The controllability analysis of a discrete-time reservoir engineering system

This paper is concerned with a discrete-time reservoir engineering consisting of a sequence of qubits which are used to manipulate the quantum state of a harmonic oscillator. We consider the controllability problem regarding such a system both in the finite time and infinite time scenarios, with detailed proofs. In more concrete terms, we discuss how the quantum state of the harmonic oscillator evolves step by step, and provide the final state after k steps. The role of entanglement involved in the reservoir is also discussed by demonstrating photon numbers. In terms of infinite time controllability analysis, we take advantage of the approximated Lindblad master equation, to explore under what circumstances can we stabilize coherent states, squeezed states and Schrödinger cat states. Stabilization of certain quantum states as well as the corresponding convergence rates are proved by means of Lyapunov functions. These results shed light on a systematic framework towards controllability analysis in quantum reservoir engineering.