Commun. Theor. Phys. (Beijing, China) 4 (004) pp. 103 109 c International Academic Publishers Vol. 4, No. 1, July 15, 004 Information Entropy Squeezing of a Two-Level Atom Interacting with Two-Mode Coherent Fields LIU Xiao-Juan 1,,3 and FANG Mao-Fa 1,, 1 Department of Physics, Hunan Normal University, Changsha 410081, China Anhui Institute of Optics and Fine Mechanics, the Chinese Academy of Sciences, Hefei 30031, China 3 Department of Physics, Hunan University of Science and Technology, Xiangtan 41101, China (Received October 15, 003; Revised November 18, 003) Abstract From a quantum information point of view we investigate the entropy squeezing properties for a two-level atom interacting with the two-mode coherent fields via the two-photon transition. We discuss the influences of the initial state of the system on the atomic information entropy squeezing. Our results show that the squeezed component number, squeezed direction, and time of the information entropy squeezing can be controlled by choosing atomic distribution angle, the relative phase between the atom and the two-mode field, and the difference of the average photon number of the two field modes, respectively. Quantum information entropy is a remarkable precision measure for the atomic squeezing. PACS numbers: 4.50.Dv Key words: information entropy squeezing, variance squeezing, information entropy uncertainty relation 1 Introduction Squeezing of the atom has been a center of attraction in many theoretical and experimental studies in the quantum optics field over the past few years due to its potential application in high-resolution spectroscopy, [1] the high-precision atomic fountain clocks, [] high-precision spin polarization measurements, [3] generation of quantum controlled few-photon states, [4] control of quantum noise, atomic spin polarization measurements, [5] and generation of squeezed light, [6] which has been recently applied to quantum information theory, for example in quantum teleportation, [7,8] cryptography, [9,10] and dense coding. [11] It is important to note that all these studies of atomic squeezing are based on the Heisenberg Uncertainty Relation (HUR), which is regarded as the standard limitation on measurements of quantum fluctuations. The HUR is formulated in terms of the variance of the system observable. However, the HUR cannot give us sufficient information on atomic squeezing in such cases, as is shown in Ref. [1]. There is an elegant entropic way of reformulating HUR. Several authors [13,14] have studied quantum uncertainty by using quantum entropy theory and obtained an entropic uncertainty relation (EUR) for position and momentum which can overcome the limitation of the HUR. One of the authors of this paper has presented EUR for the two-level atom and defined the entropy squeezing for the two-level atom by using the quantum information theory. [1] More recently, much work on atomic information entropy squeezing in one-photon and multiphoton JCM has been reported. [15 17] On the other hand, considerable attention has been paid to the study of the systems where a two-level atom interacts with the two-mode fields [18 0] via the two-photon transition owing to a good entanglement between the twomode fields and their potential application in the quantum information processing. [1 3] In Refs. [17] [0], the authors have studied the quantum reduced entropy of the atom and the relative entropy between the two-mode fields. However, less attention has been paid to the study of the squeezing for a two-level atom interacting with the two-mode coherent fields via the two-photon transition by using the quantum information entropy theory, which will be the model discussed in this paper. We shall compare its results with those of atomic squeezing based on the HUR, and show that the information entropy is a sensitive measure for the atomic squeezing. Since a three-level atom (in the configuration) with arbitrary detuning can be exactly reduced to a two-level system, [4 6] whose effective Hamiltonian is given by Eq. (1), then our results can also be fit on the entropy squeezing question of a three-level system. Model and Atomic Information Entropy Squeezing The system considered here consists of a two-level atom interacting with the two-mode coherent fields via the two-photon transition processes. The effective Hamiltonian in the rotating-wave approximation [] is ( h = 1) H = ω 0 S z + ω j a j a j + g(a 1 a S + a 1 a S + ), (1) j=1 The project supported by National Natural Science Foundation of China under Grant No. 1987400, the Natural Science Foundation of Hunan Province of China under Grant No. 01JJy3030, and the Scientific Research Fund of the Education Department of Hunan Province of China under Grant No. 01c60 Correspondence author, E-mail: mffang@sparc.hunnu.edu.cn
104 LIU Xiao-Juan and FANG Mao-Fa Vol. 4 where i = z, +, are the usual pseudo-spin operators of the two-level atom, a j (a j) is the photon creation (annihilation) operator of the field mode of frequency w j (j = 1,, and g is the coupling constant for the atomic twomode fields. For simplicity, we consider the case of twophoton resonance, that is, ω 0 = ω 1 + ω. Using standard techniques, [3] it can be shown that this Hamiltonian gives rise to the following time evolution operator in the interaction picture, [ T = cos(âgt) a 1 a sin(âgt)/â ] a 1 a sin( ˆBgt)/ ˆB cos( ˆBgt), ( where  = a 1 a a 1 a, ˆB = a 1 a a 1a. (3) We consider the case that at the time t = 0 the twolevel atom is in a coherent superposition state of the excited state e and the ground state g, ψ A (0) = cos e + exp(iϕ) sin g. (4) And the field is in the two-mode coherent state, ψ F (0) = α 1, α = F (n 1 )F (n ) n 1, n, ( F (n j ) = exp n j n 1,n =0 ) α n j j nj!, (5) where 0 θ π denotes the atomic distribution, 0 ϕ π is the phase of the atom dipole, α j = nj exp (iψ j ), and n j and ψ j represent the initial average photon number and the direction angle of the excitation for mode j (j 1,, respectively. The initial state of the system can be written as ψ F A (0) = ψ F (0) ψ A (0). (6) At any time t > 0 the state vector of the system is given by ψ F A (t) = U I (t) ψ F A (0) = D e + T g (7) with D = T = n 1,n =0 ( F (n 1 )F (n ) cos exp (i[n 1 ψ 1 + n ψ ]) cos(gtγ n1+1,n +1) n 1, n ) i sin exp( i[ϕ n 1 ψ 1 n ψ ]) sin(gtγ n1,n ) n 1 1, n 1, (8) n 1,n =0 ( F (n 1 )F (n ) cos exp(i[n 1 ψ 1 + n ψ ]) sin(gtγ n1+1,n +1) n 1 + 1, n + 1 ) + i sin exp ( i[ϕ n 1 ψ 1 n ψ ]) cos(gtγ n1,n ) n 1, n, (9) where γ n1,n = n 1, n. At any time t > 0 the density operator for the system is given by [ ] D D D T ρ F A (t) = ψ F A (t) ψ F A (t) =. (10) T D T T Tracing ρ FA (t) over the two-mode fields gives rise to the reduced atom density operator for the system [ ] ρ ρ 1 ρ A (t) = Tr F ψ FA (t) ψ FA (t) =, (11) ρ 1 ρ 11 where ρ = D D, ρ 1 = T D, ρ 1 = D T, ρ 11 = T T. (1 We now discuss the properties of atomic information entropy squeezing by using the above results. Here we define the entropy squeezing for the twolevel atom by using the quantum information entropy theory. [13] The information entropy of the Pauli operators S α (α = x, y, z) for a two-level atom system is H(S α ) = P i (S α ) ln P i (S α ), α = x, y, z, (13) i=1 where P i (S α ) = ψ αi ρ ψ αi, i = 1,, which is the probability distribution for two possible outcomes of measurements of an operator S α. H(S x ), H(S y ), and H(S z ) satisfy H(S x ) + H(S y ) ln H(S z ). (14) The inequality (14) may also be rewritten as 4 δh(s x )δh(s y ) δh(s z ), (15) where δh(s α ) = exp[h(s α )]. We define the squeezing of the atom by using the EUR (15) named entropy squeezing, which has received little attention in the past discussions. The fluctuation in the component S(α) (α x or y) of the atomic dipole is said to be squeezed in entropy if the information entropy H(S α ) satisfies the following condition E(S α ) = δh(s α ) δh(sz ) < 0 (α x or y). (16) We recall HUR, S(x) S(y) S z /. (17) Based on the HUR the fluctuation in the component S α of the atomic dipole is said to be squeezed if S(α) satisfies the condition V (S α ) = S(α) S z / < 0 (α x or y), (18)
No. 1 Information Entropy Squeezing of a Two-Level Atom Interacting with Two-Mode Coherent Fields 105 where S α = [ Sα S α ] 1/. Next we will calculate the atomic entropy squeezing in the considered system. By using the atomic reduced density operator ρ A (t) given by Eqs. (11) and (1, we obtain the information entropies of the atomic operators (S x ), (S y ) and (S z ), H(S x ) = [ 1 + Re (ρ 1(t)) ] ln [ 1 + Re (ρ 1(t)) ] [ 1 Re (ρ 1(t)) ] ln [ 1 Re (ρ 1(t)) ], H(S y ) = [ 1 + Im (ρ 1(t)) ] ln [ 1 + Im (ρ 1(t)) ] [ 1 Im (ρ 1(t)) ] ln [ 1 Im (ρ 1(t)) ], H(S z ) = ρ (t) ln ρ (t) ρ 11 ln ρ 11 (t), (19) where ρ ij (t) is given by Eq. (1. Obviously, formula (16) directly connects quantum information with quantum fluctuations, and contains all of the statistical moments orders, while equation (18) only contains the second order of statistical moments. Meaningful information can be retrieved from Eq. (16) in the case of S z = 0 since the right-hand side of Eq. (15) is always non-zero whereas the inequality Eq. (15) is trivially satisfied. We next examine some numerical results for atomic information entropy squeezing. 3 Numerical Results and Discussions On the basis of the analytical solution presented in the previous section we now numerically examine the properties of the atomic information entropy squeezing. For comparison we also consider the atomic variance squeezing, in the different initial states of the system. 3.1 Influence of Atomic Distribution Angle In Figs. 1(a) 1(d) we have plotted the time evolution of the entropy squeezing E(S x ), E(S y ) as well as the variances V (S x ) and V (S y ), while the atomic inversion is shown in Fig. (a), for the atom initially in excited state (θ = 0 ) and the field in the coherent state with the average photon number n 1 = n = 40 and the relative phase β = φ (ψ 1 +ψ ) = π between the atom and the two-mode coherent fields. Fig. 1 The time evolution of the squeezing factors. The atom is initially in the excited state and the field in the coherent state with the initial average photon number n 1 = n = 40 and the relative phase β = π between the atom and the two-mode coherent fields. Information entropy squeezing factors (a) E(S x) and (b) E(S y). Variance squeezing factors (c) V (S x) and (d) V (S y). When t (k +1)π/g (k = 0, 1,,...), we can see from Figs. 1(a) and 1(c) that there are great differences between E(S x ) and V (S x ), i.e., the former shows the atomic squeezing while the later does not exhibit any squeezing. This comes from the fact that atomic inversion W (t) = S z / = 0 (plotted in Figs. (a)) in the above time stages, in
106 LIU Xiao-Juan and FANG Mao-Fa Vol. 4 which one cannot get any information on squeezing from the HUR (17), while the definition of the information entropy squeezing (16) is untrivial and can provide sufficient information. Our study also shows that when the atom is initially in the ground state, the evolutions of these squeezing factors are exactly the same as those when the atom is initially in the excited state, with the same β, n 1, n. This means that the information entropy is a sensitive measure for the atomic squeezing. Fig. The time evolution of the atomic inversion. (a) The atom is initially in the excited state and the field in the coherent state with the initially average photon number n 1 = n = 40, the relative phase β = π between the atom and the two-mode coherent fields. (b) The atom is initially in the coherent superposition state with θ = π/, the phase ϕ = π/, and the field in the coherent state with the initial average photon number n 1 = n = 40, the relative phase β = 0 between the atom and the two-mode coherent field. Fig. 3 The time evolution of the squeezing factors. The atom is initially in the coherent superposition state with θ = π/, and the field in the coherent state with the initial average photon number n 1 = n = 40, the relative phase β = 0 between the atom and the two-mode coherent field. Information entropy squeezing factors (a) E(S x) and (b) E(S y), variance squeezing factors (c) V (S x) and (d) V (S y). From Figs. 3(a) and 3(b), we can see that both E(S x ) and E(S y ) predict squeezing when atom is in the coherent superposition state with θ = π/, the coherent field with n 1 = n = 40, and the relative phase β = 0 between the atom and the two-mode coherent fields. It is interesting to find that there is alternative information entropy squeezing in the atomic polarization components S x and S y, whereas neither V (S x ) nor V (S y ) displays variance squeezing as
No. 1 Information Entropy Squeezing of a Two-Level Atom Interacting with Two-Mode Coherent Fields 107 illustrated in Figs. 3(c) and 3(d). This can be quantitatively interpreted by using the result of Ref. [0]. At the time t = t R /4 = (k + 1)π/g (k = 0, 1,...), a quarter of the revival time of the atomic inversion S z, the atomic reduced entropy tends to 0, the atom and the two-mode coherent fields are disentangled, and the atom has achieved its pure state, [0] ( tr ) ψa 1 ( + + i ), (0) 4 which is just the eigenstate of the atomic Pauli operator S y. From Eq. (13), we obtain information entropies of atomic Pauli operators S x, S y, and S z at this time to be, respectively, H(S y ) = 0.0, H(S x ) = H(S z ) = ln. Correspondingly, δh(s y ) = 1, δh(s x ) = δh(s z ) =. By using these results we calculate entropy squeezing factors to be E(S y ) 0.414 < 0, and E(S x ) = 0.586 > 0. This shows that the operator S y exhibits optimal entropy squeezing while no entropy squeezing occurs in the operator S x at t R /4. This analytical result is in accord with the result of numerical calculation shown in Fig. 3(b). There is similar interpretation about Fig. 3(a) to that on Fig. 3(b). The difference between the results of variance squeezing and those of information entropy squeezing can be interpreted as follows. Due to the influence of the initial atomic coherence, the amplitude of the atomic inversion S z evolves to very tiny values, close to zero (shown in Fig. (b)), so the definition of the variance based on the HUR is almost no long valid. We can also control the number of components that are to be squeezed. If θ = 0 or θ = π, there is no more than one component to be squeezed, while there may be two components that are alternatively at the different times when the atom is initially in the coherent superposition state. In particularly when θ = π/, β = 0, there is optimal periodic information entropy squeezing in both S x and S y at the different times (see Figs. 3(a) and 3(b)). 3. Influence of Relative Phase Between Atom and Field An important phenomenon has been found, that is, the direction of the atomic information entropy squeezing can be changed and the features of the evolutions of E(S x ) and E(S y ) can be exchanged completely when θ, ϕ are fixed and β is changed. The characters of this kind are explored in Fig. 4. Fig. 4 The time evolutions of the information entropy squeezing factors. The atom is initially in the coherent superposition with θ = π/, the field in the coherent state with the initial average photon number n 1 = n = 40. (a) E(S x) with β = π/4; (b) E(S y) with β = π/4; (c) E(S x) with β = π/4; (d) E(S y) with β = π/4. It can be seen that, in the same time, in the case where θ = π/ and β tends to π/4, information entropy is squeezed
108 LIU Xiao-Juan and FANG Mao-Fa Vol. 4 in S x but not squeezed in S y (shown in Figs. 4(a) and 4(b)). While when β tends to π/4, there is information entropy squeezing in S y but no squeezing in S x (shown in Figs. 4(c) and 4(d)). This means that the relative phase between the atom and two-mode coherent fields determines the direction of information entropy squeezing. 3.3 Influence of Difference in Average Photon Number Between Two Field Modes Finally, the influence of the difference n ( n = n 1 n ) in the average photon number between the two-mode coherent fields on information entropy squeezing is shown in Figs. 5(a) 5(c) for θ = π/ and β = 0, we take n = 30 (see Fig. 5(a)), n = 0 (see Fig. 5(b)), and n = 0 (see Figs. 5(c)), respectively. It can be observed that there is an increase in the duration of information entropy squeezing with the decrease of n. The results show that, to obtain the longest duration of information entropy squeezing, we must take n = 0. Fig. 5 The time evolutions of the information entropy squeezing factors. The atom is initially in the coherent superposition with θ = π/, the relative phase β = 0 between the atom and the two-mode coherent field. (a) E(S x) with n = 30; (b) E(S x) with n = 0; (c) E(S x) with n = 0. 4 Conclusion In conclusion, we have investigated the information entropy squeezing for a two-level atom interacting with the two-mode coherent fields by using quantum information entropy. We can conclude, first, that quantum information entropy is a remarkable precision measure for atomic squeezing and overcomes the triviality of the variance squeezing based on HUR. In particular, when the atom is in the eigenstate of the operators S x or S y, the entropy squeezing is quite a good measure. Second, the squeezed component number of the information entropy squeezing is decided by the atomic initial distribution angle. Third, the direction of the information entropy squeezing is determined by the relative phase between the atom and the two-mode coherent fields. Finally, the duration of the information entropy squeezing can be controlled by choosing the difference of the average photon numbers for the two coherent field modes. Our results are important for the experimental observation of the atomic squeezing. By the way, since a three-level atom in the λ, Ξ, and -type configuration respectively with arbitrary detuning can be exactly reduced to a two-level system, [4 6] whose effective Hamiltonian is given by Eq. (1), then our results can also be suitable for the entropy squeezing question of a three-level system.
No. 1 Information Entropy Squeezing of a Two-Level Atom Interacting with Two-Mode Coherent Fields 109 References [1] M. Kitagawa and M. Ueda, Phys. Rev. A47 (1993) 5138. [] D.J. Wineland, J.J. Bollinger, and W.M. Itano, Phys. Rev. A50 (1994) 67; A. Sφrensen and K. Mφlmer, e-print quantph (1999) 199003044. [3] J.L. Sφrensen, J. Hald, and E.S. Polzik, Phys. Rev. Lett. 80 (1998) 3487. [4] H. Saito and M. Ueda, Phys. Rev. Lett. 79 (1997) 3869; H. Saito and M. Ueda, Phys. Rev. Lett. 59 (1999) 3059. [5] Y. Wu, Phys. Rev. A61 (000) 033803; A. Kuzmich, K. Mølmer, and E.S. Polaik, Phys. Rev. Lett. 79 (1997) 478. [6] L.M. Kuang and X. Chen, Phys. Rev. A50 (1994) 48; D.F. Walls and P. Zoller, Phys. Rev. Lett. 47 (1981) 709; K.C. Zhu, J. Mod. Opt. 43 (1996) 33; L.M. Kuang, F.B. Wang, and Y.G. Zhou, J. Mod. Opt. 41 (1994) 1307. [7] Y. Wu and X.X. Yang, Phys. Rev. Lett. 78 (1997) 3086; L. Zhou and L.M. Kuang, Phys. Lett. A30 (00 73; S.L. Braunstein and H.J. Kimble, Phys. Rev. Lett. 80 (1998) 869; G.J. Milburn and S.L. Braunstein, Phys. Rev. A60 (1999) 937. [8] A. Furusawa, J. Sorensen, S.L. Braunstein, C.A. Fuchs, H.J. Kimble, and E.S. Polzik, Science 8 (1998) 706. [9] T.C. Ralph, Phys. Rev. A61 (000) 010303R. [10] M. Hillery, Phys. Rev. A61 (000) 0309. [11] M. Ban, J. Opt. B: Quantum Semiclass. Opt. 1 (000) L9; M. Ban, J. Opt. B: Quantum Semiclass. Opt. (000) 786. [1] M.F. Fang, P. Zhou, and S. Swain, J. Opt. 47 (000) 1043. [13] I.I. Hirschman, Am. J. Math. 79 (1957) 15. [14] B.I. Bialynicki and J. Mycielski, Commun. Math. Phys. 44 (1975) 19; W. Beckner, Ann. Math. 10 (1975) 159; D. Deutsch, Phys. Rev. Lett. 50 (1983) 631. [15] M.F. Fang, P. Zhou, and S. Swain, Chin. Phys. Lett. 17 (000) 885. [16] Abdel-Aty Mahmoud, M. Sebawe Aballa, and A.S.F. Obada, J. Opt. B4 (00 13. [17] C.X. Li and M.F. Fang, Chin. Phys. 1 (003) 94. [18] H. Tohya, Phys. Rev. A63 (001) 0305. [19] C.Z. Wang and M.F. Fang, Chin. Phys. 51 (00 1989. [0] X.J. Liu and M.F. Fang, Chin. Phys. 1 (003) 0971. [1] C.H. Bemett and D.P. Divincenzo, Nature 404 (000) 47; A.K. Ekert, Phys. Rev. Lett. 67 (199 661; D. Deutsch and R. Jozsa, Proc. R. Soc. London A493 (199 553; C.H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. Wootters, Phys. Rev. Lett. 70 (1993) 1895. [] C.M Cavers and B.L. Schumaker, Phys. Rev. A31 (1985) 3068. [3] Y.R. Shen, The Principles of Nonlinear Optics, Chapter 10, Wiley, New York (1984). [4] X.X. Yang and Y. Wu, Commun. Theor. Phys. (Beijing, China) 3 (1999) 189. [5] Y. Wu and X. Yang, Phys. Rev. A56 (1997) 443. [6] Y. Wu, Phys. Rev. A54 (1996) 1586.