Long-Lived and Transient Supersolid Behaviors in Dipolar Quantum Gases

By combining theory and experiments, we demonstrate that dipolar quantum gases of both ${}^{166}\mathrm{Er}$ and ${}^{164}\mathrm{Dy}$ support a state with supersolid properties, where a spontaneous density modulation and a global phase coherence coexist. This paradoxical state occurs in a well-defined parameter range, separating the phases of a regular Bose-Einstein condensate and of an insulating droplet array, and is rooted in the roton mode softening, on the one side, and in the stabilization driven by quantum fluctuations, on the other side. Here, we identify the parameter regime for each of the three phases. In the experiment, we rely on a detailed analysis of the interference patterns resulting from the free expansion of the gas, quantifying both its density modulation and its global phase coherence. Reaching the phases via a slow interaction tuning, starting from a stable condensate, we observe that ${}^{166}\mathrm{Er}$ and ${}^{164}\mathrm{Dy}$ exhibit a striking difference in the lifetime of the supersolid properties, due to the different atom loss rates in the two systems. Indeed, while in ${}^{166}\mathrm{Er}$ the supersolid behavior survives only a few tens of milliseconds, we observe coherent density modulations for more than 150 ms in ${}^{164}\mathrm{Dy}$. Building on this long lifetime, we demonstrate an alternative path to reach the supersolid regime, relying solely on evaporative cooling starting from a thermal gas.

Popular Summary

Supersolidity is a paradoxical phase of matter where both superfluid and crystalline orders coexist. Predicted 50 years ago, the existence of such a phase has been long debated in theory and in experiments. Here, we report on evidence for hallmarks of this exotic state in ultracold dilute atomic gases.

While most work has focused on achieving supersolidity in helium, researchers have recently turned to atomic gases, in particular, those with strong dipolar interactions. Recent experiments have revealed that such gases exhibit fundamental similarities with superfluid helium. These features lay the groundwork for reaching a state with both spontaneous density modulation and global phase coherence, which are indicators of supersolidity.

We experimentally create states showing these properties in both erbium and dysprosium quantum gases, and we connect our observations to theoretical phase diagrams. While in erbium the supersolid behavior is only transient, the dysprosium realization shows an unprecedented stability. Here, the supersolid behavior is not only long-lived but also can be directly achieved via evaporative cooling, starting from a thermal sample.

Our results with dysprosium offer exciting prospects for near-future experiments and theories, as the supersolid state is little affected by dissipative dynamics or excitations, thus paving the way for probing its excitation spectrum and its superfluid behavior.

Figure 1

Phase diagram of an ${}^{166}\mathrm{Er}$ and a ${}^{164}\mathrm{Dy}$ dipolar BEC in a cigar-shaped trap. (a) Illustration of the trap geometry with atomic dipoles oriented along $z$. (b) Integrated density profile as a function of $a_s$ for an ${}^{166}\mathrm{Er}$ ground state of $N=5\times {10}^4$. In the color bar, the density scale is upper limited to $4\times {10}^4\mu {\mathrm{m}}^{-1}$ in order to enhance the visibility in the supersolid regime. (c)–(e) Exemplary density profiles for an insulating droplet state (ID) at $a_s=49a_0$, for a state with supersolid properties (SSP) at $51a_0$, and for a BEC at $52a_0$, respectively. (f),(g) Phase diagrams for ${}^{166}\mathrm{Er}$ and ${}^{164}\mathrm{Dy}$ for trap frequencies ${\omega }_{x,y,z}=2\pi \times (227,31.5,151)$ and $2\pi \times (225,37,135)\mathrm{Hz}$, respectively. The gray color identifies ground states with a single peak in $n(y)$ of large Gaussian width, ${\sigma }_y>2{\ell }_y$. The dark blue region in (f) shows the region where $n(y)$ exhibits a single sharp peak, ${\sigma }_y\le 2{\ell }_y$, and no density modulation. The red-to-blue color map shows $S$ in the case of a density-modulated $n(y)$. In (g) the color map is upper limited to use the same color code as in (f) and to enhance visibility in the $\mathrm{low}\text{−}N$ regime. The inset in (g) shows the calculated density profile for ${}^{164}\mathrm{Dy}$ at $N=7\times {10}^4$ and $a_s=91a_0$.

Figure 2

Coherence in the interference patterns: measurement and toy model. (a)–(c) Examples of single TOF absorption images at $t_h=5\mathrm{ms}$ for ${}^{166}\mathrm{Er}$ at $a_s=\{54.7(2),53.8(2),53.3(2)\}{a}_0$, respectively. Corresponding average pictures for 100 images obtained under the same experimental conditions (d)–(f) and their Fourier transform (FT) profiles (g)–(i). The gray lines show the FT norm $|\mathcal{F}[n](y)|$ of the individual profiles. The averages, $n_{\mathcal{M}}$ (blue squares) and $n_{\mathrm{\Phi }}$ (red dots), are fitted to three-Gaussian functions (blue solid line and brown dashed line, respectively). The dotted lines show the components of the total fitted function corresponding to the two side peaks in $n_{\mathrm{\Phi }}$. (j)–(l) Interference patterns from the toy-model realizations with 100 independent draws using $N_D=4$, $d=2.8\mu \mathrm{m}$, ${\sigma }_y=0.56\mu \mathrm{m}$ (see text) and for different ${\varphi }_i$ distributions: (j) ${\varphi }_i=0$, (k) ${\varphi }_i$ normally distributed around 0 with $0.2\pi$ standard deviation, (l) ${\varphi }_i$ uniformly distributed between 0 and $2\pi$. (m)–(o) Corresponding FT profiles for the toy model, same color code as (g)–(i).

Figure 3

Supersolid behavior across the phase diagram. Measured side peak amplitudes, $A_{\mathrm{\Phi }}$ (circles) and $A_{\mathcal{M}}$ (squares), with their ratio in inset (a), and calculated link strength $S$ (b) as a function of $a_s-a_s^{*}$ for ${}^{166}\mathrm{Er}$. For nonmodulated states, we set $S=0$ in theory and $A_{\mathrm{\Phi }}/A_{\mathcal{M}}=0$ in experiment (crosses in inset). In the inset, open and closed symbols correspond to $A_{\mathrm{\Phi }}/A_{\mathcal{M}}>0.8$ and $\le 0.8$, respectively. In the experiments, we probe the system at a fixed $t_h=5\mathrm{ms}$. Horizontal error bars are derived from our experimental uncertainty in $B$, vertical error bars corresponding to the statistical uncertainty from the fit are smaller than the data points. The measured and calculated critical scattering lengths are $a_s^{*}=54.9(2)a_0$ and $51.15a_0$, respectively [62]. The numerical results are obtained for the experimental trap frequencies and for a constant $N=5\times {10}^4$ [63].

Figure 4

Time evolution of the supersolid properties. Amplitudes $A_{\mathrm{\Phi }}$ (circles) and $A_{\mathcal{M}}$ (squares) in the supersolid regime as a function of the holding time in trap for (a) ${}^{166}\mathrm{Er}$ at $54.2(2)a_0$ and for (b) ${}^{164}\mathrm{Dy}$ at 2.04 G. The solid lines are exponential fits to the data. The insets show the time evolution of $A_{\mathrm{\Phi }}/A_{\mathcal{M}}$ for the above cases (filled triangles), and, for comparison, in the ID regime (empty triangles) for Er at $a_s=53.1(2)a_0$ (a).

Figure 5

Survival time of the coherent density-modulated state. $t_{\mathrm{\Phi }}$ in ${}^{166}\mathrm{Er}$ as a function of $a_s$ (a) and ${}^{164}\mathrm{Dy}$ as a function of $B$ (b). The error bars refer to the statistical uncertainty from the fit. The range of investigation corresponds to the supersolid regime for which phase-coherent density-modulated states are observed. This range is particularly narrow for ${}^{166}\mathrm{Er}$.

Figure 6

Evaporative cooling to a state with supersolid properties. ${}^{164}\mathrm{Dy}$ absorption images showing the transition to a state with supersolid properties at 2.43 G (a)–(d) and to a BEC state at 2.55 G (i)–(l), via different durations of the last evaporation step. These durations are 10 ms (a),(i), 50 ms (b),(j), 100 ms (c),(k), and 300 ms (d),(l). The density profiles (e)–(h) are integrated over the central regions of the corresponding absorption images (a)–(d). The color map indicates the atomic density in momentum space.

Figure 7

Lifetime of the supersolid properties achieved via evaporative cooling. Time evolution of the amplitudes $A_{\mathrm{\Phi }}$ (red circle) and $A_{\mathcal{M}}$ (square) after an evaporation time of 300 ms at 2.43 G and an equilibration time of 100 ms. The inset shows the time evolution of $A_{\mathrm{\Phi }}/A_{\mathcal{M}}$. At $t_h=0\mathrm{ms}$, the atom number in the phase-coherent density-modulated component is $N=2.2(2)\times {10}^4$. (b),(c) Averaged absorption images of 25 realizations after 50 and 300 ms of holding time, respectively. Note that the thermal background has been subtracted from the images. The color map indicates the atomic density in momentum space.