724 Mahmoud Abdel-Aty Vol. 37 and multiphoton instead of single mode and single photon, addition of Kerr-like medium, and Stark shift. On the other ha

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1 Commun. Theor. Phys. (Beijing, China) 37 (2002) pp 723{732 c International Academic Publishers Vol. 37, No. 6, June 15, 2002 Quantum Information and Entropy Squeezing of a Nonlinear Multiquantum JC Model Mahmoud Abdel-Aty y z Mathematics Department, Faculty of Science, South Valley University, Sohag, Egypt (Received December 10, 2001) Abstract We investigate the entropy squeezing of the nonlinear k-quantum JC model. A denition of squeezing is presented for this system based on the quantum information theory. The utility of the denition is illustrated by examining squeezing in the information entropy of a nonlinear k-quantum two-level atom. The inuence of the atomic coherence and the detuning parameter on the properties of the information entropy and squeezing of the atomic variables is examined. PACS numbers: Dv Key words: information entropy, variance squeezing, nonlinear mediums 1 Introduction Since mid of the last decade the new eld of quantum information and computation has emerged, not only oering the potential of immense practical computing power, but also suggesting deep links between the well-established disciplines of quantum theory, information theory and computer science. Theoretically quantum computers can perform some types of calculations much faster than classical computers, [1] but the technological diculties of manipulating quantum information have so far prevented researchers from constructing a quantum computer which is able to perform useful tasks. The diculty of building a quantum computer was greatly diminished when it was realized that a network of quantum phase gates operating in the product space of two qubits, single bit rotations, and single bit phase shift gates can constitute a universal quantum computer. [2 3] The eld of quantum information and computing is based on manipulation of quantum coherent states. [4 5] Existing device of quantum optics has been proposed as experimental implementation and employed to realize quantum computers. In the meantime squeezing states of light oer possibilities of improving the performance of optical devices since they can reduce uctuations in one of quadrature below the level associated with the vacuum states. [6] This situation is relevant to the optical communication networks as well as to many optical devices. Such light has recently been used in a power-recycled interferometer [7] and a phase-modulated signal-recycled interferometer, [8] aiming to improve signicantly the sensitivity of these devices. It has been shown that this light can be used to tune the resonant frequency of the cavity without actually moving the signal recycling mirror or changing the bandwidth of the interferometer without substantially decreasing the sensitivity at the resonant frequency. [8] Also we may point out that, squeezed light has been applied to quantum information theory, for example in quantum teleportation, [9 10] cryptography, [11 12] and dense coding. [13] In this respect the security in quantum cryptography [14] relies on the uncertainty relation for eld quadrature components of these states. Recently, the Shannon entropies associated with the photon number probability distribution and phase probability distribution are given respectively. [15 16] This has motivated us to study the same problem in the context of a nonlinear multiquantum Jaynes{Cummings (JC) model. From quantum information point of view we shall consider the problem of squeezing for a single two-level atom interacting with a single-mode including any form of nonlinearity of both the eld mode and the intensity-dependent atom-eld coupling. The organization of this paper is as follows. In Sec. 2 we introduce our Hamiltonian model and give exact expression for the density matrix ^(t). In Sec. 3 we employ the density matrix to investigate the properties of the entropy squeezing. Finally we devote Sec. 4 to give our discussion. 2 The Model In the last decade a huge number of papers have appeared in the literature considering the JC model [17] in great details. Most of these papers were concentrated on the statistical behavior as well as the dynamics of the model. However, to meet the experimental realization, there are several attempts to generalize and modify this model. For example, the consideration of multimode The project supported by the Project Math 1418/19 of the Institut fut Mathematik, Universitat Flensburg, Germany y The present address: Institut fur Mathematik und ihre Didaktik, Universitat Flensburg, D Flensburg, Germany z abdelaty@hotmail.com

2 724 Mahmoud Abdel-Aty Vol. 37 and multiphoton instead of single mode and single photon, addition of Kerr-like medium, and Stark shift. On the other hand, one can see that the quantized center of mass motion of a single two-level ion, interacting with a single-mode laser light eld, in a harmonic potential trap provides a good testing ground for some intriguing phenomena which we are familiar with in the cavity quantum electrodynamics. It has been shown that such a singletrapped and laser-irradiated ion can be modelled by a strongly nonlinear multiquantum JC model. [18] Furthermore, the degree of quantum control that can be achieved in a coupled system of internal and vibrational degrees of freedom was demonstrated in experiments on the generation of nonclassical states of motion of a single-trapped ion. In order to realize extended quantum registers, the attention has turned to systems of several ions with well controlled interaction between them. Although the ion-trap quantum computation introduced by Cirac and Zoller [19] is a potentially powerful technique for storage and manipulations of quantum information, however this scheme is not the only possibility of achieving dynamics which are conditioned on the internal states of dierent ions. For example a logic gate between two ions can also be realized by using two photon transitions, addressing both of the ions simultaneously and only virtually exciting the vibrational degrees of freedom. Thus, there is no doubt of a strong relation between JC model, ion trap and quantum information. This in fact has encouraged and stimulated us to study the nonlinear multiquantum JC model. The model we shall consider is consisting of a single two-level atom interacting with a single-mode including any form of nonlinearity of both the eld mode and the intensitydependent atom-eld coupling which can be expressed by rotating-wave approximation in the following form H = h 2 z + h!a a + (a y a) + hff k (a y a) a k + + a yk ; f k (a a)g : (1) We denote by (a y a) the one-mode eld nonlinearity, f k (a y a) an arbitrary intensity-dependent atom-eld coupling, and the detuning parameter between the eld frequency! and the atomic transition frequency! a where =! ;k! a : are the pseudo-spin matrices of two-level atom, z is the z-component of the Pauli spin matrix, a (a y ) is the annihilation (creation) operator of a photon. When we set (a y a) = 0, it has been shown that such a single trapped and laser-irradiated ion exhibits a strongly nonlinear multiquantum JC model dynamics described by the Hamiltonian (1), [18] where f k (a y a) = exp 1X 2r+k r!(r + k)! ; 2 r=0 cos r + + k 2 a yr a r (2) where is the Lamb{Dicke parameter. Here the quantized center-of-mass motion of the ion, which is coupled via the laser to the internal electronic degree of freedom, plays a role analogous to that played by the cavity light eld mode in the conventional JC model. Meanwhile this prediction has been conrmed experimentally [20] and modications due to micromotion have been studied. [21] In contrast to the nonlinear or multiquantum generalization of the JC model in quantum optics, the strength of the laser-induced vibronic coupling can be easily controlled by varying the intensity of the applied laser, and the typical coupling strength is much stronger. [22] We now suppose that the initial state of the atom is a superposition of the ground state and the excited state, = # 1 jeihej + # 0 jgihgj 2 S(H A ) : (3) Also we suppose that the initial state of the eld is the coherent state, $ = jihj 2 S(H F ) ji = exp ; 1 2 jj2 X l l p l! (4) where jj 2 is the intensity of the initial coherent eld. Then we set the continuous mapping Et from the state space S(H A ) of the atom to the tensor product state space S(H A )S(H F ) of the atom and eld, where Et describes the time evolution between the atom and eld dened by the unitary operator generated by H, E t : S(H A ) ;! S(H A ) S(H F ) E t = U t( $)U t U t exp ;it H : (5) h This unitary operator U t can be written as where E U t = 1X n=0 = h! e ;ite n + k 2 + j + ih + j + e ;ite ; j ; ih ; j (6) + h 2 (E e + E g ) [ + (n + k)] n (7)

3 No. 6 Quantum Information and Entropy Squeezing of a Nonlinear Multiquantum JC Model 725 are the eigenvalues with n = The eigenvectors associated with the eigenvalues E where j r 1 4 (h + < ; <(n + + k)! k))2 + 2 fk 2 (n + k)(n : (8) n! are given by + i = C 1+ jn ei + C 2+ jn + k gi j ; i = C jn ei + C jn + k gi (9) p C 1 = f k (n + k) (n + k)!=n! (10) q( < ; 12 <(n + k) n) f 2k (n + k)(n + k)!=n! 2; 1; 2 = ; + 1< ; <(n + k) n : (11) q( < ; 12 <(n + k) n) f 2k (n + k)(n + k)!=n! C Having obtained the explicit forms of the unitary operator U t, the eigenvalues and eigenfunctions for the system under consideration, we are therefore in position to discuss the entropy squeezing of the system. This will be seen in the following section. 3 Entropy Squeezing There is an elegant entropic way of reformulating the uncertainty principle of quantum mechanics, recall the Heisenberg uncertainty principle. The state is that standard deviations (C) and (D) for observables C and D must satisfy the relation (C)(D) 0:5jh j[c D]j ij (12) for a quantum system in the state j i. Therefore if we take C = X cjcihcj D = X djdihdj (13) to be spectral decompositions for C and D and dene f(c D) = max jhcjdij (14) c d to be the maximum delity between any two eigenvectors of jci and jdi, we can see for the Pauli matrices x and z, the function f(x z) is equal to 1= p 2. In this section we shall employ the uncertainty relation to study the information of squeezing entropy. Although the Heisenberg uncertainty relation cannot give us sucient information on the atomic squeezing for some cases, however it can be used as a general criterion for the squeezing in terms of information entropy of a two-level atom in the JC model. The uncertainty relation for a two-level atom characterized by the Pauli operators S x S y, and S z is given by S x S y 0:5jhS z ij (15) where S = p hs 2 i ; hs i 2. Fluctuation in the component S of the atomic dipole is said to be squeezed if S satises the condition V (S ) = (S ; p 0:5jhS z ij ) < 0 = x or y : (16) Recently in an even N-dimensional Hilbert space, the optimal entropic uncertainty relation for sets of N + 1 complementary observable with the nondegenerate eigenvalues has been investigated. [24] This can be described by the inequality N+1 X k=1 h N i [ N 2 ]h H(S k ) ln 1 + N i [1+ N 2 ] (17) 2 2 where H(S k ) represents the information entropy of variable S k. On the other hand, for an arbitrary quantum state the probability distribution for N possible outcomes of measurements of the operator S is P i (S ) = h ijj ii = x y z i = 1 2 ::: N (18) where j ii is an eigenvector of the operator S such that S j ii = i j ii = x y z i = 1 2 ::: N: (19) as The corresponding information entropies are dened H(S ) = ; NX i=1 P i (S )lnp i (S ) = x y z : (20) Thus, to obtain the information entropies of the atomic operators S x, S y, and S z for a two-level atom with N = 2, one can use the reduced atomic density operator (t): For the present case we nd that H(S x ) = ; ln([0:5 + Re eg (t)] [0:5+Re eg(t)] [0:5 ; Re eg (t)] [0:5;Re eg(t)] ) (21) H(S y ) = ; ln([0:5 + Im eg (t)] [0:5+Im eg(t)] [0:5 ; Im eg (t)] [0:5;Im eg(t)] ) (22)

4 726 Mahmoud Abdel-Aty Vol. 37 H(S z ) = ; ln( gg (t) gg(t) [1 ; gg (t)] [1; gg(t)] ) : (23) Now we nd the quantities gg (t), ge (t), ee (t), and eg (t) = ge(t). From the results of Sec. 2, it is easily to have the nal state after interaction between the atom and eld, E t U t( $)U t ( $)f e ite(m) + j (m) + ih X = f e ;ite + j + ih + j + ; e;ite j m n (m) + j + e ite(m) ; j (m) ; ih E t = X m n b n b m f 1 (t)jn eihm ej + 2 (t)jn eihm + k gj ; ih ; jg (m) ; jg (24) + 3 (t)jn + k gihm ej + 4 (t)jn + k gihm + k gjg (25) where b n = e ;jj2 =2 ( n = p n!) and the coecients i (t) (i = ) are given by 1 (t) = e ;ite(nm) ++ (#1 jc 1 + j2 jc (m) 1 + j2 + # 0 C 2 + C(m) 2 + C 1 + C(m) 1 + ) + e ;ite(nm) +; (# 1 jc 1 + j2 jc 2 ; (m) j2 + # 0 C 2 + C(m) 1 ; C 1 + C(m) 2 ; ) + e ;ite(nm) +; (# 1 jc 2 ; j2 jc (m) 1 + j2 + # 0 C1 ; C(m) 2 + C 2 ; C(m) 1 + ) + e ;ite(nm) ;; (# 1 jc 2 ; j2 jc 2 ; (m) j2 + # 0 C1 ; C(m) 1 ; C 2 ; C(m) 2 ; ) (26) 2 (t) = e ;ite(nm) ++ (#1 jc 1 + j2 C (m) 1 + C(m) # 0jC (m) 2 + j2 C 1 + C 2 + ) + e ;ite(nm) +; (# 1 jc 1 + j2 C 2 ; (m) C(m) 1 ; + # 0jC 1 ; (m) j2 C 1 + C 2 + ) + e ;ite(nm) ;+ (#1 jc 2 ; j2 C (m) 1 + C(m) # 0jC (m) 2 + j2 C 2 ; C ; ; ) + e ;ite(nm) ;; (# 1 jc 2 ; j2 C 2 ; (m) C(m) 1 ; + # 0jC (m) 2 + j2 C 1 + C 2 + ) (27) 3 (t) = e ;ite(nm) ++ (#1 jc (m) 1 + j2 C 2 + C # 0jC 2 + j2 C (m) 2 + C(m) 1 + ) + e ;ite(nm) +; (# 1 jc 2 ; (m) j2 C 2 + C # 0jC 2 + j2 C 1 ; (m) C(m) 2 ; ) + e ;ite(nm) ;+ (#1 jc (m) 1 + j2 C 1 ; C 2 ; + # 0jC 1 ; j2 C (m) 2 + C(m) 1 + ) + e ;ite(nm) ;; (# 1 jc 2 ; (m) j2 C 1 ; C 2 ; + # 0jC 1 ; j2 C 1 ; (m) C(m) 2 ; ) (28) 4 (t) = e ;ite(nm) ++ (#1 C 1 + C(m) 1 + C 2 + C(m) # 0jC 2 + j2 jc (m) 2 + j2 ) + e ;ite(nm) +; (# 1 C 1 + C(m) 2 ; C 2 + C(m) 1 ; + # 0jC 2 + j2 jc 1 ; (m) j2 ) + e ;ite(nm) ;+ (#1 C2 ; C(m) 1 + C 1 ; C(m) # 0jC 1 ; j2 jc (m) 2 + j2 ) + e ;ite(nm) ;; (# 1 C2 ; C(m) 2 ; C 1 ; C(m) 1 ; + # 0jC 1 ; j2 jc 1 ; (m) j2 ) (29) where E (nm) ij = E i ; E (m) j : This result can be used to derive a number of results for the observable quantities. From the the above equations, it is easy to compute the probability amplitudes in the following forms ee (t) = gg (t) = eg (t) = 1X n=0 1X n=0 1X n=0 he nje t jn ei hg n + kje t jn + k gi he nje t jn + k gi : (30) As one can see, it is unlikely to express the sums in the above equations in a closed form, however for reasonably large value of n, direct numerical evaluations can be performed. In the next section we shall discuss the dynamical behavior of the entropy squeezing of the present model. For a two-level atom, where N = 2, we have 0 H(S ) ln 2 (31) and hence, the information entropies of the operators S x S y S z will satisfy the inequality H(S x ) + H(S y ) 2ln2; H(S z ) : (32)

5 No. 6 Quantum Information and Entropy Squeezing of a Nonlinear Multiquantum JC Model 727 In other word if we dene H(S ) = exp[h(s )] (33) then the inequality (32) can be written as H(S x )H(S y ) 4(H(S z )) ;1 : (34) Now if H(S ) = 1 then the atom will be in a pure state. However when H(S ) takes the value 2 then the atom will be in a completely mixed state. Since the quantities H(S x ) and H(S y ) are only measuring the uncertainties of the atomic polarization components S x and S y respectively, it is clear from the entropic uncertainty relation (17) that there is impossibility of simultaneously having complete information about the observable S x and S y : We dene here the squeezing of the atom using the entropic uncertainty relation (EUR) (32), named entropy squeezing, which has received a little attention in past discussion. The uctuation in component S ( = x or y) of the atomic dipole is said to be \squeezed in entropy" if the information entropy H(S ) of S satises the condition E(S ) = H(S ) ; 2(jH(S z )j) ;1=2 < 0 = x or y : (35) Employing the results obtained here we shall be able to discuss the entropy squeezing which has received a little attention in the literature. This will be done in the following section. 4 Discussion and Conclusion Our aim of this section is to analyze and discuss the analytical solution given in the above sections within the framework of the numerical computations. Therefore we shall concentrate on examining the temporal evolution of the entropy squeezing. With a great precision an excellent accuracy for the behavior of the entropy squeezing factors has been determined. Furthermore we have employed a resolution of 10 3 point per unit of time for regions exhibiting strong uctuation. As initial condition and for all our plots we have taken the coherence parameter to be real, where its square is equal to the intensity of the initial coherent eld. We shall examine the temporal evolution of the entropy squeezing and variance squeezing factors E(S x ), E(S y ), V (S x ), and V (S y ), which are shown in Figs 1(a) 1(d) but the atomic inversion W (t) is plotted in Fig. 1(e) for the atom initially in the excited state (# = 0) with the mean photon number jj 2 = 25, in the absence of detuning parameter. Figures 1(a) and 1(c) predict no squeezing in the variable S x when the atom initially in the excited state but gures 1(b) and 1(d) present a great dierence between E(S y ) and V (S y ). E(S y ) shows entropy squeezing during the collapse for the atomic inversion W (t) as gure 1b exhibited, whilst V (S y ) predicts variance squeezing in a short duration during the atomic inversion W (t) revival, see Fig. 1(e). Fig. 1 The time evolution of the squeezing factors of a two-level atom interacting with a single-mode in the absence of the nonlinear medium and f = 1. The atom is initially in the ground state ( = ) and the eld in the coherent state with the initial average photon number n = 25, and the detuning parameter = 0 where (a) is for the entropy squeezing factor E(S x ) (b) the entropy squeezing factor E(S y ) (c) the variance squeezing factor V (S x ) (d) the variance squeezing factor V (S y ) and (e) the time evolution of the atomic inversion under the same conditions.

6 728 Mahmoud Abdel-Aty Vol. 37 Fig. 2 The time evolution of the squeezing p factors of a two-level atom interacting with a single-mode in the absence of the nonlinear medium and f = n. The atom is initially in the ground state ( = ) and the eld in the coherent state with the initial average photon number n = 25 and the detuning parameter = 0 where (a) is for the entropy squeezing factor E(S x ) (b) the entropy squeezing factor E(S y ) (c) the variance squeezing factor V (S x ) (d) the variance squeezing factor V (S y ) and (e) the time evolution of the atomic inversion under the same conditions. Fig. 3 The time evolution of the squeezing p factors of a two-level atom interacting with a single-mode in the absence of the nonlinear medium and f = n. The atom is initially in the ground state ( = ) and the eld in the coherent state with the initial average photon number n = 25 and the detuning parameter = 10 where (a) is for the entropy squeezing factor E(S x ) (b) the entropy squeezing factor E(S y ) (c) the variance squeezing factor V (S x ) (d) the variance squeezing factor V (S y ) and (e) the time evolution of the atomic inversion under the same conditions.

7 No. 6 Quantum Information and Entropy Squeezing of a Nonlinear Multiquantum JC Model 729 Fig. 4 The time evolution of the squeezing factors of a two-level atom interacting with a single mode. The nonlinearity of the single mode eld with a Kerr-type medium, i.e. (^a y^a) = ^a 2y^a 2, where is related to the third-order p nonlinear susceptibilities for the processes of self-phase modulation of the eld mode ( = 0:5) and f = n. The atom is initially in the ground state ( = ) and the eld in the coherent state with the initial average photon number n = 25 and the detuning parameter = 0 where (a) is for the entropy squeezing factor E(S x ) (b) the entropy squeezing factor E(S y ) (c) the variance squeezing factor V (S x ) (d) the variance squeezing factor V (S y ) and (e) the time evolution of the atomic inversion under the same conditions. Fig. 5 The time evolution of the squeezing factors of a two-level atom interacting with a single-mode in the absence of the nonlinear and the detuning parameter = 0. The atom is initially p in the superposition state = =2, = =4 and the eld in a coherent state with n = 25 and = =4 and f = n where (a) is for the entropy squeezing factor E(S x ) (b) the entropy squeezing factor E(S y ) (c) the variance squeezing factor V (S x ) (d) the variance squeezing factor V (S y ) and (e) the time evolution of the atomic inversion under the same conditions.

8 730 Mahmoud Abdel-Aty Vol. 37 Fig. 6 The time evolution of the squeezing factors of a two-level atom interacting with a single mode in the absence of the nonlinear medium and f = 1. The atom is initially in the ground state ( = ) and the eld in the coherent state with the initial average photon number n = 25 and the detuning parameter = 10, where (a) is for the entropy squeezing factor E(S x ) (b) the entropy squeezing factor E(S y ) (c) the variance squeezing factor V (S x ) (d) the variance squeezing factor V (S y ) and (e) the time evolution of the atomic inversion under the same conditions. Fig. 7 The time evolution of the squeezing factors of a two-level atom interacting with a single mode. The nonlinearity of the single-mode eld with a Kerr-type medium, i.e. (^a y^a) = ^a 2y^a 2, where is related to the third-order nonlinear susceptibilities for the processes of self-phase modulation of the eld mode ( = 0:5) and f = 1. The atom is initially in the ground state ( = ) and the eld in the coherent state with the initial average photon number n = 25 and the detuning parameter = 0 where (a) is for the entropy squeezing factor E(S x ) (b) the entropy squeezing factor E(S y ) (c) the variance squeezing factor V (S x ) (d) the variance squeezing factor V (S y ) and (e) the time evolution of the atomic inversion under the same conditions.

9 No. 6 Quantum Information and Entropy Squeezing of a Nonlinear Multiquantum JC Model 731 Figure 1(b) shows that at half of the revival time where t = 0:5 t R = ;1p n = 5 ;1 optimal entropy squeezing is attained, because at the time t = 0:5t R the atom has achieved almost pure state ' 2 ;1=2 (j ei+ j gi) at # = 0:5. This state is just an eigen state of the atomic operator S y, then we have S y = 0 to be the smallest possible value shown in Fig. 1(d) where no any variance squeezing around this time exhibits, since the atomic inversion satises W (t) = 0 at t = 0:5t R. The intensity-dependent coupling eects are considered in Fig. 2 where we take f = p n. In Fig. 2, the regular behavior of the entropy squeezing is noticed. In this case V (S x ) and V (S y ) show variance squeezing while no squeezing appears in both E(S x ) and E(S y ): Detuning eect results in elongating the revival time (see Fig. 3). Also the atomic system losses some of its energy to the system as can be shown from the mean value of W (t) that attains a lower value than that in the case of resonance. Squeezing in all quantities appear for this case. However the amount of squeezing is not as pronounced as the case of resonance (see Fig. 2). It is to be remarked that the oscillations become smaller as time develops, and we obtain a disappearance of the structure when the time increases further t 30. Now we will turn our attention to the eect on the entropy squeezing of the nonlinearity of the eld with a Kerr-type medium as an example, i.e., (a y a) = a y a(a y a ; 1), where is related to the third-order nonlinear susceptibility. In fact the optical Kerr eect is one of the most extensively studied phenomena in the eld of nonlinear optics because of its applications ranging from frequency conversions [23] to quantum nondemolition measurements. [24] Recently, Schmidt and co-workers [25] have proposed a nonlinear scheme based on electromagnetically induced transparency [26] to enhance the magnitude of the cross-phase modulation. A scheme depending on applications of the displacing operator and propagating a laser beam in a nonlinear Kerr medium has been proposed to perform quantum gates. [5] We show that a nonlinear interaction of the Kerr-like medium with the eld mode leads to increasing values of the variance squeezing. In this case, the eld and atom almost retain a strong entanglement in the time evolution process. At higher values of the Kerr nonlinearity there is a competition between the processes: atom + eld interaction and nonlinearity + eld interaction. As a matter of fact the latter process dominates over the former and the system goes into a kind of regular evolution, characteristic of the Kerrlike medium. One can realize that, there are periodically changes always occurring in the entropy squeezing and variance squeezing as a common property. This should be expected resultant of the existence of periodic functions in the expression of entropy squeezing and variance squeezing. In Fig. 4 we set f(^n) = p n, and the Kerr-like medium = = 0:5, while the mean photon number is taken to be n = 25, we nd that the squeezing does not appear in E(S x ), (see Fig. 4(a)). However, for the same period of time and in the absence of the nonlinearity, one can see that E(S y ) (Fig. 4(b)) is increasing in its value, while the squeezing phenomenon is noted to be pronounced. Furthermore there is increasing in the number of oscillations. With respect to the variances of squeezing we realize that there is decreasing in its value showing negative values for all the periods of the time, reecting the eects of the Kerr-like medium. But the uctuations in this case are steady at most of the periods, see Fig. 4(c). On the other hand V (S y ) shows increasing in its value compared with the case when the nonlinearity is absent, further the phenomenon of squeezing can be easy observed, see Fig. 4(d). To realize the eect of the atomic superposition on the squeezing information entropy and variance squeezing we set = =2, = =4 and the eld in a coherent state with n = 25 and = =4. It is obvious from Figs 5(a) and 5(b) that the entropies for the quadratures S x and S y show alternating squeezing whereas V (S x ) and V (S y ) as illustrated in Figs 5(c) and 5(d) display no variance squeezing and no entropy squeezing. To show and compare the eect of the Kerr-like medium and the detuning for dierent values of the intensity coupling we set f(^n) = 1 and = = 10 in Fig. 6 while = = 0:3 in Fig. 7. In Fig. 6 we show that detuning p eect results in elongating the revival time T R = 2 n + 2 =4. Also the atomic system losses some of its energy to the system as can be shown from the mean value for W (t) that attains a lower value than that in the case of resonance. Squeezing in all quantities appears for this case in contrast with the case in which f(^n) = p n shown in Figs 3 and 6. However the amount of squeezing is not as pronounced as the resonance case. The eect of the Kerr medium in the squeezing of the entropy and variances is depicted in Fig. 7. We take the same parameters of Fig. 4 and put = = 0:3 and f(^n) = 1. The eect of the Kerr medium on the atomic occupation number results in the inhibiting energy in atomic system. The more = increases the higher mean values for the W (t) as shown in Fig. 7(e). Also it results in the faster oscillations of the atomic inversion. This is also re- ected on the behavior of both the entropy and variance squeezing quantities, periodic squeezing is observed in all of these quantities. 5 Summary We have used the Heisenberg uncertainty relation as a general criterion for the squeezing in terms of information entropy of a two-level atom multiphoton process, taking into account arbitrary forms of nonlinearities of both the eld and intensity-dependent atom-eld coupling. In particular, we have explored the inuence of the various parameters of the system on the entropy squeezing. Such

10 732 Mahmoud Abdel-Aty Vol. 37 systems are potentially interesting for their ability to process information in a novel way and might nd application in models of quantum logic gates. An idealized situation when the cavity losses are negligible is considered here, however, in the case of real experiment the losses must be introduced. It can be expected that for a non-ideal but high-quality cavity our results are of relevance to the case that the Hamiltonian is appropriate for the experimental setup. Thus, the model presented here has not only demonstrated the eect on entropy squeezing of the nonlinearity of a single-mode eld theoretically, but also is of experimental importance in measuring the entropy squeezing of an atom interacting with single-eld mode in a cavity containing any kind of the nonlinearities in the future, providing some guidelines to experimentalist in the identication of the kind of unknown nonlinear medium and the utilization of the nonlinearities of atom-eld interaction. A nonlinear Kerr-type medium and detuning eects are taken into account and the dependence of the entropy squeezing and atomic inversion is considered. It is found that entropy squeezing is aected strongly when a nonlinear medium is taken into account. We have shown in our system that the eect of Kerr-like medium on the entropy is negative and the eect of detuning on the atomic variable squeezing is positive. This emphasis on the fact that the intensity coupling has a remarkable eect on the squeezing of the entropy and the system of a single twolevel atom with Kerr-like medium can have a potential application in the eld of quantum information. References [1] D. Deutsch, Proc. R. Soc. London 400 (1985) 97 R. Horodecki, M. Horodecki, and P. Horodecki, Phys. Rev. A59 (1999) 1799 A. Ekert and R. Jozsa, Rev. Mod. Phys. 68 (1996) 733. [2] A. Barenco, D. Deutsch, A. Ekert, and R. Jozsa, Phys. Rev. Lett. 74 (1995) 4083 S. Lloyd, Phys. Rev. Lett. 75 (1995) 346 N. Weaver, J. Math. Phys. 41 (2000) 240. [3] A. Rauschenbeutel, et al., Phys. Rev. Lett. 83 (1999) [4] A. Steane, Rep. Prog. Phys. 61 (1998) 117 D.P. Vincenzo, Science 270 (1995) 255 L.K. Grover, Phys. Rev. Lett. 79 (1997) 325. [5] J. Pachos and M. Chountasis, quant-ph/ [6] R. Loudon and P.L. Knight, J. Mod. Opt. 34 (1987) 709 H.J. Kimble and D.F. Walls, J. Opt. Soc. Am. B4 (1987) 10. [7] A. Brillet, J. Gea-Banacloche, C.N. Man, and J.Y. Vinet, The Detection of Gravitational Radiation, ed. D.G. Blair, Cambridge, Cambridge University Press (1991) V. Chicharmane and S.V. Dhurandhar, Phys. Rev. A54 (1996) 786. [8] V. Chicharmane, S.V. Dhurandhar, T.C. Ralph, M. Gray, H-A. Bachor, and D.E. McClelland, Phys. Rev. A57 (1998) 786. [9] S.L. Braunstein and H.J. Kimble, Phy. Rev. Lett. 80 (1998) 869 G.J. Milburn and S.L. Braunstein, Phys. Rev. A60 (1999) 937. [10] A. Furusawa, J. Sorensen, S.L. Braunstein, C.A. Fuchs, H.J. Kimble, and E.S. Polzik, Science 282 (1998) 706. [11] T.C. Ralph, Phys. Rev. A61 (2000) R. [12] M. Hillery, Phys. Rev. A61 (2000) [13] M. Ban, J. Opt. B: Quantum Semiclass. Opt. 1 (2000) L9 M. Ban, J. Opt. B: Quantum Semiclass. Opt., 2 (2000) 786. [14] S. Reynaud, A. Heidmann, E. Giacobino, and C. Fabre, Progress in Optics, ed. E. Wolf, Elsevier, Amsterdam (1992) Vol. 30. [15] G.A. Rosas, J.A. Vaccaro, and S.M. Barnett, Phys. Lett. A205 (1995) 247 A. Joshi, Phys. Lett. A270 (2000) 249. [16] S. Abe, Phys. Lett. A186 (1992) 163. [17] E.T. Jaynes and F.W. Cummings, Proc. IEEE 51 (1963) 89. [18] S. Wallentowitz and W. Vogel, Phys. Rev. A55 (1997) [19] J.I. Cirac and P. Zoller, Phys. Rev. Lett. 74 (1995) [20] C. Monroe, D.M. Meekhof, B.E. King, and D.J. Wineland, Science 272 (1996) [21] P.J. Bardro, C. Leichtle, G. Schrade, and W.P. Schleich, Act. Phys. Slov. 46 (1996) 231. [22] D.M. Meekhof, C. Monroe, B.E. King, W.M. Itano, and D.J. Wineland, Phys. Rev. Lett. 76 (1996) [23] R.W. Boyd, Nonlinear Optics, Academic Press, Boston, MA (1992) S.P. Tewari and G.S. Agarwal, Phys. Rev. Lett. 56 (1986) [24] P. Grangier and J.F. Roch, Opt. Commun. 72 (1989) 387 Q.A. Turchette, et al., Phys. Rev. Lett. 75 (1995) 4710 J.P. Poizat and P. Grangier, ibid. 70 (1993) 271. [25] H. Schmidt and A. Imamoglu, Opt. Lett. 21 (1996) 1936 A.J. Merriam, et al., Phys. Rev. Lett. 84 (2000) 5308 Opt. Lett. 24 (1999) 625 M.D. Lukin, et al., Adv. At. Mol. Opt. Phys. 42 (2000) 347 L. Deng, et al., Phys. Rev. A58 (1998) 707. [26] S.E. Harris, J.E. Field, and A. Imamoglu, Phys. Rev. Lett. 64 (1990) 1107 K.J. Boller, ibid. 66 (1991) 2593 J.E. Field, K.H. Hahn, and S.E. Harris, ibid. 67 (1991) 3062 K. Hakuta, L. Marmet, and B.P. Stoiche, Phys. Rev. Lett. 66 (1991) 596.