General Relaxation Time of the Fidelity for Isolated Quantum Thermodynamic Systems

Abstract

General evaluation of the relaxation time to equilibrium is usually considered as difficult, since it would strongly depend on the model of interest. In this paper, we provide a generic initial relaxation time of the fidelity for the isolated large systems. The decay of the fidelity is a combination of the Lorentzian and a sinusoidal oscillation. We calculate the relaxation time of the Lorentzian envelop, and the period of the oscillation. Remarkably, these two time scales are the same order when the energy range of the microcanonical state is larger than the thermal fluctuation. Also, the power law decay generally exists for long time regime.

Extremely quick thermalization in a macroscopic quantum system for a typical nonequilibrium subspace

“…The escape from a single state may provide a hint for such considerations (see also [23]). Take an arbitrary state ξ…”

“…Recently there have been several related works on the time scales for equilibration in isolated quantum systems. See, for example, [18][19][20][21][22][23].…”

Eigenstate thermalization hypothesis, time operator, and extremely quick relaxation of fidelity

“…Equation (5) shows that the energy eigenstates are superpositions of continuously many quasi eigenstates of 'time operator', which approximately satisfies the orthogonality(8) [12]. In the next section, we numerically verify how the quasi eigenstate t Y ñ | ( ) typically well represents the microcanonical state, and is considered as a relevant basis to discuss the foundation of ETH.…”

“…Several different approaches have been studied, including through a restriction on the macroscopic observables [10,11], the general evaluation of relaxation time [12][13][14][15], the Eigenstate thermalization hypothesis (ETH) [1,[16][17][18][19][20][21], and dynamical experiments in autonomous cold atomic systems [22][23][24]. Of these, we focus on the foundation of ETH in terms of the time-energy uncertainty by noting that the energy eigenstates are globally distributed in the basis of suitably defined 'time operator' as detailed below (1) and in section 2.…”

“…However, proper definitions of 'time operator' remains still controversial in general. Instead of inquiring into the best definition, we explore a foundation of ETH by introducing 'time operator' as (2) and its quasi eigenstates, which are approximately orthogonal through the extremely quick relaxation of the fidelity [12][13][14][15]. Note that such quasi orthogonality is analogous to that of the coherent state, and is used in quantum non demolition measurement [28,29].…”