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Quality factor of a matter-wave beam. François Impens SYRTE, Observatoire de Paris, CNRS, UPMC ; 61 avenue de l’Observatoire, 75014 Paris, France. (Dated: January 29, 2010) Imperfectionsindiluteatomicbeamspropagatingintheparaxialregimeandinpotentialsofcylin- drical symmetry have been characterized experimentally through the measurement of a parameter analogous to a beam quality factor [Riou et al., Phys. Rev. Lett. 96, 070404 (2006)]. We propose 0 a generalization of this parameter, which is suitable to describe dilute matter waves propagating 1 beyondtheparaxialregime andin fully general linearatom-optical systems. Thepresentedquality 0 factor shows that the atomic beam symmetry can be traded for a bettertransverse collimation. 2 PACSnumbers: 03.75.Pp,41.85.Ew,31.15.xh n a J I. INTRODUCTION physical significance in optics. For cylindrical light 9 beams, a quality factor compares the divergence of a 2 There is a strong analogy between the propagation of non-ideal beam to a standard set by a perfect Gaussian ] light and matter waves [1]. It is manifest in the paraxial mode. It conveys the idea of degradationin the collima- s waveequationforthemodeU ofanelectromagneticfield tion or, equivalently, that of diminution in phase-space ga E(x,y,z) = U(x,y,z)ei(kz−ωt)eˆ which takes the form of density. A relevant quality parameter should thus be - a bidimensional Schrödinger equation left invariant during the propagation in perfect linear t n ∂U(x,y,z) ∂2 ∂2 optical systems which preserve the beam collimation. a 2ik =− + U(x,y,z)+F(x,y)U(x,y,z), ∂z (cid:18)∂x2 ∂y2(cid:19) u Such systems are modelled by a finite set of ABCD (1) q withafunctionF(x,y)dependingonthelocalrefraction matrices which describe linear input-output relations in . at index. A spectacular achievement proving the similarity the light beam phase-space [15]. Their atomic coun- m of the atomic and light fields was the realization of a terpart is the quantum evolution under an Hamiltonian quasicontinuous matter beam analog to a laser [2–9]. It quadraticinpositionandimpulsion,whichinvolvessimi- - d has been shown recently that the extraction of a quasi- lar relations with 3×3 ABCD matrices [16, 17]. The n continuous atom laser from a Bose-Einstein condensate phase space evolution of the atomic beam then also o byaweakRF[10–13]orRaman[14]outcouplinginvolves amounts to a time-dependent map in the arguments of c thepropagationthroughadiscontinuouspotentialwhich the Wigner distribution [1] [ degrades the beam collimation. To quantify this effect, 1 Riou et al. [10] introduced a parameter analogous to 1 v W(r,p,t)=W D(r−ξ)− B(p−mφ), a quality factor. It allows one to estimate the the (cid:18) m 8 transverse width of a cylindrical atomic beam falling in e e 6 −mC(r−ξ)+A(p−mφ), t . 3 a uniform gravitational field and in the paraxial regime, 0(cid:17) 5 i.e.,withatomsstronglyacceleratedalongoneprivileged e e . direction. The stands for the transposition, the vectors r,p are 1 the position and impulsion, the matrices A,B,C,D 0 0 However, an important difference between light andevectors ξ,φ are time-dependent parameters deter- 1 and matter waves is that the latter generally do not mined in Ref.[16]. Thanks to the unimodularity of the : propagate in this paraxial regime. This is typically ABCD matrices, the propagationunder those quadratic v the case for a condensate expanding after a sudden Hamiltonians preserves the phase-space density and as i X trap shut down, which can be viewed as a pulsed atom such do not alter the atomic beam collimation. In this r laser. Even for a weakly outcoupled atom laser, the view, these Hamiltonians can be considered as perfect a paraxialapproximationisoftenvalidonlyafter acertain aberrationless atom-optical systems and should thus propagation time [10] during which the beam must be leave invariant a well-defined quality factor. treated in a more general framework. A comparison betweentheusualSchrödingerequationandtheparaxial This is not the case for the parameter used so far to wave equation (1) shows that for these matter waves, characterize the quality of matter beams [10, 14]. This time plays the role devoted to the axis of propagation is disturbing, since its current definition would some- Oz in paraxial optics, and that the transverse space is timesleadtoignoringpossibleimprovementsoftheatom three dimensional and spanned by {eˆ ,eˆ ,eˆ } instead lasercollimationthroughnoncylindricalatom-opticalel- x y z of {eˆ ,eˆ }. ements such as astigmatic atomic lenses [18]. It seems x y thususefultoproposeanewqualityfactorforatomlasers To extend the concept of matter beam quality beyond which respects this invariance requirement. This is the the paraxial regime, it is necessary to reconsider its main purpose of this paper. 2 II. QUALITY FACTOR OF A CYLINDRICAL This definition is consistent if one considers only BEAM matter-waves and atom-optical systems with cylindrical symmetry, and a propagation in the paraxial regime: Letus firstreconsiderthe definitionofaqualityfactor the parameter M12 is then unaltered by the propagation for a cylindrical light beam, before extending this con- in aberrationless systems. cept to atom optics. Partially coherent light beams can be described by means of a first-order field correlation However, these conditions are far too restrictive to functionΓrelatingthefieldamplitudeatdifferentpoints characterize the matter beams in most experiments, of planes transverse to the direction of propagation O . which generally involve potentials without any symme- z Fora cylindricalbeam, onlya singletransversedirection try and a matter-wave propagation outside the paraxial Ox needs to be considered in the correlation function regime. Athree-dimensionalqualityfactorM3,invariant under the whole set of possible ABCD matrices, is then ∗ Γ(x ,x ,z)=hE(x ,z)E (x ,z)i. required. Following previous treatments of optical and 1 2 1 2 quantummechanicalinvariants[20–22], we now examine The Wigner transform of this function provides a phase how to properly define this parameter. space picture of the beam ′ ′ W(x,k ,z)= dx′Γ(x+ x ,x− x ,z)e−ikxx′, III. QUADRATIC PROPAGATION INVARIANT x Z 2 2 FOR MATTER BEAMS which can be used to define moments in the transverse position and wave vectors OurgoalistoreplacetheparameterM1byanABCD- invariant combination of moments which still provides an insight on the matter beam divergence. In the unidi- hm(x,k ,z)i= dxdk m(x,k ,z)W(x,k ,z). x Z x x x mensional case, the price to pay to express the product ∆x| ∆k | (3) as an invariant expression (2) was to w x w Because the beam is cylindrical and transverse introduce an additional moment hx k i mixed in wave x hxi = hkxi = 0. The moments ∆x = hx2i and vector and position. This term can be viewed as an el- ∆k = hk2i are, respectively, the transverpse squared ement of a variance matrix. Such matrices have been x x width anpd wave-vectordispersions of the beam. introduced in optics [20], and they can also characterize a partially coherent atomic beam [23]: Optimalcollimationisachievedwithaperfectlycoher- ent Gaussian mode, for which the position and direction ∆rr ∆rv V = . moments verify at the waist (cid:18)∆rv ∆vv (cid:19) 1 e ∆x| ∆k | = . The matrices ∆ab are w,Gaussian x w,Gaussian 2 ha b i ha b i ha b i Partially coherent cylindrical light beams have a lessen x x x y x z collimation quantitatively estimated thanks to a quality ∆ab = haybxi haybyi haybzi  factor M2 [19]. This parameter can be expressed as a hazbxi hazbyi hazbzi   combinationofmoments left invariantduring the propa- with the vectors a,b =r,p/m. We now derive the vari- gation in aberrationless cylindrical optical systems anceevolutionwhentheatomicfieldpropagatesunderan M2 Hamiltonianwhichisquadraticinpositionandimpulsion = hx2i| hk2i| − hxk i|2. (2) 2 z x z x z p pˆβ(t)pˆ m Atthebeamwaist,thisexpressionreducestotheproduct H(ˆr,pˆ)= −ˆrα(t)pˆ− ˆrγ(t)ˆr−mg(t)·ˆr+f(t)·pˆ. 2m 2 of transverse squared width and divergence which has a e (5) e e clear interpretation in terms of phase-space dispersion α(t), β(t) and γ(t) are 3×3 matrices; f(t) and g(t) are M2 three dimensional vectors. It follows from the equations =∆x| ∆k | . (3) w x w 2 ofmotionforthepositionandimpulsionoperatorsinthe Heisenberg picture that the variance matrix V satisfies This is indeed the definition which has been adopted to expressthe qualityofamatterbeam[10,14],oncetaken dV α(t) β(t) intoaccountDeBroglierelationbetweenmomentumand =ΓV +VΓ Γ(t)= . wave vector p=~k dt (cid:18) γ(t) α(t) (cid:19) 2 The integration of this last equation involves the time- M12 = ~∆x|w∆px|w. (4) ordering operator T and the atom-optical ABCD ma- 3 trix [24]: Its expression in terms of position and impulsion momenta is A(t,t ) B(t,t ) M(t,t0) = (cid:18)C(t,t00) D(t,t00)(cid:19) M34 = 34~2 hx2ihp2xi−hxpxi2+hy2ihp2yi−hypyi2 t ′ ′ (cid:2) = T exp − dt′ α(t′) β(t′) . + hz2ihp2zi−hzpzi2+2(hxyihpxpyi−hxpyihypxi) (cid:20) Z (cid:18) γ(t) α(t)(cid:19)(cid:21) t0 + 2(hxzihpxpzi−hxpzihzpxi)+2(hyzihpypzi−hypzihzpyi)] . The evolution of the variance matrix in the quadratic (10) Hamiltonian (5) is then similar to the transformation laws in optics Thisexpressioncanberewritteninamorecompactform asasumofmomentsalongsingleandmultipledirections: V(t) = M(t,t )V(t )M(t,t ). (6) 0 0 0 1 The most straightforwardattempt tfo generalize the uni- M34 = 3[Qx+Qy+Qz−Axy−Axz −Ayz] dimensionalqualityfactorM2 tothreedimensionswould 1 4 consist in considering norms and scalar products in Qη = ~2hη2ihp2ηi Eq.(2) instead of single coordinates 8 ′ ′ Aηη′ = ~2(hηpηihη pη′i−hηη ihpηpη′i) (11) hx2+y2+z2ihp2 +p2+p2i−hr·pi2. x y z q The quantities Q correspondto the fourth powerof the η Ataninstantsatisfyinghr·pi=0,the multidimensional previousmatter-wavequalityfactorM consideredalong 1 equivalent of a “waist”, this expression would express the three spatial directions, while the terms Aηη′ reflect the greater phase space occupied by the matter beam. correlations between different directions. If one chooses Unfortunately, this quantity is not left invariant in the thecoordinatesystemalongthebeamprincipalaxis,the propagationunder a general quadratic Hamiltonian. ′ ′ position momenta satisfy hηη i = 0 for η 6= η , yielding a simpler expression for Aηη′: Nonetheless, it is possible to define a quality factor which respects the invariance requirement and still pro- 8 ′ vides a meaningful insight into the beam phase space. Aηη′ = ~2hηpη′ihη pηi WeusethefactthattheABCDmatricesassociatedwith matter wavepropagationaresymplectic, i.e., they verify These two families of parameters have different physical at all times the following relation significance and obey different constraints. The parameters Q are a manifestation of the beam divergence and M(t,t0)−1 =LM(t,t0)L⇐⇒L= M(t,t0)LM(t,t0), are bouηnded below by the Heisenberg principle: Qη ≥1. e f (7) The terms Aηη′ reflect the beam asymmetry and can with the matrix be of either sign, they cancel for spherical clouds. As previously announced, a Gaussian matter wave satisfies 0 −1 L=i . M =1. (cid:18)1 0 (cid:19) 3 Thissymplecticstructurecanindeedbeusedtogenerate This atomic beam quality factor thus reflects the de- a family of invariants [20–22]. The lowest order of this parturefromafundamentalGaussianmode: abeamwith invariant family indeed generalizes the definition of the a quality factor M ≫1 needs to be expanded onto sev- 3 current matter beam quality factor eral modes, which is likely to degrade the fringe pattern inanatomicinterferenceexperimentortoaddadditional 4m2 M4(t)= Tr[V(t)LV(t)L] . (8) noiseintheatomicbeam[25]. Inprinciple,ifthesemodes 3 3~2 were put into a fully coherent and accurately controlled The constant is adjusted to yield M =1 for a Gaussian superposition, a high value of the quality factor would 3 matter wave. Provingthe invarianceofthis parameteris merelyinduceanadditionalcomplexityinthefringepat- simple and identical to optics [20]. Combining relation tern. Thisidealsituationis,however,notencounteredin (6) describing the variance matrix propagation with the practice,sincethecoefficientsofthemodedecomposition cyclic property of the trace, the definition (8) can be are generally not accessible. For most precision interfer- recast as ometric measurement, a simple TEM mode is indeed 00 M34(t)= 43m~22TrhV(t0)M(t,t0)LM(t,t0)V(t0)M(t,t0)LM(t,t0)ihq.uigahliltyypfraecfteorrabolef.thIenltahseerLiInGvoOlveedxpiesrimmaeinntt,atinheedbaetama f f f value M ≃ 1 thanks to a mode-cleaning step which fil- The symplecticrelation(7)thengivesthe desiredinvari- tersouthighordermodes[26,27]. Thisqualityfactorre- ance of M (t): 3 quirement, reflecting the control on the beam structure, 4m2 should also apply to accurate interferometers involving M34(t)= 3~2 Tr[V(t0)LV(t0)L]=M34(t0). (9) atom lasers. 4 Itseemsingeneralmoreadvantageoustominimizethe of an atom laser. Higher order invariants propor- quantitiesQη whicharedirectlyrelatedtothebeamcolli- tional to Ik = m~ 2k Tr[V(t)LV(t)L]k could also be mation. Inthisrespect,the factthatthe invariantquan- considered to d(cid:0)esc(cid:1)ribe the atomic beam, but they do tity be M3 instead of the coefficients Qη has a practical notprovidethesameinsightintothephase-spacedensity. interest. Consideringforinstanceacylindricalbeamwith given Qx = Qy = Qr = M14 and Qz, it is possible to re- Note: Since the publication of this paper, a different duce Q while keepingthe qualityfactorM constantby r 3 nonlinear matter wave quality factor has been pro- increasingQ orbygeneratingnegativeasymmetricalpa- z posed [29], which is preserved in presence of mean-field rametersAηη′ [28]. Onecanthustradesomelongitudinal interactions. However, this nonlinear parameter is collimation or cylindrical symmetry for a better trans- suitable only for uniform and paraxial atomic beams verse collimation. The corresponding ABCD transfor- propagating in transverse potentials of cylindrical mationcouldbeimplementedbyastigmaticatom-optical symmetry. The generalizations of the matter wave lenses such as electromagnetic waves with an asymmet- quality factor presented in this paper and in Ref. [29] ricwave-front[18]. Thispossibilityofimprovementisig- are indeed complementary: they enable one to apply nored in Ref. [10, 14], which assume that the transverse the concept of beam quality beyond the paraxial and divergence measured at the waist (i.e. M2) is an upper 1 beyond the linear regime of propagation respectively. bound for the atom laser collimation in the subsequent Thenon-paraxialandthenon-linearmatterwavequality propagation. factorscanthusbefruitfullyappliedindifferentcontexts. IV. CONCLUSION A quality factor has been defined for matter waves ACKNOWLEDGEMENTS [M , Eq.(10)], which addresses the general propagation 3 of an atom laser. 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