A solid state spin-wave quantum memory for time-bin qubits

0.69 MB·English

by  Mustafa Gündoğan

A solid state spin-wave quantum memory for time-bin qubits
Mustafa Gu¨ndo˘gan,1 Patrick M. Ledingham,1,∗ Kutlu Kutluer,1
Margherita Mazzera,1,† and Hugues de Riedmatten1,2
1ICFO-Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain
2ICREA-Instituci´o Catalana de Recerca i Estudis Avanc¸ats, 08015 Barcelona, Spain
(Dated: June16, 2015)
We demonstrate the first solid-state spin-wave optical quantum memory with on-demand read-
out. UsingthefullatomicfrequencycombschemeinaPr3+:Y2SiO5crystal,westoreweakcoherent
pulses at the single-photon level with a signal to noise ratio > 10. Narrow-band spectral filtering
basedon spectralholeburningin asecond Pr3+:Y2SiO5crystalisusedtofilterouttheexcessnoise
5 created by control pulses to reach an unconditional noise level of (2.0±0.3)×10−3 photons per
1 pulse. Wealsoreportspin-wavestorageofphotonictime-binqubitswithconditionalfidelitieshigher
0 than a measure and prepare strategy, demonstrating that the spin-wave memory operates in the
2 quantumregime. Thismakesourdevicethefirstdemonstrationofaquantummemoryfortime-bin

n qubits, with on demand read-out of the stored quantum information. These results represent an
u important step for theuse of solid-state quantummemories in scalable quantumnetworks.
J
PACSnumbers: 03.67.Hk,42.50.Gy,42.50.Md
5
1

Photonic quantum memories are essential in quan- [16], with a storage time limited by the coherence of ex-
]
h tum information science (QIS) where they are used as cited states. Another solution is to transfer the optical
p
quantuminterfacesbetweenflyingandstationaryqubits. excitations to long lived collective spin excitations (or
-
t They enable the synchronization of probabilistic quan- spin-waves), using strong control pulses [31–33]. This
n
tumprocessese.g. inquantumcommunication[1,2]and gives access to much longer storage times [34, 35]. Op-
a
u computing [3]. The implementation of quantum memo- erating a solid state spin-wave memory in the quantum
q ries (QMs) for light requires strong interactions between regimehassofarremainedelusive,becauseofinsufficient
[ individual photons and matter. This can be achieved by signal-to-noiseratio(SNR)atthesingle-photonlevel[33].

2 placing individual quantum systems (e.g. single atoms) Here,usingthefullatomicfrequencycomb(AFC)pro-
v inhighfinessecavities[4]orbyusingensemblesofatoms, tocolinaPr3+:Y SiO crystalwestoreandretrieveweak
2 5
0 wherethephotonsaremappedontocollectiveatomicex-
coherent pulses on-demand with SNR > 10 for single-
8
citations. AtomicsystemsarenaturalcandidatesasQMs
9 photon-level input. Using a narrowband filter based on
[5–14], but solid state systems offer interesting perspec-
3 spectral hole burning in a second crystal, we achieve an
0 tivesfor scalabilityandintegrationinto existingtechnol- unconditional noise floor of (2.0±0.3)×10−3 photons
. ogy [15–21].
1 perpulse. Theuseofspectralholesasnarrowbandfilters
0 Rare-earth ion doped solids are promising candidates has been already demonstrated in storage schemes oper-
5 for high performance solid state QMs since they have atinginaclassicalfashion[36–38],butitishereexploited
1
excellentcoherencepropertiesatcryogenictemperatures for the first time to enter the quantum regime. Finally,
:
v [22]. Theyalsoexhibitlargestaticinhomogeneousbroad- we demonstrate storage and retrieval conditional fideli-
i eningoftheopticaltransitionswhichcanbetailoredand ties (i.e., assuming that a photon was reemitted) higher
X
used as a resource for various storage protocols, e.g. en- thanclassicalmemoriesfortime-binqubitsatthe single-
r
a abling temporally [23] and spectrally [24] multiplexed photonlevel,takingintoaccountthePoissonianstatistics
quantum memories. Recent experimental progress in- andthefiniteefficiencyofthememory. Theseresultsrep-
cludesqubitstorage[15,24–27],highlyefficientquantum resent the first demonstration of a solid state spin-wave
storageofweakcoherentstates[16],storageofentangled quantum memory, enabling on-demand read-out of the
and single photons [17, 18, 28], entanglement between stored qubits. They also provide the first example of a
two crystals [29] and quantum teleportation [30]. spin-wavequantum memory for time-bin qubits (for any
system), an essential resource in quantum communica-
Yet, nonclassical states have so far only been stored
tion [39] and processing [40].
as collective optical atomic excitations with fixed stor-
age times [17, 18, 28]. While this may provide a use- TheAFCtechnique[15,23]isbasedonspectraltailor-
ful resource if combined with massive multiplexing and ing of an inhomogeously broadened absorption line into
deterministic quantum light sources [24], the ability to a comb-shaped structure with periodicity ∆. The in-
read-out the stored state on-demand is essential for ap- put pulses resonant with the comb are mapped onto a
plications where the quantum memory is used as a syn- collective optical atomic excitation. After an initial de-
chronizing device. On-demand read-out can be achieved phasing,theexcitationsrephaseatatime1/∆givingrise
by actively controlling the optical collective excitations to a forward collective emission [15, 23]. Before the co-
2
herent emission two strong control pulses are applied to
transfer the excitation to and from a long-lived ground
statetoachievethespin-wavestorageoftheinputpulses.
The full AFC scheme requires ions with at least three
ground states, one being used as auxiliary state for op-
tical pumping [41]. The spin-wave storage efficiency is
given by η =η ×η2 ×η , where η is the ef-
SW AFC T C AFC
ficiency of the storage at the excited state and depends
on the optical depth and comb finesse [23], η accounts
C
forthe decoherenceduringthe groundstatestorage,and
ηT is the transfer efficiency of the control pulses. FIG. 1: (a) Quantum memory setup. The memory (MC)
The realization of the full AFC scheme in the single- and filter (FC) crystals are located inside a liquid-free cooler
photonregime is challengingas the strong controlpulses (Oxford V14) operating at a temperature of 2.5K. They
are both 3mm long and doped with a Pr3+ concentration
create noise which may dominate the weak signal re-
of 0.05%. The control and input beams are steered towards
trieved from the memory. Two main mechanisms are
the memory with an angle of ∼ 1.5◦, leading to a exctinc-
responsibleforthisnoise. i)Spatialleakagefromthecon- tion ratio of 10−5. The beam diameters on the crystal are
trolmodeintotheinputmodeduetoscatteringfromthe 280µm and 90µm for strong and input modes, respectively.
optical surfaces and ii) interaction of the control pulses Theweak coherent states are preparedbyattenuatingbright
with residual population in the spin storage state due to pulseswithvariableneutraldensityfilters(NDFs). Aportion
imperfectopticalpumping. Thelatterincludescollective of the input beam is picked up before the NDF and sent to
a photodiode (PD1) for the calibration of the mean photon
effects,suchasfree-inductiondecay(FID), orincoherent
numberperpulse. Amechanicalshutterisusedtoprotectthe
fluorescent emission. Note that four-wave mixing is not
SPD during the memory and filter preparation. HWP: half-
a dominant source of noise in our system, in contrast to
waveplate;AOM:acousto-opticalmodulator;DG:diffraction
Raman memories in atomic vapors [42] (see [41] for de- grating; SPD: single-photon detector. The dashed red beam
tails). To reduce the noise we employ spatial, temporal indicates thefilterpreparation mode. (b) Hyperfinesplitting
andspectralfiltering. Thespectralfilteringischallenging of the first sublevels of the ground 3H4 and the excited 1D2
inPr3+:Y2SiO5, as the input andcontrolfrequenciesare manifold of Pr3+ in Y2SiO5.
separatedbyonly10.2MHz(seeFig. 1(b)). Asanarrow-
band spectral filter we use a second Pr3+:Y SiO crystal
2 5
wherewepreparespectralholesofvariablewidth[36–38]. tal. A diffraction grating (DG) is then used to filter the
The experimental arrangement and the relevant en- noise not resonantwith the crystal. The retrievedsignal
ergy level scheme of Pr3+ at 606nm are shown in Fig. is coupled with 60% efficiency into a single-mode fiber
1. The main laser beam at 606nm (Toptica TA-SHG for connectionto the single photon detector (SPD, Pico-
pro) is split into three to be used as input mode, fil- Quant τ SPAD-20, detection efficiency ηd = 60%, dark
ter preparation, and lastly for control pulses and mem- countrate ∼10Hz). The totaltransmissionofthe input
ory preparation. They all pass through acousto-optical beam from the cryostat to the SPD is about 13%.
modulators(AOMs)indouble-passconfiguration,driven We tailor the AFC using optical pumping techniques
by an arbitrary waveformgenerator (Signadyne), to cre- as described in [33, 43]. We isolate a single class of
ate the necessary pulse sequences. The beams are then atoms[44,45]andcreatea3.5MHzwideAFCwith∆=
carried with polarization-maintaining single-mode opti- 200 kHz on the 1/2 −3/2 transition within a 14 MHz
g e
cal fibers to another optical table where the cryostat wide transparency window (see [41] for details and for a
is located. The maximum available optical powers are comb example). During the preparation of the mem-
about 20 mW, 3.5 mW and 150 µW for control, filter ory, the population removed from the comb is stored
preparation, and input modes, respectively, measured in in the auxiliary 5/2 state, while the 3/2 is emptied.
g g
front of the cryostat. The frequency of the 606nm laser To further remove unwanted residual population in the
is stabilized by Pound-Drever-Halltechnique to a home- 3/2 state, we apply an extra series of 100 pulses on
g
made temperature controlled Fabry-Perotcavity housed the 3/2 − 3/2 transition after the comb preparation.
e g
in a vacuum chamber. The input light is linearly polar- We then start the single-photon-level storage measure-
izedclosetotheopticalD axistomaximizethe interac- ments. Weak gaussian input pulses with full width at
2
tion with the Pr3+ ions. The measured optical depth half maximum (FWHM) duration of 430 ns and mean
of the Pr3+ transition at 606nm is about 7 for both photon number µ are mapped on the AFC and trans-
in
memory and filter crystal. In both cases, the inhomo- ferred to spin-waves thanks to a strong control pulse.
geneouslinewidth is about6GHz. After the storage,the ThecontrolpulseshaveaGaussiantemporalprofilewith
retrievedsignalpassesthroughdifferentdiffractionorder FWHM of 700 ns and are spectrally chirped by 5MHz
modes (-1st and 1st) of two consecutive AOMs, acting about the 3/2 −3/2 transition. For each comb prepa-
e g
as temporal gate before passing through the filter crys- ration,1000storagetrialsareperfomedwitharepetition
3
s] GateClosed GateOpen (81.7±2.6)% (assuming ηC ∼ 75% [32]). A convenient
ns bin6800Input/20 (a) µin=1.15 SWE ficigeunrceyoisf gmiveernitbtyaktihnegminintoimaucmcouµnint ntheceesnsoairsyetaonddeteeffict-
6 ControlPulses a spin-wave echo with SNR = 1, called µ1. From the
9
0.40 µin=0.45 linear fit of the experimental data of Fig. 2(b), we find
nts [420 µin=0 µw1ith=µ0in.06=91±, 0a.s00sh2.owWneinthFeing.v2a(rcy).τsTbhye cdheacanyginingtThSe
u
o 0 SNR is compatible with a spin inhomogeneous broaden-
C 0 5 10 15
ing of γ = (26±1)kHz, similar to previous measure-
Time [µs] in
ments with bright pulses [31, 32]. We still observe SNR
(b) (c)
15 =4.5±0.4 for T =18.8µs (τ =23.8µs).
S s
10
Wecanestimatethecontributionofthefiltercrystalin
R R10
N N the suppression of the noise by preparing a wider trans-
S S
parency window. For a filter width of 14MHz (squares
5
1 in Fig. 2(b)) there is no filtering at the control fre-
0 quency, and we observe an increase of the noise floor
0.1 1 10 0 10 20
to (2.3±0.6)×10−2 together with a slightly higher re-
µ T [µs]
in s trieval efficiency (η =(2.9±0.2)%), which results in
SW
µ values up to about 1. When the filter crystal is by-
1
FIG.2: (a). Timehistograms oftheretrievedphotonsmea- passed, the noise floor raises to (0.23±0.01), indicating
suredfordifferentinputphotonnumberswhenatransparency
that the inhomogeneously broadened absorption profile
window 2MHz wide is prepared in the filter crystal. The
of Pr3+ also contributes to partially filter the noise [28].
input (µin = 0.9) and the control pulses, as measured in
Nonetheless,forthissetofmeasurements,wewereableto
photon counting and from a reference photodiode (PD2 in
Fig. 1), respectively, are also displayed. The chosen 0.7µs achievehigherstorageefficiency,i.e. ηSW =(5.3±0.5)%,
widedetectionwindowisindicatedbythedashedlinesabout leading to a limited increase of the µ1 to about 4.
the three-level echo; it includes ∼ 80% of the counts in the For applications in QIS, it is crucial that the optical
full echo mode. (b). Signal-to-noise ratio (SNR) as a func- memorypreservesthecoherenceofthestoredqubits. We
tionofthenumberofinputphotonsfordifferentfilterwidth.
takeadvantageoftheintrinsictemporalmultimodalityof
Circles: 2MHz; Squares: 14MHz. The error bars (smaller
the AFC protocolto demonstratethe phasepreservation
than the data points) are evaluated with Poissonian statis-
tics. The black dotted line indicates the limit of detection in the spin-wavestorageoftime-bin qubits. This type of
SNR=1. Thedashedlinesarelinearfitsoftheexperimental encodingis widelyusedinquantumcommunicationasit
data. Panel (c). Decay of theSNR as a function of the spin- is robust against decoherence in optical fibers [39]. The
wave storage time TS with average photon number µin = 1. time-binqubitsareexpressedas|ψini=c1|ei+c2ei∆α|li,
FromthefitwithaGaussian profile,thespininhomogeneous where|ei(|li)representsaqubitintheearly(late)time-
broadeningγin =(26±1)kHzcan beextrapolated. By com- bin, ∆α is their relativephase, and c2+c2 =1. In order
paringtheSNRthatwemeasureatastoragetimeTS =7.8µs 1 2
tostoretime-binqubits, the durationofthe inputpulses
with the extrapolation at TS =0µs, we can evaluate the de-
is reduced(from430to 260ns), leadingto a reductionof
coherence term ηC to be about 75%.
η to about 2.2% and to an increase of the µ up to
SW 1
µ =0.11±0.01. We start by evaluating the fidelity of
1p
rate of ∼ 7kHz. The full cycle has a period of 700 ms, the states |ei and |li, located at the poles of the Bloch
including memory preparation and light storage. It is sphere, F and F, by storing only the early and the late
e l
synchronizedwith the cryostatcycle to reduce the effect qubits. We obtain average fidelity values for the poles
ofvibrations. The sequence isthen repeated500to 1000 ranging from F =85% to 98% for photon number per
el
times to acumulate sufficient statistics. qubit, µ , going from 0.6 to 5.9 [41].
q
Fig. 2(a) shows the time histograms of the retrieved We then storesuperpositionstates locatedonthe equa-
photons with different µ . For this measurement, the tor ofthe Blochsphere. We use the memory itself to an-
in
crystal filter has a hole width of 2 MHz. The spin-wave alyze the retrievedqubits [33] applying two partialwrite
storage time is T = 7.8 µs, leading to a total stor- pulses as depicted in Fig. 3(a). This method provides a
S
age time of τ = 1/∆+T = 12.8µs. From the trace convenientwayofanalyzingtime-binqubits,buthasthe
s S
with µ = 0, we estimate an unconditional noise floor drawbackof reducing the storage efficiency. As a matter
in
of (2.0±0.3)×10−3 photons per pulse at the memory offact,inordertoinserttwowritepulses,theirduration
crystal. For µ = 1.15, we measure SNR = 16.3±2.4. needstobereduced,whichdecreasestheirefficiency. Fig.
in
The linear scaling of the echo SNR with respect to in- 3(b)reportsexamplesofinterferencefringesforµ =1.5.
q
creasing µ is shown with blue circles in Fig. 2(b). From sinusoidal fits, we obtain a raw mean visibility of
in
Typical values of efficiencies are η = (5.6±0.3)% V =(72.5±1.3)%. Thefidelityoftheprocessiscalcu-
AFC +−
and η = (2.8±0.1)%, from which we deduce η = lated from the visibility, as F =(1+V)/2 [39]. Finally,
SW T
4
w1 w2 r |eli+|lei reached for µq >0.25 (see [41]).
(a) ∆β The very low noise probability and the ability to ob-
∆α tain µ1 ≪ 1 opens prospects for the spin-wave storage
of single-photons, as required for many applications in
QIS. In that case, the probability to have a photon be-
|ei |li |eei |lli fore the memory (i.e including all optical loss between
source and memory) needs to be higher than µ to en-
(b) 1
ter the quantum regime. In the current experiment, the
100
storageefficiencyislimitedbythe availablecomboptical
s
nt depth in our 3 mm long crystal, by the limited trans-
u
Co 50 fer efficiency (η2 = (67±4)%) and by technical issues
T
(cryostatvibrations,laserlinewidth, see [41]). Note that
muchhigher efficiencies (for storagein the excitedstate)
0
100 150 200 250 have been obtained in Pr3+:Y SiO using longer crystal
2 5
∆β [Degrees]
(69 %) [16] or impedance matched cavities (58 %) [46].
Longer storage times can also be achieved with dynam-
FIG. 3: (a) Pulse sequence to measure the time-bin qubit
ical decoupling techniques to counteract decoherence in
coherence. We apply two partial write pulses with a relative
the spin state [34, 35, 47, 48].
phase∆βinordertospliteachpulseintotwotemporallysep-
arated echoes. If the delay between the time-bins |ei and |li
is equal to the time difference between the two write pulses, 1
we can overlap the late echo of the early bin, |eli, with the
early echo of the late bin, |lei. The output of the memory
(occurring after the single read pulse r) has three time-bins,
{|eei,|eli+|lei,|lli}. Aninterferencewilloccurinthecentral
0.9
time-bin if the coherence is preserved. The input pulses, lo-
catedat|eiand|lihavearelativephasedifferenceof∆α. (b)
Interferencefringesobtainedintegratingoverthecentralout- al
ot
put time-bin (in this case ∆td = 0.5µs) as a function of the T
relativephasedifference∆β forµq =1.5. Circles: ∆α=90◦, F 0.8
V =(71.6±6.8)%;Squares: ∆α=135◦,V =(73.4±3.5)%.
0.7
we obtain a total conditional fidelity per retrieved qubit
F = 1F + 2F , where F (F ) is the average fi-
T 3 el 3 +− el +−
delityoverthepoles(equator)basis. Theobtainedvalues 0 2 4 6
are reported in Fig. 4 for different µq. We observe that µq
the fidelity decreases with µ . To explain this behavior,
q
we fit our data with a simple model taking into account FIG. 4: Total fidelity vsinput photon numberper qubit,µq.
the decrease of SNR with µ and the reduced efficiency The light orange squares are the data points with an error
q
due to the double write protocol [41]. The good agree- barof1standarddeviation. Theorangedottedlineisafitto
ment between the simple model and the data provides thedata points using Eq. (5) in [41], with thecorresponding
shadedareabeingthe1standarddeviationoftheerrorinthis
evidence that the decrease of fidelity is only due to the
fit. The solid blue (dashed green) line is the classical limit
noise created by the control pulses, and not to a loss of
obtained by a measure and prepare strategy for our memory
coherence.
efficiency of ηSW = 2.2% (ηSW = 100%) when testing the
In order to infer the quantum nature of our memory, memorywithweakcoherentstates[25]. Thedash-dottedline
the total fidelity is compared with the highest fidelity is the classical limit for testing the memory with a single-
photon Fock state (F =2/3).
achievable with a measure-and-prepare approach (solid
curveinFig. 4)takingintoaccountthePoissonianstatis-
tics of the input states and the finite memory efficiency In conclusion, we demonstrated the spin-wave storage
(2.2%) [4, 11, 25]. Using this criteria, the experimen- and on-demand retrieval of weak coherent states at the
tal data are higher than the limit for a classical mem- single-photon level in a solid state memory based on a
ory by more than one standard deviation for most µ Pr3+:Y SiO crystal. Thisisthefirstdemonstrationthat
q 2 5
investigated. With the raw data, the memory is in the solidstatespin-waveopticalmemoriescanoperateinthe
quantum regime for µ > 0.96. When correcting for the quantumregime,overcomingastronglimitationforAFC
q
lossofefficiencyinthe analysis(whichcouldbe achieved QMs. We achieved a SNR higher than 10 for single-
byanalyzingthequbitswithanexternalinterferometer), photonlevelinputpulses,highenoughtoenablethestor-
the model predicts that the quantum regime would be ageofsinglephotons. Finally,weconfirmedthequantum
5
natureofourmemorybystoringtime-binqubitsencoded
inweakcoherentstatesanddemonstratingconditionalfi-
delities for the retrieved qubits higher than what is pos-
siblewithclassicalmemories. Ourdevicethusrepresents Supplemental Material
the firstspin-wavememoryforphotonic time-binqubits.
These results open the door for long-lived storage and In the supplemental material we report details on
on-demand readout of non-classical light states in solid the atomic frequency comb preparation (section I.), the
state devices and represent an important step in view of sources of noise (section II.), the chosen strategy to sup-
usingsolidstatequantummemoriesinscalablequantum press the technical noise (section III.), the characteriza-
architectures. tion of the filter cristal (section IV.), and, finally, we de-
Acknowledgements. We acknowledge financial sup- scribeatheoreticalmodelforthefidelityofthespin-wave
portbytheERCstartinggrantQuLIMA,bytheSpanish storage of time-bin qubits (section V.).
Ministry of Economy and Competitiveness (MINECO)
andtheFondoEuropeodeDesarrolloRegional(FEDER)
through grant FIS2012-37569, by the European project Atomic Frequency Comb Preparation
CHIST-ERA QScale and by the People Programme
(Marie Curie Actions) of the EU FP7 under REA Grant To prepare the atomic frequency comb (AFC) we fol-
Agreement No. 287252. low the approach already described in [33, 43]. The
frequency of the strong beam is firstly scanned by 14
MHz in 100 ms to create a wide transparency window
(also referred to as pit) in the Pr3+ absorption line by
spectral hole burning (curve a in Fig. 5). Afterwards
a 2MHz-wide sweep is operated outside the pit to pump
theatomsbacktothe1/2 state. Thiscreatesa3.5MHz-
g
wide absorbing feature corresponding to the 1/2 −3/2
g e
transition inside the pit, but also populates the 3/2
g
state which in principle must be empty for later trans-
fer of the input field (see Fig. 1(b) in the main text).
Thus, a clean pulse with duration of 50 ms is applied
on 3/2 − 3/2 transition to empty this ground state.
g e
The clean pulse also contributes to suppress the tran-
sitions of different classes of Pr3+ ions which might be
addressed by the preparation pulses [49]. Then, a se-
quence ofsingle-frequency hole-burning pulses is applied
on the single class absorption feature resonant with the
1/2 −3/2 transition,eachtime changingthe frequency
g e
by a fixed amount, ∆. This sequence burns periodi-
callyspacedholesintheabsorbingfeaturecorresponding
to the 1/2 −3/2 transition and anti-holes at the fre-
g e
quency of the 3/2 −3/2 transition, so a short burst of
g e
cleanpulses is applied to maintain the 3/2 state empty.
g
For the AFC preparation it is crucial to have a third
ground state (5/2 ) to use as auxiliary state where to
g
store the excess of population resulting from the optical
pumping. An example of the resulting comb structure
for ∆ = 200kHz is shown in Fig. 5, curve b. With
the described procedure we are able to tailor absorp-
tion peaks as narrow as (43±3)kHz, with a comb op-
tical depth d = 4.5±0.1 and background optical depth
d =0.75±0.04. It is worth noting that the limited dy-
0
namical range of the detector might fix a higher bound
forthedetectedpeakamplitudeandthusleadtoanover-
estimated full width at half maximum.
When the memory is characterized with classical
pulses,wefindtypicalefficiencyvaluesofη =(10.3±
AFC
1.4)% and η = (4.4 ± 0.1)%. The prediction for
SW
6
7 control pulse creates spin-waves via an off-resonant Ra-
1/2 → 3/2 maninteractionandthesecondcontrolpulsereadsthem
g e
6 out as noisy photons in the same mode as the memory
4
output. This has been identified as the main source of
D noise in Raman type quantum memories using atomic
5 O
vapors [42]. However, this type of noise does not seem
2
to be, at this stage, an important source of noise in our
4
spin-wave memory. This is evidenced by the following
D
O 0 noise measurements. First, we have measured that the
3 −6 −5 −4 two control pulses contribute equally to the noise in the
Relative Frequency [MHz]
b echowindow. Thiscanbeexplainedbythefactthatthey
2 interactwiththesamenumberofionsgivingtwoequally
1/2 → 5/2 sizedincoherentfluoresecencefields(giventhetime scale
g e
1 3/2 → 3/2 ofthestoragetimecomparedtotheexcitedstatelifetime
a g e T1=164µs[51]). Iffour-wavemixingwasnotnegligible,
thenoisewouldbesuppressedbymorethanafactor2by
0
−5 0 5 switchingoffoneofthe twocontrolpulses. Furthermore,
Relative Frequency [MHz] the noise arising from four-wave mixing would decay for
increasing storage times with the same behavior of the
FIG.5: Exampleoftransparencywindow(curvea)andcomb spin-wave echo, i.e. following the inhomogeneous spin
trace prepared with ∆=200kHz(curveb). The inset shows broadening (see Fig. 2(c) of the main text). As a mat-
amagnification ofthecombcorrespondingtothe1/2g−3/2e ter of fact, the noise floor in our case is constant within
transition.
error over the whole range of storage times investigated
(see Fig. 6).
GaussianAFCpeaksisη =d˜2e−d˜e−7/F2e−d0,where
AFC
d˜= d/F is the effective comb optical depth [23]. The 3
measured value of η is compatible with the predic-
AFC
tion,η =(12.2±1.6)%,forthemeasuredvaluesof
AFC,th )
d=4.5±0.1,F =4.7±0.4andd =0.75±0.04. Forthe 3
0 −
sameopticaldepthofourcombbutassumingd =0,the 0
0 1 2
model predicts a maximal efficiency of ηAFC,th =26.5% ×
for a finesse of F = 4. Note that an even higher effi- (

r
ciency, ηAFC,th = 34.8%, is expected for square shaped o
o
combpeakandafinesseofF =3.2([50]). Inourcase,the
l
achievable contrast is limited by technical reasons, such f 1
e
as vibrations of the cryo-cooler and finite linewidth of s
i
o
the laser. When measuredinphotoncountingmode, the
n
efficiency is further reduced by the coincidence window
taken (containing ∼80% of the total counts), by loss in 0
the narrow-band filter (∼ 10%) and by instabilities due 0 5 10 15 20
to long integration times (∼10% loss). T [µs]
s
Sources of noise FIG. 6: Noise floor in the echo window as a function of the
time between the two control pulses, Ts. The error bars are
calculated taking into account Poissonian statistics.
ThefullAFC protocolforlightstorageimplies the use
of strong transfer pulses which are sources of different
kind of noise. First of all there might be spatial leakage Severalreasonscouldexplainthisdifference. First,the
fromthe preparation/controlmode into the input mode, non-collectiveemissionfromthe1D excitedstatemainly
2
essentially due to reflections and scatting from the opti- happens throughintermediate crystal-fieldmanifold and
cal surfaces. Moreover,the controlpulses might interact involves the coupling with the phonons of the YSO ma-
with the residual population left in the 3/2 state, due trix. As a matter of fact, the branching ratio of the
g
to imperfect cleaning(see section). This interactioncan spontaneous emission through the 1D →3H transition
2 4
give rise to coherent (e.g. free-induction decay) or inco- is very low (≈3% ) [51]. Furthermore, among the tran-
herentemission. Anotherpotentialsourceofnoiseinthis sitionswithinthe3H →1D thatcouldbeoff-resonantly
4 2
system may arise from four-wave mixing, where the first excited by the first control pulse, many of them are
7
characterized by relative oscillator strengths one order ) 3 1.2
−2 (a) (b)
of magnitude lower than that employed to transfer the 0
1
excitationtothegroundstate. DuringtheAFCprepara- ×2 0.8
(
tion, most of the ions are collected in the auxiliary 5/2g or µ1
state but, due to the relative oscillator strength of the flo 1 0.4
transitions starting from it, they can only be involved e
s
in the almost closed cycle transition 5/2g ↔ 5/2e [44]. noi 0 0
Those characterized by a non-negligible probability for 0 5 10 15 0 5 10 15
Filter width [MHz] Filter width [MHz]
off-resonantexcitationeither aresuppressedby the state
cleaning that we operate in the memory preparation (it ns]150Input/10 GateClosed GateOpSeWnecho
isthecaseofthetransitionsstartingfromthe3/2gstate) s bi (c) µin=3.7
or are tailored in the comb structure (transitions having 6 n100 µin=1.2
the 1/2g as initial state). In the latter case, the second 0.9 µin=0
control pulse would give rise to a coherent emission in a [4 50 ControlPulses
well separated temporal mode with respect of the spin- nts
wave echo. ou 0
C 0 5 10 15
Time [µs]
Filtering
FIG. 7: Noise floor in the echo window (a) and µ1 (b) value
as a function of the spectral hole width in the filter crystal.
To achieve a low noise floor, (2.0 ± 0.3)× 10−3, we
Filled symbols: hole centered at the input pulse frequency;
use spatial, temporal and spectral filtering. As for the
opensymbols: transparencywindowcoincidentwiththatpre-
spatial filtering, the controland input beams are sent to paredinthememorycrystal(seecurvebinFig. 5). (c)Time
the memory with a small angle, leading to an extinction histogram of the retrieved photons for different input pho-
of 10−5. Temporal filtering is achieved thanks to a tem- ton numbers, µin, and a filter width of 14MHz. The input
poral gate implemented with two AOMs. It allows us (µin =0.9)andthecontrolpulses(measuredwithareference
photodetector) are also displayed.
to block the leakage of the strong control beams in the
signal mode. This is important not to blind the single-
photon detectors, but also to avoid burning a spectral
hole in the crystal filter. A diffraction grating is em- measurementswhereweburninthefiltercrystalspectral
ployed to decrease the noise originated from incoherent holes of different width centered about the input pulse
fluorescence. The narrow-bandspectralfilteris prepared frequency (solid symbols). The noise in the echo win-
simultaneouslywiththeAFC.Thefrequencyofthefilter dow increases by one order of magnitude when the filter
mode (shown in dashed red line in Fig. 1(a)) is scanned width increases from 0.8 to 12 MHz. The open symbol
by 1.2MHz around the input light frequency for 100 ms referstotheextremecaseofa14MHz-widetransparency
whichresultsinatransparencywindowofaround2MHz window coincidentwith the one preparedinthe memory
due to the power broadening effect. In order to tempo- crystalto hostthe AFC(see curveainFig. 5). Herethe
rally discriminate the stored and retrieved weak pulses noise floor is further doubled because the transparency
from the portion of the FID leaking in the echo mode windowallowsthecontrolpulsefrequencytopassbyand
and happening at short time scales (a few µs), the op- reachthe detector. The increase in the noise affects con-
timization of the excited state storage efficiency at long sequently the µ value as shown in Fig. 7(b). Neverthe-
1
storagetimesiscrucial. Bycarefullyoptimizingthestor- less,this value alsodepends onthe echocounts,thus the
agetimes,inboththeexcitedandgroundstates,andthe narrowestfilter width does notnecessarilycorrespondto
pulsedurationsacompromisecanbereachedbetweenthe the best case. As a matter of fact, the best µ value is
1
noise level and a reasonable retrieval efficiency. measured for a filter width which allow the major por-
tionoftheechosignaltopassbybutsuppressesefficiently
the noise at the controlpulse frequency, i.e. 2MHz. The
Filter crystal characterization value of µ also depends on the echo efficiency and is
1
thus more affected by power fluctuations than the noise
We providein this sectiondetails aboutthe noise sup- floor. We believe this is the reason why its increase in
pression realized by the filter crystal. The main advan- the case of totally open transparency window is more
tageofusingasecondcrystalisthatitprovidesadynam- pronounced than that of the noise floor (compare filled
icalfilter, whose center frequency and bandwidth canbe and open symbols in panel (a) and (b) of Fig. 7). Panel
easily tuned, as it is obtained by spectral hole burning. (c) of Fig. 7 shows the time histogram of the retrieved
Fig. 7(a) represents the noise floor (noise counts per photonsfordifferentinputphotonnumbersinthecaseof
pulse at the memory crystal) as evaluated from noise a 14MHz-wide pit prepared in the filter crystal. Finally
8
we evaluate the extinction ratio of our 2MHz-wide spec-
tral filter by switching off the AFC and sending input
pulses through the transparency window in the memory
S+N
crystal firstly resonantwith the filter center, then at the
control pulse frequency. By comparing the total counts
reaching the detector in the two cases, we estimate the
extinction ratio to be about 750.
N
Modeling the Fidelity
Fig. 4 from the main document shows the measured
E L
total fidelity as a function of the mean photon number
per qubit, µ , with a corresponding fit using the model (a)
q
we develop in this section. The aim of the model is to
show that the reduction of the fidelity as the photons
per qubit are reduced is entirely due to the noise of the
S +N
memory and not due to a loss of coherence caused by α
e.g. phase noise in the laser. The goal of our model is
to express the total fidelity as a function of the same µ
1
parameter. S +N
4α
Firstly,letuslookattheoutputofthememoryforthe
two differing measurements, namely the measurementof
the |ei or |li qubit (i.e. the poles of the Bloch sphere)
and the measurement of the coherence between |ei and
|li (i.e the equators of the Bloch sphere). The former
E−E E−L + L−E L−L
(latter) are schematically depicted in Fig. 8(a) (8(b)).
For a given integration width, we sum over both E
(b)
andLforthepolesmeasurement,whileforthecoherence
measurement we integrate over the window labelled E-
FIG. 8: (a) Schematic example of the memory output for
L+L-E (see figure 8(b)). storageofan|eiqubit. (b)Schematicexampleofthememory
Considerthe example depictedin Fig. 8(a), where the outputforstorageofan|ei+|liwheretherelativephasesare
|eiqubithasbeenstoredandthenrecalled. Welabelthe such that the maximum interference is observed.
integrated counts in the window about E as S+N and
the integratedcounts aboutthe window L as N, where S
is the signal(notincluding the noise)andN is the noise.
Then, the fidelity is the ratio of the counts in E to the
total counts over E and L, i.e. the windowatthe time labelledE-EareS/4α+Nwhere
thesignalSisthesameasdefinedearlier. Here,thesignal
S+N S/N+1 SNR+1
F = = = . (1) S is reduced by a factor of 4, this is because the number
e
S+N + N S/N+2 SNR+2
of photons per qubit are distributed equally within the
where SNR is the signal to noise ratio. Provided the input time-bins (factor of 2) and the echo is distributed
efficiencyandnoisearethesameforstoringthe|liqubit, equally between the output time-bins (factor of 2). The
F =F =F . Finally,wecanexpresstheabovefidelity factor α reduction in the signal is because we use differ-
l e el
as a function of the µ of this measurement (see main ent bandwidth pulses for the double write process which
1
text for definition) by noting that SNR=µ /µ , where givesareducedefficiencyintheecho. Theamountofsig-
q 1p
µ is the µ characteristic of the poles measurement. nalinthewindowE-L+L-Eis4timesthatofthesignal
1p 1
The expression for the fidelity becomes in the L-L window provided the interference is fully co-
herent (note the same is true for the E-E window).The
µ +µ
F = q 1p . (2) resulting counts are then S/α+N. It is worth noting
el
µ +2µ
q 1p that if we instead used an unbalanced Mach Zehnder in-
and we measure µ =0.11±0.01. terferometer, we would have no reduction in the signal
1p
Now let us consider Fig. 8(b), which depicts an ar- and α=1.
bitrary qubit |ei + |li being recalled, and the relative
phases are such that the maximum interference is ob- The fidelity of the coherence measurement is ob-
served. Firstly, the integrated counts corresponding to tained from the visibility of the interference fringe by
9
F = (1 + V)/2[39]wherethe visibility V is definedas concluding our simple model for the fidelity. Taking the
measured µ , we fit the data in Fig. 9 with equation
1p
max−min S/α+N − N
5 having α as a free parameter. We get α = 2.5±0.3
V = =
max+min S/α+N + N which agrees excellently with the measured value stated
S SNR earlier. We therefore can conclude that the reduction
= = (3)
S+2Nα SNR+2α of the fidelity is due to the noise on the output of the
memory and not phase noise. For µ > 0.96 the fidelity
q
where max (min) refers to the maximum (minimum) in-
surpassesthe classicalbound, remaining in the quantum
tegratedcountsofthefringe,themaxbeingS/α+Nand
regime. Also in Fig. 9 we plot the case for α = 1 for
min being N . Note that we have V in terms of the SNR
µ = 0.11 and µ = 0.07 (the case shown in Fig. 2 of
1p 1p
ofthepolesmeasurement,sotheresultingfidelitycanbe
the main text). The fidelity exceeds the classical bound
written in terms of µ , i.e.
1p for photon numbers much less that 1 for these cases.
For completeness, we include table I which shows all
1 1 µ
q
F = + , (4) the measured fidelities for every µ tested. Also, we in-
+− 2 2 µ +2αµ q
q 1p cludeexamplesoftheoutputofthememoryforearlyand
late inputs (Fig. 10(a)) and for the input |ei+|li where
where the subscript +(−) refers to the qubit |ei ± |li
thephaseofthedouble-writeprocessisadjustedtoshow
(|ei±i|li),andweassumeF =F =F . Wemeasure
+ − +−
maximum and minimum interference in the E-L + L-E
α by comparing the µ in the L-L window of figure 8(b)
1
window.
to µ and we get α=2.5±0.6.
1p
1
µq Fel F+− FT FC
5.9 (97.9±1.5)% (97.2±1.7)% (97.4±1.2)% (93.0±0.1)%
0.9 7 1 3.2 (96.9±2.2)% (93.3±1.2)% (94.5±1.1)% (90.1±0.1)%
0 1
=0. =0. 1.5 (93.5±2.9)% (86.2±1.6)% (88.6±1.4)% (86.2±0.1)%
T µ1p µ1p 1.1 (93.3±3.1)% (85.8±1.5)% (88.3±1.4)% (84.4±0.1)%
F 1, 1,
0.8 = = 0.6 (84.9±4.2)% (72.9±1.3)% (76.9±1.6)% (81.0±0.1)%
α α
TABLEI:FidelitiesFel,F+− andFT foreachµq tested. FC
is the classical benchmark for testing the device with weak
0.7 coherentstateswhichdependsontheefficiencyofthememory
and theinput photon numberused [4, 25]. Thestated errors
are one standard deviation.
0.1 1 µ 10
q
FIG.9: Totalfidelityvsinputphotonnumberperqubit. The
light orange squares are the data points with an error bar of
1 standard deviation. The light orange dotted line is a fit to
∗ Presentaddress: DepartmentofPhysics,ClarendonLab-
the data points using eq. 5 with µ1p fixed at 0.11 and α = oratory, University of Oxford, Oxford OX13PU, United
2.5±0.3. The corresponding shaded area is the 1 standard
Kingdom
deviationoftheerrorinthisfit. Thesolidblue(dashedgreen)
† Electronic address: [email protected]
line is the classical limit obtained by a measure and prepare
[1] L.-M. Duan,M.D.Lukin,J. I.Cirac, and P.Zoller, Na-
strategy and a memory efficiency of η3LE = 2.2% (η3LE = ture 414, 413 (2001).
100%) when testing the memory with weak-coherent states
[2] N. Sangouard, C. Simon, H. de Riedmatten, and N.
[4, 25]. We see that for µq > 0.96 the fidelity is above the Gisin, Rev.Mod. Phys.83, 33 (2011).
classical benchmark. We show also the case of α = 1 and
[3] E. Knill, R. Laflamme, and G. J. Milburn, Nature 409,
µ1p =0.11, i.e. thecasewhereourdetectionofthecoherence 46 (2001).
isreplacedbyanunbalancedMachZehnderinterferometerfor
[4] H. P. Spechtet al.,Nature473, 190 (2011).
example. Furthermore,we show thecase of α=1 and µ1p = [5] C. W. Chou et al.,Nature438, 828 (2005).
0.07, i.e. theµ1 measured fortheoptimised pulsebandwidth [6] T. Chaneli`ere et al.,Nature 438, 833 (2005).
case. Thedash-dottedlineistheclassicallimitfortestingthe
[7] K.S.Choi,H.Deng,J.Laurat,andH.J.Kimble,Nature
memory with a single-photon Fock state (F =2/3).
452, 67 (2008).
[8] H. Zhang et al.,Nat Photon 5, 628 (2011).
Now,the total fidelity is givenby FT = 13Fel+32F+−, [9] M. Lettneret al., Phys.Rev.Lett. 106, 210503 (2011).
which leads to [10] S. Riedl et al.,Phys. Rev.A 85, 022318 (2012).
[11] A. Nicolas et al., Nat Photon 8, 234 (2014).
1 µq+µ1p 1 µq [12] B. Julsgaard et al.,Nature 432, 482 (2004).
F = + 1+ , (5)
T 3(cid:20)µ +2µ (cid:21) 3(cid:20) µ +2αµ (cid:21) [13] M. Hosseini et al.,Nat Phys 7, 794 (2011).
q 1p q 1p
10
]300 ]50 [25] M.Gu¨ndog˘anetal.,Phys.Rev.Lett.108,190504(2012).
s (a) s (b)
n n [26] C.Clausen,F.Bussi`eres,M.Afzelius,andN.Gisin,Phys.
bi bi40
s s Rev. Lett.108, 190503 (2012).
n200 n [27] Z.-Q. Zhou et al., Phys.Rev.Lett. 108, 190505 (2012).
6 6 30
9 9 [28] D.Riel¨anderet al.,Phys.Rev.Lett.112,040504(2014).
40. 40.20 [29] I. Usmani et al.,Nat Photon 6, 234 (2012).
s [100 s [ [30] F. Bussi`eres et al.,Nat Photon 8, 775 (2014).
nt nt10 [31] M. Afzelius et al.,Phys.Rev.Lett. 104, 040503 (2010).
u u
o o [32] M. Gu¨ndog˘an et al., New Journal of Physics 15, 045012
C 0 C 0
12 13 14 15 12 14 16 (2013).
Time [µs] Time [µs] [33] N. Timoney et al., Phys.Rev.A 88, 022324 (2013).
[34] J.J.Longdell,E.Fraval,M.J.Sellars,andN.B.Manson,
Phys. Rev.Lett. 95, 063601 (2005).
FIG.10: (a)Anexampleofthedetectedcountsattheoutput
[35] G. Heinze, C. Hubrich, and T. Halfmann, Phys. Rev.
ofthememoryfortheearly(lightorangehistogram)andlate
Lett. 111, 033601 (2013).
(darkbluehistogram) inputqubitswithµq =3. Thevertical [36] H. Zhang et al., Applied Physics Letters 100, 131102
dashedlinescorrespondtothe0.5µsintegration windowsfor
(2012).
the calculation of the fidelity. (b) An example of detected
[37] S.E.Beavan,M.P.Hedges,andM.J.Sellars,Phys.Rev.
countsattheoutputofthememoryforaninputqubit|ei+|li
Lett. 109, 093603 (2012).
with µq =5.9. Thedarkblue(lightorange) histogram shows [38] S. E. Beavan, E. A. Goldschmidt, and M. J. Sellars, J.
the case when the relative phase of the double-write process
Opt. Soc. Am.B 30, 1173 (2013).
issuchthattheinterferenceismaximized (minimized)in the
[39] I. Marcikic et al.,Nature 421, 509 (2003).
integration window (indicated by the vertical dashed line).
[40] P. Humphreys et al., Phys. Rev. Lett. 111, 150501
ThepulsefeatureontherightcorrespondstotheL-Lwindow
(2013).
ofFig. 8(b)whiletheE-Ewindowisnotdetectedhereasthe
[41] SeeSupplementalMaterial,whichincludesRefs.[49–51].
temporal gate is closed to negate the effects of the control
[42] P. S. Michelberger et al., New Journal of Physics 17,
pulses.
043006 (2015).
[43] N. Maring et al., New Journal of Physics 16, 113021
(2014).
[44] M. Nilsson et al., Phys.Rev.B 70, 214116 (2004).
[14] M. Sprague et al.,Nat Photon 8, 287 (2014).
[45] O. Guillot-No¨el et al.,Phys. Rev.B 79, 155119 (2009).
[15] H.de Riedmatten et al.,Nature 456, 773 (2008).
[46] M. Sabooni, Q. Li, S. Kr¨oll, and L. Rippe, Phys. Rev.
[16] M. P. Hedges, J. J. Longdell, Y. Li, and M. J. Sellars,
Lett. 110, 133604 (2013).
Nature465, 1052 (2010).
[47] M. F. Pascual-Winter, R.-C. Tongning, T. Chaneli`ere,
[17] C. Clausen et al.,Nature469, 508 (2011).
and J.-L. Le Gou¨et, Phys. Rev.B 86, 184301 (2012).
[18] E. Saglamyurek et al.,Nature469, 512 (2011).
[48] M. Lovric, D. Suter, A. Ferrier, and P. Goldner, Phys.
[19] E. Togan et al.,Nature466, 730 (2010).
Rev. Lett.111, 020503 (2013).
[20] H.Bernien et al., Nature497, 86 (2013).
[49] O. Guillot-No¨el et al.,Phys. Rev.B 75, 205110 (2007).
[21] D. G. England et al., Phys. Rev. Lett. 114, 053602
[50] M. Bonarota, J. Ruggiero, J.-L. Le Gou¨et, and T.
(2015).
Chaneli`ere, Phys.Rev.A 81, 033803 (2010).
[22] R.M. Macfarlane, J. Lumin. 100, 1 (2002).
[51] D. L.McAuslan, J. J. Longdell, and M. J.Sellars, Phys.
[23] M. Afzelius, C. Simon, H. de Riedmatten, and N. Gisin,
Rev. A 80, 062307 (2009).
Phys.Rev.A 79, 052329 (2009).
[24] N.Sinclair et al.,Phys. Rev.Lett. 113, 053603 (2014).