A photon-pair source with controllable delay based on shaped inhomogeneous broadening of rare-earth doped solids

0.19 MB·English

by  Pavel Sekatski

A photon-pair source with controllable delay
based on shaped inhomogeneous broadening of rare-earth doped solids
Pavel Sekatski,1 Nicolas Sangouard,1 Nicolas Gisin,1 Hugues de Riedmatten,1,2,3 and Mikael Afzelius1
1Group of Applied Physics, University of Geneva, 1211 Geneva 4, Switzerland
2ICFO-Institute of Photonic Sciences, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain
3ICREA-Instituci´o Catalana de Recerca i Estudis Avanc¸ats, 08015 Barcelona, Spain
(Dated: February 1, 2011)
Spontaneous Raman emission in atomic gases provides an attractive source of photon pairs with
a controllable delay. We show how this technique can be implemented in solid state systems by
appropriately shaping the inhomogeneous broadening. Our proposal is eminently feasible with
currenttechnology and providesa realistic solution to entangle remote rare-earth doped solids in a
1
heralded way.
1
0
2
Introduction ThecreationofcorrelatedStokes–anti-
n
Stokesphotonpairsinatomicensemblesviaspontaneous
a
J Raman emission [1] plays a central role in quantum
communication, starting with the implementation of
1
3 quantum repeaters [2]. The basic principle requires an
ensemble of lambda systems coupled to a pair of optical

] laser fields, see Fig. 1. An off-resonant laser pulse,
h
p the write pulse, produces a frequency shifted Stokes
- photon via spontaneous Raman emission. Detection
t
n of this Stokes photon in the far field, such that no
a information is revealed about which atom it came from,
u
heralds the creation of a single collective atomic spin
q
excitation. A remarkable feature of such a collective
[
atomic state is that it canbe readoutvery efficiently. A FIG. 1: a) Basic level scheme and b) pulse sequence for the
1
resonant laser pulse, the read pulse, allows one, through creation of correlated Stokes – anti-Stokes pairs via sponta-
v
acollectivespontaneousRamanemission,toideallymap neousRamanemission. Allatomsstartsing.Theinhomoge-
5
0 the collective spin excitation into an anti-Stokes photon neouslybroadenedopticaltransitionisshapedintoanatomic
0 propagating in a well defined spatio-temporal mode. frequency comb. A resonant write pulse Ωw excites the g-
e transition, making possible the spontaneous emission of a
6 This provides a photon-pair source with a very special
. Stokesphoton. ThedetectionofaStokesphoton(attimetd)
1 property : the delay between the Stokes and anti-Stokes is uniquely correlated with the creation of a single collective
0 photons can be controlled by choosing the timing atomic excitation corresponding to one excitation in s delo-
1 between the write and read pulses. Such a source has calized over all the atoms. This collective excitation can be
1
inspiredmanyexperimentsinatomicgases,includingthe mapped very efficiently into an anti-Stokes photon using a
:
v first single-photon storage in an atomic ensemble [3, 4], resonant read pulse Ωr (at time τ +td) and beneficing from
Xi the first heralded creation of entanglement between theecho-photon-typere-emission(attime2π/∆+τ)inherent
to inhomogeneous transition shaped into a frequency comb.
atomic ensembles [5] and even the implementation of
r
a the first elementary blocks of quantum repeaters [6].
Despite this impressive body of work, there are strong
All rare-earth elements have in common weak dipole
motivations to use more practical systems, e.g. in the
moments on the relevant 4f - 4f transitions and large
solid state. Rare-earth-doped solids seem naturally well
inhomogeneously broadened spectra due to the interac-
suited, at least at first sight. They are widely available
tion with the host crystal. These two properties make
thanks to their use for solid-state lasers. Thanks to
the creation of Stokes – anti-Stokes pairs in rare-earth
their particular electronic structure, they can be seen as
doped solids challenging. In hot alkali gases, where
a frozen gas of atoms, with optical and spin transitions
there is also an inhomogeneous broadening due to the
featuring excellent coherence properties. Moreover,they
Doppler effect, spontaneous Raman processes have been
have alreadyshownexcellent capability to store light for
successfully performed[4] with a write pulse far detuned
longtimes [7]withhighefficiency[8]andnegligiblenoise
from the resonance – the probability p for the emission
[8,9]. Lastbutnotleast,theyhavelargeinhomogeneous
of a Stokes photon in a given mode being enhanced by
absorption spectrum and narrow homogeneous lines
the use of large write intensities. However, in rare-earth
leading to a high temporal multimode capacity [10].
doped solids, where the dipole moments are typically
two to three orders of magnitude weaker, the required
2
intensities would be, at best, difficult to achieve. For α¯ = α π where F = ∆ /γ (see appendix). Note
F 4ln2 0
resonant write pulses, atoms are transfered into the that this shaping, which is at the heart of memories
p
excited state, leading to a higher p. But their energy based on atomic frequency comb [13], is known to pro-
difference, due to the inhomogeneous broadening,makes duce a photon echo-type of reemission in a well defined
themdistinguishable. Theycannomoreinterfere,which spatial mode at time 2π/∆ . Here, on the other hand,
0
makes the readout of the collective excitation inefficient we look at the spontaneous emission of a Stokes photon
[11]. at time t < 2π/∆ and we benefit from the efficiency
d 0
Weproposeasimplesolutiontothisproblem. Itconsists of the photon-echo-like remission for the readout of the
in shaping the spectral inhomogeneous broadening of atomic spin wave.
the optical transition so that the atomic dipoles rephase
at the readout step, leading to an efficient emission Spontaneous emission of Stokes photons Let us con-
of the anti-Stokes photon even when resonant write sider the Stokes field which propagates in the forward
pulses are used. Several shaping methods are available directionwiththe carrierfrequencyω . Its envelopisde-
s
to force the atomic dipoles to rephase. For example, scribed by the slowly varying quantum operator [15]
the inhomogeneous broadening could be shaped into a
narrowabsorptionline with a reversibleandcontrollable ˆ(z,t)= L eiωs(t−z/c) dωaˆ (t)eiωz/c. (1)
s ω
broadening [12]. In what follows, we focus on a shaping E 2πc
r Z
based on a comb-like structure composed with periodic
Under the dipole and rotating wave approximation, the
narrow peaks [13]. This provides a temporally multi-
interaction between the Stokes field and the medium is
plexed versionof a spontaneous Ramansource,similarly
governedby the Hamiltonian
to the proposal of Ref. [14] in atomic gases but without
the need for a cavity. Since spontaneous Raman based
L
protocols are well suited for entangling remote atomic Hint = ¯hg dωaˆωeiωzj/c e sj +h.c. (2)
− 2πc | ih |
ensembles in a heralded way [1] without the need for r j Z
X
ultra-narrowband pair sources, our proposal paves the
way to the implementation of the first elementary link g = ℘ ωs is the atom-field coupling constant with
2h¯ε0AL
of quantum repeaters with solid-state devices. ℘ theqdipole moment of the e s transition and A
−
the interaction section. The equations of motion for
Write step Let us start by a description of the write the Stokes field and for the atomic coherence σj (t) =
se
step. We consider a medium consisting of lambda atoms s ejeiωs(t−zj/c) are given by
| ih |
(as depicted in Fig. 1) initially prepared in the state
d d
g. The optical transition g-e is shaped into a frequency ( +c )ˆ(z,t)=igL δ(z z )σj (3)
combmadeofnarrowpeakswithacharacteristicwidthγ, dt dz Es − j se
j
separated by ∆ and spanning a large atomic frequency X
0 d
range Γ. The g-s transition is considered to be homoge- σj = i∆jσj igˆ(z ,t)σj . (4)
dt se − se− Es j z,s
neous. Aweak write laserpulse with the Rabifrequency
Ω (t), is sent through the medium to transfer a small ∆j = ωj ω is the frequency detuning and σj =
w es − s z,s
part of atoms in the excited state e in such a way that e ej s sj isapproximatedbyitsmeanvalueθ2e−α¯zj
it can spontaneously decay into the level s by emitting |inihwh|a−t|foilhlo|ws. Pluggingtheformalsolutionofeq0uation
Stokes photons. Taking the absorptioninto account, the (14) into equation (13), we obtain the expression of the
write pulse drives the atoms into Stokes field at z (see appendix)
|ψwi=jYN≥1(n1j(1− 21θ02e−α¯zj)|gji+eikwzjθ0e−nα¯jzj/2 |eji). Eˆs(z,t) =+iegα2¯LR0zθ02e−α¯ezα2¯′dRzzz′jEˆθs02(e0−,α¯tz)′dz′e−i∆jtσj (0).(5)
:=Gj :=Ej c se
j|Xzj<z
n ensures t|he norm{azlization},θ = |1 ds {zΩ (s)} 1
j 0 2 × w ≪ Weconsideredforsimplicity,thatthetransitionsg-eand
refers to the area of the write pulse at the entrance of
R s-ehavethesamedipolemoments. Sincethestateofthe
the crystal and k corresponds to its wave number. z
w j complete systemafter the write pulseis givenby Ψ =
is the position of the j-th atom within a medium of | wi
ψ ,0 where 0 is the vaccum for the electromagnetic
length L and N is the total number of atoms. α¯ is the | w i | i
field, the average number of Stokes photons emitted at
absorption per unit length of the atomic medium. It
time t in a mode of temporal duration √2π/Γ is given
is proportional to the ratio between the absorption per d
by
peak α and the finesse F of the comb, c.f. below. For
concreteness,weconsideranatomicdistributionmadeof √2πc
gaussian peaks with full width half maximum γ so that ΓL hΨw|Eˆs†(L,td)Eˆs(L,td)|Ψwi≈θ02(1−e−α¯L) (6)
3
for θ 1 (see appendix). This formula is a very depths (see appendix). Furthermore, since the readout
0
≪
useful and can easily be used in practice. Note of spontaneous Raman protocols is conditioned on the
first that the relative number of atoms transferred detection of a Stokes photon, the retrieved signal is not
into the excited state by the write pulse is given by affected by the non-unit coupling efficiency of an input
1 Lθ2e−α¯zNdz = θ02 (1 e−α¯L).Therefore,theformula photon into the memory (e.g. due to imperfect spectral
(N6)t0ells0usthaLttheaα¯vLerag−enumberofphotonsinamode filters, non-unit coupling into monomode fibers ...) (see
witRh a durationcorrespondingto the inverseofthe over- appendix). Contrary to the photon-pair source based
all spectrum, is merely the optical depth α¯L times the on rephased amplified spontaneous emission [16], our
relative number of atoms in the state e. In what follows, proposal provides highly correlated pairs even for large
we are interested in the regime where the optical depth optical depths where the retrieval efficiency is high.
α¯L is large to get high readout efficiencies but the write
pulse is weak to get a high signal-to-noise ratio for the Noise rate We now account for intrinsic noise. The
readout, c.f. below. In this case, the success probability conditional state (7) that we used to calculate the effi-
fortheemissionofaStokesphoton(6)reducestop θ2. ciency of the readout process corresponds to the ideal
≈ 0 case where a single Stokes photon has been emitted and
Readout efficiency We now calculate the efficiency of detected. However, many atoms prepared in the excited
the readoutprocess. At time τ after the detectionof the state by the write pulse, can emit Stokes photons in
Stokes photon, a read pulse resonant with the transition all the spatio-temporal modes. This unwanted emission
s-e and associated with the Rabi frequency dsΩ (s)= populates the state s so that after the interaction with
r
π goes through the atomic ensemble and exchanges the the readpulse,atomsoccupy the excitedstatee andcan
population of states s and e. For θ 1, tRhe resulting produce spontaneous noise in the anti-Stokes mode. To
0
atomic state is given by (see appendix≪) take this noise into account, we consider the worst case
where all the atoms prepared in the excited state by the
N write pulse decay on the e-s transition. Tracing over the
ψr =ζ eizj(kw−kr−ωs/c)e−α¯zj/2e−i∆jtd ej (7) Stokes photons, the atomic state after the interaction
| i | i×
j≥1 with the read pulse is well approximated by
X
Gℓe−iωgs(td+τ)|gℓi+Eℓeizℓkre−iωeℓs(td+τ)|sℓi N
ℓY6=j(cid:16) (cid:17) ̺n = ((1−θ02e−α¯zj)|gjihgj|+θ02e−α¯zj|ejihej|). (10)
j≥1
with ζ =(α¯L)21/(N(1 e−α¯L))12. Considera re-emission O
associated with an ant−i-Stokes mode ˆ (z,t) propagat- The noise is then deduced from
as
E
ing in the backward direction. Following the method
presented before (under the assumption that e ej √2πctr ˆ† (0,2π/∆+τ)ˆ (0,2π/∆+τ)ρ
g g j 1) we find at z =0 | ih | − ΓL Eas Eas n
| ih | ≈− (cid:0) θ2 (cid:1)
= 0(1 e−2α¯L)
ˆas(0,t)=e−α2¯Lˆas(L,t) 2 −
E E
+igcL e−α2¯zje−i∆j(t−(τ+td))σgje(τ +td) (8) wmhizeerde Fρn, t=he̺snig⊗na|0l-itho0-|n.oFisoerrlaatrigoeisenloowugerhbαoLunadneddobpyti-
Xzj
2 2
so that, at time 2π/∆+ τ where all the atoms are in signal-to-noise = . (11)
≥ θ2 p
phase, the efficiency of the readout process is 0
One can thus conclude that despite inhomogeneous
√2πc
Ψ ˆ† (0,2π/∆+τ)ˆ (0,2π/∆+τ) Ψ broadening inherent to solid-state systems, our protocol
ΓL h r|Eas Eas | ri achieves similar characteristics to the ones obtained in
=(1 e−αFL√4lπn2) e−2lnπ22F2. (9) cold atomic gases - the signal-to-noise can be very high
−
providedthat the number of Stokes photons per mode is
One sees that there is a tradeoffbetweenabsorptionand low.
dephasing. However,for large enough αL and optimized
F, the readout efficiency can be arbitrary close to Feasibility For concretness, we now discuss the
100%. Let us directly note some of the advantages experimental feasibility of spontaneous Raman pro-
of spontaneous Raman protocol over other schemes. cesses in rare-earth-doped materials. Pr:Y SiO is a
2 5
First, the proposal based on spontaneous Raman is very promising material for initial experiments, since
significantly more efficient than a memory where the excellent hyperfine coherence [17] and quantum memory
photon has first to be absorbed before being reemitted. efficiencies of order 70% [8] have already been demon-
It leads to a much higher efficiency for small optical strated. The main drawback of Praseodymium is the
4
small hyperfine separation(a few MHz) which limits the APPENDIX
number of peaks within the atomic comb and thus the
multimode capacity, c.f. below. If we consider the 3H
4
- 1D transition at 606 nm, one can realistically shape Atomic distribution For concreteness,we consideran
2
atomic spectral distribution shaped into a frequency
combs with individual peaks of FWHM γ 30kHz [18]
≈ comb made with gaussian peaks
over a spectral range of Γ 2MHz and an absorption
≈
per peak of order αL 10 [19] (the branching ratio is
approximated by 1). F≈or F=5, p ≈ 0.1 can be achieved Θ(∆):= ∆0 e−2∆Γ22 ∞ e−(∆+2jγ˜∆20)2. (12)
provided that the write pulse satisfy θ2 = 0.1 at the 2πγ˜Γ
0 j=−∞
entrance of the crystal. This would lead to a readout X
efficiency of 65% and a signal-to-noise ratio larger than Γ denotes the overall width, ∆ is the peak separation
0
10 under the assumption that the noise is dominated and γ˜ is the width of an individual peak.
by the spontaneous emission. Now consider a setup We are interested in the regime Γ ∆ γ˜ where
0
involving two Pr-doped solids located 1 km apart so a large number of well separated p≫eaks sp≫ans a large
that the corresponding Stokes modes are combined on spectral bandwidth. In this regime, one can check that
a beamsplitter at a central station. Further consider a d∆Θ(∆)=1.
fiber attenuation of 9 dB/km, corresponding to 606 nm Wedefinethefinesseofthecombastheratioofthepeak
photons[20]andassumeacouplingefficiencyintooptical Rseparationover the full width at half-maximum of a sin-
fibers of ηc = 50%. The average time to detect a Stokes gle peak γ, i.e.
photon after the beamsplitter and thus to entangle the
two crystals is T = 1 where η , η are the trans- ∆ 1 ∆
2rpηcηtηd t d F := 0 = 0.
mission and detection efficiencies and r is the repetition γ √8ln2 γ˜
rate. Assume η = 70%. For p = 0.05 where the fidelity
d
of the entanglement is = 1 3p(1 ηcηtηd) 0.85 Note that the Fourier transform of the atomic spectral
F − − ≈
[2], one finds T 0.1s for a repetition rate of r =1KHz distribution
≈
so that a few hours would be sufficient for performing a
full tomography. Note that the fiber lengths have to be Θ˜(t):= d∆Θ(∆)e−i∆t
actively stabilized to guarantee an interferometric phase
Z
stability on this time scale [2].
is also a series of gaussian peaks with individual peak
width1/Γ,separatedby2π/∆ andspanningtheoverall
0
Conclusion Our approach opens an avenue to-
temporal width 1/γ˜.
wards the heralded entanglement of remote solids.
Further note that we consider a perfect three-level
Beyond that, the present scheme might be useful for
system (without additional levels) and we assumed that
quantum communications since spontaneous Raman
the optical transitions g e and e s have the same
processes lead to the production of narrowband pairs − −
dipole moments. Therefore,the absorption of g e with
well suited for storage in atomic ensembles. Let us −
all the atoms in g is the same as the one associated to
finally emphasize that in our protocol, spin waves
e s if all the atoms are prepared in s.
created at different times, say t ,t ,t ,... are inde- −
d1 d2 d3
pendent and if the read pulse is sent at time τ after
Emission ofStokesphotons Wenowdetailthewayto
the first detection, these spin waves rephase at times
findthesolutionofequationsassociatedtothe dynamics
τ+2π/∆,τ+2π/∆ (t t ),τ+2π/∆ (t t ),....
− d2− d1 − d3− d1 of the Stokes field
The number of spin waves that can be stored is roughly
given by the number of peaks composing the comb d d
( +c )ˆ(z,t)=igL δ(z z )σj (13)
(see [13] and appendix) and can be merely increased dt dz Es − j se
by making use of wider range of the inhomogeneous Xj
d
broadening. In the framework of quantum repeaters, σj = i∆jσj igˆ(z ,t)σj . (14)
this temporal multiplexing has been shown to greatly dt se − se− Es j z,s
enhance the distribution rate of entanglement [21].
First, we plug the formal solution of the equation (14)
We thank T.Chaneliere,J.-L.Le Gouet,J.Minarand t
C. Simon for helpful comments and interesting discus- σsje(t)=−igσzj,s dt′e−i∆j(t−t′)Eˆs(zj,t′)+e−i∆jtσsje(0).
sions. We gratefully acknowledge support by the EU Z0 (15)
project Qurep and the Swiss NCCR Quantum Photon- intotheequation(13)andsinceweconsiderStokesmodes
ics. with a characteristic duration τ L/c, we can neglect
s
≫
the temporal retardation effects in the crystal, i.e. the
5
time derivative in the equation (13). This leads to Since initially, there is no excitation in the Stokes mode
Ψ = ψ ,0 , we obtain
w w
d gL | i | i
dzEˆs(z,t)=i c δ(z−zj)e−i∆jtσsje(0)+ ˆ(L,t ) Ψ =igL
Xj Es d | wi c
g2cL δ(z−zj)σzj,s tdt′e−i∆j(t−t′)Eˆs(zj,t′). (16) × eα2¯RzLjθ02e−α¯z′dz′e−i∆jtde−iωszj/cEj|sji
Xj Z0 Xj (cid:18)
We then replaceσj by its meanvalue calculatedonthe
z,s (G g +E e ) 0 . (20)
k k k k
tsitnauteou|sψwlimi,iti.e. σzj,s L≈Nθd02ze′−α¯∞zj adn∆dΘw(∆e)t,aNkebtehinegcothne- ×kY6=j | i | i | i
j → 0 L −∞ Notethateachtermofthesumhasonlyoneatominthe
total number of atoms and L the length of the medium.
P R R state s , so when taking the scalar product with the
The last term of the equation (16) reduces to j
| i
corresponding bra, only the term with the same atom in
g2N t sj willgiveanonzerocontribution. Underthe approx-
c θ02e−α¯zZ0 dt′Θ˜(t−t′) Eˆs(z,t′) ihma|tion |Ej|2 ≈θ02e−α¯zj, one gets
Ψ ˆ†(L,t )ˆ(L,t ) Ψ =
Sinceweareinterestedintheemissionprocesswhichhas h w|Es d Es d | wi
at typt′icoafl odrudreartioτsn.τCs,ownseidoenrliyngnetehdetroegciomnesidwehrevraeluτess of g2cL22 eα¯RzLjθ02e−α¯z′dz′θ02e−α¯zj. (21)
2π−/∆ ,onlythe centralpeakofΘ˜ contributesandΘ˜(t≪ Xj
0
−
t′) is well approximated by Finally,wereplacethesumoverj bytheintegralexpres-
sion. Since there is no term with a spectral dependence,
Θ˜(t t′) e−(t−t′)2Γ2/2. the integral over ∆ is carried out directly and gives one.
− ≈
This yields to the spatial density of photons
For Γτ >1, Θ˜(t t′) acts as a delta function
s
− 1
Ψ ˆ†(L,t )ˆ(L,t ) Ψ
Θ˜(t−t′)≈e−(t−t′)2Γ2/2 ≈ √Γ2πδ(t−t′) =Lhgc22Nw|EsLdzedα¯REzLsθ02e−dα¯z′|dz′wθi02e−α¯z
Z0
and the equation for the Stokes field reduces to
= g2N(eα¯R0Lθ02e−α¯z′dz′ 1)
d gL α¯c2 −
dzEˆs(z,t)=i c δ(z−zj)e−i∆jtσsje(0) = g2N(eθ02(1−e−α¯L) 1)
Xj α¯c2 −
+α2¯ θ02e−α¯zEˆs(z,t). (17) ≈ √2Γπcθ02(1−e−α¯L)
α¯ whichcorrespondstotheopticaldepthperunitlength, where, at the last line, we expanded the exponential to
is defined by thefirstorder. Consequently,thenumberofphotonsper
temporal mode of duration √2π/Γ is given by
√2πg2N
α¯ := . (18)
c Γ √2π c
Ψ ˆ†(L,t )ˆ(L,t ) Ψ θ2(1 e−α¯L).
Γ Lh w|Es d Es d | wi≈ 0 −
In the main text, we introduced the optical depth per (22)
unit length associated to the central peak α. It is given Note that in the general case, the number of pho-
by α:= g2N ∆0 so that α¯ = π α. tons emitted in a temporal mode √2π/Γ is given by
c γ˜Γ 4ln2F
It is then straightforward topcheck that the solution of eα¯R0Lhσz,sidz−1.Intheparticularcasewherealltheatoms
the equation (17) is given by are preparedin e, the averagenumber of Stokes photons
permode isgivenbyeα¯L 1.This agreeswiththeresult
ˆs(z,t) =eα2¯R0zθ02e−α¯z′dz′ˆs(0,t) (19) presented in Ref. [16]. −
E E
+igcL eα2¯Rzzjθ02e−α¯z′dz′e−i∆jtσsje(0). Fpuerrtwherirtenoattetetmhaptttihseginvuemnbbeyr of Stokes photons emitted
j|Xzj<z
2π c √2πΓ
Ψ ˆ†(L,t )ˆ(L,t ) Ψ θ2(1 e−α¯L)
To find the average number of Stokes photons per tem- ∆ Lh w|Es d Es d | wi≈ ∆ 0 −
0 0
poral mode, we first evaluate the ket ˆ(L,t ) Ψ . (23)
s d w
E | i
6
and is thus roughly the product of the number of peaks where σj = e ej g g j is approximated by σj
z,g | ih | −| ih | z,g ≈
Γ/∆ composing the atomic frequency comb by the 1inwhatfollows. Usingthemethodspresentedbefore,
0
−
success probability for the emission of a Stokes photon we find at z =0
per mode. It can thus be merely increased by making
use of wider range of the inhomogeneous broadening. ˆas(0,t)=e−α2¯Lˆas(L,t)
E E
gL
Atomic state prepared by the Stokes photon detec- +i c e−α2¯zje−i∆j(t−(τ+td))σgje(τ +td). (27)
tion Consider the successful event where a Stokes pho- Xzj
ton is detected at time t . The conditional atomic This solution allows one to evaluate the readout effi-
d
state ψ is obtained by calculating the evolution of ciency. Foraperfectspatialphasematching(allthewave
d
the initial state Ψ until the time t and by pro- vectorssumtozero)theket ˆ (0,t) Ψ isgivenby(up
w d as r
jecting the resulting state into 0 ˆ(L,t ). First, let to a global phase factor) E | i
s d
h |E
us directly note that the initial state can be written
as |Ψwi = Nj≥1 Gjσgjs(0)+Eje−iωszj/cσejs(0) |sji|0i igcLζ e−α¯zje−i∆j(t−τ)|gji×
whereσgjs(0)Q=|gj(cid:0)ihsj|.Sincethestate|sji|0iisa(cid:1)neigen- Xj
sintattiemaes.soFcuiartthedermtootrhe,etehigeeonpvearluateo0r,σijt smtaeyrseluyncahcqanugireeds Gℓe−iωgs(td+τ)|gℓi+Eℓeizℓkre−iωeℓs(td+τ)|sℓi |0i.
gs
the phaseterme−iωgstd. Theevolutionofσj is obtained ℓY6=j(cid:16) (cid:17)
es (28)
by plugging the field solution (19) back into the atomic
solution (15) and then taking the adjoint. For θ 1, Projecting on the corresponding bra, one can show that
0
≪
we get the spatial density of photons is well approximated by
N 1
ψd = ζ eizj(kw−ωs/c)e−α¯zj/2e−i∆jtd sj LhΨr|Eˆa†s(0,t)Eˆas(0,t)|Ψri≈
| i | i
j≥1
X g2L
× Gℓe−iωgstd|gℓi+Eℓe−iωeℓstd|eℓi c2 ζ2 e−α¯zje−i∆j(t−τ)Gj×
ℓY6=j(cid:16) (cid:17) Xj
 
with ζ = (α¯L)21 . Furthermore, at time τ after the e−α¯zℓei∆ℓ(t−τ)G∗ . (29)
(N(1−e−α¯L))12 ℓ!
ℓ
detection of the Stokes photon, a read pulse resonant X
with the transition s-e and associatedwith the Rabi fre- Eachsumisthenreplacedbyanintegralandinthelimit
quency dsΩ (s)=π goesthroughthe atomic ensemble where θ 1
r 0 ≪
and exchanges the population of states s and e. There-
fore, theRatomic state becomes e−α¯zje−i∆j(t−τ)Gj
≈
j
X
N
ψr =ζ eizj(kw−kr−ωs/c)e−α¯zj/2e−i∆jtd ej N Ldze−α¯z ∞ d∆Θ(∆)e−i∆(t−τ). (30)
| i | i× L
Xj≥1 Z0 Z−∞
Gℓe−iωgs(td+τ) gℓ +Eℓeizℓkre−iωeℓs(td+τ) sℓ . Thespatialintegraliscarriedoutdirectly,whilethespec-
| i | i
tral integral is related to the Fourier transform of the
ℓY6=j(cid:16) (cid:17)
atomic distribution. We obtain
We now have all the necessary ingredients to calculate
the efficiency of the readout process. 1 Ψ ˆ† (0,t)ˆ (0,t) Ψ
Lh r|Eas Eas | ri≈
Readoutefficiency Theoperatorcorrespondingtothe Γ (1 e−α¯L)Θ¯(t τ)2. (31)
envelop of the anti-Stokes field propagating in the back- √2πc − | − |
ward direction is given by
ByevaluatingtheFouriertransformofthechosenatomic
L distribution (12) for the first revival time, i.e. around
Eˆas(z,t)=r2πceiωas(t+z/c)Z dωaˆω(t)eiωz/c. (24) t−τ =2π/∆0, we end up with
Its evolution is given the equations of motion √2π c
Ψ ˆ† (0,2π/∆ +τ)ˆ (0,2π/∆ +τ) Ψ
d d Γ Lh r|Eas 0 Eas 0 | ri
(dt −cdz)Eˆas(z,t)=−igL δ(z−zj)σgje, (25) (1 e−α¯L)e−2lnπ22F2.
j ≈ −
X
d σj = i∆jσj +igˆ (z ,t)σj (26) The success probability to find an anti-Stokes photon
dt ge − ge Eas j z,g within a temporal mode of duration √2π/Γ centered at
7
2π/∆ +τ approachesoneforlargeenoughopticaldepth i.e. for large optical depth and optimized finesse
0
provided that the comb finesse is optimized.
Note that for a quantum memory based on atomic fre- 2
signal-to-noise .
quency comb, the efficiency has a similar expression but ≥ θ2
0
the term (1 e−α¯L) is squared because the photon has
−
firsttobeabsorbedbeforebeingreemitted. Forweakop- Note that in the case of a quantum memory, the pho-
tical depths, a spontaneous Raman scheme is thus more ton to be stored is subject to many defective manipu-
efficient than the corresponding quantum memory. For lations, e.g. to non-unit coupling efficiency into mono-
example, for αL = 0.1 and optimized finesses, the effi- mode fibers or to imperfect spectral filters. This reduces
ciencyofspontaneousRamanschemereaches1%whereas the signal and hence, strongly limits the signal-to-noise
it is limited to 0.1% for a quantum memory based on ratio. This is a significant drawback with respect to
atomic frequency comb. spontaneousRaman basedsources where the readoutef-
Further note that for an emission in the forward direc- ficiency is conditioned on the successful detection of a
tion, the readout efficiency is given by Stokes photon.
(α¯L)2e−α¯Le−2lnπ22F2 (32)
1 e−α¯L
−
andis limited to 65%by reabsorptioninsteadof54%for
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E [9] H. de Riedmatten, M. Afzelius, M.U. Staudt, C. Simon,
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(33)
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one, we get
J. Zimm-Justin eds. (North-Holland 1997).
1 [16] P.M. Ledingham et al.,Phys.Rev.A81,012301 (2010).
tr ˆ† (0,2π/∆ +τ)ˆ (0,2π/∆ +τ)ρ
L Eas 0 Eas 0 n [17] E. Fraval, M.J. Sellars, and J.J. Longdell, Phys. Rev.
Lett. 95, 030506 (2005).
(cid:0) g2L (cid:1)
= θ2e−2α¯zj [18] G. Hetet et al.,Phys. Rev.Lett.100, 023601 (2008).
c2 0 [19] A. Amari et al., Journal of Luminescence 130, 1579
j
X (2010), M. Sabooni et al., Phys. Rev. Lett. 105, 060501
and we conclude that the noise is bounded by θ02(1 (2010).
2 − [20] Note that this wavelength could be converted to tele-
e−2α¯L) by replacing the discrete sum by its integral ex-
com wavelengths, in order to profit from the optimal
pression and by taking the duration of the anti-Stokes
transmissionofopticalfibers,usinge.g.parametricdown-
mode into account. conversion with asingle-photon pumpbutastronglaser
Therefore, the signal-to-noise is bounded by stimulating into one of the two down-converted modes,
asreportedinRefs.H.Takesue,Phys.Rev.A82,013833
signal-to-noise≥ θ22((11−ee−−α2¯α¯LL))e−2lnπ22F2 [21] (C2.0S10im)o;nNe.tCaul.r,tzPhetysa.l.R,eOvp.tL.eetxtp.r9e8ss, 11980,5202309(290(0270)1.0).
0 −
8
[22] See e.g. N. Sangouard, C. Simon, M. Afzelius, and N. description of this effect.
Gisin, Phys. Rev. A 75, 032327 (2007) for a detailed