The Weakly Interacting Bose Gas

A homogeneous three-dimensional Bose gas is weakly interacting when its scattering-length gas parameter $na^3$ is small. The Gross–Pitaevskii equation and the leading energy density are saddle-point results built from the matched coupling $g=4\pi a/m$; depletion and Lee–Huang–Yang terms are fluctuation corrections, not part of that saddle.

Required background. Use ideal condensation, short-range scattering matching, and collective-field transformations.

Matched dilute-gas action

For a repulsive gas at wavelengths much longer than the interaction range $r_0$, take

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

The last equality is a low-energy matching condition. A regulated bare coupling depends on the cutoff; inserting $4\pi a/m$ before the ultraviolet subtraction in a loop calculation double counts renormalization. At the saddle, $\psi=e^{-i\mu t}\Phi(\mathbf x)$ gives

$$\left[-\frac{\nabla^2}{2m}+V_{\mathrm{ext}}+g|\Phi|^2\right]\Phi=\mu\Phi, \qquad \int \mathrm d^3x\,|\Phi|^2=N_0.$$

For a uniform condensate, $n_0=|\Phi|^2$, $\mu=gn_0$, and

$$\frac{E_{\mathrm{MF}}}{V}=\frac{g n_0^2}{2}.$$

The energy cost of a density variation balances kinetic and interaction terms at the healing length

$$\xi=\frac{1}{\sqrt{2mgn_0}}=\frac{1}{\sqrt{8\pi a n_0}}.$$

Thus the contact description requires $r_0\ll\xi$ and the dilute expansion requires $na^3\ll1$. The Gross–Pitaevskii construction and its normalization are derived in Pethick and Smith 2008, chs. 5–6.

Traps and the Thomas–Fermi limit

When the condensate varies over a scale $R\gg\xi$, neglecting the kinetic term gives

$$n_0(\mathbf x)=\frac{\mu-V_{\mathrm{ext}}(\mathbf x)}{g} \,\Theta\!\left(\mu-V_{\mathrm{ext}}(\mathbf x)\right).$$

This Thomas–Fermi profile fails in a boundary layer of thickness order $\xi$ and near vortices, where gradients are essential. It also assumes a local three-dimensional coupling; tight confinement changes the matching.

Control and handoff

The first omitted contribution to the homogeneous energy is relative order $\sqrt{na^3}$. Quantum depletion has the same characteristic small parameter Pitaevskii and Stringari 2016, chs. 4–5. Consequently a Gross–Pitaevskii solution may be spatially accurate while its energy is not accurate enough for a cancellation problem such as a droplet. The Bogoliubov page supplies quadratic fluctuations, and the dilute-expansion page performs the renormalized next order.

Exercises

Restore $\hbar$ and verify the dimensions of $g$, $\mu$, and $\xi$ in three dimensions.

Solution

$g=4\pi\hbar^2a/m$ has dimensions energy times volume. Hence $gn$ is an energy and equals the mean-field chemical potential. Balancing $\hbar^2/(2m\xi^2)=gn$ gives $\xi=\hbar/\sqrt{2mgn}$, a length.

References

• Christopher J. Pethick and Henrik Smith, Bose–Einstein Condensation in Dilute Gases, 2nd ed., Cambridge University Press (2008), chs. 5–6, doi:10.1017/CBO9780511802850.
• Lev P. Pitaevskii and Sandro Stringari, Bose–Einstein Condensation and Superfluidity, Oxford University Press (2016), chs. 4–5, doi:10.1093/acprof:oso/9780198758884.001.0001.