Bragg scattering of an ultracold dipolar gas across the phase transition from Bose-Einstein condensate to supersolid in the free-particle regime

We present an experimental and theoretical study of the response of a dipolar supersolid to a Bragg excitation at high-energy defined by the impulse approximation regime. We experimentally observe a continuous reduction of the response when tuning the contact interaction from an ordinary Bose-Einstein condensate to a supersolid state and ultimately to an incoherent array of droplets. Already in the supersolid regime, the observed reduction is faster than the one theoretically predicted by the Bogoliubov–de Gennes theory. By comparing experiments and theories, we are able to attribute this discrepancy to the presence of coherent phase dynamics induced by the crossing of the phase transition. The phase variations are found to change character along the phase diagram and become predominantly incoherent only when reaching the incoherent-droplet regime.

Figure 1

(a) Axial excitation spectrum of the transversely symmetric modes and (b) corresponding DSF of an infinitely elongated dipolar supersolid at $a_{\mathrm{s}}=51a_0$ in a harmonic trap with ${\omega }_{x,y,z}=2\pi \times (250,0,160)$ Hz. The color maps correspond to $‖u‖$ and $S(k,\omega )$, respectively. The inset shows the integrated axial density profile $n\left(y\right)$ of the ground state with mean density $4.7\times {10}^3\mu {\text{m}}^{-1}$. (c) $S\left(k\right)$ for the 3D-trapped system with ${\omega }_{x,y,z}=2\pi \times (250,31,160)$ Hz. $S\left(k\right)$ is calculated at $k=4.2\mu {\mathrm{m}}^{-1}\approx 1.8k_{\text{c}}$ (grey line) and normalized by its value at the BEC-SSP phase transition, $S^{*}$. The atom number is varied with $a_{\mathrm{s}}$ to match the experimental conditions [34]. The red (blue) line shows the result from the SIA (DAA). (upper inset) Integrated density profile of the ground state at $a_{\mathrm{s}}=54.49a_0$ and $N=5\times {10}^4$ atoms. (lower inset) Evolution of the ground state's central contrast $C$. For the infinite (3D-trapped) case, $k_{\text{c}}=2.3\mu {\mathrm{m}}^{-1}\left(2.4\mu {\mathrm{m}}^{-1}\right)$.

Figure 2

Fraction of Bragg-excited atoms as a function of $\omega$ for various $a_{\mathrm{s}}$ across the BEC-SSP-ID regimes (see labels). The spectra are vertically offset for visibility. Here and throughout the Letter, the error bars correspond to one standard error. Solid lines show the Gaussian fits to the data.

Figure 3

Experimental $\mathcal{F}$ (circles) versus $a_{\mathrm{s}}$ across the BEC-SSP-ID phases. For the lowest three $a_{\mathrm{s}}$, we do not observe a resonance and plot the standard deviation of the data as an error estimate. Horizontal error bars correspond to uncertainties of the magnetic field [34]. Theoretical $\mathcal{F}$ (lines) from the BdG calculations on the ground state (grey), from the RTE simulations (black), and from the rescaled BdG calculations that include $\mathrm{\Delta }\mathrm{\Theta }$ obtained from the RTE (blue). The gray shading corresponds to the uncertainty in $a_{\mathrm{s}}$ of the experimental phase transition (vertical line).

Figure 4

RTE simulations without Bragg excitation. (a) Time evolution of the integrated in situ density of the wave function for $a_{\mathrm{s}}=54.04a_0$. (b) ${\langle C\rangle }_{\tau }$ (triangles) and $\mathrm{\Delta }\mathrm{\Theta }$ (squares) versus $a_{\mathrm{s}}$. The grey line corresponds to the central contrast obtained from the ground-state theory. The solid blue line is a smooth interpolation of $\mathrm{\Delta }\mathrm{\Theta }$, fixed to unity at the phase transition point. The shadings give the standard deviation obtained from five simulation runs. The vertical line corresponds to the phase transition point. (c) Phase cuts corresponding to the simulation shown in (a).