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arXiv:2004.12577 · Physical Review Research

Supersolidity in an elongated dipolar condensate

P. B. Blakie, D. Baillie, Lauriane Chomaz, F. Ferlaino

Published Dec 4, 2020• 83 citations
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Abstract

We present a theory for the emergence of a supersolid state in a cigar-shaped dipolar quantum Bose gas.Our approach is based on a reduced three-dimensional (3D) theory, where the condensate wave function is decomposed into an axial field and a transverse part described variationally.This provides an accurate fully 3D description that is specific to the regime of current experiments and efficient to compute.We apply this theory to understand the phase diagram for a gas in an infinite tube potential.We find that the supersolid transition has continuous and discontinuous regions as the averaged density varies.We develop two simplified analytic models to characterize the phase diagram and elucidate the roles of quantum droplets and of the roton excitation.

DOI
10.1103/physrevresearch.2.043318

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We present a theory for the emergence of a supersolid state in a cigar-shaped dipolar quantum Bose gas.Our approach is based on a reduced three-dimensional (3D) theory, where the condensate wave function is decomposed into an axial field and a transverse part described variationally.This provides an accurate fully 3D description that is specific to the regime of current experiments and efficient to compute.We apply this theory to understand the phase diagram for a gas in an infinite tube potential.We find that the supersolid transition has continuous and discontinuous regions as the averaged density varies.We develop two simplified analytic models to characterize the phase diagram and elucidate the roles of quantum droplets and of the roton excitation.
We calculate the ground states of a dipolar Bose gas confined in an infinite tube potential. We use the extended Gross-Pitaevskii equation theory and present a numerical method to efficiently obtain solutions. A key feature of this method is an analytic result for a truncated dipole-dipole interaction potential that enables the long-ranged interactions to be accurately evaluated within a unit cell. Our focus is on the transition of the ground state to a crystal driven by dipole-dipole interactions as the short-ranged interaction strength is varied. We find that the transition is continuous or discontinuous depending upon the average system density. These results give deeper insight into the supersolid phase transition observed in recent experiments and validate the utility of the reduced three-dimensional theory developed in [Phys. Rev. Res. 2, 043318 (2020)] for making qualitatively accurate predictions.
We investigate the excitation spectrum and compressibility of a dipolar Bose-Einstein condensate in an infinite tube potential in the parameter regime where the transition between superfluid and supersolid phases occurs. Our study focuses on the density range in which crystalline order develops continuously across the transition. Above the transition the superfluid shows a single gapless excitation band, phononic at small momenta and with a roton at a finite momentum. Below the transition, two gapless excitations branches (three at the transition point) emerge in the supersolid. We examine the two gapless excitation bands and their associated speeds of sound in the supersolid phase. Our results show that the speeds of sound and the compressibility are discontinuous at the transition, indicating a second-order phase transition. These results provide valuable insights into the identification of supersolid phenomena in dipolar quantum gases and the relationship to supersolidity in spin-orbit coupled gases.
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.
We present a theoretical study of a mixture of antidipolar and nondipolar Bose-Einstein condensates confined to an infinite tube. We predict the presence of a spin roton and its associated instability, which triggers a continuous unmodulated-to-supersolid phase transition. We characterize the phase diagram of the binary system, ranging from the quasi-1D to the radial Thomas-Fermi (elongated 3D) regimes. We also present the dynamic formation of supersolids following a quench from the uniform miscible phase, which maintains phase coherence across the system.
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.
We provide a theoretical characterization of the dynamical crossing of the superfluid-supersolid phase transition for a dipolar condensate confined in an elongated trap, as observed in the recent experiment by Biagioni et al. [G. Biagioni, N. Antolini, A. Ala\~na, M. Modugno, A. Fioretti, C. Gabbanini, L. Tanzi, and G. Modugno, Phys. Rev. X 12, 021019 (2022)]. By means of the extended Gross-Pitaevskii theory, which includes the Lee-Huang-Yang quantum fluctuation correction, we first analyze the ground-state configurations of the system as a function of the interparticle scattering length, for both trap configurations employed in the experiment. Then, we discuss the effects of the ramp velocity, by which the scattering length is tuned across the transition, on the collective excitations of the system in both the superfluid and supersolid phases. We find that, when the transverse confinement is sufficiently strong and the transition has a smooth (continuous) character, the system essentially displays a (quasi) one-dimensional behavior, its excitation dynamics being dominated by the axial breathing modes. Instead, for shallower transverse trapping, when the transition becomes discontinuous, the collective excitations of the supersolid display a coupling with the transverse modes, signaling the onset of a dimensional crossover.