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arXiv:1503.02122 · Australian Control Conference

Robust mean square stability of open quantum stochastic systems with Hamiltonian perturbations in a Weyl quantization form

Arash Kh. Sichani, Igor G. Vladimirov, Ian R. Petersen

Published Nov 1, 2014• 6 citations
arXivHTMLPDF

Abstract

This paper is concerned with open quantum systems whose dynamic variables satisfy canonical commutation relations and are governed by quantum stochastic differential equations. The latter are driven by quantum Wiener processes which represent external boson fields. The system-field coupling operators are linear functions of the system variables. The Hamiltonian consists of a nominal quadratic function of the system variables and an uncertain perturbation which is represented in a Weyl quantization form. Assuming that the nominal linear quantum system is stable, we develop sufficient conditions on the perturbation of the Hamiltonian which guarantee robust mean square stability of the perturbed system. Examples are given to illustrate these results for a class of Hamiltonian perturbations in the form of trigonometric polynomials of the system variables.

DOI
10.1109/aucc.2014.7358693

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This paper is concerned with the sensitivity of invariant states in linear quantum stochastic systems with respect to nonlinear perturbations. The system variables are governed by a Markovian Hudson-Parthasarathy quantum stochastic differential equation (QSDE) driven by quantum Wiener processes of external bosonic fields in the vacuum state. The quadratic system Hamiltonian and the linear system-field coupling operators, corresponding to a nominal open quantum harmonic oscillator, are subject to perturbations represented in a Weyl quantization form. Assuming that the nominal linear QSDE has a Hurwitz dynamics matrix and using the Wigner-Moyal phase-space framework, we carry out an infinitesimal perturbation analysis of the quasi-characteristic function for the invariant quantum state of the nonlinear perturbed system. The resulting correction of the invariant states in the spatial frequency domain may find applications to their approximate computation, analysis of relaxation dynamics and non-Gaussian state generation in nonlinear quantum stochastic systems.
The paper is concerned with open quantum systems whose Heisenberg dynamics are described by quantum stochastic differential equations driven by external boson fields. The system-field coupling operators are assumed to be quadratic polynomials of the system variables, with the latter satisfying canonical commutation relations. In combination with a cubic system Hamiltonian, this leads to quasilinear quantum stochastic systems which extend the class of linear quantum systems and yet retain algebraic closedness in the evolution of mixed moments of system variables up to any order. Although such a system is nonlinear and its quantum state is no longer Gaussian, the dynamics of the moments are amenable to exact analysis, including the computation of their steady-state values. A generalized criterion is outlined for quadratic stability of the quasilinear systems.
This paper considers open quantum systems whose dynamic variables satisfy canonical commutation relations and are governed by Markovian Hudson-Parthasarathy quantum stochastic differential equations driven by external bosonic fields. The dependence of the Hamiltonian and the system-field coupling operators on the system variables is represented using the Weyl functional calculus. This leads to an integro-differential equation (IDE) for the evolution of the quasi-characteristic function (QCF) which encodes the dynamics of mixed moments of the system variables. Unlike quantum master equations for reduced density operators, this IDE involves only complex-valued functions on finite-dimensional Euclidean spaces and extends the Wigner-Moyal phase-space approach for quantum stochastic systems. The dynamics of the QCF and the related Wigner quasi-probability density function (QPDF) are discussed in more detail for the case when the coupling operators depend linearly on the system variables and the Hamiltonian has a nonquadratic part represented in the Weyl quantization form. For this class of quantum stochastic systems, we also consider an approximate computation of invariant states and discuss the deviation from Gaussian quantum states in terms of the χ2χ^2χ2-divergence (or the second-order Renyi relative entropy) applied to the QPDF. The results of the paper may find applications to investigating different aspects of the moment stability, relaxation dynamics and invariant states in open quantum systems.
This paper considers the problem of robust stability for a class of uncertain nonlinear quantum systems subject to unknown perturbations in the system Hamiltonian. The nominal system is a linear quantum system defined by a linear vector of coupling operators and a quadratic Hamiltonian. This paper extends previous results on the robust stability of nonlinear quantum systems to allow for quantum systems with dynamic uncertainties. These dynamic uncertainties are required to satisfy a certain quantum stochastic integral quadratic constraint. The robust stability condition is given in terms of a strict bounded real condition. This result is applied to the robust stability analysis of an optical parametric amplifier.
This paper considers the problem of robust stability for a class of uncertain nonlinear quantum systems subject to unknown perturbations in the system Hamiltonian. The nominal system is a linear quantum system defined by a linear vector of coupling operators and a quadratic Hamiltonian. This paper extends previous results on the robust stability of nonlinear quantum systems to allow for quantum systems with dynamic uncertainties. These dynamic uncertainties are required to satisfy a certain quantum stochastic integral quadratic constraint. The robust stability condition is given in terms of a strict bounded real condition. This result is applied to the robust stability analysis of an optical parametric amplifier.
This paper is concerned with a stochastic dissipativity theory using quadratic-exponential storage functions for open quantum systems with canonically commuting dynamic variables governed by quantum stochastic differential equations. The system is linearly coupled to external boson fields and has a quadratic Hamiltonian which is perturbed by nonquadratic functions of linear combinations of system variables. Such perturbations are similar to those in the classical Lur'e systems and make the quantum dynamics nonlinear. We study their effect on the quantum expectation of the exponential of a positive definite quadratic form of the system variables. This allows conditions to be established for the risk-sensitive stochastic storage function of the quantum system to remain bounded, thus securing boundedness for the moments of system variables of arbitrary order. These results employ a noncommutative analogue of the Doleans-Dade exponential and a multivariate partial differential version of the Gronwall-Bellman lemma.