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Deterministic secure communication via atomic momentum state
Optik 122 (2011) 1965–1969
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Deterministic secure communication via atomic momentum state A. El Allati a,b,∗ , Y. Hassouni a , F. Saif c a
Faculté des Sciences, Département de Physique, LPT-URAC-13, Université Mohammed V- Agdal Av. Ibn Battouta, B.P. 1014, Rabat, Morocco The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy c Department of Electronics, Quaid-i-Azam University, 45320 Islamabad, Pakistan b
a r t i c l e
i n f o
Article history: Received 30 July 2010 Accepted 12 December 2010
Quantum cryptography Atomic momentum state Bragg regime Raman-Nath regime Cavity
a b s t r a c t We present a secure direct communication scheme via atomic external degrees of freedom. The scheme consists of transmitting the secret message by superposition of atomic center-of-mass momentum states, quantum bit, via atomic scattering in the Bragg regime and Raman-Nath regime, respectively. The information can be coded/decoded during the transmission, without basis reconciliation, privacy amplification, and end of the communication. The security of information against any eavesdropper is ensured by the interference pattern. Crown Copyright © 2011 Published by Elsevier GmbH. All rights reserved.
1. Introduction Quantum cryptography, known quantum key distribution, is a process in which the users of communication establish a technique for sharing secret key in the quantum channel. The first protocol was invented in QKD by Charles H. Bennett and Gilles Brassard BB84 protocol in 1984 [1], using one-time pad protocol [2] with the laws of quantum mechanics. Since then, many famous protocols were proposed in the literature like as Bennett 92 protocol B92 [3] and Einstein–Podolsky–Rosen protocol EPR [4,5]. All these protocols are usually non deterministic, where the sender and receiver cannot determine the bits of information only at the end of communication. In addition, Alice and Bob used some classical algorithms for correcting the errors and increasing the security, corrector code and privacy amplification [6]. And also, they are announced the measurement bases through classical channel, in order to check the key and detect the eavesdropper presence, Eve. However, The complexity, the cost, and the security of the realization scheme of these protocols are increased by their conditions, discarded almost half the sequence of qubits, and presence an eavesdrop in the public channel. In a recent paper, different quantum cryptography schemes are proposed to improve and facility the implementation of secret communication. And also, the schemes are consisted to transmit and
∗ Corresponding author at: Faculté des Sciences, Département de Physique, LPTURAC-13, Université Mohammed V- Agdal Av. Ibn Battouta, B.P. 1014, Rabat, Morocco. E-mail address: [email protected] (A. El Allati).
secure direct information, where the the message is encoded deterministically through the quantum channel, without discarded the qubits and less expensive [7–11]. In this way, we propose a secure direct communication scheme using single atom, the scheme consists of sending two-level state atom in their ground state and in excite state through two cavities respectively. The controlling atomic-field interaction time inside the first cavity allows us to generate out atomic momentum states, qubits. A control phase introduces by the emitter for generating the different qubits. The receiver uses the second cavity for decoding the information based on interference pattern, and determines from the visibility of the interferential fringes the message. Any eavesdrop of states conducts to destruction the interference. This paper is organized as follows: in Section 2, we present a scheme of communication in detail. In Section 3, we present the protocol. In Section 4, a conclusion with some perspectives are given.
2. Scheme of communication Let us consider a beam of atom in the ground state moving with center of mass momentum P0 , towards cavity. The inner field in the cavity is an optical standing wave of wavelength . One atom is injected in the cavity at a time and the cavity is aligned along the x-axis, see Fig. 1, the atom interacts with the cavity field which is in Fock state |n. Taken large detuning between the cavity field frequency and the atomic transition frequency ω0 , the atom remains in the initial state with no spontaneous emission of photons. The result of the interaction “atom–field” is the atom may be undeflected or deflected in the direction of wave propagation by an even multiple of photon momentum h ¯ k [12].
0030-4026/$ – see front matter. Crown Copyright © 2011 Published by Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2010.12.013
1966
A. El Allati et al. / Optik 122 (2011) 1965–1969
2.1. Bragg regime P`0
Cavity field
In the Bragg regime of atomic deflection, for large detuning [14], the recoil frequency of the atom is much higher than the effective Rabi frequency, viz
x P0
Scattered beam
µ
Incident beam
P¡`0
ωrec n .
(7)
The energy conservation in the Bragg regime requires that Fig. 1. Suggested experimental of scattered atom, which illustrates by an incoming atomic beam with momentum P0 , along the field and at an angle with vertical axis. In the Bragg regime, there are two possible directions with momentum P0 and P−0 .
The total Hamiltonian of the interaction atom in the presence of its center-of-mass motion with standing wave having a wave number k, defined as k = 2/, in dipole and in rotating wave approximations is given by H=
Pˆ x2 + 12 h ¯ ω0 z + h ¯ a+ a + h ¯ g cos(kˆx)(+ a + − a+ ), 2M
(1)
where, Pˆ x and xˆ are the momentum and position operators of the atom, the operators z and ± are the Pauli matrices, and a and a+ are the annihilation and creation operators of the standing wave, respectively. g is the atom–field coupling constant, = − ω0 describes detuning between the atomic transition frequency and the field frequency. We suppose that the detuning is supposed to be large enough, such that direct atomic transitions do not occur, but the interactions between a single atom and the cavity field in a state |n do occur, off resonance, [13]. So, it is very rare to find the atom in the excited state. The system may be then be described by the effective Hamiltonian ˆ eff H
Pˆ 2 h ¯ |g|2 ˆ − + cos(2kˆx + 1), n = x − 2M 2
(2)
with + − + a+ a z = − a+ a − + . The wave function of the atom–field system after an interaction time t, in discrete momentum space is given by +∞
| (t) =
g,n
(CP (t)|Pl , g, n + CPe,n (t)|Pl , e, n), l
(3)
l
l=−∞
momentum P after interactions, and n is the number of photons in the cavity. During the atom–field interaction, the momentum transferred to the atom by the field is either 0 or 2¯hk [12], for each complete Rabi cycle, the momentum of the exciting atom is then given by P = P0 + ¯hk,
(4)
where is an even integer. The time evolution of the atom during its interaction with the cavity field is given by the Schrödinger equation as i¯h
d ˆ eff | (t), | (t) = H dt
(5)
which leads to the following equations of motion for the probability g,n amplitudes CP :
g,n
i
∂CP (t)
∂t
= ωrec ( + 0 )CPn (t) −
=h ¯ k2 /2M
n g,n g,n (t) + CP −2¯hk (t)), (C 2 P +2¯hk
(6)
where ωrec is the frequency associated with the photon recoil, n = g2 n/2 is the effective Rabi frequency, and 0 = P0 k/M is the Doppler shift. From Eq. (6), one can study atomic scattering through the cavity in any regime.
(P0 + ¯hk) , 2M
(8)
where P0 = 0 k/2¯h = P0 is the initial momentum of the atom [15] and Pout is the momentum of the atom when it emerges out of the cavity after interactions. In the Bragg condition, 0 = 2, 4, 6..., which corresponds to the first, second, third order of Bragg scattering, respectively. Eq. (8) has two solutions = 0 and = − 0 , which means that there are only two possible directions of the atomic momentum out of the cavity:
=0 = −0
undeflected, deflected.
Substituting | (t) into the Schrödinger equation yields the coupled differential equations from = 0 to = − 0 for the Bragg regime, as g,n
i
∂CP (t)
=
0
∂t
−
n g,n g,n + CP ), (C −2 2 P2
g,n
i
∂CP (t)
g,n
−2
= ωrec (−2)(−2+0 )CP (t) −
∂t ... ... ... ... g,n ∂C− (t) 0 i ∂t
−2
... ... ... ...
n g,n g,n (C (t)+CP (t)), −2 2 P2
... ... ... ...
= −
(9)
n g,n g,n (t) + CP (t)). (C −0 −2 2 P−0 +2
The adiabatically resolution of the coupled differential equations allows to obtain two coupled equations [16] (for 0 > 2) as
g,n l
=
2M
where |Pl is the atom state of transverse momentum Pl , and CP (t) and CPe,n (t) are the probability amplitudes that the atom exists with l
2
2 Pout
g,n
i
∂CP (t) 0
∂t g,n
i
∂CP
−0
g,n
g,n
= An CP (t) − 12 Bn C− (t), 0
(t)
(10)
0
g,n
g,n
= An C− (t) − 12 Bn CP (t),
∂t
(11)
0
0
where n
An = −
( 2 )
2
ωrec (0 − 2)(2)
(12)
,
and ( n )
Bn = (2ωrec )
0 2
−1
0 2
.
(13)
[(0 − 2)(0 )...4..22 ]
From Eqs. (10) and (11), the probability of the atom exiting with momentum P0 is CP0 (t), and that of exiting with momentum P−0 is C−0 (t). So, the probability amplitudes are given, for the initial g,n conditions CP (0) = 1 and CP− (0) = 0, by 0
g,n CP (t) 0
=
0
Bn g,n g,n e−iAn t [CP (0) cos( (t)) + iC− (0) sin( 12 Bn t)], 0 0 2
(14)
Bn g,n (t)) + iCP (0) sin( 12 Bn t)]. 0 2
(15)
C− (t) = e−iAn t [C− (0) cos( g,n
g,n
0
0
A. El Allati et al. / Optik 122 (2011) 1965–1969
where a and a+ are the annihilation and creation operators of the cavity mode. The effective Hamiltonian for atom–field system is given by
Bob Alice x
ÁA = 0; ¼
P0
ˆ eff = H
Screen
Pˆ x2 ¯ g cos(kˆx)(+ a + − a+ ). +h 2M
First Cavity Second Cavity
1 = √ 2
| (t)final
(16)
The probabilities that the atom exits with momentums P0 and P−0 are respectively given by
P(P−0 , t) =
sin2 ( 12 Bn t).
For an atom–cavity interaction time such as (1/2)Bn t = /4, the above probabilities are the same, and the output of cavity atom state is | (
1 ) = √ [|P0 + |P−0 ] ⊗ |g. 2Bn atom 2
1 ) = √ [ei A |P0 + |P−0 ] ⊗ |g. 2Bn 2
0
2 P−
0
⎞
(26)
t
g,n+1
g,n+1
i
∂Ce,n (t) ∂t
√ g,n+1 g,n+1 =(0 + ωrec 2 )Ce,n (t) + 12 g n + 1(C+1 (t) + C−1 (t)), (27)
g,n+1
i
∂C
(t)
∂t
g,n+1
= (0 + ωrec 2 )C
√ e,n e,n (t) + 12 g n + 1(C+1 + C−1 (t)), (28)
(19) where the initial conditions
The probability of finding the atom in either of the states |P0 or |P−0 is 1/2, which corresponds to second diffraction of Bragg, deflected/undeflected. The atom leaves the cavity with the superposition of atomic momentum states. And also one of the momentum state passes through the control phase A = 0, in the order to produce the phases between them, the state is defined as | (
P2
⎝e−i 2M¯h t (Ce,n (t)|P , e, n + Cg,n+1 (t)|P , g, n + 1)
e,n (t), C− (t) and C− (t) represent the time with Ce,n (t), C dependent probability amplitudes for the atom. The probability amplitudes to find an atom in the state |P satisfy the following Eqs. (27) and (28), which may be deduced, as in Bragg regime, from the Schrodinger equation,
(17) (18)
⎛
g,n+1 e,n +e 2M¯h (C− (t)|P− , e, n + C− (t)|P− , g, n + 1)⎠ , −i
The substitution of these conditions in Eqs. (14) and (15) allows to get the state of the scattered atom and cavity field as:
∞
=−∞
Fig. 2. Atom interference scheme with one atom passing through two cavities, respectively, and different devices as Ramsey field, control phase A and screen.
P(P0 , t) = cos2 ( 12 Bn t),
(25)
The final state of the system after an interaction time t, inside the second cavity is
Ramsey Field
| (t) = eiAn [cos( 12 Bn t)|P0 + i sin( 12 Bn t)|P−0 ]|g, n.
1967
(20)
g,n+1
Ce,n (0) 0
=
e,n C− (0) 0
= 1 and
g,n+1 C (0)
=
C− (0) = 0 are supposed. The analytical solution of the differential Eqs. (27) and (28) can be solved in the Raman-Nath regime. The Raman-Nath regime is defined as the case of very short interaction time. This regime of atomic diffraction is characteristic of situations where the recoil energy is smaller than the interaction energy [17], viz ωrec g
n + 1.
(29)
Choosing specific values of the phase A , we get the following two particular superpositions
The analytical solution of differential Eqs. (27) and (28)[18], are given by
1 |0L = √ (|P−0 + |P0 ), 2
(21)
e,n (t) = exp(−i(0 t + ))J2 ( (t)), C2
(30)
1 |1L = √ (|P−0 − |P0 ). 2
g,n+1 C2+1 (t)
(31)
(22)
The information is encoded in orthogonal momentum states, these states are carrying and securing the information between Alice and Bob. Any attempts by Eve to eavesdrop the states will be detected by the interference phenomenon which obtains by the Raman-Nath regime. 2.2. Raman-Nath regime Let us consider the previous atom, which exits the first cavity in the state (19), and excites by a Ramsey field, then travels freely towards a second cavity, see Fig. 2, in the state 1 | = √ (ei A |P0 + |P−0 ) ⊗ |e, 2
The atom–field interaction inside the second cavity is supposed to occur at exact resonance ( = 0) and is described by the Hamiltonian H=
Pˆ x2 ¯ ωz + h ¯ a+ a + h ¯ g cos(kˆx)(+ a + − a+ ), + 12 h 2M
where J is the -th-order Bessel function, (t) = √ 2g n + 1/0 sin(0 t/2), and all the other probability ampli/ 0 then 0 = / 0, the probability tudes are zero. In the case P0 = g,n+1 amplitudes Ce,n (t) and C (t) are periodic functions of t of period T=
2 = . v0 0
(24)
(32)
where = 2 /k is the wavelength of the field and v0 = P0 /M is the velocity of the atomic motion along the cavity. The same results are obtained for the atomic momentum states |P− . The final atom–field state is then given by | (t)final
(23)
= exp(−i(0 t + )( + 12 ))J2+1 ( (t)),
1 = √ 2
∞
=−∞ 2 P−
0
⎡
P2
0
g,n+1 ⎣e−i 2M¯h t (C2e,n (t)|P2 , e, n + C2+1 (t)|P2+1 , g, n + 1)
⎤
(33)
t
g,n+1 e,n +e 2M¯h (C−2 (t)|P−2 , e, n + C−(2+1) (t)|P−(2+1) , g, n + 1)⎦ , −i
this coherent superposition state of momentum gives the interference pattern at the detector plane.
1968
A. El Allati et al. / Optik 122 (2011) 1965–1969
The wave function of the atom with momentum P , coming from the initial state undiffracted beam, after interaction time t is given by
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
P02 t e,n 1 −i 1 (p , t) = √ e 2M¯h C (t)|e, n 2 P02 t g,n+1 1 −i (t)|g, n + 1 1 (p , t) = √ e 2M¯h C 2
even,
odd,
the wave function coming from initial state diffracted beam with momentum P− after interaction time t is given by
⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩
2 P− 0 t e,n 1 −i 2 (p , t) = √ e 2M¯h C− (t)|e, n 2 2 P− 0 t g,n+1 1 −i (t)|g, n + 1 2 (p , t) = √ e 2M¯h C− 2
even,
odd.
The probability to find the atomic momentum ¯hk is P(p , t) =
1 |J ( )|2 (1 + cos(20 t)). 2
(34)
The interference behavior is represented by the term cos (20 t), and Bessel function J ( ) is explained as the envelop function of the fringes. The momentum distribution of the atom in the x-direction can exhibited by the fringes in the different intensities detected on the screen. The plot in Fig. 3 shows the variation of atomic momentum distribution P(p , t) as a function √ of , for 0 = 1.63 ∗ 108 Hz and 2g n + 1/0 = 20 [19] at time as, t = (3/2)(1/0 ). 3. Protocol The correlation of superposition of atomic momentum states allows to carry the information directly and securely, where the emitter encodes the message using the first cavity, and also the receiver decodes the message using the second cavity, without using the notion of the private key [1,2]. Furthermore, the eavesdroppers can be detected from the visibility of the interferential fringes in the output of cavity. The fundamental properties of the protocol in detail: 1. Alice prepares a momentum superposition qubit, in the state |0L or |1L . Alice encodes the message using the phases A , either 0 or . Then, she transmits the atoms to Bob with regular time intervals between them. 2. Bob uses second cavity for visibility of the fringes, see Fig. 3, then decodes the message from fringes of interference. 3. Bob detects the presence of Eve by destruction of interference, he calls Alice to stop the communication.
The protocol is simple and less expensive only with one quantum channel, and, it gives some advantages. Bob does not wait the end communication to extract the key, such as a lot of protocols [1,3,4]. The protocol need not to announce the bases, and discard about half of the qubits, bases reconciliation. And also, Alice and Bob need not to sacrifice subset of secret bits publicly for increasing the security [20]. The safe of information is ensured by the measurement of the states, any eavesdropper’s attacks must disturb the quantum states. Then, the security is depended of apparition or destruction of the interference pattern, any measure or intercept of single atom introduces some errors in the screen
Fig. 3. The variation of atomic momentum state as function of between −20¯hk to 20¯hk with time 3/40 . (a) |0L = √1 (|P−0 + |P0 ) (b) |1L = √1 (|P−0 − |P0 ) 2
2
(Fig. 3), so, destruction of interference means the presence of an eavesdrop. 4. Conclusion We proposed a simple scheme for securing direct communication using atomic center-of-mass momentum state. The scheme used the ideas atom optics and cavity. The generation of superposition momentum states are realized in the out of cavity using atomic scattering by a standing wave field in a Bragg regime. The interference pattern was gotten by atomic scattering with field inside the cavity in a Raman-Nath regime. In our scheme, the secret message can be encoded directly using atomic momentum states with control of phases. So, the superposition of states used as support of carrying the information direct without public channel or discarded the qubits as classical protocols [1,3]. The scheme opens a new field for using the atomic momentum states in quantum information, such as teleportation [21], and also in the quantum computer, especially representing the Hadamard operation.
A. El Allati et al. / Optik 122 (2011) 1965–1969
Acknowledgments A.E.A. acknowledges the hospitality of the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. He would like to thank Pr. M. El Baz for previous discussions on the subject and would like to thank Pr. S. Najmi for his important remarks on the manuscript. References [1] C.H. Bennett, G. Brassard, Quantum cryptography: public key distribution and coin tossing, in: Proceedings of IEEE International Conference on Computers, Systems and Signal Processing, Bangalore, India (IEEE, New York, 1984), 1984, p. 175. [2] G.S. Vernam, Cipher printing telegraph systems for secret wire and radio telegraphic communications, J. AIEE 45 (1926) 109. [3] C.H. Bennett, Quantum cryptography using any two nonorthogonal states, Phys. Rev. Lett. 68 (1992) 3121–3124. [4] A.K. Ekert, Quantum cryptography based on Bell’s theorem, Phys. Rev. Lett. 67 (1991) 661–663. [5] C.H. Bennett, G. Brassard, N.D. Mermin, Quantum cryptography without Bell’s theorem, Phys. Rev. Lett. 68 (1992) 557–559. [6] M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, 2000. [7] A. Beige, B.G. Englert, C. Kurtsiefer, H. Weinfurter, Secure communication with a publicly known key, Acta Phys. Pol. A 101 (2002) 357–368.
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[8] K. Boström, T. Felbinger, Deterministic secure direct communication using entanglement, Phys. Rev. Lett. 89 (2002) 187902. [9] F.G. Deng, G.L. Long, Secure direct communication with a quantum one-time pad, Phys. Rev. A 69 (2004) 052319. [10] C. Wang, F.G. Deng, G.L. Long, Multi-step quantum secure direct communication using multi-particle Green–Horne–Zeilinger state, Opt. Commun. 253 (2005) 15–20. [11] A. El Allati, M. El Baz, Y. Hassouni, Quantum key distribution via tripartite coherent states, Quant. Inf. Process, DOI: 10 1007/s11128–010-0213-y. [12] A.F. Bernhardt, B.W. Shore, Coherent atomic deflection by resonant standing waves, Phys. Rev. A 23 (1981) 1290–1301. [13] C.M. Savage, S.L. Braunstein, D.F. Walls, Macroscopic quantum superpositions by means of single-atom dispersion, Opt. Lett. 15 (1990) 628–630. [14] M. Marte, S. Stenholm, Multiphoton resonances in atomic bragg scattering, Appl. Phys. B 54 (1992) 443–450. [15] P. Meystre, E. Schumacher, S. Stenholm, Atomic beam deflection in a quantum field, Opt. Commun. 73 (1989) 443–447. [16] A. Khalique, F. Saif, Engineering entanglement between external degrees of freedom of atoms via Bragg scattering, Phys. Lett. A 314 (2003) 37–43. [17] F. Saif, Fam Le Kien, M.S. Zubairy, Quantum theory of a micromaser operating on the atomic scattering from a resonant standing wave, Phys. Rev. A 64 (2001) 043812. [18] M.O. Scully, W.E. Lamb, Quantum theory of an optical maser. I. General theory, Phys. Rev. 159 (1967) 208–226. [19] J.S. Peng, G.X. Li, Intoduction to Modern Quantum Optics, World Scientific Publishing Company, Cambridge University Press, 1998, p. 471. [20] C.H. Bennett, F. Bessette, G. Brassard, L. Salvail, J. Smolin, Experimental quantum cryptography, J. Cryptol. 5 (1992) 3. [21] S. Qamar, S.Y. Zhu, M.S. Zubairy, Teleportation of an atomic momentum state, Phy. Rev. A 67 (2003) 042318.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Deterministic secure communication via atomic momentum state A. El Allati a,b,∗ , Y. Hassouni a , F. Saif c a
Faculté des Sciences, Département de Physique, LPT-URAC-13, Université Mohammed V- Agdal Av. Ibn Battouta, B.P. 1014, Rabat, Morocco The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy c Department of Electronics, Quaid-i-Azam University, 45320 Islamabad, Pakistan b
a r t i c l e
i n f o
Article history: Received 30 July 2010 Accepted 12 December 2010
Quantum cryptography Atomic momentum state Bragg regime Raman-Nath regime Cavity
a b s t r a c t We present a secure direct communication scheme via atomic external degrees of freedom. The scheme consists of transmitting the secret message by superposition of atomic center-of-mass momentum states, quantum bit, via atomic scattering in the Bragg regime and Raman-Nath regime, respectively. The information can be coded/decoded during the transmission, without basis reconciliation, privacy amplification, and end of the communication. The security of information against any eavesdropper is ensured by the interference pattern. Crown Copyright © 2011 Published by Elsevier GmbH. All rights reserved.
1. Introduction Quantum cryptography, known quantum key distribution, is a process in which the users of communication establish a technique for sharing secret key in the quantum channel. The first protocol was invented in QKD by Charles H. Bennett and Gilles Brassard BB84 protocol in 1984 [1], using one-time pad protocol [2] with the laws of quantum mechanics. Since then, many famous protocols were proposed in the literature like as Bennett 92 protocol B92 [3] and Einstein–Podolsky–Rosen protocol EPR [4,5]. All these protocols are usually non deterministic, where the sender and receiver cannot determine the bits of information only at the end of communication. In addition, Alice and Bob used some classical algorithms for correcting the errors and increasing the security, corrector code and privacy amplification [6]. And also, they are announced the measurement bases through classical channel, in order to check the key and detect the eavesdropper presence, Eve. However, The complexity, the cost, and the security of the realization scheme of these protocols are increased by their conditions, discarded almost half the sequence of qubits, and presence an eavesdrop in the public channel. In a recent paper, different quantum cryptography schemes are proposed to improve and facility the implementation of secret communication. And also, the schemes are consisted to transmit and
∗ Corresponding author at: Faculté des Sciences, Département de Physique, LPTURAC-13, Université Mohammed V- Agdal Av. Ibn Battouta, B.P. 1014, Rabat, Morocco. E-mail address: [email protected] (A. El Allati).
secure direct information, where the the message is encoded deterministically through the quantum channel, without discarded the qubits and less expensive [7–11]. In this way, we propose a secure direct communication scheme using single atom, the scheme consists of sending two-level state atom in their ground state and in excite state through two cavities respectively. The controlling atomic-field interaction time inside the first cavity allows us to generate out atomic momentum states, qubits. A control phase introduces by the emitter for generating the different qubits. The receiver uses the second cavity for decoding the information based on interference pattern, and determines from the visibility of the interferential fringes the message. Any eavesdrop of states conducts to destruction the interference. This paper is organized as follows: in Section 2, we present a scheme of communication in detail. In Section 3, we present the protocol. In Section 4, a conclusion with some perspectives are given.
2. Scheme of communication Let us consider a beam of atom in the ground state moving with center of mass momentum P0 , towards cavity. The inner field in the cavity is an optical standing wave of wavelength . One atom is injected in the cavity at a time and the cavity is aligned along the x-axis, see Fig. 1, the atom interacts with the cavity field which is in Fock state |n. Taken large detuning between the cavity field frequency and the atomic transition frequency ω0 , the atom remains in the initial state with no spontaneous emission of photons. The result of the interaction “atom–field” is the atom may be undeflected or deflected in the direction of wave propagation by an even multiple of photon momentum h ¯ k [12].
0030-4026/$ – see front matter. Crown Copyright © 2011 Published by Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2010.12.013
1966
A. El Allati et al. / Optik 122 (2011) 1965–1969
2.1. Bragg regime P`0
Cavity field
In the Bragg regime of atomic deflection, for large detuning [14], the recoil frequency of the atom is much higher than the effective Rabi frequency, viz
x P0
Scattered beam
µ
Incident beam
P¡`0
ωrec n .
(7)
The energy conservation in the Bragg regime requires that Fig. 1. Suggested experimental of scattered atom, which illustrates by an incoming atomic beam with momentum P0 , along the field and at an angle with vertical axis. In the Bragg regime, there are two possible directions with momentum P0 and P−0 .
The total Hamiltonian of the interaction atom in the presence of its center-of-mass motion with standing wave having a wave number k, defined as k = 2/, in dipole and in rotating wave approximations is given by H=
Pˆ x2 + 12 h ¯ ω0 z + h ¯ a+ a + h ¯ g cos(kˆx)(+ a + − a+ ), 2M
(1)
where, Pˆ x and xˆ are the momentum and position operators of the atom, the operators z and ± are the Pauli matrices, and a and a+ are the annihilation and creation operators of the standing wave, respectively. g is the atom–field coupling constant, = − ω0 describes detuning between the atomic transition frequency and the field frequency. We suppose that the detuning is supposed to be large enough, such that direct atomic transitions do not occur, but the interactions between a single atom and the cavity field in a state |n do occur, off resonance, [13]. So, it is very rare to find the atom in the excited state. The system may be then be described by the effective Hamiltonian ˆ eff H
Pˆ 2 h ¯ |g|2 ˆ − + cos(2kˆx + 1), n = x − 2M 2
(2)
with + − + a+ a z = − a+ a − + . The wave function of the atom–field system after an interaction time t, in discrete momentum space is given by +∞
| (t) =
g,n
(CP (t)|Pl , g, n + CPe,n (t)|Pl , e, n), l
(3)
l
l=−∞
momentum P after interactions, and n is the number of photons in the cavity. During the atom–field interaction, the momentum transferred to the atom by the field is either 0 or 2¯hk [12], for each complete Rabi cycle, the momentum of the exciting atom is then given by P = P0 + ¯hk,
(4)
where is an even integer. The time evolution of the atom during its interaction with the cavity field is given by the Schrödinger equation as i¯h
d ˆ eff | (t), | (t) = H dt
(5)
which leads to the following equations of motion for the probability g,n amplitudes CP :
g,n
i
∂CP (t)
∂t
= ωrec ( + 0 )CPn (t) −
=h ¯ k2 /2M
n g,n g,n (t) + CP −2¯hk (t)), (C 2 P +2¯hk
(6)
where ωrec is the frequency associated with the photon recoil, n = g2 n/2 is the effective Rabi frequency, and 0 = P0 k/M is the Doppler shift. From Eq. (6), one can study atomic scattering through the cavity in any regime.
(P0 + ¯hk) , 2M
(8)
where P0 = 0 k/2¯h = P0 is the initial momentum of the atom [15] and Pout is the momentum of the atom when it emerges out of the cavity after interactions. In the Bragg condition, 0 = 2, 4, 6..., which corresponds to the first, second, third order of Bragg scattering, respectively. Eq. (8) has two solutions = 0 and = − 0 , which means that there are only two possible directions of the atomic momentum out of the cavity:
=0 = −0
undeflected, deflected.
Substituting | (t) into the Schrödinger equation yields the coupled differential equations from = 0 to = − 0 for the Bragg regime, as g,n
i
∂CP (t)
=
0
∂t
−
n g,n g,n + CP ), (C −2 2 P2
g,n
i
∂CP (t)
g,n
−2
= ωrec (−2)(−2+0 )CP (t) −
∂t ... ... ... ... g,n ∂C− (t) 0 i ∂t
−2
... ... ... ...
n g,n g,n (C (t)+CP (t)), −2 2 P2
... ... ... ...
= −
(9)
n g,n g,n (t) + CP (t)). (C −0 −2 2 P−0 +2
The adiabatically resolution of the coupled differential equations allows to obtain two coupled equations [16] (for 0 > 2) as
g,n l
=
2M
where |Pl is the atom state of transverse momentum Pl , and CP (t) and CPe,n (t) are the probability amplitudes that the atom exists with l
2
2 Pout
g,n
i
∂CP (t) 0
∂t g,n
i
∂CP
−0
g,n
g,n
= An CP (t) − 12 Bn C− (t), 0
(t)
(10)
0
g,n
g,n
= An C− (t) − 12 Bn CP (t),
∂t
(11)
0
0
where n
An = −
( 2 )
2
ωrec (0 − 2)(2)
(12)
,
and ( n )
Bn = (2ωrec )
0 2
−1
0 2
.
(13)
[(0 − 2)(0 )...4..22 ]
From Eqs. (10) and (11), the probability of the atom exiting with momentum P0 is CP0 (t), and that of exiting with momentum P−0 is C−0 (t). So, the probability amplitudes are given, for the initial g,n conditions CP (0) = 1 and CP− (0) = 0, by 0
g,n CP (t) 0
=
0
Bn g,n g,n e−iAn t [CP (0) cos( (t)) + iC− (0) sin( 12 Bn t)], 0 0 2
(14)
Bn g,n (t)) + iCP (0) sin( 12 Bn t)]. 0 2
(15)
C− (t) = e−iAn t [C− (0) cos( g,n
g,n
0
0
A. El Allati et al. / Optik 122 (2011) 1965–1969
where a and a+ are the annihilation and creation operators of the cavity mode. The effective Hamiltonian for atom–field system is given by
Bob Alice x
ÁA = 0; ¼
P0
ˆ eff = H
Screen
Pˆ x2 ¯ g cos(kˆx)(+ a + − a+ ). +h 2M
First Cavity Second Cavity
1 = √ 2
| (t)final
(16)
The probabilities that the atom exits with momentums P0 and P−0 are respectively given by
P(P−0 , t) =
sin2 ( 12 Bn t).
For an atom–cavity interaction time such as (1/2)Bn t = /4, the above probabilities are the same, and the output of cavity atom state is | (
1 ) = √ [|P0 + |P−0 ] ⊗ |g. 2Bn atom 2
1 ) = √ [ei A |P0 + |P−0 ] ⊗ |g. 2Bn 2
0
2 P−
0
⎞
(26)
t
g,n+1
g,n+1
i
∂Ce,n (t) ∂t
√ g,n+1 g,n+1 =(0 + ωrec 2 )Ce,n (t) + 12 g n + 1(C+1 (t) + C−1 (t)), (27)
g,n+1
i
∂C
(t)
∂t
g,n+1
= (0 + ωrec 2 )C
√ e,n e,n (t) + 12 g n + 1(C+1 + C−1 (t)), (28)
(19) where the initial conditions
The probability of finding the atom in either of the states |P0 or |P−0 is 1/2, which corresponds to second diffraction of Bragg, deflected/undeflected. The atom leaves the cavity with the superposition of atomic momentum states. And also one of the momentum state passes through the control phase A = 0, in the order to produce the phases between them, the state is defined as | (
P2
⎝e−i 2M¯h t (Ce,n (t)|P , e, n + Cg,n+1 (t)|P , g, n + 1)
e,n (t), C− (t) and C− (t) represent the time with Ce,n (t), C dependent probability amplitudes for the atom. The probability amplitudes to find an atom in the state |P satisfy the following Eqs. (27) and (28), which may be deduced, as in Bragg regime, from the Schrodinger equation,
(17) (18)
⎛
g,n+1 e,n +e 2M¯h (C− (t)|P− , e, n + C− (t)|P− , g, n + 1)⎠ , −i
The substitution of these conditions in Eqs. (14) and (15) allows to get the state of the scattered atom and cavity field as:
∞
=−∞
Fig. 2. Atom interference scheme with one atom passing through two cavities, respectively, and different devices as Ramsey field, control phase A and screen.
P(P0 , t) = cos2 ( 12 Bn t),
(25)
The final state of the system after an interaction time t, inside the second cavity is
Ramsey Field
| (t) = eiAn [cos( 12 Bn t)|P0 + i sin( 12 Bn t)|P−0 ]|g, n.
1967
(20)
g,n+1
Ce,n (0) 0
=
e,n C− (0) 0
= 1 and
g,n+1 C (0)
=
C− (0) = 0 are supposed. The analytical solution of the differential Eqs. (27) and (28) can be solved in the Raman-Nath regime. The Raman-Nath regime is defined as the case of very short interaction time. This regime of atomic diffraction is characteristic of situations where the recoil energy is smaller than the interaction energy [17], viz ωrec g
n + 1.
(29)
Choosing specific values of the phase A , we get the following two particular superpositions
The analytical solution of differential Eqs. (27) and (28)[18], are given by
1 |0L = √ (|P−0 + |P0 ), 2
(21)
e,n (t) = exp(−i(0 t + ))J2 ( (t)), C2
(30)
1 |1L = √ (|P−0 − |P0 ). 2
g,n+1 C2+1 (t)
(31)
(22)
The information is encoded in orthogonal momentum states, these states are carrying and securing the information between Alice and Bob. Any attempts by Eve to eavesdrop the states will be detected by the interference phenomenon which obtains by the Raman-Nath regime. 2.2. Raman-Nath regime Let us consider the previous atom, which exits the first cavity in the state (19), and excites by a Ramsey field, then travels freely towards a second cavity, see Fig. 2, in the state 1 | = √ (ei A |P0 + |P−0 ) ⊗ |e, 2
The atom–field interaction inside the second cavity is supposed to occur at exact resonance ( = 0) and is described by the Hamiltonian H=
Pˆ x2 ¯ ωz + h ¯ a+ a + h ¯ g cos(kˆx)(+ a + − a+ ), + 12 h 2M
where J is the -th-order Bessel function, (t) = √ 2g n + 1/0 sin(0 t/2), and all the other probability ampli/ 0 then 0 = / 0, the probability tudes are zero. In the case P0 = g,n+1 amplitudes Ce,n (t) and C (t) are periodic functions of t of period T=
2 = . v0 0
(24)
(32)
where = 2 /k is the wavelength of the field and v0 = P0 /M is the velocity of the atomic motion along the cavity. The same results are obtained for the atomic momentum states |P− . The final atom–field state is then given by | (t)final
(23)
= exp(−i(0 t + )( + 12 ))J2+1 ( (t)),
1 = √ 2
∞
=−∞ 2 P−
0
⎡
P2
0
g,n+1 ⎣e−i 2M¯h t (C2e,n (t)|P2 , e, n + C2+1 (t)|P2+1 , g, n + 1)
⎤
(33)
t
g,n+1 e,n +e 2M¯h (C−2 (t)|P−2 , e, n + C−(2+1) (t)|P−(2+1) , g, n + 1)⎦ , −i
this coherent superposition state of momentum gives the interference pattern at the detector plane.
1968
A. El Allati et al. / Optik 122 (2011) 1965–1969
The wave function of the atom with momentum P , coming from the initial state undiffracted beam, after interaction time t is given by
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
P02 t e,n 1 −i 1 (p , t) = √ e 2M¯h C (t)|e, n 2 P02 t g,n+1 1 −i (t)|g, n + 1 1 (p , t) = √ e 2M¯h C 2
even,
odd,
the wave function coming from initial state diffracted beam with momentum P− after interaction time t is given by
⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩
2 P− 0 t e,n 1 −i 2 (p , t) = √ e 2M¯h C− (t)|e, n 2 2 P− 0 t g,n+1 1 −i (t)|g, n + 1 2 (p , t) = √ e 2M¯h C− 2
even,
odd.
The probability to find the atomic momentum ¯hk is P(p , t) =
1 |J ( )|2 (1 + cos(20 t)). 2
(34)
The interference behavior is represented by the term cos (20 t), and Bessel function J ( ) is explained as the envelop function of the fringes. The momentum distribution of the atom in the x-direction can exhibited by the fringes in the different intensities detected on the screen. The plot in Fig. 3 shows the variation of atomic momentum distribution P(p , t) as a function √ of , for 0 = 1.63 ∗ 108 Hz and 2g n + 1/0 = 20 [19] at time as, t = (3/2)(1/0 ). 3. Protocol The correlation of superposition of atomic momentum states allows to carry the information directly and securely, where the emitter encodes the message using the first cavity, and also the receiver decodes the message using the second cavity, without using the notion of the private key [1,2]. Furthermore, the eavesdroppers can be detected from the visibility of the interferential fringes in the output of cavity. The fundamental properties of the protocol in detail: 1. Alice prepares a momentum superposition qubit, in the state |0L or |1L . Alice encodes the message using the phases A , either 0 or . Then, she transmits the atoms to Bob with regular time intervals between them. 2. Bob uses second cavity for visibility of the fringes, see Fig. 3, then decodes the message from fringes of interference. 3. Bob detects the presence of Eve by destruction of interference, he calls Alice to stop the communication.
The protocol is simple and less expensive only with one quantum channel, and, it gives some advantages. Bob does not wait the end communication to extract the key, such as a lot of protocols [1,3,4]. The protocol need not to announce the bases, and discard about half of the qubits, bases reconciliation. And also, Alice and Bob need not to sacrifice subset of secret bits publicly for increasing the security [20]. The safe of information is ensured by the measurement of the states, any eavesdropper’s attacks must disturb the quantum states. Then, the security is depended of apparition or destruction of the interference pattern, any measure or intercept of single atom introduces some errors in the screen
Fig. 3. The variation of atomic momentum state as function of between −20¯hk to 20¯hk with time 3/40 . (a) |0L = √1 (|P−0 + |P0 ) (b) |1L = √1 (|P−0 − |P0 ) 2
2
(Fig. 3), so, destruction of interference means the presence of an eavesdrop. 4. Conclusion We proposed a simple scheme for securing direct communication using atomic center-of-mass momentum state. The scheme used the ideas atom optics and cavity. The generation of superposition momentum states are realized in the out of cavity using atomic scattering by a standing wave field in a Bragg regime. The interference pattern was gotten by atomic scattering with field inside the cavity in a Raman-Nath regime. In our scheme, the secret message can be encoded directly using atomic momentum states with control of phases. So, the superposition of states used as support of carrying the information direct without public channel or discarded the qubits as classical protocols [1,3]. The scheme opens a new field for using the atomic momentum states in quantum information, such as teleportation [21], and also in the quantum computer, especially representing the Hadamard operation.
A. El Allati et al. / Optik 122 (2011) 1965–1969
Acknowledgments A.E.A. acknowledges the hospitality of the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. He would like to thank Pr. M. El Baz for previous discussions on the subject and would like to thank Pr. S. Najmi for his important remarks on the manuscript. References [1] C.H. Bennett, G. Brassard, Quantum cryptography: public key distribution and coin tossing, in: Proceedings of IEEE International Conference on Computers, Systems and Signal Processing, Bangalore, India (IEEE, New York, 1984), 1984, p. 175. [2] G.S. Vernam, Cipher printing telegraph systems for secret wire and radio telegraphic communications, J. AIEE 45 (1926) 109. [3] C.H. Bennett, Quantum cryptography using any two nonorthogonal states, Phys. Rev. Lett. 68 (1992) 3121–3124. [4] A.K. Ekert, Quantum cryptography based on Bell’s theorem, Phys. Rev. Lett. 67 (1991) 661–663. [5] C.H. Bennett, G. Brassard, N.D. Mermin, Quantum cryptography without Bell’s theorem, Phys. Rev. Lett. 68 (1992) 557–559. [6] M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, 2000. [7] A. Beige, B.G. Englert, C. Kurtsiefer, H. Weinfurter, Secure communication with a publicly known key, Acta Phys. Pol. A 101 (2002) 357–368.
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