We use the Ground-State Path Integral Monte Carlo method to study a Bose-Einstein condensate of strongly interacting NaCs polar molecules under the action of a fully anisotropic double microwave shielding potential characterized by a linear and an elliptical polarization field. In particular, we analyze the ground state of the system and its structure as a function of the ellipticity angle
ξ. While for the circularly polarized case (
ξ=0) a gas phase is realized, one or more self-bound droplets are observed for small
∣ξ∣'s above a threshold value near
3∘. With increasing
ξ, the observed droplets rapidly become tightly bound and are estimated to form a superfluid array. Our results compare favorably to the experimental observations in [Zhang et al., Nature \textbf{651}, 601 (2026)] for positive
ξ, while moderate differences show up for
ξ<0 where our simulations conform to the expected symmetries of the intermolecular potential.
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.
We investigate the physics of ultracold dipolar molecules using path-integral quantum Monte Carlo simulations, and construct the complete phase diagram extending from weak to strong interactions and from small to mesoscopic particle numbers. Our calculations predict the formation of self-bound quantum droplets at interaction strengths lower than previous estimates for molecular condensates. Strikingly, we observe that, for stronger interactions, the oblate quantum droplet transitions to a two-dimensional sheet or superfluid membrane with a thickness of a single molecule. As interactions are increased further the system continually loses superfluidity while correlations develop, and is eventually found to undergo a transition to a crystalline monolayer that remains self-bound without external confinement. The spontaneous formation of such two-dimensional phases from a three-dimensional quantum gas is traced back to the peculiar anisotropic form of the dipole-dipole interaction generated by microwave dressing of rotational molecular states. For sufficiently large particle numbers, crystallization takes place for comparably low interaction strengths that do not promote two-body bound states and should thus be observable in ongoing experiments without limitations from three-body recombination.
Recent experiments on molecular droplets have opened a new frontier of self-organization in strongly dipolar quantum matter. Microwave-dressing of polar molecules permits to tune both the strength and the angular structure of long-range interactions, potentially promoting a rich spectrum of quantum phases, from superfluid droplets with varying geometry and insulating or supersolid droplet arrays to strongly correlated crystals of individual molecules. Using path-integral Monte Carlo simulations of large molecular ensembles, we demonstrate that experimentally observed droplet arrays emerge as a metastable non-equilibrium state from the quenching of a gas-droplet phase transition under entirely broken rotational symmetry of the microwave-induced interaction potential. We moreover find that a crystalline phase of molecules, predicted for antidipolar interactions, is absent under conditions of recent experiments. This is traced back to the lack of angular symmetry in currently employed microwave-dressing, which qualitatively reshapes the many-body energy landscape and cannot be captured by effective scalar interaction parameters. Our results provide the first direct comparison of ab initio simulations and experiments and establish interaction anisotropy as a key aspect of molecular quantum gases.
Anisotropic interactions profoundly affect topological excitations in quantum fluids. Motivated by recent advances in the studies of microwave-shielded polar molecules, we introduce self-trapped modes in the form of pancake-shaped quantum droplets (QDs) with embedded vorticity, which are maintained by strongly anisotropic dipole-dipole interactions. The stability region of singly charged (
S=1) vortices nearly coincides with that of the ground-state QDs (
S=0), demonstrating the robustness of the vortex states. The strong anisotropy of the system splits the core (pivot) of vortex QDs with
S=2 into separated unitary ones. The angular momentum and stability of the state with
S=2 are affected by the separation between the unitary cores. Higher-charge vortex QDs with
S>2 are stable too, for sufficiently large particle numbers and inter-core separations. On the other hand, bound vortex-antivortex pairs with
S=±1 are unstable. Head-on collisions between the vortex QDs exhibit distinct regimes, including rebound, merger, and fragmentation. Tuning the cylindrically symmetric component of the dipolar interaction reveals a pronounced sign-dependent response: positive tuning preserves the self-bound vortex, whereas negative tuning drives expansion and fragmentation. The results demonstrate that the microwave-dressed molecular QDs offer a robust platform for the realization of self-trapped vortex states, demonstrating how the strong anisotropy reshapes their structure, stability, and dynamics.
We investigate how dipolar interactions between microwave-shielded polar molecules structure the self-bound droplets formed under variation of the interaction strength. We identify the transition from droplets to crystals as a finite-size first order transition. With droplet-ring states and transitional supersolid states we predict additional structure in the crystal and droplet phases, respectively. To describe this strongly correlated regime, and in particular the reconfiguration of quantum ground states, we design a variational Monte Carlo framework based on neural quantum states. It is especially suitable to describe ground states and almost degenerate states with very different configurations. Moreover, one can easily determine the superfluid fraction. Our results reveal the sequence of finite-size structures through which dipolar interactions reorganize molecular droplets into crystals.