Nonautonomous Master Equation Research Articles

Global asymptotic stability of nonautonomous master equations: A proof of the Earnshaw–Keener conjecture

We consider nonautonomous master equations of finite-state, continuous-time Markovian jump processes with uniformly continuous and bounded transition matrix functions. The Earnshaw–Keener conjecture states that if the omega-limit set of the transition matrix function contains at least one matrix which is neither decomposable nor splitting, then the difference of any two probability distribution solutions tends to zero at infinity. The conjecture has been confirmed under the additional assumption that the transition matrix function is almost-automorphic. In this paper, we prove the conjecture in its full generality.

Dynamics of master equations with time-recurrent structure

For the nonautonomous master equation of a finite-state, continuous-time jumping-process with time-recurrent structure, we prove that any two probability distribution solutions will be asymptotic to each other provided that the transition rate matrix possesses at some time non-degenerate property (called as “neither decomposable nor splitting”). This extends the corresponding result for time-periodic cases in Earnshaw and Keener (SIAM J Appl Dyn Syst 9:220–237, 2010) to the time-recurrent structure including quasi-periodicity, almost-periodicity and almost-automorphy.

A note on global asymptotic stability of nonautonomous master equations

We present sufficient conditions to either preclude or guarantee global asymptotic stability of linear differential equations for time-dependent $\mathbb W$-matrices. These conditions are concerned with integrability or non-integrability of the matrix entries. The proofs employ differential inequalities.

On the standard quantum Brownian equation and an associated class of non-autonomous master equations

It is shown that the standard quantum Brownian equation (QBE) can violate positivity not only past the thermal correlation time, but at arbitrarily long times at high system frequencies. In an effort to improve the standard QBE, exact operator solutions are provided for a class of non-autonomous master equations. These exact solutions are used to derive sufficient positivity conditions for the coefficients of the master equations.

Invariant Manifolds of Binomial-Like Nonautonomous Master Equations

It has recently been shown that certain finite-state jump processes used to model ion channel kinetics give rise to nonautonomous master equations which possess low-dimensional invariant manifolds. In particular, for the model of an ion channel with many identical and independent subunits existing in either an open or closed state, the one-dimensional manifold of binomial distributions, whose parameter is the probability of a single subunit being in an open state, is an invariant manifold of the associated master equation, and it is the only such one-dimensional manifold. In this paper, we make explicit how the invariance of this manifold arises from the structure of the transition rate matrix of the master equation. We also show that, despite not having any apparent underlying structure or symmetry akin to the ion channel example, there exists a two-parameter family of master equations that shares the essential features of this example and therefore possesses unique, one-dimensional invariant manifolds similar to the manifold of binomial distributions. The mere existence of such a family suggests that there may be other nonautonomous master equations that are devoid of any obvious structure but nevertheless possess invariant manifolds of low dimension.

Global Asymptotic Stability of Solutions of Nonautonomous Master Equations

We provide various conditions on the transition rate matrix of the (possibly nonautonomous) master equation of a finite-state, continuous-time jump process under which every probability distribution solution is globally asymptotically stable in the set of such solutions, thereby extending van Kampen's theorem for constant transition rate matrices. We propose a classification of the set of continuous, uniformly bounded transition rate matrices and show how our results fit into this classification. By constructing counterexamples, we show that certain natural assumptions on the transition rate matrix do not in general ensure that the probability distribution solutions of the associated master equation are globally asymptotically stable.

A two-level atom coupled to a controllable squeezed vacuum field reservoir

The dissipative and decoherence properties of the two-level atom interacting with the squeezed vacuum field reservoir are investigated based on the nonautonomous master equation of the atomic density matrix in the framework of algebraic dynamics. The nonautonomous master equation is converted into a Schr\"{o}dinger-like equations and its dynamical symmetry is found based on the left and right representations of the relevant algebra. The time-dependent solution and the steady solution are obtained analytically. The asymptotic behavior of the solution is examined and the approach to the equilibrium state is proved. Based on the analytic solution the response of the system to the squeezed vacuum field reservoir is studied numerically.

Production of Squeezed State of Single Mode Cavity Field bythe Coupling of Squeezed Vacuum Field Reservoir in Nonautonomous Case

The dissipative and decoherence properties as well as the asymptoticbehaviour of the single mode electromagnetic field interacting with thetime-dependent squeezed vacuum field reservoir are investigated indetail by using the algebraic dynamical method. With the help of theleft and right representations of the relevant hw(4) algebra, thedynamical symmetry of the nonautonomous master equation of the system isfound to be su(1,1). The unique equilibrium steady solution is foundto be the squeezed state and any initial state of the system is provento approach the unique squeezed state asymptotically. Thus the squeezedvacuum field reservoir is found to play the role of a squeezing mold ofthe cavity field.

Dynamical symmetry and analytical solutions of the non-autonomous quantum master equation of the dissipative two-level system: decoherence of the quantum register

Based on the non-autonomous quantum master equation, we investigate the dissipative and decoherence properties of the two-level atom system interacting with the environment of thermal quantum radiation fields. For this system, by a novel algebraic dynamic method, the dynamical symmetry of the system is found, the quantum master equation is converted into a Schr\"{o}dinger-like equation and the non-Hermitian rate (quantum Liouville) operator of the master equation is expressed as a linear function of the dynamical u(2) generators. Furthermore, the integrability of the non-autonomous master equation has been proved for the first time. Based on the time-dependent analytical solutions, the asymptotic behavior of the solution has been examined and the approach to the equilibrium state has been proved. Finally, we have studied the decoherence property of the multiple two-level atom system coupled to the thermal radiation fields, which are related to the quantum register.