Table Of ContentInterference and dynamics of light from a distance-controlled atom pair
in an optical cavity
Andreas Neuzner, Matthias K¨orber, Olivier Morin, Stephan Ritter,∗ and Gerhard Rempe
Max-Planck-Institut fu¨r Quantenoptik, Hans-Kopfermann-Strasse 1, 85748 Garching, Germany
Interference is central to quantum physics and
occurs when indistinguishable paths exist, like in
a double-slit experiment. Replacing the two slits
with two single atoms1 introduces optical non-
linearities for which nontrivial interference phe-
nomenaarepredicted2–6. Theirobservation, how-
6 ever, has been hampered by difficulties in prepar-
1
ing the required atomic distribution, controlling
0
the optical phases and detecting the faint light.
2
Here we overcome all of these experimental chal-
n
lengesbycombininganopticallatticeforatomlo-
a
calisation, an imaging system with single-site res-
J
olution, and an optical resonator for light steer-
1
ing. We observe resonator-induced saturation of
2
resonance fluorescence7,8 for constructive inter-
] ference of the scattered light and nonzero emis-
h
sion with huge photon bunching for destructive
p
interference. The latter is explained by atomic
-
t saturation and photon pair generation3–5. Our
n
experimental setting is scalable and allows one
a
u to realize the Tavis-Cummings model9 for any
q numberofatomsandphotons, explorefundamen-
[ tal aspects of light-matter interaction10–14, and
implement new quantum information processing
1
v protocols15–18. Figure 1. Experimental setup. a, A two-dimensional
7 A multitude of non-classical radiation effects like pho- optical lattice is formed by a retroreflected, red-detuned ○1
5 tonantibunching19 andsqueezing20 werepredictedinthe andablue-detuned○2 laserbeaminahigh-finesseopticalres-
7 resonance fluorescence of single quantum emitters. The onator○3. Amicroscopeobjective○4 isusedtoimageandde-
5 experimental observations of these effects21,22 are mile- terministicallyremoveindividualatomstrappedinthelattice.
0 stones in the development of quantum optics. However, Anatompairisdrivencoherentlywitharunning-wavebeam
1. the large mismatch between the light mode driven by an ○5 propagating transversally through the resonator. Scatter-
0 ing from this beam into the single-sided cavity is studied via
emitter in free space and the light mode detected by an
6 transmissionthroughtheoutcouplingmirror○6 asafunction
observer makes such quantum effects unpractical for ap-
1 of the atomic positions. b, A typical fluorescence image of
plications. The way out is to couple the emitter to an
: an atom pair, with atoms separated by several lattice sites
v opticalresonatorwhichenhancestheinteractionstrength along the x- and y-directions. In order to determine ∆ and
i x
X oftheemitterwithatailor-madelightmode. Infact,the ∆ , Gaussian point spread functions are fitted to the indi-
y
experimental realization of well-controlled atom-cavity vidual atom images. c, A two-dimensional histogram of the
r
a systems with single atoms as emitters has propelled the difference coordinates between atom pairs reveals the lattice
application potential of quantum-optical phenomena23 geometry. Thedataiscorrectedforasmallnon-orthogonality
enormously and, in addition, has enabled the observa- ofthelattice. Theone-dimensionalprojectionsontotheaxes
show that single-site resolution is achieved.
tion of fundamentally new quantum-mechanical radia-
tion effects induced by the cavity24–26. When scaling
thesesingle-atomsystemstomultipleatoms,relativeop-
ticalphasesappearasanewdegreeoffreedom. Asthose to observe the predicted interference phenomena. Only
are determined by the spatial arrangement of the atoms, few experimental approaches towards these goals exist
both, subwavelength localisation and precise knowledge hitherto8,27, and disclosing a realistic avenue for scaling
abouttherelativepositionsoftheatoms, aremandatory towards many emitters is still a formidable challenge.
Herewereportonexperimentswithadeterministically
preparedpairof87Rbatomspermanentlycoupledtothe
mode of a 0.5mm long optical Fabry-Perot resonator.
∗ e-mail: [email protected] Both atoms are driven with a laser propagating perpen-
2
of the difference vector of 45,000 atom pairs. We achieve
unambiguoussingle-sitedetectionofthedifferencevector
between atoms.
2 An experimental run begins by loading a random dis-
s
n tribution of atoms into the lattice. A fluorescence image
o
ti oftheatomsisautomaticallyevaluatedandexcessatoms
a
cit are identified. These are subsequently removed with a
x
e resonant push-out beam that is steered by an acousto-
of 1 optical deflector and focused through the objective. In
r
e thisway,apairofatomsisdeterministicallypreparedand
b
m data is taken until consecutively recorded atom images
u
N indicate the loss of one or both atoms.
Theatompairispreparedinaregionoftheintracavity
0 latticewithaspatialextentsosmallthatthemodulusof
the light-matter coupling strength g can be regarded as
constant(seeSupplementaryInformation). Thephaseof
0 1 2
Number of atoms g alternates between 0 and π in adjacent trapping sites
alongthecavity(y-axis),whilethephaseoftherunning-
wave driving laser changes continuously along the per-
Figure 2. Energy spectrum of the atom-cavity sys-
pendicular x-axis. The difference phase φ for a pair of
tem. For a single atom, the well-known anharmonic Jaynes-
atomsisdiscreteduetotheopticallattice, andtherefore
Cummings ladder is found. For two atoms, only a state
amounts to
that is symmetric with respect to exchange of the atoms
√
|S⟩=(|eg⟩+|ge⟩)/ 2 couples to the cavity, while the anti-
√ 532
symmetric state |A⟩ = (|eg⟩−|ge⟩)/ 2 remains decoupled. φ=∆n · 2π+∆n ·π,
Inthesingly-excitedmanifold,thesymmetricstatethusforms x 780 y
collective normal modes while |0A⟩ remains at the unper-
where∆n and∆n aretheintegernumberofdifference
turbed resonance energy (cid:126)ω. In the doubly excited manifold x y
lattice sites along the transversal and the longitudinal
for two atoms, all states that contain a single excitation in
direction, respectively, oftheintracavitytrap. Notethat
the cavity are detuned. Only the antisymmetric state |1A⟩
and a superposition state with the cavity or the atoms being thesingle-siteresolutionistantamounttoidentifyingthe
doubly excited remain at resonance energy 2(cid:126)ω. exact value of the phase φ.
Optical pumping initialises the atoms in the state
|g⟩ = |5S ,F=2,m =2⟩, and the cavity length is sta-
1/2 F
dicular to the cavity axis. Collective scattering of light bilised in such a way that a longitudinal mode red
intothecavitymodeisstudiedasafunctionofthephase detuned from the intracavity lattice is resonant with
difference with which the atoms couple to both the cav- the cycling transition |g⟩ ↔ |e⟩ = |5P ,F=3,m =3⟩
3/2 F
ity mode and the driving laser. The geometry resembles at frequency ω. On this transition, our parameters
the proposed free-space situation3–5, with the difference (g,κ,γ) = 2π ×(7.6,2.8,3.0)MHz put us in the single-
that the scattering phases are controlled by the atomic atom strong coupling regime of cavity quantum electro-
pattern instead of the angle of observation. dynamics. Here,κandγ arethedecayratesofthecavity
Totightlyconfinetheatomstowell-definedpositions,a field and the atomic dipole, respectively. By polarizing
two-dimensionalopticallatticeisimplementedinsidethe thetransversaldrivingbeamorthogonaltothelinearpo-
cavitywithperiodicitiesof386nmand532nmalongand larizationofthetrappinglight,wecandrivethe|g⟩↔|e⟩
perpendicular to the cavity axis, respectively (Fig.1a). transition exclusively (see Supplementary Information).
Optical molasses cooling brings the atoms close to the We thus realise the theoretical paradigm of an exactly
mechanicalgroundstateofthelattice. Theresidualspa- known number of two-level atoms, two in our case, cou-
tial extent of 25nm renders the atoms effectively point- pled with equal strength and known optical phases to
like. Duringcooling,atomicfluorescenceiscollectedbya the single light mode of a cavity and a transversal driv-
high-numerical-apertureobjectiveforimaging. Notethat ing laser.
the close vicinity of the cavity mirrors (with a diameter TheenergyeigenstatesoftheunperturbedJaynes-and
of 1.5mm) limits the achievable numerical aperture and Tavis-CummingsHamiltonians9 areshowninFig.2upto
poses a challenge in avoiding stray light. Nevertheless, the doubly excited manifold. For two (and more) atoms,
withanexposuretimeof0.75sitispossibletoimagein- all states have well defined parity with respect to ex-
dividualatoms(Fig.1b)andobtainspotsizeswithafull change of atoms. While symmetric states |S⟩ couple to
width at half maximum of 1.3µm (see Supplementary the resonator, antisymmetric states |A⟩ are decoupled
Information). This allows us to find the origin of the and remain unshifted. The latter can be understood as
emitted light with subwavelength precision by determin- a consequence of destructive interference of the emission
ing the center of Gaussian point-spread functions fitted from individual atoms. The driving laser induces transi-
totheimages. Fig.1cshowsatwo-dimensionalhistogram tions between the collective light-matter states, and the
3
a 100 foldlargerasonemightpredictforthecoherentin-phase
)
z emissionoftwoatoms. Thesecondfeatureoccursforthe
H
k out-of-phasesituation(phasedifferenceφ=π)wherethe
( 80
s photon-emission rate drops below that of a single atom,
n
o but not even close to zero as intuitively expected. As
ot 60
h shown in Fig.3b we furthermore observe an extremely
p
strong phase-dependent effect in the normalised proba-
eof 40 bility to simultaneously emit two photons, in accordance
at with predictions made for free-space interference4,5. In
R
0.0 0.5 1.0 1.5 2.0 fact, the value of the equal-time intensity-correlation
b Phaseφ(π) function g(2)(τ = 0) rises from a value of 0.79 ± 0.16
80 for in-phase coupling of the atoms (φ = 0) to 64±7 for
out-of-phase coupling (φ = π), indicating the transition
60 fromnearlyPoissoniantostronglysuper-Poissonianpho-
) ton emission. In the following we discuss these findings.
0
= 40 The weak dependency of the output power on the
τ
(
2) number of atoms for in-phase coupling has been pre-
(g 20
dicted theoretically7,28 for high atom-cavity cooperativ-
ity C = g2/2γκ ≫ 1. A first indication was reported
0
0.0 0.5 1.0 1.5 2.0 recently8. The effect is a consequence of the fact that in
Phaseφ (π) steady-stateafieldbuildsupinsidethecavitythatinter-
feres destructively with the driving field at the positions
of the atoms. In the limit of perfectly reflecting mirrors
Figure 3. Photon-emission rate (a) and g(2)(0) pho-
(κ → 0), the driving and the cavity field are of equal
tonstatistics(b)asafunctionoftherelativephaseφ.
strength and perfectly cancel each other at the position
a,Forexcitationresonantwiththebareatomandtheempty
of the atoms. As a result, the intracavity power is inde-
cavity at a Rabi frequency of Ω = 920kHz ≪ g, the rate of
pendent of the number of atoms. For our cooperativity
photons emitted from the cavity shows a sinusoidal modula-
tion as a function of the interatomic phase φ. The dashed C = 3.4, the steady state depends weakly on the num-
black line indicates the single-atom value of 75kHz. Statis- ber of atoms. The measured value is in agreement with
tical error bars are within the symbols. b, The value of the a numerical simulation, the result of which is shown in
second-order correlation function g(2)(0) indicating a transi- Fig.3andintheremainderofthepaperassolidredlines.
tionfromalmostPoissoniantostronglysuper-Poissonianpho- All parameters of this model were determined from in-
ton emission. The solid red lines in both plots are calculated dependent single-atom experiments (see Supplementary
from a numerical model without free parameters.
Information).
When the phase between the atoms is nonzero, the
driving field not only couples the ground state |0gg⟩ to
√
value of the phase difference φ determines which tran- thesymmetricnormalmodes(|0S⟩±|1gg⟩)/ 2butalso
sitions can occur. For φ = 0 only transitions between to the antisymmetric state |0A⟩. For the particular case
statesofequalatomicsymmetryarepossible, whereasin of φ = π, the ground state is exclusively coupled to the
the case of φ = π every photon absorption changes the state |0A⟩ in the singly-excited manifold. In this state,
atomic symmetry. These different excitation pathways thecavityisnotpopulated,anddecaybacktotheground
are expected to give rise to a strong dependency of the state via emission into free space competes with further
light emitted from the cavity on the phase φ. In search excitation of the atoms to the doubly excited manifold
(cid:112) (cid:112)
for these interference effects, we record the emission of (Fig.4a). Indeed, the state 2/3|0ee⟩− 1/3|2gg⟩ can
photons from the cavity with two single-photon detec- be resonantly excited while |1A⟩ remains unpopulated
tors behind a 50:50 beam splitter. This setup allows us because it has the same symmetry as |0A⟩. The two re-
tomeasureboththeintracavityphotonnumber⟨a†a⟩and maining states in the second manifold are detuned from
thephotonnumberfluctuationsviathesecond-ordercor- the laser and therefore cannot be excited, too (Fig. 2).
relation function g(2)(τ)=⟨a†a†(τ)a(τ)a⟩/⟨a†a⟩2, where This two-step excitation path may thus lead to popula-
a (a†) are photonic annihilation (creation) operators. tion of the cavity with a photon pair without admixture
Fig.3a shows the rate of photons emitted from the of single-photon components such that correlated emis-
cavity as a function of φ. The observed interference sion is expected.
fringe displays several intriguing features which are not Fig.4b shows the g(2)(τ) correlation function for out-
expectedinasimplepictureoftwoclassicaldipolesradi- of-phase coupling (φ = π). Detection of a first photon
(cid:112) (cid:112)
ating into free space. The first one refers to the in-phase from 2/3|0ee⟩− 1/3|2gg⟩ heralds preparation of the
photon-emission rate (phase difference φ = 0) which is systemin|1gg⟩. Thesystemistheninasuperpositionof
only a factor of 1.3 larger than the rate observed for a thetwonormalmodesandundergoesacollectivevacuum
single atom (measured dashed line in Fig.3a), not four- Rabi oscillation. This manifests itself as a temporal os-
4
a b 80 c 3.0
70
2.5
60
2.0
50
τ) 40 τ)1.5
(2)g( 30 (2)g(1.0
20
0.5
10
0 0.0
0 25 50 75 100 0 25 50 75 100
τ (ns) τ (ns)
Figure 4. Emission dynamics for in- and out-of-phase coupling. a, Mechanism of light scattering into the cavity
mode in the out-of-phase configuration φ=π: From the antisymmetric state |0A⟩, the atom pair can either decay emitting a
photonintofreespaceatarate2γ orbefurtherexcitedtoafullyinvertedstate,whichdepositsapairofphotonsinthecavity.
b, Decay through the symmetric states leads to collective vacuum Rabi oscillations which are observable in the second-order
correlation function. Consistently, we observe giant photon bunching g(2)(τ = 0) = 64±7 with a revival after one period of
the collective vacuum Rabi oscillation. c, For φ = 0, the data are compatible with the theoretical expectation of g(2)(τ) ≈ 1
for an almost coherent intracavity field.
cillationintheg(2) correlationfunction. Consistentwith well as the implementation of novel entanglement and
the numerical simulation (red line), we observe a quick quantum information processing schemes with multiple
decayoftheinitialbunchingpeakfollowedbyarevivalat qubits15–18.
√
(2 2g)−1,i.e. afterafullperiodofthecollectivevacuum
Rabi cycle. In contrast, g(2)(τ) for the case of in-phase
coupling (Fig.4c) is within statistical errors consistent
with the expectation of an almost coherent intracavity
ACKNOWLEDGMENTS
field.
In summary, we demonstrated that the combination
of deterministic preparation of an atom pair together WethankA.Kochankeforthedesignoftheobjective,
withsingle-site-resolvedimagingthereofenablesustoen- F. Saworski and C. Hahn for contributions in an early
ter the new regime of phase-resolved multi-atom cavity stage of the experiment and M. Uphoff for discussions.
quantum electrodynamics. The experimental techniques This work was supported by the European Union (Col-
presentedherearemandatoryrequirementsforfutureex- laborative Project SIQS), by the Bundesministerium fu¨r
periments aiming at the exploration of collective radia- BildungundForschungviaIKT2020(QK QuORePand
tion effects predicted for multi-atom systems2,3,6,10–14 as Q.com-Q) and by the DFG via NIM.
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6
SUPPLEMENTARY INFORMATION velope of the transversal lattice. Independent measure-
ments show that the atoms are cooled to approximately
Apparatus and loading of atoms 30µK in the optical molasses, which is close to the me-
chanical ground state of the tightly confined directions.
Whentheintracavitylatticeisswitchedon,atomsthat
Our system consists of a UHV chamber that holds the
wereinitiallyonlyweaklyconfinedalongthey-axisbythe
optical cavity and a rubidium dispenser, which directly
Gaussian envelope of the transversal lattice are pinned
loadsamagneto-opticaltrap(MOT)1cmawayfromthe
cavity. The cavity is 498µm long, leading to a free spec- to random positions. The envelope of their positions in
the two-dimensional lattice (Fig.1c) has a full width at
tralrangeof301GHz,andsupportsaTEM modewith
a waist (1/e2 intensity radius) of 30µm. T0h0e finesse of half maximum (FWHM) of 3µm. The waist of the red
latticeis16µmsuchthattheresidualspreadoftheatoms
the cavity is 55,000, which results in a total field decay
resultsinanuncertaintyofthelightshiftbelow1%. The
rate of the cavity of κ = 2π·2.8MHz. The output cou-
intracavity lattice populates a longitudinal mode whose
plingmirrorhasatransmissionofT =100ppm, while
OC
frequency is 13 free spectral ranges higher than that of
thesecondmirrorhasatransmissionofonlyT =4ppm.
2
themodeusedforthedescribedexperiments. Therefore,
Thisleadstoapartialcavityfielddecayratethroughthe
a spatial beating pattern between the two modes with
output coupling mirror of κ = 2π ·2.4MHz. Decay
OC a periodicity of 38µm appears. The intersection of the
into a single free-space mode thus dominates the total
cavity losses. The polarisation decay rate for 87Rb is transversalopticallatticewiththecavitymodeisaligned
such that the atoms are trapped in a region where the
γ = 2π·3.0MHz and given the cavity mode volume V,
two longitudinal modes are exactly out of phase, such
the expected coherent coupling rate on the cycling tran-
thattheintracavitytrapconfinestheatomstoantinodes
sition|5S ,F=2,m =±2⟩↔|5P ,F=3,m =±3⟩with
1/2 F 3/2 F of the mode at the 87Rb D line. Given the small spatial
transition dipole moment µ is 2
ge
extent of the trapping area in comparison to the beating
(cid:114) ω patter, we can assume the modulus of the coupling g to
g = 2ϵ V(cid:126)µge =2π·7.8MHz>(γ,κ), be constant. Limited by collisions with the background
0 gas, the 1/e-storage time for a single atom is 20s, which
which puts us in the strong coupling regime of cavity is decreased by a factor of two for an atom pair.
QED with a cooperativity C = g2 =3.7.
2κγ
The atom-preparation sequence starts by loading the
Imaging in the 2D lattice
magneto-optical trap (MOT). Subsequently, a running-
wave dipole trap at 1064nm with its focal point halfway
between the MOT and the cavity guides the atoms into Duringrepetitionoftheactualexperiment,interleaved
the cavity after the MOT is switched off. Following a with periods of molasses cooling, atomic fluorescence
transfer time of 150ms, which corresponds roughly to is collected via a specifically designed objective with
halfacycleofthelongitudinaltrapfrequency,astanding- a working distance of 24.9mm through an 8mm thick
wavetrapfocusedinsidethecavityregionisswitchedon. fused-silica vacuum window. The numerical aperture is
Atomstrappedinthis2.6mKdeepinitialtraparecooled limited along the x-axis by the entry aperture of the ob-
by application of a lin⊥lin optical molasses red detuned jective to 0.46 and along the y-axis by the edges of the
22MHzfromtheunshifted5S ,F=2↔5P ,F=3transi- cavity mirrors to 0.32. Collected fluorescence at 780nm
1/2 3/2
tion. Atomic fluorescence is imaged onto an Andor Ixon is imaged onto the EMCCD camera. The total system
DU-897 electron multiplying CCD camera. During this has a magnification of 0.48µm/pixel and the minimum
stage, the blue and therefore repulsive intracavity lat- FWHM of an atom image limited by diffraction is 1.23
tice at a wavelength of 772.4nm is held at a low power and 1.78pixel along the x- and y-axis, respectively.
that is just sufficient to keep the cavity locked with a The exposed images have a background signal of 450
Pound-Drever-Hall technique but has no significant in- counts per pixel and a single atom accounts for approxi-
fluence on the motion of the atoms. After image evalua- mately 18,000 additional counts. Gaussian point-spread
tion,apushoutbeamissteeredthroughtheobjectivevia functions are fitted to the atom images. In case of two
anacousto-opticalmodulatortodeterministicallyremove atomsbeingtooclose, thedataisdiscarded. Histograms
unwanted atoms. After preparation, the atomic pattern of fitted widths along the x- and y-direction are shown
ispositionedatthecentreofthecavitymodebymeansof in Fig.S1a. Along both directions, the diameter of the
a tiltable glass-plate in the path of the 1064nm standing atomimagesisafactor≈1.5greaterthanthetheoretical
waveS1. diffraction limit. This discrepancy arises partly from the
Subsequently,theintracavitylatticepowerisincreased motionoftheatomsalongtheweaklyconfinedz-axiswith
20foldtoatrapdepthof0.8mKwhiletheredtransversal anextentcomparabletothefocaldepthoftheobjective.
latticeisloweredtoatrapdepthof1.7mK.Theresulting To quantify the achieved precision with which an atom’s
two-dimensional lattice has degenerate trap frequencies position can be determined and to characterise possible
alongthex-andy-axisof520kHz. Alonggravity(z-axis), geometrical distortions of the lattice, we project the x-
the atoms are only weakly confined by the Gaussian en- and y- difference coordinates of detected atom pairs, bin
7
a 34
x axis
s 3
ce yaxis 46.4 MHz
ccuren 12 MHz 59.7 MHz 75.1 MHz 79.9 MHz
O 28.7
0
1.0 1.2 1.4 1.6 1.8 5P , F=3
3/2
FWHM(µm) 384.228 115 THz σ- σ+
b 0.25
)0.20
m
µ
(0.15 5S1/2, F=2 35.2 MHz
M
H
W0.10
F m = -3 -2 -1 0 +1 +2 +3
0.05 F
-10 0 10 20 30 40 50 60 70
Rotationangle(mrad) Figure S2. Light shifts of the excited energy eigenstates
caused by the 1064nm dipole trap (not to scale). A tensor
Figure S1. a, Full width at half maximum (FWHM) of the lightshiftoftheexcited5P ,F=3stateshiftsthetransition
3/2
fitted Gaussian point-spread functions along the two Carte- from 5S ,F=2,m =2 to 5P F′=3,m =1 out of resonance
1/2 F 3/2 F
siandirections. ThesolidlinesareGaussianfits. b,FWHMof by more than 7 atomic linewidths.
the detected atom pairs’ difference positions projected onto
the two axes as a function of a global rotation applied to
the data. The solid lines are quadratic fits to the data. For following image.
the two axes, minima are found at slightly different, nonzero
rotation angles indicating a deviation of the lattice from or-
thogonality of 1.6◦ and a global rotation of the lattice with
Excitation scheme and optical pumping
respect to the camera’s coordinate system of 0.64◦.
The trapping light at a wavelength of 1064nm, which
the data and evaluate the width of individual peaks in is linearly polarised along the cavity axis, introduces a
theperiodicpattern(Fig.1c). Fig.S1bshowstheFWHM strong tensor light shift onto the excited |5P3/2,F=3⟩
ofthepeaksalongthetwoCartesiandirectionsasafunc- state. Fig.S2 shows the calculated energy structure at
tion of a global rotation applied to the coordinates. The the trapping-light intensity of 1.1·1010Wm−2, which is
twocurveshaveminimaatdifferent,nonzerorotationan- found via independent light shift spectroscopyS2. The
glesindicativeofasmallnon-orthogonalitypresentinthe degeneracy of the hyperfine Zeeman substates is lifted
lattice. To account for the global rotation and this non- by up to 51MHz. The cavity is not birefringent and
orthogonal deformation, the following transformation is when tuned into resonance with the cycling transi-
applied to the raw data: tion |5S1/2,F=2,mF=2⟩ ↔ |5P3/2,F=3,mF=3⟩, a circu-
larly polarised mode couples to the atom. The ten-
(cid:18)x′ (cid:19) (cid:18)Cβ −Sβ (cid:19)(cid:18) Cα Sα (cid:19)(cid:18)x(cid:19) sor light shift caused by the dipole trap shifts the
=
y′ 0 1 −Sα Cα y |5P ,F=3,m =1⟩ state by 46.4MHz out of resonance,
3/2 F
whichcorrespondstomorethan7atomiclinewidths. The
with (Cβ)Sβ being the (co-)sine of the skew angle β = orthogonal σ−-polarised mode is therefore far detuned
1.6◦ and (Cα)Sα being the (co-)sine of the rotation an- from the atom. The same shift allows us to couple ex-
gle α = 0.64◦. The minimum FWHM of 100nm corre- clusively to this circularly polarised atomic dipole with
sponds to a precision of 71nm for the detection of a sin- linearlypolarisedlightimpingingorthogonallytothecav-
gleatomandallowsforbasicallyunambiguoussingle-site ity axis. In the atomic reference frame, this light drives
detection. When the difference vector of an atom pair is σ+ and σ− transitions with equal strength. But again,
found in terms of discrete trapping sites, the atom pair’s the σ−-polarised transition is far detuned, such that we
measureddifferencevectorisreplacedbytheexactvalue can treat the atoms as effective two-level systems with a
givenbytheaboveequation,whichisthenusedtocalcu- well known atomic dipole, which allows for quantitative
late the phase difference φ. On average, approximately predictions.
every15sanatomleavesitstrappingsiteandisrecooled In order to prepare all atoms in the
intoanothersite,thusperformingajumpofuptoseveral |5S ,F=2,m =+2⟩ state, circular pumping light
1/2 F
µm. We detect these jumps on the images and discard resonant with the cavity is applied for 80µs in combina-
all data that was taken during exposure of the image in tionwithaweakrepumpingbeamincidentperpendicular
which the jump appeared, the preceding image and the to the cavity that brings atoms back from the F=1
8
1 0 where ∆c and ∆a are the detuning of the cavity and
t /2 p atomic transition with respect to the driving laser fre-
)
z 8 0 M H z quencyrespectively. Thetwocollapseoperatorsdescribe
H
s (k 6 35 MM HH zz dspeeccatyivoeflyt.heWceavniutymefireilcdalalyndfinodf athtoemsitceapdoylasrtisaatteioρn roe-f
n ss
to 1 0 M H z the system and calculate the expected rate of photons
o 4
f ph masoRgen=eo3u0s.4bMroHadze·nTinrg(ad†aueρstso).fiInnitoertdeemrptoeramtoudreeloifnhthoe-
te o 2 trapped atom, we average over Boltzmann-distributed
a 0 atomic detunings. Assuming classical motion and all de-
R
-2 0 -1 0 0 1 0 2 0 greesoffreedomtobethermalised,thefollowingintegral
D
D r iv in g la s e r d e tu n in g ( M H z ) is found for the expectation value:
4 (cid:90) ∞
Figure S3. Simulated photon emission rate for our cavity ⟨R(∆ )⟩= √ R(∆ +τr2)r2e−r2dr,
a π a
QEDparameters(g,γ,κ)=2π·(7.6,3.0,2.8)MHzandadriv- 0
ing Rabi frequency of Ω=2π·200kHz for different values of
where τ = αe−αgk T is a temperature parameter nor-
thetemperatureparameterτ andadetuningbetweenanatom (cid:126)αg B
at the trap bottom and the cavity that is equal to τ. malised to the relative difference in dynamic polarisabil-
ity of the excited (α ) and ground state (α ). The as-
e g
sumptionsmadeinthederivationofthisformulaareonly
ground state. A problem appears when trying to opti- partially valid. See ref.S2 for a discussion of validity.
cally pump more than a single atom while the cavity is When the cavity is tuned such that it is resonant with
held on resonance with the cycling transition. The first an atom at the bottom of the trap that experiences the
atom that couples to the cavity will lead to a normal- maximum light shift, the effect of a finite temperature is
mode splitting of the cavity resonance and suppress an asymmetric broadening towards lower transition fre-
the intracavity intensity by a factor of (1+2C)2 = 69, quencies as the atom explores spatial regions with lower
thereby effectively switching off the light and preventing trapintensity. Choosingaslightlyhigherpowersuchthat
the second atom from being pumped. To circumvent an atom at the bottom of the trap is blue detuned from
this problem, we lower the intensity of the dipole trap the cavity restores a symmetric normal-mode spectrum.
during optical pumping by 20%. Besides detuning the Fig.S3 illustrates the residual effect of a finite tempera-
cycling transition from the cavity resonance, the above turewhenthebottomofthetrapisdetunedbythevalue
mentioned tensor light shifts on the other m states are τ. The predominant qualitative change is a loss of con-
F
reduced as well, which simplifies optical pumping. The trast of the observed curve. For temperatures exceeding
first atom to appear on the cycling transition now only τ/2π ≈ 5MHz, a visible asymmetry in the form of a
leads to a relatively small attenuation of the pumping broadening of the blue resonance with respect to the red
light by a dispersive shift of the cavity such that all becomes apparent.
atomscanbeopticallypumped. Theefficiencyofoptical Fig.S4 shows experimental data taken with a single
pumping is characterised by performing cavity-reflection atom (red triangles) and with two atoms that couple in
spectroscopy and we find a single-atom efficiency of phase (black dots). The solid red line is a fit of the de-
η =0.87. scribed numerical model extended to take the non-unity
optical pumping efficiency into account. As all possible
transitions are detuned by several linewidth, when op-
Theoretical model tical pumping did not succeed to initialise the atom in
|F = 2,m = +2⟩ (Fig.S2), we assume that no light is
F
created in these cases. The pumping efficiency η thus
Thesinglecavitymodeisdescribedbythebosonican-
onlyrescalesthey-axis. Thevalidityofthismodelisfur-
nihilation (creation) operator a (a†) and the two level
ther substantiated by measuring the polarisation of the
atoms with ground state |g⟩ and excited state |e⟩ are de-
light emitted from the cavity. We find the light to be
scribed by the atomic operators σ = |g⟩⟨e| and σ+ =
right-circularly polarised with an admixture of orthogo-
|e⟩⟨g|. For a single atom in the cavity, the master equa-
nally polarised light < 1% which is close to the atten-
tion of the driven system is
uation ratio of the involved polarisation optics. For the
H =−∆ a†a−∆ σ+σ−+g(aσ++a†σ−)+H dataset shown in Fig.S4, we find the atomic detuning
c a drive
with respect to the cavity to be 2.89MHz and the tem-
Ω
H = (σ++σ−) perature parameter τ/2π =2.28MHz.
drive 2
Forapairofatomswithidenticallight-mattercoupling
d
ρ=−i[H,ρ] strengthg,themasterequationofthesystemisextended
dt
by collapse operators to describe free-space emission by
+γ(2σ−ρσ+−σ+σ−ρ−ρσ+σ−) the second atom and coupling terms to the quantised
+κ(2aρa†−a†aρ−ρa†a) cavity field and the classical driving field. By means of
9
30 z)1 0 0 P erfect pum ping, zero tem perature
P erfect pum ping, therm al broadening
z) kH 8 0 Im perfect pum ping, therm al broadening
H (
k s
( 20 n 6 0
ns to
hoto f pho 4 0
ofp 10 te o 2 0
e a 0
at R 0 .0 0 .5 1 .0 1 .5 2 .0
R f p
P h a s e ( )
0
-20 -10 0 10 20
FigureS5. Simulatedoutputphotonrateasafunctionofthe
Drivinglaserdetuning∆(MHz)
interatomic phase φ including different experimental imper-
fections. The horizontal line indicates the single-atom value
Figure S4. Cavity emission as a function of the detuning of of 75kHz.
the driving laser in case of a single (red triangles) and two
atoms (black dots) that couple in phase (φ = 0). The Rabi
frequencyisΩ=2π·300kHz. Thesolidredlineisafittothe mogeneousbroadeningreducestheindistinguishabilityof
data and the dashed black line is a numerical calculation for the atoms. An additional quarter of the observed light
the case of two atoms using the paramaters found by fitting stemsfromsingle-atomemissionincasesofimperfectop-
to the single-atom data. The collectively-enhanced coupling
√ tical pumping.
strengthg = 2g manifestsitselfinagreaterseparationof
eff
thetworesonanceswhiletheamountofemissiononthebare
atoms’ resonance increases only slightly.
REFERENCES FOR SUPPLEMENTARY
INFORMATION
gauge transformations eiϕa → a˜, the phases appearing
S1. Nußmann, S., Hijlkema, M., Weber, B., Rohde, F.,
for the different couplings are collected in front of one
term. Expectation values of intensities a˜†a˜ and fluctua- Rempe,G.&Kuhn,A.Submicronpositioningofsin-
gle atoms in a microcavity. Phys. Rev. Lett. 95,
tionsthereofareleftunaffectedbythesetransformations
173602 (2005).
and we find the following Hamiltonian:
S2. Neuzner, A., K¨orber, M., Du¨rr, S., Rempe, G. &
H =−∆ a†a−∆ σ+σ−−∆ σ+σ− Ritter,S.Breakdownofatomichyperfinecouplingin
c a1 1 1 a2 2 2
+g(cid:2)(σ++σ+)a+(σ−+σ−)a†(cid:3) a deep optical-dipole trap. Phys. Rev. A 92, 053842
1 2 1 2 (2015).
Ω
+ (σ++σ−+eiφσ++e−iφσ−)
2 1 1 2 2
The effects of imperfect optical pumping and ther-
mal averaging are taken into account in the same way
as described above for a single atom assuming identical
atomicdetunings∆ ,uncorrelatedopticalpumpingwith
a
single-atom efficiency η and thermal motion described
by τ. The emission rate from the cavity is found to be
⟨R ⟩ = η2⟨R′⟩+2η(1−η)⟨R ⟩, where R′ and R are
2 2 1 2 1
thermally averaged emission rates for two and a single
atom, respectively. The dashed black curve in Fig.S4 is
calculated applying this numerical model for the case of
φ=0 and agrees well with the data.
Fig.S5showsthedifferentcontributionsofdetrimental
experimental imperfections to the model used to calcu-
late the solid red line in Fig.3(a). The dotted line in
Fig.S5 is calculated without thermal averaging and as-
sumingperfectinitialstatepreparationoftheatompair.
Even under these perfect conditions, the observed inten-
sity is only reduced by a factor of five for destructive
interferenceoftheatoms, becauseatomicsaturationand
emission of photon pairs contribute a significant part of
the light. Taking thermal broadening into account in-
creases the intensity at φ = π by a factor of 2 as inho-
