Phase-space density of an atomic gas ($\mathcal{D}=n{\lambda }_T^3$)

The dimensionless occupancy scale $n{\lambda }_T^3$, with number density $n$ and thermal de Broglie wavelength ${\lambda }_T=h/\sqrt{2\pi m{k}_{B}T}$. Increasing it approaches the regime where quantum statistics matters. This use of degeneracy must not be confused with degeneracy of an individual energy eigenspace.

For a classical ideal gas in a three-dimensional harmonic trap, the peak phase-space density is $\mathcal{D}=N(\hslash \bar{\omega }/k_{B}T{)}^3$. At fixed trap frequencies, atom loss accompanied by $T\propto N^{\alpha }$ increases $\mathcal{D}$ only for $\alpha >1/3$.