A new narrow 3-atom loss resonance associated with an Efimov trimer crossing the 3-atom threshold has recently been discovered in a many-body system of ultracold 6Li atoms in the three lowest hyperfine spin states at a magnetic field near 895 G. O'Hara and coworkers have used measurements of the 3-body recombination rate in this region to determine the complex 3-body parameter associated with Efimov physics. Using this parameter as the input, we calculate the universal predictions for the spectrum of Efimov states and for the 3-body recombination rate in the universal region above 600 G where all three scattering lengths are large. We predict an atom-dimer loss resonance at (672 +/- 2) G associated with an Efimov trimer disappearing through an atom-dimer threshold. We also predict an interference minimum in the 3-body recombination rate at (759 +/- 1) G where the 3-spin mixture may be sufficiently stable to allow experimental study of the many-body system.
Lithium-6 atoms in the three lowest hyperfine states display universal properties when the S-wave scattering length between each pair of states is large. Recent experiments reported four pronounced features arising from Efimov physics in the atom-dimer relaxation rate, namely two resonances and two local minima. We use the universal effective field theory to calculate the atom-dimer relaxation rate at zero temperature. Our results describe the four features qualitatively and imply there is a hidden local minimum. In the vicinity of the resonance at 685 G, we perform a finite temperature calcul…
Bosonic atom-trimer scattering is studied in the unitary limit using momentum-space equations for four-particle transition operators. The impact of the Efimov effect on the atom-trimer scattering observables is explored and a number of universal relations is established. Positions and widths of tetramer resonances are determined. Trimer relaxation rate constant is calculated.
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.
An Efimov trimer near the atom-dimer threshold can increase the atom loss rate in ultracold trapped atoms through the {\it avalanche mechanism} proposed by Zaccanti et al. A 3-body recombination event creates an energetic atom and dimer, whose subsequent elastic collisions produce additional atoms with sufficient energy to escape from the trapping potential. We use Monte Carlo methods to calculate the average number of atoms lost and the average heat generated by recombination events in both a Bose-Einstein condensate and a thermal gas. We take into account the energy-dependence of the cross s…
We develop an experimental technique for rf-association of Efimov trimers from three-atoms continuum. We apply it to probe the lowest accessible Efimov energy level in bosonic lithium in the region where strong deviations from the universal behavior are expected, and provide quantitative study of this effect. Position of the Efimov resonance at the atom-dimer threshold, measured with a different experimental technique, concurs with the rf-association results.
We use an effective-field-theory framework to analyze the Efimov effect in heteronuclear three-body systems consisting of two species of atoms with a large interspecies scattering length. In the leading-order description of this theory, various three-body observables in heteronuclear mixtures can be universally parameterized by one three-body parameter. We present the next-to-leading corrections, which include the effects of the finite interspecies effective range and the finite intraspecies scattering length, to various three-body observables. We show that only one additional three-body param…
The bound states of a system consisting of two heavy identical atoms and one light electron interacting through the finite-range pairwise potentials are explored, focusing on their dependence on the electron-atom scattering length. In the case of an exact resonance in the electron-atom interaction, the binding energy of an electron yields an effective $1/r^{2}$ potential for the relative motion of the atoms; One major finding is a universal potential that depends on the polarization length which goes beyond the Efimov region. An analytic expression for that potential is extracted from numerica…
We observed an enhanced atom-dimer relaxation due to the existence of Efimov states in a three-component mixture of 6Li atoms. We measured the magnetic-field dependence of the atom-dimer loss coefficient in the mixture of atoms in state |1> and dimers formed in states |2> and |3>, and found two peaks corresponding to the degeneracy points of the |23> dimer energy level and energy levels of Efimov trimers. We found that the locations of these peaks disagree with universal theory predictions, in a way that cannot be explained by non-universal two-body properties. We constructed theoretical model…
Particles with resonant short-range interactions have universal properties that do not depend on the details of their structure or their interactions at short distances. In the three-body system, these properties include the existence of a geometric spectrum of three-body Efimov states and a discrete scaling symmetry leading to log-periodic dependence of observables on the scattering length. Similar universal properties appear in the four-body system and possibly higher-body systems as well. For example, universal four-body states have recently been predicted and observed in experiment. These …
Recent cold atom experiments report a surprising universal scaling of the first Efimov resonance position a_{-}^1 by the two-body van der Waals length r_{vdW}. The ratio C=-a_{-}^1/r_{vdW}=8.5~9.5 for identical particles appears to be a constant regardless of the atomic spin configuration, the Feshbach resonance employed to tune the scattering length, and even the atomic species, with K-39 being the only exception. This result indicates that the Efimov energy structure is insensitive to the details of the short range potential. We suggest that the universality results from the quantum reflecti…
The "avalanche mechanism" has been used to relate Efimov trimer states to certain enhanced atom loss features observed in ultracold atom gas experiments. These atom loss features are argued to be a signature of resonant atom-molecule scattering that occurs when an Efimov trimer is degenerate with the atom-molecule scattering threshold. However, observation of these atom loss features has yet to be combined with the direct observation of atom-molecule resonant scattering for any particular atomic species. In addition, recent Monte-Carlo simulations were unable to reproduce a narrow loss feature…
We study the recombination process of three atoms scattering into an atom and diatomic molecule in heteronuclear mixtures of ultracold atomic gases with large and positive interspecies scattering length at finite temperature. We calculate the temperature dependence of the three-body recombination rates by extracting universal scaling functions that parametrize the energy dependence of the scattering matrix. We compare our results to experimental data for the 40K-87Rb mixture and make a prediction for 6Li-87Rb. We find that contributions from higher partial wave channels significantly impact th…
Using the framework of effective field theory, we present a detailed study of the Efimov effect in higher partial waves for systems of two identical particles and a third distinguishable particle. Depending on the total angular momentum $L$, the two identical particles must be bosons or fermions. We derive analytical expressions for the elastic and inelastic atom-dimer scattering cross sections as well as the atom-dimer relaxation rate at the dimer breakup threshold. For the experimentally most relevant case of P-waves, we numerically calculate the atom-dimer scattering cross sections and rela…
The discrete Efimov scaling behavior, well-known in the low-energy spectrum of three-body bound systems for large scattering lengths (unitary limit), is identified in the energy dependence of atom-molecule elastic cross-section in mass imbalanced systems. That happens in the collision of a heavy atom with mass $m_H$ with a weakly-bound dimer formed by the heavy atom and a lighter one with mass $m_L \ll m_H$. Approaching the heavy-light unitary limit the $s-$wave elastic cross-section $σ$ will present a sequence of zeros/minima at collision energies following closely the Efimov geometrical law.…
Efimov physics originally refers to a system of three particles. Here we review recent theoretical progress seeking for manifestations of Efimov physics in systems composed of more than three particles. Clusters of more than three bosons are tied to each Efimov trimer, but no independent Efimov physics exists there beyond three bosons. The case of a few heavy fermions interacting with a lighter atom is also considered, where the mass ratio of the constituent particles plays a significant role. Following Efimov's study of the (2+1) system, the (3+1) system was shown to have its own critical mas…
The Efimov effect, with its ladder of weakly bound three-atomic molecules, poses intriguing questions in the theoretically controversial and experimentally demanding regime of merging of the first excited Efimov energy level with the atom-dimer continuum. Using an original experimental technique, where a coherent superposition state of diatomic and triatomic molecules is utilized, we investigate this regime and reveal a striking behavior: Instead of merging with the atom-dimer continuum the trimer energy level rebounds from it and becomes a deeper bound state again. In addition, instead of a t…
We study the two-body and three-body bound states in ultracold atomic mixtures with one of the atoms subjected to an isotropic spin-orbit (SO) coupling. We consider a system of two identical fermions interacting with one SO coupled atom. It is found that there can exist two types of three-body bound states, Efimov trimers and universal trimers. The Efimov trimers are energetically less favored by the SO coupling, which will finally merge into the atom-dimer threshold as increasing the SO coupling strength. Nevertheless, these trimers exhibit a new kind of discrete scaling law incorporating the…
We explore the properties of weakly bound bosonic states in the strongly interacting regime. Combining a correlated-Gaussian (CG) basis set expansion with a complex scaling method, we extract the energies and structural properties of bosonic cluster states with $N\le6$ for different two-body potentials. The identification of five- and six-body resonances attached to an excited Efimov trimer provides strong support to the premise of Efimov universality in bosonic systems. Our study also reveals a rich structure of bosonic cluster states. Besides the lowest cluster states which behave as bosonic…
In 1970 V. Efimov predicted a puzzling quantum-mechanical effect that is still of great interest today. He found that three particles subjected to a resonant pairwise interaction can join into an infinite number of loosely bound states even though each particle pair cannot bind. Interestingly, the properties of these aggregates, such as the peculiar geometric scaling of their energy spectrum, are universal, i.e. independent of the microscopic details of their components. Despite an extensive search in many different physical systems, including atoms, molecules and nuclei, the characteristic sp…
The multichannel Efimov physics is investigated in ultracold heteronuclear admixtures of K and Rb atoms. We observe a shift in the scattering length where the first atom-dimer resonance appears in the $^{41}$K-$^{87}$Rb system relative to the position of the previously observed atom-dimer resonance in the $^{40}$K-$^{87}$Rb system. This shift is well explained by our calculations with a three-body model including the van der Waals interactions, and, more importantly, the multichannel spinor physics. With only minor difference in the atomic masses of the admixtures, the shift in the atom-dimer …
We discuss our recent observation of an atom-dimer Efimov resonance in an ultracold mixture of Cs atoms and Cs_2 Feshbach molecules [Nature Phys. 5, 227 (2009)]. We review our experimental procedure and present additional data involving a non-universal g-wave dimer state, to contrast our previous results on the universal s-wave dimer. We resolve a seeming discrepancy when quantitatively comparing our experimental findings with theoretical results from effective field theory.
Recent experiment has observed Feshbach resonances between alkaline and alkaline-earth atoms. These Feshbach resonances are insensitive to the nuclear spin of alkaline-earth atoms. Ultilizing this feature, we propose to take this system as a candidate to perform quantum simulation of the Kondo effect. An alkaline atom can form a molecule with an alkaline-earth atom with different nuclear spins, which plays the role of spin-exchange scattering responsible for the Kondo effect. Furthermore, we point out that the existence of three-body bound state and atom-molecule resonance due to the Efimov ef…
In view of recent experiments in ultracold atomic systems, the photoassociation of Efimov trimers, composed of three identical bosons, is studied utilizing the multipole expansion. We study both the normal hierarchy case, where one-body current is dominant, and the strong hierarchy case, relevant for photoassociation in ultracold atoms, where two-body current is dominant. For identical particles in the normal hierarchy case, the leading contribution comes from the r^2 s-mode operator and from the quadrupole d-mode operator. The s-mode reaction is found to be dominant at low temperature, while …
We use a minimal zero-range model for describing the bound state spectrum of three-body states consisting of two Cesium and one Lithium atom. Using a broad Feshbach resonance model for the two-body interactions, we show that recent experimental data can be described surprisingly well for particular values of the three-body parameter that governs the short-range behavior of the atomic potentials and is outside the scope of the zero-range model. Studying the spectrum as a function of the three-body parameter suggests that the lowest state seen in experiment could be influenced by finite range co…
We study the universality of the three-body parameters for systems relevant for ultracold quantum gases with positive $s$-wave two-body scattering lengths. Our results account for finite-range effects and their universality is tested by changing the number of deeply bound diatomic states supported by our interaction model. We find that the physics controlling the values of the three-body parameters associated with the ground and excited Efimov states is constrained by a variational principle and can be strongly affected by $d$-wave interactions that prevent both trimer states from merging into…
Efimov physics is a universal phenomenon arising in quantum three-body systems. For systems with resonant two-body interactions, Efimov predicted an infinite series of three-body bound states with geometric scaling symmetry. These Efimov states were first observed in cold Cs atoms and have since been reported in a variety of atomic systems. While theories predict non-universal behavior for narrow Feshbach resonances, experiments on Efimov resonances are so far consistent with predictions based on universal theories. Here we directly compare the Efimov spectra in a $^6$Li-$^{133}$Cs mixture nea…
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-Fermi crossover in Efimov spectrum, and formation of i…
We calculate shallow three-body bound states in the universal regime, defined by Efimov, with inclusion of both scattering length and effective range parameters. The universal spectrum is recovered for the least bound states, whereas for larger binding energies we find corrections to the universal scaling laws. We recover known results for broad Feshbach resonances with small effective range, whereas in the case of narrow resonances we find a distinct non-monotonic behavior of the threshold at which the lowest Efimov trimer merges with the three-body continuum. To address the issue of the phys…
A universal dimer is subject to secondary collisions with atoms when formed in a cloud of ultracold atoms via three-body recombination. We show that in a collisionally opaque medium, the value of the scattering length that results in the maximum number of secondary collisions may not correspond to the Efimov resonance at the atom-dimer threshold and thus can not be automatically associated with it. This result explains a number of controversies in recent experimental results on universal three-body states and supports the emerging evidence for the significant finite range corrections to the fi…
Extremely weakly-bound three-boson systems are predicted to exhibit intriguing universal properties such as discrete scale invariance. Motivated by recent experimental studies of the ground and excited helium trimers, this work analyzes the three-body parameter and the structural properties of three helium atoms as the s-wave scattering length is tuned artificially. Connections with theoretical and experimental studies of the Efimov scenario as it pertains to cold atom systems are made.
Efimov states are known to have a discrete real space scale invariance, working in momentum space we identify the relevant discrete scale invariance for the scattering amplitude defining its Weierstrass function as well. Through the use of the mathematical formalism for discrete scale invariance for the scattering amplitude we identify the scaling parameters from the pole structure of the corresponding zeta function, it's zeroth order pole is fixed by the Efimov physics. The corresponding geometrical fractal structure for Efimov physics in momentum space is identified as a ray across a logarit…
Ultracold atomic gases have developed into prime systems for experimental studies of Efimov three-body physics and related few-body phenomena, which occur in the universal regime of resonant interactions. In the last few years, many important breakthroughs have been achieved, confirming basic predictions of universal few-body theory and deepening our understanding of such systems. We review the basic ideas along with the fast experimental developments of the field, focussing on ultracold cesium gases as a well-investigated model system. Triatomic Efimov resonances, atom-dimer Efimov resonances…
Few-body physics related to the Efimov effect is discussed using the functional renormalization group method. After a short review of renormalization in its modern formulation we apply this formalism to the description of scattering and bound states in few-body systems of identical bosons and distinguishable fermions with two and three components. The Efimov effect leads to a limit cycle in the renormalization group flow. Recently measured three-body loss rates in an ultracold Fermi gas $^6$Li atoms are explained within this framework. We also discuss briefly the relation to the many-body phys…
Motivated by recent experiments on $^{6}$Li-$^{133}$Cs atomic mixtures with high mass imbalance, we study the Efimov correlation in atomic system of two heavy bosons ($^{133}$Cs) immersed in a bath of light fermions ($^{6}$Li). Using the Born-Oppenheimer approximation, we identify two different regimes, depending on the Fermi momentum of light fermions ($k_F$) and the boson-fermion scattering length $a_s(<0)$, where the presence of underlying Fermi sea plays distinct roles in the Efimov-type binding of bosons. Namely, in the regime $k_F|a_s|\lesssim 1$ ($k_F|a_s|\gtrsim1$), the Fermi sea induc…
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 calculate the neutron-Calcium-60 S-wave scattering phase shifts using state of the art coupled-cluster theory combined with modern ab initio interactions derived from chiral effective theory. Effects of three-nucleon forces are included schematically as density dependent nucleon-nucleon interactions. This information is combined with halo effective field theory in order to investigate the Calcium-60-neutron-neutron system. We predict correlations between different three-body observables and the two-neutron separation energy of Calcium-62. This provides evidence of Efimov physics along the C…
We convert a strongly interacting ultracold Bose gas into a mixture of atoms and molecules by sweeping the interactions from resonant to weak. By analyzing the decay dynamics of the molecular gas, we show that in addition to Feshbach dimers it contains Efimov trimers. Typically around 8\% of the total atomic population is bound into trimers, identified by their density-independent lifetime of about 100~$μ$s. The lifetime of the Feshbach dimers shows a density dependence due to inelastic atom-dimer collisions, in agreement with theoretical calculations. We also vary the density of the gas acros…
We present a state-of-the-art reanalysis of experimental results on Efimov resonances in the three-fermion system of $^6$Li. We discuss different definitions of the 3-body parameter (3BP) for Efimov states, and adopt a definition that excludes effects due to deviations from universal scaling for low-lying states. We develop a finite-temperature model for the case of three distinguishable fermions and apply it to the excited-state Efimov resonance to obtain the most accurate determination to date of the 3BP in an atomic three-body system. Our analysis of ground-state Efimov resonances in the sa…
Universal behaviour has been found inside the window of Efimov physics for systems with $N=4,5,6$ particles. Efimov physics refers to the emergence of a number of three-body states in systems of identical bosons interacting {\it via} a short-range interaction becoming infinite at the verge of binding two particles. These Efimov states display a discrete scale invariance symmetry, with the scaling factor independent of the microscopic interaction. Their energies in the limit of zero-range interaction can be parametrized, as a function of the scattering length, by a universal function. We have f…
A general mechanism for the loss of conformal invariance is the merger and disappearance of an infrared fixed point and an ultraviolet fixed point of a renormalization group flow. We show explicitly how this mechanism works in the case of identical bosons at unitarity as the spatial dimension $d$ is varied. For $d$ between the critical dimensions $d_{\rm 1}=2.30$ and $d_{\rm 2}=3.76$, there is loss of conformality as evidenced by the Efimov effect in the three-body sector. The beta function for an appropriate three-body coupling is a quadratic polynomial in that coupling. For $d<d_{\rm 1}$ and…
We address the microscopic origin of the universal three-body parameter that fixes the spectrum of three-atom systems in the Efimov regime. We identify it with the van der Waals two-body correlation, which causes the three-atom system to deform when the three atoms come within the distance of the van der Waals length, effectively preventing them from coming closer due to the kinetic-energy cost associated with this three-body deformation. This deformation mechanism explains the universal ratio of the scattering length at the triatomic resonance to the van der Waals length observed in several e…
Motivated by recent experimental investigations of Cs-Cs-Li Efimov resonances, this work theoretically investigates the few-body properties of $N-1$ non-interacting identical heavy bosons, which interact with a light impurity through a large $s$-wave scattering length. For Cs-Cs-Cs-Li, we predict the existence of universal four-body states with energies $E_4^{(n,1)}$ and $E_4^{(n,2)}$, which are universally linked to the energy $E_3^{(n)}$ of the $n$th Efimov trimer. For infinitely large $^{133}$Cs-$^6$Li and vanishing $^{133}$Cs-$^{133}$Cs scattering lengths, we find $(E_4^{(1,1)}/E_3^{(1)})^…
Efimov physics is drastically affected by the change of spatial dimensions. Efimov states occur in a tridimensional (3D) environment, but disappear in two (2D) and one (1D) dimensions. In this paper, dedicated to the memory of Prof. Faddeev, we will review some recent theoretical advances related to the effect of dimensionality in the Efimov phenomenon considering three-boson systems interacting by a zero-range potential. We will start with a very ideal case with no physical scales, passing to a system with finite energies in the Born-Oppenheimer (BO) approximation and finishing with a general…
Physical systems characterized by a shallow two-body bound or virtual state are governed at large distances by a continuous-scale invariance, which is broken to a discrete one when three or more particles come into play. This symmetry induces a universal behavior for different systems, independent of the details of the underlying interaction, rooted in the smallness of the ratio $\ell/a_B \ll 1$, where the length $a_B$ is associated to the binding energy of the two-body system $E_2=\hbar^2/m a_B^2$ and $\ell$ is the natural length given by the interaction range. Efimov physics refers to this u…
The Efimov effect is the only experimentally realized universal phenomenon that exhibits the renormalization-group limit cycle with the three-body parameter parametrizing a family of universality classes. Recent experiments in ultracold atoms have unexpectedly revealed that the three-body parameter itself is universal when measured in units of an effective range. By performing an exact functional renormalization-group analysis with various finite-range interaction potentials, we demonstrate that the onset of the renormalization-group flow into the limit cycle is universal, regardless of short-…
We have measured the binding energy of $^7$Li Feshbach molecules deep into the non-universal regime by associating free atoms in a Bose-Einstein condensate by modulating the magnetic field. We extract the scattering length from these measurements, correcting for non-universal short-range effects using several different methods. We find that field-dependent effective range corrections agree well with the data. With this more precise determination of the scattering length vs. field we reanalyze our previous data on the location of atom loss features produced by the Efimov effect \cite{PollackSci…
We report on the measurement of four-body recombination rate coefficients in an atomic gas. Our results obtained with an ultracold sample of cesium atoms at negative scattering lengths show a resonant enhancement of losses and provide strong evidence for the existence of a pair of four-body states, which is strictly connected to Efimov trimers via universal relations. Our findings confirm recent theoretical predictions and demonstrate the enrichment of the Efimov scenario when a fourth particle is added to the generic three-body problem.
We report the observation of a three-body recombination resonance in an ultracold gas of cesium atoms at a very large negative value of the $s$-wave scattering length. The resonance is identified as the second triatomic Efimov resonance, which corresponds to the situation where the first excited Efimov state appears at the threshold of three free atoms. This observation, together with a finite-temperature analysis and the known first resonance, allows the most accurate demonstration to date of the discrete scaling behavior at the heart of Efimov physics. For the system of three identical boson…
Probability distributions for correlation functions of particles interacting via random-valued fields are discussed as a novel tool for determining the spectrum of a theory. In particular, this method is used to determine the energies of universal N-body clusters tied to Efimov trimers, for even N, by investigating the distribution of a correlation function of two particles at unitarity. Using numerical evidence that this distribution is log-normal, an analytical prediction for the N-dependence of the N-body binding energies is made.
The Efimov effect was first predicted for three particles interacting at an $s$-wave resonance in three dimensions. Subsequent study showed that the same effect can be realized by considering two-body and three-body interactions in mixed dimensions. In this work, we consider the three-body problem of two bosonic $A$ atoms interacting with another single $B$ atom in mixed dimensions: The $A$ atoms are confined in a space of dimension $d_A$ and the $B$ atom in a space of dimension $d_B$, and there is an interspecies $s$-wave interaction in a $d_{\rm int}$-co-dimensional space accessible to both …
Physics is said to be universal when it emerges regardless of the underlying microscopic details. A prominent example is the Efimov effect, which predicts the emergence of an infinite tower of three-body bound states obeying discrete scale invariance when the particles interact resonantly. Because of its universality and peculiarity, the Efimov effect has been the subject of extensive research in chemical, atomic, nuclear and particle physics for decades. Here we employ an anisotropic Heisenberg model to show that collective excitations in quantum magnets (magnons) also exhibit the Efimov effe…
The Efimov effect in heteronuclear cold atomic systems is experimentally more easily accessible than the Efimov effect for identical atoms, because of the potentially smaller scaling factor. We focus on the case of two or three heavy identical bosons and another atom. The former case was recently observed in a mixture of 133Cs and 6Li atoms. We employ the Gaussian Expansion Method as developed by Hiyama, Kino et al.. This is a variational method that uses Gaussians that are distributed geometrically over a chosen range. Supplemental calculations are performed using the Skorniakov-Ter-Martirosi…
