Entangling Atomic Spins with a Strong Rydberg-Dressed Interaction

0.88 MB·English

by  Y. -Y. Jau

Entangling Atomic Spins with a Strong Rydberg-Dressed Interaction
Y.-Y. Jau, A. M. Hankin, Tyler Keating, I. H. Deutsch, and G. W. Biedermann
Sandia National Laboratories, Albuquerque, New Mexico 87123, USA and
Center for Quantum Information and Control (CQuIC),
University of New Mexico, Albuquerque NM 87131
(Dated: January 7, 2016)
Controlling quantum entanglement between parts of a many-body system is the key to unlock-
ing the power of quantum technologies such as quantum computation, high-precision sensing, and
the simulation of many-body physics. The spin degrees of freedom of ultracold neutral atoms in
their ground electronic state provide a natural platform for such applications thanks to their long
coherence times and theability to control them with magneto-optical fields. However, the creation
6 of strong coherent coupling between spins has been challenging. Here we demonstrate a strong
1 and tunable Rydberg-dressed interaction between spins of individually trapped cesium atoms with
0 energyshifts oforder1MHzin unitsofPlanck’s constant. Thisinteraction leads toaground-state
2 spin-flipblockade,wherebysimultaneoushyperfinespinflipsoftwoatomsareinhibitedduetotheir
mutual interaction. We employ this spin-flip blockade to rapidly produce single-step Bell-state en-
n
tanglementbetweentwoatomswithafidelity 81(2)%,excludingatomloss events,and 60(3)%
a
≥ ≥
J when loss is included.

5

Pristinequantumcontrolofmany-bodycorrelationsis ploy coherent control in the ground state manifold to
]
h fundamental to realizing the power of quantum informa- create entangled Bell-states.
p tion processors (QIP). Steady progress has continued in
-
t variousplatforms rangingfrom solidstate spintronics [1]
n and superconductors [2, 3] to nanophotonics [4] and ul- 319 nm
a &
u tracold trapped atoms, both ionic [5–7] and neutral [8– atom 2 Raman
+3.3 µm
q 10]. Coldneutralatomsareparticularlyattractiveasthe
[ ability to create entanglement between atoms would al- B 18 nm

low for greatly increased precision of interferometers for
2 n E 2 µs
v applications in clocks [11–13], and force sensors [14–16]. tio
2 In addition, cold atoms provide a natural platform for osi drop recapture
6 quantum-simulationofcondensedmatterphysics[17,18] p
8
and scalable digital quantum computers [19–21]. Con-
3
trolledentanglementofneutralatoms,however,hasbeen
0
. challenging,particularlyifoneseekstunableinteractions -3.3 µm atom 1
1
that are strong, coherent, and long-range ( µm). time
0 ∼
5 Onemechanismtoachievestrong,long-rangecoupling state preparation Rydberg dressing state detection
1 is the Rydberg blockade [22]. This has been success-
v: fullyemployedforimplementingcontrolledentanglingin- FIG. 1. Experiment sequence. To achieve both a strong
i teractions between atoms [9, 10, 23] and quantum logic ground-state atom-atom interaction and high-fidelity signal
X detection we perform the following steps at different inter-
gates [24]. In the standard protocol, short pulses excite
r atomic spacings. In the experimental procedure, two Cs
thepopulationofoneatomtotheRydbergstateandop-
a atomsareinitially6.6µmapart,andheldbyopticaltweezers.
tical excitation of a second atomis blockaded because of
After qubit-state preparation, the two trapped atoms trans-
the electricdipole-dipoleinteraction(EDDI)[21]. Anal- late toward each other with an average speed of 9 mm/s (18
ternative protocol is to adiabatically dress the ground nm step every 2 µs) by ramping the modulation frequencies
state with the excited Rydberg state [25–27]. This of the AOM. At the target distance, the Rydberg dressing
Rydberg-dressedinteractionenablestunable,anisotropic laser at 319 nm turns on to illuminate the two atoms simul-
taneouslywithaRamanlaser. Thetweezersareextinguished
interactions that open the door to quantum simulations
during this step to eliminate optical perturbation. The two
ofavarietyofnovelquantumphases[26,28,29]. Inaddi-
atoms then translate back to the original positions for state
tion, it allows for quantum control of interacting atoms
detection.
based solely on microwave/rf fields whose phase coher-
ence is easily maintained. Applications include spin-
Rydberg-dressing arises from the modification of the
squeezing for metrology [13, 25], and quantum comput-
light-shift (LS) (i.e., the optical AC Stark shift) of
ing [30, 31]. While the promise of Rydberg-dressed in-
two atoms due to the EDDI. We characterize this by
teractionsisgreat,experimentaldemonstrationhasbeen
elusive. We present here a clear measurement of this the interaction strength J ≡ ∆EL(2S) − 2∆EL(1S), where
interactionbetweentwoRydberg-dressedatomsandem- ∆E(2) is the LS for two atoms in the presence of
LS
2
EDDI and ∆E(1) is the LS for a lone atom opti- strength, J, depends on the EDDI and the optical Rabi
LS
cally excited near a Rydberg level. By tuning the frequency Ω . Thus, the distance between the two
L
Rydberg excitation laser near the Rydberg resonance, Rydberg-dressed atoms and the choice of the principal
double excitation of two Rydberg atoms is blockaded quantum number are crucial experimental parameters.
when U ~Ω , where Ω is the laser Rabi fre- Ideally, we would like the atoms to be located far apart
dd L L
≫
quency and U is the EDDI energy, which scales from for individual addressing, and conversely, in close prox-
dd
1/R6 to 1/R3 depending on the interatomic distance imity to maximize J. Our particular implementation of
R. The resulting difference in the two-atom spec- an AOM (acousto-optic modulator) allows us to create
trum implies that ∆E(2) = 2∆E(1), which plateaus at two optical tweezers that trap each atom from the same
LS 6 LS
J = ~ ∆ +sign(∆ ) ∆2 +2Ω2 2 ∆2 +Ω2 , laserbysimultaneouslydrivingtheAOMattwofrequen-
2 h L L (cid:16)p L L− p L L(cid:17)i cies (see Supplementary Material). We achieve this goal
where ∆ is the laser detuning, when the blockade is
L byindependentlysweepingthevaluesofthesefrequencies
perfect [26]. In the strong dressing regime, this can lead
anddynamically translatingthe traps. The capabilityof
to a very strong interaction, J ~Ω , when Ω ∆ .
∼ L L ∼ L maximizing J at shorter interatomic distance allows us
We employ 133Cs atoms, and encode qubits in the to reduce the principalquantum number of the Rydberg
hyerfine “clock states” 0 6S1/2,F = 4,mF = 0 , level. Thus,theoscillatorstrengthfordirectexcitationis
| i ≡ | i
|1i ≡ |6S1/2,F = 3,mF = 0i, of two-atoms individually improved, which allows us to maximize ΩL for the same
trapped in optical tweezers at 938 nm and separated by Rydberglaserintensity. Thishastheaddedbenefitofre-
a few microns [23]. By choosing ∆L small compared ducing the interactionofthe environment,whichrapidly
| |
to the 9.2 GHz hyperfine splitting, in the presence of increases for high-lying Rydberg levels and is a common
EDDI, only the two-qubit state, 0,0 , receives a non- challenge in these experiments.
| i
negligible, two-atom dressing energy J. Because this
energy is a shift of the Rydberg-dressed ground state,
it can be used to control two-body interactions solely RESULTS
with mw-frequency fields. In particular, by applying
9.2 GHz radiation on the clock-state resonance (e.g., via
WedirectlymeasureJ asafunctionoftheinteratomic
two-photon Raman lasers) with Raman-Rabi frequency
distance. OurexperimentisillustratedinFig.1. Thetwo
Ωmw, two atoms initially in the state |1i are blockaded trapped 133Cs atoms are initially prepared in state 1,1
from undergoing double spin-flips, 1,1 0,0 , when | i
J/~ Ω . This is a spin-flip bloc|kadei(→Fig|. 6ai) in the and we dynamically translate them to be in close prox-
≫ mw imity at a targeted distance R. We then extinguish the
dressed-ground-state, analogous to the familiar excited-
tweezersforashorttimetoeliminatelightshiftsfromthe
state Rydberg blockade.
dipole-trap laser, and immediately apply a short pulse
Note, unlike the single atom light-shift, the ratio of of the Raman and Rydberg-dressing lasers concurrently.
dressingenergyJ to the photonscatteringrate (the fun- Afterwards,theopticaltweezersarerestoredtorecapture
damental source of decoherence) improves closer to res- the fallingatomsandtranslatethembackto the original
onance,sincefarfromresonanceJ ∆−3,whileabsorp- positions for independent state detection, which is ac-
tion followed by spontaneous decay∼scaLles as γ ∆−2. complishedbyusingthe 6S ,F =4 6P ,F′ =5
For this reason, we operate in a regime of stro∼ng RLy- D2 cyclingtransitionto|det1e/r2mine whie→the|r e3a/c2hatomisi
dberg dressing with a detuning, ∆L ΩL. For exam- instate 0 (brighttothis excitation)or 1 (darkto this
ple, one of the experimental condition≤s used in Fig. 6 is excitatio|n)i. Weusea319-nmlaserfordr|esisingthe133Cs
ΩL/2π = 4.3 MHz and ∆L/2π = 1.3 MHz, where the atom,whichcouplesatomsdirectlyfromthegroundstate
dressedstate has a strongadmixture of Rydberg charac- to the Rydberg level, 6S ,F =4 64P , in a single
1/2 3/2
ter, 0.6r +0.80 , or 64% probability in 0 . Although photontransition[23]. This avoids→unwantedpopulation
| i | i | i
thereisabout36%probabilityin r ofthedressedstate, in an intermediate, short-lived excited state that arises
| i
thelightshiftontheRydberg-dressedstateisinsensitive in the typical two-photon Rydberg excitation method,
to thermalmotion thatgives risesto a fluctuationin the which causes additional ground-state decoherence [30].
difference in the optical phase seen at the relative posi- We choose a detuning that is small compared to the
tions between the two atoms. Such thermal noise was a ground-state hyperfine splitting so that the dressing of
limiting factor in the generation of spin entanglement in 1,1 in the F = 3 manifold is negligible, but all other
theworkofWilket. al[10]. Thethermalatomicmotions g|rouind states in the logical basis, 0,0 , 0,1 , 1,0 ,
in the Rydberg-dressed interactions, however, does lead are now well described in the dress{ed| baisi|s. Tio| driiv}e
toaDopplershift,andthusnoiseintheopticaldetuning spin flips, we apply the Raman laser fields to the two
∆L,butthisishighlysuppressedwhenitisinthe strong Rydberg-dressed atoms when they are at a desired sep-
dressing regime (∆L ΩL), because ΩL dominates the aration. By sweeping the Raman (microwave)frequency
≤
energy shift of the dressed state. and measuring the resulting spin flips, we obtain a two-
The key mechanism that determines the interaction qubit energy spectrum of the form shown in Fig. 6b. To
3
FIG. 2. Rydberg-dressed ground-state interaction J and the spin-flip blockade. (a) Energy-level diagram of the
spin-flipblockadeontheRydberg-dressedtwo-qubitsublevels: ForasufficientlylargeJ,onlythetransitionfrom 1,1 (1,0
| i→ | i
or 0,1 ) is allowed and the double spin-flip transition from (1,0 or 0,1 ) 0,0 is blockaded when microwave radiation
| i | i | i → | i
(stimulatedRamantransitioninourcase)isappliedatthenoninteracting,single-atomqubitresonancefrequency. (b)Scanning
the microwave frequency of the stimulated Raman pulse applied to the two trapped Rydberg-dressed 133Cs atoms reveals the
ground-state spin-flip blockade. The excitation from 1,1 0,0 occurs via an anti-blockade two-photon transition. J/h is
| i → | i
simply twice the shift of the resonance frequency for excitation to the state 0,0 . (c) Experimental data of J versus R with
| i
two sets of parameters. The dashed curves are the calculated values based on a detailed model with no free parameters (see
SupplementaryMaterial).
detemine J, we measure the microwave resonance fre- tect such resonances in our experiment. We use a best-
quency forthe transition 1,1 (1,0 + 0,1 )/√2 and fit of the blockade shift curve of r,r from our detailed
| i→ | i | i | i
for the two-microwave-photon transition 1,1 0,0 . model[30]tocalculateJ asshowninFig.6c(seeSupple-
J/~isgivenbytwicethefrequencydifferen|ceoift→hes|etwio mentary Materialsand Fig. S3). With an ideal Rydberg
resonances. ThemicrowaveRabirateΩ andthepulse blockade,U ,wefindtheexpectedplateauinJ at
mw dd
→∞
time of the stimulated Raman laser are properly chosen short interatomic distances predicted from the two-level
toensuretheobservationofboththetwo-photonandthe model. The plateau atsmall R is characteristicof a per-
single-photon microwave resonance. The relative popu- fectRydbergblockade,andagreeswithsimpletheoretical
lations, P , P , P , and P of the four, two-qubit predictions.
1,1 1,0 0,1 0,0
computationalbasisstatescanbedirectlymeasuredwith
With large values of J, we can employ the spin-flip
thecoincidentbrightanddarksignalsdeterminedbythe
blockade to create entanglement between atomic spin
photon counting of the two APDs (avalanche photodi-
qubits. By driving a resonant Raman-pulse 1 0
odes). Fig. 6c plots the measured J as a function of in- | i → | i
simultaneously on the two atoms, we cause Rabi os-
teratomic distance for two different combinations of the
cillations between 1,1 and the entangled Bell state,
319-nmlaser detuning ∆L and the optical Rabi rate ΩL. Ψ =(0,1 + 1,0|)/√i2astheexperimentaldatashow
+
| i | i | i
While the simple two-level atom model gives a clean in Fig. 3. A signature of the spin-flip blockade is the
theoretical prediction for J as described above, the true characteristic increase in the Raman-Rabi frequency to
atomicphysicsismorecomplex. TheEDDIshifts64P √2Ω [10]. The Bell state Φ = (0,0 + 1,1 )/√2,
3/2 mw +
| i | i | i
negatively (red), thus for Rydberg-dressing we tune the the two-atom cat state, can be generated from Ψ
+
| i
319-nm laser to the blue. At short interatomic spacings, by subsequently applying a global π/2 rotation on the
however, the two-atom Rydberg energy levels strongly qubits. Alternatively,whiletheRydbergdressinglaseris
mixtoyieldaspectrumwithmolecularquality[32]. The still on, a two-microwave-photon π/2-pulse at a shifted
resulting“spaghetti”ofmolecularlevelscouldpotentially frequency, resonant with the transition from 1,1
| i →
lead to additional unwanted resonances that would ruin 0,0 , also generates Φ . This method has a lower fi-
+
| i | i
the Rydberg blockade. This may also affect the lifetime delitybecausethetwo-photonmicrowaveRabiratemust
ofthe two-atomdressedstate. Theexistenceofsuchres- be small in order to avoid excitation to the off-resonant
onances, however, depends on the oscillator strengths. Ψ state, and decoherence is more likely on this long
+
| i
With our experimental resolution, we are unable to de- time scale.
4

1.0
ms 1.0 |Y 〉 ⇒ |〈0,1|r|1,0〉| · 2 = 0.81 – 0.01
o +
0.5 e at 0.8
P gl
0.0 1 Sin 0.6
1.0
0.4
s
n P
ve populatio 001...050 1,1 ms fQParity: ()-000...022
Relati 0.5 P +P wo ato -0.4
0.0 0,1 1,0 T -0.6
1.0
-0.8
|F 〉 ⇒ |〈0,0|r|1,1〉| · 2 = 0.81 – 0.02
0.5 -1.0 +
P
0,0 0 1 2 3
0.0
0 2 4 6 8 10 Relative phase offset (f) of p /2 pulse (rad)
Time (m s)
FIG. 3. Generating entanglement directly. Top panel: FIG. 4. Entanglement verification. Aglobal π/2 pulseis
RabioscillationsofasingleRydberg-dressedCsqubit. Lower applied to the undressed system after the entangled state is
three panels: two-atom data with Rydberg-dressed spin-flip prepared. The data is obtained with the same experimen-
blockade (J/h≈750 kHz with ΩL/2π = 4.3 MHz, ∆L/2π = tal parameters used for data in Fig. 3. |Ψ+i data is fit-
1.1 MHz, R = 2.9 µm). The data points are fitted with ted with a straight line, and Φ+ data is fitted with a si-
| i
curves of damped oscillation and exponentially varied offset. nusoidal function. It shows that both Bell states generated
Rabi oscillation occurs between two spin down atoms and a fromourexperimenthaveafidelity 81(2)%. Here,ρrepre-
≥
two-qubit entangled state. There is a √2 enhancement of sentsthetwo-qubitdensitymatrix. Theparitymeasurement,
themicrowaveRabirate,Ωmw arisingfromtheblockade,and P1,1+P0,0−(P0,1+P1,0), allows direct determination ofthe
excitation to state 0,0 is strongly suppressed due to the amplitudes of the off-diagonal elements for both entangled
| i
transition blockade as shown in Fig. 6a. The maximum Bell states.
state Ψ entanglement is thusgenerated at around 2 µs.
+
| i
entanglementfidelity 81 2%forgeneratingboth Ψ
+
≥ ± | i
and Φ when both atoms are recaptured in the trap.
We measure the fidelity of Bell-state preparation as +
| i
follows. For a given J, the Raman pulse duration T is
chosen so that √2Ω T = π with Ω J/~. Follow-
mw mw
ing this procedure, the automated exper≪imental control DISCUSSION
system checks whether both atoms are still present in
the traps; if so, it counts as a “valid” operation. We de- Given that we measure the two-atom survival proba-
termine the lower bound of the entanglement fidelity by bility after the procedureto be 74 2%, the success rate
±
measuring the off-diagonal coherence between the two- of deterministically generating an entangled qubit pair
qubit logical basis states, x′,y′ ρx,y , where ρ is the is therefore 60 3%. With our current experimental
two-qubit density matrix. hFor t|hi|s, wie apply a global data rate ( ≥10 s−±1), on average we generate 6 pairs of
≈
π/2 pulse with phase φ to the entangled state. As a entangled qubits per second. The current factors lim-
function of φ, we measure the expected value of the par- iting the entanglement fidelity given a valid procedure
ity Q(φ)= σ σ =[P +P (P +P )](φ), are: the optical pumping efficiency, decay of the Raman
z z φ 1,1 0,0 0,1 1,0
h ⊗ i −
where P (φ) is the population in the logicalstate x,y Rabi oscillation of the Rydberg-dressed states (Fig. 3),
x,y
| i
after application of the π/2 pulse [33, 34]. For qubits and the strengthof J. Optical pumping efficiency deter-
prepared in the Ψ state, Q is independent of φ and mines how well we can prepare the atoms in the qubit
+
| i
alwaysapositive number; Q =1 whenthe entanglement subspace (computational space), which can be improved
is perfect. Forqubits preparedinthe Φ state, Q(φ) is with a more carefulpumping scheme. Decay of the Rabi
+
| i
an oscillating function of φ. In this case, perfect entan- oscillation impacts the fidelity of the π pulse; we can
glement corresponds to perfect oscillation visibility. The improve this fidelity by increasing Ω but only for suf-
mw
entanglementfidelity(fidelitybetweenthepreparedstate ficientlylargeJ. Oneidentifiedcontributiontothedecay
and the target Bell state) is 2 x′,y′ ρx,y , as mea- is the shorter than expected single-atom Rydberg-state
≥ |h | | i|
sured from Q(φ) [34]. When this fidelity is greater than lifetime (see Supplementary Material). A comprehensive
50%, the state is necessarily entangled. The measure- explanationofthetwo-atomRabioscillationdecayisthe
ment of Q(φ) in Fig. 4 shows that with the same exper- subject of future work.
imental parameters as used in Fig. 3, we have achieved The protocol presented here is quite robust despite
5
some technical imperfections, and we expect improve- tance R by ramping the AOM modulation frequencies
ment in future experiments. For example, currently the as shown in Fig. 1 of the main article. From the initial
Bell-stateentanglementisgeneratedinthedressedbasis, separation of 6.6 µm, we can continuously vary R down
resulting in atom loss due to the admixture of the Ry- to a minimum value of 1.5 µm, at which point the two
dberg state. In principle dressed states can be returned traps begin to merge causing atom loss. During the Ry-
to the bare ground states by adiabatically ramping the dberg dressing period, we turn off the traps to eliminate
dressinglaser’sintensityanddetuninginanoptimalway the light shift due to the trap laser. Afterward, we re-
[31], something we will implement in a future generation store the traps to recapture the falling atoms. We use
of the experiment. Finally, minimizing other experimen- a Rydberg dressing laser at 319 nm which drives direct,
tal imperfections will likely improve the Rydberg state single-photon transitions from 6S to nP . This Ry-
1/2 3/2
lifetime, improvethe deterministic entanglementfidelity, dberg excitation laser is designed to cover the principal
and open the door to more full controlof complex quan- quantum numbers ranging from n = 30 to ionization.
tum systems. Because EDDI causes a red shift of the two-atom Ryd-
berg state r,r , we blue detune the Rydberg excitation
| i
laser. With a strong bias magnetic field, we use the Ry-
SUPPLEMENTARY MATERIALS dberg state nP with magnetic sublevel m = 3/2 for
3/2 J
dressing the qubit state 0 (Fig. 5).
| i
Experimental platform
For state detection, we translate the two trapped
Cs atoms back to the original positions. The
Ourexperimentalplatform[23]is builtonlasercooled state-dependent detection is accomplished using the
133Cs atoms, singly trapped in optical tweezers (dipole 6S ,F =4 6P ,F′ =5 D2cyclingtransitionto
1/2 3/2
| i→| i
traps). We encode qubits in the clock states of the ce- determinewhethereachatomisinstate 0 (brighttothis
| i
sium 6S1/2 ground-statemanifoldwith a hyperfine split- excitation) or 1 (dark to this excitation). In the case
| i
ting ωHF/2π = 9.2 GHz. We choose as our logical basis thatthe atomisfoundtobedark,weimmediatelyapply
0 = 6S ,F = 4,m = 0 and 1 = 6S ,F =
1/2 F 1/2
| i | i | i |
3,m = 0 as shown in Fig. 5. The two atoms are
F
i
loaded from a MOT (magneto-optical-trap)into the two
optical dipole traps with a separation of 6.6 µm. These
dipole traps are generated by sending a 938-nm laser
beam to an AOM (acousto-optic modulator), which is
simultaneously modulated at two different frequencies.
This creates two beams with a well-defined angular sep-
aration that depends linearly on the AOM drive fre-
quency difference. These beams pass through a focus- 319 nm
ing lens forming two tightly-focused spots with waists
of r1/e2 = 1.29(3) µm and a well-defined spatial sep-
aration that depends on the angular separation of the
two beams. Following polarization gradient cooling, the
aattom4.8teGmptehreantutruerinssreodnu,ceadndtow≈e o2p0tiµcKal.lyApbuimaspfitehlde . . . . . .
atoms into state 6S ,F = 4,m = 0 (logical basis
1/2 F
| i
state 0 ) using a π-polarized laser at 895 nm tuned to
| i ′
6S ,F = 4 6P ,F = 4 and a repump laser
1/2 1/2
| i → | i ′
at 852 nm tuned to 6S ,F = 3 6P ,F = 4 .
1/2 3/2
| i → | i
Tsthraeys,tfiatcetiptiroeupsarmataigonneetffiiccfiieenldcypirso≈du9c5ed%,blyimvietcetdorbyligthhet . . . .
shifts fromthe dipole-traplaser. We apply atwo-photon
Ramanlaserfieldtoperformaglobalπ rotationtobring
FIG. 5. Relevant energy level diagram for the 133Cs atom.
the atoms from 0,0 1,1 . The stimulated Raman
| i → | i The qubit states 0 and 1 are encoded in the clock state
transition uses the carrier and one sideband from a laser | i | i
of the Cs hyperfine sublevels F = 4,m = 0 and F =
F
tuned50-GHzredoftheCsD2line(6S1/2 →6P3/2)and 3,mF = 0i. The Rydberg dre|ssing laser, detunied ∆L|from
modulated via a fiber-based EOM (electro-optic modu- 64P ,m = 3/2 , strongly interacts with the F = 4 hy-
| 3/2 J i | i
lator). perfinemanifold. Here,ω +∆E(1)/~isthehyperfinesplit-
HF LS
ToRydberg-dresstheatomswithastrongEDDI(elec- ting summed with the single-atom light shift dueto the319-
tric dipole-dipole interaction), we dynamically translate nm laser. A two-photon stimulated Raman transition drives
Rabioscillations between 0 and 1 at a rate Ω .
the two Cs atoms into close proximity to a targeted dis- | i | i mw
6
the repump laser simultaneously with the cycling laser In Fig. 3 of the main article, the optimal π time for
to check that the atom is indeed in state 1 by verifying generating Ψ is 2 µs. Toproduce aBellstate Φ ,
+ +
| i | i ≈ | i
its presence in the trap. Note that the detection method we simply apply a global π/2 pulse to Ψ . By fine
+
| i
identifies the entire 6S ,F =4 manifold with 0 and tuning the experimental parameters, our best entangle-
1/2
| i | i
the entire 6S ,F = 3 manifold with 1 . The fluo- mentfidelityis 81(2)%excludingtheatomlossevents,
1/2
| i | i ≥
rescence signals are detected via two APDs (avalanche and 60(3)% when loss is included. If the atom recap-
≥
photodiodes) that are coupled via optical fiber. This ture process were to completely filter all cases when the
non-destructivemethodallowsustoreuseatomswithout atomwas excitedto the Rydberg state,we wouldexpect
reloadingnew atoms fromthe MOT,increasingourdata to have a lower two-atom survival probability compared
rate to 10 s−1 from 1 s−1. The experimental pro- to the 74% recapture probabilty we present here. With
≈ ∼
tocolis carriedoutwith anFPGA (Field Programmable perfect filtering and excluding the atom loss events, the
Gate Array) control system [23]. entanglement fidelity would be much higher and eventu-
allylimitedbythe statepreparationefficiency. However,
it would still be near 60% with loss included. For tech-
Generating Bell states nical reasons, we observe that a fraction of the atoms
that are excited to the Rydberg state decay back to the
In order to obtain a strong differential light shift, and ground state within the recapture time window. This
therefore a strong dressed-interaction energy J between is only consistent with a shorter than expected Rydberg
two atoms, we require ∆ Ω (the strong dressing lifetime. The resulting increase in the two-atomsurvival
L L
≤
regime). In fact, this condition also leads to a maxi- probabilityreducesthevalueofthefidelitymeasurement
mum suppression of the optical detuning noise due to thatisconditionalontheatomlossevents. Weanticipate
the thermal motion of the atoms. The probability am- a substantial improvement of the entanglement fidelity
plitude of the ground-state 0 in the dressed state is by achieving close to the natural Rydberg-state lifetime
| i
(∆ / ∆2 +Ω2 + 1)1/2/√2, and has a value between andalso by using anadiabatic rampofthe intensity and
L p L L
1/√2 for maximal dressing and 1 for no dressing at all. detuning of the 319 nm Rydberg laser in order to return
In our experiment we choose two example conditions: the dressed state back to the bare ground state.
(Ω /2π =4.4 MHz, ∆ /2π=4 MHz) and(Ω /2π =4.3
L L L
MHz, ∆ /2π = 1.3 MHz). This leads to the dressed
L
states of 0.41r +0.910 and 0.6r +0.80 or 84% and Calculation of Rydberg-state interactions and the
64% probabil|itiy in 0 |riespective|lyi. In th|isi experiment ground-state interaction J
| i
we do not adiabatically transfer the dressed state back
to the bare ground state, which lowers the atom recap- CalculatingJ asafunctionoftheinteratomicdistance
ture probability. A detailed discussion regarding strong R requires knowledge of how the two-atom Rydberg en-
Rydberg dressing and adiabatic transfer can be found in ergylevelsshiftasafunctionofR. Themixingofatomic
Refs.[30, 31]. When there is a probability that the atom orbitalsduetointeractionsleadstomolecular-likeenergy
is excited to the Rydberg state, there is a probability levels. Weusethesamenumericalcodeswedevelopedin
that the atom will not be recaptured in the trap during our previous work [23, 30] for the calculation. Figure 6a
the time window ( 10 µs) of efficient recapture . In shows the calculated two-atom molecular energy levels
≈
the detection, we only count the data with atoms that for 64P for R from 1 µm to 10 µm with electric field
3/2
remain trapped. We track statistics concerning the lost E = 6.4 V/m and magnetic field B = 4.8 G, parame-
| | | |
atoms. terstypicalofourexperiment. Thereareatotalof14,400
AstraightforwardmethodforgeneratingtheBellstate energylevelsincludedinthe calculation,whichcoverthe
Ψ = (0,1 + 1,0 )/√2 is illustrated in Fig. 3 of principal quantum numbers from n = 60 to n = 66 and
+
| i | i | i
the main article. This shows typical experimental data the orbitals from l = 0 to l = 4. In the calculation,
representing Rabi flopping in the presence of Rydberg theRydberg-RydberginteractionsincludeEDDI,electric
dressing with the parameters n = 64, Ω /2π = 4.3 dipole-quadrupole, and electric quadrupole-quadrupole
L
MHz, ∆ /2π = 1.1 MHz, R = 2.9 µm, and J/h 750 interactions. Asonecansee,thereisa“spaghetti”struc-
L
≈
kHz. Each data point is the average of several hundred ture at very short R due to the strong mixing between
measurements but with various Raman pulse durations. allcombinationsofRydbergenergylevels. Whenexcited
With a strong J, microwaveexcitationto the state 0,0 with the 319-nm laser, the probability of exciting one of
| i
is blockaded. The microwave Rabi oscillation can only these resonances in the spaghetti depends on the oscil-
occur between 1,1 and (0,1 + 1,0 )/√2 with a very lator strength of the different molecular levels. In the
| i | i | i
small probability exciting to 0,0 . We also find a factor figure, we denote this oscillator strength by the dark-
| i
of√2enhancementofthemicrowaveRabiratecompared ness of the lines. We see that most molecularresonances
tothesingle-atomRabirateassecondaryevidenceofen- are only weaklycoupled to the Rydberg excitationlaser.
tanglement. With limited computational resources, it is numerically
7
0.7 0
(a) (b)
0.68 -100
0.66 -200
)
z
H
G0.64 -300
(
h
/0.62 ) -400
z
es H
rgi 0.6 (k -500
e h
n /
E0.58 J -600
e
v
ti0.56 -700
a
el
R
0.54 -800
0.52 -900
0.5 -1000
2 4 6 8 10 2 4 6 8 10
Interatomicdistance,R(µm) R(µm)
FIG. 6. (a) Calculated two-atom Rydberg sublevels of 64P with E = 6.4 V/m and B = 4.8 G. The vertical axis is the
3/2 | | | |
energy offset from thecenter of gravity of 64P. The black dashed curveis thefitting result of our selected two-atom Rydberg
state m = 3/2,m = 3/2 . (b) Solid curve: the calculated J as a function of R using all levels from the left panel. The
J J
| i
parameters of the dressing laser are Ω /2π = 4.3 MHz and ∆ /2π = 1.3 MHz. Dashed curve: using the black dashed curve
L L
from the left panel with thesame parameters.
intractable to calculate the entirety of multipole interac- Rydberg State Lifetime
tions in the infinite atomic basis set. Hence, the calcu-
lated spaghetti feature can never be precise. In Fig. 6b,
Decay of the microwave Rabi oscillation of the Ryd-
we plot the two calculated J(R) curves with parameters
berg dressed atoms is partially explained by the shorter
Ω /2π = 4.3 MHz and ∆ /2π = 1.3 MHz using the
L L thanexpectedlifetimeoftheRydberg-state. Thelongest
result from Fig. 6a. The J(R) represented in the solid
measuredRydberg-statelifetimeforasingleatominthis
curve uses all the energy levels we calculate. It has a
experiment is on the order of 40 µs while the expected
wiggling structure at small R mainly due to the multi-
natural lifetime due solely to vacuum and blackbody ra-
ple level crossing and mixing of the doubly-excited Ry-
diation is 150 µs. Our measurements indicate that
dberg levels. The J(R) represented in the dashed curve ≈
the Rydberg state lifetime is further reduced by photo-
is a best-fit curve for the doubly-excited Rydberg state
induced processes on the nearby ITO-coated surfaces,
ofourinterest(m =3/2,m =3/2 )torepresent r,r
J J which are for shielding the external electric-field. We
| i | i
as a function of R shown by the black-dashed curve in
performed direct measurements of the single-atom Ryd-
Fig. 6a. This simple blockade shift curve uses a form
bergstatelifetimebyusingtworesonantopticalRydberg
of C/R6. The single fitting parameter C is determined
excitationπpulseswithavariabledelaytimeinbetween.
by using the all-level calculated data with R 3.5 µm.
By measuring the probability of the atom in the ground
≥
One can see that the two results of J(R) in Fig. 6b are
state following this sequence, we can determine the Ry-
verysimilarasidefromthewigglingstructureonthesolid
dberg state lifetime. We found that the Rydberg state
curve. As mentioned before, this structure changes with
lifetime was greatly reduced when the laser light acts on
different atomic basis sets used in the calculation, and
thenearbyITOcoatedsurface. Astudyofthiseffectwas
wehavenotbeenabletoobtainaconvergedanswerwith
carriedout witha non-resonant638-nmlaser,whichwas
ourlimitedcomputerpower. Wethereforeusethedashed
originallyusedforgeneratingchargesontheITOsurfaces
J(R) curve to compare with our experimental data.
tocompensatethebackgroundelectricfields[23]. Theex-
periment indicates that stronger 638-nm laser intensity
leads to a shorter Rydberg state lifetime. We suspect
that scattered laser light impinging on the nearby sur-
face,suchas the 319-nmand938-nmlasers,canproduce
8
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