Dipolar confinement-induced molecular states in harmonic waveguides
DDipolar confinement-induced molecular states inharmonic waveguides
Gaoren Wang , Panagiotis Giannakeas , Peter Schmelcher , Zentrum f¨ur Optische Quantentechnologien, Universit¨at Hamburg, LuruperChaussee 149, 22761 Hamburg, Germany Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana47907, USA The Hamburg Centre for Ultrafast Imaging, Universit¨at Hamburg, LuruperChaussee 149, 22761 Hamburg, Germany
Abstract.
The bound states of two identical dipoles in a harmonic waveguideare investigated. In the regime of weak dipole-dipole interactions, the local frametransformation (LFT) method is applied to determine the spectrum of dipolarconfinement-induced bound states analytically. The accuracy of the LFT approachis discussed by comparing the analytical results with the numerical ones based on asolution of the close-coupling equations. It is found that close to the threshold energyin the waveguide, the LFT method needs to include more partial wave states to obtainaccurate bound state energies. As the binding energy increases, the LFT methodusing a single partial wave state becomes more accurate. We also compare the boundstates in waveguides and in free space. For the bosonic case, the s -wave dominatedbound state looks like a free-space state when its energy is below a certain value. Forthe fermionic case, the p -wave dominated bound state energies in waveguides and infree-space coincide even close to zero energy.
1. Introduction
Ultracold gases in tightly confining traps have attracted much attention particularlysince one can realize effective one- [1, 2] and two- [3] dimensional systems by tuningtheir geometry. The tight confinement modifies significantly the interparticle collisionalproperties, and specifically leads to confinement induced resonances (CIRs) which havebeen predicted theoretically [4, 5, 6] and observed experimentally [1, 7]. The capabilityto tune the two-body interaction via Feshbach resonances and/or CIRs enabled theinvestigation of strongly correlated many-body physics in low dimensions [8]. Tight trapsalso affect the two-body bound state properties, and confinement induced molecular(CIM) states have been observed for isotropic interparticle interaction [1, 2].Dipolar gases [9, 10, 11, 12, 13, 14] possess anisotropic dipole-dipole interaction(DDI) [15] which makes it interesting to investigate the influence of a tight trap on suchsystems. It has been demonstrated that adding a trap in one direction can suppress thereactive scattering collisions in a polar molecular gas [16]. In addition, one can control a r X i v : . [ phy s i c s . a t m - c l u s ] M a y ipolar confinement-induced molecular states in harmonic waveguides z axis) of the waveguide. The local frame transformation (LFT) approach[22, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36] is applied to calculate the DCIMstates when the DDI is weak. A dipolar bound state equation is derived which allowsone to determine the energies of the DCIM states analytically with the free-spacescattering information as input. Moreover, the dipolar bound state equation withinthe Born approximation shows explicitly the influence of the DDI in determining thebound state energies. By comparing the LFT results with corresponding numericalcalculations, it is found that, below threshold, the LFT approach with the single partialwave approximation is accurate even in the presence of DDI. Close to threshold, oneneeds to include higher partial wave states in the LFT approach to get accurate boundstate energies. The dependence of the DCIM states on the DDI strength is explored.Based on numerical calculations, both the weak and strong DDI regimes are investigated.We find that qualitatively the dependence of the DCIM states on the DDI is similar inthese two regimes. The DCIM state becomes increasingly bound as the DDI increases.The l > s -wave dominant states. New DCIM states can emerge by increasingthe DDI.The paper is organized as follows. Section II introduces our computational methods.The set of close coupling equations is provided in a partial wave basis, and the dipolarbound state equations based on the LFT approach are presented. In Sec. III, theproperties of the dipolar confinement induced molecular states are discussed. Both thebosonic and fermionic cases are analyzed. Sec. IV contains our conclusions.
2. Computational method
In harmonic waveguides, the center of mass motion and relative motion are separable.The Hamiltonian of the relative motion, which contains a short-range isotropicinteraction in conjunction with DDI, is expressed as H = T + V t ( r ) + V sr ( r ) + V d ( r ) , (1) ipolar confinement-induced molecular states in harmonic waveguides T is the kinetic energy. V t ( r ) is the transverse trapping potential, and is assumedto be an isotropic two-dimensional harmonic potential V t ( r ) = µω ⊥ ρ , where µ is thereduced mass, ω ⊥ is the trapping frequency, and ρ is the magnitude of the transversecomponent of the interparticle vector r . V sr ( r ) is the short-range isotropic potential,and depends on the species under consideration. V sr ( r ) is modeled by a Lennard-Jones(LJ) potential V sr ( r ) = C /r − C /r which possesses a van der Waals potential tail(note that r = | r | ). V d ( r ) is the DDI, and has the usual form V d ( r ) = d r (1 − θ )where d is the dipole moment, and θ is the angle between the z axis and the interparticlevector r . We remark that the singularity of the DDI at the origin is remedied by thecorresponding behavior of the short range potential at the origin. The threshold energyof two dipoles without the confinement is chosen to be the zero energy point. The energyof the scattering threshold in waveguides is given by E th = (cid:126) ω ⊥ .The three potential terms in Eq. (1) determine three length scales in the system.The length scale associated with the transverse confinement is the harmonic oscillatorlength a ⊥ = (cid:112) (cid:126) /µω ⊥ whereas the length scale of the short-range interaction term isthe van der Waals length given by the relation β = (2 µC / (cid:126) ) / . Finally the DDI ischaracterized by the dipole length l d = µd / (cid:126) . Below we will introduce two methods todetermine the bound state belonging to the Hamiltonian (1). One approach is the close-coupling method which solves the problem numerically. In the weak DDI regime, thedipole length and the van der Waals length are far smaller than the harmonic oscillatorlength. The local frame transformation method is in this case applied to derive theDCIM states (semi)analytically. We expand the two-body wavefunction ψ in the partial wave basis ψ ( r, θ, φ ) = 1 r (cid:88) lm f lm ( r ) Y lm ( θ, φ ) , (2)where f lm ( r ) are the radial wavefunctions, Y lm are the spherical harmonics, and l and m are the partial wave quantum number and the magnetic quantum number, respectively.Since the system under investigation is cylindrically symmetric, the magnetic quantumnumber is conserved. In the following m is set to zero, and is consequently omitted.The kinetic term T and the short-range potential V sr ( r ) are diagonal in the partial wavebasis. The dipole potential V d ( r ) and the transverse trapping potential V t ( r ) coupledifferent partial wave states. The matrix elements of V d ( r ) and V t ( r ) in the partialwave basis are given respectively by [42] V ll (cid:48) d ( r ) = < l | V d | l (cid:48) > = − d r (cid:112) (2 l + 1)(2 l (cid:48) + 1) (cid:32) l l (cid:48) (cid:33) , (3) V ll (cid:48) trap ( r ) = < l | V trap | l (cid:48) > = 13 µω ⊥ r δ ll (cid:48) − µω ⊥ r (cid:112) (2 l + 1)(2 l (cid:48) + 1) (cid:32) l l (cid:48) (cid:33) , (4) ipolar confinement-induced molecular states in harmonic waveguides δ is the Kronecker delta function, and the large curved brackets are 3-j symbols.In the partial wave basis, the Schr¨odinger equation is a set of close-coupling equationssatisfied by the radial wavefunction f l ( r ) (cid:88) l (cid:20) − (cid:126) µ d dr + V c ( r ) + V sr ( r ) (cid:21) f l ( r )+ (cid:88) ll (cid:48) (cid:16) V ll (cid:48) d ( r ) + V ll (cid:48) trap ( r ) (cid:17) f l (cid:48) ( r ) = E (cid:88) l f l ( r ) , (5)where V c ( r ) = l ( l +1)2 µr is the centrifugal term, and E is the total energy. With theboundary condition that the bound state wavefunction vanishes at r → r → ∞ ,the set of close-coupling equations given in Eq. (5) is solved numerically based on thelog-derivative algorithm [43]. To obtain the bound state energy E b and the wavefunction ψ the approach of Ref. [44] is employed.In the numerical calculation, the dimensionless version of Eq. (5) is used, in whichthe length and energy are scaled by a ⊥ and (cid:126) ω ⊥ respectively. The coefficient C in theLJ potential is fixed, such that β /a ⊥ is 0.018, which is an experimentally achievablevalue. For example, the van der Waals coefficient of two ground state Er atoms is1723 au [38], and the corresponding van der Waals length β is 151 au. In Ref. [39],a transverse trapping potential with frequency ω ⊥ ∼
600 Hz is realized experimentallyfor Er atoms. The corresponding harmonic oscillator length a ⊥ is 8518 au, and theratio β /a ⊥ amounts to 0.018. For other systems, the van der Waals lengths would bedifferent. Nevertheless one can tune the transverse trapping frequency ω ⊥ to achievethe desired value for β /a ⊥ . The coefficient C in the LJ potential is varied such thatthe scattering length of the LJ potential can be changed significantly, and moreover, thenumber of bound states supported by the LJ potential can be tuned. The partial wave probabilitydensities (PD) of the DCIM state P lb = | f l | are calculated via close-coupled method[43, 44]. By examining the partial wave PDs, regions dominated by different terms inthe Hamiltonian (1) are identified.The partial wave PDs for a bosonic DCIM state are shown in the upper panelof Fig. 1 (solid line). The energy of the DCIM state is E b / (cid:126) ω ⊥ = 0 .
5. The scaled C s = C / ( (cid:126) ω ⊥ a ⊥ ), which is dimensionless, is set to be 6.7 × − . The ratio l d /a ⊥ is0.026. In the range r/a ⊥ (cid:28)
1, the trapping potential is far smaller than the short-rangepotential and DDI, and can be neglected. As a demonstration, we performed the freespace scattering calculation by dropping the term V t ( r ) in Eq. (1). The partial wavePD P ls of the scattering state at energy E b is also shown in the upper panel of Fig. 1(dotted line). The lower panel of Fig. 1 is a zoom-in plot of the upper panel in theshort-range region r/a ⊥ (cid:28)
1, and clearly shows that P lb has the same nodal structure as P ls in this region. This indicates that the trapping potential is negligible for r/a ⊥ (cid:28) ipolar confinement-induced molecular states in harmonic waveguides s -wave component. The oscillatory behavior of P lb in the short-range region r/a ⊥ (cid:28)
1, shown in the lower panel of Fig. 1, is due to the presence of thepotential well of the interaction potential V int ( r ) = V sr ( r ) + V d ( r ). It is noted that thereare maxima for P lb ’s with l > r/a ⊥ >
1. By examining the potentialcurves in the partial wave basis, we encounter potential wells at distances r/a ⊥ > l > V cent andthe increasing trapping potential V trap ( r ) as the interparticle distance increases. Thepositions of the maxima for the high partial wave PDs coincide with the positions ofthe minima of the outer potential wells at r/a ⊥ > r/a ⊥ >
1, the short-range potential V sr ( r ) and the DDI potential V d ( r ) decay to zero, and the trapping potential V trap ( r ) becomes dominant. In thisregion the system is mainly governed by V t ( r ), and is nearly independent of V sr ( r ) and V d ( r ). In order to confirm this, we compare in Fig. 3 the partial wave PDs of two DCIMstates calculated with different short-range potential V sr ( r ) and DDI potential V d ( r ).The bound state PD P lb shown in Fig. 1 is also shown in Fig. 3 (solid line). As statedbefore, in the calculation of P lb the LJ potential supports ten bound states, and thedipole moment l d /a ⊥ is 0.026. Another bound state PD P (cid:48) lb , the energy of which is also E b , is additionally shown (dotted line). The LJ potential supports here one bound stateand l d /a ⊥ is zero in the calculation of P (cid:48) lb . Due to the different V sr ( r ) and V d ( r ) usedin the calculations, the short-range parts of the two bound state PDs are significantlydifferent. Nevertheless, in the region r/a ⊥ >
1, the two bound state PDs are nearly thesame, and are, to a large extend, determined by the trapping potential.Based on the above observation of length scale separation in the system, weintroduce the local frame transformation approach below, which can connect the two-body properties in waveguides with the scattering properties in free space analytically.
The concept of the LFT approach was introduced in Ref. [25, 26] to calculate the Starkeffect of nonhydrogenic Rydberg spectra. Subsequently the method was generalizedto study the photodetachment of negative ions in magnetic fields [27, 28] and thephotoionization of atoms [29, 30]. The application of the LFT approach to ultracoldcollisions in quasi-low dimensional geometry was pioneered in Ref. [31]. Until now, ithas been applied to understand different aspects of ultracold collisions in harmonicwaveguides, such as higher partial wave confinement-induced resonances(CIR) [32],energy dependence of the CIRs [33], multi-open-channel collisions [34] and the dipolarCIRs [22]. Ultracold collisions in other confining geometries have been discussed in[35]. Recently the LFT approach has been adopted to treat the two-body scatteringanalytically in the presence of spin-orbit coupling [37]. Here we provide a further ipolar confinement-induced molecular states in harmonic waveguides -8 -6 -4 p r ob a b ilit y d e n s it y l=0l=2l=4l=6 r/a ⊥ (a) -8 -6 -4 p r ob a b ilit y d e n s it y l=0 l=2l=4l=6 r/a ⊥ (b) Figure 1. (Upper panel) Bound state partial wave probability density P lb of twoidentical bosonic dipoles in a waveguide is shown (solid line). The free-space scatteringstate probability density P ls is also provided (dashed line). C s = 6 . × − and theLJ potential supports ten bound states. The ratio l d /a ⊥ is 0.026. The bound stateenergy and the scattering energy are the same E/ (cid:126) ω ⊥ = 0 . P ls is renormalized at r /a ⊥ = 0 .
01, so that P l =0 s ( r /a ⊥ ) = P l =0 b ( r /a ⊥ ). (Lower panel) A zoom-in pictureat short distances. application of the LFT approach, and show that one can obtain a comprehensive analysisof the DCIM states. In the following, the key idea of the LFT approach is brieflyintroduced (see Ref.[31, 32, 33] for more details) and then the bound state equationfor two dipoles in waveguides is presented. Finally, within the Born approximation, thedipolar bound state equation is simplified to show the dependence on the DDI explicitly. As shown in Fig. 1 and 3, the lengthscales of the short-range potential V sr ( r ), DDI potential V d ( r ) and the trapping potential V t ( r ) are separated in the weak DDI regime. In the region r/a ⊥ (cid:28) V t ( r ) is negligiblysmall compared to V sr ( r ) and V d ( r ), and the two dipoles effectively interact as if thereis no confinement. The partial wave basis is employed to construct the wavefunction in ipolar confinement-induced molecular states in harmonic waveguides -20246 s ca l e d po t e n ti a l e n e r gy l=0 l=2 l=4l=6l=8 r /a ⊥ V d i s Figure 2.
The scaled diagonal potential energy V s di ( r ) = V di ( r ) / (cid:126) ω ⊥ in the partialwave basis with V di ( r ) = V sr ( r ) + V d ( r ) + V t ( r ) + V c ( r ). The partial wave quantumnumber l is associated to the corresponding channel potential. -9 -6 -3 p r ob a b ilit y d e n s it y l=0l=2l=4l=6 r/a ⊥ Figure 3.
The bound state probability density P lb shown in Fig. 1 is given viathe solid line. The bound state probability density P (cid:48) lb (dotted line) is calculated for C s = 2 . × − and l d /a ⊥ = 0. The corresponding LJ potential supports one boundstate in free space. The two bound states possess the same energy. this region. The DDI couples different partial wave states, and the effect of the two-bodyinteraction is encapsulated in the free space K matrix, denoted as K . In the region r/a ⊥ (cid:29)
1, the interparticle interaction V sr ( r ) + V d ( r ) vanishes, and the system can betreated as two non-interacting particles in a waveguide. The wavefunction in this regioncan be written asΨ( r ) = F − GK D , (6)where F = diag { F , F , · · ·} and G = diag { G , G , · · ·} are diagonal matrices. F n and G n are the regular and irregular solutions of the Hamiltonian (1) without the short-range potential V sr ( r ) and the DDI potential V d ( r ). The explicit expressions for F n and G n have been given in [32] which are the product of the eigenfunctions of the twodimensional harmonic oscillator in the transverse plane and the standing wave with ipolar confinement-induced molecular states in harmonic waveguides K is the K matrix in such quasione-dimensional geometry. In the intermediate region β /a ⊥ , l d /a ⊥ < r <
1, both V sr ( r ) + V d ( r ) and V trap ( r ) are small compared to the kinetic energy. Then in thisregion, both the partial wave basis and the asymptotic basis, ie. F n and G n , can beused to describe the wavefunction. A local transformation matrix U can be definedwhich connects the two basis sets. The element of the transformation matrix U reads[33, 36] U Tl,n = √ − d a ⊥ (cid:115) l + 1 kq n P l (cid:16) q n k (cid:17) , (7)where n is the transverse harmonic oscillator quantum number. d is l/ l and( l + 1) / l . P l ( x ) is Legendre polynomial, q n is the channel momentum alongthe waveguide axis determined by ( (cid:126) q n ) µ = E − (cid:126) ω ⊥ (2 n + 1).The local frame transformation U can be used to express K in terms of K according to the relation [33] K = UK U T . (8)From the K matrix, one can deduce both bound state and scattering informationin waveguides [33]. We are interested in the bound state spectrum here. By imposingasymptotically an exponentially decaying boundary condition in the wavefunction (6),one obtains the following relation for K [32, 33]det( I − iK ) = , (9)where the roots of Eq. (9) provides us with the energies of the confinement-inducedbound state E b . Next we examine the explicit expression for the K matrix in the presence of DDI, and derive the dipolar bound state equation in termsof the K matrix. For systems consisting of atoms governed by the van der Waalsinteraction, the single partial wave approximation works quite well in the ultracoldregime [33]. In the presence of DDI, different partial wave states are coupled [40]. To bespecific, the l partial wave is coupled to the l (cid:48) partial wave with l (cid:48) = l, l ± l = l (cid:48) = 0is an exceptional case which is not coupled by the DDI. In the determination of DCIR[22], the LFT approach with three partial wave states can accurately reproduce thenumerical results in the weak DDI regime. Therefore, we include up to three lowestpartial wave states in the derivation of the bound state equation either for bosons orfermions. For identical particles, the free space K D matrix including three partial wavestates can be expressed as [22] K D = K l ,l K l ,l K l ,l K l ,l K l ,l K l ,l K l ,l , (10) ipolar confinement-induced molecular states in harmonic waveguides l , l , and l label the quantum numbers of the lowest three partial wave statesin ascending order. For identical bosons, the lowest three partial wave states are the s , d and g wave states. For identical fermions, these are the p , f and h wave states. In theexpression of K D (see Eq. (10)), direct couplings between different partial waves dueto the DDI are included, such as K l ,l , K l ,l and K l ,l , K l ,l . Since the weak DDI isconsidered here, the indirect couplings between partial wave states which are mediatedby another state are set to zero, such as K l ,l and K l ,l .From the K D matrix, the generalized scattering length a l,l (cid:48) is introduced as [41] a l,l (cid:48) = − K l,l (cid:48) /k, (11)where k = (cid:112) µE/ (cid:126) is the collisional momentum. By substituting Eqs. (8) and (10)into Eq. (9), the dipolar bound state equation including three partial wave states canbe written as a l ,l = ik ∆ N ∆ D , (12)where ∆ N = − iM l ,l K l ,l + M l ,l K l ,l + iM l ,l K l ,l − M l ,l K l ,l K l ,l − M l ,l M l ,l K l ,l K l ,l − iM l ,l M l ,l K l ,l K l ,l + M l ,l K l ,l (1 − iM l ,l K l ,l )+ M l ,l ( − M l ,l K l ,l K l ,l − M l ,l K l ,l − iM l ,l K l ,l K l ,l + M l ,l K l ,l ( − iM l ,l K l ,l ) + K l ,l ( i + M l ,l K l ,l ))+ 2 M l ,l K l ,l ( i + M l ,l K l ,l + M l ,l ( K l ,l + iM l ,l K l ,l K l ,l )) , (13)∆ D = − M l ,l M l ,l (cid:0) iK l ,l + M l ,l K l ,l − M l ,l K l ,l K l ,l (cid:1) + M l ,l (cid:0) M l ,l K l ,l − K l ,l ( i + M l ,l K l ,l ) (cid:1) + M l ,l (cid:0) − iK l ,l + M l ,l (cid:0) K l ,l − K l ,l K l ,l (cid:1)(cid:1) + M l ,l ( − iM l ,l K l ,l + iM l ,l K l ,l + M l ,l (cid:0) K l ,l − K l ,l K l ,l (cid:1) + M l ,l (cid:0) − M l ,l K l ,l + K l ,l ( i + M l ,l K l ,l ) (cid:1) ) , (14)and M l,l (cid:48) is the trace (cid:80) n U Tl,n U n,l (cid:48) over all the closed transverse harmonic oscillatormodes, and are known analytically [33]. The explicit expressions for M l,l (cid:48) are given inthe appendix for the cases of identical bosons and fermions considered in this work.In the dipolar bound state equations (12), two sets of quantities are neededto determine the DCIM state. One set is the M l,l (cid:48) , which contain the geometricalinformation of the waveguide and are known analytically (see appendix). The other setis the elements of the free-space K matrix, or equivalently the generalized scatteringlength a l,l (cid:48) which encapsulate the effect of the interparticle interaction. Eq. (12) providesus with the spectrum of the DCIM states once the free-space K is known. This isone of the main results of this paper. ipolar confinement-induced molecular states in harmonic waveguides K l ,l = K l ,l = K l ,l = 0 in Eq. (13) and (14), one can obtain the boundstate equation including two partial wave states. By setting all the other elements ofthe K matrix to zero except K l ,l , the bound state equation with a single partialwave state is obtained. One can obtain the a l,l (cid:48) by solving the free space scattering problem numerically [42]. Alternatively, theBorn approximation can be adopted to compute analytically a l,l (cid:48) away from resonances[42] a l,l = − l d (2 l − l + 3) , (15)and a l,l − = − l d (2 l − (cid:112) (2 l + 1)(2 l − . (16)We note that the Born approximation can not be used to calculate the term a ss [42].Applying the Born approximation to calculate the high partial wave elements of the K D matrix, the dipolar bound state equation (12) for the bosonic dipoles is simplifiedto a ss = ik − η B l d + η B l d + η B l d σ B + σ B l d + σ B l d , (17)where η B = 2 ik (cid:16) M dd + 77 √ M ds + 11 √ M gd + 15 M gg (cid:17) ,η B = k (cid:16) M ds − M gd + 2 M ds (cid:16) M gd + 6 √ M gg (cid:17) − √ M gd M gs + M dd (7 M gg − M gs − M ss ) (cid:17) ,η B = − ik (cid:0) M ds M gg − M ds M gd M gs + M gd M ss + M dd (cid:0) M gs − M gg M ss (cid:1)(cid:1) , (18)and σ B = − M ss ,σ B = − ik (cid:16) M ds + 11 √ M ds M gs + 15 M gs − (cid:16) M dd + 11 √ M gd + 15 M gg (cid:17) M ss (cid:17) ,σ B = − k (cid:0) M ds M gg − M ds M gd M gs + M gd M ss + M dd (cid:0) M gs − M gg M ss (cid:1)(cid:1) . (19)The dipolar bound state equation for the p -wave dominated fermionic DCIM state,which exists in the vicinity of free space resonance of a pp , is simplified within the Bornapproximation to a pp = ik − η F l d + η F l d + η F l d σ F + σ F l d + σ F l d , (20) ipolar confinement-induced molecular states in harmonic waveguides η F = 2 ik (cid:16) M ff + 429 √ M fp + 65 √ M hf + 385 M hh (cid:17) ,η F = k (cid:16) M fp − M hf + 10 M fp (cid:16) √ M hf + 22 √ M hh (cid:17) − √ M gd M gs + M dd (cid:16) M hh − √ M hp − M pp (cid:17)(cid:17) ,η F = − ik (cid:0) M fp M hh − M fp M hf M hp + M hf M pp + M ff (cid:0) M hp − M hh M pp (cid:1)(cid:1) , (21)and σ F = − M pp ,σ F = − ik M fp + 65 √ M fp M hp + 385 M hp − (1001 M ff + 65 √ M hf + 385 M hh ) M pp ) ,σ F = − k (cid:0) M fp M hh − M fp M hf M hp + M hf M pp + M ff (cid:0) M hp − M hh M pp (cid:1)(cid:1) . (22)Compared to the dipolar bound state equation (12), Eqs. (17) and (20) explicitly revealthe influence of the DDI in determining the energies of DCIM states. The influenceof the waveguide is contained in the two sets of parameters η and σ in Eqs. (17) and(20) which are expressed in terms of M ll (cid:48) . These equations allow us to investigate thedependence of DCIM states on a ss (bosonic dipoles) or a pp (fermionic dipoles) for fixedDDI analytically, which will be studied in the following section.
3. Dipolar confinement induced molecular states
In the following the dipolar confinement induced bound states are investigated. Boththe identical bosonic and fermionic dipoles are considered. In each case, two setsof calculations have been performed. In a first set, the dipole moment d is fixed, C is varied and accordingly the generalized scattering lengths a l,l (cid:48) change. Such asituation can be realized experimentally by tuning the short-range interaction V sr ( r ) viaFeshbach resonances [45] while keeping the DDI unchanged. We consider the boundstate dominated by its lowest partial wave state, i.e. the s -wave dominated bound statefor identical bosonic dipoles or p -wave dominated bound state for identical fermionicdipoles. The variation of the binding energy as a function of a ss or a pp is examined.For the second set we fix C and allow the dipole moment d to vary. This case can beachieved experimentally, for example in the case of electric dipoles, by tuning externalelectric fields. The dependence of the bound state energy on the DDI, more specificallythe dipole length, is studied. ipolar confinement-induced molecular states in harmonic waveguides For bosonic DCIM states and for a fixed DDI strength l d /a ⊥ = 0 . E s bi = E bi / (cid:126) ω ⊥ is shown in the upper panel of Fig. 4 as a function of a ⊥ /a ss . Wenote that the binding energy in waveguides is given by E bi = E th − E b . The numericaldata obtained from the close coupling method are shown as a black solid line. C is varied here in the region where the corresponding LJ potential supports either oneor no bound state in free space. We consider the DCIM states in the energy range0 < E b / (cid:126) ω ⊥ <
1. Once a bound state is determined for a specific C , then the free-space scattering calculation is performed to calculate a ss at the bound state energy.The bound state energy obtained by the LFT approach including one (green dottedline), two (blue dotted-dashed line) and three (red dashed line) partial wave states arealso shown in the upper panel of Fig. 4. In this set of LFT calculations, a ss can betreated as a parameter, and all the other generalized scattering lengths needed in thebound state equations can be calculated within the Born approximation. The LFTapproach including a single partial wave state reproduces the numerical results wellwhen the binding energy E s bi is larger than 0.1. Approaching the scattering thresholdin the waveguide E s bi →
0, there will be a large portion of the bound state wavefunctionspanning over large distances r/a ⊥ > E s bi < . s , d and s , d , g partial wave states (blue dotted-dashedline and red dashed line in Fig. 4), an avoided crossing appears in the binding energycurve in the energy region E s bi close to one, i.e. E b close to zero, which is not observed inthe numerical result and in the LFT approach with s wave state only. The lower panelof Fig. 4 shows magnification in the vicinity of the avoided crossing. The appearance ofthe avoided crossing is attributed to the inaccuracy of the local frame transformationfor higher partial wave states in the energy region E b →
0. To apply the LFT, anintermediate regime is needed where the kinetic energy dominates the interparticleinteraction potential V sr ( r )+ V d ( r ) and the trapping potential V t ( r ). As stated before weassume an intermediate region exists between β /a ⊥ < r/a ⊥ <
1. As shown in Fig. 2, thechannel potentials for higher partial wave states are positive between β /a ⊥ < r/a ⊥ < s -wave state. If the energy E is close to 0, there is no well-defined intermediate regionwhere the LFT can be applied accurately for higher partial waves. This results in theunphysical avoided crossing in the LFT approach including d and g wave states. It isworth noting that in the energy region where the avoided crossing appears for higher ipolar confinement-induced molecular states in harmonic waveguides -15 -10 -5 00.010.11 numerical dataLFT (s)LFT (sd)LFT (sdg) a ⊥ / a ss (a) -20.80.91 numerical dataLFT (s)LFT (sd)LFT (sdg) a ⊥ / a ss (b) i Figure 4. (Upper panel) The scaled binding energy E s bi of the bosonic DCIM stateswith varying a ⊥ a/ ss . The numerical results (black solid line) are shown together withthe LFT calculations including one (green dotted line), two (blue dotted-dashed line)and three (red dashed line) partial wave states. l d /a ⊥ is 0.026. (Lower panel) Azoom-in for the energy region where a spurious avoided crossing appears in the LFTcalculation including high partial wave states. partial wave states, the bound state energy based on the LFT approach with a single s -wave state agrees very well with the numerical results. In addition, these calculationshows that DCIM states also exist in the region a ss <
0. It demonstrates the impactof the confinement on the dipolar system since in free space case dipolar bound statesarise only for a ss > l d /a ⊥ . The bound statespectrum in the waveguide is calculated numerically via the close-coupling method, andthe scaled bound state energy E sb = E b / (cid:126) ω ⊥ is shown in Fig. 5 (black solid line). Thebound state energies based on the LFT approach are shown as red circles. The boundstate energies in free space without the transverse trapping potential are also obtainednumerically and are depicted as blue squares. In the numerical calculation, C s is fixed ipolar confinement-induced molecular states in harmonic waveguides l d /a ⊥ b (a) numerical bound state in waveguidebound state in waveguide based on LFTnumerical bound state in free space l d /a ⊥ b (b) Figure 5. (Upper panel) Black solid line: the numerical scaled energy E sb of thebosonic DCIM states as a function of scaled dipole length l d /a ⊥ . Red circles: scaledenergies of DCIM states determined within the LFT approach. Blue squares: numericalbound state energy of two identical bosonic dipoles in free space. C s = 2 . × − and the LJ potential supports one bound state which is very close to the thresholdin free space. (Lower panel) A magnification in the region of the second bound statefrom the left in the upper panel. to 2.18 × − . The LJ potential supports one bound state which is close to E = 0,the scattering threshold in free space. Varying the coefficient C , the general featuresof E sb as a function of l d /a ⊥ remain the same as those shown in Fig. 5. In the LFTcalculation, the dipolar bound state equation (12) including three partial waves, whichis quite accurate close to the threshold E th , is used to determine the bound state in theenergy region E b / (cid:126) ω ⊥ ∈ (1 / , E b / (cid:126) ω ⊥ ∈ (0 , / ipolar confinement-induced molecular states in harmonic waveguides l d /a ⊥ = 0 .
13 is more sensitive to the variation of the DDI comparedto the other bound states. An analysis of the wavefunction reveals that this is a g -wave dominant bound state, and all the other bound states are dominated by a s -wavecomponent.As shown in Fig. 5, the bound state energy decreases as the DDI strength increases.When the bound state energy is well below the threshold E th , the dipoles are localizedat distances r/a ⊥ (cid:28)
1, and the dipolar bound state wavefunction exponentially vanishesprior to the region where V t ( r ) becomes important. Therefore, in this case the boundstate in waveguides is essentially like a bound state in free space. This situation isexamined in Fig. 5 by comparing the scaled bound state energy in the waveguide (blacksolid line) with the scaled bound state energy in free space (blue squares). In waveguides,the bound state ends at E = (cid:126) ω ⊥ which is the scattering threshold. In contrast, thescattering threshold in free space is E = 0, and hence the free-space bound state existsin the energy region E sb <
0. We note that the DCIM state considered in this workrefers to the bound state in the waveguide and is not necessarily an additional stateinduced by the confinement. It is clearly shown in Fig. 5 that, due to the presenceof the confinement, the DCIM state exists in larger energy region and dipole momentregion compared to the free-space dipolar bound state. As shown in the upper panelof Fig. 5, the bound state energies with and without trapping potential coincide witheach other except for the energy region E sb close to 0. A magnification in the vicinity of l d /a ⊥ = 0 .
055 is shown in the lower panel. The bound state which is close to E = 0 infree space is shifted in the presence of the waveguides. As its energy lowers, this shiftbecomes smaller. When E sb = −
1, i.e. the bound state energy is − (cid:126) ω ⊥ , the effect of thewaveguide is already negligible, and the bound state energies in the waveguide and infree space are almost the same.In the strong DDI regime, where the dipole length l d is comparable to the harmonicoscillator length a ⊥ , the dependence of the scaled bound energy E sb of the DCIM stateson l d /a ⊥ is determined numerically and is depicted in Fig. 6 (black solid line). The LFTapproach can not be applied to calculate the DCIM states in the strong DDI regimesince the length scale separation does not apply. The numerical free-space dipolar boundstate energies are also shown in Fig. 6 (blue squares). In the strong DDI regime, differentpartial wave channels are strongly mixed. As shown in Fig. 6, there are more boundstates which are dominated by higher partial wave components and are sensitive to thevariation of the DDI strength. Moreover, unlike the situation in the weak DDI regime,where the DCIM states are dominated by a single partial wave component, the DCIMstates in the strong DDI regime contain a significant number of different partial wavecomponents of the same order of magnitude. For example, the first DCIM state fromthe left in Fig. 6 contains both s and d wave components significantly. The qualitative ipolar confinement-induced molecular states in harmonic waveguides b l d /a ⊥ -2-101 Figure 6.
Black solid line: The numerically obtained scaled energy E sb of the bosonicDCIM states as a function of the scaled dipole length l d /a ⊥ in the strong DDI regime.Blue squares: the scaled bound state energies in free space. C s = 2 . × − andthe LJ potential supports one bound state which is very close to the threshold in freespace. features of the dependence of the DCIM states on the DDI in the strong DDI regime aresimilar to those in the weak DDI regime. New DCIM states emerge as the DDI strengthincreases. In the following discussion on the fermionic DCIM states, we will focus onthe weak DDI regime. Let us now focus on the fermionic DCIM states. For fixed DDI strength l d /a ⊥ = 0 . a ⊥ /a pp is shown in Fig. 7. The fermionicbound state energy curve has similar features as the bosonic case. As shown in thecorresponding lower panel, the LFT approach using a single partial wave state (greendotted line) deviates from the numerical results (black solid line) only close to thethreshold. By including more partial wave states (red dashed line), the LFT approachprovides a more accurate bound state energy when E s bi tends to zero, and also producesthe spurious avoided crossing when E s bi tends to 1, see the upper panel of Fig. 7. Thedependence of the scaled fermionic bound state energy E s bi on the DDI strength, namely l d /a ⊥ , is provided in Fig. 8. Here C s = 1 . × − and the short range potential V sr supports a p -wave bound state very close to E = 0. Other choices of the value for C lead to a similar behavior. The solid line in Fig. 8 depicts the numerical bound stateenergies. A series of new bound states emerges as the DDI strength increases. Amongthe bound states shown in the upper panel of Fig. 8, the one around l d /a ⊥ = 0 .
095 isa l = 5-wave dominant bound state, and the other states are p -wave dominated. Allthese bound states are more sensitive to the variation of the DDI strength compared tothe s -wave dominant bound state shown in Fig. 5. This can be understood as follows.The potential matrix element V l,ld ( r ) vanishes for the s -wave channel, and is nonzero ipolar confinement-induced molecular states in harmonic waveguides numericalLFT (p)LFT (pfh) a ⊥ / a pp (a) -10.010.1 numericalLFT (p)LFT (pfh) a ⊥ / a pp -0.5 (b) Figure 7. (Upper panel) The scaled binding energy E s bi of the fermionic DCIM statesas a function of a ⊥ /a pp . The numerical results (black solid line) are shown togetherwith the LFT results including one (green dotted line) and three (red dashed line)partial wave states. l d /a ⊥ is 0.026. (Lower panel) A magnification in the energyregion close to the threshold. for l > l > l d /a ⊥ = 0 . E sb →
0. The reasonfor this is that there are centrifugal potential barriers for all the channels, includingthe lowest p -wave channel. The potential barrier tends to constrain the bound statewavefunction in the short-range region where the trapping potential is negligible. This ipolar confinement-induced molecular states in harmonic waveguides l d /a ⊥ b (a) l d /a ⊥ b numerical bound state energy in waveguidesbound state energy in waveguides based on LFTnumerical bound state energy in free space (b) Figure 8. (Upper panel) Black solid line: the numerical scaled energy E sb ofthe fermionic DCIM states as a function of the scaled dipole length l d /a ⊥ . Redcircles: scaled energies of the DCIM states obtained from the LFT approach. Bluesquares: numerical bound state energy of two identical fermionic dipoles in free space. C s = 1 . × − and the LJ potential supports one p -wave bound state which is veryclose to the threshold in free space. (Lower panel) A magnification of the region of thethird bound state from the left. is clearly shown in Fig. 9 where the partial wave PD of a bound state at E sb = 0 inthe waveguide is depicted. The bound state barely feels the trapping potential, and isalmost a bound state in free space.
4. Conclusions
We have investigated the dipolar confinement induced molecular (DCIM) states inharmonic waveguides. Identical bosonic and fermionic dipoles are considered. In theweak DDI regime, in which the dipole length is smaller than the harmonic oscillatorlength, the local frame transformation (LFT) approach are utilized to connect the ipolar confinement-induced molecular states in harmonic waveguides -9 -6 -3 l=1l=3l=5l=7 r/a ⊥ p r ob a b ilit y d e n s it y Figure 9.
The partial wave probability densities of the bound state at E sb = 0 fortwo fermionic dipoles in the waveguide for l d /a ⊥ = 0 . bound state in waveguides with the scattering properties in free space analytically.By examining the numerical partial wave probability densities of the DCIM states, weshow that length scale separation exists in the weak DDI regime which is crucial for theapplication of the LFT approach.The LFT dipolar bound state equation is given. The bound state energies calculatedvia LFT approach are compared with the numerical ones. Since both DDI and thetrapping potential couple different partial wave states, one expects that multiple partialwave states are involved in the LFT approach. Indeed, close to the scattering thresholdin waveguides E = (cid:126) ω ⊥ , the LFT approach including the lowest partial wave state failsto provide accurate bound state energies, and higher partial wave states are needed.However, when E tends to zero, the bound state energies based on the LFT approachincluding higher partial wave states deviate from the numerical ones. The reason isthat one can not find an intermediate region for higher partial wave channels wherethe kinetic energy is significantly larger than the interparticle interaction potential andthe trapping potential. The local frame transformation in such a case is less accurate,and this results in spurious avoided crossings. Nevertheless, one can still use the LFTapproach in this energy region since the single partial wave approximation is validaccording to the comparison with the numerical calculations.The dependence of the DCIM states on the DDI has been investigated, and both theweak and strong DDI regimes have been studied. As the DDI strength increases, a seriesof DCIM states emerges. The s -wave dominated DCIM states are less sensitive to thevariation of the DDI strength as compared to the higher partial wave ( l >
0) dominatedDCIM states. This is due to the fact that the matrix elements of the DDI potentialvanish for the s -wave channel and are nonzero for l >
0. The l > (cid:126) ω ⊥ . Forthe fermionic case, the centrifugal potential barrier in the channel potentials localizes ipolar confinement-induced molecular states in harmonic waveguides
5. Acknowledgments
G. W. acknowledges a fellowship from the Alexander von Humboldt Foundation. P. G.acknowledges financial support by the NSF through grant PHY-1306905 and from theMax-Planck Institute for the Physics of Complex Systems in Dresden. G. W. thanks V.Melezhik for fruitful discussions.
6. Appendix
For identical bosons, s , d and g wave states are involved, and the explicit expressionsfor M l,l (cid:48) read as follows M ss = − iζ (cid:2) , − (cid:15) (cid:3) (cid:113) + (cid:15) , (23) M ds = −√ iζ (cid:2) − , − (cid:15) (cid:3) (cid:0) + (cid:15) (cid:1) / + iζ (cid:2) , − (cid:15) (cid:3) (cid:113) + (cid:15) , (24) M dd = 5 − iζ (cid:2) − , − (cid:15) (cid:3) (cid:0) + (cid:15) (cid:1) / − iζ (cid:2) − , − (cid:15) (cid:3) (cid:0) + (cid:15) (cid:1) / − iζ (cid:2) , − (cid:15) (cid:3) (cid:113) + (cid:15) , (25) M gs = 3 − iζ (cid:2) − , − (cid:15) (cid:3) (cid:0) + (cid:15) (cid:1) / − iζ (cid:2) − , − (cid:15) (cid:3) (cid:0) + (cid:15) (cid:1) / − iζ (cid:2) , − (cid:15) (cid:3) (cid:113) + (cid:15) , (26) M gd = − √ iζ (cid:2) − , − (cid:15) (cid:3) (cid:0) + (cid:15) (cid:1) / + 125 iζ (cid:2) − , − (cid:15) (cid:3) (cid:0) + (cid:15) (cid:1) / + 39 iζ (cid:2) − , − (cid:15) (cid:3) (cid:0) + (cid:15) (cid:1) / + 3 iζ (cid:2) , − (cid:15) (cid:3) (cid:113) + (cid:15) , (27) M gg = 9 (cid:32) − iζ (cid:2) − , − (cid:15) (cid:3) (cid:0) + (cid:15) (cid:1) / − iζ (cid:2) − , − (cid:15) (cid:3) (cid:0) + (cid:15) (cid:1) / − iζ (cid:2) − , − (cid:15) (cid:3) (cid:0) + (cid:15) (cid:1) / − iζ (cid:2) − , − (cid:15) (cid:3) (cid:0) + (cid:15) (cid:1) / − iζ (cid:2) , − (cid:15) (cid:3) (cid:113) + (cid:15) , (28)where (cid:15) = E − (cid:126) ∗ ω ⊥ (cid:126) ω ⊥ , and E is the total energy. ζ ( a, s ) is the Hurwitz zeta function.For identical fermions, p , f and h waves are involved in the LFT calculation. The ipolar confinement-induced molecular states in harmonic waveguides M l,l (cid:48) read then M pp = 3 iζ (cid:2) − , − (cid:15) (cid:3) (cid:0) + (cid:15) (cid:1) / , (29) M fp = −√ (cid:32) − iζ (cid:2) − , − (cid:15) (cid:3) (cid:0) + (cid:15) (cid:1) / − iζ (cid:2) − , − (cid:15) (cid:3) (cid:0) + (cid:15) (cid:1) / (cid:33) , (30) M ff = 7 (cid:32) iζ (cid:2) − , − (cid:15) (cid:3) (cid:0) + (cid:15) (cid:1) / + 15 iζ (cid:2) − , − (cid:15) (cid:3) (cid:0) + (cid:15) (cid:1) / + 9 iζ (cid:2) − , − (cid:15) (cid:3) (cid:0) + (cid:15) (cid:1) / (cid:33) , (31) M hp = √ (cid:32) iζ (cid:2) − , − (cid:15) (cid:3) (cid:0) + (cid:15) (cid:1) / + 35 iζ (cid:2) − , − (cid:15) (cid:3) (cid:0) + (cid:15) (cid:1) / + 15 iζ (cid:2) − , − (cid:15) (cid:3) (cid:0) + (cid:15) (cid:1) / (cid:33) , (32) M hf = −√ (cid:32) − iζ (cid:2) − , − (cid:15) (cid:3) (cid:0) + (cid:15) (cid:1) / − iζ (cid:2) − , − (cid:15) (cid:3) (cid:0) + (cid:15) (cid:1) / − iζ (cid:2) − , − (cid:15) (cid:3) (cid:0) + (cid:15) (cid:1) / − iζ (cid:2) − , − (cid:15) (cid:3) (cid:0) + (cid:15) (cid:1) / (cid:33) , (33) M hh = 11 (cid:32) iζ (cid:2) − , − (cid:15) (cid:3) (cid:0) + (cid:15) (cid:1) / + 2205 iζ (cid:2) − , − (cid:15) (cid:3) (cid:0) + (cid:15) (cid:1) / + 3395 iζ (cid:2) − , − (cid:15) (cid:3) (cid:0) + (cid:15) (cid:1) / + 525 iζ (cid:2) − , − (cid:15) (cid:3) (cid:0) + (cid:15) (cid:1) / + 225 iζ (cid:2) − , − (cid:15) (cid:3) (cid:0) + (cid:15) (cid:1) / (cid:33) . (34) [1] Moritz H, St¨oferle T, G¨unter K, K¨ohl M and Esslinger T 2005 Phys. Rev. Lett. , 210401[2] Sala S, Z¨urn G, Lompe T, Wenz A N, Murmann S, Serwane F, Jochim S and Saenz A 2013 Phys.Rev. Lett. , 203202[3] Martiyanov K, Makhalov V and Turlapov A 2010
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