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Thesis
Rare-earth quantum memories for single photons and entanglement
USMANI, Imam
Abstract
The ability of storing and retrieving quantum states of light is an important experimental
challenge in quantum information science. A powerful quantum memory for light is required in
a quantum repeater, which would allow long distance (>500km) quantum communications. To
be implemented in such application, the quantum memory must allow on-demand readout,
with high fidelity and efficiency, and a long storage time. Additionally a multimode capacity (for
temporal or spatial modes) would allow multiplexing. Our approaches focus on rare-earth
doped crystals, i.e. solid state quantum memory. I present, in this work, our contributions for a
solid-state quantum storage, with good performances in every criteria. In particular, I present
the preservation of quantum entanglement during the storage, which paves the way for the
implementation of quantum memories in quantum repeaters.
Reference
USMANI, Imam. Rare-earth quantum memories for single photons and entanglement.
Thèse de doctorat : Univ. Genève, 2013, no. Sc. 4544
URN : urn:nbn:ch:unige-276020
DOI : 10.13097/archive-ouverte/unige:27602
Available at:
http://archive-ouverte.unige.ch/unige:27602
Disclaimer: layout of this document may differ from the published version.
UNIVERSITÉ DE GENÈVE
Groupe de Physique Appliquée - Optique
FACULTÉ DES SCIENCES
Prof. N. Gisin
Rare-earth quantum memories for single photons and
entanglement
THÈSE
présentée à la Faculté des Sciences de l'Université de Genève
pour obtenir le grade de Docteur ès Sciences, mention Physique
par
Imam Azim Usmani
du Grand-Saconnex (GE)
Thèse N◦ 4544
GENÈVE
2013
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Lo thèse doit porter lo déclorotion précédenfe et remplir les conditions énumérées
dons les "lnformotionsrelotives oux thèses de doctorot à I' U ni ver si té de Ge nè ve".
Résumé de la Thèse
L'information quantique a, depuis environ une vingtaine d'années, apporté une
vision supplémentaire à la mécanique quantique et permis d'en entrevoir des applications très prometteuses. Elle se base sur des concepts nouveaux par rapport à la mécanique classique, comme la superposition d'états quantiques ou l'intrication. Parmi
les possibles champs d'applications, on retrouve les ordinateurs quantiques, qui permettraient de résoudre certains problème beaucoup plus rapidement qu'un ordinateur
classique, ainsi que les communications quantiques. Celles-ci incluent par exemple la
cryptographie quantique qui illustre très bien comment on peut proter des particularités de la mécanique quantique pour une application concrète. En eet, le secret
d'une clé d'encryption est garanti par les lois de la physique quantique. En général,
l'information est encodée sous forme de qubits. Ceux-ci, à l'instar d'un bit, peuvent
prendre les valeurs 0 ou 1, mais aussi toute superposition de ces deux valeurs. La valeur
d'un qubit peut ainsi être représentée par une coordonnée sur une sphère de rayon unité
et contient donc plus d'information qu'un bit classique. Pour des réalisations expérimentales, il faut pouvoir générer des états quantiques, mais aussi être capable de les
modier, les mesurer et les transporter. Les photons sont idéalement adaptés pour
encoder des qubits et peuvent en particulier être aisément transportés par bre optique. Toutefois, il peut être nécessaire de les stocker pour un temps donné, c'est à dire
de transférer leur état quantique (de manière réversible) dans un système stationnaire
comme des atomes. Ceci requière comme outil une mémoire quantique pour les photons. Il faut remarquer que, contrairement à une mémoire classique, un état quantique
stocké reste inconnu. En eet, la mesure d'un système physique modierait irrémédiablement son état quantique, et il ne serait de toute façon pas possible de connaître
de manière déterministe son état. Une mémoire quantique doit évidemment préserver
l'état d'un photon, et en particulier maintenir l'intrication éventuel qu'il aurait avec
un autre photon. La réalisation d'une mémoire quantique performante promettrait de
nouvelles applications, comme par exemple les répéteurs quantiques. Ceux-ci amèneraient la possibilité d'étendre la cryptographie quantique sur de longues distances ou
de créer des réseaux quantiques. Les mémoires, placées dans les n÷uds du réseaux
auraient un rôle de synchronisation entre les diérents liens.
La recherche pour la réalisation d'une mémoire quantique est un domaine très actif
iii
et demande de trouver un système stationnaire fortement couplé à la lumière et capable
de maintenir susamment longtemps un état quantique. Dans le groupe de physique
appliquée (GAP) à Genève, la recherche se concentre sur les cristaux dopés aux terres
rares, refroidis à des température cryogénique (≈3 K). Ceci permet d'utiliser un grand
nombre d'atomes qui sont piégés naturellement par le cristal. Ces systèmes on été
employés avec succès dans le domaine des lasers, mais la recherche pour le stockage
d'état quantique est beaucoup plus récente.
Mon travail, dans le cadre de cette thèse, a été de chercher à réaliser une mémoire
quantique pour photons, qui pourrait être utilisée dans des future répéteurs quantiques.
Tout d'abord, nous avons eectué des mesures spectroscopiques dans un candidat potentiel pour une mémoire quantique, un cristal dopé au néodyme : Nd3+:Y2SiO5. Cela
a permis de trouver une conguration pour implémenter un protocole de mémoire quantique, le peigne en fréquence atomique. Nous avons alors réalisé diverses expériences
pour montrer le potentiel d'une mémoire quantique dans un cristal dopé au terres
rares. D'une part, nous avons cherché à maximiser l'ecacité de stockage dans le
Nd3+:Y2SiO5, ainsi que dans un autre cristal (Eu3+:Y2SiO5) à l'aide d'une cavité optique. Puis, nous avons démontré le stockage de plusieurs qubits dans un seul ensemble
d'atomes, et montré une capacité jusqu'à 64 modes temporels de la mémoire. Aussi,
nous avons accompli le stockage de lumière dans une onde de spin, ce qui a permis
d'allonger signicativement le temps de stockage. A chaque fois, nous avons cherché
à réaliser ces mesures avec des états cohérents avec, en moyenne, environ un photon
par impulsion, pour démontrer le caractère quantique du stockage. Cependant, nous
avons pu démontrer des tests plus forts, en stockant des vrai photons uniques, pour la
première fois dans une mémoire à état solide. Ceci a permis de montrer que le stockage préservait également l'intrication d'un photon. Nous avons également pu réaliser
la création annoncée d'intrication entre deux mémoires diérentes, séparé par 1.3 cm.
Ces expériences ont contribué à montrer qu'il est possible de réaliser une mémoire
dans les cristaux dopés aux terres rares, en atteignant tous les critères possibles pour
un répéteur quantique. Dans le futur, il faudra trouver un système qui combinera, au
moins, toutes les performances atteintes jusqu'ici dans les diérentes expériences.
iv
Contents
Résumé de la Thèse
1 Introduction
2 Theory
iii
1
5
2.1 Medium for quantum storage . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Quantum storage protocol . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Spectral hole burning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 Realization of a quantum memory
3.1
3.2
3.3
3.4
3.5
Spectroscopy of a rare earth doped crystal
Storage eciency . . . . . . . . . . . . . .
Multimode quantum storage . . . . . . . .
On-demand storage . . . . . . . . . . . . .
Testing a QM with single photons . . . . .
4 Discussions and Outlook
Bibliography
Acknowledgements
A Additional spectroscopy
List of Publications
B Published articles
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31
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35
39
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57
59
65
67
Demonstration of atomic frequency comb memory for light with spin-wave
storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Towards an ecient atomic frequency comb quantum memory . . . . . . . . 72
Mapping multiple photonic qubits into and out of one solid-state atomic ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
v
Quantum storage of photonic entanglement in a crystal . . . . . . . . . .
Heralded quantum entanglement between two crystals . . . . . . . . . . .
Atomic frequency comb memory with spin-wave storage in Eu3+:Y2SiO5
Single-photon-level optical storage in solid-state spin-wave memory . . .
vi
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92
101
114
122
Chapter 1
Introduction
A memory for classical information is a very old and basic concept. It takes a large
variety of forms, from a simple piece of paper to a hard drive. Nowadays, it is possible
to massively produce devices with impressively large capacities and fast access times.
Improving their performances or nding new systems is still an active eld, but the
concept of a classical memory remains the same. However, quantum mechanics has
changed our fundamental description of physics. It introduces new concepts such as
a quantum state, quantum superposition or entanglement. Based on that, fascinating
applications have been developed. For example, quantum cryptography [1] uses quantum key distribution (QKD) where security is guaranteed through the laws of quantum
physics. To progress experimentally though, the eld of quantum information needs
also new tools, such as a quantum memory (QM). Such a device must have the ability to store and faithfully release a quantum state. In particular, one would need an
on-demand readout, that is to say the output can be released whenever it is needed.
Quantum storage must face some intrinsic concepts of quantum mechanics which differentiates it from a classical memory. For example, we know a measure of a quantum
system will aect its quantum state in an irreversible way. Because of this, an initially
unknown state cannot be determined with 100% probability and, more generally, only
limited amount of information can be extracted from a nite quantum ensemble [2].
This particularity is actually used against a potential eavesdropper in QKD. However,
because of this, a quantum memory cannot be based on a measure and write-down
strategy.
Here, I focus on optical quantum memories [3]. Photons are indeed ideal carriers
of a quantum states and can travel through large distances. The diculty, though,
is to store them in a stationary device for a certain amount of time. A quantum
memory would be an elementary tool in many quantum optics experiments, as are
today optical bres, single photon detectors or photon sources. Moreover, there are
several potential applications in quantum information as I will describe here. In linear
optics quantum computation (LOQC) [4, 5], a QM is a unitary gate, which has the
1
role of synchronization between dierent computing channels. Also quantum memory
could be a useful component of a photon source. For example, using spontaneous
parametric down conversion (SPDC) one can herald the presence of a single photon
[6, 7]. It is however emitted at random time which makes it unsuitable for some
applications. This heralded photon could be stored in a QM and released when needed,
thus realizing an on-demand source of single photons[8]. Our main interest, though,
is the implementation of quantum memories in quantum communications (QC) [9]. I
mentioned for example quantum key distribution. Since the original idea of Benett
and Brassard [10], it has been realized experimentally, in real-eld conditions and
even commercially systems are available. While in 1992 the rst experimental proof
was demonstrated over a short link of 30 cm[11], the distance has been impressively
increased. Using polarization based QKD in free space a distance of 144 km has been
reached [12], while time-bin qubits have been used for QKD over 250 km in optical
bres [13]. One should note that a protocol can be secure even if it is not based on true
single photons. In particular, an important alternative approach based on continuous
variables uses coherent and squeezed states [14, 15].
The distance of communication is however limited to few hundreds of km. Indeed,
the signal sent through an optical bre undergoes losses increasing exponentially with
the communication distance. For example, even with ultra-low loss optical bres (-0.16
dB/km), a photon has a probability of 10−16 to travel through 1000 km. In classical
communications, this problem is solved with ampliers. It is however not a solution
in QC, since an arbitrary quantum state cannot be perfectly copied deterministically
(non-cloning theorem [16]). Fortunately, the idea of a quantum repeater was proposed
[17, 18, 19] to distribute entanglement over large distances. The principle is to split
a long communication distance into shorter links (g.1.1). Using entangled photon
pair sources, one attempts to share entanglement in each link. It is a probabilistic
process, and the losses in a link must be low enough, so that this probability is reasonable. Because this does not happen necessarily at the same time in all links, this
entanglement must be maintained in quantum memories placed at each border of a
link to allow synchronization. Afterwards, Bell measurements [20, 21] in intermediate
stations will allow to swap the entanglement between QMs separated by the complete
communication distance. In this context, we can consider more generally a quantum
network [22] where nodes generate, process and store quantum information, while photons transport quantum states from site to site and distribute entanglement over the
entire network. Additionally, i would like to point out that a quantum repeater could
even be used in an optical interferometric telescope which has the potential to image
extra-solar planets [23].
Now, to test if a QM is suitable for those dierent applications, I will use a variety
of criteria [3] that I summarize here. First, the most important for a quantum memory
is to re-emit the most faithfully a quantum that was stored. One can calculate the
delity of the output to the input state and, ultimately, it should be above a threshold
2
a) Entanglement creation
QM
QM
QM
A
B
C
QM ... QM
D
...
W
QM
QM
QM
X
Y
Z
b) First entanglement swapping
QM
A
QM ... QM
QM
... W
Z
D
c) Last entanglement swapping
QM
QM
A
Z
Figure 1.1: Quantum repeater scheme.
for possible error correction, for any given input state. In practice, it very convenient
to dene some more specic criteria, such as the eciency η which is the ratio between
the energies of the output and input states. Ideally, it should be the closest to one.
However, some classical optical storage schemes can easily reach 100% (and even more
due to light amplication processes) even though they are not suitable for a quantum
memory [24, 25]. Therefore, it is necessary to complete this criterion with a measure
of delity, which can be conditioned on the re-emission of an output. Indeed, when
working with single photon detectors, it is possible to remove the vacuum component
of the state by post-selecting on the detections. This is not the case with homodyne
and heterodyne measurements used in continuous variable techniques, since there is
always a detection. Also, a measure of the noise level helps to determine if a QM
can potentially store true single photons with a good signal-to-noise ratio. Indeed, if
there is no input, the probability that the QM emits a noise photon must be close to
zero. Additionally, a quantum memory should ideally have the capacity of storing many
dierent qubits at the same time. This multimode capacity is quantied by the number
of modes it can accept and depends strongly of the storage scheme [26]. The storage
time is another criterion and it should be long enough to perform a particular task.
While classical memories have no real limit and can always be copied if they undergo
physical damage, a quantum memory must usually face decoherence that grows with
time. Finally, we note that an optical quantum memory usually works for a specic
wavelength and frequency bandwidth. This must be taken into account for potential
applications. For example, a quantum memory working at telecom wavelengths would
be a clear advantage in quantum communications [27].
Research in quantum storage is an active eld and many realizations have already
been made in various systems. Compact overviews can be found in ref.[3, 28, 29, 18].
Nevertheless, no QMs have yet been implemented in a real application. Our longterm motivation is therefore to realize a QM so that it will full all the requirements
to achieve this, in particular to implement it in a future quantum repeater. The
3
realization of a QM requires a stationary medium that interacts strongly with light and
is capable of maintaining a quantum state, for example an atomic ensemble. Secondly,
we need to use a quantum memory protocol so that the user can map a quantum
state into the medium and release it when it is needed. This can be achieved using
electro-magnetically induced transparency (EIT), photon echoes or Raman transitions.
Finally, a QM needs to be characterized. This can be done by using some criteria, or
by storing some quantum states and measuring the delity of the storage. In the
next chapter, I will discuss the choice of a suitable medium for quantum storage and,
in particular, the properties of rare earth doped crystals which are used in this thesis.
Also, I will describe quantum storage protocols, specically the atomic frequency comb
(AFC) protocol based on photon echoes. In chapter 3, I will discuss the experimental
realization of a QM. My work in this thesis was guided by the various performance
criteria a QM needs for the implementation in a quantum repeater. In addition, we
were able to test our QMs with the storage of true single photons and a measure of
entanglement between two QMs.
4
Chapter 2
Theory
2.1 Medium for quantum storage
We discuss now the system in which a QM can be implemented. The question
of nding the ideal system is still open, as it is the case for example in quantum
computing. For QC applications we would like to work with light at optical or nearinfrared wavelength, say from 400 to 1600 nm. One can use individual systems such as
trapped ions [30] or atoms in high nesse cavities [31]. Another approach is the use of
atomic ensembles [28] featuring a strong light matter interaction due to a large optical
thickness [28]. Quantum storage was rst demonstrated in vapours of rubidium or
cesium[32, 33, 34]. A usual diculty in such system is the decoherence due to atomic
collisions. To overcome this, one solution is to cool and trap the atoms, using magnetooptical trap (MOT) [34] or optical lattices [35]. The drawback is a higher complexity
of the experiment.
Rare earth doped crystals
Promising alternatives are solid-state atomic ensembles. Specically, rare earth
ion doped crystals [29, 36] have already been widely studied in the context of laser
applications and storage of strong laser pulses. The ions are impurities with a doping
level from 10 to 1000 parts-per-millions (ppm) and since they are naturally trapped in
the host crystal, the ensemble is sometimes described as a frozen gas. Because of this
natural trapping, the crystal needs only to be cooled down with commercial cryostat
and the setup is not in principle that complex. The crystals are often the same as the
ones used in lasers, only the doping concentration is usually smaller. They are of good
optical quality and it is useful to cut it along an optical axis, so that the light does
not necessarily undergo birefringence. Rare earth elements have the particularity that
for most of them, the 4f electronic shell is incomplete and optical transitions occur in
it (see g. 2.1). We note that for free ions, such transitions would be forbidden by
5
selection rules, but they become here weakly allowed because the crystal eld changes
the wave functions of the electrons. The 4f shell is closer to the nucleus than some
(complete) outer shell (5s, 5p and 6s) which induces a screening. This isolates the
4f shell from the environment and it leads to very long coherence time. Indeed, the
optical coherence time is usually in the 1 µs to 1 ms regime below 4 K. We note that
the level structure will depend on whether it is a Kramers ion (odd number of electron
in 4f) or a non-Kramers ion (even number of electrons). Note that we here consider
trivalent rare earth ions (RE3+). For a Kramers ion, the ground state is a doublet whose
degeneracy can be lifted with an external magnetic eld. These Zeeman states can be
easily separated by tenths of GHz, and are used in quantum storage protocols. For
a non-Kramers ions, though, the angular momentum degeneracy is completely lifted
by the crystal eld. Therefore, if we need more than one ground state, we can use
ions with a nuclear spin, which will induce a hyperne structure. These electronic and
nuclear spin levels also have impressively long coherence times at cryogenic conditions,
from 1 ms to 1 s. Another point is that environment variations in the crystal induce an
inhomogeneous broadening Γinh in the optical transition, typically between 100 MHz
and 10 GHz. We note that this is much broader than the homogeneous linewidth
(γh =1kHz-1MHz) (g. 2.2). As a consequence, the dierent resonance transitions of
an ion may not be distinguished in a broadened spectrum and this leads to dierent
classes of atoms. However, this may help for the realization of a QM with a large
bandwidth and, because of a high Γγ ratio, a high multimode capacity. This is the
basis of the atomic frequency comb memory.
inh
h
2.2 Quantum storage protocol
Atomic frequency comb
To realize a quantum memory, one needs to apply a protocol that allows the
reversible mapping of light. For example, using electro-magnetically induced transparency (EIT) [37] one can slow and stop light for a certain time. Alternatively, in a
Duan-Lukin-Cirac-Zoller (DLCZ) scheme [17], through an optical excitation and the
detection of a Stokes photon, the creation of a single excitation in an atomic ensemble
is heralded. This single excitation can be released, on-demand with a π pulse, through
the emission of an anti-Stokes photon. This type of quantum memory, sometimes described as a photon pistol can be used to herald entanglement between remote atomic
ensembles [17]. The two schemes have been combined in a single experiment, both in a
warm vapour [32] or in a cold gas [34]. However, in rare earth doped crystals, the large
inhomogeneous broadening induces a fast dephasing, much faster than the coherence
time, which suppresses all coherent emissions of the atoms. Fortunately, photon echo
protocols, which have already been used for optical data storage, allow to compensate
6
Figure 2.1: An example of energy levels of rare earth elements.
7
Figure 2.2: Illustration of a inhomogeneous broadening in a rare earth doped crystal. Γinh is
typically between 100 MHz and 10 GHz, which is much larger than the homogeneous linewidth
(γh =1kHz-1MHz)
the inhomogeneous broadening. We illustrate here the principle of such protocol with
the state of the atomic ensemble after the absorption of a photon:
|Ψi =
N
X
j=1
cj eiδj t e−ikzj |g1 . . . ej . . . gN i
(2.1)
N is the number of atoms in the ensemble, cj a factor that depends on the frequency
and position of the atom j, δj its frequency detuning and zj its spatial position while k
is the wave number of the light eld. This state represents one excitation delocalized
among the N atoms. As it can be predicted from Maxwell equations, the polarization of
the atomic ensemble induces the emission of an electric eld. However, because of the
inhomogeneous broadening, δj has a large spread which will lead to a fast dephasing.
Therefore, after a short time (about the input pulse duration) the emission is incoherent, that is to say its intensity is proportional to N, and in all directions. The principle
of any photon echo protocol is to act such as the term e−iδ t is the same for all j at a
particular time. This leads to constructive interference, thus coherent emission. It will
take form of an emitted pulse (a photon echo), with intensity proportional to N2 and
in a specic direction determined by the spatial phase imprinted in the atoms e−ikz .
Because N is very large, the incoherent emission is completely negligible compared to
the photon echo. Some protocols are dened as classical, such as a two-pulse photon
echo (2PE). Because of intrinsic noise created by the optical π pulse, it can never work
as quantum memory for single photons[24], even though it can reach eciencies of
100% (or more). Protocols for quantum storage have been proposed in the last ten
years, such as controlled and reversible inhomogeneous broadening (CRIB) [38, 39, 40]
or atomic frequency combs (AFC) [41]. Our work here focuses on this last. It has the
advantage, as we will see later, of being highly multimode. Also, compared to CRIB,
it is not necessary to apply an external electric eld gradient. The principle of it, is
to tailor a periodic function in the inhomogeneous absorption prole (g. 2.3a). More
j
j
8
(a)
Atomic density
lds
l fie
Output
mode
tro
Con
Input m
ode
e
s
D
g
g
aux
Atomic detuning d
Intensity
(b)
Input
mode
Control fields
2p / D - T0
Ts
Output
mode
Time
T0
Figure 2.3: (a) schematic level structure of the atomic ensemble, with a periodic absorption
prole for AFC storage (b) An input is absorbed by the comb, transferred into a spin state
for a time Ts , and re-emitted when the atoms are in resonance.
precisely, it consists in a series of peaks of specic shapes separated by a detuning 2π∆,
which composes the atomic frequency comb. This periodicity induces that at the time
t=1/∆ after absorption of the wave packet, all atoms are in phase which forces the
emission of the echo (g. 2.3b). This intuitive explanation is conrmed in a detailed
analysis [41] and it is shown that the protocol works for any input state that is much
weaker than a π pulse. We note that from the point of view of Bonarota et al.[42], AFC
is more similar to EIT because it is based on a dispersion (caused by the variations in
the absorption prole), while CRIB is an absorbing storage protocol (the absorption
prole is at).
It is very useful to simulate or calculate numerically the eect of an AFC for various
absorption prole that cannot be solved analytically. For this purpose, one can use a
Maxwell-Bloch simulator or, alternatively, a general formula has been derived [43] which
assumes only that the susceptibility function is periodic. Here, we used a numerical
method, working for any absorption prole, that we describe here. The linear part of
the susceptibility χ for an ensemble of two-level atoms can be written as a perturbation
solution[44]:
N
X
cj
(2.2)
χ(ω) ∝
(ω − ω) − iγ
j
j=1
h
ω is the angular frequency of the light eld, ωj is the resonant frequency of atom j and
γh is the homogeneous linewidth. Hence, the complex wave number can be calculated
9
and depends strongly on ω:
p
k(ω) = ω/c 1 + χ(ω)
The absorption coecient is given by:
α = 2k 00
where k00 is the imaginary part of k. If necessary, we can adjust the amplitude in Eq.2.2
to obtain the desired absorption coecient. Now, we suppose the input eld can be
written as:
Z
Ein (t) =
Ẽ(ω)e−iωt dω
Through the medium of length L, each components propagates and acquire a phase
k(ω)L. Hence the output eld can be calculated:
Eout (t) =
Z
Ẽ(ω)ei(k(ω)L−ωt) dω
For various atomic distributions, we calculated numerically Eout and the dierent resulting parameters, such as the storage eciency, which were in good accordance with
Maxwell-Bloch simulations and analytical results. Moreover, we were able to study
some subtle eects arising, for example when the AFC is not innite.
We described here a two-level AFC protocol, for which the storage time (1/∆)must
be chosen in advance, during the AFC preparation. Moreover, it is limited by the
coherence time of the optical transition. In the goal of achieving a quantum memory
with on-demand readout and long storage time, we must use a 3-level AFC scheme [41]
which includes the transfer to an additional ground level |si (a spin state for example).
After an input has been absorbed, we apply a π pulse (control eld) on the |si − |ei
transition which moves the coherence to the |gi−|si transition (see g. 2.3). Assuming
there is no spin inhomogeneous broadening, the phase evolution in Eq.2.1 will stop.
After a time Ts, we apply a second π pulse, which moves back the coherence to the
|gi−|ei transition and the phases evolve again. This leads to the re-emission of an echo
at the time Ts + 1/∆ (g.2.3b). Therefore, a complete AFC protocol would allow to
realize an on-demand quantum memory in an inhomogeneously broadened ensemble.
Recently, a DLCZ scheme using an AFC in an inhomogeneous ensemble has been
proposed [45]. This would allow to implement a photon pairs source in the medium.
It requires the same techniques as the implementation for a complete AFC scheme.
Therefore the path to achieve both protocols are the same.
2.3 Spectral hole burning
We briey describe the process of spectral hole burning [46][47], which can be
used to tailor a specic structure in the absorption prole, or to do spectroscopic
10
(a)
1
2
Absorption
Class:
(b)
Angular frequency
Figure 2.4: Spectral hole burning in a three-level system.
measurements in a material. We rst consider a probe eld propagating in a medium
of two-level atoms. If the light is in resonance with the atoms, the light intensity
decreases exponentially with the length L of the medium:
I(L) = e−αL I(0)
is the intensity of the light at position l and α is the absorption coecient. We
often use the optical depth d = αL in the context of quantum storage. The absorption
coecient can easily be measured by the use of a probe eld transmitted through the
medium. The important point here is that it depends on the population in the ground
(Ng ) and excited state (Ne):
I(l)
α = (Ng − Ne ) σ.
where σ is the cross section. The population can be changed via optical pumping which
leads to a decrease of absorption (a spectral hole ) or an increase of it (a spectral antihole ). To illustrate this process, we consider a simple case of an atomic ensemble with
two ground states and one excited state (g.2.4a). The inhomogeneous broadening is
larger than the split ∆Eg in the ground state. The temperature makes that the two
ground states are equally populated at equilibrium. We consider the relaxation time
of the ground states (Tz ) is longer than the excited state lifetime (T1). A pumping
beam at an angular frequency ωpump is shined into the medium and, because of the
inhomogeneous broadening, it is resonant with 2 dierent types of atoms (g.2.4a).
After the optical pumping, the population in the ground states of the 2 classes of
atoms has changed, which modies the absorption spectrum (g.2.4b) by creating
spectral holes and anti-holes.
11
Many spectroscopic measurements can be done using this technique. The decay of
a spectral hole after the pumping process allows to determinate the relaxation time
of the ground states TZ . The positions of the holes and anti-holes are functions of
the energy splits, thus one can measure the g tensor by applying various magnetic
elds. Alternatively, with a more elaborate pumping beam, it is possible to tailor
any structure in the absorption prole, such as an AFC. A crucial parameter is the
eciency of the optical pumping, that is to say what fraction of atoms will be moved
from one ground state to another (the degree of spin polarization). This may indeed
eect the AFC storage eciency. A simple rate equation model of a 3-level system
allows to determine the ratio between the populations in the ground states [48]. In a
steady state, after a long optical pumping, it depends of the ratio between TZ and T1:
TZ
ρ2
=1+2 ,
ρ1
T1
where ρ1(ρ2) is the population fraction in the initial (nal) ground state. For this reason, we need the ground state relaxation time to be long compared to the excited state
lifetime. We note that we have not taken into account the branching ratio between
the dierent transitions. In the case the dierent transitions have similar strengths,
the formula is a good approximation. Another important feature is the ability to burn
narrow holes, since it determines the duration of an AFC storage in the optical transition. While, ultimately, it is limited by the homogeneous linewidth of the atoms[49],
other processes may broaden it. First, the laser linewidth must be narrow enough to
attain this limit. Also, uctuations of the resonance frequency of an atom during the
hole burning, that is to say spectral diusion, will broaden a spectral hole. Finally, to
achieve ecient optical pumping, the pumping beam must have a high intensity and
long duration, which induces power broadening[49].
We conclude by noting that, in many rare earth doped crystals, the level structure
is richer than in our simple model. There may be a lot of dierent transitions possible,
which leads to more complicate hole burning spectra but the principles introduced here
remain the same.
12
Chapter 3
Realization of a quantum memory
The realization of an optical quantum memory is a dicult and challenging task.
We want not only to realize a proof-of-principle of quantum storage, we would also like
to realize a quantum memory whose performances would allow to use it some day in a
quantum repeater [17, 18, 19]. I will present here the dierent aspects on which I worked
for such a goal. First, one needs to nd a suitable system to implement a quantum
memory. In Geneva, our research focuses on rare earth doped crystals. Even though
we believe strongly that these are promising systems [29], we need to nd the most
suitable medium for our interests. I will rst present our spectroscopic measurements
in a neodymium doped ortho-silicate crystal, Nd3+:Y2SiO5 with the goal of using it
for quantum storage. In the following sections, I will discuss the various criteria of a
QM: the storage eciency, multimode capacity and storage time. Finally, I will present
tests of QMs: storage of single photons and heralded entanglement between two QMs.
3.1 Spectroscopy of a rare earth doped crystal
Introduction
Motivation
We believe rare earth doped crystals have a high potential for the realization of a
quantum memory [29, 36]. Of course, some of them are more promising than others,
because of their dierent properties. They have been widely studied in the context of
laser applications and optical data storage, but research for quantum storage is more
recent. Therefore, spectroscopy of rare earth doped crystals is an active eld, with the
goal of nding the ideal system with the best properties for a quantum memory. It is
important to understand that it is not possible to build ourselves the ideal system by
xing the dierent parameters, such as the resonant frequency of the atoms or their
coherence time. In fact, these are xed by the laws of physics - or Nature. What we
13
can do, is to chose the host crystal and the dopant (with a particular concentration).
Then, we can play with just a few settings of the environment such as magnetic eld
or temperature (the lowest possible in principle). This doesn't mean that doing spectroscopy consists of testing randomly a large number of materials - though it can lead
to new discoveries. One should understand and model the dierent processes and interactions in these systems. With the help of dierent spectroscopic studies published
in the past, it should be possible to nd potential candidates for the realization of a
QM and study them.
Requirements
We specify now the required properties of a medium for the implementation of a
quantum memory, in particular for an AFC protocol. As depicted in g.2.3 a resonant
transition at an optical frequency is necessary to absorb an input eld. We want for
this transition a high oscillator strength (for a strong light-matter coupling) and long
coherence time. To tailor an AFC in the absorption prole, an auxiliary state |auxi
is needed to transfer some atoms through spectral hole burning. The lifetime of this
auxiliary state must be long (compared to the excited state lifetime) so that this process
is ecient. Also, an ecient AFC has a narrow structure. Hence, it is necessary to
burn very narrow spectral holes, which depends on the homogeneous linewidth and
other processes such as spectral diusion. Finally an additional ground state, usually
a spin state |si is required to include a spin-wave storage and on-demand readout. For
a long storage time, |si should have a long coherence time with respect to |gi.
Choice of Nd :Y SiO
3+
2
5
We are interested here in nding a rare earth doped crystal for a 2-level AFC protocol that would be highly multimode and ecient enough to store true single photons.
Kramers ions are suited for this, since a magnetic eld allows to use exactly two Zeeman states as |gi and |auxi. At the time this thesis was started (2008), Er3+:Y2SiO5
was a very promising material, since very long optical coherence times and low spectral
diusions were measured at low temperature and high magnetic eld [50]. However,
because of a bad branching ratio and short Zeeman state lifetime (compared to the
excited state lifetime) [51], the optical pumping was not ecient [48]. We note a realization of a CRIB protocol was nevertheless demonstrated [52], for the rst time at
the single photon level and at a telecom wavelength. The rst mapping of light at
the single photon level in a solid-state (independently of the protocol) was achieved in
2008 using Nd3+:YVO4 [53]. An advantage of this material was a high absorption of
α = 40 cm−1 . However, the storage time and the eciency were quite low. These limitations were due to inecient optical pumping and strong superhyperne interactions
between Neodymium and Vanadium ions which aected the quality of the AFC prole.
14
This interaction was also observed and problematic in Er3+:LiNbO3. With the knowledge of these previous results, we chose to investigate Nd3+:Y2SiO5. Indeed, there is
no abundant elements with a strong nuclear magnetic moment (such as niobium or
vanadium) in the host crystal that could cause a strong superhyperne interactions.
Also, the optical pumping is potentially very ecient in neodymium compared to erbium because of the short excited state lifetime (≈ 300µs for Nd, ≈10 ms for Er). It
is however necessary to nd a magnetic eld conguration in which the Zeeman state
lifetime is long compared to that.
Setup
We did a series of measurement in Nd3+:Y2SiO5crystals, cooled to 3 K, to nd a
conguration suited for an ecient AFC. We measured the absorption prole and its
inhomogeneous broadening, the optical coherence time, the Zeeman state lifetime and
the minimal width of a spectral hole. In appendix, we present a partial measure of
the g tensor in the ground and excited state. The samples had a doping concentration
of 35 ppm and were of various lengths, from 1 to 10 mm. The light was propagating
along the crystallographic axis b and its polarization was in the plane dened by D1
and D2 which are the eigen axes of the index of refraction. Magnetic elds have also
been applied in the D1-D2 plane with dierent intensities and directions. At rst, we
used permanent magnets outside the cryostat. With such a conguration, the direction
of the magnetic eld could easily be varied by changing the magnets position, but the
intensity of the eld was limited to 30 mT. To increase this, we placed permanent
magnets inside the cryostat, which allowed to apply a eld of 300 mT. However, we
could test only a few directions, since it was more complicated to change the position
of the magnets. Finally, by using another cryostat, a superconducting magnet was
available, which could generate a variable eld up to 2 T. For technical reasons, this
cryostat was not used for AFC storage, but only for spectroscopy measurements.
Absorption and inhomogeneous broadening of the optical transition
As expected [54], Nd3+:Y2SiO5has a resonance at 883.2350(6) nm corresponding to
the 4I -4F transition of the site 1 in Y2SiO5. With a probe eld, we could measure the
absorption coecient as a function of laser frequency and observe an inhomogeneous
broadening (g.3.1). It has symmetric distribution, close to a Gaussian with a full
width at half maximum (FWHM) of 4.6 GHz. The absorption is maximal when the
polarization is parallel to D1 (α = 3.43 cm−1) and minimal when its parallel to D2
(α = 1.32 cm−1). From additional measurements[55], it appears that the eigen axes of
refraction index coincide with the eigen axes of absorption.
9
2
3
2
15
Inhomogeneous Broadening
α [cm-1]
3
polarization along D1
polarization along D2
2
1
0
-7
-6
-5
-4
-3
-2
-1
0
1
Detuning [GHz]
2
3
4
5
6
7
Figure 3.1: Absorption spectrum of Nd3+:Y2SiO5. The FWHM of the inhomogeneous broad-
ening is 4.6 GHz. Also, the eigen axes of absorption coincide with the eigen axes of refraction
index D1 and D2 . The absorption coecient is α = 3.43 cm−1 and α = 1.32 cm−1 , along D1
and D2 respectively.
The absorption coecient will determine the storage eciency through the optical
depth (d = αL). For example, with a 2 cm long crystal, the eciency should reach
41% for an optimized AFC storage [56] in the forward direction. However, in some
cases the absorption can be reduced, if for instance, we apply a strong magnetic eld
or if we want to use both polarizations. This could be overcome by increasing the
doping concentration, multiple passes in the crystal or using an impedance-matched
cavity [57].
Zeeman state lifetime
Because Nd3+ is a Kramers ion, each level has a remaining degeneracy that can
be lifted with an external magnetic eld (a partial study of the g tensor can be found
in the appendix). In particular, this creates two Zeeman states (Ms = ±1/2) in the
ground level. They can be used as |gi and |auxi (2.3) for a two-level AFC protocol. For
an ecient optical pumping, thus ecient AFC storage, the Zeeman state lifetime (TZ )
must be long compared to the excited state lifetime (T1) as we have seen in section 2.3.
From ref.[54] and our own measurements, we know the excited state lifetime is about
300 µs. Therefore, we need TZ to be at least tenths of ms. We will present in this section
measures of TZ for various magnetic elds. Although the population relaxation occurs
in radio frequencies, we can measure TZ with an optical laser, through the dynamic of
16
14
12
TZ [ms]
10
8
6
4
2
0
-90 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 90
Angle of B with D2
Figure 3.2: The Zeeman lifetime has been measured for dierent orientations of the magnetic
eld (B≈ 20 mT).
a spectral hole. With an ecient optical pumping, a majority of atoms are transferred
from an initial to a nal ground state, which creates a spectral hole at the frequency
of the pump (g.2.4). However, because of population relaxation, the spectral hole is
subject to an exponential decay at a rate 1/TZ . Therefore, we measured the dynamic of
a spectral hole after the optical pumping, and the obtained decay time is TZ . We note
that if the optical pumping is inecient (as in Er3+:Y2SiO5for example), the excited
state is still populated at the end of the hole burning. Because of this, the spectral
hole dynamic would include a decay at a rate 1/T1.
To our knowledge, there are no previous measurements of TZ in Nd3+:Y2SiO5. We
started with a moderate magnetic eld (≈ 20 mT) by the use of permanent magnets
placed outside the cryostat. We measured TZ for various angles θ between the D2 axis
and the magnetic eld. We note that θ and θ+180are equivalent, since it corresponds
to an inversion of the magnetic eld. There is however an ambiguity on θ. Indeed,
although the D1 and D2 axes can be well identied, their directions are unknown
without a crystallographic study of the sample. For this reason, an angle of +θ could
be taken as −θ in another experiment. We observed a Zeeman state lifetime between
5 and 14 ms (g.3.2). The longest lifetime arises for an angle of about −40, while at
an angle of 90the spectral hole was to small to measure its dynamic. Interestingly,
D1 and D2 do not appear to be axes of symmetry as is the case for the g tensor (g.
A.4 in appendix). We must say we don't have a good explanation for this angular
dependence of TZ .
To search for longer Zeeman lifetimes, we increased the magnetic eld (up to 2 T)
17
for two particular angles, θ = −30and θ=90(g.3.3a). For both angles, TZ starts
to increase with the magnetic eld, reaches a maximum, and nishes to decrease. In
particular, we obtained a Zeeman lifetime of 150 ms for a magnetic eld of 370 mT at
an angle of -30with the D2 axis. Comparing the two angles, we note the maximum
of TZ does not arise for the same magnetic eld. However, this may be explained by
the fact that the g factor of the ground state is dierent for both angles. Actually,
while the optimal magnetic elds are dierent for the two angles, the ground state
splits are almost the same (≈ 12GHz) when TZ reaches a maximum (g.3.3b). This is
probably not a coincidence, but we are yet far of having a full theoretical understanding
of it. Many processes have been identied to explain the spin relaxation rate [50]. The
one-phonon direct process, involving the absorption or emission of a resonant phonon,
increases with the magnetic eld. The two-phonon Raman and two-phonon resonant
Orbach processes depend on the temperature but not on the magnetic eld. A process
that should decrease with the magnetic eld is the ip-op rate (exchange of spin with
a neighbour ion). At high magnetic eld, we may neglect the ip-op, and the spin
relaxation rate (R = 1/TZ ) can be written as[50]:
5
R(∆E) = αD · (∆E) coth
∆E
2kB T
gµB B
2kB T
+ αRO (T ),
,
(3.1)
where αD is an anisotropic constant, ∆E the split in the ground state, kB the Boltzmann constant, and αRO (T ) is the relaxation rate due to the Raman and Orbach
processes. In g.3.4 , we use this formula to t our data for ∆E >12 GHz. The good
agreement between the data and the tted curve indicates that the direct process must
be the cause for the increase of spin relaxation rate with the magnetic eld. However,
it is more dicult to explain the spin relaxation at low magnetic elds. Indeed, the
ip op rate should be[50]:
2
Rf f = αf f sech
where αf f is an anisotropic constant and µB the Bohr magneton. If we t the formula
to the data, g is huge (around 50), much larger than the observed g factor of the
Zeeman state (which is always less than 4 g.A.4). Therefore, it appears we don't have
a good explanation of the spin relaxation rate process at low magnetic eld.
We also would like to study TZ if we increase the temperature. This can help us to
understand the underlying process of spin relaxation. For example, the Raman process
has a dependency of T 9 and it may dominate the spin relaxation rate above a certain
temperature. Also, for practical reasons, we would like to know at which temperature
must be the crystal to operate well as a quantum memory. We measured spectral hole
decays for various temperatures (g.3.5) at a particular angle (300 mT with θ=-30).
It appears that the spin relaxation time is roughly constant between 3 and 4 K and it
18
Figure 3.3: Zeeman state lifetime for various magnetic elds. top: For both angles of B
with D2 , we see the same behaviour, an increase of lifetime followed by a decrease, but the
maximum arise for a dierent B. However, we can plot(bottom) the lifetime as a function of
the split in the ground state, and we see that the maximum arise exactly for the same split.
19
Spin relaxation rate (Hz)
2
10
1
10
0
5
10
15
20
25
30
35
40
45
Zeeman states split (GHz)
Figure 3.4: Spin relaxation rate (1/TZ ) for an angle of B with D2 of -30(blue dots) and
90(green squares). A t using eq.3.1 is in good agreement with experimental data for large
splits.
increases above this temperature. We note we observed spectral holes even above 6 K,
however with a fast decay.
Finally, we investigate the eciency of the optical pumping for the same particular
magnetic eld (which will be used for AFC storage). The laser frequency was scanned
during the optical pumping, which created a large spectral pit (g.3.6). After this,
we measured the remaining optical depth in the centre of the pit (d0) and compared
to the initial optical depth (d). The ratio d0/d was about 4%. This is a signicant
improvement if we compare to Er3+:Y2SiO5, where the optical pumping was very inefcient. We will see in section 3.2 how it does aect the storage eciency. In a simple
3-level model (sec.2.3), the ratio d0/d should be equal to the ratio between the excited
state lifetime and the Zeeman state lifetime (if T 1 TZ ). Here, T1=300µs and for this
magnetic eld TZ =120 ms, so that we have T1/TZ =0.25% which does not agree with
the measure of d0/d. We note that the branching ratio is close to 0.5 (measures in the
appendix), so that it shouldn't aect much the optical pumping eciency. We do not
have yet a good explanation for this disagreement. One possibility is that T1 is longer
than estimated. Indeed, the average time for an ions to decay from the excited state
to the ground state may be much longer than 300µs if it is trapped in a meta-stable
level.
20
population fraction in initial ground state
1
0.8
0.6
3.2 K
3.7 K
4.2 K
4. 8K
5.4 K
0.4
0.2
0
0
100
200
300
waiting time after optical pumping [ms]
400
Figure 3.5: Spectral hole decay for various temperatures. Under 4 K, the spin relaxation
rate is constant.
3.5
3
optical depth
2.5
2
1.5
1
0.5
0
-20
-10
0
10
20
30
40
50
Detuning [MHz[
Figure 3.6: Pit of absorption measured shortly after optical pumping. Even if the burning
time has been long enough, there is a residual absorption in the pit of around 4% of the initial
value.
21
B[mT]
20
77
300
20
300
θ[]
-30
-30
-30
5
30
T2[µs]
4.9±0.2
15±9
93±15
6.3±0.3
60±3
x
2
1
1
2
1
Table 3.1: Coherence time for dierent magnetic eld congurations. The value for x is not
a tting parameter. It was xed to 1 if the decay appeared to be exponential, and xed to 2
if it was not the case.
Coherence time of the optical transition
Upon absorption of a wave packet, an atomic ensemble is in a particular entangled
state where the excitation is delocalized among all the ions (eq.2.1). The coherence
between the ground and excited states is essential, since it allows a collective interference and coherent emission of a photon echo. However, because of the interaction with
environment, this coherence undergoes an exponential decay with a time constant T2,
the coherence time. This denes the homogeneous linewidth γh = πT1 and a spectral
hole cannot be narrower than 2γh [49]. In the context of AFC storage, this will aect
the storage eciency when the optical storage time approaches T2. Indeed, it would
require to tailor, in the absorption prole, narrow structures of widths approaching γh.
We present here measures of T2 by the use of two-pulse photon echoes (2PE) [58]
for various magnetic elds. The principle is to send two short but strong pulses into
the medium, separated by a time t12. A coherence is created between the ground and
excited states, which results in an emission of an echo in the same spatial mode at a
time t12 after the second pulse. The exponential decay of the echo intensity with storage
time allows to measure directly the coherence time. The echo intensity can be described
in a empirical form proposed by Mims[59]: Iecho = I0e−(4t /T ) where x ∈ [1; 2] is a
phenomenological constant resulting from spectral diusion. The obtained values of T2
for various magnetic elds are given in table 3.1.
The highest measured value is T2=93µs for a magnetic eld of 300 mT with θ =
−30. This is relatively high, and approaches the excited state lifetime T1 =300µs1 .
Also, for a high magnetic eld, the decay is almost purely exponential (x=1) which
implies low spectral diusion. With this result, it appears that it should be possible
to burn very narrow spectral holes, since the corresponding homogeneous linewidth is
γh =3.4 kHz. However, with a more precise measure, we observed a small oscillation in
the echo intensity with a period of about 1.5 µs. This is probably due to an additional
level structure and we will see in the next section how it does aect the spectral hole
2
12
1T
1
2
x
has been measured through uorescence detection and stimulated photon echoes
22
90
80
70
T2 [μs]
60
50
40
30
20
10
0
2.5
3
3.5
4
4.5
5
Temperature [K]
5.5
6
6.5
Figure 3.7: Optical coherence time with increasing temperature.
burning.
Finally, we measured T2 for higher temperatures, with B=300 mT and θ = −30(g.3.7).
Not surprisingly, the coherence time is shorter for higher temperatures. There is however still a strong photon echo at 6 K.
Narrow spectral hole burning
For ecient and long AFC storage, it is necessary to tailor narrow structures in
the absorption prole. In particular, it is necessary to burn very narrow spectral holes.
Their width is ultimately limited by the the homogeneous linewidth which has been
measured here through 2PE spectroscopy. However, after a certain time, a spectral
hole may be broadened by spectral diusion. 2PE is not much sensitive to this eect,
it will only lead to a non-exponential decay of an echo. Therefore, a reliable method
to measure spectral diusion is simply to perform spectral hole burning spectroscopy.
However, to avoid power broadening, the optical pumping power must be small, so
that only a small fraction of population is transferred. Also, the laser linewidth should
be narrow enough, so that a spectral hole width is not limited by the coherence of the
laser2.
We present here narrow spectral hole burning for a magnetic eld of 20 mT and
300 mT at an angle of -30with D2 (g. 3.8). This last conguration was chosen
because it is close to an optimum in term of optical pumping eciency and for practical
2 We
note that SHB may be used to measure the linewidth of a laser if the atoms linewidth is known
23
optical depth
3
2
1
0
-4'000
B=10mT
B=300mT
-2'000
0
detuning [kHz]
2'000
4'000
Figure 3.8: Spectral holes for two dierent magnetic elds at an angle of -30with D2. The
hole widths are 240 kHz. For B=300 mT, we observe side holes at ±640 kHz that are probably
due to a superhyperne interaction with yttrium ions.
reasons (it is dicult to get a higher B with permanent magnets). In both case, the
measured hole width is about 240 kHz. We note this is larger than the homogeneous
linewidth: γh=3.4kHz for 300 mT and γh=65 kHz for 20 mT. The broadening for the
low magnetic eld is probably due to spectral diusion, as it was expected from a
non-exponential 2PE decay. We note that this broadening is quite moderate, since
it is usually dicult to attain the limit of the homogeneous linewidth. For the high
magnetic eld though, we believe there may be here some power broadening, since we
observed narrower spectral holes. This is however negligible compared to side holes that
appears at ±650 kHz. This makes it problematic to tailor an AFC for long storage
times. Indeed, not only it adds an unwanted structure, but it appears that atoms
between the central and the side holes are also aected by optical pumping. For this
reason, we can consider the eective linewidth for this magnetic eld is much larger
than 240 kHz.
The side holes also induced an oscillation on the 2PE decay. This is the result of
a quantum beat between dierent transitions. We believe these transitions occur from
a superhyperne interaction with yttrium ions in the crystal. Indeed the positions of
the side holes depend on the external magnetic eld. For B=77 mT, they were at
24
kHz. If we assume a linear dependency, it appears to be about 2 MHz/T. This
is compatible with the gyromagnetic ratio of yttrium ions (Y=2.1 MHz/T). We note
that the magnetic moment of yttrium is small compared to the one of vanadium, which
explains why the superhyperne interaction is here less strong than in Nd3+:YVO4.
Nevertheless, yttrium is an abundant element with a magnetic moment and it appears
superhyperne interaction is inevitable. At 300 mT, the split of 650 kHz makes it
problematic, and the question remains open whether a stronger magnetic eld would
suppress this interaction or not.
±194
Conclusion on Nd3+ :Y2 SiO5 spectroscopy
In this work, we presented various spectroscopic measures on Nd3+:Y2SiO5 to nd
a suitable conguration for AFC storage. We rst measured the inhomogeneous broadening together with the absorption coecient for two axes of polarization. For various
magnetic elds and temperatures, we performed spectral hole burning, which allowed
to measure Zeeman state lifetimes. Additionally, we measured the optical pumping
eciency and the width of a spectral hole. This was completed with measures of coherence time through 2PE. Measures of g tensors are presented in the appendix. We
note that we did not test all possible magnetic congurations, and we do no have yet a
full theoretical understanding of the various mechanisms in the material. Nevertheless,
setting a magnetic eld at an angle θ=-30is, for now, the most promising for AFC
storage. With a low magnetic eld, (≈20 mT), the spectral diusion is reasonable, so
that it is possible to obtain a hole linewidth of 240 kHz. However, the optical pumping
is inecient, because of a short Zeeman states lifetime. On the contrary, the optical
pumping is ecient for a strong magnetic eld (300-400 mT). There is however a strong
superhyperne interaction with yttrium ions, that forbids to tailor narrow structures
for an AFC. We conclude that such a conguration is promising to implement a twolevel AFC storage. It is potentially multimode (sec.3.3) if we create large combs and
it may also be ecient (sec.3.2and sec.3.5) if we restrain it to short storage times.
3.2 Storage eciency
Motivation
The storage eciency η of a quantum memory is dened as the ratio between the
energies of the input and output states. We want to maximize it, since it is a crucial
parameter in any applications. For example, in the case of a quantum repeater, one
usually assumes an eciency of 90% to maintain a high entanglement distribution
rate. Additionally, a high eciency would allow us to realize elaborate experiments,
such as storage of true single photons where the diculty is to distinguish a weak
signal from the noise. Before the present work (2008), demonstrations of a quantum
25
storage protocol in solid-state materials were quite inecient. In our group, storage of
weak coherent states was achieved in Nd3+:YVO4[53] and Er3+:Y2SiO5[52] with storage
eciency of 0.5% and 0.2%, respectively.
Calculation of eciency
For ecient storage, we need a high optical depth of the medium. This allows
a strong light-matter coupling, thus a high probability of absorption and emission.
Additionally, in a photon echo quantum memory, all atoms need to be in phase at the
time of the echo. Therefore the rephasing eciency needs to be high. This is usually
achieved by tailoring narrow structures in the absorption prole. As an example, we
give the formulas of storage eciency in the case of an AFC constituted of Gaussian
peaks (see g.2.3). This was obtained by solving Maxwell-Bloch equations in the
original AFC paper [41]. If the emission is in the forward mode, we have:
7
˜
ηf = d˜2 e−d e− F 2 ,
where F is the nesse of the AFC: F = ∆/γ with ∆ as the separation between the
Gaussian peaks and γ their FWHM. d˜ is the eective optical depth and can be expressed
as d˜ ≈ d/F where d is the amplitude of a peak (in other words, the maximal optical
depth). We note that because of the re-absorbing factor e−d˜, the eciency is limited to
ηmax =4e−2 ≈54%. If the emission can be done in the backward direction, (by the use
of anti-propagating control elds for example) it is possible to reach unit eciency:
˜
7
ηb = (1 − e−d )2 e− F 2
Ideally, we would need highly absorbing (d˜) and narrow peaks (F ). However, for a
given medium, d is xed. Therefore, to increase d˜, it is necessary to decrease the
nesse. This would however aect the rephasing eciency (e− ). This implies that
there is a trade-o, for a given d, between high absorption and ecient rephasing. The
optimal nesse can be calculated analytically in this case, but also empirically if the
eciency is more dicult to predict.
Originally, the eciency was calculated for Gaussian peaks. However, depending
on the AFC preparation, the peaks have sometimes a Lorentzian shape [60, 43]. An
analytical formula for such a case has been given by Chanelière et al.[43] and it was
noted that the optimal eciency is worse if the peaks are Lorentzian rather than
Gaussian. This brings the question of the optimal shape of the peaks to maximize
the storage eciency. To answer this, we may use a formula given by Bonarota et
al.[56], which assumes only that the absorption of the medium is a periodic
in
P function
−iω/∆
frequency. If the absorption is decomposed in a Fourier series: α(ω) = n αne
,
the eciency can be formulated as:
ηf (L) = |α−1 L|2 e−α L
(3.2)
7
F2
0
26
0.6
0.35
0.3
optimized efficiency ηf
Optimized efficiency ηf
0.5
0.4
0.3
Peak shape:
Square
Super Gaussian
Gaussian
Triangle
Lorentzian
0.2
0.1
0
0.25
0.2
0.15
0.1
0.05
0
5
10
Maximal optical depth d
15
20
(a)
0
0
5
10
15
20
25
maximal optical depth d
30
35
40
(b)
Figure 3.9: (a) The storage eciency is calculated for various shapes of an AFC using
eq.3.2. It appears that square peaks are optimal.(b) If we introduce an absorbing background
proportional to the maximal optical depth (here d0 = 0.04 · d), the eciency cannot reach
54%. Instead, we observe here an optimum for d=8 of 33% and the absorbing background
dominates for higher optical depths.
Using this general formula, we calculated (numerically) the optimal eciency for various peak shapes as a function of the maximal optical depth (g.3.9(a)). It appears
that the square shape is the most ecient among the tested possibilities and it is expected it would be more ecient than any shape [56]. In principle this could be proven
analytically using formula 3.2, but an intuitive explanation can also justify it. Indeed,
the rephasing of an AFC is the most ecient if the atoms are the closest to the centre
of a peak. Since the optical depth at any frequency is limited by d, a square shape is
the solution that saturates this constraint. We note that tailoring such a square comb
is, in a way, the most dicult solution, since it would require an innite coherence to
tailor vertical edges. Therefore, the homogeneous linewidth should be small compared
to the width of a peak.
We mention also that, during the AFC preparation, the optical pumping is not
necessarily perfect. A fraction of atoms remains in their initial state, which results in
an absorbing background d0. This aects the storage eciency by a factor e−d [53].
In principle, d0 is proportional to d, assuming the preparation is not more dicult at
high absorption. For this reason, there is a trade-o between a high absorption in the
AFC and a background absorption, and this results in an optimal optical depth of the
sample(g.3.9(b)).
0
27
AFC in Nd :Y SiO
3+
2
5
We present here AFC storage in a Nd3+:Y2SiO5crystal. From the knowledge of our
spectroscopic measures, we applied a magnetic eld of 300 mT at an angle of −30with
respect D2 to obtain a long Zeeman lifetime. Because the split in the ground state is
larger than the inhomogeneous broadening, the absorption is divided by 2 compared
to the situation without a magnetic eld. Using a 1 cm long crystal and a doublepass conguration, we obtain a maximal optical depth of about 3. An AFC is tailored
(g.3.10(a)) by the use of a non-coherent preparation. It consists of scanning the laser
frequency and modulating its intensity, so that it permits a frequency-selective optical
pumping. If the pre-programmed storage time is short (less than 100 ns), the width
of a spectral hole is narrow enough to create squarish and non-overlapping peaks. We
tested the AFC by storing a classical pulse during 33 ns and observed an eciency
of 17% (3.10(b)). We note that we are here at the limit of the bandwidth of the
photo-diode and of the acousto-optic modulator (AOM) which creates the input pulse.
For this reason it was not possible to measure here the storage eciency for shorter
storage times, but we mention we reached an eciency of 21% for a storage time of
25 ns in the single photon regime [61]. To increase more the eciency, we would need a
higher optical depth. However, because of the absorbing background, it would not be
possible to reach more than 33%(g.3.9(b)). This would occur for an optical depth of
8 and a comb nesse of 4.6, and we estimate this would bring many technical diculties. Therefore, to increase signicantly the storage eciency, we would probably need
another material. For example, we reached η=35% in a Pr3+:Y2SiO5crystal in collaboration with Lund university [60] and Hedges et al.. demonstrated a storage eciency
of 69%, by the use of a gradient-echo memory [62]. Nonetheless, we demonstrated
here a signicant improvement of storage eciency, and it will help us to realize more
elaborate experiments.
AFC in Eu :Y SiO with an impedance-matched cavity
3+
2
5
In this thesis, we worked also on the storage of light in a Eu3+:Y2SiO5crystal,
which is a non-Kramers ion. In the perspective of ecient storage, it has a very good
potential: the optical pumping is extremely ecient because of an ultra long spin states
lifetime [63] and it would be possible to tailor very narrow structures because of a very
narrow homogeneous linewidth [63, 64, 65]. However, our main motivation comes from
the fact that it is a very promising system to implement a QM with an on-demand
readout and a highly multimode capacity [66]. The splits in the ground states are large
compared to Pr3+:Y2SiO5[67], but the oscillator strength is much lower. Unfortunately,
this cannot be compensated with a higher doping concentration, since it increases
a lot the inhomogenous broadening and only a little the absorption coecient [63].
Increasing the crystal length is also not a satisfying solution, since it would bring many
28
1
3.5
0.9
3
0.8
input
0.7
relative intensity
optical depth
2.5
2
1.5
0.6
0.5
0.4
transmission (23%)
0.3
1
echo (17%)
0.2
0.5
0.1
0
0
20
40
60
Detuning [MHz]
80
0
-40
100
(a)
-30
-20
-10
0
10
20
30
40
time [ns]
50
60
70
80
90
100
(b)
Figure 3.10: (a) AFC with periodicity of 30 MHz in a Nd3+:Y2SiO5crystal. (b) It is tested
by storing a classical pulse for 33 ns and we obtain an eciency of 17%.
technical diculties. An alternative has been proposed [57, 68] to place the crystal
into an asymmetric cavity (Gires-Tournoi interferometer) to reach perfect absorption
(g.3.11) . If the QM has an optical depth d˜, the impedance-matching conditions are
reached when the reectivity is R1 = e−2d˜ for the rst mirror and R2 = 1 for the second
one. Indeed, similarly to a standard Fabry-Perot cavity, in the backward direction, the
eld emitted by the cavity interferes with the eld reected by the rst mirror. In
resonance, this interference is destructive, so that the intensity is zero at the output
of the cavity and the QM has absorbed 100% of the light. Moreover, an echo will
be emitted at the expected storage time, and the eciency can now reach 100% [57].
With such a scheme, it is therefore possible to attain high eciencies, even with a
weakly absorbing medium. If the spatial interference is perfect, η is only limited by
the rephasing eciency (narrow peaks) and the re-absorption in a single pass.
We worked here to realize a cavity-enhanced storage in a Eu3+:Y2SiO5crystal, by
the use of mirrors placed outside the cryostat 3.12(a). Because the cavity is sensitive
to any environment perturbations (vibrations, sounds, temperature uctuations...), it
needs to be actively stabilized using a piezo-electric on a mirror and a probe light
to obtain an error signal. The advantage of such setup relies on the possibility to
add another crossed-beam that would not be aected by the mirrors (g.3.12(a)).
Such beam can be used to prepare the AFC, scan the absorption prole and, most
importantly, to implement a control eld to allow spin-wave storage. We could observe
the cavity modes and a very high absorption (g. 3.12(b)), but we were not yet able to
implement an AFC storage. We need to overcome few technical details, but we believe
it is only a matter of time to be able realize a cavity-enhanced AFC storage.
29
Figure 3.11: Schematic setup of a QM in an impedance-matched cavity. The light emitted
by the cavity interferes with the reection on the rst mirror. This allows to reach 100%
storage eciency, even for a weakly aborbing medium.
1.1
stabilisation
light
1
0.9
probe light
photo-diode
0.8
PID box
mirror R=70%
amplifier
reflected intensity
preparation light
(optical pumping)
0.7
empty cavity
0.6
0.5
0.4
0.3
cryostat
cavity containing
an attenuator
0.2
0.1
miror R=100%
0
laser frequency
piezo-electric
(a)
(b)
Figure 3.12: (a) Setup to realize cavity-enhanced storage in a Eu3+:Y2SiO5crystal. (b)By
scanning the laser frequency, we observe resonances in the reected light by the cavity. The
QM is simulated here with a variable attenuator. Thanks to the impedance matching conditions, it is possible to absorb about 90% of the light. It is here limited by the quality of the
spatial interference between all the modes of the cavity. If the light is out of resonance or if
the cavity is empty, all the light is rejected.
30
3.3 Multimode quantum storage
Classical memories have today the capacity to store a huge amount of information.
In analogy, we expect a QM could have a multimode capacity, that is to say, it would
store more than one qubit at the same time. The number of modes (N ) it can store is
a crucial criterion depending for many applications. For example, it can speedup the
entanglement generation in some quantum repeater architectures [69]. Of course, the
largest N would be the best, but we note that a value of N=100 is often assumed to
compare quantum repeater architectures [18]. In general, one can consider a spatial,
frequency or temporal multiplexing. The last two possibilities appear to be the most
adapted for the coupling to single mode bres. We will here consider temporal modes.
For almost any storage protocol (EIT, CRIB...), an increase of the number of modes
requires an increase of optical depth [26]. For this reason, before this work, quantum
storage demonstrations were usually not multimode. However, in an AFC protocol,
the number of temporal modes is independent of the optical depth [41, 26] and it is
considered intrinsically multimode. Indeed, N is proportional to the storage time and
to the bandwidth of a QM: the time-bandwidth product. For an AFC, the bandwidth
can be increased by increasing the number of peaks without changing their separation
∆ (g.3.13). Alternatively, on can increase the storage time by increasing the density
of peaks. Both solution does not require a higher optical depth. Therefore, to realize
a highly multimode QM, we need to create an AFC with a large number of peaks. We
note that rare earth doped crystals are well adapted to this, since the inhomogeneous
broadening is usually large compared to the homogeneous linewidth.
Here [70], we demonstrated the multimode potential of a quantum memory based
on AFC by storing up to 64 dierent time modes in a Nd3+:Y2SiO5 crystal (g.3.14).
We had to limit the storage time to roughly 2µs because of material limits (sec.3.1).
The comb bandwidth (100 MHz), however, could have been increased to few GHz
with a more elaborate optical pumping. We used weak coherent states with less than a
photon per pulse, and the detection was performed with a silicon single photon counter.
This demonstrates, in comparison to previous multimode optical data storage[71], the
potential to store true quantum states. Moreover, we also showed the preservation of
coherence by performing interference between consecutive pulses with various phase
dierences (g.3.15). For more details, we refer to the published paper [70] (appendix
B, p.79)
3.4 On-demand storage
Up to here, I presented experiments with a preprogrammed storage time with a 2level AFC. Although it could be rapidly tuned, the storage time could not be changed
after an input had been absorbed in the QM. An on-demand readout is however nec31
(a)
(b)
3.0
2.5
Efficiency [%]
Optical depth
6
2.0
1.5
5
4
3
2
1.0
1
0.5
-50
-25
0
25
0
50
5
Optical Detuning [MHz]
10
15
20
25
30
Input pulse duration [ns]
Figure 3.13: (a) By adding absorbing peaks to an initial AFC (green line), we increase its
Normalized counts
bandwidth and keep the optical depth (dashed black line) (b) This allows to store shorter
pulses with (at least) the same eciency, thus to increase the number of temporal modes.
The same principle could be applied to increase the storage time.
1.0
Input modes
Output modes x50
0.8
0.6
0.4
0.2
0.0
0.0
0.4
0.8
1.2
1.6
2.0
2.4
Time (ms)
Figure 3.14: Because of a large time-bandwidth product, it is possible to send 64 dierent
short pulses into the QM before the light is re-emitted. Each full pulse have an average
number of photons n̄ < 1.
32
(b)
140
700
120
600
100
500
80
400
Counts
Counts
(a)
60
300
40
200
20
100
0
0
Noise level
0
200
400
600
800
1000
1200
0
1
Time [ns]
2
3
4
Phase [rad]
Figure 3.15: (a) A series of pulses with various phases is stored in a medium with two AFCs
with dierent periodicities. This induces, at the output, the interference between consecutive
pulses. Thanks to the multimode capacity, it is possible to visualize a complete interference
fringe in one measurement. (b) The raw visibility (red curve) of this interference is 78±3%.
Substracting the detector dark count, the net visibility is 86±3%.)
essary in most of the applications, in particular for a quantum repeater. Additionally,
for a 2-level AFC, the storage time is limited (in the best case) to the optical coherence
time. However, quantum repeaters require long storage times. Indeed, the memory
storage time must be much larger than the time for a photon to travel through an
elementary link. This can go from 10 ms to 1 s [18]. The solution is to add 2 control
pulses to include a spin-wave storage [41]. This allows to re-emit a photon whenever
it is needed, and the storage duration is now limited by the spin coherence time which
can be extremely long. Spin-wave storage has already be implemented in solid-state
memories in an EIT protocol, and for classical light [72, 73]. The goal, here, was to
realize it with an AFC.
Spin-wave storage in Pr :Y SiO and in Eu :Y SiO
3+
2
3+
5
2
5
To realize a complete AFC protocol, we need a system with at least three long-lived
ground states and one optical excited state. Excellent candidates are non-Kramers ions
with a hyperne structure. For example, Pr3+:Y2SiO5 and Eu3+:Y2SiO5 have exactly 3
ground states at zero magnetic elds with very long coherence times. Pr3+:Y2SiO5 has a
high oscillator strength, which allowed an ecient 2-level AFC storage [60]. Therefore,
we realized the rst complete AFC protocol in this system, in collaboration with Lund
university [74] (g.3.16). It allowed us to store bright pulses up to 20µs. Later, we were
able to implement also spin-wave storage in a Eu3+:Y2SiO5crystal [75]. More details
33
Normalized intensity
Normalized intensity
14
12
10
8
16
12
8
4
0
0
6
4
1
2
3
Time (ms)
4
Detector gate off
x10
2
Control pulses
0
0
2
4
6
Time (ms)
8
10
12
Figure 3.16: Spin-wave storage in a Pr3+:Y2SiO5crystal. The inset shows a 2-level AFC
storage. When the two control pulses are added, the storage time is extended
can be found in the published papers [74, 75] (appendix B p.67 and p.114).
Storage of weak coherent states
The future challenge is to implement spin-wave storage with true single photons
as it would be in a quantum repeater. However, all the previous experiments, were
realized with bright classical pulses. Therefore, the rst step is to store weak coherent
states. This is however much more dicult than with a 2-level AFC, because of the very
bright control pulses: they may induce dierent kind of noise, such as free induction
decay, uorescence, but also after-pulses and detector blinding. In this perspective,
Eu3+:Y2SiO5 is more adapted to Pr3+:Y2SiO5, because of the larger level spacings in
the ground states. Indeed, the frequency separation between the control eld and
the echo is larger, which makes the ltering more feasible. We note also, that in an
AFC protocol, unlike EIT, the control elds is switched o during the photon echo.
Nonetheless, we had to use a ltering cavity, temporal gating and dierent spatial mode
to obtain a satisfying signal-to-noise ratio at the single photon level. Very recently,
we were able to store weak coherent states with an average of 2.5±0.6 photon per
pulses (g.3.17). We shown also a relatively low noise of 7.1 ± 2.3 · 10−3 photons per
mode. These results are not yet published, but an article is under a review process [76]
(appendix B p.122).
Spin refocusing
In all the experiment I presented here, the storage times (roughly tens of µs) were
much shorter than the spin coherence times (15.5 ms in Eu3+:Y2SiO5 [77]). This
is attributed to inhomogeneous spin dephasing. Indeed, the spin transitions have a
inhomogeneous broadening of usually tenth of kHz. This can however be compensated
34
Figure 3.17: First demonstration of a spin-wave storage in a solid-state memory with weak
coherent states. Here the average number photon per pulses is 2.5±0.6.
with spin echo techniques. An impressive demonstration was realized by Longdell et
al.. to stop light for more than 1 s in a Pr3+:Y2SiO5crystal with an EIT protocol. We
note that a dynamical decoupling sequence allowed to extend even the spin coherence
time. To achieve this, the time between the refocusing pulses must be much shorter
than the coherence time. We are now working on implementing spin refocusing in
our Eu3+:Y2SiO5 quantum memory. We were able to observe a clear extension of the
storage time (g.3.18) to roughly 10 ms. This work is however under progress and we
need to understand why we observe no echo for certain storage times.
3.5 Testing a QM with single photons
A QM must have the ability to store faithfully a quantum state, which implies
it must also preserve quantum entanglement. In previous experiments [52, 53], weak
coherent states were stored. This demonstrated that it is possible to obtain a satisfying
signal-to-noise ratio, even at the single photon level, but the preservation of a quantum
state or entanglement was not shown. We want here to test a QM by the use of a
single photon source. Additionally, this would be a step forward in the realization of
quantum repeater, which includes always the generation of quantum states. We note
that before 2011, the storage of true single photons was never realized in a solid-state
device.
Single photon sources
Single photons can be produced by an external source or by the QM itself (as in a
DLCZ scheme [17, 45]). We note that, for previous experiments in rubidium or cesium
vapours [34, 32], single photons were emitted by a separate atomic ensemble. This
35
Figure 3.18: Spin-wave storage of classical pulses including a spin refocusing technique. The
storage time has been increase to almost 10 ms, but we don't understand yet large variations
of the echo intensity.
implied that the photons had a wavelength and bandwidth perfectly adapted to the
QM. However, a QM based on AFC in a rare-earth crystal have potentially a large
bandwidth. Therefore, it opens up the possibility to use other kind of sources that
produces large bandwidth photons. For example, SPDC in a non-linear medium [6, 7]
is widely used in quantum optics experiments to create correlated photon pairs with
a high rate. Moreover, such a source is part of certain quantum repeater schemes
[69, 18]. Here we used a PPKTP waveguide pumped by a 532 nm light, which is
converted into a idler eld at a telecom wavelength (1338 nm) and into a signal eld
at the wavelength of the QM (883 nm). Both modes are ltered with cavities to match
the AFC bandwidth (120 MHz). More details and characterization of this source can
be found in the supplementary information of ref.[78] (Appendix B p.101).
Single photon storage
We used this source to demonstrate the rst storage of true single photons in a
solid-state device [61]. The signal photon is sent to a Nd3+:Y2SiO5 crystal to be stored
for up to 200 ns and the idler photons was coupled to a 50 m long telecom bre. A
histogram of the coincidence detections is shown
in g.3.19 and we measured a cross
(2)
correlation well above the classical limit of gsi =2. Additionally, we could demonstrate
that the storage process preserved the energy-time entanglement between the signal
and idler photons. By the use of a Franson interferometer [79] (g.3.20), we performed
a Bell experiment to violate a Clauser-Horne-Shimony-Holt (CHSH) inequality[80]. We
found the Bell parameter (S = 2.64 ± 0.23) to be by three standard deviation above
the classical limit (S=2). For more details about these results, I refer to the published
paper [61] (appendix B p.92). We note that Saglamyurek et al. have performed a
36
a
with AFC
b
Classical regime
Figure 3.19: (a) The histogram of coincidence detections shows that the signal photon was
stored for various durations in the QM (b) The cross-correlation is maintain well above the
classical limit for any storage time
very similar experiments with a thulium-doped lithium niobate waveguide, which was
published in the same issue[81].
Heralded entanglement between two QMs
A quantum repeater is composed of distant and entangled quantum memories.
Therefore we estimate an important test is to perform heralded entanglement between
two separated QMs, at least for a short spatial separation. Also, such experiment has
never been demonstrated for solid-state devices before we started this work. Here,
we used our photon pairs source to herald a single-photon entanglement and store
it into two Nd3+:Y2SiO5 crystals separated by 1.3 cm [78] (g.3.21(a)). After the
re-emission of the light eld, a tomography was performed on the photonic state.
This allowed to calculate the concurrence, which is a measure of entanglement. It
goes from 0 for a separable state to 1 for a maximally entangled state. We found a
lower-bound on the concurrence that was positive, which revealed the entanglement
of the QMs (g.3.21(b)). Such a setup was inspired by previous works in cold gases
[82, 83, 84]. One can found more details in the published paper [78] (appendix B p.101).
We mention a similar work has been published shortly before this, demonstrating the
entanglement of diamonds at room temperature [85].
37
a
b
Path Combinations:
short-short
long-long
long-short
short-long
Start
Long Arm
SSPD
50%
Short Arm
Stop
50%
Si APD
Source of
Entangled
Photons 883 nm
1338 nm
50
%
Long Storage Time
%
50
: Faraday Mirror
Short Storage Time
Figure 3.20: (a) Franson-type setup. It allows the interference between the |shorti|shorti
and |longi|longi modes. (b) The good visibilities of this interference (78±4% and 84±4) is
sucient to violate a Bell inequality
m
1c
Heralding
detector
MB
B
Filtering
133
88
m
5
Memory
preparation
MA
Detector 1
PBS
PPKTP
waveguide
cm
BS
DM
n
32
A
FR
m
3n
8nm
1.3
Switch
Detector 2
(a)
(b)
Figure 3.21:
(a) An SPDC source heralds the presence of a single photon which is sent into a
beamsplitter. This creates a single photon entangled state, delocalized between to spatial modes.
This state is transferred to two quantum memories that become entangled and keep the excitation
for 50 ns. To measure the amount of entanglement, the excitation is transferred back to a light eld,
and a tomography is performed with two single photon detectors.(b) This allowed to calculate a lower
bound on the concurrence. A positive value revealed the presence of entanglement.
38
Chapter 4
Discussions and Outlook
In this thesis, I worked on dierent aspects of a quantum memory with the prospect
of implementing it in a quantum repeater. First, our spectroscopic measurements
in a Nd3+:Y2SiO5 crystal allowed to nd a system for a promising implementation
of an AFC protocol. Also, we worked on the important properties of a quantum
memory: the storage eciency, the multimode capacity and the storage time (with
an on-demand readout). Finally, we tested our QMs, through the storage of single
photons, preservation of time-bin entanglement and heralding of entanglement between
two crystals.
Spectroscopy
Through our spectroscopic measures in Nd3+:Y2SiO5, we found interesting magnetic
eld congurations for a two-level AFC. It allowed us to realize many breakthroughs
for quantum storage in a solid-state device [61, 70, 78, 55], and we hope more will
come. Unfortunately, with a low magnetic eld, the optical pumping was not ecient.
For this reason, the experiments were done with a high magnetic eld. However,
a superhyperne interaction with yttrium ions limited the storage time to less than
2µs. Also, even for a high magnetic eld, the optical pumping is not perfect and
about 4% of atoms remain in the spectral hole where the AFC is created. Therefore,
it should be interesting to nd another conguration with a longer Zeeman lifetime
and less aecting superhyperne interactions. To nd this, we would need to do a
more systematic spectroscopy for various magnetic elds in the D1-D2 plane. A better
theoretical understanding of the spin relaxation process as a function of the magnetic
eld would also help for nding such conguration. Also, we observed some very
long lived spectral holes (∼1 s) which probably results from the hyperne structure in
143
Nd and 145Nd isotopes. This would promise very ecient optical pumping and an
additional ground state that could be used for a complete AFC scheme. However, the
high value of the nuclear spin (7/2), together with the 1/2 electronic spin, generates 16
39
dierent levels in the ground and excited states, thus 256 transitions! Because of this,
a hole burning spectrum is extremely dicult to read, in particular if the two isotopes
are present in a naturally doped crystal. Therefore, it would be interesting to work
with a isotopically pure sample, to measure the eciency of the optical pumping and
maybe nd a Λ system suitable for a 3-level AFC storage.
I also believe that it is necessary to search for other possible systems, that is to say
ions and host crystals. For example, Nd3+:YLiF4 isotopically pure in 7Li presents a
very small inhomogeneous broadening of 45 MHz [86]. Not only could it lead to high
absorption, but it also allows to resolve the dierent optical and hyperne transitions
without the need for spectral hole burning. With only one class of atoms, it would
be much easier to nd a Λ system. It would be interesting to continue this work by
doing coherence measurements and spectral hole burning in this system. I would also
like to mention Er3+:Y2SiO5 which has the great advantage of having its resonance
transitions at telecom wavelength. Previous work [50] demonstrated impressively long
coherence times and low spectral diusion at very low temperature and strong magnetic
eld. Unfortunately, further experiments in Geneva [51] showed this was counterbalanced by inecient optical pumping due to a short Zeeman lifetime compared to
the excited state lifetime. For this reason, there is no more research, to our knowledge,
for quantum storage in this material. However, we know that the Zeeman lifetime can
be strongly increased with a moderate magnetic eld at a particular angle, as we have
seen in Nd3+:Y2SiO5. Hence, there is still a potential in Er3+:Y2SiO5, but we would
need to search more systematically for long Zeeman lifetime for various magnetic eld
conguration. Another idea is the use of stimulated emission to increase the eciency
of optical pumping. It was already demonstrated in Er3+:Y2SiO5[48] and it could be
interesting to implement in any other materials, including Nd3+:Y2SiO5.
At present, the most promising materials for a spin-wave storage are probably nonKramers ions, in particular Pr3+:Y2SiO5 and Eu3+:Y2SiO5 because of extremely long
coherence times, very weak spectral diusion and very ecient optical pumping. Also,
at zero magnetic eld, they have exactly the 3 ground states required for a complete
AFC protocol. We note however that they have a narrower bandwidth and a wavelength
less accessible to common diode lasers. We nally note that the spin coherence time
can be dramatically increased by operating at a magnetic eld where a transition has
a ZEro First Order Zeeman (ZEFOZ) shift [87]. This has been applied successfully to
Pr3+:Y2SiO5[88] but there are many other candidates [87]. However, one must take
into account that by applying such eld, the degeneracy of the spin states is usually
lifted, which complicates the level structure and makes the spectral tailoring (e.g. AFC
preparation) more dicult.
40
Storage eciency
In this thesis, one of our goals was to increase the eciency of quantum storage,
since it helps to realize more sophisticated experiments, and it is also a crucial parameter in a future quantum repeater. As we described in sec.3.2, it depends on the
quality of the atomic frequency comb, which should have highly absorbing peaks with
square shapes and the lowest possible absorbing background (d0). These parameters, in
turn, depend on the material used, which should have a high initial absorption, narrow
homogeneous linewidth and long spin/Zeeman lifetimes. We demonstrated a storage
eciency in Nd3+:Y2SiO5 of 20% which allowed us to realize important breakthroughs
for quantum storage in a solid-state memory[61, 78]. It was possible to tailor square
peaks, but only for short storage times (<100 ns) because of the eective homogeneous
linewidth in this material. In principle, a higher initial absorption, for example with
higher doping concentration, would permit to increase the eciency. However, because
of the absorbing background, it would be limited, in the ideal case, to 33% (in a backward emission), assuming that the preparation is not more dicult at high optical
depths. This could be improved by nding a better magnetic eld conguration as we
discussed in the previous section.
Recently, our researches focused on Eu3+:Y2SiO5, which has a good potential for a
QM with spin-wave storage for single photons with a high multimode capacity. The
extremely narrow homogeneous linewidth of 122 Hz [64] is promising for long storage time in the optical transition. Our experiments, though, were restricted by the
laser linewidth (≈30 kHz), limiting our ability to create squarish peaks. The priority
for future experiments in this material should be to realize a much more stable laser,
for example by using an ultra low expansion (ULE) [89, 90] cavity in vacuum. Also,
Eu3+:Y2SiO5 has a relatively weak oscillator strength, making it dicult to obtain high
absorption. One solution is to increase the doping concentration. But this will also
increase the optical inhomogeneous broadening, thus the absorption would not much
higher. Another solution is to place the crystal in an asymmetric cavity (Gires-Tournoi
interferometer) which may allow 100% absorption using an impedance matching condition [57, 68]. Then, a weakly absorbing medium could allow very ecient storage.
Our preliminary results for such experiments indicate absorption of more than 90%,
however without performing AFC storage. To achieve this, the crucial points are alignment of the cavity (spatial mode matching), its length stabilization and reduction of
intra-cavity losses. The conguration of Sabooni et al.[91, 92], using reection coating
on the crystal surfaces, is more robust and does not require active stabilization. This
led to a cavity enhanced storage with 58% eciency. However, our setup (g.3.12(a)),
with mirrors outside the cryostat, allows to use a crossed beam, which overlaps with
the probe eld in the region of the cryostat, but which is not reected by the cavity
mirrors. Hence, a control eld, even out of resonance with the cavity, could be sent
through this crossed beam and allow spin-wave storage. Therefore, our plan in this
41
project is to rst demonstrate a cavity enhanced storage for short storage time. Then,
with a narrower laser linewidth, we hope to increase the storage time and implement
spin-wave storage.
In general, cavity-enhanced storage will be useful to compensate a weakly absorbing
medium. Therefore it opens up quantum storage for more materials, even for those
with a weak oscillator strength. Also, for a two-level AFC, it allows to reach 100%
eciency [57] while a forward echo is limited to 54% [41]. Meanwhile, the impact of
an impedance-matched cavity on other storage protocols remains unknown and could
be investigated. While a cavity-enhanced CRIB would certainly be similar, a DLCZ
scheme in a cavity would need a more detailed analysis, since the Stokes and anti-Stokes
have dierent frequencies and can have dierent spatial directions. Also, an interesting
eect has been recently observed [91]. The strong dispersion in a hole-burning medium
(slow light in a spectral hole) induces a drastic reduction of the cavity mode spacing
and bandwidth. This may be a complication for quantum storage in term of achievable
bandwidth, and thus, this eect should be more deeply studied.
Multimode storage
The multimode capacity of a QM is an important characteristic that allows the
speed up of entanglement distribution rate in some quantum repeater architectures
[69]. AFC is intrinsically multimode in the time domain, since, unlike other protocols,
the number N of temporal modes is independent of the available optical depth[26,
41]. N can be increased via a large AFC bandwidth and long two-level AFC storage
time. Also, rare earth doped materials, because of a large inhomogeneous broadening
compared to the homogeneous linewidth, are well adapted to multimode storage. Here
we demonstrated the reversible mapping of 64 temporal modes at the single photon
level in a Nd3+:Y2SiO5crystal. Through the interference of consecutive modes, the
preservation of coherence was shown. This proof-of-principle experiment demonstrated
that future QM could be highly multimode if based on AFC. The number of modes
can clearly be much higher [56, 71], but usually a value of 100 modes is typically
assumed for a quantum repeater to compete with direct transmission [18]. Therefore,
the next challenge seems to store roughly this number of modes in more sophisticated
experiments. For example, we demonstrated true single photon storage in Nd3+:Y2SiO5
[61]. However only few temporal modes were stored, because the experiment required
high eciency, thus short storage time. This could be relatively easily compensated
by increasing the AFC bandwidth to 1 GHz or more, which would allow to show a
multimode storage with true single photons in a solid. On the other hand, non-Kramers
ions are promising systems for spin-wave storage. In this case, though, a large AFC
bandwidth is dicult to reach, because the separations between the hyperne states
are small and we would need extremely high Rabi frequency to create, for the control
eld, a π pulse with a large bandwidth. Therefore, the solution would be to increase
42
the storage time for the two-level AFC echo, to several tenths of µs. We believe this
to be within reach in Eu3+:Y2SiO5, if we had a laser with narrow enough linewidth.
We note however we would be, at a certain point, limited by a temperature-dependent
spectral diusion [90].
On-demand readout & storage time
For any applications we mentioned in the introduction, a quantum memory must
include an on-demand readout. Also, the storage time must be long enough to achieve
a particular task. For example, quantum repeaters require 10 ms to 1 s depending of
the architecture [18]. This can be done with a complete AFC protocol, which includes a
spin-wave storage by the use of control elds. We realized this in a Pr3+:Y2SiO5 crystal,
in collaboration with Lund University, which was the rst realization of an AFC with
spin-wave storage in a solid-state [74]. We could later reproduce this in a Eu3+:Y2SiO5
crystal which has very promising properties for a QM adapted to a quantum repeater
[75]. Using dierent ltering stages to suppress the noise induced by the control elds,
we observed a noise of only (7.1 ± 2.3) · 10−3 photons per mode, which allowed to store
weak coherent pulses with an average of 2.5 ± 0.6 photons [76]. The phase preservation
was demonstrated through interferences between two consecutive modes. The noise is
in principle low enough to store true single photons from an external source. However,
the low storage eciency (≈ 0.4%) decreases the signal-to-noise ratio, which explains
why we could not store coherent pulses containing less than 2.5 photons in average.
Therefore, it is necessary to increase the storage eciency in a manner that would not
increase the noise. This could be done with an impedance matched cavity or a laser
with longer coherence time. Also, this system could be used to implement a photon
pairs source based on a DLCZ scheme [45]. We estimate that it should be feasible in
a short term period in our system with a few ameliorations.
The total storage time was limited by the spin inhomogeneous broadening which
induced dephasing in the spin state. This can be overcome with spin refocusing using
RF pulses [93, 94]. A sequence of pulses resonant with the spin transition allows to
compensate the inhomogeneous broadening and even extend the spin coherence time
using dynamical decoupling [93, 73]. Our current eort to implement spin refocusing in
Eu3+:Y2SiO5 indicates a total storage time of roughly 10 ms, which already represents
an increase of 3 orders of magnitude. We hope we will be able to soon combine this with
spin-wave storage of weak coherent pulses. We could then measure if the noise level
would increase with the number of RF pulses [95]. I would nally like to mention that
the storage time could be increased even more dramatically with dynamical decoupling
sequence combine to a ZEFOZ transition. Indeed, Fraval et al.. [93] have extend
the coherence time in Pr3+:Y2SiO5 from 82 ms to 30 s with such a sequence. In
Eu3+:Y2SiO5 though, it has been suggested that the coherence time (without dynamical
decoupling) could reach hundreds of second at a ZEFOZ transition [96]. Therefore, one
43
could extrapolate an extended coherence time of several hours! And this would still be
shorter than the spin lifetime that has been estimated to reach 20 days [63].
Entanglement in Quantum memories
To characterize a quantum memory, a direct method is to store quantum states, in
our case single photons, and measure the storage delity. This also goes in the direction
of realizing a complete quantum repeater, which not only requires quantum memories,
but also single photon sources, single photon detectors, large interferometers and Bell
measurements. Here we used a photon pair source based on SPDC in a non-linear crystal [61]. To match the QM bandwidth, the idler and signal photons were ltered with
an optical cavity and etalons. The idler eld, at telecom wavelength, was sent through
a 50 m long optical bre while the signal eld was stored in a Nd3+:Y2SiO5 crystal
and re-emitted using an AFC protocol. The measured cross correlations between the
idler and signal photons demonstrated the quantum character of the source. Moreover,
using a Franson interferometer, we could perform a Bell experiment and demonstrate,
through the violation of a CHSH inequality, that the energy-time entanglement between
the idler and signal eld was preserved during the storage. In another experiment [78],
we were able to herald and measure entanglement between two QMs by mapping a
single photon entangled state into two crystals. The QMs were separated by 1.3 cm
and the distance could not be increased in that setup, since additional optical losses
would decrease the amount of entanglement.
If we would like to herald entanglement between two QMs separated by a large
distance, i.e. an elementary link of a quantum repeater, a solution would be to duplicate
the photon pair source and use the scheme of Simon et al.. [69]. In that case, though,
the distance would not be very large, since the storage time in Nd3+:Y2SiO5 didn't
exceed 100 ns (although we could probably push it to 1 µs). It would be more interesting
to use a Eu3+:Y2SiO5 crystal, since we already implemented a spin-wave storage in it.
However this material requires photons with a narrow bandwith (≈1-5 MHz) which
would be dicult to produce with SPDC, in particular for this wavelength (580 nm).
It would probably be necessary to use a doubly resonant Optical Parametric Oscillator
(OPO) [97] so that additional ltering would not be necessary. We believe a more
promising solution is to use a DLCZ scheme as we discussed in the previous section,
so that the photons are directly produced by the atomic ensemble. However, photons
at 580 nm would undergo large losses in an optical bre(≈10 dB/km), thus it would
be necessary to coherently down convert them to a telecom wavelength [98]. Such
a scheme, although experimentally challenging, would allow to herald entanglement
between two QMs separated by a large distance.
44
Outlook
To sum up, I would argue that from the results presented here and from other related
experiments realized in the community, solid-state devices have a strong potential
for the realization of a quantum memory adapted to a quantum repeater. Indeed,
dierent experiments have achieved high performances in the various criteria such as
the storage eciency, storage time, noise level and multimode capacity. These numbers
have however not yet been achieved in the same system. Which leads to the question
whether there is a material in which we can combine all the required performances. We
believe that Eu3+:Y2SiO5 has the potential to do it, and our future work will focus on
it. This will however also require the development of a highly coherent laser, the use
of an impedance matched cavity and reliable spin refocusing techniques. We believe
also that, in parallel, it is still necessary to search for alternative systems and study
the potential of various rare earth doped crystals. We are now optimistic that in a few
years a prototype of a quantum repeater (or at least an elementary link of it) based on
solid-state QMs could be realized. However, it is still a huge step to realize a quantum
repeater that could compete with direct transmission. For example, we will have to
take into account all the important optical losses arising in a more complex setup.
Surely, it will take time until a quantum repeater would be commercialised as is today
QKD, but progress may be faster than we think.
45
46
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56
Acknowledgements
Cette thèse a été un long travail et, heureusement, je n'ai pas été seul pour l'accomplir.
J'aimerai en proter pour remercier tout les gens qui m'ont aidé dans mon travail, mais
aussi ceux qui m'ont soutenu ou tout simplement accompagné pour que je passe 5 très
belles années!
Avant tout, je crois que j'aimerais remercier tout le groupe du GAP-optique dans
son ensemble. En eet, l'ambiance y était excellente, et elle le sera toujours j'en suis
sûr. Ceci m'a permis de passer de très bon moments et il a été très facile de collaborer
avec tout le monde. Nicolas Gisin porte sûrement la responsabilité de cette atmosphère
positive, et je le remercie de m'avoir accueilli dans ce groupe. J'ai beaucoup apprécié
d'avoir été supervisé par Mikael Afelius. Il a su me guider tout en écoutant toutes les
idées que je pouvais proposer. En plus, nous avons passé de très bon moments au pub
et dans les gradins de la patinoire! Hugues a également été un excellent un superviseur
pour mon début de thèse, et je suis très content qu'il ait été dans mon jury. Merci
encore à Nuala qui a su être une très bonne partenaire de labo et membre du jury.
Merci également à Philippe Grangier qui a fait le déplacement, j'ai été très honoré de
sa présence.
Encore au début de la thèse, j'ai eu le plaisir d'aller à Lund pour une collaboration
très fructueuse. J'aimerais remercier Stefan Kröll de nous avoir accueilli, et il a été
très agréable de travailler avec Atia.
Je remercie tout mes collègues de labo: Björn, Christoph, Félix, Pierre et Nuala.
Cela s'est passé merveilleusement bien, et surtout nous avons pu être ami en dehors du
labo et partager plein d'activités. L'ambiance dans le bureau a aussi été parfaite (même
si ça détournait du travail..), alors merci à Cyril, Tommaso, Pierre, Emmanuel, Alexey,
Nino, Björn et Cyril B. J'ai aussi beaucoup aimé les activités annexes au bureau, cartes
et brassage! Alors merci encore à Anthony, Tomy et ce bon vieux Pavel. En fait, je
remercie simplement tous les membres du GAP, même si votre nom n'est pas là! Merci
aussi à tous les footeux avec qui j'ai passé d'excellents jeudi midi et même quelques
matchs du tournoi universitaire. Je suis content d'avoir succédé à Noé et Emmanuel
pour l'organisation du foot, et j'essaierai de revenir le plus souvent possible.
J'ai aussi beaucoup apprécié la disponibilité et l'aimabilité de toutes les secrétaires,
Isabelle, Laurence et Natalie. J'ai aussi beaucoup été aidé par tous les techniciens j'ai
57
aimé discuté avec vous tous, Jean-Daniel, Claudio, Olivier, Raphael et Mathieu.
Une petite dédicace encore à mes amis physiciens du master/bachelor. J'ai passé
d'immenses bon moments avec vous tous, et je suis heureux qu'on se voit toujours
régulièrement.
Un grand merci également à tous ceux qui étaient à la soutenance, j'ai vraiment
été très content que vous soyez là. Mais je remercie bien sûr aussi tous mes amis qui
n'étaient pas présents à ce moment, mais avec qui simplement je passe du bon temps.
Et j'aimerais nir par le plus important, un immense merci à tous les membres de
ma famille, sans qui je ne serais pas qui je suis.
58
Appendix A
Additional spectroscopy
g tensor of Nd
Nd3+:Y2SiO5is a Kramers doublets, hence the ground and the excited states behave
like a spin 12 system. The Hamiltonian of the system is :
→
−
→
−
H = µB B gµν S
→
−
where→
µB = 14GHz/T is the Bohr magneton, B the magnetic eld, g the Landé tensor
and −S a column vector with the three Pauli matrix. This 2 dimensions Hamiltonian
will give two eigen states separated by ∆E. Here we choose an external magnetic eld
B in the D1-D2 plane, at an angle θ with D2. Then, we have a Landé factor depending
of θ :
∆E = g(θ)µB B
We will label ∆Eg/e and gg/e for the ground/excited state. We want here to measure
gg/e for dierent θ. The rst technique to measure g is to do spectral hole burning. By
burning a spectral hole at a central frequency, the absorption spectrum will present
holes and anti-holes of absorption, because atoms have been moved to dierent states
(see g. A.1). The position of this holes and anti-holes depends of ∆Eg /e and can
be moved by a variable magnetic eld. Since their position is proportional to ge, gg
and gg ±ge, we could deduce the Landé factor from a linear t. For B k D2, we nd
gg (θ = 0) = 2.91 and ge (θ = 0) = 0.16.
Then, by changing only the magnetic eld orientation, we could deduce ge for all
angle θ by measuring the side-holes position (g. A.2).
We note that we could work with relatively low magnetic eld (<100 mT), because
magnetic dipole is high for electronic orbitals.
There is another method to measure the Landé factor without hole burning. If
we apply a strong magnetic eld, the splitting in the ground state and excited state
59
Hole burning spectrum
400
350
250
200
200
hole/anti-hole position [MHz]
α [m-1]
300
150
100
50
0
150
hole
anti-hole 1
anti-hole 2
anti-hole 3
-150
0
5
10
15
Magnetic field [mT]
-100
-50
20
25
0
Detuning [MHz]
50
100
Figure A.1: Absorption spectrum after hole burning with Bk D2. Because the Nd ions have
a split in the ground and excited state, there is a hole at central frequency, 2 side holes at
±∆Ee and 6 anti-holes at ±∆Eg and ±∆ Eg ±Ee . Inset: position of a side-hole and three
anti-holes in function of magnetic eld. Using a linear t, we can deduce the Landé factor in
the ground and excited state: gg (θ = 0) = 2.91 and ge (θ = 0) = 0.16
60
0.9
0.8
0.7
0.6
ge
0.5
0.4
0.3
0.2
0.1
0
-90 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 90
angle of B with D2
Figure A.2: Measure of the Landé factor in the excited state by measuring the position of
side holes for dierent orientation of magnetic eld. D1 and D2 are close of being axes of
symmetry.
become larger than the inhomogeneous broadening. Then, by looking at the absorption
spectrum(g. A.3), we can directly measure the splits for dierent magnetic elds and
calculate g using a linear t. The results are given in tab. A.1
As a useful result, we have now the values of ge for all directions of B in the D1-D2
plane. This is important to know the potential bandwidth of quantum memory in this
material. We can now predict the necessary magnetic eld for a given orientation to
have a minimal bandwidth. In any case, we note that the splits are large compare
to what we can have in non-Kramers doublets. The choice of the magnetic eld will
however depend of other important parameter such as coherence and lifetime in the
optical and spin transition.
θ
0
90
30
150
gg
ge
2.86 ±0.16
1.43 ± 0.05 0.78 ± 0.07
2.7±0.1
2.38±0.05 0.42±0.07
Table A.1: value of the Landé factor for dierent angle θ between magnetic eld and D2
using the absorption spectrum with a variable magnetic eld. The value for ge are consistent
with the measures using spectral holes in g.A.2.
61
2
α [cm-1]
1.5
1
0.5
0
Detuning
Figure A.3: Absorption spectrum for a magnetic eld of 0.95 T parallel to D1. The splits of
the ground and excited being larger than the inhomogeneous broadening, we can distinguish
the 4 transitions.
Then, we have the values for gg for a few angle. This can be compared to the work
of [99] where the whole tensor has been measured (g. A.4). The splits have probably
an inuence in the Zeeman state lifetime as we will see in the next section. Also, we
see that a strong magnetic eld divide the absorption by a factor 2 or 4 when the splits
become larger than the inhomogeneous broadening. In principle, this eect could be
compensated by an additional optical pumping.
Additionally we can estimate the branching ratio between the dierent transitions
using g. A.3. Indeed, we see the absorption is weaker for the optical transitions that
don't conserve the spin. We normalize the absorption, so that we get the coecient βi.
When both transitions are equally strong, then β1 = β2 = 0.5. We got an estimation
for two magnetic conguration in tab. A.2. For an angle of 150 with D2, β1=0.64,
which is encouraging for ecient optical pumping. For comparison, the ideal value
would be 0.5, but the branching ratio in Er3+:Y2SiO5is β ≈ 0.95, which aected badly
the optical pumping.
62
4
gg
3
2
1
-90 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20
Angle of B with D2
30 40 50
60
70 80
90
Figure A.4: Landé factor for a megnetic eld in the D1-D2 plane. We compare our results
(dots) with the work from [99] (line). It seems compatible within the errors.
θ
90
150
β1
0.76 ± 0.01
0.64 ± 0.01
Table A.2: Branching ratio for the strongest transition (conserving the spin) for two angles
between B and D2 .
63
64
List of Publications
[1] Mikael Afzelius, Imam Usmani, Atia Amari, Björn Lauritzen, Andreas Walther,
Christoph Simon, Nicolas Sangouard, Ji°í Miná°, Hugues de Riedmatten, Nicolas
Gisin, and Stefan Kröll. Demonstration of atomic frequency comb memory for
light with spin-wave storage. Phys. Rev. Lett., 104(4):040503, January 2010.
[2] A. Amari, A. Walther, M. Sabooni, M. Huang, S. Kroll, M. Afzelius, I. Usmani,
B. Lauritzen, N. Sangouard, H. de Riedmatten, and N. Gisin. Towards an ecient
atomic frequency comb quantum memory. Journal of Luminescence, 130(9, Sp.
Iss. SI):15791585, SEP 2010.
[3] Imam Usmani, Mikael Afzelius, Hugues de Riedmatten, and Nicolas Gisin. Mapping multiple photonic qubits into and out of one solid-state atomic ensemble. Nat
Commun, 1:12, April 2010.
[4] Christoph Clausen, Imam Usmani, Felix Bussieres, Nicolas Sangouard, Mikael
Afzelius, Hugues de Riedmatten, and Nicolas Gisin. Quantum storage of photonic
entanglement in a crystal. Nature, 469(7331):508511, January 2011.
[5] Imam Usmani, Christoph Clausen, Felix Bussieres, Nicolas Sangouard, Mikael
Afzelius, and Nicolas Gisin. Heralded quantum entanglement between two crystals.
Nat Photon, 6(4):234237, April 2012.
[6] N Timoney, B Lauritzen, I Usmani, M Afzelius, and N Gisin. Atomic frequency
comb memory with spin-wave storage in 153 eu 3 + :y 2 sio 5. Journal of Physics
B: Atomic, Molecular and Optical Physics, 45(12):124001, 2012.
[7] N Timoney, I Usmani, P Jobez, M Afzelius, and N Gisin. Single-photon-level
optical storage in a solid-state spin-wave memory, 2013.
65
66
Appendix B
Published articles
67
PRL 104, 040503 (2010)
PHYSICAL REVIEW LETTERS
week ending
29 JANUARY 2010
Demonstration of Atomic Frequency Comb Memory for Light with Spin-Wave Storage
Mikael Afzelius,1,* Imam Usmani,1 Atia Amari,2 Björn Lauritzen,1 Andreas Walther,2 Christoph Simon,1
Nicolas Sangouard,1 Jiřı́ Minář,1 Hugues de Riedmatten,1 Nicolas Gisin,1 and Stefan Kröll2
1
Group of Applied Physics, University of Geneva, CH-1211 Geneva 4, Switzerland
2
Department of Physics, Lund University, Box 118, SE-22100 Lund, Sweden
(Received 14 August 2009; published 27 January 2010)
We present a light-storage experiment in a praseodymium-doped crystal where the light is mapped onto
an inhomogeneously broadened optical transition shaped into an atomic frequency comb. After absorption
of the light, the optical excitation is converted into a spin-wave excitation by a control pulse. A second
control pulse reads the memory (on-demand) by reconverting the spin-wave excitation to an optical one,
where the comb structure causes a photon-echo-type rephasing of the dipole moments and directional
retrieval of the light. This combination of photon-echo and spin-wave storage allows us to store
submicrosecond (450 ns) pulses for up to 20 s. The scheme has a high potential for storing multiple
temporal modes in the single-photon regime, which is an important resource for future long-distance
quantum communication based on quantum repeaters.
DOI: 10.1103/PhysRevLett.104.040503
PACS numbers: 03.67.Hk, 42.50.Gy, 42.50.Md
A quantum memory (QM) for photons is a light-matter
interface that can achieve a coherent and reversible transfer
of quantum information between a light field and a material system [1]. A QM should enable efficient, high-fidelity
storage of nonclassical states of light, which is a key resource for future quantum networks, particularly in quantum repeaters [2–6]. In order to achieve reasonable entanglement distribution rates, it has been shown that some
type of multiplexing is required [4,5], using for instance
independent frequency, spatial or temporal modes (multimode QM).
Several types of light-matter interactions have been
proposed for building a QM, for instance electromagnetically induced transparency [7–10], Raman interactions
[11–14], or photon-echo techniques [15–21]. Photonecho techniques in rare-earth-ion doped crystals have an
especially high multimode capacity for storing classical
light [22]. Classical photon echoes are not useful, however,
for single-photon storage due to inherent noise problems
[23]. The photon-echo QM based on controlled reversible
inhomogeneous broadening [15–18] is free of these noise
problems. But this technique has a lower time-multiplexing
capacity than classical photon echoes, for a given optical
depth, due to loss of storage efficiency as the controlled
frequency bandwidth is increased [19,24]. Some of us
recently proposed a photon-echo type QM based on an
atomic frequency comb (AFC) [19] that has a storage
efficiency independent of the bandwidth, allowing optimal
use of the inhomogeneous broadening of rare-earth-doped
crystals. An AFC memory has the potential for providing
multimode storage capacity [19,24] crucial to quantum
repeaters. The few reported AFC experiments [20,21]
have been investigating the physics of the optical AFC
echo, where the memory storage time is predetermined
by the periodicity of the comb. For quantum repeaters it
is crucial to be able to choose the time of the memory
0031-9007=10=104(4)=040503(4)
readout (on-demand readout). Here we present the first
light-storage experiment where an AFC is used in combination with reversible transfer of the excitation to a spin
state [19], resulting in on-demand readout and storage
times longer than 20 s.
The underlying idea of the AFC QM is to shape an
inhomogeneously broadened optical transition jgi ! jei
into a periodic series of narrow and highly absorbing
peaks with periodicity , see Fig. 1. A photon with a
bandwidth that is matched to the width of the AFC
structure is then stored as an optical excitation delocalized over the peaks, which we can write as j c i ¼
PN
ij t ikzj
e
jg1 ej gN i where N is the number
j¼1 cj e
of atoms in the AFC, j is the detuning of atom j with
respect to the laser frequency, zj is the position, k is the
wave number of the light field, and the amplitudes cj
depend on the frequency and on the spatial position of
the particular atom j. The terms in this large superposition
state accumulate different phases due to the inhomogeneous distribution of atomic resonance frequencies, resulting
in a loss of the initially strong collective coupling to the
light mode. But the periodic AFC peak separation leads
to a rephasing of the terms after a time 1=, which restores
the strong collective coupling, leading to a photon-echo
type reemission [25], the AFC echo. The narrow and highly
absorbing peaks can theoretically absorb all the light and
completely emit the energy in the AFC echo [19]. A large
number of peaks leads to a high multimode capacity
[19,24].
In the original proposal [19] on-demand readout and
longer storage times rely on reversible transfer of the
optical excitation to a long-lived spin state by strong
control pulses (see Fig. 1). If we imagine a perfect pulse
applied at time T 0 , each term in the superposition state
becomes jg1 sj gN i; thus, we have a single collec-
040503-1
Ó 2010 The American Physical Society
PRL 104, 040503 (2010)
(b)
(a)
g
5/2
2.0
e
1.5
e
3/2
Output mode
Control fields
Input mode
0.5
2.0
0.0
0.4 0.5 0.6 0.7 0.8
1.5
s
g
3/2
s
5/2
aux
e
1.0
0.5
Absorption depth d
1.0
1/2
1/2
0.0
(c)
Intensity
0
2
4
6
8
10
Frequency (MHz)
Input
mode
Output
mode
T
week ending
29 JANUARY 2010
PHYSICAL REVIEW LETTERS
Ts
T
Time
FIG. 1 (color online). (a) The experiment was performed on
the 3 H4 ! 1 D2 transition in Pr3þ . The ground and excited state
manifolds both have three hyperfine levels denoted MI ¼ 1=2,
3=2, 5=2. The three-level lambda system was formed by the
levels labeled jgi, jsi and jei, following the notation in [19].
(b) Experimental absorption spectrum showing the AFC on the
jgi $ jei transition created within a 18 MHz wide transmission
hole using the spectral holeburning sequence described in the
text. Here the comb consists of 9 peaks with spacing ¼
250 kHz. The holeburning sequence also empties the jsi level,
whereas jauxi is used for population storage. Note that the
second AFC in the center of the spectrum is due to the weaker
transition from the ground state 1=2 to the excited state 5=2.
(c) The pulse sequence showing the input pulse to store, the two
control fields for the back-and-forth transfer to the spin state, and
the retrieved output pulse.
tive spin-wave excitation. If we assume that the spin transition is homogeneously broadened, then each term is
0
frozen with the phase term eij T due to the time spent in
the excited state [19]. In practice, inhomogeneous spin
broadening adds other phase factors and reduces the collective spin wave, as will be discussed below. After a time
Ts in the spin state, another control pulse transfers the
excitation back to the excited state and the AFC evolution
resumes, leading to a reemission after a total storage time
T 0 þ Ts þ T 00 where T 0 þ T 00 ¼ 1=.
The storage material used in this experiment is a
praseodymium-doped Y2 SiO5 crystal (Pr3þ concentration
of 0.05%) with an optical transition at 606 nm. The optical
homogeneous linewidth at cryogenic temperature is around
1 kHz, whereas the inhomogeneous broadening is about
5 GHz [26]. The ground and excited states both have a
hyperfine manifold consisting of three closely spaced levels [Fig. 1(a)], assuming no applied magnetic field. Three
ground-state levels are necessary for the experiment, see
below, which was our main motivation for choosing Pr3þ .
The different hyperfine transitions are usually hidden
within the large inhomogeneous broadening. By spectral
hole burning techniques one can, however, isolate a subensemble of atoms whose different hyperfine transitions
can be unambiguously excited. This distillation technique,
which we will summarize here, has been the subject of
several papers [27–29]. A laser beam whose frequency is
swept pumps atoms from ground levels jgi and jsi to the
auxiliary storage level jauxi; see Fig. 1, which creates a
wide transmission window within the inhomogeneous profile. In the next step a narrow absorption peak is created in
the hole by coherently transferring back atoms, within a
narrow frequency range, from jauxi to jgi [28]. In this
experiment we extended this method by transferring back
atoms at different frequencies to create a frequency comb.
In Fig. 1(b) we show the absorption spectrum recorded
after the preparation sequence. The AFC created on the
jgi $ jei transition is clearly visible. The jsi level is used
for the spin-wave storage, which means that the control
pulses will be applied on the jsi $ jei transition displaced
by 10.2 MHz with respect to the jgi $ jei transition. Note
that the absorption spectrum in Fig. 1(b) is shown only for
visualization purposes, since the fast frequency scan
method [29] we used leads to distortions for absorption
depths d * 2. The experimental setup is shown in Fig. 2.
The control pulses were both counterpropagating with
respect to the input pulse. By phase matching condition
the output pulse then copropagates with the input signal
[19]. Using this configuration we could reduce noise due to
off-resonant free-induction decay emission produced by
the strong control pulses.
In a preliminary experiment we investigate the AFC
echo on the jgi $ jei transition without applying the control pulses, see inset in Fig. 3. This allows us to optimize
the relevant comb parameters in order to obtain a strong
echo. The input pulse duration was set to 450 ns so that the
bandwidth is entirely contained in the 2 MHz wide AFC.
The efficiency of this AFC echo, which we define as the
Control
Input
Pr:Y2 SiO5
15%
Output
85%
BS
Input/Output
PD
FIG. 2 (color online). The experimental setup. The spectral
hole burning and storage pulse sequences were created using a
frequency-stabilized laser and acousto-optic modulators similar
to the setup in [28] (not shown). A beam splitter (BS) split the
light into a strong and a weak beam, whose two modes were
overlapped in the crystal cooled to 2.1 K. The detected output
pulse propagating to the right originated from the combination of
a weak input pulse incident on the crystal from the left, and two
strong control pulses incident from the right. Note that the signal
disappeared when the control pulses were blocked directly after
the BS. An AOM was used to direct only the transmitted input
pulse and the output pulse onto the photodiode (PD), effectively
working as a detector gate.
040503-2
PRL 104, 040503 (2010)
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29 JANUARY 2010
PHYSICAL REVIEW LETTERS
ratio of the AFC echo area to the input pulse area, depend
on the shape of the AFC [19]. Two critical parameters are
the peak absorption depth d and the finesse defined as F ¼
= where is the full-width at half maximum of a peak.
For instance, a high finesse leads to low decoherence
during the storage time 1=, but also to a lower effective
absorption d=F of the input pulse [19]. The peak absorption d could be controlled by the power of the laser beam
creating the peaks in the transmission hole, but a high
power also had an impact on the finesse by causing power
broadening of the peaks. The peak width was also limited
by laser frequency stability, resulting in typical widths of
100 kHz. For a periodicity of ¼ 1 MHz, the optimized efficiency was about 15%. The delay of 1 s was
not sufficiently long, however, for applying the control
pulses before the emission of the AFC echo. We therefore
set the periodicity to ¼ 250 kHz giving us 4 s to apply
the control pulses. The closer spacing of the peaks lowered
the finesse of the comb, thus lowering the efficiency to
5% (see inset in Fig. 3). This efficiency is in reasonable
agreement with numerical simulations using the experimentally estimated peak absorption depth and finesse of
the comb.
In Fig. 3 we show the main result where two control
pulses are applied on jsi $ jei to transfer the excitation to
the jsi hyperfine level. The retrieved pulse is clearly observed above the noise level. This realizes a true storage of
the input pulses, with on-demand readout. Thus, the control pulses provide a mechanism for momentarily interrupting the predetermined AFC evolution [19]. We tested this
mechanism in detail by varying the time at which the first
control pulse was applied T 0 ¼ ð1:17; 1:63; 2:23Þ s
[cf. Fig. 1(c) for notation]. This resulted in different measured durations T 00 ¼ ð2:84; 2:41; 1:85Þ s. The total time
spent in the excited state jei, however, is constant (within
the measurement error), T 0 þ T 00 ¼ ð4:01; 4:04; 4:08Þ s,
corresponding to the expected 1=.
In Fig. 4 we show storage experiments where the spinwave storage time Ts is varied. The output signal is clearly
visible up to 20 s of total storage time (1= þ Ts ). The
exponential decay of the output signal as a function of Ts
can be attributed to inhomogeneous spin dephasing, corresponding to an inhomogeneous broadening of 26 kHz
consistent with previous measurements [30]. We point out
that this can be compensated for by spin echo techniques.
With such techniques Longdell et al. [31] stopped light
during >1 sec in a Pr3þ :Y2 SiO5 crystal using electromagnetically induced transparency.
We now discuss the total storage efficiency . Clearly is bounded by the AFC echo efficiency e ¼ 4%–5% for
1= ¼ 4 s (cf. Fig. 3 inset). This can be improved by
increasing the finesse and optical depth (see discussion
above). The spin-wave storage further decreases the efficiency. By extrapolating the experimentally measured to
the limit Ts ! 0 we find that ¼ 0:5%–1%, which is a
value independent of spin dephasing. The effect of the
control pulses can now be understood by a simple model.
We assume that one pulse has a single-atom jsi ! jei
transfer efficiency T , which is constant as a function of
detuning. Then the application of the two control pulses
reduce the efficiency to ¼ e 2T . Numerical simulations
using 3-level Maxwell-Bloch equations show that this
simple model is correct if the control pulses do not induce
any additional phase decoherence. From the experimental
values given above we thus find T 0:30–0:45.
To better understand this value we have performed
numerical calculations. The control pulses were two identical complex hyperbolic secant pulses, which can achieve
Intensity (arb. units)
1.0
Normalized intensity
14
12
10
8
16
Intensity (arb. units)
Normalized intensity
5
12
8
4
0
0
6
1
2
3
Time ( s)
4
4
x10
2
Control pulses
2
4
6
8
10
3
0.6
0.4
0.2
0.0
2
0
5
10 15 20 25 30
Duration of spin storage TS ( s)
1
0
0
0
4
0.8
12
-2
Time ( s)
FIG. 3 (color online). Storage in the spin state using two
control pulses. Shown are (from left) the partially transmitted
input pulse, control pulses (strongly attenuated by the closed
optical gate), and the output pulse (magnified by a factor 10 for
clarity). Here 1= ¼ 4 s and Ts ¼ 7:6 s, resulting in a total
storage time 1= þ Ts ¼ 11:6 s. Inset: The AFC echo observed at 1= ¼ 4 s when the control pulses are not applied.
Note that the vertical scales have been normalized to 100 with
respect to the input pulse before the crystal; thus, these yield (in
percent) the efficiency for the echo and transmission coefficient
for the input pulse.
0
10
12 14 16
Time ( s)
18
20
FIG. 4 (color online). Experimental traces for spin-storage
times Ts ¼ 5:6, 7.6, 10.6, and 15:6 s. All other parameters
are the same as those in Fig. 3. The input pulses are superimposed to the left (truncated) and the output pulses for different
Ts are seen to the right. The leakage of the control pulses through
the optical gate is not shown. For clarity there is also a break in
the horizontal scale. The decay of the signal (see inset) as a
function of Ts is due to inhomogeneous spin dephasing. The
solid curve is a fitted Gaussian function corresponding to 26 kHz
(full-width at half maximum) spin broadening.
040503-3
PRL 104, 040503 (2010)
Input modes
1.0
Intensity (arb. units)
PHYSICAL REVIEW LETTERS
0.8
0.6
Control
pulses
0.4
Output modes
0.2
0.0
0
2
4
6
8
10
12
14
16
Time ( s)
FIG. 5 (color online). Storage of two temporal input modes.
The normalized input modes shown here (dashed line) are
recorded before the crystal using a reference detector. The
AFC was prepared with a periodicity of ¼ 200 kHz, corresponding to 5 s storage time on the optical transition. The spinstorage time was set to Ts ¼ 7:6 s, resulting in a total storage
time of 1= þ Ts ¼ 12:6 s.
efficient, broadband transfer of population [28] and coherence [29]. Our pulses had duration 600 ns and Rabi frequency 1:2 MHz (close to the maximal value in this
experiment), the frequency chirp being 2 MHz. These
values were found by empirically optimizing the size of
the output pulse. Using the numerical model we find T ¼
0:75, also averaged over the bandwidth of the AFC. Based
on this we would expect a total efficiency ¼
5% 0:752 ¼ 2:8% using the simplified model above,
which is significantly higher than what we observe. The
most probable explanation for the discrepancy is imperfect
spatial mode overlap between the counterpropagating input and control pulses (see Fig. 2). A larger control beam
would make a more spatially uniform Rabi frequency and
would facilitate mode alignment. Note that the theoretical
T can be improved by increasing the Rabi frequency (by a
factor of 2) and adapting the duration and chirp of the pulse
in order to achieve an efficient (T 95%) transfer.
We finally show an example of storage of two temporal
modes; see Fig. 5. Note that both modes are stored with the
same efficiency, which is a particular feature of the AFC
memory due to the fact that each mode spend the same
total time (1= þ Ts ) in the memory. The number of
modes we could store was mainly limited by the number
of peaks Np that could be created in the AFC, since the
number of input modes one can store is proportional to Np
[19]. Np could be increased by making narrower peaks
and/or a wider AFC. The width is currently limited by the
separation of the hyperfine transitions (cf. Fig. 1), which
could be increased via the nuclear Zeeman effect. The most
significant improvement can be made by creating narrower
peaks, which in principle can approach the homogeneous
linewidth of about 1 kHz. A multimode storage capacity in
the range of tens of modes appears feasible. Other rareearth-ion-doped crystals have even higher multimode potential, for instance Eu3þ :Y2 SiO5 as discussed in Ref. [19].
In conclusion, we have demonstrated the first lightstorage experiment combining an atomic frequency comb
week ending
29 JANUARY 2010
and spin-wave storage. Using this method we stored optical
submicrosecond (450 ns) pulses for up to 20 s as a spinwave in Pr3þ :Y2 SiO5 . This optical bandwidth is more than
1 order of magnitude higher than previous stopped-light
experiments demonstrated in rare-earth crystals [31]. The
spin-storage time could be greatly extended by spin echo
techniques [31].
The authors acknowledge useful discussions with Pavel
Sekatski. The work was supported by the Swiss NCCR
Quantum Photonics, the Swedish Research Council, the
Knut and Alice Wallenberg Foundation, the Crafoord
Foundation, the ERC Advanced Grant QORE, the Lund
Laser Center, and the EC projects Qubit Applications
(QAP) and FP7 Grant No. 228334.
*[email protected]
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040503-4
ARTICLE IN PRESS
Journal of Luminescence 130 (2010) 1579–1585
Contents lists available at ScienceDirect
Journal of Luminescence
journal homepage: www.elsevier.com/locate/jlumin
Towards an efficient atomic frequency comb quantum memory
A. Amari a,, A. Walther a, M. Sabooni a, M. Huang a, S. Kröll a, M. Afzelius b, I. Usmani b, B. Lauritzen b,
N. Sangouard b, H. de Riedmatten b, N. Gisin b
a
b
Department of Physics, Lund University, P.O. Box 118, SE-22100 Lund, Sweden
Group of Applied Physics, University of Geneva, CH-1211 Geneva 4, Switzerland
a r t i c l e in fo
abstract
Available online 1 February 2010
We present an efficient photon-echo experiment based on atomic frequency combs [Phys. Rev. A 79
(2009) 052329]. Echoes containing an energy of up to 35% of that of the input pulse are observed in a
Pr3 + -doped Y2SiO5 crystal. This material allows for the precise spectral holeburning needed to make a
sharp and highly absorbing comb structure. We compare our results with a simple theoretical model
with satisfactory agreement. Our results show that atomic frequency combs has the potential for highefficiency storage of single photons as required in future long-distance communication based on
quantum repeaters.
& 2010 Elsevier B.V. All rights reserved.
Keywords:
Quantum repeater
Quantum memory
Atomic frequency comb
Storage efficiency
1. Introduction
The distribution of entanglement over long distances is a
critical capability for future long-distance quantum communication (e.g. quantum cryptography) and more generally for quantum
networks. This can be achieved via so-called quantum repeaters
[1–4], which can overcome the exponential transmission losses in
optical fiber networks. Quantum memories (QM) for photons [5–
9] are key components in quantum repeaters, because the
distribution of entanglement using photons is of probabilistic
nature due to the transmission losses over long quantum
channels. QMs enables storage of entanglement in one repeater
segment until entanglement has also been established in the
adjacent sections. For quantum repeaters a QM should be able to
store single-photon states with high conditional fidelity F and
with high storage and retrieval efficiency, Z [4]. Further, it has
recently been shown that in order to reach useful entanglement
distribution rates in a repeater, QMs with multiplexing capacity
(multimode QM) are necessary [3,10].
Significant progress have been achieved lately using atomic
ensembles for manipulating the propagation and quantum state
of an optical field, see Hammerer et al. [11] for a recent review.
Storage of single photons using electromagnetically induced
transparency (EIT) has been demonstrated with warm [7] and
cold vapors [6,8] of alkali atoms. Storage of light at the single
photon level has been demonstrated also in rare-earth-ion-doped
crystals (REIC) [9]. REICs are characterized by large optical
inhomogeneous broadening which enables storage and recall of
Corresponding author. Tel.:+ 46 462229625.
E-mail address: [email protected] (A. Amari).
0022-2313/$ - see front matter & 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.jlumin.2010.01.012
coherent information by manipulating and controlling the
inhomogeneous dephasing using echo techniques. Although traditional photon echoes cannot be used in the single-photon regime
due to spontaneous-emission noise induced by the p-pulse [12],
photon echo techniques avoiding this noise have been proposed;
controlled reversible inhomogeneous broadening (CRIB) [13–17]
and more recently atomic frequency combs (AFC) [18]. The AFC
protocol may offer a breakthrough for the practical construction
of quantum repeaters capable of achieving sufficient entanglement distribution rate, since the number of modes that can be
stored in an AFC QM is independent of the memory material
absorption depth. Since the proposal of the AFC scheme, storage of
light pulses at the single photon level (so called weak coherent
states) has been shown in Nd3 + :YVO4 [9]. The fidelity of the
storage was measured by storing a time-bin qubit and performing an interference measurement on the recalled qubit. The
resulting interference fringe visibility was V= 95%, which corresponds to a fidelity F= (1 +V)/2 [19] of 97.5%. This shows that light
at the single photon level can be stored and retrieved without
introducing noise, and future experiments are likely to improve
the fidelity further. The combined storage and retrieval efficiency,
however, was only 0.5% in that experiment. A more recent
experiment in Tm3 + :YAG [20] showed improved efficiency of 9%,
also with weak coherent states. Finally, storage of 64 weak
coherent states encoded in different temporal modes has been
achieved in Nd3 + :Y2SiO5 [21], underlying the high multimode
capacity of the AFC scheme. In view of these encouraging results
in terms of fidelity and multimode storage, it is clear that increasing the efficiency is of great importance, particularly in the
perspective future long-distance quantum repeaters where QM
efficiencies of 90% are necessary with the architectures known
today [4].
ARTICLE IN PRESS
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Here we report a photon-echo experiment based on an AFC in a
Pr -doped Y2SiO5 crystal. We measure echoes containing an
energy of up to 35% of that of the input pulse, which is the highest
AFC echo efficiencies measured so far. This shows that AFC-based
schemes can be used for efficient light storage. This improvement
is possible because of a good control of the procedure that creates
the atomic frequency comb, via optical pumping techniques, and
because of a storage medium with high optical depth. Combs
with peaks of widths 100–300 kHz with peak absorption depths
approaching 10 were created inside a transparent region created
by optical pumping techniques in a part of the inhomogeneous
profile in a Pr3 þ : Y2 SiO5 crystal. We also examine parameters
related to the experimental optimization of the efficiency and
compare to a theoretical model in order to understand how to
further improve the efficiency of storage and retrieval from a
memory using the AFC scheme.
The paper is organized in the following way. In Section 2 we
give an overview of the theory of AFC. In Section 3 the
experimental setup is described. Preparation of a narrow periodic
series of absorption peaks is discussed in Section 4. In Section 5
experimental results of AFC echoes are presented and compared
with the theoretical model. Conclusions are given in Section 6.
3+
pffiffiffiffiffiffiffiffiffiffiffiffi
2 2
the decay (dephasing) is given by et g~ =2 where g~ ¼ g= 8 ln 2. For
the first echo emission at t ¼ 2p=D this dephasing factor becomes
2
2
eð1=F Þðp =4ln2Þ (note that the factor applies to the field amplitude),
where F ¼ D=g. From this observation it follows that a highfinesse grating strongly reduces the intrinsic dephasing. In general
the dephasing factor (for the field amplitude) is given by the
Fourier-transform of one peak in the comb.
Obtaining a high efficiency echo also requires a strong
interaction between the ensemble of atoms and the field, which
can be achieved by a high absorption depth, d. It is shown in Ref.
[18] that the comb absorbs uniformly over the photon bandwidth,
under the assumption that gp 4 D. The effective absorption d~
depth depend on the exact shape of the peaks in the comb, but in
general it decreases with increasing F for a given peak amplitude,
since the total number of atoms decreases. For Gaussian peaks
one finds that d~ d=F, and the fraction of the input light that is
transmitted through the AFC is given by [18]
~
T ¼ ed ;
ð1Þ
while the absorption is simply given by 1 T. For an AFC
consisting of peaks with Gaussian line shape the resulting echo
efficiency is given by (see Ref. [18] for the derivation)
2
~
1
p2
Z ¼ d~ ed eF2 2 ln 2 ;
2. Theory of AFC
We consider an ensemble of atoms with a transition jgSjeS
having a narrow homogeneous linewidth gh , but a large
inhomogeneous broadening Gin b gh . There are thus many
addressable spectral channels within the optical transition. We
also assume that there is at least one more meta-stable ground
state, jauxS, having a long population lifetime. This allows a highresolution spectral shaping of the jgSjeS transition by spectral
hole burning, where jauxS is used as population storage reservoir.
These properties are often found in rare-earth-ion-doped crystals
[22,23], which are considered here. The detailed experimental
procedure for precise spectral shaping depend on the particular
system. In Section 4 we discuss the procedure for Praseodymium
doped Y2SiO5 crystals.
We assume that the inhomogeneously broadened transition
has been shaped into a periodic series of narrow peaks, called an
atomic frequency comb, see Fig. 3. We further assume that the
light pulse to be stored has a spectral bandwidth, gp , larger than
the periodicity in the comb (gp 4 D), but smaller than the total
comb structure. The interaction between the input pulse and a
ground-state population grating versus frequency generally
results in a photon echo emission after a time 1=D, which is used
in accumulated or spectrally programmed photon echoes [24–29].
The echo emission arises from the evolution of the atomic
coherence induced by the input pulse, which periodically
rephases due to the periodicity in the atomic population grating.
In typical echo experiments only a small fraction of the input
pulse is re-emitted in the echo and the storage time is not variable
since it is set by the predetermined grating periodicity D. This is
not useful for quantum repeaters where efficiencies close to 100%
and on-demand read-out of the quantum memory is necessary
[3]. Solutions to these issues were, however, recently proposed in
Ref. [18].
In Ref. [18] it is shown theoretically that a comb-shaped
grating consisting of sharp and strongly absorbing peaks could
generate a very efficient echo. This can be understood in terms of
the Fourier-transform of the grating function, which governs the
evolution of the atomic coherence. The periodicity in frequency
results in a periodic time evolution, with an overall decay given by
the width of the peak in the comb. For a series of well-separated
Gaussian peaks, with full width at half maximum (FWHM) g,
ð2Þ
where qualitatively the first factor can be understood as the
coherent response of the sample, the second factor the
re-absorption of the echo and the last factor the previously
mentioned dephasing. For a high finesse, F, and high peak
absorption d, the efficiency tends to a maximum of 54% for an
effective absorption depth d~ ¼ 2, limited by re-absorption of the
echo. Higher efficiency can be achieved using three-level storage
and counter-propagating fields [18] (see below). In this work we
show experimental efficiencies up to 35%, which is significantly
higher than previous AFC experiments. This improvement results
from our ability to make high finesse, high absorbing comb
structures.
We also note that a solution to the predetermined storage time
was proposed in Ref. [18] (see above). It is based on coherent
transfer of the excited state amplitude to a long-lived ground
state coherence, for instance a spin coherence, before the
appearance of the echo. The memory can be read-out by
transferring back the amplitude to the excited state, after a time
determined by the user. This aspect of the proposal was recently
demonstrated experimentally [30]. A three-level system and
counterpropagating control pulses allows for a spatial reversal
of the propagation of the echo (so called backward recall). In the
absence of dephasing backward recall can reach 100% efficiency
by cancelation of the re-absorption, as discussed in Refs.
[13,15,18,31,32].
3. Experiment
The measurements were performed on the site 1 transition
D2 3H4 at 605.977 nm in a Pr3 þ : Y2 SiO5 crystal immersed in
liquid helium at a temperature close to 2.1 K. The sample was
20 10 10 mm3 and had a Pr3 + concentration of 0.05% which
gives an absorption depth in the range 60 od o80 [33] at the
center of the inhomogeneous profile. The high absorption was
critical in order to obtain highly absorbing peaks (see Section 4).
A ring dye laser (Coherent699-21) using Rhodamine 6G
pumped by Nd:YVO4 laser (Coherent Verdi) is used to give
600 mW output power at l ¼ 605:977 nm. The laser is stabilized
against a spectral hole in a second Pr3 þ : Y2 SiO5 crystal, yielding a
coherence time 4 100 ms and a frequency drift o1 kHz=s [34]. In
order to create the desired pulse shapes and to eliminate beam
1
ARTICLE IN PRESS
A. Amari et al. / Journal of Luminescence 130 (2010) 1579–1585
movement accompanying frequency shifts, the laser light was
passed twice through a 200 MHz acousto-optic modulator (AOM)
with a bandwidth of 100 MHz. A 1 GS/s arbitrary waveform
generator (Tektronix AWG520) controlled the AOM, allowing
direct control of the light pulse amplitude, phase, and frequency.
After the AOM, the light passed through a single mode optical
fiber to clean up the spatial mode. A beam sampler removed a
small percentage of the light before the cryostat to be used as a
reference beam. The rest of the beam passed through a l=2 plate,
such that the light polarization could be aligned along the
transition dipole moment direction to give maximum absorption.
The beam was then focused to a 100 mm radius at the center of the
sample, which gave Rabi frequencies of maximum 2 MHz for the
strongest transitions.
The spectral structures were measured by scanning the light
frequency across the spectral structure and recording the
intensities of both the transmitted and the reference beams
[35]. The signals from the detectors were divided to reduce the
effect of laser amplitude fluctuations. The intensity of the probe
pulses were chosen such that they did not affect the created
spectral structures during the readout process. The scan rate was
also set such that it had negligible effect on the resolution of the
recorded spectra (see discussion below).
1581
electromagnetically induced transparency (EIT) [36], slow light
[33] or quantum computing [37,38]. Since the precise control of
the absorption structure is of particular concern for this paper, we
will here make a detailed description of how to create a test
platform for AFC experiments and essentially the same technique
can be used also for the other listed experiments mentioned
above.
4.1. Pit creation
The inhomogeneous 1D2 3H4 absorption line in Pr3 þ : Y2 SiO5
is about 5 GHz and the homogeneous line width of a single Pr ion
is about a kHz at temperatures below 5 K. A chirped laser pulse
applied somewhere within the inhomogeneous profile will create
a spectral hole through optical pumping. The maximum width of
the spectral hole burnt by such a scan is given by the specific level
structure of the Pr3 þ ion [39] (see Fig. 2). After relaxation from
excited states the ion has to be in one of the three hyperfine
ground levels, which have a total separation of 27.5 MHz, but the
maximum spectral hole width is reduced by the hyperfine
splitting of the excited state levels of 9.4 MHz, yielding a final
spectral hole interval of 18.1 MHz. The scanned pulse will thus
create a simple, wide spectral hole, henceforth called a pit, and is
shown in Fig. 1a.
When light pulses are applied to perform operations inside the
pit, they also have a probability to interact with the tails of the
absorption profile of the ions outside the pit, and in particular
with the ions immediately outside, forming the walls of the pit.
Generally one would like to avoid such interactions and the
simple pit in Fig. 1a then is not optimal. Fortunately, it is possible
to shuffle many ions in the walls of the pit further away from the
pit. This is illustrated in the top right part of Fig. 1. In this figure, a
class of ions having the j0S-je1S transition at some specific
frequency just inside the pit and the transitions from the other
two ground state levels outside the pit, is displayed. For this ion
class, the simple burning pulse only targets the j0S-state, so only
this ground state will be emptied. However, it is clear that these
4. Preparation of AFC
Creating the atomic frequency comb structure, with good
control of all necessary parameters, such as peak height, width,
separation and number of peaks, can be challenging. Especially
considering that the frequency comb structure also preferably
should be well separated in frequency from all other absorbing
atoms in the material. However, in rare-earth-metal-ion-doped
crystals the inhomogeneous absorption profile can indeed be
efficiently manipulated, providing the flexibility needed to
meet all those requirements. This flexibility is useful not only
for AFC, but also for many other similar experiments, such as
Absorption
Absorption
Laser scan
10.2 MHz
1.2
1
0.8
0.6
0.4
0.2
0
-10
Absorption(αL)
Absorption(αL)
<18 MHz
-5
0
5
Frequency (MHz)
10
0
2.5
17.3 MHz
aux
1
2
1.5
1
0.5
0
-20
-10
0
10
20
30
Frequency (MHz)
Fig. 1. (color online) In (a) a simple spectral pit created only by scanning a pulse across a specific interval o 18 MHz is shown (upper part shows a schematic view and
lower part shows actual experimental data). In (b) a more optimal pit is shown, where additional burning pulses on different frequency intervals have iteratively been
applied to spectrally remove ions as far from the spectral pit as possible (see text). For an exact pulse sequence, see Appendix A. Note that the frequency scale in the two
experimental figures is different.
ARTICLE IN PRESS
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A. Amari et al. / Journal of Luminescence 130 (2010) 1579–1585
ions do not have to be in the j1S state, in fact, it would be better
if they could be further shifted outwards so that all end up in
the jauxS state. This can be done by additional burning pulses at
the j1S-je1S transitions. This will cause some ions to go to the
jauxS-state but also cause some of the ions to fall back down into
the pit, and thus, to get the optimal effect, one would have to
iterate between the pulses burning at the center of the pit and the
ones burning outside it to improve the walls. Similar techniques
can be used on the lower frequency side of pit to obtain the final
optimal pit, as shown in the lower part of Fig. 1b.
Sechyp 2
Absorption
Sechyp 1
12
|0 >
4.3. Peak creation
After a suitable pit has been created, a narrow selection of
ions is coherently burnt back into the pit. This narrow ensemble of
ions now forms an absorption peak spectrally clearly separated
from all other ions. The pulses used for this transfer are two
complex hyperbolic secant pulses (sechyp for short). The first
one targets the j5=2gS-j5=2eS transition, of ions having their
j1=2gS-j1=2eS transition at frequency zero (0) MHz. The second
pulse is applied immediately after the first, before the excited ions
can decay spontaneously, on the j5=2eS-j1=2gS transition. One
could imagine taking other routes to burn back a peak, but this
particular route is advantageous because the first pulse targets
the strongest transition, which means the exciting pulse power,
and thus power broadening effects, can be kept at a minimum.
The deexcitation pulse on the other hand then targets a weak
transition, but since this transition is inside the pit and spectrally
far away from other ions, the power can here be increased
without any power broadening effects.
Fig. 2 briefly illustrates how to create such a peak structure,
and also displays an experimentally created version. Creating a
full atomic frequency comb from this situation, is now the
relatively simple matter of adding additional coherent burn-back
pulses, with appropriate frequency offsets, creating the additional
peaks. The shape of the peaks, as well as width and absorption
height, is determined by the burn-back pulses. Changing the
spectral shape of these pulses will change the shape of the peaks,
and increasing the pulse power will cause more ions to be
transferred, which results in higher absorption peaks (as long as
there is enough ions available in the crystal at that frequency).
Absorption (αL)
5/2e
3/2e
1/2e
1/2g→1/2e
0.6
|0 >
|1 >
|aux >
|aux >
|1 >
4.2. Experimental implementation
The exact sequence of pit burning pulses differ depending on
the exact level structure of the ion used. Table 1 in Appendix A
lists the explicit pulses used to create an optimal pit in
Pr3 þ : Y2 SiO5 in this work, and the order in which we have
applied those pulses is also listed in Appendix A. When working in
other materials, essentially the same sequencing can be used, but
of course, the actual frequencies have to be changed to match the
transitions of the ion in question.
The different optical pumping BurnPitX pulses listed in Table 1
(see Appendix A) are repeated and iterated, as explained in the
previous section, in order to create good shallow walls while
maintaining no atoms inside the pit. The repetition sequence
given at the end of the Appendix A is somewhat arbitrary.
A higher number of repetitions reflects the fact that the primarily
target transition has relatively low oscillator strength. The exact
numbers can be changed a bit up or down without significant
effect on the result. There is a 1 ms waiting time after every single
BurnPitX pulse (see Appendix A), to give excited ions time to decay
back to the ground state before the next pulse arrives. The excited
state lifetime is T1 ¼ 164 ms [39].
|e >
|e >
1/2g→3/2e
0.4
0.2
1/2g→5/2e
0
-2
0
2
4
6
8
10
Frequency(MHz)
12
1/2g
14
16
|0 >
|1 >
|aux >
Fig. 2. (Color online) Top part shows the sequence and position of the pulses that
burns back a narrow ensemble of ions into an empty spectral pit. The lower (b)
part shows an experimental version of such a created peak structure. The
difference in height of the three different transition from the j0S state, comes from
the fact that these transitions have different oscillator strength.
This yields a good control over all the essential parameters of
the AFC.
4.4. Comb structure measurement
One of the goals of this work was to compare the observed AFC
echo efficiency with the one predicted by the theoretical model
discussed in Section 2. This requires a precise measurement of the
AFC structure in order to determine the shape, width and height
of the peaks. To do this the laser was slowly swept in frequency
across the created AFC structure and the transmitted light
intensity was detected after the sample. From this transmission profile of the structure, the absorption spectrum can be
calculated. The intensity of the scan pulse was chosen such that it
did not affect the created spectral structure. For most comb
measurements we chose to scan the laser over one peak only and
lowered the scan rate to a minimum. This is to reduce the effect of
the scan rate on the measurement resolution [40,35]. By varying
the rate we confirmed that the measured width was indeed
independent of the scan rate.
The comb was created on the j1=2gS-j1=2eS transition in
order to maximize the absorption. It is, however, very challenging
to measure absorptions above d= 3–4. In order to circumvent this
problem we instead measured the comb structure on the weaker
j1=2gS-j5=2eS transition, cf. Fig. 2. The ratio of these two
transitions is known from previous work [41], thus the optical
depth of the j1=2gS-j1=2eS transition is readily inferred from
the measured absorption spectrum.
5. Results and discussion
The input pulse is stored on the j1=2gS-j1=2eS transition,
which is the transition for ions in state j1=2gS with the highest
oscillator strength. This results in a comb with high optical depth
d. The bandwidth of the AFC is limited by the frequency
separation between the excited states and in the present case
this is about 4.6 MHz, as set by the j1=2eS and j3=2eS separation
ARTICLE IN PRESS
A. Amari et al. / Journal of Luminescence 130 (2010) 1579–1585
(cf. Fig. 2). Fig. 3 shows a comb containing four peaks, and where
the width of each peak is about g ¼ 150 kHz. The separation
between the peaks was set to D ¼ 1:2 MHz. The pulse to be stored
has a Gaussian shape with duration 200 ns, resulting in a
frequency power spectrum with FWHM 2 MHz (see Fig. 3).
A high-efficiency echo is shown in Fig. 4. The emitted echo is
observed after 800 ns, as expected from the comb periodicity
ts ¼ 1=D ¼ 800 ns. To be able to calculate the efficiency of the
echo, first a reference input pulse is sent through the empty pit
(no AFC prepared). This pulse is thus completely transmitted
through the sample. The AFC is then prepared inside the pit and
an identical pulse (the storage pulse) is sent in. This pulse is
partially absorbed in the medium and produces an echo. The ratio
between the area of the echo and the area of the reference pulse
gives the storage efficiency. A small part of the reference pulse as
well as the storage pulse is split off before they enter the cryostat
so unintentional input power differences between the reference
and storage pulses can be compensated for. For the data shown in
Fig. 4 we measured an efficiency of 35%. To our knowledge this is
the highest AFC echo efficiency observed up to date.
1.2 MHz
As discussed in Section 2 the efficiency depends strongly on
the shape of the AFC. Our comb structure measurements show
peaks that have near Gaussian shapes. This facilitates the
theoretical modeling since we can use the simple model discussed
in Section 2 and we need only then to measure two parameters;
the peak absorption d and peak width g (the finesse is calculated
from the relationship F ¼ D=g). In order to make a quantitative
comparison with the model, we varied the peak absorption d and
measured the resulting input pulse transmission and echo
efficiency. This was done by increasing the power of the back
burning pulses used in the peak creation (see Section 4.3). For
each power setting we also measured the comb structure to find d
and F. The width of the peaks could be varied by changing the
chirp width of the sechyp pulses used for peak creation. We did
measurements as a function of d for two settings of the chirp; 200
and 300 kHz. The measured peak widths for these two settings
were 175 and 245 kHz, respectively, corresponding to F= 6.9 and
4.9. With increasing back burning power (hence increasing d) the
peaks were slightly broadened due to power broadening, but the
observed increase was only 10–15% for the data considered here.
The widths given above are averages over all back burning powers
(hence d values).
In Fig. 5 we show measured transmission coefficients of the
input pulse and the efficiencies of the echo for the two data sets.
The data are plotted as a function of the measured peak
6
70
2
0
-3000
-2000
-1000
0
1000
2000
3000
45
F=5
F=5
F=7
F=4
F=4
60
40
35
30
50
25
40
20
30
15
20
10
Frequency (kHz)
10
5
0
0
1
2
3
4
5
6
7
8
1.0
80
F=5
70
F=4
F=5
0.6
0.4
0.2
F=4
60
25
40
F=3
30
10
5
1
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Time (us)
Fig. 4. (Color online) The dashed line shows the input pulse that is completely
transmitted through the empty pit (no absorption). The solid line shows the
partially transmitted input pulse and the subsequent echo emission with the AFC
created in the pit (cf. Fig. 3). The echo efficiency is 35% of the input pulse (see text
for details).
20
15
20
0
-0.4
35
30
50
10
0.0
45
40
F=3
0.8
9
Absorption depth d
Transmission (%)
Normalised intensity (arb. units)
Fig. 3. (Color online) From the inhomogeneous absorption profile, four peaks with
ions, all absorbing on the j1=2gS-j1=2eS transition are created. The peak width
(FWHM) is g ¼ 150 kHz and they are separated by D ¼ 1:2 MHz. The input pulse
has a Gaussian power spectrum with FWHM= 2 MHz.
Efficiency (%)
4
F=7
Efficiency (%)
80
150 kHz
Transmission (%)
Absorption depth
8
1583
2
3
4
5
6
7
8
9
0
10
Absorption depth d
Fig. 5. (Color online) Measured transmission of the input pulse (open circles and
left axis) and echo efficiency (closed circles and right axis) as a function of the
measured optical depth for two different experimental values of the finesse, (a)
F= 6.9 and (b) F = 4.9. The dashed lines are theoretically calculated transmission
coefficients and solid lines are calculated efficiencies for (a) F = 4,5 and 7, and for
(b) F = 3,4 and 5 (see text for details).
ARTICLE IN PRESS
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A. Amari et al. / Journal of Luminescence 130 (2010) 1579–1585
absorption d as extracted from the AFC spectra. Theoretical
transmission and efficiency curves are also shown. These were
calculated using the experimental d values, for different values of
finesse. It is observed that the transmission coefficient is very
sensitive to the finesse, whereas the efficiency is less sensitive up
to d =4 5. In general the best agreement for the transmission is
obtained for a finesse lower than the one measured from the
comb spectra (see above). We can also see that the best-fit finesse
is lower for the 300 kHz data set, than for the 200 kHz, which is
to be expected. The echo efficiency shows a reasonably good
agreement with all three values of the finesse up to d= 4 5. But
for d Z 5 the discrepancy between the experimental and
theoretical values becomes significant for F= 4 and 5, while for
F= 3 it is still satisfactory.
The general trend, for both transmission and efficiency, is that
our data fit better with a finesse lower than the measured one. This
comparison is, however, made within the theoretical framework of
Section 2 where a comb of Gaussian peaks was assumed. Both the
transmission and echo efficiency are strongly dependent on the
actual peak shape [18,20]. For instance, the same comparison with
a Lorentzian model [20] yields very different best-fit values for the
finesse. Although much effort was devoted to the precise
measurement of the comb structure, it is still conceivable that
the actual peak shape deviate from a pure Gaussian shape.
Particularly since the sechyp pulses used for peak creation have
power spectra with more super-Gaussian shape [35]. Another
source of error could be imperfections in the peak creation pulses,
where the high power needed to obtain high optical depth might
generate an increase in the absorption background due to offresonant excitation. Such an additional absorption will reduce both
the experimental transmission and efficiency [9] compared to the
theoretical model in Section 2 where such an absorption background is neglected. This would particularly affect the high d range,
where indeed we observe a larger discrepancy.
Appendix A
The optical pumping pulses used for creating the pit structure
in Fig. 1b are presented in Table 1.
Table 1
List of pulses used for the pit burning sequence, with start and end frequencies.
Pulse
nstart ðMHzÞ
nend ðMHzÞ
Orel
BurnPit1
BurnPit2
BurnPit3
BurnPit4
BurnPit5
BurnPit6
BurnPit7
BurnPit8
BurnPit9
BurnPit10
+ 31.85
+ 23.85
+ 15.95
+ 23.85
16.85
8.85
+ 15.95
+7.35
1.10
+7.65
+ 24.15
+ 16.15
+ 7.65
+ 16.15
9.15
1.15
+ 7.65
1.10
+ 7.35
+ 15.95
3=2g -1=2e
3=2g -5=2e
3=2g -5=2e
3=2g -5=2e
5=2g -5=2e
5=2g -1=2e
3=2g -5=2e
3=2g -5=2e
5=2g -1=2e
5=2g -1=2e
This set of pulses will create the pit in Fig. 1b with zero absorption (no absorbing
ions) from 1.2 up to 16.2 MHz. The frequency scale is defined by denoting the
j1=2gS-j1=2eS transition, for an arbitrarily selected ion class, as zero MHz. The
column Orel lists the primary target transition of the scan for the purpose of
knowing what light intensity to choose in order to match the Rabi frequency to the
relative oscillator strength. Note that pulse numbers 2 and 4 are the same, as is
pulses 3 and 7.
These pulses are then repeated in an iterative sequence in the
following manner
(1)
(2)
(3)
(4)
Repeat
Repeat
Repeat
Repeat
60
30
20
30
times:
times:
times:
times:
BurnPit5, BurnPit6.
BurnPit1-4, BurnPit6-10.
BurnPit1-4, BurnPit6.
BurnPit7-10.
Finally yielding the pit in Fig. 1b.
6. Conclusions
We have in detail described optical pumping and preparation
procedures for creating AFC structures in Pr3 þ : Y2 SiO5 . We were
able to make comb structures yielding 4 30% AFC echo efficiency,
which are the most efficient AFC echoes observed up to date. We
believe that further progress will be possible, by carefully optimizing
the comb parameters. It should thus be possible to approach the
theoretical limit of 54% used in the present forward-propagation
configuration. In order to make a significant further progress, recall
in the backward direction would be necessary, in which case 100%
efficiency is theoretically possible in the absence of dephasing.
In this work we also compared the experimentally observed
efficiencies to a theoretical model. Considerable care has been put
into determining the line shape and line width of the generated
AFC structure with good precision in order to be able to theoretically model the experimental efficiencies. Still, at high optical
densities the finesse required to theoretically reproduce the
experimental echo efficiency are lower than those measured
experimentally. Nevertheless, the present results indeed show
that high efficiency QMs can be created using the AFC technique.
Acknowledgments
This work was supported by the Swedish Research Council, the
Knut and Alice Wallenberg Foundation, the Swiss NCCR Quantum
Photonics, the European Commission through the integrated
project QAP, and the ERC Advanced Grant QORE.
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ARTICLE
Received 18 Feb 2010 | Accepted 5 Mar 2010 | Published 12 Apr 2010
DOI: 10.1038/ncomms1010
Mapping multiple photonic qubits into and
out of one solid-state atomic ensemble
Imam Usmani1, Mikael Afzelius1, Hugues de Riedmatten1 & Nicolas Gisin1
The future challenge of quantum communication is scalable quantum networks, which
require coherent and reversible mapping of photonic qubits onto atomic systems (quantum
memories). A crucial requirement for realistic networks is the ability to efficiently store multiple
qubits in one quantum memory. In this study, we show a coherent and reversible mapping of
64 optical modes at the single-photon level in the time domain onto one solid-state ensemble
of rare-earth ions. Our light–matter interface is based on a high-bandwidth (100 MHz) atomic
frequency comb, with a predetermined storage time of ⲏ1 μs. We can then encode many qubits
in short ( < 10 ns) temporal modes (time-bin qubits). We show the good coherence of mapping
by simultaneously storing and analysing multiple time-bin qubits.
1
Group of Applied Physics, University of Geneva, CH-1211 Geneva 4, Switzerland. Correspondence and requests for materials should be addressed to
H.d.R. (email: [email protected]) or to M.A. (email: [email protected]).
NATURE COMMUNICATIONS | 1:12 | DOI: 10.1038/ncomms1010 | www.nature.com/naturecommunications
1
ARTICLE
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aux
| W 〉 = ∑ cnei 2pd nt e −ikzn | g en g 〉
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Results
The light–matter interface. An AFC is based on a periodic
modulation (with periodicity Δ) of the absorption profile of an
inhomogeneously broadened optical transition |g〉→|e〉 (see Fig. 1).
The modulation should ideally consist of sharp teeth (with full-width
at half-maximum γ ) having a high peak absorption depth d
(see Fig. 1b). Such a modulation can be created by optical pumping
techniques (see Experiment section and Methods). This requires
an atomic ensemble with a static inhomogeneous broadening and
many independently addressable spectral channels. This can be
found in RE-doped solids in which inhomogeneous broadening is
of the order of 1–10 GHz and the homogeneous linewidth is of the
order of 1–100 kHz when cooled to < 4 K. When a weak photonic
coherent state |α〉L with n– < 1 is absorbed by the atoms in the comb,
the state of the atoms can be written as |α〉A = |G〉 + α|W〉 + O(α2).
Here, |G〉 = |g1…gN〉 represents the ground atomic state and
Δ
2.5
e
generally d modes can encode a qudit. Time-bin qubits are widely
used in fibre-based quantum communication1,2 because of their
resilience against polarization decoherence in fibres.
Temporal multimode storage is difficult, however, because of the
scaling of the number of stored modes Nm as a function of optical
depth d of the storage medium21,27. For EIT and Raman interactions,
Nm scales as √d, making it very difficult to store many modes27.
Recently, we proposed21 a multimode storage scheme based on
atomic frequency combs (AFC) with a high intrinsic temporal multimode capacity21,27. Using this method, we recently showed22 that a
weak coherent state |α〉L with mean photon number n– = |α|2 < 1 can
be coherently and reversibly mapped onto a YVO4 crystal doped
with neodymium ions. Later experiments28–31 in other RE-doped
materials have improved the overall storage efficiency (35%) and
storage time (20 μs). Yet, in these experiments, only a maximum
of four modes have actually been stored at the single-photon level;
thus, the predicted21,27 high multimode capacity has yet to be shown
experimentally. In this study, we show reversible mapping of 64
temporal modes containing weak coherent states at the singlephoton level onto one atomic ensemble in a single spatial mode
using an AFC-based light–matter interface.
10
Optical detuning (MHz)
20
Transmitted
photons
Efficiency (%)
Q
uantum communication1 offers the possibility of secure
transmission of messages using quantum key distribution2
and teleportation of unknown quantum states3. Quantum
communication relies on the creation, manipulation and transmission of qubits in photonic channels. Photons have proven to
be robust carriers of quantum information. Yet, the transmission
of photons through a fibre link, for instance, is inherently a lossy
process. This leads to a probabilistic nature of the outcome of
experiments. In large-scale quantum networks4, the possibility of
synchronizing independent and probabilistic quantum channels
is required for scalability5,6. A quantum memory enables this by
momentarily holding a photon and then releasing it when another
part of the network is ready. To reach reasonable rates in a realistic
network, it is necessary to use multiplexing7, which demands quantum memories capable of storing many single photons in different
modes.
A quantum memory requires a coherent medium with strong
coupling to a light mode. Strong and coherent interactions can be
found in ensembles of atoms8, for instance, alkali atoms or rare-earth
(RE) ions doped into crystals. The latter are attractive for quantum
storage applications, as they provide solid-state systems with a
large number of stationary atoms having excellent coherence properties. Optical coherence times of up to milliseconds9 and spin
coherence times greater than seconds10 have been shown at low
temperature (ⱗ4 K).
A quantum memory also requires a scheme for achieving efficient and reversible mapping of the photonic qubit onto the atomic
ensemble. Techniques investigated include stopped light based on
electromagnetically induced transparency (EIT)11–13, Raman interactions14–17 or photon-echo-based schemes18–22. Much progress has
been made in terms of quantum memory efficiency13,23 and storage
time24,25. Storage of multiple qubits is challenging, however, because
it necessitates a quantum memory that can store many optical modes
into which qubits can be encoded. A mode can be defined in time,
space or frequency. Lan et al.26 recently showed that a multiplexed
quantum memory can be achieved by storing several single excitations in separate spatial modes, each functioning as an independent
memory. Time multiplexing, as used in classical communication,
has the great advantage of requiring only a single spatial mode7,21,27,
hence a single quantum memory. Moreover, each pair of temporal
modes can be used to encode different time-bin qubits1, or more
0.075
0.050
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5
4
3
2
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0
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Emitted photons
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2.5
0.000
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0.4 0.6 0.8
Time (μs)
1.0
Figure 1 | AFC storage scheme. (a) Simplified level scheme of the Nd ions doped into Y2SiO5. We use the optical transition at 883 nm between the 4I9/2
ground state and 4F3/2 excited state. The former is split into two Zeeman levels by a 0.3 T magnetic field (|g〉 and |aux〉). The experiment is conducted on
|g〉 − |e〉, where the absorption profile is shaped into an AFC by optically pumping atoms into |aux〉. The basic idea is to send in a pulse sequence on |g〉 − |e〉
that has a periodic spectral density (due its Fourier spectrum). Some of the excited atoms have a certain probability to spontaneously de-excite to |aux〉.
The atoms left behind in |g〉 form the grating (see panel (b)). To build up a deep grating, the sequence is repeated many times (up to the timescale of
the population relaxation between |g〉 and |aux〉). More details on the preparation can be found in the Experimental setup and Methods section. (b) An
example of a generated comb with periodicity Δ = 10 MHz. The relevant AFC parameters defined in the text are indicated. (c) Mapping of weak coherent
states with n– = 0.5 (in a single temporal mode) onto the Nd-doped crystal. Shown are two different experiments with Δ = 10 MHz (dashed line offset
vertically) and 1 MHz (solid line). The photons that are transmitted without being absorbed are detected at t = 0, whereas the absorbed and reemitted
photons are detected around t = 1/Δ. The vertical scale has been normalized such that it yields efficiency. Inset: The overall write and read efficiency as
a function of 1/Δ.
2
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represents one induced optical excitation delocalized over all the
N atoms in the comb. In equation (1), zn is the position of atom n, k
is the wave number of the single-mode light field, δn is the detuning
of the atom with respect to light frequency and the amplitude cn
depends on the frequency and spatial position of the particular
atom n. The initial (at t = 0) collective strong coupling between the
light mode and atoms is rapidly lost because of inhomogeneous
dephasing caused by exp(i2πδnt) phase factors. If we assume that
the peaks are narrow compared with the periodicity (that is, a
high comb finesse F = Δ/γ), then δn≈mnΔ (where mn is an integer)
and the W state will rephase after a preprogrammed time T = 1/Δ.
The rephased collective state W will cause a strong emission in the
forward direction (as defined by the absorbed light).
This type of photon-echo emission is also observed in accumulated or spectrally programmed photon echoes32–35, which inspired
our proposal. Spectral atomic gratings have also been proposed36
and shown37 for coherent optical delay of streams of strong classical pulses. The interest in spectral gratings was recently renewed
in the context of quantum memories, when it was realized how to
achieve a much more efficient spectral grating than previously possible. In fact, a 100% efficient echo process is theoretically possible
in a backward emission configuration21. This is possible because of
the highly absorbing and sharp peaks in the AFC structure. In practice, the finite finesse of the comb still needs to be accounted for,
which causes a partial loss of the collective state. However, in ref. 21
we show theoretically that F = 10 induces a negligible loss, which in
combination with a high optical depth d makes the AFC scheme
very efficient. High-efficiency mapping using high-finesse combs
have recently been shown experimentally28,30,31. These experiments
and the present study store light for a predetermined time given by
1/Δ. We thus emphasize that we also proposed21 and experimentally
showed29 a way to achieve on-demand readout by combining AFC
with spin-wave storage. On-demand readout is a crucial resource
for applications in quantum networks to render different quantum
channels independent.
The multimode property of an AFC memory can easily be understood qualitatively. For a periodicity Δ and Np peaks, its total bandwidth is of the order of ~NpΔ, indicating that a pulse of duration
τ~1/(NpΔ) can be stored. The multimode capacity stems from the
fact that the grating can absorb a train of weak pulses before the first
pulse is reemitted after T = 1/Δ (see Fig. 1c). This simple calculation
results in a multimode capacity Nm∝T/τ∝Np. Thus, a comb with
many peaks, Np, allows us to create a highly multimode memory in
the temporal domain. In this context, RE-doped solids are particularly interesting because of their high spectral channel density.
Experiment. In this study, we work with a neodymium-doped
Y2SiO5 crystal, cooled to 3 K, having a transition wavelength at
883 nm with good coherence properties (see Methods for spectro-
scopic information). This wavelength is convenient as we can work
with a diode laser and silicon-based single-photon counters having
low noise (300 Hz) and high efficiency (32%).
The comb is prepared on the |g〉 − |e〉 transition by frequencyselective pumping of atoms into an auxiliary state |aux〉 (see Fig. 1).
There are different techniques for achieving this. For instance, by
creating a large spectral hole and then transferring back atoms
from an auxiliary state to create a comb, as used in ref. 29. Here,
we use a technique similar to that employed in ref. 22, in which a
series of pulses separated by a time, T, pump atoms from |g〉 to |aux〉
(through |e〉) with a power spectrum having a periodicity 1/T = Δ.
This technique is also frequently used in accumulated photon-echo
techniques32,37. Here, each pulse sequence consisted of three pulses
in which the central pulse is π-dephased (see Fig. 3c). This sequence
has a power spectrum with ‘holes’. A Fourier analysis shows that
the width of the holes in the power spectrum decreases when the
number of pulses in the sequence increases, resulting in a higher
comb finesse. In this experiment, three pulses were enough to reach
the optimal comb finesse (F≈3) to achieve the maximal efficiency
for our optical depth. We refer to the Methods section for more
details on the preparation sequence.
The experimental sequence is divided into two parts: the preparation of AFC and the storage of weak pulses. To increase the efficiency of optical pumping during the preparation, and thus the
depth of the comb, the pulse sequence was repeated 2,000 times,
with a delay of 16 μs between each sequence. This is followed by a
delay of 5 ms (≈17T1) to avoid fluorescence noise from atoms left in
the excited state. During the storage sequence, 1,000 independent
trials are performed at a repetition rate of 200 kHz. The entire preparation and storage sequence is then repeated with a rate of 5 Hz. An
overview of the experimental setup is given in Fig. 2.
In Fig. 1c, we show storage experiments with predetermined
storage times of T = 100 ns and 1 μs, for a single temporal mode. The
overall in–out mapping efficiencies, defined as the ratio of the output pulse counts to the input pulse counts, are ~6 and ~1%, respectively (see inset of Fig. 1c). In the Methods section, we present a
theoretical analysis of the efficiency performance. The efficiency for
single-mode storage is currently lower than that achieved in bestperformance single-mode memories, for example, those given in
refs 13, 16, 17, 20, 28, 30, 38. However, as explained later, our interface compares very favourably with these experiments in terms of
potential multimode storage efficiency.
Multimode storage results. The main goal of this study is to show
high multimode storage. Following the discussion above, we should
maximize the number of peaks in the comb. This can be carried out
by increasing the density of peaks in a given spectral region (that is,
increasing the storage time T) or by changing the width of the AFC
(that is, increasing the bandwidth). Here, we fix the storage time
Figure 2 | Experimental setup. Description of the optical setup used for multiqubit storage experiments. Indicated are the polarization beam splitter (PBS),
λ/2 plates, polarization controllers (PC), Faraday rotator (FR), double-pass acousto-optic modulators (AOM), the fast electro-optic amplitude modulator
(EOM), neutral density (ND) filters, mechanical choppers (MC) and the silicon single-photon counter (APD). See Methods for more details.
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Time
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f1 =–40 MHz
f2 = –20 MHz
f3=0 MHz
f4=20 MHz
f5=40 MHz
Normalized counts
Figure 3 | Increasing spectral bandwidth. (a) Experimental combs created using preparation sequences with either single (solid line) or five (dashed line)
simultaneous pump frequencies. The frequency-shifted sequences allow us to enlarge the frequency range over which the optical pumping is efficient,
thereby creating a wide 100 MHz comb. (b) Efficiency as a function of the duration (full-width at half-maximum, FWHM) of the input pulse for a single(circles) and five (squares)-frequency preparation. As the duration decreases, the bandwidth of the input pulse increases. The decrease in efficiency for
short pulses is due to bandwidth mismatch for large bandwidths when using a single-preparation frequency. This experiment clearly illustrates the gain in
bandwidth in the extended preparation sequence for which only a small decrease in efficiency is observed. (c) Pulse sequence for atomic frequency comb
preparation (see text). To increase the bandwidth, pulses are repeated with shifted frequencies f = 0, ±20 and ±40 MHz. This pulse sequence was used for
most of our experiments. Here, it creates a comb of 100 MHz bandwidth and a periodicity of 1 MHz. The total sequence takes 16 μs.
1.0
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Output modes ×50
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Figure 4 | Storage of 64 temporal modes. The input (left part) is a random sequence of full and empty time bins, in which the mean photon number
in the full ones is n– ⱗ 1. The output (right part) clearly preserves the amplitude information to an excellent degree. The predetermined storage time was
T = 1.32 μs; the duration of the input, 1.28 μs; the mode separation, 20 ns; and the mode duration (full-width at half-maximum, FWHM), 5 ns. The output
has been multiplied with a factor of 50 for clarity. The average storage and retrieval efficiency was 1.3%. Other examples of multimode storage, for
example, with all time bins filled, can be found in the Supplementary Information.
to T = 1.3 μs, by which we reach an efficiency of ⲏ1%, and concentrate our efforts on increasing the bandwidth. The spectral width
of the grating is essentially given by the width of the power spectrum of the preparation sequence, which, using the pulse sequence
described above, only results in a width of about 20–30 MHz. We
can, however, substantially increase the total width by inserting
more pulses in the preparation sequence, which are shifted in frequency (see Fig. 3c). We thus optically pump atoms over a much
larger frequency range. Note that the frequency shift should be a
multiple of Δ to form a grating without discontinuities. In this way,
we managed to extend the bandwidth of the interface to 100 MHz,
as shown in Fig. 3a, without significantly affecting the AFC echo
efficiency. This is illustrated in Fig. 3b, in which we show storage
efficiency as a function of the duration of the input pulse when
the preparation sequence contains a single or five frequencies. The
maximum bandwidth allows us to map short, ⱗ5 ns pulses into
memory.
4
In addition to the present motivation for multimode storage, a
large bandwidth is equally interesting for interfacing a memory with
non-classical single-photon or photon-pair sources. These usually
have a large intrinsic bandwidth that requires extensive filtering for
matching bandwidths. In this case, our extended bandwidth (×5)
would require a corresponding factor of less filtering.
We show the high multimode capacity of our interface by storing 64 temporal modes during a predetermined time of 1.3 μs (see
Fig. 4), with an overall efficiency of 1.3%. This capacity is more than
an order of magnitude higher than that previously achieved for
multiplexing a quantum memory in a single spatial mode16,22. As
shown, we can store a random sequence of weak coherent states.
Storage of random trains of single-photon states has been proposed
for multiplexing long-distance quantum communication systems
on the basis of the so-called quantum repeaters5,6. The maximum
rate of communication would then be proportional to the number
of modes that can be stored7. Our experiment clearly shows the gain
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Figure 5 | Multimode coherence measurement. (a) The output signal (solid line) generated by the double-AFC scheme, which causes an interference
between consecutive modes. The input sequence (not shown) is a series of weak coherent states (n̄ ≈1.8) of equal amplitude ck2, in which the relative phase
difference between consecutive modes φk + 1 − φk ranges from − π/6 to 8π/6 with a step of π/6. This allows us to capture a complete interference fringe in
one measurement. It also clearly shows the preservation of coherence over the complete multimode output. By changing the detuning of the centre of the
second AFC with respect to the carrier frequency of the light, we can impose an additional relative π-phase21 on the corresponding output (see Methods).
This shifts the interference fringe with half a period (dashed line). (b) The corresponding net interference visibilities are 86±3% (open circles) and 85±3%
(filled squares), with detector noise (dashed line) subtracted. The uncorrected raw visibilities were 78±3% and 76±2%. Error bars represent statistical
uncertainties of photon counts (±1 s.d.).
Discussion
For temporal multimode storage, the efficiency of our interface
would outperform the current EIT and Raman-based quantum
memories in homogeneously broadened media, although impressive
efficiencies have been achieved for single-mode storage13,23. This is
because of the poor scaling of efficiency as a function of the number
of modes for a given optical depth27. It would also compare favourably with the recent few modes storage experiment16 using gradient
echo memory, another echo-based storage scheme, also because of
the scaling of mode capacity for a given optical depth (Nm~d)7,27.
Nevertheless, an increase in storage efficiency and on-demand readout are necessary for applications in quantum communication.
The next grand challenge is to combine multimode storage,
high efficiency30 and on-demand readout29 in one experiment. The
immediate efforts will most probably be devoted to praseodymiumand europium-doped Y2SiO5 crystals, in which the crystal-field
Input
Output
0.25
0.20
Normalized counts
that can be made using an AFC-based quantum memory. It thus
opens up a route towards achieving efficient quantum communication using quantum repeaters.
It is now possible to use consecutive temporal modes |k〉 and
|k + 1〉 to encode time-bin qubits ck | k 〉 + ck +1ei(fk +1 −fk ) | k + 1〉 ,
in which case, a good coherence between modes is crucial. The coherence can be measured by preparing superposition states and performing projective measurements using an interferometric setup.
Projective measurements on time-bin qubits are usually performed
using an imbalanced Mach–Zehnder interferometer, in which consecutive time bins interfere1. We can perform the same task with
our light–matter interface using a double-AFC scheme (with Δ1 and
Δ2) as shown in ref. 22. In short, the difference in delay 1/Δ1 − 1/Δ2
is the delay in an imbalanced Mach–Zehnder interferometer. The
technique used for preparing a double-AFC structure is explained
in Methods. As shown in Fig. 5, we observe excellent coherence over
all modes with an average visibility of V = 86±3%, corresponding to
a conditional qubit fidelity of F = (1 + V)/2≈93%.
To further illustrate our ability to store multimode light states, we
create a light pulse with a random amplitude modulation. As shown
in Fig. 6, we can faithfully store this kind of light pulse. The possibility of storing weak arbitrary light states using photon-echo-based
schemes was pointed out already by Kraus et al. (ref. 19). We believe
that this work, in which complex phase and amplitude information
are reversible and coherently mapped onto one atomic ensemble,
is the first experimental realization showing these properties at the
single-photon level.
0.15
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0.00
0.2
0.4
0.6
0.8
1.0
Time (μs)
Figure 6 | Mapping of a randomly amplitude-modulated pulse. As seen,
the overlap between the normalized input (dashed line) and output (solid
line) pulses is excellent. The total average number of photons in the 1 μs
long input pulse is n– ≈4.
ground state has the necessary number of spin levels (three levels)
for implementing the on-demand readout. The recent achievements
in praseodymium-doped Y2SiO5 crystals are very encouraging29,30,
although the bandwidth was limited to a few MHz because of the
hyperfine level splitting. Europium-doped Y2SiO5 has the potential
of offering higher bandwidths (up to 70 MHz) and narrower comb
peaks, which result in a higher multimode capacity21. To exploit the
high-bandwidth results reported in this work, using neodymiumdoped crystals, one needs to find a third spin level with a long spin
coherence lifetime. An interesting path forward is to investigate
neodymium isotopes with a hyperfine structure (143Nd and 145Nd)39.
Recent results on a similar system40, 167Er3 + :CaWO4, show coherence
times approaching 100 μs for hyperfine transitions. Clearly, this
path requires extensive spectroscopic studies to optimize the spin
population and coherence lifetimes. However, it is very interesting,
as it opens up several material candidates (for example, doped with
erbium41 and neodymium) for quantum memory applications.
To summarize, we have shown the reversible mapping of up
to 64 optical temporal modes at the single-photon level onto one
solid-state atomic ensemble. We have shown that the quantum
coherence of the stored modes is preserved to a high extent. The
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different modes can then be used to encode multiple time-bin
photonic qubits. Alternatively, they could also be considered as
high-dimensional qudit states. This opens up possibilities to store
higher dimensional quantum states such as entangled qudits
encoded in time-bin bases42. Our experiment opens the way to
multiqubit quantum memories, which are a crucial requirement for
realistic quantum networks.
Methods
Sample. The sample is a 10 mm-long neodymium-doped yttrium orthosilicate
crystal (Nd3 + :Y2SiO5) with a low Nd3 + concentration of 30 p.p.m. The inhomogeneous broadening of the 4I9/2–4F3/2 absorption line is around 6 GHz and the
optical depth is 1.5 for this sample (in single-pass configuration), when the
polarization of the light is along the D1 crystallographic axis. We measured an
excited state lifetime of T1 = 300 μs using fluorescence spectroscopy and stimulated
photon echoes. With conventional photon echoes (two-pulse), we measured a
homogeneous linewidth of 3.5 kHz (T2 = 90 μs) for a magnetic field of 300 mT and
temperature of 3 K. Each level is a Kramer’s doublet that splits into two spin states
in a magnetic field. For the field orientation (at 30° relative to the D2 axis) used in
this experiment, we measured g-factors of gg = 2.6 and ge = 0.5. In a 300 mT
magnetic field, the ground and excited states were thus separated by 10.9 and
2.1 GHz. We measured a ground state Zeeman population relaxation lifetime
of around T1Z = 100 ms by spectral hole burning. In the spectral hole-burning
measurements, we also observed a superhyperfine interaction of neodymium ions
with yttrium. This causes additional spectral side holes at around 640 kHz (for the
present magnetic field), thus the effective homogeneous linewidth is around 1 MHz.
This was our main limitation for the efficiency of our light–matter interface, as it
affected our ability to create a good comb for longer storage times (1/Δ≈1 μs).
Experimental setup. We present more details on the optical setup shown in
Fig. 2. We used a diode laser that is actively frequency stabilized ( < 100 kHz), using
a spectral hole as reference. The output was split into two beams using a polarization beam splitter. Each beam could be amplitude, frequency and phase modulated
using double-pass acousto-optic modulators. One beam was used for creating
preparation pulses and the other one for creating the weak pulses to be stored
(strongly attenuated using neutral density filters). In the weak path, an additional
electro-optic amplitude modulator was used to create short-input pulses for the
multimode storage experiments. The paths were mode overlapped using a fibrecoupled beam combiner. The light was sent through the crystal, again in free space,
in a double-pass setup to double the optical depth to d = 3. Using a Faraday rotator
and a λ/2 plate before the crystal, we could separate the input and output modes
on the polarization beam splitter, while keeping a constant linear polarization
optimized for maximum absorption in the crystal. Output light was collected with
a multimode fibre and detected by a silicon single-photon counter. Two
synchronized mechanical choppers blocked the detector during the preparation
sequence and blocked the preparation beam during the storage sequence. The
transmission between the input of the cryostat and the detector was typically
between 25 and 30%.
Comb preparation. We now explain in more detail the preparation sequence
allowing us to create the desired comb. The goal is to optically pump atoms from
|g〉 to |aux〉 in a frequency-selective manner (see Fig. 1a), wherein the atoms left in
|g〉 will form the comb. This can be achieved by two pulses of duration τ separated
by time T, as done in ref. 22, which has a power spectrum of width ~1/τ with a
sinusoidal modulation of periodicity 1/T. To create a sharper comb structure
having higher finesse, it is useful to have a wide power spectrum with sharp ‘holes’
instead of the sinusoidal modulation above. This can be achieved by increasing
the number of pulses in the sequence in which the central pulse is π-dephased and
has a field amplitude corresponding to the sum of the amplitudes of the side pulses
(see Fig. 3c). In frequency space, the short and intense central pulse interferes
destructively with the periodic spectrum of the side pulses, creating a wide power
spectrum with the desired holes. From this simple Fourier argument, it is clear that
the width of these holes in the spectrum is proportional to the number of pulses.
In our study, the optimal finesse is close to 3 for the optical depth of our material,
in which case, three pulses were enough to achieve this finesse.
For the interference experiment, shown in Fig. 5, one temporal input mode
must be split into two possible temporal output modes. For this purpose,
we can create two superimposed combs, with a different periodicity Δi. An
incoming photon then has an equal probability of being absorbed by either of the
combs and will consequently be reemitted as a coherent superposition of the two
corresponding output modes. To create two combs with periodicity Δ1 and Δ2, we
use the sequence shown in Supplementary Fig. S3. A sequence with N = 3 pulses
separated by τ1 is superposed with another sequence of N = 3 pulses separated by
τ2, with the central pulse of each sequence being centred at t = 0. Thus, the total
number of pulses will be 2N − 1 = 5, and the pulse area of the central one should
be equal to the sum of the others in order to achieve the appropriate interference
effect in the power spectrum.
6
It is also possible to impose a phase change φ on the output mode related to a
particular comb. This is achieved by frequency shifting the corresponding comb
by a fraction of the comb spacing Δ, as discussed in ref. 21. To do so using our
preparation method, we must add a phase to each pulse preparing the corresponding comb (see Supplementary Fig. S3). If we label the pulses with k = ±1, ±2, ±3…
(excluding the central pulse), then the phase of each pulse should be kφ. It is
straightforward to show by Fourier analysis that the corresponding power
spectrum will see a frequency shift in the position of the holes, which in turn will
cause a shift in the position of the created comb.
Storage efficiency analysis. Efficiency can be calculated theoretically using the
2
formula21,22 h ≈ (d / F )2 e−d / F e−7 / F e−d0 . The different terms can be given a qualitative understanding. The first term represents collective coupling, the second the
reabsorption of the reemitted light, the third is an intrinsic dephasing factor
due to finesse and the last term is a loss due to an absorption background d0. For
the comb with Δ = 10 MHz, we measure d≈1.7, F≈2.7 and d0≈0.5 (see Fig. 1b),
resulting in a theoretical efficiency of η ≈ 5% in close agreement with the experiment (see Fig. 1c). The major limiting factor here is d0 (caused by an imperfect
preparation of the comb) and then the optical depth of the comb d (the finesse
being close to optimum for this d, see ref. 21). The decrease in efficiency for longer
storage time periods (see inset of Fig. 1c) is principally due to an increase in background absorption d0 and an accompanying decrease in peak absorption d. This
in turn is caused by the effective spectral resolution of 1 MHz in optical pumping,
which is a limitation of the present material (see Methods above). Yet, the storage
efficiency is between one and two orders of magnitude higher than that which
we achieved in the material Nd:YVO422, which we attribute to an improvement
in optical pumping in this neodymium-doped material.
References
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Acknowledgments
We acknowledge financial support from the Swiss NCCR Quantum Photonics, the EC
projects Qubit Applications (QAP) and the ERC Advanced Grant (QORE). We also
acknowledge useful discussions with Christoph Simon and Nicolas Sangouard.
Author contributions
I.U. built the experimental setup, carried out the experiment and analysed the data,
which were supervised by M.A. and H.d.R. All authors contributed to the conception
and analysis of the experiment and to the writing of the paper.
Additional information
Supplementary Information accompanies this paper on www.nature.com/
naturecommunications.
Competing financial interests: The authors declare no competing financial interests.
Reprints and permission information is available online at http://npg.nature.com/
reprintsandpermissions/
How to cite this article: Usmani, I. et al. Mapping multiple photonic qubits into and out
of one solid-state atomic ensemble. Nat. Commun. 1:12 doi: 10.1038/ncomms1010 (2010).
NATURE COMMUNICATIONS | 1:12 | DOI: 10.1038/ncomms1010 | www.nature.com/naturecommunications
7
Supplementary Information Mapping multiple photonic qubits onto one solid-state atomic ensemble
Imam Usmani,Mikael Afzelius,Hugues de Riedmatten and Nicolas Gisin
Group of Applied Physics, University of Geneva, CH-1211 Geneva 4, Switzerland
(Dated: March 5, 2010)
In this supplementary material we provide a figure (Fig. S1) of the comb structure for 1 µs storage
time. We also show additional results (Fig. S2) of multimode storage experiments and an additional
figure (Fig. S3) describing the preparation sequence for double-AFC experiments.
2
Optical depth
2.0
1.5
1.0
0.5
-50
-25
0
25
Optical Detuning [MHz]
Figure S 1: Experimental atomic frequency combs. We here show experimental combs with 1MHz peak separation
(corresponding to 1 µs storage time) for 20 MHz (green line) and 100 MHz (black line) bandwidths, respectively. The bandwidth
enlargement technique used in this work provides a larger bandwidth without deteriorating other comb properties such as optical
depth d or finesse F . The AFC echo efficiency is thus unaffected (cf. Figure 3 in the article). However, when compared to combs
with larger peak separation (eg. shorter storage time) as shown in Fig. 1b, the peak height d decreases, and the absorption
background d0 increases. This explains the lower storage efficiency for 1 µs storage time, as shown in Fig. 1c (see Methods for
a discussion on storage efficiency modeling). As explained in the paper, the number of modes is proportional to the number of
peaks in the comb. Here we create more than 100 peaks which allows us to store 64 modes with negligible mode overlap. Note
that the variation in optical depth as a function of frequency is caused by laser intensity variations during the measurement.
normalized counts
3
1.0
Input mode
Output mode x50
0.8
0.6
0.4
0.2
0.0
0.0
0.5
1.0
1.5
2.0
2.5
2.0
2.5
normalized counts
time [ s]
1.0
Input mode
Output mode x50
0.8
0.6
0.4
0.2
0.0
0.0
0.5
1.0
1.5
time [ s]
Figure S2: Storage of 64 optical modes. Show are two more examples of storage of 64 temporal modes, with different input
sequences as compared to Fig. 4 in the article. In the upper trace, all the modes are full, which allows us in principle to store
32 time-bin qubits. An efficiency of 1.4% was measured, with a mean photon number per input pulse of n̄ ≈ 0.5. In the lower
trace, we modulated the amplitude of the input pulses. An efficiency of 1.6% was measured, with a mean photon number in
the most intense input pulses of n̄ ≈ 0.9
4
Figure S 3: Double-AFC preparation technique. Pulse sequence for the preparation of two atomic frequency combs, as
required for the double read out in the interference experiments. In this example, the two AFCs have a periodicity 1/T1 and
1/T2 . The phase change ±φ on the pulses creating the 1/T2 comb will cause a corresponding phase change φ of the AFC echo.
See Methods section in article for details.
NEWS & VIEWS RESEaRch
a mere first — albeit promising — step on a
fresh path in the emerging field of cognitiveenhancement research.
These outstanding questions notwithstanding, the current study2 has yet another interesting twist. The genes encoding IGF-II and
IGF-IIR are known to be imprinted (they display a dichotomous effect on growth in most
tissues studied8). The IGF-II gene — like many
other genes for which only the paternal copy
is expressed — favours growth. Conversely,
the IGF-IIR gene — like most maternally
expressed genes — suppresses growth. Chen
and co-workers’ data are not only the first to
attribute a clear-cut function to these genes in
the brain, but also to show that, at least in the
rat hippocampus, their protein products cooperate to enhance memory. But, given that the
spatial and developmental patterns of IGF-II
and IGF-IIR expression in the brain are highly
complex9, more work is needed to decipher
their exact regulation.
One process by which the expression of
maternal and paternal copies of IGF-II and
IGF-IIR is regulated is DNA methylation8 —
an epigenetic mechanism that is dynamically
altered after learning10. So it would be interesting to know whether learning per se can also
trigger epigenetic changes and so the subsequent alterations in the expression of IGF-II and
IGF-IIR. Moreover, ‘natural cognition enhancers’ such as environmental enrichment, which
are known to act through epigenetic mechanisms11, might also epigenetically regulate
IGF-II and IGF-IIR. If so, for those in search of a
memory boost, natural enhancers would be an
attractive alternative to purely pharmacological
agents. ■
QUAN TU M I NF ORM ATI O N
Entanglement on ice
The ability to store entangled photons in a solid-state memory, and to retrieve
them while preserving the entanglement, is a required step on the way to practical
quantum communication. This step has now been taken. See Letters p.508 & p.512
JEVON LONGdELL
C
ommunication over long distances
needs repeaters — for classical communication these are devices that receive
input data and retransmit them. In an important move towards long-distance communication of quantum information, two groups1,2
(pages 508 and 512 of this issue) have demonstrated the basic building block of a solid-state
quantum repeater. The authors have managed
to put entangled photons ‘on ice’. That is, they
were able to show that they could keep a pair
of photons entangled, even after storing and
retrieving one of the photons in a cryogenically
cooled crystal.
Two quantum-mechanical systems are
entangled when it is impossible to properly
describe the quantum-mechanical state of one
of them in isolation. The concept was introduced by Einstein, Podolsky and Rosen in their
famous 1935 paper3. Quantum mechanics predicts completely random, but perfectly correlated, results for some measurements of the
two entangled systems. It also states that the
results of these measurements are not determined until one of the systems is measured.
However, the fact that these correlations still
existed when the systems were too far apart
for any signalling between them to be possible, led Einstein and colleagues3 to conclude
that quantum mechanics was not a complete
description of entangled systems. They argued
that the measurement results must have been
determined by some ‘hidden variable’ when
the entangled pair was created, and not at the
time of measurement.
Since 1935, entangled states have been
produced, and the strange predictions that
quantum mechanics makes about them have
been validated. Furthermore, Bell showed4
that all hidden-variable theories will give
predictions that, in certain circumstances,
differ from those of quantum mechanics. It
was therefore possible to do experiments that
supported quantum mechanics over hiddenvariable theories. Today, entanglement is at the
heart of quantum information. The challenge
in achieving secure, long-distance communication of quantum information is entangling
systems that are spatially far apart.
In many ways, photons are ideal carriers
of quantum information: they are easy to
manipulate with optics, can be sent long distances either through optical fibres or in free
space, and good single-photon detectors are
also available. There are downsides, however. Unless speeding through a vacuum,
photons have very short lives. This problem
is exacerbated by the fact that most methods
for creating, and many methods for manipulating, entangled photons are probabilistic.
And this is why the ability to store entangled
photons in a memory and then recall them,
while preserving the entanglement, which
has now been demonstrated by Clausen et al.1
and Saglamyurek et al.2, is so important. It
enables much more sophisticated operations
Johannes Gräff and Li-Huei Tsai are at the
Picower Institute for Learning and Memory,
Department of Brain and Cognitive Sciences,
Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139, USA.
e-mails: [email protected]; [email protected]
Maher, B. Nature 452, 674–675 (2008).
Chen, D. Y. et al. Nature 469, 491–497 (2011).
Russo, V. C. et al. Endocr. Rev. 26, 916–943 (2005).
Sonntag, W. E. et al. Neuroscience 88, 269–279
(1999).
5. Dudai, Y. Annu. Rev. Psychol. 55, 51–86 (2004).
6. Milekic, M. H. & Alberini, C. M. Neuron 36, 521–525
(2002).
7. Feinberg, A. P. Nature 447, 433–440 (2007).
8. Reik, W. & Walter, J. Nature Rev. Genet. 2, 21–32
(2001).
9. Gregg, C. et al. Science 329, 643–648 (2010).
10. Day, J. J. & Sweatt, J. D. Nature Neurosci. 13,
1319–1323 (2010).
11. Fischer, A., Sananbenesi, F., Wang, X., Dobbin, M. &
Tsai, L.-H. Nature 447, 178–182 (2007).
1.
2.
3.
4.
through synchronization of these probabilistic
processes.
An example of these more sophisticated
operations is long-distance quantum communication. Entanglement, and therefore
secure communication, protected by quantum
mechanics5, can be achieved over moderate
distances (tens of kilometres) by simple propagation of entangled pairs of photons. But at
larger distances, photon loss quickly becomes
a significant problem, preventing the distribution of entanglement. One way to solve this
problem is to break the link between the two
parties into a small number of sub-links joined
together with quantum repeaters6. Entangled
photons would then be transmitted between
the repeaters. The memories in the repeaters
can store entangled photon pairs associated
with successful transmission along part of the
link until other parts of the link are also successful. In this way, quantum repeaters prevent
the effective bit rates for transferring information between the two parties from dropping off
exponentially (or worse) with distance. This is
because, with repeaters, entanglement can be
achieved without one photon having to travel
along the whole link, which for long links can
be very unlikely.
When implementing quantum repeaters,
it has been the memory aspect that has been
the most troublesome. But now Clausen et al.1
and Saglamyurek et al.2 have given us proof-ofprinciple demonstrations. They have produced
entangled photons by means of spontaneous
parametric downconversion. In this process,
an entangled pair of two lower-energy photons
is created from a higher-energy photon in an
optically nonlinear crystal. The wavelength
of the ‘pump’ lasers that produce the higherenergy photon was chosen by both groups1,2
such that one member of the entangled photon
pair had a suitable wavelength for being stored
in a quantum memory based on a cryogenic
crystal doped with rare-earth ions; and, as
required by quantum repeaters, the wavelength
2 7 JA N UA Ry 2 0 1 1 | VO L 4 6 9 | N AT U R E | 4 7 5
© 2011 Macmillan Publishers Limited. All rights reserved
RESEaRch NEWS & VIEWS
of the other member of the pair was suitable for
long-distance transmission in optical fibre.
Mapping of photonic entanglement into and
out of a quantum memory has been demonstrated already with trapped-atom systems7.
However, the new work1,2 is the first to achieve
it using a solid-state memory. The use of the
solid state offers certain practical advantages.
For example, Saglamyurek and colleagues2
formed the memory in an optical waveguide,
which could enable integrated devices to be
built. More significantly though, the authors’
approach1,2 to quantum memories using cryogenic rare-earth-ion-doped crystals is rapidly
developing and has already surpassed the
storage bandwidths2, capacities8, efficiencies9
and storage times10 of other approaches.
Such cryogenic rare-earth-ion-doped
systems have already been studied for classical optical signal processing because of these
systems’ large ratio of inhomogeneous to
homogeneous broadening11 for their optical
absorption lines. That is, the optical absorption linewidth of each dopant atom is very
narrow, whereas the linewidth of the ensemble
of dopants can be very large. This makes them
very suitable systems for photon echoes,
which is where the dopants emit a pulse of
light (echo) in response to earlier applied light
pulses. In particular, photon echoes allow signal processing with simultaneous large bandwidth, determined by ensemble linewidth,
and high resolution, determined by singledopant linewidth. The rapid advances of rareearth quantum memories have been, in large
part, due to the development of photon-echo
techniques, which are suitable for preserving
quantum states of light.
Although the storage of entanglement in a
solid is a significant step, the efficiencies and
storage times in the entanglement-storage
experiments of Clausen et al.1 and Saglamyurek et al.2 need to be improved; they are currently inferior to those that can be achieved in a
small spool of optical fibre. And whereas good
efficiency, storage time and bandwidth have
all been demonstrated by others, in separate
demonstrations, the next challenge awaiting
researchers is to achieve all these performance
metrics for the same memory. This should
CI RCA dI AN RHY TH M S
Redox redux
Oscillations in gene transcription that occur in response to biological daily clocks
coordinate the physiological workings of living organisms. But turnover in cellular
energy may be sufficient to make the clock tick. See Article p.498 & Letter p.554
J O S E P H B A S S & J O S E P H S . TA K A H A S H I
L
ast spring, a visitor at the biennial
meeting of the Society for Research
on Biological Rhythms in Florida
approached the geneticist Sydney Brenner
inquiring as to what it was that scientists
studying circadian rhythms actually do. With
a glimmer in his eye, Brenner responded that
the meeting concerned “those things that
only happen once each day”. Indeed, all forms
of life undergo circadian (roughly 24-hour)
fluctuations in energy availability that are tied
to alternating cycles of light and darkness.
Biological clocks organize such internal energetic cycles through transcription–translation
feedback loops. But two papers1,2 in this issue
show that, in both humans and green algae,
rhythmic cycles in the activity of peroxiredoxin enzymes can occur independently of
transcription.
Biological circadian oscillators have long
been recognized as a self-sustained phenomenon, their 24-hour length being both invariant over a wide range of temperatures and
responsive to light. Early indications that genes
underlie the clocks came3 from the isolation
of mutant fruitflies carrying altered, and yet
heritable, circadian rhythms. This and subsequent work4,5 established that endogenous
molecular clocks consist of a transcription–
translation feedback loop that oscillates
every 24 hours in cyanobacteria, plants, fungi
and animals.
Although the specific clock genes are not
evolutionarily conserved across distinct phyla,
their architecture is similar. The forward limb
of the clock involves a set of transcriptional
activators that induce the transcription of
a set of repressors. The latter comprise the
negative limb, which feeds back to inhibit the
forward limb. This cycle repeats itself every
24 hours (Fig. 1).
Energetic cycles are one type of physiological process that shows transcription-dependent circadian periodicity6,7; such cycles include
the alternating oxygenic and nitrogen-fixing
phases of photosynthesis, and the glycolytic
and oxidative cycles in eukaryotes (organisms
with nucleated cells). The idea that biochemical
flux per se may couple circadian and energetic
cycles was first suggested by McKnight and
colleagues8, who showed that varying the redox
state of the metabolic cofactor NAD(P) affects
the activity of two clock proteins, and it gained
further support from subsequent studies9–14.
4 7 6 | N AT U R E | VO L 4 6 9 | 2 7 JA N UA Ry 2 0 1 1
© 2011 Macmillan Publishers Limited. All rights reserved
open up new capabilities and technologies
that will stretch quantum mechanics in a way
that we have not yet been able to. Who knows,
it might break. ■
Jevon Longdell is at the Jack Dodd Centre for
Quantum Technology, Department of Physics,
University of Otago, Dunedin 9054,
New Zealand.
e-mail: [email protected]
1. Clausen, C. et al. Nature 469, 508–511 (2011).
2. Saglamyurek, E. et al. Nature 469, 512–515 (2011).
3. Einstein, A., Podolsky, B. & Rosen, N. Phys. Rev. 47,
777–780 (1935).
4. Bell, J. S. Rev. Mod. Phys. 38, 447–452 (1966).
5. Ekert, A. K. Phys. Rev. Lett. 67, 661–663 (1991).
6. Briegel, H.-J., Dür, W., Cirac, J. I. & Zoller, P. Phys. Rev.
Lett. 81, 5932–5935 (1998).
7. Choi, K. S., Deng, H., Laurat, J. & Kimble, H. J. Nature
452, 67–71 (2008).
8. Usmani, I., Afzelius, M., de Riedmatten, H. & Gisin,
N. Nature Commun. 1, 12 (2010).
9. Hedges, M. P., Longdell, J. J., Li, Y. & Sellars, M. J.
Nature 465, 1052–1056 (2010).
10. Longdell, J. J., Fraval, E., Sellars, M. J. & Manson, N. B.
Phys. Rev. Lett. 95, 063601 (2005).
11. Barber, Z. W. et al. J. Lumin. 130, 1614–1618
(2010).
But exactly how transcriptional and nontranscriptional cycles may be interrelated was
still not fully understood.
To address this relationship, O’Neill and
Reddy1 (page 498) examined the rhythmic
properties of human red blood cells (RBCs).
In their mature form, these cells lack both a
nucleus and most other organelles, including
energy-producing mitochondria. They function mainly as oxygen shuttles, utilizing the
protein haemoglobin as the delivery vehicle.
Some of the most abundant proteins in
mature RBCs are the evolutionarily conserved
enzymes of the peroxiredoxin family, which
can inactivate reactive oxygen species (ROS).
Class-2 peroxiredoxins contain a cysteine
amino-acid residue in their active site that
undergoes oxidation when ROS accumulate.
This results in the enzyme’s transition from
a monomeric to a dimeric state. Excess ROS
accumulation induces the formation of even
higher-order oligomers. Peroxiredoxin function is essential for RBC survival, as defects in
the expression or activity of these enzymes lead
to the breakdown of the cells.
A previous survey15 searching for proteins
that show circadian rhythms of expression in
liver identified peroxiredoxins. In their study,
O’Neill and Reddy1 monitored the monomer–
dimer transition of these proteins in RBCs
from three humans. They observed two main
circadian features in these enucleated cells.
First, the oligomerization pattern was selfsustained over several cycles within an approximate 24-hour period and was not affected by
temperature. Second, peroxiredoxin oxidation
cycles were synchronized in response to temperature cycles, a property called entrainment
that is a hallmark of circadian oscillators.
LETTER
doi:10.1038/nature09662
Quantum storage of photonic entanglement in a
crystal
Christoph Clausen1*, Imam Usmani1*, Félix Bussières1, Nicolas Sangouard1, Mikael Afzelius1, Hugues de Riedmatten1,2,3
& Nicolas Gisin1
intensity correlations between the two photons still exist after storage
and retrieval. We then show, through a violation of a Bell inequality, that
the storage process creates a light–matter entangled state. In addition,
these results represent the first successful mapping of energy–time
entangled photons onto a quantum memory.
Our experiment consists of a coherent solid-state quantum memory
and a source of entangled photons. A schematic of the experiment is
shown in Fig. 1. The source is based on non-degenerate SPDC in a
nonlinear waveguide pumped by continuous wave light at 532 nm.
This yields energy–time entangled photons with the signal photon at
Quantum memory
e
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AFC preparation
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Nd:Y2SiO5
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532-nm
laser
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3
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PPKTP
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8
3
,3
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Grating
1
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Source of
entangled
photons
FBG
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90
Entanglement is the fundamental characteristic of quantum physics—
much experimental effort is devoted to harnessing it between various
physical systems. In particular, entanglement between light and
material systems is interesting owing to their anticipated respective
roles as ‘flying’ and stationary qubits in quantum information technologies (such as quantum repeaters1–3 and quantum networks4).
Here we report the demonstration of entanglement between a photon
at a telecommunication wavelength (1,338 nm) and a single collective
atomic excitation stored in a crystal. One photon from an energy–
time entangled pair5 is mapped onto the crystal and then released into
a well-defined spatial mode after a predetermined storage time. The
other (telecommunication wavelength) photon is sent directly
through a 50-metre fibre link to an analyser. Successful storage of
entanglement in the crystal is proved by a violation of the Clauser–
Horne–Shimony–Holt inequality6 by almost three standard deviations (S 5 2.64 6 0.23). These results represent an important step
towards quantum communication technologies based on solid-state
devices. In particular, our resources pave the way for building multiplexed quantum repeaters7 for long-distance quantum networks.
Although single atoms8,9 and cold atomic gases10–15 are currently
some of the most advanced light–matter quantum interfaces, there is
a strong motivation to control light–matter entanglement with more
practical systems, such as solid-state devices16. Solid-state quantum
memories for photons can be implemented with cryogenically cooled
crystals doped with rare-earth-metal ions17, which have impressive
coherence properties at temperatures below 4 K. They have the
advantage of simple implementation because rare-earth-metal-doped
crystals are widely produced for solid-state lasers, and closed-cycle
cryogenic coolers are commercially available. Important progress
has been made over the last years in the context of light storage into
solid-state memories, including long storage times18, high efficiency19
and storage of light at the single photon level with high coherence and
negligible noise19–23. Yet these experiments were realized with classical
bright pulses or weak coherent states of light. Although this is sufficient to characterize the performance of the memory, and even to
infer the quantum characteristics of the device19,20, it is not sufficient
for the implementation of more sophisticated experiments involving
entanglement, as required for most applications in quantum information science. For this purpose, it is necessary to store non-classical
light, in particular individual photons that are part of an entangled
state (generated, for example, through spontaneous parametric downconversion, SPDC), similar to previous demonstrations using electromagnetically induced transparency in cold atomic gases14,15. In
addition, for quantum communication applications, the other part
of the entangled state should be a photon at telecommunication wavelength in order to minimize loss during transmission in optical fibres.
In this Letter, we report on an experiment in which a photon from an
entangled pair is stored in a quantum memory based on a rare-earthmetal-doped crystal. More specifically, we show that non-classical
InGaAs APD
Filtering cavity
Figure 1 | Experimental set-up. The experimental set-up can be divided into
three parts: the Nd:Y2SiO5 crystal serving as quantum memory, the laser system
for the preparation of the AFC in the crystal, and the source of entangled
photons with associated spectral filtering. During the experiment we
periodically switch between 15 ms of AFC preparation and frequency
stabilization and a 15-ms measurement phase, in which single photons are
stored. During the preparation, the comb structure is prepared by frequencyselective optical pumping of atoms from the ground state | gæ to the auxiliary
state | auxæ using light from an 883-nm diode laser in combination with an
acousto-optic modulator (AOM). The fibre optic switch is in the upper
position, and the silicon avalanche photodiode (Si APD) is protected from the
bright light by a chopper. During the measurement phase, the positions of
switch and chopper are reversed. Now, photon pairs are generated in the
periodically poled potassium titanyl phosphate (PPKTP) waveguide via SPDC.
The two photons in a pair are spatially separated by a diffraction grating and
then strongly filtered by two etalons, a cavity and a fibre Bragg grating (FBG).
Photons at 883 nm are sent through the crystal in a double-pass configuration
to increase the absorption probability, and are afterwards detected by the Si
avalanche photodiode. Photons at 1,338 nm are directed towards a
superconducting single photon detector (SSPD) located in another laboratory
50 m away. All relevant quantities are extracted from the coincidence statistics
of the two detectors. Details of the frequency stabilization and the filtering
system are given in the Methods. PBS, polarizing beam splitter; FR, Faraday
rotator.
1
Group of Applied Physics, University of Geneva, CH-1211 Geneva 4, Switzerland. 2ICFO—Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860 Castelldefels, Barcelona, Spain. 3ICREA—
Institució Catalana de Recerca i Estudis Avançats, 08015 Barcelona, Spain.
*These authors contributed equally to this work.
5 0 8 | N AT U R E | VO L 4 6 9 | 2 7 J A N U A RY 2 0 1 1
©2011 Macmillan Publishers Limited. All rights reserved
LETTER RESEARCH
the memory wavelength of 883 nm, and the idler photon at the telecom
wavelength of 1,338 nm. Both photons initially have a spectral width of
approximately 1.5 THz, a factor of 104 larger than the 120-MHz
bandwidth of the memory. Hence, strong filtering is crucial14 to
achieve signal-to-noise ratios sufficiently large to reveal the presence
of entanglement during storage. After filtering, the signal photon is
sent to the memory, and the idler photon is coupled into a fibre leading
to a detector in another laboratory 50 m away. Owing to the low loss at
telecommunication wavelengths, this distance could, in principle, be
extended to several kilometres without significantly affecting the
results presented here.
The quantum memory is a 1-cm-long Y2SiO5 crystal, impuritydoped with neodymium ions having a resonance at 883 nm with good
coherence properties23. It is based on a photon-echo-type interaction
using an atomic frequency comb (AFC) (see ref. 24 and Supplementary
Information). In an AFC, the absorption profile of the atomic
ensemble is shaped into a comb-like structure by optical pumping.
A photon is then, with some efficiency, absorbed and re-emitted into
a well-defined spatial mode due to a collective rephasing of the atoms
in the comb structure. The time of re-emission depends on the period
of the comb and is predetermined. We have previously shown that this
kind of memory can store multiple temporal modes23 and is therefore
perfectly suited for storing energy–time entangled photons. For the
work presented here, we have significantly improved the storage efficiency to obtain sufficiently large signal-to-noise ratios. Indeed, using a
new optical pumping scheme for the preparation of the AFC (see
Supplementary Information), the efficiency was increased by a factor
of three for storage times below 200 ns, now reaching values up to 21%
(see results below).
In a first experiment we verified that the non-classical nature of the
intensity correlations between the signal (883 nm) and idler (1,338 nm)
modes is preserved after the storage and retrieval process. If we assume
ð2Þ
second-order auto-correlations of signal and idler gx (where x 5 ‘s’
ð2Þ
for signal or ‘i’ for idler) satisfying 1ƒgx ƒ2, then non-classicality is
ð2Þ
proved by measuring a cross-correlation gsi ~Psi =Ps Pi greater than 2
(see ref. 25). Here, Ps (or Pi) is the probability of detecting a signal (or
idler) photon, and Psi the probability of a coincidence detection (see
Methods).
ð2Þ
We first measured gsi as a function of the pump power of the
source, as shown in Fig. 2a. We find an optimum around a pump power
ð2Þ
ð2 Þ
of 3 mW, where gsi <115 without the AFC memory, and gsi < 30
after a 25-ns storage, thus proving the quantum character of the storage
(note that all results presented in this Letter are without any subtraction of background noise). The reduction in the cross-correlation with
a
b
35
140
the storage is due to limited efficiency (21%), which effectively
increases the contribution of accidental coincidences stemming from
dark counts and multiple pair emissions. Next, we measured the memory
efficiency and the cross-correlation for different storage times, as shown
in Fig. 2b and c.
We now turn our attention towards a particular kind of quantum
correlation, namely entanglement. By performing a two-photon
quantum interference experiment, we show that the entanglement of
the photon pair is preserved when the signal photon is stored in the
crystal.
Photon pairs generated by our source are energy–time entangled,
that is, the two photons in a pair are created simultaneously to ensure
energy conservation, but the pair-creation time is uncertain to within
the coherence time of the pump laser. We wish to reveal the presence of
this entanglement using a Franson-type set-up5. As detailed in the
Supplementary Information, the correlations can be interpreted as
stemming from local measurements performed on a post-selected
time-bin entangled state: p1ffiffi2 ðjEs Ei izjLs Li iÞ, where the early and late
time bins jEs,iæ and jLs,iæ are separated by a time of 25 ns set by the
analysing interferometer (see Fig. 3a). In our experiment, however, the
state of the signal photon is stored as a collective atomic excitation in
the quantum memory before the measurement. Moreover, using a
double AFC scheme20,23, the memory is used not only to store the
entangled photon, but also to analyse it as part of the measurement.
More precisely, the incident time-bins jEsæ and jLsæ are mapped to
distinct AFC modes jEQMæ and jLQMæ, respectively (where subscript
QM denotes quantum memory). Storage of the entangled signal
photon then creates a light–matter entangled state:
1
pffiffiffi ðjEQM Ei izjLQM Li iÞ
ð1Þ
2
The predetermined storage times of jEQMæ and jLQMæ are 75 ns and
50 ns, respectively. After absorption, both AFCs coherently re-emit the
stored excitation into the same well-defined temporal and spatial
mode with a relative phase Dws. This re-emission, followed by detection, constitutes the measurement of the state of the memory. The idler
photon is measured using a fibre-based time-bin qubit analyser with a
25-ns delay and a relative phase Dwi between the short and long arms.
The coincidence detection probability is given by:
Psi !1zV cosðDws zDwi Þ
where V is the visibility of interference. Figure 3b shows the measured
coincidence rate as a function of Dws for two values of Dwi. The raw
visibilities are V 5 (84 6 4)% and (78 6 4)%.
Storage
time
200
c
50
25
20
80
60
15
40
10
20
0
5
15
10
Pump power (mW)
20
5
100 ns
66 ns
100
50 ns
33 ns
50
40
25 ns
0
0
50
Figure 2 | Non-classical correlations and storage efficiency. a, Crossð2Þ
correlation gsi as a function of the pump power incident on the wave guide. Data
points shown were taken with an AFC memory storage time of 25 ns (brown
square symbols), and for comparison, with the crystal prepared with a 120-MHzwide transmission window, that is, without AFC (blue circle symbols). The size of
the coincidence window is 10 ns. b, Coincidence histograms for different
predetermined storage times, vertically offset for clarity. For comparison, the
lowest histogram was taken without AFC. The pump power was 3 mW. c, Cross-
100
150
Delay (ns)
200
Efficiency (%)
133 ns
150
gsi(2) with AFC
gsi(2) without AFC
25
100
gsi(2) with AFC
30
Coincidences in one hour
200 ns
120
ð2Þ
250
30
20
15
10
5
0
0
20
50
100 150 200
10
0
0
50
100
150
Storage time (ns)
200
ð2Þ
correlation gsi as a function of storage time with 10-ns coincidence window,
extracted from b. For storage times up to 200 ns the correlations stay well above
ð2Þ
the classical limit of gsi ~2 (dashed line). The inset shows the storage efficiency
for the same range of storage times. With increasing storage times, limiting
factors in the storage medium degrade the comb shape and reduce the
efficiency and cross-correlation (see Supplementary Information). However,
the latter stays well above the classical limit for storage times up to 200 ns. Error
bars show 61 standard deviation (s.d.).
2 7 J A N U A RY 2 0 1 1 | VO L 4 6 9 | N AT U R E | 5 0 9
©2011 Macmillan Publishers Limited. All rights reserved
RESEARCH LETTER
Start
SSPD
50%
Faraday Δφi
mirrors
50%
Idler
Coincidences in two hours
18
b
Coincidence window
for post-selection
16
14
80
12
10
8
Stop
6
4
2
0
–40
Source of
entangled
photons
–20
0
20
40
Tstop – Tstart (ns)
60
80
100
Si APD
Signal
Coincidences in two hours
a
60
40
20
EQM (75 ns)
Es
Ls
Δφs
0
LQM (50 ns)
Storage
–240
–180
–120
Measurement
–60
0
Δφs (°)
60
120
180
240
Figure 3 | Storage of photonic entanglement in a crystal. a, Franson-type
set-up used to reveal the entanglement. A qubit analyser consisting of an
unbalanced, fibre-based Michelson interferometer with 25-ns delay and
relative phase Dwi is inserted before the SSPD used to detect the idler photon
(see also Fig. 1). The signal photon is stored in the crystal, yielding a light–
matter entangled state. The state of the memory is measured through reemission and detection of the photon in the time-window at zero-time delay
(central peak) of the coincidence histogram (inset). This post-selects
measurement on the entangled state of equation (1). The relative phase Dws can
be reliably set to any desired value (see the Supplementary information).
b, Number of coincidences in the central peak in two hours as a function of the
relative phase Dws for two values of Dwi. The pump power was 5 mW, and the
size of the coincidence window 10 ns. The solid and dashed lines result from fits
to equation (2) and respectively give visibilities of V 5 (78 6 4)% and
(84 6 4)%. The visibilities are mainly limited by the level of accidental
coincidences (cross symbols). The fit also gives a difference between the two
values of Dwi of 75u 6 10u. These values closely match settings necessary for a
maximal violation of the CHSH inequality. Error bars are 61 s.d.
Quantum entanglement can be revealed by a violation of the Clauser–
Horne–Shimony–Holt (CHSH) inequality6. The possibility of violating
this inequality, that is, of finding a CHSH parameter Sp
.ffiffiffi2, can be
inferred indirectly from a visibility larger than 1 2<70:7%.
Nevertheless, we performed the measurements necessary for a direct
violation of the inequality and obtained S 5 2.64 6 0.23. This proves the
presence of entanglement between the idler photon and the matter qubit
in the crystal, provided the effect of the memory on single photons is
appropriately described as storage followed by measurement (see
Supplementary Information). This description is correct within the
theory of AFC memories24, which is supported by experiments storing
weak coherent states of light11,21–23. Note also that we do not claim any
demonstration of nonlocal correlations. Indeed, besides the usual
locality and detection loopholes, here the measurement setting has
to be chosen before the photonic qubit is mapped onto the crystal.
This could have been avoided by adding an interferometer after the
memory, the latter being used for storage only. We did not do so
because we think that it is elegant and simple to use the memory also
as a small quantum processor that performs the measurement.
A particularly intriguing situation arises when post-selecting on the
case where only jEsæ is stored in the crystal for 25 ns using a single AFC
scheme, while jLsæ is directly transmitted. Indeed, the imbalance
between the storage efficiency and the transmission probability offers
a well-suited qubit analyser for a violation of the CHSH inequality
using bases lying in the x–z plane of the Bloch sphere. We performed
such a measurement and observed S 5 2.62 6 0.15 (see Supplementary Information). This implies that the initial photon–photon entangled
state is mapped onto a state of the form:
pffiffiffiffiffiffiffiffi
gabs jEQM Ei izjLs Li i
ð3Þ
using optical fibres3. To achieve this long-term goal, several future
advances are required. The user must be able to trigger the re-emission
of the memory, whereas in our experiment the duration of the storage is
pre-determined. We have proposed24 and demonstrated26 a method for
achieving on-demand re-emission using so-called spin-wave storage.
This has the additional benefit of allowing longer storage times owing to
the more robust spin coherence. Another crucial aspect is the efficiency,
which is directly linked to the optical depth of the material24. It can be
increased by using longer crystals19 or optical cavities27,28.
The creation of entanglement between a single photon and a
macroscopic object—in this case a single collective atomic excitation
delocalized over a 1-cm-long crystal—is fascinating in itself. Beyond its
fundamental interest, we believe that our demonstration of storage of
entanglement in a crystal represents an important step towards
quantum repeaters based on solid-state quantum memories.
We note that, parallel to this work, Saglamyurek et al. have demonstrated storage and retrieval of an entangled photon using a thuliumdoped lithium niobate waveguide29.
where gabs is the absorption efficiency. This is an entangled state
between a telecommunication-wavelength qubit and a light–matter
hybrid qubit. We note that this kind of hybrid qubit is the key ingredient
of an efficient quantum repeater protocol based on atomic ensembles
and linear optics3.
This work is part of the effort towards implementing a quantum
repeater, which could provide a solution to the distance limit (due to
intrinsic loss) for entanglement distribution and quantum cryptography
METHODS SUMMARY
Spectral filtering and detection. The bandwidth of the photon pairs is reduced by
a factor of 104 in several steps. Pump, signal and idler photons are spatially
separated by a diffraction grating (see Fig. 1). In combination with coupling into
single-mode fibres, this reduces the bandwidth to tens of gigahertz. A subsequent
passage through a Fabry–Perot cavity reduces the bandwidth of the idler photon to
45 MHz (corresponding to a coherence time of about 4 ns), and a fibre Bragg
grating blocks all but one of the longitudinal cavity modes. The signal photon is
filtered by two etalons with a linewidth of 600 MHz each, and different free spectral
ranges. The detector efficiency is 8% for the idler photon with 10-Hz dark counts,
and 30% with 100-Hz dark counts for the signal photon.
Frequency stabilization. We must ensure, for the whole duration of a measurement, that the central frequency of the optical filtering system at 1,338 nm and of the
AFC at 883 nm both satisfy the energy conservation of the SPDC process. To do this,
a small fraction of the light at 883 nm is overlapped with the light of the 532-nm laser
that pumps the PPKTP waveguide. This leads to the creation of light at 1,338 nm by
difference frequency generation (DFG). Using this DFG signal, the frequency of the
532-nm light is adjusted such that the detection rate on a separate InGaAs avalanche
photodiode (see Fig. 1) stays constant, which means that the 1,338-nm DFG light is
in resonance with the filtering cavity. Long-term stability of the 883-nm laser itself is
achieved by continuously referencing it to a Fabry–Perot cavity.
5 1 0 | N AT U R E | VO L 4 6 9 | 2 7 J A N U A RY 2 0 1 1
©2011 Macmillan Publishers Limited. All rights reserved
LETTER RESEARCH
Full Methods and any associated references are available in the online version of
the paper at www.nature.com/nature.
Received 24 August; accepted 9 November 2010.
Published online 12 January 2011.
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Supplementary Information is linked to the online version of the paper at
www.nature.com/nature.
Acknowledgements We thank R. Locher for help during the early stages of the
experiment. We are grateful to A. Beveratos and W. Tittel for lending us avalanche
photodiodes. This work was supported by the Swiss NCCR Quantum Photonics, the
Science and Technology Cooperation Program Switzerland–Russia, as well as by the
European projects QuRep and ERC-Qore. F.B. was supported in part by FQRNT.
Author Contributions All authors contributed extensively to the work presented in this
paper.
Author Information Reprints and permissions information is available at
www.nature.com/reprints. The authors declare no competing financial interests.
Readers are welcome to comment on the online version of this article at
www.nature.com/nature. Correspondence and requests for materials should be
addressed to M.A. ([email protected]).
2 7 J A N U A RY 2 0 1 1 | VO L 4 6 9 | N AT U R E | 5 1 1
©2011 Macmillan Publishers Limited. All rights reserved
RESEARCH LETTER
METHODS
Spectral filtering and detection. The narrowband filtering of the SPDC photons
consists of several steps (see Fig. 1). First, a diffraction grating spatially separates
the pump, signal and idler photons and, in combination with coupling into singlemode fibres, reduces the bandwidth of the photons at 883 nm (or 1,338 nm) to
90 GHz (or 60 GHz). Photons at 1,338 nm are then coupled through a Fabry–Perot
cavity with linewidth 45 MHz and free spectral range of 23.9 GHz. Subsequently, a
fibre Bragg grating with 16 GHz bandwidth ensures that only a single longitudinal
cavity mode remains.
Filtering one of the photons in the pair is the same as filtering the photon pair as a
whole, because energy conservation guarantees that photons measured in coincidence
have the same bandwidth. However, uncorrelated photons would then contribute
significantly to the accidental coincidences. Therefore, complementary filtering at
883 nm was necessary. To do this, we used one solid and one air-spaced etalon, both
with bandwidths around 600 MHz. Different free spectral ranges of 42 and 50 GHz
eliminate uncorrelated longitudinal modes. Additionally, outside the 120-MHz bandwidth of the AFC, the absorption of the crystal with an inhomogeneous linewidth of
6 GHz provides a final filtering step.
We used detectors with 30% detection efficiency and approximately 100 Hz
dark counts at 883 nm, and detectors with 8% detection efficiency and approximately 10 Hz at 1,338 nm. Together with a transmission of the filtering system for
the signal (or idler) photon of 45% (or 14%), and 4% (or 14%) for the remainder of
the optical set-up, we reached an overall detection efficiency of 0.5% (or 0.15%)
(see also Supplementary Information). These numbers could, in principle, be
significantly improved through optimized optical alignment, the use of antireflection-coated elements, and so on.
Frequency stabilization. In the experiment, coincidence rates are typically a few
per minute. With accumulation times thus reaching several hours, a high degree of
frequency stability of the lasers and filtering elements is indispensable. In particular,
frequency drifts of the AFC preparation laser with respect to the pump laser of the
SPDC source have to be eliminated. Otherwise, the photon-pair frequencies v883 1
v1338 5 v532 imposed by energy conservation in the SPDC would not simultaneously match the centre of the AFC and that of the filtering system at 1,338 nm.
Drifts were eliminated using the following method. First, the long-term stability of
the 883-nm laser was dramatically increased by locking it to a temperature-stabilized Fabry–Perot cavity. Second, during the 15-ms preparation cycle, we injected a
fraction of the 883-nm light into the waveguide. The frequency of the light created at
1,338 nm via difference frequency generation (DFG) was tuned by controlling the
frequency of the pump laser at 532 nm. Using a side-of-fringe technique, we could
then lock the frequency of the DFG signal to the transmission peak of the filtering
cavity. As a result, long-term frequency deviations between the centre of the AFC
structure and the filtered photon pairs were reduced to about 1 MHz over several
hours.
For measurements involving the unbalanced Michelson interferometer for the
idler photon, the phase of the interferometer was also stabilized using the highly
coherent DFG light.
Photon correlations in SPDC. Neglecting the exact frequency dependence, the
state of the photons created in the SPDC process is described by
pffiffiffi
j0s , 0i iz pj1s , 1i izOð pÞ, where the subscript ‘s’ (or ‘i’) indicates the signal
(or idler) mode at 883 nm (or 1,338 nm). Here, the pair creation probability p is
assumed to be small and proportional to the pump power. In such a state, the
signal and idler modes individually exhibit the statistics of a classical thermal field,
ð2Þ
that is, their auto-correlations are gx ~2 for x 5 s or i. We stress, however, that
the criterion for non-classicality of the cross-correlation that we used, namely
ð2Þ
ð2Þ
gsi ~Psi =Ps Pi w2, requires only that 1ƒgx ƒ2, which is always fulfilled by
non-degenerate photon pairs created through SPDC. In practice, Psi (or PsPi) is
determined by the number of coincidences in a certain time window centred on (or
away from) the coincidence peak. For low pump powers, the measured crosscorrelation is usually limited by detector dark counts, and at high pump powers it
is reduced by the contribution of multiple pairs, that is, higher-order terms in p.
©2011 Macmillan Publishers Limited. All rights reserved
SUPPLEMENTARY INFORMATION
doi:10.1038/nature09662
Supplementary Information: Quantum Storage of Photonic Entanglement in a Crystal
Christoph Clausen, Imam Usmani, Félix Bussières, Nicolas Sangouard,
Mikael Afzelius, Hugues de Riedmatten, and Nicolas Gisin
Group of Applied Physics, University of Geneva, CH-1211 Geneva 4, Switzerland
(Dated: November 4, 2010)
ATOMIC FREQUENCY COMB
The atomic frequency comb (AFC) memory is a
photon-echo based scheme, where the absorption profile
of the crystal is shaped into a comb-like structure using
optical pumping, thus making it possible to take full advantage of the high atomic density in a doped crystal,
despite of inhomogeneous broadening [1].
When a photon enters the crystal, with a spectral
bandwidth covering a large part of the AFC spectrum,
it is absorbed, provided that the optical depth is sufficient. After the absorption, the photon is stored in a
single atomic excitation delocalized over all the atoms,
corresponding to a collective Dicke type state,
N

j
cj e−ikzj ei2πδj t |g1 · · · ej · · · gN ,
(1)
where zj is the position of atom j (note that it is sufficient
to consider only the forward propagating spatial mode),
k is the wave-number of the light field, δj the detuning
of the atom with respect to the laser frequency and the
amplitudes cj depend on the frequency and on the spatial position of the particular atom j, which can be in
the ground state |gj  or the excited state |ej . In the
AFC the distribution of atomic detunings δj is periodic
with period ∆. After a time ts = 1/∆, the components
of the state (1) are in phase and lead to the reemission
of a photon in a well defined spatial mode k with high
probability.
Here we used as storage medium a 1 cm long Y2 SiO5
crystal doped with Nd3+ ions to a concentration of
30 ppm, and cooled to 3 K. This crystal has already
shown a high multimode capacity [2] with storage and
retrieval efficiency between 6% (for ts = 100 ns) and
1% (ts = 1.5 µs). In the work presented here we have
improved the storage efficiency in the range ts = 25
to 200 ns by creating comb peaks with an approximate
square shape. This has been shown to be the optimal
shape in terms of storage efficiency [3]. Note also that
the AFC structure is prepared by using an incoherent
optical pumping technique consisting of simultaneously
sweeping the laser frequency and modulating its intensity, as also used in [4]. The light used for preparation
and storage is focussed onto the crystal, with a beam
waist diameter of 40 µm. As a result we observe an efficiency of 21% for ts = 25 ns and 12% for ts = 100
ns. The decay of the efficiency with longer storage time
FIG. S1. Experimental absorption spectrum of an
AFC prepared by an incoherent optical pumping
technique. The peaks have a width larger than the effective linewidth of the material, so it is possible to use the
whole available optical depth and to have almost squareshaped peaks, which optimizes the rephasing. The dashed
lines delimit the 120 MHz bandwidth of the AFC.
(thus for closer comb spacing ∆) is due to limitations in
the coherence properties of the material, which deteriorates the comb shape for closely spaced peaks. We refer
to Ref. [2] for a more detailed discussion on this optical
pumping issue. For the shortest storage time of 25 ns,
however, this problem was almost negligible. Hence, the
obtained efficiency is mainly limited by the absorption
depth of the crystal. An example of a close to optimal
comb for ts = 25 ns is shown in Fig. S1. Using the measured comb as an input to a numerical Maxwell-Bloch
simulation gave an efficiency close to the measured one.
CHSH TEST AS WITNESS FOR LIGHT-MATTER
ENTANGLEMENT
Here, we discuss the way we create and measure the
entanglement between a photon at telecommunication
wavelength and a collective atomic excitation stored in a
crystal.
Our experiment starts with a source of photon pairs
(signal-idler) entangled in energy and time [5]. To reveal
such an entanglement, the usual way consists of using a
w w w . n a t u r e . c o m / NATURE | 1
RESEARCH SUPPLEMENTARY INFORMATION
2
Franson setup (see Fig. S2a) involving two unbalanced
Mach-Zehnder interferometers, one for each of the signal
and idler photons. By choosing appropriately the arm
length of the interferometers to avoid single photon interferences, but such that the delay is shorter than the
coherence time of the pump, the uncertainty in the creation time leads to a quantum interference between the
two path combinations short-short and long-long. The
non-local character of this interference can be revealed
through the violation of a Bell inequality by properly
choosing the phases and the beamsplitters of the interferometers [6].
Since our experiment presents strong similarities with
the Franson experiment, let us analyze in detail the violation of the CHSH inequality [7]. Instead of energytime entanglement, consider for simplicity time-bin entanglement (obtained by post-selection in the Franson
setup) involving two particular modes, the early E and
late L time bins separated by the delay of the interferometers [8],
1
√ (|Es Ei  + |Ls Li ).
2
(2)
It is instructive to describe the interferometers in the
Franson setup as a tool allowing (i) to transfer the entanglement between temporal modes to an entanglement
of spatial modes and (ii) to measure these modes in the
appropriate bases. Let us detail these two points separately.
(i) First, the signal mode is sent on a 1 × 2 optical
switch such that the entanglement (2) becomes
1
√ (|longs , Ei  + |shorts , Li )
2
(3)
where long and short describe two spatial modes corresponding to the long and short arms of the interferometer. Note that, in practice, the optical switch is usually replaced by a 50/50 beamsplitter followed by a postselection, as in Fig. S2a.
(ii) Once the time-bin encoding is mapped into a spatial encoding, the signal mode is measured, e.g. on the
equator of the Bloch sphere, by choosing the relative
phase of the short and long modes to be either ∆φ1s = π/4
or ∆φ2s = 3π/4 and then by combining the short and long
modes on a 50/50 beamsplitter. This corresponds to the
measurement bases
1
1
X1 = √ (σx + σy ), X2 = √ (−σx + σy ).
2
2
where σx and σy are Pauli matrices (see Fig. S3a).
If the same transformation and a similar measurement
(with either ∆φ1i = π/2 or ∆φ2i = 0) are performed
on the idler photon, corresponding to the measurement
bases (Fig. S3a)
Y1 = σy , Y2 = σx ,
2 | WWW . NATURE . COM / NATURE
a)
spatial
ps
bs
bs
bs
bs
energy time
measurement
measurement
b)
ps
measurement
crystal - photon
ps
bs
bs
memory
energy time
measurement
FIG. S2. a) The Franson setup. The energy-time entanglement carried by a photon pair is first mapped onto a spatial entanglement. A projective measurement is then performed using a phase shifter (ps), a beamsplitter (bs) and a
single-photon detector (half-circle). b) Our experiment. A
crystal, with two AFCs, replaces one of the interferometers.
The energy-time entanglement is mapped onto crystal-photon
entanglement, i.e. an entanglement involving two collective
atomic excitation modes and two spatial photonic modes.
the CHSH correlators are given by
1
Ē(X1 , Y1 ) = Ē(X2 , Y1 ) = − √
2
1
−Ē(X1 , Y2 ) = Ē(X2 , Y2 ) = − √
2
leading to a maximal CHSH parameter of
S = |Ē(X1 , Y1 ) + Ē(X2 , Y1 ) − Ē(X1 , Y2 ) + Ē(X2 , Y2 )|
√
= 2 2,
which violates the local upper bound of S = 2.
We now analyze our experiment when using the double
AFC scheme [2], where one of the Mach-Zehnder interferometers is replaced by a light-matter interface with
two atomic frequency combs (Fig. S2b), which differ by
their periodicity. This interface is equivalent to two distinct atomic ensembles capable of storing, with close-toequal efficiency, a temporal mode for either τ1 = 50 ns or
τ2 = 75 ns. Consider the state (2) as starting point. By
sending the signal mode to the crystal and by focusing
on memory outputs resulting either from a long storage
of an early mode or from a late photon stored for a short
time, we post-select the following light-matter entangled
state,
1
√ (|EQM Ei  + |LQM Li ) .
2
(4)
where EQM (LQM ) denotes a single collective atomic excitation (see Eq. (1)) for the comb associated with the
SUPPLEMENTARY INFORMATION RESEARCH
3
τ2 (τ1 ) storage time. This is the heart of our result, the
entanglement between a photon and a single atomic excitation delocalized over a 1 cm long crystal. To measure
this entanglement, we perform a projection onto
1
√ (LQM | + e−i∆φs EQM |)
2
a
b
(5)
by sending the photon emitted coherently by the two
combs to the detector. This requires that both AFCs can
reemit the stored excitation with close-to-equal probability. This emission merely plays the role of the beamsplitter measurement in the Franson setup. To perform the
CHSH test previously described the relative phase ∆φs is
reliably set to either ∆φ1s or ∆φ2s . This is done by shifting
the comb patterns with respect to the central frequency
of the signal photon [1]. This phase is very stable over
the storage time, resulting in a direct measurement of
the matter qubit with high fidelity. Note that, on the
idler side, the fiber interferometer is actively stabilized
to maintain a desired relative phase.
In the experiment, the projection (5) is achieved by
post-selecting events in the central coincidence peak (see
Fig. 3a in the main text). In the figure, the left sidepeak
corresponds to the detection of |Es  that was stored in
|LQM , and the right sidepeak to the detection of |Ls 
that was stored in |EQM .
Since we had only one detector on each side, we had
to add a π-phase shift every time we wanted to access
the second outcome of the chosen basis. For this reason,
we had to make 4 measurements per correlator, i.e. 16
for the whole Bell test. For a pump power of 3 mW we
obtained
Ē(X1 , Y1 ) = 0.68 ± 0.12,
Ē(X2 , Y1 ) = 0.79 ± 0.10,
Ē(X1 , Y2 ) = −0.60 ± 0.10,
Ē(X2 , Y2 ) = 0.57 ± 0.14,
leading to S = 2.64 ± 0.23. Note that the pump power
used for this measurement is lower than the 5 mW used
for the visibility measurements (Fig. 3b in the main text),
resulting in a higher signal-to-noise ratio (c.f. Fig. 2a in
the main text). Here, the uncertainties are standard deviations related to the poissonian statistics of the coincidence events. This is a clear violation of the CHSH
inequality that proves, regardless of experimental noise,
that the idler photon was entangled with a collective excitation in the crystal.
FIG. S3. Measurement bases for the violation of the
CHSH inequality. Arrows indicate signal (idler) bases X1
and X2 (Y1 and Y2 ). (a) For the violation of the CHSH inequality, all four bases lie on the equator of the Bloch sphere
with the appropriate angles between them. (b) The hybrid
qubit represents a state with unequal weights between the basis states |EQM  and |Ls , and optimal violation can only be
obtained on a plane including the poles of the Bloch sphere.
spanned by the states |EQM , |Ls , associated to the signal mode.
Contrary to the treatment above, we only use a single
AFC. In the post-selection we consider only those cases
where the early signal mode is stored in the AFC mode
|EQM  for a time of exactly τ = 25 ns. The detection
of this mode, which we will hereafter refer to as echo,
is thus made indistinguishable from the detection of the
late mode |Ls , which is directly transmitted through the
memory. Taking the echo efficiency ηecho and the transmission probability ηtrans into account, this corresponds
to a projection onto the vector
cos θ/2Ls | + e−i∆φs sin θ/2EQM |,
(6)


ηtrans
ηecho
with cos θ/2 = ηtrans
+ηecho , and sin θ/2 =
ηtrans +ηecho .
The phase factor is controlled via the AFC structure [1]
and is chosen to be either ∆φ1s = 0 or ∆φ2s = π. This
corresponds to measuring
X1 = sin θ σx + cos θ σz ,
X2 = − sin θ σx + cos θ σz .
Fifty metres away, the idler photon, at telecommunication wavelength, is projected either onto the z-direction
(corresponding to the operator Y1 = σz ) by measuring
the time of arrival at the detector or onto the x-direction
(Y2 = σx ) using a Michelson interferometer. One then
finds for the correlators,
CHSH INEQUALITY WITH A HYBRID QUBIT
Ē(X1 , Y1 ) = Ē(X2 , Y1 ) = cos θ,
Ē(X1 , Y2 ) = −Ē(X2 , Y2 ) = sin θ.
The violation of the CHSH inequality using the hybrid qubit is done in analogue to the violation described
above. However, we now consider the entanglement between the idler photon and a hybrid light-matter qubit,
so that the CHSH polynomial S  = 2 cos θ√+ 2 sin θ is
√
−ηecho
maximized to 2 2 for cos θ = ηηtrans
= 22 , i.e. for a
trans +ηecho
w w w . n a t u r e . c o m / NATURE | 3
RESEARCH SUPPLEMENTARY INFORMATION
4
ratio between the echo and transmitted pulses
1
ηecho
,
≈
ηtrans
5.8
(7)
which corresponds to the bases indicated in Fig. S3b.
In the experiment, the AFC structure is modified to
approximate this requirement (we measured ηtrans ≈
0.36, ηecho ≈ 0.05, giving a ratio of 1/7.2). Under the
assumption that the marginals are the same for the idler
photon, independent of the result of the measurement on
the signal mode, we measured
Ē(X1 , Y1 ) = 0.68 ± 0.05
Ē(X2 , Y1 ) = 0.71 ± 0.06
the contributions of the two terms in Eq. (8), thereby
yielding optimal CHSH violation from a non-maximally
entangled state.
SPECTRAL FILTERING OF PHOTON PAIRS
Here we give an overview of bandwidth and efficiency
for the elements of the optical setup, where the efficiency
for optical elements equals the peak transferred intensity normalized to incoming intensity. The free spectral
ranges (FSR) of etalons and cavity, as well as the dark
count rate (DC) of the detectors, are given in parenthesis.
Wavelength Element
Ē(X1 , Y2 ) = 0.63 ± 0.09
883 nm
Ē(X2 , Y2 ) = −0.60 ± 0.09
leading to S  = 2.62 ± 0.15, a violation of the CHSH inequality by more than 4 standard deviations. This clearly
shows that when the early mode was stored in our crystal, the two qubit state involving the hybrid qubit and
the idler qubit was entangled.
Taking a closer look, it might at first be surprising
that the violation of the Bell inequality can be maximal,
since directly after the absorption the system is described
by the post-selected, non-maximally entangled state (neglecting noise)
√
1
(α|EQM Ei  + |Ls Li ) ,
1 + α2
90 GHz
600 MHz
600 MHz
120 MHz
–
70%
80%
80%
–
4%
–
120 MHz
30%
0.5%
Grating
Filter Cavity (FSR 24 GHz)
Fiber Bragg Grating
Fiber coupling, circulator, polarization controller, fiber beam
splitter and mirrors
Detector (DC 10 Hz)
Total
60 GHz
45 MHz
16 GHz
–
90%
30%
50%
14%
–
45 MHz
8%
0.15%
(8)
where α is related to the absorption efficiency ηabs of the
√
quantum memory by α = ηabs . (The absorption efficiency corresponds to the absorption by the comb peaks
and does not include the absorption by residual atoms
whose resonance frequencies fall between the peaks, so
that ηtrans = 1 − ηabs .) The explanation is that the
memory based measurement is a generalized measure2
ment. More precisely, assuming that ηecho = ηabs
η [9],
the measurement consists of two generalized measurements {X1 , X2 } made from {Πs+1 , Πs−1 , 1 − Πs+1 − Πs−1 },
i.e. projections onto the non-orthogonal vectors
√
√
Πs+1 : ηtrans Ls | + e−i∆φs ηηabs EQM |,

ηtrans
√
EQM |,
Πs−1 : e−i(∆φs +π) ηηabs Ls | +
ηabs
where φs is controlled from the AFC structure to be either φs,1 = 0◦ or φs,2 = 180◦ for X1 and X2 respectively.
Under the fair sampling assumption where inconclusive
results are discarded, one can only take successful projections on Πs+1 and Πs−1 into account. If one assigns
the value +1 (−1) to a conclusive projection into Πs+1
(Πs−1 ), one finds random marginals and a Bell violation
√
of 2 2 provided that the condition (7) is fulfilled. This
is analogue to the distillation of entanglement reported
in ref. [10] where a non unitary filtering process equalizes
4 | WWW . NATURE . COM / NATURE
1338 nm
Bandwidth Efficiency
Grating
Solid etalon (FSR 42 GHz)
Air-spaced etalon (FSR 50 GHz)
AFC
Fiber coupling, polarization controller, fiber optic switch, mirrors, lenses and windows
Detector (DC 100 Hz)
Total
[1] Afzelius, M., Simon, C., de Riedmatten, H. & Gisin, N.
Multimode quantum memory based on atomic frequency
combs. Phys. Rev. A 79, 052329 (2009).
[2] Usmani, I., Afzelius, M., de Riedmatten, H. & Gisin,
N. Mapping multiple photonic qubits into and out of one
solid-state atomic ensemble. Nat Commun 1, 1–7 (2010).
[3] Bonarota, M., Ruggiero, J., Le Gouët, J.-L. &
Chanelière, T. Efficiency optimization for atomic frequency comb storage. Phys. Rev. A 81, 033803 (2010).
[4] Lauritzen, B. et al. Telecommunication-wavelength solidstate memory at the single photon level. Phys. Rev. Lett.
104, 080502 (2010).
[5] Franson, J. D. Bell inequality for position and time. Phys.
Rev. Lett. 62, 2205–2208 (1989).
[6] Tittel, W., Brendel, J., Zbinden, H. & Gisin, N. Violation
of bell inequalities by photons more than 10 km apart.
Phys. Rev. Lett. 81, 3563–3566 (1998).
[7] Clauser, J. F., Horne, M. A., Shimony, A. & Holt, R. A.
Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett. 23, 880–884 (1969).
[8] Brendel, J., Gisin, N., Tittel, W. & Zbinden, H. Pulsed
energy-time entangled twin-photon source for quantum
communication. Phys. Rev. Lett. 82, 2594–2597 (1999).
[9] η describes decoherence in the memory due to the finite
finesse of the comb. Note that this equation can be used
in the limit where re-absorption is negligible (see Ref. [1]),
which is the case in our experiment.
[10] Kwiat, P. G., Barraza-Lopez, S., Stefanov, A. & Gisin,
N. Experimental entanglement distillation and ‘hidden’
non-locality. Nature 409, 1014–1017 (2001).
news & views
light sources in optoelectronic circuits for
communications, opens up completely new
avenues of research in nanoscale sensing or
simply gives us a better understanding of
how light and matter interact.
❒
Rupert F. Oulton is at the Department of Physics at
Imperial College London, The Blackett Laboratory,
Experimental Solid State Physics, Imperial College
London, Prince Consort Road, London SW7 2AZ, UK
e-mail: [email protected]
References
1.
2.
3.
4.
Oulton, R. F. Mater. Today 15, 592–600 (January/February 2012).
Berini, P. & De Leon, I. Nature Photon. 6, 16–24 (2012).
Khurgin, J. B. & Sun, G. Appl. Phys. Lett. 100, 011105 (2012).
Wang, F. & Shen, Y. R. Phys. Rev. Lett. 97, 206806 (2006).
5. Johnson, P. B. & Christy, R. Phys. Rev. B 6,
4370–4379 (1972).
6. Ma, R.‑M. et al. Nature Mater. 10, 110–113 (2010).
7. Noginov, M. A. et al. Nature 460, 1110–1112 (2009).
8. Hill, M. T. et al. Opt. Express 17, 11107–11112 (2009).
9. Genov, D. A. et al. Phys. Rev. B 83, 245312 (2011).
10.Gontijo, I. et al. Phys. Rev. B 60, 11564–11567 (1999).
11.Ma, R.‑M. et al. Laser Photon. Rev. http://dx.doi.org/10.1002/
lpor.201100040 (2012).
12.Ding, K. et al. Phys. Rev. B 85, 041301 (2012)
QUANTUM OPTICS
Linking crystals with a single photon
Linking distant quantum memories with light has been a goal of the quantum information community for many
years. A team at the University of Geneva has now demonstrated that memories made from rare-earth-ion-doped
crystals can be connected using a single photon.
Steven Olmschenk
T
he ability to transmit quantum states
over large distances is a goal that is
being hotly pursued by researchers
across the globe. Although distances as
great as 144 km have been successfully
spanned by single photons1, connections
over arbitrary distances will probably require
the use of quantum memories (capable of
storing quantum states) to allow repeaters to
overcome the detrimental losses associated
with long transmission distances.
Although simple amplification of a
quantum signal using a conventional optical
amplifier is ruled out by the no-cloning
theorem2, which prohibits copying an
unknown quantum state, quantum memories
make it possible to break the link up into
several shorter segments. The deployment
of quantum memories may not only enable
long-distance quantum communication for
ultrasecure information transfer, but might
also be used to perform distributed quantum
information processing (or quantum
cloud computing), allowing access to
computational problems that are inaccessible
through classical approaches.
For these reasons, significant effort is
currently being directed towards linking
quantum memories with photons to
establish quantum connections between
distant quantum memories. Entanglement
is widely viewed as an essential resource for
creating such a quantum connection in both
quantum communication and quantuminformation processing. A particularly useful
type of entanglement-creating operation is
one that is ‘heralded’, whereby a ‘heralding
event’ (such as the detection of a photon)
signals the creation of entanglement
between the quantum memories. Heraldedentanglement operations between quantum
memories have been realized in a number of
a
b
Figure 1 | Classical and quantum shell games. a,
In a classical shell game, a single pea is hidden
under one of two shells. b, In a quantum shell
game, the pea is under both shells simultaneously
in the form of a superposition — this is analogous
to the form of entanglement used in the work of
Usmani et al.
physical implementations, including atomic
ensembles3 and trapped atomic ions4. Now,
writing in Nature Photonics, Usmani et al.
experimentally demonstrate how a heralded
single photon can create entanglement
between two rare-earth-ion-doped crystals
separated on the centimetre scale5. One of the
advantages of this system is that established
fabrication techniques for these solid-state
devices may assist scaling for future advanced
operations in quantum information.
The basic form of entanglement employed
by Usmani et al. is a bit like a quantum
version of a shell game (Fig. 1). In a classic
shell game, a single ‘pea’ is placed under
one of multiple shells and the player guesses
where the pea resides. The quantum version
of this game would involve a single pea that
is in a superposition of being under all shells
simultaneously. Once a measurement is
made, the quantum superposition collapses
and the pea is found under one of the
shells. The entangled state produced by the
experiment presented in this issue of Nature
Photonics is analogous to this situation,
where a single photon replaces the pea, and
crystals replace the shells.
Usmani et al. began by producing a pair
of photons through the standard quantumoptics technique of spontaneous parametric
downconversion. Because the generation of
this photon pair is probabilistic, detecting one
of the two photons heralds the presence of the
other, thus giving researchers a guarantee that
they are sending just one photon into their
experiment. The latter of these photons is the
‘pea’ in the quantum shell game. Usmani et al.
sent this photon to a beamsplitter and
directed the two output paths towards
two separate crystals (Fig. 2). The photon
is absorbed by one of the crystals, where
it excites one of the many optically active
rare-earth ions confined therein. However,
the inherent lack of information regarding
which path the photon travelled — and thus
by which crystal it was absorbed — results in
the creation of an entangled state between the
two crystals. One of the two crystals ought
to contain an excitation, but the excitation is
in an entangled superposition state between
the two crystals — this is analogous to the
pea being under both shells simultaneously.
Usmani et al. were able to store this
superposition state for up to 33 ns.
Because directly measuring the excitation
of the crystal is a significant experimental
challenge, Usmani et al. verified the
entanglement by using an echo technique
to write the stored entangled state back to a
photon. Correlation measurements between
the two paths of this re-emitted photon
established a lower bound for the quality of
NATURE PHOTONICS | VOL 6 | APRIL 2012 | www.nature.com/naturephotonics
© 2012 Macmillan Publishers Limited. All rights reserved.
221
news & views
the entanglement between the two crystals.
Using these measurements, Usmani et al.
were able to demonstrate that their single
heralded photon indeed created an entangled
state between the two crystals.
Of course, there are plenty of challenges
that must be overcome before this system can
be used to establish a large quantum network.
One of the biggest challenges — one faced
by all such probabilistic entangling schemes
demonstrated so far — will be increasing the
overall efficiency of the system, which limits
the rate of entanglement generation and thus
the rate of information transfer. Another
necessary improvement is increasing the
distance between the crystals — Usmani et al.
employed a spacing of just 1.3 cm to avoid the
use of multiple cryostats. Other issues include
improving the quality of the entanglement
and lengthening the storage duration of the
quantum state.
However, researchers have already
demonstrated impressive progress towards
tackling many of these issues, including
demonstrations of long-lived coherences
in crystals6 for potentially storing received
information and proposed architectures
to increase the distance between crystal
quantum memories7. Moreover, the
intriguing possibility of fabricating structures
Crystals
Beamsplitter
Heralded photon
Mirror
Figure 2 | Creating entanglement between two quantum memories. A heralded single photon (the ‘pea’)
first passes through a beamsplitter. The outputs of the beamsplitter are directed to the two crystals (the
‘shells’), where the photon is absorbed to produce a single excitation. The lack of information regarding
the path of the photon creates an entangled state between the two crystals.
directly into the crystals to improve the
overall efficiency of the system adds to
the appeal of these solid-state devices.
Combining such advances with the spatial
and temporal multiplexing abilities of
these crystals, which might also be used to
improve the entanglement rate, will make this
system a strong contender for the scalable
technology needed to implement large
quantum networks.
❒
Steven Olmschenk is at the Joint Quantum
Institute, a research partnership between the
University of Maryland and the National Institute
of Standards and Technology, College Park,
Maryland 20742, USA.
e-mail: [email protected]
References
1. Ursin, R. et al. Nature Phys. 3, 481–486 (2007).
2. Wootters, W. K. & Zurek, W. H. Nature 299, 802–803 (1982).
3. Sangouard, N., Simon, C., de Riedmatten, H. & Gisin, N.
Rev. Mod. Phys. 83, 33–80 (2011).
4. Duan, L.‑M. & Monroe, C. Rev. Mod. Phys. 82, 1209–1224 (2010).
5. Usmani, I. et al. Nature Photon. 6, 234–237 (2012).
6. Fraval, E., Sellars, M. J. & Longdell, J. J. Phys. Rev. Lett. 95,
030506 (2005).
7. Simon, C. et al. Phys. Rev. Lett. 98, 190503 (2007).
VIEW FROM... SPIE PHOTONICS WEST 2012
Photons, neurons and wellbeing
Techniques for the targeted optical stimulation of neurons may offer new ways to tackle medical problems such as
heart defects, epilepsy, Parkinson’s, blindness and hearing loss.
David Pile
U
sing light to control biological
processes is a relatively new joint
research direction of the biological
and physical sciences. Yet there is already
a great deal of motivation for developing
such technologies, as made clear from the
“Photons and Neurons” sessions at SPIE
Photonics West 2012, held in San Francisco,
USA, on 21–26 January 2012.
Over a billion people worldwide
suffer from brain disorders such as
stroke, depression, migraine, epilepsy,
Parkinson’s, chronic pain and blindness.
According to Edward Boydon, an expert
on brain disorders from the Massachusetts
Institute of Technology Media Lab,
few of these disorders are effectively
treatable by drugs or neurosurgical
procedures. Furthermore, the treatments
that do exist are often partial or present
undesirable side effects. Part of the problem
222
is the complexity of the dense neural circuits
in the brain.
“Ideally we would be able to hone in on
the precise circuits that can best contribute
to the remedy of disease, and then use
those circuits as drug targets, or as targets
for neurosurgeons to implant electrodes to
reduce symptoms,” Boydon explained. “To
do this, we invented a new technology that
allows us to control specific cells embedded
within dense neural circuits with light. We
do this by exploiting the photosynthetic
and photosensory proteins found in many
biological species, which convert light into
electrical energy.”
Installing photosensory cells in a
particular region of the brain modifies the
neuronal cells to respond to light while
leaving the surrounding cells unaffected.
Boydon’s team deliver the genes that encode
for these proteins to targeted neurons
by employing gene therapy viral vectors
currently used for gene therapy trials in
humans. They deliver the light by inserting
compact optical probes such as optical
fibres attached to small LEDs or lasers into
the brain. Boydon and colleagues are now
working on three-dimensional (3D) lightdelivery devices — arrays of waveguides — to
achieve improved control over 3D neural
circuits. Light delivery methods in the field
of optogenetics (the stimulation of nerve
cells with light) was one of the key discussion
points at the meeting.
Partrick Degenaar from Newcastle
University in the UK explained that the
rise of optogenetics has many exciting
applications to neuroprosthesis. Future
developments will require researchers to
find ways of efficiently delivering light at the
required intensity and depth — particularly
because biological tissue is a strong scatterer
NATURE PHOTONICS | VOL 6 | APRIL 2012 | www.nature.com/naturephotonics
© 2012 Macmillan Publishers Limited. All rights reserved.
LETTERS
PUBLISHED ONLINE: 4 MARCH 2012 | DOI: 10.1038/NPHOTON.2012.34
Heralded quantum entanglement between
two crystals
Imam Usmani, Christoph Clausen, Félix Bussières, Nicolas Sangouard, Mikael Afzelius* and
Nicolas Gisin
Quantum networks must have the crucial ability to entangle
quantum nodes1. A prominent example is the quantum repeater2–4, which allows the distance barrier of direct transmission
of single photons to be overcome, provided remote quantum
memories can be entangled in a heralded fashion. Here, we
report the observation of heralded entanglement between
two ensembles of rare-earth ions doped into separate crystals.
A heralded single photon is sent through a 50/50 beamsplitter,
creating a single-photon entangled state delocalized between
two spatial modes. The quantum state of each mode is
subsequently mapped onto a crystal, leading to an entangled
state consisting of a single collective excitation delocalized
between two crystals. This entanglement is revealed by
mapping it back to optical modes and by estimating the
concurrence of the retrieved light state5. Our results highlight
the potential of crystals doped with rare-earth ions for
entangled quantum nodes and bring quantum networks based
on solid-state resources one step closer.
Quantum entanglement challenges our intuition about
physical reality. At the same time, it is an essential ingredient in
quantum communication6, quantum precision measurements7
and quantum computing8,9. In quantum communication, photons
are naturally used as carriers of entanglement using either freespace or optical-fibre transmission. However, even with ultralowloss telecommunication optical fibre, the transmission probability
decreases exponentially with distance, limiting the achievable
communication distance to a few hundred kilometres10. A potential
solution is to use quantum repeaters2 based on linear optics and
quantum memories3,4 with which the entanglement distribution
time scales polynomially with the transmission distance, provided
entanglement between quantum memories in remote locations
can be heralded. In this context, one can more generally consider
prospective quantum networks1 where nodes generate, process
and store quantum information, while photons transport
quantum states from site to site and distribute entanglement over
the entire network. An essential step towards the implementation
of such potential technologies is to create entanglement between
two quantum memories in a heralded manner3,4.
Experimental observation of heralded entanglement between
two independent atomic systems for quantum networks has been
achieved using cold gas ensembles involving either two distinct
ensembles5 or two spatial modes in the same ensemble11–14.
Complete elementary links of a quantum repeater (based on
heralded entanglement) have been implemented in cold gas
systems15,16 and with two trapped ions17. For the realization of
scalable quantum repeaters, solid-state devices are technologically
appealing18. In this context, important results have already been
obtained by using crystals doped with rare-earth ions (REs) as
quantum memories. These results have demonstrated light storage
times greater than one second19, a storage efficiency of 69%
(ref. 20) and quantum storage of 64 independent optical modes in
one crystal21 (see also ref. 22). Recent achievements23,24 include the
storage, in a single RE-doped crystal, of photonic time-bin entanglement generated through spontaneous parametric downconversion
(SPDC). Heralded entanglement between two RE-doped crystals
using an SPDC source, a common resource in quantum optics,
represents an important step towards the implementation of
quantum repeater architectures based on solid-state devices25.
Here, we present the observation of heralded quantum entanglement between two neodymium ensembles doped in two yttrium
orthosilicate crystals (Nd3þ:Y2SiO5) separated by 1.3 cm (Fig. 1
and Methods). A nonlinear optical waveguide is pumped to
produce photon pairs by means of SPDC. The resulting idler
(1,338 nm) and signal (883 nm) photons are strongly filtered to
match the working bandwidth of the crystals, yielding a coherence
time of 7 ns. In the limit where the probability of creating a single
pair is much smaller than one, the detection of an idler photon
heralds the presence of a single signal photon. By sending the
latter through a balanced beamsplitter, one heralds,
√ neglecting
optical losses, a single-photon entangled state26 1/ 2(|1lA |0lB +
|0lA |1lB ) between the two spatial output modes A and B. In each
of the modes a crystal acts as quantum memory, denoted MA or
MB. Upon absorption of the single photon27, the detection of an
idler photon heralds the creation of a single collective √
excitation
delo
calized between the two crystals, written as 1/ 2(|WlA |0lB +
|0lA |WlB ). Here, |WlA(or |WlB) is the Dicke-like state created by
the absorption of a single photon in MA (or MB). To determine the
presence of entanglement between the memories, we use a
photon-echo technique based on an atomic frequency comb27–31
that reconverts the collective excitation into optical modes A and B
after a preprogrammed storage time of 33 ns (with a total efficiency
of 15%). The resulting fields can then be probed using single-photon
detectors to reveal heralded entanglement between the memories. As
the entanglement cannot increase through local operations on the
optical modes A and B, the entanglement of the retrieved light
fields provides a lower bound for entanglement between the
two memories.
The photonic state retrieved from the memories is described by a
density matrix r, including loss and noise, expressed in the Fock state
basis. To reveal entanglement in this basis, a tomographic approach
based on single-photon detectors can be used5,13,14. Specifically, from
the values of the heralded probabilities pmn of detecting m [ {0,1}
photons in mode A and n [ {0,1} in mode B, combined with the
magnitude of the coherence between these modes, a lower bound
on the concurrence C of the detected fields is obtained by
√
C ≥ max(0, V( p01 + p10 ) − 2 p00 p11 )
(1)
Group of Applied Physics, University of Geneva, CH-1211 Geneva 4, Switzerland. * e-mail: [email protected]
234
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© 2012 Macmillan Publishers Limited. All rights reserved.
NATURE PHOTONICS
LETTERS
DOI: 10.1038/NPHOTON.2012.34
1 cm
1.3
c
Heralding
detector
MB
B
Filtering
1,33
8n
m
88
3
A
m
MA
FR
nm
BS
DM
5
Detector 2
PBS
m
n
32
PPKTP
waveguide
Memory
preparation
Detector 1
Switch
Figure 1 | Experimental set-up. Quantum memories MA and MB are implemented using neodymium ions doped into yttrium ortho-silicate crystals
(Nd3þ:Y2SiO5) separated by 1.3 cm and cooled to 3 K using a cryostat (see ref. 23 for details). The total efficiency of each memory (used in a double-pass
configuration) is 15%. A fibre-optic switch is used to alternate between a 15-ms-long preparation of the two neodymium ensembles as atomic frequency
combs on the 4I9/2 4F3/2 transition, followed by attempts at entanglement creation for another 15 ms. The preparation includes a 4 ms waiting time to
avoid fluorescence from atoms left in the excited state. For entanglement creation, continuous-wave light at 532 nm is coupled into a periodically poled (PP)
KTP waveguide, leading to the production of pairs of photons at wavelengths of 883 nm and 1,338 nm through SPDC. Photons from each pair are separated
on a dichroic mirror (DM) and frequency filtered to below the 120 MHz bandwidth of the quantum memories. Detection of an idler photon at 1,338 nm
(using a low-noise superconducting single photon detector38) heralds the presence of a signal photon at 883 nm. The signal photon now traverses the
switch, a polarizing beamsplitter (PBS) and a Faraday rotator (FR), before a 50/50 beamsplitter (BS) creates single-photon entanglement between
spatial modes A and B. This entanglement is, upon absorption, mapped onto crystals MA and MB. After a preprogrammed storage time of 33 ns, the
photons are re-emitted and pass through the beamsplitter again. Depending from which output mode of the beamsplitter they emerge, they either
reach detector 2 (silicon-based single-photon detector) or are rotated in polarization by the Faraday rotator and reflected by the polarizing beamsplitter
towards detector 1.
(concurrence is a measure of entanglement, ranging from 0 for a
separable state to 1 for a maximally entangled state). The term V is
the interference visibility obtained by recombining optical modes
A and B on a 50/50 beamsplitter and is directly proportional to
the coherence between the retrieved fields in modes A and B. To
obtain a large concurrence, one should maximize V (the coherence)
and p10 þ p01 (the probability to detect the heralded photon), and
minimize p00 and p11 (the probabilities of detecting separable states
|0lA|0lB and |1lA|1lB stemming from a lost signal photon and from
two signal photons, respectively). To estimate V, p00 , p10 , p01 and
p11 , we used the set-up of Fig. 1 (see Methods for details). For p11 ,
in particular, we used two different methods, as described in
the following.
In the first method, we use a direct measurement of threefold
coincidences, that is, involving all three detectors (see
Supplementary Information). With a pump power of 16 mW,
we obtained C (MLE) ¼ 6.3+3.8 × 1025 using a maximum likelihood
estimation (MLE) of the threefold coincidence probability, and
C (CE) ¼ 3.9+3.8 × 1025 using a more conservative estimation
(CE). Both estimations yield a concurrence that is greater than 0
by at least one standard deviation, which is consistent with the presence of entanglement between the two crystals. This measurement
required 166 h, a period in which two threefold coincidences were
observed. The prohibitively long integration time of this method
prevented us from attempting it with lower pump powers (that is,
for a lower probability of creating more than one pair). Hence, to
study how the concurrence changes with pump power, we used a
second method based on twofold coincidences, which we
now describe.
In the second method, p11 is estimated using the fact that
our SPDC source produces a state that is very close to a
two-mode-squeezed state (TMSS) (see Methods and
Supplementary Information). We therefore assume that the
measured zero-time cross-correlation g s,i is consistent with a
TMSS and can be written as g s,i = 1 + 1/p, where p and p 2 are
interpreted as the probabilities of creating one and two photon
pairs, respectively (where p ≪ 1). In practice, g s,i is taken as
the average value of gAs,i and gBs,i, where gAs,i (or gBs,i) is obtained by
blocking mode B (or mode A). From the relationship between p
and p11 , we find
p11 =
4p10 p01
g s,i − 1
(2)
In the Methods and Supplementary Information, we present
additional measurements that support our assumption about our
source, and show that it leads to a lower bound on the concurrence.
We performed a series of measurements for several values of the
pump power, which is proportional to p provided p ≪ 1.
Figure 2a shows the interference of the delocalized single photon
retrieved from the memories. The visibility does not depend on
the pump power, and has an average value of 96.9+1.5%.
Figure 2b shows the measured g s,i , from which it can be seen that
reducing the pump power increases the cross-correlation, as
expected. Using additional measurements, we estimated the transmission loss, memory efficiency, dark count probability and pair
creation probability of our set-up. These values were then used in
a theoretical model (shaded region in Fig. 2b) that is in excellent
agreement with the measured values of g s,i , providing additional
evidence that the estimated value of p11 yields a lower bound
on the concurrence (see Supplementary Information). Figure 2c
shows the lower bound on the concurrence for all pump powers,
calculated using equations (1) and (2). The concurrence decreases
with pump power because p11 increases, but all other terms in
equation (1) depend on photon loss only and hence are approximately constant. Nevertheless, the concurrence remains larger
than zero for all used pump powers, consistent with heralded entanglement between the atomic ensembles inside the two crystals. The
results of the measurement of the concurrence based on threefold
coincidences are plotted in Fig. 2c and agree, within uncertainty,
with the results of the method based on measurement of the
cross-correlation. The observed values of the concurrence are
lower bounds on the amount of entanglement of the detected
fields and are almost entirely determined by optical loss. Factoring
out the detector inefficiency and interferometer loss yields a lower
NATURE PHOTONICS | VOL 6 | APRIL 2012 | www.nature.com/naturephotonics
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235
LETTERS
NATURE PHOTONICS
a
1.0
0.9
Coincidences
Visibility
800
0.8
0.7
0.6
600
400
200
0
0.5
0.25
0.30
0.35
0.40
Phase (a.u.)
0.45
0.4
b
0
2
4
6
8
10
Pump power (mW)
12
14
16
0
2
4
6
8
10
Pump power (mW)
12
14
16
0
2
4
6
8
10
Pump power (mW)
12
14
16
30
25
g—s, i
20
15
10
5
0
c
1.6
Concurrence (10−4)
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
Figure 2 | Results. a, Visibility as a function of pump power. Visibility is
approximately constant with an average of 96.5+1.2% (green shaded
region). Inset: visibility curves at 16 mW measured with detectors 1 (blue
filled circles) and 2 (red open squares) (15 min acquisition time per point;
error bars are smaller than the symbols). The different amplitudes result
from non-uniform loss after recombination of modes A and B on the
beamsplitter. The visibilities agree, within uncertainty, to the fits. b, Zerotime cross-correlation gs,i as a function of pump power. The decreasing
values agree well with a theoretical model (shaded region; see
Supplementary Information). c, Lower bound on the concurrence estimated
using the cross-correlation measurement as a function of pump power (blue
filled circles), calculated using equations (1) and (2), and the mean visibility
of a. The concurrence decreases with pump power, as expected, but remains
positive up to 16 mW. The shaded region corresponds to the model in b.
All measured values of p10 , p01 and p11 are given in the Supplementary
Information. All values are based on raw counts (that is, without subtraction
of dark counts and accidental coincidences) and were obtained with
coincidence detection windows of 10 ns. Uncertainties are obtained
assuming Poissonian detection statistics. The lower bound on the
concurrence at 16 mW obtained from measured threefold coincidences using
either the MLE (green open diamond) or the CE (green filled diamond) are
also shown (horizontally offset for clarity).
236
DOI: 10.1038/NPHOTON.2012.34
bound of 0.01 for the concurrence of the field retrieved just after
the crystals (see Methods).
The high interference visibility indicates that the stored entanglement does not significantly decohere during the 33 ns, and the
coherent nature of the storage endures for longer times, as shown
previously using storage of weak coherent states21,28. Increasing
the storage time does, however, lower the cross-correlation function
g s,i , as measured in ref. 23, and this should consequently lower the
concurrence of the stored entanglement (this is essentially due to
the decrease of the memory efficiency with increasing storage
time, as shown in the Supplementary Information and in ref. 23).
With Nd3þ:Y2SiO5 one could, in principle, store entanglement for
up to 1 ms (ref. 21). A promising approach to go beyond these
limits, and to allow on-demand retrieval of the stored entanglement,
is to implement spin-wave storage27 to increase the storage times (as
demonstrated recently29 in Pr3þ:Y2SiO5), and to place the crystal
inside an impedance-matched cavity to increase the efficiency32,33.
Such improvements are necessary for the development of a
quantum repeater based on solid-state quantum memories, and
are the subject of current research.
In conclusion, we have reported an experimental observation of
heralded quantum entanglement between two separate solid-state
quantum memories. We emphasize that although the entangled
state involves only one excitation, the observed entanglement
shows that the stored excitation is coherently delocalized among
all the neodymium ions in resonance with the photon, meaning
1 × 1010 ions in each crystal. Our results demonstrate that RE
ensembles, naturally trapped in crystals, have the potential to
form compact, stable and coherent quantum network nodes.
Moreover, single-photon entanglement is a simple form of entanglement that can be used for teleportation34 and entanglement
swapping operations3 and can also be purified using linear optical
elements35. It is also a critical resource in several proposals for
quantum repeaters3,25,36 as it is less sensitive to transmission loss
and detector inefficiencies4. This, however, is counterbalanced by
the extra requirement of phase stabilization over long distances,
and the duplication of the resources for post-selecting entanglement
shared between two photons, which is needed to implement qubits
measurements4. Our experimental approach is based on solid-state
devices, the key components being the PPKTP chip (the photon
source) with an integrated waveguide and the crystal memory. We
believe that this approach opens up possibilities for the integration
of components, such as frequency filtering directly on the chip37 or
waveguide quantum memories24. The prospect of combining solidstate photon sources and quantum memories is therefore attractive
for practical future quantum networks. One important challenge in
this context is to create entanglement between two remote solids in a
heralded way using two distant sources of photon pairs and a central
station performing a single-photon Bell state measurement25, that is,
the realization of an elementary link for quantum repeaters.
Methods
Experimental set-up and concurrence estimation. To reveal entanglement in
the Fock state basis, one cannot resort to violating a Bell inequality given solely
inefficient and noisy single-photon detectors. Instead, quantum state tomography
using single-photon detectors was developed5,13,14. Implicit to this method are
the assumptions that (i) the creation of more than two pairs is negligible and (ii) the
off-diagonal elements of r with different number of photons vanish (this is valid
because no local oscillator providing an aphase reference was used; ref. 5). To
estimate V, p00 , p10 and p01 , we used the set-up of Fig. 1 in the following way. First,
the visibility V was measured by allowing the re-emitted delocalized photon to
interfere with itself using a balanced Michelson interferometer for which the
phase was actively stabilized. We then blocked spatial mode B to estimate p10 by
summing the number of detections on detectors 1 and 2, conditioned on a heralding
signal. This is justified, as the probability of creating two photon pairs is much
smaller than the probability of creating a single one. Probability p01 is estimated
similarly by blocking mode A instead of B. Alternatively, we could also estimate the
sum p10 þ p01 directly by randomizing the phase of the interferometer, with both
arms unblocked (further clarifications and justifications on these measurements are
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NATURE PHOTONICS
LETTERS
DOI: 10.1038/NPHOTON.2012.34
given in the Supplementary Information). Then, p00 is estimated through
normalization of the total probability, p00 þ p10 þ p01 ≈ 1, which is justified by the
fact that p11 ≪ p10 þ p01 ≪ p00.
The estimation of p11 using the cross-correlation, which is motivated by the
results of ref. 13, assumes that all the observed detections stem from a TMSS. In the
Supplementary Information, we show that this assumption is conservative and that
it yields a lower bound for the concurrence. Using the cross-correlation to estimate
p11 requires only the measurement of twofold coincidences, rather than threefold
coincidences as for the other method. For our specific set-up, this resulted in a
reduction of the measurement time by a factor of 106 to achieve a similar statistical
confidence on the concurrence. Moreover, this method requires no physical
modifications to the optical circuit to measure the different components of the
retrieved fields, which simplifies its implementation. In the Supplementary
Information, we present additional measurement that provide evidence that our
source produces a state that is very close to an ideal TMSS, and we summarize
the results here. We measured the second-order autocorrelation of the signal
(2)
(or idler) mode without storage to be g(2)
s,s (0) ¼ 1.81(2) (or gi,i (0) ¼ 1.86(9)), which
is very close to the maximal value of 2 associated with the thermal photon
statistics (the lower observed value can be entirely attributed to the finite temporal
resolution of our detectors). We also measured the zero-time second-order
autocorrelation function of the heralded signal photon just before storage and
(2)
obtained gs,s|i
(0) ¼ 0.061(4) for a pump power of 8 mW, which is consistent
with p ≪ 1.
The observed values of the concurrence are lower bounds on the amount of
entanglement of the detected fields and are almost entirely determined by optical
loss. To see this, we first note that when both the multi-pair creation probability and
the total transmission probability h of thesignal
photon are small, the concurrence
is approximately given by C ≈ h(V − 2/ g s,i − 1). At 8 mW of pump power,
we measured gs,i ≈ 10, p00 ¼ 0.9997831(71) and p11 ¼ 5.18(40) × 1029, and hence
C ≈ 0.3h. With h ≈ p10 þ p01 ¼ 2.2 × 1024, we see that C ≈ 6.6 × 1025 ≪ 0.3.
Using this, an estimate of the concurrence of the fields retrieved just after the crystals
is obtained by noticing that the transmission h is the product of the probability to
find the signal photon inside the fibre per heralding signal (20%), the memory
efficiency (15%), the transmission in the interferometer (2.4%) and detector
efficiency (30%). Factoring out the detector inefficiency and the interferometer loss
yields a lower bound of 0.01 for the concurrence of the field retrieved just after
the crystals.
All results were derived from raw counts, that is, without the subtraction of dark
counts and accidental coincidences, and were obtained with coincidence detection
windows of 10 ns. We note that there is no measurable temporal overlap between
stored and transmitted photons, and that the transmitted photons do not contribute
to the entanglement (see Supplementary Information for details, where histograms
showing the temporal profiles of detected photons are shown).
Received 24 August 2011; accepted 27 January 2012;
published online 4 March 2012
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Acknowledgements
The authors thank H. de Riedmatten, P. Sekatski and J. Laurat for stimulating discussions,
and A. Korneev for help with the superconducting detector. This work was supported by
the Swiss National Centres of Competence in Research (NCCR) project ‘Quantum Science
Technology (QSIT)’, the Science and Technology Cooperation Program Switzerland–
Russia, the European Union FP7 project 247743 ‘Quantum repeaters for long distance
fibre-based quantum communication (QUREP)’ and the European Research Council
Advanced Grant ‘Quantum correlations (QORE)’. F.B. was supported in part by le Fond
Québécois de la Recherche sur la Nature et les Technologies.
Author contributions
All authors conceived the experiment. I.U., C.C. and F.B. performed the measurements.
I.U., C.C., F.B., N.S. and M.A. analysed the data. All authors contributed to the writing
of the manuscript. I.U., C.C. and F.B. contributed equally to this work.
Additional information
The authors declare no competing financial interests. Supplementary information
accompanies this paper at www.nature.com/naturephotonics. Reprints and permission
information is available online at http://www.nature.com/reprints. Correspondence and
requests for materials should be addressed to M.A.
NATURE PHOTONICS | VOL 6 | APRIL 2012 | www.nature.com/naturephotonics
© 2012 Macmillan Publishers Limited. All rights reserved.
237
SUPPLEMENTARY INFORMATION
doi: 10.1038/nphoton.2012.34
Supplementary information for “Heralded quantum entanglement between two
crystals”quantum entanglement between two
Supplementary information for “Heralded
crystals”
Imam Usmani, Christoph Clausen, Félix Bussières, Nicolas Sangouard, Mikael Afzelius, and Nicolas Gisin
Group of Applied Physics, University of Geneva,
Chemin de Pinchat 22, CH-1211 Geneva 4, Switzerland
Group of Applied Physics, University of Geneva,
(Dated: January 26, 2012)
Chemin de Pinchat 22, CH-1211 Geneva 4, Switzerland
(Dated: January 26, 2012)
Imam Usmani, Christoph Clausen, Félix Bussières, Nicolas Sangouard, Mikael Afzelius, and Nicolas Gisin
I.
OVERVIEW
I.
OVERVIEW
This appendix provides details on the method we used
to This
estimate
the threefold
probability
p11used
usappendix
providescoincidence
details on the
method we
ing
either
the
cross-correlation
g
or
by
a
direct
meas,i
to estimate the threefold coincidence probability p11 ussurement
thecross-correlation
threefold coincidence
probability.
Specifing eitherofthe
gs,i or
by a direct
meaically,
we
present
in
section
II
a
model
describing
our
surement of the threefold coincidence probability. Specifsetup
composed
of
the
source
of
photon
pairs
and
ically, we present in section II a model describing the
our
memories,
and includes
the effect
of darkpairs
counts.
setup composed
of the source
of photon
and We
the
provide
evidence
that the the
method
on the
measurememories,
and includes
effectbased
of dark
counts.
We
ment
of evidence
gs,i to estimate
yields based
a lower
on the
provide
that thep11
method
onbound
the measureconcurrence
at
detection
and,
consequently,
on
the
ment of gs,i to estimate p11 yields a lower bound onconthe
currence
of the
excitation
stored inonthe
concurrence
at delocalized
detection and,
consequently,
thequancontum
memories.
Finally, in section
III, stored
we give
on
currence
of the delocalized
excitation
in details
the quanour
method
to
estimate
p
using
threefold
coincidences,
11
tum memories. Finally, in section III, we give details on
we
the uncertainties
calculated
and we
ourdiscuss
methodhow
to estimate
p11 using are
threefold
coincidences,
show
the results.
we discuss
how the uncertainties are calculated and we
show the results.
II.
A.
CONCURRENCE USING THE
CROSS-CORRELATION
II. CONCURRENCE USING THE
CROSS-CORRELATION
Characterization of the source of photon pairs
A.
Characterization of the source of photon pairs
Let us begin by a description of the source of photonLet
pairs
we used
our experiment.
Photonof pairs
us begin
by ain description
of the source
phoare
produced
by
spontaneous
parametric
downconversion
ton pairs we used in our experiment. Photon pairs
(SPDC),
yielding
a two-modeparametric
squeezed state
[1]:
are produced
by spontaneous
downconversion
(SPDC), yielding a +∞
two-mode squeezed state [1]:
cosh−2 r +∞ tanh2n r |ni ns ni ns |.
cosh−2 rn=0 tanh2n r |ni ns ni ns |.
(A1)
(A1)
n=0
Here, ni and ns correspond to the number of photons
in
the idler
andn signal
modes (and are both equal to n).
Here,
ni and
s correspond to the number of photons
The
coherences
between
terms(and
withare
different
number
of
in the idler and signal modes
both equal
to n).
pairs
are
lost
due
to
the
lack
of
a
shared
phase
reference.
The coherences between terms with different number of
The
r can
be lack
written
αPpump ,reference.
where α
pairsparameter
are lost due
to the
of a as
shared
P phase
is
the
downconversion
efficiency
and
is
the
con-α
pump
The parameter r can be written as αPpump , where
tinuous
wave
pump
power.
It
is
convenient
to
consider
is the downconversion efficiency
and Ppump is the con2
the
parameter
p = tanh
r, such
the probability
to
tinuous
wave pump
power.
It isnthat
convenient
to consider
2(1 − p)p , which corresponds to
create
n
pairs
is
given
by
the parameter p = tanh r, such that the probability to
acreate
thermal
photon
number
When
the pump
n pairs
is given
by (1distribution.
− p)pn , which
corresponds
to
power
is
small
such
that
r,
p
1,
then
the
probabilities
a thermal photon number distribution. When the pump
of
creating
one or
twothat
pairsr,are,
a very
approxpower
is small
such
p with
1, then
thegood
probabilities
2
imation,
given
by
p
and
p
,
respectively.
In
this
situation
of creating one or two pairs are, with a very good approxwe
have given by p and p2 , respectively. In this situation
imation,
we have
(A2)
p = tanh r2 ≈ r2 = αPpump (p 1).
p = tanh
r2 ≈ r2 = αPpump
nature photonics
| www.nature.com/naturephotonics
(p 1).
(A2)
The signal and idler modes of the two-mode squeezed
state,
individually,
also exhibit
therThe when
signalconsidered
and idler modes
of the two-mode
squeezed
mal
statistics.
Hence, a measurement
of their
zero-time
state,
when considered
individually, also
exhibit
ther(2)
(0)zero-time
should
second-order
functionof gtheir
mal statistics.auto-correlation
Hence, a measurement
yield
a value of
2 [1]. In the limit
whereg (2)
p (0)
1,
this
should
second-order
auto-correlation
function
can
be
tested
directly
using
a
Hanbury
Brown
&
Twiss
yield a value of 2 [1]. In the limit where p 1, this
setup
obtained
by using
inserting
a 50/50Brown
beam &splitter
can be[2]tested
directly
a Hanbury
Twiss
after
the
frequency
filter
of
the
signal
field
and,splitter
simisetup [2] obtained by inserting a 50/50 beam
larly,
of
the
idler
field.
Note
that
the
thermal
statistics
after the frequency filter of the signal field and, simiregime
experimentally
accessible
herethermal
becausestatistics
the colarly, ofisthe
idler field. Note
that the
herence
time
of
the
filtered
signal
and
idler
fields
(0.9
regime is experimentally accessible here because theand
co7herence
ns, corresponding
to 350signal
and 43
MHz
retime of the filtered
and
idlerlinewidths,
fields (0.9 and
spectively)
are larger to
than
temporal
of the
7 ns, corresponding
350the
and
43 MHzresolution
linewidths,
refree-running
single
photon
detectors
used
for
detection
spectively) are larger than the temporal resolution of the
(less
than 350single
ps for photon
both thedetectors
signal and
idler
free-running
used
fordetectors).
detection
Hence,
the
detection
system
approximately
selects
a sin(less than 350 ps for both the signal and idler detectors).
gle
temporal
mode
of
the
SPDC
process.
Hence, the detection system approximately selects a singleThe
temporal
modeauto-correlation
of the SPDC process.
measured
of the idler field,
(2)
(τ ), measured
is shown on
Fig. A1-a. Atofτ the
= 0,idler
we have
gi,iThe
auto-correlation
field,
(2)
(2)
ggi,i
(0)
=
1.86(9)
with
a
320-ps
temporal
binning.
The
(τ
),
is
shown
on
Fig.
A1-a.
At
τ
=
0,
we
have
i,i
(2) points are compared against a theoretical curve obdata
gi,i (0) = 1.86(9) with a 320-ps temporal binning. The
tained
assuming
a 350 ps detection
and acurve
43 MHz
data points
are compared
against a jitter
theoretical
obwide
spectrum
(full
width
at
half
maximum).
The
tained assuming a 350 ps detection jitter and a 43agreeMHz
ment
is good, despite
the large
observed
fluctuations.
In
wide spectrum
(full width
at half
maximum).
The agreeparticular,
the
fluctuations
give
the
impression
that
the
ment is good, despite the large observed fluctuations. In
data
points the
around
the maximum
on average
lower
particular,
fluctuations
give theare
impression
that
the
than
the
theoretical
curve.
These
fluctuations
are
bedata points around the maximum are on average lower
lieved
to be
statistical,
but they
also be are
caused
than the
theoretical
curve.
Thesecould
fluctuations
beby
an
underestimation
of
the
temporal
resolution.
The
lieved to be statistical, but they could
also
be
caused
(2)
autocorrelation
of the signal
gs,s (τresolution.
), is shownThe
on
by an underestimation
of thefield,
temporal
(2)g (2) (0) = 1.81(2),
Fig.
A1-b and has
maximum
value gofs,s
s,s), is shown on
autocorrelation
ofathe
signal field,
(τ
(2)
in
with the
theoretical
ob= 1.81(2),
Fig.excellent
A1-b andagreement
has a maximum
value
of gs,s (0) curve
tained
assuming
a
350
MHz
wide
(full
with
at
half
maxin excellent agreement with the theoretical curve obimum)
spectrum.a In
cases,
tained assuming
350both
MHz
widethe
(fullvalue
withat
atmaximum
half maxis
slightly
lower
than
2;
this
is
attributed
entirely
to the
imum) spectrum. In both cases, the value at maximum
finite
temporal
resolution
of
our
detection.
Furthermore,
is slightly lower than 2; this is attributed entirely to the
we
note
that theresolution
value at maximum
is lowerFurthermore,
than in the
finite
temporal
of our detection.
idler
case
simply
because
the
temporal
spectrum
we note that the value at maximum is lower thanisinnarthe
rower,
hence
accentuating
thetemporal
impact of
the finite
residler case
simply
because the
spectrum
is narolution
on thisaccentuating
reduction. Overall,
the ofmeasured
rower, hence
the impact
the finiteautorescorrelations
are
consistent
with
the
assumption
thatautoour
olution on this reduction. Overall, the measured
source
produces
a state of the
Eq. (A1). that our
correlations
are consistent
withform
the of
assumption
source
produces
a state
of the form
of source
Eq. (A1).
Another
important
parameter
of the
is the zerotime
auto-correlation
of
the
signal
conditioned
thezerodeAnother important parameter
of the source isonthe
(2)
(0).
This
quantifies
the
tection
of
an
idler
photon,
g
time auto-correlation of the signal
s,s|i conditioned on the desub-poissonnian
nature
of theg (2)
heralded signal field, hence
tection of an idler
photon,
s,s|i (0). This quantifies the
its
single-photon
character
[1].
For a two-mode
squeezed
sub-poissonnian nature of the heralded
signal field,
hence
its single-photon character [1]. For a two-mode squeezed1
© 2012 Macmillan Publishers Limited. All rights reserved.
supplementary information
a
doi: 10.1038/nphoton.2012.34
2
2.0
Memory
g (2)
i,i (τ )
1.8
Pair
source
1.6
not reemitted
( : absorbed)
Decoherence
1.4
Loss
Not absorbed
1.2
1.0
5
10
0
Delay τ (ns)
5
−20 −15 −10 −5
b
0
15
20
2.0
g (2)
s,s (τ )
1.8
1.6
1.4
1.2
1.0
−10
−5
10
FIG. A1. Measurements of second-order auto-correlation
functions of a, the idler photon (1338 nm), and b, the signal
photon (883 nm) before the crystals. The values at zero delay
in a and b are close to the ideal value of 2. The red lines are
theoretical curves based on the spectra of the optical filtering
elements and the jitter of the detection system. Data was
accumulated for 72 hours with a pump power of 16 mW in a,
and for 12 hours at 8 mW in b. Error bars are due to Poisson
counting statistics.
(2)
state, the relation between gs,s|i (0) and the parameter
FIG. A2. Model of our experimental setup. The source produces idler-signal pairs described by a two-mode squeezed
state. The memory is represented by three beamsplitters
(to account for the absorption, the decoherence and the reemission respectively). Each temporal mode after the memory
ds comes either from a signal photon emitted at an early time,
then stored and later retrieved (mode se ), or from a photon
produced at a late time and directly transmitted through the
memory (mode s ). To take both contributions into account,
a virtual source creating idler-signal pairs in a late time bin
has been introduced. The correlations between the modes
ds and di are characterized through the measurement of the
second-order cross-correlation function.
later retrieved; or from a photon created at a later time
and not absorbed by the medium. Hence, we need to
consider two temporal modes. The storage process is a
linear operation, mapping the light field onto an atomic
state. In our model, it can be represented by a beamsplitter with a transmission corresponding to the absorption
probability, c.f. Fig. A2. Another beamsplitter is introduced to account for the decoherence, inherent to the
storage process [3]. Finally, a last beamsplitter accounts
for the reemission of the photon. It is identical to the first
one and combines the two temporal modes into a single
temporal mode ds . The output of the last beamsplitter
is detected to access the cross-correlation function
2
p = tanh r is
(2)
gs,s|i (0)
ḡs,i =
2
=2−
≈ 4p for p 1.
(1 + p)2
(A3)
For p 1, this conditional zero-time auto-correlation
of the signal photon can be measured using a Hanbury
(2)
Brown & Twiss setup. We measured gs,s|i (0) = 0.061(4)
with a pump power of 8 mW. This shows the very good
single photon character of the heralded signal field, and
hence that p 1.
B.
Modelling the source and the quantum memory
Let us continue with the description of our model by focusing on the quantum memory. For now we will consider
only the case of a single memory, which corresponds to
the situation with one arm of the interferometer blocked.
Due to non-unit absorption efficiency, each signal photon detection after the memory stems either from a photon created at an early time, stored in the medium, and
2
d†i di d†s ds d†i di d†s ds (A4)
between detection on the early idler mode di and signal
mode ds . Note that the overall transmission (in intensity)
of the three beamsplitters corresponds to the efficiency
of the storage and retrieval processes ηecho , while the reflection of the last beamsplitter is the transmission of the
atomic ensemble ηtrans .
Using the Heisenberg picture
d†s ds = 0|U † d†s U U † ds U |0
where
√
ηecho cosh r se + sinh r i†e
√
+ ηtrans cosh r s + sinh r i†
U † ds U =
leading to
d†s ds = (ηtrans + ηecho ) sinh2 r.
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© 2012 Macmillan Publishers Limited. All rights reserved.
3
supplementary information
doi: 10.1038/nphoton.2012.34
Furthermore, we have
leading to
TABLE A1. Conditional probabilities p01 , p10 , p11 and the
ratio ηtrans /ηecho for various pump powers.
U † di U = cosh r ie + sinh r s†e
d†i di = sinh2 r.
Following similar lines, we find
d†i di d†s ds = sinh2 r sinh2 r (ηtrans + ηecho )
+ cosh2 r ηecho .
Therefore, the cross-correlation function between the
idler-signal modes is given by
1+
1
tanh2 r 1 +
ηtrans
ηecho
.
C.
Note that the effect of the memories is to add the term
(1 + ηηtrans
), which decreases the cross-correlation. One
echo
can further take the detector noise into account by adding
an additional source of noise with a Poissonian photon
distribution. We finally find
gs,i = 1 +
2
tanh r 1 +
1
ηtrans
ηecho
+
Pump
power
1 mW
2 mW
3 mW
4 mW
8 mW
13 mW
16 mW
Mean
ηdark
pc
(A5)
where ηdark is the probability to get a dark count within
the detection window and pc is the conditional probability of detecting the signal photon retrieved from
the memory. We recall that for small pump powers,
tanh2 r ≈ αPpump .
To support the model presented above, we performed
additional measurements to determine the values of α,
ηtrans /ηecho , and pc . First, we estimated α = 2.71(8) ×
10−3 pairs/mW (per 10 ns within a 43 MHz wide spectral
window) from a measurement of the cross-correlation as
function of pump power with one crystal prepared as a
transmission window (no storage) while the path of other
crystal was blocked. Then, ηtrans /ηecho was measured
from the ratio of the probabilities of detecting the heralded signal photon when it is either transmitted through
the memory (no storage), or stored and retrieved from
the memory. This measurement was performed twice –
once for each arm of the interferometer (with the other
arm blocked) – and the outcomes were averaged. The
conditional probability pc was taken as the sum of probabilities p10 and p01 of detecting the heralded signal photon retrieved from the memories (see main text and Table A1). Finally, we measured a dark count probability
of ηdark = 2 × 10−6 per 10 ns detection window for the
signal mode detector. Idler photons were detected using
a superconducting nanowire single photon detector with
a dark count rate of 20 Hz. This low dark count rate is
neglected in our calculations. Using these parameters, we
compared the measured cross-correlation ḡs,i with values
of gs,i predicted by Eq. (A5). The comparison is shown
on Fig. 2B in the main text. We find excellent agreement
within the statistical uncertainties, thereby supporting
our model for the source and the memory.
p01 (10−4 ) p10 (10−4 ) p11 (10−9 )
1.04(14)
1.193(75)
0.952(72)
1.105(72)
1.185(51)
1.247(56)
1.146(47)
1.123(30)
0.82(12)
0.809(63)
0.878(70)
0.902(66)
0.984(50)
1.131(52)
1.175(48)
0.957(27)
1.33(30)
1.63(19)
1.61(20)
2.82(31)
5.18(40)
8.79(66)
9.56(64)
ηtrans
ηecho
2.84(33)
3.03(17)
2.59(17)
3.35(19)
3.13(12)
2.86(11)
2.748(93)
2.936(69)
Lower bound on the concurrence from the
cross-correlation
We now provide evidence, based on our model, that
the method we used to estimate the threefold coincidence probability from the measurement of the crosscorrelation leads to an underestimation of the concurrence. For this, we consider a hypothetical situation
where photons are detected directly in modes A and B,
that is, before the modes are recombined on the beamsplitter (see Fig. 1 of the main text). Hence, singlephoton interference and two-photon bunching do not
arise. The overall transmission for each memory is given
by p01 and p10 , including detection (we recall that in our
experiment, p10 is the probability per heralding signal to
detect a photon at detector 1 or 2 when the arm of crystal B is blocked, and similarly for p01 ). We then compare
the threefold coincidence probability pth
11 predicted by our
model with the one obtained from the method based on
the measured cross-correlation ḡs,i . Let us first concentrate on pth
11 . Using our model for the source and the
memory, we can predict the threefold coincidence probability should be given by
ηtrans
ηdark
th
p11 = 4p10 p01 αPpump 1 +
+
. (A6)
2ηecho
pc
In the derivation of this formula we considered the probability to detect an idler photon to be very small (it is of
order 10−3 per 10 ns). Furthermore, only contributions
of single and double pairs were included. This approximation is well supported since the single pair creation
probability p is of order 10−3 per 10 ns.
The above expression should be compared with the
threefold coincidence probability estimated using ḡs,i =
1 + 1/p, i.e. assuming detections stem from a two-mode
squeezed state. Combining this method with our model
(Eq. A5) yields
p11 =
4p10 p01
ḡs,i − 1
= 4p10 p01 αPpump
ηtrans
1+
ηecho
ηdark
+
.
pc
One can see that the ratio ηtrans /ηecho enters in p11 with-
3
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© 2012 Macmillan Publishers Limited. All rights reserved.
supplementary information
doi: 10.1038/nphoton.2012.34
4
out the prefactor of one-half, such that p11 > pth
11 . From
Eq. (1) of the main text, we directly see that an overestimation of p11 leads to a lower bound on the concurrence
at detection, and hence on the concurrence of the entanglement stored in the memories.
300
250
200
ḡsi = 14.91(76)
350
ḡsi = 1.44(23)
Note that both expressions for
and p11 are obtained
by assuming that the double pair creation probability is
p2 , consistent with a thermal statistics of a two-mode
squeezed state. One can naturally ask if our lower bound
is still valid when the photon number distribution is not
exactly thermal, but possibly in between thermal and
Poisson statistics. The Poissonnian case is obtained when
the temporal resolution is much larger than the coherence time of the photons, yielding a highly multimode
case and second-order auto-correlations of the signal and
idler modes equal to 1 [4, 5]. A temporal resolution that
is of the same order as the coherence time of the photons therefore yields a double pair creation probability
equal to ap2 , where 1/2 < a < 1 [4], and hence an autocorrelation function that is approximately given by 2a,
comprised between 1 and 2, as observed here. In this
case, the expression 4p10 p01 in both pth
11 and p11 above
should be replaced with 4ap10 p01 < 4p10 p01 . Therefore,
we see directly that assuming that we have thermal statistics (a = 1), even if this is not exactly the case as in this
experiment, can only overestimate p11 , and hence underestimate the concurrence.
400
Counts in 2h
pth
11
450
150
100
50
0
−100
−50
0
50
Time difference (ns)
100
FIG. A3. Coincidence histogram of the measurements taken
with a pump power of 4 mW. The x-axis corresponds to the
time difference between the detection of a signal photon and
an idler photon. The photons that are transmitted through
the memory (without storage) give rise to the leftmost peak at
a zero time difference. The rightmost peak at 33 ns stems from
stored and retrieved photons. The light-blue region marks
the coincidence window used for the calculation of the crosscorrelation gs,i and of the concurrence. For comparison, the
value of gs,i with a coincidence window positioned half-way
between the two peaks is shown (light-red).
III. CONCURRENCE USING A DIRECT
MEASUREMENT OF THE THREEFOLD
COINCIDENCE PROBABILITY
A.
D.
Coincidence window selection
Figure A3 shows the histogram of the time difference
between the detection of a signal photon (at either detector 1 or 2) and an idler photon and with the arm of
memory B blocked, added to the similar histogram obtained with the arm of memory A blocked and B opened.
We see two clear peaks corresponding to signal photons
that are not absorbed by the crystal, and those that are
stored and reemitted 33 ns later. Also shown in the figure
is the 10 ns coincidence window, centered on the second
peak, used in the calculation of the concurrence. The
peaks are well separated, and have no measurable overlap at 33 ns. Hence, the transmitted photons did not
contribute the observed entanglement. In fact, shifting
the coincidence window to a delay of half of the storage
time reduces the cross-correlation to gs,i ≈ 1.4 (instead
of ≈ 15 at 33 ns), yielding a concurrence of 0, and hence
no entanglement. Hence, photons detected around 17 ns
do not show entanglement.
4
Method
We now describe how we estimate the probability p11
from the measurement of threefold coincidences. For this,
we use the experimental setup shown on Fig. A4; this is
the same setup shown on Fig. 1 of the main text. Unlike
our estimation based on the measurement of the crosscorrelation (section II), this direct method makes use of
the experimental setup in its entirety, that is, with recombination of the modes A and B on the beamsplitter. Furthermore, it does not require assumptions on the
statistics of the source of photon pairs and on how the
memories operate.
Let us consider the two-photon part, denoted ρ2 , of the
complete density matrix describing the state of the retrieved fields just after the crystals. If the relative phase
between the spatial modes is randomized, then the coherences are zero and we can write
ρ2 = q11 |1111| + q20 |2020| + q02 |0202|.
(A7)
Here, q11 is the probability per heralding signal that the
fields retrieved from crystals A and B each contain one
photon, q20 (q02 ) the probability per heralding signal that
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supplementary information5
FIG. A4. Experimental setup used to estimate p̄11 directly. This setup is the same of Fig. 1 of the main text. The relative
phase between the arms of the interferometer was randomized by letting the interferometer (see Fig. A4) drift by itself over the
total measurement period of more than 100 hours. PBS: polarizing beamsplitter; FR: Faraday rotator; BS: beamsplitter; MA
and MB are crystal memories A and B. The detection efficiency from the output modes of the beamsplitter to detectors 1 and 2
and η1 and η2 , respectively. The amplitude transmission and reflection coefficients of modes a and b right after the crystals,
through the beamsplitter, are shown.
the fields retrieved from crystals A and B each contain
2 and 0 (0 and 2) photons, respectively. Estimation of
p11 can be performed by measuring the q11 probability
in Eq. A7, as in Ref. [6]. We show here that one can also
estimate p11 by measuring q20 and q02 .
We first derive useful relations between the occupation probabilities. We use the following definitions: PH
is the probability to herald the presence of a signal photon (i.e. the probability to detect an idler photon durA
B
ing the detection time window); ηecho
(ηecho
) is the storage and retrieval efficiency of crystal A (B); P2 is the
probability that the source of photon pair emits 2 pairs
of photons during the detection time window; R and T
are the intensity reflection and transmission coefficients
of the beamsplitter when photons are incident from the
left hand side, satisfying R + T ≤ 1 and R, T > 0 (see
Fig. A4). Using these definitions, and the linear nature
of the optical storage process, we have
1
A
B
2RT ηecho
ηecho
P2 ,
PH
1 2 A 2
=
T (ηecho ) P2 ,
PH
1 2 B 2
=
R (ηecho ) P2 ,
PH
0 < αt2 , αr2 , βt2 , βr2 ≤ 1/2, and we note that R = βr2 and
T = αt2 . Using these notations, we can show that the
probability to get a coincidence at detectors 1 and 2 per
heralding signal is given by
p̄11 = (a11 q11 + a20 q20 + a02 q02 )η1 η2 ,
where η1 (η2 ) is the transmission from output mode a
(b) – after the beamsplitter – to detector 1 (detector 2),
including the detector efficiency. We also have
2
β2
a11 = βr2 αr − t
,
αr
a20 = 2αt2 αr2 ,
a02 = 2βt2 βr2 .
We characterized our beamsplitter and obtained αt2 =
T = 0.479, αr2 = 0.422, βt2 = 0.482 and βr2 = R = 0.409.
These values yield
q11 =
q20
q02
a11 = 0.0028,
a20 = 0.394,
a02 = 0.404.
with which we can write q20 and q02 as a function of q11 :
q20 = q11 (R/2T ),
q02 = q11 (T /2R).
(A8)
We use these expressions later.
We now consider recombination at the beamsplitter.
We describe the effect of the beamsplitter on mode operators a and b as follows:
a † → α t a † + α r b† + α l l † ,
b † → β r a † + β t b† + β l l † ,
where mode l represents loss. We set all coefficients to be
real. Energy conservation yields αr2 + αt2 + αl2 = 1 (and
similarly for the coefficients of mode b). Also, unitarity
imposes that αr αt + βr βt = 0. Finally, we suppose that
(A9)
(A10)
We immediately notice that a11 can be neglected with respect to a20 and a02 . This essentially means that whenever a photon is retrieved from each crystal, they bunch
with almost certainty and do not significantly contribute
to threefold coincidences. It also requires indistinguishability of the retrieved photons, which is consistent with
the large single photon interference visibility observed
(96.9%). Combining Eq. A9 and Eq. A10, and writing
a20 ≈ a02 ≈ 0.4, we get
p̄11 ≈ 0.4 η1 η2 (q20 + q02 ).
(A11)
Finally, using Eq. A8, we get
T
R
+
q11
q20 + q02 =
2T
2R
= 1.012 q11
≈ q11
5
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supplementary information
and thus
p̄11 ≈ 0.4 η1 η2 q11 .
(A12)
We now have to determine how to use Eq. A12 in the
calculation of the concurrence. This method should give
a result consistent with the way we measured p10 + p01 ,
which was done as follows. We let the phase of the interferometer drift to randomize it, and measured the probability to get a detection at detector 1 or 2 per heralding
signal. This is equivalent to, first, measure crystal A
(with the path of crystal B blocked) using an effective
detector of efficiency ηA = αt2 η1 + αr2 η2 , and then measure crystal B (with the path of crystal A blocked) using
an effective detector of efficiency ηB = βr2 η1 + βt2 η2 , and
then add the results together. Hence, if both crystals
were measured simultaneously and separately, the threefold coincidence probability per heralding signal would
be
q11 ηA ηB .
(A13)
We can now find the relation between Eq. A12 and A13.
For this, we use the fact that, in our experiment, η2 was
larger than η1 , which is apparent from the visibility plot
in the main text (inset of Fig. 2-A). Hence, by setting
η1 = η2 /2, we get q11 ηA ηB ≈ 0.454 η22 q11 . Therefore, we
immediately see that if we multiply Eq. A12 by 2.27, we
get Eq. A13.
In summary, to estimate the probability p11 used in
the formula of the concurrence, namely
√
C ≥ max(0, V (p10 + p01 ) − 2 p00 p11 ),
(A14)
we multiply the measured p̄11 by 2.27. This multiplicative factor was obtained using approximations, and hence
has some uncertainty associated to it. Nevertheless, as
explained in the next subsection, this uncertainty can
be neglected in front of the large statistical uncertainty
of p̄11 .
B.
Statistical uncertainty
Here we provide details on how the statistical uncertainty on p̄11 was calculated in the method based on the
measurement of threefold coincidences (subsection III A).
We denote NH the number of heralding signals during
the measurement period T , P̄11 the stochastic variable
representing the probability to register a threefold coincidence per heralding signal, and p̄11 our estimation of
P̄11 obtained from the measurement.
Since p̄11 1 and NH 1, the distribution of the
number of coincidences between detectors 1 and 2 per
heralding signal (i.e. threefold coincidences) is accurately
described by a Poisson distribution. Specifically, the
probability to get n coincidences given NH heralding signals is given by
P (n|p̄11 ) = e−NH p̄11
6
n
(NH p̄11 )
.
n!
(A15)
6
doi: 10.1038/nphoton.2012.34
The probability P (n|p̄11 ) is the likelihood function of p̄11
given n. Hence, P̄11 can be estimated by maximizing this
likelihood, which yields
√
n
n
(MLE)
=
±
,
p̄11
NH
NH
where MLE stands for “maximum likelihood estimation”.
This method yields an appropriate estimate of P̄11 , unless
one measures n = 0, in which case one would conclude
that p̄11 = 0 ± 0.
A more conservative estimate can be obtained in the
following way. Using Bayes’ law, we can write the probability density function Ω(p̄11 |n) of getting P̄11 = p̄11
given n measured threefold coincidences:
Ω(p̄11 |n) = 1
0
P (n|p̄11 )
P (n|p̄11 ) dp̄11
.
The denominator of Eq. A15 can be calculated:
1
n
n
k
NH
1
−NH p̄11 (NH p̄11 )
−NH
dp̄11 =
1−e
e
n!
NH
k!
0
k=0
1
.
≈
NH
The last equality holds because n NH . Therefore:
Ω(p̄11 |n) = NH P (n|p̄11 ).
(A16)
Using Eq. A16, we can calculate the expected value of
P̄11 and its second moment, respectively denoted P̄11 2
and P̄11
:
1
n+1
,
p̄11 Ω(p̄11 |n) dp̄11 ≈
P̄11 =
NH
0
1
(n + 1)(n + 2)
2
P̄11
=
p̄211 Ω(p̄11 |n) dp̄11 ≈
.
2
NH
0
The standard deviation of P̄11 is therefore
√
n+1
2
2
.
P̄11 − P̄11 =
NH
Hence, given n measured threefold coincidences and
NH heralding signals, this conservative method yields
√
n+1
n+1
(CE)
±
,
p̄11 =
NH
NH
where CE stands for “conservative estimation”. In our
results, we estimate the concurrence with both the maximum likelihood and the conservative estimations of P̄11 .
C.
Experimental results
The estimation of p00 , p10 + p01 and p11 proceeded as
follows. For a time T = 166 hours, we let the interferometer drift to randomize the phase. During that period of
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supplementary information7
doi: 10.1038/nphoton.2012.34
time, we recorded the total number of heralded twofold
coincidences between the heralding detector and detector 1 (N1|H ), and between the heralding detector and
detector 2 (N2|H ), and calculated N2 = N1|H + N2|H .
We also recorded the total number of heralded threefold
coincidences, denoted N12|H , between all three detectors.
We used a coincidence time window of 10 ns; this is the
same value we used in the estimation of the concurrence
based on the cross-correlation. We calculated
p10 + p01
and the CE:
(CE)
p11
= 2.27
N12|H + 1
±
NH
(MLE)
p11
(MLE)
p11
= 2.27
N12|H
±
NH
.
= 2.9 ± 2.1 × 10−9
and
(CE)
C (MLE) ≥ 6.3 ± 3.8 × 10−5
p11
For p11 , we used both the MLE, yielding
N12|H + 1
NH
Finally, we estimated p00 = 1 − p10 − p01 − p11 .
With a 10 ns coincidence window, we obtained p10 +
p01 = 1.7777(34) × 10−4 , V = 96.9 ± 1.5%, and observed
two threefold coincidences (N12|H = 2). This yields
√
N2
N2
=
±
.
NH
NH
= 3.9 ± 2.2 × 10−9
C (CE) ≥ 3.9 ± 3.8 × 10−5
N12|H
NH
,
[1] L. Mandel and E. Wolf, Optical Coherence And Quantum
Optics (Cambridge University Press, Cambridge, 1995).
[2] R. H. Brown and R. Q. Twiss, Nature 178, 1046 (1956).
[3] M. Afzelius, C. Simon, H. de Riedmatten, and N. Gisin,
Phys. Rev. A 79, 052329 (2009).
[4] H. D. Riedmatten, V. Scarani, I. Marcikic, A. Acin,
Both the MLE and CE yield a lower bound for the concurrence that is greater than 0 by at least one standard
deviation.
W. Tittel, H. Zbinden, and N. Gisin, J. Mod. Opt. 51,
1637 (2004).
[5] P. Sekatski, N. Sangouard, F. Bussières, C. Clausen,
N. Gisin, and H. Zbinden, Preprint: arXiv:1109.0194
(2011).
[6] C. W. Chou, H. de Riedmatten, D. Felinto, S. V. Polyakov,
S. J. van Enk, and H. J. Kimble, Nature 438, 828 (2005).
7
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© 2012 Macmillan Publishers Limited. All rights reserved.
IOP PUBLISHING
JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS
doi:10.1088/0953-4075/45/12/124001
J. Phys. B: At. Mol. Opt. Phys. 45 (2012) 124001 (8pp)
Atomic frequency comb memory with
spin-wave storage in 153Eu3+:Y2SiO5
N Timoney, B Lauritzen, I Usmani, M Afzelius and N Gisin
Group of Applied Physics, University of Geneva, CH-1211 Geneva 4, Switzerland
E-mail: [email protected]
Received 16 December 2011, in final form 8 February 2012
Published 8 June 2012
Online at stacks.iop.org/JPhysB/45/124001
Abstract
153
Eu3+ :Y2 SiO5 is a very attractive candidate for a long-lived, multimode quantum memory
due to the long spin coherence time (∼15 ms), the relatively large hyperfine splitting
(100 MHz) and the narrow optical homogeneous linewidth (∼100 Hz). Here we show an
atomic frequency comb memory with spin-wave storage in a promising material
153
Eu3+ :Y2 SiO5 , reaching storage times slightly beyond 10 μs. We analyse the efficiency of
the storage process and discuss ways of improving it. We also measure the inhomogeneous
spin linewidth of 153 Eu3+ :Y2 SiO5 , which we find to be 69 ± 3 kHz. These results represent a
further step towards realizing a long-lived, multimode solid-state quantum memory.
(Some figures may appear in colour only in the online journal)
1. Introduction
echo is referred to as a two-level echo in the remainder of
this paper. The efficiency of the echo in the forward direction,
assuming a comb which can be described by a sum of Gaussian
functions, is given by [14]
2
d
2
η≈
e−7/F e−d/F e−d0 ,
(1)
F
where d (= αL, α is the absorption coefficient and L the
sample length) is the optical depth and F is the comb finesse.
More specifically, F = /γ , where is as before and γ is
the comb tooth full-width at half-maximum (FWHM). Here
we also include an additional loss factor (the last factor) due
to an absorbing background d0 , which often occurs due to
imperfect preparation of the comb. Note that although we
consider Gaussian-shaped teeth here, which fit well with
our experimental data, other shapes have been considered
elsewhere [15, 16]. Important results have been obtained using
such a two-level AFC scheme, where a heralded single photon
was stored in a crystal [17, 18] or heralded entanglement
was generated between two crystals [19]. Two-level echo
efficiencies using an AFC scheme of 15–25% are seen in many
experiments [13, 16, 17, 20, 21]. In these papers and the work
presented here, the inhomogeneously broadened medium is
an inorganic crystal weakly doped with rare-earth ions. Such
crystals placed in commercially available cryostats cooled to
less than 4 K have impressive coherence properties on optical
and spin transitions [9, 22]. We refer to [8] for a comprehensive
review of these materials in terms of quantum memories. In
Quantum communication [1] provides resources and
capabilities, such as quantum key distribution [2], that are
not possible to obtain using classical communication. A
major challenge to quantum communication, however, is
to overcome the inherent losses of quantum channels, e.g.
optical fibres. A solution to the problem of long-distance
quantum communication is the quantum repeater [3–6], which
in principle can work over arbitrary distances. To implement
a quantum repeater, quantum memories and effective delay
lines of variable duration are required. Atom-based memories
are attractive candidates; indeed much research has been
done in recent years to implement them [7, 8]. In addition
to faithfully reproducing the input mode and storing for
potentially long times [9, 10], memories should also have a
multimode capacity [11, 12] and an on-demand readout [13]
to realize a good quantum memory for quantum repeaters. The
quantum memory based on an atomic frequency comb (AFC)
[14] has the potential of achieving these ambitious goals in
one memory.
An AFC memory is one which is based on an
inhomogeneously broadened medium which has been tailored
to contain a frequency comb of narrow peaks (teeth) of atomic
population. The frequency spacing between the narrow peaks
() dictates the storage time of the memory: coherent reemission of the input mode, an echo, is seen in the same
forward direction a time 1/ after the input mode. Such an
0953-4075/12/124001+08$33.00
1
© 2012 IOP Publishing Ltd
Printed in the UK & the USA
J. Phys. B: At. Mol. Opt. Phys. 45 (2012) 124001
N Timoney et al
the memory in an auxiliary state. In terms of rare-earthdoped crystals, praseodymium and europium are interesting
candidates. Their nuclear spin of I = 5/2, results in three
hyperfine levels at zero applied magnetic field [9]. A full AFC
scheme was demonstrated for the first time in praseodymium
[13]. The nature of the material used in these results
places a limit of the number of modes to tens of modes.
This limit is due to the hyperfine splitting of the material
(approximately 10 MHz) and the optical homogeneous width
of the material—1 kHz. Europium on the other hand has
a larger hyperfine splitting (approximately 100 MHz) and
an optical homogeneous linewidth of the order of 100 Hz
[22]; with such a material it should be possible to create a
multimode memory with at least an order of magnitude more
modes [14]. Also, larger frequency separation of the input
and control frequencies will be useful for spectral filtering,
a likely requirement for future single-photon storage. The
drawback of europium-doped materials is the low oscillator
strength, which results in the low Rabi frequency of the control
fields and the low optical depth. Both these factors are serious
limitations in our present experiment, as we will describe. We
note, however, that a material with a small optical depth does
not necessarily have a poor efficiency, as equation (1) would
imply. By placing the crystal in an impedance-matched cavity,
it should be possible to achieve high efficiencies despite a low
optical depth [25, 26].
e
s
g
Figure 1. An illustration indicating the time order of an AFC
involving spin-wave storage. The input mode is in resonance with
the |g → |e transition; the control pulses are applied on the
|e → |s transition. The time 1/ is defined by the periodic
frequency separation of the teeth of the AFC.
this particular work, we use 153 Eu3+ :Y2 SiO5 , which we believe
has the potential of fulfilling the requirements of a quantum
memory for quantum repeaters [5, 14].
The scheme described above, however, is not the complete
AFC scheme as proposed in [14]. The conversion of the optical
excitation to a spin excitation is missing. This requires the
presence of another ground state. For a complete scheme, the
input mode is followed by a control pulse which transfers
the optical coherence between |g and |e to a spin coherence
between |g and |s. The time line and an illustration of the
relevant levels is shown figure 1. This control pulse ‘stops the
clock’ of the predefined memory time of 1/. A second control
pulse ‘restarts the clock’ by reverting to the optical coherence
between |g and |e. The time between the control pulses is not
predefined, such that the application of the second control pulse
allows an on-demand readout of the memory. The explanation
above is simplified since it does not take into account spin
dephasing due to inhomogeneous broadening, which leads to
a decay in storage efficiency as a function of spin storage
time. This dephasing can, however, be compensated for by
using spin-echo techniques, allowing long storage times only
limited by the spin coherence time.
An interesting aspect of the AFC scheme is that it is
possible to implement a multimode memory in the time domain
where the multimode capacity is not dependent on increasing
optical depth [14, 23]. Rather the limit on the number of
temporal modes which can be stored depends only on the
number of peaks in the AFC. This is restricted, in turn, by
the width of the AFC, generally limited by the hyperfine
transition spacing and the smallest comb tooth width which
can be obtained. Another restriction, of a technical nature, is
imposed by the frequency bandwidth of the control pulses,
which must spectrally cover the entire AFC spectrum. To
transfer a large bandwidth, the control pulses would most
likely take the form of chirped-shaped pulses as discussed
in [24].
To implement a full AFC scheme, the atomic element
must have three levels in the ground state, so that the atoms
not required for the comb can be spectrally separated from
2. Europium
Europium-doped Y2 SiO5 has been the subject of several
spectroscopic studies, measuring for instance absorption
coefficients, inhomogeneous and homogeneous broadenings
and hyperfine level spacings [22, 27–29]. Until recently,
however, the ordering of the hyperfine levels and the
transition branching ratios between these were unknown
in 153 Eu3+ :Y2 SiO5 . We thus conducted spectroscopic
investigations of an isotopically enriched 153 Eu3+ :Y2 SiO5
crystal [30] to determine a suitable -system in which to
perform a complete AFC scheme. In this work, we use one
of the potential -systems identified in our previous work,
shown in figure 2. We use the stronger ±|3/2g → ±|3/2e
transition, with a greater optical depth, for the input mode,
while the control fields are applied on the weaker ±|5/2g →
±|3/2e transition. It should be noted that from [30] three
other potential -systems could be identified, which could
all work as well or even better than the selected one (i.e.
have larger oscillator strengths). This particular configuration
was chosen for a first proof-of-principle demonstration due
to available frequency shifts by our frequency shifters. Other
configurations will be tested in future work.
Our crystal has a 153 Eu3+ doping level of 100 ppm, an
inhomogeneous broadening of 700 MHz and an absorption
coefficient of 1.2 cm−1 on the 7 F0 → 5 D0 transition. We
note that larger optical depths are possible with increased
doping [27].
2
J. Phys. B: At. Mol. Opt. Phys. 45 (2012) 124001
±5/2e
5
N Timoney et al
figure 3. The losses due to these AOMs are compensated for
with the amplifier after the 3 × 3 coupler; such recuperation
could not be obtained at 580 nm.
Once the light is produced at 580 nm, it passes through an
AOM (AOM:M) which determines the intensity and duration
of the light in the crystal. AOM:M is in double pass, which
allows small frequency scans of at least 15 MHz to be made
uniformly in intensity. In fact more than one spatial mode is
required for the spin-wave storage measurements performed
in this paper. This is due to the low amplitude of the observed
echo signal, compared to the strong control pulse. In addition,
we observed that the control pulse causes free induction decay
in its spatial mode, presumably due to off-resonant excitation,
which will act as a background noise on the weak echo. To
avoid this noise, we use two spatial modes: one for the comb
preparation and the control modes, and another for the input
pulse. In order to have fast independent control over each of
the spatial modes, we use two AOMs, one for each mode. The
zero order from the first pass of AOM:M is thus diverted to
another AOM (AOM:I) which operates at the same frequency.
AOM:I is also in a double-pass configuration. The resulting
light from AOM:I and AOM:M crosses the europium crystal.
Before the cryostat, there can be up to 53 mW of light from
AOM:M, depending on the radio frequency amplitude applied
to AOM:M. This light is focussed in the crystal with a beam
waist of 60 μm. The light from AOM:I is significantly weaker;
it can only reach a maximum power of 3 mW. The light hitting
the detector can be attenuated using AOM:D in single pass
after the cryostat.
The light at 580 nm is frequency stabilized using a second
continuous wave SFG source. This source is not shown in
figure 3. Light from both diode lasers is taken before the
amplifiers, combined on a dichroic mirror and sent through
a second PPKTP waveguide. A cavity with a free spectral
range of 1 GHz and finesse of 600 is used in a Pound–Drever–
Hall configuration. The cavity mirrors are separated by an
invar spacer and the structure is temperature stabilized. The
correction for the error signal is applied to the 930 nm laser.
The frequency stabilization removed slow frequency drifts and
narrowed the laser linewidth to slightly less than 50 kHz as
measured by spectral hole burning.
260 MHz
D0 ±3/2e
194 MHz
580 nm
±1/2e
7
±1/2 g
F0 ±3/2 g
±5/2g
90 MHz
119 MHz
Figure 2. Hyperfine structure of the 7 F0 → 5 D0 transition in
153
Eu3+ :Y2 SiO5 . The blue and green arrows indicate the -system
used for this work, where ±|3/2g → ±|3/2e acts as the |g → |e
transition for the input mode, and ±|5/2g → ±|3/2e the |e → |s
transition for the control fields. Although a -system containing two
ground states is all that is required for the complete AFC scheme, a
third ground state (an auxiliary state) is required to store the atoms
which are not part of the AFC. For the -system used in this paper,
the auxiliary state is ±|1/2g. The ±|1/2g → ±|5/2e transition is
used in the comb preparation stage, as explained in the text.
3. Experimental description
The 7 F0 → 5 D0 transition requires a light source at 580 nm.
This wavelength is not covered by diode lasers at present,
so we have chosen to generate it using sum frequency
generation (SFG) of two wavelengths: 1540 nm and 930 nm.
The nonlinear medium used is a PPKTP waveguide, which
produces 110 mW of light at 580 nm at the output where there
is 1.5 W of light at 1540 nm and 400 mW of light at 930 nm at
the input. To reduce the losses of the light produced at 580 nm,
the three frequencies required for the measurements shown
in this paper are produced using acousto optical modulators
(AOMs) on the 1540 nm light. This light is recombined in a
3 × 3 coupler; the polarization of each AOM can be controlled
before it is combined, thus adjusting the polarization required
for the waveguide. The polarization control is suppressed in
930 nm
T.Amp
AOM
0
1540 nm
Er Amp
AOM
Er Amp
AOM:I
1
Europium
Crystal
1
AOM:M
PPKTP
AOM:D
AOM
Figure 3. A very basic illustration of the experiment. The sum frequency process in the PPKTP waveguide generates the light at 580 nm
from the 1540 and 930 nm diode lasers. The light from both lasers is amplified; a tapered amplifier is used for the light at 930 nm and two
erbium amplifiers for the light at 1540 nm. The AOMs at 1540 nm select the atomic transition used. Two AOMs (AOM:M and AOM:I) at
580 nm before the europium crystal determine the amplitude and the duration of light in either the control or the input mode. These two
modes overlap in the europium crystal. Further explanation of the experiment can be found in the text.
3
J. Phys. B: At. Mol. Opt. Phys. 45 (2012) 124001
N Timoney et al
1
10
0.7
Input mode
Two level echo (η = 1.5 %)
0.6
0
10
Intensity, au
Optical depth
0.5
0.4
0.3
0.2
−1
10
−2
10
0.1
0
−3
−0.5
0
Frequency, MHz
10
0.5
−1
0
1
2
Times, μs
3
4
5
Figure 4. (a) A sample comb created with a periodicity of 0.5 MHz. The blue solid line shows the measured comb. This comb is obtained by
measuring the absorption of ±|3/2g → ±|3/2e with a small scan of <2 MHz over 200 μs. The comb shown is an average of 20
measurements. The green-dashed line shows a Gaussian comb with γ = 165 kHz (corresponding to F = 3.03), d0 = 0.04 and d = 0.54.
(b) The corresponding echo, where the efficiency is measured to be 1.5%. It was necessary to change the detector gain for the two
measurements shown, which results in more noise on the background of the input mode.
structure of ions. A fraction of the excited ions in ±|5/2e
will decay to |3/2g, thus forming the desired spectral comb of
ions on ±|3/2g → ±|3/2e . Since only a fraction of the ions
decay to |3/2g, it is necessary to repeat the comb preparation
many times. Ions that fall down in |5/2g must also be removed
by optical pumping, forcing most of these ions into the desired
|3/2g state. In the experiments shown here, 80 repetitions are
used. More details on all the steps used in the comb preparation
can be found in [30].
A sample comb is shown in figure 4(a) where the
frequency separation of the comb is = 0.5 MHz (this
corresponds to a two-level AFC storage time of 2 μs). The
comb measurement is performed by measuring the absorption
of ±|3/2g → ±|3/2e . The maximum optical depth that
we could obtain on this transition is d = 0.8, higher than
the maximum peak height shown in figure 4(a), where the
maximum value is d = 0.54. Peaks with higher absorption are
obtainable with more power in the pulse stream; however, this
results in broader peaks. Broader peaks reduce the finesse (F)
of equation (1), thus lowering the maximum echo efficiency
possible. The green-dotted line shown in figure 4(a) represents
a Gaussian comb in frequency (ν) of the form
The europium crystal itself is housed in a pulse tube
cooler, where the cold finger has a temperature of 2.8 K. We
observe spectral broadening of the teeth in the comb when the
cryostat cooler is switched on compared to when it is switched
off. The spectral broadening was very shot-to-shot dependent,
indicating that it depends on where the measurement is
performed within the pulse tube cycle (with a period of
700 ms). It is likely that this spectral broadening is caused
by crystal movement in the same direction as the beam
propagation, induced by the vibrations from the cryostat
compressor or rotary valve. Although the exact physical
mechanism leading to this broadening remains unclear, we
believe it to be due to phase noise in the atom–light interaction
induced by the modulation of the laser–crystal distance,
leading to an effective laser line broadening. By triggering the
experimental sequence on the vibrations in the cooling tubes
using a piezo, the observed jitter on the width of a comb tooth
is reduced by a factor of 3, from 150 to 50 kHz. We conclude
this section by noting that the technical spectral broadening
due to laser linewidth and cooler vibrations adds up to about
100 kHz.
j=+4
4. Comb preparation and two-level echo efficiency
n(ν) = d
j=−4
The frequency comb required for an AFC quantum memory
can be prepared using a stream of pulses where the inverse of
the time separation gives the frequency separation () of the
comb produced [11, 16]. Spectral tailoring techniques similar
to those used in [13] and [30] are necessary before the comb
creation to isolate an atomic system such as that described in
figure 2. Note that the spectral tailoring is performed over a
certain frequency range, in our case over roughly 10 MHz.
The spectral tailoring includes a spin polarization process,
which prepares the atoms in one of the ground states, in
this case in the auxiliary state ±|1/2g (see figure 2). The
comb is then prepared by sending a stream of 15 pulses on
the ±|1/2g → ±|5/2e transition, exciting a spectral comb
2
− (ν− j)
2
e
2γ̄
+ d0 ,
(2)
√
where the peak width is given by γ̄ = γ / 8 ln 2 (all other
parameters are defined as before). The finesse of each comb is
obtained by measuring the width of the peaks created. The
minimum width of a peak in the comb is limited by the
laser linewidth and the frequency noise induced by vibrations
from the cryostat. Additional contributions include possible
power broadening from the peak preparation pulses and the
inhomogeneous spin linewidth of the material.
The echo which can be seen using the comb in figure 4(a)
is shown in figure 4(b). The efficiency of the emitted two-level
echo is obtained by comparing the area of the input mode when
there is no absorption on the input mode transition (the blue
4
N Timoney et al
2.5
5
2
4
1.5
3
1
2
0.5
1
0
1.5
2
2.5
3
3.5
Storage time (μs)
4
4.5
length between the laser source and the crystal; this could be
done using an interferometric setup. In addition, due to our
preparation method, the inhomogeneous spin linewidth is also
likely to contribute towards the minimum linewidth. This issue
can, however, easily be resolved by changing the preparation
method.
The maximum efficiency obtained at 2 μs storage time
is limited by the low optical depth of our material. Higher
efficiencies can certainly be reached using a crystal with higher
doping concentration. Indeed a peak optical depth three times
higher has been obtained [27], which ideally would result in a
tenfold increase in the efficiency (see equation (1)). Another
interesting solution would be to place a cavity around the
crystal [25, 26]. Both methods might be required in order to
approach unit efficiency. We conclude this section by noting
that whilst not dictating an absolute limit, these efficiencies
do represent a serious limitation in our present system, which
is relevant to the following results of spin-wave storage in the
complete AFC scheme.
Finesse
Echo efficiency %
J. Phys. B: At. Mol. Opt. Phys. 45 (2012) 124001
0
5
Figure 5. The decay in the echo efficiency of a two-level AFC echo.
The squares show the efficiencies using a comb preparation
described in the text for a range of AFC storage times (1/). The
finesse expected from γ = 165 kHz is plotted using a solid red line
(right-hand y axis).
5. Spin-wave storage
solid line) to the area of echo signal (the green solid line). The
efficiency measured here is 1.5%, which is higher than that
expected (1.2%) using equation (1) and the values of d and d0
obtained in figure 4(b) (cf figure 5). We estimate an error of
±5% on the efficiencies shown due to the nonlinear reaction
of changing the gain on the detector. Additionally, the optical
depth measured in the comb measurement of figure 4 is likely
to be too low. First, the optical depth has been reduced by the
input pulse which was not suppressed for this measurement.
Second, the rate of the scan, 2 MHz in 200 μs, implies a scan
resolution of roughly 100 kHz, large enough to detrimentally
affect the values extracted for d and d0 .
Using the same preparation method, the time separation of
the comb preparation pulses and thus the frequency separation
of the comb teeth () was varied. The two-level echo efficiency
for a storage time of up to 5 μs was measured and the results
are plotted in figure 5. A comb has been measured for each
efficiency shown in figure 5. For longer storage times, we do
not deem this comb measurement to be accurate. The effects
of our scan resolution, the vibrating cryostat and the laser
linewidth are hard to separate from the actual comb. Instead
we merely extract a value of γ from these plots of 165 kHz,
and show the measured finesse on the right-hand axis. The
trend of decreasing finesse accounts for the trend of decreasing
efficiency.
The optimum storage time in our system is currently
2 μs. But these results do not represent a fundamental limit
to the maximum storage time or two-level efficiency in this
material. The storage time is clearly limited by the AFC tooth
width (γ ). The minimum width obtained was limited by our
laser linewidth and the effective linewidth broadening probably
due to crystal movement induced by the cryostat cooler. In
order to improve the storage time, we should improve the
laser frequency stabilization and reduce the presumed effect
of crystal movement. The latter can be done by employing
a low-vibration pulse tube cooler or by stabilizing the path
In order to perform spin-wave storage, we spatially separate
the input mode from the control mode using a cross
beam configuration, as already discussed in the experimental
description section (see also figure 3). While reducing the
free induction decay noise caused by the control fields [13],
the crossed beam configuration also has a negative effect; it
reduces the efficiency of the two-level echo. A sample twolevel echo is shown in figure 6. This echo is measured without
the control pulses, but in a crossed mode configuration where
the comb is prepared using AOM:M. The efficiency of the twolevel echo is measured to be 0.24% at 1/ = 4 μs. This should
be compared to the single spatial mode measurement, where
the efficiency was measured to be 0.6% (see figure 5). The
two-level echo efficiency is thus reduced by almost a factor of
3. This is attributed to imperfect overlap of the two beams.
In the introduction, we talked about using chirped pulses
for our control pulses. Such pulses would allow us to efficiently
transfer a large bandwidth, potentially all of the teeth in
the AFC spectrum. But these pulses are generally of longer
duration as compared to a π -pulse [24]. There is however a
limit on the time which we can use to perform the control
pulses. The duration and shape of the input pulse and one
control pulse cannot be longer than the predefined two-level
echo time 1/. In addition, the duration of the input pulse
defines the bandwidth which the control pulse must transfer.
In our current experimental setup, the Rabi frequency of the
control pulse transition, ±|5/2g → ±|3/2e , is estimated to
be of the order of 300 2π kHz. To transfer a bandwidth of
300 kHz, using a π -pulse, 1.7 μs are necessary at this Rabi
frequency. If we set 1/ = 2 μs to obtain the highest two-level
echo efficiency, the input pulse would have to be shorter than
0.3 μs. The spectral bandwidth of such an input pulse would
be >3 MHz, the majority of which will not be transferred by a
1.7 μs control pulse. This illustrates that a compromise must
be made between the efficiency of the two-level echo and the
5
J. Phys. B: At. Mol. Opt. Phys. 45 (2012) 124001
N Timoney et al
2
10
Input mode
Two level echo (η = 0.24 %)
Three level echo (η = 0.02%)
Control pulses
1
γ
Intensity au
Intensity, au
10
0.03
0
10
IS
0.025
= 69 ±3 kHz
0.02
0.015
−1
10
0.01
−2
10
0
−3
10
−2
0
2
4
6
Times, μs
8
10
2
4
6
8
Ts (μs)
10
12
14
Figure 7. Inhomogeneous spin linewidth measurement. Each point
represents the maximum of the Gaussian function which has been
fitted to each trace. Some traces and fits are shown; the rest are
suppressed for clarity. The formula used to fit the data is described
in the text and yields an inhomogeneous linewidth of 69 ± 3 kHz.
12
Figure 6. An input mode (blue), its two-level echo with no control
pulses (green) and a spin stored echo, a three-level echo, when the
control pulses are applied (red). Compared to the two-level echoes
shown in figure 5, the input mode is not in the same spatial mode as
the preparation of the AFC. This is to reduce free induction decay
background noise on the three-level echo. The input mode and the
preparation beams overlap in the crystal. The control fields are in the
same mode as that of the preparation. Imperfect overlap of the two
modes accounts for the efficiency of the two-level echo dropping by
almost a factor of 3 compared to figure 5. Changing the gain on the
detector distorts the pulse shape of the echo. All of the
measurements represent an average of 20 traces.
where Ts defines the spin storage time, A is a constant and γIS
is the inhomogeneous spin linewidth. An echo of increasing
spin storage time is shown in figure 7. The complete memory
time is given by Ts + 1/. Each recorded echo is fitted with
a Gaussian function, the maximum of which is used to fit
equation (3). We obtain an inhomogeneous spin linewidth of
69 ± 3 kHz. Some echo traces and their fits are shown, but
most have been suppressed for the sake of clarity.
Armed with the knowledge of the inhomogeneous spin
linewidth, we can estimate the efficiency of the transfer pulses
used. For the example shown in figure 6, the measured
efficiency can be extrapolated to Ts = 0, resulting in an
estimated 0.04% efficiency without spin dephasing. Hence,
the estimated three-level echo at Ts = 0 is 16% of the twolevel echo (0.04%/0.24% = 0.16). Following the simple model
discussed in [13], we can then calculate the transfer efficiency
per pulse, which we find to be 40%. It should be noted that
since the bandwidth of the control pulses is not much larger
than the input pulse bandwidth, we should take this value as
an effective average over the bandwidth. It is also difficult to
estimate the effect of imperfect beam overlap on the efficiency.
However, considering that we observe a strong decrease in the
two-level echo due to insufficient overlap, it is likely that the
overlap has a non-negligible effect on the estimated transfer
pulse efficiency.
We emphasize that it is possible to refocus the spin
coherence using radio frequency pulses (spin echo), which
would allow us to store for durations of the order of the spin
coherence time. This was measured to be 15.5 ms for the
151
Eu3+ isotope in Y2 SiO5 at zero magnetic field [31]. One can
expect a similar value for 153 Eu3+ . As a future perspective, one
can also think of applying a magnetic field to cancel the firstorder Zeeman effect, creating a memory which for europium
could be of the order of many seconds [32]. This is similar to
the work performed in praseodymium [9]. The excellent spin
coherence properties are a clear strength of europium-doped
materials.
efficiency of the transfer. We use the final echo efficiency as
an indicator of the best compromise.
The input pulse had approximately a Gaussian shape,
while the identical control pulses were square shaped. We set
the two-level storage time to 1/ = 4μs. The durations of the
input pulse and the control pulses were optimized by looking
at the efficiency of the three-level echo. The highest efficiency
was reached with an input pulse with a full-width at halfmaximum of 1.3 μs and control pulses of duration 1.55 μs.
The resulting echo with spin-wave storage is shown in figure 6,
where the spin storage time is roughly 5 μs, leading to a total
storage time of 9 μs. We checked that the echo is not present
if we remove the first control pulse or if we remove the input
mode, as expected. In figure 6, we also show the associated
two-level echo (no control pulses applied) obtained with an
identical comb structure. Note that AOM:D is used to gate the
detector around the measured echoes, as the input mode would
saturate the detector at the gain required to see the echoes.
Maintaining this 1/ = 4 μs AFC, it is possible to vary
the time between the control pulses (Ts ). This is the on-demand
storage time of the full AFC scheme. For a rare-earth-doped
crystal, we expect to see a decay given by the inhomogeneous
spin linewidth of the material [13]. The storage time of the
complete AFC scheme as described in the introduction of this
paper is limited by this linewidth. If we assume a Gaussian
distribution for the spin broadening, the decay of the stored
echo is given by
−Ts2 γ 2 π 2 IS
echo height = A e 2ln2 ,
(3)
6
J. Phys. B: At. Mol. Opt. Phys. 45 (2012) 124001
N Timoney et al
6. Conclusion, discussion and outlook
Acknowledgments
Here we show for the first time an AFC memory with
spin-wave storage in a europium crystal. Europium is an
interesting material for quantum memories thanks to its long
spin coherence time and its potential for multimode storage.
The experiments we report here represent a first step in this
direction. Yet, several important aspects of the memory must
be significantly improved.
First, the efficiencies which we report are very low, indeed
europiums’ handicap lies in its low optical depth. There exist
promising proposals to increase the optical depth by placing
the crystal in a cavity thus increasing the efficiency of a twolevel echo. The measurements shown in this paper do not
represent a fundamental limit of the efficiency nor of the
storage time of a two-level echo. The latter has an impact on
the multimode capacity of the memory. The storage time could
be improved by changing the comb preparation method to one
such as that found in [11], which would remove the effect of
the inhomogeneous linewidth from the minimum peak width
obtainable. Indeed peaks as narrow as 1 kHz can be found in
the recent publication [33]. Further technical improvements
on the laser linewidth or a cryostat with less vibration would
also improve the storage time shown in this paper, allowing
for multimode storage and the use of more efficient, chirped
control pulses.
Another technical difficulty is as follows: the relatively
small dipole moment of this transition means that large
amounts of power are required to transfer a relatively
small bandwidth. To increase the bandwidth of the transfer,
significantly larger amounts of power are required. Recent
developments of powerful lasers for sodium-based systems at
589 nm [34] can also be applied to developing narrowband,
powerful and compact lasers at 580 nm.
In the spin-wave storage experiments presented here, we
could store for up to about 10 μs, limited by inhomogeneous
spin dephasing. From these measurements, we could estimate
the spin linewidth, which we found to be 69 ± 3 kHz.
The inhomogeneous spin linewidth is not a restriction,
however, on the maximum storage time of the medium. The
storage time can be increased using spin refocussing pulses
such as [9].
An important milestone for quantum memories based
on rare-earth-doped crystals would be to store an optical
pulse on the single-photon level as a spin-wave excitation,
a milestone that has not yet been reached in any rare-earthbased memory. The low overall efficiencies obtained in these
experiments currently make this a very challenging experiment
in 153 Eu3+ :Y2 SiO5 . The possible improvements that we have
detailed, however, should make it possible in the near future.
In this context, we recently proposed a method of generating a
photon pair source with variable delay [35], using the same
resources used for spin-wave storage-based AFC quantum
memory. The overall efficiency of this source also has a less
strong dependence on optical depth and control pulse transfer
efficiency. An advantage of 153 Eu3+ :Y2 SiO5 in this context is
the large hyperfine splitting, which facilitates spectral filtering
as compared to praseodymium-doped Y2 SiO5 .
We would like to thank Claudio Barreiro for technical
assistance and N Sangouard and H de Riedmatten for useful
discussions. We are also grateful to Y Sun, R L Cone and R M
Macfarlane for kindly lending us the 153 Eu3+ doped Y2 SiO5
crystal. This work was financially supported by the Swiss
NCCR-QSIT and by the European projects QuRep, Q-Essence
and CIPRIS (FP7 Marie Curie Actions).
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Single-photon-level optical storage in a solid-state spin-wave memory
N. Timoney, I. Usmani, P. Jobez, M. Afzelius,∗ and N. Gisin
arXiv:1301.6924v1 [quant-ph] 29 Jan 2013
Group of Applied Physics, University of Geneva, CH-1211 Geneva 4, Switzerland
A long-lived quantum memory is a firm requirement for implementing a quantum repeater scheme.
Recent progress in solid-state rare-earth-ion-doped systems justifies their status as very strong candidates for such systems. Nonetheless an optical memory based on spin-wave storage at the singlephoton-level has not been shown in such a system to date, which is crucial for achieving the long
storage times required for quantum repeaters. In this letter we show that it is possible to execute
a complete atomic frequency comb (AFC) scheme, including spin-wave storage, with weak coherent
pulses of n̄ = 2.5 ± 0.6 photons per pulse. We discuss in detail the experimental steps required to
obtain this result and demonstrate the coherence of a stored time-bin pulse. We show a noise level
of (7.1 ± 2.3) · 10−3 photons per mode during storage, this relatively low-noise level paves the way
for future quantum optics experiments using spin-waves in rare-earth-doped crystals.
Quantum communication if rigorously executed provides us with a provably secure method of communication
[1]. However, inherently lossy channels limit the distance
over which the communication can be performed, which
today is roughly 250 km [2, 3]. A quantum repeater
which can in principle allow quantum communication
over longer distances [4–6], provided that the required
quantum memories are developed. Prime candidates for
quantum memories are atomic systems, which are capable of maintaining the coherence of stored excitations for
long times. Atomic systems that are currently investigated range from individual quantum systems[7, 8], lasercooled atomic gases [9, 10], room-temperature atomic
vapours [11–13], to rare-earth-ion-doped crystals [14, 15].
Crystals doped with rare-earth-ion impurities have
attractive coherence properties when cooled < 4K, in
particular hyperfine states can have coherence times
which can approach seconds [16]. This has provided a
strong motivation for developing quantum memories using such systems. Following the first storage experiment
at the single-photon level [17], a succession of experiments demonstrated storage of single photons [18, 19],
generation of light-matter [14, 15] and matter-matter entanglement using crystals [20]. The quantum memory
performances have also been strongly developed, particularly in terms of storage efficiency [21, 22], multimode capacity [23, 24] and polarization qubit storage [19, 25, 26].
These experiments were performed for short storage
times (in the 10 ns to few µs regime) using an optical coherence, rather than exploiting long spin coherence
times. Spin storage experiments require strong optical
control fields to convert the initial optical coherence to
a spin coherence. Photon noise is induced by such an
operation, which has been nonetheless shown to work for
alkali atomic systems [7–10, 12, 13]. In rare-earth-iondoped solids the task is complicated since there is less
spectral separation between the weak signal field and the
optical control field (roughly 100 times less). Scattering
from the control field is thus more likely, as it propagates
through a dense solid-state crystal.
Two quantum memory schemes were specifically pro-
posed for solid-state ensembles; the controlled and reversible inhomogeneous broadening (CRIB) memory (see
[27] and references therein) and the atomic frequency
comb (AFC) memory [28]. The AFC has a particularly
high multimode capacity, which is the ability to store
trains of single photon pulses [28, 29]. This is crucial for
speeding-up quantum repeater protocols [5]. The AFC
scheme is based on an echo induced by a regular spectral
grating of periodicity ∆, in the absorption profile of an
atomic ensemble. An AFC echo is emitted a time defined by 1/∆, unless the optical coherence is transferred
(written) to a spin coherence before the time 1/∆ has
elapsed. Reversing the transfer retrieves an optical pulse
(referred to as an AFC spin-wave echo). AFC memories
which only use the optical coherence are delay lines unless combined with spin-wave storage [28], which allow
for on-demand read out and significantly longer storage
times. Only a few AFC spin-wave storage experiments
have been reported, all involving storage of bright classical pulses [30–32].
Here we demonstrate storage of an optical pulse containing a few photons on average, using an AFC memory
combined with spin-wave storage in a europium doped
Y2 SiO5 crystal. We apply a strategy of filtering in space,
time and frequency in order to reduce unwanted emission
from the crystal at the moment the weak pulse is recovered from the crystal. To quantify the degree of noise we
measure the unconditional noise floor [12],which is the
probability for the memory to produce a noise photon
when the memory is read. We report that the unconditional noise floor can be reduced to (7.1 ± 2.3) · 10−3
by our filtering strategy, which is low enough to allow
for a range of quantum information schemes that require
manipulation of spin coherence. Using the ability of the
AFC memory to store multiple time bins, we also store
and analyse a time-bin pulse with higher photon numbers, showing the high coherence of our quantum memory.
Europium is a promising candidate for quantum memories due to its fine coherence properties at T < 6K [33–
35], which ultimately could lead to an extremely long-
2
(a)
(b)
FIG. 1. (a)The atomic level scheme of the optical transition
7
F0 →5 D0 in 151 Eu3+ :Y2 SiO5 . (b) A schematic of the experimental setup around the memory, the rest of the experiment
has been suppressed for simplicity. The control and preparation beam is in single pass (wide labelled arrow). The input
mode (thin dashed line) is in double pass, with the help of a
Faraday Rotator (FR) and a polarizing beamsplitter (PBS).
On return from the crystal the input mode passes through a
Fabry-Perot(FP) cavity (bandwidth of 7.5 MHz). A classical
detector (Sd) and 10 µW of horizontally polarized light (thin
dotted line) is used to actively and intermittently stabilize
the cavity to the frequency of the input mode. An accousto
optical modulator (AOM) in double pass acts as a detector
gate.
lived [36] and multimode memory [28]. In this work we
use the optical 7 F0 →5 D0 transition at 580 nm. The crystal is isotopically pure 151 Eu3+ :Y2 SiO5 (100ppm). At a
temperature of around 3 K we measure an overall absorption coefficient of α = 1.5 cm−1 and an optical inhomogeneous linewidth of 500 MHz. The relevant energy
diagram is shown in Fig. 1a. Our input and control fields
excite two optical-hyperfine transitions separated by 35.4
MHz.
The schematic of the experimental set up (Fig. 1b)
shows only the optics around the cryostat containing the
151
Eu3+ :Y2 SiO5 crystal of length L=1 cm. The storage
mode crosses the control and preparation mode through
the crystal. Given the measured angular separation of
the beams before the cryostat we estimate a spatial mode
overlap of 95 %. A double-pass configuration was implemented on the storage mode to increase the optical depth
[23], while the control mode was in single pass. The laser
and the acousto-optic modulators (AOMs) used for spectral control are not shown in Fig. 1b. The laser at 580
nm is a commercially available system based on an amplified diode laser at 1160 nm and a frequency doubling
stage. Before the cryostat the intense control pulses had
peak powers of up to 300 mW. The diode laser is stabilized to have a spectral linewidth of approximately 30
kHz.
The AFC comb structures are created with frequency
selective optical pumping techniques, which are now wellestablished techniques for spectral shaping of inhomoge-
neously broadened transitions, see for instance [30, 37]. A
particular feature of our preparation sequence is that we
first pump all ions into the ±|1/2ig state, and then create
the comb-structure by removing atoms from this state.
This has the benefit of reducing the effect of the inhomogeneous spin linewidth, which could otherwise limit
the minimum tooth width in the comb structure [31].
The maximum optical depth we can achieve on the input
transition is αL = 2.4, in double-pass configuration.
We first characterize our memory using bright input
pulses of many photons and detecting the pulses with a
linear photodiode. We observe AFC echo efficiencies of
more than 5% for 1/∆ = 6 µs, and AFC spin-wave echo
efficiencies of 1% for spin-wave storage time TS of 18
µs. The reduction in efficiency is mostly due to imperfect control pulses. We estimate the transfer efficiency
per control pulse to be 0.49. By measuring the decay of
the spin-wave echo as a function of TS , we estimate the
inhomogeneous spin linewidth to be 8 kHz. This measurement will be further detailed in a future publication.
The 8 kHz linewidth is surprisingly low, a factor of 8 less
than for the 153 Eu3+ :Y2 SiO5 (100 ppm) sample we previously used [31]. This results in a spin-wave memory
time of about 50 µs, defined by the point where the efficiency is reduced to exp(−1), the longest so far obtained
in an AFC memory. By applying spin refocussing techniques we can expect to increase it further, up to the spin
coherence time of 15 ms [35].
AFC spin-wave storage for weak coherent pulses with
average photon numbers between n̄ = 2.5 ± 0.6 and
n̄ = 11.2 ± 0.6 are shown in Fig. 2. The input pulse
is 2 µs long, the memory parameters are 1/∆=6 µs and
TS =21 µs, leading to a total storage time of 27 µs. The
duration and shape of the control pulses were optimized
for the highest signal-to-noise ratio (SNR), see discussion
below. These measurements are performed, as all of the
measurements shown in this letter, without the cryostat
switched on to reduce the effect of vibrations on the comb
structure [31].
There are two principal mechanisms which are responsible for the noise created by the bright control pulses.
One is scattering of the laser light itself from optical surfaces. Another is emission from the atoms which have
been excited by the pulses, this includes incoherent fluorescence, coherent free-induction-decay (FID) type emission and an unexpected off resonantly excited echo.
Spatial separation of the input and control modes is
used to shield the single photon counting detector from
scattered light, but this did not lead to sufficient suppression. A double-pass AOM (shown in Fig. 1b) is used as
a detector gate in time, exploiting the temporal separation between the control fields and the emitted spin-wave
echo, providing a suppression of roughly 106 . This proved
sufficient to prevent detector blinding or significant afterpulsing.
The emission noise is, however, also present in the tem-
3
(b)
Input
Input
100
C1
x 30
50
Counts
Counts
150
0
(c)
150
C2
10
20
Time ( µs)
30
250
100
0
25
C1
50
x 30
Output
0
500
Counts
200
(d)
C2
Output
0
0
10
20
Time ( µs)
30
30
Time ( µs)
35
5
15
10
SNR
(a)
5
0
0
n̄
10
FIG. 2. Storage of a weak coherent pulse with (a) n̄ = 2.5 ± 0.6 and (b) n̄ = 11.2 ± 0.6. The input mode recorded with no comb,
the position of the control fields (C1 and C2), and finally a magnified (x 30) signal of the AFC spin-wave echo (blue curve), the
associated noise without an input pulse (green curve) and the detector dark counts (red curve) are shown. Note that the total
measurement time differs between the data sets in (a) and (b). (c) The same echo data of (a) with the temporally separated
off-resonant echo (OREO). (d) SNR for different n̄. Shown is also a fitted linear slope fixed to 1 at n̄ = 0 by definition.
poral mode of the output mode. A diffraction grating and
a Fabry Perot (FP) cavity are used to spectrally filter this
noise. The FP cavity is necessary in particular to reduce noise originating from FID. Sharp spectral features
about the control transition, by products of the spectral
tailoring required to prepare the AFC, cause the FID.
We could reduce this noise by altering our preparation
sequence, to increase the transparency window around
the control field transition. The frequency of this noise
is close to that of the control field, a fact which we observe by changing the frequency at which we lock our FP
cavity.
In addition to the fluorescence and FID noise, we also
observed an unexpected noise source at the input frequency. We believe its presence to be due to off-resonant
excitation from the control fields, we thus call this an
off-resonant echo (OREO) (see Fig. 2c). The OREO is
observed a time 1/∆ after C2, supporting this hypothesis. We also observe a strengthening of the echo if C1
is present. We explain this by supposing that the offresonant excitation of C1 is combined with transfer to
the spin state. C2 then reads out the excitation in the
same manner which it does the single-photon-level input
pulse. The observed 1/∆ dependence of the OREO, also
fits to this explanation. Although the OREO is considerably larger than the AFC spin-wave echo which we are
seeking to retrieve, the two echoes occur in temporally
separated modes (see Figs 2a and c). We could reduce
the impact of temporal mode leakage by carefully tuning
the shape of the control fields, which is consistent with
an off-resonant excitation mechanism. Note that since
the FID and the OREO are coherent processes, the corresponding emission should only be strong in the control
mode. Scattering inside the crystal does however introduce significant cross talk between the spatial modes.
The temporal shape of the remaining noise we observe
in Fig. 2a is indicative of FID noise. This gives us rea-
son to believe that a more efficient filtering system would
permit us to increase the power in the control fields thus
increasing their efficiency. The remaining noise, in this
particular measurement, amounts to (5.1 ± 1.3) · 10−3
photons per mode emitted at the crystal. The SNR up
to n̄ = 11.2 ± 0.6 is shown in Fig. 2d. These measurements were taken on a range of different days for the
same experimental parameters. The SNR follows a linear dependence within the experimental errors, see fitted
linear slope in Fig. 2d. Measurements carried out for
higher average photon numbers (not shown) confirmed
this behaviour.
The final memory efficiency in the photon counting experiment was significantly lower than for the bright pulse
storage. The optimization of the duration and shape of
the control pulses led to a lower transfer efficiency. Furthermore, a photon counting experiment requires time
averaging, for example, the measurement for n̄ = 2.5±0.6
was taken over the course of three hours. This challenges
the stability of the experiment, in particular, laser fluctuations create reduced quality combs, which negatively
affect the AFC echo efficiency. Averaging over all the
measurements shown in Fig. 2d, we obtain a global memory efficiency of (3.8 ± 1.5) · 10−3 and an unconditional
noise floor of (7.1 ± 2.3) · 10−3 .
Finally we show the coherence of the AFC spin-wave
echo. To do this we store a time-bin pulse in the memory
where we vary the phase of one of the time bins. We then
self interfere the time-bin pulse using a temporal beamsplitter and examine the interference curve. The visibility of the curve gives a measure of coherence preservation
in the memory. For the measurement shown in this letter, the temporal beamsplitter comes in the form of the
control pulses. The scheme is pictorially shown in Fig.
3a.
To store and analyse the time-bin pulse, we need clean
temporal separation between the retrieved pulses and
4
(a)
(b)
(c)
300
100
V
fit
80
= 0.87 ±0.06
Counts
Sum counts
400
200
100
0
0
60
40
20
0.5
1
1.5
Phase shift, π
2
0
26
28
30
Time, µs
32
FIG. 3.
(a) The method used to measure the coherence
of the AFC spin-wave echoes. The single write operation of
Fig. 2a,b (C1) is replaced by a double write operation(W1,
W2). If the temporal separation (T) of the input mode is
equal to that of the double write operation , the first echo
of the second write operation and second echo of the first
operation interfere. (b) The visibility curve for two pulses
with n̄ = 176 ± 8 . (c) The signal of a constructive and
destructive case. The thick dashed lines show the temporal
window which was used to obtain the interference curve. The
detector gate has cut some of the first echo and the OREO is
not shown in this temporal slice.
enough time to see the triple pulse structure shown in
Fig. 3c after the final control field. To do this we extend the AFC time from 6 to 8 µs, and reduce the pulse
width of the input pulses and the entire pulse length of
the control pulses. These measures further reduce the
efficiency with which we can store in the memory to ηs =
(6.3±0.1)·10−4 for each mode, including the reduction in
storage efficiency due to the double write operation. TS
was set to 21 µs in this experiment, yielding a total memory time of about 29 µs. A visibility curve for n̄ = 176±8
is shown in figure 3b, where we measure V = 0.87 ±0.06.
We suspect that laser phase noise contributes negatively
to our visibility curve. A simple calculation shows that
frequency noise with σf = 25 kHz reduces the baseline
to V = 0.95. Together with the noise level this accounts
for the visibility we measure. For n̄ = 51 ± 3 we observe
a further drop in visibility to V = 0.71 ±0.1. This is due
to the increasingly important role of noise in determining
the minimum of the visibility curve. We note that with
higher storage efficiency, it should be possible to obtain
high visibilities for lower photon numbers.
The unconditional noise floor achieved in our experiment should in principle allow us to store a single-photonlevel optical pulse with high SNR. The limited SNR obtained at a few photons is entirely given by the low overall
memory efficiency. Future experiments should therefore
aim at increasing the efficiency, while we consider the
filtering to be sufficient for quantum applications. The
memory inefficiency is due to two major factors; 1) insufficient optical depth and 2) insufficient control field transfer efficiency. To increase the optical depth we will implement an impedance-matched cavity around the memory, as proposed in [38, 39]. Indeed recent results using
such an impedance-matched cavity have shown an optical
AFC efficiency of 58% using a crystal with optical depth
comparable to ours if in a single-pass configuration [22].
2) The control field transfer efficiency can most easily be
improved by using longer adiabatic transfer pulses [40].
Such long temporal windows can be created by increasing
the AFC echo time (1/∆). Using the narrowest measured
optical homogeneous linewidth measured with europium
of 122 Hz [33], 1/∆ times of around 1 ms are in principle
possible. Furthermore increasing 1/∆ will also allow us
to exploit the multimode capability of the AFC scheme.
Currently, however, our laser linewidth represents serious
technical obstacle in increasing 1/∆ towards this limit,
beyond the shown 6-8 µs. We also note that employing
long adiabatic control fields should reduce off resonant
excitation, further decreasing the remaining noise.
To conclude, we have demonstrated the first optical
storage as a spin-wave in a solid-state memory, in the
regime of a few photons per input pulse. This was
made possible by a strategy of extensive filtering and
by carefully shaping the temporal envelope of the strong
control pulses. The final unconditional noise floor of
(7.1 ± 2.3) · 10−3 is low enough to allow for quantum
schemes using spin-wave storage and manipulation, such
as the generation of quantum-correlated spin-wave and
photonic excitations using variant of the DLCZ [41] approach adapted to the solid-state [42–44]. These schemes
will, in turn, allow for generation of entanglement between light and matter and entanglement of solid state
remote quantum memories, a basic building block for
quantum repeaters.
We would like to thank C. Barreiro for technical assistance, and N. Sangouard and P. Goldner for useful discussions. We gratefully acknowledge R. Cone and R.M.
MacFarlane for lending us the isoptically pure 151 Eu crystal. This work was financially supported by the Swiss
National Centres of Competence in Research (NCCR)
project Quantum Science Technology (QSIT) and by the
European projects QuRep (FET Open STREP), CIPRIS
(FP7 Marie Curie Actions) and Q-Essence (FET Proactive Integrated Project).
∗
[email protected]
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