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This paper considers one-mode open quantum harmonic oscillators with a pairof conjugate position and momentum variables driven by vacuum bosonic fieldsaccording to a linear quantum stochastic differential equation. Such systemsmodel cavity resonators in quantum optical experiments. Assuming that thequadratic Hamiltonian of the oscillator is specified by a positive definiteenergy matrix, we consider a modified version of the quantum Karhunen-Loeveexpansion of the system variables proposed recently. The expansion employseigenvalues and eigenfunctions of the two-point commutator kernel for linearlytransformed system variables. We take advantage of the specific structure ofthis eigenbasis in the one-mode case (including its connection with theclassical Ornstein-Uhlenbeck process). These results are applied to computingquadratic-exponential cost functionals which provide robust performancecriteria for risk-sensitive control of open quantum systems.

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