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Optik
jou rn al homepage: www.elsev ier .de / i j leo
ntropy squeezing of atom and the generation of Schrödinger-cat statesn the three-photon Jaynes–Cummings model
inghong Liaoa,∗, Muhammad Ashfaq Ahmadb
Department of Electronic Information Engineering, Nanchang University, Nanchang 330031, ChinaDepartment of Physics, COMSATS Institute of Information Technology, Lahore 54000, Pakistan
a r t i c l e i n f o
rticle history:eceived 16 October 2011ccepted 27 February 2012
ACS:
a b s t r a c t
In this work, we investigate the effect of the initial phase of the field and the atomic coherence onthe time evolution of entropy squeezing of the atom and the atomic inversion in the three-photonJaynes–Cummings model. Moreover, the evolution of the von Neumann entropy of the atom, Q func-tion and the generation of the Schrödinger-cat states are examined. The results show that the entropysqueezing of the atom and the atomic inversion are sensitive to the initial phase of the field and the initial
2.50.−p2.50.Dv
eywords:ntropy squeezingon Neumann entropy
atomic state. It is also found that the Schrödinger-cat states of the cavity field can be obtained at one halfof the revival time in the large photon number approximation.
The Jaynes–Cummings model (JCM) is a standard and importantodel which describes the interaction of a two-level atom and a
ingle mode of the radiation field [1,2]. JCM reveals many importanturely quantum properties such as sub-Poissonian statistics, anti-unching, squeezing, collapse and revival phenomenon [3]. Atomicqueezing has shown a great interest [4–7] owing to its potentialpplications in high-resolution spectroscopy [8], the high precisiontomic fountain clock [9,10], high-precision spin polarization mea-urements [11], generation of squeezed light [12], etc. The entropyqueezing technique has been applied to the single two-level atomnteracting with a single-mode or two modes, i.e., the JCM [13–19].
ore recently, considerable attention has been paid to the entropyqueezing and entanglement in solid-state circuit QED [20–22], inhich a nano-resonator serves as a cavity. Such systems possess
reat potential applications in quantum information processing23–25].
Furthermore, it is interesting that for the one-photon JCM26–28], the cavity field is created in a Schrödinger cat states atntervals corresponding to one half of the revival time. Buzek and
ladky [29] have investigated the generation of Schrödinger cat
tates of the field in the two-photon JCM. They have found that athe quarter of the revival time the field is produced in Schrödinger
cat states. Fu and Solomon [30] have examined the dynamics ofa four-photon JCM for large photon number approximation. Theyhave shown that at certain times the cavity field is in superpositionof two Kerr states. In this paper we investigate the dynamics andquantum characteristics of the three-photon JCM in the large pho-ton number approximation and it is shown that the cavity field canbe generated in the Schrödinger-cat states at one half of the revivaltime.
The organization of the paper is arranged as follows: we intro-duce the three-photon JCM and present the matrix elements ofreduced density operator of the atom in Section 2. The influenceof the initial phase of the field and the initial state of the atom onthe time evolution of the atomic squeezing and atomic inversionis investigated in Section 3. In Section 4 we study the time evolu-tion of the von Neumann entropy of the atom, the dynamics of theQ function and the analysis of the production of the Schrödinger-cat states in this model. The main results are summarized inSection 5.
2. The three-photon Jaynes–Cummings model
We consider a two-level atom interacting with a single-modequantized field for three-photon transitions. The effective Hamil-tonian of the model with the rotating-wave approximation can be
here a† and a are the creation and annihilation operators of theeld of frequency ω, Sz and S± are the Pauli spin operators of thetom. ω0 is the atomic transition frequency and g is the couplingonstant between the field and the atom. Now if the atom is initiallyn the coherent superposition state of the exited state | + 〉 and theround state | − 〉, then the initial state vector of the atom is
�, �〉 = cos
(�
2
)|+〉 + sin
(�
2
)exp(i�)|−〉, (2)
here � denotes the distribution of the initial atom ranging from 0o � and � is the relative phase of the atomic levels. For the excitedtate we take � = 0. The wavefunction describes the atom in theround state when we take � = �.
Assuming that at t = 0 the field is previously prepared in theoherent state
˛〉 =∞∑n=0
bn|n〉, (3)
n = exp(inϕ) exp
(−n2
)nn/2
√n!, (4)
here n ang ϕ are the average photon number and the phase anglef the field, respectively. For simplicity, we consider the resonantase (ω0 = 3ω). The solution of the Schrödinger equation in thenteraction picture is given by
(t)〉 =∞∑n=0
[An(t)|n, +〉 + Bn(t)|n, −〉], (5)
here the coefficients An(t) and Bn(t) are
n(t) = bn cos
(�
2
)cos(�nt) − i sin
(�
2
)exp(i�)bn+3 sin(�nt),
(6)
n(t) = bn sin
(�
2
)exp(i�) cos(�n−3t) − i cos
(�
2
)
× bn−3 sin(�n−3t), (7)
here �n is the generalized Rabi frequency
n = g√
(n + 1)(n + 2)(n + 3). (8)
rom Eqs. (5)–(7), one can obtain the matrix elements of reducedensity operator �(t) of the atom to be
ee(t) =∞∑n=0
|An(t)|2, (9)
eg(t) =∞∑n=0
An(t)B∗n(t) = �∗
ge(t), (10)
gg(t) =∞∑n=0
|Bn(t)|2. (11)
y employing Eqs. (9)–(11), we are in a position to discuss the prop-rties of the entropy squeezing of the atom and the atomic inversionf the system. This will be done in Section 3.
. Entropy squeezing of the atom
The main purpose of this section is to discuss the squeezing ofnformation entropy for the system under consideration. We giveefinitions of the atomic squeezing in terms of the information
24 (2013) 1083– 1088
entropy and the atomic variances. For a two-level atom, charac-terized by the Pauli operators Sx, Sy and Sz, the uncertainty relationcan be expressed as
SxSy ≥ |〈Sz〉|2, (12)
where (S) =√
〈S2〉 − 〈S〉2. Fluctuation in the component S of
the atomic dipole is said to be squeezed if S satisfies the condition
V(S) = (S) −√
|〈Sz〉|/2 < 0, ≡ x or y. (13)
However, the value of 〈Sz〉 is dependent on the atomic states whichare used to perform the average, and Eq. (12) cannot give sufficientinformation on the atomic squeezing when 〈Sz〉 = 0 [13]. In order tosolve the problem, the squeezing in terms of information entropyhas been presented [13] as a measure of the squeezing of the spinsystem.
In an even N-dimensional Hilbert space, an optimal EUR for setsof N + 1 complementary observables with non-degenerate eigen-values can be described by the inequality [32,33]. It takes the form
N+1∑k=1
H(Sk) ≥ N
2ln
(N
2
)+
(N
2+ 1
)ln
(N
2+ 1
), (14)
where H(Sk) represents the information entropy of the variable Sk.The quantity H(Sk) can be described as follows: for an arbitraryatomic system described by the density matrix �, the probabil-ity distribution of N possible outcomes of measurements of theoperator S is
Pi(S) = 〈 i|�| i〉, ≡ x, y, z and i = 1, 2, . . . , N, (15)
where | i〉 is an eigenvector of the operator S such that
S| i〉 = �i| i〉, ≡ x, y, z and i = 1, 2, . . . , N. (16)
The corresponding information entropies are defined as
H(S) = −N∑i=1
Pi(S) ln Pi(S), ≡ x, y, z. (17)
For a two-level atom, N = 2, then 0 ≤ H(S) ≤ ln 2. The informationentropies of the operators, Sx, Sy and Sz satisfy
H(Sx) + H(Sy) ≥ 2 ln 2 − H(Sz), (18)
This inequality may also be written as
ıH(Sx)ıH(Sy) ≥ 4ıH(Sz)
, (19)
where
ıH(S) ≡ exp[H(S)]. (20)
It is evident that ıH(S) = 1(ıH(S) = 2) corresponds to the atombeing in a pure (mixed) state. Eq. (19) shows the impossibilityof simultaneously having complete information about the observ-ables Sx and Sy where ıH(Sx) and ıH(Sy), respectively, measure theuncertainties of the polarization components Sx and Sy.
From Eq. (19) the components S( ≡ x or y) are said to besqueezed with respect to the information entropy if one or bothof them satisfy the condition
2
E(S) = ıH(S) − √
ıH(Sz)< 0, ≡ x or y. (21)
Using the matrix elements of the atomic reduced density oper-ator �(t), which is given by expressions (9)–(11), (15) and (17), we
Q. Liao, M.A. Ahmad / Optik 124 (2013) 1083– 1088 1085
0 0.2 0.4 0.6
−0.4
−0.2
0
0.2
0.4
0.6
gt
E(S
x)(t) (a)
0 0.2 0.4 0.6
−0.4
−0.2
0
0.2
0.4
0.6
gt
E(S
y)(t) (b)
0 0.2 0.4 0.6
−0.4
−0.2
0
0.2
0.4
0.6
gt
V(S
x)(t) (c)
0 0.2 0.4 0.6
−0.4
−0.2
0
0.2
0.4
0.6
gt
V(S
y)(t) (d)
Fig. 1. The time evolution of the squeezing factors for the atom initially in the excitedstate (� = 0, � = 0) and the field initially in coherent state with initial mean photonnt(
ca
H
tttiVtwi
Fϕff
0 0.2 0.4 0.6
−0.4
−0.2
0
0.2
0.4
0.6
gt
E(S
x)(t) (a)
0 0.2 0.4 0.6
−0.4
−0.2
0
0.2
0.4
0.6
gt
E(S
y)(t) (b)
0 0.2 0.4 0.6
−0.4
−0.2
0
0.2
0.4
0.6
gt
V(S
x)(t) (c)
0 0.2 0.4 0.6
−0.4
−0.2
0
0.2
0.4
0.6
gt
V(S
y)(t) (d)
Fig. 3. The time evolution of the squeezing factors for the atom initially in super-position state (� = �/2, � = 0) and the other parameters are the same as in Fig. 1. (a)
umber (n = 64) and initial phase ϕ = 0. (a) The entropy squeezing factor E(Sx)(t), (b)he entropy squeezing factor E(Sy)(t), (c) the variance squeezing factor V(Sx)(t), andd) the variance squeezing factor V(Sy)(t).
an obtain the information entropies of the atomic operators Sx, Sy
nd Sz in the form.
H(Sx) = −[
12
+ Re(�ge(t))]
ln[
12
+ Re(�ge(t))]
−[
12
− Re(�ge(t))]
ln[
12
− Re(�ge(t))],
(22)
H(Sy) = −[
12
+ Im(�ge(t))]
ln[
12
+ Im(�ge(t))]
−[
12
− Im(�ge(t))]
ln[
12
− Im(�ge(t))],
(23)
(Sz) = −�ee(t) ln �ee(t) − �gg(t) ln �gg(t). (24)
Employing the results obtained here we shall be able to discusshe effect of atomic coherence and the initial phase of the field onhe dynamical behavior of the squeezing and atomic inversion ofhe system. To do so we have plotted both the entropy squeez-ng factors E(Sx)(t) and E(Sy)(t), and the variance squeezing factors
(Sx)(t) and V(Sy)(t) against the scaled time gt for the atom is ini-ially in the different states and the field is in the coherent stateith initial average photon number n = 64. Moreover, the atomic
nversion Sz(t) against the scaled time gt is plotted in Fig. 4.
0 0.2 0.4 0.6
−0.4
−0.2
0
0.2
0.4
0.6
gt
E(S
x)(t) (a)
0 0.2 0.4 0.6
−0.4
−0.2
0
0.2
0.4
0.6
gt
E(S
y)(t) (b)
0 0.2 0.4 0.6
−0.4
−0.2
0
0.2
0.4
0.6
gt
V(S
x)(t) (c)
0 0.2 0.4 0.6
−0.4
−0.2
0
0.2
0.4
0.6
gt
V(S
y)(t) (d)
ig. 2. The time evolution of the squeezing factors for the initial phase of the field = �/2 and the other parameters are the same as in Fig. 1. (a) The entropy squeezing
actor E(Sx)(t), (b) the entropy squeezing factor E(Sy)(t), (c) the variance squeezingactor V(Sx)(t), and (d) the variance squeezing factor V(Sy)(t).
The entropy squeezing factor E(Sx)(t), (b) the entropy squeezing factor E(Sy)(t), (c)the variance squeezing factor V(Sx)(t), and (d) the variance squeezing factor V(Sy)(t).
Fig. 1 displays the time evolution of the squeezing factors forthe atom initially in the excited state (� = 0, � = 0) and the initialphase of the field ϕ = 0. It is obvious from Fig. 1 (a) and (c) thatboth E(Sx)(t) and V(Sx)(t) predict no squeezing in the variable Sx
when the atom is initially in the excited state, whereas Fig. 1(b)and (d) present a great differences between the squeezing fac-tors E(Sy)(t) and V(Sy)(t). E(Sy)(t) displays entropy squeezing (seeFig. 1(b)) during the collapse for the atomic inversion Sz(t) as illus-trated in Fig. 4(a), while V(Sy)(t) predicts no variance squeezingat all, see Fig. 1(d). The results show that the entropy squeezinggives better information on the atomic system than the atomicvariance.
To show the effect of the initial phase of the field on the squeez-ing we have plotted in Fig. 2 the time evolutions of the squeezingfactors for initial phase of the field ϕ = �/2 and the other parame-ters are the same as in Fig. 1. From Fig. 2 one can observe that thetime evolution of the squeezing factors is opposite in the variableSx and Sy compared with the case in Fig. 1. Moreover, the entropy
squeezing can be seen in the first quadrature E(Sx)(t). This demon-strates that the initial phase of the field ϕ determines the squeezingdirection.
Fig. 4. The time evolution of the atomic inversion Sz(t) as a function of the scaledtime gt. The field is initially in coherent state with initial mean photon number(n = 64) and the atom is initially in different state. (a) � = 0, � = 0, ϕ = 0; (b) � = 0,� = 0, ϕ = �/2; (c) � = �/2, � = 0, ϕ = 0.
1 ptik 124 (2013) 1083– 1088
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086 Q. Liao, M.A. Ahmad / O
In order to demonstrate the influence of the atomic coherencen the squeezing we assume the atom is initially in superpositiontate (� = �/2, � = 0) and all the other parameters are the same as inig. 1. The outcome is presented in Fig. 3. It is obvious from Fig. 3(a)nd (b) that there is alternate entropy squeezing in the atomicolarization components Sx and Sy, whereas V(Sx)(t) and V(Sy)(t),s illustrated in Fig. 3(c) and (d), display no variance squeezing.
In Fig. 4(a)–(c), we plot the atomic inversion Sz(t) against thecaled time gt corresponding to the three-photon case in Figs. 1–3.n the one or two photon cases atomic inversion exhibits the usualeriodic collapse and revival phenomenon. The three-photon sys-em also shows this behavior and the revival time tR = 2�/(3g
√n)
34]. However, due to the nonlinearity of the three-photon system,his effect does not last as long as in the one or two photon cases.urther from Fig. 4(c) the amplitude of the fluctuation is very smallhen the atom is initially in superposition state.
. von Neumann entropy and Q function
In this section we assume that the atom is initially prepared inhe excited state | + 〉. We shall study the time evolution of the voneumann entropy of the atom. Moreover, the dynamics of the Q
unction and the analysis of the production of the Schrödinger-cattates in the three-photon JCM are examined.
For the atom initially in the excited state and an arbitrary time > 0, the coupled atom-field wave function is found from Eq. (5) toe
| (t)〉 =∞∑n=0
{bn cos
[√(n + 1)(n + 2)(n + 3)gt
]|+〉
− ibn−3 sin[√
n(n − 1)(n − 2)gt]
|−〉}
|n〉,(25)
q. (25) is an exact solution for an initial atomic excited state | + 〉nd an initial coherent field. Following the work of Gea-Banacloche26], for large photon number case (n 1), considering the prop-rty of Poissonian distribution, one can make some approximations34]: bn−3 ≈ bn, and
n ≈ n3/2
[1 + 3(n − n + 3)
2n
], (26)
n−3 ≈ n3/2
[1 + 3(n − n)
2n
]. (27)
hen we can derive an approximation solution for the system
| (t)〉 ≈{
cos
[n3/2
(1 + 3(3 − n)
2n
)gt
]|+〉
+ i sin(
12n3/2gt
)|−〉
} ∞∑n=0
bn cos(
3n2nn3/2gt
)|n〉
−{
sin
[n3/2
(1 + 3(3 − n)
2n
)gt
]|+〉
+ i cos(
12n3/2gt
)|−〉
} ∞∑n=0
bn sin(
3n2nn3/2gt
)|n〉.
(28)
Now we study the evaluation of the von Neumann entropy,efined as [35]
= −Tr[� ln �], (29)
here � is the density operator for a given quantum system. S = 0or a pure state and S > 0 for a mixed state. As shown by Knight and
o-workers [36–40] the von Neumann quantum entropy is a con-enient measure of the entanglement of two interacting quantumubsystems, which automatically includes all moments of the den-ity operator. For the system in which both the atom and the cavity
Fig. 5. The time evolution of the von Neumann entropy of the atom Sa(t) as a functionof the scaled time gt. The other parameters are the same as in Fig. 1.
field mode start from decoupled pure states, the atomic and fieldentropy are equal and can be expressed in terms of the eigenvalues+(t) and −(t) of the reduced atom density operator by means ofEqs. (9)–(11)
Employing the matrix elements of reduced atom density opera-tor given by Eqs. (9)–(11), we investigate the properties of the vonNeumann entropy of the atom. Fig. 5 shows the temporal evolutionof the von Neumann entropy of the atom as function of the scaledtime gt for the initial mean photon number n = 64. From Fig. 5 it isobserved that the interaction is switched on, i.e., the interchange ofenergy between the coherent state field and the atom occurs, thebipartite exhibits immediately entanglement. As the interactionproceeds, the von Neumann entropy goes gradually to the max-imum value (0.693) and the atom is strongly entangled with thefield. The von Neumann entropy almost tends to zero at the half ofthe revival time (tR = �/(12g)) where the atom is disentangled withthe field and the system of the atom or field returns almost to apure state. Next we will see that the Schrödinger-cat states of thecan be generated at this time.
Now, we give our attention to the dynamics of the Q function andthe analysis of the production of the Schrödinger-cat states in thethree photon JCM. The von Neumann entropy and the Q function arevery useful tools for analyzing the formation of the Schrödinger-catstates. When the field mode is in a state with the minimum valueof the field entropy and the Q function is composed of equal twopeaks, we can infer that such a field-mode state is a Schrödinger-cat states and the state vector of the system can be expressed ina factored form at this time. The Q function of the field mode isdefined in terms of the diagonal elements of the density operatorin the coherent state basis. It takes the form [41]
Q (ˇ) = 1�
〈ˇ|�f |ˇ〉, (32)
where �f is the reduced density operator of the cavity field and |ˇ〉is a coherent state. Taking the trace in the atom space, one finds forthe reduced density operator of the cavity field from Eq. (5)
∞∑∗
�f =
m,n=0
[bnbm cos(�nt) cos(�mt)|n〉〈m|
+bn−3b∗m−3 sin(�n−3t) sin(�m−3t)|n〉〈m|],
(33)
Q. Liao, M.A. Ahmad / Optik 1
Re(β)
Im(β)
(a)
−10 −5 0 5 10−10
−8
−6
−4
−2
0
2
4
6
8
10
Re(β)
Im(β)
(b)
−10 −5 0 5 10−10
−8
−6
−4
−2
0
2
4
6
8
10
Fig. 6. Contour plots of the Q function. The field is initially in coherent state withia
Io
Fftsthbsfu
Ctt
5
oepocr
(
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
nitial mean photon number (n = 64), the atom is initially in excited state � = 0, � = 0nd the time (a) t = 0 and (b) t = tR/2 = �/24.
nserting Eq. (33) into Eq. (32) we can easily obtain the Q-functionf the cavity field
Q (ˇ) = exp(−|ˇ|2)�
⎛⎝
∣∣∣∣∣∞∑n=0
ˇ∗n
(n!)1/2bn cos(�nt)
∣∣∣∣∣2
+∣∣∣∣∣
∞∑n=0
ˇ∗n
(n)!1/2bn−3 sin(�n−3t)
∣∣∣∣∣2⎞⎠ .
(34)
ig. 6 displays the contour plots of the Q function given by Eq. (34)or the field initially in coherent state with n = 64 and ϕ = 0. When
= 0, the cavity field is in a coherent state and its Q-function is Pois-onian, as can be seen from Fig. 6(a). For particular choices of theime: t = (tR/2) = (�/(3g
√n)) = (�/24g) which corresponds to one
alf of the revival time of the atomic inversion. From Fig. 6(b) it cane seen that, at t = tR/2, the Q function splits into two blobs with theame amplitude but opposite phase. At this time the state vectoror the system can be written in a factored form approximately bysing Eq. (28)
| (t = tR
2
)〉 ≈
{−1 +
√3
2|+〉 + i
1 +√
32
|−〉} ∞∑n=0
bn cos(n�
2
)|n〉
= | a(t = tR
2
)〉 ⊗ | f
(t = tR
2
)〉.
(35)
ombining this with the fact that the field entropy at this time tendso zero (see Fig. 5), we conclude that the field states generated athis time are shown to be a Schrödinger-cat states.
. Conclusions
In this paper we have investigated the effect of the initial phasef the field and the atomic coherence on the time evolution ofntropy squeezing of the atom and the atomic inversion in three-hoton JCM, and studied the evolution of the von Neumann entropyf the atom and Q function and the production of the Schrödinger-at states when the atom is initially in excited state. The obtainedesults are summarized as follows:
1) The results show that the entropy squeezing gives better infor-mation on the atomic system than the atomic variance and theinitial phase of the field determines the squeezing direction.
[
24 (2013) 1083– 1088 1087
Moreover, the initial state of the atom plays an important rolein the entropy squeezing.
(2) The atomic inversion exhibits the usual periodic collapse andrevival phenomenon. It is however a short-lived phenomenondue to the effects of nonlinearity.
(3) The von Neumann entropy almost tends to zero at the half of therevival time and the Q function splits into two peaks with thesame amplitude but opposite phase. We conclude that the fieldstates generated at this time are shown to be a Schrödinger-catstates.
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