Table Of ContentCollapses and revivals of stored orbital angular momentum of light in a cold atomic
ensemble
D. Moretti, D. Felinto, and J. W. R. Tabosa
Departamento de F´ısica, Universidade Federal de Pernambuco,
Cidade Universit´aria, 50670-901 Recife, PE, Brazil
(Dated: January 8, 2009)
We report on the storage of orbital angular momentum of light in a cold ensemble of cesium
atoms. We employ Bragg diffraction to retrieve the stored optical information impressed into the
atomic coherence by the incident light fields. The stored information can be manipulated by an
appliedmagnetic field andwewereable toobservecollapses andrevivalsduetotherotation of the
stored atomic Zeeman coherence for times longer than 15 µs.
9
PACSnumbers: 42.50.Gy, 42.50.Ex,42.50.Va
0
0
2
Light beams carrying orbital angular momentum [21, 22, 23, 24]. For instance, a light vortex was stored
n
(OAM) have attracted an enormous recent interest ow- in a hot vapor for hundreds of microseconds and its ro-
a
J ing to the possibility of encoding quantum information bustness against diffusion demonstrated [21]. However,
in a multidimensional state space [1], to their use to ex- to date the storage and manipulation of superpositions
7
cite vortices in Bose-Einstein condensates [2], as well as of OAM states into an atomic ensemble, as well as the
] to a number of others interesting applications [3]. One characterizationof the retrievedstates, has not yet been
h
important family ofthese beams, the Laguerre-Gaussian achieved.
p
- (LG) modes of the electromagnetic field [4], possesses In this work, we report the storage of superpositions
nt wave-fronts dislocation or vortices specified by a topo- of OAM states as well as its manipulation through an
a logical charge m, which sets its OAM along the propa- applied transverse magnetic field, which reveals the col-
u gation direction as being equal to m¯h per photon. The lapses and revivals of the stored information mapped in
q
coherent and nonlinear interaction of light beams carry- theatomicensemble. Sinceanyquantumprotocolswould
[
ingOAMwithatomicsystemshavebeenreportedprevi- involve necessarily arbitrary multimode coherent super-
1 ously using different experimental schemes [5, 6, 7, 8, 9]. positions of these quantum states, its storage and ma-
v
Fromtheperspectiveofquantuminformationprocessing, nipulation, even in the classical limit, constitute a prove
9
the use of multidimensional state space has a promising of principle of such requirement. Differently from pre-
3
9 prospect to achieve higher quantum efficiency [10]. In- vious schemes used to store images in thermal atomic
0 deed, entanglement between photons with OAM and a ensembles,weemployadelayedbackwardfour-wavemix-
1. cold atomic ensemble was already reported in [11] and ing (FWM) configurationwhich allowsus to retrieve the
0 more recently the generation of twin light beams with stored information in a different direction, thus strongly
9 OAMwasachievedviafour-wavemixingin ahotatomic facilitating its imagery.
0 vapor [12].
For the experiment we used the Zeeman sublevels
:
v However, further development in this field is strongly of the degenerate two-level system associated with the
i
X conditioned to our capability of reversiblystore and ma- cesium D2 line cycling transition 6S1/2(F = 3)
′ ↔
r nipulatethesehigherdimensionalquantumstatesoflight 6P3/2(F = 2). The cesium atoms were obtained from
a intolong-livedatomiccoherences. Thelightstorage(LS) a magneto-optical trap (MOT) operating in the closed
′
in an electromagnetically induced transparency (EIT) transition 6S1/2(F = 4) 6P3/2(F = 5) with a recy-
↔
medium [13], which allow us to obtain later information cling repumping beam resonantwith the open transition
′
of a previously stored light pulse, is a well understood 6S1/2(F = 3) 6P3/2(F = 3). The atoms were pre-
↔
phenomenon and was originally described in terms of pared in the state 6S1/2(F = 3) by switching off the re-
a mixed two component light-matter excitation, called pumpingbeamforaperiodofabout1mstoallowoptical
dark state polariton [14]. However, in a simpler alter- pumpingbythetrappingbeamsvianon-resonantexcita-
′
native picture, LS can be described as being due to the tion to the excited state F = 4. The MOT quadrupole
creation of a ground state coherence grating which con- magnetic field is also switched off during these optical
tains information on the amplitude and phase of an op- pumping period. Typically, the measured optical den-
tical field and which survives after the switching off of sity of the sample of cold atoms in the F = 3 ground
the incident fields. To date, several experimental ob- state is approximately equal to 3 for appropriate MOT
servations of this phenomenon were realized in different parameters. Three pairs of Helmholtz coils are used to
systems [15, 16, 17, 18, 19, 20]. Recent theoretical and compensate for residual magnetic fields. In Fig. 1 (a),
experimentalworkhavealsoaddressedthestorageofspa- (b) we show a generic Λ three-level system consisting
tial structures of light beams (images) in atomic vapors of two degenerate ground states and one excited state
2
belonging to the corresponding hyperfine Zeeman man-
ifold, coupled with the incident beams at two different
instants of time, according to the time sequence showed
(a) (b)
in Fig. 1-(c) and the beam geometry of Fig. 1 (d). As
dlaebpeilclteeddains tWhesaenfidgWure′,s,htahveetwopopionsciitdeenctirwcurliatirnpgoblaearimzas-, 6P3 /2, F’=2 mF mF
tion and are incident on the MOT forming a small angle W, s + W´, s - D, s + R, s -
θ 30. These beams will therefore excite a ground-
sta≈te coherence grating into the atomic ensemble. The 6S1 /2, F=3 mF- 1 mF+1 mF- 1 mF+1
induced coherence grating is probed by a reading beam
R, which is counter-propagatingto the writing beam W (c)
andhasacircularpolarizationoppositetothatbeam. In
thecontinuouswave(cw)excitationoftheensemble,this W, W’ “ON” “OFF”
corresponds to the well-known backward FWM configu-
ration [25]. For the time sequence shown in Fig. 1(c), R “OFF” “ON”
theInreRtreife.v2e0dwsiegnsahlocwoerdretshpaotndthsetoditffhreacBtreadggfiedldiffDracgteionn-
oerfabteeadmbyRtinhteoatthoemsitcoreendsecmohbelreeniscegigvreantinbgyatnhdeacsohsheroewnnt ts=0 Time
′
bsuemlowoficsocnoturinbtuerti-opnrospfraogmatainllgetxocitthede awtroimteicbedaipmolWes,.with (d)
such dipoles obtained from the expression for the slowly CCD
varying coherences
g (t)e−γtsΩ (~r)Ω (~r)Ω∗ (~r) M BS PD
σ2,1a(~r,t)= R ΩW(~r)R2+ ΩWW′(~r)2W′ , (1) L1 LG01 BS D
| | | | HM LG00 L2 BS W’ MOT
where, according to Fig. 1 (a) and (b) we labelled 2 and
1a the excited state m and the ground state m W r R
| Fi | F − /l2 PBS l/2 PBS /l4 B l/4 PBS /l2
1 , respectively. In this equation, Ω (~r) represents the
i
i
Rabi frequency associated with beam i in an atom at
FIG. 1: (a) Generic Zeeman three-level system, showing the
position~r,beinggiventhenbyΩi(~r)=µa,b i(~r)ei~ki·~r/¯h, coupling with the grating writing beams W and W′. Here
E
with µa,b the dipole moment between the levels a and b thebeam W has amuchhigher intensitythan thebeam W′,
coupledbyΩi, i(~r)theelectricfieldenvelopeofbeami, so theatoms are optically pumped into the highest mF level
E
and~k its wave-vector. The term e−γts is a consequence and this level scheme would correspond to mF =2. (b) The
i
couplingofthereadingbeamRwiththecoherentlyprepared
of the effective decay of the ground state coherence due
three-level atom, generating the diffracted beam D. (c) The
to a residual inhomogeneous magnetic field, with γ the
switching time sequence for the writing and reading beams.
decay rate and t the “storage time”. Finally, g (t) is a
s R (d) Simplified experimental scheme. All the beams are pro-
function describing the shape of the light pulse in mode vided by a diode laser locked to the F =3↔F′ =2 transi-
D retrieved by the read pulse R: tion,andareintensitymodulatedusingacousto-opticalmod-
ulators not shown in the scheme. B~ : Magnetic field; PBS:
e−γ1tsinh(γ2t) Polarizingbeamsplitter;BS:Beamsplitter;HM:Holographic
g (t)= , (2)
R γ2 mask; CCD: Camera; PD: Photodetector; L: Lens.
with γ1 = Γ22/2+γ, γ2 = p(Γ22/2−γ)2−4|ΩR|2/2, approximated by
and Γ22 is the spontaneous decay rate of level 2. In the
arebaodviengeqpuualtsieo.nsTthe=re0trcieovrerdesppounlsdessthoaptheeanstdarittsosfatthue- σ2,1a(~r,t)=AgR(t)e−γtsEW∗ ′(~r)e−i~kW′·~r, (3)
ration and decay dynamics were extensively studied in with A a constant in position and time, depending only
Ref. 20. on the pumping-beams intensities and the dipole mo-
Inour presentexperiment, fields W andRcanbe well mentsoftheinvolvedtransitions. Equation(3)explicitly
approximatedbyplanewavesexcitingtheatomicensem- indicatesthat,inourexperimentalconditions,theDfield
ble. On the other hand, since the waist of beam W′ is propagates in the direction ( ~kW′) opposite to W′, and
− ∗
considerably smaller than the cloud of atoms, the trans- have a transverse beam profile given by (~r). As we
′ EW′
verse profile of beam W is what actually defines the will show, for a Laguerre-Gaussianmode in W′(~r), this
E
excited-ensemble shape. Taking this into account, the implies that the OAM of the retrieved beam will be in-
factthat~k = ~k ,andthattheintensityofW ismuch vertedwithrespectedtotheW′ beam. Equation(3)also
W R
− ′
larger than the one of W , the expressionfor the excited shows that the retrieved pulse amplitude decays with a
coherenceresponsiblefortheretrievedfieldDcanbewell rate determined by the ground-state-coherencelifetime.
3
We consider now explicitly the case where the inci- thetopologicalchargeoftheretrievedbeam,weconsider
′
dentwritingbeamW containsLaguere-Gaussianmodes as a reference wave the diffracted beam associated with
with topological charge m, i.e., LGm, described, in po- a incident Gaussian beam W′ = LG0, whose retrieved
0 0
lar coordinates (r,φ,z) in the plane z = 0, by a field profile is shown in Fig. 2 (d). In Fig. 2 (c) we show the
amplitude W′ (√2r/ω0)mexp( r2/ω02)e−imφ, where image for the incident superposition W′ = LG10+LG00,
E ∝ −
ω0 is the minimum beam waist. As shown in Fig. 1 obtained by retro-reflecting it with a mirror. In Fig. 2
′
(d), the writing beam W can be composed either by (e), (f) and (g) we show the corresponding retrieved im-
a Laguerre-Gaussian mode with a topological charge agesobtained for the different storagetimes indicated in
m=1,i.e.,LG1,generatedbytheholographicmaskHM each image. The retrieved signal intensity, as measured
0
and the lens L1, or by a zero charge Laguerre-Gaussian simultaneously with the photodetector, decays exponen-
mode, or by a superposition of these two modes. These tially with a time constant of approximately 3µs, which
twobeams,haveapproximatelythesamepowerof20µW we attribute mainly to non-compensated magnetic field
and are focused by the lens L2 into the MOT with a di- gradients [20]. Clearly, these images revealthat incident
ametermuchsmallerthanthe sizeofthe trappedatomic andtheretrievedbeamshavethesamesingletopological
cloud( 2mm). The writingbeamW isasimple Gaus- charge. However, in order to determine its correspond-
sianmo≈de (LG0), hasa powerof 2.0mW anda diam- ing sign, we carefully have to take into account that the
0
≈
eteroftheorderofthetrappedcloud. Apairofacousto- sense of rotation of the spiral in the interference pattern
optical modulators (AOM) is used to scan the common (which is determined by the sign of m) changes upon re-
′
frequency of the writing beams W and W around the flection on a mirror and after passing through a focus
atomic resonance, as well as to allow their fast switch- [6]. This leads to two consecutive reversionsin the sense
ing. ThereadingbeamR,hasGaussianintensityprofile, of the spiral associated with the imagery of the incident
′
power,anddiameterofthe sameorderofbeamW,pass- W superposition. Accordingtotherecordedinterference
ing also through another pair of AOM’s. patternsshowninFig. 2(c)and(e)-(g),wecantherefore
conclude that the incident and the retrievedbeams have
(a) (b) the same sign for its topological charge. Moreover, as
these two beams propagates in opposite directions they
shouldcarryoppositeOAM.Theseobservationsnotonly
demonstrates that superpositions of OAM states can be
ts=1.0m mmm s stored into the atomic ensemble, but also the reversible
transfer of OAM between light and atoms.
(c) (d)
In another series of experiments, we applied an exter-
nal dc magnetic field of magnitude B 0.6 G, nearly
≈
orthogonal to the plane defined by the incident beams,
andthereforeperpendiculartothe quantizationaxis,de-
ts=1.0m mmm s finedalongthebeamspropagationdirection. Inthiscase
(e) (f) (g) weobserveaseriesofcollapsesandrevivalsofthe stored
Zeeman coherence as can be seen in Fig. 3, which shows
the amplitude of the retrievedsignalassociatedwith the
input writing beam W′ = LG1+LG0 for different stor-
0 0
age times. It is worth noticing that a similar collapses
ts=1.0m mmm s ts=2.0m mmms ts=3.0m mmms
and revivals observation has been reported before and
FIG. 2: (a) The incident writing beam intensity profile for was interpreted as being due to the Larmor precession
the case W′ = LG10; (b) and (d) The retrieved beam associ- of the collective spin excitation aroundthe applied mag-
ated respectively with an incident writing beam with modes netic field [27]. Differently from the results presented in
W′ =LG10 and W′ =LG00; (c) Incident writing beam profile Fig. 2, we have observed approximately a four-fold in-
corresponding to a beam superposition W′ = LG10 +LG00. crease in the decay time of the stored coherence grating
Thisimage is obtainedbyretro-reflectingtheincident super-
in the presence of the applied magnetic field. Probably
position right after the MOT. (e), (f) and (g) the retrieved
thisisassociatedtoareducedeffectofthemagneticfield
signal corresponding to the incident superposition shown in
(c) for the different storage times indicated in thefigure. All inhomogeneityduetothemuchstrongerapplieduniform
this set of measurements is obtained without magnetic field. magnetic field.
For the applied magnetic field, the associated Larmor
Under the EIT resonant condition, we have recorded period T = 2π/Ω , where Ω = g µ B/¯h is the Lar-
L L L F B
the image of the retrieved beam D for different incident mor frequency, g the Lande factor for the lowerground
F
′
W modes: In Fig. 2 (a) and (b) we show respectively state F = 3 and µ the Bohr magneton, is T 5µs
B L
theimageoftheincidentW′ =LG1 modeandtheimage [26]. Thus, revivals of the stored coherence should≈occur
0
ofthe correspondingretrievedbeam. Inorderto analyze forintegermultiples ofthe Larmorperiod. Furthermore,
4
at half the Larmor period the Zeeman coherence and our knowledge,this constitute the first demonstrationof
populations are just reversed and therefore we should such manipulation of OAM stored in atomic coherences.
also expect partial revivals of the retrieved signal Insummary,wehaveexperimentallydemonstratedthe
for odd multiples of half the Larmor period. Theses storage of OAM of light in a cold ensemble of cesium
considerations are completely consistent with the series atoms, as well as the reversible exchange of OAM be-
of revivals observed in Fig. 3. We have also recorded tween light and atoms. The manipulation of the stored
OAM was also demonstrated via Larmor precession of a
spatially distributed ground state coherence in an exter-
(cid:1)(cid:2)(cid:3) nal magnetic field. In particular, collapses and revivals
of the stored coherence were directly observed and re-
veal the control of a collective light-matter excitation.
We believe our results fill up one of the gaps challenging
the implementation of quantum information processing
(cid:1)(cid:3)(cid:4)(cid:3)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:2)
(cid:2) encoded in a multidimensional state space.
(cid:1)(cid:4)(cid:3)
We gratefully acknowledge fruitful discussion with A.
Lezama. ThisworkwassupportedbytheBrazilianAgen-
(cid:1)(cid:2)(cid:3) cies CNPq/PRONEX, CNPq/Inst. Milˆenio and FINEP.
(cid:1)(cid:3)(cid:4)(cid:3)(cid:12)(cid:6)(cid:13)(cid:3)(cid:8)(cid:2)
(cid:1)(cid:4)(cid:3) (cid:2)
(cid:1)(cid:5)(cid:3)
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