Structure and Occurrence of Three-Body Halos in Two Dimensions

 The quantum-mechanical three-body problem is reformulated in two dimensions by use of hyperspherical coordinates and an adiabatic expansion of the Faddeev equations. The effective radial potentials are calculated and their large-distance asymptotic behavior is derived analytically for short-range two-body interactions. Energies and wave functions are computed numerically for various potentials. An infinite series of Efimov states does not exist in two dimensions. Borromean systems, i.e. bound three-body systems without bound binary subsystems, can only appear when a short-range repulsive barrier at finite distance is present in the two-body interaction. The corresponding Borromean state is never spatially extended. For a system of three weakly interacting identical bosons we find two bound states with both binding energies proportional to the two-body binding energy. In the limit of small binding these states are spatially located at the very large distances characterized by the scattering length. Their properties are universal and independent of the details of the potential. We compare throughout with the corresponding properties in three dimensions. Received September 25, 1998; accepted for publication January 30, 1999     

Borromean three-body heteroatomic resonances

We present an approach to analyze recent experimental evidences of Efimov resonant states in mixtures of ultracold gases, by considering two-species three-body atomic systems bound in a Borromean configuration, where all the two-body interactions are unbound. For such Borromean three-body systems, it is shown that a continuum three-body s-wave resonance emerges from an Efimov state as a scattering length or a three-body scale is moved. The energy and width of the resonant state are determined from a scaling function with arguments given by dimension-less energy ratios relating the two-body virtual state subsystem energies with the shallowest three-body bound state. The peculiar behavior of such resonances is that their peaks are expected to move to lower values of the scattering length, with increasing width, as one raises the temperature. For Borromean systems, two resonant peaks are expected in ultralow-temperature regimes, which will disappear at higher energies. It is shown how a Borromean-Efimov excited bound state turns out to a resonant state by tuning the virtual two-body subsystem energies or scattering lengths, with all energies written in units of the next deeper shallowest Efimov state energy. The resonance position and width for the decay into the continuum are obtained as universal scaling functions (limit cycle) of the dimensionless ratios of the two and three-body scales, which are calculated numerically within a zero-range renormalized three-body model.     

Approaching universality in weakly bound three-body systems

Atom-dimer scattering below the three-body breakup threshold is studied for a system of three identical bosons. The atom-dimer scattering length and the energy of the most weakly bound three-body state are shown to be strongly correlated. An appropriate rescaling of the observables reveals the subtlety of the correlation and serves to identify universal trends in the unitary limit of divergent two-body scattering length. The correlation provides a new quantitative measure of the degree of universality in three-body systems with short-ranged interactions, as well as a consistency check of effective field theories and other theoretical models.     

Condensate fraction and critical temperature of interacting Bose gas in anharmonic trap

The spectral flow of three-body (trimer) states consisting of two heavy (impurity) particles sitting in a condensate of light bosons is considered. Assuming that the condensate is weakly interacting and that an impurity and a boson have a resonant zero-range two-body interaction, we use the Born-Oppenheimer approximation to determine the effective three-body potential. We solve the resulting Schrödinger equation numerically and determine the trimer binding energies as a function of the coherence length of the light bosonic condensate particles. The binding energy is found to be suppressed by the presence of the condensate when the energy scale corresponding to the coherence length becomes of order the trimer binding energy in the absence of the condensate. We find that the Efimov scaling property is reflected in the critical values of the condensate coherence length at which the trimers are pushed into the continuum.     

Three-Body Physics in a Finite Volume

Three particles with large two-body scattering lengths display universal properties including a spectrum of three-body bound states called “Efimov trimers”. I calculate the spectrum of three identical bosons inside a finite cubic box below the three-body breakup threshold. The dependence of the spectrum on the box size and the effects of the breakdown of spherical symmetry are investigated using effective field theory. The renormalization of the effective field theory in the finite volume is explicitly verified. The study of the three-nucleon system inside a finite cubic volume provides a tool for the understanding of Lattice QCD results. I study the triton in a finite volume at physical and unphysical pion masses.     

Two-and three-body decay modes of SUSY Higgs particles

We summarize the dominant decay modes of the neutral and charged Higgs bosons in the Minimal Supersymmetric extension of the Standard Model. While two-body decays are in general dominating, the branching ratios for three-body decays of the heavy scalar, pseudoscalar and charged Higgs bosons can be large below the thresholds if top quarks, W/Z bosons or heavy scalar bosons are involved. Analytical expressions have been derived for the partial decay widths and the physical implications of these decay modes are discussed.     

Tunneling dynamics of a few bosons with both two-and three-body interactions in a double-well potential

We investigate the tunneling dynamics of a few bosons with both two-and three-body interactions in a doublewell potential. Uncorrelated tunneling of Rabi oscillation with the minimum period can happen only when the two-and three-body interactions satisfy a critical condition, i.e., the effective interaction energy is minimized. When the atomic interactions are slightly away from the critical condition in the weak interaction regime, the uncorrelated tunneling exhibits collapse-revival character. When the atomic interactions are strong and far away from the critical condition, the correlated tunneling with Rabi oscillation occurs. The tunneling period(the period of collapse-revival) increases(decreases) when the rate between the two-body and three-body interactions is away from the corresponding critical condition or when the number of bosons increases. Further, the tunneling properties are understood with the help of the energy spectrum of the system. Eventually, the effect of the initial configuration on the tunneling dynamics of a few bosons for both odd and even numbers of bosons is studied, which results in intriguing consequences.     

Measurements of Tan's contact in an atomic Bose-Einstein condensate

A powerful set of universal relations, centered on a quantity called the contact, connects the strength of short-range two-body correlations to the thermodynamics of a many-body system with zero-range interactions. We report on measurements of the contact, using rf spectroscopy, for an (85)Rb atomic Bose-Einstein condensate (BEC). For bosons, the fact that contact spectroscopy can be used to probe the gas on short time scales is useful given the decreasing stability of BECs with increasing interactions. A complication is the added possibility, for bosons, of three-body interactions. In investigating this issue, we have located an Efimov resonance for (85)Rb atoms with loss measurements and thus determined the three-body interaction parameter. In our contact spectroscopy, in a region of observable beyond-mean-field effects, we find no measurable contribution from three-body physics.     

Three-body entanglement induced by spontaneous emission in a three two-level atoms system

We study three-body entanglement induced by spontaneous emission in a three two-level atoms system by using the entanglement tensor approach. The results show that the amount of entanglement is strongly dependent on the initial state of the system and the species of atoms. The three-body entanglement is the result of the coherent superposition of the two-body entanglements. The larger the two-body entanglement is, the stronger the three-body entanglement is. On the other hand, if there exists a great difference in three two-body entanglement measures, the three-body entanglement is very weak. We also find that the maximum of the two-body entanglement obtained with nonidentical atoms is greater than that obtained with identical atoms via adjusting the difference in atomic frequency.     

Non-quantum liquefaction of coherent gases

We show that a gas-to-liquid phase transition at zero temperature may occur in a coherent gas of bosons in the presence of competing nonlinear effects. This situation can take place in atomic systems like Bose-Einstein condensates in alkali gases with two-body and three-body interactions of opposite signs, as well as in laser beams which propagate through optical media with Kerr (focusing) and higher order (defocusing) nonlinear responses. The liquefaction process takes place in the absence of any quantum effect and can be formulated in the framework of a mean field theory, in terms of the minimization of a thermodynamic potential. We study from a thermodynamic point of view all the stationary solutions of the cubic-quintic nonlinear Schrödinger equation which describes our system. We show that solitonic localized solutions connect the gaseous and liquid phases. Furthermore, we also perform a numerical simulation in the presence of linear gain and three-body recombination where a rich dynamics, including the emergence of self-organization behavior, is found.     

Correlated trapped bosons and the many-body Efimov effect

We study two-body correlations in systems of identical bosons. We use a Faddeev type of decomposition of the wave function where all pairs of particles are treated equally. At large scattering length appears a series of Efimov-like states, i.e., spatially extended, model-independent many-body bound states inside the trap. They may play a role in the decay of Bose-Einstein condensates.     

Three-Body Recombination with Two-Channel Contact Interactions

We use a two-channel contact interaction model to describe a system of three identical bosons. The two-channel model quantitatively describes the phenomena of Feshbach resonance in agreement with the phenomenological expression relating scattering length to magnetic detuning. The model also has a finite effective range. We investigate finite range effects in three-body recombination. The simpler one-channel contact interaction model predicts a characteristic geometric scaling of minima in the recombination coefficient as a function of scattering length with scaling parameter 22.7. We show that this factor is reduced when the effective range is included. We compare calculations to experiment.     

Effective Field Theory Analysis of Three-Boson Systems at Next-To-Next-To-Leading Order

We use an effective field theory for short-range forces (SREFT) to analyze systems of three identical bosons interacting via a two-body potential that generates a scattering length, a, which is large compared to the range of the interaction, ?. The amplitude for the scattering of one boson off a bound state of the other two is computed to next-to-next-to-leading order (N²LO) in the ?/a expansion. At this order, two pieces of three-body data are required as input in order to renormalize the amplitude (for fixed a). We apply our results to a model system of three Helium-4 atoms, which are assumed to interact via the TTY potential. We generate N²LO predictions for atom-dimer scattering below the dimer breakup threshold using the bound-state energy of the shallow Helium-4 trimer and the atom-dimer scattering length as our two pieces of three-body input. Based on the convergence pattern of the SREFT expansion, as well as differences in the predictions of two renormalization schemes, we conclude that our N²LO phase- shift predictions will receive higher-order corrections of < 0.2 %. In contrast, the prediction of SREFT for the binding energy of the “deep” trimer of Helium-4 atoms displays poor convergence.     

Efimov Physics in Small Bosonic Clusters

We study small clusters of bosons, A = 2, 3, 4, 5, 6, characterized by a resonant interaction. Firstly, we use a soft-gaussian interaction that reproduces the values of the dimer binding energy and the atom-atom scattering length obtained with LM2M2 potential, a widely used ⁴He-⁴He interaction. We change the intensity of the potential to explore the clusters’ spectra in different regions with large positive and large negative values of the two-body scattering length and we report the clusters’ energies on Efimov plot, which makes the scale invariance explicit. Secondly, we repeat our calculation adding a repulsive three-body force to reproduce the trimer binding energy. In all the region explored, we have found that these systems present two states, one deep and one shallow close to the A ? 1 threshold, and scale invariance has been investigated for these states. The calculations are performed by means of Hyperspherical Harmonics basis set.     

Universality of Three-Body Systems in 2D: Parametrization of the Bound States Energies

Universal properties of mass-imbalanced three-body systems in 2D are studied using zero-range interactions in momentum space. The dependence of the three-particle binding energy on the parameters (masses and two-body energies) is highly non-trivial even in the simplest case of two identical particles and a distinct one. This dependence is parametrized for ground and excited states in terms of supercircles functions in the most general case of three distinguishable particles.     

Accumulation of three-body resonances above two-body thresholds

We calculate resonances in the three-body system with attractive Coulomb potential by solving homogeneous Faddeev-Merkuriev integral equations for complex energies. The equations are solved using the Coulomb-Sturmian separable expansion method. This approach allows us to study the exact behavior of the three-body Coulomb systems near the threshold. A negatively charged positronium ion is used as a test case. In addition to locating all previously known S-wave resonances of the positronium ion, we also find a large number of new resonant states that accumulate just slightly above the two-body thresholds. The pattern of accumulation of resonant states above the two-body thresholds suggests that probably they are infinite in number. We conjecture that this may be a general property of the three-body system with an attractive Coulomb potential.     

Sensitivity of the three-body calculations to different effects

The bound state of few-body systems in light nuclei is studied as a three-body problem. The three-body problem is solved following the different approaches of the Faddeev formalism as well as the unitary pole approximation. Separable approximations are introduced to reduce the three-body problem to a set of coupled integral equations. Numerical calculations are carried out for the resulting integral equations and the separable expansion. In the present work, we calculate the ground-state binding energy of the bound three-nucleon system³H. The main interest of the present work is to investigate the sensitivity of the three-body binding energy to different effects in the problem. For this reason, we study the dependence of the three-body binding energy of different forms of local and separable two-body potentials, on the effective range of the two-body potentials, and on the percent of theD state in the deuteron wave function. Also, we test the sensitivity of the three-body binding energy to the considered number of terms from the separable expansion.     

Universality in few-body systems

Low-energy universality in atomic few-body systems as a result of a large two-body scattering length has gained a lot of attention recently. Here, I discuss recent progress in describing the three-body recombination of cold atoms in terms of a finite set of universal scaling functions and review results for the recombination length of ¹³³Cs atoms obtained with these functions. Furthermore, I will consider the inclusion of effective range corrections and the relevance for further calculations in atomic and nuclear physics.     

Tjon Lines and Scaling Limit in Four-Body Systems

The fixed-slope correlation between tetramer and trimer binding energies, observed by Tjon in the context of nuclear physics, is mainly a manifestation of the dominance of the two-nucleon force in the nuclear potential, which makes the four-body scale on the order of the three-body one. In a more general four-boson case, the correlation between tetramer and trimer binding energies has a non-fixed slope, which expresses the dependence on the new scale. The associated scaling function generates a family of Tjon lines. This conclusion relies on a recent study with weakly-bound four identical bosons, within a renormalized zero-range Faddeev-Yakubovsky formalism.     

Liberating Efimov Physics from Three Dimensions

When two particles attract via a resonant short-range interaction, three particles always form an infinite tower of bound states characterized by a discrete scaling symmetry. It has been considered that this Efimov effect exists only in three dimensions. Here we review how the Efimov physics can be liberated from three dimensions by considering two-body and three-body interactions in mixed dimensions and four-body interaction in one dimension. In such new systems, intriguing phenomena appear, such as confinement-induced Efimov effect, Bose?CFermi crossover in Efimov spectrum, and formation of interlayer Efimov trimers. Some of them are observable in ultracold atom experiments and we believe that this study significantly broadens our horizons of universal Efimov physics.