Quality factor of a matter-wave beam

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by  François Impens

Quality factor of a matter-wave beam.
Franc¸ois 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 Schr¨odinger 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
betweentheusualSchr¨odingerequationandtheparaxial 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 mo-
menta 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 parame-
ters 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. It generalizes the currently adopted
beam quality factor [M , Eq. (4)], indeed only appro- The author is greatly indebted to Christian J. Bord´e
1
priate to describe the propagation of matter-waves in for enlightening discussions on atom optics. He thanks
the paraxial regime and in cylindrical potentials, and EmericDeClercqformanuscriptreading. Thisworkwas
which canleadto underestimate the optimalcollimation supported by DGA.
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