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arXiv:2112.12749 · Physical Review A

Moment of inertia and dynamical rotational response of a supersolid dipolar gas

S. M. Roccuzzo, Alessio Recati, S. Stringari

Published Feb 16, 2022• 19 citations
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Abstract

We show that knowledge of the time-dependent response of a trapped gas, subject to a sudden rotation of a confining harmonic potential, allows for the determination of the moment of inertia of dipolar supersolid configurations. While in the presence of one-dimensional arrays of droplets the frequency of the resulting scissors oscillation provides accurate access to the value of the moment of inertia, two-dimensional-like configurations are characterized by low-frequency resonances in the rotating signal, reflecting the presence of significant rigid-body components in the rotational motion. Using the formalism of response-function theory and simulations based on the so-called extended time-dependent Gross-Pitaevskii equation, we point out the crucial role played by the low-frequency components in the determination of the moment of inertia and of its deviations from the irrotational value. We also propose a protocol based on the stationary rotation of the trap, followed by its sudden stop, which might provide a promising alternative to the experimental evaluation of the moment of inertia.

DOI
10.1103/physreva.105.023316

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We show that knowledge of the time-dependent response of a trapped gas, subject to a sudden rotation of a confining harmonic potential, allows for the determination of the moment of inertia of dipolar supersolid configurations. While in the presence of one-dimensional arrays of droplets the frequency of the resulting scissors oscillation provides accurate access to the value of the moment of inertia, two-dimensional-like configurations are characterized by low-frequency resonances in the rotating signal, reflecting the presence of significant rigid-body components in the rotational motion. Using the formalism of response-function theory and simulations based on the so-called extended time-dependent Gross-Pitaevskii equation, we point out the crucial role played by the low-frequency components in the determination of the moment of inertia and of its deviations from the irrotational value. We also propose a protocol based on the stationary rotation of the trap, followed by its sudden stop, which might provide a promising alternative to the experimental evaluation of the moment of inertia.
Distinctive features of supersolids show up in their rotational properties. We calculate the moment of inertia of a harmonically trapped dipolar Bose-Einstein condensed gas as a function of the tunable scattering length parameter, providing the transition from the (fully) superfluid to the supersolid phase and eventually to an incoherent crystal of self-bound droplets. The transition from the superfluid to the supersolid phase is characterized by a jump in the moment of inertia, revealing its first order nature. In the case of elongated trapping in the plane of rotation, we show that the moment of inertia determines the value of the frequency of the scissors mode, which is significantly affected by the reduction of superfluidity in the supersolid phase. The case of an in-plane isotropic trapping is instead well suited to study the formation of quantized vortices, which are shown to be characterized, in the supersolid phase, by a sizeable deformed core, caused by the presence of the surrounding density peaks.
A key manifestation of superfluidity in liquids and gases is a reduction of the moment of inertia under slow rotations. Nonclassical rotational effects have also been considered in the context of the elusive supersolid phase of matter, in which superfluidity coexists with a lattice structure. Here, we show that the recently discovered supersolid phase in dipolar quantum gases features a reduced moment of inertia. Using a dipolar gas of dysprosium atoms, we studied a peculiar rotational oscillation mode in a harmonic potential, the scissors mode, previously investigated in ordinary superfluids. From the measured moment of inertia, we deduced a superfluid fraction that is different from zero and of order of unity, providing direct evidence of the superfluid nature of the dipolar supersolid.
Quantum Gases When a bucket of water is rotated, the water rotates with the vessel, contributing to the total moment of inertia. If such an experiment were done with a superfluid, it would decouple from the vessel and would not contribute to rotation. Tanzi et al. studied an intermediate case, a supersolid, which is predicted to only partially decouple, resulting in a moment of inertia smaller than the classical value. Whereas previous such experiments were done with helium, the authors used a gas of highly magnetic dysprosium atoms in an optical trapping potential that was suddenly changed, causing the gas to oscillate. Measuring the frequency of these oscillations provides evidence for a reduced moment of inertia. Science , this issue p. [1162][1] [1]: /lookup/doi/10.1126/science.aba4309
We show that a sudden change in the polarization direction of the magnetic dipole moments of the atoms in a dipolar Bose-Einstein condensate (BEC) can serve as a useful dynamical probe to sense its superfluid and solid-like properties. We find that for small angular deviation of the polarization direction, actuated for instance by modifying an external magnetic field, the superfluid state undergoes an undamped scissors mode oscillation, a characteristic signature of superfluidity. In contrast, both the droplet and supersolid states exhibit a scissors-mode oscillation, which is effectively damped due to multiple closely spaced frequency components. Notably, we find that this damping rate provides a qualitative measure for the rigidity of different phases of a dipolar BEC. Furthermore, there exists a maximum angular deviation of the polarization direction, beyond which the droplet and the supersolid states undergo a permanent deformation i.e., we find an analog of the usual elastic to plastic phase transition of solids. We characterize this transition numerically using the fidelity of the condensate wavefunction with the ground state as well as the droplet width and periodicity of the supersolid density of the condensate which are experimentally accessible. Thus, the dynamical protocol introduced here can be an important experimental benchmark to identify and characterize the superfluid and solid properties of different phases of dipolar BECs.
We study dipole-mode and scissors-mode oscillations of a harmonically trapped dipolar supersolid, composed of dipolar droplets arranged on a one-dimensional (1D) or a 2D lattice, to establish the robustness of its crystalline structure under translation and rotation, using a beyond-mean-field model including a Lee-Huang-Yang interaction. The dipolar atoms are polarized in the zzz direction with the supersolid crystalline structure lying in the $x\text{\ensuremath{-}}y$ plane. A stable dipole-mode oscillation is possible in the case of both quasi-1D and quasi-2D dipolar supersolids, whereas a sustained angular scissors-mode oscillation is possible only in the case of a quasi-1D dipolar supersolid between a maximum and a minimum of trap anisotropy in the $x\text{\ensuremath{-}}y$ plane. In both cases there is no visible deformation of the crystalline structure of the dipolar supersolid during the oscillation. The theoretical estimate of the scissors-mode-oscillation frequency is in good agreement with the present results and the agreement improves with an increase in the number of droplets in the supersolid and also with an increase in the confining trap frequencies. The results of this study can be tested experimentally with present knowhow.
We analyze the pinning of vortices for a stationary rotating dipolar supersolid along the low-density paths between droplets as a function of the rotation frequency. We restrict ourselves to the stationary configurations of vortices with the same symmetry as that of the array of droplets. In particular, such an analysis clearly reveals that vortices are not only pinned at local density minima, but instead their coordinates are smooth functions of the rotation frequency. Our approach to explaining such a behavior exploits the fact that the wave function of each rotating droplet acquires a linear phase on the coordinates. Hence, the relative phases between the nearest neighboring droplets allow us to predict the position of the vortices in the intermediate low-density region. Here, we show that, for a droplet distribution forming a triangular lattice, the phases of three neighboring droplets are needed for the correct description of the vortex location. In particular, for our confined system, we demonstrate that the estimate accurately reproduces the extended Gross-Pitaevskii results in the spatial regions where the neighboring droplets are well defined.