Ab initio determination of Bose-Hubbard parameters for two ultracold atoms in an optical lattice using a three-well potential

We calculate numerically the energy spectrum of the six dimensional problem of two interacting Bosons in a three-well optical lattice. The particles interact via a full Born-Oppenheimer potential, which can be adapted to model the behavior of the $s$-wave scattering length at Feshbach resonances. By adjusting the parameters of the corresponding Bose-Hubbard (BH) Hamiltonian the deviation between the numerical energy spectrum and the BH spectrum is minimized. This defines the optimal BH parameter set, which we compare to the standard parameters of the BH model. The range of validity of the BH model with these parameter sets is examined, and an analytical prediction of the interaction parameter is introduced. Furthermore, an extended BH model and implications due to the energy dependence of the scattering length and couplings to higher Bloch bands at a Feshbach resonance are discussed.

Figure 1

(Color online) Visualization of the “hopping” parameter $J$, the onsite energies ${ϵ}_i$ $\left(i=0,\pm 1\right)$, and the interaction strength $U$. The potentials $V_{22}\left(\vec{r}\right)$ (blue solid), $V_{\text{OL}}\left(\vec{r}\right)$ (black dashed), and $V_{\text{conf}}\left(\vec{r}\right)$ (brown dash-dotted) are depicted for $y=z=0$.

Figure 2

(Color online) Extents of the lowest four energy bands of the one-dimensional optical lattice as a function of the lattice depth $V_0$. The energy of the first band can be approximated by the harmonic ground-state energy $\frac{1}{2}\hslash \omega$ (dashed line). A better approximation includes a constant energy offset of $E_r/4$ (see the text).

Figure 3

(Color online) Analytical energy spectrum of relative motion $l=0$ states of two particles interacting by the potential ${\widehat{V}}_{\text{ps}}$ [Eq. (7)] in a harmonic trap as a function of the scattering length. The spectrum defines the energy offsets $\Delta E_{\text{harm}}^0$ and $\Delta E_{\text{harm}}^1$ relative to the ground-state energy in the absence of interaction, $\frac{3}{2}\hslash \omega$.

Figure 4

(Color online) The correction factor $\mathcal{A}$ as a function of the lattice depth $V_0$. Inset: Comparison of $U^{\text{BH}}$, $U^{\text{harm}}$, and $U^{\text{corr}}$ for $V_0=\hslash \omega$ as a function of the scattering length. In accordance with the definition of $\mathcal{A}$ the behavior of $U^{\text{BH}}$ and $U^{\text{corr}}$ matches for small $a_{\text{sc}}$.

Figure 5

(Color online) Onsite energies as a function of the lattice depth $V_0$: the optimal values together with the BH representation for the middle well $\left({ϵ}_0\right)$ and the outer wells $\left({ϵ}_{\pm 1}\right)$ for $k_0{x}_B=3\pi /2$. Inset: In order to examine the impact of the confining potential ${\widehat{V}}_{\text{conf}}$ the value of $\delta {ϵ}_i={ϵ}_i-⟨w_0|\frac{{\widehat{p}}^2}{2m}+{\widehat{V}}_{\text{OL}}|w_0⟩$ is depicted, where the value of the onsite energy for no confinement is subtracted. Clearly, for $V_0>1.5\hslash \omega$ both results are converged.

Figure 6

(Color online) (a) Hopping parameter $J$ as a function of the lattice depth $V_0$: the optimal value together with the BH representation for $k_0{x}_B=\frac{3}{2}\pi ,2\pi ,\infty$. (b) The six eigenenergies in the rescaled form [see Eq. (15)] for $a_{\text{sc}}=0$ and $U=0$ as a function of the lattice depth $V_0$. The BH energies are obtained for onsite energies with boundaries at $k_0{x}_B=3\pi /2$ and a hopping parameter with boundaries at $k_0{x}_B=2\pi$.

Figure 7

(Color online) (a) The interaction parameter as a function of the lattice depth $V_0$ in the weak interacting regime with $a_{\text{sc}}\approx -180a_0$. This results in $0.02\le \left|a_{\text{sc}}/a_{\text{ho}}\right|\le 0.08$ in this graph. (b) The six eigenenergies in the rescaled form [see Eq. (16)] as a function of the lattice depth $V_0$.

Figure 8

(Color online) (a) The interaction parameter as a function of the lattice depth $V_0$ in the strong interacting regime with $a_{\text{sc}}\approx -4600a_0$. This results in $0.5\le \left|a_{\text{sc}}/a_{\text{ho}}\right|\le 2.2$ in this graph. (b) The six eigenenergies in the rescaled form [see Eq. (16)] as a function of the lattice depth $V_0$.

Figure 9

(Color online) The interaction parameter for different scattering lengths at $\frac{V_0}{\hslash \omega }=1.7$. To observe the behavior at the resonance, the dependence on $\frac{a_{\text{ho}}}{a_{\text{sc}}}$ is shown.

Figure 10

(Color online) Eigenenergies as a function of the scattering length at $\frac{V_0}{\hslash \omega }=1.7$. (a) Sketch of energy spectrum (see text for details). (b) Energy spectrum obtained from a full numerical calculation. The three thick (red) lines indicate energies which are closest to the energies predicted by the BHM.

Figure 11

(Color online) Energy difference $E_5-E_2$ in different orders of approximation as a function of the lattice depth $V_0$ for $a_{\text{sc}}\approx -180a_0$. The value of $E_5-E_2$ is shown relative to the same energy difference predicted by the BHM in BH representation. The BH representation of ${ϵ}_0,{ϵ}_{\pm 1}$ was determined for $k_0{x}_B=3\pi /2$ and the value of $J$ and $J_2$ for $k_0{x}_B=2\pi$.