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Statistical mechanical aspects of a spin-12 entropy system
Chaos, Solitons and Fractals 33 (2007) 1635–1641 www.elsevier.com/locate/chaos
Statistical mechanical aspects of a spin-12 entropy system Isa A. Al-Khayat Mathematics Department, College of Science, Bahrain University, P.O. Box 32038, Bahrain Accepted 6 March 2006
Abstract In this paper we consider some statistical properties of the entropy due to the time development of a two-level system. Analytic expression for the density matrix is obtained to investigate the influence of the mean photon number on the atomic inversion, field and atomic Wehrl entropies. The results show that the general features of both entropies are similar. Ó 2006 Elsevier Ltd. All rights reserved.
1. Introduction The manipulation of simple systems such as spin-1/2 [1] has opened new possibilities regarding not only the investigation of foundations of quantum mechanics, but also applications in quantum information. In such a system, internal degrees of freedom of an atomic ion may be coupled to the electromagnetic field as well as to the motional degrees of freedom of the ion’s center of mass. Under certain circumstances, in which full quantization of the three sub-systems becomes necessary, we have an interesting combination of two bosonic systems coupled to a spin-like system. A possibility is to place the ion inside a high finesse cavity in such a way that the quantized field gets coupled to the atom. The Hamiltonian of the Jaynes–Cummings (JC)-model [2] is the commonly preferred framework for describing the interaction of an atom (or ion or molecule) with the electromagnetic field (or the harmonic oscillation of the center of mass of the atom in an external field). Also, the interaction between a two-level atom and the radiation field is a quantum optical problem that lies at the heart of many problems in laser physics [3]. Quantum entanglement began to be seen not only as a puzzle, but also as a resource to be manipulated for communication, information processing and quantum computing, such as in the investigation of quantum teleportation, dense coding, decoherence in quantum computers and the evaluation of quantum cryptographic schemes [4]. A number of entanglement measures have been discussed in the literature, such as the von Neumann reduced entropy, the relative entropy of entanglement [5], the socalled entanglement of distillation and the entanglement of formation [6]. Several authors proposed physically motivated postulates to characterize entanglement measures [7,8]. These postulates (although they vary from author to author in the details) have in common that they are based on the concepts of the operational formulation of quantum mechanics [9]. The main purpose of this paper is to show how Wehrl entropy approach of two-level systems allows one to know all information about the entanglement between the two subsystems. Our strategy can be stated in very simple terms:
E-mail address: [email protected] 0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.03.040
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starting from the exact solution of the Schro¨dinger equation for the two-level model, obtaining in this way what we call the final state of the system. Here we fully investigate the relation between the field entropy and Wehrl entropy and present numerical evidences of their relations with the collapse-revival phenomena. The organization of the paper is as follows: in Section 2 we introduce our Hamiltonian model and give exact expression for the density matrix. In Section 3 we employ the density matrix to investigate the properties of the atomic inversion. We devote Section 4 to study the field entropy while the Wehrl entropy will be presented in Section 5. Finally we give our conclusion in Section 6.
2. Spin-12 system In the interest of retaining as much clarity as possible, we first recall some well-known facts [2] about the system we wish to treat here, which consists of a two-level system. As usual, to describe this system we use the Palui operators rz and r ±, where 1 rz ¼ ðjeihej jgihgjÞ; rþ ¼ jeihgj and r ¼ jgihej: 2
ð1Þ
The JC Hamiltonian describing the interaction of a spin-1/2 (two-level atom) interacting with a single cavity field mode in the rotating wavw approximation (RWA) is given by 1 þ xA rz þ kð^ay r þ ^arþ Þ ðh ¼ 1Þ: ð2Þ H ¼ xF ^ay ^a þ 2 Here xF is the field frequency and xA is the transition frequency between the excited state jei and ground state jgi of the atom, k is the linear atom-field coupling strength, D = xA xF is the detuning parameter. The field creation (annihilation) operator is a+ (a), they satisfy the commutation relation [a, a+] = 1. Initially we assume that the field start from a coherent state jai and the atom initially in the superposition state, then the initial state of the total system reads as 1 X # # iu jn; ei þ sin e jn; gi : jWð0Þi ¼ bn cos ð3Þ 2 2 n¼0 Here, 0 6 # 6 p, denotes the initial distribution of the two level and 0 6 u 6 2p, is the relative phase between the upper and lower states of a two-level atom. In order to find the final state of the system, we solve the Schro¨dinger equation for the state vector jW(t)i which is given by i
djWðtÞi ¼ H jWðtÞi: dt
ð4Þ
For any time t > 0, the solution of the Schro¨dinger equation under the Hamiltonian (2) can be written as [3] 1 X jWðtÞi ¼ fAðn; tÞjn; ei þ Bðn; tÞjn; gitg:
ð5Þ
n¼0
The coefficients A(n, t) and B(n, t) are given by pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi a sinð#=2Þeiu pffiffiffiffiffiffiffiffiffiffiffi sin½kt n þ 1 ; Aðn; tÞ ¼ Cn cosð#=2Þ cos½kt n þ 1 þ i nþ1 pffiffiffi p ffiffi ffi pffiffiffi cosð#=2Þ n Bðn; tÞ ¼ Cn sinð#=2Þeiu cos½kt n þ i sin½kt n : a
ð6Þ
The expression of the state vector (5) clearly display the entanglement of the interacting system, atom and the field. The time evolution of the subsystems described by the density operators of the atom and the field, respectively qA ðtÞ ¼ trF fjWðtÞihWðtÞjg ¼ EðtÞjeihej þ GðtÞjeihej þ FðtÞjeihgj þ F ðtÞjeihgj and qF ðtÞ ¼ trA fjWðtÞihWðtÞjg ¼
X fAðn; tÞB ðm; tÞ þ Aðn; tÞB ðm; tÞgjnihmj: n;m
ð7Þ
ð8Þ
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Then, the time-dependent coefficients are given by X jAðn; tÞj2 ; EðtÞ ¼ n
GðtÞ ¼
X
jBðn; tÞj2 ;
ð9Þ
n
FðtÞ ¼
X
A ðn; tÞBðn; tÞ:
n
3. Populations inversion For many purposes, the coherent control of the atom-field dynamics plays a crucial role. We consider the atomic inversion from which the phenomenon of collapses and revivals can be observed. The expressions jA(n, t)j2 and jB(n, t)j2 represent the probabilities that at a time t the field has n photons present and the atom is in level jei and jgi, respectively. The probability P(n, t) that there is n photon in the field at time t is therefore obtained by taking the trace over the atomic states, i.e., P ðn; tÞ ¼ jAðn; tÞj2 þ jBðn; tÞj2 ;
ð10Þ
where P(n, 0) is the probability to find n photons in the field at time t = 0 which is given for a coherent state by P(n, 0) = j bnj2. Another important quantity one may consider, is the atomic inversion hrz(t)i which is related to the probability amplitudes jA(n, t)j2 and j B(n, t)j2 by the expression 1 1X ½jAðn; tÞj2 jBðn; tÞj2 : ð11Þ hrz ðtÞi ¼ 2 n¼0 Since the resulting series cannot be analytically summed in a closed form, we will evaluate them numerically. In Fig. 1 we plot the atomic inversion as a function of scaled time kt for the mean photon number n ¼ 20. The dynamical behavior of the atomic inversion in the JC-model with the cavity field initially prepared in the photon number state jni has been found to be purely sinusoidal [3]. Due attention has been paid to the case when the cavity field is prepared in 1
Atomic inversion
0.75 0.5 0.25 0 -0.25 -0.5 -0.75 10
20
a
30
40
50
60
50
60
Scaled time 1
Atomic inversion
0.75 0.5 0.25 0 -0.25 -0.5 -0.75 10
b
20
30
40
Scaled time
Fig. 1. The evolution of the atomic inversion as a function of the scaled time kt. Calculations assume that (- = 0, / = 0) and the field in the coherent state with different values of the initial average photon number n where (a) n ¼ 20 and (b) n ¼ 30.
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the coherent state. In this case the approximate analytical expression for the atomic inversion has been found to show that thepperiodic exchange of the collapses and revivals of the time evolution of the atomic inversion occur at ffiffiffi ktr ¼ 2p n (see Fig. 1a). We know that the collapses are caused by the dephasing of the various terms in the sums in Eq. (11). Thus we can calculate the time in whichpthe revivalspwill ffiffiffiffiffiffiffiffiffiffiffi ffiffiffi occur by estimating the time that neighbor terms nÞ 2p. Fig. 1b is representing different value of the n þ 1 2k in the sums will be in phase again (for n n): T R ð2k mean photon number, where the values of the parameter n is equal to 30, and the other parameters have the same values as in Fig. 1a. One observes that the inversion shows similar behavior but the revival occurs at later times. The spectrum of the Rabi frequencies is nonlinear in n. Let us treat this frequency as a continuous quantity and expand the dispersion pffiffiffiffiffiffiffiffiffiffiffi ð1Þ ð2Þ ðrÞ curve ln around the point n. Let us write n þ 1 ¼ ln , and ln ¼ ln þ ln ðn nÞ þ ln ðn nÞ2 þ . . . ; ln ¼ k k ð1=k!Þd ln =dn jn¼n . The first term of the ln expansion is responsible for rapid oscillations of the model while the remaining terms are responsible for their envelope.
4. Entropy Now we will turn our attention to the effect on the entropy of the mean photon number. The entropy S of a quantum-mechanical system is a measure of how close the system is to a pure state and is defined by [7] S ¼ tr½q ln q; ð12Þ where q is the density operator of the quantum system and the Boltzmann constant is assumed to be unity. In this case, S = 0, for a pure state and S > 0, for a mixed state. In our model, the initial state is prepared in a pure state, so the whole atom-field system remains in a pure state at any time t > 0, and its entropy is always zero. However, due to the entanglement of the atom and the cavity field at t > 0, both the atom and the field are generally in mixed states, although at certain times the field and the atomic subsystems are ‘almost’ in pure states. Since the initial state is a pure state, the entropy of the field Sf equals the entropy of the atom Sa [7]. The entropy Sf or Sa, which is referred to as the entanglement of the total system in quantum information, is used to measure the amount of entanglement between the two subsystems. When Sf = Sa = 0, the system is disentangled or separable and both the field and atomic subsystems are in pure states. The field and atomic entropy Sa = tra{qalnqa} can be written as S a ¼ S f ¼ kþ lnðkþ Þ k lnðk Þ; where k± are the eigenvalues of the atomic reduced field density operator qa qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 k ¼ 1 ðEðtÞ GðtÞÞ2 þ 4jFðtÞj2 ¼ hrz ðtÞi2 þ hrx ðtÞi2 þ hry ðtÞi2 : 2 2 In the case of a disentangled pure joint state Sf(t) is zero, and for maximally entangled states it gives ln2 0.693. Supplemental to the analytical solution presented above for the field entropy, we shall devote the present discussion to analyze the numerical results of the field entropy and entanglement. Here we would like to point out that in order to ensure an excellent accuracy, the behavior of the field entropy function Sf(t) has been determined with great precision, where a resolution of 103 point per unit of scaled time has been employed for regions exhibiting strong fluctuation. In our consideration to the behavior of the field entropy as a function of the scaled time kt. In Fig. 2 we have plotted the field entropy as a function of the scaled time kt the parameters n ¼ 20, t = p or 0, / = 0. It is easy to observe the existence of the usual collapse and revival of Rabi oscillations. It is remarkable to point out that, the first maximum of the field entropy at t > 0 is achieved in the collapse time, while at one-half of the revival time the entropy reaches its local minimum. It is well known that the evolution of two-level atom is governed by the collapse and revivals for both slow and fast oscillations, where the atom and the field are never stop exchanging energy, so that the convergence to a final state does not take place. When we consider larger values of the mean photon number, similar feature to that observed in the atomic inversion is observed also here. Also we can see that the amplitude of the oscillation becomes smaller, in agreement with the previous work [10]. It should be expected that as a result of the longtime evolutions of the field entropy, there are certain amounts of the mean photon number would make the Rabi oscillations of the atom exhibit a superstructure, see Fig. 2b. In this case the degree of entanglement between the field and the atom reduces and almost pure state are reached. 5. Wehrl entropy In the next discussion we turn our attention to interesting non-classical phenomenon emerging as a direct consequence of quantum interference between component states of the field. Namely, we will analyze oscillations in the quasiprobability distribution. This probability distribution is defined as [11,12]
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0.8 0.7
Entropy
0.6 0.5 0.4 0.3 0.2 0.1 10
20
30
40
50
60
40
50
60
Scaled time
a 0.8 0.7
Entropy
0.6 0.5 0.4 0.3 0.2 0.1 10
b
20
30
Scaled time
Fig. 2. The evolution of the function S(t) in a perfect cavity as a function of the scaled time kt. The parameters are the same as Fig. 1.
Qa ðH; U; tÞ ¼
1 hH; Ujqa ðtÞjH; Ui; p
ð13Þ
where qa(t), is the reduced density operator of the atom and qa(0) = j#, uih#, uj and j#, ui represents the atom initially in the superposition state or in general mixed state j#; ui ¼ cosð#=2Þjei þ sinð#=2Þeiu jgi: Also, we assume that the initial atomic coherent state is defined as follows: jH; Ui ¼ cosðH=2Þjei þ sinðH=2ÞeiU jgi: Then we recast Eq. (14) 1 1 Qa ðH; U; tÞ ¼ þ hrz ðtÞi cos H þ ½hrx ðtÞi cos U þ hry ðtÞi sin U sin H ; p 2 where hrx(t)i, hry(t)i and hrz(t)i are given by rffiffiffiffiffiffiffiffiffiffiffi X pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi n 2 sin # cos u cos kt n cos kt n þ 1 þ sin kt n sin kt n þ 1 ; jbn j hrx ðtÞi ¼ 2 nþ1 n hry ðtÞi ¼
rffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi sin # sin u n cos kt n cos kt n þ 1 sin kt n sin kt n þ 1 2 nþ1 n pffiffiffi p ffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi p ffiffi ffi pffiffiffi a n cos2 ð#=2Þ sin kt n cos kt n þ 1 ; þ pffiffiffiffiffiffiffiffiffiffiffi sin2 ð#=2Þ sin kt n þ 1 cos kt n a nþ1 X
ð14Þ
ð15Þ
ð16Þ
jbn j2
2 2 pffiffiffiffiffiffiffiffiffiffiffi X jbn j2 pffiffiffiffiffiffiffiffiffiffiffi n pffiffiffi 2 pffiffiffi a sin kt n þ 1 2 2 2 cos2 kt n hrz ðtÞi ¼ cos ð#=2Þ cos kt n þ 1 2 sin kt n sin ð#=2Þ a nþ1 2 n pffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi sin # sin u a sin 2kt n þ 1 n pffiffiffiffiffiffiffiffiffiffiffi þ sin 2kt n . þ a 2 nþ1
ð17Þ
ð18Þ
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The atomic Wehrl density defined as S q ðtÞ ¼ Qa ðH; U; tÞ ln Qa ðH; U; tÞ:
ð19Þ
We see from (19), the atomic Wehrl entropy is dependent on the superposition parameters H and U. Also, one can see that the atomic Wehrl entropy is a Shannon entropy for the atomic Q-function. Thus it can be defined as Shannon Wehrl entropy. Also, we see from Eq. (15) that if the atom is initially in the excited state then QA(0, U, t) is dependent only on the atomic variables hrz(t)i through the following relation 1 1 þ hrz ðtÞi ; ð20Þ Qa ð0; 0; tÞ ¼ p 2 then the atomic Wehrl entropy connected with the expectation value of the population inversion by the following relation 1 1 1 1 S q ð0; 0; tÞ ¼ þ hrz ðtÞi ln þ hrz ðtÞi : ð21Þ p 2 p 2 We note that distinct patterns for the Wehrl entropy will arise depending on the values of n and initial state parameters, determining the revival/collapse times. In order to appreciate those different situations, we show, in what follows, some plots of the atomic Wehrl entropy for different values of n (see Fig. 3). In fact the atomic Wehrl entropy is a good measure of the strength of the coherent component and it is clearly distinguishes coherent states, in another word the Wehrl entropy measures how close a given state is to the coherent states or how much coherence a given state has. Wehrl entropy is very sensitive to the phase space dynamics (such as, e. g., spreading) of the Qa(H, U, t) representation. It extracts from the Qa(H, U, t) function essential information about the investigated system. The Wehrl entropy cannot be negative. This follows from the fact that 0 6 QA(H, U, t) 6 0.3 and the QA(H, U, t) function can never be so concentrated as to make Sa(H, U, t) negative. On the contrary, classical distributions can be arbitrarily concentrated in phase space and classical entropies can take on negative values. In Fig. 3, we have plotted the time evolution of atomic Wehrl density Sq(t) against time 0 6 kt 6 60, for h = p/2, pffiffiffi and / = g = p/4. It is to noted that the atomic Wehrl entropy evolves a minn ¼ 12p. To see that the atomic Wehrl entropies influenced by increasing the imum value at the revival time tR ¼ 2p k mean photon number, we set n ¼ 30 in Fig. 3b. We see that the revival of the atomic Wehrl entropy occurs at later times as the mean photon number n increased.
0.4
Wehrl entropy
0.35 0.3 0.25 0.2 0.15 0.1 0.05 10
20
30
40
50
60
40
50
60
Scaled time
a 0.4
Wehrl entropy
0.35 0.3 0.25 0.2 0.15 0.1 0.05 10
b
20
30
Scaled time
Fig. 3. The evolution of the function Sq(t) in a perfect cavity as a function of the scaled time kt. The parameters are the same as Fig. 1.
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6. Conclusion We have established the general formalism for the field entropy and atomic Wehrl entropy in the two-level atom. In particular, we have explored the influence of the various values of the mean photon number on the atomic inversion, the field entropy, entanglement and the evolution of the Wehrl entropy. An idealized situation when the cavity losses are negligible is considered here; however, in the case of real experiment the losses must be introduced. It can be expected that for a non-ideal but high-quality cavity our results are of relevance in the case the Hamiltonian is appropriate for the experimental setup. We considered atomic inversion and found that the phenomenon of periodic collapse and revival occurs; it is however a short-lived phenomenon due to the effects of large values of the mean photon numbers. It is found that the Wehrl entropy can be used as an entanglement measure of the coupled system [13].
Acknowledgements I am indebted to S.S. Hassan and M. Abdel-Aty for critical observations, which in turn has led to the investigation above.
References [1] Wineland DJ, Monroe C, Itano WM, Leibfried D, King BE, Meekhof DM. NIST J Res 1998;103:259. [2] Jaynes ET, Cummings FW. Proc IEE 1963;51:89. [3] Sergent III M, Schully MO, Lamb WE. Laser physics. Reading, MA: Addison-Wesley; 1974; Allen L, Eberly JH. Optical resonance and the two level atom. New York: Wiley; 1975. [4] DiVincenzo DP. Science 1995;270:255; Bennett CH, Brassard G, Crepeau C, Jozsa R, Peres A, Wootters WK. Phys Rev Lett 1993;70:1895; Albeverio S, Fei SM. Phys Lett A 2000;276:811; Bennett CH, Wiesner SJ. Phys Rev Lett 1992;69:2881. [5] Plenio MB, Vedral V. Phys Rev A 1998;57:1619. [6] Bennett CH, DiVincenzo DP, Smolin JA, Wootters WK. Phys Rev A 1997;54:3824. [7] Phoenix SJD, Knight PL. Phys Rev A 1991;44:6023; Phoenix SJD, Knight PL. Phys Rev Lett 1991;66:2833; Phoenix SJD, Knight PL. Ann Phys (New York) 1988;186:381. [8] Vedral V, Plenio MB, Rippin MA, Knight PL. Phys Rev Lett 1997;78:2275; Horodecki M, Horodecki P, Horodecki R. Phys Rev Lett 2000;84:2014. [9] Kraus K. States, elects and operations. Berlin: Springer; 1983. [10] Abdel-Aty M. J Math Phys 2003;44:1457; Abdalla MS, Abdel-Aty M, Obada A-SF. Physica A 2003;326:203. [11] Wehrl A. Rev Mod Phys 1978;50:221; Wehrl A. Rep Math Phys 1979;16:353. [12] Miranowicz A, Bajer J, Wahiddin MRB, Imoto N. J Phys A: Math Gen 2001;34:388; Miranowicz A, Matsueda H, Wahiddin MRB. J Phys A: Math Gen 2000;33:5159. [13] Obada A-SF, Abdel-Khalek S. J Phys A: Math Gen 2004;37:6573.
Statistical mechanical aspects of a spin-12 entropy system Isa A. Al-Khayat Mathematics Department, College of Science, Bahrain University, P.O. Box 32038, Bahrain Accepted 6 March 2006
Abstract In this paper we consider some statistical properties of the entropy due to the time development of a two-level system. Analytic expression for the density matrix is obtained to investigate the influence of the mean photon number on the atomic inversion, field and atomic Wehrl entropies. The results show that the general features of both entropies are similar. Ó 2006 Elsevier Ltd. All rights reserved.
1. Introduction The manipulation of simple systems such as spin-1/2 [1] has opened new possibilities regarding not only the investigation of foundations of quantum mechanics, but also applications in quantum information. In such a system, internal degrees of freedom of an atomic ion may be coupled to the electromagnetic field as well as to the motional degrees of freedom of the ion’s center of mass. Under certain circumstances, in which full quantization of the three sub-systems becomes necessary, we have an interesting combination of two bosonic systems coupled to a spin-like system. A possibility is to place the ion inside a high finesse cavity in such a way that the quantized field gets coupled to the atom. The Hamiltonian of the Jaynes–Cummings (JC)-model [2] is the commonly preferred framework for describing the interaction of an atom (or ion or molecule) with the electromagnetic field (or the harmonic oscillation of the center of mass of the atom in an external field). Also, the interaction between a two-level atom and the radiation field is a quantum optical problem that lies at the heart of many problems in laser physics [3]. Quantum entanglement began to be seen not only as a puzzle, but also as a resource to be manipulated for communication, information processing and quantum computing, such as in the investigation of quantum teleportation, dense coding, decoherence in quantum computers and the evaluation of quantum cryptographic schemes [4]. A number of entanglement measures have been discussed in the literature, such as the von Neumann reduced entropy, the relative entropy of entanglement [5], the socalled entanglement of distillation and the entanglement of formation [6]. Several authors proposed physically motivated postulates to characterize entanglement measures [7,8]. These postulates (although they vary from author to author in the details) have in common that they are based on the concepts of the operational formulation of quantum mechanics [9]. The main purpose of this paper is to show how Wehrl entropy approach of two-level systems allows one to know all information about the entanglement between the two subsystems. Our strategy can be stated in very simple terms:
E-mail address: [email protected] 0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.03.040
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I.A. Al-Khayat / Chaos, Solitons and Fractals 33 (2007) 1635–1641
starting from the exact solution of the Schro¨dinger equation for the two-level model, obtaining in this way what we call the final state of the system. Here we fully investigate the relation between the field entropy and Wehrl entropy and present numerical evidences of their relations with the collapse-revival phenomena. The organization of the paper is as follows: in Section 2 we introduce our Hamiltonian model and give exact expression for the density matrix. In Section 3 we employ the density matrix to investigate the properties of the atomic inversion. We devote Section 4 to study the field entropy while the Wehrl entropy will be presented in Section 5. Finally we give our conclusion in Section 6.
2. Spin-12 system In the interest of retaining as much clarity as possible, we first recall some well-known facts [2] about the system we wish to treat here, which consists of a two-level system. As usual, to describe this system we use the Palui operators rz and r ±, where 1 rz ¼ ðjeihej jgihgjÞ; rþ ¼ jeihgj and r ¼ jgihej: 2
ð1Þ
The JC Hamiltonian describing the interaction of a spin-1/2 (two-level atom) interacting with a single cavity field mode in the rotating wavw approximation (RWA) is given by 1 þ xA rz þ kð^ay r þ ^arþ Þ ðh ¼ 1Þ: ð2Þ H ¼ xF ^ay ^a þ 2 Here xF is the field frequency and xA is the transition frequency between the excited state jei and ground state jgi of the atom, k is the linear atom-field coupling strength, D = xA xF is the detuning parameter. The field creation (annihilation) operator is a+ (a), they satisfy the commutation relation [a, a+] = 1. Initially we assume that the field start from a coherent state jai and the atom initially in the superposition state, then the initial state of the total system reads as 1 X # # iu jn; ei þ sin e jn; gi : jWð0Þi ¼ bn cos ð3Þ 2 2 n¼0 Here, 0 6 # 6 p, denotes the initial distribution of the two level and 0 6 u 6 2p, is the relative phase between the upper and lower states of a two-level atom. In order to find the final state of the system, we solve the Schro¨dinger equation for the state vector jW(t)i which is given by i
djWðtÞi ¼ H jWðtÞi: dt
ð4Þ
For any time t > 0, the solution of the Schro¨dinger equation under the Hamiltonian (2) can be written as [3] 1 X jWðtÞi ¼ fAðn; tÞjn; ei þ Bðn; tÞjn; gitg:
ð5Þ
n¼0
The coefficients A(n, t) and B(n, t) are given by pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi a sinð#=2Þeiu pffiffiffiffiffiffiffiffiffiffiffi sin½kt n þ 1 ; Aðn; tÞ ¼ Cn cosð#=2Þ cos½kt n þ 1 þ i nþ1 pffiffiffi p ffiffi ffi pffiffiffi cosð#=2Þ n Bðn; tÞ ¼ Cn sinð#=2Þeiu cos½kt n þ i sin½kt n : a
ð6Þ
The expression of the state vector (5) clearly display the entanglement of the interacting system, atom and the field. The time evolution of the subsystems described by the density operators of the atom and the field, respectively qA ðtÞ ¼ trF fjWðtÞihWðtÞjg ¼ EðtÞjeihej þ GðtÞjeihej þ FðtÞjeihgj þ F ðtÞjeihgj and qF ðtÞ ¼ trA fjWðtÞihWðtÞjg ¼
X fAðn; tÞB ðm; tÞ þ Aðn; tÞB ðm; tÞgjnihmj: n;m
ð7Þ
ð8Þ
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Then, the time-dependent coefficients are given by X jAðn; tÞj2 ; EðtÞ ¼ n
GðtÞ ¼
X
jBðn; tÞj2 ;
ð9Þ
n
FðtÞ ¼
X
A ðn; tÞBðn; tÞ:
n
3. Populations inversion For many purposes, the coherent control of the atom-field dynamics plays a crucial role. We consider the atomic inversion from which the phenomenon of collapses and revivals can be observed. The expressions jA(n, t)j2 and jB(n, t)j2 represent the probabilities that at a time t the field has n photons present and the atom is in level jei and jgi, respectively. The probability P(n, t) that there is n photon in the field at time t is therefore obtained by taking the trace over the atomic states, i.e., P ðn; tÞ ¼ jAðn; tÞj2 þ jBðn; tÞj2 ;
ð10Þ
where P(n, 0) is the probability to find n photons in the field at time t = 0 which is given for a coherent state by P(n, 0) = j bnj2. Another important quantity one may consider, is the atomic inversion hrz(t)i which is related to the probability amplitudes jA(n, t)j2 and j B(n, t)j2 by the expression 1 1X ½jAðn; tÞj2 jBðn; tÞj2 : ð11Þ hrz ðtÞi ¼ 2 n¼0 Since the resulting series cannot be analytically summed in a closed form, we will evaluate them numerically. In Fig. 1 we plot the atomic inversion as a function of scaled time kt for the mean photon number n ¼ 20. The dynamical behavior of the atomic inversion in the JC-model with the cavity field initially prepared in the photon number state jni has been found to be purely sinusoidal [3]. Due attention has been paid to the case when the cavity field is prepared in 1
Atomic inversion
0.75 0.5 0.25 0 -0.25 -0.5 -0.75 10
20
a
30
40
50
60
50
60
Scaled time 1
Atomic inversion
0.75 0.5 0.25 0 -0.25 -0.5 -0.75 10
b
20
30
40
Scaled time
Fig. 1. The evolution of the atomic inversion as a function of the scaled time kt. Calculations assume that (- = 0, / = 0) and the field in the coherent state with different values of the initial average photon number n where (a) n ¼ 20 and (b) n ¼ 30.
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the coherent state. In this case the approximate analytical expression for the atomic inversion has been found to show that thepperiodic exchange of the collapses and revivals of the time evolution of the atomic inversion occur at ffiffiffi ktr ¼ 2p n (see Fig. 1a). We know that the collapses are caused by the dephasing of the various terms in the sums in Eq. (11). Thus we can calculate the time in whichpthe revivalspwill ffiffiffiffiffiffiffiffiffiffiffi ffiffiffi occur by estimating the time that neighbor terms nÞ 2p. Fig. 1b is representing different value of the n þ 1 2k in the sums will be in phase again (for n n): T R ð2k mean photon number, where the values of the parameter n is equal to 30, and the other parameters have the same values as in Fig. 1a. One observes that the inversion shows similar behavior but the revival occurs at later times. The spectrum of the Rabi frequencies is nonlinear in n. Let us treat this frequency as a continuous quantity and expand the dispersion pffiffiffiffiffiffiffiffiffiffiffi ð1Þ ð2Þ ðrÞ curve ln around the point n. Let us write n þ 1 ¼ ln , and ln ¼ ln þ ln ðn nÞ þ ln ðn nÞ2 þ . . . ; ln ¼ k k ð1=k!Þd ln =dn jn¼n . The first term of the ln expansion is responsible for rapid oscillations of the model while the remaining terms are responsible for their envelope.
4. Entropy Now we will turn our attention to the effect on the entropy of the mean photon number. The entropy S of a quantum-mechanical system is a measure of how close the system is to a pure state and is defined by [7] S ¼ tr½q ln q; ð12Þ where q is the density operator of the quantum system and the Boltzmann constant is assumed to be unity. In this case, S = 0, for a pure state and S > 0, for a mixed state. In our model, the initial state is prepared in a pure state, so the whole atom-field system remains in a pure state at any time t > 0, and its entropy is always zero. However, due to the entanglement of the atom and the cavity field at t > 0, both the atom and the field are generally in mixed states, although at certain times the field and the atomic subsystems are ‘almost’ in pure states. Since the initial state is a pure state, the entropy of the field Sf equals the entropy of the atom Sa [7]. The entropy Sf or Sa, which is referred to as the entanglement of the total system in quantum information, is used to measure the amount of entanglement between the two subsystems. When Sf = Sa = 0, the system is disentangled or separable and both the field and atomic subsystems are in pure states. The field and atomic entropy Sa = tra{qalnqa} can be written as S a ¼ S f ¼ kþ lnðkþ Þ k lnðk Þ; where k± are the eigenvalues of the atomic reduced field density operator qa qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 k ¼ 1 ðEðtÞ GðtÞÞ2 þ 4jFðtÞj2 ¼ hrz ðtÞi2 þ hrx ðtÞi2 þ hry ðtÞi2 : 2 2 In the case of a disentangled pure joint state Sf(t) is zero, and for maximally entangled states it gives ln2 0.693. Supplemental to the analytical solution presented above for the field entropy, we shall devote the present discussion to analyze the numerical results of the field entropy and entanglement. Here we would like to point out that in order to ensure an excellent accuracy, the behavior of the field entropy function Sf(t) has been determined with great precision, where a resolution of 103 point per unit of scaled time has been employed for regions exhibiting strong fluctuation. In our consideration to the behavior of the field entropy as a function of the scaled time kt. In Fig. 2 we have plotted the field entropy as a function of the scaled time kt the parameters n ¼ 20, t = p or 0, / = 0. It is easy to observe the existence of the usual collapse and revival of Rabi oscillations. It is remarkable to point out that, the first maximum of the field entropy at t > 0 is achieved in the collapse time, while at one-half of the revival time the entropy reaches its local minimum. It is well known that the evolution of two-level atom is governed by the collapse and revivals for both slow and fast oscillations, where the atom and the field are never stop exchanging energy, so that the convergence to a final state does not take place. When we consider larger values of the mean photon number, similar feature to that observed in the atomic inversion is observed also here. Also we can see that the amplitude of the oscillation becomes smaller, in agreement with the previous work [10]. It should be expected that as a result of the longtime evolutions of the field entropy, there are certain amounts of the mean photon number would make the Rabi oscillations of the atom exhibit a superstructure, see Fig. 2b. In this case the degree of entanglement between the field and the atom reduces and almost pure state are reached. 5. Wehrl entropy In the next discussion we turn our attention to interesting non-classical phenomenon emerging as a direct consequence of quantum interference between component states of the field. Namely, we will analyze oscillations in the quasiprobability distribution. This probability distribution is defined as [11,12]
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0.8 0.7
Entropy
0.6 0.5 0.4 0.3 0.2 0.1 10
20
30
40
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a 0.8 0.7
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Fig. 2. The evolution of the function S(t) in a perfect cavity as a function of the scaled time kt. The parameters are the same as Fig. 1.
Qa ðH; U; tÞ ¼
1 hH; Ujqa ðtÞjH; Ui; p
ð13Þ
where qa(t), is the reduced density operator of the atom and qa(0) = j#, uih#, uj and j#, ui represents the atom initially in the superposition state or in general mixed state j#; ui ¼ cosð#=2Þjei þ sinð#=2Þeiu jgi: Also, we assume that the initial atomic coherent state is defined as follows: jH; Ui ¼ cosðH=2Þjei þ sinðH=2ÞeiU jgi: Then we recast Eq. (14) 1 1 Qa ðH; U; tÞ ¼ þ hrz ðtÞi cos H þ ½hrx ðtÞi cos U þ hry ðtÞi sin U sin H ; p 2 where hrx(t)i, hry(t)i and hrz(t)i are given by rffiffiffiffiffiffiffiffiffiffiffi X pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi n 2 sin # cos u cos kt n cos kt n þ 1 þ sin kt n sin kt n þ 1 ; jbn j hrx ðtÞi ¼ 2 nþ1 n hry ðtÞi ¼
rffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi sin # sin u n cos kt n cos kt n þ 1 sin kt n sin kt n þ 1 2 nþ1 n pffiffiffi p ffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi p ffiffi ffi pffiffiffi a n cos2 ð#=2Þ sin kt n cos kt n þ 1 ; þ pffiffiffiffiffiffiffiffiffiffiffi sin2 ð#=2Þ sin kt n þ 1 cos kt n a nþ1 X
ð14Þ
ð15Þ
ð16Þ
jbn j2
2 2 pffiffiffiffiffiffiffiffiffiffiffi X jbn j2 pffiffiffiffiffiffiffiffiffiffiffi n pffiffiffi 2 pffiffiffi a sin kt n þ 1 2 2 2 cos2 kt n hrz ðtÞi ¼ cos ð#=2Þ cos kt n þ 1 2 sin kt n sin ð#=2Þ a nþ1 2 n pffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi sin # sin u a sin 2kt n þ 1 n pffiffiffiffiffiffiffiffiffiffiffi þ sin 2kt n . þ a 2 nþ1
ð17Þ
ð18Þ
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The atomic Wehrl density defined as S q ðtÞ ¼ Qa ðH; U; tÞ ln Qa ðH; U; tÞ:
ð19Þ
We see from (19), the atomic Wehrl entropy is dependent on the superposition parameters H and U. Also, one can see that the atomic Wehrl entropy is a Shannon entropy for the atomic Q-function. Thus it can be defined as Shannon Wehrl entropy. Also, we see from Eq. (15) that if the atom is initially in the excited state then QA(0, U, t) is dependent only on the atomic variables hrz(t)i through the following relation 1 1 þ hrz ðtÞi ; ð20Þ Qa ð0; 0; tÞ ¼ p 2 then the atomic Wehrl entropy connected with the expectation value of the population inversion by the following relation 1 1 1 1 S q ð0; 0; tÞ ¼ þ hrz ðtÞi ln þ hrz ðtÞi : ð21Þ p 2 p 2 We note that distinct patterns for the Wehrl entropy will arise depending on the values of n and initial state parameters, determining the revival/collapse times. In order to appreciate those different situations, we show, in what follows, some plots of the atomic Wehrl entropy for different values of n (see Fig. 3). In fact the atomic Wehrl entropy is a good measure of the strength of the coherent component and it is clearly distinguishes coherent states, in another word the Wehrl entropy measures how close a given state is to the coherent states or how much coherence a given state has. Wehrl entropy is very sensitive to the phase space dynamics (such as, e. g., spreading) of the Qa(H, U, t) representation. It extracts from the Qa(H, U, t) function essential information about the investigated system. The Wehrl entropy cannot be negative. This follows from the fact that 0 6 QA(H, U, t) 6 0.3 and the QA(H, U, t) function can never be so concentrated as to make Sa(H, U, t) negative. On the contrary, classical distributions can be arbitrarily concentrated in phase space and classical entropies can take on negative values. In Fig. 3, we have plotted the time evolution of atomic Wehrl density Sq(t) against time 0 6 kt 6 60, for h = p/2, pffiffiffi and / = g = p/4. It is to noted that the atomic Wehrl entropy evolves a minn ¼ 12p. To see that the atomic Wehrl entropies influenced by increasing the imum value at the revival time tR ¼ 2p k mean photon number, we set n ¼ 30 in Fig. 3b. We see that the revival of the atomic Wehrl entropy occurs at later times as the mean photon number n increased.
0.4
Wehrl entropy
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a 0.4
Wehrl entropy
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Fig. 3. The evolution of the function Sq(t) in a perfect cavity as a function of the scaled time kt. The parameters are the same as Fig. 1.
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6. Conclusion We have established the general formalism for the field entropy and atomic Wehrl entropy in the two-level atom. In particular, we have explored the influence of the various values of the mean photon number on the atomic inversion, the field entropy, entanglement and the evolution of the Wehrl entropy. An idealized situation when the cavity losses are negligible is considered here; however, in the case of real experiment the losses must be introduced. It can be expected that for a non-ideal but high-quality cavity our results are of relevance in the case the Hamiltonian is appropriate for the experimental setup. We considered atomic inversion and found that the phenomenon of periodic collapse and revival occurs; it is however a short-lived phenomenon due to the effects of large values of the mean photon numbers. It is found that the Wehrl entropy can be used as an entanglement measure of the coupled system [13].
Acknowledgements I am indebted to S.S. Hassan and M. Abdel-Aty for critical observations, which in turn has led to the investigation above.
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