Exploring the Cooling Limit of Quantum Mechanical Oscillators via Optimization

In this work, we have studied the optimal cooling limit when cooling a quantum mechanical oscillator by coupling it to an auxiliary. Since this limit is necessarily confined by the energy transfer rate of the control, i.e., the magnitude of the spectrum of the control Hamiltonian, we search for the optimal control Hamiltonian that can achieve the maximal cooling fidelity, given a bounded energy transfer rate. First we analyze and present a class of Hamiltonian which generates the perfect cooling result in the absence of decoherence. Then with perturbation analysis, we further show that such Hamiltonian is also optimal at least to the first order of the dynamical dissipative terms. With the help of numerical optimization, we have studied cases with different damping rates of the system and the auxiliary, and derived the plot of the optimal cooling fidelity curve versus time as well as the optimal time point to achieve the maximal fidelity.