Analytic expressions for Hubbard models with arbitrary structures in programmable optical lattices

We investigate the use of programmable optical lattices for quantum simulation of Hubbard models, determining analytic expressions for the hopping and Hubbard $U$, finding that they are suitable for emulating strongly correlated systems with arbitrary structures, including those with a multiple site basis and impurities. Programmable potentials are highly flexible, with the ability to control the depth and shape of individual sites in the optical lattice dynamically. Quantum simulators of Hubbard models with (1) an arbitrary basis are required to represent many real materials of contemporary interest, (2) broken translational symmetry are needed to study impurity physics, and (3) dynamical lattices are needed to investigate strong correlations out of equilibrium. We derive analytic expressions for Hubbard Hamiltonians in programmable potential systems. We find experimental parameters for quantum simulation of Hubbard models with an arbitrary basis, concluding that programmable optical lattices are suitable for this purpose. Finally, we demonstrate how programmable optical lattices can be used for quantum simulation of ionic Hubbard models, and thus the onset of charge transfer insulating states, in parameter regimes that are currently inaccessible to experiment.

Figure 1

(a) Superposition of spot and pancake potentials. The magenta (dark-gray) bar across the plot is the pancake, which also extends into the $y$ direction. The yellow (light-gray) peaks are the spot potential. (b) Same as (a), plotted without the pancake potential.

Figure 2

Effect of increasing beam waist on a periodic array of painted potentials. For small waist, spots have the form of an array of independent Gaussians. As the waist increases, they overlap to form sinusoids.

Figure 3

Correction to ${\omega }_{xy}$ given by Eq. (29). Corrections are negligible until $w_0/a\sim 0.45$.

Figure 4

Comparison of integrand and approximate integrand for computing $T_{\mathit{m}\mathit{n}}$, when $w_0/a=0.15$, $V_{0,\mathrm{s}}=8{\hslash }^2/2M{a}^2$, $z_R=a$. Integrand corresponding to Eq. (32) (integrand full), $T_{\mathit{m}\mathit{n}}^{\prime }+T_{\mathit{n}\mathit{m}}^{\prime }$ with lattice potential only at initial and final hopping sites, consistent with Eq. (36) (integrand), and $T_{\mathit{m}\mathit{n}}^{\prime }+T_{\mathit{n}\mathit{m}}^{\prime }$ with the approximations in Eq. (38) (approximation). The bottom panel shows the difference between the full and approximate integrands, which is small (residual).

Figure 5

Same as Fig. 4, with $w_0/a=0.3$, $V_{0,\mathrm{s}}=8{\hslash }^2/2M{a}^2$, $z_R=a$. The residual between the full and approximate integrands is tiny.

Figure 6

Variation of the hopping $t$, with $w_0$, $z_R$, and $V_{0,\mathrm{s}}$. For the calculations, we set $M=1,a=1,\hslash =1$, leading to results with the units ${\hslash }^2/2M{a}^2$. A wide range of hopping parameters can be achieved by modifying the spot properties.

Figure 7

Variation of the Hubbard $U$ with $w_0/a$ and $z_R/a$. For the calculations, we set $g=1$, $M=1,a=1,\hslash =1$, leading to results with the units ${\hslash }^2/2M{a}^2$. For $g=1$, the Hubbard $U$ has a magnitude of a few $t$. $U$ scales linearly with $g$, so is easily tuned by changing the magnetic field associated with the Feshbach resonance.

Figure 8

Hopping vs $w_0/a$ calculated using Eq. (42), for an array of identical Gaussian traps of spot size (a) $w_0=0.5\mu \mathrm{m}$, (b) $w_0=0.7\mu \mathrm{m}$, and trap depths $V_{0,\mathrm{s}}=400,V_{0,\mathrm{s}}=700$ and $V_{0,\mathrm{s}}=1000\mathrm{nK}$.

Figure 9

Parameter space of the ionic Hubbard model when $w_{\mathit{A}}=500\mathrm{nm}$ and $(V_A+V_B)/2=400\mathrm{nK}$. All key parts of the parameter space can be accessed: $U_A/U_B$ ranges from $\sim 0.5$ to $\sim 5$, its inverse $U_B/U_A$ ranges from $\sim 0.2$ to $\sim 2$, and $\left|\mathrm{\Delta }\right|/t$ ranges between 0 and $\sim 5$ for the parameter ranges shown.

Figure 10

Same as Fig. 9, but with $w_{\mathit{A}}=700\mathrm{nm}$. Similar ranges of $\mathrm{\Delta }/t$ can be explored, although hopping is smaller and a reduced range of Hubbard $U$ ratios can be explored compared to the case with $w_{\mathit{A}}=500\mathrm{nm}$.

Figure 11

Examples of systems that either require or would benefit from a painted-potential approach.