Critical temperature of a Bose gas in an optical lattice

We present theory for the critical temperature of a Bose gas in a combined harmonic lattice potential based on a mean-field description of the system. We develop practical expressions for the ideal-gas critical temperature and corrections due to interactions, the finite-size effect, and the occupation of excited bands. We compare our expressions to numerical calculations and find excellent agreement over a wide parameter regime.

Figure 1

Exact (solid) and quadratic shape approximation (dash) 3D cubic TIL ground-band density of states for (a) $V=0$ and (b) $V=5E_R$, where $V\equiv {\prod }_j{V}_j^{1/d}$, $E_R\equiv h^2/8m{a}^2$, and $a\equiv {\prod }_j{a}_j^{1/d}$.

Figure 2

(a) Width of the 3D cubic TIL ground band, using Eq. (2) (solid), maximum less minimum energies (dash) and the tight-binding approximation [1, 23] (dot). (b) Spread function $S\left(w\right)$ using Eq. (12) (solid) and the quadratic shape approximation with Eq. (11) (dash).

Figure 3

(Color) 3D cubic CHL results. [(a) and (b)] Comparison of ideal finite-size effect between quadratic shape approximation (solid) and numerical results (ground band in green and all bands in blue), effective-mass Eq. (18) (dash-dot), and $w\to 0$ limit [Eq. (16)] (dash). [(c) and (d)] Comparison of quadratic shape approximation (solid) to numerical results (dot), using $a=425\text{nm}$, $a_s=5.77\text{nm}$ and ignoring the finite-size effect. Both ideal (ground band in green, all bands in blue) and interacting results (ground band in red, all bands in magenta) are shown. For these parameters, the Mott insulator transition for unit occupancy is at $V\approx 13E_R$. [(a) and (c)] $\omega =2\pi \times 32\text{Hz}$, $N={10}^5$. [(b) and (d)] $\omega =2\pi \times 64\text{Hz}$, $N={10}^6$.

Figure 4

Comparison of actual (dot) to effective-mass (dash) and quadratic shape approximation (solid) critical temperature for an ideal Bose gas in a 3D cubic CHL with (a) $\omega =2\pi \times 16\text{Hz}$, $N={10}^3$, (b) $\omega =2\pi \times 64\text{Hz}$, $N={10}^6$.