Past exam of the quantum matter course of the University of Oxford / 2025 / 2 / e

The Hartree product state underlying the Gross–Pitaevskii equation has the factorized one-body coherence of a Hartree condensate. Choosing its nodeless ground state orbital real givesIn particular, the Thomas–Fermi approximation for a condensate from part (d) predictsThe curve starts at $\mu /g$, decreases with zero slope at the origin, and reaches zero with an infinite slope at $R$. Restoring the kinetic energy in the full Gross–Pitaevskii equation rounds that edge and produces a nonzero tail. This corrects the Thomas–Fermi approximation for a condensate, but the resulting one-body density matrix still factorizes.

The additional missing physics is quantum depletion. Interactions scatter pairs of bosons into orbitals outside the Bose-Einstein condensate even at zero temperature. In terms of the true Bose-Einstein condensate orbital ${\phi }_0$ the exact one-body density matrix decomposes asThe single-orbital Hartree product state sets $N_0=N$ and omits ${\rho }_{\mathrm{ex}}$ altogether. In a weakly interacting Bose gas, Bogoliubov theory adds a comparatively short-range noncondensed contribution near the origin, together with reduced weight in the broad Bose-Einstein condensate component. The central value remains the total number density, not just the condensate density. Its numerical shift relative to the original Gross–Pitaevskii equation solution also depends on the interaction-induced redistribution of number density; it is not fixed by the condensate fraction alone.