Bogoliubov excitations of a Bose-Einstein condensate in the presence of the non-Hermitian skin effect

The non-Hermitian skin effect is one of the most remarkable topological features of non-conservative physics. It causes a macroscopic accumulation of states at the boundaries of a confined system, opening new ways for controlling and directing waves in non-Hermitian settings. This effect has recently been observed in experiments with Bose-Einstein condensates of exciton polaritons and ultracold bosonic quantum gases. The underlying physics is captured by a mean-field Hamiltonian with a trapping potential and an imaginary gauge field. While prior work has examined the effect of the imaginary gauge field term on the condensate itself, its effect on the collective excitations on top of the condensate remained unexplored. Here, we develop a new theory for the Bogoliubov excitations in a non-Hermitian framework and systematically study their spectrum across different interaction regimes. By analysing the position expectation values, we find that while the repulsive interactions oppose the localisation of the condensate due to the skin effect, the excitations remain strongly affected by the imaginary gauge field. As a consequence, in the presence of harmonic confinement, we predict a strong shift in the dipole oscillation frequency, which can be observed in experiments. This is in contrast to Hermitian systems, where this frequency is fixed by the trap dimensions. Furthermore, we develop a consistent second-quantised field theory of the collective excitations in this non-Hermitian system. Strikingly, we find that a second-quantised treatment removes the shift of the dipole oscillation frequency entirely. This intriguing discrepancy between the two theoretical approaches to the treatment of elementary excitations in a non-Hermitian system can be resolved in experiments.