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Rotational predissociation of extremely weakly bound atom-molecule complexes produced by Feshbach resonance association
Abstract
We study the rotational predissociation of atom-molecule complexes with very small binding energy. Such complexes can be produced by Feshbach resonance association of ultracold molecules with ultracold atoms. Numerical calculations of the predissociation lifetimes based on the computation of the energy dependence of the scattering matrix elements become inaccurate when the binding energy is smaller than the energy width of the predissociating state. We derive expressions that represent accurately the predissociation lifetimes in terms of the real and imaginary parts of the scattering length and effective range for molecules in an excited rotational state. Our results show that the predissociation lifetimes are the longest when the binding energy is positive, i.e., when the predissociating state is just above the excited state threshold.
... Controlled energy transfer in molecular arrays may also be used for the study of controlled chemical interactions for a class of reactions stimulated by energy excitation of the reactants. Directing energy to a particular lattice site containing two or more reagents can be used to induce a chemical interaction [69], an inelastic collision or predissociation [70] with the complete temporal and spatial control over the reaction process. ...
An elementary excitation in an aggregate of coupled particles generates a
collective excited state. We show that the dynamics of these excitations can be
controlled by applying a transient external potential which modifies the phase
of the quantum states of the individual particles. The method is based on an
interplay of adiabatic and sudden time scales in the quantum evolution of the
many-body states. We show that specific phase transformations can be used to
accelerate or decelerate quantum energy transfer and spatially focus
delocalized excitations onto different parts of arrays of quantum particles. We
consider possible experimental implementations of the proposed technique and
study the effect of disorder due to the presence of impurities on its fidelity.
We further show that the proposed technique can allow control of energy
transfer in completely disordered systems.
At temperatures close to absolute zero, the molecular reactions and collisions are dominantly governed by quantum mechanics. Remarkable quantum phenomena such as quantum tunneling, quantum threshold behavior, quantum resonances, quantum interference, and quantum statistics are expected to be the main features in ultracold reactions and collisions. Ultracold molecules offer great opportunities and challenges in the study of these intriguing quantum phenomena in molecular processes. In this article, we review the recent progress in the preparation of ultracold molecules and the study of ultracold reactions and collisions using ultracold molecules. We focus on the controlled ultracold chemistry and the scattering resonances at ultralow temperatures. The challenges in understanding the complex ultracold reactions and collisions are also discussed.
Quantum-mechanical scattering calculations of Feshbach resonances arising from van der Waals molecule formation are used to determine vibrational and rotational predissociation lifetimes. A multichannel effective range theory is used to establish the relationship between predissociation and the zero-temperature limit of collisional quenching for resonances lying close to the thresholds for dissociation. The elastic scattering Feshbach resonances may be measurable in ultracold atom-molecule experiments.
We give a snapshot of the rapidly developing field of ultracold polar molecules and walk the reader through the papers appearing in this Topical Issue.
Vibrational relaxation of trapped molecules due to collisions with cold atoms is investigated using the results of quantum-mechanical scattering calculations. Trap loss is analyzed using an exactly solvable kinetic model that includes direct collisional quenching and an indirect process of vibrational predissociation. At low atom density, the relaxation is due primarily to collisional quenching. At high atom density, the relaxation involves additional time scales due to the formation and decay of van der Waals complexes. It is shown that the most weakly bound state of the van der Waals complex for a given diatomic vibrational level controls the relaxation at all atom densities. Possible experiments using trapped molecules are discussed.
Feshbach resonances are the essential tool to control the interaction between atoms in ultracold quantum gases. They have found numerous experimental applications, opening up the way to important breakthroughs. This review broadly covers the phenomenon of Feshbach resonances in ultracold gases and their main applications. This includes the theoretical background and models for the description of Feshbach resonances, the experimental methods to find and characterize the resonances, a discussion of the main properties of resonances in various atomic species and mixed atomic species systems, and an overview of key experiments with atomic Bose-Einstein condensates, degenerate Fermi gases, and ultracold molecules.
We show that an ensemble of 2Σ molecules in the ro-vibrational ground state trapped on an optical lattice exhibits collective spin excitations that can be controlled by applying superimposed electric and magnetic fields. In particular, we show that the lowest energy excitation of the molecular ensemble at certain combinations of electric and magnetic fields leads to the formation of a magnetic Frenkel exciton. The exciton bandwidth can be tuned by varying the electric or magnetic fields. We show that the exciton states can be localized by creating vacancies in the optical lattice. The localization patterns of the magnetic exciton states are sensitive to the number and distribution of vacancies, which can be exploited for engineering many-body entangled spin states. We consider the dynamics of magnetic exciton wavepackets and show that the spin excitation transfer between molecules in an optical lattice can be accelerated or slowed down by tuning an external magnetic or electric field.
We show that an ensemble of polar molecules trapped in an optical lattice can
be considered as a controllable open quantum system. The coupling between
collective rotational excitations and the motion of the molecules in the
lattice potential can be controlled by varying the strength and orientation of
an external DC electric field as well as the intensity of the trapping laser.
The system can be described by a generalized Holstein Hamiltonian with tunable
parameters and can be used as a quantum simulator of excitation energy transfer
and polaron phenomena. We show that the character of excitation energy transfer
can be modified by tuning experimental parameters.
In the present work, we demonstrate the possibility of controlling by an external field the dynamics of collective excitations (excitons) of molecules on an optical lattice. We show that a suitably chosen two-species mixture of ultracold polar molecules loaded on an optical lattice forms a phononless crystal, where exciton-impurity interactions can be controlled by applying an external electric field. This can be used for the controlled creation of many-body entangled states of ultracold molecules and the time-domain quantum simulation of disorder-induced localization and delocalization of quantum particles.
We report on the observation of an elementary exchange process in an optically trapped ultracold sample of atoms and Feshbach molecules. We can magnetically control the energetic nature of the process and tune it from endoergic to exoergic, enabling the observation of a pronounced threshold behavior. In contrast to relaxation to more deeply bound molecular states, the exchange process does not lead to trap loss. We find excellent agreement between our experimental observations and calculations based on the solutions of three-body Schrödinger equation in the adiabatic hyperspherical representation. The high efficiency of the exchange process is explained by the halo character of both the initial and final molecular states.
We explore the potential energy surfaces for NH molecules interacting with
alkali-metal and alkaline-earth atoms using highly correlated ab-initio
electronic structure calculations. The surfaces for interaction with
alkali-metal atoms have deep wells dominated by covalent forces. The resulting
strong anisotropies will produce strongly inelastic collisions. The surfaces
for interaction with alkaline-earth atoms have shallower wells that are
dominated by induction and dispersion forces. For Be and Mg the anisotropy is
small compared to the rotational constant of NH, so that collisions will be
relatively weakly inelastic. Be and Mg are thus promising coolants for
sympathetic cooling of NH to the ultracold regime.
How does a chemical reaction proceed at ultralow temperatures? Can simple quantum mechanical rules such as quantum statistics,
single partial-wave scattering, and quantum threshold laws provide a clear understanding of the molecular reactivity under
a vanishing collision energy? Starting with an optically trapped near–quantum-degenerate gas of polar 40K87Rb molecules prepared in their absolute ground state, we report experimental evidence for exothermic atom-exchange chemical
reactions. When these fermionic molecules were prepared in a single quantum state at a temperature of a few hundred nanokelvin,
we observed p-wave–dominated quantum threshold collisions arising from tunneling through an angular momentum barrier followed
by a short-range chemical reaction with a probability near unity. When these molecules were prepared in two different internal
states or when molecules and atoms were brought together, the reaction rates were enhanced by a factor of 10 to 100 as a result
of s-wave scattering, which does not have a centrifugal barrier. The measured rates agree with predicted universal loss rates
related to the two-body van der Waals length.
This article presents a review of the current state of the art in the research field of cold and ultracold molecules. It serves as an introduction to the Special Issue of the New Journal of Physics on Cold and Ultracold Molecules and describes new prospects for fundamental research and technological development. Cold and ultracold molecules may revolutionize physical chemistry and few body physics, provide techniques for probing new states of quantum matter, allow for precision measurements of both fundamental and applied interest, and enable quantum simulations of condensed-matter phenomena. Ultracold molecules offer promising applications such as new platforms for quantum computing, precise control of molecular dynamics, nanolithography, and Bose-enhanced chemistry. The discussion is based on recent experimental and theoretical work and concludes with a summary of anticipated future directions and open questions in this rapidly expanding research field. Comment: 82 pages, 9 figures, review article to appear in New Journal of Physics Special Issue on Cold and Ultracold Molecules
We report here on the production of an ultracold gas of tightly bound ${\mathrm{Rb}}_{2}$ triplet molecules in the rovibrational ground state, close to quantum degeneracy. This is achieved by optically transferring weakly bound ${\mathrm{Rb}}_{2}$ molecules to the absolute lowest level of the ground triplet potential with a transfer efficiency of about 90%. The transfer takes place in a 3D optical lattice which traps a sizeable fraction of the tightly bound molecules with a lifetime exceeding 200 ms.
A quantum gas of ultracold polar molecules, with long-range and anisotropic interactions, not only would enable explorations of a large class of many-body physics phenomena but also could be used for quantum information processing. We report on the creation of an ultracold dense gas of potassium-rubidium (40K87Rb) polar molecules. Using a single step of STIRAP (stimulated Raman adiabatic passage) with two-frequency laser irradiation, we coherently transfer extremely weakly bound KRb molecules to the rovibrational ground state of either the triplet or the singlet electronic ground molecular potential. The polar molecular gas has a peak density of 10(12) per cubic centimeter and an expansion-determined translational temperature of 350 nanokelvin. The polar molecules have a permanent electric dipole moment, which we measure with Stark spectroscopy to be 0.052(2) Debye (1 Debye = 3.336 x 10(-30) coulomb-meters) for the triplet rovibrational ground state and 0.566(17) Debye for the singlet rovibrational ground state.
We investigate the resonance properties of ultracold ground state 6Li+6Li, 7Li+7Li, and 23Na+23Na collisions. The locations of various resonances and their corresponding error bounds due to the uncertainty of the interatomic potentials are presented. Also, the resonance widths are computed using rigorous coupled-channel calculations, as well as a modified version of Feshbach theory valid for strong fields.
We observe magnetically tuned collision resonances for ultracold Cs2 molecules stored in a CO2-laser trap. By magnetically levitating the molecules against gravity, we precisely measure their magnetic moment. We find an avoided level crossing which allows us to transfer the molecules into another state. In the new state, two Feshbach-like collision resonances show up as strong inelastic loss features. We interpret these resonances as being induced by Cs4 bound states near the molecular scattering continuum. The tunability of the interactions between molecules opens up novel applications such as controlled chemical reactions and synthesis of ultracold complex molecules.
We have generalized the BOUND and MOLSCAT packages to allow calculations in basis sets where the monomer Hamiltonians are off-diagonal and used the new capability to carry out bound-state and scattering calculations on 3He-NH and 4He-NH as a function of magnetic field. Following the bound-state energies to the point where they cross thresholds gives very precise predictions of the magnetic fields at which zero-energy Feshbach resonances occur. We have used this to locate and characterize two very narrow Feshbach resonances in 3He-NH. Such resonances can be used to tune elastic and inelastic collision cross sections, and sweeping the magnetic field across them will allow a form of quantum control in which separated atoms and molecules are associated to form complexes. For the first resonance, where only elastic scattering is possible, the scattering length shows a pole as a function of magnetic field and there is a very large peak in the elastic cross section. For the second resonance, however, inelastic scattering is also possible. In this case the pole in the scattering length is dramatically suppressed and the cross sections show relatively small peaks. The peak suppression is expected to be even larger in systems with stronger inelasticity. The results suggest that calculations on ultracold molecular inelastic collisions may be much less sensitive to details of the potential energy surface than has been believed.
In the absence of inelastic scattering, Feshbach resonances produce poles in scattering lengths and very large peaks in elastic cross sections. However, inelastic scattering removes the poles. Whenever the resonant state is coupled comparably to the elastic and inelastic channels, the scattering length exhibits only a small oscillation and peaks in cross sections are dramatically suppressed. A resonant scattering length is defined to characterize the amplitude of the oscillation, and is shown to be small for many collisions of ultracold molecules. The results suggest that cross sections for ultracold molecular inelastic collisions are much less sensitive to details of the potential than has been believed.
Magnetically tunable Feshbach resonances were employed to associate cold diatomic molecules in a series of experiments involving both atomic Bose as well as two spin component Fermi gases. This review illustrates theoretical concepts of both the particular nature of the highly excited Feshbach molecules produced and the techniques for their association from unbound atom pairs. Coupled channels theory provides the rigorous formulation of the microscopic physics of Feshbach resonances in cold gases. Concepts of dressed versus bare energy states, universal properties of Feshbach molecules, as well as the classification in terms of entrance- and closed-channel dominated resonances are introduced on the basis of practical two-channel approaches. Their significance is illustrated for several experimental observations, such as binding energies and lifetimes with respect to collisional relaxation. Molecular association and dissociation are discussed in the context of techniques involving linear magnetic field sweeps in cold Bose and Fermi gases as well as pulse sequences leading to Ramsey-type interference fringes. Their descriptions in terms of Landau-Zener, two-level mean field as well as beyond mean field approaches are reviewed in detail, including the associated ranges of validity.
In this article we shall discuss the methods available for the calculation of the rates of atomic collision processes. We shall exclude from consideration any methods which are of specific application to nuclear or high energy collisions as these are discussed in Vols. XXXIX, XL, XLI and XLIII of this Encyclopedia though some of the methods which will be discussed are applicable to such collisions. No mention will be made of the rates of chemical reactions as these are adequately dealt with in chemical textbooks.
The rotational structure in the HF stretching bands of the HF dimer has been recorded with nearly Doppler-limited resolution using a tunable difference-frequency laser spectrometer and a long-path cell held at low temperatures and pressures. Two bands are observed; the higher frequency band, corresponding principally to the ‘‘free’’ hydrogen stretch, has the appearance of a B-type or perpendicular band with two prominent Q subbranches RQ0 and PQ1; the lower band is an A-type or parallel band arising primarily from the ‘‘bonded’’ hydrogen vibration. The K=0 subbands of both vibrations have been fully assigned and fit with polynomial expansions in J(J+1) to yield ground state constants in excellent agreement with a previous microwave resonance molecular beam study of the HF dimer. The subbands exhibit doubling and 10:6 intensity ratios for alternate J indicative of the internal rotation tunneling motion proposed in the microwave study. A pressure-independent broadening of the lines to about twice the Doppler width, attributed to vibrational predissociation, is observed for the bonded hydrogen stretching band.
We show how the linear reference potential method for solution of the close‐coupled equations, which arise in inelastic scattering theory, can be reformulated in terms of an ‘‘imbedding‐type’’ propagator. Explicit expressions are given for the blocks of the propagator matrix in terms of Airy functions. By representing these functions in terms of moduli and phases, in both classically allowed and classically forbidden regions, one can evaluate the propagator without any numerical difficulty. The resulting algorithm is tested on a highly pathological problem—the rotationally inelastic scattering of a polar molecule by a spherical ion at extremely low kinetic energy—and found to be completely stable.
It is shown that in the presence of inelastic scattering, zero energy elastic and inelastic scattering can be characterized by a complex scattering length, the imaginary part of which is related directly to the total inelastic scattering cross section. Collisions between H atoms and vibrationally excited H2 molecules are investigated. Zero energy cross sections for all vibrational states of H2 and the corresponding complex scattering lengths are reported using accurate quantum mechanical calculations. We obtain large values of the elastic cross sections which we attribute to s-wave bound and quasi-bound states of H⋯H2(v). The energies and lifetimes of the quasi-bound states are extracted from the complex scattering lengths.
Widths and energies for predissociating levels of H2–inert-gas Van der Waals molecules are calculated by solving the close-coupling equations for accurate potential-energy surfaces. These exact results are used to test several approximate schemes previously employed for calculating level energies and widths. None of these is found to be entirely satisfactory, and a new method is proposed which gives more accurate results. In this new method (SEPTOC) the secular equations are solved in matrix form for the closed-channel manifold, and the open channels are treated by perturbation theory. The limited form of SEPTOC tested here gives very promising results.
Accurate close-coupling calculations are performed for vibrationally predissociating states of Hâ-Ar, Dâ-Ar, and HD-Ar, using the best potential energy surface available. All the states examined have very small widths (GAMMA 20 ..mu..s). There is a pronounced tendency for predissociation to yield rotationally hot diatomic molecules, even for the Hâ-Ar and Dâ-Ar complexes where the present potential has no anisotropic terms of higher order than Pâ(cos theta). This near-resonant effect is particularly strong for HD-Ar, where all Legendre terms are present in the potential; in this case, about 50% of the HD products are formed in the highest two accessible rotational levels. There is some evidence for a rotational rainbow effect in the product rotational state distributions. Perturbation theory calculations which attempt to reproduce the accurate calculations are also reported. They successfully model the qualitative features of the close-coupling results, but are not quantitatively accurate even for these weakly coupled systems. It appears that this inadequacy is due to the need for a very accurate representation of the bound state wave function and to the neglect of important couplings between the different open channels. This conclusion is supported by the observation that very large basis sets are required to obtain convergence of the close-coupling calculations. 4 figures, 7 tables.
A new method for solving multichannel scattering problems is presented.
The key to the method is an efficient algorithm for numerically solving the
matrix Ricatti equation for the logarithmic derivative of the wave function.
Previous calculalions on He/sup +/-Ne collision at 70.9 eV were repeated. (JFP)
We explore a recent proposal [Fedichev et al., Phys. Rev. Lett. 77, 2913 (1996)] for altering the mean interaction strength between ultracold atoms using an appropriately detuned laser. Although care must be taken to minimize laser-driven loss processes, we find large ranges of intensities and detunings where useful changes might be affected. Accordingly, we present simple formulas for the effects of laser light that should prove useful in designing specific experiments. We demonstrate the validity of these formulas by comparison with exact close-coupling models. In particular, we find that useful changes of the mean-field interaction require sufficiently high laser intensities that the rate of laser-induced stimulated emission exceeds the natural spontaneous emission rate.
Internal rotational predissociative levels of OH–Ar (A 2Σ+) have been identified lying up to 350 cm−1 above the OH A 2Σ+ (v=0, 1)+Ar dissociation limit. The predissociative level energies, lifetimes, and OH A 2Σ+ rotational product distributions have been measured. Complexes prepared in many of these predissociative levels are long lived with lifetimes ≥50 ps. A novel variation of stimulated emission pumping has enabled quantitative OH A 2Σ+ rotational distributions to be obtained following OH–Ar predissociation. The OH product distributions are highly selective. The highest energetically available channel is always populated, yet in many cases, low rotational levels are conspicuously absent. The OH–Ar predissociative levels have been assigned nearly good quantum numbers based on a rotational contour analysis of the predissociative features and/or the OH A 2Σ+ rotational product distributions. A two‐step mechanism involving Coriolis coupling and the potential anisotropy has been proposed to describe the predissociation process. A comparison between the experimentally measured and theoretically calculated observables provides a guide for further refinements of the OH A 2Σ++Ar potential energy surface.
We report an extensive computational study of rotationally predissociating metastable states of the Ar⋅HCl van der Waals complex, using a highly realistic empirical intermolecular potential recently proposed by Hutson and Howard. The states are characterized by fully converged, close‐coupled, scattering calculations. Resonance energies, widths, and partial widths are extracted by fitting the energy dependence of S matrices. Total angular momenta of 0 and 1 are studied, and the calculations span an energy range from 0 to 1400 cm−1. The resonance widths vary from <10−4 to ≳5 cm−1, and it is shown that the isolated narrow resonance approximation is of poor validity for the wider resonances. Comparison of the close‐coupling results with approximate calculations enables assignment of approximate quantum numbers to the metastable states. Physical explanations are suggested for the strong trends in resonance parameters as a function of the intermolecular stretching, diatom rotation, and molecule‐fixed angular momentum projection quantum numbers. A changeover from a near‐molecule‐fixed to a near‐space‐fixed coupling scheme, as angular momentum is increased, is clearly demonstrated. The results are of considerable relevance to the design of experiments and the development of approximate computational methods in this area.
In this paper we advance a quantum mechanical colinear model for vibrational predissociation on a single electronic potential surface of a linear triatomic van der Waals molecule X⋅⋅⋅BC, where BC is a conventional diatomic, while X represents a rare‐gas atom. The zero‐order states of the system are represented as products of an eigenfunction of the vibrating BC bond and a function describing the (bound or unbound) motion of X relative to the center of mass of BC bond which is frozen at its equilibrium configuration. The residual interaction representing the deviation between the interaction potential of X with the vibrating BC molecule, and the interaction of X with the frozen diatomic, induces discrete–continuum and continuum–continuum coupling. On the basis of the analysis of these coupling terms we assert that the zero‐order basis provides a reasonable description of the initial and final states. We have also demonstrated that the zero‐order resonance widths are small relative to their spacings and, furthermore, we have shown that continuum–continuum couplings prevail essentially only between adjacent continua. The dynamics of vibrational predissociation were reduced to the problem of the decay of a single resonance into a manifold of adjacently coupled continua. Closed analytical expressions for the rate of vibrational predissociation and for the vibrational distribution of the products were derived incorporating the effects of discrete–continuum and continuum–continuum coupling. We have explored the dependence of the rate of vibrational predissociation on the frequency of the BC molecule establishing a new energy gap law for this process. We have also investigated the dependence of the rate on the potential parameter of van der Waals bond and on the mass of the rare‐gas atom. Finally, a study of the nature of the final vibrational distribution of the diatomic fragment resulting from the vibrational predissociation process was provided.
Effective range theory is developed for systems of many coupled two-body channels with angular momenta li. Derivatives of the amplitudes Mij (where M is essentially the inverse of the K matrix) are formed. Quite in analogy with one channel effective range theory, the diagonal elements Mii are accurately given, by an expression quadratic in the momentum ki. The coefficients Rii of ki2 are effective range type integrals which are interpretable in terms of the range of forces and can be taken to be energy independent to the same extent as in the one channel theory. The non-diagonal elements Mij are, to a good approximation, energy independent, even for Rii greatly different from Rjj and li ≠ lj. The case of two coupled channels is studied in detail: a computer experiment was performed to test the validity of the theory; for l = 0, the properties of narrow resonances including the interactions which can lead to them are thoroughly investigated. The positions of the poles of the T matrix are briefly considered. Comparison is made between the effective range type of parametrization and Breit-Wigner theory. The present theory is contrasted to the effective range theory for the eigenphase shifts; the eigenphase shift theory is shown, in principle, to be less accurate. Some possible applications are briefly discussed.
The threshold properties of a two-body system which is coupled to other two-body channels are investigated. In particular, we examine the new channel when it is coupled to only one other, open, channel. At the threshold of the new channel we consider three real parameters (which describe the scattering amplitudes): the complex scattering length, a = A − iB, in the new channel and the diagonal K matrix element in the old channel, c. We first establish the quantitative behavior of these parameters with respect to certain averaged strengths of the interactions. An interesting application is, for example, a relation between the three actual parameters and the c that would apply if the coupling between the channels were turned off. Effective range expansions exist for the three parameters. (These can be easily generalized to the problem of many old channels.) These have the same form as the familiar one-channel expansion, except that the coefficient, r0, of the quadratic term in the expansion is now an effective range type intergral multiplied by a factor depending on a and c. For example, in the effective range expansion of a, we find that if is large (compared to a kinematical factor, roughly the momentum, κ1, in the old channel) r0 will be large compared to the range of forces. A similar result applies for the effective range expansion of the length . We then illustrate the usefulness of some of the relations by solving, exactly, the problem in which the 2-channel system is described by 2 coupled Schrödinger equations with square well potentials. The scattering length is also examined by means of a complex potential for the situation where there are many coupled channels.
We carry out calculations on $M$-changing collisions of NH ($^3\Sigma^-$)
molecules in magnetically trappable states using a recently calculated
potential energy surface. We show that elastic collision rates are much faster
than inelastic rates for a wide range of fields at temperatures up to 10 mK and
that the ratio increases for lower temperatures and magnetic fields. If NH
molecules can be cooled to temperatures approaching 10 mK and brought into
contact with laser-cooled Mg then there is a good prospect that sympathetic
cooling can be achieved.
This review describes recent experimental and theoretical advances in forming molecules in ultracold gases of trapped alkali metal atoms, both by magnetic tuning through Feshbach resonances and by photoassociation. Molecular Bose-Einstein condensation of long-range states of both boson dimers and fermion dimers was achieved in 2002-3. Condensates of boson dimers were found to be short-lived, but long-lived condensates of fermion dimers have been produced. Signatures of triatomic and tetraatomic molecules have recently been observed. Both homonuclear and heteronuclear molecules have been formed by photoassociation, mostly in very high vibrational levels. Recent attempts to produce ultracold molecules in short-range states (low vibrational levels) are described. Experimental and theoretical work on collisions of ultracold molecules is discussed.
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