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arXiv:1612.06942 · Physical Review A

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

Michael L. Wall, Nirav P. Mehta, Rick Mukherjee, Shah Saad Alam, Kaden R. A. Hazzard

Published Apr 25, 2017• 16 citations
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Abstract

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, Do\ifmmode \mbox{\c{c}}\else \c{c}\fi{}aj 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 Do\ifmmode \mbox{\c{c}}\else \c{c}\fi{}aj 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 Do\ifmmode \mbox{\c{c}}\else \c{c}\fi{}aj 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.

DOI
10.1103/physreva.95.043635

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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 sss-wave scattering length governs the collisional physics. However, Do\ifmmode \mbox{\c{c}}\else \c{c}\fi{}aj 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 Do\ifmmode \mbox{\c{c}}\else \c{c}\fi{}aj 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 Do\ifmmode \mbox{\c{c}}\else \c{c}\fi{}aj 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.
We derive effective lattice models for ultracold bosonic or fermionic nonreactive molecules (NRMs) in an optical lattice, analogous to the Hubbard model that describes ultracold atoms in a lattice. In stark contrast to the Hubbard model, which is commonly assumed to accurately describe NRMs, we find that the single on-site interaction parameter U is replaced by a multichannel interaction, whose properties we elucidate. Because this arises from complex short-range collisional physics, it requires no dipolar interactions and thus occurs even in the absence of an electric field or for homonuclear molecules. We find a crossover between coherent few-channel models and fully incoherent single-channel models as the lattice depth is increased. We show that the effective model parameters can be determined in lattice modulation experiments, which, consequently, measure molecular collision dynamics with a vastly sharper energy resolution than experiments in a free-space ultracold gas.
We calculate the parameters of the recently derived many-channel Hubbard model that is predicted to describe ultracold nonreactive molecules in an optical lattice, going beyond the approximations used by Do\ifmmode \mbox{\c{c}}\else \c{c}\fi{}aj et al. [A. Do\ifmmode \mbox{\c{c}}\else \c{c}\fi{}aj et al., Phys. Rev. Lett. 116, 135301 (2016)]. Although those approximations are expected to capture the qualitative structure of the model parameters, finer details and quantitative values are less certain. To set expectations for experiments, whose results depend on the model parameters, we describe the approximations' regime of validity and the likelihood that experiments will be in this regime, discuss the impact that the failure of these approximations would have on the predicted model, and develop theories going beyond these approximations. Not only is it necessary to know the model parameters in order to describe experiments, but the connection that we elucidate between these parameters and the underlying assumptions that are used to derive them will allow molecule experiments to probe new physics. For example, transition state theory, which is used across chemistry and chemical physics, plays a key role in our determination of lattice parameters, thus connecting its physical assumptions to highly accurate experimental investigation.
Ultracold atomic gases in optical lattices provide an unique framework to study quantum phenomena in strongly correlated systems. In addition to the precise control over all relevant parameters in the experiment, these experiments can be mapped to a fundamental model from solid-state physics. For the bosonic version of this model, the so-called Bose-Hubbard model, a phase transition from a superfluid to a Mott insulator was theoretically predicted and later experimentally observed in an ultracold gas of Rb-87 atoms in three-dimensional as well as in one-dimensional optical lattices. Apart from homogeneous optical lattices one can introduce more complex lattice topologies such as two-color superlattices which give rise to a rich phase diagram including more exotic phases like the Bose-glass. We employ various powerful many-body techniques like exact diagonalization in complete and truncated Hilbert spaces and the Density-Matrix Renormalization Group (DMRG) algorithm to study the phase diagrams of the one-dimensional Bose-Hubbard and the one-dimensional Bose-Fermi-Hubbard Hamiltonian. Most theoretical studies of these systems discuss the phase diagrams with respect to the generic parameters of the Hubbard model. These Hubbard parameters, however, depend non-trivially on the control parameters used in experiments. The focus of this work is on the ab-initio calculation of the phase diagram of ultracold Rb-87 in one-dimensional optical superlattices starting directly from the experimental setup. To this end, we first employ band-structure calculations to extract the Hubbard parameters from the experimental parameters. Then, we use state-of-the-art DMRG techniques to solve the many-body problem for realistic particle numbers and lattice sizes that occur in experiments. Our results show that by using the intensities of the two laser fields forming the two-color superlattice as control parameters while keeping all other experimental parameters fixed, it is possible to access all relevant quantum phases of the system. Furthermore, we found out that the critical values of the laser intensities for the different phase transitions depend strongly on a third parameter that has to be included for a realistic description of the experiment. This third parameter is the strength of a harmonic trapping potential which accounts for the Gaussian shape of the laser fields and an additional magnetic potential used to confine the atoms in the center of the trap.
We study polar alkali-metal dimer scattering in a quasi-one-dimensional geometry for both reactive and nonreactive species. Elastic and reactive rates are computed as a function of the amplitude of a static electric field within a purely long-range model with suitable boundary conditions at shorter range. We describe the diatomic molecules as rigid rotors and results are compared to the fixed-dipole approximation. We show in particular that for molecules with a sufficiently strong induced dipole moment oriented perpendicular to the trap axis, the long-range repulsive interaction leads to the suppression of short-range processes. Such shielding effect occurs for both reactive and nonreactive molecules, preventing two-body reactions as well as losses due to complex-mediated processes [M. Mayle et al., Phys. Rev. A 85, 062712 (2012)] from occurring. The present results demonstrate the possibility to suppress loss rates in current ultracold molecule experiments using one-dimensional confinement.
We numerically study the problem of two fermions in a three-dimensional optical lattice interacting via a zero-range Feshbach resonance and display the dispersions of the bound states as a two-particle band structure with unique features compared to typical single-particle band structures. We show that the exact two-particle solutions of a projected Hamiltonian may be used to define an effective two-channel, few-band model for the low-energy, low-density physics of many fermions at arbitrary s-wave scattering length. Our method applies to resonances of any width and can be adapted to multichannel situations or higher-ℓ pairing. In strong contrast to usual Hubbard physics, we find that pair hopping is significantly altered by strong interactions and the presence of the lattice, and the lattice induces multiple molecular bound states.
We present a theory for ultracold atomic gases in an optical lattice near a Feshbach resonance. In the single-band approximation the theory describes atoms and molecules that can both tunnel through the lattice. Moreover, an avoided crossing between the two-atom and the molecular states occurs at every site. We determine the microscopic parameters of the generalized Hubbard model that describes this physics, using the experimentally known parameters of the Feshbach resonance in the absence of the optical lattice. As an application we also calculate the zero-temperature phase diagram of an atomic Bose gas in an optical lattice.