Beyond-Mean-Field Effects in Rabi-Coupled Two-Component Bose-Einstein Condensate

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-coupling frequency $\mathrm{\Omega }$, the BMF energy density crosses over from the nonanalytic Lee-Huang-Yang scaling $\propto n^{5/2}$ to an expansion in integer powers of density, where, in addition to a two-body BMF term $\propto n^2\sqrt{\mathrm{\Omega }}$, there emerges a repulsive three-body contribution $\propto n^3/\sqrt{\mathrm{\Omega }}$. We experimentally evidence these two contributions, thanks to their different scaling with $\mathrm{\Omega }$, in the expansion of a Rabi-coupled two-component ${}^{39}\mathrm{K}$ condensate in a waveguide. 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 the vanishing mean field.

Figure 1

Minimal Thomas-Fermi radius of the condensate after 75 ms of expansion as a function of the Rabi coupling strength (for $\delta /\mathrm{\Omega }=0.23$). Each point is obtained by averaging 15 fluorescence images and the error bars correspond to the single shot standard deviation. The curves correspond to quasi-1D extended Gross-Pitaevskii simulations (see text) with or without magnetic field noise and assuming different forms of the BMF term. Black solid curve: full BMF term with noise; black dashed curve: full BMF without noise; green dotted curve: two-body BMF term without noise; red dash-dotted curve: two-body plus three-body BMF term without noise. Inset: Experimental Thomas-Fermi radius of the condensate after 75 ms of expansion as a function of $\delta /\mathrm{\Omega }$, for a Rabi frequency $\mathrm{\Omega }/2\pi =12.29\mathrm{kHz}$. The curve is a parabolic guide to the eye.

Figure 2

Density profiles for $|\uparrow ⟩$ (black) and $|\downarrow ⟩$ (red) atoms after 75 ms of expansion at $\mathrm{\Omega }=0$. The blue dashed curve is a 1D Thomas-Fermi fit of the density profile in state $|\downarrow ⟩$.