Particle fluctuations in nonuniform and trapped Bose gases

The problem of particle fluctuations in arbitrary nonuniform systems with Bose-Einstein condensate is considered. This includes the case of trapped Bose atoms. It is shown that the correct description of particle fluctuations for any nonuniform system of interacting atoms always results in thermodynamically normal fluctuations.

💡 Research Summary

The paper addresses a long‑standing question in the theory of Bose‑Einstein condensates (BECs): how particle‑number fluctuations behave in spatially non‑uniform, trapped systems of interacting bosons. While ideal, uniform Bose gases exhibit “anomalous” fluctuations that grow faster than the total particle number, real experiments always involve interatomic interactions and external confinement, which dramatically modify the fluctuation spectrum. The authors set out to prove that, for any arbitrary external potential and any realistic interaction, the fluctuations remain thermodynamically normal, i.e., the variance ⟨δN²⟩ scales linearly with the mean particle number N.

The analysis begins with the second‑quantized Hamiltonian that includes a generic trapping potential V(r) and a two‑body interaction U(r−r′). Working in the grand‑canonical ensemble, the chemical potential μ is fixed and the average particle number ⟨N̂⟩ and its variance are expressed through density‑density correlation functions. The central theoretical tool is the Bogoliubov transformation. The field operator ψ̂(r) is split into a macroscopic condensate wavefunction ψ₀(r) and a fluctuation operator ϕ̂(r). By retaining terms up to second order in ϕ̂, the authors obtain the Bogoliubov‑de Gennes (BdG) equations for the quasiparticle amplitudes u_j(r) and v_j(r) and the corresponding excitation energies ε_j.

Because the system is non‑uniform, the authors invoke the local‑density approximation (LDA). Within LDA each point r is treated as if it were part of a homogeneous gas characterized by a local density n(r) and a local sound speed c_s(r)=√