Quantum storage via refractive index control

0.19 MB·English

by  Alexey Kalachev

Quantum storage via refractive index control
Alexey Kalachev1,2 and Olga Kocharovskaya1
1Institute for Quantum Studies and Department of Physics,
Texas A&M University, College Station, TX 77843–4242, USA,
2
Zavoisky Physical-Technical Institute of the Russian Academy of Sciences, Sibirsky Trakt 10/7, Kazan, 420029, Russia
(Dated: January 5, 2011)
Off-resonant Raman interaction of a single-photon wave packet and a classical control field in
an atomic medium with controlled refractive index is investigated. It is shown that a continuous
change of refractive index during the interaction leads to the mapping of a single photon state to
a superposition of atomic collective excitations (spin waves) with different wave vectors and visa
versa. The suitability of refractive index control for developing multichannel quantum memories is
discussed and possible schemes of implementation are considered.
1
1 PACSnumbers: 42.50.Fx,42.50.Gy,32.80.Qk
0
2
n I. INTRODUCTION multiplexing is currently most demanded from the view
a point of fiber optical communication. Being combined
J withanyapproachtoquantumstoragementionedabove,
Duringthe pastdecadethe opticalquantummemories
4 therefractiveindexcontrolprovidesanadditionaldegree
have became one of the active areas of research in the
of freedom for multiplexing thereby improving capacity
field of quantum optics and quantum information (see
]
of a multimode quantum storage device or allowing op-
h the Reviews [1–4]). Such devices are considered as basic
p ingredient for scalable linear-optical quantum comput- eration in a multichannel regime. Such a multiplexing
- methodiscloselyconnectedwiththeangular[28]orholo-
ers and efficient quantum repeaters. For practical quan-
t
n tum information applications, it is necessary to develop graphic [31] ones since it also resorts to phase-matching
a conditions in an extended atomic ensemble, but it does
memorieswhichcouldstorequantumstatesoflightwith
u not exploit different spatial modes of the field. In effect,
close to 100% efficiency and fidelity, and provide long
q
the additionalmultiplexing capacityis basedonthe pos-
[ and controllable storagetimes or delay-bandwidth prod-
sibility to use frequency and wavelength of the field in a
ucts. In this respect significant experimental progress
1 storage material as independent parameters.
has been achieved in demonstration of optical quantum
v The paper is organized as follows. In Sec. II, we
storage using electromagnetically induced transparency
8
analyze the storage and retrieval of single-photon wave
5 [5–8], photon echo induced by controlled reversible in-
packets via refractive index control during off-resonant
6 homogeneous broadening [9–13] or by atomic-frequency
Raman interaction. In Sec. III, suitability of refractive
0 comb [14–20], and off-resonance Raman interaction [21–
. 23]. Optical quantum memories are usually assumed to indexcontrolfordevelopingmultichannelquantummem-
1
ories is discussed. In Sec. IV, we consider possible ways
0 storeandrecallopticalpulses,suchassingle-photonwave
of refractive index manipulation and some implementa-
1 packets,exploitinginhomogeneousbroadenedtransitions
1 or modulated control fields. In the present work we sug- tion issues.
v: gest one more possibility. By considering quantum stor-
i age based on off-resonant Raman interaction, we show
X
thatmanipulationofrefractiveindexinathree-levelreso- II. STORAGE AND RETRIEVAL OF
r nantmediumallowsonetostoreandrecallsingle-photon SINGLE-PHOTON WAVE PACKETS
a
wave packets without using inhomogeneous broadening
of the atomic transitions or manipulating the amplitude As a basic model we consider cavity-assisted quan-
of the control field. A single-photon wave packet may tum storage, which is motivated by the following rea-
be reversibly mapped to a superposition of atomic col- sons. First, enclosing an atomic ensemble in a cavity
lective excitations with different wave vectors, which is makes it possible to achieve high efficiency of quantum
analoguestothatoforthogonalsubradiantstatescreated storage with optically thin materials. This may be espe-
in an extended atomic ensemble [24]. cially useful for considered off-resonant Raman interac-
As wellas providinganinteresting possibility for stor- tion since the cross-section of the two-photon transition
age, the refractive index control may also be useful for is usually small. Second, there is no need for backward
optimizing multiplexing regimes of multimode quantum retrieval when optically thin materials are used, which
memories, development of which is important in the relieves one of having to perform phase conjugation of
prospect of both quantum communication [25, 26] and the atomic states used for storage.
computation[27]. Particularly,multimode memoriescan We consider a system of N 1 identical three-level
≫
significantly increase the quantum communication rate atomswhichareplacedina single-endedringcavityand
for short storage times. Different ways of multiplexing interact with a weak quantum field (single-photon wave
has been suggested [28–31], among which time domain packet) to be stored and with a strong classical control
2
is the Rabi frequency of the classical field, and g is the
Input Output coupling constant between the atoms and the quantized
field mode. The values of n and n are considered below
M1 c
asparameterschangingintime. Sincevariationsofthem
are supposed to be much less than 1, we leave such time
dependence only in phase factors and ignore it in the
M3 M2 factors Ω and g as functions of refractive indices.
Atoms In the Heisenberg picture, we define the following
slowly varying atomic operators: P = σj eiωt, S =
j 13 j
Control σj ei(ω−ωc)t,andcavityfieldamplitude =aeiωt. From
12 E
the input-output relations for the cavity field [32, 33] we
have
3

(t)=√2κ (t) (t), (4)
out in
E E −E
Input/Output Control where 2κ is the cavity decay rate and in ( out) is the
E E
,k ,k input(output)field(asingle-photonwave-packet). From
c c
the Heisenberg—Langevin equations, assuming that all
the populationis inthe groundstate initially andtaking
2 into account that the quantum field is weak, we find
1
P˙ = (γ +i∆)P +iΩS eikczj +ig eikzj, (5)
j P j j
− E
FIG. 1: (Color online) Schematic of a quantum memory de- S˙ = (γ +i∆ )S +iΩ∗P e−ikczj, (6)
j S S j j
−
vice (above) and atomic level structure (below). Mirrors M2
andM3areperfectlyreflectingforthesingle-photonfieldand ˙ = κ +ig Pje−ikzj +√2κ in. (7)
E − E E
fullytransmittingforthecontrolfield,M1isapartiallytrans- Xj
mittingmirror. Thedifferenceofwavevectorsk−kc ismod-
Hereγ andγ aretheratesofdephasing,∆=ω ω is
ulatedviarefractiveindexcontrolduringoff-resonantRaman P S 3
−
interaction. a one-photon detuning, and ∆S =ω2+ωc ω is a two-
−
photon detuning. We have not included the Langevin
noise atomic operators since they make no contribution
field (Fig. 1). The atoms have a Λ-type level structure, to normally ordered expectation values in consistence
thefieldsareRamanresonanttothelowest(spin)transi- with the approximation that almost all atoms remain in
tion,andthecavityisresonanttothequantumfield. We the ground state (see, e.g., [34] for discussion). Finally,
assume that the atoms are not moving being impurities in the Raman limit, when the single-photon detuning is
embedded in a solid state material. The interaction vol- sufficientlylarge,adiabaticallyeliminatingP inEqs. (5)
j
ume is supposed to have a large Fresnel number, which and (6), and going to the collective atomic operators
allows us to take advantage of one-dimensional approx-
1
imation. The Hamiltonian of the three-level system in S = S e−iqzj, (8)
q j
the dipole and rotating wave approximations is √N Xj
H =H0+V, (1) we obtain
where S˙ = γ′S +ig′√Nφ(k k q) , (9)
q q c
− − − E
N ˙ = κ +√2κ in
H =~ωa†a+ ~ω σj +~ω σj , (2) E − E E
0 2 22 3 33 +ig′∗√N φ(q+k k)S . (10)
Xj=1(cid:16) (cid:17) c− q
Xq
Here φ(q) = 1/N N exp(iqz ) is the diffraction func-
N j=1 j
V = ~ Ω(t)σj eikczj−iωct+gaσj eikzj +H.c. tion,γ′ =γ +γ PΩ2/∆2,g′ =gΩ∗/∆,andtheresulting
− Xj=1(cid:16) 32 31 (cid:17) frequencyshSift∆P′|=|∆S+ Ω2/∆hasbeencompensated
| |
(3) by tuning the coupling field frequency. The wave vec-
Here σj = m n are the atomic operators, n is torsq aremultiples of2π/L,whereListhe lengthofthe
mn | jih j| | ji
the nth state (n = 1,2,3) of jth atom with the energy atomic medium.
~ω (ω = 0 < ω < ω ), z is the position of the jth Thephasemismatchingfactorsφ(q),whichareusually
n 1 2 3 j
atom, a isthe photonannihilationoperatorinthe cavity ignored on the assumption that a single spatial mode of
mode, k = ω n /c and k = ωn/c are the wave vectors the spin coherence is excited and phase-matching is per-
c c c
of the classical and quantum fields, respectively, n and fect, are now considered. Suppose that we can manipu-
c
n are refractiveindices atthe frequencies ω and ω, Ω(t) late the difference q+k k by refractive index control
c c
−
3
withoutchangingfrequenciesandpropagationdirections achievedbyoff-resonantinteractionoftheatomicsystem
of the interacting fields. We discuss possible ways of im- withthe controlfieldwhenthe values ofn thatusedfor
c
plementation below. For now it is sufficient to consider storage are scanned again. Suppose, e.g., that the time
the case when one of the wave vectors, say k of the dependence of refractive index during the time interval
c
controlfield,ischangedlinearlyintime duringtheinter- [0,T], when (t) = 0, is reversed. In this case, instead
in
E
action so that q+k k = (ω /c)n˙ (t t ), where t is of Eq. (15) we have
c c c q q
− −
the moment when q+k k =0 for a given q. Then
c
− ˙(t)= κ (t)
φ[ (q+k k)]=e±iβ(t−tq)sinc[β(t t )], (11) E − E
± c− − q |g′|2N√2κ sinc[β(t+t )]F ( T,0, )e−γ′t.
q q in
where sinc(x) = sin(x)/x and β = (ωc/c)(L/2)n˙c. The − κ+Γ Xq − E
phasefactorse±iβtmaybecompensatedbylinearorsaw- (18)
tooth phase modulation of the control field. In such a
situation, the phase of the controlfield remains constant For slow-varying in this equation takes the form
E
at the point z =L/2 during the refractive index change.
As a result, Eqs. (9) and (10) take the form ˙(t)= κ (t) Γ√2κ ( t)e−2γ′t. (19)
in
E − E − κ+ΓE −
S˙ = γ′S +ig′√Neiβtqsinc(β(t t )) , (12)
q − q − q E Finally,afteradiabaticeliminationofthecavityfieldand
˙ = κ +√2κ using Eq. (4) we obtain
in
E − E E
+ig′∗√NXq e−iβtqsinc(β(t−tq))Sq. (13) Eout(t)=−κ2+ΓΓEin(−t)e−2γ′t. (20)
Now it is possible to consider storage and retrieval of The solution Eq. (20) is exactly the same as that in the
a single-photon wave packet. Let the atomic system in- cavity-assisted storage with inhomogeneous broadening
teracts with the quantum field during the time interval [35, 36]. The output field becomes time-reversed replica
[ T,0] with the initial condition S ( T) = 0, q. Then of the input field provided that the duration of wave
q
f−rom Eqs. (12) and (13) we have − ∀ packet is much smaller that the decay time 1/γ′, and
the efficiency of the storagefollowed by retrieval is max-
S (t)=ig′√NF ( T,t, )eiβtq−γ′t, (14) imum under impedance-matching condition κ = Γ. The
q q
− E
only difference is the collective absorption/emissionrate
˙(t)= κ (t)+√2κ (t)
E − E Ein Γ, which in our case takes the form
g′ 2N sinc[β(t t )]F ( T,t, )e−γ′t,
−| | − q q − E g2N Ω2 π
Xq Γ= | | . (21)
(15) ∆2 2β
Itmeansthatthetimeintervalδ =π/β,whichisactually
where F ( T,t, ) = t (τ)sinc[β(τ t )]eγ′τdτ. If
q − E −T E − q the time interval between two adjacent orthogonal spin
the cavity field varieRs slowly than δ =π/β, and γ′δ statescreateduponrefractiveindexcontrol,isanalogous
E ≪
1, then Eq. (15) takes the form to inhomogeneous life-time, i.e., reversal inhomogeneous
linewidth. We see that a single-photon wave packet can
˙ = κ +√2κ in Γ , (16) be effectively stored and reproduced via refractive index
E − E E − E
control in a three-level system without inhomogeneous
where Γ = g′ 2Nδ/2. Then the cavity field can be
| | broadening and without modulating the Rabi frequency
adiabatically eliminated provided that Γ + κ is much
of the control field during the interaction. It is also im-
greaterthanthebandwidthoftheinputfield,whichgives
portantthat the time dependence of the refractiveindex
=√2κ(κ+Γ)−1 , and from Eq. (14) we find
E Ein neednotbereversedduringtheretrieval. Ifthevaluesof
n during retrieval are orderedlike those during storage,
ig′√N√2κ c
S (0)= F ( T,0, )eiβtq Eq. (20) is replaced by
q q in
κ+Γ − E
= ig′√N√2κπ (t )e(iβ+γ′)tq. (17) out(t)= 2Γ in(t T)e−γ′T. (22)
in q E −κ+ΓE −
κ+Γ βE
Thus a single-photon wave packet may be reconstructed
Equation (17) describes the mapping of an input single-
without time reversal so that its temporal shape be not
photon wave packet to a superposition of collective exci-
deformed by the rephasing process.
tations (spin waves) with different wave vectors. If the
The total change of refractive index during storage or
rateofrefractiveindexchangeissufficientlylargesothat
retrieval is
δ issmallerthanthefastesttimescaleoftheinputpulse,
then the temporal shape (t) is imprinted on the am- T λ
plitude Sq(0) as a functionEionf wavevector q. Retrieval is ∆n=n˙cT =(cid:18)δ(cid:19)L, (23)
4
|E(t)|2, 1 III. MULTICHANNEL QUANTUM STORAGE
a.u. Input
0.8 Pulse
InthepreviousSection,wewereinterestedinthesitua-
tionwhenthebandwidthofthephotonislargerthanthe
0.6
inhomogeneous linewidth of the Raman transition. Now
we consider the opposite case, when the storage and re-
0.4
trieval are implemented, for example, by manipulating
the inhomogeneous broadening, which should be larger
0.2 thanthe photonbandwidth, andrefractiveindex control
isusedforrealizingmultichannelregimeofquantumstor-
−06 −4 −2 0 2 4 6 t/τp age. The idea is that different wave vectors of the spin
wavescorrespondto different channelsofstorageandre-
FIG. 2: (Color online) Storage and retrieval of a Gaussian trieval just as in the case of angular multiplexing. For
pulse for different values of δ = π/β. The black solid curve example, considerthe Raman echo memoryscheme with
is the input pulse of duration τp. Other curves represent the controlled reversible inhomogeneous broadening that is
outputfieldforthecases δ=2τp (bluecircles), δ=τp (green switched in time periodically [11]. The multichannel
dashed line), δ = τp/2 (red solid line). These curves were regime can be achieved by assigning different wave vec-
obtained by numerically solving Eqs. (12) and (13), treated
tors k to the control field (in our case — by refractive
as complex numberequations, with the condition κ=Γ and c
′ index control) during different dephasing/rephasing cir-
γ τp ≪1. Thesignofdkc/dtischangedatthemomentt=0.
cles. In a similar way, we can consider memory schemes
based on resonant interaction. Let the storage and re-
trievalbeimplementedusingatomicfrequencycomb[30],
whereλ=2πc/ω . NumericsshowthataGaussianpulse
c which dephases and then rephases after a time T, and
with the duration (FWHM) as short as 2δ can be stored
π-pulses transferring the optical coherence to/from the
and recalled with the efficiency 0.99 provided that γ′ is
spin coherence are used [15]. Then we can make differ-
small enough (Fig. 2). Therefore, taking T/δ 1 and
λ/L 10−5 we have ∆n 10−5, which may be∼consid- ent n for different π-pulses thereby creating orthogonal
∼ ∼ spinwavesondifferentstorage/retrievalcyclesorwe can
ered as the minimum refractive index increment needed
change refractive index for the weak field to be stored
for storage of a single pulse under typical experimental
so that δ = T, which leads to the same result. In any
conditions. The ratio between the total accessible range
case, such multichannel regimes enable one to process
ofrefractiveindexchangeandthisminimumvaluedeter-
new quantumstateswhile preservingthose storedbefore
mines thenumberofpulsescanbe storedinaseries,i.e.,
and provides access to all states kept in store in any or-
the mode capacity of quantum memory.
der. Itisalsoimportantthatthephasemodulationofthe
TheimpedancematchingconditionΓ=κmaybewrit- control field, which is required to store and recall pulses
ten in the following form: without resort to inhomogeneous broadening, becomes
needless in the case of such channel division. Regard-
Cγ′δ/2=1, (24) ing the impedance matching condition, it takes the form
C = 1 since δ/2 is replaced by inhomogeneous life-time
whereC =g2N/κγ = g′ 2N/κγ′ isthecooperativitypa- T2∗ =1/γ′.
rameter, which could |als|o be expressed in terms of the Apart from the finite storage time due to irreversible
relaxation,the maximumnumberofchannelsisalsolim-
cavity finesse and resonantabsorptioncoefficient α as
C = αL /2πF, provided that the resonant medium fills ited by the signal-to-noise ratio. The latter can be esti-
F mated in the following simple way. Consider two chan-
the cavity. Thus to store and recall pulses broader than
the Raman linewidth γ′, we need to increase C above nels corresponding to wave vectors with the difference
1 appropriately. In general case, γ′ consists of homoge- ∆km = 2πm/L, where m is an integer. During retrieval
neous and inhomogeneous contributions. If we do noth- fromoneofthemtheprobabilityofretrievalfromanother
isdeterminedbyphasemismatchingandproportionalto
ingwiththelatter,thedelay-bandwidthproductofquan-
tum storage (or multimode capacity) provesto be of the Pm =[sinc(∆kmL/2)]2,whichmaybe equalto zeroonly
formonochromaticfield. Nowitisnecessarytotakeinto
order of C. It may be increased by choosing materials
with a narrowerline-width or shorter pulses with appro- account the bandwidth δk = δωn/c of the retrieved sig-
nals(herenistheaveragevalueofrefractiveindexduring
priate increasing the cooperative parameter. But if we
are able to reverse the inhomogeneous broadening, the storage or retrieval). If δk k, then
≪
delay-bandwidth product may be increased significantly
depending on a residual noncontrollable broadening, say 1 δk/2 Lx 2 1 δωL n 2
P dx= . (25)
γ′ , of the Raman transition. As a result, the mode m ≈ δk Z (cid:20)2πm(cid:21) 12(cid:20) ω λm(cid:21)
hom −δk/2
capacityofquantumstoragemaybe C times largerthan
thatachievablewithoutrefractiveindexcontrol,whichis We see that the noise from another channel is quadrat-
determined by the ratio γ′/γ′ . ically proportional to the bandwidth of the photons δω
hom
5
and length of the sample L. As an example, let L/λ = no amplification. One way of avoiding the gain is to
105, ω/2π=2 1014 Hz, and n=2. If δω/2π 50MHz, use excited state absorption [47]. Finally, we would like
then P . 10·−4 for m 1 so that 100 cha∼nnels pro- to note that some possibilities of refractive index con-
m
≥
vide total signal-to-noise ratio of the order of 100. In trol are provided even by a frequency shift of an absorp-
ordertomaintainthisratioforbroadersignals,thevalue tion structure relative to the Λ-type one. At cryogenic
of m should be increased proportionally to the band- temperaturesopticaltransitionsofimpurity crystalsand
width,whichmeansincreasingrefractiveindexdifference especially of rare-earth ion-doped ones have very nar-
between adjacent channels. rowhomogeneouslinessothatintheneighborhoodofan
inhomogeneous profile there might be rather strong fre-
quencydependenceofrefractiveindexandyetverysmall
IV. REFRACTIVE INDEX CONTROL absorption. In this respect, spectral hole-burning tech-
niques can be very useful for preparing inhomogeneous
Letus discusspossiblewaysofrefractiveindex manip- profileswithsharpedgesaswellasforcreatingΛ-systems
ulationthataresuitedtoquantumstoragedevices. First, onanonabsorbingbackground. Amoredetailedanalysis
ifadopednonlinearcrystalisusedasastoragemedium, of such an approach, which invokes specific information
we can take advantage of the linear electro-optic effect. about resonant materials, will be presented elsewhere.
For example, the maximum value of the index change in
LiNbO ,whichislimitedbythebreakdownelectricfield,
3
is of the order of 10−3. Although attaining this value by
V. CONCLUSION
applyingamoderatevoltageispossibleonlyforawaveg-
uide configuration, the doped nonlinear materials, par-
ticularly LiNbO :Er3+, hold promise in quantum stor- It is shown that single-photon wave packets can be
3
age applications, and therefore traditional electro-optic stored and recalled in a resonant three-level medium by
techniques of refractive index manipulation might be of means of refractive index controlwithout recourse to in-
helpful. Second, the resonant enhancement of the re- homogeneous broadening and modulating the amplitude
fractive index with vanishing absorption can be realized of control fields. Such a scheme for quantum storage
via quantum interference effects [37–42]. Although the can be combined with other techniques to increase the
maximum enhancement demonstrated experimentally in mode capacity of quantum memories or to develop mul-
a gaseous medium is 10−4 [43, 44], there is good reason tichannel storage devices. Although implementation of
toexpectmuchlargervalues,suchas10−2,forsolidstate refractiveindexcontrolisstillachallengingexperimental
materials [45, 46]. It should be noted that manipulating problem, we hope that significant progress can be made
refractive index either on the frequency of the quantum in this field by taking advantage of solid-state materials,
field to be stored or on the frequency of the strong con- which are also promising for quantum storage applica-
trol field should not only guarantee small losses but also tions.
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