Exploring molecular supersolidity via exact and mean-field theories: single microwave shielding

Abstract

Ultracold polar molecules with microwave shielding provide a powerful platform for exploring quantum many-body physics with strong, anisotropic interactions. We develop an extended Gross-Pitaevskii framework for bosonic molecules under single microwave shielding, incorporating effective interactions and quantum fluctuations, and benchmark it against exact Quantum Monte Carlo simulations. In the regime of positive scattering lengths, our approach captures superfluid, supersolid, and droplet phases with excellent accuracy. We show that elliptic microwave polarization induces direction-dependent superfluidity--absent in cylindrically symmetric systems. A quasi-1D theory reveals that roton softening and instabilities can be controlled via ellipticity, consistent with recent experiments. Furthermore, we find that the nature of the superfluid-to-supersolid transition is strongly influenced by the ellipticity: the transition is sharp at low ellipticity and continuous at higher values. This tunability offers a potential route for low entropy preparation of molecular supersolids via adiabatic ramps. While double shielding is often used experimentally, our results demonstrate that single-shielded molecules already offer rich, controllable behavior, laying the groundwork for future studies with more complex shielding schemes.

Figures (11)

• Figure 1: Spontaneous density modulation through microwave-shielding induced interaction shaping. Top row: Long-range interaction potential $V_3(\bm{r})$ in three-dimensional space, for increasing ellipticity $\xi$. Bottom row: Schematic density distributions of ultracold molecules in a cigar-shaped trap, showing a superfluid, supersolid, and independent droplet phase, induced through increasing ellipticity.
• Figure 2: Efficacy of the pseudo-potential approximation. (a) s-wave scattering length $a_s$ as a function of $a_\mathrm{dd}$ and $\xi$. The grey-dashed line highlights the position of a field-linked resonance. (b) Plot of $t$-matrix amplitudes $t^{mm'}_{\ell\ell'}$ for $a_\mathrm{dd}=2.4 R_\Omega$: $t_{00}^{00}$ (black), $t_{02}^{00}$ (red), $t_{20}^{00}$ (blue) and $t_{02}^{02}$ (orange). Solid lines correspond to the original potential, while dotted lines correspond to the pseudo-potential $V_\text{SMS}'$. By construction, $t_{00}^{00}$ is the same for $V_\mathrm{SMS}$ and $V_\mathrm{SMS}'$.
• Figure 3: Agreement between mean-field theory and Monte Carlo methods in the superfluid-to-droplet regime, with $a_\mathrm{dd}=5.11\times10^{-2}a_\perp$. (a) Condensate widths as a function of ellipticity in the three cartesian directions. Solid lines correspond to eGPE-M, diamonds to PIMC. The dashed line shows $\sigma_z$ from the eGPE-M without the quantum fluctuation correction ($\eta_\text{QF}=0$), leading to collapse for $\xi>20^\circ$. (b) Fidelity, see Eq. \ref{['eqn:fid']}. The background color of (a) and (b) corresponds to $\eta_\text{dd}$. Panels (c) and (d) show linear densities for two representative values of $\xi$; colors and styles are the same as in (a) and (b). Column densities in the $yz$ plane corresponding to (c) are shown in (e) for eGPE-M and in (f) for PIMC; those corresponding to (d) are shown in (g) for eGPE-M and in (h) for PIMC. Peak densities for the highlighted states are (c) $5.03a^{-3}_\perp$ and (d) $90.88a^{-3}_\perp$.
• Figure 4: Agreement between mean-field theory and Monte Carlo methods in the superfluid-to-droplet regime, with $a_\mathrm{dd}=9.82\times10^{-2}a_\perp$. Note, there are no stable solutions when $\eta_\text{QF}=0$. Peak densities for highlighted states are (c) $18.42a^{-3}_\perp$ and (d) $171.9a^{-3}_\perp$. Panels are the same as in Fig. \ref{['fig:superfluid']}.
• Figure 5: Fully anisotropic superfluidity in molecular Bose gases. (a) Excitation spectrum from Eq. \ref{['eqn:exspec3D']} with $n_0a_\text{dd}^3 = 0.2$, $\xi = 12^\circ$, and $\eta_\text{dd} = 2$. Each curve $\epsilon (k_j)$ is evaluated along one axis (as indicated in the legend) with momenta in the orthogonal directions set to zero, i.e., $\epsilon(k_x) = \epsilon(k_x, 0, 0)$, etc. (b) Directional speeds of sound $c_j$. The dashed line indicates the $\xi$ value used in (a). All other parameters are as in (a). We have defined the frequency scale $\omega_\text{dd} = \hbar/m a_\text{dd}^2$ and corresponding velocity scale $c_\text{dd} = a_\text{dd}\omega_\text{dd}$.