On-site correlations in optical lattices: Band mixing to coupled quantum Hall puddles

We extend the standard Bose-Hubbard model to capture arbitrarily strong on-site correlations. In addition to being important for quantitatively modeling experiments, for example, with rubidium atoms, these correlations must be included to describe more exotic situations. Two such examples are when the interactions are made large via a Feshbach resonance and when each site rotates rapidly, making a coupled array of quantum Hall puddles. Remarkably, even the mean field approximation to our model includes all on-site correlations. We describe how these on-site correlations manifest themselves in the system’s global properties: modifying the phase diagram and depleting the condensate.

Figure 1

(a) On-site energy with noninteracting energies subtracted off, $I_n=E_n-(3\hslash \omega /2-\mu )n$, and energies scaled by $E_{\mathrm{R}}=\hslash {}^2{\pi }^2/(2m{d}^2)$, using typical ${}^{87}\mathrm{Rb}$ parameters: lattice spacing $d=532$ nm, scattering length $a=5.32$ nm. Dashed curve, neglecting on-site correlations; solid curves, including correlations. Bottom to top curves: $n=2,3,4,5$, respectively. (b) Representative hopping matrix elements with on-site correlations, relative to those neglecting on-site correlations, ${\tau }^{(\mathit{mn})}\equiv t^{(\mathit{mn})}/[t\sqrt{(m+1)n}]$, as a function of lattice depth $V_0$ on a scale. Bottom to top lines: $t^{(03)},t^{(31)},t^{(05)},t^{(35)}$, respectively. (c) For comparison, $t^{(01)}/E_{\mathrm{R}}$ calculated from the exact Wannier states (top curve) along with our Gaussian approximation to it with and without nonorthogonality corrections (second- and third-highest curves, respectively). Also shown is the next-nearest-neighbor hopping matrix element (bottom curve). The effective Hamiltonian parameters are calculated perturbatively in $a/d$ for a Gaussian ansatz.

Figure 2

(a) On-site two-particle energy as a function of scattering length $a$ rescaled by the on-site harmonic oscillator energy $\hslash \omega =2\sqrt{V_0{E}_{\mathrm{R}}}$, for the two lowest-energy branches. The corresponding characteristic length is $\ell =\sqrt{\hslash /m\omega }$. (b) Log plot of rescaled hopping matrix elements ${\tau }^{(\mathit{mn})}\equiv t^{(\mathit{mn})}/[t\sqrt{(m+1)n}]$. Solid and dashed curves are $t^{(11)}/(\sqrt{2}t)$ and $t^{(12)}/(2t)$, respectively. We have chosen the lattice depth $V_0=15E_{\mathrm{R}}$; this affects only the horizontal scale. In the ordinary Bose-Hubbard model, $t^{(\mathit{mn})}/[\sqrt{(m+1)n}t]=1$ for all $m,n$, as confirmed by this figure’s $a=0^{-}$ (lower, red) branch and $a=0^{+}$ (upper, blue) branch limits. The resonance at $-d/a=0$ separates the molecular side (a) from the atomic side (b). Note that $t^{(10)}/t$ (not shown) is universally equal to unity regardless of interaction strength, since interatomic correlations are absent when there is a single particle per site.

Figure 3

Representative Gutzwiller mean-field theory phase diagrams, showing constant density (black, roughly horizontal) and constant $\xi \equiv {\zeta }_1+{\zeta }_2+{\zeta }_3$ (red, roughly vertical) contours. $\xi$, similar to the condensate density, is a combination of the mean fields ${\zeta }_m$, defined after Eq. (5). Density contours are $n=\{0.01,0.2,0.5,0.8,0.99,1.01,1.2,\dots \}$ and order-parameter contours are $\xi =\{0.2,0.4,\dots \}$, except in (d), where we take contours $\xi =\{0.02,0.04,\dots \}$. The phase diagrams are functions of ${\mu }_{\mathrm{eff}}\equiv \mu /E_{\mathrm{R}}$ and $t_{\mathrm{eff}}\equiv \exp (-\sqrt{V_0/E_{\mathrm{R}}})$, where the lattice depth $V_0$ is the natural experimental control parameter. We plot versus $t_{\mathrm{eff}}$, instead of $V_0$, as this is closer to the Hamiltonian matrix elements and more analogous to traditional visualizations of the Bose-Hubbard phase diagram. (a) Ordinary Bose-Hubbard model for $a=0.01d$, (b) lattice bosons restricted to fillings $n=0,1,2$ with $a=0.01d$, on the next-to-lowest energy branch on the $a>0$ side of resonance, (c) lattice boson model with $a=d$, and (d) fractional quantum Hall puddle array model, taking $\omega -\Omega =0.1E_{\mathrm{R}}$ (see text for details). Panels (b) and (c) use the Hamiltonian parameters from the exact two-particle harmonic well solution.

Figure 4

Single lobe of the Mott-insulator–superfluid boundary. Complete characterization of the mean-field lobe shape in the $z\mathrm{t̄}$-$\mathrm{\mu ̄}$ plane for all possible $t^{\pm }$’s [see Eq. (7) for definitions]. (a) Fix $t^{-}=0.5$, vary $t^{+}$ from 1 (outer curve) to 21 (inner curve) in steps of 2. (b) Fix $t^{+}=1.5$, vary $t^{-}$ from 0 (outer curve) to 1 (inner curve) in steps of $0.15$.