Adiabatic elimination in compound quantum systems with feedback - art. no.013803

Feedback in compound quantum systems is effected by using the output from o ne subsystem (''the system'') to control the evolution of a second subsyste m ("the ancilla") that is reversibly coupled to the system. In the limit wh ere the ancilla responds to fluctuations on a much shorter time scale than does the system, we show that it can be adiabatically eliminated, yielding a master equation for the system alone. This is very significant as it decr eases the necessary basis size for numerical simulation and allows the effe ct of the ancilla to be understood more easily. We consider two types of an cilla: a two-level ancilla (e.g., a two-level atom) and an infinite-level a ncilla (e.g.. an optical mode). For each, we consider two forms of feedback . coherent (for which a quantum-mechanical description of the feedback loop is required) and incoherent (for which a classical description is sufficie nt). We test the master equations we obtain using numerical simulation of t he full dynamics of the compound system. For the system (a parametric oscil lator) and feedback (intensity-dependent detuning) we choose, good agreemen t is found in the limit of heavy damping of the ancilla. We discuss the rel ation of our work to previous work on feedback in compound quantum systems, and also to previous work on adiabatic elimination in general.