Fluctuation-Dissipation Theorem in an Isolated System of Quantum Dipolar Bosons after a Quench

Abstract

We examine the validity of fluctuation-dissipation relations in isolated quantum systems taken out of equilibrium by a sudden quench. We focus on the dynamics of trapped hard-core bosons in one-dimensional lattices with dipolar interactions whose strength is changed during the quench. We find indications that fluctuationdissipation relations hold if the system is nonintegrable after the quench, as well as if it is integrable after the quench if the initial state is an equilibrium state of a nonintegrable Hamiltonian. On the other hand, we find indications that they fail if the system is integrable both before and after quenching. PACS numbers: 05.30.Jp, 03.75.Kk, The fluctuation-dissipation theorem (FDT) [1-3] is a fundamental relation in statistical mechanics which states that typical deviations from the equilibrium state caused by an external perturbation (within the linear response regime) dissipate in time in the same way as random fluctuations. The theorem applies to both classical and quantum systems as long as they are in thermal equilibrium. Fluctuation-dissipation relations are not, in general, satisfied for out-of-equilibrium systems. In particular, if a system is isolated, it is not clear whether once taken far from equilibrium fluctuation-dissipation relations apply at any later time. Studies of integrable models such as a Luttinger liquid [4] and the transverse field Ising chain [5] have shown that the use of fluctuation-dissipation relations to define temperature leads to values of the temperature that depend on the momentum mode and/or the frequency being considered. More recently, Essler et al. [6] have shown that for a subsystem of an isolated infinite system, the basic form of the FDT holds, and that the same ensemble that describes the static properties also describes the dynamics.The question of the applicability of the FDT to isolated quantum systems is particularly relevant to experiments with cold atomic gases [7,8], whose dynamics is considered to be, to a good approximation, unitary [9]. In that context, the description of observables after relaxation (whenever relaxation to a time-independent value occurs) has been intensively explored in the recent literature [10]. This is because, for isolated quantum systems out of equilibrium, it is not apparent that thermalization can take place. For example, if the system is prepared in an initial pure state |φ ini that is not an eigenstate of the HamiltonianĤ (Ĥ|ψ α = E α |ψ α ) (as in Ref. [9]), then the infinite-time average of the evolution of the observableÔ can be written as Ô (t) = ∑ α |c α | 2 O αα ≡ O diag , where c α = ψ α |φ ini , O αα = ψ α |Ô|ψ α , and we have assumed that the spectrum is nondegenerate. The outcome of the infinite-time average can be thought of as the prediction of a "diagonal" ensemble [11]. O diag depends on the initial state through the c α 's (there is an exponentially large number of them), while the thermal predictions depend only on the total energy φ ini |Ĥ|φ ini ; i.e., they need not agree.The lack of thermal...

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