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arXiv · 1909.07377

A Karhunen-Loeve expansion for one-mode open quantum harmonic oscillators using the eigenbasis of the two-point commutator kernel

Igor G. Vladimirov·Matthew R. James·Ian R. Petersen
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Abstract

This paper considers one-mode open quantum harmonic oscillators with a pair of conjugate position and momentum variables driven by vacuum bosonic fields according to a linear quantum stochastic differential equation. Such systems model cavity resonators in quantum optical experiments. Assuming that the quadratic Hamiltonian of the oscillator is specified by a positive definite energy matrix, we consider a modified version of the quantum Karhunen-Loeve expansion of the system variables proposed recently. The expansion employs eigenvalues and eigenfunctions of the two-point commutator kernel for linearly transformed system variables. We take advantage of the specific structure of this eigenbasis in the one-mode case (including its connection with the classical Ornstein-Uhlenbeck process). These results are applied to computing quadratic-exponential cost functionals which provide robust performance criteria for risk-sensitive control of open quantum systems.

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Igor G. Vladimirov, Matthew R. James, Ian R. Petersen. 2019-09-16. A Karhunen-Loeve expansion for one-mode open quantum harmonic oscillators using the eigenbasis of the two-point commutator kernel. https://arxiv.org/abs/1909.07377

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