Experimental observation of transient velocity-selective coherent population trapping in one dimension

0.19 MB·English

by  Frank Vewinger

APS/123-QED
Experimental observation of transient velocity-selective coherent population trapping
in one dimension
Frank Vewinger∗ and Frank Zimmer†
Fachbereich Physik
Technische Universit¨at Kaiserslautern
D-67663 Kaiserslautern
(Dated: February 1, 2008)
Wereporttheobservationoftransientvelocity-selectivecoherentpopulationtrapping(VSCPT)in
abeamofmetastableneonatoms. Theatomicmomentumdistributionresultingfromtheinteraction
with counterpropagating σ+ and σ− radiation which couples a Jg = 2 ↔ Je = 1 transition is
measured via the transversal beam profile. This transition exhibits a stable VSCPT dark state
formed by the two |J = 2,m = ±1 > states, and a metastable dark state containing the |J =
5 2,m = ±2 > and |J = 2,m = 0 > states. The dynamics of the formation and decay of stable
0 and metastable dark states is studied experimentally and numerically and the finite lifetime of the
0 metastable dark state is experimentally observed. We compare the measured distribution with a
2
numerical solution of themaster equation.

n
a PACSnumbers: 32.80.Pj,42.50.Vk,33.80.Ps
J

7
2 Velocity-selective coherent population trapping in the metastable state 3P2.
(VSCPT) occurs by optical pumping of atoms into The coupling scheme for the Jg = 2 ↔ Je = 1 tran-
1 states with well defined momenta, which are decoupled sition driven by σ+ and σ− irradiation with equal fre-
v from the light field. In a Jg = 1 Je = 1 transition quencyisshowninFigure1. Thesystemischaracterized
4 a dark state exists, which is a tra↔pping state for zero by the Hamiltonian,
6
momentum, leading to efficient cooling below the
1
one-photon recoil limit [1, 2], which may be assisted by p2
01 polarization-gradient precooling [3]. For ground state H = 2M +~ω0Pe+VAL, (1)
5 angular momenta Jg > 1 several dark states may exist.
0 These are not necessarily eigenstates of the kinetic where Pe is the projector onto the excited states,
/ energy Hamiltonian and thus they are transient [4], not p the momentum operator and M is the atomic
h
p leading to population trapping. The stability of those mass. The interaction Hamiltonian VAL = VΛ + VIW
- states may be recovered by introducing m-dependent consists of two parts. One describes the Λ-system
nt shifts of the Zeeman sublevels by a dc stark field [5] or g−1,e0,g+1 , the other the inverted-W configuration
{ }
a additional laser fields [6]. The extension of VSCPT on g−2,e−1,g0,e+1,g+2 . Intherotatingwaveapproxima-
{ }
u three dimensions has also been discussed [7]. The exis- tion the two terms read
q
tence of transient dark states has for instance been used
: ~
iv tino cdeesmiuomns[t8r]a.teFuartmhuerltmipolree,bfeoarmspaetcoimficicpoinlaterrizfeartoiomnesteorf VΛ = X 2q130{Ω+|e0,qihg−1,q−~k| (2)
X q
r thhaesblaeseenr sfiheoldwsnt[h9e].eRxiescteenntcleyosftuhdigiehs-voeflovecliotycidtya-rskelsetcattivees + Ω−|e0,qihg+1,q+~k|}exp(−iωLt)+h.c.,
a ~
ctroohmeraegnntetpicoaplulylaitniodnucterdaptrpainngspearnednciyts(EreIlTa)tioanndtoatoemlecic- VIW = Xq 2hq160Ω+|e−1,q−~kihg−2,q−2~k| (3)
entanglement where reported [10].
+q110 Ω−|e−1,q−~ki+Ω+|e+1,q+~ki hg0,q|
For a J =2 J =1 transition coupled by counter- (cid:2) (cid:3)
g e
propagatingσ+↔and σ− beams the existence of transient +q160Ω−|e+1,q+~kihg+2,q+2~k|ie−iωLt+h.c.,
VSCPThasbeenshowntheoretically[4],buttothe best
of our knowledge experimental momentum distributions where Ω+ (Ω−) is the Rabi frequency of the coupling
have not been reported. In particular the metastability laser for the transition g0 e+1 (g0 e−1 ).
ofsomeofthedarkstateshasnotbeendirectlyshownin TheinteractionHami|ltoin↔ian|V ise|paira↔tes| theiatomic
AL
the experiment. We present measurements that clearly statesintotwosetsofmomentumfamilies,whicharecou-
show the signature of a stable anda transientdark state pled to each other only by spontaneous emission. These
in the momentum distribution of a beam of neon atoms are
FΛ(q) = e0,q , g−1,q ~k , q+1,q+~k (4)
{| i | − i | i}
∗Electronicaddress: [email protected] FIW(q) = {|g−2,q−2~ki,|e−1,q−~ki,|g0,qi,
†Electronicaddress: [email protected] e+1,q+~k , g+2,q+2~k . (5)
| i | i}
2
a ) b )
1 5 5 E n e rg y [ x 1 0 3 c m - 1 ]
3P
1 5 0 1 2 p 5 3 p Polarizer
l/4
e -1 g 0 e +1 3P 1 1 4 5
1 4 0
atomic beam
6 3 1 1 3 6 1 3 5 1P 1 2 p 5 3 s detector
3P 2 3P 3P 3P 2 l/4
g -2 g -1 g 0 g +1 g +2 0 1
V U V
1S 1 s 2 2 s 2 2 p 6 FIG. 2: Schematic setup of the experiment. The cylindrical
0
lenseshaveafocallengthof250mmandtheyareinaconfocal
arrangement. Furtherdetails can be found in thetext.
FIG. 1: a) Coupling scheme for the J = 2 ↔ J = 1 transi-
tion driven by σ+ and σ− polarized light. The numbers are
the squares of the ratios of the Clebsch-Gordan coefficients.
b) Level scheme of neon including the relevant levels for the tion of about 0.15~k. The population of the metastable
experiment. 3P0 stateisdepletedbyopticalpumpingbeforetheatoms
interact with the circular polarized laser beams in a re-
gion 20 cm downstream of the second slit. The spatial
TheFΛ(q)familyinvolvestwostateswhichhaveforq =0 distributionoftheatomsinthestates3P0 and3P2 ismea-
the same energy. Thus two-photon resonance within
sured 120 cm downstream of the interaction zone using
the Λ-system can be maintained using a single laser fre-
a movable channeltrondetector behind a entrance slit of
quency and the dark state ΨΛ (q = 0) formed by the
members of this family will|beNsCtable. Tihe FIW(q) fam- 2m5omµmentwuimdtohf, ∆lepadin0g.2t~oka. Irnestohleutiinotneroafcttihoen trreagniosnvetrhsee
ily involves states with different momenta. Thus two- ≈
magneticfieldisactivelycompensatedtolessthan1µT,
photon resonance between the states |g±2i and |g0i can to assure the degeneracy of all Zeeman states to within
notbemaintainedwithasinglelaserfrequency. Further-
better than 130 kHz. The laser beam passes through
more the dark state ΨIW formed within this family is
no eigenstate of the k|inNetCicienergy Hamiltonian p2/2M. a polarizer, two quarter wave plates and optionally two
cylindrical lenses before being retroreflected (Fig. 2), es-
Thelifetimeτ ofthisstatehasbeencalculatedpertur-
IW tablishing a counterpropagating σ+ σ−-configuration.
batively, assuming that all Clebsch-Gordan coefficients −
In order to realize different interaction times three dif-
are equal [4]. The two dark states read
ferentsetups for the width of the laser beams were used.
The first setup uses cylindrical lenses in a confocal ar-
1
|ΨΛNCi = Ω2++Ω2− rnaenagretmheenfotcwuist.hTthheetartaonmsiitctbimeaemofcrtohsesaintgomthsethlarsoeurgbhetahme
q
× Ω−|g−1,q−~ki−Ω+|g+1,q+~ki , (6) ltaosΓertbea1m0,iwshesetriemΓaitsedthteowbiedathfeowft1h0e0trnasn,sciotriorensbpeotnwdeineng
|ΨINWCi = (cid:2) Ω4 +6Ω12Ω2 +Ω4 hΩ2−|g−2,q−(cid:3)2~ki tshiteio3nP≈1ofanthde3Pat2omstaicteb.eTamheilsasneortbmeaemasuprreodfildeiraetctthlyebpuot-
+ + − −
q inferred from the dimensions of the optical setup. The
√6Ω+Ω− g0,q +Ω2+ g+2,q+2~k . (7) laser beams are parallel to within 10−5 rad. The peak
− | i | ii Rabifrequencyis in the orderof500MHz andthe lasers
Forinteractiontimesτ <τIW bothdarkstatesappearas are tuned from the |g0i↔|e0i resonance by 100 MHz to
trappingstatesforq =0,givingrisetoamomentumdis- reduce the influence of stray light from the windows. To
tribution with peaks at 2~k, ~k,0,~k and 2~k. For increase the interactionbetween the atoms and the light
interaction times τ > τ− the−contribution of ΨIW field the cylindrical lenses can be removed,leading to an
vanishes and only two peIaWks at ~k remain in th|e NmCoi- interaction time of Γt = 200. Using a telescope in front
mentum distribution. ± of the polarizer the beam diameter can be increased to
In the experiment, a beam of neon atoms emerges 8 mm, leading to an interaction time of Γt=800.
from a liquid nitrogen cooled discharge nozzle source. Figure 3 shows the initial momentum distribution
A fraction of the order of 10−4 of the atoms is in the (grey area) and the result for a short interaction time
metastable states 3P0 or 3P2 of the 2p53s electronic con- of Γt 10 (squares). Five peaks at 2~k, ~k and zero
≈ ± ±
figuration [11]. The flow velocity of the atoms is about momentum are clearly resolved. The asymmetry of the
470 ms−1 with a width of the velocity distribution of momentum distribution is due to different intensities of
about 100 ms−1 (FWHM). The beam is collimated by a the σ+ and σ− beams: The retroreflected beam passes
50 µm and a 10 µm slit positioned 144 cm apart, corre- twice through a window and the (uncoated) cylindrical
sponding to a width of the transversal velocity distribu- lensbeforecrossingtheatomicbeam,resultinginalower
3
u.] u.]
a. a.
nts [ nts [
u u
o o
C C
−4 −3 −2 −1 0 1 2 3 4 −4 −3 −2 −1 0 1 2 3 4
momentum [hk/2π] momentum [hk/2π]
FIG.3: Transverseatomicmomentumprofileafterashortin- FIG. 4: Transversal momentum distribution after the inter-
teraction (Γt<10). The grey area shows the initial momen- action with a tilted retroreflected laser. The dashed lines are
tumdistribution(scaleddown),thedashedlinesaregaussian gaussian fits to the individual peaks, the solid line is their
fits to the individual peaks. The full line is the sum of the sum. The depopulation of the peaks with negative momen-
gaussian fits. tum is clearly visible.
intensity in the retroreflected beam. Thus the Rabi fre-
quenciesofthe twolaserbeams arenotequal,Ω+ =Ω−,
6
therefore we find an asymmetric population distribution
withinthedarkstates(6)and(7). Thisisalsoconfirmed
bynumericalsimulationsoftheprocess. Thepeakatzero u.]
momentum also contains contributions from population a.
inthe3P0state,whichispopulatedbyspontaneousemis- nts [
sion from the upper state 3P1 during the interaction. ou
C
In order to do a supplementary test that we ob-
serve velocity-selective coherent population trapping the
retroreflectedbeamwasslightlytilted. Asufficientover-
lap of both beams was sustained but the retroreflected
beaminteractswiththe atomsafterthe darkstateshave
been populated. Due to optical pumping the population
−4 −3 −2 −1 0
of the states |g−2i, |g−1i and |g0i is depleted, and the momentum [hk/2π]
height of the peaks at negative momenta decrease, since
the internal and external states in the dark states are
strongly correlated. The measurement with tilted lasers FIG. 5: Transverse atomic momentum profile after an inter-
is shown in figure 4, which shows good agreement of ex- action of 8 µs (Γt ≈ 800). The lines are gaussian fits to the
perimental and calculated profiles. peaks. The peaks at p = ±~k reflect the stable dark-state
|ΨΛ (q=0)i.
When increasing the laser beam diameter with a tele- NC
scope to 8 mm the interaction time is of the order
Γt 800. We then observe the transversal beam pro-
file≈shown in figure 5. The peaks at 2~k are no longer levels outside the system. The Bloch equation reads
szfleeeercnotemadnobdmyoetnnhtleyumptheaeakpssptaeaabtrletshbdeeamcrakoumssteeanttehtae|±Ψ±stΛN~aCkte.i3TsPuh0reviipsveepaso,kpruaet-- ρ˙ = −Γ~i [H,ρ]
latedbyspontaneousemissionduringtheprocessofdark ∆+ ∆− ρ+ρ ∆− ∆+
− 2 · ·
state preparation. (cid:2)(cid:0) (cid:1) (cid:0) (cid:1)(cid:3)
We compare the measured data to a solution of the + 3Γ d2Ω ∆+ ǫ †exp( ikn R)ρ (8)
8π Z · − ·
generalized optical Bloch equations in the family mo- Xǫ⊥n(cid:0) (cid:1)
mentum basis [2], neglecting spontaneous decay to other exp(ikn R) ∆+ ǫ ,
× · ·
(cid:0) (cid:1)
4
process of stimulated raman transitions causes an effi-
cientpopulationtransfertothedarkstate ΨIW(q =0) ,
| NC i
characterized by a momentum distribution located at
p = 2~k and p = 0. Absorption of a photon fol-
±
lowedby spontaneous emission results in a randomwalk
inmomentumspaceduetotheemissionofphotonsinan
arbitrary direction. This diffusive process successively
populates the stable dark state ΨΛ (q = 0) which is
characterized by the two peaks a|t mNComentumip = ~k
±
in the momentum representation. Due to the statisti-
cal nature of this process the population of the dark
state ΨΛ (q =0) increasesforlongerinteractiontimes,
| NC i
Γt>200.
The calculations show good agreement with the mea-
sured data. The calculations show a structure with five
peaksat 2~k,...,2~kforΓt 100 200,whilethemea-
− ≈ −
surements yield this structure for Γt 10 20. The
≈ −
values for the interaction time are not directly compara-
FIG. 6: Dynamics of the momentum distribution, derived ble, as the calculations where done for a constant Rabi
from a numerical solution of the Bloch equations (8). The
frequency, while in the experiment the light fields have
population is encoded by thegreyscale given on theright.
a gaussian shape. Furthermore the widths of the mea-
sured peaks is smaller than expected from the simula-
tions, which is due to neglecting spontaneous emission
where ∆ are the lowering and raising parts of the re-
± out of the system 3P1,3P2 .
duceddipoloperatorandǫisthepolarizationvector[12]. { }
The measurements for a long interaction time of Γt=
Risthepositionoperatorofthecenterofmasswhichacts
800 (Fig. 5) show good agreement with the numerical
onlyontheexternalvariables. Thesecondtermdescribes
results. The population of the stable dark state (6) is
the decrease of the excited-state populations and coher-
rising, while the population of other states is decaying
ences resulting from spontaneous emission; the term un-
intothis darkstate viathe diffusionin momentumspace
dertheintegraloverallsolidangles(thirdterm)describes
due to the spontaneous emission of photons.
the feeding of the ground-state population and coher-
ences by spontaneous emission of a fluorescence photon Inthisworkwehavepresentedmeasurementswhichdi-
into the solid angle around the direction n, with energy rectly demonstrate the existence of transient dark states
~ckandpolarizationǫ n. Theexplicitformofthediffer- with a well defined momentum distribution in a m-state
entialequationsforthe⊥densitymatrixelementsaregiven manifold of a Jg =2-levelcoupled to a level with Je =1
in [13]. The equations were integrated stepwise with a by counterpropagating σ+ and σ− radiation. The mea-
resolution of 1/50Γ−1 in time and ~k/20 in momentum sured data show good agreement with quantum density
spaceonanintervalof[ 8~k,8~k]. Thecalculationswere matrixcalculationsforthe velocitydistributionforshort
done for a Rabi freque−ncy of 0.3Γ with no time depen- as well as longer interaction times. For a quantitative
dence. The initial momentum distribution is given by a analysismoredetailedexperimentsaswellascalculations
gaussianprofile of width ∆q =0.15~k centered at p=0, includingspontaneousemissionintootherstatesthanthe
whichis takenfromthe experiment. Initially allZeeman 3P2-state are needed.
states in the 3P2 manifold are equally populated. The We thank K. Bergmann and M. Fleischhauer for
results are shown in Fig. 6. their support, discussions and helpful comments on the
For a short interaction time of Γt<15, the atoms ab- manuscript. We thank M. Heinz for his contribution
sorb a photon from one of the laser beams, followed by to the experiments. This work was supported by the
stimulated emission into the other beam. This leads to Deutsche Forschungsgemeinschaft (Project Be 623/32)
a change in momentum by ∆p = 2~k, and peaks in and under the Graduiertenkolleg 792 ’Nichtlineare Op-
the momentum distribution at p =± 2~k appear. This tik und Ultrakurzzeitphysik’.
±
[1] A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, [3] M.S.Shahriar,P.R.Hemmer,M.G.Prentiss,P.Marte,
and C. Cohen-Tannoudji, Physical Review Letters 61, J.Mervis,D.P.Katz,N.P.Bigelow,andT.Cai,Physical
826 (1988). Review A 48, R4035 (1993).
[2] A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, [4] F. Papoff, F. Mauri, and E. Arimondo, Journal of the
and C. Cohen-Tannoudji, Journal of theOptical Society Optical Society of America B 9, 321 (1992).
of America B 6, 2112 (1989). [5] M. A.Ol’shanii, 24, L583 (1991).
5
[6] C. Menotti, G. Morigi, J. Muller, and E. Arimondo, 53, 946 (1996).
Physical Review A (Atomic, Molecular, and Optical [10] M.KiffnerandK.-P.Marzlin (2005), quant-ph/0501092.
Physics) 56, 4327 (1997). [11] J. M. Weber, K. Hansen, M.-W. Ruf, and H. Hotop,
[7] M. Ol’shanii and V. Minogin, Optics Communications Chem. Phys.239, 271 (1998).
89, 393 (1992). [12] Y. Castin, H. Wallis, and J. Dalibard, Journal of the
[8] M.Weitz,T.Heupel,andT.W.H¨ansch,PhysicalReview Optical Society of America B 6, 2047 (1989).
Letters 77, 2356 (1996). [13] H. Wallis, Physics Reports 255, 203 (1995).
[9] M. Widmer, M. R. Doery, M. J. Bellanca, W. F. Buell,
T. H. Bergeman, and H. J. Metcalf, Physical Review A