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Teleportation of atomic states via resonant atom–field interaction
15 August 1999
Optics Communications 167 Ž1999. 111–113 www.elsevier.comrlocateroptcom
Teleportation of atomic states via resonant atom–field interaction Shi-Biao Zheng Department of Electronic Science and Applied Physics, Fuzhou UniÕersity, Fuzhou 350002, China Received 1 March 1999; received in revised form 13 May 1999; accepted 26 May 1999
Abstract A scheme is proposed for the teleportation of an unknown atomic state. It is based on resonant atom–field interactions. q 1999 Elsevier Science B.V. All rights reserved. PACS: 03.65.Bz; 42.50.Dv
In recent years, much attention has been paid to quantum entanglement. The measurement on one of two entangled systems not only gives information on the other system, but also provides possibilities for manipulating it. This striking feature is useful for realizing quantum cryptography w1x and computers w2x. Recently, Bennett et al. w3x have shown that the quantum entanglement can also be used to teleport an unknown quantum state. In the first step of the teleportation process, two spin-1r2 particles are prepared in the maximally entangled state. Then, a joint measurement is performed on the particle to be teleported and one of the correlated pair. Finally, the information of the joint measurement is transmitted to the other observer through a classical channel and thus the initial state of the teleported particle on the second particle of the correlated pair can be reconstructed. Davidovich et al. w4x have presented a scheme for the teleportation of an unknown atomic state between two high-Q cavities initially prepared in an entangled photon-number states. Cirac et al. w5x have made another cavity QED proposal for the realization of quantum teleportation of an atomic state by
using two additional atomic levels of one of the correlated pair. We w6x have also presented a scheme for teleporting an unknown atomic state. In the scheme, Raman and Jaynes–Cummings atom–field interactions are used. More recently, schemes for teleporting certain cavity field states have been proposed w7,8x. In this paper, we propose an alternative scheme for the teleportion of an unknown atomic state. Consider a Rydberg atom having one excited state < e : and two less excited states < g : and < g X : with different magnetic quantum numbers. As will be shown, the two lower levels do not need to have the same energy. Let the atom interact with a single-mode cavity field. We assume the cavity field is sq polarized so that the atomic state < g X : is not affected and thus only transition < e : ™ < g : occurs during the atom–field interactions w9,10x. Suppose the atomic transition is resonant with the cavity field. Then, in the interaction picture, the atom–cavity interaction is described by the Jaynes–Cummings Hamiltonian HI j s g Ž aq< g :² e < q a < e :² g < . ,
0030-4018r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 9 . 0 0 2 8 2 - 5
Ž 1.
S.-B. Zhengr Optics Communications 167 (1999) 111–113
112
where aq and a are the creation and annihilation operators for the cavity mode, and g is the atom– cavity coupling strength. Assume that the atom Žatom a. to be teleported is of the above mentioned type and in the state < f :a s c e < e :a q c g < g :a ,
< c :1 s
< : < : < : < : '2 w e b 0 y i g b 1 .
Ž 4.
Now the state of the whole system combined by two atoms and one cavity can be expanded as 1
2
q
q:
where
":
1
< :< : < :< : '2 Ž e a 0 " i g a 1 . ,
'2
Ž yi < e :a <1: " < g :a <0: . .
Ž 7.
< g :a ™
1
'2
'2 1
'2
Ž < g :a y < e :a . .
'2 Ž 8.
2 1 2
Ž < g :a q < g X :a . Ž <0: . i <1: . ,
Ž 11 .
Ž y< g :a q < g X :a . Ž yi <1: . <0: . .
Ž 12 .
1
'2 1
'2
< g :a Ž <0: . i <1: . ,
Ž 13 .
< e :a Ž yi <1: . <0: . .
Ž 14 .
< g :aX Ž <0: . i <1: . ™
™ 1
Ž < e :a q < g :a . ,
1
Now send an atom initially in the state < g :aX through the cavity. Choose the interaction time appropriately so that the cavity field state is replicated onto this atom. Then let the atom cross a sq polarized classical field, undergoing a pr2 pulse. Thus we obtain
Now let atom a cross a sq polarized classical field, undergoing the transition < e :a ™
Ž 10 .
Next, let the atom cross a sy polarized classical field, undergoing the transition < g X :a ™ < e :a , and then cross a sq polarized classical field, undergoing the transition of Eq. Ž8.. Thus we have
1
1 s
Ž 6.
yi Ž < g :a q < g X :a . <1:
2
Now let this atom cross the cavity and choose the interaction time tX appropriately so that gtX s p. Then we obtain the evolution of the Bell states
Ž 5.
Ž 9.
" Ž < g :a y < g X :a . <0: .
Ž c e < g :b q c g < e:b .
q
Ž < g :a q < g X :a . <0:
2
1
1
"i Ž < g :a y < g X :a . <1: ,
Ž 3.
where t is the interaction time. We choose the atomic velocity carefully so that gt s pr4 is fulfilled. Then we have 1
Ž 2.
where c e and c g are unknown coefficients. The atom Žatom b . to receive the teleported state is initially prepared in the state < e : b . We first send this atom through an initially empty sq polarized cavity. When the atom exists, the cavity the state of the system is given by < c : 1 s cos Ž gt . < e : b <0: y i sin Ž gt . < g : b <1: ,
Then let it cross a sy classical field, undergoing the transition < e :a ™ < g X :a . This leads to
1
< : < : < : '2 Ž g a . e a . 0 X
½
X
y< e :aX <0: , < g :aX <0:
< g :aX Ž yi <1: . <0: . ™ y
™
½
Ž 15 .
1
< : < : < : '2 Ž e a " g a . 0 X
y< g :aX <0: . y< e :aX <0:
X
Ž 16 .
S.-B. Zhengr Optics Communications 167 (1999) 111–113
Thus the evolution for the Bell states is given by
½ ½
y< g :a < e :aX <0: , < g :a < g :aX <0:
Ž 17 .
y< e :a < g :aX <0: . y< e :a < e :aX <0:
Ž 18 .
Hence, the joint measurement can be achieved by detecting atoms a and aX separately. With the outcome of the joint measurement on atoms a and aX transmitted to the receiver, an appropriate rotation to atom b to reconstruct the initial state of atom a can be applied. It is necessary to compare our scheme with previous schemes. Like the scheme of Ref. w5x, our scheme also only involves resonant atom–field interactions. Thus, the interaction time required to disentangled Bell states is much shorter than that of the schemes of Refs. w4,6x. For example, using dispersive atom– field interaction w12x, the transformation 1 Ž< g :a q
'2 < e :a . ™
1
Ž< g :a q < e :a . is completed in a time t s
113
tion times of atoms a, aX , and b with the cavity field are prg s 2 = 10y5 s, prŽ2 g . s 10y5 s, and prŽ4 g . s 0.5 = 10y5 s, respectively. The travelling times of these atoms can thus be assumed to be 2 = 10y4 s, 10y4 s, and 0.5 = 10y4 s, respectively. Therefore, the time required to complete the whole procedure is about 3.5 = 10y4 s, much shorter than Tr . A cavity with a quality factor Q s 10 9 is experimentally achievable w14x. In the present case, the cavity field frequency is n s 51.099 GHz. Thus, the photon lifetime is Tc s QrŽ2 pn . , 3.0 = 10y3 s, much longer than the required time. Therefore, based on cavity QED techniques presently or soon to be available, the proposed scheme might be realizable.
Acknowledgements This work was supported by the Science Research Foundation of Education Committee of Fujian Province and Funds from Fuzhou University.
'2
pDrg 2 , with D being the detuning between the atomic transition frequency and the cavity mode frequency. Setting D s 10 g, we obtain t s 10 prg. In the present scheme, such a transformation is completed in a time of t s prg. The reduction of the operation time can suppress the decoherence effects. When compared with the scheme of Ref. w5x, our scheme has some advantages. Firstly, the present scheme only uses one cavity, while the scheme of Ref. w5x requires two cavities. Furthermore, in order to perform a joint measurement, the scheme of Ref. w5x requires techniques to distinguish the four states of a four-level atom by a single measurement. This might be experimentally problematic. This problem does not appear in our scheme. Finally, we briefly address the experimental feasibility of the proposed scheme. For Rydberg atoms with principal quantum numbers 50 and 51, the radiative time is Tr s 3 = 10y2 s, and the coupling constant is g s 2 p = 25 kHz w13x. Thus, the interac-
References w1x A.K. Ekert, Phys. Rev. Lett. 67 Ž1991. 661. w2x D. Deutsch, R. Jozsa, Proc. R. Soc. London A 439 Ž1992. 553. w3x C.H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, W. Wootters, Phys. Rev. Lett. 70 Ž1993. 1895. w4x L. Davidovich, N. Zagury, M. Brune, J.M. Raimond, S. Haroche, Phys. Rev. A 50 Ž1994. R895. w5x J.I. Cirac, A.S. Parkins, Phys. Rev. A 50 Ž1994. R4441. w6x S.B. Zheng, G.C. Guo, Phys. Lett. A 232 Ž1997. 171. w7x M.H.Y. Moussa, Phys. Rev. A 54 Ž1996. 4661. w8x S.B. Zheng, G.C. Guo, Phys. Lett. A 236 Ž1997. 180. w9x J.I. Cirac, A.S. Parkins, Phys. Rev. A 50 Ž1994. R4441. w10x J.I. Cirac, P. Zoller, Phys. Rev. Lett. 74 Ž1995. 4091. w11x S.L. Braunstein, A. Mann, M. Revzen, Phys. Rev. Lett. 68 Ž1992. 3259. w12x M. Brune, S. Haroche, J.M. Raimond, L. Davidovich, N. Zagury, Phys. Rev. A 45 Ž1992. 5193. w13x M. Brune, F. Schmidt-Kaler, A. Maali, J. Dreyer, E. Hagley, J.M. Raimond, S. Haroche, Phys. Rev. Lett. 76 Ž1996. 1800. w14x J.M. Raimond, M. Brune, S. Haroche, Phys. Rev. Lett. 79 Ž1997. 1964.
Optics Communications 167 Ž1999. 111–113 www.elsevier.comrlocateroptcom
Teleportation of atomic states via resonant atom–field interaction Shi-Biao Zheng Department of Electronic Science and Applied Physics, Fuzhou UniÕersity, Fuzhou 350002, China Received 1 March 1999; received in revised form 13 May 1999; accepted 26 May 1999
Abstract A scheme is proposed for the teleportation of an unknown atomic state. It is based on resonant atom–field interactions. q 1999 Elsevier Science B.V. All rights reserved. PACS: 03.65.Bz; 42.50.Dv
In recent years, much attention has been paid to quantum entanglement. The measurement on one of two entangled systems not only gives information on the other system, but also provides possibilities for manipulating it. This striking feature is useful for realizing quantum cryptography w1x and computers w2x. Recently, Bennett et al. w3x have shown that the quantum entanglement can also be used to teleport an unknown quantum state. In the first step of the teleportation process, two spin-1r2 particles are prepared in the maximally entangled state. Then, a joint measurement is performed on the particle to be teleported and one of the correlated pair. Finally, the information of the joint measurement is transmitted to the other observer through a classical channel and thus the initial state of the teleported particle on the second particle of the correlated pair can be reconstructed. Davidovich et al. w4x have presented a scheme for the teleportation of an unknown atomic state between two high-Q cavities initially prepared in an entangled photon-number states. Cirac et al. w5x have made another cavity QED proposal for the realization of quantum teleportation of an atomic state by
using two additional atomic levels of one of the correlated pair. We w6x have also presented a scheme for teleporting an unknown atomic state. In the scheme, Raman and Jaynes–Cummings atom–field interactions are used. More recently, schemes for teleporting certain cavity field states have been proposed w7,8x. In this paper, we propose an alternative scheme for the teleportion of an unknown atomic state. Consider a Rydberg atom having one excited state < e : and two less excited states < g : and < g X : with different magnetic quantum numbers. As will be shown, the two lower levels do not need to have the same energy. Let the atom interact with a single-mode cavity field. We assume the cavity field is sq polarized so that the atomic state < g X : is not affected and thus only transition < e : ™ < g : occurs during the atom–field interactions w9,10x. Suppose the atomic transition is resonant with the cavity field. Then, in the interaction picture, the atom–cavity interaction is described by the Jaynes–Cummings Hamiltonian HI j s g Ž aq< g :² e < q a < e :² g < . ,
0030-4018r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 9 . 0 0 2 8 2 - 5
Ž 1.
S.-B. Zhengr Optics Communications 167 (1999) 111–113
112
where aq and a are the creation and annihilation operators for the cavity mode, and g is the atom– cavity coupling strength. Assume that the atom Žatom a. to be teleported is of the above mentioned type and in the state < f :a s c e < e :a q c g < g :a ,
< c :1 s
< : < : < : < : '2 w e b 0 y i g b 1 .
Ž 4.
Now the state of the whole system combined by two atoms and one cavity can be expanded as 1
2
q
q:
where
":
1
< :< : < :< : '2 Ž e a 0 " i g a 1 . ,
'2
Ž yi < e :a <1: " < g :a <0: . .
Ž 7.
< g :a ™
1
'2
'2 1
'2
Ž < g :a y < e :a . .
'2 Ž 8.
2 1 2
Ž < g :a q < g X :a . Ž <0: . i <1: . ,
Ž 11 .
Ž y< g :a q < g X :a . Ž yi <1: . <0: . .
Ž 12 .
1
'2 1
'2
< g :a Ž <0: . i <1: . ,
Ž 13 .
< e :a Ž yi <1: . <0: . .
Ž 14 .
< g :aX Ž <0: . i <1: . ™
™ 1
Ž < e :a q < g :a . ,
1
Now send an atom initially in the state < g :aX through the cavity. Choose the interaction time appropriately so that the cavity field state is replicated onto this atom. Then let the atom cross a sq polarized classical field, undergoing a pr2 pulse. Thus we obtain
Now let atom a cross a sq polarized classical field, undergoing the transition < e :a ™
Ž 10 .
Next, let the atom cross a sy polarized classical field, undergoing the transition < g X :a ™ < e :a , and then cross a sq polarized classical field, undergoing the transition of Eq. Ž8.. Thus we have
1
1 s
Ž 6.
yi Ž < g :a q < g X :a . <1:
2
Now let this atom cross the cavity and choose the interaction time tX appropriately so that gtX s p. Then we obtain the evolution of the Bell states
Ž 5.
Ž 9.
" Ž < g :a y < g X :a . <0: .
Ž c e < g :b q c g < e:b .
q
Ž < g :a q < g X :a . <0:
2
1
1
"i Ž < g :a y < g X :a . <1: ,
Ž 3.
where t is the interaction time. We choose the atomic velocity carefully so that gt s pr4 is fulfilled. Then we have 1
Ž 2.
where c e and c g are unknown coefficients. The atom Žatom b . to receive the teleported state is initially prepared in the state < e : b . We first send this atom through an initially empty sq polarized cavity. When the atom exists, the cavity the state of the system is given by < c : 1 s cos Ž gt . < e : b <0: y i sin Ž gt . < g : b <1: ,
Then let it cross a sy classical field, undergoing the transition < e :a ™ < g X :a . This leads to
1
< : < : < : '2 Ž g a . e a . 0 X
½
X
y< e :aX <0: , < g :aX <0:
< g :aX Ž yi <1: . <0: . ™ y
™
½
Ž 15 .
1
< : < : < : '2 Ž e a " g a . 0 X
y< g :aX <0: . y< e :aX <0:
X
Ž 16 .
S.-B. Zhengr Optics Communications 167 (1999) 111–113
Thus the evolution for the Bell states is given by
½ ½
y< g :a < e :aX <0: , < g :a < g :aX <0:
Ž 17 .
y< e :a < g :aX <0: . y< e :a < e :aX <0:
Ž 18 .
Hence, the joint measurement can be achieved by detecting atoms a and aX separately. With the outcome of the joint measurement on atoms a and aX transmitted to the receiver, an appropriate rotation to atom b to reconstruct the initial state of atom a can be applied. It is necessary to compare our scheme with previous schemes. Like the scheme of Ref. w5x, our scheme also only involves resonant atom–field interactions. Thus, the interaction time required to disentangled Bell states is much shorter than that of the schemes of Refs. w4,6x. For example, using dispersive atom– field interaction w12x, the transformation 1 Ž< g :a q
'2 < e :a . ™
1
Ž< g :a q < e :a . is completed in a time t s
113
tion times of atoms a, aX , and b with the cavity field are prg s 2 = 10y5 s, prŽ2 g . s 10y5 s, and prŽ4 g . s 0.5 = 10y5 s, respectively. The travelling times of these atoms can thus be assumed to be 2 = 10y4 s, 10y4 s, and 0.5 = 10y4 s, respectively. Therefore, the time required to complete the whole procedure is about 3.5 = 10y4 s, much shorter than Tr . A cavity with a quality factor Q s 10 9 is experimentally achievable w14x. In the present case, the cavity field frequency is n s 51.099 GHz. Thus, the photon lifetime is Tc s QrŽ2 pn . , 3.0 = 10y3 s, much longer than the required time. Therefore, based on cavity QED techniques presently or soon to be available, the proposed scheme might be realizable.
Acknowledgements This work was supported by the Science Research Foundation of Education Committee of Fujian Province and Funds from Fuzhou University.
'2
pDrg 2 , with D being the detuning between the atomic transition frequency and the cavity mode frequency. Setting D s 10 g, we obtain t s 10 prg. In the present scheme, such a transformation is completed in a time of t s prg. The reduction of the operation time can suppress the decoherence effects. When compared with the scheme of Ref. w5x, our scheme has some advantages. Firstly, the present scheme only uses one cavity, while the scheme of Ref. w5x requires two cavities. Furthermore, in order to perform a joint measurement, the scheme of Ref. w5x requires techniques to distinguish the four states of a four-level atom by a single measurement. This might be experimentally problematic. This problem does not appear in our scheme. Finally, we briefly address the experimental feasibility of the proposed scheme. For Rydberg atoms with principal quantum numbers 50 and 51, the radiative time is Tr s 3 = 10y2 s, and the coupling constant is g s 2 p = 25 kHz w13x. Thus, the interac-
References w1x A.K. Ekert, Phys. Rev. Lett. 67 Ž1991. 661. w2x D. Deutsch, R. Jozsa, Proc. R. Soc. London A 439 Ž1992. 553. w3x C.H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, W. Wootters, Phys. Rev. Lett. 70 Ž1993. 1895. w4x L. Davidovich, N. Zagury, M. Brune, J.M. Raimond, S. Haroche, Phys. Rev. A 50 Ž1994. R895. w5x J.I. Cirac, A.S. Parkins, Phys. Rev. A 50 Ž1994. R4441. w6x S.B. Zheng, G.C. Guo, Phys. Lett. A 232 Ž1997. 171. w7x M.H.Y. Moussa, Phys. Rev. A 54 Ž1996. 4661. w8x S.B. Zheng, G.C. Guo, Phys. Lett. A 236 Ž1997. 180. w9x J.I. Cirac, A.S. Parkins, Phys. Rev. A 50 Ž1994. R4441. w10x J.I. Cirac, P. Zoller, Phys. Rev. Lett. 74 Ž1995. 4091. w11x S.L. Braunstein, A. Mann, M. Revzen, Phys. Rev. Lett. 68 Ž1992. 3259. w12x M. Brune, S. Haroche, J.M. Raimond, L. Davidovich, N. Zagury, Phys. Rev. A 45 Ž1992. 5193. w13x M. Brune, F. Schmidt-Kaler, A. Maali, J. Dreyer, E. Hagley, J.M. Raimond, S. Haroche, Phys. Rev. Lett. 76 Ž1996. 1800. w14x J.M. Raimond, M. Brune, S. Haroche, Phys. Rev. Lett. 79 Ž1997. 1964.