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2D Schr\"odinger symmetry and 3D breathers/Kelvin-ripple complexes as quasi-massive-Nambu-Goldstone modes

Daisuke A. Takahashi, Toshiaki Fujimori, Keisuke Ohashi, Muneto Nitta

Published Jun 29, 2017• 0 citations
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Abstract

Bose-Einstein condensates (BECs) confined in a two-dimensional (2D) harmonic trap are known to possess a hidden 2D Schrodinger symmetry, that is, the Schrodinger symmetry modified by a trapping potential. Spontaneous breaking of this symmetry gives rise to a breathing motion of the BEC, whose oscillation frequency is robustly determined by the strength of the harmonic trap. In this paper, we demonstrate that the concept of the 2D Schrodinger symmetry can be applied to predict the nature of three dimensional (3D) collective modes propagating along a condensate confined in an elongated trap. We find three kinds of collective modes whose existence is robustly ensured by the Schrodinger symmetry, which are physically interpreted as one breather mode and two Kelvin-ripple complex i.e., composite modes in which the vortex core and the condensate surface oscillate interactively. We provide analytical expressions for the dispersion relations (energy-momentum relation) of these modes using the Bogoliubov theory [D. A. Takahashi and M. Nitta, Ann. Phys. 354, 101 (2015)]. Furthermore, we point out that these modes can be interpreted as quasi-massive-Nambu-Goldstone (NG) modes, that is, they have the properties of both quasi-NG and massive NG modes: quasi-NG modes appear when a symmetry of a part of a Lagrangian, which is not a symmetry of full a Lagrangian, is spontaneously broken, while massive NG modes appear when a modified symmetry is spontaneously broken.

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Bose-Einstein condensates (BECs) confined in a two-dimensional (2D) harmonic trap are known to possess a hidden 2D Schrodinger symmetry, that is, the Schrodinger symmetry modified by a trapping potential. Spontaneous breaking of this symmetry gives rise to a breathing motion of the BEC, whose oscillation frequency is robustly determined by the strength of the harmonic trap. In this paper, we demonstrate that the concept of the 2D Schrodinger symmetry can be applied to predict the nature of three dimensional (3D) collective modes propagating along a condensate confined in an elongated trap. We find three kinds of collective modes whose existence is robustly ensured by the Schrodinger symmetry, which are physically interpreted as one breather mode and two Kelvin-ripple complex i.e., composite modes in which the vortex core and the condensate surface oscillate interactively. We provide analytical expressions for the dispersion relations (energy-momentum relation) of these modes using the Bogoliubov theory [D. A. Takahashi and M. Nitta, Ann. Phys. 354, 101 (2015)]. Furthermore, we point out that these modes can be interpreted as quasi-massive-Nambu-Goldstone (NG) modes, that is, they have the properties of both quasi-NG and massive NG modes: quasi-NG modes appear when a symmetry of a part of a Lagrangian, which is not a symmetry of full a Lagrangian, is spontaneously broken, while massive NG modes appear when a modified symmetry is spontaneously broken.
Bose-Einstein condensates (BECs) confined in a two-dimensional (2D) harmonic trap are known to possess a hidden 2D Schr\"odinger symmetry, that is, the Schr\"odinger symmetry modified by a trapping potential. Spontaneous breaking of this symmetry gives rise to a breathing motion of the BEC, whose oscillation frequency is robustly determined by the strength of the harmonic trap. In this paper, we demonstrate that the concept of the 2D Schr\"odinger symmetry can be applied to predict the nature of three-dimensional (3D) collective modes propagating along a condensate confined in an elongated trap. We find three kinds of collective modes whose existence is robustly ensured by the Schr\"odinger symmetry, which are physically interpreted as one breather mode and two Kelvin-ripple complex modes, i.e., composite modes in which the vortex core and the condensate surface oscillate interactively. We provide analytical expressions for the dispersion relations (energy-momentum relation) of these modes using the Bogoliubov theory [D. A. Takahashi and M. Nitta, Ann. Phys. 354, 101 (2015)]. Furthermore, we point out that these modes can be interpreted as ``quasi-massive-Nambu-Goldstone (NG) modes'', that is, they have the properties of both quasi-NG and massive NG modes: quasi-NG modes appear when a symmetry of a part of a Lagrangian, which is not a symmetry of a full Lagrangian, is spontaneously broken, while massive NG modes appear when a modified symmetry is spontaneously broken.
The Gross-Pitaevskii or nonlinear Schr\"odinger equation relevant to ultracold atomic gaseous Bose-Einstein condensates possesses a modified Schr\"odinger symmetry in two spatial dimensions, in the presence of a harmonic trapping potential, an (artificial) constant magnetic field (or rotation), and an (artificial) electric field of a quadratic electrostatic potential. We find that a variance and a center of a trapped gas with or without a vorticity can be regarded as massive Nambu-Goldstone (NG) modes associated with spontaneous breaking of the modified Schr\"odinger symmetry. We show that the Noether theorem for the modified Schr\"odinger symmetry gives universal equations of motion which describe exact time evolutions of the trapped gases such as a harmonic oscillation, a cyclotron motion, and a breathing oscillation with frequencies determined by the symmetry independent of the details of the system. We further construct an exact effective action for all the NG modes.
We show that quasi-Nambu-Goldstone (NG) modes, which play prominent roles in high energy physics but have been elusive experimentally, can be realized with atomic Bose-Einstein condensates. The quasi-NG modes emerge when the symmetry of a ground state is larger than that of the Hamiltonian. When they appear, the conventional vacuum manifold should be enlarged. Consequently, topological defects that are stable within the conventional vacuum manifold become unstable and decay by emitting the quasi-NG modes. Contrary to conventional wisdom, however, we show that the topological defects are stabilized by quantum fluctuations that make the quasi-NG modes massive, thereby suppressing their emission.
The state of a quantum particle confined to a spherical surface is commonly described using spherical harmonics, which play a crucial role in solving many scientific problems featuring spherical symmetry. A mean-field description of a system of many interacting quantum particles confined to a spherical shell naturally leads to nonlinear generalizations of spherical harmonics. In this study, we analyze the emergence of such nonlinear modes in a shell-shaped Bose-Einstein condensate with both repulsive and attractive two-particle interactions. We find several classes of nonlinear modes. The first kind is an analytical continuation of a single spherical harmonic or their superposition for nonvanishing atom-atom interaction. Some of these solutions have planar counterparts in the form of Townes solitons and Townes vortices, which either appear stable even in the case of attractive interactions or can be stabilized by three-particle interactions. The second kind originate from symmetry-breaking bifurcations in the case of a repulsive interaction, and therefore have no linear counterparts. Moreover, we demonstrate the existence of a breathing dynamics that oscillates primarily between an unstable soliton and a stable vortex necklace state. Our findings show that the physics of nonlinear matter waves on curved geometries differs substantially from that in planar systems, suggesting the feasibility of observing these unconventional wave patterns experimentally.
We consider rotating finite-size vortex arrays in Bose-Einstein condensates that are confined by cylindrically symmetric external potentials. We show that such systems possess two exact Nambu-Goldstone zero modes associated with two spontaneously broken continuous symmetries of the system. We verify our analytical result via direct numerical diagonalizations of the Bogoliubov--de Gennes equations. We conclude by comparing rotating vortex lattices in superfluids to supersolids and discrete time crystals.
We have calculated collective-mode spectra for three-dimensional, rotating Bose-Einstein condensates in oblate harmonic traps using the microscopic Bogoliubov--de Gennes field theory. For condensates with Nv{N}_{v}Nv​ vortices, Nv{N}_{v}Nv​ Kelvin-Tkachenko-mode branches are obtained. The features of these modes are compared with those predicted by a classical point-vortex model. We have created movies to visualize the motion of the vortices corresponding to the Kelvin-Tkachenko waves.