Transient Supersolid Properties in an Array of Dipolar Quantum Droplets

We study theoretically and experimentally the emergence of supersolid properties in a dipolar Bose-Einstein condensate. The theory reveals a ground state phase diagram with three distinct regimes—a regular Bose-Einstein condensate and incoherent and coherent arrays of quantum droplets. The coherent droplets are connected by a background condensate, which leads—in addition to the periodic density modulation—to a robust phase coherence throughout the whole system. We further theoretically demonstrate that we are able to dynamically approach the ground state in our experiment and that its lifetime is limited only by three-body losses. Experimentally we probe and confirm the signatures of the phase diagram by observing the in situ density modulation as well as the phase coherence using matter wave interference.

Popular Summary

A supersolid is a paradoxical state of matter in which atoms assume a rigid repeating pattern, as in many solids, and yet flow without friction, as in a superfluid. This means that a supersolid should have phonon excitations, as in a crystal, as well as a frictionless flow of atoms through that crystal. It may be possible to create such exotic behavior by leveraging the intrinsic interactions of the atoms. Here, we do just that by studying theoretically and experimentally the emergence of supersolid properties in a trapped dipolar gas.

A dipolar gas consists of magnetic atoms or polar molecules chilled to near absolute zero. The dipoles exert long-range forces on one another that can lead to many unusual behaviors. Here, we first show theoretically that it is possible to create a state very close to the actual supersolid ground state, with its lifetime limited only by inelastic chemistry. We then prepare a dipolar Bose-Einstein condensate of highly magnetic dysprosium atoms and confirm these predictions by directly observing spatial ordering into a droplet array as well as robust phase coherence across the array. Both of these features are hallmarks of the supersolid state.

Moving forward, we will need to prove the actual superfluidity of the state and attempt to extend its lifetime beyond the observed 20 ms.

Figure 1

Phase transition from BEC to immersed droplets to isolated droplets. (a) Schematic of the experimental sequence and corresponding ground states. We start from a BEC and then change the scattering length to its final value, where we observe three different regimes: at high scattering lengths a BEC, at low scattering lengths isolated droplets, and in between droplets immersed in a superfluid background. We compare the dynamical simulations to the corresponding ground states. The pictures show the phase in color scale weighted by the density distribution. (b) To quantify the transition we calculate the ratio of the first minimum compared to the center peak height of the ground state containing $3.5\times {10}^4$ atoms. (c) Ground state phase diagram of this calculated ratio for different atom numbers and scattering lengths. Note that with higher atom number, the number of droplets, and, as a consequence, also their overlap increases. The dashed black line is a guide to the eye.

Figure 2

Overlap of the calculated dynamical state with the ground state. (a) Time evolution of the density difference $\parallel n_{\mathrm{g}.\mathrm{s}.}-n(t){\parallel }^2/\parallel n_{\mathrm{g}.\mathrm{s}.}{\parallel }^2$, where $n(t)$ is the dynamical density, $n_{\mathrm{g}.\mathrm{s}.}$. is the ground state density, and $\parallel \cdots \parallel$ is the Euclidean norm. We observe little density overlap for the isolated droplets (red) and (except for a residual breathing mode) high density overlap for the coherent droplet state (blue). (b) Time evolution of the phase difference ${\varphi }_0-{\varphi }_1$ between the central droplet and its nearest neighbor for the dynamical simulation. For the coherent droplet the phase difference caused during the formation process is rapidly compensated leading to a vanishing phase difference between the two neighboring droplets. On the other hand, the phase difference of isolated droplets increases almost linearly after the formation, which is due to their difference in chemical potential. Including three-body losses we see a constant (experimentally measured $L_3$, dashed line) or slowly decreasing ($L_3/4$, dotted line) phase difference during the lifetime of the droplets. The arrows indicate the respective formation time of the droplets for the two scattering lengths.

Figure 3

Evolution from in situ density distribution to time-of-flight interference. On the left-hand side, we show an exemplary in situ image together with the integrated density distribution for the phase-coherent droplet regime, revealing a clear density modulation. Towards the right-hand side, we show the expansion dynamics for different times of flight, which exhibits the characteristic increase of the fringe spacing of expanding matter waves. On the very right, we show the interference pattern after 30 ms time of flight together with the corresponding integrated momentum distribution. In these interference patterns a clear substructure at half the principle fringe spacing can be seen. While the principle interference peaks yield information about the nearest-neighbor coherence, the additional peaks correspond to next-nearest-neighbor coherence. The individual images shown result from independent experimental realizations.

Figure 4

Evaluation of the coherence between neighboring droplets. Absolute value of the Fourier transform of the integrated interference patterns after 30 ms time of flight, showing signs of the multiple frequencies of the interference due to nearest-neighbor, next-nearest-neighbor, and even higher-order coherence. This is schematically shown in the inset of (a). The gray lines correspond to single-shot realizations and the blue lines to the mean of all available realizations for a given atom number for $a_s=92.5a_0$ [(a), blue] and $a_s=89a_0$ [(b), red]. In the latter case, the position of the side peaks changes randomly (b), while in the coherent droplet regime (a), the side peaks appear at the same position in every realization and show very little variance in their amplitude. This provides clear evidence for the phase coherence between the droplets (a), with the variance of the peak height of the first side peak corresponding to nearest-neighbor coherence, the second peak corresponding to next-nearest-neighbor coherence, and the third to next-next-nearest-neighbor coherence. The shown exemplary Fourier transforms correspond to the red points in Figs. 5 and 5.

Figure 5

In situ modulation and phase coherence reveal signatures of the theoretical phase diagram. Spectral weight of the observed in situ modulation (a), nearest-neighbor coherence (b), and next-nearest-neighbor coherence (c). Only in the range where we observe a density modulation in (a) we also see interference patterns emerging in time of flight (b), (c). For a narrow range of the contact interaction strength, we see clear evidence for phase coherence up to the next-nearest neighbor. The red points labeled “a” and “b” correspond to the exemplary Fourier transforms shown in Figs. 4 and 4. The dashed black lines are the same guide to the eye that was shown in Fig. 1.