Birth, Life, and Death of a Dipolar Supersolid

In the short time since the first observation of supersolid states of ultracold dipolar atoms, substantial progress has been made in understanding the zero-temperature phase diagram and low-energy excitations of these systems. Less is known, however, about their finite-temperature properties, particularly relevant for supersolids formed by cooling through direct evaporation. Here, we explore this realm by characterizing the evaporative formation and subsequent decay of a dipolar supersolid by combining high-resolution in-trap imaging with time-of-flight observables. As our atomic system cools toward quantum degeneracy, it first undergoes a transition from thermal gas to a crystalline state with the appearance of periodic density modulation. This is followed by a transition to a supersolid state with the emergence of long-range phase coherence. Further, we explore the role of temperature in the development of the modulated state.

Figure 1

Evaporation trajectory through the finite-temperature phase diagram. (a) At $T=0$ (bottom plane), the phase diagram for a gas of dipolar atoms is spanned by the $s$-wave scattering length $a_s$ and the condensate atom number $N_c$. In an elongated trap, it features a BEC (white) and independent droplet (ID, black) phases, separated in places by a supersolid state (SSS, gray scale). The plotted lightness in the $T=0$ phase diagram represents the droplet link strength across the system (cf. Ref. [16]). Away from $T=0$, the phase diagram is not known. We explore this region through evaporation into [red, near (i)] and out of [blue, near (ii)] the SSS, along a trajectory represented schematically by the colored arrow. (b) Single-shot image of the optical density (OD) of the sample in trap. Here, a system of four “droplets” within the SSS region is shown, together with its projected density profile. (c) Single-shot matter-wave interference pattern after 35 ms TOF expansion (OD) and the corresponding projected profile. The color scale is truncated for visual clarity. The background clouds of thermal atoms present are not visible in the color scales of (b) or (c); for 35 ms TOF and around 50 nK [as in (c)], the thermal atoms show an approximately isotropic 2D Gaussian distribution of mean width $\overline{\sigma }\sim 55\mu \mathrm{m}$.

Figure 2

Growth and spread of density modulation during evaporation. (a) Averaged in-situ density profiles (no recentering, approximately 20 shots per time step) along the long trap axis as a function of hold time $t_h$ after the fast ramp reduction of trap depth (see main text). (b) The density correlator $C^{\prime }(d)$ (solid black line) is fitted by a cosine-modulated Gaussian function (dashed red line) to extract the correlation length $L$. Gray regions are strongly influenced by imaging noise and excluded from fits. Correlators are displayed for $t_h=50\mathrm{ms}$ (upper) and $t_h=300\mathrm{ms}$ (lower). (c) Density-density correlation length $L$ versus $N_c$, for the same time steps shown in (a). Horizontal error bars are the standard deviation over repetitive shots, vertical error bars reflect the correlator fit uncertainty, red points correspond to the correlators of (b). The dashed line indicates the simple atom number scaling of the Thomas-Fermi radius of a harmonically trapped BEC, $\propto N_c^{1/5}$.

Figure 3

Development of modulation and coherence while evaporating into the supersolid state. (a) Sample temperature $T$ (left ordinate, bullets), total ($N$, right ordinate, dashed red line), and coherent atom number ($N_c$, solid red line) as a function of the ramp crop time $t_c$. The shadings reflect the respective confidence intervals. (b) The phasors $P_i$ (black dots), representing the magnitude and phase coherence of modulation for selected $t_c$ (dotted lines; same radial scale for all polar plots). The red shading reflects mean and variance of the distribution. (c) Evolution of the Fourier amplitude means $A_M$ (filled markers) and $A_{\mathrm{\Phi }}$ (open markers).

Figure 4

Life cycle of a supersolid state. Density modulation $M$ (from in-situ images) during the evaporation process (left ordinate, bullets; the vertical error bars reflect the propagated uncertainty returned by the fitting routine). The sample temperature decreases during the hold time $t_h$ and is encoded by the color filling. $N_c$ (from TOF images) is the number of coherent atoms over $t_h$ (right ordinate, red line; the light red shading reflects the measurement standard deviation). At two times where $N_c\sim 1.1\times {10}^4$ (vertical dashed lines), but at which the atoms have different temperatures, $M$ differs substantially. The corresponding averaged in-situ images below confirm a higher level of modulation at earlier $t_h$. Inset: the observed modulation $M$ plotted versus $N_c$.