Entropic uncertainty in the Jaynes–Cummings model in presence of a second harmonic generation

Optics Communications 244 (2005) 431–443

www.elsevier.com/locate/optcom

Entropic uncertainty in the Jaynes–Cummings modelin presence of a second harmonic generation

M. Sebawe Abdalla a,*, S.S. Hassan b, M. Abdel-Aty c

a Mathematics Department, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabiab Mathematics Department, College of Science, University of Bahrain, P.O. Box 32038, Bahrainc Mathematics Department, Faculty of Science, South Valley University, 82524 Sohag, Egypt

Received 8 June 2004; received in revised form 15 September 2004; accepted 17 September 2004

Abstract

In the present communication we investigate the usual Jaynes–Cummings Hamiltonian model, describing two-level

atom interacting with an electromagnetic field, in the presence of the second harmonic generation (degenerate paramet-

ric amplifier). Exact solutions of the wave function in the Schrodinger picture have been obtained for two different

cases. In the first case the field frequency x is not equal to the splitting photon frequency e, where the canonical trans-formation has been invoked to obtain the solution of the wave function. In the second case, we considered both fre-

quencies are equal (e = x) and the system is taken to be at exact resonance. Both solutions have been used to

discuss the atomic inversion as well as the entropy squeezing. It has been shown that the system is sensitive to any

change in the coupling parameter responses of the second harmonic generation as well as to the atomic phase angle.

� 2004 Published by Elsevier B.V.

PACS: 03.67.�a; 03.65.Ta; 42.50.Dv

1. Introduction

Advanced experimental (as well theoretical) re-

search in the topics of micromasers and high cavity

0030-4018/$ - see front matter � 2004 Published by Elsevier B.V.

doi:10.1016/j.optcom.2004.09.051

* Corresponding author.

E-mail addresses: [email protected] (M.S. Abdalla),

[email protected] (S.S. Hassan), abdelatyquant@

yahoo.co.uk (M. Abdel-Aty).

quantum electrodynamics (e.g. [1–3]) has been

inspired, in the first place, by the fundamental Jay-

nes–Cummings (JC) model of atom–field interac-

tion [4]. The JC model describes the interaction

of a single two-level atom with a single mode of

quantized cavity radiation field in the absence of

any dissipation process by the atom or the field.Extension of the JC model has been carried

out to include effects like: multi-atom system,

multi-level atom, multi-mode cavity field, time/

mailto:[email protected]

mailto:[email protected]

mailto:abdelatyquant@

432 M.S. Abdalla et al. / Optics Communications 244 (2005) 431–443

field-dependent coupling constant, dissipation

processes by the atom/the field (see [5], and refer-

ences therein).

The aim of the present paper is to examine the

JC model in the presence of a second harmonicgeneration process by the same cavity field

(namely degenerate parametric amplification).

Specifically, we examine two main quantum as-

pects of the suggested model: Collapse and revival

in the atomic inversion and entropic uncertainty

relations. The model we adopt here is given by

H�h¼ xayaþ nðtÞay2 þ nastðtÞay þ x0

2rz

þ kðay þ aÞðr� þ rþÞ; ð1Þ

where ay and a are the creation and annihilation

operators for the cavity mode such that

½a; ay� ¼ 1, and x and x0 are the field and the

atomic transition frequencies, respectively, while

k is the coupling constant between the field and

the atom. The operators r+(r�) and rz are theusual raising (lowering) and inversion operators

for the two-level atomic system, satisfying

[rz,r±] = ± 2r± and [r+,r�] = rz. The time-depend-

ent complex function n(t) is a response of the sec-

ond harmonic generation (degenerate parametric

amplifier) and is given by

nðtÞ ¼ ik2expð�2ietÞ; ð2Þ

where k is an arbitrary constant and e is the fre-

quency of the split photon. The above model

may be compared with a Hamiltonian model de-scribes a long-lived mesoscopic superposition

states in cavity quantum electrodynamics which

depends on parametric amplification and an engi-

neered squeezed-vacuum reservoir for cavity-field

states [6]. Also the above Hamiltonian may be re-

garded as a generalization of the degenerate para-

metric amplifier model in absence of the quantum

fluctuations [7], where light with a number ofinteresting properties is conserved in the process

of frequency conversion when pumped by light

of frequency 2x. It should be noted that the sec-

ond harmonic generation is represented in the

Hamiltonian (1) by the terms ay2 and a2 while

the last term describes the atom–field interaction

in the usual JC model outside the rotating wave

approximation (RWA). This approximation will

be applied to the Hamiltonian, however, at later

steps of deriving our solution in Section 2. Discus-

sion related to the atomic inversion is given in

Section 3, followed by examination of the entropyand variances squeezing in Section 4, while our

conclusion is given in Section 5.

2. The wave function

To study the dynamics of the system we have to

obtain the exact expression of the time-dependentwave function in the Schrodinger picture. There

are two different cases to consider: the first case

when the split photon frequency e is not equal tothe field frequency x, while the second case when

e = x.

Case I (x 6¼ e). To deal with this case we introduce

the time-dependent operators

A ¼ a expðietÞ; and Ay ¼ ay expð�ietÞ: ð3Þ

Thus, if we substitute Eq. (3) into the Hamiltonian

(1), taking care with the generating function, then

we have

H�h¼ dA

yAþ i

k2ðAy2 � A

2Þ� �

þ x0

2rz þ kðA expð�ietÞ

þ AyexpðietÞÞðr� þ rþÞ; ð4Þ

where d = (x � e), and x 6¼ e. Moreover, if we in-

voke the canonical transformation [8]

Ay ¼ B

ycosh/þ iB sinh/;

A ¼ B cosh/� iBysinh/;

ð5Þ

where B and Bysatisfy the commutator ½B; By� ¼ 1,

then Eq. (4) becomes

H�h¼ XB

yBþ x0

2rz þ k½ðB cosh/� iB

ysinh/Þ

� expð�ietÞ þ ðBycosh/þ iB sinh/Þ

� expðietÞ�ðr� þ rþÞ; ð6Þ

where

/ ¼ 1

2tanh�1ðk=dÞ; X ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid2 � k2

p: ð7Þ

M.S. Abdalla et al. / Optics Communications 244 (2005) 431–443 433

In the interaction picture the Hamiltonian (6) is

given by

V IðtÞ�h

¼ kfBðe�iet cosh/þ i sinh/eietÞe�iXt

þ Byðeiet cosh/� i sinh/e�ietÞeiXtg

� ðr�e�ix0t þ rþe

ix0tÞ: ð8Þ

Now, if we apply the RWA,where we neglect the

energy non-conserving terms Br� and Byrþ, then

Eq. (8) reduces to

V IðtÞ ¼ �hk Byr�F ðtÞ þ BrþF �ðtÞ

h i; ð9Þ

where

F ðtÞ exp½�iXt� ¼ cosh/ exp½�iðx0 � eÞt�� i sinh/ exp½�iðx0 þ eÞt�: ð10Þ

It should be noted that to avoid any appearance of

non-conservative terms we have applied the RWA

to the rotated operators BðByÞ not to the physical

operators AðAyÞ. Further, the rapidly oscillating

terms, exp[±i(x0 + e)t], are neglected within the

RWA and then the interaction Hamiltonian (9)

takes the form

V IðtÞ�h

¼ k cosh/fByr� exp½�iðx0 � eÞt� exp½iXt�

þ Brþ exp½iðx0 � eÞt� exp½�iXt�g: ð11Þ

Note that, the transformed Hamiltonian (11) as a

JC form of interaction attained by RAW, but with

the canonically transformed field operators B, By.

The Schrodinger equation for the interaction

Hamiltonian (11) is given by

i�ho

ot

��wðtÞi ¼ V IðtÞjwðtÞi; ð12Þ

where with the usual procedure for example [9] the

state |w(t)æ is expanded as

jwðtÞi ¼X1n¼0

½Ca;nðtÞja; ni þ Cb;nþ1ðtÞjb; nþ 1i�:

ð13Þ

The time-dependent function Ca,n(t) in the above

equation is the probability amplitude that the

atom is in its upper state |aæ and the field has n

photons, while the function Cb,n+1(t) refers to the

probability amplitude that the atom is in its lower

state |bæ and the field has n + 1 photons. In this

case, we have the following coupled differential

equations:

idCa;nðtÞ

dt¼ k

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðnþ 1Þ

pcosh/ exp½�iðD� XÞt�Cb;nþ1;

and

idCb;nþ1ðtÞ

dt¼ k


pcosh/ exp½iðD� XÞt�Ca;n;

ð14Þ

where D = (x0 � e) is the detuning parameter. The

general solution of the above system can be written

as

Ca;nðtÞ ¼ exp � i

2ðD� XÞt

� �

�Ca;nð0Þ cos gnt þ i ðD�XÞ

2gnsin gnt

� ��ið~g

ffiffiffiffiffiffiffiffiffiffiffinþ 1

p=gnÞ sin gntCb;nþ1ð0Þ

24

35;

and

Cb;nþ1ðtÞ ¼ expi

2ðD�XÞt

� �

�Cb;nþ1ð0Þ cosgnt� i ðD�XÞ

2gnsingnt

� ��ið~g

ffiffiffiffiffiffiffiffiffiffiffinþ 1

p=gnÞ singntCa;nð0Þ

24

35;ð15Þ

where gn is the Rabi frequency given by

gn ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½~g2ðnþ 1Þ þ 1

4ðD� XÞ2

r;

~g ¼ k cosh/: ð16Þ

It is easy to check that the probability amplitude

jCa;nðtÞj2 þ jCb;nþ1ðtÞj2 ¼ 1 ð17Þis always satisfied for all tP 0.

Case II (x = e). In order to consider the case in

which x = e, one may think of the limiting case,however this is a crucial way to do so. This can

be noted from the inconsistency which would ap-

pear resultant of the singularity in Eq. (7). Now,

if we set d = 0 then we can rewrite the Hamiltonian

(1) in the form


H�h¼ xayaþ nðtÞay2 þ n�ðtÞa2 þ x0

2rz

þ kðay þ aÞðr� þ rþÞ; ð18Þ

where

nðtÞ ¼ ik2expð�2ixtÞ:

We follow up the same procedure as before and

define new operators A1 and Ay1 such that

A1 ¼ a expðixtÞ; and Ay1 ¼ ay expð�ixtÞ; ð19Þ

then Eq. (18) takes the form

H�h¼ x0

2rz þ kðA1rþ expð�ixtÞ þ A

y1r�

� expðixtÞÞ þ ik2ðAy2

1 � A2

1Þ: ð20Þ

In terms of the interaction picture we have

V IðtÞ�h

¼ kf½A1 cosh kt þ Ay1 sinh kt�rþ expð�iDtÞ

þ ½Ay1 cosh kt þ A1 sinh kt�r� expðiDtÞg;

ð21Þ

where D = (x � x0) is the detuning parameter.Here, we point out that as a result of the degener-

ate parametric amplifier terms the interaction

Hamiltonian acquired rapid oscillating terms

which we ignore within the RWA. Hence Eq.

(21) reduces to

V IðtÞ�h

¼ k cosh kt A1rþ expð�iDtÞ þ Ay1r� expðiDtÞ

� �:

ð22ÞNow using Eqs. (12), (13) and (22), we get

idCa;nðtÞ

dt¼ k cosh kt


pexp½�iDt�Cb;nþ1;

ð23Þand

idCb;nþ1ðtÞ

dt¼ k cosh kt


pexp½iDt�Ca;n: ð24Þ

It is not easy task to find analytical solution for

the above system of equations. However, if we

consider the exact resonance case D = 0, then we

have

d2Ca;n

dt2� k tanh kt

dCa;n

dtþ k2ðnþ 1Þcosh2ktCa;n ¼ 0;

ð25Þwhich has a solution

Ca;nðtÞ ¼ Ca;nð0Þ cosðg sinh ktÞ� i sinðg sinh ktÞCb;nþ1ð0Þ;

and

Cb;nþ1ðtÞ ¼ Cb;nþ1ð0Þ cosðg sinh ktÞ� i sinðg sinh ktÞCa;nð0Þ; ð26Þ

where g ¼ ðk=kÞffiffiffiffiffiffiffiffiffiffiffinþ 1

pis the modified Rabi fre-

quency. For computational purpose it is conven-

ient to rewrite the above solution in the

following form:

Ca;nðtÞ ¼ I0ðgÞþ 2X1r¼1

ð�ÞrI2rðgÞcoshð2rktÞ !

Ca;nð0Þ

� 2iX1r¼0

ð�ÞrI ð2rþ1ÞðgÞ sinh½ð2rþ 1Þkt� !

�Cb;nþ1ð0Þ;

and

Cb;nþ1ðtÞ¼ I0ðgÞþ2X1r¼1


Cb;nþ1ð0Þ

�2iX1r¼0

ð�ÞrI ð2rþ1ÞðgÞsinh½ð2rþ1Þkt� !

�Ca;nð0Þ; ð27Þ

where In(Æ) is the modified Bessel function of order

n. Having obtained the exact solutions of the wave

function in the mentioned two cases, Eqs. (15) and

(27), we investigate both the atomic inversion and

the entropy squeezing in the following two

sections.

3. Atomic inversion

The atomic population inversion can be written

as

hrzðtÞi ¼1

2TrqaðtÞfjeihej � jgihgjg; ð28Þ

where qa(t) is the atomic density matrix.

� �


To derive the density matrix operator qðtÞ let usconsider at time t = 0 the effective two-level atom

is in a coherent superposition state of the excited

state |eæ and ground state |gæ, such that

jh;Ui ¼ cos hjei þ e�iU sin hjgi; ð29Þ

where U is the relative phase of the two atomic lev-

els and h is the atomic distribution. Note that

h = 0,(p/2) corresponds to atom in its excited

(ground) state, while the field is initially in coher-

ent state

jai ¼X1n¼0

Cnjni; Cn ¼ exp � 1

2jaj2

� �anffiffiffiffin!

p : ð30Þ

Therefore, after some minor algebra the density

matrix q can be written as

qðtÞ ¼ cos2hX1n¼0

X1m¼0

½AnA�mjn; eihe;mj þBnþ1B�

mþ1jnþ 1;gi

� hg;mþ 1j þAnB�mþ1jn; eihg;mþ 1j þA�

mBnþ1jm; ei

� hg;nþ 1j� þ sin2hX1n¼0

X1m¼0

½AnA�mjn; ei

� he;mj þBnþ1B�mþ1jnþ 1;gihg;mþ 1j

Fig. 1. The time evolution of the atomic inversion as a function of the

for: (a) initially excited atom (h = 0) with / = 0, and D = 0, (b) as (a)

D = 5k.

þ AnBmþ1jn;gihg;mþ 1j þ AnBmþ1jnþ 1;gi

� hg;mj� þ 1

2eiU sin 2h

X1n¼0

X1m¼0

½AnA�mjn; ei

�hg;mj þ Bnþ1B�mþ1jnþ 1; eihg;mþ 1j

þ AnB�mþ1jn; eihg;mþ 1j þ A�

mBnþ1jm; ei

� hg;nþ 1j� þ 1

2e�iU sin 2h

X1n¼0

X1m¼0

½AnA�mjn; ei

� he;mj þ Bnþ1B�mþ1jnþ 1;gihg;mþ 1j

þ Bnþ1A�mjnþ 1;gihe;mj þ AnB�

mþ1jn;gi� he;mþ 1j�; ð31Þ

where An(t) and Bn+1(t) for the first case (x 6¼ e)are given by

AnðtÞ ¼ Cn cos gnt þ iðD� XÞ2gn

sin gnt� �

� exp � i

2ðD� XÞt

� �;

Bnþ1ðtÞ ¼ �i~gffiffiffiffiffiffiffiffiffiffiffinþ 1

p

gnCn sin gnt exp

i

2ðD� XÞt

� �;

ð32Þ

scaled time kt (Case I (x 6¼ e)) for �n ¼ 25, d = 0.5k, k = 0.3k, andbut for intermediate atomic state, h = p/3, (c) as (a) but h = p/3,


while for the second case (x = e) we have

AnðtÞ¼Cn I0ðgÞþ2X1r¼1


;

Bnþ1ðtÞ¼�2iCn

X1r¼0

ð�ÞrI ð2rþ1ÞðgÞsinh½ð2rþ1Þkt� !

:

ð33Þ

Now, we examine the computational plots of

the atomic inversion according to Eqs. (28), (31)

and Eq. (32) in the two cases:

Case I (x 6¼ e). In Fig. 1(a) we plotted the atomic

inversion against the scaled time kt, for an initially

excited atom (h = 0) with phase U = 0 at exact res-

onance D = 0 and for mean photon number

n ¼ 25, and the difference between the field and

the photon splitting frequency d/k = 0.5, whilethe second harmonic generation coupling parame-

ter k/k = 0.3. The atomic inversion behaves similar

to that of the usual JC model ([5], and references

therein) where it fluctuates around zero with

amplitude extrema around ±0.75. For an interme-

diate atomic state with h = p/3 (Fig. 1(b)) and for

the same values of the other parameters, the

Fig. 2. As Fig. 1, but for Case II (x = e), �n ¼ 25, and for: (a) h = U =

(d) as (a) but for h = p/3.

behavior is similar to Fig. 1(a) but with reduced

amplitude and fluctuations and the function is

slightly shifted above the zero value. In presence

of detuning parameter D/k = 5 (Fig. 1(c)) the func-

tion shifts its fluctuations around �0.25 with nochange it its amplitude.

Case II (x = e). In this case, the field frequency x is

equal to the second harmonic generation fre-

quency e, while the detuning parameter D is zeroaccording to our solutions, see Eq. (27). For ini-

tially excited atom (h = 0) with phase U = 0,

n ¼ 25, and small value of k = 0.1 k (Fig. 2(a))

the atomic inversion shows behavior similar to

that of the usual JC model. In this particular case

we observe in the first period of collapse a regular

fluctuations with decreasing amplitude as the time

develops. This is followed by a long period of revi-val where a strong interaction between the atom

and the field is expected. For the second period

of collapse we realize a rapid fluctuations with

decreased amplitude. For large values of k/k =

0.5,0.8 (Figs. 2(b) and (c)) the atomic inversion

shows rapid fluctuations after the onset of the

interaction where the function amplitude reaches

its maximum value (nearly one). Also the function

0, k = 0.1k, (b) as (a) but for k = 0.5k, (c) as (a) but for k = 0.8k,


shows short period of collapse followed with revi-

val period shorter than that in Fig. 2(a). In the sec-

ond period of collapse the function shows rapid

fluctuations with interference patterns for all peri-

ods of the considered time, while the second periodof revival is too limited. This means that the inter-

action between the field and the atom gets weaker.

The same behavior can be realized for the interme-

diate state case where h = p/3 (Fig. 2(d)); however,

the amplitude of the function decreases considera-

bly and the extrema just occur after the onset of

the interaction between �0.4 and 0.5.

4. Entropy and variances squeezing

The argument for using entropic uncertainty

relations for two-level systems rather than Hei-

senberg uncertainty relations to investigate

quantum fluctuations was recently discussed in

[10] and references therein. Now for two physi-cal observable represented by the Hermitian

operators A and B satisfying the commutation

relation ½A; B� ¼ iC, Heisenberg uncertainty rela-

tion reads

hðDAÞ2ihðDBÞ2i P 1

4jhCij2; ð34Þ

where hðDAÞ2i ¼ ðhA2i � hAi2Þ. Consequently, theuncertainty relation for a two-level atom charac-

terized by the Pauli operators Sx, Sy and Sz, satis-

fying the commutation ½Sx; Sy � ¼ iSz can also be

written as DSxDSy P 12jhSzij.

Fluctuations in the component Sa of the atomic

dipole is said to be squeezed if Sa satisfies the

condition

V ðSaÞ ¼ DSa �

ffiffiffiffiffiffiffiffiffiffiffihSzi2

��

vuut0@

1A < 0; a ¼ x or y:

ð35Þ

In an even N-dimensional Hilbert space, the inves-

tigation of the optimal entropic uncertainty rela-

tion for sets of N + 1 complementary observablewith the non-degenerate eigenvalues can be de-

scribed by the inequality [11]

XNþ1

k¼1

HðSkÞ PN2ln

N2

� �þ 1þ N

2

� �ln 1þ N

2

� �;

ð36Þ

where HðSkÞ represents the information entropy of

the variable Sk. On the other hand, for an arbitrary

quantum state the probability distribution for N

possible outcomes of measurements of the opera-

tor Sa is P iðSaÞ ¼ hWaijqjWaii, where jWaiæ is an

eigenvector of the operator Sa such that

SajWaii ¼ kaijWaii; a ¼ x; y; z; i ¼ 1; 2; . . . ;N . The

corresponding Shannon information entropies

are then defined as

HðSaÞ ¼ �XNi¼1

P iðSaÞ ln P iðSaÞ; a ¼ x; y; z: ð37Þ

To obtain the Shannon information entropies of

the atomic operators Sx; Sy and Sz for a two-level

atom, with N = 2 , one can use the reduced atomic

density operator qðtÞ. Thus, we have the followingexpression:

HðSxÞ ¼ � 1

2qxðtÞ þ 1Þ½ � ln 1

2qxðtÞ þ 1Þ½ �

� �

� 1

21� qxðtÞ½ � ln 1

21� qxðtÞ½ �

� �;

HðSyÞ ¼ � 1

2iqyðtÞ þ 1Þ

ln1

2iqyðtÞ þ 1Þ � �

� 1

21� iqyðtÞ

ln1

21� iqyðtÞ � �

;

HðSzÞ ¼ � 1

2qzðtÞ þ 1Þ½ � ln 1

2qzðtÞ þ 1Þ½ �

� �

� 1

21� qzðtÞ½ � ln 1

21� qzðtÞ½ �

� �:

ð38Þ

Since the uncertainty relation of the entropy can

be used as a general criterion for the squeezing

of an atom, therefore for a two-level atom

where N = 2, we have 0 6 HðSaÞ 6 ln 2, andhence, the Shannon information entropies of

the operators Sx; Sy ; Sz will satisfy the

inequality

HðSxÞ þ HðSyÞ þ HðSzÞ P 2 ln 2: ð39Þ


In other words if we define dHðSaÞ ¼exp½HðSaÞ�, then we can write [12–14],

dHðSxÞdHðSyÞdHðSzÞ P 4: ð40ÞIt is interesting to mention that the above

inequality was previously obtained and estab-

lished to be optimal. The fluctuation in compo-nent Sa ða ¼ x or yÞ of the atomic dipole are

said to be ‘‘squeezed in entropy’’ if the Shannon

information entropy HðSaÞ of Sa satisfies the

condition

EðSaÞ ¼ dHðSaÞ �2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

dHðSzÞq < 0 ða ¼ x; or yÞ;

ð41Þ

where EðSaÞ is the entropy squeezing factor.

Now, we employ the results obtained here to

discuss the entropy squeezing as well as variances

squeezing.

Case I (x 6¼ e). In Fig. 3 we have plotted both the

entropy squeezing factors EðSxÞ and EðSyÞ, and the

variance squeezing factors V ðSxÞ and V ðSyÞ againstthe scaled time kt, for an initially excited atom

(h = 0) with phase U = 0. The mean photon num-

Fig. 3. The time evolution of the squeezing factors as functions of the

= 0, and the field in the coherent state with �n ¼ 25, d = 0.5, k/k = 0.3. (

factor EðSyÞ; (c) the variance squeezing factor V ðSxÞ; and (d) the vari

ber n ¼ 25, and the coupling parameter response

of the second harmonic generation k/k = 0.3, while

the difference between the field frequency x and

the splitting frequency e is d = 0.5. At exact reso-

nance (D = 0) the results are essentially those ofthe usual JC model [15] where we observe squeez-

ing in the first quadrature EðSxÞ at several periodsof the time. The quadrature EðSyÞ shows no

squeezing at all, but its value reduced slightly twice

during the interaction period. The variance squeez-

ing factors V ðSx;yÞ show very little squeezing taking

with rapid fluctuations. For the intermediate state

h = p/3 (Fig. 4), while the degree of squeezing inEðSxÞ is the same as in Fig. 3(a), the component

EðSyÞ starts to show squeezing at t = 0 and both

components EðSx;yÞ show fluctuations with lesser

interference pattern. The variance components

V ðSx;yÞ (Figs. 4(c) and (d)) show more squeezing

than in Figs. 3(c) and (d) with still irregular

fluctuations.

For atomic phase value U = p/4 (Fig. 5) EðSxÞshows several periods of rapid fluctuations com-

pared with zero phase value (U = 0) in Fig. 4(a).

For the quadrature EðSyÞ, squeezing only occurs

after the onset of the interaction Fig. 5(b) and

scaled time kt. The atom is initially in the excited state h = 0, /a) The entropy squeezing factor EðSxÞ; (b) the entropy squeezingance squeezing factor V ðSyÞ.

Fig. 5. The same as in Fig. 3 but h = p/3, / = p/4.

Fig. 4. The same as in Fig. 3 but h = p/3.


the function has two periods of fluctuations with

interference pattern. The behavior of the variance

component V ðSxÞ in Fig. 5(c) is similar to that inFig. 4(c) but the component V ðSyÞ shows squeez-

ing for longer period of time (Fig. 5(d)) as com-

pared with the zero phase value (U = 0) in Fig.

4(d). Thus, atomic phase is responsible for increas-

ing the squeezing amount in the component V ðSyÞ.In the off-resonance case (D/k = 5), see Fig. 6,

pronounced change occurs in both entropy and

Fig. 6. The same as in Fig. 5 but D = 5k.


variance squeezing factors: EðSxÞ still shows

squeezing but for lesser period of time compared

with Fig. 5 while EðSyÞ shows no squeezing at

Fig. 7. The time evolution of the squeezing factors as functions of the

= 0, and the field in the coherent state with �n ¼ 25, k = 0.5k. (a) The eEðSyÞ; (c) the variance squeezing factor V ðSxÞ; and (d) the variance sq

all. Also, the component V ðSyÞ shows more

squeezing than that for V ðSxÞ, see Figs. 6(c)

and (d).

scaled time gt. The atom is initially in the excited state h = 0, /ntropy squeezing factor EðSxÞ; (b) the entropy squeezing factor

ueezing factor V ðSyÞ.

Fig. 8. The same as in Fig. 7 but h = p/3.


Case II (x = e). For the field frequency x isequal to the second harmonic generation fre-

quency e, and for the same set of parameters

as in Case I, we see a small amount of squeezing

occurs in EðSxÞ at several periods of time, see

Fig. 7(a) with irregular rapid fluctuations around

the maximum value for considerable period of

time. The component EðSyÞ shows no squeezing

and it fluctuates irregularly around its maximumvalue (0.6), (Fig. 7(b)). For the variance compo-

nents V ðSx;yÞ squeezing occurs after the onset of

the interaction with irregular fluctuations (Figs.

7(c) and (d)). For large value of k = 0.8k and

for initially excited atom (h = 0) we have very

similar behavior to that in Fig. 7. For the inter-

mediate state (h = p/3), Fig. 8 the component

EðSxÞ shows squeezing after it reaches its maxi-mum value, while the component EðSyÞ shows

squeezing at t = 0 and once more for t > 0 (for

very shorter time period).

The variance component V ðSx;yÞ has not shownany different behavior from that in Figs. 7(c) and

(d). So, away from the excited state the noise de-

creases and consequently the entropic uncertainty

function shows pronounced squeezing in one ( or

two) of its two quadrature measures EðSxÞ andEðSyÞ.

5. Conclusion

We have studied the effect of the second or-

der harmonic generation (degenerate parametric

amplifier) in the usual JC model, namely, atwo-level atom in constant coupling with a sin-

gle cavity mode of the electromagnetic field.

The solution of the wave function is obtained

within rotating wave approximation for two

different cases. In the first case the field fre-

quency x is not equal the frequency of the

splitting photon e, while in the second case

both frequencies are equal. A canonical trans-formation has been used in the first case to

obtain the wave function solution, while in

the second case the solution is obtained at ex-

act resonance (field frequency = atomic transi-

tion frequency). Our investigations for the

collapse and revival phenomena in the atomic

inversion and for the entropic uncertainty rela-

tions in the presence of the second harmonic


generation field (k 6¼ 0) as compared with the

usual JC model (k = 0) are summarized as

follows:

(i) The atomic inversion in Case I shows that forinitially intermediate atomic state (0<h < p/2)the pattern of collapse and revival is qualita-

tively similar to that of the JC model but with

reduced amplitude. In Case II for increasing

value of the parameter k and for initially

excited atom the usual pattern in the JC model

of collapse and revival in the atomic inversion

changes to rapid fluctuations of interferencepatterns for all time considered.

(ii) More squeezing shows in the entropic uncer-

tainty component EðSyÞ and the variance

component V ðSyÞ in Case I at exact reso-

nance (D = 0) and for initially intermediate

atomic state (h 6¼ 0). The non-zero value

(U 6¼ 0) for the atomic phase induces more

rapid fluctuations behavior. For D 6¼ 0,more squeezing is observed in the variance

component V ðSyÞ but not in the entropic

component EðSyÞ, while the component

EðSxÞ shows squeezing for lesser period of

time. In Case II: For initially excited atom(h = 0) squeezing occurs for lesser period

of time with irregular rapid fluctuations

about the maximum value for all entropic

and variance components EðSx;yÞ; V ðSx;yÞ.For initially intermediate atomic state

(h 6¼ 0) the noise in the system is less and

squeezing occurs in one (or the two) of

the entropic uncertainty measure EðSx;yÞ.Also, with atom initially in atomic state

(0 6 h < p/2) the presence of the second har-

monic field results in the fluctuation pattern

in all components EðSx;yÞ; V ðSx;yÞ for most

the time considered. Finally, we note that

our model is the usual JC model in a

high-Q cavity but with the presence of the

second harmonic generation field of the sin-gle cavity mode [16] not coupled to the

atom, i.e., the atom couples only to the first

harmonic of the single mode cavity and the

coupling with this ‘‘background’’ second

harmonic field component can be ignored

within rotating approximation.

Acknowledgment

S.S. Hassan acknowledges the hospitality of

the ICTP (Trieste Italy) as Senior Associate

where the final version of the paper wasprepared.

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Entropic uncertainty in the Jaynes–Cummings model in presence of a second harmonic generation

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