Quantum networks hold the promise for revolutionary advances in information
processing with quantum resources distributed over remote locations via
quantum-repeater architectures. Quantum networks are composed of nodes for
storing and processing quantum states, and of channels for transmitting states
between them. The scalability of such networks relies critically on the ability
to perform conditional operations on states stored in separated quantum
memories. Here we report the first implementation of such conditional control
of two atomic memories, located in distinct apparatuses, which results in a
28-fold increase of the probability of simultaneously obtaining a pair of
single photons, relative to the case without conditional control. As a first
application, we demonstrate a high degree of indistinguishability for remotely
generated single photons by the observation of destructive interference of
their wavepackets. Our results demonstrate experimentally a basic principle for
enabling scalable quantum networks, with applications as well to linear optics
quantum computation.Comment: 10 pages, 8 figures; Minor corrections. References updated. Published
  at Nature Physics 2, Advanced Online Publication of 10/29 (2006

A Kuzmich

B Julsgaard

BB Blinov

C Santori

C Simon

C. W. Chou

CK Hong

CW Chou

D Felinto

D. Felinto

DN Matsukevich

E Knill

E. W. Schomburg

H de Riedmatten

H-J Briegel

H. de Riedmatten

H. J. Kimble

J Beugnon

J Fiurášek

J Laurat

J Volz

J. Laurat

JD Franson

JI Cirac

JK Thompson

L Mandel

L-M Duan

MA Nielsen

MD Eisaman

P Zoller

T Chanelière

T Legero

V Balić

English

arXiv

Crossref

Conditional control of the quantum states of remote atomic memories for quantum networking

Quantum networks hold the promise for revolutionary advances in information processing with quantum resources distributed over remote locations via quantum-repeater architectures. Quantum networks are composed of nodes for storing and processing quantum states, and of channels for transmitting states between them. The scalability of such networks relies critically on the ability to carry out conditional operations on states stored in separated quantum memories. Here, we report the first implementation of such conditional control of two atomic memories, located in distinct apparatuses, which results in a 28-fold increase of the probability of simultaneously obtaining a pair of single photons, relative to the case without conditional control. As a first application, we demonstrate a high degree of indistinguishability for remotely generated single photons by the observation of destructive interference of their wave packets. Our results demonstrate experimentally a basic principle for enabling scalable quantum networks, with applications also to linear optics quantum computation

Felinto, D.

Chou, C. W.

Laurat, J.

Schomburg, E. W.

de Riedmatten, H.

Kimble, H. J.

Caltech Authors - Main

Supplementary Information –
Conditional control of the quantum states of remote atomic memories for quantum
networking
D. Felinto, C. W. Chou, J. Laurat, E. W. Schomburg, H. de Riedmatten, and H. J. Kimble
Norman Bridge Laboratory of Physics 12-33, California Institute of Technology, Pasadena, California 91125, USA
(Dated: September 25, 2006)
Supporting documentation is provided for our manuscript Ref. [1].
I. DECOHERENCE
Figure 1 shows the variation of the conditional probabilities of detecting one (p c
2
) and two (p c
22
) photons in fields
2L and 2R, once the two ensembles are ready to fire, as functions of the number N of trials that occur between the
two field-1 detections. This figure was obtained from the same raw data as Fig. 2 of Ref. [1]. Fields 2L and 2R have
then orthogonal polarizations, and are combined at a beam splitter as shown in Fig. 1b of Ref. [1]. In order to obtain
the quantities in Fig. 1, we divided the number of coincidences in field 2 that followed two temporally separated
detections in field 1 by the number of times the two ensembles were prepared with that specific time separation. The
solid lines are fittings considering an exponential decay of the conditional probability pc for the second photon from
either of the two ensembles, once a detection has occurred in field 1. We assumed the same pc and decay time for
the two systems. Note that the two systems were actually set up to have similar pc and similar Raman linewidths
for transitions between the hyperfine ground states [2], which should correspond to the system coherence time. The
expressions used to fit were then
p c
22
(N) =
p2c
2
e−N/Nc , (1)
p c
2
(N) =
pc + pce
−N/Nc
2
− p c
22
(N) . (2)
0 5 10 15 20
0
2
4
 D2a & D2b
c
c
p 2
2 
(x1
03
)
N
0
5
10
 
 D2a
 D2b
p
2
 (x10
2)
FIG. 1: Conditional probabilities pc2 and p
c
22 of measuring one (red circles and green triangles) and two (black squares) photons
in field 2, respectively, once the two ensembles are ready to fire. The red and black curves are fits using Eqs. (1) and (2), as
discussed in the text. The green line is 0.95× the red line, as the field 2 level measured by D2b is always 5% lower than for
D2a, indicating a possible difference of detection efficiency of this magnitude.
We assume above that p c
22
always involves one photon coming from an excitation stored during N trials. In this way,
we are neglecting the two-photon component of field 2, as well as diverse sources of background. For p c
2
, the first two
terms take into account that the conditioned detection event can be originated from either an ensemble that has just
been excited, or a stored (for N trials) excitation. The third term subtracts the probability of having a joint detection
© 2006 Nature Publishing Group 
 
2on field 2 (p c
2
gives the probability of detecting an event on one detector and zero on the other). From the fitting, we
then obtain pc = 0.091 and Nc = 18, corresponding to a coherence time τc = Nc× 0.525µs = 9.5µs. Keeping in mind
the simplicity of the above expressions, which do not take into account any background in field 2 or its two-photon
component, the inferred conditional probability pc is then consistent with the independently measured values of about
0.085 for each ensemble.
From the above discussion, it is then straightforward to obtain an expression taking into account decoherence for
the measured p1122, presented in Fig. 2 of Ref. [1]. Note first that, in the ideal case of very long coherence time
(Nc →∞), p1122 can be obtained from the p11 expression presented in the Methods section of Ref. [1] by multipling
it by the conditional probability p c
22
(0) of obtaining a pair of photons with effectively zero delay (N/Nc → 0) between
them:
pideal
1122
≈
(2N − 1)p2
1
p2c
2
, when p1 << 1 . (3)
Considering the measured value of p2
1
and the value of pc obtained from the fits of Fig. 1, we then obtain the green
line plotted in Fig. 2. In order to introduce decoherence in this analysis, each term of the p11 expression deduced in
the Methods section should be multiplied by the proper p c
22
(N) as defined in Eq. 1:
p1122(N) = p1
{
p1p
c
22
(0) + 2
[
(1− p1)p1p
c
22
(1) + (1− p1)
2p1p
c
22
(2) + · · ·+ (1− p1)
N−1p1p
c
22
(N − 1)
] }
. (4)
A plot of this expression, considering the pc and Nc obtained from the fits in Fig. 1, is shown as the red curve in Fig. 2.
The quite reasonable agreement with the experimental data (filled squares) indicates then that the experimentally
observed increase in p1122 can be understood by the increase in p11 provided by the circuit combined with the
decoherence of the stored collective excitation.
0 5 10 15 20
0
1
2
3
 
 
p 1
12
2 
(x1
07
)
N
  experiment
  ideal case
  corrected for decoherence
FIG. 2: Probability p1122 of coincidence detection as functions of the number N of trials waited between the independent
preparations of the two ensembles (L,R) with 1 excitation each. Filled squares give the measured joint probability p1122 of
preparing the two ensembles and detecting a pair of photons, one in each output of the beam splitter, in fields 2L,2R. Error bars
indicate statistical errors. The polarizations for fields 2L,2R were set to be orthogonal. These experimental results were also
presented in Fig. 2 of Ref. [1]. The green curve gives the theoretically expected pideal1122 for the ideal case of very long coherence
time, as given by Eq. 3. The red curve gives the theoretical p1122 for the case of a finite coherence time, Eq. 4.
Note finally that p1122 times the number of trials per second gives the rate of conditional joint detections in fields
2L, 2R. In this way, from Fig. 2 we can see that a larger coherence time could still enhance this detection rate by up
to a factor of 1.6 for N = 23 (enhance the factor F1122 from 28 to 45). An increase on the conditional probability pc
would also greatly improve this rate, since it scales with p2c . As discussed in the text, we infer that the probability
qc of extracting the photon from the ensemble is about 34% for our experimental conditions. Thus, considering the
same amount of losses on the field 2 pathways and the same detection efficiencies, we infer that ideally, if one achieved
qc = 1, pc can still be increased by up to a factor of 3, which would increase the joint detections rate by 9. This
indicates a good prospect for further optimizations of our system in the future.
© 2006 Nature Publishing Group 
 
3II. VISIBILITY AS A FUNCTION OF w
In the case of two indistinguishable single-photon wavepackets combined at a 50/50 beam splitter (BS), no co-
incidences should be observed at the two ouputs of the BS [3]. However, if the combined fields are not perfectly
overlapping single photons, coincident detections at the output of the BS can occur due to the two-photon component
in each input port. In this section, we evaluate the loss of visibility (as defined in Ref. [1]) due to this effect with a
simple model.
Suppose that the two ensembles are prepared each with stored excitation, as heralded by a detection in field 1 for
both ensembles. Let’s denote Pn the probability of finding n photons in field 2, and assume, for simplicity, the various
Pn are the same for both ensembles. In each field (before the BS), the two-photon suppression is characterized by the
parameter w [4]:
w =
2P2
P 2
1
, (5)
so that the two-photon component can be written as:
P2 =
wP 2
1
2
. (6)
Let us now combine the two fields at the BS. The probability to have one photon at each output of the BS, when the
two wavepackets do not overlap (e.g., if they have orthogonal polarizations) is given by:
p⊥ =
P2
2
+
P2
2
+
P 2
1
2
=
wP 2
1
4
+
wP 2
1
4
+
P 2
1
2
, (7)
where the two first terms corresponds to the terms with two photons in one input mode of the BS, and the third term
to the case with one photon in each input mode. The factor 1/2 corresponds to the 50% chance that the photons
split at the BS. In this simplified calculation, we neglect the case where we have two photons in one input and one in
the other one, whose probability is on the order of P 3
1
.
If the two fields overlap perfectly at the BS (parallel polarizations), the term with one photon in each input does
not lead to coincidences, and the probability to have one photon in each output is then:
p‖ =
wP 2
1
4
+
wP 2
1
4
. (8)
Taking Eqs. (7) and (8) into account, we find that the visibility can be written as:
V =
p⊥ − p‖
p⊥
=
1
1 + w
. (9)
In our case, we have g12 ≈ 23 for the two ensembles, from which we estimate w ≈ 0.17 [5]. This leads to a maximal
visibility of Vmax = 0.85 for a perfect overlap ξ = 1.0 between the fields. From our measured visibility of 0.77, we
then estimate an overlap ξ = 0.90.
III. JOINT-DETECTION LEVELS FOR EVENTS IN DIFFERENT TRIALS
In Fig. 3b, we show how the conditional probability of detecting two photons, when the (L,R) systems are ready,
decreases as a function of the delay between the two detections for the situation where the fields 2L,2R are combined
with the same (red) or orthogonal (black) polarizations. It corresponds then to Fig. 3 of Ref. [1]. The time td of
the detections is obtained from the recording of events in our acquisition card, and it refers to a fixed reference that
marks the beginning of the 525 ns repetition periods. From this list of detection times, we obtain the relative delay
τ when the two detections occur in the same trial.
We also show in Fig. 3, the cases where the two detections in D2a and D2b occur in different trials, with the event
in one detector occuring when the two ensembles are ready and the event in the other detector occuring the next time
the ensembles are ready. Figures 3a and 3c give the cases in which one detector or the other registers an event first.
The fact that the signal level is similar in both cases, with different polarizations, indicates that there is no large
misalignment when the half-wave plate is turned to switch between the two polarization configurations. Even though,
if we integrate the curves in Figs. 3a and 3c, the value obtained for orthogonal polarization is about 0.08 lower than
© 2006 Nature Publishing Group 
 
4the one obtained for parallel polarization. We confirmed this value by calculating it also from other different-trials
peaks (for detection events separated by up to 5 ready signals). The curves with different polarizations were taking at
alternate cycles of half-hour data taking, just turning a single half-wave plate between them. In this way, we believe
the decrease for orthogonal polarizations comes just from a small misalignment in the fiber input introduced by this
operation.
The level of the peak in (b) obtained with orthogonal polarizations should be half that observed in (a),(c) for
the case where pure single photons arrive at the beam splitter. The experimental observed ratio r is found to be
r = 0.60 ± 0.05. This ratio can be explained by the two-photon component of our generated state. As previously
done in section II, let’s denote P1 and P2 the probabilities of finding respectively one or two photons in each field 2.
For the sake of simplicity, these probabilities are taken equal for the two ensembles. Including two-photon events, the
probability for coincidence in the two detectors is given for the center peak (Fig. 2b) by:
1
2
P 2
1
+ P2 (10)
The first term takes into account the two cases where the single photons are both reflected or transmitted at the
beam splitter. The second one corresponds to the case where two photons arriving at one input of the beam splitter
are split into the two arms. Higher-order cases, which involve for instance two photons in each input, or two photon
in one input and one in the other, are neglected.
-40 0 40
0.0
0.2
0.4
-40 0 40 -40 0 40
c
 
τ (ns)
p 2
2 
(x1
03
)
 
 
(a)
 
 
 
(b)
τ (ns) τ (ns)
 
 
 
(c)
FIG. 3: Conditional joint-detection probability pc22(τ) of registering two events in D2a and D2b, once a ready signal is generated
as a result of the two ensembles being ready to fire, as a function of the time difference τ between the two detections. (b) The
two detections occur within the same trial. (a) Detector D2a fires first and D2b fires after the next ready signal. (c) Same as
(a), but with the detector order inverted. The red circles (black squares) provide the results for field 2 from the two ensembles
having orthogonal (parallel) polarizations.
In a similar way, the probability for the others peaks, where detections occur in different trials, can be written as
(P1 + 2P2)
2
(11)
This expression corresponds basically to the mean photon number going to one detector times the mean photon
number going to the other one.
© 2006 Nature Publishing Group 
 
5With P2 = wP
2
1
/2, and by neglecting higher-order terms, the ratio becomes:
r =
1
2
1 + w
1 + 2wP1
(12)
With P1 = 0.085 and w = 0.17, this expression gives then an expected value r = 57%, which is consistent with the
observed one.
IV. VISIBILITY AS A FUNCTION OF INTEGRATION WINDOWS
In Fig. 4 we show the results of the measurement of V for different integration windows around τ = 0. We see
that for an integration window around the center, from τ = −6 ns to τ = 6 ns, the visibility is 80± 10%, while the
integration using the whole window gives V = 77± 6%, indicating that the suppression is roughly uniform for all τ ,
which is also consistent with having close to transform-limited wavepackets for both fields 2L,2R.
0 10 20 30 40
0.6
0.7
0.8
0.9
1.0
 
 
V
τ
w
 (ns) 
FIG. 4: Visibility V of the two-photon coincidence suppression as a function of the integration window for the time difference
between the two field-2 detections. The integration is made from −τw to τw.
V. TIME WINDOWS
The electronic time windows used for field 1 and field 2 detections were 80 ns and 90 ns long, respectively, positioned
around the center of the respective wavepackets. In order to analyze the conditional field-2 wavepackets overlap in
Fig. 3 of Ref. [1], and also in the above Figs. 3 and 4, we introduced in the analysis an additional time window, only
44 ns long around the conditional field 2. As can be seen in Fig. 4 of Ref. [1], this corresponds to consider the whole
conditional field-2 wavepackets.
[1] Felinto, D. et al. manuscript submitted to Nature Physics (2006).
[2] Felinto, D., Chou, C.W., de Riedmatten, H., Polyakov, S.V. & Kimble, H.J. Control of decoherence in the generation of
photon pairs from atomic ensembles. Phys. Rev. A 72, 053809 (2005).
[3] Optical Coherence and Quantum Optics, Mandel, L. & Wolf, E. (Cambridge University Press, 1995).
[4] Chou, C.W., Polyakov, S.V., Kuzmich, A., & Kimble, H.J. Single-photon generation from stored excitation in an atomic
ensemble. Phys. Rev. Lett. 92, 213601 (2004).
[5] Laurat, J. et al. Efficient retrieval of a single excitation stored in an atomic ensemble. Opt. Express 14, 6912-6918 (2006).
© 2006 Nature Publishing Group 
 


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Conditional control of the quantum states of remote atomic memories for
  quantum networking

Conditional control of the quantum states of remote atomic memories for quantum networking

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