Efimov Effect Research Articles

Renormalization of three-body interaction in DDK system

In systems comprising three identical particles, the Efimov effect emerges in the unitary limit. This article, utilizing the method of effective field theory along with the effective range expansion (ERE) and low-momentum expansion, reproduces the Efimov effect beyond unitary limit and identical case and applies to DDK system. It also links the high-energy scale with the ultraviolet cutoff in the three-body system. We discuss a method for obtaining exact solutions for the three-body wave function below the threshold of dimer production, that is D and Ds0⁎(2317). It turns out that introducing this scale is a necessary condition for the emergence of DDK three-body bound state. Since without this scale, the wave function would be solved below threshold with arbitrary binding energy, which violates the conditions of unitarity and energy quantization. The calculation involving this scale results in a discrete spectrum below the threshold and restores unitarity, making the effects induced by this scale significant for the formation of DDK bound state. In order to establish this in a complete formalism, the paper further gives an analytical expression for the three-body coupling constant, and compares it to the numerical calculation in DDK system.

Density Effects on the Interferometry of Efimov States by Modulating Magnetic Fields

Density Effects on the Interferometry of Efimov States by Modulating Magnetic Fields

Accurate Simulation of Efimov Physics in Ultracold Atomic Gases with Realistic Three-Body Multichannel Interactions

We give a detailed and self-contained description of a recently developed theoretical and numerical method for the simulation of three identical bosonic alkali-metal atoms near a Feshbach resonance, where the Efimov effect is induced. The method is based on a direct construction of the off-shell two-body transition matrix from exact eigenfunctions of the embedded two-body Hamiltonians, obtained using realistic parameterizations of the interaction potentials which accurately reproduce the molecular energy levels. The transition matrix is then inserted into the appropriate three-body integral equations, which may be efficiently solved on a computer. We focus especially on the power of our method in including rigorously the effects of multichannel physics on the three-body problem, which are usually accounted for only by various approximations. We demonstrate the method for 7Li, where we recently showed that a correct inclusion of this multichannel physics resolves the long-standing disagreement between theory and experiment regarding the Efimovian three-body parameter. We analyze the Efimovian enhancement of the three-body recombination rate on both sides of the Feshbach resonance, revealing strong sensitivity to the spin structure of the model thus indicating the prevalence of three-body spin-exchange physics. Finally, we discuss an extension of our methodology to the calculation of three-body bound-state energies.

Universality of Efimov states in highly mass-imbalanced cold-atom mixtures with van der Waals and dipole interactions

We study three-body systems in a mass-imbalanced, two-component, cold-atom mixture, and we investigate the three-body parameter of their Efimov states for both bosonic and fermionic systems, with a major focus on the Er-Er-Li Efimov states. For a system interacting solely via van der Waals interactions, the van der Waals universality of the three-body parameter is analytically derived using the quantum defect theory. With the addition of a perturbative dipole interaction between the heavy atoms, the three-body parameters of the bosonic and fermionic Efimov states are found to behave differently. When the dipole interaction is as strong as the van der Waals interaction, corresponding to realistic Er-Er-Li Efimov states, we show that the van der Waals universality persists once the effects of the nonperturbative dipole interaction are into the s-wave and p-wave scattering parameters between the heavy atoms. For a dipole interaction much stronger than the van der Waals interaction, we find that the universality of the Efimov states can be alternatively characterized by a quasi-one-dimensional scattering parameter due to a strong anisotropic deformation of the Efimov wave functions. Our work thus clarifies the interplay of isotropic and anisotropic forces in the universality of the Efimov states. Based on the renormalized van der Waals universality, the three-body parameter is estimated for specific isotopes of Er-Li cold-atom mixtures. Published by the American Physical Society 2024

The Movement of Efimov States in 2n Halo Nucleus: probing the roles of mass asymmetry and short range of interaction

This brief communication presents the results of our work to study the movement of Efimov states in 2-neutron halo nuclei. We discuss the emergence and evolution of Efimov states in to resonances beyond the two-body (n-Core) breakup threshold and also the movement of the Efimov state in a 2-neutron halo nucleus to the region of the so called, Thomas Collapse. We especially, probe the roles of mass asymmetry of the three-body system (n-n-core) and the short range of the n-core interaction in the movement of the Efimov states. The analysis has been carried out within the framework of a three-body model, assuming two-body, non-local, separable potentials for the binary sub-systems, namely, the n-n and n-core systems. The primary motivation has been to scan the parameter space of the range parameter and determine the value(s) for which the system can support the Efimov states. The analysis shows that the value of the range parameter $\beta$ = 5.2$\alpha$ ( $\alpha$ being the energy parameter given in terms of the deuteron binding energy ) produces the realistic three-body binding energy for $^{20}$C and also generates the excited Efimov states. However, as the value of $\beta$ is increased corresponding to decrease in the range of the two-body (n-core) interaction the ground and excited state binding energies increase drastically. In other words, as the range of the two-body interaction tends to zero the system enters the Thomas Collapse region away from the Efimov region.

Effects of p-wave interactions on Borromean Efimov trimers in heavy–light Fermi systems

We investigate the effects of p-wave interactions on Efimov trimers in systems comprising two identical heavy fermions and a light particle, with mass ratios larger than 13.6. Our focus lies on the Borromean regime where the ground-state trimer exists in the absence of dimers. Using pair-wise Lennard–Jones potentials and concentrating on the Lπ=1− symmetry, we explore the critical value of the interspecies s-wave scattering length a c at which the Borromean state appears in several two-component particle systems. Our exploration encompasses the universal properties of a c and the influence of p-wave fermion–fermion interactions on its value. We find that, in the absence of p-wave fermion–fermion interactions, a c is determined universally by the van der Waals radius and mass ratio. However, the introduction of p-wave fermion–fermion interactions unveiled a departure from this universality. Our calculations show that the critical interspecies scattering length a c now depends on the details of the fermion–fermion p-wave interaction. And, the presence of p-wave fermion–fermion interactions favors the formation of the Borromean state. Additionally, our investigation reveals that Efimov effect in the 1− symmetry persist even when the fermion–fermion interaction reaches the p-wave unitary limit.

Evolution of Efimov states

Evolution of Efimov states

Reshaped three-body interactions and the observation of an Efimov state in the continuum

Efimov trimers are exotic three-body quantum states that emerge from the different types of three-body continua in the vicinity of two-atom Feshbach resonances. In particular, as the strength of the interaction is decreased to a critical point, an Efimov state merges into the atom-dimer threshold and eventually dissociates into an unbound atom-dimer pair. Here we explore the Efimov state in the vicinity of this critical point using coherent few-body spectroscopy in 7Li atoms using a narrow two-body Feshbach resonance. Contrary to the expectation, we find that the 7Li Efimov trimer does not immediately dissociate when passing the threshold, and survives as a metastable state embedded in the atom-dimer continuum. We identify this behavior with a universal phenomenon related to the emergence of a repulsive interaction in the atom-dimer channel which reshapes the three-body interactions in any system characterized by a narrow Feshbach resonance. Specifically, our results shed light on the nature of 7Li Efimov states and provide a path to understand various puzzling phenomena associated with them.

Interferometry of Efimov states in thermal gases by modulated magnetic fields

Interferometry of Efimov states in thermal gases by modulated magnetic fields

Rigorous derivation of the Efimov effect in a simple model

We consider a system of three identical bosons in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^3$$\\end{document} with two-body zero-range interactions and a three-body hard-core repulsion of a given radius a>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ a > 0$$\\end{document}. Using a quadratic form approach, we prove that the corresponding Hamiltonian is self-adjoint and bounded from below for any value of a. In particular, this means that the hard-core repulsion is sufficient to prevent the fall to the center phenomenon found by Minlos and Faddeev in their seminal work on the three-body problem in 1961. Furthermore, in the case of infinite two-body scattering length, also known as unitary limit, we prove the Efimov effect, i.e., we show that the Hamiltonian has an infinite sequence of negative eigenvalues En\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E_n$$\\end{document} accumulating at zero and fulfilling the asymptotic geometrical law En+1/En→e-2πs0forn→+∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\;E_{n+1} / E_n \\; \\rightarrow \\; e^{-\\frac{2\\pi }{s_0}}\\,\\; \\,\ ext {for} \\,\\; n\\rightarrow +\\infty $$\\end{document} holds, where s0≈1.00624\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s_0\\approx 1.00624$$\\end{document}.

Universal geometry of two-neutron halos and Borromean Efimov states close to dissociation

The geometry of Borromean three-body halos, such as two-neutron halo nuclei or triatomic molecules close to dissociation, is investigated using a three-body model. This model enables to analytically derive the universal geometric properties found recently within an effective-field theory for halos made of a core and two resonantly-interacting particles [Phys. Rev. Lett., 128, 212501 (2022)]. It is shown that these properties not only apply to the ground three-body state, but also to all the excited (Efimov) states where the core-particle interaction is resonant. Furthermore, a universal geometry persists away from the resonant regime between the two particles, for any state close to the three-body threshold. This “halo universality”, which applies equally to all states, is different from the Efimov universality, which is only approximate for the ground state. It is explained by the separability of the hyper-radius and hyper-angles close to the three-body dissociation threshold.

The Faddeev and Schrödinger approaches to Efimov states—a numerical comparison

We compare effective hyperradial three-body potentials calculated using the S-wave part of the Faddeev equations to calculations using the full Schrödinger equation. As two-body model potential we test both a short-range potential and a Lennard-Jones potential with van der Waals tail. In the former case we find excellent agreement between the two methods for the lowest adiabatic state, indicating that the Faddeev method can be a useful tool also for numerical computations. For excited states the two methods show important differences, but agree for hyperradii larger than about five times the range of the potential (independent of the value of the scattering length and of the number of bound states). For the van der Waals potential, we focus on how well the Faddeev method reproduces the so-called van der Waals universality. We find that indeed the universality is manifest also using this method, but at a slightly different value of the universal parameter . We further derive an efficient method to solve the integro-differential equation arising in the Faddeev method.

Single-particle momentum distribution of Efimov states in noninteger dimensions

Single-particle momentum distribution of Efimov states in noninteger dimensions

Formation of Cluster Systems in Chaotic Condensed Media

Purpose. The study of cluster formation in a system of chaotically moving and interacting particles taking into account the Efimov effect and the "golden" section.Methods. Methods of mathematical modeling, quantum mechanics, a model of solid spheres, and a cluster model were used.Results. Within the framework of the proposed work, it is noted that in a three-particle system of particles, it is possible to form their spatial configuration in the form of a "golden" triangle, and in the case of an excited state of two particles, the third particle is far enough away from the other two, it is this configuration that corresponds to the conditions for the occurrence of the Efimov effect in a three-particle system.Based on the mathematical formalism of the description of self-organization processes in the work, it is shown that in chaotic environments within the framework of the Efimov model, with the involvement of the "golden" section in the mutual arrangement of three interacting particles, it is possible to form disk-shaped clusters containing a "magic" number of particles. In the structure of these clusters, the formation of quantum-dimensional regions in the form of a torus is possible. The parameters of such areas are defined.Conclusion. The described model of the formation and decay of disk-shaped clusters, taking into account the Efimov effect and the "golden" section rule, allows us, without resorting to a complex solution of equations in the three-body problem, to obtain important relations following from strict theories. The proposed approach implies the possibility of self-organization of clusters and the formation of quantum-dimensional regions in their structure, for example, in the form of a torus with a potential well, capable of capturing charged particles and determining their energy spectrum, as well as explaining the appearance of spectral bands in the IR spectra of substances.The proposed approach may be of practical importance, for example, for predicting the IR spectra of liquids, the presence of quantum dots in liquids with a wide spectrum of excitation from UV to IR radiation.

Compensating three-dimensional field inhomogeneity in cold atom Efimov-state search by a time-averaged optical potential.

The Efimov effect and its universal property are of paramount importance in quantum few-body physics. Despite this, the predicted ground state Efimov resonance has not yet been observed in 39,40,41K-87Rb mixtures within the currently available observation window. Cooling atoms in the microgravity environment of outer space might overcome this limitation, whereas the residual curvature of the strong magnetic fields may result in significant atom leakage. In this work, we propose an optical method based on far-detuned time-averaged dipole potential to counteract the three-dimensional inhomogeneous field. The target intensity distribution can be conveniently obtained by modulating the central position of the quasi-1D print beam using acoustic optical modulators. Within a volume of 300 × 300 × 400µm3, the residual potential fluctuations can be reduced by two orders of magnitude to less than 100 pK. The proposed approach offers a realistic prospect of investigating the Efimov-type resonance in the 40K-87Rb Bose-Fermi mixture.

Multichannel hyperspherical model for Efimov physics with van der Waals interactions controlled by a Feshbach resonance

Here we present a four-channel model that incorporates a magnetically tunable Feshbach resonance in a system of three atoms that interact via pairwise van der Waals interactions. Our method is designed to model recent experiments where the tunability of the scattering length has been used to study three-body Efimov states, which appear in the limit of a diverging two-body scattering length. Using this model, we calculate three-body adiabatic and effective potential curves and study how the strength (or width) of the Feshbach resonance affects the three-body potential that is connected to the Efimov effect. We find that the position of the repulsive barrier, which has been used to explain the so-called van der Waals universality in broad resonances, is slightly shifted as the narrow resonance limit is approached and that this shift is correlated to the appearance of two avoided crossings in the adiabatic energy landscape. More importantly, the attractive well is markedly shifted upward in energy and is extremely shallow for the narrowest resonance. We argue that this behavior is connected to the breakdown of van der Waals universality for weak (narrow) resonances.

Erratum: Single-particle momentum distributions of Efimov states in mixed-species systems [Phys. Rev. A 87 , 062702 (2013)

Erratum: Single-particle momentum distributions of Efimov states in mixed-species systems [Phys. Rev. A <b>87</b> , 062702 (2013)

Loosely bound few-body states in a spin-1 gas with near-degenerate continua

A distinguishing feature of ultracold collisions of bosonic lithium atoms is the presence of two near-degenerate two-body continua. The influence of such a near degeneracy on the few-body physics in the vicinity of a narrow Feshbach resonance is investigated within the framework of a minimal model with two atomic continua and one closed molecular channel. The model allows analysis of the spin composition of loosely bound dimers and trimers. In the two-body sector the well-established coupled-channel calculation phenomenology of lithium is qualitatively reproduced, and its particularities are emphasized and clarified. In the three-body sector we find that the Efimov trimer energy levels follow a different functional form as compared to a single continuum scenario while the thresholds remain untouched. This three-channel model with two atomic continua complements our earlier developed three-channel model with two molecular channels [Yudkin and Khaykovich, Phys. Rev. A 103, 063303 (2021)] and suggests that the experimentally observed exotic behavior of the first excited Efimov energy level (Yudkin, Elbaz, and Khaykovich, arXiv:2004.02723) is most probably caused by the short-range details of the interaction potential.

P -wave Efimov physics implications at unitarity

Equal mass fermionic trimers with two different spin components near the unitary limit are shown to possess a universal van der Waals bound or resonance state near $s$-wave unitarity, when $p$-wave interactions are included between the particles with equal spin. Our treatment uses a single-channel Lennard-Jones interaction with long range two-body van der Waals potentials. While it is well-known that there is no true Efimov effect that would produce an infinite number of bound states in the unitary limit, we demonstrate that another type of universality emerges for the symmetry $L^{\Pi}=1^{-}$. The universality is a remnant of Efimov physics that exists in this system at $p$-wave unitarity, and it leads to modified threshold and scaling laws in that limit. Application of our model to the system of three lithium atoms studied experimentally by Du, Zhang, and Thomas [Phys. Rev. Lett. {\bf 102}, 250402 (2009)] yields a detailed interpretation of their measured three-body recombination loss rates.

Tuning of Efimov states in non-integer dimensions

The purpose of this paper is to show that, by combining Feshbach resonances with external confining potentials, the energy scale factor of neighboring Efimov states can be tremendously reduced. The Efimov conditions can be reached for systems made of three different particles. For the case of two identical light particles and a heavy particle, the energy factor can be reduced by many orders of magnitude, and the Efimov states are in this way more easily reachable experimentally. The equivalence between external potentials and the formulation in terms of non-integer dimensions, d, is exploited. The technically simpler d-method is used to derive analytic expressions for two-component relative wave functions describing two short-range square-well interacting particles. The two components express one open and one closed channel. The scattering length is obtained after phase shift expansion, providing an analytic form for the Efimov condition. We illustrate the results by means of systems made of ^7Li, ^{39}K, and ^{87}Rb, with realistic parameters. The related pairs of dimension and magnetic field are shown and discussed. The results are universal as they only rely on large-distance properties.Graphic abstract