Entanglement swapping of atomic states through the photonic Faraday rotation
aa r X i v : . [ qu a n t - ph ] M a y Entanglement swapping of atomic states through the photonic Faraday rotation
W.P. Bastos, W.B. Cardoso, A.T. Avelar, and B. Baseia
Instituto de F´ısica, Universidade Federal de Goi´as, 74.001-970, Goiˆania, Goi´as, Brazil
We propose an entanglement swapping of atomic states confined by cavities QED using a photonic Faradayrotation. Two schemes are considered in which we use three and four cavities, respectively, plus an additionalcircularly-polarized photon. After interacting with an atom trapped inside the cavity the system evolves to anentangled atom-photon state. The entanglement swapping is then achieved by a Bell-state measurement uponthe entire atom-photon state.
PACS numbers: 03.67.Bg, 03.67.Hk, 42.50.Ex
In quantum computation and quantum communication [1]the entanglement performs a fundamental resource for manyprotocols, such as quantum teleportation [2], quantum densecoding [1], and distributed quantum computation [3]. In thiscontext, entanglement swapping plays an important role inseveral protocols of quantum information transfer, in partic-ular, it is arguably one of the most important ingredients forquantum repeaters and quantum relays [4, 5], as well as inteleportation of entangled states. For the entanglement swap-ping protocol, two pairs of particles are usually used with eachpair previously entangled, and a Bell-state measurement madeupon a particle of each pair leads the remaining particles to anentangled state, even if they have never interacted previouslywith each other.Due to the importance of entanglement swapping vari-ous experimental results have been presented recently [6–13]. In Ref. [6] the authors employed two pairs of polar-ized entangled photons and making a Bell-state measurementupon the photon of each pair, they displayed an entangle-ment of freely propagating particles that have never inter-acted or dynamically coupled by any other means. Entan-glement swapping operations have been reported in [7] vianuclear magnetic resonance quantum-information processing,over long distances in optical fibers [9], and uncondition-ally for continuous variables [8]. The entanglement swap-ping for continuous-variable has been used to realize quan-tum teleportation beyond the no-cloning limit [10]. Multi-stage entanglement swapping in photonic system [11], an ion-trap quantum processor through entanglement swapping [12],and the first experimental demonstration of the Greenberger-Horne-Zeilinger entanglement swapping [13] have also beenreported.On the other hand, a lot of theoretical schemes have ap-peared in the recent literature [14–21] generalizing the stan-dard entanglement swapping for: multiparticle systems [14];multi-qudit systems [15], where the authors have extended thescheme originally proposed for two pairs of qubits and an ar-bitrary number of systems composed by an arbitrary numberof qudits; d-level systems in a generalized cat state [16], use-ful for protocol of secret sharing. Concerning with secret shar-ing in cavity QED [19], multiparty secret sharing of quantuminformation [17] and classical information [20], secure mul-tiparty quantum communication by Bell states [18] and byentangled qutrits [21] based on entanglement swapping havealso been proposed. In addition to the list of entanglement swapping applica-tions, the purification of entangled states by local actions, us-ing a variant of entanglement swapping, was studied in Refs.[22–24]. This issue was extended for continuous variables in[25]. The quantum key distribution schemes [26] and telepor-tation of a two-particle entangled state [27, 28] employing en-tanglement swapping have been reported, as well as entangle-ment swapping without joint measurement [29]. In the QED-cavity context, a scheme based on two atoms and two cavitiesinitially prepared in two pairs of atom-photon nonmaximallyentangled states, was considered in [30] to create maximallyentangled photon-photon and atom-photon states via entan-glement swapping, with atomic states in either a three-levelcascade or lambda configuration in [31], with resonant inter-action of a two-mode cavity with a λ -type three-level atominvolving only a single measurement in [32] and, in [33], analternative scheme to implement the entanglement swapping.More recently, an entanglement swapping in the two-photonJaynes-Cummings model was proposed in [34].Here, taking advantage of the quantum regime of strong in-teractions between single atoms and photons present in a mi-crotoroidal resonator [35], as employed in [36] for quantuminformation processing, we propose an entanglement swap-ping of states of atoms confined in distant low QED cavitiesusing photonic Faraday rotations. The main idea is to makeuse of the Faraday rotation produced by single-photon-pulseinput and output process regarding low-Q cavities [37]. Inview of our applications, we revisited the input-output rela-tion for a cavity coherently interacting with a trapped two-level atom, recently considered in Ref. [36]. We considera three-level atom interacting with a single mode of a low-Qcavity pumped by photonic emission of a single photon sourcevia optical fibers. Fig. 1 shows the atomic levels of the atomtrapped inside the cavities. Each transition is described by theJaynes-Cummings model.One can use the quantum Langevin equation of the cav-ity mode a driven by the corresponding cavity input operator a in ( t ) and the atomic lowering operator are [38], ˙ a ( t ) = − [ i ( ω c − ω p ) + κ a ( t ) − gσ − ( t ) − √ κa in ( t ) , (1a) ˙ σ − ( t ) = − [ i ( ω − ω p ) + γ σ − ( t ) − gσ z ( t ) a ( t ) + √ γσ z ( t ) b in ( t ) , (1b)respectively, where a ( a † ) is the annihilation (creation) op-erator of the cavity field with frequency ω c ; σ z and σ + ( σ − ) are, respectively, inversion and raising (lowering) oper-ators of the two-level atom with frequency difference ω be-tween these two-levels. κ and γ are, respectively, the cavitydamping rate and the atomic decay rate. The vacuum inputfield b in ( t ) felt by the two-level atom satisfies the commuta-tion relation [ b in ( t ) , b † in ( t ′ )] = δ ( t − t ′ ) . The input and out-put fields of the cavity are related by the intracavity field as a out ( t ) = a in ( t ) + √ κa ( t ) [38].In this way, considering a large enough κ to be sure that wehave a weak excitation by the single-photon pulse on the atominitially prepared in the ground state, i.e., keeping h σ z i = − throughout our operation, as shown in [36] one can adiabati-cally eliminate the cavity mode and arrive at the input-outputrelation of the cavity field, r ( ω p ) = [ i ( ω c − ω p ) − κ ][ i ( ω − ω p ) + γ ] + g [ i ( ω c − ω p ) + κ ][ i ( ω − ω p ) + γ ] + g , (2)where r ( ω p ) ≡ a out ( t ) /a in ( t ) is the reflection coefficient ofthe atom-cavity system. Now, considering the case of g = 0 and an empty cavity we have [38] r ( ω p ) = i ( ω c − ω p ) − κ i ( ω c − ω p ) + κ . (3)According to [36] the transitions | e i ↔ | i and | e i ↔ | i are due to the coupling to two degenerate cavity modes a L and a R with left (L) and right (R) circular polarization, respec-tively. For the atom initially prepared in | i , the only possibletransition | i → | e i implies that only the L circularly polar-ized single-photon pulse | L i will work. Hence Eq. (2) leadsthe input pulse to the output one as | Ψ out i L = r ( ω p ) | L i ≈ e iφ | L i with φ the corresponding phase shift being determinedby the parameter values. Note that an input R circularly po-larized single-photon pulse | R i would only sense the emptycavity; as a consequence the corresponding output governedby Eq. (3) is | Ψ out i R = r ( ω p ) | R i = e iφ | R i with φ aphase shift different from φ . Therefore, for an input linearlypolarized photon pulse | Ψ in i = √ ( | L i + | R i ) , the outputpulse is | Ψ out i − = 1 √ e iφ | L i + e iφ | R i ) . (4)This also implies that the polarization direction of the reflectedphoton rotates an angle Θ − F = ( φ − φ ) / with respect to thatof the input one, called Faraday rotation [37]. If the atom isinitially prepared in | i , then only the R circularly polarizedphoton could sense the atom, whereas the L circularly polar-ized photon only interacts with the empty cavity. So we have, | Ψ out i + = 1 √ e iφ | L i + e iφ | R i ) , (5)where the Faraday rotation is Θ + F = ( φ − φ ) / . Case - Firstly, we assume the previously entangledstate between the atoms confined in the cavities A and B , L R|1|0 |e
FIG. 1: Atomic configuration of the three-level atom trapped in thelow-Q cavities. (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) PS QWP
PD AB |ψ AB |ψ C FIG. 2: (Color online) Schematic diagram of the entanglement swap-ping procedure for the case 1 using three three-level atoms trappedin cavities A , B , and C , a single photon source (PS), a quarter-waveplate (QWP), and a photodetector of polarization (PD). given by | ψ i AB = 1 √ | i AB + | i AB ) . (6)In another spatial position, an entanglement has been previ-ously prepared in atom confined in cavity C with a photon,named , in Faraday rotated state, written in the form | ψ C i = 1 √ | i C | η i − + | i C | η i + ) , (7)where | η i − = ( e iφ | L i + e iφ | R i ) / √ and | η i + =( e iφ | L i + e iφ | R i ) / √ . This entanglement is constructedvia interaction of the polarized photon (in the state | ψ i =( | L i + | R i ) / √ ) with a three-level atom (previously pre-pared in | ψ i C = ( | i C + | i C ) / √ ). Fig. 2 shows the entireprocedure of the case 1.The entanglement swapping is realized through a Bell-statemeasurement on the system composed by the photon and theatom confined in the cavity B . To this end, we send the photonthrough the cavity B to interact with the atom, leading thestate of the whole system given as | ψ ′ i = 12 √ h | i AC ( e i ( φ + φ ) | L i + e i ( φ + φ ) | R i ) | i B + | i AC ( e iφ | L i + e iφ | R i ) | i B + | i AC ( e iφ | L i + e iφ | R i ) | i B + | i AC ( e i ( φ + φ ) | L i + e i ( φ + φ ) | R i ) | i B i . (8)Next, assuming φ = π and φ = π/ , plus the application ofa Hadamard operation upon the state of the atom B and the MAPS ESR AO | L i B | i AC + | i AC σ x | L i B | i AC − | i AC iσ y | R i B | i AC − | i AC σ z | R i B | i AC + | i AC I TABLE I: Atomic rotations completing the entanglement swappingprocedure for the case 1. The first column represents the measure-ment in the atom B and the photon states (MAPS), second columnis the result of entanglement swapping resulting (ESR), and the thirdis the atomic operation (AO) considering the atom A (local) repre-sented by Pauli operators with I being the identity operator. (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) PS QWP
PD AB CD |ψ |ψ
CDAB
FIG. 3: (Color online) Schematic diagram of the entanglement swap-ping procedure for de case 2, using the same notation of Fig. 2 plusan additional atom confined in cavity D . photon via an external laser beam and a quarter-wave plate(QWP), respectively, the state of the entire system evolves to | ψ ′′ i = 12 √ − i | L i B ( | i AC + | i AC )+ i | L i B ( | i AC − | i AC ) − | R i B ( | i AC − | i AC )+ | R i B ( | i AC + | i AC )] . (9)Finally, appropriate detections of the state of the atom B andthe polarized photon state conclude the entanglement swap- ping. Table I summarizes the atomic rotations to complete theentanglement swapping. Case - Now, we start by considering two pairs of atoms,trapped inside the cavities A , B , C , and D , and previouslyentangled in the following states | ψ i AB = 1 √ | i AB + | i AB ) , | ψ i CD = 1 √ | i CD + | i CD ) . (10)The scheme for entanglement swapping is summarized in Fig.3. Firstly, an auxiliary photon is sent to interact with the atomconfined in the cavity D , leading the state of the entire atom-photon system to the form | ϕ i = 12 ( | i AB + | i AB ) ⊗ ( | i CD | η i + + | i CD | η i − ) . (11)Next, considering a Hadamard operation upon the atom D , wehave | ϕ ′ i = 14 (cid:2) e iφ | L i ABC ( | i D − | i D )+ e iφ | L i ABC ( | i D + | i D )+ e iφ | R i ABC ( | i D − | i D )+ e iφ | R i ABC ( | i D + | i D )+ e iφ | L i ABC ( | i D − | i D )+ e iφ | L i ABC ( | i D + | i D )+ e iφ | R i ABC ( | i D − | i D )+ e iφ | R i ABC ( | i D + | i D ) (cid:3) . (12)In sequence, the photon emerging form the cavity D is sentto interact with the atom in the cavity B and, soon after, aHadamard operation upon the atom B and upon the photonstates, transforms the state of the whole system to the form | ϕ ′′ i = 18 (cid:8) [( e iφ + e iφ ) | L i − ( e iφ − e iφ ) | R i ] | i AC ( | i BD − | i BD − | i BD + | i BD )+ [( e iφ + e iφ ) | L i + ( e iφ − e iφ ) | R i ] | i AC ( | i BD + | i BD + | i BD + | i BD )+ 2 e i ( φ + φ ) | L i | i AC ( | i BD + | i BD − | i BD − | i BD )+ 2 e i ( φ + φ ) | L i | i AC ( | i BD − | i BD + | i BD − | i BD ) o , (13)and with the single choice φ = π and φ = π/ , we obtain | ϕ ′′′ i = 14 [ | R i BD ( | i AC − | i AC ) + | R i BD ( | i AC + | i AC )+ | R i BD ( | i AC + | i AC ) + | R i BD ( | i AC − | i AC ) − i | L i BD ( | i AC + | i AC ) − i | L i BD ( | i AC − | i AC )+ i | L i BD ( | i AC − | i AC ) + i | L i BD ( | i AC + | i AC )] . (14) MAPS ESR AO | R i BD | i AC − | i AC − iσ y | R i BD | i AC + | i AC σ x | R i BD | i AC + | i AC σ x | R i BD | i AC − | i AC − iσ y | L i BD | i AC + | i AC I | L i BD | i AC − | i AC σ z | L i BD | i AC − | i AC σ z | L i BD | i AC + | i AC I TABLE II: Atomic rotations completing the entanglement swappingprocedure for the case 2. The first column represents the measure-ment in the atoms BD and the photon states (MAPS); second columnis the result of entanglement swapping resulting (ESR), and the thirdis the atomic operation (AO) considering the atom A (local) repre-sented by Pauli operators with I being the identity operator. So, with a detection of the photon polarization plus separated atomic measurements in the atoms B and D , one concludesthe entanglement swapping. Table II presents the atomic op-eration to reconstruct the initial state.In conclusion, we presented an entanglement swapping ofatomic states confined in cavities QED using photonic Fara-day rotation. This scheme involves only virtual excitations ofthe atoms and the cavities considered here are low-Q cavities,i.e., with large decay. We also have considered the schemewith ideal detectors and fibers without absorption. Two pro-tocols for entanglement swapping were demonstrated and inview of the current status of the technology our scheme is fea-sible. This scheme can also be modeled in a quantum-dot sys-tem, by simply replacing the atoms by excitons [39]. Acknowledgement
We thank the CAPES, CNPq, and FUNAPE-GO, Brazilianagencies, for the partial support. [1] M. A. Nielsen and I. L. Chuang,
Quantum Computationand Quantum Information (Cambridge, Cambridge UniversityPress, 2000).[2] C. H. Bennett et al. , Phys. Rev. Lett. , 1895 (1993).[3] V. Buzek et al. , Phys. Rev. A , 3327 (1997).[4] M. Zukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert, Phys.Rev. Lett. , 4287 (1993).[5] H. J. Briegel, W. Dur, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. , 5932 (1998); W. Dur, H. J. Briegel, J. I. Cirac, and P. Zoller,Phys. Rev. A , 169 (1999).[6] J.-W. Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger,Phys. Rev. Lett. , 3891 (1998).[7] N. Boulant et al. , Phys. Rev. A , 032305 (2003).[8] X.-J. Jia et al. , Phys. Rev. Lett. , 250503 (2004).[9] H. de Riedmatten et al. , Phys. Rev. A , 050302(R) (2005).[10] N. Takei, H. Yonezawa, T. Aoki, and A. Furusawa, Phys. Rev.Lett. , 220502 (2005).[11] A. M. Goebel et al. , Phys. Rev. Lett. , 080403 (2008).[12] M. Riebe et al. , Nature Phys. , 839 (2008).[13] C.-Y. Lu, T. Yang, and J.-W. Pan, Phys. Rev. Lett. , 020501(2009).[14] S. Bose, V. Vedral, and P. L. Knight, Phys. Rev. A , 822(1998).[15] J. Bouda and V. Buzek, J. Phys. A - Math. Gen. , 4301 (2001).[16] V. Karimipour, A. Bahraminasab, and S. Bagherinezhad, Phys.Rev. A , 042320 (2002).[17] Y. M. Li, K. S. Zhang, and K. C. Peng, Phys. Lett. A 324, 420(2004).[18] J. Lee, S. Lee, J. Kim, and S. D. Oh, Phys. Rev. A , 032305(2004).[19] Y.-Q. Zhang, X.-R. Jin, and S. Zhang, Phys. Lett. A , 380(2005).[20] Z.-J. Zhang and Z.-X. Man, Phys. Rev. A , 022303 (2005).[21] Y.-B. Zhan, L.-L. Zhang, and Q.-Y. Zhang, Opt. Comm. , 4633 (2009).[22] S. Bose, V. Vedral, and P. L. Knight, Phys. Rev. A , 194(1999).[23] B.-S. Shi, Y.-K. Jiang, and G.-C. Guo, Phys, Rev. A , 054301(2000).[24] M. Yang, Y. Zhao, W. Song, and Z.-L. Cao, Phys. Rev. A 71,044302 (2005).[25] R. E. S. Polkinghorne and T. C. Ralph, Phys. Rev. Lett , 2095(1999).[26] D. G. Song, Phys. Rev. A , 034301 (2004).[27] H. Lu and G.-C. Guo, Phys. Lett. A , 209 (2000).[28] W. B. Cardoso and N. G. de Almeida, Phys. Lett. A , 201(2009).[29] M. Yang, W. Song, and Z. L. Cao, Phys. Rev. A , 034312(2005).[30] Z.-Z. Wu, M.-F. Fang, and C.-L. Jiang, Comm. Theor. Phys. ,553 (2006).[31] E. S. Guerra and C. R. Carvalho, J. Mod. Opt. , 865 (2006).[32] Z.-B. Yang, Comm. Theor. Phys. , 649 (2007).[33] Z. He, C.-Y. Long, S.-J. Qin, and G.-F. Wei, Int. J. QuantumInf. , 837 (2007).[34] A. D. dSouza, W. B. Cardoso, A. T. Avelar, and B. Baseia, Phys.Scr. , 065009 (2009).[35] B. Dayan et al. , Science , 1062 (2008).[36] J.-H. An, M. Feng, and C. H. Oh, Phys. Rev. A , 032303(2009).[37] B. Julsgaard, A. Kozhekin, and E. S. Polzik, Nature , 400(2001).[38] D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, Berlin, 1994).[39] M. N. Leuenberger, M. E. Flatt´e, and D. D. Awschalom, Phys.Rev. Lett.94