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arXiv:1005.1226 · Physical Review A

Variational properties of a pumped dynamical system

Erika Andersson, Stig Stenholm

Published Feb 16, 2010• 0 citations
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Abstract

We earlier constructed a generalized entropy concept to show the direction of time in an evolution following from a Markov generator. In such a dynamical system the entity found changes in a monotonic way starting from any initial state of the system. In this article, we generalize the treatment to the case when population is pumped into the system from levels not explicitly considered. These populations then pass through the coupled levels and exit by decay to levels outside the system. We derive the form of the equation of motion and relate it to our earlier treatments. It turns out that the formalism can be generalized to the new situation. Its physically relevant features are demonstrated and the behavior obtained is illustrated by a numerical treatment of the standard two-level system with pumping and relaxation included.

DOI
10.1103/physreva.81.022111

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We earlier constructed a generalized entropy concept to show the direction of time in an evolution following from a Markov generator. In such a dynamical system the entity found changes in a monotonic way starting from any initial state of the system. In this article, we generalize the treatment to the case when population is pumped into the system from levels not explicitly considered. These populations then pass through the coupled levels and exit by decay to levels outside the system. We derive the form of the equation of motion and relate it to our earlier treatments. It turns out that the formalism can be generalized to the new situation. Its physically relevant features are demonstrated and the behavior obtained is illustrated by a numerical treatment of the standard two-level system with pumping and relaxation included.
We consider the Lindblad-type master equation of an open system. We address the question how to construct a functional of the quantum state which displays a monotonic behavior in time. This thus defines uniquely the direction of time in the system. As the generator of time evolution is not a Hermitian operator, the theory requires the considerations of right and left eigenstates. In this paper we assume them to form two complete bases, which allows us to construct the desired quantity. This can be interpreted as a generalized entropy functional. We show how the construction is carried out in the general case, and we illustrate the theory by solving the case of an externally driven and damped two-level system. The treatment is related to earlier work in the field, and its possible relation to time inversion is discussed.
We consider a two-level system, , coupled to a general n level system, , via a random matrix. We derive an integral representation for the mean reduced density matrix ρ( t ) of in the limit n → ∞, and we identify a model of which possesses some of the properties expected for macroscopic thermal reservoirs. In particular, it yields the Gibbs form for ρ(∞). We also consider an analog of the van Hove limit and obtain a master equation (Markov dynamics) for the evolution of ρ( t ) on an appropriate time scale.
We present a general theory that connects the pump process in multilevel lasers with the statistics of the laser field. The key ingredient in our approach is the derivation of an effective master equation that involves only the two laser levels and that contains a non-Markovian pump term. This pump term gives rise to a narrowing of the photon-number distribution in steady state. The qualitative features of this dynamic noise reduction can be inferred from the eigenvalues of the pump term, which are known analytically. As the mechanism that is responsible for the noise reduction, we identify a correlated excitation process in which the effective excitation rate from the lower to the upper laser level depends on the photon number. This correlated excitation process is very different from an excitation at equidistant times, such as in lasers with periodic external injection. We stress that our approach is general and the treatment nonlinear because we do not resort to approximations such as an adiabatic elimination of atomic variables or the linearization of a Fokker-Planck equation. It is therefore particularly relevant for the description of systems where the active medium consists of a few atoms only and where the field losses are not negligible on the time scale of the atomic relaxations. This would be the case in an ion-trap laser. For such a situation our results are substantially different from those obtained by a linearized treatment. \textcopyright{} 1996 The American Physical Society.
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The dynamics and thermodynamics of a quantum time-dependent field coupled to a two-level system, well known as the Jaynes-Cummings Hamiltonian, is studied, using the maximum entropy principle. In the framework of this approach we found three different infinite sets of relevant operators that describe the dynamics of the system for any temporal dependence. These sets of relevant operators are connected by isomorphisms, which allow us to consider the case of mixed initial conditions. A consisent set of initial conditions is established using the maximum entropy principle density operator, obtaining restrictions to the physically feasible initial conditions of the system. The behavior of the population inversion is shown for different time dependencies of the Hamiltonian and initial conditions. For the time-independent case, an explicit solution for the population inversion in terms of the relevant operators of one of the sets is given. It is also shown how the well known formulas for the population inversion are recovered for the special cases where the initial conditions correspond to a pure, coherent, and thermal field.