SQUEEZING AND AMPLITUDE-SQUARED SQUEEZING IN THE MODEL OF TWO NONIDENTICAL TWO-LEVEL ATOMS

January 19, 2007 10:20 WSPC/140-IJMPB 03653

International Journal of Modern Physics BVol. 21, No. 2 (2007) 145–157c© World Scientific Publishing Company

SQUEEZING AND AMPLITUDE-SQUARED SQUEEZING

IN THE MODEL OF TWO NONIDENTICAL

TWO-LEVEL ATOMS

E. K. BASHKIROV

Department of General and Theoretical Physics, Samara University,

1 Academ. Pavlov Str. Samara, Russia

[email protected]

Received 30 May 2005Revised 5 April 2006

Squeezing and amplitude-squared squeezing for two two-level nonidentical atoms in loss-less cavity has been investigated assuming the field to be initially in the coherent state.The time-dependent squeezing parameters have been calculated. The influence of therelative difference of two coupling constants on the squeezing parameters has beenanalyzed.

Keywords: Squeezing; amplitude-squared squeezing; two nonidentical atoms.

1. Introduction

The squeezed states of light were investigated intensively over the last few decades

both from theoretical and experimental points of view1 and attract considerable

attention because of their possible practical applications for high-precision opti-

cal measurements, optical communications and optical processing.2 A variety of

schemes for producing squeezed states has been proposed: four wave mixing, op-

tical parametric processes, second harmonic generation, Kerr effect, atom-cavity

coupling, diode lasers, twin photon beams, etc.3–5 The first observation of squeezed

states generated by four-wave mixing in an optical cavity has been performed by

Slusher et al.6 More than 40 different experiments have been reported at present.

The review of modern experiments and applications of squeezed light has been done

in a recent Guide of Bachor and Ralph,7 and the theoretical aspects of squeezing

have been discussed in a review of Dodonov.8

The possibility of squeezing phenomenon in Jaynes–Cummings model (JCM)

was analyzed by several authors starting with the Meystre and Zubairy.9–16 The

multiphoton, nondegenerate two-mode and two-atom generalizations of JCM have

also been demonstrated to produce squeezing.17–20 The field squeezing in the two-

atom JCM with one and multiphoton transitions has been investigated in several

papers for initial coherent, squeezed, vacuum and thermal field input.18–22 Last

145

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146 E. K. Bashkirov

years some interest has arisen in a higher-order squeezing.23 One type of higher-

order squeezing, namely, squeezing of the square of the field amplitude or in brief the

amplitude-squared squeezing (ASS) has been proposed by Hillery.24 The ASS has

been shown to exist in one- and multiphoton JCM25,26 and two-atom JCM.21,22,29

The experiments for generating a light squeezing in atom-cavity system have

been performed only for high densities of atoms in cavity. Placing the atoms inside

a cavity with modes tuned close to the atomic resonance makes a coupled system

with a strong nonlinearity. In one-atom case, the generation of motional squeezed

state of a trapped atom has been observed in the experiments with 9Be+ ion con-

fined in a Paul trap.32 At present there are two distinct sets of experiments that

are successful in generating quadrature squeezed light in atom-cavity systems.7 The

first experiments by Orozco et al. used a Doppler-free Na beam.30 The atoms were

optically pumped into sublevels and entered a cavity, which was tuned close to res-

onance with the transition. A systematic search for the optimum squeezing found

that the best condition was a cavity that is close to bistable operation. In the exper-

iment the cavity is illuminated by a single mode of light from a tunable dye laser.

The laser frequency and the two squeezing sidebands are all within a single cav-

ity resonance. The experimental results show the best measured noise suppression

of −1.1 dB. An independent search for the optimum parameters of atom-coupling

squeezing was carried out by the group in Canberra.31 In their experiment Ba was

used, which allowed the simplification that the atoms have a two-level structure

and thus optical pre-pumping is no longer necessary. This investigation found a

second, completely separate case for the generation of squeezed light, namely the

high transmission regime of an atom-cavity system, which is tuned close to bistable

operation. In the experiment, a high temperature, high density Ba atomic beam

generates an optically thick sample of atoms inside a single sided optical cavity.

The intensity of the squeezed light is sufficient to cause interference with the local

oscillator, the power at each of the two detectors varies when the local oscillator

phase is scanned. The data from the time scan were converted into a squeezing

ellipse. In the considered experiment the noise suppression is found to be up to

20%. The obtained results cannot be modeled successfully as a standard Kerr in-

teraction and the whole complex interaction of the atoms with the cavity has to be

considered.31 While these experiments are an interesting example of the complexity

of atom-cavity coupling, they have not produced strong reliable noise suppression.

Recently, simple experiment in one-atom cavity quantum electrodynamics have

been proposed by Lutterbach and Davidovich33 to generate and detect the highly

squeezed states of the electromagnetic field. The basic suggested experimental

scheme is of close connection with the experiments of Haroche et al.34–36 The

squeezed states are proposed to construct using a superposition of coherent states.

The probability of getting the state which exhibits a squeezing in this scheme is

equal to the probability of detection of the atom in the ground state. The authors

are able to achieve the high values of squeezing using a few atoms.

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The Model of Two Nonidentical Two-Level Atoms 147

In this paper, we are concerned about the dynamics of squeezing in the simple

collective model with two nonidentical two-level atoms in lossless cavity. This model

has recently attracted considerable attention in the study of the collective atom-field

interaction. The exact solutions of this model for the lossless cavity and the exact

atom-field resonance has been calculated first for one-photon transitions by Zubairy

et al.37–39 for two-photon transitions by Jex40 and for m-photon transitions by Xu

et al.41 Based on these solutions both the collapse-revival phenomenon of the atomic

coherence for initial coherent,37–39 binomial42 and squeezed field state41 and the

photon statistics41,43 have been considered. The entanglement of two nonidentical

atoms, interacting with the thermal and coherent fields in the lossless cavity has

been studied in Refs. 44–46. Agarwal and co-authors have investigated the two-

photon absorption47 and large two-photon vacuum Rabi oscillations48 in a system

of two nonidentical atoms, taking into account the detuning. Kielich and co-authors

have considered the squeezing in the two-atom collective spontaneous emission and

resonance fluorescence.49–51 We will consider the second-order squeezing and ASS in

the system of two atoms with different coupling constants, which interacts with one-

mode coherent field in lossless cavity and analyze the dependence of the squeezing

on the relative difference of two coupling constants.

Let us consider a system of two nonidentical two-level atoms interacting with

a single-mode quantized electromagnetic field in a lossless resonant cavity via the

one-photon-transition mechanism. The Hamiltonian of the system considered in the

rotating wave approximation is

H = ~ωa+a +

2∑

i=1

~ω0Rzi +

2∑

i=1

~gi(R+i a + R−

i a+) , (1)

where a+ and a are the creation and annihilation operators of photons of the cavity

field, R+i and R−

i are the raising and the lowering operators for the ith atom, ω

and ω0 are the frequencies of the field mode and the atoms, gi is the coupling

constant between the ith atom and the field. We assume the field to be at one-

photon resonance with the atomic transition, i.e. ω0 = ω.

We denote by |+〉 and |−〉 the excited and the ground states of a single atom

and by |n〉 the Fock state of the electromagnetic field. The two-atom wave function

can be expressed as a combination of state vectors of the form |v1, v2〉 = |v1〉|v2〉,where v1, v2 = +,−. Let the atoms are initially in the ground state |−,−〉 and the

field is initially in a coherent state

|α〉 =

∞∑

n=0

exp

(

−|α|22

)

αn

√n!

,

where α = |α|eıϕ and n = |α|2 is the initial mean photon number or dimensionless

intensity of the cavity field.

The time-dependent wave function of the total system |Ψ(t)〉 obeys the

Schrodinger equation

ı~|Ψ(t)〉 = H |Ψ(t)〉 . (2)

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148 E. K. Bashkirov

Using the Hamiltonian (1) the wave function is found to be

|Ψ(t)〉 =

∞∑

n=0

exp[−ı(n − 1)ωt] exp

(

−|α|22

)

αn

√n!

× [C(n)1 (t)|+, +; n − 2〉 + C

(n)2 (t)|+,−; n − 1〉

+ C(n)3 (t)|−, +; n − 1〉 + C

(n)4 (t)|−,−; n〉] . (3)

With the help of formulas (1)–(3) we can obtain the equations of motion for

probability coefficients Cni (t). These equations must be written separately for n = 0,

n = 1 and n ≥ 2:

C(0)i = 0 (i = 1, 2, 3, 4) ; (4)

C(1)1 = 0 , C

(1)2 = −ıg1C

(1)4 , C

(1)3 = −ıg2C

(1)4 ,

C(1)4 = −ı(g1C

(1)2 + g2C

(1)3 ) ;

(5)

and for n ≥ 2

C(n)1 = −ı(g2

√n − 1C

(n)2 + g1

√n − 1C

(n)3 ) ,

C(n)2 = −ı(g2

√n − 1C

(n)1 + g1

√nC

(n)4 ) ,

C(n)3 = −ı(g1

√n − 1C

(n)1 + g2

√nC

(n)4 ) ,

C(n)4 = −ı(g1

√nC

(n)2 + g2

√nC

(n)3 ) .

(6)

For atoms initially prepared in the ground state we have the following initial

conditions for probability coefficients

C(n)4 (0) = 1 , C

(n)1 (0) = C

n)2 (0) = C

(n)3 (0) = 0 (n = 0, 1, 2, . . .) . (7)

The solutions of Eqs. (4)–(6) with initial conditions (7) are found to be

C(0)1 (t) = C

(0)2 (t) = C

(0)3 (t) = 0 , C

(0)4 (t) = 1 ; (8)

C(1)1 (t) = 0 , C

(1)2 (t) =

−ı sin(√

1 + R2t)√1 + R2

,

C(1)3 (t) =

−ıR sin(√

1 + R2t)√1 + R2

,

C(1)4 (t) = cos(

√1 + R2t)

(9)

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and for n ≥ 2

C(n)1 (t) =

2R√

(n − 1)n

β[cos(λ+t) − cos(λ−t)] ,

C(n)2 (t) =

−4ıR2(n − 1)√

n

β

{

λ2+ + (1 − R2)n

λ+[β − (1 + R2)]sin(λ+t)

− λ2− + (1 − R2)n

λ−[β + (1 + R2)]sin(λ−t)

}

,

C(n)3 (t) =

−4ıR(n− 1)√

n

β

{

λ2+ − (1 − R2)n

λ+[β − (1 + R2)]sin(λ+t)

− λ2− − (1 − R2)n

λ−[β + (1 + R2)]sin(λ−t)

}

,

C(n)4 (t) =

8R2(n − 1)n

β

[

cos(λ+t)

β − (1 + R2)+

cos(λ−t)

β + (1 + R2)

]

,

(10)

where

λ± =√

(1 + R2)(2n − 1) ± β/√

2 ,

β =√

(2n − 1)2(1 + R2)2 − 4(n − 1)n(1 − R2)2 ,

R = g2/g1 .

To investigate the second-order field squeezing, we introduce the slowly varying

quadrature components X1, X2:

X1 =1

2(aeıωt + a+e−ıωt) ,

X2 =1

2ı(aeıωt − a+e−ıωt) .

Thus [X1, X2] = ı/2, which implies the uncertainty relation (∆X1)2(∆X2)

2 ≥ 1/16,

where (∆Xi)2 = 〈X2

i 〉 − 〈Xi〉2 (i = 1, 2) are variances of quadrature components.

Normal squeezing occurs when variances satisfy the relation (∆Xi)2 < 1/4 (i = 1 or

2). The condition for squeezing one can write in the form Si < 0, where squeezing

parameters are

Si =(∆Xi)

2) − 1/4

1/4= 4(∆Xi)

2 − 1 (i = 1, 2) .

The value Si = −1 corresponds to 100% squeezing in ith quadrature component.

In terms of photon creation and annihilation operators we can rewrite squeezing

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150 E. K. Bashkirov

parameters in the form

S1 = 2〈a+a〉 + 2 Re〈a2e2ıωt〉 − 4(Re〈aeıωt〉)2 , (11)

S2 = 2〈a+a〉 − 2 Re〈a2e2ıωt〉 − 4(Im〈aeıωt〉)2 . (12)

Using Eq. (3) we can obtain

〈a+a〉 = n −[

2

∞∑

n=2

pn|C(n)1 |2 +

∞∑

n=1

pn(|C(n)2 |2 + |C(n)

3 |2)]

= A0 ,

eıωt〈a〉 = α

{

∞∑

n=2

pn(C(n)1 )∗C

(n+1)1

√

n − 1

n + 1+

∞∑

n=1

pn[(C(n)2 )∗C

(n+1)2

+ (C(n)3 )∗C

(n+1)3 ]

√

n

n + 1+

∞∑

n=0

pn(C(n)4 )∗C

(n+1)4

}

= αA1 ,

e2ıωt〈a2〉 = α2

{

∞∑

n=2

pn(C(n)1 )∗C

(n+2)1

√

(n − 1)n

(n + 1)(n + 2)

+

∞∑

n=1

pn[(C(n)2 )∗C

(n+2)2 + (C

(n)3 )∗C

(n+2)3 ]

√

n

n + 2

+

∞∑

n=0

pn(C(n)4 )∗C

(n+2)4

}

= α2A2 .

(13)

The parameter α of initial coherent state is α =√

n exp iϕ. Let below ϕ = 0. Then,

for squeezing parameters S1 and S2 one can write

S1 = 2A0 + 2nA2 − 4nA21 , (14)

S2 = 2A0 − 2nA2 . (15)

To define the squeezing of the square of the field amplitude or amplitude-squared

squeezing (ASS) we can introduce the quantities24

Y1 =1

2(a2e2ıωt + a+2e−2ıωt) ,

Y2 =1

2ı(a2e2ıωt − a+2e−2ıωt) .

The operators Y1 and Y2 correspond to the real and imaginary parts, re-

spectively, of the field amplitude squared and obey the commutation relation

[Y1, Y2] = i(2n + 1), where n = a+a. The uncertainty relation for these two quanti-

ties has the form

(∆Y1)2(∆Y2)

2 ≥ 〈n + 1/2〉2 .

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The ASS in Yi exists if (∆Yi)2 < 〈n + 1/2〉. Then, we can introduce the squeezing

parameters for ASS in the following form

Qi =(∆Yi)

2 − 〈n + 1/2〉〈n + 1/2〉 = 〈n + 1/2〉−1((∆Yi)

2 − 1) .

The ASS is obtained whenever Qi < 0 for i = 1 or i = 2 and Qi = −1 will

correspond to 100% ASS. In terms of photon creation and annihilation operators

we can rewrite ASS parameters in the form29

Q1 =1

4〈n + 1/2〉−1[2〈a+2a2〉 + 2 Re〈a4e4ıωt〉 − 4(Re〈a2e2ıωt〉)2] , (16)

Q2 =1

4〈n + 1/2〉−1[2〈a+2a2〉 − 2 Re〈a4e4ıωt〉 − 4(Im〈a2e2ıωt〉)2] . (17)

From Eq, (3) we have

〈a+2a2〉 =

∞∑

n=4

pn(n − 2)(n − 3)|C(n)1 |2

+

∞∑

n=3

pn(n − 1)(n − 2)[|C(n)2 |2 + |C(n)

3 |2] +

∞∑

n=2

pn|C(n)4 |2 = A3 ,

e4ıωt〈a2〉 = α2

{

∞∑

n=2

pn(C(n)1 )∗C

(n+4)1

√

(n − 1)n

(n + 3)(n + 4)

+∞∑

n=1

pn[(C(n)2 )∗C

(n+4)2 + (C

(n)3 )∗C

(n+4)3 ]

√

n

n + 4

+

∞∑

n=0

pn(C(n)4 )∗C

(n+4)4

}

= α2A4 . (18)

Taking into account Eqs. (13), (16)–(18) we can rewrite the ASS parameters Q1

and Q2 in the form

Q1 =1

4〈n + 1/2〉−1[2A3 + 2n2A4 − 4n2A2

2] , (19)

Q2 =1

4〈n + 1/2〉−1[2A3 − 2n2A4] . (20)

Using the expressions (11)–(20), we have calculated the squeezing parameters Si

and Qi for various initial photon numbers n and relative difference of two coupling

constants R.

Figure 1 presents the long time behavior of parameters S1 and S2 for n = 0.2

and R = 0.5. For small field intensities n as soon as t > 0 we observe negative values

of S1 (squeezing in the first field quadrature component) and positive values of S2.

As time goes on, S1 and S2 break into oscillation and reverse sign. The maximum

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152 E. K. Bashkirov

4 8 12g1t

-0.2

0.2

0.4

0.6

S1,S2

Fig. 1. Long-time behavior of the squeezing parameters S1 (solid line) and S2 (dashed line) formodel with n = 0.2 and R = 0.5.

degree of subsequent squeezing may be larger than that for the first squeezing. With

increasing n the degree of squeezing in S1 decreases.

Figures 2–5 present the short time behavior of squeezing parameter S1 (the

first squeezing) for different small field input intensities n and values of relative

difference of two coupling constants R. Obviously, that for the case R = 0 we deal

with a single atom and the case R = 1 corresponds to identical atoms. For small

input intensities n (let us say 0 ≤ n ≤ 0.3) the degree of first squeezing increases

with decreasing R (as R decreases from 1 to 0 the maximum degree of squeezing

increases from 20% to 27% for n = 0.2). For field intensities n ≈ 0.3 the maximum

degree of squeezing is insensitive to choice of R. But for larger intensity input (let us

say n > 0.3) the dependence of the degree of squeezing from R is reversed. When,

for instance, n = 0.4 the increasing R from 0 to 1 leads to increasing the degree of

squeezing from 18% to 28%. Note that at the beginning of time scale the squeezing

parameter S1 for model with two nonidentical atoms takes the positive values in

contrast to that for single atom or two identical atoms and the first squeezing of

S1 is reached with some delay time. But this feature is distinct only for relative

1 2 3 4g1t

-0.1

-0.2

-0.3

S1

Fig. 2. Short-time behavior of the squeezing parameter S1 for model with n = 0.2 and R = 0(solid line), 0.5 (dashed line) and 1 (dotted line).

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1 2 3 4g1t

-0.1

-0.2

-0.3

S1

Fig. 3. Same as Fig. 2 but with n = 0.4.

1 2 3 4g1t

-0.1

-0.2

-0.3

S1


0.5 1 1.5g1t

-0.05

-0.025

0.05

S1

Fig. 5. Short-time behavior of the squeezing parameter S1 for model with n = 1.0 and R = 0.1(solid line), 0.3 (dashed line), 0.5 (dashed line with small stroke) and 0.7 (dotted line).

large initial intensities. For n > 0.8 the R -dependence of the degree of squeezing

has nonmonotone character. Note that for large input intensities, the parameter S1

exhibits weak first squeezing and with increasing n the squeezing is vanished first

for intermediate values of R (see Fig. 5).

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154 E. K. Bashkirov

2 4 6 10 12g1t

-0.025

-0.05

0.025

0.05

Q1,Q2

Fig. 6. Long-time behavior of the ASS parameters Q1 (solid line) and Q2 (dotted line) for modelwith n = 0.8 and R = 0.5.

Figure 6 presents the long time behavior of ASS parameters Q1 and Q2 for

n = 0.2 and R = 0.5. These parameters for small input intensities are carried out

in much the same way as S1 and S2 but the amount of squeezing for ASS is less than

that for second-order squeezing. The maximum degree of squeezing in Q1 decreases

with increasing the parameter R. The dependence Q1 and Q2 from intensity n have

the more complicated character but for large intensities n the ASS is weak in both

components.

Figures 7–9 present the short time behavior of squeezing parameter Q1 (the

first ASS) for different field intensities n and different values of relative difference

of two coupling constants R. For small input intensities n (let us say 0 ≤ n ≤ 0.7)

the degree of first ASS increases with decreasing R (as R decreases from 1 to 0

the maximum degree of squeezing increases from 1.5% to 5% for n = 0.4). For

n > 0.7 the R -dependence of the degree of squeezing has nonmonotone character.

In particular, for model with n = 0.8 the maximum of ASS is equal to 6% when

R = 0.5. Similarly to second-order squeezing the first ASS appears with some delay

time when 0 < R < 1 and with an increase in the input intensity the ASS is

vanished primarily for intermediate values of R.

1 2g1t

-0.025

-0.05

Q1

Fig. 7. Short-time behavior of the squeezing parameter Q1 for model with n = 0.4 and R = 0(solid line), 0.5 (dashed line) and 1 (dotted line).

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0.5 1 1.5 2g1t

-0.05

-0.025

Q1


0.5 1g1t

-0.04

-0.02

0.02

0.04

Q1

Fig. 9. Same as Fig. 7 but n = 1.2.

Thus, we have considered the effects of second-order squeezing and amplitude-

squared squeezing of the cavity field mode in the model with two nonidentical

atoms. The squeezing holds for the atoms prepared initially in the ground state. We

have demonstrated that nonidentical atoms field squeezing can outperform identical

atoms field squeezing.

Acknowledgments

The author is indebted to Prof. V. L. Derbov for helpful discussions and support.

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