Theory of relaxation and pumping of quantum oscillator non-resonantly coupled with the other oscillator

Abstract

The paper shows mechanisms of both the pumping and energy decay of an "isolated" oscillator. The oscillator is only non-resonantly coupled with the adjacent oscillator which resonantly interacts with the thermal bath environment. Under these conditions the "isolated" oscillator begins interacting with the thermal bath environment of the adjacent oscillator. The conclusion is based on the kinetic equation derived relative to anti-rotating terms of the initial Hamiltonian, with the latter being the Hamiltonian of two oscillators and environment of one of them.Quantum oscillators simulate photon systems in micro resonators, which can be either coupled with each other on mirrors or with pump and vacuum (thermal bath) fields of the thermal bath, providing an example of an open quantum system. Interaction with thermal baths leads to the attenuation of quantum oscillators. Effective research of attenuating quantum systems were made possible due to the introduction of the notion of a kinetic equation (master equation) into the mathematical apparatus of nonlinear and quantum optics [1][2][3].In [4][5], the general form of the kinetic equation is established, that is now commonly referred to as the Lindblad-type of the kinetic equation. Many recently published papers on the analysis of open quantum systems dynamics start with the definition (as initial ones) of precisely kinetic equations in the Lindblad form with a predefined Lindblad operators.In view of this, researchers consider atomic systems interacting with electromagnetic fields of various nature [6-9], photon systems consisting of photons of cavity modes interacting with other cavity systems, with intracavity and boundary atoms [10][11][12][13], and other optical problems.Since the relaxation channels are already defined by the appropriate terms and Lindblad operators in the initial equations, the majority of further approximations, and, in particular, dispersion approximation do not adequately describe the case under study. In fact, in optics, the initial Hamiltonian of two non-resonantly interacting atoms has both rapidly and slowly varying terms. This is distinctly seen from the Hamiltonians of the basic models of quantum optics in the interaction representation [14,15]. The rapidly varying terms in the interaction representation are commonly referred to as anti-rotating ones. It was observed that in optical problems the success of the approach based on kinetic equations in the Lindblad form is due to the neglect of antirotating terms [16]. However, the neglect is possible only in resonant processes, and not always in all cases [17,18]. Thus it is vital to take into account the anti-rotating terms in deriving the kinetic equation for resonant, quasi-resonant and non-resonant processes, as well as to analyze their significance in optical effects.The present paper considers the dynamics of a quantum oscillator that is non-resonantly coupled to another oscillator. The connection of the kind is often neglected, assuming that the given oscillator can be considered t...

Effective Gibbs state for averaged observables

“…where • 0 is the average with respect to the Gibbs state for the free Hamiltonian. So nonnegativity for ∆S in the second order of perturbation theory agrees with (12). Moreover, it could be calculated (see App.…”

“…5.2] or as result [3,4] of perturbation theory with Bogolubov-van Hove scaling [5,6] (see also corrections beyond the zeroth order in [7]). Also there is wide discussion of applicability of rotating wave approximation (RWA) and systematic perturbative corrections to it in the literature [8,9,10,11,12,13,14,15,16,17]. But in this work we consider such averaging not as approximation but as restriction of our observation capabilities.…”

Long-time Markovianity of multi-level systems in the rotating wave approximation

“…This model and its particular cases are widely used as a test model for many approaches to the open quantum systems [16,17,18,19,20,21,22,15]. But let us remark that it omits non-RWA effects which could be important for real physical systems [24,23,25].…”

Non-perturbative effects in corrections to quantum master equation arising in Bogolubov-van Hove limit

“…So we choose to study the perturbation theory with Bogolubov-van Hove scaling by its example. We also do not discuss the validity of the RWA which was called into question in the literature [68][69][70][71][72].…”