Critical temperature of a Rashba spin-orbit coupled Bose gas in harmonic traps

We investigate theoretically Bose-Einstein condensation of an ideal, trapped Bose gas in the presence of Rashba spin-orbit coupling. Analytic results for the critical temperature and condensate fraction are derived, based on a semi-classical approximation to the single-particle energy spectrum and density of states, and are compared with exact results obtained by explicitly summing discrete energy levels for small number of particles. We find a significant decrease of the critical temperature and of the condensate fraction due to a finite spin-orbit coupling. For large coupling strength and finite number of particles $N$, the critical temperature scales as $N^{2\mathrm{/}5}$ and $N^{2\mathrm{/}3}$ in three and two dimensions, respectively, contrasted to the predictions of $N^{1\mathrm{/}3}$ and $N^{1\mathrm{/}2}$ in the absence of spin-orbit coupling. Finite size corrections in three dimensions are also discussed.