  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.
