Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2203.01133 · Physical Review A

Two-body continuum states in noninteger geometry

E.R. Christensen, Eric Garrido, A. S. Jensen

Published Mar 15, 2022• 3 citations
arXivHTMLPDF

Abstract

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 noninteger 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 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.

DOI
10.1103/physreva.105.033308

Related papers

Three-body continuum states and Efimov physics in noninteger geometry

Eric Garrido, E.R. Christensen, A. S. Jensen

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.

arXiv:2206.13106•Physical Review A•Jul 8, 2022•6 citations

Tuning of Efimov states in non-integer dimensions

Eduardo Garrido, Aksel Stenholm Jensen

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

arXiv:2303.03845•The European Physical Journal D•Mar 1, 2023•2 citations

Three bosons in one dimension with short-range interactions: Zero-range potentials

Nirav P. Mehta, James R. Shepard

We consider the three-boson problem with $\ensuremath{\delta}$-function interactions in one spatial dimension. Three different approaches are used to calculate the phase shifts, which we interpret in the context of the effective range expansion, for the scattering of one free particle off a bound pair. We first follow a procedure outlined by McGuire in order to obtain an analytic expression for the desired $S$-matrix element. This result is then compared to a variational calculation in the adiabatic hyperspherical representation, and to a numerical solution to the momentum-space Faddeev equations. We find excellent agreement with the exact phase shifts, and comment on some of the important features in the scattering and bound-state sectors. In particular, we find that the $1+2$ scattering length is divergent, marking the presence of a zero-energy resonance which appears as a feature when the pairwise interactions are short range. Finally, we consider the introduction of a three-body interaction, and comment on the cutoff dependence of the coupling.

arXiv:nucl-th/0504049•Physical Review A•Sep 28, 2005•16 citations

Three-body scattering area of identical bosons in two dimensions

Junjie Liang, Hongye Yu, Shina Tan

We study the wave function $φ^{(3)}$ of three identical bosons scattering at zero energy, zero total momentum, and zero orbital angular momentum in two dimensions, interacting via short-range potentials with a finite two-body scattering length $a$. We derive asymptotic expansions of $φ^{(3)}$ in two regimes: the 111-expansion, where all three pairwise distances are large, and the 21-expansion, where one particle is far from the other two. In the 111-expansion, the leading term grows as $\ln^3(B/a)$ at large hyperradius $B=\sqrt{(s_1^2+s_2^2+s_3^2)/2}$. At order $B^{-2}\ln^{-3}(B/a)$, we identify a three-body parameter $D$ with dimension of length squared, which we term the three-body scattering area. This quantity should be contrasted with the three-body scattering area previously studied for infinite or vanishing two-body scattering length. If the two-body interaction is attractive and supports bound states, $D$ acquires a negative imaginary part, and we derive its relation to the probability amplitudes for the production of two-body bound states in three-body collisions. Under weak modifications of the interaction potentials, we derive the corresponding shift of $D$ in terms of $φ^{(3)}$ and the changes of the two-body and three-body potentials. We also study the effects of $D$ and $φ^{(3)}$ on three-body and many-body physics, including the three-body ground-state energy in a large periodic volume, the many-body energy and the three-body correlation function of the dilute two-dimensional Bose gas, and the three-body recombination rates of two-dimensional ultracold atomic Bose gases.

cond-mat.quant-gasCross-listedarXiv:2601.20509•Jan 29, 2026•0 citations

Confinement of N-Body Systems and Non-integer Dimensions

Eric Garrido, A. S. Jensen

Abstract The squeezing process of a three-dimensional quantum system by use of an external deformed one-body oscillator potential can also be described by the d-method, without external field and where the dimension can take non-integer values. In this work we first generalize both methods to N particles and any transition between dimensions below 3. Once this is done, the use of harmonic oscillator interactions between the particles allows complete analytic solutions of both methods, and a direct comparison between them is possible. Assuming that both methods describe the same process, leading to the same ground state energy and wave function, an analytic equivalence between the methods arises. The equivalence between both methods and the validity of the derived analytic relation between them is first tested for two identical bosons and for squeezing transitions from 3 to 2 and 1 dimensions, as well as from 2 to 1 dimension. We also investigate the symmetric squeezing from 3 to 1 dimensions of a system made of three identical bosons. We have in all the cases found that the derived analytic relations between the two methods work very well. This fact permits to relate both methods also for large squeezing scenarios, where the brute force numerical calculation with the external field is too much demanding from the numerical point of view, especially for systems with more than two particles.

arXiv:2403.06519•Few-Body Systems•Apr 6, 2024•4 citations

Efimov effect in non-integer dimensions induced by an external field

Eric Garrido, A. S. Jensen

• 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.

arXiv:2008.07392•Physics Letters A•Oct 26, 2020•9 citations
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 ddd method employing a noninteger dimension ddd. Using a square-well potential, we derive analytic expressions for scattering lengths and phase shifts. In particular, we consider the dimension, dE{d}_{E}dE​, for infinite scattering length, where the Efimov effect occurs by addition of a third particle. We give explicitly the equivalent continuum wave functions in ddd and ordinary three-dimensional space, and show that the phase shifts are the same in both methods. Consequently the ddd method can be used to obtain low-energy two-body elastic cross sections in an external field.
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 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
We consider the three-boson problem with $\ensuremath{\delta}$-function interactions in one spatial dimension. Three different approaches are used to calculate the phase shifts, which we interpret in the context of the effective range expansion, for the scattering of one free particle off a bound pair. We first follow a procedure outlined by McGuire in order to obtain an analytic expression for the desired SSS-matrix element. This result is then compared to a variational calculation in the adiabatic hyperspherical representation, and to a numerical solution to the momentum-space Faddeev equations. We find excellent agreement with the exact phase shifts, and comment on some of the important features in the scattering and bound-state sectors. In particular, we find that the 1+21+21+2 scattering length is divergent, marking the presence of a zero-energy resonance which appears as a feature when the pairwise interactions are short range. Finally, we consider the introduction of a three-body interaction, and comment on the cutoff dependence of the coupling.
We study the wave function φ(3)φ^{(3)}φ(3) of three identical bosons scattering at zero energy, zero total momentum, and zero orbital angular momentum in two dimensions, interacting via short-range potentials with a finite two-body scattering length aaa. We derive asymptotic expansions of φ(3)φ^{(3)}φ(3) in two regimes: the 111-expansion, where all three pairwise distances are large, and the 21-expansion, where one particle is far from the other two. In the 111-expansion, the leading term grows as ln⁡3(B/a)\ln^3(B/a)ln3(B/a) at large hyperradius B=(s12+s22+s32)/2B=\sqrt{(s_1^2+s_2^2+s_3^2)/2}B=(s12​+s22​+s32​)/2​. At order B−2ln⁡−3(B/a)B^{-2}\ln^{-3}(B/a)B−2ln−3(B/a), we identify a three-body parameter DDD with dimension of length squared, which we term the three-body scattering area. This quantity should be contrasted with the three-body scattering area previously studied for infinite or vanishing two-body scattering length. If the two-body interaction is attractive and supports bound states, DDD acquires a negative imaginary part, and we derive its relation to the probability amplitudes for the production of two-body bound states in three-body collisions. Under weak modifications of the interaction potentials, we derive the corresponding shift of DDD in terms of φ(3)φ^{(3)}φ(3) and the changes of the two-body and three-body potentials. We also study the effects of DDD and φ(3)φ^{(3)}φ(3) on three-body and many-body physics, including the three-body ground-state energy in a large periodic volume, the many-body energy and the three-body correlation function of the dilute two-dimensional Bose gas, and the three-body recombination rates of two-dimensional ultracold atomic Bose gases.
Abstract The squeezing process of a three-dimensional quantum system by use of an external deformed one-body oscillator potential can also be described by the d-method, without external field and where the dimension can take non-integer values. In this work we first generalize both methods to N particles and any transition between dimensions below 3. Once this is done, the use of harmonic oscillator interactions between the particles allows complete analytic solutions of both methods, and a direct comparison between them is possible. Assuming that both methods describe the same process, leading to the same ground state energy and wave function, an analytic equivalence between the methods arises. The equivalence between both methods and the validity of the derived analytic relation between them is first tested for two identical bosons and for squeezing transitions from 3 to 2 and 1 dimensions, as well as from 2 to 1 dimension. We also investigate the symmetric squeezing from 3 to 1 dimensions of a system made of three identical bosons. We have in all the cases found that the derived analytic relations between the two methods work very well. This fact permits to relate both methods also for large squeezing scenarios, where the brute force numerical calculation with the external field is too much demanding from the numerical point of view, especially for systems with more than two particles.
• 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.