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arXiv:quant-ph/0702027 · Physical Review A

Quantum thermalization with couplings

Hang Dong, Shuo Yang, X. F. Liu, C. P. Sun

Published Oct 23, 2007• 29 citations
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Abstract

We present an exactly solvable model to study the role of the system-bath coupling for the generalized canonical thermalization, which reduces almost all the pure states of the ``universe'' (formed by a system $S$ plus its surrounding heat bath $B$) to a canonical equilibrium state of $S$. It is found that, for the overwhelming majority of the ``universe'' states (they are entangled at least), the diagonal canonical typicality remains robust with respect to finite interactions between $S$ and $B$. Particularly, a decoherence mechanism is utilized here to account for the vanishing of the off-diagonal elements of the reduced density matrix of $S$. The nonvanishing off-diagonal elements due to the finite size of the bath and the stronger system-bath interaction might offer more to quantum thermalization.

DOI
10.1103/physreva.76.044104

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We present an exactly solvable model to study the role of the system-bath coupling for the generalized canonical thermalization, which reduces almost all the pure states of the universe'' (formed by a system $S$ plus its surrounding heat bath $B$) to a canonical equilibrium state of $S$. It is found that, for the overwhelming majority of the universe'' states (they are entangled at least), the diagonal canonical typicality remains robust with respect to finite interactions between SSS and BBB. Particularly, a decoherence mechanism is utilized here to account for the vanishing of the off-diagonal elements of the reduced density matrix of SSS. The nonvanishing off-diagonal elements due to the finite size of the bath and the stronger system-bath interaction might offer more to quantum thermalization.
The role of the system-bath coupling is studied for the generalized canonical thermalization of a system S surrounded by a heat bath B, which was most recently shown [S. Popescu, et al., Nature Physics 2,754(2006) and S. Goldstein et al., Phys. Rev. Lett. 96, 050403(2006)] as a kinematic reduction to a canonical equilibrium state from almost all the pure states of the universe formed by S plus B. This kinematic process is revisited by explicitly considering the energy shell deformation due to the interaction between S and B with an exactly solvable model. By calculating the state numbers of the universe and its subsystems S and B in the deformed energy shells, it is found that, for the overwhelming majority of the universe states (they are entangled at least), the diagonal canonical typicality remains robust with respect to finite interactions between S and B. Particularly, the kinematic decoherence is utilized here to account for the vanishing of the off-diagonal elements of the reduced density matrix of S. It is pointed out that the non-vanishing off-diagonal elements due to the finiteness of bath and the stronger system-bath interaction might offer novelties of the quantum thermalization.
In this review, we discuss the decoherence and thermalization of a quantum spin system interacting with a spin bath environment, by numerically solving the time-dependent Schrödinger equation of the whole system. The effects of the topologic structure and the initial state of the environment on the decoherence of the two-spin and many-spin system are discussed. The role of different spin-spin coupling is considered. We show under which conditions the environment drives the reduced density matrix of the system to a fully decoherent state, and how the diagonal elements of the reduced density matrix approach those expected for the system in the microcanonical or canonical ensemble, depending on the character of the additional integrals of motion. Our demonstration does not rely on time-averaging of observables nor does it assume that the coupling between system and bath is weak. Our findings show that the microcanonical distribution (in each eigenenergy subspace) and canonical ensemble are states that may result from pure quantum dynamics, suggesting that quantum mechanics may be regarded as the foundation of quantum statistical mechanics. Furthermore, our numerical results show that a fully decoherent quantum system prefers to stay in an equilibrium state with a maximum entropy, indicating the validity of the second law of thermodynamics in the decoherence process.
We solve the time-dependent Schrodinger equation for the combination of a spin system interacting with a spin bath environment. In particular, we focus on the time development of the reduced density matrix of the spin system. Under normal circumstances we show that the environment drives the reduced density matrix to a fully decoherent state, and furthermore the diagonal elements of the reduced density matrix approach those expected for the system in the canonical ensemble. We show one exception to the normal case is if the spin system cannot exchange energy with the spin bath. Our demonstration does not rely on time-averaging of observables nor does it assume that the coupling between system and bath is weak. Our findings show that the canonical ensemble is a state that may result from pure quantum dynamics, suggesting that quantum mechanics may be regarded as the foundation of quantum statistical mechanics.
We study measures of decoherence and thermalization of a quantum system SSS in the presence of a quantum environment (bath) EEE. The entirety S+ES+ES+E is prepared in a canonical-thermal state at a finite temperature; that is, the entirety is in a steady state. Both our numerical results and theoretical predictions show that measures of the decoherence and the thermalization of SSS are generally finite, even in the thermodynamic limit, when the entirety S+ES+ES+E is at finite temperature. Notably, applying perturbation theory with respect to the system-environment coupling strength, we find that under common Hamiltonian symmetries, up to first order in the coupling strength it is sufficient to consider SSS uncoupled from EEE, but entangled with EEE, to predict decoherence and thermalization measures of SSS. This decoupling allows closed-form expressions for perturbative expansions for the measures of decoherence and thermalization in terms of the free energies of SSS and of EEE. Large-scale numerical results for both coupled and uncoupled entireties with up to 40 quantum spins support these findings.
It is well known that a system weakly coupled to a heat bath is described by the canonical ensemble when the composite S + B is described by the microcanonical ensemble corresponding to a suitable energy shell. This is true for both classical distributions on the phase space and quantum density matrices. Here we show that a much stronger statement holds for quantum systems. Even if the state of the composite corresponds to a single wave function rather than a mixture, the reduced density matrix of the system is canonical, for the overwhelming majority of wave functions in the subspace corresponding to the energy interval encompassed by the microcanonical ensemble. This clarifies, expands, and justifies remarks made by Schrödinger in 1952.
A system put in contact with a large heat bath normally thermalizes. This means that the state of the system ρℐ(t) approaches an equilibrium state ρ eq ℐ , the latter depending only on macroscopic characteristics of the bath (e.g. temperature), but not on the initial state of the system. The above statement is the cornerstone of the equilibrium statistical mechanics; its validity and its domain of applicability are central questions in the studies of the foundations of statistical mechanics. In the present contribution we discuss the recently proven general theorems about thermalization and demonstrate how they work in exactly solvable models. In particular, we review a necessary condition for the system initial state independence (ISI) of ρ eq ℐ , which was proven in our previous work, and apply it for two exactly solvable models, the XX spin chain and a central spin model with a special interaction with the environment. In the latter case we are able to prove the absence of the system ISI. Also the Eigenstate Thermalization Hypothesis is discussed. It is pointed out that although it is supposed to be generically true in essentially not-integrable (chaotic) quantum systems, it is how-ever also valid in the integrable XX model.