Phase modulation by quantum gates of two microwave pulses in the framework of spin-boson model

As one of the most successful platforms of quantum control, trapped ions can be modulated by sequential microwave pulses to realize high-fidelity quantum logic gates, and dephasing noise may lead to invalidation of phase locking. In this work, we utilize the Dirac-Frenkel time-dependent variational approach with Davydov ansatz to simulate spin echo dynamics in the framework of spin-boson model. As the essential modulation parameters, pulse duration and waiting time have been comprehensively investigated to optimize the phase gates by two microwave pulses. We find that, as spin orientation undergoes periodic changes, the phase difference by acting one and two pulses exhibits spontaneous locking following time evolution, which indicates the robustness of the quantum phase gates. Spectrum of the environmental noise that is appropriate for the phase locking is also determined.