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arXiv:1206.2408 · Physical Review B

Towards equilibration and thermalization between finite quantum systems: Unitary emulation of dephasing effects and inelastic interactions

Manas Kulkarni, Kunal L. Tiwari, Dvira Segal

Published Oct 12, 2012• 10 citations
arXivHTMLPDF

Abstract

Using a unitary time evolution scheme, we demonstrate an approach towards a Gibbs-like equilibrium state, with a common temperature and a chemical potential, of two metallic grains connected by a weak link prepared with a different number of noninteracting electrons. This unitary equilibration is achieved by coupling the link to a third finite mesoreservoir that can emulate dephasing and inelastic processes. The two quantum systems equilibrate but do not evolve towards a canonical state when mimicking dephasing effects only. Exact numerical results are supported by analytic calculations based on an extension of the quantum Langevin equation approach, providing the dynamics of the total density matrix.

DOI
10.1103/physrevb.86.155424

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Using a unitary time evolution scheme, we demonstrate an approach towards a Gibbs-like equilibrium state, with a common temperature and a chemical potential, of two metallic grains connected by a weak link prepared with a different number of noninteracting electrons. This unitary equilibration is achieved by coupling the link to a third finite mesoreservoir that can emulate dephasing and inelastic processes. The two quantum systems equilibrate but do not evolve towards a canonical state when mimicking dephasing effects only. Exact numerical results are supported by analytic calculations based on an extension of the quantum Langevin equation approach, providing the dynamics of the total density matrix.
Two identical finite quantum systems prepared initially at different temperatures, isolated from the environment, and subsequently brought into contact are demonstrated to relax towards Gibbs-like quasi-equilibrium states with a common temperature and small fluctuations around the time-averaged expectation values of generic observables. The temporal thermalization process proceeds via a chain of intermediate Gibbs-like states. We specify the conditions under which this scenario occurs and corroborate the quantum equilibration with two different models.
The problem of mutual equilibration between two finite, identical quantum systems, A and B , prepared initially at different temperatures is elucidated. We show that the process of energy exchange between the two systems leads to accurate equipartition within energy shells in the Hilbert space of the total non-interacting, composite system, A ⊗ B . This scenario occurs under the general condition of a weak interaction between the systems. We predict that the sole hypothesis of such equipartition is sufficient to obtain a relaxation of the peers, A and B , towards a common thermal-like state. This conjecture is fully corroborated by an exact diagonalization of several quantum models.
A system evolving unitarily does not dephase or thermalize by definition. Yet, if one considers a part of a unitary system, situation is different. Here we analyze an isolated set of coupled qubits taking one qubit at a time as a subsystem, and writing self-consistently a weak-coupling master equation for it. Weakly anharmonic oscillators and ideal qubits (two-level systems) demonstrate dephasing, i.e. the diagonal elements of the density matrix reach a steady-state in the long-time limit, and the off-diagonal elements of ideal qubits decay exponentially. For the linear coupling between the qubits, only the degenerate ones interact, and the system does not reach thermal distribution. However, non-linearity, in our analysis in form of three-wave mixing, ensures a thermal distribution in the long time limit, where the temperature is determined uniquely by the energy in the initial non-thermal state. Finally, we present a potential experimental scheme based on a superconducting quantum circuit.
Two identical finite quantum systems prepared initially at different temperatures, isolated from the environment, and subsequently brought into contact are demonstrated to relax towards Gibbs-like quasiequilibrium states with a common temperature and small fluctuations around the time-averaged expectation values of generic observables. The temporal thermalization process proceeds via a chain of intermediate Gibbs-like states. We specify the conditions under which this scenario occurs and corroborate the quantum equilibration with two different models.
The problem of mutual equilibration between two finite, identical quantum systems, A and B, prepared initially at different temperatures is elucidated. We show that the process of energy exchange between the two systems leads to accurate equipartition within energy shells in the Hilbert space of the total non-interacting, composite system, A \otimes B. This scenario occurs under the general condition of a weak interaction between the systems. We predict that the sole hypothesis of such equipartition is sufficient to obtain a relaxation of the peers, A and B, towards a common thermal-like state. This conjecture is fully corroborated by an exact diagonalization of several quantum models.
For open quantum systems coupled to a thermal bath at inverse temperature β, it is well known that under the Born, Markov, and secular approximations, the system density matrix will approach the thermal Gibbs state with the bath inverse temperature β. We generalize this to systems where there exists a conserved quantity (e.g., the total particle number), where for a bath characterized by inverse temperature β and chemical potential μ, we find equilibration of both temperature and chemical potential. For couplings to multiple baths held at different temperatures and different chemical potentials, we identify a class of systems that equilibrates according to a single hypothetical average but in general nonthermal bath, which may be exploited to generate desired nonthermal states. Under special circumstances, the stationary state may again be described by a unique Boltzmann factor. These results are illustrated by several examples.