Vibrational resonance in a one-dimensional dissipative Bose-Josephson junction

We investigate the linear and nonlinear response of a one-dimensional dissipative Bose-Josephson junction subjected simultaneously to a weak low-frequency probe and a rapidly oscillating high-frequency external drive. Starting from the dissipative two-mode Bose-Josephson equations, we derive an effective higher-order nonlinear equation for the population imbalance by retaining the leading nonlinear correction. Using time-scale separation and perturbative analysis, we obtain analytical expressions for both the linear response at the fundamental frequency and the nonlinear response at the second harmonic. We show that the high-frequency modulation modifies the effective potential landscape and dynamically breaks the symmetry around the stationary state, giving rise to a finite second-harmonic response that is absent without the rapidly oscillating field. Both the linear and nonlinear response amplitudes exhibit resonance-like enhancement for optimal values of the high-frequency driving strength. We further analyze the dependence of the resonance characteristics on interaction strength, dissipation, and driving parameters in both the zero-phase and $\pi$-phase modes and compare the analytical predictions with direct numerical simulations. Our results demonstrate a controllable mechanism toward realizing linear and nonlinear vibrational resonance in a one-dimensional dissipative Bose-Josephson junction and open new possibilities for controlling collective dynamics in driven ultracold bosonic systems.