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Russian Mathematical Surveys

Quantization of linear systems of differential equations with a quadratic invariant in a Hilbert space

Valerij V. Kozlov

Published Apr 1, 2021• 0 citations
Publisher
DOI
10.1070/rm9992

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arXiv:1711.03503•Nov 9, 2017•1 citation
Abstract Linear systems of differential equations with an invariant in the form of a positive definite quadratic form in a real Hilbert space are considered. It is assumed that the system has a simple spectrum and the eigenvectors form a complete orthonormal system. Under these assumptions, the linear system can be represented in the form of the Schrödinger equation by introducing a suitable complex structure. As an example, we present such a representation for the Maxwell equations without currents. In view of these observations, the dynamics defined by some linear partial differential equations can be treated in terms of the basic principles and methods of quantum mechanics.
In this paper we solve for the quantum propagator of a general time-dependent system quadratic in both position and momentum, linearly coupled to an infinite bath of harmonic oscillators. We work in the regime where the quantum optical master equation is valid. We map this master equation to a Schr\"odinger equation on super-Hilbert space and utilize Lie-algebraic techniques to solve for the dynamics in this space. We then map back to the original Hilbert space to obtain the solution of the quantum dynamics. The Lie-algebraic techniques used are preferable to the standard Wei-Norman methods in that only coupled systems of first-order ordinary differential equations and purely algebraic equations need only be solved. We look at two examples.
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.