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arXiv:2211.17178 · Physical Review A

Ground-state stability and excitation spectrum of a one-dimensional dipolar supersolid

Tobias Ilg, Hans Peter Büchler

Published Jan 31, 2023• 20 citations
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Abstract

We study the behavior of the excitation spectrum across the quantum phase transition from a superfluid to a supersolid phase of a dipolar Bose gas confined to a one-dimensional geometry. Including the leading beyond-mean-field effects within an effective Hamiltonian, the analysis is based on Bogoliubov theory with several order parameters accounting for the superfluid as well as solid structure. We find fast convergence of the ground-state energy in the supersolid with the number of order parameters and demonstrate a stable excitation spectrum with two Goldstone modes and an amplitude mode in the low-energy regime. Our results suggest that there exists an experimentally achievable parameter regime for dysprosium atoms, where the supersolid phase exhibits a stable excitation spectrum in the thermodynamic limit and the transition into the supersolid phase is of second order driven by the roton instability.

DOI
10.1103/physreva.107.013314

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We study the behavior of the excitation spectrum across the quantum phase transition from a superfluid to a supersolid phase of a dipolar Bose gas confined to a one-dimensional geometry. Including the leading beyond-mean-field effects within an effective Hamiltonian, the analysis is based on Bogoliubov theory with several order parameters accounting for the superfluid as well as solid structure. We find fast convergence of the ground-state energy in the supersolid with the number of order parameters and demonstrate a stable excitation spectrum with two Goldstone modes and an amplitude mode in the low-energy regime. Our results suggest that there exists an experimentally achievable parameter regime for dysprosium atoms, where the supersolid phase exhibits a stable excitation spectrum in the thermodynamic limit and the transition into the supersolid phase is of second order driven by the roton instability.
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.
In one dimension, quantum fluctuations prevent the appearance of long-range order in a supersolid, and only quasi-long-range order can survive. We derive this quantum critical behavior and study its influence on the superfluid response and properties of the solid. The analysis is based on an effective low-energy description accounting for the two coupled Goldstone modes. We find that the quantum phase transition from the superfluid to the supersolid is shifted by quantum fluctuations from the position where the local formation of a solid structure takes place. For current experimental parameters with dipolar atomic gases, this shift is extremely small and cannot be resolved yet, i.e., current observations in experiments are expected to be in agreement with predictions from mean-field theory based on the extended Gross-Pitaevskii formalism.
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.
Motivated by a recent experiment [L. Chomaz et al., Nat. Phys. 14, 442 (2018)], we perform numerical simulations of a dipolar Bose-Einstein condensate (BEC) in a tubular, periodic confinement at T=0T=0T=0 within density functional theory, where the beyond-mean-field correction to the ground-state energy is included in the local density approximation. We study the excitation spectrum of the system by solving the corresponding Bogoliubov--de Gennes equations. The calculated spectrum shows a roton minimum, and the roton gap decreases by reducing the effective scattering length. As the roton gap disappears, the system spontaneously develops a periodic linear structure formed by denser clusters of atomic dipoles immersed in a dilute superfluid background. This structure shows the hallmarks of a supersolid system, i.e., (i) a finite nonclassical translational inertia along the tube axis and (ii) the appearance of two gapless modes, i.e., a phonon mode associated with density fluctuations and resulting from the translational discrete symmetry of the system, and a Nambu-Goldstone gapless mode corresponding to phase fluctuations, resulting from the spontaneous breaking of the gauge symmetry. A further decrease in the scattering length eventually leads to the formation of a periodic linear array of self-bound droplets.
We study the spectrum of elementary excitations of a dipolar Bose gas in a three-dimensional anisotropic trap across the superfluid-supersolid phase transition. Theoretically, we show that, when entering the supersolid phase, two distinct excitation branches appear, respectively associated with dominantly crystal and superfluid excitations. These results confirm infinite-system predictions, showing that finite-size effects play only a small qualitative role, and connect the two branches to the simultaneous occurrence of crystal and superfluid orders. Experimentally, we probe compressional excitations in an Er quantum gas across the phase diagram. While in the Bose-Einstein condensate regime the system exhibits an ordinary quadrupole oscillation, in the supersolid regime we observe a striking two-frequency response of the system, related to the two spontaneously broken symmetries.
We theoretically investigate the spectrum of elementary excitations of a trapped dipolar quantum gas across the BEC-supersolid phase transition. Our calculations reveal the existence of distinct Higgs amplitude and Nambu-Goldstone modes that emerge from the softening roton modes of the dipolar BEC at the phase transition point. On the supersolid side of the transition, the energy of the Higgs amplitude mode increases rapidly, leading to a strong coupling to higher-lying modes. Our Letter highlights how the symmetry-breaking nature of the supersolid state translates to finite-size systems.