Weakly Interacting Bose Gas

The weakly interacting Bose gas describes nonrelativistic bosons whose low-energy collisions are characterized by a scattering length. In three spatial dimensions, its dilute repulsive regime supports a condensate with sound-like collective excitations. The model connects an interatomic scattering measurement to a many-body excitation spectrum.

Contact interaction and dilute regime

Consider a uniform gas at zero temperature, with no external potential. For atomic mass $m$ and bosonic field $\psi$, the grand-canonical action is

$$S=\int dt\,d^3x\left[ i\psi^*\partial_t\psi-\frac{|\nabla\psi|^2}{2m} +\mu|\psi|^2-\frac g2|\psi|^4 \right], \qquad g=\frac{4\pi a}{m}.$$

Here $a>0$ is the low-energy $s$-wave scattering length and $\mu$ fixes the density. The contact coupling is matched to two-body scattering: a cutoff-dependent bare coupling inside a loop integral cannot simply be identified with $4\pi a/m$. Assume total density $n$ satisfies $na^3\ll1$ and momenta are well below the inverse interaction range.

The field has a global $U(1)$ symmetry and conserves atom number. This is a nonrelativistic theory with a preferred rest frame of the gas. A positive scattering length specifies the repulsive gas regime; it does not exclude molecular bound states or guarantee the absolute stability of a particular atomic species.

Condensate and Bogoliubov spectrum

For a constant condensate $\psi=\sqrt{n_0}$, stationarity gives $\mu=gn_0$ at leading order. The corresponding energy density is $gn_0^2/2$. Expanding $\psi=\sqrt{n_0}+\delta\psi$ to quadratic order mixes $\delta\psi$ with its complex conjugate. The mode frequencies are the positive eigenvalues of

$$\begin{pmatrix} \epsilon_k+gn_0 & gn_0\\ -gn_0 & -\epsilon_k-gn_0 \end{pmatrix}, \qquad \epsilon_k=\frac{k^2}{2m}.$$

Its characteristic equation gives

$$E_k^2=\epsilon_k(\epsilon_k+2gn_0).$$

This is the uniform-gas Bogoliubov dispersion Dalfovo et al. 1998, §IV.1, eq. (68). In the long-wavelength limit,

$$E_k\simeq c_s k, \qquad c_s=\sqrt{\frac{gn_0}{m}}, \qquad \xi=\frac{1}{\sqrt{2mgn_0}}.$$

The healing length $\xi$ locates the crossover: $k\xi\ll1$ gives sound, while $k\xi\gg1$ gives $E_k=\epsilon_k+gn_0+\cdots$ within the contact theory’s momentum range. Setting $g\to0$ at fixed $n_0$ recovers the free quadratic spectrum.

Observables and limits

Density-response measurements probe this dispersion; the compressibility provides a second check through $mc_s^2=n\,\partial\mu/\partial n$ at leading order, where $n_0\simeq n$. Quantum depletion makes $n_0$ differ from $n$, so the mean-field relations are not exact at finite interaction strength.

The calculation assumes a weakly depleted condensate. Strong interactions, appreciable interaction range, loss dynamics, or low-dimensional infrared fluctuations require additional treatment. In particular, changing to one or two spatial dimensions changes both scattering and the conditions for long-range order.

Develop the calculation

The Weakly Interacting Bose Gas develops the condensate approximation; Bogoliubov Quasiparticles derives the mode transformation. Use Contact Interactions and Scattering Length when relating bare couplings to measured scattering. The Bose–Hubbard Model instead retains a spatial lattice.

References

• Dalfovo, Franco, Stefano Giorgini, Lev P. Pitaevskii, and Sandro Stringari. Theory of Trapped Bose-Condensed Gases. Preprint (1998), arXiv:cond-mat/9806038v1. Open text.