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arXiv:1512.08751

Evolution of quasi-characteristic functions in quantum stochastic systems with Weyl quantization of energy operators

Igor G. Vladimirov

Published Dec 29, 2015• 6 citations
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Abstract

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 $χ^2$-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.

DOI
10.48550/arxiv.1512.08751

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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 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.
This paper is concerned with a class of open quantum systems whose dynamic variables have an algebraic structure, similar to that of the Pauli matrices pertaining to finite-level systems. The system interacts with external bosonic fields, and its Hamiltonian and coupling operators depend linearly on the system variables. This results in a Hudson--Parthasarathy quantum stochastic differential equation (QSDE) whose drift and dispersion terms are affine and linear functions of the system variables. The quasilinearity of the QSDE leads to tractable dynamics of mean values and higher-order multipoint moments of the system variables driven by vacuum input fields. This allows for the closed-form computation of the quasi-characteristic function of the invariant quantum state of the system and infinite-horizon asymptotic growth rates for a class of cost functionals. The tractability of the moment dynamics is also used for mean square optimal Luenberger observer design in a measurement-based filtering problem for a quasilinear quantum plant, which leads to a Kalman-like quantum filter.
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 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.
This paper considers a class of open quantum systems with an algebraic structure of dynamic variables, including the Pauli matrices for finite-level systems as a particular case. The Hamiltonian and the operators of coupling of the system to the external bosonic fields depend linearly on the system variables. The fields are represented by quantum Wiener processes which drive the system dynamics in the form of a quasilinear Hudson-Parthasarathy quantum stochastic differential equation whose drift vector and dispersion matrix are affine and linear functions of the system variables. This quasilinearity leads to a tractable evolution of the two-point commutator matrix of the system variables (and their multi-point mixed moments in the case of vacuum input fields) involving time-ordered operator exponentials. The resulting exponential decay in the two-point commutation relations is a manifestation of quantum decoherence, caused by the dissipative system-field interaction and making the system lose specific unitary dynamics features which it would have in isolation from the environment. We quantify the decoherence in terms of the rate of the commutation relations decay and apply system theoretic and matrix analytic techniques, such as algebraic Lyapunov inequalities and spectrum perturbation results, to the study of the asymptotic behaviour of the related Lyapunov exponents in the presence of a small scaling parameter in the system-field coupling. These findings are illustrated for finite-level quantum systems (and their interconnections through a direct energy coupling) with multichannel external fields and the Pauli matrices as internal variables.
This paper is concerned with a filtering problem for a class of nonlinear quantum stochastic systems with multichannel nondemolition measurements. The system-observation dynamics are governed by a Markovian Hudson-Parthasarathy quantum stochastic differential equation driven by quantum Wiener processes of bosonic fields in vacuum state. The Hamiltonian and system-field coupling operators, as functions of the system variables, are represented in a Weyl quantization form. Using the Wigner-Moyal phase-space framework, we obtain a stochastic integro-differential equation for the posterior quasi-characteristic function (QCF) of the system conditioned on the measurements. This equation is a spatial Fourier domain representation of the Belavkin-Kushner- Stratonovich stochastic master equation driven by the innovation process associated with the measurements. We also discuss a more specific form of the posterior QCF dynamics in the case of linear system-field coupling and outline a Gaussian approximation of the posterior quantum state.