We show a mechanism that projects a pair of neutral two-level atoms from an
initially uncorrelated state to a maximally entangled state while they remain
spacelike separated. The atoms begin both excited in a common electromagnetic
vacuum, and the radiation is collected with a partial Bell-state analyzer. If
the interaction time is short enough and a certain two-photon Bell state is
detected after the interaction, a high degree of entanglement, even maximal,
can be generated while one atom is outside the light cone of the other, for
arbitrary large interatomic distances.Comment: v2: version accepted in Phys. Rev. 

C. Cohen-Tannoudji

Carlos Sabín

Juan León

English

arXiv

4 pages, 4 figures.-- PACS nrs.: 03.67.Bg; 03.65.Ud; 42.50.Ct.-- ArXiv pre-print available at: http://arxiv.org/abs/0807.2711We show a mechanism that projects a pair of neutral two-level atoms from an initially uncorrelated state to a maximally entangled state while they remain spacelike separated. The atoms begin both excited in a common electromagnetic vacuum, and the radiation is collected with a partial Bell-state analyzer. If the interaction time is short enough and a certain two-photon Bell state is detected after the interaction, a high degree of entanglement, even maximal, can be generated while one atom is outside the light cone of the other, for arbitrary large interatomic distances.This work was supported by Spanish MEC FIS2005-05304 and CSIC 2004 5 OE 271 projects.Peer reviewe

León, Juan

Sabín, Carlos

Digital.CSIC

Entanglement swapping between spacelike-separated atomsJuan León* and Carlos Sabín†Instituto de Física Fundamental, CSIC, Serrano 113-B, 28006 Madrid, SpainReceived 16 July 2008; published 10 November 2008We show a mechanism that projects a pair of neutral two-level atoms from an initially uncorrelated state toa maximally entangled state while they remain spacelike separated. The atoms begin both excited in a commonelectromagnetic vacuum, and the radiation is collected with a partial Bell-state analyzer. If the interaction timeis short enough and a certain two-photon Bell state is detected after the interaction, a high degree of entangle-ment, even maximal, can be generated while one atom is outside the light cone of the other, for arbitrary largeinteratomic distances.DOI: 10.1103/PhysRevA.78.052314 PACS numbers: 03.67.Bg, 03.65.Ud, 42.50.CtI. INTRODUCTIONEntanglement between distant atoms is a key resource forquantum information and computation. There are mainly twodifferent known ways to generate it: by interaction betweenthe atoms for instance, 1 or by detection of the emittedphotons 2–6. Some of these proposals have been realizedexperimentally for instance, 7. For the latter cases, inprinciple, there is no reason to expect that the swapping 8of atom-photon to atom-atom entanglement can only beginto occur when one atom enters into the light cone of theother.The possibility of entanglement generation betweenspacelike-separated atoms is of both theoretical and practicalinterest, and was addressed from different points of view in9–12. In 12, we analyze this issue perturbatively in asimple model of a pair of two-level atoms interacting locallywith the electromagnetic field, initially in the vacuum state13. Tracing over the field states, the atoms are only classi-cally correlated, but applying n n n=0,1 ,2 being thenumber of photons up to second order in perturbationtheory, the atoms get entangled. For n=0 the entanglementis generated by the interaction term and therefore is onlyrelevant when one atom enters into the light cone of theother, despite the finiteness of the Feynman propagator be-yond that region. But for n=1,2 entanglement may be siz-able, although small, if the interatomic distance is shortenough. In 10, the trace over the field states was consideredin a model with a pair of two-level detectors coupled to ascalar field. The detectors may get entangled if a suitabletime-dependent coupling is introduced, and this was appliedto a linear ion trap in 11. In 9, only the vacuum case whent→0 was analyzed, and no entanglement measures wereconsidered.There are two possible interpretations for these effects: asa transfer of preexisting entanglement of the vacuum 10,11or as a consequence of the propagation of virtual quantaoutside the light cone 9. Both are compared and discussedin 9.In this paper we will go one step further and consider thatthe photons are detected with definite momenta and polariza-tions. We show that, in principle, a high degree of entangle-ment, even maximal, can be generated between spacelikeseparated atoms if a Bell state of the emitted photons is de-tected. We will consider a pair of neutral two-level atomsseparated by a fixed and arbitrary distance and study theevolution of an initially uncorrelated state under local inter-action with the electromagnetic field. We focus on the two-photon emission which, although it has a smaller probabilityof success, shows a larger fidelity of the projected state withthe desired state and has an entanglement robust to atomicrecoil 14. The photons pass through a partial Bell-state ana-lyzer 15, and we use entanglement measures to study theevolution of entanglement in the projected atomic states afterdetection of the different photonic Bell states. The resultsshow that interaction times must be short, but interatomicdistances can be as large as desired. The interaction time isindependent of the photodetection time, which is only relatedwith the distance from the atoms to the detectors. That dis-tance can be such that the photodetection can occur while theatoms remain spacelike separated.The results can be interpreted as a transfer of part of thevacuum entanglement after a postselection process. If nomeasurement were performed the atoms would have classicalcorrelations transferred by the vacuum. In 10 the classicalcorrelations may become entanglement with a suitable time-dependent coupling. The postselection process can be seen asan alternative way to achieve the entanglement transference.While the results in 9,10 are mainly theoretical, these couldbe probed experimentally, and would show for the first timethe possibility of transfer entanglement from the vacuumstate of the quantum field to spacelike-separated atoms.II. ENTANGLEMENT SWAPPING BETWEEN SPACELIKE-SEPARATED ATOMSTo address the atom-field interactions, we assume that therelevant wavelengths and the interatomic separation aremuch larger than the atomic dimensions. The dipole approxi-mation, appropriate to these conditions, permits the splittingof the system Hamiltonian into two parts H=H0+HI that areseparately gauge invariant. The first part is the Hamiltonianin the absence of interactions other than the potentials that*leon@imaff.cfmac.csic.es; URL: http://www.imaff.csic.es/pcc/QUINFOG/†csl@imaff.cfmac.csic.es; URL: http://www.imaff.csic.es/pcc/QUINFOG/PHYSICAL REVIEW A 78, 052314 20081050-2947/2008/785/0523144 ©2008 The American Physical Society052314-1keep A and B stable and the self-interaction terms that can beremoved when radiative corrections are considered 16,H0=HA+HB+Hfield. The second contains all the interactionsof the atoms with the field HI=−10n=A,Bdnxn , t ·Dxn , t,where D is the electric displacement field, and dn=ie	d3xiExi−xnG is the electric dipole moment ofatom n, which we will take as real and of equal magnitudefor both atoms d=dA=dB, E and G being the excitedand ground states of the atoms, respectively.In what follows we choose a system given initially by theproduct state 0= EE0 in which atoms A and B are in theexcited state E and the field in the vacuum state 0. Thesystem then evolves under the effect of the interaction duringa lapse of time t, and, up to order e2, 0, 1, or 2 photons maybe emitted. If after that a two-photon state is detected, = photon 1,photon 2 = k,k ,,ckk ,,k,k 1being k, k momenta and ,  polarizations, the pro-jected state, up to order e2, can be given in the interactionpicture asphotons,atom 1,atom 2t =  f EE + gGGN  , 2wheref = + 12TSA+SA− + SB+SB−0, g = + TSB−SA−0 ,3and N=f 2+ g2, being S=− 1	0t dtHIt S=S++S−, andT the time ordering operator. Here, g describes single-photonemission by both atoms, while f corresponds to two-photonemission by a single atom. The sign of the superscript isassociated to the energy difference between the initial andfinal atomic states of each emission. In quantum optics, f isusually neglected by the introduction of a rotating wave ap-proximation RWA, but as we will see later, for very shortinteraction times f and g may be of similar magnitude. Ac-tually, a proper analysis of this model can be performed onlybeyond the RWA 17–19. Without RWA vacuum entangle-ment cannot be transferred to the atoms with this particularpostselection process, but the trace over the field stateswould be a classically correlated state, as in 12. In thatcase, a one-photon postselection process would entangle theatoms.Equation 3 can be written asf = + 12t1 − t2SA+t1SA−t2 + SB+t1SB−t20 ,g = + SB−t1SA−t20 . 4Finally, in the dipole approximation the actions S inEq. 4 reduce toS = − i0tdteitd · Ex,t , 5where =	E−	G is the transition frequency, and we areneglecting atomic recoil. This depends on the atomic prop-erties  and d, and on the interaction time t. In our calcula-tions we will take d /ec=510−3, which is of the sameorder as the 1s→2p transition in the hydrogen atom, con-sider t1, and analyze the cases L /ct1 around thelight cone, L being the interatomic distance. We will use thestandard mode expansion for the electric field: Ex= i c2023	d3kkeik·xk ,ak−e−ik·x*k ,ak† , withak ,ak† =3k−k.The photons pass through a partial Bell-state analyzer15 consisting of a beam splitter BS and two polarizationbeam splitters PBS with four single-photon detectors attheir output ports. If two detectors, one at one output port ofone PBS and one at an output port of the other, click at thesame time, a state − is detected, while if the two clicks arein the two output ports of only one PBS, the state is +. Ifone of the four detectors emits a double click, the state canbe + or −. Taking into account momenta and symme-trization, the Bell states can be written as =12 k↓,k ↑ + k ↑,k↓ k↑,k ↓ + k ↓,k↑ , =12 k↓,k ↓ + k ↓,k↓ k↑,k ↑ + k ↑,k↑ ,6where ↑ and ↓ are the photon polarizations, with polarizationvectorsk,↑ = −12 k,1 + k,2andk,↓ = 12 k,1 − k,2 ,wherek,1 = cos k cos k,cos k sin k,− sin kandk,2 = − sin k,cos k,0 .Here k ,k =ak† ak† 0.We will use the concurrence 20 C to compute the en-tanglement of the atomic states when the different Bell statesare detected. The concurrence of the atomic part of a statelike Eq. 2 is just given byC =2fg*N2. 7We assume that the atoms A ,B are along the y axis, at y=L /2, respectively, and the dipoles are parallel along the zaxis, corresponding to an experimental setup in which theJUAN LEÓN AND CARLOS SABÍN PHYSICAL REVIEW A 78, 052314 2008052314-2dipoles are induced by suitable external fields 9. We alsotake k= k = /c.Under that condition, the first remarkable thing is that for− and −, we have f =g=0. Therefore, at least whileonly E1 transitions are considered, in this model the Bell-state analyzer is complete: if two different detectors click thestate is +, while if one detector clicks twice, the state is+. First, we focus on +. Considering Eqs. 4–6, withthe mode expansion for the electric field and the commuta-tion relation for the creation and annihilation operators, astandard computation leads tof = K,t,2jtcos z2h+, ,g = K,t,cos z2h−, , 8with K , t ,= cd2t222e2 sin k sin k  being the fine-structure constant, jt= −1+e2it1−2itt2 , h ,= sin k sin ksin k sin k, k ,k corresponding to kˆand k ,k to kˆ , and z=L /c. Notice that jt decreasesas t grows, and eventually vanishes as t→ as required byenergy conservation. The L dependence is a result of theindividual dependence on the position of each atom, not onthe relative distance between them.Taking into account Eqs. 7 and 8 the concurrence isgiven byC =4cos z2h+,cos z2h−,cos2 z2h+, jt + cos2 z2h−, 4jt. 9Now, we assume that the 50:50 BS is at y ,z= 0,L /2,the two PBS are at d /22, L /2+d /22, and the fourdetectors are at d /2, L /2+d /2 and d /2, L /2 seeFig. 1. Equation 8 will not depend on the value of d, whichis the distance traveled by the photon to any detector afterleaving the BS. Notice that, with this setup, h−=0 and h+=2.In Fig. 2 we represent Eq. 9 under these conditions as afunction of x=L /ct for three different values of z differentvalues of L. Notice that a high degree of entanglement,maximal for x large enough short enough interaction timest, can be achieved in all cases when one atom is beyond thelight cone of the other x1. As t→ x→0, the concur-rence eventually vanishes, in agreement with the fact that theonly atomic state allowed by energy conservation is just theseparable state GG.In Fig. 3 we represent Eq. 9 as a function of z for threedifferent values of z /x=t, to give an alternative descrip-tion. The mutual light cone corresponds to the region zt in each case. The concurrence oscillates with the posi-tion of the atoms, and eventually vanishes at z=2n+1 /2 n=0,1 ,2 , . . . , as a consequence of the vanishingof cosz /2. For a given interaction time t, the maximum ofthe concurrence can be achieved for interatomic distances aslarge as desired. In particular, a maximally entangled state isgenerated for t=1, which corresponds to t10−15 s.In Fig. 4 we sketch Eq. 9 as a function of =k=k forgiven values of x and z. Notice that the maximum values forthe entanglement are around =n /2 n=0,1 ,2 , . . . ,  /2corresponding to the setup of Fig. 1.So far, we have focused on +, but, in principle, +could be detected as well. The coefficients f and g wouldhave the opposite sign to those of + and therefore theconcurrence would be the same. But, due to the interactiontimes considered here, the relaxation time of a single detec-tor must be extremely short in order to emit a double click.III. CONCLUSIONSIn conclusion, we have shown that, in principle, two neu-tral two-level atoms can evolve from an initially uncorrelatedA BBSPBS PBSFIG. 1. Schematic setup for the entanglement swapping de-scribed in the text. The atoms A and B are at y ,z= L /2,0. Theemitted photons pass through a 50:50 BS at 0,L /2 and two PBSat d /22, L /2+d /22, and there are four single-photon detec-tors at the outport ports of the two PBS, at d /2, L /2+d /2 andd /2, L /2. Taking into account that − and − are forbiddenin our model, a + is detected when there are coincidence clicksin two detectors and + when there is a double click in one de-tector. Then the atoms are projected into the atomic part of the state2.0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.20.40.60.81.0xCFIG. 2. Concurrence for the atomic state when a Bell state +or + of the photons is detected, as a function of x=L /ct for z=L /c=1 solid, 5 dashed, and 10 dotted. The light cone is atx1. For x1 the interaction time is short enough to have a sig-nificant amount of entanglement.ENTANGLEMENT SWAPPING BETWEEN SPACELIKE-… PHYSICAL REVIEW A 78, 052314 2008052314-3state to a highly entangled state in a time shorter than thetime required for the light to travel between them. At theinitial time, both atoms are excited in a common electromag-netic vacuum. They are allowed to interact with the field dueto an induced dipole during a time t and, up to second orderin perturbation theory, n=0,1 ,2 photons may be emitted.After that, the emitted radiation passes through a partial Bell-state analyzer. For interaction times t10−15 s and if a two-photon Bell state + or + the other two are forbidden inthis model is detected after that, the atoms are projected intoan entangled state, which may be maximally entangled forshort enough t. For a given t, the degree of entanglementoscillates periodically with the distance and the maximumdegree available can be achieved for interatomic distances Las large as desired. Notice that the interaction time t, whichmust be t10−15 s, is absolutely independent of the time tat which the photodetection takes place. Since the distancetraveled by the photons from the atoms to the detector isL /2+d, d being arbitrary, the photodetection can occur af-ter a time tŒL /c. A suitable choice of d is necessary inorder to ensure that the atoms may remain spacelike sepa-rated. The degree of entanglement is independent of d.ACKNOWLEDGMENTSThis work was supported by Spanish MEC FIS2005-05304 and CSIC 2004 5 OE 271 projects.1 S. J. van Enk, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 78,4293 1997.2 C. Cabrillo, J. I. Cirac, P. García-Fernández, and P. Zoller,Phys. Rev. A 59, 1025 1999.3 X. L. Feng, Z. M. Zhang, X. D. Li, S. Q. Gong, and Z. Z. Xu,Phys. Rev. Lett. 90, 217902 2003.4 L. M. Duan and H. J. Kimble, Phys. Rev. Lett. 90, 2536012003.5 C. Simon and W. T. M. Irvine, Phys. Rev. Lett. 91, 1104052003.6 L. Lamata, J. J. García-Ripoll, and J. I. Cirac, Phys. Rev. Lett.98, 010502 2007.7 D. L. Moehring, P. Maunz, S. Olmschenk, K. C. Younge, D. N.Matsukevich, L. M. Duan, and C. Monroe, Nature London449, 68 2007.8 M. Żukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert,Phys. Rev. Lett. 71, 4287 1993.9 J. D. Franson, J. Mod. Opt. 55, 2117 2008.10 B. Reznik, A. Retzker, and J. Silman, Phys. Rev. A 71, 0421042005.11 A. Retzker, J. I. Cirac, and B. Reznik, Phys. Rev. Lett. 94,050504 2005.12 J. León and C. Sabín, e-print arXiv:0804.4641.13 E. Fermi, Rev. Mod. Phys. 4, 87 1932.14 S. Zippilli, G. A. Olivares-Rentería, G. Morigi, C. Schuck, F.Rhode, and J. Eschner, New J. Phys. 10, 103003 2008.15 K. Mattle, H. Weinfurter, P. G. Kwiat, and A. Zeilinger, Phys.Rev. Lett. 76, 4656 1996.16 C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interactions Wiley Interscience, New York, 1998.17 E. A. Power and T. Thirunamachandran, Phys. Rev. A 56,3395 1997.18 P. W. Milonni, D. F. V. James, and H. Fearn, Phys. Rev. A 52,1525 1995.19 A. K. Biswas, G. Compagno, G. M. Palma, R. Passante, and F.Persico, Phys. Rev. A 42, 4291 1990.20 S. Hill and W. K. Wootters, Phys. Rev. Lett. 78, 5022 1997.0 2 4 6 8 100.00.20.40.60.81.0zCFIG. 3. Concurrence for the atomic state when a Bell state +of the photons is detected, as a function of z=L /c for t=1solid, 4 dashed, and 7 dotted. The light cone for each curve isat zt.0 1 2 3 4 5 60.00.20.40.60.8ΦCFIG. 4. Concurrence for the atomic state when a Bell state +of the photons is detected, as a function of  for z=L /c=5 andx=Lc / t=2.5.JUAN LEÓN AND CARLOS SABÍN PHYSICAL REVIEW A 78, 052314 2008052314-4

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Entanglement swapping between spacelike-separated atoms

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Entanglement swapping between spacelike separated atoms

Entanglement swapping between spacelike separated atoms

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