Confinement of two-body systems and calculations in d dimensions
CConfinement of two-body systems and calculations in d dimensions. E. Garrido and A.S. Jensen Instituto de Estructura de la Materia, IEM-CSIC, Serrano 123, E-28006 Madrid, Spain Department of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C, Denmark (Dated: September 10, 2019)A continuous transition for a system moving in a three-dimensional (3D) space to moving in alower-dimensional space, 2D or 1D, can be made by means of an external squeezing potential. Asqueeze along one direction gives rise to a 3D to 2D transition, whereas a simultaneous squeezealong two directions produces a 3D to 1D transition, without going through an intermediate 2Dconfiguration. In the same way, for a system moving in a 2D space, a squeezing potential along onedirection produces a 2D to 1D transition. In this work we investigate the equivalence between thiskind of confinement procedure and calculations without an external field, but where the dimension d is taken as a parameter that changes continuously from d = 3 to d = 1. The practical caseof an external harmonic oscillator squeezing potential acting on a two-body system is investigatedin details. For the three transitions considered, 3D → → → d is found. This relationis well established for infinitely large 3D scattering lengths of the two-body potential for 3D → →
1D transitions, and for infinitely large 2D scattering length for the 2D →
1D case.For finite scattering lengths size corrections must be applied. The traditional wave functions forexternal squeezing potentials are shown to be uniquely related with the wave functions for specificnon-integer dimension parameters, d . I. INTRODUCTION
The properties of quantum systems depend cruciallyon the dimension of the space where they are allowed tomove. A clear example of this is the centrifugal barrierin the radial Schr¨odinger equation, which, for zero totalangular momentum, is negative for a two-body systemin two (2D) dimensions, while it is zero in three (3D)dimensions [1]. An immediate consequence of this is thatany infinitesimal amount of attraction produces a boundstate in 2D, whereas in 3D a finite amount of attraction isnecessary for binding a system [2]. As an exotic exampleof recent interest we can mention, at the three-body level,the occurrence in 3D of the Efimov effect [3]. In 2D, thiseffect does not occur, neither for equal mass three-bodysystems [4] nor for unequal mass systems [5].The transition from a three-dimensional to a lower-dimensional space is commonly investigated by means ofan external trap potential that confines the system un-der investigation in a certain region in the space. In thisway, in [6] a harmonic oscillator trap potential is used,and binary atomic collisions are investigated under dif-ferent confinement regimes. This work focus on scatter-ing properties in confined spaces, not necessarily similarto the asymmetric squeezing of only one or two spatialdimensions. More recently, the particle physics formal-ism has been specifically extended to connect d = 3 and d = 2 by continuously compactifying, by means of an in-finite potential well, one of the dimensions [7–9]. Work-ing in momentum space, the intentions were to studythree-body physics, but two-body subsystems are thennecessary ingredients. The physical interpretation of thesqueezing parameter in this procedure is a problem nec-essary to be addressed to connect properly to measure-ments. The same compactifying procedure along one or two directions has been employed to investigate the S -matrix for two-body scattering [10], and again the fo-cus is on scattering under confining conditions. Anotherstructure-related investigation has appeared in the liter-ature, that is the superfluid phase transition temperaturein the crossover from three to two dimensions [11]. Thisis necessarily a many-body effect although prompted bytwo-body properties.In all these works the procedure has been to performgenuine 3D calculations where the external potential en-ters explicitly in order to limit the space available. How-ever, an alternative can be to employ an abstract formu-lation where the dimension d can take different valuesdescribing the different possible scenarios. For instance,in [12] an expansion in terms of 1 /d is performed, where d is thought of as an integer, allowing extrapolations be-tween integers. The philosophy has been to extrapolateobtainable results as function of 1 /d for very large d andfirst down to d = 3 for two or more particles [13]. Goingfurther down towards d = 2 is probably going too far [13],both because 1 /d = 1 / d = 3to d = 2. Non-integer dimensions have also been em-ployed in various subfields of mathematics and physics,see e.g. [14, 15]. For many particles even mixed dimen-sions have been used to study exotic structures [9]. Apractical continuous connection between integer dimen-sions is interesting in order to understand the relatedstructure variations.In this work the approach would be formulation in co-ordinate space by simple analytic continuation of the ab-stract formulation in terms of the dimension parameter d assuming non-integer values. The limits of d = 1 , , a r X i v : . [ phy s i c s . a t m - c l u s ] S e p of deformed external fields squeezing one or more dimen-sions to zero spatial extension corresponding to infinitelyhigh zero point energy. This method was used in a recentwork [17], where the continuous confinement of quantumsystems from three to two dimensions was investigated.The confinement in [17] was treated by use of two dif-ferent procedures. In the first one the particles are putunder the effect of an external trap potential acting ona single direction. This potential continuously limits themotion of the particles along that direction, in such away that for infinite squeezing the system moves in a 2Dspace. In the second method the external potential isnot used, and the problem instead is solved directly in d dimensions, where d is a parameter that changes contin-uously from 3 to 2. This formulation has the advantagethat the numerical effort required is similar to solving theordinary problem for integer dimensions.In [17] the 3D to 2D confinement was investigated fortwo-body systems and external harmonic oscillator con-fining potentials. The purpose of this work is to extendthe investigation to squeezing up to one dimension (1D).This can be done in two different ways. In the first onewe consider a simultaneous squeezing along two direc-tions, in such a way that the external potential pushesthe system, initially moving in 3D, to moving in 1D with-out going through an intermediate 2D geometry. Thesecond procedure consists on two consecutive squeezingprocesses along one direction, giving rise to a 3D to 2Dsqueezing followed by a 2D to 1D squeezing. The obvi-ous, but complicated, extension to systems made of morethan two particles is left for a forthcoming work.In all the confinement scenarios (3D → → → d as a parameter that changes continuously from d = 3to d = 2 or d = 1. One of the main goals is then, for allthe cases, to establish the equivalence between a givenvalue of the confining harmonic oscillator frequency andthe dimension d describing the same physical situation.The connection between the harmonic oscillator pa-rameter and the dimension d should preferentially beuniversal in the sense of being independent of the de-tails of the potential. It is well known that necessaryingredients for the appearance of universal properties ofquantum systems are the existence of two-body interac-tions with large scattering lengths, and, to a large extent,the preponderance of relative s -waves between the con-stituents. The existence, under these conditions, of auniversal connection between the harmonic oscillator pa-rameter and the dimension will be investigated. Here it isclear that large squeezing confining a wave function to beinside the two-body potential must depend on potentialdetails. However, comparing the two methods, it can stillresult in the same universal dependence, since both aresubject to the same potential. To be practical, we haveestablished such a highly desirable connection betweenthe wave functions obtained in the two methods. The overall purpose of the present work is therefore tostudy the a number of different transitions between inte-ger dimensions, and to establish the universal connectionbetween the d -parameter results and those of the bruteforce three dimensional calculation with a deformed ex-ternal field. The connection must allow the numericallysimpler d -method to be self-sufficient, that is in itselfproviding full information including correspondence toexternal field and three dimensional wave function. Thepaper is organized as follows. In section II we describe theprocedure used to confine a two-body system by use ofan external harmonic oscillator potential. In section IIIwe briefly describe the method used to solve the two-body problem in d dimensions. Section IV presents an-alytic results in the large squeezing limit, that is closeto one or two dimensions. Sections V and VI presentand discuss the numerical results and section VII givesthe universal translation between the two methods. Fi-nally, Section VIII contains a summary and the futureperspectives are briefly discussed. A mathematical con-nection between wave functions from the two methodsare given in an appendix. II. HARMONIC OSCILLATOR SQUEEZING
A simple way to confine particles is to put them un-der the effect of an external potential with steep wallsthat forces them to move in a confined space. Therefore,the problem to be solved is the usual Schr¨odinger equa-tion, but where, together with the interaction betweenthe particles, the confining one-body potential has to beincluded.In this work we shall consider an external harmonicoscillator potential whose frequency will be written as ω = (cid:126) m ω b ho , (1)where m ω is some arbitrary mass. Obviously, the smallerthe harmonic oscillator length b ho , the more confined theparticles are in the corresponding direction.In the following we describe how this harmonic oscil-lator potential is treated for the three confinement cases,3D → → → A. 3D → In this case the external harmonic oscillator poten-tial is assumed to act along the z -direction. Therefore,the problem to be solved here will be the usual three-dimensional two-body problem but where, on top of thetwo-body interaction, each of the two particles feels theeffect of the external trap potential: V ( i ) trap = 12 m i ω r i cos θ i = 12 m i (cid:126) m ω b ho r i cos θ i , (2)where Eq.(1) has been used, and where r i and θ i are theradial coordinate and polar angle associated to particle i with mass m i . Eventually, for b ho = 0 the particles canmove only in the two dimensions of the xy -plane.As usual, the two-body wave function can be expandedin partial waves as:Ψ( r ) = (cid:88) (cid:96)m u (cid:96) ( r ) r Y (cid:96)m ( θ, ϕ ) , (3)where r = r − r is the relative coordinate betweenthe two particles whose direction is given by the polarand azimuthal angles θ and ϕ , respectively. For simplic-ity, in the notation we shall assume spinless particles, al-though the generalization to particles with non-zero spinis straightforward.For two particles with masses m and m and coordi-nates r and r we have that: r = m µ r + m µ r − m + m µ r cm , (4)where µ is the reduced mass and r cm is the position ofthe two-body center of mass. This expression permits usto write the full trap potential as:12 m ω r cos θ + 12 m ω r cos θ =12 µω r cos θ + 12 ( m + m ) ω r cm cos θ cm , (5)where θ cm is the polar angle associated to r cm . Theexpression above implies that, after removal of the centerof mass motion, the squeezing potential to be used in therelative two-body calculation takes the form: V trap ( r, θ ) = 12 µω r cos θ, (6)whose ground state energy is E ho = (cid:126) ω/ u (cid:96) in Eq.(3) are the solutions ofthe radial Schr¨odinger equation (cid:20) ∂ ∂r − (cid:96) ( (cid:96) + 1) r − µ (cid:126) V b ( r ) + 2 µE tot (cid:126) (cid:21) u (cid:96) − µ (cid:126) (cid:88) (cid:96) (cid:48) m (cid:48) (cid:104) Y (cid:96)m | V trap ( r, θ ) | Y (cid:96) (cid:48) m (cid:48) (cid:105) Ω u (cid:96) (cid:48) = 0 , (7)where V b ( r ) is the two-body interaction (assumed to becentral), (cid:104)(cid:105) Ω indicates integration over the angles only,and E tot is the total relative two-body energy. Theenergy E of the two-body system will be obtained af-ter subtraction of the harmonic oscillator energy, i.e. E = E tot − (cid:126) ω/ (cid:96) , is not conserved. In otherwords, the trap potential is not diagonal in (cid:96)(cid:96) (cid:48) . In par-ticular we have: (cid:104) Y (cid:96)m | V trap ( r, θ ) | Y (cid:96) (cid:48) m (cid:48) (cid:105) Ω = 12 µω r (cid:104) Y (cid:96)m | cos θ | Y (cid:96) (cid:48) m (cid:48) (cid:105) Ω , (8) and (cid:104) Y (cid:96)m | cos θ | Y (cid:96) (cid:48) m (cid:48) (cid:105) Ω = δ mm (cid:48) ( − m (cid:88) L (2 L + 1) √ (cid:96) + 1 √ (cid:96) (cid:48) + 1 ×× (cid:18) L (cid:19) (cid:18) (cid:96) L (cid:96) (cid:48) (cid:19) (cid:18) (cid:96) L (cid:96) (cid:48) − m m (cid:19) , (9)where the brackets are 3 j -symbols, L can then obviouslyonly take the values L = 0 and L = 2, and therefore (cid:96) + (cid:96) (cid:48) has to be an even number (so, the parity is well-defined). Note that the angular momentum projection m actually remains as a good quantum number throughoutthe transition to 2D. Therefore, the value taken for m in Eq.(9) characterizes the solution in 2D obtained afterinfinite squeezing.In the 3D-limit ( V trap = 0) the partial waves decouple,and the 3D wave function has, of course, a well-definedorbital angular momentum. In the calculations reportedin this work the 3D wave function will be assumed tohave (cid:96) = 0, which therefore means that m = 0.Note that if we take m ω = µ , and we make use ofEqs.(1) and (6), the coupled equations (7) can be writtenas: (cid:20) ∂ ∂r b − (cid:96) ( (cid:96) + 1) r b − V b b ( r ) + 2 E btot (cid:21) u (cid:96) (10) − r b ( b bho ) (cid:88) (cid:96) (cid:48) m (cid:48) (cid:104) Y (cid:96)m | cos θ | Y (cid:96) (cid:48) m (cid:48) (cid:105) Ω u (cid:96) (cid:48) = 0 , where, taking b as some convenient length unit, we havedefined r b = r/b , b bho = b ho /b , V b b = V b / ( (cid:126) /µb ), and E btot = E tot / ( (cid:126) /µb ). In other words, when taking b and (cid:126) /µb as length and energy units, respectively, thetwo-body radial equation, Eq.(10), is independent of thereduced mass of the system. B. 2D → The confinement from two to one dimensions can bemade in a similar way as done in the previous subsec-tion for the 3D →
2D case, that is, solving the two-dimensional two-body problem with an external squeez-ing potential along one direction (that we choose alongthe y -axis).As before, when working in the center of mass frame,the external potential will be given by V trap = 12 µω r sin ϕ, (11)where, as in Eq.(6), the radial coordinate r is the relativedistance between the two particles, but where now thepolar angle ϕ is such that x = r cos ϕ and y = r sin ϕ . Inthis way, after infinite squeezing, the particles are allowedto move along the x -axis only.In 2D the partial wave expansion of the wave function,analogous to Eq.(3), is given by:Ψ( r ) = (cid:88) m u m ( r ) √ r Y m ( ϕ ) , (12)where the angular functions Y m ( ϕ ) = 1 √ π e imϕ (13)are the eigenfunctions of the 2D angular momentum op-erator − i (cid:126) ∂/∂ϕ , whose eigenvalue (cid:126) m can take positiveand negative values, where m is an integer.Using the expansion in Eq.(12), the 2D radialSchr¨odinger equation then reads: (cid:20) ∂ ∂r + − m r − µ (cid:126) V b ( r ) + 2 µE tot (cid:126) (cid:21) u m − µ (cid:126) (cid:88) m (cid:48) (cid:104) Y m | V trap ( r, ϕ ) | Y m (cid:48) (cid:105) Ω u m (cid:48) = 0 , (14)which is equivalent to Eq.(7). Using Eq.(13), it is notdifficult to see that (cid:104) Y m | V trap ( r, ϕ ) | Y m (cid:48) (cid:105) Ω = 12 µω r (cid:104) Y m | sin ϕ | Y m (cid:48) (cid:105) Ω , (15)and (cid:104) Y m | sin ϕ | Y m (cid:48) (cid:105) Ω = 12 δ m,m (cid:48) − δ m,m (cid:48) ± , (16)which implies that, as in the 3D →
2D case, the squeezingpotential is again mixing different angular momentumquantum numbers.The procedure shown up to here is completely analo-gous to the one described in the previous subsection for3D →
2D squeezing. However, in this case the start-ing point is the 2D Schr¨odinger equation Eq.(14), whichshows the important feature that for s -waves ( m = 0) the“centrifugal” barrier is actually attractive, and it takesthe very particular form of − / r . This barrier hap-pens to be precisely the critical potential giving rise tothe “falling to the center” or Thomas effect [18]. As aconsequence, the numerical resolution of the differentialequation (14) can encounter difficulties associated to thispathological behavior.To overcome this numerical problem, it is more conve-nient to face the 2D →
1D squeezing problem solving theSchr¨odinger equation directly in Cartesian coordinates: (cid:20) − (cid:126) µ (cid:18) ∂ ∂x + ∂ ∂y (cid:19) + V ( x, y ) + V trap ( y ) − E tot (cid:21) Ψ = 0 . (17)This can be easily made after expanding the two-bodyrelative wave function, Ψ, in an appropriate basis setwhere the Hamiltonian can be diagonalized. A possiblechoice for the basis could be {| ψ n x ( x ) ψ n y ( y ) (cid:105)} , with ψ n being the harmonic oscillator eigenfunctions. Needless to say, solving either Eq.(14) or (17) is completely equiva-lent, although in (17) the problematic attractive barrieris not present, and the dependence on the angular mo-mentum m also disappears.Again, as discussed in Eq.(10), after taking m ω = µ in Eq.(1), and b and (cid:126) /µb as length and energy units,respectively, Eq.(14) (or (17)) becomes µ -independent. C. 3D → If the confinement procedures 3D →
2D and 2D → → x and the y directions simultaneously. The trap potential felt byeach of the particles is then: V ( i ) trap = 12 m i ω x x i + 12 m i ω y y i , (18)where x i and y i are the x and y coordinates of particle i , and ω x and ω y are the harmonic oscillator frequenciesof each of the two external potentials. These frequenciesdetermine the independent squeezing on each of the di-rections, and of course the infinitely many possible valuesof the ω x /ω y -ratio determine the infinitely many possibleways of squeezing from 3D into 1D.Let us here consider the simplest case in which ω x = ω y = ω . After using spherical coordinates, we triviallyget that: V ( i ) trap = 12 m i ω r i sin θ i = 12 m i (cid:126) m ω b ho r i sin θ i , (19)which, as one could expect, is identical to Eq.(2) butreplacing cos θ i by sin θ i . This simply means that thevector coordinate r i is not projected on the z -axis, buton the xy -plane.Therefore, the discussion below Eq.(2) still holds here,but replacing cos θ by sin θ all over, which leads again toEq.(7) but where now (cid:104) Y (cid:96)m | V trap ( r, θ ) | Y (cid:96) (cid:48) m (cid:48) (cid:105) Ω = 12 µω r (cid:104) Y (cid:96)m | sin θ | Y (cid:96) (cid:48) m (cid:48) (cid:105) Ω , (20)with (cid:104) Y (cid:96)m | sin θ | Y (cid:96) (cid:48) m (cid:48) (cid:105) Ω = δ (cid:96)(cid:96) (cid:48) δ mm (cid:48) − (cid:104) Y (cid:96)m | cos θ | Y (cid:96) (cid:48) m (cid:48) (cid:105) Ω , (21)and where the last matrix element is given by Eq.(9).Since in this case we have included two harmonic os-cillator potentials, the energy provided by them, still as-suming ω x = ω y = ω , will be E ho = (cid:126) ω , and therefore E = E tot − (cid:126) ω . III. TWO-BODY SYSTEMS IN d -DIMENSIONS An alternative to the continuous squeezing of the par-ticles by means of external potentials can be to solve thetwo-body problem in d dimensions, where d can take anyvalue within the initial and final dimensions (3 ≥ d ≥ ≥ d ≥
1, or 3 ≥ d ≥ b ho , it is then possible to associatethis particular squeezing parameter to some specific non-integer value of the dimension, such that the propertiesof the system can be obtained by solving the, in generalsimpler, d -dimensional problem. The basic properties oftwo-body systems in d dimensions are described in Ap-pendix C of Ref.[1]. For this reason, in this section wejust collect the key equations relevant for the work pre-sented here. A. Theoretical formulation
Let us consider a two-body system where the rela-tive coordinate between the two constituents is givenby r . In principle the components of the vector r ina d -dimensional space will be given by the d Carte-sian coordinates ( r , r , · · · , r d ). As it is well known,when dealing with central potentials, it is however muchmore convenient to use the set of generalized sphericalcoordinates, which contain just one radial coordinate r = (cid:112) r + r + · · · + r d and d − d = 3). In this way,the two-body wave function can be expanded in terms ofthe generalized d -dimensional spherical harmonics, whichdepend on the d − d ( r ) = 1 r d − (cid:88) ν R ( d ) ν ( r ) Y ν (Ω d ) , (22)where ν represents the summation over all the requiredquantum numbers, Ω d collects the d − (cid:82) Y ∗ ν Y ν (cid:48) d Ω d = δ νν (cid:48) .Of course, what written above makes full sense pro-vided that d takes integer values. However, when d isnot an integer, which obviously will happen when chang-ing the dimension continuously from the initial down tothe final dimension, the meaning of the d − s -waves only, which implies that the wave function (22)is angle independent, and it can actually be written as:Ψ d ( r ) = 1 r ( d − / R d ( r ) Y d , (23)where the constant s -wave spherical harmonic, Y d , can beobtained simply by keeping in mind that in d dimensionswe have that [19]: (cid:90) d Ω d = 2 π d/ Γ (cid:0) d (cid:1) , (24) which, making use of the fact that Y ∗ d Y d (cid:82) d Ω d = 1, im-mediately leads to: Y d = (cid:34) Γ (cid:0) d (cid:1) π d/ (cid:35) / . (25)Finally, the radial wave function, R d ( r ), in Eq.(23)is obtained as the solution of the d -dimensional radialSchr¨odinger equation, which for s -waves reads [1]: (cid:20) ∂ ∂r − ( d − d − r − µ (cid:126) ( V b ( r ) − E ) (cid:21) R d ( r ) = 0 , (26)where, since the only potential entering is just the two-body interaction, the energy E is the true two-body rel-ative energy.It is important to note that if we write d = 2 + x with − ≤ x ≤
1, the barrier in Eq.(26) takes the form( x − / r , which indicates that the equation to besolved is the same for d = 2 − x and d = 2 + x . Thismight suggest that the bound state solutions of Eq.(26)should be symmetric around d =2. For instance, for d =1and d =3 the barrier disappears, and one could expect thesame solution in the two cases. However, as discussedbelow, this is not really like this.Note that the centrifugal barrier in Eq.(26) can alsobe written in the usual way, (cid:96) ∗ ( (cid:96) ∗ + 1) /r , simply bydefining (cid:96) ∗ = ( d − /
2. This means that Eq.(26) is for-mally identical to the usual radial two-body Schr¨odingerequation with angular momentum (cid:96) ∗ . Therefore, as it iswell known, Eq.(26) has in general two possible solutions,each of them associated to a different short-distance be-havior: R (1) d ( kr ) kr → −→ krj (cid:96) ∗ ( kr ) kr → −→ r (cid:96) ∗ +1 = r d − , (27) R (2) d ( kr ) kr → −→ krη (cid:96) ∗ ( kr ) kr → −→ r − (cid:96) ∗ = r − d , (28)where j (cid:96) ∗ and η (cid:96) ∗ are the regular and irregular spher-ical Bessel functions, respectively, and k = (cid:112) µ | E | / (cid:126) .Within the dimension range 1 ≤ d ≤
3, these two ra-dial solutions go to zero for kr →
0, except for d = 1and d = 3, where one of the solutions goes to a constantvalue.However, as shown in Eq.(23), the full radial wavefunction is actually R d ( r ) /r ( d − / . After dividing bythe phase space factor we immediately see that the so-lution in Eq.(27) leads to a constant value of the fullradial wave function at r = 0 no matter the dimension d . Thus, the solution (27) is valid in the full dimensionrange 1 ≤ d ≤
3. However, after dividing by the phasespace factor, the radial solution obtained from Eq.(28)behaves at short distances as r − d . This means that thissolution is regular at the origin only for d <
2, whereasit has to be disregarded for d >
2. For d > d = 2 is not real. The two-body problem has two solu-tions with physical meaning for d <
2, and only one for d -2-1.5-1-0.50 B i nd i ng e n e r gy Gaussian Potential I
Figure 1: Binding energies obtained from Eq.(26) as a func-tion of d . The two-body potential used is the Gaussian po-tential indicated as potential I in Table I. The dashed linecorresponds to the states to be discarded due to the diver-gence of the solution at the origin. d >
2. Furthermore, the solution to be disregarded for d > d = 1 than for d = 3. This is illustrated in Fig. 1, wherewe show the two computed binding energies arising fromEq.(26) as a function of the dimension d .Although not relevant at this stage, let us mention forcompleteness that the two-body potential used in the cal-culation is the one that later on will be called Gaussianpotential I. The branch indicated in Fig. 1 by the dashedcurve for d > d = 2 the twosolutions merge into a single one, as expected due to thefact that the short-distance behavior (27) and (28) is thesame in this case. B. Interpretation of the wave function
Once the d -dimensional radial wave function, R d ( r ),has been obtained, it is possible to compute different ob-servables in d dimensions, as for instance the root-mean-square radius, which is given by the simple expression r d = (cid:90) ∞ r | R d ( r ) | dr. (29)However, the reliability of a direct use of a non-integer d -dimensional wave function, Ψ d , to compute a givenobservable, that unavoidably is measured in a three- ortwo-dimensional space, is not obvious. It could actuallylook more convenient to use instead the wave functionobtained with an external squeezing potential, which, al-though very often more difficult to compute, is, in thestrict sense, a three- or two-dimensional wave function. In order to exploit the simplicity in the calculation ofthe Ψ d wave function, it is necessary to obtain a pro-cedure to translate the non-integer d -dimensional wavefunction into the ordinary three- or two-dimensionalspace.To do so, let us start by noticing that a squeezing ofthe system by an external field acting along one (for the3D →
2D and 2D →
1D cases) or two (for the 3D → d dimensions, Ψ d , as corresponding toa deformed three-dimensional system (for the 3D → →
1D cases), or a deformed two-dimensional sys-tem (for the 2D →
1D case). The simplest way to ac-count for this is to deform the radial coordinate r alongthe squeezed direction(s). In this way, if we consider theusual Cartesian coordinates { x, y, z } for an initial 3D sys-tem, or the { x, y } coordinates for an initial 2D system, wecan interpret Ψ d as an ordinary three- or two-dimensionalwave function, but where the radial argument, r , is re-placed by ˜ r , which is defined as: r → ˜ r ≡ (cid:112) x + y + ( z/s ) ≡ (cid:113) r ⊥ + ( z/s ) , (30) r → ˜ r ≡ (cid:112) ( x + y ) /s + z ≡ (cid:112) ( r ⊥ /s ) + z , (31) r → ˜ r ≡ (cid:112) x + ( y/s ) , (32)for the 3D → →
1D and 2D →
1D cases, respec-tively.In the expressions above, s is a scale parameter, as-sumed to be independent of the value of the squeezedcoordinate, and which, in principle, lies within the range0 ≤ s ≤
1. For s = 1, the relative radial coordinate ˜ r isthe usual one in spherical or polar coordinates, and thesystem is not deformed. For s = 0, only z = 0 in Eq.(30), r ⊥ = 0 in Eq.(31), and y = 0 in Eq.(32) are possible, oth-erwise ˜ r = ∞ no matter the value of the non-squeezedcoordinate, and, since we are dealing with bound sys-tems, Ψ d (˜ r ) = Ψ d ( ∞ ) = 0. Therefore, the s = 0 situ-ation corresponds to a completely squeezed system intotwo dimensions when (30) is used (3D → →
1D or2D → d -dimensional wave function, Ψ d , has to be normalizedin the new three-dimensional space for the 3D →
2D and3D →
1D cases:2 π (cid:90) r ⊥ dr ⊥ dz | ˜Ψ d ( r ⊥ , z, s ) | = 1 , (33)or in the new two-dimensional space for the 2D → (cid:90) dxdy | ˜Ψ d ( x, y, s ) | = 1 , (34)where ˜Ψ d ∝ Ψ d denotes the normalized wave function.With this interpretation, where ˜Ψ d is a wave functionin the ordinary 3D (or 2D) space, one can obtain theexpectation value of any observable F ( r ) in the usualway, that is: (cid:104)F ( r ) (cid:105) s = (cid:90) r ⊥ dr ⊥ dzdϕ F ( r ) | ˜Ψ d ( r ⊥ , z, s ) | (35)or (cid:104)F ( r ) (cid:105) s = (cid:90) dxdy F ( r ) | ˜Ψ d ( x, y, s ) | . (36)The remaining point here is how to determine the valueof the scale parameter, s , corresponding to a specificsqueezing produced by an external field with a given os-cillator parameter, b ho . The interpretation of Ψ d as anordinary (deformed) wave function using a constant s is very tempting. We believe this must be correct toleading order. If the Schr¨odinger equations are approx-imately solved by variation using single-gaussian solu-tions we can directly identify the matching scale fac-tor s . For instance, in case of squeezing along the z -direction, the two single-gaussian solutions will have theform R ext ∝ e − r ⊥ / b − z / b z when the external field isused, and R d ∝ e − r / b d after the d -calculation. If, fol-lowing Eq.(30), we interpret R d as a function of ˜ r , weeasily get that R ext and R d are the same if b = b d and s = b z /b d . One way to improve is to allow s to be afunction of the squeezed coordinate. The assumption ismostly that this dependence on the squeezed coordinateis rather smooth, in such a way that the function canbe safely expanded around some constant average value,which therefore is the leading term.In any case, the scale parameter s has to be a functionof the non-integer dimension parameter, d , or equiva-lently, the squeezing length parameter, b ho . The value ofthe average scale parameter, s , then has to be obtainedby comparison of the wave functions from a full externalfield, Ψ b ho ( r ), with r = r ⊥ + z (when squeezing from3D) or r = x + y (when squeezing from 2D), and thenormalized d -dimensional wave function, ˜Ψ d (˜ r ), definedabove.Being more precise, we define the overlaps: O D ( s ) = 2 π (cid:90) r ⊥ dr ⊥ dz ˜Ψ d ( r ⊥ , z, s )Ψ b ho ( r ) , (37)and O D ( s ) = (cid:90) dxdy ˜Ψ d ( x, y, s )Ψ b ho ( r ) , (38)which are valid for initial 3D and 2D spaces, respectively.The scale parameter, s , is then determined such that theoverlap (37) for the 3D →
2D and 3D →
1D cases, or theoverlap (38) for the 2D →
1D case, is maximum.In Appendix A we show, Eqs.(A7), (A12) and (A16),that the scale factor, s , is actually given by s = (cid:18) (cid:104) z (cid:105) s (cid:104) z (cid:105) s =1 (cid:19) / ; s = (cid:18) (cid:104) r ⊥ (cid:105) s (cid:104) r ⊥ (cid:105) s =1 (cid:19) / ; s = (cid:18) (cid:104) y (cid:105) s (cid:104) y (cid:105) s =1 (cid:19) / , (39) for the 3D → → →
1D squeezingcases, respectively, where (cid:104)(cid:105) s are expectation values asdefined in Eqs.(35) and (36). This result says that s isnothing but the ratio between the expectation value ofthe squeezing coordinate for that value of s , and the oneobtained without deforming the wave function ( s = 1).These expressions make evident that the scale parameter, s , is a measure of the deformation along the squeezingdirection(s). IV. LARGE SQUEEZING REGIME
The large-squeezing region corresponds to very smallvalues of the oscillator parameter, b ho , which implies thatthe squeezing potential dominates over the two-body in-teraction along the squeezing direction(s). As a conse-quence, the root-mean-square value of a given squeezedcoordinate u is, for large squeezing, essentially given bythe one corresponding to the harmonic oscillator poten-tial. In particular this implies that, for a squeezing pro-cess d i D → d f D we can write, in a compact way: u rms = (cid:104) u (cid:105) / b ho → −→ (cid:114) d i − d f b ho , (40)where u can be z , r ⊥ , or y for the 3D → → →
1D cases, respectively, and d i and d f indicatethe initial and final dimension.Let us now focus on the radius, r d , defined in Eq.(29).To simplify ideas, let us consider first the case of a3D →
2D squeezing along the z -coordinate, and write r d as r d = (cid:104) r (cid:105) = (cid:104) r ⊥ (cid:105) + (cid:104) z (cid:105) . (41)Since the squeezing takes place along the z direction,the expectation value (cid:104) z (cid:105) is the one feeling the squeezingeffect, whereas (cid:104) r ⊥ (cid:105) to leading order does not. FollowingEq.(39) we can then write: r d = (cid:104) r ⊥ (cid:105) + s (cid:104) z (cid:105) s =1 , (42)where (cid:104) z (cid:105) s =1 is the expectation value of (cid:104) z (cid:105) when Ψ d isinterpreted as non-deformed standard 3D wave function.Furthermore, for a spherical 3D wave function we knowthat (cid:104) x (cid:105) = (cid:104) y (cid:105) = (cid:104) z (cid:105) , which means that (cid:104) r ⊥ (cid:105) = (cid:104) x (cid:105) + (cid:104) y (cid:105) = 2 (cid:104) z (cid:105) s =1 , and we can then write: r d = (cid:112) s (cid:104) z (cid:105) / s =1 = √ s s (cid:104) z (cid:105) / , (43)for 3D → →
1D case, ex-cept for the fact that now in Eq.(41) the squeezing is feltby (cid:104) r ⊥ (cid:105) and not by (cid:104) z (cid:105) . Therefore, using again Eq.(39),we can write (cid:104) r ⊥ (cid:105) = s (cid:104) r ⊥ (cid:105) s =1 , and (cid:104) z (cid:105) = (cid:104) r ⊥ (cid:105) s =1 / → r d = (cid:114) s (cid:104) r ⊥ (cid:105) / s =1 = (cid:114) s s (cid:104) r ⊥ (cid:105) / . (44)Finally, similar arguments for the 2D →
1D case, where r d = (cid:104) r (cid:105) = (cid:104) x (cid:105) + (cid:104) y (cid:105) , assuming the squeezing along y ,lead to: r d = (cid:112) s (cid:104) y (cid:105) / s =1 = √ s s (cid:104) y (cid:105) / . (45)From Eqs. (43), (44), and (45) it is easy to obtain that,to leading order: s ≈ (cid:115) d f d i − d f u rms /r d (cid:113) − ( u rms /r d ) , (46)where u rms ≡ (cid:104) u (cid:105) / , and u represents either z , r ⊥ , or y , depending on what squeezing process we are dealingwith.At this point it is easy to replace in Eq.(46) thelarge squeezing behavior of (cid:104) z (cid:105) , (cid:104) r ⊥ (cid:105) , and (cid:104) y (cid:105) givenin Eq.(40), and obtain the following expression for thescale parameter, s , in case of large squeezing: s b ho → −→ d f (cid:16) b ho r d (cid:17) − ( d i − d f ) (cid:16) b ho r d (cid:17) / , (47)where d i and d f again denote the initial and final dimen-sion. V. POTENTIALS
Let us start the numerical illustration by first spec-ifying the chosen potentials along with a few of theircharacteristic properties. In the following subsections wecontinue to present results of the two methods describedformally in Sections II and III.
A. General properties
To allow general, and hopefully universal conclusions,we shall use two different radial shapes for the two-bodypotential: A Gaussian potential, V b ( r ) = S g e − r /b , anda Morse-like potential, V b ( r ) = S m ( e − r/b − e − r/b ).The range of the interaction, b , will be taken as the corre-sponding (in principle different) length unit. Therefore,as discussed in section II A, taking m ω = µ in Eq.(1)and (cid:126) /µb as energy unit, the Schr¨odinger equation isindependent of the reduced mass.For each of the potential shapes, three different interac-tions will be considered, potentials I, II, and III. The po-tential parameters for each of them are chosen such thatthe 3D scattering length, a D , is the same for both, theGaussian and the Morse shapes. The a D values are in allthe cases positive (therefore holding a 3D bound state),and change from comparable to the potential range, b ,for potential I, to about 20 times b for potential II, up toa value of about 40 times b for potential III. Pot. I Pot. II Pot. III S g − . − . − . E D − . − . · − − . · − E D − . − . − . E D − . − . − . a D a D a D r D r D r D S m .
294 0 .
474 0 . E D − . − . · − − . · − E D − . − . · − − . · − E D − . − . − . a D a D ∼ − a D r D r D r D S g ,and Morse, S m , potentials used. For each of them we givethe s -wave two-body binding energies of the ground state inthree, two, and one dimensions ( E D , E D , and E D ), thecorresponding s -wave scattering lengths a D , a D and a D ,and the root-mean-square radii r D , r D , and r D . All theenergies are given in units of (cid:126) /µb and the lengths in unitsof b , where b is the range of either the Gaussian or the Morsepotential. The details of the potentials are given in Table I forthe employed Gaussian and Morse shapes, where thelengths are in units of b , and the energies, includingthe strengths, S g and S m , are in units of (cid:126) /µb . Thecharacterizing s -wave scattering lengths, a D , a D , and a D , are obtained in three-dimensional, two-dimensional,and one-dimensional calculations, respectively. The two-dimensional scattering length a D is defined as given inEq.(C.6) in Ref.[1]. Furthermore, we give in Table Ithe corresponding s -wave ground state binding energies, E D , E D , and E D together with their root-mean-squareradii, r D , r D , and r D .All the potentials given in Table I give rise to only onebound state, i.e., the ground state. The only exceptionis Potential I with Morse shape in one dimension. Thetwo-body system described by this potential has a weaklybound excited state, such that the large value of the scat-tering length a D is in this case related to the appearanceof this second state, whose energy can be approximatedby some constant divided by a D (Eq.(C.9) in Ref.[1]). B. External harmonic oscillator potential
Let us start with the case of confinement by meansof an external (harmonic oscillator) potential. The pro-
Figure 2: Two-body energies (after subtracting the harmonicoscillator energy), in units of (cid:126) /µb , for the Gaussian poten-tials in a 3D →
2D squeezing, as a function of the (cid:96) max valueincluded in the expansion (3). The results for b bho = b ho /b =0.1, 0.25, 0.5, 1, 2, and 10 are shown. Panels ( a ), ( b ), and( c ) refer to potentials I, II, and III, respectively. On eachpanel, the upper and lower horizontal dashed lines indicatethe two-body energies in the 3D and 2D cases, respectively,as indicated by the “3D” and “2D” labels. cedure is as described in section II, where it is shownhow the trap potential, which is not central, mixes dif-ferent values of the relative orbital angular momentum.This is made evident in Eqs.(9), (16), and (21) for thethree different squeezing processes (3D → → → →
2D case and call (cid:96) max the maxi-mum value of (cid:96) included in the expansion (3). In Fig. 2awe show the convergence of the two-body energy as afunction of (cid:96) max for the Gaussian potential I (Table I),and for different values of b bho = b ho /b in the squeezingpotential. The energy shown, E , is the two-body en- ergy obtained after subtracting the harmonic oscillatorenergy, i.e., E = E tot − E ho . The horizontal dashed linesare the two-body energies obtained after a 3D and a 2Dcalculation, respectively, which are given in Table I.As expected, for small values of the oscillator parame-ter we recover the computed 2D-energy, whereas for largevalues of b bho the 3D-energy is approached. We can alsosee that the smaller b bho , the larger the (cid:96) max value neededto get convergence. Partial waves with (cid:96) -values up toaround 80 are at least needed for b bho = 0 .
1, for whichwe get a converged energy of − . − .
908 obtained in a 2D-calculation. For largevalues of b bho the convergence is obviously much faster.For b bho = 10 we obtain an energy of − . − .
269 corresponding to the 3D calculation.This result is already obtained including the (cid:96) = 0 com-ponent only.In Fig. 2b we show the same as in Fig. 2a for the Gaus-sian potential II. The general features are the same asbefore, although there are some remarkable differencesarising from the fact that now the scattering length isabout 9 times bigger. First, for b bho = 10 we obtain aconverged energy of − . · − , which, although in thefigure seems to be very close to 3D energy, it differs bymore than a factor of three ( E D = − . · − ). Toget a better agreement with the 3D-energy, b bho valuesof a few times the 3D scattering length are needed (asin fact observed for potential I). The second importantdifference is that convergence is now slower than before,and higher values of (cid:96) max are needed to get convergencefor small oscillator lengths.These facts are more emphasized when using the Gaus-sian potential III, whose corresponding curves are shownin Fig. 2c. In this case we have obtained for b bho = 10 aconverged energy of − . · − , about an order of mag-nitude more bound than the 3D energy ( − . · − ).For b bho = 0 .
1, an (cid:96) max value of at least 140 is neededin order to get convergence. This is of course related tothe large 3D scattering length. In three dimensions thebound two-body system is clearly bigger than with theother two potentials, and, consequently, it starts feelingthe confinement sooner than in the other cases.The convergence features for 2D →
1D and 3D → b bho (little squeezing) a small number of partialwaves are enough to get convergence and the energy inthe initial dimension, 2D or 3D, is approached, whereasfor small values of b bho (large squeezing) a higher num-ber of partial waves is required, and the energy in thefinal dimension, 1D or 2D, is recovered. For this reasonwe consider it to be unnecessary to show the correspond-ing figures. In any case, as discussed in section II B, for2D →
1D squeezing it is more convenient to solve directlyEq. (17), where the partial wave expansion does not enterexplicitly.In Figs. 3a, 3b, and 3c we show, for the 3D → → →
1D cases, respectively, the con-0 -3 -2 -1 Potential IPotential IIPotential III -2 -1 -4 -3 -2 -1 Figure 3: Converged two-body energies (normalized to theenergy in the final dimension) for the three Gaussian (thickcurves) and Morse (thin curves) potentials as a function of b bho .Panels (a), (b), and (c) correspond to 3D → → →
1D squeezing, respectively. verged values of the two-body energy E for the Gaussianpotentials (thick curves) and the Morse potentials (thincurves), as a function of the oscillator parameter, b bho .The energy is normalized to the energy in the final di-mension, either E D or E D , given in Table I. Therefore,for small values of b bho all the curves go to 1. C. Two-body energy in d dimensions As described in Section III, the continuous squeezingof the system from some initial dimension d i to some finaldimension d f can also be made by solving the two-bodyproblem in d dimensions, where d f ≤ d ≤ d i . We cantherefore compute the same observable as in Fig. 3 butsolving the two-body Schr¨odinger equation (26), wherethe dimension d is taken as a parameter. It is importantto remember that for d > κr → d < d − d f ) / ( d i − d ) for the squeezing cases 3D →
2D (panel (a)),2D →
1D (panel (b)), and 3D →
1D (panel (c)). Again,the thick and thin curves correspond to the results withthe Gaussian and Morse potentials, respectively. Thechoice of the abscissa coordinate is such that the curvescan be easily compared to the ones in Fig. 3. In fact, a -3 -2 -1 Potential IPotential IIPotential III -4 -3 -2 -1 -4 -3 -2 -1 Figure 4: Two-body energies (normalized to the energy in thefinal dimension) for the three Gaussian (thick curves) and thethree Morse (thin curves) potentials used as a function of ( d − d f ) / ( d i − d ), where d is the dimension varying continuouslyfrom the initial dimension d i to the final dimension d f . Panels(a), (b), and (c) correspond to 3D → → →
1D squeezing, respectively. simple eye inspection of both figures makes evident theexistence of a univocal connection between b bho and d . VI. COMPARING THE TWO METHODS
Although the parameters used on each of the two meth-ods, b ho and d , have a very different nature, they bothare used to describe the same physics process of squeezingthe system into a lower dimensional space. We shall firstconnect these parameters by use of the energies leadingfrom initial to final dimension in the squeezing processes.Then we turn to the crucial comparison of the relatedwave functions which require an interpretation and a de-formation parameter, as described in Section III B. A. Relation between b bho and d The relation between b bho and d obtained directly fromFigs. 3 and 4 is shown in Fig. 5 for all the potentials.Panels (a), (b), and (c) correspond to the 3D → → →
1D cases, respectively. As in theprevious figures, the thick and thin curves are, respec-tively, the results obtained with the Gaussian and Morsepotentials.1
Potential IPotential IIPotential III -1 Figure 5: Values of d as a function of b ho /r ∞ d f D , obtained bymatching the energies in Figs. 3 and Fig. 4, for the potentialsin Table I. The cases of 3D → →
1D and 3D → r ∞ d f D in the root-mean-square radius in the finaldimension, d f , obtained with a potential such that a D = ∞ . In the figure we show d as a function of b ho /r ∞ d f D ,where r ∞ d f D is the root-mean-square radius of the boundtwo-body system in the final dimension obtained with aninteraction such that a D = ∞ . In particular r ∞ D and r ∞ D take the values 0.769 and 1.474, in units of b , forthe Gaussian shape, and 1.478 and 3.128 for the Morseshape, respectively. This is a way to normalize the size ofthe bound state in the final dimension, d f , to the valuecorresponding to the potential, which, in principle, is ex-pected to provide a universal connection between d and b ho . In fact, as shown in the figure, for each of the two po-tential shapes, the curves corresponding to the potentialswith large scattering length, potentials II and III, are al-most identical to each other. Furthermore, the curvesfor these two potentials corresponding to the Gaussian(thick curves) and Morse (thin curves) potentials are notvery different. Only the cases corresponding to potentialI give rise to curves clearly different to the other ones.This result is consistent with the idea of relating uni-versal properties of quantum systems to the presenceof relative s-waves and large scattering lengths. Thishas been established as a universal parameter describ-ing properties of weakly bound states without reference Potential IPotential IIPotential III -1 Figure 6: The scale parameter s as a function of b bho for thepotentials in Table I and the three squeezing scenarios con-sidered in this work. to the responsible short-range attraction. For this rea-son, the translation between b ho and d shown in Fig. 5for the potentials with large scattering length should bevery close to the desired universal relation between thetwo parameters. B. Scale parameter
As discussed in subsection III B, the wave function in d dimensions can be interpreted as an ordinary wave func-tion in three dimensions (in the 3D →
2D or 3D → →
1D case),but with a deformation along the squeezing direction, asshown in Eqs. (30), (31), and (32). The value of the scaleparameter, s , is obtained as the one maximizing the over-lap, O D , in Eq.(37) for the 3D →
2D or 3D →
1D cases,or the overlap, O D , in Eq.(38) for the 2D →
1D case.These overlaps, which are functions of the scale param-eter, s , are just the overlap between the wave functionobtained with the external squeezing potential, Ψ b ho , andthe renormalized wave function, ˜Ψ d , obtained in d dimen-sions, where b ho and d are related as shown in Fig. 5.The results obtained for the scale parameter are shownin Fig. 6 for the three squeezing processes and the usualthree potentials for both the Gaussian (thick curves) andMorse (thin curves) shapes. In all the cases the maxi-mized overlap value is very close to 1. In fact, in the2most unfavorable computed case ( b bho =0.1), the overlapvalue is, for all the cases, at least 0.98. As expected,a large squeezing ( b bho →
0) implies a small value of s ( s → b bho ) corre-sponds to s →
1. In fact, for b ho = ∞ , Eqs.(7) and (14)are identical to Eq.(26) for d = 3 and d = 2 respectively.This means that the wave functions, Ψ b ho and Ψ d , areidentical, and the corresponding overlaps, O D or O D ,are trivially maximized and equal to 1 for s = 1.Another feature observed in Fig. 6 is that when thesqueezing begins, for relatively large values of b bho , thescale parameter, s , can be bigger than 1. This is es-pecially true for potentials II and III in the 3D → d veryclose to the initial dimension) the interpretation of the d -wave function as the three-dimensional wave function,˜Ψ d (˜ r ), gives rise to a state with the particles a bit tooconfined along the squeezing direction, in such a way thatmaximization of the overlap (37) or (38) requires a smallrelease of the confinement by means of a scale factor big-ger than 1. This is very likely a consequence of using aconstant scale parameter, or equivalently, that the per-pendicular and squeezing directions are not completelydecoupled for these short-range potentials.The differences between the curves shown in Fig. 6 arerelated to the size of the two-body system in the initialdimension. In panels (a) and (c), potential III describes atwo-body system in 3D clearly bigger than the other po-tentials (see Table I), and therefore the curve correspond-ing to this potential is the first one feeling the squeezing,i.e., it is the first one for which s deviates from 1 whenthe squeezing parameter, b bho decreases. For the same rea-son the second potential feeling the squeezing is potentialII, and for potential I the deviation from s = 1 starts foreven smaller values of b bho . For the 2D →
1D case, Fig. 6b,the curves corresponding to potentials II and III are verysimilar, since these two potentials describe systems witha very similar size in 2D (see Table I).For the same reason there is a clear dependence on thepotential shape. In general, given a squeezing parameter,the root-mean-square radius is clearly bigger with theMorse potential than in the Gaussian case, as seen forinstance in Table I with the r D and r D values. Thisfact implies that, for a given b bho , the scale parameter incase of using the Morse potential is clearly smaller thanwhen the Gaussian potential is used.A simple way to account for these size effects is to plotthe scale parameter, s , as a function of b ho /r d , where r d is the root-mean-square radius of the system for di-mension d as given in Eq.(29). This is shown in Fig. 7,where we can see that for all the three potentials andthe Gaussian and Morse shapes, the curves collapse intoa single universal curve. The only discrepancy appearsin panel (c) in the region where the squeezing begins toproduce some effect, where the bump shown by poten-tials II and III is not observed in the case of potential I.This also happens, although to a much smaller extent, inthe 3D →
2D case shown in panel (a). We also show in
Potential IPotential IIPotential III -1 Figure 7: The scale parameter s as a function of b ho /r d , seeEq.(29), for the potentials in Table I and the three squeezingscenarios considered in this work. Except for panel (c) in thevicinity of s = 1, the curves corresponding to the Gaussian(thick curves) and Morse (thin curves) potentials are, to alarge extent, indistinguishable. The circles show the analyti-cal expression (47) valid for large squeezing. Fig. 7 (dots) the analytical expression given in Eq.(47),which gives the relation between s and b ho /r d in the caseof large squeezing, i.e., in the case of small b ho values.As we can see, the analytical expression can be used forvalues of b ho /r d (cid:46) . s ,as a function of u rms /r d , where u rms = (cid:104) u (cid:105) / , and u corresponds to z , y , or r ⊥ depending on what squeezingprocess we are dealing with, 3D → → → × (cid:112) ( d i − d f ) /d f ) is plottedas a function of u rms /r d . As we can see, all the curvesfor all the squeezing cases are very similar to each other.The main difference appears in the low squeezing region.In fact, in the case of no squeezing the value of u rms /r d is different for each case, 1 / √
3, 1 / √
2, or (cid:112) /
3, as in-dicated by the arrows in the figure. The dotted curve isthe analytical form in Eq.(46), which for u rms /r d (cid:46) . b ho /r d and u rms /r d , simply by connectingthe values of these two quantities corresponding to the3 Pot. I, GaussianPot. II, GaussianPot. III, GaussianPot. I, MorsePot. II, MorsePot. III, Morse -2 -1 Figure 8: The scale parameter s ( × (cid:112) ( d i − d f ) /d f ), as a func-tion of u rms /r d for the potentials in Table I, where u = z , u = y , and u = r ⊥ for 3D → → → -2 -1 -2 Potential I, GaussianPotential II, GaussianPotential III, GaussianPotential I, MorsePotential II, MorsePotential III, Morse
Figure 9: Value of b ho /r d ( × (cid:112) d i /
2) as a function of u rms /r d ( × (cid:112) d i / ( d i − d f )) for the potentials in Table I, where u = z , u = y , and u = r ⊥ for 3D → → → same value of the scale parameter. For large squeezing,this relation should in fact be determined by Eq.(40).The result is shown in Fig. 9, where the factors multi-plying u rms /r d and b ho /r d ( (cid:112) d i / (cid:112) d i / ( d i − d f ))have been chosen in such a way that all the curves fol-low very much a rather universal curve. Only some ofthe curves show some discrepancy in the region of verysmall squeezing. In particular this is what happens withpotentials II and III in the 3D →
1D case, which pro-duce the bump that differs from the rest of the curvesfor (cid:112) d i / ( d i − d f ) (cid:104) r ⊥ (cid:105) / /r d ≈
1. This is the same de-viation from the universal curve observed in Fig. 7c forthese two potentials. This universal behaviour is quite d r d (cid:104) z (cid:105) / s b bho (cid:104) z (cid:105) / , the scale parameter s , and thesqueezing parameter b bho for the Morse potential II, obtainedfrom the universal curves as shown in Fig. 10 for the dimen-sions d = 2 . d = 2 .
50, and d = 2 .
75. Their corresponding r d values are given in the second column of the table. All thelengths are given in units of the range of the interaction.Thenumbers within parenthesis are the values obtained from thecalculations. well reproduced using Eq.(40), which, as shown by thedotted line, follows the computed curves almost up to theregion where the discrepancy mentioned above shows up. VII. UNIVERSAL RELATIONS
Some of the results shown in the previous sectionshow what we could consider a universal behaviour. Thecurves shown in Fig. 7 are very much independent of thescattering length of the potential, and of the shape ofthe potential. Furthermore, the curves shown in Fig. 8,and specially the ones in Fig. 9, can also be consideredindependent of the squeezing process.One could then think that from these universal curvesit should be possible to determine the dimension d thatshould be used to mimic the squeezing process producedby an external field with squeezing parameter, b ho . Thisis however not so simple, since, for instance in Fig. 7, werelate s not just with b ho , but with b ho /r d , and r d is theroot-mean-square radius in the d -calculation, Eq.(29),where d must be the dimension associated to the squeez-ing parameter, b ho . In other words, use of the universalcurves in Fig. 7 to obtain the s value corresponding tosome squeezing parameter, b ho , requires previous knowl-edge of the relation between d and b ho . The same hap-pens in Fig. 8, where u rms is the root-mean-square ra-dius in the squeezing direction, which can be computedonly after knowing the scale parameter, obtained aftermaximization of Eqs.(37) or (38), which again requiresprevious knowledge of the value of b ho associated to agiven dimension. The same problem appears in Fig. 9.However, Figs. 7 to 9 can be used to estimate the rela-tion between d and b ho in an indirect way. For instance,in Fig. 10a we show the universal curves shown in Fig. 8for the 3D →
2D case. On top we plot for three differ-ent dimensions, d = 2 . d = 2 .
50 and d = 2 .
75, thecurves (squares) showing (cid:104) z (cid:105) / /r d as a function of thescale parameter, s , for one of the potentials used in thiswork, in particular for the Morse potential II. The pointswhere these curves cut the universal curve determine thespecific values of s and (cid:104) z (cid:105) / corresponding to each di-mension (note that r d is simply given by Eq.(29), andit does not depend on s ). Due to the numerical uncer-4 -1 -1 d=2.25d=2.50d=2.75 Potential IPotential IIPotential III
Figure 10: ( a ): The same as in Fig. 8 for the 3D →
2D case,where we show (squares) the computed values of (cid:104) z (cid:105) / /r d asa function of the scale parameter s for three different dimen-sions, d = 2 .
25, 2 .
50, and 2 .
75. ( b ): The same as in Fig. 7bwhere the arrows indicate the values of b ho /r d correspondingto the scale parameter s where the squared lines in the upperpart cut the universal curve. tainty in the universal curve, we have also considered theuncertainty (light-blue rectangles) in where the crossingis actually taking place. The results of the estimate forthese three dimensions is given in Table II, where the sec-ond column shows the r d value for each dimension, andthe third and fourth columns give the estimated rangeobtained from Fig. 10a for (cid:104) z (cid:105) / and s , respectively.Within parenthesis we give the precise value obtainedfrom the calculation. As one can see, the estimate isreasonably good.Once the s value is known, we can use Fig. 7 (or Fig. 9),as shown in Fig. 10b, to determine the values of b ho thatcorrespond to each of the dimensions considered. Theresults obtained are given in the last column of Table II,together with the computed values which are given withinparenthesis.In any case, it is obvious that a direct connection be-tween d and b ho is highly desirable. In fact, as shown Potential IPotential IIPotential III -1 Figure 11: The same as Fig. 5, but after the transformationdefined in Eq.(48). c c c → − .
28 0.78 0.622D →
1D 1.34 0.24 0.253D → − .
41 1.12 0.54Table III: Parameters used in the numerical fit given inEq.(49) giving rise to the curves indicated by the trianglesin Fig. 11. in Fig. 5, at least the curves corresponding to poten-tials II and III (the ones having a large scattering length)show very much the same behaviour for a given poten-tial shape, but even if we consider the results with theGaussian and Morse potentials, the curves are not far ofbeing universal.An attempt of making the curves in Fig. 5 fully uni-versal was introduced in [17], which can be generalizedto a general d i D → d f D squeezing process as˜ b ho = b ho (cid:118)(cid:117)(cid:117)(cid:116) b ho + r d f D a d i D + r d f D . (48)The result of this transformation is shown in Fig. 11.As we can see, the effect of the scattering length beingcomparable to the range of the potential is corrected to alarge extent, and all the curves follow a rather universalcurve for each of the d i D → d f D squeezing processes.In Fig. 11 we also show an analytical fit (triangles) thatreproduces very well the universal curve for each of the5squeezing scenarios. Since the curve is model indepen-dent, the special form of the fitting function is unimpor-tant provided it gives a sufficiently accurate connectionbetween ˜ b ho and d . We have different options but onepossibility is˜ b ho r ∞ d f D = c (cid:18) d − d f d i − d (cid:19) d i / + c tan (cid:18)(cid:18) d − d f d i − d f (cid:19) c π (cid:19) , (49)which is a combination of two functions, each of thembeing equal to zero at d = d f , and to ∞ at d = d i , andwhose relative weight is used to fit the curves betweenthese two limits. The computed fitting constants for eachof the three squeezing processes are given in Table III.It is important to mention that, in principle, instead ofthe two-body energy, as shown in Figs. 3 and 4, one couldhave used a different observable in order to determine theconnection between d and the squeezing parameter, b ho .We have checked that when the root-mean-square radiiare used, the same universal relation as the one shownin Fig. 11 is obtained. The main practical problem inthis case is that, in general, the convergence of the com-puted radii when the external squeezing is considered, isclearly slower than the convergence of the energy. Evenlarger values of the two-body relative angular momentaare needed in the expansion (3), which actually is a sourceof numerical inaccuracies, especially for large squeezingscenarios. VIII. SUMMARY AND CONCLUSIONS
We investigate in details how a dimension-dependentcentrifugal barrier can be the substitute for an externalone-body potential. We choose the ground state of asimple two-body system with Gaussian and Morse short-range interactions. The dimension parameter is integerin the initial formulations, which in this report are ana-lytically continued to allow non-integer dimensional val-ues. The external potential is chosen as the both, ex-perimentally and theoretically, practical harmonic oscil-lator which in the present context necessarily must beanisotropic or deformed. A well defined unique transfor-mation between the parameters of the two methods thenmakes each of them complete with precise predictions ofresults from the other method. The simpler centrifugalbarrier computations are then sufficient to provide ob-servables found with an external potential.The overall idea is then to start with an ordinary inte-ger dimension of 3, 2, or perhaps 1, and apply an in-creasingly confining external potential in one or morecoordinates. This is equivalent to increasing frequency,or decreasing oscillator length in the corresponding di-rections while other coordinates are left untouched. Theprocess leads from one integer dimension to another lowerone. The results are compared with calculations withoutexternal potential but with a dimension-dependent cen-trifugal barrier where the same initial and final config- urations are assumed and mathematically correct. Theaim is to establish a desired unique relation between thedimension parameter and the oscillator squeezing length.We first describe in details how the harmonic oscilla-tor confinement is implemented in the investigated tran-sitions, 3D → →
1D and 3D → Appendix A: The scale parameter and expectationvalues in the squeezing direction1. 3D → In this case the radial coordinate, r , is redefined asgiven in Eq.(30): r → ˜ r ≡ (cid:112) x + y + ( z/s ) ≡ (cid:113) r ⊥ + ( z/s ) . (A1)The normalization of the wave function (23) requirescalculation of N s = (cid:90) r ⊥ dr ⊥ dzdϕ | Ψ d (˜ r ) | , (A2)which, after defining u = z/s , can be rewritten as: N s = s (cid:90) r ⊥ dr ⊥ dudϕ | Ψ d (˜ r ) | = s I , (A3)where ˜ r = r ⊥ + u and the integral, I , is independentof the scale parameter. Note that for d = 3, since Ψ d =3 is already in the 3D space we then trivially have that N s =1 = 1.Therefore, the wave function, ˜Ψ d = Ψ d / √N s , is nor-malized to 1 in the ordinary three-dimensional space. Af-ter this normalization we can now compute the expecta-tion value, (cid:104) z (cid:105) s , which given by (cid:104) z (cid:105) s = (cid:90) z r ⊥ dr ⊥ dzdϕ | ˜Ψ d (˜ r ) | , (A4)which, again under the transformation, u = z/s , takesthe form: (cid:104) z (cid:105) s = s N s (cid:90) u r ⊥ dr ⊥ dudϕ | Ψ d (˜ r ) | = s N s I , (A5)where I is independent of s .Making now use of Eq.(A3) we get: (cid:104) z (cid:105) s = s I I , (A6)from which we get the final expression for the scale pa-rameter: s = (cid:18) (cid:104) z (cid:105) s (cid:104) z (cid:105) s =1 (cid:19) / . (A7)
2. 3D → In this case the radial coordinate, r , is redefined asgiven in Eq.(31): r → ˜ r ≡ (cid:112) ( x + y ) /s + z ≡ (cid:112) ( r ⊥ /s ) + z . (A8) We then proceed exactly as in the 3D →
2D case, butusing the transformation, u = r ⊥ /s . In this way thenormalization constant (A2) reads now: N s = s (cid:90) ududzdϕ | Ψ d (˜ r ) | = s I . (A9)In the same way, under the same transformation, theexpectation value (cid:104) r ⊥ (cid:105) s = (cid:90) r ⊥ dr ⊥ dzdϕ | ˜Ψ d (˜ r ) | , (A10)can be rewritten as: (cid:104) r ⊥ (cid:105) s = s N s (cid:90) u dudzdϕ | Ψ d (˜ r ) | = s N s I , (A11)which again, by use of Eq.(A9) leads to: s = (cid:18) (cid:104) r ⊥ (cid:105) s (cid:104) r ⊥ (cid:105) s =1 (cid:19) / . (A12)
3. 2D → In this case the radial coordinate, r , is redefined asgiven in Eq.(31): r → ˜ r ≡ (cid:112) x + ( y/s ) , (A13)and the normalization constant is given by: N s = (cid:90) dxdy | Ψ d (˜ r ) | = s (cid:90) dxdu | Ψ d (˜ r ) | = s I , (A14)where now u = y/s .The expectation value, (cid:104) y (cid:105) s , is now: (cid:104) y (cid:105) s = (cid:90) y dxdy | ˜Ψ d (˜ r ) | = (A15)= s N s (cid:90) u dxdu | Ψ d (˜ r ) | = s I I . As before, since I and I are s -independent, we thenget the analogous result: s = (cid:18) (cid:104) y (cid:105) s (cid:104) y (cid:105) s =1 (cid:19) / . (A16) Acknowledgments
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