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Entanglement dynamics and quasiprobability distribution for the degenerate Raman process
Optik 125 (2014) 1739–1744
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Optik journal homepage: www.elsevier.de/ijleo
Entanglement dynamics and quasiprobability distribution for the degenerate Raman process Qinghong Liao a,b,c,∗ , Ye Liu a , Qiurong Yan a , Muhammad Ashfaq Ahmad d a
Department of Electronic Information Engineering, Nanchang University, Nanchang 330031, China Key Laboratory of Beam Technology and Material Modification of Ministry of Education, Beijing Normal University, Beijing 100875, China School of Materials Science and Engineering, Nanchang University, Nanchang 330031, China d Department of Physics, COMSATS Institute of Information Technology, Lahore 54000, Pakistan b c
a r t i c l e
i n f o
Article history: Received 29 April 2013 Accepted 14 September 2013
PACS: 42.50.-p 42.50.Dv
a b s t r a c t In this paper, we consider the model which consists of a degenerate Raman process involving two degenerate Rydberg energy levels of an atom interacting with a single-mode cavity field. The influence of the atomic coherence on the von Neumann entropy of the atom and the atomic inversion is investigated. It is shown that the atomic coherence decreases the amount of atom-field entanglement. It is also found that the collapse and revival times are independent of the atomic coherence, while the amplitude of the revivals is sensitive to this coherence. Moreover, the Q function and the entropy squeezing of the field are examined. Some new conclusions can be obtained. © 2013 Elsevier GmbH. All rights reserved.
Keywords: Von Neumann entropy Entanglement Q function Raman interaction
1. Introduction Quantum entanglement is one of the most remarkable features of quantum theory [1,2]. It plays an essential role in the quantum information such as quantum key distribution [3] quantum computing [4], teleportation [5], cryptographic [6,7], dense coding [8,9] and entanglement swapping [10–12]. An investigation of the atom-field entanglement for Jaynes–Cummings (JC) model has been initiated by Phoenix and Knight [13,14] and Gea-Banacloche [15,16]. The time evolution of the field (atomic) entropy reflects the time evolution of the degree of entanglement between the atom and the field. The higher the entropy, the greater the entanglement. Many papers have focused on the properties of the entanglement for a quantized field interacting with a two-level atom in the JC model [17–23] and the various extensions for the JC model [24–28]. The squeezing effect [29–31] of an optical field has become an attractive subject in quantum optics. Since Yuen [32] proposed the concept of the squeezed state in 1976. It has been shown that the squeezing of light can be realized through the interaction of an
∗ Corresponding author at: Department of Electronic Information Engineering, Nanchang University, Nanchang 330031, China. Tel.: +86 791 83969670. E-mail address: [email protected] (Q. Liao). 0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.09.055
optical field and atoms, so the investigation of the field squeezing in an atom-field interaction system is significant. The entropy uncertainty relation [33–35] and entropy squeezing [36,37] are possible highly-sensitive measures of the field squeezing effect. Abdel-Aty et al. [38] have investigated both entropy squeezing and variance squeezing in the framework of Shannon information entropy for a single Rydberg atom having two degenerate levels interacting with the radiation field in a single-mode ideal cavity. They have shown that there is a new kind of quantum squeezing in the entropy using the entropic uncertainty relation. In this paper, we investigate the von Neumann entropy of the atom, the atomic inversion, Q function and the entropy squeezing of the field for a atom having two degenerate levels interacting with the radiation field in a singlemode ideal cavity. The transition between the levels is carried out by a -type degenerate two-photon process via a third level far away from single-photon resonance. The organization of the paper is arranged as follows: we introduce the model and the basic equations for the system under consideration in Section 2. We investigate the effects of the atomic coherence on the von Neumann entropy of the atom and the atomic inversion in Section 3. The dynamics of the Q function and the relationship between the quasiprobability distribution and the atomic inversion are examined in Section 4 The entropy squeezing of the field are analyzed in Section 5. In Section 6, the main results are summarized.
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2. The model and the basic equations
0.7
H = ga a(S− + S+ )
( = 1),
(1)
where a† and a are the creation and annihilation operators of the cavity field of frequency ω which obey [a, a† ] = 1. S± are the usual Pauli spin operators, while g is the coupling constant between the atom and the field mode for the two-photon interaction. Now if the atom is initially in the coherent superposition state of the exited state | + and the ground state | − , then the initial state vector of the atom is
2
|, = cos
2
|+ + sin
exp(−i)|−,
(2)
where denotes the distribution of the initial atom ranging from 0 to and is the relative phase of the atomic levels. For the excited state we take = 0. The state vector describes the atom in the ground state when we take = . Furthermore, we assume that the field is initially in coherent state |˛ =
∞
bn |n,
(3)
n=0
bn = exp(inϕ) exp
−n 2
n/2
n √
n!
,
(4)
where n and ϕ are the average photon number and the phase angle of the field respectively. The solution of the Schrödinger equation in the interaction picture is given by | (t) =
∞
[An (t)|n, + + Bn (t)|n, −],
(5)
n=0
with the coefficients An (t) and Bn (t) are
An (t) = bn cos
Bn (t) = −ibn cos
2
cos(gnt) − i exp(−i) sin
2
2
sin(gnt) , (6)
sin(gnt) + i exp(−i) sin
2
cos(gnt) . (7)
From the Eqs. (5)–(7), one can obtain the matrix elements of reduced density operator (t) of the atom to be ++ (t) =
∞
|An (t)|2 ,
(8)
n=0
+− (t) =
∞
∗ An (t)Bn∗ (t) = −+ (t),
(9)
n=0
−− (t) =
∞ n=0
|Bn (t)|2 .
(10)
Sa(t)
a
S (t)
0.4
0.3
0.3
0.2
0.2
0.1
0.1 0
1
2
3
4
5
6
7
8
9
0
10
0
1
2
3
4
gt 0.7
0.7
(c)
0.6
6
7
8
9
10
6
7
8
9
10
(d)
0.6
0.5
0.5 0.4
a
0.4 0.3
0.3
0.2
0.2
0.1
0.1
0
5
gt
Sa(t)
†
0.5
0.4
0
(b)
0.6
0.5
S (t)
The model under the consideration consists of a single Rydberg atom having two degenerate levels interacting with the radiation field in a single-mode ideal cavity. This model has played an important role in several physical phenomena of interest, for example, super-radiance, coherent Raman and Brillouin scattering, as well as stimulated emission of radiation. The atomic levels are labeled as the excited state | + , the ground state | − , and the transition between | − and | + is carried out by a -type degenerate two-photon process via a third level far away from single-photon resonance. This can be represented by an effective Hamiltonian, in a frame rotating at frequency ω [39–41]:
0.7
(a)
0.6
0
1
2
3
4
5
6
7
8
9
10
0
0
1
2
3
4
5
gt
gt
Fig. 1. The time evolution of the von Neumann entropy of the atom Sa (t) as a function of the scaled time gt. The field is initially in coherent state with initial mean photon number (n = 25) and the atom is initially in different state with = 0 (a) = 0, (b) = /6, (c) = /4 and (d) = /3.
By employing the Eqs. (8)–(10), we are in a position to discuss the properties of the von Neumann entropy of the atom and the atomic inversion of the system. This will be seen in Section 3. 3. von Neumann entropy of the atom In this section we shall examine the influence of the atomic coherence on the von Neumann entropy and the collapses and revivals of the atomic inversion. We start the investigation with the von Neumann entropy. We use the von Neumann entropy as a measurement of the degree of entanglement between the atom and the field of the system under consideration. Quantum mechanically, the von Neumann entropy is defined as [42] S = −Tr[ ln ],
(11)
where is the density operator for a given quantum system. If describe a pure state, then S = 0, and if describes a mixed state, / 0. As shown by Knight and co-workers [13–16] the von then S = Neumann quantum entropy is a convenient and sensitive measure of the entanglement of two interacting quantum subsystems, which automatically includes all moments of the density operator. The time evolution of the atomic entropy carries the information about the degree of atom-field entanglement. For the system in which both the atom and the cavity field mode start from decoupled pure states, the atomic and field entropy are equal and may be expressed in terms of the eigenvalues + (t) and − (t) of the reduced atom density operator. Sa (t) = Sf (t) = −[ + (t) ln + (t) + − (t) ln − (t)], where ± (t) =
1 2
1±
1 − 4[++ (t)−− (t) − |+− (t)|2 ]
(12)
.
(13)
Employing the matrix elements of reduced atom density operator given by Eqs. (8)–(10), we investigate the properties of the von Neumann entropy of the atom. Fig. 1 displays the time evolution of the von Neumann entropy of the atom for the atom is initially in different state with = 0 (a) = 0, (b) = /6, (c) = /4, (d) = /3. In our computation, we have taken the initial mean photon number of the field equal to n = 25 and initial phase ϕ = 0. It is remarkable that the evolution
Q. Liao et al. / Optik 125 (2014) 1739–1744 0.7
1
(a)
0.6
1
(a)
z
S (t)
0.4 0.3
(b)
0.5
Sz(t)
0.5
0.5
Sa(t)
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0
0
0.2
−0.5
0.1 0
1
2
3
4
5
6
7
8
9
10
−1
gt
0
2
4
6
8
10
−1
0
2
4
gt
6
8
10
6
8
10
gt
1
0.7
1
(b)
0.6
(c)
0.5
0.5 0.4
z
S (t)
Sa(t)
−0.5
0.3
(d)
0.5
Sz(t)
0
0
0
0.2
−0.5
0.1 0
0
1
2
3
4
5
6
7
8
9
10
gt
of the von Neumann entropy is rather different compared with the normal two photon JCM case [43,44]. As seen from Fig. 1(a) the entropy is a periodic function of time and reaches its maximum at the middle of the revival time (in the case of the normal two-photon JCM entropy is minimized at the quarter of the revival time and maximized at the half of the revival time [43,44]). At these times the entropy evolves to the maximum value (0.693) and the atom is strongly entangled with the field. The long living entanglement can be observed. Furthermore, it is observed that the von Neumann entropy evolves reaches its minimum values at gt(n) = n(n = 0, 1, 2, 3, . . .) and the atom is completely disentangled from the field. The influence of the atomic coherence on the von Neumann entropy is plotted in Fig. 1(b), (c) and (d). From these figures it is shown that the evolution period is independent of the atomic coherence, while the maximum of the entropy decreases as the increases. That is, the atomic coherence decreases the amount of entanglement between the atom and the field. In particular, if the atom is initially in the superposition state (14)
Then the state vector of the atom-field system at t > 0, can be expressed as | (t) =
∞ b
n
√ {[cos(gnt) − i sin(gnt)]|n, + + [cos(gnt) − i sin(gnt)]|n, −}, 2
n=0
= |˛e
−igt
0
2
4
6
8
10
−1
0
2
gt
Fig. 2. The time evolution of the von Neumann entropy of the atom Sa (t) as a function of the scaled time gt. The atom is initially in excited state = 0, = 0 and the field is initially in the coherent state with the average photon number (a) n = 5 and (b) n = 25.
1 | (0)A = √ (|+ + |−). 2
−1
−0.5
(15)
1 √ (|+ + |−). 2
From Eq. (15) it follows that if the atom is initially in the superposition state (Eq. (14)), then at t > 0 the atom and the field are completely disentangled. That is, the von Neumann entropy of atom is equal to zero for all the periods of the considered time. To show the influence of the intensity of the initial coherent field on the von Neumann entropy of the atom. Fig. 2(a) and (b) displays the time evolution of the entropy as a function of the scaled time gt for the initial mean photon number n = 5, 25 respectively. From these figures it is observed that the higher the intensity of the initial coherent field the smaller the value of the entropy at t(n) . At the high enough intensities of the cavity field the entropy at t(n) is equal to zero (see Fig. 2(b)). Now, we give our attention to the dynamics of the atomic inversion. The influence of the atomic coherence on the atomic inversion
4
gt
Fig. 3. The time evolution of atomic inversion Sz (t) as a function of the scaled time gt. The parameters are the same as in Fig. 1.
is plotted in Fig. 3. From Fig. 3, it is observed that the collapse and revival phenomenon of the Rabi oscillations of the inversion are independent of the initial state of the atom, while the amplitude of the revivals decreases as the increases. From Eqs. (14) and (15), it is also shown that the system can exhibit atomic trapping, i.e, the atomic inversion is equal to zero in the overall time evolution process when the atom is initially in the superposition state (Eq. (14)). This is in agreement with the result discussed in Ref. [41]. 4. Q function In this section we assume that the atom is initially prepared in the excited state | + . Then the state vector of the system at t > 0 has the form | (t) =
∞
bn [cos(gnt)|n, + − i sin(gnt)|n, −],
n=0
(16)
1 1 = [(|˛eigt + |˛e−igt )|+] − [(|˛eigt − |˛e−igt )|−]. 2 2 We shall study the dynamics of the Q function and the relationship between the quasiprobability distribution and the atomic inversion. The quasiprobability distribution functions [45–47] are important tools to discuss the statistical description of a microscopic system, and also to provide insight into the nonclassical features of the radiation field. It is well known that Q-function is positive definite at any point in the phase space for any quantum state. More than just a theoretical curiosity, Q function can be constructed in homodyne experiments [48,49]. The Q function of the field mode is defined in terms of the diagonal elements of the density operator in the coherent state basis. It takes the form [47] Q (ˇ) =
1 ˇ|f (t)|ˇ,
(17)
where f (t) is the reduced density operator of the cavity field and |ˇ is a coherent state. Taking the trace in the atom space, one finds for the reduced density operator of the cavity field from the Eq. (5). f (t) =
∞
[bn b∗m cos(gnt) cos(gmt)|nm| + bn b∗m sin(gnt) sin(gmt)|nm|].
(18)
m,n=0
Inserting Eq. (18) into Eq. (17) we can easily obtain the Q-function of the cavity field
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Q. Liao et al. / Optik 125 (2014) 1739–1744 0.1
(a)
6
0.08
4 2
t
1
t3
−2
t1
t2
−4
n=2 5
0.02
t0
0
x
0
n=1 5
0.04
δ (t)
Im(β)
0.06
t2
−0.02 −0.04 −0.06
−6 −0.08
−5
0
5 −0.1
Re(β) t 1
0
t
1
t
0
4
5
6
7
8
9
10
Fig. 5. The time evolution of the position entropy squeezing factor ıx (t) as a function of the scaled time gt. The atom is initially in excited state = 0, = 0. and the field is initially in the coherent state with the average photon number n = 15 (solid curve), n = 25 (dotted curve).
5. Entropy squeezing of the field
0
z
3
gt
0.5
S (t)
2
3
(b)
The position and momentum entropies of the field are defined as [33–35,50]:
−0.5
∞
Sx (t) = − −1
1
t
2
∞
0
1
2
3
4
5
6
7
gt
(21)
p|f (t)|plnp|f (t)|pdp,
(22)
∞
Sp (t) = −
Fig. 4. (a) Contour plots of the Q function. The field is initially in coherent state with initial mean photon number (n = 25), the atom is initially in excited state = 0, = 0 at the time gt0 = 0, gt1 = /4, gt2 = /2, gt3 = . (b) The atomic inversion Sz (t) as a function of the scaled time gt. The times t0 , t1 , t2 , t3 are indicated by the vertical bars.
x|f (t)|xlnx|f (t)|xdx,
∞
where the density matrix element can be obtained from Eq. (18),
2 2 ∞ ∞ x|f (t)|x = An (t)x|n + Bn (t)x|n . n=0
We obtain p|f (t)|p by replacing x with p in Eq. (23). In the position and momentum representations, the expressions for the Fock state |n of the field take the form
⎛ 2 ∞ exp(−|ˇ| ) ˇ∗n ⎝ Q (ˇ) = bn cos(gnt) 1/2 (n!)
2
x|n =
n=0
2 ⎞ ∞ ∗n ˇ + bn sin(gnt) ⎠ . (n)!1/2
(19)
n=0
The contour lines of the Q function are plotted in Fig. 4(a). As seen at t0 , the initial distribution of the Q function is a shifted √ single Gaussian distribution centered around n = 5. Near the initial time the atomic inversion shows pronounced Rabi oscillation (see Fig. 4(b)). Then the distribution splits √ into two counterrotating peaks moving along the circle |ˇ| = n = 5 to the left side (as seen at t1 and t2 in Fig. 4(a)). During the existence of the two peaks the collapse of the atomic inversion occurs (see Fig. 4(b)). At the collision of the two peaks (as seen at t3 in Fig. 4(a)) a revival of the atomic inversion is observed (see Fig. 4(b)). At this time t = the von Neumann entropy is equal to zero and the atom is completely disentangled from the field (see Fig. 1(a)). The state vector for the system can be written in a factored form by using Eq. (16) | (t = ) = |˛ei |+, = | − ˛|+.
(23)
n=0
(20)
This corresponds to the field’s being in the state |˛ei and the atom in the state | + .
p|n =
1 in
exp(−x2 ) Hn (x), √ n 2 n!
exp(−p2 ) Hn (p), √ n 2 n!
(24)
(25)
respectively, where Hn (x) and Hn (p) are the Hermite polynomials. The entropy uncertainty relation [33,35,50] of the position and momentum is exp[Sx (t)] exp[Sp (t)] e.
(26)
Now we here introduce two quantities, which we call entropy squeezing factors √ ıx (t) = exp[Sx (t)] − e, (27) ıp (t) = exp[Sp (t)] −
√ e.
(28)
When ıx (t) < 0 (ıp (t) < 0), the position (momentum) component of the field is squeezed. In what follows, we shall discuss the effect of the intensity of the initial coherent field on the dynamical behavior of the entropy squeezing of the field for the system under consideration. In Fig. 5, we present the position entropy squeezing factor ıx (t) as a function of the scaled time gt for different values of average photon number n = 15 and n = 25. The atom is initially in excited state. From this figure it is clearly seen that the position component of the field exhibits periodic squeezing with a period /g,
Q. Liao et al. / Optik 125 (2014) 1739–1744
regardless of the intensity of the initial coherent field. Furthermore, the maximum squeezing of the position component occurs at gt = (2m + 1)/2 (m = 0, 1, 2, 3, . . .). For the larger intensity of the intensity of the initial coherent field, i.e., the dotted curve, one can observe that the squeezing depth is smaller than the case of the weak intensity of the initial coherent field shown in Fig. 5. The comparison of the solid curve and the dotted curve in Fig. 5 shows that the entropy squeezing depth is determined by the intensity of the initial coherent field. It is also shown that the higher the intensity of the initial coherent field, the smaller the value of the entropy squeezing depth. 6. Conclusions In this work, we have investigated a degenerate Raman process involving two degenerate Rydberg energy levels of an atom interacting with the radiation field in the single mode ideal cavity. In the frame work of the von Neumann quantum entropy, we have investigated atom-field entanglement and examined the atomic coherence on the von Neumann entropy of the atom and the atomic inversion. Furthermore, the Q function and the entropy squeezing of the field were analyzed using a numerical approach. The obtained results are summarized as follows. (1) The periodic long living entanglement is achieved. The atomic coherence decreases the amount of entanglement between the atom and the field. (2) The collapse and revival phenomenon of the Rabi oscillations of the atomic inversion are independent of the atomic coherence, while the amplitude of the revivals is sensitive to this coherence. (3) The entropy squeezing depth of the field is determined by the intensity of the initial coherent field. The higher the intensity of the initial coherent field, the smaller the value of the entropy squeezing depth. Acknowledgements This project was supported by National Natural Science Foundation of China (grant nos. 11247213, 61368002, 10664002, 11264030), China Postdoctoral Science Foundation (grant no. 2013M531558), Jiangxi Postdoctoral Research Project (grant no. 2013KY33), the Natural Science Foundation of Jiangxi Province (grant no. 20122BAB201031), the Foundation for Young Scientists of Jiangxi Province (Jinggang Star) (grant no. 20122BCB23002), and the Research Foundation of the Education Department of Jiangxi Province (grant nos. GJJ13051, GJJ13057). References [1] A. Peres, Quantum Theory: Concepts and Methods, Kluwer Academic, Dordrecht, The Netherlands, 1993. [2] G. Alber, T. Beth, M. Horodecki, P. Horodecki, R. Horodecki, M. Rötteler, W. Weinfurter, R. Werner, A. Zeilinger, Quantum Information: an Introduction to Basic Concepts and Experiments, Springer, Berlin, 2001. [3] A. Ekert, Quantum cryptography based on Bells theorem, Phys. Rev. Lett. 67 (1991) 661–663. [4] G. Benenti, G. Casati, G. Strini, Principle of Quantum Computation and Information, World Scientific, Singapore, 2005. [5] C.H. Bennet, G. Brassard, C. Crepeau, R. Jozsa, A. Peresand, W.K. Wootters, Teleporting an unknown quantum state via dual classical and EinsteinPodolsky-Rosen channels, Phys. Rev. Lett. 70 (1993) 1895–1899. [6] J.I. Cirac, N. Gisin, Coherent eavesdropping strategies for the four state quantum cryptography protocol, Phys. Lett. A 229 (1997) 1–7. [7] C.A. Fuchs, N. Gisin, R.B. Griffiths, C.S. Niu, A. Peres, Optimal eavesdropping in quantum cryptography. I. Information bound and optimal strategy, Phys. Rev. A 56 (1997) 1163–1172. [8] L. Ye, G.C. Guo, Scheme for implementing quantum dense coding in cavity QED, Phys. Rev. A 71 (2005) 034304.
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Knight, Squeezed light, J. Mod. Opt. 34 (1987) 709–759. [32] H.P. Yuen, Two-photon coherent states of the radiation field, Phys. Rev. A 13 (1976) 2226–2243. [33] I.I. Hirshman Jr., A note on entropy, Am. J. Math. 79 (1957) 152–156. [34] S. De Nicola, R. Fedele, M.A. Man’ko, V.I. Man’ko, New uncertainty relations for tomographic entropy: application to squeezed states and solitons, Eur. Phys. J. B 52 (2006) 191–198. [35] I. Bialynicki-Birula, Entropic Uncertainty Relations in Quantum Mechanics, Springer, Berlin-Heidelberg, 2006. [36] E. Majernikova, V. Majernik, S. Shpyrko, Entropic uncertainty measure for fluctuations in two-level electron-phonon models, Eur. Phys. J. B 38 (2004) 25–35. [37] M.S. Abdalla, S.S. Hassan, M. Abdel-Aty, Entropic uncertainty in the Jaynes–Cummings model in presence of a second harmonic generation, Opt. Commun. 224 (2005) 431–443. [38] M. Abdel-Aty, I.A. Al-Khayat, S.S. Hassan, Shannon information and entropy squeezing of a single-mode cavity QED of a Raman interaction, Int. J. Quant. Inform. 4 (2006) 807–814. [39] P.L. Knight, Quantum fluctuations and squeezing in the interaction of an atom with a single field mode, Phys. Scr. T12 (1986) 51–55. [40] S.D.J. Phoenix, P.L. Knight, Periodicity, phase, and entropy in models of 2-photon resonance, J. Opt. Soc. Am. B 7 (1990) 116–124. [41] J.L. Cirac, L.L. Sfinchez-Soto, Population trapping in two-level models: Spectral and statistical properties, Phys. Rev. A 44 (1991) 3317–3324. [42] J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, 1955. ´ Macroscopic superposition states of light via 2-photon res[43] V. Buˇzek, B. Hladky, onant interaction of atoms with cavity field, J. Mod. Opt. 40 (1993) 1309–1324. [44] H. Fu, Y. Feng, A.I. Solomon, States interpolating between number and coherent states and their interaction with atomic systems, J. Phys. A: Math. 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Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Entanglement dynamics and quasiprobability distribution for the degenerate Raman process Qinghong Liao a,b,c,∗ , Ye Liu a , Qiurong Yan a , Muhammad Ashfaq Ahmad d a
Department of Electronic Information Engineering, Nanchang University, Nanchang 330031, China Key Laboratory of Beam Technology and Material Modification of Ministry of Education, Beijing Normal University, Beijing 100875, China School of Materials Science and Engineering, Nanchang University, Nanchang 330031, China d Department of Physics, COMSATS Institute of Information Technology, Lahore 54000, Pakistan b c
a r t i c l e
i n f o
Article history: Received 29 April 2013 Accepted 14 September 2013
PACS: 42.50.-p 42.50.Dv
a b s t r a c t In this paper, we consider the model which consists of a degenerate Raman process involving two degenerate Rydberg energy levels of an atom interacting with a single-mode cavity field. The influence of the atomic coherence on the von Neumann entropy of the atom and the atomic inversion is investigated. It is shown that the atomic coherence decreases the amount of atom-field entanglement. It is also found that the collapse and revival times are independent of the atomic coherence, while the amplitude of the revivals is sensitive to this coherence. Moreover, the Q function and the entropy squeezing of the field are examined. Some new conclusions can be obtained. © 2013 Elsevier GmbH. All rights reserved.
Keywords: Von Neumann entropy Entanglement Q function Raman interaction
1. Introduction Quantum entanglement is one of the most remarkable features of quantum theory [1,2]. It plays an essential role in the quantum information such as quantum key distribution [3] quantum computing [4], teleportation [5], cryptographic [6,7], dense coding [8,9] and entanglement swapping [10–12]. An investigation of the atom-field entanglement for Jaynes–Cummings (JC) model has been initiated by Phoenix and Knight [13,14] and Gea-Banacloche [15,16]. The time evolution of the field (atomic) entropy reflects the time evolution of the degree of entanglement between the atom and the field. The higher the entropy, the greater the entanglement. Many papers have focused on the properties of the entanglement for a quantized field interacting with a two-level atom in the JC model [17–23] and the various extensions for the JC model [24–28]. The squeezing effect [29–31] of an optical field has become an attractive subject in quantum optics. Since Yuen [32] proposed the concept of the squeezed state in 1976. It has been shown that the squeezing of light can be realized through the interaction of an
∗ Corresponding author at: Department of Electronic Information Engineering, Nanchang University, Nanchang 330031, China. Tel.: +86 791 83969670. E-mail address: [email protected] (Q. Liao). 0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.09.055
optical field and atoms, so the investigation of the field squeezing in an atom-field interaction system is significant. The entropy uncertainty relation [33–35] and entropy squeezing [36,37] are possible highly-sensitive measures of the field squeezing effect. Abdel-Aty et al. [38] have investigated both entropy squeezing and variance squeezing in the framework of Shannon information entropy for a single Rydberg atom having two degenerate levels interacting with the radiation field in a single-mode ideal cavity. They have shown that there is a new kind of quantum squeezing in the entropy using the entropic uncertainty relation. In this paper, we investigate the von Neumann entropy of the atom, the atomic inversion, Q function and the entropy squeezing of the field for a atom having two degenerate levels interacting with the radiation field in a singlemode ideal cavity. The transition between the levels is carried out by a -type degenerate two-photon process via a third level far away from single-photon resonance. The organization of the paper is arranged as follows: we introduce the model and the basic equations for the system under consideration in Section 2. We investigate the effects of the atomic coherence on the von Neumann entropy of the atom and the atomic inversion in Section 3. The dynamics of the Q function and the relationship between the quasiprobability distribution and the atomic inversion are examined in Section 4 The entropy squeezing of the field are analyzed in Section 5. In Section 6, the main results are summarized.
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2. The model and the basic equations
0.7
H = ga a(S− + S+ )
( = 1),
(1)
where a† and a are the creation and annihilation operators of the cavity field of frequency ω which obey [a, a† ] = 1. S± are the usual Pauli spin operators, while g is the coupling constant between the atom and the field mode for the two-photon interaction. Now if the atom is initially in the coherent superposition state of the exited state | + and the ground state | − , then the initial state vector of the atom is
2
|, = cos
2
|+ + sin
exp(−i)|−,
(2)
where denotes the distribution of the initial atom ranging from 0 to and is the relative phase of the atomic levels. For the excited state we take = 0. The state vector describes the atom in the ground state when we take = . Furthermore, we assume that the field is initially in coherent state |˛ =
∞
bn |n,
(3)
n=0
bn = exp(inϕ) exp
−n 2
n/2
n √
n!
,
(4)
where n and ϕ are the average photon number and the phase angle of the field respectively. The solution of the Schrödinger equation in the interaction picture is given by | (t) =
∞
[An (t)|n, + + Bn (t)|n, −],
(5)
n=0
with the coefficients An (t) and Bn (t) are
An (t) = bn cos
Bn (t) = −ibn cos
2
cos(gnt) − i exp(−i) sin
2
2
sin(gnt) , (6)
sin(gnt) + i exp(−i) sin
2
cos(gnt) . (7)
From the Eqs. (5)–(7), one can obtain the matrix elements of reduced density operator (t) of the atom to be ++ (t) =
∞
|An (t)|2 ,
(8)
n=0
+− (t) =
∞
∗ An (t)Bn∗ (t) = −+ (t),
(9)
n=0
−− (t) =
∞ n=0
|Bn (t)|2 .
(10)
Sa(t)
a
S (t)
0.4
0.3
0.3
0.2
0.2
0.1
0.1 0
1
2
3
4
5
6
7
8
9
0
10
0
1
2
3
4
gt 0.7
0.7
(c)
0.6
6
7
8
9
10
6
7
8
9
10
(d)
0.6
0.5
0.5 0.4
a
0.4 0.3
0.3
0.2
0.2
0.1
0.1
0
5
gt
Sa(t)
†
0.5
0.4
0
(b)
0.6
0.5
S (t)
The model under the consideration consists of a single Rydberg atom having two degenerate levels interacting with the radiation field in a single-mode ideal cavity. This model has played an important role in several physical phenomena of interest, for example, super-radiance, coherent Raman and Brillouin scattering, as well as stimulated emission of radiation. The atomic levels are labeled as the excited state | + , the ground state | − , and the transition between | − and | + is carried out by a -type degenerate two-photon process via a third level far away from single-photon resonance. This can be represented by an effective Hamiltonian, in a frame rotating at frequency ω [39–41]:
0.7
(a)
0.6
0
1
2
3
4
5
6
7
8
9
10
0
0
1
2
3
4
5
gt
gt
Fig. 1. The time evolution of the von Neumann entropy of the atom Sa (t) as a function of the scaled time gt. The field is initially in coherent state with initial mean photon number (n = 25) and the atom is initially in different state with = 0 (a) = 0, (b) = /6, (c) = /4 and (d) = /3.
By employing the Eqs. (8)–(10), we are in a position to discuss the properties of the von Neumann entropy of the atom and the atomic inversion of the system. This will be seen in Section 3. 3. von Neumann entropy of the atom In this section we shall examine the influence of the atomic coherence on the von Neumann entropy and the collapses and revivals of the atomic inversion. We start the investigation with the von Neumann entropy. We use the von Neumann entropy as a measurement of the degree of entanglement between the atom and the field of the system under consideration. Quantum mechanically, the von Neumann entropy is defined as [42] S = −Tr[ ln ],
(11)
where is the density operator for a given quantum system. If describe a pure state, then S = 0, and if describes a mixed state, / 0. As shown by Knight and co-workers [13–16] the von then S = Neumann quantum entropy is a convenient and sensitive measure of the entanglement of two interacting quantum subsystems, which automatically includes all moments of the density operator. The time evolution of the atomic entropy carries the information about the degree of atom-field entanglement. For the system in which both the atom and the cavity field mode start from decoupled pure states, the atomic and field entropy are equal and may be expressed in terms of the eigenvalues + (t) and − (t) of the reduced atom density operator. Sa (t) = Sf (t) = −[ + (t) ln + (t) + − (t) ln − (t)], where ± (t) =
1 2
1±
1 − 4[++ (t)−− (t) − |+− (t)|2 ]
(12)
.
(13)
Employing the matrix elements of reduced atom density operator given by Eqs. (8)–(10), we investigate the properties of the von Neumann entropy of the atom. Fig. 1 displays the time evolution of the von Neumann entropy of the atom for the atom is initially in different state with = 0 (a) = 0, (b) = /6, (c) = /4, (d) = /3. In our computation, we have taken the initial mean photon number of the field equal to n = 25 and initial phase ϕ = 0. It is remarkable that the evolution
Q. Liao et al. / Optik 125 (2014) 1739–1744 0.7
1
(a)
0.6
1
(a)
z
S (t)
0.4 0.3
(b)
0.5
Sz(t)
0.5
0.5
Sa(t)
1741
0
0
0.2
−0.5
0.1 0
1
2
3
4
5
6
7
8
9
10
−1
gt
0
2
4
6
8
10
−1
0
2
4
gt
6
8
10
6
8
10
gt
1
0.7
1
(b)
0.6
(c)
0.5
0.5 0.4
z
S (t)
Sa(t)
−0.5
0.3
(d)
0.5
Sz(t)
0
0
0
0.2
−0.5
0.1 0
0
1
2
3
4
5
6
7
8
9
10
gt
of the von Neumann entropy is rather different compared with the normal two photon JCM case [43,44]. As seen from Fig. 1(a) the entropy is a periodic function of time and reaches its maximum at the middle of the revival time (in the case of the normal two-photon JCM entropy is minimized at the quarter of the revival time and maximized at the half of the revival time [43,44]). At these times the entropy evolves to the maximum value (0.693) and the atom is strongly entangled with the field. The long living entanglement can be observed. Furthermore, it is observed that the von Neumann entropy evolves reaches its minimum values at gt(n) = n(n = 0, 1, 2, 3, . . .) and the atom is completely disentangled from the field. The influence of the atomic coherence on the von Neumann entropy is plotted in Fig. 1(b), (c) and (d). From these figures it is shown that the evolution period is independent of the atomic coherence, while the maximum of the entropy decreases as the increases. That is, the atomic coherence decreases the amount of entanglement between the atom and the field. In particular, if the atom is initially in the superposition state (14)
Then the state vector of the atom-field system at t > 0, can be expressed as | (t) =
∞ b
n
√ {[cos(gnt) − i sin(gnt)]|n, + + [cos(gnt) − i sin(gnt)]|n, −}, 2
n=0
= |˛e
−igt
0
2
4
6
8
10
−1
0
2
gt
Fig. 2. The time evolution of the von Neumann entropy of the atom Sa (t) as a function of the scaled time gt. The atom is initially in excited state = 0, = 0 and the field is initially in the coherent state with the average photon number (a) n = 5 and (b) n = 25.
1 | (0)A = √ (|+ + |−). 2
−1
−0.5
(15)
1 √ (|+ + |−). 2
From Eq. (15) it follows that if the atom is initially in the superposition state (Eq. (14)), then at t > 0 the atom and the field are completely disentangled. That is, the von Neumann entropy of atom is equal to zero for all the periods of the considered time. To show the influence of the intensity of the initial coherent field on the von Neumann entropy of the atom. Fig. 2(a) and (b) displays the time evolution of the entropy as a function of the scaled time gt for the initial mean photon number n = 5, 25 respectively. From these figures it is observed that the higher the intensity of the initial coherent field the smaller the value of the entropy at t(n) . At the high enough intensities of the cavity field the entropy at t(n) is equal to zero (see Fig. 2(b)). Now, we give our attention to the dynamics of the atomic inversion. The influence of the atomic coherence on the atomic inversion
4
gt
Fig. 3. The time evolution of atomic inversion Sz (t) as a function of the scaled time gt. The parameters are the same as in Fig. 1.
is plotted in Fig. 3. From Fig. 3, it is observed that the collapse and revival phenomenon of the Rabi oscillations of the inversion are independent of the initial state of the atom, while the amplitude of the revivals decreases as the increases. From Eqs. (14) and (15), it is also shown that the system can exhibit atomic trapping, i.e, the atomic inversion is equal to zero in the overall time evolution process when the atom is initially in the superposition state (Eq. (14)). This is in agreement with the result discussed in Ref. [41]. 4. Q function In this section we assume that the atom is initially prepared in the excited state | + . Then the state vector of the system at t > 0 has the form | (t) =
∞
bn [cos(gnt)|n, + − i sin(gnt)|n, −],
n=0
(16)
1 1 = [(|˛eigt + |˛e−igt )|+] − [(|˛eigt − |˛e−igt )|−]. 2 2 We shall study the dynamics of the Q function and the relationship between the quasiprobability distribution and the atomic inversion. The quasiprobability distribution functions [45–47] are important tools to discuss the statistical description of a microscopic system, and also to provide insight into the nonclassical features of the radiation field. It is well known that Q-function is positive definite at any point in the phase space for any quantum state. More than just a theoretical curiosity, Q function can be constructed in homodyne experiments [48,49]. The Q function of the field mode is defined in terms of the diagonal elements of the density operator in the coherent state basis. It takes the form [47] Q (ˇ) =
1 ˇ|f (t)|ˇ,
(17)
where f (t) is the reduced density operator of the cavity field and |ˇ is a coherent state. Taking the trace in the atom space, one finds for the reduced density operator of the cavity field from the Eq. (5). f (t) =
∞
[bn b∗m cos(gnt) cos(gmt)|nm| + bn b∗m sin(gnt) sin(gmt)|nm|].
(18)
m,n=0
Inserting Eq. (18) into Eq. (17) we can easily obtain the Q-function of the cavity field
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Q. Liao et al. / Optik 125 (2014) 1739–1744 0.1
(a)
6
0.08
4 2
t
1
t3
−2
t1
t2
−4
n=2 5
0.02
t0
0
x
0
n=1 5
0.04
δ (t)
Im(β)
0.06
t2
−0.02 −0.04 −0.06
−6 −0.08
−5
0
5 −0.1
Re(β) t 1
0
t
1
t
0
4
5
6
7
8
9
10
Fig. 5. The time evolution of the position entropy squeezing factor ıx (t) as a function of the scaled time gt. The atom is initially in excited state = 0, = 0. and the field is initially in the coherent state with the average photon number n = 15 (solid curve), n = 25 (dotted curve).
5. Entropy squeezing of the field
0
z
3
gt
0.5
S (t)
2
3
(b)
The position and momentum entropies of the field are defined as [33–35,50]:
−0.5
∞
Sx (t) = − −1
1
t
2
∞
0
1
2
3
4
5
6
7
gt
(21)
p|f (t)|plnp|f (t)|pdp,
(22)
∞
Sp (t) = −
Fig. 4. (a) Contour plots of the Q function. The field is initially in coherent state with initial mean photon number (n = 25), the atom is initially in excited state = 0, = 0 at the time gt0 = 0, gt1 = /4, gt2 = /2, gt3 = . (b) The atomic inversion Sz (t) as a function of the scaled time gt. The times t0 , t1 , t2 , t3 are indicated by the vertical bars.
x|f (t)|xlnx|f (t)|xdx,
∞
where the density matrix element can be obtained from Eq. (18),
2 2 ∞ ∞ x|f (t)|x = An (t)x|n + Bn (t)x|n . n=0
We obtain p|f (t)|p by replacing x with p in Eq. (23). In the position and momentum representations, the expressions for the Fock state |n of the field take the form
⎛ 2 ∞ exp(−|ˇ| ) ˇ∗n ⎝ Q (ˇ) = bn cos(gnt) 1/2 (n!)
2
x|n =
n=0
2 ⎞ ∞ ∗n ˇ + bn sin(gnt) ⎠ . (n)!1/2
(19)
n=0
The contour lines of the Q function are plotted in Fig. 4(a). As seen at t0 , the initial distribution of the Q function is a shifted √ single Gaussian distribution centered around n = 5. Near the initial time the atomic inversion shows pronounced Rabi oscillation (see Fig. 4(b)). Then the distribution splits √ into two counterrotating peaks moving along the circle |ˇ| = n = 5 to the left side (as seen at t1 and t2 in Fig. 4(a)). During the existence of the two peaks the collapse of the atomic inversion occurs (see Fig. 4(b)). At the collision of the two peaks (as seen at t3 in Fig. 4(a)) a revival of the atomic inversion is observed (see Fig. 4(b)). At this time t = the von Neumann entropy is equal to zero and the atom is completely disentangled from the field (see Fig. 1(a)). The state vector for the system can be written in a factored form by using Eq. (16) | (t = ) = |˛ei |+, = | − ˛|+.
(23)
n=0
(20)
This corresponds to the field’s being in the state |˛ei and the atom in the state | + .
p|n =
1 in
exp(−x2 ) Hn (x), √ n 2 n!
exp(−p2 ) Hn (p), √ n 2 n!
(24)
(25)
respectively, where Hn (x) and Hn (p) are the Hermite polynomials. The entropy uncertainty relation [33,35,50] of the position and momentum is exp[Sx (t)] exp[Sp (t)] e.
(26)
Now we here introduce two quantities, which we call entropy squeezing factors √ ıx (t) = exp[Sx (t)] − e, (27) ıp (t) = exp[Sp (t)] −
√ e.
(28)
When ıx (t) < 0 (ıp (t) < 0), the position (momentum) component of the field is squeezed. In what follows, we shall discuss the effect of the intensity of the initial coherent field on the dynamical behavior of the entropy squeezing of the field for the system under consideration. In Fig. 5, we present the position entropy squeezing factor ıx (t) as a function of the scaled time gt for different values of average photon number n = 15 and n = 25. The atom is initially in excited state. From this figure it is clearly seen that the position component of the field exhibits periodic squeezing with a period /g,
Q. Liao et al. / Optik 125 (2014) 1739–1744
regardless of the intensity of the initial coherent field. Furthermore, the maximum squeezing of the position component occurs at gt = (2m + 1)/2 (m = 0, 1, 2, 3, . . .). For the larger intensity of the intensity of the initial coherent field, i.e., the dotted curve, one can observe that the squeezing depth is smaller than the case of the weak intensity of the initial coherent field shown in Fig. 5. The comparison of the solid curve and the dotted curve in Fig. 5 shows that the entropy squeezing depth is determined by the intensity of the initial coherent field. It is also shown that the higher the intensity of the initial coherent field, the smaller the value of the entropy squeezing depth. 6. Conclusions In this work, we have investigated a degenerate Raman process involving two degenerate Rydberg energy levels of an atom interacting with the radiation field in the single mode ideal cavity. In the frame work of the von Neumann quantum entropy, we have investigated atom-field entanglement and examined the atomic coherence on the von Neumann entropy of the atom and the atomic inversion. Furthermore, the Q function and the entropy squeezing of the field were analyzed using a numerical approach. The obtained results are summarized as follows. (1) The periodic long living entanglement is achieved. The atomic coherence decreases the amount of entanglement between the atom and the field. (2) The collapse and revival phenomenon of the Rabi oscillations of the atomic inversion are independent of the atomic coherence, while the amplitude of the revivals is sensitive to this coherence. (3) The entropy squeezing depth of the field is determined by the intensity of the initial coherent field. The higher the intensity of the initial coherent field, the smaller the value of the entropy squeezing depth. Acknowledgements This project was supported by National Natural Science Foundation of China (grant nos. 11247213, 61368002, 10664002, 11264030), China Postdoctoral Science Foundation (grant no. 2013M531558), Jiangxi Postdoctoral Research Project (grant no. 2013KY33), the Natural Science Foundation of Jiangxi Province (grant no. 20122BAB201031), the Foundation for Young Scientists of Jiangxi Province (Jinggang Star) (grant no. 20122BCB23002), and the Research Foundation of the Education Department of Jiangxi Province (grant nos. GJJ13051, GJJ13057). References [1] A. Peres, Quantum Theory: Concepts and Methods, Kluwer Academic, Dordrecht, The Netherlands, 1993. [2] G. Alber, T. Beth, M. Horodecki, P. Horodecki, R. Horodecki, M. Rötteler, W. Weinfurter, R. Werner, A. Zeilinger, Quantum Information: an Introduction to Basic Concepts and Experiments, Springer, Berlin, 2001. [3] A. Ekert, Quantum cryptography based on Bells theorem, Phys. Rev. Lett. 67 (1991) 661–663. [4] G. Benenti, G. Casati, G. Strini, Principle of Quantum Computation and Information, World Scientific, Singapore, 2005. [5] C.H. Bennet, G. Brassard, C. Crepeau, R. Jozsa, A. Peresand, W.K. Wootters, Teleporting an unknown quantum state via dual classical and EinsteinPodolsky-Rosen channels, Phys. Rev. Lett. 70 (1993) 1895–1899. [6] J.I. Cirac, N. Gisin, Coherent eavesdropping strategies for the four state quantum cryptography protocol, Phys. Lett. A 229 (1997) 1–7. [7] C.A. Fuchs, N. Gisin, R.B. Griffiths, C.S. Niu, A. Peres, Optimal eavesdropping in quantum cryptography. I. Information bound and optimal strategy, Phys. Rev. A 56 (1997) 1163–1172. [8] L. Ye, G.C. Guo, Scheme for implementing quantum dense coding in cavity QED, Phys. Rev. A 71 (2005) 034304.
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