We theoretically calculate and experimentally measure the beyond-mean-field (BMF) equation of state in a coherently-coupled two-component Bose-Einstein condensate (BEC) in the regime where averaging of the interspecies and intraspecies coupling constants over the hyperfine composition of the single-particle dressed state predicts the exact cancellation of the two-body interaction. We show that with increasing the Rabi frequency, the BMF energy density crosses over from the nonanalytic Lee-Huang-Yang (LHY) scaling ∝ n 5/2 to an expansion in integer powers of density, where, in addition to a two-body BMF term ∝ n 2 , there emerges a repulsive three-body contribution ∝ n 3 . We work in a Rabi-coupled two-component 39 K condensate which is released in a waveguide. Its expansion dynamics is governed by the BMF energy allowing for its quantitative measurement. By studying the expansion with and without Rabi coupling, we reveal an important feature relevant for observing BMF effects and associated phenomena in mixtures with spin-asymmetric losses: Rabi coupling helps preserve the spin composition and thus prevents the system from drifting away from the point of vanishing mean field.
The existence of quantum droplets in binary Bose-Einstein condensate mixtures rely on beyondmean field effects, competing with mean-field-effects. Interestingly, the beyond-mean field effects change from repulsive in three dimension (3D) to attractive in 1D leading to drastically different behaviors. We quantitatively model quantum droplets in the beyond-mean-field crossover from 1D to 3D in the relevant case of an elongated harmonic trap and give realistic numbers for experimental realizations. We identify and quantify two main limiting factors: three-body losses and tiny energy scales. The crossover region is appealing as it offers a trade-off between these two main limitations opening the possibility of observing stable flat-top density profiles, a yet unobserved, characteristic feature of quantum droplets. It would permit testing beyond-mean-field theories to an unprecedented precision.
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