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Coherent state path integral and Langevin equation of interacting fermions
Physica A 312 (2002) 431 – 446
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Coherent state path integral and Langevin equation of interacting fermions B. Mieck1 Fachbereich Physik, Universitat-Gesamthochschule Essen, 45117 Essen, Germany Received 16 February 2002
Abstract Interacting fermions, electrons and holes in a semiconductor, are coupled to a thermal reservoir of bosons which yield the -uctuating noise. We use a coherent state path integral formulation on the time contour for non-equilibrium systems in terms of anticommuting variables which replace the fermionic creation- and annihilation operators in the time development operator. An auxiliary commuting 1eld x (tp ), de1ned on the time contour, is introduced by a Hubbard– Stratonovich transformation. In terms of this new 1eld, a Langevin equation is derived which is similar to a saddle-point equation with a random force fx (t). In comparison to the bosonic case of excitons in a semiconductor previously described, one obtains a di4erent expression for the Langevin equation and, especially, a di4erent relation for the probability distribution of the noise term fx (t). The calculation for density matrix elements with the Langevin approach can also be interpreted as an average over modi1ed non-equilibrium Green functions with the appropriately c 2002 Published by Elsevier Science B.V. derived probability distribution of the noise. PACS: 05.40.−a; 05.10.Gg; 02.70.Lq Keywords: Langevin equation; Coherent states; Anticommuting variables
1. Introduction A Langevin equation consists of a relation for the mean kinetics and a -uctuating force term, describing noise of a surrounding medium. This concept has already been applied for Brownian motion [1– 4] where the noise is introduced into Newton’s equation by a random variable. Noise is also important in electronic circuits [5] and 1
Supported by the DFG. E-mail address: [email protected] (B. Mieck).
c 2002 Published by Elsevier Science B.V. 0378-4371/02/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 2 ) 0 0 9 6 2 - 7
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B. Mieck / Physica A 312 (2002) 431 – 446
for the transition of an incoherent system to coherence, as for example in a laser [6 –10] or in a Bose–Einstein condensate [11,12]. In the case of a bosonic system it is rather obvious to identify the classical mean kinetics and the -uctuations caused by a thermal reservoir, as investigated recently by a coherent state path integral [13]. The -uctuations of a complex-numbered coherent state 1eld F x (t), the average of the two 1elds x (t+ ), x (t− ) on the time contour, are driven by an inhomogenous c-numbered quantity which can be generated by a random sequence of numbers. In essence, one obtains the following, classical Langevin equation with a c-numbered 1eld F x (t) and a random force term Fx (t) (compare Eq. (25) in Ref. [13]): t @F x (t) + Fx (t) = dt K(t; x; t ; x )F x (t ) @t t0 1 − ˝
x
1 1 w F ∇x F x (t) + Ex (t) − | (t)|2 F x (t) : ˝ 2V x
(1)
According to the general -uctuation–dissipation theorem, the noise is also accompanied by a dissipative term K(t; x; t ; x ). If we try to transfer this concept to fermions, also coupled to a thermal reservoir (e.g. phonons in a semiconductor), Grassmann-valued, anticommuting variables have inevitably to be introduced in the coherent state path integral on the time contour (for an introduction to fermionic coherent states and their application see Refs. [14 –16]; the time contour for non-equilibrium phenomena is introduced in Refs. [17–23,15]). It is then not obvious how to identify the -uctuating numbers and the random variable which has also to be a Grassmann number. Since anticommuting numbers cannot be computed with a c-numbered random generator, the concept of a classical Langevin equation with a random, numerically de1ned noise breaks down. 2 Therefore, the 1rst part in Ref. [13], the classical Langevin equation with an inhomogenous random number, is not applicable for fermionic systems. However, combinations of fermionic operators can have classical limits, as e.g. the densities and the polarization composed of two Fermi-operators. This approach has been used in [8,7] and rather recently in Ref. [24]. In this paper we follow a di4erent approach which has also been examined for a bosonic system of excitons, coupled to a thermal reservoir in a semiconductor (see second part in Ref. [13]). Auxiliary real-valued variables x (t+ ), x (t− ) are introduced on the time contour by a Hubbard– Stratonovich transformation [25,26] which reduces the Grassmann variables of the coherent state path integral to bilinear forms which can be integrated out. The remaining generating function is given in terms of the auxiliary quantities x (tP ) whose mean value Fx (t) of the time contour can be identi1ed for the -uctuating variable of the interacting fermions coupled to a thermal reservoir. The aim of the paper is a derivation of a Langevin equation with this auxiliary variable Fx (t) after the Hubbard–Stratonovich transformation. The -uctuating number Fx (t) can be regarded as a -uctuating 2 However, it cannot be excluded that algebraic computer programs can determine the -uctuations of Grassmann valued, fermionic 1elds.
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potential or self-energy, depending on the point of view. This will be seen in the detailed equations. By way of example, we consider for the derivation of a Langevin-type equation for fermions a semiconductor in the electron–hole picture, coupled to a thermal reservoir of phonons [27–29]. The following equations with the various Hamiltonians will be speci1ed below: H(
+ x; ;
+ x; ; B ; B ; t)
= HF + HE (t) + HFF + HB + HB; F ;
1 ∇x x; (a b HE (t) = d · E(t) HF =
+ x;
x;;
HFF =
x;
;
+ + x; x;
(3) + b a
x; x; )
+ + x1 ; x2 ; s s V|x1 −x2 | x2 ; x1 ;
x1 ; x2 ; ;
;
;
(6)
sh = −1 ;
a ; b : ae = +1;
ah = 0;
(4) (5)
E(t) = E0 (t − t0 ) cos(!(t − t0 )) ; s : se = +1;
(2)
(7) be = 0;
bh = −1 :
(8)
The Hamiltonian is de1ned in terms of fermionic operators x; , x;+ in which the index labels the various electron and hole bands, and in which the variable index x always denotes a three-dimensional vector ˜x for brevity. The purely fermionic part of the Hamiltonian (2) (compare [29]) consists of the kinetic term HF (3) with the single particle energies ( 1 ∇x ) in parabolic band approximation, the fermion–fermion interaction HFF (5) with matrix element V|x1 −x2 | and the time-dependent interaction HE (t) (4) with the coherent laser 1eld E(t). The array s takes the values +1 or −1 for particles (electrons) and antiparticles (holes), respectively, so that the correct sign of e.g. the Coulomb potential in HFF (5) is considered. The values ae = +1, ah = 0 and be = 0, bh = −1 ensure the correct form of the pair creation and annihilation in HE (t). The momentum of the classical laser 1eld E(t) in HE (t) can be neglected and the dipole moment d is approximated by a constant value. 3 Furthermore, bosonic operators B , B+ of a thermal reservoir are introduced whose states are labelled by the index and whose dispersion relation in terms of the states is given by the frequencies ! in HB : ˝! B+ B ; (9) HB =
HB; F =
(B+ g; x
+ x; s x;
+
+ ∗ x; s x; gx; B )
:
(10)
x; ; 3 The inclusion of a 1nite momentum of the photon and a momentum-dependent dipole moment d ˜k; does not alter the 1nal Langevin equation qualitatively.
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B. Mieck / Physica A 312 (2002) 431 – 446
The bosonic operators of the reservoir can be related to acoustic and longitudinal optical phonons of a semiconductor [27,28]. The acoustic phonons yield the noise in the electron–hole system and are especially important if one studies condensation of excitons near the band edge of electrons in a semiconductor [30]. For our purposes these bosons need not be explicitly speci1ed, but it is known that acoustic or longitudinal optical phonons couple to electrons and holes, as generically described by the Hamiltonian HB; F (10) with the coupling g; x between reservoir states and Fermi 1elds. In comparison to Ref. [13] with bosonic excitons, a linear coupling as B+ g; x bx cannot be applied for the electrons and holes because fermion conservation allows only a coupling to pairs of Fermi 1elds. This di4erence leads to other expressions for the Hubbard–Stratonovich transformation and for the 1nal Langevin equation. The aim is to derive a Langevin equation for density matrix elements y+1 ; (t) y2 ; (t). Therefore, the following expression with the time development operator U (t; t0 ) is considered: y+1 ; (t) y2 ; (t) = Tr B &B 0 |U (t0 ; t) y+1 ; U (t; tN )U (tN ; t) y2 ; U (t; t0 )|0 ; (11) t U (t; t0 ) = T exp − d)H( ˝ t0
+ x; ;
+ x; ; B ; B ; ))
;
&B = exp{−*(HB − B NB )}=ZB :
(12) (13)
The reservoir is taken into account by the trace with the statistical operator &B (13) over the states . At and below temperatures of ≈ 300 K, it is a good approximation to use the pure vacuum state |0 for the electrons and holes without any fermions present. They are created by the laser pulse E(t) in HE (t) (4). In order to obtain a coherent state path integral for y+1 ; (t) y2 ; (t), one has to separate the time development into N time steps Kt between initial time t0 and 1nal time tN y+1 ; (tj ) y2 ; (tj ) = Tr B e−*(HB −B NB ) =ZB ×0 |e Kt=˝·H(t1 ) : : : e Kt=˝·H(tj ) × e− Kt=˝·H(tN ) : : : e− Kt=˝·H(tj+1 ) Kt = (tN − t0 )=N :
+ Kt=˝·H(tj+1 ) y1 ; e y2 ; e
: : : e Kt=˝·H(tN )
− Kt=˝·H(tj )
: : : e− Kt=˝·H(t1 ) |0 ; (14)
At every time step an overcomplete set of coherent states for the fermions is inserted, whereas the operators B+ , B of the reservoir are kept for a relation of a generating functional to be developed below (23). In the case of fermions the coherent state 1elds .x; are Grassmann-valued and are contained in the exponential with the creation operator x;+ . 4 According to the time development in forward “+” U (t; t0 ) and backward 4 Note that the Gaussian factor is contained in the completeness relation and not in the coherent state |. as it is sometimes preferred.
B. Mieck / Physica A 312 (2002) 431 – 446
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“−” U (t0 ; t) direction, a contour time tp has to be introduced for the anticommuting coherent state 1elds + |. = exp − (15) .x; x; |0; x; |. = .x; |. ; x;
p = ±t+ ; t− ;
.y; → .y; (tp ); 1=
x;
(16)
.x;∗ (tp ).x; (tp ) |.(tp ).(tp )| : d.x;∗ (tp )d.x; (tp ) exp −
(17)
x;
Since we are investigating a system under non-equilibrium conditions, contour integrals have also to be considered in the path integral for the forward dt+ and backward dt− direction of integration [17–23,15] ∞ t0 (1) (1) (1) dtp1 : : : = dt+ : : : + dt− ::: : (18) C
t0
∞
The fermionic coherent states |. are eigenstates of the annihilation operators (15) so that we can replace the fermionic operators x; , x;+ by their eigenvalues .x; (tp ), .x;∗ (tp ) in the time development operator at the corresponding contour time step. This is possible because the Hamilton operator in the exponentials of U (t + Kt; t) is in normal order (for a detailed derivation and introduction to fermionic coherent states see Refs. [14 –16]). In terms of the fermionic coherent states on the time contour, the generating function for the density matrix is given by the following expression and is explained further below @ Z[J˜ ] = d[.x; (tp )] exp − dtp .x;∗ (tp ) .x; (tp ) @tp C x;
×exp − ˝
dtp
x;
×exp − ×exp −
×
C
dtp
C
x;;
˝
C
dtp
.x;∗ (tp )
×exp −
C
1 ∇x .x; (tp )
Ex (tp )[a b .x;∗ (tp ).x;∗ (tp )
+ b a .x; (tp ).x; (tp )]
x1 ;x2 ;1 ;2
.x∗1 ;1 (tp ).x∗2 ;2 (tp )
dtp(1)1 dtp(2)2
s1 s2 V|x1 −x2 | .x2 ;2 (tp ).x1 ;1 (tp )
x1 ;x2 ;1 ;2
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B. Mieck / Physica A 312 (2002) 431 – 446
×
.x∗1 ;1 (tp(1)1 )J˜ (tp(1)1 ; x1 ; 1 ; tp(2)2 ; x2 ; 2 ).x2 ;2 (tp(2)2 )
dtp (B+ (tp )g; x .x;∗ (tp )s .x; (tp ) ×Tr B &B TC exp − ˝ C
x;;
+ .x;∗ (tp )s .x; (tp )gx;∗ B (tp )) ×exp − dtp(1)1 dtp(2)2 .x∗1 ;1 (tp(1)1 )s1 .x1 ;1 (tp(1)1 ) C x1 ;x2 ;1 ;2 ;
× (gx∗1 ; =˝)GB; (tp(1)1 ; tp(2)2 )(g; x2 =˝).x∗2 ;2 (tp(2)2 )s2 .x2 ;2 (tp(2)2 ) ; B (t) = e− ! (t−t0 ) B (t0 ) ;
(19) (20)
Ex (tp ) = d · E(tp )=˝ + jx (tp );
Ex (t+ ) = Ex (t− ) :
(21)
The 1rst factor in the generating function Z[J˜ ] with the time derivative follows from the Gaussian function in the completeness relations and the overlap of neighbouring contour time steps. The second, third and fourth factor in Z[J˜ ] are related to the kinetic energy HF , the laser 1eld term HE (t) and the fermion–fermion interaction HFF . Since the coherent state 1elds .x; (tp ) are eigenvalues of the annihilation operators + ∗ x; , the anticommuting 1elds .x; (tp ), .x; (tp ) just replace the operators x; , x; in HF , HE (t) and HFF [14 –16]. A source term jx (t) has also been added to the laser 1eld E(t) for a spontaneous symmetry breaking and has been combined with E(t) to the term Ex (tp ) which has equal values on the two branches of the time contour (21). The 1fth factor in (19) contains the source term J˜ for generating density matrix elements y+1 ; (t) y2 ; (t) (11; 14) by taking appropriate derivatives. The sixth and last factor of Z[J˜ ] with the trace Tr B over the operators B (tp ), B+ (tp ) can be evaluated explicitly and is given under the braces by the exponential term with the contour Green’s function GB; (tp(1)1 ; tp(2)2 ) of the reservoir. This last relation becomes apparent if the pairs of the coherent state 1elds which couple to the reservoir operators B (tp ), B+ (tp ) are abbreviated by the source k (tp ): k (tp ) = (g; x =˝).x;∗ (tp )s .x; (tp ) ; (22)
x;
Tr B &B TC exp −
C
dtp
(B+ (tp )k (tp )
+
k∗ (tp )B (tp ))
B. Mieck / Physica A 312 (2002) 431 – 446
437
(1) (2) (2) (1) (2) ∗ (1) k (tp1 )GB; (tp1 ; tp2 )k (tp2 ) : = exp − dtp1 dtp2 C
(23)
Then the trace with B (tp ), B+ (tp ) over the reservoir states in Z[J˜ ] can be identi1ed with the de1nition for the generating function of GB; (tp(1)1 ; tp(2)2 ) which is also determined by the contour time ordered product of B (tp(1)1 ) and B+ (tp(2)2 ) (24). According to the contour time ordered product of the reservoir operators, GB; (tp(1)1 ; tp(2)2 ) can be calculated by using B (t) = exp(− ! (t − t0 ))B (t0 ) (20) GB; (tp(1)1 ; tp(2)2 ) = Tr B [&B TC (B (tp(1)1 )B+ (tp(2)2 ))] ; GB; (tp(1)1 ; tp(2)2 )
=e
− ! (t (1) −t (2) )
(24)
2(t (1) − t (2) ) + n! ;
n! ;
1 + n! ;
2(t (2) − t (1) ) + n!
! ; (25)
n! = (exp{*(˝! − B )} − 1)−1 : In coordinate space the non-equilibrium reservoir Green’s function is given by (gx∗1 ; =˝)GB; (tp(1)1 ; tp(2)2 )(g; x2 =˝) : GB (tp(1)1 ; x1 ; tp(2)2 ; x2 ) =
(26)
(27)
These special forms of the non-equilibrium Green functions GB; (tp(1)1 ; tp(2)2 ) (25) and GB (tp(1)1 ; x1 ; tp(2)2 ; x2 ) (27) yield the probability distribution of the noise and the dissipative part in the Langevin equation as can be seen in the next section [15].
2. Derivation of the Langevin equation The generating function Z[J˜ ] (19) consists of four terms, which are bilinear in the Grassmann 1elds, and two terms, which are quartic in the anticommuting variables. The two quartic terms are the interaction V|x1 −x2 | of HFF (5) and the contour time Green’s function GB (tp(1)1 ; x1 ; tp(2)2 ; x2 ) (27). These two terms are combined in the contour time matrix A(tp(1)1 ; x1 ; tp(2)2 ; x2 ) for which the delta function 5(tp(1)1 − tp(2)2 ) and the additional sign array 6p1 for the two branches of the contour time are introduced A(tp(1)1 ; x1 ; tp(2)2 ; x2 ) = GB (tp(1)1 ; x1 ; tp(2)2 ; x2 ) + 6p1 5(tp(1)1 − tp(2)2 )V|x1 −x2 | ; (28) ˝ +1; p = +; 6p = dtp(1)1 5(tp(1)1 − tp(2)2 ) = 6p2 : (29) −1; p = −; C The aim is to reduce the quartic term with the matrix A to bilinear terms. This can be obtained with a Hubbard–Stratonovich transformation which leads to an additional
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B. Mieck / Physica A 312 (2002) 431 – 446
auxiliary 1eld x (tp ) on the time contour. In order to simplify notations, the contour integral and the summation over space points is abbreviated by the multiplication symbol ⊗. The summation over electron–hole bands of pairs of anticommuting 1elds is expressed by (.∗ s.)(tp ; x) ⊗ = dtp ::: (30) C
∗
x
(. s.)(tp ; x) =
.x;∗ (tp )s .x; (tp ) :
(31)
Including these notations, the Hubbard–Stratonovich transformation of the quartic term with the contour time matrix A(tp(1)1 ; x1 ; tp(2)2 ; x2 ) takes the form [25,26] exp{−(.∗ s.)(tp(1)1 ; x1 ) ⊗ A(tp(1)1 ; x1 ; tp(2)2 ; x2 ) ⊗ (.∗ s.)(tp(2)2 ; x2 )} = d[x (tp )] exp{− ⊗ A ⊗ } ×exp{− ⊗ A ⊗ (.∗ s.)} exp{− (.∗ s.) ⊗ A ⊗ } :
(32)
The auxiliary real 1eld x (tp ) does not depend on the electron–hole band label . This is a property of the fermion–fermion HFF and reservoir–fermion HB; F interactions which both contain the sign array s for electrons (particles) and holes (antiparticles). The independence of x (tp ) on the band label follows because one can sum over the bands in the bilinear pairs .∗ s. of the quartic interaction terms. Since the generating function Z[J˜ ] (19) has been transformed to a relation with bilinear anticommuting 1elds .x; (tp ), they can be integrated out so that a path integral with the auxiliary real variable x (tp ) remains. However, the driving laser 1eld term, related to HE (t), has pairs of .x; h (tp ) .x; e (tp ) and .x;∗ e (tp ) .x;∗ h (tp ) 1elds, but not .x;∗ h (tp ) .x; e (tp ) and .x;∗ e (tp ) .x; h (tp ), as in comparison to the other terms. Therefore, the bilinear Grassmann variables have to be reordered in the exponential of Z[J˜ ] (19) according to the new 1eld x; (tp ) .x; e (tp ); = e; x; (tp ) = (33) .x;∗ h (tp ); = h : F 2; F : : : for the contour Notations are further simpli1ed by using the barred numbers 1; time, 3-D space vector and band label and by abbreviating " the contour integral and summation over spatial coordinates and band index with 1F ::: = dtp(1)1 ::: : (34) 1F = (tp(1)1 ; x1 ; 1 ); 1F
C
x1 ;1
Applying the simplifying notations (30) and (34), the generating function, reordered with the 1elds x; (tp ), is given by the following relation: Z[J ] = N d[x (tp )] exp{− ⊗ A ⊗ } d[x; (tp )]
B. Mieck / Physica A 312 (2002) 431 – 446
F F 2) F + 8(1; F 2) F + J (1; F 2))( F F ×exp − + (1)(S( 1; 2) :
439
(35)
F 2F 1;
F 2), F 8(1; F 2) F and J (1; F 2) F on the time contour are related to the kiThe matrices S(1; netic energy and laser 1eld term, to the self-energy from the Hubbard–Stratonovich transformation and to the source term @ (1) (2) (1) (2) F F S(1; 2) = S(tp1 ; x1 ; 1 ; tp2 ; x2 ; 2 ) = 6p1 5x1 ;x2 5(tp1 − tp2 ) 51 ;2 @tp(2)2 ! 1 + Ex2 (tp(2)2 )(a1 b2 + b1 a2 ) ; (36) + s 1 1 ∇x ˝ 2 F 2) F = 6p1 5x1 ;x2 51 ;2 5(tp(1) − tp(2) ) 8(1; 1 2 × ⊗ A(: : : ; tp(2)2 ; x2 ) + A(tp(1)1 ; x1 ; : : :) ⊗ :
(37)
Having integrated over the anticommuting 1elds, we obtain a determinant or exp{Tr ln : : :} term in the generating function with the variable x (tp ) on the time contour Z[J ] = N d[x (tp )] exp{− ⊗ A ⊗ } exp{Tr ln(S + 8 + J )} : (38) The trace “Tr” over the logarithm of the sum of matrices S, 8, J has to be de1ned as a contour integral and summation over space coordinates and band index Tr[Matrix] = dtp Matrix(tp ; x; ; tp ; x; ) : (39) C
x;
In order to derive the Langevin equation, the mean part Fx (t) and the di4erence 5x (t) of the plus–minus branches of the auxiliary variable x (tp ) are introduced x (t+ ) = Fx (t) +
1 5x (t); 2
1 Fx (t) = (x (t+ ) + x (t− )); 2
x (t− ) = Fx (t) −
1 5x (t) ; 2
5x (t) = x (t+ ) − x (t− ) :
(40) (41)
As in the bosonic case (compare Eqs. (49) and (50) in [13]), one performs an exchange of integration variables from x (t+ ), x (t− ) to Fx (t), 5x (t) in Z[J ] and expands the exponential of Z[J ] up to second order in 5x (t). The 1rst order term of 5x (t) multiplies the noise fx (t), and the second-order term gives after integration over the -uctuating part 5x (t) the probability distribution of the noise term fx (t). These steps will be brie-y described in the remainder of this section. The transformation of the Gaussian part exp{− ⊗ A ⊗ } from the contour time variable x (tp ) to Fx (t), 5x (t) is straightforward. One has to apply the relation (28) for A(tp(1)1 ; x1 ; tp(2)2 ; x2 ) and the special form of the non-equilibrium Green’s function
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B. Mieck / Physica A 312 (2002) 431 – 446
GB; (tp(1)1 ; tp(2)2 ) (25) for the reservoir states . The result is as follows exp{− ⊗ A ⊗ } = exp −
∞
t0
×exp
dt dt
∞
×exp − 2 ˝ (2)
(1)
dt dt
(1)
(1)
(2)
(1)
(2)
(2)
5x1 (t )M (t ; x1 ; t ; x2 )5x2 (t )
(2)
(1)
∞
t0
dt
1
(2)
K(t ; x1 ; t ; x2 ) = 2 Im
5x1 (t)V|x1 −x2 | Fx2 (t)
(42)
x1 ; x2
gx∗ ;
(2)
5x1 (t )K(t ; x1 ; t ; x2 )Fx2 (t )
x1 ; x2
M (t ; x1 ; t ; x2 ) =
(1)
(2)
x1 ; x2
t0
(1)
(1)
˝
e
− ! (t (1) −t (2) )
gx∗ ; 1
˝
2(t
(1)
n!
(2)
1 + 2
− t )e
g; x2 ; ˝
(43)
− ! (t (1) −t (2) ) g; x2
˝
! :
(44)
The matrix M (t (1) ; x1 ; t (2) ; x2 ) (43) in the second-order term with 5x (t) determines a part of the probability distribution for the noise and K(t (1) ; x1 ; t (2) ; x2 ) (44) already is a dissipative part in the 1nal Langevin equation. Both terms are related by the general -uctuation–dissipation theorem (cf. Eqs. (35) and (36) in Ref. [13]). However, it is not suQcient to expand only the Gaussian exp{− ⊗ A ⊗ } in the generating function Z[J ] (38). The trace-logarithm term of the sum of matrices S, 8, J in (38) has also F 2) F is the only to be expanded up to second order in 5x (t). Since the self-energy 8(1; term in (38) which contains the contour time variable x (tp ), we introduce a mean F 1; F 2) F and the di4erence 58(1; F 2) F in accordance to the mean value Fx (t) and value 8( the di4erence 5x (t) (see Eqs. (40) and (41)). F 2; F : : :, In the following relations one has to distinguish between the barred numbers 1; which abbreviate a contour time, a space vector and band index, and the primed num bers 1 ; 2 ; : : : which refer to the ordinary time t (1) ; t (2) ; : : : and spatial " vector x1 ; x2 ; : : : without the band index. This also holds for summation symbols; "1F is written for a contour integral, spatial summation including band label, whereas 1 just replaces the ordinary time integral and summation of space points F 1; F 2) F = 8(t F p(1) ; x1 ; 1 ; tp(2) ; x2 ; 2 ) = 6p1 5x1 ;x2 51 ;2 5(tp(1) − tp(2) ) 8( 1 2 1 2 ∞ × dt Fx (t )6r [A(tr ; x ; tp(2)2 ; x2 ) + A(tp(1)1 ; x1 ; tr ; x )] t0
1 = t (1) ; x1 ;
(45)
x r=±
1
::: =
∞
t0
dt (1)
x1
:::
(46)
B. Mieck / Physica A 312 (2002) 431 – 446
F = 58(1)
441
˜ 1; F 1 )5(1 ) ; 58(
1
(47)
˜ p(1) ; x1 ; 1 ; t (1) ; x1 ); ˜ 1; F 1 ) = 58(t 58( 1 = [A(tr(1) ; x1 ; tp(1)1 ; x1 ) + A(tp(1)1 ; x1 ; tr(1) ; x1 )]6p1 : 2 r=±
(48)
Applying relations (45) – (48), we obtain for the Tr ln : : : term in (38) up to second F 2) F the following expressions: order in 5x (t) and 1rst order in J (1; exp{Tr ln(S + 8 + J )} = exp{Tr ln(S + 8)} ×exp{Tr ln(1 + (S + 8)−1 J )} ;
(49)
F exp{Tr ln(S + 8)} = exp{Tr ln(S + 8)} F −1 (1; F 1)58( F F ×exp (S + 8) 1) 1F
1 F −1 (1; F 2)58( F F F −1 (2; F 1)58( F F ×exp − (S + 8) 2)(S + 8) 1) : 2 FF
(50)
1;2
exp{Tr ln(1 + (S + 8)−1 J )} = exp
F −1 (1; F 2)J F (2; F 1) F (S + 8)
F 2F 1;
F −1 (1; F 2)58( F F F −1 (2; F 3)J F (3; F 1) F ×exp − (S + 8) 2)(S + 8) F 2; F 3F 1;
×exp
F −1 (1; F 2)58( F F F −1 (2; F 3)58( F F (S + 8) 2)(S + 8) 3)
F 2; F 3; F 4F 1;
F −1 (3; F 4)J F (4; F 1) F × (S + 8)
:
(51)
In the next step one has to reorder the complex relations (50) and (51) according to F 2). F Furthermore, the terms of 5x (t) of the Gaussian their order in 5x (t) and J (1; exp{− ⊗ A ⊗ } with matrices M (t (1) ; x1 ; t (2) ; x2 ) and K(t (1) ; x1 ; t (2) ; x2 ) have to be included. As the result we obtain a matrix B0 (1 ; 2 ) for the second moment of the noise, F 4) F which multiplies the source J (4; F 1) F and which shifts the matrix a matrix B1 (1 ; 2 ; 1; B0 (1 ; 2 ) in the exponent of Z[J ]. The 1rst-order term in 5x (t) multiplies the noise fx (t) whose expression is the 1nal Langevin equation. Moreover, there is a matrix F 3) F which is combined with the source term J . All the mentioned terms B0 (1 ; 2 ), g(1 ; 2;
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F 4), F fx (t) and g(1 ; 2; F 3) F follow from the expansion of the exponent of Z[J ] B1 (1 ; 2 ; 1; (38) up to second order in the di4erence 5x (t) of the plus–minus contour branches F 2). F The Jacobian for the change of x (tp ) and up to 1rst order in the source term J (1; of variables from x (t+ ), x (t− ) to Fx (t), 5x (t) is unity. Therefore, the expansion of Z[J ] up to second order in 5x (t) yields the following relations:
B0 (1 ; 2 ) = B0 (t (1) ; x1 ; t (2) ; x2 ) = M (1 ; 2 ) +
1 F −1 (2; F 1)5 F 8( ˜ 1; F 2 ) ; F 2)5 F 8( ˜ 2; F 1 )(S + 8) F −1 (1; (S + 8) 2 FF
(52)
1;2
F 4) F = B1 (1 ; 2 ; 1;
F −1 (1; F 2)5 F 8( ˜ 2; F 1 ) (S + 8)
F 3F 2;
F 3)5 F 8( ˜ 3; F 2 )(S + 8) F −1 (3; F 4) F ; F −1 (2; ×(S + 8)
fx1 (t (1) ) = f(1 ) = −2
V|x −x | 1
x2
˝
2
Fx2 (t (1) )
F −1 (1; F 1)5 F 8( ˜ 1; F 1 ) ; K(1 ; 2 )(2 F ) − (S + 8) +
(54)
1F
2
F 3) F = g(1 ; 2;
(53)
F −1 (2; F 1)5 F 8( ˜ 1; F 1 )(S + 8) F −1 (1; F 3) F ; (S + 8)
(55)
1F
Z[J ] = N
F d[Fx (t)] d[5x (t)] exp{Tr ln(S + 8)}
F 4)J F (4; F 1) F 5(2 ) ×exp − 5(1 ) B0 (1 ; 2 ) − B1 (1 ; 2 ; 1; F 4F 1;
1 ;2
F 3)J F (3; F 2) F ×exp 5(1 ) f(1 ) + g(1 ; 2; 1
×exp
F 2F 1;
F 3F 2;
F −1 (1; F 2)J F (2; F 1) F (S + 8)
:
(56)
It remains to perform the integration over 5x (t) and the change of variables from Fx (t) to the noise fx (t). Since the Langevin equation (54) has a causal structure (fx (t) only depends on previous and equal times of Fx (t)), the Jacobian for the exchange of variables from Fx (t) to fx (t) is a constant which is absorbed in the normalization factor N. The generating function Z[J ] (56), from which the density matrix elements
B. Mieck / Physica A 312 (2002) 431 – 446
443
y;+ (t) y; (t) (11) and (14) follow by taking appropriate derivatives of the source F 2), F is therefore given in terms of the noise fx (t) by the expression term J (1; F Z[J ] = N d[f(1 )] exp{Tr ln(S + 8)}
×exp
F 2)J F (2; F 1) F F −1 (1; (S + 8)
F 2F 1;
1 F 4)J F (4; F 1) F ×exp − tr ln B0 (: : : ; : : :) − B1 (: : : ; : : : ; 1; 2 F F 1;4
1 F 3)J F (3; F 2) F f(1 ) + ×exp − g(1 ; 2; 4 FF 1 ;2
2;3
× B0 (: : : ; : : :) −
−1 F 4)J F (4; F 1) F B1 (: : : ; : : : ; 1;
(1 ; 2 )
F 4F 1;
× f(2 ) +
F 6F 5;
F (6; F 5) F 6)J F : g(2 ; 5;
(57)
By taking derivatives of Z[J ] (57) with respect to the source term, one can generate the density matrix elements. 3. Results and discussion The 1nal Langevin equation with the noise fx (t) and -uctuating self-energy or potential Fx (t) is determined by Eq. (54). The -uctuating variable Fx (t) is now a F 1; F 2) F functional of the noise fx (t). This also holds for the self-energy matrix 8( Fx (t) = Fx [t; fx (t )]; 2
V|x −x | 1
x2
=
˝
2
2
F 8F = 8[f] ;
(58)
Fx2 (t (1) )
K(1 ; 2 ) (2 F ) −
F −1 (1; F 1) F 58( ˜ 1; F 1 ) − f(1 ) : (S + 8)
(59)
1F
If we compare the Langevin equation (59) with the saddle point equation (68) in Ref. [29], the di4erence is found in the noise term f(1 ) and dissipative part K(1 ; 2 ) of (59). The 1nal Langevin equation (59) is obviously a modi1ed saddle point equation as in Ref. [29], but where the noise fx (t) includes now contributions around the saddle
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B. Mieck / Physica A 312 (2002) 431 – 446
point of the time contour so that thermal -uctuations of a reservoir (e.g. phonons) are considered. According to the general dissipation–-uctuation theorem, the noise is accompanied by dissipation [27]. This leads to the term with K(1 ; 2 ) and the special F 1; F 2) F in the Langevin equation. form of 8( The probability distribution P[f(1 )] for the noise follows by performing appropriate derivatives of Z[J ] (57); e.g. in order to calculate the density matrix element y;+e (t) y; e (t), the derivative of Z[J ] (57) has to be computed with respect to J (t− ; y; e; t+ ; y; e) and by setting J to zero. This yields the relations ' Z[J ] '' @ + y; e (t) y; e (t) = @J (t− ; y; e; t+ ; y; e) Z[0] 'J =0 1 F d[f(1 )] exp{Tr ln(S + 8[f])} = Z[0] −1 F ×(S + 8[f]) (t+ ; y; e; t− ; y; e);
P[f(1 )]
(60)
1 P[f(1 )] ˙ [det(B0 )]−1=2 exp − f(2 )B0−1 (2 ; 3 )f(3 ) 4
;
(61)
2 ;3
where P[f(1 )] in the integrand is the probability distribution for the noise f(1 ). The numerical computation of density matrix elements as (60) is given by the following procedures: (1) Random numbers f(1 ) have to be generated according to the distribution P[f(1 )] (61). If one considers the exact distribution as given in (61), the probability distribution becomes non-Gaussian because the matrix B0−1 (2 ; 3 ) contains the F self-energy 8F = 8[f]. By performing continued fractions on the Langevin equation (59) for an arbitrary noise term fx (t), the non-Gaussian probability distribution can be determined. (2) A random sequence of numbers for fx (t) according to P[f(1 )] is applied to the Langevin equation (59) which is then solved numerically by continued fraction in order to obtain Fx (t) from (59). (3) The realized, -uctuating potential Fx (t) is used to compute the terms exp{Tr ln(S + F (S + 8) F −1 (t+ ; y; e; t− ; y; e) in the integrand of (60) with the special form for 8)} the self-energy 8F given by Eq. (45). F (S + 8) F −1 (t+ ; y; e; t− ; y; e) over (4) One has to average the terms exp{Tr ln(S + 8)} many di4erent realizations according to the probability distribution P[f(1 )] (61). Other density matrix elements as y;+h (t) y; h (t) and y; h (t) y; e (t) follow by taking appropriate derivatives of (57) with respect to the source term J ' Z[J ] '' @ + y; h (t) y; h (t) = − @J (t+ ; y; h; t− ; y; h) Z[0] 'J =0
B. Mieck / Physica A 312 (2002) 431 – 446
=−
1 Z[0]
445
F d[f(1 )] exp{Tr ln(S + 8[f])}
−1 F (t− ; y; h; t+ ; y; h) ×(S + 8[f])
P[f(1 )] ;
(62)
' Z[J ] '' @ y; h (t) y; e (t) = @J (t− ; y; h; t+ ; y; e) Z[0] 'J =0 1 F d[f(1 )] exp{Tr ln(S + 8[f])} = Z[0] −1 F ×(S + 8[f]) (t+ ; y; e; t− ; y; h)
P[f(1 )] :
(63)
The same probability distribution P[f(1 )] as in (60), with which one has to average the remaining integrand, is obtained. The computation for the various density matrix elements di4ers only in the calculation of the corresponding matrix elements of (S + −1 F F 8[f]) . The inverse of the matrix (S + 8[f]) is related to a modi1ed contour time Green’s function [17–23]. Cum grano salis, one averages in relations (60), (62), (63) for density matrix elements over modi1ed non-equilibrium Green functions [15] so that noise and dissipation are considered on a quantum level. If one wants to calculate higher order correlation functions, which e.g. require the second derivative of Z[J ] with respect to the source term, one has to expand the exponent of the integrand of Z[J ] (38) up to second order in 5x (t) and has also to include second-order terms of the source J in the expansion of exp{Tr ln(1 + (S + 8)−1 J )} (51).
Acknowledgements This work has been supported by the Sonderforschungsbereich 237 “Unordnung und groTe Fluktuationen”.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
R. Brown, Philos. Mag. 4 (1828) 161. R. Brown, Ann. Phys. Chem. 14 (1828) 294. A. Einstein, Ann. Phys. 17 (1905) 549. A. Einstein, Ann. Phys. 19 (1906) 289. W.T. Co4ey, Y.P. Kalmykov, J.T. Waldron, The Langevin Equation with Applications in Physics, Chemistry and Electrical Engineering, World Scienti1c, Singapore, 1996. M. Lax, Phys. Rev. 145 (1966) 110. H. Haken, Laser Theory, Handbuch der Physik, Vol. XXV=2c, Springer, Berlin, 1970. H. Haug, H. Haken, Z. Phys. 204 (1967) 262. H. Haug, Z. Phys. 200 (1967) 57. H. Haug, Phys. Rev. 184 (1969) 338. R. Graham, J. Mod. Opt. 47 (2000) 2615. R. Graham, J. Stat. Phys. 101 (2000) 243.
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[13] B. Mieck, Physica A 294 (2001) 96. [14] J.W. Negele, H. Orland, Quantum Many-Particle Systems, Addison-Wesley, Reading, MA, 1988. [15] H. Kleinert, Pfadintegrale in Quantenmechanik, Statistik und Polymerphysik, BI Wissenschaftsverlag, Mannheim, 1993. [16] B. Mieck, Rep. Math. Phys. 47 (2001) 139. [17] J. Schwinger, J. Math. Phys. 2 (1961) 407. [18] P.M. Bakshi, K.T. Mahanthappa, J. Math. Phys. 4 (1963) 1. [19] P.M. Bakshi, K.T. Mahanthappa, J. Math. Phys. 4 (1963) 12. [20] L.P. Keldysh, Sov. Phys. JETP 20 (1965) 1018. [21] R.A. Craig, J. Math. Phys. 3 (1968) 605. [22] D.F. DuBois, in: W.E. Brittin, A.D. Barut (Eds.), Lectures in Theoretical Physics, Vol. 9c, Gordon Breach, New York, 1967, p. 469. [23] Landau-Lifshitz, Theoretical Physics, Vol. 10, Pergamon Press, New York, 1981 (Chapter 10). [24] M.I. Muradov, Phys. Rev. B 58 (1998) 12 883. [25] J. Hubbard, Phys. Rev. Lett. 3 (1959) 77. [26] R.L. Stratonovich, Sov. Phys. Dokl. 2 (1958) 416. [27] H. Haug, S.W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, World Scienti1c, Singapore, 1993. [28] H. Haug, A.-P. Jauho, Quantum Kinetics in Transport and Optics of Semiconductors, Springer, Berlin, 1996. [29] B. Mieck, Physica A 269 (1999) 455. [30] O.M. Schmitt, D.B. Tran Thoai, L. Banyai, P. Gartner, H. Haug, Phys. Rev. Lett. 86 (2001) 3839.
www.elsevier.com/locate/physa
Coherent state path integral and Langevin equation of interacting fermions B. Mieck1 Fachbereich Physik, Universitat-Gesamthochschule Essen, 45117 Essen, Germany Received 16 February 2002
Abstract Interacting fermions, electrons and holes in a semiconductor, are coupled to a thermal reservoir of bosons which yield the -uctuating noise. We use a coherent state path integral formulation on the time contour for non-equilibrium systems in terms of anticommuting variables which replace the fermionic creation- and annihilation operators in the time development operator. An auxiliary commuting 1eld x (tp ), de1ned on the time contour, is introduced by a Hubbard– Stratonovich transformation. In terms of this new 1eld, a Langevin equation is derived which is similar to a saddle-point equation with a random force fx (t). In comparison to the bosonic case of excitons in a semiconductor previously described, one obtains a di4erent expression for the Langevin equation and, especially, a di4erent relation for the probability distribution of the noise term fx (t). The calculation for density matrix elements with the Langevin approach can also be interpreted as an average over modi1ed non-equilibrium Green functions with the appropriately c 2002 Published by Elsevier Science B.V. derived probability distribution of the noise. PACS: 05.40.−a; 05.10.Gg; 02.70.Lq Keywords: Langevin equation; Coherent states; Anticommuting variables
1. Introduction A Langevin equation consists of a relation for the mean kinetics and a -uctuating force term, describing noise of a surrounding medium. This concept has already been applied for Brownian motion [1– 4] where the noise is introduced into Newton’s equation by a random variable. Noise is also important in electronic circuits [5] and 1
Supported by the DFG. E-mail address: [email protected] (B. Mieck).
c 2002 Published by Elsevier Science B.V. 0378-4371/02/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 2 ) 0 0 9 6 2 - 7
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for the transition of an incoherent system to coherence, as for example in a laser [6 –10] or in a Bose–Einstein condensate [11,12]. In the case of a bosonic system it is rather obvious to identify the classical mean kinetics and the -uctuations caused by a thermal reservoir, as investigated recently by a coherent state path integral [13]. The -uctuations of a complex-numbered coherent state 1eld F x (t), the average of the two 1elds x (t+ ), x (t− ) on the time contour, are driven by an inhomogenous c-numbered quantity which can be generated by a random sequence of numbers. In essence, one obtains the following, classical Langevin equation with a c-numbered 1eld F x (t) and a random force term Fx (t) (compare Eq. (25) in Ref. [13]): t @F x (t) + Fx (t) = dt K(t; x; t ; x )F x (t ) @t t0 1 − ˝
x
1 1 w F ∇x F x (t) + Ex (t) − | (t)|2 F x (t) : ˝ 2V x
(1)
According to the general -uctuation–dissipation theorem, the noise is also accompanied by a dissipative term K(t; x; t ; x ). If we try to transfer this concept to fermions, also coupled to a thermal reservoir (e.g. phonons in a semiconductor), Grassmann-valued, anticommuting variables have inevitably to be introduced in the coherent state path integral on the time contour (for an introduction to fermionic coherent states and their application see Refs. [14 –16]; the time contour for non-equilibrium phenomena is introduced in Refs. [17–23,15]). It is then not obvious how to identify the -uctuating numbers and the random variable which has also to be a Grassmann number. Since anticommuting numbers cannot be computed with a c-numbered random generator, the concept of a classical Langevin equation with a random, numerically de1ned noise breaks down. 2 Therefore, the 1rst part in Ref. [13], the classical Langevin equation with an inhomogenous random number, is not applicable for fermionic systems. However, combinations of fermionic operators can have classical limits, as e.g. the densities and the polarization composed of two Fermi-operators. This approach has been used in [8,7] and rather recently in Ref. [24]. In this paper we follow a di4erent approach which has also been examined for a bosonic system of excitons, coupled to a thermal reservoir in a semiconductor (see second part in Ref. [13]). Auxiliary real-valued variables x (t+ ), x (t− ) are introduced on the time contour by a Hubbard– Stratonovich transformation [25,26] which reduces the Grassmann variables of the coherent state path integral to bilinear forms which can be integrated out. The remaining generating function is given in terms of the auxiliary quantities x (tP ) whose mean value Fx (t) of the time contour can be identi1ed for the -uctuating variable of the interacting fermions coupled to a thermal reservoir. The aim of the paper is a derivation of a Langevin equation with this auxiliary variable Fx (t) after the Hubbard–Stratonovich transformation. The -uctuating number Fx (t) can be regarded as a -uctuating 2 However, it cannot be excluded that algebraic computer programs can determine the -uctuations of Grassmann valued, fermionic 1elds.
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potential or self-energy, depending on the point of view. This will be seen in the detailed equations. By way of example, we consider for the derivation of a Langevin-type equation for fermions a semiconductor in the electron–hole picture, coupled to a thermal reservoir of phonons [27–29]. The following equations with the various Hamiltonians will be speci1ed below: H(
+ x; ;
+ x; ; B ; B ; t)
= HF + HE (t) + HFF + HB + HB; F ;
1 ∇x x; (a b HE (t) = d · E(t) HF =
+ x;
x;;
HFF =
x;
;
+ + x; x;
(3) + b a
x; x; )
+ + x1 ; x2 ; s s V|x1 −x2 | x2 ; x1 ;
x1 ; x2 ; ;
;
;
(6)
sh = −1 ;
a ; b : ae = +1;
ah = 0;
(4) (5)
E(t) = E0 (t − t0 ) cos(!(t − t0 )) ; s : se = +1;
(2)
(7) be = 0;
bh = −1 :
(8)
The Hamiltonian is de1ned in terms of fermionic operators x; , x;+ in which the index labels the various electron and hole bands, and in which the variable index x always denotes a three-dimensional vector ˜x for brevity. The purely fermionic part of the Hamiltonian (2) (compare [29]) consists of the kinetic term HF (3) with the single particle energies ( 1 ∇x ) in parabolic band approximation, the fermion–fermion interaction HFF (5) with matrix element V|x1 −x2 | and the time-dependent interaction HE (t) (4) with the coherent laser 1eld E(t). The array s takes the values +1 or −1 for particles (electrons) and antiparticles (holes), respectively, so that the correct sign of e.g. the Coulomb potential in HFF (5) is considered. The values ae = +1, ah = 0 and be = 0, bh = −1 ensure the correct form of the pair creation and annihilation in HE (t). The momentum of the classical laser 1eld E(t) in HE (t) can be neglected and the dipole moment d is approximated by a constant value. 3 Furthermore, bosonic operators B , B+ of a thermal reservoir are introduced whose states are labelled by the index and whose dispersion relation in terms of the states is given by the frequencies ! in HB : ˝! B+ B ; (9) HB =
HB; F =
(B+ g; x
+ x; s x;
+
+ ∗ x; s x; gx; B )
:
(10)
x; ; 3 The inclusion of a 1nite momentum of the photon and a momentum-dependent dipole moment d ˜k; does not alter the 1nal Langevin equation qualitatively.
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The bosonic operators of the reservoir can be related to acoustic and longitudinal optical phonons of a semiconductor [27,28]. The acoustic phonons yield the noise in the electron–hole system and are especially important if one studies condensation of excitons near the band edge of electrons in a semiconductor [30]. For our purposes these bosons need not be explicitly speci1ed, but it is known that acoustic or longitudinal optical phonons couple to electrons and holes, as generically described by the Hamiltonian HB; F (10) with the coupling g; x between reservoir states and Fermi 1elds. In comparison to Ref. [13] with bosonic excitons, a linear coupling as B+ g; x bx cannot be applied for the electrons and holes because fermion conservation allows only a coupling to pairs of Fermi 1elds. This di4erence leads to other expressions for the Hubbard–Stratonovich transformation and for the 1nal Langevin equation. The aim is to derive a Langevin equation for density matrix elements y+1 ; (t) y2 ; (t). Therefore, the following expression with the time development operator U (t; t0 ) is considered: y+1 ; (t) y2 ; (t) = Tr B &B 0 |U (t0 ; t) y+1 ; U (t; tN )U (tN ; t) y2 ; U (t; t0 )|0 ; (11) t U (t; t0 ) = T exp − d)H( ˝ t0
+ x; ;
+ x; ; B ; B ; ))
;
&B = exp{−*(HB − B NB )}=ZB :
(12) (13)
The reservoir is taken into account by the trace with the statistical operator &B (13) over the states . At and below temperatures of ≈ 300 K, it is a good approximation to use the pure vacuum state |0 for the electrons and holes without any fermions present. They are created by the laser pulse E(t) in HE (t) (4). In order to obtain a coherent state path integral for y+1 ; (t) y2 ; (t), one has to separate the time development into N time steps Kt between initial time t0 and 1nal time tN y+1 ; (tj ) y2 ; (tj ) = Tr B e−*(HB −B NB ) =ZB ×0 |e Kt=˝·H(t1 ) : : : e Kt=˝·H(tj ) × e− Kt=˝·H(tN ) : : : e− Kt=˝·H(tj+1 ) Kt = (tN − t0 )=N :
+ Kt=˝·H(tj+1 ) y1 ; e y2 ; e
: : : e Kt=˝·H(tN )
− Kt=˝·H(tj )
: : : e− Kt=˝·H(t1 ) |0 ; (14)
At every time step an overcomplete set of coherent states for the fermions is inserted, whereas the operators B+ , B of the reservoir are kept for a relation of a generating functional to be developed below (23). In the case of fermions the coherent state 1elds .x; are Grassmann-valued and are contained in the exponential with the creation operator x;+ . 4 According to the time development in forward “+” U (t; t0 ) and backward 4 Note that the Gaussian factor is contained in the completeness relation and not in the coherent state |. as it is sometimes preferred.
B. Mieck / Physica A 312 (2002) 431 – 446
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“−” U (t0 ; t) direction, a contour time tp has to be introduced for the anticommuting coherent state 1elds + |. = exp − (15) .x; x; |0; x; |. = .x; |. ; x;
p = ±t+ ; t− ;
.y; → .y; (tp ); 1=
x;
(16)
.x;∗ (tp ).x; (tp ) |.(tp ).(tp )| : d.x;∗ (tp )d.x; (tp ) exp −
(17)
x;
Since we are investigating a system under non-equilibrium conditions, contour integrals have also to be considered in the path integral for the forward dt+ and backward dt− direction of integration [17–23,15] ∞ t0 (1) (1) (1) dtp1 : : : = dt+ : : : + dt− ::: : (18) C
t0
∞
The fermionic coherent states |. are eigenstates of the annihilation operators (15) so that we can replace the fermionic operators x; , x;+ by their eigenvalues .x; (tp ), .x;∗ (tp ) in the time development operator at the corresponding contour time step. This is possible because the Hamilton operator in the exponentials of U (t + Kt; t) is in normal order (for a detailed derivation and introduction to fermionic coherent states see Refs. [14 –16]). In terms of the fermionic coherent states on the time contour, the generating function for the density matrix is given by the following expression and is explained further below @ Z[J˜ ] = d[.x; (tp )] exp − dtp .x;∗ (tp ) .x; (tp ) @tp C x;
×exp − ˝
dtp
x;
×exp − ×exp −
×
C
dtp
C
x;;
˝
C
dtp
.x;∗ (tp )
×exp −
C
1 ∇x .x; (tp )
Ex (tp )[a b .x;∗ (tp ).x;∗ (tp )
+ b a .x; (tp ).x; (tp )]
x1 ;x2 ;1 ;2
.x∗1 ;1 (tp ).x∗2 ;2 (tp )
dtp(1)1 dtp(2)2
s1 s2 V|x1 −x2 | .x2 ;2 (tp ).x1 ;1 (tp )
x1 ;x2 ;1 ;2
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×
.x∗1 ;1 (tp(1)1 )J˜ (tp(1)1 ; x1 ; 1 ; tp(2)2 ; x2 ; 2 ).x2 ;2 (tp(2)2 )
dtp (B+ (tp )g; x .x;∗ (tp )s .x; (tp ) ×Tr B &B TC exp − ˝ C
x;;
+ .x;∗ (tp )s .x; (tp )gx;∗ B (tp )) ×exp − dtp(1)1 dtp(2)2 .x∗1 ;1 (tp(1)1 )s1 .x1 ;1 (tp(1)1 ) C x1 ;x2 ;1 ;2 ;
× (gx∗1 ; =˝)GB; (tp(1)1 ; tp(2)2 )(g; x2 =˝).x∗2 ;2 (tp(2)2 )s2 .x2 ;2 (tp(2)2 ) ; B (t) = e− ! (t−t0 ) B (t0 ) ;
(19) (20)
Ex (tp ) = d · E(tp )=˝ + jx (tp );
Ex (t+ ) = Ex (t− ) :
(21)
The 1rst factor in the generating function Z[J˜ ] with the time derivative follows from the Gaussian function in the completeness relations and the overlap of neighbouring contour time steps. The second, third and fourth factor in Z[J˜ ] are related to the kinetic energy HF , the laser 1eld term HE (t) and the fermion–fermion interaction HFF . Since the coherent state 1elds .x; (tp ) are eigenvalues of the annihilation operators + ∗ x; , the anticommuting 1elds .x; (tp ), .x; (tp ) just replace the operators x; , x; in HF , HE (t) and HFF [14 –16]. A source term jx (t) has also been added to the laser 1eld E(t) for a spontaneous symmetry breaking and has been combined with E(t) to the term Ex (tp ) which has equal values on the two branches of the time contour (21). The 1fth factor in (19) contains the source term J˜ for generating density matrix elements y+1 ; (t) y2 ; (t) (11; 14) by taking appropriate derivatives. The sixth and last factor of Z[J˜ ] with the trace Tr B over the operators B (tp ), B+ (tp ) can be evaluated explicitly and is given under the braces by the exponential term with the contour Green’s function GB; (tp(1)1 ; tp(2)2 ) of the reservoir. This last relation becomes apparent if the pairs of the coherent state 1elds which couple to the reservoir operators B (tp ), B+ (tp ) are abbreviated by the source k (tp ): k (tp ) = (g; x =˝).x;∗ (tp )s .x; (tp ) ; (22)
x;
Tr B &B TC exp −
C
dtp
(B+ (tp )k (tp )
+
k∗ (tp )B (tp ))
B. Mieck / Physica A 312 (2002) 431 – 446
437
(1) (2) (2) (1) (2) ∗ (1) k (tp1 )GB; (tp1 ; tp2 )k (tp2 ) : = exp − dtp1 dtp2 C
(23)
Then the trace with B (tp ), B+ (tp ) over the reservoir states in Z[J˜ ] can be identi1ed with the de1nition for the generating function of GB; (tp(1)1 ; tp(2)2 ) which is also determined by the contour time ordered product of B (tp(1)1 ) and B+ (tp(2)2 ) (24). According to the contour time ordered product of the reservoir operators, GB; (tp(1)1 ; tp(2)2 ) can be calculated by using B (t) = exp(− ! (t − t0 ))B (t0 ) (20) GB; (tp(1)1 ; tp(2)2 ) = Tr B [&B TC (B (tp(1)1 )B+ (tp(2)2 ))] ; GB; (tp(1)1 ; tp(2)2 )
=e
− ! (t (1) −t (2) )
(24)
2(t (1) − t (2) ) + n! ;
n! ;
1 + n! ;
2(t (2) − t (1) ) + n!
! ; (25)
n! = (exp{*(˝! − B )} − 1)−1 : In coordinate space the non-equilibrium reservoir Green’s function is given by (gx∗1 ; =˝)GB; (tp(1)1 ; tp(2)2 )(g; x2 =˝) : GB (tp(1)1 ; x1 ; tp(2)2 ; x2 ) =
(26)
(27)
These special forms of the non-equilibrium Green functions GB; (tp(1)1 ; tp(2)2 ) (25) and GB (tp(1)1 ; x1 ; tp(2)2 ; x2 ) (27) yield the probability distribution of the noise and the dissipative part in the Langevin equation as can be seen in the next section [15].
2. Derivation of the Langevin equation The generating function Z[J˜ ] (19) consists of four terms, which are bilinear in the Grassmann 1elds, and two terms, which are quartic in the anticommuting variables. The two quartic terms are the interaction V|x1 −x2 | of HFF (5) and the contour time Green’s function GB (tp(1)1 ; x1 ; tp(2)2 ; x2 ) (27). These two terms are combined in the contour time matrix A(tp(1)1 ; x1 ; tp(2)2 ; x2 ) for which the delta function 5(tp(1)1 − tp(2)2 ) and the additional sign array 6p1 for the two branches of the contour time are introduced A(tp(1)1 ; x1 ; tp(2)2 ; x2 ) = GB (tp(1)1 ; x1 ; tp(2)2 ; x2 ) + 6p1 5(tp(1)1 − tp(2)2 )V|x1 −x2 | ; (28) ˝ +1; p = +; 6p = dtp(1)1 5(tp(1)1 − tp(2)2 ) = 6p2 : (29) −1; p = −; C The aim is to reduce the quartic term with the matrix A to bilinear terms. This can be obtained with a Hubbard–Stratonovich transformation which leads to an additional
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B. Mieck / Physica A 312 (2002) 431 – 446
auxiliary 1eld x (tp ) on the time contour. In order to simplify notations, the contour integral and the summation over space points is abbreviated by the multiplication symbol ⊗. The summation over electron–hole bands of pairs of anticommuting 1elds is expressed by (.∗ s.)(tp ; x) ⊗ = dtp ::: (30) C
∗
x
(. s.)(tp ; x) =
.x;∗ (tp )s .x; (tp ) :
(31)
Including these notations, the Hubbard–Stratonovich transformation of the quartic term with the contour time matrix A(tp(1)1 ; x1 ; tp(2)2 ; x2 ) takes the form [25,26] exp{−(.∗ s.)(tp(1)1 ; x1 ) ⊗ A(tp(1)1 ; x1 ; tp(2)2 ; x2 ) ⊗ (.∗ s.)(tp(2)2 ; x2 )} = d[x (tp )] exp{− ⊗ A ⊗ } ×exp{− ⊗ A ⊗ (.∗ s.)} exp{− (.∗ s.) ⊗ A ⊗ } :
(32)
The auxiliary real 1eld x (tp ) does not depend on the electron–hole band label . This is a property of the fermion–fermion HFF and reservoir–fermion HB; F interactions which both contain the sign array s for electrons (particles) and holes (antiparticles). The independence of x (tp ) on the band label follows because one can sum over the bands in the bilinear pairs .∗ s. of the quartic interaction terms. Since the generating function Z[J˜ ] (19) has been transformed to a relation with bilinear anticommuting 1elds .x; (tp ), they can be integrated out so that a path integral with the auxiliary real variable x (tp ) remains. However, the driving laser 1eld term, related to HE (t), has pairs of .x; h (tp ) .x; e (tp ) and .x;∗ e (tp ) .x;∗ h (tp ) 1elds, but not .x;∗ h (tp ) .x; e (tp ) and .x;∗ e (tp ) .x; h (tp ), as in comparison to the other terms. Therefore, the bilinear Grassmann variables have to be reordered in the exponential of Z[J˜ ] (19) according to the new 1eld x; (tp ) .x; e (tp ); = e; x; (tp ) = (33) .x;∗ h (tp ); = h : F 2; F : : : for the contour Notations are further simpli1ed by using the barred numbers 1; time, 3-D space vector and band label and by abbreviating " the contour integral and summation over spatial coordinates and band index with 1F ::: = dtp(1)1 ::: : (34) 1F = (tp(1)1 ; x1 ; 1 ); 1F
C
x1 ;1
Applying the simplifying notations (30) and (34), the generating function, reordered with the 1elds x; (tp ), is given by the following relation: Z[J ] = N d[x (tp )] exp{− ⊗ A ⊗ } d[x; (tp )]
B. Mieck / Physica A 312 (2002) 431 – 446
F F 2) F + 8(1; F 2) F + J (1; F 2))( F F ×exp − + (1)(S( 1; 2) :
439
(35)
F 2F 1;
F 2), F 8(1; F 2) F and J (1; F 2) F on the time contour are related to the kiThe matrices S(1; netic energy and laser 1eld term, to the self-energy from the Hubbard–Stratonovich transformation and to the source term @ (1) (2) (1) (2) F F S(1; 2) = S(tp1 ; x1 ; 1 ; tp2 ; x2 ; 2 ) = 6p1 5x1 ;x2 5(tp1 − tp2 ) 51 ;2 @tp(2)2 ! 1 + Ex2 (tp(2)2 )(a1 b2 + b1 a2 ) ; (36) + s 1 1 ∇x ˝ 2 F 2) F = 6p1 5x1 ;x2 51 ;2 5(tp(1) − tp(2) ) 8(1; 1 2 × ⊗ A(: : : ; tp(2)2 ; x2 ) + A(tp(1)1 ; x1 ; : : :) ⊗ :
(37)
Having integrated over the anticommuting 1elds, we obtain a determinant or exp{Tr ln : : :} term in the generating function with the variable x (tp ) on the time contour Z[J ] = N d[x (tp )] exp{− ⊗ A ⊗ } exp{Tr ln(S + 8 + J )} : (38) The trace “Tr” over the logarithm of the sum of matrices S, 8, J has to be de1ned as a contour integral and summation over space coordinates and band index Tr[Matrix] = dtp Matrix(tp ; x; ; tp ; x; ) : (39) C
x;
In order to derive the Langevin equation, the mean part Fx (t) and the di4erence 5x (t) of the plus–minus branches of the auxiliary variable x (tp ) are introduced x (t+ ) = Fx (t) +
1 5x (t); 2
1 Fx (t) = (x (t+ ) + x (t− )); 2
x (t− ) = Fx (t) −
1 5x (t) ; 2
5x (t) = x (t+ ) − x (t− ) :
(40) (41)
As in the bosonic case (compare Eqs. (49) and (50) in [13]), one performs an exchange of integration variables from x (t+ ), x (t− ) to Fx (t), 5x (t) in Z[J ] and expands the exponential of Z[J ] up to second order in 5x (t). The 1rst order term of 5x (t) multiplies the noise fx (t), and the second-order term gives after integration over the -uctuating part 5x (t) the probability distribution of the noise term fx (t). These steps will be brie-y described in the remainder of this section. The transformation of the Gaussian part exp{− ⊗ A ⊗ } from the contour time variable x (tp ) to Fx (t), 5x (t) is straightforward. One has to apply the relation (28) for A(tp(1)1 ; x1 ; tp(2)2 ; x2 ) and the special form of the non-equilibrium Green’s function
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B. Mieck / Physica A 312 (2002) 431 – 446
GB; (tp(1)1 ; tp(2)2 ) (25) for the reservoir states . The result is as follows exp{− ⊗ A ⊗ } = exp −
∞
t0
×exp
dt dt
∞
×exp − 2 ˝ (2)
(1)
dt dt
(1)
(1)
(2)
(1)
(2)
(2)
5x1 (t )M (t ; x1 ; t ; x2 )5x2 (t )
(2)
(1)
∞
t0
dt
1
(2)
K(t ; x1 ; t ; x2 ) = 2 Im
5x1 (t)V|x1 −x2 | Fx2 (t)
(42)
x1 ; x2
gx∗ ;
(2)
5x1 (t )K(t ; x1 ; t ; x2 )Fx2 (t )
x1 ; x2
M (t ; x1 ; t ; x2 ) =
(1)
(2)
x1 ; x2
t0
(1)
(1)
˝
e
− ! (t (1) −t (2) )
gx∗ ; 1
˝
2(t
(1)
n!
(2)
1 + 2
− t )e
g; x2 ; ˝
(43)
− ! (t (1) −t (2) ) g; x2
˝
! :
(44)
The matrix M (t (1) ; x1 ; t (2) ; x2 ) (43) in the second-order term with 5x (t) determines a part of the probability distribution for the noise and K(t (1) ; x1 ; t (2) ; x2 ) (44) already is a dissipative part in the 1nal Langevin equation. Both terms are related by the general -uctuation–dissipation theorem (cf. Eqs. (35) and (36) in Ref. [13]). However, it is not suQcient to expand only the Gaussian exp{− ⊗ A ⊗ } in the generating function Z[J ] (38). The trace-logarithm term of the sum of matrices S, 8, J in (38) has also F 2) F is the only to be expanded up to second order in 5x (t). Since the self-energy 8(1; term in (38) which contains the contour time variable x (tp ), we introduce a mean F 1; F 2) F and the di4erence 58(1; F 2) F in accordance to the mean value Fx (t) and value 8( the di4erence 5x (t) (see Eqs. (40) and (41)). F 2; F : : :, In the following relations one has to distinguish between the barred numbers 1; which abbreviate a contour time, a space vector and band index, and the primed num bers 1 ; 2 ; : : : which refer to the ordinary time t (1) ; t (2) ; : : : and spatial " vector x1 ; x2 ; : : : without the band index. This also holds for summation symbols; "1F is written for a contour integral, spatial summation including band label, whereas 1 just replaces the ordinary time integral and summation of space points F 1; F 2) F = 8(t F p(1) ; x1 ; 1 ; tp(2) ; x2 ; 2 ) = 6p1 5x1 ;x2 51 ;2 5(tp(1) − tp(2) ) 8( 1 2 1 2 ∞ × dt Fx (t )6r [A(tr ; x ; tp(2)2 ; x2 ) + A(tp(1)1 ; x1 ; tr ; x )] t0
1 = t (1) ; x1 ;
(45)
x r=±
1
::: =
∞
t0
dt (1)
x1
:::
(46)
B. Mieck / Physica A 312 (2002) 431 – 446
F = 58(1)
441
˜ 1; F 1 )5(1 ) ; 58(
1
(47)
˜ p(1) ; x1 ; 1 ; t (1) ; x1 ); ˜ 1; F 1 ) = 58(t 58( 1 = [A(tr(1) ; x1 ; tp(1)1 ; x1 ) + A(tp(1)1 ; x1 ; tr(1) ; x1 )]6p1 : 2 r=±
(48)
Applying relations (45) – (48), we obtain for the Tr ln : : : term in (38) up to second F 2) F the following expressions: order in 5x (t) and 1rst order in J (1; exp{Tr ln(S + 8 + J )} = exp{Tr ln(S + 8)} ×exp{Tr ln(1 + (S + 8)−1 J )} ;
(49)
F exp{Tr ln(S + 8)} = exp{Tr ln(S + 8)} F −1 (1; F 1)58( F F ×exp (S + 8) 1) 1F
1 F −1 (1; F 2)58( F F F −1 (2; F 1)58( F F ×exp − (S + 8) 2)(S + 8) 1) : 2 FF
(50)
1;2
exp{Tr ln(1 + (S + 8)−1 J )} = exp
F −1 (1; F 2)J F (2; F 1) F (S + 8)
F 2F 1;
F −1 (1; F 2)58( F F F −1 (2; F 3)J F (3; F 1) F ×exp − (S + 8) 2)(S + 8) F 2; F 3F 1;
×exp
F −1 (1; F 2)58( F F F −1 (2; F 3)58( F F (S + 8) 2)(S + 8) 3)
F 2; F 3; F 4F 1;
F −1 (3; F 4)J F (4; F 1) F × (S + 8)
:
(51)
In the next step one has to reorder the complex relations (50) and (51) according to F 2). F Furthermore, the terms of 5x (t) of the Gaussian their order in 5x (t) and J (1; exp{− ⊗ A ⊗ } with matrices M (t (1) ; x1 ; t (2) ; x2 ) and K(t (1) ; x1 ; t (2) ; x2 ) have to be included. As the result we obtain a matrix B0 (1 ; 2 ) for the second moment of the noise, F 4) F which multiplies the source J (4; F 1) F and which shifts the matrix a matrix B1 (1 ; 2 ; 1; B0 (1 ; 2 ) in the exponent of Z[J ]. The 1rst-order term in 5x (t) multiplies the noise fx (t) whose expression is the 1nal Langevin equation. Moreover, there is a matrix F 3) F which is combined with the source term J . All the mentioned terms B0 (1 ; 2 ), g(1 ; 2;
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B. Mieck / Physica A 312 (2002) 431 – 446
F 4), F fx (t) and g(1 ; 2; F 3) F follow from the expansion of the exponent of Z[J ] B1 (1 ; 2 ; 1; (38) up to second order in the di4erence 5x (t) of the plus–minus contour branches F 2). F The Jacobian for the change of x (tp ) and up to 1rst order in the source term J (1; of variables from x (t+ ), x (t− ) to Fx (t), 5x (t) is unity. Therefore, the expansion of Z[J ] up to second order in 5x (t) yields the following relations:
B0 (1 ; 2 ) = B0 (t (1) ; x1 ; t (2) ; x2 ) = M (1 ; 2 ) +
1 F −1 (2; F 1)5 F 8( ˜ 1; F 2 ) ; F 2)5 F 8( ˜ 2; F 1 )(S + 8) F −1 (1; (S + 8) 2 FF
(52)
1;2
F 4) F = B1 (1 ; 2 ; 1;
F −1 (1; F 2)5 F 8( ˜ 2; F 1 ) (S + 8)
F 3F 2;
F 3)5 F 8( ˜ 3; F 2 )(S + 8) F −1 (3; F 4) F ; F −1 (2; ×(S + 8)
fx1 (t (1) ) = f(1 ) = −2
V|x −x | 1
x2
˝
2
Fx2 (t (1) )
F −1 (1; F 1)5 F 8( ˜ 1; F 1 ) ; K(1 ; 2 )(2 F ) − (S + 8) +
(54)
1F
2
F 3) F = g(1 ; 2;
(53)
F −1 (2; F 1)5 F 8( ˜ 1; F 1 )(S + 8) F −1 (1; F 3) F ; (S + 8)
(55)
1F
Z[J ] = N
F d[Fx (t)] d[5x (t)] exp{Tr ln(S + 8)}
F 4)J F (4; F 1) F 5(2 ) ×exp − 5(1 ) B0 (1 ; 2 ) − B1 (1 ; 2 ; 1; F 4F 1;
1 ;2
F 3)J F (3; F 2) F ×exp 5(1 ) f(1 ) + g(1 ; 2; 1
×exp
F 2F 1;
F 3F 2;
F −1 (1; F 2)J F (2; F 1) F (S + 8)
:
(56)
It remains to perform the integration over 5x (t) and the change of variables from Fx (t) to the noise fx (t). Since the Langevin equation (54) has a causal structure (fx (t) only depends on previous and equal times of Fx (t)), the Jacobian for the exchange of variables from Fx (t) to fx (t) is a constant which is absorbed in the normalization factor N. The generating function Z[J ] (56), from which the density matrix elements
B. Mieck / Physica A 312 (2002) 431 – 446
443
y;+ (t) y; (t) (11) and (14) follow by taking appropriate derivatives of the source F 2), F is therefore given in terms of the noise fx (t) by the expression term J (1; F Z[J ] = N d[f(1 )] exp{Tr ln(S + 8)}
×exp
F 2)J F (2; F 1) F F −1 (1; (S + 8)
F 2F 1;
1 F 4)J F (4; F 1) F ×exp − tr ln B0 (: : : ; : : :) − B1 (: : : ; : : : ; 1; 2 F F 1;4
1 F 3)J F (3; F 2) F f(1 ) + ×exp − g(1 ; 2; 4 FF 1 ;2
2;3
× B0 (: : : ; : : :) −
−1 F 4)J F (4; F 1) F B1 (: : : ; : : : ; 1;
(1 ; 2 )
F 4F 1;
× f(2 ) +
F 6F 5;
F (6; F 5) F 6)J F : g(2 ; 5;
(57)
By taking derivatives of Z[J ] (57) with respect to the source term, one can generate the density matrix elements. 3. Results and discussion The 1nal Langevin equation with the noise fx (t) and -uctuating self-energy or potential Fx (t) is determined by Eq. (54). The -uctuating variable Fx (t) is now a F 1; F 2) F functional of the noise fx (t). This also holds for the self-energy matrix 8( Fx (t) = Fx [t; fx (t )]; 2
V|x −x | 1
x2
=
˝
2
2
F 8F = 8[f] ;
(58)
Fx2 (t (1) )
K(1 ; 2 ) (2 F ) −
F −1 (1; F 1) F 58( ˜ 1; F 1 ) − f(1 ) : (S + 8)
(59)
1F
If we compare the Langevin equation (59) with the saddle point equation (68) in Ref. [29], the di4erence is found in the noise term f(1 ) and dissipative part K(1 ; 2 ) of (59). The 1nal Langevin equation (59) is obviously a modi1ed saddle point equation as in Ref. [29], but where the noise fx (t) includes now contributions around the saddle
444
B. Mieck / Physica A 312 (2002) 431 – 446
point of the time contour so that thermal -uctuations of a reservoir (e.g. phonons) are considered. According to the general dissipation–-uctuation theorem, the noise is accompanied by dissipation [27]. This leads to the term with K(1 ; 2 ) and the special F 1; F 2) F in the Langevin equation. form of 8( The probability distribution P[f(1 )] for the noise follows by performing appropriate derivatives of Z[J ] (57); e.g. in order to calculate the density matrix element y;+e (t) y; e (t), the derivative of Z[J ] (57) has to be computed with respect to J (t− ; y; e; t+ ; y; e) and by setting J to zero. This yields the relations ' Z[J ] '' @ + y; e (t) y; e (t) = @J (t− ; y; e; t+ ; y; e) Z[0] 'J =0 1 F d[f(1 )] exp{Tr ln(S + 8[f])} = Z[0] −1 F ×(S + 8[f]) (t+ ; y; e; t− ; y; e);
P[f(1 )]
(60)
1 P[f(1 )] ˙ [det(B0 )]−1=2 exp − f(2 )B0−1 (2 ; 3 )f(3 ) 4
;
(61)
2 ;3
where P[f(1 )] in the integrand is the probability distribution for the noise f(1 ). The numerical computation of density matrix elements as (60) is given by the following procedures: (1) Random numbers f(1 ) have to be generated according to the distribution P[f(1 )] (61). If one considers the exact distribution as given in (61), the probability distribution becomes non-Gaussian because the matrix B0−1 (2 ; 3 ) contains the F self-energy 8F = 8[f]. By performing continued fractions on the Langevin equation (59) for an arbitrary noise term fx (t), the non-Gaussian probability distribution can be determined. (2) A random sequence of numbers for fx (t) according to P[f(1 )] is applied to the Langevin equation (59) which is then solved numerically by continued fraction in order to obtain Fx (t) from (59). (3) The realized, -uctuating potential Fx (t) is used to compute the terms exp{Tr ln(S + F (S + 8) F −1 (t+ ; y; e; t− ; y; e) in the integrand of (60) with the special form for 8)} the self-energy 8F given by Eq. (45). F (S + 8) F −1 (t+ ; y; e; t− ; y; e) over (4) One has to average the terms exp{Tr ln(S + 8)} many di4erent realizations according to the probability distribution P[f(1 )] (61). Other density matrix elements as y;+h (t) y; h (t) and y; h (t) y; e (t) follow by taking appropriate derivatives of (57) with respect to the source term J ' Z[J ] '' @ + y; h (t) y; h (t) = − @J (t+ ; y; h; t− ; y; h) Z[0] 'J =0
B. Mieck / Physica A 312 (2002) 431 – 446
=−
1 Z[0]
445
F d[f(1 )] exp{Tr ln(S + 8[f])}
−1 F (t− ; y; h; t+ ; y; h) ×(S + 8[f])
P[f(1 )] ;
(62)
' Z[J ] '' @ y; h (t) y; e (t) = @J (t− ; y; h; t+ ; y; e) Z[0] 'J =0 1 F d[f(1 )] exp{Tr ln(S + 8[f])} = Z[0] −1 F ×(S + 8[f]) (t+ ; y; e; t− ; y; h)
P[f(1 )] :
(63)
The same probability distribution P[f(1 )] as in (60), with which one has to average the remaining integrand, is obtained. The computation for the various density matrix elements di4ers only in the calculation of the corresponding matrix elements of (S + −1 F F 8[f]) . The inverse of the matrix (S + 8[f]) is related to a modi1ed contour time Green’s function [17–23]. Cum grano salis, one averages in relations (60), (62), (63) for density matrix elements over modi1ed non-equilibrium Green functions [15] so that noise and dissipation are considered on a quantum level. If one wants to calculate higher order correlation functions, which e.g. require the second derivative of Z[J ] with respect to the source term, one has to expand the exponent of the integrand of Z[J ] (38) up to second order in 5x (t) and has also to include second-order terms of the source J in the expansion of exp{Tr ln(1 + (S + 8)−1 J )} (51).
Acknowledgements This work has been supported by the Sonderforschungsbereich 237 “Unordnung und groTe Fluktuationen”.
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