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Physical Review A

Kelvon–Bogoliubov-phonon interaction dynamics in a superfluid quantum vortex

Kazunari Ochi, Junichi Takahashi, Eiji Nakano

Published Feb 3, 2025• 0 citations
Publisher

Abstract

Elementary excitation in the presence of a bosonic superfluid vortex line includes a kelvon and a Bogoliubov phonon as Nambu-Goldstone bosons, associated with spontaneous breaking of translational and $\text{U}(1)$ symmetries, respectively. Using the Bogoliubov approximation, we construct an effective Hamiltonian to describe the interplay among kelvons and Bogoliubov phonons in a microscopic manner and apply it for the decay process of an excited kelvon with Bogoliubov-phonon emission. Such a decay process governs the late-time behavior of vortex-excitation dynamics observed in cold-atom experiments and also in follow-up numerical simulations using the time-dependent Gross-Pitaevskii equation. We show our numerical result of the decay process and compare it with these previous studies. Our formalism provides the interaction Hamiltonian in terms of wave functions and energy spectra of the elementary excitations, thus potentially being useful in the description of other low-energy phenomena caused by the kelvon--Bogoliubov-phonon interaction.

DOI
10.1103/physreva.111.023302

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Elementary excitation in the presence of a bosonic superfluid vortex line includes a kelvon and a Bogoliubov phonon as Nambu-Goldstone bosons, associated with spontaneous breaking of translational and U(1)\text{U}(1)U(1) symmetries, respectively. Using the Bogoliubov approximation, we construct an effective Hamiltonian to describe the interplay among kelvons and Bogoliubov phonons in a microscopic manner and apply it for the decay process of an excited kelvon with Bogoliubov-phonon emission. Such a decay process governs the late-time behavior of vortex-excitation dynamics observed in cold-atom experiments and also in follow-up numerical simulations using the time-dependent Gross-Pitaevskii equation. We show our numerical result of the decay process and compare it with these previous studies. Our formalism provides the interaction Hamiltonian in terms of wave functions and energy spectra of the elementary excitations, thus potentially being useful in the description of other low-energy phenomena caused by the kelvon--Bogoliubov-phonon interaction.
Abstract It is generally accepted that phonons in a superfluid Bose gas are Goldstone bosons. This is justified by spontaneous symmetry breaking (SSB), which is usually defined as follows: the Hamiltonian of the system is invariant under the U ( 1 ) transformation Ψ ^ ( r , t ) → e i α Ψ ^ ( r , t ) , whereas the order parameter Ψ ( r , t ) is not. However, the strict definition of SSB is different: the Hamiltonian and the boundary conditions are invariant under a symmetry transformation, while the ground state is not. Based on the latter criterion, we study a finite system of spinless, weakly interacting bosons using three approaches: the standard Bogoliubov method, the particle-number-conserving Bogoliubov method, and the approach based on the exact ground-state wave function. Our results show that the answer to the question in the title is ‘no’. Thus, phonons in a real-world (finite) superfluid Bose gas are similar to sound in a classical gas: they are not Goldstone bosons, but quantized collective vibrational modes arising from the interaction between atoms. In the case of an infinite Bose gas, however, the picture becomes paradoxical: the ground state can be regarded as either infinitely degenerate or non-degenerate, making the phonon both similar to a Goldstone boson and different from it.
Superfluids are inviscid flows in which vorticity is supported on filaments with quantized circulation. Such objects, known as quantum vortices, exhibit a hydrodynamical behavior. Experimentally, the dynamics of superfluids has been studied by using particles, which nowadays have become the main tool for visualizing quantum vortices. In this Thesis, we study numerically and analytically the dynamics of active and finite-size particles in superfluids. The superfluid is modeled with the Gross–Pitaevskii equation, while the particles are implemented as moving repulsive potentials coupled with the macroscopic wave function describing the superfluid. Firstly, the model is used to investigate the interaction between particles and quantum vortices at very low temperatures. This part aims to give a theoretical background to the current experiments in which macroscopic particles are used to sample superfluid vortices and quantum turbulence. Specifically, we address the following problems: the capture of a particle by a quantum vortex, the reconnections of vortex filaments and the propagation of Kelvin waves in presence trapped particles and the dynamics of particles in decaying quantum turbulence. In the last part of the manuscript, finite temperature effects are studied in the Fourier-truncated Gross–Pitaevskii model. The goal is to characterize the dynamics of impurities immersed in a thermal bath and how their presence modifies the statistical properties of the fluid. In particular, the random motion of the impurities and the temperature dependence of the friction coefficient are studied. Finally, the clustering of impurities and its effect on the phase transitions of the condensate are investigated.
Kelvin waves---helical waves on quantized vortex lines---are the normal modes of vortices in a superfluid. At zero temperature, the only dissipative channel of vortex dynamics is phonon emission. Starting with the hydrodynamic action, we derive the Hamiltonian of vortex-phonon interaction, thereby reducing the problem of the interaction of Kelvin waves with sound to inelastic elementary excitation scattering. On the basis of this formalism, we calculate the rate of sound radiation by superfluid turbulence at zero temperature and estimate the value of short-wavelength cutoff of the turbulence spectrum.
We study the decay of Bogoliubov quasiparticles in one-dimensional Bose gases. Starting from the hydrodynamic Hamiltonian, we develop a microscopic theory that enables one to systematically study both the excitations and their decay. At zero temperature, the leading mechanism of decay of a quasiparticle is disintegration into three others. We find that low-energy quasiparticles (phonons) decay with the rate that scales with the seventh power of momentum, whereas the rate of decay of the high-energy quasiparticles does not depend on momentum. In addition, our approach allows us to study analytically the quasiparticle decay in the whole crossover region between the two limiting cases. When applied to integrable models, including the Lieb-Liniger model of bosons with contact repulsion, our theory confirms the absence of the decay of quasiparticle excitations. We account for two types of integrability-breaking perturbations that enable finite decay: three-body interaction between the bosons and two-body interaction of finite range.
We discuss the mechanism of spontaneous symmetry breaking and the elementary excitations for a weakly-interacting Bose gas at a finite temperature. We consider both the non-relativistic case, described by the Gross-Pitaevskii equation, and the relativistic one, described by the cubic nonlinear Klein-Gordon equation. We analyze similarities and differences in the two equations and, in particular, in the phase and amplitude modes (i.e., Goldstone and Higgs modes) of the bosonic matter field. We show that the coupling between phase and amplitude modes gives rise to a single gapless Bogoliubov spectrum in the non-relativistic case. Instead, in the relativistic case the spectrum has two branches: one is gapless and the other is gapped. In the non-relativistic limit we find that the relativistic spectrum reduces to the Bogoliubov one. Finally, as an application of the above analysis, we consider the Bose-Hubbard model close to the superfluid-Mott quantum phase transition and we investigate the elementary excitations of its effective action, which contains both non-relativistic and relativistic terms.