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arXiv:2303.03845 · The European Physical Journal D

Tuning of Efimov states in non-integer dimensions

Eduardo Garrido, Aksel Stenholm Jensen

Published Mar 1, 2023• 2 citations
arXivHTMLPDF

Abstract

Abstract 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 $$^7$$ 7 Li, $$^{39}$$ 39 K, and $$^{87}$$ 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

DOI
10.1140/epjd/s10053-023-00628-2

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Abstract 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 7^77 7 Li, 39^{39}39 39 K, and 87^{87}87 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
• Induced Efimov effect. • Confined three-body systems. • Non-integer dimensions. The Efimov effect can be induced by means of an external deformed one-body field that effectively reduces the allowed spatial dimensions to less than three. To understand this new mechanism, conceptually and practically, we employ a formulation using non-integer dimension, which is equivalent to the strength of an external oscillator field. The effect most clearly appears when the crucial two-body systems are unbound in three, but bound in two, dimensions. We discuss energy variation, conditions for occurrence, and number of Efimov states, as functions of the dimension. We use practical examples from cold atom physics of 133 Cs- 133 Cs- 133 Cs, 87 Rb- 87 Rb- 87 Rb, 133 Cs- 133 Cs- 6 Li, and 87 Rb- 87 Rb- 39 K. Laboratory tests of the effect can be performed with two independent parameters, i.e. the external one-body field and the Feshbach two-body tuning. The scaling and (dis)appearance of these Efimov states occur precisely as already found in three dimensions.
Continuum structures of three short-range interacting particles in a deformed external one-body field are investigated. We use the equivalent ddd method employing noninteger dimension, ddd, in a spherical calculation with a dimension-dependent angular momentum barrier. We focus on dimensions close to the critical dimension, d=dEd={d}_{E}d=dE​, 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<3d<3d<3 with definite ddd-dependent power of energy.
Abstract 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 Cs– 133 Cs– 6 Li system as the limit cycle of the correlation between the energies of two consecutive Efimov states as a function of D and the heavy-light binding energy E D 2 . In addition, we found that the energy of the ( N + 1)th excited state reaches the two-body continuum independently of the dimension D when E 2 D / E 3 ( N ) = 0.89 , where E 3 ( N ) is the N th excited three-body binding energy.
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 reflection of the Efimov wavefunciton by the short-range molecular potential. Assuming Born-Oppenheimer approximation and strong quantum reflection, we obtain an analytic result of C=9.475... for three identical particles. We suspect the exceptional case of K-39 is a result of resonant coupling between the Efimov state and a short-range molecular state.
We consider a system of three identical bosons near a Feshbach resonance in the universal regime with large scattering length usually described by model-independent zero-range potentials. We employ the adiabatic hyperspherical approximation and derive the rigorous large-distance equation for the adiabatic potential for finite-range interactions. The effective range correction to the zero-range approximation must be supplemented by a new term of the same order. The nonadiabatic term can be decisive. Efimov physics is always confined to the range between effective range and scattering length. The analytical results agree with numerical calculations for realistic potentials.
We solved analytically the three-body mass-imbalanced problem embedded in DDD dimensions for zero-range resonantly interacting particles. We derived the negative energy eigenstates of the three-body Schr\"odinger equation by imposing the Bethe-Peierls boundary conditions in DDD 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 close to the Efimov limit.