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Von Neumann entropy and phase distribution of two mode parametric amplifier interacting with a single atom
Annals of Physics 318 (2005) 266–285 www.elsevier.com/locate/aop
Von Neumann entropy and phase distribution of two mode parametric amplifier interacting with a single atom M. Sebawe Abdalla
a,* ,
A.-S.F. Obada b, M. Abdel-Aty
c
a
b
Mathematics Department, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia Department of Mathematics, Faculty of Science, Al-Azher University, Nasr City, 11884 Cairo, Egypt c Mathematics Department, Faculty of Science, South Valley University, 82524 Sohag, Egypt Received 30 October 2004; accepted 4 January 2005 Available online 26 February 2005
Abstract In the present article, we introduce a Hamiltonian model that consists of two modes of the field in a perfect cavity to interact with a single two-level atom. The interaction between the fields has been taken into account and considered to be in the parametric amplifier form. The model in one hand can be regarded as a generalization of the Jaynes–Cummings model (JCM), however, in the other hand it can be considered as a generalization of the parametric amplifier model. Under a certain condition the exact solution to the Schro¨dinger equation is obtained. Employing this solution and for chosen values of different parameters we discuss numerically the atomic occupation probabilities as well as the degree of entanglement through the entropy of the field. The system shows superstructure phenomenon similar to that appeared from the effect of the Kerr-like medium on the Jaynes–Cummings model. The von Neumann entropy and phase distribution for both two-mode correlated and uncorrelated coherent states cases are also considered. 2005 Elsevier Inc. All rights reserved.
*
Corresponding author. Fax: +96614676484. E-mail address: [email protected] (M.S. Abdalla).
0003-4916/$ - see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.aop.2005.01.002
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1. Introduction The Jaynes–Cummings model (JCM) represents one of the most fundamental models in the field of quantum optics [1]. This in fact is not only due to the mathematical solvability of the model, but also to its richness with many different interesting phenomena, such as collapse-revival phenomenon [2], squeezing [3], antibunching [4], chaos [5], and trapping states [6–9], etc. The model is the simplest quantum electrodynamic model of an atom–field interaction describing the interaction of an undamped two-level atom with a single non-decaying electromagnetic field mode [10–13]. It was first used to examine the classical aspects of spontaneous emission and to reveal the existence of Rabi oscillations in the atomic excitation probabilities for fields [14]. However, it has been discovered that the JCM atomic population presented direct evidence for the discreteness of photons [15]. At the beginning a limiting number of extensions of the Jaynes–Cummings model have been carefully studied. The coupling of N two-level atoms to a single quantized radiation mode has been solved. Finite algorithms for the eigenfrequencies were given for the case of resonance (detuning D = 0) [16]. The coupling of a single twolevel atom to M equal-frequency modes has been used as a simple model of decay processes [17]. However, as a result of the development of low-temperature high-Q cavities, and the use of Rydberg atoms, the idealized situation of a single two-level atom interacting with a single-mode of quantized radiation field in a lossless cavity has been experimentally realized [18]. These advances in turn have certainly provided incentive for extending and generalizing this model. Thus many authors have been encouraged and stimulated to modify the original model and to generalize it into different directions. For example, it has been generalized and extended to include the effects of cavity mode decay as well as black-body radiation fields [19]. Also it has been extended further for multi-atom [20] and multi-level systems [21], as well as multi-photon and multi-mode [22]. In the meantime effects of spatial mode structure of the cavity and the transient effects arising from a time-dependent atom–field coupling coefficient in JCM evolution have been reported [23]. Furthermore the effects of stochastic phase fluctuations in the atom–field coupling coefficient have also been considered, where the fluctuations are modeled by random telegraph process [24,25]. The aim of the present work is to generalize this model further, however, in a different direction rather than those that have been considered earlier. For example, if we consider a single atom injected within a cavity pumped simultaneously by two different fields, then the interaction will occur between the atom and the fields as well as between the fields themselves. This type of the interaction has been ignored in the literature and therefore one of our goals is to fill this gap and to consider such system, see for example [22]. Therefore, if we assume that the later interaction is of the parametric amplifier form, then one can construct the effective Hamiltonian as follows: ^ ¼H ^A þ H ^ AF þ H ^ FF ; H ^ A is the atomic part where H
ð1Þ
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^ A ¼ 1 ^z H hx 0 r 2
ð2Þ
^ AF is the atom–field interaction and H ^ AF ¼ hk1 ð^ ^ Þ þ ^ Þ; H ay1 þ ^ a1 Þð^ rþ þ r hk2 ð^ ay2 þ ^ a2 Þð^ rþ þ r
ð3Þ
^ FF is the Hamiltonian which represents the field–field interaction in a non-dewhile H generate parametric amplifier form. In terms of creation and annihilation operators the lossless Hamiltonian describing such a system can be written as ^ FF ¼ hx1 ^ H hx 2 ^ hð^ ay1 ^ a1 ^ ay1 ^ a1 þ ay2 ^ a2 þ k ay2 þ ^ a2 Þ:
ð4Þ
In the above formulae xi and ki, i = 1, 2 are the bichromatic field frequencies, and the coupling constants between the atom and each mode, while k is the parametric amplifier constant. x0 is the natural transition frequency of the atom and the operators r+ (r), and rz are the usual raising (lowering) and inversion operators for the atomic system, satisfying [rz, r±] = ±2r± and [r+, r] = rz, while ^ayi ð^ai Þ; ði ¼ 1; 2Þ is the creation (annihilation) operator for the ith quantized bichromatic field satisfying the commutation relations ½^ ai ; ^ ayj ¼ dij ð¼ 1 if i ¼ j and zero otherwiseÞ. Before we ^ FF . During the quango further let us point out the derivation of the Hamiltonian H tization between the atom and the field within cavity it is possible to see an interaction between the fields themselves. In this case to provide coupling among the various cavity modes one may consider the dielectric constant varies with time. Barring accidental degeneracies, we therefore assume that only two nondegenerate modes are coupled by the pump field at frequency x1 + x2 and refer to them as the signal and idler modes. Therefore, if one adjusts the phase pump to be time-de^ FF can be immedipendent such as (x1 + x2) t = / (t) (say) then the Hamiltonian H ately obtained. It is to be noted that in (5) we have kept the counter rotating terms. Here we would like to say that the Hamiltonian (1) has two advantages; one of them it can be regarded as a generalization of the usual JCM which represents the interaction between one mode and a single two-level atom. While the second it can be regarded as a generalization of the field–field interaction in a parametric amplifier form. Thus we have two kinds of operations: atom–field interaction which represents energy exchange between the atom and the fields, while the other one represents some sort of amplification within cavity. Recently, some experiments have been carried out by the authors of [26,27]. Those authors considered the cavity quantum electrodynamics (QED) experiments involving Rydberg atoms crossing superconducting cavities with different frequency regimes. The entanglement in this case resultant from the interaction of a two-level atom with a cavity field mode. In these experiments the authors introduced an important step towards the realization of more complex entanglement in cavity QED by involving in their experiments two independent cavity modes. They have shown that a single atom can be used to entangle the two modes, while a second atom being employed later to reveal this entanglement. This in fact somehow in agreement with our idea of considering a Hamiltonian model consists of a two-level atom in interaction with two modes. However, in addition we have taken into consideration the interaction between the fields in the parametric amplifier form. The main purpose of the present work is to examine the
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degree of entanglement for the above system. This will be achieved for two different cases. First when the fields are considered to be correlated, while the second when we regard both fields are uncorrelated. To reach our goal we have to calculate the wave function in the Schro¨dinger picture. This will be introduced in Section 1, followed by Section 2.1, where we discuss the atomic occupation probabilities for both correlated and uncorrelated cases. In Section 3, we shall consider von Neumann entropy to discuss the entropy of the present system and consequently the degree of entanglement. Our results will be discussed in Section 4 for two different cases; one is the correlated and the other is the uncorrelated case. In Section 5, we shall discuss the phase distribution for both correlated and uncorrelated cases. Finally, our conclusion will be given in Section 6. 2. The wave function The main purpose of the present section is to give the exact expression of the timedependent wave function in Schro¨dinger picture. To reach this goal let us introduce the canonical transformation: y
^ a1 ¼ ^ b1 cosh f ^ b2 sinh f; y
^ b2 cosh f ^ b1 sinh f; a2 ¼ ^
ð5Þ
where
1 2k f ¼ tanh1 ð6Þ 2 x1 þ x2 y ^i ð ^ and the operators b bi Þ; i ¼ 1; 2 have properties similar to that of the operators y ^ ai ð^ ayi Þ; i ¼ 1; 2 such that ½^ ai ; ^ ayj ¼ ½^ bi ; ^ bj ¼ dij . Now if we substitute Eq. (7) into Eq. (1) and after some calculations then the Hamiltonian is transformed to the form 2 h i X y y hx0 ^ ¼ ^þ þ ^ ^ Þ ; ^z þ H h Xi ^ bi r bi r ð7Þ bi ^ b i þ li ð ^ r 2 i¼1 where, i = 1, 2 and: X1 ¼ x1 cosh2 f þ x2 sinh2 f k sinh 2f; X2 ¼ x2 cosh2 f þ x1 sinh2 f k sinh 2f; l1 ¼ k1 cosh f k2 sinh f; l2 ¼ k2 cosh f k1 sinh f:
ð8Þ
It should be noted that to avoid any appearance of non-conservative terms in obtaining Eq. (9) we have applied the rotating wave approximation with respect to the rotated y operators ^ bi and ^ bi , not to the original (physical) operators ^a and ^ay . To continue our progress we have to find the solution of the wave function, in the interaction picture V I ðtÞjwi ¼ ih
o jwi; ot
ð9Þ
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where the time-dependent interaction Hamiltonian V I (t) is given by V I ðtÞ ¼
2 X
y
^þ expðiDj tÞ þ ^ ^ expðiDj tÞ; lj ½^ bj r bj r
ð10Þ
j¼1
where Dj ¼ ðXj x0 Þ; j ¼ 1; 2 is the detuning parameter. For the present system it is not an easy task to obtain an explicit solution of the wave function. However, if we adjust the coupling parameter k such that x1 þ x2 k ¼ k1 k2 2 ; ð11Þ k1 þ k22 then Eq. (9) reduces to the form y y y hx0 ^ ¼ ^z þ hn ^ ^þ þ ^b2 r ^ : H hX 1 ^ r hX2 ^ b2 r b1 ^ b1 þ b2 ^ b2 þ ð12Þ 2 This means that the restrictive condition (13) in this case impliesq the coupling paramffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
eter l1 fi 0 while the other coupling l2 survives and equals n ¼ k22 k21 . Moreover, the restrictive condition means also that the coupling parameter k between the field– field has been chosen to be stronger than the coupling parameter between the field and the atom provided that ki xi, i = 1, 2. However, this is not realistic to be applicable in the experimental work. Therefore to make the restrictive condition consistent with the recent experiments, one may adjust the coupling parameter k to be less than one of the atom–field coupling. In the present case we have to select k < k2, which means k1 (x1 + x2), and consequently the other coupling parameter qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 will be larger than k1 ðx1 þ x2 Þ k21 . This of course would lead to strong interaction between the atom and the second field mode which is in agreement with the experimental observation. y ^1 ¼ ^ It is easy to realize that the operator N b1 ^ b1 in Eq. (14) is a constant of the motion and this allows us to find the explicit solution of the final state matrix of the density operator Et q in the form Et q ¼ jwðtÞihwðtÞj; where jwðtÞi ¼
ð13Þ
1 X 1 X
ut
n1 ¼0 n2 ¼0
vt
ðn1 ;n2 Þ
ðn1 ;n2 Þ
! jn1 ; n2 ib
and the probability amplitudes are given by: ! ! ! ðn ;n Þ ðn ;n Þ 1 ;n2 Þ cos h2 Bt 1 2 Aðn ut 1 2 t ; ¼ ðn ;n Þ ðn ;n 1Þ expði/Þ sin h2 vt 1 2 Bt 1 2 Atðn1 ;n2 1Þ 1 ;n2 Þ Aðn t
1 ;n2 Þ Bðn t
D2 sin gn2 þ1 t ¼ cos gn2 þ1 t i qn1 ;n2 ; 2 gn2 þ1 pffiffiffiffiffiffiffiffiffiffiffiffiffi sin gn þ1 t 2 ¼ in n2 þ 1 qn1 ;n2 ; gn2 þ1
ð14Þ
ð15Þ
ð16Þ
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qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gn2 ¼ D22 þ n2 n2 . The modified Rabi frequency gn2 contains the coupling parameter k which reflects the effect of the parametric amplifier term. This can be noticed from the modification in the detuning parameter D2. It would be interesting to compare this situation with that of the JCM when the system evolves under the influence of the third-order nonlinearity of a Kerr-medium [28]. This means that the parametric amplifier plays a role similar to that of the Kerr-like medium. In fact the term multiplied by the coupling parameter k represents the nonlinear exchange through simultaneous annihilation or creation of a photon in both the subharmonic modes at the expense of pumping. Without doubt nondegenerate parametric amplifiers can perfectly provide single-mode and two-mode squeezing, moreover it has been employed in experiments. For example, the fourth-order interference effects arise when pairs of photons produced in parametric amplifier are injected into Michelson interferometers [29,30]. Also the second-order interference is observed in the superposition of signal photons from two coherently pumped parametric amplifiers when the paths of the idler photons are aligned [31], for more details see [32]. Here we may point out that the initial atomic state is considered to be a combination between the excited state and the ground state, while the field is in a combination of the Fock states of the two modes with amplitude qn1 ;n2 , i.e. ! 1 X 1 X cos h2 jwð0Þi ¼ qn1 ;n2 jn1; n2 ib ; ð17Þ sin h2 ei/ n1 ¼0 n2 ¼0 where h is the angle of coherence and / is the relative phase of the two atomic levels, while |n1, n2æb is the field state of the two modes in the rotated frame. It is obvious from the above equation that the excited state can be obtained if we take h fi 0 while the ground state results when we take h fi p. In our consideration we assume that both fields are in the two-mode uncorrelated coherent state |b1, b2æ = |b1æ |b2æ such that X 1 X 1 1 bn11 bn22 pffiffiffiffiffiffiffiffiffiffiffi jb1 ; b2 ib ¼ exp ½jb1 j2 þ jb2 j2 jn1 ; n2 ib 2 n1 !n2 ! n1 ¼0 n2 ¼0 ¼
1 X 1 X
qn1 ;n2 jn1 ; n2 ib :
ð18Þ
n1 ¼0 n2 ¼0
However, to relate the state of the physical two modes (of the ^a’s) and the rotated ones (of the ^ b’s) we use Eq. (7). Thus, we realize that the vacuum state with respect to ayi Þ; i ¼ 1; 2 is in fact the squeezed vacuum state for the rotated physical operators ^ ai ð^ y operators ^ bi ð^ bi Þ; i ¼ 1; 2, where y y j0; 0ia ¼ exp f^ b1 ^ ð19Þ b2 f ^ b1 ^ b2 j0; 0ib and whence the uncorrelated two-mode coherent states ja1 ; a2 ia ¼ ja1 ia1 ja2 ia2 for the physical operators becomes a combination of correlated displaced Fock states given by
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y y ja1 ; a2 ia ¼ exp f^ b1 ^ b2 f ^ b1 ^ b2 jb1 ; b2 ib ¼ ðcosh nÞ1
1 X ðtanh nÞn jb1 ; n; b2 ; nib ;
ð20Þ
n¼0
where |b1, næb is a displaced Fock state given by rffiffiffiffiffi (X 1 1 n! mn mn 2 2 jb1 ; nib ¼ exp jb1 j b1 Ln ðjb1 j Þjmib 2 m! m>n ) rffiffiffiffiffi 1 X m! X 2 nm nm ðb1 Þ Lm ðjb1 j Þjmib ¼ þ qm ðb1 ; nÞjmib n! m
ð21Þ
and therefore the coefficient of |m1, m2æb in Eq. (22) after substituting (23) in it represents the amplitude for the state in the correlated case. Furthermore the relations between the amplitudes a 0 s and b 0 s is given by: b1 ¼ a1 cosh f þ a2 sinh f; b2 ¼ a2 cosh f þ a1 sinh f:
ð22Þ
The states |m1, m2æa are eigenstates of the operators a^yi ^ai ; ði ¼ 1; 2Þ ^ ayi ^ ai jmi ia ¼ mi jmi ia ;
i ¼ 1; 2
ð23Þ
and similarly for the rotated states y y ^ bj ^ bj jnj ib ¼ nj jnj ib ;
j ¼ 1; 2:
ð24Þ
In the following, we shall employ the results obtained here to consider the atomic occupation probabilities related to the present system namely to examine the effect of the parametric amplifier term on Jaynes–Cummings model. 2.1. Atomic occupation probabilities The phenomenon of collapses and revivals represents one of the most important nonclassical phenomenon in the field of quantum optics. The observation of this phenomenon occurred during the course of interaction between the field and the atom within cavity. In the present section we shall concentrate on studying this phenomenon to see how the field–field interaction in the parametric amplifier form affects this phenomenon. This can be achieved by considering the atomic occupation probabilities. Generally the matrix elements of the reduced atomic density operator are given by the expression 1 X 1 X qA ðtÞ ¼ hn1 ; n2 ; ijwðtÞihwðtÞjn1 ; n2 ; ji; ð25Þ ij n1 ¼0 n2 ¼0
where, i, j = e, g. In fact the atomic occupation probabilities represent another important quantity which is nearly equal and parallel to the atomic inversion Ærz (t)æ. This latter function can be obtained from Eq. (27), and it takes the form
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A hrz ðtÞi ¼ qA ee ðtÞ qgg ðtÞ. It should be noted that the atomic occupation probabilities results are drastically different from the predictions of the semiclassical theory where the atom in the excited state cannot make a transition to the lower level in the absence of a driving field, however, in the fully quantum mechanical treatment, the transition from the upper level to the lower level in the vacuum is possible due to the spontaneous emission. To analyze the effect of the parametric amplifier term on the atomic occupation probabilities for the present system we consider two different cases. One when the fields are correlated while the second case when the fields are uncorrelated using Eq. (20). This will be seen in the next subsections.
2.2. Uncorrelated case Let us start this subsection by discussing the behavior of the atomic occupation probabilities function qA ee ðtÞ (the probability of finding the atom in the excited state). We shall consider the case in which both fields are in uncorrelated two-mode coherent state, namely the state given by Eq. (20). For this reason we have plotted the function qA ee ðtÞ against the scaled time k1t in Fig. 1. In these figures, we have taken the ratio of the coupling parameters
Fig. 1. The evolution of the occupation probability Pe (t) as functions of the scaled time k1t. Calculations assume xi/k1 = 1, (A) h = 0, / = 0, k2/k1 = 2, a1 = 2, a2 = 4.5, (B) h = 0, / = 0, k2/k1 = 2, a1 = 4.5, a2 = 2, (C) h = p/3, / = 0, k2/k1 = 2, a1 = 2, a2 = 4.5, and (D) h = 0, / = 0, k2/k1 = 5, a1 = 2, a2 = 4.5.
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k2/k1 = 2, and the ratio of the frequencies relative to the coupling parameter k1 such that xi/k1 = 1, i = 0, 1, 2, while h = / = 0. This means that the detuning parameter D2 „ 0 and affects the system behavior. In this case and for fixed values of a1 = 2 and a2 = 4.5 we can realize from Fig. 1A that just after the onset of the interaction the function qA ee ðtÞ fluctuates between nearly 0 and 1 (around 0.5) but for short period of the time showing the collapse behavior. This short period of collapse is followed by a long period of revival which indicates a strong correlation occurring between the atom and the fields. However, we can see another period of collapse occurring just half of the value of the considered time, where the maximum and minimum values of the fluctuations in this case occurred between 0.75 and 0.25, respectively. This is followed by a period of revival shorter than the previous one. The collapse phenomenon starts again to appear but with amplitude less than that of the previous one. As soon as we exchange the values of the mean photon numbers such that a1 = 4.5 and a2 = 2 we can see some changes occurring in the function behavior. In this case we observe at t > 0 a short period of collapse immediately after onset of the interaction followed by long period of revival showing a strong interaction between the atom and the field. Subsequent periods of collapses and revivals can be observed, however, with a decrease in the maximum and the minimum values of the fluctuations. We also noted in this case that the interference between the fluctuations pattern during the collapses period are more pronounced than that in the previous case, see Fig. 1B. This in fact indicates the effect of the nonlinearity resultant of the field–field interaction. It is interesting to compare this behavior with superstructure phenomenon which has been realized during the examination of the Jaynes–Cummings model in the presence of Kerr-like medium, see [28]. For further investigation we have considered the case in which the angle h changes to take the value h = p/3 (intermediate state case), in this case we note reduction in the fluctuations value, see Fig. 1C. However, the second period of revival is more pronounced than in the excited state case when a1 = 2 and a2 = 4.5. As the time goes on more decrease in the amplitude of the fluctuations can be observed in both second and third period of collapses as should be expected. In this case, we can say that the population clearly exhibits the characteristic collapses and revivals and provides us with information about discrete nature of the quantized atom-cavity field interaction. Finally, when we increase the values of the coupling parameters ratio such that k1/k2 = 0.2, which means the value of the effective coupling parameter k would be decreased, then we can observe a drastic change occurring in the function behavior. For instance we realize that after the onset of the interaction there is a small period of collapse followed by short period of revival (smaller than all the previous cases). In the meantime we observe an increase in the collapse period which is parallel to decreasing in the revival period (indicates a weak interaction). However, compared to the previous case (strong coupling case) we can observe during the collapse period there is no interference occurring between the fluctuations pattern (but more revivals can be seen). Further the maximum and the minimum values of the fluctuations occur faster than the strong coupling case, see Fig. 1D.
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2.3. Correlated case Here we turn our attention to consider the case in which both fields are in the twomode correlated coherent state taking into account Eq. (22) to represent the field state. For this reason we have plotted the function qA ee ðtÞ against the scaled time k1t in Fig. 2. To make a comparison between the correlated and the uncorrelated cases we have to take the values of the parameters similar to that of the previous case. Therefore, we shall consider the case in which h = / = 0, and the coupling parameters ratio k1/k2 = 0.5, also the fields and the atomic frequencies are taken to be equal, while the ratio between the frequencies and the coupling parameter k1 is xi/k1 = 1, i = 0, 1, 2. In this case and providing the amplitudes a1 = 2 and a2 = 4.5, we find the function reduces its value to be around 0.26. Furthermore, we realize there is no revival period for all values of time, however, the function fluctuates with irregular pattern, see Fig. 2A. In the meantime if we exchange the values of the mean photon numbers such a1 = 4.5 and a2 = 2, we observe no big change in the figure shape, except some decreases in the fluctuations number with a slight revival behavior. Furthermore the function reduces its value to be around 0.22. This in fact indicates an increase in the correlation between the atom and the fields, where the nonlinear effect of the parametric amplifier (field–field interaction) in this case is weaker, see Fig. 2B. As in the uncorrelated fields state case the same observation has been noticed when we have considered the intermediate state case by taking h = p/3, where the function reduces its fluctuations values, see Fig. 2C. The function shows rapid fluctuations as soon as we increase the values of the coupling parameters ratio, such that
Fig. 2. The same as in Fig. 1 but for the correlated case.
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k2/k1 = 5. Furthermore it also shows collapses for all values of the time which refers to weak interaction between the fields and the atom resultant of the parametric amplifier term. This of course is in agreement with Eq. (13) where the value of the effective coupling parameter k in this case is reduced to be approximately 0.4, and consequently the nonlinear effect will be apparent, see Fig. 2D. Thus, we may conclude that as a result of the existence of the field–field interaction term in the present system, the value of the atomic occupation probabilities in the correlated case is less than its value in the uncorrelated case. Also the general behavior of the function in the uncorrelated case is similar to that of Jaynes–Cummings model, however, with the observation of the superstructure phenomenon. On the other hand and for the two-mode correlated coherent state case we find the effect of the parametric amplifier is apparent (superstructure phenomenon). Finally, we would like to refer to the drastic change occurring in the function behavior resultant of exchanging the values of the mean photon numbers. This in fact means that during the course of the interaction the correlation between one of the field (second mode) and the atom becomes stronger more than that of the correlation between the atom and the other field (first mode). This is consistent with our application of the canonical transformation to the Hamiltonian model, where one of the rotated mode behaves as a free oscillator (first mode), see Eq. (14). 3. Von Neumann entropy It is well known that under the restriction that it is possible to recover the source with fidelity close to 1, Schumacher noiseless channel coding theorem quantifies the resources required to do quantum data compression. In the case of a source producing orthogonal quantum states |wjæ with probabilities pj this theorem reduces to telling us that the source may be compressed down too but not beyond the classical limit H (pj), where H (pj) represents the information entropy of the variable pj. However, in the more general case of non-orthogonal states being produced by the source, SchumacherÕs theorem refers to how much a quantum source may be compressed, and the answer in this case is definitely not Shannon entropy. Instead a new entropic quantity the von Neumann entropy turns out to be the correct answer. In general the von Neumann entropy agrees with the Shannon entropy if and only if the states |wjæ are orthogonal. Otherwise, the von Neumann entropy for the source pj, |wjæ is in general strictly smaller than the Shannon entropy H (pj). Thus the definition of von Neumann for the entropy is more general than the Shannon entropy. This in fact encouraged us to devote the present section to use von Neumann entropy to consider the degree of entanglement for the present system. Due to mutual entropy the degree of entanglement by means of measuring the dif F A F ference from the disentangled state qA t qt ðqt TrF Et q; qt TrA Et qÞ; F A F I Et q ðqA t ; qt Þ ¼ TrEt qðlog Et q log qt qt Þ
ð26Þ
can be obtained. Here we may point out that in the pure state case this degree of entanglement becomes only twice the von Neumann entropy of the field, i.e.,
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F F I Et q ðqA t ; qt Þ ¼ 2Sðqt Þ. Therefore, we derive an explicit analytical expression for the von Neumann entropy in terms of the final state elements. This in fact does not involve the diagonalization of the final state. More precisely, the von Neumann entropy is obtained as a single integral of a rational function on the real line which is determined by the final state elements. The final state matrix qA t fulfills the characteristic identity [32] 2
A ðqA t Þ ðqt Þ þ gI ¼ 0;
ð27Þ
where g is given by g ¼ det½qA t :
ð28Þ
As a first step we have to analyze the problem under our hand. To do so let us characterize when we consider the derivatives of the von Neumann entropy which kind of equations may appear in this case. To do this we expand the logarithm as " A # 1 X qt k A A A A A Sðqt Þ ¼ Trðqt log qt Þ ¼ Tr½qt logð1 þ qt 1Þ ¼ Tr ; ð29Þ k k¼1 where the factors ðqA t Þk are defined by k
A A ðqA t Þk ¼ qt ð1 qt Þ :
ð30Þ
Using the above equations we can write F þ þ I Et q ðqA t ; qt Þ ¼ 2ðc log c þ c log c Þ;
ð31Þ
where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 1 A 2 2qee ðtÞ 1 þ 4jqA c ¼ ð32Þ eg ðtÞj : 2 2 Eq. (34) will be the basis of the numerical investigation. Interesting features resulting from the different parameters are discussed in the following section.
4. Discussion of the results In this section, we shall analyze and discuss the degree of entanglement due to the von Neumann entropy. This will be done based on the results obtained in the above section. Since we have considered in our numerical computations the uncorrelated and the correlated cases, therefore we shall deal with each case separately. This would enable us to make a comparison between these two different cases. 4.1. Uncorrelated case To discuss the uncorrelated case we have depicted the entropy function in Fig. 3 for different values of the effective parameters. For example, we have taken h = / = 0, and the ratio of the coupling parameters k1/k2 = 0.5, while the fields and the
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atomic frequencies ratio relative to the coupling parameter k1 are xi/k1 = 1, i = 0, 1, 2. In this case and for fixed values of a1 = 2 and a2 = 4.5 we can see from Fig. 3A that after the onset of the interaction the entropy function increases to reach its maximum showing strong entanglement. However, its value decreases after a short period of the time to reach its minimum. The function starts to increase its value again however with rapid fluctuations showing a strong interference between the pattern resultant of the field–field interaction term. This behavior is also observed for the other intervals of the considered period, and we may consider it as the superstructure phenomenon. It is interesting to compare this phenomenon with the superstructure phenomenon which has been mentioned above (see for example [28]). Thus we may conclude that the effect of the field–field interaction in the parametric amplifier form on the atom–field system is similar to that of the Kerr-like medium effect on the same system. On the other hand if we exchange the values of the mean photon numbers such that a1 = 4.5 and a2 = 2 we realize there is a change in the figure shape, for example the entropy function reaches its maximum just after the onset of the interaction showing strong entanglement. Then its value starts to decrease to reach the minimum, however, this occurs after a period longer than in the previous case. In this case we can also see a slight appearance of the fluctuations during the time in which the function decreases to its minimum. This is followed by another increasing period of the function value where we can see rapid fluctuations with interference between the patterns. Further it is also realized that in the second decreasing period the function does not reach its minimum, however, it fluctuates more rapidly than in the first
F Fig. 3. The evolution of the quantum von Neumann entropy I Et q ðqA t ; qt Þ as functions of the scaled time nt. x0 = x1 = x2 = 1, (A) h = 0, / = 0, k2/k1 = 0.1, a1 = 2, a2 = 4.5, (B) h = 0, / = 0, k2/k1 = 0.1, a1 = 4.5, a2 = 2, (C) h = p/3, /=0, k2/k1 = 0.1, a1 = 2, a2 = 4.5, and (D) h = 0, / = 0, k2/k1 = 10, a1 = 2, a2 = 4.5.
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period, see Fig. 3B. This behavior is also observed during the rest of the considered period where more interference between the fluctuations patterns are exhibited. Although the mean photon numbers value have been taken similar to that of the first case, however, when we take the angle h = p/3 (the intermediate state case) the function immediately reduces its value without any change in the general behavior (compare with the pervious case, Fig. 3B) showing weak entanglement, see Fig. 3C. As soon as we increase the values of the coupling parameters ratio, such that k2/k1 = 5, the function shows some revivals in the fluctuations compared to the first case (Fig. 3A), but without essential changes in the figure shape. This may be interpreted as follows: when the values of the ratio between the coupling parameters k1 and k2 are increased then the value of the effective coupling parameter k is decreased and this of course leads to weak effect of the nonlinearity part. This means that the system would lead to entanglement between the field and the atom stronger than that of the first case for some periods of the time, see Fig. 3D. 4.2. Correlated case Since the case of the uncorrelated coherent states case is entirely different from that of the correlated coherent state case, therefore to complete our study we shall consider in this subsection the degree of entanglement for the correlated case. For this purpose we have plotted Fig. 4 taking into consideration the values of all parameters as the same as in the uncorrelated coherent state case. For instance we have depicted the en-
Fig. 4. The same as in Fig. 3, but for the correlated case.
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tropy function in Fig. 4A for k1/k2 = 0.5, and the atomic and fields frequencies are taken such that xi/k1 = 1, i = 0,1,2, while h = / = 0. In this case and for fixed values of a1 = 2 and a2 = 4.5 we can see the entropy function is fluctuating around one showing strong entanglement between the fields and the atom for all periods of the considered time. Although the effect of the nonlinearity is apparent (the superstructure phenomenon), however, the correlation between the fields and the atom in this case is stronger than that of the uncorrelated case. For this reason we can easily realize the entropy function nearly has a fixed value during the considered time. Thus we can say that the effect of the non-linearity on the system in this case is weak, which means the atom interacts strongly with both fields far away from the field–field interaction which is nearly ineffective. When we exchange the values of the mean photon numbers such that a1 = 4.5 and a2 = 2, then the entropy function in this case shows some revivals occurred in its behavior, but without any essential variation in the degree of the entanglement, see Fig. 4B. As should be expected the function for the intermediate state case (h = p/3) reduces its fluctuations values, however, we can see a slight collapses in its behavior compared to the other case, see Figs. 4A and C. Increasing the value of the coupling parameters ratio such that k2/k1 = 5, more collapses can be seen with interference between the fluctuations patterns. Also we have observed an increase in the fluctuations without decreasing in the degree of entanglement. This means that the interaction between the atom and the field is still strong but not enough to avoid the effect of the non-linearity on the system during the course of the interaction (where superstructure phenomenon can be seen), see Fig. 4D. Finally to identify and compare the results presented above for the degree of entanglement with another accepted entanglement measure such as concurrence or fidelity [33]: One, possibly not very surprising, principal observation is that the numerical calculations corresponding to the same parameters, which have been considered in Figs. 3 and 4, give nearly the same behavior. This means that the entanglement due to the von Neumann entropy and other measures are qualitatively the same for the present system. We must emphasis, however, that no single measure alone is enough to quantify the entanglement [34].
5. Phase distribution To complete our work for examining the effect of parametric amplifier on the modified Jaynes–Cummings model, we shall discuss in the present section the properties of the phase distribution. In fact much attention has been paid to the phase dependence of quantum noise in squeezed light after the experimental realization of optical homodyne tomography allowing quantum phase mean values to be calculated from the measured field density matrix. For the phase description there are different techniques to deal with some of them based on a Hermitian quantum phase operator or associated with quasiprobability distribution functions in a phase space [35–37]. On the other hand one can find the approach which is based on operational definition of quantum phase. The phase eigenstates for a single-mode field are defined as
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j#i ¼
1 X
expðin#Þjni;
p 6 # 6 p
281
ð33Þ
n¼0
which in fact represents the eigenstates of the Susskind–Glogower exponential phase operator given by 1=2 b ð1 þ b b aÞ a U ay b
ð34Þ
with eigenvalue exp (in#), i.e., b Uj#i ¼ expði#Þj#i:
ð35Þ
The states |#æ are neither normalizable nor orthogonal, however, it forms a complete set and resolve unity according to Z p 1 d#j#ih#j ¼ I: ð36Þ 2p p Since we are dealing with two-mode system therefore we have to generalize the phase states to include the two-mode states, in this case we have 1 X 1 X j#1 ; #2 i ¼ expðin#1 Þ expðim#2 Þjn; mi; ð37Þ n¼0 m¼0
which are eigenstates of the two-mode phase operator y b 12 ð1 þ b ay b a Þ1=2 b a ð1 þ b b b U bÞ1=2 b b
ð38Þ
with eigenvalues exp (i [#1 + #2]). Thus the phase distribution P (#1, #2) is then defined as P ð#1 ; #2 Þ ¼
1 ð2pÞ
2
2
jh#1 ; #2 jwðtÞij ;
ð39Þ
where the phase distributions P (#1) and P (#2) of the individual modes are given by Z p P ð#1 ; #2 Þ d#j ; ð40Þ P ð#i Þ ¼ p
i, j = 1, 2 and i „ j. Since we discuss the phase distribution for the present system, therefore we shall handle the two different cases; the case in which both fields are in uncorrelated two-mode coherent state, namely the state given by Eq. (20). While the other one is the case in which both fields are in correlated two-mode coherent state, corresponding to the state given by Eq. (22). For this reason we have plotted Figs. 5 and 6, corresponding to the uncorrelated and correlated cases, respectively. 5.1. Uncorrelated case To discuss the uncorrelated two-field coherent state we have plotted in Fig. 5 the phase probability distribution P (#2, t) as a function of the scaled time k1t and #2 taking into consideration the value of the ratio xi/k1 = 1, i = 0, 1, 2, while h = / = 0,
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Fig. 5. The evolution of the phase probability distribution P (q, t). Calculations assume xi/k1 = 1, h = 0, / = 0, (A) k2/k1 = 2, a1 = 4.5, a2 = 2, (B) k2/k1 = 2, a1 = 2, a2 = 4.5, (C) k2/k1 = 3, a1 = 2, a2 = 4.5, and (D) k2/k1 = 4, a1 = 2, a2 = 4.5.
and the ratio between the coupling parameters k2/k1 = 2, with the amplitude values a1 = 4.5 and a2 = 2. Also we have considered the case in which the amplitude values are a1 = 2 and a2 = 4.5. For example at time t = 0 we realize from Fig. 5A the phase distribution P (#2, t) starts with a single-peaked structure at #2 = 0 corresponding to the initial coherent state. Then as the time develops the peak splits into two peaks moving into two opposite directions. However, the amplitudes of the split peaks fluctuate in time giving a tooth-like shape until the two peaks reach the values #2 = ±p at middle of the revival time but in this range the amplitudes of the peaks do not show any fluctuations. The picture changes greatly as time develops further where we find that the two-peak profile breaks up into multi-peak with reduction of the amplitudes of these peaks. Thus the phase distribution shows diffusion as well as bifurcation. This phenomenon can be compared with the effect of Stark shifts and Kerr-like medium, see [29]. However, when we change the amplitudes of the coherent states (case of strong field) the revival time is longer and the fluctuations in the amplitudes in this case are smaller and hence irregularity is not shown as pronounced as in the earlier case. Increasing the value of the coupling parameter ratio such that k2/k1 = 3 leads to
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Fig. 6. The same as in Fig. 1 but for correlated case.
make the diffusion and bifurcation phenomenon sets in at earlier times than in the previous case, (compare Figs. 5A and B with C and D). 5.2. Correlated case Here we consider the case in which both fields are in correlated two-mode coherent state, namely the state given by Eq. (23). In this case we restrict our consideration for the same fixed values of the parameters of the uncorrelated case discussed earlier. It is easy to realize that for the correlated state case the situation is entirely different from that of the uncorrelated case, this can be seen from Fig. 6. For example, when we plotted the phase probability distribution P (#2, t) as a function of the scaled time k1t and #2, using the same value of parameters we find two peaks at ±p/2 with many different smaller peaks in between. As time develop each peak split into two peaks diverging into opposite directions until they reach the borders at #2 = ±p. In their motion outwards the height of the peaks fluctuate showing tooth-like shape in one direction and decaying on the amplitudes in the other direction. By increasing the time of interaction the peaks converge to the initial position of the peak. Increasing time further shows bifurcation of each peak into multi-peaks and disorder in the fluctuations. This disorder sets in faster as we increase the ratio
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k2/k1, this can be realized from the irregular behavior of the peaks. When compared with the uncorrelated case the phase distribution here is far more complicated especially the disorder phenomenon which is pronounced in the correlated case.
6. Conclusion In the present paper, we have introduced a new Hamiltonian model. This model consists of two fields interacting with a two-level atom within perfect cavity. The field–field interaction has been taken into account and considered to be in the parametric amplifier form. The model can be regarded as a generalization of two different systems. In one hand it represents the parametric amplifier model and in other hand it represents the Jaynes–Cummings model. Under certain condition an analytic solution to Schro¨dinger equation has been obtained and employed to discuss the atomic occupation probabilities. The general solution is also applied to the analysis of the entanglement in the context of quantum von Neumann entropy giving an explicit expression for the entanglement degree. The features of the entanglement shows superstructure phenomenon for certain values of parameters of the atom–fields interaction resultant of the parametric amplifier term. Finally, we have considered the behavior of the phase distribution function for the present model. The case of the correlated state shows additional aspects to the uncorrelated case namely the disorder phenomenon which increases by increasing the field–field interaction.
References [1] E.T. Jaynes, F.W. Cummings, Proc. IEEE 51 (1963) 89. [2] J.H. Eberly, N.B. Narozhny, J.J. Sanchez-Mondragan, Phys. Rev. Lett. 44 (1980) 1323; N.B. Narozhny, J.J. Sanchez-Mondragan, J.H. Eberly, Phys. Rev. A 23 (1981) 236. [3] P. Meystre, M.S. Zubairy, Phys. Lett. A 89 (1982) 390; J.R. Kuklinski, J.L. Madajczyk, Phys. Rev. A 37 (1988) 3175. [4] R.G. Short, L. Mandel, Phys. Rev. Lett. 51 (1983) 384; F. Diedrich, H. Walther, Phys. Rev. Lett. 58 (1987) 203. [5] P.W. Milonni, J.R. Ackerhalt, H.W. Galbraith, Phys. Rev. Lett. 50 (1983) 966. [6] J.J. Slosser, P. Meystre, S.L. Braunstein, Phys. Rev. Lett. 63 (1989) 934. [7] J.J. Slosser, P. Meystre, Phys. Rev. A 41 (1990) 3867. [8] J. Gea-Banacloche, Phys. Rev. Lett. 65 (1990) 3385; Phys. Rev. A. 44 (1991) 5913. [9] S.J.D. Phoenix, P.L. Knight, Phys. Rev. A 44 (1991) 6023; Phys. Rev. Lett. 66 (1991) 2833; Ann. Phys. (NY) 186 (1988) 381. [10] S. Haroche, J. Raimond, Adv. Atom. Mol. Phys. 20 (1985) 347. [11] P. Filipowicz, P. Meystre, G. Rempe, H. Walther, Optica Acta 32 (1985) 1105. [12] G. Rempe, F. Schmidt-Kaler, H. Walther, Phys. Rev. Lett. 64 (1990) 2783. [13] B.W. Shore, P.L. Knight, J. Mod. Opt. 40 (1993) 1195. [14] T.R. Gentile, B.J. Hughey, D. Kleppner, Phys. Rev. A 40 (1989) 5103. [15] A. Bandilla, H.-H. Ritze, IEEE J. Quant. Electron. 24 (1988) 1338. [16] M. Tavis, F.W. Cummings, Phys. Rev. 170 (1966) 379.
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[17] R. Weiss, Helv. Phys. Acta 46 (1973) 546; G. Compagno, F. Persico, Phys. Rev. A 15 (1977) 2032. [18] G. Rempe, H. Walther, N. Klein, Phys. Rev. Lett. 58 (1985) 353. [19] P.L. Knight, S. Barnett, Phys. Rev. A 90 (1987) 342; R.R. Puri, G.S. Agarwal, Phys. Rev. A 35 (1987) 3433. [20] Z. Deng, Opt. Commun. 54 (1983) 222; A. Joshi, S.V. Lawande, J. Mod. Opt. 38 (1991) 1407. [21] T.M. Makhviladze, L.A. Shelepin, Phys. Rev. A 9 (1974) 538; N.N. Bogolubov Jr., F. Le Kien, A.S. Shumovski, Phys. Lett. A 101 (1984) 201; A.M. Abdel-Hafez, M.M. Ahamed, A.-S.F. Obada, Phys. Rev. A 34 (1987) 1634. [22] M.S. Abdalla, M.M. Ahamed, A.-S.F. Obada, Phys. A 162 (1990) 215; Phys. A 170 (1991) 393; M.S. Abdalla, M. Abdel-Aty, A.-S.F. Obada, Opt. Commun. 211 (2002) 225; J. Opt. B: Quantum Semiclass. Opt. 4 (2002) 396; M.A. Marchiolli, R.J. Missori, J.A. Roversi, J. Phys. A 36 (2003) 12275; A.-S.F. Obada, M.S. Abdalla, E.M. Khalil, Physica A 336 (2004) 433. [23] A. Joshi, S.V. Lawande, Phys. Rev. A 48 (1993) 2276. [24] J.I. Cirac, R. Zoller, W.D. Phillips, Phys. Rev. A 46 (1992) 2688. [25] C.A. Blockey, D.F. Walls, H. Risken, Europhys. Lett. 17 (1992) 509. [26] A. Rauschenbeutel, P. Bertet, S. Osnaghi, G. Nogues, M. Brune, J.M. Raimond, S. Haroche, Phys. Rev. A 64 (2001) 050301(R). [27] J.M. Raimond, M. Brune, S. Haroche, Rev. Mod. Phys. 73 (2001) 565. [28] M. Abdel-Aty, J. Math. Phys. 44 (2003) 1457 and Virtual J. Quant. Infor. April 2003, Volume 3, Issue 4, see also M. Kozierowski, J. Mod. Optics 48 (2001) 773. [29] P.G. Kwiat, W.A. Vareks, C.K. Hong, H. Nathel, R.Y. Chiao, Phys. Rev. A 41 (1990) 2910. [30] Z.Y. Ou, X.Y. Zou, L.J. Wang, L. Mandel, Phys. Rev. Lett. 65 (1990) 321. [31] X.Y. Zou, L.J. Wang, L. Mandel, Phys. Rev. Lett. 67 (1991) 318. [32] F.A.A. El-Orany, J. Perina, M.S. Abdalla, Opt. Commun. 187 (2001) 199. [33] M.A. Martn-Delgado, Available from: (2002); F. Verstraete, et al., Phys. Rev. Lett. 92 (2004) 087201. [34] E.F. Galvao, M.B. Plenio, S. Virmani, J. Phys. A 33 (2000) 8809; M.S. Kim, et al., Phys. Rev. A 65 (2002) 040101 (R). [35] A.-S.F. Obada, G.M. Abd Al-Kader, J. Mod. Opt. 46 (1999) 263; S.-C. Gou, J. Steinbach, P.L. Knight, Phys. Rev. A 55 (1997) 3719. [36] S.M. Chumakov, K.-E. Hellwig, A.B. Klimov, Int. J. Theo. Phys. 37 (1998) 471. [37] A.-S.F. Obada, A.M. Abdel-Hafz, M. Abdel-Aty, Eur. Phys. J. D 3 (1998) 289.
Von Neumann entropy and phase distribution of two mode parametric amplifier interacting with a single atom M. Sebawe Abdalla
a,* ,
A.-S.F. Obada b, M. Abdel-Aty
c
a
b
Mathematics Department, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia Department of Mathematics, Faculty of Science, Al-Azher University, Nasr City, 11884 Cairo, Egypt c Mathematics Department, Faculty of Science, South Valley University, 82524 Sohag, Egypt Received 30 October 2004; accepted 4 January 2005 Available online 26 February 2005
Abstract In the present article, we introduce a Hamiltonian model that consists of two modes of the field in a perfect cavity to interact with a single two-level atom. The interaction between the fields has been taken into account and considered to be in the parametric amplifier form. The model in one hand can be regarded as a generalization of the Jaynes–Cummings model (JCM), however, in the other hand it can be considered as a generalization of the parametric amplifier model. Under a certain condition the exact solution to the Schro¨dinger equation is obtained. Employing this solution and for chosen values of different parameters we discuss numerically the atomic occupation probabilities as well as the degree of entanglement through the entropy of the field. The system shows superstructure phenomenon similar to that appeared from the effect of the Kerr-like medium on the Jaynes–Cummings model. The von Neumann entropy and phase distribution for both two-mode correlated and uncorrelated coherent states cases are also considered. 2005 Elsevier Inc. All rights reserved.
*
Corresponding author. Fax: +96614676484. E-mail address: [email protected] (M.S. Abdalla).
0003-4916/$ - see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.aop.2005.01.002
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1. Introduction The Jaynes–Cummings model (JCM) represents one of the most fundamental models in the field of quantum optics [1]. This in fact is not only due to the mathematical solvability of the model, but also to its richness with many different interesting phenomena, such as collapse-revival phenomenon [2], squeezing [3], antibunching [4], chaos [5], and trapping states [6–9], etc. The model is the simplest quantum electrodynamic model of an atom–field interaction describing the interaction of an undamped two-level atom with a single non-decaying electromagnetic field mode [10–13]. It was first used to examine the classical aspects of spontaneous emission and to reveal the existence of Rabi oscillations in the atomic excitation probabilities for fields [14]. However, it has been discovered that the JCM atomic population presented direct evidence for the discreteness of photons [15]. At the beginning a limiting number of extensions of the Jaynes–Cummings model have been carefully studied. The coupling of N two-level atoms to a single quantized radiation mode has been solved. Finite algorithms for the eigenfrequencies were given for the case of resonance (detuning D = 0) [16]. The coupling of a single twolevel atom to M equal-frequency modes has been used as a simple model of decay processes [17]. However, as a result of the development of low-temperature high-Q cavities, and the use of Rydberg atoms, the idealized situation of a single two-level atom interacting with a single-mode of quantized radiation field in a lossless cavity has been experimentally realized [18]. These advances in turn have certainly provided incentive for extending and generalizing this model. Thus many authors have been encouraged and stimulated to modify the original model and to generalize it into different directions. For example, it has been generalized and extended to include the effects of cavity mode decay as well as black-body radiation fields [19]. Also it has been extended further for multi-atom [20] and multi-level systems [21], as well as multi-photon and multi-mode [22]. In the meantime effects of spatial mode structure of the cavity and the transient effects arising from a time-dependent atom–field coupling coefficient in JCM evolution have been reported [23]. Furthermore the effects of stochastic phase fluctuations in the atom–field coupling coefficient have also been considered, where the fluctuations are modeled by random telegraph process [24,25]. The aim of the present work is to generalize this model further, however, in a different direction rather than those that have been considered earlier. For example, if we consider a single atom injected within a cavity pumped simultaneously by two different fields, then the interaction will occur between the atom and the fields as well as between the fields themselves. This type of the interaction has been ignored in the literature and therefore one of our goals is to fill this gap and to consider such system, see for example [22]. Therefore, if we assume that the later interaction is of the parametric amplifier form, then one can construct the effective Hamiltonian as follows: ^ ¼H ^A þ H ^ AF þ H ^ FF ; H ^ A is the atomic part where H
ð1Þ
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^ A ¼ 1 ^z H hx 0 r 2
ð2Þ
^ AF is the atom–field interaction and H ^ AF ¼ hk1 ð^ ^ Þ þ ^ Þ; H ay1 þ ^ a1 Þð^ rþ þ r hk2 ð^ ay2 þ ^ a2 Þð^ rþ þ r
ð3Þ
^ FF is the Hamiltonian which represents the field–field interaction in a non-dewhile H generate parametric amplifier form. In terms of creation and annihilation operators the lossless Hamiltonian describing such a system can be written as ^ FF ¼ hx1 ^ H hx 2 ^ hð^ ay1 ^ a1 ^ ay1 ^ a1 þ ay2 ^ a2 þ k ay2 þ ^ a2 Þ:
ð4Þ
In the above formulae xi and ki, i = 1, 2 are the bichromatic field frequencies, and the coupling constants between the atom and each mode, while k is the parametric amplifier constant. x0 is the natural transition frequency of the atom and the operators r+ (r), and rz are the usual raising (lowering) and inversion operators for the atomic system, satisfying [rz, r±] = ±2r± and [r+, r] = rz, while ^ayi ð^ai Þ; ði ¼ 1; 2Þ is the creation (annihilation) operator for the ith quantized bichromatic field satisfying the commutation relations ½^ ai ; ^ ayj ¼ dij ð¼ 1 if i ¼ j and zero otherwiseÞ. Before we ^ FF . During the quango further let us point out the derivation of the Hamiltonian H tization between the atom and the field within cavity it is possible to see an interaction between the fields themselves. In this case to provide coupling among the various cavity modes one may consider the dielectric constant varies with time. Barring accidental degeneracies, we therefore assume that only two nondegenerate modes are coupled by the pump field at frequency x1 + x2 and refer to them as the signal and idler modes. Therefore, if one adjusts the phase pump to be time-de^ FF can be immedipendent such as (x1 + x2) t = / (t) (say) then the Hamiltonian H ately obtained. It is to be noted that in (5) we have kept the counter rotating terms. Here we would like to say that the Hamiltonian (1) has two advantages; one of them it can be regarded as a generalization of the usual JCM which represents the interaction between one mode and a single two-level atom. While the second it can be regarded as a generalization of the field–field interaction in a parametric amplifier form. Thus we have two kinds of operations: atom–field interaction which represents energy exchange between the atom and the fields, while the other one represents some sort of amplification within cavity. Recently, some experiments have been carried out by the authors of [26,27]. Those authors considered the cavity quantum electrodynamics (QED) experiments involving Rydberg atoms crossing superconducting cavities with different frequency regimes. The entanglement in this case resultant from the interaction of a two-level atom with a cavity field mode. In these experiments the authors introduced an important step towards the realization of more complex entanglement in cavity QED by involving in their experiments two independent cavity modes. They have shown that a single atom can be used to entangle the two modes, while a second atom being employed later to reveal this entanglement. This in fact somehow in agreement with our idea of considering a Hamiltonian model consists of a two-level atom in interaction with two modes. However, in addition we have taken into consideration the interaction between the fields in the parametric amplifier form. The main purpose of the present work is to examine the
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degree of entanglement for the above system. This will be achieved for two different cases. First when the fields are considered to be correlated, while the second when we regard both fields are uncorrelated. To reach our goal we have to calculate the wave function in the Schro¨dinger picture. This will be introduced in Section 1, followed by Section 2.1, where we discuss the atomic occupation probabilities for both correlated and uncorrelated cases. In Section 3, we shall consider von Neumann entropy to discuss the entropy of the present system and consequently the degree of entanglement. Our results will be discussed in Section 4 for two different cases; one is the correlated and the other is the uncorrelated case. In Section 5, we shall discuss the phase distribution for both correlated and uncorrelated cases. Finally, our conclusion will be given in Section 6. 2. The wave function The main purpose of the present section is to give the exact expression of the timedependent wave function in Schro¨dinger picture. To reach this goal let us introduce the canonical transformation: y
^ a1 ¼ ^ b1 cosh f ^ b2 sinh f; y
^ b2 cosh f ^ b1 sinh f; a2 ¼ ^
ð5Þ
where
1 2k f ¼ tanh1 ð6Þ 2 x1 þ x2 y ^i ð ^ and the operators b bi Þ; i ¼ 1; 2 have properties similar to that of the operators y ^ ai ð^ ayi Þ; i ¼ 1; 2 such that ½^ ai ; ^ ayj ¼ ½^ bi ; ^ bj ¼ dij . Now if we substitute Eq. (7) into Eq. (1) and after some calculations then the Hamiltonian is transformed to the form 2 h i X y y hx0 ^ ¼ ^þ þ ^ ^ Þ ; ^z þ H h Xi ^ bi r bi r ð7Þ bi ^ b i þ li ð ^ r 2 i¼1 where, i = 1, 2 and: X1 ¼ x1 cosh2 f þ x2 sinh2 f k sinh 2f; X2 ¼ x2 cosh2 f þ x1 sinh2 f k sinh 2f; l1 ¼ k1 cosh f k2 sinh f; l2 ¼ k2 cosh f k1 sinh f:
ð8Þ
It should be noted that to avoid any appearance of non-conservative terms in obtaining Eq. (9) we have applied the rotating wave approximation with respect to the rotated y operators ^ bi and ^ bi , not to the original (physical) operators ^a and ^ay . To continue our progress we have to find the solution of the wave function, in the interaction picture V I ðtÞjwi ¼ ih
o jwi; ot
ð9Þ
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where the time-dependent interaction Hamiltonian V I (t) is given by V I ðtÞ ¼
2 X
y
^þ expðiDj tÞ þ ^ ^ expðiDj tÞ; lj ½^ bj r bj r
ð10Þ
j¼1
where Dj ¼ ðXj x0 Þ; j ¼ 1; 2 is the detuning parameter. For the present system it is not an easy task to obtain an explicit solution of the wave function. However, if we adjust the coupling parameter k such that x1 þ x2 k ¼ k1 k2 2 ; ð11Þ k1 þ k22 then Eq. (9) reduces to the form y y y hx0 ^ ¼ ^z þ hn ^ ^þ þ ^b2 r ^ : H hX 1 ^ r hX2 ^ b2 r b1 ^ b1 þ b2 ^ b2 þ ð12Þ 2 This means that the restrictive condition (13) in this case impliesq the coupling paramffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
eter l1 fi 0 while the other coupling l2 survives and equals n ¼ k22 k21 . Moreover, the restrictive condition means also that the coupling parameter k between the field– field has been chosen to be stronger than the coupling parameter between the field and the atom provided that ki xi, i = 1, 2. However, this is not realistic to be applicable in the experimental work. Therefore to make the restrictive condition consistent with the recent experiments, one may adjust the coupling parameter k to be less than one of the atom–field coupling. In the present case we have to select k < k2, which means k1 (x1 + x2), and consequently the other coupling parameter qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 will be larger than k1 ðx1 þ x2 Þ k21 . This of course would lead to strong interaction between the atom and the second field mode which is in agreement with the experimental observation. y ^1 ¼ ^ It is easy to realize that the operator N b1 ^ b1 in Eq. (14) is a constant of the motion and this allows us to find the explicit solution of the final state matrix of the density operator Et q in the form Et q ¼ jwðtÞihwðtÞj; where jwðtÞi ¼
ð13Þ
1 X 1 X
ut
n1 ¼0 n2 ¼0
vt
ðn1 ;n2 Þ
ðn1 ;n2 Þ
! jn1 ; n2 ib
and the probability amplitudes are given by: ! ! ! ðn ;n Þ ðn ;n Þ 1 ;n2 Þ cos h2 Bt 1 2 Aðn ut 1 2 t ; ¼ ðn ;n Þ ðn ;n 1Þ expði/Þ sin h2 vt 1 2 Bt 1 2 Atðn1 ;n2 1Þ 1 ;n2 Þ Aðn t
1 ;n2 Þ Bðn t
D2 sin gn2 þ1 t ¼ cos gn2 þ1 t i qn1 ;n2 ; 2 gn2 þ1 pffiffiffiffiffiffiffiffiffiffiffiffiffi sin gn þ1 t 2 ¼ in n2 þ 1 qn1 ;n2 ; gn2 þ1
ð14Þ
ð15Þ
ð16Þ
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qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gn2 ¼ D22 þ n2 n2 . The modified Rabi frequency gn2 contains the coupling parameter k which reflects the effect of the parametric amplifier term. This can be noticed from the modification in the detuning parameter D2. It would be interesting to compare this situation with that of the JCM when the system evolves under the influence of the third-order nonlinearity of a Kerr-medium [28]. This means that the parametric amplifier plays a role similar to that of the Kerr-like medium. In fact the term multiplied by the coupling parameter k represents the nonlinear exchange through simultaneous annihilation or creation of a photon in both the subharmonic modes at the expense of pumping. Without doubt nondegenerate parametric amplifiers can perfectly provide single-mode and two-mode squeezing, moreover it has been employed in experiments. For example, the fourth-order interference effects arise when pairs of photons produced in parametric amplifier are injected into Michelson interferometers [29,30]. Also the second-order interference is observed in the superposition of signal photons from two coherently pumped parametric amplifiers when the paths of the idler photons are aligned [31], for more details see [32]. Here we may point out that the initial atomic state is considered to be a combination between the excited state and the ground state, while the field is in a combination of the Fock states of the two modes with amplitude qn1 ;n2 , i.e. ! 1 X 1 X cos h2 jwð0Þi ¼ qn1 ;n2 jn1; n2 ib ; ð17Þ sin h2 ei/ n1 ¼0 n2 ¼0 where h is the angle of coherence and / is the relative phase of the two atomic levels, while |n1, n2æb is the field state of the two modes in the rotated frame. It is obvious from the above equation that the excited state can be obtained if we take h fi 0 while the ground state results when we take h fi p. In our consideration we assume that both fields are in the two-mode uncorrelated coherent state |b1, b2æ = |b1æ |b2æ such that X 1 X 1 1 bn11 bn22 pffiffiffiffiffiffiffiffiffiffiffi jb1 ; b2 ib ¼ exp ½jb1 j2 þ jb2 j2 jn1 ; n2 ib 2 n1 !n2 ! n1 ¼0 n2 ¼0 ¼
1 X 1 X
qn1 ;n2 jn1 ; n2 ib :
ð18Þ
n1 ¼0 n2 ¼0
However, to relate the state of the physical two modes (of the ^a’s) and the rotated ones (of the ^ b’s) we use Eq. (7). Thus, we realize that the vacuum state with respect to ayi Þ; i ¼ 1; 2 is in fact the squeezed vacuum state for the rotated physical operators ^ ai ð^ y operators ^ bi ð^ bi Þ; i ¼ 1; 2, where y y j0; 0ia ¼ exp f^ b1 ^ ð19Þ b2 f ^ b1 ^ b2 j0; 0ib and whence the uncorrelated two-mode coherent states ja1 ; a2 ia ¼ ja1 ia1 ja2 ia2 for the physical operators becomes a combination of correlated displaced Fock states given by
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y y ja1 ; a2 ia ¼ exp f^ b1 ^ b2 f ^ b1 ^ b2 jb1 ; b2 ib ¼ ðcosh nÞ1
1 X ðtanh nÞn jb1 ; n; b2 ; nib ;
ð20Þ
n¼0
where |b1, næb is a displaced Fock state given by rffiffiffiffiffi (X 1 1 n! mn mn 2 2 jb1 ; nib ¼ exp jb1 j b1 Ln ðjb1 j Þjmib 2 m! m>n ) rffiffiffiffiffi 1 X m! X 2 nm nm ðb1 Þ Lm ðjb1 j Þjmib ¼ þ qm ðb1 ; nÞjmib n! m
ð21Þ
and therefore the coefficient of |m1, m2æb in Eq. (22) after substituting (23) in it represents the amplitude for the state in the correlated case. Furthermore the relations between the amplitudes a 0 s and b 0 s is given by: b1 ¼ a1 cosh f þ a2 sinh f; b2 ¼ a2 cosh f þ a1 sinh f:
ð22Þ
The states |m1, m2æa are eigenstates of the operators a^yi ^ai ; ði ¼ 1; 2Þ ^ ayi ^ ai jmi ia ¼ mi jmi ia ;
i ¼ 1; 2
ð23Þ
and similarly for the rotated states y y ^ bj ^ bj jnj ib ¼ nj jnj ib ;
j ¼ 1; 2:
ð24Þ
In the following, we shall employ the results obtained here to consider the atomic occupation probabilities related to the present system namely to examine the effect of the parametric amplifier term on Jaynes–Cummings model. 2.1. Atomic occupation probabilities The phenomenon of collapses and revivals represents one of the most important nonclassical phenomenon in the field of quantum optics. The observation of this phenomenon occurred during the course of interaction between the field and the atom within cavity. In the present section we shall concentrate on studying this phenomenon to see how the field–field interaction in the parametric amplifier form affects this phenomenon. This can be achieved by considering the atomic occupation probabilities. Generally the matrix elements of the reduced atomic density operator are given by the expression 1 X 1 X qA ðtÞ ¼ hn1 ; n2 ; ijwðtÞihwðtÞjn1 ; n2 ; ji; ð25Þ ij n1 ¼0 n2 ¼0
where, i, j = e, g. In fact the atomic occupation probabilities represent another important quantity which is nearly equal and parallel to the atomic inversion Ærz (t)æ. This latter function can be obtained from Eq. (27), and it takes the form
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A hrz ðtÞi ¼ qA ee ðtÞ qgg ðtÞ. It should be noted that the atomic occupation probabilities results are drastically different from the predictions of the semiclassical theory where the atom in the excited state cannot make a transition to the lower level in the absence of a driving field, however, in the fully quantum mechanical treatment, the transition from the upper level to the lower level in the vacuum is possible due to the spontaneous emission. To analyze the effect of the parametric amplifier term on the atomic occupation probabilities for the present system we consider two different cases. One when the fields are correlated while the second case when the fields are uncorrelated using Eq. (20). This will be seen in the next subsections.
2.2. Uncorrelated case Let us start this subsection by discussing the behavior of the atomic occupation probabilities function qA ee ðtÞ (the probability of finding the atom in the excited state). We shall consider the case in which both fields are in uncorrelated two-mode coherent state, namely the state given by Eq. (20). For this reason we have plotted the function qA ee ðtÞ against the scaled time k1t in Fig. 1. In these figures, we have taken the ratio of the coupling parameters
Fig. 1. The evolution of the occupation probability Pe (t) as functions of the scaled time k1t. Calculations assume xi/k1 = 1, (A) h = 0, / = 0, k2/k1 = 2, a1 = 2, a2 = 4.5, (B) h = 0, / = 0, k2/k1 = 2, a1 = 4.5, a2 = 2, (C) h = p/3, / = 0, k2/k1 = 2, a1 = 2, a2 = 4.5, and (D) h = 0, / = 0, k2/k1 = 5, a1 = 2, a2 = 4.5.
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k2/k1 = 2, and the ratio of the frequencies relative to the coupling parameter k1 such that xi/k1 = 1, i = 0, 1, 2, while h = / = 0. This means that the detuning parameter D2 „ 0 and affects the system behavior. In this case and for fixed values of a1 = 2 and a2 = 4.5 we can realize from Fig. 1A that just after the onset of the interaction the function qA ee ðtÞ fluctuates between nearly 0 and 1 (around 0.5) but for short period of the time showing the collapse behavior. This short period of collapse is followed by a long period of revival which indicates a strong correlation occurring between the atom and the fields. However, we can see another period of collapse occurring just half of the value of the considered time, where the maximum and minimum values of the fluctuations in this case occurred between 0.75 and 0.25, respectively. This is followed by a period of revival shorter than the previous one. The collapse phenomenon starts again to appear but with amplitude less than that of the previous one. As soon as we exchange the values of the mean photon numbers such that a1 = 4.5 and a2 = 2 we can see some changes occurring in the function behavior. In this case we observe at t > 0 a short period of collapse immediately after onset of the interaction followed by long period of revival showing a strong interaction between the atom and the field. Subsequent periods of collapses and revivals can be observed, however, with a decrease in the maximum and the minimum values of the fluctuations. We also noted in this case that the interference between the fluctuations pattern during the collapses period are more pronounced than that in the previous case, see Fig. 1B. This in fact indicates the effect of the nonlinearity resultant of the field–field interaction. It is interesting to compare this behavior with superstructure phenomenon which has been realized during the examination of the Jaynes–Cummings model in the presence of Kerr-like medium, see [28]. For further investigation we have considered the case in which the angle h changes to take the value h = p/3 (intermediate state case), in this case we note reduction in the fluctuations value, see Fig. 1C. However, the second period of revival is more pronounced than in the excited state case when a1 = 2 and a2 = 4.5. As the time goes on more decrease in the amplitude of the fluctuations can be observed in both second and third period of collapses as should be expected. In this case, we can say that the population clearly exhibits the characteristic collapses and revivals and provides us with information about discrete nature of the quantized atom-cavity field interaction. Finally, when we increase the values of the coupling parameters ratio such that k1/k2 = 0.2, which means the value of the effective coupling parameter k would be decreased, then we can observe a drastic change occurring in the function behavior. For instance we realize that after the onset of the interaction there is a small period of collapse followed by short period of revival (smaller than all the previous cases). In the meantime we observe an increase in the collapse period which is parallel to decreasing in the revival period (indicates a weak interaction). However, compared to the previous case (strong coupling case) we can observe during the collapse period there is no interference occurring between the fluctuations pattern (but more revivals can be seen). Further the maximum and the minimum values of the fluctuations occur faster than the strong coupling case, see Fig. 1D.
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2.3. Correlated case Here we turn our attention to consider the case in which both fields are in the twomode correlated coherent state taking into account Eq. (22) to represent the field state. For this reason we have plotted the function qA ee ðtÞ against the scaled time k1t in Fig. 2. To make a comparison between the correlated and the uncorrelated cases we have to take the values of the parameters similar to that of the previous case. Therefore, we shall consider the case in which h = / = 0, and the coupling parameters ratio k1/k2 = 0.5, also the fields and the atomic frequencies are taken to be equal, while the ratio between the frequencies and the coupling parameter k1 is xi/k1 = 1, i = 0, 1, 2. In this case and providing the amplitudes a1 = 2 and a2 = 4.5, we find the function reduces its value to be around 0.26. Furthermore, we realize there is no revival period for all values of time, however, the function fluctuates with irregular pattern, see Fig. 2A. In the meantime if we exchange the values of the mean photon numbers such a1 = 4.5 and a2 = 2, we observe no big change in the figure shape, except some decreases in the fluctuations number with a slight revival behavior. Furthermore the function reduces its value to be around 0.22. This in fact indicates an increase in the correlation between the atom and the fields, where the nonlinear effect of the parametric amplifier (field–field interaction) in this case is weaker, see Fig. 2B. As in the uncorrelated fields state case the same observation has been noticed when we have considered the intermediate state case by taking h = p/3, where the function reduces its fluctuations values, see Fig. 2C. The function shows rapid fluctuations as soon as we increase the values of the coupling parameters ratio, such that
Fig. 2. The same as in Fig. 1 but for the correlated case.
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k2/k1 = 5. Furthermore it also shows collapses for all values of the time which refers to weak interaction between the fields and the atom resultant of the parametric amplifier term. This of course is in agreement with Eq. (13) where the value of the effective coupling parameter k in this case is reduced to be approximately 0.4, and consequently the nonlinear effect will be apparent, see Fig. 2D. Thus, we may conclude that as a result of the existence of the field–field interaction term in the present system, the value of the atomic occupation probabilities in the correlated case is less than its value in the uncorrelated case. Also the general behavior of the function in the uncorrelated case is similar to that of Jaynes–Cummings model, however, with the observation of the superstructure phenomenon. On the other hand and for the two-mode correlated coherent state case we find the effect of the parametric amplifier is apparent (superstructure phenomenon). Finally, we would like to refer to the drastic change occurring in the function behavior resultant of exchanging the values of the mean photon numbers. This in fact means that during the course of the interaction the correlation between one of the field (second mode) and the atom becomes stronger more than that of the correlation between the atom and the other field (first mode). This is consistent with our application of the canonical transformation to the Hamiltonian model, where one of the rotated mode behaves as a free oscillator (first mode), see Eq. (14). 3. Von Neumann entropy It is well known that under the restriction that it is possible to recover the source with fidelity close to 1, Schumacher noiseless channel coding theorem quantifies the resources required to do quantum data compression. In the case of a source producing orthogonal quantum states |wjæ with probabilities pj this theorem reduces to telling us that the source may be compressed down too but not beyond the classical limit H (pj), where H (pj) represents the information entropy of the variable pj. However, in the more general case of non-orthogonal states being produced by the source, SchumacherÕs theorem refers to how much a quantum source may be compressed, and the answer in this case is definitely not Shannon entropy. Instead a new entropic quantity the von Neumann entropy turns out to be the correct answer. In general the von Neumann entropy agrees with the Shannon entropy if and only if the states |wjæ are orthogonal. Otherwise, the von Neumann entropy for the source pj, |wjæ is in general strictly smaller than the Shannon entropy H (pj). Thus the definition of von Neumann for the entropy is more general than the Shannon entropy. This in fact encouraged us to devote the present section to use von Neumann entropy to consider the degree of entanglement for the present system. Due to mutual entropy the degree of entanglement by means of measuring the dif F A F ference from the disentangled state qA t qt ðqt TrF Et q; qt TrA Et qÞ; F A F I Et q ðqA t ; qt Þ ¼ TrEt qðlog Et q log qt qt Þ
ð26Þ
can be obtained. Here we may point out that in the pure state case this degree of entanglement becomes only twice the von Neumann entropy of the field, i.e.,
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F F I Et q ðqA t ; qt Þ ¼ 2Sðqt Þ. Therefore, we derive an explicit analytical expression for the von Neumann entropy in terms of the final state elements. This in fact does not involve the diagonalization of the final state. More precisely, the von Neumann entropy is obtained as a single integral of a rational function on the real line which is determined by the final state elements. The final state matrix qA t fulfills the characteristic identity [32] 2
A ðqA t Þ ðqt Þ þ gI ¼ 0;
ð27Þ
where g is given by g ¼ det½qA t :
ð28Þ
As a first step we have to analyze the problem under our hand. To do so let us characterize when we consider the derivatives of the von Neumann entropy which kind of equations may appear in this case. To do this we expand the logarithm as " A # 1 X qt k A A A A A Sðqt Þ ¼ Trðqt log qt Þ ¼ Tr½qt logð1 þ qt 1Þ ¼ Tr ; ð29Þ k k¼1 where the factors ðqA t Þk are defined by k
A A ðqA t Þk ¼ qt ð1 qt Þ :
ð30Þ
Using the above equations we can write F þ þ I Et q ðqA t ; qt Þ ¼ 2ðc log c þ c log c Þ;
ð31Þ
where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 1 A 2 2qee ðtÞ 1 þ 4jqA c ¼ ð32Þ eg ðtÞj : 2 2 Eq. (34) will be the basis of the numerical investigation. Interesting features resulting from the different parameters are discussed in the following section.
4. Discussion of the results In this section, we shall analyze and discuss the degree of entanglement due to the von Neumann entropy. This will be done based on the results obtained in the above section. Since we have considered in our numerical computations the uncorrelated and the correlated cases, therefore we shall deal with each case separately. This would enable us to make a comparison between these two different cases. 4.1. Uncorrelated case To discuss the uncorrelated case we have depicted the entropy function in Fig. 3 for different values of the effective parameters. For example, we have taken h = / = 0, and the ratio of the coupling parameters k1/k2 = 0.5, while the fields and the
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atomic frequencies ratio relative to the coupling parameter k1 are xi/k1 = 1, i = 0, 1, 2. In this case and for fixed values of a1 = 2 and a2 = 4.5 we can see from Fig. 3A that after the onset of the interaction the entropy function increases to reach its maximum showing strong entanglement. However, its value decreases after a short period of the time to reach its minimum. The function starts to increase its value again however with rapid fluctuations showing a strong interference between the pattern resultant of the field–field interaction term. This behavior is also observed for the other intervals of the considered period, and we may consider it as the superstructure phenomenon. It is interesting to compare this phenomenon with the superstructure phenomenon which has been mentioned above (see for example [28]). Thus we may conclude that the effect of the field–field interaction in the parametric amplifier form on the atom–field system is similar to that of the Kerr-like medium effect on the same system. On the other hand if we exchange the values of the mean photon numbers such that a1 = 4.5 and a2 = 2 we realize there is a change in the figure shape, for example the entropy function reaches its maximum just after the onset of the interaction showing strong entanglement. Then its value starts to decrease to reach the minimum, however, this occurs after a period longer than in the previous case. In this case we can also see a slight appearance of the fluctuations during the time in which the function decreases to its minimum. This is followed by another increasing period of the function value where we can see rapid fluctuations with interference between the patterns. Further it is also realized that in the second decreasing period the function does not reach its minimum, however, it fluctuates more rapidly than in the first
F Fig. 3. The evolution of the quantum von Neumann entropy I Et q ðqA t ; qt Þ as functions of the scaled time nt. x0 = x1 = x2 = 1, (A) h = 0, / = 0, k2/k1 = 0.1, a1 = 2, a2 = 4.5, (B) h = 0, / = 0, k2/k1 = 0.1, a1 = 4.5, a2 = 2, (C) h = p/3, /=0, k2/k1 = 0.1, a1 = 2, a2 = 4.5, and (D) h = 0, / = 0, k2/k1 = 10, a1 = 2, a2 = 4.5.
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period, see Fig. 3B. This behavior is also observed during the rest of the considered period where more interference between the fluctuations patterns are exhibited. Although the mean photon numbers value have been taken similar to that of the first case, however, when we take the angle h = p/3 (the intermediate state case) the function immediately reduces its value without any change in the general behavior (compare with the pervious case, Fig. 3B) showing weak entanglement, see Fig. 3C. As soon as we increase the values of the coupling parameters ratio, such that k2/k1 = 5, the function shows some revivals in the fluctuations compared to the first case (Fig. 3A), but without essential changes in the figure shape. This may be interpreted as follows: when the values of the ratio between the coupling parameters k1 and k2 are increased then the value of the effective coupling parameter k is decreased and this of course leads to weak effect of the nonlinearity part. This means that the system would lead to entanglement between the field and the atom stronger than that of the first case for some periods of the time, see Fig. 3D. 4.2. Correlated case Since the case of the uncorrelated coherent states case is entirely different from that of the correlated coherent state case, therefore to complete our study we shall consider in this subsection the degree of entanglement for the correlated case. For this purpose we have plotted Fig. 4 taking into consideration the values of all parameters as the same as in the uncorrelated coherent state case. For instance we have depicted the en-
Fig. 4. The same as in Fig. 3, but for the correlated case.
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tropy function in Fig. 4A for k1/k2 = 0.5, and the atomic and fields frequencies are taken such that xi/k1 = 1, i = 0,1,2, while h = / = 0. In this case and for fixed values of a1 = 2 and a2 = 4.5 we can see the entropy function is fluctuating around one showing strong entanglement between the fields and the atom for all periods of the considered time. Although the effect of the nonlinearity is apparent (the superstructure phenomenon), however, the correlation between the fields and the atom in this case is stronger than that of the uncorrelated case. For this reason we can easily realize the entropy function nearly has a fixed value during the considered time. Thus we can say that the effect of the non-linearity on the system in this case is weak, which means the atom interacts strongly with both fields far away from the field–field interaction which is nearly ineffective. When we exchange the values of the mean photon numbers such that a1 = 4.5 and a2 = 2, then the entropy function in this case shows some revivals occurred in its behavior, but without any essential variation in the degree of the entanglement, see Fig. 4B. As should be expected the function for the intermediate state case (h = p/3) reduces its fluctuations values, however, we can see a slight collapses in its behavior compared to the other case, see Figs. 4A and C. Increasing the value of the coupling parameters ratio such that k2/k1 = 5, more collapses can be seen with interference between the fluctuations patterns. Also we have observed an increase in the fluctuations without decreasing in the degree of entanglement. This means that the interaction between the atom and the field is still strong but not enough to avoid the effect of the non-linearity on the system during the course of the interaction (where superstructure phenomenon can be seen), see Fig. 4D. Finally to identify and compare the results presented above for the degree of entanglement with another accepted entanglement measure such as concurrence or fidelity [33]: One, possibly not very surprising, principal observation is that the numerical calculations corresponding to the same parameters, which have been considered in Figs. 3 and 4, give nearly the same behavior. This means that the entanglement due to the von Neumann entropy and other measures are qualitatively the same for the present system. We must emphasis, however, that no single measure alone is enough to quantify the entanglement [34].
5. Phase distribution To complete our work for examining the effect of parametric amplifier on the modified Jaynes–Cummings model, we shall discuss in the present section the properties of the phase distribution. In fact much attention has been paid to the phase dependence of quantum noise in squeezed light after the experimental realization of optical homodyne tomography allowing quantum phase mean values to be calculated from the measured field density matrix. For the phase description there are different techniques to deal with some of them based on a Hermitian quantum phase operator or associated with quasiprobability distribution functions in a phase space [35–37]. On the other hand one can find the approach which is based on operational definition of quantum phase. The phase eigenstates for a single-mode field are defined as
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j#i ¼
1 X
expðin#Þjni;
p 6 # 6 p
281
ð33Þ
n¼0
which in fact represents the eigenstates of the Susskind–Glogower exponential phase operator given by 1=2 b ð1 þ b b aÞ a U ay b
ð34Þ
with eigenvalue exp (in#), i.e., b Uj#i ¼ expði#Þj#i:
ð35Þ
The states |#æ are neither normalizable nor orthogonal, however, it forms a complete set and resolve unity according to Z p 1 d#j#ih#j ¼ I: ð36Þ 2p p Since we are dealing with two-mode system therefore we have to generalize the phase states to include the two-mode states, in this case we have 1 X 1 X j#1 ; #2 i ¼ expðin#1 Þ expðim#2 Þjn; mi; ð37Þ n¼0 m¼0
which are eigenstates of the two-mode phase operator y b 12 ð1 þ b ay b a Þ1=2 b a ð1 þ b b b U bÞ1=2 b b
ð38Þ
with eigenvalues exp (i [#1 + #2]). Thus the phase distribution P (#1, #2) is then defined as P ð#1 ; #2 Þ ¼
1 ð2pÞ
2
2
jh#1 ; #2 jwðtÞij ;
ð39Þ
where the phase distributions P (#1) and P (#2) of the individual modes are given by Z p P ð#1 ; #2 Þ d#j ; ð40Þ P ð#i Þ ¼ p
i, j = 1, 2 and i „ j. Since we discuss the phase distribution for the present system, therefore we shall handle the two different cases; the case in which both fields are in uncorrelated two-mode coherent state, namely the state given by Eq. (20). While the other one is the case in which both fields are in correlated two-mode coherent state, corresponding to the state given by Eq. (22). For this reason we have plotted Figs. 5 and 6, corresponding to the uncorrelated and correlated cases, respectively. 5.1. Uncorrelated case To discuss the uncorrelated two-field coherent state we have plotted in Fig. 5 the phase probability distribution P (#2, t) as a function of the scaled time k1t and #2 taking into consideration the value of the ratio xi/k1 = 1, i = 0, 1, 2, while h = / = 0,
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Fig. 5. The evolution of the phase probability distribution P (q, t). Calculations assume xi/k1 = 1, h = 0, / = 0, (A) k2/k1 = 2, a1 = 4.5, a2 = 2, (B) k2/k1 = 2, a1 = 2, a2 = 4.5, (C) k2/k1 = 3, a1 = 2, a2 = 4.5, and (D) k2/k1 = 4, a1 = 2, a2 = 4.5.
and the ratio between the coupling parameters k2/k1 = 2, with the amplitude values a1 = 4.5 and a2 = 2. Also we have considered the case in which the amplitude values are a1 = 2 and a2 = 4.5. For example at time t = 0 we realize from Fig. 5A the phase distribution P (#2, t) starts with a single-peaked structure at #2 = 0 corresponding to the initial coherent state. Then as the time develops the peak splits into two peaks moving into two opposite directions. However, the amplitudes of the split peaks fluctuate in time giving a tooth-like shape until the two peaks reach the values #2 = ±p at middle of the revival time but in this range the amplitudes of the peaks do not show any fluctuations. The picture changes greatly as time develops further where we find that the two-peak profile breaks up into multi-peak with reduction of the amplitudes of these peaks. Thus the phase distribution shows diffusion as well as bifurcation. This phenomenon can be compared with the effect of Stark shifts and Kerr-like medium, see [29]. However, when we change the amplitudes of the coherent states (case of strong field) the revival time is longer and the fluctuations in the amplitudes in this case are smaller and hence irregularity is not shown as pronounced as in the earlier case. Increasing the value of the coupling parameter ratio such that k2/k1 = 3 leads to
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Fig. 6. The same as in Fig. 1 but for correlated case.
make the diffusion and bifurcation phenomenon sets in at earlier times than in the previous case, (compare Figs. 5A and B with C and D). 5.2. Correlated case Here we consider the case in which both fields are in correlated two-mode coherent state, namely the state given by Eq. (23). In this case we restrict our consideration for the same fixed values of the parameters of the uncorrelated case discussed earlier. It is easy to realize that for the correlated state case the situation is entirely different from that of the uncorrelated case, this can be seen from Fig. 6. For example, when we plotted the phase probability distribution P (#2, t) as a function of the scaled time k1t and #2, using the same value of parameters we find two peaks at ±p/2 with many different smaller peaks in between. As time develop each peak split into two peaks diverging into opposite directions until they reach the borders at #2 = ±p. In their motion outwards the height of the peaks fluctuate showing tooth-like shape in one direction and decaying on the amplitudes in the other direction. By increasing the time of interaction the peaks converge to the initial position of the peak. Increasing time further shows bifurcation of each peak into multi-peaks and disorder in the fluctuations. This disorder sets in faster as we increase the ratio
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k2/k1, this can be realized from the irregular behavior of the peaks. When compared with the uncorrelated case the phase distribution here is far more complicated especially the disorder phenomenon which is pronounced in the correlated case.
6. Conclusion In the present paper, we have introduced a new Hamiltonian model. This model consists of two fields interacting with a two-level atom within perfect cavity. The field–field interaction has been taken into account and considered to be in the parametric amplifier form. The model can be regarded as a generalization of two different systems. In one hand it represents the parametric amplifier model and in other hand it represents the Jaynes–Cummings model. Under certain condition an analytic solution to Schro¨dinger equation has been obtained and employed to discuss the atomic occupation probabilities. The general solution is also applied to the analysis of the entanglement in the context of quantum von Neumann entropy giving an explicit expression for the entanglement degree. The features of the entanglement shows superstructure phenomenon for certain values of parameters of the atom–fields interaction resultant of the parametric amplifier term. Finally, we have considered the behavior of the phase distribution function for the present model. The case of the correlated state shows additional aspects to the uncorrelated case namely the disorder phenomenon which increases by increasing the field–field interaction.
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