Quantum decoherence and thermalization at finite temperature within the canonical-thermal-state ensemble

Abstract

We study measures of decoherence and thermalization of a quantum system S in the presence of a quantum environment (bath) E. The whole system is prepared in a canonical thermal state at a finite temperature. 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 the uncoupled system to predict decoherence and thermalization measures of S. This decoupling allows closed form expressions for perturbative expansions for the measures of decoherence and thermalization in terms of the free energies of S and of E. Numerical results for both coupled and decoupled systems with up to 40 quantum spins validate these findings.PACS numbers: 03.65. Yz, 75.10.Jm, 75.10.Nr, 05.45.Pq Decoherence and thermalization are two basic concepts in quantum statistical physics [1]. Decoherence renders a quantum system classical due to the loss of phase coherence of the components of a system in a quantum superposition via interaction with an environment (or bath). Thermalization drives the system to a stationary state, the (micro) canonical ensemble via energy exchange with a thermal bath. As the evolution of a quantum system is governed by the time-dependent Schrödinger equation, it is natural to raise the question how classicality could emerge from a pure quantum state. This question is becoming more important technologically, for example in designing quantum computers [2] where decoherence effects are a major impediment in engineering implementations.Various theoretical and numerical studies have been performed, trying to answer this fundamental question, e.g., the microcanonical thermalization of an isolated quantum system [3][4][5][6], canonical thermalization of a system coupled to a (much) larger environment [3,[7][8][9][10][11][12][13][14][15][16][17], and of two identical quantum systems at different temperatures [18,19]. In our earlier work [20], we found that at infinite temperature the degree of decoherence of a system S scales with the dimension of the environment E if the state of the whole system S + E is randomly chosen from the Hilbert space of the whole system. We showed that in the thermodynamic limit, the system S decoheres thoroughly.In this Letter, we investigate for the first time quantitatively how classicality emerges from a pure quantum state when S + E is of a finite size and at a finite temperature. We assume that the whole system S + E is in a canonical thermal state, a pure state at finite inverse temperature β [21][22][23], and investigate measures of the decoherence and the thermalization of S. This canonical thermal state could be the result of a thermalization process of the whole system S + E coupled to a large heat bath, which we do not consider any further. The state of the system S is described by the reduced density matrix. The degree of decoherence of the system S is measured in terms of σ , defined below in terms of measures of the offdiagonal c...

Relaxation, thermalization, and Markovian dynamics of two spins coupled to a spin bath

“…From our simulation results, it is an empirical fact that Eq. (36), which clearly is of the Markovian type, describes the TDSE data of the two-spin system rather well. On the other hand, the exact relations Eq.…”

“…, 15 and for all t ≥ 0 it follows from Eqs. (35) and (36) by inspection that M(τ) = M(τ). From our simulation results, it is an empirical fact that Eq.…”

Finite thermal reservoirs and the canonical distribution

“…Four important papers have recently dealt with key aspects of this question [7][8][9][10]. While we are primarily concerned with the equilibrium features of the statistical mechanics of finite systems, these papers explored the dynamical approach to equilibrium under the deterministic time development of Newton's equations for classical systems [7] and Schrödinger's equation for quantum systems [8][9][10]. A consequence of their computations was that for both classical and quantum systems the projection of the joint probability distribution of two finite systems onto one of the systems was increasingly well approximated by the canonical distribution as the system sizes increased.…”

“…In this paper, we take a simpler approach to a more limited problem than was treated in Refs. [7][8][9][10]. We restrict ourselves to the equilibrium statistical mechanics of finite systems, so we ignore the time development of the microscopic states.…”

“…Section IV then defines δ, which was introduced in Refs. [7][8][9][10] a measure of the difference between the joint probability distribution and the canonical approximation. A system of simple harmonic oscillators is used to illustrate how a finite reservoir can generate an excellent approximation to the canonical distribution -even if it is smaller than the system of interest.…”