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arXiv:2412.15552 · Physical Review A

Supersolid spectroscopy

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

Published May 7, 2025• 7 citations
arXivHTMLPDF

Abstract

We develop a linear-response theory to provide a unified description of two recent spectroscopy protocols for probing one-dimensional supersolid states realized in cold-atom systems. Both protocols involve applying a periodic optical potential to excite the supersolid and determine its excitation frequencies and density response characteristics. This information can be used to estimate the superfluid fraction. We validate our linear-response theory against nonlinear mean-field simulations of the dynamics for ring, box, and harmonically trapped cases. A key focus is the behavior at the band edge---the regime occurring when the optical potential used to excite the system has a wavelength that is twice the value of the supersolid lattice constant. Here symmetry can be used to selectively excite a mode from one of the two low-energy gapless excitation bands. We also consider the application of the spectroscopy protocols to determine the superfluid fraction, showing the relationship to hydrodynamic theory and a Josephson-junction array model. Our analysis and results suggest that the phase protocol we introduce has potential advantages for experimental application over the density protocol.

DOI
10.1103/physreva.111.053305

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We develop a linear-response theory to provide a unified description of two recent spectroscopy protocols for probing one-dimensional supersolid states realized in cold-atom systems. Both protocols involve applying a periodic optical potential to excite the supersolid and determine its excitation frequencies and density response characteristics. This information can be used to estimate the superfluid fraction. We validate our linear-response theory against nonlinear mean-field simulations of the dynamics for ring, box, and harmonically trapped cases. A key focus is the behavior at the band edge---the regime occurring when the optical potential used to excite the system has a wavelength that is twice the value of the supersolid lattice constant. Here symmetry can be used to selectively excite a mode from one of the two low-energy gapless excitation bands. We also consider the application of the spectroscopy protocols to determine the superfluid fraction, showing the relationship to hydrodynamic theory and a Josephson-junction array model. Our analysis and results suggest that the phase protocol we introduce has potential advantages for experimental application over the density protocol.
High-energy scattering spectroscopy is a widely established technique for probing the characteristic properties of complex physical systems. Motivated by the recent observation of long-sought supersolid states in dipolar quantum Bose gases, I investigate the general relationships existing between the density contrast, the superfluid fraction, and the response to a high-energy scattering probe of density-modulated states within a classical-field approach. I focus on the two extreme regimes of shallow and deep supersolids, which are of particular interest in describing the phase transitions of the supersolid to a uniform superfluid and an incoherent crystal state, respectively. Using relevant Ans\"atze for the fields of dipolar supersolid states in these regimes, I specify and illustrate the scaling laws relating the three observables. This work was first prompted to develop an intuitive understanding of a concomitant study based on experiments and mean-field numerical simulations. Beyond this specific application, this works provides a simple and general framework to describe density-modulated states and in particular the intriguing case of supersolids. It describes key properties characterizing the supersolid order and highlights possibilities for probing such properties based on high-energy scattering response.
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.
High-energy scattering spectroscopy is a widely-established technique for probing the characteristic properties of complex physical systems. Motivated by the recent observation of long-sought supersolid states in dipolar quantum Bose gases, I investigate the general relationships existing between the density contrast, the superfluid fraction, and the response to a high-energy scattering probe of such exotic states within a simple model. I focus on the scaling laws relating these three observables in the two extreme regimes of shallow and deep supersolids, which are of particular interest in describing the phase transitions of the supersolid to a uniform superfluid and an incoherent crystal state, respectively. I illustrate and specify the general formulae using relevant ansatzes for the supersolid states of dipolar gases in these regimes. This work brings an intuitive understanding of a concomitant study based on experiments and mean-field numerical simulations. It generally highlights new possibilities for probing supersolid states while providing a simple framework to describe such intriguing states.
We propose a protocol to excite the Goldstone modes of a supersolid dipolar Bose-Einstein condensed gas confined in a ring geometry. By abruptly removing an applied periodic modulation proportional to cos(φ), where φ is the azimuthal angle, we explore the resulting oscillations of the gas by solving the extended Gross-Pitaevskii equation. The value of the two longitudinal sound velocities exhibited in the supersolid phase are analyzed using the hydrodynamic theory of supersolids at zero temperature, which explicitly takes into account both the superfluid and the crystal nature of the system. This approach allows for the determination of the layer compressibility modulus as well as of the superfluid fraction, f_{S}, in agreement with the Leggett estimate of the nonclassical moment of inertia.
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.
Motivated by the recent rapid development of the field of quantum gases in optical lattices, we present a comprehensive study of the spectrum of ultracold atoms in a one-dimensional optical lattice subjected to a periodic lattice modulation. Using the time-dependent density matrix renormalization group method, we study the dynamical response due to lattice modulations in different quantum phases of the system with varying density. For the Mott-insulating state, we identify several excitation processes, which provide important information about the density profile of the gases. For the superfluid, the dynamical response can be well described in a local density approximation. This simplification can be valuable in understanding the strong-correlated superfluid in a slow-varying harmonic potential. All these spectroscopic features of an inhomogeneous system can be used as a test for the validity of the Bose-Hubbard model in a parabolic trapping potential.