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arXiv:2211.17251 · Physical Review Research

Quantum fluctuations in one-dimensional supersolids

Chris Bühler, Tobias Ilg, Hans Peter Büchler

Published Aug 9, 2023• 9 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevresearch.5.033092

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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 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.
It is intuitive to think that heating any material leads to an increase in random thermal motion and, thereby, usually counteracts any ordering efforts. For example, heating ice leads to the formation of a liquid and ultimately to a gas. However, experiments with dipolar Bose-Einstein condensates suggested that this general phenomenology does not always take place [1]. In order to describe the surprising superfluid-supersolid transition upon an increase in temperature at fixed condensed particle number we employ Bogoliubov theory with local density approximation [2], [3] and, indeed, find that such behaviour can also be expected theoretically [Fig. 1(a)]. With this approach we find [3] that the scaling with respect to density of the shift in energy due to thermal fluctuations is opposite to that of quantum fluctuations and, therefore, promotes the formation of high density regions [Fig. 1(b)]. Consistent with that we find that an increase in temperature promotes roton-softening [Fig. 1(c)]. Furthermore, we find good agreement between the experimentally measured density and the theoretical prediction including thermal fluctuations, whereas not taking thermal corrections into account yields a different result [Fig. 1(d)]. These findings call for further investigation of the surprising thermodynamic behaviour of ultracold quantum ferrofluids.
Abstract The properties of a supersolid state (SS) in quasi-one-dimensional dipolar Bose–Einstein condensate is studied, considering two possible mechanisms of realization—due to repulsive three-body atomic interactions and quantum fluctuations in the framework of the Lee–Huang–Yang theory. The role of both mechanisms in the formation of SS properties has been emphasized. The proposed theoretical model, based on minimization of the energy functional, allows evaluating the amplitude of the SS for an arbitrary set of parameters in the extended Gross–Pitaevskii equation (eGPE). To explore the dynamics of the SS first we numerically construct its ground state in different settings, including periodic boundary conditions, box-like trap and parabolic potential, then impose a perturbation. In oscillations of the perturbed supersolid we observe the key manifestation of SS, namely the free flow of the superfluid fraction through the crystalline component of the system. Two distinct oscillation frequencies of the supersolid associated with the superfluid fraction and crystalline components of the wave function are identified from numerical simulations of the eGPE.
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 derive a controlled expansion into mean field plus fluctuations for the extended Bose-Hubbard model, involving interactions with many neighbors on an arbitrary periodic lattice, and study the superfluid-supersolid phase transition. Near the critical point, the impact of (thermal and quantum) fluctuations on top of the mean field grows, which entails striking effects, such as negative superfluid densities and thermodynamical instability of the superfluid phase---earlier as expected from mean-field dynamics. We also predict the existence of long-lived ``supercooled'' states with anomalously large quantum fluctuations.
Quantum fluctuations play a pivotal role in determining the ground state properties and phase transitions of ultra-cold bosonic systems. This study presents a comprehensive theoretical investigation of fluctuation effects beyond mean-field theory in dilute Bose gases, examining their impact on the superfluid–Mott insulator transition, quantum droplet formation, and critical phenomena. Beginning with the Gross–Pitaevskii mean-field framework , we systematically incorporate quantum corrections through Bogoliubov theory, yielding the Lee–Huang–Yang (LHY) energy correction scaling as . The quantum depletion quantifies the fraction of atoms driven from the condensate by zero-point fluctuations. For the Bose–Hubbard model , quantum Monte Carlo simulations establish the superfluid–Mott insulator critical point at , shifted 15% from the mean-field prediction. Critical exponents and confirm 3D XY universality. Quantum droplets in binary mixtures are stabilized by the repulsive LHY correction against mean-field collapse, with equilibrium densities matching experimental observations in K mixtures. Finite-size scaling analysis and the Ginzburg criterion delineate regimes where fluctuation effects dominate. All theoretical predictions are validated against precision measurements in ultracold atomic gases.