Aspects of arbitrarily oriented dipoles scattering in plane: short-range interaction influence
aa r X i v : . [ phy s i c s . a t o m - ph ] N ov Aspects of arbitrarily oriented dipoles scattering in plane: short-range interactioninfluence
Eugene A. Koval ∗ and Oksana A. Koval † Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research,Dubna, Moscow Region 141980, Russian Federation A.M. Obukhov Institute of Atmospheric Physics,Russian Academy of Sciences, Moscow 119017, Russian Federation (Dated: November 17, 2020)The impact of the short-range interaction on the resonances occurrence in the anisotropic dipolarscattering in a plane was numerically investigated for the arbitrarily oriented dipoles and for a widerange of collision energies. We revealed the strong dependence of the cross section of the 2D dipolarscattering on the radius of short-range interaction, which is modeled by a hard wall potential and bythe more realistic Lennard-Jones potential, and on the mutual orientations of the dipoles. We definedthe critical (magic) tilt angle of one of the dipoles, depending on the direction of the second dipole forarbitrarily oriented dipoles. It was found that resonances arise only when this angle is exceeded. Incontrast to the 3D case, the energy dependencies of the boson (fermion) 2D scattering cross sectiongrows (is reduced) with an energy decrease in the absence of the resonances. We showed that themutual orientation of dipoles strongly impacts the form of the energy dependencies, which beginto oscillate with the tilt angle increase, unlike the 3D dipolar scattering. The angular distributionsof the differential cross section in the 2D dipolar scattering of both bosons and fermions are highlyanisotropic at non-resonant points. The results of the accurate numerical calculations of the crosssection agree well with the results obtained within the Born and eikonal approximations.
I. INTRODUCTION
The three-dimensional (3D) dipolar scattering problemhas been studied extensively (e.g. [1–10]), whereas thetwo-dimensional (2D) dipolar scattering was separatedto the stand-alone topic, that has been actively devel-oped and has attracted increasing interest in a numberof theoretical [1, 2, 11–19] and experimental [11, 12, 17–19] studies recently.Dipolar gases are more stable in a quasi-two-dimensional geometry, in contrast with a 3D case, due tothe absence of the ”head-to-tail” instability [11, 20, 21].Optical lattices with 2D geometries are prospective can-didates for a dipolar gas stabilization, trapping, and dy-namics controlling because the dipole-dipole interaction(DDI) is isotropic and repulsive in the case of dipolemoments polarized along the frozen direction, whereastilting of the polarization axis leads to a controllableanisotropy of the interaction [2, 11, 12]. Molecules colli-sions in one layer of a pancake-shaped trap are modelledby a 2D dynamics of the molecules [12, 22–27]. The in-vestigations of a dipolar diatomic molecules interactionin a plane are highly relevant due to prospects for one ofthe possible realization of a qubit and application to thequantum computing schemes [28, 29].As noted in Refs. [17, 22], the ongoing experiments callfor a deeper understanding of the short-range physics in-fluence on dipolar scattering in a plane. From the exper-imental point of view, the dipoles’ short-range interac- ∗ [email protected] † [email protected] tion control is possible via external fields and Feshbachresonances mechanisms [2, 20]. There is a number ofsignificant differences of a 2D dipolar scattering in con-trast to the 3D scattering, e.g. an s − wave divergence inthe low-energy limit and an existence of a weakly boundstate for any attractive potential [30]. The known re-sults, describing shape resonances in dependencies of thecross section on the short-range interaction (SRI) radius(cut-off radius) for 3D space [4], are inapplicable to the2D dipolar scattering. The dependencies of the 2D dipo-lar scattering cross section for the fixed SRI radius werestudied in Refs. [23, 27]. The two-dimensional dipolarscattering cross section dependencies on the SRI radiusfor the arbitrary orientation of the dipole moments havenot been studied. This research fills this gap.We determined, that for the scattering in a plane crosssections for fermions are substantially smaller, than thosefor bosons at low energies of collisions, like in the low-energy 3D dipolar scattering [4]. Cross sections of a 2Ddipolar scattering of fermions at high energies are compa-rable with the cross sections of bosons. We demonstratedthe resonances’ occurrence and an increase of their num-ber at the decrease of SRI radius below the “threshold”values obtained in this paper. In the absence of the res-onances the energy dependencies of the boson (fermion)dipolar scattering cross section grows (is reduced) withenergy decrease in 2D case, in contrast to the 3D case,where it has the form of a plateau for both bosons andfermions [4]. We showed that the mutual orientation ofdipoles strongly impacts the form of the resonant andnon-resonant energy dependencies, which begin to oscil-late with the tilt angle increase, unlike the 3D dipolarscattering [4, 31], where there is no such oscillations.In contrast to a large amount of the papers, devoted tothe scattering of aligned dipoles [2, 4, 22, 23, 32–34] weconsider the scattering of arbitrarily oriented dipoles in aplane. Anisotropic collisions of two dipolar Bose-Einsteincondensates [35] or collisions of slow polar molecules pre-pared in a “cryofuge” [36], make the two-body differentialscattering cross section detectable. The collided dipoleswill have different orientations of dipole moments if theinitial samples of dipolar gases are prepared under dif-ferently directed external electric fields. In this paper,the strong changes of the differential cross section an-gular distributions in resonant and non-resonant pointswere revealed for the scattering of the arbitrarily orienteddipoles in a plane, not only for fermions but for bosonsas well. So, this paper is highly relevant for an interpre-tation of such experiments.The dipoles interaction potential at short-range is ap-proximated here by two types of potentials: by a hardwall with the radius ρ SR [4, 22, 23, 27, 32, 33] and witha more realistic Lennard-Jones potential [34, 37], whileat long-range it is modelled by the DDI [38].The description for the 2D dipolar scattering problemand the improved numerical algorithm (with the bet-ter convergence) for its solution are briefly described inSec. II. The analysis of the numerically obtained resultsand their comparison with the results within the Bornand eikonal approximations are presented in Sec. III.Sec. IV contains the conclusion remarks. II. QUANTUM SCATTERING OFARBITRARILY ORIENTED DIPOLES IN PLANE
The 2D Shr¨odinger equation for describing quantumdipolar scattering in a plane on anisotropic potential U ( ρ, φ ) in polar coordinates ( ρ, φ ) reads: h − ~ µ (cid:18) ρ ∂∂ρ (cid:18) ρ ∂∂ρ (cid:19) + 1 ρ ∂ ∂φ (cid:19) + U ( ρ, φ ) − E i Ψ ( ρ, φ ) = 0(1)with boundary condition in the asymptotic region ρ → ∞ : Ψ ( ρ, φ ) → e i qρ + f ( q, φ, φ q ) e iqρ √− iρ . (2)The relative momentum q is defined by the collision en-ergy E with q = √ µE/ ~ and µ denotes the reducedmass of the system. The incoming wave direction q /q isdefined by the φ q angle.The interaction potential has the form: U ( ρ, φ ) = V SR ( ρ ) + V dd ( ρ, φ ) , (3)The long-range interaction potential of the two arbi-trarily oriented dipoles V dd ( ρ, φ ) in a plane reads: V dd ( ρ , d , d ) = 1 ρ (cid:18) ( d d ) − d ρ )( d ρ ) ρ (cid:19) , (4) Figure 1. The scheme of mutual orientation of two arbitrarilyoriented dipoles d and d , moving in the XY plane. where d i ( i = 1 ,
2) – dipole moments, and ( d i ρ ) /ρ denotetheir projections on the axis, that connect dipole cen-tres of mass. The expression (4) in polar coordinates isrepresented as: V dd ( ρ, φ ; α, β, γ ) = d d ρ [sin( α ) sin( γ ) cos( β )++ cos( α ) cos( γ ) − α ) sin( γ ) cos( φ ) cos( φ − β )] , (5)where angles α and γ define the dipole tilt with respectto the Z -axis and the angle β determines the spatial ori-entation of the Zd and Zd planes. The scheme of themutual orientation of two arbitrarily oriented dipoles isrepresented in Fig. 1.In this paper, for an approximation of the repulsiveSRI V SR ( ρ ) we use two different types of potentials: thehard wall potential with the width of ρ SR (so that thewave function is equal to zero at ρ SR ): V SR ( ρ ) = (cid:26) ∞ , ρ ρ SR , ρ > ρ SR , (6)which has been previously applied by other authors [4,23, 33], andthe more realistic Lennard-Jones (LJ) potential [34, 37] V SR ( ρ ) = C ρ − C ρ . (7)A fixed value of the C parameter of the Lennard-Jones potential was taken for polar molecules, e.g., C = 1 . a.u. for the N a Rb polar molecule [37,39]. The dipolar length scale D for polar molecules( D ≈ N a Rb ) is much larger thanthe van der Waals length scale R = (2 µC / ~ ) / ( µ = 100167 a.u.; R ≈
740 a.u. for N a Rb ) [37] andthe Lennard-Jones potential is effectively used as theshort-range repulsion potential for the dipole-dipole in-teraction potential. Increasing the C at the fixed C leads to an increase of the short-range part of the U ( ρ, φ ).So we vary C to reproduce the change of ρ SR .In order to compare the results of two types of poten-tial with each other, we need to relate the SRI radius ρ SR with the Lennard-Jones C , C parameters. Wedefine ρ SR for the Lennard-Jones potential as ρ SR =min( ρ ( φ )), where ρ ( φ ) is the positions of the zerosof the potential U ( ρ, φ ). From the physical point ofview, min( ρ ( φ )) — the minimal distance, that moleculescould reach at low energies of collision. For the con-sidered polar molecules the term − C /ρ is small com-pared to V dd . Thus, ρ SR is found under the condition C /ρ + V dd ( ρ, β/
2) = 0, from which the next relationfollows: ρ SR = " C ( E D D ) − sin( α ) sin( γ ) β )2 − cos( α ) cos( γ ) . (8)For the hard wall potential (6), the equation (1) withpotential (3) in the dipole units of length D and energy E D : D = µd / ~ , E D = ~ /µ d , could be written in a dimensionless form: (cid:20) − (cid:18) ∂ ∂ρ + 1 ρ ∂∂ρ + 1 ρ ∂ ∂φ (cid:19) + U ( ρ, φ ) (cid:21) Ψ = E Ψ (9)For the N a Rb polar molecule the dipole moment d = 1 .
35 a.u. [39], so E D = 2 . × − a.u.. We note,that in case the hard wall potential is chosen as V SR ,the dipolar scattering properties can be scaled for theparameters of the particular system [31]. For the consid-eration of the Lennard-Jones potential, the dipolar scat-tering properties depend on the relative strength of thevan der Waals and dipolar interactions. The Lennard-Jones potential conversion to the dipolar units reads: V SR ( ρ ) = 1 E D (cid:18) C ρ − C ρ (cid:19) . (10)A scattering differential cross section is defined by thecalculated scattering amplitude f ( q, φ, φ q ) dσ ( q, φ, φ q )/ d Ω= | f ( q, φ, φ q ) | , (11)where d Ω = dφ q dφ . A total cross section is obtained byaveraging over incoming wave directions ( φ q ) and inte-gration over scattering angle φ : σ ( q )= 12 π π Z π Z dσd Ω dφ q dφ. (12)A transition to scattering of identical bosons(fermions) is done with the symmetrization ǫ = +1 (an-tisymmetrization ǫ = −
1) of the wave function:Ψ ( ρ, φ ) → e i qρ + ǫ e − i qρ + f ( φ ) e iqρ √− iρ (13)as well as the differential cross section: dσ ( φ ) /d Ω = | f ( φ, φ q ) | = | f ( φ ) + ǫf ( | ◦ − φ | ) | . (14) The definitions (11) and (14) show, that it leads toan increase for bosons and a decrease for fermions of thedifferential cross section with respect to the case of dis-tinguishable particles for some scattering angles, e.g., for φ + φ q = 90 ◦ : dσ B ( φ + φ q ) d Ω = 4 dσ ( φ + φ q ) d Ω (for bosons) ,dσ F ( φ + φ q ) d Ω = 0 (for fermions) . To tackle the problem (1)–(2), along with interac-tion potential (3) we apply the numerical scheme, thatwe applied to the quantum anisotropic scattering in aplane [27].The following wave function expansion is applied:Ψ ( ρ, φ ) ≈ √ ρ M X m = − M M X j =0 ξ m ( φ ) ξ − mj ψ j ( ρ ) , (15)where ξ − mj = π M +1 ξ ∗ jm = √ π M +1 e − im ( φ j − π ) — is the in-verse matrix to the square matrix (2 M + 1) × (2 M + 1) ξ jm = ξ m ( φ j ), that is defined on the uniform angular grid φ j = πj M +1 (where j = 0 , , ..., M ). In the angular grid’snodes φ j : Ψ ( ρ, φ j ) ≈ ψ j ( ρ ) / √ ρ . ξ m ( φ ) = ( − m √ π e imφ are the eigenfunctions of the operator h (0) ( φ ) = ∂ ∂φ andserve as a basis of functions for the wave function expan-sion over the angular variable.In representation (15), the 2D Schr¨odinger equationtransforms in the system of (2 M + 1) coupled second-order differential equations: d ψ j ( ρ ) dρ + 2 µ ~ (cid:18) E − U ( ρ, φ j ) + ~ µρ (cid:19) ψ j ( ρ )+ − ρ X j ′ M X j ′′ = − M j ′′ ξ jj ′′ ξ − j ′′ j ′ ψ j ′ ( ρ ) = 0 . (16)The seven-point finite difference approximation of six-order accuracy is used for the derivatives discretization.An obtained on each iteration matrix problem is tackledwith the matrix modification of the sweep algorithm forthe band matrix.The original numerical algorithm [27] was modi-fied. For a differential grid over a radial variable( { ρ k } ; ( k = 0 , , ..., N )) we use the non-uniform grid: ρ k = ρ + ( ρ N − ρ ) t k , (17) t k = k/N ; t k ∈ [0 , . (18)and which is similar to the quasi-uniform grids [40]. Thismodification allows us to substantially decrease the num-ber of differential grid nodes N over the radial variable ρ and to increase the algorithm’s convergence rate, whichhas saved a lot of computation time. β = ° β = ° β = ° β = ° β = ° γ deg ) c β , γ ) d e g ) β ( deg ) (cid:2) c (cid:0) β , γ = (cid:1) )( d e g ) Figure 2. The dependence of the critical tilt angle α c ( β, γ )of the dipole d on the rotation angle β and on the tilt angle γ of the dipole d (see the scheme in Fig. 1). The insetpresents the critical tilt angle α c ( β, γ ) as a function of therotation angle β for the case of the dipoles with equal tiltangles ( γ = α ). III. RESULTSA. Critical (magic) angle for scattering ofarbitrarily oriented dipoles
Here we consider the reshaping of the dipole-dipoleinteraction potential V dd with increasing the tilt angle α of the dipole d , with respect to the direction of thedipole d : with the tilt angle γ and rotation angle β between the planes Zd and Zd (see Fig. 1).Domains of the attractive dipolar interaction arisearound the points φ ′ , that are defined by the expres-sion: ∂V dd ( ρ,φ ) ∂φ (cid:12)(cid:12)(cid:12) φ ′ = 0. The critical tilt angle α c ( β, γ ) isdefined as the angle α (see Fig. 1), above which valuesof V dd potential become negative in the points φ = φ ′ : V dd ( ρ, φ ) <
0. Thus, the condition: V dd ( ρ, φ ′ ) = 0 , (19)determines the dependence of the critical tilt angle α = α c ( β, γ ) of the dipole d on the rotation angle β and thetilt angle γ of the dipole d : α c ( β, γ ) = arctan (cid:18) γ )3 + cos( β ) (cid:19) , (20)presented in Fig. 2. The critical tilt angle α c ( β, γ ) in-creases as β → ◦ (e.g. at γ = 45 ◦ the angle α c in-creases from 26 . ◦ to 45 ◦ ). It should be noted, that at β = 180 ◦ the critical tilt angle can be found from a plainratio α c ( β, γ ) = 90 ◦ − γ , which is indicated in Fig. 2 bythe solid red line.When the dipoles’ tilt angles are equal ( γ = α ) andthe second dipole is rotated along the Z axis, the critical tilt angle as a function of the rotation angle β has theform: α c ( β, γ ) = arctan s
23 + cos( β ) , (21)(see the inset of Fig. 2). When one considers the aligneddipoles case γ = α and β = 0 ◦ , the expression (21) re-produces the known value of the critical (magic) angle α c ( β, γ ) = arctan √ ≈ . ◦ or 90 ◦ − α c ( β, γ ) = 54 . ◦ (if the tilt angle is defined with respect to the plane XY ),as previously mentioned [23, 41, 42]. The maximal valueof the critical angle α c ( β, γ ) at γ = α is equal to 45 ◦ andit is reached at β = 180 ◦ . B. SRI-induced resonances in low-energy 2Dscattering of boson and fermion dipoles
We numerically calculated the accurate values of thetotal cross section σ ( ρ SR ) of a 2D scattering as a func-tion of a SRI radius ρ SR at the low collision energy E = 5 × − E D . The obtained results for the identi-cal boson and identical fermion scattering are illustratedin Figs. 3 and 4, respectively.Fig. 3 presents the dependence of the total cross sec-tion of the dipolar scattering of identical bosons on theSRI radius ρ SR in the collision of two aligned β = 0 ◦ ( a, c ) and unaligned β = 180 ◦ ( b, d ) dipoles (at γ = α ),tilted at the angle α = 45 ◦ ( a, b ), as well as for a limit-ing case of dipoles lying in the scattering plane α = 90 ◦ ( c, d ). The results for the hard wall potential are markedby a black dashed line, whereas for the Lennard-Jonespotential by the blue solid line. The distinct narrow res-onances in the dependence of the calculated total crosssection σ B ( ρ SR ) for bosonic dipoles occur for the scat-tering of the two aligned dipoles ( β = 0 ◦ ; γ = α ) at tiltangles α = 45 ◦ , presented in Fig. 3(a). The number ofresonances in the cross section dependence on the ρ SR is quadrupled with increasing tilt angle α from 45 ◦ upto 90 ◦ . The calculated dependence of the σ B ( ρ SR ) forthe two dipoles, that are directed in the scattering plane XY ( α = 90 ◦ ) is presented in Fig. 3(c). The analysisof Fig. 3(c) shows, that as ρ SR decreases the resonancesincreases in number while simultaneously becoming nar-rower.The cross section does not depend on the SRI potentialand there are no dipolar scattering resonances at α ≤ α c ( β, γ ) and ρ SR /D ≪
1, that is shown in Fig. 3(b)for α = 45 ◦ ( γ = α ). Thus, the rotation of the dipolemoment vector d around the Z axis by β → ◦ causesthe narrowing of the domains, where the dipolar potentialis attractive, and the number of the resonances decreasesuntil they disappear at β → ◦ . It should be noted,that the resonances emerge also in the case of oppositelyoriented dipoles ( β = 180 ◦ ) lying in the scattering planeat α → ◦ ( γ = α , α > α c ), as illustrated in Fig. 3(d).The increase of tilt angles α, γ of dipoles and arising of B ( S R ) / D B ( S R ) / D B ( S R ) / D B ( S R ) / D SR /D ( a )( b )( c )( d ) Figure 3. The dependence of the total cross section of thedipolar low-energy scattering σ B of identical bosons on theSRI radius ρ SR at collisions of two aligned β = 0 ◦ ( a, c ) andunaligned β = 180 ◦ ( b, d ) dipoles (at γ = α ), tilted to theangle α = 45 ◦ ( a, b ), as well as for a limiting case of dipoleslying in the scattering plane α = 90 ◦ ( c, d ). The resultsobtained with the use of the hard wall potential are markedwith a black dashed line, whereas those obtained by the useof the Lennard-Jones potential — by the blue solid line. ( a )( b )( c )( d ) Figure 4. The same as in Fig. 3 for identical fermions . TheBorn approximation is marked by a green bold line. domains with attractive dipolar potential leads to thestrong dependence of the dipolar scattering cross sectionon the radius ρ SR of the SRI potential.The distinguishable particles scattering in low-energylimit has also been investigated and the obtained resultsare almost matching within a factor of 4 with the ones for the scattering of identical bosons, due to the s -wavecontribution, that grows with an energy decrease.The analysis of our calculations results has showed,that the fermion scattering is significantly different fromthe boson scattering at low energies. As expected accord-ing to the Wigner threshold law [43] the fermion scatter- Eik /D SR /D = 0.076 SR /D = 0.032 SR /D = 0.15 SR /D = 0.05 ( a ) Eik /D SR /D = 0.073 SR /D= 0.15 SR /D = 0.0313 ( c ) Eik /D SR /D = 0.1925 SR /D = 0.54 SR /D = 0.235 SR /D = 0.07 ( b ) Eik /D ( d ) Figure 5. The energy dependencies of the total cross section of the dipolar scattering of identical bosons σ B ( E ) ( α = 45 ◦ (a),90 ◦ (b)) and identical fermions σ F ( E ) ( α = 45 ◦ (c), 90 ◦ (d)) for aligned dipoles configuration β = 0 ◦ and γ = α . The tilt angleexceeds the critical angle α > α c ( α c = 35 . ◦ ) for such dipole mutual orientations. The curves corresponding to the resonancepoints in Figs. 3 and 4 are indicated by a red solid line, the non-resonant curves by a blue dashed line; the Born approximationby a green dotted line, the eikonal approximation by a gray dashed line. ing is suppressed compared with the boson scattering.The calculated total cross section of dipolar scatteringof identical fermions as a function of ρ SR is presentedin Fig. 4 for equal angles γ = α of dipoles, tilted to theangle 45 ◦ for aligned β = 0 ◦ ( a ) and unaligned β = 180 ◦ ( b ) configurations, and also for the limiting case of twodipoles lying in the scattering plane α = 90 ◦ for par-allel β = 0 ◦ ( c ) and antiparallel β = 180 ◦ ( d ) dipolemoments. The comparative analysis of Fig. 3 and Fig. 4demonstrates, that for the case of fermion collisions, theamplitudes of the resonances are two orders of magnitudesmaller, than those for bosons, in the case of low energyof the collision.The shapes of resonant curves in Figs. 3(c) and 4(c)for the particular case of 2D scattering of aligned dipoleslying in the scattering plane ( α = 90 ◦ ) have the similarprofile with the results of Ref. [4], obtained for the low-energy 3D dipolar scattering (see Fig.3 of Ref. [4]). In our opinion, this fact of the similar profile is due to thehigh probability of the “head-to-tail” dipole collisions.We have also calculated the dependencies σ B ( ρ SR ) , σ F ( ρ SR ) when using the realistic Lennard-Jones potential as V SR . The results are presented inFigs. 3 and 4 and are marked by a blue solid line. TheLennard-Jones potential models short-range repulsionmore physically, in the sense of introducing correlationsbetween the different partial waves short-range phases,which shift narrow resonances in high partial waves, asseen in Figs. 3 and 4. But the resonances’ structureremains qualitatively the same as when using the hardwall potential, because the resonances are due to the s -wave ( p -wave) dominance in the scattering of bosons(fermions). The comparison of the calculation results,which were obtained by using the hard wall potentialand the Lennard-Jones potential, shows their equalefficiency. The results of calculations are qualitativelyand quantitatively similar. So we make a conclusionabout a good applicability of the hard wall potential forthe 2D dipolar scattering characteristic calculations andhereafter we present the results, which were obtainedusing the hard wall potential.We reveal, that the 2D dipolar scattering cross sec-tion dependencies on the SRI radius for both bosons andfermions are strongly changed for different mutual ori-entations of arbitrarily directed dipoles. Figs. 3 and 4illustrate this fact. C. Energy dependencies of cross section of bosonicand fermionic dipoles scattering in plane
The energy dependencies of the cross section of thedipolar scattering in a plane were studied in Ref. [22]for the isotropic repulsive case of dipolar interaction.Here we present the results of the calculations of theenergy dependencies for the anisotropic dipolar interac-tion potential V dd . The energy dependencies of the totalcross section of the dipolar scattering of identical bosons σ B ( E ) ( α = 45 ◦ (a), 90 ◦ (b)) and identical fermions σ F ( E )( α = 45 ◦ (c), 90 ◦ (d)) for aligned dipoles configuration β = 0 ◦ ; γ = α are illustrated in Fig. 5. The tilt angleexceeds the critical angle α > α c ( α c = 35 . ◦ ) for suchdipole mutual orientations. The curves corresponding tothe resonance points in Figs. 3 and 4 are indicated by ared solid line; the non-resonant curves by a blue dashedline; the Born approximation by a green dotted line, theeikonal approximation by a gray dashed line.The analysis of the dependencies shows that in a low-energy limit, the cross section of resonant cases is atleast an order of magnitude greater than the values fornon-resonant cases. The cross section of identical bosonsscattering increases in the E → s -wave, which is caused by the divergence in2D space [30]. It should be noted, that in the vicinityof resonances the cross section is an order of magnitudegreater than those at the absence of resonances. All res-onant curves of the cross section σ F ( E ) for the dipolarscattering of fermions demonstrate a peak shape in thelow-energy limit, in contrast to the non-resonant curves,that monotonically decrease. These peaks shift to thelower energies with the growing value of its maximumsat an increase of ρ SR . The resonances for fermions arenarrower than for bosons, due to potential barriers forhigh partial waves, that suppress the partial cross sec-tion in the low-energy limit.The boson (fermion) dipoles 2D scattering cross sec-tion in the absence of resonances increases (decreases)in the low-energy limit in contrast to the 3D scattering,where the cross section in the absence of resonances hasthe form of a plateau in the low-energy limit for bothbosons and fermions (see Fig. 2(a,b) in Ref. [4] or Fig. 1in Ref. [31]).We also showed that the mutual orientation of dipolesstrongly impacts the form of the energy dependencies. Figure 6. The dependence of the total cross section for dipolarscattering of fermions on the tilt angle α in degrees (at γ = α )and on the rotation angle β in degrees obtained within theBorn Approximation. Thus, at an increase of the angle α from 45 ◦ to 90 ◦ (see Fig. 5) the resonant and non-resonant dependencies σ B ( E ) , σ F ( E ) begin to oscillate, unlike the 3D dipolarscattering [4, 31], where there is no such oscillations. D. Born approximation
For weak dipole moments, the Born approximation(BA) [44] was used to estimate the ultracold dipoles scat-tering amplitude in Refs. [22, 23, 32, 45–47]. A goodagreement with close-coupling calculations [22, 23, 32,45–47] was obtained away from resonances. However, inthe vicinity of the resonances the Born approximation isnot valid and one has to numerically calculate the scat-tering cross sections [48].We generalize the expression of the total cross sectionof aligned dipoles, obtained within the Born approxima-tion in Ref. [23] for the long-range part of the poten-tial (5) by a summation of three series of partial crosssection, for the case of arbitrarily oriented dipoles ( iden-tical fermions ): σ BA /D = 8 p E/E D " π ( π − (cid:16) sin( α ) sin( γ ) cos( β )++ cos( α ) cos( γ ) −
32 sin( α ) sin( γ ) cos( β ) (cid:17) + (cid:18) π ( π + 2) + 329 − (cid:19) (cid:18)
34 sin( α ) sin( γ ) (cid:19) . (22)The dependence (22) of the BA on the the tilt angle α (at γ = α ) and the rotation angle β at the energy E = 5 × − E D is illustrated in Fig. 6.As seen in Fig. 4, the σ F obtained within the Bornapproximation (BA), that is marked with a bold greenline, roughly agrees with the calculated values of σ F ( ρ SR )away from resonances, but it is inapplicable when theresonant structure of the σ F ( ρ SR ) dependence is dense.The BA predicts the same cross section values for the β = 0 ◦ and β = 180 ◦ at α = 90 ◦ . On the basis of theanalysis of Fig. 6, one could expect that the scatteringcross section should be less dependent on the tilt angle α as β → ◦ and should approach the value of α = 0 ◦ case, which is confirmed only for the α < α c . Thus,the calculated dependencies show, that the BA (22) isapplicable to the two-dimensional dipolar scattering offermions when the tilt angle of the dipoles is less thancritical angle α < α c , when the DDI potential does notsupport bound states.The BA is not directly applicable to the scatteringof identical bosons or distinguishable particles, due toa divergence of the s − wave contribution in the regionof the short-range potential, which must be explicitlyaccounted, since there is no centrifugal barrier for the s − wave. The divergence could be avoided by the use ofthe pseudopotential with a “potential strength” [32]. Al-though, in order to obtain the “potential strength” it isnecessary to numerically solve the full two-dimensionalShr¨odinger equation, so it is useful only for the followingmany-body calculations. That is why we compare theBA only with the scattering of fermions.In the low-energy limit the values of σ BA ( E ), ob-tained within the BA, marked by a green dotted line inFig. 5( c , d ), agrees well with the non-resonant dependen-cies σ F ( E ) for the fermion scattering. Note, that thereis no such agreement for the resonant curves in Fig. 5. E. Eikonal approximation
In the high energy regime E ≫ E D , the semi-classicaleikonal approximation [22, 23, 49]: σ Eik /D = 4 √ π (2 E/E D ) / , (23)is applicable to estimate the scattering cross section. For E > E D the dependencies of the 2D dipolar scatteringcross section on the collision energy of bosons, σ B ( E ),and fermions, σ F ( E ), oscillate around the curve of theeikonal approximation σ Eik ( E ), marked by a grey dot-dashed line in Fig. 5( c , d ). The dipolar scattering crosssections, obtained within the eikonal approximation, arein an excellent agreement with presented in Fig. 5 numer-ically calculated cross sections of dipolar scattering σ B and σ F , that correspond to the absence of resonances,at E > E D . This is explained as follows: the morepartial waves are involved in the scattering with an in-crease of the collision energy, the more the scatteringcross section approaches to semiclassical estimates. How-ever, the cross sections of dipolar scattering of bosonsand fermions obtained numerically, which correspond to the occurrence of resonances, are strongly different fromthose of σ Eik , which indicates a significant impact of the s − and p − waves on the scattering of dipoles even for thehigh energies E ≫ E D . F. Impact of SRI on high-energy 2D scattering ofarbitrarily oriented dipoles
We also analyzed the SRI impact on the total crosssection for the high energies of the dipolar collisions in aplane ( E = 50 E D ), that were studied in Refs. [23, 27] atfixed values of ρ SR . The calculated dependencies of thetotal cross section on the dipole tilt angle α (at γ = α ),for variable ρ SR and rotation angle β are presented inFig. 7 for the case of identical bosons and fermions. Thegraph analysis demonstrates the existence of “threshold”value ρ SR /D = 0 .
3, below which the strong oscillations(with resonances number increase) occur in the depen-dencies σ B ( α ), σ F ( α ). The number and magnitude ofthe σ B ( α ), σ F ( α ) oscillations decrease with an increaseof the rotation angle β → ◦ . Note, that oscillationsand resonances in the dependencies σ B ( α ), σ F ( α ) disap-pear at the tilt angle α less than the critical angle α c ( β, γ )at γ = α (21) and ρ SR /D < .
1. That is, in the consid-ered range ρ SR /D < the SRI effects dipolar scatteringif the tilt angle α is greater than the critical angle . TheDDI could be seen as a perturbation of the isotropic re-pulsive part at ρ SR /D ≫
1, and this regime will be stud-ied in future works. The SRI radius ρ SR decreases withthe tilt angle α larger than the critical angle α c whichleads to the occurrence of oscillations and resonances inthe dipolar scattering cross section of bosons as well asfermions. Occurrence of the oscillations and resonancesat the angle α achieving the critical angle α c is demon-strated in Fig. 7 for the range ρ SR /D ≤ .
1, when bothdipolar interaction and SRI contribute to the scattering(in Fig. 7(a,b) α c = 35 . ◦ , in Fig. 7(c,d) α c = 39 . ◦ , inFig. 7(e,f) α c = 45 ◦ ). G. Angular distributions of differential crosssection
We revealed the strong dependence of the angular dis-tributions of the 2D dipolar scattering differential crosssection dσ/d
Ω on the value of SRI radius ρ SR . The dif-ferential cross sections of the dipolar scattering of bosonsfor resonant case are presented in Fig. 8(a,c) at the points ρ SR /D = 0 . , .
056 and for non-resonant case are pre-sented in Fig. 8(b,d) at the points ρ SR /D = 0 . , . Q BX and Q BY denote dσ B /d Ω cos( φ ) and dσ B /d Ω sin( φ ) respectively.The differential cross section angular distributions forbosons exhibit circular shape in the resonant ρ SR pointsboth for α = 45 ◦ and α = 90 ◦ , indicating s − wave dom-inance in the resonance emergence. At dipole tilt an-gles, which are larger than a critical angle, the dσ/d Ω . . (deg) SR /D (cid:176) B / D ( a ) . . . . F / D (cid:176) SR /D (deg) ( b ) . . (deg) SR /D B / D (cid:176) ( c ) . . . . (cid:176) F / D (deg) SR /D ( d ) . . (deg) SR /D B / D (cid:176) ( e ) . . . . (deg) SR /D (cid:176) F / D ( f ) Figure 7. The dependencies of the total cross section on the tilt angle α of the dipoles (with respect to the normal to thescattering plane XY ) for the case of the equal dipole tilt angles γ = α at various values of ρ SR in the 2D scattering of identical bosons ( β = 0 ◦ ( a ) , ◦ ( c ) , ◦ ( e )) and fermions ( β = 0 ◦ ( b ) , ◦ ( d ) , ◦ ( f )). The critical angle α c ( β, γ ) = 35 . ◦ , . ◦ , ◦ for β = 0 ◦ , ◦ , ◦ , respectively, and γ = α . σ ( ρ SR ) minimum, that is demonstrated inFig. 8(b) for α = 45 ◦ . Whereas at the tilt angle α = 90 ◦ angular distributions of dσ/d Ω are strongly anisotropicat the points of a minimum of total cross section depen-dence σ B ( ρ SR ), indicating that the s − wave contributionis suppressed and the scattering is governed by higherpartial waves. So, in contrast to the central potentials,the 2D low-energy dipolar scattering of bosons is stronglyanisotropic and its properties are highly sensitive to theSRI radius as well as dipoles mutual orientation.The angular distributions of differential cross section dσ/d Ω of the dipolar scattering of fermions are alwaysanisotropic. As shown in Fig. 8(e,g) at the points ρ SR /D = 0 . , . α = 45 ◦ (e) and 90 ◦ (g) at the po-sitions of SRI-resonances; while at non-resonant ρ SR points the differential cross section dσ/d Ω changes withincreasing α , as illustrated in Fig. 8(f,h) at the points ρ SR /D = 0 . , .
39 respectively. In Fig. 8(e,f,g,h) Q FX and Q FY denote dσ F /d Ω cos( φ ) and dσ F /d Ω sin( φ ) re-spectively. The angular distributions of the 2D dipolarscattering differ significantly from the angular distribu-tions of differential cross sections of the 3D dipolar scat-tering [47]. Dipolar fermions can scatter more stronglythan dipolar bosons in the 3D case [47], whereas in a 2Dcase the cross section of dipolar scattering of fermions isseveral orders of magnitude less than the scattering crosssection of bosons at low energies. The dependencies ofthe differential cross section of the boson and fermion 2Dscattering on the incident angle φ q also differ. In contrastto the 3D dipolar scattering [47], the scattering of bosonsin a plane does not depend on the incident angle φ q atthe resonant points, while it could be highly sensitive tothe incident angle in non-resonant points. The 2D scat-tering of fermions always depends on the incident angle φ q . IV. CONCLUSION
The impact of the short-range interaction on the res-onances occurrence in the anisotropic dipolar scatteringin a plane was numerically investigated for different ori-entations of the dipoles and for a wide range of collisionenergies. We revealed the strong dependence of the crosssection on the radius of short-range interaction, which ismodeled by a hard wall potential and by the more realis-tic Lennard-Jones potential. Both potentials showed analmost equal applicability. The analysis of the obtainedresults showed, that the short-range potential replace-ment does not change of the resonance structure, leadingonly to the slight shifts of resonance positions. The re-sults of the accurate numerical calculations of the crosssection agree well with the results obtained within theBorn and eikonal approximations.It was found, that the s − wave ( p − wave) dominates inthe angular distributions of the differential cross section at resonance points in the 2D dipolar scattering of iden-tical bosons (identical fermions), whereas the higher par-tial waves dominate at non-resonant points and the dif-ferential cross sections are highly anisotropic. The crosssections for the fermion scattering in a plane are sub-stantially smaller, than those for bosons at low energiesof collisions, like in the low-energy 3D dipolar scatter-ing [4]. At high energies cross sections of a 2D dipolarscattering of fermions are comparable with the cross sec-tions of bosons.For both low and high collision energies, we reveal the“threshold” values of the radius of the short-range inter-action, below which there has been an appearance and agrowth of the number of resonances. We also defined thecritical (magic) tilt angle of one of the dipoles, depend-ing on the direction of the second dipole for arbitrarilyoriented dipoles. It was found that resonances arise onlywhen this angle is exceeded. In the absence of the res-onances the energy dependencies of the boson (fermion)dipolar scattering cross section grows (is reduced) withenergy decrease in 2D case, in contrast to the 3D case,where it has the form of a plateau for both bosons andfermions [4]. We also showed that the mutual orienta-tion of dipoles strongly affects the form of the energydependencies. Thus, at an increase of the dipoles tiltangle α → ◦ the resonant and non-resonant cross sec-tion energy dependencies begin to oscillate, unlike the 3Ddipolar scattering [4, 31], where there is no such oscilla-tions.Obtained data on the 2D scattering cross sections of ar-bitrarily oriented dipoles allow us to conclude that the oc-currence and the resonances’ number could be controlled also by varying the radius of the short-range interactionand the dipoles mutual orientation. ACKNOWLEDGMENTS
The authors acknowledge the support of the RussianFoundation for Basic Research, Grant No. 19-32-80003.1 ( a ) ( b )( c ) ( d )( e ) ( f )( g ) ( h ) Figure 8. The dependencies of the differential cross sections dσ/d
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