Ultracold polar molecules offer electric dipole moments, rich rotational structure and long coherence times in a single quantum gas, giving access to interaction regimes that are difficult to reach with atoms. However, realizing Bose-Einstein condensation in these systems has remained difficult because two-body collisional losses usually prevent efficient evaporative cooling. Here we produce a condensate of ground-state sodium-rubidium molecules using dual microwave shielding, in which two microwave fields suppress loss while allowing control of the long-range interactions. Starting from an optically trapped gas of ground-state sodium-rubidium molecules, we cool the molecules to quantum degeneracy and obtain condensates containing about 500 molecules. By tuning the dipolar interactions, we also observe both gas-phase condensates and a self-bound quantum droplet, with the gas-to-droplet transition identified from time-of-flight expansion. These results establish sodium-rubidium molecules as a platform for studying strongly dipolar quantum matter with tunable long-range interactions.

Dual-microwave shielding has emerged as a powerful tool for stabilizing ultracold polar molecules while tuning their intermolecular interactions. However, the two microwave fields are generally not perfectly orthogonal in experiments. Such misalignment introduces an in-plane component of the linearly polarized microwave, whose frequency differs from that of the elliptically polarized field. This component prevents complete cancellation of the dipole-dipole interaction and, more critically, renders the single-molecule dressed state intrinsically time-dependent, so that the conventional time-independent scattering framework is no longer available. Here we develop a Floquet theory that yields an analytic effective potential and enables accurate scattering calculations for polar molecules in non-orthogonal dual microwave fields. We find that, though misalignment weakens the shielding moderately, inelastic losses remain strongly suppressed under experimentally relevant conditions. Meanwhile, misalignment provides additional tunability of the interaction anisotropy and strength, which has been directly applied to recent experimental observations on the gas-to-droplet transition [Z. Shi et al, arXiv:2508.20518 (2025)] and Fermi-surface deformation in microwave-shielded molecular gases [S. Biswas et al, arXiv:2602.22447]. The framework is not restricted to dual-microwave shielding and can be generalized straightforwardly to arbitrary multi-frequency driving, providing a versatile tool for manipulating ultracold polar molecules under complex microwave configurations.

Rydberg atom arrays composed of multiple atomic species stand as a highly promising platform for quantum computation. However, the underlying physics of these systems as quantum many-body systems remains poorly understood, owing to the intricate competition between attractive and repulsive interactions, phenomena that entirely defy the Rydberg blockade mechanism. We systematically calculate the ground-state phase diagram of alternating two-species atom arrays and their quench dynamics. Our findings reveal several novel quantum states absent in traditional cold-atom platforms, such as the period-4 product state |1100...⟩, the period-6 product state |111000...⟩, and an order-disorder mixed phase. In the quench dynamics, we confirm ℤ_2 ordered state qualify as novel quantum many-body scars. Based on our perturbation analysis, the underlying physics ought to be described by a series of Cooper pair states spanning the entire energy spectrum, rather than the PXP low-energy effective model. A detailed analysis is also provided regarding the experimental preparation of those product states. Numerical evidence demonstrates that the proposed scheme exhibits robustness against typical experimental imperfections, thereby confirming its experimental feasibility. Moreover, the ground-state problem of the two-species array naturally maps to more general combinatorial optimization problems, extending the class of optimization tasks accessible to programmable neutral-atom quantum processors. Our work paves a new way for quantum simulation of novel quantum many-body states, which emerge from the interplay between competing interactions among different atom species and quantum fluctuations.

Dual-species Rydberg atom arrays extend single-species platforms by introducing competing interaction scales and enhanced quantum fluctuations, enabling phenomena beyond homogeneous settings. In this work, we study the ground-state phase diagram of a one-dimensional dual-species Rydberg atom ladder using large-scale density-matrix renormalization group calculations. We identify disordered phases, multiple ordered phases with ℤ_2, ℤ_3, and ℤ_4 symmetry, as well as floating phases characterized by incommensurate wave vectors and algebraically decaying correlations. Importantly, we observe a smooth crossover between distinct ℤ_2-ordered regimes, reflecting a reorganization of low-energy degrees of freedom rather than a true phase transition, which is absent in single-species Rydberg arrays. We further uncover a multi-critical point at the boundary between the ℤ_2 ⊗ℤ_2 and ℤ_3 ⊗ℤ_3 ordered phases, where Ising, chiral, and first-order transition lines intersect. Our results demonstrate that dual-species Rydberg atom arrays provide a unique platform for realizing crossover physics and multi-critical behavior inaccessible in single-species architectures.

We investigate the thermodynamic properties of lattice Bose gases in optical cavities in the Mott-insulator limit. We find that the system assumes anomalous thermodynamic behavior that can be traced back to the breaking of fundamental additivity by its infinite-long-range interaction. Specifically, the system shows striking ensemble inequivalence between the canonical ensemble and the grand canonical one, sharply manifesting in the distinct anomalous structure of the thermodynamic phase diagram in the canonical ensemble. In particular, in the temperature regime at around half of the on-site energy, the system manifests negative compressibility and anomalous reentrant phase transitions where the ordered charge density wave phase revives from the homogenous phase upon increasing the temperature. Direct experimental observation of the anomalous behavior can be realized in the current experiments with well-controlled total particle-number fluctuations.

ABSTRACT Mimicking the synergistic proton conduction in biology remains a formidable challenge for synthetic materials. Here, we report a strategic approach by engineering dynamic proton “hubs” within hydrogen‐bonded organic frameworks (HOFs). These hubs are supramolecular secondary building units (SSBUs) formed by charge‐assisted hydrogen bonds between ammonium and sulfonate groups, which are programmed into the frameworks to direct proton traffic. The resulting ammonium‐sulfonate HOF, BPDS_NH 4 , exhibits a highly competitive proton conductivity of 0.21 S cm −1 (90°C, 90% RH). The exceptional performance is governed by the dual function of the proton hubs: their high charge density and hydrophilicity create a highway for vehicular transport of solvated proton clusters (H + (H 2 O) n ), while the dense, dynamic H‐bonded network within each hub serves as a confined arena for ultrafast Grotthuss hopping. These processes are synergistically coupled, as verified by activation energy analysis and H/D isotope effect. This work establishes the construction of supramolecular proton hubs as a versatile blueprint for the rational design of advanced proton‐conducting materials.

Ultracold atoms coupled to optical cavities offer a powerful platform for studying strongly correlated many-body physics. Here, we propose an experimental scheme for creating biatomic molecules via cavity-enhanced photoassociation from an atomic condensate. This setup realizes long-range three-body interactions mediated by tripartite cavity-atom-molecule coupling. Beyond a critical pump strength, a self-organized square lattice phase for molecular condensate emerges, resulting in hybrid atom-molecule superradiance with spontaneous U(1) symmetry breaking. Distinct from previously observed ultracold bosonic (fermionic) atomic superradiance, our findings demonstrate bosonic enhancement characterized by a cubic scaling of steady-state photon number with total atom number. Additionally, strong photon-matter entanglement is shown to effectively characterize superradiant quantum phase transition. Our findings deepen the understanding of quantum superchemistry and exotic many-body nonequilibrium dynamics in cavity-coupled quantum gases.

We investigate the finite-temperature phase diagram of polar molecules shielded by dual microwave fields using the path integral Monte Carlo method combined with the worm algorithm. We determine the critical temperature T_c for Bose-Einstein condensations (BECs) and identify two distinct phases below T_c: the expanding gas (EG) phase and the self-bound gas (SBG) phase. We further analyze the temperature and interaction-strength dependence of the condensate and superfluid fractions. Notably, in contrast to dilute atomic BECs, the SBG phase displays a low condensate fraction and a high superfluid fraction, resembling the behavior of strongly correlated ^4He superfluids. These significant many-body correlations arise from the interplay between long-range dipole-dipole interactions and the short-range shielding potential. Furthermore, we demonstrate that the aspect ratio of the gas provides a characteristic geometric signature to accurately determine the EG-to-SBG transition, robust against external trapping potentials. Our findings provide unbiased and numerically exact results to guide upcoming experiments with polar molecules.

We investigate the ground-state phases of ultracold gases composed of bosonic microwave-shielded polar molecules (MSPMs). Using a translational symmetry-breaking variational ansatz with Jastrow correlations, we characterize the many-body correlations arising from the large shielding core of the two-body potential in a dense gas. We show that the molecular gases are always stabilized by the shielding potential and support a self-bound gas phase and an expanding gas phase. Furthermore, we find that, analogous to liquid ^{4}He, the condensate fraction is significantly reduced when the size of the shielding core of the two-body potential becomes comparable to the intermolecular distance. For experimental detection, we also identify a bimodal feature in the momentum distribution. Our work invalidates the application of the Gross-Pitaevskii equation to molecular gases and establishes a universal framework to reveal the many-body correlations in dense molecular gases.

We propose a novel scheme to realize the supersolid phase in ultracold gases of microwave-shielded polar molecules by engineering an additional anisotropy in inter-molecular dipolar interaction via an elliptically polarized microwave. It is shown through quantum Monte-Carlo calculations that the interplay of the anisotropies between the interaction and trapping potential gives rise to rich quantum phases. Particularly, it is found that the supersolid phase emerges in the parameter regime accessible to current experiments. Our study paves the way for exploring the properties of supersolid phases in ultracold gases of polar molecules.

Realizing Bose-Einstein condensation of polar molecules is a long-standing challenge in ultracold physics and quantum science due to near-universal two-body collisional losses. Here, we report the production of a Bose-Einstein condensate of ground-state sodium-rubidium molecules via high efficiency evaporative cooling, with losses suppressed using the dual microwave shielding technique. The ability to tune the dipolar interaction between these ultracold polar molecules is crucial for producing the condensate and enables exciting prospects for future applications. We explore different regimes of dipolar interactions, realizing both the gas phase and the quantum droplet phase of the molecular condensate. This work opens new avenues for investigating quantum matter with strong dipolar interactions and for quantum simulation of long-range many-body systems.

While binary atomic Bose-Einstein condensates are typically prone to collapse under strong interspecies attraction, it has been shown that higher-order fluctuation corrections, known as Lee-Huang-Yang corrections, can stabilize the mixture. In this work, we demonstrate an alternative stabilization mechanism based on kinetic energy. Specifically, we consider a one-dimensional mixture of a quasi-BEC and a Tonks-Girardeau gas, and show that the kinetic energy of the TG component can counteract the interspecies attraction, thereby preventing collapse. This balance leads to the formation of a self-bound quantum droplet, which exhibits two distinct regimes: a low-density and a high-density droplet. We argue that these regimes are smoothly connected by a crossover. Furthermore, an analysis of the derivatives of the ground state energy reveals that the transition from a miscible mixture to the droplet phase is of third order. Our findings extend the theoretical understanding of quantum droplets in low-dimensional quantum gases, and the proposed system is experimentally accessible within current ultracold atom platforms.

We investigate the two- and many-body physics of ultracold polar molecules dressed by dual microwaves with distinct polarizations. Using Floquet theory and multichannel scattering calculations, we identify a regime with the largest elastic-to-inelastic scattering ratio, which is favorable for performing evaporative cooling. Furthermore, we derive and subsequently validate an effective interaction potential that accurately captures the dynamics of microwave-shielded polar molecules (MSPMs). We also explore the ground-state properties of the ultracold gases of MSPMs by computing physical quantities such as gas density, condensate fraction, momentum distribution, and second-order correlation. It is shown that the system supports a weakly correlated expanding gas state and a strongly correlated self-bound gas state. Since the dual-microwave scheme introduces an additional control knob and is essential for creating ultracold Bose gases of polar molecules, our work pave the way for studying the two- and many-body physics of the ultracold polar molecules dressed by dual microwaves. The authors report on a Floquet-theoretical approach to microwave shielding of bosonic ultracold molecules. This clarifies the underlying mechanism behind recently developed dual-microwave schemes and identifies the optimal parameter regime for evaporation.

We propose an experimental scheme to realize phase transition from dark superradiance to conventional superradiance in a microwave cavity coupled to polar molecules. The competition between cavity-mediated infinite-range repulsions and finite-range attractive dipolar interactions stabilizes a variety of exotic quantum phases, including vortex, vortex anti-vortex pairs, and superradiant phase, all emerging without external driving fields. In vortex phase associated with dark superradiance, cavity remains in vacuum state while profoundly reshaping the condensate's ground-state wave functions. In particular, the spin configuration locally parallel but globally anti-parallel is a direct consequence of competing for two nonlocal interactions. Beyond Dicke paradigm, dipolar dressing of condensate enables access to an unexplored regime of repulsion-dominated superradiance. A Bogoliubov analysis of low-energy excitation spectrum confirms that the condensate remains stable, avoiding roton-maxon induced collapse even in strongly dipolar regime.

Ultracold atoms with cavity-mediated long-range interactions offer a promising platform for exploring emergent quantum phenomena. Building on recent experimental progress, we propose a novel scheme to create supersolid square and plane wave phases in spin-1/2 condensates. We demonstrate that the self-ordered supersolid phase supports an undamped gapless Goldstone mode across a broad parameter regime. This proposal is comprehensively described by the two-component Tavis-Cummings model with hosting a U(1) symmetry. By exploiting the superradiant photon-exchange process, our approach also constructs the cavity-mediated spin-momentummixing interactions between highly correlated spin and momentum modes, which may open avenues for exploring spin-momentum squeezing and spatially distributed multipartite entanglement. (c) 2025 Chinese Laser Press

We investigate the static and dynamic properties of tetratomic molecules formed by two microwave-shielded polar molecules across field-linked resonances. In particular, we focus on two-body physics and experimental techniques unexplored in the recent experiment [X.-Y. Chen et al., Nature 626, 283 (2024)]. We show that, compared to the lowest tetramer state, higher tetramer states typically have longer lifetimes, which may facilitate a further cooling of tetramer gases towards quantum degeneracy. To detect tetramers, we identify the distinctive time-of-flight images from ramp dissociation, which can be observed by lowering the ramp rate of the microwave. Remarkably, in the modulational dissociation of tetramers, we find that multi-photon processes induce dissociation even below the threshold modulation frequency when the modulation amplitude is sufficiently high. Given the universal form of the inter-molecular potential for microwave-shielded polar molecules, our results also apply to other molecular gases widely explored in recent experiments.

We investigate the ground-state properties of the quasi-one-dimensional dipolar gases using continuous matrix product states techniques. Making use of the first- and second-order correlation functions, we find that the system supports the superfluid, super-Tonks-Girardeau, and quasicrystal phases according to the Luttinger liquid theory. We also map out the phase diagram on the parameter plane consisting the contact and dipolar interaction strengths. Furthermore, we compute the Luttinger parameter, the structure factor, and the momentum distribution of the system. Finally, we show that the predicted dipolar effect can potentially be observed in quasi-one-dimensional gases of polar molecules.

Multiple quantum coherences are often employed to describe quantum many-body dynamics in nuclear spin systems and recently, to characterize quantum phase transitions in trapped ions. Here we investigate the multiple-quantum-coherence dynamics of a spin-1 Bose-Einstein condensate. By adjusting the quadratic Zeeman shift, the condensate exhibits three quantum phases. Our numerical results show that the spectrum of multiple quantum coherence does indeed catch the quantum critical points. More importantly, with only a few low-order multiple quantum coherences, the spin-1 condensate exhibits rich signals of the many-body dynamics, beyond conventional observables. The experimental implementation of such multiple quantum coherence protocol is also discussed.

Ultracold polyatomic molecules offer intriguing new opportunities in cold chemistry, precision measurements, and quantum information processing, thanks to their rich internal structure. However, their increased complexity compared to diatomic molecules presents a formidable challenge to employ conventional cooling techniques. Here, we demonstrate a new approach to create ultracold polyatomic molecules by electroassociation in a degenerate Fermi gas of microwave-dressed polar molecules through a field-linked resonance. Starting from ground state NaK molecules, we create around $1.1\times 10^3$ tetratomic (NaK)$_2$ molecules, with a phase space density of $0.040(3)$ at a temperature of $134(3)\,\text{nK}$, more than $3000$ times colder than previously realized tetratomic molecules. We observe a maximum tetramer lifetime of $8(2)\,\text{ms}$ in free space without a notable change in the presence of an optical dipole trap, indicating these tetramers are collisionally stable. The measured binding energy and lifetime agree well with parameter-free calculations, which outlines pathways to further increase the lifetime of the tetramers. Moreover, we directly image the dissociated tetramers through microwave-field modulation to probe the anisotropy of their wave function in momentum space. Our result demonstrates a universal tool for assembling ultracold polyatomic molecules from smaller polar molecules, which is a crucial step towards Bose--Einstein condensation (BEC) of polyatomic molecules and towards a new crossover from a dipolar Bardeen-Cooper-Schrieffer (BCS) superfluid to a BEC of tetramers. Additionally, the long-lived FL state provides an ideal starting point for deterministic optical transfer to deeply bound tetramer states.

In quantum metrology, measurement and estimation schemes are vital for achieving higher precision, along with initial state preparation. This article presents the compound measurement of parity and particle number, which is optimal for a broad range of states named equator states (ESs). ES encompasses most pure input states used in current studies and, more significantly, a wide range of mixed states. Moreover, the ES can be prepared directly using non-demolition parity measurement. We thus propose an improved quantum phase estimation protocol applicable to arbitrary input states, ensuring precision consistently surpassing that of the standard protocol. The proposed scheme is also demonstrated using a nonlinear interferometer, with the realization of the non-demolition parity measurement in atomic condensates.