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Teleportation of entangled states without Bell-state measurement via a two-photon process
Optics Communications 284 (2011) 1086–1089
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Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Teleportation of entangled states without Bell-state measurement via a two-photon process A.D. dSouza, W.B. Cardoso ⁎, A.T. Avelar, B. Baseia Instituto de Física, Universidade Federal de Goiás, 74.001-970, Goiânia, GO, Brazil
a r t i c l e
i n f o
Article history: Received 23 March 2010 Received in revised form 4 October 2010 Accepted 8 October 2010 Keywords: Teleportation Without Bell-state measurement Two-photon process
a b s t r a c t In this letter we propose a scheme using a two-photon process to teleport an entangled field state of a bimodal cavity to another one without Bell-state measurement. The quantum information is stored in a zero- and twophoton entangled state. This scheme requires two three-level atoms in a ladder configuration, two bimodal cavities, and selective atomic detectors. The fidelity and success probability do not depend on the coefficients of the state to be teleported. For convenient choices of interaction times, the teleportation occurs with fidelity close to the unity. © 2010 Elsevier B.V. All rights reserved.
Quantum entanglement [1] is the cornerstone of exotic phenomena in quantum mechanics. It radically differs from ingredients of classical physics and plays an important role to demonstrate fundamental aspects of the theory. Entangled states constitute useful resources to perform tasks that cannot be realized by classical states such as superdense code [2], entangled-based quantum cryptography [3,4], and quantum teleportation [5]. In particular, quantum teleportation provides a mechanism to transfer, from a system to another, the quantum information contained in the state of one or more qubits using a quantum channel (entangled state) plus a classical channel to transfer an additional classical information required to reconstruct the teleported state. Besides being useful for quantum communication via quantum computers, quantum teleportation is fundamental for universal quantum computation [6]. Quantum teleportation has been experimentally proved in various physical contexts, such as in traveling waves [7], optical continuousvariables [8], nuclear magnetic resonance [9], photons in waveguides [10], trapped ions [11], etc. Nonetheless, in the important scenario of microwave cavity QED it remains as a challenge yet. In the theoretical realm, Davidovich et al. [12] proposed a scheme to teleport an unknown atomic state between two high-Q cavities initially prepared in entangled photon number states. An alternative scheme proposed by Cirac and Parkins [13] employed two additional atomic levels of one of a correlated pair to teleport atomic states. Other proposals can also be found in [14–17]. In Ref. [18] Zheng proposed a scheme for approximately and conditionally teleporting an unknown atomic state in a cavity QED, a
⁎ Corresponding author. E-mail address: [email protected] (W.B. Cardoso). 0030-4018/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.10.032
procedure known as “teleportation without Bell-state measurement”. In it, only one particle of the entangled pair should be detected, which projects the other particle in a known state and simplifies the reconstruction of the teleported state. So, the use of a single atomic detection and an appropriate atom–field interaction allows one to distinguish a specific Bell-state among four possibilities. From the experimental point of view, the simplicity of the apparatus is achieved at the expense of a reduction of the success probability. After Ref. [18] various schemes of teleportation without Bell-state measurement were proposed [19–21]. In Ref. [19] a one-photon process described by the Jaynes–Cumming model was used to teleport entangled states from a bimodal cavity to another. In [20] the scheme was extended for teleportation of GHZ-states. Here we propose a scheme using a two-photon process to teleport an entangled bimodal cavity-field state consisting of a zero- and twophoton from a cavity to another without Bell-state measurement. It is worth mentioning that the two-photon process has been demonstrated in [22] for a microwave QED cavity and offers some advantages in comparison with the one-photon process, as the reduction of interaction times due to the increasing of the atom–field coupling strength and the lower decoherence induced by stray fields [23]. In addition, two-photon process can be easily obtained with Rydberg atoms with principal quantum number n N 89, which can be statesensitively detected using tunneling field ionization with quantum efficiencies above 80% and an ionization efficiency above 98% [24]. To describe the two-photon process we will use the two-photon Jaynes– Cummings model in the full microscopical Hamiltonian approach (FMHA) as explained in Ref. [25]. Different from the effective Hamiltonian approach, the FMHA is also valid for small average photon number. In Ref. [26] the reader will find a more detailed discussion about the validity of the effective Hamiltonian approach
A.D. dSouza et al. / Optics Communications 284 (2011) 1086–1089
and its connection with the FMHA, concerned with quantum teleportation of atomic states. We will consider a three-level atom that interacts with a single mode of a cavity field. In the absence of a driven field acting upon the atom, the model describing the atom–field interaction is given by the FMHA. In the interaction picture the Hamiltonian reads [25] HFMHA
−iδt † iδt = ħg1 a je〉〈f je + a j f 〉〈e j e
pffiffiffiffiffiffiffiffiffiffiffiffiffi g1 n + 1 iδt sinðΛn t Þe 2 Ce Cn Λn
Cf ;n+1 ðt Þ = −i
+
−i
ð1Þ
cosðΛn t Þ−
where g1 and g2 stand for the one-photon coupling constant with respect to the transitions |e〉 ↔ | f 〉 and | f 〉 ↔ |g〉, respectively. The detuning δ is given by δ = Ω− ωe −ωf = ωf −ωg −Ω;
h
i j ψðt Þ〉 = ∑ Ce;n ðt Þj e; n〉 + Cf ;n ðt Þj f ; n〉 + Cg;n ðt Þj g; n〉 ; n
ð3Þ
where the |k, n〉, with k = e, f, g, indicate the atom in the state |k〉 and the field in the Fock state |n〉. The coefficients Ck, n(t) stand for the corresponding probability amplitudes. The insertion of Eqs. (1) and (3) in the time dependent Schrödinger equation furnishes the coupled first-order differential equations for the probability amplitudes pffiffiffiffiffiffiffiffiffiffiffiffiffi −iδt dCe;n ðt Þ = −ig1 Cf ;n+1 ðt Þ n + 1e ; dt p ffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi iδt dCf ;n+1 ðt Þ iδt = −ig1 Ce;n ðt Þ n + 1e −ig2 Cg;n+2 ðt Þ n + 2e ; dt pffiffiffiffiffiffiffiffiffiffiffiffiffi −iδt dCg;n+2 ðt Þ = −ig2 Cf ;n+1 ðt Þ n + 2e : dt
ð4Þ
As usually, we consider the entire atom–field system as decoupled at the initial time t = 0, Ce;n ð0Þ = Ce Cn ð0Þ; Cb;n+1 ð0Þ = Cf Cn+1 ð0Þ; Cc;n+2 ð0Þ = Cg Cn+2 ð0Þ;
ð5Þ
iδ iδt sinðΛn t Þ e 2 Cf Cn+1 2Λn
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðn + 1Þðn + 2Þ γn ðt ÞCe Cn Λn α 2n
pffiffiffiffiffiffiffiffiffiffiffiffiffi g2 n + 2 −iδt sinðΛn t Þe 2 Cf Cn+1 Λn
where δ iδt −iδt γn ðt Þ = Λn cosðΛn t Þ + i sinðΛn t Þ−Λn e 2 e 2 ; 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi δ2 + α 2n ; Λn = 4 αn =
ð6Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g12 ðn + 1Þ + g22 ðn + 2Þ;
ð12Þ
Λn being the Rabi frequency. The substitutions n → n − 1 in Eq. (8) and n → n − 2 in Eq. (9) allow one to obtain the Cf, n(t) and Cg, n(t), respectively. Next, we first consider an entangled state of zero- and twophotons previously prepared in modes “3” and “4” of the cavity C2, as follows j ϕð0Þ〉34 = α j0; 2〉34 + β j 2; 0〉34 ;
ð13Þ
where α and β are unknown coefficients, with |α|2 + |β|2 = 1. For details see Ref. [27], where we have recently shown how to generate the EPR and W entangled states of zero- and two-photons. The nonlocal channel is constructed by two three-level atoms (designed by subindex a and b) in a ladder configuration (Fig. 1) and the modes “1” and “2” of the cavity C1. This nonlocal channel is prepared with the atoms previously prepared in their excited states (|e〉a, b) and the cavity-field modes in the vacuum state (|0, 0〉12). Then, the atom a is sent to interact only with the mode 1 and soon after the atom b is led
|e Ω
(ωe − ωf) δ
From the solution of these coupled differential equations with the initial conditions in Eq. (5) we get the time dependent coefficients as " # pffiffiffiffiffiffiffiffiffiffiffiffiffi g 2 ðn + 1Þ g1 n + 1 −iδt γ ð t Þ + 1 C C −i sinðΛn t Þe 2 Cf Cn+1 Ce;n ðt Þ = 1 n e n 2 Λ Λn α n n " # pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð7Þ g g ðn + 1Þðn + 2Þ ð Þ + 1 2 γ t C C ; n g n + 2 Λn α 2n
ð10Þ
ð11Þ
where the Cn(0) stand for the amplitudes of the arbitrary initial field state and the Ca are atomic amplitudes of the (normalized) initial atomic state j χ〉 = Ce j e〉 + Cf j f 〉 + Cg j g〉:
ð9Þ
" # g22 ðn + 2Þ + γn ðt Þ + 1 Cg Cn+2 ; Λn α 2n
ð2Þ
where Ω is the cavity-field frequency and ωe, ωf, and ωg are the frequencies associated with the atomic levels |e〉, |f〉, and |g〉, respectively. In what follows we present a brief review of our work in [26]. The state that describes the combined atom–field system is written as
g1 g2
−i
ð8Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffi g2 n + 2 iδt sinðΛn t Þe 2 Cg Cn+2 ; Λn
iδt † −iδt ; + ħg2 a jf 〉〈g j e + a j g〉〈f j e Cg;n+2 ðt Þ =
1087
Ω
|f
(ωf − ωg) |g
Fig. 1. Schematic diagram of the three-level atom interacting with a single mode of a cavity field.
1088
A.D. dSouza et al. / Optics Communications 284 (2011) 1086–1089
to interact with the mode 2 of the cavity C1. Thus, the state that composes the quantum channel is written as ðe;0Þ ðe;0Þ ðe;0Þ j ψ〉12ab = Ce;0 ðt1 Þ Ce;0 ðt1 Þj 0; 0; e; e〉12ab + Cf ;1 ðt1 Þj 0; 1; e; f 〉12ab ðe;0Þ + Cg;2 ðt1 Þj 0; 2; e; g〉12ab ðe;0Þ ðe;0Þ ðe;0Þ + Cf ;1 ðt1 Þ Ce;0 ðt1 Þj 1; 0; f ; e〉12ab + Cf ;1 ðt1 Þj 1; 1; f ; f 〉12ab ðe;0Þ ð14Þ + Cg;2 ðt1 Þj 1; 2; f ; g〉12ab ðe;0Þ ðe;0Þ ðe;0Þ + Cg;2 ðt1 Þ Ce;0 ðt1 Þj 2; 0;;g; e〉12ab + Cf ;1 ðt1 Þj 2; 1; g; f 〉12ab ðe;0Þ + Cg;2 ðt1 Þj 2; 2; g; g〉12ab where the Cji are the time dependent coefficients given by Eqs. (7)–(9); the index i connects the time evolution with the initial state and the index j represents the populated state. The atom–field interaction time inside the cavity C1 for these two atoms are considered as being the same, t1. Next, the atom a crosses the cavity C2 interacting only with the mode 3 and is later detected in its excited state. Soon after, the atom b crosses the same cavity to interact with the mode 4, being also detected in its excited state. In the cavity C2 these two atoms also interact during the same time, t2. Likewise, after some algebra we obtain the state of the whole system in the form ( 2 ðe;0Þ ðe;0Þ ðe;2Þ Ce;0 ðt1 Þ Ce;0 ðt2 ÞCe;2 ðt2 Þj 0; 0; 0; 2〉1234
j Φ〉1234 = N α
ðe;0Þ
ðe;0Þ
ðe;0Þ
ð f ;2Þ
ðe;0Þ
ðe;0Þ
ðe;0Þ
ð g;2Þ
+ Ce;0 ðt1 ÞCf ;1 ðt1 ÞCe;0 ðt2 ÞCe;1 ðt2 Þj 0; 1; 0; 1〉1234
Fig. 2. Success probability versus the interaction times t1 and t2. An appropriate choice of the times t1 and t2 optimizes the probability. This choice is not appropriate to optimize the fidelity (see Fig. 3).
coupling constant of these atoms with the cavity is of approximately g = 17.5 MHz. In our simulations we use this value and a detuning of 10 g. As an example, with the choices t1 = 2 μs and t2 = 9 μs one gets the success probability P ≃ 4.4% and the fidelity F ≃ 97.8%. These values are close to those using one-photon transition, as found in Ref. [19]. Then, for this case the whole interaction time is about T = 2 × t1 + 2 × t2 = 22 μs. Now, taking into account the atomic velocity va ≃ 750 m/s and the length of the apparatus L ≃ 15 cm [28], we obtain a flight time around Tf ≃ 4 × 10− 4. Considering a cavity decay lifetime of Tc = 0.1 s [29] we will have Tc/Tf ≃ 250, which shows the feasibility of our scheme. In summary, we presented a scheme to teleport an entanglement of zero- and two- photon states using a two-photon process described
+ Ce;0 ðt1 ÞCg;2 ðt1 ÞCe;0 ðt2 ÞCe;0 ðt2 Þj 0; 2; 0; 0〉1234 +β
ð15Þ
2 ðe;0Þ ðe;2Þ ðe;0Þ Ce;0 ðt1 Þ Ce;2 ðt2 ÞCe;0 ðt2 Þ j 0; 0; 2; 0〉1234
ðe;0Þ
ðe;0Þ
ðe;0Þ
ð f ;2Þ
ðe;0Þ
ðe;0Þ
ðe;0Þ
ð g;2Þ
+ Cf ;1 ðt1 ÞCe;0 ðt1 ÞCe;0 ðt2 ÞCe;1 ðt2 Þj 1; 0; 1; 0〉1234 + Ce;0 ðt1 ÞCg;2 ðt1 ÞCe;0 ðt2 ÞCe;0 ðt2 Þj 2; 0; 0; 0〉1234
) ;
where N is a normalization factor. The success probability to detect the atoms a and b in their excited states is obtained as ðe;0Þ 4 ðe;0Þ 2 ðe;2Þ 2 Pe;e = jCe;0 ðt1 Þj j Ce;0 ðt2 Þj j Ce;2 ðt2 Þj ðe;0Þ
2
ðe;0Þ
2
ðe;0Þ
ð16Þ 2
ð f ;2Þ
2
+ j Ce;0 ðt1 Þ j j Cf ;1 ðt1 Þj j Ce;0 ðt2 Þj j Ce;1 ðt2 Þj ðe;0Þ 2 ðe;0Þ 2 ðe;0Þ 2 ð g;2Þ 2 + j Ce;0 ðt1 Þj j Cg;2 ðt1 Þj j Ce;0 ðt2 Þj j Ce;0 t2 j : Now, from the Eq. (15) for an appropriate choice of the times t1 and t2 we obtain the teleported state in the approximated form |Φ〉1234 ≃ (α|0, 2〉12 + β|2, 0〉12)|0, 0〉34, which allows us to get the fidelity F = ||12〈Ψ|Φ〉1234||2 of the teleported state (with |Ψ〉12 given by Eq. (13)). In this way we obtain ðe;0Þ
F=
ðe;0Þ
ðe;0Þ
ð g;2Þ
jCe;0 ðt1 Þ j 2 j Cg;2 ðt1 Þj 2 j Ce;0 ðt2 Þj 2 j Ce;0 ðt2 Þ j 2 Pe;e
:
ð17Þ
Fig. 2 shows the probability (15) versus the interaction times t1 and t2, while Fig. 3a and b shows the fidelity (16) versus t2 considering an appropriate choice t1 = 2 μs. Note that both, the success probability and fidelity, do not depend on the coefficient α. Let us make a comment on the experimental feasibility of the present scheme. The devices used here are based on those of Ref. [23] that uses Rydberg atoms with quantum number n N 89. The
Fig. 3. The fidelity versus t1 and t2 is shown in (a). In (b) we display the success probability (black-solid line) and fidelity (red-dashed line) of the teleported state given respectively by Eqs. (15) and (16) versus t2, for a fixed time t1 = 2 μs.
A.D. dSouza et al. / Optics Communications 284 (2011) 1086–1089
by two-photon Jaynes–Cummings model in the full microscopical Hamiltonian approach. On the one hand, this scheme is advantageous in comparison with the teleportation of a single qubit [18] since the success probability and the fidelity do not depend on the coefficients of the state to be teleported. On the other hand, the time spent in the entire procedure is about 180 μs (assuming the atoms emerging one by one from the apparatus), which is much smaller than the decoherence time of the cavity (0.1 s) [29], showing the experimental feasibility of the scheme. Acknowledgements We thank the CAPES, CNPq, and FUNAPE/GO, Brazilian agencies, for the partial supports. References [1] R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, Rev. Mod. Phys. 81 (2009) 865. [2] C.H. Bennett, S.J. Wiesner, Phys. Rev. Lett. 69 (1992) 2881. [3] A.K. Ekert, Phys. Rev. A 67 (1991) 661. [4] N. Gisin, G. Ribordy, W. Tittel, H. Zbinden, Rev. Mod. Phys. 74 (2002) 145. [5] C.H. Bennett, et al., Phys. Rev. Lett. 70 (1993). [6] D. Gottesman, I.L. Chuang, Nature (London) 402 (1999) 390. [7] D. Bouwmeester, et al., Nature 390 (1997) 575; D. Boschi, et al., Phys. Rev. Lett. 80 (1998) 1121; E. Lombardi, F. Sciarrino, S. Popescu, F. De Martini, Phys. Rev. Lett. 88 (2002) 070402; J.W. Pan, et al., Nature 421 (2003) 721. [8] A. Furusawa, et al., Science 282 (1998) 706. [9] M.A. Nielsen, E. Knill, R. Laflamme, Nature 396 (1998) 52.
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[10] I. Marcikic, et al., Nature 421 (2003) 509. [11] M. Riebe, et al., Nature 429 (2004) 734; M.D. Barrett, et al., Nature 429 (2004) 737. [12] L. Davidovich, et al., Phys. Rev. A 50 (1994) R895. [13] J.I. Cirac, A.S. Parkins, Phys. Rev. A 50 (1994) R4441. [14] M.H.Y. Moussa, Phys. Rev. A 54 (1996) 4661; Phys. Rev. A 55 (1997) R3287. [15] M.H.Y. Moussa, B. Baseia, Mod Phys Lett B 12 (1998) 1209. [16] G. Pires, N.G. de Almeida, A.T. Avelar, B. Baseia, Phys. Rev. A 70 (2004) 025803. [17] G. Pires, A.T. Avelar, B. Baseia, N.G. de Almeida, Phys. Rev. A 71 (2005) 060301(R); I.P. de Queirós, S. Souza, W.B. Cardoso, N.G. de Almeida, Phys. Rev. A 76 (2007) 034101. [18] S.B. Zheng, Phys. Rev. A 69 (2004) 064302. [19] W.B. Cardoso, A.T. Avelar, B. Baseia, N.G. de Almeida, Phys. Rev. A 72 (2005) 045802. [20] W.B. Cardoso, Int. J. Theor. Phys. 47 (2008) 977. [21] Z.L. Cao, Y. Zhao, M. Yang, Physica A 360 (2006) 17; M. Yang, Z.L. Cao, Int. J. Quantum Inf. 4 (2006) 341; Z.B. Yang, J. Phys. B: At. Mol. Opt. Phys. 39 (2006) 603; W.B. Cardoso, A.T. Avelar, B. Baseia, J. Phys. B: At. Mol. Opt. Phys. 40 (2007) 1089; A.D. dSouza, W.B. Cardoso, A.T. Avelar, B. Baseia, Commun. Theor. Phys. 52 (2009) 41; A.D. dSouza, W.B. Cardoso, A.T. Avelar, B. Baseia, Commun. Theor. Phys. 52 (2009) 241. [22] M. Brune, et al., Phys. Rev. Lett. 59 (1987) 1899. [23] P.J. Blythe, B.T.H. Varcoe, New J. Phys. 8 (2006) 213. [24] M. Tada, et al., Phys. Lett. A 303 (2002) 285; Y. Kishimoto, et al., Phys. Lett. A 303 (2002) 279. [25] A.H. Toor, M.S. Zubairy, Phys. Rev. A 45 (1992) 4951. [26] A.D. dSouza, W.B. Cardoso, A.T. Avelar, B. Baseia, Physica A 388 (2009) 1331. [27] W.B. Cardoso, W.C. Qiang, A.T. Avelar, B. Baseia, J. Phys. B: At. Mol. Opt. Phys. 43 (2010) 155502. [28] J.M. Raimond, M. Brune, S. Haroche, Rev. Mod. Phys. 73 (2001) 565. [29] S. Gleyzes, et al., Nature 446 (2007) 297.
Contents lists available at ScienceDirect
Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Teleportation of entangled states without Bell-state measurement via a two-photon process A.D. dSouza, W.B. Cardoso ⁎, A.T. Avelar, B. Baseia Instituto de Física, Universidade Federal de Goiás, 74.001-970, Goiânia, GO, Brazil
a r t i c l e
i n f o
Article history: Received 23 March 2010 Received in revised form 4 October 2010 Accepted 8 October 2010 Keywords: Teleportation Without Bell-state measurement Two-photon process
a b s t r a c t In this letter we propose a scheme using a two-photon process to teleport an entangled field state of a bimodal cavity to another one without Bell-state measurement. The quantum information is stored in a zero- and twophoton entangled state. This scheme requires two three-level atoms in a ladder configuration, two bimodal cavities, and selective atomic detectors. The fidelity and success probability do not depend on the coefficients of the state to be teleported. For convenient choices of interaction times, the teleportation occurs with fidelity close to the unity. © 2010 Elsevier B.V. All rights reserved.
Quantum entanglement [1] is the cornerstone of exotic phenomena in quantum mechanics. It radically differs from ingredients of classical physics and plays an important role to demonstrate fundamental aspects of the theory. Entangled states constitute useful resources to perform tasks that cannot be realized by classical states such as superdense code [2], entangled-based quantum cryptography [3,4], and quantum teleportation [5]. In particular, quantum teleportation provides a mechanism to transfer, from a system to another, the quantum information contained in the state of one or more qubits using a quantum channel (entangled state) plus a classical channel to transfer an additional classical information required to reconstruct the teleported state. Besides being useful for quantum communication via quantum computers, quantum teleportation is fundamental for universal quantum computation [6]. Quantum teleportation has been experimentally proved in various physical contexts, such as in traveling waves [7], optical continuousvariables [8], nuclear magnetic resonance [9], photons in waveguides [10], trapped ions [11], etc. Nonetheless, in the important scenario of microwave cavity QED it remains as a challenge yet. In the theoretical realm, Davidovich et al. [12] proposed a scheme to teleport an unknown atomic state between two high-Q cavities initially prepared in entangled photon number states. An alternative scheme proposed by Cirac and Parkins [13] employed two additional atomic levels of one of a correlated pair to teleport atomic states. Other proposals can also be found in [14–17]. In Ref. [18] Zheng proposed a scheme for approximately and conditionally teleporting an unknown atomic state in a cavity QED, a
⁎ Corresponding author. E-mail address: [email protected] (W.B. Cardoso). 0030-4018/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.10.032
procedure known as “teleportation without Bell-state measurement”. In it, only one particle of the entangled pair should be detected, which projects the other particle in a known state and simplifies the reconstruction of the teleported state. So, the use of a single atomic detection and an appropriate atom–field interaction allows one to distinguish a specific Bell-state among four possibilities. From the experimental point of view, the simplicity of the apparatus is achieved at the expense of a reduction of the success probability. After Ref. [18] various schemes of teleportation without Bell-state measurement were proposed [19–21]. In Ref. [19] a one-photon process described by the Jaynes–Cumming model was used to teleport entangled states from a bimodal cavity to another. In [20] the scheme was extended for teleportation of GHZ-states. Here we propose a scheme using a two-photon process to teleport an entangled bimodal cavity-field state consisting of a zero- and twophoton from a cavity to another without Bell-state measurement. It is worth mentioning that the two-photon process has been demonstrated in [22] for a microwave QED cavity and offers some advantages in comparison with the one-photon process, as the reduction of interaction times due to the increasing of the atom–field coupling strength and the lower decoherence induced by stray fields [23]. In addition, two-photon process can be easily obtained with Rydberg atoms with principal quantum number n N 89, which can be statesensitively detected using tunneling field ionization with quantum efficiencies above 80% and an ionization efficiency above 98% [24]. To describe the two-photon process we will use the two-photon Jaynes– Cummings model in the full microscopical Hamiltonian approach (FMHA) as explained in Ref. [25]. Different from the effective Hamiltonian approach, the FMHA is also valid for small average photon number. In Ref. [26] the reader will find a more detailed discussion about the validity of the effective Hamiltonian approach
A.D. dSouza et al. / Optics Communications 284 (2011) 1086–1089
and its connection with the FMHA, concerned with quantum teleportation of atomic states. We will consider a three-level atom that interacts with a single mode of a cavity field. In the absence of a driven field acting upon the atom, the model describing the atom–field interaction is given by the FMHA. In the interaction picture the Hamiltonian reads [25] HFMHA
−iδt † iδt = ħg1 a je〉〈f je + a j f 〉〈e j e
pffiffiffiffiffiffiffiffiffiffiffiffiffi g1 n + 1 iδt sinðΛn t Þe 2 Ce Cn Λn
Cf ;n+1 ðt Þ = −i
+
−i
ð1Þ
cosðΛn t Þ−
where g1 and g2 stand for the one-photon coupling constant with respect to the transitions |e〉 ↔ | f 〉 and | f 〉 ↔ |g〉, respectively. The detuning δ is given by δ = Ω− ωe −ωf = ωf −ωg −Ω;
h
i j ψðt Þ〉 = ∑ Ce;n ðt Þj e; n〉 + Cf ;n ðt Þj f ; n〉 + Cg;n ðt Þj g; n〉 ; n
ð3Þ
where the |k, n〉, with k = e, f, g, indicate the atom in the state |k〉 and the field in the Fock state |n〉. The coefficients Ck, n(t) stand for the corresponding probability amplitudes. The insertion of Eqs. (1) and (3) in the time dependent Schrödinger equation furnishes the coupled first-order differential equations for the probability amplitudes pffiffiffiffiffiffiffiffiffiffiffiffiffi −iδt dCe;n ðt Þ = −ig1 Cf ;n+1 ðt Þ n + 1e ; dt p ffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi iδt dCf ;n+1 ðt Þ iδt = −ig1 Ce;n ðt Þ n + 1e −ig2 Cg;n+2 ðt Þ n + 2e ; dt pffiffiffiffiffiffiffiffiffiffiffiffiffi −iδt dCg;n+2 ðt Þ = −ig2 Cf ;n+1 ðt Þ n + 2e : dt
ð4Þ
As usually, we consider the entire atom–field system as decoupled at the initial time t = 0, Ce;n ð0Þ = Ce Cn ð0Þ; Cb;n+1 ð0Þ = Cf Cn+1 ð0Þ; Cc;n+2 ð0Þ = Cg Cn+2 ð0Þ;
ð5Þ
iδ iδt sinðΛn t Þ e 2 Cf Cn+1 2Λn
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðn + 1Þðn + 2Þ γn ðt ÞCe Cn Λn α 2n
pffiffiffiffiffiffiffiffiffiffiffiffiffi g2 n + 2 −iδt sinðΛn t Þe 2 Cf Cn+1 Λn
where δ iδt −iδt γn ðt Þ = Λn cosðΛn t Þ + i sinðΛn t Þ−Λn e 2 e 2 ; 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi δ2 + α 2n ; Λn = 4 αn =
ð6Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g12 ðn + 1Þ + g22 ðn + 2Þ;
ð12Þ
Λn being the Rabi frequency. The substitutions n → n − 1 in Eq. (8) and n → n − 2 in Eq. (9) allow one to obtain the Cf, n(t) and Cg, n(t), respectively. Next, we first consider an entangled state of zero- and twophotons previously prepared in modes “3” and “4” of the cavity C2, as follows j ϕð0Þ〉34 = α j0; 2〉34 + β j 2; 0〉34 ;
ð13Þ
where α and β are unknown coefficients, with |α|2 + |β|2 = 1. For details see Ref. [27], where we have recently shown how to generate the EPR and W entangled states of zero- and two-photons. The nonlocal channel is constructed by two three-level atoms (designed by subindex a and b) in a ladder configuration (Fig. 1) and the modes “1” and “2” of the cavity C1. This nonlocal channel is prepared with the atoms previously prepared in their excited states (|e〉a, b) and the cavity-field modes in the vacuum state (|0, 0〉12). Then, the atom a is sent to interact only with the mode 1 and soon after the atom b is led
|e Ω
(ωe − ωf) δ
From the solution of these coupled differential equations with the initial conditions in Eq. (5) we get the time dependent coefficients as " # pffiffiffiffiffiffiffiffiffiffiffiffiffi g 2 ðn + 1Þ g1 n + 1 −iδt γ ð t Þ + 1 C C −i sinðΛn t Þe 2 Cf Cn+1 Ce;n ðt Þ = 1 n e n 2 Λ Λn α n n " # pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð7Þ g g ðn + 1Þðn + 2Þ ð Þ + 1 2 γ t C C ; n g n + 2 Λn α 2n
ð10Þ
ð11Þ
where the Cn(0) stand for the amplitudes of the arbitrary initial field state and the Ca are atomic amplitudes of the (normalized) initial atomic state j χ〉 = Ce j e〉 + Cf j f 〉 + Cg j g〉:
ð9Þ
" # g22 ðn + 2Þ + γn ðt Þ + 1 Cg Cn+2 ; Λn α 2n
ð2Þ
where Ω is the cavity-field frequency and ωe, ωf, and ωg are the frequencies associated with the atomic levels |e〉, |f〉, and |g〉, respectively. In what follows we present a brief review of our work in [26]. The state that describes the combined atom–field system is written as
g1 g2
−i
ð8Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffi g2 n + 2 iδt sinðΛn t Þe 2 Cg Cn+2 ; Λn
iδt † −iδt ; + ħg2 a jf 〉〈g j e + a j g〉〈f j e Cg;n+2 ðt Þ =
1087
Ω
|f
(ωf − ωg) |g
Fig. 1. Schematic diagram of the three-level atom interacting with a single mode of a cavity field.
1088
A.D. dSouza et al. / Optics Communications 284 (2011) 1086–1089
to interact with the mode 2 of the cavity C1. Thus, the state that composes the quantum channel is written as ðe;0Þ ðe;0Þ ðe;0Þ j ψ〉12ab = Ce;0 ðt1 Þ Ce;0 ðt1 Þj 0; 0; e; e〉12ab + Cf ;1 ðt1 Þj 0; 1; e; f 〉12ab ðe;0Þ + Cg;2 ðt1 Þj 0; 2; e; g〉12ab ðe;0Þ ðe;0Þ ðe;0Þ + Cf ;1 ðt1 Þ Ce;0 ðt1 Þj 1; 0; f ; e〉12ab + Cf ;1 ðt1 Þj 1; 1; f ; f 〉12ab ðe;0Þ ð14Þ + Cg;2 ðt1 Þj 1; 2; f ; g〉12ab ðe;0Þ ðe;0Þ ðe;0Þ + Cg;2 ðt1 Þ Ce;0 ðt1 Þj 2; 0;;g; e〉12ab + Cf ;1 ðt1 Þj 2; 1; g; f 〉12ab ðe;0Þ + Cg;2 ðt1 Þj 2; 2; g; g〉12ab where the Cji are the time dependent coefficients given by Eqs. (7)–(9); the index i connects the time evolution with the initial state and the index j represents the populated state. The atom–field interaction time inside the cavity C1 for these two atoms are considered as being the same, t1. Next, the atom a crosses the cavity C2 interacting only with the mode 3 and is later detected in its excited state. Soon after, the atom b crosses the same cavity to interact with the mode 4, being also detected in its excited state. In the cavity C2 these two atoms also interact during the same time, t2. Likewise, after some algebra we obtain the state of the whole system in the form ( 2 ðe;0Þ ðe;0Þ ðe;2Þ Ce;0 ðt1 Þ Ce;0 ðt2 ÞCe;2 ðt2 Þj 0; 0; 0; 2〉1234
j Φ〉1234 = N α
ðe;0Þ
ðe;0Þ
ðe;0Þ
ð f ;2Þ
ðe;0Þ
ðe;0Þ
ðe;0Þ
ð g;2Þ
+ Ce;0 ðt1 ÞCf ;1 ðt1 ÞCe;0 ðt2 ÞCe;1 ðt2 Þj 0; 1; 0; 1〉1234
Fig. 2. Success probability versus the interaction times t1 and t2. An appropriate choice of the times t1 and t2 optimizes the probability. This choice is not appropriate to optimize the fidelity (see Fig. 3).
coupling constant of these atoms with the cavity is of approximately g = 17.5 MHz. In our simulations we use this value and a detuning of 10 g. As an example, with the choices t1 = 2 μs and t2 = 9 μs one gets the success probability P ≃ 4.4% and the fidelity F ≃ 97.8%. These values are close to those using one-photon transition, as found in Ref. [19]. Then, for this case the whole interaction time is about T = 2 × t1 + 2 × t2 = 22 μs. Now, taking into account the atomic velocity va ≃ 750 m/s and the length of the apparatus L ≃ 15 cm [28], we obtain a flight time around Tf ≃ 4 × 10− 4. Considering a cavity decay lifetime of Tc = 0.1 s [29] we will have Tc/Tf ≃ 250, which shows the feasibility of our scheme. In summary, we presented a scheme to teleport an entanglement of zero- and two- photon states using a two-photon process described
+ Ce;0 ðt1 ÞCg;2 ðt1 ÞCe;0 ðt2 ÞCe;0 ðt2 Þj 0; 2; 0; 0〉1234 +β
ð15Þ
2 ðe;0Þ ðe;2Þ ðe;0Þ Ce;0 ðt1 Þ Ce;2 ðt2 ÞCe;0 ðt2 Þ j 0; 0; 2; 0〉1234
ðe;0Þ
ðe;0Þ
ðe;0Þ
ð f ;2Þ
ðe;0Þ
ðe;0Þ
ðe;0Þ
ð g;2Þ
+ Cf ;1 ðt1 ÞCe;0 ðt1 ÞCe;0 ðt2 ÞCe;1 ðt2 Þj 1; 0; 1; 0〉1234 + Ce;0 ðt1 ÞCg;2 ðt1 ÞCe;0 ðt2 ÞCe;0 ðt2 Þj 2; 0; 0; 0〉1234
) ;
where N is a normalization factor. The success probability to detect the atoms a and b in their excited states is obtained as ðe;0Þ 4 ðe;0Þ 2 ðe;2Þ 2 Pe;e = jCe;0 ðt1 Þj j Ce;0 ðt2 Þj j Ce;2 ðt2 Þj ðe;0Þ
2
ðe;0Þ
2
ðe;0Þ
ð16Þ 2
ð f ;2Þ
2
+ j Ce;0 ðt1 Þ j j Cf ;1 ðt1 Þj j Ce;0 ðt2 Þj j Ce;1 ðt2 Þj ðe;0Þ 2 ðe;0Þ 2 ðe;0Þ 2 ð g;2Þ 2 + j Ce;0 ðt1 Þj j Cg;2 ðt1 Þj j Ce;0 ðt2 Þj j Ce;0 t2 j : Now, from the Eq. (15) for an appropriate choice of the times t1 and t2 we obtain the teleported state in the approximated form |Φ〉1234 ≃ (α|0, 2〉12 + β|2, 0〉12)|0, 0〉34, which allows us to get the fidelity F = ||12〈Ψ|Φ〉1234||2 of the teleported state (with |Ψ〉12 given by Eq. (13)). In this way we obtain ðe;0Þ
F=
ðe;0Þ
ðe;0Þ
ð g;2Þ
jCe;0 ðt1 Þ j 2 j Cg;2 ðt1 Þj 2 j Ce;0 ðt2 Þj 2 j Ce;0 ðt2 Þ j 2 Pe;e
:
ð17Þ
Fig. 2 shows the probability (15) versus the interaction times t1 and t2, while Fig. 3a and b shows the fidelity (16) versus t2 considering an appropriate choice t1 = 2 μs. Note that both, the success probability and fidelity, do not depend on the coefficient α. Let us make a comment on the experimental feasibility of the present scheme. The devices used here are based on those of Ref. [23] that uses Rydberg atoms with quantum number n N 89. The
Fig. 3. The fidelity versus t1 and t2 is shown in (a). In (b) we display the success probability (black-solid line) and fidelity (red-dashed line) of the teleported state given respectively by Eqs. (15) and (16) versus t2, for a fixed time t1 = 2 μs.
A.D. dSouza et al. / Optics Communications 284 (2011) 1086–1089
by two-photon Jaynes–Cummings model in the full microscopical Hamiltonian approach. On the one hand, this scheme is advantageous in comparison with the teleportation of a single qubit [18] since the success probability and the fidelity do not depend on the coefficients of the state to be teleported. On the other hand, the time spent in the entire procedure is about 180 μs (assuming the atoms emerging one by one from the apparatus), which is much smaller than the decoherence time of the cavity (0.1 s) [29], showing the experimental feasibility of the scheme. Acknowledgements We thank the CAPES, CNPq, and FUNAPE/GO, Brazilian agencies, for the partial supports. References [1] R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, Rev. Mod. Phys. 81 (2009) 865. [2] C.H. Bennett, S.J. Wiesner, Phys. Rev. Lett. 69 (1992) 2881. [3] A.K. Ekert, Phys. Rev. A 67 (1991) 661. [4] N. Gisin, G. Ribordy, W. Tittel, H. Zbinden, Rev. Mod. Phys. 74 (2002) 145. [5] C.H. Bennett, et al., Phys. Rev. Lett. 70 (1993). [6] D. Gottesman, I.L. Chuang, Nature (London) 402 (1999) 390. [7] D. Bouwmeester, et al., Nature 390 (1997) 575; D. Boschi, et al., Phys. Rev. Lett. 80 (1998) 1121; E. Lombardi, F. Sciarrino, S. Popescu, F. De Martini, Phys. Rev. Lett. 88 (2002) 070402; J.W. Pan, et al., Nature 421 (2003) 721. [8] A. Furusawa, et al., Science 282 (1998) 706. [9] M.A. Nielsen, E. Knill, R. Laflamme, Nature 396 (1998) 52.
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