Physical Realizability Conditions for Mixed Bilinear-Linear Quantum Cascades with Pure Field Coupling

This paper aims to provide conditions under which a quantum stochastic differential equation can serve as a model for interconnection of a bilinear system evolving on an operator group SU(2) and a linear quantum system representing a quantum harmonic oscillator. To answer this question we derive algebraic conditions for the preservation of canonical commutation relations (CCRs) of quantum stochastic differential equations (QSDE) having a subset of system variables satisfying the harmonic oscillator CCRs, and the remaining variables obeying the CCRs of SU(2). Then, it is shown that from the physical realizability point of view such QSDEs correspond to bilinear-linear quantum cascades.