Particle scattering by harmonically trapped Bose and Fermi gases
PParticle scattering by harmonically trapped Bose and Fermi gases
Ankita Bhattacharya , , Samir Das , and Shyamal Biswas ∗ School of Physics, University of Hyderabad, C.R. Rao Road, Gachibowli, Hyderabad-500046, India Present Address: Institute of Theoretical Physics, TU Dresden, 01069 Dresden, Germany (Dated: February 19, 2018)We have analytically explored the quantum phenomenon of particle scattering by harmonicallytrapped Bose and Fermi gases with the short ranged (Fermi-Huang δ p [1]) interactions among theincident particle and the scatterers. We have predicted differential scattering cross-sections andtheir temperature dependence in this regard. Coherent scattering even by a single boson or fermionin the finite geometry gives rise to new tool of determining energy eigenstate of the scatterer.Our predictions on the differential scattering cross-sections, can be tested within the present dayexperimental setups, specially, for (i) 3-D harmonically trapped interacting Bose-Einstein condensate(BEC), (ii) BECs in a double well, and (iii) BECs in an optical lattice. PACS numbers: 03.65.Nk, 67.85.-d, 03.65.-w
I. INTRODUCTION
In the existing literature, quantum scattering theoryis discussed both for classical scatterers (which are ei-ther fixed or having classical motions in space [2]) andquantum scatterers e.g. quantum scattering by atoms,molecules, nuclei, etc [3]. ‘Particle’[47] can be scatteredcoherently from each and every point of the region ofspace of the quantum scatterer if it is fired onto the re-gion, and carries information about the state of the scat-terer after being scattered. There are some theoreticaldiscussions on quantum scattering for unfixed quantumscatterer(s) bounded in a region of space, e.g. diffrac-tion of atoms from a standing-wave Schrodinger field [4],scattering of slowly moving atoms by a 3-D harmoni-cally trapped BEC within Bogoliubov-de Gennes formal-ism [5], particle scattering by a weakly interacting BEC[6–11], transport of atoms across interacting BECs in a1-D optical lattice [12], a nondestructive method to probea complex quantum system using multi-impurity atomsas quantum probes [13], particle scattering by quantumscatterers in restricted geometries [14], etc .In none of the previous works, related to the quan-tum scatterers, temperature dependence of the scatter-ing amplitude or that of the differential scattering cross-section was studied except that for scatterer(s) in boxgeometries or in array of boxes [14] and for scatterersin 3-D harmonic trap in the thermodynamic limit [8].Thus, we naturally take up discussion on quantum scat-tering to introduce quantum scattering with quantizedmotions of the scatterers in thermal equilibrium in finitegeometries of harmonic trap as probe for Fermi-Huang δ p [1] interactions (among the ‘incident’ particle and thescatterers), which although are easy to deal with, havehuge applications in the field of ultra-cold atoms [15, 16]. ∗ Electronic address: sbsp [at] uohyd.ac.in
We are specially interested in temperature dependenceof differential scattering cross-section for scatterers inthe harmonically trapped geometry in this regard, as be-cause, thermodynamic properties of ultra-cold gases inharmonic traps are of growing interest [16–19].If a plane wave ( e ikz ) associated with a free particle(‘particle’) of a given momentum ( p = (cid:126) k ˆ k ) is scatteredby a fixed scatterer (at r = 0) with an interacting po-tential ( V int ( r )), then a spherical wave ( e ikr r ) goes outof the scatter with a scattering amplitude ( f ( θ, φ )) to aparticular direction ( θ and φ ) with respect to the ini-tial direction of incidence (ˆ k ). If the scatterer is notfixed, say, the scatterer is a particle in a 1-D simple har-monic oscillator ( −∞ < x < ∞ ), then, according tothe superposition principle, the ‘particle’ would be scat-tered coherently from all the positions ( { x } ) with therespective probability density | ψ n ( x ) | , where ψ n ( x )( n = 0 , , , .... ) is the normalized energy eigenstate ofthe scatterer. In this situation, spherical waves ( e ikr (cid:48) r (cid:48) )go out after scattering form all the source (of scattering)points ( { x } ). All the outgoing spherical waves ( { e ikr (cid:48) r (cid:48) } )interfere, at a distance r = x + r (cid:48) to a particular direc-tion ( θ, φ ) from the center of the oscillation, with differ-ent phases and give rise to a coherent scattering ampli-tude f n ( θ, φ ) which now depends on the quantum state( | ψ n > ) of the scatterer. Small angle neutron scatter-ing by quantum dots was investigated by Pinero et al without precisely probing quantized motions of the scat-terers in them [20]. Particle scattering by coherent mediawas also studied experimentally by Chikkatur et al [21]and Bromley et al [22]. While Bromley et al did notprobe quantized motions of the scatterers in the densemedium, Chikkatur et al , could probe quantized motionof the scatterers, to a certain extent, in a BEC; thoughthey did not probe angular dependence of the scatteringamplitude. However, electron scattering by harmonicallytrapped BEC [23] and Fermi gas [24] was studied theo-retically for T →
0. Although temperature dependence a r X i v : . [ qu a n t - ph ] F e b in particle scattering by a BEC was studied by Montina[8], he considered quantized motions of the bosonic scat-terers in the thermodynamic limit within Thomas-Fermiapproximation. About light scattering by a BEC or byultracold atoms in optical trap, the experimental work ofSchneble et al in Ref. [25], the recent theoretical worksof Ezhova et al in Ref. [26], Zhu et al in Ref. [27] andKozlowski et al in Ref. [28], the review work of Mekhovand Ritsch in Ref. [29], and the references therein, arequite interesting. Above all, temperature dependence ofthe differential scattering cross-section, for particle scat-tering with quantized motion(s) of the scatterer(s) in har-monic/optical trap, has not been studied so far.This article begins with revisiting of the quantum the-ory of particle scattering for a fixed (classical) scattererwith Fermi-Huang potential (i.e. regularized δ poten-tial: V int ( r ) = gδ p ( r ) = gδ ( r ) ∂∂r r ). Then we havegeneralized the theory for quantum scatterer(s) in re-stricted geometries, in particular, for bosonic/fermionicscatterer(s) in a (i) 1-D harmonic trap, (ii) 2-D harmonictrap, and (iii) 3-D harmonic trap. Then we have cal-culated the scattering amplitudes, and have plotted thedifferential scattering cross-sections for all the cases. Wealso have investigated temperature dependence of the dif-ferential scattering cross-sections for the above cases, andspecially emphasized on the differential cross-section forparticle scattering by BEC(s) in the 3-D harmonic trap[30], double well trap, and the optical lattice. II. PARTICLE SCATTERING BY A SINGLESCATTERER IN A HARMONIC TRAP
In quantum scattering theory of particle scattering wedeal with the time independent Schrodinger equation (cid:18) − (cid:126) m ∇ + V int ( r ) (cid:19) ψ ( r ) = Eψ ( r ) . (1)Scattering of an incident particle of mass m and the givenmomentum (cid:126) k ˆ k , is recast, as scattering of a ‘particle’ (i.e.scattering of the plane wave ψ in ≡ e ikz ) by the inter-acting potential V int ( r ), into an outgoing spherical wave ψ out ≡ e ikr r . General form of the solution to Eqn.(1), inthe radiation zone, takes the form [2] ψ ( r ) = ψ ( r, θ, φ ) (cid:39) A (cid:20) e ikz + f ( θ, φ ) e ikr r (cid:21) , (2)where | A | is proportional to the intensity of the incident‘particle’. From this information, we can find out thescattering (probability) amplitude ( f ( θ, φ )) of the out go-ing spherical wave to a particular direction ( θ, φ in usualconvention) with respect to the direction of the incidence.The scattering amplitude, for V int ( r ) = gδ p ( r ), with allorders of the Born series, takes the form [16, 31–33] f ( θ, φ ) = − mg π (cid:126) (1 + ik mg π (cid:126) ) . (3) We have considered the scatterer to be fixed for Eqn.(3).If the scatterer is not fixed, rather having a relative mo-tion with the incident ‘particle’ keeping the interactingpotential unaltered, then the scattering amplitude wouldtake the form f ( θ, φ ) = − ¯ µg π (cid:126) (1 + ik ¯ µg π (cid:126) ) , (4)where ¯ µ = mMm + M is the reduced mass and M is the massof the scatterer [16]. The scattering amplitude is inde-pendent of θ and φ for low energy scattering, so that,s-wave scattering length can be conveniently defined, forlow energy scattering, as a s = lim k → − f ( θ, φ ). Thus,we quantify the coupling constant, as g = π (cid:126) a s ¯ µ . A. For a single scatterer in a 1-D harmonic trap If r be the position of the incident particle, such thatthe center of the trapped potential ( r = ) is the origin,then the δ p interaction between the incident particle at r and the scatterer at x ˆ i can be expressed as V int ( r ) = gδ p ( r − x ˆ i ) . (5)Eqn.(3) can be recast for this problem by using Eqn.(5)as f ( θ, φ ) = − mg k π (cid:126) e i ( k − k (cid:48) ) · x ˆ i = − mg k π (cid:126) e − i k (cid:48) · x ˆ i (6)where g k = g ika s m/ ¯ µ . Eqn.(6) is correct only if thescattering has happened only from r = x ˆ i , and will notbe correct if x ˆ i is not a fixed point.Although we can consider Eqn.(6) where there is a rel-ative motion (between the particle and scatterer) by re-placing mass of the ‘particle’ by the reduced mass, yetthe scatterer is still classical as we have not quantizedthe motion of the scatterer. Let us now consider quan-tized motion of the scatterer(s) into the theory of quan-tum scattering, and begin with the scatterer as a parti-cle in a 1-D harmonic trap potential V ( r ) = M ω x x where ω x is the angular frequency of oscillations, M )is the mass and r = x ˆ i is the position of the scat-terer such that −∞ < x < ∞ . We again considerthe scatterer to scatter the incident ‘particle’, Ae ikz , bythe interacting potential V int ( r ) = gδ p ( r − x ˆ i ). Thoughthe incident ‘particle’ and the scatterer are charge-less,they are distinguished by their mass and spin. The in-cident ‘particle’ is having zero spin, and it does not feelthe trap potential except the contact potential with thescatterer. The scatterer is of nonzero spin (and mag-netic moment), so that, it can be trapped by an in-homogeneous magnetic field with the potential energy V ( r ) = M ω x x [16]. Scattering amplitude, if the scat-terer is fixed at r = x ˆ i , would be the same as that in π π π π θ D n x ( θ , ϕ )/ s Solid -> n x =
0, Dashed -> n x =
1, Dotted -> n x = ϕ = ( π ) in the right ( left ) side FIG. 1: Intensity distribution ( D n x ( θ, φ ) = | f n x ( θ, φ ) | ) alonga line parallel to the x -axis for scattering of a ‘particle’ ( e ikz )by the 1-D harmonic oscillator along x -axis for a s as unitlength, ka s = 5, l x /a s = 1, and m/M = 0 .
1. Plots followfrom Eqn.(8).
Eqn.(6) as f ( θ, φ ) = − mg k π (cid:126) e − i k (cid:48) · x ˆ i . Normalized energyeigenstate of the scatterer, corresponding to the energyeigenvalue E n x = ( n x + 1 / (cid:126) ω x , can be written, as [2] ψ n x ( x ) = (cid:115) √ πl x √ n x n x ! H n x ( x /l x ) e − x / l x , (7)where l x = (cid:112) (cid:126) /M ω x is the confining length scale of thescatterer, and H n x ( x /l x ) is the Hermite polynomial ofdegree n x = 0 , , , ... . Now, the quantum scattering ishappening from all the points −∞ < x < ∞ simulta-neously with respective probability density {| ψ n ( x ) | } .Thus, the scattering amplitude for the scatterer in thequantum state | ψ n > , can be written, using Eqn.(6), as f n x ( θ, φ ) = − mg k π (cid:126) (cid:90) ∞−∞ e − i k (cid:48) · x ˆ i | ψ n x ( x ) | dx = − mg k π (cid:126) e − q x l x L n x (2 q x l x ) , (8)where q x = k sin( θ ) cos( φ ) / L n x (2 q x l x ) is the Laguerre polynomial of degree n x [34].We show the profile of the differential scattering cross-section ( D n x ( θ, φ ) = | f n x ( θ, φ ) | ) for the 1-D case in FIG.1 for different quantum numbers. B. For a single scatterer in a 2-D harmonic trap
For 2-D case, the trap potential would be V ( r ) = M ω x x + M ω y y where ω y is the angular frequencyof oscillations along y direction, and r = x ˆ i + y ˆ j is theposition of the scatterer such that −∞ < y < ∞ . Weagain consider, that, the scatterer to scatter the incident ‘particle’, Ae ikz , by the interacting potential V int ( r ) = gδ p ( r − x ˆ i − y ˆ j ). Thus, scattering amplitude, for the2-D case, would be, in the separable form f n x ,n y ( θ, φ ) = − mg k π (cid:126) e − q x l x − q y l y L n x (2 q x l x ) L n y (2 q y l y )(9)where ψ n x ,n y ( x , y ) is the normalized energy eigenstateof the scatterer with energy eigenvalue E n x ,n y = ( n x +1 / (cid:126) ω x + ( n y + 1 / (cid:126) ω y and n y = 0 , , , ... . C. For a single scatterer in a 3-D harmonic trap
Above generalization, however, is not obvious for thescatterer in a 3-D harmonic trap potential V ( r ) = M ω x x + M ω y y + M ω z z as because we furtherhave to consider momentum transfer mechanism for themotion of the scatterer along the z direction since theincident ‘particle’ has momentum only along the z direc-tion. For this reason, generalization Eqn.(6), for an ar-bitrary fixed position r = x ˆ i + y ˆ j + z ˆ k in 3-D, wouldbe f ( θ, φ ) = − mg k π (cid:126) e − i k (cid:48) · ( x ˆ i + y ˆ j )+ i ( k − k (cid:48) ) · z ˆ k . Thus, 3-Dgeneralization of Eqn.(9) would be in the separable form f n x ,n y ,n z ( θ, φ ) = − mg k π (cid:126) (cid:90) e − i k (cid:48) · ( x ˆ i + y ˆ j )+ i ( k − k (cid:48) ) · z ˆ k ×| ψ n x ,n y ,n z ( x , y , z ) | d r = − mg k π (cid:126) e − q x l x − q y l y − ¯ q z l z × L n x (2 q x l x ) L n y (2 q y l y ) L n z (2¯ q z l z ) , (10)where ψ n x ,n y ,n z ( x , y , z ) is the normalized energyeigenstate of the quantum scatterer in the 3-D harmonictrap with energy eigenvalue E n x ,n y ,n z = ( n x + 1 / (cid:126) ω x +( n y + 1 / (cid:126) ω y + ( n z + 1 / (cid:126) ω z , n z = 0 , , , ... , ω z is theangular frequency of oscillation of the scatterer along the z direction, l z = (cid:112) (cid:126) /M ω z , and ¯ q z = − k (1 − cos θ ) / − k sin ( θ/
2) which acts like an obliquity factor. Differ-ential scattering cross-section for the 3-D harmonic scat-terer, can be obtained from Eqn.(10), as D n x ,n y ,n z ( θ, φ ) = (cid:12)(cid:12)(cid:12)(cid:12) mg k π (cid:126) e − q x l x − q y l y − ¯ q z l z × L n x (2 q x l x ) L n y (2 q y l y ) L n z (2¯ q z l z ) (cid:12)(cid:12)(cid:12)(cid:12) . (11)Eqn.(11) though goes beyond the first Born approxima-tion, it is fully consistent (for ka s (cid:28)
1) with the re-sult obtained by Bodefeld and Wilkens after truncatingthe Lippmann-Schwinger equation to the level of the firstBorn approximation [4]. Averaging over the position ofthe scatterer in Eqn. (10) (and that in the precedingtwo as well) is justified by the fundamental principle ofsuperposition[48], that, if we do not know the initial po-sition of the scatterer rather know only its energy eigen-state | ψ n x ,n y ,n z > , then the scattering takes place fromall the points { r } of the scatterer with the respectiveprobability densities {| ψ n x ,n y ,n z ( x , y , z ) | } . We areconsidering the energy eigenstate | ψ n x ,n y ,n z > to be un-altered in the process of scattering. Energy eigenstatewould change in the process of inelastic scattering [4].We will discuss about the reasons in the concluding sec-tion to justify less probability of the inelastic scatteringin the context of thermal and many-body effects [5]. III. PARTICLE SCATTERING BY BOSE ANDFERMI GASES IN THERMODYNAMICEQUILIBRIUM IN 3-D HARMONIC TRAPS
Let us now consider N identical ideal scatterers in the3-D harmonic trap [16, 17]. Above expression in Eqn.(10)can be generalized for these scatterers, all of which scat-ter the incident ‘particle’ ( Ae ikz ) by the same delta po-tential ( V int ( r ) = (cid:80) Nj =1 gδ p ( r − r j )), as f n , n ,..., n N ( θ, φ ) = − mg k π (cid:126) e −|| ¯ q · l || n j = N (cid:88) n j =1 L n jx (2 q x l x ) × L n jy (2 q y l y ) L n jz (2¯ q z l z ) , (12)where n j = ( n jx , n jy , n jz ) represents quantum numberscorresponding the energy eigenstate of the j th oscillator,and || ¯ q · l || = q x l x + q y l y + ¯ q z l z . Eqn.(12), however, is ap-plicable not only for distinguishable scatterers, but alsofor Bose and Fermi scatterers as all the energy eigenstatesare orthogonal.Let us now consider the ideal scatterers in thermody-namic equilibrium with its surroundings at temperature T and chemical potential µ . Scattering amplitude for thescatterers would now depend upon the temperature andchemical potential, and can be written, as¯ f T ( θ, φ ) = − mg k π (cid:126) e −|| ¯ q · l || ( ∞ , ∞ , ∞ ) (cid:88) n =(0 , , ¯ n n L n x (2 q x l x ) × L n y (2 q y l y ) L n z (2¯ q z l z ) , (13)where ¯ n n = e ( E n − µ ) /kBT ∓ represents no. of scat-terers in the single-particle quantum state ψ n ( r ) = ψ n x ,n y ,n z ( x , y , z ) for Bose ( − ) or Fermi (+) scatterers,and E n = E n x ,n y ,n z = ( n x + 1 / (cid:126) ω x + ( n y + 1 / (cid:126) ω y +( n z + 1 / (cid:126) ω z . Eqn.(13) is our prediction for the scat-tering amplitude for a harmonically trapped ideal Boseor Fermi gas at any temperature. For a single particle,¯ n n in Eqn.(13) can be replaced by the Boltzmann proba-bility P n = e − E n /k B T /Z where Z = (cid:80) n e − E n /k B T is thepartition function. We show temperature dependence of¯ D T ( θ, φ ) = | ¯ f T ( θ, φ ) | for a single particle in FIG. 2. Wealso show its statistics dependence in the FIG. 2 (inset).For T →
0, all ( N ) the Bose scatterers occupy theground state. Differential scattering cross-section, in this π π π π θ D T ( θ ,0 )/ s Dashed: T -> = - K, Solid : T -> ∞ ; k s = l x = l y = l z = s ; ω = Hz, s = π π π π θ D T ( θ ,0 )/ N s Bose, Fermi & Cl. [ N = , T -> ] FIG. 2: Intensity distribution for scattering of a ‘particle’( e ikz ) by a 3-D isotropic harmonic oscillator for a s as unitlength and m/M = 0 .
1. Plots follow from the right hand sideof Eqn.(13) with ¯ n n replaced by P n . Dashed, dotted and solidlines correspond to D T ( θ,
0) for T → T → − K, and T →∞ , respectively. In the inset, the parameters remain same,except the temperature and no. scatterers. Dashed, dottedand solid lines in the inset are linked to Bose gas (Eqn.(14)),Fermi gas (Eqn.(16)), and classical scatterers (Eqn.(13) for n → ∞ limit) in the 3-D isotropic harmonic trap. situation, takes the form, from Eqn.(13), as¯ D T → ( θ, φ ) = | ¯ f T → ( θ, φ ) | = | N a k | e − || ¯ q · l || , (14)where a k = mg k π (cid:126) = a s m/ ¯ µ ika s m/ ¯ µ . We plot Eqn.(13) inFIG. 4(b) for relevant values of parameters. Eqn.(14)leads to the scattering cross-section, for k →
0, as σ = (cid:90) π dθ (cid:90) π dφ ¯ D T → ( θ, φ ) sin θ = 4 π | N a s m/ ¯ µ | . (15)On the other hand, for T →
0, all the ( N ) Fermi scat-terers (of the same spin component, say spin up) willoccupy the first N single particle states. Thus, for large N and isotropic case, modulus squared of the r.h.s. ofEqn.(13), takes the form[49], for harmonically trappedFermi gas as¯ D T → ( θ, φ ) = | N a k | e − || ¯ q · l || N / . (16)For classical scatterers ( n x , n y , n z → ∞ ), in contrary tothe above, the scattering amplitude in Eqn.(13) would beinfinitely narrow[50] as shown in FIG. 2 (inset). A. Weak interparticle interactions and finite sizeeffects for Bose scatterers in a 3-D harmonic trap
Temperature dependence of the scattering amplitudecomes from the triple summation in Eqn.(13). The sum-mation, in the thermodynamic limit, followed by the Tay-lor expansions of the Laguerre polynomials about k = 0with q = ( q x + q y + ¯ q z ), ω x = ω y = ω z = ω , ¯ l = (cid:112) (cid:126) /M ω , t = k B T (cid:126) ω , z = e µ/k B T and N = t Li ( z ) takes theform S = t Li ( z ) − q ¯ l t Li ( z ) + ( q ¯ l ) [12 t Li ( z ) +3 t Li ( z )] + O ( q ) for the Bose gas above the condensa-tion point ( T > T c = (cid:126) ωk B [ N/ζ (3)] / )[51]. For T < T c ,similar form also appears with non-condensate fraction( t/t c ) where t c = k B T c (cid:126) ω . Condensate part shows tem-perature dependence only in the form of the condensatefraction N N = 1 − ( t/t c ) . Temperature dependence of¯ D T ( θ, φ ) = | ¯ f T ( θ, φ ) | , for the Bose gas, with the ap-propriate temperature dependence of the chemical po-tential [35], is shown in FIG. 3. For Fermi gas, onlychange would be the replacement of the Bose-Einsteinintegrals ( Li j ( z )) by the Fermi integrals ( − Li j ( − z )) ∀ j .For anisotropic trap, forms of the bulk quantities aremostly unaltered with the replacement ω = ( ω x ω y ω z ) / .For the finite size of the trap and weak inter-scattererinteractions ( π (cid:126) ˜ a s M (cid:80) i,j T c . However, effect of the interactions,for T →
0, may not necessarily be perturbative, and canbe better described within Thomas-Fermi approximation[5].In the FIG. 4(b) we show the scaling result for the an-gular dependence ( θ ) of the differential scattering cross-section for the weakly interacting case of the BEC for T →
0, and finite temperature and size effects over thisresult within the Hartree-Fock approximation accordingto the prescription described above. It is quite clear fromthe plots in the FIG. 4(b), that repulsive interactions leadto narrowing down the profile of the differential scatter-ing cross-section around θ = 0 as the condensate broad-ens up around θ = 0. This is quite natural, as because,the scattering amplitude for the extended object (BEC)is Fourier decomposed at all the source points of scat-tering. However, if temperature increases, probability ofexcited states being occupied by the scatterers increases,which in turn increases probability of scattering to somelarger angles like that shown in FIG. 1. Thus, increaseof temperature leads to large angle scattering. However,coherency get reduced if scatterers are found in different TT c D T ( θ ,0 )/ N s Dotted: θ =
0, Dashed: θ = π /
2, Solid : θ = π ; k s = l x = l y = l z = s = l , ω = Hz, N = TT c D T ( π ,0 )/ N s ( b ) : H - F result 0.2 0.4 0.6 0.8 1.0 TT c ℓ ˜/ l ( a ) : For ˜ s / l = FIG. 3: Temperature dependence of the differential scatteringcross-section along the forward ( θ = 0, dotted line), perpen-dicular ( θ = π/
2, dashed line) and backward ( θ = π , solidlines) directions for 3-D harmonically trapped isotropic idealBose gas for the relevant parameters as mentioned above.Plots follow from Eqn.(13) for m/M = 0 .
1. Inset-a repre-sents finite temperature scaling (¯ l → ˜ (cid:96) ) of ¯ l for the samesystem (within the 4th order in ˜ (cid:96)/ ¯ l − π (cid:126) ˜ a s M with˜ a s = 90 a = 0 . l for Rb atoms [39]. Dotted line in theinset-b represents finite size and inter-scatterer effects withinthe H-F approximation over the solid line which also repre-sents backward scattering in the main figure. energy eigen states other than the ground state at a finitetemperature. It results reduction of the scattering cross-section with the increase of temperature. This is true ingeneral. This is also apparent in FIG. 2 both for idealBose and Fermi scatterers in harmonic traps. We willalso investigate the same for interacting BECs in othertrapped geometries like double-well trap and optical lat-tice trap.
IV. PARTICLE SCATTERING BY BOSESCATTERS IN OTHER 3-D OPTICAL TRAPSA. For Bose scatterers in a double-well potential
Let us now consider an ideal gas of N + N Bose scatter-ers in a 3-D double-well potential V ( r ) = − M ω x x / M ω x x / d + 2 M ω y y / M ω z z /
2, such that, fre-quency of oscillation is the same as that in the previouscase, and the minima of double-well are separated along x -axis by a distance d [16, 40]. In thermodynamic equilib-rium, for T →
0, all the particles condense to the groundstate. Within the tight-binding approximation (which isvery good for d (cid:29) l x ), there would be two distinct con-densates of N scatterers in each well, such that each ofthe condensates scatters the incident ‘particle’ ( Ae ikz )like that in Eqn.(14). However, net scattering amplitudewould be the superposition of the scattering amplitudescorresponding to the individual condensate as the setupis analogue of the double slit experiment [16, 41]. Thus,scattering from the two condensates would interfere, as¯ D T → ( θ, φ ) = | N a k | e − || ¯ q · l || (cid:20) (cid:0) πd sin( θ ) λ (cid:1)(cid:21) . (17)Here we did not consider any Josephson oscillation as d (cid:29) l x [42, 43]. We plot the differential cross-sectionin FIG. 4(c) for relevant values of parameters. In thesame figure we further present scaling results for weaklyinteracting Bose scatterers in the double-well trap wellbelow the condensation point and finite temperature andsize effects within the Hartree-Fock approximation on topof the tight binding approximation in a similar way asprescribed in the previous section for the Bose scatterersin the harmonic trap. B. For Bose scatterers in a 1-D optical lattice
Let us now consider N (cid:48) x -axis by the lattice spacing d . Entire system is in thermodynamic equilibrium. For T →
0, all the condensates have the same ( N ) number ofparticles. So, the system essentially is a 1-D grating of 3-D condensates. Within the tight-binding approximation,there would be N (cid:48) distinct condensates of N scatterers ineach well, such that each of the condensates scatters theincident ‘particle’ ( Ae ikz ) like that in Eqn.(14) [16, 45].However, net scattering amplitude would be the super-position of the scattering amplitudes corresponding tothe individual condensate as the setup is now analogueof the 1-D grating experiment. Thus, scattering from the N (cid:48) condensates would interfere, as¯ D T → ( θ, φ ) = | N a k | e − || ¯ q · l || (cid:20) sin( N (cid:48) πd sin( θ ) λ )sin( πd sin( θ ) λ ) (cid:21) . (18)Since d (cid:29) l x , Eqn.(18) is good for the Mott insula-tor phase of the condensates. We plot ¯ D T → ( θ, φ ) inFIG. 4(d) for relevant values of parameters. In the samefigure we further present scaling results for weakly in-teracting Bose scatterers in the optical lattice trap wellbelow the condensation point and finite temperature andsize effects within the Hartree-Fock approximation on topof the tight binding approximation in a similar way asprescribed in the previous section for the Bose scatterersin the harmonic trap.Again we see, in FIGs. 4(c) and (d), according to ourexpectation, that, repulsive interactions lead to narrow-ing down the profile of the differential scattering cross-section around θ = 0 as the condensates broaden uparound θ = 0. Increase of temperature, as expected andexplained before, leads to large angle scattering also for π π π π θ D n x , n y , n z ( θ ,0 )/ s ( a ) : For a scatterer in a 3 - D harmonic trap 0 π π π π θ D T ( θ ,0 )/( N s ) ( b ) : For a BEC in a 3 - D harmonic trap0 π π π θ D T ( θ ,0 )/( N s ) ( c ) : For BECs in a double well 0 π π π π θ D T ( θ ,0 )/( N s ) ( d ) : For BECs in 1 - D optical lattice
FIG. 4: Total differential scattering cross-section for quantumscatterer(s) in trapped geometry. For all the figures, we haveconsidered the following: l x = l y = l z = a s , ks s = 2, and m/M = 0 .
1. Solid, dotted and dashed lines in FIG. 4 (a)follow Eqn.(11) for n x = 5 , n y = 1 , n z = 0; n x = 0 , n y =0 , n z = 0; and n x = 20 , n y = 20 , n z = 20 respectively.Solid line in FIG. 4 (b) represents Eqn.(14), the solid line inFIG. 4 (c) represents Eqn.(17) for d = 10 l x , and the solidline in FIG. 4 (d) represents Eqn.(18) for d = 10 l x and N (cid:48) =10. While the solid lines in the last three (4 b,c,d) figuresrepresent non interacting BEC(s), the dotted lines in the samefigures represent scaling results for interacting BEC(s) with˜ a s = 0 . l as set in FIG. 3, and dashed lines represent finitetemperature and size effects over the dotted lines within theH-F approximation for T /T c = 0 . the scatterers in the double-well trap and the optical lat-tice trap. Coherency would be lost in presence of thedisorders in the BECs. In this situation the differentialscattering cross-sections in the FIGs. 4(b), (c) and (d)would be infinitesimally narrow like that shown by thesolid lines in the FIG. 2. V. CONCLUSIONS
To conclude, we have presented quantum theory ofparticle scattering by quantum scatterers in quantizedbound states in harmonically trapped geometry forFermi-Huang δ p [1] interactions (between the incidentparticle and the scatterers), which although are easy todeal with have huge applications in the field of ultra-cold atoms [15, 16]. Particle scattering by the quan-tum scatterer(s) in thermal equilibrium in finite geom-etry of optical traps has not been investigated before usexcept for T → D T ( θ, φ )s at T = T c , except for the for-ward scattering as shown in FIG. 3, can be used to detectoccurrence of BEC by particle scattering method. Ourpredictions can be tested within the present day experi-mental setups.Just by looking into the scattering intensity-patternfor sufficiently large energy of the incident ‘particle’, asshown in FIG. 1, and counting the maximum number ofthe zeros of the differential scattering cross-section along x -axis, one can easily determine energy eigenstate of 1-Dharmonic oscillator, as number of the zeros along the x axis is equal to quantum number n x . From the highestpossible peak height of the forward differential scatteringcross-section for a scatterer in the harmonic oscillator,one can easily determine scattering length ( a s ) of theincident ‘particle’ as the height, for low energy of theincident ‘particle’, is proportional to a s .We have constructed our theory for a single incident‘particle’. For a beam of ¯ N incident ‘particles’, A inEqn.(2) would be replaced by √ ¯ N A , and all the resultswhich depend on ‘ A ’ would be scaled accordingly. How-ever, the scattering amplitude, the differential scatteringcross-section, and the total scattering cross-section areindependent of ‘ A ’. So, all our result would be unalteredunder this scaling.Parameters used for plotting the figures are not spe-cific to a particular scattering problem. However, weset m/M = 0 . K as(fermionic) scatterer and He as the scattered particle.The ratio of m/M though would be even less (0 . Rb (bosonic scatterer) and He(scatterer particle) our results would not change much, as m/ ¯ µ for both the cases are approximately 0 .
91 and 0 . a s / ¯ l = 1 and ˜ a s / ¯ l = 0 . Rb atoms) to show stronger effect dueto the particle scattering than that due to inter-scattererinteractions. Values of ω and ka s are set 1000 and 2respectively to clearly show effect of temperature on par-ticle scattering by a harmonic oscillator in the ultra-coldregime ( T ∼ − K). If k increases, number of max-ima and minima increases in the profile of the differen-tial scattering cross-section. The number of maxima andminima further increases if the quantum number (i.e. thenodes in the wave function of the scatterer) increases. Weset N = 10 to show a significant difference between theparticle scattering by a Bose gas and that by a Fermigas in a harmonic trap. The later one shifts towards theclassical limit if N increases.Here, we have considered only elastic scattering. Ele-mentary excitations over the BEC leads to inelastic scat-tering involving inelastic processes where the trappedparticles in scattering out-states are found in differentharmonic oscillator states than those in the scatteringin-states. Hence, inelastic scattering is less probabilisticat finite temperatures. Moreover, differential scatteringcross-section in inelastic channels decays exponentially with the number of scatterers beyond a certain value [5].Particle scattering by weakly interacting harmonicallytrapped BEC was already studied, for T →
0, by Idzi-aszek et al with consideration of the first Born approx-imation for gδ ( r ) potential [5]. One may suspect theirresult, as, gδ ( r ) can not truly scatter a ‘particle’ exceptin 1-D [32, 33]. However, the first Born approximation,for gδ ( r ) interaction, surprisingly gives correct result for ka s → Acknowledgments
S. Das acknowledges financial support (JRF) of theUGC, India. S. Biswas acknowledges financial supportof the DST, Govt. of India under the INSPIRE Fac-ulty Award Scheme [No. IFA-13 PH-70]. We are in-debted to Prof. C. Timm, TU Dresden, Germany for hisvaluable critical comments. We are thankful to the re-viewers for their thorough reviews and highly appreciatetheir comments and suggestions, which significantly con-tributed to improving the quality of the paper. Usefulcomments from Prof. J.K. Bhattacharjee, IACS, Indiaare also gratefully acknowledged. We also thank Prof.K. Rzazewski, CTPPAS, Poland for introducing the Ref.[5] to our knowledge. [1] E. Fermi, Ricerca Sci. , 13 (1936); K. Huang and C. N.Yang, Phys. Rev. , 767 (1957)[2] D. J. Griffiths, Introduction to Quantum Mechanics , 2nded., Pearson Education, Singapore (2005)[3] E. Timmermans and R. Cote, Phys. Rev. Lett. , 3419(1998)[4] S. Bodefeld and M. Wilkens, Quantum Semiclass. Opt. , 511 (1996)[5] Z. Idziaszek, K. Rzazewski, and M. Wilkens, J. Phys. B:At. Mol. Opt. Phys. , L205 (1999)[6] A. Wynveen, A. Setty, A. Howard, J. W. Halley, and C.E. Campbell, Phys. Rev. A , 023602 (2000)[7] U. V. Poulsen, Ph.D. Thesis, University of Aarhus (2002)[8] A. Montina, Phys. Rev. A , 023609 (2002)[9] I. Haring, Ph.D. Thesis, TU Dresden (2003)[10] U. V. Poulsen and K. Molmer, Phys. Rev. A , 013610(2003)[11] J. Brand, I. Haring, and J.-M. Rost, Phys. Rev. Lett. ,070403 (2003)[12] R. A. Vicencio, J. Brand, and S. Flach, Phys. Rev. Lett. , 184102 (2007)[13] M. Streif, A. Buchleitner, D. Jaksch, and J. Mur-Petit,Phys. Rev. A , 053634 (2016)[14] A. Bhattacharya and S. Biswas, Quant. Phys. Lett. , 5(2017)[15] T. Busch, B. G. Englert, K. Rzazewski, and M. Wilkens,Foun. of Phys. , 549 (1998)[16] L. Pitaevskii and S. Stringari, Bose-Einstein Condensa-tion , Oxford Sc. Pub. (2003)[17] F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari,Rev. Mod. Phys. , 463 (1999)[18] I. Bloch et al , Rev. Mod. Phys. , 885 (2008)[19] S. Giorgini et al , Rev. Mod. Phys. , 1215 (2008)[20] M. Pinero, N. de la Rosa-Fox, R. Erce-Montilla, L. Es-quivias J. Sol-Gel Sci. Tech. , 527 (2003)[21] A. P. Chikkatur, A. Gorlitz, D.M. Stamper-Kurn, S. In-ouye, S. Gupta, and W. Ketterle, Phys. Rev. Lett. ,483 (2000)[22] S. L. Bromley et al , Nat. Commun. , 11039 (2016)[23] H.-J. Wang, X. Yi, X. Ba, and C. Sun, Phys. Rev. A ,043604 (2001)[24] H.-J. Wang and W. Jhe, Phys. Rev. A , 023610 (2002)[25] D. Schneble, Y. Torii, M. Boyd, E. W. Streed, D. E.Pritchard, W. Ketterle, Science , 475 (2003)[26] V. Ezhova, L. Gerasimov, and D. Kupriyanov, J. Phys.:Conf. Ser. , 012045 (2016)[27] B. Zhu, J. Cooper, J. Ye, and A. M. Rey, Phys. Rev. A , 023612 (2016)[28] W. Kozlowski, S. F. Caballero-Benitez, and I. B. Mekhov,Phys. Rev. A , 013613 (2015) [29] I. B. Mekhov and H. Ritsch, J. Phys. B: At. Mol. Opt.Phys. , 102001 (2012)[30] K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. vanDruten, D. S. Durfee, D. M. Kurn, and W. Ketterle,Phys. Rev. Lett. , 3969 (1995)[31] For Born series for the δ p potential, please, see PHYS852Quant. Mech. II, Spring 2010: HW Assignment 10.[32] C. N. Friedman, J. Functional Analysis , 346 (1972);R. M. Cavalcanti, Rev. Bras. Ens. Fis. , 336 (1999)[33] I. Mitra, A. DasGupta, and B. Dutta-Roy, Am. J. Phys. , 1101 (1998)[34] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals,Series, and Products , 7th ed., p.806, Elsevier, AM (2007)[35] S. Biswas and D. Jana, Eur. J. Phys. , 1527 (2012)[36] S. Giorgini, L. Pitaevskii, and S. Stringari, Phys. Rev. A , R4633 (1996)[37] S. Biswas, Phys. Lett. A , 1574 (2008)[38] S. Biswas, Eur. Phys. J. D , 653 (2009)[39] J. R. Ensher, D. S. Jin, M. R. Matthews, C. E. Wieman,and E. A. Cornell, Phys. Rev. Lett. , 4984 (1996)[40] M. R. Andrews et al , Science , 637 (1997)[41] I. Bloch, T. W. Hansch, and T. Esslinger, Nature ,166 (2000)[42] S. Levy, E. Lahoud, I. Shomroni, and J. Steinhauer, Na-ture , 579 (2007)[43] S. Raghavan, A. Smerzi, S. Fantoni, and S. R. Shenoy,Phys. Rev. A , 620 (1999)[44] B. P. Anderson and M. A. Kasevich, Science , 1686(1998)[45] P. Pedri et al , Phys. Rev. Lett. , 220401 (2001)[46] M. Greiner, M. O. Mandel, T. Esslinger, T. Hansch, andI. Bloch, Nature , 39 (2002)[47] By ‘particle’, we mean, wave associated with the particle.[48] The superposition principle is often applied in a simi-lar way for the light scattering (diffraction) by a doubleslit. Please see R. P. Feynman, R. B. Leighton, and M. L.Sands, The Feynman Lectures on Physics: Quantum Me-chanics , Vol. 3, Chapter 1, Addison-Wesley, MA (1965)for the same.[49] As because, we can approximate (cid:80) ∞ n =0 e − n/N L n ( x ) = e N + x − e N − e N for N (cid:29)
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