (or, similarly, when the nonlinearity of a planar waveguide is self-focusing), a constant amplitude continuous wave is unstable due to the modulational instability [see, e.g., Hasegawa (1989)], and breaks down into a sequence of localized pulses (or beams for the spatial domain). These pulses are bright solitons. Propagation of bright solitons in optical fibers has been verified in a number of elegant experiments, see, e.g., the pioneering paper by Mollenauer et al. (1980), as well as more recent investigations of long-distance soliton transmission in periodically amplified fibers [e.g., Mollenauer et al. (1990)]. These results have been presented in several review papers and books [e.g., Hasegawa (1989), Agrawal (1989), Hasegawa and Kodama (1995) and Haus and Wong (1996)]. In the case of normal group-velocity dispersion in fibers (or a self-defocusing nonlineariy in waveguides), bright solitons do not exist, instead initial pulses (or spatially localized beams) undergo enhanced dispersion (or diffraction) induced broadening and chirping. In this case a constant amplitude wave is modulationally stable and localized pulses can appear only as “holes” on a continuous wave (cw) background, i.e., as dark solitons. Interest in the behaviour of such dark solitons has been motivated by several experimental observations of temporal dark solitons in optical fibers (Emplit et al., 1987; Kro¨kel et al., 1988; Weiner et al., 1988) and spatial dark solitons in bulk media and waveguides (Andersen et al., 1990; Swartzlander et al., 1991; Allan et al., 1991; Skinner et al., 1991; Luther-Davies and Yang, 1992a,b; Duree et al., 1995; Taya et al., 1996; Z. Chen et al., 1996b). Although there has now been many successful experiments in which dark solitons have been observed in optical systems because of the relative ease of producing high intensity light beams or short (ps or fs) optical pulses, it should be remembered that the basic physics behind dark-soliton propagation on a modulationally stable background wave is quite fundamental and it applies to nonlinear problems with a different physical context. By way of illustration we note here some other experimental results including the excitation of nonpropagating n-kink surface modes in a long channel of shallow liquid driven parametrically (Denardo et al., 1990), the observation of dark-soliton standing waves in a discrete mechanical system [Denardo et al. (1992); see also the theory of this phenomenon presented by Kivshar (1993a)], the observation of high-frequency dark solitons during pulse propagation in thin magnetic films (M. Chen et al., 1993), and so on. Optical dark solitons have been investigated in many theoretical and experimental papers and several years ago the early results in this field were summarized in two review papers (Weiner, 1992; Kivshar, 1993b). However, recent experimental achievements have increased interest in the potential applications of optical dark solitons. For example, it was demonstrated (Luther-Davies and Yang, 1992b) that various types of all-optical switches may be ‘written’ using structures created during propagation and interaction of dark spatial solitons. As was demonstrated earlier for bright solitons (Reynaud and Barthelemy, 1990), these induced structures can guide a weak probe beam of a different frequency or polarization thus acting as light induced structured waveguides. These kinds of devices have very interesting properties, e.g., they may conserve transverse velocity — the key characteristic used in dark spatial soliton switching — even in the presence of two-photon absorption (Yang et al., 1994); the effect which can have a dramatic destructive influence on bright spatial solitons (Silberberg, 1990a). The purpose of this review paper is to describe, in the framework of an unified approach, the basic physics of the dark-soliton propagation using the examples taken from nonlinear waveguide optics. We present a systematic analysis of the properties of scalar dark solitons, in the framework of the generalized NLS equations, and vector dark solitons and their generalizations such as

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bright—dark solitons, in the framework of coupled NLS equations. Although throughout we relate the theory to possible applications in guided wave optics, the analytical results are rather general and are applicable to other fields. First, we discuss the physical origin and properties of dark solitons and similar localized structures in the nonlinear systems with no or small dissipation. We consider the most interesting examples, with applications in nonlinear guided wave optics, and discuss the effect of perturbations and the stability of dark solitons, and their waveguiding properties. As a generalization of the concept of dark solitons, we include a summary of results on dark solitons in quadratic or s(2) media; vector dark solitons; and dark—bright solitons; polarization domain walls; (2#1)dimensional dark solitons of circular symmetry; etc. One of the important parts of our review is a summary of the experimental results demonstrating the generation and propagation of (1#1)and (2#1)-dimensional dark solitons. We emphasize that in presenting the analytical results, we avoid the traditional restrictions associated with consideration of only the cases of integrable models (i.e., the cubic NLS equation, for scalar solitons, or the Manakov equation, for vector solitons), as has been done in many previous review papers and books on optical solitons. Instead, we concentrate on the physics of the underlying phenomena and more realistic (generally nonintegrable) physical models. This involves naturally discussions of soliton stability and instabilityinduced evolution of dark solitons, since in nonintegrable models solitary waves can become unstable. At the same time, we do not discuss here some phenomena which can be also related to the physics of dark solitons, such as vectorial dark solitons of the ¹M type [e.g., Y. Chen (1991a,b)], different types of envelope shock waves connecting background of nonequal intensities [e.g., Christodoulides (1991), Kivshar and Malomed (1993), Kivshar and Turitsyn (1993), Cai et al. (1997)], dark surface modes in slab waveguide structures with defocusing nonlinearity [e.g., Andersen and Skinner (1991a,b), Miranda et al. (1992) and Y. Chen and Atai (1992)], dark solitons and dark-profile modes in discrete lattices (Kivshar, 1993a; Kivshar et al., 1994a,c), dark gap solitons in the systems with periodically varying parameters such as optical waveguides with grating [e.g., Feng and Kneubu¨hl (1993), Kivshar (1995) and Kivshar et al. (1995)]. We believe these, and some other related topics still require further analysis and deeper insight into stability as well as experimental verifications. The structure of our review paper is as follows. In Section 2 we provide a kind of framework for the remaining chapters. First, we discuss the basic equations describing the physics of dark solitons in nonlinear optics. This includes nonlinear pulse propagation in optical fibers (temporal solitons), where weak nonlinearity can be described in the framework of the Kerr effect, and self-guided optical beams (spatial solitons) that require to introduce generalized phenomenological models of non-Kerr nonlinearities. Next, we describe the physical origin of dark solitons and discuss their difference from bright solitons by analyzing the results of modulational instability of continuous waves. As we demonstrate, this leads to a different choice of mathematical tools for analyzing these two types of solitons, and a specific role of the integrals of motion and, in the case of dark soliton, the soliton phase. To discuss solitary waves in more realistic physical models described by a perturbed cubic NLS equation (temporal solitons) or in media with non-Kerr optical response (spatial solitons), we present a summary of the results of the perturbation theory developed for dark solitons (Section 3) and also discuss the characteristic scenarios of the instability-induced dynamics dark solitons (Section 4). In particular, we analyse some effects important for optical applications,

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e.g. dark-soliton jitter (Section 3.3); the effect of third-order dispersion on dark solitons (Section 3.4); and briefly discuss the role of a finite-width background in experimental observations of dark solitons (Section 3.5). In Section 5 we present several important multi-component generalizations of dark solitons for systems describing incoherent and parametric interactions between optical polarization modes or harmonics. Section 6 gives a summary of experimental results on dark solitons in optical fibers and the (1#1)-dimensional dark spatial solitons. Extensions of the concept of dark soliton to higher dimensions are presented in Section 7, where we discuss also the theory and experimental demonstrations of optical vortex solitons. Finally Section 8 concludes the review being served as a guide to some open, unsolved problems.

2. Dark vs. bright solitons 2.1. Optical fibers To discuss the physics of dark solitons in optical fibers, we should start from the basic dynamical equation for the complex envelope amplitude of the electric field which can be derived taking into account the weak nonlinearity arising from the Kerr effect in a silica glass. This derivation is well-known and it is exactly the same as in the case of bright solitons [see, e.g., Agrawal (1989), Hasegawa (1989) and Hasegawa and Kodama (1995)]. The derivation is based on the well-known Maxwell’s equations for a dielectric medium in which is assumed that the electric displacement vector D can be split up into two parts, linear and nonlinear ones. The nonlinearity arises from the Kerr effect alone, and the nonlinear part D may be presented in the form, D "n DED2E, where E is nl nl 2 the electric field, and n is the Kerr coefficient. 2 The next important step of the derivation is the use of the fact that wave envelope function E(z, t) is a slowly varying function in the propagation coordinate z and retarded time t, which can be expanded using the Fourier space variable Du"u!u . This represents a small frequency shift of 0 the side band from the carrier frequency u , which in turn induces a small shift pf the carrier wave 0 number, Dk"k!k . The expansion of the wave number k(u) around k can be therefore 0 0 presented as a standard Fourier expansion, ­k k!k " 0 ­u

K

1 ­2k (u!u )# 0 2 ­u2

K

(u!u )2#2 , 0

u/u0 u/u0 where the second-order derivative describes the wave dispersion. Expanding the field envelope E and taking into account simultaneously both temporal (or group-velocity) dispersion and weak Kerr nonlinearity, we arrive at the well-known cubic nonlinear Schro¨dinger (NLS) equation. In an appropriate system of normalized coordinates, this equation becomes i

­u p ­2u ! #DuD2u"0 , ­z 2 ­x2

(2.1)

where p"$1 stands for the sign of the second-order derivative of k(u). This sign corresponds to two distinct types of the fiber group-velocity dispersion, namely anomalous, when p"!1, or normal, when p"#1. The meaning of other values is the following: u is the normalized amplitude

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of the electric field envelope E describing the pulse, z is the normalized distance along the fiber, and the time variable t is a retarded time measured in the reference frame moving along the fiber with the group velocity. The normalization units are well known and, as a matter of fact, they are the same as for bright solitons [see, e.g. Hasegawa (1989)], zPz/Z , x"(t!z/» )/t . 0 ' # The most frequently used normalized units are pc¹2 ¹ Z "0.322 , t" , 0 # 1.763 j2D 0 2p d2k dk D" , »~1"2 . ' j2 du2 du 0 u/u0 u/u0 Here the parameter ¹ represents the full width at half maximum (FWHM) of the pulse intensity, and the pulse propagation is considered in the reference frame moving with the group velocity » . '

G H

G H

2.2. Nonlinear waveguides 2.2.1. Why temporal and spatial solitons are different Usually, the stationary beam propagation in planar waveguides is considered as the phenomenon similar to the pulse propagation in fibers referring to the so-called spatio-temporal analogy in wave propagation [Akhmanov et al. (1967); see also Svelto (1974)]. This means that the propagation coordinate z is treated as the evolution variable and the spatial beam profile along the transverse direction, for the case of waveguides, is similar to the temporal pulse profile, for the case of fibers. This analogy has been developed by many researchers, and it is based on a simple notion that both beam and pulse propagation can be described by the cubic NLS equation [see, e.g., Boardman and Xie (1993), Chiao et al. (1993)]. However, contrary to the accepted opinion, there exists a crucial difference between these two phenomena. Indeed, in the case of the nonstationary pulse propagation in fibers, the operation wavelength is usually selected near the zero of the group-velocity dispersion. This means that the absolute value of the fiber dispersion is small enough to be compensated by a weak nonlinearity such as that produced by the (very weak) Kerr effect in optical fibers which leads to a nonlinearity-induced change in the refractive index by the order of 10~10. Therefore, nonlinearity in fibers is always weak and it is well modeled by the cubic NLS equation, which is known to be integrable by means of the inverse scattering transform [Zakharov and Shabat (1971, 1973); see also Zakharov et al. (1980)]. However, for very short (fs) pulses the cubic NLS equation should be corrected to include some additional (but still small) effects such as higher-order dispersion, induced Raman scattering, etc. (e.g., Hasegawa (1989) and Hasegawa and Kodama (1995)]. Thus, in fibers nonlinear effects are very weak and they become important on large distances (of order of hundred meters or even kilometers). Contrary to the pulse propagation in optical fibers, the physics underlying the stationary beam propagation in planar waveguides and a bulk medium is different. In this case the nonlinear change in the refractive index should compensate for the beam spreading caused by diffraction which is not a small effect. That is why to observe spatial solitons much larger nonlinearities are usually required, and very often such nonlinearities are not of the Kerr type (e.g. they saturate at higher

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intensities). This leads to the models of generalized non-linearities (see Section 2.2.3 below) with the properties of solitary waves different from those described by the integrable cubic NLS equation. For example, unlike the solitons of the integrable cubic NLS equation, solitary waves of generalized nonlinearities may be unstable, they also show some interesting properties such as fusion due to collision, etc. Propagation distances usually involved into the phenomenon of the beam self-focusing and spatial soliton propagation are of order of millimeters or centimeters. Nevertheless, the physics of spatial solitary waves is very rich and it should be understood in the framework of nonintegrable models. 2.2.2. Basic equations First, we consider the propagation of a monochromatic scalar electric field E in a bulk optical medium with an intensity-dependent refractive index, n"n #n (I), where n is the linear 0 /0 refractive index, and n (I) describes the variation in the index due to the field with the intensity /I"DED2. The function n (I) is assumed to be dependent only on the light intensity, and it is usually /introduced phenomenologically. Solutions of the governing Maxwell’s equation can be presented in the form E(R , Z; t)"E(R , Z)e*b0Z~*ut#c.c. , (2.2) M M where c.c. denotes complex conjugate, u is the source frequency, and b "k n "2pn /j is the 0 0 0 0 plane-wave propagation constant for the uniform background medium, in terms of the source wavelength j"2pc/u, c being the free-space speed of light. Further, we assume a (2#1)dimensional model, so that the Z-axis is parallel to the direction of propagation, and the X- and ½-axis are two transverse directions. The function E(R , Z) describes the wave envelope which in the absence of nonlinear and M diffraction effects E would be a constant. If we substitute Eq. (2.2) into the two-dimensional, scalar wave equation, we obtain the generalized nonlinear parabolic equation,

A

B

­E ­2E ­2E 2ik n # # #2n k2n (I)E"0 . 0 0 ­Z 0 0 /­X2 ­½2

(2.3)

In dimensionless variables, Eq. (2.3) becomes the well-known generalized NLS equation, where local nonlinearity is introduced by the function n (I). /For the case of the Kerr (or cubic) nonlinearity we have n (I)"n I, n being the coefficient of /2 2 the Kerr effect of an optical material. Now, introducing the dimensionless variables, i.e. measuring the field amplitude in the units of k Jn Dn D and the propagation distance in the units of k n , we 0 0 2 0 0 obtain the (2#1)-dimensional NLS equation in the standard form, i

A

B

­u 1 ­2u ­2u # # $DuD2u"0 , ­z 2 ­x2 ­y2

(2.4)

where the sign ($) is defined by the type on nonlinearity, self-defocusing (‘minus’, for n (0) or 2 self-focusing (‘plus’, for n '0). 2 For propagation in a slab waveguide, the field structure in one of the directions, say y, is defined by the linear guided mode of the waveguide. Then, the solution of the governing Maxwell’s

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equation has the structure n z~*ut#2 , E(R , Z; t)"E(X, Z)A (½)e*b(0) M n

(2.5)

where the function A (½) describes the corresponding fundamental mode of the slab waveguide. n Similarly, substituting this ansatz into Maxwell’s equations and averaging over ½, we come again to the renormalized equation of the form of Eq. (2.4) with the ½-derivative omitted, which in the dimensionless form becomes the standard cubic NLS equation i

­u 1 ­2u # $DuD2u"0 . ­z 2 ­x2

(2.6)

As has been discussed above, Eq. (2.6) coincides formally with Eq. (2.1) of Section 2.1, which has been derived in the theory of pulse propagation in dispersive nonlinear optical fibers. 2.2.3. Models of optical nonlinearities The generalized NLS Eq. (2.3) has been considered in many papers for analyzing the beam self-focusing and properties of spatial bright and dark solitons [see, e.g., Zakharov et al. (1971), Zakharov and Synakh (1975), Kaplan (1985a,b), Enns and Mulder (1989), Mulder and Enns (1989), Gatz and Herrmann (1991, 1992), Herrmann (1992), Bass et al. (1992), Snyder and Sheppard (1993), Kro´likowski and Luther-Davies (1992, 1993), Kro´likowski et al. (1993), Valley et al. (1994), Christodoulides and Carvalho (1995), Pelinovsky et al. (1996a,b), Micallef et al. (1996)]. All types of non-Kerr nonlinearities discussed in relation with the existence of solitary waves in nonlinear optics can be divided, generally speaking, into three main classes: (i) competing nonlinearities, e.g. focusing (defocusing) cubic and defocusing (focusing) quintic nonlinearity [see, e.g., Kaplan (1985a,b), Gatz and Herrmann (1991); Kro´likowski et al. (1993)] and also generalization to a power nonlinearity [e.g., Pelinovsky et al. (1996a) and Micallef et al. (1996)]; (ii) saturable nonlinearities [see, e.g., Snyder and Sheppard (1993), Kro´likowski and Luther-Davies (1992, 1993), Valley et al. (1994) and Christodoulides and Carvalho (1995)], and (iii) transiting nonlinearities [see, e.g., Kaplan (1985a,b), Enns and Mulder (1989) and Bass et al. (1992)]. Usually, the nonlinear refractive index of an optical material deviates from the linear (Kerr) dependence for larger light intensities. Nonideality of the nonlinear optical response is known for semiconductor (e.g., AlGaAs, CdS, CdS Se ) waveguides and semiconductor-doped glasses [see, 1~x x e.g., Roussignol et al. (1987), Acioli et al. (1990) and Lederer and Biehlig (1994)]. Larger deviation from the Kerr nonlinearity is observed for nonlinear polymers. For example, recently the measurements of a large nonresonant nonlinearity in single crystal PTS (p-toluene sulfonate) at 1600 nm (Lawrence et al., 1994a,b) revealed a variation of the nonlinear refractive index with the input intensity which can be modeled by competing, cubic-quintic nonlinearity, n (I)"n I#n I2 . /2 3

(2.7)

This model describes a competition between self-focusing (n '0), at smaller intensities, and 2 self-defocusing (n (0), at larger intensities. Similar models are usually employed to describe the 3 stabilization of wave collapse in the (2#1)-dimensional NLS equation [e.g., Josserand and Rica (1997) and references therein].

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In a more general case, the models with competing nonlinearities can be described by power-law dependence on the beam intensity, n (I)"n Ip#n I2p , (2.8) /p 2p where p is a positive constant and usually n n (0. p 2p Models with saturable nonlinearities are the most typical ones in nonlinear optics. For higher powers saturation of nonlinearity has been measured in many materials and consequently the maximum refractive index change has been reported [see, e.g., Coutaz and Kull (1991)]. We do not linger on the physical mechanisms behind the saturation but merely note that it exists in many nonlinear media being usually described by phenomenological models introduced more than 25 years ago [see, e.g., Gustafson et al. (1968), Reichert and Wagner (1968) and Marburger and Dawes (1968)]. The effective generalized NLS equation with saturable nonlinearity is also the basic model to describe the recently discovered (1#1)-dimensional photovoltaic dark solitons in photovoltaic—photorefractive materials as LiNbO [see Valley et al. (1994)]. Unlike the phenomenological 3 models usually used to describe saturation of nonlinearity, for the case of photovoltaic solitons this model finds its rigorous justification (Valley et al., 1994; Christodoulides and Carvalho, 1995). There exist several models of the nonlinearity saturation. From a general point of view, the function n (I) describing the nonlinearity saturation should be characterized by three independent /parameters: the saturation intensity, I , the maximum change in the refractive index, n , and the 4!5 = Kerr coefficient n which appears for small I. In particular, the phenomenological model 2 1 n (I)"n 1! , (2.9) /= (1#I/I )p 4!5 satisfies these criteria, provided n "n p/I . In the particular case p"1, the model defined by 2 = 4!5 Eq. (2.9) reduces to the well-known expression derived from the two-level model, which is used most frequently. For the case p"2 the model Eq. (2.9) possesses localized solutions for bright and dark solitons in an explicit analytical form (Kro´likowski and Luther-Davies, 1992, 1993). At last, bistable solitons introduced by Kaplan (1985a,b) usually require a special type of the intensity-dependent refractive index which changes from one type to another one, e.g., it varies from one kind of the Kerr nonlinearity, for small intensities, to another kind with different value of n , for larger intensities. This type of nonlinearity is known to support bistable dark solitons (Enns 2 and Mulder, 1989; Mulder and Enns, 1989) as well. One of the simplest models of such transiting nonlinearities describes a change from one type of the Kerr dependence, to the other one, i.e.,

G

H

G

n(1)I I(I #3 n (I)" 2 (2.10) /n(2)I I'I . 2 #3 A smooth transition of this kind can be modeled by the function (Enns and Mulder, 1989) n (I)"n IM1#a tanh[c(I2!I2 )]N , (2.11) /2 #3 where for I;I , n (I)Kn(1)I, where n(1)"n [1!a tanh2(cI2 )], and for I
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2.3. When the NLS equation fails The scalar NLS equation discussed above for both temporal and spatial wave propagation is considered to be rather universal model. It is derived from the first principles on the basis of very general assumptions about dispersive (or diffractive) and nonlinear properties of physical systems. However, the NLS model may fail in a number of cases, and therefore one should be aware of the validity limits of this simple model. Here we discuss only two main circumstances when the NLS equation fails [see also Gibbon (1990), for examples from other fields]. First of all, a standard derivation of the NLS equation is based on the so-called multi-scale asymptotic technique, sometimes called reductive perturbation method [e.g., Jeffrey and Kawahara (1982) and Taniuti and Nishihara (1983)]. It assumes nonresonant nonlinearities when the most important effects are described by an envelope of the field of the fundamental frequency u propagating with the carrier wave number k. All higher-order harmonics, even being excited, are assumed to be very small and, therefore, they do not modify the field evolution on the main frequency which, in the case of the cubic nonlinearity, is described by the NLS equation. However, when some multiple frequencies are generated, they may strongly affect the wave propagation at the fundamental harmonic provided the so-called matching conditions are satisfied. For example, strong interaction between the main frequency u and two other frequencies u and u occurs 1 2 provided u"u #u and the phase mismatch Dk"k!(k #k ) vanishes. This kind of three1 2 1 2 wave mixing is possible in a medium where the lowest-order nonlinearity is quadratic. When the medium nonlinearity is cubic, the wave coupling is possible in the form of a four-wave mixing process. When any of such resonance conditions is satisfied, the envelope of the fundamental field becomes strongly coupled to a secondary field (or more than one field) and a single NLS equation becomes not valid. Coupling between the modes may also support multi-component solitary waves which might differ very much from the conventional solitons of the scalar NLS equation (see Section 5.4 below). Additionally, the intermode interaction may lead to instability of solitary waves (see, e.g., Section 5.3 below). The second class of problems when the NLS model fails is closely related to optical spatial solitons described by non-Kerr nonlinearities. Indeed, it is well known that the NLS equation with nonlinearity stronger than cubic, e.g. a power-law focusing nonlinearity DuD2qu, has localized solutions which blowup, so that a singularity appears in finite z. This phenomenon occurs for negative values of the system Hamiltonian under the condition qD52, where q is the nonlinearity power and D stands for the (D#1)-dimensional model [e.g., Zakharov (1972)]. Blowup (or collapse) in finite z means that the NLS model of this dimension fails as an envelope equation since it breaks the scales on which it was derived in the framework of the multi-scale asymptotic technique. For spatial solitons this condition means that if D"2, than the cubic nonlinearity DuD2u is already sufficient to induce collapse. If D"1, then one needs the quintic (or higher-order) nonlinearity to induce collapse. Blowup indicates also that the primary NLS model should be corrected, e.g. by taking into account the effects of nonparaxiality in the beam self-focusing [Feit and Fleck (1988); Fibich (1996)]. Therefore, we can conclude that the NLS equation has a multiple of uses even though its applicability must be treated with care. In nonlinear optics, it is not only generic model for describing self-guided beams or spatial solitons, in waveguides, or bulk media and pulses or temporal solitons, in optical fibers, but it allows also to describe the beam self-focusing possessing

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the properties of blowup solutions. When the generalized NLS equation is valid as the main approximation, all corrections to it can be treated within the perturbation theory. However, near resonances the inclusion of higher-order corrections is very often meaningless and requires another approach [see, e.g., Oughstun and Xiao (1997), as an example of the failure of the envelope approximation for ultrashort pulses in a dispersive attenuative medium]. 2.4. Modulational instability and solitons The NLS equation has the simplest solution in the form of a continuous wave (cw) given, e.g. for the Eq. (2.6), by the expression, u"u e*kx~*bz , 0

b"1 k2Gu2 , 2 0

(2.12)

where the sign stands for the type of nonlinearity. Let us investigate the linear stability of the exact solution, Eq. (2.12), against small perturbations. To do so, we follow the standard procedure and look for solutions describing small variations around the exact solution, Eq. (2.12), in the form, u"(u #m)e*kx~*bz`*t , 0

(2.13)

where the function m and derivatives of the phase t are assumed to be small. Substituting Eq. (2.13) into the NLS Eq. (2.6), we come to a system of two coupled linear equations for m and t. Looking for solutions to these functions in the form, m, t&exp(iXz!iQx), we obtain the dispersion relation, (X!kQ)2"Q2($u2#1 Q2) , 0 4

(2.14)

which shows that small excitations on the cw background, Eq. (2.12), are absolutely stable only for the case of the defocusing medium (the sign ‘plus’), whereas they become unstable for the focusing nonlinearity provided Q2(4u2. In the former case, the small-amplitude waves can propagate 0 along the background and these waves are characterized by the minimum (‘sound’) velocity c2"u2 . 0

(2.15)

This property of modulational instability of the cw background is closely connected with the existence and type of solitary wave solutions of the NLS equation. Namely, spatially localized solutions with vanishing asymptotics are possible only for the case when the plane wave solution is unstable, i.e., only for the focusing nonlinearity. Similarly, the NLS Eq. (2.1) displays the same properties of the plane wave stability and, therefore, divides localized solutions into two different classes depending on the sign parameter p. For example, the defocusing nonlinearity, in the spatial problem, corresponds to the normal group-velocity dispersion and p"#1, in the temporal problem. Thus, in the case p"!1 (anomalous dispersion) the cw solution in unstable and therefore the appropriate boundary conditions to Eq. (2.1) is DuDP0 at xP$R. For these conditions Zakharov and Shabat (1971) showed that Eq. (2.1) [and, therefore, Eq. (2.6)] is exactly integrable by means of the inverse scattering transform, and it possesses localized solutions called bright

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solitons. General solution for the bright soliton has the form, ae*vx@2~*(v2@4~a2)z , u(z, x)" cosh[a(x!vz)]

(2.16)

where a is the soliton amplitude and v is its velocity. At v"0 this soliton has a simplified structure referred to as the fundamental bright soliton (see Fig. 2a) ae*a2z . u(z, x)" cosh(ax)

(2.17)

For p"#1 (normal dispersion) the cw solution DuD"u is always stable against small 0 modulations of its shape, and, as a result, Eq. (2.1) has soliton solutions in the form of localized ‘dark’ pulses created on the cw background. The NLS equation with the boundary conditions DuDPu is also exactly integrable by the inverse scattering technique (Zakharov and Shabat, 1973) 0 and its one-soliton solution for a single dark soliton can be written in the form, u(z, x)"u [B tanh H#iA] e*u20z , 0 where

(2.18)

H"u B(x!Au z) . (2.19) 0 0 Parameters A and B are connected by a simple relation, A2#B2"1, so that instead of two parameters we can use only one, introducing A"sin / and B"cos /. The effective angle / corresponds to the total phase shift across the dark soliton, 2/. Soliton solution, Eqs. (2.18) and (2.19), has, unlike the bright soliton, Eq. (2.16), the only parameter / and the function B2"cos2 / characterizes the soliton intensity at the center, i.e., the minimum soliton intensity relative to the

Fig. 2. Intensity and phase as functions of normalized coordinate x for bright (a), black (b) and (c) gray solitons [adapted from Tomlinson et al. (1989)].

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background intensity. The soliton intensity profile DuD2 has the dip-like shape (see Fig. 2b and Fig. 2c)

A

B

cos2 / . DuD2"u2 1! 0 cosh2 H

(2.20)

The value u sin / has the meaning of the soliton velocity in the x-space, i.e. the relative velocity 0 between the dark soliton and the background. We distinguish two limit cases of the dark soliton solution, Eq. (2.18). The so-called fundamental dark soliton (at /"0) (2.21) u(z, x)"u tanh(u x) e*u20z , 0 0 is the anti-symmetric function of x with the n phase shift and zero intensity at its center (see Fig. 2b). Another limit case, when cos2/(1, corresponds to the so-called gray solitons (see Fig. 2c) when the minimum intensity does not drop to zero [for the small-amplitude limit, see, e.g., discussions in Section 2.5.2 and also Kivshar (1993b)]. As may be seen in Fig. 2a—c, one of the major differences between bright and dark solitons is in the phase dependence: Bright solitons have a constant phase across the localized region but dark solitons have a nontrivial distribution of their phase, so that there exists a finite phase jump across the localized region. Below, we relate this difference with the different physics of these two types of solitary waves. A dark-soliton solution of the generalized (1#1)-dimensional NLS equation i

­u 1 ­2u # #g(DuD2)u"0 , ­z 2 ­x2

(2.22)

propagating on a cw background with the phase velocity b/k can be always found in quadratures and presented in the following form, u(z, x)"u [G(x)#iF(x)] e*kx~*bz , 0 where the real and imaginary parts satisfy the boundary condition

(2.23)

G2($R)#F2($R)"1 , u being the background amplitude. This kind of solution can be presented as a (rather complic0 ated) curve on the plane of the complex variable K defined by its real part, G, and imaginary part, F, i.e. K"G#iF, with the spatial variable x treated as an internal parameter [see Kosevich and Kovalev (1989) and Kro´likowski et al. (1993)]. In the particular case of the cubic NLS equation, the solution for a dark soliton is given by Eq. (2.18), so that F"const and the corresponding curve on the K-plane is a straight line (see Fig. 3). For some particular nonlinear functions g(DuD2), dark soliton solutions of the generalized NLS Eq. (2.22) can be found in an explicit analytical form. There exist only a few such cases: two types of dark solitary waves in the model of cubic—quintic nonlinearity, Eq. (2.7) [e.g., Barashenkov and Makhankov (1988), Gagnon (1989), Makhankov (1990) and Kivshar et al. (1996)], dark solitons of the so-called threshold nonlinearity (Snyder et al., 1993) and, at last, dark solitons in a saturable medium described by Eq. (2.9) at p"2 (Kro´likowski and Luther-Davies, 1993). As has been mentioned above (see also Section 2.5.1 below), the special case of the cubic NLS Eq. (2.1) at p"#1 or Eq. (2.6) with the sign ‘minus’ is exactly integrable by the inverse scattering transform

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Fig. 3. Schematic presentation of a dark soliton solution of the defocusing cubic NLS equation on a complex plane K. Angle / defines the soliton ‘grayness’.

(Zakharov and Shabat, 1973; Iizuka et al., 1991) meaning that not only single dark solitons but also their superposition describing elastic collisions of N dark solitons can be formally found in an explicit analytic form [see some examples in the papers by Iizuka et al. (1991), Gagnon (1993) and Miller (1996)]. 2.5. Dark solitons: mathematical tools 2.5.1. Inverse scattering transform As is well known, the basic mathematical method to analyse any kind of wave propagation in linear media is the famous Fourier transform method. For definiteness, our discussion below assumes the case of the stationary beam propagation and spatial solitons, though it equally applies to the pulse propagation in fibers. So, making decomposition of the input beam u(z"0, x) into a set of linear Fourier modes, we reduce the problem of the beam propagation to a trivial oscillatory evolution of the Fourier components (or harmonics), and then construct the beam shape at any propagation distance z. In the case of a homogeneous linear medium, the only type of the Fourier modes are given by nonvanishing periodic functions, and the spectrum of the corresponding eigenvalues is continuous. However, the Fourier method is rather general and it can be also applied to analyse wave propagation in inhomogeneous linear media. In this case, additionally to the periodic modes with the continuous spectrum, there appear the so-called spatially localized (or guided) modes existing due to a spatial variation of the properties of the linear medium. This localized or guided modes are possible however only for special discrete eigenvalues. Therefore, a complete set of linear eigenstates which can be obtained by the Fourier method includes both discrete and continuous modes. As is well known, the Fourier transform method cannot be used for nonlinear systems where the superposition principle does not exist. However, one may try to invent another kind of (nonlinear) decomposition to obtain nonlinear modes whose evolution will be reduced to trivial oscillations similar to that given by the Fourier transform. Such a decomposition is known to exist only for some special nonlinear equations (the so-called exactly integrable equations), and it is called the inverse scattering transform method. In some sense, the inverse scattering transform provides an inherent similarity between the modes of linear inhomogeneous systems and those of nonlinear

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homogeneous systems where both types of localized modes, i.e., guided waves, for the former case, and solitary waves, for the latter case, correspond to a certain set of discrete eigenvalues. The main idea of the nonlinear decomposition based on the inverse scattering transform is to find an appropriate linear eigenvalue problem which includes the input beam u(z"0, x) as an effective potential. The main property of this linear eigenvalue problem is: In spite of an explicit dependence of the potential on z, the eigenvalues are conserved quantities provided the wave field u(z, x) satisfies the primary nonlinear equation. Finding the eigenvalue problem with this property, i.e., eigenvalues remain invariant when the potential evolves according to a (integrable) nonlinear evolution equation, is a common feature of the integrability technique based on the inverse scattering transform method. The property that the eigenvalues remain invariant means that if they are found (or known) for u(z"0, x), they remain the same for any z. For a given u(z"0, x), the eigenvalue problem can be solved as a scattering problem for an auxiliary eigenfunction W(x; j), j being a spectral problem eigenvalue. The scattering data are the amplitude a(j) of a transmitted wave, the amplitude b(j) of a reflected wave, the eigenvalues of the discrete spectrum Mj N, and the n normalized coefficients b of the eigenfunctions of the discrete spectrum. Similar to the machinery n of the standard Fourier transform method, the evolution of the scattering data Ma(j), b(j), j , b N is n n trivial, and the solution of the primary nonlinear equation is then found using results of the inverse scattering method to find the function u(z, x) through the z-dependent scattering data. The similar method is well-established in quantum mechanics, however here this method is used for zdependent potential provided the conservation constrains are satisfied. Importantly, each eigenvalue of the discrete spectrum of the scattering problem governs a localized solution, i.e., soliton, after such a reverse procedure. The soliton beam is of conserved shape due to invariance of the eigenproblem spectrum. Therefore, the stationary nature of the eigenvalues provides the important property of the stability of the soliton beams when undergoing collisions. Hence, solitons are important not only as a particular solutions of the nonlinear equation, but as an unique solution whose stability is guaranteed by the invariant property of the corresponding eigenvalue problem. Furthermore, using the scattering data one may decompose any localized input beam into a set of normal (nonlinear) modes (like linear Fourier modes) and the soliton modes are dominant nonlinear modes in such a decomposition, i.e., asymptotically a localized input beam will be transformed into a set of solitons emitting an energy excess as radiation. As was found by Zakharov and Shabat (1971, 1973), the NLS Eq. (2.6) is exactly integrable and it may be therefore analysed by means of the inverse scattering transform method for any sign of nonlinearity (or, for temporal solitons, for any sign of the dispersion coefficient). Mathematical tools of the inverse scattering transform allow us to solve the Cauchy problem for the NLS Eq. (2.6) if the input beam profile is given. According to this method, to find which type of input beam u(0, x) generates solitons and to determine soliton parameters, one has to solve the eigenvalue Zakharov—Shabat problem for the auxiliary two-component eigenfunction (W , W )T, 1 2 ­W 1"ijW !iu(0, x)W , 1 2 ­x

­W 2"!ijW !iu* (0, x)W , 2 2 ­x

(2.24)

where for the sign ‘minus’ in Eq. (2.6) the boundary conditions are: u(0, x)Pu , for xP#R, and 0 u(0, x)Pu e*h, for xP!R, where the asterisk stands for complex conjugation, u is the 0 0 background amplitude, and h is a constant phase.

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For the NLS Eq. (2.6) with the sign ‘minus’, the spectral problem, Eq. (2.25), has a discrete spectrum lying in the region of real values DjD(u , and the discrete spectrum is invariant in z when 0 the coordinate evolution is included. Each real discrete eigenvalue j "u sin / corresponds to n 0 n a dark soliton beam with the amplitude of the dip in the background defined by u cos / that 0 n moves with the transverse 2 sin / . Thus, the asymptotic evolution of the input localized beam is n described by the eigenvalues of the discrete spectrum of the scattering problem, Eq. (2.25), so that any input beam will be transformed into a certain set of solitons for large enough propagation distances z. From the viewpoint of experimental studies of the soliton generation by an arbitrary input beam, it is useful to know the solution of the scattering problem (and, therefore, the number and parameters of the generated solitary waves) for certain types of input beam profile. For example, let us take the shape of the input beam as the tanh-like profile, i.e., u(0, x)"u tanh(ax) , (2.25) 0 where the ratio u /a is arbitrary. Then, as has been shown by Zhao and Bourkoff (1989a,b), the 0 eigenproblem, Eq. (2.25), can be solved exactly and the eigenvalues of the discrete spectrum are defined as j "0 , j "!j "Ju2!w2 , where n"1, 2 ,2, N , (2.26) 1 2n 2n`1 0 n 0 and positive w are defined as n na w "u 1! . n 0 u 0 Maximum value N is the largest integer number satisfying the condition N (u /a. The first, zero 0 0 0 eigenvalue corresponds to a black soliton with the zero intensity at its center, which is always created by the input beam, Eq. (2.25). The even number of the secondary eigenvalues, Eq. (2.26), correspond to N symmetric pairs of gray solitons propagating to the left and right. Thus, the total 0 number of dark solitons created by the input beam, Eq. (2.25), is N"2N #1, and it depends on 0 the ratio u /a. 0 The inverse scattering transform can be employed, in principle, to analyze the soliton generation in the cubic NLS equation from any type of the input profile. Several input profiles were analyzed in the literature additionally to the tanh profile, Eq. (2.25). For example, Gredeskul and Kivshar (1989b) analyzed the so-called step-like profile, u(0, x)"u e*h1, for x(0, and u(0, x)"u e*h2, for 0 0 x'0, showing that it always generates a single dark soliton corresponding to the eigenvalue j "!u cos[1(h !h )]. To generate several dark solitons by the similar modulation of the 1 1 0 2 2 background phase, one need to use an input pulse with a several phase steps. In particular, N phase steps can generate N dark solitons in the asymptotic region provided (Kivshar and Gredeskul, 1990)

A

B

Dq '(2u )~1D cot(h /2)#cot(h /2)D , j 0 j`1 j where Dq is the distance between the ( j#1)th and jth steps, and h is the value of the jth phase j j jump. In particular, two steps of different signs always produce two dark solitons of opposite velocities (Gredeskul et al., 1990). Similar results can be obtained for the modulation of the background amplitude and the simple case of a boxlike input was analyzed by Zakharov and

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Shabat (1973), for zero intensity in the box, and by Gredeskul and Kivshar (1989b), for a general case. The most interesting feature of dark solitons, i.e., their thresholdless generation, was demonstrated and rigorously proven by Gredeskul and Kivshar (1989a) for the cubic NLS equation. Indeed, as is well known, generation of bright solitons requires to exceed a certain threshold defined through the input profile. Namely, bright solitons will be created by a localized input pulse provided (Kivshar, 1989) :`=Du(0, x)D dx'p/2. Unlike bright solitons, dark solitons can be created ~= by an arbitrary initial small dip on a cw background, for example, when the input pulse has the following general form: u P0 at DxDPR. 1

u(0, x)"u e*h#u (x) , 0 1

(2.27)

As was shown by Gredeskul and Kivshar (1989a), for an arbitrary but small u (x) (which falls off 1 fast enough at DxDPR), and for

G P

D,Re e~*h

H

`= u (x) dx (0 , 1 ~=

(2.28)

there always exist two eigenvalues of the discrete spectrum of the spectral problem, Eq. (2.25), j "$j ,$u (1!1D2), which correspond to a pair of dark solitons with equal amplitudes 1,2 0 0 2 u D and the opposite velocities $2j . As a consequence of this result, an even input pulse always 0 0 produces at least a pair of dark solitons. 2.5.2. Small-amplitude approximation The case B2"cos2/;1 in Eq. (2.18) corresponds to a shallow or small-amplitude dark soliton. This is the case when the soliton transverse velocity v is close to the limit speed c of linear waves propagating on the cw background. Importantly, in this limit we can apply a reductive perturbation technique to obtain analytical results even for the nonintegrable genreralized NLS Eq. (2.22) because in this case the dynamics of dark solitons is described by an effective Korteweg de Vries (KdV) equation [e.g., Makhankov (1990) and Kivshar (1993b)]. To discuss the dark-soliton dynamics in the small-amplitude limit, we look for solutions of the self-defocusing NLS Eq. (2.6) in the form, u(z, x)"Mu #a(z, x)N e*u20ze*((z,x) . 0

(2.29)

Substituting Eq. (2.29) into the NLS Eq. (2.6), we obtain two equations for the functions / and a which are not defined yet. Now if one will use new variables which allow us ‘to split’ the propagation directions, m"e(x!cz) ,

f"e3z ,

(2.30)

e being an arbitrary small parameter connected with the soliton amplitude, and look for solutions in the form of the asymptotic series in e, a"e2a #e4a #2, 0 1

/"e/ #e3/ #2, 0 1

(2.31)

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then in the lowest approximation those equations yield the simple equation for the beam phase, ­/ ca 0"! 0 . ­m u 0

(2.32)

and the KdV equation for the beam amplitude, 2c

­a ­a 1 ­3a 0#6u a 0! 0"0 . 0 0 ­f ­m 4 ­m3

(2.33)

Eq. (2.33) has a sech2-type solution describing a soliton, so that the dark soliton in this smallamplitude limit can be obtained as an asymptotic expansion, Eqs. (2.29) and (2.30), calculated with the help of the KdV soliton solution [e.g., Kivshar (1993b)]. For the NLS Eq. (2.6) the established link between the small-amplitude dark solitons and solitons of the KdV Eq. (2.33) is not a remarkable fact because both these models are exactly integrable, and we can obtain, in principle, analytical results for them separately with the same completeness. However, the most important result is that such a property is also valid for the generalized NLS Eq. (2.22). For example, for the generalized NLS equation we can derive [e.g., Kivshar et al. (1993)] the following KdV equation: 2c

­a ­a 1 ­3a 0! 0!2u [3g@(u2)#u2gA(u2)]a 0"0 , 0 0 0 0 0 ­m ­f 4 ­m3

(2.34)

where c2"u2g@(u2) 0 0 is the limit speed of linear waves in the NLS model of generalized nonlinearity. Additionally, the approach based on the small-amplitude approximation of dark solitons is very useful to analyse the influence of different perturbations on the dark-soliton dynamics using the known analytical results for the KdV solitons [see, e.g., Kivshar and Afanasjev (1991a) and Kivshar (1993b)]. This approach was also extended to the case of the anomalous dispersion in the perturbed NLS equation (Frantzeskakis, 1996). 2.6. Integrals of motion A difference between bright and dark solitons can be also seen in the definition of the integrals of motion of the NLS model for vanishing (bright) and non-vanishing (dark) boundary conditions. Being mostly interested in the case of spatial solitons, we consider the cubic NLS equation in the form of Eq. (2.6) with the self-defocusing (negative) nonlinearity. Eq. (2.6) can be treated as the Euler equation which follows from the Lagrangian with the density,

A

B K K

i ­u ­u* 1 ­u 2 1 L" u* ! ! DuD4 , u ! 2 2 ­z ­z 2 ­x

(2.35)

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corresponding to the system Hamiltonian,

P GK K

H

1 `= ­u 2 H " #DuD4 dx . (2.36) 505 2 ­x ~= Because the cubic NLS Eq. (2.6) is exactly integrable, it possesses an infinite number of integrals of motion. However, our primary interest is in the nonintegrable generalized NLS models, so that we are interested only in the fundamental invariants which exist also in a general case and have a clear physical meaning. Because the system described by the NLS equation is conservative, the total energy defined by Eq. (2.36) is conserved. Additionally, we consider also the field momentum,

P A

B

­u* ­u i `= u !u* dx , M " 505 2 ­x ­x ~= and the power,

P

(2.37)

`=

DuD2 dx . (2.38) ~= Invariants, Eqs. (2.36), (2.37) and (2.38), can be formally introduced for any types of the boundary conditions to Eq. (2.6). However, it is easy to see that in the case of nonzero boundary conditions, the values defined by Eqs. (2.36), (2.37) and (2.38) are divergent. Indeed, let us calculate these values for the exact cw solution, Eq. (2.12), and obtain, P " 505

P "u2¸ , #8 0

M "ku2¸"kP , #8 0

P2 M2 H " # , #8 2¸ 2P

(2.39)

where ¸ defines a spatial extension of the cw beam. It is clear that these values are not finite provided ¸PR. Obviously, the similar problem appears if we use the invariants, Eqs. (2.36), (2.37) and (2.38), for the case of dark soliton which corresponds to the same nonvanishing boundary conditions. However, for a dark soliton itself it is possible to introduce finite (or renormalized) expressions for these invariants. We consider the most general form of a dark soliton propagating on the cw background wave, Eq. (2.12). The solution of Eq. (2.6) describing a dark soliton with the velocity v moving on a running background with the propagation constant b can be written in the following form u(x, z)"u MB tanh[u B(x!vz)]#iAN e*kx~*bz`*h0 , (2.40) 0 0 where, as above, the parameter b"1k2#u2 characterizes the dispersion relation for the back2 0 ground wave, h(0) is a constant phase, and the soliton and background parameters, A, B, and v are connected by the relations, u A"v!k , A2#B2"1 . (2.41) 0 In such a form, the dark soliton solution, Eq. (2.40), is characterized by three independent parameters, two of them, u and k, describe the amplitude and wave number of the cw background, 0 while only one, e.g., A, characterizes the dark soliton itself. The asymptotics of the solution, Eq. (2.40), coincide with those of the cw solution, Eq. (2.12). However, the presence of a dark soliton on the background manifests itself in different phases at xP$R, i.e., the plane wave

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soliton is shifted in phase, and the total phase shift across the dark soliton is

C

Dh"2 tan~1

AB D

AB

p A B ! "!2 tan~1 . 2 B A

(2.42)

Let us define the renormalized integrals of motion which characterize the dark soliton itself. It is clear that the integrals of motion, Eqs. (2.36), (2.37) and (2.38), describe a complex object ‘background#dark soliton’, and we should modify the integrals to remove the corresponding contributions of the cw background. After such a renormalization procedure the integrals of motion calculated for the solution, Eq. (2.40), become finite. We will present this renormalization for the simpler case k"0, i.e. when the cw background is at rest. It is clear that the power, Eq. (2.38), should be renormalized calculating a difference between the total power, Eq. (2.38), and the corresponding value, Eq. (2.39), contributed from the background

P

P" 3

=

~=

dx (u2!DuD2) . 0

(2.43)

Calculating P for the solution, Eq. (2.40), we find P "2u B. This value is often called complimentr r 0 ary power of a dark soliton. The similar procedure can be used to renormalize the system Hamiltonian,

P

1 = dx H" r 2 ~=

GK K

H

­u 2 #(DuD2!u2)2 , 0 ­x

(2.44)

and for the dark soliton solution, Eq. (2.40), it takes the form H "4 (c2!v2)3@2 . r 3

(2.45)

Renormalization of the field momentum is a bit tricky. From the integral of motion, Eq. (2.37), we should take away a contribution of the background which this time is given by the phase difference produced by the presence of the dark soliton, Eq. (2.42). Because the momentum of the cw background has the form M"ku2¸ [see Eq. (2.39)], this phase difference gives a nonzero 0 contribution even at k"0. Indeed, this contribution is calculated as u2:k(x) dx"u2Dh, where k(x) 0 0 describes a local change of the background wavenumber. As a result, the renormalized momentum of a dark soliton is defined as

P

A

B

i = ­u* ­u M" !u* !u2Dh , dx u 3 2 0 ­x ­x ~=

(2.46)

where Dh is given by Eq. (2.42). Substituting the solution, Eq. (2.40), at k"0 into Eq. (2.46), we find M "!2vJc2!v2#2c2 tan~1 3 where c2 is defined in Eq. (2.15).

A

B

Jc2!v2 , v

(2.47)

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Differentiating the formulas, Eqs. (2.47) and (2.45), over the soliton velocity v, we recover the simple relation ­H 3 "v , (2.48) ­M 3 which indicates that the renormalized integrals of motion satisfy the standard expression of classical mechanics so that a dark soliton, similar to a bright soliton, can be regarded as an effective particle. For the case of the generalized NLS Eq. (2.22), we can also introduce the renormalized invariants which are free of any divergence and characterize the dark soliton itself. In a similar way, we define the complementary power P , 3 `= P" (u2!DuD2) dx (2.49) 3 0 ~= the renormalized field momentum,

P

P A

BA

B

i `= ­u* ­u u2 M" u !u* 1! 0 dx , 3 2 ­x ­x DuD2 ~= and the renormalized Hamiltonian,

P GK K P

(2.50)

H

`= 1 ­u 2 @u@2 [g(u2)!g(I)] dI dx . (2.51) # 0 2 ­x 2 0 ~= u All these values remain finite for any type of the nonlinear function g(I) in the generalized NLS Eq. (2.22). H" 3

2.7. Physical interpretation of dark solitons As has been mentioned above, the Fourier transform method is a useful tool for linear systems where it employs the familiar superposition principle. This method can be also applied to analyse linear wave propagation in inhomogeneous media when, additionally to the periodic modes with the continuous spectrum, there appear spatially localized (or guided) modes existing due to spatial variations of the properties of the linear medium. This localized or guided modes are possible however only for special discrete eigenvalues. Therefore, a complete set of linear eigenstates which can be obtained by the Fourier method includes both discrete and continuous modes. For nonlinear systems the Fourier method cannot be used and the superposition principle is not valid. However, as was discussed in Section 2.5.1, the inverse scattering transform for a certain class of nonlinear equations provides a qualitative similarity between the modes of linear inhomogeneous systems and those of nonlinear homogeneous systems in the sense that both types of localized modes, i.e. guided waves, for the former case, and solitary waves, for the latter case, correspond to a certain set of discrete eigenvalues. We may wonder if this kind of similarity can be formulated in a direct way, when a nonlinear equation does not allow the application of the inverse scattering transform. As was advanced by Snyder and coauthors (Snyder et al., 1993, 1995), such a similarity does exist for a rather wide class

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of stationary (and, sometimes, periodic) solutions of the Maxwell’s equations and it provides a complete analogy between guided waves as modal solutions of the waveguide optics, from one side, and solitary waves of generalized nonlinearities, from the other side. To demonstrate this property, we follow the papers by Snyder et al. (1991, 1993, 1995) and consider the stationary (1#1)-dimensional evolution of the TE field E(x, z)"E(x)e~*bz in a homogeneous medium with the intensity-dependent refractive index n. Then, the envelope function E(x) is a solution of the scalar wave equation,

G

H

d2 #k2n2(DED2)!b2 E(x)"0 , 0 dx2

(2.52)

which is more general than the generalized NLS Eq. (2.22) derived from Eq. (2.52) in the paraxial approximation. Now, as was pointed out by Snyder et al. (1991), a spatially localized solitary wave solution E(x) of Eq. (2.52) can be treated as a guided mode of the effective linear waveguide that it induces. Indeed, let us define the linear waveguide using the so-called self-consistency relation (Snyder et al., 1991, 1993) n2 (x),n2(DE(x)D2) , (2.53) -*/ provided the solution E(x) of the nonlinear Eq. (2.52) is known. Then, it immediately follows that E(x) is also a solution of the linear eigenvalue problem, Eq. (2.52), with the spatial-dependent refractive index defined by Eq. (2.53). This simple notion allows to borrow from the literature on linear optical waveguides [e.g., Snyder and Love (1973)] to find analytical solutions and the nonlinear media corresponding to them. For example, a slab waveguide is found to correspond to the so-called threshold nonlinearity [e.g., Snyder et al. (1991)]. Additionally, the interpretation of solitary waves as modes of the induced waveguides provides a simple physics which, for example, can explain why some kind of solitary waves are possible (Snyder et al., 1995). As an example, we consider a linear waveguide with a refractive index profile of the form n2 (x)"n2 #(n2!n2 ) sech2(x/a) , (2.54) -*/ = 0 = where a is the characteristic profile half-width. All modes of this waveguide can be expressed with the help of Legendre functions [e.g., Snyder and Love (1973)]. In particular, the fundamental bound mode is given by the familiar expressions E(x)"E sechs(x/a) , (2.55) 0 with the propagation constant b2"(k n )2#(s/a)2. Self-consistency condition, Eq. (2.53), allows 0 = to conclude that the mode, Eq. (2.54), corresponds to a bright soliton of the Kerr medium provided s"1 at a fixed value of the waveguide parameter, »2,(k a)2(n2!n2 )"2 . 0 0 = Thus, a bright solitary wave can be treated as the bound mode of the linear waveguide it induces. As for dark solitons, it is clear from their unbounded fields that, if a dark soliton is a mode of its induced linear waveguide, than it must be a radiation mode, that is, a mode of the continuous spectrum. Dark solitary waves constructed from modes of linear waves were discussed by

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Snyder et al. (1993), who provided a simple physical picture of dark solitons through reflectionless plane-wave scattering from a linear dielectric waveguide. For the sech-type waveguide, Eq. (2.54), the fields of radiation modes consist of an incident plane wave and its reflections, except for integer values of the parameter s introduced as »2"s(s#1), where » is the waveguide parameter defined above. For these particular values of s, there is no reflection wave. For example, for s"1 and the waveguide parameter »"J2, the radiation reflectionless mode of the sech-type waveguide, Eq. (2.54), has the form of a dark soliton of the cubic (or Kerr) medium, E(x)"E

e*qx [tanh(x/a)!iQ] , 0 (1!iQ)

(2.56)

where Q"aq, and q is a real continuous variable (0(q(k n ) related to the direction of the 0 = incident plane wave, and it also defines the propagation constant, b2"(k n )2!q2. Thus, the 0 = elementary physics of reflectionless plane-wave scattering from a linear waveguide allows a deeper insight into the theory of dark solitons.

3. Perturbation theory and applications 3.1. Equations for soliton parameters 3.1.1. Direct method Renormalized integrals of motion for dark solitons allows us to apply a straightforward technique for describing the soliton perturbation-induced dynamics. The approach based on the invariants is the simplest one, and it can be regarded as a direct method whereas more involved technique is based on the inverse scattering transform [e.g., Kivshar and Malomed (1989)]. First, we consider the case of a constant background when a perturbation does not change the parameters of the cw solution. We consider the perturbed NLS equation, i

­u 1 ­2u # !DuD2u"eP(u) , ­z 2 ­x2

(3.1)

where the term eP(u) in the right-hand side stands for a small perturbation. Because the perturbation is assumed do not change the cw background, it should vanish as DxDPR. To find the equation for the evolution of the soliton parameters, first we introduce the new function v(x, z) to make the renormalization procedure more straightforward, u(x, z)"u v(x, z) e~*u20z , 0 and obtain the equation for v, i

­v 1 ­2v # #(1!DvD2)v"ePI (v) , ­f 2 ­m2

(3.2)

(3.3)

where ePI (v) is a renormalized perturbation eP(u). Considering only the case of a nonpropagating background wave, we may now analyze how the parameters of the dark soliton solution be

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changed due to the perturbation from the right-hand side of Eq. (3.3). At e"0, the dark soliton can be written in the form, v(f, m)"cos / tan H#i sin / ,

H"g(m!Xf) ,

(3.4)

where f"u2z, m"u x, g"cos /, and X"sin /. This solution is characterized by the soliton 0 0 phase angle / (D/D(p/2) which describes the ‘darkness’ of the soliton through the simple relation cos2 / . DuD2"1! cosh2 H

(3.5)

To treat analytically the influence of a small perturbation ePI (v) on the parameters of the dark soliton, Eq. (3.4), we use the so-called adiabatic approximation of the perturbation theory for solitons [see, e.g., Kivshar and Malomed (1989)]. According to this approach, the parameters of the dark soliton, Eq. (3.4), are considered as slowly varying in z but with the functional shape which remains unchanged, i.e., it is assumed to be described by Eq. (3.4), where we should modify Z as the following:

C P

D

H"cos /(f) m! df@ sin /(f@) .

(3.6)

To derive the equation for the perturbation-induced evolution of the soliton phase /(f), we start from the renormalized Hamiltonian of the unperturbed system,

P

GK K

H

­v 2 1 = dm #(DvD2!1)2 , H" 3 2 ­m ~= which for the soliton solution, Eq. (3.4), takes the value H "4 cos3/ [cf. Eq. (2.45)]. Calculating 3 3 the derivative of H over f and using Eq. (3.3), we find the result (Kivshar and Yang, 1994a) 3 ­v* e d/ `= dm PI (v) " Re , (3.7) ­f df 8 cos2/ sin / ~= where the functions in the right-hand side of Eq. (3.7) should be also calculated in the adiabatic approximation using the solution given by Eq. (3.4). The similar equation can be obtained by the equivalent approach based on the Lagrangian technique, as was shown by Kivshar and Kro´likowski (1995a). If the perturbation ePI (v) in Eq. (3.7) does not vanish at DxDPR, it will certainly affect the background wave. This is the standard case of dissipative perturbations which produce a slow decay of the background amplitude (Kivshar and Yang, 1994a; Yang et al., 1994). Taking the limit DxDPR and interesting in the evolution of the nonpropagating background u (z) (i.e., that which " does not depend on x), we obtain the equation for the background amplitude u " du i "!Du D2u "eP(u ) . (3.8) " " " dz

GP

H

Eq. (3.8) allows to find the law describing the background evolution in the presence of perturbations. Generally, a solution of Eq. (3.8), u (z), may be presented in the form, u (z)"u (z) exp[ih(z)], " " 0 where the function u (z) and h(z) characterize the change of the background amplitude and phase, 0

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respectively. To describe now the evolution of a dark soliton on this varying background we should remove the background by the transformation [cf. Eq. (3.2)] u(z, x)"u (z) e*h(z)v(x, z) , (3.9) 0 and to find an effective nonlinear equation for the function v(x, z). In many cases (e.g., see below) such an equation can be transformed into a perturbed NLS Eq. (3.3) after a change of the variables, so that this will alow to apply immediately the result given by Eq. (3.7). 3.1.2. Method based on the inverse scattering technique To derive the adiabatic equation for the soliton parameters, we can employ the perturbation theory based on the inverse scattering transform, similar to that developed for bright solitary waves and also for some other nonlinear models [e.g., Kivshar and Malomed (1989)]. The first effort to derive the adiabatic equations for the parameters of a dark soliton by means of the inverse scattering transform was attempted by Konotop and Vekslerchik (1994). However, these authors based their theory of the assumption that the phase of a dark soliton is fixed by the boundary condition and it does not change. This assumption led to an artificial singularity and required to introduce one more parameter for describing the soliton evolution. As is clear from the results presented above (see also Section 4.1), the local phase of the soliton is not a conserved quantity and, therefore, the assumption made by Konotop and Vekslerchik (1994) is, generally speaking, wrong. That is why the equations derived by those authors have narrow applicability limits and can be used for very special classes of perturbations only. A correct approach should take into account the evolution of the background which supports a dark soliton, similar to the invariant-based method discussed above. Here we give a sketch of another way to derive the equations for the soliton parameters. We consider the perturbed NLS equation in the following form: i

­u 1 ­2u # #(u2!DuD2)u"eP(u) , 0 ­z 2 ­x2

(3.10)

where we keep explicitly the background intensity u2 because, in general, it may vary under the 0 action of perturbations. We assume the validity of the adiabatic approximation when the structure of the dark soliton solution remains unchanged but its parameters become dependent on the evolution variable z. This means that the main relation between the soliton parameters, j2#w2"u2, still remains valid for z, even when the background varies. Differentiating this n n 0 relation, we come to the equation w dw u du dj n"! n n# 0 0 , (3.11) j dt j dt dt n n which defines the evolution of the soliton velocity. However, this equation should be completed by two others. First, the evolution of the spectral parameter can be determined by the technique of the inverse scattering similar to that developed by Konotop and Vekslerchik (1994),

P

`= dw ie n" MP(u)W2(x, j; z)!P*(u)W2(x, j; z)N dx , 2 1 dt 2a@(j )b n n ~=

(3.12)

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where W (x, j; z) are the so-called Jost functions introduced in Section 2.5.1. As for the back1,2 ground, its evolution cannot be self-consistently determined by the soliton parameters and, to derive the corresponding equations, we should analyze the perturbed NLS equation in the asymptotic limit DxDPR. Taking this limit in the NLS Eq. (3.10), we obtain two equations for the background intensity u and phase h, 0 du 0"R sin h#R cos h , (3.13) 1 2 dz 1 dh " (R cos h!R sin h) , (3.14) 2 dz u 1 0 where R ,ReMeP(u e*h)N and R ,InMeP(u e*h)N. Eqs. (3.13) and (3.14) are equivalent to Eq. (3.8) 1 0 2 0 obtained in Section 3.1.1 and, together with Eqs. (3.11) and (3.12) present a full system of equations for the background and soliton parameters. A development of a complete perturbation theory for dark solitons based on the inverse scattering transform, which will take into account not only the evolution of the soliton parameters but also radiation generated under the action of an external perturbation [see discussions of the importance of radiative effects in Burtsev and Camassa (1997)] still remains an open problem. 3.2. Physical applications In this section we apply the adiabatic equations derived above to analyze several particular cases which correspond to physically important perturbations. From these examples it follows how to take properly into account the effect of the varying background on the dynamics of dark solitons. 3.2.1. Effect of gain and loss As is well-known, in the problem of the propagation of spatial solitons nonlinearity is usually associated with two-photon absorption which, in fact, appears as a by-product of enhanced nonlinearity [see, e.g., Silberberg (1990a) and references therein]. In the presence of either twophoton absorption (TPA) or nonlinear gain, the stationary self-localized states of a light wave are no longer possible but in the case of a combined effect, when two-photon absorption is compensated by a gain, the stationary solution in the form of a fundamental dark soliton can exist. To describe the effect of TPA on dark solitons, we consider the NLS equation modified as follows [see, e.g., Silberberg (1990a) and Yang et al. (1994)], i

­u 1 ­2u # !DuD2u"!iKDuD2u , ­z 2 ­x2

(3.15)

where K is the normalized TPA coefficient defined as K"b/2k n , where k is the free-space 0 2 0 wave vector, b and n are the intensity-dependent absorption and refractive index coefficients, 2 respectively. As is well known, in the absence of TPA (i.e., at K"0), Eq. (3.15) describes the case of a defocusing Kerr nonlinearity and the background wave u"u exp(!iu2z) is modulationally 0 0 stable. Nonlinear absorption, even small, leads to an attenuation of the cw background so that its

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amplitude and phase become slowly dependent on Ku2(0)z according to 0 u (0) 0 , (3.16) u (z)" 0 J1#2Ku2(0)z 0 1 z ln[1#2Ku2(0)z] . (3.17) h(z)" u2(z@) dz@" 0 0 2K 0 To separate the evolution of the background and dark soliton, we apply the following transformation:

P

u(z, x)"u (z) e*h(z)v(z, x) , (3.18) 0 where u (z) and h(z) change according to Eqs. (3.16) and (3.17). Then, the function v satisfies the 0 equation i

­v 1 ­2v # !(DvD2!1)v"!iK(DvD2!1)v , ­f 2 ­m2

(3.19)

where f and m are new coordinates which are connected with z and x by the following differential relations, df"u2(z) dz and dm"u (z) dx. After such a transformation, the resulting equation has 0 0 a vanishing perturbation and it can be treated by the perturbation theory for solitons. The equation for the soliton phase angle / in the primary variables takes the form d/ 1 " Ku2(z) sin(2/) , dz 3 0

(3.20)

where the background amplitude u (z) evolves according to Eq. (3.16). Eq. (3.20) may be easily 0 integrated to give the result /(z)"tan~1Mtan /(0)[1#2Ku2(0)z]1@3N . (3.21) 0 One of the main characteristics of the dark-soliton switching devices is the so-called soliton steering angle (Luther-Davies and Yang, 1992b). It is easy to see that the total shift of the dark soliton along the x-axis is given by the relation :z dz@ u (z@) sin /(z@), so that the steering angle X may be defined 0 through the local transverse velocity, ¼(z)"tan s"u (z) sin /(z) . (3.22) 0 The important conclusion based on Eq. (3.22) is the following: When the dark soliton propagates in the presence of TPA on a decaying background u (z), the function sin /(z) grows slowly keeping, at 0 least for small /(0), the product of Eq. (3.22) almost constant (Yang et al., 1994). This simply means that the steering angles for switching devices based on dark soliton propagation are almost preserved in a Kerr nonlinear medium in the presence of TPA. From the physical point of view, this important property simply follows from the nature of nonlinear absorption: the background intensity decays faster than the central minimum in the soliton forcing the soliton. Fig. 4a shows the evolution of the background u (z), the soliton phase angle sin /(z), and the 0 transverse soliton velocity ¼(z)"u (z) sin /(z). The analytical results (solid curves) based on 0 Eqs. (3.16) and (3.21) are in perfect agreement with the results of the numerical simulations (opened diamond marks) and, as may be seen, the steering angle is almost preserved provided /(0) is small.

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Fig. 4. (a) Evolution of the background amplitude, u (z), the function sin /(z), and the transverse velocity of the dark 0 soliton, ¼(z) in a Kerr medium with TPA at K"0.05. The solid curves are from Eqs. (3.20) and (3.21), the diamond marks are obtained from a numerical simulation of Eq. (3.15) for /(0)"0.1p. (b) Contour plots corresponding to the evolution presented in (a) (Kivshar and Yang, 1994a).

Small deviations of the numerical data from the adiabatic relationship are caused by transition radiation which slightly changes the intensity of the background (see Fig. 4b). It is important to compare the result of Eq. (3.22) with the corresponding result of linear absorption described by the contribution eP(u)"!icu on the right-hand side of Eq. (3.15) instead of the term !iKDuD2u. As follows from the corresponding perturbed NLS equation, in this case the background wave decays exponentially u (z)"u (0) e~cz , 0 0

P

h(z)"

z

0

u2(0) u2(z@) dz@" 0 (1!e~2cz) . 0 2c

(3.23)

As in the case of the TPA-induced dynamics, first of all we remove the background evolution by the transformation, Eq. (3.18), where this time the functions u (z) and h(z) are defined by Eq. (3.23). The 0 important result of such a transformation is that the effective Eq. (3.19) for the function v(f, m) is the NLS equation without any perturbation. This immediately implies that the transformation, Eq. (3.18), does allow to exclude the effect of the linear absorption considering the pulse evolution in the new reference frame, so that the soliton phase angle does not change, d//dz"0. The similar conclusion follows from the analysis presented by Lisak et al. (1991); see also Giannini and Joseph (1990), who however used a different method which does not allow a generalization to the case of nonlinear loss. 3.2.2. Stabilization of dark solitons Linear or/and nonlinear absorption affects the background but it can be compensated by a gain. In the presence of both linear gain and TPA, the background can be stabilized but the dark soliton is still unstable (Kivshar and Yang, 1994a). This result of the perturbation theory is in agreement with the existence of the exact solution of the complex Ginzburg—Landau equation which is known to be unstable (Sakaguchi, 1991; Y. Chen, 1992b). However, as was recently pointed out by Ikeda et al. (1995) [see also Ikeda et al. (1997)], a dark soliton can be stabilized by a nonlinear gain.

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Indeed, as we demonstrated above, the effect of a linear gain is trivial, so that it does not allow to compensate for TPA. If, however, we assume the gain to be strong enough to include some higher-order terms, i.e., eP(u)"du#c DuD2u#c DuD4u , 1 2 the effective equation for the dark-soliton angle becomes (Ikeda et al., 1995, 1997)

G

H

2c c d/ " 1 u2(z)! 2 u4(z) [2 cos2 /!5] sin(2/) , 0 15 0 3 dz

(3.24)

where u2(z) satisfies the equation for the background field, 0 du 0"!(du #c u2#c u4) u , 0 1 0 2 0 0 dz which shows that u is stably stationary at u "1 provided d#c #c "0 and c #2c '0. This 0 0 1 2 1 2 condition can be made consistent with the stability of the dark soliton defined by Eq. (3.24), allowing to control dark solitons by varying nonlinear gain (Ikeda et al., 1995, 1997). This result was verified by Ikeda et al. (1995, 1997) by means of direct numerical simulations of the perturbed NLS equation. Maruta and Kodama (1995) suggested a completely different idea of the dark-soliton stabilization by means of a synchronized phase modulation. In their approach, the effective perturbation of the NLS equation appears in the form eP(u)"k cos(ux)u, where k is the phase-modulation coefficient, and u is the modulation frequency synchronized with the initial pulse separation of a sequence of dark solitons. As was shown by Maruta and Kodama (1995), this perturbation leads to a nontrivial variation of the soliton phase which can display a stable dynamics near the fixed point /"0 (corresponding to a black soliton with the zero minimum amplitude at its center) due to the effect of phase-locking between the soliton and periodically varying perturbation. An interesting method to compensate for the amplitude variation of a dark soliton in a lossy medium was suggested by Kim et al. (1996). They demonstrated that dark solitons are stabilized when phase-sensitive amplification and spectral filtering are used together. Indeed, in a periodically amplified system, the spectral filtering are used together. Indeed, in a periodically amplified system, the spectral filtering was shown to inhibit the sideband instabilities typical for both cw waves and solitons, including dark solitons as earlier demonstrated by Allen et al. (1994, 1995). From the other side, a phase-sensitive amplification inhibits to the destabilization of the constant-intensity background wave caused by the filtering (Kim et al., 1996). 3.2.3. Raman self-frequency shift Perturbation theory for dark solitons can be also effectively applied for analyzing the dark solitons in optical fibers. One of the important perturbations acting on dark solitons in fibers appears when the pulse duration in fibers reaches the subpicosecond regime. Then, it becomes necessary to include higher-order dispersion effects that are presented by higher-order derivatives in the effective NLS equation for the wave envelope. Stimulated Raman scattering (SRS) is known to be one of the dominant effects for very short optical pulses. For bright solitons this effect causes the so-called self-frequency shift [Mitschke and Mollenauer (1986), Gordon (1986), see also

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Hasegawa and Kodama (1995)], whereas for dark solitons, a self-frequency shift at the initial stage of the pulse evolution (Weiner et al., 1989) leads finally to a decay of dark solitons (Kivshar, 1990a,b; Kivshar and Afanasjev, 1991b; Uzunov and Gerdjikov, 1993; Kivshar and Yang, 1994a). From the physical point of view, SRS originates from the noninstantaneous, delayed response of the fiber nonlinearity. This effect may be described in the time domain by a response function that has a form of a decaying sinusoidal oscillations (Stolen et al., 1989). The Raman contribution to the nonlinear refractive index may be taken into account in a rather general form,

C

P

n DuD2Pn (1!a)DuD2#a 2 2

D

t

dt@ Du(t@)D2 f (t!t@) , (3.25) ~= where a is the fraction of the total (low frequency) nonlinearity with a delayed response, and f (t) is the Raman response function [see, e.g., Stolen et al. (1989)]. The Raman response function of fused silica is extremely short, so that Eq. (3.25) may be considered in local approximation expanding the function Du(t!s)D2 in the integrand of Eq. (3.25) (here s"t!t@) in the Taylor series around t to obtain a local perturbation to the NLS equation, eP(u)"eu

­ (DuD2) , ­t

(3.26)

e being proportional to the Raman gain parameter a. In such a form, the effect of SRS may be analysed as a perturbation to the standard NLS dynamics and for dark solitons it was investigated numerically (Weiner et al., 1989; Kivshar and Afanasjev, 1991b; Zhao and Bourkoff, 1992) and analytically (Kivshar, 1990a; Kivshar and Afanasjev, 1991b). As a matter of fact, the general formula describing the dark-soliton propagation in the presence of SRS was obtained by Uzunov and Gerdjikov (1993), whereas its small-amplitude limit was derived even earlier by Kivshar (1990a) within the asymptotic approach based on the perturbed Korteveg—de Vries equation. In general, the effect is described by the perturbed NLS equation

P

­u 1 ­2u t Du(t@)D2G(t!t@) dt@ . ! #DuD2u"!eJ u ­z 2 ­t2 ~= To write Eq. (3.27), we assume that the first term of the expansion

P

P

(3.27)

P

­ `= sf (s) ds , (3.28) (DuD2) ­t ~= ~= is already included into the main Kerr-type nonlinearity DuD2u so that the “renormalized” response function G(s) has the property :`= G(s) ds"0. If the parameter eJ max(G) is small, we may treat the ~=a perturbation. Applying the perturbation theory to the dark right-hand side of Eq. (3.27) as soliton, Eq. (3.4)) [with the change zP!z because of the difference in the signs for Eqs. (2.25) and (3.27)], and changing the order of integration, one may come to the following equation for the soliton phase angle: u(t) ds f (s)Du(t!s)D2PDuD2u

P

A

`=

f (s) ds!u

B

d/ `= s ds G "eJ u cos / F(s) , 0 dz u cos / ~= 0

(3.29)

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where 2[s(3!tanh2 s)!3 tanh s] F(s)" . sinh2 s tanh2 s

(3.30)

As follows from Eq. (3.30), the local approximation may be used only in the case when the Raman response function is short in comparison with the soliton width (u cos /)~1. In this case we may 0 simply expand the function F(s) into the Fourier series as F(s)KF(0)#sF@(0) and obtain the resulting equation in the form of a perturbed NLS equation as an expansion in the inverse width of the soliton (u cos /), this value is assumed to be small in comparison with the inverse characteristic 0 decay ¹ of the response function. The result is (Uzunov and Gerdjikov, 1993; Kivshar and Yang, 1994a) d/ 4 " eu3 cos3 / , dz 15 0

(3.31)

where e,eJ

P

`=

yG(y) dy .

(3.32)

~= The small-amplitude limit of the result, Eq. (3.32), was obtained earlier by Kivshar (1990a) with the help of the KdV soliton approximation. Fig. 5 shows the evolution of a dark soliton with an initial negative (a) or positive (b) velocity in the presence of the SRS contribution at e"0.1. Comparison of the numerical simulation results with Eq. (3.31) is presented in Fig. 5c as the soliton phase angle given by the function sin / vs the propagation distance at two different initial values, /(0)"!0.2p and /(0)"#0.2p. As may be seen from those figures, Eq. (3.31) describes rather well the soliton dynamics which, as a matter of

Fig. 5. (a) Change of the phase angle of a dark soliton under the action of the Raman self-scattering effect for e"0.1, /"!0.2p and /"0.2p. (b), (c) Contour plots for the same initial data, /"!0.2p and /"0.2p, respectively (Kivshar and Yang, 1994a).

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fact, corresponds to transformation of a dark soliton with arbitrary parameters into a smallamplitude soliton due to the continuous SRS-induced frequency and position shift. 3.3. Dark-soliton jitter For long-distance propagation of temporal bright solitons, losses should be compensated for by a periodic amplification. As was firstly predicted by Gordon and Haus (1986), an undesirable effect of periodic amplification will cause, through amplified spontaneous emission, a random frequency shift of the solitons that in turn results in a temporal jitter at the output of the fiber links. The jitter parameters observed experimentally (Mollenauer et al., 1990) showed excellent agreement with the model of Gordon and Haus (1986). For bright solitons, the main result of the Gordon—Haus effect can be demonstrated in a few lines. We start from the NLS Eq. (2.1) with p"!1 and present its bright soliton solution in the form [cf. Eq. (2.16)] a u (t, z)" exp (iXt!iqz) , 4 cosh m

(3.33)

where m"a(t!Xz) and q"(X2!a2)/2. As can be seen from Eq. (3.33), the bright soliton has two independent parameters, the soliton amplitude a and frequency shift X which are defined, in particular, by two invariants, the power P and momentum M , 505 505 i `= ­u ­u* `= DuD2 dt , M " P " u* !u , 505 2 505 ­t ­t ~= ~= which for the soliton solution, Eq. (3.33), take the values P "2a and P "!2aX. 4 4 Following the main ideology of Gordon and Haus (1986), we consider now the effect of adding small perturbation du to the soliton field u . Such a nonsolitonic field produces a change in the 4 soliton parameter, radiation corrections being of the second order in du. The frequency fluctuations dX can be then found from the energy and momentum conservation (Gordon and Haus, 1986). Let us now consider that the nonsolitonic correction is a noise arising from a broadband amplifier. If the amplifier bandwidth is greater than the spectral width of the solitons, then we may consider that its effective noise contribution is white. To describe such a noise, we take the addendum du as a stochastic complex filed with the statistical properties Sdu(t)T"Sdu*(t)T"0 and the only nonzero pair correlator, Sdu(t)du*(t@)T"Dd(t!t@). Parameter D is the mean value of the random fluctuations, and it is proportional to the mean number of photons per mode at the amplifier output (Gordon and Haus, 1986). Averaging directly the calculated value dX with the help of this correlator, we find that

P

P A

P K

B

K

D `= ­u 2 SdX2T" i #Xu dt . (3.34) 2a2 ­t ~= Calculating the right-hand side of Eq. (3.34) for the soliton solution, we recover the famous result of Gordon and Haus (1986), SdX2T"1Da . 3

(3.35)

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We now consider the case of dark solitons that occur in the normal fiber dispersion regime described by Eq. (2.1) at p"#1. We follow the original paper by Kivshar et al. (1994a). To simplify the problem, we first apply the transformation u"u v(t, z) exp(iu2z), so that the function 0 0 v(t, z) satisfies the NLS equation of the form i

­v 1 ­2v ! #u2(DvD2!1)v"0 , 0 ­z 2 ­t2

(3.36)

which has a dark soliton solution of the form of Eq. (2.18). To calculate the jitter, we use two invariants, the renormalized Hamiltonian H and renor3 malized momentum M defined is Section 2.6. Calculating the values of M and H for the soliton 3 3 3 solution, one can find the results M "2 sin / cos / and H "4u cos3 /. As above, we consider 3 3 3 0 the effect of adding a small perturbation dv to the soliton field v , which for the primary field 4 u means the selection of the fluctuations in the form: u"u (v #dv) exp(iu2z). The frequency 0 4 0 fluctuations dX, where X"u sin /, depend on the fluctuations of the background amplitude du 0 0 and of the internal phase angle d/. From the conservation identities resulting from the Hamiltonian and the renormalized momentum, one can be find that dX"a(/)dH #b(/)dM , i.e., 3 3 ­v* ­v* `= dv a(/) dX"2 Im !b(/) dt , (3.37) ­z ­t ~= where

GP

C

D H

3 sin / u (3 sin2/#cos2/) a(/)"! , b(/)" 0 . 4 cos3/ 4 cos3/ Applying the same reasoning as in the case of the bright soliton, we obtain the following result (Kivshar et al., 1994a)

P K

K

2D `= ­v ­v 2 SdX2T" a(/) !b(/) dt , (3.38) u2 ­z ­t 0 ~= where, as above, we have assumed that the field dv is a stochastic complex noise with the only nonzero correlator D Sdv(t)dv*(t@)T" d(t!t@) . u2 0 Substituting the exact soliton solution into Eq. (3.38), we finally obtain the result

(3.39)

8D D SdX2T" cos3/[a(/)X!b(/)]2" cos / . (3.40) 3u2 6 0 Validity of this result was discussed by Kivshar et al. (1994a). To make a comparison between the cases of bright and dark solitons, we must consider a black soliton of the same amplitude as the bright one. We then set u "a"1 and /"0. For this choice 0 of parameters Eqs. (3.35) and (3.40) may be now compared to yield the simple relation . SdX2T "1SdX2T "3*')5 $!3, 2

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Let us now consider the consequences of this result for soliton-based transmission systems. We assume a fiber link of length ¸ equipped with N amplifiers. Because the noise of amplified spontaneous emission, each amplifier induces a random frequency shift dX that slightly modifies the velocity of the solitons that propagate over the length z "¸/N of the fiber sections. The L frequency parameter X represents, in soliton units, the inverse group velocity of the soliton. It is possible to show (Gordon and Haus, 1986) that the distribution of the times of arrival of the solitons at the output of the link has a variance jitter given by Sdt2T"¸3SdX2T/3z . Introducing L the temporal jitter of the transmission as being the standard deviation of this distribution, Sdt2T1@2, we obtain the following relation: 1 Sdt2T1@2 " Sdt2T1@2 . $!3, J2 "3*')5

(3.41)

This result shows that the temporal jitter of the fundamental dark solitons is J2 lower than that obtained with bright solitons. This result is in perfect agreement with the conclusions of the numerical investigation reported earlier by Hamaide et al. (1991). The calculations presented above have been recently extended by Panoiu et al. (1995) to include the SRS effect on the jitter of dark solitons. The SRS effect was taken into account as a local perturbation, Eq. (3.26), to the NLS equation. It was shown that the simultaneous action of random fluctuations and SRS effect can, in some cases, compensate each other leading to the lowering jitter. This is, in particular, the case of the background fluctuations when there exists a distance where the dark-soliton jitter completely vanishes (Panoiu et al., 1995). 3.4. Effect of third-order dispersion Experimental demonstration of long-distance transmission and data coding with the help of dark solitons [see Section 6.1 below and also Nakazawa and Suzuki (1995a,b)] has led to renewed efforts to explore this type of soliton for optical communications. In order to reduce the operating power, the transmission wavelength should be selected closer to the zero of the group-velocity dispersion where, however, the soliton propagation is influenced by the third-order dispersion. The effect of the third-order dispersion on bright solitons is now well understood (Wai et al., 1986, 1990; Menyuk and Wai, 1992). Under the action of the third-order dispersion, a bright soliton develops a nonvanishing asymptotics in the form of a tail (Menyuk and Wai, 1992). Wai et al. (1990) showed that the tail amplitude A is exponentially small in the third-order dispersion coefficient b, A&exp(!1/b), and can be calculated using asymptotic analysis ‘beyond all orders’. As was pointed out, this result is consistent with Menyuk’s robustness hypothesis (Menyuk, 1993), according to which autonomous, homogeneous, Hamiltonian deformations of integrable equations lead to solitary waves that radiate beyond all orders if they radiate at all. The effect of third-order dispersion on dark solitons was investigated first in the framework on the small-amplitude approximation (Kivshar, 1991a,b; Kivshar and Afanasjev, 1991a) and reviewed by Kivshar (1993b) [see also the recent paper by Frantzeskakis (1996), where the more general case was analyzed). It was shown that, in the framework of the small-amplitude approximation when an effective KdV equation is valid, the third-order dispersion does not affect strongly the existence and properties of gray solitons. However, Kivshar and Afanasjev (1991a) found an

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interesting feature of dark solitons near the zero point of the group velocity dispersion. Namely, dark solitons exist as ‘humps’ (instead of ‘dips’) on the background of the amplitude u in the 0 region, 1(d(4, where d,a3@2/bu , a and b are dispersion coefficients defined below. Such 0 antidark solitons match exactly the conventional dark solitons at the critical points, where they are described by one of two modified versions of the KdV equation (Grimshaw et al., 1997). Since, the dark and antidark solitons can exist in the same region of the soliton parameters, Afanasjev and Kivshar (1991a) suggested that a head-on collision between them can occur. This head-on collision was investigated analytically by Huang and Velarde (1996) who applied an asymptotic expansion technique to show that the soliton can preserve their identities only to the second order. Huang and Velarde (1996) calculated also the phase shift of the solitons after the collision. Effect of the third-order dispersion on dark solitons of large or moderate amplitude was analyzed by Afanasjev et al. (1996) [see also Karpman (1993)] who also carried out numerical simulations to confirm the phenomenon of the dark-soliton resonance decay predicted by the theory. In this section, discussing the effect of the third-order dispersion on dark solitons, we follow the work by Afanasjev et al. (1996). As is well known [e.g., Hasegawa (1989)] the pulse propagation in optical fibers near the zero of the group-velocity dispersion is described by the perturbed NLS equation which we write here in the form, i

­u ­2u ­3u !a #2DuD2u"ib , ­z ­t2 ­t3

(3.42)

where a and b are the corresponding dispersion coefficients. When bO0, the information about the existence of localized solutions can be obtained by analyzing the asymptotics. We take u"(u #m) exp(2iu2z) and linearize Eq. (3.42) for small m. Then, for a stationary wave moving with 0 0 the velocity v, we seek a solution in the form v&(m #im ) exp(ikf), where f"t!vz, and obtain the 3 * equation for i,k2(v), (v!bi)2"a(ai#4u2) . 0

(3.43)

Quadratic Eq. (3.43) has two roots. Following Afanasjev et al. (1996), we consider v2(c2, where DcD"2Jau is the effective speed of sound. Then, one root, say i , is always negative, and it 0 ~ corresponds to exponentially decaying soliton tails. However, for any bO0 Eq. (3.43) has an additional, positive root, say i , which describes a nonvanishing oscillating tail of the dark soliton. ` Existence of nonvanishing tails can be usefully viewed as a resonant generation of linear waves, which takes place provided the speed of the solitary wave v coincides with the phase velocity » of 1) the linear waves. Indeed, the condition » "v leads immediately to Eq. (3.43). From the physical 1) point of view, this result implies that the solitary wave acts as a source generating trailing oscillations which with the leading front propagating with the wave group velocity » . This process ' is demonstrated in Fig. 6a and Fig. 6b for the case of a black soliton and b"0.18. At the initial stage of the soliton evolution some transition radiation is excited. This radiation propagates to the left, moving with the sound speed c, and quickly separates from the dark soliton, as is seen in Fig. 6a. The radiation creates an additional, small-amplitude dark soliton (see Fig. 6b) which is, according to Kivshar and Afanasjev (1991a), is stable in the presence of the third-order dispersion.

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Fig. 6. Formation of a nonvanishing oscillating tail for a black soliton at b"0.18. (a) Grey-scale plot, in which the white lines show the propagation with the sound velocity c. (b) Intensity profile at z"10 (Kivshar and Afanasjev, 1996).

An oscillating tail is formed from the right of the soliton, and its front propagates with the group velocity » which is different from the velocity of the dark soliton v and sound velocity c. As a result ' of the generation of a continuously growing tail, the soliton amplitude decreases and its velocity increases, i.e., the soliton decays. Fig. 7 shows subsequently the evolution of the field intensity at the soliton center, I "u2v2/c2, vs. propagation distance z, which varies from zero to approxim.*/ 0 ately 0.8. Similar dynamics was also reported for initially grey solitons of moderate amplitudes (Afanasjev et al., 1996).

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Fig. 7. Long-term adiabatic decay of a black soliton in the presence of the third-order dispersion. Shown is the evolution of the minimum soliton intensity I (z) at different values of b (Kivshar and Afanasjev, 1996). .*/

Calculation of the oscillation amplitude is a delicate task which requires all orders of the asymptotic expansion (Wai et al., 1990; Kivshar and Malomed, 1991). However, a qualitatively correct result can be obtained considering the linear equation for the soliton perturbations m"u!u . As was shown by Karpman (1993) and Afanasjev et al. (1996), the corresponding 4 solution can be found in a cumbersome form, but its general structure is given by the expression: m"Ah(f)h(!f#» z) sin (Ji f#/) , ' ` where h(x)"1 for x'0 and h(x)"0 for x(0, and f"t!vz. The dependence of the tail amplitude A on b and v, can be presented as the following (Afanasjev et al., 1996)

A

B

pJa Ji , (3.44) A&CB(i )csch ` ` u cos h 0 where C is a constant and B is an algebraic function of i , a positive root of Eq. (3.43). Using the ` first-order expansion for small b, k"Ji +a/b, one can easily show that the tail amplitude ` depends on b exponentially, i.e. in the similar fashion as for bright solitons (Wai et al., 1986, 1990). However, the special feature of dark solitons is the algebraic factor J1!v2/c2 in the exponent which demonstrates that the radiation amplitude becomes exponentially small for any fixed b but in the limit v2Pc2. This explains the validity of the small-amplitude approximation used earlier by Kivshar and Afanasjev (1991a). Indeed, as the amplitude of the dark soliton decreases, for a fixed value of b the amplitude of the oscillating tail decays exponentially fast. This means that the oscillation amplitude becomes beyond all orders of the asymptotic expansion in the soliton amplitude, and in this limit the soliton radiation and decay are negligible. 3.5. Background of finite extent In experiments reporting dark solitons, localized beams (or pulses) have been created on a background beam (or pulse) of a finite extend (Emplit et al., 1987; Kro¨kel et al., 1988;

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Weiner et al., 1988; Swartzlander et al., 1991; Allan et al., 1991; Luther-Davies and Yang, 1992a,b). Therefore, the interpretation of the experimental results as the observation of dark-soliton propagation could be questionable because the background, being only in several times longer than the dark beams or pulses observed, spreads significantly due to dispersion, in the temporal case, or diffraction, in the spatial case. However, Tomlinson et al. (1989) demonstrated, by means of direct numerical simulations, that optical dark solitons superimposed upon backgrounds only 10 times wider than the soliton width can exhibit stable soliton-like propagation for relatively short distances. During the propagation, the background spreads, reduces its intensity, and develops a frequency chrip (in the temporal case) but, nevertheless, dark pulses created on such a finite-width background adiabatically maintain their soliton characteristics. As has been pointed out by Gredeskul et al. (1990), for the finite-width background the corresponding eigenvalue problem of the associated IST (see Section 2.5.1 above) has no eigenvalues of the discrete spectrum and the dark pulses created on a vanishing background correspond instead to the so-called quasi-stationary states of the eigenvalue problem, Eq. (2.24). This simply means that the dark beams created on a finite-width background never become proper solitons, and they should disappear as soon as the propagation distances increase. However, these results are correct from the mathematical point of view but they do not give a clear physical explanation to the following fact: ¼hy dark pulses, even being not proper solitons, do not change significant when the background itself spreads, reduces its intensity, and develops a frequency chirp?. The answer to this question follows from the analysis done by Kivshar and Yang (1994b) which we briefly discuss below. It is known that for vanishing boundary conditions the NLS equation describes the spreading pulses (or beams) which undergo enhanced broadening and chirping. Let us take such a quasilinear background in a rather general form, u(x, z)"u (x, z) e*h(x,z) , 0

(3.45)

where we just introduce the background amplitude, u (x, z), and phase, h(x, z). We consider now 0 the evolution of a dark pulse superimposed upon such a spreading background, looking for a solution of the NLS equation in the form, u(x, z)"u (x, z) e*h(x,z)v(x, z) , 0

(3.46)

where v(x, z) falls off fast as x increases. Substituting Eq. (3.46) into the NLS equation and assuming that the function u (x, z) exp[ih(x, z)] is a solution of the NLS equation, we obtain the following 0 equation for the function v(x, z), i

­v 1 ­2v ­ ­v # !u2(DvD2!1)v"!2 [ln(u e*h)] , 0 0 ­z 2 ­x2 ­x ­x

(3.47)

where u ,u (x, z) is the background amplitude introduced above. Introducing slow variables 0 0 f and m according to the relations, df+u2(x, z) dz, dm+u (x, z) dx, and taking the pulse v(f, m) as 0 0 a dark soliton, Eq. (3.4), the pulse evolution due to such a perturbation can be effectively analysed

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with the help of Eq. (3.7), giving the evolution equation for the soliton phase angle,

P

A

B

d/ 1 `= dH 1 ­u 0 , " cos2/ df 2 cosh2H u ­H 0 ~=

(3.48)

where u "u (Z, f) is considered in the reference frame moving with the soliton [see Eq. (3.6)]. 0 0 Eq. (3.48) is valid for an arbitrary background, and it clearly shows that the evolution of a dark soliton does not depend on the background phase, so that the enhanced frequency chirp developed by the background does not affect the dark pulse and it maintains adiabatically its properties. Additionally, as follows from Eq. (3.48), the change of the soliton angle depends not on the extension and intensity of the background but rather on its local slope. In particular, these results indicate that properly selecting the input background shape we may keep the steering angle of the dark soliton, ¼"u sin / almost unchanged because the effect of the 0 background decay can be completely compensated by the internal dynamics of the dark soliton. This effect was indeed demonstrated numerically (Kivshar and Yang, 1994b) comparing the soliton evolution for the constant and varying backgrounds.

4. Instability-induced soliton dynamics 4.1. Stability of dark solitons Optical dark solitons are of both fundamental and technological importance if they are stable under propagation. For temporal solitons in optical fibers, nonlinear effects are weak and the model based on the cubic NLS equation is valid is most of the cases. Therefore, being described by the integrable or nearly integrable models, solitons are always stable, or their dynamics can be affected by (generally small) external perturbations which can be treated in the framework of perturbation theory. For spatial solitons in waveguides or bulk media, higher powers are usually required, so that real optical materials demonstrate essentially non-Kerr change of the nonlinear refractive index with the increase of the light intensity. Generally speaking, the nonlinear refractive index always deviates from Kerr for larger input powers, e.g., it saturates. Therefore, models with a more general intensity-dependent refractive index are employed to analyze spatial dark solitons and, as we discuss here, dark solitary waves in such non-Kerr materials can become unstable. Stability of bright solitons of the generalized NLS Eq. (4.2) has been extensively investigated for many years, and the criterion for the soliton stability has been derived by different methods (see, e.g., Vakhitov and Kolokolov (1973), Weinstein (1985), Kuznetsov et al. (1986), Kusmartsev (1989) and Mitchell and Snyder (1993)]. Stability of bright solitons in the generalized NLS equation of any dimension is given by the simple criterion (Vakhitov and Kolokolov, 1973) dP d " db db

AP

B

DE(r, z)D2 dr '0 , V

where P is the total soliton power and b is the soliton propagation constant. Stability of self-guided in nonlinear waveguide structures can be also given in some cases by the same criterion (e.g., Jones

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and Moloney (1986) and Mitchell and Snyder (1993)] but, generally, it is more complicated and depends on a particular structure and the type of nonlinearity. Different scenarios of the instability-induced dynamics of bright solitons have been found and analyzed both analytically and numerically [see, for a review, e.g., Kuznetsov et al. (1986)]. Recently, Pelinovsky et al. (1996a) developed an asymptotic analytical approach to describe the dynamics of unstable solitons (e.g. diffraction-induced decay, collapse, or switching to a novel stable state). All these results allow to say, as we believe, that the picture of the instabilities of bright solitons is complete. In contrast to bright solitons, the stability criterion for dark solitons of the generalized NLS equation has not been understood until recently and, even more, this issue created a lot of misunderstanding in the past. In particular, we notice some efforts to employ, by analogy with bright solitons, the soliton complementary power [see, e.g., Enns and Mulder (1989)] and to use this value for analyzing the soliton bistability (Mulder and Enns, 1989; Bass et al., 1992), a statement that a black dark soliton (a dark soliton with zero intensity at its center) is always stable (Mitchell and Snyder, 1993), etc. However, as was already noticed by Kro´likowski et al. (1993), the complimentary power does not define stability of dark solitons. From a historical point of view, the first efforts to analyze the stability of dark solitons were stimulated by numerical simulations done by Barashenkov and Kholmurodov (1986) who observed instability of the so-called ‘bubbles’, localized waves of rarefaction of the Bose gas condensate. These nontopological solitary waves belong, in the (1#1)-dimensional case, to the family of dark solitons of the NLS equation with the cubic—quintic nonlinearity, Eq. (2.7), and they survive in higher dimensions [see, e.g., Barashenkov and Makhankov (1988)]. Although quiescent bubbles were found to be always unstable regardless of dimension [Barashenkov et al. (1989), see also a proof given by De Bouard (1995)], in numerical simulations it was revealed that the moving bubbles can be stabilized at nonzero velocities (Barashenkov and Kholmurodov, 1986). Later Bogdan et al. (1989) [see also Barashenkov and Panova (1993)] explained this phenomenon through the multivalued dependence of the bubble energy vs. the renormalized bubble momentum. However, it was believed for a long time that kink-type dark solitons (e.g., black solitons) are always stable [see, e.g., Mitchell and Snyder (1993)]. Instability of black solitons was observed for the first time by Kivshar and Kro´likowski (1995b) in numerical simulations of the NLS equation with the saturable nonlinearity, Eq. (2.9), at p"2. These authors used the variational principle to link the instability to the existence of multivalued solution in terms of the system invariants, and also derived the instability threshold condition by using the asymptotic expansions. Even it was known for some time that the stability criterion for dark solitons should be defined through the renormalized soliton momentum, dM d 3" dv dv

GP A

BA

­u* ­u i `= u !u* ­x ­x 2 ~=

B H

u2 1! 0 dx '0 , DuD2

and this was shown to be consistent with the results of numerical simulations (Barashenkov and Kholmurodov, 1986; Barashenkov et al., 1989; Kivshar and Afanasjev, 1996) and the variational principle for bubble-type (Bogdan et al., 1989) and kink-type (Kivshar and Kro´likowski, 1995b) dark solitons, the rigorous proof of this stability criterion was presented only recently by Barashenkov (1996), with the help of the Lyapunov function, and Pelinovsky et al. (1996b), with the help of

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the asymptotic expansions near the instability threshold. The first approach does not allow to describe the instability itself but it is more general, whereas the second method is valid in the vicinity of the instability threshold being however sufficient to determine the instability domain. We would like to note that the stability of dark solitons can be also formulated by using another definition of the renormalized momentum M , e.g., that introduced for any dimension by Jones and 3 Roberts (1982), i M" 3 2

P

[(u!1)+u*!(u*!1)+u] dr . V This invariant was used by Kuznetsov and Rasmussen (1995) to analyze the transverse instability of a dark-soliton stripe and vortex line in a cubic NLS equation. In this chapter we follow the approach based on the asymptotic multiscale analysis developed by Pelinovsky et al., 1996b) which allows to describe both linear and nonlinear regimes of the instability-induced dynamics of dark solitons. We also describe the results of a detailed numerical simulations of different scenarios of the evolution of unstable dark solitons. 4.2. Asymptotic approach 4.2.1. Stationary solutions We consider the generalized NLS Eq. (2.22) and look for stationary solution on a cw background in the form u(x, z)"t(x, z) e*g(q)z ,

(4.1)

where q"u2 is the background intensity and the function t satisfies the conditions t(x, z)Pq for 0 xP$R. Substituting Eq. (4.1) into Eq. (2.22), we obtain the equation i

­t 1 ­2t # #[g(DtD2)!g(q)]t"0 . ­z 2 ­x2

(4.2)

Now, dark soliton t is defined as a localized travelling-wave solution of Eq. (4.2), 4 t (m)"U(m) e*h(m) , (4.3) 4 where m"x!vz and two real functions, U,U(m; v, q) and h,h(m; v, q), depend on two parameters, the soliton velocity v and the intensity q of the cw background. These functions satisfy the following ordinary differential equations,

A

B

dh q "v 1! , dm U2

A

(4.4)

B

q2 1 d2U v2 # U! #[g(U2)!g(q)]U"0 . 2 U3 2 dm2

(4.5)

Eqs. (4.3), (4.4) and (4.5) describe a dark soliton defined by two parameters, the soliton velocity v and the background intensity q. For the soliton solution of the form of Eq. (4.3) we can define the renormalized invariants, the renormalized or complementary power, Eq. (2.49), renormalized

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momentum, Eq. (2.50), and renormalized Hamiltonian, Eq. (2.51). Defined for the soliton solution, Eq. (4.3) with the index ‘s’, these invariants take the form

P

1 `= (U2!q) dm , (4.6) P (v, q)" 4 2 ~= `= (U2!q)2 M (v, q)"!v dm , (4.7) 4 U2 ~= U2 `= 1 dU 2 v2 (U2!q)2 # [g(q)!g(I)] dI dm . (4.8) H (v, q)" # 4 2 dm 2 U2 q ~= Besides, we can find the analytical expression for the total phase shift S of the background wave 4 across the dark soliton,

P P GA B

S (v, q)"v 4

P A `=

~=

P

H

B

q 1! dm . U2

(4.9)

4.2.2. Equation for soliton velocity As has been shown by Pelinovsky et al. (1996b), the analysis of stability of dark soliton solutions, Eqs. (4.3), (4.4) and (4.5) can be carried out in the framework of the perturbation theory if a change of the soliton parameters is slow in time. Soliton instability is weak near the instability threshold when the velocity v of the unstable dark soliton is close to a critical value v defined by the # instability threshold equation ­M /­vD #"0. Moreover, we suppose that the amplitude of 4 v/v instability-induced perturbations remains small for an extended time interval and the localized wave is close to a dark soliton t with slowly (adiabatically) varying parameters. Therefore, we can 4 introduce a small parameter e which characterizes a small perturbation of the unstable dark soliton, and look for solutions t to Eq. (4.2) in the form of the following asymptotic (multi-scale) expansion, t"Mt (m; v, q)#et (m; v, q; X, ¹)#e2t (m; v, q; X, ¹)#O(e3)N e*R(X,T) , 4 1 2 where

P

1 T m"x! X (¹) , X (¹)" v(¹@) d¹@ , X"ex , ¹"ez , 4 e 4 0 and e;1. Here the functions v(¹) and R(x, ¹) describe the slowly varying soliton velocity and local phase, respectively, X and ¹ stand for ‘slow’ variables, and X (¹) is the coordinate of the soliton 4 center (minimum amplitude) with respect to the X-axis. Generally speaking, solutions to the linear inhomogeneous equations for t ,2 may diverge 1 exponentially along the inner interval as mPR. Such divergences usually break down the asymptotic expansion procedure. However, in the vicinity of the instability threshold, where ­M /­v&O(e), the first-order correction t can be expressed in an implicit form (Pelinovsky et al., 4 1 1996b) and then a bounded solution of the second-order perturbation correction t has to be 2 found. In this way, the function t does not display exponentially diverging terms if the velocity of 2

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the perturbed dark soliton satisfies a certain differential equation, solvability condition [see Eq. (4.10) below]. Using the method of matched asymptotics, Pelinovsky et al. (1996b) have shown that near the instability threshold the evolution of the soliton velocity v as derived from the conservation of momentum, is described by the following equation:

C

D

A B

d 1 dv 2 dv , M (v, q)#k (v, q) "K (v, q) 4 4 d¹ e 4 d¹ d¹

(4.10)

where M is the soliton renormalized momentum, Eq. (4.7), and the coefficients are defined as 4 q ­S 2 2c ­P 2 4 # 4 , (4.11) k (v, q)" 4 2c ­v q ­v

A B

A B

and

C A B

A BD

vq ­S 2 2cv ­P 2 ­P ­S 1 4 #2c 4 4# 4 , K (v, q)" 4 2c ­v ­v ­v (c2!v2) q ­v

(4.12)

where P and S are defined by Eqs. (4.6) and (4.9), respectively, and c is the limiting velocity of 4 4 linear waves, c2"qg@(q). Similar calculations based on the conservation of the energy (Hamiltonian) show that the first-order variations of the renormalized momentum dM and energy dH of a perturbed dark soliton are related by the equation (see Section 2.6), vdM#dH"0 .

(4.13)

However, neither momentum nor energy of the perturbed dark soliton is conserving and this reveals essentially dissipative character of the instability-induced dynamics of unstable dark solitons. Such a dissipative dynamics of the dark soliton instability is explained by generation of the radiation fields propagating away from the perturbed dark soliton to the right and to the left. The profile of the radiation fields can be obtained in the following explicit form (Pelinovsky et al., 1996b), dv ºB"f (v, q) at X"X (¹) , B 4 d¹

(4.14)

where

A

B

­P q ­S 1 4 . c 4$ f (v, q)"! B ­v 2 ­v c(cGv)

(4.15)

Asymptotic Eq. (4.10) cannot be generally integrated. Nevertheless, it describes a rather simple dynamics of the dark soliton instability and related evolution of the radiation fields of Eq. (4.14). The specific features of this dynamics are analysed below for three particular cases. Here we describe more general features of the instability-induced dark-soliton dynamics. First, we consider a linear approximation of the asymptotic Eq. (4.10) substituting v"v #v ejT, where v is the (initial) velocity of the unperturbed dark soliton and v is its small 0 1 0 1 deviation caused by an initial) perturbation. Neglecting nonlinear terms in Eq. (4.10) we find the

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instability growth rate j,

A B

1 ­M 4 j"! . ek (v , q) ­v 4 0 v/v0

(4.16)

The result of Eq. (4.16) gives the criterion of the dark soliton instability, and it proves that dark solitons become unstable provided ­M /­vD 0(0 [we notice that the coefficient k (v , q) is always 4 v/v 4 0 positive, see the definition in Eq. (4.11)]. Therefore, the bifurcation analysis implies that all dark solitons with negative slope of the renormalized momentum M (v) are unstable. 4 Next, we analyse the general conditions when the instability of dark solitons can occur. Let us first consider the small-amplitude limit, when DvDPc. In this limit dark solitons can be described by an effective KdV equation (see Section 2.5.2), and the derivative (­M /­v) can be found explicitly 4 (Pelinovsky et al., 1996b), and it is always positive. Therefore, small-amplitude dark solitons of the generalized NLS equation are always stable and the instability can occur only for dark solitons of larger amplitude (or smaller velocity v). However, in the limit of small velocities, vP0, the slope ­M /­v can become negative provided 4

A BK ­S 4 ­v

K

2 ( P (0 . q 4 v/0 v/0

For many models, the total phase shift S is given by a monotonic function rising from the value 4 !p, at vP0 (‘black’ soliton) to zero, at vPc (small-amplitude or ‘grey’ solitons), see Fig. 8a and Fig. 8b. For example, this situation is typical for the Kerr and power-law nonlinearity, g(I)&Ip, as well as for the generalized Kerr model with the nonlinearity g(I)"I# bI2. For these models the slope ­S /­v is always positive and instabilities of dark solitons cannot occur. However, for some 4 more complicated but still physically important models the instability of dark solitons does take place and the general form of the renormalized momentum for such cases is shown in Fig. 8a and Fig. 8b. We discuss these instabilities below. Nonlinear dynamics of an unstable dark soliton and radiation fields emitted have been analyzed by Pelinovsky et al. (1996b). In a region of small velocities near the critical value v , one can apply # a small-amplitude (but still nonlinear) approximation substituting v"v #e»(¹), in order to 0 simplify Eq. (4.10) and reduce it to the form,

A B

d» 1 ­M 4 k (v , q) # 4 0 d¹ e ­v

A B

1 ­2M 4 »# »2"0 . 2 ­v2 v/v0 v/v0

(4.17)

This equation resembles the motion equation of an effective particle of mass k and velocity 4 » under the action of a nonlinear dissipative force. Type of the instability scenario depends essentially on a sign of the initial perturbation and the particular form of the dependence M (v). In 4 a general case, dark solitons of larger intensity and smaller velocity are unstable in the linear approximation while small-amplitude solitons with the velocities close to the limit velocity c are stable. Therefore, for this type of the functions M (v) the derivative (­2M /­v2)D 0 in Eq. (4.17) is 4 4 v/v positive and any perturbation with »(0)'0 leads to a bounded scenario of the dark soliton instability. This process corresponds to a transformation of an unstable dark soliton into a stable soliton of larger velocity and smaller amplitude, described by a simple solution of Eq. (4.17)

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Fig. 8. Schematic presentation of the renormalized momentum M (v) of the dark soliton for two distinct cases: (a) the 4 minimum intensity vanishes when vP0, and (b) the minimum intensity remains finite for vP0. In both the cases the negative slope indicates unstable dark solitons (Pelinovsky et al., 1996b).

(see Pelinovsky et al., 1996b), » » 0 & »" , (4.18) (» !» ) e~jT#» & 0 0 where j is defined by Eq. (4.16) (j'0), » is the initial velocity of the unstable dark soliton, and 0 » is the final velocity of stable solitons defined as & ­2M ~1 2 ­M 4 4 . (4.19) » "! & ­v2 e ­v v/v0 v/v0 This result is valid only if the renormalized momentum of the perturbed dark soliton is a conserved quantity during the soliton transformation. However, this quantity does not conserve beyond the quadratic approximation and the change DM between the value of the renormalized momentum M for the final stable dark soliton and that for the initial unstable soliton, M , can be calculated & 0 directly from Eq. (4.10) as follows,

A BK GA BK H

DM"e

P

`=

A B

dv 2 e3j(D»)2 K (v, q) K (v , q) , d¹" 4 4 0 d¹ 6

(4.20) ~= where the coefficient K is defined above in Eq. (4.12). We note that this coefficient can have, in 4 general, an arbitrary sign and, therefore, transitions from unstable to stable dark solitons can lead

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to either increasing or decreasing of the value of the soliton renormalized momentum. As a matter of fact, the type of the momentum change is determined by a balance between the radiation field º` propagating to the same direction as the perturbed dark soliton and the field º~ propagating to the opposite direction. These radiation fields can be also calculated analytically with the help of Eq. (4.14),

C

D

jD» j ºB" f (v , q) sech2 (XGc¹) , (4.21) 4 B 0 2(cGv ) 0 which coincide, with an accuracy of the amplitude factor, with the sech2-type profile of the stationary dark soliton solutions to the generalized NLS equation in the small-amplitude approximation (see Section 2.5.2). Evolution of the radiation field given by Eq. (4.21) obeys asymptotically the KdV equations with a'0. It is well-known that the sech2-type initial condition in the KdV equation leads to a generation of solitons only if the wave amplitude is negative. In the opposite case, i.e., when the input amplitude of the localized pulse is positive, the initial profile, Eq. (4.21), transforms into linear dispersive waves. A simple analysis indicates (Pelinovsky et al., 1996b), that in the limit v P0 the coefficient f is positive while the coefficient f is negative. Moreover, it is 0 ` ~ possible to show that the sign of the coefficient f remains unchanged throughout the instability ~ region so that the counter-propagating radiation field, described by the function º~, should always generate a stable (shallow) dark soliton as a result of the transformation of the primary unstable dark soliton. On the other hand, the radiation field, described by the function º`, decays into dispersive waves if f (v , q)'0 or it can also produce an additional (stable) dark soliton ` 0 provided f (v , q)(0. ` 0 4.3. Examples of non-Kerr dark solitons 4.3.1. Competing nonlinearities In the case of competing nonlinearities, e.g. focusing plus defocusing, the dark solitons of Eq. (4.2) display features different from those for dark solitons of the cubic NLS equation. Due to self-focusing at smaller intensities, the minimum amplitude of a dark soliton may not reach zero even at v"0, provided the parameter of the cubic nonlinearity is large enough. As a result, the total phase shift S (v) and, therefore, the renormalized momentum M (v) tend to zero in both the limits, 4 4 vP0 and vPc. This leads to the appearance of a negative slope of M (v) for small v and, 4 correspondingly, to instability of dark solitons, see Fig. 8b. This phenomenon is observed for the generalized NLS Eq. (4.2) with competing power-law nonlinearity, Eq. (2.8), which we write here in the dimensionless form, g(I)"1(aIp!bI2p) , (4.22) 2 where a and b are both positive. The first term describes the self-focusing effect for smaller intensities, and therefore it may prevent the existence of black solitons with zero minimum intensity. For p"1 the generalized NLS Eq. (4.2) with Eq. (4.22) corresponds to the focusing cubic and defocusing quintic nonlinearity which describes a deviation from the Kerr medium of an optical material (see also Introduction). Remarkably, the model, Eq. (4.2), with competing nonlinearity, Eq. (4.22), at p"1 possesses an explicit analytical solution for dark soliton [probably first found by Barashenkov and Makhankov (1988)]. Therefore, although the general analysis of the

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competing nonlinearities is qualitatively valid for any value of p, below we restrict ourselves by the case p"1 when the results can be obtained analytically. Exact solution for a dark soliton of the cubic-quintic nonlinearity can be found in the form 2k2 U2(m)"1! , a#b cosh(2km)

(4.23)

a"(4b!1) , b"Ja2!4bk2 , 3 3

(4.24)

where, for simplicity, we take q"1 and a"2. Soliton amplitude k is related to the soliton velocity v as k2#v2"b!1. First, the condition k2'0 yields DvD(c(b)"Jb!1. This shows that the dark soliton, Eqs. (4.23) and (4.24), exists only for b'1 (see Fig. 9a). Then, we use the instability criterion defined above and evaluate the slope of the function M (v), the dark soliton becomes unstable for 4 ­M (v)/­v(0. We check that the negative slope of M (v) appears only for 1(b(1.5 where the 4 4 dark soliton at v"0 has a nonzero amplitude at the minimum, U2(0)"(3!2b)/b. The function M (v) for the particular case b"1.2 is shown in Fig. 9b. The instability region is shown in Fig. 9a. 4

Fig. 9. (a) Regions of existence, DvD(c(b), and instability, DvD(v (b), of the dark soliton Eqs. (4.23) and (4.24). #3 (b) Renormalized momentum M (v) for the dark soliton, Eqs. (4.23) and (4.24), at b"1.2. Thick dashed and solid 4 branches correspond to unstable (v(v ) and stable (v'v ) dark solitons, respectively. Thin solid curve depicts the #3 #3 change of the minimum soliton intensity I . Arrows 1 and 2 correspond to the evolution of the unstable soliton .*/ presented in Figs. 10 and 11, respectively (Pelinovsky et al., 1996b).

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To study the evolution of unstable dark solitons, Pelinovsky et al. (1996b) performed numerical simulations. The dark soliton, Eqs. (4.23) and (4.24), was perturbed t (m)"MU(m)#e[1!U2(m)]N e*h(m) , 1%35 which does not change the soliton phase. Initial velocity v of the unstable soliton was chosen in the 0 unstable region, while e was taken both positive and negative in the interval 0.0001(DeD(0.02. The numerical simulations revealed two completely different scenarios of the dynamics of the unstable dark solitons depending on the sign of e. Some of these features, including the splitting of an unstable dark soliton, were reported earlier by Barashenkov and Kholmurodov (1986).

Fig. 10. Splitting of an unstable dark soliton, Eqs. (4.23) and (4.24), for b"1.2, v "0.02, and e"#0.005. (a) Intensity 0 profiles at z"0 (solid curve) and z"100 (dashed curve), and the corresponding (b) contour plot and (c) propagation dynamics (Pelinovsky et al., 1996b).

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The first scenario is observed for e'0, when initially the soliton amplitude is slightly decreased. Effectively, this corresponds to a ‘push’ of the unstable soliton toward the stable branch of M (v) in 4 Fig. 9b (curve 1) existing for larger values of v. Example of such simulations for the case v "0.04 is 0 shown in Fig. 10a—c where the soliton splitting is observed. The second scenario of the soliton instability takes place for e(0. In this case, the unstable dark soliton is ‘pushed’ deeper into the instability region (see curve 2 in Fig. 9b). The corresponding simulations for the case v "0.04 are presented in Fig. 11a—c where we observe the formation of 0 two kinks propagating in the opposite directions. This scenario of the soliton instability can be called ‘collapse of dark solitons’.

Fig. 11. ‘Collapse’ of the unstable dark soliton, Eqs. (4.23) and (4.24), into two kinks for b"1.2, v "0.02, and 0 e"!0.005. (a) Intensity profiles at z"0 (solid curve) and z"100 (dashed curve), and the corresponding (b) contour plot and (c) propagation dynamics. Thin solid curve in (a) presents, for a comparison, the exact kink solution, Eq. (4.25) (Pelinovsky et al., 1996b).

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Fig. 12. Change of the minimum soliton intensity I for two scenarios of the soliton instability presented in Figs. 10 .*/ and 11: splitting (upper solid, 1, and dashed, 2, curves), and decay into kinks (collapse) (lower solid curve). The dotted line displays the critical intensity I , which corresponds to the instability threshold v"v and it is defined in Fig. 9b #3 #3 (Pelinovsky et al., 1996b).

Thus, the dark solitons evolve asymmetrically in dependence of the type of the initial perturbation. Fig. 12 shows the change of the minimum soliton intensity for both the scenarios. In the first case (e'0), the initial exponential growth of the perturbation amplitude (thick solid curve) saturates at approximately z"55, and the unstable dark soliton splits into two stable solitons of smaller amplitudes (see curves 1 and 2 in Fig. 12) which move after the splitting into the opposite directions as shown in Fig. 10b and Fig. 10c. When the initial soliton velocity is selected far from the threshold value v , more than two secondary solitons are generated. In the other case (e(0), # the exponential growth of the initial perturbation allows the minimum intensity to reach zero (see Fig. 12, thin solid line). Then, the region of zero intensity starts to spread out while the background intensity increases outside the localized wave (see Fig. 11b). Finally, this process results in formation and steady-state propagation of two kink structures. These kinks are described by the exact solutions to the generalized NLS Eqs. (4.2) and (4.22) at p"1, Jq # t (m, z)" e*u#z , k J1#eBDm

(4.25)

where q "3a/4b, u "2bq2, and D2"3a2/4b. The kink of Eq. (4.25) connects two stable back# # # ground waves, the cw background of a selected intensity q with the zero-intensity background. # 4.3.2. Saturable nonlinearity In this section we consider the generalized NLS model, Eq. (4.2), with the saturable nonlinearity, Eq. (2.9), which we write in the following dimensionless form:

C

D

1 1 !1 , g(I)" 2 (1#aI)p

(4.26)

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where the parameter a has the meaning of a ratio of the maximum intensity I to the saturation .!9 intensity, I , i.e., a"I /I , and the parameter p is the saturation index. This type of nonlin4!5 .!9 4!5 earity in the generalized NLS Eq. (4.2) is used to analyze the effect of saturation of the nonlinear refractive index at larger intensities (see also Introduction). In the case p"1 the nonlinearity, Eq. (4.26), appears also in the theory of photovoltaic bright and dark solitons (Valley et al., 1994; Christodoulides and Carvalho, 1995). On the other hand, the model, Eqs. (4.2) and (4.26), at p"2 is known to exhibit explicit soliton solutions in the form of bright and dark solitons (Kro´likowski and Luther-Davies, 1992, 1993). With the help of these exact solutions, it has been recently revealed that dark solitons supported by the saturable nonlinearity have a total phase shift larger than the limit value n realized at the black soliton at v"0 (Kro´likowski et al., 1993). These solitons are clearly observed in numerical simulations to be stable in some region of the parameter region. Later the instability of dark solitons has been pointed out exactly for the same model (Kivshar and Kro´likowski, 1995b). Here, we present the results for the case p"2 which displays essentially the same dynamics of the dark soliton instabilities as the model, Eqs. (4.2) and (4.26), for other values of p. Fig. 13a presents the regions of existence, v(c(a), and instability, v(v (a), of the dark solitons #3 in the model defined by Eqs. (4.2) and (4.26) at p"2 and q"1. The dashed line in Fig. 13a depicts

Fig. 13. (a) Regions of existence, v(c(a), and stability, v(v (a), of dark solitons in the saturable model, Eqs. (3.42) and #3 (4.26) at p"2. Dashed curve shows the region where a dark soliton has a total phase shift larger than p. Renormalized momentum M (v) (b) and the total phase shift S (v) (c) calculated for dark solitons at two values of the saturation 4 4 parameter a: a"6 (curves 1) and a"12 (curves 2). Critical velocity v corresponds to the instability threshold #3 (Pelinovsky et al., 1996b).

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the region of the parameter plane where the dark solitons have the total phase shift larger than n [see also discussions in Kro´likowski et al. (1993)]. the typical dependencies of the renormalized momentum M (v) and the total phase shift S (v) are shown for a"6 and a"12 in Fig. 13b and 4 4 Fig. 13c, respectively. It is clearly seen that the appearance of a large phase shift of the largeamplitude dark solitons serves as a pilot of their instability. However, among the dark solitons with the phase shift larger than n there exist both stable solitons, realized for smaller amplitudes and larger velocities, and unstable solitons, realized for larger amplitudes and smaller velocities (see Fig. 13a). Next, the development of the dark soliton instability in the saturable model described by Eqs. (4.2) and (4.26) at p"2 is presented in Fig. 14a—c. If follows that the instability-induced soliton dynamics in this model displays features which are different from those for the case of competing nonlinearities. Indeed, being perturbed by a small perturbation, the unstable black soliton transforms symmetrically into a stable grey soliton which corresponds to a positive slope of the function M (v) as shown in Fig. 14a. Despite the change of the soliton velocity is small for the 4 nonlinearity (4.26) at p"2 the change of the minimum soliton intensity (see Fig. 14b) and the soliton position (see Fig. 14c) clearly indicate an initial, exponential growth of the perturbations at

Fig. 14. Dynamics of an unstable black soliton in the model, Eqs. (3.42) and (4.26) at p"2 and a"12. Shown are: (a) renormalized momentum M (v) and transitions corresponding to a transformation of an unstable ‘black’ soliton 4 into a stable ‘grey’ soliton depending on a sign of the initial perturbation, (b) change of the minimum soliton intensity I , .*/ (c) change of the soliton position (Pelinovsky et al., 1996b).

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Fig. 15. Stability threshold curve for dark solitons of the generalized model of saturable medium, Eq. (4.26), as the dependence of the critical saturation intensity a~1 vs. the parameter p (Kivshar and Afanasjev, 1996). #3

the unstable black soliton and, then, their saturation at the level which corresponds to a stable grey soliton. The development of the instability of a grey soliton occurs basically in the same manner. The novel feature is asymmetric transitions to the right and to the left (see Pelinovsky et al., 1996b). The similar instability exists for any p in the model described by Eqs. (4.2) and (2.9). Fig. 15 presents the threshold of the instability of a black soliton for the model, Eqs. (4.2) and (4.26), as a function of the saturation parameter p (Kivshar and Afanasjev, 1996). 4.3.3. Transiting nonlinearity In this section we consider one more example of the model of optical solitons described by the generalized NLS Eq. (4.2). This is the case of bistable solitary waves introduced, for the first time, for bright solitons by Kaplan (1985a,b). In the case of dark solitons, bistability can appear when the branch of unstable solitons is found for intermediate values of the soliton velocity allowing transitions between two (or more) types of solitons belonging to stable branches. For the first time, the bistability of dark solitons was discussed by Enns and Mulder (1989), Mulder and Enns (1989) and Bass et al. (1992), who however used the complementary power P for discussing the bistability 4 regimes. As has been clarified above, the complementary power can possess both positive and negative slopes, but it does not characterize the soliton stability and therefore bistable regimes should be analyzed in the framework of the renormalized soliton momentum [see Pelinovsky et al. (1996b)]. The typical nonlinearity displaying soliton bistability was suggested by Enns and Mulder (1989) and it is a kind of defocusing transiting nonlinearity, g(I)"!IM1#a tanh[c(I2!I2)]N , 0

(4.27)

which describes a smooth transition from one linear dependence for small intensities, I;I , when 0 g(I)"[a tanh(cI2)!1]I, to the other linear dependence for large intensities, I
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g(I)&!(1#a)I. Parameters a, I in Eq. (4.27) characterize the amplitude and threshold intensity 0 of the nonlinearity transition, while c~1@2 determines the characteristic width of the transition region. The renormalized soliton momentum M (v) for dark solitons of the model, Eq. (4.2), with the 4 transiting nonlinearity, Eq. (4.27), at a"0.5, c"10, and varying I , is shown in Fig. 16a. There 0 exists a rather narrow region of the soliton velocities where the momentum M (v) has a negative 4 slope indicating a possibility of bistable dark solitons. Fig. 16b presents an enlarged part of one of the dependencies M (v) of Fig. 16a which displays 4 stable (solid) and unstable (dashed) branches. The instability region, v(1)(v(v(2), where, at #3 #3 I "0.6, v(1)+0.955 and v(2)+1.014, is rather narrow, and it is defined by the criterion 0 #3 #3 ­M (v)/­v(0. The characteristic feature of the transiting nonlinearity is the stability of black 4 solitons, which belong to one of the stable branches. Thus, in this case the unstable branch corresponds to grey solitons. Numerical simulations of the instability-induced dynamics of dark solitons in the model, Eqs. (4.2) and (4.27), have been performed for a dark soliton with the initial velocity v "0.96 (see 0 Pelinovsky et al., 1996b). The dynamics displays indeed two types of the transitions (switching) of a dark soliton from the unstable to one of the stable (1 or 2) branches of the stationary solutions. The first type of the soliton switching describes a transition to a stable dark soliton with larger value of the minimum intensity and larger velocity, as shown by the curves 1 in Fig. 17a—c.

Fig. 16. (a) Renormalized momentum M (v) for dark solitons supported by the transiting nonlinearity, Eq. (4.27) for 4 a"0.5, c"10 and different values of the parameter I : 0.5, 0.6, and 0.7, shown next to the curves. (b) Stable (thin solid) 0 and unstable (dashed) branches of the soliton renormalized momentum M (v) for the case a"0.5, c"10, and I "0.6. 4 0 The thick solid curve displays a change of the minimum soliton intensity I vs. soliton velocity v (Pelinovsky et al., .*/ 1996b).

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Fig. 17. Dynamics of the unstable dark soliton in the model, Eq. (3.42), with the transiting nonlinearity, Eq. (4.27), for a"0.5, c"10, and I "0.6. Soliton initial velocity v "0.96. Shown are (a) evolution of the minimum soliton intensity I , and (b) 0 0 .*/ change of the relative position of a soliton on a moving background, for both the unperturbed soliton (curves 0) and two types of the bounded scenario for the evolution of a perturbed dark soliton (curves 1 and 2). Transitions from the unstable branch to the stable branches (1 or 2) are shown in (c) by arrows on the plot of the renormalized momentum (Pelinovsky et al., 1996b).

The second type describes a transition to another stable dark soliton, with smaller value of the minimum intensity I and smaller velocity v, as shown by the curves 2 in Fig. 17a—c. In the latter .*/ case, the transition is observed with negligible radiation and, therefore, at almost unchanged value of the soliton renormalized momentum.

5. Multi-component dark solitons 5.1. Mode interaction: general overview Solitary waves discussed above are described by one-component scalar fields. This is the simplest case of single-frequency pulses in optical fibers or scalar TE electric field in optical waveguides

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under the condition of nonresonant optical nonlinearities. If either or both of these conditions is not satisfied, the scalar generalized NLS equation is not valid (see, e.g., Section 2.3) and the field of the main frequency (or main polarization) becomes coupled to other components. This is the case of the so-called multi-mode wave propagation which usually leads to multi-component solitary waves. For the pulse propagation in a nonlinear optical fiber, such a situation appears in multi-mode fibers due to coupling between the modes through nonlinearities and nonuniformities of the optical material, and the pulse propagation is then described by a system of coupled NLS equations [see, e.g., Crosignani and DiPorto (1981, 1982)]. The same physics is behind the birefringency effect which leads to a coupling between two polarizations (Menyuk, 1989). Different in physics, the interaction between pulses or beams of two different frequencies lead to the similar system of coupled NLS equations, provided the four-wave mixing is neglected (see, e.g., Ryskin (1994), and references therein). Nonlinear interaction between two polarizations arising from the tensor nature of the nonlinear s(3) susceptibility was first discussed by Maker et al. (1964) who showed that the nonlinear polarization in an isotropic medium may be written in the form P "s(3)[A(u)E(E ) E*)#B(u)E*(E ) E)] . NL The effect of this vectorial interaction between perpendicular polarizations on modulational instability and solitary waves was first considered in the pioneering paper by Berkhoer and Zakharov (1970). Generally speaking, there exist five principally different cases of the coupling between two modes in a Kerr medium and corresponding vectorial solitary waves (for simplicity, we assume the case of the temporal solitons and focusing nonlinearity): f Bright solitons, each in the mode with the anomalous dispersion (vector bright solitons) [e.g., Berkhoer and Zakharov (1970), Manakov (1974), Tratnik and Sipe (1988), Christodoulides and Joseph (1988) and Menyuk (1989)]; f Bright soliton in the mode with the anomalous dispersion coupled to a dark soliton in the mode with normal dispersion (normal dark—bright pair) [e.g., Afanasjev et al. (1989a,b), Wang and Yang (1990), Hong et al. (1991), Kivshar (1992) and Buryak et al. (1996a)]; f Bright soliton in the mode with normal dispersion exists due to mutual coupling to a dark soliton in the mode with anomalous dispersion (the so-called inverted dark—bright pair) [e.g., Trillo et al. (1988) and Afanasjev et al. (1989a)]; f Two dark solitons, each in the mode with the normal dispersion (vector dark solitons) [e.g., Kivshar and Turitsyn (1993), Haelterman and Sheppard (1994d,e), Radhakrishnan and Lakshmanan (1995) and Sheppard and Kivshar (1997)]; f Bright pulse supported by a dark soliton, both modes are with the normal dispersion (solitoninduced waveguides, in the linear limit, or dark—bright pair, in a nonlinear regime) [e.g., Christodoulides (1988) and Sheppard and Kivshar (1997)]. All these cases are described by two NLS equations, coupled due to crossphase modulation. These coupled equations become asymmetric for the interaction between envelope of different carrier frequencies or some additional coupling terms, e.g. due to four-wave mixing effect, may appear. There is a little known about vectorial dark solitons and dark—bright in non-Kerr materials and in the systems described by the equations different from the coupled NLS equations. We would like

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to mention dark-soliton pairs in a nonlinear coupler (Ankiewicz et al., 1994), which are probably all unstable, families of unstable multimode dark—bright solitons predicted for a photorefractive medium with a refractive index change of opposite sign for different polarization (Kro´likowski et al., 1996), and incoherently coupled dark—bright solitons pairs in a biased photorefractive medium (Christodoulides et al., 1996). Stability of this kind of vectorial solitons is one of the most important issues and, after very recent advances in the stability of multi-parameter solitary waves (Buryak et al., 1996b) we can expect some progress in the near future. There exists a variety of different problems involving multi-component wave interaction and, as a matter of fact, a systematic approach to multi-component dark solitons as solitons of nonlinear equations describing such interaction is absent. That is why, in this section we discuss only a few the most characteristic examples of solitary waves existing on a nonvanishing background (e.g., dark—bright soliton pairs, polarization domain walls, and dark solitons in s(2) materials) just to show how the concept of dark soliton can be generalized to include more than one field. However, we notice that there exist more open questions than solved problems in the theory of multicomponent dark solitons. 5.2. Dark—bright solitons 5.2.1. Model and exact solutions We consider the interaction between either (i) two waves of different frequencies u and u , or 1 2 (ii) two waves of the same frequency u but belonging to two different polarizations. Then, for the slowly varying wave envelopes u and u , the most important nonresonant interaction between the 1 2 waves is due to the cross-phase modulation effect, and the problem can be described by the system of two coupled NLS equations i

i

A A

B B

­u ­2u ­u 1#g (Du D2#pDu D2)u "0 , 1#d 1 #c 1 ­m2 1 1 2 1 ­m ­f

(5.1)

­2u ­u ­u 2#g (Du D2#pDu D2)u "0 . 2!d 2 #c 2 ­m2 2 2 1 2 ­f ­m

(5.2)

In nonlinear optics, Eqs. (5.1) and (5.2) appear in two different physical situations. In the first, or spatial case, these equations describe interaction between two continuous waves of distinct frequencies u and u with m standing for the transverse coordinate, and c "1/2k(u ) take into 1 2 j j account the mode diffraction. In the second, or temporal case, m stands for time, d"1(d !d ), 2 2 1 where d "dk(u )/du are the modal group velocities, and c "!1d2k(u )/du2 describe the modal j j j j 2 j j dispersions (see Sections 2.1 and 2.2.2). We are interested below in stationary solutions of Eqs. (5.1) and (5.2), when the (small) groupvelocity mismatch is compensated due to the nonlinearity-induced shift of the carrier frequency, so that we apply the transformation, JDc D 2 e*k1m~*u1f , u "¼ 1 Jg Dc D 2 1

º u " e*k2m~*u2f , 2 g 1

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where u "$dk #c k2 ( j"1, 2) are the nonlinearity-induced shifts of the wave frequency, and j j j j k "$d/2c . Then, the dimensionless envelopes ¼ and º satisfy the following system of coupled j j NLS equations: i

­¼ ­2¼ #s #(gD¼D2#pDºD2)¼"0 , ­t ­x2

io

A

B

­º ­2º DºD2 #r # #pD¼D2 º"0 , ­t ­x2 g

(5.3) (5.4)

where t"f, x"m/JDc D, o"Dc D/Dc D, and g"g /og . The sign functions r,sgn(c )"$1 and 1 1 2 1 2 1 s,sgn(c )"$1 describe the type of the group-velocity dispersion, negative or positive. This is 2 the most general system describing two incoherently coupled nonlinear modes in a Kerr medium. The parameters g and o produce an effective asymmetry between the modes for the case of the interaction between waves of the different frequencies, e.g., g"(u /u )2 and o"(u /u )3. The 1 2 1 2 coefficient p in Eqs. (5.3) and (5.4) is the parameter of the cross-phase modulation which can be defined in terms of parameters of the corresponding physical problems [see, e.g., Agrawal (1989), and Menyuk (1989)], and usually 0(p42. For example, in fiber optics the system of Eqs. (5.3) and (5.4) describes interaction between either two waves of the same carrier frequency but belonging to different polarizations, or two waves of the same polarization but different frequencies. For two waves of different frequencies u and u , one usually has (g"u2/u2) p"2, whereas 1 2 1 2 for two waves of different polarizations in a birefringent optical medium, (g"1) p"2 (Menyuk, 3 1989). As has been mentioned in Section 5.1, Eqs. (5.3) and (5.4) allow the existence of five different types of localized solutions for solitary waves. In this section we are interested in finding coupled soliton states, dark—bright vector solitons, which combine dark (in the ¼-component) and bright (in the º-component) solitary waves, so that we look for stationary solutions of the system of Eqs. (5.3) and (5.4) in the form, ¼(f, q)"w(x, t) e*gt , º(f, q)"u(x, t) e*bot e*h ,

(5.5)

where x"f!vq, t"q, h"ro(Vx#v2t), w"w #iw and u are complex and real functions, 4 r i 2 respectively. The following boundary conditions are posed: Dw(x)D2P1 and u(x)P0 as xP$R. Substitution of Eq. (5.5) into Eqs. (5.3) and (5.4) yields the system of equations, i

­w ­2w ­w !iv ! #[g(DwD2!1)#pDuD2]w"0 , ­x ­x2 ­t

A

B

­u DuD2 ­2u io !r !bu# #pDwD2 u"0 , ­t g ­x2

(5.6)

which defines a family of stationary two-parameter solitons, w"w (x)"w (x)#iw (x) and 0 03 0* u"u (x), provided that the t-derivatives are omitted. These localized solutions are defined by two 0 independent parameters, the propagation constant b and soliton velocity v. Because Eq. (5.6) are not integrable, most localized solutions can be found only numerically. However, there exists a special sub-class of analytical solutions of Eq. (5.6) describing a

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one-parameter family of dark—bright solitons, w (x)"cos u tanh(cx)#i sin u , u (x)"a sech(cx) , (5.7) 0 0 where c2"g(1!p2)/(2#2gp)!v2/4, a2"2c2(g#p)/(1!p2), tan u"!v/(2c), and b"c2#p. These analytical solutions exist only for p2(1 (and at r"#1) and they can be characterized by one continuous parameter (e.g., the soliton velocity ») generalizing the zero-parameter family of solitons found earlier by Afanasjev et al. (1989a,b). However, the exact solutions give only a subset of a more general family of vector solitons of Eq. (5.6). Each of these stationary solutions generates the corresponding nonstationary solitons through the standard Galilean transformation. 5.2.2. Soliton-induced waveguides To explain the existence and some of the features of the bright—dark localized solutions of the model described by Eqs. (5.3) and (5.4) we study bifurcations from the one-component solutions. This analysis also allows to introduce an important physical concept of the soliton-induced waveguides and light guiding. First, we consider the solution of Eqs. (5.3) and (5.4) as a one-component dark soliton and add small perturbations writting, º"O(e), and ¼"¼ (x, t)#O(e2), where ¼ is the one-component s s dark soliton (s"!1), ¼ (x, t)"e*gt tanh(xJg/2) , (5.8) s with the background amplitude ¼ "1. It is easy to verify from Eqs. (5.3) and (5.4) that the term 0 O(e) in the expansion for ¼ is zero. Substituting this expansion into Eqs. (5.3) and (5.4), we obtain, to the leading order, two decoupled equations, of which the equation for º is io

­2º ­º #r #pD¼ D2º"0 . s ­x2 ­t

(5.9)

Stationary solutions of Eq. (5.9) can be presented in the form º(x, t)"e*(K@o)t f (X) ;

X"xJg/2 ,

(5.10)

where the function f (X) is a solution of the following eigenvalue problem r

2p 2 d2f ! f" (K!p) f , g dX2 g cosh2 X

(5.11)

with K to be determined. The eigenvalue problem, Eq. (5.11), is well known in many branches of physics. In particular, in optical waveguide theory it is the model equation of the sech2-profile graded index planar waveguide (Snyder and Love, 1973). For r"!1 the eigenvalue problem of Eq. (5.11) has solutions f (X) that decay as XP$R only if K(p (the so-called guide or bound modes). The number of such bound modes depends on the parameter p/g. In particular, for p/g"2 we have two localized solutions [see, e.g., Snyder and Love (1973)]. The lowest-order (symmetric) solution º "sechlX exists for l"1.56 and it appears at the critical value K"K "0.78. This corres1 1 ponds to a bifurcation of the vector soliton (¼, º) in which a symmetric function º branches from

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the one-component dark soliton ¼ , and it describes a linearized limit of the coupled dark and s bright solitons. In contrast, for r"#1 there exist no localized eigenfunctions of the problem described by Eq. (5.11) which in this case describes the so-called ‘anti-waveguide’. This means that, if coupled bright and dark solitons are possible, they should correspond to a nonzero threshold energy and they have no linear limit described by an effective linear guided mode. The phenomenon described above is usually referred to as soliton-induced waveguiding of a weak probe beam, º;D¼ D, and it was widely discussed in the literature. Probably for the first time, the s use of an effective waveguide created by the refractive index change due to an intense electromagnetic beam was discussed by Askar’yan (1962) who analyzed the effect of the beam gradient on electrons and atoms and suggested it for directing electrons and plasma transport. In application to the light guiding by solitary beams, there exist many papers discussing basically the same idea for both spatial and temporal solitons [see, e.g., Manassah (1991)]. The use of dark solitons as induced waveguides was suggested and verified experimentally by Luther-Davies and Yang (1992a,b). For temporal solitons in fibers, the phenomenon of the soliton-induced waveguide can be used for compression of bright pulses by dark solitons. This effect was suggested by Jin et al. (1993) who however did not mention the physical origin of this effect due to waveguiding properties of dark solitons. Jin et al. (1993) demonstrated numerically that bright pulses can be compressed in the region of normal dispersion (which is inaccessible in the soliton compression scheme) by coupling it to a dark soliton of a finite width background. The best compression ratio of this scheme was reported to be 3.3 (Jin et al., 1993). The effect of pulse walk-off on the compression of bright pulses by dark solitons was discussed by Cao and Zhang (1996) who demonstrated that the walk-off is crucial leading to asymmetric pulses, much lower compression ratios and longer compression lengths. They also pointed out that asynchronous coupling of the bright pulse with a dark soliton can improve the compression scheme. When the amplitude of the probe beam º grows, it cannot be described any longer by the effective linear Eq. (5.11) and therefore the full system of two incoherently coupled NLS equations should be considered. In the framework of this model, the case of the soliton-induced waveguide at r"!1 corresponds to the linear limit of the dark-bright solitary wave. General solutions of this kind have been not investigated yet in details, including the stability analysis similar to the case of the opposite dispersions discussed in Section 5.3. However, it was recently found that the integrable case of the Manakov equations (p"o"g"1) for defocusing nonlinearity allows a detailed analysis (Radhakrishnan and Lakshmanan, 1995; Sheppard and Kivshar, 1997) and the existence of N-soliton solutions describing interaction of dark—bright vector solitons (Sheppard and Kivshar, 1997). How the main features of these solitary waves survive in a more general nonintegrable model has not been understood yet, and this issue requires the further analysis. 5.3. Modes with opposite dispersions For the case of opposite dispersions, i.e, at r"#1, the dark soliton behaves like an antiwaveguide and therefore it cannot guide a small-amplitude probe beam. The important question then is to find if the soliton guiding will become possible in a nonlinear regime, which is described by the coupled NLS equations, with the dark—bright vector solitary waves. Also, stability of these solitary waves seems to be the important issue to be analyzed.

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To find localized solutions of the model of Eq. (5.6) describing the coupled bright and dark solitons in the case of opposite dispersions, Buryak et al. (1996a) used a numerical scheme based on a relaxation technique to solve three coupled ordinary differential equations of the second order. As a result, they found that in a physically important domain of the system parameters g and p, i.e. g&1 and 0(p42, they system of Eq. (5.6) possesses a two-parameter family of stationary solutions with the asymptotics Dw(x)D2P1 and u(x)P0 at xP$R. Such solitons exist in a certain region, b'b "p and DvD(v (b) [see Buryak et al. (1996a)]. The exact solutions given by #3 .!9 Eq. (5.7) correspond to a parabolic curve in this plane. This dark—bright solitons was characterized by Buryak et al. (1996a) by two integral quantities, the power of the bright soliton in the u-component,

P

P (b; v)" u

`=

DuD2 dx ,

(5.12)

~= and the renormalized momentum of the dark soliton in the w-component which is defined as follows:

P G

HG

H

i `= ­w ­w* 1 M (v; b)" w* !w 1! dx . (5.13) w 2 ­x ­x DwD2 ~= Note that the power P is an integral of motion of the system, Eq. (5.6), whereas the second u conserved quantity is the total momentum M "M #M , where 505 u w ­u ­u* `= i u* !u dx . (5.14) M (v; b)" o w ­x ­x 2 ~= Typical examples of dark—bright solitons and the change of the functions P (b) at v"0 and u M (v) at b"1.5 are displayed in Fig. 18a and Fig. 18b, respectively. An important conclusion w following from these plots is that, for almost all values of b and v within the region of the soliton existence soliton bistability is observed. The lowest order (i.e., one-soliton) solutions belong to two main branches which differ by the value of the phase jump across the dark-component (see inserts in Fig. 18b). In particular, for v"0 one of the branches corresponds to a dark soliton with the (maximum) phase jump equal exactly to p (point and insert E in Fig. 18b) whereas the other one, to a dark soliton with no phase jump (point D). For any nonzero velocity v, the solitons of the lower branch also possess a phase jump (see the change of the soliton profiles in Fig. 18b, when going along the sequence DPFPH). As the velocity v increases, the difference between the phase jumps of the solitons of the two types becomes smaller and exactly at DvD"v (b) the two branches .!9 merge (point H in Fig. 18b). The critical point H, where the derivative ­M /­v tends to infinity, w separates the solutions into two distinct families. As b approaches its minimum value, b "p, the #3 soliton amplitudes decrease (see the insert C in Fig. 18a), and we can apply the small-amplitude approximation suggested by Kivshar (1991b). For example, for p2(1 we obtain the asymptotic dependence

P G

H

4g(2g!v2)Jb!p P (b; v)+ , u [2g(1!p2)!v2] shown in Fig. 18a for the case v"0 as a dashed curve.

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Fig. 18. (a) Characteristic dependence of the energy Q , Eq. (5.12), vs. b for »"0, p"2, and g"1. Upper branch u 3 describes a transition from one-hump (e.g., points E and A) to two-hump (e.g., point B) solitons. Dashed curve is the analytical result (Kivshar, 1992). (b) Characteristic dependence of the renormalized momentum M , Eq. (5.13), vs. w » (»'0) for b"1.5. Inserts show the profiles of the bright component (vanishing at xP$R) and dark component, both real (antisymmetric) and imaginary (symmetric) parts (Buryak et al., 1996a,b).

The general stability criterion for dark—bright solitons in the model, Eq. (5.6), has to be obtained analytically by Buryak et al. (1996a,b) using the multi-scale asymptotic method proposed earlier (Pelinovsky et al., 1995). This criterion, as has been revealed recently, is a particular case of the general criterion of instability for two-parameter solitary waves investigated in details for the case of three-wave resonant interaction in a diffractive medium (Buryak et al., 1996b). The threshold of the soliton instability is defined by the condition ­P ­M ­P ­M u 505! u 505"0 . ­b ­v ­v ­b

(5.15)

To simplify the discussion of stability, Buryak et al. (1996a) considered the case v"0 when the first term in Eq. (5.15) vanishes because ­P /­vP0 for vP0. This factorizes the general u stability condition into two conditions which can be understood as symmetric (or bright-type) and

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antisymmetric (or dark-type) instabilities. For the solitons of the lower branch in Fig. 18a, ­P /­b'0, and thus these dark—bright solitons are always stable with respect to brightu type instability which results in a symmetric growth of the amplitude of the u-component. In contrast, for the solitons of the upper branch, the derivative ­P /­b can be negative (see, u e.g., Fig. 18a), and thus these solitons can become unstable with respect to symmetric perturbations. The analysis of the dark-type instability shows that the solitons of the upper branch (see Fig. 18b) are always unstable to antisymmetric perturbations. On the other hand, because of the relation ­M /­v"­M /­v!r(o2/2)P , the stability criterion defined by the second condition 505 w u becomes ­M /­v(r(o2/2)P , and it predicts instability for the solitons of the lower branch if the w u parameter o is small enough, i.e., for o(o (p, g, b), and it predicts stability otherwise. Stability #3 diagram was found and also verified numerically by Buryak et al. (1996a), and it is shown in Fig. 19. This diagram proves that stable dark—bright solitons can exist even in the case of the opposite dispersions when there exist no linear guided modes trapped by a dark soliton. These dark—bright solitons can be however stable only if the power of a bright component is not too high and the corresponding dark component does not have a phase-jump. This condition is not restricted by a particular choice of p, but with increase of p in Eqs. (5.3) and (5.4), the instability threshold o also rapidly increases. #3 5.4. Polarization instability and domain walls An interesting class of localized solutions with nonvanishing boundary condition was shown to exist in the case of vectorial interaction between polarizations in an isotropic medium [see, e.g., Haelterman and Sheppard (1994a—e); Malomed (1994)]. Following Haelterman and Sheppard (1994c,e), we consider propagation of polarized light in an isotropic Kerr medium which in dimensionless units can be described by a system of symmetrically

Fig. 19. Stability diagram for the dark—bright solitons of the lower branch of Fig. 18a. Stability threshold is given by the o (b)-dependence (Buryak et al., 1996a,b). #3

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coupled NLS equations: i

1 ­2E ­E `#(DE D2#pDE D2)E "0 , `! ` ~ ` 2 ­x2 ­z

(5.16)

i

­E 1 ­2E ~! ~#(DE D2#pDE D2)E "0 , ~ ` ~ ­z 2 ­x2

(5.17)

where E and E are the circular polarization components and p"(1#B)/(1!B) is the ` ~ coefficient of the cross-phase modulation. These equations can also describe the case of a highly birefringent optical fibers for which p is a function of the fiber parameters and E are the B amplitudes of elliptically polarized fiber eigenmodes (Menyuk, 1989). The simplest solution of Eqs. (5.16) and (5.17) describes the steady state of (linearly polarized) cw mode E "E "E expMi(1#p) E2zN . ` ~ 0 0 The linear stability analysis of this solution was carried out by several authors [e.g., Berkhoer and Zakharov (1970), Agrawal (1987) and Haelterman and Sheppard (1994c)]. We use the standard ansatz E "(E #a ) expMi(1#p)E2zN, and linearize the NLS equation for small a looking for B 0 B 0 B the solution in the form a "a exp(jz) cos(Xx). This leads to a characteristic polynomial of the B B fourth degree in j, and determines the instability eigenvalue j "XJ(p!1)P !X2/4 , (5.18) 1 0 where P is the power of the cw polarization components, P "E2. The maximum gain of the 0 0 0 modulational instability is found to be j "(p!1)P which corresponds to the optimal fre.!9 0 quency X "[2(p!1)P ]1@2. This kind of modulational instability exist for the normal dispersion . 0 regime, when there is no instability of a single NLS equation. That is why it can be called extended, or polarization modulational instability. Importantly, if the cross-phase modulation becomes smaller than self-phase modulation, i.e., p(1, the instability no longer occurs. It is usually believed that modulational instability should be associated with the existence of localized wave solutions. Indeed, as has been mentioned in Section 2.4, scalar bright solitons are associated with the existence of modulational instability of a scalar focusing NLS equation. Using this analogy, Agrawal (1987) suggested the existence of solitary waves of the coupled system of Eqs. (5.16) and (5.17) associated with the extended modulational instability which can exist even in a defocusing medium (or normal dispersion regime) due to intermode interaction. The link between the extended modulational instability discussed above and solitary waves was demonstrated for the first time by Haelterman and Sheppard (1994e) who demonstrated the existence of the so-called polarization domain walls as solitary waves associated with this kind of polarization modulational instability. Exact stationary localized solutions of the system of Eqs. (5.16) and (5.17) can be found only numerically (Haelterman and Sheppard, 1994c,e). We look for solutions of the form, E (x, z)"u(x) e*bz , E (x, z)"v(x) e*bz , ` ~ where the functions u(x) and v(x) are real and b is the propagation constant. Substituting these expressions into Eqs. (5.16) and (5.17) brings us to the system of two coupled ordinary differential

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equations: 1 d2u "!bu#u3#pv2u , 2 dx2

(5.19)

1 d2v "!bv#v3#pu2v , 2 dx2

(5.20)

Eqs. (5.19) and (5.20) describe the motion of a unit mass in the (u, v) plane in the potential º(u, v)"b(u2#v2)!1(u4#v4)!pu2v2 . 2 The solitary wave solutions correspond to the separatrix trajectories of this potential. The separatrices that connect the pairs of opposite maxima correspond to the circularly polarized NLS dark solitons, u(x)"Jb tanh (Jbx) , u(x)"0 ,

v(x)"0 ,

v(x)"Jb tanh(Jbx) .

The separatrices connecting opposite saddle points correspond to the linearly polarized NLS dark solitons, u(x)"$v(x)"Jb/(1#p) tanh(Jbx). The separatrices connecting adjacent maxima of the potential º(u, v) can be found only numerically for each value of p because the model described by Eqs. (5.19) and (5.20) is generally nonintegrable. They correspond to the kink shape localized solutions, shown in Fig. 20a—d. The solution of this kind connects two domains of orthogonal stable eignepolarizations of the Kerr medium and that is why it can be called a polarization domain wall. It describes a change of the field ellipticity q"(u!v)/(u#v) from q"#1 to q"!1, while q"0 corresponds to a linear polarization state.

Fig. 20. Envelopes u(x) and v(x) of the polarization domain wall found numerically as localized solutions of Eqs. (5.19) and (5.20) for b"1 and (a) p"1.2, (b) p"2, (c) p"7, and (d) p"40.

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Fig. 21. Numerical simulation of the propagation of the polarization domain wall on a broader square-shaped background. The curves show the total intensity profile of the field I"E2 #E2 . The polarization state switches from ` ~ the mode E to the node E around x"0 (Haelterman and Sheppard, 1994e). ` ~

Haelterman and Sheppard (1994e) demonstrated that the polarization domain wall can be treated as the limiting case of a periodic solution and it can be interpreted as a solitary wave associated with the polarization modulational instability of the coupled NLS Eqs. (5.16) and (5.17). They also checked the stability of this solution by direct numerical simulations of Eqs. (5.16) and (5.17). Fig. 21 shows the example of propagation of the total intensity profile I"E2 (x)#E2 (x) ` ~ associated with the polarization domain wall excited on a background beam of a finite extention. It is important to note that the localized solutions describing domain walls are also known in other fields. Some examples are domain walls between convection patterns of different symmetry being described by the similar system of coupled Ginzburg—Landau equations [see, e.g., Malomed et al. (1990) and Aranson and Tsimring (1995)], and the so-called self-induced gap solitons which separate different standing waves in a discrete lattice (Kivshar, 1993a). 5.5. Parametric dark solitons in s(2) media As has been mentioned in Section 2.3, the NLS equation is not valid near resonances where the wave of the fundamental frequency becomes coupled to some other frequency. This means that the solitary waves of the cubic nonlinearity can be drastically modified near such resonant points e.g. due to third-harmonic generation [see, e.g., Sammut et al. (1997)]. Importantly, the similar parametric coupling between waves of different frequencies can lead to solitary waves even in the cases when they are not supported by the leading nonresonant nonlinearity. As has been understood recently, this is the key mechanism leading to solitary waves in optical materials with quadratic (or s(2)) nonlinearities. It was shown long time ago [see, e.g., Armstrong et al. (1962) and Ostrovsky (1967)] that a product of second-order nonlinearities can lead to an effective third-order process which resembles a cubic nonlinearity of a Kerr medium. However, only recently experiments have

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confirmed (DeSalvo et al., 1992; Nitti et al., 1994) that large nonlinearity-induced phase shift, self-focusing and self-diffraction can be observed in materials with s(2) susceptibility due to cascaded s(2) : s(2) second-order parametric processes. The experimental results have stimulated further efforts in analysing various nonlinearity-induced effects which can be observed due to cascading. In particular, it has been shown that such cascaded nonlinearities can support the intensity-dependent light propagation in the form of spatial (or temporal) bright optical solitons [see Karamzin and Sukhorukov (1974), Schiek (1993), Hayata and Koshiba (1993b), Werner and Drummond (1993, 1994) and Buryak and Kivshar (1994, 1995a—c)]. In particular, it was shown that, for a very special choice of the system parameters, an exact analytical solution exists (Karamzin and Sukhorukov, 1974). Recently, Buryak and Kivshar (1995a) have demonstrated the existence of a family of two-wave bright solitons; this family includes, as a special case, the exact solution found earlier (Karamzin and Sukhorukov, 1974; Hayata and Koshiba, 1993a,b). Dark solitons in s(2) materials have been first discussed by Werner and Drummond (1994) for two particular cases when an effective NLS equation can be derived. These cases are (i) the limit of the large phase mismatch between harmonics and (ii) the zero dispersion of the second harmonics. Hayata and Koshiba (1994) found one particular exact solution for a dark soliton beyond the NLS approximation. The first theory of two-wave dark solitons in dispersive optical materials with quadratic nonlinearities was elaborated by Buryak and Kivshar (1995b,c). This theory describes all the cases discussed earlier and also presented families of localized solutions with nonvanishing asymptotics. However, this theory is still not complete because it does not include grey solitons and it does not allow arbitrary value of the walk-off between the harmonics. In this section, we discuss two-wave dark solitons in a s(2) medium following the original results obtained by Buryak and Kivshar (1995a—c). Considering interaction between the first (u "u) and second (u "2u) harmonics in a disper1 2 sive/diffractive dielectric medium with s(2) nonlinear susceptibility, we derive the system of two nonlinear equations for slowly varying envelopes E and E coupled through components s(2) of 1 2 ijk the nonlinear second-order susceptibility tensor, i

­E ­E ­2E 1#id 1#c 1#s E*E e*Dkz"0 , 1 1 1 1 2 ­z ­m ­m2

i

­E ­E ­2E 2#id 2#c 2#s E2e~*Dkz"0 , 2 ­m 2 ­m2 2 1 ­z

(5.21)

where s ,(4pu2/k c2)s(2)(u; 2u, !u) and s ,(8pu2/k c2)s(2)(2u; u, u), z is the propagation 1 1 2 2 distance, and Dk,(2k !k ) is the wave vector mismatch between the harmonics. The system of 1 2 Eq. (5.21) generalizes the standard equations of the second-harmonic generation, and it describes two different physical situations. In the first, spatial case, the difference (d !d ) describes the 1 2 so-called spatial walk-off effect, m stands for the transverse coordinate, and c "1/2k , ( j"1, 2), so j j that Eq. (5.21) take into account the effect of diffraction. In the second, temporal case, m stands for time, d "­k /­u and c "!1 ­2k /­u2 describe group velocities and group-velocity dispersions, j j j j 2 j j respectively. For stationary solutions of Eq. (5.21), when the walk-off effect and the wave-vector mismatch are compensated due to the nonlinearity-induced phase-locking effect, the following transformation can be applied, E "w(i/J2ps s ) exp(ib z#iXm), E "u(i/s ) exp(ib z#2iXm), where b and 1 1 2 1 2 1 2 1

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b ,2b !Dk are the nonlinearity-induced shifts of the propagation constant, p"Dc D/Dc D, 2 1 1 2 i"b #d X#c X2, and X"(d !d )/2(2c !c ). Then, equations for w and u take the univer1 1 1 1 2 2 1 sal dimensionless form: i

­w ­2w #r !w#w*u"0 , ­f ­q2

ip

­u ­2u 1 #s !au# w2"0 , ­f ­q2 2

(5.22)

where f"iz, q"(DiD/Dc D)1@2(m!lz), l"(2c d !c d )/(2c !c ), r"sign(ic ), s"sign(ic ), 1 2 1 1 2 2 1 1 2 and a"(b #2d X#4c X2)p/i. 2 2 2 For large a, when the derivatives in the equation for u can be neglected, it is possible to show that u+w2/2a and Eq. (5.22) can be formally reduced to a single NLS equation for w (Werner and Drummond, 1993) for which dark solitons exist at r"!1. Similar results may be found in the case when the dispersion of the second harmonic vanishes (Werner and Drummond, 1994). Stationary solutions are described by Eq. (5.22) with f-derivatives omitted. The real functions w(q) and u(q) can be then considered as two coordinates in the corresponding mechanical problem with Hamiltonian H"rp2 /2#sp2/2#º(w, u), where p "dw/dq, p "du/dq, and º(w, u) is the w u w u two-dimensional potential, º(w, u)"1 (w2u!au2!w2). For r"!1 and large a, dark-soliton 2 solutions can be found in the form of the asymptotic series in a~1 (Buryak and Kivshar, 1994; Buryak, 1996) (J2Z!tanh Z) #O(a~3@2) , w"J2a tanh Z# Ja cosh2 Z

(5.23)

[5!4cosh2 Z#(Z/2)tanh Z] #O(a~2) , u"tanh2 Z# a cosh4 Z

(5.24)

where Z"q/J2. The asymptotic series, Eqs. (5.23) and (5.24), does not display any difference between the cases of normal and anomalous dispersions, i.e. between the cases s"1 and s"!1. However, as was pointed out by Buryak and Kivshar (1995a), the type of the stationary solutions of the system, Eq. (5.22), depends crucially on the sign of dispersion s. The importance of dispersion may be already seen in the analysis of modulational instability of two-wave interaction in a s(2) medium first by Buryak and Kivshar (1995b,c) and Trillo and Ferro (1995). Such cw solutions are given by the nonlinear stationary eigenmodes of s(2) media (Trillo et al., 1992; Kaplan, 1993), and here they appear as asymptotic tails of dark solitons which connect two cw solutions of the same amplitude but different phases. The analysis of the linear stability of the simplest cw solution w "$J2a, u "1 against modulations &exp(iqf#ioq) reveals two 0 0 branches for the dispersion relation (Buryak and Kivshar, 1995b; Buryak, 1996), q2 (o2)"[!B$JB2!D] , 1,2

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where B"[(so2#a)2#4ap#ro2p2(ro2#2)]/(2p2) , D"o2(2r#sa#o2)(o4#sao2!2rsa)/p2 . These two spectrum branches resemble ‘optical’ and ‘acoustic’ modes of a diatomic lattice and they exist due to relative and collective dynamics of w and v components, respectively. It is important that instabilities can appear due to both acoustic (as for a single NLS equation) and optical branches (parametric modulational instability). As was first found by Buryak and Kivshar (1995a) the cw solution can be modulationally stable only for the case r"1 and s"#1 for some p and a. Therefore, stable dark solitons of Eq. (5.22) can be expected only when dispersion coefficients of the fundamental and second harmonic modes have opposite signs, so that there exist no stable spatial dark solitons supported by v(2) nonlinearity. Indeed, in the case r"!1, s"#1 and positive a, a continuous family of dark one-soliton solutions has been found by Buryak and Kivshar (1995b). The dependence u(0) vs. a and two characteristic representatives of this family (with monotonous and nonmonotonous tails) are

Fig. 22. Characteristic profiles of two-wave dark solitons at a"1.0 and a"10.0. Dependence u(0) vs. a characterizes the whole continuous family of localized solutions found numerically whereas the dashed curve u(0)"1/a corresponds to the NLS asymptotic solution (Buryak and Kivshar, 1995b).

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shown in Fig. 22 where the dashed curve corresponds to the similar dependence u(0)"1/a found for the asymptotic solution, Eqs. (5.23) and (5.24). Dark solitons of this family exist for every positive value of a. However, the modulational instability analysis predicts that at r"!1 and s"#1 the cw background, which supports these dark solitons, can be stable only provided a'2 and p'p (p +1.689). Additionally, as was found by Buryak and Kivshar (1995b), the dark #3 #3 solitons may possess nonmonotonous tails if 2(a(8. The existence of the nonmonotony tails leads to important consequences for the soliton interaction in the adiabatic approach when dark solitons are treated as effective particles interacting through exponentially decaying forces, such nonmonotonous tails produce local minima in the effective interaction potential of weakly overlapping solitons, and, therefore, a dark soliton with nonmonotonous tails can trap another dark soliton creating multi-hole dark soliton bound states (Buryak and Kivshar, 1995c). For the case r"s"!1 the corresponding results are summarized in Fig. 23 (Buryak and Kivshar, 1995c). In this case, instead of a continuous family, there is only a discrete set of dark-soliton solutions. The simplest one (shown as a black circle) is the exact solution found by Hayata and Koshiba (1994): w(q)"$J2u(q)"$J2[1!(3/2)sech2 (q/2)], which may be

Fig. 23. Discrete set of stationary localized solutions of Eq. (5.22). Dashed curve u "!1/a is given by the asymptotic .*/ soliton of the NLS equation. The filled circle corresponds to the exact solution found by Werner and Drummond (1993) which are unstable (Buryak and Kivshar, 1995b).

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considered as a dark soliton of the lowest order. Importantly, as follows from the analysis presented above, all dark solitons of this type are modulationally unstable, and this result has been confirmed by direct numerical simulations. It is important to note that all solutions found in this second case (r"s"!1) are very different from each other, and each of them exists only at a particular value of a. This phenomenon was explained as being produced by a trapped radiation when a single dark soliton does not exist as a stationary object but two- (or multi-) soliton bound states are still possible provided some resonant conditions are satisfied [see Buryak and Kivshar (1995a) and Buryak (1995, 1996)].

6. Experimental verifications 6.1. Dark solitons in fibers 6.1.1. General remarks It is well known that optical bright solitons can be used for long-distance optical communications to drastically increase the bit rate of fiber transmission systems (see, e.g., Hasegawa (1989), Hasegawa and Kodama (1995) and Haus and Wong (1996)]. Similar uses have been proposed for dark solitons. The technical difficulties to be considered when dark solitons are used for long-distance optical communications can be split into two main areas, generation and modulation, and propagation, with concerns which overlap from one area to the other. Generation is less straightforward than for bright solitons because dark solitons require both a local change in phase as well as amplitude of the background wave (to avoid a large amount of radiation). Various techniques have been proposed and demonstrated for generating single or multiple dark soliton pulses. The first experiments involved either the generation of single dark solitons (Emplit et al., 1987; Weiner et al., 1988) or a pair of dark solitons (Kro¨kel et al., 1988) on a bright pulse background. Several authors studied the generation of quasi-continuous trains of dark solitons by colliding two bright pulses in an optical fibre (Rothenberg and Heinrich, 1992; Williams et al., 1994). Although it was suggested some years ago that continuous dark soliton pulse trains could be generated using electro-optic modulators (Zhao and Bourkoff, 1990), until recently such pulse trains had only been generated using either temporal shaping by high resolution spectral filtering (Haelterman and Emplit, 1993) or by beating together two cw signals in a dispersion tapered fibre (Richardson et al., 1994). For a long time there was no method for encoding the dark soliton train to form a data sequence. Recently, however, Nakazawa and Suzuki (1995a,b) used an electro-optic modulator as an effective technique for simultaneously generating and modulating a stream of dark solitons. Their impressive results will no doubt stimulate further the work in this direction. Propagation studies are implicit in much of the above work and include, for example, the observation of reshaping of even or odd symmetry dark pulses into grey or black dark solitons in the presence of positive group-velocity dispersion in optical fibres (Kro¨kel et al., 1988; Weiner et al., 1988). Raman scattering has been shown to lead to decay of the dark soliton pulses at high optical powers (Weiner et al., 1989), whilst recently Foursa and Emplit (1996a) used Raman gain to compensate for linear fibre losses encountered during dark soliton transmission. Observation

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of dark soliton propagation over distances relevant to telecommunications was reported by Emplit et al. (1993), whilst Nakazawa and Suzuki (1995b) recently demonstrated transmission of a 10Gbit/s pseudorandom dark soliton data train over 1200 km. Other interesting features of dark soliton propagation, not yet experimentally demonstrated, include their improved stability with respect to Gordon-Haus jitter (Hamaide et al., 1991; Kivshar et al., 1994a) and interactions over a longer length scale than bright solitons (Zhao and Bourkoff, 1989b). However, because the overall phase of the dark soliton rotates at twice the rate of the bright soliton phase, dark solitons are more sensitive to periodic perturbations than the equivalent bright soliton over the same absolute distance. There remain many open questions to be answered about temporal dark solitons. The main reason to study them is the amount of fibre in the ground optimized for transmission in the normal dispersion regime. Other reasons include the development of 1.3 lm amplifiers using praseodymium, and the fact that dark solitons can tolerate a smaller mark : space ratio than bright. This means that the average power needed to transmit bright and dark solitons can be of a similar order of magnitude. 6.1.2. Generation of dark solitons Although both bright and dark solitons in optical fibers were predicted in 1973 by Hasegawa and Tappert (1973a,b) almost simultaneously and bright solitons were already experimentally observed by Mollenauer, Stolen, and Gordon in 1980 (Mollenauer et al., 1980), dark solitons remained a mathematical curiosity until quite recently. This is because of the relative difficulty of generating a short ‘dark pulse’ on a continuous or long background pulse as a seed to the evolution of a dark soliton with well-defined phase. The first attempt to study dark pulse propagation in optical fibers experimentally was made by Emplit et al. (1987). They created odd-symmetry dark pulses using a spectral filtering technique from a beam containing a p-phase step. They observed that the dark pulses had properties similar to those of a fundamental dark soliton although their experimental results did not demonstrate dark soliton propagation because of the rather long dark pulses that were used (&5 ps at 600 nm). As a result, fiber losses could not be neglected because the characteristic length for soliton propagation (&220 m) was greater than the fiber attenuation length of &140 m. Thus, these experiments did not provide clear enough evidence of dark-soliton generation. Later Kro¨kel et al. (1988) provided the first experimental results on dark soliton propagation when they showed experimentally that an even dark pulse evolved into a symmetric pair of low-constant (small-amplitude) dark pulses that propagate unmistakably as solitons in accordance with earlier numerical results by Blow and Doran (1985) and a general theory of this pair generation developed by Gredeskul and Kivshar (1989a,b). In Kro¨kel et al. experiments 0.3 ps duration dark pulses were generated on a 100 ps duration 532 nm background pulse from a frequency doubled Nd : YAG laser by using a 15 mm long, optical Kerr effect fibre modulator. The pulses were launched into a 10 m single mode polarization preserving fiber and the output signals measured using the autocorrelator at different input power levels. At high powers ('9 W) the input pulse evolved into a pair of dark soliton pulses with opposite velocities relative to the background and these could be clearly distinguished from the structures produced during linear propagation. Their results were in very good agreement with numerical solutions of the NLS equation.

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The creation of a pair of dark solitons with opposite velocities and equal amplitudes as observed by Kro¨kel et al. (1988) is a simple consequence of the fact that the input pulse did not contain the phase jump needed to create a single dark soliton (see Section 2.4). Any input pulse with the same phase on its leading and trailing edge will lead to the creation of a symmetric pair of dark solitons (Gredeskul and Kivshar, 1989a). To create a single dark soliton, one needs to prepare a background pulse containing an appropriate phase jump as was achieved in the attempt by Emplit et al. (1987). The first successful experiments in which single dark soliton propagation was confirmed were reported by Weiner et al. (1988). Using a pulse tailoring technique that relies on spatial filtering within a standard grating decompressor they propagated 185 fs duration antisymmetric input dark pulses, which closely corresponded to the form of the fundamental dark soliton, through a 1.4 m length of single-mode optical fiber. Their experimental results are presented in Fig. 24a—e by dotted lines. Fig. 24a shows an intensity cross-correlation measurements of the input pulse. The duration of the central hole was 185 fs (intensity FWHM), and the background duration 1.76 ps FWHM. Cross-correlation traces of the output pulses from the 1.4 m fiber are plotted in Fig. 24b—e for various power levels. At the lowest power (1.5 W peak input power), propagation was almost linear. As the power was increased, the background pulse broadened and acquired a square profile because of the combined effects of nonlinearity and dispersion. Same time, the width of the output dark pulse decreased. At 300 W peak input power, the output dark pulse was of essentially the same duration as the input. Thus the dark pulse underwent soliton-like propagation and emerged from the fiber almost unchanged, even in the presence of significant broadening and chirping of the finite-duration background pulse. Computer solutions of the NLS

Fig. 24. Measured (dotted lines) and calculated (solid lines) cross-correlation data for the odd-symmetry dark pulse. (a) Input dark pulse. (b)—(e) Pulses emerging from the fiber for peak input power of (b) 1.5, (c) 52.5, (d) 150, and (e) 300 W (Weiner et al., 1988).

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equation, shown in Fig. 24a—e by solid curves, were in qualitative agreement with the experimental data. By changing the spatial mask in the decompressor, even symmetry pulses could be obtained. The results in this case are plotted in Fig. 25a—e. Fig. 25a shows a measurement of the input pulse, and cross-correlation measurements of the output pulses from the fiber are shown in Fig. 25b—e. The trends are similar to those reported by Kro¨kel et al. (1988). At low power the background pulse was reshaped by interference with the chirped, temporally broadened dark pulse. As the power was increased, one may observe the formation of two low-contrast holes, separated by +2.3 ps at 285 W peak input power. Again, the data agree closely with numerical solutions to the NLS equation shown as solid lines in Fig. 25a—e. Therefore, the experiments confirmed the crucial importance of the phase profile of the input pulse: an odd dark pulse propagates undistorted as a black soliton, while an even dark pulse splits into a pair of gray solitons. Nonlinear propagation of 5.3 ps odd-symmetry dark pulses through a 1 km long single-mode fiber at 850 nm was reported by Emplit et al. (1993). The choice of this wavelength allowed them to study dark soliton propagation in conditions more compatible with telecommunications. It was demonstrated that the soliton-like propagation is possible in a fiber 2.5 times longer than the soliton characteristic distance z despite a finite background and fiber loss. A quantitative 0 agreement with numerical simulations was found for both temporal and spectral measurements. Weiner et al. (1989) also reported the effect of temporal and spectral self-shifts of dark solitons in fibers. Such an effect is well-known for bright optical solitons [see, e.g., Hasegawa (1989) and Hasegawa and Kodama (1995)]. In the case of dark solitons these shifts become increasingly pronounced as the intensity and the fiber length are increased. As has been shown, the experimental

Fig. 25. The same as in Fig. 24, but for the even-symmetry dark pulse. (a) Input pulse. (b)—(e) Pulse emerging from the fiber for peak input power of (b) 2.5, (c) 50, (d) 150, and (e) 285 W (Weiner et al., 1988).

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data are in a good agreement with numerical simulations made in the framework of the modified NLS equation that includes the Raman contribution to the nonlinear refractive index. Analytical description of the effect was proposed by Kivshar (1990a,b) (see also Section 3.2.2 above where another version of the analytical approach is presented). The Raman effect was employed to advantage in the work by Foursa and Emplit (1996a) where Raman amplification was used to compensate for fibre losses. A highly Ge-doped single mode fibre, 395 m long, was used as the gain medium into which was launched a 2.7 ps long dark pulse imposed upon a 39 ps duration 883 nm background. The output pulses were analysed using a streak camera. A counter-propagating pump pulse at 850 nm with an average power of 190 mW was used to produce a Raman gain of up to 3 dB compensating fully the 6 dB/km absorption losses in the fibre. Although these single pulse propagation studies confirmed the basic properties of dark solitons, the generation of pulse trains is essential for application in optical communications. In initial experiments dark soliton trains were created by colliding two right pulses launched with a time delay between them into an optical fibre in conditions of normal dispersion (Rothenberg and Heinrich, 1992; Williams et al., 1994). This process for creating dark soliton trains involves three stages. In the first stage the two pulses broaden nonlinearity forming rectangular frequency-chirped pulses. The chirped pulses eventually broaden sufficiently that they start to overlap and linearly interfere forming a pulse with sinusoidal density modulation and alternating phase. Finally the nonlinearity acts on this modulation producing a train of dark solitons in a similar way to that which occurs during amplification (Dianov et al., 1989). The first experimental observation of this effect was reported by Rothenberg and Heinrich (1992) following the earlier numerical predictions of Rothenberg (1991). A 2 ps duration pulse from a dye laser was split into an input pulse pair using an interferometer and focussed into an optical fibre 100 m long. The output pulses were analysed using a streak camera. By varying the pulse separation and power the transition from nearly sinusoidal modulation to a characteristic dark soliton-like structure was observed. The later work of Williams et al. (1994) extended the process to create high repetition dark pulse trains (up to 60 GHz) and propagated these trains for distances up to 2 km. The generation of dark solitons on a background containing a noise component was reported by Grudinin et al. (1988). Such an experimental observation is related to the work by Gredeskul et al. (1990) [see also Gredeskul and Kivshar (1989a)] who demonstrated analytically that any dip (deterministic or random) on a cw background will evolve into at least one pair of dark solitons. The first high repetition rate continuous train of dark soliton pulses was generated by Richardson et al. (1994). In their work the CW beams from a pair of single frequency DFB lasers temperature tuned to have their frequencies separated by 100 GHz were focussed into a 1.5 km length of fibre with slowly decreasing normal dispersion. Dark soliton pulses 1.6 ps in duration were measured at the output of the fibre using an autocorrelator. The technique is similar to that already predicted (Dianov et al., 1989) and demonstrated (Chernikov et al., 1992) for the generation of bright soliton trains. So far the problem of encoding information onto dark soliton trains had remained unsolved. However, quite recently Nakazawa and Suzuki (1995b) demonstrated both encoding and detection of a pseudorandom data train of dark solitons. Fundamental to their work was the availability of a fast Lithium Niobate push—pull electro-optic modulator. An non-return-to-zero (NRZ) data

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pattern and clock pattern were combined in an ‘and’ gate. The ‘and’ data stream as then converted to a Q and QM outputs to drive the Mach—Zehnder interferometer using a T-flip-flop circuit, see Fig. 26a and Fig. 26b. Dark solitons about 50 ps long were generated. Detection (Fig. 27a and Fig. 27b) involved the separation of the incoming dark soliton train into two arms of Mach—Zehnder interferometer with a one-bit shift in time introduced between them at

Fig. 26. Generation scheme for pseudorandom binary sequence dark soliton train: (a) block diagram, (b) operating principle (Nakazawa and Suzuki, 1995b).

Fig. 27. Detection scheme for pseudorandom binary sequence dark soliton train using one-bit-shifting with MachZehnder interferometer: (a) block diagram, (b) operating principle (Nakazawa and Suzuki, 1995b).

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the output. Interference between these two signals created an inverted version of the original NRZ data train. In subsidiary experiments (Nakazawa and Suzuki, 1995a) propagated the dark soliton trains through dispersion shifted optical fibre demonstrating a 2.5 dB power penalty was required to maintain a BER or 10~10 after 1200 km. These last results clearly indicate that most of the basic issues involved with the generation, encoding and detection of dark soliton pulse trains can be overcome although higher bit rates will require more sophisticated equipment. Hence, the basic possibility of using dark soliton pulse transmission in normally dispersive fibre has now been established. 6.1.3. Interaction between dark solitons The first experimental investigation of the interaction between black and grey solitons in an optical fiber has been recently reported by Foursa and Emplit (1996b). Experiments were carried out using a 1 km single-mode fiber with an effective core area of 28 lm2 and dispersion of 90 ps/nm/km. The fiber length equaled to approximately 7 soliton distances. A pair of dark solitons with adjustable ‘blackness’ were generated using a pair of phase plates inserted in the frequency dispersed part of the beam within a grating decompressor. The coupled signal power slightly exceeded the level of the fundamental dark soliton in order to partially compensate for the fiber losses. The output pulses were analysed with a streak camera with the interaction between the pair of solitons investigated by measuring the change in arrival time relative to the interaction-free case. The streak camera images shown in Fig. 28a—c show temporal profiles of various inputs (left) and the fiber outputs (right). In Fig. 28a a single grey pulse with the grayness parameter B2"0.78 (see Section 2.3) was observed to walk from the leading edge (!10 ps) to the trailing

Fig. 28. Streak camera images of the pulses at the shaping output (left) and the fiber output (right) for the grayness parameter B2"0.78. (a) Single gray pulse, (b) single black pulse, and (c) gray and black pulse pair. Black solitons are marked by solid cursors and gray pulses at the fiber output by dotted lines (Foursa and Emplit, 1996b).

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edge (#4 ps) of the background pulse. At the same time the carrier pulse lengthened in time because of the effects of dispersion and self-phase modulation. The soliton also became broader after propagation because the fiber losses decrease the amplitude of the background pulse. Fig. 28 shows the case of a black soliton positioned originally at the centre of the background pulse where no relative motion was expected or observed. The temporal measurements when both dark and gray solitons were present at the same time, showed the following behaviour. The gray soliton walked through the black pulse during propagation through the fibre but as a result of the interaction its walk off increased by 3 ps (compare Fig. 28a and Fig. 28c). At the same time dark pulse also changes its position towards the leading edge of the carrier pulse by 2 ps (see Fig. 28b and Fig. 28c). These results provided evidence of the basic repulsive nature of the interaction between the coherent dark solitons and were well reproduced by numerical simulations based on the scalar cubic NLS equation carried out by Foursa and Emplit (1996b) [see also Thurston and Weiner (1991) and Diankov and Uzunov (1995)]. 6.2. Spatial dark solitons 6.2.1. General remarks Experimental studies of bright spatial solitons in self-focussing glasses and semiconductors had already demonstrated the main features predicted by the cubic NLS equation (Aitchinson et al., 1990; Reynaud and Barthelemy, 1990) before any experimental work on dark spatial solitons had been attempted. The first experimental investigations of dark spatial solitons in self-defocusing media were performed almost simultaneously for a bulk medium, by Andersen et al. (1990) and Swartzlander et al. (1991), and for planar waveguides, by Allan et al. (1991) and Skinner et al. (1991). Andersen et al. (1990) and Swartzlander et al. (1991) reported the generation of dark soliton stripes, grids and crosses on the transverse cross-section of an optical beam propagating through a bulk self-defocusing medium. Although it is well known that such quasi-one-dimensional structures should suffer from a transverse modulational instability and hence do not exist as stable analytical solutions for dark solitons in the (2#1)-dimensional NLS equation, the experimental and numerical data presented in these papers using various two-dimensional amplitude and phase masks provided strong evidence that the observed phenomenon were indeed due to the creation of dark spatial solitons. A number of experimental factors contributed to the absence of the long wavelength transverse modulational instability in these experiments. Firstly, the moderate beam intensities led to stabilization due to the finite size of the laser beam (that is the fastest growing unstable mode had a period much larger than the beam size). Secondly, in experiments using thermal media (Swartzlander et al., 1991), the diffusive nature of the nonlinearity led to stabilization, the effect known also to suppress collapse of light beam [see, e.g., Turitsyn (1985) and Suter and Blasberg (1993)]. The transverse modulational instability will not, however, always be suppressed. Quite recently (Mamaev et al., 1996a; Tikhonenko et al., 1996a) it was observed in experiments where higher nonlinearity with an essentially local response was available. The instability led to a snake like distortion of the soliton stripe and eventually breakup to form pairs of optical vortex solitons (Tikhonenko et al., 1996a).

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By analogy with the case of grey temporal solitons, grey spatial solitons have a finite transverse velocity relative to the background beam that supports them. A convenient way of understanding this motion is to recognize that a nonzero on-axis intensity can only occur if a travelling crosses the soliton. Since the background is a plane wave (i.e., it is characterized by a single k-vector) then the soliton must propagate at some angle to this background wave to generate the travelling wave component. The direction of the soliton is determined by its grayness which in turn is determined by the change of phase across the soliton. It follows that adjusting the phase change across the soliton allows it to be scanned relative to the background wave. This was first demonstrated by Luther-Davies and Yang (1992b) for single dark solitons and later by Bosshard et al. (1994) and Mamyshev et al. (1994) for an array of dark solitons [see also West and Kennedy (1993) where dark soliton trains were investigated analytically with the help of the inverse scattering transform]. The ability to change the propagation direction of a dark spatial soliton is useful when they are considered for use in optical switching. Although large number of concepts for optical switching have been suggested, reconfigurable light induced waveguides is of particular interest. Spatial solitons can be thought of as self-guided waves, that is they ‘write’ a waveguide into the nonlinear material in which they are created and propagate as modes of that waveguide. Bright solitons are bound modes of the waveguides they induce whilst dark solitons are reflectionless radiation modes (Snyder et al. (1993), see also Section 2.6). As a result of this concept, quite a lot of attention has been paid to ‘writing’ ideal structured waveguides into nonlinear materials using dark spatial solitons. The creation of a pair of grey solitons from an amplitude hole in a pulse is well-known from the results on temporal solitons. In the spatial domain a pair of solitons which emerge from a square intensity dip in an input beam provides an optical structure similar to a waveguide ½-junction. This was first demonstrated by Luther-Davies and Yang (1992a) and recently by Z. Chen et al. (1996a) and Taya et al. (1996) in photorefractive materials. Another interesting property of dark spatial solitons is their behaviour during collisions. Numerical simulations have demonstrated that the soliton induced X-junctions created by such collisions form lossless transmission junctions (Luther-Davies and Yang, 1996a). Experimentally the X-junction is difficult to realize in materials with diffusive nonlinearities but recently they have been observed in photorefractive materials (Segev et al., 1996). Dark and bright solitons in photorefractive media have attracted considerable attention in recent years (Valley et al., 1994; Taya et al., 1995; Iturbe-Castillo et al., 1995; Taya et al., 1996; Z. Chen et al., 1996a,b; Segev et al., 1996). The work on photorefractive solitons will be dealt with in one of the following sections. 6.2.2. Dark soliton stripes and grids A report of the first experiments on dark spatial solitons were published by Swartzlander et al. (1991). A cw frequency-stabilized dye laser with a beam of power P &100 mW was passed */ through a wire mesh and then imaged with a lens into a ¸"18 mm-long cell containing &1012 atoms/cm3 of sodium vapor. By tuning the laser below the atomic D resonance, a strong 2 defocussing response with an effective nonlinearity n up to K!3]10~7 cm2/W was obtained. 2 Far-field intensity profiles were recorded at the distance of '1 m from the output of the cell. When the laser was detuned far from the resonance the normal Fraunhofer pattern due to linear

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diffraction from the grid was observed. However as the D resonance was approached this pattern 2 underwent a remarkable transformation to form a well-organised array of square dots. Numerical simulations of the NLS equation showed that the formation of regular patterns of dark stripes within the background field was a universal phenomenon, most likely associated with the production of multiple dark spatial solitons. From these far-field patterns the authors observed that these (2#1)-dimensional dark stripes behaved very similarly to the analytical (1#1)-D dark solitons. To confirm that dark solitons were indeed being created, a further series of experiments was undertaken this time using a weakly absorbing liquid as a thermally defocussing nonlinear material. Near field images at the output of the medium were recorded at high power (nonlinear propagation) and low power (linear propagation). The use of either an amplitude mask (a wire cross) or a p phase mask (again in the form of a cross) led respectively to the creation of pairs of or single crossed dark soliton-like stripes on a uniform background in the nonlinear regime. The transverse velocity of the pair of stripes generated from the amplitude mask varied with the width of the wire in the manner predicted for (1#1)-D dark spatial solitons. A similar result had been published by Andersen et al. (1990) from experiments where a single wire was used to mask an input beam to a thermally defocussing nonlinear medium and multiple (higher order) dark soliton stripes with transverse velocities predictable from the theory of Zakharov and Shabat (1973), were created. Numerical study of controllable branching of optical beams by dark soliton stripes depending on a proper choice of the initial phase profile and the width of the crossed dark stripes and the background beam intensity was analyzed by Neshev et al. (1997b). Although such dark-soliton stripes should be unstable to transverse modulations (see Section 7.2 below), the experimental data described above and numerical simulations carried out for a Kerr medium did not show any manifestation of this instability. The natural explanation is that the finite-width beam used by Andersen et al. (1990) and Swartzlander et al. (1991) prevented the development of long-wavelength instabilities. Nevertheless, for stronger light intensities stripe breakup should be observed with the subsequent generation of vortex paris, as we describe below in Section 7.4.1. Almost simultaneously with the work of Swartzlander et al. (1991), there appeared two reports of dark soliton generation in ZnSe bulk semiconductors (Allan et al., 1991; Skinner et al., 1991). In these experiments 30 ps duration pulses from a frequency doubled Nd : YAG laser were passed through phase or amplitude masks into a 5]5]2 mm single crystal of ZnSe. ZnSe has an instantaneous defocussing component of the nonlinear refractive index at j"532 nm, due to the dispersive change associated with two-photon absorption. The output from the crystal was imaged onto the slit of a streak camera aligned orthogonal to the axis of the expected dark soliton stripe to provide time resolution. Interferograms of the beam emerging from the crystal were also recorded. When a p phase mask was used the streak records showed intensity dependent narrowing of the central dark zone accompanied by defocussing of the background indicating the creation of a dark soliton. Allan et al. (1991) were able to confirm that the soliton constant (the product of the square of the width and the peak dark irradiance) was conserved in their experiments. Interferograms of the output beam for p phase or amplitude masks showed the phase structures expected for black and grey solitons respectively. The results thus gave quantitative evidence of the fact that the irradiance minimum propagates as a dark spatial soliton.

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6.2.3. Dark soliton induced waveguides All spatial solitons are self-guided waves and hence it is useful to think of them as optical modes of the waveguides they induce in the nonlinear medium [see, e.g., Snyder et al. (1993)]. Using this concept it is immediately clear that soliton induced waveguides could be used as light guides for co-propagating beams (De la Fuente et al., 1991). In the case of dark spatial solitons the guided beam is localized as a bound mode within the dark zone of the background wave. When the power in the guided mode is infinitesimally small, it does not perturb the dark soliton, whereas at high powers it is possible for the two waves to form a bound dark—bright soliton pair (see Sections 5.1 and 6.3). The first results on dark soliton induced waveguides with co-propagating guided modes were reported by Luther-Davies and Yang (1992a,b). In those experiments the beam from a multiline argon-ion laser was passed through a dispersing prism to separate the different wavelengths. The 514 nm radiation passed unfocused through a polarizing beam splitter into a 10 cm long cell containing a thermally nonlinear liquid. The output beam from the cell was imaged onto a screen by using a 50 mm focal-length lens and a CCD camera and a frame grabber used to capture the images. The on-axis beam intensity at the input to the cell was +30 W/cm2 for 514 nm powers of 1 W. The 476 nm argon line was routed by an attenuator and injected into the nonlinear medium through the polarizing beam splitter coaxial with the 514 nm beam. The probe beam was focused to a spot size suitable for launching into the various soliton structures. Two types of perturbation were used to create structured dark soliton induced waveguides using the 514 nm beam. The first was a simple p phase jump formed by evaporating a 530 nm thick SiO 2 layer across one half of a glass slide. The phase jump was positioned across the middle of the 514 nm beam at different distances from the input to the cell. The separation of mask and cell determined the width of the zone over which the p phase change occurred in the input field by varying the amount of linear diffraction. When the width of this zone was much greater than the width of the dark soliton in the medium, a transition region was created as the dark soliton formed, and this region had the shape of a tapered waveguide. The experimental data, supported by numerical simulations, showed that this tapered waveguide was able to adiabatically couple an input beam launched coaxial with the dark zone at the input into a bound mode of the dark soliton induced waveguide, thus forming an ideal adiabatic taper. In fact it is generally the case that structured waveguides formed by the creation or interaction between spatial solitons have almost ideal optical properties. This is qualitatively due to the absence of abrupt interfaces in these light induced structures. A second type of input perturbation was also used: an amplitude jump formed by placing a fine ("40 lm) diameter wire across the beam 1 cm from the input to the nonlinear medium. In this case, the lowest-order soliton solution corresponds to the formation of a diverging pair of dark solitons. However the solitons only form some distance from the input and the overall structure linking the pair of diverging dark solitons to the single dark input stripe is equivalent to a waveguide ½-junction. Numerical simulations showed that the ½-junction should be essentially lossless, and this was supported by the experimental results. The experimental measurements of the input and output fields from the ½-junction are shown in Fig. 29. Note that the use of dark soliton stripes in this experiment meant that divergence in the vertical direction was not controlled. The overall transmission of the ½-junction structure was '85% and it is possible this figure to be further improved by more careful attention to the launch conditions.

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Fig. 29. Experimental beam profiles: (a) the input soliton-forming-beam pattern, (b) the output dark-soliton pattern, (c) the input probe-beam pattern, (d) the output probe-beam pattern (Luther-Davies and Yang, 1992b).

A useful feature of soliton induced structures is that they can be reconfigured by changing the launch conditions. An important example which creates steerable waveguides is based on the use of grey solitons whose phase can be controlled. It is well known that dark solitons propagate at an angle, determined by their phase, to the plane wave background in which they are embedded. Luther-Davies and Yang (1992b) demonstrated that an adjustable phase mask within the input beam could, therefore, be used to scan the position of a dark soliton at the output of a nonlinear medium over several soliton widths (limited only by the available nonlinear change in the refractive index and the propagation distance). It was also pointed out that different dark soliton channels could cross (forming X-junctions) and that the signals propagating in those channels would pass without loss across those junctions. This would in principle allow a large number of cross-connects between an array of input and output channels without cross talk. The theory of the dark solitons arrays was developed by West and Kennedy (1993) who used the technique of the inverse scalleting transform [see also, Swartzlander (1992) and Lundquist et al. (1995), where the multiple dark soliton generation was also discussed]. Experimentally, the generation and steering of arrays of dark solitons and the beams guided within them was reported by Mamyshev et al. (1994) and Bosshard et al. (1994). To create the dark soliton array the principal proposed for generating dark soliton pulse trains in optical fibres by beating two laser frequencies in the presence of adiabatic amplification was extended to the spatial case. A beam from an Argon ion laser was spatially filtered and expanded before being split into two beams that were directed at different angles into a nonlinear medium to form an interference pattern with a period of 100 lm. A cylindrical lens was used to focus the beam along the axis parallel to the fringes, thereby increasing the beam intensity as a function of propagation distance through the nonlinear medium and providing effective amplification. The experimental data confirmed that this procedure generated the new spatial structures in the far field pattern of the emerging beam which were essential for dark soliton creation and good agreement was observed between the experiments and numerical solution of the (1#1)-dimensional NLS equation. It was demonstrated that these soliton arrays could guide a co-propagating beam as a bound mode of the array. Steering of the

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array as reported by Bosshard et al. (1994) required only a relative change in the intensity of the two beams. Although induced X-junctions based on collision of dark solitons are also of potential importance, only results from simplified models (Sheppard, 1993; Miller, 1996) or numerical simulations (Sheppard, 1993; Yang et al., 1993) have so far been reported. A major reason for this has been the reliance for the creation of dark spatial solitons on nonlinear media with a diffusive nature where X-junctions do not appear to form. 6.2.4. Dark solitons in photorefractive materials In the past two years, interest in spatial solitons has increased markedly through the use of photorefractive nonlinear materials. For the first time, dark solitons were mentioned in connection with the photorefractive effect by Belanger and Mathieu (1987) who used the concept of dark solitons for a qualitative explanation of the optical branching effect observed by Jerominek et al. (1985, 1986) in Ti : LiNbO slab waveguide. However, that time the experimental results were not 3 sufficient to make that explanation more convincing. Now, it is well established that in these materials, three different types of photorefractive response can lead to the beam self-trapping and localized structures. The first type of self-localization occurs when diffraction is suppressed by two-wave mixing involving refractive index grating within the photorefractive material in the presence of an applied field. The shift in the grating phase due to the external field provides phase coupling between incident and scattering waves necessary to compensate for diffraction (note that in the absence of the field two-wave mixing leads to energy coupling which cannot compensate for diffraction). The effect is transient (lasting a fraction of a second to several seconds depending on the materials properties) since the external field will eventually be screened by the background conductivity. The effective nonlinearity in this situation is nonlocal, and can be changed from self-focussing to defocussing by changing the direction of the externally applied field. An important property of the corresponding self-localized beams is their independence of the absolute light intensity (Segev et al., 1992; Crosignani et al., 1993; Duree et al., 1993). The second and third types of photorefractive response give a birth to spatial solitons known as screening solitons (Segev et al., 1994a,b) and photovoltaic solitons (Valley et al., 1994), respectively, result from nonuniform screening (the former) or from photovoltaic fields (the latter) which appear in the crystal in steady-state conditions. Unlike the first type of photorefractive soliton, the effective nonlinearity is local and hence the shape and width of screening or photovoltaic solitons depend on intensity. It is a characteristic of photorefractive solitons that they can generally be modelled using a conventional NLS model with a saturating nonlinearity [see, e.g., Christodoulides and Carvalho (1995) and Segev et al. (1996)]. The first experimental observation of photorefractive dark solitons of the nonlocal type was reported by Duree et al. (1995). To observe planar dark solitons, they launched a dark notch associated with a p phase jump in the center of an Argon ion laser beam into a rhodium-doped strontium barium niobate [SBN] crystal. An external electric field of !400 V/cm was applied parallel to the c-axis which also corresponded to the trapping direction. In the absence of the field the 21 lm wide notch at the input diffracted to form a 35 lm wide dark region at the exit face of the crystal. With the field turned on, diffraction was fully compensated and defocussing of the background also occurred. An important signature of the nonlocal photorefractive solitons is their insensitivity to the absolute light intensity. To verify this the authors varied the input power over

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2 orders of magnitude, from 3 to 300 lW (intensities of 0.3—30 W/cm2), and observe no change in the shape or the size of the dark soliton (Duree et al., 1995). In the same paper, the observation of vortex dark solitons was also reported (see Section 7.4). Screening solitons were first reported by Iturbe-Castillo et al. (1995). The physical process involved is best understood by considering a narrow notch in an otherwise uniform infinite plane wave propagating in a biassed photorefractive medium. In the illuminated regions the conductivity increases and the resistivity decreases. As a result, the voltage drop occurs primarily across the dark region leading to a local increase in the field and a corresponding local change in the refractive index via the pockels effect. Iturbe-Castillo et al. (1995) reported the creation of dark soliton stripes about 20 lm wide when a barium titanate [BTO] crystal was illuminated with He—Ne laser beam in the presence of an applied field up to 7.5 kV/cm and an incoherent background beam. They also demonstrated probe beam guidance by the photorefractive-soliton-induced waveguide. In fact it has been a common theme in experiments on photorefractive solitons to demonstrate their use as light induced waveguides. This is in part due to the fact that the photorefractive response is rather slow and leads to quasipermanent structures. This is particularly true in the case of photovoltaic spatial solitons where illumination of the nonlinear medium leads to photovoltaic currents which transport charge away from the illuminated region preferentially along the c-axis of a ferroelectric crystal. Photovoltaic solitons only exist when there is a component of the intensity gradient along the c-axis. Since the physical mechanism for soliton generation involves the separation and trapping of charge, the index perturbation persists in the dark and may be useful in creating semipermanent waveguides for wavelengths where the material is insensitive to light. Recently, planar dark solitons due to the bulk photovoltaic effect in lithium niobate were observed (Taya et al., 1995). A 488 nm beam from an argon ion laser containing a p phase step illuminated a nominally undoped sample of lithium niobate at low power level (20 mW) normal to the crystal c-axis. At intensities of the order of 10 W/cm2 dark solitons with widths of approximately 20 lm formed after 15 min of exposure. It was demonstrated that solitons only formed when the intensity gradient was parallel to the c-axis as predicted and the induced waveguides could trap 514 nm radiation as a bound mode. These photovoltaic waveguides lasted up to 39 h after the soliton forming beam was removed. In recent work (Taya et al., 1996) also demonstrated the formation of a waveguide ½-junction using photovoltaic solitons. Much of the most recent experiments with photorefractive materials has concentrated on screening solitons (Z. Chen et al., 1996a,d; Mamaev et al., 1996a—c). Uniform illumination to the edges of the crystal is important in creating the localized screening region and this can be achieved either by the use of an incoherent background illumination (Iturbe-Castillo et al., 1995) or by extending the soliton background field to cover the full crystal aperture (Z. Chen et al., 1996a). Most recently screening solitons have been used to demonstrate the creation of waveguide ½-junctions and higher order solitons (Z. Chen et al., 1996b), dark-bright solitons pairs (Z. Chen et al., 1996d), the transverse breakup of dark soliton stripes (Mamaev et al., 1996a,b) and instability and splitting of vortex solitons (Mamaev et al., 1996c). Many exciting results have now been reported using photorefractive materials. The main impediment to applications in photonics is the current switching speed which in many cases is limited by the response time of the material to several seconds or longer. Some prospects exist to meet that challenge and in some materials response speeds can be improved dramatically by changing carrier and trap densities, whilst retaining the high photorefractive sensitivity.

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6.3. Coupled dark—bright solitons Experimental observation of the simultaneous propagation of dark and bright solitons of different wavelength (1064 and 532 nm) interacting due to the cross-phase modulation effect in a Kerr medium of the focusing type has been reported by Shalaby and Barthelemy (1992). Dark pulse was created by reshaping of a longer pulse, and both pulses with parallel polarizations were launched in a cell of 15 mm length filled with CS . The propagation distance corresponded 2 approximately to four Rayleigh lengths associated with the bright wave. The intensity ratio between the two soliton beams was varied by rotating a help wave plate followed by a polarizer in the infrared path. When the dark infrared beam and the bright visible beam are simultaneously launched, the interaction manifests itself by both the narrowing of the central hole in the exiting infrared profile and the reshaping of the green output into a fundamental soliton like wave. The experimental results were compared to the numerical simulations of the linear and nonlinear propagation, with a qualitatively good agreement. The dark beam launched alone was completely destroyed during its propagation in a focusing medium. The first observation of incoherently coupled dark—bright spatial soliton pairs in a biased bulk strontium barium niobate photorefractive crystal has been reported by Z. Chen et al. (1996d). This kind of solitary waves have been predicted in the context of photorefractive materials by Christodoulides et al. (1996) where it was shown that a coupled bright—dark soliton pair can propagate in a biased photorefractive crystal provided that the two pairing beams share the same polarization and wavelength and are mutually incoherent. As a matter of fact, such a soliton pair involves two steady-state photorefractive screening solitons which propagate collinearly in the crystal and experience a refractive-index modulation induced by both beams. The coupled soliton pair can be in dark—bright as well as bright—bright and dark—dark realizations. A stable dark—bright photorefractive soliton pair can only be realized using a self-defocussing nonlinearity and when the peak intensity of dark component is higher than that of the bright component. In earlier experiments (Z. Chen et al., 1996c) the coupling and decoupling between two bright screening solitons was demonstrated. First, Z. Chen et al. (1996d) generated a coupled dark—bright soliton pair, see Fig. 30a and Fig. 30b. The width of dark notch was 14 mm and that of the bright beam was 11 mm, see Fig. 30a.

Fig. 30. Photographs showing a coupled fundamental dark—bright soliton pair: (a) input, (b) output (normal diffraction), and (c) output coupled soliton pair (Z. Chen et al., 1996d).

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Fig. 31. Photographs showing (a) the output dark and (b) the bright beams when the pair is decoupled. (c) Bright beam guided in the triple-soliton induced waveguide (Z. Chen et al., 1996d).

Without the external field, each beam diffracts after 5 mm of propagation in the crystal, as shown in Fig. 30b. By applying a voltage of !400 V (negative relative to the c-axis) between the two electrodes (spaced by 4.5 mm), Z. Chen et al. (1996c) observed that the output beams coupled to a steady-state dark—bright soliton pair. They monitored the bright and dark components of the soliton pair by blocking one and sampling the other within a time interval of 0.1 s, so the refractive index modulation induced by the coupled soliton pair cannot respond in such a short time to the change due to blocking of one beam. Fig. 30c shows the photographs of the output dark and bright components taken by Z. Chen et al. (1996d) immediately (less than 0.1 s) after the pairing beam is blocked. In this way, it is possible to distinguish between the components despite that they have the same frequency and polarization. The two beams are coupled to form a fundamental soliton pair. Z. Chen et al. (1996d) presented also an elegant proof that each of the two beams cannot maintain the form of a fundamental soliton when they are decoupled, i.e., when one of the beams is blocked and a new steady-state is reached. Fig. 31a and Fig. 31b show photographs of the dark and bright beam, respectively, taken after the pairing beam is blocked for a time (1 min) much longer than the crystal response time. Since the polarity of the applied voltage is not appropriate for the bright soliton, the bright beam alone diffracts and experiences self-defocussing, see Fig. 31b. In the case of dark beam alone, the applied voltage is too high to maintain the fundamental dark soliton without the presence of the bright beam. The dark beam, instead, evolves into a triplesoliton structure shown in Fig. 31a when the bright beam is absent and all other experimental conditions remain unchanged. Interestingly, as Z. Chen et al. (1996d) unblocked the bright beam, guidance of the bright beam into the triple channels (waveguide) induced by the high-order dark soliton was observed (Fig. 31c). This waveguide persisted for a few seconds before the bright beam gets defocussed by the bias field. Eventually, a fundamental dark soliton was retrieved without the bright beam by readjusting the voltage to !250 V.

7. Dark solitons in higher dimensions 7.1. Introductory remarks In the waveguide geometry light beams are confined by a slab waveguide and, with a good accuracy, guided waves can be regarded as (1#1)-dimensional objects. However, in a bulk medium there exists one more, transverse, direction and therefore the concept of solitary waves should be extended to higher dimensions. The simplest way to do this is just to consider quasi-one-dimensional dark solitons in a bulk medium. This idea leads to different kinds of

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superpositions of such plane solitary waves, including grids and crosses of dark solitons (Swartzlander et al., 1991), a superposition of bright and dark plane solitons (Hayata and Koshiba, 1993a), etc. In spite of the fact that such plane solitary waves are expected to be similar to (1#1)-dimensional solitons, the freedom to extend to the other direction brings a completely new physics: It is generally believed that all kinds of plane solitary waves are unstable to transverse modulation with a finite-width instability band [see, e.g., Kuznetsov et al. (1986) and references therein]. In the case of dark solitons, the linear stability analysis was first developed for a defocussing Kerr medium (i.e., for the cubic NLS equation) by Kuznetsov and Turitsyn (1988) who demonstrated that plane dark solitons (i.e., dark soliton stripes), are unstable to transverse long-wavelength modulations. This instability can be suppressed provided a stripe is bent to form a loop (or ring dark soliton) with the radius less than the smallest possible wavelength of the instability domain (Kivshar and Yang, 1994c). However, such a ring dark soliton cannot exist as a stationary object, and it expands or collapses depending on initial conditions. An important physical question then is: ¼hat kind of stable stationary structures can exist in the (2#1)-dimensional geometry? In the focussing case, it is known that transverse instability of a plane, (1#1)-dimensional bright soliton leads to the creation of (2#1)-dimensional bright solitons of circular symmetry that can be stable in a non-Kerr bulk medium [see, e.g., Kuznetsov et al. (1986)]. A similar scenario is expected for dark solitons. Indeed, numerical simulations (Law and Swartzlander, 1993; McDonald et al., 1993; Josserand and Pomeau, 1995; Tikhonenko et al., 1996a; Mamaev et al., 1996a,b), asymptotic analytical theory (Pelinovsky et al., 1995), and recent experimental results (Tikhonenko et al., 1996a; Mamaev et al., 1996a,b) have demonstrated that the transverse instability of plane dark solitons leads to a generation of pairs of optical vortex solitons with alternate polarities. Optical vortex solitons are the only stable (2#1)-dimensional stationary structures which have been reported up to now to exist in a bulk optical medium with the nonlinear defocussing refractive index. In the context of optics, the vortex solitons have been predicted theoretically by Snyder et al. (1992) as ‘stable black self-guided beams of circular symmetry’. As a matter of fact, these objects have been known much earlier, since the pioneering paper by Pitaevsky (1961) [see also Ginzburg and Pitaevsky (1958)], as topological excitations of an imperfect Bose gas in the theory of superfluids. Experimental observation of optical vortex solitons has been already reported by several groups and in the following sections we discuss generation, properties, and experimental observation of optical vortex solitons. Furthermore, vortex solitons themselves can be generalized to include polarization properties of light. There exist two types of such generalization. First, we can allow the field of different polarizations to have a vortex-like structure. This kind of double vortex solitons is known in other fields [see, e.g., Perivolaropoulos (1993) and Pismen (1994a,b) and references therein] and it was recently introduced in nonlinear optics (Law and Swartzlander, 1994; Velchev et al., 1996). Second, a vortex of one polarization can guide the field of the other polarization giving rise to the (2#1)-dimensional generalization of the bright—dark solitons discussed in Section 5.1 [see, e.g., Pismen (1994b) and Sheppard and Haelterman (1994)]. To conclude this introductory section, we would like to mention that the (2#1)-dimensional beam propagation can be generalized to include one more dimension. This is the case when we allow nonstationary beam propagation and introduce time as an extra variable. In fact, this

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extension leads to the concept of light bullets introduced by Silberberg (1990b) for the (3#1)dimensional generalization of bright solitons. In the case of dark solitons, a vortex soliton in the (3#1)-dimensional space (i.e. two transverse and one propagation coordinates plus time) forms the so-called vortex line which itself is unstable to transverse modulations (Kuznetsov and Rasmussen, 1995). It is expected that this instability should lead to the formation of ‘dark light bullets’ which are known to exist in the anomalous group-velocity dispersion regime (Y. Chen and Atai, 1995) [while for the normal dispersion the situation is far from clear, even for a focussing medium, e.g., Chernev and Petrov (1992)]. These interesting objects still require the further analysis of their structure, stability, and interaction. A scenario of the decay of a vortex line as well as the existence of localized multidimensional waves in the case of the normal dispersion still remain unknown. 7.2. Transverse instability of plane solitons In the case of two transverse dimensions, i.e., for a stationary propagation in a bulk medium, dark solitons can be observed experimentally as dark stripes or grids with the properties similar to those of one-dimensional dark solitons (see Section 6.2.2 above). However, the linear stability analysis shows that a plane dark soliton is unstable to transverse long-wavelength modulations [Kuznetsov and Turitsyn (1988); see also Kuznetsov and Rasmussen (1995)]. Numerical calculations show that, as a result of the development of this instability, a dark stripe may decay into a sequence of optical vortex solitons of alternative polarities [e.g., Law and Swartzlander (1993), McDonald et al. (1993), Tikhonenko et al. (1996a) and Mamaev et al. (1996a,b)]. From the mathematical point of view, the problems mentioned above can be described in the framework of the cubic NLS equation with a transverse coordinate y included, i

­u 1 # + 2u#(1!DuD2)u"0 , ­z 2

(7.1)

where the dimensionless, slowly varying field amplitude u has nonzero asymptotics, DuDP1 for x, yP$R, and the vector operator + is defined as +"(­/­x, ­/­y), so that + 2,+ ) +. As has been shown in Section 2.3, the NLS Eq. (7.1) has an exact one-dimensional solution describing a dark soliton stripe parallel to the y-axis, which we write here in the following form: u (x, z)"k tanh[k(x!vz)]#iv , 4

(7.2)

where the amplitude parameter k (0(k(1) and the transverse velocity v (v241) are coupled through the relation k2#v2"1. According to Kuznetsov and Turitsyn (1988), the plane dark soliton, Eq. (7.2), is unstable against transverse perturbations &cos(py) with the wave numbers p(p (k) where #3 p2 (k)"k2!2#2Jk4!k2#1 . #3

(7.3)

In the parameter plane (p, k), the instability domain is bounded by the curve p"p (k), so that if we #3 apply a periodic perturbation with any wave number p(p (k), the amplitude of a plane dark #3

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soliton, i.e., a dark-soliton stripe, will grow exponentially in the transverse direction according to the linear stability analysis. Nonlinear regimes of such an instability have been investigated numerically [e.g., Law and Swartzlander (1993) and McDonald et al. (1993)] and also analytically, by means of the asymptotic technique valid near the threshold of the soliton instability (Pelinovsky et al., 1995). In particular, it was demonstrated that a plane dark soliton may decay into a chain of optical vortices of the opposite polarities. This kind of the instability-induced evolution, as was shown by Pelinovsky et al. (1995), is one of the main scenarios of the instability of a plane black soliton, whereas an unstable plane grey soliton may not decay into vortices, instead it displays long-lived oscillations accompanying by emission of linear waves propagating along the background. Recently, the transverse instability of a plane dark soliton in a strongly saturable optical medium has been studied analytically and numerically by Kivshar et al. (1997b) in the framework of the generalized NLS equation. By employing an asymptotic expansion technique for perturbations with small wavenumbers, Kivshar et al. (1997b) derived an analytical expression for the growth rate j of the transverse modulations (p2;1), M ­P 1@2 4 H j2 "a2p2 , a" , (7.4) M 4 ­v

CA B D

where P (v) and H (v) are the renormalized invariants, momentum and Hamiltonian, introduced in 4 4 Section 2.5. The index ‘s’ stands for those values calculated for the exact (1#1)-dimensional solution for a plane dark soliton. The analytical result (7.4) and numerical simulations carried out by Kivshar et al. (1997b) display the main effect of the nonlinearity saturation: it leads to an effective suppression of the dark-soliton transverse instability. Fig. 32 presents the numerically calculated maximum growth rate (j ) of the transverse instability of a black (i.e., the dark soliton M .!9 at v"0) plane soliton in a standard model of saturable nonlinearity which clearly displays the suppression of instability.

Fig. 32. Maximum growth rate of the transverse instability of a plane black soliton in a saturable defocussing medium vs. the dimensionless saturation parameter s. Crosses are numerical simulation results whereas the solid curve is a numerical approximation (Kivshar et al., 1997b).

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7.3. Vortex solitons: theory 7.3.1. Stationary solutions It is well known that in linear optics and acoustics vortex can appear as a particular mode described by the wave equation and associated with a wavefront screw dislocation (or phase singularity) of the linearly diffracting field (Nye and Berry, 1974; Bazhenov et al., 1992; Heckenberg et al., 1992; Basistiy et al., 1995). These dislocations can be generated, for example, by the wave scattering from a rough surface [e.g., Baranova et al. (1981) and Freund (1994)]. In a selfdefocussing nonlinear medium this screw dislocation can create a stationary beam structure with a phase singularity known as a vortex soliton. Existence of vortex solutions of the (2#1)-dimensional defocussing NLS equation can be established from an analogy between optics and fluid mechanics. Indeed, using the so-called Madelung transformation [see, e.g., Spiegel (1980), Donelly (1991) and Nore et al. (1993)] u(r, z)"o(r, z)er(r, z) , we transform the NLS Eq. (7.1) to the following form: ­o2 #+ ) (o2+u)"0 , ­z

(7.5)

­u 1 + 2o # (+u)2"1!o2# . ­z 2 2o

(7.6)

Eqs. (7.5) and (7.6) can be treated as the equations of conservation for mass and momentum of a compressible inviscid fluid of density p"o2 and velocity V"+u with the pressure defined as p"p2/2. Importantly, this kind of analogy between optics and fluid mechanics still remains valid for the generalized NLS equation with the nonlinearity function g(DuD2), in this case the effective pressure is defined as

P

p(p)" dx p

d g(p) . dx

This analogy is however is not exact because, additionally to the standard pressure Eq. (7.6) includes the second term, so-called ‘quantum mechanical pressure’, which has no analog in fluid mechanics. The Madelung transformation is singular at the points where o"0 and around such points on the plane (x, y) the circulation of V is equal to 2p. These points are topological defects of the scalar field called vortices. To find the structure of the stationary solution for the vortex soliton (or dark soliton with the circular symmetry), we look for the solutions of the normalized NLS Eq. (2.3) for the cubic nonlinearity in the polar coordinates r and h, u(r, h; z)"u º(r)e*mhe*u20z , 0

(7.7)

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where u is the background intensity, the integer m is the so-called winding number (or the vortex 0 charge), and the modulus function º(r) satisfies the (normalized) boundary value problem, d2º 1 dº m2 # ! º#(1!º2)"0 , dr2 r dr r2

(7.8)

for positive r and the boundary conditions, º(0)"0 , º(R)"1 . The continuity of u at r"0 forces the first condition, while º(R)"1 is consistent with a locally uniform state as rPR. Asymptotic behavior of º(r) may be established directly from Eq. (7.8): º(r)&ar@m@#O(r@m@`2) , as rP0 , m2 º(r)&1! #O(1/r4) , as rPR. 2r2 Fig. 33 depicts profiles of the normalized function º(r) found numerically for four different values m [e.g., Neu (1990); see also Velchev et al. (1997)]. The neighborhood of r"0 where the function º is significantly less than one is called the vortex core. Generally speaking, the structure of the vortex soliton can be also defined in a similar way for any nonlinear medium by solving numerically the equation for the amplitude function º(r) similar to Eq. (7.8). This analysis was carried out for the NLS model with nonlinearity saturation (Y. Chen, 1992a,b; Tikhonenko et al., 1997). No qualitatively new features were discovered. As was mentioned by Tikhonenko et al. (1997), in a saturable medium the effective diameter of the vortex core

Fig. 33. Numerical solution for the vortex profile in a Kerr medium for different values of the vortex charge, m"1, 2, 3, and 4 (Neu, 1990).

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increases almost linearly with the dimensionless saturation parameter defined as s"I /I , where 0 4 I and I are the saturation and maximum background intensities. 4 0 Stability of vortex solitons in a generalized NLS equation has been not addressed so far. However, it is usually believed that the vortices with the winding numbers m"$1 are topologically stable, whereas those with larger values of DmD are unstable against decay into DmD single-charge vortices. This later statement requires the further analysis because, as was recently demonstrated by Aranson and Steinberg (1996) in the context of a model of superflow, the multicharged vortex solitons are surprisingly very long-lived objects, contrary to the accepted opinion. 7.3.2. Vortex rotation and drift Being excited experimentally on a diffracting background beam of finite extend, a vortex soliton displays a nontrivial dynamics. The effects such as the vortex rotation and radial drift relatively to the Gaussian carrier-wave beam are indeed observed experimentally, and they cannot be predicted from the analysis of stationary solutions. Describing these effects theoretically requires the development of special analytical techniques for analyzing the vortex motion. This was done in the papers by Christou et al. (1996) and Kivshar et al. (1997a). Following Kivshar et al. (1997a), we consider the propagation of a monochromatic scalar electric field E in a bulk optical medium with an intensity-dependent refractive index, n"n #n (I), 0 /where n is the linear refractive index of the unperturbed medium, and n (I) describes the change in 0 /the index due to the field intensity I"DED2. In the so-called paraxial approximation, Maxwell’s equations can be reduced to the generalized NLS equation for the slowly varying envelope E(z, r)"E(z, r)exp(!ik n z) of the electric field [cf. Eq. (2.3)], 0 0 ­E 2ik n #+ 2E#g(I)E"0 , 0 0 ­z

(7.9)

where k is the free-space wave number, and the gradient operator + was defined above. The 0 function g(I)"2n k2n (I) describes the nature of nonlinearity, and it is determined by the 0 0 /intensity-dependent correction n (I) to the refractive index. Analysing the interaction of a vortex /soliton with a background field, we look for solutions of Eq. (7.9) in the form (Kivshar and Yang, 1994b) E(z, r)"JI e*h" u(z, r) " and assume that both the background intensity I (z, r) and phase h (z, r) satisfy Eq. (7.9). This " " yields the equation for the auxiliary field u(z, r), ­u #+ 2u#[g(I DuD2)!g(I )]u"!+u ) f , 2ik n " " 0 0 ­z

(7.10)

where the complex vector f is defined by gradients of the background field, f"f 3#if *,+ ln I #2i+h , " " and the boundary condition DuDP1 applies for large r.

(7.11)

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In the particular case of a defocussing Kerr medium, i.e., when n (I)"!Dn DI, Eq. (7.10) takes /2 the form of a perturbed NLS equation [cf. Eq. (7.1)] ­u 2ik n #+ 2u#2k2n Dn DI (1!DuD2)u"!+u ) f . 0 0 ­z 0 0 2 "

(7.12)

Kivshar et al. (1997a) applied the method of matched asymptotic expansions to analyze a slow vortex motion in a shallow-gradient background field. The expansions near and far form the vortex core followed from the asymptotic matching at an intermediate distance. We mention here the most important steps of this derivation which follows a similar analysis (Rubinstein and Pismen, 1994) that, in its turn, draws on the application of the same technique in other settings (Neu, 1990; Pismen and Rodriguez, 1990; Pismen and Rubinstein, 1991; Rubinstein and Pismen, 1994). We assume that the function u describes a vortex with the centre coordinate r (z), and the fields 0 I and h vary slowly in comparison with the vortex scales. Then, the problem is to describe " " a change of the position of the vortex (i.e., the so-called vortex drift) for the field u under the action of these slowly varying background fields. To clarify the idea of a ‘background field’, one may picture it as the field which would exist if the vortices were somehow removed. We rescale to dimensionless coordinates by using the value of the background field at the vortex centre, I "I (r ), so that zPz/(k n Dn DI ) and rPr/(k n J2Dn DI /n ), then set f"eF where DFD"O(1). 0 " 0 0 0 2 0 0 0 2 0 0 In this case Eq. (7.12) becomes ­u I #+ 2u# " (1!DuD2)u"!e+u ) F , (7.13) ­z I 0 where F has also been rescaled. The vortex velocity, in this new coordinate scale, is assumed to be small, producing the same small parameter e as the background gradients, so that i

dr 0"w"eV , dz

(7.14)

where DV D"O(1). Next, we need to solve Eq. (7.13) in the vicinity of the vortex core in the reference frame moving with the vortex drift velocity w. Since the background field does not change significantly on the scale of the core, the term I /I can be expanded as " 0 I /I +1#r ) + ln I Dr r0,1#er ) F3 , (7.15) " 0 " / 0 where the rescaled complex vector F is calculated at the position of the vortex, F ,FDr r0. Thus, 0 0 / Eq. (7.13) becomes

+ 2u#(1!DuD2)u"e[(iV!F ) ) +u!r ) F3 (1!DuD2)u] . (7.16) 0 We expand also the field u as u"u #eu #2 and substitute it into Eq. (7.16). In the 0 1 zero-order approximation in e we find the standard stationary NLS equation for the vortex + 2u #(1!Du D2)u "0 , (7.17) 0 0 0 so that, in the polar coordinates of the moving frame, its solution is given by the expression u "o(r)e*m( 0

(7.18)

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where m"$1 is the vortex charge (polarity), and the function o(r) that verifies

A

B

d2o 1 do 1 # # 1! !o2 o"0 dr2 r dr r2

(7.19)

is the well known vortex amplitude profile, first studied in the superfluid context, and known numerically [Pitaevsky (1961); see also Donelly (1991) and references therein]. The first-order approximation yields the inhomogeneous equation, L(u , u*)"W(r) 1 1 with the homogeneous part

(7.20)

L(u , u* )"+ 2u #u !2Du D2u !u2u* , 1 1 1 1 0 1 0 1 and the right-hand-side

(7.21)

W(r)"(iV!F ) ) +u !r ) F3 (1!Du D2)u . (7.22) 0 0 0 0 0 Solvability condition of the linear Eq. (7.20) can be used to derive the equation of motion for the vortex core. These conditions follow from the orthogonality of the inhomogeneous part, W(rN, to the two components of the translational eigenfunction, +u*, of the adjoint homogeneous equation. 0 Using the method of matched asymptotics allows to present the solvability conditions in the following vector form (Kivshar et al., 1997a) w"f * #mJf 3 ln (1cD f 3 Dec) (7.23) 0 0 4 0 where c is found numerically (for the cubic nonlinearity c+1.126), c is the Euler constant (c+0.577), J is the operator of rotation by p/2 defined by the matrix

A

J"

B

0 !1 1

0

,

and the force components, f, are defined by Eq. (7.11), but evaluated at the vortex core. In the dimensional units which are required to compare with the experimental results (see Section 7.4.2 below), the motion equation for the vortex core can be presented, by rescaling Eq. (7.23), as the following:

A

dr m 0" !+h # CJ+ ln I k n 0 0 dz " 2 "

BK

, (7.24) /r0 where J is the operator introduced above, and C is slowly varying function of I . In the particular " case of the Kerr medium, the coefficient C has been derived above:

A

B

r

cecD+I D " 4k n J2n I /n 0 0 2 0 0 but in general case of non-Kerr medium we can expect a similar form of the motion equation. Indeed, for a general function g(I) in Eq. (7.10), the coordinates are rescaled by g(I ) rather than 0 C"!ln

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I as before, and Eq. (7.13) becomes 0 ­u g(I ) g(DuD2I ) " u"!e+u ) F . i #+ 2u# " 1! (7.25) ­z g(I ) g(I ) 0 " Expanding as in Eq. (7.15) and proceeding exactly as in the Kerr case gives in the zero-order a modified equation for the stationary vortex profile

C

D

+ 2u #(1!G(Du D)2)u "0 , (7.26) 0 0 0 where G(x)"g(I x)/g(I ). In the first-order approximation Eq. (7.20) is again obtained, but with 0 0 a modified coefficient on the r ) F3 term. Thus, all results are obtained as before with a modified 0 numerical coefficient c and with a rescaled coefficient

A

B

cecD+ I D " 4k n J2n (I )/n 0 0 /- 0 0 In particular, for the important case of a saturable nonlinearity we take C"!ln

I g(I)" , 1#sI where the dimensionless parameter s characterizes, as above, the inverse saturation intensity I relative to the background intensity I ; larger values of s correspond to a stronger saturation of 4 0 the nonlinearity. Calculation of the coefficient c for this model yields c"1.126 at s"0 (nonsaturable case), c"1.412 at s"1, and c"1.639 at s"2 (Kivshar et al., 1997a). As follows from the analysis presented above, to describe the vortex drift induced by the diffracting background field, one should know the evolution of this field a priori, so that the radial and angular velocity components for the vortex motion can be calculated according to Eq. (7.24). It is assumed that the background field in the absence of the vortex evolves in approximately the same manner as it would when hosting a vortex. Thus, even qualitative knowledge of the propagation behaviour of a field may be used, along with the vortex equation of motion, to predict the action of a vortex subsequently nested in that field. The following example of a vortex nested in a Gaussian beam serves to illustrate how vortex motion may be simply predicted (Christou et al., 1996). The transverse ‘velocity’ of a vortex, according the model, Eq. (7.24), has two components arising separately from the transverse phase and intensity gradients of its background field, evaluated at the position of the vortex. The first component, !+h , is directed normal to the wavefront of the " background, that is in the direction of energy flow in the background field, giving rise to radial motion of a vortex in a Gaussian beam [see also Christou et al. (1996)]. The second component, 1mCJ+ ln I , is directed along the intensity contour (isophote) of the background upon which the 2 " vortex is positioned, with the sense of direction given by the vortex charge, m"$1. For a Gaussian background, the isophote in any transverse plane is a circle, and so the second component of velocity describes the angular motion of the vortex, first observed experimentally by Luther-Davies et al. (1994). The flattening of the intensity profile under nonlinear action reduces the intensity gradient and thus subtracts from the rotation experienced by a vortex in linear propagation. For flatter intensity profiles (plane waves), the motion of the vortex becomes more

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dependent on the background wavefront solely, and results of other work which have examined this behaviour (Staliunas, 1994a,b; Roux, 1995) may be recovered. A simple analysis can be made for ‘beam-like’ field, employing a Gaussian ansatz to approximate the evolution of the background field in a self-defocussing medium (Christou et al., 1996). Using the Gaussian ansatz, we can make explicit calculations in Eq. (7.24), and the resulting equations for the vortex core can be integrated to yield w(z) r (z)" r (0) , 0 w(0) 0

(7.27)

P

mC z df , /(z)"/(0)# w2(f) k n 0 0 0

(7.28)

where the polar coordinates r and / characterize the vortex position at a propagation distance z. 0 Here w(z) is the beam radius which can be calculated by various methods [e.g., Butylkin et al. (1989)]. These equations have been found to effectively characterize aspects of vortex behaviour important to the problem of vortex steering (Christou et al., 1996; Kivshar et al., 1997a). Although linear propagation is outside the parameters set in the derivation of the equation of motion, exact agreement is obtained, in this case, with vortex dynamics calculated by other methods (Indebetouw, 1993), provided that CP1. Vortex interaction is also adequately accounted for by considering the host beam for one vortex as being comprised of the underlying background field along with the remaining vortices. A single vortex has circular isophotes centred on the vortex core and also circular energy flow. Thus one vortex interacting with the background field generated by another, will move in the direction normal to the line connecting its core with the background vortex. The situation is exactly the same reversing the roles of the vortices in the pair. The resultant motion of the vortex pair can therefore be only circular or parallel, depending on the vortex chirality. It is also possible to include the effect of a non-planar background in this picture, in order to estimate any influence on the observed interactions between vortices, as observed experimentally by Luther-Davies et al. (1994). A simple physical argument underlies this form for the vortex equation of motion used in the above examples, which may clarify the mechanisms underlying vortex behaviour. Consider the ‘momentum’, :I+h, of a small element of the transverse field surrounding the vortex core. In the first case, assume that the intensity is uniform, then the momentum of the element is proportional to the sum of the vortex phase gradient around the core, which is zero, and the sum of the background phase gradient around the core. Thus the element around the core has a momentum approximately proportional to the background phase gradient at the vortex position, giving rise to the first velocity component in the equation of motion. In the second case, assume that the background phase is uniform, then the background phase gradient around the core is zero. The momentum of the element is thus proportional to the sum of the vortex phase gradient weighted by the background intensity, around the core. Using the ideas of vector summation, it is apparent that an imbalance in background intensity, over the small region around the vortex core, gives rises to a net momentum component in the direction normal to the intensity gradient, i.e., along the isophote, sourcing the second velocity component in the equation of motion. That the origin of vortex dynamical effects may be justified in this way imparts a generality to the equation of motion,

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suggesting that it may retain some validity even outside the specific parameters set in its formal derivation. 7.4. Vortex solitons: experiments 7.4.1. Transverse instability of dark soliton stripes Most of the experiments on dark spatial solitons reported in Section 6 involved the creation of dark soliton stripes in a (2#1)-dimensional geometry. As noted, such stripes should be unstable due to transverse modulational instability which leads to stripe breakup (Kuznetsov and Turitsyn, 1988) and the eventual creation of optical vortex solitons (Law and Swartzlander, 1993; McDonald et al., 1993). The instability was avoided in the early experiments by the use of finite-sized background beams and weak nonlinearity. By increasing the nonlinearity, however, the transverse instability should be observed even with finite sized beams. Experiments to verify the existence of this transverse instability, and through it the creation of optical vortex solitons, have been performed by Tikhonenko et al. (1996a) using a continuous wave, Ti : sapphire laser and a nonlinear medium comprised of atomic rubidium vapour. Very similar observations, with less evidence of the stripe decay into vortex solitons, were performed simultaneously by Mamaev et al. (1996a,b) for photorefractive dark solitons. In the experiments using rubidium vapor (Tikhonenko et al., 1996a), the laser output was a linearly polarized slightly elliptical Gaussian beam with a wavelength tuned close to the rubidium atom resonance line at 780 nm. A p phase jump was imposed across the beam centre using a mask and the resulting beam imaged into the nonlinear medium. The rubidium vapour concentration could be increased up to 1013 cm~3 by changing the cell temperature. Images of the beam at the output of the cell were recorded by a CCD camera and frame capture system. A schematic of this experimental arrangement, used also to observed optical vortex solitons, is shown in Fig. 34. The important step in observing the instability was to resonantly enhance the value of nonlinearity of the medium by tuning the laser frequency close (!0.4 GHz to !1.0 GHz) to the rubidium atom D line and the use of the maximum vapour pressure consistent with tolerable 2 absorption. The power in the beam at the input face of the cell was 240 mW with a 1/e2 waist of 0.3 mm. A maximum nonlinear refractive index change of order of 10~4 was achieved. Fig. 35 shows a series of output intensity profiles, calculated numerically (left column) and observed experimentally (right column), with increasing cell temperature and with the detuning fixed at approximately 0.85 GHz. For vanishingly small vapour concentration, the beam underwent linear propagation through the medium, leading to the output intensity profile shown in Fig. 35a. With increasing temperature (i.e., increasing nonlinearity), the output beam developed a vertically uniform dark soliton stripe, as shown in Fig. 35b. Further increase in the temperature

Fig. 34. Schematic presentation of the experimental setup for the observation of the dark soliton decay into vortex solitons and the measurement of the vortex parameters.

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Fig. 35. Output beam intensity profiles demonstrating the instability of a dark soliton stripe as the nonlinearity is increased. The vapor concentration increases from vanishingly small in (a) to the order of 1013 cm~2 in (f ). The cell temperatures were (a) 40, (b) 72, (c) 82, (d) 90, (e) 112, and (f ) 125. ¸eft: Results of numerical simulations. Right: Experimental results. Laser detuning was held constant at 0.85 GHz, and the power of the beam at the cell was 240 mW, corresponding to a maximum intensity of approximately 170 W/cm2 (Tikhonenko et al., 1996a).

led to the growth of a periodic modulation of the uniformity of the stripe (see Fig. 35c). As the temperature was further increased, the breakup of the stripe began, initially appearing as a growing, ‘snake-type’ bending (Fig. 35d), then as breaking, with field coalescing into dark spots at the

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inflection points in the bends (Fig. 35e). At the highest nonlinearity the dark spot assumed close to circular symmetry consistent with the predicted formation of a pair of optical vortex solitons (Fig. 35f and insert). The process of breakup was found to be sensitive to the size of the phase step on the input beam. Misalignment of the phase mask from the optimum (which provided a phase jump closest to p) caused the final stages in evolution of the instability involving the creation of optical vortex pairs, to disappear. Under such conditions, only the snake-type bending of the soliton stripe was observed. This supported the theoretical prediction (see Section 7.2) that the growth rate of the instability should be lower for grey soliton stripes. Tikhonenko et al. (1996a) also carried out numerical simulations based on the generalized NLS equation including saturation and dissipative effects for comparison with the experimental results (see Fig. 35, left column). The calculated output intensity distributions showed the same dynamics observed experimentally with increasing nonlinearity. The beam defocussing, power depletion and instability growth rates seen in experiments appeared to be well approximated by the simulations. However, the period of the transverse perturbation corresponding to the maximum growth rate appeared to be smaller than that seen in experiment, by a factor of 1.5—2. This discrepancy is most likely due to (i) the physically complicated nonlinear response of rubidium vapour, which was only approximated by the model used in the simulations; (ii) the difficulty in accurately characterizing the initial field which was found to sensitively affect the simulations. It should be noted that this sensitivity was not observed in the experiments, suggesting that the breakup process may have been partially stabilized by some physical mechanisms not included in the model (e.g., nonlocality in the form of diffusion). Similar experimental results were reported by Mamaev et al. (1996a) [see also Mamaev et al. (1996b), where the instability of a bright-soliton stripe was also reported]. In their experiments a biassed photorefractive SBN crystal, irradiated with a 10 mW He—Ne laser beam containing a phase step, was used as the nonlinear medium. Numerical simulations using the generalized NLS equation with a saturable nonlinearity demonstrated a similar breakup of the initial stripe into a set of optical vortex solitons as reported by Tikhonenko et al. (1996a,b). The effectively nonlinearity could be varied by increasing the bias voltage on the crystal. When zero voltage was applied, the dark stripe spread due to diffraction as did the background beam. As the applied voltage increased the background beam underwent self-defocussing and a dark-stripe soliton was clearly formed. A further increase in the voltage up to 990 V [the maximum voltage reported was 1410 V (Mamaev et al., 1996a) and 2000 V (Mamaev et al., 1996b)] led to the appearance of the snake-like bending of the dark soliton stripe. However, the final state was markedly different from that shown in Fig. 34f because it did not present clear evidence of the creation of optical vortex soliton pairs. However, the authors reported the observation of zeroes in the electromagnetic field from interferometric measurements of the output beam with the distances between the zeroes being about 40 lm. This measurement indicates that wave-front dislocations similar to single vortices were being formed. 7.4.2. Optical vortex solitons The seminal work of Nye and Berry (1974) on wavefront dislocations in optical beams has led to a substantial growth of interest in screw dislocations (optical vortices) in linearly diffracting beams [see, e.g., Heckenberg et al. (1992) and Basistiy et al. (1993)]. In a self-defocussing nonlinear

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medium screw dislocations can form dark optical vortex solitons as predicted by Snyder et al. (1992) and first demonstrated experimentally by Swartzlander and Law (1992, 1993). In Swartzlander and Law’s experiment (1992) a thermally nonlinear medium was irradiated with an Argon laser beam which had passed through a mask used to impose an approximation to the helical phase structure of an optical vortex. This mask contained regions of ‘0’, ‘p’ and ‘2p’ phase thickness surrounding a single point in the plane of the mask. At the exit from the medium a dark spot localized at this point was observed and interferometric measurements demonstrated the presence of a phase dislocation at that point, supporting the idea that an optical vortex soliton had formed. The authors also demonstrated that the (2#1)-dimensional induced waveguide existed in the vicinity of the vortex by co-propagating a He—Ne beam as a guided mode through the induced structure. A more practical method of creating the helical vortex field in the input beam was suggested by Heckenberg et al. (1992). By numerically calculating the interference pattern between either an on-axis spherical wave or off-axis plane wave and the optical vortex they computer-generated diffracting masks which could be used to create single optical vortices in the first order diffracted beams from a Gaussian input. This method was used in the work by Swartzlander and Law (1993), Luther-Davies et al. (1994), Tikhonenko et al. (1995), Tikhonenko et al. (1996a,b), and Christou et al. (1996). The advantage with this method is that beams with arbitrarily placed numbers of vortices with selectable chirality (clockwise or anti-clockwise) can be generated. The final method of vortex creation recognizes that far field of a single vortex is identical to the p"1, GaussLaguerre mode of an optical resonator (Duree et al., 1995). A diffracting mask was used to create pairs of like-charged vortices in a single beam by Luther-Davies et al. (1994) to demonstrate rotation of the solitons around beam axis at the output of a nonlinear medium with increasing nonlinearity. An off-axis vortex rotates around the axis of a Gaussian beam by 90° as it propagates from the beam waist to infinity (Indebetouw, 1993). This rotation exactly matches the so-called Guoy shift which characterizes the change of on-axis phase of the Gaussian beam with propagation relative to that of a plane wave. The defocussing action of the nonlinear medium acts to flatten the phase fronts of the background beam which reduces the effective Guoy shift at a given distance. Hence the action of the nonlinearity is to subtract from the natural rotation the vortex experiences during linear propagation in a nonuniform Gaussian beam (the effect is absent in a uniform plane wave). Using Rb vapour pumped resonantly using a Ti : sapphire laser near 780 nm as the nonlinear medium and a focussed Gaussian beam with its confocal parameter much shorter than the cell length, it was demonstrated that a change in vortex rotation of &90° could be obtained with increasing nonlinearity with little change in radial position of the vortex. It was suggested that this motion could be used to create an optical rotary switch using the light guiding properties of the dark vortex soliton. A number of other papers have dealt with the creation, dynamics and waveguiding properties of optical vortex solitons in saturating nonlinear media (Law and Swartzlander, 1993; Segev et al., 1994a,b; Duree et al., 1995; Tikhonenko and Akhmediev, 1996; Tikhonenko et al., 1996a, 1997; Christou et al., 1996; Luther-Davies et al., 1997). Segev et al. (1994a,b) and Duree et al. (1995) provided the first experimental demonstration of the creation of optical vortex solitons in photorefractive media using the nonlocal response in a biassed SBN crystal. They used an input beam which was either the coherent ‘donut’ mode of a laser, or

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constructed the ‘donut’ mode by summing two beams: one with a vertical notch and the other with a horizontal notch, with an appropriate p/2 relative phase between them. Both methods produced a Gauss—Laguerre beam that possessed the desired azimuthal phase dependence to create an optical vortex soliton at the output of the medium. In spite of the anisotropy of the nonlinear response in the photorefractive material nearly circular vortex solitons were obtained although it was noted that this would depend on the input conditions. Spatial dynamics of single-charged vortices in an anisotropic photorefractive medium has been also studied by Mamaev et al. (1996c) which results a bit different from those reported earlier by Duree et al. (1995). In particular, the experimental and numerical results presented by Mamaev et al. (1996c) indicated a stretching of the vortex in a media with an anisotropic nonlocal nonlinearity, and its subsequent decay. At the same time, Mamaev et al. (1996c) demonstrated the existence of a bound state of a counterrotating vortex pair created from a single vortex soliton. All these results clearly indicate a high sensitivity to the initial conditions for generating vortex solitons in photorefractive media, and also the importance of a finite-width background that can lead to a chance of the vortex charge. Definitely, further experimental studies will be necessary to clarify these issues. Tikhonenko and Akhmediev (1996) studied the creation of single optical vortices in Rubidium vapor [for experimental details, see Luther-Davies et al. (1994, 1997)]. They compared numerical predictions of the output field distributions and demonstrated that saturation and absorption must be taken into account to obtain a reasonable agreement. They also described a new method of steering solitons, and hence the waveguides they induce, based on an interaction with a weak coherent background wave. This idea is closely related to a steering phenomenon earlier described for vortices nested in linearly propagating beams (Basistiy et al., 1993). Christou et al. (1996) demonstrated this steering method experimentally, using a coherent background wave whose intensity was &20% of that of a beam containing an off-axis vortex. By adjusting the relative phase of the background wave, the position of the vortex could be moved to any selectable angular position in the output beam. Christou et al.’s work (1996) [see also Kivshar et al. (1997a)] also introduced a simple analytic model to describe the vortex motion as has been discussed above in Section 7.3.2. It showed that the position of the vortex could be described as a result of a radial drift proportional to the gradient of the phase of the background beam and an azimuthal drift proportional to the gradient of the intensity [this effect was observed earlier in numerical simulations by McDonald et al. (1992)]. Using a Gaussian approximation for the defocussing background wave, this resulted in a magnification of the absolute radial displacement of the vortex in proportion to the ratio of the beam radii with and without defocussing. Experimental results confirmed this prediction, see Fig. 36a and Fig. 36b, even under the conditions where the background deviated from the assumed Gaussian form. To assess the affect of a nonGaussian background, simulations of the beam propagation were carried out which yielded an excellent fit with the experimental and numerical data. Interestingly a small but definite saturation of the transfer function can be observed when the vortex is launched very near the edges of the beam. A detailed comparison between the theoretical and experimentally measured vortex diameters in a saturable defocusing medium was reported by Tikhonenko et al. (1997). First, the authors developed an analytical theory to find the stationary, radially symmetric localized solitons of the generalized NLS equation with a saturating nonlinearity. Saturation was characterized by

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Fig. 36. Radial (a) and angular (b) position of the vortex at the output of the cell as a function of the detuning below the resonance. Two sets of the data show results for a cell temperature 88C (boxes) and 108C (crosses). Solid curves show the interpolated analytical results (Kivshar et al., 1997a,b).

a dimensionless saturation parameter s"I /I , I being the background intensity and I , the 0 4!5 0 4!5 saturation intensity. It was noticed that the vortex profile and diameter depended strongly on the degree of saturation and that the FWHM diameter of the vortex increased almost linearly with s. To link the theory, which deals with stationary solutions on an infinite uniform background, and the experiments, where an input beam with a somewhat arbitrary intensity profile and helical phase is used, Tikhonenko et al. (1997) analyzed the transition from several typical input profiles to a vortex soliton. For comparison the dynamics of the vortex propagation was investigated experimentally using Rb vapour as a medium with a variable saturating nonlinearity. Some novel phenomena were reported such as rotation of initially elliptic vortex core as the soliton formed. Measurements of the vortex diameter as a function of the effective saturation showed (see Fig. 37) that the almost linear growth with s, predicted by the theory, could only be observed in the region of high saturation where the effective propagation distance, shown in Fig. 37 by a dotted curve, was long enough to approximate to the stationary case. For lower saturations, the measured vortex diameters were observed to be less than the corresponding stationary values (see Fig. 37) indicating the experimentally observed vortex was still far from a steady-state regime. At last, as one of the future possible practical application of vortex solitons, we would like to mention the recently demonstrated three-dimensional trapping of low-index particles (20 lm

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Fig. 37. Comparison between the theory and experiment for the vortex diameters in a saturable medium. Shown are the experimentally measured mean value of the vortex diameter (solid curve with marks) as a function of the dimensionless saturation s. The similar value obtained from the stationary vortex profile is shown as a dashed line. Dotted line displays the variation of the dimensionless propagation distance at the output of the cell (Tikhonenko et al., 1997).

diameter hollow glass spheres in water) by using a single, strongly focused, Gaussian beam containing an optical vortex (Gahagan and Swartzlander, 1996). Transverse trapping was attributed to gradient directed toward the vortex core, that allowed to trap and form ring patterns of high-index particles. In their studies, Gahagan and Swartzlander (1996) used the computergenerated holographic technique to generate an optical vortex. 7.5. Ring dark solitons As has been shown in Sections 7.2 and 7.4.1, dark-soliton stripes are unstable to transverse long-wavelength modulations. The instability band is characterized by the maximum modulation wave number p , so that a soliton stripe is stable to the transverse periodic perturbations of the #3 short wavelength j provided j (j , where j ,2p/p . Let us consider a dark-soliton loop M M #3 #3 #3 which is formed by a quasi-two-dimensional dark soliton of the length ¸. From the physical point of view it becomes clear that such transverse instabilities should be suppressed provided the condition ¸(j holds. The loop of the lowest energy is expected to have a circular symmetry, #3 therefore, one may expect the instability to be stabilized for ring dark solitary waves in selfdefocussing materials. This simple reasoning allowed Kivshar and Yang (1994c) to introduce the so-called ring dark solitons.

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Following Kivshar and Yang (1994c) [see also Kivshar and Yang (1994d)], we treat radially symmetric solutions of the (2#1)-dimensional NLS equation on the unit-intensity background in the form of a dark soliton ring with slowly varying parameters, u(z, r)"e~*z[cos /(z) tanh H#i sin /(z)] ,

(7.29)

H"cos /(z)Mr!R(z)N ,

(7.30)

where /(z) (D/D(p/2) and R(z) are the slowly varying soliton angle and its centre coordinate, respectively. The physical meaning of these parameters is the following: the soliton angle / describes the contrast of a ring dark soliton, B"cos2/, and it is connected with the phase jump across the ring, 2/, if we calculate the phase difference between the outer and inner regions separated by the ring. The effective variable R(z) is the radius of the ring at the propagation distance z. For large values of the soliton radius, the ring soliton can be regarded as quasi-one dimensional object with the curvature treated as a perturbation. This allows the evolution of the soliton parameters to be analysed using the adiabatic approximation of the perturbation theory for dark solitons (see Section 3.1 above), where the term &r~1 is an effective perturbation. The resulting evolution equations derived from the perturbation-induced dynamics of the system Hamiltonian, take the form (Kivshar and Yang, 1994c,d) dR d/ (D!1) " cos / , "sin / . 3R dz dz

(7.31)

Combining these equations, we find the radial velocity of the ring dark soliton as a function of its radius R,

C

A B

D

dR R(0) (2@3)(D~1) 1@2 ¼, "i 1!cos2/(0) , dz R

(7.32)

where i"sgn[sin /(0)]"$1, /(0) and R(0) being the initial values of the parameters. Eq. (7.32) shows that the minimum radius of the collapsing ring dark soliton is defined by the initial conditions, R "R(0)[cos /(0)]3@(D~1), (7.33) .*/ and at R"R the dark soliton has the maximum contrast. Depending on the initial value /(0) of .*/ the soliton phase /, the dark soliton can collapse to reach R , or it diverges decreasing its .*/ contrast. The linear stability analysis (Kuznetsov and Turitsyn, 1988) predicts that the dark soliton stripe is stable when the condition (in our notations) (7.34) p 'p (/)"[2Jsin4/#cos2/!(1#sin2/)]1@2 M #3 is satisfied, p being the wave number of the transverse perturbations. The result of Eq. (7.34) shows M that the instability band vanishes for small amplitude dark solitons when cos /P0. Thus, when the limit length of the ring 2pR is smaller than the minimum wavelength 2p/p (0) for the .*/ #3 instability region, we expect the ring dark soliton to be stable because, being expanded, it gets ‘grey’

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and even more stabilized. This gives the condition R p (0)(1 . (7.35) .*/ #3 Numerical simulations of the NLS equation for both cases mentioned above were carried out by Kivshar and Yang (1994c) and excellent agreement with the analytical results was demonstrated. One of the cases of the evolution of a ring dark soliton at D"2 is shown in Fig. 38a—d. When the dark ring soliton collapses, it reaches its minimum radius, see Fig. 38c, and then it expands again (Fig. 38d). At the turning point, the validity of the adiabatic approximation breaks down, and the dark ring expands along the trajectory slightly different from that predicted by the theory. Nevertheless, this solitary wave is robust and it perfectly conserves its radial symmetry as is shown in Fig. 38a—d. Importantly, in the small-amplitude approximation discussed in Section 2.5.2 for (1#1)-dimensional dark solitons, the ring dark solitons can be described by the so-called cylindrical Korteweg de Vries equation (Kivshar and Yang, 1994c) which is known to be exactly integrable. A very interesting idea has been suggested by Dreischuh et al. (1996b) who studied theoretically the guiding of multiple bright signal beams by a single ring dark soliton. The authors proposed to use a (1#1)-dimensional bright dark soliton as an induced all-optical cable to transmit (2#1)dimensional bright pulses, see Fig. 39a. It is clear that, even not mentioned explicitly in that work, this idea equally applies to the transmission of light bullets through waveguides ‘written’ by dark solitons. However, since a dark soliton stipe suffers from a transverse modulational instability,

Fig. 38. Evolution of a ring dark soliton in a Kerr medium for collapsing regime, (a) z"0 and (b) z"3.0, and divergence regime: (c) z"6.0 and (d) z"9.0 (Kivshar and Yang, 1994c).

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Fig. 39. Scheme of the (2#1)-dimensional dark soliton guiding: (a) a planar soliton waveguide, (b) a curved waveguide formed by a ring dark soliton, and (c) multi-soliton guiding by a ring dark soliton. The positions of elliptical signal beams are shown as bright spots (Dreischuh et al., 1996b).

Dreischuh et al. (1996b) suggested to use a ring dark soliton as a curved waveguide shown in Fig. 39b. This idea is a nontrivial generalization of the concept of the soliton-induced waveguide discussed above for (1#1)-dimensional dark solitons, and it allows to use more than one single beam. For example, Fig. 39c shows the grayscale image of a ring dark soliton that guides 8 elliptic bright signal beams. Dreischuh et al. (1996b) demonstrated also a misalignment stability of the guiding scheme, similar to the inherent stability of a bright—dark soliton pair. Experimental results demonstrating the existence of these ring dark solitons were reported by Baluschev et al. (1995a,b). To provide the initial conditions, the authors placed an amplitude mask consisting of opaque dots ranging from 50 to 250 lm in diameter in front of a thermally nonlinear medium (ethanol containing a red absorbing dye). A copper vapour laser (P"4 W) was used to produce the background beam. Single and double (at higher power) dark ring structures separating regions of roughly uniform intensity confirmed that ring dark solitons were created. Baluschev et al. (1995b) also measured the dependence of the transverse velocity of a ring on the intensity for a fixed value of the dark-beam diameter. The results were qualitatively in good agreement with the theory and numerical simulations reported earlier by Kivshar and Yang (1994c) [see also Kanemov et al. (1997)]. Quite recently the phase profile of ring dark solitons has also been measured (Dreischuh et al., 1996a; Neshev et al., 1997a) and it was shown that the rings correspond to regions where a phase step (p occurs in the background beam, as expected for a ring dark soliton.

8. Conclusion and open problems We have presented different aspects of the physics of optical dark solitons, including analytical, numerical, and experimental results. They demonstrate a number of very interesting properties of these nonlinear waves which can exist on a (modulationally stable) background being characterized

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by a nontrivial phase distribution of the field. In many cases discussed in this review, these dark solitary waves can be not only described analytically and numerically for a variety of nonlinear models, but, and this is the most amazing fact, they can be also verified, with a relatively good accuracy, by direct measurements of the pulse propagation in optical fibers and beam propagation through a defocussing nonlinear medium. As has been already mentioned, the application of dark solitons to optical communications still remains unclear, it is likely that the answer on this question depends on a general success of the application of bright optical solitons to long-distance fiber communications. As for spatial solitons, many problems involving the light-guiding-light concept are still not solved experimentally, and this definitely will require more efforts to be done making this concept a practical reality. Searching and employing new materials with strong nonlinear properties may sufficiently speed up this process. As the list of important theoretical problems which still remain unsolved, we would like to mention the following. (a) Stability theory for dark solitons and vortex solitons is not completed yet, in the sense of a solution of the corresponding linear eigenvalue problem. In fact, the completeness of eigenfunctions is still questionable. (b) Classification and stability of vector dark solitons calls for numerical solutions of the coupled NLS equations with nonequal (nonzero) boundary conditions. The analytical solution found by Kivshar and Turitsyn (1993) is just one member of the whole family of localized solutions not yet known. It is expected that the general solution of this kind will involve two independent parameters, the soliton velocity and the difference between the propagation constants, and it will describe a dark soliton on a rotating background. Stability of this vector solitons is also an open problem. (c) Dark solitons in s(2) media have been shown to be unstable for the case of the mode dispersions of the same sign. This is also the important case of spatial dark solitons. Can parametric interactions in a s(2) medium support spatial dark solitons? We expect that the effect of the next order, cubic nonlinearity or nondegenerated three-wave mixing on dark solitons may suppress this instability and finally lead to suggestions for experimental verifications. (d) Dark solitons created by resonantly interacting waves in a diffractive (or dispersive) medium have been not investigated theoretically yet. When the effect of the wave walk-off, in the spatial domain, or the group-velocity mismatch, in the temporal domain, are important, only coupled states of bright—dark solitons have been found. More general types of solitary waves on a background field are expected when diffraction becomes important. This also includes dark solitons in the systems of three and four resonantly interacting waves and the effect of harmonic generation near the parametric coupling between the waves provided the matching condition is satisfied. (e) The use of dark spatial solitons for light guiding light is an important theoretical concept useful for physical applications. However, the analysis of the soliton-based devices such as X- and ½-junctions is usually restricted by integrable models. The work to extend this concept to more realistic physical situations has been just started. Also, increasing the amplitude of a guided beam we break the validity of the linearized equations describing the probe beam guidance, the problem becomes naturally nonlinear, and it describes dark—bright soliton pairs. Properties of this kind of soliton become more and more clear after recent theoretical and experimental works on this topic,

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but a general approach to this problem is still absent. This includes the classification and stability analysis of bright—dark vector solitons. In particular, we expect that the polarization domain walls should appear as a special limit case of the general solution for the bright—dark solitons. Among numerous experimental problems related to dark solitons, it is worth to mention just a few. (a) Polarization properties of dark solitons and vortex solitons require an experimental study. Observation of vectorial properties of dark solitons was, to the best of our knowledge, never reported. This also includes incoherent interaction between dark solitons. In particular, some theoretical papers described polarization-induced instability (Law and Swartzlander, 1994) and interaction between vortex solitons of different backgrounds (Velchev et al., 1996). (b) Possibility of steerable light guiding by dark solitons in a bulk medium, e.g. by plane, ring, or vortex solitons, has been discussed theoretically but never realized in practice. This will require, in fact, an additional theoretical analysis of guiding properties of dark solitons in higher dimensions and (3#1)-dimensional generalizations of dark solitons (‘dark bullets’ and ‘donut modes’). (c) Recent demonstrations, presented by Nakazawa and Suzuki (1995a,b), of the potential use of dark solitons for data coding and signal transmission should open room for experimental activity in this direction. Increased interest in the effect of nonlinearity on a standard non-return-to-zero (NRZ) communication schemes [see, e.g., Kodama and Wabnitz (1995)] will also require some efforts to understand if dark solitons might ‘coexist’ with the NRZ transmission [see, e.g., interesting ideas in the paper by Ngo et al. (1996)]. ¼ill spatial and temporal dark solitons be useful in the future? For spatial dark solitons, with their relatively popular appeal, the answer may be yes. But for temporal dark solitons, which are less popular than their bright relatives, the answer — though closer to no — is harder to predict.

Acknowledgements It is a great pleasure for us to thank all colleagues and students with whom we had discussed and worked on this subject for the last several years. We should like to name especially Vsevolod Afanasjev, Philippe Emplit, Sergei Gredeskul, Marc Haelterman, Wieslaw Kro´likowski, Len Pismen, Moti Segev, Allan Snyder, Vladimir Tikhonenko, and also our great team of students: Alexander Buryak, Jason Christou, Dmitry Pelinovsky, Adrian Sheppard, Vika Steblina, and Yang Xiaoping. This work was supported by the Australian Photonics Cooperative Research Centre through the theoretical and experimental projects on nonlinear waveguides and light guiding light.

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LASERS AS A BRIDGE BETWEEN ATOMIC AND NUCLEAR PHYSICS

Sergei MATINYAN!," ! Physics Department, Duke University, P.O. Box 90305, Durham, NC 27708-0305, USA. [email protected] " Yerevan Physics Institute, Yerevan 375036, Armenia

AMSTERDAM — LAUSANNE — NEW YORK — OXFORD — SHANNON — TOKYO

Physics Reports 298 (1998) 199—249

Lasers as a bridge between atomic and nuclear physics Sergei Matinyan!," ! Physics Department, Duke University, P.O. Box 90305, Durham, NC 27708-0305, USA. [email protected] " Yerevan Physics Institute, Yerevan 375036, Armenia Received August 1997; editor: G.E. Brown Contents Preface 1. Introduction 1.1. Simple estimate 1.2. Nuclear Compton effect 2. Atomic shells in the ‘‘Compton” excitation of nuclei 3. Laser-induced nuclear anti-Stokes transitions 3.1. b-transitions of nuclei 3.2. Electromagnetic transitions of nuclei 4. Laser-assisted internal conversion 4.1. Theoretical background 4.2. Some examples: N vs. I 4.3. Experimental problems 5. Electron Bridge mechanism as a source of deexcitation of nuclei 5.1. Experimental observation 5.2. The role of lasers in the EB mechanism of deexcitation of nuclei 5.3. Theoretical considerations

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5.4. IEB mechanism of nuclear excitation 5.5. Some estimates 6. Lasers in the study of anomalously low-lying isomeric nuclear states: (229mTh) 6.1. Introductory remarks 6.2. Deexcitation of isomeric state by the EB mechanism 6.3. Pumping isomers by laser resonance 6.4. Pumping efficiency 6.5. Modified IEB mechanism in the study of 229mTh 7. Summary and conclusions Appendix A. Parameters characterizing the interaction of laser radiation with electrons (dimensional arguments) Appendix B. Two representations of the interaction of radiation with matter Appendix C. Notation index References

0370-1573/98/$19.00 Copyright ( 1998 Elsevier Science B.V. All rights reserved PII S 0 3 7 0 - 1 5 7 3 ( 9 7 ) 0 0 0 8 4 - 7

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Abstract This paper reviews the application of visible and ultraviolet laser radiation to several topics in low-energy nuclear physics. We consider laser-induced nuclear anti-Stokes transitions, laser-assisted and laser-induced internal conversion, and the electron bridge and inverse electron bridge mechanisms as tools for deexcitation and excitation of low-lying nuclear isomeric states. A study of the anomalously low-lying nuclear isomeric states (in the case of the 229Th nucleus) is presented in detail. ( 1998 Elsevier Science B.V. All rights reserved. PACS: 21.10.!k; 23.20.Lv; 23.20.Nx; 32.80.!t; 32.80.Wr; 42.62.!b Keywords: Laser-assisted and laser-induced nuclear transitions; Electron bridge and inverse electron bridge mechanisms; Excitation and deexcitation of nuclear levels; Nuclear isomers; Anomalously low-lying nuclear levels

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Preface The present paper reviews the direct application of laser radiation to several topics in nuclear physics. Here the term “direct” refers to the use of laser radiation in the visible (optical) and UV region (energy of laser photons +u &1—10 eV (j "10~4—10~5 cm)) for studying low-energy nuclear L L phenomena and related processes. The “traditional” approach of using laser radiation in nuclear physics is well known. The Compton backscattering of laser photons off accelerated electrons produces intense, highly (&100%) polarized c-ray beams with excellent energy resolution. This very effective method of employing conventional lasers in particle and nuclear physics was proposed long ago [1] and was implemented successfully in various accelerator facilities (SLAC, BNL, LURE (Orsay), ETL (Tsukuba)). The recent completion of the free electron laser (FEL) Facility at Duke University gives the possibility of yielding c-ray fluxes of +107—109 c/s (a factor of 103 higher than is obtainable with conventional lasers in the energy range from 2—200 MeV [2] for studying phenomena of low and intermediate energy nuclear and particle physics. However, in the present survey these kinds of topics in Nuclear Physics will be completely avoided. The aim here is to consider the wide range of very low-energy nuclear physics topics where the “direct” (i.e., not transformed into c-rays by Compton backscattering) laser beam1 interaction with electronic atomic shells serves as a tool for studying the properties of low-lying nuclear levels and accompanying dynamical processes. The theoretical study of these phenomena has a long and sometimes controversial history. It is interesting to point out that the history of these studies dates back to the 1920s when Einstein discussed the possibility that radioactivity could be induced by the action of optical quanta [3]. The literature on the subject of the present review is comprised of more than one hundred studies published mostly in American and former Soviet physics journals. To my knowledge there has not been an attempt to review this field exhaustively. Unfortunately, in the review literature, the main papers from the former USSR, which have contributed significantly to the field, are not discussed appropriately. The short review paper [4] suffers from one-sidedness. Motivated by these reasons and stimulated by some of my colleagues from Triangle Universities Nuclear Laboratory (TUNL) and the Physics Department of Duke University, I present a rather complete review of the field covering the “direct” use of laser radiation in the energy range of the FEL Facility at Duke University (E [ 12 eV) for studying low-energy nuclear physics phenomena. L Due to the diversity of the topics and methods, the present survey has to be rather schematic, but this will be compensated for by providing a detailed list of references. The paper is organized as follows: After introductory remarks concerning the ineffectiveness of laser radiation in the optical range for the unmediated study of properties of nuclei, the role of the atomic shell as a mediator is shown in the case of Compton excitation of nuclei. In Section 3, the laser-induced nuclear anti-Stokes transitions are investigated and electromagnetic transitions are compared with b-transitions. It is

1 For qualitative features of the interaction of laser radiation with electrons, see Appendix A.

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shown that the laser-induced electromagnetic anti-Stokes transitions are effective only for very strong laser fields (I Z 1019 W/cm2). Section 4 is devoted to the laser-assisted internal conversion process. The technique for calculating the internal conversion coefficient for this process is presented. The laser plays an important role in specific nuclei where the internal conversion in the laser-free case is forbidden by energy conservation. The technique for calculating the internal conversion coefficient for this process is presented. Experimental issues related to this process are discussed. In Section 5 we study the Electron Bridge mechanism as a means for deexciting nuclei and we discuss the role of laser radiation in enhancing the deexcitation. The Inverse Electron Bridge mechanism is presented as a tool for exciting nuclear states to their isomeric states. Section 6 is devoted to the study of the anomalously low-lying nuclear isomeric states in the 229mTh nucleus. Methods introduced in Sections 4 and 5 are used for a detailed study of this phenomenon. Finally, Section 7 gives a summary. In Appendix A, dimensional arguments are used to give some useful information about the role that high intensity plays in the interaction of radiation with electrons and atoms. Finally, Appendix B presents two alternative ways of describing the interaction of radiation with matter.

1. Introduction Q. Is it possible to obtain induced radioactivity by bombardment of matter with quanta of light? A. First of all, I have to say that, probably, there exists radioactivity of matter induced by the action of the light quanta; the difficulty of the observation of such phenomenon, if it exists, is that the effect which has to be observed is very small. The confirmation of this effect is hard but possible.2

1.1. Simple estimate Since the wavelength of laser radiation jL is much larger than the nuclear size RA [& (2—10) ] 10~13 cm], the direct coupling strength of laser photons to a nuclear system considered as a quantum state with level spacing DEA of the order of one hundred keV, is very small, even at extremely high radiation intensity. It is worthwhile, to appreciate this smallness, to make the following simple estimate. The interaction of electromagnetic radiation with a nucleus is more conveniently described by the Hamiltonian3 in the so-called Go¨ppert—Mayer gauge [5] Z HLA"!e + r1 ) EL(t) , p/1

(1)

2 On 16 April 1925, the National Academy of Science of Argentina held a special session to award A. Einstein the Academy’s honorary member diploma. At the reception, Einstein answered the questions of the Academy members. Four years later, details of the Session and the “interview” were published in the Proceedings of the Academy (Reception de la doctor Alberto Einstein en la session especial de la Akademia el dia 16 de april de 1925; Annales Sociedad Cientifica Argentina, 107 (1929) 337—347) [3]. 3 The problem of relating the dipole from Eq. (1) to the commonly used form of the matter-field interaction Hamiltonian (1/2m)(p!eA)2 P !(e/m)p ) A#(e2/2m)A2, has a long and controversial history which is resolved in the paper of Ref. [6]. See Appendix B for some details.

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Fig. 1. Feynman diagrams describing the Compton excitation of a nucleus.

where r is the radius vector of individual protons in the nucleus and E is the electric field of the 1 L laser radiation. DE D is proportional to I1@2 where I is the laser beam intensity which will be L measured hereafter in units of W/cm2. From Eq. (1) the following estimate for the magnitude of the matrix element can be derived: M + (10~12—10~13)I1@2 eV .

(2)

Using for the nuclear level spacing DE Z 1 keV, we obtain for the coupling strength in the A best-case scenario (e.g., when the shielding of the atomic shells is ignored) M [ (10~15—10~16)I1@2 . (3) DE A Therefore, the situation for the direct use of laser radiation remains hopeless even in the case of very low-lying levels (e.g., for 229Th, where DE is on the order of a few eV. See Section 6). A 1.2. Nuclear Compton effect Here we will show that the direct excitation of nuclear levels by X-rays or by hypothetical c-ray lasers does not look encouraging either. Consider, for example, nuclear excitation by the Compton effect at resonance (see Fig. 1 for the corresponding Feynman diagrams). For the cross section we have in the one resonance level approximation: C C C IN NF pA " N , (4) cc k2 (E !+u )2#C2/4 L IN L where C and C are the widths of level N associated with the transition to the initial state, I, and IN NF to the final state, F, respectively. Here, k (u ) is the wave number (frequency) of the incident laser L L photon, C is the statistical weight of the intermediate state N, and C is the total width of the level N N. E ,E !E , where E is the energy of the state I(N). IN I N I(N) Estimates [7] with E !E !+u + 105 eV give for the cross section F I L pA + 10~39—10~40 cm2 , cc which practically is unobservably small. These introductory examples show the ineffectiveness of direct excitation of nuclear levels by radiation.

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2. Atomic shells in the “Compton” excitation of nuclei The example considered in the Introduction suggests the following modification to the Compton mechanism involving the excitation of low-lying nuclear levels: when the Compton effect occurs on the electron bound in the atomic shell, excitation of the nucleus takes place instead of scattering of the photon. The mechanism is described by the diagrams in Fig. 2a and Fig. 2b. We note that if one exchanges the direction of the photon line (absorption of a photon is changed to emission of a photon in the final state) one arrives at the so-called internal Compton effect where the photon from the deexcitation of the nucleus is scattered on a bound electron. This last mechanism is the basis of the electron bridge (EB) mechanism whereas the first one (described by the diagrams in Fig. 2) corresponds to the inverse electron bridge (IEB) mechanism. The EB mechanism can be considered as an alternative to the well-known internal conversion (IC) process where the deexcitation of the nucleus results in the ejection of an electron out of the atomic shell without emission of a photon. We will see below how laser radiation can assist in influencing the EB and IC mechanisms by resonating at the corresponding atomic level or by eliminating the mismatch between the nuclear (DE) and atomic (De) level differences. Here, two aspects of laser radiation are very important: 1. The dipole character of the interaction with the electronic shell which, of course, is common to all kinds of electromagnetic radiation. This property leads to an effective reduction of possible multipolarities inherent, in general, to nuclear radiation; 2. unique, due to the lasers’ very high intensity, is the possibility of multiphoton absorption (emission). This fact facilitates the resonance matching between DE and De. These two essential aspects of laser radiation will be focused on below in the study of the role of laser radiation in low-energy nuclear physics.

Fig. 2. Diagrams describing the “Compton” effect on the bound electron (IEB mechanism).

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Turning now to the excitation of low-lying nuclear states (generally nonresonant) by c-rays, we write the matrix elements M and M corresponding to the diagrams in Fig. 2a and Fig. 2b: a b S f FDH DInTSnDH DiT */5 c , (5a) M "+ a e !e !+u n i L n S f DH DnTSnFDH DiIT c */5 M "+ . (5b) b e !e #+u n f L n We have used the following notations: DiT, DnT, D f T describe the initial, intermediate, and final states of an electron with energies e , e , e , respectively. I and F denote the initial and final states of i n f the nucleus with energies E and E . H is the Hamiltonian describing the interaction of an I F */5 electron with a nucleus and H is the Hamiltonian describing the interaction of a photon with an c electron. Conservation of energy gives: e #+u "e #E (E ,E !E ). i L f FI FI F I We confine ourselves here to the non-covariant perturbation theory (see Eqs. (5a) and (5b)) since the exact calculation requires construction of the relativistic Green’s function of an electron in the field of an atom. The physically simpler treatment above is transparent and effective. We concentrate on the case where the laser photon energy is much smaller than the rest energy of the electron, +u (mc2 or, more precisely, smaller than the electron binding energy in the shell, L +u [(aZ)2mc2, where a"e2/+c is the fine-structure constant, m is the electron mass, and Z is the L atomic number. Inspection of the denominators in the matrix elements (Eqs. (5a) and (5b)) shows that the main contribution to M results from the summation over positive energy e values of the states DnT, a n whereas the main contribution to M comes from negative values of e . That means that the b n excitation of a nucleus described by Eq. (5a) occurs mainly through inelastic scattering of a free electron (e ' 0) (free—free transitions). Therefore, Eq. (5a) gives the main contribution at suffin ciently high incident photon energies (+u <(aZ)2mc2). On the other hand, Eq. (5b) is more L important when a nucleus is excited in transitions of an electron between bound—bound and bound—free atomic states. This indicates that this diagram gives the main contribution at the energies we are interested in (+u [ (aZ)2mc2). Calculations [7] indeed confirm these physical L reasonings. In the calculation of the cross section 2p 1 1 p" + + + DM D2o(e ) , (6) c " f + 2J #1 2J #1 I i i mimf mImF (Here o(e ) is the density of states of the final electron, m , m and m , m are magnetic quantum f i f I F numbers of the initial and final states of the electron and the nucleus, respectively; the summation index i denotes the occupied states in the (initial) electron shell) the approximation of one resonance level n [7] is used which is more effective and reasonable in the case of bound—bound transitions. If the energies of the nuclear excitation and of the electronic transition are sufficiently close, a significant increase of the cross section is achievable. The resulting cross section includes the probability of c-radiation of multipolarity ¸ emitted by the nucleus, and the well-known matrix elements for the electronic electromagnetic transitions M(E¸) and M(M¸) (E and M in M refer to electric and magnetic transitions, respectively). For

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Table 1 Parameters of nuclear (E (keV), J PJ ) and atomic (e , j Pj ) bound—bound transitions, D"E !E #e !e FI I F jn,ji i n I F jn ji "!E #e , and calculated Compton excitation cross sections for some long-lived isotopes [7] FI jnji Isotope

E , keV FI

J PJ I F

j Pj i n

e , keV jnji

D, keV

p , lb c

119Sn 50 129J 53 125Te 52 161Dy 66 185Tm 76 189Os 76 193Ir 77 197Au 79 201Mg 80 193Pt 78

24 27 36.5 43.8 8.42 69.60 73.1 77 1.57 12.7

1`P3` 2 2 7`P5` 2 2 1`P3` 2 2 1`P3` 2 2 1`P3` 2 2 3~P5~ 2 2 3`P1` 2 2 3`P1` 2 2 3~P5~ 2 2 1~P3~ 2 2

2s P1s 1@2 1@2 2s P1s 1@2 1@2 3s P1s 1@2 1@2 2s P1s 1@2 1@2 3s P2s 1@2 1@2 3s P1s 1@2 1@2 3s P1s 1@2 1@2 3s P1s 1@2 1@2 2p P2p 3@2 1@2 4s P2s 1@2 1@2

24.74 27.88 30.81 44.74 7.81 70.82 72.85 77.4 1.93 13.04

0.74 0.88 !4.7 0.44 !0.61 1.22 !0.25 0.4 0.36 0.34

4 5 0.14 110 196 26 9 10 17 62

higher multipolarities, the cross section drops sharply (&(R /a)2L, where R is the radius of the A A nucleus and a is the radius of the electron shell). Thus, the maximum value of p occurs for c transitions with the lowest possible multipolarity ¸. Expressions for the matrix elements M(E(M), ¸) are given in the literature (see, e.g. Ref. [8]). The summation in Eq. (6) is carried out over the occupied states in the initial and final electron shells, and the choice of the characteristics of the intermediate state DnT (e.g., its angular momentum j , etc.) depends on the initial state DiT. In n the calculations the widths of the atomic and nuclear levels are usually neglected in comparison with the parameter D,E !E #e !e . This parameter enters in the cross section in the form I F n i [(E !E )/D]2. From this, it is evident that for the bound—free transition case, e (e , D+E !E , I F i n I F i.e., the resonance condition is not fulfilled. The largest cross section occurs for the bound—bound atomic transitions. This case is most effective if E is sufficiently close to the energy of the electronic transition (D is small). M1 IF transitions are very important. For example, for the cross section of Compton excitation of the isotype 110Sn with the M1 transition 1` P 3`, E "24 keV, the atomic transition 2s P 1s 50 2 2 FI 1@2 1@2 and D"0.74 keV, one obtains p "4 lb. In Table 1, we show the parameters for nuclear and c atomic transitions and the resulting cross sections p for several long-lived isotopes. The energies of c the nuclear transitions have been taken from Ref. [9] and those of the electronic transitions from Ref. [10]. We see that the cross sections are fairly large (&(10—102) lb). The case of the bound—free transitions is less effective. Here, p is of the order of (10~2—1) lb. For c example [7]: 57Fe, p "5]10~2 lb; 83Kr, p "10~1 lb; 129Xe, p "1 lb. 26 c 36 c 54 c The conclusion from the above considerations is that the relatively low-lying nuclear vibrational, collective states can be excited at a rate of (10~7—10~9) per photon. Of course, the experimental situation is very difficult, due to the high background associated with the intense primary and secondary fluxes of X-rays. The fact that the energies of the X-rays and the nuclear radiations do not coincide (the difference reaches several keV) can perhaps be used to detect and to study low-lying states in nuclei. Concluding, we mention for the sake of

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completeness, that in the case of higher energy c-rays (+u <(aZ)2mc2), where Eq. (5a) gives the L main contribution, the cross sections are much lower, typically of the order of 10~34 cm2. The above examples which do not always have a direct connection with the nuclear physics conducted by UV and optical lasers (the main topic of interest in the present work), teaches us that atomic physics is merged with nuclear physics due to the coupling of electromagnetic radiation with electronic shells. Lasers with their high intensity have the potential to enhance and sometimes induce nuclear (weak and electromagnetic) transitions. Below we will illustrate these issues in several examples (Sections 3—6). 3. Laser-induced nuclear anti-Stokes transitions 3.1. b-transitions of nuclei Lasers, like other sources of electromagnetic radiation, provide the possibility of enhancing transitions from excited nuclear states and of obtaining information on specific nuclear transitions unaccessible otherwise. Consider, as an example, a three-level nuclear system as illustrated in Fig. 3. First, we concentrate on the b-transition: The nucleus initially is in the isomeric state a from which it b-decays (very slowly) to its ground state b with the rate cw . Under the influence of an external electromagnetic ab field with frequency u one can induce a two-step b-decay of a to b: a virtual excitation of the L nucleus to the level c by absorption of a single (or, sometimes multiple) photon with energy +u and L rate c%- of the laser field and subsequent b-decay from cPb with rate cw . ca cb This scheme gives not only the possibility of increasing the rate of the b-decay to the state b (if cw ' cw ; of course, cw ;c%- ) but provides information on the b-transition cPb which is somecb ab cb ca times not accessible [11]. Before turning to the more interesting cases where all the transitions discussed above are of electromagnetic nature, thus eliminating the condition cw ;c%- (unavoidable for b-transitions), it is cb ca worth mentioning that claims (1977—1983) about strong modifications of forbidden and even allowed b-decay total rates [12] by very intense, long-wavelength, coherent electromagnetic radiation were not confirmed, although their discussion has attracted the interest of many authors. The latest investigations (1987) have demonstrated that there is no laser-field-dependent enhancement of the total b-decay rates in the long-wavelength limit, although the spectra are greatly modified (for a more complete review and the resolution of the controversies in some previous studies see Refs. [13,14]. However, we note that, in principle, the rate of another nuclear b-process, the orbital electron capture by a nucleus (so-called K-capture) is naturally influenced by the laser radiation, because the probability of this process depends on the electron density in the vicinity of the nucleus. Therefore, the K-capture process can be sensitive to the influence of a strong electromagnetic field. Particularly, in contrast to the case of b-decay where the total rate does not depend on the nuclear spin orientation,4 for the orbital electron capture, due to the hyperfine 4 Of course, due to the parity non-conservation, the probability of the particle (e~ or l ) emission parallel to the nuclear e spin in b decay differs from the probability for the emission in the opposite direction. However, the total rate does not depend on nuclear spin orientation simply because of the isotropy of space (rotational invariance).

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Fig. 3. Scheme of an anti-Stokes b-transition on the nucleus.

interaction between the electron and the nucleus, the total rate of K-capture depends critically on the orientation of the electron and nuclear spins, and therefore, on temperature [15]. The rate of the orbital electron capture from an unfilled s orbital of a free atom or ion at low temperature (k¹;d, where d is the hyperfine splitting) depends on the sign of the hyperfine constant h [15]. The application of a laser field with frequency u "d/+ can influence the rate. In L the particular case of negative h, when the orbital electron capture practically does not take place, it is possible to induce this process by applying the resonant radiation with u "d/+. Large effects L are expected especially for hydrogen-like ions since the competing decays from other bound electrons are absent.5 In principle, properly chosen radiation could cause a modulation of the electron capture rate and the subsequent characteristic X-ray emission could serve as a monitor for observing this process. As a justification of the old wisdom that “the novel is well forgotten old knowledge” and for the sake of completeness, we note that a very similar idea was proposed almost two decades ago [16] for modifiying the population of hyperfine structure sub-levels of k-mesic atoms through resonant laser radiation, and the subsequent change in the nuclear k-capture probability. 3.2. Electromagnetic transitions of nuclei Turning now to the case of electromagnetic nuclear transitions, we again consider the three-level nuclear system with electromagnetic transition rates c , c , and c and an applied laser field of ab ca cb frequency u to promote the aPc transition (Fig. 4) from the isomeric state ma to the level c. L It is easy to recognize that this scheme resembles the scheme of the famous Lamb—Retherford experiment on the Lamb shift measurement of the hydrogen atom where level a corresponds to the 2s atomic state, level b corresponds to the 1s ground state, and level c corresponds to the 2p 1@2 1@2 1@2 state which is excited by electron beam bombardment. The fact that, in contrast to the b-decay, c + c , opens new possibilities. The state c can decay ca cb to states a and b by the emission of radiation and also by internal conversion (IC) to other, not specified states with total decay rate c "c #c #c #2. c ca cb IC 5 The orbital electron capture probability rapidly decreases with an increase in the number of competing atomic shells from which an electron can be captured.

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Fig. 4. Scheme of an anti-Stokes electromagnetic transition on the nucleus.

We take the interaction between the external radiation field and the system from Eq. (1) with E "E cos u t. The laser field can be considered homogeneous (spatially uniform) as its L 0 L wavelength is much larger than atomic (and, of course, nuclear) dimensions. The equations of motion for the probability amplitudes of states a and c in the interaction representation (so-called, Bethe—Lamb equations) [17—19,6] are: aR "!1c a!(i/2+)» e~*Dt, cR "!1c c!(i/2+)» e*Dt , (7) 2 ab ca 2 c ac where » is the transition dipole matrix element between the states a and c, and ac D"(E !E )/+!u is a detuning frequency. In the following, we concentrate on the most c a L interesting case where the spontaneous decay of state c to the isomeric state a is negligible (c + 0, ca no backcoupling cPa). Solving then Eq. (7) with the initial conditions a(0)"1, c(0)"0, we have: Da(t)D2"e~cabt,

A B

Dc(t)D2"

D» D 2 e~cabt#e~cct!2e~ct cos Dt ac , 2+ D2#d2

(8)

with

AB c

(c $c ) c . " ab 2 d

(9)

Since we are interested in the case where the state c decays much faster than the isomeric state a (c
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the expression [19]

A B

g"

D» D 2 c 1 ac cb . 2+ c D2#c2/4 ab c

(10)

For the matrix element » in Eq. (10), assuming that the nucleus is spinless, one finds ac (11) D» D2"1(eE )2DScDrDaTD2 ac 3 0 with E2"4p+/j /, where / is the photon flux and j is the wavelength of the radiation. 0 L L The nuclear dipole matrix element ScDrDaT can be expressed by the lifetime c~1 using a semiclassica cal calculation of Einstein’s A-coefficient in the condition of thermal equilibrium [18]. The resulting relation coincides with the result of the quantum mechanical Weisskopf—Wigner theory for the spontaneous decay [20]: + j3 caC / D» D2" ac 2p j ca L with j "2p+c/(E !E ) and the width C "+c for the transition cPa. ca c a ca ca Finally, one obtains for g:

(12)

A B

+/ j 3j2 C C ca L cb ca g" (13) C j 8p (+D)2#C2/4 ab L c where we introduced, instead of the rates c, the corresponding widths, C"+c. The factor (j /j )3 in Eq. (13), which has been left out of the first theoretical study of this process, ca L makes the experimental situation very difficult. For instance, for laser photon energies of +u "10 eV and a nuclear transition level difference E !E "100 keV this factor is 10~12, which L c a leads to very small g values for moderate laser intensities. The effect is noticeable only for small nuclear level spacings ([ 10—10~2 eV). The situation can be improved using resonant laser radiation with +D;C /2. As can be seen from Eq. (10), in this case and under the conditions of c moderate laser intensities (D» D(C , but D» DZ(C C )1@2, i.e., IZ1014 W/cm2) g can be of the ac c ac c ab order of unity. This means that the rates of two-step and one-step processes are equal. For higher intensities (D» D5C ) the anti-Stokes process dominates. This condition corresponds to ac c IZ1019 W/cm2 (see Ref. [19] for details). Under these conditions, Eq. (7) must be solved precisely. The resulting picture looks as follows (again D;c /2, c ;c ). The system oscillates between c ab c the levels a and c. In contrast to the weak field case (D» D;C /2), where level c, as well as a, decays ac c with the slower rate c , now this laser-excited state decays with the faster rate c + c /2. The long ab c lived isomer a is depleted via c to b on a very short time scale. For the case of the isomeric state 111mPd one obtains c~1 + 1.5]10~11 s (C "2.3]10~6 eV (E1 transition), C "4.3]10~5 eV c ca cb (E2 transition), C + 4.5]10~5 eV). The condition D» DZC corresponds to intensities c ac c IZ1.3]1019 W/cm2. The total probability for the anti-Stokes process in a very strong field approaches its maximal value C C 1 D» D2 ac N!S(¹)" cb + cb , ¹< . j 2C C C #D» D2 2C c c ab c c c ac

(14)

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Hence, a pulse of + 0.1 ns with the above intensity ensures that all irradiated isomeric nuclei decay. The main experimental problem with the resonant excitation is the severe condition +DDD;C /2 which imposes very strong constraints on the bandwidth Du/u of the radiation c necessary for g to exhibit a resonant peak. The associated requirements on the beam monochromaticity can be somewhat compensated for at the expense of high intensity. It is important to note that the theoretical treatment with only a single intermediate atomic state is reliable only very close to the resonance. Concluding this section, we notice that similar results to those obtained here in the case of a strong radiation field (D» DZC ) apply to the nuclear b-decay [11] where, as mentioned above, ac c c is always much larger than c . For the c-transitions of nuclei one can find nuclei with c ;c . ca cb ca cb In this specific case the spontaneous decay of the intermediate state c tends not to go back to the isomeric state a but leads into the final ground state b.

4. Laser-assisted internal conversion 4.1. Theoretical background Internal conversion (IC) is a clear example where nuclear physics (deexcitation of a nucleus) is merged with an atomic process (ejection of an electron from an atomic shell). Thus, in the spirit of the present review, it is quite appropriate to consider IC in detail, especially because the theoretical approaches developed for studying laser-assisted IC are common to some other nuclear-atomic processes where lasers play an important role. In this process, as already pointed out in the Introduction, the effective reduction of the multipolarity of nuclear gamma-ray transitions plays a crucial role. However, there exists another issue where lasers have a substantial influence: changing the atomic surroundings of a nucleus which affects IC (we already encountered this kind of laser influence in Section 3 when we considered the role of lasers in the orbital electron capture). The influence of a laser field on IC has been investigated firstly just from this point of view [21]. For weakly bound electrons participating in the IC, the laser-assisted removal of one of the electrons that significantly contribute to IC leads to a significant decrease of the coefficient of IC.6 We consider here the more direct and efficient role of the intense laser radiation field on the IC process. In general, if the IC process is non-zero in the laser free case (a&3%% O 0) the change of IC a due to the laser influence is small (a&3%%
6 The coefficient of IC a is defined as the ratio of the probability ¼ for IC and the radiative transition probability IC IC ¼ for a given nuclear transition E(M)¸ with electric (magnetic) radiation multipolarity ¸. IC coefficients can be c calculated by well-developed methods [8].

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not take place (a&3%%"0). If the photon energy of the laser beam +u is of the order of DDD, the IC L originally forbidden IC process may occur via absorption of the necessary numbers of laser quanta. Thus, the electromagnetic radiation (laser) essentially accelerates the nuclear c-decay rate and, more importantly, induces IC which is absent otherwise. We first consider some theoretical issues connected with the treatment of laser-assisted IC. The theoretical description of the electronic states under the joint action of the Coulomb potential of the nucleus and the intense radiation field is only approximate; no precise analytical solution is known. We confine ourselves here to a simplified non-relativistic model [23—27] for the system “nucleus#electron#intense radiation field” with Hamiltonian7 H"H #H , where H " 0 I 0 H #H is the sum of the Hamiltonians for the unperturbed nucleus H (the explicit form of 0N 0% 0N H does not enter in the following treatment) and a single electron H participating in IC, 0N 0% H "!Ze2/R#(1/2m)(p!(e/c)A)2 . (15) 0% Here, A is the vector potential of the external laser field which will be specified later; R, p, m denote electron coordinates, momentum and mass, respectively. The interaction between the nucleus and electron is of the Coulomb type Z e2 Ze2 H "! + # , (16) I DR!x D R p p/1 where x denotes the nuclear pth proton radius vector. Relations (15) and (16) reflect a strong p simplification, first of all, because the interaction between the electron and the intense radiation field is comparable to the binding energy of the electrons in the atomic shell. However, for the inner electrons (K, L-shells) the Coulomb potential of the nucleus still dominates and the modification of the initial electron states by the laser field can be treated perturbatively. For the final (free) electrons one can use the well-known Volkov solution for the charged particle in an electromagnetic field [28]. Furthermore, since in this model one only uses Eq. (16) for the electron—nucleus interaction [29], the photon exchange between the nucleus and the electron shell does not appear explicitly. Such a simplified treatment emphasizes the corrections which at least are necessary to improve the theory of this process: 1. Use of the relativistic Dirac Hamiltonian for the bound electronic state “dressed” by the intense radiation field. This improvement requires mainly numerical calculations. 2. Taking into account the effect of the nuclear size, and 3. The screening of the Coulomb potential of the nucleus by inner-shell electron clouds. Item (1) is most important and has no exact analytic solution (and the same holds for Eqs. (15) and (16)). Thus, approximations are unavoidable, and we will be confined here to the simplifications described above. Since the final free electron is influenced by the intense laser field, the

7 There are several other approximate models in the literature, e.g., a model where the Coulomb potential of the nucleus is replaced by a spherical harmonic oscillator [22].

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corresponding wave function is based on Volkov’s solution of the Dirac [28] and Klein—Gordon [30] equations for an electromagnetic plane wave with specially chosen initial conditions. Volkov’s paper has not received much attention for a long time, presumably due to the absence of appropriate radiation sources at that time. However, with the invention of the laser, the theoretical interest has been renewed, and different exact and approximate solutions have been given with various choices of initial conditions and different methods for the theoretical description (classical relativistic and non-relativistic, quantum mechanical and quantum electrodynamical [31—35]). We give here the original Volkov solution for the Klein—Gordon (KG) and Dirac (D) equations. With the gauge for the radiation field vector-potential A (x)"0, we have: 0

C

P

D

ie b t (x)"e~*p>xexp ! db@ [2pA(b@)!eA2(b@)] , KG 2p ) q ~= t (x)"t (x)[1#e(q ) A)/(2p ) q)]u(p) , D KG

(17)

where p(e, p) is the four-momentum of the charged particle, q(u, q) is one of the laser photons, u(p) is a Dirac spinor, and b,q ) x"ut!q ) x. The scalar product p ) q"eu(1!€ ) q/DqD) is in the nonrelativistic case (v;1) simply given by mu, and t (x) + t (x)u(p), and finally D KG p<eA"eE /u, where E is the amplitude of the laser electric field. 0 0 Under these conditions, the initial and final electronic states appear as two different approximations of the solution of the Hamiltonian (Eq. (16)): t (R, t)"/ (R)e~*EBt@+ exp((ie/+c)A ) R) , i njj

C

PC

D D

2 i t e p! A(t@) dt@ u(~)(R) . t "exp ! c f 2m+ c

(18) (19)

Here / (R) is a hydrogen-type solution with quantum numbers n, j, j and energy eigenvalue njj E and u(~)(R) is an electron Coulomb wave function. In these relations (Eqs. (18) and (19)) we B C restored, temporarily, + and c. For the vector potential of the radiation field we write for the circular (c) and linear (l) polarization:

A

Acl"

B

a[eL cos b#eL sin b] 1 2 . aeL cos b 3

(20)

The unit vectors eL , eL , eL , perpendicular to each other, define the frame of reference, and 1 2 3 a"cE /u. 0 Inspection of Eqs. (18) and (19) shows that if we add to the condition p<eE /u"eE j /2pc 0 0 L (with momentum p"+k (k — wave number)) the condition (e/+c) A ) R;1, which is well satisfied for the inner shells, we achieve a great simplification: (i) the space-dependent and time-dependent parts of the S-matrix element can be separated, (ii) the space-dependent part is the same as in the laser-free case. Notice that the first condition (p<eE j /2pc) can be rewritten as the condition that the kinetic 0 L energy of the final electron is larger than r Ij2/2p2 + 10~14Ij2 + 10~26Ij2 eV where r is the 0 L L 0

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classical radius of the electron, and the laser intensity I is measured in W/cm2 and j in cm. Thus, L this number is extremely small (j &10~5 cm) for any reasonable intensity. (See Appendix A for L a dimensional analysis of scales characterizing the laser interaction with an electron.) Using the above approximation, one is able to calculate the laser-induced IC coefficient aL,-!4 for a transition njj of multipolarity ¸ and for an electronic state of quantum numbers n, j, j. The calculations include the point nucleus approximation (Dx D(DRD), the expansion 1/DR!x D of Eq. (16) in terms of p p spherical harmonics, and the Wigner—Eckart theorem which permits to express the matrix element of the electric multipole moment of order l, m through its reduced matrix element between the nuclear states I and F [36]. Because this reduced matrix element decreases rapidly with increasing l, we are confined to the lowest l determined by the multipolarity ¸ of the nuclear transition, as usual. The further evaluation involves the use of the addition theorem for spherical harmonics and the orthogonality of the 3j symbols that enter here, and the averaging (and summing) over the magnetic quantum numbers of the initial (final) nuclear and electronic states, etc. The approximation of neglecting the screening of the nuclear Coulomb potential is improved somewhat by using instead of the charge eZ the effective charge eZ (n)"en(DE D/R )1@2, where %&& B y R "e2/2a , n is the principal quantum number of the electronic state with binding energy E , and y B B Bohr radius a . The effective charge eZ corresponds to the charge of the hydrogen-like nucleus B %&& where the electron has the same binding energy as in the real atom with the same quantum numbers. As we emphasized above, we are interested in the situation where a&3%%"0, i.e., D(0. For our IC purposes, of course, the most important case is the case near threshold: +u [DE D. The laser field IF B must be so intense that the interaction energy of the electrons with the laser field becomes comparable to the binding energy of the electrons in the shell. This gives a condition for the laser field intensity

A B

I5

Z 4 %&& [+u (eV)]26.31]108 W/cm2 . L n

(21)

If one uses the small electron momentum assumption pna /Z ;1, then DDD;DE D. B %&& B The final result for aL,-!4 depends on the following parameters: Nuclear (and atomic) parameters: njj E "E !E , ¸ — multipolarity of the nuclear c-rays, Z , and D"E !DE D. The laser parameters c I F %&& c B enter in the following combinations: r "D/+u , I, and r"r !2e2I/m+u3. The second term in 0 L 0 L the parameter r multiplied by +u gives the laser pondermotive potential which in some cases L reduces the effectivity at very high intensities I. For the case D(0, the aL,-!4 is expressed in terms of the threshold IC coefficient aL,T) which is njj njj theoretically calculable. Thus, if +u is comparable to DDD, then, for the laser-free forbidden case, IC L can start after absorption of the corresponding number of laser photons demanded by energy conservation. As a result, we have [24] aL,-!4"aL,T)¹ c , njj njj (l )

(22)

where ¹ c " + ¹ c (b ) (l ) N;~r (l ) N

(23)

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with index c(l) corresponding to circular (linear) polarization. N in Eq. (26) is a positive integer. Here b "b (N#r)1@2 with N 0

S

A

B

eE +u 1 I 1@2 L "1.07]10~6 b " 0 . (24) 0 mu2 R a (+u )3 L y B L As before we measure I in W/cm2 and +u in eV. The summation in Eq. (23) goes over the L number of laser photons absorbed by the electronic shell. In Eq. (23),

P

1 bN ¹ (b )" J (x) dx , (25) c N 2N 2b N 0 where J (x) is the Bessel function. The corresponding formula for ¹l(b ) includes the generalized N 2N Bessel functions J (b, d): N d 1 ¹l(b )" J2 b x,! dx . (26) N N N 4 0 For weak laser fields (b (1, or I(1012(+u)3), applying the small argument expression for the 0 Bessel and generalized Bessel functions, we obtain

P A

A B A B

B

b .*/ 2N.*/ (2N )!! N .*/ , (27) 2 (N !)2(2N #1)!! .*/ .*/ b .*/ 2N.*/ 1 , (28) ¹l" N (N !)2(2N #1) 2 .*/ .*/ where N is the minimum number of laser photons necessary for the onset of the originally .*/ forbidden IC process. As b2 & I,¹ c & IN.*/. N () Curves exist in the literature [24]l where the dependence of ¹(b , r ) (Eqs. (23) and (25)) on the 0 0 laser intensity is given for different materials and lasers with I(1014 W/cm2. Unfortunately, for higher I ([ 1020 W/cm2) such curves do not exist. It is important to note that for larger values of I ('1014~16 W/cm2) the hindering effect of the laser pondermotive potential [37] must be taken into account. This potential 2e2I/m+u3 ) +u (+2 eV at I"1013 W/cm2) that enters into the L L expression for r"r !2e2I/m+u3 leads to the necessity to increase N in the summation of 0 L .*/ Eq. (23) and, thus, decreases the probability for this process. Fig. 5 shows the above mentioned curves for ¹(b , r ). 0 0 The “(anti) efficiency” of the pondermotive forces depends on the precise knowledge of the ionization thresholds of the electron states. The values for the ionization thresholds calculated in the non-relativistic approximation can be changed appreciably in more precise calculations. The actual ionization threshold may be larger than the value suggested by the simplified model. Particularly, it was demonstrated [38] in a non-relativistic calculation for the hydrogen atom that the electron stays attached to the atom even at very high laser intensity. This phenomenon certainly needs more work to be understood. Recently, more refined calculations reduced the magnitude in the laser intensity [39]. Thus, a more detailed study of the ionization process and its effects on the electronic states and their widths is required. ¹" c

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Fig. 5. The quantity ¹(b, D/+u ) defined by Eq. (25) (circular polarization) is defined by the relation I"8.73 ] 1011b2. L 0 The curves a—d correspond to +u "5 eV. The nuclei 235U and 183W are denoted by their electronic shell (see Table 2). L The curves c and g denoted by K correspond to the 105mAg (K shell) [24].

4.2. Some examples: N vs. I Consider as an example for the above considerations [25] an isomeric E3 transition, 105mAg, with E "25.470 keV. The binding energy DE D for the K shell is 25.524 keV [40]. Thus, IF B D"!44 eV, and IC is forbidden for the laser-free case. In the model considered above it is possible to calculate the laser threshold IC coefficient a3T) "152 [25]. 1 11 0 The comparison of this calculation with the general, 2relativistic IC theory [41] leads to an agreement within 20%. For a laser with +u "6.42 eV, i.e., for the number of laser photons N"7 necessary to start IC, L and I"2.3 ] 1014 W/cm2 (b "1) one obtains a3L!4 + 10~13, an unobservably small value. For 0 1 121 0 the hypothetical value of +u "30 eV which would lead to N between 1 and 2 and with L I"2.4 ]1016 W/cm2 we would have a3IC "0.16 (N"2). 1 121 0 Consider now the two characteristic cases with D(0: the isomeric nuclei 235mU and 183mW, for which atomic and nuclear data exist [40]. In Table 2, the necessary data [40] and the calculated

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Table 2 Atomic and nuclear data and results of the calculations of aL,-!4 [24] for +u "5 eV, I"b2(+u )38.73]1011W/cm2, and njj L 0 L b "eE /u J2/+u ) mc2 0 0 L L Atom

235mU, E "73.5 eV IF

Electron shell Binding energy DE D, eV B D"E !DE D, eV IF B n I1@2 b"1.07]10~6 Z (+u )3@2 %&& L b 0 I, W/cm2 ¹ aL,-!4 njj

5d (O ) 3@2 4 105 !31.5

183mW, E "544 eV IF 5d

(O ) 5@2 5 96 !22.5

0.55

0.58

5 2.7]1015 0.27 2.1]1015

5 2.7]1015 1.2 2.7]1015

4s (N ) 1@2 1 592 !48 0.44 9.5 1016 3.6 3.3]102

results for aL [24] are given. For 235mU, E "73.5 eV; for 183mW, E "544 eV. An energy of njj IF IF +u "5 eV was taken for the laser photons. L The increase of the laser intensity (e.g., to I"1020~21 W/cm2) will lead, for the near threshold cases, D(0, to a drastic increase of the laser-assisted IC coefficients aL,-!4. The estimates lead to njj enormous values for aL!4. Unfortunately, these estimates did not properly take into account the role njj of laser pondermotive forces with their hindering role at very intense fields. The detailed and accurate study at very high laser intensity will unavoidably demand the aid of modern, powerful computers. Before turning to the experimental issues related to laser assisted (and induced) IC we would like to mention that via the electron bridge mechanism it is also possible to accelerate the IC process and to obtain higher values for the IC coefficients. This subject will be considered in Section 5 which is devoted to the study of this mechanism. 4.3. Experimental problems The main problems faced in experimental studies of laser-assisted and induced IC are the production and collection of a sufficient amount of isomeric nuclei and their irradiation by intense (I51014 W/cm2) and relatively long laser pulses. Typically, one experiences a high background of ions and especially, electrons. Due to the high electron background, it may be more useful to detect the soft X-rays which are emitted in the recombination processes when the vacancies left by the conversion electrons are occupied by electrons from higher shells, rather than to detect the slow IC electrons. The number of emitted soft X-ray photons N can be estimated as c N "mAt a-!4/aT , (29) c IC IC where aT is the total laser free IC coefficient, a-!4 is the laser induced IC coefficient for the shell IC IC under study (see Table 2), A is the activity of the sample, t is the total irradiation time (duration of the laser pulse in an ordinary case), and m is the efficiency for soft X-ray detection.

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Taking, e.g., a-!4/aT + 0.2 (for 183mW, a-!4"3.3 ] 102, see Table 2), m"10~2, t + 10~9 s, we obtain the necessary activity to obtain N + 1 in one laser pulse A + 6 ] 1011 Bq. The c problem of producing a sample of low density but high activity can be solved using a method similar to the one proposed in Ref. [42] to sort out the isomeric nuclei embedded in molecules, e.g., in our case, WO [24]. This method uses the fact that some isomeric nuclei have angular momenta 3 higher than the ground state which leads to different molecular spectra due to the difference in fine interactions. The 183mW isomer can be produced by thermal neutron capture: 182W(n, c)183mW(p & 2 mb) (the abundance of 182W is 26.3%). Besides the fast isomer separation one needs a soft X-ray detection method of high enough resolution in order to be able to select soft X-rays from the recombination of the atomic shells (N , N , N in the case of 183mW). 1 2 3 Finally, the irradiation time t may be increased to force the laser light to run back and forth through the sample multiple times, while a laser active material compensates for the losses, thus keeping the intensity approximately constant. Concluding this section, we notice that the specific IC process, where the shell electron remains in the discrete state, can be considered also from the point of view of the Electron Bridge mechanism which we will study in the next section. Thus, the laser’s role in the elimination of the mismatch between atomic and nuclear level differences (resonant IC) will be clearly seen.

5. Electron Bridge mechanism as a source of deexcitation of nuclei 5.1. Experimental observation A systematic way to use the electron shell as a mediator between the laser beam and the nucleus for studying nuclear low-energy properties is provided by the so-called Electron Bridge (EB) mechanism. The idea was born long ago, in 1958, in the former USSR [43] and has been developed more recently into a useful tool [44,45]. The EB mechanism effectively transfers the energy of a nuclear transition to the atomic shell which, passing through excited intermediate state(s), emits monoenergetic c-rays, thus providing the nuclear deexcitation. It is a third order process with respect to the electromagnetic interaction (see Fig. 6 for the corresponding Feynman diagram). The energy of the emitted c-rays E is E !(e !e ) with e "!B , where B is the binding energy of c IF f i i(f) i(f) i(f) the electron in the atomic shell DiT(D f T). When B 'B (e 'e ), E (E , E !E , one obtains i f f i c IF I F the so-called “Stokes line” in the deexcitation spectrum. The inverse EB (denoted as IEB below) mechanism corresponds to the analogous diagrams shown in Fig. 6 with the external radiation absorbed by the electron of the initially unperturbed shell, thus, providing a way to excite nuclei. The first experimental observation of the nuclear deexcitation via an EB mechanism was achieved in 1985 by the Zagreb—Ottawa group [46]. In Ref. [46] the deexcitation of the 30.7 keV (lifetime ¹ & 13.6 y) isomeric level of 93Nb was studied and 28.2 keV energy photons were 1@2 observed as a result of the EB mechanism involving the initial ¸-electron states (with binding energies B 1"2.675 keV, B 2"2.426 keV, B 3"2.368 keV) and the final state N (B 5(30 eV) or L L L 5 N

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Fig. 6. Feynman diagrams corresponding to the Electron Bridge mechanism.

Fig. 7. (a) The spectrum obtained after subtraction of contributions from room background, source impurities, and external bremsstrahlung. The full circles represent the experimental data while the full line represents the total estimated spectrum N(E ). The dashed line represents the ICE contribution in the region of interest; at lower energies there are only c ICE contributions. (b) The spectrum produced by the inelastic-electronic-bridge effect with 28.2 keV photons [46].

higher electron states. The higher L states were used instead of the K state, since excitation of the K shell gives photons with energies below the niobium K X-ray energy, and the associated background is too high.

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It is necessary to point out that the EB processes are usually accompanied by processes connected with IC channels8 which, as a second-order process, are more probable, in general, than the third-order process under study (e.g., the external bremsstrahlung associated with IC of the 30.7 keV level). There are also other contributions from the internal Compton scattering produced in the deexcitation of the initial nuclear level, etc. However, in Ref. [46], these sources of background were found to be negligible. Fig. 7 shows a spectrum of c-rays after background subtraction. The first theoretical estimate given by the authors of Ref. [46] involves the numerical solution of the inhomogeneous Dirac equation for an electron in the nuclear Coulomb field and hydrogen-like atomic wave functions. The ratio g calculated for the probability ¼(3) of the third-order transition based on the diagrams c Fig. 6 and the probability ¼(1) for the direct nuclear deexcitation was found to be c g "¼(3)/¼(1)"0.19, whereas the experimental result was g"0.070 $ 0.018. Later, more 5) c c detailed theoretical calculations gave g "0.069 [47]. 5) There exists a second experimental result [48] based on the study of the decay of the isomeric state 193mIr with energy 80.27 keV (¹ "10 d). This isomeric state was obtained by ir1@2 radiation of a 99% enriched sample of 192Os with thermal neutrons via the chain 192Os(n,c) P 193Os b(0.3¤) P 193mIr. Experiments conducted under unfavorable conditions due to the T1@2/30 ) high background associated with the IC channel (+105¼(3)) gave for g the value + 0.21. c Theoretical estimates for L electrons yielded g"0.18 which should be taken only as the lower bound due to the neglect of the additional contributions from higher shells (M, N,2). Estimates in Ref. [48] give for g(M, N,2) & 0.1—0.2. Again, more precise calculations are in agreement with the experimental value 0.21 [47]. Before turning to the role of lasers in the nuclear deexcitation process provided by the EB mechanism, it is worthwhile to note the following. We have already mentioned that the probability for IC increases with the effectiveness of the EB mechanism. In turn, the existence of the EB channel leads to a significant decrease in the IC coefficient with respect to its so-called “tabulated” value where the EB mechanism is not taken into account. For instance, for the isomeric state 235mU, the “tabulated” IC coefficient (a (E3)+1019) is reduced by a factor of +105 [47]. IC 5.2. The role of lasers in the EB mechanism of deexcitation of nuclei Now, we will discuss the laser-assisted EB process where an electron of an atom containing an isomeric nucleus absorbs the nuclear transition photon of energy E "E and emits a number c IF N of atomic transition photons, thus eliminating the mismatch between E and e (the energy IF ni difference between the intermediate and initial atomic states). The resonantly excited electronic states finally decay by X-ray emission.

8 As it was already stressed in the pioneering papers [43] on the EB mechanism, such third order effects in a should be substantial if the IC coefficients are very large. That reflects the close connection between the IC process (with an electron in the continuum, or in the discrete state) and the EB mechanism where an additional photon is emitted.

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Fig. 8. Scheme explaining the elimination of the misfit between the atomic and nuclear level energy differences.

The schematic diagram of Fig. 8 explains the process for the case of the emission of two photons. In this scheme, an electron from the initial shell DiT “dressed” by the intense laser field absorbs a nuclear radiation quantum with energy E , emits two laser photons with energy 2+u , and then c L occupies resonantly the intermediate state DnT with energy e (e "!B , where B is the binding n n n n energy of the electron in shell DnT), and finally decays to the final state D f T (" DiT in this particular case) with emission of an X-ray photon with energy +u . Due to the dipole character of emission 9 (absorption) of the laser quanta (l"1), the laser also “transfers” angular momentum and therefore effectively reduces the multipolarity ¸ of the nuclear transition radiation to ¸"1 (¸"3 for the case of 235mU considered below), thus enhancing the probability of the process. Fig. 9 presents the Feynman diagrams describing this process. At resonance, the first diagram is dominating. We have E !E "e !e #N+u , I F n i L or, for the detuning D,

(30)

D"E !E #e !e . (31) I F i n The well-known and theoretically studied case is the isomeric nucleus 235mU with energy E "73.5 eV (angular momentum J "1`, ¹ "26 min) which has an E3 transition to the I I 2 1@2 ground state F with J "7~. F 2 Among the numerous electronic orbits of the uranium atom we choose the orbit 6s with 1@2 binding energy 71 eV (e "!71 eV), which is closest to E , as the initial electronic shell DiT. We i c describe the complex electronic structure of the uranium atom in the one-electron approximation, i.e., we neglect the splitting of shells with definite principal quantum number. At resonance this approximation is appropriate. As an intermediate atomic shell, we select the 8s shell with binding energy 2.14 eV 1@2 (e "!2.14 eV). Thus, the detuning D is 4.64 eV. The resonance will be achieved by two (four) laser n photon emission with energy 2.32 (1.16) eV, respectively. The final atomic one-electron state D f T is supposed to be the same as DiT [49]. All electronic states (DiT, DnT, D f T) are “dressed” states.

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Fig. 9. Feynman diagram describing a laser-assisted resonance deexcitation of the nucleus.

Due to the laser, not only the resonance condition is fulfilled, but the originally E3 c transition is converted into the emission of an E1 X-ray photon, thus increasing the rate of the X-ray emission significantly. The laser beam makes it possible to avoid the selection rules for X-ray emissions which are acting in the laser free case. The reader may notice the arbitrariness in the selection of the intermediate state DnT and its energy E . However, the (assumed) value of E is not so important for the phenomenon under study n n due to the associated small changes in energy, whereas a change in the order of the resonance (i.e., the number N of laser photons needed to fulfill the resonance conditions, D"N+u is crucial. L 5.3. Theoretical considerations The theoretical study of EB processes is analogous to the calculations of the laser assisted IC which were presented in Section 4. The theoretical framework was developed in the papers [26,27,49—52]. The study [49,51] is based on the Hamiltonians H and H (Eqs. (15) and (16)), and we take as 0 I substitute for Eq. (15) the dipole form (1): The interaction of the linearly polarized laser field with the electron is described classically by a Hamiltonian of type (1) H "!er ) E (t) and the L% L interaction of the emitted (quantized) X-ray field is described by the same kind of Hamiltonian H "!er ) E (t), where %9 9 E (t)"E eL cos u t , (32) L 03 L 2pu 1@2 9 E (t)"i + (33) e(ae~*u9t!a`e*u9t) . 9 » u9,e Here u and e are the frequency and unit polarization vector of the emitted radiation, a and a` are 9 the annihilation and creation operators, and » is the normalization volume, respectively. Again, as before, the influence of the other electrons is taken into account by the use of the effective nuclear charge Z (n) in the one—electron wave functions. Also, due to the expression %&& (Eq. (16)) that describes the electron—nucleus interaction, photon exchange between the nucleus and the electron shell does not appear [29]. In the treatment of the joint influence of the nuclear Coulomb and laser fields on the dressed electron in the subshell DnT, it is convenient to use parabolic coordinates [49] for the shell of

A B

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principle quantum number n, n"n #n #DmD#1, where n , n are the parabolic quantum 1 2 1 2 numbers [53], and m is the magnetic quantum number. The wave function describing the “dressed” electron state can approximately be written as [54] `= t(n, n , n , m)"/ 1 2 + J (j 1 2)e~*(En`N+uL)t@+ , (34) 1 2 nnm N nn N/~= where / 1 2 is a hydrogen-type solution in parabolic coordinates, J is a Bessel function of the first nnm N kind, N is the number of absorbed or emitted laser photons, E "e !iC /2 is the complex energy n n n of the intermediate electronic state DnT with energy e and width C , and finally the argument of the n n Bessel function is 3n(n !n )E ea 1 2 0 B. " (35) n1n2 Z +u %&& L This solution is used as a basis for the construction of the electron Green’s function and S-matrix for the process. The further procedure is analogous to the one we described for IC (Section 4). The initial and final state is taken as is- and fs-type, respectively, where i( f ) is the principal quantum number for the DiT(D f T) state R (r)½ . Here, R is the nonrelativistic radial part of the hydrogen-type i(f)0 00 i(f)0 solution with principal quantum number i( f ) and orbital angular momentum l"0, and ½ is the 00 corresponding spherical harmonics. Z for the DiT state (n"6) is equal to Z (6)"13.71, and for %&& %&& the DnT state (n"8), Z (8)"3.17. %&& As a final result of this procedure, we obtain for the ratio g"¼-!4/¼41 of the probability ¼-!4 of fi fi fi the laser assisted resonant EB mechanism and the probability ¼41 of the spontaneous c decay [29], fi the following expression: j

C D

DJ D2DI D2 c 2L~2 2e2 [(2¸!1)!!]2 ni L,ni F, (36) g" 9 (¸#1)(2¸#1) (D!N+u )2#(C2/4) u L n IF where C "C #C , and C is the natural linewidth of the state DnT for the laser-free case and n n0 nf n0 C is the laser field contribution to the power broadened linewidth, nf

P P

J " dr r3R (Z (i), r)R (Z (n), r) , ni i0 %&& n1 %&& (37)

I " dr r1~LR (Z (i), r)R (Z (n), r) . L,ni i0 %&& nL %&& For given values of i"6, n"8, ¸"3, Z (i), Z (n) we have for J and I %&& %&& ni L,ni J "!0.51a , I "4.4 ] 10~51/a4 . 86 B 3,86 B The quantity F in Eq. (36) is

A B

u 3 9k F "+ f , F"+ K K u IF K K

(38)

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225

Fig. 10. F vs. I. (a) Two photon case (+u "2.32 eV), (b) Four photon case (+u "1.32 eV). Curves in both figures are K L L numbered by K [51].

where u "u !Nu #Ku , and K is an integer. The quantities f are quadratic combinations 9K IF L L K of products of the Bessel function J (j 1 2) ) J (j 1 2) with the proper Clebsh—Gordon coefficients. K nn K nn The quantities f are maximal at K"0,$1. K Figs. 10 and 11 show the intensity I and the K dependence of F ( f ) [51]. K K At the resonance (D"N+u, N"2 or 4, +u "2.32 or 1.16 eV, respectively) assuming an L intensity I + 1012 W/cm2, and an estimated value of C + 10~5 eV (from a 2p P 1s transition of n the same energy) we obtain for the 235U isomeric state [49]. g + 6.7 ] 109 .

(39)

If one takes into account the fact that the intense laser broadens the linewidth C [51], the nf magnitude of g decreases but still remains of the order of 109, leading to expectations that the affect is accessible to experimental study, though the IC coefficient is much, much higher. We note that above we concentrated on the special case of the bound—bound atomic transitions where the final electron shell is the same as the initial one (DiT"D f T). In this case the IC strongly dominates over the EB process. Concluding this part of our consideration of the EB processes and the laser’s role in the acceleration of the radiative deexcitation of the nucleus, we emphasize that the above example clearly shows again the advantage of the use of intense lasers in this field: f due to the high intensity, lasers provide the fulfillment of the resonant conditions ensured by the multiphoton absorption (emission).

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Fig. 11. f vs K. (a) and (b) indicate different channel structure of f at I"1011.5 W/cm2 and 102 W/cm2 [51]. K K

f multipolar nuclear c-transitions are converted into the emission (absorption) of electric dipole radiation. Laser radiation transfers not only the energy to the shell to eliminate the mismatch between the atomic and nuclear transition energies, but also “transfers” angular momentum to provide the dipole character of the final radiation, which leads to an enhancement in the probability of X-ray radiation by the final electron. Turning now to the case of the resonant atomic transitions DiT P DnT P D f T O DiT, we use the simple estimates based on the argument that the EB mechanism can be considered as a kind of IC process where the shell electron, absorbing the c-radiation of the nucleus, undergoes a transition not to the continuous spectrum but rather to the discrete spectrum. We shall refer to this conversion mechanism as “discrete conversion”. Just as lasers assist the ordinary IC process, we also can speak here about the “laser-assisted discrete internal conversion”. In the case of tuning by laser quanta to remove the mismatch between nuclear and atomic transition frequencies, we refer to the “resonant discrete internal conversion”. This way of treating the acceleration of the nuclear deexcitation process by resonant laser photons presents an alternative method of the theoretical study of the EB mechanism [26,27]. It is quite useful since it provides parallels to the well developed and accepted methods used in studies of IC phenomena. The extension of the theory of IC (and “resonant” IC) to the discrete case introduces a new issue since the formal determination of the “discrete IC” coefficient a leads to a quantity with $ the dimension of energy, so that the physical interpretation of the IC coefficient as the ratio of conversion and radiative transition probabilities is lost. The reason for that is clear since the normalizations of the final state wave functions for the continuous and discrete states are different. This view of the EB process as an IC process for discrete electron final state gives for the discrete conversion coefficient g$"¼$ /¼(1) defined as the ratio of the probabilities for discrete converIC c sion ¼$ and direct nuclear deexcitation ¼(1) introduced previously, the simple expression which IC c does not depend on the nuclear matrix element: a (M(E)¸)C n , g$" $ 2p(D2#C2/4) n

(40)

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where a for the E-type transition (in our case of the 235U nucleus) has the form: $ a¸ (2j#1)[(2¸!1)!!M]2u~2L~1 , a (E¸) + IF $ ¸#1

227

(41)

and M is the matrix element for the atomic transition DiT P D f T

P

M"S f Dr~L~1DiT"

=

g (r)g (r)r1~L dr , (42) i f 0 where g is the non-relativistic (large component) radial wave function for the electron in the i(f) DiT(D f T) state. For the 235mU isomeric state, in the laser free case of a large resonant defect, D
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Fig. 12. Feynman diagram corresponding to the laser-assisted “resonant discrete IC” with DiT O D f T.

that at high intensities, when the ionization broadening of the electronic states is higher than their radiative widths and where the broadening increases faster than linearly with I, g$ will be reduced. The estimates of this section demonstrate that, in principle, the decay rate of the isomer 235mU to the ground state can be controlled by the use of intense laser radiation. Particularly, the EB process or the “resonance” discrete and continuous conversion can be used as a method of producing vacancies in the atomic shells. The coherent radiation resulting from filling these vacancies, interesting enough itself, can be used for the detection and study of the resonant EB processes. 5.4. IEB mechanism of nuclear excitation The discussions presented in the preceding subsection in connection with the “resonant discrete” IC providing a control mechanism for the decay rate of nuclear isomers apply also to the reverse process: a nucleus is excited by an electron transition when the laser radiation resonantly eliminates the mismatch between the electron level energy difference e "De !e D and the nuclear fi f i level energy difference E "E !E . FI F I The situation where the external radiation (e.g., X-rays) transfers, through the excitation of the atomic shell, energy to a nucleus which is initially in its ground state is called IEB mechanism. If the energy of the electron transition e is close to the energy of the nuclear transition E , then the fi FI resonantly enhanced excitation of a nucleus can be achieved by the absorption or emission — depending on whether E or e is larger — of a number of laser photons in addition to an X-ray FI fi photon. Again, the excitation of the electronic shell by the combined action of external radiation and resonant laser fields (first stage) can be more effective than the direct excitation of the nucleus by external radiation due to the dipole character of the interaction of the electron shell with the long wavelength radiation field. In the second stage the input energy is converted into energy of high multipolarity radiation needed for nuclear excitation. Of course, the applied laser field must be tuned to resonance to fulfill the requirement of energy conservation in this stage. The level scheme (a) and corresponding Feynman diagram (b) are presented in Fig. 13. The (approximate) theoretical treatment of this process is very similar to the laser-assisted nuclear deexcitation considered above. This is reflected in the use of the Hamiltonian (Eq. (16)), expressions of the type (1) for the Hamiltonians H and H describing the interactions of the %L %9 electron with the laser field and the X-ray field, respectively, Eqs. (32)—(34) for the wave function of

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Fig. 13. (a) Scheme of laser-assisted resonance excitation of a nucleus. (b) Feynman diagram corresponding to the scheme of Fig. 14.

Fig. 14. The effective Feynman diagram describing the laser-assisted resonant nuclear excitation.

the electron state “dressed” by the intense laser field, as well as Eq. (35), etc. All these approximations mean that the diagram in Fig. 13b must be replaced by the diagram in Fig. 14, where only the intermediate atomic state is dressed according to the sum of the diagrams shown in Fig. 15 and is described by the wave function (Eq. (34)), although, strictly speaking, both DiT and D f T states have to be considered as dressed states, too.

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Fig. 15. A “dressed” (by laser radiation) intermediate electronic state.

The dressed intermediate electronic state is able to absorb the laser photons tuned to resonance before the nuclear excitation takes place. 5.5. Some estimates We omit detailed calculations which are similar to the previous one for obtaining Eqs. (36)—(38). Let us take again, as an example, the 235mU isomeric state [10,40,55] with E "73.5 eV, ¸"3 IF and O (5d )(E "105 eV) and P (6p )(E "32.3 eV) shells as DiT and D f T states. The intermedi4 3@2 B 3 3@2 B ate state DnT of binding energy E "2.14 eV can be excited from the initial O shell by the B 4 absorption of a soft X-ray photon with energy +u "103 eV if no additional laser photon is 9 absorbed (N"0 case). The effective charges and the principal and angular momentum quantum numbers of the electronic states are: DiT: Z "13.89, n "5, l "2( j "3/2), (E "105 eV) , %&& i i i B DnT: Z "3.173, n"8, l"3 or 1, (E "2.14 eV) , %&& B D f T: Z "9.245, n "6, l "1( j "3/2), (E "32.3 eV) . %&& f f f B The energy mismatch is 0.8 eV, i.e., tuning of the laser photon around +u +1 eV (N"1) will L result in a resonant excitation of 235U from the ground state to its isomeric state 235mU. Using an intensity of the laser field of I"1011 W/cm2 (to avoid the power broadening of the atomic widths) one obtains for g defined as the ratio of the yield for nuclear excitation by c laser-assisted IEB process to the yield of direct nuclear excitation by c-ray absorption, the value g + 3 ] 1012 [54]. c We conclude with the statement that the combined application of an X-ray source and an intense laser beam for nuclear excitation can be more effective than the direct c-excitation if among the atomic electron shells there are two with an energy difference close to the nuclear excitation energy. The exact tuning of the electron transition to resonance is established by applying an intense laser of appropriate photon energy and intensity. Under these conditions, the laser-assisted IEB mechanism opens a realistic method for nuclear excitation with energy differences E between the IF excited and ground states of the order of several keV. Concluding this section, we mention, for the sake of completeness, that there exists an alternative way to induce the nuclear transitions by producing electron holes in the atomic shells via a laser-induced plasma [56]. For the example of 235U considered above, for instance, a hole is produced in the 5d shell, then the electron of the above mentioned 6p state transits to this hole 3@2 3@2 (6p (e "!32.5 eV) P 5d (e "!103.1 eV)); the emitted photon excites the ground state of 3@2 i 3@2 f 235º(7)~ to the isomeric state with JP"1` (73.5 eV, ¹ "26 min). The laser produced plasma 2 2 1@2 has a high electron density ( Z 1019 cm~3).

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The calculations are analogous to the ones developed in this section for the study of the laser-assisted IEB mechanism as a means for exciting nuclei. However, the difference is that the photon spectrum of the self radiation of the plasma is continuous, with a Planck frequency distribution: 2 u2 Pn(u )" , (45) Pp exp(u /k¹)!1 Pwhere ¹ is the plasma temperature and k is the Boltzmann constant. The excitation of nuclei in a plasma via the IEB mechanism was investigated theoretically in Ref. [57]. Two cases are interesting here. If the plasma temperature ¹ is so high that complete ionization of the atomic levels which normally would participate in the IC process, takes place, then the IEB will occur through the discrete part of the spectrum. If, however, ¹ is too low to provide the complete ionization of such levels (i.e., E
6. Lasers in the study of anomalously low-lying isomeric nuclear states: (229mTh) 6.1. Introductory remarks In this section we use the ideas and theoretical treatments which were already discussed and applied in some detail in the previous sections. We think that the specific role of the phenomenon of anomalously low-lying nuclear state (+ several eV) justifies a detailed discussion. The problem has a relatively long history. In 1989 the results of the first relatively accurate experimental determination of energies and intensities of c transitions populating the ground state and first excited state of 229Th obtained from the a decay of 233U (233U P229 Th#a) were reported [63]. Earlier, in the study of the rotational-band structure of 229Th it was concluded [64] that the excited 3`[6 3 1]9 state is located 2 quite close (within 100 eV) to the 229Th ground state (5`[6 3 3]). This statement was indirectly 2 supported by the study of the 230Th(d, t)229Th reaction [65,66]. Later it was established that the

9 Set [Nn K] are the Nilsson model quantum numbers. Z

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energy difference between the isomeric state and ground state of 229Th does not exceed 10 eV [67]. In the paper quoted above [63] this upper limit was reduced to 5 eV ((!1 $ 4) eV). Unfortunately, the techniques used in these investigations did not make it possible to determine the energy of the isomeric state E more accurately or to measure its half-life time. Furthermore, it is worth *4 mentioning that from these studies it cannot definitely be excluded that the state 3`[6 3 1] is the 2 ground state of 229Th as opposed to the state 5`[6 3 3] [27,63]. 2 In a recent paper [68], in an attempt to improve the value for the energy of the 229Th isomeric level, the authors of Ref. [63] remeasured the energies of a number of c-rays associated with the c-decay of 233U. Compared with their earlier study, they considered more c-rays in the 229Th spectrum, used more well-measured energy calibration and reference lines, and more detectors with better energy resolution. They were able to more closely match the counting rates in the c-ray peaks whose relative energies had to be measured, and to reduce systematic errors. More than 111 c-ray spectra were measured, and a value of 3.5 $ 1.0 eV was determined for the energy of the low-lying isomeric level of 229Th. In these undoubtedly improved measurements, again, it was assumed that the 3` state lies above the 5` state. 2 2 The importance of the existence of low-lying isomeric states, unusually low on a nuclear energy scale, is obvious. This includes not only nuclear physics itself, but also optics, solid-state physics, lasers, plasma, etc. For example, considering the great sensitivity of these levels to the electronic stucture, the lifetimes of such states are expected to depend on the chemical and physical environment in which these isomers are embedded. Thus, the knowledge of the lifetime, e.g., of 229mTh in different chemical and physical environments could provide valuable information for atomic and condensed matter physics. It is important to emphasize that closely spaced levels with energy spacing of several tens eV are encountered quite frequently at excitation energies of the order of & 102 eV. However, they have vanishingly small probabilities in comparison to the background connected with decays to low-lying states or to the ground state. These transitions can be studied only in the isomer decays of low-lying levels. The frequently considered example of 235U is typical from this point of view. However, its isomer, as we discussed already, decays via the IC channel, despite the fact that the efficiency of the EB (or IEB) processes is very high for 235U (g<1; see Section 5) it is small on the scale of the IC process, which dominates here (a + 1019). For other examples considered above (e.g., 93mNb or 193mIr), g(1 and again IC is the IC dominant process. From this point of view, the situation of 229Th is entirely different: the ionization potential of the thorium atom is 6.08 eV, so that a transition of the isomeric nucleus 229mTh to the ground state via IC with a final unbound electron is impossible (at least, for an isolated atom). These considerations are the basis for the high interest in laser radiation as a means to accurately study the properties of 229Th (energy level differences, lifetime of isomeric nucleus, etc.). 6.2. Deexcitation of isomeric state by the EB mechanism First, we consider the deexcitation of the isomeric state of the 229Th nucleus to its ground state using the generally accepted ordering of the levels shown in Fig. 16. Fig. 6 gives the Feynman diagram describing the EB process. The dominant nuclear transition is of M1 type. The allowed E2 transition is damped by the ratio (R /j )2, where j Z2.5 ] 10~5 cm is the wavelength of the nuclear radiation and R is the nuclear A N N A radius.

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Fig. 16. Scheme of the two lowest levels of 229Th.

In calculating the EB process probability, as before, we made a number of simplifications: we considered only the direct diagram; here, as in many other cases (see, e.g., Ref. [47]) the exchange diagram is small. Second, we used again the single-level approximation which is here well justified by the concrete analysis of the atomic level structure [52]. The final goal is the calculation of the ratio g for the probabilities ¼(3) and ¼(1) introduced earlier (see Section 5, Eqs. (36)—(38) using the c c traditional method or (Eqs. (40)—(44)) based on the “discrete IC” [26,52]). In Ref. [52] the initial electronic state was taken as DiT"6d , with energy e "!4.2 eV, intermediate DnT"7p , 3@2 i 1@2 e "!2.9 eV (values for e , e are theoretically estimated in Ref. [52]). n i n Calculations [52] have shown that g is less than unity, i.e., the direct nuclear deexcitation is the dominant channel, if the nuclear transition energy difference E "E !E is smaller than *4 I F e !e "1.3 eV, leading to the expected lifetime of the isomer ¹ Z 10 d. n i 1@2 If the energy of the nuclear transition is at resonance with the energy of one of the allowed atomic transitions, D , E !E !(e !e ) + 0 (e.g., 6d P 6d or 7s P 6d ), the EB probability I F n i 3@2 5@2 1@2 3@2 increases sharply. Of course, ensuring the condition D"0 demands a much better knowledge of E than is available today ((3.5$1) eV [68]). For the range 24E 45 eV theoretical calculations *4 *4 give ¹ Z 10 min [52], whereas the authors of [68] state that ¹ Z 45 h for E "3.5 eV (M1 1@2 1@2 *4 transition). Taking into account the $1 eV uncertainty in the excited state energy, ¹ could be 1@2 as long as +120 h or as short as +20 h. However, one has to realize that the half-lifetime of an anomalously low-lying nuclear level (E +several eV) is a very subtle quantity. It is influenced by the interaction not only with the *4 electronic shell, but, due to the extremely small value of the isomeric state energy, it will be affected also by the physical and chemical properties of the sample containing the atoms of 229Th. Experimental and theoretical studies of such a phenomena may open new interesting directions into atomic and condensed matter physics. We now turn to the question of the laser-assisted deexcitation of the isomeric 229Th state to its ground state using the resonant, discrete IC with bound electrons, applying a laser field with the appropriate frequency. As we saw above and in Section 5, this leads to a drastic acceleration of the nuclear isomeric decay [52,27]. Very recently, the probability for this process was recalculated [69]. As atomic levels, in contrast of [52], the states 7s and 8s were considered with energies 1@2 1@2 taken from Refs. [70] and [27], respectively. The energy difference of 3.713 eV between these states is very close to the 3.5 eV nuclear decay energy. This electronic transition (M1 excitation) requires a photon energy of 0.1065 eV in a resonant two-photon process. The experimental signal for the excitation could be the photon emission during the decay of the excited 8s state via the 7p and 1@2 1@2 7p states. 3@2

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Fig. 17. Deexcitation probability for the 3` state in the 229Th nucleus for the 7s P 8s electronic transitions as 2 1@2 1@2 a function of the laser intensity. The photon energy is fixed to the energy of a resonant two-photon process for a nuclear excitation energy of 3.5 eV (solid line), 4.0 eV (long-dashed line), and 3.0 eV (short-dashed line) or the resonant four-photon process for an energy of 3.5 eV (dotted line) [69].

There is another possibility [27,52] based on the absorption of a single laser photon which excites the 8s state to the 8p final state (first-order process in the laser interaction whereas the 1@2 1@2 first scheme corresponds to the second-order process). In this case the necessary photon energy is 0.712 eV. Because the probability increases in the first-order case only linearly with intensity, the secondorder (in the laser field) process was chosen in Ref. [69] since its probability has a quadratic increase and it is not small in absolute value. Use of a resonant laser field is crucial since in the laser free case the deexcitation of the 3` nuclear state with excitation of the electron states given above 2 has only a very small probability (&5 ] 10~13) [69]. Fig. 17 [69] shows the strong dependence of the deexcitation probability of the nuclear isomeric 3` state of 229Th for the 7s P 8s M1 atomic transition on the laser intensity I. The laser 2 3@2 1@2 photon energy is fixed to the energy of a resonant two-photon process with E "3.5 eV (solid line), *4 4.0 eV (long-dashed line), and 3.0 eV (short-dashed line) or for the resonant four-photon process for an energy of 3.5 eV (dotted line). We observe a strong increase with I, until the ionization thresholds of the electron states are reached where the probability drops strongly. As seen in Fig. 17, the absolute value of the deexcitation probability depends on the laser photon energy. One can also notice the quadratic dependence of the atomic shell excitation probability on I for the two-photon process and the fourth power of the intensity increase (with smaller probability). In the paper [69] several examples (161Dy, 189Os, 193Ir, 197Au, 235U, 237Np) are considered from the point of view of the IEB mechanism leading to the excitation of nuclei. The paper considers laser intensities up to 1024 W/cm2. We highly recommend this paper to the interested reader.

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235

6.3. Pumping isomers by laser resonance The role of lasers is important also for the inverse process, i.e., the pumping of the ground-state nuclei 229Th to its isomeric state. As we stressed before, the E "E !E is known today with *4 F I poor precision, which makes the direct application of the laser light to the nucleus in the ground-state ineffective. The solution is to use the IEB mechanism which was discussed in Section 5 and which provides the non-radiative excitation of the nucleus in an electron transition induced by the laser radiation field. It is necessary to tune the laser light to the wavelengths of well-known atomic transitions. In this case, even if there is a significant difference between the energies of the atomic (e ) and nuclear (E ) transitions involved, the excitation of the nucleus has in IF a large enough probability to provide for an efficient pumping of the isomeric state, which in turn opens the possibility to measure the energy and the lifetime of the low-lying nuclear isomeric state. The (third order in the electromagnetic interaction) process is described by the Feynman diagrams in Figs. 14 and 15. At the resonance (+u "e !e ) which provides more effective L n i pumping, the (direct) diagram of Fig. 14 dominates and the single level approximation is reliable. These features essentially simplify the calculations. Below, we follow the considerations given in the papers [71,72]. For the wave functions of the initial (i), intermediate (n), and final ( f ) shell electrons with energies e and widths C we take i,n,f i,n,f C t (R, t)"e~*(ei,n,f~i i,n,f@2 )tt (R) , (46) i,n,f i,n,f where the initial electronic shell DiT of the thorium atom is taken as 6d27s2 with e "!6.08 eV, and i C is taken to be zero. i Because the transition is of magnetic type (M1), we use the relativistic electronic wave functions t(R) including the large g(R) and small f (R) component with the normalization condition

P

dR R2(g2(R)#f 2(R))"1 .

The nuclear wave functions of the initial stationary state with energy E and the final isomeric state I with energy E and width C are F *4 C t (r, t)"e~*EItt (r) , t (r, t)"e~*(EF~i *4 @2 )tt (r) . (47) I I F F A matrix element of the third-order process described by the direct diagram of Fig. 14 includes, in addition to the wave functions for the electrons and the nucleus, the electromagnetic interactions of the photon with the electrons, jk(R)A (R), and nucleus Jk(r)A (r), respectively, where k k jk (R)"etM (R)ckt (R) (48) if f i is the electron electromagnetic current and Jk (r)"et`(r)JK kt (r) , (49) IF F I where JK k is the four-vector of the nuclear electromagnetic current not specified here. Furthermore, the matrix element includes the photon, D , and the intermediate electron propagators. kl The standard calculations include the expansion of the photon vector potentials and propagator in terms of multipoles (see, e.g., Refs. [73,74]). Furthermore, the matrix element of the nuclear current (Eq. (49)) is expressed in terms of the reduced matrix elements [29], taking into account the

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selection rules for the nuclear transition, etc. Finally, one obtains the following expression for the cross section p(3) of the IEB process [72] at resonance with the atomic DiT P DnT transition (+u "e !e ) (C "0, C ;C ): L n i i *4 n j2 C C3 (u ; iPn, M(E)1) E2 (E !E ; M1; nP f, IP F) A L */5 F I p(3)" L 1# f . (50) p 2C C [e !e ! (E !E )]2#C2(1#C /2C )2 n n n f F I n f n This expression is valid when the characteristic width of the laser line is comparable in magnitude with the atomic intermediate state width C : Du +C . Here, C3 is the width of the atomic n L n A radiative transition iPn of multipolarity M(E)I, and E2 is the (energy)2 of the interaction of the */5 electron transition current jk (R) (48) and the nuclear current Jk (r) (49) averaged over the initial nf IF and summed over the final states:

A

B

KP

K

2 (51) E2 "+ d3r d3Rjk (R)D (u; R!r) ) Jl (R) , nf kl IF */5 IF nf where D (u; R!r) is the Greens function of the photon in the coordinate-frequency representakl tion [75] !g exp[iuDR!rD] kl D (u; R!r)" . kl DR!rD With Du ) RD;1, Eq. (51) is reduced to the form IF E2 (u ; M1)"1C3 (u ; nPf; M1) ) C3 (u ; IPF, M1)(1#d2) , */5 FI 4 A IF N IF where C3 is the probability for the radiative nuclear transition and N Im[M (u )] M1 IF d" Re[M (u )] M1 IF is the analog of the corresponding well-known quantity in the theory of IC, and

P

M (u)"(, #, ) M1 n f

=

(52)

(53)

(54)

dr r2h(1)(ur)[g (r) f (r)#f (r)g (r)] , (55) 1 n f n f 0 where h(1)(x) is the spherical Hankel function and ,"(l!j)(2j#1), and l and j are the orbital and 1 total angular momentum of the electron in the corresponding shell. The matrix element M (u) is M1 calculated numerically in Refs. [71,72]. Typical values for the important quantity E2 vary for */5 different atoms in the range (10~10—10~12) eV2. In the already mentioned recent paper [69] this quantity was calculated for several nuclei on the basis of the adiabatic description of the dressed electronic states in the laser field. The values obtained there for E2 coincide within one order of */5 magnitude with the results of Refs. [71,72]. However, in the case of the popular 235U nucleus the result is in drastic disagreement with the calculation of Refs. [71,72]. In the situations where the width Du of the laser is larger than the widths of the atomic states L C , it is necessary to carry out the integration of Eq. (50) obtained for monochromatic laser n,f photons over the shape of the laser line g(u !u ) with :g(u !u ) du "1. L 0 L 0 L

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237

At the resonance, u "u , with the conditions C ;+Du , and C , C ;C ;e !e !E , 0 in n,f L f *4 n n f FI one has

G

C3 (iPn; M(E)1)E2 (u ; M1; nPf, IPF) 1/p */5 IF p(3) + j2 A 3%4 L [e !e !(E !E )]2 (ln 2/p)1@2 n f F I

for L , for G ,

(56)

where L stands for Lorentzian and G for Gaussian line shapes. This expression, along with Eqs. (50), (53) and (55), is the basis for the estimates below. 6.4. Pumping efficiency The arguments given in the previous subsection showed that the stimulation of the pumping of the isomeric state 229mTh by the IEB mechanism requires a careful study of the structure and properties of the atomic levels of the thorium atom. However, information about the 229Th atom is not complete. In Ref. [76] only the energies of the levels are given, while in Ref. [77] energies, spins, and parities for some of the states are presented. Two cases are interesting in the IEB mechanism for the thorium atom [71,72]. 1. The atomic transition DiTPDnT has the multipolarity M1 (i.e., the same as the isomeric nuclear transition DITPDFT) and the final state of the electron shell, D f T, coincides with the initial DiT one (“elastic” IEB). The essential parameter C3 , defining the cross section p(3) of the IEB mechanism A (see Eq. (50)) is under these conditions &(10~1—10~2)C . This situation is realized in the case, n for example, DnT"6d37s(5F ), with e "!5.15 eV [78]. 3 n 2. The DiTPDnT transition is of E1 or M1 type but the final state does not coincide with the initial one (“inelastic” IEB mechanism). One possible realization is DnT"6d7s27p(3F ), e "!4.74 eV 2 n and D f T"5f 6d7s2(3F ), e "!5.06 eV [78]. Under favorable conditions, for an atomic E1 2 f transition DiTPDnT and small mismatch between E ! E and the energy of one of the atomic F I transitions DnTPD f T (+ 0.1 eV), the excitation cross section due to the IEB mechanism may reach a value of the order p(3) + 10~20—10~21 cm2 for ordinary lasers with Du /u +10~6. At 3%4 L L less favorable conditions, i.e., fixing the value for the mismatch DI "E !e between the FI nf atomic and nuclear transitions in the energy denominators of Eqs. (50) and (56) to 1 eV, for u "1!5 eV and Du /u +10~6, and taking for E2 values between 10~10 and 10~12 it was L L L */5 found that p(3) + 10~23—10~25 cm2. By moving “up” in the energies of the excited atomic states, one can find suitable atomic levels which should be stimulated in the case E '2 eV. Notice that the value for p(3) found is a lower *4 estimate. Due to the relatively high density of the excited atomic levels of 229Th, the actual value of DI may be much less than 1 eV. For the elastic or inelastic M1 atomic transitions DiTPDnT, the cross section is p(3)+ 10~25—10~26 cm2 due to the small value of C3 (M1) (+ a2C3 (E1)). A A Defining the excitation efficiency m as the ratio of the number of produced isomeric nuclei 229mTh to the number of thorium atoms exposed to the laser light, we can write m"p(3)q/ , L

(57)

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where q is the irradiation time and / is the flux density of the laser photons, / "I/+u "o c, and L L L c o is the laser photon density. To lower the background from the natural a-decay of the thorium c nuclei with the activity A (a)"3.2 ] 10~12N(Th) decays per second, where N(Th) is the number T) of thorium nuclei, let us estimate the induced activity for a very small amount of thorium atoms taking a sample of mass 10~8 g evaporated as a layer of thickness 10 A_ (N(Th)"1013) onto a backing of diameter 1 mm. Under these conditions only a fairly small absorption of the photons emitted in the decay of the 229mTh isomers in the sample will occur. Irradiation of this target with a laser beam of power 100—200 mW focused onto +1 mm for the q"102—103 s gives the efficiency m +0.01—1 with p(3)"10~23—10~24 cm2. As a result, the cactivity of the isomer 229mTh will be 105—107 Bq (for the ¹*4 +10 d). Thus, using a tunable laser 1@2 one can stimulate a suitable atomic state with energy greater than the energy of the isomeric transition E !E , and the nuclear excitation will take place via the decay channels of this atomic F I state. In conclusion, the example of the 229Th nuclei shows that the optical laser pumping of low-lying isomeric nuclear states via the IEB mechanism is a realistic and relatively simple tool to measure and study lifetimes and energy levels. 6.5. Modified IEB mechanism in the study of 229m¹h There exists a modification to the use of the IEB mechanism for the excitation and study of the properties of the isomeric state of the thorium nucleus [79]. Below, we follow the paper [79]. It is connected with the study of the photon energy which emerges during the nuclear excitation. Schematically, it looks as follows (Fig. 18): A laser beam excites the atomic electron to a state of angular momentum defined by the selection rules in the subsequent deexcitation of the shell. The electronic deexcitation leads to the excitation of the nucleus and, simultaneously, to the photon emission. In the following, we assume that the electron has already reached the excited state DnT. The nuclear quadrupole transition 5`P3` (or 3`P5`, if the nuclear ground state of 229Th has 2 2 2 2 angular momentum 3` and not 5` as it is usually claimed), with photon emission in the dipole 2 2 regime, imposes the condition j #j 53. Thus, a new line appears in the optical spectrum of the i n laser excited 229Th spectrum, and its frequency is defined by the relation +u"+u !+u . ni FI That opens the possibility of the

(58)

1. precise determination of the energy of the 229mTh isomeric nucleus by measuring the photon frequency u; 2. solution of the uncertainty concerning the question of the angular momenta of the nuclear ground and isomeric states (5` (or 3`) and 3` (or 5`), corresponding to the sign of the difference 2 2 2 2 u !u k0). Furthermore, once the magnitude of the emitted photon frequency u is known, the F I process can be accelerated by the resonant application of the laser with frequency u "u. The L simplified calculations are very similar to the calculations of Sections 4 and 5 for the IC and EB processes. The role of the initial electronic state DiT plays here the laser-excited, dressed electronic state DnT. Again, emission of the photon frequency u is treated quantummechanically.

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Fig. 18. Scheme of the laser-assisted energy and angular momentum transfer from the atomic shell to the 229Th nucleus.

The nuclear transition is a quadruple type, ¸"2. The initial, excited atomic state is taken as i : DnT"8p7s2, n "8, l "1 . i i We take the final electronic state D f T as f : D f T"6d27s2 with n "6, l "2 satisfying the above-mentioned condition about the angular momenta inf f volved. Since the initial electronic state in our scheme is the excited state, we have to take into account the energy distribution of this state around its central energy e with width c (line shape), i0 i c 1 o(e )" i (59) i 2p (e !e )2#c2/4 i i0 i as a weight factor in the integration (averaging) over the initial electronic state. In the resonance case we have e "e #E !E #+u . The integration gives a factor (2/p)c~1. The laser induced i0 f F I L i IEB process at u "u can be written as L 1 1 (60) ¼L " ¼ /j2 , fi 2p fi L c i where / is the laser flux (Nc/»)(" I/+u ) (»-volume), and ¼ is the transition probability (per L fi unit time) of the spontaneous photon emission due to the IEB: ¼ "C¼° DSFEQK EITD2DInili f fD2 , fi fi L L,n l where

AB

(61)

r 2L e6 8p2(¸#1) C (¸) 1 (62) ¼° " 0 fi a m2c3a4 (2¸#1) (2l #1) +u B B i and DSFEQK EITD2 is the squared reduced matrix element of the multipole moment of order L ¸ between the nuclear states DIT and DFT, and is expected to have a value between 10~2 and 102 (see, Ref. [80]).

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In Eq. (61), Inili f f"aL`2:r1~LR i i(Zi , r)R f f(Zf , r) dr (cf. Eq. (37)) which for our case (n "8, B n l %&& n l %&& i L,n l l "1, ¸"2, n "6, l "2, Zi "2.17, Zf "4.65) gives DI81 D2"4.49 ] 10~4. C, C (¸) are i f f %&& %&& 2,62 1 constants expressed in terms of 3j-symbols. For ¼° we obtain fi ¼° "8.7 ] 10~51/+u s~1 (63) fi and ¼ "1.7 ] 10~8DSFEQK EITD21/+u s~1 . (64) fi L The ground state of 229Th has a half-lifetime ¹ + 2.3 ] 1011 s [40]. A sample of N 229Th atoms 1@2 has a radioactivity A"3 ] 10~12N (in units of Bq). Thus, for a sample with N"1018 atoms we have A"3 ] 106 Bq. If we suppose that 1% of the atoms can be populated in the desired electronic initial state by the laser field, then the photon emission rate R (in s~1) induced by the IEB process is R"¼ 1016"1.7 ] 108DSFEQK EITD21/+u (65) fi L which is larger than 106 s~1. It is a lower bound since one can expect that the state DFT has a much smaller half-lifetime than the state DIT, thus the activity of the sample will be much larger. If we compare these estimates with estimates discussed at the end of Section 6.4 for a somewhat different process, where N was 1013 and the obtained activity was 105—107 Bq for an irradiation time &103 s, we obtain here an induced rate R&108—1010 s~1 or, for N"1018, as used above, R+1013—1015 s~1. These numbers look encouraging and show that the laser-driven IEB mechanism can offer a reliable method for determining accurately the energy difference and other properties such as lifetime and angular momentum of ground and isomeric states, etc., and controlling a radioactive decay rate. One can go further and use the small energy difference of the 3` and 5` levels of the 229Th 2 2 nucleus to modify the a-decay rate of the nucleus in a laser driven resonant process where these two nuclear levels are mixed by the magnetic field of the laser field [4]. As a result, the a-decay rate of the 229Th nucleus to one of the levels of the daughter nucleus can be written at resonance as R"R #lR , (66) 5@2 3@2 where R is the a-decay rate for the 5 (3) state in the laser free case, 5@2(3@2) 2 2 p2e2I l" bq , (67) M2c3Du L where M is the nucleon mass, Du is the bandwidth of the laser, q is the irradiation time and b is the L reduced M1 transition matrix element in units of the Bohr magneton. Here, b is of order 10~4 (see, e.g. Ref. [74]). Assuming that the lifetime of the 229mTh(3`) is much shorter than that of the ground 2 state (5`) and taking Du + 5.7 ] 106 s~1, b & 10~6, q"1 h, I"103 W/cm2, one obtains l"0.19 2 [4]. This last example10 shows once more the potency of the laser assisted EB and IEB mechanisms 10 Such kind of effects, of course, necessitate further detailed and accurate investigation. Particularly, in the considered example the screening of the magnetic field by atomic shells seems essential for the level mixing.

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for modifying and controlling nuclear radioactivity by the influence of light — a topic which goes back to Einstein, as shown in the epigraph to the present paper.

7. Summary and conclusions In the present paper we have tried to review several applications of optical and UV-lasers for studying low-energy properties of nuclei. The most effective tool for these applications is the use of the electronic shells of atoms as mediators between the laser field of appropriate frequency and the nuclei of interest. All effects considered above are united by the realization of two unique advantages of lasers (besides other important properties like monochromaticity, polarization, etc., shared with other sources of electromagnetic radiation): 1. High intensity of the laser beam makes the multiphoton absorption (emission) by atomic shells possible, thus providing the effective elimination of the mismatch between the energy differences of atomic and nuclear transitions and, furthermore, leading to resonantly enhanced effects. This possibility of using multiples of the laser frequency u to provide the transition energy is the L leading principle for the use of intense laser beams for the study of low-energy nuclear processes and properties; 2. The dipole character of the interaction of the laser radiation with the atomic shells provides a reduction of the (usually high) multipolarity of the nuclear electromagnetic radiation, thus effectively enhancing the transition probability (the laser light transfers not only energy, but also angular momentum). It was shown that for intense laser beams (I51019 W/cm2), the so-called “anti-Stokes” transition, although not so probable as it was expected before, provides the effective mechanism for the deexcitation of nuclear levels located close enough to the nuclear ground state. This mechanism can be used to study specific nuclear levels and their transitions unaccessible otherwise. It was shown that the laser-assisted (induced) internal conversion (IC) is effective only for nuclei where IC is forbidden by energy conservation in the laser-free cases. Close and below the threshold for IC the role of optical and UV-lasers is very important and leads, in principle, to observable laser induced IC coefficients. Unfortunately, for very intense lasers the hindering effects of pondermotive forces have not been taken into account yet. Thus, the theoretical study of the IC processes at very high I is important. Generally, this case demands the elimination of several simplifications that we used for the study of this process. Here, time-consuming numerical calculations are required. We have studied the role of the EB and IEB mechanisms in processes of resonant deexcitation and excitation of isomeric nuclei with emphasis on measuring their energies and lifetimes. Special attention was paid to the study of the low-lying isomeric states of the nucleus 229mTh where the use of optical and UV-lasers seems the most promising. The existence of anomalously low-lying excited nuclear levels (as 229mTh) opens a real possibility to control and modify nuclear processes by optical and UV-lasers. Furthermore, the great sensitivity of these low-lying nuclear levels to the physical and chemical environment could provide valuable information for atomic and condensed matter physics, optics, plasma physics, lasers, etc.

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Almost all processes considered here were treated using many simplified approximations, thus one cannot expect to achieve precise quantitative conclusions. The present values for the transition probabilities of many processes are rather uncertain and may change considerably if more exact calculations of the electronic transition matrix elements become available in the future. Of particular concern is the assumption of independent one-electron levels used in all cases above, which limits the predictive power of the calculations considerably. Due to many-body effects, any perturbation of neighboring electronic shells by the intense laser radiation will have an influence on the binding energy and wave function of each electron. However, we believe that the essential aspects of the laser-assisted and laser-induced nuclear processes studied here will turn out to be correct and will provide useful directions for future studies. From the experimental point of view, it is necessary to search for the most favorable low multipolarity transitions between inner electron states to achieve a strong nucleus—electron shell coupling. In present (and future) applications, especially when the fulfillment of the resonance conditions by laser fine tuning becomes experimentally feasible, the knowledge of the precise energies of the nuclear and atomic states involved is of fundamental importance. Acknowledgements It is a pleasure to thank Werner Tornow, who attracted the author’s attention to the subject, for discussions, support, and interest in the present work. I am grateful to Vladimir Litvinenko for explanations and discussions of the operation, present status and future of the Duke University FEL Facility, to Berndt Mu¨ller for discussion of theoretical topics, interest and encouragement, and to Karl Straub, who attracted my attention to the role of laser radiation in the orbital electron capture process by nuclei, and for his general interest in the present work. I also thank Holly Pulis for helping to prepare this manuscript. This work was supported in part by the US Department of Energy, Office of High Energy and Nuclear Physics, under Grant No. DEFG05-91ER40619. Appendix A. Parameters characterizing the interaction of laser radiation with electrons (dimensional arguments) Here we use dimensional arguments to show the role of the most important parameter of the intense laser field, the intensity, in interaction processes with electrons [81]. The intensity I"+u oc (o is the number density of laser photons) must, of course, play the L essential, nontrivial role in such physical quantities as transition probability, cross section, etc. However, its inclusion should not change the dimensionality of the expressions for such quantities. This means that the density o must always enter in the combination o¸3 where ¸ is some length characteristic to the effect under consideration. There are several lengths (for free-electron and monochromatic radiation): f classical radius of the electron r "e2/mc2"2.8 ] 10~13 cm 0

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f wavelength of the laser radiation ,j /2p j (+ 10~4!10~5 cm), or L L L f (reduced) electron Compton wavelength "+/mc"3.86 ] 10~11 cm . c The first two of them are classical quantities, is of quantum nature. Following Ref. [81], it is c illuminating to separate the most interesting dimensionless combinations of o (which is a quantum quantity since it measures the numbers of photons (o"I/+u c)) and these three lengths into L classical and quantum parameters: We have (retaining for clarity + and c): Classical dimensionless parameters: o r2"Ir2/u mc2 , c0 0 L 2/mc3"10~10j2I , o r j "Ir L c0 L 0 L I 2 L . C : o j2" c L u mc2 3 L Quantum dimensionless parameters: C : 1 C : 2

Q : o 3"I 3/+u c , 1 L L L Q : o 2r "I 2r /c1/+u . 2 L0 L0 L The parameter C is extremely small. The parameter C has an interesting physical meaning, 1 2 defining the ratio of the electromagnetic energy r 2I/c of the wave with volume r 2 passing by 0 0 (in “contact” with) the electron of size r , to the electron rest energy. We see that the relativistic 0 effects described by C "10~10j2I are small for the j of interest here (j &10~4—10~5 cm) unless 2 L L L I51010 W/j2 + 1020 W/cm2. The parameter C has the same meaning as C but with respect to a much larger volume 3. It 3 2 seems to play no role since it is difficult to imagine an electron to be probed in the interaction region as large as 3. The interesting quantum parameter is Q which measures the ratio of the 2 electromagnetic energy r 2I/c to the photon energy +u , i.e., defines the number of the laser 0 L photons in the interaction region r 2. In other words, just this parameter describes the non-linear, 0 multiphoton interactions with free electrons. We note that these kinds of effects were important in several processes described in the text, the only difference is that such multiphoton processes referred to bound electrons. The parameter Q has the same meaning as Q , again with respect to 1 2 the larger volume 3. The above arguments were applied to the interaction of laser radiation with free electrons. In some sense, the dimensional arguments are useful for atomic systems even under the influence of very intense lasers with an electric field strength E of the order of (1—5)e/a2 (i.e., 0 B I"(7 ] 1016—2 ] 1018) W/cm2). With laser photon energy +u "5 eV the amplitude of the L motion of the electrons is larger than eE /mu2 & 30a . 0 L B These arguments were the basis for the authors of Refs. [82,22] to construct an approximate theory of nuclear excitations due to the motion of the shell electrons in the very intense field of the

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optical and UV lasers when a laser driven highly localized current of collectively oscillating electrons induces the nuclear excitation. Each electron in this current moves on its own classical trajectory (see also Refs. [83—85]). Such a model gives the upper limit for the effect of nuclear excitations [82]. In conclusion, it is worthwhile to remark that for atomic electrons there are additional characteristic lengths such as the Bohr radius a or a /Z, so one could expect that the correspondB B ing important parameters C and Q will be much larger (see also Section 4 which is devoted to the 2 2 laser-assisted and induced IC processes). Thus, the interaction of laser radiation of optical and UV range with a simple atomic system would offer a very promising way to study the nonlinear effects intrinsic to the intense laser field. This issue was illustrated in the text from the point of view of low-energy nuclear physics.

Appendix B. Two representations of the interaction of radiation with matter As it was pointed out in the Introduction (footnote 1), the question of relating the dipole form (1) as the Hamiltonian of the matter-radiation field interaction to the general gauge invariant (non-relativistic) Hamiltonian H" 1 (pL !eA)2(c"1) which gives the interaction term 2m !(e/m)pL ) A#(e2/2m)A2 (B.1) has a long and controversial history (see Ref. [6]). At first glance, the problem is simply solved by applying to the wave function t in the Schro¨dinger equation i ­t/­t"Ht the unitary transformation t(r, t)"expMiA(t) ) rN/(r, t)

(B.2)

(in the dipole approximation the vector potential A does not depend on r). We obtain for /(r, t) the equation i ­//­t"[(1/2m)pL 2#r ) AQ (t)]/

(B.3)

which leads, using E"!AQ (t), to the Go¨ppert—Mayer form (1). Obviously, if A depends on r, i.e., we avoid the dipole approximation, this correspondence is not valid. This equivalence is well known. But this is not the whole story. First of all, one needs to be careful if in actual studies (practically unavoidable) approximations (besides the dipole one) have to be made. Eqs. (B.3) and (1) are equivalent in the dipole approximation if the wave functions t and / are exact solutions of the corresponding Schro¨dinger equations. If approximations are permitted, the equivalence may be destroyed. This situation leads, for example, to the conclusion [86] that when studying model systems with a finite number of levels, one must choose the dipole interaction in the form (1), rather than in the form (Eq. (B.3)). However, even if the t and / are the exact solutions of the Schro¨dinger equation, one needs to be cautious if the problem includes damping which, in the case, e.g., of the two-level system (described by a 2] 2 matrix) has off-diagonal elements for the p ) A interaction. It makes the dipole form !er ) E more convenient, although, if one takes into account the proper change of the damping matrix under the transformation (Eq. (B.2)), they are equivalent.

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Since this point is not emphasized in textbooks we give here the derivation [6] of the above statement for the two-level system studied in Section 3. Consider the Bethe—Lamb equations (Eq. (7)) for two states DaT and DbT with decay constants c , E !E "+u !+u "+u, under a(b) a b a b influence of the radiation field E(t)"E sin lt with Hamiltonian in the dipole form: H"!eE ) r, 0 we have aR "!(c /2)a#(i/+)eE ) r sin lt e*utb , bQ "!(c /2)b#(i/+)eE ) r sin lt e*uta a 0 ab b 0 ba (r "SaDrDbT). These equations are derivable from the Schro¨dinger equation ab i+ ­t/­t"[H !1i+C!eE(t) ) r]t , 0 2 where H is the unperturbed Hamiltonian H "p/2m#º(r) with eigenstates t and / : 0 0 a b H t "+u t , H t "+u t 0 a a a 0 b b b and C is the Weisskopf—Wigner-type decay operator with eigenvalues c , c : a b Ct "c t . a(b) a(b) a(b) Obviously, if we insert the wave function of our two-level system

(B.4)

(B.5)

(B.6)

(B.7)

(B.8) t(r, t)"a(t)e~*uatt (r)#b(t)e~*ubtt (r) a b into (Eq. (B.5)) we obtain the Bethe—Lamb equations (Eq. (B.4)). Now, to replace the E ) r term in Eq. (B.5) by the p ) A-type interaction (Eq. (B.1)) and we make the transformation (Eq. (B.2)): t(r, t)"¹`(r, t)/(r, t) ,

(B.9)

with ¹(r, t)"eie@+A(t)> r (we retain here +), thus obtaining the equation for /: i+ ­//­t"MH !(i+/2)C@(r, t)!(e/m)p ) A#(e2/2m)A2N/ . (B.10) 0 Thus, we arrive at the important result that in the equivalence stated above between the two forms of interactions (in dipole approximation) it is necessary to treat the transformed damping operator C@ C@"¹C¹` .

(B.11)

In contrast to Eq. (B.4), C@ depends on r and t and is not diagonal with respect to the eigenstates t of H . Indeed, we have a(b) 0 c 0 (B.12) C" a 0 c b c cos2 /#c sin2 / !i(c !c )sin / cos / b a b C@ " a , (B.13) i(c !c )sin / cos / c sin2 /#c cos2 / a b a b where /"eA(t) ) r /+. ba These results show that it is wrong to simply replace the E ) r interaction by the p ) A form; the damping matrix has to be changed in addition. The off-diagonal elements of the new damping

A A

B

B

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matrix C@ contribute both to the coupling of the t and t states. Finally, it is instructive to write a b the Bethe—Lamb-type equations for the amplitudes a and b in the expansion / in terms of t and a t: b aR "!(1C@ (t)#i/+(e2/2m)A2(t))a!(1C@ (t)!i/+(e/m)p ) A(t))e*utb , 2 ab ab 2 aa (B.14) bQ "!(1C@ (t)#i/+(e2/2m)A2(t))b!(1C@ (t)!i/+(e/m)p ) A(t))e*uta , 2 bb 2 ba ba where p "SaD pDbT. ab We note that in the calculations of the probability amplitudes in the two representations the initial conditions for the equations of motion (Eq. (B.14)) are in general different from the initial conditions of the Bethe—Lamb equations (Eq. (B.4)) [6]. A last remark concerns a practical issue. For higher-order processes it becomes more and more cumbersome to solve the p ) A equations (Eq. (B.14)) due to the transformed damping matrix elements. Therefore, for practical calculations even for a two-level atom the use of the Bethe—Lamb equations (Eq. (B.4)) is more convenient.

Appendix C. Notation index j L L Dj L +u L j ab u L k L R A DE A H a"e2/+c R r p E L E 0 I / M p c C N DIT DFT DNT E I

laser radiation wave length measured in nm or cm "j /2p L laser radiation linewidth energy of laser photon measured in eV ,2p+c/(E !E ) a b frequency of laser photon wave number of laser photon nuclear radius nuclear level spacing Hamiltonian fine structure constant radius vector of atomic electron radius vector of the pth nuclear proton electric field of laser radiation amplitude of laser electric field laser beam intensity measured in units of W/cm2 photon flux matrix element cross section level-decay rate "+c—level width number of laser photons initial nuclear state final nuclear state intermediate nuclear state energy of initial nuclear state

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E F E N DiT, DnT, D f T e ,e ,e i n f E IF e if E *4 EB IEB IC DE De m(M) Z Z %&& E B c A o(e ) f ¸ l m n o r "e2/mc2"2.8 ] 10~13 0 "+/mc"3.86 ] 10~11 cm c a "+2/me2"5.3 ] 10~9 cm B R "e2/2a y B p"+k k E(M)¸ J (J ) I(F) i(f) m (m ) I(F) i(f) D ¹ a IC a $ n, j, j e ,e ,e 1 2 3

247

energy of final nuclear state energy of intermediate nuclear state initial, intermediate, and final atomic electronic state, respectively energy of initial, intermediate, and final atomic electrons, respectively ,E !E I F ,e !e i f energy of nuclear isomeric states electron bridge inverse electron bridge internal conversion nuclear level energy difference atomic electron level energy difference electron (proton) mass atomic number effective atomic number electron binding energy in atomic shell, sometimes we use DE D,B B speed of light vector potential of electromagnetic field density of final electron states multipolarity of nuclear radiation orbital angular momentum magnetic quantum number principal quantum number number density of laser photons classical electron radius electron Compton wavelength Bohr radius Rydberg potential electron momentum wave vector electric (magnetic) transition with multipolarity ¸ nuclear (atomic) angular momentum for initial (final) states nuclear (atomic) magnetic quantum numbers for initial (final) states energy mismatch; detuning time interval, temperature coefficient of internal conversion coefficient of IC for discrete final electron state quantum numbers of hydrogen-type solution unit vectors defining the photon polarization

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[42] V.S. Letokhov, C.P. Chebotayev, Nonlinear Laser Spectroscopy, vol. 4 of Springer Series in Optical Sciences, Chap. 10, Section 2.3, Springer, Berlin, 1977, pp. 428—431. [43] V.A. Krutov, Izv. Akad. Nauk SSSR Ser. Fiz. 22 (1958) 162. [44] V.A. Krutov, V.N. Fomenko, Ann. Phys. (Leipzig) 21 (1968) 291. [45] B. Crasemann, Nucl. Instr. Meth. 112 (1973) 33. [46] D. Kekez, A. Ljubicy icy , K. Pisk, B.A. Logan, Phys. Rev. Lett. 55 (1985) 1366. [47] V.M. Kolomietz, V.N. Kondrat’ev, Sov. J. Nucl. Phys. 51 (1990) 400. [48] V.A. Zheltonozhskii, P.N. Muzalev, A.F. Novgorodov, M.A. Ukhin, Sov. Phys. JETP 67 (1988) 10. [49] P. Ka´lma´n, Phys. Rev. A 43 (1991) 2603. [50] D. Hinneburg, Z. Phys. A 300 (1981) 129. [51] P. Ka´lma´n, T. Keszthelyi, Phys. Rev. A 44 (1991) 4761. [52] V.F. Strizhov, E.V. Tkalya, Sov. Phys. JETP 72 (1991) 387. [53] L.D. Landau, E.M. Lifshitz, Quantum Mechanics, Pergamon Press, Oxford, 1977. [54] P. Ka´lma´n, T. Keszthelyi, Phys. Rev. A 47 (1993) 1320. [55] M.R. Shmorak, Nuclear Data Sheets 69 (1993) 375. [56] M. Morita, Progr. Theor. Phys. 49 (1973) 1574. [57] E.V. Tkalya, Dokl. Akad. Nauk. SSSR 315 (1990) 1373. [58] R.V. Arutyunyan, C.A. Bol’shov, Dokl. Akad. Nauk. SSSR 305 (1989) 839. [59] Y. Izawa, C. Yamanaki, Phys. Lett. B 88 (1979) 59. [60] V.I. Goldanskii, V.A. Namiot, JETP Lett. 23 (1976) 451. [61] V.I. Goldanskii, V.A. Namiot, Phys. Lett. B 62 (1976) 393. [62] V.I. Goldanskii, V.A. Namiot, Sov. J. Nucl. Phys. 33 (1981) 169. [63] C.W. Reich, R.G. Helmer, Phys. Rev. Lett. 64 (1990) 271; see also C.W. Reich et al., Int. J. Appl. Radiat. Isot. 35 (1984) 185. [64] L.A. Kroger, C.W. Reich, Nucl. Phys. A 259 (1976) 29; L.A. Kroger, PhD Thesis, University of Wyoming, 1971 (unpublished). [65] K.S. Toth, Nucl. Data Sheets 24 (1978) 263. [66] D.G. Burke et al., Phys. Rev. C 42 (1990) 499. [67] Y.A. Akovali, Nucl. Data Sheets 58 (1989) 555. [68] R.G. Helmer, C.W. Reich, Phys. Rev. C 49 (1994) 1845. [69] S. Typel, C. Leclercq-William, Phys. Rev. A 53 (1996) 2547. [70] K.N. Huang et al., At. Data Nucl. Data Tables 18 (1976) 243. [71] E.V. Tkalya, Pis’ma Zh. Eksp. Theor. Fiz. 55 (1992) 216; JETP Lett. 55 (1992) 211. [72] E.V. Tkalya, Yad. Fiz. 55 (1992) 2881; Sov. J. Nucl. Phys. 55 (1992) 1611; Nucl. Phys. A 539 (1992) 203. [73] A.I. Akhiezer, V.B. Berestetcskii, Quantum Electrodynamics, Wiley, New York, 1965. [74] J.M. Eisenberg, W. Greiner, Excitation Mechanisms of the Nucleus, vol. 2 of Nuclear Theory, North-Holland, Amsterdam, 1970. [75] V.B. Berestetskii, E.M. Lifshitz, L.P. Pitaevskii, Quantum Electrodynamics, 2nd edn., Pergamon Press, Oxford, 1982. [76] C.H. Corliss, U. Borman, Transition probabilities and oscillator strengths of 70 elements, Mir, Moscow, 1968, p. 416 (in Russian). [77] A. Giachetti et al., J. Res. Nat. Bur. Stand A 78 (1974) 247. [78] A.A. Radtsig, B.M. Smirnov, Properties of Atoms and Atomic Ions, Energoatomizdat, Moscow, 1986 (in Russian). [79] P. Ka´lma´n, T. Keszthelyi, Phys. Rev. C 49 (1994) 324. [80] J. Eisenberg, W. Greiner, Nuclear Models, vol. 1 of Nuclear Theory, North-Holland, Amsterdam, 1970, p. 24. [81] J. Eberly, in: E. Wulf (Ed.), Progress in Optics, vol. VII, North-Holland, Amsterdam, 1969. [82] J.F. Barger, D. Gogny, M.S. Weiss, J. Quantum Spectrosc. Radiat. Transfer 40 (1988) 717. [83] J.C. Solem, L.C. Biedenharn, J. Quant. Spectrosc. Radiat. Transfer 40 (1988) 707. [84] G.A. Rinker, J.C. Solem, L.C. Biedenharn, Soc. Photo-Opt. Instr. Eng. 875 (1988) 92. [85] G.A. Rinker, J.C. Solem, L.C. Biedenharn, Adv. Laser Sci. 1 (1987) 75. [86] N.B. Delone, V.P. Krainov, Springer Series in Chemical Physics, Atoms in Strong Light Fields, vol. 28, Springer, Berlin, 1985.

SOME GEOMETRICAL AND TOPOLOGICAL PROBLEMS IN POLYMER PHYSICS

A.L. KHOLODENKO!, T.A. VILGIS" ! 375 H.L. Hunter Laboratories, Clemson University, Clemson, SC 29634-1905, USA " Max-Planck Institut fu( r Polymerforschung, Postfach 3148, D-55021, Mainz, Germany

AMSTERDAM — LAUSANNE — NEW YORK — OXFORD — SHANNON — TOKYO

Physics Reports 298 (1998) 251—370

Some geometrical and topological problems in polymer physics A.L. Kholodenko!, T.A. Vilgis" ! 375 H.L. Hunter Laboratories, Clemson University, Clemson, SC 29634-1905, USA " Max-Planck Institut fu( r Polymerforschung, Postfach 3148, D-55021, Mainz, Germany Received June 1997; editor: I. Procaccia Contents 1. Introduction 2. Relevance of entanglements (some experimental facts and related theoretical works) 2.1. Some properties of ring polymers in dilute solutions and in melts 2.2. Polymer dynamics and topology 2.3. Polymer networks 3. Single chain problems which involve entanglements (general considerations) 3.1. Topological persistence length and the probability of knot formation 3.2. Knot complexity and the average writhe 3.3. The unknotting number and the number of distinct knots for polymer of given length N 4. Methods of describing knots (links) 4.1. Differential geometric approach 4.2. Path integral approach via Abelian and non-Abelian Chern—Simons field theory 4.3. Algebraic (group-theoretic) description of knots (links) via knot polynomials 4.4. Unifying link between different approaches 5. Probability of knotting: the detailed treatment 5.1. Planar Brownian motion in the presence of a single hole. The role of finite size effects 5.2. Quantum groups and planar Brownian motion 5.3. Jones polynomial, Temperley—Lieb algebra and statistical mechanics of knots (links)

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5.4. Probability of knotting and the role of finite size effects 6. Single chain problems which involve geometrical and topological constraints 6.1. Semi-flexible polymer chain in the nematic environment 6.2. Semi-flexible polymers confined between the parallel plates and in the half space 6.3. Polymers confined into semi-flexible tubes 6.4. Configurational statistics of the planar random walks restricted by the area constraint 7. Knot complexity — detailed treatment 7.1. Calculation of the topological persistence length 7.2. Calculation of the averaged writhe 7.3. Calculation of the knot complexity 7.4. Calculation of the unknotting number and the number of distinct knots as a function of polymers length N 7.5. Some physical applications 7.6. Link energy and the probability of entanglement between two ring polymers 8. Polymer dynamics: an interplay between topology and geometry 8.1. Statistical mechanics of a melt of polymer rings 8.2. Statistical mechanics of planar rings in an array of obstacles (the replica approach) 8.3. Statistical mechanics of planar rings in an array of obstacles (the Riemann surface approach)

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Appendix A A.1. Planar Brownian motion in the presence of two holes A.2. Spatial Brownian motion in the presence of knots (links) References

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Abstract In this work we discuss some problems of polymer physics which require use of the geometrical and topological methods for their solution. Selection of problems is made to provide some balanced view between the real physical situations and the mathematical methods which are required for their understanding. We consider both static and dynamic properties of polymer solutions which depend on the presence of entanglements. These include: problems related to polymer collapse, statics and dynamics of individual circular polymers and concentrated polymer solutions, problems related to elasticity of rubbers and gels, motion of polymers through pores, etc. This work serves both as an introduction to the field and as a guide for further study. ( 1998 Elsevier Science B.V. All rights reserved. PACS: 61.41#e; 02.40.Pc; 05.90.#m Keywords: Polymer entanglements; Knots and links; Path integrals; Differential geometry of curves; Statistical mechanics

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1. Introduction Knot theory was born in Scotland around the year of 1867. Two Scotsmen living in Edinburg: J.C. Maxwell and P.G. Tait and one Irishman living in Glasgow: W. Thomson (Lord Kelvin) were the founders of what has become a knot theory. According to Thomson’s theory of chemical elements all atoms are made of small knots formed by vortex lines of ether, Knott (1911), which have to be “kinetically stable”. Hundred years later Sakharov (1972), following ideas of Wheeler, Lee and Yang, had suggested that the elementary particles are made of knots. Whether this is true or not remains to be seen but what is known to be true is that, starting from the work of Symanzik (1969), all quantum field theories admit polymer representation. This means that, for some reason, polymer and particle physics are very closely related. Moreover, recently Ashtekar (1996) had argued that polymer representation plays an important role in gravity. Since the nonperturbative gravity involves knots (Gambini and Pullin, 1996), the circle of ideas which are more than hundred years old appears to be closed (or, may be, even “knotted”!). More seriously, the interplay between the knot theory and physical phenomena is not at all a recent feature. In a series of papers (reproduced in “Knots and Applications” Edited by Kauffman, 1995) Kelvin (W. Thomson) had formulated hydrodynamics of knotted vortex rings with such degree of completeness, that hundred years later his results have not lost their significance (Ricca and Berger, 1996). At the same time, the role of topology in quantum mechanics had been recognized much later by Aharonov and Bohm (1959) and Finkelstein and Rubinstein (1968). Since polymer physics and quantum mechanics/quantum field theory are closely related to each other (Symanzik, 1969; de Gennes, 1979), evidently, that the same (or very similar) topological problems should occur in polymer physics as well. For example, the Aharonov—Bohm effect (Kleinert, 1995), has its analogue in the statistics of planar Brownian walks in the presence of a hole. (For a quick introduction to this topic, please, see the Appendix.) It is not our purpose in this review to provide the reader with a chronological list of developments both in the knot theory and in polymer physics. Anyone who would like to make such a list is going to run inevitably into the dilemma: how to keep a balance between the genuinely mathematical developments in knot theory and truly physical, chemical or biological applications of knot theory. At this moment, to our knowledge, there is a series of monographs on “Knots and Everything” edited by L. Kauffman, which, has no less than seven volumes to date starting with “Knots and Physics” by Kauffman himself (1993). At the same time, there is yet another series entitled “Proceedings of Symposia in Applied Mathematics” by the American Mathematical Society. These proceedings, e.g. Vols. 45 and 51, contain also very valuable information about the applications of knot theory to various natural phenomena. To these proceedings one may add series such as “Regional Conference Series in Mathematics”. In particular, a very nice summary of the results by Jones is published in Vol. 80 of this series. In addition, the series “Advances in the Mathematical Physics” and the “Journal of Knot Theory and its Ramifications” occasionally also contain applied information. Unfortunately, even this list of references is not sufficient if one wants to work actively in this rapidly developing field of research. To keep up to date on the developments related to knots and links, perhaps, it is not too unusual to use the already existing electronic databases. These are at Duke University http://eprints.math.duke.edu/archive.html; at the Los Alamos National Laboratory http://xxx.lanl.gov; at the Geometry Centers of the

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University of Minnesota, http://www.umn.edu and the University of Massachusetts, http://www.gang.umass.edu. With all this information the question arises: Is it possible to say something new (or different) on the subject of knots (links)? We believe that the answer is “yes”. It is possible to say something new, provided, that one can keep a delicate balance between the mathematical rigor and the physical reality. We hope, that this work serves exactly this purpose. That is, we tried as much as we could to provide a sufficient mathematical background which is truly needed for the development but, at the same time, we tried to use a language which is familiar to the researchers in polymer and, more general, in condensed matter physics so that, hopefully, the reader will not find himself (herself) lost in mathematics. Lately, we had become aware of similar efforts, e.g. see Murasugi (1996) and Nechaev (1996). These works are more mathematical and have a little or no overlap with the content of this review. Selection of the material for this review is based mainly on our own original works and, whence, necessarily reflects our vision of this field. Nevertheless, we wholeheartedly encourage the reader to develop his (or her) own opinion about the field and, for this purpose, to look at other sources of information. This work is organized as follows. In Section 2 we provide some illustrative examples of the relevance of entanglements to various phenomena in polymer physics. We use the examples and the language which is commonly accepted in this field. We hope that by choosing such style people of various fields, tastes and skills should be able to decide for themselves how far they want to go into this boundless field. We apologize to those who would like to see this review to be more mathematical and to those who think that it is too mathematical. Whence, immediately, beginning from Section 3, we tend to be more mathematically precise without loosing physics from our sight. In particular, the content of our Section 3, incidentally, is closely related to the latest published results of Stasiak et al. (1996) and Katrich et al. (1996) on the average writhe and the average crossing number for biological knots and by Zurer (1996) on the probability of knotting in proteins. The average crossing number is of interest in connection with the mobility of knotted DNA in gels under electrophoresis or upon centrifugation. We discuss these issues in Sections 2 and 7. In Section 4 we provide a background needed for the actual calculation of these observables. In particular, we emphasize the role of differential geometric as well as algebraic and field-theoretic concepts needed for computations which involve real physical knots. We also provide a unifying link between different approaches. It is important to keep in mind that the very concept of a knot is dimension-dependent. This precisely means that all nontrivial knots in 3-dimensions are trivial unknots in 4 dimensions (Bing and Klee, 1964). This implies that e-expansions used in physics literature are, strictly speaking, not permissible for problems which involve knots. We do not consider higher dimensional knotting in this review. For example, if a usual knot is just an embedding of a circle S1 into R3 (or, more generally, S3"R3XMRN) one can think more generally about embedding(s) of Sp into Sq, p(q (Rolfsen, 1976). By the way, the opposite embeddings are also possible and are known as Hopf mappings (or Hopf fibrations), e.g. see Ono (1994). Example of such mapping is only briefly discussed in Section 6. Some physical applications of the Hopf fibrations could be found, e.g. in Monastyrsky (1993). We also do not discuss the case when S1 is not embedded but immersed into S3. In this case we should allow the self-interaction of the knot/link-segments between themselves. Such situation would require us to consider the Vassiliev invariants, Murasugi (1996). As it was shown very recently by Bar-Natan (1996) the Vassiliev invariants are related to more traditionally used

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invariants (e.g. HOMFLY or Jones polynomials defined in Section 4) through use of quantum group methods (Chari and Pressley, 1995). Since we touch upon these methods only very gently in Section 5, we do not elaborate on this very physically important subject. But we have decided to mention about it in this review since we anticipate potentially significant physical applications of Vassiliev invariants in the future, e.g. see Deguchi and Tsurusaki (1994) for steps in this direction. Extension of the notion of linking and self-linking to higher dimensional manifolds is also not only of academic interest. For example, extension of the concept of self-linking, Section 4.2, to higher dimensional manifolds leads to its connection with the Euler’s characteristics for these manifolds (Guillemin and Pollack, 1974). Moreover, a simple extension of this connection leads to the Lefschetz fixed point theory which is an extension of the famous Brower fixed point theorem dealing with the question of how many roots, the equation f (x)"x, could have. The questions of this sort are being frequently asked in the context of quantum field theories (Zinn-Justin, 1993), in connection with problems which involve stochastic quantization. Moreover, since the Lefschetz fixed point theory (which is aimed at the calculation of the Lefschetz index) is closely connected with the Morse theory, this leads quantum mechanically to the consideration of various kinds of supersymmetric problems (Witten, 1981). We mention these facts to the reader who is interested in physical applications of the apparently exotic concepts development by mathematicians. If Section 3 only introduces some basic knot observables while Section 4 provides some basic tools to describe these observables, Section 5 already provides the first application of these results. It deals with the long standing problem formulated by Delbru¨ck (1962) about the probability of knot formation P as a function of polymer length N. This problem was solved, in part, by Sumners N and Whittington (1988) and Pippenger (1989) who produce for the quantity f "1!P an N N estimate given by Eq. (3.2) with c being some undetermined constant, c(1. In Section 5 we determine this constant while in Section 7 we calculate the topological persistence length N which T also enters the result for f , e.g. see Eq. (3.5). Solution of the Delbru¨ck problem has profound N implications on all aspects of polymer physics since, according to Delbru¨ck (1962) (and now proven), for NPR and in absence of the excluded volume effects almost all polymers are knotted or quasi-knotted. In the last case, following Delbru¨ck, one can (at least in our imagination) “close” the ends of otherwise linear polymers with some straight line so that the resulting circular polymer will be almost surely knotted. If SR2T is the mean square end-to-end distance, then at h conditions SR2T&N so that the ratio JSR2T/NPN~1@2P0, i.e. for NPR all polymers at h conditions could be considered as effectively closed and, whence, effectively knotted. In order to obtain additional results about knotted polymers, the information presented in Section 4 turns out to be insufficient. Whence, in Section 6 we provide an additional geometrical background which is needed for solutions of the physical problems presented in Sections 7 and 8. The material of Section 6 is by no means exhaustive since we have selected only those geometrical problems which are directly used later. The reader should be warned, however, at this point, that the material of this section is so comprehensive that only a small portion of it, e.g. that presented in Section 6.2, could serve as an introduction to the whole field of surface-related phenomena, e.g. see Eisenriegler (1993). Moreover, the delicate interplay between the topological and geometrical effects discussed in Section 6.1 could also be readily generalized (Kholodenko, 1990, 1995), and is related to the statistical mechanics of semiflexible polymers. Usefulness of the Dirac propagators (Kholodenko, 1990, 1995), for the description of conformational properties of semiflexible polymers has been proven recently experimentally by Hickl et al. (1997) in a series of measurements of the static

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scattering function S(k) for polymers of arbitrary flexibility based on the theoretical calculations of S(k) which involve the Dirac propagator (Kholodenko, 1993). Unlike the traditionally used Kratky—Porod propagators (Kleinert, 1995), which do not allow to obtain S(k) in closed analytic form, use of the Dirac propagators for this purpose creates no computational difficulties. In addition, use of the Dirac-like propagators is essential for the theory of semiflexible polymers to account for the hairpin effects (see, de Gennes, 1982; Kholodenko and Vilgis, 1995; and Section 6.1). Confinement of polymers into tubes, discussed in Section 6.3, is not an intrinsic feature of polymer physics and has some similarities with motion of electrons in quasi-one-dimensional conductors. We provide some information in this regard in Sections 6.3 and 8.6. Already this observation makes some aspects of polymer physics, e.g. reptation, closely connected with the theory of quantum chaos. A simple extension of the problem which was first discussed by Levi (1965) about the planar Brownian walk which encloses a prescribed area A, presented in Section 6.4 and further used in Section 8, leads to very deep results connected with Selberg’s trace formula. Incidentally, the recently published book by Grosche (1996), could serve as an excellent supplement to some of the results presented in Section 8. Unlike Grosche’s book, however, the results of Section 8 are targeted towards polymer applications.The results of Section 6 are also being extensively used in Section 7 where we provide details of calculations of observables introduced and discussed in Sections 2 and 3. In this section it is possible to push calculations to the extent that all our results can be compared against available numerical data. The material of this section could be especially useful for biological applications as discussed, e.g. in Vologodskii et al. (1979) or Stasiak et al. (1996). At the same time, the results of Section 7.6 may also play an important role in the development of the theory of entangled polymer networks (Everaers and Kremer, 1996; Kholodenko and Vilgis, 1997; Vilgis and Otto, 1997). The reader who is interested mainly in biological applications may not read any further since Section 8 deals with a typical polymer problem about the rheological properties of dense polymer networks. The effects of topology and geometry on these properties was always suspected, e.g. see Doi and Edwards (1986), but, to our knowledge, were not properly implemented so that the many-body topological and geometrical effects remained hidden in the tube which surrounds the “reptating” polymer chain, de Gennes (1979). The existence of such a tube was postulated and the transition from the reptation regime, where the tube is expected to be well defined, to the Rouse regime, where it ceases to exist, was poorly understood. Since the experimental data which accompany such type of transition are readily available, e.g. see Fetters et al. (1994), we compare these data against our theoretical predictions in Tables 1 and 2. Earlier accounts of our theoretical results could be found in Kholodenko and Vilgis (1994), and Kholodenko (1996a,b,c). It is important, that the reader understands that the results of this section are valid in both static and dynamic conditions since they mainly involve topological arguments. For the reader’s convenience we provide some essentials of these arguments in Appendix A.1. Appendix A.1 should be read very much independently of the main text and has a value on its own. We provide in it some arguments which are unobscured by technical or polymer-related details so that the topological issues should become more obvious. Since we do not expect that most of our potential readers are familiar with some specialized mathematical literature, the emphasis is made on concepts rather than on rigorous definitions, etc. Nevertheless, we provide a sufficient number of references in order to make our presentation sufficiently serious. In particular, we argue that the natural logic of development of topological ideas goes from considering the planar Brownian motion in the presence of just one

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hole through generalization of this problem to include many holes and, then, through discussion of the Brownian motion in three-dimensional space in the presence of a knot. The last topic is briefly discussed in Appendix A.2. All these problems are interrelated and, in the last case, the potential for biological applications should be apparent. Since most living DNAs are knotted the Brownian motion in the vicinity of such knotted DNAs can, in principle, recognize the different knotted structures. This fact should be taken into consideration in all theories of molecular recognition. Unfortunately, to cover just these subjects in sufficient depth would require reviews even longer than ours. Hence, if our readers make an effort in these directions, we would feel that our goals are achieved.

2. Relevance of entanglements (some experimental facts and related theoretical works) 2.1. Some properties of ring polymers in dilute solutions and in melts The role of circular polymers in biology is well documented, e.g. see Wasserman and Cozzarelli (1986), while synthetically the ring-shaped polystyrenes were obtained relatively recently, e.g. see ten Brinke and Hadziioannou (1987) and references therein. Their synthesis had led to a number of interesting experimental studies which we shall briefly discuss in this section and in more detail in the rest of this paper. There are several conditions for the ring polymers which need to be added to the list of conditions of synthesis for the linear polymers. These include: (a) conditions under which the rings can be formed (e.g. in good solvent the chances of ring formation should be much smaller due to the excluded volume effects); (b) conditions under which the rings could be knotted; (c) conditions under which the rings can be interlocked. All these conditions were qualitatively analyzed in the past. For example, the dynamics of ring closure was analyzed by Wilemski and Fixman (1974), by Szabo et al. (1980) and, more recently, by Pastor et al. (1996). The role of solvent quality on ring formation was analyzed by de Gennes (1990b) and, independently, by von Rensburg and Wittington (1990). Conditions under which the rings could be knotted were analyzed by Sumners and Whittington (1988), by Pippenger (1989) and by Kholodenko (1991, 1994).These results will be discussed in more detail below in Sections 3—5. In addition, there are related problems, e.g. how knot formation is affected by the polymer stiffness (this defines the topological persistence length, Section 3), how many different knots can be made of linear polymers of length N (e.g. see Sections 3 and 7), how one can recognize these different knots (the rest of this paper), and to what extent topologically different knots behave physically different (Section 7 and Appendix A.2.). The important issue of link formation which was initially discussed in the pioneering work by Frisch and Wasserman (1961), raises several additional questions. For example, assume that we have a solution of both linear and ring polymers of equal concentrations and we are interested in forming a simple link (a catenane), e.g. see Fig. 10. Following Frisch and Wasserman (1961), we may be interested to know the conditional probability bM that the threading of a particular ring by a given linear chain (with subsequent cyclization) will result in a stable catenane. The probability p that a given ring and now cyclized but initially open forms 12 a catenane is bM -times the probability of overlap of their segmental distributions, i.e. the ratio of their

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spherical covolume 4p(R #R )3 to the total volume », that is 3 1 2 p "bM 4p(R #R )3/» , (2.1) 12 3 1 2 where R and R are the corresponding radii of gyration. Frisch and Wasserman (1961) had made 1 2 a plausible assumption that bM K1 which provides an yield ½ of catenanes per cyclized chain as 2 ½"o2p(R #R )3 . (2.2) 3 1 2 This produces for the total concentration of catenanes C the result K C "o2B , (2.3) K where the density o"n/» with n being the total number of rings (or linear chains), and B is defined by ½/o and has a meaning of the (topological!) second virial coefficient. The most spectacular outcome of these simple calculations lies in the fact that the subsequent Monte Carlo results of Vologodskii et al. (1975), indeed, had produced B which is in remarkable agreement with simple qualitative analysis by Frisch and Wasserman (1961). In Section 7 we reproduce analytically the result for B using path integral methods. In the same section we also reproduce the Monte Carlo results for the probability of linking (entanglement) between two ring polymers. This result has some implications for calculation of the elastic moduli of the crosslinked entangled polymer networks to be discussed below and in Section 7. Biological applications of the results related to catenanes can be found in recent papers by Levene et al. (1995) and Vologodskii and Cozzarelli (1993) while the real experiments on knotting of DNA molecules are discussed by Rybenkov et al. (1993), and Shaw and Wang (1993). The above results include only static properties of rings. New additional effects arise when dynamical effects are considered. Since these effects are being understood much less than static effects, we shall only briefly discuss some recent theoretical and experimental results for completeness of our presentation. They are naturally going to be only qualitative and should serve only as a starting point of further more systematic investigations. To begin we would like to recall the statement made in the classical paper by Brochard and de Gennes (1977). “At this stage it appeared natural to extend the analysis toward the case of theta solvents, where the static conformations of the chains become nearly ideal. We decided to do this and found, to our great surprise, that theta solvents are considerably more difficult than good solvents!2.In a good solvent, the chain is very much swollen and makes no knots on itself. In a poor solvent, it is more compact and makes many self-knots2. ¹he single-chain analysis in the entangled (i.e. h-point) regime is the most delicate exercise in dynamical scaling and requires very long explanations2. Thus, after a long reflection, we decided to restrict the present discussion to the many-chain problem (semidilute solutions) at the h-point; this remains comparatively simple, because the fluctuation modes are plain waves”. Since 1977 not much had changed as we shall demonstrate shortly. For the recent experimental results in this field, please, see Brulet et al. (1996). Subsequently, de Gennes (1984) had noticed that concentrated polymer solutions (melts) also present a puzzle if their dynamics is of interest. This happens, for instance, if one can rapidly quench the melt by abruptly changing the melt temperature below the temperature of crystallization. If then one measures the relaxation time q which is required to bring the melt back to its initial state, R one then observes that this time is much longer than the terminal time q JM3.3 (where M is the 5 molecular weight of the chain). This could be understood (qualitatively) if one recognizes that in the

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melt the individual chains are Gaussian-like with R JJM. Since the length N of the polymer is ' proportional to M, the ratio JN/N goes asymptotically to zero for NPR, i.e. the melt could be viewed as a solution of randomly interlocked (quasi)rings (see below). Since most of these rings will be (quasi)knotted (see Sections 3—6) the rapid temperature quenching (from above) will leave this melt in a glassy-like state, since the ring of length N could be in K(c(N)) different topological states (Section 7) due to the fact that the number c(N) of crossings in the knot projection (see Sections 3—7) which characterizes the knot complexity grows rapidly with N. When the temperature is rapidly raised (from below) the tight knots could be readily formed, de Gennes (1984), thus causing an enormous relaxation time q which is associated with their untightening. Moreover, since not R only knots but the links could be formed as well during quenching, this process could provide an additional strong contribution to the observed effect. We calculate the probability of link formation in Section 7, and in this section we shall describe how this quantity is related to the elastic moduli of the crosslinked entangled networks. An attempt to understand the dynamics of the collapse of the individual polymer chain was also made by de Gennes (1985). His results were subsequently refined by Grosberg et al. (1988), Rabin et al. (1995) and others. Some numerical results related to these works could be found in the paper by Ma et al. (1995) which also provides references on the related numerical work. The main outcome of this work is the consensus that for the linear polymers the dynamics of collapse is two-stage process. This has been recently confirmed experimentally, e.g. see Chu et al. (1995), Ueda and Yoshikawa (1996). However, there is a considerable disagreement, e.g. see Chu and Ying (1996) and Chu et al. (1995), about the role of knotting in the dynamics of the collapse process. For instance, in the de Gennes (1985) paper there is no mentioning of knots; in the Grosberg et al. (1988) paper there is an argument in favor of tight knot formation at the second stage of the two-stage collapse process, while in Chu et al. (1995), based on the experimentally observed comparability of the relaxation times for both stages, the suggestion is made that the knotting effects could be important at the first stage as well. Chu and Ying (1996) argue, however, that the interpretation of experimental data suggests that knotting plays no role (or dominant role) in the kinetics of individual chain collapse. Finally, according to Grosberg et al. (1988) the collapse of an unknotted ring polymer should be a one-stage process. Since there are no experimental data available on collapse of rings (knotted or unknotted), no further discussion on this topic is possible at the time this review is written. 2.2. Polymer dynamics and topology Although we have discussed some dynamical aspects of ring polymers and melts in Section 2.1, we would like to present here some additional (less controversial) results related to dynamics of individual circular polymer chains and to dynamics of melts. Let us begin with the paper by Brinke and Hadziioannou (1987). These authors had performed extensive Monte Carlo calculations for ring polymers. They had taken into account the topology effects so that their calculations provided data for both knotted and unknotted rings. Calculation of the radius of gyration R as well as scattering form factor S(q) for both knotted and unknotted ' rings, and comparison with real experimental data indicates that the difference between the knotted and the unknotted observables is marginal. That is, although the dimensions of knotted

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Fig. 1. Example of a daisy-like ring system.

rings are slightly smaller for rings (as compared with the linear polymers of the length N), the critical exponents (in good solvent regime) are the same and are independent of the knot type (i.e. the same as for unknots). The same conclusion had been reached in the subsequent work by von Rensburg and Wittington (1991). We are not discussing here more recent Monte Carlo results by Orlandini et al. (1996) which provide exponents depending upon the knot type. These latest results should await some experimental verification, since they are not related to observables such as R or ' S(q). Both S(q) and R can be used in hydrodynamical calculations (e.g. calculation of the diffusion ' coefficient D of the macromolecule). Comparison with real experimental data indicates that dynamical data (e.g. for D) are in accord with static data, i.e., the value(s) of critical exponent(s) (e.g. for the hydrodynamic radius) are the same for both linear and circular polymers, with the overall dimensions of the circular polymers being uniformly smaller as compared with the linear polymers of the same molecular weight. The above results can be explained qualitatively based on recent arguments by Quake (1994) (please, see also Section 7). Quake makes the assumption that, independent of knot complexity, the fundamental scaling law for polymers, R JNl, is retained. Then, a knot Kof length N with c[K] ' essential crossings (e.g. see Section 3.2) is considered as c[K] loops each of length N/c[K], e.g. see Fig. 1 and Burkchard et al. (1996). Each loop has a radius of gyration R J(N/c[K])l so that the ' total volume » of K is »Jc » Jc(N/c[K])3l. Whence, the radius of gyration R (K) for -001 ' K should scale as R (K)J»1@3JNl[c[K]]1@3~l . '

(2.4)

If l is taken to be of Flory-type, i.e. l"3, then the above estimate provides for R (K) the following 5 ' result: R (K)JN3@5[c[K]]~4@15 . '

(2.5)

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In Sections 3 and 7 we are going to demonstrate that c[K] is an actually N-dependent quantity, so that the estimate given by Eq. (2.5) is, strictly speaking, inconsistent with the initial assumption about the behavior of R . Nevertheless, Quake’s arguments could be somehow repaired if, instead ' of c[K] we would use the average writhe SD¼ [K]DT which is directly related to c[K], e.g. 3 see Sections 3 and 7. Since, as we have demonstrated (Kholodenko and Vilgis, 1996), SD¼ [K]DTJJN, we obtain, instead of Eq. (2.5), the following estimate for R : 3 ' R (K)JN3@5N~0.13 . (2.6) ' The obtained qualitative results explain why knotted rings are always smaller than the linear polymers of the same length. Alternative results based on the concept of porosity P(N) are presented in Section 7. Based on these static results, Quake was able to provide an estimate for the relaxation time q based on the assumption that the Rouse model can adequately describe the R dynamics of knotted rings. The argument is rather standard (Kremer and Binder, 1988), and goes as follows. The fundamental relaxation time q is a long distance relaxation time which is determined R when the center of mass of the polymer has moved a distance of the order R . When it is interpreted ' in terms of local monomer—solvent interaction, each flip of the monomer changes position of the center of mass by a factor of 1/N. Since the flips are uncorrelated, they add up as in the case of random walk, i.e. (DR)2J(1/N)2. During the Rouse time q there are q N such displacements, so R R that the total displacement is (1/N)2q NKR2 . R ' From here we obtain

(2.7)

q JN2l`1[c[K]]2@3~2l . (2.8) R We have used c[K] in Eq. (2.8) just to be in accord with Quake (1994). Evidently, for consistency reasons, c[K] should be replaced by SD¼ [K]DT or by P(N). This is especially true in view of the fact 3 that Monte Carlo data provided by Quake cannot be directly used to plot q as a function of N. In R the absence of excluded volume interactions we have 2l"1 and Eq. (2.8), indeed, produces the Rouse time (if c[K] is independent of N). The above results are relevant only to very dilute solutions of knotted rings in good or h-solvents. Below the h-point the dynamics of the collapsed individual linear chains was recently studied by Monte Carlo methods by Milchev and Binder (1994). Even for the linear chains the obtained results are inconclusive (e.g. dynamical critical exponents are temperature-dependent, etc.). We hope that this fact will stimulate more research in this area in the future. In the opposite limit of polymer melts the situation is relatively better, since the reptation theory of de Gennes (1971) and Doi and Edwards (1978) provides rather satisfactory qualitative explanation of the viscoelastic properties of melts of linear polymers. As for melts of ring polymers, an attempt had been made (see e.g. Kholodenko, 1991; Obukhov et al., 1994), to extend the existing linear polymer theory. Since the experimental data by McKena et al. (1989) strongly indicate that the results for rings parallel that for the linear polymers (just like in the dilute regime), we tend to believe that the linear theory can be used for melts of rings as well (Kholodenko, 1991). This can be understood if we recall, e.g. see Section 2.1, that even linear polymers at h-conditions are asymptotically closed, since JSR2T/NP0 for NPR. This argument could be traced back to Delbru¨ck

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(1962) and Kholodenko (1994). The fact that polymer melt of linear polymers can be also viewed as melt of randomly linked quasi-rings has some profound effect on the individual chain(s) in such melt, to be discussed in Sections 7 and 8. Here we would like to provide only some qualitative arguments. Following Doi and Edwards (1978), and de Gennes (1990a), we shall assume that “every chain, at a given instant, is confined within a ‘tube’ as it cannot intersect the neighboring chains. The chain thus moves inside the tube like a snake” (i.e. reptates). The diffusive motion of such trapped chain is Rouse-like so that the diffusion coefficient D (N) scales like R D~1(N)"Nq /b2 , (2.9) R 0 where q is solvent relaxation time, while b is the characteristic parameter of the Rouse model of the 0 order of the size of the individual bead, e.g. see Eq. (2.7). The length of the tube ¸ and its radius a are assumed to be related to the length Nb of the trapped chain of N effective beads via the simple relation ¸aKNb2 .

(2.10)

The characteristic time q needed for the chain to leave the domain of space of order ¸ can be 5 estimated via

A B

AB

¸2 Nb2 2Nq b 2 0&q N3 q& & , (2.11) 5 D (N) 0 a b2 a R while for the translational self-diffusion coefficient D , Doi and Edwards (1986) provide an estimate T b2 a2 a2 a & . (2.12) D &D & T R¸ q N Nb2 q N2 0 0 The last result is in remarkable agreement with experiments on monodisperse melts while for q experimental data suggest q &N3.4. There are many attempts to “repair” the simple arguments 5 5 leading to an estimate of Eq. (2.11). In Sections 6 and 8 we shall discuss in detail some of these attempts, while here we restrict ourselves only by the following remarks. The fact that the chain “cannot intersect the neighboring chains”, de Gennes (1990a), makes its “motion” quasi-onedimensional. The very fact that the “motion” is restricted, naturally breaks the symmetry between the longitudinal and the transversal diffusive motions of the chain (Section 6.3) causing the effective additional stiffness for the longitudinal component of “motion”. The mechanism(s) by which the longitudinal “motion” becomes more stiff have both the topological (Kholodenko, 1991), and the geometrical (Kholodenko, 1995, 1996a, 1996b, 1996c), origins. But, irrespective to the underlying mechanism, it is possible to carry out scaling analysis analogous to that given by Eqs. (2.11) and (2.12), which includes the anticipated effects of longitudinal stiffening. This analysis was performed by Tinland et al. (1990) and, independently, by Kholodenko (1991). Stiffening of the longitudinal “motion” was also advocated in more recent papers by Perico and Selifano (1995) and Wang (1995). To incorporate the stiffness effects into the scaling analysis, we would like to notice that the diffusion coefficient D (N) for the Rouse chain, Eq. (2.9), and the translational diffusion coefficient R

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D (N) for the rigid rod G ln N D (N)K (2.13) G 3pbg bK N 4 will look identical if formally we put b2/q in Eq. (2.9) equal to (ln N)/3pbg bK , where b~1 is the usual 0 4 Boltzmann’s temperature factor, g is the solvent’s viscosity and bK is the diameter of the rod. Now, 4 instead of Eq. (2.11), we can write

A B

AB

¸2 q Nb2 3 b 4 q& & 0 &q N3 , (2.14) 5 D (¸) ab2 a 0 a G where we had assumed, that D~1(¸)K¸q /b2a. Since, according to Eq. (2.12), the result for D is G 0 T b-independent, the replacement of D by D will produce no change and, accordingly, the R G translational self-diffusion coefficient will remain the same, i.e. proportional to N~2. At the same time, Eq. (2.11) will change. Since, according to Eq. (2.13), b2J ln N. If we now formally put ln N"Nu, then for experimentally used values of N(1054N4106) we obtain 0.194u40.21. By combining this result with Eq. (2.11), we obtain, q&q N3`2u , (2.15) 0 where 2u lies in the range of 0.3842u40.42. The obtained result is in excellent agreement with the experimental data presented in the book by Doi and Edwards (1986). The extreme case of rigid rod diffusion coefficient given by Eq. (2.13) should not be taken too literally since the stiffness of the chain is scale-dependent property. This means that the effective persistent length &a is expected to be larger than b (which is in accord with Doi and Edwards, 1986). If a/b<1, then ¸/b;N, according to Eq. (2.10) taken from Doi and Edwards (1986). The results discussed above are also relevant to the description of the viscoelastic properties of crosslinked polymer networks, gels, etc. (de Gennes, 1979) which we would like to discuss briefly now. 2.3. Polymer networks Study of the role of topology in polymer networks (rubbers, gels, glasses, etc.) was initiated in seminal work by Edwards (1967a,b; 1968). More detailed study of this topic could be found in the subsequent works by Deam and Edwards (1976), and Edwards and Vilgis (1988). More recent developments are summarized in the recent work by Panykov and Rabin (1996), where many additional relevant references could be found. Polymer melts and polymer networks have many things in common. For example, in both systems there are entanglements which constrain motion of individual chain(s). The presence of entanglements alone is sufficient for the formation of tubes. The concept of a tube had been put forward in the work by Edwards (1967b) in the context of polymer networks and had been successfully used by de Gennes (1971) in connection with the reptation model discussed in Section 2.2. The tube can be formed only if the length of the chain N exceeds some characteristic length N (the contour length between two successive entanglements along the polymer’s back% bone). The parameter N is related somehow to the monomer density, as will be explained in %

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Section 8. The role of topology in both polymer melts and polymer networks is thus effectively reduced to the description of the individual polymer chains inside the fictitious tube. The philosophy of such approach is in complete accord with similar mean field calculations in quantum mechanics, e.g. Hartree or Hartree—Fock type of approximation(s), etc. Unlike the case of quantum mechanics, in the present case the attempts to systematically reduce a well-posed microscopic problem which explicitly accounts for entanglements, e.g. see Deam and Edwards (1976), to the mean field tube model, had only been partially successful. In Section 8 we provide an alternative treatment of this problem, which takes topological effects explicitly into account, and compare our theoretical results against recent experimental data of Fetters et al. (1994). In case of networks, there is another characteristic length scale N (the contour 4 length between two successive crosslinks along the polymer’s backbone). Whence, it is reasonable to consider the situations when N 'N and N (N . In the first case the presence of tube(s) 4 % 4 % should be important (Edwards and Vilgis, 1988), while in the second the effect caused by the tube existence should become unimportant. In reality both N and N are fluctuating quantities which 4 % depend on the polymer/monomer density in a nontrivial way, e.g. see Duering et al. (1994), which most of the time is not well understood. This is caused by the conditions of preparation of the networks, e.g. by vulcanization or by radiation crosslinking. In both cases the final product contains a wide distribution of strand lengths and a large number of dangling ends. The dangling ends are expected to slow down any relaxation significantly, but are not believed to actively support stress. These factors make any attempts of rigorous theoretical treatment quite difficult (Mark and Erman, 1992; Iwata and Edwards, 1988, 1989). The technical complications come as well from the fact that the polymer melt can be viewed as an annealed system while a network is certainly quenched. This means that, in general, one has to use the replica trick methods similar to that used in the theory of spin glasses, Mezard et al. (1988), in order to calculate the observables (Edwards and Vilgis, 1988). Recently, an attempt to by-pass the replica trick procedure was made (Solf and Vilgis, 1995, 1996, 1997). In the regime when N 'N the presence of topological % 4 entanglements can be ignored and then the quenched disorder can be dealt with analytically without replicas. Development of these results to the regime N 'N remains a challenging problem. 4 % In order to understand better the complexities associated with entanglements one can, following de Gennes (1979), think of polymer networks made of concatenated rings , the so-called “olympic” gel. In such a system, no permanent crosslinks are present, and the elasticity is determined exclusively by the topology of concatenated rings. The properties of such networks are expected to be (Vilgis and Otto, 1997) very different from that known for the conventional rubbers, Treloar (1975). An “olympic” gel model is a limiting case of a more complicated model proposed by Graessly and Pearson (1977). In this model the network is made out of polymer loops which may be entangled pairwise at random. It is possible to calculate the shear modulus G for such model (see below) even in the presence of the permanent crosslinking since the topological G and the 5 crosslinking G parts of G are expected to enter into the total modulus G additively (Kramer and C Ferry, 1975; Everaers and Kremer, 1996). The underlying assumptions of Graessly—Pearson (G-P) model are: (a) the polymer loops are randomly distributed in space so that the number of loops per unit volume is o (defined in Eq. (2.3)); (b) the contributions of these loops to the entropy of deformation are independent and pairwise additive;

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(c) if DRD is the distance between the centers of mass of some loop pair, then f (DRD) is the N probability of this pair to be linked, and N is the contour length of the polymer, as before; (d) only the affine deformations are considered so that after the deformation the new displacement vector R@"R for all loop pairs. Whence, if G"G #G , then 5 C G "F[ f (x)]oJ k ¹ , (2.16) 5 B where k ¹ is the usual temperature factor, f (x),f (x), B N = dx x2f (x) , (2.17) oJ "2poR3(N) L 0 1 = dx x4( f @(x))2[ f (x)[1!f (x)]]~1 (2.18) F[ f (x)]" 15 0 while f @(x)"df/dx. The entanglement radius, R (N) is, to some extent, an adjustable parameter of L a G—P model but, according to Everaers and Kremer (1996), can be estimated from the selfconsistency equation

P P

P

4p 1 R3(N)" d3r f (DrD) . N 3 L 2

(2.19)

Whence, if the probability of linking is known, the topological contribution G to the elastic sheer 5 modulus can be calculated according to Eq. (2.16). This probability was estimated by Monte Carlo methods by Vologodskii et al. (1975) and was recently reobtained by Everaers and Kremer (1996) who compared their Monte Carlo data for G with G—P result, Eq. (2.16). The comparison was 5 made using two independent methods. First, G was estimated numerically without any reference to 5 Eq. (2.16). The results of these simulations are nicely summarized by the equation (G!G )/oJ "0.85 k ¹ (2.20) C B which indicates that the topological contribution to the shear modulus is independent of chain length N. Then, the linking probability f (DRD) was estimated numerically for the simplest link, e.g. N see Fig. 10, and is found to be in complete agreement with Vologodskii et al. (1975). It was found that f (DRD)"A expM!c(R/R )uN , (2.21) N L where both A and c are numerical constants, AK0.6 and c"A/2, while R"DRD. The exponent u was found to be equal to 3 but, following G—P, we argue that it can, in principle, have values lower than 3. Substitution of thus obtained f (DRD) into Eq. (2.16) have produced N G!G C"1.3 k ¹ (2.22) B oJ which is in excellent agreement with the independent result given by Eq. (2.20). In Section 7 we reobtain the distribution function analytically. In order to compare our results with existing data in literature, several comments need to be made. First, already in the paper by

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Graessly and Pearson several trial distribution functions were tested, all in the form of Eq. (2.21), but with the exponent u ranging between 1 and 3. The exponent 3 was taken from the work of Vologodskii et al. (1975) while the exponent 2 appears in the analytical calculations of Prager and Frisch (1967) of the entanglement probability between the planar Brownian walk and the infinite rod perpendicular to the plane. For arbitrary m,R/R they obtained f (DRD)"erf c(m), which for R N large m’s produces f (DRD)JexpM!m2N. The above result was also independently effectively obN tained by Helfand and Pearson (1983) who provided an estimate of the entanglement probability for a closed polymer loop trapped into an array of obstacles (meant to represent other chains). We provide some related results on this subject in Section 8 and Appendix A.1. In Section 7.6 we demonstrate analytically that the exponent u in Eq. (2.21) can take only the values between 2 and 3. To understand this and other facts discussed in this section, we need to rely on solid mathematical background about knots and links which begins with the next section.

3. Single chain problems which involve entanglements (general considerations) 3.1. Topological persistence length and the probability of knot formation In his seminal papers, Edwards (1967a,b) had noticed that “treating polymer as a random path clearly must fail at small distances when the precise molecular structure dominates 2. It is not clear, however, whether the question of whether random path contain a knot is at all meaningful in the mathematical idealization of infinitesimal steps. One would guess that such questions are not meaningful, getting into unresolved, perhaps unresolvable, questions of measure 2. since a random path permitting infinitesimal steps will be ‘infinitely knotted’ .” With these remarks in mind, it is obvious that the cut-off must somehow be introduced into any kind of discussion which involves real polymers which may be topologically entangled. This cut-off can be introduced both in the continuous and in the lattice polymer models, e.g. see de Gennes (1979). When a flexible polymer is modeled on the lattice, the lattice unit step length can be conveniently chosen to be a unity. In the continuum, such a choice is also permissible if the total polymer length N is being measured in the units of Kuhn’s length l. In various models of polymers (Kholodenko, 1995), the role of l is being played by the persistence length lK . More precise definitions will be provided later in the text. Both l and lK do not have a topological origin, but they do affect the topological properties of polymers. For instance, let us consider a closed random walk on some three-dimensional lattice. It is reasonable to anticipate that there should be a minimal number of steps N (which depends upon the geometry of lattice) in order for the first non-trivial T knot to be formed. Accordingly, for closed walks of less than N on the lattice, no knots can be T formed. The idea about estimating N originated some time ago in the work by Delbru¨ck (1962), T but was rigorously developed only recently. Diao (1993, 1994) using rather sophisticated combinational arguments had found that for a simple cubic lattice N "24. In Section 7 we shall provide T much simpler derivation of this result using path integrals. In the mean time, we would like to notice that, along with N which we call “topological persistence length”, there is a related quantity, T f , which is the probability for a closed walk of N steps to remain unknotted. Frisch and N

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Wasserman (1961) and Delbru¨ck (1962) put forward a conjecture that lim P ,1!f P1 , (3.1) N N N?= i.e. for NPR the probability P for a closed walk to be knotted tends to unity. This conjecture N had been proven only recently by Sumners and Whittington (1988) and by Pippenger (1989). A very nice account of these results could be found in the monograph by Welsh (1993). The above authors had shown that lim (f )1@N"c (3.2) N N?= where the constant c(1 had remained undetermined. Recently, Kholodenko (1991, 1994) had been able to provide an estimate of the constant c. By analyzing Monte Carlo data by Windwer (1990), who tried to fit his results for f by using the ansatz N f "cJ kJ NNa (3.3) N with a"0, kJ "0.9949 and cJ "1.2325, Kholodenko (1991, 1994) had found that it is sufficient to determine only cJ . Indeed, using Eq. (3.3) we obtain 1"f T"cJ kJ NT . N This produces at once

AB

1 N@NT f "cJ N cJ

(3.4)

(3.5)

so that if N is known, f is determined by cJ . Eq. (3.5) is in agreement with Eq. (3.2) with c in T N Eq. (3.2) being cJ ~1 in Eq. (3.5) (for NPR). In Section 5 the analytical derivation of the result(s) of Eq. (3.3) (or Eq. (3.5)) will be provided. For completeness, we would like to mention that, in addition to N , there is another number, T called the edge number, e(K). For a given knot, it is defined as the minimal number of edges required to represent the given knot K as a polygon in three-dimensional space (Randell, 1994). e(K) is a topological invariant similar to the minimal crossing (unknotting) number u(K) to be further discussed in Sections 3.3 and 7.4. Unfortunately, as far as we can see, e(K) is of little importance for polymers. Indeed, it can be shown that for the unknot e(K)"3 and for the trefoil knot e(K)46, etc. To obtain these numbers in the case of polymers, one needs to use rather unrealistic freely jointed chain model of polymers. This model provides satisfactory description of polymers at larger scales (in h solvent regime), but is much less realistic at the smaller scales where the bond angles and the torsional bond energies should be taken into account. But, unlike N , e(K) T can be used in the continuum, i.e., in the off-lattice calculations. Whence, if the polymer is made of rather long rigid rods connected by the freely flexible joints, e(K) can be used, in principle. 3.2. Knot complexity and the average writhe It is rather remarkable that the notion of knot complexity came to knot theory at its birth (Harpe et al., 1986). One of the cofounders of knot theory, Tait, had formulated the main tasks of knot

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Fig. 2. Sign convention for the oriented crossing.

theory among which he expected “to establish a hierarchy among knots relying on some notion of complexity”. As it will be discussed in Section 4, there are two ways to describe knots: differential-geometric and via planar diagrams. In the last case we are dealing with 4-valent planar graphs where at each crossing the decision should be made about how this crossing must be resolved, e.g. see Fig. 2. If we disregard this resolution and just count the number of vertices c(K) for a given knot K projection into some plane, we obtain the knot complexity (Kholodenko and Rolfsen, 1996). c(K) is not a topological invariant and is not the only quantity which measures the knot complexity. Other quantities are discussed in Sections 3.3 and 7. They are all interrelated. For instance, let e(p)"$1 where p is some vertex in the planar knot diagram. Then, it is possible to define the writhe ¼ [K] for a given knot via 3 ¼ [K]" + e(p) , (3.6) 3 o p S(K) where S[K] denotes the set of crossings on some knot diagram K (Kauffman, 1987a). In case when knots are generated on some 3D lattices, the question arises how the knot complexity c(K) and the writhe ¼ [K] of the knot K depend on the number of steps N which are 3 required to form this knot. Evidently, the very same knot can be placed onto the lattice in many ways. Whence, it makes sense to introduce the averaged complexity Sc[K]T and the averaged writhe S¼ [K]T where S2T means the averaging over the possible arrangements of a given knot 3 K on the lattice. Alternatively, one can think of generating some knot K and changing the orientations of the plane into which it is projected. This strategy was chosen in the recent numerical simulations by Whittington et al. (1993, 1994a, b). These authors have found that Sc[K]TJNac ,

(3.7)

where a K1.122$0.005 and c SD¼ [K]DTJNa , (3.8) 3 where a50.5. At the same time, S¼ [K]T"0, by the symmetry arguments as it will be explained 3 below, in Section 7.2. The results of Whittington (1994a) indicate that the obtained values for a are not sufficiently reliable. c These authors argue (without proof!) that actually 1(a (2. Recently, Arteca (1994, 1995) c

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had performed independent detailed numerical simulations and found that a &1.40$0.04. The c situation with the averaged writhe is more reliable since in Whittington (1993) the exponent a was found analytically to be 0.5. This result is also supported by a completely different calculation by Yor (1992) and by much earlier Monte Carlo results by Chen (1981), Le Bret (1980) and Vologodskii et al. (1979). In Sections 7.2 and 7.3 we shall rederive the results Eqs. (3.7) and (3.8) using path integrals. We shall rigorously demonstrate that 1(a (1.5 and that the inclusion of c the excluded volume effects lowers the upper bound for a from 1.5 to less than 1.4. Obtained results c are in excellent agreement with the numerical results of Arteca (1994, 1995). 3.3. The unknotting number and the number of distinct knots for polymer of given length N From the previous discussion it is intuitively expected that the knot complexity c(K) should be associated with the unknotting number u(K) which is the minimal number of self-crossings which will turn knot into unknot (Kholodenko and Rolfsen, 1996). The question arises how c[K] is related to u(K). Moreover, the unknotting number u(K) is a topological invariant, Rolfsen (1976), while we have noticed that the averaged c[K] is N-dependent. The answer to this question will be provided in Section 7. Here we only notice that u[K] is intrinsically connected with the fact that our knot, i.e. the circle S1, is embedded into R3 (or S3, i.e. R3XMRN). If, instead, we would consider the embedding of our knot into R4 (or S4), then it can be shown (Bing and Klee, 1964), that any nontrivial knot in R3 becomes an unknot in R4. This fact is reminiscent of the fact that any self-avoiding walk in R3 becomes effectively Gaussian in R4 (de Gennes, 1979). The above theorem of Bing and Klee makes use of the e-expansions in knot problems questionable. The relation between u(K) and c(K) is known in literature as Bennequin conjecture (Bennequin, 1983; Menasco, 1994), and mathematically can be stated as 1(D¼ [K]D!nL #1)4u[K]41(c[K]!nL #1) , (3.9) 2 3 2 where it is assumed that the knot is made of a closure of a braid of nL strings (see Section 5 for precise definitions of braids). The above inequality can be understood using the following arguments (Gilbert and Porter, 1994). Any knot projection can be decomposed into Seifert circles by deleting crossings and glueing the reminding arcs in such a way that they form a set of circles as depicted in Fig. 3. The two arcs and the parts of the crossing removed make up a rectangle. If our knot projection was given an orientation, then the Seifert circles also acquire an orientation as well as the rectangles. Let us now twist these rectangles (as if we would make a Mo¨bius strip) and reglue them back to the circles. Obviously, instead of a knot, this time we shall obtain a surface. The boundary of this surface is our knot K. This surface has a genus g[K] and by means of a very simple argument (Gilbert and Porter, 1994, pp. 92—93), it can be shown that g[K]41(c[K]!s#1) , (3.10) 2 where s is the number of Seifert circles. In Kholodenko and Rolfsen (1996) it is shown that s and nL are interrelated (see also Section 4). By comparing inequalities Eqs. (3.9) and (3.10) we conclude that u[K]Kg[K]

(3.11)

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271

Fig. 3. Formation of Seifert circles for figure-eight knot.

and, since u[K] is a topological invariant, g[K] is also an invariant of a knot K. From the above discussion it follows that the number of distinct knots should somehow be dependent on u[K] (or g[K]). According to Tutte (1963), the number ¹[n] of different planar graphs with n edges is estimated to be ¹[n]42]12n .

(3.12)

In Freedman et al. (1994) it is argued that the correspondence MD, nNcrossingsPMG, with n edgesN

(3.13)

is at most 2n to 1. Here, D is a knot diagram while G is a planar graph, so that the number of knot diagrams with exactly n crossings is bounded by 2n¹(n)42(24n). Given this result, the number K(n) of knot diagrams with at most n crossings must satisfy 2n4K(n)42(24)n .

(3.14)

Whence, if n is known then K(n) can be related (identified) with the number of distinct knots for the knot diagram with n crossings. Moreover, since n&c(K) as was shown in Freedman et al. (1994), we can replace the above inequality with 2c*K+4K(n)42(24)c*K+ .

(3.15)

Whence, knowledge of c[K] provides us with some information about u[K] and K[n]. These facts are going to be fully exploited in Section 7.

4. Methods of describing knots (links) 4.1. Differential geometric approach From the point of view of differential geometry knots are just closed curves in three-dimensional Euclidean space. As is well known, (see, e.g. Dubrovin et al., 1985), every nonplanar curve is being fully described by its local curvature and torsion. Frenchel (1951) had noticed that for any closed

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curve (knotted or not) of length N

P

N dq Dk(q)D52p , (4.1) 0 where k(q) is the local curvature of the curve. This result resembles the famous Gauss—Bonnet theorem for surfaces

P

1 2p

K dS"s(M2) , (4.2) M2 where s(M2) is the Euler characteristic of the manifold M2 (Monastyrsky, 1993), and, indeed, was motivated by the result of Eq. (4.2). More surprising is the result of Milnor (1950) who had shown that for the knotted curve

P

N dq Dk(q)D'4p . (4.3) 0 This result was generalized for surfaces by Langevin and Rosenberg (1976) who had proven that for the unknotted torus

P

1 DKD dS"4 2p

(4.4)

(to be compared with Eq. (4.2)) and if the torus is knotted, then

P

1 DKD dS58 . 2p

(4.5)

This result was subsequently refined by Kuiper and Meeks (1984) and by Willmore (1982) who had demonstrated that if the surface is unknotted and H is the extrinsic curvature (i.e. H"1(k #k ), 2 1 2 where k and k are principal curvature radii), then 1 2 1 H2 dS5p , (4.6) 2p 2 M while for the knotted surface

P P

1 2p

H2 dS'4 . (4.7) M2 Although in this work we shall not touch the topic of knotted surfaces, we believe, that the above results deserve attention, especially in light of the results presented in Section 7. Besides the result Eq. (4.3), Milnor (1950) had also obtained additional results for closed curves

P

P

N N dq Dk(q)D# dq Di(q)D52pn (4.8) 0 0 where i(q) is the torsion of the curve. For the unknot, n"1. This result along with Eq. (4.3) should be taken into consideration when the path integrals for semi-flexible polymers are calculated

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(Kholodenko, 1990, 1995). We shall discuss some of the implications of these constraints on path integrals in Section 7. Use of these constraints will allow us to calculate the topological persistence length N defined in T Section 3 and, in principle, affects other observables such as SD¼ [K]DT, Sc[K]T, etc. also intro3 duced in Section 3. 4.2. Path integral approach via Abelian and non-Abelian Chern—Simons field theory Beginning from the seminal works of Edwards (1967a,b, 1968), topological entanglements in polymers are being described by the constrained path integrals which effectively employ the observables of the Abelian Chern—Simons field theory (ACSFT). The non-Abelian variant of these path integral calculations, to our knowledge, was used for polymer problems only in Kholodenko (1994). As was noticed already in Section 3.3, use of the field-theoretic methods for knot problems should be performed with extreme caution since e-expansions are, strictly speaking, illegitimate for problems which involve knots (links). The most attractive feature of the non-Abelian variant of the Chern—Simons field theory (NACSFT) lies in its ability to connect knots (links) of different complexities via skein (recurrence) relations (Guadagnini, 1993). This allows effectively to disentangle knotted polymer configurations, thus reducing the problem with complicated constraints to that without constraints. This does not imply that the information about entanglements is lost during this disentanglement process. The disentangled partition function will still remember its initial state as it is explained in Section 4.4. To demonstrate how the above general statements are implemented, let us consider the simplest situation of n interlocked polymer rings. This problem was considered before in Section 2.3, but now we would like to emphasize the mathematical aspects of the problem. If we ignore the excluded volume effects, the partition function Z for an assembly of simple circular polymer chains in three-dimensions can be written as

P

AP

B G

P

H

P

AP

B G

P

H

n Ni 3 n Ni Z" < D[r(q )]d (4.9) dq rR (q ) exp ! + dq rR 2 . i i i i 2l 0 i/1 i/1 0 where rR "dr/dq. For an assembly of n interlocked rings we can write, using Eq. (4.9), the following result: n Ni 3 n Ni dq rR (q ) exp ! + dq rR 2 Z" < D[r(q )]d i i i i i i 2l 0 i/1 i/1 0 n ]d c!+ lk(i, j) , i,j where

A

Q

Q

B

AK

KB

(4.10)

1 dl e (4.11) j j r !r i j Ci Cj and dl "rR dq , r "r (q), etc. The constant c in Eq. (4.10) should be an integer thus making the i * i i i d-function to be the Kronecker’s delta. The microcanonical formulation given by Eq. (4.10) is somewhat inconvenient, because it does not readily allow the standard field-theoretic treatment. 1 lk(i, j)" 4p

dl ] i

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To clarify this point, let us introduce the abelian CS action SA . Following Guadagnini (1993), we C~S have k SA [A]" #~4 8p

P

d3x ekloA ­ A , k l o

(4.12)

3

M where the constant prefactor k/8p in front of the action is chosen for further convenience and M3"R36MRN"S3. Define now the Abelian Wilson loop ¼(C) via

GQ H

¼(C)"exp ie

dl ) A

(4.13)

and consider the average for the set of n loops forming a link ¸,

T

Q

U

T

U

n n " < ¼(C ) , S¼(¸)T " < expMie dl ) A N #~4 i i i i Ci #~4 #~4 i/1 i/1 where the average S T is defined by #~4

P

S T "NK D[A] expMiSA [A]N2 #~4 #~4

(4.14)

(4.15)

with normalization constant NK being chosen in such a way that S1T "1. In view of Eq. (4.12), #~4 the average in Eq. (4.14) is easily computable since it involves the calculation of Gaussian-like integrals. The result of this averaging procedure produces:

G A B

H

2p n (4.16) S¼[¸]T "exp !i + e e lk(i, j) . #~4 i j k i,j The sum in the exponent of Eq. (4.16) contains the “undesirable” self-interaction terms (for i"j). Calculation of these terms is nontrivial (Guadagnini, 1993), and the final result depends upon how the limiting procedure iPj was performed in Eq. (4.11). Let us consider this procedure in some detail since we will use these results in Sections 5—8. For the linking number, given by Eq. (4.11), we can write an equivalent expression as follows:

Q Q

1 lk(i, j)" 4p

dxk

C

i

C

j

(x!y)o dyle . kvoDx!yD3

(4.17)

Let now yk(q)"xk(q)#enk(q), eP0`, Dn(s)D"1 .

(4.18)

By combining Eqs. (4.17) and (4.18) we obtain,

P P

1 (x(s)!x(q)!en(q))o 1 1 lk(i, i)"lk (i)"lim , ds dq e xR k(xR l#enR l) & klo Dx(s)!x(q)!en(q)D3 4p 0 0 e?0

(4.19)

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275

where the subscript f stands for framing. Depending upon the orientation of nk (Witten, 1989b; Calugareanu, 1961), one may obtain lk (i)"0 , & which is known as the standard(s) framing, or

(4.20)

lk (i)"¼ [i] , (4.21) & 3 where ¼ [i] was defined in Section 3 (one should identify i with K). The last case is known as 3 vertical (v) framing. More details on the framing procedure can be found in Bar-Natan (1995) and Aldinger et al. (1995). Using these results, one can claim that, at least for the case of standard framing, ACSFT can be used to obtain the partition function for the interlocked rings, Eq. (4.10), if instead of the microcanonical the grand canonical ensemble is used. Evidently, in this case, instead of Eq. (4.14), one should write

T G A B

HU

2p n SS¼(¸)T T " exp !i (4.22) + e e lk(i, j) #~4 1 i j k 1 i,j with S2T being defined by 1 n Ni 3 n Ni S2T " + D[r (q )] exp ! + (4.23) dq rQ 2 d dq rR (q ) 2 . 1 i i i i i i i 2l 0 i/1 i/0 0 The specifics of polymer problems, as compared with the standard field theory, lies in the fact that it is always necessary to perform a double average as in the case of Eq. (4.22). Moreover, since (for e "e) the combination e2(2p/k) is not an integer in general (and should be self-consistently i determined as it is always done in the grand canonical calculations), the polymer average in Eq. (4.22) is quite nontrivial. We illustrate this by considering an auxiliary problem of calculation of the double average for the polymer ring placed on the multiply connected plane (polymer ring entangled with array of rigid rods of infinite length). This problem is discussed in Appendix A.1 and in Section 8 in connection with the theory of reptation. Use of ACSFT does not allow us to relate the problem of an assembly of n interlocked rings to that of n!1 rings, etc., since it does not involve the skein relations (recursion relations relating knots (links) of different complexity). The situation can be dramatically improved if the NACSFT is considered instead. In this case, instead of the action, given by Eq. (4.12), we have to consider the “improved” action given by (Guadagnini, 1993)

P

k S [A]" #~4 4p

G

P

A

P

H AP

B

B

2 d3x eklo Tr A ­ A #i A ­ A , (4.24) k l o 3 k l o M3 where k is some integer and A (x)"Aa ¹a with ¹a being infinitesimal generators of some Lie group k k G, which obey commutation relations of the corresponding Lie algebra: [¹a,¹b]"if abc¹c

(4.25)

and, in addition, Tr[¹a¹b]"1dab . 2

(4.26)

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Instead of the Abelian Wilson loop, Eq. (4.13), now we have to use its non-Abelian generalization given by

C G Q HD

¼ (C)"Tr P exp ie o

dl ) A , (4.27) c where o specifies the type of the representation for ¹a’s and P denotes the path ordering operator (along the C-curve), while Tr denotes the operation of taking the trace. Using thus defined ¼ (C) we can now consider the averaged products, like that given by o Eq. (4.14), with the averaging being performed with the help of Eq. (4.15), where, instead of the action given by Eq. (4.12), we have to use now the action given by Eq. (4.24). The most spectacular difference between the Abelian and the non-Abelian variants of Chern—Simons field theory lies in the fact that different link averages in the last case become related to each other. This is the source of various knot polynomials. 4.3. Algebraic (group-theoretic) description of knots (links) via knot polynomials To understand how the recursion (skein) relations originate, we have to consider in some detail calculation of averaged ¼ (C) defined by Eq. (4.27). For this purpose we need to expand the o exponent in Eq. (4.27) first, thus producing

C Q Q P Q P P

x ¼ (C)"Tr I#i dxk A ! dxk dyl A (x)A ( y) o k l a k C C x y !i dxk dyl dzoA (x)A (y)A (z)#2 , k l o C where

Q P

P P

D

(4.28)

P P

x 1 s 1 1 1 dxk dyl" ds dq xR k(s)xR l(q)" ds dq xR k(s)xR l(q) . (4.29) 2 C o o o o Following Guadagnini et al. (1990), let us choose for G the group Sº(NK ), then, upon averaging with the help of Eqs. (4.15) and (4.24) we obtain for an assembly of n interlocked loops forming a link ¸ the following perturbative result:

G A A A

A BA B B A B BA B B A B

NK 2!1 n + lk (C ) & i 2NK i/1 2p 2 NK 2!1 n 1 2p 2 NK 2!1 2 n NK + o(C )! + lk2(C ) # i & i k 2NK 2 k 2NK i/1 i/1 2p 2 NK 2!1 2 n + lk2(C )lk (C ) ! & i & j k 2NK iEj 2p 2 1 NK 2!1 n ! + lk2(C ,C )#O(k~3) , i j k 2NK 2NK iEj where f denotes a type of framing: standard(s) or vertical (v). 2p S¼(¸)T "NK n 1!i & k

A BA

B

H

(4.30)

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277

Here o(C )"o (C )#o (C ) , i 1 i 2 i

(4.31)

where

Q P P

1 o (C )"! 1 i 32p2

dxk

Ci

x y dyl dzo eabce e e Ipjq(x, y, z) , kap lbj ocq

Q P P P

1 o (C )" 2 i 8p2

dxk

x

dyl

Ci

y z dzo dwp e

(w!y)a(z!x)b e pla okb Dw!yD3Dz!xD3

(4.32)

(4.33)

with

P

(w!x)p(w!y)j(w!z)q Ipjq(x, y, z)" d3w . Dw!xD3Dw!yD3Dw!zD3

(4.34)

The value of o(C ) is independent of the choice of framing and, in particular, for the unknot º , it i 0 was explicitly calculated with the result o(º )"! 1 . 0 12 For the case of standard framing Eq. (4.30) acquires a much simpler form. In particular, for just one loop, we obtain

G A BA

B

H

2p 2 NK 2!1 o(¸)#O(k~3) . S¼(¸)T "NK 1# 4 k 2

(4.35)

As it was shown by Witten (1989a) and, independently, by Fro¨lich and King (1989), for the case of unknot º the average S¼(º )T can be calculated exactly with the result (for Sº(NK )) 0 0 4 sin(pNK /(k#NK )) . S¼(º )T " 0 4 sin(p/(k#NK ))

(4.36)

In the Abelian case, NK "1, and S¼(º )T "1. This result is in agreement with Eq. (4.16) in view of 0 4 Eq. (4.20). To compare Eqs. (4.36) and (4.35) it is sufficient to replace ¸ by º in Eq. (4.35) and use 0 the Taylor series expansion of Eq. (4.36). Through second order in k~1 we obtain

G A

B

H

1 p 2 S¼(º )T "NK 1! (NK 2!1)#O(k~2) . 0 4 6 NK #k

(4.37)

If now k is replaced by k#NK in Eq. (4.35), then the complete agreement is reached between Eqs. (4.35) and (4.37) (since o(º )"! 1 ). The need to replace k by k#NK was shown, e.g. in 0 12 careful perturbative calculations by Shifman (1991).

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Consider now the ratio

G A BA B B A B A BC B A B A B

2p NK 2!1 n S¼(¸)T & "NK n~1 1!i + lk (C ) G& " & i L S¼(º )T k 2NK 0 & i/1 n 2p 21 NK 2!1 2 n 2p 2 NK 2!1 ! o(¸ )! + o(C ) + lk2(C )! NK 0 i & i k 2 2NK k 2 i/1 i/1 2p 2 NK 2!1 2 n 2p 2 1 NK 2!1 2 n ! + lk (c )lk (C )! + lk2(C ,C )#O(k~3) . & i & j i j k 2NK k 2NK 2NK iEj iEj (4.38)

A B A A BA

D

H

The higher order terms in the above expansion had been formerly considered by Guadagnini et al. (1989). For two unlocked rings use of the standard framing in Eq. (4.38) produces

G

A B

H

1 2p 2 (NK 2!1)#O(k~3) . G4 "NK 1! L 24 k

(4.39)

Comparison between Eqs. (4.39) and (4.36) (with kÁk#NK ) immediately produces S¼(¸)T "S¼(º )T S¼(º )T 4 0 4 0 4 in view of Eq. (4.37). Evidently, for n disconnected rings we would obtain

(4.40)

S¼(¸)T "(S¼(º )T )n . (4.41) 4 0 4 This result is in agreement with that obtained nonperturbatively by Witten (1989a) and is central for developing the theory of polynomials for knots and links. Indeed, following Harpe et al. (1986) let us consider three oriented knots (links) ¸ , ¸ and ¸ . Their projections onto an arbitrary ` ~ 0 plane differing from each other by just one crossing is shown in Fig. 4. Let us define axiomatically a link invariant P[¸]. Evidently, for the unknot º we should require 0 P[º ]"const . (4.42) 0 in view of Eq. (4.36). Let a , a and a be some, yet undetermined, constants. Then, we impose the ` ~ 0 condition (skein relation): a P[¸ ]#a P[¸ ]#a P[¸ ]"0 . ` ` ~ ~ 0 0 In particular, let ¸ ,¸ and ¸ be three link projections as depicted in Fig. 5. ` ~ 0 Then, using Eq. (4.43) we obtain

(4.43)

a P[º ]#a P[º ]#a P[º2]"0 , (4.44) ` 0 ~ 0 0 0 where º2 denotes a union of two unlocked rings (i.e. º2"º Xº ). By analogy with Eq. (4.40), we 0 0 0 0 can impose a requirement that P[º2]"P[º ]P[º ] . 0 0 0

(4.45)

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279

Fig. 4. Projections of three knots (links) which differ by just one crossing.

Fig. 5. A special case of three oriented links which differ just by one crossing.

Using this result in Eq. (4.43) we obtain at once a #a ~,S¼(º )T . P[º ]"! ` 0 0 4 a 0

(4.46)

Comparison with Eq. (4.36) allows us to obtain immediately the skein relation for P[¸]:

C

D

C C

D DB

ipNK ipNK P[¸ ]!exp # P[¸ ] exp ! ` ~ k#NK k#NK

A C

D

ipNK ipNK " exp ! !exp # k#NK k#NK

P[¸ ] 0

(4.47)

which coincides exactly with that obtained by Fro¨lich and King (1989) who used completely different methods to obtain this result. If we divide both sides of this equation by P[º ] for a single 0 loop, then we obtain the skein relation for the HOMFLY polynomial, P[¸]/P[º ]"G4 (Gilbert 0 L and Porter, 1994), which can be described axiomatically via a set of relations G4 0"1 , (4.48a) U uG4 `!u~1G4 ~"z G4 , (4.48b) L L L (4.48c) if G4 &G4I , then G4 "G4I , L L L L where & means that two knots ¸ and ¸I are ambient isotopic, i.e. that their projections are invariant with respect to all three Reidemeister moves as depicted in Fig. 6.

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Fig. 6. Three local Reidemeister moves. The variance with respect to moves (b) and (c) guarantees the regular (i.e. in the plane) isotopy, while the invariance with respect to (a) guarantees ambient (i.e. in space) isotopy.

Thus defined HOMFLY polynomials do not require NACSFT for their justification, e.g. see Harpe et al. (1986). At the same time, the actual values of constants u and z in Eq. (4.48b) remain undetermined, while Eq. (4.47) provides u"qNK @2 ,

1 z"Jq! , Jq

G

q"exp !i

H

2p . k#NI

(4.49)

Naively, we can obtain the Jones polynomial » from HOMFLY if we put u"t~1 and L z"Jt!(1/Jt) in Eq. (4.48b), e.g. see Gilbert and Porter (1994). Thus, we obtain the skein relation,

A

B

1 1 1 » ! » " Jt! » t L` t L~ Jt L0

(4.50)

supplemented with the normalization condition » 0"1 . (4.51) U Comparison with Eq. (4.47) and use of Eq. (4.49) leads us to the only choice: NK "2 and Jt"!1/Jq. If we recognize that the expansion of Eq. (4.30) can be also considered for the case of vertical framing, then taking into account that o(C ) are framing-independent, and using the i definition of ¼ [K] given by Eq. (3.6) (along with Eq. (4.21)), we notice the following. For a knot 3 K"KI X¸ the writhe ¼ [KI X¸ ]"¼ [KI ]#1 while if K"KI X¸ , then ¼ [KI X¸ ]" ` 3 ` 3 ~ 3 ~

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281

¼ [KI ]!1. Using these facts, we obtain for the unknot depicted in Fig. 6a, 3 (4.52) where

A BA G A

a"1!i

B A BA BH

B

1 2p 2 NK 2!1 NK 2!1 ! #O(k~3) 2 k 2NK 2NK

2p k

2p NK 2!1 +exp !i 2NK k

.

(4.53)

Using this result, it can be also shown (Guadagnini, 1990), that, in general, S¼(¸)T "aW3*L+S¼(¸)T v 4 or, in view of Eq. (4.38),

(4.54)

G4 "a~W3*L+G7 . L L Let us now use this fact and substitute Eq. (4.54) into Eq. (4.48b). We obtain

(4.55)

ua~W3*L`+Gv `!u~1a~W3*L~+Gv ~"za~W3*L0+G7 0 . (4.56) L L L Using known properties of ¼ [¸], just mentioned, we obtain from Eq. (4.56) the following result: 3 bG7 `!b~1G7 ~"zG7 0 , where b"ua~1 . (4.57) L L L This skein relation should be considered along with Eq. (4.52) (and its conjugate), i.e. (4.58a) G7 `"aG7 0 , L L G7 ~"a~1G7 0 . (4.58b) L L The results just obtained are in accord with the results obtained by Cotta-Ramusino et al. (1990). Evidently, by construction, e.g. see Eq. (4.38), G7 0"1. Comparison between this result and e.g. U Eq. (4.48a) dictates, in view of Eq. (4.55), that for the unknot ¼ (º )"0. This happens to be a very 3 0 important fact which allows us to obtain various polynomials using Eqs. (4.57), (4.58a) and (4.58b) and the normalization condition for G7 0. To make a connection with the field theory, we have to U remember that the actual values of constants a, b, u and z are not arbitrary, e.g. see Eq. (4.49). Using Eqs. (4.53) and (4.57) we obtain a"q(NK ~1)@2NK ,

b"ua~1"q1@2NK ,

z"q1@2!q~1@2 .

(4.59)

To obtain the Jones polynomial, let us consider as before a special case of Sº(2), i.e. NK "2. Then, using Eq. (4.59) we obtain a"q3@4,

b"1@3,

z"q1@2!q~1@2

(4.60)

and also, in view of Eq. (4.55), we have »(¸, q1@2)"q~3W3*L+@4G7 . (4.61) L These results are in complete accord with Guadagnini (1993), where they were obtained in a different way. The important thing to remember is that »(¸, q1@2) is an invariant of an ambient

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Fig. 7. A projection for the link composition ¸1d¸2.

(i.e. three-dimensional) isotopy while G7 is only regular, i.e. two-dimensional (in the plane), isotopy. L The differential geometric properties of ¼ [¸] which can make »(¸, q1@2) ambient-isotopic will be 3 fully exploited in Sections 6 and 7. To make our discussion complete, we would like to notice the apparent difference between the field- theoretic and the existing mathematical formulations of various knots (link) polynomials. This difference can be seen most vividly if we return back to our discussion related to the skein relation, Eq. (4.43). In physics literature (Witten, 1989a,b; Guadagnini, 1993), Eq. (4.45) is obtained using the physical arguments (see, e.g. Eqs. (4.39) and (4.40)). At the same time, mathematicians, see, e.g. Harpe et al. (1986) or Lickorish and Millet (1987), discuss a somewhat different relation, e.g. a #a ~P[¸ ]P[¸ ] . P[¸ X¸ ]"! ` 1 2 1 2 a 0

(4.62)

If ¸ and ¸ are both unknots, then 1 2 a #a ~ P[º2]"! ` 0 a 0

(4.63)

since, by definition P[º ]"1. This needs to be contrasted with Eq. (4.45). If we specialize to 0 HOMFLY polynomial, e.g. see Eqs. (4.58a) and (4.48c), then we obtain (Harpe, 1986; Lickorish and Millet, 1987), u!u~1 G 1X 2" G 1G 2 L L L L z

(4.64)

G 1 2"G 1G 2 , L jL L L

(4.65)

and

where the link composition d is graphically defined in Fig. 7. Obviously, both the Eqs. (4.62) and (4.65) are in formal disagreement with Eq. (4.46) for the unknot. Moreover, Eq. (4.63) implies the normalization condition G 0"1. The factorization U property given by Eq. (4.45) and leading to (Eq. (4.46)) is physically very important, Witten (1989a), but formally is in contradiction with Eq. (4.62). To resolve the existing difficulty, let us assume,

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Fig. 8. Three related links.

following Guadagnini (1993) and Witten (1989a) that, instead of Eq. (4.62), the following result is correct P[¸ X¸ ]"P[¸ ]P[¸ ] . (4.66) 1 2 1 2 Specializing to HOMFLY skein relation given by Eq. (4.48b) we would obtain for links ¸ ,¸ and ` ~ ¸ depicted in Fig. 8 the following result: 0 uP(¸ d¸ )!u~1P(¸ d¸ )"zP[¸ X¸ ] . (4.67) 1 2 1 2 1 2 This would immediately imply z P[¸ ]P[¸ ] P(¸ d¸ )" 1 2 1 2 u!u~1

(4.68)

to be compared with Eq. (4.64). Since, according to Eqs. (4.47) and (4.48b) we should have u!u~1 P[º ]" , 0 z

(4.69)

this produces our final result: P[º ] P[¸ d¸ ]"P[¸ ]P[¸ ] (4.70) 0 1 2 1 2 which is obviously consistent with Eq. (4.68). For the unknot(s), Eq. (4.70) produces an identity. Evidently, it is possible to change the rest of the arguments, e.g. in Lickorish and Millet (1987), in order to justify the HOMFLY-like skein relation, Eq. (4.48b), but with normalization condition, Eq. (4.48a), being replaced by Eq. (4.69). Eqs. (4.66) and (4.70) are crucial for the applications of NACSFT to polymer problems. Since the Jones polynomial is a special case of HOMFLY, the arguments presented above are related to the Jones polynomial as also can be seen from the work by Witten (1989a) where Eq. (4.70) was also obtained (see, e.g. his Eqs. (4.55) and (4.56)) by using completely different set of arguments. 4.4. Unifying link between different approaches In Section 3.3 we had shown that the knowledge of the crossing number c[K] allows to estimate the unknotting number u(K) as well as the number of distinct knots with n crossings (n&c[K]). The question arises: how the crossing number is related to the characteristics of various

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polynomials introduced in Section 4.3? In addition, it is of interest to know how the differentialgeometric description of knots is related to their group-theoretic (algebraic) description. In this subsection we are going to address mainly the first question, and the detailed answer to the second question will be provided in Section 7. Consider now once again the HOMFLY polynomial defined by Eqs. (4.48a), (4.48b) and (4.48c). For a given knot (link) ¸ we will obtain (by using the skein relations) the polynomial in z or Laurent polynomial in u. Let G4 ,P (u, z), then we have either L L n/M P (u, z)" + b (u)zn (4.71) L n n/m or n/E P (u, z)" + a (z)un . (4.72) L n n/e In the first case, by definition, we have b (u)O0Ob (u) while in the second, a (z)O0Oa (z). Let m M e E us define u-span (P )"E!e and z-span (P )"M!m. Using these definitions it can be shown, L L Murasugi and Przytycki (1993), that 1[u!span (P )]4b[¸]!14nL !1 , (4.73) 2 L where nL was defined after Eq. (3.9) and b[¸] is known as the braid index. For the unknot º , b[º ]"1, while in general 0 0 b[¸]4s[D]!ind[D] , (4.74) where s(D) is the number of Seifert circles (see, e.g. Eq. (3.10)) for the planar diagram D for some knot (link) ¸. The index of the diagram D, ind [D], is defined in Murasugi and Przytycki (1993) and its general definition is rather complicated. Whence, we would like to avoid its explicit use (Kholodenko and Rolfsen, 1996), by relating the inequality (Eq. (4.73)) to other inequalities whose physical meanings are more transparent. Following Morton (1986), we also have M4c[¸]!(s[D]!1)

(4.75)

while Eq. (4.74) can be rewritten as (b[¸]!1)#ind[D]4s[D]!1 .

(4.76)

By rewriting Eq. (4.75) as (s[D]!1)#M4c[¸]

(4.77)

and using Eq. (4.76) we arrive at the inequality (b[¸]!1)#ind[D]#M4(s[D]!1)#M4c[¸] .

(4.78)

This inequality allows us to write ind[D]#M4c[¸]!(b[¸]!1) .

(4.79)

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Comparison between this result and the Bennequin’s inequality (Eq. (3.9)) and additionally assuming that ind[D]4M, produces M41(c[¸]!(b[¸]!1)) . 2 Whence, the Bennequin inequality (Eq. (3.9)) is equivalent to the assertion MKu[¸] .

(4.80)

(4.81)

It is possible actually to prove a stronger result (Kauffman, 1987b; Murasugi, 1987; Thistlethwaite, 1987; Turaev, 1987) z!span(P )44c[¸] . (4.82) L Moreover, for the alternating knots the above inequality becomes an equality (a knot (link) is considered to be alternating if traveling along the knot diagram one meets crossings alternatively at overpasses and underpasses). Unfortunately, not all knots have an alternating projection diagrams. According to Thistlethwaite (1987) “Amongst the 12 965 unoriented prime knot types of up to 13 crossings, precisely 6236 are non-alternating”. For the alternating knots, it is possible to obtain in addition a much stronger result (Murasugi, 1987). Indeed, since the Jones polynomial is a special case of HOMFLY, one can define as well a t-span for » defined by Eqs. (4.50) and (4.51). Then, it L may be possible to prove that t!span(» )"c[¸] , (4.83) L i.e. the crossing number of an alternating knot (link) is exactly the span of its Jones polynomial. This fact is quite remarkable since the Jones polynomial is directly related to the Potts model of statistical mechanics as will be shown in Section 5. This means that, at least for the alternating knots (links), the averaged crossing number, defined by Eq. (3.7) can be sytematically calculated using known tools of statistical mechanics! The “thermodynamic” nature of the crossing number c[¸] for the alternating knot (link) can be seen from the following extensive property of c[¸] (Murasugi, 1987) c[¸ d¸ ]"c[¸ ]#c[¸ ] , (4.84) 1 2 1 2 where the operation d was defined in Fig. 7. This property means that, at least for the alternating knots (links), one can apply blob-like analysis in the style of de Gennes (1979). Unfortunately, this property no longer holds for the nonalternating knots (Adams, 1994). Some attempts to analyze this, more general, situation are presented in Soteros et al. (1992) in connection with the problem of the proper choice of a good measure for the knot complexity. We urge the interested reader to consult these references for more details.

5. Probability of knotting: the detailed treatment 5.1. Planar Brownian motion in the presence of a single hole. ¹he role of finite size effects The planar Brownian motion in the presence of a single hole is known in quantum mechanics, e.g. see Kleinert (1995), in connection with the Aharonov—Bohm effect. In the context of polymer

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problems related to statistical mechanics of rubber and glasses, the use of the Aharonov—Bohm effect was originally considered by Edwards (1967a,b). Although this single hole problem can be solved exactly, it does not allow a straightforward generalization to the case of Brownian motion in the presence of many (even two!) holes. Some ways of solving this, more general, problem are discussed in Section 8 and Appendix A.1. Here we restrict ourselves only to the one-hole case. By analogy with Eq. (4.10), we can write down the constraint path integral (Edwards, 1967a,b) as follows:

P

r

(N)/r

A

P

B G P

H

1 N xyR !yxR 1 N dq dq rR 2 D[r(q)]d w! exp ! x2#y2 2p l r r 1 0 0 (0)/ where the winding number w defined by G(r , r ; w)" 1 2

(5.1)

P

1 N xyR !yxR dq (5.2) w" x2#y2 2p 0 is the two-dimensional analogue of the linking number lk(i, j) defined in Eq. (4.11) (the hole can be considered as a point of intersection of another closed polymer (of infinite length!) with the plane). In the absence of a constraint, the path integral of Eq. (5.1) can be easily calculated with the result

G H

R2 G (r , r ; h)"C exp ! , N 1 2 Nl

(5.3)

where R2"r2#r2!2r r cos h 1 2 12 and the constant C is fixed by normalization. Since it is known that

(5.4)

= (5.5) exp(z cos h)" + I (z)e*/ h , n n/~= where I (z) is the modified Bessel function, I (z)"I (z), we can rewrite Eq. (5.3) in the following n n ~n form:

G

H

G

H

= + e*mhI (z) , (5.6) m m/~= where z"2r r /Nl. To make a connection with the Aharonov—Bohm effect it is sufficient, 12 following Wilczek (1990), to rewrite Eq. (5.6) as follows: r2#r2 2 G (r , r ; h)"C exp ! 1 N 1 2 Nl

= + e*mhI (z) . (5.7) @m`aL @ m/~= When the flux aL O0 and aL being noninteger the r.h.s. of Eq. (5.7) describes the “free” propagator in the plane in the presence of the “magnetic flux” tube which is perpendicular to the plane and goes through the hole. The presence of an extra “flux”, for polymer problems is discussed in Sections 8.2 and 8.3. Here we shall assume that aL P0. Following Wilczek (1990), it is convenient to represent r2#r2 2 GaL (r , r ; h)"C exp ! 1 N 1 2 Nl

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Eq. (5.6) in the equivalent form

P P

= = G (r , r ; h)" + dx dj d(h!x#2pm)e*jxQ (r , r ; N) N 1 2 @j@ 1 2 ~= m/~= which can be easily obtained with the help of another known identity (Kleinert, 1995): = = + d(m!x)" + e*nx2p , m/~= m/~= where

G

(5.8)

(5.9)

H

r2#r2 2 I (z) . Q (r , r ; N)"C exp ! 1 @j@ @j@ 1 2 Nl

(5.10)

Using the results just obtained, Eq. (5.6) acquires the following equivalent form: = G (r , r ; h)" + G (r , r ; h, w) , N 1 2 N 1 2 w/~= where

P

G (r , r ; h, w)" N 1 2

=

dj e*j(h`2pw)Q (r , r ; N) . @j@ 1 2 ~= Following Saito and Chen (1973) this result can be conveniently restated in the form

(5.11)

(5.12)

G (r , r ; h, w),"G (r , r ; h) f (h, z) , (5.13) N 1 2 N 1 2 w with f (h,z) being given by w = dj e*j(h`2pw)I (z) . (5.14) f (h, z)"e~z #04 h @j@ w ~= Thus defined function f (h, z) has remarkable properties. In particular, it satisfies the skein relation w f (h#2p, z)"f (h, z) (5.15) w w~1 analogous to the skein relations discussed in Section 4. Successive use of Eq. (5.15) permits us to write as well

P

f (h, z)"f (h#2pw, z) . (5.16) w 0 Whence, in complete agreement with the results of knot theory, the problem of computation of f with arbitrary winding number w can be always reduced to the computation of f . w 0 One of the important quantities of interest is the a priori probability p that a ring-shaped w polymer is wrapped w times around a hole (or another polymer). This probability can be defined with the help of Eqs. (5.3), (5.11), (5.12) and (5.13). For this purpose we define Z as

P P P P

Z" dr dr G (r , r ; h"2p) , 1 2 N 1 2

(5.17)

Z(w)" dr dr G (r , r ; h"2pw) , 1 2 N 1 2

(5.18)

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so that Z(w) p " w Z

(5.19)

and, by construction, = + p "1 . (5.20) w w/~= Fortunately, it is possible to calculate p explicitly. To do so we have to use the known identity w 1 = dx e~axI (x)" (a#(Ja2!1))~l . (5.21) l Ja2!1 0 By combining Eqs. (5.6), (5.17) and (5.21) we now obtain the following result:

P

P

Z" d2r G (r "r "r; h"2p) N 1 2

P

Nl = = dx e~axI (x) C + " lim @m@ ` 2 0 m/~= a?1 Nl = 1 " C + (1#J2e)~@m@, e"a!1 . 2 J2e m/~= Similarly, we also obtain

P

(5.22)

Z(w)" d2r G (r "r "r; h"2pw) N 1 2

P

P

Nl = = dj e*j2pw dx e~axI (x) " lim C @j@ 2 ~= 0 a?1` Nl C = " dj e~k@j@`*j2pw , (5.23) 2 J2e ~= where in the last line we have introduced k"ln (1#J2e). Using Eqs. (5.23), (5.22) and (5.19) we obtain

P

:= dj e*j2pw~k@j@ . (5.24) p " lim ~= w e~k@m@ += /~= e?0` m Use of Eq. (5.8) indicates that p obeys the normalization condition given by Eq. (5.20) as required. w It is very important to notice that p is independent of the length of the chain N as well as of l. This w fact underscores the topological nature of p . In reality, however, p may depend upon the physical w w characteristics of the polymer involved in our problem. Indeed, let the diameter of our hole be of order l. Then, if the length of the polymer chain is N, the winding number w cannot be larger than N/l. Consider now the denominator of Eq. (5.24) with such restriction. We obtain e~k!e~Nlk +@ e~k@m@"1#2 . 1!e~k m

(5.25)

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The limiting procedure eP0`, lP0` with the constraint k/lPc so that Nc41 brings the above expression to the following form: 2 2 +@ e~k@m@K eNc(e~Nc!e~2Nc)K eNc Nc . (5.26) k k m Finally, let us consider p which is the probability that our ring is not entangled with the hole. 0 Evidently p is a two-dimensional analogue of f introduced in Section 3.1. To calculate p we only 0 N 0 have to put w"0 in the numerator of Eq. (5.24) and to combine the result with Eq. (5.26). Whence, we obtain p "e~Nc/Nc . 0 Let !c"ln kJ then we can rewrite p as 0 1 kJ N,cJ (N)kJ N . p " 0 N ln(1/kJ )

(5.27)

(5.28)

This result, indeed, resembles Eq. (3.3) (since a"0 in Eq. (3.3)). The main conclusion of this derivation lies in acknowledging that the functional form of p reflects the role of the finite size 0 corrections in the topological problem . We had introduced already N in Section 3.1 which is also T a nonuniversal and lattice-dependent quantity to be calculated in Section 7. Whence, in dealing with real polymers the topological and nontopological properties are essentially interrelated. The result for p obtained above has very suggestive thermodynamic appearance. We would like to 0 demonstrate that, indeed, this expression (as well as Eq. (3.3)) has well-defined thermodynamic (statistical mechanics) meaning. To see this, we need to take another look at the whole problem discussed in Section 5.1. 5.2. Quantum groups and planar Brownian motion To develop an alternative approach to the whole problem the following identity:

P

=

G

HA B

1 a2#b2 ab dx xe~ox2J (ax) J (bx)" exp ! I , l l l 2o 2o 4o

(5.29) 0 where the modified Bessel function I (z) is related to the usual Bessel function J (z) via l l I (z)"e~*lp@2J (z), is the most helpful. l l By comparing Eqs. (5.5) and (5.29) we immediately obtain:

P

= = dx xe~Nl4 x2J (r x)J (r x) , (5.30) G (r , r ; h)"cL + e*mh m 1 m 2 N 1 2 0 m/~= where, as before, the constant cL is fixed by normalization. Let us recall (Vilenkin, 1968) that the functions e*m aJ (Rr) are the eigenfunctions of the two-dimensional Laplace operator +2 (written in m polar coordinates) corresponding to the eigenvalue !R2. If Nl/4 in Eq. (5.30) is interpreted as Euclidean time, then Eq. (5.30) can be interpreted in terms of the usual Green’s functions so that Z"2p

P

=

0

dr r G (r "r "r; h"2p) N 1 2

(5.31)

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is the partition function in the usual statistical mechanics sense. Whence, in order to obtain Z it is essential to know the eigenfunctions and the eigenvalues of the corresponding “Hamiltonian” operator which, in turn, can be obtained in a group-theoretic fashion (Vilenkin, 1968). Such a derivation is in complete accord with the group-theoretic formulation of knot theory (Gilbert and Porter, 1994; Chari and Pressley, 1995). Let us begin with the observation that each point p(x, y) in two-dimensional plane can be carried by the motion in the plane into the point p(x1, y1), where x1"x cos a!y sin a#a ,

y1"x sin a#y cos a#b .

(5.32)

The parameters, a,b,a which uniquely determine the motion are given by !R(a(R ,

!R(b(R ,

04a(2p .

To proceed, we introduce the vectors n"(a, b), x"(x, y) and the matrix A so that Eq. (5.32) can be rewritten as x1"A x#n .

(5.33)

The transformation in the plane is fully determined by the pair (A, n). Consider now two successive transformations in the plane. Then, their composition ° is given by (A, n) ° (B, g)"(A ) B, n#A ) g)

(5.34)

which defines a semidirect product of the groups of additive translations E and the group of 2 rotations SO(2) (Vilenkin, 1968). Instead of working with the semidirect product of two groups it is more convenient to enlarge the vector space x to make it three dimensional. For such enlarged space it is possible to introduce the matrix T(g) given by

A

B

cos a !sin a a

T(g)" sin a 0

cos a

b .

0

1

(5.35)

For such a defined matrix T(g) a usual group composition law holds T(g ) ) T(g )"T(g ) g ) , 1 2 1 2 where ) denotes the usual matrix multiplication. The matrix T(g) provides a representation of the group M(2). If * denotes a semidirect product, then M(2)"E *SO(2). The Lie algebra of the above 2 group is formed by three elements a , a and a , which obey the following commutation relations 1 2 3 (Vilenkin, 1968): [a , a ]"0 , [a , a ]"a , [a , a ]"a . (5.36) 1 2 2 3 1 3 1 2 where [a,b]"ab!ba, as usual. This Lie algebra is very similar to that used for the angular momentum in quantum mechanics and, whence, the subsequent steps of analysis are the same. One introduces the raising and lowering operators aB"a $ia so that eigenfunctions can be de1 2 scribed in terms of two quantum numbers n and R (see, e.g. Eq. (5.30)), where the quantum number R is defined according to the equation, a`a~f (R)"!R2f (R) n n

(5.37)

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while the quantum number n is defined with the help of a f (R)"inf (R) , 3 n n

(5.38)

etc. ¹he task now is to demonstrate that the commutation relations (Eq. (5.36)) can be obtained from the NACSF¹. To this purpose, using results of Sections 4.2 and 4.3 and choosing instead of the group Sº(NK ) another group, e.g. GSº(2), Armand-Ugon et al. (1996), we can obtain, instead of HOMFLY, the Dubrovnik polynomial which is characterized by the following set of skein relations (Kauffman, 1990): (5.39) (5.40) D[º ]"(a!a~1)/z#1 . (5.41) 0 Actually, the last result for the unknot is taken from the Armand-Ugon et al. (1996). It is motivated by the same arguments as were presented in Section 4.3, i.e. by the difference in normalization: in mathematics literature (Kauffman, 1990, 1996), (5.42) D[º ]"1 , 0 and in physics literature, see, e.g. Eq. (4.46). Dubrovnik polynomial is an invariant of regular isotopy. As before, see, e.g. Eq. (4.55), one can make an invariant of ambient isotopy using D via K (Kauffman, 1987) G "a~W3*K+D . K K

(5.43)

Let ¼ (c) be a Wilson loop operator, e.g. see Eq. (4.27). Following Martin (1989) and Turaev o (1989) one can define a Poisson bracket via M¼ (c),¼ (c@)N "+ Sc{{ ¼ (c@@) , (5.44) o o P.B. cc{ o c{{ where, in case of Dubrovnik polynomial, the structure constants Sc{{ are defined by the symbolic cc{ rule (5.45) Evidently, Eqs. (5.39) and (5.44) are equivalent and, whence, nonsurprisingly this fact is used in the nonperturbative quantum gravity (Martin, 1989; Armand-Ugon et al., 1996). For us it is important only that one can formally quantize such “mechanics” via the usual rule 1 M,N N [,] , P.B. i+

(5.46)

i.e. it is possible to define a Lie algebra where, instead of the Poisson brackets, the usual commutators are being used. With these remarks, it is easy now to make connection between the

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knot polynomials and the commutator algebra given by Eq. (5.36). Indeed, to this purpose, let us consider yet another invariant of regular isotropy, the Kauffman polynomial, which also can be obtained with the help of NACSFT (Guadagnini, 1993). It is defined axiomatically through the following set of skein relations: (5.47) (5.48) SLXKT"dI SKT ,

(5.49)

SLT"1 .

(5.50)

The last relation is, of course, unnecessary if the NACSFT is used. The constants A, B, dI are arbitrary in the above axiomatic formulation, but are fixed in NACSFT and by the requirements of invariance with respect to the Reidemeister moves, Fig. 6, as will be further explained below in this section. By combining Eqs. (5.47) and (5.48) we obtain (5.51) This skein relation looks exactly the same as for the Dubrovnik polynomial, Eq. (5.39) so that, indeed these two polynomials are interrelated (Kauffman, 1990). At the same time, by combining Eqs. (5.47) and (5.48) in a different way we obtain as well (5.52) (5.53) where ZK "(A/B!B/A). Using results of Eqs. (5.43) and (5.45) we conclude that, upon proper rescaling and quantization, Eqs. (5.52) and (5.53) can be identified with the second and the third of Eq. (5.36). Notice that, on one hand, [a , a ]"0, but, on the other, using the Jacobi identity, we 1 1 can obtain [a , [a , a ]]"![a , [a , a ]]![a , [a , a ]] . 1 2 3 2 3 1 3 1 2 From here and in view of Eq. (5.36) we conclude that there are two options: [a , a ]"0 1 2

(5.54)

or [a , a ]"a . (5.55) 1 2 3 The first option leads to the complete set of commutators given by Eq. (5.36) while the second leads to the Lie algebra S¸(2,C), which can be obtained from the Virasoro algebra defined by the set of relations c [¸ ,¸ ]"(n!m)¸ # (n3!n)d . n n n`m 12 n`m,0

(5.56)

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The S¸(2,C) algebra is obtained from the Virasoro if we put the central charge c"0. In this case, instead of Eq. (5.56) we obtain [¸ , ¸ ]"¸ , (5.57) 1 0 1 [¸ , ¸ ]"!¸ , (5.58) ~1 0 ~1 [¸ , ¸ ]"2¸ . (5.59) 1 ~1 0 Let now a "¸ !¸ , a "!(¸ #¸ ) and a "¸ . Then we obtain 1 ~1 1 2 ~1 1 3 0 [a , a ]"a , [a , a ]"a , [a , a ]"4a . (5.60) 2 3 1 3 1 2 1 2 3 This Lie algebra can now be compared with that given by Eq. (5.36). It is actually possible to obtain Eq. (5.36) from Eq. (5.60). To do so, several steps are needed (Vilenkin, 1968), which are similar to that used in the theory of quantum groups (Chari and Pressley, 1995). Notice that if we rescale the operators a and a in Eq. (5.60), e.g. a PAa , a PAa , then the first two commutators in 1 2 1 1 2 2 Eq. (5.60) will not change while the third is going to change into [a ,a ]"A~24a . (5.61) 1 2 3 Let now APR so that A~2"e2P0`. Let us introduce a "a(0)#ea(1) and a "a(0)#ea(0). 1 1 1 2 2 2 Then, in the limit eP0` the operators a(0),a(0) and a will satisfy the Lie algebra of Eq. (5.36) while 1 2 3 the operators a(1),a(1) and a will obey Eq. (5.60). In the limit eP0` their contribution can be 1 2 3 ignored. From the results presented above, several conclusions can be drawn: (a) the probability p given by Eq. (5.24) is of purely topological origin since it is independent of w the length of the chain N or the cut-off length l; (b) the probability p can be determined if the eigenfunctions and the eigenvalues of the w “topological” Hamiltonian could be obtained in order to construct the statistical mechanics partition function; (c) the Hamitonian is expected to be some Casimir operator of the corresponding Lie algebra which is directly obtainable by the quantization procedure from the skein relations for some knot polynomials; (d) in order to utilize the information contained in the partition function, the finite size effects should be properly taken into account. All the above statements are general enough to be used in more complicated three-dimensional calculations to be discussed below. 5.3. Jones polynomial, ¹emperley—¸ieb algebra and statistical mechanics of knots (links) In Section 4.4 we have discussed the remarkable properties of the Jones polynomial which are reflected in Eq. (4.83). We had emphasized that Eq. (4.83) could be very useful in statistical mechanical calculations. Now we would like to develop this statement somewhat deeper. Before doing so, we would like to mention that the property of » based on Eq. (4.83) is not the only one L which makes » so special. Indeed, Jones (1985) (see also Welsh, 1993) had shown that, in addition, L » has the following basic properties: L

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(1) For any link ¸ » (1)"(!2)n(L)~1 , L where n(¸) is the number of components of ¸. (2) If ¸ is a knot, then » (e2p*@3)"1 . L (3) If ¸ is a knot, then

K

(5.62)

(5.63)

K

d» (t) L dt

"0 . (5.64) t/1 (4) For the connected sum ¸ d¸ , see e.g. Fig. 7, the polynomial » obeys the rule 1 2 L » 1 2(t)"» 1(t)» 2(t) (5.65) L jL L L to be compared with Eqs. (4.65) and (4.84). The above properties of the Jones polynomial should be taken into account in any statistical mechanics application, see, e.g. Wu (1992). To develop statistical mechanics, we shall follow the strategy developed in Section 5.2. Following this strategy, we have to find a set of commutation relations and to look for their representations in order to find the eigenvalues and the eigenfunctions of the corresponding Hamiltonian. Alternatively, we may start with the non-Abelian variant of an expression, like Eq. (4.22), and to use skein relations (and the properties defined by Eqs. (4.66) and (4.70) of knot polynomials) in order to disentangle the Chern—Simons and the polymer averages. After that, we will arrive at some kind of knot polynomial (times the disentangled polymer partition function). Since we will be interested in the ratios, e.g. like that given by Eq. (5.19), the disentangled partition function drops out and we need to analyze only the knot polynomial. Because the Dubrovnik polynomial is related to the Kauffman, we need to study now the Kauffman polynomial. Let us begin with the observation that the Kauffman polynomial for a given knot K, introduced in the previous section, can be written as (Kauffman, 1990, 1993) SKT"+ AnA(S)BnB(S)dI @S@~1,Z , (5.66) M N S where the summation MSN takes place over 2c(K) states of the planar 4-valent graph with c(K) crossings and n (S), n (S) correspond to the number of splices of the A- and B-type defined by A B Clearly, polynomial SKT thus defined has an appearance of the statistical mechanical partition function Z (Baxter, 1982; Wu, 1992), for some two-dimensional spin model defined on such 4-valent graph. The constants A and B in Eq. (5.66) can be considered as some fugacities in the grand canonical formalism. In contrast the partition function Z defined by Eq. (5.31) is written in the canonical formalism. For example, the average winding number SwT can be obtained with help of

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Eq. (5.19) as = SwT" + wp w w/~= while the average number of splices of the A and B-types could be assigned by

(5.67)

Sn T"A(­/­A) ln Z , (5.68a) A Sn T"B(­/­B)ln Z . (5.68b) B So that if Sn T and Sn T are given, then the fugacities A and B are implicitly determined via A B Eqs. (5.68a) and (5.68b). Unfortunately, the given knot cannot be uniquely characterized by Sn T A and Sn T since, as it was discussed in Section 3.3, the combination B Sn T#Sn T"c[K] (5.69) A B is not a topological invariant (unlike u[K]!). Moreover, as it is written SKT is not even an invariant of a regular isotopy. Kauffman (1988) had demonstrated that SKT becomes an invariant of a regular isotropy only if AB"1 and dI "!(A2#B2). To make an ambient isotopic invariant f out of SKT it is sufficient to write f (A)"a~W3*K+SKT , a"!A3 , K to be compared with Eq. (4.55). This means, that in case of standard framing (see, e.g. Eq. (4.20)) f (A)"SKT. Since, according to Kauffman, the Jones polynomial » (t) is related to the Kauffman K L polynomial via the simple relation » (t)"f (t~1@4) , L K one concludes that, at least for the case of standard framing, one can write

(5.70)

» (t)"SKT L provided that AB"1, dI "!(A2#B2) and A"t~1@4. So, formally, by using Eqs. (4.83), (5.68a), (5.68b) and (5.69) we obtain

A

c[K]"t!span »(t)" A

B

K

­ ­ #B ln Z B/A~1 2 ­A ­B dI /~(A `B2) 1 A/t~4

(5.71)

In Section 3 we have learned that for a given c[K], the number of possible knots K(n) is bound by the inequality (Eq. (3.14)). Since in Eq. (3.7) we had shown (see also Section 7) that Sc[K]TJNac we conclude that by prescribing particular value for c[K], the statistical mechanics of knots becomes quite well defined through Eqs. (5.66), (5.68a), (5.68b), (5.69), (5.70) and (5.71). Eqs. (5.68a) and (5.68b) are three-dimensional analogues of Eq. (5.67). In Section 5.1 we have discussed the role of the finite size effects. They will show up in Eq. (5.67) if we realize that for the polymer chain of length N and Kuhn’s length l the winding number cannot take infinite values and

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is always restricted by N/l. The quantity c[K] plays the same role in three dimensions as N/l in two. This is so because Sc[K]TJNac, i.e. by assigning the effective polymer length N/N , where N was T T defined in Section 3.1, we automatically introduce a physical cut-off into the problem. The question arises: what is the three-dimensional analogue of p (Eq. (5.28)) and how is it 0 calculated. We claim that f introduced in Eq. (3.3) is the three-dimensional analog of p . To N 0 calculate f , several steps are still required. First, by analogy with Eq. (5.19) we can write N Z(º ) 0 . f " (5.72) N Z But Z(º )"» 0"1 according to Eq. (4.51). Whence, we can write as well (for NPR) 0 U f~1"Z . (5.73) N Next, following Kauffman (1988), it is convenient to define the bracket polynomial [K] which is related to SKT in a straightforward way SKT"dI ~1[K] .

(5.74)

For this polynomial the relations of Eqs. (5.47), (5.48) and (5.49) hold (with S2T being replaced by [2]) but Eq. (5.50) is being replaced by [L]"dI .

(5.75)

This makes the polynomial [K] more physical, e.g. see the discussion leading to Eq. (4.69), since it can be obtained from the NACSFT. Alternatively, [K] can also be obtained from the graph expansion for the Potts model (Kauffman, 1987a,b). In this case it is sufficient only to assign a particular value to the coefficients A, B and dI for the defining relations for the bracket polynomial [K]. Specifically, instead of Eqs. (5.47), (5.48), (5.49) and (5.50), following Kauffman (1987a), we write (5.76) (5.77) [LK]"!q1@2[K] ,

[L]"!q1@2,

(5.78)

where in the last two equations we have changed the sign “#” to “!” (as compared to Kauffman, 1987a) for reasons which will become clear shortly below. For an alternative explanation of the switch of signs, please consult Kauffman and Saleur (1993). The physical meaning of constants q and v will also be explained shortly below. At this stage it is convenient to introduce the concept of braids and braid generators p . Braids i are formed when n points on a horizontal line are connected by n “strings” to n points on another horizontal line being directly below the first n points. A general n-braid is constructed from the trivial braid by successive applications of the braid generators p , i"1,2, n. By regarding the i trivial n-braid as an identity operator (generator) a set of generators p define the braid group B . i n Given a particular n-braid, some link (or knot) can be formed by tying the opposite ends (i.e. connecting the inputs of the strings with the outputs in some prescribed way: normally the

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Fig. 9. Diagrammatic representation of the braid group generators pi and associated with them the Temperly—Lieb generators hi.

beginning of a given string is tied with its end, thus forming a closed braid). According to Wadati et al. (1989), any link (knot) can be represented by a closed braid. This representation is highly nonunique, however. The equivalent braids expressing the same link are mutually transformed by successive applications of Markov moves of types I and II, which for braids play the same role as the Reidemeister moves, see, e.g. Fig. 6, for knots. Following Kauffman (1990), it is more useful to introduce a set of generators h related to p as it is depicted in Fig. 9. Let b represent some braid i i and [b] represent the value of the bracket polynomial on a closed braid bM , i.e. [b]"[bM ]. Using Eqs. (5.76a) and (5.77a), we obtain

(5.79) With the help of Eq. (5.79) we can write p "q~1@2vh #1 , p~1"h #q~1@2v1 . i i i i i i The Temperly—Lieb (TL) algebra can now be written as follows:

(5.80)

h2"q1@2h , h h h "h , h h "h h , Di!jD52 . (5.81) i i i i`1 i i i j j i By simple rescaling of TL generators: h "q1@2e we arrive at the form of TL algebra discussed by i i Jones (1985): e2"e , e e e "(1/q)e , e e e e , Di!jD52 . (5.82) i i i i`1 i i i j/ j i The TL algebra, Eq. (5.82), replaces the Lie algebra of the group M(2), Eq. (5.36), and is directly related to the Potts model, (Baxter, 1982; Kauffman and Saleur, 1993), on one hand, and to the Jones polynomial » (t), (Jones, 1985) on the other. In particular, following the original work of K (Jones, 1985), q~1 in the second relation of Eq. (5.82) corresponds to his t/(t#1)2 where variable t is the same as in the skein relation given by Eq. (4.50). At the same time, q represents the number

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of states of the Potts model (Baxter, 1982; Kauffman and Saleur, 1993) defined by the following partition function:

G

H

Z (q, v)"+ exp K + d(s , s ) "+ < (1#vd(s , s )) , i j i j G M N M N s s Wi,jX Wi,jX

(5.83)

where v"expMKN!1, Si, jT denotes the nearest neighbors and MsN denotes the summation over q states at each site (vertex) of some planar graph G with the total number of sites c[K]. It is possible to introduce an alternating link (knot) associated with G (Kauffman, 1988; Baxter, 1982). In terms of such alternating link (or rather its planar projection), the partition function Z (q, v) can G be written as Z (q, v)"qc*K+@2[K(G)] G

(5.84)

with [K(G)] being defined through Eqs. (5.74)—(5.78). Finally, let us recall that, according to the results of Section 4, the NACSFT provides us with the value of Jt"expMip/(k#NK )N, see, e.g. Eqs. (4.49) and (4.50). This implies that

A

B

A

B

(1#t)2 1 2 p q" " Jt# . "4 cos2 t k#NK Jt

(5.85)

We had shown before, see, e.g. Eqs. (4.47) and (4.49), that NK "2. Because k can take only integer values (Witten, 1989a), we obtain the following Beraha numbers (Saleur, 1990) for q: k"0, q"0 (random resistors network, Wu (1982)); k"1, q"1 (percolation, Wu (1982)); k"2, q"2 (Ising model, Wu (1982)); kPR, q"4 (four colors problem, Wu (1982)). In view of Eqs. (5.70), (5.74) and (5.85), we observe that since » (t) is the invariant of ambient K isotopy, we must require [K(G)] to be at least an invariant of a regular isotopy (i.e. an invariant under the 2nd and 3rd type of Reidemeister moves, e.g. see Fig. 6). Using Eqs. (5.75)—(5.78) we have

(5.86) The equation obtained leads us to the conclusion that in order for K[(G)] to become an invariant of a regular isotopy, we must require q~1v2#1!v"0 .

(5.87)

Using methods analogous to that described in Kauffman (1988), it can be also easily shown that the invariance of [K(G)] under the 3rd Reidemeister move will also hold if the requirement of Eq. (5.86)

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is satisfied. Using Eq. (5.86) we obtain v2!qv#q"0 or (5.88) v "(q/2)$J(q2/4)!q . 1,2 From here, it follows that q54 in order for v to be real. But, according to Eq. (5.85), we have q44. Hence, the only choice we have is q"4. Because of the connection between [K(G)] and the Potts model given by Eq. (5.84), there is yet another reason for q to be equal to 4. For the Potts model, the criticality condition is given by the following equation (Baxter, 1982): q~1v2"1 .

(5.89)

Combining Eq. (5.87) with Eq. (5.89) produces v"2 and, whence, according to Eq. (5.89), we obtain again q"4. Notice that the above derivation is possible only with the choice of signs indicated in Eqs. (5.76)—(5.78). From here we conclude that in order for [K(G)] to be a regular isotopy invariant, we have to require q"4 and for the corresponding Potts model to be critical. Moreover, at criticality, in view of Eq. (5.70), the above polynomial is also an invariant of ambient isotopy and, whence, is directly connected with the Jones polynomial. According to Wadati et al. (1989), the restricted 8-vertex SOS model at criticality can also be solved by TL algebra, Eq. (5.82), exhibiting the exponents * which are directly obtainable from the Virasoro algebra, Eq. (5.56), with c"1!6/k(k#1) and *"(p2!1)/2k(k#1) with k"2, 3, 42; 14p4k!1. Whence, in principle, TL and Virasoro algebras are interrelated and, in part, this was demonstrated in Section 5.2. This circumstance allows us to transfer results and the experience of two-dimensional problems discussed in Section 5.2 to three dimensions. 5.4. Probability of knotting and the role of finite size effects We would like to remind the reader at this point that the probability f for the closed circular N polymer of the effective length N/N to remain unknotted was introduced in Section 3 while T Eq. (5.73) provides the explicit link with statistical mechanics thus allowing us to calculate f . N Using Eqs. (5.73)—(5.78) we obtain as well (!1) f~1" [K(G)] . N q1@2

(5.90)

The factor (!1) is chosen in accord with Eq. (5.78). This means that after unknotting the knot K with the help of skein relations, Eqs. (5.76) and (5.77), we shall obtain a polynomial times the unknot so that the minus sign disappears. More generally, we expect the product of unknots times the polynomial. Each of these unknots will carry a factor of (!1)q1@2. For the graph of c[K] vertices we will have a factor of (!1)c*K+qq*K+. Looking at Eq. (5.84) and, in view of Eq. (5.83), we know (Adams, 1994; Baxter, 1982), that Z (q, v)"qc*K+#2 . G This requires us to write, instead of Eq. (5.90),

K

Z (q, v) c f~1" . N (!1)c*K+~1qc*K+`1 v2 2 q/ /4

(5.91)

(5.92)

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The exact solution for the Potts model at criticality ( for q"4) is known (Baxter, 1982; Wu, 1982). Depending upon the geometry of the lattice we obtain

G

CC D D CC DD

C(1) 4 ln 1 4 (square lattice) 8 C(3) 1 4 f" ln Z " (5.93) G c[K] C(7) C(1) 3 3 ln 2 6 (triangular lattice) C(2) C(5) 3 6 In order to use this result in Eq. (5.92) we have to remember that Eq. (5.93) was actually obtained in the thermodynamic limit, i.e. for c[K]PR. In reality, in order to make a comparison with numerical simulations discussed in Section 3, e.g. see Eq. (3.3), Eq. (5.93) should be augumented with the finite size corrections. Let our lattice contain c[K]"¸]M sites, then the partition function Z can be written G (Karowski, 1988; William, 1991), as

G

¸ p Z Kexp !¸Mf# c c M6

H

(5.94)

where f is the corresponding free energy defined in Eq. (5.93) while c is the central charge, e.g. see Eq. (5.56), which, for the Potts model with q"4, is equal to one (Dotsenko and Fateev, 1984). If MK¸, then by combining Eqs. (3.3), (5.92) and (5.94) we obtain,

G H G C G H

DH

p 1 f "Jq exp ! c exp c[K] f# ln q N 6 2 p "Jq exp ! c kJ c*K+ . 6

(5.95)

The last equation defines kJ . Since kJ can be eliminated according to Eqs. (3.4) and (3.5) we are left with expression for cJ defined in Eq. (3.3):

G H

p cJ "2 exp ! . 6

(5.96)

The numerical value of cJ "1.1847697 differs from the numerically obtained cJ "1.2325 by Windwer (1990) with error of 4%. The result of Eq. (5.96) is in qualitative accord with that obtained by Sumners and Whittington (1988) and by Pippenger, Eq. (3.2). It is documented in Kleinert (1995) and was earlier announced in Kholodenko (1991, 1994).

6. Single chain problems which involve geometrical and topological constraints 6.1. Semi-flexible polymer chain in the nematic environment The conformational statistics of fully flexible chains is well described by the Klein—Gordon like Gaussian propagators, Zinn-Justin (1993). The situation changes dramatically when the effects of

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rigidity should be taken into account. As it was argued in Kholodenko (1995), there is an infinity of propagators which are capable, in principle, of describing semi-flexible chains. Here we shall not go into the full details of this very broad subject. The interested reader may consult Kholodenko (1995) and Ambjorn et al. (1977) and references therein. Here we shall illustrate only some aspects of conformational statistics of semi-flexible chains which involve the geometrical and (or) topological constraints. In particular, we would like to dicuss how the choice of framing affects the conformational statistics of polymers. To illustrate its effect on chain flexibility, let us recall that the traditionally used Kratky—Porod (KP) model of semiflexible chains (Kleinert, 1995) involves the KP propagator given by

P

G P A BH

c N (N)/uf du 2 D[u(q)] d(u2!1)exp ! (6.1) dq 2 dq u (0)/ui 0 with c being some phenomenological rigidity parameter. The geometrical constraint u2"1 in the path integral measure in Eq. (6.1) converts the Brownian motion in R3 into the Brownian motion on the unit sphere S2. Analogously to the flat R3 case, the diffusion on S2 can be described in terms of the corresponding Schro¨dinger-like diffusion equation which in quantum mechanics is known as an equation for the rigid rotator. Since we are going to use the results for the rigid rotator in Section 7, we shall describe here the properties of the rigid rotator model which we shall use later. To do so, we would like to simplify our task by considering, instead of the diffusion on the sphere, the diffusion on the circle. In this case the Hamiltonian HK is given by (Kholodenko and Vilgis, ( 1995) u

G(u , u ; N)" f i

c d2 HK "! ( 2 d/2

(6.2)

with dimensionless eigenvalues E given by E "1l2, l"0,$1,$2,2 and eigenfunctions l l 2 1 exp(il/). This form of the eigenfunctions comes from the periodic boundary condition t" l J2p requirement: t (/#2p)"t (/) . (6.3) l l As we shall see in Section 6.2, there are physical situations when it is possible to generalize the condition of Eq. (6.3) by requiring instead t (/#2p)"exp(!ih)t (/) , l l where 04h(2p. This leads to the wave function

(6.4)

t (/)"(1/2p) exp Mi(l!h/2p)/N l which produces the eigenvalue

(6.5)

E "1[l!h/2p]2 . (6.6) l 2 Instead of dealing with the untraditional boundary condition, e.g. Eq. (6.4), it is possible to replace it by the conventional one, Eq. (6.2), provided that the Hamiltonian of Eq. (6.1) is being replaced by HK h "(c/2)[i(d/d/)#h/2p]2 . (

(6.7)

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At the classical level this Hamiltonian comes from the Lagrangian ¸ given by h ¸ "(c/2)[(d//dt)2#(h/2p)d//dt] . (6.8) h Since the last term in Eq. (6.8) is a total time derivative, it seems natural that it may be discarded. This is not exactly the case at the quantum level in view of Eq. (6.6). The corresponding Euclidean path integral which describes the diffusion on the unit circle is given by

P

G(u ,u ;N)" f i

(N)/uf

u

(0)/ui

D[u(q)] d(u2!1)

u

G P A B

P

C DH

c N du 2 h N d u (6.9) ]exp ! dq #i dq tan~1 y . 2 dq 2p dq u x 0 0 The last term in the exponent is just the winding number introduced earlier in Eq. (5.2) as can be easily shown, e.g. see Kholodenko and Vilgis (1994). Since this path integral is related to the completely solvable quantum mechanical problem, see, e.g. Eqs. (6.5) and (6.6), we can analyze its properties in some detail. To this purpose let us introduce the generating function F( p) given by

T A P

BU

T A P P

BU

N u(q) dq , (6.10) 0 where the average S2T is performed with help of Eq. (6.9) provided that u "u "u. Using i f Eq. (6.10) we obtain the following expansion for F( p): F( p)" exp i p )

F( p)" exp i p )

N

N u(q) dq 0

P P N

q

dq@ Su (q)u (q@)T . (6.11) a b 0 0 0 This expansion was analyzed by Polyakov (1990), and additional details are provided in Kholodenko and Vilgis (1995). By symmetry, the first nontrivial term in Eq. (6.11) vanishes while for the second term we have "1#i p )

dqSu(q)T!p p a b

dq

Su (q)u (q@)T" + S0Du DlTSDu D0Te~(El~E0)@q~q{@c a b a b lE0 1 c h "d + exp ! l l! Dq!q@D . (6.12) ab l2 2 p lE0 For h"$p and l"$1 the term in the exponent vanishes and, for Dq!q@DPR such term becomes the leading term in the series expansion for the correlator. Using this fact Eq. (6.11) produces in this limit:

G A B

T A P

F( p)" exp i p )

BU

N u(s) dq 0

H

p2 K1! N2#O(p4) . 2

(6.13)

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The same result can be obtained following Polyakov’s ingenious trick (Polyakov, 1988) which later on was proven rigorously by Alekseev and Shatashvili (1988). The trick lies in the fact that, instead of u-averaging, defined in Eq. (6.10), one can perform spin averaging by using the properties of Pauli matrices p . Since i Tr p "0 but Tr p2"1 i i we can formally write

T A P

BU

N u(s) dq ,Sexp (i p ) rN)T , (6.14) p 0 where S2T ,Tr(2). p When the r.h.s. of Eq. (6.14) is expanded and the traces over p ’s are taken, the result of Eq. (6.13) i is recovered. The Laplace transform of the r.h.s. of Eq. (6.14) produces exp i p )

P

=

T

1 dN e~sNSexp (i p ) rN)T " p i p ) r!s

U

. (6.15) 0 p Obviously, Eq. (6.15) describes the Dirac propagator in d"2. The above analogy with the Dirac propagator exists, however, only for a special value of h: h"p. For hOp the analogy is lost but the relevant physical results, e.g. that given by Eq. (6.12), are not much affected. Indeed, if we are interested in calculating SR2T, then using Eq. (6.12) and keeping only the DlD"1 term, which is permissible in the limit Dq!q@DPR, we obtain Su (q)u (0)TKd 2 exp M!c/2Ncosh ((ch/2)DqD) . (6.16) a b ab The qualitative and quantitative analysis of this result performed in Kholodenko and Vilgis (1995) indicates that the presence of the h term in Eq. (6.16) makes the chain effectively more stiff. The physical origin of an additional stiffness could be different in general. In this section we shall consider the effects of the nematic environment on a single chain while in Section 6.2 the boundary effects will be considered. In the case of polyelectrolytes, i.e. polymer chains which have charges on their backbones, the stiffness could have an electrostatic origin (Kholodenko, 1995). This is a subject of an ongoing research which we shall not touch in this review. Warner et al. (1985) (WGB) and independently Rusakov and Shiliomis (1985) have considered the conformational and thermodynamic properties of a single semi-flexible chain inserted into the already existing nematic environment. At the path integral level, the action for such a chain is given by

P CA B

D

1 N du 2 S " (6.17) dq c #gJ [3(u )2!1] , WGB 2 z dq 0 where u is the z component of the unit vector u. The corresponding Schro¨dinger-like equation for z the propagator of Eq. (6.1) (with action given by Eq. (6.17)) can be written (in dimensionless units) as [d2/dx2!g cos2 x]t"Et (d"2)

(6.18)

304

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or [d2/dx2#cot x(d/dx)!g cos2 x]t"Et (d"3) ,

(6.19)

where 04x(p, x"/. In both cases the cosine term can be formally expanded and, for d"2, we find a familiar double-well potential characteristic for the hairpin-like problems as discussed by de Gennes (1982). A hairpin is an immediate return (or sharp bend) of a chain in the nematic ordering field. In d"3 the presence of an extra term (cot x(d/dx)) formally destroys the nice hairpin picture suggested by de Gennes (1982). Since hairpins were recently observed experimentally (Li et al., 1994), it is essential that the existing models of the semi-flexible chains account for their existence. In Kholodenko and Vilgis (1995) this problem was solved in the following way. By substituting t"z(x)Jsin x into Eq. (6.19) it is converted into (6.20) [d2/dx2#cot x(d/dx)!g cos2 x!1!1/ (4 sin x)] z(x)"Ez(x) . 4 This equation differs from Eq. (6.19) considered by WGB by the presence of two extra terms. These two extra terms are not the artefacts of our substitution. Indeed, let us consider the threedimensional version of the path integral given by Eq. (6.9). Following Polyakov (1990) and Kholodenko and Vilgis (1995) the corresponding path integral can be written as follows:

P

(N)/uf D[u(q)]d(u2!1) u (0)/ui c N du 2 N dq ]exp ! #iH dq C[u(q)] , 2 dq 0 0 u

G(u , u ; N)" f i

G P A B

P

(6.21)

H

where

P

C[u(q)]"

N

dq@ u )

C

D

du du ' . dq dq@

(6.22)

0 Unlike the two-dimensional case given by Eq. (6.9), in three dimensions the value of the constant H in Eq. (6.21) is not arbitrary. Dirac (1931) had shown that 2H"0,$1,$2,$3,2. In Kholodenko and Vilgis (1995) it was shown that for a special value of H: H"1, the Schro¨dingerlike equation for the propagator of Eq. (6.21) can be written as [d2/dx2#cot x(d/dx)!1 cot2 x]t(x)"Et(x) . (6.23) 4 Eq. (6.23) is directly related to Eq. (6.20) (for g"0) and, hence, the double-well model by de Gennes (1982) for hairpins is just a special case of the Dirac monopole model in the external field (Dirac, 1931). In Section 4 we had considered a problem of “framing” for the self-linking number lk(i, i), e.g. see Eq. (4.19). Using Calugareanu-White theorem, Kauffman (1987a), one can write lk(i, i)"¹ [i]#¼ [i] w r where, according to Pohl (1968),

P

!1 N ¹ [i]" dq C[u(q)]#const. w 2p 0

(6.24)

(6.25)

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Evidently, Eqs. (6.24) and (6.25) provide the analytical expression for the writhe of the curve: ¼ [i]"lk(i, i)!¹ [i] . (6.26) 3 w More details about the differential-geometric properties and calculation of ¼ [i] can be found in 3 Aldinger et al. (1995). Evidently, Eq. (6.25) can be used in the path integral of Eq. (6.21), unlike the formal expression of Eq. (3.6). Nevertheless, both expressions are equivalent. As it was explained in Kholodenko and Rolfsen (1996), ¼ [i] also has the meaning of a winding number so that the path 3 integral of Eq. (6.21) is the correct three dimensional extension of the path integral of Eq. (6.9). To demonstrate that ¼ [i] has, indeed the meaning of the winding number, the following 3 arguments are helpful. Let r(q) describe an embedding of a circle S1 into R. For two circles embedded into R3 one can construct a unit vector r(q)!r(q@) n(q,q@)" Dr(q)!r(q@)D

(6.27)

which provides a map S1]S1PS2 known as the Gauss map. The degree of such defined Gauss mapping for S1]S1 (diagonal) PS2 is the winding number which is conventionally known as writhe and is analytically given by

P P A

1 ¼ [K]" 3 4p

dq

1

dq@

1

B

dr dr (r(q)!r(q@)) ' ) . dq dq@ Dr(q)!r(q)@D3

(6.28)

S S More details on the degree of mapping can be found in Dubrovin et al. (1985). We shall use the above facts in Section 7 where we are going to calculate the average writhe. Here we only notice that Eq. (6.19) (with monopole term, e.g. like that in Eq. (6.20)) has additional unexpected properties, if besides the nematic perturbation (described by cos2 x term), there is a perturbation of nonnematic origin (e.g. constant electric field, polymer in the flow, etc.). In this case, one has to consider an equation of the form [!(c/2)d2/dx2#g cos2 x#f cos x]t"Et

(6.29)

which is known as Whittaker—Hill (WH) equation. In the context of polymer problems it was first discussed by Ja¨hnig (1979). WH equation has interesting mathematical properties, e.g. see Magnus and Winkler (1966) or Urwin and Arscott (1970). Most important for us is the fact that for nonzero values of g the force constant f can have only discrete values. This means that if in the nematic environment we would like to stretch the polymer chain, then the application of a given force will not necessarily cause the stretching, i.e. the stretching will have a stepwise character. Recently, experiments were conducted to check the elastic properties of DNA molecules in solution, see, e.g. Cluzel et al. (1996). In the case of finite DNA concentrations, the experimental force—extension curve (e.g. see Fig. 2B of Cluzel et al.) shows the characteristic stepwise extension in complete agreement with the predictions based on the study of the WH model (Kholodenko and Vilgis, 1995). 6.2. Semi-flexible polymers confined between the parallel plates and in the half space In Kholodenko et al. (1994) this problem was treated for chains of arbitrary flexibility. Because the obtained results are rather bulky if one is interested in chains of arbitrary flexibility, we provide

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here only the summary of the results which are surprisingly closely connected with that presented in Section 6.1. In the presence of some external potential »(r) the distribution function G for the Gaussian-like polymer obeys the following equation of “motion”:

C

D

­ l ! + 2r #»(r) G(r, r@; N)"d(N)d(r!r@) . ­N 6

(6.30)

To treat chains of arbitrary flexibility it is sufficient to replace the “non-relativistic” Schro¨dingerlike Eq. (6.30) with that for the Dirac propagator, as it is explained in Kholodenko et al. (1994) and Kholodenko and Borsali (1995). In the case of half-space it is natural to consider separately the “longitudinal”, i.e. perpendicular to the wall, and the “transversal” “motions” of the chain. The longitudinal “motion” is one dimensional. The presence of the surface is being modeled by some sort of d-like potential, i.e. »(r)"d d(x). With this potential Eq. (6.30) is easily solvable with 0 the result

C

D

1 d 0 G(x, x@; s)" exp(!JsDx!x@D)! exp(!JsDxD!Dx@DJs) , 2Js d #2Js 0

(6.31)

where the Laplace variable s is conjugate to the polymeric length N. The result thus obtained corresponds to the so-called penetrable surface model. In this case, the chain can legitimately “tunnel” through the interface, e.g. between two liquids. At the same time, the impenetrable surface restricts the chain to the half-space regardless of the strength of the polymer-surface interaction. It can be demonstrated that the propagator for the impenetrable case can be obtained with the help of penetrable, given by Eq. (6.31), with the result (x and x@'0):

C

D

Js!d J 1 e~Js@x~x{@# e~ s(x`x{) , G(x, x@; s)" Js#d 2Js

(6.32)

where d"d /2. Consider now two limiting cases: (a) d P0 and (b) d P$R. In the first and the 0 0 0 second case we obtain, respectively, G(x, x@; s)"(1/2Js)[exp (!JsDx!x@D)#exp(!Js(x#x@))] ,

(6.33a)

G(x, x@; s)"(1/2Js)[exp (!JsDx!x@D)!exp (!Js(x#x@))] .

(6.33b)

Comparison with similar problems in quantum mechanics (Kleinert, 1995) indicates that Eq. (6.33a) can be interpreted as a Euclidean-type version of the two-particle relative amplitude for two bosons while Eq. (6.33b) represents the two-particle relative amplitude for fermions. This means that for the arbitrary strength of the interaction parameter d we have the case of an 0 intermediate (or fractional) statistics! This fact is going to be exploited in Section 8. The case of parallel plates can be easily obtained with help of the half-space result, Eq. (6.32), so that we

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provide here only the final answer:

C

= exp(!JsDx!x@!2MdI D) + M/~= Js!d exp(!JsDx#x@#2MdI D) , # Js#d

1 G (x, x@; s)" , 2Js

D

(6.34)

where dI is the distance between the plates. Consider now the inverse Laplace transform of the auxiliary Green’s function given by 1 = GI (x, x@; s)" + exp (!JsDx!x@!2MdI D) . 2Js M/~=

(6.35)

We obtain

G

H

1 = 1 + exp ! (x!x@!2MdI )2 GI (x, x@; N)" Nl 2JN M/~=

(6.36)

Jp , H(xJ !xJ @; iq) , 2dI where H(x,y) is the elliptic theta function (Mumford, 1983) and q"pNl/4dI 2, xJ "x/2dI . Whence, at least for dP0 and dP$R, we can write for G (x, x@; N) the following simple result: , 2dI G (x, x@; N)"H(xJ !xJ @, iq)$H(xJ #xJ @iq) . (6.37) , But the theta function is just the Green’s function for the particle moving on the circle (Kleinert, 1995) which we just had discussed in Section 6.1! This means that the condition for the wave function given by Eq. (6.4) just reflects the strength of the interactions between the polymer and the parallel plates. Whence, we have just demonstrated that (a) the half-plane and the parallel plates problems are interrelated in the sense that, at least for the infinitely repulsive walls, and (or) zero interactions with walls, the parallel plates problems are obtainable from the half-space problem by a simple replacement: (1/JpN) exp M!x2/NlNP(1/dI )H(xJ , iq) ;

(6.38)

(b) the interaction between the polymer and the walls is responsible for the fractional statistics for both the half space and the parallel plates problems; (c) the explicit connection between the phase h in Eq. (6.4) and the strength of interaction between the polymer and the wall(s) can be worked out, in principle, based on the work by Gaveau and Schulman (1986). For mathematically rigorous justification of all these results the reader may consult Aldaya et al. (1996). 6.3. Polymers confined into semi-flexible tubes Brownian motion of “particles” on the manifolds was considered by mathematicians some time ago, e.g. see McKean (1969). In physics literature study of this problem was initiated by da Costa

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(1981). da Costa had discussed the problem of how to develop the quantum mechanics for “particles” constrained to the manifolds which are embedded into R3. That is to say: how the extrinsic and intrinsic curvature properties of the manifold affect the Brownian motion. We had discussed already the motion on the circle and on the sphere S2. Let us now consider, instead of S2, an arbitrary reasonable smooth surface S which is parametrically given as r"r(q , q ), where 1 2 q and q are the local coordinates at some point on the surface which has a spatial position r. To 1 2 constrain the “motion” of our ficticious particle to the surface it requires to have some “squeezing” potential. To define such potential, one has to introduce the vector R"R (q , q , q ) so that 1 2 3 R(q , q , q )"r(q , q )#q n(q , q ) , (6.39) 1 2 3 1 2 3 1 2 where n is the continuous unit vector normal to S. The “squeezing” potential » is evidently a function of q , e.g. one may write 3 0, q "0 , 3 »(q )" (6.40) 3 R, q '0 . 3 If we want to develop quantum mechanics (or statistical physics), such “hard core” potential is somewhat inconvenient. To avoid this problem, the harmonic-like potential is usually used: » (q )"1mj2q2 where j is defined through the relation: Sq2T"+/mj, where + is Planck’s constant 3 3 j 3 2 and m is the mass of the particle. In polymer physics, the above combination may be associated with the tube radius a2, e.g. see Doi and Edwards (1978) and Section 8. If g is the metric tensor of ij the surface, e.g.

G

­r ­r g " ) , i, j"1, 2 ij ­q ­q i j and h is the second fundamental form of the surface, i.e. ij d2r h "n ) , i, j"1, 2 , ij ­q ­q i j where the vector n is defined in Eq. (6.39), then for the metric tensor G defined by ij ­R ­R G " ) , i, j"1—3 ij ­q ­q i j it is possible to obtain G "g #[ag#(ag)T] q #(agaT) q2 , ij ij ij 3 ij 3 G "G "0 , i"1, 2 , G "!1 , i3 3i 33 where the matrix elements a are defined via ij a "(1/g)(g h !g h ) , 11 12 21 22 11 etc. and g"det(g ). ij

(6.41)

(6.42)

(6.43)

(6.44) (6.45)

(6.46)

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Using the above results, one can write, in case of polymers, the diffusion equation in the curvilinear coordinates (q , q , q ) as follows: 1 2 3 ­ l 3 1 ­ ­ W" + JG(G)ij W!» (q )W , (6.47) j 3 ­N 6 ­q ­q i i/1 JG i where Gij is defined through the equation

A

B

3 + GikG "di , (6.48) kj j k/1 and G"det(G ). To proceed, one needs to use Eqs. (6.43), (6.44), (6.45) and (6.46) in Eq. (6.47) and ij to insure that the function W is properly normalized, i.e. to require

P

DW(q , q , q )D2 d»"1 , 1 2 3

(6.49)

where d»"dS f (q , q , q ) dq and 1 2 3 3 f (q , q , q )"1#Tr(a )q #det(d )q2 , (6.50) 1 2 3 ij 3 ij 3 (6.51) dS"Jg dq dq . 1 2 By introducing a new “wave” function via s(q , q , q ; N)"Jf W(q , q , q , N) it is possible to 1 2 3 1 2 3 rewrite Eq. (6.47) as follows:

A

B

­ l l ­2 ­s l 2 1 ­ Jggij s# ([1Tr(a )]2!det(a ))s# s!» (q )s" . (6.52) + 2 ij ij j 3 ­q 6 6 ­q2 ­N 6 ­q Jg j i 3 i,j/1 This equation naturally admits the separation of variables. By writing s"s s where s "s (q ) M , , N 3 and s "s (q , q ) one obtains easily M N 1 2 l d2 ­ s " s !» (q )s , j 3 , 6 dq2 , ­N 3 and

A

B

l 2 1 ­ ­ 1 ­ s " + Jggij s # ([1Tr(a )]2!det(a ))s . ij ij M ­g M 6 2 ­N M 6 ­g j i,j/1 Jg i For the curves (tubes) one needs to replace Eq. (6.39) by

(6.53)

R(q , q , q )"r(q )#q n (q )#q n (q ) , (6.54) 1 2 3 1 2 2 1 3 3 1 where n and n are the usual Serret—Frenet unit vectors, e.g. see Kholodenko (1990). By repeating 2 3 the same steps as before, for the surfaces, we finally arrive at the desired result:

A

B

­2 w2 ­ l ­2 # s ! (q2#q2)s " s , M 2 3 M 6 ­N M 6 ­q2 ­q2 3 2 ­s l d2 s #k2(q )s " , . 1 , ­N 6 dq2 , 1

(6.55) (6.56)

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In arriving at this result we have used the “soft” harmonic oscillator-like confining potential and used w2, instead of j, since with this notations Eq. (6.55) coincides exactly with that discussed by Doi and Edwards (1978). Unlike Doi and Edwards (1978), however, the longitudinal part contains an extra potential term which comes from the effects of the local curvature k(q ) of the tube. Since 1 k(q ) is, in general, the random variable and since in Eq. (6.56) its contribution always comes with 1 the positive sign, this implies that the presence of the curvature term may lead to the localization. This is obviously an undesirable conclusion which contradicts the main postulates of the reptation theory (Doi and Edwards, 1978) which were briefly discussed in Section 2.2 and will also be discussed in Section 8. To repair the situation, we have to assume that the effects of the local curvature are relatively small, which is possible only if the tube conformation deviates only slightly from that of the straight line. This leads to the effective rigidification of the polymer chain backbone constrained into the tube. The scaling analysis of such rigidified effective primitive chain (already mentioned in Section 2.2) presented in Kholodenko (1991) is discussed in Section 8 in connection with de Gennes—Doi—Edwards reptation theory (Doi and Edwards, 1978; de Gennes, 1971). The above derivation of the confining equations developed by da Costa (1981) had been improved by many authors, e.g. see Matsutani (1992), Burgess and Jensen (1993), Kugler and Shtrikman (1988), Ao and Thouless (1994), Maraner (1995), Clark and Bracken (1996), etc. The above cited papers provide some additional condensed matter applications which could be potentially useful in polymer physics. The additional polymer related discussion could also be found in Kholodenko (1996c). As it follows from Eq. (6.55), the transversal part of the Brownian motion takes place in the oscillator-like potential. This feature is characteristic of the problem about the probability for the random walk of N steps to enclose the prescribed area A. Since we shall employ some of the results related to this problem in Section 8, we would like now to provide some essentials in Section 6.4. 6.4. Configurational statistics of the planar random walks restricted by the area constraint Following Kholodenko (1996a), let us consider the calculation of the probability density given by

P

A KP A G P A BH

1 N dy dy dq x !y D[r(q)] d A! 2 dq dq r 0 (0)/r(N) 1 N dr 2 dq ]exp ! l dq 0 to be compared with Eq. (5.1). The algebraic area P(A, N)"pNl

P A

B

BKB

(6.57)

1 N dy dx dq x !y A" (6.58) 2 dq dq 0 suffers from several deficiencies. First, unlike the “true”, or geometric, area it can be both positive and negative, that is why the modulus sign is included in Eq. (6.57). Second, if the curve which encloses the area A has self-intersections, then the total algebraic area might be much smaller (or even zero!) than the corresponding geometric areas enclosed by the subloops. Levi (1965) had analyzed these problems in connection with computation of P(A, N). He found that although

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classically (i.e. geometrically) the area given by Eq. (6.58) is ill-defined, stochastically the area is well defined and leads to meaningful results (e.g. see Theorems 55.1 and 55.2 in his book). Although Levi had solved this problem long ago, in physics this problem was solved even earlier and it is known to be directly connected (Sondheimer and Wilson, 1951), with Landau diamagnetism problem (Landau, 1930). The probability given by Eq. (6.57) is normalized according to the usual prescription

P

=

dA P(A, N)"1 . (6.59) 0 In reality, however, this prescription is physically unrealistic since the polymer chain of length N cannot enclose the area greater than N2/4p (i.e. the area of the circle that has perimeter length equal to N. This means that the correct normalization should be

P

N2@4p dA P(A, N)"1 . (6.60) 0 Moreover, if the modulus sign in Eq. (6.57) is absent, then instead of the normalization defined by Eq. (6.60) another normalization is used (Duplantier, 1989):

P

=

dA P(A, N)"1 . (6.61) ~= The physical consequences of these different normalizations as well as extension of Eq. (6.57) to the case when the circular polymers are of arbitrary rigidity is being treated in Kholodenko (1996a). Here we shall not go into details of these calculations and only restrict ourselves with the results which will be used in Section 8. It is convenient to introduce the Fourier transform of P(A, N) via

P

1 = P(A, N)" dk P(k, N) exp (ikA) . 2p ~= The modulus sign in Eq. (6.57) can be eliminated with the help of the identity

(6.62)

P

1 dk e~*u@T@ 0 e~*k0T , " (6.63) 2pi K2 2u C where K2"u2!k2 and C is an appropriately chosen contour in the complex plane. By combin0 ing Eqs. (6.57), (6.62) and (6.63) we obtain

P P

G P

H

N dk k 0 D[r(q)]exp ! dq L[r(q)] , P(k, N)" piG(0, N) K2 r 0 c (N)/r(0) where

AB

A

B

1 dr 2 ik dy dy L[r(q)]" , # 0 x !y l dq dq dq 2 K2"k2!k2 and G(0, N)"(pNl)~1. 0

(6.64)

(6.65)

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The path integral of Eq. (6.64) was calculated by Khandekar and Wiegel (1988). However, the same result was obtained much earlier by Sondheimer and Wilson (1951) by recognizing that the Lagrangian of Eq. (6.65) corresponds to that for the particle in the constant magnetic field H such that H"+]A and A"M!(H/2)y, (H/2)xN .

(6.66)

With help of this observation, the action in the path integral of Eq. (6.64) can be rewritten as

P GA B

H

N m dr 2 dr dq (6.67) #ie ) A[r(q)] , 2 dq dq 0 where, in case of polymers, the mass m"2/l, the charge e"1, +"c"1 and H"k . Using such 0 defined dictionary, we can use directly Wilson and Sondheimer’s results in order to obtain S[r(q)]"

P(A, N)"(2p/Nl)[cosh(2pA/Nl)]~2.

(6.68)

The average area can now be obtained as

P

SAT"

=

dA P(A, N)"(Nl/2p)ln 2 . (6.69) 0 In principle, we can calculate any moment, e.g. SAnT since we have closed form explicit expression for P(A, N) given by Eq. (6.68). More interesting, however, is to calculate the generating function

P

= dA e~kAP(A, N) (6.70) 0 which is just the Laplace transform of P(A, N). If we combine Eqs. (6.57) and (6.70), we immediately discover that the delta function in Eq. (6.57) disappears, and the problem of calculation of F(k) is reduced to that given by Eq. (6.64) (with k being replaced by k). It was shown in Kholodenko (1996a,b) that the constant k has also a physical meaning: !k"Dp, where Dp is the pressure difference between the inside and outside of the two dimensional vesicle (circle). Whence, one can think about calculating SAT for the prescribed pressure difference Dp. In this case, one obtains after some calculation the following result for SAT: F(k)"

SAT"1/Dp!(Nl/2)cot (DpNl/2) .

(6.71)

This result is going to be used in Section 8.

7. Knot complexity – detailed treatment 7.1. Calculation of the topological persistence length The topological persistence length N was introduced in Section 3.1. It is defined as a minimal T number of steps on some three-dimensional lattice required for the first nontrivial knot to be formed. Evidently, N is non-universal quantity which depends on the lattice type. T

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In this section we shall provide an estimate of N using the existing path integrals for T semi-flexible polymers, e.g. that given by Eq. (6.1). Since N is a lattice-dependent quantity, while T the path integral is defined in the continuum, our result for N will depend upon the discretization T procedure which is used to arrive at the continuum limit for the path integral we are going to use (Kholodenko, 1995). Such dependence, however, is not too strong as we shall demonstrate shortly. In order to obtain the result for N we need to use the Milnor’s inequality given by Eq. (4.3). This T inequality should be combined with the Schwarz inequality (Frenchel, 1951) valid for any closed curve, and given by

AP

B

P

N 2 N dq Dk(q)D 4N dq k2(q) , (7.1) 0 0 where k(q), as before, is the local curvature of the curve. If we think of the curve as made of real physical material, e.g. polymer, then using polymer methodology, e.g. see Section 6, we have to perform the statistical average of Eq. (7.1) with help of the path integral(s) for the semi-flexible chains. The statistical average S2T in terms of such path integral(s) can be in the simplest case defined as (2p)24

P

G P A BH

c N du 2 dq D[u(q)]d(u2!1)exp ! (7.2) 2, 2 dq u 0 (0)/u(N) where the normalization constant NI is chosen in such a way that S1T"1 and the constant c was defined in Eq. (6.1). In the fully flexible limit, cP0, the polymer chain behaves as Gaussian. It is known (Kholodenko, 1993) that in this limit the polymer Kuhn’s step length l"2c. We can associate the length l with the unit step length of the random walk on some regular (e.g. cubic) three-dimensional lattice (de Gennes, 1979). Such identification should be done with some caution, however, since the discrete analogue of the path integral of Eq. (7.2) is expected to exist and to be well defined. As results of Kholodenko (1995) indicate, the lattice-dependent factors (like J2 for cubic lattice, etc.) are likely to occur when the identification between the discrete and the continuum formulations are made. These factors are responsible for some numerical differences in final results for N . From the experimental point of view, the measured combination T 2cN"lN"SR2T does not allow to measure separately l and N in one measurement. Some independent measurements are required (Kholodenko, 1993; Hickl et al., 1997), which inevitably introduce some errors. Whence, both the discrete and the continuum formulations can provide only the upper and the lower bounds for N as will be further explained below. T By combining Eqs. (7.1) and (7.2) we obtain S2T"NI

TAP

B U TP

U

TAP

B U TP

U

N 2 N 4N (7.3) dqDk(q)D dq k2(q) . 0 0 This inequality should be valid for any closed polymer configuration. At the same time, by combining inequalities Eqs. (4.3), (7.1) and (7.3) we obtain the following result for the knotted curves: (2p)24

(4p)24

N 2 4N dqDk(q)D 0

N dq k2(q) 0

.

(7.4)

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Since any three-dimensional curve is being characterized by the local curvature k(q) and torsion i(q) (Kholodenko, 1990), it is not a priori clear that the path integral of Eq. (7.2) contains knotted three-dimensional configurations (at least within the semiclassical level of approximation) since the action in the path integral contains only the curvature part. Fortunately, Langer and Singer (1984a,b) and subsequently Bryant and Griffiths (1986), and, even more recently, Kholodenko and Nesterenko (1995), have shown that this is indeed the case. More specifically, Langer and Singer (1984a,b) had considered a three-dimensional variational problem for the functional of the type

P

F [C]" LS

N ds (k2(s)#m2) , 0

(7.5)

where ds is the length element along the contour C and the Lagrange multiplier m2 accounts for the fact that the length of the curve is fixed. Kholodenko and Nesterenko (1995) had shown that the variational problem for the functional F [c] produces trajectories which are identical to those LS obtained from the functional

P

1 N F [c]" ds k2(s) KN 2 0

(7.6)

defined on the space of constant curvature (e.g. sphere). This result is in accord with those obtained earlier by Griffiths (1983). The numerical value of the constant curvature is directly related to m2, (Kholodenko and Nesterenko, 1995). Langer and Singer (1984a,b) had demonstrated that “there is a countable infinity of (similarity classes of ) closed nonplanar elasic curves in R3. All such elasicae are embedded and lie on the embedded tori of revolution (see, e.g. Fig. 23) infinitely many are knotted and the knot types which occur are precisely the (n, m) torus knots (see, e.g. Fig. 23 and Rolfsen, 1976) satisfying m'n. The integers m and n determine the elasticae uniquely (up to similarity).” To actually perform the averaging, several steps are required. First, we would like to notice that for the semi-flexible polymers it is the dimensionless combination N/c which actually determines how stiff the polymer chain is. In terms of the Kuhn’s length, we have u"N/l"N/2c. Using this result, the action functional in Eq. (7.2) can be rewritten as

P

P

c N 1 1 S" dq k2(q)" dq k2(q) , 2 4u 0 0

(7.7)

where in arriving at the last equality we have taken into account that in the case of natural parametrization, n2"1, we have k2(q)"(dn/dq)2 and n"dr/dq where r(q) is the spatial position of the polymer segment at contour position q. By combining Eq. (7.2) with Eqs. (7.4) and (7.7) we obtain,

TP

N

U TP

N dq k2(q) " 0

1

0

U

dq k2(q) "!4

­ ln I[u] , ­u~1

(7.8)

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where

P P

I[u]" du

G P

H

1 1 dq k2(q) D[u(q)] d(u2!1) exp ! 4u u u u i/ f/ 0

= " + (2n#1) exp (!u(n#1)n) . (7.9) n/0 In arriving at the last line in Eq. (7.9), we have used the results of Kholodenko and Vilgis (1996). As in this reference, we would like to replace the summation by integration (which corresponds to the semi-classical approximation) in the last line. This then produces:

P

I[u]K

=

dx 2x exp (!ux2) . 0 Combining Eq. (7.8) with Eq. (7.10) produces (within the approximations made)

TP

1

U

dq k2(q) "4u .

(7.10)

(7.11)

0 Combining this result with the inequality (Eq. (7.4)) we obtain (4p)24 4u or (2p)24u .

(7.12)

Since (2p)2+40 and since u is the effective number of steps on the lattice we obtain u540 .

(7.13)

This result is in excellent agreement with the numerical results of Windwer (1990), see, e.g. Eq. (3.4). Indeed, by using the experimental values for kJ and cJ in Eq. (3.4) one obtains u"40.884. At the same time, if we would choose the rescaled length: NPNJ2 (or, equivalently, the rescaled Kuhn’s length lPl/2) we would obtain instead u528

(7.14)

which is in very good agreement with Diao’s rigorous calculations, (Diao, 1993, 1994) which provide N "24 for knots on the cubic lattice. Since factors like J2 reflect the symmetry of the T cubic lattice and naturally emerge in the discretized models for the semi-flexible polymers (Kholodenko, 1995), the results of Eqs. (7.13) and (7.14) represent the upper and the lower bound estimates for N on the cubic lattice. Evidently, if we would choose a different lattice, the results for T N might be somewhat different. T 7.2. Calculation of the averaged writhe In Section 3.2 we had discussed the numerical simulations which produce for the averaged writhe the result given by Eq. (3.8). To obtain this result analytically, i.e. with help of the existing

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path integral methods, we have to use Eq. (6.24) along with the auxiliary Eqs. (6.24)—(6.28). Use of these equations permits us to write the partition function Z(g) which is very similar to that given by Eq. (7.9), i.e.,

P P

Z[g]" du

D[u(q)]d(u2!1)

(0)/ (N)/ 1 1 dq [k2(q)#ig ¼ [u(q)]] . (7.15) ]exp ! 3 4u 0 With the help of such defined partition function we can obtain the averaged writhe according to the usual rule: u i

u f

u

G P

1 d ln Z(g) . S¼ [K]T" 3 i dg

H

(7.16)

The imaginary factor i"J!1 in Eq. (7.15) is introduced for the sake of convenience: to show the correspondence with the exactly solvable quantum problem. Thus defined average writhe is nonzero only for the fixed orientation of the closed curve C. This means that it can be both positive or negative depending upon the orientation of the curve (e.g. see the definition of ¼ [K] given by 3 Eq. (3.6)). In those cases, when in simulations both orientations of the curve are being considered (Whittington et al., 1993, 1994a,b), one needs to calculate SD¼ [K]DT instead of Eq. (7.16). This 3 causes us to replace ¼ [u(q)] in the exponent of Eq. (7.15) by the value of its modulus. The presence 3 of the modulus sign in the exponent of the partition function in Eq. (7.15) causes no additional computational problems since the identity, Eq. (6.63), allows us to reduce this, apparently more complicated problem, to that without the modulus sign which is known to be soluble (Kholodenko and Vilgis, 1996). Indeed, in Section 6.1 we had mentioned already, that the propagator of Eq. (6.21) can be obtained by solving the Schro¨dinger-like Eq. (6.23) for the Dirac monopole (Dirac, 1931). In Kholodenko and Vilgis (1995, 1996) we had provided all technical details needed for proof of this fact. Additional details can be found in the paper by Dunne (1992) and Aitchinson (1987). To calculate Z(g) given by Eq. (7.15) we need to know only the spectrum of the corresponding Schro¨dinger-like operator and its degeneracy. Following Dunne (1992), we easily obtain = Z[g]" + (2DgD#2n#1) expM!uE(n)N , n/0 where E(n) is given by E(n)"n(n#1)#DgD(2n#1) .

(7.17)

(7.18)

For g"0 we obtain back the result Eq. (7.9) as required. For gO0 we have to use formally Eq. (7.16). This leads us to the problem. Eq. (7.17) does not contain an imaginary part while Eq. (7.16) contains the imaginary factor of i so that, if we formally use Eq. (7.17) in Eq. (7.16) we will obtain, seemingly, physically wrong imaginary result for the averaged writhe. The mistake in performing such formal manipulation lies in our so far formal treatment of the path integral of Eq. (7.15). Much more careful treatment, see, e.g. Kholodenko and Vilgis (1995, 1996) reveals that

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the imaginary i-factor in Eq. (7.15) is not an artefact but an essential ingredient of the whole Dirac monopole problem. This can be seen already from our treatment of the two-dimensional analogue of the Dirac monopole problem discussed in Section 6.1. The transition from two to three dimensions is not merely a replacement of the winding number, Eq. (5.2), by that given by Eq. (6.28). It also involves the quantization of the coupling constant g in Eq. (7.15) known already to Dirac (1931). In three dimensions g"2pH where 2H"0,$1,$2,2. Since g is quantized, the differentiation, e.g. like that given in Eq. (7.16), should be performed with some caution. The same caution should be exercised in view of the modulus sign for g in Eq. (7.15). The existence of this sign can be naturally associated with the possibility to have two orientations for the closed contour C as we have mentioned already. If the orientation of the contour is fixed, then by choosing e.g., g'0, we can formally define the average writhe as

C

D

1 = = SD¼ [K]DT" 2 + expM!uE(n)N# + (2n#1)(2g#1#2n)expM!uE(n)N . (7.19) 3 Z[g] n/0 n/0 To compare this result with Eq. (3.8), it is sufficient now to let gP0` and to replace the summation by integration, e.g. as it was done in Eq. (7.10). This then produces

CS

N SD¼ [K]DT" 3 2c

D S

A B

c3@2 2cp N 1 #const. J #O . N3@2 N c JN

(7.20)

In arriving at this result we have used the definition of u, Eq. (7.7), and required gP0` to reach an agreement with the numerical result of Eq. (3.8). Should we choose instead gP0~, we would obtain, instead of Eq. (7.20), SD¼ [K]DTJ!JN so that, indeed, the algebraic sum of this result 3 and that given by Eq. (7.20) produces zero in complete agreement with writhe definition given in Section 3.2. The fact that only one result was considered, while Eq. (7.19) provides the results for arbitrary (albeit discrete!) values of g, is associated with the specificity of the numerical simulations leading to the result of Eq. (3.8). In the limit gP0` the average, Eq. (7.16), represents a kind of Kubo-like result, where the average is made over the unperturbed “equilibrium” system. In the future numerical simulations one might be willing to study the general case: when both u and g are allowed to wary. This could be especially relevant for studying the supercoiled DNA, etc. 7.3. Calculation of the knot complexity We had defined the knot complexity in Section 3.2 as a number of vertices c[K] in the planar graph for a given knot K. If the vertices are not resolved, e.g. like indicated in Fig. 2, then different knots could have, in principle, the same complexity. Whence, c[K] is not a topological invariant for a given knot K. At the same time, the unknotting number u[K] is. Use of the Bennequin conjecture, Eq. (3.9), allows to provide some bounds for u[K] if both ¼ [K] and c[K] are known. Using the 3 definition of ¼ [K] given by Eq. (3.6) we can define c[K] via 3 c[K]" + D e(p)D . p | S(K)@

(7.21)

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At the same time, using the definition of ¼ [K] given by Eq. (6.28), we can write as well 3 (r(q)!r(q1)) 1 dr dr dq dq1 ' ) . (7.22) c[K]" dq dq1 Dr(q)!r(q1)D3 4p 1 S S1 Such defined analytical expression for the knot complexity c[K] is in agreement with that proposed by Arnold (1986). One may naively think that if one substitutes Eq. (7.22) into Eq. (7.15) (instead of ¼ [º(q)]) the corresponding path integral could be easily solved as well. Unfortunately, 3 use of the identity, Eq. (6.63), cannot help in this case so that to obtain the Schro¨dinger-like expression for the partition function Z[g] is not an easy task. Instead, there is another method of computation (Freedman et al., 1994; Freedman and Z-Hu 1991) which uses the notion of the knot energy. The knot energy E[K] can be defined as follows:

P P KA

P

P

K

G

H

1 r`N@2 1 ! , (7.23) dq dq@ Dr(q)!r(q@)Da Dq!q@Da ~N@2 q~N@2 where the arch-length parametrization is used (i.e. Ddr/dqD"1) and a is some constant, 1(a 43. As the above authors had shown, E[K]"

N@2

B

c[K]#2/p4(1/2p)E[K] .

(7.24)

The constant 2/p is obtained only for a special value of a: a"2. This value of a has some physical significance associated with the reparametrization invariance of the r.h.s. of Eq. (7.23). To see this, let us consider a special case of an unknot º : a circle c of radius R. Then, the energy of a circle can 0 0 be calculated as

P P p

G

H

1 q`p 1 dq dq@ ! . (7.25) [2R sinDq!q@D/2]a [RDq!q@D]a ~p ~q~p For a"2, E[c ] becomes independent of the radius R and, whence, of the length of the curve N. 0 For any other a we obtain evidently E[c ]"R2 0

E[c ]J R2~aJ N2~a . (7.26) 0 If we assume that the above N-dependence persists also for more complicated (knotted) situations, then using Eq. (7.24) and ignoring the factor of 2/p (for large N) we obtain c[K] 4N2~a

(7.27)

which would require a to be less than or equal to one in order to be in qualitative agreement with the result of Eq. (3.7). This, however, is not permissible in view of the definition of E[K], Eq. (7.23), which requires a to be between 1 and 3. The resolution of this contradiction can be found if we analyze the averaged value of E[K]. The averaged knot energy is defined by

P P T

U

N N 1 , (7.28) dq dq@ Dr(q)!r(q@)Da 0 0 where we have disregarded the singular counterterm present in Eq. (7.23) for reasons which will become clear shortly below. To this purpose we have to decide what kind of the averaging SE[K]T"

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319

procedure can be used in Eq. (7.28). For N
PP

G P A BH

3 N dr 2 dq D[r(q)] exp ! ))) . (7.29) 2l dq r r r 0 (0)/ (N)/ The normalization constant N is chosen, as before, to satisfy S1T"1. Replacement of Eq. (7.2) G by Eq. (7.29) is done mainly for technical (computational) reasons. Evidently, Eq. (7.2) (or more complicated path integral for semiflexible chains) can be used as well, in principle. Our choice of averaging allows us to disregard the singular counterterm present in Eq. (7.23), since the averages of the type given by the r.h.s. of Eq. (7.28) are known in literature (Feynman, 1972; Kholodenko, 1992). Before we engage ourself into calculation of the averaged knot energy, we would like to provide some justification for the word “energy” used by mathematicians. To this purpose, let us rewrite E[K], given by Eq. (7.23), into the following equivalent form: S2T"N dr G

P

E[K]"

N dq f (q) , 0

(7.30)

where

P G

H

1 N 1 dq@ ! . (7.31) Dr(q)!r(q@)Da Dq!q@Da 0 For a very large and very stiff circle, at least locally, the conformation of the contour is very close to the rigid rod limit (i.e. k(q)P0). In this case our problem resembles that known in the theory of polyelectrolyte chains (Kholodenko, 1995), i.e. polymer chains which carry some charges along their contours. When charges are unscreened, the cumulative electrostatic repulsion between the different segments along the chain is given by Eq. (7.30) with a"1. In the case of knots, a'1, and the electrostatic analogy cannot be used straightforwardly. Nevertheless, we can employ similar methods of analysis of E[K]. To this purpose, for k(q)P0, we can use the following Taylor series expansion f (q)"

dr 1 d2r r(q@)"r(q)# (q!q@)# (q!q@)2#2 . dq 2 dq2

(7.32)

Use of the Secret—Frenet formulas from the differential geometry of estaic curves (Dubrovin et al., 1995; Kholodenko, 1990), allows us to obtain after some algebra

C

D

s2 1@2 , Dr(q)!r(q@)DKs 1! k2(q) 12

(7.33)

where s"Dq!q@D. For s;J12/Dk(q)D we obtain

A

B

1 1 s2 a a@2 1 K 1# k2(q) + # s2~ak2(q)#2 . Dr(q)!r(q@)Da sa 12 sa 24

(7.34)

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The first term in this expansion cancels with second (counter) term in Eq. (7.31) while the second term for a"2 becomes s-independent. In this limit we can safely write

P

N N dq k2(q) (7.35) E[K]" 12 0 to be compared with Eq. (7.7). Obviously, for a"2 the knot energy has the same physical meaning as the elastic bending energy. For the unknot º , Freedman et al. (1994) had calculated E[K] and 0 found (for a"2): E[º ]"6p#4"22.84964. Whence, we can write also for the unknot: 0 N N dq k2(q)+22.84954 . (7.36) E[º ]" 0 12 0 When this result is combined with the inequality Eq. (7.3) and Eq. (7.7) we obtain

P

P

p2 1 1 dq k2(q)5 6p#4"E[º ]5 (7.37) 0 3 12 0 in complete agreement with the results discussed in Section 7.1. With these observations we are ready now to calculate the averaged knot energy given by Eq. (7.28). To perform an average in Eq. (7.28), let us formally define the Fourier transform of the potential DrD~a via

P

P

4p 4p = dr r1~a sin kr" v (k)" drDrD~ae*k > r" const(a) , (7.38) a k3~a k 0 where const(a)":=dx x1~a sin x. 0 The const(a) is well defined only for 1(a(3, and this result is in a complete agreement with Freedman et al. (1994), where the same bounds were obtained by using completely different arguments. Using Eq. (7.28) we can write as well

P

1 1 v (r)" " dk e~*k > rv (k) . a a DrDa (2p)3

(7.39)

By combining Eq. (7.28) with Eq. (7.39), we obtain

P

1 dk v (k) S(k) , SE[K]T" a (2p)3

(7.40)

where S(k) is defined by

P P

N N dq dq@Se~*k >(r(q)~r(q{))T . (7.41) 0 0 This quantity (up to numerical prefactor) is the static scattering form-factor for the circular Gaussian-like polymers. This quantity was calculated by Casassa (1965), and it is for this reason we have chosen the averaging procedure specified by Eq. (7.29). The action in the exponent of Eq. (7.29) is not reparametrization-invariant while the energy, Eq. (7.23), is (for a"2). S(k)"

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The lack of reparametrization invariance for this and related action(s), and its consequences for the calculation of physical observables was recently discussed in Kholodenko (1995). The experience with flexible polymers suggests, nevertheless, that for a large u the Gaussian approximation defined by Eq. (7.29) is quite adequate (the excluded volume effects can be easily incorporated into Eq. (7.29), if necessary, as it is explained below). For large u the difference between the circular and the linear polymers becomes unimportant when computing S(k), see, e.g. Feynman (1972). This fact allows us to write at once the result for S(k):

P P

1 1 dy dy@ e~(lN@6)@y~y{@ . 0 0 Combining this result with Eq. (7.40) we obtain, S(k)"N2

P

(7.42)

P P

N2 = 1 1 dk ka~1 dy dy@ e~(lN@6)@y~y{@k2 SE[K]T" (4p)2const(a) (2p)3 0 0 0 "const@(a)N2~a@2 ,

(7.43)

where const@(a) is defined by the first line of Eq. (7.43) (with appropriately rescaled k). The result of Eq. (7.43) should be compared against Eq. (7.26) and against the numerical result given by Eq. (3.7) (in view of Eq. (7.24)). For aK1 we have SE[K]TJN3@2 while for a"3 we obtain SE[K]TJN1@2. While the first value lies within the domain of the expected values of a , see, e.g. # Eq. (3.7), the second value is considerably lower. To sharpen our estimates, let us take now a closer look at the value of const@(a) in Eq. (7.43). We have, upon proper rescaling,

P P P P P P

1 dy@ e~k2@y~y{@ 0 0 0 1 1 1 = dk ka~1 e~k2 . J dy dy@ Dy!y@Da@2 0 0 0 The last integral is manifestly nonsingular only for 1(a(2 which produces at once const@(a)J2

=

dk ka~1

1

dy

(7.44)

c N4SE[K]T4c@ N3@2 , (7.45) a a where c and c@ are some constants depending on a. Using Eq. (7.24) and ignoring the factor of 2/p a a which is permissible for large N’s we obtain 1 Sc[K]T4 SE[K]T . 2p

(7.46)

By combining Eqs. (7.45) and (7.46) we conclude that the observed value(s) of a , defined by # Eq. (3.7), should lie within the following bounds: 1#d(a (1.5!d (7.47) # with dP0`. The lower bound for a is also in accord with the result of Eq. (4.84) for the alternating c knot(s) (link(s)).

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The above bounds are obtained without taking account of the excluded volume effects. The experience with similar types of calculations (Kholodenko, 1992), suggests that the upper bound in Eq. (7.47) can be noticeably lowered thus bringing our estimate for Sc[K]T closer to the experimentally observed, see, e.g. Eq. (3.7), and the discussion related to it. To this purpose we would like to recall that the mean end-to-end distance SR2T scales like N2l where l"1 for Gaussian chains 2 and lK3 for the chains with excluded volume (de Gennes, 1979). Whence, the result of Eq. (7.43) 5 can be equivalently rewritten (for the Gaussian chains) as SE[K]TJN2~al,Na#

(7.48)

and, by continuity, we expect it to be correct also in the good solvent regime. For a close to 1 this would produce a "2!al(1.4 (7.49) # while for the upper permissible value of a"2, the lower bound for the exponent a should remain # unchanged, i.e. a "1, in view of Eq. (7.44). Hence, account for the excluded volume effects, brings # our results much closer to the experimentally observed (Arteca, 1994, 1995). 7.4. Calculation of the unknotting number and the number of distinct knots as a function of polymer length N The unknotting number of u[K] was defined in Section 3.2 as the minimal number of selfcrossings which will turn knot K into an unknot. Unfortunately, there is no known analytical expression for u[K]. The Bennequin inequality (conjecture) Eq. (3.9) provides some bounds for u[K]. We had demonstrated in Section 4.4 that u[K] is of the order of the highest degree of the corresponding HOMFLY polynomial. By combining the inequality Eq. (4.80) with the conjecture Eq. (4.81), we obtain MKu[K]41c[K] . 2 Accordingly, for the averaged quantities we obtain 1 u[K]4 SE[K]T 4p

(7.50)

(7.51)

where we have used the fact that u[K] is a topological invariant while c[K] is not and its average is bound by SE[K]T according to the inequality (7.46). Obtained estimate for u[K] deserves some additional comments. Indeed, since for a given knot K, u[K] is a topological invariant, it should be independent of the polymer length N. At the same time, the average knot complexity Sc[K]T as well as SE[K]T exhibit strong N-dependence. Moreover, the averaged Sc[K]T makes physical sense only with respect to the length of the polymer. This can be seen already from our calculations of N . T For N(N we may still anticipate to observe crossings in knot projection(s) onto some chosen T plane(s). The minimal number of crossings to produce a non-trivial knot should be at least 3 (Rolfsen, 1976). Whence, for NKN we expect to have at least 3 crossings. This naturally T reintroduces the lower cut off into the knot problem. At the same time, if the number of crossings c[K]Kn is fixed but NPR, then the knot complexity Sc[K]TKn does not mean much, because n/NP0. By performing the average in Eq. (7.51) we actually replace the problem related to a given

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323

knot K to that related to all possible knots of length N. Since Sc[K]T grows faster than the length N, using the results of the previous subsection we obtain n/N&Na#~1PR. The obtained infinity is not physically relevant, however. Indeed, if we would ignore (for the moment only), the excluded volume effects, then we would need to consider Sc[K]T crossings in the volume »& [JSR2T]3&N3@2. This creates the ratio P"Sc[K]T/N3l&N2~l(3`a)

(7.52)

(with l"1 in the absence of excluded volume effects) which we shall call the “packing capacity” of 2 a knot. According to our estimate, Eq. (7.47), this ratio will go to at most a constant for a"1. This would require to have no more than one crossing per unit volume (if we use the system of units where l"1) which is physically sensible. Whence, the lower bound for the exponent a, a"1, can be obtained based on simple physical arguments. The upper bound for a can be also simply obtained if we formally consider the collapsed state for which JSR2TJN1@3, i.e. l"1. Using this result in 3 Eq. (7.52) and requiring P to be a constant, we obtain a"3 in complete accord with Eq. (7.23). Consider now the problem of calculation of the number of distinct knots for the polymer of length N. If the given knot K has n crossings, n"c[K], then, according to Eq. (3.15), the average number of different knots SK(n)T with n crossings is bound by 2Wc*K+X4SK(n)T4 2(24)Wc*K+X ,

(7.53)

where Sc[K]T was estimated in Section 7.3. Use of this result allows us to introduce an additional entropy term (not present for the linear polymers) via S[K]"k ln SK(n)T , (7.54) B where k is Boltzmann’s constant. This extra entropy term leads to some measurable effects B discussed in the next subsection. 7.5. Some physical applications Roovers and Toporowski (1983) and Roovers (1985) have found that ring polystyrene in cyclohexane has a relatively large second virial coefficient A at H temperature compared to the 2 linear polymers of the same chemical composition. At the same time, they found that H temperature for the ring polymers (H ) is noticeably lower than that for the linear polymers of the same 3 chemical composition. These effects can be qualitatively explained with help of the entropic contribution defined by Eq. (7.54). Following Grosberg and Khokhlov (1989), one can introduce the swelling ratio d"JSR2T/lN. For the linear chain the free energy can be written as

A B

AB

F BJN C 1 -*/%!3+d2# # d~3# d~6 , (7.55) l3 k ¹ l6 d2 B where BJ(¹!H)/H and the constant C is responsible for the strength of 3-body interactions. It is important to realize that H temperature can be defined in several ways (Kholodenko and Freed, 1984b). For example, it can be defined as a temperature at which the chain is exactly Gaussian, or

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as a temperature at which the second virial coefficient A vanishes. Since in the experiments by 2 Roovers and Toporowski (1983), Roovers (1985) the second definition is used, we shall adopt it in this work too. Minimization of the free energy with respect to d produces the following equation for the linear chain: d5!d"x#y d~3 ,

(7.56)

where x"BJN/l3 and y"C/l6. This equation produces physically meaningful qualitative results in good, H and poor solvent regimes (Grosberg and Khokhlov, 1989). In particular, under H-conditions, if we choose d"1 as a solution to Eq. (7.56) (which corresponds to the onset of collapse, Kholodenko and Freed, 1984b), this would require us to write x"!y, or B"!C/l31/JN .

(7.57)

Using the definition of B, we obtain the known shift of the H-temperature (without the logarithmic corrections, which only could be obtained field-theoretically, see, e.g. Kholodenko and Freed, 1984b): H!¹JN~1@2 .

(7.58)

For rings we have to account for the extra entropy term introduced in Eq. (7.54). Since the entropy is always defined up to an additive constant, it is convenient to measure entropy with respect to H conditions. In this case, repeating the same steps as for the linear polymers, we arrive at the following equation: d5!d"x#yd~3#const. du`3 ,

(7.59)

where u"a /l, in view of Eq. (7.48), and the actual value of non-negative const. is unimportant in # these qualitative calculations. For d"1 we have now x"!y!const., or B"!(C/l3#l3const.)1/JN .

(7.60)

This result indicates that the rings should have lower temperature H at which they behave as ideal. 3 This means that at H temperature for the linear polymers, the ring polymers will have d'1, i.e. the topological entropy, Eq. (7.54), produces the same effect on rings as if they would have an additional excluded volume-type interaction which makes the second virial coefficient for rings effectively larger than that for the linear polymers. This is in complete accord with the observations by Roovers and Toporowski (1983). Our explanation of these effects differs, however, from that provided by Iwata (1989). Consider now another application. Our calculation of the averaged knot energy, Eq. (7.28), is very similar to the calculation of the diffusion coefficient D for the individual polymer chain. Within the Kirkwood approximation the calculation of D involves the averages like

P P T

U

k ¹ N dq N dq@ 1 D" B , 6p g N N Dr(q)!r(q@)D 4 0 0

(7.61)

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see, e.g. Kholodenko (1992). Here, g is the viscosity of the solvent. For the Gaussian-like chains the 4 average in Eq. (7.61) is easily computable with the result (for D): k ¹ B , (7.62) D" 6pg JSR2T 4 where SR2TJN. Hence, the diffusion coefficient for polymers formally resembles that for the hard spheres if the sphere radius R is approximated by JSR2T. Not surprisingly, the porocity (the packing capacity, P which we had introduced earlier in Eq. (7.52)) indicates that, indeed, at H conditions our ring polymer acts as if it is a hard impenetrable sphere. This result, surprisingly, comes also from the completely different type of calculations performed by Oono and Kohmoto (1982). Evidently in solvents other than H-solvents the concept of porocity can be used as well. In detailed calculations Wiegel (1980) and later Starting and Wiegel (1994) had shown how to perform calculations for solutions of polymers modelled as porous hard spheres. The central idea for these calculations is a phenomenological Darcy’s law which can be written as g F"! 4V , k

(7.63)

where F is the force exerted by the fluid on the medium (per unit volume) and V is the velocity of fluid while the phenomenological constant k is related to our porocity P as can be seen from Wiegel (1980). The microscopic origin of the Darcy law is studied in some detail by Bear (1972). Unlike Wiegel’s work in which the porocity was introduced as a phenomenological parameter, in the present case its origin is known so that the additional refinements could be made, see, e.g. the results by Quake (1994) discussed in Section 2. Finally, let us notice, following Delbru¨ck (1962), that the difference between the truly circular and the linear polymers is not so significant, at least at H conditions where JSR2TKJN so that the ratio JSR2T/NP0, for NP0. I.e. the distance between the ends of an open polymer chain is much smaller than the contour distance. Under these conditions, the rapid temperature quench could bring our polymer into one of SK(n)T globular-like (glassy) states as was first noticed by de Gennes (1984). Since, nevertheless, such a glassy state is not a state of true equilibrium, there is a difference between the kinetics of collapse for linear and circular polymers (de Gennes, 1985; Grosberg et al., 1988; Ma et al., 1995). Still, additional applications of the obtained results could be made for problems which involve vortices in superfluid helium, classical turbulence, superconductivity, etc., see, e.g. Akao (1996). We deliberately avoid discussions of these applications with hope that the interested reader can easily restore the details, based on the results which we describe in this review, if necessary. 7.6. Link energy and the probability of entanglement between two ring polymers The formalism developed above can be easily extended for the links. In case of links, new questions could be posed (as discussed in Section 2) in addition to that presented in the previous subsection. Vologodskii et al. (1975) being influenced by much earlier work by Frisch and Wasserman (1961), had discussed the following problem. Consider two closed random walks which are independently generated on some cubic lattice. For each walk the position of center of mass (for

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Fig. 10. The simplest (Hopf) link considered by Vologodskii et al. (1975) in connection with calculation of the probability P2(R).

unit masses at the ends of each bond segment) was determined with respect to some coordinate system. Then, thus generated “polymer” rings were brought to the proximity of each other so that the distance between their centers becomes R. This motion of one ring with respect to other was made by means of a simple parallel translations (i.e. without deforming “polymer chains”) so that one ring could go through another without the excluded volume restrictions (phantom chains). The probability P (R) that the rings are not entangled was determined as well as the probability P (R) of 0 i entanglement into a topological state i (where i was determined through usage of knot polynomials (e.g. see Section 4)). The possibility of having rings both knotted and entangled was disregarded, and only entanglements between rings were counted. As a result of these numerical simulations, P (R) was determined to behave as 0 P (R)"1!A exp(!a R3) , (7.64) 0 0 0 where A is some constant (of order unity for long chains) and a JN~1.7. For the simplest link 0 0 depicted in Fig. 10 the results of computer simulation had produced the following result for P (R): 2 P (R)"A exp (!a R3) , (7.65) 2 2 2 where the polymer length dependence of constants A and a was not given explicitly. This 2 2 dependence was estimated only quite recently by Everaers and Kremer (1996) who found a +a/2R3 with aK0.6 and R KJN. These results are going to be reproduced below with help 2 L L of the link energy E (Freedman et al., 1994), defined by (for the n-component link) L n 1 n (7.66) E ,E(MK N)" + E[K ,K ]# + E[K ,K ] , i j L i i i 2 i/1 i,j/1 iEj where E[K ,K ],E[K ] , (7.67) i i i with E[K ] being defined in Eq. (7.23), while E[K ,K ] is being defined as i i j 1 N N E[K ,K ]" dq dq . (7.68) i j Dr(q )!r(q )Da i j i j 0 0 The arch-length parametrization is being used here in complete agreement with Eq. (7.23). Since, according to Vologodskii et al. (1975), the individual rings are assumed to be knotless, the first form

P P

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in Eq. (7.66) can be omitted. As before, we need to calculate the averaged link energy. This averaging can be performed with help of the averaging procedure defined by Eq. (7.29) (this time, for two ring-type polymers). The reason why we are interested in the averaged link energy could be seen from the standard statistical mechanics arguments, Hill (1956), which connects the pair correlation function g(r , r )"g(Dr !r D) with the potential of the mean force º(Dr !r D): 1 2 1 2 1 2 º(Dr !r D) 1 2 g(Dr !r D)"exp ! , (7.69) 1 2 k ¹ B where, in case of m point-like interacting particles, º(Dr !r D) is given by 1 2 1 m +@ »(Dr !r D) m{ i j º(Dr !r D)" (7.70) < dr + »(Dr r D) exp ! i,j/1 1 2 i`2 i~ j Z k ¹ B i/1 i,j with »(Dr !r D) being the microscopic (two-body) potential and Z is just the partition function i j itself. The prime(s) indicate the absence of self-interaction terms, i.e. iOj, in Eq. (7.70). In case of the entangled polymers, the correlation function g(Dr !r D),g(R) can be identified 1 2 with P (R) introduced earlier. ¹his is effectively done in Vologodskii et al. (1975). Whence, calcui lation of P (R) is reduced to the calculation of the potential of the “mean force”, i.e. to the i calculation of the averaged link energy. The attempts to perform such calculation (but without use of Eq. (7.68)!) were made in the past, see, e.g. Tanaka (1982) or Iwata and Kimura (1981). No agreement has been reached between these calculations and Monte Carlo simulations by Vologodskii et al. (1975). The calculation of the averaged link energy is very similar to the calculation of the second virial coefficient A for the dilute polymer solutions, see, e.g. Kholodenko and Freed (1983). 2 The only novelty of the present calculation, as compared to the calculation of A , lies in the 2 additional constraint

A

B

P

P

G

H

P

1 N 1 N R! dq r(q )# dq r(q )"0 (7.71) 1 1 2 2 N N 0 0 which needs to be inserted into the corresponding path integral measure thus reflecting the fact that the distance between the center of masses of rings should be equal to R. The calculation of A involves the Feynman diagram depicted in Fig. 11. Here the solid closed line(s) indicate(s) the 2 propagator(s) for the ring polymer(s), the wavy line indicates the interaction between the polymers and the volume factor »~1 is needed to make A volume-independent. 2 For completeness, we provide some details of this simpler calculation which are needed in the more difficult case involving the constraint, Eq. (7.71). In the system of units, in which the Kuhn’s lenth l is chosen to be 2d, the propagator G (R, q) for the open chain can be written as 0 ddk G (R, q)" e~k2q`*k > R . (7.72) 0 (2p)d

P

Since the calculation of the loop(s) in the denominator of Fig. 11 involves integrals like :ddR G (R, N), we obtain two volume factors coming from two rings. In the numerator the extra 0 volume factor will come because of the translational invariance of the interaction potential. These considerations explain the presence of the volume factor in Fig. 11. Calculation of the numerator

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Fig. 11. Feynman diagram for the calculation of the second virial coefficient A2.

involves calculation of the following expression:

P P P P P P

N N dq dq@ dr dr dr@ dr@ G (r !r ,q) 1 2 1 2 0 1 2 0 0 ]G (r !r , N!q)»(Dr (q)!r@ (q@)D) 0 2 1 2 2 ]G (r@ !r@ ,q@)G (r@ !r@ , N!q@) , (7.73) 0 1 2 0 2 1 where »(Dr !r D) is the polymer—polymer interaction potential. Substitution of Eq. (7.72) into 1 2 Eq. (7.73) and account of translational invariance immediately produce the following result for I: I"

P

I"N2[G (0,N)]2»3 dR »(DRD) 0

(7.74)

so that the result for A follows: 2 A "N2»(k"0) , 2 where »(k"0) is obtained by noticing that

(7.75)

P

»(k)" dR e*k > R»(DRD) .

(7.76)

To account for the constraint given by Eq. (7.71) it is useful to recalculate I using a different method, Feynman (1972). For this purpose, we have to consider the calculation of the following auxiliary functional integral for the closed path:

P P

G P

H

1 N N dq dq@rR 2#ik ) r(q) . D[r(q@)] exp ! (7.77) 4 r 0 0 (0)/r(N) To calculate such an integral it is very useful to introduce the following Fourier decomposition of r(q): IK "

A B

= npq r(q)"a # + a cos . 0 n N n/1

(7.78)

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Using this decomposition the action in the exponent of the path integral of Eq. (7.77) can be written as (Kholodenko and Quian, 1988)

A B

p2 = = npq S" . (7.79) + a2 n2!ik ) + a cos n n 8N N n/1 n/0 The integral over the zero mode a produces d(k) so that the second term in Eq. (7.79) (with 0 components a other than zero) vanishes upon k-integration. The first term in Eq. (7.79) has the n same structure as that calculated for the closed paths (Feynman, 1972), and, whence will produce the same result as Eq. (7.72) (for R"0). The presence of d(k) is important since if we represent the interaction potential as

P P

ddk ddk@ k r k r »(r , r )" e* > 1`* {> 2»(k, k@) 1 2 (2p)d (2p)d

(7.80)

so that »(k,k@)"»(k)d(k#k@), then evidently, zero modes coming from two rings will remove k and k@ integrations in Eq. (7.80) thus producing the volume factor d(0) coming from d(k#k@). Collecting all terms together, we arrive again at the result of Eq. (7.75) as required. Consider now the more complicated case which involves the constraint of Eq. (7.71). In this case, we have to substitute into the path integral measure the d-factor given by

P

C A

P

P

BD

d3K N N exp iK ) R!1/N dq r(q)#1/N dq@r(q@) . (2p)3 0 0 The presence of this factor changes the action in the exponent of Eq. (7.77) into d"

P

P

1 N 1 N dq@rR 2!ik ) r(q)#i K ) dq@ r(q@) . S" N 4 0 0 Use of the Fourier expansion, Eq. (7.78), converts the above action into

(7.81)

(7.82)

A B

p2 = = npq S@" #i K ) a . (7.83) + a2n2!ik ) + a cos 0 n n 8N N n/1 n/0 Integration of the zero mode produces now the d-constraint: d(k!K). By integrating over k (see, e.g. Eq. (7.80)) we are left with the following action:

C

A BD

p2 = npq S@" . (7.84) + a2 n2!i K ) a cos n n 8N N n/1 Performing the Gaussian integration over each of a modes we obtain now the following result n including both rings and discarding factors like d(0):

P

P P C A B

d3K K R N N º(DRD) " e* > » (K) dq dq@ a (2p)3 k ¹ 0 0 B 2NK2 = 1 npq npq@ cos2 #cos2 ]exp ! + p2 N N n2 n/1

G

A BDH

.

(7.85)

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Following Feynman (1972), we can replace the summation in the exponent of Eq. (7.85) by the integration via the rule:

P

= N = + 2P dx2 . p 0 n/1

(7.86)

This produces after a little calculation (upon completion of q integrations and rescaling):

A B P

A B

º(DRD) 2 const (a) R 4~a = sin xy dx xa~5(1!e~x2@R2)2 " , p k ¹ xy JN B 0

(7.87)

where const(a) was defined after Eq. (7.38) and y~1"JN. Straightforward convergence analysis indicates that the obtained integral is convergent for 14a43 in complete agreement with the results of Section 7.3. In order to actualy use Eq. (7.87) we notice that for y;1 we can subdivide the domain of x-integration into two parts, e.g. from 0 to 1 and from 1 to R. We then can appropriately Taylor series expand the integrand in each subdomain of integration by taking into account that (in chosen system of units) R251. For a"1 and R;N we obtain, in view of Eq. (7.69), the result of Everaers and Kremer (1996) given by Eq. (7.65) while for a"2 we obtain the result of Helfand and Pearson (1983). Comparison between these results and Eq. (7.44) indicates that the exponent a in Eq. (7.87) is likely to be bounded by the inequality 14a42 for any kind of entanglement of a given polymer with other polymers (or with itself). This observation leads us to Eq. (2.21) where, accordingly, we obtain 24u43. Obtained results allow us to calculate several additional quantities. For example, in view of Eq. (7.64), one can calculate the topological second virial coefficient AT between two non-entangled 2 polymers. Following Vologodskii et al. (1975), we obtain

P C

A

BD

1 F (R) AT" d3r 1!exp ! 0 2 2 k ¹ B

,

(7.88)

where F (R) is related to P (R) via 0 0 F "!k ¹ ln P (R) . 0 B 0

(7.89)

Substitution of Eq. (7.89) into Eq. (7.88) produces, in view of Eqs. (7.64) and (2.21), the following result for AT (for aK1): 2 AT"4pR3 . 2 3 L

(7.90)

It is quite remarkable that this result was obtained with help of only qualitative arguments by Frisch and Wasserman (1961) as discussed in Section 2. The existence of non-negative AT causes 2 additional repulsion between the polymer rings (not to be confused with depression of H temperature discussed in Section 7.5) thus leading to the effective reduction of their sizes. More quantitative analysis of this phenomenon is provided in Section 8.

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8. Polymer dynamics: an interplay between topology and geometry 8.1. Statistical mechanics of a melt of polymer rings Already in Section 2 we have noticed that in the melt of linear polymers of length N the ratio JSR2T/NP0 for NPR. I.e. dynamically, for times q(q the melt of polymer rings and linear 5 polymers should behave very similar which is indeed the case, see, e.g. McKena et al. (1989) (especially Fig. 20 of this reference). This experimental observation is very important for the development of the dynamics of polymer melts of both linear and ring polymers. The entanglements which are inevitably present in such melts in the form of (quasi) links are not only responsible for the formation of the effective tube which surrounds the given polymer chain (which is well documented experimentally, Straube et al., 1995) but affect also the stiffness of the trapped chain. We have discussed this fact, in part, in Section 6.3 from the geometrical point of view. Here we would like to discuss the same problem from the topological point of view. To this purpose, let us consider again the partition function given by Eq. (4.10). Following the work of Brereton and Vilgis (1995), we shall concentrate our attention on a single ring placed in a melt of other rings. The many body problem which involves different rings is going to be reduced effectively to the one-body problem for the ring which is being singled out. To this purpose, let us rewrite Eq. (4.10) in the following form:

T

U

n n Z(Mc N,Mm N)" < d(lk(a, b),m ) < d(lk(b, b@), m ) (8.1) a bb{ ab bb{ bEa b;b{Ea where d(x,y) is, as before, the Kroneker’s delta and S2T denotes the polymer averaging, e.g. like that given by Eq. (7.29), of all n!1 chains, except one, which we denote as a. Since the Kroneker’s delta can be written as

P

2p dg expMig(x!y)N , 2p 0 Eq. (8.1) can be equivalently rewritten according to Brereton and Vilgis (1995) as d(x,y)"

(8.2)

P

n 2p dg bb{Z(Mc N;Mg N) expM!ig m N (8.3) Z(Mc N;Mm N)" < a bb{ bb{ bb{ a bb{ 2p bb{/1 0 where (ua(q)]ub(!q)) ) q 1 Z(Mc N,Mgbb{N)" exp + (g #g ) a ab ba q2 X bEa (ub(q)]ub{(!q)) ) q 1 ]exp + g , (8.4) bb{ q2 X bb{Ea and X is the volume of the system. In arriving at the result given by Eq. (8.3) the linking number, defined by Eq. (4.11), has been transformed with help of identities

T G G

Q

dlad(ra!r) ,

ua(r)"

c 1 qk rk " d3q exp(iq ) r) . q2 DrD3 2p2i a

P

HU

H

(8.5) (8.6)

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Such transformation allows to calculate the partition function, Eq. (8.1), in formally closed form given by

P

P

n 1 p 2p~abb{ ds expMi2s m N Z(Mc N;Mm N)" < da bb{ bb{ bb{ a bb{ bb{ p2 abb{ 0 bb{/1 1 ]exp ! + [A (q;Ms N)l(q;Mc N)!B (q;Ms N)/(q;Mc N)] aa bb b aa bb{ a X q

G

H

(8.7)

where the matrices

G C G C

D H D H

n C2 ~1 A (q;Ms N)" C2 1# aa bb{ c(q) q2 n C3 C2 ~1 B (q;Ms N)" 1# aa bb{ c(q) q2 q2

,

aa ,

with matrix C being given by ab C ,[C(q)] "(1/n)s c(q) ; ab ab ab while c(q)"o

QQ QQ QQ

(8.8)

aa

dl ) dl@SexpMiq ) (r!r@)NT ,

(8.9)

(8.10)

ca ca

l(q;Mc N)" a

/(q;Mc N)" a

dl · dl@

ca ca ca c

expMiq ) (r(l)!r(l@))N , q2

expMiq ) (r(l)!r(l@))N dl]dl@ ) q q2 a

(8.11) (8.12)

with o"Nn/X. Evidently, in order to calculate Eq. (8.7) explicitly, some approximations should be made. These are discussed in Brereton and Vilgis (1995). The results can be considerably simplified if all linking numbers m in Eq. (8.1) are being put equal to zero which corresponds to the description of the ab melt of unlinked rings. In this case, after some algebra, one arrives at the result Z(Mc N, M0N)"expMiplk(a, a)Nexp[!(1/l )E[Mc N]] (8.13) a %&& a with the self-linking number lk(a,a) being defined by Eq. (4.19) while the knot energy E(Mc N) is a being given by

QQ

dl · dl@ E[Mc N]"lim a Dr(l)!r(l@)D1`e e?0 ca ca with l "l(6/ol3p2). In the original paper of Brereton and Vilgis (1995) a different terminology for %&& E[Mc N] is being used (they call it the “self-inductance”). This choice of terminology is due to a chronological reasons: the paper by Brereton and Vilgis had appeared in 1995 while Kholodenko

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and Rolfsen’s had been published in 1996. Both terms in the exponents of Eq. (8.13) were already discussed earlier in this work. The first term is associated with the choice of framing, see, e.g. Section 4.2, and causes the polymer chain to be more stiff, Section 6.1. The presence of the second term is essential if the polymer chain is effectively knotted as discussed in Sections 3, 5 and 7. Implicitly, it can also be associated with the probability for a given chain not to be entangled with other chains as discussed in Section 7.6. When the r.h.s. of Eq. (8.13) is substituted into the path integral, e.g. Eq. (7.2), for the ring C , this leads to the delicate competition between the stiffening a and softening. In Kholodenko (1991) only the stiffening effect was taken into account which amounts to the assumption (also implicitly present in de Gennes (1971) and Doi and Edwards (1978) treatment of reptation) that the chain trapped into the tube is knotless. If the rigidity wins, then one can use the scaling analysis of Section 2.2 in order to arrive at famous result: q JN3.4 for 5 the viscosity. Since, however, according to Eqs. (3.5) and (5.95), for NPR the fraction of the unknotted rings is completely negligible, the presence of the second term in the exponent of Eq. (8.13) is quite natural and effectively counterbalances the stiffening leading to the noticeable contraction of the ring in the polymer melt (Mu¨ller et al. 1996), in qualitative accord with calculations of Brereton and Vilgis (1995). Since the topological effects alone are unable to make the trapped polymer backbone more stiff, the geometrical factors discussed in Section 6.3 should be taken into account. They are responsible for making the longitudinal part of the trapped polymer motion more stiff so that the scaling analysis of Section 2.2 could be used. The transversal part of this motion requires additional discussion since it is responsible for the transition from the Rouse to the reptation regime of the dynamics of polymer melts (Kholodenko, 1996b,c; Kholodenko and Vilgis, 1994). 8.2. Statistical mechanics of planar rings in an array of obstacles (the replica approach) The transversal motion of the trapped polymer is usually described by the oscillator-like Schro¨dinger equation, see, e.g. Doi and Edwards (1978) and Eq. (6.55). We have demonstrated in Kholodenko and Vilgis (1994), that this oscillator-like Schro¨dinger problem can be reinterpreted in terms of magnetic language. In this language we are dealing with the quantum Landau-diamagnetism-like problem about the planar “motion” of charged particles placed in the constant magnetic field. Such reinterpretation allows us to look at the whole problem of chain confinement from a much wider perspective. In Kholodenko (1996a,b) it was shown that the Landau diamagnetism problem is also isomorphic to the problem about the planar random walk which encloses the fixed prescribed area A, see, e.g. Section 6.4. Now, we want to introduce some complications into this problem. Specifically, let us assume that our closed planar walk takes place at the punctured plane where the punctures are meant to represent the cross sections of other chains, or tubes. In the case of chains the punctures have infinitely small radius while in the case of tubes they have a finite radius. Topologically, however, this fact makes no difference, see, e.g. Kholodenko (1996b,c) and Appendix A.1. Whence, we may want to calculate the probability of enclosing a given area A by the planar random walk of N steps in the presence of impurities with some prescribed surface density oL "n /A where n is the total number of cross sections (punctures). We would like to 5 5 impose an additional constraint that no impurities are allowed to be inside the contour which encloses the area A. The presence of randomly distributed impurities introduces some sort of quenched (or annealed) disorder into the problem which is normally being treated with the use of replicas. Use of

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replicas can be bypassed based on topological considerations, see, e.g. Kholodenko (1996b,c) and below, but it is of interest to compare the observables which can be calculated in both ways. To this purpose, we would like to consider a closely related problem about the properties of the closed planar walk of N steps which is entangled with a random array of cross sections, e.g. twodimensional analogue of Eq. (4.10) where the constant c is now a random variable with prescribed probability distribution. This problem was considered by Tanaka (1984) and, more recently, by Otto and Vilgis (1996). Related results were earlier obtained by Nechaev and Rostiashvili (1993) and Rostiachvili et al. (1993) based on the fundamental earlier work by Brereton and Shah (1980). To develop our results, let us recall a useful identity (Fulton, 1995), dz dx#i dy x dx#y dy !y dx#x dy d ln z" " " #i z x#iy x2#y2 x2#y2 ,d ln r#i dw(h) ,

(8.14)

where, according to Eq. (5.2), :N dw(h)"w and we used polar coordinates: x"r cos h, y"r sin h in 0 the last of our equations. Evidently, we can consider as well a combination +n5 d ln (z!a ) which would place singulari/1 i ities (punctures) of the complex z-plane at points a . Obviously, the total winding number w5 can be i written now as (Fulton, 1995),

C P

D

N n5 w5" + Im dq d ln(z!a ) i 0 i/1 N , dq rR (q) ) A[r(q)] , (8.15) 0 where the vector potential A[r(q)] is given by A"(A ,A ) with x y n5 n5 A "! + (y!a i)r~1, A " + (x!a i)r~1 and r2"(x!a i)2#(y!a i)2 . x x i y y i i x y i/1 i/1 For a single closed polymer chain of length N which is entangled with punctures the partition function Z(c) (with account of the excluded volume effects) can be written as

P

P

AP B A P

Z(c)" D[r(q)]d

N

0

dq rR d c!

B

N dq rR ) A[r(q)] expM!S[r(q)]N , 0

(8.16)

where

P

P P

1 N a2 N N S[r(q)]" dq rR 2# dq dq@d(r(q)!r(q@)) l2 2 0 0 0 with a2 being the two-dimensional excluded volume parameter. Since both the locations a of punctures as well as the total winding number w5,c are i fluctuating variables it is necessary to perform some sort of averaging of Z(c) in order to calculate the obsevables (e.g. SR2T, etc.). It is assumed, that the disorder associated with the location of punctures could be considered as annealed while the disorder associated with w5 as quenched. To

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perform the average for the case of quenched disorder normally requires the use of replicas. This can be accomplished in several steps. First, we can rewrite the annealed average of Z(c) as

P

P

AP B

SZ(g)T " D[A]d(+ ) A) D[r(q)]d A

G

P

P

N dq rR 0

H

N 1 ]exp ! d2x(+]A)2!ig dq rR ) A!S[r(q)] , (8.17) 2u 0 0 where the parameter u is related to the distribution of obstacles which is assumed to be 0 Gaussian-like. The function Z(g) is related to Z(c) via Fourier transform:

P

= dg expMigcNSZ(g)T . A 2p ~= Second, upon introduction of the “current” J via Z(c)"

(8.18)

P

N dq rR (q)d(r!r(q)) 0 use of the Hubbard—Stratonovich transformation in Eq. (8.17) allows us to eliminate the A-field. This produces the following result for SZ(g)T : A N u g2 SZ(g)T " D[r(q)]d dq rR exp !S[r(q)]! 0 A[r(q)] . (8.19) A 2 0 Here, following Cardy (1994), we have introduced an area J(r)"

P

AP B G

H

P P

A[r(q)]" d2r d2r@SA (r)A (r@)TJ (r)J (r@) k k k k

(8.20)

which has the same meaning as the expression introduced earlier, see, e.g. Eq. (6.58). Upon the substitution of an identity 1":dA d(A!A[r(q)]) inside the path integral, Eq. (8.19) it is possible to rearrange terms so that the result for SZ(g)T now looks like this A u g2 1 SZ(g)T " D[A]d(+ ) A) dA daJ exp ! d2x(+]A)2! 0 !iaJ A A 2 2

P

P P

]Z(e, A) ,

G P

A

BH

(8.21)

where Z(e, A) is defined by

P

G

P

H

Z(e, A)" D[r(q)] exp !S[r(q)]!ie d2r A[r(q)] ) J(r(q)) .

(8.22)

with e"J2iaJ . Following Nechaev and Rostiashvili (1993), the last expression can be rewritten with the help of replicas in terms of the n-component complex scalar field theory path integral:

P P

Z(e, A)"lim Du Du* expM!S[u,u*]N , n?0

(8.23)

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where u"Mu ,2,u N and 1 n n l2 ¸a2 n S[u,u*]" + u* m2! (+!ieA)2 u # + Du D2Du D2 . i i i j 4 4 i/1 i.j/i In this expression the mass variable m2 is &N~1 while ¸ is the average size of the polymer in the direction perpendicular to the plane. Substitution of Eq. (8.23) into Eq. (8.21) and integration over the field A produces in the replica symmetric approximation, i.e. +nDu D2"nDuD2, the following final i i result:

A

B

G P

H

, SZ(e, A)T "exp !n d2r ¸ %&& A

(8.24)

where

A

A B

B

l2 l2 ¸a2 DuD2 ¸ "iaJ ! DuD2 ln # DuD2 #(m2!¸a2M2)DuD2# DuD4 %&& 4p 2p 4 M2 with M2 being an arbitrary mass which appears as a result of regularization of the one-loop corrections coming from A-integration. By introduction of the “free energy” f (aJ ) via

CP

1 ­ f (aJ )" » ­n

d2r ¸

%&&

D

(8.25)

"¸

%&&

if it is possible to rewrite Eq. (8.21) as

P A

A B

B

l2 l2 DuD2 SZ(g)T " dA d A# »DuD2ln ! »DuD2 e~Vf(A,g) A 4p 2p M2

(8.26)

where »":d2r and f (A,g) is defined by f (A,g)"(u g2/2»)/A#(m2!¸a2M2)DuD2#(¸a2/4)DuD4 . (8.27) 0 Use of this result in Eq. (8.18) with account that c is Gaussianly distributed random variable allows to calculate the average Z(c). Actual calculations of this quantity can be only performed with help of the saddle point approximation which produces the following consistency conditions: o "(¸/a)2(l2/4p) (u /D )(1!c2/D ) , # 0 # 0 # 1/N "(¸a2/2)o ln[¸3o ] , ¸3o '1 , # # # #

(8.28) (8.29)

and (8.30) A "(»/2p)o (1!1ln[¸3o ]) # # # 2 with o"DuD2 and parameters D and c characterizing the average total winding number (c is the # 0 0 mean winding number) and D is the dispersion of the winding number while u is the mean density # 0 of obstacles. For the fixed value of parameters u ,c and * the above results determine the critical 0 0 # length N so that below the critical length the polymer acts as if it is still fully flexible (Gaussian# like) while above N it collapses to the conformational state of branched polymer. Indeed, #

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according to Eq. (8.26), we have A"»(l2/2p)(o!(1/2)oln(o¸3)) . If now we take into account that o"N/», this result can be written as (for N'N ): # A"(l2/2p)N(1!1 ln N#2) 2 K(l2/2p) N1~1@2 .

(8.31)

(8.32)

Since A&SR2T we obtain immediately JSR2TJlN1@4 which is the scaling law for the branched polymer without excluded volume. The main conclusions of the calculations just presented could be summarized as follows: 1. The problem about conformational properties of a planar ring trapped (entangled) in an array of obstacles was actually reduced to the problem about the calculation of the effective area which such a ring encloses. 2. It was shown that the problem is well defined only above a certain threshold (in parameter space). 3. Below this threshold the ring polymer collapses and acquires the shape of the branched polymer (the last result is being independently used by Obukhov et al. (1994) to describe the dynamics of rings in gels as discussed in Section 2). Below we shall reproduce these results using completely different (topological) methods which do not rely on use of replicas. By doing this some new aspects of the “trapping problem” will be revealed. 8.3. Statistical mechanics of planar rings in an array of obstacles (the Riemann surface approach) In Section 6.4 we had discussed configurational statistics of the planar random walks restricted by the area constraint. Surprisingly, the problem about the random entanglements considered in Section 8.2 happens to be very closely related to this area problem. In this section we will try to clarify why, indeed, such connections exist. As it was already noticed in Section 6.4, the planar area constraint problem is essentially equivalent to the standard Landau diamagnetism problem (Landau, 1930), in case the plane is not punctured. In such a (standard) case, the problem lies in quantum (and statistical mechanics) treatment of motion of the electron in the presence of constant magnetic field H defined by the vector potential A, see, e.g. Eq. (6.66). In Kholodenko (1996a) full analysis of this problem is given for both nonrelativistic and relativistic electrons (since this problem happens to be isomorphic to the problem of statistical description of deformable planar droplets of arbitrary rigidity). We shall discuss only the nonrelativistic limit in this review. The relativistic effects are briefly discussed in Kholodenko (1996a). In the nonrelativistic limit the solution is reduced to that known for the quantum harmonic oscillator with frequency depending upon the strength of the magnetic field H. Whence, for arbitrary small H we still have an infinite tower of equidistant discrete energy levels. The situation changes dramatically if the motion of an electron is considered on the punctured plane. In this case we may have, depending upon the surface density oL of punctures, a finite number of bound states or even no bound states at all (Kholodenko, 1996b,c). Whence, we may anticipate, that there is some threshold oL so that above (below) oL there will (will not) be bound states. The above picture # #

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Fig. 12. (a) Fusion of two punctured spheres produces a sphere again. (b) Fusion of two punctured tori produces a new surface of genus 2.

can be now recast into polymer language (Kholodenko and Vilgis, 1994; Kholodenko, 1996c). The transversal part of the diffusive motion, Eq. (6.55), is isomorphic to the Landau diamagnetism problem (Kholodenko and Vilgis, 1994). In the presence of planar punctures Eq. (6.55) should be modified. Upon such modification the tube existence and stability will be determined by the number of available bound states. The transition from zero to finite number of bound states is discontinuous. We formulate our results in such a way that the numerical predictions of our theory related to the onset of tube creation (destruction) associated with transition from the Rouse (no tubes) to the reptation (tube assisted) regime could be directly compared with experimental data (Kholodenko, 1996c), and demonstrate very satisfactory agreement with the experiment. Quantitative results obtained below are in qualitative accord with the results of Otto and Vilgis (1996) discussed in Section 8.2. Let us begin with the following auxiliary example. Following Arnold (1978) (see, e.g. Appendix A.1), let us consider the classical motion of a particle in a square with periodic boundary conditions (i.e. on the torus). We shall complicate matters by putting inside a square another circle (hole) so that our particle can elastically scatter out of this hole and the walls of the square. The classical motion in such billard takes place actually on a Riemann surface which is known as a double torus (i.e. sphere with two handles). The double torus is obtained by gluing two copies of the usual torus with a hole in it as depicted in Fig. 12. The gluing is done around the circumference of a hole. It is well known that the Riemann surfaces represent the case of surfaces of constant negative curvature. The classical motion on such surfaces is chaotic (Arnold, 1978). To bring this auxiliary problem closer to our original tube problem, let us consider, instead of just one hole, many (with some surface density oL introduced in Section 8.2). Then, it is intuitively clear that we will end up with the Riemann surface of genus g (sphere with g handles) where the genus g is determined by the density of obstacles (holes). All this can be made quite rigorous by considering homotopy of the paths on the punctured plane with periodic boundary conditions and by using the van Kampen theorem as explained, e.g., in Massey (1967), Gilbert and Porter (1994) or Fulton (1995). We deliberately would like to avoid all these mathematical complications unfamiliar to most of the readers trained in

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polymer physics. Instead, we would like to use more intuitive examples (including that of Arnold) which have some physical appeal. But since the van Kampen theorem tells us that the punctured plane is effectively the Riemann surface (irrespective of the classical mechanics example discussed above), one can exploit this fact “quantum mechanically”. Let us recall that the conformational properties of flexible chains in the external random potential » are described with the help of the end-to-end distribution function G(r, r@; N) which obeys the “equation of motion” (in three dimensions)

A

B

l ­ ! +r2#»(r) G(r, r@; N)"d(r!r@)d(N) . ­N 6

(8.33)

Upon the decomposition of this equation into longitudinal and transversal parts (as discussed in Section 6.3) we are left with effectively two independent Schro¨dinger-like equations. The transversal (planar) problem could be treated, in principle, with the help of the methods described in Section 8.2. Following the seminal work of de Gennes (1971) on reptation (see, e.g. his Eqs. (2.4) and (2.5)), the random environment can be modeled, however, with the help of a Smoluchovskitype equation for G given by ­ ­ ­ G"D G!c G . ­r2 ­r ­N

(8.34)

The actual values of constants D and c depend on the microscopic model used to arrive at Eq. (8.34), For example, in Nechaev et al. (1987) the “motion”on the regular lattice is considered (see also Nechaev (1990)) while in Nechaev (1988) “motion” on the Bethe lattice is being considered. Eq. (8.34) appears to be universal (Helfand and Pearson, 1983; Rubinstein and Helfand, 1985; Mehta et al. 1991) and independent of the dimensionality of the embedding space. In case the “motion” takes place on the regular lattice D"2pq, c"q!p, p"z~1, q"1!p and z is the coordination number of the lattice. The above equation should be actually supplemented with initial and boundary conditions, e.g. G(r, N"0)"d(r) ,

D(­G/­r)D !cG(0, N)"0 . (8.35) r/0 As it was argued in Kholodenko (1996b), to obtain the general solution of Eq. (8.35) it is permissible initially to ignore the boundary conditions: once the general solution is obtained, it will be forced to satisfy the specific boundary conditions. So far, we have not made any connection(s) between Eq. (8.34) and the topological properties of the underlying two-dimensional punctured plane. To do so, we would like to pose the question: is it possible to rewrite Eq. (8.34) in the form of diffusion-type equation on some curved manifold? The answer to this question is “yes”. To prove this, let us first bring Eq. (8.34) to the dimensionless form. If one chooses a"D/c2 and b"D/c, then one obtains the dimensionless analogue of Eq. (8.34) given by ­ ­2 ­ G(x,q)" G! G . ­q ­x2 ­x

(8.36)

Let us demonstrate that this equation can be rewritten in an equivalent form as (­/­q)G"(1/Jg)­ (gabJg­ )G a b

(8.37)

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for some metric tensor g . If we choose: g "1, g "g "0, g "e~2x, then we can obtain: ab xx rx xr rr g"e~2x, grr"1, gxx"e2x so that we get

A

B

A

B

1 ­ ­ ­ ­ ­ (g Jg­ 2)"ex ex 2 #ex ex 2 . b g a ab ­x ­x ­u ­u

(8.38)

If G(r,q) is u-independent, then use of Eq. (8.38) in Eq. (8.37) produces back Eq. (8.36) as required. Once we have obtained the metric tensor g of surface, we can find out what kind of surface it ab determines. The first fundamental form of the surface (the length) can be written now, based on the above results, as ds2"dx2#e~2xdu2 .

(8.39)

By introducing a new variable y"ex we obtain dy"ex dx and, therefore, Eq. (8.39) can be rewritten as ds2"dx2#e~2x du2"(dy2#du2)/y2 .

(8.40)

In mathematical literature the metric given by the last expression of Eq. (8.40) is known as the hyperbolic metric (Arnold, 1978; Stillwell, 1992). The Poincare´ model H consists of a subset of the complex plane C defined by H"Mz"u#iy3CDy'0N

(8.41)

supplemented with hyperbolic metrics given by Eq. (8.40) (Poincare, 1882; Buser, 1992). If we would use complex variables z and zN , then Eq. (8.40) could be rewritten as ds2"(dz)2/(Im z)2 .

(8.42)

For finite distances d between z and z@ in this model we could obtain with the help of Eq. (8.42) the following result: Dz!z@D2 cosh d(z, z@)"1# 2 Im z Im z@

(8.43)

to be compared with the usual Euclidean distance d (z, z@)"Dz!z@D . (8.44) E With the help of d just defined, the solution of Eq. (8.36) (without boundary effects) is known to be (Buser, 1992; Kholodenko, 1996b)

P

1 = dx x e~x2@4q G (z, z@; q)" e~q@4 . (8.45) H 2(2pq)1@2 d(z,z{)Jcosh x!cosh d(z, z@) Earlier, when we have discussed Arnold’s billiard, the claim was made that the actual motion takes place on the Riemann surface (i.e. sphere with g handles) instead of H-plane (also known as the Lobachevski plane). There is no contradiction, however, between the earlier claim and the results just obtained since the Lobachevski plane is the universal covering surface for the Riemann

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341

Fig. 13. The planar representation of a torus is just a square with opposite sides properly identified. In this representation the homotopy of paths around a puncture inside a square is equivalent to the homotopy of paths around a puncture near one of the corners.

surface of any genus g'1, Stillwell (1992). The notion of the covering surface could be easily understood using the following example. Let C be a discrete translation in the complex plane C (or R2), then a given square S can be obtained as an image of some fundamental square SK upon q q translation, i.e. S "CSK . The torus can be obtained as a quotient R2/C so that R2 is the universal q q covering surface for the torus. Analogously, every Riemann surface can be constructed from some fundamental 4g-gon on the H-plane (g'1), see, e.g. Figs. 13 and 14 for g"2 surfaces. Whence, any Riemann surface is just a quotient H/C where C is some generator of discrete translations in H-plane (Buser, 1992). The genus g of the surface is directly connected with the number of punctures in the plane and this fact is completely independent of whether these punctures are frozen or not. Moreover, the hyperbolicity will remain even if we remove the restriction that the motion should be strictly planar: because tubes are entangled in three dimensions (Kholodenko and Vilgis, 1994), and form quasi-knotted configurations, Brownian motion in the presence of such quasiknots will remain hyperbolic (Thurston, 1979). To understand intuitively how this happens we refer the reader to the Appendix. For the moment, let us consider again the simplest Arnold’s billiard which is just a union of two punctured toruses glued along the circumference of the punctures, see, e.g. Fig. 12. To construct such a billiard we need two copies of the Riemann sphere each having three holes. We can glue together two holes on each sphere thus converting it into punctured torus and, then, we can glue the resulting objects together to make the final product. It can be shown (Buser, 1992) that every Riemann surface of genus g'1 is just a collection of thrice punctured spheres along with the gluing prescription, which is used for their assembly. Once we recognized this fact, we can construct a finite square lattice made of m2 copies of the Arnold square. By gluing these squares together it is possible to insert yet another set of k holes into this lattice (Buser, 1992), thus forming a surface of genus g"1#1(m2#k) , 2

(8.46)

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Fig. 14. Two punctured tori in planar representation: (a) can be transformed into (b) and, then, glued together along c , 1 and c thus resulting in the octagon (c). 2

k"M0,2,2,2mN if m is even or k"M1,3,5,2,2m!1N if m is odd. For m"1 using Eq. (8.46) we obtain g"2 in agreement with (Arnold 1978). Let, as in Section 8.2, oL "n /A, then, the logic of the previous discussion suggests us to choose 5 n "m2#k (8.47) 5 so that the filling fraction l can be defined now as (Kholodenko and Vilgis, 1994), l"pa2oL , i.e. we can think of cross sections of tubes as a planar gas of disk of area pa2. Suppose that there is some sort of interaction between such disks. Then, by analogy with other models of statistical mechanics, it is natural to expect that the system of such disks can undergo a phase transition (e.g. solid—liquid-like) which is controlled by l so that for some critical value l"l* we would have l*"pa2(o*)oL (o*) ,

(8.48)

where o* is the critical monomer density (recall, that o&N/»). The explicit dependence of a and oL on o is unknown in general (but, since oL "n /A, it is expected that oL &o) and should be 5 dependent upon the details of the model which is used.

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Since the punctures in our plane have finite sizes, and since Eq. (8.37) accounts for the curvature effects only, we need to complicate this equation so that it can account for the finite sizes of cross sections. To this purpose we begin with planar case discussed in Sections 6.4 and 8.2. In particular, using Eq. (6.55) we can formally define the averaged size of our tube cross section as Sq2#q2T"l/w,a2 . (8.49) 2 3 This definition is in accord with that given in Doi and Edwards (1978) (with S2T being the usual polymer average, see, e.g. Eq. (8.89) below). By continuously changing w we obtain continuous changes in a2. This will no longer be true if the above average is considered in the multiply connected plane which is effectively the Riemann surface. To demonstrate this, we would like to reobtain Eq. (8.49) in a more systematic way which was outlined in Section 6.4. Using Eq. (6.71) we obtain in the limit of small Dp the following result for SAT: SATK 1 Dp(Nl)2 . (8.50) 12 For a circle of circumference N the area is N2/4p. It is the maximal area which can be enclosed by the walk of length N. Whence, one can rewrite Eq. (8.50) in equivalent form as SAT Ka2 , Dpl2

(8.51)

where on the r.h.s. the area a2 is identified with that given in Eq. (8.50). Since the smallest area SAT of the circle cannot be smaller than Kl2, then, evidently, in this extreme case we would have 1/DpKa2 .

(8.52)

Since [Dp]"[A~1] while [w]"[l~1] we, indeed, have reobtained Eq. (8.49). Use of the area (or magnetic language) formalism to determine the tube cross section is more advantageous, as compared with Eq. (8.49), since it allows to consider problems related to random walks with the area constraint on the Riemann surfaces. To this purpose, the following key observation is helpful (Kholodenko, 1996a,b). The probability P(A, N) for a random walk to enclose an area A is given by the ratio P(A, N)"Z(A, N)/Z(0, N) ,

(8.53)

where Z(A, N) is given by

P

Z(A, N)" dr G(r "r "r, NDA) 1 2

(8.54)

with G(r "r "r, NDA) being given by the r.h.s. of Eq. (6.57) (divided by pNl) and Z(0, N) is the 1 2 same but with A"0. The r.h.s. of Eq. (8.54) is just the usual statistical mechanical partition function (Feynman, 1972). Whence, by analogy with Section 7.2, to obtain Z(A, N) we need to know only the eigenvalues (and their degeneracies) of the corresponding Schro¨dinger-like operators. The spectrum of such operators on the Riemann surfaces can be also obtained where the partition function Z(A, N) is known in mathematical literature as Selberg’s trace formula (Buser, 1992).

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Let us consider this topic in some detail. Since in the flat case the spectrum of Landau “electron” e is known to be just that for the harmonic oscillator (Kholodenko, 1996b), n e "Dpl(n#1) (8.55) n 2 with degeneracy g "DpNl/4, the partition function is easily obtainable n x = , (8.56) Z(Dp, N)" + g e~Nen" n sin x n/0 where x"DpNl/2. The function Z(Dp, N) is related to Z(A, N) via

P

Z(Dp, N)"

N2@4p

dA eADpZ(A, N)

(8.57)

0 and is even more convenient since

(8.58) SAT lim (­/­Dp)ln Z(Dp,N) . D p?0 Generalization of the result of Eq. (8.56) to the Riemann surface of genus g can be now accomplished without any problems with the result g!1 Z(Dp, N)" + (2DpNlR2!2n!1) expM!e NN n 4R2 1 0yny@b@~2

(8.59)

with

C

A

BD

1 1 1 2 !b2! n# !b e" n 4R2 4 2

(8.60)

and b"DpNlR2, R2"s2/l2 with s being an average distance between the obstacles in the plane. As it was noticed in Kholodenko (1996b) the parameter R plays the role of a radius of curvature of the manifold: for R2PR (flat case) one obtains: DpNl Z(Dp, N)+ sin (DplN/2)

(8.61)

which effectively coincides with Eq. (8.56). But for finite R@s one has to require that the partition function Z(Dp, N) remains nonnegative and well defined. The nonnegativity of Z(Dp, N) requires 2DpNlR2!2n!150

(8.62)

while for the sum in Eq. (8.59) to be well defined we have to require as well (8.63) DbD!1!n50 . 2 Taking into account the definition of b given after Eq. (8.60) we conclude that both inequalities Eqs. (8.62) and (8.63) are equivalent. In particular, the inequality (8.63) implies that the reduced “magnetic field” DbD should exceed a certain threshold, in our case, DbD51 2

(8.64)

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345

in order for the tube to exist. Moreover, the theory which produced the result given by Eq. (8.59) dictates, yet another set of constraints:

A

a JB2!4DcD "arctanh lR B

B

,

(8.65)

b!2R2Jc"n#1 , (8.66) 2 where B"b/R2. The constant DcD can be eliminated from these equations thus producing tanh

AB S

S

a n#1/2 2b!n!1/2 " . lR b b

(8.67)

If DcD"0 in Eq. (8.65) we obtain tanh (a/(lR))"1

(8.68)

which leads to the requirement aPR for fixed R. In this case the tube does not exist. Hence, for the tube to exist one must require DcDO0 and (a/lR)41. A crude estimate for b can now be obtained from the following self-consistent equation for b which follows from Eq. (8.67) (for n"0) (8.69) b tanh 1KJb!1 , 4 e.g. it is assumed that a+R (or the size of the tube is of the order of the distance between the obstacles). Numerical solution of Eq. (8.69) produces bK0.86(1$0.643). From the theory of the operators on Riemann surfaces (Kholodenko, 1996b), it is known that f bK , 2(g!1)

(8.70)

where f"0,$1,$2,2 and g!1 is given by Eqs. (8.46) and (8.47). By combining these equations we obtain f bK "0.86(1$0.643) . n 5 If A"fpa2, then oL "n /A can be rewritten as 5 poL a2Kn /f . 5 By combining this result with Eqs. (8.49) and (8.71) produces l*K0.708 .

(8.71)

(8.72)

(8.73)

This result is too high as compared to the estimate l*K0.0286 which was obtained in Kholodenko and Vilgis (1994) with help of other methods to be discussed below. To improve the above estimate we can, e.g., require, by analogy with the theory of coil—globule transitions (Kholodenko and Freed, 1984a), that in addition to n"0 (i.e. to the ground state) there is at least one more discrete state, e.g. n"1. Using Eq. (8.67) and repeating previous calculations, produces l*K0.341 .

(8.74)

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This result is considerably in better agreement with the earlier obtained as we shall demonstrate now by direct comparison of this result with the experimental data. To do so, we need to recall some basic facts from chemistry. Let n be the number of moles, then the total number of “particles” (polymers) NI is given by NI "nN where N is Avogadro’s number. A A Let ¼ be the total weight (mass) of polymer(s) of molecular weight M, then n"¼/M while the density o is defined by o"¼/» where », as before, is the total volume. Let » "AK SR2T3@2 be the 0 ' volume occupied by a single polymer chain. Here AK is some unknown constant, of order unity according to Fetters et al. (1994), and SR2T is related to M via ' SR2T"cL M , (8.75) ' where SR2T"1SR2T and cL is some proportionality factor. In writing Eq. (8.75) it is being assumed ' 6 that the individual chains in the melt are effectively at h point conditions (Fetters et al., 1994) Consider now a combination » oN /M,NI /(»/» )"cJ . By construction, this combination is just 0 A 0 a fraction cJ of “lattice sites” (of total number »/» ) which are occupied by polymers. Evidently, 0 04cJ 41. If, following Fetters et al. (1994), we assume cJ +1 (i.e. polymer melt), then we obtain 1KAK 6~3@2M1@2cL 3@2N o . (8.76) A If M is the molecular weight of the segment of polymer chain between the entanglements, then % Eq. (8.76) produces M "o~263(AK cL 3@2)~2N~2 . % A Using Eq. (8.75) we can eliminate the constant cL from Eq. (8.77) thus producing

AT UB

M Ko~2AK ~2 %

R2 M

~3 63N~2. A

(8.77)

(8.78)

In view of Eq. (8.75), it is reasonable to assume that a2KcL M , which then produces % SR2T a2 " (8.79) M M % in agreement with Fetters et al. (1994). Alternative expression was found by He and Porter (1992), who obtain instead M SR2T"28pa2M. The numerical factor of 28p cannot be further checked % using the data from Fetters et al. (1994) and, whence, we shall use the result of Eq. (8.79) in order to produce the final numbers. By combining Eqs. (8.78) and (8.79) we can estimate a as

S

A B

62@3 SR2T M MK . (8.80) % AK oN SR2T M A The last result is in complete accord with Eq. (3.3) of Fetters et al. (1994). According to this reference, Eqs. (8.78) and (8.80) could be used for the independent measurements of M and % a provided that o and SR2T/M are known. Let us now have another look at these results in the light of the discussion presented earlier in this section. Using Eqs. (8.79) and (8.80) we obtain aK

A B

62@3 M % aK AK oN a2 A

(8.81)

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Table 1 Molecular characteristics of polymers at ¹"413 K Polymer

o (g cm~3)

PE PEO PEB-11.7

0.784 1.064 0.793

PMMA PEE

1.13 0.807

a"d5/2 (A_ )

AK

828 1624 1815

16.4 18.75 21.3

10013 11084

33.5 39.3

5.865 5.672 5.802 1 5.77 5.543

M%

Table 2 Molecular characteristics of polymers at ¹"298 K Polymer

o (g cm3)

M%

a"d5/2 (A_ )

AK

PEB-14 HHPP PEE PMA 65-MYRC

0.860 0.878 0.866 1.11 0.891

1522 3347 9536 8801 24874

18.45 24.2 35.0 30.35 44.05

6.903 6.59 6.29 6.94 8.00

or, equivalently, N a3o 1 A " . (8.82) AK 63@2M % This result can be compared now against Eq. (8.48) which can now be equivalently rewritten as pa3(oL /a)"l .

(8.83)

Taking into account the definitions of oL and o we can now identify oN /63@2M with oL /a and, A % hence, l/p with AK ~1. The theoretically obtained l* given by Eq. (8.74) can be used now to obtain AK K9.21. This result can be compared against the experimental data of Fetters et al. (1994). Based on the data from Tables 1 and 2 of Fetters et al. the Tables 1 and 2 of Kholodenko (1996c) are reproduced here (in units and notations used by Fetters et al., 1994). In calculating AK with help of Eq. (8.82) the conversion factor coming from the combination N o/63@2 is estimated to be 0.04082, based on N "6]1023, 1 A_ "10~8 cm. Also, the tube A A diameter d "2a since a is the tube radius. The results of Tables 1 and 2 are in good agreement with 5 our theoretical estimate AK "9.21 based on Eq. (8.74). In addition, by combining Eqs. (8.75) and (8.80) we can also write ao"const.

(8.84)

This result is obtained theoretically in Kholodenko and Vilgis (1994) using the analogy with quantum Hall effect (QHE) formalism. Independently, the same result was obtained by Kavasalis and Noolandi (1988) based on the packing model of reptation. The result of Eq. (8.84) is supported

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by numerical simulations of Wittmer and Binder (1992) and Kremer and Grest (1994). Eq. (8.84) is based on the assumption that the combination SR2T/M is constant. Theoretical calculations performed by Lachowski et al. (1988) and Hutnik et al. (1991) indicate that this should be the case and provide for the above ratio the result: SR2T/M+1.03—1.1. This result is only in qualitative agreement with the data from Tables 1 and 2 of Fetters et al. (1994) as is acknowledged by the authors. Accordingly, use of the data given in Tables 1 and 2 of this work for computation of the product oa leads to less satisfactory results for AK : the results for AK are uniformly smaller (roughly by a factor of 6) than those given in Tables 1 and 2. This is not too surprising since the result given by Eq. (8.82) was obtained without restrictions on the ratio to be fixed and universal. Since the independent numerical data of Wittmer and Binder (1992) and Kremer and Grest (1994) and the theory of Kholodenko and Vilgis (1994) strongly support Eq. (8.84), we would like to present these theoretical arguments in favor of Eq. (8.84) in Section 8.4. 8.4. Statistical mechanics of planar rings in an array of obstacles (QHE approach) In Appendix A.1 we have discussed complications which arise from considering the Brownian motion at the twice punctured plane as compared to the well understood one puncture case discussed in Section 5.1. Here, we would like to develop the results of Appendix A.1 in order to illuminate some additional physical aspects of the whole problem. In physics literature, study of path integrals in multiply connected spaces was initiated to our knowledge by Shulman (1971) and later developed by many authors, see, e.g. Levay et al. (1996) and references therein. In the mathematics literature, the same problem was studied by Pitman and Yor (1986, 1989) who use methods which are noticeably different from that used in physics literature. It would be interesting to make a detailed comparison of these approaches in the future. The most typical (hydrogen atom-like) problem which is well studied is the problem related to the quantum mechanics of the particle on a circle which we had discussed in Section 6.2. The key idea of solving the circular problem lies in realizing the fact that the universal covering space for a circle S1 is just a straight line R1. Since the path integral for R1 is well known, then the path integral for S1 can be obtained by some sequence of operations leading from S1 to R1 and back to S1 (Tanimura and Tsutsui, 1995) (very much in accord with the results of Appendix A.1). Now, if we have a hole in the plane, the closed paths around a hole are homotopic to S1 (Fulton, 1995). If we would have some interaction between the Brownian particle and the hole (which could be just a world line of another particle), then this would be equivalent to having fractional statistics (with the strength of interaction d interpolating between the Bose and the Fermi statistics as it was explained in Section 6.2). Let us now have two holes instead, then we have to consider instead of S1 a product S1]S1 as depicted in Fig. 15. The universal covering space for the “figure eight” is known to be (see, e.g. Dubrovin et al., 1985), a four-valent Bethe lattice as depicted in Fig. 16. This explains why, e.g. Nechaev et al. (1987) and others had used a Bethe lattice to study the reptation. The “figure eight” can be also obtained by considering paths on the once punctured torus (which was discussed earlier in connection with Arnold’s billiard) as depicted in Fig. 17. If we make a puncture in a sphere S2 and glue two copies of S2 together, the result will be S2 as depicted in Fig. 12a, but if we glue two punctured tori together, as depicted in Fig. 12b, we shall obtain a surface of genus 2. At the same time, if we think about the torus as a square with sides properly identified, then the punctured torus will look like that in

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Fig. 15. Homotopy of paths on the twice punctured plane (“figure eight”).

Fig. 16. Universal covering space of the “figure eight”.

Fig. 13. If we glue together these polygons as depicted in Fig. 14 we shall obtain a double torus in the planar representation. There are four distinct paths on this torus as depicted in Fig. 18b so that if we make cuts along these paths we re-obtain Fig. 14c. At the same time, if we would think of homotopy of these paths, we would obtain a bouquet of four circles (instead of two as in Fig. 15). Two out of our four circles had originated from the

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Fig. 17. Topological structure of the once punctured torus.

Fig. 18. If the sides of the octagon are glued together in an order shown in (a), then the resulting surface (b) coincides with that depicted in Fig. 12b. Alternatively, if the cuts are made along a , a and b , b on the double torus, we will 1 2 1 2 obtain again Fig. 14c.

periodic boundary conditions (see, e.g. Fig. 14a) and were left unaccounted in Fig. 15. Evidently, the Bethe lattice structure of Fig. 16 becomes more complicated when the surfaces of higher genus are being considered. But, in any case, the Bethe lattice calculations, e.g. like that discussed in Nechaev (1990), are effectively calculations on the universal covering surface for the Riemann surface of given genus (Stillwell, 1993), so that S1PR1PS1 calculational procedure for the circle is replaced now by the H/CP¹PH/C where ¹ is the corresponding Bethe lattice. The Bethe lattice calculations are not readily extendable to account for the “magnetic field” effects and, hence, the

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351

Riemann surface approach described in the previous subsection is more advantageous. Moreover, the QHE picture of tube stability developed in Kholodenko and Vilgis (1994) is in complete accord with the Riemann surface approach as we are going to demonstrate shortly. The QHE model of tube stability is based on the following observations. According to Section 6.4 the Landau Hamiltonian HK for an “electron” of “mass” m placed in the “magnetic” field H"e]A is known to be (Sondheimer and Wilson, 1951), 1 1 A2 HK " + 2# A ) e # , r r 2m mi 2m

(8.85)

so that the Bloch equation for the density matrix o can be written as !(­/­b)o"HK o

(8.86)

provided that o(r, r@; bP0)"d(r!r@). Using results of Sections 6.3 and 6.4 the last equation can be equivalently rewritten as

A

A

B

B

1 H ­ ­ H2 ­ ! + 2! x !y # (x2#y2) o"0 . r 2mi ­y Lx 8m ­b 2m

(8.87)

The corresponding polymer problem is obtained by the following replacements: b¢N,

1 l H2 w2 ¢ , ¢ , 2m 6 8m 6

and, if one considers only the states with the total angular monumentum zero, then the last equation coincides exactly with Eq. (6.55) while the partition function Z, given by

P

(8.88)

P

(8.89)

Z" dr o(r"r@"r; b)

coincides with that given by Eq. (8.54) (with obvious redefinitions of w or Dp). If W (r) is the n eigenfunction of the Schro¨dinger-like operator given by Eq. (8.87), then the size of the tube can be estimated according to Eq. (8.49) as a2" d2z [W (z, zN )]2DzD2 , 0

where use was made of the planarity of the magnetic Schro¨dinger problem which allows us to introduce complex variables z"x#iy and zN "x!iy so that, upon rewriting the whole problem in terms of z and zN , one obtains for the lowest Landau level wave function W (z, zN ) the following 0 result:

A

w W (z, zN )"NI exp ! DzD2 0 l

B

(8.90)

with NI being a normalization constant. The validity of the approximation for a2 rests on the assumption that for large N’s the density matrix o can be approximated by o(r, r@; N)Ke~e0N W* (r)W (r@) . 0 0

(8.91)

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The presence of holes (other polymers) in the plane can be accounted for by introducing the mutual winding number h constraint into the corresponding path integrals. In the absence of the ij “magnetic” field the functional integral for an assembly of topological interacting planar Brownian walks is given by

P

G P C

DH

n5 N n5 1 U n5 d G" < D[r(q )]exp ! dq + rQ 2#i + h(r (q)) i l i 2p dq ij 0 i/1 i/1 i:j where

A B

y(q) , h(r (q))"tan~1 ij x(q)

,

(8.92)

(8.93)

n was defined e.g. in Eq. (8.47) and U is some constant which is related to the filling fraction l, 5 defined before Eq. (8.48) (for details, please, consult Kholodenko and Vilgis, 1994) as l"4p/U .

(8.94)

The result given by Eq. (8.92) should be compared now against earlier discussed Eq. (6.9). The presence of the “magnetic” field which describes the tube(s) cross section(s) can be accounted easily now by analogy with Eq. (6.67). At the same time, although at the classical level the interaction term in the exponent of Eq. (8.92) is the total “time” derivative and, hence, can be discarded, it cannot be ignored at the quantum level as we have explained in Section 6.1. For U"0 the total Hamiltonian HK is just a collection of single particle Hamiltonians, i.e.

A

B AG

HB

n5 1 ­ 1 ­ HK " + HK r , ,H r , . i i ­r i i ­r i i i/1 Topological interactions change HK into

AG

(8.95)

HB

­ U r , ! + +rih(r !r ) (8.96) i ­r i j 4p i iEj so that the Schro¨dinger-like equation (with or without magnetic field) can be written now as HK "H 5

­ W(Mr N, t)"HK (Mr N, t)W . i 5 i ­t

(8.97)

Elementary examples of the above procedure were discussed in Section 6.1. Eq. (8.97) by design assumes that all “particles” are moving in the same “time” t (in case of polymers N). In the theory of Brownian motion there is no need, however, to make such an assumption (McKean, 1969). In case of polymers this was recognized by des Cloizeaux and Jannink (1990). The sychronized “time” is used in the theory of directed polymers (Kardar and Zhang, 1987), without explicitly acknowledging this fact (e.g. see also Blatter et al., 1994). Use of one “time” (instead of many) is equivalent of saying that all polymers are of the same length and are indistinguishable. des Cloizeaux and Jannink (1990) had carefully analyzed this issue for polymers and found that this assumption may sometimes lead to wrong results. The indistinguishability is also closely associated with statistics as we have demonstrated in Section 6.2. Extension of the “anionic philosophy” to the case of distinguishable particles was recently made by Liguori and Mintchev (1995) and Isakov et al.

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353

(1995). The Riemann surface approach presented in Section 8.3 does not require these assumptions and, hence, is more consistent with the traditional language used in polymer physics, see, e.g. des Cloizeaux and Jannink (1990). Nevertheless, the time peculiarity just described is not a major stumbling block towards obtaining physically meaningful results in the present case. Indeed, if instead of the correct single valued function W in Eq. (8.97) we would use the multi-valued functions

G

H

U (8.98) WI (Mr N,t)"exp ! + h(r !r ) W(Mr N, t) i j i i 2p i:j then this equation will be replaced by an equivalent single-particle Schro¨dinger equation for WI ­ WI "HK WI , ­t

(8.99)

where HK is given by Eq. (8.95). The reader is referred again to Section 6.1 for illustrative elementary example of such transformation. Use of complex variables z and zN and ground state dominance assumption, Eq. (8.91), allow us to write the many-body wave function WI for our Landau-like problem in the form

G

H

w n5 (8.100) WI (Mz N, MzN N)"NI < (z !z )U@2p exp ! + Dz D2 . i j i 0 i i l i:j i For U"0 we obtain back the product of Landau wavefunctions (see, e.g. Eq. (8.90)), while for nonzero U we obtain, instead of Eq. (8.89), the following result for a2:

P

a2" d2z DzD2oL (z, zN ) */5

(8.101)

with

P

n5 (8.102) oL (z, zN )" < d2z DWI (Mz N,MzN N)D2 . i 0 i i */5 i/2 The combined use of Eqs. (8.100), (8.101) and (8.102) reduces the problem of computation of a2 to the calculation of the classical statistical mechanical average

P

1 n5 a2" < d2z Dz D2 expM!HI [z, zN ]N , i 1 Z i/1 where, in view of Eq. (8.94), we have

(8.103)

2w !4 + ln Dz !z D# + Dz D2 (8.104) HI [z, zN ]" i j i l l i:j i and Z is a normalization constant (partition function). The Hamiltonian HI is known in the literature as describing the one-component plasma (OCP), (Caillol et al., 1982), while the wave function of Eq. (8.100) is known as Laughlin wave function (Laughlin, 1983), used in the theory of quantum Hall effect (QHE). For n <1 one can try to 5

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calculate a2 in Eq. (8.103) by using the saddle point method. By introducing the new variable x"z/Jn the Hamiltonian HI can be rewritten now as 5 HI [x,xN ] m "+ Dx D2! + ln Dx !x D2 , (8.105) i i j n 2n 5 5 i:j i where m"4/l. Minimization of Eq. (8.103) produces the following saddle point equations: m 1 xN " + , (8.106) i n x !x 5 j i j (iEj) 1 m . (8.107) x" + i n xN !xN i j 5 j (iEj) To solve these equations, we need to multiply Eq. (8.106) by x while Eq. (8.107) by xN . By writing i i 2pik x "R exp , (8.108) k n 5 i.e. by assuming that the “particles” are located on concentric rings of radius R, we obtain,

G H

m m n !1 m 2 1 R2" + " 5 K " , (8.109) n 2 l 1!expM2pij/n N n 2 5 j 5 5 where in the second line the sum rule was used (Kogan et al., 1992) to arrive at the final result. Using Eq. (8.109) the average distance SrT between “particles” can be calculated according to equation 2pR SrTK "(2p/n )J2/l . 5 n 5 Using back z-variable (instead of x) the above result can be written now as

(8.110)

(8.111) SrT"2pJ(2/ln )a . 5 Since we can always write SrT"const. a (where const.52), we obtain using Eq. (8.111) the following result: ln "const@ . 5 or, in view of Eq. (8.48),

(8.112)

pa2oL n "const@ . 5 Since oL &n , see, e.g. Eq. (8.48), we obtain 5 ao"const@@ .

(8.113)

and this result coincides with Eq. (8.84)!

(8.114)

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The equation of state for OCP is known to be (Hague and Hemmer, 1971) P"o(k ¹!g2/4) . (8.115) B Then, for ¹'¹ , where ¹ "g2/4k , the system is in a “gas” phase while for ¹(¹ it is in # # B # a liquid phase. For ¹(¹ the pressure P in the above equation becomes negative and the system # undergoes gas—liquid transition. We have shown in Kholodenko and Vilgis (1994), that for our polymer problem g2"U/p and l"4p/U. Caillol et al. (1982) show, by using the Monte Carlo methods, that the OCP can also undergo yet another transition: from “liquid” to “solid” phase. This happens for l*+0.0286 to be compared with our earlier result, Eq. (8.74). Evidently, the difference in these critical values could be understood using the following physical arguments. The result, Eq. (8.74), is related to the onset of tube formation while the result l*"0.0286 is related to the regime when the tube already exists and is well defined. Comparison between Eq. (8.73) (obtained for n"0 in Eqs. (8.67) and (8.74)) ‘(obtained for n"1 in Eq. (8.67)) indicates that for large enough n we can expect the result for l* coming from OCP calculations. The above qualitative arguments can be made mathematically more precise. This is accomplished in Section 8.5. 8.5. Connections with theories of quantum chaos In this section we want to demonstrate that QHE and Riemann surface approaches to reptation are not only interconnected but also could be viewed from the broader angle provided by theories of quantum chaos and quantum mesoscopic systems. Following Kholodenko (1996b), let us take another look at Eq. (8.37). By using the conformal transformation w"(z!i)/(z#i) , z3H ,

(8.116)

where H was defined in Eq. (8.41), the metric, Eq. (8.42), can be transformed into 4((du)2#(dv)2) ds2" , [1!(u2#v2)]2

(8.117)

where w"u#iv. The above metric converts the Poincare H-plane model into the Lobachevski unit circle model. Evidently, both descriptions are equivalent (Stillwell, 1992), but use of the unit circle formulation allows us to obtain some additional information a bit easier. To this purpose let now u"sinh h cos u, v"sinh h sin u and r"tanh (h/2) where 04h(R, 0(u42p. In terms of such parametrization Eq. (8.37) can be rewritten now as 1+ 2 P(r, u; q)"[1/(1!r2)] (­/­q)P(r, u; q) 4 r,r where ­2 1­ 1 ­2 +2 " # # . r,r ­r2 r ­r r2 ­u2

(8.118)

(8.119)

Eq. (8.118) coincides with Eq. (3.6) of Nechaev (1988) (if the last one is rewritten in dimensionless units). Notice that if instead of r we would use the original h-variable, then Eq. (8.118) would

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acquire the following form (which we shall use in subsequent discussion):

A

A

B

B

­ 1 ­ 1 ­ ­2 sinh # P(h, u; q)" P(h, u; q) . ­h sinh2h ­h2 ­q sinh h ­h

(8.120)

The resulting equation looks almost familiar: if instead of the sinh function we would use the sin function the above equation would describe the Brownian motion on a sphere. In the present case, however, we are dealing with the case of pseudosphere. As in Section 8.3, we assume that the distribution function P is u-independent. Then Eq. (8.120) can be converted into

A

¸K P(x!x@; q),

B

d ­ d2 #coth2x P(x!x@; q)" P(x!x@; q) , dx ­q dx2

(8.121)

where x"h. Evidently the distribution function P(x,q) can be found if the eigenvalue problem for the operator ¸K is solved. In connection with this eigenvalue problem, let us consider a seemingly unrelated eigenvalue problem related to the particle in a potential g2 sinh~2x with g being some adjustable constant. This eigenvalue problem can be formulated, as usual, as

C

D

1 d2 HK u " ! #g2 sinh~2x u "Eu . n n n 2 dx2

(8.122)

It can be shown (Olshanetsky and Perelomov, 1985) that the function u "(sinh x)k (8.123) 0 is the solution of Eq. (8.122) with the eigenvalue E "!k2/2 provided that k(k!1)"2g2. This 0 function is not normalizable however, since it is increasing for xPR. Hence, E does not belong 0 to the spectrum of the operator HK . Upon substitution of u "u / into Eq. (8.122), it is converted n 0 n into the equation ¸K / "!(k2#n2)/ , (8.124) n n where the operator ¸K is the same as in Eq. (8.121). Hence, Eqs. (8.121) and (8.124) have the same eigenfunctions. The one-body Hamiltonian HK defined by Eq. (8.122) can be easily generalized to the many-body case and is known in the literature as Calogero—Sutherland (CS) Hamiltonian (Felder and Veselov, 1994). It is given by n 1 d2 n u HK "! + #a + , (8.125) CS 2 dx2 sinh2(x !x )u i j i i/1 i:j where n is the number of “particles” in the (one-dimensional) system and a and u are some constants. For uP0, Eq. (8.125) is known as Calogero Hamiltonian. To use the CS Hamiltonian in the reptation problem, several issues need to be resolved. First, Eq. (8.120) describes the transversal part of the “motion” of an individual primitive chain (so far in the absence of the “magnetic field”). For noninteracting chains the total Hamiltonian should be evidently just a sum of one-body Hamiltonians. Therefore, naively, the total Hamiltonian HK T should look like

C

D

n 1 d2 g2 HK " + ! # #HK . CS */5 2 dx2 sinh2x i i i/1

(8.126)

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Eq. (8.92) suggests that the nature of interaction between tubes is topological so that HK cannot */5 have a small parameter and cannot be considered as just a small perturbation. The eigenvalue problem for CS Hamiltonian can be solved exactly (Cherednik, 1994). As was noticed already by Sutherland (1985), to find the eigenfunctions and eigenvalues of HK is sufficient to know only the CS two-body scattering phase shift. Because the two-body problem is always reduceable to the one-body problem, we are essentially coming back to Eq. (8.122) (and, hence, to Eq. (8.121)!). Mathematicians have demonstrated rigorously that the wave functions of the CS and Knizhnik—Zamolodchikov (KZ) equations are practically the same. But in Kholodenko—Vilgis (1994) it was demonstrated that the Laughlin wave function, given by Eq. (8.100), obeys the KZ equation. Hence, we are left with the conclusion that the CS Hamiltonian (perhaps, with an extra quadratic term to account for the “magnetic field”, (see, e.g. Azuma and Ito, 1994) can be also used to describe the interacting tube model which was discussed in Section 8.4. Hence, the Riemann surface and the QHE models of reptation are effectively isomorphic to each other. Since the CS Hamiltonian is widely used in the theory of chaotic/disordered systems, see, e.g. Mucciolo et al. (1994), Beenakker and Rejaei (1994), etc., one can think of reptation as yet another illustration of the universality of the description of chaotic systems based on CS-type models. 8.6. Connections with theories of mesoscopic systems The formalism developed above closely resembles that developed to describe the conductivity in quasi-one-dimensional metallic wires (Datta, 1995). This theory is known to produce an additional effect, e.g. quantization of conductance: when the cross section of a quasi-one-dimensional wire changes continuously the conductivity changes discontinuously (see, e.g. Figs. 1 and 2 of Jascual et al. (1995) and Fig. 46 of Beenaker and van Houten (1991)). In addition, the same discontinuity effect can be achieved by continuously varying the voltage between the ends of the wire (see, e.g. Fig. 44 of Beenakker and van Houten (1991)). In Kholodenko (1996b) and Kholodenko and Vilgis (1994) it was emphasized that, “although tubes can apparently appear and disappear, at time scales shorter than the terminal relaxation time q &N3.4 (see, e.g. Section 2) the melt of flexible polymers could be viewed as porous continuum T with tubes (pores) being randomly distributed in it, Teraoka et al. (1992)”. The importance of this point of view was recently emphasized by Milchev and Binder (1994) while Krupenkin and Taylor (1995a,b) had considered a “motion” of polymers through pores using ideas similar to that discussed in Section 6.3. The quantization of force—extension relation ( just like quantization of conductance) was discussed in Section 6.1 and is in qualitative agreement with the experimental data of Cluzel et al. (1996) as discussed also in Section 6.1. In the case of polyelectrolytes, the electric field is used in gel electrophoresis to separate molecules of different sizes. It would be very interesting to develop closer links between the theory of gel electrophoresis (Rubinstein, 1987; Duke, 1989; Prahofer and Spohn, 1996; etc.) and the theory of electronic conduction in mesoscopic systems. The asymmetry of descriptions of the longitudinal and transversal motions of the polymer along the tube leads to a controversy which is also known in the theory of the QHE. Specifically, if we modify Eq. (8.86) by adding some sort of random potential »K to the Hamiltonian defined by Eq. (8.85), then, even in the presence of weak disorder, all electronic states are localized in two dimensions (i.e. “conductivity” is zero in QHE language). Once the magnetic field is turned on, the

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conductivity reappears. Moreover, the value of Hall conductance is independent of the degree of disorder. The shape of the conductance curve for the case of QHE is similar to the shape of conductance curve for quasi-one-dimensional conductors discussed above. Using polymer language, the above situation can be restated as follows: In the absence of an area constraint, i.e. *p"0, in Eq. (6.70) the propagator of Eq. (8.45) produces the following result for the averaged radius of gyration (Nechaev et al., 1987), z J2p SR2TK lJN ' z!2 8

(8.127)

which is in complete accord with our result in Eq. (8.32). Incidentally, the same result was obtained by de Gennes (1971) (e.g. see Section 4 of this paper) by completely different methods. If time t is associated with N (as de Gennes did), then the diffusion coefficient D can be estimated as D&dSR2T/dN&1/JN. Because the diffusion and conductivity are related via Einstein relation ' (Kholodenko, 1985), it is obvious that the result given by Eq. (8.127) leads to the localization (i.e. absence of conductivity in QHE language). At the same time, according to the main postulates of reptation theory (Doi and Edwards, 1986), the statistics of the primitive path (chain) for large N should be Gaussian-like, i.e. “conduction” should take place. In Section 6.3 we have discussed the longitudinal part of primitive chain “motion” and had indicated that resolution of the localization paradox lies in the assumption that the motion is facilitated by the existence of tube domains which are almost linear (rigid-rod-like). Alternatively, if instead of initially fully flexible (Gaussian) chains, the Dirac (semi-flexible) chains are used (Kholodenko, 1990, 1995), then the localization may be prevented. In the context of QHE this was recently demonstrated by Ludwig et al. (1994). If the Dirac chains are used, then the results of Section 8.3 should be reanalyzed since one should study in this case spinors and Dirac-type equations on the Riemann surfaces, see, e.g. Gilkey (1995). These problems are similar so that studied in the theory of superstings, see, e.g. Green et al. (1987). Finally, we would like to mention that Hess (1988) had provided a very important alternative dynamical treatment of reptation emphasizing the difference between the longitudinal and the transversal parts of motion of the trapped chain. His treatment, however, does not involve topological considerations and these are quite independent of dynamics as we demonstrate in Appendix A.1.

Acknowledgements The authors had benefitted from discussions with many people. Professors Kurt Binder (Univ. Mainz), Kurt Kremer (MPI, Mainz), Harry Frisch (Univ. Albany, NY) were instrumental in providing many useful references related to experimental and numerical results which involve entanglements. Dr. Jack Douglas (NIST) had attracted our attention to works of McKean and collaborators. Professors Hagen Kleinert (Freie Universita¨t Berlin) and Dale Rolfsen (Univ. of British Columbia) provided helpful comments related to the Jones polynominals, Chern—Simons field theories, etc. Professor Louis Kauffman (UIC) had provided us with historical and technical

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references related to applications of knot theory to various physical problems. Generous support of this project by the Max-Planck Gesellschaft and the staff of MPI (Mainz) is also highly appreciated.

Appendix A. A.1. Planar Brownian motion in the presence of two holes McKean and Sullivan (1984) and Lyons and McKean (1984) have studied this problem in some detail. For the case of n punctures (holes) the problem was studied by Pitman and Yor (1986, 1989) (please, see also Yor’s book, Yor (1992)). Since their presentation is highly mathematical, it may not be readily accessible to most of physics educated readers. Hence, we would like to present here only the intuitive arguments. As it was noticed in Lyons and McKean (1984) (Section 2 of their paper), the Brownian motion in the presence of two holes in the plane is essentially the same as the Brownian motion on the thrice punctured sphere. This is so because, as is well known, the sphere is isomorphic to R2XMRN, where MRN is the point at infinity. In Section 8.3 it is argued that the thrice punctured sphere is the main building block of any Riemann surface. The nontrivial case of Brownian motion on the thrice punctured sphere can be readily understood from the point of view of homotopy theory. To this purpose, let us recall the case of just one puncture we have considered in Section 5.1. When we have two (or three) holes (punctures), the single winding number constraint used in Section 5.1 can no longer be used. The above single hole problem can be generalized if we take into account that the winding number constraint can be replaced (via Hubbard—Stratonovich transformation) by an electromagnetic-like field so that effectively, one has to consider the propagator for the nonrelativistic “charged” particle in the presence of the Abelian gauge field. This approach was discussed in Sections 4.2 and 8.2. The fact that the field is Abelian is caused by the group structure of the rotations around the circle which is Abelian. In physics literature the above type of problem is usually associated with the Aharonov—Bohm effect (Kleinert, 1995). This well-developed picture breaks down as soon as we include the second hole. In this case, one has to consider three types of homotopically distinct paths: (a) around the first hole, (b) around the second hole and (c) around both holes (see, e.g. Figs. 15 and 16). Let g , g and g be the generators of the above motions, then, according to 1 2 = McKean and Sullivan (1984), they are not independent since they are subject to the constraint: g g g "1. This constraint makes the homotopy group non-Abelian. Hence, one has to calculate = 2 1 path integrals for the planar Brownian motion of “charged” particles in the presence of the non-Abelian gauge field (Balachandran et al., 1991). Alternatively, following McKean (1969), one can develop a completely different approach by noticing the following. For the case of three punctures it is possible to find the conformal transformation of the interior of the punctured triangle into the upper Poincare half-plane H which was defined in Eq. (8.41). Normally, the conformal transformation will transform the interior of the punctured triangle into H so that the vertices A, B, C of the original triangle will be transformed into the points a, b, c on the x axis of the H model. Since, however, we are interested not only in the interior of the triangle but in the mapping of the entire twice (or thrice) punctured complex plane into H, the standard approach via

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Fig. 19. Sequence of operations leading from R3CMno. of linesN to R2CMa1,2,an5N.

Schwarz—Christofel conformal transformation (Nechaev, 1988) cannot be used. Following McKean (1969), let us reconsider the once punctured plane results. McKean argues that, instead of considering the Brownian motion on the punctured plane, i.e. on R2CM0N, one can consider the Brownian motion on the Riemann surface (i.e. universal covering surface for R2CM0N). It happens that if z3R2CM0N, then w"ln z defines the desired mapping into the Riemann surface for the logarithm. For the case of the twice punctured plane, the functional mapping analogous to z"exp w is found to be z"j(w), where

A

B

= 1#q2n 8 j(w)"16q < 1#q2n~1 n/1

(A.1)

with q"exp(ipw) and w"u#iv. The above transformation maps the fundamental domain of the H plane (w"u#iv, v'0) so that the entire H plane is covered by translations (tessellations) of this domain as discussed in Section 8.3 into the whole z-plane which contains cuts along the real axis from !R to 0 and from 1 to #R. The inverse function w"j~1(z) is regular in the cut z-plane and maps it conformally to the fundamental domain in the H plane. It can be shown (e.g. Dubrovin et al., 1985) that, homotopically, R3CMlineN is equivalent to R2CM0N. This means that in the case of many lines in R3, the space R3CMno. of linesN is homotopically equivalent to R2CMa ,2, a 5N where a ,2, a 5 are punctures in R2 plane. This can be intuitively understood with 1 n 1 n help of Fig. 19. More details can be found in Stillwell (1993) or Massey (1967). The above results provide the required topological justification for the separation of “motion” of the trapped polymer chain into the transversal and the longitudinal parts as discussed in the main text in Sections 6.3, 8.3 and 8.4. Evidently, these results are valid for both static and dynamic treatments of polymer melts. We have argued in Section 8.3 that the planar Brownian motion in the presence of many holes can be described by the diffusion on the hyperbolic surface of constant negative curvature. We just had seen that this fact remains unchanged in the presence of a set of lines in R3. The question arises: Will the motion in R3 remain hyperbolic if R3 contains some knot? Below we provide some evidence in favor of the hyperbolicity leaving practical applications of this fact outside the scope of this review.

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Fig. 20. Geometry of loops ai around each crossing. Here ai, aj and ai`1 denote arcs which join different neighboring crossings.

Fig. 21. Disentangled paths around each crossing according to the convention explained in the text.

A.2. Spatial Brownian motion in the presence of knots (links) Three-dimensional description of knots (links) can be made very similar to that we have just encountered in Section A.1. Specifically, in the planar case, we have chosen some reference point p3R2 and have encircled each hole with paths which begin and end at p. The same procedure is possible to generalize to R3. One point compactification of R3(or C) produces S3 (as much as one point compactification of R2 (or C) produces S2). So, we embed our knot K into S3 and study homotopy of paths in S3CK with respect to some point p inside S3 which do not belong to K. Let us choose an orientation for K and then encircle each knot crossing by loops a as depicted in Fig. 20. i More advantageous, however, is to disentangle these loops as depicted in Fig. 21. To this purpose the convention must be introduced that, e.g., the lower arc a into the crossing is i followed by the arch a out of the crossing. This leads to the result that for the crossing of type (1) i`1 the curve a a~1a~1 a contracts to the point, hence, producing the relation i j j`1 j a a~1a~1 a "1 , (A.2) i j j`1 j or a a "a a . j i i`1 j

(A.3)

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Fig. 22. Figure eight knot together with the basic loop generators a and b.

Alternatively, for the crossing of type (2) we have a a "a a . (A.4) i j j i`1 Let, in general, Ma N denote a set of loops for K. They are called generators of the knot group G(K). i Every element of G(K) can be written as G(K)"an1an22an55 , (A.5) 1 2 n where each n is a nonzero integer. The type of Eq. (A.2) or Eq. (A.4) are called relations (in general, i they are denoted by r ). A presentation for G(K) consist of finite sets Ma N and Mr N and is usually i i i written as (Gilbert and Porter, 1994) (A.6) G(K)"(a ,2, a 5 D r ,2, r ) . m 1 n i Following Milnor (1982, 1994) and Riley (1975) consider now the figure eight knot as shown in Fig. 22. The generators a and b are subject to a single relation (ab~1a~1b)a"b(ab~1a~1b) .

(A.7)

The generators a and b will correspond to some matrices A and B which must satisfy a single matrix equation. It is found that these matrices belong to the group PS¸(2,C)"S¸(2,C)/M$1N. Recall (Kholodenko, 1996b), that in the case of H-model PS¸(2,R) group was used so that if

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Fig. 23. A representative example of a torus knot.

c3S¸(2,R) and if z3H, then az#b cz" cz#d

(A.8)

with a, b, c, d being real and M$1N factor are 2]2 identity matrices responsible for reflections. The planar H-model, Eq. (8.41), can be easily extended to the R3 model defined by ` R3 M(x, y, t)3R3; z"x#iy"C, t'0N . `

(A.9)

Then, instead of z used in H, one can use z#jt in R3 where ` z#jt"x#iy#jt#k0

(A.10)

is a quaternion. The transformation c now becomes a transformation CK which has the same form as Eq. (A.8) but with complex coefficients. The quotient R3 /CK which is an hyperbolic 3-manifold is ` very much like the H/C which is a hyperbolic 2-manifold discussed in Section 8.3. For an introduction to the theory of 3-manifolds we recommend Meyerhoff (1992). Thurston (1979, 1982) had shown that S3CK has a hyperbolic structure if K is not a torus knot and does not contain satellites. An example of a torus knot is given in Fig. 23 while Fig. 24 illustrates the concept of a satellite knot. From the previous discussion it follows that (a) one can think about knots in terms of their S3CK complements (Gordon and Luecke, 1989); (b) if in the planar case the Aharonov—Bohm effect is a “hydrogen atom” model for all fractional, QHE, etc. features, in the three-dimensional case, the Brownian motion in the presence of a knot K should play the same role. Classical dynamics of such a motion was considered by Goodman (1983) while the study of the Brownian motion in the presence of a knot was initiated in the work of Varopoulos (1985) (please,

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Fig. 24. Some knot and its satellite.

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