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arXiv:2107.06680 · Physical Review Letters

Two-Dimensional Supersolid Formation in Dipolar Condensates

Thomas Bland, Elena Poli, Claudia Politi, Lauritz Klaus, Matthew A. Norcia, Francesca Ferlaino, Luis Santos, R. N. Bisset

Published May 13, 2022• 107 citations
arXivHTMLPDF

Abstract

Dipolar condensates have recently been coaxed to form the long-sought supersolid phase. While one-dimensional supersolids may be prepared by triggering a roton instability, we find that such a procedure in two dimensions (2D) leads to a loss of both global phase coherence and crystalline order. Unlike in 1D, the 2D roton modes have little in common with the supersolid configuration. We develop a finite-temperature stochastic Gross-Pitaevskii theory that includes beyond-mean-field effects to explore the formation process in 2D and find that evaporative cooling directly into the supersolid phase-hence bypassing the first-order roton instability-can produce a robust supersolid in a circular trap. Importantly, the resulting supersolid is stable at the final nonzero temperature. We then experimentally produce a 2D supersolid in a near-circular trap through such an evaporative procedure. Our work provides insight into the process of supersolid formation in 2D and defines a realistic path to the formation of large two-dimensional supersolid arrays.

DOI
10.1103/physrevlett.128.195302

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Dipolar condensates have recently been coaxed to form the long-sought supersolid phase. While one-dimensional supersolids may be prepared by triggering a roton instability, we find that such a procedure in two dimensions (2D) leads to a loss of both global phase coherence and crystalline order. Unlike in 1D, the 2D roton modes have little in common with the supersolid configuration. We develop a finite-temperature stochastic Gross-Pitaevskii theory that includes beyond-mean-field effects to explore the formation process in 2D and find that evaporative cooling directly into the supersolid phase-hence bypassing the first-order roton instability-can produce a robust supersolid in a circular trap. Importantly, the resulting supersolid is stable at the final nonzero temperature. We then experimentally produce a 2D supersolid in a near-circular trap through such an evaporative procedure. Our work provides insight into the process of supersolid formation in 2D and defines a realistic path to the formation of large two-dimensional supersolid arrays.
Dipolar condensates have recently been coaxed into supersolid phases supporting both superfluid and crystal excitations. While one-dimensional (1D) supersolids may be prepared via a roton instability, we find that such a procedure in two dimensions (2D) leads to greater heating. We go on to show that 2D roton modes have little in common with the supersolid configuration: instead, unstable centralized rotons trigger a process of nonlinear crystal growth. By evaporatively cooling directly into the supersolid phase--hence bypassing the first-order roton instability--we experimentally produce a 2D supersolid in a near-circular trap. We develop a stochastic Gross-Pitaevskii theory that includes beyond-meanfield effects to further explore the formation process. We calculate the static structure factor for a 2D supersolid, and compare to a 1D array. These results provide insight into the process of supersolid formation in 2D, and define a realistic path to the formation of large two-dimensional supersolid arrays.
We characterize the role of roton instability in the formation of a supersolid state of an elongated dipolar condensate, following a quench of the contact interactions across the superfluid-supersolid transition, as observed in recent experiments. We perform dynamical simulations by means of the extended Gross-Pitaevskii equation including quantum corrections, for different final values of the $s\text{\ensuremath{-}}\mathrm{wave}$ scattering length. The corresponding excitation spectrum is computed using an effective one-dimensional description, which provides a reasonably accurate prediction for the growth rate of the most unstable mode observed in the simulations. To analyze the behavior of the system, we employ the inverse participation ratio, which conveniently characterizes the different degree of localization in the superfluid and supersolid phases. By means of a suitable effective ansatz for the density, we derive a simple yet effective expression for the formation time of the supersolid. This expression provides valuable insights regarding its scaling behavior with respect to the sss-wave scattering length.
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.
In a recent experiment, it has been found that by tuning the interaction strength of a dipolar Bose-Einstein Condensate confined in a quasi-1D geometry orthogonal to the polarization direction, the roton gap vanishes, while density modulations are induced in the trapped gas. This suggests the possibility of realizing a supersolid in dipolar quantum gases. Inspired by this experiment, we simulate a dipolar BEC in a ring geometry by confining the dipoles in a ”tube” orthogonal to the polarization direction and enforcing periodic boundary conditions. Our calculations are based on a Density Functional approach where the total energy density functional includes the beyond-mean-field correction of Lee-Huang-Yang, which partially accounts for quantum fluctuations. We study the ground state properties of the system by evolving in imaginary time the EulerLagrange equation derived from the variational principle. We also study the excitation spectrum of the system by solving the corresponding Bogoliubov-de Gennes equations. We find that the calculated excitation spectrum shows indeed a roton minimum, and the roton gap can become vanishingly small by reducing the effective scattering length. As the roton gap disappears, we find that the system spontaneously develops in its ground-state a periodic structure formed by denser clusters of atomic dipoles immersed in a dilute superfluid background. We find that this structure shows the hallmarks of a supersolid system, i.e. (i) a finite non-classical translational inertia and (ii) the appearance, besides the phonon mode, of the Nambu-Goldstone gapless mode corresponding to fluctuations in the phase, and related to the spontaneous breaking of the gauge symmetry.
Superfluids with a tendency towards periodic crystalline order have both a phonon and roton like spectrum of collective modes. The softening of the roton spectrum provides one route to a supersolid. We show that roton softening occurs in 87Rb spinor condensates once dipolar interactions and spin dynamics are taken into account. By including the effects of a quasi-two-dimensional geometry and rapid Larmor precession, we show a dynamical instability develops in the collective mode spectrum at finite wavevectors. We construct phase diagrams showing a variety of instabilities as a function of the direction of the magnetic field and strength of the quadratic Zeeman shift. Our results provide a possible explanation of current experiments in the Berkeley group Phys. Rev. Lett. 100:170403 (2008).
We theoretically investigate supersolidity in three-dimensional dipolar Bose-Einstein condensates. We focus on the role of trap geometry in determining the dimensionality of the resulting droplet arrays, which range from one-dimensional to zigzag, through to two-dimensional supersolids in circular traps. Supersolidity is well established in one-dimensional arrays, and may be just as favorable in two-dimensional arrays provided that one appropriately scales the atom number to the trap volume. We develop a tractable variational model---which we benchmark against full numerical simulations---and use it to study droplet crystals and their excitations. We also outline how exotic ring and stripe states may be created with experimentally feasible parameters. Our work paves the way for future studies of two-dimensional dipolar supersolids in realistic settings.