Approach to Equilibrium in Nano-scale Systems at Finite Temperature

Abstract

We study the time evolution of the reduced density matrix of a system of spin-1/2 particles interacting with an environment of spin-1/2 particles. The initial state of the composite system is taken to be a product state of a pure state of the system and a pure state of the environment. The latter pure state is prepared such that it represents the environment at a given finite temperature in the canonical ensemble. The state of the composite system evolves according to the time-dependent Schrödinger equation, the interaction creating entanglement between the system and the environment. It is shown that independent of the strength of the interaction and the initial temperature of the environment, all the eigenvalues of the reduced density matrix converge to their stationary values, implying that also the entropy of the system relaxes to a stationary value. We demonstrate that the difference between the canonical density matrix and the reduced density matrix in the stationary state increases as the initial temperature of the environment decreases. As our numerical simulations are necessarily restricted to a modest number of spin-1/2 particles (<36), but do not rely on time-averaging of observables nor on the assumption that the coupling between system and environment is weak, they suggest that the stationary state of the system directly follows from the time evolution of a pure state of the composite system, even if the size of the latter cannot be regarded as being close to the thermodynamic limit.

Quantum decoherence scaling with bath size: Importance of dynamics, connectivity, and randomness

“…This lack of thermalization at low temperatures for small systems is supported by simulations in Ref. [17].…”

“…The canonical ensemble is given by the diagonal elements of the reduced density matrix ρ if the offdiagonal elements [as measured by σ (t)] can be neglected [1,17]. As long as E has a finite Hilbert space D E our scaling results can be used to argue that in a strict sense, the system will not be in the canonical state unless D E → ∞.…”

“…In previous work [16,17], we numerically demonstrated that a quantum system interacting with an environment at high temperature relaxes to a state described by the canonical ensemble. In this paper we focus on investigating the dynamic properties of the decoherence of a quantum system S, being a subsystem of the whole system S + E. We do this both with a theoretical prediction and by simulating the dynamics of a relatively large system S + E of spin-1/2 particles using a time-dependent Schrödinger equation (TDSE) solver [18].…”

“…In the canonical ensemble the diagonal elements of the reduced density matrix are related to the terms in the canonical partition function, in particular,ρ ii = e −βE i /Z [16,17]. Therefore we have a connection between the quantum purity P and how close the system is to a canonical ensemble.…”

“…In general, based on the fact that the system decoheres, i.e., the off-diagonal elements of the reduced density matrix approach zero, we expect that the diagonal elements take (approach to) the form of the canonical distribution exp(−βE i ) where β = 1/k B T with T denoting the temperature and k B Boltzmann's constant, which is taken to be one in this paper, and where E i 's denote the eigenvalues of H S [16,17]. The difference between the diagonal elements ρ ii (t) and the canonical distribution is conveniently characterized by If the system relaxes to its canonical distribution both δ(t) and σ (t) are expected to vanish, b(t) converging to the effective inverse temperature b.…”