Dual role of squeezed reservoir in quantum phase synchronization: Boosting and blockade

This study explores the dual role of a squeezed reservoir in controlling the quantum phase synchronization of a driven two-level system. We first demonstrate, through a Liouvillian eigen-spectrum analysis, that the squeezed reservoir can induce a stable limit cycle, transforming the passive TLS into a genuine self-sustained oscillator. This enables a qualitative transition from a weak ``forced response" to a robust, high-quality synchronization (or entrainment). This enhancement is characterized not only by a greater degree of phase locking but also by an increased frequency selectivity, manifested as a narrower Arnold tongue. More strikingly, we reveal that the squeezing angle acts as a control parameter to actively suppress synchronization. By tuning this angle, the reservoir can drive the system into a classical mixed state, inducing a quantum synchronization blockade via the quenching of steady-state coherence. Our findings establish squeezed-reservoir engineering as a versatile strategy for actively modulating quantum synchronization, with feasible implementations in circuit quantum electrodynamics.