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The Excitation Spectra of a Supersolid

Lily Platt

Published Sep 20, 2024• 0 citations
OpenAlex

Abstract

This thesis explores the excitation spectra of supersolids - materials that remarkably manifest both lattice structure and superfluidity. We focus on an ultracold dilute gas exhibiting one-dimensional crystal structure with two gapless excitation branches. Two distinct supersolid systems are examined: a one dimensional Bose-Einstein condensate with softcore interactions, and a three dimensional dipolar Bose Einstein condensate confined within an infinitely long tube-shape potential. We investigate the crystal-like excitations of the supersolid through the use of the linear chain model and by introducing an elastic constant derived from ground-state calculations. Subsequently, we adopt a hydrodynamic approach based on an effective Lagrangian, which depends upon three fields, corresponding to lattice deformations, defect density current, the global phase fluctuations and four elastic parameters. This description of the supersolid allows for the determination of low-energy excitations, as well as the behaviour of the associated density and phase fluctuations of the two gapless branches. We validate our predictions for the low energy excitations by comparison to results from directly solving the Bogoliubov-de-Gennes equations, finding excellent agreement. This thesis provides a deeper understanding of the different types of supersolid systems, and how their low-energy behaviour depends upon their characteristic elastic energy scales.

Related papers

Sound waves and fluctuations in one-dimensional supersolids

L. M. Platt, D. Baillie, P. B. Blakie

We examine the low-energy excitations of a dilute supersolid state of matter with a one-dimensional crystal structure. A hydrodynamic description is developed based on a Lagrangian, incorporating generalized elastic parameters derived from ground state calculations. The predictions of the hydrodynamic theory are validated against solutions of the Bogoliubov--de Gennes equations, by comparing the speeds of sound, density fluctuations, and phase fluctuations of the two gapless bands. Our results are presented for two distinct supersolid models: a dipolar Bose-Einstein condensate in an infinite tube and a dilute Bose-Einstein condensate of atoms with soft-core interactions. Characteristic energy scales are identified, highlighting that these two models approximately realize the bulk incompressible and rigid lattice supersolid limits.

arXiv:2403.19151•Physical Review A•Aug 16, 2024•13 citations

Excitations of a two-dimensional supersolid

Elena Poli, D. Baillie, Francesca Ferlaino +1 more

We present a theoretical study of the excitations of the two-dimensional supersolid state of a Bose-Einstein condensate with either dipole-dipole interactions or soft-core interactions. This supersolid state has three gapless excitation branches arising from the spontaneously broken continuous symmetries. Two of these branches are related to longitudinal sound waves, similar to those in one-dimensional supersolids. The third branch is a transverse wave arising from the nonzero shear modulus of the two-dimensional crystal. We present the results of numerical calculations for the excitations and dynamic structure factor characterizing the density fluctuations and study their behavior across the discontinuous superfluid to supersolid transition. We show that the speeds of sound are described by a hydrodynamic theory that incorporates generalized elastic parameters, including the shear modulus. Furthermore, we establish that dipolar and soft-core supersolids manifest distinct characteristics, falling into the bulk-incompressible and rigid-lattice limits, respectively.

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Supersolid behavior of a dipolar Bose-Einstein condensate confined in a tube

S. M. Roccuzzo, Francesco Ancilotto

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 $T=0$ 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.

arXiv:1810.12229•Physical Review A•Apr 2, 2019•125 citations

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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.

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Hydrodynamic theory of rotating ultracold Bose–Einstein condensates in supersolid phase

Rashi Sachdeva, Sankalpa Ghosh

Abstract Within the mean field Gross–Pitaevskii framework, ultracold atomic condensates with long-range interaction are predicted to have a supersolid-like ground state beyond a critical interaction strength. Such a mean field supersolid-like ground state has periodically modulated superfluid density which implies the coexistence of superfluid and crystalline order. An ultracold atomic system in such a mean field ground state can be subjected to an artificial gauge field created either through rotation or by introducing space dependent coupling among hyperfine states of the atoms using Raman lasers. Starting from this Gross–Pitaevskii energy functional that describes such systems at zero temperature, we construct a hydrodynamic theory to describe the low-energy long-wavelength excitations of a rotating supersolid of weakly interacting ultracold atoms in two spatial dimensions for a generic type of long-range interaction. We treat the supersolidity in such a system within the framework of the well known two-fluid approximation. Considering such a system in the fast rotation limit where a vortex lattice in superfluid coexists with the supersolid lattice, we analytically obtain the dispersion relations of collective excitations around this equilibrium state. The dispersion relation gives the modes of the rotating supersolid which can be experimentally measured within the current technology. We point out that this can clearly identify an ultracold atomic supersolid phase in an unambiguous way.

arXiv:1308.1592•Journal of Physics B Atomic Molecular and Optical Physics•Apr 15, 2015•2 citations

Acoustic fluctuations of two-dimensional dipolar supersolids

P. B. Blakie

We investigate the microscopic structure of the long-wavelength acoustic excitations of two-dimensional dipolar supersolids with triangular and honeycomb crystal order. Using Bogoliubov--de Gennes calculations, we determine the density, phase, current, and lattice-displacement fluctuations associated with the three acoustic modes. We develop a procedure to extract the lattice displacement directly from the microscopic density perturbation, allowing the modes to be identified as one transverse and two longitudinal branches. The displacement field further separates the density fluctuations into contributions from crystal strain and particle transport relative to the lattice. This reveals that the lower longitudinal mode can have large strain and defect-density fluctuations that substantially cancel in the total density response. We show that the long-wavelength fluctuation amplitudes agree with hydrodynamic theory, which is specified entirely by the elastic coefficients of the supersolid. Our analysis provides a microscopic connection between Bogoliubov excitations and the crystalline and superfluid degrees of freedom of two-dimensional supersolids, and identifies their signatures in density- and current-sensitive probes.

arXiv:2608.06800•Aug 7, 2026•0 citations
This thesis explores the excitation spectra of supersolids - materials that remarkably manifest both lattice structure and superfluidity. We focus on an ultracold dilute gas exhibiting one-dimensional crystal structure with two gapless excitation branches. Two distinct supersolid systems are examined: a one dimensional Bose-Einstein condensate with softcore interactions, and a three dimensional dipolar Bose Einstein condensate confined within an infinitely long tube-shape potential. We investigate the crystal-like excitations of the supersolid through the use of the linear chain model and by introducing an elastic constant derived from ground-state calculations. Subsequently, we adopt a hydrodynamic approach based on an effective Lagrangian, which depends upon three fields, corresponding to lattice deformations, defect density current, the global phase fluctuations and four elastic parameters. This description of the supersolid allows for the determination of low-energy excitations, as well as the behaviour of the associated density and phase fluctuations of the two gapless branches. We validate our predictions for the low energy excitations by comparison to results from directly solving the Bogoliubov-de-Gennes equations, finding excellent agreement. This thesis provides a deeper understanding of the different types of supersolid systems, and how their low-energy behaviour depends upon their characteristic elastic energy scales.
We examine the low-energy excitations of a dilute supersolid state of matter with a one-dimensional crystal structure. A hydrodynamic description is developed based on a Lagrangian, incorporating generalized elastic parameters derived from ground state calculations. The predictions of the hydrodynamic theory are validated against solutions of the Bogoliubov--de Gennes equations, by comparing the speeds of sound, density fluctuations, and phase fluctuations of the two gapless bands. Our results are presented for two distinct supersolid models: a dipolar Bose-Einstein condensate in an infinite tube and a dilute Bose-Einstein condensate of atoms with soft-core interactions. Characteristic energy scales are identified, highlighting that these two models approximately realize the bulk incompressible and rigid lattice supersolid limits.
We present a theoretical study of the excitations of the two-dimensional supersolid state of a Bose-Einstein condensate with either dipole-dipole interactions or soft-core interactions. This supersolid state has three gapless excitation branches arising from the spontaneously broken continuous symmetries. Two of these branches are related to longitudinal sound waves, similar to those in one-dimensional supersolids. The third branch is a transverse wave arising from the nonzero shear modulus of the two-dimensional crystal. We present the results of numerical calculations for the excitations and dynamic structure factor characterizing the density fluctuations and study their behavior across the discontinuous superfluid to supersolid transition. We show that the speeds of sound are described by a hydrodynamic theory that incorporates generalized elastic parameters, including the shear modulus. Furthermore, we establish that dipolar and soft-core supersolids manifest distinct characteristics, falling into the bulk-incompressible and rigid-lattice limits, respectively.
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.
Abstract Within the mean field Gross–Pitaevskii framework, ultracold atomic condensates with long-range interaction are predicted to have a supersolid-like ground state beyond a critical interaction strength. Such a mean field supersolid-like ground state has periodically modulated superfluid density which implies the coexistence of superfluid and crystalline order. An ultracold atomic system in such a mean field ground state can be subjected to an artificial gauge field created either through rotation or by introducing space dependent coupling among hyperfine states of the atoms using Raman lasers. Starting from this Gross–Pitaevskii energy functional that describes such systems at zero temperature, we construct a hydrodynamic theory to describe the low-energy long-wavelength excitations of a rotating supersolid of weakly interacting ultracold atoms in two spatial dimensions for a generic type of long-range interaction. We treat the supersolidity in such a system within the framework of the well known two-fluid approximation. Considering such a system in the fast rotation limit where a vortex lattice in superfluid coexists with the supersolid lattice, we analytically obtain the dispersion relations of collective excitations around this equilibrium state. The dispersion relation gives the modes of the rotating supersolid which can be experimentally measured within the current technology. We point out that this can clearly identify an ultracold atomic supersolid phase in an unambiguous way.
We investigate the microscopic structure of the long-wavelength acoustic excitations of two-dimensional dipolar supersolids with triangular and honeycomb crystal order. Using Bogoliubov--de Gennes calculations, we determine the density, phase, current, and lattice-displacement fluctuations associated with the three acoustic modes. We develop a procedure to extract the lattice displacement directly from the microscopic density perturbation, allowing the modes to be identified as one transverse and two longitudinal branches. The displacement field further separates the density fluctuations into contributions from crystal strain and particle transport relative to the lattice. This reveals that the lower longitudinal mode can have large strain and defect-density fluctuations that substantially cancel in the total density response. We show that the long-wavelength fluctuation amplitudes agree with hydrodynamic theory, which is specified entirely by the elastic coefficients of the supersolid. Our analysis provides a microscopic connection between Bogoliubov excitations and the crystalline and superfluid degrees of freedom of two-dimensional supersolids, and identifies their signatures in density- and current-sensitive probes.