Condensed Matter Theory of Dipolar Quantum Gases

Observation of Bose-Einstein condensation of dipolar molecules

Ensembles of particles governed by quantum mechanical laws exhibit intriguing emergent behaviour. Atomic quantum gases1,2, liquid helium3,4 and electrons in quantum materials5-7 all exhibit distinct properties because of their composition and interactions. Quantum degenerate samples of ultracold dipolar molecules promise the realization of new phases of matter and new avenues for quantum simulation8 and quantum computation9. However, rapid losses10, even when reduced through collisional shielding techniques11-13, have so far prevented evaporative cooling to a Bose-Einstein condensate (BEC). Here we report on the realization of a BEC of dipolar molecules. By strongly suppressing two- and three-body losses via enhanced collisional shielding, we evaporatively cool sodium-caesium molecules to quantum degeneracy and cross the phase transition to a BEC. The BEC reveals itself by a bimodal distribution when the phase-space density exceeds 1. BECs with a condensate fraction of 60(10)% and a temperature of 6(2) nK are created and found to be stable with a lifetime close to 2 s. This work opens the door to the exploration of dipolar quantum matter in regimes that have been inaccessible so far, promising the creation of exotic dipolar droplets14, self-organized crystal phases15 and dipolar spin liquids in optical lattices16.

Efficient Creation of Ultracold Ground State ^{6}Li^{40}K Polar Molecules

We report the creation of ultracold ground state ^{6}Li^{40}K polar molecules with high efficiency. Starting from weakly bound molecules, stimulated Raman adiabatic passage is adopted to coherently transfer the molecules to their singlet rovibrational ground state |X^{1}Σ^{+},v=0,J=0⟩. By employing a singlet stimulated Raman adiabatic passage pathway and low-phase-noise narrow-linewidth lasers, we observed a one-way transfer efficiency of 96(4)%. Held in an optical dipole trap, the lifetime of the ground state molecules is measured to be 5.0(3) ms. The large permanent dipole moment of LiK is confirmed by applying a dc electric field on the molecules and performing Stark shift spectroscopy of the ground state. With recent advances in the quantum control of collisions, our work paves the way for exploring quantum many-body physics with strongly interacting ^{6}Li^{40}K molecules.

Quantum chemical calculations of electron affinities of alkaline earth metal atoms (Ca, Sr, Ba, and Ra)

We performed high-level ab initio quantum chemical calculations, incorporating higher-order excitations, spin-orbit coupling (SOC), and the Gaunt interaction, to calculate the electron affinities (EAs) of alkaline earth (AE) metal atoms (Ca, Sr, Ba, and Ra), which are notably small. The coupled-cluster singles and doubles with perturbative triples [CCSD(T)] method is insufficient to accurately calculate the EAs of AE metal atoms. Higher-order excitations proved crucial, with the coupled-cluster singles, doubles, and triples with perturbative quadruples [CCSDT(2)Q] method effectively capturing dynamic electron correlation effects. The contributions of SOC (ΔESOs) to the EAs calculated using the multireference configuration interaction method with the Davidson correction, including SOC, positively enhance the EAs; however, these contributions are overestimated. The Dirac-Hartree-Fock (DHF)-CCSD(T) method addresses this overestimation and provides reasonable values for ΔESO (ΔESO-D). Employing additional sets of diffuse and core-valence correlation basis sets is critical for accurately calculating the EAs of AE metal atoms. The contributions of the Gaunt interaction (ΔEGaunt) to the EAs of AE metal atoms are negligible. Notably, the CCSDT(2)Q with the complete basis set limit + ΔESO-D + ΔEGaunt produced EA values for Ca, Sr, and Ba that closely aligned with experimental data and achieved accuracy exceeding the chemical accuracy. Based on our findings, the accurately proposed EA for Ra is 9.88 kJ/mol.

Atomic physics on a 50-nm scale: Realization of a bilayer system of dipolar atoms

Controlling ultracold atoms with laser light has greatly advanced quantum science. The wavelength of light sets a typical length scale for most experiments to the order of 500 nanometers (nm) or greater. In this work, we implemented a super-resolution technique that localizes and arranges atoms on a sub-50-nm scale, without any fundamental limit in resolution. We demonstrate this technique by creating a bilayer of dysprosium atoms and observing dipolar interactions between two physically separated layers through interlayer sympathetic cooling and coupled collective excitations. At 50-nm distance, dipolar interactions are 1000 times stronger than at 500 nm. For two atoms in optical tweezers, this should enable purely magnetic dipolar gates with kilohertz speed.

Superlattice Quantum Solid of Dipolar Excitons

We study dipolar excitons confined at 330 mK in a square electrostatic lattice of a GaAs double quantum well. In the dipolar occupation blockade regime, at 3/2 filling, we evidence that excitons form a face-centered superlattice quantum solid. This phase is realized with high purity across 36 lattice sites, in a regime where the mean interaction energy exceeds the depth of the electrostatic lattice confinement. The superlattice solid then closely relates to Wigner crystals.

Ultracold field-linked tetratomic molecules

Ultracold polyatomic molecules offer opportunities1 in cold chemistry2,3, precision measurements4 and quantum information processing5,6, because of their rich internal structure. However, their increased complexity compared with diatomic molecules presents a challenge in using conventional cooling techniques. Here we demonstrate an approach to create weakly bound ultracold polyatomic molecules by electroassociation7 (F.D. et al., manuscript in preparation) in a degenerate Fermi gas of microwave-dressed polar molecules through a field-linked resonance8-11. Starting from ground-state NaK molecules, we create around 1.1 × 103 weakly bound tetratomic (NaK)2 molecules, with a phase space density of 0.040(3) at a temperature of 134(3) nK, more than 3,000 times colder than previously realized tetratomic molecules12. We observe a maximum tetramer lifetime of 8(2) ms in free space without a notable change in the presence of an optical dipole trap, indicating that these tetramers are collisionally stable. Moreover, we directly image the dissociated tetramers through microwave-field modulation to probe the anisotropy of their wavefunction in momentum space. Our result demonstrates a universal tool for assembling weakly bound ultracold polyatomic molecules from smaller polar molecules, which is a crucial step towards Bose-Einstein condensation of polyatomic molecules and towards a new crossover from a dipolar Bardeen-Cooper-Schrieffer superfluid13-15 to a Bose-Einstein condensation of tetramers. Moreover, the long-lived field-linked state provides an ideal starting point for deterministic optical transfer to deeply bound tetramer states16-18.

Highly efficient creation and detection of deeply bound molecules via invariant-based inverse engineering with feasible modified drivings

Stimulated Raman Adiabatic Passage (STIRAP) and its variants, such as M-type chainwise-STIRAP, allow for efficiently transferring the populations in a multilevel system and have widely been used to prepare molecules in their rovibrational ground state. However, their transfer efficiencies are generally imperfect. The main obstacle is the presence of losses and the requirement to make the dynamics adiabatic. To this end, in the present paper, a new theoretical method is proposed for the efficient and robust creation and detection of deeply bound molecules in three-level Λ-type and five-level M-type systems via "Invariant-based shortcut-to-adiabaticity." In the regime of large detunings, we first reduce the dynamics of three- and five-level molecular systems to those of effective two- and three-level counterparts. By doing so, the major molecular losses from the excited states can be well suppressed. Consequently, the effective two-level counterpart can be directly compatible with two different "Invariant-based Inverse Engineering" protocols; the results show that both protocols give a comparable performance and have a good experimental feasibility. For the effective three-level counterpart, by considering a relation among the four incident pulses, we show that this model can be further generalized to an effective Λ-type one with the simplest resonant coupling. This generalized model permits us to borrow the "Invariant-based Inverse Engineering" protocol from a standard three-level Λ-type system to a five-level M-type system. Numerical calculations show that the weakly bound molecules can be efficiently transferred to their deeply bound states without strong laser pulses, and the stability against parameter variations is well preserved. Finally, the detection of ultracold deeply bound molecules is discussed.

Laser Scheme for Doppler Cooling of the Hydroxyl Cation (OH+)

We report on a cycling scheme for Doppler cooling of trapped OH+ ions using transitions between the electronic ground state X3Σ- and the first excited triplet state A3Π. We have identified relevant transitions for photon cycling and repumping, have found that coupling into other electronic states is strongly suppressed, and have calculated the number of photon scatterings required to cool OH+ to a temperature where Raman sideband cooling can take over. In contrast to the standard approach, where molecular ions are sympathetically cooled, our scheme does not require co-trapping of another species and opens the door to the creation of pure samples of cold molecular ions with potential applications in quantum information, quantum chemistry, and astrochemistry. The laser cooling scheme identified for OH+ is efficient despite the absence of near-diagonal Franck-Condon factors, suggesting that broader classes of molecules and molecular ions are amenable to laser cooling than commonly assumed.

Manipulating Growth and Propagation of Correlations in Dipolar Multilayers: From Pair Production to Bosonic Kitaev Models

We study the nonequilibrium dynamics of dipoles confined in multiple stacked two-dimensional layers realizing a long-range interacting quantum spin 1/2 XXX model. We demonstrate that strong in-plane interactions can protect a manifold of collective layer dynamics. This then allows us to map the many-body spin dynamics to bosonic models. In a bilayer configuration we show how to engineer the paradigmatic two-mode squeezing Hamiltonian known from quantum optics, resulting in exponential production of entangled pairs and generation of metrologically useful entanglement from initially prepared product states. In multilayer configurations we engineer a bosonic variant of the Kitaev model displaying chiral propagation along the layer direction. Our study illustrates how the control over interactions, lattice geometry, and state preparation in interacting dipolar systems uniquely afforded by AMO platforms such as Rydberg and magnetic atoms, polar molecules, or trapped ions allows for the control over the temporal and spatial propagation of correlations for applications in quantum sensing and quantum simulation.

Entanglement propagation and dynamics in non-additive quantum systems

The prominent collective character of long-range interacting quantum systems makes them promising candidates for quantum technological applications. Yet, lack of additivity overthrows the traditional picture for entanglement scaling and transport, due to the breakdown of the common mechanism based on excitations propagation and confinement. Here, we describe the dynamics of the entanglement entropy in many-body quantum systems with a diverging contribution to the internal energy from the long-range two body potential. While in the strict thermodynamic limit entanglement dynamics is shown to be suppressed, a rich mosaic of novel scaling regimes is observed at intermediate system sizes, due to the possibility to trigger multiple resonant modes in the global dynamics. Quantitative predictions on the shape and timescales of entanglement propagation are made, paving the way to the observation of these phases in current quantum simulators. This picture is connected and contrasted with the case of local many body systems subject to Floquet driving.

Temperature-Dependent Contact of Weakly Interacting Single-Component Fermi Gases and Loss Rate of Degenerate Polar Molecules

Motivated by the experimental realization of single-component degenerate Fermi gases of polar ground state KRb molecules with intrinsic two-body losses [L. De Marco et al., A degenerate Fermi gas of polar molecules, Science 363, 853 (2019).SCIEAS0036-807510.1126/science.aau7230], this work studies the finite-temperature loss rate of single-component Fermi gases with weak interactions. First, we establish a relationship between the two-body loss rate and the p-wave contact. Second, we evaluate the contact of the homogeneous system in the low-temperature regime using p-wave Fermi liquid theory and in the high-temperature regime using the second-order virial expansion. Third, conjecturing that there are no phase transitions between the two temperature regimes, we smoothly interpolate the results to intermediate temperatures. It is found that the contact is constant at temperatures close to zero and increases first quadratically with increasing temperature and finally-in agreement with the Bethe-Wigner threshold law-linearly at high temperatures. Fourth, applying the local-density approximation, we obtain the loss-rate coefficient for the harmonically trapped system, reproducing the experimental KRb loss measurements within a unified theoretical framework over a wide temperature regime without fitting parameters. Our results for the contact are not only applicable to molecular p-wave gases but also to atomic single-component Fermi gases, such as ^{40}K and ^{6}Li.

Observation of Rydberg Blockade Due to the Charge-Dipole Interaction between an Atom and a Polar Molecule

We demonstrate Rydberg blockade due to the charge-dipole interaction between a single Rb atom and a single RbCs molecule confined in optical tweezers. The molecule is formed by magnetoassociation of a Rb+Cs atom pair and subsequently transferred to the rovibrational ground state with an efficiency of 91(1)%. Species-specific tweezers are used to control the separation between the atom and molecule. The charge-dipole interaction causes blockade of the transition to the Rb(52s) Rydberg state, when the atom-molecule separation is set to 310(40) nm. The observed excitation dynamics are in good agreement with simulations using calculated interaction potentials. Our results open up the prospect of a hybrid platform where quantum information is transferred between individually trapped molecules using Rydberg atoms.

Theoretical Investigation of Spectroscopic Properties of the Alkaline-Earth-Metal Monohydrides toward Laser Cooling and Magneto-Optical Trapping

Alkaline-earth-metal monohydrides MH (M = Be, Mg, Ca, Sr, Ba) have long been regarded as promising candidates toward laser cooling and trapping; however, their rich internal level structures that are amenable to magneto-optical trapping have not been completely explored. Here, we first systematically evaluated Franck-Condon factors of these alkaline-earth-metal monohydrides in the A²Π_1/2 ← X²Σ⁺ transition, exploiting three respective methods (the Morse potential, the closed-form approximation, and the Rydberg-Klein-Rees method). The effective Hamiltonian matrix was introduced for MgH, CaH, SrH, and BaH individually in order to figure out their molecular hyperfine structures of X²Σ⁺, the transition wavelengths in the vacuum, and hyperfine branching ratios of A²Π_1/2(J' = 1/2,+) ← X²Σ⁺(N = 1,-), followed by possible sideband modulation proposals to address all hyperfine manifolds. Lastly, the Zeeman energy level structures and associated magnetic g factors of the ground state X²Σ⁺(N = 1,-) were also presented. Our theoretical results here not only shed more light on the molecular spectroscopy of alkaline-earth-metal monohydrides toward laser cooling and magneto-optical trapping but also can contribute to research in molecular collisions involving few-atom molecular systems, spectral analysis in astrophysics and astrochemistry, and even precision measurement of fundamental constants such as the quest for nonzero detection of electron's electric dipole moment.

Formation of Ultracold Molecules by Merging Optical Tweezers

We demonstrate the formation of a single RbCs molecule during the merging of two optical tweezers, one containing a single Rb atom and the other a single Cs atom. Both atoms are initially predominantly in the motional ground states of their respective tweezers. We confirm molecule formation and establish the state of the molecule formed by measuring its binding energy. We find that the probability of molecule formation can be controlled by tuning the confinement of the traps during the merging process, in good agreement with coupled-channel calculations. We show that the conversion efficiency from atoms to molecules using this technique is comparable to magnetoassociation.

Effective Potential and Superfluidity of Microwave-Shielded Polar Molecules

We analytically show that the effective interaction potential between microwave-shielded polar molecules consists of an anisotropic van der Waals-like shielding core and a modified dipolar interaction. This effective potential is validated by comparing its scattering cross sections with those calculated using intermolecular potential involving all interaction channels. It is shown that a scattering resonance can be induced under microwave fields reachable in current experiments. With the effective potential, we further study the Bardeen-Cooper-Schrieffer pairing in the microwave-shielded NaK gas. We show that the superfluid critical temperature is drastically enhanced near the resonance. As the effective potential is suitable for exploring the many-body physics of molecular gases, our results pave the way for studies of the ultracold gases of microwave-shielded molecular gases.

Optical Trapping of a Polyatomic Molecule in an ℓ-Type Parity Doublet State

We report optical trapping of a polyatomic molecule, calcium monohydroxide (CaOH). CaOH molecules from a magneto-optical trap are sub-Doppler laser cooled to 20(3)  μK in free space and loaded into an optical dipole trap. We attain an in-trap molecule number density of 3(1)×10^{9}  cm^{-3} at a temperature of 57(8)  μK. Trapped CaOH molecules are optically pumped into an excited vibrational bending mode, whose ℓ-type parity doublet structure is a potential resource for a wide range of proposed quantum science applications with polyatomic molecules. We measure the spontaneous, radiative lifetime of this bending mode state to be ∼0.7  s.

Globally Accurate Gaussian Process Potential Energy Surface and Quantum Dynamics Studies on the Li(2S) + Na2 → LiNa + Na Reaction at Low Collision Energies

The LiNa2 reactive system has recently received great attention in the experimental study of ultracold chemical reactions, but the corresponding theoretical calculations have not been carried out. Here, we report the first globally accurate ground-state LiNa2 potential energy surface (PES) using a Gaussian process model based on only 1776 actively selected high-level ab initio training points. The constructed PES had high precision and strong generalization capability. On the new PES, the quantum dynamics calculations on the Li(2S) + Na2(v = 0, j = 0) → LiNa + Na reaction were carried out in the 0.001–0.01 eV collision energy range using an improved time-dependent wave packet method. The calculated results indicate that this reaction is dominated by a complex-forming mechanism at low collision energies. The presented dynamics data provide guidance for experimental research, and the newly constructed PES could be further used for ultracold reaction dynamics calculations on this reactive system.

Ultracold Gas of Dipolar NaCs Ground State Molecules

We report on the creation of bosonic NaCs molecules in their absolute rovibrational ground state via stimulated Raman adiabatic passage. We create ultracold gases with up to 22 000 dipolar NaCs molecules at a temperature of 300(50) nK and a peak density of 1.0(4)×10^{12}  cm^{-3}. We demonstrate comprehensive quantum state control by preparing the molecules in a specific electronic, vibrational, rotational, and hyperfine state. We measure the ground state ac polarizability at 1064 nm along with the two-body loss rate, which we find to be universal. Employing the tunability and strength of the permanent electric dipole moment of NaCs, we induce dipole moments of up to 2.6 D at a dc electric field of 2.1(2)  kV/cm and demonstrate strong microwave coupling between the two lowest rotational states with a Rabi frequency of 2π×45  MHz. A large electric dipole moment, accessible at relatively small electric fields, makes ultracold gases of NaCs molecules well suited for the exploration of strongly interacting phases of dipolar quantum matter.

Raman laser induced self-organization with topology in a dipolar condensate

We investigate the ground states of a dipolar Bose-Einstein condensate (BEC) subject to Raman laser induced spin-orbit coupling with mean-field theory. Owing to the interplay between spin-orbit coupling and atom-atom interactions, the BEC presents remarkable self-organization behavior and thus hosts various exotic phases including vortex with discrete rotational symmetry, stripe with spin helix, and chiral lattices with C₄ symmetry. The peculiar chiral self-organized array of square lattice, which spontaneously breaks both U(1) and rotational symmetries, is observed when the contact interaction is considerable in comparison with the spin-orbit coupling. Moreover, we show that the Raman-induced spin-orbit coupling plays a crucial role in forming rich topological spin textures of the chiral self-organized phases by introducing a channel for atoms to turn on spin flipping between two components. The self-organization phenomena predicted here feature topology owing to spin-orbit coupling. In addition, we find long-lived metastable self-organized arrays with C₆ symmetry in the case of strong spin-orbit coupling. We also present a proposal to observe these predicted phases in ultracold atomic dipolar gases with laser-induced spin-orbit coupling, which may stimulate broad theoretical as well as experimental interest.

Field-linked resonances of polar molecules

Scattering resonances are an essential tool for controlling the interactions of ultracold atoms and molecules. However, conventional Feshbach scattering resonances¹, which have been extensively studied in various platforms^1-7, are not expected to exist in most ultracold polar molecules because of the fast loss that occurs when two molecules approach at a close distance^8-10. Here we demonstrate a new type of scattering resonance that is universal for a wide range of polar molecules. The so-called field-linked resonances^11-14 occur in the scattering of microwave-dressed molecules because of stable macroscopic tetramer states in the intermolecular potential. We identify two resonances between ultracold ground-state sodium-potassium molecules and use the microwave frequencies and polarizations to tune the inelastic collision rate by three orders of magnitude, from the unitary limit to well below the universal regime. The field-linked resonance provides a tuning knob to independently control the elastic contact interaction and the dipole-dipole interaction, which we observe as a modification in the thermalization rate. Our result provides a general strategy for resonant scattering between ultracold polar molecules, which paves the way for realizing dipolar superfluids¹⁵ and molecular supersolids¹⁶, as well as assembling ultracold polyatomic molecules.

Dual-density waves with neutral and charged dipolar excitons of GaAs bilayers

Strongly correlated quantum particles in lattice potentials are the building blocks for a wide variety of quantum insulators-for instance, Mott phases and density waves breaking lattice symmetry^1-3. Such collective states are accessible to bosonic and fermionic systems^2,4-10,11,12. To expand further the spectrum of accessible quantum matter phases, mixing both species is theoretically appealing because density order then competes with phase separation^13-16. Here we manipulate such a Bose-Fermi mixture by confining neutral (boson-like) and charged (fermion-like) dipolar excitons in an artificial square lattice of a GaAs bilayer. At unitary lattice filling, strong inter- and intraspecies interactions stabilize insulating phases when the fraction of charged excitons is around (1/3, 1/2, 2/3). We evidence that dual Bose-Fermi density waves are then realized, with species ordered in alternating stripes. Our observations highlight that dipolar excitons allow for controlled implementations of Bose-Fermi Hubbard models extended by off-site interactions.

Probing site-resolved correlations in a spin system of ultracold molecules

Synthetic quantum systems with interacting constituents play an important role in quantum information processing and in explaining fundamental phenomena in many-body physics. Following impressive advances in cooling and trapping techniques, ensembles of ultracold polar molecules have emerged as a promising platform that combines several advantageous properties^1-11. These include a large set of internal states with long coherence times^12-17 and long-range, anisotropic interactions. These features could enable the exploration of intriguing phases of correlated quantum matter, such as topological superfluids¹⁸, quantum spin liquids¹⁹, fractional Chern insulators²⁰ and quantum magnets^21,22. Probing correlations in these phases is crucial to understanding their properties, necessitating the development of new experimental techniques. Here we use quantum gas microscopy²³ to measure the site-resolved dynamics of quantum correlations of polar ²³Na⁸⁷Rb molecules confined in a two-dimensional optical lattice. By using two rotational states of the molecules, we realize a spin-1/2 system with dipolar interactions between particles, producing a quantum spin-exchange model^21,22,24,25. We study the evolution of correlations during the thermalization process of an out-of-equilibrium spin system for both spatially isotropic and anisotropic interactions. Furthermore, we examine the correlation dynamics of a spin-anisotropic Heisenberg model engineered from the native spin-exchange model by using periodic microwave pulses^26-28. These experiments push the frontier of probing and controlling interacting systems of ultracold molecules, with prospects for exploring new regimes of quantum matter and characterizing entangled states that are useful for quantum computation^29,30 and metrology³¹.

Two-dimensional electronic spectroscopy of an ultracold gas

Femtosecond coherent multidimensional spectroscopy is demonstrated for an ultracold gas.  A setup for phase modulation spectroscopy is used to probe the 3²S_1/2-2²P_1/2,3/2 transition in an 800μK-cold sample of ⁷Li atoms confined in a magneto-optical trap. The observation of a double quantum coherence response, a signature of interparticle interactions, paves the way for detailed investigations of few- and many-body effects in ultracold gases using this technique. The experiment combines a frequency resolution of 3 GHz with a potential time resolution of 200 fs, which allows for high-resolution studies of ultracold atoms and molecules both in the frequency and in the time domain.

Ultracold Sticky Collisions: Theoretical and Experimental Status

Collisional complexes, which are formed as intermediate states in molecular collisions, are typically short-lived and decay within picoseconds. However, in ultracold collisions involving bialkali molecules, complexes can live for milliseconds, completely changing the collision dynamics. This can lead to unexpected two-body loss in samples of nonreactive molecules. During the past decade, such "sticky" collisions have been a major hindrance in the preparation of dense and stable molecular samples, especially in the quantum-degenerate regime. Currently, the behavior of the complexes is not fully understood. For example, in some cases, their lifetime has been measured to be many orders of magnitude longer than recent models predict. This is not only an intriguing problem in itself but also practically relevant, since understanding molecular complexes may help to mitigate their detrimental effects. Here, we review the recent experimental and theoretical progress in this field. We treat the case of molecule-molecule as well as molecule-atom collisions.

Dipolar physics: a review of experiments with magnetic quantum gases

Since the achievement of quantum degeneracy in gases of chromium atoms in 2004, the experimental investigation of ultracold gases made of highly magnetic atoms has blossomed. The field has yielded the observation of many unprecedented phenomena, in particular those in which long-range and anisotropic dipole-dipole interactions (DDIs) play a crucial role. In this review, we aim to present the aspects of the magnetic quantum-gas platform that make it unique for exploring ultracold and quantum physics as well as to give a thorough overview of experimental achievements. Highly magnetic atoms distinguish themselves by the fact that their electronic ground-state configuration possesses a large electronic total angular momentum. This results in a large magnetic moment and a rich electronic transition spectrum. Such transitions are useful for cooling, trapping, and manipulating these atoms. The complex atomic structure and large dipolar moments of these atoms also lead to a dense spectrum of resonances in their two-body scattering behaviour. These resonances can be used to control the interatomic interactions and, in particular, the relative importance of contact over dipolar interactions. These features provide exquisite control knobs for exploring the few- and many-body physics of dipolar quantum gases. The study of dipolar effects in magnetic quantum gases has covered various few-body phenomena that are based on elastic and inelastic anisotropic scattering. Various many-body effects have also been demonstrated. These affect both the shape, stability, dynamics, and excitations of fully polarised repulsive Bose or Fermi gases. Beyond the mean-field instability, strong dipolar interactions competing with slightly weaker contact interactions between magnetic bosons yield new quantum-stabilised states, among which are self-bound droplets, droplet assemblies, and supersolids. Dipolar interactions also deeply affect the physics of atomic gases with an internal degree of freedom as these interactions intrinsically couple spin and atomic motion. Finally, long-range dipolar interactions can stabilise strongly correlated excited states of 1D gases and also impact the physics of lattice-confined systems, both at the spin-polarised level (Hubbard models with off-site interactions) and at the spinful level (XYZ models). In the present manuscript, we aim to provide an extensive overview of the various related experimental achievements up to the present.

Evaporation of microwave-shielded polar molecules to quantum degeneracy

Ultracold polar molecules offer strong electric dipole moments and rich internal structure, which makes them ideal building blocks to explore exotic quantum matter^1–9, implement quantum information schemes^10–12 and test the fundamental symmetries of nature¹³. Realizing their full potential requires cooling interacting molecular gases deeply into the quantum-degenerate regime. However, the intrinsically unstable collisions between molecules at short range have so far prevented direct cooling through elastic collisions to quantum degeneracy in three dimensions. Here we demonstrate evaporative cooling of a three-dimensional gas of fermionic sodium–potassium molecules to well below the Fermi temperature using microwave shielding. The molecules are protected from reaching short range with a repulsive barrier engineered by coupling rotational states with a blue-detuned circularly polarized microwave. The microwave dressing induces strong tunable dipolar interactions between the molecules, leading to high elastic collision rates that can exceed the inelastic ones by at least a factor of 460. This large elastic-to-inelastic collision ratio allows us to cool the molecular gas to 21 nanokelvin, corresponding to 0.36 times the Fermi temperature. Such cold and dense samples of polar molecules open the path to the exploration of many-body phenomena with strong dipolar interactions.

A general and efficient approach to evaporatively cool ultracold polar molecules through elastic collisions to create a degenerate quantum gas in three dimensions is demonstrated using microwave shielding.

Density Matrix Renormalization Group for Continuous Quantum Systems

We introduce a versatile and practical framework for applying matrix product state techniques to continuous quantum systems. We divide space into multiple segments and generate continuous basis functions for the many-body state in each segment. By combining this mapping with existing numerical density matrix renormalization group routines, we show how one can accurately obtain the ground-state wave function, spatial correlations, and spatial entanglement entropy directly in the continuum. For a prototypical mesoscopic system of strongly interacting bosons we demonstrate faster convergence than standard grid-based discretization. We illustrate the power of our approach by studying a superfluid-insulator transition in an external potential. We outline how one can directly apply or generalize this technique to a wide variety of experimentally relevant problems across condensed matter physics and quantum field theory.

Electric-Field-Controlled Cold Dipolar Collisions between Trapped CH_{3}F Molecules

Reaching high densities is a key step toward cold-collision experiments with polyatomic molecules. We use a cryofuge to load up to 2×10^{7} CH_{3}F molecules into a boxlike electric trap, achieving densities up to 10^{7}/cm^{3} at temperatures around 350 mK where the elastic dipolar cross section exceeds 7×10^{-12}  cm^{2}. We measure inelastic rate constants below 4×10^{-8}  cm^{3}/s and control these by tuning a homogeneous electric field that covers a large fraction of the trap volume. Comparison to ab initio calculations gives excellent agreement with dipolar relaxation. Our techniques and findings are generic and immediately relevant for other cold-molecule collision experiments.

Reactions between layer-resolved molecules mediated by dipolar spin exchange

Microscopic control over polar molecules with tunable interactions enables the realization of distinct quantum phenomena. Using an electric field gradient, we demonstrated layer-resolved state preparation and imaging of ultracold potassium-rubidium molecules confined to two-dimensional planes in an optical lattice. The rotational coherence was maximized by rotating the electric field relative to the light polarization for state-insensitive trapping. Spatially separated molecules in adjacent layers interact through dipolar spin exchange of rotational angular momentum; by adjusting these interactions, we regulated the local chemical reaction rate. The resonance width of the exchange process vastly exceeded the dipolar interaction energy, an effect attributed to thermal energy. This work realized precise control of interacting molecules, enabling electric field microscopy on subwavelength scales and allowing access to unexplored physics in two-dimensional systems.

The effect of spin-orbit coupling on molecular properties: Potential energy curve, transition dipole moment and laser cooling scheme of NH

The influence of spin-orbit coupling on the cooling of NH molecular laser is investigated based on the ab initio theory. The potential energy curves (PECs) and spectral constants for four Λ-S states (X³Σ^-, a¹Δ, b¹Σ⁺ and A³Π) and eight Ω states [Formula: see text] a¹Δ₂, [Formula: see text] and [Formula: see text] ) of NH molecule are obtained by the multi-reference configuration interaction (MRCI) plus Davidson correction. The spectroscopic constants (R_e, ω_e, ω_eχ_e, B_e, D_e) are obtained by solving the one dimensional radial Schrödinger equation, and the results are almost the same as the previously reported data. In addition, the transition dipole moment, permanent dipole moment, Franck-Condon factors, and radiative lifetime of NH molecule are also acquired. Also, the effects of the intermediate state on the [Formula: see text] , [Formula: see text] and [Formula: see text] transitions are considered. The feasible laser cooling schemes using a single laser are formulated. In the proposed cooling scheme, there wavelengths for the [Formula: see text] are used, the main pump lasers are λ₀₀ = 335.74 nm. The feasible transition is based on this. It is found that spin-orbit coupling has a significant effect on potential energy curves, permanent dipole moments, transition dipole moments and vibration energy levels.

Empirical LiK excited state potentials: connecting short range and near dissociation expansions

We report on a high-resolution spectroscopic survey of ⁶Li⁴⁰K molecules near the 2S + 4P dissociation threshold and produce a fully empirical representation for the B¹Π potential by connecting available short- and long-range data. The purpose is to identify a suitable intermediate state for a coherent Raman transfer to the absolute ground state, and the creation of a molecular gas with dipolar interactions. Starting from weakly bound ultracold Feshbach molecules, the transition frequencies to twenty-six vibrational states are determined. Our data are combined with long-range measurements [Ridinger et al., EPL, 2011, 96, 33001], and near-dissociation expansions for the spin-orbit coupled potentials are fitted to extract the van der Waals C₆ dispersion coefficients. A suitable vibrational level is identified by resolving its Zeeman structure and by comparing the experimentally attained g-factor to our theoretical prediction. Using mass-scaling of the short-range data for the B¹Π [Pashov et al., Chem. Phys. Lett., 1998, 292, 615620] and an updated value for its depth, we model the short- and the long-range data simultaneously and produce a Rydberg-Klein-Rees curve covering the entire range.

Kink Dynamics and Quantum Simulation of Supersymmetric Lattice Hamiltonians

We propose a quantum simulation of a supersymmetric lattice model using atoms trapped in a 1D configuration and interacting through a Rydberg dressed potential. The elementary excitations in the model are kinks or (in a sector with one extra particle) their superpartners-the skinks. The two are connected by supersymmetry and display identical quantum dynamics. We provide an analytical description of the kink (skink) quench dynamics and propose a protocol to prepare and detect these excitations in the quantum simulator. We make a detailed analysis, based on numerical simulation, of the Rydberg atom simulator and show that it accurately tracks the dynamics of the supersymmetric model.

Hilbert Space Shattering and Disorder-Free Localization in Polar Lattice Gases

Emerging dynamical constraints resulting from intersite interactions severely limit particle mobility in polar lattice gases. Whereas in absence of disorder hard-core Hubbard models with only strong nearest-neighbor interactions present Hilbert space fragmentation but no many-body localization for typical states, the 1/r^{3} tail of the dipolar interaction results in Hilbert space shattering, as well as in a dramatically slowed down dynamics and eventual disorder-free localization. Our results show that the study of the intriguing interplay between disorder- and interaction-induced many-body localization is within reach of future experiments with magnetic atoms and polar molecules.

Berezinskii-Kosterlitz-Thouless Phase Transitions with Long-Range Couplings

The Berezinskii-Kosterlitz-Thouless (BKT) transition is the paradigmatic example of a topological phase transition without symmetry breaking, where a quasiordered phase, characterized by a power-law scaling of the correlation functions at low temperature, is disrupted by the proliferation of topological excitations above the critical temperature T_{BKT}. In this Letter, we consider the effect of long-range decaying couplings ∼r^{-2-σ} on the BKT transition. After pointing out the relevance of this nontrivial problem, we discuss the phase diagram, which is far richer than the corresponding short-range one. It features-for 7/4<σ<2-a quasiordered phase in a finite temperature range T_{c}<T<T_{BKT}, which occurs between a symmetry broken phase for T<T_{c} and a disordered phase for T>T_{BKT}. The transition temperature T_{c} displays unique universal features quite different from those of the traditional, short-range XY model. Given the universal nature of our findings, they may be observed in current experimental realizations in 2D atomic, molecular, and optical quantum systems.

Strongly correlated excitonic insulator in atomic double layers

Excitonic insulators (EIs) arise from the formation of bound electron-hole pairs (excitons)^1,2 in semiconductors and provide a solid-state platform for quantum many-boson physics^3-8. Strong exciton-exciton repulsion is expected to stabilize condensed superfluid and crystalline phases by suppressing both density and phase fluctuations^8-11. Although spectroscopic signatures of EIs have been reported^6,12-14, conclusive evidence for strongly correlated EI states has remained elusive. Here we demonstrate a strongly correlated two-dimensional (2D) EI ground state formed in transition metal dichalcogenide (TMD) semiconductor double layers. A quasi-equilibrium spatially indirect exciton fluid is created when the bias voltage applied between the two electrically isolated TMD layers is tuned to a range that populates bound electron-hole pairs, but not free electrons or holes^15-17. Capacitance measurements show that the fluid is exciton-compressible but charge-incompressible-direct thermodynamic evidence of the EI. The fluid is also strongly correlated with a dimensionless exciton coupling constant exceeding 10. We construct an exciton phase diagram that reveals both the Mott transition and interaction-stabilized quasi-condensation. Our experiment paves the path for realizing exotic quantum phases of excitons⁸, as well as multi-terminal exciton circuitry for applications^18-20.

Metastability and discrete spectrum of long-range systems

Long-lived quasi-stationary states (QSSs) are a signature characteristic of long-range interacting systems both in the classical and in the quantum realms. Often, they emerge after a sudden quench of the Hamiltonian internal parameters and present a macroscopic lifetime, which increases with the system size. Despite their ubiquity, the fundamental mechanism at their root remains unknown. Here, we show that the spectrum of systems with power-law decaying couplings remains discrete up to the thermodynamic limit. As a consequence, several traditional results on the chaotic nature of the spectrum in many-body quantum systems are not satisfied in the presence of long-range interactions. In particular, the existence of QSSs may be traced back to the finiteness of Poincaré recurrence times. This picture justifies and extends known results on the anomalous magnetization dynamics in the quantum Ising model with power-law decaying couplings. The comparison between the discrete spectrum of long-range systems and more conventional examples of pure point spectra in the disordered case is also discussed.

Theoretical investigation of laser cooling for BN- anion by ab inito calculation

A theoretical investigation for the feasibility of laser cooling BN^-anion is presented. An ab initio calculation on the three low-lying states Χ²Σ⁺, Α²Π and Β²Σ⁺ are performed at the CASSCF/MRCI + Q level. The calculated spectroscopic constants are in good agreement with the available theoretical and experimental data. Radiative properties including Franck-Condon factor, Einstein coefficients and radiative lifetimes are determined. The calculation shows that the transition B²Σ⁺(v^')↔X²Σ⁺(v^'') has highly diagonal FCFs, especially f₀₀ = 0.9898, and enough short radiative lifetimes. A cooling scheme by three laser beams is proposed, which requires one main pumping laser(λ₀₀ = 474.67 nm) and two repumping lasers (λ₀₁ = 514.64 nm, λ₁₂= 514.90 nm). The population dynamics of cooling is investigated with the rate equation approach. The simulation demonstrates that the population does not remain trapped within the intermediate Α²Π state. The resultant scattered photons are about2.5×10⁴, which is expected to stop BN^-anion molecule in a cryogenic beam theoretically.

Assembly of a Rovibrational Ground State Molecule in an Optical Tweezer

We demonstrate the coherent creation of a single NaCs molecule in its rotational, vibrational, and electronic (rovibronic) ground state in an optical tweezer. Starting with a weakly bound Feshbach molecule, we locate a two-photon transition via the |c^{3}Σ_{1},v^{'}=26⟩ excited state and drive coherent Rabi oscillations between the Feshbach state and a single hyperfine level of the NaCs rovibronic ground state |X^{1}Σ,v^{''}=0,N^{''}=0⟩ with a binding energy of D_{0}=h×147044.63(11)  GHz. We measure a lifetime of 3.4±1.6  s for the rovibronic ground state molecule, which possesses a large molecule-frame dipole moment of 4.6D and occupies predominantly the motional ground state. These long-lived, fully quantum-state-controlled individual dipolar molecules provide a key resource for molecule-based quantum simulation and information processing.

Dynamical Generation of Spin Squeezing in Ultracold Dipolar Molecules

We study a bulk fermionic dipolar molecular gas in the quantum degenerate regime confined in a two-dimensional geometry. Using two rotational states of the molecules, we encode a spin 1/2 degree of freedom. To describe the many-body spin dynamics of the molecules, we derive a long-range interacting XXZ model valid in the regime where motional degrees of freedom are frozen. Because of the spatially extended nature of the harmonic oscillator modes, the interactions in the spin model are very long ranged, and the system behaves close to the collective limit, resulting in robust dynamics and generation of entanglement in the form of spin squeezing even at finite temperature and in the presence of dephasing and chemical reactions. We discuss how the internal state structure can be exploited to realize time reversal and enhanced metrological sensing protocols.

Resonant collisional shielding of reactive molecules using electric fields

Full control of molecular interactions, including reactive losses, would open new frontiers in quantum science. We demonstrate extreme tunability of ultracold chemical reaction rates by inducing resonant dipolar interactions by means of an external electric field. We prepared fermionic potassium-rubidium molecules in their first excited rotational state and observed a modulation of the chemical reaction rate by three orders of magnitude as we tuned the electric field strength by a few percent across resonance. In a quasi-two-dimensional geometry, we accurately determined the contributions from the three dominant angular momentum projections of the collisions. Using the resonant features, we shielded the molecules from loss and suppressed the reaction rate by an order of magnitude below the background value, thereby realizing a long-lived sample of polar molecules in large electric fields.

Entanglement via rotational blockade of MgF molecules in a magic potential

Diatomic polar molecules are one of the most promising platforms of quantum computing due to their rich internal states and large electric dipole moments. Here, we propose entangling rotational states of MgF molecules in an optical tweezer array via strong electric dipole-dipole interactions. We employ two rotational states with the projection quantum number of the total angular momentum MF = 0 to maximize the dipole-dipole interaction with a given separation distance. The splitting of 1.27 kHz between two entangled states is predicted for MgF molecules separated by 1 μm. The resolution of the entangled states can be achieved in a magic optical potential where the rotational states have the same trap frequencies. The magic potential can be formed by tuning the angle between the molecules' quantization axis and the linear polarization of trapping light to a "magic angle". We calculate the magic angle for MgF molecules under reasonable experimental conditions and obtain that the trap frequencies of the two involved states can be matched within a few 10s of Hz. By establishing an entanglement scheme for the molecules, our results provide a first step towards quantum computing using MgF molecules.

New states of matter with fine-tuned interactions: quantum droplets and dipolar supersolids

Quantum fluctuations can stabilize Bose-Einstein condensates (BEC) against the mean-field collapse. Stabilization of the condensate has been observed in quantum degenerate Bose-Bose mixtures and dipolar BECs. The fine-tuning of the interatomic interactions can lead to the emergence of two new states of matter: liquid-like self-bound quantum droplets and supersolid crystals formed from these droplets. We review the properties of these exotic states of matter and summarize the experimental progress made using dipolar quantum gases and Bose-Bose mixtures. We conclude with an outline of important open questions that could be addressed in the future.

Magnetic Dipolar Interaction between Hyperfine Clock States in a Planar Alkali Bose Gas

In atomic systems, clock states feature a zero projection of the total angular momentum and thus a low sensitivity to magnetic fields. This makes them widely used for metrological applications like atomic fountains or gravimeters. Here, we show that a mixture of two such nonmagnetic states still displays magnetic dipole-dipole interactions comparable to the one expected for the other Zeeman states of the same atomic species. Using high-resolution spectroscopy of a planar gas of ^{87}Rb atoms with a controlled in plane shape, we explore the effective isotropic and extensive character of these interactions and demonstrate their tunability. Our measurements set strong constraints on the relative values of the s-wave scattering lengths a_{ij} involving the two clock states.

Mott-Glass Phase of a One-Dimensional Quantum Fluid with Long-Range Interactions

We investigate the ground-state properties of quantum particles interacting via a long-range repulsive potential V_{σ}(x)∼1/|x|^{1+σ} (-1<σ) or V_{σ}(x)∼-|x|^{-1-σ} (-2≤σ<-1) that interpolates between the Coulomb potential V_{0}(x) and the linearly confining potential V_{-2}(x) of the Schwinger model. In the absence of disorder the ground state is a Wigner crystal when σ≤0. Using bosonization and the nonperturbative functional renormalization group we show that any amount of disorder suppresses the Wigner crystallization when -3/2<σ≤0; the ground state is then a Mott glass, i.e., a state that has a vanishing compressibility and a gapless optical conductivity. For σ<-3/2 the ground state remains a Wigner crystal.

Dipolar evaporation of reactive molecules to below the Fermi temperature

The control of molecules is key to the investigation of quantum phases, in which rich degrees of freedom can be used to encode information and strong interactions can be precisely tuned1. Inelastic losses in molecular collisions2-5, however, have greatly hampered the engineering of low-entropy molecular systems6. So far, the only quantum degenerate gas of molecules has been created via association of two highly degenerate atomic gases7,8. Here we use an external electric field along with optical lattice confinement to create a two-dimensional Fermi gas of spin-polarized potassium-rubidium (KRb) polar molecules, in which elastic, tunable dipolar interactions dominate over all inelastic processes. Direct thermalization among the molecules in the trap leads to efficient dipolar evaporative cooling, yielding a rapid increase in phase-space density. At the onset of quantum degeneracy, we observe the effects of Fermi statistics on the thermodynamics of the molecular gas. These results demonstrate a general strategy for achieving quantum degeneracy in dipolar molecular gases in which strong, long-range and anisotropic dipolar interactions can drive the emergence of exotic many-body phases, such as interlayer pairing and p-wave superfluidity.

Coherently forming a single molecule in an optical trap

Ultracold single molecules have wide-ranging potential applications, such as ultracold chemistry, precision measurements, quantum simulation, and quantum computation. However, given the difficulty of achieving full control of a complex atom-molecule system, the coherent formation of single molecules remains a challenge. Here, we report an alternative route to coherently bind two atoms into a weakly bound molecule at megahertz levels by coupling atomic spins to their two-body relative motion in a strongly focused laser with inherent polarization gradients. The coherent nature is demonstrated by long-lived atom-molecule Rabi oscillations. We further manipulate the motional levels of the molecules and measure the binding energy precisely. This work opens the door to full control of all degrees of freedom in atom-molecule systems.

Tune-Out and Magic Wavelengths for Ground-State ^{23}Na^{40}K Molecules

We demonstrate a versatile, state-dependent trapping scheme for the ground and first excited rotational states of ^{23}Na^{40}K molecules. Close to the rotational manifold of a narrow electronic transition, we determine tune-out frequencies where the polarizability of one state vanishes while the other remains finite, and a magic frequency where both states experience equal polarizability. The proximity of these frequencies of only 10 GHz allows for dynamic switching between different trap configurations in a single experiment, while still maintaining sufficiently low scattering rates.

Forming a Single Molecule by Magnetoassociation in an Optical Tweezer

We demonstrate the formation of a single NaCs molecule in an optical tweezer by magnetoassociation through an s-wave Feshbach resonance at 864.11(5) G. Starting from single atoms cooled to their motional ground states, we achieve conversion efficiencies of 47(1)%, and measure a molecular lifetime of 4.7(7) ms. By construction, the single molecules are predominantly [77(5)%] in the center-of-mass motional ground state of the tweezer. Furthermore, we produce a single p-wave molecule near 807 G by first preparing one of the atoms with one quantum of motional excitation. Our creation of a single weakly bound molecule in a designated internal state in the motional ground state of an optical tweezer is a crucial step towards coherent control of single molecules in optical tweezer arrays.