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

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.