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We characterize the low-lying excitations of a quantum vortex in aquasi-two-dimensional Bose-Einstein condensate (BEC) using the standarddefinition of the density of states (DOS) and a modified version that issensitive to complementary aspects of the excitation's spectrum. The latterproves to be particularly relevant to studying the polaronic state realizedwhen an impurity is embedded in a quantum vortex. We establish that theimpurity becomes sensitive to the transversal fluctuations of the vortex, viaits remnant kelvon mode, and to the phase fluctuations of the BECNambu-Goldstone mode. The presence of the vortex yields an anomalous excitationspectrum with a finite energy gap and non-linear DOS at low energies. We findthat the high sensitivity of the kelvon mode to external potentials provides achannel of quantum-level control over impurities trapped in a vortex. Thisextra control channel may be of practical use for the proposal of usingvortex-trapped impurities as qubit units for quantum information processing.

Hydrodynamic signatures and spectral properties of the quantum vortex