Forty years of Efimov physics: How a bizarre prediction turned into a hot topic

Bose–Einstein Condensate Dark Matter That Involves Composites

Improving the Bose–Einstein condensate model of dark matter through the repulsive three-particle interaction to better reproduce observables such as rotation curves reveals both different thermodynamic phases and few-particle correlations. Using the numerically found solutions of the Gross–Pitaevskii equation for averaging the products of local densities and for calculating thermodynamic functions at zero temperature, it is shown that the few-particle correlations imply a first-order phase transition and are reduced to the product of single-particle averages with a simultaneous increase in pressure, density, and quantum fluctuations. Under given conditions, dark matter exhibits the properties of an ideal gas with an effective temperature determined by quantum fluctuations. Characteristics of oscillations between bound and unbound states of three particles are estimated within a simple random walk approach to qualitatively model the instability of particle complexes. On the other hand, the density-dependent conditions for the formation of composites are analyzed using chemical kinetics without specifying the bonds formed. The obtained results can be extended to the models of multicomponent dark matter consisting of composites formed by particles with a large scattering length.

Asymmetric Lineshapes of Efimov Resonances in Mass-Imbalanced Ultracold Gases

The resonant profile of the rate coefficient for three-body recombination into a shallow dimer is investigated for mass-imbalanced systems. In the low-energy limit, three atoms collide with zero-range interactions, in a regime where the scattering lengths of the heavy–heavy and the heavy–light subsystems are positive and negative, respectively. For this physical system, the adiabatic hyperspherical representation is combined with a fully semi-classical method and we show that the shallow dimer recombination spectra display an asymmetric lineshape that originates from the coexistence of Efimov resonances with Stückelberg interference minima. These asymmetric lineshapes are quantified utilizing the Fano profile formula. In particular, a closed-form expression is derived that describes the width of the corresponding Efimov resonances and the Fano lineshape asymmetry parameter q. The profile of Efimov resonances exhibits a q-reversal effect as the inter- and intra-species scattering lengths vary. In the case of a diverging asymmetry parameter, i.e., |q|→∞, we show that the Efimov resonances possess zero width and are fully decoupled from the three-body and atom–dimer continua, and the corresponding Efimov metastable states behave as bound levels.

Low-temperature chemistry using the R-matrix method

Techniques for producing cold and ultracold molecules are enabling the study of chemical reactions and scattering at the quantum scattering limit, with only a few partial waves contributing to the incident channel, leading to the observation and even full control of state-to-state collisions in this regime. A new R-matrix formalism is presented for tackling problems involving low- and ultra-low energy collisions. This general formalism is particularly appropriate for slow collisions occurring on potential energy surfaces with deep wells. The many resonance states make such systems hard to treat theoretically but offer the best prospects for novel physics: resonances are already being widely used to control diatomic systems and should provide the route to steering ultracold reactions. Our R-matrix-based formalism builds on the progress made in variational calculations of molecular spectra by using these methods to provide wavefunctions for the whole system at short internuclear distances, (a regime known as the inner region). These wavefunctions are used to construct collision energy-dependent R-matrices which can then be propagated to give cross sections at each collision energy. The method is formulated for ultracold collision systems with differing numbers of atoms.

Ultracold Heteronuclear Three-Body Systems: How Diabaticity Limits the Universality of Recombination into Shallow Dimers

The mass-imbalanced three-body recombination process that forms a shallow dimer is shown to possess a rich Efimov-Stückelberg landscape, with corresponding spectra that differ fundamentally from the homonuclear case. A semianalytical treatment of the three-body recombination predicts unusual spectra with intertwined resonance peaks and minima and yields in-depth insight into the behavior of the corresponding Efimov spectra. In particular, the patterns of the Efimov-Stückelberg landscape are shown to depend inherently on the degree of diabaticity of the three-body collisions, which strongly affects the universality of the heteronuclear Efimov states.

Generalized Efimov Effect in One Dimension

We study a one-dimensional quantum problem of two particles interacting with a third one via a scale-invariant subcritically attractive inverse square potential, which can be realized, for example, in a mixture of dipoles and charges confined to one dimension. We find that above a critical mass ratio, this version of the Calogero problem exhibits the generalized Efimov effect, the emergence of discrete scale invariance manifested by a geometric series of three-body bound states with an accumulation point at zero energy.

Efimov Resonance and Three-Body Parameter in a Lithium-Rubidium Mixture

We study collisional heating in a cold ^{7}Li-^{87}Rb mixture near a broad Feshbach resonance at 661 G. At the high field slope of the resonance, we find an enhanced three-body recombination rate that we interpret as a heteronuclear Efimov resonance. With improved Feshbach spectroscopy of two further resonances, a model for the molecular potentials has been developed that now consistently explains all known Feshbach resonances of the various Li-Rb isotope mixtures. The model is used to determine the scattering length of the observed Efimov state. Its value of -1870a_{0} Bohr radii supports the currently discussed assumption of universality of the three-body parameter also in heteronuclear mixtures.

Microscopic origin and universality classes of the Efimov three-body parameter

The low-energy spectrum of three particles interacting via nearly resonant two-body interactions in the Efimov regime is set by the so-called three-body parameter. We show that the three-body parameter is essentially determined by the zero-energy two-body correlation. As a result, we identify two classes of two-body interactions for which the three-body parameter has a universal value in units of their effective range. One class involves the universality of the three-body parameter recently found in ultracold atom systems. The other is relevant to short-range interactions that can be found in nuclear physics and solid-state physics.

Three-body bound states in atomic mixtures with resonant p-wave interaction

We employ the Born-Oppenheimer approximation to find the effective potential in a three-body system consisting of a light particle and two heavy ones when the heavy-light short-range interaction potential has a resonance corresponding to a nonzero orbital angular momentum. In the case of an exact resonance in the p-wave scattering amplitude, the effective potential is attractive and long range; namely, it decreases as the third power of the interatomic distance. Moreover, we show that the range and power of the potential, as well as the number of bound states, are determined by the mass ratio of the particles and the parameters of the heavy-light short-range potential.

Observation of excitonic N-body bound states: polyexcitons in diamond

We have found a series of resonances associated with the bound state (polyexcitons, PE(N)s) of N excitons up to N=6 in the emission spectra of diamond under two-photon excitation at around 10 K. Time-resolved spectra show a stepwise formation of PE(N)s with smaller to larger N, as well as a successive decay from larger to smaller N. At higher excitation levels, the transformation of PE(N)s into a condensed phase of electron-hole droplets occurs. The binding energies of the PE(N)s, normalized to the exciton Rydberg energy, agree well with those of silicon, suggesting the universality of the phenomena.

THE HYLLERAAS BINDING ENERGY OF HYDRIDE AND ELECTRON AFFINITIES

The normalized electron affinity of the hydrogen atom, is the fundamental measure of anionic electron correlation. The three-body H (−) and AB(−) systems analogous to Efimov three-body bosons support multiple excited states. The first complete set of ground state electron affinities of the main group atoms and homonuclear diatomic molecules are reported using the Hylleraas variational binding energy of the hydride anion. Thermal electron affinities and activation energies for the formation of the 27 bonding states of O2(−) are reported from electron capture detector and atmospheric pressure negative ion mass spectrometry. These are iterated through magnetron, flame, swarm, electron impact, photodetachment, and negative ion photoelectron spectra to obtain more precise self-consistent values. Electron affinities for NO are similarly reported. These data are used to calculate Herschbach ionic Morse Person electron curves for the 54 O2(−) and 87 NO (−) states predicted by adiabatic correlation rules. A new ground state adiabatic electron affinity of SF63.00(10) eV is determined from negative ion mass spectra.

Super Efimov effect of resonantly interacting fermions in two dimensions

We study a system of spinless fermions in two dimensions with a short-range interaction fine-tuned to a p-wave resonance. We show that three such fermions form an infinite tower of bound states of orbital angular momentum ℓ=±1 and their binding energies obey a universal doubly exponential scaling E(3)((n))∝exp(-2e(3πn/4+θ)) at large n. This "super Efimov effect" is found by a renormalization group analysis and confirmed by solving the bound state problem. We also provide an indication that there are ℓ=±2 four-body resonances associated with every three-body bound state at E(4)((n))∝exp(-2e(3πn/4+θ-0.188)). These universal few-body states may be observed in ultracold atom experiments and should be taken into account in future many-body studies of the system.

Many bosons in a narrow magnetic Feshbach resonance

The many-boson problem in the presence of an asymptotically narrow Feshbach resonance is considered. The low energy properties are investigated using a two-channel Hamiltonian. The energy spectrum of this model is shown to be bounded from below in the limit of a zero range interaction. This implies the promising possibility of achieving a strongly interacting bosonic phase in a dilute regime where the details of the actual interatomic forces are irrelevant. The integral relation between the energy and the one-body momentum distribution is derived.

Control of resonant interaction between electronic ground and excited states

We observe magnetic Feshbach resonances in a collision between the ground and metastable states of two-electron atoms of ytterbium (Yb). We measure the on-site interaction of doubly occupied sites of an atomic Mott-insulator state in a three-dimensional optical lattice as a collisional frequency shift in a high-resolution laser spectroscopy. The observed spectra are well fitted by a simple theoretical formula, in which two particles with an s-wave contact interaction are confined in a harmonic trap. This analysis reveals a wide variation of the interaction with a resonance behavior around a magnetic field of about 1.1 G for the energetically lowest magnetic sublevel of 170Yb, as well as around 360 mG for the energetically highest magnetic sublevel of 174Yb. The observed Feshbach resonance can only be induced by an anisotropic interatomic interaction. This scheme will open the door to a variety of studies using two-electron atoms with tunable interaction.

Universal four-body states in heavy-light mixtures with a positive scattering length

The number of four-body states known to behave universally is small. This work adds a new class of four-body states to this relatively short list. We predict the existence of a universal four-body bound state for heavy-light mixtures consisting of three identical heavy fermions and a fourth distinguishable lighter particle with a mass ratio κ>/~9.5 and short-range interspecies interaction characterized by a positive s-wave scattering length. The structural properties of these universal states are discussed, and finite-range effects are analyzed. The bound states can be experimentally realized and probed by utilizing ultracold atom mixtures.

Few-body physics with ultracold atomic and molecular systems in traps

Few-body physics has played a prominent role in atomic, molecular and nuclear physics since the early days of quantum mechanics. It is now possible-thanks to tremendous progress in cooling, trapping and manipulating ultracold samples-to experimentally study few-body phenomena in trapped atomic and molecular systems with unprecedented control. This review summarizes recent studies of few-body phenomena in trapped atomic and molecular gases, with an emphasis on small trapped systems. We start by introducing the free-space scattering properties and then investigate what happens when two particles, bosons or fermions, are placed in an external confinement. Next, various three-body systems are treated analytically in limiting cases. Our current understanding of larger two-component Fermi systems and Bose systems is reviewed, and connections with the corresponding bulk systems are established. Lastly, future prospects and challenges are discussed. Throughout this review, commonalities with other systems such as nuclei or quantum dots are highlighted.

Trimer liquids and crystals of polar molecules in coupled wires

We investigate pairing and crystalline instabilities of bosonic and fermionic polar molecules confined to a ladder geometry. Combining analytical and numerical techniques, we show that gases of composite molecular dimers as well as trimers can be stabilized as a function of the density difference between the wires. A shallow optical lattice can pin both liquids, realizing crystals of composite bosons and fermions. We show that these exotic quantum phases are robust against conditions of confinement of the molecular gas to harmonic finite-size potentials.

Scaling properties of universal tetramers

We evidence the existence of a universal correlation between the binding energies of successive four-boson bound states (tetramers), for large two-body scattering lengths (a), related to an additional scale not constrained by three-body Efimov physics. Relevant to ultracold atom experiments, the atom-trimer relaxation peaks for |a|→∞ when the ratio between the tetramer and trimer energies is ≃4.6 and a new tetramer is formed. The new scale is also revealed for a < 0 by the prediction of a correlation between the positions of two successive peaks in the four-atom recombination process.

Efimov physics from a renormalization group perspective

We discuss the physics of the Efimov effect from a renormalization group viewpoint using the concept of limit cycles. Furthermore, we discuss recent experiments providing evidence for the Efimov effect in ultracold gases and its relevance for nuclear systems.

Three body recombination of ultracold dipoles to weakly bound dimers

We use universality in two-body dipolar physics to study three-body recombination. We present results for the universal structure of weakly bound two-dipole states that depend only on the s-wave scattering length (a). We study threshold three-body recombination rates into weakly bound dimer states as a function of the scattering length. A Fermi golden rule analysis is used to estimate rates for different events mediated by the dipole-dipole interaction and a phenomenological contact interaction. The three-body recombination rate in the limit where a≫D contains terms which scale as a{4}, a{2}D{2}, and D4, where D is the dipolar length. When a≪D, the three-body recombination rate scales as D4.