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Optik 125 (2014) 1739– 1744

Contents lists available at ScienceDirect

Optik

jou rn al homepage: www.elsev ier .de / i j leo

ntanglement dynamics and quasiprobability distribution for theegenerate Raman process

inghong Liaoa,b,c,∗, Ye Liua, Qiurong Yana, Muhammad Ashfaq Ahmadd

Department of Electronic Information Engineering, Nanchang University, Nanchang 330031, ChinaKey Laboratory of Beam Technology and Material Modification of Ministry of Education, Beijing Normal University, Beijing 100875, ChinaSchool of Materials Science and Engineering, Nanchang University, Nanchang 330031, ChinaDepartment of Physics, COMSATS Institute of Information Technology, Lahore 54000, Pakistan

r t i c l e i n f o

rticle history:eceived 29 April 2013ccepted 14 September 2013

ACS:2.50.-p

a b s t r a c t

In this paper, we consider the model which consists of a degenerate Raman process involving two degen-erate Rydberg energy levels of an atom interacting with a single-mode cavity field. The influence of theatomic coherence on the von Neumann entropy of the atom and the atomic inversion is investigated. Itis shown that the atomic coherence decreases the amount of atom-field entanglement. It is also foundthat the collapse and revival times are independent of the atomic coherence, while the amplitude of therevivals is sensitive to this coherence. Moreover, the Q function and the entropy squeezing of the field

2.50.Dv

eywords:on Neumann entropyntanglement

function

are examined. Some new conclusions can be obtained.© 2013 Elsevier GmbH. All rights reserved.

aman interaction

. Introduction

Quantum entanglement is one of the most remarkable featuresf quantum theory [1,2]. It plays an essential role in the quan-um information such as quantum key distribution [3] quantumomputing [4], teleportation [5], cryptographic [6,7], dense cod-ng [8,9] and entanglement swapping [10–12]. An investigation ofhe atom-field entanglement for Jaynes–Cummings (JC) model haseen initiated by Phoenix and Knight [13,14] and Gea-Banacloche15,16]. The time evolution of the field (atomic) entropy reflectshe time evolution of the degree of entanglement between the atomnd the field. The higher the entropy, the greater the entanglement.any papers have focused on the properties of the entanglement

or a quantized field interacting with a two-level atom in theC model [17–23] and the various extensions for the JC model24–28].

The squeezing effect [29–31] of an optical field has become an

ttractive subject in quantum optics. Since Yuen [32] proposed theoncept of the squeezed state in 1976. It has been shown that thequeezing of light can be realized through the interaction of an

∗ Corresponding author at: Department of Electronic Information Engineering,anchang University, Nanchang 330031, China. Tel.: +86 791 83969670.

E-mail address: nculqh@163.com (Q. Liao).

030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved.ttp://dx.doi.org/10.1016/j.ijleo.2013.09.055

optical field and atoms, so the investigation of the field squeezing inan atom-field interaction system is significant. The entropy uncer-tainty relation [33–35] and entropy squeezing [36,37] are possiblehighly-sensitive measures of the field squeezing effect. Abdel-Atyet al. [38] have investigated both entropy squeezing and variancesqueezing in the framework of Shannon information entropy for asingle Rydberg atom having two degenerate levels interacting withthe radiation field in a single-mode ideal cavity. They have shownthat there is a new kind of quantum squeezing in the entropy usingthe entropic uncertainty relation. In this paper, we investigate thevon Neumann entropy of the atom, the atomic inversion, Q func-tion and the entropy squeezing of the field for a atom having twodegenerate levels interacting with the radiation field in a single-mode ideal cavity. The transition between the levels is carried outby a �-type degenerate two-photon process via a third level faraway from single-photon resonance.

The organization of the paper is arranged as follows: we intro-duce the model and the basic equations for the system underconsideration in Section 2. We investigate the effects of the atomiccoherence on the von Neumann entropy of the atom and the atomicinversion in Section 3. The dynamics of the Q function and the rela-

tionship between the quasiprobability distribution and the atomicinversion are examined in Section 4 The entropy squeezing of thefield are analyzed in Section 5. In Section 6, the main results aresummarized.

1 k 125 (2014) 1739– 1744

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�

�

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7(a)

gt

S a(t)

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7(b)

gt

S a(t)

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7(c)

gt

S a(t)

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7(d)

gt

S a(t)

Fig. 1. The time evolution of the von Neumann entropy of the atom Sa(t) as a function

740 Q. Liao et al. / Opti

. The model and the basic equations

The model under the consideration consists of a single Rydbergtom having two degenerate levels interacting with the radiationeld in a single-mode ideal cavity. This model has played an impor-ant role in several physical phenomena of interest, for example,uper-radiance, coherent Raman and Brillouin scattering, as wells stimulated emission of radiation. The atomic levels are labeleds the excited state | + 〉, the ground state | − 〉, and the transi-ion between | − 〉 and | + 〉 is carried out by a �-type degeneratewo-photon process via a third level far away from single-photonesonance. This can be represented by an effective Hamiltonian, in

frame rotating at frequency ω [39–41]:

= ga†a(S− + S+) (� = 1), (1)

here a† and a are the creation and annihilation operators of theavity field of frequency ω which obey [a, a†] = 1. S± are the usualauli spin operators, while g is the coupling constant between thetom and the field mode for the two-photon interaction.

Now if the atom is initially in the coherent superposition state ofhe exited state | + 〉 and the ground state | − 〉, then the initial stateector 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 state vector describes the atom in the groundtate when we take � = �. Furthermore, we assume that the field isnitially in coherent state

˛〉 =∞∑n=0

bn|n〉, (3)

n = exp(inϕ) exp

(−n2

)nn/2

√n!, (4)

here n and ϕ are the average photon number and the phase anglef the field respectively. The solution of the Schrödinger equationn the interaction picture is given by

(t)〉 =∞∑n=0

[An(t)|n, +〉 + Bn(t)|n, −〉], (5)

ith the coefficients An(t) and Bn(t) are

n(t) = bn

[cos

(�

2

)cos(gnt) − i exp(−i�) sin

(�

2

)sin(gnt)

], (6)

n(t) = −ibn[cos

(�

2

)sin(gnt) + i exp(−i�) sin

(�

2

)cos(gnt)

].

(7)

rom the Eqs. (5)–(7), one can obtain the matrix elements ofeduced density operator �(t) of the atom to be

++(t) =∞∑n=0

|An(t)|2, (8)

+−(t) =∞∑n=0

An(t)B∗n(t) = �∗

−+(t), (9)

−−(t) =∞∑n=0

|Bn(t)|2. (10)

of the scaled time gt. The field is initially in coherent state with initial mean photonnumber (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 theproperties of the von Neumann entropy of the atom and the atomicinversion 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 atomiccoherence on the von Neumann entropy and the collapses andrevivals of the atomic inversion.

We start the investigation with the von Neumann entropy. Weuse the von Neumann entropy as a measurement of the degree ofentanglement between the atom and the field of the system underconsideration. Quantum mechanically, the von Neumann entropyis 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,then S /= 0. As shown by Knight and co-workers [13–16] the vonNeumann quantum entropy is a convenient and sensitive mea-sure 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 informationabout the degree of atom-field entanglement. For the system inwhich both the atom and the cavity field mode start from decoup-led pure states, the atomic and field entropy are equal and maybe expressed in terms of the eigenvalues +(t) and −(t) of thereduced atom density operator.

Sa(t) = Sf (t) = −[+(t) ln +(t) + −(t) ln −(t)], (12)

where

±(t) = 12

{1 ±

√1 − 4[�++(t)�−−(t) − |�+−(t)|2]

}. (13)

Employing the matrix elements of reduced atom density operatorgiven by Eqs. (8)–(10), we investigate the properties of the vonNeumann 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, wehave taken the initial mean photon number of the field equal ton = 25 and initial phase ϕ = 0. It is remarkable that the evolution

Q. Liao et al. / Optik 125 (2014) 1739– 1744 1741

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7(a)

gt

S a(t)

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7(b)

gt

S a(t)

Fig. 2. The time evolution of the von Neumann entropy of the atom Sa(t) as a functionoin

onetJmtice3ipttibi

|

Te

Fpci

otgtcez

s

0 2 4 6 8 10−1

−0.5

0

0.5

1

(a)

gt

S z(t)

0 2 4 6 8 10−1

−0.5

0

0.5

1

(b)

gt

S z(t)

0 2 4 6 8 10−1

−0.5

0

0.5

1

(c)

gt

S z(t)

0 2 4 6 8 10−1

−0.5

0

0.5

1

(d)

gt

S z(t)

f the scaled time gt. The atom is initially in excited state � = 0, � = 0 and the fields initially in the coherent state with the average photon number (a) n = 5 and (b)

= 25.

f the von Neumann entropy is rather different compared with theormal two photon JCM case [43,44]. As seen from Fig. 1(a) thentropy is a periodic function of time and reaches its maximum athe middle of the revival time (in the case of the normal two-photonCM entropy is minimized at the quarter of the revival time and

aximized at the half of the revival time [43,44]). At these timeshe entropy evolves to the maximum value (0.693) and the atoms strongly entangled with the field. The long living entanglementan be observed. Furthermore, it is observed that the von Neumannntropy evolves reaches its minimum values at gt(n) = n�(n = 0, 1, 2,, . . .) and the atom is completely disentangled from the field. The

nfluence of the atomic coherence on the von Neumann entropy islotted in Fig. 1(b), (c) and (d). From these figures it is shown thathe evolution period is independent of the atomic coherence, whilehe maximum of the entropy decreases as the � increases. Thats, the atomic coherence decreases the amount of entanglementetween the atom and the field. In particular, if the atom is initially

n the superposition state

(0)A〉 = 1√2

(|+〉 + |−〉). (14)

hen the state vector of the atom-field system at t > 0, can bexpressed as

| (t)〉 =∞∑n=0

bn√2

{[cos(gnt) − i sin(gnt)]|n, +〉 + [cos(gnt) − i sin(gnt)]|n, −〉},

= |˛e−igt 〉 1√2

(|+〉 + |−〉).(15)

rom Eq. (15) it follows that if the atom is initially in the super-osition state (Eq. (14)), then at t > 0 the atom and the field areompletely disentangled. That is, the von Neumann entropy of atoms equal to zero for all the periods of the considered time.

To show the influence of the intensity of the initial coherent fieldn the von Neumann entropy of the atom. Fig. 2(a) and (b) displayshe time evolution of the entropy as a function of the scaled timet for the initial mean photon number n = 5, 25 respectively. Fromhese figures it is observed that the higher the intensity of the initialoherent field the smaller the value of the entropy at t(n). At the high

nough intensities of the cavity field the entropy at t(n) is equal toero (see Fig. 2(b)).

Now, we give our attention to the dynamics of the atomic inver-ion. The influence of the atomic coherence on the atomic inversion

Fig. 3. The time evolution of atomic inversion Sz(t) as a function of the scaled timegt. The parameters are the same as in Fig. 1.

is plotted in Fig. 3. From Fig. 3, it is observed that the collapse andrevival phenomenon of the Rabi oscillations of the inversion areindependent of the initial state of the atom, while the amplitude ofthe 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 evolutionprocess 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 inthe excited state | + 〉. Then the state vector of the system at t > 0 hasthe form

| (t)〉 =∞∑n=0

bn[cos(gnt)|n, +〉 − i sin(gnt)|n, −〉],

= 12

[(|˛eigt〉 + |˛e−igt〉)|+〉] − 12

[(|˛eigt〉 − |˛e−igt〉)|−〉].(16)

We shall study the dynamics of the Q function and the rela-tionship between the quasiprobability distribution and the atomicinversion. The quasiprobability distribution functions [45–47] areimportant tools to discuss the statistical description of a micro-scopic system, and also to provide insight into the nonclassicalfeatures of the radiation field. It is well known that Q-functionis positive definite at any point in the phase space for any quan-tum state. More than just a theoretical curiosity, Q function can beconstructed in homodyne experiments [48,49]. The Q function ofthe field mode is defined in terms of the diagonal elements of thedensity 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 findsfor the reduced density operator of the cavity field from the Eq. (5).

�f (t) =∞∑

[bnb∗m cos(gnt) cos(gmt)|n〉〈m| + bnb

∗m sin(gnt) sin(gmt)|n〉〈m|]. (18)

m,n=0

Inserting Eq. (18) into Eq. (17) we can easily obtain the Q-functionof the cavity field

1742 Q. Liao et al. / Optik 125 (2014) 1739– 1744

Re(β)

Im(β)

t0

t1

t1

t2

t2

t3

(a)

−5 0 5

−6

−4

−2

0

2

4

6

0 1 2 3 4 5 6 7−1

−0.5

0

0.5

1

(b)

gt

S z(t)

t0

t1

t2

t3

Fig. 4. (a) Contour plots of the Q function. The field is initially in coherent statewith 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) asa function of the scaled time gt. The times t0, t1, t2, t3 are indicated by the verticalbars.

Q

sst(istcavds

Ti

0 1 2 3 4 5 6 7 8 9 10

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

gt

δ x(t)

n=1 5

n=2 5

Fig. 5. The time evolution of the position entropy squeezing factor ıx(t) as a function

(ˇ) = exp(−|ˇ|2)�

⎛⎝

∣∣∣∣∣∞∑n=0

ˇ∗n

(n!)1/2bn cos(gnt)

∣∣∣∣∣2

+∣∣∣∣∣

∞∑n=0

ˇ∗n

(n)!1/2bn sin(gnt)

∣∣∣∣∣2⎞⎠ . (19)

The contour lines of the Q function are plotted in Fig. 4(a). Aseen at t0, the initial distribution of the Q function is a shiftedingle Gaussian distribution centered around

√n = 5. Near the ini-

ial time the atomic inversion shows pronounced Rabi oscillationsee Fig. 4(b)). Then the distribution splits into two counterrotat-ng peaks moving along the circle |ˇ| =

√n = 5 to the left side (as

een at t1 and t2 in Fig. 4(a)). During the existence of the two peakshe collapse of the atomic inversion occurs (see Fig. 4(b)). At theollision of the two peaks (as seen at t3 in Fig. 4(a)) a revival of thetomic inversion is observed (see Fig. 4(b)). At this time t = � theon Neumann entropy is equal to zero and the atom is completelyisentangled from the field (see Fig. 1(a)). The state vector for theystem can be written in a factored form by using Eq. (16)

| (t = �)〉 = |˛ei�〉|+〉,(20)

= | − ˛〉|+〉.

his corresponds to the field’s being in the state |˛ei�〉 and the atomn the state | + 〉.

of the scaled time gt. The atom is initially in excited state � = 0, � = 0. and the field isinitially in the coherent state with the average photon number n = 15 (solid curve),n = 25 (dotted curve).

5. Entropy squeezing of the field

The position and momentum entropies of the field are definedas [33–35,50]:

Sx(t) = −∫ ∞

∞〈x|�f (t)|x〉ln〈x|�f (t)|x〉dx, (21)

Sp(t) = −∫ ∞

∞〈p|�f (t)|p〉ln〈p|�f (t)|p〉dp, (22)

where the density matrix element can be obtained from Eq. (18),

〈x|�f (t)|x〉 =∣∣∣∣∣

∞∑n=0

An(t)〈x|n〉∣∣∣∣∣2

+∣∣∣∣∣

∞∑n=0

Bn(t)〈x|n〉∣∣∣∣∣2

. (23)

We obtain 〈p|�f(t)|p〉 by replacing x with p in Eq. (23). In the positionand momentum representations, the expressions for the Fock state|n〉 of the field take the form

〈x|n〉 =√

exp(−x2)√�2nn!

Hn(x), (24)

〈p|n〉 = 1in

√exp(−p2)√�2nn!

Hn(p), (25)

respectively, where Hn(x) and Hn(p) are the Hermite polynomials.The entropy uncertainty relation [33,35,50] of the position andmomentum is

exp[Sx(t)] exp[Sp(t)] � �e. (26)

Now we here introduce two quantities, which we call entropysqueezing 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 ofthe field is squeezed. In what follows, we shall discuss the effect ofthe intensity of the initial coherent field on the dynamical behaviorof the entropy squeezing of the field for the system under consid-eration.

In Fig. 5, we present the position entropy squeezing factor ıx(t)

as a function of the scaled time gt for different values of averagephoton number n = 15 and n = 25. The atom is initially in excitedstate. From this figure it is clearly seen that the position compo-nent of the field exhibits periodic squeezing with a period �/g,

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egardless of the intensity of the initial coherent field. Further-ore, the maximum squeezing of the position component occurs

t gt = (2m + 1)�/2 (m = 0, 1, 2, 3, . . .). For the larger intensity of thentensity of the initial coherent field, i.e., the dotted curve, one canbserve that the squeezing depth is smaller than the case of theeak intensity of the initial coherent field shown in Fig. 5. The com-arison of the solid curve and the dotted curve in Fig. 5 shows thathe entropy squeezing depth is determined by the intensity of thenitial coherent field. It is also shown that the higher the intensityf the initial coherent field, the smaller the value of the entropyqueezing depth.

. Conclusions

In this work, we have investigated a degenerate Raman processnvolving two degenerate Rydberg energy levels of an atom inter-cting with the radiation field in the single mode ideal cavity. Inhe frame work of the von Neumann quantum entropy, we havenvestigated atom-field entanglement and examined the atomicoherence on the von Neumann entropy of the atom and the atomicnversion. Furthermore, the Q function and the entropy squeezingf the field were analyzed using a numerical approach. The obtainedesults are summarized as follows.

1) The periodic long living entanglement is achieved. The atomiccoherence decreases the amount of entanglement between theatom and the field.

2) The collapse and revival phenomenon of the Rabi oscillationsof the atomic inversion are independent of the atomic coher-ence, while the amplitude of the revivals is sensitive to thiscoherence.

3) The entropy squeezing depth of the field is determined by theintensity of the initial coherent field. The higher the intensityof the initial coherent field, the smaller the value of the entropysqueezing depth.

cknowledgements

This project was supported by National Natural Science Foun-ation of China (grant nos. 11247213, 61368002, 10664002,1264030), China Postdoctoral Science Foundation (grant no.013M531558), Jiangxi Postdoctoral Research Project (grant no.013KY33), the Natural Science Foundation of Jiangxi Provincegrant no. 20122BAB201031), the Foundation for Young Scientistsf Jiangxi Province (Jinggang Star) (grant no. 20122BCB23002), andhe Research Foundation of the Education Department of Jiangxirovince (grant nos. GJJ13051, GJJ13057).

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