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.

Scheme for Deterministic Loading of Laser-Cooled Molecules into Optical Tweezers.

We propose to repeatedly load laser-cooled molecules into optical tweezers, and transfer them to storage states that are rotationally excited by two additional quanta. Collisional loss of molecules in these storage states is suppressed, and a dipolar blockade prevents the accumulation of more than one molecule. Applying three cycles loads tweezers with single molecules at an 80% success rate, limited by residual collisional loss. This improved loading efficiency reduces the time needed for rearrangement of tweezer arrays, which would otherwise limit the scalability of neutral molecule quantum computers.

Quantum state-resolved molecular dipolar collisions over four decades of energy.

Collisions between cold polar molecules represent a fascinating research frontier but have proven hard to probe experimentally. We report measurements of inelastic cross sections for collisions between nitric oxide (NO) and deuterated ammonia (ND3) molecules at energies between 0.1 and 580 centimeter-1, with full quantum state resolution. At energies below the ~100-centimeter-1 well depth of the interaction potential, we observed backward glories originating from peculiar U-turn trajectories. At energies below 0.2 centimeter-1, we observed a breakdown of the Langevin capture model, which we interpreted in terms of a suppressed mutual polarization during the collision, effectively switching off the molecular dipole moments. Scattering calculations based on an ab initio NO-ND3 potential energy surface revealed the crucial role of near-degenerate rotational levels with opposite parity in low-energy dipolar collisions.

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 platforms1-7, are not expected to exist in most ultracold polar molecules because of the fast loss that occurs when two molecules approach at a close distance8-10. Here we demonstrate a new type of scattering resonance that is universal for a wide range of polar molecules. The so-called field-linked resonances11-14 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.

Resonances in Non-universal Dipolar Collisions.

Scattering resonances due to the dipole-dipole interaction between ultracold molecules, induced by static or microwave fields, are studied theoretically. We develop a method for coupled-channel calculations that can efficiently impose many short-range boundary conditions, defined by a short-range phase shift and loss probability as in quantum defect theory. We study how resonances appear as the short-range loss probability is lowered below the universal unit probability. This may become realizable for nonreactive ultracold molecules in blue-detuned box potentials.

Symmetry Breaking in Sticky Collisions between Ultracold Molecules.

Ultracold molecules undergo "sticky collisions" that result in loss even for chemically nonreactive molecules. Sticking times can be enhanced by orders of magnitude by interactions that lead to nonconservation of nuclear spin or total angular momentum. We present a quantitative theory of the required strength of such symmetry-breaking interactions based on classical simulation of collision complexes. We find static electric fields as small as 10 V/cm can lead to nonconservation of angular momentum, while we find nuclear spin is conserved during collisions. We also compute loss of collision complexes due to spontaneous emission and absorption of black-body radiation, which are found to be slow.

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 matter1-9, implement quantum information schemes10-12 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.

Correlated rotational excitations in NO-CO inelastic collisions.

We present a joint experimental and theoretical study of rotationally inelastic collisions between NO (X2Π1/2, ν = 0, j = 1/2, f) radicals and CO (X1Σ+, ν = 0, j = 0) molecules at a collision energy of 220 cm-1. State-to-state scattering images for excitation of NO radicals into various final states were measured with high resolution by combining the Stark deceleration and velocity map imaging techniques. The high image resolution afforded the observation of correlated rotational excitations of NO-CO pairs, which revealed a number of striking scattering phenomena. The so-called "parity-pair" transitions in NO are found to have similar differential cross sections, independent of the concurrent excitation of CO, extending this well-known effect for collisions between NO and rare gas atoms into the realm of bimolecular collisions. Forward scattering is found for collisions that induce a large amount of rotational energy transfer (in either NO, CO, or both), which require low impact parameters to induce sufficient energy transfer. This observation is interpreted in terms of the recently discovered hard collision glory scattering mechanism, which predicts the forward bending of initially backward receding trajectories if the energy uptake in the collision is substantial in relation to the collision energy. The experimental results are in good agreement with the predictions from coupled-channels quantum scattering calculations based on an ab initio NO-CO potential energy surface.

Control of reactive collisions by quantum interference.

In this study, we achieved magnetic control of reactive scattering in an ultracold mixture of 23Na atoms and 23Na6Li molecules. In most molecular collisions, particles react or are lost near short range with unity probability, leading to the so-called universal rate. By contrast, the Na + NaLi system was shown to have only ~4% loss probability in a fully spin-polarized state. By controlling the phase of the scattering wave function via a Feshbach resonance, we modified the loss rate by more than a factor of 100, from far below to far above the universal limit. The results are explained in analogy with an optical Fabry-Perot resonator by interference of reflections at short and long range. Our work demonstrates quantum control of chemistry by magnetic fields with the full dynamic range predicted by our models.

Glory scattering in deeply inelastic molecular collisions.

For molecular collisions, the deflection of a molecule's trajectory provides one of the most sensitive probes of the interaction potential and there are general rules of thumb that relate the direction of deflection to precollision conditions. Following intuition, forward scattering results from glancing collisions, whereas near head-on collisions result in back scattering. Here we present the observation of forward scattering in inelastic processes that defies this common wisdom. For deeply inelastic collisions between NO radicals and CO or HD molecules, we observed forward scattering in fully resolved pair-correlated differential cross-sections, despite the low impact parameters that are needed to induce a sufficient energy transfer. We rationalized these findings by extending the textbook model of hard-sphere scattering-taking inelastic energy transfer into account-and attribute the forward scattering to glory-type trajectories caused by attractive forces. This phenomenon, which we refer to as hard-collision glory scattering, is predicted to be ubiquitous. We derive under which conditions hard-collision glory scattering occurs and retrospectively identify such behaviour in previously studied systems.

Observation of microwave shielding of ultracold molecules.

Harnessing the potential wide-ranging quantum science applications of molecules will require control of their interactions. Here, we used microwave radiation to directly engineer and tune the interaction potentials between ultracold calcium monofluoride (CaF) molecules. By merging two optical tweezers, each containing a single molecule, we probed collisions in three dimensions. The correct combination of microwave frequency and power created an effective repulsive shield, which suppressed the inelastic loss rate by a factor of six, in agreement with theoretical calculations. The demonstrated microwave shielding shows a general route to the creation of long-lived, dense samples of ultracold polar molecules and evaporative cooling.

Resonant Dipolar Collisions of Ultracold Molecules Induced by Microwave Dressing.

We demonstrate microwave dressing on ultracold, fermionic ^{23}Na^{40}K ground-state molecules and observe resonant dipolar collisions with cross sections exceeding 3 times the s-wave unitarity limit. The origin of these interactions is the resonant alignment of the approaching molecules' dipoles along the intermolecular axis, which leads to strong attraction. We explain our observations with a conceptually simple two-state picture based on the Condon approximation. Furthermore, we perform coupled-channel calculations that agree well with the experimentally observed collision rates. The resonant microwave-induced collisions found here enable controlled, strong interactions between molecules, of immediate use for experiments in optical lattices.

Imaging the onset of the resonance regime in low-energy NO-He collisions.

At low energies, the quantum wave-like nature of molecular interactions results in distinctive scattering behavior, ranging from the universal Wigner laws near 0 kelvin to the occurrence of scattering resonances at higher energies. It has proven challenging to experimentally probe the individual waves underlying these phenomena. We report measurements of state-to-state integral and differential cross sections for inelastic NO-He collisions in the 0.2 to 8.5 centimeter-1 range with 0.02 centimeter-1 resolution. We studied the onset of the resonance regime by probing the lowest-lying resonance dominated by s and p waves only. The highly structured differential cross sections directly reflect the increasing number of contributing waves as the energy is increased. Only with CCSDT(Q) level of theory was it possible to reproduce our measurements.

Rotational-vibrational resonance states.

Resonance states are characterized by an energy that is above the lowest dissociation threshold of the potential energy hypersurface of the system and thus resonances have finite lifetimes. All molecules possess a large number of long- and short-lived resonance (quasibound) states. A considerable number of rotational-vibrational resonance states are accessible not only via quantum-chemical computations but also by spectroscopic and scattering experiments. In a number of chemical applications, most prominently in spectroscopy and reaction dynamics, consideration of rotational-vibrational resonance states is becoming more and more common. There are different first-principles techniques to compute and rationalize rotational-vibrational resonance states: one can perform scattering calculations or one can arrive at rovibrational resonances using variational or variational-like techniques based on methods developed for determining bound eigenstates. The latter approaches can be based either on the Hermitian (L2, square integrable) or non-Hermitian (non-L2) formalisms of quantum mechanics. This Perspective reviews the basic concepts related to and the relevance of shape and Feshbach-type rotational-vibrational resonance states, discusses theoretical methods and computational tools allowing their efficient determination, and shows numerical examples from the authors' previous studies on the identification and characterization of rotational-vibrational resonances of polyatomic molecular systems.

Photoinduced Two-Body Loss of Ultracold Molecules.

The lifetime of nonreactive ultracold bialkali gases was conjectured to be limited by sticky collisions amplifying three-body loss. We show that the sticking times were previously overestimated and do not support this hypothesis. We find that electronic excitation of NaK+NaK collision complexes by the trapping laser leads to the experimentally observed two-body loss. We calculate the excitation rate with a quasiclassical, statistical model employing ab initio potentials and transition dipole moments. Using longer laser wavelengths or repulsive box potentials may suppress the losses.

Six-dimensional potential energy surface for NaK-NaK collisions: Gaussian process representation with correct asymptotic form.

Constructing accurate global potential energy surfaces (PESs) describing chemically reactive molecule-molecule collisions of alkali metal dimers presents a major challenge. To be suitable for quantum scattering calculations, such PESs must represent accurately three- and four-body interactions, describe conical intersections, and have a proper asymptotic form at the long range. Here, we demonstrate that such global potentials can be obtained by Gaussian Process (GP) regression merged with the analytic asymptotic expansions at the long range. We propose an efficient sampling technique, which allows us to construct an accurate global PES accounting for different chemical arrangements with <2500 ab initio calculations. We apply this method to (NaK)2 and obtain the first global PES for a system of four alkali metal atoms. The resulting surface exhibits a complex landscape including a pair and a quartet of symmetrically equivalent local minima and a seam of conical intersections. The dissociation energy found from our ab initio calculations is 4534 cm-1. This result is reproduced by the GP models with an error of less than 3%. The GP models of the PES allow us to analyze the features of the global PES, representative of general alkali metal four-atom interactions. Understanding these interactions is of key importance in the field of ultracold chemistry.

Microwave Shielding of Ultracold Polar Molecules.

We use microwaves to engineer repulsive long-range interactions between ultracold polar molecules. The resulting shielding suppresses various loss mechanisms and provides large elastic cross sections. Hyperfine interactions limit the shielding under realistic conditions, but a magnetic field allows suppression of the losses to below 10^{-14} cm^{3} s^{-1}. The mechanism and optimum conditions for shielding differ substantially from those proposed by Gorshkov et al. [Phys. Rev. Lett. 101, 073201 (2008)PRLTAO0031-900710.1103/PhysRevLett.101.073201], and do not require cancellation of the long-range dipole-dipole interaction that is vital to many applications.

Energy dependent parity-pair behavior in NO + He collisions.

Colliding molecules behave fundamentally differently at high and low collision energies. At high energies, a collision can be described to a large extent using classical mechanics, and the scattering process can be compared to a billiard-ball-like collision. At low collision energies, the wave character of the collision partners dominates, and only quantum mechanics can predict the outcome of an encounter. It is, however, not so clear how these limits evolve into each other as a function of the collision energy. Here, we investigate and visualize this evolution using a special feature of the differential cross sections for inelastic collisions between NO radicals and He atoms. The so-called "parity-pair" transitions have similar differential cross sections at high collision energies, whereas their cross sections are significantly different in the quantum regime at low energies. These transitions can be used as a probe for the quantum nature of the collision process. The similarity of the parity-pair differential cross sections at high energies could be theoretically explained if the first-order Born approximation were applicable. We found, however, that the anisotropy of the NO-He interaction potential is too strong for the first-order Born approximation to be valid, so higher-order perturbations must be taken into account.

Publisher Correction: O2-O2 and O2-N2 collision-induced absorption mechanisms unravelled.

In the version of this Article originally published, Figures 3 and 4 were erroneously swapped, this has been corrected in all versions of the Article.

Correlated energy transfer in rotationally and spin-orbit inelastic collisions of NO(X2Π1/2, j = 1/2f) with O2(X3Σg-).

We present a combined experimental and theoretical study of state-to-state inelastic scattering of NO(X2Π1/2, j = 1/2f) with O2(X3Σg-) molecules at a collision energy of 480 cm-1, focusing in particular on the observation and interpretation of correlated excitations in both NO and O2. Various final states of the NO radical, in both spin-orbit manifolds, were measured with high resolution using a crossed molecular beam apparatus which employs a combination of Stark deceleration and velocity map imaging. Velocity map imaging directly measures both the angular distribution and the radial velocity distribution of the scattered NO molecules, which probes the kinetic energy uptake or release and hence correlated excitations of NO-O2 pairs. Simultaneous excitations of NO and O2 were resolved for all studied final states of NO. In all cases, the experimental results excellently agree with the results of simulations based on quantum scattering calculations. Trends are discussed by analyzing the scattering wave functions from the calculations.