Short-time kinetics and initial correlations in nonideal quantum systems

There are many reasons to consider quantum kinetic equations describing the evolution of a many-particle system on ultrashort time scales, e. g. correct energy conservation in nonideal plasmas, buildup of correlations (bound states), and kinetics of ultrafast processes in short-pulse laser plasmas...

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Дата:	2006
Автори:	Kremp, D., Semkat, D., Bornath, Th.
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Опубліковано:	Інститут фізики конденсованих систем НАН України 2006
Назва видання:	Condensed Matter Physics
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Цитувати:	Short-time kinetics and initial correlations in nonideal quantum systems / D. Kremp, D. Semkat, Th. Bornath // Condensed Matter Physics. — 2006. — Т. 9, № 3(47). — С. 431–446. — Бібліогр.: 42 назв. — англ.

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spelling	irk-123456789-1213682017-06-15T03:04:36Z Short-time kinetics and initial correlations in nonideal quantum systems Kremp, D. Semkat, D. Bornath, Th. There are many reasons to consider quantum kinetic equations describing the evolution of a many-particle system on ultrashort time scales, e. g. correct energy conservation in nonideal plasmas, buildup of correlations (bound states), and kinetics of ultrafast processes in short-pulse laser plasmas. We present a quantum kinetic theory including initial correlations and being valid on arbitrary time scales. Various examples of short-time relaxation processes, such as relaxation of the distribution function and the energy, are shown. Furthermore, gradient expansions of the general equations and the role of bound states are discussed. Є багато причин щоб розглядати квантовi кiнетичнi рiвняння, що описують еволюцiю багаточастинкової системи в ультракоротких часових масштабах, наприклад збереження енергiї у неiдеальнiй плазмi, наростання кореляцiй (зв’язанi стани) та кiнетика ультрашвидких процесiв у короткоiмпульснiй лазернiй плазмi. Представлено квантову кiнетичну теорiю, що включає початковi кореляцiї та є справедливою на довiльних часових масштабах. Показано рiзнi приклади короткочасових релаксацiйних процесiв, такi як релаксацiя функцiї розподiлу та енергiї. Крiм того обговорюється градiєнтний розклад загальних рiвнянь та роль зв’язаних станiв. 2006 Article Short-time kinetics and initial correlations in nonideal quantum systems / D. Kremp, D. Semkat, Th. Bornath // Condensed Matter Physics. — 2006. — Т. 9, № 3(47). — С. 431–446. — Бібліогр.: 42 назв. — англ. 1607-324X PACS: 05.20.Dd, 52.25.Dg, 52.27.Gr DOI:10.5488/CMP.9.3.431 http://dspace.nbuv.gov.ua/handle/123456789/121368 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	There are many reasons to consider quantum kinetic equations describing the evolution of a many-particle system on ultrashort time scales, e. g. correct energy conservation in nonideal plasmas, buildup of correlations (bound states), and kinetics of ultrafast processes in short-pulse laser plasmas. We present a quantum kinetic theory including initial correlations and being valid on arbitrary time scales. Various examples of short-time relaxation processes, such as relaxation of the distribution function and the energy, are shown. Furthermore, gradient expansions of the general equations and the role of bound states are discussed.
format	Article
author	Kremp, D. Semkat, D. Bornath, Th.
spellingShingle	Kremp, D. Semkat, D. Bornath, Th. Short-time kinetics and initial correlations in nonideal quantum systems Condensed Matter Physics
author_facet	Kremp, D. Semkat, D. Bornath, Th.
author_sort	Kremp, D.
title	Short-time kinetics and initial correlations in nonideal quantum systems
title_short	Short-time kinetics and initial correlations in nonideal quantum systems
title_full	Short-time kinetics and initial correlations in nonideal quantum systems
title_fullStr	Short-time kinetics and initial correlations in nonideal quantum systems
title_full_unstemmed	Short-time kinetics and initial correlations in nonideal quantum systems
title_sort	short-time kinetics and initial correlations in nonideal quantum systems
publisher	Інститут фізики конденсованих систем НАН України
publishDate	2006
url	http://dspace.nbuv.gov.ua/handle/123456789/121368
citation_txt	Short-time kinetics and initial correlations in nonideal quantum systems / D. Kremp, D. Semkat, Th. Bornath // Condensed Matter Physics. — 2006. — Т. 9, № 3(47). — С. 431–446. — Бібліогр.: 42 назв. — англ.
series	Condensed Matter Physics
work_keys_str_mv	AT krempd shorttimekineticsandinitialcorrelationsinnonidealquantumsystems AT semkatd shorttimekineticsandinitialcorrelationsinnonidealquantumsystems AT bornathth shorttimekineticsandinitialcorrelationsinnonidealquantumsystems
first_indexed	2025-07-08T19:43:50Z
last_indexed	2025-07-08T19:43:50Z
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fulltext	Condensed Matter Physics 2006, Vol. 9, No 3(47), pp. 431–446 Short-time kinetics and initial correlations in nonideal quantum systems D.Kremp, D.Semkat, Th.Bornath Universität Rostock, Institut für Physik, D–18051 Rostock, Germany Received March 16, 2006, in final form May 12, 2006 There are many reasons to consider quantum kinetic equations describing the evolution of a many-particle system on ultrashort time scales, e. g. correct energy conservation in nonideal plasmas, buildup of correlations (bound states), and kinetics of ultrafast processes in short-pulse laser plasmas. We present a quantum kinetic theory including initial correlations and being valid on arbitrary time scales. Various examples of short-time relaxation processes, such as relaxation of the distribution function and the energy, are shown. Furthermore, gradient expansions of the general equations and the role of bound states are discussed. Key words: kinetic theory, real-time Green’s functions, short-time relaxation PACS: 05.20.Dd, 52.25.Dg, 52.27.Gr 1. Introduction Nonequilibrium properties of many-particle systems are successfully described by kinetic equa- tions of the Boltzmann type. In the quantum case, the Boltzmann equation has the shape { ∂ ∂t + ∂Ea ∂p ∂ ∂R − ∂Ea ∂R ∂ ∂p } fa(pRt) = Ia(pRT ), (1) where fa is the Wigner distribution. The most interesting term in this equation is the collision integral Ia(pRT ) given here in binary collision approximation I(p1RT ) = 2π ~ ∫ dp2dp̄1dp̄2 (2π~)9 δ ( E12 − Ē12 ) ∣ ∣〈p1p2\|T + (E12 + iε) \|p̄2p̄1〉 ∣ ∣ 2 × ( f̄1f̄2 (1 ± f1) (1 ± f2) − f1f2 ( 1 ± f̄1 ) ( 1 ± f̄2 )) , (2) and having the typical features of a Boltzmann-like collision integral: the kinetic energy conserving delta function, the transition probability in two-particle collisions given by the on-shell T-matrix and the usual combination of Wigner functions describing the occupation at the beginning and the end of the collision. In spite of the fundamental character of Boltzmann-like kinetic equations, which describe the irreversible relaxation of an arbitrary initial Wigner distribution to the equilibrium distribution and, moreover, being the basic equations of transport theory, there exist many principle shortcomi- ngs. This was pointed out already by Prigogine [1] and Resibois [2], Kadanoff and Baym [3] and by Klimontovich [4]. The Boltzmann equation 1. cannot describe the short-time behavior (t < tcorr) correctly, 2. conserves the kinetic energy 〈T 〉 only instead of the total energy 〈T 〉+〈V 〉 which is unphysical, especially for strongly correlated many-particle systems, 3. does not describe, because of the on-shell T-matrix, the influence of bound states on the kinetics, 4. gives as equilibrium solution the distribution for ideal particles. c© D.Kremp, D.Semkat, Th.Bornath 431 D.Kremp, D.Semkat, Th.Bornath These shortcomings follow from a certain assumption necessary for the derivation of the Boltz- mann equation from the basic equations of statistical mechanics. Let us consider this idea more in detail. The relaxation of a strongly correlated many-particle system from the initial state to the equilibrium state runs over several stages as shown in figure 1. In the vicinity of the initial time, the evolution is strongly influenced by the initial condition, especially by the initial correlations. Figure 1. Stages of relaxation of a many-particle system. We observe in this initial stage the buildup of correlations, and the kinetics is a non-Markovian one, i. e., there are memory effects. For times bigger than the correlation time1, the time evolution of the system is determined only by the one-particle density operator. All higher density operators are now functionals of the latter one. Initial correlation and memory effects are damped out, the kinetics is Markovian. Finally, after the collision time, we have local equilibrium and, therefore, a reduction of the phase space to the configuration space description. Let us first consider a simple description of the relaxation process. We start from the Bogolyubov hierarchy for the reduced density operators in binary collision approximation. In this approximation there follows a closed system of equations i~ dF1 dt − [HHF 1 , F1] = nTr [V12, g12], (3) i~ dg12 dt − [HHF 12 + V12, g12] = [V12, F1F2]. (4) It is easy to find a formal solution of the second equation g12(t) = U(t, t0)g12(t0)U(t0, t) + 1 i~ ∫ t−to 0 dτU (t, t− τ) [V12, F (t− τ)F (t− τ)]U (t− τ, t) . (5) Here, the first term describes the influence of the initial correlations on the time evolution of the system. The U(t, t0) are two-particle propagators. The second term describes the non-Markovian buildup of correlations, i. e., the value of the correlation operator at the time t is determined by all previous times. Inserting now (5) into the first equation (3) leads to a closed non-Markovian equation for the one-particle density operator which describes the short-time behavior and the influence of initial correlations. The Boltzmann equation follows from this equation under very restrictive assumptions: i) We apply the Bogolyubov condition of weakening of initial correlations (no correct short-time behavior), i. e. lim t0→−∞ g12(t0) = 0. (6) ii) We neglect the retardation (destruction of the correct energy conservation strongly influences the short-time behavior). With condition (6), the initial stage is completely neglected. But there are many reasons to extend the description into the initial region. We here mention only, for example, correct energy conservation in nonideal plasmas, buildup of correlations, and kinetics of ultrafast processes, e. g. in short pulse laser plasmas. 1The correlation time can be defined as the duration of the interaction of two particles, i. e., the range of the interaction potential over the characteristic particle velocity (e. g. the thermal velocity). For plasmas, tcorr is equal to the inverse plasma frequency. 432 Short-time quantum kinetics Of course the pure binary collision approximation is too simple, especially in describing the damping in the propagators U and, therefore, the weakening of initial correlations and other many-particle effects like Pauli blocking, self-energy, screening of the Coulomb potential in plasmas, lowering of the ionization energy etc. A generalization of the equations (3,4) is given in [5] and [6]. In this paper we shall consider these problems from a more general point of view of the powerful real-time Green’s functions [3,7–10]. The original scheme of real-time Green’s functions contains no contribution of initial correlati- ons. In order to include them, several methods have been used, e. g. analytical continuation to real times which permits to include equilibrium initial correlations [3,11,12], and perturbation theory with initial correlations [13,14]. A convincing solution has been presented by Danielewicz [8]. He developed a perturbation theory for a general initial state and derived generalized Kadanoff-Baym equations that take into account arbitrary initial correlations. A straightforward and very intuitive method which is not based on perturbation theory uses the equations of motion for the Green’s functions [15–18]. It will be presented in the next sections. Some effects of the initial stage can be taken into account already by expanding the non- Markovian equation with respect to the time retardation and spatial nonlocality [1,3,19–21]. There- fore, we shall consider such an expansion in the last section of the paper in order to discuss the transition from the two-time Kadanoff-Baym equations (KBE) to the usual Markovian kinetic equation for the Wigner function. 2. Martin-Schwinger hierarchy. Initial correlation The formalism of real-time Green’s functions describes the many-particle system by a system of single-, two-, and more-particle Green’s functions. We consider the definition of the single-particle Green’s function as an average over the time-ordered product of field operators ψ(1) and ψ†(1′), g(1, 1′) = 1 i~Tr { %TC { S(t0, t0)ψ(1)ψ†(1′) }} Tr {TCS(t0, t0)} (7) with S(t0, t0) = TC exp { −i ∫ C d2d2̄U(2, 2̄)ψ†(2)ψ(2̄) } . TC denotes the ordering along the Keldysh contour [7,22]. The field operators are given in the interaction picture with respect to a formal external potential U(1, 1′). Their equation of motion is ( i~ ∂ ∂t1 + ~ 2∇2 1 2m ) ψ(1) = ± ∫ d2V (1 − 2)ψ†(2)ψ(2)ψ(1)\|t1=t2 . (8) The density operator is in general a nonequilibrium one. In this case the adiabatic theorem is not valid and, therefore, we have a causal and anticausal time ordering. In order to handle this problem for nonequilibrium systems, the time ordering on the physical time axis has to be replaced by the ordering along the upper and lower branch of the Keldysh contour, see figure 2. Figure 2. Keldysh time contour. Now we introduce the following notation: times t located on the upper (+) branch are denoted by t+, times located on the lower branch (–) by t−. The expression (7) is a very compact representation of the causal and anticausal single-particle Green’s functions gc(a)(1, 1′) and the two correlation 433 D.Kremp, D.Semkat, Th.Bornath functions g>(<)(1, 1′). In dependence on the position of the two times t and t′ on the Keldysh contour, we get these functions by g(1+, 1 ′ +) = gc(1, 1′), g(1−, 1 ′ +) = g>(1, 1′), g(1−, 1 ′ −) = ga(1, 1′), g(1+, 1 ′ −) = g<(1, 1′). (9) Of special interest is the two-time correlation function g<(1, 1′). The time-diagonal part of this function is just the Wigner function. In order to derive the equations of motion for the Green’s functions (7) on the Keldysh contour, we have to introduce the equation of motion for the field operators (8) into the definition of the single-particle Green’s function (7). Then follows immediately ∫ C d1̄ { g−1 0 (1, 1̄) − U(1, 1̄) } g(1̄, 1′;U) = δ(1 − 1′) ± i~ ∫ d2V (1 − 2)g12(12, 1′2+;U) , (10) with g−1 0 (1, 1′) = ( i~ ∂ ∂t1 + ~ 2∇2 1 2m ) δ(1 − 1′) . This equation is not closed. Due to the interaction between the particles, there is a coupling with the two-particle Green’s function and so on. Equation (10) is therefore a member of an infinite chain of equations for many-particle Green’s functions, well known as Martin-Schwinger hierarchy [23]. To find a closed equation for the one-particle Green’s function, we proceed in well-known manner. First, we define the self-energy by ∫ C d1̄ Σ(1, 1̄)g(1̄, 1′) = ±i~ ∫ d2V (1 − 2)g12(12, 1′2+) = ±i~ ∫ d2V (1 − 2) { ± δg(1, 1′;U) δU(2+, 2) + g(1, 1′)g(2, 2+) } . (11) Here we took into account that g12 can be derived from g by means of functional derivation. Next, we introduce this definition into the first equation of the Martin-Schwinger hierarchy. Then follows immediately ∫ C d1̄ { g−1 0 (1, 1̄) − U(1, 1̄) − Σ(1, 1̄)) } g(1̄, 1′) = δ(1 − 1′). (12) This equation is a Dyson equation for nonequilibrium systems which was derived for the first time by Kadanoff and Baym [3] and by Keldysh [7]. Of course, all problems are now transferred into the self-energy. Therefore, one has to consider the properties of this function and to find a possibility to determine appropriate approximations for Σ. Let us first consider the dependence of Σ on the initial value of the two-particle Green’s function. From (11) it is obvious that, in the limit t→ t0, the self-energy has to fulfill the important relation lim t1=t′ 1 =t0 ∫ C d1̄Σ(1, 1̄)g(1̄, 1′) = ±i~ ∫ dr2 V (r1 − r2) g12(r1r2, r ′ 1r2; t0) = ±i~ ∫ dr2 V (r1 − r2) {g(1, 1 ′)g(2, 2+)t0 ± g(1, 2+)g(2, 1′)t0 + c(r1r2, r ′ 1r2; t0) } , (13) which means that Σ depends on an arbitrary initial value g12(r1r2, r ′ 1r2; t0) of the two-particle cor- relation function. Consequently, it is necessary to introduce initial conditions in order to determine the self-energy uniquely. However, for a regular integrand, the integral over the Keldysh contour vanishes in the limit t, t′ → t0. Therefore, equation (13) can be fulfilled only if the self-energy has a contribution proportional to a δ-function with respect to the times. Such a term is the Hartree- Fock contribution ΣHF. But this term produces only the uncorrelated contribution to the initial 434 Short-time quantum kinetics binary density matrix. This means that the self-energy must contain a second part with a δ-like singularity. The general structure of the self-energy is thus given by Σ(1, 1′) = ΣHF(1, 1′) + Σc(1, 1′) + Σin(1, 1′), Σin(1, 1′) = Σin(1, r′1t0)δ(t ′ 1 − t0). (14) It has to be noted that in the adjoint Dyson equation there occurs the self-energy Σ̂ having another initial contribution Σin [17]. In order to discuss the influence of the initial correlations on the time evolution of the single- particle Green’s function, we insert (14) into equation (12) and fix the time arguments to opposite branches on the contour. Then we get the Kadanoff-Baym equations (KBE) for the correlation functions ( i~ ∂ ∂t1 + ~ 2∇2 1 2m ) g≷(1, 1′) − ∫ d1̄U(1, 1̄)g≷(1̄,1′) = ∫ dr̄1 ΣHF(1, 1̄)g≷(1̄,1′) − t1 ∫ t0 d1̄ [Σ>(1, 1̄) − Σ<(1, 1̄)]g≷(1̄,1′)+ t 1′ ∫ t0 d1̄ [ Σ≷(1, 1̄) + Σin(1, 1̄) ] [ g<(1̄,1′) − g>(1̄,1′) ] . (15) In contrast to the original KBE [3], there are two new properties: (i) the equations are valid for arbitrary initial times t0, (ii) they contain an additional self-energy contribution Σin which describes the influence of initial correlations. These equations are very general and lead, in the time-diagonal form and for special approximations for the self-energy, to all well-known standard kinetic equations for the Wigner distribution. It is very useful to introduce retarded and advanced quantities by FR/A = ±Θ[±(t − t′)] {F>(t, t′) − F<(t, t′)}. Then the KBE take the convenient form ( i~ ∂ ∂t1 + ~ 2∇2 1 2m ) g≷(1, 1′) − ∫ d1̄U(1, 1̄)g≷(1̄,1′) − ∞ ∫ −∞ d1̄ΣR(1, 1̄)g≷(1̄, 1′) = ∞ ∫ −∞ d1̄ [ Σ≷(1, 1̄) + Σin(1, 1̄) ] gA(1̄, 1′) , (16) ( i~ ∂ ∂t1 + ~ 2∇2 1 2m ) gR/A(1, 1′)− ∫ d1̄U(1, 1̄)gR/A(1̄,1′) − ∞ ∫ −∞ d1̄ΣR/A(1, 1̄)gR/A(1̄, 1′) = δ(1−1′). (17) In order to determine the self-energy uniquely, boundary or initial conditions are necessary. Under the condition tcorr < tcoll the Bogolyubov condition of weakening of initial correlations, lim t0→−∞ g12(12, 1′2′)\|t0 = [g(1, 1′)g(2, 2′) ± g(1, 2′)g(2, 1′)]t0 , may be applied. With this condition follows Σin = 0, and we get the original KBE. This was shown in [15]. The most general and natural idea to fix the solution of the hierarchy for real times, however, is an initial condition g12(12, 1′2′)\|t0 = g(1, 1′)g(2, 2′)\|t0 ± g(1, 2′)g(2, 1′)\|t0 + c(r1r2, r ′ 1r ′ 2; t0) . (18) Here, c(r1r2, r ′ 1r ′ 2; t0) is the initial correlation. 3. Self-energy and initial correlations The central problem is to determine the self-energy including initial correlations. In order to find the self-energy function, we start with (11) and, therefore, need the functional derivative δg/δU . 435 D.Kremp, D.Semkat, Th.Bornath Fortunately, a simple procedure for the calculation of this quantity is available [3]. Equation (12) can be written for t1, t ′ 1 > t0 in the form ∫ C g−1(1, 1̄)g(1̄, 1′)d1̄ = δ(1 − 1′) . (19) Here, the inverse Green’s function g−1 is given by g−1(1, 1̄) = g−1 0 (1, 1̄) − U(1, 1̄) − Σ(1, 1̄) . (20) Functional differentiation of (19) with respect to the external potential yields ∫ C d1̄ δg−1(1, 1̄) δU(2′, 2) g(1̄, 1′) = − ∫ C d1̄ g−1(1, 1̄) δg(1̄, 1′) δU(2′, 2) . (21) Using (20), the general solution of this equation and its adjoint can be found: δg(1, 1′) δU(2′, 2) = ±L(12, 1′2′) = g(1, 2′)g(2, 1′) + ∫ C d1̄d1 g(1, 1̄) δ [ Σ(1̄, 1) + Σin(1̄, 1) ] δU(2′, 2) g(1, 1′) ±C(12, 1′2′), (22) where C is an arbitrary function which obeys the homogeneous equation, i. e. ∫ C d1̄ g−1(1, 1̄)C(1̄2, 1′2′) = 0, (23) and where Σin is the initial correlation contribution to Σ̂. There are three similar conditions to (23). One condition follows from the crossing symmetry (1 ↔ 2) and two conditions follow from the adjoint equation of equation (12). The function C(12, 1′2′) makes it possible to take into account initial correlations. To show this we consider equation (22) in the limit t1, t ′ 1 → t0. Since the integral over the contour vanishes, it follows directly L(r1, r2, r ′ 1, r ′ 2, t0) = c(r1, r2, r ′ 1, r ′ 2, t0) ± g(r1, r ′ 2, t0)g(r2, r ′ 1, t0). (24) Hence, the function c(t0) has to be identified with initial binary correlations. In order to explore the temporal evolution of the function C(12, 1′2′), we solve equation (23) (and the three analogous relations) together with the initial condition C(12, 1′2′)\|t1=t2=t′ 1 =t′ 2 =t0 = C(t0). (25) The result is C(12, 1′2′) = ∫ C d1̄d2̄d1d2 g(1, 1̄)g(2, 2̄) c(1̄2̄, 1 2) g(2, 2′)g(1, 1′), (26) with c(1̄2̄, 1 2) = c(r̄1t0, r̄2t0, r1t0, r2t0)δ(t̄1 − t0)δ(t̄2 − t0)δ(t1 − t0)δ(t2 − t0). (27) For the self-energy Σ we find, introducing equations (22) and (26) into (11), the following functional equation Σ(1, 1′) = ±i~ ∫ d2V (1 − 2)    ± ∫ C d1̄ g(1, 1̄) δ[Σ(1̄,1′) + Σin(1̄,1′)] δU(2+, 2) + δ(1 − 1′)g(2, 2+) ± δ(2 − 1′)g(1, 2+) + ∫ C d1̄d2̄d2 g(1, 1̄)g(2, 2̄) c(1̄2̄,1′2) g(2, 2+)    . (28) 436 Short-time quantum kinetics For Σ̂, a similar equation follows readily [17]. With equation (28), the self-energy is given as a functional of the interaction, the initial correlations, and the one-particle Green’s function. From the definition of c, equation (27), it is obvious that the last contribution on the r.h.s. is local in time with a δ-type singularity at t = t′. Additional terms of this structure arise from the functional derivative. A further important property of the self-energy follows from comparing Σ, equation (28), with the corresponding expression for Σ̂. One verifies that Σ = Σ̂ for all times t1, t ′ 1 > t0, which means, in particular, that for these times, a well defined inverse Green’s function does exist. Equation (28) is well suited to derive approximations for the self-energy. By iteration, a per- turbation series for Σ in terms of g, V , and c can be derived which begins with Σ1(1, 1′) = ±i~δ(1 − 1′) ∫ d2V (1 − 2)g(2, 2+) ± i~ ∫ d2V (1 − 2) × ∫ C d1̄d2̄d2 g(1, 1̄)g(2, 2̄) c(1̄2̄,1′2) g(2, 2+) + exchange, (29) or, in terms of Feynman diagrams, Second order contributions are evaluated straightforwardly, too, with the result In contrast to the conventional diagram technique, the initial correlation appears now as a new basic element, drawn by a shaded rectangle. The analysis of the iteration scheme shows that all contributions to the self-energy (all diagrams) fall into two classes: (i) the terms ΣHF und Σc which begin and end with a potential, and (ii) Σin beginning with a potential, but ending with an initial correlation. This is in accordance with the structure of the self-energy, equations (14), discussed in the previous section. The same result was obtained by Danielewicz based on his perturbation theory for general initial states [8]. If one considers the first two iterations for the self-energy more in detail, it becomes evident that in the initial correlation contribution – in front of the function c – just the ladder terms appear which lead to the buildup of the two-particle Green’s function. Obviously, the iteration “upgrades” the product of retarded one-particle propagators in the function C to a full two-particle propagator, in the respective order, i. e., Σin is of the form2 Σin(1, 1′) = ±i~ ∫ d2V (1 − 2) ∫ dr̄1dr̄2dr1dr2 ×gR 12(12, r̄1t0, r̄2t0)c(r̄1t0, r̄2t0, 1 ′, r2t0)g A(r2t0, 2 +)δ(t′1 − t0). (30) The analytical properties of gR 12(12, r̄1t0, r̄2t0) give rise to a damping γ12 leading to a decay of the initial correlation for times t > 1/γ12 ∼ tcorr. Thus, after the initial stage of relaxation, the Bogolyubov condition of weakening of initial correlations is reproduced automatically. The method of including the initial binary correlations presented here is, undoubtedly, math- ematically rigorous. The question arises which initial conditions are physically meaningful, i. e., which restrictions exist concerning the form of the function c(t0). The answer to that question has, in principle, to be given by the actual experimental situation. It can be shown, however, that a sufficient condition for the initial state is that it can be produced by a preceding evolution from another, in particular an uncorrelated, state, see [24]. The latter statement is again tangent to another question – why one does not include the description of such a preceding evolution instead 2Explicit expressions for Σ and Σin in T-matrix approximation have been given in [17]. 437 D.Kremp, D.Semkat, Th.Bornath of constructing an initial correlation? The decision, of course, depends on the physical situation. However, there exist a variety of cases where the formation of the “initial” correlation is much more complicated than the result itself, consider, for example, the generation of a dense partially ionized plasma from a solid state target by an intense laser pulse. In such cases, the formalism developed above can be of high interest and usability. 4. Numerical illustration and discussion Let us come back to the Kadanoff-Baym equations (15). Analytical solutions of these complex equations are available only for free particles. But also numerical solutions are possible up to now only for relatively simple approximations for the self-energy [25]. A short review of the numerical scheme for the solution of the Kadanoff-Baym equations and some results are given, e. g., in papers of Köhler et al. [26,27], and in the textbook [28]. Here we present the results for the numerical solution of the Kadanoff-Baym equations in second Born approximation for the self-energy. VD denotes the statically screened Coulomb potential. The initial correlation contribution is given by Σin(p1; t1, t ′ 1) = −i~5 ∫ dp2 (2π~)3 dq (2π~)3 VD(q)gR(p1 + q; t1, t0)g R(p2 − q; t1, t0) × c(p1 + q,p2 − q,p1,p2; t0)g A(p2; t0, t1)δ(t ′ 1 − t0). (31) The initial correlations were chosen in the form of the Debye pair correlation: c(p1,p2;p1 + q,p2 − q; t0) = − 1 (i~)2 VD(q) kBT f(p1)f(p2)[1 − f(p1 + q)][1 − f(p2 − q)] ∣ ∣ ∣ ∣ t0 . (32) Im g < Ht,t’L 0 0.5 1 1.5 t@fsD 0 0.5 1 1.5 t’@fsD -0.00005 0 0.00005 0.0001 0 0.5 1 1.5 t@fsD -0.00005 0 0.00005 0.0001 Figure 3. Temporal evolution of the imaginary part of g<(t, t′), momentum k = 0.8/aB , for electrons in hydrogen. The initial distribution is a Fermi distribution with T = 10000 K and n = 1021 cm−3. As a model system, electrons in a hydrogen plasma are considered with the density n = 1021 cm−3 and the initial temperature T0 = 10000 K. Figure 3 shows the result of the numerical compu- tation for the full time dependence of Im g<(p, t, t′) for a fixed momentum. The time evolution of the Wigner function for this fixed momentum is given on the time diagonal t = t′. Perpendicular to the diagonal, the spectral properties become apparent. There are oscillations which are connected with the single-particle energy and a damping of the amplitude which characterizes the damping of the single-particle states. Macrophysical quantities are directly computed from the correlation function. For example, the kinetic energy is given by 〈T 〉(t) = V ∫ dp (2π~)3 p2 2m (−i~)g<(p, t, t) (33) 438 Short-time quantum kinetics and the potential energy follows from 〈V 〉(t) = 1 4 V ∫ dp (2π~)3 {( i~ ∂ ∂t − i~ ∂ ∂t′ ) − p2 m } (−i~) g<(p, t, t′)t=t′ . (34) Here, V is the system volume. 0.0 0.4 0.8 1.2 1.6 Time T[fs] -1 0 1 2 E ne rg y[ 10 -5 R yd /a B 3 ] Figure 4. Temporal evolution of the kinetic energy without (narrow-dotted line) and with ini- tial correlations (wide-dotted line), potential energy w/o (narrow-dashed line) and with initial correlations (wide-dashed line) and total energy without (solid line) and with initial correlations (dash-dotted line) for electrons in a hydrogen plasma. Parameters see in figure 3. Figure 4 shows the time evolution of kinetic, potential, and total energies for electrons in a hydrogen plasma for two different initial states: an initially correlated and an uncorrelated case [18]. The following analogies and differences can be observed: • In spite of the same initial density and temperature, the initial states are principally different. In the initially uncorrelated case, the correlation energy is zero at t0 = 0. If there exist initial correlations, it has already a finite value, leading, e. g., to a different total energy. • The behavior for short times (t < tcorr) is completely different, too. In both cases, this stage is determined by the correlation buildup. In the case of nonzero initial correlations, however, this buildup is superposed by their decay. Therefore, the value of the correlation energy is decreased. • The evolution in the kinetic stage, i. e. after the correlation time tcorr, is independent of the initial correlation. Thus, Bogolyubov’s weakening condition [cf. equation (6)] comes out automatically in that stage without postulating it, i. e., the system has “forgotten” about the initial state. • Due to total energy conservation, different final kinetic energies (and therefore, final temper- atures) are reached. Thus, the equilibrium state is effected by the initial correlation due to its contribution to the total energy. • The equilibrium correlations agree approximately. They are nearly independent of the initial state. We can conclude that the energy evolution on short time scales strongly depends on the initial state of the system, especially on the initial binary correlations. In the example shown above, the initial state is overcorrelated and, thus, causes a reduction of the kinetic energy. It can be easily verified by a simple energy balance that an overcorrelated initial state will always cause a decrease of kinetic energy while an undercorrelated initial state leads to its increase. Apart from contributions due to correlations built up in the system, the kinetic energy is closely connected 439 D.Kremp, D.Semkat, Th.Bornath with the temperature. Thus, we can conclude that, by preparing an overcorrelated initial state, it should be possible to cool down a system [29–31]. This cooling effect can be of experimental importance e. g. for ultracold ions in traps in order to reach still lower temperatures. 5. Kinetic equation in first order gradient expansion. Bound states Let us now discuss the transition from the Kadanoff-Baym equations (KBE) to the usual kinetic equations for the Wigner function, f(p,R, t) = ∫ dω 2π (±i)g<(p, ω,R, t), (35) with g<(p, ω,R, T ) being the Fourier transform of g<(1, 1′) = g<(r, t,R, T ) with respect to the variables r = r1 − r′1 and t = t1 − t′1, respectively. R and T are determined by R = (r1 + r′1)/2 and T = (t1 + t′1)/2. An equation for the Wigner function follows from (15) and its adjoint equation in the time diagonal case t1 = t′1 = T , ( ∂ ∂T + p ma ∇R −∇RU eff a (R, t)∇p ) fa(p,R, T ) = T ∫ t0 dt̄ {[ Σ<(T, t̄) + Σin(T, t̄) ] ×gA(t̄, T ) + ΣR(T, t̄)g<(t̄, T ) − g<(T, t̄)ΣA(t̄, T ) − gR(T, t̄) [ Σ<(t̄, T ) + Σin(t̄, T ) ]} . (36) The time-diagonal KBE is the most general quantum kinetic equation. Like in the case of the two-time KBE, we retained still (i) the effect of initial correlations, (ii) the effect of retardation, and (iii) the validity of conservation laws. The r.h.s. of equation (36) represents a far-reaching generalization of usual collision integrals. However, the kinetic equation for the Wigner function is not closed because the single-particle Green’s function arises on the r.h.s. In order to get a closed kinetic equation in explicit form, one still has to solve two problems: (i) The self energy Σ has to be determined as a functional of g≷ by appropriate approximations. Standard approximations are, e. g., the T-matrix approximation and the RPA. (ii) The two-time correlation functions have to be reconstructed from the Wigner distribution. This so-called reconstruction problem is often solved using the Kadanoff-Baym ansatz (KBA). A more general solution was proposed by Lipavský, Špička, and Velický [32] who derived an integral equation for the reconstruction. Usually only the first iteration of this equation, the generalized Kadanoff-Baym ansatz (GKBA) ±g<(t1, t ′ 1) = gR(t1, t ′ 1)f(t′1) − f(t1)g A(t1, t ′ 1) (37) is used. With the GKBA one gets from (36) a closed non-Markovian kinetic equation for the Wigner distribution [5,6,10,17]. We shall not follow this line here. Some effects of the initial stage can be taken into account already by expansions with respect to the retardation. The KBE (16) are nonlocal equations in space and time. Applying the gradient expansion [3,9,10] to (16) and the adjoint equation, one obtains for the correlation functions g≷ the following equations {RegR−1 , ig≷} − {iΣ≷,RegR} = g<Σ> − g>Σ<. (38) For the corresponding spectral function a = i(g> − g<) = i(gR − gA) the difference of the two above equations gives {RegR−1 , a} − {Γ,RegR} = 0. (39) 440 Short-time quantum kinetics All quantities in (38) and (39) are functions of p, ω,R, T . The damping Γ = i(Σ>−Σ<) = −2Im ΣR was introduced, and Poisson brackets were defined by {A,B} = ∂A ∂ω ∂B ∂T − ∂A ∂T ∂B ∂ω − ∂A ∂p ∂B ∂R + ∂A ∂R ∂B ∂p . (40) The equations for the retarded and advanced Green’s functions on the same level of approximation read ( ω ± iε− p2 2m − ΣR/A(p, ω,R, T ) ) gR/A(p, ω,R, T ) = 1 . (41) The KBE (38) in gradient approximation are the basic equations for our further consideration. We shall follow the line of the papers [21,33,34]. The first bracket on the l.h.s. of (38) is a general drift term, and the r.h.s. describes collisions. The second contribution on the l.h.s. of equation (38) is not so easy to interprete. If damping is small, it may be neglected, cf. (39). In general, this term is off-shell with respect to quasiparticle energies. If this term is neglected – no damping – one gets with RegR−1 = g−1 0 −ΣHF −ReΣR corr for the spectral function the following expression a(ω, p) = 2πδ(ω − p2/2m− ReΣR) = Z 2πδ(ω − E(p)) (42) with the quasiparticle energy E(p) = p2/2m+ ReΣR(ω = E) and the renormalization factor Z−1 = 1 − ∂ReΣR ∂ω ∣ ∣ ∣ ∣ ω=E . (43) The drift term in the corresponding kinetic equation is called, therefore, on-shell drift. The different approximations for the on-shell drift are connected with the assumption of the smallness of the gradient contributions. In dependence on the approximation of RegR−1 , the free particle, the mean field and the quasiparticle drifts may be obtained as it was carried out in detail in [35]. The resulting kinetic equation describes ideal (quasi)particles. Retaining the off-shell contribution, however, enables us to overcome the shortcomings of that equation, cf. the discussion of Boltzmann-like equations (1) in the introduction. One is able to find nonideality contributions and to avoid inconsistencies like violation of sum rules by (42) etc. The derivatives on the l.h.s. can be rearranged to give a more useful form ∂ ∂T [ ig≷ − ∂ReΣR ∂ω ig≷ + iΣ≷ ∂RegR ∂ω ] + ∂ ∂ω [ ∂ReΣR ∂T ig≷ − iΣ≷ ∂RegR ∂T ] = g<Σ> − g>Σ<. (44) For the sake of clarity, we dropped the derivatives with respect to p and R. A kinetic equation for the Wigner function (35) follows immediately by an integration of (44) with respect to ω ∂f ∂T − ∂ ∂T ∫ dω 2π [ (±i)g< ∂ReΣR ∂ω − (±i)Σ< ∂RegR ∂ω ] = ± ∫ dω 2π ( g<Σ> − g>Σ< ) . (45) Using the dispersion relations for Reg and ReΣ, this equation can be transformed to ∂f ∂T = I0 (p, T ) + I1 (p, T ) (46) with I0 (p1T ) = ∫ dω 2π [ (±i)Σ<(ωT )ig>(ωT ) − iΣ>(ωT )(±i)g<(ωT ) ] , (47) I1 (p1T ) = ∂ ∂T ∫ dω1dω2 (2π)2 P ′ ω1 − ω2 [ iΣ> (ω1T ) (±i)g< (ω2T ) − (±i)Σ< (ω1T ) ig> (ω2T ) ] . (48) 441 D.Kremp, D.Semkat, Th.Bornath Equation (46) represents a kinetic equation for the full Wigner distribution function. On the r.h.s., the collision integrals are given in terms of the correlation functions g≷ and the self-energy functions Σ≷[g≷]. The latter are, again, functionals of the g≷. Therefore, a reconstruction formula is needed in I0 and I1 in order to get a closed kinetic equation for the Wigner distribution function f . The additional collision integral I1 in (46) is clearly a new element and reflects nonideality corrections. However, also the term I0 contains in the self-energy functions further gradient (nonideality) corrections besides the usual Boltzmann-like terms. For further progress in deriving a kinetic equation, the connection between the correlation functions g≷ and the Wigner distribution will be investigated. The starting point is the equation (44). The real part of the retarded Green’s function can be determined using the dispersion relation. In the limit Γ → 0 it follows that RegR(p, ω,R, T ) = P ω − p2 2m − ReΣR(p, ω,R, T ) . (49) Taking this approximation for RegR in equation (44), one gets ∂ ∂T [ ( 1 − ∂ReΣR ∂ω ) ( ig≷ + iΣ≷ P ′ ω − p2 2m − ReΣR )] + ∂ ∂ω [ ∂ReΣR ∂T ( ig≷ + iΣ≷ P ′ ω − p2 2m − ReΣR )] = g<Σ> − g>Σ<. (50) The structure of this equation now suggests to define a new quantity Q≷ by iQ≷ = ig≷ + iΣ≷ P ′ ω − p2 2m − ReΣR , (51) which leads to a kinetic equation for the Q≷ ∂ ∂T [( 1 − ∂ReΣ ∂ω ) iQ≷ ] + ∂ ∂ω [ ∂ReΣ ∂T iQ≷ ] = Q<Σ> −Q>Σ< . (52) The physical meaning of the special correlation functions Q≷ becomes obvious if we introduce a spectral function according to the usual definition by AQ = i(Q> − Q<). The equation for AQ following from (52) ∂ ∂T [( 1 − ∂ReΣ ∂ω ) i(Q> −Q<) ] + ∂ ∂ω [ ∂ReΣ ∂T i(Q> −Q<) ] = 0 has the solution AQ = i(Q> −Q<) = 2πδ ( ω − p2 2m − ReΣR ) . (53) From the representation (53), it turns out that the quantity Q describes quasiparticles [21,33,34]. It is useful to introduce a quasiparticle distribution function by ±iQ< = 2πδ ( ω − p2 2m − ReΣR ) fQ(p,R, T ) ±X(p, ω,R, T ), (54) iQ> = 2πδ ( ω − p2 2m − ReΣR ) ( 1 ± fQ(p,R, T ) ) +X(p, ω,R, T ) (55) with a function X(p, ω,R, T ) arbitrary so far. The determination of the quantity X is the point where the problem of an ansatz arises. As stated above, the correlation functions Q≷ describe undamped quasiparticles. If these functions 442 Short-time quantum kinetics are considered in the time domain, an ansatz similar to the generalized Kadanoff-Baym ansatz can be used: ±Q<(t, t′) = GR Q(t, t′)fQ(t′) − fQ(t)GA Q(t, t′) (56) with fQ and G R/A Q being the quasiparticle distribution function and the quasiparticle popagators, respectively. Introducing the difference time τ and the centered time T in the usual way and performing an expansion with respect to the retardation in the quasiparticle distribution, we get after Fourier transformation ±iQ<(ω, T ) = AQ(ω, T )fQ(T ) − ∂ ∂ω ReGR Q(ω, T ) ∂ ∂T fQ(T ) . (57) The correlation function Q> is given in a similar way by iQ>(ω, T ) = AQ(ω, T )(1 ± fQ(T )) − ∂ ∂ω ReGR Q(ω, T ) ∂ ∂T (1 ± fQ(T )) . (58) Comparison with (54,55) gives X(ω, T ) = − ∂ ∂ω ReGR Q(ω, T ) ∂ ∂T ( ±fQ(T ) ) . (59) If we insert (57) in (51), the following expression in terms of the quasiparticle distribution function can be derived for g< [21] ±ig<(ω, T ) = 2πδ ( ω − p2 2m − ReΣR ) fQ(T ) − P ′ ( ω − p2 2m − ReΣ ) (±i)Σ<(ω, T ) − P ′ ω − p2 2m − ReΣ(ω) ∂ ∂T (±fQ(T )) . (60) A similar expression, but without the last term on the r.h.s was found in [33,34], see also [35]. As it can be seen from this expression, the correlation functions consist of a pole contribution and an off-pole part. In order to get an equation for fQ, one can use equation (52) together with (60). As the quantity X (59) is of first order in the derivatives (it consists of a collision term), it can be neglected on the l.h.s. of (52). The δ-function in (60) leads to the quasiparticle energies. Then ReΣ does not only depend on the variables R or p explicitly but also implicitly via ω = E(p,R, T ) [3]. With an integration with respect to ω and the reconstruction formulae (57) and (58) there follows a kinetic equation for determining the quasiparticle’s distribution function [ ∂ ∂T + ∂E ∂p ∂ ∂R − ∂E ∂R ∂ ∂p ] fQ(p,R, T ) = Z [ (±i)Σ<(p, E,R, T )(1 ± fQ(p,R, T )) − iΣ>(p, E,R, T )fQ(p,R, T ) ] . (61) This equation is well known as Landau-Silin equation in the theory of Fermi liquids. Now it is easy to obtain the full Wigner function. Integration of (60) with respect to ω and expansion of the renormalization factor Z gives [21,35] f(p,R, T ) = fQ(p,R, T ) ( 1 + ∂ ∂ω ReΣ(p, ω,R, T )\|ω=E ) − ∫ dω 2π P ′ ( ω − p2 2m − ReΣ ) (±i)Σ<(ω, T ) . (62) With equation (62), an interesting relation was obtained which makes it possible to determine the Wigner function f from the quasiparticle distribution function fQ which may be obtained from the 443 D.Kremp, D.Semkat, Th.Bornath Landau-Silin equation. The first term on the r.h.s. describes ideal quasiparticles and the second one stands for the scattering contribution of quasiparticles. Finally we have to express the correlation function g≷ in terms of the Wigner function f instead of the quasiparticle distribution fQ. This can be done in the way that the quasiparticle distribution function in (60) is substituted by the full Wigner function using equation (62) [21] ±ig<(ω, T ) = 2πδ (ω − E) f(T ) − P ′ ω − E ∂ ∂T f(T ) − P ′ (ω − E) (±i)Σ<(ω, T ) + 2πδ (ω − E) ∫ dω̄ 2π P ′ (ω̄ − E) (±i)Σ<(ω̄, T ) . (63) In the case of the thermodynamic equilibrium, Σ< = Γ f(ω) with f(ω) being Bose- or Fermi-like functions. Then from (63) the extended quasiparticle approximation follows [10,36]. Now we are able to find a closed kinetic equation for the Wigner function in first order gradient approximation. We start with equation (48). On the r.h.s., the collision integrals are given in terms of the one particle correlation functions g≷ and the self-energy functions Σ≷. The latter are, again, functionals of the g≷. As an example we consider the self-energy in a ladder approximation, i. e., it is given by the T-matrix T≷ Σ≷(p1ω, T ) = ∫ dp2 (2π)3 dω̄ 2π 〈 p1p2 ∣ ∣ ∣ T≷(ω + ω̄, T ) ∣ ∣ ∣ p2p1 〉 (±i)g≶(p2ω̄, T ) . (64) Expressing the correlation functions of the T-matrix via the so-called optical theorem, however, leads to further gradient terms. Using further the reconstruction formula (63) in I0 and I1 to the appropriate order, one gets a kinetic equation for the Wigner distribution function f . We obtain the Boltzmann equation in first order retardation [5,21,37] ( ∂ ∂T + p ma ∇R −∇RU eff a (R, T )∇p ) fa(p,R, T ) = lim ε→0 ( 1 + 1 2 d dT d dε ) IB(p, ε) = IB(p) + IR(p) . (65) Here, IB(p, ε)\|ε→0 = IB(p) is the Boltzmann collision integral with IB(ε) = − ∫ dp2dp̄1dp̄2 ∫ dω 1 π ε (ω − E12)2 + ε2 1 π ε (ω − Ē12)2 + ε2 × ∣ ∣ 〈 p1p2\|T R(ω + iε)\|p̄2p̄1 〉∣ ∣ 2 [ f̄1f̄2 (1 ± f1) (1 ± f2) − f1f2 ( 1 ± f̄1 ) ( 1 ± f̄2 )] . (66) The first order retardation contributions from the derivative optical theorem for the T-matrix and the reconstruction formulae used in I0 as well as the term I1 are collected in IR [21]. Due to the additional term IR, the kinetic equation (65) conserves, in contrast to the usual Boltzmann equation, the density and the energy of a nonideal system [21,35]. It is interesting to have a closer look at the term I1 (48) in the T-matrix approximation I1 = ∂ ∂T ∫ dp2 (2π)3 { ∂ ∂ω Re 〈 p1p2\|T R(ω)\|p2p1 〉 \|ω=E(p1)+E(p2) f1f2 − ∫ dω 2π 〈 p1p2\|T <(ω + ε(p2))\|p2p1 〉 P ′ ω − E(p1) (1 − f1 − f2) } . (67) It is important to remark that the second term in (67) is determined by the off-shell T-matrix. Therefore, it can be expected that this term contains bound state contributions. In order to separate the bound state parts, the off-shell term T< is written as a bilinear expansion with respect to the bound and scattering eigenstates of the two-particle Hamiltonian [10,21] I1 = I1 scatt + ∂ ∂T ∫ dp2 (2π)3 ∑ jP 〈p1p2\|jP 〉〈Pj\|p2p1〉Nj(P ). (68) 444 Short-time quantum kinetics Bound states can be considered now as a new species of composed particles. Then it is obvious to introduce the distribution function of the bound states fB(p1) and the free particles by the relations [21,38] fB(p1) = ∫ dp2 (2π~)3 ∑ jP 〈 p1p2\|Ψ jP 〉 〈Ψ̃jP \|p2p1〉Nj(P ) , (69) f(p1) = fB(p1) + fF (p1). (70) Using these definitions, we get for the kinetic equation for the free particle distribution function fF (p1) [21,39] ( ∂ ∂T + p ma ∇R −∇RU eff a (R, T )∇p ) fF a (p,R, T ) = IB + IR scatt. (71) This equation is the basic equation of the kinetic theory with reactions, e. g. ionization and re- combination, in plasmas. But we have to take into account that the kinetic equation is effected by bound states only if there are changes of the bound state occupation Nj(P ) with respect to the time. That is the case only in collisions of at least three particles or for external fields. The approximation of the self-energy is therefore to be extended to a cluster expansion up to three or more particles, see e. g. [10,38,40–42]. 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Є багато причин щоб розглядати квантовi кiнетичнi рiвняння, що описують еволюцiю багаточастин- кової системи в ультракоротких часових масштабах, наприклад збереження енергiї у неiдеальнiй плазмi, наростання кореляцiй (зв’язанi стани) та кiнетика ультрашвидких процесiв у короткоiмпуль- снiй лазернiй плазмi. Представлено квантову кiнетичну теорiю, що включає початковi кореляцiї та є справедливою на до- вiльних часових масштабах. Показано рiзнi приклади короткочасових релаксацiйних процесiв, такi як релаксацiя функцiї розподiлу та енергiї. Крiм того обговорюється градiєнтний розклад загальних рiвнянь та роль зв’язаних станiв. Ключовi слова: Кiнетична теорiя, дiйсночасовi функцiї Грiна, короткочасова релаксацiя PACS: 05.20.Dd, 52.25.Dg, 52.27.Gr 446

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