Finite-temperature theory of superfluid bosons in optical lattices

A practical finite-temperature theory is developed for the superfluid regime of a weakly interacting Bose gas in an optical lattice with additional harmonic confinement. We derive an extended Bose-Hubbard model that is valid for shallow lattices and when excited bands are occupied. Using the Hartree-Fock-Bogoliubov-Popov mean-field approach, and applying local-density and coarse-grained envelope approximations, we arrive at a theory that can be numerically implemented accurately and efficiently. We present results for a three-dimensional system, characterizing the importance of the features of the extended Bose-Hubbard model and compare against other theoretical results and show an improved agreement with experimental data.

Figure 1

(Color online) Density of states for the 3D cubic lattice: $b=0$ (black solid curve), 001 (red dashed curve; the integers specify the components $b_x,b_y,b_z$), 011 (green dashed-dotted curve), and 002 (blue dotted curve) for (a) $V=5E_R$ and (b) $V=10E_R$.

Figure 2

(Color online) Tight-binding (black solid curve) and actual (red dashed curve) density of states for $V=5E_R$ for (a) the 1D and (b) the 2D square lattice.

Figure 3

(Color online) Tight-binding (black solid curve) and actual (red dashed curve) 3D cubic-lattice ground-band density of states for (a) $V=2E_R$ and (b) $V=5E_R$

Figure 4

(Color online) 3D cubic-lattice density of states for $V=15E_R$ (black solid curve); the free-particle density of states shifted by the minimum-energy eigenvalue ($\frac{\pi }{4}\sqrt{\left[K-K_0\left(\mathbf{0}\right)\right]/E_R}$, blue dashed curve) and by the spatially averaged energy of the lattice [$\frac{\pi }{4}\sqrt{\left(K-\frac{1}{2}{\sum }_j{V}_j\right)/E_R}$, red dashed-dotted curve].

Figure 5

(Color online) Combined harmonic cubic-lattice density of states in three dimensions for [(a),(b)] $V=5E_R$ and [(c),(d)] $V=15E_R$ from the full diagonalization (black dotted curve), LDA (red solid curve), and $E_{\text{cr}}$ (dashed-dotted curve). Shown are [(a),(c)] the ground to the first excited bands with the LDA ground band (lower red solid line) for reference; [(b),(d)] many bands and the high-energy approximation (63) (dashed curve). The LDA is so good that it is obscured by the full diagonalization results in all cases.

Figure 6

(Color online) Procedure for LDA calculation.

Figure 7

Condensate fraction for a noninteracting combined harmonic cubic lattice in three dimensions with $N=1000$, $\omega =0.02{\omega }_R$, and $V=15E_R$, comparing full diagonalization (solid curve), LDA with $\mu \le {\mu }_{\text{fs}}$ (dashed curve), and LDA with $\mu \le 0$ (dashed-dotted curve).

Figure 8

Noninteracting condensate fraction for $N={10}^5$, $\omega =0.01{\omega }_R$, (a) $V=2E_R$, and (b) $V=5E_R$. The full diagonalization curve (solid curve) is almost obscured by the all-neighbor result (dashed curve) and is appreciably different from the nearest-neighbor result (dashed-dotted curve).

Figure 9

Ratio of number of noncondensate atoms in first three (solid curve) and beyond first three (dashed curve) excited bands to noncondensate atoms in the ground band at the critical temperature for the experimental setup of [12].

Figure 10

Quantum depletion of ${}^{23}\text{N}\text{a}$ in a 3D optical lattice. The data points with error bars give the experimental quantum depletion. The curves give quantum depletion calculated by [46] (solid curve) and our calculated quantum depletion (dashed curve).

Figure 11

Condensate and quantum depletion fractions for the parameters of [12]; (a) $V=5E_R$ and (b) $V=10E_R$. Results are for the HFBP method for the condensate only (solid curve), for the condensate plus quantum depletion (dashed curve) and for the Hartree-Fock method (dashed-dotted curve).

Figure 12

(Color online) Spatial densities for the parameters of [12] at $T=0.8T_c$, (a) $V=5E_R$, and (b) $V=10E_R$. Results are for the HFBP method: total (black solid curve), condensate (cyan dashed curve), quantum depletion (green filled circles) and the Hartree-Fock method: total (blue dashed-dotted curve) and condensate (red dotted curve).

Figure 13

Ground-band Wannier functions (solid curve) compared to the Gaussian approximation (dashed curve) for (a) $V=2E_R$ and (b) $V=15E_R$.

Figure 14

Wannier function for the (a) first and (b) second excited bands for $V=5E_R$ (solid curve) compared to the harmonic-oscillator approximation (dashed curve).

Figure 15

(Color online) Fourier series for the 1D translationally invariant lattice spectrum for [(a),(b)] $V=E_R$, [(c),(d)] $V=5E_R$, [(a),(c)] ground band, and [(b),(d)] first excited band, using all neighbors (black solid curve), nearest, and next-nearest neighbors (red dashed curve) and nearest neighbors (blue dashed-dotted curve).

Figure 16

Ratio of beyond nearest-neighbor to nearest-neighbor hopping. $J_{b,j}^2/J_{b,j}^1$ (solid curve), $J_{b,j}^3/J_{b,j}^1$ (dashed curve), and $J_{b,j}^4/J_{b,j}^1$ (dashed-dotted curve) for (a) ground and (b) first excited bands.

Figure 17

Error due to (a) on-site variation of the trap, $I_1={\int }_{-\infty }^{\infty }dx{\left(x/a_x\right)}^2\left[{\left|w_1\left(x\right)\right|}^2-{\left|w_0\left(x\right)\right|}^2\right]$ (this form is chosen, so that the error is in units of $\frac{1}{2}m{\omega }_x^2{a}_x^2$), (b) contribution from adjacent sites, $I_2=\left|{\int }_{-\infty }^{\infty }dx{\left(x/a_x\right)}^2{w}_b^{\ast }\left(x\right)w_b\left(x-a_x\right)\right|$, ground band (solid curve), and first excited band (dashed curve), and (c) Wannier function overlap between the ground and first excited bands, $I_3={\int }_{-\infty }^{\infty }dx\left(x/a_x\right)w_0^{\ast }\left(x\right)w_1\left(x\right)$.

Figure 18

Interaction coefficients in three dimensions. (a) Ground band. On site (dotted curve). All sites: noncondensate $U^{\prime }$ (D3) (solid curve) and condensate $U^{″}$ (D5) (dashed curve). No-lattice limit: all sites (D6) $(\times )$ and on site (D7) (○). (b) Excited band. On site: 000,001 (these integers specify the components $b_x,b_y,b_z$ of each band) (solid curve), 001,001 (dashed curve), and 010,001 (dashed-dotted curve); corresponding all sites (dotted curve).