Hubbard physics with ultracold polar molecules: on-site interaction energies for shielded molecules

Technical Summary: On-site Interaction Energies for Shielded Ultracold Polar Molecules

Problem Statement
Ultracold polar molecules offer a unique platform for quantum simulation and many-body physics due to their long-range, tunable dipole-dipole interactions. A critical regime for exploring exotic phases (such as pair superfluidity and supersolids) is the Hubbard model with multiple site occupancy. However, standard experimental approaches often exclude double occupancy because unshielded molecules on the same lattice site undergo rapid collisional loss. While shielding techniques (using static electric or microwave fields) can suppress these losses by creating repulsive barriers, the resulting on-site interaction energy ($U$) is not well understood.

Existing theoretical frameworks for calculating $U$, primarily based on Wannier functions and contact potentials (valid for ultracold atoms), fail for shielded molecules. These methods neglect strong on-site correlations induced by the large repulsive core of the shielding potential and the anisotropic nature of the dipole-dipole interaction. Consequently, the dependence of $U$ on lattice strength and scattering parameters for shielded molecules remains an open question, particularly regarding whether $U$ can be tuned to allow stable multiple occupancy.

Methodology
The authors employ a rigorous numerical approach that explicitly accounts for on-site correlations, moving beyond the single-band Wannier function approximation.

• Model System: The study focuses on pairs of ultracold polar molecules (specifically NaCs) shielded by circularly polarized microwaves. The interaction is modeled using 2-dimensional effective potentials ($V(R, \theta)$) that exhibit a large repulsive core at short range and dipole-dipole character at long range.
• Hamiltonian: The problem is separated into center-of-mass and relative motions within a harmonic trap. The relative motion is described by the Schrödinger equation including the reduced mass, relative angular momentum, the effective interaction potential $V(R)$, and the harmonic trapping potential $V_{trap}(R)$.
• Calculation: The authors solve the coupled-channel Schrödinger equation numerically using the BOUND package to obtain exact relative energies ($E_{rel}$). The on-site interaction energy is defined as $U = E_{rel} - \frac{3}{2}\hbar\omega$, where $\frac{3}{2}\hbar\omega$ is the relative energy of two non-interacting particles in separate sites. Scattering lengths ($a$) are calculated independently using the MOLSCAT package for comparison.
• Parameters: Calculations span a range of Rabi frequencies ($\Omega$) and detunings ($\Delta$) to cover moderate positive to large negative scattering lengths. The lattice strength is varied to explore the ratio $U/t$ (interaction to tunneling energy), a key parameter for Hubbard physics.

Key Results
The study reveals that the behavior of $U$ for shielded molecules is fundamentally distinct from that of ultracold atoms:

1. Breakdown of Atom-Like Models: For shielded molecules, $U$ is not a monotonic function of the trap frequency $\omega$, nor is it simply proportional to the scattering length $a$. The standard approximation $U \propto a\omega$ (Eq. 1) fails significantly, even for small $|a|$. This deviation arises because the wavefunction is "squeezed" between the hard repulsive core of the shielding potential and the confining trap, raising the energy above contact-potential predictions.
2. Unbound States: For pairs that are unbound at zero trap frequency, $U$ can be negative at low $\omega$ but crosses zero to become positive as $\omega$ increases. At lattice strengths suitable for Hubbard physics ($V_0 \gtrsim 5E_r$), $U$ is typically large and positive ($U/t > 60$), effectively imposing a hard-core constraint that prevents multiple occupancy.
3. Bound States: Crucially, the shielding potentials support 2-body bound states with binding energies of a few kHz. For these states, $U(\omega)$ initially becomes more negative as $\omega$ increases (due to the relative energy of separated particles rising faster than the bound state energy), but eventually crosses zero to positive values at higher $\omega$.
4. Tunability: The zero-crossing of $U$ for bound states occurs within experimentally accessible ranges of lattice strength. By varying the lattice depth or the shielding parameters (Rabi frequency), $U/t$ can be tuned continuously from negative values (attractive) to large positive values (repulsive), covering the range $-10 < U/t < 50$ required for rich Hubbard physics.

Significance and Claims
The paper claims that understanding and controlling $U$ for shielded molecules is essential for realizing the full potential of ultracold molecules in many-body physics. The primary significance lies in demonstrating that:

• Multiple Occupancy is Feasible: Unlike unshielded molecules or atoms with specific scattering lengths, shielded molecules in bound states allow for the tuning of $U$ to values that permit stable double occupancy without fast collisional loss.
• Access to New Physics: This capability removes the "hard-core" constraint often assumed in molecular Hubbard models. It opens the door to experimental studies of models with multiple site occupancy, enabling the exploration of exotic phases such as pair superfluidity, solid, and supersolid phases, which are governed by the competition between tunneling and on-site interactions.
• Universality: While specific calculations are performed for NaCs, the results are presented as universal for polar molecules when energies and frequencies are scaled by the molecule-dependent dipolar energy scale $E_3$.

The authors conclude that the ability to tune $U$ across the zero-crossing for bound states provides a controllable pathway to explore strongly dipolar Hubbard physics that was previously inaccessible.