Rabi oscillations and excitation trapping in the coherent excitation of a mesoscopic frozen Rydberg gas

0.2 MB·English

by  Markus Reetz-Lamour

Rabi oscillations and excitation trapping
in the coherent excitation of a mesoscopic frozen Rydberg gas
M. Reetz-Lamour, T. Amthor, J. Deiglmayr, and M. Weidemu¨ller∗
Physikalisches Institut der Universit¨at Freiburg, Hermann-Herder-Str. 3, D-79104 Freiburg, Germany
(Dated: February 2, 2008)
Wedemonstratethecoherentexcitationofamesoscopicensembleofabout100ultracoldatomsto
Rydbergstates by driving Rabi oscillations from the atomic ground state. We employ a dedicated
beam shaping and optical pumpingscheme to compensate for the small transition matrix element.
8 We study the excitation in a weakly interacting regime and in the regime of strong interactions.
0 Whenincreasingtheinteractionstrengthbypairstateresonancesweobserveanincreasedexcitation
0 rate through coupling to high angular momentum states. This effect is in contrast to theproposed
2 andpreviouslyobservedinteraction-inducedsuppressionofexcitation,theso-calleddipoleblockade.

n
a PACSnumbers: 03.65.Yz,32.80.Pj,32.80.Rm,34.20.Cf,34.60.+z
J

0
Rydbergatoms,withtheirrichinternalstructure,have (a) (b)
3 er
] btuereyn[i1n].theInfotchuesloafstatdoemcaicdpehRysyidcbsefrogr mphoyresictshawneareceenx-- m nS1/2/ nD5/2 pump
ph tended to laser-cooled atomic gases which allowed the 480 n re actlooumd
- study of a frozen system with controllable, strong inter- d F=3 5P
m actions and negligible thermal contributions. This has F=2 3/2 100mm excitation
… 780 nm
o opened a wide field in both experiment and theory cov- m
at eringsuchdiverseareasasresonantenergytransfer[2,3], 780 n e4x8c0i tnamtion pumper
cs. ptulamsmraanfdorommawtiaolnk[s4[,85,],9]e.xoInticadmdoitleiocnu,letsh[e6s,e7]f,roaznednqRuyand-- FF==21 5S1/2 ex(coivtaetriloanp vreogluiomne)
i bergsystemshavebeenproposedasapossiblecandidate
s
y for quantum information processing [10, 11]. However, FIG. 1: (a) Relevant level scheme for two-photon excitation
h the coherent excitation of Rydberg atoms, an impor- of rubidium. (b) Preparation of a mesoscopic subensemble.
p tantprerequisitefor quantuminformationprotocols,has All atoms are pumped to the F=1 hyperfine component of
[ the ground state. Only a tight tube of atoms is pumped
proven a challenging task. While Rabi oscillations be-
to F=2 wherefrom the excitation originates. A mesoscopic
2 tween different Rydberg states have been demonstrated
subensemble containing about 100 atoms within this tube is
v andthoroughlyanalyzedbefore[12],Rabioscillationsbe-
transferred to a Rydberg state by near-resonant two-photon
1
tweenthegroundstateandRydbergstatesofatomshave excitationwithtwocounterpropagatinglaserbeamsat780nm
2
3 not been observed directly so far, mainly owing to the and 480nm, respectively. The excitation laser at 480nm has
4 small transition matrix element. a flat-top intensity profile shown in the inset. This ensures a
. InthisLetterwereportontheexperimentalrealization constant Rabifrequency overthe excitation volume.
1
1 and observation of Rabi oscillations between the ground
7 and Rydberg states of ultracold atoms. We demonstrate
0 how strong interatomic interactions influence the coher- diffractive optical element that produces a flattop beam
:
v ent excitation of a mesoscopic cloud of atoms. We show profilewhichischaracterizedwithanadaptedCCDcam-
Xi that an interaction-induced coupling to a larger num- era with a spatial resolution of 5.6µm. The measured
ber of internal states can be used to trap the excitation. flattop is depicted in Fig. 1. Further details will be de-
r
a Employed in a controlled way, this effect offers future scribed elsewhere [14]. This setup allows a tightly fo-
applications in the experimental realization of quantum cussed beam with high intensity and at the same time
random walks with exciton trapping [9]. overcomes the limits of a Gaussian beam shape. Gaus-
Ourexperimentsareperformedwithamagneto-optical sian beams have been used for stimulated adiabatic pas-
trap (MOT) of about 107 87Rb atoms at densities of sage to high Rydberg states [13, 15] but do not allow
1010cm−3 and temperatures below 100µK. Rydberg ex- theobservationofsynchronousRabifloppingsinameso-
citation is achieved with two counterpropagating laser scopic sample due to the wide range of intensities across
beams at 780nm and 480nm (see Fig. 1). The laser at the beam [13]. In order to address only atoms in the
780nmiscollimatedtoawaistof1.1mmensuringacon- flattop region we use spatially selective optical pumping
stant Rabi frequency of 2π×55MHz over the excitation (see Fig. 1): The magnetic field of the MOT is turned
volumeasdeterminedfromAutler-Townessplittings[13]. off3msbeforeexcitation[27]. Afterturningoffthetrap-
Thelaserat480nmisreferencedtoatemperaturestabi- ping lasers, all atoms are pumped to the F=1 hyper-
lized Zerodur-resonator and its beam is shaped with a fine component of the 5S ground state within 300µs.
1/2
2
0.6 1.5 andthat allatoms performidentical Rabioscillations si-
bility0.5 DVns multaneously. The solid line in Fig. 2 is a theoretical
oba0.4 1 @al predictionwhichtakesthe measuredtime dependance of
pr gn the excitation pulses into account. It incorporates three
excitation000...123 0.5detectorsi sitnhotueerrflcmaeste-dtoiofaptdebeesptahamatsepi,nrtgoh:fieletr,heeasindrduemaalafiiinnntiitenengsliaatsydemrdbiixsattnruidbrweutiodioftnhthooeff
0 0 2.4MHz. Withincreasingimportancethesethreesources
0 50 100 150 200 250 300 350
leadtotheobserveddampingofthemeasuredRabioscil-
excitationtime@nsD
lations. A detailed discussion together with quantitative
FIG.2: Rabioscillations betweenthe5S1/2 and31D5/2 state comparisons between measured and calculated Rabi fre-
of 87Rb. Each dot is an average of measurements over 28 quencies and other systematic verifications will be pub-
experimental repetition cycles. The solid line shows the sim- lishedelsewhere[14]. Forlongertimes (&200ns)the ex-
ulated excitation probability taking the measured intensity citation probability reaches a steady state value of 1/2.
distribution,theresidualadmixtureoftheintermediate5P3/2 This allows us to calibrate the detector: by overlapping
state,andourfinitelaserlinewidthintoaccount. Thescaling
the model with the measured data we obtain a scaling
between detector signal and simulated excitation probability
factorbetweensingleatomexcitationprobabilityandde-
is a free parameter that represents thedetector efficiency.
tector signal [28].
The observation of Rabi oscillations between ground
and Rydberg states is a first step towards the imple-
A small tube of atoms perpendicular to the excitation
mentation of quantum information protocols with cold
beams is transferred to the F=2 hyperfine component
Rydberg gases. The most promising protocols further
with a 1µs pulse of a tightly focused repumping beam
rely on the so-called dipole blockade [11]. The dipole-
(waist 10µm, 10nW). Only these atoms are resonant
blockade describes a system in which the excitation of
withthe excitationlasers. They areopticallypumped to
more than one atomis detuned fromthe single-atomex-
the stretched F=2, m =2 state by a 1µs pulse of a σ+-
F citation due to interaction-induced energy shifts, which
polarized pumper beam superimposed with the excita-
suppressesmultipleexcitations. Itallowstostoreasingle
tion lasers. From this state we can excite both stretched
excitationinamesoscopiccloudofatomsandisaccompa-
nSandnDstatesdepending onthe helicityoftheexcita-
nied by an increased collective Rabi frequency, evidence
tion laser polarization; n denotes the principal quantum
of which was observed only recently [17]. The first ex-
number. By maintaining the two-photon resonance but
perimental signatures of the blockade were observed as
detuning from the intermediate state 5P by typically
3/2 a suppression of excitation [18, 19] due to the extraordi-
δ/2π =140MHz this state experiences negligible popu-
narily strong van der Waals interactions at large values
lationandatoms aredirectly transferredto the Rydberg n. Recently,a similar effectwasobservedatlowervalues
state [13]. The overlapbetween the repumping laser and of nbyusingpairstateresonancestoswitchfromvander
the 480nm laser defines the excitation volume of about
Waals to the stronger resonant dipole-dipole interaction
10µm×10µm×100µm, corresponding to ∼100 atoms
with an external electric field [20].
(see Fig. 1.(b)). Rydberg atoms are detected by field- Forrubidium,nDstatesoffersimilarresonances. Fig.3
ionizationandaccelerationontoamicrochannelplatede-
shows the electric field dependance of the involved pair
tector. Thereisnosignalforelectricfieldsbelowtheion-
states. At vanishing electric field two 47D atoms ex-
ization threshold, therefore we can rule out the presence 5/2
perience van der Waals interaction. At an electric field
of spurious ions during excitation. After detection the
of ∼ 260mV/cm the 47D –47D pair state becomes
MOT is turned back on and the whole cycle is repeated 5/2 5/2
energeticallydegeneratewiththedipole-coupled49P –
every 70ms. Our setup allows to compensate electric 3/2
45F (J=5/2, 7/2) pair state. This changes the interac-
J
fields E in a direction perpendicular to the excitation
⊥ tion to long-range resonant dipole interaction. In con-
lasers. Field components parallel to the field plates can-
trast to the picture of the dipole blockade, however, the
notbecompensatedandaredeterminedfromStarkshifts
increased interaction strength does not translate into a
to be E =160mV/cm [16]. The field values given in the
k reduced excitation probability at the atomic resonance:
text represent the total field E =(E2+E2)1/2.
k ⊥ Fig. 4 shows the temporal evolution of the 47D5/2 pop-
Fig.2showsthe measuredfractionofexcitedRydberg ulation for two different densities. For the lower density
atomsinthe31D stateasafunctionofexcitationtime. of4.7×109cm−3 (graypoints) we observedamped Rabi
5/2
Rabioscillationsareclearlyvisible. Weobservethesame floppings similar to low-n states. For the higher density
temporal dependance for both nS and nD states with (1010cm−3,blackpoints)weobserveanincreasedexcita-
n<40atbothlowandhighatomdensities. Thisindicates tionthatclearlyexceeds0.5. Thismeansthatpartofthe
that for these small n values the observed Rabi oscilla- atomsremaininRydbergstateswithoutbeingstimulated
tions are not affected by Rydberg–Rydberg interactions back to the ground state. The effect is clearly density-
3
2000
49P+45G+ | DD | PF 0.8
1500 d WF D(E) y
DMHz 1000 WR g G obabilit0.6
@shift | gD onpr0.4
Stark 500 49P+45F xcitati
e
0.2
0
47D+47D
-500
100 200 300 400 500 0 200 400 600 800
electricfield@mV(cid:144)cmD excitationtime@nsD
FIG. 3: Stark shift of dipole coupled pair states around the FIG. 4: Temporal evolution of the excitation probability for
47D5/2+47D5/2 asymptote. The two arrows mark the total the 47D5/2 state at an electric field E = 160mV. For the
electric field at which thetwospectra ofFig. 5 aretaken. At smallerdensityof4.7×109cm−3(graydots)weseeadamped
an electric field of ∼260mV/cm the47D5/2+47D5/2 state is Rabi oscillation similar to Fig. 2. For the higher density of
degenerate with the 49P3/2+45FJ (J=5/2, 7/2) state which 1010cm−3 (black dots) thestimulated emission is suppressed
leadstoresonantdipole-dipoleinteraction. Theinsetshowsa leading to a constantly increasing Rydberg population. The
simplified levelschemeforthissystem involvingthecoherent lines are the solution of the model schematically shown in
excitation of a ground state/Rydberg state atom pair |gDi Fig. 3 for weak (dashed) and strong (solid) interactions.
to a pair of two Rydberg atoms |DDi with a Rabi frequency
Ω ,alaserdetuningδandaneffectivelinewidthγ. Thestate
R
|DDiisdipolecoupledtoanalmostdegeneratepair|PFiwith
aninteraction energy~Ω andanenergy difference~∆ that level system and can be fully described by the accord-
F
depends on the electric field E. The dephasing of the dipole ing optical Bloch equations [21]. In the case of coher-
coupling(seetext)isphenomenologicallydescribedbyadecay ent couplings (Γ = 0), this system describes the dipole
rate Γ out of thethree-levelsystem. blockade as an analog to the Autler-Townes splitting in
the optical system. By including incoherent coupling
(Γ>0),we reproducethe increasedexcitationmeasured
dependent and must therefore be interaction-induced. in Fig. 4: The solid line corresponds to model calcula-
Asourdetectioncannotdistinguishbetween47D,49P, tions with parameters for the excitation that are scaled
and45F,weassumeanaccumulationofatomsinthelat- frommeasuredlow-nRabioscillations. Wehaveset∆to
ter states. However, the increased excitation is in con- the theoretical value ∆(E = 160mV) = −2π×150MHz
trast to the previously observed excitation blockade at (see Fig. 3). This leaves two free parameters: the cou-
another pair state resonance [20]. We attribute the dif- pling strength Ω = 2π × 40MHz and the dephasing
F
ference to the fact that the pair state resonance used in rate Γ = 2π ×30MHz where chosen to achieve a good
Ref. [20] comprises well-defined pair states (nP –nP overlapwiththeexperimentaldata. Bothvaluesarecon-
3/2 3/2
and nS –(n+1)S with |m| = 1/2) while in our case sistent with the theoretical value Ωmax = 2π×64MHz
1/2 1/2 F
the involved 45F state is highly degenerate. As each for the strongest coupling between the stretched states
J
sub-statehasadifferentcouplingstrengththisleadstoa (m = J = ℓ+1/2) at the most probable interatomic
J
dephasing and a coupling back to the original 47D state distanceof2.6µm. Thedashedline correspondstoalow
is suppressed. The residual electric field in our setup density model where Ω and Γ are scaled down propor-
F
enhances this dephasing as it mixes different m -levels tionaltothedensity(asexpectedforan1/R3-dependent
J
and thus leads to a dipole coupling with all possible val- interaction).
ues of mJ, comprising 14 sub-states of the two values of Withthe samevaluesforΩF andΓ ourmodelalsoex-
J =5/2 and 7/2 for the 45FJ state and 6 sub-states for plainsspectralmeasurementsshowninFig.5. Bychang-
the 47D5/2 state. ingtheelectricfieldwecantunetheenergydifferencebe-
Wecanheuristicallymodelthe dephasingbetweendif- tween the |DDi and |PFi pair states. Fig. 5(a) shows a
ferent pair states with an analytical model illustrated in measured spectrum of the 47D line at E=160mV/cm
5/2
Fig. 3. It incorporates the coherent excitation of a pair (E =0) together with model calculations. All model
⊥
|gDiofagroundstate atom|gianda47Datom|Dito a parameters are the same as those for the excitation
pair|DDioftwo47Datomsaswellasthecouplingtothe time scan in Fig. 4. Fig. 5(b) shows a spectrum at
|PFi state with a field-dependent energy shift relative to E=260mV/cm corresponding to ∆=0. The dashed line
|DDi. The dephasing is introduced phenomenologically showsthemodelresultinthelow-densitylimit(Ω =0),
F
by a decay rate Γ out of the three level system. This the solid line includes a coupling with Ω =2π×5MHz
F
system is an analog of the excitation of an optical three and Γ = 2π × 30MHz, both chosen to yield the best
4
Thiscanresultinpairstateswhicharenotdipole-coupled
0.8
to other pair states thus stopping spatial diffusion. This
HaL
y
bilit0.6 effect can be exploited as an exciton trap for continuous
ba time quantumrandomwalk experiments [9]. It alsocon-
o
pr0.4 stitutes another constraint to states that are suited for
n
o dipole blockade experiments besides the already identi-
citati0.2 fied zeroes in Rydberg–Rydberg interactions for certain
x
e internal states [24] and atom pair alignments [25].
0
We thankT.Pohlforfruitful discussionsandC.Pruss
0.8 and W. Osten from the Institute for Technical Optics
y HbL in Stuttgart for the production of the beam shaping el-
bilit0.6 ement. The project is supported by the Landesstiftung
a
b
o Baden-Wu¨rttemberg within the ”Quantum Information
pr0.4
n Processing”program,andbyagrantfromtheMinistryof
o
ati Science, Research and Arts of Baden-Wu¨rttemberg (Az:
cit0.2
x 24-7532.23-11-11/1).
e
After submission of this article we became aware of
0
-15 -10 -5 0 5 10 15 a complementary work which demonstrates Rabi oscil-
laserfrequency@MHzD lation with an interaction-dependent damping in micro-
scopic samples of 1 to 10 atoms [26].
FIG.5: (a)Spectrumofthe47D5/2 resonancelineatanelec-
tric field of E=160mV/cm at densities of 1010cm−3 (blue)
and 4.7×109cm−3 (red). For the higher density we observe
a line shift and broadening combined with an increased ex-
∗
[email protected]
citation probability. The solid and dashed lines are model
[1] T. F. Gallagher, Rydberg Atoms (Cambridge University
calculations. The line shift is a result of an energy differ-
Press, Cambridge, 1994).
ence between the |DDi and |PFi pair states. (b) Same for
[2] W. R. Anderson et al., Phys. Rev.Lett. 80, 249 (1998).
E=260mV/cm. The line shift is reduced to zero, while line
[3] I. Mourachko et al., Phys.Rev.Lett. 80, 253 (1998).
broadening and an increased excitation remain. All spectra
[4] M. P. Robinson et al., Phys.Rev.Lett. 85, 4466 (2000).
are taken by scanning the 480nm laser (fixed detuning from
[5] T. Amthoret al., Phys. Rev.Lett. 98, 023004 (2007).
theintermediate level) with an excitation time of 400ns.
[6] C. Boisseau et al., Phys. Rev.Lett. 88, 133004 (2002).
[7] C. H.Greene et al., Phys. Rev.Lett. 85, 2458 (2000).
[8] R. Cˆot´e et al., New J. Phys. 8, 156 (2006).
overlap with the experiment. We attribute the reduced [9] O. Mu¨lken et al., Phys.Rev. Lett. 99, 090601 (2007).
couplingstrengthΩF totheincreasedelectricfieldwhich [10] D. Jaksch et al., Phys.Rev.Lett. 85, 2208 (2000).
splits the involved states, effectively reducing ΩF from [11] M. D.Lukin et al., Phys.Rev.Lett. 87, 037901 (2001).
40MHz at 160mV to 5MHz at 260mV. There are three [12] T. R.Gentile et al., Phys. Rev.A 40, 5103 (1989).
distinctfeaturesofthemeasuredspectraathigherdensi- [13] J. Deiglmayr et al., Opt.Comm. 264, 293 (2006).
[14] M. Reetz-Lamouret al., arXiv:0801.3999 (2008).
ties: linebroadening,anenhancedexcitationprobability,
[15] T. Cubel et al., Phys. Rev.A 72, 023405 (2005).
andanelectric fielddependent shift. The modelqualita-
[16] K. Singer et al., J. Phys.B 38, S321 (2005).
tively describes all of these features. [17] R.Heidemannetal.,Phys.Rev.Lett.99,163601(2007).
Wewanttoemphasizethattheexcitationtrappingde- [18] D. Tong et al., Phys.Rev.Lett. 93, 063001 (2004).
scribed here is not due to state redistribution by colli- [19] K. Singer et al., Phys.Rev.Lett. 93, 163001 (2004).
sions with ions [22] since we have verified that there are [20] T. Vogt et al., Phys.Rev. Lett. 97, 083003 (2006).
no ions produced during the short excitation time. Fur- [21] R. M. Whitley and C. R. Stroud Jr., Phys. Rev. A 14,
1498 (1976).
thermore the excitation volume is too small to support
[22] S. K. Duttaet al., Phys. Rev.Lett. 86, 3993 (2001).
avalancheionization. Theexcitationenhancementisalso
[23] C. Ateset al., Phys.Rev. Lett. 98, 023002 (2007).
different from the antiblockade for the case where the [24] T. G. Walker and M. Saffman, J. Phys. B 38, S309
Autler-Townes splitting of the lower transition matches (2005).
theRydberg–Rydberginteractionenergy[23]. Themech- [25] A. Reinhard et al., Phys.Rev.A 75, 032712 (2007).
anism presented here can be seen as an energy diffusion [26] T. A.Johnson et al., arXiv:0711.0401 (2007).
inside the atom rather than a spatial diffusion which [27] Weusethethermalexpansionofthecloudtochangethe
groundstatedensitybyincreasingthistimeupto15ms.
relies on internal state swapping between atoms and is
[28] Thisfactorisinverseproportionaltothenumberofatoms
the sourceofadifferentdephasingmechanismforenergy
intheexcitationvolumeandthereforealsotothedensity.
transfer processes [2, 3]. The internal energy dissipation
Bydeterminingthisfactor wecanthuscalibraterelative
can even counteract spatial diffusion by transferring the densities. Wedothisfor low-n states. Absolutedensities
excitation into states with different angular momenta. are determined by absorption imaging.