Efimov Effect Research Articles

Energetics and Efimov states of three interacting bosons and mass-imbalanced fermions in a three-dimensional spherical harmonic trap

We consider a system of three particles, either three identical bosons or two identical fermions plus an impurity, within a three-dimensional isotropic trap interacting via a contact interaction. Using two approaches, one using an infinite sum of basis states for the wavefunction and the other a closed form wavefunction, we calculate the allowable energy eigenstates of the system as a function of the interaction strength, including the strongly and weakly interacting limits. For the fermionic case this is done while maintaining generality regarding particle masses. We find that the two methods of calculating the spectrum are in excellent agreement in the strongly interacting limit. However the infinite sum approach is unable to uniquely specify the energy of Efimov states, but in the strongly interacting limit there is, to a high degree of accuracy, a correspondence between the three-body parameter required by the boundary condition of the closed form approach and the summation truncation order required by the summation approach. This specification of the energies and wavefunctions forms the basis with which thermodynamic variables such as the virial coefficients or Tan contacts, or dynamic phenomena like quench dynamics can be calculated.

Potential experimental evidence of an Efimov state in 12C and its influence on astrophysical carbon creation

We describe the search of the Efimov state in 12C performed by analysing a large set of 3-α particles coincidence data measured at the LNS laboratories with the CHIMERA detector. The data are compatible with the presence of an Efimov state at the excitation energy of 7.485 MeV, with an excitation probability smaller than 0.3% of the population probability of the Hoyle state. The Efimov state gamma decay probability should be around 30%, much larger than the one of the Hoyle state. The population of this level is compatible with the Suda limit for the 3-α reaction rate in stars because its population should be possible only through direct 3-α reaction. More investigations improving the detector angular resolution are needed to confirm these preliminary observations.

Effects of Efimov states on quench dynamics in a three-boson trapped system

We investigate the effects of Efimov states on the postquench dynamics of a system of three identical bosons with contact interactions in a spherically symmetric three-dimensional harmonic trap. The quench we consider is in the $s$-wave contact interaction, and we focus on quenches from the noninteracting to strongly interacting regimes and vice versa. The calculations use the hyperspherical solutions of the three-body problem and enable us to evaluate the semianalytical results of the Ramsey and particle separation, postquench. In the case where the interactions are quenched from the noninteracting to strongly interacting regime we find convergent aperiodic solutions for both the Ramsey signal and the particle separation. In contrast, for quenches from the strongly interacting regime to the noninteracting regime both the Ramsey signal and particle separation are periodic functions. However, in this case we find that the solutions for the particle separation diverge, indicating that in such a system large oscillations may be observable.

Fate of multiparticle resonances: From Q -balls to He3 droplets

We consider a system of $N$ nonrelativistic particles which form a near-threshold resonance. Assuming no subset of these particles can form a bound state, the resonance can only decay through an "explosion" into $N$ particles. We show that the decay width of the resonance scales as $E^{\Delta-5/2}$ in the limit when the energy $E$ of the resonance goes to zero, where $\Delta$ is the ground state energy of a system of $N$ particles in a spherical harmonic trap with unit frequency. The formula remains valid when some pairs of final particles have zero-energy $s$-wave resonance, but the Efimov effect is not present. In the limit of large $N$, we show that the final particles follow a Maxwell-Boltzmann distribution if they are bosons, and a semicircle-like law if they are fermions. We argue that metastable $^3$He droplets exist with the lifetime varying over many orders of magnitudes ranging from a fraction of a nanosecond to values greatly exceeding the age of the Universe.

Efimov effect for two particles on a semi-infinite line

The Efimov effect (in a broad sense) refers to the onset of a geometric sequence of many-body bound states as a consequence of the breakdown of continuous scale invariance to discrete scale invariance. While originally discovered in three-body problems in three dimensions, the Efimov effect has now been known to appear in a wide spectrum of many-body problems in various dimensions. Here, we introduce a simple, exactly solvable toy model of two identical bosons in one dimension that exhibits the Efimov effect. We consider the situation where the bosons reside on a semi-infinite line and interact with each other through a pairwise δ-function potential with a particular position-dependent coupling strength that makes the system scale invariant. We show that, for sufficiently attractive interaction, the bosons are bound together, and a new energy scale emerges. This energy scale breaks continuous scale invariance to discrete scale invariance and leads to the onset of a geometric sequence of two-body bound states. We also study the two-body scattering off the boundary and derive the exact reflection amplitude that exhibits discrete scale invariance.

Quatrièmes coefficients d’amas et du viriel d’un gaz unitaire de fermions pour un rapport de masse quelconque

We calculate the fourth cluster coefficients of the homogeneous unitary spin 1/2 Fermi gas as functions of the internal-state mass ratio, over intervals constrained by the 3- or 4-body Efimov effect. For this we use our 2016 conjecture (validated for equal masses by Hou and Drut in 2020) in a numerically efficient formulation making the sum over angular momentum converge faster, which is crucial at large mass ratio. The mean cluster coefficient, relevant for equal chemical potentials, is not of constant sign and increases rapidly close to the Efimovian thresholds. We also get the fourth virial coefficients, which we find to be very poor indicators of interaction-induced 4-body correlations. We obtain analytically for all $n$ the cluster coefficients of order $n$ + 1 for an infinity-mass impurity fermion, matching the conjecture for $n=3$. Finally, in a harmonic potential, we predict a non-monotonic behavior of the 3 + 1 cluster coefficient with trapping frequency, near mass ratios where this coefficient vanishes in the homogeneous case.

Three-body continuum states and Efimov physics in noninteger geometry

Continuum structures of three short-range interacting particles in a deformed external one-body field are investigated. We use the equivalent $d$ method employing noninteger 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 2 and 3, 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 the wave functions, scattering lengths, phase shifts, and elastic-scattering cross sections. The procedure and the results are universal, valid for all short-range potentials, and for large scattering lengths. We discuss the properties and validity of the derived expressions by means of the simplest system of three identical bosons. The derived expressions are particularly useful for very small energies, where full numerical calculations are often not feasible. For energies where the numerical calculations can be performed, a good agreement with the analytic results is found. These model results may be tested by scattering experiments for three particles in an equivalent external deformed oscillator potential. The cross sections all vanish in the zero-energy limit for $d<3$ with definite $d$-dependent power of energy.

Three-body resonances in pionless effective field theory

We investigate the appearance of resonances in three-body systems using pionless effective field theory at leading order. The Faddeev equation is analytically continued to the unphysical sheet adjacent to the positive real energy axis using a contour rotation. We consider both, the three-boson system and the three-neutron system. For the former, we calculate the trajectory of Borromean three-body Efimov states turning into resonances as they cross the three-body threshold. For the latter, we find no sign of three-body resonances or virtual states at leading order. This result is validated by exploring the level structure of three-body states in a finite volume approach.

Universal Properties of Weakly Bound Two-Neutron Halo Nuclei.

We construct an effective field theory of a two-neutron halo nucleus in the limit where the two-neutron separation energy B and the neutron-neutron two-body virtual energy ε_{n} are smaller than any other energy scale in the problem, but the scattering between the core and a single neutron is not fine-tuned, and the Efimov effect does not operate. The theory has one dimensionless coupling which formally runs to a Landau pole in the ultraviolet. We show that many properties of the system are universal in the double fine-tuning limit. The ratio of the mean-square matter radius and charge radius is found to be ⟨r_{m}^{2}⟩/⟨r_{c}^{2}⟩=Af(ε_{n}/B), where A is the mass number of the core and f is a function of the ratio ε_{n}/B which we find explicitly. In particular, when B≫ε_{n}, ⟨r_{m}^{2}⟩/⟨r_{c}^{2}⟩=2/3A. The shape of the E1 dipole strength function also depends only on the ratio ε_{n}/B and is derived in explicit analytic form. We estimate that for the ^{22}C nucleus higher-order corrections to our theory are of the order of 20% or less if the two-neutron separation energy is less than 100keV and the s-wave scattering length between a neutron and a ^{20}C nucleus is less than 2.8fm.

Bose polaron and the Efimov effect: A Gaussian-state approach

The authors study a light impurity immersed in a weakly interacting Bose-Einstein condensate using a numerical variational technique based on Gaussian states. By showing how the polaron state becomes unstable due to the decay into the Efimov cluster, the authors unveil an interesting interplay between polaron physics and Efimov physics.

Holographic tetraquarks and the newly observed Tcc+ at LHCb

We describe a heavy and exotic tetraquark state as a holographic molecule, by binding the lightest heavy-light meson $({0}^{\ensuremath{-}},{1}^{\ensuremath{-}})$ multiplet to a flavored sphaleron, in the bulk of the Witten-Sakai-Sugimoto model. Bound tetraquark states emerge as Efimov states in the heavy quark limit, with a binding energy for a charm tetraquark comparable to the ${T}_{cc}^{+}$ recently reported by the LHCb collaboration, but with a substantially smaller width, for a large but finite 't Hooft coupling. Fixing the parameters of the model at the empirical mass of ${T}_{cc}^{+}$, allows for a prediction of the bindings of the undiscovered bottom-charm and bottom tetraquarks.

Two-body continuum states in noninteger geometry

Wave functions, phase shifts and corresponding elastic cross sections are investigated for two short-range interacting particles in a deformed external oscillator field. For this we use the equivalent $d$-method employing a non-integer dimension $d$. Using a square-well potential, we derive analytic expressions for scattering lengths and phase shifts. In particular, we consider the dimension, $d_E$, for infinite scattering length, where the Efimov effect occurs by addition of a third particle. We give explicitly the equivalent continuum wave functions in $d$ and ordinary three dimensional (3D) space, and show that the phase shifts are the same in both methods. Consequently the $d$-method can be used to obtain low-energy two-body elastic cross sections in an external field.

Crossover polarons in a strongly interacting Fermi superfluid

We investigate the zero-temperature quasiparticle properties of a mobile impurity immersed in a strongly interacting Fermi superfluid at the crossover from a Bose-Einstein condensate (BEC) to a Bardeen--Cooper--Schrieffer (BCS) superfluid, by using a many-body $T$-matrix approach that excludes Efimov trimer bound states. Termed BEC-BCS crossover polaron, or crossover polaron in short, this quasiparticle couples to elementary excitations of a many-body background and therefore could provide a useful probe of the underlying strongly interacting Fermi superfluid. Due to the existence of a significant pairing gap $\Delta$, we find that the repulsive polaron branch becomes less well-defined. In contrast, the attractive polaron branch is protected by the pairing gap and becomes more robust at finite momentum. It remains as a delta-function peak in the impurity spectral function below a threshold $2\Delta$. Above the threshold, the attractive polaron enters the particle-hole continuum and starts to get damped. We predict the polaron energy, residue and effective mass for realistic Bose-Fermi mixtures, where the minority bosonic atoms play the role of impurity. These results are practically useful for future cold-atom experiments on crossover polarons.

Efimovian states of three charged particles

When three particles in three dimensions interact with a short-range potential fine-tuned to an infinite scattering length, they form an infinite sequence of loosely bound states obeying discrete scale invariance known as Efimov states. Here we show that analogous states are formed by three charged particles carrying two equal charges and one opposite charge in one, two, and three dimensions without any fine-tuning. Our finding is based on the Born-Oppenheimer approximation, where an effective inverse-square attraction is induced as a consequence of the dipole-charge interaction between a hydrogenlike heavy-light atom and a far-separated heavy particle. Because the resulting Efimovian states emerge toward the second or higher dissociation threshold, they are to be realized as quasibound states and may be observed by exciting hydrogen molecular ions and trions in excitonic systems. We also consider the same system but with a logarithmic Coulomb potential relevant to quantum vortices in two-dimensional superfluids, where the Efimovian states are shown to emerge as genuine bound states toward the first dissociation threshold.

Search for rare 3-α decays in the region of the Hoyle state of 12C

We performed a refined analysis of the data collected with the CHIMERA detector with the aim of searching rare 3-α decay modes near the Hoyle state of 12C. In particular we investigated the direct and sequential decay modes of the Hoyle state including also the recently suggested Efimov state. The comparison of selected events with simulations constrains the precision achievable with measurements in different experimental conditions. We observe good sensitivities to different decay scenarios also in the CHIMERA spherical region notwithstanding its limited angular resolution. The search for the decay of the Efimov state in 12C was performed by using new selection strategies as suggested by simulations. The obtained results are not conclusive, however, we are able to exclude the presence of a sequential decay of this state, while our data could be compatible with its direct decay.

Efimov physics implications at p -wave fermionic unitarity

Efimov physics at $p$-wave unitarity for three equal mass fermions in multiple symmetries interacting via Lennard-Jones potentials is predicted to modify the long-range interaction potential energy, but without producing a true Efimov effect. This analysis treats the total orbital angular momenta and parities, ${J}^{\mathrm{\ensuremath{\Pi}}}={0}^{+},{1}^{+},{1}^{\ensuremath{-}}$, and ${2}^{\ensuremath{-}}$, for either three spin-polarized fermions ($\ensuremath{\uparrow}\ensuremath{\uparrow}\ensuremath{\uparrow}$) or two spin-up and one spin-down fermion ($\ensuremath{\downarrow}\ensuremath{\uparrow}\ensuremath{\uparrow}$). Our results for the long-range interaction in some of those cases agree with previous work by Werner and Castin and by Blume et al., namely, in cases where the $s$-wave scattering length goes to infinity. The present results extend those calculated interaction energies to small and intermediate hyperradii comparable to the van der Waals length, and we consider additional unitarity scenarios where the $p$-wave scattering volume approaches infinity. The crucial role of the diagonal hyperradial adiabatic correction term is identified and characterized.

Bose-Einstein Condensation of Efimovian Triples in the Unitary Bose Gas.

In an atomic Bose-Einstein condensate quenched to the unitary regime, we predict the sequential formation of a significant fraction of condensed pairs and triples. At short distances, we demonstrate the two-body and Efimovian character of the condensed pairs and triples, respectively. As the system evolves, their size becomes comparable to the interparticle distance, such that many-body effects become significant. The structure of the condensed triples depends on the size of Efimov states compared with density scales. Unexpectedly, we find universal condensed triples in the limit where these scales are well separated. Our findings provide a new framework for understanding dynamics in the unitary regime as the Bose-Einstein condensation of few-body composites.

The absence of the Efimov effect in systems of one- and two-dimensional particles

We study virtual levels of $N$-particle Schr\"odinger operators and prove that if the particles are one-dimensional and $N\ge 3$, then virtual levels at the bottom of the essential spectrum correspond to eigenvalues. The same is true for two-dimensional particles if $N\ge 4$. These results are applied to prove the non-existence of the Efimov effect in systems of $N\ge 4$ one-dimensional or $N\ge 5$ two-dimensional particles.

Three-boson stability for boosted interactions towards the zero-range limit

We study the three-boson bound-state mass and wave functions for ground and excited states within the three-body relativistic framework with Kamada and Glöcke boosted potentials in the limit of a zero-range interaction. We adopt a nonrelativistic short-range separable potential, with Yamaguchi and Gaussian form factors, and drive them towards the zero-range limit by letting the form factors' momentum scales go to large values while keeping the two-body binding fixed. We show that the three-boson relativistic masses and wave functions are model-independent towards the zero-range limit, and the Thomas collapse is avoided, while the nonrelativistic limit kept the Efimov effect. Furthermore, the stability in the zero-range limit is a result of the reduction of boosted potential with the increase of the virtual pair center of mass momentum within the three-boson system. Finally, we compare the present results with Light-Front and Euclidean calculations.

Formation of ultracold heteronuclear polyatomic molecule: multi-path scheme and interference effect

We investigate the creation of stable heteronuclear polyatomic molecules based on the three-body and higher order Efimov state via a generalized stimulated Raman adiabatic passage scheme. Within the mean-field approximation, we establish the multi-path conversion models and derive the dark state solutions. The multi-path atom–polymer conversion dynamics and interference effect are studied. We show that the multi-path constructive interference permits higher conversion efficiency, resulting in a state that is closer to the ideal dark state solutions. The effects on the conversion efficiency of the single-photon detuning, the strength of the Rabi pulse, the spontaneous emission from intermediate states and the interaction between the particles are also discussed. Our results not only include the previous ultracold molecule formation techniques, such as the single-path polyatomic molecule, double-path pentamer and triple-path trimer, but also open a possible route for the multi-path formation of more complex polyatomic molecules.