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Proceedings of the National Academy of Sciences

Dissipative Properties of Quantum Systems

A.P. Grécos, I. Prigogine

Published Jun 1, 1972• 25 citations
Publisher

Abstract

We consider the dissipative properties of large quantum systems from the point of view of kinetic theory. The existence of a nontrivial collision operator imposes restrictions on the possible collisional invariants of the system. We consider a model in which a discrete level is coupled to a set of quantum states and which, in the limit of a large "volume," becomes the Friedrichs model. Because of its simplicity this model allows a direct calculation of the collision operator as well as of related operators and the constants of the motion. For a degenerate spectrum the calculations become more involved but the conclusions remain simple. The special role played by the invariants that are functions of the Hamiltonion is shown to be a direct consequence of the existence of a nonvanishing collision operator. For a class of observables we obtain ergodic behavior, and this reformulation of the ergodic problem may be used in statistical mechanics to study the ergodicity of large quantum systems containing a small physical parameter such as the coupling constant or the concentration.

DOI
10.1073/pnas.69.6.1629

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We consider the dissipative properties of large quantum systems from the point of view of kinetic theory. The existence of a nontrivial collision operator imposes restrictions on the possible collisional invariants of the system. We consider a model in which a discrete level is coupled to a set of quantum states and which, in the limit of a large "volume," becomes the Friedrichs model. Because of its simplicity this model allows a direct calculation of the collision operator as well as of related operators and the constants of the motion. For a degenerate spectrum the calculations become more involved but the conclusions remain simple. The special role played by the invariants that are functions of the Hamiltonion is shown to be a direct consequence of the existence of a nonvanishing collision operator. For a class of observables we obtain ergodic behavior, and this reformulation of the ergodic problem may be used in statistical mechanics to study the ergodicity of large quantum systems containing a small physical parameter such as the coupling constant or the concentration.
It is difficult to incorporate the concept of interacting physical particles in the frame of conventional quantum mechanics. It comes from the fact that in this formalism the Hamiltonian plays a dual role as the generator of motion and as the observable associated with the energy. As a result it is easy to define either a representation in which particles interact but are not well defined (a representation in which the Hamiltonian is not diagonal) or a representation in which the units are well defined but do not interact. We expect that in the search for a representation in which we may speak of both well defined and still interacting particles, the distinction between reversible and irreversible processes should play a basic role. Certainly we do not want to eliminate irreversible processes such as scattering, production or decay which are basically related to the mechanisms through which we observe particles. To distinguish between reversible and irreversible processes we have to work in a representation such that a Liapounov function (the entropy) may be ascribed to the time evolution of the system. The introduction of such a representation through a specific class of non-unitary transforms is described. We then propose newmore » quantum rules directly applicable to dissipative systems with a Liapounov function. For systems with a discrete spectrum these new quantum rules reduce trivially to the usual quantum rules as formulated by Born, Jordan and Heisenberg. The theory is applied to a simple field theoretical model - the so-called Friedrichs model-for which all calculations can be performed exactly.« less
The problem of calculation of time-correlation functions and relaxation phenomena in interacting quantum many-body systems or quantized fields is put into the framework of ergodic theory and dynamical systems. In this spirit we discuss spectral analysis of the adjoint propagator in a suitable Hilbert space of quantum observables in Heisenberg picture as an alternative approach to characterize many-body dynamics. We illustrate our approach by working out the time-autocorrelation functions at infinite temperature of a large class of observabIes for a spin s = 1/2 Ising chain in a periodically kicking transverse magnetic field, and show that they decay to their asymptotic values as ex: t3/ 2 General understanding of time-correlation functions and relaxation phenomena of interacting quantum many-body systems is a challenging problem with only very partial existing solutions. Situation is much better if one considers the motion of a one-body or few-body systems where the classical ergodic theory (see e.g. [1]) describes a variety of different types of motion, ranging from quasi-periodic integrable motion characterized by a discrete spectrum of the (Liouville) evolution operator, through KAM quasi-integrability, to ergodicity, mixing and chaos characterized by a continuous evolution spectrum. Mixing, which is equivalent of saying that correlation functions of an arbitrary pair of observables (A,B) decay in time, limt-HXJ ((A(t)B) (A)(B)) = 0, is the necessary dynamical property needed to justify the relaxation to equilibrium (micro/grand/canonical) state and the laws of statistical mechanics, such as the fluctuation-dissipation theorem and transport laws. Mixing in quantum many-body system has been investigated by numerical experiment only quite recently [2]. In another numerical experiment [3], a suggestive possibility of dynamical phase transition from non-ergodic/non-mixing to ergodic/mixing dynamics in a non-integrable periodically kicked chain of interacting spinless fermions (kicked t-V model) has been proposed. General absence of mixing in completely integrable quantum many-body systems (such as those solvable by Bethe ansatz), being the consequence of an infinite sequence of conservation 317 J. Bonca et al. (eds.), Open Problems in Strongly Correlated Electron Systems. 317-322. © 2001 Kluwer Academic Publishers. 318 OPEN PROBLEMS IN STRONGLY CORRELATED ELECTRON SYSTEMS laws, is directly responsible for finite temperature anomalous transport properties of such systems [4, 5]. Quantum mixing property is defined (here) relative with respect to a certain (Hilbert) (sub)space 6 of observables and is most conveniently discussed in Heisenberg picture where the equations of motion for quantum observable A(t) are addressed. Our approach may be interpreted also as a kind ofgeneralization ofLee's recurrence-relations methodof constructing continued-fraction representation of time-correlation functions [6]. After a short outline of the main ideas we will illustrate the method by describing a recent calculation [8] of the time-correlation functions for Ising s = 1/2 spin chain periodically kicked with transverse magnetic field. Quantum dynamics in Heisenberg picture is governed by the Heisenberg equation (d/dt)A(t) = i( adH(t))A(t), where (adH)A:= [H, A1 =HA AH. Our aim is to include description of dynamics of periodically time-dependent (e.g. kicked) quantum systems so we will adopt Floquet formalism and integrate equations of motion over one period of time T (for autonomous systems, T may be arbitrary, e.g. infinitesimally small), defining the so called quantum Floquet map or one-step propagatorU := T exp(-i J; dtH(t) )which induces the adjoint Floquet map U ad acting on the (sub)space 6 of observables A(t +T) = U ad A(t), U A:= utAU. We should require the subspace 6 to be invariat under dynamics, namely A E 6 :::} U ad A E 6. Linear space 6 is turned into the Hilbert space by introducing an innner product of two extensive observables in thermodynamic limit (size L -+ 00) as (AlB) = (AtB), (A):= lim .!.. tr A £--+00 L tr 1 (1) (2) which has a meaning of a canonical average at infinite temperature. 1 We note that the adjoint Floquet map U ad is a unitary operator in 6 w.r.t. inner product (1), or equivalently, that ad H is selfadjoint. Thus a pair (U ad , 6) defines a unitary evolution to which we apply standard terminology and results of spectral theory of dynamical systems (see e.g. part III of ref. [1]). To any observable A E 6 we assign spectral measure dI-'A(fJ), where the spectral parameter lies on a unit circle () E 8 1 == [0, 21f), which is directly related to the time-autocorrelation function of A (t) (AtA(mT)) = (AI[U 1 A) = / dI-'A(fJ)e • So, ergodic properties of the unitary evolution are dual to its spectral properties: IThis is the simplest choice which makes sense also for time-dependent systems. For autonomous systems one may work with more general finite-temperature canonical inner product (see e.g. [6]). Ergodic Properties ofQuantum Spin Chains: Kicked Transverse Ising Model 319 • Quantum dynamics is ergodic in 6 iff 1 is (at most) simple (nondegenerate) eigenvalue of Uad (with eigenvector 1 if 1 E 6). Then limT-too ~ JOT dt {(A(t)IB) (All)(IIB)} = O. • Quantum dynamics is mixing in 6 iflimt-too(A(t)IB) = (All)(IIB) for arbitrary A, B E 6. Then U ad has an absolutely continuous spectrum, except possibly for a simple eigenvalue 1. • If 1 is a multiply degenerate eigenvalue of U ad then the corresponding eigenvectors Qn are the conservation laws of 6, U ad Qn = Qn Let Qn be a complete set of orthonormalized (QnIQm) = dnm eigenvectors with eigenvalue 1. The averaged time-autocorrelation function is the spectral weight at point 1 1 jT DA:= lim -T dt(A(t)IA) = L. I(QnIA )1 • T-too 2 -T n (3) For example, if A = J is the particle/spin current D J is proportional to the charge/spin stiffness at infinite temperature and D J > 0 indicates ballistic transport. Even when the set of known conservation laws is incomplete, the above relation gives rigorous lower bound on DA [7, 5]. Next we propose a simple but nontrivial application of the above picture. We consider aHilbert space 6 ofobservables over infinite spin 1/2 chains described by Pauli operators a}, s E {x, y, z}, j E Z, which is spanned by two infinite sequences of selfadjoint observables Un and Vn,nEZ, namely n ~ 1, n =0, n <-1 ,
By using the general transport theory of a normal many-fermion system recently developed by R\'esibois, the collision term is calculated up to fourth order in a coupling parameter. It is shown that the higher order terms can be arranged into the form of the quasiparticle collision term suggested by Landau in the Fermi-liquid theory, except for the transformation operator introduced by R\'esibois to connect the particle description to the quasiparticle one. We also analyze the three-body collision operator in transport theory from the same quasiparticle point of view to the lowest nontrivial order in the coupling constant. It is shown that the anomalous terms in this operator contain the renormalization effects on the usual two-body collision operator.
We investigate the effects of dissipation on the quantum dynamics of many-body systems at quantum transitions, especially considering those of the first order. This issue is studied within the paradigmatic one-dimensional quantum Ising model. We analyze the out-of-equilibrium dynamics arising from quenches of the Hamiltonian parameters and dissipative mechanisms modeled by a Lindblad master equation, with either local or global spin operators acting as dissipative operators. Analogously to what happens at continuous quantum transitions, we observe a regime where the system develops a nontrivial dynamic scaling behavior, which is realized when the dissipation parameter uuu (globally controlling the decay rate of the dissipation within the Lindblad framework) scales as the energy difference $\mathrm{\ensuremath{\Delta}}$ of the lowest levels of the Hamiltonian, i.e., $u\ensuremath{\sim}\mathrm{\ensuremath{\Delta}}$. However, unlike continuous quantum transitions where $\mathrm{\ensuremath{\Delta}}$ is power-law suppressed, at first-order quantum transitions $\mathrm{\ensuremath{\Delta}}$ is exponentially suppressed with increasing the system size (provided the boundary conditions do not favor any particular phase).
Abstract Dynamics and features of quantum systems can be drastically different from classical systems. Dissipation is understood as a general mechanism through which quantum systems may lose part or all of their quantum aspects. Here we discuss a method to analyze behaviors of dissipative quantum systems in an algebraic sense. This method employs a time-dependent product between system’s observables which is induced by the underlying dissipative dynamics. We argue that the long-time limit of the algebra of observables defined with this product yields a contractive algebra which reflects the loss of some quantum features of the dissipative system, and it bears relevant information about irreversibility. We illustrate this result through several examples of dissipation in various Markovian and non-Markovian systems.