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We study the coherent and dissipative aspects of a driven spin-1Bose-Einstein condensate (BEC) when the Zeeman energy is modulated around astatic bias value. Resonances appear when the bias energy matches an integernumber of modulation quanta. They constitute the atomic counterpart of Shapiroresonances observed in microwave-driven superconducting Josephson junctions.The population dynamics near each resonance corresponds to slow and non-linearsecular oscillations on top of a rapid `micromotion'. At long times and in anarrow window of modulation frequencies around each resonance, we observe arelaxation to asymptotic states that are unstable without drive. Thesestationary states correspond to phase-locked solutions of the Josephsonequations generalized to include dissipation, and are analogous to thestationary states of driven superconducting junctions. We find that dissipationis essential to understand this long-time behavior, and we propose aphenomenological model to explain quantitatively the experimental results.Finally, we demonstrate hysteresis in the asymptotic state of the driven spinorBEC when sweeping the modulation frequency across a Shapiro resonance.

Relaxation and hysteresis near Shapiro resonances in a driven spinor condensate