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

TL;DR

This paper develops a specialized quantum Karhunen-Loeve expansion for one-mode open quantum harmonic oscillators, leveraging the eigenbasis of the two-point commutator kernel to facilitate robust control performance analysis.

Contribution

It introduces a modified quantum Karhunen-Loeve expansion tailored for one-mode systems using the eigenbasis of the two-point commutator kernel, connecting quantum and classical processes.

Findings

Eigenbasis simplifies the analysis of quantum harmonic oscillators.

Application to quadratic-exponential cost functionals for control.

Connection with classical Ornstein-Uhlenbeck process.

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.