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arXiv:2607.06130

Self-Bound Droplets of Ultracold Dipolar Molecules under Tunable Double Microwave Shielding

Roger Melero, Jordi Boronat, Ferran Mazzanti

Published Jul 7, 2026• 0 citations
arXivHTMLPDF

Abstract

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 $3^\circ$. 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.

DOI
10.48550/arxiv.2607.06130

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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 ξξξ. While for the circularly polarized case (ξ=0ξ=0ξ=0) a gas phase is realized, one or more self-bound droplets are observed for small ∣ξ∣|ξ|∣ξ∣'s above a threshold value near 3∘3^\circ3∘. With increasing ξξξ, 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 ξξξ, while moderate differences show up for ξ<0ξ<0ξ<0 where our simulations conform to the expected symmetries of the intermolecular potential.
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.
This thesis presents a computational study of self-bound quantum droplets of microwave-shielded polar molecules using the Path Integral Ground State (PIGS) method. Extending previous work on cylindrically symmetric systems, this research investigates the effects of anisotropy induced by an elliptically polarized microwave field on the ground state of these droplets. The results demonstrate that introducing anisotropy enhances droplet stability, reducing the critical particle number required for self-binding. The most significant finding is the discovery of a structural crossover at large elliptic angles, where the uniform droplets develop an internal “stripe” phase characterized by a stable density modulation. The number of these stripes is shown to depend systematically on the particle number and interaction strength.
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.
Abstract Ultracold polar molecular gases provide a powerful platform for exploring quantum many-body physics with strong, long-range, and anisotropic interactions. In this work, we develop an extended Gross-Pitaevskii approach tailored to bosonic dipolar molecules under single-microwave shielding, incorporating their effective interactions and adapting the quantum fluctuation corrections. We benchmark this beyond-mean-field theory against exact path-integral Quantum Monte Carlo simulations. Focusing on the regime of positive scattering lengths, we find excellent agreement across a range of quantum phases, including superfluid, supersolid, and droplet states. We show that elliptic microwave polarization induces anisotropic superfluidity with direction-dependent sound velocities along each spatial axis—an effect absent in atomic dipolar gases. A quasi-one-dimensional theory captures roton softening and predicts roton instabilities tunable via ellipticity. While most experiments rely on double-microwave shielding to reduce losses, we demonstrate that single-shielded molecules already support rich and tunable many-body behavior. Our framework is readily extendable to the double-shielded case. This work establishes a versatile theoretical foundation for ultracold molecular gases and opens the door to future studies with more advanced shielding protocols.
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 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.