Balancing coherent and dissipative dynamics in a central-spin system

The average time required for an open quantum system to reach a steady state (the steady-state time) is generally determined through a competition of coherent and incoherent (dissipative) dynamics. Here, we study this competition for a ubiquitous central-spin system, corresponding to a central-spin-1/2 coherently coupled to ancilla spins and undergoing dissipative spin relaxation. The ancilla system can describe N spins-1/2 or, equivalently, a single large spin of length I=N/2. We find exact analytical expressions for the steady-state time in terms of the dissipation rate, resulting in a minimal (optimal) steady-state time at an optimal value of the dissipation rate, according to a universal curve. Due to a collective-enhancement effect, the optimized steady-state time grows only logarithmically with increasing N=2I, demonstrating that the system size can be grown substantially with only a moderate cost in steady-state time. This paper has direct applications to the rapid initialization of spin qubits in quantum dots or bound to donor impurities, to dynamic nuclear-spin polarization protocols, and may provide key intuition for the benefits of error-correction protocols in quantum annealing.