Improving fidelity in atomic state teleportation via cavity decay

0.24 MB·English

by  Grzegorz Chimczak

Improving fidelity in atomic state teleportation via cavity decay
Grzegorz Chimczak and Ryszard Tana´s
Nonlinear Optics Division, Physics Institute, Adam Mickiewicz University, 61-614 Poznan´, Poland∗
(Dated: February 9, 2008)
Weproposeamodifiedprotocol ofatomicstateteleportation fortheschemeproposedbyBoseet
al. (Phys. Rev. Lett. 83,5158(1999)). Themodifiedprotocolinvolvesanadditionalstageinwhich
quantum information distorted during the first stage is fully recovered by a compensation of the
damping factor. The modification makes it possible to obtain a high fidelity of teleported state for
cavities that are much worse than that required in the original protocol, i.e., their decay rates can
be over 25 times larger. The improvement in the fidelity is possible at the expense of lowering the
probability of success. Weshow that themodified protocol is robust against dark counts.
7
0 I. INTRODUCTION twoordersofmagnitudebelowofwhatiscurrentlyavail-
0 able [16, 17, 18, 19, 20, 21, 22, 23].
2 Here, we present a protocol that reduces the effect of
n Quantum teleportation [1] is considered to be a per- cavity decay on the fidelity. This protocol makes it pos-
a fect way of transferring qubits over long distances. It sibletousecavitieswithlargerdecayrateswithoutwors-
J is particularly important to teleport qubits represented ening the fidelity but at the expense of lowering success

0 by the atomic states, which can store quantum infor- rates.
3
mation for sufficiently long time as to make it available

for further quantum processing. However, in contrast
1 II. MODEL
to the teleportation of photonic states, the teleportation
v
4 of atomic states over long distances is a difficult task.
3 As yet, the longest distance achieved experimentally for The teleportationprotocolthat we proposein this pa-
2 atomic states is of the order of micrometers [2, 3] while per is designed for the same device which Bose et al. [5]
1 for photonic states is of the order of kilometers [4]. It is considerintheirscheme. ThedeviceisdepictedinFig.1.
0 obvious that the distance of atomic states teleportation It is composed of two cavities CA and CB, a 50-50beam
7
has to be orders of magnitude greater to make the tele-
0
portationuseful inquantumcommunication. Inorderto
/
h makethisdistancegreater,itisnecessarytoemploypho-
p
tons, which are the best long distance carriers of quan-
-
t tum information, to establish quantum communication
n
betweentheremoteatomsandcompletetheatomicstate
a
u teleportation. Such a scheme of atomic state teleporta-
q tion has been presented by Bose et al. [5]. They have
: proposedanadditionalstageofteleportationprotocol—
v
the preparationstage, in which the state of sender atom
i
X is mapped onto the sender cavity field state and there-
r forecanbeteleportedinthenextstageusingwellknown
a
linear optics techniques. The possibility of operating
on atomic qubits with linear optics elements is the rea-
son why a combination of atomic states and cavity field
stateshasbeenrecentlysuggestedinmanyproposals,not
only in proposals of teleportation protocols [6, 7, 8] but
also in other schemes of quantum information process-
ing [9, 10, 11, 12, 13, 14, 15]. Unfortunately, the state FIG. 1: (Color online) Schematic representation of thesetup
mapping and whole preparation stage is not perfect be- torealizelongdistanceteleportationofatomicstatesviapho-
cause of a destructive role played by cavity decay. The tons.
cavity decay reduces the fidelity of teleported state and
the probability of success. Bose et al. [5] have suggested splitter,twolasersLA andLB andtwosingle-photonde-
a way to minimize a destructive role of this imperfection tectorsD+andD−. Thereceiver,Bob,hasthecavityCB
by assuming very small cavity decay rate. However, the andthelaserLB. Theotherelementsofthedeviceareat
value of cavity decay rate required by their protocol is the side of the sender — Alice. Inside each cavity there
is one trapped atom, modeled by a three-level Λ system
withtwostablegroundstates 0 and 1 ,andoneexcited
| i | i
state 2 as shown in Fig. 2. Only the excited state de-
| i
∗Electronicaddress: [email protected] caysspontaneously, therefore the groundstates are ideal
2
the expressions a more compact form we use the nota-
tion jn j n to describe the state of
atom mode
| i ≡ | i ⊗ | i
the atom-cavity system. During the whole teleportation
process the time evolution of the system is restricted to
the subspace spanned by the states: 00 , 10 and 01 .
| i | i | i
The state 00 experiences no dynamics because there is
| i
nooperatorintheHamiltonian(2)whichcanchangethis
state. Timeevolutionoftheothertwostatesisdescribed
by
FIG. 2: Level scheme of the Λ atom interacting with the
classical laser field and thequantized cavity mode. e−iHt 10 = eiδte−κ2t i2δ sin Ωκt 01
| i Ω 2 | i
κ
h (cid:16) (cid:17)
Ω t κ Ω t
κ κ
+ cos + sin 10 ,
candidates for an atomic qubit. The spontaneous decay
2 Ω 2 | i
κ
rate of the excited state is denoted by γ. Operations (cid:16) (cid:16) (cid:17) (cid:16) (cid:17)(cid:17) i
on the qubit coded in the superposition of both ground e−iHt 01 = eiδte−κ2t i2δ sin Ωκt 10
| i Ω 2 | i
states are possible using two transitions: 0 2 and κ
| i ↔ | i hΩ t (cid:16)κ (cid:17) Ω t
1 2 . Firstofthetransitionsiscoupledtothecavity + cos κ sin κ 01 ,
m| iod↔ew| iiththecouplingstrengthg whilethesecondtran- 2 − Ωκ 2 | i
(cid:16) (cid:16) (cid:17) (cid:16) (cid:17)(cid:17) i
sitioniscoupledtoaclassicallaserfieldwiththecoupling (4)
strength Ω. Since we want a population of the excited
stateto be negligible,the laserfieldandthe cavitymode where Ωκ = √4δ2 κ2. There are two important local
are detuned from the corresponding transition frequen- operationswecanp−erformonthesystemstateviae−iHt.
cies by ∆. Beside the spontaneous decay of the excited First of them is to map the atomic state onto the cav-
atomicstatethereisanotherdecaymechanism. Onemir- ity mode and second is the generation of the maximally
ror in each cavity is partially transparent and therefore entangled state of the atom and the cavity mode. The
photons leak out of the cavities through these mirrors atomicstatemappingonecanobtainbyturningthelaser
at a rate κ. The evolution of each atom-cavity system on for time tA is given by
i(sh¯g=ov1erhneerde abnydthinetehffeecftoilvloewninong)-Hermitian Hamiltonian 10 ieiδtAe−κt2A 01 , (5)
| i → | i
H = (∆−iγ)σ22+(Ωσ21+gaσ20+H.c.)−iκa†a, wthheemreatxAim=al(l2y/eΩnκt)a[nπg−ledarscttaatne(Ωthκe/lκa)s]e.rIsnhoorudlderbteotcurrenaetde
(1)
on for time t =(2/Ω )arctan(Ω /(2δ κ))
B κ κ
−
whereadenotestheannihilationoperatorforAlice’scav-
ity mode (aA) or Bob’s cavity mode (aB). In (1) we 10 eiδtBe−κt2B 2δ sin ΩκtB (10 +i01 ).(6)
introduce the flip operators σij i j . Both our pro- | i → Ωκ 2 | i | i
tocol as well as the Bose et al.≡p|riohto|col work in the (cid:16) (cid:17)
When the laser is turned off then Ω = 0, and the
low saturation limit (g2/∆2, Ω2/∆2 1) and there- HamiltoniangoesoverintoH = δa†aσ iκa†a. Then
≪ 00
fore the excited atomic state can be adiabatically elimi- − −
all the terms of the Hamiltonian correspond to the di-
nated [24, 25, 26, 27, 28, 29, 30, 31]. Either of them re-
agonal elements in matrix representation, and the non-
quiresmallvalues ofthe spontaneousdecayrate(∆ γ
unitary Schr¨odinger equation can be easily solved. The
≫
and γg2/∆2, γΩ2/∆2 κ) [32] which makes it possible
evolution of the states 10 and 01 , when the laser is
≪
toneglectγ asafirstapproximation. Finally,weassume | i | i
turned off, are thus given by
the conditionΩ=g whichleads toa verysimple formof
the Hamiltonian e−iHt 10 = 10 ,
| i | i
H = δσ δa†aσ (δaσ +H.c.) iκa†a, (2) e−iHt 01 = eiδte−κt 01 . (7)
11 00 10 | i | i
− − − −
whereδ =g2/∆. Theevolutiondescribedby(2)isinter-
rupted by collapses. Photon decays registered by detec- III. TELEPORTATION PROTOCOL
tors correspond to the action of the collapse operator
Both protocolsstart with the same initial state — the
C =√κ(a +iǫa ), (3)
A B
unknown state that Alice wants to teleport, which is
where ǫ is equal to 1 for photon detection in D and stored in her atom. Bob’s atom is prepared in the state
+
equal to 1 for photon detection in D . 1 and the field modes of both cavities are empty, so we
−
− | i
The simple form of the Hamiltonian (2) allows for an- have
alytical solutions of the non-unitary Schr¨odinger equa-
tion and get expressions for the time evolution of quan- ψ A = α00 A+β 10 A, (8)
| i | i | i
tum states which are used in both protocols. To give ψ = 10 . (9)
B B
| i | i
3
The teleportation protocol with improved fidelity con- B. Detection stage I
sists of five stages: (A) the preparation stage, (B) the
detection stage I, (C) the compensation stage, (D) the When the quantum information is mapped onto the
detection stage II and (E) the recovery stage. state of Alice’s cavity field and the maximally entangled
state of Bob’s cavity field and the target atom is cre-
ated, then the joint measurement of both cavity fields
A. Preparation stage can be performed. During this stage Alice and Bob per-
form the joint measurement just by waiting with their
The preparation stage is necessary because the quan- lasers turned off. The teleportation is successful if the
tuminformationencodedinitially inAlice’satomis tele- detectorsregisteroneandonlyonephoton. Insuccessful
ported by performing joint measurement on the field casesthejointstateofAlice’sandBob’ssystemsbecomes
state of both cavities. Before of the detection stage Al-
ice has to map the quantum information onto her cavity φ(td) = (iǫα00 B+eiδtAe−κt2Aβ 10 B)00 A
| i | i | i | i
field state while Bob has to create the maximally entan- +ieiδt1e−κ2t1βe−κtdeiδtd
gledstateofhisatomandhiscavityfield. AliceandBob e
(01 00 +iǫ00 01 ), (14)
achieve their goals by switching their lasers on for times × | iB| iA | iB| iA
t and t , respectively. After the preparation stage the
A B where t is the time of this detection stage. Until now
d
state of Alice’s atom-cavity system is given by
theoperationsinbothprotocolsareexactlythesame. In
the protocol of Bose et al. it is assumed that time t is
ψ A =α00 A+ieiδtAe−κt2Aβ 01 A, (10) much longer than κ−1 and thus all unwanted statesdin
| i | i | i
expression (14) can be neglected. Finally, after remov-
and Bob’sesystem state becomes ing a phase factor, the state of Bob’s atom is given by
α0 +e−κtA/2β 1 . It is obvious that the fidelity of
B B
ψ B =e−κt2B 2δ sin ΩκtB (10 B+i01 B). (11) te|leiportedstatew|illineverreachunitybecauseofthefac-
| i Ωκ 2 | i | i tor e−κtA/2. Moreover,in the protocol of Bose et al. the
(cid:16) (cid:17)
fidelityofteleportedstatedecreaseswithincreasingκ. In
e
This first stage is successful only under the absence of
our protocol, we use one of the unwanted states to com-
photondetectionevent. Theprobabilitythatnocollapse
pensate for the factor e−κtA/2. This compensation can
occurs during Alice’s operation is given by the squared
bedoneifwechoosethetimeofthisdetectionstagesuch
norm of the state vector
that eiδtd = 1. Then expression (14) can be rewritten
−
as
P = α2+e−κtA β 2. (12)
A
| | | | φ(td) = iǫα00 B 00 A+eiδtAe−κt2Aβe−κtdǫ00 B 01 A
Similarly, we can obtain appropriate expression for the | i | i | i | i | i
probability of no collapse during Bob’s operation +eiδtAe−κt2Aβ(10 B ie−κtd 01 B)00 A.(15)
e | i − | i | i
8δ2 Ω t
PB =e−κtB Ω2 sin2 κ2B . (13) C. Compensation stage
κ
(cid:16) (cid:17)
Itisevidentthatthestatemappingisnotperfectbecause In the compensation stage Bob compensates for the
of the damping factors that appear in expression (12) factore−κtA/2 byturninghislaseronfortimetc. During
for P and in expression (10) for the state ψ . These the operationAlice’s laserremainsturnedoff. Oncondi-
A A
damping factors reduce both the probabili|tyi that the tion that no photon detection occurs during time tc, the
state mapping is successful and the fidelityeof this op- unnormalizedjoint state at the end of this stage is given
eration. The quantum information after the mapping by
operationis alsomodified by the phase factor ieiδtA but,
incontrasttodampingfactors,thephasefactorscanlater φ(tc) = eiδ(tA+tc)e−κt2Aβe−κ(td+tc)ǫ00 B 01 A
| i | i | i
bdueceeastihlye cfiodmelpiteyn.saItnedorfdorerantodmthaekreefothreetphreoybadboilnitoytPre- e −ieiδ(tA+tc)βe−κ(tA2+tc)ϕ(tc)|01iB|00iA
and the fidelity close to unity Bose et al. assume thaAt +eiδ(tA+tc)βe−κ(tA2+tc)ϑ(tc)10 B 00 A
| i | i
Ω κ, which means that both κ and t values are
κ A +iǫα00 00 , (16)
smal≫l and the damping factor e−κtA/2 is close to unity. | iB| iA
Generally,however,the damping factoris notunity even where
for very small κ and t and, in consequence, the fidelity
A Ω t 2δ+κe−κtd Ω t
of the teleported state is diminished. Since high fideli- ϕ(t ) = e−κtdcos κ c sin κ c ,
c
tiesarerequiredbyquantumcomputationalgorithms,we 2 − Ωκ 2
will show how to compensate for this factor in the next Ω t(cid:16) κ(cid:17)+2δe−κtd Ω t(cid:16) (cid:17)
κ c κ c
ϑ(t ) = cos + sin . (17)
stages of the protocol. c 2 Ω 2
κ
(cid:16) (cid:17) (cid:16) (cid:17)
4
It is seen that this operation transfers population from value of t by finding numerically t satisfying the
dmax d
the state 01 00 , which is unwanted, to the state condition
B A
| i | i
10 00 . Of course, we want the transfer to compen-
B A
s|atei f|or ithe factor e−κtA/2 and therefore tc has to fulfill e−κ(tA+tcmax)/2ϑ(tcmax)=1. (21)
the condition
The problem of choosing t is much simpler when we
d
e−κ(tA2+tc)ϑ(tc) = 1. (18) want to compensate for the factor e−κtA/2 for as large
κ as possible. From figure 3 one can see that the limit
40
D. Detection stage II
35
30
The population of one of the two unwanted states is D
s25
already reduced after the previous stage, but it cannot Μ
@
beneglectedyet. Moreover,thepopulationofthesecond 20
x
unwanted state is still considerable. Presence of the two ma15
d
unwanted states decreases the teleportation fidelity, so t10
in the fourth stage of the protocol Alice and Bob have 5
to eliminate them. All they have to do to achieve this
0.02 0.04 0.06 0.08 0.1 0.12
goal is simply to wait for a finite time t κ−1. Af-
D Κ(cid:144)2Π @MHzD
≫
ter time t the populations of both unwanted states are
D
negligible and unnormalized joint state can be very well
FIG. 3: The value of tdmax as a function of κ for
approximated by
(∆;Ω;g)/2π = (100;10;10) MHz calculated numerically us-
ing condition (21).
φ(t ) = (iǫα00 +eiδ(tA+tc)β 10 )00 .(19)
D B B A
| i | i | i | i
t decreases with increasing κ. Therefore we should
dmax
e
choose the smallest value of t by setting m to zero.
E. Recovery stage d
Finally, Bob has to remove the phase shift factor IV. NUMERICAL RESULTS
iǫe−iδ(tA+tc) to recoverthe originalAlice’s state. To this
end Bob adds to the state 1 an extra phase shift
| iatomB Let us now compare both protocols. For this purpose
withrespectto the state 0 usingthe Zeemanevo-
| iatomB we compute the average probability of success and the
lution [5]. After this operation the state of Bob’s atom
averagefidelity ofteleported state for the same values of
is exactly the same as the initial state of Alice’s atom,
the detuning and both coupling strengths as in Ref. [5],
i.e., α0 + β 1 , and thus the teleportation
| iatomB | iatomB i.e., (∆;Ω;g)/2π = (100;10;10) MHz. It is necessary to
fidelity of this protocol can be very close to unity. This
takeaveragevaluesoverallinputstatesbecausetheprob-
completes the teleportation protocol.
ability of success in both protocols as well as the fidelity
Now it is time to explain in detail how to choose the in the Bose et al. protocol all depend on the unknown
time t . The condition eiδtd = 1 leads to many so-
d − moduli of the amplitudes α and β of the initial state.
lutions given by t = π(2m+1)/δ, where m is a non-
d The fidelity in our protocol seems to be independent of
negative integer. However, we cannot set m arbitrary
the amplitudes of initial state and should be equal to
because ϑ(t ) and the probability of success in second
c unity. However,this is onlytrue forthe simplifiedmodel
stage are functions of t . It is obvious that the proba-
d forwhichtheexcitedstateiseliminated. Inmoregeneral
bility of observing one photon during detection time t
d model described by the Hamiltonian (1) the population
increaseswith increasingt . On the other hand, we can-
d oftheexcitedstatehasanonzerovalueduringtheevolu-
not choosethis detectiontime too long because the pop-
tion givenby (4) evenif the atomis initially preparedin
ulation of unwanted state 01 00 can then be too
small to compensate for the| faicBto|r eiA−κtA/2. Thus, t is its ground state. However,the population of the excited
d stateremainszerofortheinitialstate 00 ofatom-cavity
limited by some time t . Let us now estimate t . | i
dmax dmax systembecausethestateexperiencesnodynamics. Ifthe
Expression e−κ(tA+tc)/2ϑ(t ) takes its maximal value for
c initialstateis asuperpositiongivenby(8)thenthe pop-
the time of the compensation stage given by
ulationofthe excitedstatedepends onthe moduliofthe
amplitudes α and β. Since the population of the excited
2 2δΩ e−κtd
t = arctan κ . (20) state reduces the fidelity, it is also necessary to average
cmax Ωκ Ω2κ+κ(κ+2δe−κtd)! the fidelity in our protocol over all input states.
We compute all the averages numerically using the
The factor e−κtA/2 can be compensated for only under method of quantum trajectories [33, 34] together with
the condition that e−κ(tA+tcmax)/2ϑ(tcmax) 1. Since the Monte Carlo technique. Each trajectory starts with
≥
both ϑ(t ) and t depend on t , we can estimate the a random initial state and evolves according to a chosen
c cmax d
5
teleportation protocol. If measurement indicates success 1
then we calculate the fidelity of teleported state at the
end of the protocol. Otherwise, we reject a trajectory 0.995
y
as unsuccessful. After generating 20 000 trajectories we t
i
averagethe fidelityoveralltrajectoriesandcalculatethe l 0.99
e
averageprobabilityofsuccessasaratioofthenumberof d
i
successful trajectories to the number of all trajectories. F
0.985
Therearesomeproblemsthatappearwhenweusethe
Hamiltonian (1) to simulate performance of our proto-
50 100 150 200 250
col. First, the fidelity is sensitive to the inaccuracy in
Κ(cid:144)2Π @kHzD
calculations of phase shift factors. The compensation of
the factor e−κtA/2 requires the phase shift of the state
FIG. 4: The average fidelity of teleportation in the new pro-
01 00 relative to the state 10 00 to be equal
| iB| iA | iB| iA tocol (diamonds) and in Bose et al. protocol (open squares)
to i as shown in (15). Therefore the time t of the de-
d as functions of the cavity decay rate for (∆;Ω;g;γ)/2π =
tec−tion stage I has to satisfy the condition eiδtd = 1. (100;10;10;0) MHz. Theaverages are takenover 20 000 tra-
−
However, the analytical expression for δ is derived from jectories.
theHamiltonian(2)andthusexp(iδt )isonlyanapprox-
d
imationtotherealphaseshiftfactor. Unfortunately,the
populationtransferthattakesplaceinthe compensation t can be calculated analytically as in the Bose et al.
B
stage leads to an unknown extra phase shift in the final protocol. However, the analytical expressions are func-
state(19)whenthephaseshiftbetweenstates 01 B 00 A tions of κ, so, for different values of κ the population of
| i | i
and 10 B 00 A differs from the expected value i. Of the excited state andthe fidelities ofoperationstake dif-
| i | i −
course,thisunknownphaseshiftcannotbe compensated ferent values. If we want to stabilize the average fidelity
forintherecoverystage,whichmeansthatthefidelityin at a high level for different values of κ then we have to
our protocol can even be smaller than the fidelity in the compute t and t numerically.
A B
Bose et al. protocol. To overcome this problem we use
Tobeginwithourcalculations,wesetthespontaneous
a numerical optimization procedure which finds, for the
decay rate of excited state to zero because we want to
moregeneralmodel,sucht thatthejointstateofAlice’s
d know how close to unity is the fidelity in the ideal case
andBob’ssystemsbecomesasclosetotheexpectedstate
in which there is no possibility of photon emission to
given by (15) as possible.
modes other than the cavity modes. Fig. 4 shows that
Second, a question arises: how to estimate the biggest the modified protocolreallystabilizes the fidelity oftele-
value of κ for which the compensation is still possible? ported state at a high level. The fidelity is reduced only
This value is very important because we want to know by the nonzero population of the excited state and does
how good (or rather bad) cavities can be used for effec- not decrease with increasing κ until κ/2π is about 0.25
tivehighfidelityteleportation. Inthesimplifiedmodelof MHz. The fidelity of teleported state in the protocol of
our protocol governed by the Hamiltonian (2) this value Bose et al. is reduced by the population of excited state
can be computed from (21) and is about κ/2π 0.17 as well as by the factor exp( κt /2) and, as expected,
≈ − A
MHz. However, the population of the excited atomic itdecreaseswith increasingκ. Itis seenfromFig.4 that
state changes this value because of the transfer of popu- there are discontinuous jumps of the fidelity values. The
lation from the state 20 B 00 A to the state 10 B 00 A discontinuities come from the numerical procedure find-
| i | i | i | i
in the compensation stage. To estimate the acceptable ing such t for which the mapping fidelity is maximal.
A
value of κ, we plot the average fidelity and the average The time t of the mapping operationis a function ofκ,
A
probability of success as functions of κ. The population andthemappingfidelity reachesitsmaximalvaluewhen
of the excited atomic state changes also the time tc for the population of the excited state reaches its minimal
which the improvement of the fidelity in our protocol is value. Sincethepopulationoftheexcitedstateoscillates
thebestoneandthusthevalueoftc calculatedfrom(18) thenumericallycalculatedtA jumps,asκincreases,from
canbeusedonlyasastartingpointinthenumericalcom- one value for which the population of the excited state
putation of this time. From numerical results presented is minimal after the mapping operation to the next such
in Fig. 4 we find that there is a plateau in the fidelity value. Thus, the factor exp( κt /2) and the fidelity of
A
−
of the modified protocol up to κ/2π 0.25 MHz after the teleported state also exhibit discontinuous behavior.
≈
which the fidelity jumps down. We consider the value of Fig.5 shows thatthe probability ofsuccess in the proto-
κ at the jump as the biggest value of κ. col with improved fidelity is always less than the proba-
Third,thepopulationoftheexcitedatomicstateoscil- bilityofsuccessintheprotocolofBoseetal. Fortunately,
lates. Sincethepopulationoftheexcitedstatediminishes there is only a small difference between the probabilities
the fidelity of operations periodically, it is necessary to of both protocols for the biggest cavity decay rate for
compute numerically, for all operations, such times that whichcompensationisstillpossible,i.e.,forκ/2π 0.25
≈
minimize the population simultaneously maximizing the MHz.
fidelity. Until now we have assumed that times t and So far we have assumed that there is no possibility of
A
6
0.5 dark counts. These imperfections lead to lowering the
average fidelity in both teleportation protocols because
0.4
y ofrandomnesswhichthey introducetothe measurement
t
i outcome. Thereisnowaytodistinguishtheunsuccessful
l0.3
i caseoftwophotonemissionsfromthedesiredcaseofone
b
ba0.2 photon emission when only one of the two emitted pho-
o
r tons is detected. It is also not possible to recognize the
P0.1 unsuccessfulcaseofno emissionif onedarkcountoccurs
during the detection stage. The quantum information
50 100 150 200 250 that Alice wants to teleport is destroyed in the unsuc-
Κ(cid:144)2Π @kHzD
cessful cases. If one cannot reject such cases then the
average fidelity is reduced. Therefore it is necessary to
FIG.5: Theaverageprobabilityofsuccessfulteleportationas
usedetectorswithveryhighefficiencyηandalowenough
a function of the cavity decay rate. The diamonds show the
darkcountrate. As faras we know, the highest detector
average probability of success in thenewprotocol. Theopen
efficiency has been reported by Takeuchi et al. [36] and
squares correspond to the average probability in Bose et al.
is equal to η = 0.88. To study the effect of the detector
protocol. Theaveragesaretakenover20000trajectories. The
inefficiency on the protocols under discussion, we have
parameters regime is (∆;Ω;g;γ)/2π=(100;10;10;0) MHz.
performed numerical calculations under the assumption
that there are not dark counts first. We have used the
same parameters as previously, i.e., (∆;Ω;g;γ;κ)/2π =
photon emission to modes other than the cavity mode.
(100;10;10;1;0.265)MHz and we have found that both
Let us now relax this assumption and investigate the in-
protocols are sensitive to the detector inefficiency. The
fluence ofthe spontaneousemissiondecayrate ofthe ex-
average fidelity is reduced to 0.894 in the Bose et al.
cited state on both teleportation protocols. The sponta-
protocol and to 0.905 in the modified protocol. Success
neousatomicemissiondestroysthequantuminformation
rates remain almost unchanged — 0.353 in the Bose et
which Alice wants to teleport to Bob. Such runs of the
al. protocol and 0.306 in the modified protocol. It is
teleportation protocols are unsuccessful and should be
obvious that the reliable teleportation requires detectors
rejected. However,aneventofspontaneousatomicemis-
efficiency η = 0.88 or higher. Unfortunately, the dark
sion cannot be detected in both schemes and therefore
count rate of the detector increases roughly exponen-
the spontaneous decay rate of excited state reduces the
tially with the efficiency [36] and is as high as 20 kHz
average fidelities. We can only suppress this imperfec-
at the highest efficiency reported by Takeuchiet al. [36],
tion by taking γg2/∆2, γΩ2/∆2 κ [32]. The biggest κ
≪ i.e., η = 0.88. The high efficiency of the detector means
forwhichthecompensationisstillpossibleallowsforthe
also the high rate ofdark counts, which are not goodfor
choice of γ/2π=1 MHz. We have generated 20 000 tra-
teleportation. To clarify the situation, we have also in-
jectories to compute the average fidelities and the aver-
vestigated the influence of the dark count rate on both
age probabilities for the parameters (∆;Ω;g;γ;κ)/2π =
teleportation protocols. Surprisingly, the protocol with
(100;10;10;1;0.265)MHz. As a result we have obtained
improvedfidelityhasappearedtobelesssensitivetothis
the average fidelity of 0.972 and the average probabil-
ity of 0.36 for the Bose et al. protocol and the average imperfection than the Bose et al. protocol. The average
fidelity in the Bose et al. protocol appeared to be equal
fidelity of0.978andthe successrateof0.31forthe mod-
to 0.801 while the average fidelity in the modified pro-
ified protocol. The results indicate that the inability to
tocol to be equal to 0.897, for the parameters η = 0.88
distinguish the runs of protocols, in which spontaneous
and the dark count rate 20 kHz. The difference between
emission occurs, reduces only slightly the average fideli-
the two protocolsis quite impressive,but it has a simple
ties when γ/2π = 1 MHz. The average probabilities of
explanation. In either protocol there is only one stage
success remain unchanged.
when the detection of one photon is expected — the de-
Other two important imperfections, which we have to tection stage in the Bose et al. protocol and the detec-
take into account, are a finite detection quantum effi-
tion stage I in the modified protocol. Only in these two
ciency and the presence of dark counts. It is necessary
stages occurrence of the dark count can be erroneously
to include such sources of noise in our numerical calcu-
accepted as a successful measurement event because all
lations because they areintroducedby all realdetectors.
otherstagesrequirenophotondetectiontobesuccessful.
So far we have assumed in our analysis perfect detectors
Thus,onecaneasilyunderstandwhythe influenceofthe
that are able to registerall collectedphotons and do not
dark counts on both protocols is different by comparing
produce any signal in the absence of photons. In prac-
the times of the two crucial stages — the time of the
tice, this assumption is not valid. The probability that detection stage of the Bose et al. protocol (in our calcu-
a single photon reaching the detector is converted into
lations we set t =10κ−1) that is much longer than the
the measurable signal, which is called the quantum effi- D
time of the detection stage I (t =πδ−1) of the modified
ciency and denoted by η, is less than unity for all real d
protocol. This means that there are many more rejected
detectors [33, 35]. Moreover, there are clicks, for all real dark counts in the modified protocolthan in the Boseet
detectors, even in the absence of light. They are called
7
al. protocol. A bigger number of rejected runs with the 1
dark count events leads to an increased average fidelity
0.9
andatthe sametime toadecreasedsuccessrate. There-
y
fore, the success rate is reducedmore significantly in the t0.8
modifiedprotocol(0.237)thanintheBoseetal. protocol li
e
(0.331). d0.7
i
F
Finally,wegeneralizeourcalculationstoincludelosses 0.6
in the mirrors and during the propagation. The absorp-
tion in the mirrors can be taken into account by making
0.2 0.4 0.6 0.8 1
the replacement κ = κ′ + κ′′ in the Hamiltonian (1), 1-Η’
where κ′ is the decay rate corresponding to the photon
transmission through the mirror and κ′′ is the photon FIG. 6: The average fidelity including the effects of photon
lossratedue to absorptioninthe mirrors. The evolution losses as a function of the overall detection inefficiency for
of the system is conditional, so we need also the collapse (∆;Ω;g;γ;κ)/2π = (100;10;10;1;0.265) MHz and the dark
operators corresponding to the absorption of photons in count rate 20 kHz. The diamonds correspond to the new
the mirrors. The additional collapse operators are given protocol and the open squares correspond to the Bose et al.
by C = √2κ′′a and C = √2κ′′a . As before, the protocol.
A A B B
collapseoperatorsdescribingphotondetectionsaregiven
by (3) but with κ replaced by κ′. So, we now have two
extra collapse operators describing evolution of the sys- 0.4
tem. However,itcanbe checkedthat suchevolutioncan 0.35
be described without using the extra collapse operators y 0.3
t
when we make the replacement κ = κ′ +κ′′ in the col- i0.25
l
lapse operators given by equation (3) and multiply the bi 0.2
probability of photon detection by η = (κ′/κ), which a
a b0.15
o
is the probability that a photon is detected despite the Pr 0.1
fact that there is absorption in the mirrors. The proba-
0.05
bility of detection in the presence of absorption is then
P′ = η P . The presence of absorption means effec- 0.2 0.4 0.6 0.8 1
D a D
tively lower efficiency of the detector. 1-Η’
In the same way, we easily can take into account all FIG. 7: Average probabilities that measurement indicates
photon losses during the propagation between the cav- success for the modified protocol (diamonds) and for the
ities and the detectors [5, 37]. All we need to include Bose et al. protocol (open squares) as functions of the
such losses into consideration is to introduce additional overall detection inefficiency. The parameter regime is
efficiency factor η . Multiplying all the factors, we find (∆;Ω;g;γ;κ)/2π = (100;10;10;1;0.265) MHz, the dark
p
the overall detection efficiency η′ = η η η. To visual- count rate is 20 kHz.
a p
ize the effect of such losses, we have plotted the average
fidelity and the average probability for both protocols
as functions of the overall detection inefficiency, i.e., as
functions of 1 η′. In order to make the average values V. CONCLUSIONS
−
reliable, we have generated 100 000 trajectories for each
η′. From figure 6 it is clear that with increasing photon
We have presented the teleportation protocol for the
losses the average fidelity is reduced for both protocols.
device proposed by Bose et al. that improves the fi-
However, the advantage of the modified protocol to be
delity of teleported state. The improvement is obtained
less sensitive to the dark counts and the compensation
by compensating for the factor e−κtA/2 which appears
for the factor e−κtA/2 result in the fidelity improvement
in the teleportation protocols. We have shown that this
that is clearly visible for almostall values of η′. The dif-
compensation makes it possible to stabilize the fidelity
ference between both protocols disappears only for such
at a high level despite the increase in the cavity decay
a small η′ that most of the trajectories for which mea-
rate. The fidelity is stabilized until κ/2π 0.25 MHz.
surement indicates success are unsuccessful cases due to ≈
This means that the high fidelity teleportation can be
dark counts. Of course, in such a case the final state of
performed for the values of the cavity decay rates over
Bob’s atom is random and the averagefidelity is 0.5.
25 times larger than the values assumed by Bose et al..
From Fig. 7 it is visible that higher fidelity can be The price we have to pay for more realistic values of the
achieved by accepting lower success rates. The average cavitydecayratesisthatwehavetoacceptlowersuccess
probability of success in the modified protocol is always rates. We have also shown that the modified protocol
less than the averagesuccess rate in the Bose et al. pro- is less sensitive to the dark counts of detectors than the
tocol. This isthe pricewehaveto payforhigherfidelity. original protocol of Bose et al.
8
ACKNOWLEDGMENTS No. 1 P03B 064 28.
This work was supported by the Polish Min-
istry of Science and Higher Education under Grant
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