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arXiv:2505.13367 · Journal of Physics A Mathematical and Theoretical

Reflection phase shifts of bouncing Bogoliubov waves

Carsten Henkel

Published Aug 5, 2025• 0 citations

Abstract

Abstract The Bogoliubov–de Gennes equations for the elementary excitations of a Bose gas are solved for an inhomogeneous, one-dimensional system in the vicinity of a turning point, addressing the full continuous spectrum. A basis change in the space of the two Bogoliubov ‘particle’ and ‘hole’ amplitudes is introduced that decouples them approximately. We find a spatially extended mode that governs mainly excitations in the condensate phase, while another mode is spatially localised to the boundary zone where the condensate density gradient is significant. An analytical and numerical discussion of the phase shift is provided that incident matter waves suffer upon reflection at the turning point, forming standing waves. We find phase shifts with relatively strong energy dispersion that show significant differences among four model potentials. As applications, we discuss the boundary-induced change in the density of states and compute eigenfrequencies in a gravitational trap, without recourse to the local density approximation. The non-condensate density at finite temperature and its quantum depletion are discussed in a companion paper.

DOI: 10.1088/1751-8121/adf7e9

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Abstract The Bogoliubov–de Gennes equations for the elementary excitations of a Bose gas are solved for an inhomogeneous, one-dimensional system in the vicinity of a turning point, addressing the full continuous spectrum. A basis change in the space of the two Bogoliubov ‘particle’ and ‘hole’ amplitudes is introduced that decouples them approximately. We find a spatially extended mode that governs mainly excitations in the condensate phase, while another mode is spatially localised to the boundary zone where the condensate density gradient is significant. An analytical and numerical discussion of the phase shift is provided that incident matter waves suffer upon reflection at the turning point, forming standing waves. We find phase shifts with relatively strong energy dispersion that show significant differences among four model potentials. As applications, we discuss the boundary-induced change in the density of states and compute eigenfrequencies in a gravitational trap, without recourse to the local density approximation. The non-condensate density at finite temperature and its quantum depletion are discussed in a companion paper.

We consider an ultracold Bose gas in a half-open potential well, otherwise confined to a quasi-one-dimensional geometry. The Bogoliubov equations for its elementary excitations are solved in the continuous spectrum to explore the particle and energy content near the edge of the gas, beyond the often applied local density approximation. In particular, gradients in the condensate density enhance density-dominated excitations in the border region. We discuss excess (missing) particles and their energy by comparing to suitable reference solutions for quasi-homogeneous systems. The density profile near the edge shows no Friedel oscillations, but a dipolar feature from the spill-out of thermally excited particles. The calculations are performed in the grand-canonical ensemble and in the thermodynamic limit.

Bogoliubov–de Gennes equations (BdGEs) for collective excitations from a trapped Bose–Einstein condensate described by a spatially smooth ground-state wavefunction can be treated analytically. A new class of closed solutions for the BdGEs is obtained for the one-dimensional (1D) and 3D spherically harmonic traps. The solutions of zero-energy mode of the BdGEs are also provided. The eigenfunctions of the excitations consist of zero-energy mode, zero-quantum-number mode and entire excitation modes when the approximate ground state is a background Bose gas sea.

The calculation of properties of Bose-Einstein condensates with dipolar interactions has proven a computationally intensive problem due to the long range nature of the interactions, limiting the scope of applications. In particular, the lowest lying Bogoliubov excitations in three-dimensional harmonic trap with cylindrical symmetry were so far computed in an indirect way, by Fourier analysis of time-dependent perturbations, or by approximate variational methods. We have developed a very fast and accurate numerical algorithm based on the Hankel transform for calculating properties of dipolar Bose-Einstein condensates in cylindrically symmetric traps. As an application, we are able to compute many excitation modes by directly solving the Bogoliubov-De Gennes equations. We explore the behavior of the excited modes in different trap geometries. We use these results to calculate the quantum depletion of the condensate by a combination of a computation of the exact modes and the use of a local density approximation.

We propose a method for finding analytical solutions of the Bogoliubov--de Gennes (BdG) equations for the low-lying collective excitations in a harmonically trapped Bose-Einstein condensate beyond the Thomas-Fermi limit. We first use a simple variational wave function for ground state to eliminate the divergence at the boundary layer of the condensate, which appears in the Thomas-Fermi approximation. We then solve the BdG equations analytically and obtain explicit and divergence-free expressions for the eigenvalues and eigenfunctions of the excitations for the traps with spherical and cylindrical symmetries. The solutions of the zero-energy mode of the BdG equations are also presented.

We develop a finite-temperature perturbation theory (beyond the mean field) for a Bose-condensed gas and calculate temperature-dependent damping rates and energy shifts for Bogolyubov excitations of any energy. The theory is generalized for the case of excitations in a spatially inhomogeneous (trapped) Bose-condensed gas, where we emphasize the principal importance of inhomogeneity of the condensate density profile and develop the method of calculating the self-energy functions. The use of the theory is demonstrated by calculating the damping rates and energy shifts of low-energy quasiclassical excitations, i.e., the quasiclassical excitations with energies much smaller than the mean-field interaction between particles. In this case the boundary region of the condensate plays a crucial role and the result for the damping rates and energy shifts is completely different from that in spatially homogeneous gases. We also analyze the frequency shifts and damping of sound waves in cylindrical Bose condensates and discuss the role of damping in the recent MIT experiment on the sound propagation.