Efimov states of three unequal bosons in non-integer dimensions
aa r X i v : . [ phy s i c s . a t m - c l u s ] S e p Noname manuscript No. (will be inserted by the editor)
Esben Rohan Christensen · A.S. Jensen · E. Garrido
Efimov states of three unequal bosons in non-integerdimensions
Received: date / Accepted: date
Abstract
The Efimov effect for three bosons in three dimensions requires two infinitely large s -wavescattering lengths. We assume two identical particles with very large scattering lengths interactingwith a third particle. We use a novel mathematical technique where the centrifugal barrier contains aneffective dimension parameter, which allows efficient calculations precisely as in ordinary three spatialdimensions. We investigate properties and occurrence conditions of Efimov states for such systemsas functions of the third scattering length, the non-integer dimension parameter, mass ratio betweenunequal particles, and total angular momentum. We focus on the practical interest of the existence,number of Efimov states and their scaling properties. Decreasing the dimension parameter from 3towards 2 the Efimov effect and states disappear for critical values of mass ratio, angular momentumand scattering length parameter. We investigate the relations between the four variables and extractdetails of where and how the states disappear. Finally, we supply a qualitative relation between thedimension parameter and an external field used to squeeze a genuine three dimensional system. The Efimov effect in three dimensions is characterized by a three-body system, where the two-body s -wave scattering lengths for at least two of the three pairs are infinitely large, and consequently infinitelymany bound three-body states can be found [1; 2]. This is a rigorous mathematical definition whichcan never be precisely obeyed, neither for systems found in nature nor for constructions in laboratories.Therefore it is essential to distinguish between occurrence conditions for the effect and appearance ofa finite number of states that can be classified as Efimov states.The Efimov effect has by now been demonstrated in many laboratories [3; 4; 5; 6], but only onedirect measurement exists of an Efimov state [3]. All other observations are convincing but indirectand restricted to derived relative features of at most three different Efimov states. We shall thereforebe concerned with the properties of only the lowest few Efimov states. This is already challenging aswitnessed by the difficulties in laboratory tuning to the very well-known mathematical conditions. This work has been partially supported by the Spanish Ministerio de Econom´ıa y Competitividad under ProjectFIS2014-51971-PEsben Rohan ChristensenDepartment of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C, DenmarkE-mail: [email protected]. JensenDepartment of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C, DenmarkE-mail: [email protected]. GarridoInstituto de Estructura de la Materia, IEM-CSIC, Serrano 123, E-28006 Madrid, SpainE-mail: [email protected]
Fig. 1: Sketch of the coordinates of the three-body system. The particles are labeled ( i, j, k ), that ispermutations of (1 , , µ jk and µ i,jk are given in Eq.(1).The experimental techniques are crucial but fortunately very developed over the last decade. Thedecisive feature is tuning of the effective two-body interaction by use of the Feshbach resonance tech-nique [7]. The central ingredient is the controllable external fields which are directly varied to place atwo-body state at zero energy corresponding to infinite s -wave scattering length. Clearly this tuningcan only be approximate with a resulting finite scattering length.The orders of the achievable control are such that cold atoms or molecules are the only candidatesfor this artificial tuning. On the other hand, then the technique is both efficient and flexible withexternal fields varying continuously from spherical to extremely deformed. This makes it practicallypossible to squeeze by use of a deformed external field, which effectively continuously reduces thespatial dimensions. The effects of this dimension variation has been investigated recently in variousways [8; 9; 10; 11].A change of dimension from 3 D to 2 D is known to produce qualitative structure changes as ev-idenced by the fact that the Efimov effect exists in 3 D but not in 2 D [1]. Furthermore, the massdependence and the dependence on the number (two or three) of contributing large scattering lengthsare extremely important for scaling properties in 3 D [12], and in addition the third finite scatteringlength may have substantial effect [13]. Most likely then the variation with dimension would be crucialfor these dependencies. In this report, we focus on the Efimov effect, and the number and structure ofEfimov states as function of the dimension parameter, d , varying between 2 and 3.In section II we first provide details of the theoretical formulation and section III describes thenumerical procedure and pertinent basic properties. In section IV we discuss occurrence of the Efimoveffect and provide characteristic properties of the corresponding states. In section V we discuss themeaning of the dimension parameter, and indicate an interpretation in terms of an external field.Finally, in section VI we conclude and point out some perspectives of this work. The
Efimov effect is the appearance of an infinite number of bound states in a three-body systemwithout bound two-body subsystems [14]. This occurs when at least two of the two-body s -wavescattering lengths, a and a , are infinitely large. We shall focus on the third scattering length, a ,since the properties depend substantially on its value between −∞ and ∞ [13]. We shall first specifywhich systems to study, then define key quantities, give equations to determine them, and discussschematic numerical results.2.1 Specifying system and purposeThe system consists of three point-like particles of masses, m i , and coordinates r i , where i = 1 , , m = m , insuch a way that a = a , which are, respectively, the scattering lengths associated to the interactionbetween particles 1 and 3, and between particles 1 and 2. We shall use hyperspherical coordinates built on the Jacobi sets in Fig. 1, where the mass factors µ jk and µ i,jk are given by: µ jk = r m j m k m ( m j + m k ) ; µ i,jk = s m i ( m j + m k ) m ( m i + m j + m k ) , (1)where m is an arbitrary normalization mass, chosen in this work to be m .The only length is the hyperradius, ρ , for example defined by mρ ≡ M X i 3) can be written " b + b (cid:18) ρµ jk a i (cid:19) ( d − A i = X j = i b F ( a, b, c, x ij ) x Lij A j , (8)where µ jk is given in Eq.(1), and the arguments of the hypergeometric function , F , are given by a = − ν , b = ν + L + d − , c = d/ L , (9) x = x = 1 √ q m m , (10) x = 11 + m m , (11)using ν as the natural variable which in turn provides λ and the potential through Eqs.(7) and (3). Thehypergeometric function F , is a special function that can be represented by a power series called thehypergeometric series. It is straight forward to incorporate in a numerical routine. For more informationsee appendix A in [1]. The b i -quantities are abbreviations depending on ν , d and L , and they are definedby b ( ν, d ) = − sin( π ( ν + d/ πd/ b ( ν, d, L ) = Γ ( d/ Γ ( ν + 1) Γ ( ν + d/ L ) sin( πν ) π ( d/ − Γ ( ν + d/ Γ ( ν + d − L ) b ( ν, d, L ) = Γ ( d/ Γ ( ν + d/ L ) Γ ( d/ L ) Γ ( ν + d/ , (12)where Γ is the complex gamma function.With all these definitions we are ready to work on Eq.(8), where the ρ -dependence only appearsthrough the combination ρ/ ( µ jk a i ). We emphasize that the scattering lengths entering in all theseexpressions are defined in d dimension. The procedure is to find the boundary matching constants, A i , ( i = 1 , , ν .Our focus is on the Efimov effect which requires two very large scattering lengths. We thereforeassume | a | = | a | = ∞ leading to the simpler equations where only the combined dependence of ρ and a , ρ/ ( µ a ), remains. In reality this means that ρ ≪ | a | . The corresponding two masses are alsoequal, i.e. m = m , which has the important consequence that dependence on masses is collected inonly one variable, namely the ratio of masses m /m = m /m . In [1] Eq. (8) is derived and limiting cases of three identical particles, a = 0 and a = ∞ arestudied. Here we explore the consequences of finite values of a , while still maintaining infinite valuesof the other two scattering lengths. With these assumptions we demand that the determinant of thesystem of equations in Eq. (8) should be zero and choose the interesting solution depending on L , asdescribed in [1], to obtain: c (cid:0) b − ( − L c (cid:1) − c = 0 , (13)where the coefficients, c i , are defined by c = b + b (cid:18) ρµ a (cid:19) ( d − c = b F ( a, b, c ; x ) x L c = b F ( a, b, c ; x ) x L . (14)The procedure is then to solve Eq. (13) to get ν for each d and L as function of ρ/ ( µ a ), and relateto λ through Eq. (7) and the hyperradial potential in the hamiltonian Eq. (3). The mass dependenceenter though µ , x and x . Mass ratio and dimension parameter are the main variables. We concentrate on the dependence on a ,while maintaining infinitely large | a | and | a | , which are necessary conditions for occurrence of theEfimov effect. We sketch the numerical technique and demonstrate the validity range of the analyticbut transcendental equations in Eq. (13). In the following subsection we give the basic quantities tobe discussed in the later sections.3.1 Validity-range of the solutionsEq. (13) is conceptually an equation for ν through all the abbreviations in the preceding definingequations. The adiabatic potentials are then determined through Eq. (13) as functions of the ratio, ρ/ ( µ a ). In principle this is done by first computing ν , then using Eq. (7) to find λ , and finallyfinding the potential in Eq. (3) expressed by λ , d and ρ . However, by inspection it is clear that the ρ -dependence can be simply isolated and in fact expressed as a function of ν . In turn, ν can be expressedas function of λ . Thus, it is numerically easy to find ρ as function of λ , and subsequently read thecurve in the opposite direction. The initially chosen λ -interval decides in this way which of the manyadiabatic potentials are obtained as function of ρ .We follow this procedure throughout this report, but first we demonstrate the validity of theapproximation of infinite (large) values of a = a . We notice that ρ and a in Eq. (13) only enter astheir ratio, which therefore is conveniently chosen as the distance parameter determining the λ -values.The numerical results for specific cases are shown in Fig. 2 for different (large) values of | a | and | a | .We choose large negative a and a -values to avoid the inevitable divergence for large ρ correspondingto two-body bound states which are irrelevant in connection with possible existence of Efimov states.The lowest λ ′ s in Fig. 2 are given relative to the Efimov critical value, λ c , since this later in this reporthas to be the reference point.The potentials related to Fig. 2 are found by numerical solutions of the full three-body equationswith finite-range gaussian two-body potentials between all pairs of particles. Since we intend to use thezero-range approximation we only consider lengths and distances much larger than the gaussian range, b g , which is used in the figure as length unit. Only the two lowest λ ’s are necessary where the firstand second carry the possible Efimov states, when a < a > 0, respectively. For our purposewe only use these (approximately) decoupled solutions, as they are responsible for all possible Efimovstates. When a > λ corresponds at large distances to a two-body bound state betweenthe identical particles. The rapid variation of this potential for very small ρ is due to the finite rangepotential causing the discontinuity at ρ = 0. To be specific, the calculation has been performed fortwo different values of | a | , | a | = 10 and 20, and two different values of a = a , a = a = − and -15 -10 -5 0 5 10 15 ρ/ ( µ a ) -4-20246 λ−λ c d=2.5 d=2.5d=3 d=3 m / m =1 Fig. 2: Computed lowest λ -solutions for d = 3 and d = 2 . 5, for L = 0, m /m = 1, and gaussiantwo-body interactions. The range of the gaussians is taken as length unit. The scattering length values | a | = 10 , 20 and a = a = − , − are considered. The left and right parts of the figure (red andblue curves) correspond to a < a > 0, for which the lowest and two lowest λ ’s are shown,repectively. For ρ/ ( µ | a | ) −→ ∞ , the two λ -solutions both converge towards the limit for ( a = 0)(black dashed). The green line indicates the Efimov condition, λ − λ c = 0. -15 -10 -5 0 5 10 15-4-20246 Fig. 3: The lowest λ -solutions for d = 3 and d = 2 . L = 0, and m /m = 1, where the lowest isred and the second lowest is blue. The two λ -solutions both converge towards the limit for ( a = 0)(black dashed), that is ρ/ ( µ | a | ) −→ ∞ . The green line indicates the Efimov condition, λ − λ c = 0. − . Therefore, each of the curves shown in the figure for a given d and a given sign of a containsactually four curves, corresponding to the four possible combinations of the values chosen for | a | and a = a .From Fig. 2 we conclude that the limit, | a | = | a | = ∞ , is reached for all ρ ≪ | a | . This means thatdeviations only would be visible about two orders of magnitude larger than the scale on the figure. Thisobservation is reassuring, but far from surprising in the hyperspherical adiabatic expansion method,where hyperradius and scattering lengths are the decisive properties at large-distances, that is outsidethe short-range potentials [1]. This genuine universal validity criterion is essentially independent ofthe variables in our investigation. Therefore it is not necessary to confirm this conclusion by showingresults for more values of d , m /m , and L . We proceed by use of the zero-range approximation and the solutions to Eq. (13). Examples com-patible with Fig. 2 are shown in Fig. 3 for zero-range interactions. The extreme limits of very large | a | = | a | from Fig. 2 are reproduced to test the numerical procedure. In this case, the lowest λ -branchfor a < a > a > 0, corresponding to the bound state, is the same forzero-range and finite-range potentials, compare Figs. 2 and 3.These figures are meant to establish the somewhat unusual presentation, where ρ/a is the distancecoordinate while we have assumed | a | = | a | = ∞ . The variation with a is substantial but onlyappearing through the ratio to ρ . The extreme limits of a = 0 are shown in Fig. 3 as the straighthorizontal dashed lines. The values for a = ±∞ are the ultimate optimum for occurrence of theEfimov effect. Comparing to Fig. 2, we emphasize that these features are only valid for ρ ≪ | a | .The left hand side of Fig. 3 is the lowest λ for negative a . This function continues smoothlythrough ρ = 0 on the right hand side where a is positive. The divergence towards −∞ correspondsto the two-body bound state between the identical particles. It is irrelevant in connection with theEfimov state with completely different structure. Instead the possible Efimov states for a > λ which for this zero-range interaction is diving down and approaching thesame value (black dashed line, a = 0 or infinite ρ ) above λ c as for large ρ and a < 0. The qualitativebehavior is the same for the two chosen examples of d = 2 . , . λ independent of distance. If this constantvalue is lower than the critical λ c infinitely many states occur, whereas a value higher than λ c does notsupport any bound state. Generalizing this schematic description we conclude that the Efimov effectwith infinitely many bound three-body states occur if and only if λ is below λ c in an infinitely largeregion of space. This is seen to be the case for both signs of a in Fig. 3 as it should for L = 0 for twoinfinitely large scattering lengths in three spatial dimensions. We emphasize that the λ -solutions forboth positive and negative a meet the Efimov condition (below the green line) for sufficiently large ρ/ ( µ | a | ).These conclusions and their dependencies on d , L , m /m , and a to be discussed later in thisreport. However, before proceeding with this we shall present a few other crucial properties. The wavefunctions, f , arising from solving Hf ( ρ ) = Ef ( ρ ) are, provided that λ < λ c and λ is constant in asufficiently large region of ρ , given by f n ( ρ ) ∝ √ ρK iξ ( κ n ρ ) , ~ κ n = − mE n , (15)where K iξ is the modified Bessel function of second kind. When the energy approaches zero thisexpression reduces to f n ( ρ ) ∝ √ ρ sin (cid:18) ξ ln (cid:18) ρρ (cid:19) + δ (cid:19) , (16)where δ is a phase depending on the boundary condition at ρ = ρ . The value of ρ is either thesmall-distance range limit or the small ρ where the potential crosses the threshold value correspondingto λ c . The many bound states arise for different energies, E n , through the number of oscillations ofthe Bessel function for this energy. This form of the wave function implies scaling properties betweenneighboring states h ρ i n +1 h ρ i n = E n E n +1 = exp(2 π/ξ ) . (17)Clearly infinitely many bound states correspond to infinitely many oscillations, that is for an in-finitely large ρ -space. In the present context the large ρ -limit is given by the scattering length | a | = | a | where the potential falls off exponentially, that is, it ceases to behave as 1 /ρ . Thus we can ratheraccurately estimate the number of bound states in this available interval by counting the correspondingnumber, N , of possible oscillations. We get N ≃ ξπ ln (cid:18) | a | ρ (cid:19) , (18) which diverge as | a | = | a | → ∞ . This discussion implies that the rigorously defined Efimov effectwith infinitely many bound states only can occur for | a | = | a | = ∞ . However, even for finite a = a there may be a finite number of bound states with precisely the same wave function properties. Thus,the number of such Efimov states is much more important than occurrence of the strict Efimov effect.The number of states is directly proportional to ξ , which is a measure of the distance below the criticalthreshold for occurrence of the effect.These derivations and considerations are strictly only valid for ρ -independent λ , but rather accuratewhen the λ -variation is weak over intervals extended over more than a few oscillation of the wavefunction in Eq. (16). It is here important to appreciate the scale on the abscissa of Fig. 3, where ρ isin units of the scattering length. The size of the variation of the curves over appropriate intervals arethen easily visually deceiving, since the spatial extension of the states may be smaller than the scaleof the variation. We maintain this x -axis because it contains the full dependence on both ρ and a . Inany case the qualitative features are correctly described.So far the discussion has only rephrased known properties for three spatial dimensions ( d ), totalangular momentum zero ( L ), and two infinite scattering lengths. After having defined the essentialingredients in the known cases, we shall extend to other d and L values as well as to arbitrary massratios, m /m . We shall still concentrate on the a -dependence while assuming the other two scatteringlengths are equal and numerically very large. The qualitative occurrence conditions can be elaborated to extract quantitative values of the theoreticalvariables. This means first of all relations between critical values for d , L , and m /m , and secondnumber of states and scaling properties. After the qualitative discussion we continue and present morequantitative results. We first connect to the more studied cases of d = 3, and afterwords we describethe dependence on smaller and non-integer values of d . We shall present results with focus on theimportant but relatively unknown a -dependence.4.1 Occurrence of the Efimov effectFrom Fig. 3 we confirm that the Efimov effect is always present for this case of two infinitely largescattering lengths in d = 3, independent of masses and size and sign of a . The monotonous hypersher-ical angular eigenvalues at infinity both approach the same value below the critical number, λ c . Thenthere is necessarily an infinite interval below λ c and the Efimov effect exists in this case. However, thisconclusion is crucially depending on d , m /m , and L .In Figs. 4 we show how the angular eigenvalues vary for different choices of these variables. Forsimplicity we have plotted all the relevant λ ’s as functions of ρ/ ( µ | a | ), that is reflected in ρ = 0 fornegative a . The overall picture is as in Fig. 3 with two curves both approaching the same asymptoticlarge-distance value from below and above, respectively. The decisive feature is whether this asymptoticvalue of a = 0 is either below or above λ c , which determines the existence or not of the Efimov effect.The behavior of λ is monotonous not only as function of ρ/ ( µ | a | ), but also as function of eachof the variables, d , L , and m /m . The asymptotic λ values for large ρ are significant indicators. Itmoves upwards in Figs. 4 when L increases, and when d and m /m decrease. This last fact indicatesthat large values of m /m (two identical heavy particles and one light) are clearly more convenientfor the appearance of the Efimov states than just the opposite (small m /m values, corresponding totwo light particles and one heavy).The figures also reveal very flat (red) curves corresponding to the unbound case of a < 0. Butalso the blue curves for a > ρ = 0 starting points.Here it is important that the x -axes are the hyperradius in units of the scattering length a . Theapproximations of constant λ are therefore very appropriate.In the limit of d = 2 the Efimov effect is not present for any choice of variables. This possibilitytherefore disappears somewhere between d = 2 and d = 3. The precise critical d -value depends on both L and m /m . For given d the curves are moved upwards with increasing L and decreasing m /m .Thus for sufficiently large d , there are critical values of both L and m /m corresponding to crossingof λ c . Fig. 4: The same quantities as Fig. 3 for different values of d = 3 . , . , . , . m /m = 1 , , , L = 0 , , 2. Here we only keep the λ ’s relevant for Efimov states, that is the first λ for a < ρ = 0 or as functions of ρ/ ( µ | a | ), and the second λ for a > d = 3 and from top to bottom for L = 0 , , L -values and mass ratio 50 for different d -values given on the curves. -20 -10 0 10 2000.511.522.53 Fig. 5: The scaling parameter, ξ , for both positive and negative a for different mass ratios. The numberof Efimov states is proportional to ξ , which increases with mass ratio. The ξ -solutions for a < a > 0, but for large ρ they converge towards the same value.The conclusion is that to rigorously have the Efimov effect the asymptotic λ -value must be below λ c , and then the effect exists for both signs of a . However, in practice these schematic curves must falloff much faster when ρ becomes comparable to | a | = | a | . The infinite series of bound Efimov states arethen abruptly cut off. Still a finite number of bound states may be present and furthermore they mayhave properties (like universality and scaling) precisely (or approximately) as genuine Efimov states.This might happen at relatively small distances on the unbound branch even when the asymptotic λ is above λ c . Thus, properties of a finite small number of states are even more interesting than strictoccurrence of the effect.4.2 Dependence for d = 3Three dimensions are very well studied. We shall therefore only use this limit to set the stage whileemphasizing the aspects of interest in the present context. The Efimov effect is present for two infinite s -wave scattering lengths for all masses for L = 0. On the other hand, the effect only exists for non-zero L -values when the mass ratio between the two heavy and the light particle exceeds critical values. Wefind ( m /m ) crit = (0 , , , , L = (0 , , , , L -values are probably only of academic interest and L = 0 , | a | = | a | but outside both the radius of theshort-range potential and the critical crossing point, clearly seen on Fig. 3 in the a > ξ and the related logarithm. The crucial dependence is shown in Fig. 5 for both signsof a and for different mass ratios.The value of ξ is the distance from λ ( ρ ) to λ c on figures similar to Fig. 3. The curves on the positive a -side corresponding to a bound heavy-heavy subsystem increase dramatically before saturating atlarge ρ at values increasing with mass ratio. The threshold values of zero correspond to ρ -values ofthe crossing point where λ = λ c in Fig. 3. On the unbound (negative) a -side we observe the oppositebehavior of a slightly increasing ξ with decreasing size of ρ or equivalently increasing | a | . This counterintuitive behavior is due to extension of the available space towards smaller distance, that is the earliercrossing of λ c , see Figs. 3 and 4. The limit here is the short-range radius.In all cases the size of ξ is unpleasantly small even if multiplied by a sizable value of a to producethe number of bound states in the accessible region of space. A realistic estimate could be a /ρ = 10 n ,where n is 4 − 6, resulting in only a few Efimov states for all a . Closely related to the number of states is the scaling between the possible states, see Eq. (17). Unless a is negative and very large,the scaling factor is very large for moderate mass ratios. Thus, the relative position of a few Efimovstates would depend strongly on the precise value of a , which in practice prevent universal predictionof these positions.4.3 Dependence for d = 3Decreasing d from 3 we know that the Efimov effect has disappeared all together in the d = 2 limit,independent of any choice of variables. The questions are therefore where the disappearance takesplace depending on mass ratio and angular momentum. We know from Figs. 4 that larger mass ratioincreases the distance from the critical λ c and consequently the ξ -value increases. The same trendremains for all other d , where the existence of the Efimov effect depends on the masses. Fig. 6: The critical mass ratio as a function of the dimension parameter for different values of L . Thestrict criteria are extracted from the large ρ -limit (blue for small a ), and for small ρ (red for largenegative a ). The mathematically well defined results for d > d for different L . Since themathematics applies to any d we also give results for d > 3. We notice the almost symmetric behavioraround d = 3 for L = 0. The minimum critical mass around d = 3 allows the existence only in a regionaround d = 3. For higher values of L , the minimum critical mass shifts to higher values of d . Theprecise numbers for L = 0 , , d = 3 are already given above but now we also see the variationsas steep increases as d decreases. The higher L the more compensation is needed from increasing themass ratio. Higher L -values could also be calculated but we refrain.The strict limit for a = 0 is less favorable than the condition for large | a | . This is obvious alreadyfor physical reasons since these limits a = 0 (large ρ ) and | a | = ∞ (small ρ ) on the plots correspondto either three or two contributing subsystems, respectively. In Fig. 6 we present the critical masses asfunctions of d for both a = 0 and the ultimate limit of a = ±∞ which can support more states. Thedifference between these estimates gives an interval for occurrence of Efimov states depending slightlyon the third scattering length. The small bump on the a = 0 curve is numerical inaccuracy in thisextreme limit.For equal masses and L = 0 the difference is that the critical value of d is pushed from d = 2 . a = 0) to d = 2 . a = ±∞ ). For other given mass ratios we may read off the critical d -valueallowing Efimov states in both these extreme limits. When d decreases towards 2 the critical massesincrease dramatically. A closer inspection of the governing equations in Eq. (13) reveals a correspondingunlimited increase, that is sufficiently large mass ratio allow Efimov states for any d > For L = 1 , ρ are almost indistinguishable. This may be understoodfrom the fact that Efimov states are only allowed for s -waves between at least two subsystems. Thisimplies that finite angular momenta must be attached to the remaining subsystem in the present worklabeled by 1. Then the corresponding s -wave scattering length a does not enter the Efimov equations. -10 -5 0 5 1000.511.522.53 Fig. 7: The scaling parameter, ξ for different values of d . The mass ratio is fixed at m /m = 50. Forthis mass ratio, tn the case of d = 2 . 25, no states are available on the bound side, and only a few onthe unbound side.As discussed for d = 3, we can also in the general cases calculate the expected number of statesand the corresponding scaling from Eqs. (17) and (18). In Fig. 7, we show examples for different d -parameters as functions of the ρ/ ( µ a ) variable. The appearance and explanations are the same asin Fig. 5 for both bound and unbound cases, threshold and saturation behavior. The numbers arelarger because we have chosen a much larger mass ratio. Still the conclusions hold about only a fewEfimov states and an unfortunate large scale parameter with huge variation close to the threshold atthe bound side. However, this variation has essentially no physical impact since the interval is toosmall to support bound states. d The dimension parameter, d , is appealingly a measure of the spatial dimension varying between 2 and3. While this is rigorously correct in the two limits, it is unfortunately much more complicated forintermediate non-integer values. A proper physical interpretation, or a direct translation, is so far onlyavailable for two particles [16]. The problem can be formulated in ordinary three spatial dimensionswhere the effective dimension, d , has to be related to confinement by an external potential. Then onecoordinate is squeezed by such external walls varying from being at infinity to practically zero, whichmeans much smaller than the short-range potential responsible for the properties.It is physically obvious that the walls of the external potential have no effect on a system of muchsmaller spatial extension, and, vice versa, crucial for a system of larger natural extension. Thus thesize of the system is decisive, as described in [16; 9], where a universal result is given for two particlesheld together by a spherical gaussian short-range potential and squeezed by a one dimensional externaloscillator potential defined by a length parameter, b ext .The resulting relation for b ext /b g ( b g is the gaussian range) as function of d is shown in Fig. 8 wherethe size of the system enters through the factor (1 + b ext /a ), where a ( > 0) is the scattering length of d -1 a/b g =2.033a/b g =18.122a/b g =40.608 Analytical fit Fig. 8: The derived dependence of b ext /b g as function of d for two-body short-range potentials withscattering lengths divided by the potential range, a/b g = (2 . , . , . b ext /a .the two-body potential. This relation is parameterized as b ext b g (cid:18) b ext a (cid:19) = c | ln(3 − d ) | c + c tan (cid:18) ( d − c πc (cid:19) , (19)where the constants ( c , c , c , c , c ) take the values (0 . , . , , . , . b ext , is not noticed by thesystem, which therefore lives in just the ordinary three-dimensional space, i.e., d = 3. In the oppositelimit, when the squeezing length approaches zero, the system is very much confined along the squeezingdirection, corresponding therefore to d = 2. These two limits are connected by the curve in Fig. 8,which we emphasize is only strictly valid for two-particle systems. The generalization to three particlesis not available at the moment, since an elaborate set of calculations are necessary for three particles inexternal fields implying complications as for a four-body problem. However, we anticipate qualitativelysimilar correspondence between d and an external field. For now the two-body relation shown in Fig. 8and Eq.(19) is sufficiently indicative for the investigations in the present report. We use the hyperspherical expansion method with one uncoupled single adiabatic potential and thedimension parameter, d , in the centrifugal barrier. We assume two identical bosons with two infinitely(equal) large s -wave scattering lengths against the third particle, a and a , which allow existence of theEfimov effect in three dimensions. We then investigate dependence on the finite scattering length ( a )between the two identical bosons, while varying the dimension parameter, the mass ratio, and totalangular momentum. The Efimov effect, the scaling and number of Efimov states are large-distancephenomena and independent of short-range attraction. We consequently use the simple zero-rangeformulation.We distinguish between occurrence of the Efimov effect and the finite number of practically acces-sible Efimov states. Both, the effect and all the corresponding states, disappear as the dimension isdecreased towards two dimensions. When all masses are equal and the angular momentum is zero theeffect disappears for d = 2 . d = 2 . We discuss specifically the qualitative difference between results for different signs of a correspond-ing to bound or unbound identical two-boson system. If the effect exists, the number of Efimov states ison the unbound side both proportional to the scaling parameter and limited to the number of possiblewave function oscillations before reaching the large scattering length. On the bound side the numberof states is given in the same way except for an additional restriction to be outside a radius varyingweakly with a . If the Efimov effect strictly does not exist, still a few Efimov states might be allowedon the unbound side while forbidden on the bound side.Finally, we provide a qualitative relation between dimension parameter and a length parameter ofan external field used to squeeze the spatial dimension of the system from 3 to 2. The precise size ofthe external field in a three-body calculation is at present only estimated. However, a firm conclusionis that a reduction of the available space parameterized by d unambiguously leads to disappearanceof both, effect and all Efimov states. We provide at the moment only a qualitative estimate of thefunction translating the d -parameter into precise size and shape of the external squeezing potential.In summary, we have calculated occurrence conditions and properties of Efimov states dependingon dimension, the third scattering length, masses, and angular momentum. All results are possible totest in practice in present day laboratories. References 1. E. 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