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Quantum entropy of isotropic coupled oscillators interacting with a single atom
1 October 2002
Optics Communications 211 (2002) 225–234 www.elsevier.com/locate/optcom
Quantum entropy of isotropic coupled oscillators interacting with a single atom M. Sebawe Abdalla a, Mahmoud Abdel-Aty b,*, A.-S.F. Obada c a
Mathematics Department, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia b Mathematics Department, Faculty of Science, South Valley University, 82524 Sohag, Egypt c Mathematics Department, Faculty of Science, Al-Azher University, Nacer City, Cairo, Egypt Received 7 March 2002; received in revised form 23 July 2002; accepted 23 July 2002
Abstract In this communication we introduce a new model of Hamiltonian which represents the interaction between a twolevel atom and two electromagnetic fields injected simultaneously within a cavity. By using the canonical transformation the model can be regarded as a generalization of the Jaynes–Cummings model (JCM). The relationship between the original and new states has been established, and the exact expression for the wave function in Schr€ odinger picture is obtained. The atomic inversion and the degree of entanglement are considered. We find that the revival time is different from what has been observed in the JCM case. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 32.80.)t; 42.50.Ct; 3.65.Ud; 3.65.Yz Keywords: Two-level system; Entropy; Entanglement
1. Introduction It is well known that the simplest fully quantum-mechanical model involving the interaction of radiation and matter is almost certainly the interaction of one mode of a quantized radiation field with a single charged harmonic oscillator. An extremely simple but still unsolved model problem is posed if the harmonic oscillator is replaced by a two-level atom. However under the so-called ‘ro-
*
Corresponding author. E-mail address: abdelaty@uni-flensburg.de (M. Abdel-Aty).
tating wave approximation’ (RWA), where the terms that resonate with the electromagnetic field are kept; the problem is reduced to an exactly solvable model. This model is known as the Jaynes–Cummings model of a two-level atom interacting with a single mode of a cavity field [1]. Although, almost four decades have passed since its appearance in the scientific community, however it is still at the heart of many fields of interest. Despite its superficial simplicity, the model can under appropriate conditions, display a surprising variety of complex behaviors. For example the non-classical effects such as vacuum-field Rabi splitting, sub-Poissonian statistics, antibunching,
0030-4018/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 2 ) 0 1 8 5 4 - 0
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collapse-revival phenomenon [2], squeezing [3], chaos [4] and trapping states [5], have been exhibited while its complex dynamical evolution and its fully quantum nature have turned it into a laboratory for theorists and the basis of many more elaborate models. The recent remarkable advances in cavity quantum electrodynamics experiments involving single atoms have allowed experimental observations of the main dynamical features of the model [6]. Reviews of this model can be seen from time to time, see for example [7]. The Hamiltonian model that describes such a system is given by H^ x0 ¼ x^ ay a^ þ r^z þ kð^ ay r^ þ a^r^þ Þ; ð1Þ h 2 where x is the field frequency, a^y and a^ are the field creation and annihilation operators satisfying the commutation relation ½^ a; a^y ¼ 1 while k is the coupling constant, r and rz are the Pauli matrices describing the two-level atom. In the last decades a huge number of papers have appeared in the literature considering this model in great details. Most of these papers were concentrated on the statistical behavior as well as the dynamics of the model. However, to meet the experimental realization, there are several attempts to generalize and modify this model. For example, the consideration of multimode and multiphoton instead of single mode and single photon [8], addition of Kerr-like medium, and Stark shift [9] have been performed. We may also refer to the Tavis– Cummings model (TCM) as a generalization to (JCM), where the angular momentum operators replace the Pauli matrices to study multi-level atom instead of two-level atom [10]. On the other hand, one can see the quantized center of mass motion of a single two-level ion, interacting with a single-mode laser light field, in a harmonic potential trap provides a good testing ground for some intriguing phenomena that appear in the cavity quantum electrodynamics [11–19]. It has been shown that such a single-trapped ion and laser-irradiated ion can be modeled by a strongly nonlinear multiquantum Jaynes–Cummings model [12,14]. In this case the quantized center of mass motion of the ion, which is coupled via the laser to the internal (electronic) degree of freedom, plays a role similar to that
played by the cavity light field mode in the conventional Jaynes–Cummings model. Furthermore, the degree of quantum control that can be achieved in a coupled system of internal and vibrational degrees of freedom was demonstrated in experiments on the generation of non-classical states of motion of a single-trapped ion. In order to realize extended quantum registers, the attention has turned to systems of several ions with well-controlled interaction between them. Although the ion trap quantum computation introduced by Cirac and Zoller is a potentially powerful technique for storage and manipulations of quantum information [20], however this scheme is not the only possibility of achieving dynamics which are conditioned on the internal states of different ions. For example a logic gate between two ions can also be realized by using two photon transitions, addressing both of the ions simultaneously and only virtually exciting the vibrational degrees of freedom [21,22]. Also one can see, strong coupling to an optical cavity mode is employed to entangle the internal states of the ions [23]. For two ions, their state-dependent recoil may be used to achieve a splitting of their spatial wave function, resulting in entanglement of position and internal state of the ions, where local laser excitation is then sufficient to generate a conditional evolution [24]. Thus, there is no doubt of a strong relation between Jaynes–Cummings model, ion trap and quantum information. This in fact has encouraged and stimulated us to modify the Jaynes– Cummings model by introducing a new Hamiltonian model. The model we shall introduce is consisting of a single atom interacting with two modes of the field pumped simultaneously within perfect cavity. In the present model we consider the interaction between the two fields is negligible and the system is isotropic, such that the two fields frequencies are equal. In this case the model we shall consider takes the form 2 X H^ x0 ¼x ay1 r^ þ a^1 r^þ Þ a^yi a^i þ r^z þ k1 ð^ h 2 i¼1
ay2 r^ þ a^2 r^þ Þ; þ k2 ð^
ð2Þ
where x is the frequency of both field modes, and x0 is the natural transition frequency of the atom,
M.S. Abdalla et al. / Optics Communications 211 (2002) 225–234
while ki ; i ¼ 1; 2 are the coupling constants between the modes and the atom. Here the r’s are the usual Pauli matrices satisfying ½rz ; r ¼ 2r ;
½rþ ; r ¼ rz ;
ð3Þ
a^yj
and a^i and are the Bose operators for the quantized modes which obey ½^ ai ; a^yj ¼ dij , where dij ¼ 1 if i ¼ j, and zero otherwise. In fact the Hamiltonian (2) can be regarded as a generalization of the Jaynes–Cummings model, this is quite obvious from the comparison between Eqs. (1) and (2) where the effect of the second mode is apparent from the existence of the coupling constant k2 . Our aim of the present paper is to consider the dynamics of the system and to study the atomic inversion and also to study the degree of entanglement for the Hamiltonian model given by Eq. (2). This implies that the quantum entropies should be addressed which in fact are generally difficult to compute. To reach our goal we have to find the exact time-dependent expressions for the dynamical operators a^i and r’s. This can be achieved by solving the Heisenberg equations of motion, however to deal with the quantum entropies it will be more convenient to use the solution of the wave function in the Schr€ odinger picture. This will be considered in Section 2. Section 3 is devoted to consider the probability amplitude as well the atomic inversion. In Section 4 we shall discuss the quantum entropy, followed by our conclusion in Section 5.
We devote the present section to find the explicit expression for the wave function in Schr€ odinger picture. However, to make our presentation clearer we shall adopt the evolution operator method. For this reason let us introduce the following transformation: b^2 ¼ a^2 cosðfÞ þ a^1 sinðfÞ;
and
ð4Þ
where f ¼ tan1
k1 ; k2
then we find the Hamiltonian (2) takes the form H^ x0 ¼ r^z þ x b^y1 b^1 þ b^y2 b^2 þ g b^y2 r^ þ b^2 r^þ ; h 2 ð6Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ^ where g ¼ k1 þ k2 , and the operators bi and b^yj satisfy the commutation relation ½b^i ; b^yj ¼ dij . It is noted that 1 N^1 ¼ b^y1 b^1 and N^ ¼ b^y1 b^1 þ b^y2 b^2 þ r^z ; ð7Þ 2 are constants of motion, this implies that D C^ ¼ g b^y2 r^ þ b^2 r^þ þ r^z ; ð8Þ 2 is also a constant of motion, the quantity D ¼ x0 x represents the detuning parameter. It should be noted that if we define the photon number n^i ¼ a^yi a^i , i ¼ 1; 2 corresponding to the physical (original) operators and N^i ¼ b^yi b^i , i ¼ 1; 2 corresponding to the artificial (rotated) operators then the relation between both photon numbers are 1 n^1 ¼ N^1 cos2 f þ N^2 sin2 f þ ðb^y1 b^2 þ b^1 b^y2 Þ sin 2f; 2 1 n^2 ¼ N^2 cos2 f þ N^1 sin2 f ðb^y1 b^2 þ b^1 b^y2 Þ sin 2f; 2 from which we have n^1 þ n^2 ¼ N^1 þ N^2 : To find the explicit expression for the wave function jwðtÞi we shall employ Eqs. (7) and (8). For this reason let us first write the solution of the wave function in Schr€ odinger picture,
2. The wave function
b^1 ¼ a^1 cosðfÞ a^2 sinðfÞ;
227
ð5Þ
o H^ jwðtÞi ¼ ih jwðtÞi; ot in the form ! iH^ t jwðtÞi ¼ exp jwð0Þi h ¼ expðixN^ tÞ expðiC^tÞjwð0Þi;
ð9Þ
ð10Þ
where jwð0Þi is the initial state of the system. Now if we assume that the atom is initially in a coherent superposition of the excited state jei and ground state jgi (i.e., a coherent excitation), then we can describe the state jh; /i by
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M.S. Abdalla et al. / Optics Communications 211 (2002) 225–234
h h jh; /i ¼ cos jei þ expði/Þ sin jgi; 2 2 ð11Þ where / is the relative phase of the two atomic levels. For the excited state we have to take h ! 0 and for the ground state we take h ! p. Since we have used the canonical transformations equation (4) to split out one of the fields operator, it will be therefore more convenient for us to work with the bases of the new operators. In this case the initial wave function jwð0Þi can be written as
1 X 1 X h jwð0Þi ¼ qn;m cos je; n; mib 2 n¼0 m¼0 h jg; n; mib ; ð12Þ þ expð i/Þ sin 2 where jn; mib are eigenstates of the operators N^1 and N^2 such that N^1 jn; mib ¼ njn; mib and N^2 jn; mib ¼ mjn; mib . Now if we assume that jn1 ; n2 ia are the eigenstates of the operators n^1 and n^2 then the relationship between the states jn1 ; n2 ia and the state jn; mib can be written as 1=2 n1 X n2 X n1 n2 n!m! m2 jn1 ; n2 ia ¼ ðÞ n1 !n2 ! m1 m2 m ¼0 m ¼0 1
2
ðcos fÞ
n2 m2 þm1
ðsin fÞ
n1 m1 þm2
jn; mib ; ð13Þ
where n ¼ m1 þ m2 and m ¼ n1 þ n2 m1 m2 . Furthermore we can introduce the new coherent states namely jb1 ; b2 i, corresponding to the new operators b^i and b^yi . Assuming both fields are in the two-mode uncorrelated coherent state jb1 ; b2 i ¼ jb1 i jb2 i such that 1 jb1 ; b2 i ¼ exp ½jb1 j2 þ jb2 j2 2 1 X 1 X bn1 bm2 pffiffiffiffiffiffiffiffiffi jn; mib n!m! n¼0 m¼0 1 X 1 X ¼ qn;m jn; mib : ð14Þ n¼0 m¼0
Thus if we use Eq. (4) together with Eq. (14) then it will be easy to establish that b1 ¼ a1 cos f a2 sin f b2 ¼ a2 cos f þ a1 sin f;
and
where ai is the eigenvalue of the physical operators a^i ; i ¼ 1; 2 with respect to the usual coherent states jai i. Since we can write the exponential operator expðiC^tÞ as expðiC^tÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2 2 y 6 6 6 ¼6 6 4
sin
b Q sin
ig
ig
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D2 þg2 b ^y b^2 4 2
t
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 D 4
y þg2 b^2 b^2
b^y2
D 4
t
þg2 b^2 b^2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 y D 4
þg2 b^2 b^2
by Q
b^2 7 7 7 7; 7 5 ð15Þ
where
0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 b ¼ cos @t D þ g2 b^2 b^y A Q 2 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 sin t D4 þ g2 b^2 b^y2 D qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i ; 2 D2 2b ^2 b^y þ g 2 4
ð16Þ
therefore, after some calculations we are able to write the wave function in the form 1 X 1 X An;m ðtÞjn; mib jei jwðtÞi ¼ n¼0 m¼0 þ Bn;m ðtÞjn; mib jgi ;
ð17Þ
where An;m ðtÞ and Bn;m ðtÞ are given by sin g t h i/ pffiffiffiffiffiffiffiffiffiffiffiffi m An;m ðtÞ ¼ i sin g m þ 1qn;mþ1 e 2 gm h D sin gm t þ cos cos gm t i qn;m 2 2 gm
1 exp ixt n þ m þ ; ð18Þ 2 h i/ D sin Jm t cos Jm t þ i Bn;m ðtÞ ¼ sin e qn;m 2 2 Jm h sin Jm t pffiffiffiffi i cos g mqn;m1 2 Jm
1 exp ixt n þ m ; ð19Þ 2 and gm and Jm are given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D2 D2 þ g2 ðm þ 1Þ; Jm ¼ þ g2 m : gm ¼ 4 4
ð20Þ
M.S. Abdalla et al. / Optics Communications 211 (2002) 225–234
When the two modes in the b^i bases are uncorraleted then the sum over n can be factored out and we are left with a single sum over m only. In this case the model becomes a modified JCM in the q jmiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b basis ffi with the vacuum Rabi frequency
g ¼ k21 þ k22 , and effective mean photon number N 2 (equals to jb2 j2 initially for the coherent state). From Eqs. (18) and (19) it is easy to show for all values of t P 0 that the probability condition 1 X jAn;m ðtÞj2 þ jBn;m ðtÞj2 ¼ 1; ð21Þ
m;n¼0
holds. Having obtained the explicit form of the wave function, we are therefore in a position to discuss some statistical properties of the system.
3. Atomic inversion We mainly devote the present section to consider the atomic inversion from which the phenomenon of collapse and revival can be observed [2,25], and see how it is affected in the present 2 2 model. The expressions jAn;m ðtÞj and jBn;m ðtÞj represent the probabilities that at time t, the fields have n and m photons present in the rotated modes and the atom is in level jei and jgi, respectively. The probability P ðn; mÞ that there are n and m photons in the fields (in the Ni bases) at time t is therefore obtained by taking the trace over the atomic states, i.e., 2
2
P ðn; m; tÞ ¼ jAn;m ðtÞj þ jBn;m ðtÞj :
ð22Þ
The P ðn; m; 0Þ is the probability that there are m and n photons present in the field at time t ¼ 0 which is given for two coherent states for the fields by h i jb j2n jb2 j2m 2 2 P ðn; m; 0Þ ¼ 1 exp jb1 j þ jb2 j : m!n! ð23Þ As another important quantity one may consider the atomic inversion W ðtÞ which is related to the 2 2 probability amplitudes jAn;m ðtÞj and jBn;m ðtÞj by the expression
W ðtÞ ¼
1 X 1 h X
i jAn;m ðtÞj2 jBn;m ðtÞj2 :
229
ð24Þ
n¼0 m¼0
It should be pointed out here that the atomic inversion result is 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. As it has been mentioned the model is transformed to a decoupled mode (the b1 mode) and an effective mode (the b2 mode). For the b2 mode the interaction becomes a modified JCM with effective mean photon number N 2 . When the initial two modes are coherent states with eigenvalues a1 and a2 the effective mode is a coherent state jb2 i with b2 given by (14). The effective mean photon number can be made at well by justifiable choice of the parameters a1 and a2 with k1 and k2 : Thus N 2 can be taken equal to zero, i.e., vacuum state by taking a2 ¼ k1 a1 =k2 or it can be taken equal to n1 þ n2 (the sum of the mean photon numbers in the initial two modes) by choosing k1 a2 ¼ k2 a1 : Thus when k1 and k2 are fixed, then the whole features of the JCM from the vacuum until the strong mode can be produced by suitable choice of the coherent state amplitude and phases of a1 and a2 . In what follows we shall choose the parameters to show the vacuum, the weak field, and the strong field regimes. Thus if we take a1 ¼ a2 and k1 ¼ k2 in this case b2 is equal to zero and as it may be expected the atomic inversion is oscillatory with period of oscillation gt ¼ p (see Fig. 1(a)). However if we take the ratio of the coupling parameters k1 =k2 ¼ 102 and a1 ¼ a2 ¼ 3, then we find b2 3:03 and the mean photon number in the effective mode is N 2 9:2 which is almost equal to the mean photon in one of the original modes. In this case the atomic inversion shows the collapses p andrevivals phenomenon ffiffiffiffiffiffi [2] with revival time 2p N 2 see Fig. 1. On the other hand the strong field case can be shown when we choose a1 ¼ a2 ¼ 3 and the coupling parameters k1 ¼ k2 . In this case the effective mode would have a mean photon number N 2 ¼ 18 which is equal to the sum of the two mean photon
230
M.S. Abdalla et al. / Optics Communications 211 (2002) 225–234
Fig. 1. The evolution of the atomic inversion as a function of the scaled time gt. Calculations assume that ðh ¼ 0; / ¼ 0Þ and the field in the coherent state and the detuning parameter D has zero value, where: (a) a1 ¼ 3, a2 ¼ 3, k1 =k2 ¼ 1; (b) a1 ¼ a2 ¼ 3, k1 =k2 ¼ 0:01.
Fig. 2. The evolution of the atomic inversion as a function of the scaled time gt. Calculations assume that ðh ¼ 0; / ¼ 0Þ and the field in the coherent state and the detuning parameter D has zero value, where: (a) k1 =k2 ¼ 0:5, a1 ¼ a2 ¼ 3; (b) k1 =k2 ¼ 1, a1 ¼ 3, a2 ¼ 5.
numbers in the physical two fields. The collapses and revivals phenomenon is more clear especially the second collapses as seen in Figs. 1(a) and 2(b) for different values of ai and ki . As soon as we take the detuning effects into consideration it is easy to realize the amplitudes of the oscillations are decreased by increasing the value of the detuning parameter, where the revival tune is elongated to ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
completely in its upper state) whatever the value of the ratio of the coupling parameters.
have the value sR ’ 2p ðD2 =4g2 Þ þ N 2 and the atomic inversion oscillates around a value shifted upwards, see Fig. 3(a). Furthermore if we take the parameter ðD=gÞ large enough then one can see the atom most of the time in its upper state, see Fig. 3(b). However, any change of the detuning parameter leads to changing in the atomic inversion and consequence, increasing the detuning parameter D leads the atomic inversion to approach its maximum value which is W ðtÞ ¼ 1 (i.e., the atom is
4. Degree of entanglement The quantum entanglement has become a subject for intensive study among those interested in the foundations of quantum theory, see for example [26–30]. Since the quantum dynamics described by the Hamiltonian (2) leads to an entanglement between the field and the atom, therefore in the forthcoming part of the present paper we shall use the field entropy as a measure for the degree of entanglement between the fields and the atom of the system under consideration. To achieve this goal let us first define the system entropy as S ¼ Trfq ln qg;
ð25Þ
M.S. Abdalla et al. / Optics Communications 211 (2002) 225–234
qAðFÞ ðtÞ ¼ TrFðAÞ jwðtÞihwðtÞj;
231
ð27Þ
where we have used the subscript A(F) to denote the atom (field), respectively. We should note here that if the atom–field system is initially in a pure state, then at any time t > 0, the entropies of the fields and the atomic subsystems are precisely equal. Thus from Eq. (17) the reduced field density operator takes the form qF ðtÞ ¼ jDðtÞihDðtÞj þ jSðtÞihSðtÞj;
ð28Þ
where jDðtÞi ¼ jSðtÞi ¼
1 X 1 X n¼0 m¼0 1 X 1 X
An;m ðtÞjn; mib ; ð29Þ
Bn;m ðtÞjn; mib :
n¼0 m¼0
In order to derive a calculation formalism of the field entropy, we must obtain the eigenvalues and eigenstates of the reduced field density operator. A general method has been developed to calculate the various field eigenstates in a simple way [5]. By using this method we obtain the eigenvalues and eigenstates of the reduced density operator Fig. 3. The evolution of the atomic inversion as a function of the scaled time gt. Calculations assume that ðh ¼ 0; / ¼ 0Þ and the field in the coherent state with a1 ¼ a2 ¼ 3, and k2 =k1 ¼ 0:5, and different values of the detuning parameter D, where: (a) D=g ¼ 5; (b) D=g ¼ 10.
where q is the density operator for a given quantum system. It should be noted that the Boltzmann’s constant k has been taken equal to unity. If the density matrix q describes a pure state, then S ¼ 0, however if q describes a mixed state, then S 6¼ 0: But according to Araki–Lieb theorem a system (whose entropy is S) consisting of two subsystem (whose entropies are SA and SF ), then these entropies satisfy the inequalities jSA SF j 6 S 6 SA þ SF . Now suppose the fields and the atom are treated as a separate system, then the entropy can be defined through the corresponding reduced density operators by n o SAðFÞ ¼ TrAðFÞ qAðFÞ ln qAðFÞ ; ð26Þ provided we treat both atom and fields separately. The density matrix qAðFÞ ðtÞ in the above equation is given by
k f ðtÞ ¼ hDðtÞ j DðtÞi exp½.jhDðtÞ j SðtÞij ¼ hSðtÞ j SðtÞi exp½ .jhDðtÞ j SðtÞij; 1 jw f ðtÞi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f exp½ði# .Þ=2jDðtÞi 2kf ðtÞ coshð.Þ exp½ ði# .Þ=2jSðtÞig; where . ¼ sinh
1
hDðtÞjDðtÞi hSðtÞjSðtÞi ; 2jhDðtÞjSðtÞij
ð30Þ
ð31Þ
and hDðtÞjSðtÞi ¼ jhDðtÞjSðtÞij expði#Þ. We can express the field entropy SF ðtÞ in terms of the eigenvalue k f ðtÞ of the reduced field density operator þ SF ðtÞ ¼ kþ f ðtÞ lnðkf ðtÞÞ kf ðtÞ lnðkf ðtÞÞ:
ð32Þ
In the case of a disentangled pure joint state SF ðtÞ is zero, and for maximally entangled states it gives ln 2. It does not appear possible to express the sums in Eq. (32) in closed form, but for not too large values of (the mean photon numbers), direct numerical evolutions can be performed. The case
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M.S. Abdalla et al. / Optics Communications 211 (2002) 225–234
Fig. 4. The evolution of the quantum field entropy as a function of the scaled time gt. Calculations assume that ðh ¼ p or 0, / ¼ 0Þ and the field in the coherent state and the detuning parameter D has zero value, where: (a) a1 ¼ 3, a2 ¼ 3, k1 =k2 ¼ 1; (b) a1 ¼ a2 ¼ 3, k1 =k2 ¼ 0:01:
Fig. 5. The evolution of the quantum field entropy as a function of the scaled time gt. Calculations assume that ðh ¼ 0; / ¼ 0Þ and the field in the coherent state and the detuning parameter D has zero value, where: (a) k1 =k2 ¼ 0:5, a1 ¼ a2 ¼ 3; (b) k1 =k2 ¼ 1, a1 ¼ 3, a2 ¼ 5.
of the effective vacuum is quite interesting where in this case the entropy oscillates between zeros and its maximum value (see Fig. 4(a)). In fact the entropy attains the zero value (i.e., disentanglement) when the atom is either in its upper or lower states (i.e., pure state) while strong entanglement occurs when the inversion is equal to zero (coherent atomic state). It is to be noted from the numerical calculations and figures that changing the values of N 2 do not affect the general behavior of the variation of entropy significantly. Although increasing the mean photon number leads to strong entanglement (maximum value of entanglement), however the maximum value of the entanglement also varies and occurs for some short period of time. As it is expected when we take one of the coupling constants very small compared to the other, see Fig. 4. This is quite obvious for the case in which
the coupling parameters ratio is 102 where the effect of one of the coupling parameter is very weak and we have the case of weak field, see Fig. 4(b). However when we increase this ratio for example k2 =k1 ¼ 0:5 we can easy realize the difference between Figs. 4(b) and 5(a) where the fluctuations are decreased, this in fact due to the increase of the effective mean photon number in this case. However we note that the detuning parameter affects the maximum values of the entropy by bringing them down. This can be attributed to the fact that on having large detuning, the atomic system is weakly coupled to the fields, and hence the degree of entanglement is weak (see Fig. 6). Furthermore, the minimum values of the entropy attain larger amounts as the detuning increases. Here we may point out that if we increase the value of the detuning parameter then we find the
M.S. Abdalla et al. / Optics Communications 211 (2002) 225–234
233
parameters play a role of shifting the photon numbers from the uncoupled field to the effective mode. The solution of Schr€ odinger equation when the atom starts in a coherent superposition of the excited and ground states (i.e., a coherent excitation) is obtained. Statistics of the atomic inversion has been considered for different values for the mean photon numbers in the two modes and the coupling constants. The detuning effect is taken into account and the dependence of the atomic inversion is considered. The degree of entanglement has been investigated through the calculation of the entropy of the system. It is found that the entanglement is affected strongly when detuning is taken into account.
Acknowledgements
Fig. 6. The evolution of the quantum field entropy as a function of the scaled time gt. Calculations assume that ðh ¼ 0; / ¼ 0Þ and the field in the coherent state with a1 ¼ a2 ¼ 3, and k2 =k1 ¼ 0:5, and different values of the detuning parameter D, where: (a) D=g ¼ 5, (b) D=g ¼ 10.
maximum value of the field entropy is always lower than the resonant case where the maximum value of the field entropy is always less than ln 2.
5. Conclusion In this article we have considered a linear interaction between two modes and a single two-level atom rather than a nonlinear form which has been earlier considered, see for example [8]. The model has been transformed to a q modified ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi JCM with modified Rabi frequency g ¼ k21 þ k22 and a decoupled mode. We noted that the photon number in the effective model which enters the JCM depends on the photon number of the physical modes, and the coupling constants k1 and k2 . In this case it is to be realized that the coupling
We are grateful to the referees for their critical reviews and for suggesting various improvements in the manuscript. Also, one of us (M.S. Abdalla) is grateful for the financial support from the project Math 1418/19 of the Research center, College of Science, King Saud University.
References [1] E.T. Jaynes, F.W. Cummings, Proc. IEEE 51 (1963) 89. [2] H.J. Eberly, N.B. Narozhny, J.J. Sanchez-Mondragan, Phys. Rev. Lett. 44 (1980) 1323. [3] P. Meystre, M.S. Zubairy, Phys. Lett. A 89 (1982) 390; J.R. Kuklinski, J.L. Madajczyk, Phys. Rev. A 37 (1982) 3175. [4] P.W. Milonni, J.R. Ackerhalt, H.W. Galbraith, Phys. Rev. Lett. 50 (1983) 966; Phys. Rev. Lett. 51 (1983) 1108. [5] J.J. Slosser, P. Meystre, S.L. Braunstein, Phys. Rev. Lett. 63 (1989) 934; J.J. Slosser, P. Meystre, Phys. Rev. A 41 (1990) 3867; J. Gea-Banacloche, Phys. Rev. Lett. 65 (1990) 3385; Phys. Rev. A 44 (1991) 5913; S.J.D. Phoenix, P.L. Knight, Phys. Rev. A 44 (1991) 6023; Phys. Rev. Lett. 66 (1991) 2833; Ann. Phys. (N.Y.) 186 (1988) 381. [6] S. Haroche, J. Raimond, Adv. Atom. Mol. Phys. 20 (1985) 347; P. Filipowicz, P. Meystre, G. Rempe, H. Walther, Opt. Acta 32 (1985) 1105; G. Rempe, F. Schmidt-Kaler, H. Walther, Phys. Rev. Lett. 64 (1990) 2783.
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[7] B.W. Shore, P.L. Knight, J. Mod. Opt. 40 (1993) 1195. [8] M.S. Abdalla, M.M. Ahmed, A.-S.F. Obada, Physica A 162 (1990) 215; Physica A 170 (1991) 393. [9] A. Joshi, R.R. Puri, Phys. Rev. A 45 (1992) 5056; M.F. Fang, X. Liu, Phys. Lett. A 210 (1996) 11. [10] M. Tavis, F.W. Cummings, Phys. Rev. 170 (1968) 379. [11] C.A. Blockley, D.F. Walls, H. Risken, Eur. Phys. Lett. 17 (1992) 509. [12] J.I. Cirac, A.S. Parkins, R. Blatt, P. Zoller, Phys. Rev. Lett. 70 (1993) 556. [13] J.I. Cirac, R. Blatt, A.S. Parkins, P. Zoller, Phys. Rev. Lett. 70 (1993) 762. [14] W. Vogel, R.L. de Matos Filho, Phys. Rev. A 52 (1995) 4214. [15] R.L. de Matos Filho, W. Vogel, Phys. Rev. Lett. 76 (1996) 4520. [16] S. Wallentowitz, W. Vogel, I. Siemers, P.E. Toschek, Phys. Rev. A 54 (1996) 943. [17] D.M. Meekhof, C. Monro, B.E. King, W.M. Itano, D.J. Wineland, Phys. Rev. Lett. 76 (1996) 1796.
[18] C. Monro, D.M. Meekhof, B.E. King, D.J. Wineland, Science 272 (1996) 1131. [19] P.J. Bardroff, C. Leichtle, G. Schrade, W.P. Schleich, Acta Phys. Slov. 46 (1996) 231. [20] J.I. Cirac, P. Zoller, Phys. Rev. Lett. 74 (1995) 4091. [21] K. Molmer, A. Sorensen, Phys. Rev. Lett. 82 (1999) 1835. [22] A. Sorensen, K. Molmer, Phys. Rev. Lett. 82 (1999) 1971. [23] T. Pellizzari, S.A. Gardiner, J.I. Cirac, P. Zoller, Phys. Rev. Lett. 75 (1995) 3788. [24] J.F. Poyatos, J.I. Cirac, P. Zoller, Phys. Rev. Lett. 81 (1998) 1322. [25] H.I. Yoo, J.H. Eberly, Phys. Rep. 118 (1985) 241. [26] M. Horodecki, P. Horodecki, R. Horodecki, Phys. Rev. Lett. 84 (2000) 2014. [27] G. Vedral, J. Mod. Opt. 47 (2000) 355. [28] M. Abdel-Aty, J. Phys. B: At. Mol. Opt. Phys. 28 (2000) 2665; A.-S.F. Obada, M. Abdel-Aty, Acta Phys. Pol. B 31 (2000) 589. [29] V. Vedral, M.B. Plenio, Phys. Rev. A 57 (1998) 1619. [30] C.H. Bennett, D.P. DiVincenzo, J.A. Smolin, W.K. Wootters, W. K. Phys. Rev. A 54 (1996) 3824.
Optics Communications 211 (2002) 225–234 www.elsevier.com/locate/optcom
Quantum entropy of isotropic coupled oscillators interacting with a single atom M. Sebawe Abdalla a, Mahmoud Abdel-Aty b,*, A.-S.F. Obada c a
Mathematics Department, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia b Mathematics Department, Faculty of Science, South Valley University, 82524 Sohag, Egypt c Mathematics Department, Faculty of Science, Al-Azher University, Nacer City, Cairo, Egypt Received 7 March 2002; received in revised form 23 July 2002; accepted 23 July 2002
Abstract In this communication we introduce a new model of Hamiltonian which represents the interaction between a twolevel atom and two electromagnetic fields injected simultaneously within a cavity. By using the canonical transformation the model can be regarded as a generalization of the Jaynes–Cummings model (JCM). The relationship between the original and new states has been established, and the exact expression for the wave function in Schr€ odinger picture is obtained. The atomic inversion and the degree of entanglement are considered. We find that the revival time is different from what has been observed in the JCM case. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 32.80.)t; 42.50.Ct; 3.65.Ud; 3.65.Yz Keywords: Two-level system; Entropy; Entanglement
1. Introduction It is well known that the simplest fully quantum-mechanical model involving the interaction of radiation and matter is almost certainly the interaction of one mode of a quantized radiation field with a single charged harmonic oscillator. An extremely simple but still unsolved model problem is posed if the harmonic oscillator is replaced by a two-level atom. However under the so-called ‘ro-
*
Corresponding author. E-mail address: abdelaty@uni-flensburg.de (M. Abdel-Aty).
tating wave approximation’ (RWA), where the terms that resonate with the electromagnetic field are kept; the problem is reduced to an exactly solvable model. This model is known as the Jaynes–Cummings model of a two-level atom interacting with a single mode of a cavity field [1]. Although, almost four decades have passed since its appearance in the scientific community, however it is still at the heart of many fields of interest. Despite its superficial simplicity, the model can under appropriate conditions, display a surprising variety of complex behaviors. For example the non-classical effects such as vacuum-field Rabi splitting, sub-Poissonian statistics, antibunching,
0030-4018/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 2 ) 0 1 8 5 4 - 0
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collapse-revival phenomenon [2], squeezing [3], chaos [4] and trapping states [5], have been exhibited while its complex dynamical evolution and its fully quantum nature have turned it into a laboratory for theorists and the basis of many more elaborate models. The recent remarkable advances in cavity quantum electrodynamics experiments involving single atoms have allowed experimental observations of the main dynamical features of the model [6]. Reviews of this model can be seen from time to time, see for example [7]. The Hamiltonian model that describes such a system is given by H^ x0 ¼ x^ ay a^ þ r^z þ kð^ ay r^ þ a^r^þ Þ; ð1Þ h 2 where x is the field frequency, a^y and a^ are the field creation and annihilation operators satisfying the commutation relation ½^ a; a^y ¼ 1 while k is the coupling constant, r and rz are the Pauli matrices describing the two-level atom. In the last decades a huge number of papers have appeared in the literature considering this model in great details. Most of these papers were concentrated on the statistical behavior as well as the dynamics of the model. However, to meet the experimental realization, there are several attempts to generalize and modify this model. For example, the consideration of multimode and multiphoton instead of single mode and single photon [8], addition of Kerr-like medium, and Stark shift [9] have been performed. We may also refer to the Tavis– Cummings model (TCM) as a generalization to (JCM), where the angular momentum operators replace the Pauli matrices to study multi-level atom instead of two-level atom [10]. On the other hand, one can see the quantized center of mass motion of a single two-level ion, interacting with a single-mode laser light field, in a harmonic potential trap provides a good testing ground for some intriguing phenomena that appear in the cavity quantum electrodynamics [11–19]. It has been shown that such a single-trapped ion and laser-irradiated ion can be modeled by a strongly nonlinear multiquantum Jaynes–Cummings model [12,14]. In this case the quantized center of mass motion of the ion, which is coupled via the laser to the internal (electronic) degree of freedom, plays a role similar to that
played by the cavity light field mode in the conventional Jaynes–Cummings model. Furthermore, the degree of quantum control that can be achieved in a coupled system of internal and vibrational degrees of freedom was demonstrated in experiments on the generation of non-classical states of motion of a single-trapped ion. In order to realize extended quantum registers, the attention has turned to systems of several ions with well-controlled interaction between them. Although the ion trap quantum computation introduced by Cirac and Zoller is a potentially powerful technique for storage and manipulations of quantum information [20], however this scheme is not the only possibility of achieving dynamics which are conditioned on the internal states of different ions. For example a logic gate between two ions can also be realized by using two photon transitions, addressing both of the ions simultaneously and only virtually exciting the vibrational degrees of freedom [21,22]. Also one can see, strong coupling to an optical cavity mode is employed to entangle the internal states of the ions [23]. For two ions, their state-dependent recoil may be used to achieve a splitting of their spatial wave function, resulting in entanglement of position and internal state of the ions, where local laser excitation is then sufficient to generate a conditional evolution [24]. Thus, there is no doubt of a strong relation between Jaynes–Cummings model, ion trap and quantum information. This in fact has encouraged and stimulated us to modify the Jaynes– Cummings model by introducing a new Hamiltonian model. The model we shall introduce is consisting of a single atom interacting with two modes of the field pumped simultaneously within perfect cavity. In the present model we consider the interaction between the two fields is negligible and the system is isotropic, such that the two fields frequencies are equal. In this case the model we shall consider takes the form 2 X H^ x0 ¼x ay1 r^ þ a^1 r^þ Þ a^yi a^i þ r^z þ k1 ð^ h 2 i¼1
ay2 r^ þ a^2 r^þ Þ; þ k2 ð^
ð2Þ
where x is the frequency of both field modes, and x0 is the natural transition frequency of the atom,
M.S. Abdalla et al. / Optics Communications 211 (2002) 225–234
while ki ; i ¼ 1; 2 are the coupling constants between the modes and the atom. Here the r’s are the usual Pauli matrices satisfying ½rz ; r ¼ 2r ;
½rþ ; r ¼ rz ;
ð3Þ
a^yj
and a^i and are the Bose operators for the quantized modes which obey ½^ ai ; a^yj ¼ dij , where dij ¼ 1 if i ¼ j, and zero otherwise. In fact the Hamiltonian (2) can be regarded as a generalization of the Jaynes–Cummings model, this is quite obvious from the comparison between Eqs. (1) and (2) where the effect of the second mode is apparent from the existence of the coupling constant k2 . Our aim of the present paper is to consider the dynamics of the system and to study the atomic inversion and also to study the degree of entanglement for the Hamiltonian model given by Eq. (2). This implies that the quantum entropies should be addressed which in fact are generally difficult to compute. To reach our goal we have to find the exact time-dependent expressions for the dynamical operators a^i and r’s. This can be achieved by solving the Heisenberg equations of motion, however to deal with the quantum entropies it will be more convenient to use the solution of the wave function in the Schr€ odinger picture. This will be considered in Section 2. Section 3 is devoted to consider the probability amplitude as well the atomic inversion. In Section 4 we shall discuss the quantum entropy, followed by our conclusion in Section 5.
We devote the present section to find the explicit expression for the wave function in Schr€ odinger picture. However, to make our presentation clearer we shall adopt the evolution operator method. For this reason let us introduce the following transformation: b^2 ¼ a^2 cosðfÞ þ a^1 sinðfÞ;
and
ð4Þ
where f ¼ tan1
k1 ; k2
then we find the Hamiltonian (2) takes the form H^ x0 ¼ r^z þ x b^y1 b^1 þ b^y2 b^2 þ g b^y2 r^ þ b^2 r^þ ; h 2 ð6Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ^ where g ¼ k1 þ k2 , and the operators bi and b^yj satisfy the commutation relation ½b^i ; b^yj ¼ dij . It is noted that 1 N^1 ¼ b^y1 b^1 and N^ ¼ b^y1 b^1 þ b^y2 b^2 þ r^z ; ð7Þ 2 are constants of motion, this implies that D C^ ¼ g b^y2 r^ þ b^2 r^þ þ r^z ; ð8Þ 2 is also a constant of motion, the quantity D ¼ x0 x represents the detuning parameter. It should be noted that if we define the photon number n^i ¼ a^yi a^i , i ¼ 1; 2 corresponding to the physical (original) operators and N^i ¼ b^yi b^i , i ¼ 1; 2 corresponding to the artificial (rotated) operators then the relation between both photon numbers are 1 n^1 ¼ N^1 cos2 f þ N^2 sin2 f þ ðb^y1 b^2 þ b^1 b^y2 Þ sin 2f; 2 1 n^2 ¼ N^2 cos2 f þ N^1 sin2 f ðb^y1 b^2 þ b^1 b^y2 Þ sin 2f; 2 from which we have n^1 þ n^2 ¼ N^1 þ N^2 : To find the explicit expression for the wave function jwðtÞi we shall employ Eqs. (7) and (8). For this reason let us first write the solution of the wave function in Schr€ odinger picture,
2. The wave function
b^1 ¼ a^1 cosðfÞ a^2 sinðfÞ;
227
ð5Þ
o H^ jwðtÞi ¼ ih jwðtÞi; ot in the form ! iH^ t jwðtÞi ¼ exp jwð0Þi h ¼ expðixN^ tÞ expðiC^tÞjwð0Þi;
ð9Þ
ð10Þ
where jwð0Þi is the initial state of the system. Now if we assume that the atom is initially in a coherent superposition of the excited state jei and ground state jgi (i.e., a coherent excitation), then we can describe the state jh; /i by
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M.S. Abdalla et al. / Optics Communications 211 (2002) 225–234
h h jh; /i ¼ cos jei þ expði/Þ sin jgi; 2 2 ð11Þ where / is the relative phase of the two atomic levels. For the excited state we have to take h ! 0 and for the ground state we take h ! p. Since we have used the canonical transformations equation (4) to split out one of the fields operator, it will be therefore more convenient for us to work with the bases of the new operators. In this case the initial wave function jwð0Þi can be written as
1 X 1 X h jwð0Þi ¼ qn;m cos je; n; mib 2 n¼0 m¼0 h jg; n; mib ; ð12Þ þ expð i/Þ sin 2 where jn; mib are eigenstates of the operators N^1 and N^2 such that N^1 jn; mib ¼ njn; mib and N^2 jn; mib ¼ mjn; mib . Now if we assume that jn1 ; n2 ia are the eigenstates of the operators n^1 and n^2 then the relationship between the states jn1 ; n2 ia and the state jn; mib can be written as 1=2 n1 X n2 X n1 n2 n!m! m2 jn1 ; n2 ia ¼ ðÞ n1 !n2 ! m1 m2 m ¼0 m ¼0 1
2
ðcos fÞ
n2 m2 þm1
ðsin fÞ
n1 m1 þm2
jn; mib ; ð13Þ
where n ¼ m1 þ m2 and m ¼ n1 þ n2 m1 m2 . Furthermore we can introduce the new coherent states namely jb1 ; b2 i, corresponding to the new operators b^i and b^yi . Assuming both fields are in the two-mode uncorrelated coherent state jb1 ; b2 i ¼ jb1 i jb2 i such that 1 jb1 ; b2 i ¼ exp ½jb1 j2 þ jb2 j2 2 1 X 1 X bn1 bm2 pffiffiffiffiffiffiffiffiffi jn; mib n!m! n¼0 m¼0 1 X 1 X ¼ qn;m jn; mib : ð14Þ n¼0 m¼0
Thus if we use Eq. (4) together with Eq. (14) then it will be easy to establish that b1 ¼ a1 cos f a2 sin f b2 ¼ a2 cos f þ a1 sin f;
and
where ai is the eigenvalue of the physical operators a^i ; i ¼ 1; 2 with respect to the usual coherent states jai i. Since we can write the exponential operator expðiC^tÞ as expðiC^tÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2 2 y 6 6 6 ¼6 6 4
sin
b Q sin
ig
ig
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D2 þg2 b ^y b^2 4 2
t
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 D 4
y þg2 b^2 b^2
b^y2
D 4
t
þg2 b^2 b^2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 y D 4
þg2 b^2 b^2
by Q
b^2 7 7 7 7; 7 5 ð15Þ
where
0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 b ¼ cos @t D þ g2 b^2 b^y A Q 2 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 sin t D4 þ g2 b^2 b^y2 D qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i ; 2 D2 2b ^2 b^y þ g 2 4
ð16Þ
therefore, after some calculations we are able to write the wave function in the form 1 X 1 X An;m ðtÞjn; mib jei jwðtÞi ¼ n¼0 m¼0 þ Bn;m ðtÞjn; mib jgi ;
ð17Þ
where An;m ðtÞ and Bn;m ðtÞ are given by sin g t h i/ pffiffiffiffiffiffiffiffiffiffiffiffi m An;m ðtÞ ¼ i sin g m þ 1qn;mþ1 e 2 gm h D sin gm t þ cos cos gm t i qn;m 2 2 gm
1 exp ixt n þ m þ ; ð18Þ 2 h i/ D sin Jm t cos Jm t þ i Bn;m ðtÞ ¼ sin e qn;m 2 2 Jm h sin Jm t pffiffiffiffi i cos g mqn;m1 2 Jm
1 exp ixt n þ m ; ð19Þ 2 and gm and Jm are given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D2 D2 þ g2 ðm þ 1Þ; Jm ¼ þ g2 m : gm ¼ 4 4
ð20Þ
M.S. Abdalla et al. / Optics Communications 211 (2002) 225–234
When the two modes in the b^i bases are uncorraleted then the sum over n can be factored out and we are left with a single sum over m only. In this case the model becomes a modified JCM in the q jmiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b basis ffi with the vacuum Rabi frequency
g ¼ k21 þ k22 , and effective mean photon number N 2 (equals to jb2 j2 initially for the coherent state). From Eqs. (18) and (19) it is easy to show for all values of t P 0 that the probability condition 1 X jAn;m ðtÞj2 þ jBn;m ðtÞj2 ¼ 1; ð21Þ
m;n¼0
holds. Having obtained the explicit form of the wave function, we are therefore in a position to discuss some statistical properties of the system.
3. Atomic inversion We mainly devote the present section to consider the atomic inversion from which the phenomenon of collapse and revival can be observed [2,25], and see how it is affected in the present 2 2 model. The expressions jAn;m ðtÞj and jBn;m ðtÞj represent the probabilities that at time t, the fields have n and m photons present in the rotated modes and the atom is in level jei and jgi, respectively. The probability P ðn; mÞ that there are n and m photons in the fields (in the Ni bases) at time t is therefore obtained by taking the trace over the atomic states, i.e., 2
2
P ðn; m; tÞ ¼ jAn;m ðtÞj þ jBn;m ðtÞj :
ð22Þ
The P ðn; m; 0Þ is the probability that there are m and n photons present in the field at time t ¼ 0 which is given for two coherent states for the fields by h i jb j2n jb2 j2m 2 2 P ðn; m; 0Þ ¼ 1 exp jb1 j þ jb2 j : m!n! ð23Þ As another important quantity one may consider the atomic inversion W ðtÞ which is related to the 2 2 probability amplitudes jAn;m ðtÞj and jBn;m ðtÞj by the expression
W ðtÞ ¼
1 X 1 h X
i jAn;m ðtÞj2 jBn;m ðtÞj2 :
229
ð24Þ
n¼0 m¼0
It should be pointed out here that the atomic inversion result is 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. As it has been mentioned the model is transformed to a decoupled mode (the b1 mode) and an effective mode (the b2 mode). For the b2 mode the interaction becomes a modified JCM with effective mean photon number N 2 . When the initial two modes are coherent states with eigenvalues a1 and a2 the effective mode is a coherent state jb2 i with b2 given by (14). The effective mean photon number can be made at well by justifiable choice of the parameters a1 and a2 with k1 and k2 : Thus N 2 can be taken equal to zero, i.e., vacuum state by taking a2 ¼ k1 a1 =k2 or it can be taken equal to n1 þ n2 (the sum of the mean photon numbers in the initial two modes) by choosing k1 a2 ¼ k2 a1 : Thus when k1 and k2 are fixed, then the whole features of the JCM from the vacuum until the strong mode can be produced by suitable choice of the coherent state amplitude and phases of a1 and a2 . In what follows we shall choose the parameters to show the vacuum, the weak field, and the strong field regimes. Thus if we take a1 ¼ a2 and k1 ¼ k2 in this case b2 is equal to zero and as it may be expected the atomic inversion is oscillatory with period of oscillation gt ¼ p (see Fig. 1(a)). However if we take the ratio of the coupling parameters k1 =k2 ¼ 102 and a1 ¼ a2 ¼ 3, then we find b2 3:03 and the mean photon number in the effective mode is N 2 9:2 which is almost equal to the mean photon in one of the original modes. In this case the atomic inversion shows the collapses p andrevivals phenomenon ffiffiffiffiffiffi [2] with revival time 2p N 2 see Fig. 1. On the other hand the strong field case can be shown when we choose a1 ¼ a2 ¼ 3 and the coupling parameters k1 ¼ k2 . In this case the effective mode would have a mean photon number N 2 ¼ 18 which is equal to the sum of the two mean photon
230
M.S. Abdalla et al. / Optics Communications 211 (2002) 225–234
Fig. 1. The evolution of the atomic inversion as a function of the scaled time gt. Calculations assume that ðh ¼ 0; / ¼ 0Þ and the field in the coherent state and the detuning parameter D has zero value, where: (a) a1 ¼ 3, a2 ¼ 3, k1 =k2 ¼ 1; (b) a1 ¼ a2 ¼ 3, k1 =k2 ¼ 0:01.
Fig. 2. The evolution of the atomic inversion as a function of the scaled time gt. Calculations assume that ðh ¼ 0; / ¼ 0Þ and the field in the coherent state and the detuning parameter D has zero value, where: (a) k1 =k2 ¼ 0:5, a1 ¼ a2 ¼ 3; (b) k1 =k2 ¼ 1, a1 ¼ 3, a2 ¼ 5.
numbers in the physical two fields. The collapses and revivals phenomenon is more clear especially the second collapses as seen in Figs. 1(a) and 2(b) for different values of ai and ki . As soon as we take the detuning effects into consideration it is easy to realize the amplitudes of the oscillations are decreased by increasing the value of the detuning parameter, where the revival tune is elongated to ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
completely in its upper state) whatever the value of the ratio of the coupling parameters.
have the value sR ’ 2p ðD2 =4g2 Þ þ N 2 and the atomic inversion oscillates around a value shifted upwards, see Fig. 3(a). Furthermore if we take the parameter ðD=gÞ large enough then one can see the atom most of the time in its upper state, see Fig. 3(b). However, any change of the detuning parameter leads to changing in the atomic inversion and consequence, increasing the detuning parameter D leads the atomic inversion to approach its maximum value which is W ðtÞ ¼ 1 (i.e., the atom is
4. Degree of entanglement The quantum entanglement has become a subject for intensive study among those interested in the foundations of quantum theory, see for example [26–30]. Since the quantum dynamics described by the Hamiltonian (2) leads to an entanglement between the field and the atom, therefore in the forthcoming part of the present paper we shall use the field entropy as a measure for the degree of entanglement between the fields and the atom of the system under consideration. To achieve this goal let us first define the system entropy as S ¼ Trfq ln qg;
ð25Þ
M.S. Abdalla et al. / Optics Communications 211 (2002) 225–234
qAðFÞ ðtÞ ¼ TrFðAÞ jwðtÞihwðtÞj;
231
ð27Þ
where we have used the subscript A(F) to denote the atom (field), respectively. We should note here that if the atom–field system is initially in a pure state, then at any time t > 0, the entropies of the fields and the atomic subsystems are precisely equal. Thus from Eq. (17) the reduced field density operator takes the form qF ðtÞ ¼ jDðtÞihDðtÞj þ jSðtÞihSðtÞj;
ð28Þ
where jDðtÞi ¼ jSðtÞi ¼
1 X 1 X n¼0 m¼0 1 X 1 X
An;m ðtÞjn; mib ; ð29Þ
Bn;m ðtÞjn; mib :
n¼0 m¼0
In order to derive a calculation formalism of the field entropy, we must obtain the eigenvalues and eigenstates of the reduced field density operator. A general method has been developed to calculate the various field eigenstates in a simple way [5]. By using this method we obtain the eigenvalues and eigenstates of the reduced density operator Fig. 3. The evolution of the atomic inversion as a function of the scaled time gt. Calculations assume that ðh ¼ 0; / ¼ 0Þ and the field in the coherent state with a1 ¼ a2 ¼ 3, and k2 =k1 ¼ 0:5, and different values of the detuning parameter D, where: (a) D=g ¼ 5; (b) D=g ¼ 10.
where q is the density operator for a given quantum system. It should be noted that the Boltzmann’s constant k has been taken equal to unity. If the density matrix q describes a pure state, then S ¼ 0, however if q describes a mixed state, then S 6¼ 0: But according to Araki–Lieb theorem a system (whose entropy is S) consisting of two subsystem (whose entropies are SA and SF ), then these entropies satisfy the inequalities jSA SF j 6 S 6 SA þ SF . Now suppose the fields and the atom are treated as a separate system, then the entropy can be defined through the corresponding reduced density operators by n o SAðFÞ ¼ TrAðFÞ qAðFÞ ln qAðFÞ ; ð26Þ provided we treat both atom and fields separately. The density matrix qAðFÞ ðtÞ in the above equation is given by
k f ðtÞ ¼ hDðtÞ j DðtÞi exp½.jhDðtÞ j SðtÞij ¼ hSðtÞ j SðtÞi exp½ .jhDðtÞ j SðtÞij; 1 jw f ðtÞi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f exp½ði# .Þ=2jDðtÞi 2kf ðtÞ coshð.Þ exp½ ði# .Þ=2jSðtÞig; where . ¼ sinh
1
hDðtÞjDðtÞi hSðtÞjSðtÞi ; 2jhDðtÞjSðtÞij
ð30Þ
ð31Þ
and hDðtÞjSðtÞi ¼ jhDðtÞjSðtÞij expði#Þ. We can express the field entropy SF ðtÞ in terms of the eigenvalue k f ðtÞ of the reduced field density operator þ SF ðtÞ ¼ kþ f ðtÞ lnðkf ðtÞÞ kf ðtÞ lnðkf ðtÞÞ:
ð32Þ
In the case of a disentangled pure joint state SF ðtÞ is zero, and for maximally entangled states it gives ln 2. It does not appear possible to express the sums in Eq. (32) in closed form, but for not too large values of (the mean photon numbers), direct numerical evolutions can be performed. The case
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Fig. 4. The evolution of the quantum field entropy as a function of the scaled time gt. Calculations assume that ðh ¼ p or 0, / ¼ 0Þ and the field in the coherent state and the detuning parameter D has zero value, where: (a) a1 ¼ 3, a2 ¼ 3, k1 =k2 ¼ 1; (b) a1 ¼ a2 ¼ 3, k1 =k2 ¼ 0:01:
Fig. 5. The evolution of the quantum field entropy as a function of the scaled time gt. Calculations assume that ðh ¼ 0; / ¼ 0Þ and the field in the coherent state and the detuning parameter D has zero value, where: (a) k1 =k2 ¼ 0:5, a1 ¼ a2 ¼ 3; (b) k1 =k2 ¼ 1, a1 ¼ 3, a2 ¼ 5.
of the effective vacuum is quite interesting where in this case the entropy oscillates between zeros and its maximum value (see Fig. 4(a)). In fact the entropy attains the zero value (i.e., disentanglement) when the atom is either in its upper or lower states (i.e., pure state) while strong entanglement occurs when the inversion is equal to zero (coherent atomic state). It is to be noted from the numerical calculations and figures that changing the values of N 2 do not affect the general behavior of the variation of entropy significantly. Although increasing the mean photon number leads to strong entanglement (maximum value of entanglement), however the maximum value of the entanglement also varies and occurs for some short period of time. As it is expected when we take one of the coupling constants very small compared to the other, see Fig. 4. This is quite obvious for the case in which
the coupling parameters ratio is 102 where the effect of one of the coupling parameter is very weak and we have the case of weak field, see Fig. 4(b). However when we increase this ratio for example k2 =k1 ¼ 0:5 we can easy realize the difference between Figs. 4(b) and 5(a) where the fluctuations are decreased, this in fact due to the increase of the effective mean photon number in this case. However we note that the detuning parameter affects the maximum values of the entropy by bringing them down. This can be attributed to the fact that on having large detuning, the atomic system is weakly coupled to the fields, and hence the degree of entanglement is weak (see Fig. 6). Furthermore, the minimum values of the entropy attain larger amounts as the detuning increases. Here we may point out that if we increase the value of the detuning parameter then we find the
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parameters play a role of shifting the photon numbers from the uncoupled field to the effective mode. The solution of Schr€ odinger equation when the atom starts in a coherent superposition of the excited and ground states (i.e., a coherent excitation) is obtained. Statistics of the atomic inversion has been considered for different values for the mean photon numbers in the two modes and the coupling constants. The detuning effect is taken into account and the dependence of the atomic inversion is considered. The degree of entanglement has been investigated through the calculation of the entropy of the system. It is found that the entanglement is affected strongly when detuning is taken into account.
Acknowledgements
Fig. 6. The evolution of the quantum field entropy as a function of the scaled time gt. Calculations assume that ðh ¼ 0; / ¼ 0Þ and the field in the coherent state with a1 ¼ a2 ¼ 3, and k2 =k1 ¼ 0:5, and different values of the detuning parameter D, where: (a) D=g ¼ 5, (b) D=g ¼ 10.
maximum value of the field entropy is always lower than the resonant case where the maximum value of the field entropy is always less than ln 2.
5. Conclusion In this article we have considered a linear interaction between two modes and a single two-level atom rather than a nonlinear form which has been earlier considered, see for example [8]. The model has been transformed to a q modified ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi JCM with modified Rabi frequency g ¼ k21 þ k22 and a decoupled mode. We noted that the photon number in the effective model which enters the JCM depends on the photon number of the physical modes, and the coupling constants k1 and k2 . In this case it is to be realized that the coupling
We are grateful to the referees for their critical reviews and for suggesting various improvements in the manuscript. Also, one of us (M.S. Abdalla) is grateful for the financial support from the project Math 1418/19 of the Research center, College of Science, King Saud University.
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