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arXiv:1907.09315 · Physical Review Letters

Fate of the Amplitude Mode in a Trapped Dipolar Supersolid

Jens Hertkorn, Fabian Böttcher, Mingyang Guo, Jan-Niklas Schmidt, Tim Langen, Hans Peter Büchler, Tilman Pfau

Published Nov 6, 2019• 67 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.123.193002

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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.
We theoretically investigate elementary excitations of dipolar quantum gases across the superfluid to supersolid phase transition in a toroidal trap. We show how decoupled first sound, second sound, and Higgs modes emerge by following their origin from superfluid modes across the transition. The structure of these excitations reveals the interplay between crystal and superfluid oscillations. Our results unify previous notions of coupled Goldstone and Higgs modes in harmonic traps, allowing us to establish a correspondence between excitations of trapped and infinitely extended supersolids. We propose protocols for selectively probing these sound and amplitude modes, accessible to state-of-the-art experiments. Published by the American Physical Society 2024
We theoretically investigate the ground states and the spectrum of elementary excitations across the superfluid to droplet crystallization transition of an oblate dipolar Bose-Einstein condensate. We systematically identify regimes where spontaneous rotational symmetry breaking leads to the emergence of a supersolid phase with characteristic collective excitations, such as the Higgs amplitude mode. Furthermore, we study the dynamics across the transition and show how these supersolids can be realized with standard protocols in state-of-the-art experiments.
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.
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.
Dipolar Bose-Einstein condensates are excellent platforms for studying supersolidity, characterized by coexisting density modulation and superfluidity. The realization of dipolar mixtures opens intriguing new scenarios, most remarkably the possibility of realizing a double supersolid, composed of two interacting superfluids. We analyze the complex excitation spectrum of a miscible trapped dipolar Bose mixture, showing that it provides key insights about the double-supersolid regime. We show that this regime may be readily probed experimentally by monitoring the appearance of a doublet of superfluid compressional modes, linked to the different superfluid character of each component. Additionally, the dipolar supersolid mixture exhibits a nontrivial spin nature of the dipolar rotons, the Higgs excitation, and the low-lying Goldstone modes. Interestingly, the analysis of the lowest-lying modes allows for monitoring the transition of just one of the components into the incoherent-droplet regime, whereas the other remains coherent, highlighting their disparate superfluid properties.
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.