Density Fluctuations across the Superfluid-Supersolid Phase Transition in a Dipolar Quantum Gas

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Density Fluctuations across the Superfluid-Supersolid Phase Transition in a Dipolar Quantum Gas

J. Hertkorn, J.-N. Schmidt, F. Böttcher, M. Guo, M. Schmidt, K. S. H. Ng, S. D. Graham, H. P. Büchler, T. Langen, M. Zwierlein, and T. Pfau

Phys. Rev. X 11, 011037 – Published 23 February 2021

Abstract

Phase transitions share the universal feature of enhanced fluctuations near the transition point. Here, we show that density fluctuations reveal how a Bose-Einstein condensate of dipolar atoms spontaneously breaks its translation symmetry and enters the supersolid state of matter—a phase that combines superfluidity with crystalline order. We report on the first direct in situ measurement of density fluctuations across the superfluid-supersolid phase transition. This measurement allows us to introduce a general and straightforward way to extract the static structure factor, estimate the spectrum of elementary excitations, and image the dominant fluctuation patterns. We observe a strong response in the static structure factor and infer a distinct roton minimum in the dispersion relation. Furthermore, we show that the characteristic fluctuations correspond to elementary excitations such as the roton modes, which are theoretically predicted to be dominant at the quantum critical point, and that the supersolid state supports both superfluid as well as crystal phonons.

Received 15 October 2020
Accepted 8 January 2021

DOI:https://doi.org/10.1103/PhysRevX.11.011037

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Dipolar gases Dipolar interaction Exotic phases of matter

Bose-Einstein condensates Supersolids

Atomic, Molecular & OpticalCondensed Matter, Materials & Applied Physics

Authors & Affiliations

J. Hertkorn ^1,*, J.-N. Schmidt^1,*, F. Böttcher ¹, M. Guo¹, M. Schmidt¹, K. S. H. Ng¹, S. D. Graham¹, H. P. Büchler², T. Langen ¹, M. Zwierlein³, and T. Pfau^1,†

¹5. Physikalisches Institut and Center for Integrated Quantum Science and Technology, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
²Institute for Theoretical Physics III and Center for Integrated Quantum Science and Technology, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
³MIT-Harvard Center for Ultracold Atoms, Research Laboratory of Electronics, and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

^*These authors contributed equally to this work.
^†t.pfau@physik.uni-stuttgart.de

Popular Summary

A supersolid is a recently discovered phase of matter in which atoms arrange themselves in a rigid crystalline structure and yet have the ability to flow without friction. A key open question is how the crystalline structure can emerge out of a superfluid to realize the supersolid’s counterintuitive properties. Here, we investigate, via direct high-resolution imaging, the density fluctuations of a dipolar quantum gas as it transitions from superfluid to supersolid. This allows us to extract the characteristic fluctuation patterns of the system, which are intimately connected to the spectrum of elementary excitations.

In close analogy to seminal work on superfluid helium, we identify important roton excitations in this spectrum, which are the characteristic fluctuations that emerge as precursors to the crystallization transition. We image the spatial pattern of these roton excitations directly, reveal their dominant role in the crystallization, and link them to the enhancement of fluctuations across the phase transition. Moreover, in the supersolid regime, we observe the presence of both superfluid and crystalline phonons, collective excitations that provide clear evidence for the simultaneous solid and superfluid nature of a supersolid.

Our study highlights the connection between fluctuations and the excitation spectrum and lays the foundations for the study of thermodynamic properties and out-of-equilibrium dynamics of the supersolid state.

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Vol. 11, Iss. 1 — January - March 2021

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Images

Figure 1
(a),(b) Schematic of the dispersion relation $ℏ ω (k)$ and associated static structure factor $S (k)$ of an elongated and strongly dipolar BEC [22]. A decrease in scattering length $a_{s}$ causes a roton minimum to emerge in the excitation spectrum, associated with a characteristic peak in the static structure factor. The roton momentum $k_{rot}$ is indicated where the roton minimum drops near zero. (c) For a given scattering length $a_{s}$ , we observe a large number of in situ densities $n_{j} (r)$ and calculate their mean $⟨ n (r) ⟩$ and the density fluctuations $δ n_{j} (r) = n_{j} (r) - ⟨ n (r) ⟩$ as the deviation of the in situ images from their mean. Investigating the mean power spectrum of the fluctuations $⟨ {| δ n (k) |}^{2} ⟩$ for different scattering lengths across the transition allows us to directly observe the static structure factor as the system passes from BEC to supersolid to isolated droplet states. The colored arrow on top indicates the direction toward lower scattering lengths, passing from BEC to supersolid to isolated droplet regimes. The color map used for the images shows the normalized amplitude of densities, density fluctuations, and mean power spectra, respectively, with the color bar referring to each row separately.
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Figure 2
Experimentally determined structure factor in units of $S_{\max} = 260$ for different scattering lengths $a_{s}$ . The dashed line on the left indicates the smallest momentum $k_{\min} / 2 π ≃ 0.08 μ m^{- 1}$ available due to the finite size of the system along the $x$ direction. The dash-dotted line at $k_{x} / 2 π ≃ 0.3 μ m^{- 1}$ roughly indicates the inverse droplet spacing in the droplet regime, which coincides with the roton momentum at the transition point. Errors are indicated as increased thickness of the lines and are obtained by the bootstrapping method [45, 54]. For illustration purposes, the lines for smaller $a_{s}$ are shifted up. The momentum axis is sampled with a spacing of $Δ k_{x} / 2 π ≃ 0.007 μ m^{- 1}$ . The colored arrow indicates the direction toward lower scattering lengths, passing from BEC to supersolid to isolated droplet regimes.
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Figure 3
Experimentally determined excitation energy $ω (k_{x})$ according to Eq. (1) assuming a temperature of 20 nK, for scattering lengths above the phase transition point. A clear roton minimum at finite momentum is observed that softens toward the transition point.
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Figure 4
(a),(b) Spatial structure of two principal components with the largest weight dominating the experimental dataset. (c),(d) Projections of (a),(b) onto the $x$ axis (blue), with a comparison to the antisymmetric and symmetric roton density fluctuations from our BdG calculation [40] (gray). (e) Mean absolute weight $W$ of the symmetric and antisymmetric roton, normalized to the weight of the antisymmetric roton mode at the transition point, which is the maximum weight of all principal components over the whole scattering length range. The roton modes are dominant, and their weights are almost degenerate from the BEC ( $a_{s} = 104.7 a_{0}$ ) leading up to the transition point ( $a_{s} = 98.4 a_{0}$ ). The gray area indicates the supersolid region previously determined [30]. Error bars indicate the standard error of the mean.
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Figure 5
Higher principal components corresponding to the phonons of the crystal and the BEC (a)–(c) including their Fourier transforms (d)–(f) and weights (g). (a) Quadrupole mode of the BEC. The antisymmetric (b) and the symmetric (c) breathing mode of the crystal show a clear splitting in their Fourier transform (e),(f). (g) Comparison of the mean absolute weights $W$ of these modes featuring a clear overlap region that indicates a supersolid phase. The gray shaded area shows the previously determined supersolid region [30]. Error bars indicate the standard error of the mean. The weights are normalized to the weight of the antisymmetric roton mode at the transition point, which is the maximal weight of all principal components over the whole scattering length range.
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