Efimov effect in non-integer dimensions induced by an external field
EEfimov e ff ect in non-integer dimensions induced by an external field E. Garrido a , A.S. Jensen b a Instituto de Estructura de la Materia, IEM-CSIC, Serrano 123, E-28006 Madrid, Spain b Department of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C, Denmark
Abstract
The Efimov e ff ect can be induced by means of an external deformed one-body field that e ff ectively reduces the allowedspatial dimensions to less than three. To understand this new mechanism, conceptually and practically, we employa formulation using non-integer dimension, which is equivalent to the strength of an external oscillator field. Thee ff ect 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 Cs-
Cs-
Cs, Rb- Rb- Rb,
Cs-
Cs- Li, and Rb- Rb- K. Laboratory tests of the e ff ect can be performed with two independent parameters, i.e. the external one-bodyfield and the Feshbach two-body tuning. The scaling and (dis)appearance of these Efimov states occur precisely asalready found in three dimensions. Keywords:
Efimov e ff ect, confinement of quantum systems, d -dimensional calculations
1. Introduction
The Efimov e ff ect was suggested theoretically aboutfifty years ago for three-body systems [1]. For the oc-currence, at least two of the three pair-interactions musthave nearly zero energy. Occurrence is optimized byhaving identical constituents, where only one two-bodyinteraction is involved. However, although in principlepossible, to find in nature two particles, identical or not,bound by zero energy would be very rare.In cold atom physics this problem was overcomeabout 25 years ago with an original method, that isthe technique of controlled tuning of the e ff ective two-body interaction by coupling via a Feshbach resonance[2, 3, 4, 5]. Subsequently, properties and consequencesof the Efimov e ff ect have been extensively studied the-oretically [6, 11, 10, 7, 8, 9] and also early establishedexperimentally [12, 13, 14, 15, 16, 17, 18].An important fact concerning the Efimov e ff ect isthat, whereas present in three dimensions (3D), is ab-sent in two (2D) [22, 23, 19, 20, 21]. The reason isthat in 3D a finite attraction is required for binding atwo-body system, but in 2D an infinitesimal attractionis su ffi cient [24, 25]. These points emphasize the vari-ation between dimensions, and in fact triggered previ- Email address: [email protected] (E. Garrido) ous investigations of dimensional transitions by di ff er-ent methods [26, 27, 28, 29]. In [30] it is shown howa two-species Fermi gas where one species is confinedin a two- or one-dimensional space, while the otherone is free in 3D, can lead, depending on the mass ra-tio, to systems showing the Efimov e ff ect. This is theconfinement-induced Efimov e ff ect introduced in [31],although no details are given on the influence from con-tinuous external squeezing. A short review of the e ff ectsof the confinement in mixed dimensions is given in [21].A novel three-body method applicable to non-integerdimensions, d , was presented about two decades ago[6]. One spectacular prediction was that the Efimove ff ect is only possible in dimensions between 2 . .
8. This d -method was recently implemented to in-vestigate the confinement of two- and three-body sys-tems [32, 33, 34, 35]. Practical calculations for non-integer dimensions are precisely as easy, or di ffi cult, asin the usual 3D-space. The necessary relation to ordi-nary physics of integer-based dimensions is also avail-able [32, 34]. The translation involves an external de-formed field [36], which is shown to be equivalent tothe non-integer dimensional treatment [35].A detailed comparison of the actual wave functionobtained introducing explicitly the external field and thedeformed wave function extracted from the d -method isalso given in [35] for di ff erent two-body potentials. It is Preprint submitted to Physics Letters A October 26, 2020 a r X i v : . [ phy s i c s . a t m - c l u s ] O c t hown how both wave functions are very much equiva-lent for both, well-bound and weakly bound, three-bodysystems. Thus, we can employ the simple method andinterpret in terms of a deformed external one-body field.The purpose of this letter is to demonstrate the powerof the d -method by exhibiting a spectacular conse-quence of applying an external manageable [37] oscilla-tor field to a three-body system. Assume the two-bodyattraction in 3D is too weak to bind. We then employa squeezing one-body oscillator field on the z -direction,while leaving the x and y -coordinates untouched. Thetwo-body systems are then forced to move towards 2D,where they at some point become bound through theunchanged two-body attractions. At this point of zerobinding, the Efimov e ff ect appears, which is at a non-integer dimension, or equivalently, at a certain strengthof the external field.Of course, the Feshbach resonance technique is stillavailable, and the combination with an independent tun-ing of the deformed external field might prove conve-nient. The two independent parameters can be used toapproach the optimal situation with three di ff erent sub-systems at zero energy. This is not possible with onlyone parameter, neither Feshbach nor external field.In this letter, we first describe the d -method and giveenough details to allow calculations and estimates. Wethen present results for use on specific cold atom gases,where the Efimov e ff ect can be manipulated to appearfor non-integer dimensions.
2. The d -method In Efimov’s original formulation the key to un-derstanding is that a long-distance e ff ective attractivepotential with an inverse-distance-square dependencearises for three particles with a strength more negativethan a given critical value. The crucial potential termhas the same form as a centrifugal barrier, which in or-dinary three-dimensional space is positive. How andwhen this occurs is well established by some patholog-ical conditions obeyed by the three constituent particlesand their interactions. Although these occurrence con-ditions are special, they have been simulated in labo-ratories and a number of derived consequences experi-mentally tested.To describe and understand in details it is convenientto adopt the formalism of the hyperspherical adiabaticexpansion method, where the hyperradius, ρ , becomesthe crucial length coordinate entering in the correspond-ing di ff erential Schr¨odinger equation. The essential re-sult, derived in many previous publications, e.g. [6], takes the following form for each adiabatic channel: − ∂ ∂ρ + λ ( d ) ( ρ ) + ( d − − ρ − mE ( d )3 b (cid:126) f ( d ) ( ρ ) = , (1)where d is the dimension parameter (2 ≤ d ≤ E ( d )3 b isthe three-body energy in d dimensions, m is a normal-ization mass disappearing in all observable quantities,and λ ( d ) is the ρ -dependent eigenvalue from the angularSchr¨odinger (or Faddeev) equation. The reduced, f ( d ) ,and total, F ( d ) , radial wave functions are related by f ( d ) ( ρ ) = ρ d − / F ( d ) ( ρ ) . (2)Only the diagonal terms are included in Eq.(1), sincethe couplings between the adiabatic channels are unim-portant as they vanish at the decisive large distances.Here we should remember that all such formula refer toone decoupled adiabatic channel.The Efimov e ff ect occurs for each channel for the par-ticular dimension d = d E for which the numerator of thee ff ective potential in Eq.(1) is, in the large- ρ limit, con-stant and less than − /
4. From Eq.(1) we see that thiscondition means that ξ d E <
0, or ξ d E = i | ξ d E | , where wehave defined ξ d E = λ ( d E ) ∞ + ( d E − , (3)with λ ( d E ) ∞ = λ ( d E ) ( ρ = ∞ ).The bound state solutions to Eq.(1) for constant λ ( d E ) = λ ( d E ) ∞ are f ( d E ) ( ρ ) ∝ √ κ d E ρ K i | ξ dE | ( κ d E ρ ), with κ d E = (cid:113) − mE ( d E )3 b / (cid:126) . The modified Bessel functionof second kind, K i | ξ dE | , decreases exponentially at largedistances as it should for a bound state. For small dis-tances we have instead K i | ξ dE | ∝ sin( | ξ d E | ln( κ d E ρ )).It is important to keep in mind that in actual cal-culations the λ ( d E ) -function is constant only over a ρ -interval limited by a scattering length, | a avd E | , in prac-tice finite, and defined as an average of the three d -dimensional two-body scattering lengths involved in thethree-body system [10]. The condition ξ d E <
0, or λ ( d E ) ∞ < − ( d E − , and therefore the Efimov e ff ect,requires that at least two of the two-body scatteringlengths are numerically very large, which is equivalentto having close to zero energy in at least two of the two-body subsystems. For ρ larger than | a avd E | the λ -functionscan no longer support bound states. The number, N E ,of Efimov states that can be held by a large, but finite,scattering length a avd E , can be estimated by counting thenumber of nodes in K i | ξ dE | between the ground state size, ρ , and | a avd E | , that is [6] N E ≈ | ξ d E | π ln (cid:0) | a avd E | ρ (cid:1) . (4)2he usual Efimov scaling for energies and root-mean-square (rms) radii still applies, which is [6] E n E n + = (cid:104) ρ (cid:105) n + (cid:104) ρ (cid:105) n = e π/ | ξ dE | , (5)where n labels the di ff erent states in the Efimov series.
3. External field translation
The formulation in terms of the exceedingly intuitivedimension parameter, d , is very e ffi cient and convenientfor theoretical calculations. However, a relation to lab-oratory controlled observable variables is needed. Thishas recently become available in investigations wherethe particles are confined, say in the z -coordinate, by anexternal one-body oscillator field.These calculations, where the deformed squeezingexternal oscillator potential with frequency ω ho is in-cluded explicitly, are performed in a three-dimensionalspace, and they are related to the above d -results asshown in Ref.[35], which is: ω pp ω ho = d − d − − d ) , (6)where ω pp is the frequency of the equivalent two-bodyoscillator interaction that, when used in the three-bodycalculation, gives rise to the same rms radius r D = (cid:112) (cid:126) / ( M ω pp ), where M is the total mass, as the orig-inal potentials. In other words, in actual calculationswith arbitrary two-body potentials, the computed valueof r D permits to obtain ω pp to be used in Eq.(6).It was already shown in [6] that for s -waves and threeidentical bosons the appearance of the Efimov e ff ectrequires d > . ω ho < . ω pp , which means that a squeezing frequency largerthan 1 . ω pp cannot produce an Efimov e ff ect for threeidentical bosons. For one light and two identical heavyparticles the limit value can move up or down dependingon the mass ratio, as shown in [33].After a d -calculation, the equivalent deformed three-dimensional wave function is constructed by scalingdown the z -direction by a factor, s , such that ρ → ˜ ρ = ρ ⊥ + ρ z / s . (7)The constant scaling, s , is approximately given by [35]1 s = (cid:115) + ω ho ω pp = (cid:115) + (cid:32) ( d − − d )2( d − (cid:33) , (8)which connects the dimension d and the deformation,determined by the scale parameter s , of the actual three-dimensional wave function. We then replace f ( d ) ( ρ ) by m H / m L / / / E (2 bd )2 D − . − . − . a D a D − . − . − . E (3 gr )2 D − . − . − . E (3 ∗ )2 D − . − . E (3 ∗∗ )2 D − . r (3 gr )2 D r (3 ∗ )2 D r (3 ∗∗ )2 D Table 1: We consider a three-body system made of two heavy iden-tical particles with mass m H and a light particle with mass m L . Theheavy-heavy interaction is zero. The heavy-light Gaussian interactionis with range, b , and strength S = − . (cid:126) / ( µ b ), where µ is the re-duced mass of the system. For m H / m L = m H / m L =
1, 133 /
6, and87 /
39 we give two-body ground state energies, E (2 bd )2 D , and scatteringlengths, a D , and a D in 2D and 3D. The lower part of the table shows,in 2D, the three-body energies and rms radii of the ground state andthe existent excited states. Lengths are in units of b , and energies inunits of (cid:126) / ( µ b ). f ( d ) ( ˜ ρ ), which, after the necessary renormalization, al-lows computations of any desired observable investi-gated in the laboratory from knowledge entirely from d -calculations, see [34, 35] for details.
4. Interactions and properties
Realistic numerical investigations need finite-rangetwo-body potentials. We have chosen a potential suchthat the two-body systems are unbound in 3D, whichwill later permit to highlight the emergence of Efimovstates for 2 ≤ d ≤
3. For our purpose any potential ful-filling these conditions could be used. The conclusionsare independent of the specific potential shape.For simplicity we choose a Gaussian radial shape, S exp( − r / b ). This potential shape has been frequentlyused to investigate the Efimov e ff ect, from the earlydays of three-body calculations [38], to much morerecent works [39, 40]. Taking (cid:126) / ( µ b ) as energyunit, the properties of the two-body systems are mass-independent. In 2D the two-body systems have only onebound state, whose energy, E (2 bd )2 D , is given in the secondrow of Table 1. The following two rows give a D and a D , in units of b , which are the scattering lengths, a d ,for d = d =
3, respectively.We consider three-body systems made of two iden-tical heavy particles with mass m H and a light particlewith mass m L . The interaction between the two heavyparticles is put equal to zero, except for the mass ratio m H / m L =
1, where the three particles are considered3 -2 -1 -6 -5 -4 -3 -2 -1 m H /m L =1 (3 ident.)m H /m L =133/6m H /m L =87/39Two-body energy -2 -1 -4 -3 -2 -1 BorromeanExcited states d E =2.75 d E =2.75 Figure 1: For the three-body states given in the lower part of Ta-ble 1, absolute value of the bound three-body energies, | E d | in units of (cid:126) / ( µ b ), as a function of the d -function in Eq.(6). The dotted bluecurves is the absolute energy of the bound two-body state. The outerand inner panels show, respectively, the evolution of the ground stateand the excited states. The black arrow indicates the region where thebound three-body states are borromean systems ( d > d E ). identical. As shown later, the asymmetric m H / m L = a D . Incontrast, in the three cases shown in Table 1, the three-body system has one well bound ground state in 2D.Furthermore, also in 2D, the mass ratios m L / m H = m H / m L = /
5. Energy variation
Starting from d =
3, when the dimension is de-creased, i.e., when the system is progressively squeezed,the bound three-body states for each system appear fromthe continuum as shown in Fig. 1, where we showthe binding energies as functions of the d -function inEq.(6). The outer and inner panels show, respectively,the evolution of the ground and excited state energiesfor the di ff erent mass ratios. In both panels the dottedcurve gives the evolution of the heavy-light two-bodyenergy. The vertical dotted line at d E = .
75 marks thedimension below which the two-body system is bound.Therefore, all the three-body bound states located to theright of this vertical line have borromean character.The overall behavior of the energy curves in Fig. 1 issimilar for all the cases. When confining from 3D to 2D,moving from right to left in the figure, di ff erent bound ρ -3-2-10 λ ( ρ ) + ( d - ) ξ E = −2.679ξ E = −0.858ξ E = −0.270 m H /m L =1 (3 ident.) m H /m L =133/6m H /m L =87/39 d=2.755d=d E =2.75d=2.745 Figure 2: The functions λ ( d ) n + ( d − for the lowest diagonal poten-tials as functions of ρ for d = d E = .
75 (solid), d = d = .
745 (dot-dashed), for mass ratios m H / m L = / /
6, and 87 /
39. The arrows indicate the asymptoticvalue of ξ d E for d = d E . states appear, that is the ground states for d = . m H / m L =
1, 133 /
6, and 87 /
39, re-spectively. From this d -value, where the systems areborromean, an increase of the confinement (decrease of d ) gives rise to a fast increase of the energy, which stabi-lizes following the same trend as the two-body energy.The behavior is similar for the excited states, shown inthe inner panel, although the three-body energies aremuch closer to the two-body ones. In particular, the en-ergy of the first excited state for m H / m L = m H / m L = /
6, appearingboth at d = .
76, follow very closely the two-body en-ergy. The first excited state for m H / m L = / d = . d E = .
75, corresponding to zero two-body energy, is the Efimov point, where infinitely manythree-body bound states also emerge, and soon after dis-appear again. However, this d -value is rather arbitraryand the three coinciding Efimov points are only due tothe choice of interactions. Increasing the two-body at-traction, or equivalently | a D | , would move the corre-sponding Efimov point to the right on Fig 1. Eventu-ally the well known Efimov condition would be reachedin 3D with infinitely many states corresponding to zerotwo-body binding energy at d = d E =
3. By weaken-ing the attractive two-body potentials, the curves wouldmove to the left towards 2D.
6. Asymptotic potentials
To understand the mechanism, we look at the e ff ec-tive potentials in Eq.(1). In Fig. 2 the solid curves show λ ( d ) ( ρ ) + ( d − as functions of ρ , for the three co-inciding Efimov points, d = d E = .
75, where | a d | = ∞ .4 .74 2.75 2.76 d N E m H /m L =1 (3 ident.)m H /m L =133/6m H /m L =87/39 Figure 3: For m H / m L = / / /
39 (dot-dashed), estimated number, N E , of bound excited states according toEq.(4) with ρ = b as functions of d . The narrow peaks, which occurfor the Efimov point of d E = .
75, are zoomed in the inset.
The functions are asymptotically constant, approach-ing the values ξ d E ≈ − . − . − . m H / m L = /
1, 133 /
6, 87 /
39, respectively. These neg-ative constants are criteria for occurrence of the Efimove ff ect in all the three cases.The dashed and dot-dashed curves are the same func-tions for the neighboring d -values, d = .
755 and d = . d > d E , as for d = . λ ( d ) functions reproduce asymptotically the hyperspher-ical spectrum K ( K + d −
2) [6], which means that ξ d becomes positive for su ffi ciently large ρ . When d < d E ,as for d = . λ ( d ) functions diverge parabolicallyto −∞ [6], and the large negative asymptotic values pre-vent the appearance of bound states of large radii. Thus,there is no Efimov-like states for d outside a very narrowinterval around d E .
7. Number of Efimov states
As seen in Fig. 2, the variation with d of the e ff ec-tive potentials in Eq.(1) is very fast. The consequenceis that the Efimov e ff ect, i.e., the Efimov states, ap-pear and disappear equally fast. We illustrate this inFig. 3, where we show, as a function of d , the esti-mated number, N E , of bound states across the Efimovpoint, as given in Eq.(4). We notice the extremely fastincrease and decrease of the number of bound states,as highlighted in the inset. The tiny dimension inter-val, 2 . < d < . ff ective potentialsand ξ d E <
0, are only fulfilled for d = d E . We also em-phasize that these features are precisely as extreme asobserved in the established Efimov scenario for d = d E -2 Thick: |ξ d E |Thin: Energy scale d E =2.75 Figure 4: For m H / m L = / / /
39 (dot-dashed), value of | ξ d E | as a function of d E (thick) and the energy scalefactor (thin) as given in Eq.(5). In our calculations all the three Efimov points havearbitrarily been chosen to be d = d E = .
75. However,the properties of the Efimov states depend on the valueof d E . Fortunately, the large-distance constants, ξ d E , canbe found independent of the interactions through a tran-scendental equation as explained in subsection 5.2.3 ofRef. [6]. As seen in Fig. 4 (thick curves), decreasing d E produces decreasing | ξ d E | , which eventually becomes 0for some d E = d L limit value. The Efimov e ff ect can-not occur for d ≤ d L , which for three identical bosons is d L = . d L oscillates depending on the mass ratio [33]. Inparticular, d L ≈ . m H / m L (cid:29)
1, while d L increaseswhen m H / m L decreases. For m H / m L = / d L ≈ .
3, as in the case of three identical particles, and d L ≈ .
54 for m H / m L = / d E decreases with in-creasing m H / m L ratio in the mass asymmetric case.This crucial fact allows easier numerical calculationsof the Efimov states and facilitates as well experimen-tal detection. This is evident in our case, d E = . m H / m L = / /
39 are 46.5 and 180214, respectively, whereas weget 882.8 for the symmetric case of three identical par-ticles ( m L / m H = N E in Fig. 3 is always above and below the other two curvesfor m H / m L = / m H / m L = /
39, respectively.From the figure we can then conclude that, in theasymmetric case, the smaller the m H / m L ratio the closer | ξ d E | to zero and, therefore, the larger the energy scalefactor between the Efimov states. A reduction of the m H / m L ratio makes soon the Efimov states essentiallyunreachable both, theoretically and experimentally. Asan example, for the asymmetric case with m H / m L = -3 -2 -1 (d-2)/(d E -d) -9 -8 -7 -6 -5 -4 -3 -2 -1 |E d | E /E = 43.7E /E = 46.6E /E = 46.5E /E = 46.5 |E ||E ||E ||E ||E ||E | Figure 5: Absolute values of the bound three-body energies, | E d | , inunits of (cid:126) / ( µ b ), for the system of mass ratio, 133 /
6, as functions of( d − / ( d E − d ), where d E = .
75 is the Efimov point for this interac-tion. The dotted curve correspond to the two-body binding energy. the energy scales with a factor of about 2 . · .
8. Efimov energies
The energies of the Efimov states appearing in thetiny d -interval above d E towards the Efimov point areshown in Fig. 5 for the mass ratio m H / m L = / d < d E ,where the two identical two-body subsystems are al-ways bound. In order to expand the small intervalfor visibility, we show the energies as functions of( d − / ( d E − d ). For comparison we also show the two-body binding energies in the figure (dotted curve).The third and fourth excited states disappear alongthe two-body threshold, whereas the ground state andthe two first excited states evolve as already shown bythe dashed curves in Fig. 1. The energies of the fivecomputed Efimov states for d = d E are, in units of (cid:126) / ( µ b ), − . · − , − . · − , − . · − , − . · − , and − . · − , respectively. The en-ergy scale factors are then equal to 43.7, 46.6, 46.5, and46.5, in very good agreement with the value ( ≈ . ξ d E = − .
679 (see Fig.2).In the higher side of the Efimov point, d > d E , wherethe two identical two-body subsystems are unbound, theEfimov states disappear in the continuum, as shown inFig. 1 for the two lowest states. Beyond d E the energyof the di ff erent states drops rapidly, and eventually, forthe interaction chosen in this work, the three-body statesbecome unbound.
9. Conclusions
In this work we have explained a novel procedurethat permits to obtain the Efimov conditions for a giventhree-body system. The tool is an external, directionalsqueezing, one-body potential. The usual external con-fining field, always present in cold atom experiments,must be deformed for our purpose, where a su ffi cientlystrong deformation would squeeze the system contin-uously into two dimensions. The particles must theneventually adjust to a two-dimensional world, whereeven a weak two-body attraction supports a bound state.Thus, for an interaction unable to bind the two-bodysystem in three dimensions, at some intermediate defor-mation the two-body system must change from unboundto being bound. This Efimov point, with zero two-body binding energy, corresponds to an infinite scatter-ing length, which leads to the infinite series of three-body Efimov states. As in three dimensions, this struc-ture will emerge for any ground, or excited-state van-ishing, two-body energy. The characteristic features areprecisely the same as in three dimensions, and thereforeopen to the same type of tests.The advantage now is that two independent control-lable parameters are then available for tuning to theEfimov condition, that is Feshbach tuning and externalfield squeezing. This must in any case be more flexi-ble than exploiting only one of these degree-of-freedom.It could be that an additional two-body subsystem canbe made to contribute, perhaps the Feshbach tuning canbe less precise, perhaps the Efimov scaling can be op-timized to a denser spectrum with more opportunities,perhaps di ff erent systems can be studied, or perhaps ap-plications on di ff erent problems present advantages.We employed the recently formulated d -method,which is precisely as easy, or di ffi cult, as an ordinarythree-body calculation. It is equivalent to the bruteforce method of applying an external field, where onemore three dimensional degree-of-freedom is requiredin more complicated calculations. The present resultsare obtained for identical bosons and for distinguish-able particles. Additional applications are abundant, asfor example more particles, di ff erent quantum symme-tries, squeezing more than one dimension, or asymmet-ric squeeze of the dimensions. Acknowledgments
This work has been partially supported by the Span-ish Ministry of Science, Innovation and UniversityMCIU / AEI / FEDER,UE (Spain) under Contract No.PGC2018-093636-B-I00.6 eferencesReferences [1] V. M. Efimov, Phys. Lett. B 33, 563 (1970).[2] T. K¨ohler, K. G¨oral, and P. S. Julienne, Rev. Mod. Phys. 78,1311 (2006).[3] I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. 80, 885(2008).[4] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Rev.Mod.Phys.82, 1225 (2010).[5] S. Deng, Z.-Y. Shi, P. Diao, Q. Yu, H. Zhai, R. Qi, and H. Wu,Science 353, 371 (2016).[6] E. Nielsen, D.V. Fedorov, A.S. Jensen, and E. Garrido, Phys.Rep. 347, 373 (2001).[7] E. Braaten and H.-W. Hammer, Phys. Rep. 428, 259 (2006).[8] E. Garrido, Few-Body Syst. 59, 17 (2018).[9] M. Mikkelsen, A. S. Jensen, D. V. Fedorov, and N. T.Zinner,Journal of Physics B: Atomic, Molecular and Optical Physics48, 085301 (2015).[10] Fedorov D. V. and Jensen A. S. Europhys. Lett. 62, 336 (2003).[11] A.S. Jensen, K. Riisager, D.V. Fedorov, and E. Garrido, Rev.Mod. Phys. 76, 215 (2004).[12] T. Kraemer, M. Mark, P. Waldburger, J. G. Danzl, C. Chin, B.Engeser, A. D. Lange, K. Pilch, A. Jaakkola, H.-C. N¨agerl,andR. Grimm, Nature (London) 440, 315 (2006).[13] M. Zaccanti, B. Deissler, C. D’Errico, M. Fattori, M. Jona-Lasinio, S. M¨uller, G. Roati, M. Inguscio, and G. Modugno,Nature Phys. 5, 586 (2009).[14] S. E. Pollack, D. Dries, and R. G. Hulet, Science 326, 1683(2009).[15] N. Gross, Z. Shotan, S. Kokkelmans, and L. Khaykovich,Phys.Rev. Lett. 103, 163202 (2009). Phys.Rev. Lett. 105,103203 (2010).[16] J. H. Huckans, J. R. Williams, E. L. Hazlett, R. W. Stites, andK. M. O’Hara, Phys. Rev. Lett. 102, 165302 (2009).[17] T. B. Ottenstein, T. Lompe, M. Kohnen, A. N. Wenz, and S.Jochim, Phys. Rev. Lett. 101, 203202 (2008).[18] A. N. Wenz, T. Lompe, T. B. Ottenstein, F. Serwane, G. Z¨urn,and S. Jochim, Phys. Rev. A80, 040702(R) (2009).[19] E. Nielsen, D.V. Fedorov, and A.S. Jensen, Phys. Rev. A 56,3287 (1997).[20] A. G. Volosniev, D.V. Fedorov, A.S. Jensen, N.T. Zinner, Eur.Phys. J. D 67, 95 (2013).[21] P. Naidon and S. Endo, Rep. Prog. Phys. 80, 056001 (2017).[22] L.W. Bruch and J.A. Tjon, Phys. Rev. A 19, 425 (1979).[23] T.K. Lim and B. Shimer, Z. Phys. A 297,185 (1980).[24] B. Simon, Ann. Phys. 97, 279 (1976).[25] A. G. Volosniev, D. V. Fedorov, A. S. Jensen, N. T. Zinner, Phys.Rev. Lett. 106, 250401 (2011).[26] J. Levinsen, P. Massignan, and M.M. Parish, Phys. Rev. X 4,031020 (2014).[27] M.T. Yamashita, F.F. Bellotti, T. Frederico, D.V. Fedorov, A.S.Jensen, N. T. Zinner, J. Phys. B: At. Mol. Opt. Phys. 48, 025302(2015).[28] J.H. Sandoval, F.F. Bellotti, A.S. Jensen, and M.T. Yamashita, J.Phys. B: At. Mol. Opt. Phys. 51, 065004 (2018).[29] D.S. Rosa, T. Frederico, G. Krein, and M.T. Yamashita, Phys.Rev. A 97, 050701(R) (2018).[30] Y. Nishida and S. Tan, Phys. Rev. Lett. 101, 170401 (2008).[31] Y. Nishida and S. Tan, Phys. Rev. A 79, 060701(R) (2009).[32] E. Garrido, A.S. Jensen, and R. ´Alvarez-Rodr´ıguez, Phys. Lett.A 383, 2021 (2019).[33] E. R. Christensen, A.S. Jensen, and E. Garrido, Few-Body Syst59, 136 (2018). [34] E. Garrido and A.S. Jensen, Phys. Rev. Research 1, 023009(2019).[35] E. Garrido and A.S. Jensen, Phys. Rev. Research 2, 033261(2020).[36] F. S. Møller, D. V. Fedorov, A. S. Jensen, N. T. Zinner, J. Phys.B: At. Mol. Opt. Phys. 52, 145102 (2019).[37] J.R. Armstrong, A.G. Volosniev, D.V. Fedorov, A.S. Jensen, andN.T. Zinner, J. Phys. A: Math. Theor. 48, 085301 (2015).[38] D.V. Fedorov, A.S. Jensen, and K. Riisager, Phys. Rev. Lett. 73,2817 (1994).[39] A. Kievsky and M. Gattobigio, Phys. Rev. A 87, 052719 (2013).[40] L. Happ, M. Zimmermann, S.I. Betelu, W.P. Schleich, and M.A.Efremov, Phys. Rev. A 100, 012709 (2019).[41] F. F. Bellotti, T. Frederico, M. T. Yamashita, D. V. Fedorov, A.S. Jensen, N. T. Zinner, Phys. Rev. 85, 025601 (2012)[1] V. M. Efimov, Phys. Lett. B 33, 563 (1970).[2] T. K¨ohler, K. G¨oral, and P. S. Julienne, Rev. Mod. Phys. 78,1311 (2006).[3] I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. 80, 885(2008).[4] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Rev.Mod.Phys.82, 1225 (2010).[5] S. Deng, Z.-Y. Shi, P. Diao, Q. Yu, H. Zhai, R. Qi, and H. Wu,Science 353, 371 (2016).[6] E. Nielsen, D.V. Fedorov, A.S. Jensen, and E. Garrido, Phys.Rep. 347, 373 (2001).[7] E. Braaten and H.-W. Hammer, Phys. Rep. 428, 259 (2006).[8] E. Garrido, Few-Body Syst. 59, 17 (2018).[9] M. Mikkelsen, A. S. Jensen, D. V. Fedorov, and N. T.Zinner,Journal of Physics B: Atomic, Molecular and Optical Physics48, 085301 (2015).[10] Fedorov D. V. and Jensen A. S. Europhys. Lett. 62, 336 (2003).[11] A.S. Jensen, K. Riisager, D.V. Fedorov, and E. Garrido, Rev.Mod. Phys. 76, 215 (2004).[12] T. Kraemer, M. Mark, P. Waldburger, J. G. Danzl, C. Chin, B.Engeser, A. D. Lange, K. Pilch, A. Jaakkola, H.-C. N¨agerl,andR. Grimm, Nature (London) 440, 315 (2006).[13] M. Zaccanti, B. Deissler, C. D’Errico, M. Fattori, M. Jona-Lasinio, S. M¨uller, G. Roati, M. Inguscio, and G. Modugno,Nature Phys. 5, 586 (2009).[14] S. E. Pollack, D. Dries, and R. G. Hulet, Science 326, 1683(2009).[15] N. Gross, Z. Shotan, S. Kokkelmans, and L. Khaykovich,Phys.Rev. Lett. 103, 163202 (2009). Phys.Rev. Lett. 105,103203 (2010).[16] J. H. Huckans, J. R. Williams, E. L. Hazlett, R. W. Stites, andK. M. O’Hara, Phys. Rev. Lett. 102, 165302 (2009).[17] T. B. Ottenstein, T. Lompe, M. Kohnen, A. N. Wenz, and S.Jochim, Phys. Rev. Lett. 101, 203202 (2008).[18] A. N. Wenz, T. Lompe, T. B. Ottenstein, F. Serwane, G. Z¨urn,and S. Jochim, Phys. Rev. A80, 040702(R) (2009).[19] E. Nielsen, D.V. Fedorov, and A.S. Jensen, Phys. Rev. A 56,3287 (1997).[20] A. G. Volosniev, D.V. Fedorov, A.S. Jensen, N.T. Zinner, Eur.Phys. J. D 67, 95 (2013).[21] P. Naidon and S. Endo, Rep. Prog. Phys. 80, 056001 (2017).[22] L.W. Bruch and J.A. Tjon, Phys. Rev. A 19, 425 (1979).[23] T.K. Lim and B. Shimer, Z. Phys. A 297,185 (1980).[24] B. Simon, Ann. Phys. 97, 279 (1976).[25] A. G. Volosniev, D. V. Fedorov, A. S. Jensen, N. T. Zinner, Phys.Rev. Lett. 106, 250401 (2011).[26] J. Levinsen, P. Massignan, and M.M. Parish, Phys. Rev. X 4,031020 (2014).[27] M.T. Yamashita, F.F. Bellotti, T. Frederico, D.V. Fedorov, A.S.Jensen, N. T. Zinner, J. Phys. B: At. Mol. Opt. Phys. 48, 025302(2015).[28] J.H. Sandoval, F.F. Bellotti, A.S. Jensen, and M.T. Yamashita, J.Phys. B: At. Mol. Opt. Phys. 51, 065004 (2018).[29] D.S. Rosa, T. Frederico, G. Krein, and M.T. Yamashita, Phys.Rev. A 97, 050701(R) (2018).[30] Y. Nishida and S. Tan, Phys. Rev. Lett. 101, 170401 (2008).[31] Y. Nishida and S. Tan, Phys. Rev. A 79, 060701(R) (2009).[32] E. Garrido, A.S. Jensen, and R. ´Alvarez-Rodr´ıguez, Phys. Lett.A 383, 2021 (2019).[33] E. R. Christensen, A.S. Jensen, and E. Garrido, Few-Body Syst59, 136 (2018). [34] E. Garrido and A.S. Jensen, Phys. Rev. Research 1, 023009(2019).[35] E. Garrido and A.S. Jensen, Phys. Rev. Research 2, 033261(2020).[36] F. S. Møller, D. V. Fedorov, A. S. Jensen, N. T. Zinner, J. Phys.B: At. Mol. Opt. Phys. 52, 145102 (2019).[37] J.R. Armstrong, A.G. Volosniev, D.V. Fedorov, A.S. Jensen, andN.T. Zinner, J. Phys. A: Math. Theor. 48, 085301 (2015).[38] D.V. Fedorov, A.S. Jensen, and K. Riisager, Phys. Rev. Lett. 73,2817 (1994).[39] A. Kievsky and M. Gattobigio, Phys. Rev. A 87, 052719 (2013).[40] L. Happ, M. Zimmermann, S.I. Betelu, W.P. Schleich, and M.A.Efremov, Phys. Rev. A 100, 012709 (2019).[41] F. F. Bellotti, T. Frederico, M. T. Yamashita, D. V. Fedorov, A.S. Jensen, N. T. Zinner, Phys. Rev. 85, 025601 (2012)