linear entropy and squeezing of the interaction between two quantum system described by su 1 1 and su 2 lie group in presence of two external terms

Linear entropy and squeezing of the interaction between two quantum system described by su 1, 1 and su2 Lie group in presence of two external terms M.. Infig.1awe consider the case of ab

described by su (1, 1) and su(2) Lie group in presence of two external terms

M Sebawe Abdalla, E M Khalil, A S.-F Obada, J Peřina, and J Křepelka

Citation: AIP Advances 7, 015013 (2017); doi: 10.1063/1.4973916

View online: http://dx.doi.org/10.1063/1.4973916

View Table of Contents: http://aip.scitation.org/toc/adv/7/1

Published by the American Institute of Physics

Linear entropy and squeezing of the interaction between two quantum system described by su (1, 1) and su(2) Lie group in presence of two external terms

M Sebawe Abdalla,1, aE M Khalil,2,3A S.-F Obada,2J Peˇrina,4

and J Kˇrepelka5

1Mathematics Department, College of Science, King Saud University, P.O Box 2455,

Riyadh 11451, Saudi Arabia

2Mathematics Department, Faculty of Science, Al-Azher University, Nassr City,

Cairo 11884, Egypt

3Mathematics Department, Faculty of Science, Taif University, P.O Box 888 Taif 21974,

Saudi Arabia

4Department of Optics and Joint Laboratory of Optics of Palack´y University and Institute

of Physics of AS CR, Faculty of Science, Palack´y University, 17 listopadu 12,

771 46 Olomouc, Czech Republic

5Joint Laboratory of Optics of Palack´y University and Institute of Physics of Physics of Academy

of Sciences of the Czech Republic, 17 listopadu 50a, 771 46 Olomouc, Czech Republic

(Received 2 October 2016; accepted 28 December 2016; published online 9 January 2017)

A Hamiltonian, that describes the interaction between a two-level atom (su(2) algebra) and a system governed by su(1, 1) Lie algebra besides two external interaction, is con-sidered Two canonical transformations are used, which results into removing the exter-nal terms and changing the frequencies of the interacting systems The solution of the equations of motion of the operators is obtained and used to discuss the atomic inver-sion, entanglement, squeezing and correlation functions of the present system Initially the atom is considered to be in the excited state while the other systems is in the Perelo-mov coherent state Effects of the variations in the coupling parameters to the external systems are considered They are found to be sensitive to changing entanglement,

variance and entropy squeezing © 2017 Author(s) All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license ( http://creativecommons.org/licenses/by/4.0/ ) [http://dx.doi.org/10.1063/1.4973916]

I INTRODUCTION

It is well known that the interaction between atom and electromagnetic field plays crucial role in the quantum optics On the other hand the interaction between three electromagnetic fields represents

an important nonlinear parametric interaction, which has played a significant role in several physical phenomena of interest, such as stimulated and spontaneous emissions of radiation, coherent Raman and Brillouin scattering In Brillouin scattering one finds that an intense monochromatic laser source induces parametric coupling between the two scattered electromagnetic fields and the acoustical phonons in the scattering medium In Raman scattering a similar coupling occurs between the scattered Stokes and anti-Stokes waves and the optical phonons of a Raman active medium.1 3Note that the interaction between three electromagnetic fields (which is of nonlinear type) can be transformed into either parametric amplifier or parametric frequency converter; the first type leads to amplification

of the system energy while the second type leads to the energy exchanges between modes.1 8 This depends on the nature of the used approximation The most familiar Hamiltonian representing such

a system is given by

ˆ

H

} = ω1ˆa†ˆa+ ω2ˆb†ˆb+ ω3ˆc†ˆc + g

ˆa†ˆbˆc + ˆaˆb†

a E-mail: m.sebaweh@physics.org

2158-3226/2017/7(1)/015013/12 7, 015013-1 © Author(s) 2017

where ωi , i = 1,2,3 are the field frequencies and g is the coupling parameter, while ˆa†, ˆb†, ˆc† ˆa, ˆb, ˆc

are creation (annihilation) operators fulfilling the relations

f

ˆa, ˆa†g = fˆb, ˆb†g = fˆc, ˆc†g = 1 (2) Now if we consider the Schwinger transformations ˆJ+= ˆa†ˆb, ˆJ−= ˆaˆb†and ˆJ z=1

2(ˆn b−ˆn a ), ˆn a

= ˆa†

ˆa, ˆn b = ˆb†ˆb then we have

f ˆJ+, ˆJ−g = 2 ˆJ z, f ˆJ z, ˆJ±g = ± ˆJ± (3) ˆ

J±and ˆJ z are the angular momentum operators which belongs to su(2) Lie algebra In this case the Hamiltonian1is converted to the Tavis-Cummings model

ˆ

H

} = ω2ˆc†ˆc+ω0

2 Jˆz + g

ˆc†Jˆ−+ ˆc ˆJ+. (4)

This Hamiltonian is a generalization of the two-level atom interaction with a single mode field, known

as the Jaynes-Cummings model, described by the following form

ˆ

H

} = ω3ˆa†ˆa+ω0

2 ˆσz + g

ˆa†ˆσ−+ ˆa ˆσ+, (5) where ˆσ± and ˆσz are the standard Pauli operators which satisfy [ ˆσ+, ˆσ−]= ˆσz, [ ˆσz, ˆσ±]= ±2 ˆσ± Thus the interaction between the fields is transformed into atoms-field interaction

On the other hand if we define9

ˆ

K+= ˆb†

ˆc†, Kˆ−= ˆbˆc, Kˆz=1

2  ˆb†ˆb + ˆc†

ˆc+ 1

where

f ˆK+, ˆK−g = −2 ˆK z, f ˆK z, ˆK±g = ± ˆK± (7) and ˆK±and ˆK zare the generators of the su(1, 1) Lie algebra, the Casimir operator of which is given by

ˆ

K2= ˆK2

2  ˆK+Kˆ−+ ˆK−Kˆ+

, f ˆK2, ˆK z g = 0, f ˆK2, ˆK±g = 0 (8) Then the Hamiltonian (1) is transformed into the form describing the interaction between a field and an su(1,1) group system, given by

ˆ

H

} = ω4ˆa†ˆa + g

ˆa†Kˆ−+ ˆa ˆK+ (9)

In fact the su(1, 1) Lie algebra plays an important role in the description of linear dissipative processes in the Liouville space.10,11

Now if we use the Holstein-Primakoff representation12

ˆ

K+= p2s − 1 + ˆn c ˆc†, Kˆ−= ˆcp2s − 1 + ˆn c, Kˆz = s + ˆn c, (10)

where ˆn c = ˆc†ˆc and s is a c-number, the Hamiltonian (5) takes the form

ˆ

H

} = ω ˆK z+ω0

2 ˆσz+ λ ˆK+ˆσ−+ ˆK−ˆσ+, (11) where λ= g(2s − 1 + ˆn c)−1 It has to be mentioned that (11), without the rotating wave approximation,

is converted to

ˆ

H

} =ω0

2 ˆσz + ω ˆK z+ λ ˆK+− ˆK−

The main purpose of this communication is to modify the above model to include two classical terms, one of them is described by su(1, 1) Lie algebra operators, while the other obeys the su(2) Lie algebra These two extra classical terms can be interpreted as the exhibition of the effect of the parametric amplification represented by su(1, 1) Lie algebra, while the other term plays the role of external driving force in sense of su(2) Lie algebra This will be achieved by studying the degree

of entanglement and the entropy as well as the different types of squeezing.13 In fact quantum

entanglement represents the correlations between physical systems that cannot be accounted for using classical physics For example, quantum information is qualitatively different from its classical counterpart, which is the origin of new ideas of quantum cryptography and quantum computing Information is defined in terms of the probabilities for certain events to occur, and the difference between classical and quantum information arises from the different ways in which the probabilities are calculated It is widely recognized that the control of quantum entanglement leads to new classes

of measurements, communication and computational systems, and in some cases it dramatically changes non-quantum analogies The research on quantifying entangled states has been considered

by several authors, for example.14–17Also in the present paper we shall consider the Glauber second order correlation function in order to examine the classical and non-classical effects of the system However, we start our study of the system by considering the atomic inversion where we can discuss the collapses and revival phenomenon and show how it differs from the standard case The suggested model can be realized in a cavity containing a two-level atom and nonlinear optical parametric medium, both of which are pumped by external strong coherent beams The organization of the paper

is as follows: in the sec.IIwe modify the Hamiltonian (12) to include two external terms Sec.III

is devoted to the introduction of the Heisenberg equations of motion and finding their solutions In sec.IVwe consider the atomic inversion behavior examining the collapses and revivals phenomenon The discussion of the degree of entanglement is given in sec.V, which is followed by a study of squeezing, introducing three kinds of the squeezing in sec.VI The correlation function is studied in sec.VIIand our conclusions are given in sec.VIII

II THE HAMILTONIAN MODEL

We devote this section to introduce the Hamiltonian model which represents the interaction between an su(1, 1) and an su(2) under the action of two classical terms To do so we consider the Hamiltonian which represents the interaction between two fields of the parametric amplifier type This Hamiltonian is given by

ˆ

H

} = ω ˆK z+ iλ ˆK+− ˆK−

Now suppose that a two-level atom is injected into the cavity Then the system will be affected

by the atom and Hamiltonian takes the form of eq.(12) On the other hand when we consider the effect of two external quantum systems, one of su(1, 1) and the other of su(2), the Hamiltonian is given by the following form

ˆ

H

} =ω0

2 ˆσz + ω ˆK z+ λ ˆK+− ˆK−

 ( ˆσ−−ˆσ+)+ λ1 ˆK++ ˆK− + λ2( ˆσ−+ ˆσ+) (14) The first and the last terms describe the two-level atom and its interaction with a classical field, while the second and the fourth term display the su(1, 1) system, and finally the third term describes the interaction between the two-level atom and the su(1, 1) amplifier system

As one can see it is difficult to tackle such problem, therefore to simplify matters we eliminate the terms multiplied by λi (i = 1,2) For this reason we introduce two kinds of the transformations,

the first is

*

,

ˆ

K+

ˆ

K−

ˆ

K z

+ /

-= *

,

cosh2ϑ sinh2ϑ −sinh 2ϑ sinh2ϑ cosh2ϑ −sinh 2ϑ

−1

2sinh 2ϑ −12sinh 2ϑ cosh 2ϑ

+ /

-*

,

ˆ

R+

ˆ

R−

ˆ

R z

+ /

-(15) and the second transformation is given by

*

,

ˆσ+

ˆσ−

ˆσz

+

-= *

,

cos2η − sin2η 1

2sin 2η

−sin2η cos2η 1

2sin 2η

−sin 2η − sin 2η cos 2η

+ /

-*

,

ˆ

S+

ˆ

S−

ˆ

S z

+ /

here we take

ϑ =1

2tanh

−1 2λ1 ω

! , η =1

2tan

−1 2λ2 ω

!

From eqs (15) and (16) taking together, the Hamiltonian (14) is simplified to

ˆ

H

} =Ω0

2 Sˆz+ Ωz Rˆz+ λ ˆR+Sˆ−+ ˆR−Sˆ+

(18) with

Ωz=qω2−4λ21 and Ω0=qω2

0+ 4λ2

From eq (19) we note that the value of the frequency Ωz is less than ω due to the coupling parameter λ1, while the frequency Ω0 increases compared to ω0 by the coupling parameter λ2 Consequently we can control the behavior of the present system through the couplings to the external fields λ1and λ2

In Sec.IIIwe employ the Heisenberg equations of motion with the Hamiltonian (18) to derive the dynamical operators

III THE EQUATIONS OF MOTION AND THEIR SOLUTION

The Heisenberg equation of motion for any dynamical operator ˆQ is given by

d

dt Qˆ = 1 i}f ˆQ, ˆ Hg +∂ ˆQ ∂t (20) Therefore the Heisenberg equations of motion for the operators appearing in eq (18) lead to the following equations:

d ˆR z

dt = −iλ ˆR+Sˆ−− ˆR−Sˆ+

, d ˆS z

dt = 2iλ ˆR+Sˆ−− ˆR−Sˆ+

,

d ˆR+

dt = iΩz Rˆ++ 2iλ ˆR z Sˆ+, d ˆS+

dt = iΩ0Sˆ+−iλ ˆR+Sˆz,

d ˆR−

dt = −iΩz Rˆ−−2iλ ˆR z Sˆ−, d ˆS−

dt = −iΩ0Sˆ−+ iλ ˆR−Sˆz. (21) These equations can be converted to the second order differential equations

d2Rˆz

dt2 + 4 ˆC2Rˆz = 4 ˆC2N − ∆ ˆˆ C, d

2Sˆz

dt2 + 4 ˆC2Sˆz = 2∆ ˆC. (22) Also we have

d2Sˆ+

dt2 −2iΩz + ˆCd ˆS+

dt −

"

2λ2 N −ˆ 1

2

! + 2Ωz Cˆ +Ωz

2

! # ˆ

S+= 0, (23) and

d2Sˆ−

dt2 + 2i

Ωz− ˆCd ˆS−

dt +

"

2λ2 Nˆ +1

2

! + 2Ωz C −ˆ Ωz

2

! # ˆ

S−= 0 (24) Furthermore

d2Rˆ+

dt2 −2iΩz + ˆC d ˆR+

dt −

"

2λ2 N −ˆ 1

2

! + 2Ωz Cˆ+Ωz

2

! # ˆ

R+= 0, (25) finally

d2Rˆ−

dt2 + 2i

Ωz− ˆC d ˆR−

dt +

"

2λ2 Nˆ+1

2

! + 2Ωz C −ˆ Ωz

2

! # ˆ

R−= 0 (26) Here we defined

ˆ

C=∆

2Sˆz+ λ ˆR+Sˆ−+ ˆR−Sˆ+

, Nˆ= ˆR z+1

2Sˆz, ∆= Ω0− Ωz (27)

It is to be noted that ˆC and ˆ N commute with each other and with ˆ H, hence they are constants of

motion In the following we look for the solution ˆC of these equations.

A The general solution

In this subsection we introduce the general solution of the above equations in terms of the new operators ˆR z (t) and ˆ R±(t) as well as ˆ S z (t) and ˆ S±(t) From the above equations we have the following

solutions

ˆ

R z (t) = ˆR z(0) cos 2 ˆCt − iλ

"

 ˆR+(0) ˆS−(0) − ˆR−(0) ˆS+(0) sin 2 ˆCt

2 ˆC

#

− ˆR−(0) ˆS+(0)+ 2 N −ˆ ∆

4Cˆ

! sin2Ctˆ ˆ

and

ˆS z (t) = ˆS z(0) cos 2 ˆCt+ iλ

"

 ˆR+(0) ˆS−(0) − ˆR−(0) ˆS+(0) sin 2 ˆCt

2 ˆC

#

− ˆR−(0) ˆS+(0)+ ∆ ˆCsin2Ctˆ

ˆ

For the other operators we get

ˆ

R−(t)= e−i(Ωz− ˆC)t

"

cos ˆµ1t − i Cˆ

ˆ

µ1 sin ˆµ1t

! ˆ

R−(0) − 2i λ

ˆ

µ1 sin ˆµ1t ˆ R z(0) ˆS−(0)

#

(30) and

ˆ

S−(t)= e−i(Ωz− ˆC)t

"

cos ˆµ1t − i(∆+ ˆC)

ˆ

µ1 sin ˆµ1t

! ˆ

S−(0)+ iλsin ˆµ1t

ˆ

µ1 Rˆ−(0) ˆS z(0)

#

The Hermitan conjugate quantities for the last two equations are given by

ˆ

R+(t)= ei(Ωz + ˆC)t" cos ˆµ2t − i Cˆ

ˆ

µ2sin ˆµ2t

! ˆ

R+(0)+ 2i λ

ˆ

µ2sin ˆµ2t ˆ R z(0) ˆS+(0)

#

(32) and

ˆ

S+(t)= ei(Ωz + ˆC)t" cos ˆµ2t+ i(∆ − ˆC)

ˆ

µ2 sin ˆµ2t

! ˆ

S+(0) − iλsin ˆµ2t

ˆ

µ2 Rˆ+(0) ˆS z(0)

#

where

ˆ

µ2

1= ˆC2+ 2λ2 Nˆ +1

2

! , µˆ2

2= ˆC2−2λ2 N −ˆ 1

2

!

The discrete representation of the su(1, 1) Lie algebra satisfies

ˆ

R2|m; ki = k (k − 1) |m; ki, Kˆz|m; ki = (m + k) |m; ki,

ˆ

R+|m; ki=√(m + 1) (m + 2k)|m + 1; ki,

ˆ

R−|m; ki=√m (m + 2k − 1)|m − 1; ki,

(35)

where |m; ki are the state vectors, for more details see ref ?.

On the other hand the inverse transformation of eq (15) takes the form

*

,

ˆ

R+

ˆ

R−

ˆ

R z

+ /

-= *

,

cosh2ϑ sinh2ϑ sinh 2ϑ sinh2ϑ cosh2ϑ sinh 2ϑ

1

2sinh 2ϑ 12sinh 2ϑ cosh 2ϑ

+ /

-*

,

ˆ

K+

ˆ

K−

ˆ

K z

+ /

In Sec.IVwe consider the atomic inversion so that we can discuss the phenomenon of collapse and revival in the present system

IV THE ATOMIC INVERSION

The main task of this section is to discuss the phenomenon of collapse and revival This is achieved by calculating the expectation value of the operator ˆσz To do so we use the modified Perelomov coherent state as the initial state for su(1, 1) given by

|R, mi= exp β+Rˆ++ β0Rˆz+ β−Rˆ−

|0, ki, where β0= z∗−z
sinh 2ϑ,

β+= zcosh2ϑ − z∗

sinh2ϑ, and β−= −z∗

cosh2ϑ + z sinh2ϑ (37) The exponential term in the state is factorized to take the form

exp β+Rˆ++ β0Rˆz+ β−Rˆ− = exp A+Rˆ+exp

f

ln(A0) ˆR0gexpA−Rˆ−

where

A±= β±sinh φ

"

φ cosh φ − β0

2 sinh φ

#−1

, A0=

"

cosh φ − β0

2φsinh φ

#−2

with φ=

s

β0 2

!2

In this case we have

|R, mi= exp

A+Rˆ+expfln(A0) ˆR0g|0, ki,

|R, mi= cosh φ − β0

2φsinh φ

!−2k ∞

X

Γ (2k + m) m!Γ (2k)

!1

A m+|m, ki. (40)

To reach our goal we consider the atom to be initially in its excited state while the quantum system is in the generalized coherent state given by the eq (40), this would enable us to find the atomic inversion using the equation

hˆσz(τ)i= h ˆS z(τ)i cos 2η − h ˆS x(τ)i sin 2η, (41)

from which we can discuss the phenomenon of collapses and revivals For this reason we plot fig.1 for different values of the coupling parameters λ1 and λ2, and for fixed value of the other

FIG 1 Atomic inversion for a two-level atom system initially in the excited state in interaction with su(1, 1) quantum system

which is initially in Peremelov coherent state for fixed values of r = 7, θ = π/4, k = 2 and ω = ω0 = 40, where a) λ 1 = λ 2 = 0, b) λ /λ = 20 and λ = 0, c) λ = 0, λ /λ = 19.99 and d) λ /λ = 20 and λ /λ = 19.99.

parameters For example, we take ω= ω0= 40λ, k = 2 and z = r exp(iθ) with r = 7 and θ = π/4 In

fig.1awe consider the case of absence of both external fields in which λ1= λ2= 0 where the function shows periods of collapses as well as sharp revivals As one can see the function shows collapse period after onset of the interaction, this is followed by a sharp revival where the function fluctuates around

zero between1 and 0.5 In the second period of the revival the function shifts its amplitude and fluctuates between0.6 and 0.9, as the time increases the function repeats this behavior We also note that the revivals are very short On the other hand when we consider absence of the su(1,1), λ2/λ = 20, while λ1= 0, different observation is seen, in this case the function decreases its amplitude and shows regular slowly oscillating envelope, however during the revivals period there are small spread of the fluctuations which increases as the time increases This means that the energy is almost concentrated

in the quantum system, see fig 1b For the case of absence of the su(2) external term in which

λ1/λ = 19.99 while λ2= 0, the function shows dense fluctuations with small amplitudes (pseudo-collapses) and slight shift up and displays a dense spread of fluctuations around the revival periods This is due to the atomic effect of the external su(1,1) term, during these periods, see fig.1c Finally we discuss the effect of both external terms λ2/λ = 20 and λ1/λ = 19.99, in this case it also shows slowly oscillatory envelope as well as revival periods For instance the function decreases its amplitude and starts with a short period of revival but the revival gets pronounced at the maximum points of the envelope curve of‘ partial collapse However, the function has larger amplitude at the second period

of revival which occurs at the minimum point of the envelope curve This behavior is repeated as the time increases, see fig.1d It may be concluded that the external su(2) term adds an oscillating while the su(1,1) external term makes very dense small fluctuations making pseudo-collapses

V THE ENTANGLEMENT

We devote this section to investigate the entanglement in the system through the linear entropy More precisely, we demonstrate the effect of the existence of the coupling parameters λ1and λ2on the entanglement To quantify the entanglement we write down the definition of the linear entropy

as18 , 19

where ξ(τ) is the well-known Bloch sphere radius, which has the form

ξ2

(τ)= hˆσx(τ)i2+ hˆσy(τ)i2+ hˆσz(τ)i2

= h ˆS x(τ)i2+ h ˆS y(τ)i2+ h ˆS z(τ)i2 (43) The Bloch sphere is a tool in quantum information, where the simple qubit state is successfully represented, up to an overall phase, by a point on the unit sphere, whose coordinates are the expectation values of the atomic set operators of the system This means that the entanglement is strictly related

to the behavior of observables of clear physical meaning The function χ(τ) ranges between 0 for disentangled bipartite and 1 for maximally entangled ones

In order to discuss the degree of entanglement we plot fig.2against the scaled time τ= λt For

this reason we apply the same parameters used to examine the atomic inversion When we consider the case λ1= λ2= 0 the function displays maximum entanglement located near the revivals of the atomic inversion as well as disentanglement through located around middle of the collapses and at the revivals periods of the atomic inversion (see fig.1a) for all the considered time, however we can see small periods of partial entanglement, see fig.2a For λ2/λ = 20 while λ1= 0 the case of absence

of the su(1,1) external term, the function decreases its amplitude and shows partial entanglement and does not reach its maximum value (full entanglement) or minimum value (disentanglement), see fig.2b This means that the interaction between atom and quantum system gets stronger than the previous case which is result of the existence of λ1 Also we examine the case in which λ2/λ = 19.99 and λ1= 0, as one can see the function reduced its minimum and shows disentanglement as well partial entanglement However, as the time increases the minimum value increases, we also noted that there are rapid fluctuations when the function approaches the minimum value and shows disentanglement, see fig.2c Finally we discuss the case when λ1/λ = 20 and λ2/λ = 19.99 In this case the function decreases its amplitude and shows partial entanglement, as the time increases the minimum value gets

FIG 2 The linear entropy against the scaled time τ to quantify the entanglement for a two-level atom system initially in the excited state in interaction with su(1, 1) quantum system which is initially in Peremelov coherent state for fixed values of

r = 7, θ = π/4, k = 2, ω = ω0 = 40 and where a) λ 1 = λ 2 = 0, b) λ 2 /λ = 20 and λ 1 = 0, c) λ 2 = 0, λ 1 /λ = 19.99 and d) λ 2 /λ = 20 and λ 1 /λ = 19.99.

far from the disentanglement points where the energy concentrated at both minimum and maximum values of the function, see fig.2d

VI THE SQUEEZING

In this section we discuss one of the important nonclassical phenomenon in the field of quantum optics, that is the squeezing phenomenon Here we consider three different types of squeezing, namely variance squeezing, entropy squeezing and normal squeezing

A Variance squeezing

The variance squeezing is built up on the concept of the uncertainty relations that discuss the quantum fluctuations In this case we have to use the entropic uncertainty relations for two-level system instead of the Heisenberg uncertainty relations The discussion of this argument is given in the refs.20,21As is well known for the quantum mechanical system with two physical observables represented by the Hermitian operators ˆA and ˆ B satisfying the commutation relation [ ˆ A, ˆ B] = i ˆC, the

Heisenberg uncertainty takes the form

h(∆ ˆA)2ih(∆ ˆB)2i ≥1

4|h ˆCi|

where h(∆ ˆA)2i= (hˆA2i − h ˆAi2) Consequently, the uncertainty relation for a two-level atom charac-terized by the Pauli operators ˆσx, ˆσyand ˆσz, satisfying the commutation relation [ ˆσx, ˆσy]= iˆσz, can also be written as ∆ ˆσx∆ˆσy≥1

2|hˆσzi| Fluctuations in the component ∆ ˆσα of the atomic dipole is said to be squeezed if ˆσα satisfies the condition

V ( ˆσα)= *

,

∆ˆσα−

s

hˆσzi 2

+

In this case we use different data to those in the previous sections, for example we consider a

fixed value of k = 2 and take ω= ω0= 1, consequently we have to change the value of the coupling parameter, this is to avoid the discontinuity which would appear in the augmented frequency To

discuss the variance squeezing we plot fig.3against this scaled time τ= λt for different values of λ1

and λ2 For λ1= λ2= 0 there is no squeezing where the quadrature V x(τ) (red solid line) reaches only

the value zero for a short period of time, while the other quadrature V y(τ) (blue dashed line) is far

away even from the zero value, see fig.3a When we consider λ2/λ = 0.3 and λ1= 0, the squeezing is observed in the second quadrature three times for a short period and no squeezing occurs in the first quadrature, see fig.3b When we examine the case in which λ1/λ = 0.3, although the first quadrature

approaches zero, however, the squeezing occurred in the second quadrature once at the middle of

the considered time, see fig.3c Different observation is in the case λ1/λ = 0.3 and λ2/λ = 0.3 where

the squeezing can be seen in both quadratures, but it starts in V y(τ) and as the time increases it is

observed in V x(τ) until it gets to be pronounced and then disappeared before the end of the considered time, see fig.3d The examinations show that increasing the coupling of the external system results

in increasing the amounts of squeezing in both quadratures

B Entropy squeezing

We now turn our attention to consider another kind of squeezing that is the entropy squeezing which is one of the important nonclassical phenomena To examine the entropic squeezing we use the Shannon information entropies of the two-level atom operators given by22 , 23

H( ˆσα)= −1

2( ˆσα+ 1) ln" 1

2( ˆσα+ 1)

#

−1

2(1 − ˆσα) ln

" 1

2(1 − ˆσα)

# , α = x, y, z. (46) The fluctuations in component ˆσα(α= x, y) of the atomic dipole are said to be squeezed in entropy

if the information entropy H( ˆσα) of ˆσαsatisfies the condition24 , 25

E( ˆσα)= *

,

δH( ˆσα) −p 2

δH( ˆσ z) +

-< 0, with δH( ˆσα)= exp H(ˆσα), where α= x or y. (47)

We plot fig.4against the scaled time τ= λt using the same data in Subsection VI A For the case in the absence of λ1 and λ2, the squeezing occurs in both quadrature E x(τ) (red solid line) and

E y(τ) (blue dashed line) where we can see exchange between the squeezing in the quadratures, see fig.4a Similar behavior is occurred when we consider λ1/λ = 0.3 and λ2= 0, however the squeezing

FIG 3 The variance squeezing against the scaled time τ for a two-level atom system initially in the excited state in interaction

with su(1, 1) quantum system which is initially in Peremelov coherent state for fixed value of r = 7, k = 2, ω= ω 0 = 1 and

θ = π/4 where a) λ = λ = 0, b) λ /λ = 0.3 and λ = 0, c) λ = 0 and λ /λ = 0.3 and d) λ /λ = 0.3, λ /λ = 0.3.

2< sup>( ˆσα+ 1) ln" 1< /sup>

2< sup>( ˆσα+ 1)

#

? ?1

2< sup> (1 − ˆσα) ln

"

2< sup> (1. .. pronounced and then disappeared before the end of the considered time, see fig.3d The examinations show that increasing the coupling of the external system results

in increasing the amounts of squeezing. .. types of squeezing, namely variance squeezing, entropy squeezing and normal squeezing

A Variance squeezing< /b>

The variance squeezing is built up on the concept of the uncertainty