Teleporting entanglements of cavity-field states

Geisa Pires,1 N. G. de Almeida,2,1,* A. T. Avelar,3 and B. Baseia11Instituto de Física, Universidade Federal de Goiás, 74.001-970, Goiânia (GO), Brazil

2Núcleo de Pesquisas em Física, Universidade Católica de Goiás, 74.605-220, Goiânia (GO), Brazil3Instituto de Física, Universidade de Brasília, 70.919-970, Brasília (DF), Brazil

(Received 15 May 2004; published 23 August 2004)

We present a scheme to teleport an entanglement of zero- and one-photon states from one cavity to another.The scheme, which has 100% success probability, relies on two perfect and identical bimodal cavities, acollection of two kinds of two-level atoms, a three-level atom in a ladder configuration driven by a classicalfield, Ramsey zones, and selective atomic-state detectors.

DOI: 10.1103/PhysRevA.70.025803 PACS number(s): 42.50.Dv, 03.65.Ud

I. INTRODUCTION

Since the proposition by Bennettet al. [1], the teleporta-tion phenomenon has received great attention, and a numberof protocols have been suggested to implement the process indifferent contexts, such as for trapped ions[2], runningwaves[3–5], and cavity QED[6]. Experimentally, teleporta-tion has been demonstrated for discrete variables[7–9] andfor a single-mode electromagnetic field of continuous vari-ables[10]. The main experimental challenge consists in theso-called Bell-state measurement, performed on the Bell op-erator basis for the particle whose state is to be teleportedplus its partner composing the quantum channel[1]. In theexperiment of Bowmeesteret al. [7], designed to manipulateonly the polarization state of single-photon pulses, only oneof the four Bell states is discriminated, resulting in a successrate of 25%. In the experiment of Boschiet al. [8], employ-ing the entanglement between the spatial and polarizationdegrees of freedom of a photon, it was possible to distinguishthe four Bell states, allowing a success rate of 100%. In theexperiment of Kimet al. [9], following exactly the originalprotocol by Bennettet al. [1] and taking advantage of non-linear interactions to accomplish the Bell-state measure-ments, a success rate of 100% can also be achieved in prin-ciple for teleporting a polarization state. Teleportation ofcontinuous variables was demonstrated in the experiment ofFurusawaet al. [10], with a fidelity of 0.58±0.02, higherthan the critical 0.5 classical bound attainable in the absenceof quantum correlation. Teleportation of an entangled qubitwith a success rate of 25% was recently reported[11], andefforts toward 50% success probability are in progress[11].

Teleportation of entanglements employing linear opticalelements, at the expense of decreasing the success probabil-ity to 50%, has also been studied[5,12], and a proposal forachieving high fidelity in the process of teleportation of en-tangled states in running waves has recently been addressed[13], with success probability near 100%. Teleportation(anddecoherence) of entangled coherent states in running waveshas also been studied[14], and a success probability of 50%was found.

In the realm of cavity QED, schemes have been proposedfor teleportation of two-particle entangled states[15], multi-particle entangled atomic states[6], and also entangled fieldstates inside high-Q cavities [16]. In Ref. [16] the authorspropose a scheme to teleport an entangled state composed oftwo qubits (readily generalized for teleportation of anN-qubit field state) from a pair of high-Q cavities to anotherpair of high-Q cavities. As usual, Ramsey zones and theStark shift effect are employed. Also, the authors suggest toemploy long-lived Rydberg atoms in circular states withprincipal quantum number around 50 for the controlledatom-field interaction times, in such a way that the wholelosses due to atomic spontaneous emission and dissipation inhigh-Q cavities were neglected.

Here we propose a scheme to teleport a two-mode en-tanglement of zero- and one-photon states from one cavity toanother. Unlike the scheme in[16], in which all six high-Qcavities required to perform the teleportation contain a singlemode, our scheme just needs two high-Q (identical) bimodalcavities[17]. Also, our scheme requires a collection of twokinds of two-level atoms, a three-level atom in a ladder con-figuration driven by a classical field, Ramsey zones, and se-lective atomic-state detectors. The success probability is100%. As assumed in[16], all the losses due to atomic spon-taneous emission and dissipation in the cavities are ne-glected. In fact, since the decoherence time is of the sameorder as the lifetimes of the qubits in a high-Q cavity and asthe (spontaneous) atomic decay, the experimental implemen-tation of the present scheme should be realized during 10−2 s,a typical time for both decoherence and damping of atomicand cavity qubits[18].

II. CONTROLLED INTERACTIONS

To perform teleportation of zero- and one-photon en-tangled states between two high-Q cavities we will need thefollowing operators:

Hon = "lss−a† + s+ad, s1d

Hof f ="l2

da†auelkeu, s2d

R± = Î1/2sI ± isyd, s3d*Corresponding author. Electronic address: nortongomes@ucg.br

(NGdeAlmeida)

PHYSICAL REVIEW A 70, 025803(2004)

1050-2947/2004/70(2)/025803(4)/$22.50 ©2004 The American Physical Society70 025803-1

Hc = "xsa†b + ab†d. s4d

Equation(1) is the usual Jaynes-Cumming model[19] anddescribes a resonant interaction of modea represented bycreation and annihilation operatorssa,a†d with a two-levelatom represented by the Pauli operatorss; l is the atom-fieldcoupling parameter. Equation(2) stands for the dispersiveatom-field interaction[19] and can be implemented via aStark shift;d=sv−v0d is the detuning between the field fre-quencyv and the atomic frequencyv0. Equation(3) repre-sents the action of the Ramsey zone properly adjusted toproduce eitherR+ or R− [18]. Equation(4) was recently ob-tained for both a three-level atom in ladder configuration anda two-level atom interacting with two modes of a cavity field[20], x being the coupling between the three- or two-levelatom and the field modes. Using the operators defined inEqs.(1)–(4), it is easy to verify the following evolutions:

R±Hugl → Î1/2suel ± ugld ; u ± l,

uel → Î1/2suel 7 ugld ; u 7 l,s5d

Uonsa,bd5uglau0lb → uglau0lb,

uelau0lb → − i uglau1lb,

uglau1lb → − i uelau0lb,

s6d

Uof fsa,bd5u + lau1lb → − u− lau1lb,

u− lau1lb → − u + lau1lb,

u + lau0lb → u + lau0lb,

u− lau0lb → u− lau0lb,

s7d

Ucsa,bdHu0lau1lb → C1u0lau1lb + iC2u1lau0lb,

u1lau0lb → C1u1lau0lb + iC2u0lau1lb.s8d

We note that the evolutionsUonsa ,bd andUof fsa ,bd areobtained by adjusting the interaction times aslt=p /2 andl2t /d=p from Uonsa ,bd=expf−iHont /"g and Uof fsa ,bd=expf−iHof ft /"g, respectively. The subindexa stands for at-oms in the ground and excited statessg,ed, andb representsthe modessa,bd and cavities(1,2).

Here we are assuming that the operationUs1,1ad leavesmode 1b unaffected. Indeed, this is a crucial point, sincerelative phases will necessarily appear from atom-field inter-actions. Next, let us prove that these phases are irrelevant. Asis readily seen from the Hamiltonian model for a two-levelatom interacting resonantly with, say, modeb and off-resonantly with modea, the effective Hamiltonian leads toan energy shift corresponding to modea. This shifting isinversely proportional to the detuningd=sva−v0d betweenthe field frequencyva of modea and the atom frequencyv0[see Eq.(2)]. Thus, if d is large enough compared with theatom-field coupling parameterl, we can neglect the effect ofthe off-resonant interaction. To estimate the additionalphases gained due to the dispersive interaction between oneof the two modes(va, vb) of the cavity and the atomic fre-quency of the two-level atom, let us consider the atom-fieldinteractions such thatsv0=vb.vad andlt=p /2. Thus, themode a will be shifted by the phasef=l2t / sv0−vad

=pl /2sv0−vad. Next, let us suppose a Stark shift on modeb leading to vb=v0+d such thatl2t /d=p. In this case,modea will be shifted by the phasef8=l2t / fsv0−vad±dg.Taking the experimental values[18] v0−va.1010 Hz, l.105 Hz, andd=10l, thenf.f8.10−5 rad, which can besafely neglected.

III. TELEPORTATION

A. Preparing the state to be teleported in cavity 1

First, a two-level atom 1,uel1, interacts resonantly withmode a, leaving cavity 1 with one photon. Then, a three-level atom [21] completes the preparation by entanglingmodesa and b (in this stage there is no further use for theatoms, so they are thrown away):

Uons1,1aduel1u0l1au0l1b → ugl1u1l1au0l1b, s9d

Ucs1a,1bdu1l1au0l1b → sC1u1lau0lb + C2u0lau1lbd ; ucl1a1b,

s10d

ucl1a1b = sC1u0̄l1a1b + C2u1̄l1a1bd, s11d

where we have definedu0̄li j ;u1liu0l j,u1̄li j ;u0liu1l j, and in-corporated the factori in the constantC2.

B. Preparing the nonlocal channel

The nonlocal channel is composed of cavity 2 and a two-level atom 2. The same procedure described to prepare thestate to be teleported in cavity 1 is now repeated for cavity 2with the modificationuC1u= uC2u=1/Î2. Next, the two-levelatom 2,uel2, crosses a Ramsey zoneR− and cavity 2 interactsoff-resonantly[Uof fs2,2bd with modeb], as indicated by Eq.(5) and Eq.(7). The result is

uxlNL =1Î2

su + l2u1l2au0l2b − i u− l2u0l2au1l2bd ;1Î2

su

+ l2u0̄l2a2b − i u− l2u1̄l2a2bd. s12d

The undesirable phases−id can easily be eliminated fromthe nonlocal channel by sending an excited two-level atominteracting off-resonantly with modea (or modeb) with theinteraction time adjusted tol2t /d=p /2. After that, the statedescribing the whole system composed of atom 2, cavity 1,and cavity 2 reads

uClT ; uxlNLucl1a1b =1Î2

hC1u0̄l1a1bu + l2u0̄l2a2b − C1u0̄l1a1bu

− l2u1̄l2a2b + C2u1̄l1a1bu + l2u0̄l2a2b − C2u1̄l1a1bu

− l2u1̄l2a2bj. s13d

If we define the states of the Bell basis composed of atom2 and modesa,b of cavity 1 as

uC±l =1Î2

su0̄l1a1bu + l2 ± u1̄l1a1bu− l2d, s14d

BRIEF REPORTS PHYSICAL REVIEW A70, 025803(2004)

025803-2

uF±l =1Î2

su0̄l1a1bu− l2 + u1̄l1a1bu + l2d, s15d

Eq. (13) can be expanded as

uClT =1Î2

hsC1u0̄l2a2b − C2u1̄l2a2bduC+l + sC1u0̄l2a2b

+ C2u1̄l2a2bduC−l + sC2u0̄l2a2b − C1u1̄l2a2bduF+l

− sC2u0̄l2a2b + C1u1̄l2a2bduF−lj. s16d

C. Bell measurement

To complete the teleportation process, we have to performa joint measurement on the stateuClT involving both atom 2and cavity 1 as represented by one of the four resultsuC+l,uC−l, uF+l, or uF−l. To do that, the atom 2 is left to interactoff-resonantly with modeb in cavity 1 and then crossesR+ asindicated by Eq.(2) and Eq. (3). When the stateuC±l isfound the result is

uC±l → 1Î2

su0̄l1a1b ± u1̄l1a1bduel2. s17d

On the other hand, when finding the stateuF±l we have

uF±l → 1Î2

su0̄l1a1b ± u1̄l1a1bdugl2. s18d

Thus, measuring atom 2 allows one to distinguish be-tween the statesuC±l and uF±l. Next, we have to discern thephases±d. This is done by sending successively two moretwo-level atoms in the ground statesugla, uglbd resonantlywith modesa,b. The atomugla, after interacting resonantlywith modea as indicated by Eq.(1), crossesR+, and evolvesaccording to Eq.(3). After that, atomuglb enters the cavity,interacting resonantly with modeb as indicated by Eq.(1)and crossesR−, evolving also in accord with Eq.(3). Theresult is

su0̄l1a1b ± u1̄l1a1bd → 51Î2

suelauelb + uglauglbd if s+ d,

1Î2

suelauglb + uglauelbd if s− d.

s19d

Therefore, when the atomsugla and uglb are measuredboth in either the ground or the excited state, the phase iss+d; if one of them is found to be in the ground state whilethe other is in the excited state, the phase iss−d. We note

from Eq. (16) that the only prompt result that completes theteleportation process from cavity 1 to cavity 2 is a jointmeasurement resulting in the stateuC−l. If the joint measure-ment results inuC+l, an atom interacting dispersively withmodea (or modeb), as indicated by Eq.(2), can be adjusted

sl2t /d=pd to repair the phase:sC1u0̄l2a2b−C2u1̄l2a2bduelb

→ sC1u0̄l2a2b+C2u1̄l2a2bduelb, the atom uelb dropping out.When a joint measurement leads to the stateuF−l, the tele-portation process can be completed by sending a three-levelatom as indicated by Eq.(8), with C1 equal to zero. On theother hand, when the joint measurement results inuF+l theprevious procedure must be followed by a two-level atominteracting dispersively[cf. Eq. (2) with l2t /d=p], whichcompletes the teleportation process.

IV. CONCLUSIONS

We have presented a scheme to teleport an entanglementof zero- and one-photon states from one high-Q cavity toanother. Our scheme is based on the following operations:(i)control of the interaction times between two-level atoms andone of the two modes sustained by the two high-Q cavities;(ii ) entanglement of the two modes of the high-Q cavities(cf.Refs.[20,21]); (iii ) use of Stark shifts and Ramsey zones. Inaddition, we have computed the relative phase gained bymodea when the interaction between the atom and modebtakes place. This phase is shown to be irrelevant. Assumingthe time spent in experiments to be less than the typical timesof damping for the qubits employed in this scheme, whichare of the same order as the decoherence time of these qubits,allows one to neglect both atomic spontaneous decay andlosses in the high-Q cavities. Our scheme is able to discerneach one of the four Bell states with 100% success probabil-ity. The present scheme is more economical than others inthe literature since it reduces the number of cavities neces-sary to perform the experiment; this diminishes the decoher-ence effect caused by unavoidable interactions between thesystem and its environment. In addition, it is more advanta-geous than those based on single-mode teleportation when itis desired that the final teleported state be entangled. In thiscase, just teleporting independently(and coherently) eachcavity mode will require further operations to entangle thetwo modes, thus implying additional procedures and even-tual sources of errors.

ACKNOWLEDGMENTS

We thank the VPG/Vice-Reitoria de Pós-Graduação ePesquisa-UCG(N.G.A.), the CAPES(G.P.), and the CNPq(N.G.A., A.T.A., B.B), Brazilian Agencies, for partially sup-porting this work.

BRIEF REPORTS PHYSICAL REVIEW A70, 025803(2004)

025803-3

[1] C. H. Bennettet al., Phys. Rev. Lett.70, 1895(1993).[2] E. Solano, C. L. Cesar, R. L. de Matos Filho, and N. Zagury,

Eur. Phys. J. D13, 121 (2001).[3] S. L. Braunstein and H. J. Kimble, Phys. Rev. Lett.80, 869

(1998).[4] C. J. Villas-Bôas, N. G. de Almeida, and M. H. Y. Moussa,

Phys. Rev. A60, 2759(1999).[5] H. W. Lee and J. Kim, Phys. Rev. A63, 012305(2000).[6] L. Davidovichet al., Phys. Rev. A50, R895(1994); J. I. Cirac

and A. S. Parkins,ibid. 50, R4441(1994); N. G. de Almeida,L. P. Maia, C. J. Villas-Bôas, and M. H. Y. Moussa, Phys. Lett.A 241, 213 (1998); N. G. de Almeida, R. Napolitano, and M.H. Y. Moussa, Phys. Rev. A62, R010101(2000).

[7] D. Bowmeesteret al., Nature(London) 390, 575 (1997).[8] D. Boschiet al., Phys. Rev. Lett.80, 1121(1998).[9] Y. H. Kim et al., Phys. Rev. Lett.86, 1370(2001).

[10] A. Furusawaet al., Science282, 706 (1998).[11] E. Lombardi, F. Sciarrino, S. Popescu, and F. De Martini,

Phys. Rev. Lett.88, 070402(2002).[12] H. W. Lee, Phys. Rev. A64, 014302(2001).[13] R. M. Serra, C. J. Villas-Bôas, N. G. de Almeida, and M. H. Y.

Moussa, J. Opt. B: Quantum Semiclassical Opt.4, 316(2002).[14] S. J. van Enk and O. Hirota, Phys. Rev. A64, 022313(2001).

[15] M. H. Y. Moussa, Phys. Rev. A55, R3287(1997).[16] M. Ikram, S. Y. Zhu, and M. S. Zubairy, Phys. Rev. A62,

022307(2000).[17] R. Arun and G. S. Agarwal, Phys. Rev. A66, 043812(2002);

R. Arun, G. S. Agarwal, M. O. Scully, and H. Walther,ibid.62, 023809(2000); Asoka Biswas and G. S. Agarwal, e-printquant-ph/0401099.

[18] M. Brune et al., Phys. Rev. Lett. 77, 4887 (1996); M.Weidinger, B. T. H. Varcoe, R. Heerlein, and H. Walther,ibid.82, 3795 (1999); J. M. Raimond, M. Brune, and S. Haroche,Rev. Mod. Phys.73, 565 (2001).

[19] L. Mandel and E. Wolf,Optical Coherence and Quantum Op-tics (Cambridge University Press, Cambridge, England, 1995);M. O. Scully and M. S. Zubairy,Quantum Optics(CambridgeUniversity Press, Cambridge, England, 1997).

[20] F. O. Prado, N. G. de Almeida, C. J. Villas-Bôas, and M. H. Y.Moussa(unpublished); R. M. Serra, C. J. Villas-Bôas, N. G. deAlmeida and M. H. Y. Moussa, e-print quant-ph/0306126.

[21] See the experimental work by A. Rauschenbeutel, P. Bertet, S.Osnaghi, G. Nogues, M. Brune, J. M. Raimond, and S.Haroche, Phys. Rev. A64, R050301(2001), reporting thepreparation of an entangled state between two modes of ahigh-Q cavity using a single circular two-level Rydberg atom.

BRIEF REPORTS PHYSICAL REVIEW A70, 025803(2004)

025803-4