Microscopic derivation of multichannel Hubbard models for ultracold nonreactive molecules in an optical lattice

Recent experimental advances in the cooling and manipulation of bialkali-metal dimer molecules have enabled the production of gases of ultracold molecules that are not chemically reactive. It has been presumed in the literature that in the absence of an electric field the low-energy scattering of such nonreactive molecules (NRMs) will be similar to atoms, in which a single $s$-wave scattering length governs the collisional physics. However, Doçaj et al. [Phys. Rev. Lett. 116, 135301 (2016)] argued that the short-range collisional physics of NRMs is much more complex than for atoms and that this leads to a many-body description in terms of a multichannel Hubbard model. In this work we show that this multichannel Hubbard model description of NRMs in an optical lattice is robust against the approximations employed by Doçaj et al. to estimate its parameters. We do so via an exact, albeit formal, derivation of a multichannel resonance model for two NRMs from an ab initio description of the molecules in terms of their constituent atoms. We discuss the regularization of this two-body multichannel resonance model in the presence of a harmonic trap and how its solutions form the basis for the many-body model of Doçaj et al.. We also generalize the derivation of the effective lattice model to include multiple internal states (e.g., rotational or hyperfine). We end with an outlook to future research.

Figure 1

Schematic of effective model (2). Ultracold NRMs in open-channel internal states $s$ tunnel through the lattice at a rate $J$ when moving to an empty site. When an NRM tunnels onto an occupied site, the tunneling is reduced by the open-channel weight ${\mathcal{O}}_{\alpha }^{s,s^{\prime }}$, the matrix element of two open-channel molecules on the two-body on-site eigenstates $|\alpha \rangle$. Energy shifts due to collisional resonances for two NRMs on a site are described by interaction energies $U_{\alpha }$.

Figure 2

Complex scattering of ultracold nonreactive molecules. At intermolecular separations large compared to a characteristic potential range $R_{\mathrm{vdW}}$, molecules propagate ballistically. These ballistic trajectories are curved in a regular fashion in an intermediate range comparable to the potential length but still larger than a characteristic short-range length $r_{\mathrm{sr}}$. Below $r_{\mathrm{sr}}$, many internal molecular states (e.g., rotations and vibrations) become strongly mixed as the constituent atoms undergo chaotic dynamics. These complex dynamics can be recast in terms of a resonance model with a large density of rovibrational resonances [see Eq. (6)].

Figure 3

Eigenvalues $E_{\alpha }$ (top row) and effective interactions $U_{\alpha }$ (bottom row) as a function of trap frequency $\omega$. (a) Generic structure illustrated by a two-channel (single-bound-state) model. Shown on top is the relative coordinate eigenenergy $E_{\alpha }$ showing an avoided crossing between a bimolecular collisional complex at fixed energy $2\pi \times 1.0$ kHz and the harmonic-oscillator ground state that is linearly dependent on $\omega$ (diagonal). The opacity at each plotted point (darkness of the line) is set to be ${\mathcal{O}}_{\alpha }$ of the corresponding eigenstate, indicating its importance to the lattice model. Shown on bottom is the on-site interaction in the lowest band (harmonic-oscillator ground state) $U_{\alpha }$. The avoided crossing in $E_{\alpha }$ manifests as a strong multivalued interaction near the crossing. Also shown are (b) $E_{\alpha }$ (top) and $U_{\alpha }$ (bottom) for realistic parameters for RbCs. Isolated resonances at small $\omega$ give way to coupled resonances at larger $\omega$.

Figure 4

Derivation of the regularized model (40). (b) The regularized zero-range limit $H_{\text{rel}}$ of (a) the true physical model $H_p$ is computed by ensuring the agreement of the forms of (c) their low-energy effective Hamiltonians $H_{\text{eff}}$, which govern all of the low-energy observables. Even though the true physical Hamiltonian cannot be computed, its form can be determined [see Eq. (39)] and shown to match that obtained from the regularized model for appropriately chosen $W_{nb}$ and ${\nu }_b$.