A non-adiabatic approach to entanglement distribution over long distances

0.65 MB·English

by  Mohsen Razavi

APS/123-QED
A non-adiabatic approach to entanglement distribution over long distances
Mohsen Razavi∗ and Jeffrey H. Shapiro
Research Laboratory of Electronics
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139 USA
(Dated: February 1, 2008)
Entanglementdistributionbetweentrapped-atomquantummemories,viz. singleatomsinoptical
cavities, is addressed. In most scenarios, the rate of entanglement distribution depends on the
efficiencywithwhichthestateoftravelingsinglephotonscanbetransferredtotrappedatoms. This
loading efficiency is analytically studied for two-level, V-level, Λ-level, and double-Λ-level atomic
configurations by means of a system-reservoir approach. An off-resonant non-adiabatic approach
to loading Λ-level trapped-atom memories is proposed, and the ensuing trade-offs between the
atom-light coupling rate and input photon bandwidth for achieving a high loading probability are
7
identified. The non-adiabatic approach allows a broad class of optical sources to be used, and
0
in some cases it provides a higher system throughput than what can be achieved by adiabatic
0
loading mechanisms. The analysis is extendedto thecase of two double-Λtrapped-atom memories
2
illuminated by a polarization-entangled biphoton.

n
a PACSnumbers: 03.67.Hk,03.67.Mn,42.50.Ct,42.50.Pq
J

8
I. INTRODUCTION sired state. In this paper, we study the loading problem

2 in the context of trapped-atom quantum memories, viz.
v single atoms trapped in high-Q optical cavities, and find
The future of quantum information science—which
7 analytical solutions for the loading probability in differ-
1 holdsthepromiseofrevolutionaryimprovementsincom-
ent scenarios in which either a single photon or a pair
0 putation [1, 2], secure communication [3], and precision
of entangled photons are driving the quantum memory
1 measurement [4]—depends on our ability to develop its
1 demanding technology. Some of the required building units. That analysis results in a novel non-adiabatic ap-
6 proachtoloadingquantummemorieswithΛanddouble-
blocks have already been developed, e.g., first genera-
0 Λ atomic configurations. We also investigate the effect
tion quantumkey distributionsystems are nowcommer-
/ of various system parameters on the loading probability,
h ciallyavailable[5]. Withrecentadvancesinsingle-photon
p technology—the development of high-flux sources of en- including the atom-light coupling rate, the input light
- bandwidth,thecavity/atomicdecayrate,andtimingoff-
tangledphotons[6,7]aswellasefficient(photon-number
t
n resolving)single-photondetectors[8,9,10]—itisforesee- sets. Atomic-ensemble memories will not be explicitly
a addressed in this paper, however, most of the analytical
ablethatthenextgenerationofsecurelong-distancecom-
u resultsthatweobtainaredirectlyapplicabletosuchsys-
munication systems will be within reach in a few years.
q
tems. We elaborate more on this issue in the discussion
: Thekeyelement,whichisstillmissingfromsuchsystems,
v section.
isareliablequantummemorythatcanstoreandmanipu-
i
X latequantuminformation. Whereastrapped-ionsystems
Entanglement is a quantum resource by which two
arepromisingchoicesforquantumprocessing,thelackof
r parties share a joint state that cannot be written in
a an efficient optical interface hinders their being used in
a tensor-product form. This state provides a stronger-
long-distance quantum communication systems. Some
than-classical correlation between two quantum systems
neutralalkaliatoms,however,e.g. rubidiumandcesium,
by which one can perfectly infer quantum measurement
are appropriate choices for capturing the states of single
results made on one system by observing the results of
photons and storing them in their metastable hyperfine
quantum measurements made on the other. This quan-
levels. In order to enhance the interaction of single pho-
tum correlation is behind various applications in quan-
tons and atomic systems, we have to either trap single
tum communication [21] and distributed quantum pro-
atoms in high-finesse optical cavities [11, 12, 13, 14] or
cessing [22]. In teleportation, entanglement serves as a
use ensembles of atoms [15, 16, 17, 18, 19, 20]. In ei-
quantum wire. Once established, it can be used for one-
ther case, it is an interesting problem to quantify how
timecommunicationbetweenthetransmitterandthere-
efficiently we can transfer the state of a single photon
ceiver. In distributed quantumprocessors,entanglement
to an atomic quantum memory, or, in other words, how
serves as a computational resource. The main trick in
efficiently we can load a quantum memory with a de-
both of these applications is that we have connected our
parties,in advance of any subsequent actions,via entan-
glement. Thisway,weneednotworryaboutthe channel
∗Electronic address: [email protected]; Now with the Institute loss or physical transportation of a qubit once we have
established entanglement. We ought to worry, however,
for Quantum Computing, University of Waterloo, Waterloo, ON,
CanadaN2L3G1. about how to generate, distribute, and maintain entan-
2
(a) Moreover, for trapped-atom quantum memories, there
Quantum Entanglement Quantum are nondestructive loading verification techniques that
Memory Unit Source Memory Unit allow us to detect photon loss [11]. Hence, throughout
entangled (b) entangled the paper, we assume that the system is lossless,and we
focus on the interaction between the source output and
Quantum Entanglement Quantum the memory modules.
Memory Unit Swapping Memory Unit
The interaction between a trapped atom and light is
(c) governed by several factors, among them are the input
Quantum Quantum photon bandwidth, the cavity decay rate, and the atom-
Memory Unit Entanglement Entanglement Memory Unit
lightcouplingrate. Inordertorealizeastrongatom-light
generation conversion
interaction,thephotonsmusthavebandwidthscompara-
FIG. 1: (Color online) Three architectures for entanglement bletoatomiclinewidths,whicharetypicallyontheorder
distribution: (a) To-the-memory architecture, in which two of tens of MHz. Higher optical bandwidths may afford
entangled photons, generated at the source, carry the entan- increasedsourcegenerationrates—asisthecaseforspon-
gled state tothequantummemory unitsandload themwith taneousparametricdownconversionsourceswhoseband-
this state. (b) From-the-memory architecture, in which, by
widths are on the order of THz [27, 28, 29, 30]—and,
the use of a common source, each memory generates a fly-
hence, increased total throughput. However, if higher
ing qubit entangled with itself. Entanglement swapping is
source bandwidth comes with lower spectral brightness,
then accomplished by means of a Bell-state measurement on
throughput may be decreased, owing to a decrease in
the photons, at the midpoint, which leaves the memories in
the loading probability. It may be possible to main-
an entangled state. (c) Memory-to-memory configuration, in
which a flying qubit entangled with one of the memories is tain a high loading probability by employing a higher
generated that then propagates to and loads the other mem- atom-light coupling rate. This last approach requires a
ory,i.e., transfers its state to that memory. more demanding implementation, however, because, for
trapped-atom quantum memories, the coupling rate is
increased by reducing the cavity length. All these is-
glementoverlongdistances,i.e.,distancesoverwhichour sues necessitate developing a quantitative loading analy-
physical system undergoes decoherence. sis that quantifies the interplay between the input pho-
There are three commonly suggested approaches for tons’ bandwidths and the quantum memory parameters,
long-distanceentanglementdistribution, all ofwhich use i.e.,cavitydecayrateandatom-lightcouplingrate. This
photonsforthetransmissionofquantuminformationand paperseekstoanswerthesequestions,whichareofprac-
atomicsystemsforitsstorage. Theyareshownschemat- ticalimportancein designingquantummemory systems,
ically in Fig. 1. In the first approach, we produce a pair by quantifying the trade-offs inherent in adiabatic and
of entangled photons at an optical source, and let the non-adiabatic loading mechanisms.
photons travel to and be stored in the quantum mem- Another important issue in dealing with the interac-
ory units [23]. In the second approach, we use entan- tion of trapped atoms and light is decoherence. There
glement swapping [24] to distribute the entanglement. are a variety of mechanisms by which our system may
Wefirstgenerateentanglementbetweenastandingqubit decohere. Decayofphotons outofthe cavityandatomic
and a flying qubit at each end of Fig. 1(b). We then let spontaneousemissionarethetwofactorsthatwillbecon-
the photons propagate to the midpoint between memo- sideredinouranalysis. Thelattereffectcanbealleviated
ries where a Bell-state measurement apparatus destroys byusingoff-resonanttransitions,whichwillbeemployed
them,leavingthequantummemoryunitsinanentangled in many scenariosin this paper. There are other sources
state [15, 25]. The last architecture, shown in Fig. 1(c), of decoherence—some related to the employed trapping
is a hybrid of the previous ones. In this approach, we andcoolingschemes[31],andsomeduetotheoversimpli-
generateasinglephotonentangledwithoneofthequan- fied picture that we use for atomic systems [32]—which
tummemories,andletthisphotonpropagatetoandload will not be addressed in our paper.
the other memory, i.e., transfer its state to that memory The rest of this paper is organized as follows. In Sec-
[19, 26]. tion II, we study the loading process for single trapped-
In all of these entanglement distribution schemes, a atom quantum memories that are illuminated by single
figure ofmerit for system performanceis its throughput, photons. The methodology that we employ in this sec-
viz. how often we obtain an entangled pair of quantum tion will be applicable to systems that use the memory-
memories. Throughput is a function of different system to-memory configuration for entanglement distribution.
parameters. For instance, in the to-the-memory archi- Westartwiththebasiccaseofatwo-levelatom,andthen
tecture, Fig. 1(a), it depends on the source generation extend our results to Λ- and double-Λ-level atoms. We
rate of entangled photons, the path and coupling losses, proposeanon-adiabaticloadingmechanismforthelatter
and the loading efficiency. System loss has the unavoid- cases, and compare it to the previously proposed adia-
able effect of reducing the throughput. Loss, however, batic loading techniques [26, 33]. In Section III we treat
does not affect the source generation rate or the nature an example of the to-the-memoryconfigurationin which
ofinteractionbetweenasinglephotonandasingleatom. the source output is a polarization-entangled biphoton,
3
e and κ can both be functions of time. Here, we assume
T
(cid:37) that κ is a constant. Suppose that this trapped-atom
g (cid:88)a module, with the atom initially in its ground state and
(cid:88)
t 0 nophotoninthecavity,isilluminatedbyasinglephoton,
as shown in Fig. 2. This driving source can be modeled
(cid:168) d(cid:88)(cid:71)((cid:88)) 1(cid:88) = (cid:168) dt(cid:39)(t) 1t (cid:76) g byasetofannihilationoperatorsaˆ ,eachcorresponding
ω
to a different temporal (spectral) mode of frequency ω,
FIG. 2: (Color online) A single two-level atom trapped in that satisfy [aˆ ,aˆ† ] = δ(ω ω′) [36]. We assume that
a single-ended high-finesse optical cavity. A single photon thedrivingfielωdisωi′nitiallyin−the followingsingle-photon
illuminates the cavity at the proper frequency to excite the
state
atom to its upperstate.
ψ = dωφ(ω) 1
0 ω
| i Z | i
and the trapped atoms have double-Λ configurations.
This is a close approximation to the architecture pro- = dtΦ(t) 1t , (1)
Z | i
posedbyresearchersfromthe MassachusettsInstitute of
Technology and Northwestern university (termed MIT- where dω φ(ω)2 = 1, and 1 = aˆ† 0 is the
NU hereafter) [11, 23, 34]. Thus the results presented | | | ωi ω| iR
multi-mRode state representing one photon at frequency
here comprise the first loading analysis of the MIT-NU
ω and the vacuum state for all other modes. Here,
system that accounts for the intracavity atoms. 0 is the reservoir’s multi-mode vacuum state. We
R
| i
canequivalently describe the initial statein the time do-
main by introducing the temporal pulse shape Φ(t) =
II. TRAPPED-ATOM QUANTUM MEMORIES dωe−iωtφ(ω)/√2π associated with the incoming pho-
DRIVEN BY SINGLE PHOTONS
tRon, and single-photon states 1 = dωeiωt 1 /√2π
t ω
| i | i
associated with time t, as shown in ERq. (1). Here and
In this section, we evaluate the loading probability for
elsewhere in this paper, we assume that the initial time
trapped-atom quantum memories, viz. single atoms in
isbeforeanypossibleinteractionbetweenthe sourceand
high-finesse optical cavities, driven by single photons.
the memory. For simplicity, and without loss of general-
The loading protocolis successful if we transfer the pho-
ity, we assume that this initial time is 0. We will relax
ton state to the corresponding atomic levels. We do
this assumption after obtaining our final results.
not deal with the complexity associated with trapping
The Hamiltonian for the abovesystem is givenby [37]
neutral atoms, and assume that the single atoms can be
trappedatafixedpointinthecavityforatimesufficient Hˆ =Hˆ +~ω σˆ +~g(ˆb†σˆ +ˆbσˆ ), (2)
for loading to occur [12]. Throughout the paper, opti- a cc a ee ge eg
cal cavities are assumed to be lossless and single ended.
where σˆ = i j , i,j g,e , and
ij
The external photon illuminates the cavity through its | ih | ∈{ }
partially reflecting mirror in a specific spatialand polar-
ization mode matched to the cavity’s mode of interest. Hˆcc =~Z dωωaˆ†ωaˆω +~ω0ˆb†ˆb+~ΓZ dω(aˆ†ωˆb+ˆb†aˆω),
We assume that the incoming light has such a narrow
(3)
bandwidth that it only interacts with a single longitudi-
with Γ κ/π being the coupling constant that con-
nal mode of the cavity. To model the temporal content ≡
nects the epxternal world to the cavity [35]. Because of
of the incoming photon, we consider a reservoir of har-
the source’s narrow bandwidth, we can and will assume
monicoscillatorscorrespondingtoacontinuumofmodes,
that all integrals in Eq. (3) run from to + .
assuggestedin[35]. Thisisalsoinaccordwiththequan- −∞ ∞
Neglecting for now the decoherence mechanisms that
tum representationof traveling-wavelight [36]. We then
influencethetimeevolution,thequantumsystemconsist-
employ a system-reservoirapproachto analyze the load-
ing of the cavity, the atom, and the reservoir is closed,
ing dynamics.
with exactly one excitation at any time t. A general
quantum state of the system at time t can thus be writ-
ten as
A. Two-level Trapped Atoms
ψ(t) = dωα (t)1 0 g +e−iω0tβ(t)0 1 g
ω ω b R b
Suppose that there is a single two-level atom, with | i Z | i| i | i | i | i | i
excited state e and ground state g , inside a single- +e−iω0tc (t)0 0 e , (4)
| i | i e | iR| ib| i
ended high-Q cavity with decay rate κ. We assume that
thefrequency-ω atomictransitionbetween e and g is where k representsthek-photonFockstateofthe cav-
a | i | i | ib
coupled, at coupling rate g, to the cavity field operator ity mode.
ˆb of frequency ω . This implies a detuning ∆=ω ω , The goalofthis section is to provideanalyticalresults
0 0 a
−
which is assumed to be much less than ω . In general, g for the memory loading probability—the probability of
0
4
absorbing the external photon by the trapped atom— Note that Eqs. (6) and (7), with initial values c (0)=
e
c (t)2. The atom in our system is connected to the ex- β(0)=0,arelinearinresponseto theinput pulseshape.
e
| |
ternalsinglephotonviathecavitymode. Theinteraction For instance, assuming that Φ (t) = Φ (t)+Φ (t), β(t)
b 1 2
of the reservoir and the cavity mode can be analyzed by will be equal to β (t)+β (t), where β (t), for i = 1,2,
1 2 i
applying the Schr¨odinger equation, i~ψ˙(t) = Hˆ ψ(t) , is given by Eq. (8) with Φ (t) replaced by Φ (t). Simi-
a b i
| i | i
to Eq. (4), to obtain larly, we can think of the initial state in Eq. (1) as a su-
perposition of its infinitesimal constituents dωφ(ω)1 ,
ω
α˙ω(t)= i(ωαω(t)+Γe−iω0tβ(t)), (5a) to which, from Eq.(9), we can associate slowly-var|yinig
−
β˙(t)= iΓeiω0t dωαω(t). (5b) probability amplitudes ce,ω(t) given by
−
R
c (t)dω =(g κ/π/ξ)
Solvingforα (t)inEq.(5a),andthenpluggingtheresult e,ω
ω ×
into Eq. (5b), we obtain tdτdωφ(ω)ep−i(ω−ω0)τ(e−κ+(t−τ) e−κ−(t−τ)).(11)
0 −
R
β˙(t)= igce(t) i√2κΦb(t) κβ(t), (6) We then have ce(t)= dωce,ω(t) as obtained in Eq. (9).
− − −
We use this result lateRr in Section III.
where Φb(t) = eiω0tΦ(t) is the baseband input pulse Thesystem-reservoircalculationgivesamorecompact
shape provided that the center frequency of the incom- form for the loading probability, c (t)2, than what we
e
| |
ing light is ω . The second term on the right-hand side obtained in [38] via the Heisenberg-Langevin equations
0
of the above equation accounts for the effect of the in- [35] for the case of on-resonance illumination at γ = 0.
put photon. The last term accounts for the decay of the The two results are nevertheless equivalent, and their
photonoutofthe cavity,andin essence,is similarto the equivalencecanbe verifiedbyatediousalgebraicmanip-
decay term in the Weisskopf-Wigner theory of sponta- ulation that we shall omit.
neous emission [37], in which a two-level system initially
in its excited state interacts with a reservoir initially in
its vacuum state. In this way, we can incorporate the 1. Numerical results
overall spontaneous decay, with rate γ, from the excited
state of the atom to the modes other than the cavity’s Thereareseveralphysicalparametersthatareofprac-
mode of interest, and thus obtain tical importance in loading a trapped-atom quantum
memory: cavity decay rate, spontaneous decay rate,
c˙ (t)=i∆c (t) igβ(t) γc (t), (7)
e e e atom-light coupling rate, and input bandwidth. How-
− −
ever,fromEq.(9), itcanbeseenthatthe loadingproba-
wherethefirsttwotermsontheright-handsideofEq.(7)
bilityisgovernedbytheratiosoftheseparameters. Thus,
represent the coherent evolution of c (t) in accord with
e for ∆=0, Eq. (9) can be written in terms of the dimen-
the Schr¨odinger equation i~ψ˙(t) =Hˆ ψ(t) .
a sionless parameters κT, g/κ, γ/g, and t/T, as follows
| i | i
Equations (6) and (7) model the loading dynamics of
a two-level trapped-atom memory illuminated by a sin- √2κT t/T
c (t) = duΦ′(u)
gle photon with pulse shape Φ(t). For the general case e ξ/g Z b ×
0
of time-dependent system parameters, these equations
can be solved numerically. For the special case of time- e−κT(κ+/κ)(t/T−u) e−κT(κ−/κ)(t/T−u) ,(12)
h − i
independent g, κ, γ, and ∆, we can solve the above
where T is the effective width of Φ (t), and Φ (t) =
equations analytically using the Laplace-transformtech- b b
niques, which yield Φ′b(t/T)/√T, i.e., Φ′b(u) is a compressed/stretched ver-
sion of Φ (t), which has unity width, in its normal-
b
i√2κ t ized time coordinate, and has been normalizedto satisfy
β(t)= − ξ Z0 dτΦb(τ)(κ′+e−κ+(t−τ)−κ′−e−κ−(t−τ)) du|Φ′b(u)|2 =1.
(8) R For a constant value of g, what we expect from the
and atom-photon interaction is a damped Rabi oscillation.
The incoming photon is initially transferred to the cav-
g√2κ t ity mode, which interacts with the atom and drives a
c (t)= dτΦ (τ)(e−κ+(t−τ) e−κ−(t−τ)), (9)
e ξ Z b − g -to-e transition. That will be followed by the re-
0 | i | i
verse transition which releases a photon into the cav-
where ity. These oscillations continue until the photon leaks
outofthecavityortheatomspontaneouslyemitsapho-
κ =(κ+γ′ ξ)/2,κ′ =κ γ′,
± ± ± ±− ton into a non-cavity mode. Figure 3 shows the loading
ξ = (κ γ′)2 4g2,γ′ =γ i∆. (10) probability for the two-level atom of Fig. 2 in the ideal
p − − − case of γ = ∆ = 0 for a hyperbolic secant pulse shape
Thelowerlimitzerointhe aboveintegralsrepresentsthe Φ′(u)=√2sech[4(u 1)]. Inthisfigure,wehavefixedthe
b −
initial time of interaction. It can be changed to accom- valueofκT to2andvariedthevalueofg/κ. Counterintu-
modate the input pulses that start before t=0 as well. itively, it can be seen that increasing the coupling rate g
5
(cid:76)g =1 (cid:39)b(cid:97)(u)1.4 exspe.c rhising
y0.8 1.5 0.8
t 1.2
ili ty exp.
bab0.6 abili 0.6 1.0 decaying
o b
Pr0.4 g =0.75 Pro 0.8 rectangular
ing 2 (cid:76) ng 0.4 g =1 0.6
d i
oa0.2 oad (cid:76) 0.4
L L 0.2 (cid:76)T =2
(cid:76)T =2 0.2
1 2 3 4
t /T 0 1 t2/T 3 4 (cid:14)1 0 u 1 2
FIG.3: (Coloronline)Loadingprobabilityversusnormalized
FIG. 5: (Color online) Left: loading probability at κT = 2,
time for κT =2 and a hyperbolic secant pulse shape at γ =
g/κ=1, and γ =∆=0 for various pulse shapes. Right: the
∆ = 0. Upon arrival of the photon, there is a peak in the
baseband pulse shapes. In order of their maximum loading
loadingprobability,whichoccursatthetimethatthephoton
probabilitiestheyare: thehyperbolicsecant(withthehighest
is most likely to haveabsorbed bythe atom. This maximum
peak), rectangular, exponentially-rising, and exponentially-
loading probability is a function of g/κ, and there exists an
decaying pulseshapes.
optimumvalueforthisparameterthatmaximizesthechance
of loading.
ticeable in this figure. First, for a fixed κ, it can be seen
3 thattherequiredvalueforthecouplingratewilldecrease
(g/(cid:76))
optimum
2.5 (cid:72)/(cid:76)=0.0 as we increase the pulse width, or equivalently, when
(cid:72)/(cid:76)=0.1 our source’s bandwidth decreases. The decrease that we
(cid:72)/(cid:76)=0.2 achieve,however,by doing so is exponentially smaller as
2
wegoto largervaluesofκT. Forinstance(notshownon
the graph),to get (g/κ) =0.1, we need κT to be
1.5 optimum
approximately100. Itcanalsobeseenthattheoptimum
max(Loading Prob)
1 coupling rate will not change significantly in the pres-
ence of decoherence. Nonzero spontaneous decay rates,
0.5 however,willreducethemaximumloadingprobability—
which could be above 90% otherwise—as shown in the
0
0 2 4 6 8 10 lower part of Fig. 4. Longer pulses, which require longer
(cid:76)T
loadingtimes,aremoreexposedtothedecayprocess,and
so,theirmaximumloadingprobabilitiesarelower. Spon-
FIG. 4: (Color online) The optimum coupling rate, and the
taneous decay can be alleviated by using large enough
maximumloadingprobabilityachievedatthisrate,versusκT
detunings in a Λ-level atomic system.
for loading a two-level atom by a single photon with hyper-
Figure5showstheloadingprobabilityasafunctionof
bolic secant pulse shape on resonance. Several spontaneous
decay rates, γ, are considered. timeforg/κ=1andκT =2,fordifferentpulseshapesof
approximately the same effective width and γ = ∆ = 0.
The four differentshapes we haveused are: a hyperbolic
doesnotnecessarilyimprovethe loadingprobability,but secantpulse; arectangularpulse; anexponentially-rising
there exists an optimum value of g/κ, which maximizes pulse; and an exponentially-decaying pulse. The differ-
theloadingprobability. Higherratiosthanthisoptimum encebetweentheirloadingprobabilitiesisseentobevery
value just increasethe number of Rabiflops betweenthe minor. It seems that having matched the input band-
two atomic levels, making it harder to find the atom in width to the cavity parameters g and κ, we can achieve
its excited state with high probability. However, by this high loading probability regardless of the photon pulse
method we cannothold the atom in its excited state un- shape. For the rest of this paper, we only consider the
lesswedrivethevalueofgtozeroatanappropriatepoint hyperbolic secant pulse shape in our calculations.
in the process, t = T , when the loading probability
Load
attains its maximum. In the next section, we show how
we can turn off the coupling in a Λ-level atom, in which B. Λ-level Trapped Atoms
we can use a control field to vary g.
Figure 4 shows the optimum value of g/κ versus κT The two-level atom that we analyzed in the previous
for severalvalues of γ and ∆=0. Severalthings are no- section, while the easiest to study, is not a practical so-
6
E e(cid:14)i(cid:88)zt alldecoherencemechanisms,with no initialexcitationin
z
the cavity and assuming that the atom is initially in its
r
groundstate g , the most generaltime evolutionfor the
T (cid:37)1 (cid:37)2 state of the sy|sitem takes the following form
(cid:88)0 (cid:56) (cid:88)z
(cid:168) dte(cid:14)i(cid:88)0t(cid:39)b(t)1t t g gc e |ψ(t)i =+ Ze−idωω0tαβω((tt))G|1ω+i|0ei−b|igωi0tcr(t)R
| i | i
+ e−i(ω0−ωz)t+iφz(t)c (t)E , (14)
e
| i
FIG. 6: (Color online) A Λ-level trapped atom illuminated
by a single photon. The cavity mode corresponding to the where
single photon drives the atom from its ground state |gi to
the auxiliary state |ri. A z-polarized beam then shelves the G 0 R 1 b g , (15a)
| i≡| i | i | i
atominthemetastablelevel|ei. Here,gc isthevacuumRabi R 0 0 r , (15b)
R b
frequency for the |gi-to-|ri transition, and Ω is the Rabi fre- | i≡| i | i | i
E 0 0 e . (15c)
quencyassociatedwiththecontrolfield,whichisproportional R b
| i≡| i | i | i
taond|E∆z2|.,rTeshpeescetitvrealnys,iatisondsefianreedoiffn-rtehseonteaxntt.bTyhedectounntirnoglsfi∆eld1 Applying the Schr¨odinger equation i~ψ˙(t) = Hˆb ψ(t)
| i | i
may also include a time-varyingphase φz(t). to Eq. (14), we get
β˙(t)= ig c (t) i√2κΦ (t) κβ(t) (16a)
c r b
lution for quantum storage. For long lifetime storage, − − −
c˙ (t)= ig β(t)+i∆ c (t) iΩ(t)c (t) γ c (t)(16b)
r c 1 r e r r
we have to store the quantum information in metastable − − −
atomic states that have low energy levels as well as low c˙e(t)= iΩ(t)cr(t) i(∆2 ∆1+φ˙z(t))ce(t), (16c)
− − −
decay rates, e.g. the hyperfine levels in the 52S or-
1/2
where we used Eq. (6) in the first equation and included
bitaloftherubidiumatom. Whereasthedirecttransition
a non-cavity decay rate γ for the upper state. We as-
betweensuchmetastable levelsareusuallydipole forbid- r
sume that the corresponding decay rate to the state e
den,wecanconnectthesestates,viaathirdexcitedlevel, | i
isnegligibleforthepurposeofloading;itcomesintoplay
by inducing a Raman transition driven by a single pho-
if we needto determine the storagetime of the quantum
ton on one leg and a classical control field on the other.
memory, which is not a topic of interest in this paper.
The spontaneous decay from the upper state can then
For large enough detunings, i.e., ∆ ,∆ Ω,g , we can
be mitigatedbyincluding largedetunings. The resulting 1 2 ≫ c
adiabatically eliminate the upper state R by assuming
Λ-levelatomisatthecoreofallinterestingneutral-atom | i
that c˙ (t)=0. As a result, we obtain
quantummemoryunits. This sectionstudies theloading r
problem in the context of trapped-atom memories with
g /∆ Ω(t)/∆
Λ-level configurations. c (t)= c 1β(t)+ 1c (t) (17)
r e
Γ Γ
Figure 6 shows a trapped Λ-level atom and its corre- r r
sponding driving beams. Here, we assume that an ex-
where Γ 1+iγ /∆ . Plugging the above equation
r r 1
ternal single-photon beam, in the initial state given by ≡
into Eq. (16), we then obtain
Eq. (1), is spatially matched to a cavity mode with an-
nihilation operator ˆb and resonance frequency ω0. This β˙(t) = igc2/∆1β(t) ig(t)c (t)
cavitymodedrivesthe g -to-r transitionwithcoupling − Γ − Γ e
| i | i r r
rate g . There may exist a detuning ∆ ω ω
c 1 ≡ 0 − gr i√2κΦ (t) κβ(t), (18a)
in this transition, where ωij denotes the transition fre- − b −
quency between levels i and j , for i,j g,r,e . The g(t) Ω2(t)/∆1
secondbeamisassumed| itobea|zi-polarize∈dc{lassica}lwave c˙e(t) = −i Γ β(t)−i Γ ce(t)
r r
with frequency ωz and a possibly time-dependent phase i(∆ ∆ +φ˙ (t))c (t), (18b)
2 1 z e
φ (t). This field’s amplitude, which is under our control − −
z
and may vary with time, determines the Rabi frequency where g(t)=g Ω(t)/∆ is the effective coupling rate be-
c 1
Ω(t). Itcouples r and e viathefollowingHamiltonian, tween g and e intheabsenceofdecay. Theseequations
whichisobtained| uinder|thierotating-waveapproximation resemb|leiEqs.|(6i) and (7), which governthe evolution of
[37]: a two-level system. To see this equivalence, we define
new variables β′(t) exp[ig2t/(∆ Γ )]β(t) and c′(t)
Hˆb = Hˆcc+~ωgrσˆrr+~ωgeσˆee+~gc(ˆb†σˆgr +σˆrgˆb) exp[ig2t/(∆ Γ )]c (t≡). Assumc ing t1hart γ /∆ 1e, an≡d
+ ~Ω(t)(e−i[ωzt+φz(t)]σˆ +ei[ωzt+φz(t)]σˆ ), (13) thus acpprox1imratineg Γ 2 by one, we thenr ob1ta≪in
re er r
| |
where Ω(t) and φz(t) are assumed to be real. The r - β˙′(t)= ig(t)c′(t) i√2κΦ′(t) κβ′(t), (19a)
to-e transition may also be off-resonant, by a detun|inig − Γr e − b −
∆ | iω ω . Similartotheprevioussection,neglecting c˙′(t)= ig(t)β′(t)+i∆c′(t) γc′(t), (19b)
2 ≡ z− er e − Γr e − e
7
where shape Φ (t)= 2/Tsech(4t/T), it turns out that [33]
b
p
Φ′b(t)=egc2γrt/∆21eigc2t/∆1Φb(t), (20a) gcsech(4t/T)
Ω(t)= .
∆=[g2 Ω2(t)]/∆ +∆ ∆ φ˙ (t), (20b) [1+tanh(4t/T)][tanh(4t/T)+κT/2 1]
c − 1 1− 2− z −
γ =g(t)γ [Ω(t)/g g /Ω(t)]/∆ . (20c) p (23)
r c c 1
− In order for this Ω(t) expression to yield a positive real
Equations(19a)and(19b)aresimilartoEqs.(6)and(7) Rabifrequency,wemusthaveκT 4. This,ontheother
≥
ifweusethenewvaluesforγ and∆asgivenbyEq.(20), hand,impliesthatforasuccessfuladiabatictransfer,our
and replace g and Φ (t), respectively, with g(t)/Γ and input pulse must be long enough so that we can slowly
b r
Φ′(t). Theloadingprobabilityisthengivenby c (t)2 change the quantum memory state.
b | e | ≈
ein−d2ugc2cγerst/a∆c21o|cn′es(tta)n|2t.fNreoqtueenthcaytsthhifettteormtheeiignc2ct/o∆m1ining Φlig′bh(tt). loaHdeinreg,aweΛp-lreovpeolsaetoamsi,mwphleicnhodno-aedsianboattiimc appopseroaancyh froer-
Wecantiltthecenterfrequencyoftheinputphotonfrom strictionsontheinputpulseshapeanddoesnotneedany
itsoriginalvalueω tocompensateforthisinducedshift. adiabaticpulseshapingforthecontrolfield. Ourmethod
0
Fromnowon,weassumethatthiscompensationhasbeen is based on what we observed in Section IIA for a two-
employed. level atom with a constant coupling rate g. There, we
For our loading problem, we are looking for particular realized that a maximum loading probability of greater
pulse shapes for Ω(t) and φ (t) that help the g -to-e than 90% was achievable, provided that we could turn
z
transition occur with high probability. The tr|adiition|ail off the atom-light coupling at t = TLoad. Here, we show
solutiontothis problememploysadiabatic-transfertech- that this is indeed possible to do for a Λ-level atom,
niques[26,33,39,40],inwhichweslowlychangethecon- by applying a constant control field for an appropriate
trol field so that the system follows its dark-state eigen- finite-duration time interval. In other words, we assume
state. In our case, for ∆1 = ∆2+φ˙z(t) and γr = 0, the φz(t)=0 and take Ω(t) to be
system’s dark state is given by,
Ω, t T
Ω(t)= ≤ Load . (24)
D = cosθ(t)G +sinθ(t)E . (21) (cid:26) 0, t>TLoad
| i − | i | i
where cosθ(t) Ω(t)/Ω , sinθ(t) g /Ω , and Ω
0 c 0 0 FromEq.(19),theeffectivecouplingrate,g =g Ω/∆ ,is
≡ ≡ ≡ c 1
Ω2(t)+g2. This state has the desired property that proportionalto the controlfield’s amplitude. Byturning
c
fpor Ω(t) ≫ gc, |Di ≈ −|Gi, but for Ω(t) ≪ gc, |Di ≈ off the control field at t = TLoad, the coupling rate g
E . So, if we start with a high value of Ω when the vanishes for t > T . Hence, if we are in the state e
|phiotonarrives,andthenslowlyreduceΩ tozero,wecan at t = T , weLwoaildl stay there until a decay proce|sis
Load
adiabatically transfer the system from G to E . The returns the atomto its groundstate. Solving for c′(t) in
| i | i e
timingisofcrucialimportance,becauseifweturnoffthe Eq. (19b) using Eq. (9), the loading probability is then
pump either before or long after the photon arrives we given by
lose the chance of absorbing the photon. There is also a
slight chance of jumping into states that are orthogonal 2κg2
c (t)2
to D ,i.e., R or B =sinθ(t)G +cosθ(t)E ,inwhich | e | ≈ ξ 2 ×
cas|e,ithe loa|diing p|roicess has co|mipletely fail|edi. | |
t 2
ityUonfdjeurmthpeindgatrok-estitahteercoRndiotironB,iniswnheicahrtzheerop,rwobeacbailn- (cid:12)Z dτe−gc2γr(t−τ)/∆21Φb(τ)(e−κ+(t−τ)−e−κ−(t−τ))(cid:12) ,
| i | i (cid:12) 0 (cid:12)
show that our loading problem is equivalent to loading (cid:12) for t T , ((cid:12)25)
(cid:12) Load (cid:12)
a cold cavity, viz. a cavity with no atom inside, with a ≤
time-varying decay rate κcos2θ(t). This can be shown where κ and ξ are given by Eq. (10) using γ and ∆
±
by paralleling the derivation used in [33] and shall be as given by Eq. (20). The loading probability is then
omitted. To maximize the loading probability, we need given by c (T )2. The effective detuning, ∆=(g2
to find an optimum assignment for cosθ(t) that maxi- Ω2)/∆ +| e∆ Loa∆d |, can be forced to zero by propecrl−y
1 1 2
−
mizes the loading probability and satisfies our adiabatic choosing ∆ and ∆ , thus compensating for the induced
1 2
conditions. In order to achieve a maximum transfer of Stark shift and enabling us to obtain a higher loading
free-field photons into the cavity mode, we need to min- probability.
imize the outgoing field components by destructively in-
terfering the directly reflected and the circulating fields.
Fleischhaueretal. haveshownthatanecessarycondition 1. Adiabatic versus non-adiabatic loading: numerical
for destructive interference is [33]: comparison
d d
lncosθ(t)+ lnΦb(t)=κcos2θ(t). (22) Inthissection,wecompareourproposednon-adiabatic
− dt dt
scheme to loading mechanisms that use adiabatic-
For our particular example of a hyperbolic secant pulse transfer techniques. Adiabatic transfer is a well-studied
8
problem in the literature. To make our comparison ex- 1
plicit, we will consider the particular examples of adia-
batic loading that have been proposed in [26] and [33]. 0.9 (cid:76)T =10
Ifenrr[i2n6g],ththeesatauttehoorfsahatvreapdpeevdi-saetdoma mqueathnotudmfomr etmraonrsy- ability0.8 75 (cid:76)T=5 2 (cid:56)(cid:97)(t/T)
b
to another trapped-atom quantum memory. Their ap- Pro0.7 (cid:76)T=7
proachis based onadiabatically transferringthe state of g (cid:76)T=10
n
otoneanmdelmoaodrys tthoeaotshinegrlempemhootroyn.,Twhheiicrhlotahdeinngprporpoacegsasteiss oadi0.6 0.5
L
facilitatedbyforcingthephoton’spulseshapetobesym- 0.5 (cid:14)2 (cid:14)1 0 1 2
t/T
metric, so that the receiver can employ a time-reversed
0.4
version of the control pulse that was used at the trans- 0 1 2 3 4 5 6
mitter. The desired pulse shape for the controlfield, un- gc2/((cid:76)(cid:37)1)
der the dark-state condition at zero effective detuning,
can be found numerically by solving the corresponding FIG. 7: (Color online) The loading probability for the adi-
Schr¨odinger equations. However,the approach from [26] abatic method as a function of κT versus gc2/(κ∆1), at the
is not suitable for an incoming photon with an arbitrary two-photon resonance ∆1 = ∆2 with φz(t) = 0, γr = 0, and
pulse shape. In [33], the authors have employed the adi- several values of κT. These curveswere obtained by numeri-
abatictransfertoloadanatomicensemblewiththestate callysolvingEq.(18)usingthecontrolpulseshapes,shownin
of a single photon. They have employed the dark-state the inset, that correspond to a hyperbolic secant input pulse
shape.
approximation under the two-photon resonance condi-
tion with a constant-phase control field. Their approach
sets certain limitations on the length of the input pulse
shape. Infact,inordertofulfillthedark-statecondition, the couplingrateinthe adiabaticcaseis to usea shorter
the input pulse shape must be longer than a threshold cavity, which yields a larger gc.
value. Unlike these two adiabatic mechanisms, our non- For nonzero values of γ /∆ , there is another param-
r 1
adiabaticapproachputsnoconstraintsontheinputpulse eter that we can vary to optimize the loading proba-
shape. bility in our non-adiabatic scheme: it is g /Ω, which
c
Several issues make our work in this paper, and the appears in the exponents −[gc2γr/∆21 + κ±](t − τ) =
numerical results presented in this section, distinct from [κ+g(gc/Ω+Ω/gc)(γr/∆1) ξ](t τ)/2inEq.(25). To
− ± −
previous work on loading quantum states into neutral increase the loading probability, for fixed values of g/κ
atoms. The first issue is our accounting for a nonzero andγr/∆1,weneedlowvaluesoftheseexponents. These
probability for populating the bright state B . We will can be achieved at gc/Ω 1. The exact optimum value
see in this section how the nonzero probabi|litiy of being forgc/Ωthat minimizes t≈he decay effectalsodepends on
inthe brightstateaffectsthe loadingperformance. That the values of γ and ξ. For the sake of comparison, how-
alsomakesitpossibletoscrutinizethedependenceofthe ever, we assume that γr/∆1 is sufficiently small that we
loading probability, in both adiabatic and non-adiabatic can neglect these subtleties, and we only focus on the
mechanisms, on the key system parameters, e.g., g /κ interplay between the coupling rate and the input band-
c
and κT. This evaluation is one of the original contribu- width. So, for the restof this section, we assume γr =0.
tions ofthis paper. Finally, ouruse ofaconstantcontrol It is interesting to find the dependence of the loading
fieldallowsustoobtainanalyticalresultsforloadingdy- probability, for the adiabatic scheme proposed in [33],
namics in the non-adiabatic case. on the coupling rate g . For this purpose, we have nu-
c
merically solved the Schr¨odinger equations in Eq. (18),
It is of practical importance to know at what values
for Ω(t) given by Eq. (23), at the two-photon resonance
of input bandwidth, representedby κT,and atomic cou-
∆ = ∆ and for φ (t) = 0. In Fig. 7 we have plotted
pling rate, represented by g /κ, a desired system perfor- 1 2 z
c c (5T)2 versus an effective coupling rate g′ g2/∆ .
mancecanbe achieved. Forournon-adiabaticapproach, | e | ≡ c 1
Choosing t = 5T ensures that the loading process has
atγ =∆=0,systemperformanceisgovernedbytheef-
r
ended, and therefore c (5T)2 is effectively the loading
fectivecouplingrateg =g Ω/∆ . The optimumvalueof e
c 1 | |
probability. This figure shows that for the two-photon
g/κcanthenbe obtainedby makingappropriatechoices
resonance case, higher g values yield higher loading
for g , Ω, and ∆ . The only conditions that we need c
c 1 probabilities. The effectivecouplingrateg′ thatweneed
to satisfy are ∆ γ and g ,Ω ∆ . That allows us
1 r c 1
≫ ≪ is about 2κ for 90% loading probability at κT = 5. It
someflexibilitytopickasmallerg ,whichdeterminesthe
c
can be seen that there is an advantage to using pulses
cavitylength,andalargerΩ,thanisthecasefortheadi-
with higher values of κT, because our adiabatic scheme
abaticapproachproposedin[33]. Forinstance,usingthe
is more efficient for longer input pulses.
hyperbolic secant pulse shape and the optimum control
pulseshape,givenbyEq.(23),wehaveg Ω(t)=g2Ω′(t), We can also solve the Schr¨odinger equations in
c c
where Ω′(t) is only a function of κT and not g . There- Eq. (18), for Ω(t) given by Eq. (23), at zero effective
c
fore, for a fixed value of κT, the only way to increase detuning, i.e., when φ˙ (t)=∆ ∆ +[g2 Ω2(t)]/∆ ,
z 1− 2 c − 1
9
3 1
(cid:76)T =9
2.25 (cid:157)(cid:158)(cid:158)(cid:158)(cid:159)(cid:76)g(cid:37)c(cid:56)1(cid:172)(cid:174)(cid:173)(cid:173)(cid:173)opt NoAnd-iaadbiaatbicatic bability0000....6789 (cid:76)g(cid:37)c21 (cid:120)1 (cid:76)g(cid:37)c(cid:56)1 (cid:120)0.4
1.5 (cid:157)(cid:158)(cid:158)(cid:158)(cid:158)(cid:159)(cid:76)g(cid:37)c21(cid:172)(cid:174)(cid:173)(cid:173)(cid:173)(cid:173)opt ng Pro00..45
1 Ploading oadi0.3
0.5 Ploading L0.2
0.1
00 2 4 (cid:76)T 6 8 10 0-1 -0.5 T0 /T0.5 1 1.5
offset
FIG. 8: (Color online) The optimum coupling rate and
FIG. 9: (Color online) Loading probability versus normal-
the maximum loading probability achieved at this rate for
izedtimingoffset inthephotonarrival. Intheadiabaticcase
bothadiabaticandnon-adiabaticapproaches. (Non-adiabatic
(theleft curve),thistimingerrorresultsinapplyingthecon-
curves start from κT = 0.5; adiabatic curves start from
trol field earlier or later than the correct time. In the non-
κT = 4.5.) For all curves, we assume that ∆ = γr = 0. A
adiabatic case (the right curve), it results in stopping the
hyperbolicsecantpulseshapehasbeenusedfortheincoming
photon. control field earlier or later than TLoad. Both curves assume
∆ = γr = 0 for their respective optimum coupling rates for
hyperbolicsecant input pulse shapes.
as proposed in [26]. Note that the control field’s Rabi
frequencygivenbyEq.(23)hasbeenobtainedunderthe rate for system A is four times that of system B, be-
two-photon resonance and dark-state conditions, and it cause we are allowing a larger output bandwidth in the
is therefore not the optimum pulse shape for the zero- formercase. FromFig.8, the maximumachievableload-
effective-detuning case. It can be verified, numerically, ingprobabilityforsystemAis above75%,while thatfor
that this control field provides a close-to-optimum per- system B is at most one. It folllows that system A can
formance. This is in accord with the results reported in have a 4 0.75=3 times higher througput than system
×
[33] implying that the loading probability will only be B.
slightly affected by small deviations from the optimum Finally, Fig. 9 shows the sensitivity of both schemes
control field pulse shape. For zero-effective detuning, in to timing offsets. Here, Toffset refers to the time that we
analogy to the non-adiabatic case, we observe the exis- stopthe controlfield in the non-adiabaticapproach,and
tence of an optimum value for g′/κ. Figure 8 shows the it represents a time shift in the adiabatic approach, i.e.,
optimumcouplingratesandthe maximumloadingprob- usingΩ(t Toffset)insteadofΩ(t). Bothcasesmayoccur
−
abilities, P , achieved at these rates as functions of if we have an inaccurate estimate of the photon arrival
loading
κT for the adiabatic and non-adiabatic approaches us- time. This figure shows that both schemes have almost
ing a hyperbolic secant pulse shape. Here, (g′/κ) is a the same tolerance for timing offsets. A 50% loading
opt
near-unityconstant,forallvaluesofκT,whereas(g/κ) probablity (3 dB loss) is achievable even if we are about
opt
decreasestowardzerowithincreasingκT. Themaximum T/2 off from the correct loading time. Although it is
±
loadingprobabilitythatcanbeachievedbytheadiabatic not shown in this figure, the same result holds if we use
approachasymptoticallygoestoone asκT increases. Its the adiabatic approach under the two-photon-resonance
value is about 75% for κT = 4.5, but we should bear in condition.
mind that the systemwill be morevulnerable to sponta-
neous decay when we use long pulses.
C. Double-Λ Trapped Atoms
Because the non-adiabatic approachhas no constraint
on input pulse width, it allows a larger class of single-
The results we obtained above are directly applicable
photon sources to be used in quantum communication
to several other problems. For instance, consider the
systems, which can result in higher throughputs. For in-
double-Λ atom shown in Fig. 10(a). Here, we are trying
stance, consider system A, which uses a downconverter
to load the memory with a single photon in an arbitrary
single-photon source at κT =1, and non-adiabatic load-
polarization state:
ing at the optimum coupling rate. We will compare it
with system B, which uses adiabatic loading with the
same source but at κT = 4, the minimum required for |ψ0i=Z dωφ(ω)(α|1ωiσ+ +β|1ωiσ−), (26)
the adiabatic case in our examples. Both systems use
identical quantum memory modules, so their κ values where α2 + β 2 = 1, and 1 refers to a single-
| | | | | ωiσ±
are the same. The total throughput is proportional to photon state at frequency ω in the σ (right/left cir-
±
the source rate and the loading probability. The source cular) polarization. The goal of the loading process is
10
E e(cid:14)i(cid:88)zt proves the equivalence of loading a V-level atom to the
(a) z
T s r loadingof a two-levelatomwhen both aredrivenby sin-
glephotons. Thisresultalsoextendstocasesinwhichthe
(cid:84)(cid:14) z-pol. two legs of the atomic system are nonlocal—e.g., when
t
(cid:168) d(cid:88)(cid:71)((cid:88))((cid:66) 1 +(cid:67) 1 ) (cid:76) e(cid:14) g(cid:84)+ e+ aa ppahiortoofn-tnwuom-lebveerletnratapnpgelde-datsotmatem.eSmuocrhiesaasryestdermiveins bany
(cid:88) (cid:84)+ (cid:88) (cid:84)(cid:14)
example of the to-the-memory configuration for entan-
(b) e e glement distribution, and it can be used in teleportation
(cid:14) +
systems.
Adiabatic g g Linear
Elimination Superposition
g
(cid:76)
III. MIT-NU ENTANGLEMENT
(c)
e DISTRIBUTION
+
T
g
t With the results that we have developed for trapped-
(cid:66)(cid:168) d(cid:88)(cid:71)((cid:88)) 1(cid:88) (cid:84)+ (cid:76) g atom quantum memories in the previous section, we can
+ addresstheperformanceofquantumcommunicationsys-
e tems that use this type of storage. Our focus on this
T (cid:14)
sectionwillbe onthe MIT-NUarchitecturefor quantum
g
communication[23]. TheMIT-NUconstructusestheto-
t
(cid:67)(cid:168) d(cid:88)(cid:71)((cid:88)) 1(cid:88) (cid:84)(cid:14) (cid:76) g tshhea-rmeepmoloarryizcaotniofinguernattainognleinmeFnitg.b1e(taw)eteondtiwstorirbuubtiediaunmd
atoms. Because the rubidium atoms can be treated as
FIG. 10: (Color online) (a) Loading a double-Λ atom driven
double-Λquantummemories[11],MIT-NUentanglement
by a polarization-state single photon. By adiabatically elim-
distributionisessentiallyaloadingproblem,inwhichwe
inating the upper states, this system becomes equivalent to
are interested in transferring an entangled state to two
theV-levelsystemshownin(b). (c)ReductionoftheV-level
loading problem to two two-level loading problems by means trapped atoms. All previous analyses of the MIT-NU
of linear superposition. architecture [23, 34, 41], however,have employed a cold-
cavityapproachinwhicheachopticalcavity—thatwould
hold an 87Rb atom in the actual implementation—is re-
to absorb this photon, by properly choosing z-polarized gardedasempty,andtheloadingprobabilityiscalculated
control fields—according to either the adiabatic or non- by determining the probability that the state of the in-
adiabatic mechanisms—and store its polarization infor- tracavityphoton fields at the end of a loading intervalis
mationinthe metastablelevels e and e thatcorre- the desired singlet. In this section, we provide the first
+ −
| i | i
spond to the σ and σ polarizations,respectively. The loadinganalysisoftheMIT-NUsystemthatincludesthe
+ −
loading probability is therefore determined by the prob- presence of atoms within the quantum memory units.
ability of ending up in e αe +β e . Figure11(a)showsaschematicoftheMIT-NUsystem:
+ −
| i≡ | i | i
Inordertocalculatetheloadingprobability,wecanuse QM and QM are trapped rubidium atom quantum
1 2
the results of the previous subsection to reduce each Λ- memories, each L km away—in opposite directions—
0
configuration leg of the atom to an equivalent two-level from a dual optical parametric amplifier (OPA) source.
system, as shown in Fig. 10(b). The resulting config- Eachopticalparametricamplifierinthedual-OPAsource
uration is a V-level atom, illuminated by an arbitrary- is a continuous-wave, type-II phase matched, doubly-
polarized single photon. This problem can be solved resonant amplifier operating at frequency degeneracy
by noting that the full Schr¨odinger equations—for the ω =ω . Its signal (S) and idler (I) outputs comprise a
S I
state of the reservoirs, cavity modes, and the atom— stream of orthogonally-polarized photon pairs that are
thatmodelthesystemevolutionarelinearinresponseto in a joint Gaussian state [23]. By coherently pump-
the superposition state in Eq. (26). Therefore, each leg ing two of these OPAs, π-rad out of phase, and com-
of the V-level atom is only driven by the correspond- bining their outputs on a polarizing beam splitter, as
ing part in the initial state, as shown in Fig. 10(c). shown in Fig. 11(b), we obtain signal and idler beams
In fact, for α dωφ(ω)1 as the input, c (t)—the that are polarization entangled [7]. Moreover, double-
| ωiσ+ −
probability amRplitude of being in the state e− —will resonant operation greatly enhances the brightness of
| i
be zero. We only get a nonzero c (t) from the input these beams, making this entanglement source compati-
−
β dωφ(ω)1 , in which case the c (t) evolution co- ble with loading atomic memories [6], something that is
| ωiσ− −
inRcides with that of a two-level system. For instance, if not the case for standard THz bandwidth parametric
∼
weareusingthenon-adiabaticloadingmechanism,c (t) down-converter sources. The signal and idler beams are
−
is given by βc (t), using c (t) from Eq. (25). The same routed down separate optical fibers to the trapped-atom
e e
argument holds for c (t), so that the overallprobability quantummemories. Todosoefficiently,theirwavelength
+
of ending up in e is (α2+ β 2)c (t)2 = c (t)2. This is chosen to lie in the low-loss 1.55µm band. However,
e e
| i | | | | | | | |