We demonstrate the possibility of existence of meta-stable N-body Efimov states in trapped Bose systems with large scattering length. We calculate spectra of trapped systems of N=3,4,5,6, and 7 bosons using a stochastic variational method with a restricted correlated Gaussian basis. For each system the calculations reveal a series of Efimov states where the energy and the r.m.s. radius exhibit the characteristic exponential dependence upon the state number. We also estimate the contribution of these states to the recombination rate of Bose-Einstein condensates.
Four-boson Efimov physics is well known in the negative energy regime but far less above the four-body breakup threshold. The part of this region with negative two-boson scattering length is studied solving rigorous four-particle scattering equations for transition operators in the momentum space. Moving away from the unitary limit the Efimov tetramers evolve from unstable bound states into resonances. Their energies and widths are studied as functions of the two-boson scattering length; a universal behavior is established and given in a dimensionless representation. The Efimov tetramers have …
We investigate theoretically and experimentally the heteronuclear Efimov scenario for a three-body system that consists of two bosons and one distinguishable particle with positive intraspecies scattering lengths. The three-body parameter at the three-body scattering threshold and the scaling factor between consecutive Efimov resonances are found to be controlled by the scattering length between the two bosons, approximately independent of short-range physics. We observe two excited-state Efimov resonances in the three-body recombination spectra of an ultracold mixture of fermionic $^6 $Li and…
Three-boson Efimov physics is well known in the bound-state regime, but far less in the three-particle continuum at negative two-particle scattering length where Efimov states evolve into resonances. They are studied solving rigorous three-particle scattering equations for transition operators in the momentum space. The dependence of the three-boson resonance energy and width on the two-boson scattering length is studied with several force models. The universal limit is determined numerically considering highly excited states; simple parametrizations for the resonance energy and width in terms…
Recent theoretical developments in the four-boson system with resonant interactions are described. Momentum-space scattering equations for the four-particle transition operators are used. The properties of unstable tetramers with approximate dimer-atom-atom structure are determined. In addition, the three- and four-cluster recombination processes in the four-boson system are studied.
This article reviews theoretical and experimental advances in Efimov physics, an array of quantum few-body and many-body phenomena arising for particles interacting via short-range resonant interactions, that is based on the appearance of a scale-invariant three-body attraction theoretically discovered by Vitaly Efimov in 1970. This three-body effect was originally proposed to explain the binding of nuclei such as the triton and the Hoyle state of carbon-12, and later considered as a simple explanation for the existence of some halo nuclei. It was subsequently evidenced in trapped ultra-cold a…
We use exact four-boson scattering equations in the momentum-space framework to study the universal properties of shallow Efimov tetramers and their dependence on the two-boson scattering length. We demonstrate that, in contrast to previous predictions, the shallow tetramer in a particular experimentally unexplored regime is not an unstable bound state but an inelastic virtual state. This leads to a resonant behaviour of the atom-trimer scattering length and thereby to a resonant enhancement of the trimer relaxation in ultracold atom-trimer mixtures.
The existence of the Efimov effect is drastically affected by the dimensionality of the space in which the system is embedded. The effective spatial dimension containing an atomic cloud can be continuously modified by compressing it in one or two directions. In the present article we determine for a general $AAB$ system formed by two identical bosons $A$ and a third particle $B$ in the two-body unitary limit, the dimensionsality $D$ for which the Efimov effect can exist for different values of the mass ratio $\mathpzc{A}=m_B/m_A$. In addition, we provide a prediction for the Efimov discrete sc…
We study Efimov resonances via three-body loss in an ultracold two-component gas of fermionic $^6$Li and bosonic $^{133}$Cs atoms close to a Feshbach resonance at 843~G, extending results reported previously [Pires \textit{et al.}, Phys. Rev. Lett. 112, 250404 (2014)] to temperatures around 120~nK. The experimental scheme for reaching lower temperatures is based upon compensating the gravity-induced spatial separation of the mass-imbalanced gases with bichromatic optical dipole traps. We observe the first and second excited Li-Cs-Cs Efimov resonance in the magnetic field dependence of the thre…
We study a three-body system, formed by two identical heavy bosons and a light particle, in the Born-Oppenheimer approximation for an arbitrary dimension $D$. We restrict $D$ to the interval $2\,<\,D\,<\,4$, and derive the heavy-heavy $D$-dimensional effective potential proportional to $1/R^2$ ($R$ is the relative distance between the heavy particles), which is responsible for the Efimov effect. We found that the Efimov states disappear once the critical strength of the heavy-heavy effective potential $1/R^2$ approaches the limit $-(D-2)^2/4$. We obtained the scaling function for the $^{133}$C…
We study three- and four-body Efimov physics in a heteronuclear atomic system with three identical heavy bosonic atoms and one light atom. We show that exchange of the light atom between the heavy atoms leads to both three- and four-body features in the low-energy inelastic rate constants that trace to the Efimov effect. Further, the effective interaction generated by this exchange can provide an additional mechanism for control in ultracold experiments. Finally, we find that there is no true four-body Efimov effect - that is, no infinite number of four-body states in the absence of two- and t…
A Fermi-Fermi mixture of 40K and 6Li does not exhibit the Efimov effect in a free space, but the Efimov effect can be induced by confining only 40K in one dimension. Here the Efimov's three-body parameter is controlled by the confinement length. We show that the three-body recombination rate in such a system in the dilute limit has a characteristic logarithmic-periodic dependence on the effective scattering length with the scaling factor 22.0 and can be expressed by formulas similar to those for identical bosons in three dimensions. The ultracold mixture of 40K and 6Li in the 1D-3D mixed dimen…
We study Efimov physics for three identical bosons interacting via a pairwise square-well potential, analyze the validity of the separable approximation as a function of the interaction strength, and investigate what is needed to improve this approximation. We find separable approximations to be accurate for potentials with just one (nearly) bound dimer state. For potentials with more bound or almost bound dimer states, these states need to be included for an accurate determination of the Efimov spectrum and the corresponding three-body observables. We also show that a separable approximation …
Armed with a new solution to the four-body problem, we reanalyze the recent Innsbruck observation of an Efimov resonance in three-body collisions of cesium atoms, and show that it provides evidence of new, universal four-body physics as well. Our theoretical treatment of four bosonic atoms in the limit of strong two-body attraction suggests a ubiquitous result: namely, that resonantly-bound states of all four atoms should occur whenever the two-body scattering length is a specific fraction (0.43) of the value where a three-body Efimov resonance occurs. A hitherto unanalyzed feature in the 2006…
Efimov physics relates to 3-body systems with large 2-body scattering lengths a and small effective ranges r. For many systems in nature the assumption of a small effective range is not valid. The present report shows binding energies E of three identical bosons calculated with 2-body potentials that are fitted to scattering data and momentum cut-offs (L) by inverse scattering. Results agree with previous works in the case of r<<a. While energies diverge with momentum cut-off L for r=0, they converge for r>0 when L=~10/r. With 1/a=0 the converged energies are given by E(n) =C(n)/r*r with n lab…
We study the manifestations of universal four-body physics in ultracold dimer-dimer collisions. We show that resonant features associated with three-body Efimov physics and dimer-dimer scattering lengths are universally related. The emergence of universal four-boson states allows for the tunability of the dimer-dimer interaction, thus enabling the future study of ultracold molecular gases with both attractive and repulsive interactions. Moreover, our study of the interconversion between dimers and Efimov trimers shows that $B_{2}+B_{2}\to B_{3}+B$ rearrangement reactions can provide an efficie…
Spinor condensates have proven to be a rich area for probing many-body phenomena richer than that of an ultracold gas consisting of atoms restricted to a single spin state. In the strongly correlated regime, the physics controlling the possible novel phases of the condensate remains largely unexplored, and few-body aspects can play a central role in the properties and dynamics of the system through manifestations of Efimov physics. The present study solves the three-body problem for bosonic spinors using the hyperspherical adiabatic representation and characterizes the multiple families of Efi…
Continuum structures of three short-range interacting particles in a deformed external one-body field are investigated. We use the equivalent $d$-method employing non-integer dimension, $d$, in a spherical calculation with a dimension-dependent angular momentum barrier. We focus on dimensions close to the critical dimension, $d=d_E$, between two and three, defined by zero two-body energies, where the Efimov effect can occur. We design for this dimension region a schematic, long-distance realistic, square-well based, three-body spherical model, which is used to derive analytic expressions for t…
Motivated by the prospect of observing Efimov and Bose polaron physics in ultracold mixtures of bosonic atoms with large mass imbalance, this work investigates the magnetic Feshbach resonances between $^{7}\text{Li}$ and $^{133}\text{Cs}$. The resonances are predicted at the 1 gauss level using the model of Pires et al. [Phys. Rev. A, 90, 012710] obtained from experimental observations of resonances between $^{6}\text{Li}$ and $^{133}\text{Cs}$. It is found that a few resonances in a practical range of magnetic field intensity could be used to tune the scattering length between $^{7}\text{Li}$…
We calculated, using seven realistic 4He-4He potentials in the literature, the Efimov spectra of the 4He trimer and tetramer and analyzed the universality of the systems. The three-(four-)body Schroedinger equations were solved fully nonadiabatically with the high-precision calculation method employed in our previous work on the 4He trimer and tetramer [Phys. Rev. A 85, 022502 (2012); 85, 062505 (2012)]. We found the following universality in the four-boson system: i) The critical scattering lengths at which the tetramer ground and excited states couple to the four-body threshold are independe…
The influence of boson-boson and boson-fermion interactions on the stability of a binary mixture of bosonic and fermionic atoms is investigated. The density profiles of the trapped mixture are obtained from direct numerical solution of a modified Gross-Pitaevskii equation that is self-consistently coupled to the mean-field generated by the interaction with the fermionic species. The fermions which in turn feel the mean-field created by the bosons are treated in Thomas-Fermi approximation. We study the effects of different combinations of signs of the boson-boson and the boson-fermion scatterin…
We perform exact calculations of collisional frequency shifts for several fermions or bosons using a singlet and triplet basis for pairs of particles. The "factor of 2 controversy" for bosons becomes clear - the factor is always 2. Decoherence is described by singlet states and they are unaffected by spatially uniform clock fields. Spatial variations are critical, especially for fermions which were previously thought to be immune to collision shifts. The spatial variations lead to decoherence and a novel frequency shift that is not proportional to the partial density of internal states.
We compute numerically the yrast line for harmonically trapped boson systems with a weak repulsive contact interaction, studying the transition to a vortex state as the angular momentum L increases and approaches N, the number of bosons. The L=N eigenstate is indeed dominated by particles with unit angular momentum, but the state has other significant components beyond the pure vortex configuration. There is a smooth crossover between low and high L with no indication of a quantum phase transition. Most strikingly, the energy and wave function appear to be analytical functions of L over the en…
The structure of binary mixtures of bosonic and fermionic atoms in an external trapping potential at zero temperature is studied on the basis of a modified Gross-Pitaevskii equation for the bosonic component which self-consistently includes the mean-field interaction generated by the fermionic cloud. The density of the fermionic component is described within the Thomas-Fermi approximation. The influence of the boson-boson and the boson-fermion s-wave interaction on the density profiles and the stability of the mixture is investigated systematically. Critical particle numbers for the mean-field…
