Theoretical tools for atom laser beam propagation
J.-F. Riou, Y. Le Coq, F. Impens, W. Guerin, C. J. Bordé, A. Aspect, P. Bouyer
aa r X i v : . [ c ond - m a t . o t h e r] F e b Theoretical tools for atom laser beam propagation
J.-F. Riou ∗ , † Y. Le Coq, F. Impens, W. Guerin ‡ , C. J. Bord´e, A. Aspect, and P. Bouyer Laboratoire Charles Fabry de l’Institut d’Optique, CNRS et Universit´e Paris Sud 11Campus Polytechnique, RD 128, 91127 Palaiseau, France SYRTE, Observatoire de Paris, CNRS, UPMC61 avenue de l’Observatoire, 75014 Paris, France (Dated: November 5, 2018)We present a theoretical model for the propagation of non self-interacting atom laser beams. Westart from a general propagation integral equation, and we use the same approximations as in photonoptics to derive tools to calculate the atom laser beam propagation. We discuss the approximationsthat allow to reduce the general equation whether to a Fresnel-Kirchhoff integral calculated byusing the stationary phase method, or to the eikonal. Within the paraxial approximation, we alsointroduce the
ABCD matrices formalism and the beam quality factor. As an example, we applythese tools to analyse the recent experiment by Riou et al. [Phys. Rev. Lett. , 070404 (2006)]. PACS numbers: 03.75.Pp, 39.20.+q, 42.60.Jf,41.85.Ew
Introduction
Matter-wave optics, where a beam of neutral atoms isconsidered for its wave-like behavior, is a domain of con-siderable studies, with many applications, ranging fromatom lithography to atomic clocks and atom interferome-ter [66]. The experimental realization of coherent matter-wave - so called atom lasers [1, 2, 3, 4, 5, 6, 7] - which fol-lowed the observation of Bose-Einstein condensation puta new perspective to the field by providing the atomicanalogue to photonic laser beams.Performant theoretical tools for characterizing thepropagation properties of matter waves and their ma-nipulation by atom-optics elements are of prime inter-est for high accuracy applications, as soon as one needsto go beyond the proof-of-principle experiment. In thescope of partially coherent atom interferometry, and forrelatively simple ( i.e. homogenous) external potentials,many theoretical works have been developed [8, 9, 10, 11]and applied successfully [12, 13]. All these tools essen-tially address the propagation of an atomic wavepacket.For fully coherent atom-laser beams, most theoretical in-vestigations focused on the dynamics of the outcoupling[14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27] andthe quantum statistical properties of the output beam[28, 29, 30, 31, 32, 33, 34, 35]. Some works specifi-cally addressed the spatial shape of the atom laser beam[36, 37], but rely essentially on numerical simulations orneglect the influence of dimensionality and potential in-homogeneity. For realistic experimental conditions, the3D external potential is inhomogeneous, and full numer- ∗ Present address: Physics Department of Penn State University,104 Davey Laboratory, Mailbox 002, University Park, PA 16802,U.S.A. ‡ Present address: Institut Non Lin´eaire de Nice, 1361 route desLucioles, 06560 Valbonne, France † Electronic address: [email protected] ical simulation become particularly cumbersome. Onethus needs a simplified analytical theoretical frameworkto handle the beam propagation.Following our previous work [38, 39], we present here indetail a simple but general framework for the propagationof atom laser beams in inhomogeneous media. We showhow several theoretical tools from classical optics can beadapted for coherent atom-optics. We address three ma-jor formalisms used in optics : the eikonal approximation,the Fresnel-Kirchhoff integral, and the
ABCD matricesformalism in the paraxial approximation.The first part of the paper gives an overview of thesetheoretical tools for atom laser beam propagation. Inthe first section, we introduce the integral equation ofthe propagation and its time-independent version. Wepresent in the second section different ways of dealingwith the time-independent propagation of the matterwave. First, the time-independent propagator is com-puted using the stationary phase approximation. Then,we show that two approximations -the eikonal and theparaxial approximation-, which apply in different physi-cal contexts, can provide a more tractable treatment thanthe general integral equation. In the second part, weshow in practice how to use these methods in the ex-perimental case of [38] with a rubidium radiofrequency-coupled atom laser. Some of these methods have recentlybeen used also for a metastable helium atom laser [40] aswell as for a Raman-coupled atom laser [41].
I. ANALYTICAL PROPAGATION METHODSFOR MATTER WAVESA. Matter wave weakly outcoupled from a source
1. Propagation equation
We consider a matter wave ψ ℓ ( r , t ) outcoupled from asource ψ s ( r , t ). We note V i ( r , t ) ( i = { ℓ, s } ), the externalpotential in which each of them evolves. We also intro-duce a coupling term W ij ( r , t ) between ψ i and ψ j . Inthe mean-field approximation, such system is describedby a set of two coupled Gross-Pitaevskii equations, whichreads i ~ ∂ t ψ i = − ~ m ∆ + V i + X k = ℓ, s g ki | ψ k | ψ i + W ij ψ j . (1)In this equation, g ik is the mean-field interaction strengthbetween states i and k . The solution of such equationsis not straightforward, mainly due to the presence of anonlinear mean-field term. However, in the case of prop-agation of matter waves which are weakly outcoupledfrom a source, one can greatly simplify the treatment[27]. Indeed, the weak-coupling assumption implies thetwo following points: • The evolution of the source wave-function is unaf-fected by the outcoupler, • the extracted matter wave is sufficiently diluted tomake self-interactions negligible.The former differential system can then be rewrittenas: i ~ ∂ t ψ s = (cid:20) − ~ m ∆ + V s + g ss | ψ s | (cid:21) ψ s , (2) i ~ ∂ t ψ ℓ = (cid:20) − ~ m ∆ + V ℓ + g s ℓ | ψ s | (cid:21) ψ ℓ + W ℓ s ψ s . (3)The source wave-function ψ s ( r , t ) now obeys a singledifferential equation (2), and can thus be determined in-dependently. The remaining nonlinear term | ψ s | in Eq.(3), acts then as an external potential for the propagationof ψ ℓ . This last equation is thus a Schr¨odinger equationdescribing the evolution of the outcoupled matter wavein the total potential V ( r , t ) in presence of a source term ρ ( r , t ), i ~ ∂ t ψ ℓ = H r ψ ℓ + ρ , (4)where H r = − ~ m ∆ r + V , (5a) V = V ℓ + g s ℓ | ψ s | , (5b) ρ = W ℓ s ψ s . (5c)
2. Integral equation
The evolution between times t and t ( t > t ) of thesolution ψ ℓ of Eq. (4) in a given volume V delimited by a surface S , is expressed by an implicit integral [42] ψ ℓ ( r , t ) = Z V d r ′ G ( r , r ′ , t − t ) ψ ℓ ( r ′ , t )+ i ~ m Z tt dt ′ Z S d S ′ · [ G ( r , r ′ , t − t ′ ) ∇ r ′ ψ ℓ ( r ′ , t ′ ) − ψ ℓ ( r ′ , t ′ ) ∇ r ′ G ( r , r ′ , t − t ′ )]+ 1 i ~ Z tt dt ′ Z V d r ′ G ( r , r ′ , t − t ′ ) ρ ( r ′ , t ′ ) , (6)where d S ′ is the outward-oriented elementary normalvector to the surface S . We have introduced the time-dependent Green function G ( r , r ′ , τ ) which verifies[ i ~ ∂ τ − H r ] G = i ~ δ ( τ ) δ ( r − r ′ ) , (7)and is related to the propagator K of the Schr¨odingerequation via a Heaviside function Θ ensuring causality, G ( r , r ′ , τ ) = K ( r , r ′ , τ ) Θ( τ ) . (8)Eq. (6) states that, after the evolution time t − t , thevalue of the wave function is the sum of three terms, thephysical interpretation of which is straightforward. Thefirst one corresponds to the propagation of the initialcondition ψ ℓ ( r ′ , t ) given at any position in the volume V . The second one takes into account the propagationof the wave function taken at the surrounding surface S ,and is non-zero only if V is finite. This term takes intoaccount any field which enters or leaks out of V . Finally,the last term expresses the contribution from the source.Eq. (6) can be successfully applied to describe thepropagation of wavepackets in an atom interferometeras described in [43]. Nevertheless, the propagation of acontinuous atom laser, the energy of which is well defined,can be described with a time-independent version of Eq.(6), that we derive below.
3. Time-independent case
We consider a time-independent hamiltonian H r anda stationnary source ρ ( r , t ) = ρ ( r ) exp ( − iEt/ ~ ) . (9)We thus look for stationnary solutions of Eq. (4) with agiven energy E , ψ ℓ ( r , t ) = ψ ℓ ( r ) exp ( − iEt/ ~ ) . (10)When t → −∞ , Eq. (6) then becomes time-independent: ψ ℓ ( r ) = 1 i ~ Z V d r ′ G E ( r , r ′ ) ρ ( r ′ )+ i ~ m Z S d S ′ · [ G E ( r , r ′ ) ∇ r ′ ψ ℓ ( r ′ ) − ψ ℓ ( r ′ ) ∇ r ′ G E ( r , r ′ )] , (11)where G E is the time-independent propagator related to K via G E ( r , r ′ ) = Z + ∞ dτ K ( r , r ′ , τ ) e i Eτ ~ . (12)Note that the first term of Eq. (6) vanishes in the time-independent version of the propagation integral equationas K ( r , r ′ , τ ) → τ → ∞ . The second term of Eq.(11) is the equivalent for matter waves of what is knownin optics as the Fresnel-Kirchhoff integral [44]. B. Major approximations for atom laser beampropagation
1. Independent treatment of a succession of potentials
As an optical wave can enter different media (freespace, lenses...) separated by surfaces, matter waves canpropagate in different parts of space, where they expe-rience potentials of different nature. For instance, whenone considers an atom laser outcoupled from a conden-sate as in the example of part II, the beam initially inter-acts with the Bose-condensed atoms and abruptly prop-agates in free space outside of the condensate. The ex-pression of the propagator in whole space would then beneeded to use the equation (11). Most generally, suchcalculation requires to apply the Feynmann’s path inte-gral method, either numerically or analytically [45]. Forexample, the time-dependent propagator K can be ana-lytically expressed in the case of a continuous potentialwhich is at most quadratic, by using the Van Vleck’s for-mula [46], or the ABCD formalism [43]. However suchexpressions fail to give the global propagator value for apiecewise-defined quadratic potential.As in classical optics, we can separate the total evo-lution of a monochromatic wave in steps, each one cor-responding to one homogeneous potential. This step-by-step approach stays valid as long as one can neglect anyreflection on the interface between these regions as wellas feedback from one region to a previous one. In thisapproach, each interface is considered as a surface sourceterm for the propagation in the following media. It al-lows us to calculate K explicitly in every part of spaceas long as the potential in each region remains at mostquadratic, which we will assume throughout this paper.
2. The time-independent propagator in the stationary phaseapproximation
Whereas the expression of G E is well known for freespace and linear potentials [8, 37], to our knowledge,there is no analytical expression for the inverted har-monic potential, which plays a predominant role in anatom laser interacting with its source-condensate. We thus give in the following a method to calculate the time-independent propagator G E in any up to quadratic po-tential.Since K is analytically known in such potentials, weuse the definition of G E as its Fourier transform (Eq.12). The remaining integral over time τ is calculated viaa stationary phase method [44], taking advantage that K is a rapidly oscillating function. We write the time-dependent propagator as K ( r , r ′ , τ ) = A ( τ ) exp [ iφ ( r , r ′ , τ )] . (13)We introduce τ n as the positive real solution(s) of ∂ τ φ ( r , r ′ , τ n ) = − E/ ~ , (14)which correspond(s) to the time(s) spent on classicalpath(s) of energy E connecting r ′ to r . We develop φ to the second order around τ n , φ ( τ ) ≃ φ ( τ n ) + ∂φ∂τ (cid:12)(cid:12)(cid:12) τ n ( τ − τ n ) + ∂ φ∂τ (cid:12)(cid:12)(cid:12) τ n ( τ − τ n ) . (15)Using the last development in the integral (12), and as-suming that the enveloppe A ( τ ) varies smoothly around τ n , we can express G E as G (1) E ≃ X n s iπφ ′′ ( τ n ) K ( τ n ) exp (cid:18) i Eτ n ~ (cid:19) . (16)Such approach is valid as long as stationary points τ n exist and their contribution can be considered indepen-dently: Eq. 16 fails if the stationary points are too closeto each other. We can estimate the validity of our ap-proach by defining an interval I n = [ τ n − θ n ; τ n + θ n ]in which the development around τ n contributes to morethan β = 90% to the restricted integral. For θ largeenough, we can use [47], (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z θ − θ dx exp (cid:20) iz x (cid:21) − r iπz (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∼ | z θ | , (17)and obtain θ n θ n = 11 − β s πφ ′′ ( τ n ) . (18)The validity condition is thus | τ n − τ n +1 | ≥ θ n + θ n +1 .If (16) is not valid, a better approximation consiststhen in developing φ to higher order around a pointwhich is inbetween successive τ n . The simplest choiceis to take the one which cancels φ ′′ , and to choose sta-tionnary points τ k which verify ∂ τ φ ( r , r ′ , τ k ) = 0 . (19)We thus develop φ to the third order around τ k , whichleads to the following expression of G E , G (2) E ≃ π K ( τ k ) exp (cid:0) i Eτ k ~ (cid:1) p − φ (3) ( τ k ) / φ ′ ( τ k ) + E/ ~ p φ (3) ( τ k ) / ! , (20)where Ai is the Airy function of the first kind [48].In practice, combining the use of G (2) E and G (1) E depend-ing on the values of r ′ and r gives a good estimate of thetime-independent propagator, as we will see in part II.Although the above approach is quite general, furtherapproximations can be made. In the region where diffrac-tion can be neglected, one can describe the propagationwith the eikonal approximation. When the propagationis in the paraxial regime, it is more appropriate to de-scribe it with the paraxial ABCD matrices, instead ofusing the general Kirchhoff integral.
3. Eikonal propagation
The purpose of this method, equivalent to the WKBapproximation, is to give a semi-classical description ofthe propagation from a matter wave, given its value ona surface. Let us consider that we know the value of thewave function of energy E on the surface S ′ . To calculateits value on any other surface S , the eikonal considersclassical paths connecting S and S ′ . Let us write thewave function as ψ ℓ ( r ) = A ( r ) exp [ iS ( r ) / ~ ] . (21)The Schr¨odinger equation on ψ ℓ reduces to [49] |∇ r S | = ~ λ , ∇ r · (cid:0) A ∇ S (cid:1) = 0 , (22)where we have introduced the de Broglie wavelength λ ( r ) = ~ p m ( E − V ( r )) . (23)The first equation is known in geometric optics as theeikonal equation [44, 50]. The calculation consists in in-tegrating the phase along the classical ray of energy E connecting r ′ to r , to obtain the phase on r , S ( r ) = Z rr ′ d u ~ λ ( u ) + S ( r ′ ) . (24)The second equation of system (22) corresponds to theconservation of probability density flux, and is equivalentto the Poynting’s law in optics. Again, after integrationalong the classical path connecting r ′ to r , one obtainsthe amplitude on S A ( r ) = A ( r ′ ) exp (cid:18) − Z rr ′ d u ∆ S ( u ) λ ( u )2 ~ (cid:19) . (25)Note that interference effects are included in this formal-ism: if several classical paths connect r ′ to r , their respec-tive contributions add coherently to each other. Also, ifsome focussing points exist, dephasings equivalent to theGouy phase in optics appear and can be calculated fol-lowing [44]. Such semiclassical treatment is valid as long as onedoes not look for the wave function value close to classi-cal turning points, and as long as transverse diffractionis negligible : transverse structures of size ∆ x must belarge enough not to diffract significantly, i.e. ∆ x ≫ ( ~ t/ m ) . This condition restricts the use of the eikonalto specific regions of space where the matter wave doesnot spend a too long time t . For instance, this is the casefor the propagation in the small region of overlap withthe BEC.The eikonal can thus be used to deal with the first termof Eq. (11), and is equivalent to the development of thisintegral around classical trajectories [67].
4. Paraxial propagation
The paraxial regime applies as soon as the transversewavevector becomes negligible compared to the axialone. It is for instance the case after some propagationfor gravity-accelerated atom-laser beams. We can thentake advantage of methods developped in optics and usethe paraxial atom-optical
ABCD matrices formalism[39, 43], instead of the general Kirchhoff integral, andcharacterize globally the beam with the quality factor M [51, 52]. a. The paraxial equation: We look for paraxial solu-tions to the time-independent Schr¨odinger equation, H r ψ ℓ ( r ) = Eψ ℓ ( r ) . (26)We decompose the wave function and the potential ina transverse (“ ⊥ ”) and parallel (“ // ”) component, taking z as the propagation axis, ( ψ ℓ ( x, y, z ) = ψ ⊥ ( x, y, z ) ψ // ( z ) ,V ( x, y, z ) = V ⊥ ( x, y, z ) + V // ( z ) , (27)where V // ( z ) = V (0 , , z ). We express the solution ψ // tothe one dimensional equation − ~ m ∂ ψ // ∂z + V // ψ // = Eψ // , (28)by using the WKB approximation, ψ // ( z ) = s m F p ( z ) exp (cid:20) i ~ Z zz d u p ( u ) (cid:21) . (29)In this expression, F is the atomic flux through any trans-verse plane, p ( z ) = q m (cid:0) E − V // ( z ) (cid:1) is the classical mo-mentum along z and z is the associated classical turn-ing point verifying p ( z ) = 0. Using these expressions,and assuming an envelope ψ ⊥ slowly varying along z ,we obtain the paraxial equation of propagation for thetransverse profile, (cid:20) i ~ ∂ ζ + ~ m ( ∂ x + ∂ y ) − V ⊥ ( x, y, ζ ) (cid:21) ψ ⊥ ( x, y, ζ ) = 0 , (30)where ζ ( z ) = R zz dz m/p ( z ) is a parameter correspondingto the time which would be needed classicaly to prop-agate on axis from the turning point z . The equation(30) can thus be solved as a time-dependent Schr¨odingerequation, ψ ⊥ ( x, y, ζ ) = Z S ′ dx ′ dy ′ K ( x, y ; x ′ , y ′ ; ζ − ζ ′ ) ψ ⊥ ( x ′ , y ′ , ζ ′ ) . (31)The use of the paraxial approximation allows us to focusonly on the evolution of the transverse wave function,reducing the dimensionality of the system from 3D to2D, as the third dimension along the propagation axis z is treated via a semi-classical approximation (Eq. 29). b. ABCD matrices:
In the case of a separable trans-verse potential independent of z , the paraxial approxima-tion restricts to two independent one dimensional equa-tions. Let us consider a potential V x at most quadraticin x . One can then write the propagator K x by using theVan Vleck formula, or equivalently the general ABCD matrix formalism [8], K x = r α πiB exp (cid:20) iα B (cid:0) Ax ′ + Dx − xx ′ (cid:1)(cid:21) . (32)The coefficients A , B , C , D verifying AD − BC = 1, arefunctions of ζ − ζ ′ , and α is an arbitrary factor depend-ing on the definition of the ABCD coefficients. Theseones are involved in the matrix describing the classicaldynamics of a virtual particle of coordinate X and speed V in the potential V x ( X ) (cid:18) X ( ζ ) αV ( ζ ) (cid:19) = (cid:18) A ( ζ − ζ ′ ) B ( ζ − ζ ′ ) /ααC ( ζ − ζ ′ ) D ( ζ − ζ ′ ) (cid:19) (cid:18) X ( ζ ′ ) αV ( ζ ′ ) (cid:19) . (33)Different choices of α can be made and popular values inthe atomoptic literature are α = 1 [8] or α = m/ ~ [38,39]. We take the last convention and, by introducing thewavevector K = mV / ~ , use throughout this paper thefollowing definition for the ABCD coefficients in whichis included the value of α , (cid:18) X ( ζ ) K ( ζ ) (cid:19) = (cid:18) A ( ζ − ζ ′ ) B ( ζ − ζ ′ ) C ( ζ − ζ ′ ) D ( ζ − ζ ′ ) (cid:19) (cid:18) X ( ζ ′ ) K ( ζ ′ ) (cid:19) . (34) c. Propagation using the Hermite-Gauss basis: Tocalculate the propagation along the x axis ψ x ( x, ζ ) = Z dx ′ K x ( x ; x ′ ; ζ − ζ ′ ) ψ x ( x ′ ,ζ ′ ) , (35)it is useful to use the Hermite-Gauss basis of functions(Φ n ) n ∈ N Φ ( x, { X, K } ) = (2 π ) − / √ X exp (cid:16) i KX x (cid:17) , (36)Φ n ( x, { X, K } ) = Φ ( x ) √ n n ! | X | n X n H n h x √ | X | i . (37) H n is the n th order Hermite polynomial, and the twoparameters ( X, K ) ∈ C , which define univocally the basisset, must verify the normalization condition KX ∗ − K ∗ X = i , (38)so that this basis is orthonormalized.These functions propagate easily via K x , asΦ n ( x, { X, K } ( ζ )) = Z dx ′ K x ( x ; x ′ ; ζ − ζ ′ ) Φ n ( x ′ , { X, K } ( ζ ′ )) , (39) i.e. the integral is calculated by replacing X ( ζ ′ ) and K ( ζ ′ ) by their value at ζ through the algebraic relation(34).Thus, the propagation of the function ψ x between twopositions z ( ζ ′ ) and z ( ζ ) is obtained by first decomposingthe initial profile on the Hermite-Gauss basis ψ x ( x, ζ ′ ) = X n c n Φ n ( x, { X, K } ( ζ ′ )) , (40)where c n = Z dx Φ ∗ n ( x, { X, K } ( ζ ′ )) ψ x ( x, ζ ′ ) . (41)The profile after propagation until z ( ζ ) is then ψ x ( x, ζ ) = X n c n Φ n ( x, { X, K } ( ζ )) . (42)The high efficiency of this method comes from the factthat, once the decomposition (40) is made, the profile atany position z ( ζ ) is obtained by calculating an algebraicevolution equation: the ABCD law (Eq. 34). Such com-putational method is then much faster than the use ofthe Kirchhoff integral, which would need to calculate anintegral for each considered position.Note that the initial choice of { X, K } ( ζ ′ ) is a prioriarbitrary as soon as it verifies the normalization con-dition (38). However, one can minimize the number offunctions Φ n needed for the decomposition if one chooses { X, K } ( ζ ′ ) as a function of the second-order momentsof the profile. d. Moments and quality factor: Let us define thesecond-order moments of ψ x , h xx ∗ i = Z dx x ψ x ψ ∗ x , (43) h kk ∗ i = Z dx ∂ x ψ x ∂ x ψ ∗ x , (44) h xk ∗ + x ∗ k i = i Z dx x [ ψ x ∂ x ψ ∗ x − ψ ∗ x ∂ x ψ x ] , (45)where we have used that ψ x is normalized( R dx | ψ x | = 1). We also define the wavefront cur-vature C [53] as C = h xk ∗ + x ∗ k i h xx ∗ i . (46)The three moments follow also an ABCD law duringpropagation. By introducing the matrix M ( ζ ) = (cid:18) h xx ∗ i h xk ∗ + x ∗ k i / h xk ∗ + x ∗ k i / h kk ∗ i (cid:19) , (47)this law is expressed as M ( ζ ) = (cid:18) A BC D (cid:19) M ( ζ ′ ) (cid:18) A BC D (cid:19) t . (48)This relation allows to derive propagation laws on thewavefront second-order moments, such as the r.m.s trans-verse size (Rayleigh law). As det( M ) is constant, thislaw also exhibits an invariant of propagation, the beamquality factor M , related to the moments and curvatureby h xx ∗ i (cid:0) h kk ∗ i − C h xx ∗ i (cid:1) = (cid:18) M (cid:19) . (49)The physical meaning of the M factor becomes clearby taking the last equation at the waist, i.e. where thecurvature C is zero: p h xx ∗ i h kk ∗ i = M . (50)The M factor is given by the product of the spatialand momentum widths at the beam waist and indicateshow far the beam is from the diffraction limit. Becauseof the Heisenberg uncertainty relation, the M factor isalways larger than one and equals unity only for a perfectgaussian wavefront.Finally, the determination of the second order mo-ments and the M factor from an initial profile allows usto choose the more appropriate values of { X ( ζ ′ ) , K ( ζ ′ ) } to parameterize the Hermite-Gauss basis used for the de-composition at z ( ζ ′ ) (Eq. 40). Indeed, these parametersare closely related to the second order moments of theHermite-Gauss functions Φ n ( x, { X, K } ) by h xx ∗ i Φ n = (2 n + 1) | X | , (51) h kk ∗ i Φ n = (2 n + 1) | K | , (52) h xk ∗ + x ∗ k i Φ n = (2 n + 1) ( XK ∗ + X ∗ K ) . (53)From this we obtain that the M factor of the mode Φ n is M n = (2 n + 1) and that all the modes have the samecurvature C Φ n = C = ( XK ∗ + X ∗ K ) / | X | . (54)It is thus natural to choose the parameters { X, K } , sothat the curvature of the profile (Eq. 46) equals C .This last condition, together with the choice | X | = h xx ∗ i /M , the normalization condition (38) and thechoice of X real (the phase of X is a global phase overthe wavefront), lead to the univocal determination of theparameters { X ( ζ ′ ) , K ( ζ ′ ) } associated with the Hermite-Gauss basis, so that the decomposition of the initial pro-file ψ x ( x, ζ ′ ) needs a number of terms of the order of M . II. APPLICATION TO ARADIOFREQUENCY-OUTCOUPLED ATOMLASER
We apply the previous framework to the radiofre-quency (rf) outcoupled atom laser described in [38]where a Bose-Einstein condensate (BEC) of rubidium87 (mass m ) is magnetically harmonically trapped (fre-quencies ω x = ω z = ω ⊥ and ω y ) in the ground state | F = 1 , m F = − i , and is weakly outcoupled to the un-trapped state | F = 1 , m F = 0 i . The BEC is consideredin the Thomas-Fermi (TF) regime described by the time-independent wave function φ s ( r ), with a chemical poten-tial µ and TF radii R ⊥ ,y = q µ/mω ⊥ ,y [54]. The exter-nal potential experienced by the beam is written V i ( r ) = µ − mω ⊥ σ − m (cid:2) ω ⊥ ( x + z ) + ω y y (cid:3) (55)inside the BEC region, and V o ( r ) = 12 mω σ q − mω (cid:2) x + ( z + σ q ) (cid:3) . (56)outside. The expulsive quadratic potential of V i origi-nates from the mean-field interaction (independent of theZeeman substates for Rb) between the laser and thecondensate, whereas that of V o (frequency ω ) is due tothe second order Zeeman effect. We have noted σ = g/ω ⊥ and σ q = g/ω the vertical sags due to gravity − mgz for m F = − m F = 0 states respectively. The rf cou-pling (of Rabi frequency Ω R ) between the condensate andthe beam is considered to have a negligible momentumtransfer and provides the atom-laser wave function witha source term ρ = ~ Ω R / φ s ( r ).In the following, we consider a condensate elongatedalong the y axis ( ω ⊥ ≫ ω y ), so that the laser dynam-ics is negligible along this direction [68]. We thus studyindependently the evolution in each vertical ( x, z ) planeat position y . We calculate the beam wave function intwo steps corresponding to a propagation in each regiondefined by V i and V o (see Fig. 1). The wave function atthe BEC frontier is calculated in section II A using theeikonal approximation. Then, in section II B, we calcu-late the wave function at any position outside the BEC,with the help of the Fresnel-Kirchhoff formalism and theparaxial ABCD matrices. A. Propagation in the condensate zone
In this section, we determine the beam wave function ψ ℓ ( r ) in the condensate zone by using the eikonal formal-ism described in section I B 3. This formalism is appro-priate in this case as the time necessary for the laser toexit the BEC region ( ≈ ≈ R ⊥ ). M zy Rf knife x Kirchhoff Int.ABCD matrix ¡ r r f BECoutputsurfaceEikonal
FIG. 1: Principle of the calculation : the wave function iscalculated from the rf knife (a circle of radius r centered atthe frame origin) using the eikonal. A general radial atomictrajectory starting at zero speed from r crosses the BECborder at r f . Once the matter wave has exited the condensateregion, the wave function is given by the Fresnel-Kirchhoffintegral, allowing to compute the wave function at any pointfrom the BEC output surface. In the paraxial approximation,we calculate the propagation using ABCD matrices.
1. Atomic rays inside the BEC
One first needs to calculate the atomic paths followedby the atom laser rays from the outcoupling surface (therf knife) to the border of the BEC. The rf knife is anellipso¨ıd centered at the magnetic field minimum (chosenin the following as the frame origin, see figure 1). Itsintersection with the ( x, z ) plane at position y is a circlecentered at the frame origin. Its radius r depends onthe rf detuning δν = m h (cid:2) ω ⊥ (cid:0) r − σ (cid:1) + ω y y (cid:3) . As weneglect axial dynamics and consider zero initial momen-tum, the classical equations of motion give for the radialcoordinate r = √ x + z , r ( t ) = r cosh ω ⊥ t allowing tofind a starting point r on the rf knife for each point r f on the BEC output surface, i.e. the BEC border belowthe rf knife [55].
2. Eikonal expression of the wave function
We now introduce a ⊥ = r ~ mω ⊥ , R = ra ⊥ , ǫ = − (cid:18) r a ⊥ (cid:19) , (57)which are respectively the size of the harmonic potential,the dimensionless coordinate and energy associated with the atom laser. Following Eq. (24) and (25), we obtain S ( R ) = ~ " R p R + ǫ + ǫ ln R + √ R + ǫ √− ǫ ! ,A ( R ) = B φ s ( r ′ )[ R ( R + ǫ )] / . (58) B is proportional to the coupling strength, and is notdirectly given by the eikonal treatment [69]. The atomlaser beam amplitude A ( R ) [70] is proportional to theBEC wave function value at the rf knife φ s ( r ). Thewave function at the BEC output surface is then ψ ℓ ( R f ) = A ( R f ) exp [ iS ( R f ) / ~ ] . (59) B. Propagation outside the condensate
Once the matter wave has exited the condensate re-gion, the volume source term ρ vanishes and the beamwave function is given by the second term of Eq. 11 only, i.e. the Fresnel-Kirchhoff integral for matter waves, al-lowing to compute the wave function at any point fromthe wave function on the BEC output surface. In thissection, we calculate the propagation using an analyti-cal expression for the time-independent propagator andapply the ABCD formalism in the paraxial regime.
1. Fresnel-Kirchhoff Integral
We perform the Fresnel-Kirchhoff integral in the ( x, z )plane at position y : ψ ℓ = i ~ m I Γ d l ′ · (cid:2) G E ∇ ψ ℓ − ψ ℓ ∇G E (cid:3) , (60)where ψ ℓ is non zero only on the BEC output surfaceas seen in section II A. The surface S of equation (11)is here reduced to its interserction contour Γ with thevertical plane. It englobes the BEC volume and is closedat infinity.Using the expression of G E calculated in appendix A,we compute equation (60) and the result is shown in Fig.2 for four different outcoupling rf detunings. When cou-pling occurs at the top of the BEC, the propagation ofthe beam exhibits a strong divergence together with awell-contrasted interference pattern. The divergence isdue to the strong expulsive potential experienced by thebeam when crossing the condensate, and interferencesoccur because atomic waves from different initial sourcepoints overlap during the propagation.Comparison with a numerical Gross-Pitaevskii simula-tion shows good agreement. We also compare the resultsobtained by using at the BEC surface either Eq. 59 orEq. B9. The eikonal method fails when coupling at thevery bottom of the BEC [Fig. 2(a)], since the classicalturning point is too close to the BEC border, whereas (a) (b)(c) (d)010 × × ¹ m) x ¹ m) x ¹ m) x ¹ m) xà ` à ` à ` à ` × × FIG. 2: Density profiles obtained at 150 µ m = z − σ belowthe BEC center. We consider the vertical plane y = 0 andhave normalised | ψ ℓ | to unity. We have drawn the resultsobtained by using as input of the Kirchhoff integral the profilecalculated using the eikonal (Eq. 59, dotted line) or exactsolutions of the inverted harmonic oscillator (Eq. B9, fullline), and compare them to a full numerical integration of thetwo-dimensional Gross-Pitaevskii evolution of the atom laser(dashed line). The used rf detunings are: (a) δν = 8900 Hz,(b) δν = 6500 Hz, (c) δν = 2100 Hz, (d) δν = − the method using the exact solutions of the inverted har-monic potential agrees much better with the numericalsimulation for any rf detuning. Finally, for very highcoupling in the BEC [Fig. 2(d)], our model slightly over-estimates the fringe contrast near the axis.
2. Propagation in the paraxial regime
Since the atom laser beam is accelerated by gravity, itenters quickly the paraxial regime. In the case consideredin [38], the maximum transverse energy is given by thechemical potential µ whereas the longitudinal energy ismainly related to the fall height z by E z ≈ mgz . For µ typically of a few kHz, one enters the paraxial regimeafter approximately 100 µ m of vertical propagation. Forlarger propagation distances, we can thus take advantageof the paraxial approximation presented in Sec. I B 4.To proceed, we start from the profile ψ ℓ ( x ) calculatedafter 150 µ m of propagation via the Kirchhoff integral.Using Eqs. (43)-(46), we extract the widths < xx ∗ > , < kk ∗ > and the beam curvature C at this position.From these parameters we calculate the beam qualityfactor M by using the general equation (49). Following the procedure presented in Sec. I B 4, we can choose theappropriate Hermite-Gauss decomposition of ψ ℓ ( x ) andthe propagation of each mode is then deduced from the ABCD matrix corresponding to the transverse part ofthe potential described in Eq. 56, V ⊥ ( x ) = − m ω x .The ABCD matrix then reads (cid:18)
A BC D (cid:19) = (cid:18) cosh ω ( ζ − ζ ′ ) ~ mω sinh ω ( ζ − ζ ′ ) mω ~ sinh ω ( ζ − ζ ′ ) cosh ω ( ζ − ζ ′ ) (cid:19) . (61)As explained in Sec. I B 4, the propagation is parame-terized by the time ζ , given by the classical equation ofmotion of the on-axis trajectory in the longitudinal partof the potential V k (˜ z ) = − m ω ˜ z , where ˜ z = z + σ q .The ABCD matrices formalism allows also to extractglobal propagation laws on the second order moments X ( ζ ), K ( ζ ) and evaluate the wavefront curvature C ( ζ ) = ℜ (cid:16) K ( ζ ) X ( ζ ) (cid:17) associated with the wavefront ψ ℓ ( x, ζ ).By considering the paraxial evolution of the r.m.s. size σ of ψ ℓ ( x, ζ ), we then obtain a generalized Rayleigh for-mula : σ ( ξ ) = σ cosh ( ωξ ) + (cid:18) M ~ mω (cid:19) sinh ( ωξ ) σ , (62)involving the M factor, and where σ = X ( ζ ) and ξ = ζ − ζ . We have introduced the focus time ζ so that C ( ζ ) = 0. The relation (62) has been fruitfully usedin [38] and [41] to extract the beam quality factor fromexperimental images. Conclusion
Relying on the deep analogy between light waves andmatter waves, we have introduced theoretical tools todeal with the propagation of coherent matter waves : • The eikonal approximation is the standard treat-ment of geometrical optics. It is valid when diffrac-tion, or wave-packet spreading, is negligible. It canbe fruitfully used to treat short time propagation,as we show on the exemple of an atom laser beamcrossing its source BEC. • The Fresnel-Kirchhoff integral comes from the clas-sical theory of diffraction. It is particularly power-ful as it allows to deal with piecewise defined poten-tial in two or three dimensions together with takinginto account diffraction and interference effects. • The
ABCD matrices formalism can be used as soonas the matter wave is in the paraxial regime. Thiswidely used technique in laser optics provides sim-ple algebraic laws to propagate the atomic wave-front, and also global laws on the second order mo-ments of the beam, as the Rayleigh formula. Thoseresults are especially suitable to characterize atomlaser beams quality by the M factor.The toolbox developed in this paper can efficientlyaddress a diversity of atom-optical setups in the limitwhere interactions in the laser remain negligible. It canbe suited for beam focussing experiments [59, 60] andtheir potential application to atom lithography [61]. Italso provides a relevant insight on beam profile effectsin interference experiments involving atom lasers or tocharacterize the outcoupling of a matter-wave cavity[62]. It could also be used in estimating the couplingbetween an atom laser beam and a high finesse opticalcavity [63]. Further developments may be carried outto generalize our work. In particular, the M factorapproach could be generalized to self interacting atomlaser beam in the spirit of [64] or to more general casesof applications, such as non-paraxial beams or morecomplex external potential symmetries [65]. Acknowledgments
The LCFIO and SYRTE are members of the InstitutFrancilien de Recherche sur les Atomes Froids (IFRAF).This work is supported by CNES (No. DA:10030054),DGA (Contracts No. 9934050 and No. 0434042), LNE,EU (grants No. IST-2001-38863, No. MRTN-CT-2003-505032 and FINAQS STREP), ESF (No. BEC2000+ andQUDEDIS).
APPENDIX A: TIME-INDEPENDENTPROPAGATOR IN AN INVERTED HARMONICPOTENTIAL
The time-dependent propagator of the inverted har-monic potential can be straightforwardly deduced fromits expression for the harmonic potential [45] by chang-ing real trapping frequencies to imaginary ones ( ω → iω ). We derive here an analytic evaluation of its time-independent counterpart G E , by using the results of sec-tion I B 2.We consider a potential in dimension d , characterizedby the expulsing frequency ωV ( r ) = V ( ) − X j ∈ [[1 ..d ]] mω r j . (A1)By introducing the reduced time s = ωτ and the har-monic oscillator size σ o = p ~ /mω , G E is expressed as G E ( r , r ′ ) = Z ∞ ds H ( s ) e iφ ( r , r ′ ,s ) , (A2)with H ( s ) = m/ (2 πi ~ sinh s ), and φ = (cid:2)(cid:0) r + r ′ (cid:1) cosh s − r · r ′ (cid:3) σ sinh s + ( E − V ( )) ~ ω s. (A3) The first-order stationary times s ± verifycosh s ± = − b ± p b + 4( E − V ( )) c E − V ( )) , (A4)where b = mω r · r ′ and c = E − V ( ) + mω ( r + r ′ ) / s ± , G E reads (eq.16) G (1) E ( r , r ′ ) = X s ± > s iπ∂ φ/∂s (cid:12)(cid:12) s ± H ( s ± ) e iφ ( s ± ) . (A5)Otherwise, the relevant stationary point s (eq. 19)verifiescosh s = r + r ′ + q ( r + r ′ ) ( r − r ′ ) r · r ′ . (A6) s is the time associated with the classical trajectory con-necting r ′ and r with the closest energy to E . If theangle beetween r and r ′ is above π/
2, then accordingto equation (A6), the absolute value of the first deriva-tive of φ is never minimal, so that e iφ ( s ) quickly oscil-lates over [0; + ∞ ) and one can take G E ( r , r ′ ) = 0. Inother cases, where the solution is unique, one developsthe phase around s and G E finally expresses as (eq. 20) G (2) E ( r , r ′ ) = 2 π H ( s ) κ e iφ ( s ) Ai (cid:18) − κ ∂φ∂s (cid:12)(cid:12)(cid:12) s (cid:19) , (A7)where κ = (cid:16) − ∂ φ∂s (cid:12)(cid:12) s (cid:17) / . APPENDIX B: EXACT SOLUTIONS OF THETWO-DIMENSIONAL INVERTED HARMONICOSCILLATOR AND RELATION WITH THEEIKONAL
In this appendix, we give an analytical expression forthe eigenfunctions of the inverted harmonic potential inthe BEC region. The use of such solutions enable us toavoid any divergence of the eikonal solution close to theturning point.Using dimensionless parameters introduced in Eq.(57), the time-independent Schr¨odinger equation in theBEC region reads − (cid:18) ∂ ψ∂R + 1 R ∂ψ∂R + 1 R ∂ ψ∂α (cid:19) − R ψ = ǫψ. (B1)Introducing the angular momentum L α = ~ i ∂ψ∂α , one candecompose the solution of this equation as the productof a radial part and an angular part ψ ( R, α ) = φ ( R ) e ilα , (B2)with l ∈ Z and ~ l is the angular momentum of the wavefunction. The general solution φ is given by φ ( R ) = c R M (cid:18) − i ǫ l iR (cid:19) + c R W (cid:18) − i ǫ l iR (cid:19) . (B3)0M( µ, ν, z ) and W( µ, ν, z ) are Whittaker functions (re-lated to the confluent hypergeometric functions of thefirst and second kind) [48] whereas c and c are com-plex coefficients.In general, the wave function must be decomposed onthe basis of the different solutions φ ( R ) parameterized by l and ǫ . However, in the following, we restrict ourselvesto the study of a solution that connects asymptoticallyto the eikonal. Thus, we are only interested in the wavefunction describing a dynamics without any transversespeed or diffraction, i.e. with l = 0. Since the waveprogresses from the rf knife R to the outer part of thepotential, we also only look for “outgoing wave” typesolutions [56]. Such solutions behave as progressive wavesin the asymptotic limit ( R → ∞ ). One can express theWhittaker functions in term of hypergeometric functions[48] for any complex parameter µ and z M( µ, , z ) = e − z/ √ z F (cid:18) − µ ; 1; z (cid:19) , (B4)W( µ, , z ) = e − z/ z µ F (cid:18) − µ, − µ ; ; − z (cid:19) . (B5)For | z | → ∞ , these functions are asymptotically ex- panded as [57] F ( a ; b ; z ) ∼ Γ( b )Γ( b − a ) ( − z ) − a F (cid:18) a, a − b + 1; ; − z (cid:19) + Γ( b )Γ( a ) e z z a − b F (cid:18) b − a, − a ; ; 1 z (cid:19) , (B6)and F (cid:18) a, b ; ; 1 z (cid:19) −→ O (cid:18) z (cid:19) . (B7)One thus obtains an asymptotic formula for equation(B3) in which terms proportional to e iR / or e − iR / appear. 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