A Perturbative Method for Nonequilibrium Steady State of Open Quantum Systems

Abstract

We develop a method of calculating the nonequilibrium steady state (NESS) of an open quantum system that is weakly coupled to reservoirs in different equilibrium states. We describe the system using a Redfield-type quantum master equation (QME). We decompose the Redfield QME into a Lindblad-type QME and the remaining part R. Regarding the steady state of the Lindblad QME as the unperturbed solution, we perform a perturbative calculation with respect to R to obtain the NESS of the Redfield QME. The NESS thus determined is exact up to the first order in the system-reservoir coupling strength (pump/loss rate), which is the same as the order of validity of the QME. An advantage of the proposed method in numerical computation is its applicability to systems larger than those in methods of directly solving the original Redfield QME. We apply the method to a noninteracting fermion system to obtain an analytical expression of the NESS density matrix. We also numerically demonstrate the method in a nonequilibrium quantum spin chain.

Markovian quantum master equation beyond adiabatic regime

“…Since τ min R ≤ τ R (t) holds by definition, the sufficient condition for the WCA [Eq. (53)] results in…”

“…This is because the differences of the Bohr frequencies correspond to the beat frequencies of the intrinsic oscillation of the off-diagonal density matrix elements. If the condition is not satisfied, the SA sometimes leads to unphysical results and one often has to use Ďt or alternatively approximated Lindblad QME [5,6,46,53].…”

Theory of entropy production in quantum many-body systems

“…since ρr is expressed in terms of thermodynamic observables. When the thermodynamic limit is taken for each subsystem, we have ∆ → 0, and then the parameters x l λ should vary with time so slowly that the state ̺r can be interpreted as "moving" in a locus of equilibrium states, so that ∂ t ̺r → 0 in (21). These are precisely the quasistatic (or reversible) transformations for which the first law involving thermodynamic entropy changes applies, hence the superindex "r" standing for reversible, and the systematic omission of the time subindex in the variables.…”

“…[9][10][11] Other aspects, more in the spirit of traditional nonequilibrium statistical mechanics, 12 include thermalization in isolated quantum systems, [13][14][15][16][17] and the establishment of steady states in open quantum systems. [18][19][20][21][22][23][24] A unified treatment along the lines of the classical theory of nonequilibrium thermodynamics is of crucial importance for a clear identification of the quantum-to-classical correspondence and the new features brought about by fully quantum-mechanical nonequilibrium behavior.…”

“…Note that we have kept the superindex "r" (although not strictly with its original connotation) in (24) because, even though the thermodynamic limit is not taken for each susbsytem, which would make ∆ → 0 and the processes necessarily quasistatic, the fact that the volume elements, δv l , are macroscopic at the atomic scale still implies that ∆ is very small and hence, from (21), that the variations ∂ tρ r should correspondingly be very small in the norm. As mentioned in section II, we then see why the classical theory works well for slow processes, i.e.…”