Quantum illumination uses entangled light that consists of signal and idler modes to achieve higher detection rate of a low-reflective object in noisy environments. The best performance of quantum illumination can be achieved by measuring the returned signal mode together with the idler mode. Thus, it is necessary to prepare a quantum memory that can keep the idler mode ideal. To send a signal towards a long-distance target, entangled light in the microwave regime is used. There was a recent demonstration of a microwave quantum memory using microwave cavities coupled with a transmon qubit. We propose an ordering of bosonic operators to efficiently compute the Schrieffer-Wolff transformation generator to analyze the quantum memory. Our proposed method is applicable to a wide class of systems described by bosonic operators whose interaction part represents a definite number of transfer in quanta.
Keywords: Schrieffer–Wolff transformation; equivalent circuit; quantum illumination; transmon-cavity quantum memory.