Tunable ion-photon entanglement in an optical cavity

0.88 MB·English

by  A. Stute

Tunable Ion-Photon Entanglement in an Optical Cavity
A. Stute,1 B. Casabone,1 P. Schindler,1 T. Monz,1 P. O. Schmidt,2,3
B. Brandst¨atter,1 T. E. Northup,1 and R. Blatt1,4
1Institut fu¨r Experimentalphysik, Universita¨t Innsbruck,
Technikerstraße 25, 6020 Innsbruck, Austria
2QUEST Institute for Experimental Quantum Metrology,
Physikalisch-Technische Bundesanstalt, 38116 Braunschweig, Germany
3Institut fu¨r Quantenoptik, Leibniz Universita¨t Hannover, 30167 Hannover, Germany
4O¨sterreichischen Akademie der Wissenschaften,
Technikerstraße 21a, 6020 Innsbruck, Austria
3
(Dated: December 11, 2013)
1
0 Publication reference: Nature 485, 482–485 (2012), doi:10.1038/nature11120
2
n Proposed quantum networks require both a quantum interface between light
a and matter and the coherent control of quantum states [1, 2]. A quantum interface
J can be realized by entangling the state of a single photon with the state of an
2 atomic or solid-state quantum memory, as demonstrated in recent experiments with
trapped ions [3, 4], neutral atoms [5, 6], atomic ensembles [7, 8], and nitrogen-vacancy

] spins [9]. The entangling interaction couples an initial quantum memory state to
h two possible light-matter states, and the atomic level structure of the memory
p determines the available coupling paths. In previous work, these paths’ transition
-
parameters determine the phase and amplitude of the final entangled state, unless the
t
n memory is initially prepared in a superposition state [4], a step that requires coherent
a control. Here we report the fully tunable entanglement of a single 40Ca+ ion and the
u
polarization state of a single photon within an optical resonator. Our method, based
q
on a bichromatic, cavity-mediated Raman transition, allows us to select two coupling
[
paths and adjust their relative phase and amplitude. The cavity setting enables

1 intrinsically deterministic, high-fidelity generation of any two-qubit entangled state.
v This approach is applicable to a broad range of candidate systems and thus presents
5 itself as a promising method for distributing information within quantum networks.
7
2
0 Opticalcavitiesareoftenproposedasameans high-finesse cavity, we implement full tomogra-
. toimprovetheefficiencyofatom-photonentan- phyofthejointatom-photonstateandgenerate
1
0 glement generation. Experiments using single maximallyentangledstateswithfidelitiesupto
3 emitters [3–5, 9] collect photons over a limited 97.4(2) %.
1 solid angle, with only a small fraction of entan-
: In initial demonstrations of atom-photon en-
v glement events detected. However, by placing
tanglement, the amplitudes of the resulting
i theemitterinsidealow-losscavity,itispossible
X state are fixed by atomic transition amplitudes
to generate photons with near-unit efficiency in
r [3, 5, 6, 9, 11]. If the final atomic states are not
a thecavitymode[1,10]. Neutralatomsinares-
degenerate, as in the case of a Zeeman split-
onatorhavebeenusedtogeneratepolarization-
ting, the phaseof the atomicstate after photon
entangled photon pairs [6, 11], but this has not
detectionisdeterminedbythetimeatwhichde-
yet been combined with coherent operations on
tectionoccurs. Incontrast,wecontrolbotham-
the atomic state. Trapped ions have the ad-
plitude and phase via two simultaneous cavity-
vantage of well-developed methods for coher-
mediated Raman transitions. The bichromatic
ent state manipulation and readout [12, 13].
Raman fields ensure the independence of the
Using a single trapped ion integrated with a
atomic state from the photon-detection time;
2
their relative amplitude and phase determine a APD1
the state parameters. Within a quantum net-
work, such a tunable state could be matched APD0
729
to any second state at a remote node, gener-
ating optimal long-distance entanglement in a
quantum-repeater architecture [14]. PBS
L/4
Atunablestatehaspreviouslybeenemployed L/2
as the building block for teleportation [4] and a
Magnetic Field
heralded gate between remote qubits [15]. In
393
this case, tunability of the entangled state is
inherited from control over the initial state of
the atom. The photonic qubit is encoded in
frequency, and as a result, integration with a
cavity would be technically challenging. The
entangling process is intrinsically probabilistic,
Cavity
with efficiency limited to 50% even if all emit-
ted photons could be collected. In the scheme b
presented here, the entangling interaction itself 42P
3/2
istunable,andnocoherentmanipulationofthe P
∆
inputstateisrequired. Foratomicsystemswith ∆ 1
2
a complex level scheme in which several transi-
tion paths are possible, the two most suitable 1
paths can be selected. V H 32D
5/2
Our experimental apparatus (Fig. 1(a)) con- D
sists of a linear Paul trap storing a single D' Ω2 Ω1
40Ca+ ion within a 2 cm optical cavity [16, 17]. 2 3
The cavity has a waist of 13 µm and finesse
42S
1/2
of 77,000 at 854 nm, the wavelength of the
S
42P −32D transition. The rates of coher-
3/2 5/2
ent atom-cavity coupling g, cavity-field decay FIG. 1. Experimental apparatus and entan-
glement sequence. a, An ion is confined in a
κ,andatomicpolarizationdecayγ aregivenby
Paul trap (indicated by two endcaps) at the point
(g,κ,γ) = 2π×(1.4,0.05,11.2) MHz. The ion
of maximum coupling to a high-finesse cavity. A
is located in both the waist and in an antin-
393-nmlasergeneratesatom-photonentanglement,
ode of the cavity standing wave, and it is local- characterizedusinga729-nmlaser. Photons’polar-
ized to within 13±7 nm along the cavity axis izationexitingthecavityisanalyzedusinghalf-and
[17]. Entanglement is generated via a bichro- quarter-waveplates (L/2, L/4), a polarizing beam-
maticRamanfieldat393nmandreadoutusing splitter cube (PBS), and fiber-coupled avalanche
photodiodes(APD0,APD1). b,AbichromaticRa-
a quadrupole field at 729 nm.
manpulsewithRabifrequenciesΩ ,Ω anddetun-
1 2
A magnetic field of 2.96 G is applied along ings ∆ ,∆ couples |S(cid:105) to states |D(cid:105) and |D(cid:48)(cid:105) via
1 2
the quantization axis zˆ and perpendicular to two cavity modes H and V (1), generating a single
the cavity axis. The cavity supports degener- cavity photon. To read out entanglement, |D(cid:48)(cid:105) is
atehorizontal(H)andvertical(V)polarization mapped to |S(cid:105) (2), and coherent operations on the
modes, where H is defined parallel to zˆ. At the S−D transition (3) prepare the ion for measure-
ment.
cavityoutput,themodesareseparatedonapo-
larizing beamsplitter and detected at avalanche
photodiodes. A half- and a quarter-waveplate
3
prior to the beamsplitter allow us to set the a Real part Imaginary part
measurement basis of the photon [18].
The entangling process is illustrated in Fig.
1(b). Following a Doppler-cooling interval, the 0.4 0.4
ion is initialized via optical pumping in the 0.2 0.2
0.0 0.0
state |S(cid:105) ≡ |42S ,m = −1/2(cid:105). In or-
1/2 S -0.2 -0.2
der to couple |S(cid:105) simultaneously to the two -0.4 -0.4
states |D(cid:105) ≡ |32D ,m = −3/2(cid:105) and |D(cid:48)(cid:105) ≡ |D´,V> |D´,V>
5/2 D
|32D ,m = −5/2(cid:105), we apply a phase-stable |D´,H> |D´,H>
5/2 D |D,H> |D,H>
bichromatic Raman field, detuned by ∆1 and |D,V> |D,V> |D,V> |D,V>
∆2 from the |S(cid:105) − |P(cid:105) transition. Here, the |D´,H>|D´,V>|D,H> |D´,H>|D´,V>|D,H>
intermediate state |P(cid:105)≡ |42P ,m =−3/2(cid:105)
3/2 P b
500 V photon
is used. The cavity is stabilized at detuning H photon
∆c ≈ −400 MHz from the |P(cid:105) − |D(cid:105) transi- n 400
1 bi
tion and ∆c =∆c+∆ from the |P(cid:105)−|D(cid:48)(cid:105) er 300
transition,2where1 ∆ D,Di(cid:48)s the Zeeman split- nts p 200
ting between |D(cid:105) anDd,D|D(cid:48) (cid:48)(cid:105). When ∆c and ∆ Eve 100
i i
satisfytheRamanresonanceconditionforboth 00 5 10 15 20 25 30 35
i = (1,2), population is transferred coherently c p) 0.04 Time bin (ms)
from |S(cid:105) to both |D(cid:105) and |D(cid:48)(cid:105), and a single nce ( 00..0032
photon is generated in the cavity [19–22]. ere 0.01
diff 0.00
The effective coupling strength of each of the e -0.01
as-0.02
tHweore,traΩn1sitainondsΩis2 gairveenthbey agmieffpli=tudΩeisGiogf/∆thie. Ph-0.03 0-3 3-5 5-8 8-11 11-15Tim15e-1 9bin (m19-s24) 24-37
Raman fields; G and G are the products of
1 2
theClebsch-Gordoncoefficientsandtheprojec- FIG. 2. Quantum state tomography of the
tions of laser and vacuum-mode polarizations joint ion-photon state, containing ∼ 40,000
events. a, Real and imaginary parts of all density
onto the atomic dipole moment [17]. In free
matrix elements for Raman phase ϕ = 0.25, from
space, these two pathways generate π- and σ+-
whichafidelityF =97.4(2)%iscalculated. Colors
polarizedphotons,respectively. Withinthecav-
forthedensitymatrixelementscorrespondtothose
ity, the π photon is projected onto H and the used in Figs. 3a and 4a. b, Temporal pulse shape
σ+ photon onto V [16, 17]. Ideally, the bichro- of H and V cavity photons. Error bars represent
matic fields generate any state of the form one s.d. based on Poissonian photon statistics. c,
Phase of the ion-photon state vs. photon-detection
|ψ(cid:105)=cosα|DH(cid:105)+eiϕsinα|D(cid:48)V(cid:105), time. Arrows indicate time-bin intervals of the to-
mography data. Error bars represent one s.d. (see
whereα≡tan−1(geff/geff)andϕisdetermined Methods).
2 1
by the relative phase of the Raman fields. To
determinethetheoverlapofthemeasuredstate
with |ψ(cid:105), we perform quantum state tomogra- the{S,D}states[12,13]. Wethenperformad-
phyoftheion-photondensitymatrixρforgiven ditional coherent operations to select the mea-
values of α and ϕ. Ion and photon are mea- surementbasisanddiscriminatebetweenS and
suredinallninecombinationsofionPaulibases D via fluorescence detection [13]. Each se-
{σ ,σ ,σ } and photon polarization bases quence lasts 1.5 ms and consists of 800 µs of
x y z
{H/V,diagonal/antidiagonal,right/left} [18]. Doppler cooling, 60 µs of optical pumping, a
Inordertomeasuretheioninallthreebases, 40 µs Raman pulse, an 4 µs mapping pulse, an
we first map the superposition of {D(cid:48),D} onto optional 4.3 µs rotation, and 500 µs of fluores-
4
cence detection. The probability to detect a a 0.6
photoninasinglesequenceis5.7%;wethusde- 0.4
tect on average 40.5 events/s. Note that the es 0.2
photon is generated with near-unit efficiency, erenc 0.0
oh -0.2
and detection is primarily limited by the prob- C
-0.4
ability for the photon to exit the cavity (16%)
-0.6 0.5 1.0 1.5 2.0
and the photodiode efficiencies (40%).
b 0.980
In a first set of measurements, we choose the 0.975
caseα=π/4,correspondingtoamaximallyen- elity 0.970
tangled state |ψ(cid:105). From the tomographic data, Fid 0.965
the density matrix is reconstructed as shown
0.960
in Fig. 2(a). Here we have tuned Ω and
1 0.955 0.5 1.0 1.5 2.0
Ω2 so as to produce both photon polarizations Raman phase (p)
with equal probability, corresponding to maxi-
mal overlap of the temporal pulse shapes of H FIG. 3. State tomography as a function of
Raman phase (∼ 340,000 events). a, Re(ρ )
and V photons (Fig. (b)). In order to demon- 14
(bluecircles)andIm(ρ )(reddiamonds)asafunc-
strate that the photon-detection time does not 14
tion of Raman phase. Errorbars are smaller than
determinethephaseofthestate,weextractthis
the size of the symbols. Each value is extracted
phase from state tomography as a function of fromafullstatetomographyofρasinFig2a. Both
the photon time bin (Fig. 2(c)). Because the curves are fitted simultaneously, with phase offset
frequency difference of the bichromatic fields constrained to π/2. The fit contrast is 95.6(4)%.
∆ −∆ is equal to the level spacing between b, Fidelities of the eight states, with a dashed line
1 2
|D(cid:105)and |D(cid:48)(cid:105),thephaseϕ=0.25πremainscon- indicatingthemeanvalue. Errorbarsrepresentone
s.d. (see Methods).
stant. Further details are given in the Methods
section.
Tomography over all time bins yields a fi-
delity of F ≡ (cid:104)ψ|ρ|ψ(cid:105) = 97.4(2) % with re- seven additional values of the relative Raman
spect to the maximally entangled state, plac- phase. As a function of ϕ, the real and imagi-
ing our system definitively in the nonclassical narypartsofthecoherenceρ ≡(cid:104)DH|ρ|D(cid:48)V(cid:105)
14
regime F > 50%. Another two-qubit entangle- vary sinusoidally as expected (Fig. 3(a)). The
ment witness is the concurrence [23], which we fidelity has a mean value of 96.9(1) % and does
calculate to be 95.2(5) %. The observed entan- notvarywithinerrorbarsoveralltargetphases
glement can also be used to test local hidden- (Fig. 3(c)).
variable models (LHVMs) via the violation of A second measurement set demonstrates
the Clauser-Horne-Shimony-Holt (CHSH) Bell control over the amplitudes cosα and sinα
inequality[24]. Entanglementofahybridatom- of the entangled ion-photon state. After
photon system holds particular interest since it selecting three target amplitudes cosα =
√ √ √
could be used for a loophole-free test of a Bell- {1/ 2,1/ 3,1/ 8}, we generate each corre-
type inequality [25]. While LHVMs require the sponding state by adjusting the Raman field
Bell observable of the CHSH-inequality to be amplitudes, since α is a function of the ra-
less than 2, we measure a value 2.75(1)> 2, tio Ω /Ω . The density matrix for each state
2 1
where quan√tum mechanics provides an upper is then measured. In Fig. 4(a), we see that
bound of 2 2. the populations ρ ≡ (cid:104)DH|ρ|DH(cid:105) and ρ ≡
11 44
We now establish that we can prepare |ψ(cid:105) (cid:104)D(cid:48)V|ρ|D(cid:48)V(cid:105) for the three target amplitudes
with high fidelity over the full range of the Ra- agree well with theoretical values. The fideli-
man phase ϕ. We repeat state tomography for ties of the asymmetric states (Fig. 4(b)) are as
5
a 1.0 tered by the mirror coatings, and only 16% en-
ns 0.8 ter the output mode. However, using mirrors
atio 0.6 with state-of-the-art losses and a highly asym-
ul
op 0.4 metric transmission ratio, an output coupling
P
0.2 efficiency exceeding 99% is possible (see Meth-
0.00.70 0.75 0.80 0.85 0.90 0.95 1.00 ods). In contrast, without a cavity, using a lens
ofnumericalaperture0.5tocollectphotons,the
b 0.985
0.980 efficiency would be 6.7%. In addition, the in-
delity00..997750 fsruairteeddwtoavfiebleenrgtdhisotrfitbhuetioount,peuntapbhliontgonlsonisgwdeilsl--
Fi
0.965
tancequantumnetworks. Wenotethatafaster
0.960
detection rate could be achieved by triggering
0.955
0.70 0.75 0.80 0.85 0.90 0.95 1.00
cos(a) ion-state readout on the detection of a photon.
FIG. 4. State tomography for three values We have demonstrated full control of the
of amplitude cosα. a, The density matrix ele- phaseandamplitudeofanentangledion-photon
mentsρ11(orangesquares)andρ44(greentriangles) state,whichopensupnewpossibilitiesforquan-
are plotted for the three target amplitudes cosα=
√ √ √ tum communication schemes. In contrast to
{1/ 2,1/ 3,1/ 8}. Errorbars are smaller than
monochromatic schemes, evolution of the rel-
the size of the symbols. Solid lines represent the
ativephaseoftheatomicstateafterphotonde-
amplitudesofthetargetstates. b,Thecorrespond-
ing fidelities are F = {96.3(3), 96.8(3), 98.0(4)}. tection is determined only by the start time of
Adashedlineindicatesthemeanvalue. Errorbars theexperimentandnotbythephoton-detection
represent one s.d. (see Methods). time. The state |ψ(cid:105) is in this sense prede-
termined and can be stored in, or extracted
from,aquantummemoryinatime-independent
manner. The bichromatic Raman process em-
highasthoseofthemaximallyentangledstates
ployed here provides a basis for a coherent
and are limited by the populations, that is, by
atom-photon state mapping as well as one- or
errors in tuning the Raman fields to match the
two-dimensional cluster state generation [26].
target values.
Errors in atomic state detection [5, 25], We thank J. Barreiro, D. Nigg, K. Ham-
atomic decoherence [11] and multiple excita- merer,andW.Rosenfeldforhelpfuldiscussions.
tions of the atom [3] reduce the fidelity of the This work was supported by the Austrian Sci-
atom-photon entangled state by (cid:28)1%. Imper- ence Fund (FWF), the European Commission
fect initialization and manipulation of the ion (AQUTE),theInstitutfu¨rQuanteninformation
due to its finite temperature and laser intensity GmbH,andaMarieCurieInternationalIncom-
fluctuations decrease the fidelity by 1%. The ing Fellowship within the 7th European Frame-
two most significant reductions in fidelity are work Program.
due to dark counts of the APDs at a rate of
36 Hz (1.5%) and imperfect overlap of the tem- Experiments were performed by A.S., B.C.,
poral pulse shapes (1%). andT.E.N.,withcontributionsfromP.S.tothe
To our knowledge, this measurement repre- setup. Data analysis was performed by A.S.,
sents both the highest fidelity and the fastest B.C., and T.M. The experiment was conceived
rate of entanglement detection to date between byP.O.S.andR.B.andfurtherdevelopedindis-
a photon and a single-emitter quantum mem- cussions with A.S., B.B., B.C., and T.E.N. All
ory. This detection rate is limited by the fact authors contributed to the discussion of results
that most cavity photons are absorbed or scat- and participated in manuscript preparation.
6
METHODS {|S,n(cid:105), |D,n(cid:105), |D(cid:48),n(cid:105)}, wheren={0,1}isthe
photon number in either of the two degenerate
Detection and state tomography cavity modes. The excited state has been adia-
batically eliminated, so that geff couples |S,0(cid:105)
1
to |D,1(cid:105) and geff couples |S,0(cid:105) to |D(cid:48),1(cid:105). Af-
The cavity output path branches at a polar- 2
ter transformation into a rotating frame U =
izingbeamsplitterintotwomeasurementpaths,
and the detection efficiencies of these paths are eiω1t|S(cid:105)(cid:104)S|ei(ω1−ω2)t|D(cid:48)(cid:105)(cid:104)D(cid:48)|, the Hamiltonian is
unequal. We compensate for this imbalance by
(ω −ω )|S(cid:105)(cid:104)S| +ω |D(cid:105)(cid:104)D|
performingtwomeasurementsforagivenchoice S 1 D
ofionandphotonbasisandsumtheresults;be- +(ωD(cid:48) −(ω1−ω2))|D(cid:48)(cid:105)(cid:104)D(cid:48)| +ωC|1(cid:105)(cid:104)1|
tween the measurements, a rotation of the out- +(cid:0)geff|D,1(cid:105)(cid:104)S,0| +geff|D(cid:48),1(cid:105)(cid:104)S,0| +h.c.(cid:1),
1 2
put waveplates swaps the two paths.
At each measurement setting, we record on where ¯h = 1, {ω ,ω ,ω(cid:48) } are the state fre-
S D D
average 4722 events in which a single photon quencies, ω is the cavity frequency, and terms
C
has been detected. While a photon is de- rotating at |ω − ω | (cid:29) geff are omitted[29].
1 2 i
tected in 5.7% of sequences, the atom is al- In this frame, the couplings geff are time-
i
ways measured. Correlations of the photon po- independent, and the states |D(cid:105) and |D(cid:48)(cid:105)
larization and the atomic state are the input are degenerate. Therefore, the phase between
for maximum likelihood reconstruction of the |D,1(cid:105) and |D(cid:48),1(cid:105) remains fixed during Raman
most likely states[27]. Error bars are one stan- transfer, and the phase between |D,0(cid:105) and
dard deviation derived from non-parametric |D(cid:48),0(cid:105) stays constant after photon detection.
bootstrapping[28] assuming a multinomial dis-
tribution.
Cavity parameters
Time independence The cavity mirrors have transmission T =
1
13ppmandT =1.3ppm,withcombinedlosses
2
The phase of the entangled atom-photon of 68 ppm. State-of-the-art combined losses at
state is inferred from the measurements of pho- this wavelength are L=4 ppm[30]. In our cav-
ton polarization and atomic-state phase. In ity, these losses would correspond to an output
the experiments of Refs.[3, 6, 9], although the coupling efficiency of T1/(T1+T2+L) = 71%.
phase of the entangled state is time indepen- To improve this efficiency, an output mirror
dent before photon detection, the phase of the with higher transmission T1 could be used; for
atomic state after photon detection evolves due example, T1 = 500 ppm corresponds to an effi-
toLarmorprecession. Itisthusnecessarytofix ciency of 99%. The cavity decay rate κ would
the time between photon detection and atomic alsoincrease,butsingle-photongenerationwith
state readout in order to measure the same near-unit efficiency is valid in the bad-cavity
ϕ for all realizations of the experiment. In regime[10].
contrast, for the case of Raman fields Ω1eiω1t
and Ω2eiω2t, the correct choice of frequency
ω −ω =ω −ω meansthatboththephaseof
1 2 D(cid:48) D
the entangled atom-photon state before photon
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