Bogoliubov excitations in trapped condensates

Write the condensate field as its stationary profile multiplied by the chemical-potential phase, plus a small fluctuation. A complex fluctuation generally contains both positive- and negative-frequency amplitudes. Expanding the density-dependent GP term couples them because a density change depends on the field and its conjugate. Neglecting that coupling recovers a particle-like approximation but loses the low-energy collective behavior that the theory was designed to capture.

For a uniform repulsive condensate, the coupled two-component eigenvalue problem yields a positive excitation energy whose square is a product of free-particle kinetic energy and that energy plus twice the mean field. The background is weakly depleted and the fluctuation is small, so the derivation is not a nonlinear saturation theory. Finite temperature, damping from quasiparticle interactions, and strongly correlated regimes require corrections beyond the elementary real Bogoliubov spectrum.

At wavevectors much smaller than inverse healing length, free-particle kinetic energy is small compared with the mean field. The square root then becomes linear in momentum and the mode is a phonon with the condensate sound speed. At large wavevector, the leading behavior is quadratic kinetic energy with a mean-field offset. A crossover mode is neither accurately described by the pure phonon line nor exactly a bare free particle.

The healing-length convention fixes where factors of two enter this comparison. With the conventional kinetic balance used previously, free-particle energy divided by mean field equals the square of k times ξ. A measured sound speed can therefore test the interaction energy if density and scattering length are calibrated. It does not alone prove that the entire observed spectrum follows uniform theory, particularly when excitation wavelengths approach cloud dimensions or the local density varies significantly.

In a slowly varying bulk region, a local-density approximation can use the uniform spectrum with local density when the wavelength is short compared with the profile variation scale. A global sloshing or breathing oscillation instead samples the entire trap and needs a trapped eigenmode calculation or a controlled hydrodynamic limit. Averaging one chosen density into the uniform formula does not generally reproduce a discrete collective frequency.

For perfectly harmonic confinement and translation-invariant interparticle interactions, center-of-mass motion decouples and the dipole frequency equals the corresponding trap frequency. This gives a useful calibration and numerical consistency check. Other modes do shift with interactions and geometry: in a spherical three-dimensional repulsive condensate deep in the Thomas–Fermi limit, the lowest monopole breathing frequency is square root of five times the trap frequency. Anharmonicity, anisotropy, and a thermal component can change these predictions.