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 Innsbruck experiment supports this universal prediction, and it provides the first known evidence to date that a direct four-body recombination process plays a role. While we confirm the conclusion reached previously by some studies that no true Efimov effect exists for four particles, we demonstrate the existence of a universal class of four-body states that are intimately connected with three-body Efimov states.
Using a generalized Bohr model and the hyper-spherical formalism for a three-body system, we derive the Thomas theorem assuming a simple interaction depending on the range of the potential. We discuss the conditions for which an unbound two-body system produces a bound three-body system and derive universal energy functions. We apply our model to $^{4}$He and Triton atoms as well as to the triton nucleus. Using their scattering lengths and effective ranges, we are able to reproduce the two-body or the three-body binding energies with only one parameter fitted. Prediction for excited (Efimov) l…
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…
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…
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 propose to observe the Efimov effect experimentally by applying an external electric field on atomic three-body systems. We first derive the lowest order effective two-body interaction for two spin zero atoms in the field. Then we solve the three-body problem and search for the extreme spatially extended Efimov states. We use helium trimers as an illustrative numerical example and estimate the necessary field strength to be less than 2.7 V/angstrom.
We present a theoretical analysis of rf association of Efimov trimers in a 2-component Bose gas with short-range interactions. Using the adiabatic hyperspherical Green's function formalism to solve the quantum 3-body problem, we obtain universal expressions for 3-body rf association rates as a function of the s-wave scattering length $a$. We find that the association rates scale as $a^{-2}$ in the limit of large $a$, and diverge as $a^3 a_{ad}^{3}$ whenever an Efimov state crosses the atom-dimer threshold (where $a_{ad}$ stands for the atom-dimer scattering length). Our calculations show that …
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…
We derive an analytic formula which describes the final bound state dependence in ultracold three-body recombination. Using an energy-dependent loss parameter, the recently observed broad resonance in an ultracold gas of $^{6}$Li atoms [T. B. Ottenstein {\it et al.}, Phys. Rev. Lett. 101, 203202 (2008)l J. H. Huckans {\it et al.}, Phys. Rev. Lett. 102, 165302 (2009)] is described quantitatively. We also provide an analytic and approximation for the three-body recombination rate which encapsulates the underlying physics of the universal three-body recombination process.
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 the impact of three-body physics in quenched unitary Bose gases, focusing on the role of the Efimov effect. Using a local density model, we solve the three-body problem and determine three-body decay rates at unitary, finding density-dependent, log-periodic Efimov oscillations, violating the expected continuous scale-invariance in the system. We find that the breakdown of continuous scale-invariance, due to Efimov physics, manifests also in the earliest stages of evolution after the interaction quench to unitarity, where we find the growth of a substantial population of Efimov states …
We report on the observation of triatomic Efimov resonances in an ultracold gas of cesium atoms. Exploiting the wide tunability of interactions resulting from three broad Feshbach resonances in the same spin channel, we measure magnetic-field dependent three-body recombination loss. The positions of the loss resonances yield corresponding values for the three-body parameter, which in universal few-body physics is required to describe three-body phenomena and in particular to fix the spectrum of Efimov states. Our observations show a robust universal behavior with a three-body parameter that st…
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 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.
We measure the critical scattering length for the appearance of the first three-body bound state, or Efimov three-body parameter, at seven different Feshbach resonances in ultracold 39K atoms. We study both intermediate and narrow resonances, where the three-body spectrum is expected to be determined by the non-universal coupling of two scattering channels. We observe instead approximately the same universal relation of the three-body parameter with the two-body van der Waals radius already found for broader resonances, which can be modeled with a single channel. This unexpected observation su…
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 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 show that four heavy fermions interacting resonantly with a lighter atom (4+1 system) become Efimovian at mass ratio 13.279(2), which is smaller than the corresponding 2+1 and 3+1 thresholds. We thus predict the five-body Efimov effect for this system in the regime where any of its subsystem is non- Efimovian. For smaller mass ratios we show the existence and calculate the energy of a universal 4+1 pentamer state, which continues the series of the 2+1 trimer predicted by Kartavtsev and Malykh and 3+1 tetramer discovered by Blume. We also show that the effective-range correction for the ligh…
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…
In this paper we discuss the recent discovery of the universality of the three-body parameter (3BP) from Efimov physics. This new result was identified by recent experimental observations in ultracold quantum gases where the value of the s-wave scattering length, $a=a_-$, at which the first Efimov resonance is created was found to be nearly the same for a range of atomic species --- if scaled as $a_-/r_{\rm vdW}$, where $r_{\rm vdW}$ is the van der Waals length. Here, we discuss some of the physical principles related to these observations that emerge from solving the three-body problem with v…
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…
Three oriented bosonic dipoles are treated using the hyperspherical adiabatic representation, providing numerical evidence that the Efimov effect persists near a two-dipole resonance and in a system where angular momentum is not conserved. Our results further show that the Efimov features in scattering observables become universal, with a known three-body parameter, i.e. the resonance energies depend only on the two-body physics, which also has implications for the universal spectrum of the four-dipole problem. Moreover, the Efimov states should be long-lived, which is favorable for their crea…
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 …
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…
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…
We analyse the recent experiments investigating the low-energy physics of three lithium 6 atoms in different internal states with resonant two-body scattering lengths. All observed features are qualitatively consistent with the expected Efimov effect, i.e. the effective universal three-body attraction that arises for large values of the scattering lengths. However, we find that a quantitative description at negative energy requires non-universal two- and three-body corrections due to presently unknown behaviour at short distance. An attempt to implement these corrections is made through energy…
We find that universal three-body physics extends beyond the threshold regime to non-zero energies. For ultracold atomic gases with a negative two-body $s$-wave scattering length near a Feshbach resonance, we show the resonant peaks characteristic of Efimov physics persist in three-body recombination to higher collision energies. For this and other inelastic processes, we use the adiabatic hyperspherical representation to derive universal analytical expressions for their dependence on the scattering length, the collision energy, and --- for narrow resonances --- the effective range. These expr…
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 demonstrate the emergence of universal Efimov physics for interacting photons in cold gases of Rydberg atoms. We consider the behavior of three photons injected into the gas in their propagating frame, where a paraxial approximation allows us to consider them as massive particles. In contrast to atoms and nuclei, the photons have a large anisotropy between their longitudinal mass, arising from dispersion, and their transverse mass, arising from diffraction. Nevertheless, we show that in suitably rescaled coordinates the effective interactions become dominated by s-wave scattering near thres…
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…
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…
We consider the Efimov trimer theory as a possible framework to explain recently observed losses by inelastic three-body collisions in a three-hyperfine-component ultracold mixture of lithium 6. Within this framework, these losses would arise chiefly from the existence of an Efimov trimer bound state below the continuum of free triplets of atoms, and the loss maxima (at certain values of an applied magnetic field) would correspond to zero-energy resonances where the trimer dissociates into three free atoms. Our results show that such a trimer state is indeed possible given the two-body scatter…
We study the quench of a degenerate ultracold Bose gas to the unitary regime, where interactions are as strong as allowed by quantum mechanics. We lay the foundations of a cumulant theory able to capture simultaneously the three-body Efimov effect and ergodic evolution. After an initial period of rapid quantum depletion, a universal prethermal stage is established characterized by a kinetic temperature and an emergent Bogoliubov dispersion law while the microscopic degrees of freedom remain far-from-equilibrium. Integrability is then broken by higher-order interaction terms in the many-body Ha…
Momentum Distribution and Contact Parameters of a mass-imbalanced three-body system across the Efimov-Unatomic transition
Instituto Tecnológico de Aeronáutica（巴西航空研究院）；Université Paris-Saclay（巴黎萨克雷大学）；CNRS/IN2P3, IJCLab（法国国家科学研究中心/法国国家核物理和粒子物理研究所，IJCLab实验室）；Universidade Estadual Paulista（圣保罗州立大学）
Using a combination of bound-state spectroscopy and loss spectroscopy, we pinpoint eight intrastate Feshbach resonances in $^{39}$K, as well as six previously unexplored interstate ones. We also perform a detailed characterization of four of the intrastate resonances and two of the interstate ones. We carry out coupled-channel scattering calculations and find good agreement with experiment. The combination of experiment and theory provides a faithful map of the scattering length $a$ and permits precision measurements of the signatures of Efimov physics across four intermediate-strength resonan…
We perform precise studies of two- and three-body interactions near an intermediate-strength Feshbach resonance in $^{39}\mathrm{K}$ at $33.5820(14)\thinspace$G. Precise measurement of dimer binding energies, spanning three orders of magnitude, enables the construction of a complete two-body coupled-channel model for determination of the scattering lengths with an unprecedented low uncertainty. Utilizing an accurate scattering length map, we measure the precise location of the Efimov ground state to test van der Waals universality. Precise control of the sample's temperature and density ensure…
We solved analytically the three-body mass-imbalanced problem embedded in D dimensions for zero-range resonantly interacting particles. We derived the negative energy eigenstates of the three-body Schrodinger equation by imposing the Bethe-Peierls boundary conditions in D-dimensions for zero-energy two-body bound states. The solution retrieves the Efimov-like discrete scaling factor dependence with dimension. The analytical form of the mass-imbalanced three-body bound state wave function can be used to probe the effective dimension of asymmetric cold atomic traps for Feshbach resonances tuned …
We present a comprehensive collection of ultracold three-body collisions properties near overlapping Feshbach resonances. Our results incorporate variations of all scattering lengths and demonstrate novel collisional behavior, such as atom-molecule interference effects. Taking advantage of the unique ways in which these collisions reflect Efimov physics, new pathways to control atomic and molecular losses open up. Further, we show that overlapping resonances can greatly improve the chances of observing multiple Efimov features in an ultracold quantum gas for nearly any system.
We investigate the energy dependence of inelastic processes in systems which possess Efimov states. We consider the three-body recombination rate $K_3$ where three free atoms interact to produce an atom--dimer pair, and the relaxation rate $K_{\rm rel}$ where an atom quenches a weakly bound state of a dimer near an Efimov resonance to more deeply bound levels. Using a model capturing the key features of the Efimov problem, we identify new energy regimes for $K_3$, namely the NTR (Near Threshold Resonance) regime behavior $E^{-2}$ for negative scattering lengths and the NTS (Near Threshold Supp…
We consider a system of three helium-4 atoms, which is so far the simplest realistic three-body system exhibiting the Efimov effect, in order to analyse deviations from the universal Efimov three-body spectrum. We first calculate the bound states using a realistic two-body potential, and then analyse how they can be reproduced by simple effective models beyond Efimov's universal theory. We find that the non-universal variations of the first two states can be well reproduced by models parametrized with only three quantities: the scattering length and effective range of the original potential, a…
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 $^4$He-$^4$He 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 calcula…
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 …
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…
A recent rejuvenation of experimental and theoretical interest in the physics of few- body systems has provided deep, fundamental insights into a broad range of problems. Few-body physics is a cross-cutting discipline not restricted to conventional subject ar- eas such as nuclear physics or atomic or molecular physics. To a large degree, the recent explosion of interest in this subject has been sparked by dramatic enhancements of experimental capabilities in ultracold atomic systems over the past decade, which now permit atoms and molecules to be explored deep in the quantum mechanical limit w…
For a system of three identical bosons interacting via short-range forces, when two of the atoms are about to form a two-body s-wave dimer, there exists an infinite number of three-body bound states. This effect is the well-known Efimov effect. These three-body states (Efimov states) are found to be universal for ultracold atomic gases and the lowest Efimov state crosses the three-body break-up threshold when the s-wave two-body scattering length is $a \approx -9.73 r_{\rm vdW}$, $r_{\rm vdW}$ being the van der Waals length. This article focuses on a generalized version of this Efimov scenario…
We report a thorough investigation of finite-temperature effects on three-body recombination near a triatomic Efimov resonance in an ultracold gas of cesium atoms. Our measurements cover a wide range from a near-ideal realization of the zero-temperature limit to a strongly temperature-dominated regime. The experimental results are analyzed within a recently introduced theoretical model based on a universal zero-range theory. The temperature-induced shift of the resonance reveals a contribution that points to an energy-dependence of the three-body parameter. We interpret this contribution in te…
Systems of three particles show a surprising feature in their bound state spectrum: a series of geometrically scaled states, known as Efimov states. These states have not yet been observed directly, but many recent experiments show indirect evidence of their existence via the so-called recombination process. The theories that predict the Efimov states also predicts either resonant enhancement of the recombination process or suppression by destructive interference, depending on the sign of the interaction between the particles. The theories predict universal features for the Efimov states, for …
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…
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…
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…
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…
