Wigner function and transition amplitude of three mutually coupled oscillators

ARTICLE IN PRESS

Physica B 392 (2007) 159–172 www.elsevier.com/locate/physb

Wigner function and transition amplitude of three mutually coupled oscillators M.M. Nassar, M. Sebawe Abdalla Mathematics Department, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia Received 28 July 2006; received in revised form 1 November 2006; accepted 10 November 2006

Abstract A full quantum mechanical treatment of three electromagnetic fields is considered. The proposed model consists of three different coupling parameters for which the rotating and counter-rotating terms are retained. An exact solution of the wave function in the Schro¨dinger picture is obtained and the connection with the coherent states wave function is given. The symmetrical ordered quasiprobability distribution function (W-Wigner function) is calculated via the wave function in the coherent states representation. The squeezing phenomenon is also examined for both single mode and squared-amplitude, where the collapse and revival phenomena are observed. For the case in which l3 ¼ 0 and o1 ¼ o2 ¼ o3 (exact resonances) we find that the late phenomenon is apparent but only after long period of the time considered. The transition amplitude between two different coherent states (a state in which all the coupling parameters are involved and a state when the coupling parameter l3 ¼ 0) is calculated. It is shown that the probability amplitude is sensitive to the variation of the mean photon numbers, and the coupling parameters as well as to the field frequencies. r 2006 Elsevier B.V. All rights reserved. PACS: 03.65.Ge; 42.50.Dv Keywords: Quasi-distribution; Wigner function; Solutions of wave equations; Bounded states

1. Introduction The problem of coupled oscillators is still regarded as one of the most fundamental problems in the field of quantum optics [1–4]. More precisely we refer to the problem of field–field interaction, where two or more electromagnetic fields are coupled. In fact this problem represents an important nonlinear parametric interaction which is associated with a large variety of physical phenomena. For example, in an optical regime we can see two different processes which attracts much attention; one is called a parametric amplifier and the other is known as a frequency converter [5–7]. The parametric frequency conversion occurs in a number of well-known phenomena [8–12]. These include the production of anti-Stokes radiation in coherent Raman and Brillouin scattering and up conversion of light signals in nonlinear media. MeanCorresponding author.

E-mail address: [email protected] (M. Sebawe Abdalla). 0921-4526/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2006.11.012

while one can see that the frequency splitting of light beams is an example of parametric amplification in which both of the coupled modes are electromagnetic. The most familiar form of the parametric amplifier is designed to amplify an oscillating signal by means of a particular coupling of the mode in which it appears to a second mode of the oscillation, the idler mode. The coupling parameter is made to oscillate with time in a way which gives rise to a steady increase of the energy in both the signal and idler modes. It is worthwhile to refer to electrical engineering applications, where microwave versions of the parametric amplifier and frequency converter have been used. For example we can find at optical frequencies, the spontaneous emission of quanta, which is not predicated by classical theory, is an important contribution. Finally we refer to the recent experiment by Geordiades et al. [13], where the squeezed light is generated by nondegenerate parametric down conversion in order to excite a twophoton transition in atomic caesium. On the other hand the

ARTICLE IN PRESS M.M. Nassar, M. Sebawe Abdalla / Physica B 392 (2007) 159–172

160

combination between these two types of interaction (the parametric amplifier and the frequency converter) leads also to an interesting problem [14–16]. This can be seen in the case of quantum nondemolition measurement where the back action evading amplifiers can be constructed by combining a parametric amplifier and a parametric frequency converter with two different coupling parameters [17–19]. Thus, a theoretical description of the amplification and frequency conversion of light must take quantum effects into account. As is well-known, there is a strong relationship between these two types of interaction and the observation of some physical phenomena such as superradiance, coherent Raman and Brillouin scattering. This in fact stimulated and encouraged us to study such types of interaction. In the present communication we introduce a quadratic Hamiltonian model that represents the mutual interaction between three electromagnetic fields. To derive this model we use the quantization of the cavity modes. This is achieved, however, without applying what is called the rotating wave approximation (RWA). This means that both of the rotating and the counter rotating terms are retained. The Hamiltonian we adopt here can be derived as follows: Since the total energy of the field is Z 1 H^ 0 ¼ ðE2 þ H2 Þ dV , (1.1) 8p cavity where E and H are the electric and magnetic fields, respectively, and  is the dielectric constant, while V is the cavity volume. Then, in terms of boson operators, we have  X  y 1 H^ 0 ¼ _ ol a^ l a^ l þ , (1.2) 2 l where  has been taken to be unity. a^ yl and a^ l , l ¼ 1; 2; 3, are the creation and annihilation operators satisfying the commutation relation ½a^ j ; a^ yk  ¼ djk ¼ 1 if j ¼ k and zero otherwise, while ol is the field frequency. In order to provide coupling between the various cavity modes we consider that the dielectric constant varies as X ðr; tÞ ¼ 1 þ Df ðrÞ cosðoi t þ fi ðtÞÞ, (1.3) i

where f ðrÞ is a function of the position vector r, D51, and fi ðtÞ is an arbitrary time-dependent phase pump. The total Hamiltonian including the interaction terms becomes H ¼ H 0 þ H 1 , where H 0 is given by Eq. (1.2) and H 1 is X H^ 1 ðtÞ ¼ _ klj cosðoi t þ fi ðtÞÞða^ yj þ a^ j Þða^ yl þ a^ l Þ. (1.4) i;j;l

The coupling coefficients klj are Z D 1=2 klj ¼ ðo o Þ f ðrÞul ðrÞuj ðrÞ dV , l j 16pc2 cavity where ul ðrÞ are the normal modes satisfying o 2 l ul ðrÞ. r ^ r ^ ul ðrÞ ¼ c

(1.5)

(1.6)

Now, if we choose f ðrÞ so that klj a0, this leaves an infinite number of modes coupled. Therefore, by a proper choice of the pump frequency oi , the interacting modes are limited to two modes. However, the pump frequency can also be adjusted to obtain an Hamiltonian describing three modes or even more. For the three modes case we have   3 X ^ HðtÞ 1 ¼ ol a^ yl a^ l þ _ 2 l¼1 þ

3 X

gi ðtÞða^ yj þ a^ j Þða^ yk þ a^ k Þ;

iajak,

ð1:7Þ

i;j;k¼1

where gi ðtÞ is the time-dependent coupling parameter. We introduce the usual definition of the Dirac operator a^ j ðtÞ ¼ ð2_oj Þ1=2 ðoj q^ j ðtÞ þ ip^ j ðtÞÞ,

(1.8)

where p^ j and q^ j are the usual momentum and coordinate operators which satisfy the commutation relations ½q^ i ; p^ j  ¼ i_dij , where dij ¼ 1, if i ¼ j and zero otherwise. In this case the Hamiltonian (1.7) can be written in the form ^ ¼1 HðtÞ 2

3 X i¼1

ðp^ 2i þ o2i q^ 2i Þ þ

3 X

li ðtÞq^ j q^ k ;

iajak,

i;j;k¼1

(1.9) where li ðtÞ is the coupling parameter which in general depends on the time. It should be noted that, if one applies the RWA to the present system, then it is possible to deduce several models of the Hamiltonian. Each one would describe a different physical problem, (for more details one may consult Refs. [20–28]). Furthermore, as one can see, the model consists of three fields that mutually interact with each other. Therefore it would be interesting to refer to the work of Garrett et al. [29], which reported an experimental demonstration of phonon squeezing in a macroscopic system by exciting a KT a O3 crystal with an ultrafast pulse of light. As another example of using the above model is the case of crystal optical activity where the authors of Refs. [30,31] used quantum mechanical model of three coupled linear harmonic oscillators to represent the elementary cell of the crystals belonging to the space groups of symmetry D43 or D63 . This in fact emphasizes the richness, the importance, as well as the wide validity of the present model. Since the Hamiltonian (1.9) is explicitly time dependent, it is not an easy task to deal with such a system. Therefore, in the present work we shall restrict ourselves to considering the coupling parameter to be constant. This can be achieved if one adjusts the phase ¯  oi tÞ, where f ¯ is an arbitrary constant. pump fi ðtÞ ¼ ðf There is no doubt that this choice would in general affect the system. However, the model still connects with many different phenomena even after we consider this particular case. This is seen in the forthcoming sections. Here we may refer to the authors of Ref. [32] who considered in detail the most general form of the

ARTICLE IN PRESS M.M. Nassar, M. Sebawe Abdalla / Physica B 392 (2007) 159–172

quadratic Hamiltonian system in position and momentum Hamiltonian. They concentrated on the generic expressions for a multidimensional anisotropic oscillator (nonstationary system) to find the Green’s function, the wave function as well as the Wigner function for some interesting states [33–35]. One of our main tasks is to consider the W -Wigner distribution functionas well as the transition amplitude between the present system and some other coupledoscillator system. This implies that we have to find either the solution of the Heisenberg equations of motion or the wave function in the Schro¨dinger picture. The organization of the paper is as follows: In Section 2, we derive the solution of the wave function in the Schro¨dinger picture. Section 3 is devoted to give the closed form expression of the wave function in the coherent-state representation. The consideration of the W-Wigner function is given in Section 4. The Green’s function is introduced in Section 5, followed by a discussion related to squeezing phenomenon in Section 6. Finally, we give our conclusion in Section 7.

161

and the Euler’s angles (the transformation angles) such that qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 1 2 l21 þ l22 A, y ¼ tan1 @ 2 fðo2  o21 Þ sin 2f þ 2l3 cos 2fg   l1 f ¼ tan1 , l2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 4ðl21 þ l22 Þ þ fðo22  o21 Þ sin 2f þ 2l3 cos 2fg2 , fðo21  o22 Þ cos 2f þ 2l3 sin 2fg2

1 c ¼ tan1 2

ð2:4Þ then the Hamiltonian (1.9) is immediately diagonalized and takes the form 1 H^ ¼ 2

3 X ðP^ 2i þ O2i Q^ 2i Þ,

(2.5)

i¼1

where O21 ¼ o21 l 21 þ o22 m21 þ o23 n21 þ l1 m1 n1 þ l2 n1 l 1 þ l3 l 1 m1 ,

2. Equations of the motion The solution of equations of motion is the key to discussing many different statistical aspects for any dynamical system. Therefore, we devote this section to finding the solution of the Heisenberg equations of motion. For the present system we have to solve a second order differential equations either for coordinates or momenta in the from ðd2 =dt2 þ BÞ A ¼ 0, where A and B are two matrices given by 2

3 q^ 1 6 7 A ¼ 4 q^ 2 5; q^ 3

2

o21

6 B ¼ 4 l3 l2

l2

o22 l1

l1 7 5. o23

(2.1)

To do so one firstly simplify the matter and diagonalize the Hamiltonian (1.9). This can be done if we introduce the transformation 2 ^ 3 2 l1 Q1 6 Q^ 7 6 l 4 25 ¼ 4 2 l3 Q^ 3

m1 m2 m3

32

3

q^ 1 n1 7 6 n2 54 q^ 2 7 5, q^ 3 n3

(2.2)

where l i , mi , and ni , i ¼ 1; 2; 3, are the direction cosines of the well-known Euler’s angles ðy; f; cÞ. A similar expression can be found for the momentum, P ¼ ðP^ 1 ; P^ 2 ; P^ 3 Þ, such that ½q^ i ; p^ j  ¼ ½Q^ i ; P^ j  ¼ i_dij , where dij is the Kronecker delta. Now suppose we arrange the frequency of the third-field mode o3 to be in the form

o3 ¼

o21 l21 þ o22 l22  2l1 l2 l3 ðl21 þ l22 Þ

O23 ¼ o21 l 23 þ o22 m23 þ o23 n23 þ l1 m3 n3 þ l2 n3 l 3 þ l3 l 3 m3 .

Note that the explicit expressions for l i , mi and ni in terms of Euler’s angles are

m1 ¼ sin f cos c þ cos f sin c cos y, m2 ¼  sin f sin c þ cos f cos c cos y, l 3 ¼ sin f sin y; m3 ¼  cos f sin y, n1 ¼ sin y sin c;

(2.3)

n2 ¼ sin y cos c;

n3 ¼ cos y.

ð2:7Þ

Now the Heisenberg equations of motion are easily obtained, and in this case we have dQ^ j ¼ P^ j ; dt

dP^ j ¼ O2j Q^ j ; dt

j ¼ 1; 2; 3.

(2.8)

Straightforward calculations lead us to write the timedependent solution of the physical coordinates, q^ k ðtÞ, k ¼ 1; 2; 3, as follows: ! 3 X q^ 1 ðtÞ ¼ ½ei ðtÞq^ i ð0Þ þ e¯i ðtÞp^ i ð0Þ , i¼1

!1=2 ,

ð2:6Þ

l 1 ¼ cos f cos c  sin f sin c cos y, l 2 ¼  cos f sin c  sin f cos c cos y,

3

l3

O22 ¼ o21 l 22 þ o22 m22 þ o23 n22 þ l1 m2 n2 þ l2 n2 l 2 þ l3 l 2 m2 ,

q^ 2 ðtÞ ¼

! 3 X ½2i ðtÞq^ i ð0Þ þ 2¯ i ðtÞp^ i ð0Þ , i¼1

ARTICLE IN PRESS M.M. Nassar, M. Sebawe Abdalla / Physica B 392 (2007) 159–172

162 3 X

q^ 3 ðtÞ ¼

! ½ti ðtÞq^ i ð0Þ þ t¯ i ðtÞp^ i ð0Þ ,

(2.9)

i¼1

where e1 ðtÞ ¼

3 X

l 2j cos Oj t;

e2 ðtÞ ¼

j¼1

e3 ðtÞ ¼

3 X

3 X

l j mj cos Oj t,

j¼1

l j nj cos Oj t

ð2:10Þ

j¼1

Rt and e¯i ðtÞ ¼ 0 ei ðtÞ dt, i ¼ 1; 2; 3. Also we find the function 21 ðtÞ ¼ e2 ðtÞ, while 2j ðtÞ, j ¼ 2; 3, are 22 ðtÞ ¼

3 X

m2j cos Oj t and

j¼1

23 ðtÞ ¼

3 X

mj nj cos Oj t.

ð2:11Þ

j¼1

Rt

Furthermore 2¯ 1 ðtÞ ¼ e¯ 2 ðtÞ, and 2¯ j ðtÞ ¼ 0 2j ðtÞ dt, j ¼ 2; 3.PFinally t1 ðtÞ ¼ e3 ðtÞ, and t2 ðtÞ ¼ 23 ðtÞ, while t3 ðtÞ ¼ 3j¼1 n2j cos Oj t. The remaining R t functions are t¯ 1 ðtÞ ¼ e¯3 ðtÞ, t¯ 2 ðtÞ ¼ 2¯ 3 ðtÞ, and t¯ 3 ðtÞ ¼ 0 t3 ðtÞ dt. The expression for the momentum can be obtained from dq^ k ðtÞ ¼ p^ k ðtÞ; dt

k ¼ 1; 2; 3.

(2.12)

Having obtained the exact expression for both q^ k ðtÞ and p^ k ðtÞ, then we are in a position to discuss the statistical properties related to the present system. This is considered in the subsequent sections. 3. Wigner function We devote this section to one of the most important quasi-distribution functions that is the Wigner function. The concept of the Wigner function can be regarded as a possible extension of a classical phase space. In fact it represents the signal in the space and frequency domains simultaneously. It has a straightforward physical interpretation and is an elegant and powerful tool for the study of optical propagation. The W-Wigner function is one among three fundamental quasi-probability distribution functions that correspond to symmetric order. The other two functions are the P-Glauber Sudershan and HusimiKano Q-function corresponding to the normal and antinormal forms, respectively. These functions can also be regarded as crucial tools to giving insight into the statistical description of a quantum mechanical system. For instance one can employ the negative values of the W-function, stretching of the Q-function and high singularity in the P-function, for obtaining more details see Refs. [2–14]. From a practical point of view one may use homodyne tomography to measure these functions. In order to calculate the Wigner function we have either to use the

solution of the Heisenberg equations of motion or to employ the wave function in the coherent-states representation. To reach our goal we use the latter method and calculate the wave function in the coherent states representation. After straightforward calculation we obtain the following expression: " !#   3 O1 O2 O3 1=4 1 X 2 2 ca ðq; tÞ ¼ exp  fjai j þ ai ðtÞg 2 i¼1 _3 p3 " # 3 pffiffiffiffiffiffiffiffiffiffi 1 X 2  exp  ðOi Qi  2 2_Oi ai ðtÞQi Þ , ð3:1Þ 2_ i¼1 where Qi  Qi ðq1 ; q2 ; q3 Þ, i ¼ 1; 2; 3, is given by Eq. (2.1) and ai ðtÞ ¼ aj ð0Þ expðiOj tÞ is the eigenvalue of the annihilation operator A^ j ðtÞ, j ¼ 1; 2; 3, with respect to the coherent states 3 1X jaj ; ti ¼ exp  jai j2 2 i¼1

!

n 1 X ai j ðtÞ pffiffiffiffiffi jnj i; nj ! nj ¼1

j ¼ 1; 2; 3. (3.2)

It should be noted that the operator, A^ j ðtÞ, is given in the Dirac representation by A^ j ðtÞ ¼ ð2Oj _Þ1=2 ðOj Q^ j þ iP^ j Þ, can be regarded as the accurate definition from which we are able to diagonalize the Hamiltonian (1.9). Alternatively, if one uses Eqs. (3.1) and (3.2), we obtain the wave function in the number state representation. Thus   O1 O2 O3 1=4 n1 þn2 þn3 cn ðq; tÞ ¼ ½2 n1 !n1 !n1 !1=2 3 3 _p rffiffiffiffiffiffi ! rffiffiffiffiffiffi ! rffiffiffiffiffiffi ! O1 O2 O3 H n1 Q H n2 Q H n3 Q _ 1 _ 2 _ 3 " # 3 1 X  exp  ½Oi Q2i þ 2i_Oi ni t . ð3:3Þ 2_ i¼1 Having obtained the wave function in both the coherent and number-state representation, we are therefore in a position to find the explicit expression of Wigner’s function. This can be achieved if one calculates the integral [36,37] W a ðq; p; tÞ ¼

ZZ 1 Z 1 ca ðq1  x1 ; q2  x2 ; q3  x3 ; tÞ p3 1 ca ðq1 þ x1 ; q2 þ x2 ; q3 þ x3 ; tÞ " # 3 X ðpk xk Þ dx1 dx2 dx3 . ð3:4Þ  exp 2i k¼1

Now if we insert Eq. (3.1) together with its complex conjugate into Eq. (3.4), then after lengthy but straightforward

ARTICLE IN PRESS M.M. Nassar, M. Sebawe Abdalla / Physica B 392 (2007) 159–172

calculations we have W a ðq; p; tÞ " # !1 3 2 O 1 O 2 O3 1X 2  exp  ðak þ ak ðtÞÞ ¼ 2 k¼1 _3 p6 ð4f 1 f 2  f 212 ÞE " !# 3 X 2 f i qi þ ½f 12 q1 q2 þ f 13 q1 q3 þ f 23 q2 q3   exp 2 i¼1

! 1 2  exp 2 ðRe gi Þqi  ðp þ Im g1 Þ 2f 1 1 i¼1 " !# D2 ½2f 1 ðp2 þ Im g2 Þ  ðp1 þ Im g1 Þf 12 2 þ  exp  , 4E 2f 1 ð4f 1 f 2  f 212 Þ 3 X

ð3:5Þ where D is given by the expression 

 2f 1 ðp3 þ Im g3 Þ  f 13 ðp1 þ Im g1 Þ D¼ f1 " # 2½ðp1 þ Im g1 Þf 13  2f 1 ðp2 þ Im g2 Þð2f 1 f 23 þ f 12 f 13 Þ þ 2f 1 ð4f 1 f 2  f 212 Þ

ð3:6Þ while E is " # ð4f 1 f 3  f 213 Þð4f 1 f 2  f 212 Þ  ð2f 1 f 23 þ f 12 f 13 Þ2 E¼ , 2f 1 ð4f 1 f 2  f 212 Þ (3.7) In the above equations we have used the abbreviations 3 1 X f1 ¼ Oi l 2i ; 2_ i¼1

f2 ¼

f3 ¼

3 1 X Oi m2i ; 2_ i¼1

3 1 X Oi n2i ; 2_ i¼1

rffiffiffi 3 pffiffiffiffiffi 2X g1 ¼ ai ðtÞ Oi l i , _ i¼1

g2 ¼

rffiffiffi 3 pffiffiffiffiffi 2X ai ðtÞ Oi mi , _ i¼1

rffiffiffi 3 pffiffiffiffiffi 2X g3 ¼ ai ðtÞ Oi ni , _ i¼1

while we defined f ij as f 12 ¼

3 1X Oi l i mi ; _ i¼1

f 23 ¼

3 1X O i ni m i . _ i¼1

f 13 ¼

function’s behavior is almost Gaussian. This can be deduced from the exponential terms in the functional structure. However, to be more specific about this behavior we have plotted the Wigner function against q1 and p1 corresponding to Re a~ and Im a~ (say), where a~ is the function parameter. Fig. 1 and for fixed value of all ai ¼ 0, while o1 ¼ 1, o2 ¼ 0:8 and l1 ¼ 0:2, l2 ¼ l3 ¼ 0:1, the function shows Gaussian behavior center it at the middle (see Fig. 1(a)). However, if we take a3 ¼ 5, keeping all other parameters unchanged, the function starts to move aside and to decrease its value, see Fig. 1(b). For the case in which a1 ¼ 5 the function moves to the opposite side, however, with further decrease in its value, see Fig. 1(c). Thus we can conclude that the variation of the initial mean photon number leads to a change in the direction of the Wigner function. 4. The squeezing phenomenon We now turn our attention to consider one of the nonclassical phenomena in the field of quantum optics that is the squeezing phenomenon [38,39]. Squeezed light is a radiation field without a classical analogue. One of its quadratures has lower fluctuations than those for a vacuum state at the expense of increased fluctuations in the other quadrature, so that the Heisenberg uncertainty relation is fulfilled. The system would possess squeezing if one of the quadratures satisfies the inequality hðDq^ j Þ2 io12 or hðDp^ j Þ2 io12. In fact the squeezed light relates to several applications in optical communication networks [40], to interferometric techniques [41] and to optical waveguide tap [42,43]. It should be noted that such light can be measured by homodyne detection, where the signal is superimposed on a strong coherent beam of the local oscillator. There are different kinds of squeezing; for example, normal squeezing [39], amplitude-squared squeezing [44], and principal squeezing [14,45]. In the present paper we shall demonstrate normal squeezing as well as amplitude-squared squeezing. For normal squeezing we define the following quadratures: 1 q^ j ¼ pffiffiffiffiffiffiffiffi ða^ yj þ a^ j Þ 2oj

(3.8)

3 1X O i l i ni _ i¼1

ð3:9Þ

As one can see, the expression of the W-Wigner function is too complicated. However, it is easy to observe that the

163

and

p^ j ¼ i

rffiffiffiffiffi oj y ða^  a^ j Þ. 2 j

(4.1)

It is interesting to point out that the two quadratures are related to the conjugate electric and magnetic field ^ and satisfy the commutation relation operators, E^ and H, ^ ½q^ j ; p^ j  ¼ iI, where I^ is the unitary operator and the uncertainty relation has the form hðDq^ j Þ2 ihðDp^ j Þ2 iX14. It should be noted that the relationship between the nonvanishing component of the electric field E^ and the dimension of coordinates q^ j is given by ^ tÞ ¼ Eðz;

X j

q^ j ðtÞ

2o2j V 0

!1=2 sin K j z,

(4.2)

ARTICLE IN PRESS M.M. Nassar, M. Sebawe Abdalla / Physica B 392 (2007) 159–172

164

Fig. 1. (a) The Wigner function W a ðq; p; tÞ, against the quadratures q1 and p1 with parameters t ¼ 0, o1 ¼ 1, o2 ¼ 0:8, l1 ¼ 0:2, l2 ¼ 0:1, l3 ¼ 0:1, while a1 ¼ a2 ¼ a3 ¼ 0 also q2 ¼ q3 ¼ p2 ¼ p3 ¼ 0; (b) the Wigner function W a ðq; p; tÞ, against the quadratures q1 and p1 with the same value as in (a) but a3 ¼ 5; (c) the Wigner function W a ðq; p; tÞ, against the quadratures q1 and p1 with the same value as in (a) but a1 ¼ 5.

while the relation between the nonvanishing component of ^ and the momentum p^ is the magnetic field H j  2 1=2 X  2o 0 ^ tÞ ¼ cos K j z, (4.3) p^ j ðtÞ Hðz; K V 0 j where K j is the wave number and V is the cavity volume, for more details see for example Ref. [46]. To measure squeezing we have to find the explicit timedependent Dirac creation and annihilation operators. These can be constructed from the explicit expression for coordinates and momenta obtained in Section 2. These take the form a^ 1 ðtÞ ¼

3 1X 1 pffiffiffiffiffiffiffiffiffiffiffi ½fðo1  oj Þej ðtÞ 2 j¼1 o1 oj

þ ið_ej ðtÞ þ o1 oj e¯j ðtÞÞga^ yj ð0Þ þ fðo1 ej ðtÞ þ oj e_¯j ðtÞÞ þ ið_ej ðtÞ  o1 oj e¯j ðtÞÞga^ j ð0Þ,

ð4:4Þ

where the dash indicates the first derivative with respect to time. It is quite difficult for one to see any signature of squeezing from the variances of the quadratures qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hðDq^ j ðtÞÞi ¼ hq^ 2j ðtÞi  hq^ j ðtÞi2 and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hðDp^ j ðtÞÞi ¼ hp^ 2j ðtÞi  hp^ j ðtÞi2 , ð4:5Þ

for such a complicated system. Therefore, to examine the squeezing phenomenon we have plotted several figures of the variances of the quadratures hðDq^ j ðtÞÞi and hðDp^ j ðtÞÞi, against time for different values of the parameters. In the meantime we have assumed that the system is initially in the vacuum state. In these figures we display the variances of the quadratures hðDq^ j ðtÞÞ2 i (solid line) and hðDp^ j ðtÞÞ2 i (dashed line), for three different field modes, assuming that o1 ¼ 1 and o2 ¼ 0:9, while the coupling parameter l1 ¼ l2 ¼ 0:1, and l3 ¼ 0:0001. For the case of the first mode we can see besides the regular fluctuations that squeezing occurs in both quadratures. However, it is more pronounced in hðDp^ 1 ðtÞÞ2 i. We also observe an increment of the amount of squeezing in the first quadrature, hðDq^ 1 ðtÞÞ2 i, corresponding to a reduction in the second quadrature hðDp^ 1 ðtÞÞ2 i, see Fig. 2(a). The same behavior is observed for the second mode, but the squeezing in this case is pronounced in the quadrature hðDq^ 2 ðtÞÞ2 i, see Fig. 2(b). The situation is different for the third mode where we can see an increment in the amount of squeezing as the time increases in both quadratures. In addition there is a periodic exchange between the variances of the quadrature hðDq^ 3 ðtÞÞ2 i and hðDp^ 3 ðtÞÞ2 i, see Fig. 2(c). Moreover, when we increase the period of the interaction, an interesting phenomenon can be seen in both quadrature variances, see Figs. 2(d,e). This phenomenon is known as collapse and revival, which have been reported during the discussion of the atomic inversion in the atom-field interaction system,

ARTICLE IN PRESS M.M. Nassar, M. Sebawe Abdalla / Physica B 392 (2007) 159–172

Δq1,Δp1

a

165

Δq2,Δp2

b

0.52 0.52 0.51

0.51 2

4

6

8

10

12

14

2

4

6

8

10

12

14

0.49

0.49

0.48

0.48

c

Δq3,Δp3

Δq3

d

0.53 0.52

0.52

0.51

0.51 2

4

6

8

10

12

14

0.49

0.49

0.48

0.48

e

20

40

80

100

60

80

100

Δp3

0.52 0.51

20

40

60

0.49 0.48

Fig. 2. (a) The quadratures variances hðDq^ 1 ðtÞÞi, and hðDp^ 1 ðtÞÞi against the time, and the system is initially in the vacuum state, where o1 ¼ 1, o2 ¼ 0:9, while l1 ¼ 0:1, l2 ¼ 0:1, and l3 ¼ 0:0001; (b) the quadratures variances hðDq^ 2 ðtÞÞi, and hðDp^ 2 ðtÞÞi against the time, with the same value as in (a); (c) the quadratures variances hðDq^ 3 ðtÞÞi, and hðDp^ 3 ðtÞÞi against the time, with the same value as in (a); (d) the quadratures variances hðDq^ 3 ðtÞÞi, against the time, with the same value as in (a); (e) the quadratures variances hðDp^ 3 ðtÞÞi against the time, with the same value as in (a).

see Ref. [47] and the references therein. Also the same phenomenon has been observed during the examination of the effect of the Kerr-like medium on the two and threefield interaction, (see for instance Refs. [21,22]). On the other hand, when the value of the coupling parameters is increased (more precisely l3 ¼ 0:5), a greater amount of squeezing built up and the maximum value of squeezing in this case occurred at the half period of the time considered. Further the squeezing starts to appear in the second quadrature for the first and second modes, hðDp^ j ðtÞÞ2 i, j ¼ 1; 2, while it starts in the first quadrature, hðDq^ 3 ðtÞÞ2 i, for the third-field mode. We also can see regular fluctuations with exchange between the variances of the quadrature, see Fig. 3(a)–(c). At exact resonance o1 ¼ o2 ¼ 5 and o3 ’ 5, the collapse and revival phenomenon is also observed in all quadratures, but only for a short period of revival, see Fig. 3(d). As another example of a nonclassical effect we introduce amplitude-squared squeezing. This kind of squeezing arises in a natural way in the second harmonic generation and its

quadrature components are defined as 1 d^1 ¼ ða^ 2j þ a^ y2 j Þ; 2 j ¼ 1; 2; 3,

1 d^2 ¼ ða^ 2j  a^ y2 j Þ, 2i ð4:6Þ

while their variances of the quadrature satisfy the uncertainty relation D2 d^1  D2 d^ 2 Xðhn^ 2j i þ 12Þ2 .

(4.7)

The system is said to be amplitude-squared squeezing if one of the quadratures S1ðjÞ ¼ D2 d^1 or S2ðjÞ ¼ D2 d^2 (say) has negative value, where D2 d^1 ¼ hn^ 2j i  hn^ j i þ Reha^ 4j i  2 Reðha^ 2j iÞ2 , D2 d^2 ¼ hn^ 2j i  hn^ j i  Reha^ 4j i þ 2 Imðha^ 2j iÞ2 .

(4.8)

ðjÞ , In Fig. 4 we have displayed the quadratures S1;2 j ¼ 1; 2; 3, against time t, however, for different values of the involved parameters. We have also considered the

ARTICLE IN PRESS M.M. Nassar, M. Sebawe Abdalla / Physica B 392 (2007) 159–172

166

a

Δq1,Δp1

0.6 0.575 0.55 0.525 0.475 0.45 0.425

Δq2,Δp2

b 0.6 0.575 0.55 0.525

2

4

6

8

10

Δq3,Δp3

c

2

0.475 0.45 0.425

4

6

8

10

60

80

Δp1

d 0.502

0.54

0.501

0.52

20 2

4

6

8

40

100

10 0.499

0.48 0.498 Fig. 3. (a) The quadratures variances hðDq^ 1 ðtÞÞi,and hðDp^ 1 ðtÞÞi against the time, and the system is initially in the vacuum state, where o1 ¼ 1, o2 ¼ 1, while l1 ¼ 0:2, l2 ¼ 0:199, and l3 ¼ 0:5; (b) the quadratures variances hðDq^ 2 ðtÞÞi, and hðDp^ 2 ðtÞÞi against the time, with the same value as in (a); (c) the quadratures variances hðDq^ 3 ðtÞÞi, and hðDp^ 3 ðtÞÞi against the time, with the same value as in (a); (d) the quadratures variances hðDp^ 1 ðtÞÞi against the time, and the system is initially in the vacuum state, where o1 ¼ 5, o2 ¼ 5, while l1 ¼ 0:5, l2 ¼ 0:6. and l3 ¼ 0:01.

a

b

1.5

0.1

1

0.05

0.5

-0.5

2

4

6

8

10

2

4

6

8

10

-0.05

-1 -0.1

c 0.4 0.2

2

4

6

8

10

-0.2 -0.4 Fig. 4. (b) the

(a) The quadratures S11 , S12 against the ð2Þ quadratures Sð2Þ 1 , S 2 against the time,

time, with the parameters o1 ¼ 1:0, o2 ¼ 0:9, while l1 ¼ 0:3, l2 ¼ 0:2, l3 ¼ 0:1 and a1 ¼ 5, a2 ¼ a3 ¼ 0; with the same value as in (a); (c) the quadratures S31 , S32 against the time, with the same value as in (a).

system to be initially in the coherent states such that a1 ¼ 5 and a2 ¼ a3 ¼ 0. In this case and for the field frequencies, o1 ¼ 1 and o2 ¼ 0:9, while the coupling parameters, l1 ¼ 0:3, l2 ¼ 0:2, and l3 ¼ 0:1, squeezing phenomenon is apparent in all quadrature variances of each field mode. However, there are some differences between them, for example in the first-mode case we observe that the squeezing starts in the first quadrature after the onset of the interaction. Also there are a regular fluctuations in each quadrature with periodic exchange between them. This of

course is due to the existence of the sinusoidal function in the structure of each quadrature, see Fig. 4(a). On the other hand we can see there is a delay for the appearance of the squeezing phenomenon in the other quadratures, S ð2Þ 1;2 and Sð3Þ . Furthermore, the squeezing in both quadratures of the 1;2 field-modes 2 and 3 starts to increasing after a considerable amount of the time. However, it is apparent for the second mode case. Also we realize that the amount of squeezing in ð2Þ Sð3Þ 1;2 is greater than that of S 1;2 for the same period of time, see Fig. 4(b), (c). It is of interest to observe that the

ARTICLE IN PRESS M.M. Nassar, M. Sebawe Abdalla / Physica B 392 (2007) 159–172

a

167

b

1.5 1

1

0.5

0.5 20

40

60

80

-0.5

100 -0.5

20

40

60

80

100

-1

-1

Fig. 5. (a) The quadratures S21 against the time, with the same value as in Fig. 4(a); (b) the quadratures S22 against the time, with the same value as in Fig. 4(a).

a

b 1

1.5

0.5

1 0.5

-0.5

20

40

60

80

100

-1

-0.5 -1

-1.5

-1.5

20

40

60

80

100

Fig. 6. (a) The quadratures S21 against the time, with the parameters o1 ¼ 1:0, o2 ¼ 0:9, while l1 ¼ 0:3, l2 ¼ 0:2, l3 ¼ 0:1 and a2 ¼ 5, a1 ¼ a3 ¼ 0; (b) the quadratures S22 against the time, with the same value as in (a).

a

b

0.3

0.3

0.2 0.1

0.2

-0.1

0.1 100

200

300

400

-0.2 -0.3

-0.1

100

200

300

400

-0.2 -0.3

Fig. 7. (a) The quadratures S11 against the time, with the parameters o1 ¼ o2 ¼ 1:0 while l1 ¼ 0:1, l2 ¼ 0:199, l3 ¼ 0:0001 and a1 ¼ 5, a2 ¼ a3 ¼ 0; (b) the quadratures S12 against the time, with the same value as in (a).

collapses and revivals in this case are pronounced in the quadratures of the second mode, Sð2Þ 1;2 . Compare with both ð3Þ S ð1Þ and S , see Fig. 5(a), (b). The same observation 1;2 1;2 can be reported when we consider a1 ¼ a3 ¼ 0 and a2 ¼ 5 while the other values of the parameters are unchanged. This means that an increase in the mean photon number would lead to a considerable change in the squeezing amount, in addition to the collapse and revival phenomenon, see Fig. 6(a), (b). In the meantime the maximum value of squeezing occurs for the case in which a1 ¼ a2 ¼ a3 ¼ 5. Also it is noted that the collapse and revival phenomenon become more pronounced at exact resonance case (all field frequencies are the same), while the value of the parameter ai determines which quadrature it is, see Fig. 7(a), (b).

5. Transition amplitude It is well-known that transition between two states occurs if the systems have two different energies. This can be detected from the model under consideration. For instance, if we consider two different states one represents the Hamiltonian (1.9) and the other represents the same model in absence of one of the coupling parameters. In this case we have two different systems with two different energies and consequently the transition between the states would occur. In the present section we calculate the transition amplitude between the wave function in the coherent state representation for the Hamiltonian (1.9) and the wave function for the same Hamiltonian when one of the coupling parameters is absent, namely l3 . Further, we

ARTICLE IN PRESS M.M. Nassar, M. Sebawe Abdalla / Physica B 392 (2007) 159–172

168

shall consider the absolute value of the transition amplitude which lead us to find the transition probability. This will only be achieved for the coherent states representation case. To calculate the transition amplitude we have to use either the Green’s function or the wave function in the Schro¨dinger picture. However, in the present communication we employ the wave function in the coherent states representation given by Eq. (3.1) together with the wave function in the same state when l3 ¼ 0. In this case the wave function in absence of l3 can be written as " !#   3 ¯ 1O ¯ 2O ¯ 3 1=4 O 1 X 2 2 ca¯ ðq; tÞ ¼ exp  fj¯ai j þ a¯ i ðtÞg 2 i¼1 _3 p3 " # qffiffiffiffiffiffiffiffiffiffi 3 1 X 2 ¯ i  2 2_O ¯ iQ ¯ i a¯ i ðtÞQ ¯ i Þ , ð5:1Þ  exp  ðO 2_ i¼1 ¯ i , i ¼ 1; 2; 3, take the forms where the frequencies O sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o21 l21 þ o22 l22 ¯1 ¼ o O , ¯3 ¼ ðl21 þ l22 Þ ¯ 22;3 ¼ 1 ½ðo21 þ o22 Þ  O 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðo21  o22 Þ2 þ 4ðl21 þ l22 Þ.

where B and R are given by

B ¼ ðr1 s3  s1 r13 Þðr1 r2  r212 Þ þ ðr1 s2  s1 r12 Þðr12 r13  r1 r23 Þ ,



R ¼ ðr1 r3  r213 Þðr1 r2  r212 Þ  ðr12 r13  r1 r23 Þ2 .

In the above equation we have used the following abbreviations 3 1 X ¯ i ¯l2i Þ, ðOi l 2i þ O 2_ i¼1 rffiffiffi 3 qffiffiffiffiffi pffiffiffiffiffi 2X ¯ i ¯li Þ, ðai ðtÞ Oi l i þ a¯ i ðtÞ O s1 ¼ _ i¼1

r1 ¼

3 1 X ¯ im ðOi m2i þ O ¯ 2i Þ, 2_ i¼1 rffiffiffi 3 qffiffiffiffiffi pffiffiffiffiffi 2X ¯ im ðai ðtÞ Oi mi þ a¯ i ðtÞ O s2 ¼ ¯ i Þ, _ i¼1

r2 ¼

3 1 X ¯ i n¯ 2i Þ, ðOi n2i þ O 2_ i¼1 rffiffiffi 3 qffiffiffiffiffi pffiffiffiffiffi 2X  ¯ i n¯ i Þ. ðai ðtÞ Oi ni þ a¯ i ðtÞ O s3 ¼ _ i¼1

r3 ¼ (5.2)

while Euler’s angles given by Eq. (2.4) become !   ðl21 þ l22 Þ3=2 1 1 l1 y ¼ tan ; f ¼ tan , l2 l1 l2 ðo21  o22 Þ

3 1 X ¯ i ¯li m ðOi l i mi þ O ¯ i Þ, 2_ i¼1

(5.3)

r13 ¼

3 1 X ¯ i ¯li n¯ i Þ, ðOi l i ni þ O 2_ i¼1

Then the Hamiltonian (1.9) immediately reduces to the usual simple form

r23 ¼

3 1 X ¯ i n¯ i m ðOi ni mi þ O ¯ i Þ and 2_ i¼1

1 H^ ¼ 2

3 X ¯ 2i Þ. ¯ 2i Q ¯ 2i þ O ðP

ð5:8Þ

The other quantities involved are given by r12 ¼

!1=2 1 4ðl21 þ l22 Þ 1 2 2 tan 2f þ 2 sec 2f . c ¼ tan 2 ðo1  o22 Þ2

(5.7)

(5.4)

i¼1

To calculate the transition amplitude we have to use the formula ZZ 1 Z h¯a1 ; a¯ 2 ; a¯ 3 ja1 ; a2 ; a3 i ¼ ca¯ ðq; tÞca ðq; tÞ dq1 dq2 dq3 . 1

(5.5) After lengthy calculations we get the transition amplitude between the states, a¯ i and ai , in the form h¯a1 ; a¯ 2 ; a¯ 3 ja1 ; a2 ; a3 i   ¯ 1O ¯ 2O ¯ 3 1=4 pffiffiffiffiffiffiffiffiffiffiffi O1 O2 O3 O ¼ r1 =R _6    1 2 B2 2 2 s ðr1 r2  r12 Þ þ ðr1 s2  s1 r12 Þ þ  exp 4k 1 R ! 3 1X 2 2  exp  ½a ðtÞ þ a¯ 2 ai j2  , ð5:6Þ i ðtÞ þ jai j þ j¯ 2 i¼1 i

k ¼ r1 ðr1 r2  r212 Þ.

ð5:9Þ

The resultant probability amplitude of the transition between two different coherent states can be obtained from the relation Pð¯ai ; ai ; tÞ ¼ jh¯ai jai ij2 . Therefore to discuss the transition amplitude between two different coherent states for the present system we have to use Eq. (5.6). Our main tasks is to analyze the above complicated results, however it is not possible to do so without using numerical computations. For this reason we have plotted several figures to display the behavior of the absolute value of the transition amplitude for different values of the parameters involved. This is performed by examining the change which would occur in the functional behavior resultant of the variation in both initial mean photon numbers and the coupling parameters. Therefore, we have plotted the function against the scaled time, l1 t, for fixed values of o1 ¼ 1, o2 ¼ 0:9, l2 ¼ 0:19 and l3 ¼ 0:2, see Fig. 8. When we take the initial mean photon numbers, n¯ i , i ¼ 1; 2; 3, such that n¯ 1 ¼ n¯ 3 ¼ 0, and n¯ 2 ¼ 1, the function starts from the value one at t ¼ 0 (as should be expected). For t40 we

ARTICLE IN PRESS M.M. Nassar, M. Sebawe Abdalla / Physica B 392 (2007) 159–172

a

b

c

1.1

1.1

1.1

1.0

1.0

1.0

0.9

0.9

0.9

0.8

0.8

0.8

0.7

0.7

0.7

0.6

0.6

0.6 β1=β2=β3=1

0.5

β1=β3=1

0.5 0.4

0.4

0.3

0.3

0.3

0.2

0.2

0.2

0.1

0.1

0.1

0.0

0.0

0.0

-0.1

-0.1

-0.1

50

100

150

200

0

50

100

λ1t

150

β1=1

0.5

0.4

0

169

200

0

50

100

150

200

λ1t

λ1t

Fig. 8. (a) Transition amplitude against the time, with the parameters o1 ¼ o11 ¼ 1:0, o2 ¼ o22 ¼ 0:9 while l2 ¼ 0:19, l3 ¼ 0:2 but b1 ¼ 1, b2 ¼ b3 ¼ 0; (b) transition amplitude against the time, with the same value as in (a) but b3 ¼ 1; (c) transition amplitude against the time, with the same value as in (b) but b2 ¼ 1.

a

b

1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 -0.1

1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 -0.1

β2=β3=1

0

200

400

600

800

λ1t

β1=β3=1

0

200

400

600

800

λ1t

Fig. 9. (a) Transition amplitude against the time, with the same value as in Fig. 8(b) but for a long time; (b) transition amplitude against the time, with the same value as in Fig. 8(a) but b2 ¼ b3 ¼ 1, b1 ¼ 0.

can observe a considerable reduction in its value only after the onset of the interaction. This is followed by another period where the value of the function increases without reaching its maximum (where no transition can occur). This behavior is seen for subsequent periods of the time where the function decreases its value as the time increases, see Fig. 8(a). Same behavior is also observed if we consider the case in which n¯ 1 ¼ n¯ 3 ¼ 1 and n¯ 2 ¼ 0; for a short period of irregular fluctuations. This is also followed by a long period of stability between the two coherent states where the maximum value of the transition has occurred (minimum value of the function), see Fig. 8(b). In the meantime when we take all the mean photon numbers into account such as n¯ i ¼ 1, more periods of stability can be

seen in the behavior of the function where the transition in this case reaches its maximum, see Fig. 8(c). To display long term behavior of the function we have considered the case in which n¯ 1 ¼ n¯ 3 ¼ 1 and n¯ 2 ¼ 0, see Fig. 9(a). In this case we see more fluctuations with the interference between the patterns. This in fact refers to instability in the transition amplitude between the two states. Same behavior is repeated as the time increases. However, the value of the function increases without reaching its maximum. Here we may report that the interference phenomenon between the pattern disappeared for the case in which n¯ 2 ¼ n¯ 3 ¼ 1 and n¯ 1 ¼ 0. However, regular fluctuations can be seen as before, see Fig. 9(b). On the other hand when we increase the value of the mean

ARTICLE IN PRESS M.M. Nassar, M. Sebawe Abdalla / Physica B 392 (2007) 159–172

170

photon numbers, the period of stability gets more pronounced: see for example Fig. 10(a) for n¯ 1 ¼ n¯ 3 ¼ 2 and n¯ 2 ¼ 0 and Fig. 10(b) for: n¯ 2 ¼ n¯ 3 ¼ 2 and n¯ 1 ¼ 0, respectively. This means that by increasing the values of the mean photon numbers we are able to get more steady exchange of the energy between the two coherent states.

Finally we have plotted Fig. 11 to examine the variation of the function behavior when the value of the coupling parameters changes. In this case the function exhibits disturbance in its behavior where irregular fluctuations can be seen when the value of the coupling parameters increases (strong interaction case). This phenomenon is

a

b

1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 -0.1

1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 -0.1

β2=β3=2

0

200

400

600

β1=β3=2

0

800

200

λ1t

400

600

800

λ 1t

Fig. 10. (a) Transition amplitude against the time, with the same value as in Fig. 8(a) but b1 ¼ b3 ¼ 2, b2 ¼ 0; (b) transition amplitude against the time, with the same value as in Fig. 8(a) but b2 ¼ b3 ¼ 2, b1 ¼ 0.

a

b

1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 -0.1

1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 -0.1

β1=β3=1

0

50

100 λ1t

150

200

β3=1

0

50

100 λ1t

150

200

d

c 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 -0.1

1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 -0.1

β3=1

0

50

100 λ1t

150

200

β2=1

0

50

100

150

200

λ1t

Fig. 11. (a) Transition amplitude against the time, with the same value as in Fig. 8(a) but l3 ¼ 0:6, b3 ¼ 1, b2 ¼ b1 ¼ 0; (b) transition amplitude against the time, with the same value as in (a) but b1 ¼ 1; (c) transition amplitude against the time, with the same value as in (a) but o1 ¼ o11 ¼ 2:0, o2 ¼ o22 ¼ 1:0, also b2 ¼ 1, b3 ¼ 0; (d) transition amplitude against the time, with the same value as in (c) but b3 ¼ 1, b2 ¼ 0.

ARTICLE IN PRESS M.M. Nassar, M. Sebawe Abdalla / Physica B 392 (2007) 159–172

pronounced for the case in which l3 ¼ 0:6, and n¯ 1 ¼ n¯ 2 ¼ 0 and n¯ 3 ¼ 1, while the other parameters are kept without change, see Fig. 11(a). Using the same values of the coupling parameters and taking n¯ 1 ¼ n¯ 3 ¼ 1 and n¯ 2 ¼ 0, it is easy to observe regular fluctuations in addition to an increase in the function values at a certain period of time, see Fig. 11(b). This means that there is a reduction in the transition rate for these periods of time. Before we end this section we draw the attention to consider the effect of the field frequency on the transition amplitude, for instance when we consider o1 ¼ 2 and keep the other parameters without change we can report two different behaviors; the first is when n¯ 1 ¼ n¯ 3 ¼ 0 and n¯ 2 ¼ 1, and the second is when n¯ 1 ¼ n¯ 2 ¼ 0 and n¯ 3 ¼ 1. For the first case the function shows parabolic shape with rapid fluctuations and the transition slows down. Furthermore, we observe that the function reaches its minimum after a considerable period of time, see Fig. 11(c). In fact this behavior is contrary to all the previous cases considered where the system has been considered to be near resonance ðo1 ’ o2 Þ. This behavior can be interpreted as follows: since the field frequency for the first and the second modes are far-resonance, o1 ¼ 2o2 , then a considerable modification in the values of the augmented frequencies of each mode would occur, see Eq. (5.6). In the meantime, the construction of the time-dependent photon numbers consists of the initial mean photon numbers multiplied by the augmented frequencies. This leads to a change in the value of the mean photon numbers n¯ i , i ¼ 1; 2; 3, and consequently the above-mentioned behavior would appear. Also for the off-resonance case and more precisely when n¯ 2 ¼ 2, and n¯ 1 ¼ n¯ 3 ¼ 0 where the timedependent photon number increases we can realize that there is an increase in the rate of exchange of the energy between the states. The same analysis can be applied to the second case for which n¯ 1 ¼ n¯ 2 ¼ 0 and n¯ 3 ¼ 2, where the situation in such case is different and the effective initial mean photon number is just n¯ 3 . In such case the function shows regular fluctuations in a steady manner and the exchange of the energy between the states has a different rate for each period of time (two values of the extreme), see Fig. 11(d). This is due to a decrease in the time-dependent photon number value resulting from the reduction in the ¯ 3. augmented frequency O 6. Conclusion In the previous sections of this paper we have introduced a Hamiltonian model to describe the mutual interaction between three electromagnetic fields. The model is considered without applying what is called rotating wave approximation (RWA) where the rotating and counterrotating terms are kept. Furthermore, we have assumed the system to be far-resonance (anisotropic case). With its use of Euler’s angles we have managed to diagonalize the Hamiltonian model (1.9) and obtained three uncoupled harmonic oscillators. This led us to find the explicit

171

expression of the wave function in both number and coherent states representation. The W-Wigner function is obtained via the wave function in the coherent state for which we find that it has Gaussian structure in general. The squeezing phenomenon is also considered as an example of the nonclassical effect where the observation of the collapse and revival phenomenon are reported. Finally we discussed in detail the probability amplitude between two different coherent states. The first state is for the whole system and the second is the state of the system in absence of the coupling parameter, l3 . It has been shown that the rate of exchange of the energy between the states is always sensitive to the variation in the initial mean photon numbers in addition to the field frequencies and the coupling parameters. Acknowledgment The authors express their gratitude to Professor M. AlGawiz for his critical reading of the manuscript. One of us (M.S.A.) is grateful for the financial support from the project Math 2005/32 of the Research Center, College of Science, King Saud University. M.M.N is also thankful the Research Center, College of Science, King Saud University for the grant ST/Math/2006/02. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

[15] [16] [17] [18] [19] [20]

[21] [22] [23]

W.H. Louisell, A. Yariv, A.E. Siegman, Phys. Rev. 124 (1961) 1646. B.R. Mollow, R.J. Glauber, Phys. Rev. 160 (1967) 1076. B.R. Mollow, R.J. Glauber, Phys. Rev. 160 (1967) 1097. J. Tucker, D.F. Walls, Ann. Phys. 52 (1960) 1. D.F. Walls, R. Barakat, Phys. Rev. A 1 (1970) 446. E.A. Mishkin, D.F. Walls, R. Barakat, Phys. Rev. 185 (1969) 1618. E.Y.C. Lu, Phys. Rev. A 8 (1973) 1053. V. Perˇ inova´, Opt. Acta 28 (1981) 747, see also V. Perˇ inova´, J. Perˇ ina, Opt. Acta 28 (1981) 769. S. Carusotto, Phys. Rev. A 40 (1989) 1848; A. Bandilla, G. Drobny, I. Jex, Phys. Rev. A 53 (1996) 507. A.S. Shumovsky, B. Tanatar, Phys. Rev. A 48 (1993) 4735. J. Fiura´sˇ ek, J. Krˇ epelka, J. Perˇ ina, Opt. Commun. 167 (1999) 115. G. Ariunbold, J. Perˇ ina, Opt. Commun. 176 (2000) 149. N.Ph. Geordiades, E.S. Polzik, K. Edamatsu, H.J. Kimble, Phys. Rev. Lett. 75 (1995) 3426. J. Perina, Quantum Statistics of Linear and Nonlinear Optical Phenomena, second ed., Kluwer Academic Publishers, Dordrecht, 1991. A. Luksˇ , V. Perˇ inova´, J. Perˇ ina, Opt. Commun. 67 (1988) 149. A. Luksˇ , V. Perˇ inova´, Z. Hradil, Acta. Phys. Pol. 74 (1988) 713. B. Yurke, J. Opt. Soc. Am. B 2 (1985) 732. R.M. Shelby, M.D. Levenson, Optics Commun. 64 (1987) 553. A. La Porta, R.M. Shelby, B. Yurke, Phys. Rev. Lett. 62 (1989) 28, see also M.S. Abdalla, Acta Phys. Slovaca 47 (1997) 353. M.S. Abdalla, M.M.A. Ahmed, S. Al-Homidan, J. Phys. A: Math. Gen. 31 (1998) 3117, see also M.S. Abdalla, Phys. Rev. A 35 (1987) 4160. F.A.A. El-Orany, M.S. Abdalla, J. Perˇ ina, J. Opt. B: Quantum Semiclass Opt. 67 (2004) 460. F.A.A. El-Orany, M.S. Abdalla, J. Perˇ ina, Eur. J. Phys. D 33 (2005) 453. M.S. Abdalla, F.A.A. El-Orany, J. Perˇ ina, J. Phys. A: Math. Gen. 32 (1999) 3457.

ARTICLE IN PRESS 172

M.M. Nassar, M. Sebawe Abdalla / Physica B 392 (2007) 159–172

[24] M.S. Abdalla, F.A.A. El-Orany, J. Perˇ ina, Acta Phys. Slovaca 50 (2000) 613. [25] F.A.A. El-Orany, J. Perˇ ina, M.S. Abdalla, Phys. Scr. 63 (2001) 128. [26] F.A.A. El-Orany, J. Perˇ ina, M.S. Abdalla, Int. J. Mod. Phys. B 15 (2001) 2125. [27] M.S. Abdalla, F.A.A. El-Orany, J. Perˇ ina, Eur. Phys. J. D 13 (2001) 423. [28] F.A.A. El-Orany, J. Perˇ ina, M.S. Abdalla, J. Opt. Quantum Semiclass Opt. B 3 (2001) 66. [29] G.A. Garrett, A.G. Rojo, A.K. Sood, J.F. Whitaker, R. Merlin, Science 275 (1997) 1638. [30] Ivo. Vysˇ in, M. Slinta´kova´, Physica 37 (1998) 87. [31] Ivo. Vysˇ in, K. Sva´cˇkova´, J. Rˇ~ha, ı Opt. Commun. 174 (2000) 149. [32] V.V. Dodonov, V.I. Man’ko, in: Invariants and Evolution of NonStationary Quantum System, Trudi FIAN (P.N. Lebedev Phys. Inst. Proc.), vol. 183, Nova Science Publishers, Commack, NY, 1989, p. 263. [33] V.V. Dodonov, V.I. Man’ko, D.L. Ossipov, Physica A 168 (1990) 1055. [34] E.A. Akhundova, V.V. Dodonov, V.I. Man’ko, Physica A 115 (1982) 215. [35] V.V. Dodonov, V.I. Man’ko, O.V. Shakhmistova, Phys. Lett. A 102 (1984) 259.

[36] G. Schrade, V.I. Man’koff, W.P. Schleich, R.J. Glauber, Quantum Semiclass Opt. 7 (1995) 307. [37] W.P. Schleich, Quantum Optics in Phase Space, Wiley-Vch Verlag, Berlin, 2001. [38] See Special Issue, J. Opt. Soc. Am. B 4, Squeezed States of the Electromagnetic Field, 1987. [39] H.P. Yuen, Phys. Rev. A 40 (1976) 3147. [40] H.P. Yuen, J.H. Shapiro, IEEE Trans. Inform. Theory IT24 (1978) 657; H.P. Yuen, J.H. Shapiro, IEEE Trans. Inform. Theory IT26 (1978) 78. [41] J.H. Shapiro, H.P. Yuen, M.J.A. Machado, IEEE Trans. Inform. Theory IT25 (1979) 179. [42] M.J. Collett, C.W. Gardiner, Phys. Rev. A 30 (1984) 1386; M.J. Collett, R. Loudon, J. Opt. Soc. Am. B 4 (1987) 1525. [43] C.M. Caves, B.L. Schumaker, Phys. Rev. A 31 (1985) 3068; B.L. Schumaker, C.M. Caves, Phys. Rev. A 31 (1985) 3093. [44] M. Hillery, Phys. Rev. A 13 (1989) 2226. [45] M. Hillery, Opt. Commun. 62 (1987) 135. [46] M. Sargent, M.O. Scully, W.E. Lamb Jr., Laser Physics, AddisonWesley, Reading, MA, 1974. [47] B.W. Shore, P.L. Knight, J. Mod. Opt. 40 (1993) 1195.