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Infinite Dimensional Analysis Quantum Probability and Related Topics

Non-equilibrium steady states of a Markov generator of weak coupling limit type modeling absorption-emission of m and n photons

M. A. Cruz de la Rosa, Roberto B. Quezada

Published Nov 29, 2016• 3 citations
Publisher

Abstract

We study detailed balance and non-equilibrium steady states of a Markov generator of weak coupling limit type, modeling absorption and simultaneous emission of [Formula: see text]- and [Formula: see text]-photons, with [Formula: see text]. In the case [Formula: see text], under natural constraints on the absorption and emission rates, there exist infinitely many non-equilibrium steady states which are convex linear combination of even and odd states.

DOI
10.1142/s0219025716500235

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We study detailed balance and non-equilibrium steady states of a Markov generator of weak coupling limit type, modeling absorption and simultaneous emission of [Formula: see text]- and [Formula: see text]-photons, with [Formula: see text]. In the case [Formula: see text], under natural constraints on the absorption and emission rates, there exist infinitely many non-equilibrium steady states which are convex linear combination of even and odd states.
The quantum Markov semigroup of the two-photon absorption and emission process has two extremal normal invariant states. Starting from an arbitrary initial state it converges toward some convex combination of these states as time goes to infinity (approach to equilibrium). We compute the exact exponential rate of this convergence showing that it depends only on the emission rates. Moreover, we show that off-diagonal matrix elements of any initial state go to zero with an exponential rate which is smaller than the exponential rate of convergence of the diagonal part. In other words quantum features of a state survive longer than the relaxation time of its classical part.
We give an explicit entropy production formula for a class of quantum Markov semigroups, arising in the weak coupling limit of a system coupled with reservoirs, whose generators [Formula: see text] are sums of other generators [Formula: see text] associated with positive Bohr frequencies [Formula: see text] of the system. As a consequence, we show that any such semigroup satisfies the quantum detailed balance condition with respect to an invariant state if and only if all semigroups generated by each [Formula: see text] do so with respect to the same invariant state.
For a quantum Markov semigroup [Formula: see text] on the algebra [Formula: see text] with a faithful invariant state ρ, we can define an adjoint [Formula: see text] with respect to the scalar product determined by ρ. In this paper, we solve the open problems of characterizing adjoints [Formula: see text] that are also a quantum Markov semigroup and satisfy the detailed balance condition in terms of the operators H, Lk in the Gorini–Kossakowski–Sudarshan–Lindblad representation [Formula: see text] of the generator of [Formula: see text]. We study the adjoint semigroup with respect to both scalar products 〈a, b〉 = tr (ρa*b) and 〈a, b〉 = tr (ρ1/2a*ρ1/2 b).
In this paper we study a class of quantum Markov semigroups whose restriction to an abelian sub-algebra coincides, on the configurations with finite support, with the exclusion type semigroups introduced in Liggett's book14 of exchange rates [Formula: see text] not symmetric in the index site r, s. We find a sufficient condition for the existence of infinitely many faithful diagonal (or classical) invariant states for the semigroup, that satisfy a quantum detailed balance condition. This class of semigroups arises naturally in the stochastic limit of quantum interacting particles in the sense of Accardi and Kozyrev.1 We call these semigroups asymmetric exclusion quantum Markov semigroups and the associated processes asymmetric exclusion quantum processes.