Abstract
We theoretically study the resonant phenomenon in a spin-1 Bose-Einstein condensate periodically driven by a quadratic Zeeman coupling. This phenomenon is closely related to the Shapiro steps in superconducting Josephson junctions, and the previous experimental work [Evrard $et al.,$ Phys. Rev. A 100, 023604 (2019)] for a spin-1 bosonic system observed the resonant dynamics and then called it Shapiro resonance. In this work, using the spin-1 Gross-Pitaevskii equation, we study the Shapiro resonance beyond the single-mode approximation used in the previous work, which assumes that all components of the spinor wavefunction have the same spatial configuration. Considering resonant dynamics starting from a polar state, we analytically calculate the Floquet-Lyapunov exponents featuring an onset of the resonance under a linear analysis and find that spin waves with finite wavenumbers can be excited. This kind of non-uniform excitation cannot be described by the single-mode approximation. Furthermore, to study the long-time resonant dynamics beyond the linear analysis, we numerically solve the one-dimensional spin-1 Gross-Pitaevskii equation, finding that the nonresonant hydrodynamic variables also grow at wavelengths of even multiples of the resonant one due to the nonlinear effect.
Highlights
The engineering of quantum systems by periodic driving has drawn great attention for over a decade, and ultracold atoms have become a promising platform for realizing such driven quantum systems due to their high experimental controllability [1,2,3]
Using the Gross-Pitaevskii equation (GPE) [17,18], we investigate the resonant dynamics starting from a polar state by periodically modulating the quadratic Zeeman (QZ) coupling
II, we introduce the GPE and the spin hydrodynamic equations for a spin-1 spinor Bose-Einstein condensate (BEC)
Summary
The engineering of quantum systems by periodic driving has drawn great attention for over a decade, and ultracold atoms have become a promising platform for realizing such driven quantum systems due to their high experimental controllability [1,2,3]. Applying various periodic modulations to ultracold atoms, recent experiments have realized several topological models such as the Haldane model and the Hofstadter-Harper model [4,5,6], and have observed exotic phases of matter such as a time crystal [7,8]. Such engineering by external driving was recently utilized to generate quantum entanglement in a spin-1 Bose-Einstein condensate (BEC) [9], which is comprised of spin-1 bosons characterized by the three magnetic sublevels m = 1, 0, and −1 [10,11].
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