Typical, finite baths as a means of exact simulation of open quantum systems

Abstract

There is presently considerable interest in accurately simulating the evolution of open systems for which Markovian master equations fail. Examples are systems that are time dependent and/or strongly damped. A number of elegant methods have now been devised to do this, but all use a bath consisting of a continuum of harmonic oscillators. While this bath is clearly appropriate for, e.g., systems coupled to the electromagnetic field, it is not so clear that it is a good model for generic many-body systems. Here we explore a different approach to exactly simulating open systems: using a finite bath chosen to have certain key properties of thermalizing many-body systems. To explore the numerical resources required by this method to approximate an open system coupled to an infinite bath, we simulate a weakly damped system and compare to the evolution given by the relevant Markovian master equation. We obtain the Markovian evolution with reasonable accuracy by using an additional averaging procedure, and elucidate how the typicality of the bath generates the correct thermal steady state via the process of "eigenstate thermalization."

Quantum Thermodynamics, Entropy of the Universe, Free Energy, and the Second Law

“…[1] where we found that the model universe evolves toward a thermal distribution with a temperature T . This is consistent with computational studies of thermodynamics in a variety of other models [3,14,15,38].…”

Fermi’s golden rule, the origin and breakdown of Markovian master equations, and the relationship between oscillator baths and the random matrix model

“…The "random matrix model" (RMM) of a quantum bath is constructed as follows [7][8][9][10][11][12][13]. One defines as the "bath" a system with N closely-spaced energy levels, and chooses these N levels so that, when arranged in order of increasing energy, their density with respect to energy increases exponentially with energy.…”

“…Second, because the Hamiltonian of the golden rule can be solved (almost) exactly [3][4][5][6], this approach provides insight into the distinct ways in which the evolution induced by an oscillator bath deviates from that of the master equation as the various conditions are relaxed. Third, it makes simple the relationship between the oscillator bath and another model of a thermal bath, the so-called "random matrix model" [7][8][9][10][11][12][13]. We show that while the oscillator bath generates a master equation in which the transition rates are independent of the initial state of the system, the random matrix model cannot do so.…”

Asymmetric temperature equilibration with heat flow from cold to hot in a quantum thermodynamic system

“…In a recent paper [4], we introduced a computational model for such a bath and showed that it comes to thermal equilibrium with a system, while exhibiting quantum thermodynamic effects related to the finite size of the bath. The variable temperature bath generalized earlier work [1,2,[5][6][7][8] on quantum thermodynamic simulations that used a constant temperature bath. The body of our work in Refs.…”

“…[1,2,4,5] and the current article are built around a largely self-contained exposition in the unpublished dissertation of P. C. L., available online [9]. This work is part of a broad program reexamining the foundations of statistical mechanics in the context of quantum pure states evolving in time [1,2,[4][5][6][7][8]. There have been other attempts at formulating the second law for quantum pure states [31][32][33], but to our knowledge none of these has yet been associated with new types of quantum thermodynamic behavior such as we have here with S Q univ in the second law.…”