High-fidelity atomic-state teleportation protocol with non-maximally-entangled states

Grzegorz Chimczak and Ryszard TanaśDepartment of Physics, Nonlinear Optics Division, Adam Mickiewicz University, 61-614 Poznań, Poland

�Received 20 August 2008; published 8 April 2009�

We propose a protocol of the long-distance atomic-state teleportation via cavity decay, which allows forhigh-fidelity teleportation even with currently available optical cavities. The protocol is based on the schemeproposed by Bose et al. �Phys. Rev. Lett. 83, 5158 �1999�� but with one important modification: it employsnon-maximally-entangled states instead of maximally entangled states.

DOI: 10.1103/PhysRevA.79.042311 PACS number�s�: 03.67.Hk, 03.67.Mn

I. INTRODUCTION

Recent years witnessed considerable progress both in the-oretical and experimental quantum information science. Thelong-range goal in the field is the realization of quantumnetworks composed of many nodes and channels. Thepresent status of the research in the field has been reviewedin �1�. The nodes of the quantum network require quantumsystems that can store quantum information for sufficientlylong time and quantum channels which should allow for fasttransfer of quantum information between the nodes. A singleatom �or ion� can be considered as a perfect quantummemory—qubit can be stored in atomic states even for 10 s�2�. Thus, trapped atoms are candidates for being compo-nents of quantum registers or nodes of quantum networks.Fast connections between the nodes can be realized withphotonic qubits which are the best carriers of quantum infor-mation. To transfer quantum information stored in one nodeto another node through the photonic channel, it is necessaryto have effective methods for mapping atomic states intofield states and back �3–7�. A number of schemes for creatingentanglement and performing quantum teleportation hasbeen proposed �8–19�. Next step would be to accomplish thelong-distance atomic-state teleportation mediated by pho-tons, but this task appears to be very challenging and has notbeen experimentally achieved yet.

A pretty simple way to complete a long-distance telepor-tation of atomic states mediated by photons was proposed byBose et al. �11�. Some modifications of this protocol can alsobe found in �20–22�. The teleportation scheme of Bose et al.�11� consist of two atom-cavity systems, a 50:50 beam split-ter, and two detectors as depicted in Fig. 1. With this devicethe teleportation can be carried out by just performing thejoint detection of both cavities fields if, before detection, thesender �Alice� maps the state of her atom onto the field stateof her cavity, and the receiver �Bob� creates the maximallyentangled state of his atom and his cavity field. Recentprogress in technology allows for such state mapping �6,7�and performing the joint detection �23�. Creation of themaximally entangled state of the atom-cavity system alsoshould be possible with the current technology. However, theBose et al. protocol �11� is hardly feasible because the fidel-ity of state mapping is drastically reduced by large dampingvalues of the currently available cavities.

In this paper we propose a modification of the Bose et al.scheme �11� consisting in exploiting, instead of the maxi-

mally entangled state, a non-maximally-entangled state withthe amplitudes chosen in such a way that the damping factorsintroduced by the state mapping are fully compensated for.With this modification of the protocol, it should be possibleto achieve high teleportation fidelities even with currentlyavailable cavities. The price we have to pay for the higherfidelities is a lower probability of success.

II. TELEPORTATION PROTOCOL WITH NON-MAXIMALLY-ENTANGLED STATES

First, let us present the main idea in a simplified way—comparing it to the standard teleportation protocol �24,25�.In the standard teleportation protocol Alice has unknown toher �and to Bob� qubit ���=��0�+��1� and one qubit of theEinstein-Podolsky-Rosen �EPR� pair. The second qubit of theEPR pair is on Bob’s site. Suppose, however, that we havethe situation depicted in Fig. 1. The state to be teleported isinitially stored in the Alice atom and next is mapped usingthe laser to the cavity field qubit, but the mapping is not

FIG. 1. �Color online� The teleportation device and level schemeof the � atom interacting with the classical laser field with couplingstrength � and with the quantized cavity mode with the couplingstrength g. Both fields are detuned from the corresponding transi-tion frequencies by �.

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perfect, and the initial state is slightly distorted. Let the stateof the Alice cavity field takes the form

���� = N���0� + ���1�� , �1�

where N=1 /����2+ ���2���2 is the normalization factor and �is a parameter that measures to what degree the original statehas been distorted. If there is no distortion �=1, and the stateis just the original state. It is important that the parameter �does not depend on the original state �it does not depend on� and �� but depends solely on the mapping procedurewhich is known for both parties of the protocol. Both partiescan agree on the details of the procedure in advance. Nowthe question arises: can we use our knowledge of � to im-prove the fidelity of the teleported state?

The standard teleportation protocol would teleport the dis-torted state �Eq. �1�� to Bob. However, if we choose thenon-maximally-entangled state, instead of the maximally en-tangled state, in the teleportation protocol, we can correct theimperfections introduced by the mapping procedure by usinga slightly modified teleportation protocol. The teleportationcircuit for this protocol is illustrated in Fig. 2.

The first qubit is the Alice cavity field qubit, the state ofwhich is initially the state ���� given by Eq. �1�, and the state��� is the non-maximally-entangled state given by

��� = a�10� + b�01� . �2�

The overall initial state is thus

�0� = ���� � ��� = N���0� + ���1�� � �a�10� + b�01��

= N���0��a�10� + b�01�� + ���1��a�10� + b�01��� . �3�

It is easy to show using the circuit from Fig. 2 that justbefore the measurements, the state is given by

�1� =N�2

��00��b��1� + a���0�� + �11��b��1� − a���0�� + �01�

�a��0� + b���1�� + �10��a��0� − b���1��� . �4�

Now, we see that if we prepare the non-maximally-entangledstate �Eq. �2�� in such a way that a=�b we obtain

�1� =Nb

�2���00����1� + �2��0�� + �11����1� − �2��0���

+ ���01����0� + ��1�� + �10����0� − ��1��� . �5�

When Alice performs the measurement on her two qubits,there are two cases when only one of the detectors registers aphoton, and the state is projected either to �01� or �10�. Sincewe assume that the beam splitter is used in the measuringapparatus, only the two outcomes are considered as success-ful because the beam splitter can only distinguish two states

from the Bell basis. The other two outcomes are rejected asunsuccessful. Alice next communicate to Bob, using the clas-sical channel, the results of her measurement �two classicalbits�, and Bob applying the postmeasurement operationsshown in Fig. 2, can recover the original Alice’s state ���with the perfect fidelity.

Of course, the teleportation scheme depicted in Fig. 2works perfectly well as the standard teleportation protocolwhen the measuring device can distinguish all four Bellstates, the original undistorted state ��� is initially on the firstqubit ��=1�, and the shared entangled state ��� is the maxi-mally entangled state �a=b=1 /�2�.

III. PHYSICAL MODEL

In the first stage of teleportation protocol, when Alice hasto map the initial state of her atom ���=��0�+��1� onto thefield state of her cavity and when Bob has to create an en-tangled state of his atom and his cavity field, the most im-portant role in the teleportation protocol play the two atom-cavity systems. Let us first describe them in more detail.Alice and Bob can change the state of their own atom-cavitysystem by switching their lasers on. When the laser illumi-nates the atom trapped inside the cavity then the evolution ofthe atom-cavity system is governed by the effective non-Hermitian Hamiltonian ��=1 here and in the following�,

H = �� − i�� 22 + �� 21 + ga 20 + H.c.� − i�a†a , �6�

where ij �i��j� denote the atomic flip operators and a de-notes the annihilation operator of the cavity field mode. Onemirror in each cavity is partially transparent to allow for thejoint measurement of the fields leaking out from both cavi-ties. Of course, the transparency of the mirror leads to adamping of the cavity field mode. We assume that photonsleak out of the cavity at a rate 2�. For simplicity, we neglectthe spontaneous decay rate of the excited atomic state �. Thisapproximation is valid if conditions ��g, �, � and�g2 /�2 ,��2 /�2�� are fulfilled �26�. We can further sim-plify Hamiltonian �6� assuming that �=g. Then, after adia-batic elimination of the excited atomic state, the Hamiltoniantakes the form

H = − � 11 − �a†a 00 − ��a 10 + H.c.� − i�a†a , �7�

where �=g2 /�. Using Hamiltonian �7� one can easily getanalytical expressions describing evolution of the initialquantum states �0�atom�0�mode and �1�atom�0�mode. First of thestates experiences no dynamics because there is no operatorin Hamiltonian �7� which can change this state. The evolu-tion of the second state is given by

e−iHt�10� = ei�te−�t/2�ia�t��01� + b�t��10�� , �8�

where we abbreviate the atom-cavity state �j�atom � �n�mode to�jn� and we use

|Φ〉{|φ′〉 • H • ��������

�������

�������

• •

X X��������

�������

�������

X X •

X Z |φ〉

FIG. 2. Modified teleportation circuit.

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a�t� =2�

��

sin���t

2 ,

b�t� = cos���t

2 +

�

��

sin���t

2 , �9�

where ��=�4�2−�2. If the laser is turned off ��=0� thenthe Hamiltonian takes the form H=−�a†a 00− i�a†a andthen the time evolution of the system can be obtained usingthe relations

e−iHt�10� = �10� ,

e−iHt�01� = ei�te−�t�01� . �10�

Equations �10� are needed to describe evolution of the devicestate during the second stage, in which the joint measure-ment of both cavities fields is performed. At this stage of theprotocol the most important role play the detectors D+ andD− together with the beam splitter BS. Registration of thephoton emission by one of the detectors corresponds to theaction of the collapse operator on the joint state of Alice’sand Bob’s systems. The collapse operator has the form

C = ���aA + i�aB� , �11�

where � is 1 for photon detection in D+ and −1 for photondetection in D−.

IV. TELEPORTATION VIA CAVITY DECAY WITH NON-MAXIMALLY-ENTANGLED STATES

Now, we can analyze the modified teleportation protocolwhich makes it possible to compensate fully for the destruc-tive effect of cavity decay and, as we believe, it could berealized even with currently available cavities. The telepor-tation protocol consists of three stages, so it is as simple asthe original teleportation protocol of Bose et al. �11�. Thethree stages are �A� the preparation stage, �B� the detectionstage, and �C� the recovery stage. At the beginning of theprotocol Alice’s atom is prepared in a state, which is un-known for Alice. Bob’s atom is prepared in the state �1�atom.Initially the field modes of both cavities are empty, so thestates of both atom-cavity systems are given by

���A = ���atom � �0�mode = ��00�A + ��10�A, �12�

���B = �10�B. �13�

As we have mentioned above, Alice has to map the statestored in her atom onto the field state of her cavity in thepreparation stage. She can do it by just turning her laser onfor the time tA= �2 /�����−arctan��� /��� �11,27�. After thisoperation her atom-cavity system is found to be in the state

��̃�A = ��00�A + iei�tAe−�tA/2��01�A. �14�

It is seen that the state mapping is done although it is imper-fect because of the damping factor e−�tA/2. We cannot avoidthis damping factor, but we can show that it is possible tocompensate for it. To this aim, in the modified teleportation

protocol, Bob creates a non-maximally-entangled state in-stead of creating maximally entangled state as it is done inthe standard teleportation protocol. He turns his laser on fortime tB changing his system state to

��̃�B = e−�tB/2�ia�tB��01�B + b�tB��10�B� . �15�

The expression for tB will be given later. Now, we have toderive the expression for probability that the first stage issuccessful. The preparation stage will succeed only under theabsence of photon detection event. Probabilities that no col-lapse occurs during Alice’s and Bob’s operations are givenby the squared norms of the state vectors �14� and �15�,respectively. They are given by

PA = ���2 + e−�tA���2,

PB = e−�tB��a�tB��2 + �b�tB��2� . �16�

Alice and Bob complete their actions in the same instant oftime. Then they turn the lasers off and the detection stagestarts. Alice during the second stage just waits for a finitetime tD��−1 registering events of photon detection. Thisstage and the whole teleportation protocol is successful whenAlice registers one, and only one, photon. In other cases,when Alice registers no photon or when she registers twophotons, the initial Alice’s state is destroyed. Until the timeof photon detection tj the evolution of the state of both atom-cavity systems is given by �10�, and at time tj both systemsstates are described by

��̃�tj��A =1

�PA

�iei��tA+tj�e−��tA+2tj�/2��01�A + ��00�A� ,

�17�

��̃�tj��B =e−�tB/2

�PB

�ia�tB�ei�tje−�tj�01�B + b�tB��10�B� . �18�

The probability of no photon emission before time tj is givenby PD�tj�= PA�tj�PB�tj�, where

PA�tj� =���2 + e−��tA+2tj����2

���2 + e−�tA���2,

PB�tj� =�a�tB��2e−�2tj + �b�tB��2

�a�tB��2 + �b�tB��2. �19�

At time tj one of the detectors registers a photon emission,what corresponds to the change in the joint state of bothatom-cavity systems. After the collapse the joint state isgiven by ��̃�tj��=C���tj��A � ���tj��B. The probability that thecollapse occurs in the time interval tj to tj +dt can be calcu-lated from 2��̃�tj� � �̃�tj��dt. After the collapse we have tonormalize the state ��̃�tj��→ ���tj�� and then the evolution ofthe joint state can again be determined using Eq. �10�, and atthe end of the detection stage, at time tD, the state is given by��̃�tD��=exp�−iH�tD− tj�����tj��. For tD��−1 the normalizedjoint state can be very well approximated by ���tD��= �00�A� ��B�tD��, where

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��B�tD�� =ei�tAe−�tA/2�b�tB��10�B + ��ia�tB��00�B

�e−�tA���2�b�tB��2 + ���2�a�tB��2. �20�

From Eq. �20� it is seen that the unwanted damping factore−�tA/2 disappears if the condition

e−�tA/2b�tB� = a�tB� �21�

is satisfied. We can now give the expression for the time tB;

tB =2

���arctan� ��e−�tA/2

2� − e−�tA/2� + n�� . �22�

The time given by expression �22� is the key parameter,which must be known to Bob to create the non-maximally-entangled state �2�. Time tB is the function of tA, so boththese times must be known to Bob. Note, however, that nei-ther time tB nor time tA depend on the amplitudes of theteleported state. If in the preparation stage Bob turns his laseron for time tB then, after the detection stage, the state of hissystem becomes identical to the initial unknown Alice’s stateexcept for the phase factor

��B�tD�� = ��00�B − i�ei�tA��10�B. �23�

Fortunately, the phase factor can be removed using the Zee-man evolution �11�, what Bob performs in the last stage afterreceiving classical information about Alice’s measurement.At the end of the whole protocol Bob has the original Alice’sstate stored in his atom ���=��0�atom+��1�atom. It turns outthat the fidelity of this teleportation protocol can be close tounity even for realistic cavity decay rates. Figure 3 showsteleportation fidelities of both protocols �the modified withnon-maximally-entangled state and the Bose et al. protocol�as functions of the cavity decay rate.

One can see that for real cavity decay rate � /2�=3.8 MHz �6� the fidelity of teleported state is still equalunity while the fidelity of the original protocol does not ex-ceed the value 2/3. This result is quite impressive but one caneasily note that the high fidelity is not for free. Since PAdepends on the damping factor, the probability that the tele-portation protocol will be successful is lowered by the in-creasing cavity decay rates. Let us now estimate the prob-ability of success for currently available cavities. Theprobability that all stages of the protocol will succeed has thefollowing form:

Psuc = PAPB�0

tD

PA�tj�PB�tj���̃�tD���̃�tD��2��̃�tj���̃�tj��dtj .

�24�

For tD��−1, the probability of success can be very wellapproximated by

Psuc = e−�tBa�tB�2. �25�

We can use this simple formula to estimate the value of thesuccess probability for the experimental parameters of Ref.�6�, i.e., we take �g ,�� /2�= �16,3.8� MHz. The protocol re-quires, however, bigger values of the detuning than that ofRef. �6�, so we take � /2�=100 MHz. With this set of pa-rameters we get the probability of success of about 0.005,and this is the price we have to pay for getting fidelity closeto unity. In Fig. 4 we plot the probability of success as afunction of the cavity decay rate � for the modified protocoland compare it to the corresponding dependence for theoriginal protocol of Bose et al. �11� for the parameter values�� ,g� / �2��= �100,16� MHz. It is seen that the probabilityof success for the modified protocol goes to zero faster thanthat for the original protocol, but it still has considerablevalues for realistic decay rates.

Let us now take into account the important imperfectionwhich is present in all real setups, i.e., finite detection effi-ciency. This imperfection is caused by absorption in the mir-rors, photon losses during the propagation between the cavi-ties and the detectors, and by nonunity detectors efficiency.In Ref. �28� the overall detection efficiency is only �=0.05.Therefore, with such efficiency only a small fraction of allsuccessful runs will be detected. Moreover, the case of twophotons emissions will be erroneously counted as a success-ful case if only one photon is detected. Of course, this effectwould lead to lowering of the fidelity. The two-photon caseis also very important if detectors cannot distinguish a singlephoton from two photons, since both emitted photons arealways collected by the same detector �23�, it is not possibleto reject such unsuccessful runs. If we want to estimate thereal values for the teleportation fidelity and the success prob-ability, we have to include the efficiency � and the two-photon case in our calculations. The probability that twophotons will be emitted from the cavities during the telepor-tation protocol and only one of them will be detected in thedetection stage is given by

1 2 3 4 5Κ�2Π �MHz�

0.6

0.7

0.8

0.9

1

1.1

F

FIG. 3. The teleportation fidelity as a function of cavity decayrate � for the modified protocol �solid line� and the original protocol�dashed line� for �� ,g� / �2��= �100,16� MHz.

1 2 3 4 5Κ�2Π �MHz�

0.1

0.2

0.3

0.4

0.5

Probability

FIG. 4. Probability of success versus cavity decay rate � for themodified protocol �solid line� and the original protocol �dashed line�for �� ,g� / �2��= �100,16� MHz.

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P2em��� = ���2e−�tA��1 − ��� , �26�

where �=1 for photon-number-resolving detectors and �=1− Psuc for conventional photon detectors. This probability de-pends on the modulus of the amplitude � which is in generalunknown. Hence, it is necessary to compute the averageprobability of two photon emissions taken over all possibleinput states. Such an average probability takes the form

P̄2em��� = e−�tA��1 − ���/2. �27�

The average probability that the measurement will indicatesuccess is then given by

P̄suc��� = �Psuc + P̄2em��� . �28�

In the case of two-photon emissions Bob’s atom is in thestate �0�. If we cannot reject all runs in which two photonswere emitted then the final state of Bob’s atom is a mixtureof �0� and ���, i.e.,

� =�Psuc������ + P2em����0��0�

�Psuc + P2em���. �29�

We can calculate the average fidelity using the density matrix�. The average fidelity of the teleportation protocol is givenby

F̄��� = 1/2 + Psuc/B − �Psuc/B�2ln�1 + B/Psuc� , �30�

where B=exp�−�tA��1−���. It is now evident that there isone more important feature of the protocol. Expressions �28�and �30� indicate that for large cavity decay rates, it is almostirrelevant if the detectors can distinguish a single photonfrom two photons. For sufficiently large cavity decay ratesPsuc is small, and therefore, � for conventional detectors isclose to unity. For example, the parameter’s regime�� ,� ,g ,�� / �2��= �100,16,16,3.8� MHz leads to �=0.995.Hence, possible implementations of the protocol with cur-rently available cavities do not require detectors with thesingle-photon resolution.

V. NUMERICAL RESULTS

The analysis of experimental feasibility of this protocolrequires taking into account another imperfection, which isthe spontaneous emission from the atom. We have done itusing numerical calculations. In the following we presentsome details of the calculations. We have calculated the av-erage fidelity and the average probability of success usingHamiltonian �6� and the quantum trajectories method�29,30�. Unfortunately, the evolution of the atom-cavity sys-tem is different than that described by Eq. �8� for the param-eters of Ref. �6�: the population of the excited state �2� can-not be neglected and the periodic behavior of the system islost because of the damping present in the system. Neverthe-less, we can choose parameters close to that of Ref. �6� forwhich the average fidelity of the teleportation protocol is stillhigh. For the well chosen parameters, times tA and tB shouldnot be too long as compared to �−1 and �−1. If we want tosatisfy this condition, we have to set � to be small enough.

Then, however, we get considerable population of the ex-cited state �2�. Fortunately, this population oscillates, and wecan use the fine tuning technique �31� to minimize its effect.Applying this technique we have chosen�� ,� ,g ,� ,�� / �2��= �62.5,16,16,4 ,2.6� MHz. For theseparameters analytical expressions for tA and tB are not pre-cise enough, and therefore we have used numerically opti-mized times tA=0.1058 �s and tB=0.0131 �s and not toolong detection time tD=4�−1�0.16 �s. The detection timetD=4�−1 is long enough to get a quite high value of thefidelity �11�. We do not set longer times of the detectionstage to make the influence of the dark counts on the proto-col negligible. For the dark count rate of 50 s−1 �32� themean time between dark counts �10 ms for both detectors� ismuch larger than the time window for detection in the pro-tocol tA+ tD�0.3 �s. Thus the dark counts can be neglected.Nevertheless, we have taken them into account in our nu-merical calculations. Results obtained from quantum trajec-tory approach are presented in Figs. 5 and 6.

In order to analyze the experimental feasibility of the pro-tocol and abilities of improving the fidelity, we plot in Fig. 5the average fidelity as a function of the overall detection

0.2 0.4 0.6 0.8 1Η

0.7

0.75

0.8

0.85

0.9

0.95

1

F

FIG. 5. The average fidelity of teleportation given by Eq. �30�for photon-number-resolving detectors �solid curve� and for con-ventional single-photon detectors �dashed curve� as a function ofthe overall detection efficiency. The diamonds and open squaresrefer to numerical results for photon-number-resolving and conven-tional detectors, respectively. The numerical results include sponta-neous emission from excited atomic state and dark counts. Theparameters are �� ,� ,g ,� ,�� / �2��= �62.5,16,16,4 ,2.6� MHz.

0.2 0.4 0.6 0.8 1Η

0.02

0.04

0.06

0.08

0.1

P

FIG. 6. The average probability of success vs the overall detec-tion efficiency. For detectors, which are able to distinguish a singlephoton from two photons analytical results given by Eq. �28� arepresented by the solid curve and numerical results are presented bydiamonds. The dashed curve and open squares correspond to ana-lytical and numerical results, respectively, for conventional single-photon detectors. The parameters are �� ,� ,g ,� ,�� / �2��= �62.5,16,16,4 ,2.6� MHz.

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efficiency �. As it is evident from the figure, the averagefidelity tends to 0.794 with decreasing �. So, the averagefidelity significantly exceeds the value 2/3 even for the realoverall detection efficiency �=0.05. Of course, the sponta-neous decay rate � /2�=2.6 MHz and dark counts of 50 s−1

reduce the average fidelity, but it is still well above the limitof 2/3. It is important because the average fidelity of theteleportation based on classical resources only cannot exceedthis limit �33,34�. Note that the protocol makes it possible toachieve values of the fidelity much higher than 0.794. Inprinciple, we can obtain the fidelity even close to unity, but itwould require better than currently available overall detec-tion efficiencies. The effect of the overall detection ineffi-ciency on the teleportation protocol is much stronger thanthat of other imperfections present in real experimental set-ups. Also, the probability of success is lowered by nonidealoverall detection efficiency, as it is evident from Fig. 6. Theprobability of success tends to zero with decreasing �. Forthe currently available efficiency of 0.05, the success rate hasthe value of 0.005, which means that it takes on averagehundreds of runs to get successful teleportation. Such smallprobability of success means that this teleportation protocolwill not have commercial applications for currently availablecavities and detectors. However, this probability is bigenough to perform long-distance teleportation of atomicstates and test it. With present day technology 2000 trials ofprotocol that consists state mapping stage and detection stagelast 360 ms �6� only. Therefore all data required can be col-lected in a reasonable time.

From Figs. 5 and 6 it is also seen that the expensivephoton-number-resolving detectors are not necessary for theparameters used in our computations. For � close to unity,there is only a small difference between the fidelity obtainedwith the assumption that the detectors have the ability todistinguish a single photon from two photons and the fidelityobtained with assumption that the detectors have not suchability. For the real overall detection efficiency �=0.05 thedifference is indistinguishable.

VI. EXPERIMENTAL FEASIBILITY OF THE PROTOCOL

Finally, we shortly discuss the realizability of our telepor-tation protocol. As mentioned above, almost all parametersused in our computations are feasible with current technol-ogy. The only parameter the value of which may be demand-ing for present technology is the detuning. In our numericalcalculations we have chosen � / �2��=62.5 MHz, which isthe value six times greater than that of Ref. �6�. Moreover, sofar we have assumed that the laser pulses have rectangular

shapes. This assumption makes it possible to examine theproposed teleportation protocol analytically and numerically.However, the shortest rising time of such pulse has duration100 ns �6,35�. Therefore, real pulses that are approximatelyrectangular cannot be shorter than 1 �s. The pulses durationtimes used in our numerical calculations are much shorter:tA=0.1058 �s and tB=0.0131 �s. So, it is rather unrealisticto implement experimentally the protocol in its present form.Nevertheless, the protocol can be easily adapted for usingother shapes of laser pulses. All what is actually needed tocomplete this teleportation protocol is the ability to performthe state mapping

��00� + ��10� → ��00� + e−�t/2��01� , �31�

and the ability to generate the non-maximally-entangled state

�10� → a�t��01� + b�t��10� , �32�

with small �a�t��2. First of these operations have already beendemonstrated experimentally �6�. The second operation canbe achieved with short Gaussian pulses.

VII. CONCLUSIONS

In conclusion, we have presented a modified protocol that,in principle, should allow for atomic-state teleportation viacavity decay using currently available optical cavities. Wehave shown that the destructive influence of large cavity de-cay on the fidelity of teleported state can be minimized byusing in the teleportation protocol the non-maximally-entangled state instead of the maximally entangled state.This happens despite the fact that both of them separatelylead to lowering of the teleportation fidelity �36�. Advantageof using non-maximally-entangled states has been indicatedalso for other quantum information protocols �37�.

We have also shown that there are two other distinguish-ing features of the protocol presented here which make iteasier to implement experimentally. First is the possibility ofusing conventional single-photon detectors instead of thephoton-number-resolving detectors. Second is the average fi-delity exceeding the limit 2/3 even for very small values ofthe overall detection efficiency. However, the high fidelitiesof teleported states for real cavities can be achieved with theprotocol at the expense of accepting low success rates.

ACKNOWLEDGMENT

This work was supported by the Polish research networkLFPPI.

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