Microscopic theory of the Hubbard interaction in low-dimensional optical lattices

TL;DR

This work presents an exact two-body solution for two atoms in quasi-1D and quasi-2D optical lattices with transverse harmonic confinement, enabling a microscopic derivation of the on-site Hubbard interaction $U$ that includes higher Bloch bands. By matching the exact low-energy scattering amplitude to the Hubbard model, the authors uncover lattice-induced resonances in $U$ as a function of the 3D scattering length $a$ and lattice parameters, and demonstrate strong agreement with experimental measurements in a quasi-2D system. The approach unifies low-dimensional renormalization (linking $a$ to effective $a_{ m 1D}$ and $a_{ m 2D}$) with a full lattice treatment, providing a numerically tractable framework and revealing fundamental limits of the single-band Hubbard description near resonances. The results have immediate relevance for Hubbard-model simulations with quantum gas microscopes and open avenues for extensions to multi-band, anisotropic lattices and moiré-like platforms.

Abstract

The Hubbard model is a paradigmatic model of strongly correlated quantum matter, thus making it desirable to investigate with quantum simulators such as ultracold atomic gases. Here, we consider the problem of two atoms interacting in a quasi-one- or quasi-two-dimensional optical lattice, geometries which are routinely realized in quantum-gas-microscope experiments. We perform an exact calculation of the low-energy scattering amplitude which accounts for the effects of the transverse confinement as well as all higher Bloch bands. This goes beyond standard perturbative treatments and allows us to precisely determine the effective Hubbard on-site interaction for arbitrary $s$-wave scattering length (see source code available at [1]). In particular, we find that the Hubbard on-site interaction displays lattice-induced resonances for scattering lengths on the order of the lattice spacing, which are well within reach of current experiments. Furthermore, we show that our results are in excellent agreement with spectroscopic measurements of the Hubbard interaction for a quasi-two-dimensional square optical lattice in a quantum gas microscope. Our formalism is very general and may be extended to multi-band models and other atom-like scenarios in lattice geometries, such as exciton-exciton and exciton-electron scattering in moiré superlattices.

Figures (5)

• Figure 1: Illustration of the low-dimensional geometries under consideration for two interacting atoms. (a) In quasi-1D, we have an optical lattice along the $z$ direction, with a transverse cylindrically symmetric harmonic confinement (i.e., along the $x$ and $y$ directions). (b) In quasi-2D, we instead consider a square lattice in the $x$-$y$ plane, with transverse harmonic confinement along the $z$ direction. The lattice strength can differ along the $x$ and $y$ directions. (c) Along a particular axis, the two atoms can either experience an identical potential (left) or a state-dependent lattice (right, where the orange spin-$\downarrow$ atom experiences the orange potential). We will use the "dimensions of confinement" to refer to $x,\, y$ in quasi-1D and to $z$ in quasi-2D. Meanwhile, the "dimensions of the lattice" refer to $z$ in quasi-1D and to $x,\, y$ in quasi-2D.
• Figure 2: (a,b) Effective on-site interaction $U$ and (c,d) two-body spectrum as a function of the 3D scattering length $a$. We show these for (a,c) quasi-1D and (b,d) quasi-2D lattices, where in both cases $v^i_\sigma = 12 V_r$, $l \simeq 0.13 d$, $R^*=0$, and we consider zero CM quasimomentum. (a,b) The on-site interaction (blue line) is compared with the predictions of perturbation theory (gray dashed line), i.e., Eq. \ref{['eq:U1Dpert']} and \ref{['Eq:U2DHubbard']} for quasi-1D and quasi-2D, respectively, along with the commonly used expression in Eq. \ref{['Eq:OldHubbardUDefn']} (gray, dashed line). (c,d) In the spectra, from bottom to top, we show the first three Bloch bands as well as the continuum (colored gray). We also show the two-body bound-state energies where we color the fully even (other) parity bound states solid orange (dot-dashed green). The bound-state energies are compared alongside those predicted by the Hubbard model (black, dashed line), where $U$ is determined by its corresponding value in (a,b). Note that the odd parity bound states do not couple to the lowest Bloch band and are thus independent of the interaction $U$. In the spectra, all energies are measured relative to the threshold energy of the lattice $E_0 \equiv E_{\vectorbold{0},\vectorbold{0};\vectorbold{0},\vectorbold{0};0}$. For clarity, we only plot the first ten two-body bound states in panel (c) and the first 20 in panel (d). Note that the lowest energy state is always even parity and is outside the range plotted in (d).
• Figure 3: The on-site interaction $U$ at unitarity (i.e., $1/a=0$) as a function of the harmonic oscillator length in (a) quasi-1D and (b) quasi-2D. Remarkably, even at unitarity the lattice can induce resonances in $U$, which can be extremely narrow as seen for $l/d \sim 0.05$. In both cases we consider a broad Feshbach resonance ($R^*=0$) and take $v^i_\sigma = 12V_r$.
• Figure 4: The on-site interaction $U$ as a function of 3D scattering length $a$ for a state-dependent lattice in (a) quasi-1D and (b) quasi-2D, where $v^i_\uparrow = 12V_r$, $v^i_\downarrow=10V_r$, $l\simeq 0.13d$ and $R^*=0$. The state-dependent lattice displays highly narrow resonances, which peak at both finite and infinite values. The inset in panel (a) shows a zoomed in region of a highly narrow, finite resonance.
• Figure 5: The on-site interaction $U$ as calculated using our theory (blue line) and experimentally extracted using lattice modulation spectroscopy greifSiteresolvedImagingFermionic2016 (olive points) for $v^x_\sigma = 12.5 V_r$, $v^y_\sigma = 15.9 V_r$, $\omega \simeq 3.71 V_r$ and $R^*=0$. Panel (a) shows the comparison, alongside perturbation theory, Eq. \ref{['Eq:OldHubbardUDefn']} (gray, dashed line), on an experimentally relevant scale of scattering lengths. Panel (b) extends to larger scattering lengths showing that the first resonance is located at roughly twice the largest scattering length considered in experiment.