Two-Atom Rydberg Blockade using Direct 6S to nP Excitation

0.67 MB·

by  A. M. Hankin

Two-Atom Rydberg Blockade using Direct 6S to nP Excitation
A. M. Hankin,1,2 Y.-Y. Jau,1 L. P. Parazzoli,1 C. W. Chou,1
D. J. Armstrong,1 A. J. Landahl,1,2 and G. W. Biedermann1,2,∗
1Sandia National Laboratories, Albuquerque, New Mexico 87185, USA
2Center for Quantum Information and Control (CQuIC),
Department of Physics and Astronomy, University of New Mexico 87131, USA
(Dated: April 22, 2014)
We explore a single-photon approach to Rydberg state excitation and Rydberg blockade. Using
detailed theoretical models, we show the feasibility of direct excitation, predict the effect of back-
groundelectricfields,andcalculatetherequiredinteratomicdistancetoobserveRydbergblockade.
4 We then measure and control the electric field environment to enable coherent control of Rydberg
1 states. With this coherent control, we demonstrate Rydberg blockade of two atoms separated by
0
6.6(3) µm. When compared with the more common two-photon excitation method, this single-
2
photon approach is advantageous because it eliminates channels for decoherence through photon

r scattering and ac Stark shifts from the intermediate state while moderately increasing Doppler
p
sensitivity.
A

1
I. INTRODUCTION two-photon experiments [9]. Minimizing photon scat-
2
tering is of particular importance when using Rydberg-

] The large polarizability of high principal quantum dressed atoms to create tunable, long-lived, many-body
h number n Rydberg states gives rise to exotic many- interactionsinaquantumgas[10]. Forexample,theadi-
p
body interactions as well as an extreme sensitivity to abatic quantum optimization protocol described in [11],
-
m the electric field environment. Precision spectroscopy is predicted to achieve a substantially higher fidelity in
of such states allows for a variety of exciting demon- the absence of photon scattering from the intermediate
o
t strationsinmetrology,fundamentalquantummechanics, state. At present, studies of Rydberg blockade with a
a and quantum information. For example, cold Rydberg single-photon transition are rare. Previous work utilized
.
s atoms employed as near-surface electric field sensors en- single-photon excitation with pulsed UV lasers to per-
c able characterization of both field amplitude and source. form high-resolution spectroscopy of Rydberg states [12]
i
s This includes experiments that explore near-surface field andtodetectRydbergblockadeasabulkeffect[13]. Still,
y
spectral density [1], induced dipole moments for sur- direct excitation using a continuous-wave (cw) UV laser,
h
p face adatoms [2], and insulator charging on an atom which would allow for coherent control of single atoms,
[ chip [3]. Large Rydberg state polarizabilities also enable has not been demonstrated.

long-rangeelectricdipole-dipoleinteractions(EDDIs)be- Here, we show coherent control of blockaded 84P
4 3/2
v tween Rydberg atoms, yielding strongly correlated sys- states of two single 133Cs atoms using a cw UV laser at
tems through the Rydberg blockade effect. Recent ex-
1 319 nm. Construction of this UV laser is informed by
9 perimentsuseRydbergblockadetoobserveentanglement calculations for the required wavelength and intensity.
1 betweenneutralatoms[4,5],acontrolled-NOTquantum
With over 300 mW of 319-nm light at the output of the
2 gate [6], and collective many-body Rabi oscillations [7].
laser, we demonstrate a Rabi frequency of over 2 MHz
1. Theseadvancesparallelanever-evolvingapproachtoRy- with this approach, in agreement with our predictions
0 dberg state control. In this paper, we demonstrate Ry- for resonance frequency and oscillator strength. Given
4 dberg blockade using a unique single-photon excitation
this success, we further develop our model to determine
1 approach to precision Rydberg spectroscopy.
a regime for observing Rydberg blockade between two
:
v The ionization threshold for ground-state alkali-metal atoms, and we observe and analyze Rydberg blockade
i atomsrangesfrom3.9to5.4eV,settingtheenergyscale
X for the 84P3/2 state.
for excitation to high-lying Rydberg states. In prac-
r tice, this is commonly accomplished with two-photon Thispaperisorganizedinthefollowingway: InSec.II,
a
excitation, where the ground and Rydberg states cou- weestablishadetailedmodelforsingle-photonexcitation
ple together through an intermediate state [8]. Two- toRydbergPstatesthatincludespredictionsfortheRy-
photon excitation avoids deep, ultraviolet (UV) wave- dberg spectrum and oscillator strengths. We next use
lengths making the implementation technologically sim- this model to design a cw UV laser system, the details
pler. However, photon scattering and ac Stark shifts of which are found in Sec. III. In Sec. IV, we describe
from the intermediate state introduce avenues for deco- the experimental technique used to trap and control two
herence, frequency noise, and dipole forces complicating atoms in close proximity. In Sec. V, we use our single-
atom control in combination with the UV laser system
to measure the background electric field inherent to our
apparatus. We then implement active control and sup-
∗ [email protected] pression of the electric field to enable coherent control of
2
where (cid:96) is the orbital angular quantum number, j is the
1.0
z) total angular momentum quantum number, E∞ is the
H
ionizationthresholdenergy,R istheRydbergconstant
P 0.8 Cs
( for cesium, and δ is the quantum defect. A method for
y
nc 0.6 calculating δ(n,(cid:96),j) is found in [15] where the observed
e
u Rydberg spectra are fit to a power series in n. The spec-
q
re 0.4 trum of 133Cs calculated with Eq. (1) is found in Fig. 1.
f
on The optical frequency for excitation directly from 6S1/2
ati 0.2 to 84P3/2 is calculated to be 941030 GHz.
cit (a) Given the transition frequency, we require the transi-
Ex 0.0 tion oscillator strength f to determine if single-photon
excitationisfeasiblewithcurrenttechnology. Asemiem-
0 (S) 1 (P) 2 (D)
piricalmethodforcalculatingf isfoundin[16]. Comput-
Orbital angular momentum
ingf requiresknowledgeoftheradialwavefunctionsand
the associated radial matrix elements (cid:104)n(cid:48)(cid:96)(cid:48)j(cid:48)|r|n(cid:96)j(cid:105). The
) 20 86S
z radialwavefunctioniscalculatedbysubstitutingtheen-
H
G 85P 84D ergiespredictedbyQDTintoSchr¨odinger’sequationand
(/2 10 85S (cid:104)n(cid:48)(cid:96)(cid:48)j|r|n(cid:96)j(cid:105) is then computed through numerical inte-
84P3 0 84P 83D gtora8ti4oPn.trWanesifitinodnsthaereofsc(i6llSator →stre8n4gPths)fo=r t6he×61S01−/28
1/2 3/2
m
o 84S and f(6S1/2 → 84P1/2) = 2 × 10−12. The 104 order
r
f -10 82D ofmagnitudedifferencebetweenthecalculatedoscillator
g 83P
n strengths is a property that is unique to cesium when
tuni -20 83S (b) compared with the other alkali-metal atoms. The diver-
De gence of the principal-series doublet (6S1/2 →nP3/2,1/2)
0 (S) 1 (P) 2 (D) oscillator strength ratio for large n is a well-known phe-
nomenon that arises with the inclusion of spin-orbit ef-
Orbital angular momentum
fects and the core polarizability [16–19]. This result fa-
vors exciting to nP states in the interest of reducing
FIG. 1. (Color online) Spectrum of cesium using QDT. (a) 3/2
laser power requirements.
Spectrum from the ground state through n = 100. The ex-
perimentalworkinthispaperfocusesonthe6S →84P While the oscillator strength determines the scaling of
1/2 3/2
transition, labeled by the blue arrow. (b) Detailed spectrum RabifrequencyΩwithlaserintensity,itdoesnotdirectly
near 84P . Fine structure splitting in the nP and nD set a lower limit on laser power. Instead, we must con-
3/2 j j
states is included. nP states appear broader due to splitting siderlimitsontheRydberglaserwaistandthecoherence
between nP1/2 and nP3/2. Fine structure between the nD timebetweentheRydbergstate|r(cid:105)andthegroundstate
states is ∼ 100 MHz and therefore not well resolved on this |g(cid:105)setbyexperimentalconditions. Forareasonableatom
scale. temperatureof10µKandatrapwaistof1µm,theatom
velocityspreadlimitsthelinewidthtoorder100kHzand
the spatial spread to order 1µm. We target a Rabi fre-
the atom. In Sec. VI, we present our study of the Ry-
quency on the order of Ω/2π =1 MHz and a laser waist
dberg blockade, including a model of the Rydberg spec-
of 10µm to avoid decoherence and intensity fluctuations
trum for two atoms as a function of interatomic spacing
duetoatommotion. CombiningthelimitationonΩwith
andanexperimentaldemonstrationoftheblockadeeffect
the targeted waist, we find that 16 mW of 319 nm light
in this system. We conclude with applications where the
is sufficient to observe state evolution that is dominated
single-photon excitation approach is expected to excel.
by coherent dynamics. The design for the cw UV laser
described in the following section surpasses this require-
ment.
II. SINGLE-PHOTON EXCITATION MODEL
We use a theoretical model for single-photon excita-
III. RYDBERG LASER
tion ofhigh-n states to accuratelycalculate the Rydberg
spectrumand6S →nP transitionoscillatorstrengths.
1/2
The cw UV laser is constructed using sum frequency
Thespectraofalkali-metalatomsarepredictedwithhigh
generation(SFG)followedbyfrequencydoubling. Asim-
precision by quantum defect theory (QDT) [14]. Using
ilar approach tailored for 313 nm is found in [20]. We
QDT,theenergiesofboundelectronicstatesE aregiven
first produce 638 nm light using SFG and then gener-
by
ate the 319-nm light via frequency doubling. The SFG
R begins with 1574- and 1071-nm fiber laser sources with
E(n,(cid:96),j)=E − Cs , (1)
∞ (n−δ(n,(cid:96),j))2 18- and 60-mW output powers, respectively. Both lasers
3
PMF PMF COL ISO Lens DM PPLN & Oven
NP Photonics FL IPG Photonics FA
1574 nm, 18 mW 1574 nm, 5 W
T = 197 K
NP Photonics FL IPG Photonics FA
1071 nm, 60 mW 1071 nm, 5 W
COL
FL-FRQ Window
Control
AOM EOM
DP-AOM
λ/4 PBS 200 MHz 50 MHz COL
100 MHz
COL
Stable Laser Systems SFG Output
PD
HF-ULE Cavity 638 nm, 1.1 W
DF
RF 3
50 MHz
SF
HF Cavity
AM Laser Lock Circuit
Servo PMF In Coherent, Inc.
Out RF 1 VCO AM PI Doubling Cavity
(BBO)
LF Cavity and SF
Lock Circuitry FD AM
Cylindrical
ULE Cavity RF 2 Lens
FB In
UV Output
319 nm, 300 mW
To Experiment
AOM, 180 MHz AOM, 200 MHz
Switch Noise Eater
FIG. 2. (Color online) Diagram of UV laser system. Frequency summing in a PPLN crystal of two fiber laser systems
yields 638 nm light and frequency doubling of this light in BBO generates UV light at 319 nm. Laser frequency is stabilized
using a multistaged servo with an ultra-low expansion (ULE), high finesse (HF) cavity as the primary reference. λ/4—
quarter-wave plate, AM—voltage amplifier, COL—fiber collimation package, DF—difference frequency, DM—dichroic mirror,
DP-AOM—double-pass acousto-optic modulator system, EOM—electro-optic modulator, FA—fiber amplifier, FB—feedback,
FD—frequency doubling, FL—fiber laser, FL-FRQ—voltage control of fiber laser frequency, ISO—optical isolator, LF—low
finesse,PD—photodiode,PI—proportional-integralfeedback,PMF—polarizationmaintainingfiber,SF—sumfrequency,RF—
radio frequency source, VCO—voltage controlled oscillator.
seed commercial 5-W fiber amplifiers and the resulting periencing a maximum intensity of 60 kW/cm2 after ac-
lightiscombinedandpassedthroughaperiodicallypoled counting for the losses incurred at each optic.
lithiumniobate(PPLN)crystalgenerating1.1Wof638-
The frequency of the Rydberg laser is stabilized to
nm light. The output of the PPLN crystal is frequency
an ultra-low expansion, high finesse (HF) cavity [21] at
doubled from 638 to 319 nm with a BBO (β-BaB O )
2 2 638nmviaamultistageservoarchitecture(Fig.2). Adi-
crystalandresultsingreaterthan 300mWatthiswave-
rectlocktothecavityfrequencyreferenceisprecludedby
length. From the spectrum shown in Fig. 1, we pre-
characteristicsofthefiberlasersources. Frequencynoise
dict the laser’s frequency can be tuned to reach 84P
on the 638 nm light exceeds the 75 kHz cavity linewidth
through 120P. Upon exiting the doubling cavity, the
as well as the bandwidth of the frequency control of the
beam is shaped into a Gaussian profile, passed through
fiber lasers. We overcome this by dividing low and high
two AOMs for intensity stabilization and switching, and
bandwidth frequency stabilization into two paths.
focused down on the two atoms with a measured 1/e2
radius of 12.9(3)µm. This results in the two atoms ex- Low-bandwidth frequency control is implemented by
firststabilizingthe638-nmlighttoalowfinesse(LF)cav-
4
x ITO coated y
636 nm
glass plate
y charging x charging laser
z laser z ITO coated
asphere
2.38 mrad dipole trap
lasers
aspheric
lens
319 nm
dipole trap ITO coated
Rydberg laser
dipole traps lasers aluminum windows
6.6 μm separation 319 nm lens mount
Rydberg laser
(a)topview (b)sideview
FIG. 3. (Color online) Diagram of the atom trapping region. Two collimated 938-nm dipole trap beams with a 2.38-mrad
relative angle pass through an aspheric lens resulting in two traps separated by 6.6(3) µm at the focal plane. A 319-nm laser,
used to excite to Rydberg states, is focused down to a 12.9(4)µm waist at the location of the atoms. The aspheric lens has a
112-nmITOcoatingonthesidefacingthedipoletrapsandananti-reflection(AR)coatingfor852nmontheoppositeside. An
aluminumcylinderisfixedconcentrictotheAR-coatedsidetoshieldagainstchargingofthisdielectric. Theresultingassembly
is fixed between two ITO-coated glass plates with a vacuum compatible, conductive epoxy. Each plate has an ITO coating on
the side closest to the traps. The entire assembly is grounded. The top ITO plate is not shown in (a) for clarity. A 636-nm
charging laser beam generates controlled charging on the ITO plates, which grants leverage over the background electric field
environment. Figures are not to scale.
ity with a 5 MHz linewidth. This frequency stabilization in the high-power 638-nm beam path, we choose a feed-
stagenarrowsthelaserlinewidththroughfeedbacktothe forward approach in favor of maximizing the UV power.
1071nmfiberlaser. Next,wesplitasmallfractionofthe Thefeed-forwardcontrolisaccomplishedbysplittingthe
638-nm light along an optical path used to monitor and radio frequency (RF) signal used to stabilize the 638-nm
stabilize the light’s frequency to the HF cavity. Because laser to the HF cavity along a second path. This new
the LF cavity lock has narrowed the 638-nm linewidth, path applies high-bandwidth corrections to the 319-nm
we are able to directly lock the light along this path to laser frequency by modulating the drive frequency of an
the HF cavity. The HF cavity lock system consists of an AOMintheUVbeampath. Thecircuitcompensatesfor
electro-optic modulator (EOM), an AOM, and the HF thefrequencydoublingthatoccursintheopticaldomain
cavity. The locking error signal is generated using the attheBBOcrystalbyplacingafrequencydoublerinthe
Pound-Drever-HalltechniquewheretheEOMmodulates secondRFpath. Thisfeed-forwardarchitecturetransfers
the phase of the light while the response is monitored the high-frequency content of the HF cavity lock to the
in reflected cavity signal. We then feedback to the drive UV light.
frequency of the AOM to stabilize the frequency of the
light to the cavity. However, because this AOM is not
placed in the primary, high-power 638-nm beam path, IV. EXPERIMENTAL APPARATUS
the laser frequency at the atom does not directly benefit
from the servo. This leaves the primary laser beam path We create two optical dipole traps for quantum con-
susceptibletodriftsintheLFcavitylengthwithchanges trol of single cesium atoms in an all glass, ultra high
in temperature, pressure, and humidity. We avoid this vacuum cell. The traps are generated inside a partial
issue using a low-bandwidth feedback loop that adjusts Faraday cage to control stray electric fields that perturb
theLFcavitylengthtostabilizethefrequencyofthe638- theRydbergstates. Rydbergexcitationoftheseatomsis
nm light to the HF cavity resonance. The result of this detectedviaatomlossasRydbergstatesareejectedfrom
portionofthelockingsystemisa638-nmlinewidthofno the trap. By carefully tuning the optical parameters to
more than 200 kHz along the high-power beam path. minimize heating, we are able to reuse the same atoms
The linewidth of the 319-nm light is narrowed further for multiple experiments. The experimental system used
using high-bandwidth feed-forward control. While it is to trap and probe two single 133Cs atoms was built by
possible to use a closed-loop servo to directly stabilize modifying the setup described in [22]. The atoms are
the laser frequency by placing the control AOM directly confined with a well defined separation, in two far off-
5
resonant dipole traps using 938 nm light. The trapping
4.0
light passes first through an acousto-optical modulator
(AOM) and then through an in-vacuum aspheric lens 3.5
with a 2.76 mm focal length. By driving the AOM at B =0.0 G
two frequencies (74.6 and 85.4 MHz) we generate two y 3.0
t
82.3m8Wmrbadea.mTshiwshAoOseMp-rloenpsagsaytsitoenmdrieresuctltiosnisndtwevoiadtiepoblye abili 2.5
traps separated by 6.6(3)µm [23]. Both traps have a ob Bz =1.1 G
1.26(1)µm waist and a 21.1(1) MHz trap depth for the pr 2.0
n
atomic ground state. Once trapped, the atoms have a o
vacuum limited trap lifetime of approximately 7 s. We ati 1.5 B =1.5 G
t y
sourcethe938nmlightfromadistributedfeedbacklaser ci 1.0
x
diode and find it necessary to filter elements of 852 and E
895 nm from this laser (D and D transitions in 133Cs) 0.5
2 1
B =4.8 G
to avoid excessive heating that inhibits stable trapping. x
0.0
The atoms are trapped 2.16 mm from the lens surface
where background electric fields can be problematic for -30 -20 -10 0 10 20 30
coherent control of Rydberg atoms [24]. We suppress
Frequency offset (MHz)
these fields by coating the surface of the lens closest to
theatomwithan112nmlayerofindiumtinoxide(ITO).
This transparent yet conductive coating is grounded to
FIG. 4. (Color online) Spectroscopy of the 84P state
3/2
dissipate charging. To further protect against the influ-
for various bias magnetic fields B. From top to bottom the
ence of external electric fields, we surround the trapping spectrum is shown for B = (0,0,0), (0,0,B ), (0,B ,0), and
z y
region with a partial Faraday cage in vacuum by mount- (B ,0,0) where the x, y, and z axes are labeled in Fig. 3.
x
ing the lens between two parallel glass plates that are The solid blue line represents a model for the spectrum that
alsocoatedwithITO(Fig.3). Usingfiniteelementanal- includes perturbations to the state due to the presence of
ysis to approximate the solution to Laplace’s equation magnetic and electric fields. A fit of the model to the data
for the electric potential, we calculate that this geome- using the magnitude and direction of E as free parameters
indicatesthepresenceofa6.35(5)V/melectricfieldcollinear
try suppresses electric fields external to the system by a
with the z axis (normal to the dipole trap lens) with a ±20◦
factor of 1000.
uncertainty. Thespectraareoffsetbymultiplesof1.0onthe
The atoms are loaded into the dipole traps from a y axis of the plot for clarity.
magneto-optical trap (MOT). The dissipative scattering
forcegeneratedbytheMOTcoolsatomsintotheconser-
vative pseudo-potential of these traps. Once captured,
period that allows for the magnetic field environment to
the atoms continue to fluoresce on the D transition of
2 stabilizeandtheMOTcloudtodissipate,wepreparethe
133Cs(6S →6P ),andwespatiallydiscriminatethis
1/2 3/2 same atoms for the single-photon excitation experiment.
signaltodetectaloadingevent. Thislightiscollectedby
the same aspheric lens used to produce the dipole traps Before excitation, we further cool the atoms and pre-
andadichroicmirrorseparatesthe938nmtrappinglight pare them in |6S1/2,F = 4,mF = 0(cid:105). The atoms are
fromthis852nmfluorescenceoftheD transition. After cooled to 16.1(1)µK using polarization-gradient cooling
2
reflecting off the dichroic, the two beams of fluorescence [27]. Theexperimentaldetailsforthiscoolingprocessare
are imaged at a plane coincident with a gold knife edge. foundin[22]. Forstatepreparationinto|F =4,mF =0(cid:105),
Theknifeedgeispositionedsuchthattheimagefromone a quantization axis is defined with a 4.8 G magnetic
atom is reflected off of the gold surface, while the image fieldandtheatomsareilluminatedwithπ-polarizedlight
from the other passes. Next, the fluorescence of each resonant with F = 4 → F(cid:48) = 4 on the D1 transi-
atom is coupled into separate 9µm core fibers [25] that tion (6S1/2 → 6P1/2). We find the large bias magnetic
feed separate avalanche photodiodes (APDs). We find fieldnecessarytoobtainanopticalpumpingefficiencyof
that this core size yields a near-optimal signal-to-noise 95(2)% into the target state [28].
ratio for single-atom detection in our apparatus. The The cold, |F = 4,m = 0(cid:105) atoms are the starting
F
technique we use to split the fluorescence beams is simi- point for the direct excitation experiments. These ex-
lartotheonedevelopedby[5]. WeadjusttheMOTcloud periments begin 1 ms following state preparation using
density to operate both traps in the collisional blockade the 319 nm laser described in Sec. III. Just before ex-
regime such that loading is limited to a maximum of one citation, the atoms are released into free-flight to avoid
atom [26]. By waiting for a coincidence of bright fluo- perturbationsintheground-stateenergyfromthedipole
rescence signals from both APDs, we load single atoms trap. They are then recaptured 1µs after the excitation
simultaneously in both traps. Once loaded, we switch pulse extinguishes. The results of these experiments are
off the loading process by extinguishing the MOT lasers foundinSecs.VandVI.WedetectpopulationintheRy-
and the quadrupole magnetic field. After a 10-ms wait dbergstatebytakingadvantageofthedifferenttrapping
6
0 (a) 1.0 (b)
ty nochargingbeam
Hz) -20 bili 0.8 chargingbeam
M -40 ba
( o 0.6
ft -60 pr
hi n 0.4
s o
k -80 ti
r a
a t 0.2
St -100 xci
E
-120 0.0
0 1 2 3 4 5 6 7 8 -120 -100 -30 -20 -10 0
Electric field (V/m) Laser frequency offset (MHz)
FIG. 5. (Color online) Electric field reduction via controlled surface charging with the charging laser. (a) Calculated Stark
shift in the frequency of the 84P state with no applied bias magnetic field. (b) Observed spectrum with (green points)
3/2
and without (red points) the charging laser beam. Measured resonance frequencies are plotted directly on (a) using the same
color scheme. The presence of the charging beam reduces the electric field magnitude to 1.5(1) V/m. Error bars shown are
representative.
forces experienced by the ground and Rydberg states for mum of 70 Hz for single-atom experiments. The logic
an atom in a red-detuned optical dipole trap; while the implementsahighspeedflowchartthatrespondsappro-
ground state of the atom is trapped, a Rydberg state priately to outcomes of the detection sequence. When
will experience an anti-trapping potential [8] causing it the atom is not detected during the check sequence, a
to quickly eject from the dipole trap and allowing atom morerobustcheckforatompresenceisperformedimme-
loss to signal Rydberg excitation [4]. diately. The robust check stage consists of a maximum
We check for atom loss 1 ms after excitation, allowing ofthreeMOTbeampulsesidenticaltothepulseusedfor
ample time for a Rydberg state to leave the trapping re- detection. Iftheatomsareobservedduringthedetection
gion. For this check, we probe for atom fluorescence on pulseoranyofthepulsesduringthechecksequence,then
the D cycling transition (F = 4 → F(cid:48) = 5) for 500µs the entire experimental cycle is repeated, skipping the
2
with two-counter propagating detection lasers that are rate-limitingtrap-loadingphase. Otherwise, theMOTis
directed along the y-axis (Fig. 3) superimposed with re- repopulated to reload the traps. Heating induced by the
pump light on the F = 3 → F(cid:48) = 4 transition. A de- detectionbeamisminimizedbyusingtheminimumnum-
tection beam detuning of 5.0 MHz and an intensity of berofpulsesrequiredtoverifyatompresence. Theactual
17mW/cm2 optimallytradesheatingrateandsignal-to- dataratedependsonthespecificsoftheexperiment. We
noise ratio. The APD that monitors fluorescence during achieve a 70-Hz data rate when performing single-atom,
detection measures an average number of photon counts state-selective, lossless detection experiments similar to
of 8.4 and 0.51 for the bright (atom present) and dark those described in [29, 30]. When implementing experi-
state(noatom)respectively. Withadiscriminatorequat- mentswithloss-baseddetectiontechniques,thedatarate
ing the bright-state with a measurement of greater than isloweranddependsontheexperimentspecificlossrate.
two counts, we achieve single-shot atom presence detec- For example, the average data rate for the experiments
tion with a 95% fidelity. We optimize the position of the detailed in Secs. V and VI are 2 Hz (Fig. 4), 0.7 Hz
gold knife edge to homogenize the response of the two (Fig. 7, single atom), and 0.3 Hz (Fig. 7, two atoms).
traps. The measured mean bright-state counts differ by The data rate in these experiments can, in principle, be
5% between the two traps and the atom presence detec- greatly improved by combining lossless detection tech-
tion fidelity differs by 0.5%. While increasing the de- niques with the transfer of Rydberg state population to
tection pulse duration improves fidelity it also increases |6S1/2,F =3,mF =0(cid:105) [5].
heating due to photon scattering which leads to atom
loss. Choosingashorterdetectionpulsereducestheprob-
ability to lose an atom allowing for the reuse of an atom V. ELECTRIC FIELD ENVIRONMENT
over multiple iterations.
The trap-loading phase of the experiment can limit Rydberg electron wave functions scale to extremely
our bandwidth by consuming around 97% of the duty large sizes with increasing n. Consequently, dipole ma-
cycle. To mitigate this reduced bandwidth we capital- trix elements between adjacent states grow as well, scal-
ize on atom reuse by employing a field-programmable ing like n2a e [31], where a is the Bohr radius and e is
0 0
gate array (FPGA) based control system for high-speed theelementarycharge. Thisinturnimpliesextremesen-
Boolean logic. The FPGA control system allows us to sitivitytodcelectricfieldsduetoincreasinglylargeelec-
increase our data rate from the 1 Hz level to a maxi- tric polarizabilities. We calculate that the 84P state
3/2
7
polarizabilityα isontheorderof1011 timeslargerthan
r
that of the ground state. We use this large polarizability
to measure the electric field environment at the dipole 40 levels
traps by studying the spectrum of the Rydberg state. laser
TheRydbergspectrumwemeasurefor84P3/2isshown z) 20
in Fig. 4. The excitation experiment uses a 580-ns UV H
M
laser pulse over a 50-MHz laser frequency scan range. 0
(
Wextheirlenawlepeerxtpuerbctataionsisn,gtlheepoebaskerivnedthsepeacbtsreunmcecoonfsaisntys ffset -20 4P3/2 (cid:11)
o 8
obfretawkoinngofnodretgheenezreartoempeaagknse.ticThfieeldobcsoenrvdeitdiodneg(Bene=rac0y) ncy -40 4P3/2
in Fig. 4 is the result of a background electric field that ue 8 (cid:12)(cid:12)
q -60
shiftstheresonancesthroughthedcStarkeffect. Tofur- e
r
F
ther our understanding of the background electric field
-80
source, we characterize the Rydberg spectrum at several (a) (b)
different bias magnetic field directions and compare the
-100
result with a detailed model.
6 7 8 9 1011 6 7 8 9 10 11
WemodelthesplittinginthefourRydbergresonances
R (µm) R (µm)
by including the perturbing effects of electric and mag-
netic fields. The relative splitting is calculated by diago-
nalizing the matrix of the total Hamiltonian, FIG.6. (Coloronline)Numericalcalculationforthe84P3/2+
84P spectrum versus interatomic separation R. The laser
3/2
H =H +H +H . (2) is resonant with the unperturbed (R→∞) state |rr(cid:105), where
atom Stark Zeeman
r → {84P ,m = 3/2}. (a) Frequency offset of all states
3/2 j
Here, H is the unperturbed Hamiltonian of a sin- with respect to r. The excitation range of the Rydberg laser
atom
gle atom and it’s matrix elements can be constructed is represented by the position and width (∼ 1 MHz) of the
using QDT [15]. The final two terms are given by red line. (b) States in (a) with line darkness weighted by
the oscillator strength to the ground state (6S , m = 0)
H = −µ ·E and H = −µ ·B. Here the 1/2 F
Stark d Zeeman m for linearly polarized light on the y-axis (Fig. 3). We choose
electricandmagneticdipolemomentoperatorsaregiven
to operate at R = 6.6(3)µm to obtain a 6.4-MHz blockade
by µ and µ respectively, E is a dc electric field, and
d m shift. The calculation includes a background electric field,
B is a dc magnetic field. While the strength and direc-
|E| = 1.6 V/m, and a bias magnetic field, B = 4.8 G as is
x
tionofBiscontrolledusingthreesetsofHelmholtzcoils, used in our experiment.
the electric field is a background intrinsic to our system.
We diagonalize Eq. (2) for a set of states large enough
toensureconvergenceoftheeigenvaluesandeigenvectors fluctuationsinE,westabilizethedipoletraplaserpower
of H over the chosen electric field range. This includes and cesium vapor pressure. Even so, this does not elim-
stateswherenrangesfrom81to89, and(cid:96)rangesfrom0 inate the observed fluctuations in Rydberg state energy.
to6. ThematrixelementsforH andH canbe The calculated electric field perturbation approximately
Stark Zeeman
calculated with techniques described in [32, 33]. A com- follows the quadratic Stark effect as is shown in Fig. 5.
parison of this theoretical model with experimental data Noise in the Stark shift, δEStark, increases linearly with
is shown in Fig. 4. Using the direction and strength of electric field noise, δE, or
theelectricfieldasafreeparameter,wefindthatthereis
δE =2α |E|δE.
a 6.35(5) V/m electric field at the location of the atom, Stark r
pointed along a direction perpendicular to the lens sur- Therefore,toreduceresonancefluctuations,weintroduce
face (Fig. 3). This field direction indicates charging of acharginglaserbeamtotheexperimenttogainleverage
the dipole trap lens. over the electric field environment.
We observe variation in surface charging with changes The charging beam generates charge on the ITO glass
in background cesium vapor pressure and dipole trap platesatapositionchosentocounteracttheelectricfield
laser intensity. While the dipole trap laser drives charge at the atom as shown in Fig. 3. The exact position and
production,whosesteady-statevalueincreaseswithlaser intensityofthechargingbeamisfinelytunedtominimize
intensity, the cesium vapor pressure modifies properties the electric field at the atom. The result of this process
of the charging process. We observe that increasing va- is found in Fig. 5(b), where the spectrum shifts blue and
porpressuredecreasesthechargingtimeconstant,which the Stark splitting is reduced. From the measured Stark
ranges from minutes to hours, and additionally reduces splitting,weestimatethattheelectricfieldattheatomis
the field strength. It is known that the density of ce- 1.5(1)V/m. Withtheintroductionofthechargingbeam,
sium coverage on a surface modifies the work function the measured Rydberg resonance has a full width at half
[34, 35]. Changes to the work function would affect the maximum of 440(50) kHz. Additionally, measurements
laser-induced charging andis a likely explanationfor the of the spectrum over a 9-h time period indicate a char-
observed trend with cesium vapor pressure. To reduce acteristic resonance drift of 20 kHz over 30 min. We find
8
that the combination of the ITO coated surfaces enables (a)
1.0
this stability in our system for high principal quantum y
t
number and correspondingly high electric field sensitiv- bili 0.8
ity. Thisisa100-foldimprovementwhencomparedwith a
b
our previous studies that utilized an identical lens with o 0.6
r
p
AR coating on the surface closest to the atom. This im-
n 0.4
proved stability allows for coherent excitation of block- o
aded Rydberg atoms as is demonstrated in Sec. VI. ati atom1
t 0.2
ci atom2
x bothatoms
E
0.0
VI. RYDBERG BLOCKADE 0 0.5 1 1.5 2 2.5 3 3.5 4
Laser pulse time (µs)
The Rydberg blockade effect occurs when the EDDI
1.0 (b)
potential energy U between nearby atoms shifts the
int state= gg
doubly excited Rydberg state |rr(cid:105) out of resonance with | i
the excitation laser, blocking multiple Rydberg excita-
tions. Producing efficient blockade entails maximizing 0.0
the interaction potential energy so that Uint (cid:29) Ω. At ty 1.0
lvaarngedeirntWeraaatlosmpioctesneptiaarlaotfiotnheRfortmheUinintte=racCti6o/nR6o.beFyrsoma babili state=|gri
this we can increase U by either increasing principal o
int r 0.0
quantum number since C ∝ n11 [36] or decreasing R. p
6 t 1.0
While it is attractive to maximize U by increasing n, n
int e
theelectricpolarizabilityalsorisesasn7 causingthesys- m state=|rgi
e
tem to be more susceptible to stray electric fields. We ur
consequently target a value of n with manageable dc as 0.0
e
Stark shift and significant blockade at interatomic dis- M 1.0
tances that are optically resolvable. While we have al- state= rr
| i
ready shown that the former condition is satisfied for
n = 84 in Sec. V, the latter can be determined numer-
0.0
ically. Therefore, to select a Rydberg state and inter-
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
atomic separation that satisfies U (cid:29) Ω, we numeri-
int
Laser pulse time (µs)
cally calculate the doubly excited Rydberg state spec-
trum nP +nP as a function of interatomic separation.
Thenumericalcalculationdeterminestheenergiesand FIG. 7. (Color online) Observation of coherent excitation
and Rydberg blockade. (a) Excitation probability for single,
transition oscillator strengths of the sublevels contained
noninteracting atoms (atoms 1 and 2) and two interacting
within nP +nP as functions of R. The calculation diag-
atoms (both atoms). The noninteracting data are labeled
onalizes a Hamiltonian that contains electric dipole and
atom 1 and atom 2, where the number represents exclusive
quadrupole interactions in a chosen subspace of nl+n(cid:48)l(cid:48)
loading of either the first or second dipole trap. Here, the
state pairs where the primed variables refer to the state excitation probability P is defined as P = 1−P(|g(cid:105)) for
e e
of the second atom. As an example, we perform the cal- single atoms and Pe = 1√−P(|gg(cid:105)) for the two-atom case.
culation for the 84P3/2 states. Because interaction chan- This plot highlights the 2 increase in excitation Rabi fre-
nelsnl+n(cid:48)l(cid:48) withenergiesclosestto84P +84P are quencyoftwoatomsinthestrongblockaderegime. (b)Mea-
3/2 3/2
the largest contributors to the EDDI [8], we perform the suredevolutionoftheblockadedtwo-atomsysteminthebasis
calculation for states that span a 40-GHz range centered {|gg(cid:105),|gr(cid:105),|rg(cid:105),|rr(cid:105)}. Error bars shown are representative.
around this target state and restrict l and l(cid:48) to 0–5. We
find the inclusion of the high angular momentum states
necessary as the spectrum becomes heavily mixed and here. Thedifferencearisesbecauseweareconcernedwith
thereforechoosethelargestrangeallowedforbyourcur- valuesofRwhereconvergenceofthecalculationrequires
rent computational resources. Techniques for computing multiplenl+nl→n(cid:48)l(cid:48)+n(cid:48)(cid:48)l(cid:48)(cid:48) channels, whereas[39]em-
U are found in [36–38], and the result closely resem- phasizes the most significant term at large interatomic
int
bles[38]. Wecalculatethatblockadebecomessignificant separations(vanderWaalsregime). Withaninteraction
below 7µm for Ω/2π = 1 MHz as is shown in Fig. 6. potential on the order of 1 MHz, signatures of Rydberg
For the experiment we choose to use a mean separation blockade should be present.
of 6.6(3)µm resulting in U /2π ∼ 6.4 MHz. The cal- Using our calculation as a guide, we experimentally
int
culation of nonzero blockade for all sublevels appears to identify signatures of Rydberg blockade. For the experi-
be in contradiction with the prediction of F¨orster zeros ment,weresonantlyexcite|84P ,m =3/2(cid:105)withvari-
3/2 j
(unshiftedstates)[39]forthenP +nP statestudied able laser pulse duration. We use a 319-nm laser inten-
3/2 3/2
9
sity of 9.3 kW/cm2 to achieve Ω = 0.816(4) MHz and citationtonP Rydbergstatesanddemonstratetheac-
3/2
apply a 4.8-G bias magnetic field to break the degen- curacyofourcalculationsfortheRydbergspectrumand
eracy in m . The measured Rydberg excitation prob- oscillator strength with single-atom spectroscopy. These
j
ability with single as well as two interacting atoms is Rydberg atoms are employed as electric field sensors to
shown in Fig. 7(a). The measured ratio of the Ryd- study laser-induced charging of nearby surfaces, and we
berg excitation Rabi frequencies Ω /Ω is 1.42(2), utilizethisinformationtomitigatenoiseontheRydberg
√ 2-atoms
which is consistent with√ 2. An increase in the excita- resonance frequency due to the dc Stark effect. Finally,
tion Rabi frequency of 2 is expected in the strongly wemodelEDDIsfortwo84P3/2Rydbergatomsasafunc-
blockaded regime where the system oscillates between tion of interatomic separation and demonstrate Rydberg
the ground state and a state that collectively shares a blockade through an increase in the collective excitation
single Rydberg excitation, as was first observed by Gae- Rabi frequency.
tan et al. [4]. Additional evidence of Rydberg blockade In principle, this single-photon approach offers an ad-
is shown in Fig. 7(b) where we plot the two-atom evo- vantage over two-photon Rydberg excitation by elimi-
lution in the basis {|gg(cid:105),|gr(cid:105),|rg(cid:105),|rr(cid:105)}. Here, we show nating the need for an intermediate state, thus avoid-
that population transfer between the ground state and ing channels for photon scattering, frequency noise, and
the singly excited state dominates the system evolution, dipole forces. Reducing photon scattering is especially
whereas excitation to |rr(cid:105) is strongly suppressed. Both attractive for the study of dipolar interactions between
plotsillustratecoherentcontroloftwostronglyblockaded Rydbergdressedstates,whereallowingthesystemtore-
atoms. lax to equilibrium can take on order of 1 ms [10, 11].
The coherent dynamics of this system, with or with- However, this fundamental limit on the photon scatter-
outtwo-atomblockade,indicatesdecoherencedominated ing rate remains to be demonstrated in our experiment.
by population relaxation Γ out of the Rydberg state. ThemoderateincreaseinDopplersensitivitywhenusing
loss
We measure Γ = 1.2(1) MHz, which is substantially single-photon Rydberg excitation can be addressed with
loss
broaderthanthecalculatedstatelinewidthof4kHz[40]. more advanced cooling techniques, such as ground-state
The trend in the evolution of the atom towards excita- cooling [28, 41].
tionto|r(cid:105), showninFig.7, occursbecauseourdetection
method can not differentiate between population in |r(cid:105)
VIII. ACKNOWLEDGEMENTS
and other atom-loss mechanisms. Examples of possible
loss sources include, an applied force on the center of
massoftheatomfromanelectricfieldgradientandare- We thank Mark Saffman, Antoine Browaeys, James
ducedstatelifetimeduetobackgroundRFfields. Order- Shaffer,SteveRolston,IvanDeutsch,TylerKeating,and
of-magnitude estimates suggest that the latter example Rob Cook for helpful discussions and suggestions. We
is more likely. In future work, we aim to investigate the would also like to thank George Burns, Peter Schwindt,
source of Γ and mitigate its decohering effects. Michael Mangan, Cort Johnson, and Andrew Ferdi-
loss
nand for contributions to the experiment. We acknowl-
edgeLaboratoryDirectedResearchandDevelopmentfor
funding this work. Sandia National Laboratories is a
VII. SUMMARY AND OUTLOOK
multi-programlaboratorymanagedandoperatedbySan-
dia Corporation, a wholly owned subsidiary of Lockheed
In summary, we present experiments and theoretical MartinCorporation,fortheU.S.DepartmentofEnergy’s
models that focus on Rydberg blockade with a single- National Nuclear Security Administration under Con-
photontransition. WeconstructaUVlaserfordirectex- tract No. DE-AC04-94AL85000.
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