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arXiv:1304.7279 · Physical Review Letters

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

Ehsan Khatami, Guido Pupillo, Mark Srednicki, Marcos Rigol

Published Jul 30, 2013• 131 citations
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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 fluctuation-dissipation 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.

DOI
10.1103/physrevlett.111.050403

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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 fluctuation-dissipation 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.
We study the validity of the fluctuation-dissipation theorem for an isolated quantum system of harmonically trapped dipolar molecules taken out of equilibrium by means of a quench, a sudden change in the Hamiltonian parameters. We find that the integrability of the system plays a crucial role in the validity of the fluctuation-dissipation theorem. Namely, the system thermalizes according to the eigenstate thermalization hypothesis and the theorem holds if the system is nonintegrable after the quench. However, it fails if the system is integrable, unless the initial state is an eigenstate of a nonintegrable Hamiltonian, in which case the system still thermalizes despite the eigenstate thermalization hypothesis failing to describe it.
We study the dynamics and the resulting state after relaxation in a quasidisordered integrable lattice system after a sudden quench. Specifically, we consider hard-core bosons in an isolated one-dimensional geometry in the presence of a quasiperiodic potential whose strength is abruptly changed to take the system out of equilibrium. In the delocalized regime, we find that the relaxation dynamics of one-body observables, such as the density, the momentum distribution function, and the occupation of the natural orbitals, follow, to a good approximation, power laws. In that regime, we also show that the observables after relaxation can be described by the generalized Gibbs ensemble, while such a description fails for the momentum distribution and the natural orbital occupations in the presence of localization. At the critical point, the relaxation dynamics is found to be slower than in the delocalized phase.
We review exact approaches and recent results related to the relaxation dynamics and description after relaxation of various one-dimensional lattice systems of hard-core bosons after a sudden quench. We first analyze the integrable case, where the combination of analytical insights and computational techniques enable one to study large system sizes. Thermalization does not occur in this regime. However, after relaxation, observables can be described by a generalization of the Gibbs ensemble. We then utilize full exact diagonalization to study what happens as integrability is broken. We show that thermalization does occur in finite nonintegrable systems provided they are sufficiently far away from the integrable point. We argue that the onset of thermalization can be understood in terms of the eigenstate thermalization hypothesis.
We study properties of isolated integrable quantum systems after a sudden quench starting from thermal states. We show that, even if the system is initially in thermal equilibrium at finite temperature, the diagonal entropy after a quench remains a fraction of the entropy in the generalized ensembles introduced to describe integrable systems after relaxation. The latter is also, in general, different from the entropy in thermal equilibrium. Furthermore, we examine the difference in the distributions of conserved quantities in the thermal versus the generalized ensembles after a quench and show that they are also, in general, different from each other. This explains why these systems fail to thermalize. A finite-size scaling analysis is presented for each quantity, which allows us to make predictions for thermodynamically large lattice sizes.
We demonstrate the role of interactions in driving the relaxation of an isolated integrable quantum system following a sudden quench. We consider a family of integrable hard-core lattice anyon models that continuously interpolates between noninteracting spinless fermions and strongly interacting hard-core bosons. A generalized Jordan-Wigner transformation maps the entire family to noninteracting fermions. We find that, aside from the singular free-fermion limit, the entire single-particle density matrix and, therefore, all one-body observables relax to the predictions of the generalized Gibbs ensemble (GGE). This demonstrates that, in the presence of interactions, correlations between particles in the many-body wave function provide the effective dissipation required to drive the relaxation of all one-body observables to the GGE. This relaxation does not depend on translational invariance or the tracing out of any spatial domain of the system.
We study the out-of-equilibrium dynamics of dipolar bosons and fermions after a sudden change in the interaction strength from zero to a finite repulsive value. We simulate the interaction quench on the initial state which is the ground state of harmonic potential with noninteracting bosons and fermions. We solve the time-dependent many-boson Schrödinger equation exactly using numerical methods. To understand the many-body dynamics we analyze several measures of many-body information entropy, monitoring their time evolution and assessing their dependence on interaction strength. We establish that for weak interaction quench the dynamics is statistics independent, both dipolar bosons and fermions do not relax, whereas it is significantly different for dipolar bosons from that of dipolar fermions in the stronger interaction quench. While dipolar bosons exhibit concurrent signature of relaxation in all entropy measures, dipolar fermions fail to relax, exhibiting modulated oscillation in all entropy dynamics. For dipolar bosons and larger interaction quench, the many-body information entropy measures dynamically approach the value predicted for the Gaussian orthogonal ensemble of random matrices, implying statistical relaxation and ergodicity. The entire relaxation process and signature of ergodicity are established by three unique features: dynamical delocalization in Hilbert space, violent orbital fragmentation, and saturation of all entropy measures to the maximum entropy states in the long-time dynamics. It highlights the importance of many-body effects with a possible exploration in quantum simulation with ultracold atoms.