Phase properties of one- and two-photon Jaynes–Cummings models with a Kerr medium

Phase properties of one- and two-photon Jaynes–Cummings models with a Kerr medium

Ts Gantsog† k, A Joshi‡ and R Tana´s§

† Max-Planck-Institut f¨ur Quantenoptik, Hans-Kopfermann-Straße 1, D-85748 Garching, Germany

‡ Department of Mathematics, UMIST, PO Box 88, Manchester M60 1QD, UK

§ Nonlinear Optics Division, Institute of Physics, Adam Mickiewicz University, 60-780 Pozna´n, Poland

k Department of Theoretical Physics, National University of Mongolia, 210646 Ulaanbaatar, Mongolia

Received 4 April 1995

Abstract. The phase properties of the cavity field of one- as well as two-photon

in a Kerr medium are studied by means of a Pegg–Barnett Hermitian phase operator formalism. The time evolution of the phase probability distribution, the phase fluctuations and the number–phase correlation are obtained. The effect of nonlinear interaction of a Kerr-like medium on the phase properties is analysed by comparing our results with those of usual

.

1. Introduction

The quantum electrodynamic model involving the interaction of a single quantized mode of the electromagnetic field with a two-level atom known popularly as the Jaynes–Cummings model ( JCM ) [1] in the literature is a useful paradigm because it is simple and exactly solvable for arbitrary atom–field coupling coefficients and field strengths. Several predictions of the

JCM are well approximated by some recent cavity quantum electrodynamics experiments involving the passage of a single atom through superconducting microwave cavities [2, 3].

The JCM has predicted many unexpected and non-trivial results, e.g. the well known phenomenon of collapses and revivals of Rabi oscillations—unquestionable evidence of the quantum nature of the electromagnetic field [4]. The other non-trivial and fascinating quantum features displayed by the JCM are squeezing [5], generation of sub-Poissonian photon statistics [6], chaotic behaviour in the semiclassical limit [7] and its sensitivity on photon statistics [8].

In the recent past, the JCM has been generalized to the case of multilevel atoms [9]

interacting with one or more discrete field modes, or to the case involving a multiphoton transition through the intermediary of virtual levels. Specifically, attention has been given to the case in which an atom undergoes a non-resonant two-photon process (known as the two-photon JCM ) and the study of collapse and revival phenomenon of Rabi oscillation is reported [10]. Further, both one- and two-photon JCM s have been extended to include the effects of a Kerr-like medium [11] and some significant results for the dynamical evolution of the atom, the quasiprobability distribution of the cavity field and the emission spectrum are reported.

1355-5111/96/030445+12$19.50 1996 IOP Publishing Ltd c 445

In the present paper we investigate the phase properties of the one- and two-photon JC

models in a Kerr-like medium within the Pegg–Barnett Hermitian phase-operator formalism.

In particular, we study the phase probability distribution, the phase variance, and number- phase correlation function. The effects of nonlinear interaction, i.e. with the Kerr-like medium, on the phase properties are discussed by comparing these results with the usual JC

model results.

2. The models and their solutions

2.1. An atom undergoing one-photon transition or one-photon JCM

A single two-level atom in a single-mode cavity is surrounded by a Kerr-like medium which can be modelled as an anharmonic oscillator [11]. The two-level atom undergoing one-photon transition is coupled to the cavity field which has a nonlinear interaction with the Kerr medium. Let ω _a and ω ₀ be the atomic transition frequency and the cavity field frequency, respectively. The effective Hamiltonian of the system in the rotating-wave approximation can be written as [11]

H = ω ⁰ a ^† a + ω ^a S z + g a S ⁺ + a ^† S ⁻
+ χ a ^†2 a ² (1) where a and a ^† are the cavity field annihilation and creation operators, S _z and S ^± are the atomic pseudo-spin operators, g is the atom–field coupling constant and χ is the third-order susceptibility representing the dispersive part of the third-order nonlinearity of the Kerr-like medium. The Hamiltonian (1) consists of two obvious constants of motion

H = H ⁰ + H ^I [H 0 , H I ] = 0 (2)

where

H 0 = ω ⁰ a ^† a + S ^z

(3) H I = 1 S ^z + g a S ⁺ + a ^† S ⁻
+ χ a ^†2 a ² (4)

1 = ω ^a − ω ⁰ . (5)

This allows us to factor out exp( −iH ⁰ t ) from the evolution operator and, in fact, to drop it altogether. In effect, the resulting state of the system can be written as

|9(t)i = exp(−iH ^I t ) |9(0)i (6)

where |9(0)i is the initial state vector of the system.

Let the atom be initially in the excited state |ei and the cavity field be in the coherent state

|αi = X ∞ n =0

Q _n exp(inϕ) |ni Q _n = exp(− ¯n/2) ¯n ^n/2

√ n! (7)

where ¯n = |α| ² is the average photon number of the initial coherent field and ϕ is the phase angle (α = |α| exp(iϕ)), the initial state vector of the system (assuming atom and field are decoupled) then reads as

|9(0)i = |ei ⊗ |αi = X

n

Q n exp(inϕ) |e, ni . (8)

With the initial conditions (8) the resulting state (6) is given by [11]

|9(t)i = X ∞ n =0

Q n e ^inϕ exp( −iχtn ² )[C n (t ) |g, n + 1i + D ⁿ (t ) |e, ni] (9)

where C n (t ) and D n (t ) are

C n (t ) = − i sin 2β ⁿ sin  n t D n (t ) = cos  ⁿ t − i cos 2β ⁿ sin  n t (10) with

sin 2β n = g √ n + 1

 n

cos 2β n = 1/2 − χn

 n

 ² _n = (1/2 − χn) ² + g ² (n + 1) . (11)

2.2. Atom undergoing two-photon transition or two-photon JCM

The other very interesting case in cavity quantum electrodynamics is a single atom interacting with a single-cavity mode via a two-photon process. Let there be a Kerr- like medium in the cavity and let ω a and ω 0 represent the atomic transition frequency and the cavity field frequency, respectively. The Hamiltonian for this system is then given by [10, 11]

H = ω 0 a ^† a + ω a S _z + g a ² S ⁺ + a ^†2 S ⁻
+ χ a ^†2 a ² (12) where exact resonance means ω a = 2ω ⁰ . In equation (12) g is the atom–field coupling constant and χ is the dispersive part of the third-order nonlinearity of the Kerr medium.

Again the Hamiltonian (12) can be written as a sum of two constants of motion

H = H ⁰ + H ^I [H 0 , H I ] = 0 (13)

where

H 0 = ω ⁰ (a ^† a + 2S ^z ) (14)

H I = 1S ^z + g a ² S ⁺ + a ^†2 S ⁻
+ χ a ^†2 a ² (15)

1 = ω ^a − 2ω ⁰ . (16)

Assuming the atom to be initially in the excited state |ei and the cavity field to be in the coherent state (defined by equation (7)), with the initial state vector as defined in equation (8), it is straightforward to write the state vector of the system at any time t as

|9(t)i = X ∞ n =0

Q n e ^inϕ exp[ −iχt (n ² + n + 1)][C ⁿ (t ) |g, n + 2i + D ⁿ (t ) |e, ni] (17) in which

C n (t ) = − i sin 2β ⁿ sin  n t D n (t ) = cos  ⁿ t − i cos 2β ⁿ sin  n t sin 2β n = g √

(n + 1)(n + 2)

 n

cos 2β n = 1/2 − χ(2n + 1)

 n

(18)

 ² _n = [1/2 − χ(2n + 1)] ² + g ² (n + 1)(n + 2) .

3. Phase properties

To describe quantum phase properties of the JCM with a Kerr medium we apply the Hermitian

phase operator formalism introduced by Pegg and Barnett [12]. This formalism is based

on introducing a finite (s + 1)-dimensional space 9 spanned by the number states |0i, |1i,

. . . , |si, for a given mode of the field. The Hermitian phase operator operates on this finite

space and, after all necessary expectation values have been calculated in 9, the value of s

is allowed to tend to infinity. A complete orthonormal basis of (s + 1) states is defined on 9 as

|θ ^m i ≡ 1

√ s + 1

s

X

n =0

exp(inθ m ) |ni (19)

where

θ m ≡ θ ⁰ + 2π m

s + 1 m = 0, 1, . . . , s . (20)

The value of θ 0 is arbitrary and defines a particular basis set of (s + 1) mutually orthogonal phase states. The Hermitian phase operator is defined as

ˆφ θ ≡

s

X

m =0

θ m |θ ^m ihθ ^m | (21)

where the subscript θ indicates the dependence on the choice of θ 0 . The phase states (19) are eigenstates of the phase operator (21) with the eigenvalues θ m restricted to lie within a phase window between θ 0 and θ 0 + 2π.

The expectation value of the kth power of the phase operator (21) in a state described by the density operator % is given by

h ˆφ θ ^k i = Tr { % ˆφ ^k θ } =

s

X

m =0

θ _m ^k hθ ^m | % |θ ^m i (22)

where hθ ^m | % |θ ^m i gives a probability of being found in the phase state |θ ^m i. The density of phase states is (s + 1)/2π, so in the continuum limit as s tends to infinity, we can write equation (22) as

h ˆφ θ ^k i =

θ

+2π

Z

θ

θ ^k P (θ ) dθ (23)

where the continuum phase distribution P (θ ) is introduced by P (θ ) = lim _s

→∞

s + 1

2π hθ ^m | % |θ ^m i

= 1 2π

X ∞ n,n

=0

%(n, n ⁰ ) exp[ −i(n − n ⁰ )θ ] (24) with θ m being replaced by the continuous phase variable θ . Here %(n, n ⁰ ) are the matrix elements of the density operator in the number state basis. Once the phase distribution function P (θ ) is known, all the quantum mechanical phase expectation values can be calculated with this function in a classical-like manner by integrating over θ . The choice of θ ₀ defines a particular window of phase values.

Of particular interest in the description of the phase properties of the field is the phase variance that can be calculated according to the formula

(1 ˆ φ) ² = Z

(2π )

θ ² P (θ ) dθ −

Z

(2π )

θ P (θ ) dθ

 2

. (25)

The phase probability distribution illustrates the phase structure of the field mode while the variance provides a good understanding of the evolution of the field phase fluctuations.

We use these field characteristics defined above to study the quantum phase properties

of the JCMs with a Kerr medium.

3.1. One-photon JCM

We can recast the state vector given by (9) as

|9(t)i = |9 ^g (t ) i |g i + |9 ^e (t ) i |e i (26) where

|9 ^g (t ) i = − i X

n

Q n e ^inϕ e ^{−iχt n}

sin 2β n sin  n t |n + 1i

|9 ^e (t ) i = X

n

Q n e ^inϕ e ^{−iχt n}

(cos  n t − i cos 2β ⁿ sin  n t ) |ni .

(27)

Now we can easily find the reduced density operator for the field by tracing over the atomic variables

%(t ) = Tr ^A |9(t)ih9(t)| = |9 ^g (t ) ih9 ^g (t ) | + |9 ^e (t ) ih9 ^e (t ) | . (28) Using equation (28) the phase probability distribution of the cavity field can thus be written as

P (θ, t ) = 1 2π



1 + 2 X

k>j

Q k Q j A kj (t ) cos (k − j)θ + χt (k ² − j ² ) 

+2 X

k>j

Q k Q j B kj (t ) sin (k − j)θ + χt (k ² − j ² ) 



(29) in which

A kj (t ) = cos( ^k t ) cos( j t ) + cos[2(β ^k − β ^j )] sin( k t ) sin( j t )

B _kj (t ) = cos(2β j ) cos( _k t ) sin( _j t ) − cos(2β k ) cos( _j t ) sin( _k t ) (30) A kj (t ) = A ^{j k} (t ) B kj (t ) = − B ^{j k} (t ) − π 6 θ 6 π .

An illustration of the time evolution of the phase probability distribution P (θ, t) (equation (29)) is shown in figure 1 for ¯n = |α| ² = 10, 1/g = 0, and for various values of χ /g. When t = 0, the phase distribution P (θ, t) has a single-peak structure corresponding to the initial coherent state. We know from the standard JCM [13] (i.e. χ /g = 0) that, if the cavity field is initially in a coherent state, then during the evolution of the system the single-peaked P (θ, t) function splits up into two peaks which move away from each other gradually (see figure 1(a)). The peaks are symmetric about θ = 0 so that the mean phase always remains equal to zero. The time behaviour of the phase probability distribution carries some information about the collapse and revival of Rabi oscillations [13]. When the phase peaks are well separated the Rabi oscillations collapse and each time as the peaks meet (at θ = 0 and/or ±π) they produce a revival (actually the region of gt where the phase peaks overlap at θ = 0 is not shown in figure 1(a)).

Added to a Kerr medium, the situation is completely changed. As shown in [14], during the propagation of a coherent field in a Kerr medium the phase distribution not only shifts, but also broadens. The same feature is also clearly visible in the case under consideration:

due to the Kerr nonlinearity the two peaks shift and broaden. However, as we observe

from figure 1(b), the diffusion rate is different for these two distributions. For χ /g = 0.1

(figure 1(c)) one of the two peaks practically disappears due to the Kerr nonlinearity. The

larger the χ , the stronger the phase diffusion which occurs (please note the different scale

of the gt axis in figure 1(c)). The close relation between the behaviour of the phase peaks

and the collapse and revival phenomena, observed for χ /g = 0, is lost.

Figure 1. Plots of the phase distribution P (θ, t) of the one-photon

with a Kerr medium as functions of scaled time gt for n = 10, 1 = 0, and various χ/g:

(a) χ /g = 0, (b) χ/g = 0.01, and (c) χ/g = 0.1.

The Hermitian phase operator expectation value and the phase fluctuation of the field can be expressed as

h ˆφi = 2 X

k>j

Q k Q j

( −1) ^{k −j}

k − j A kj (t ) sin χt (k ² − j ² )  − B ^kj (t ) cos χt (k ² − j ² ) 

(31) (1 ˆ φ) ² = π ²

3 + 4 X

k>j

Q k Q j

( −1) ^{k −j} (k − j) ²

× A kj (t ) cos χt (k ² − j ² )  + B ^kj (t ) sin χt (k ² − j ² )  − h ˆφi ² . (32)

We have plotted hni, h ˆφi and (1 ˆφ) ² in figure 2. When χ /g = 0 (figure 2(a)), the behaviour

is the usual JCM , h ˆφi = 0 and (1 ˆφ) ² oscillates around π ² /3. The behaviour of (1 ˆ φ) ² has

a direct bearing on the nature of P (θ, t): it reaches its extremum when the phase peaks

collide, and consequently when the Rabi oscillations revive [13]. With χ 6= 0 (χ/g = 0.01,

figure 2(b)), however, h ˆφi 6= 0 and regularity of (1 ˆφ) ² is lost, which becomes more

pronounced as χ /g increases further. Figure 2(c) shows the irregularity in h ˆφi and (1 ˆφ) ²

when χ /g = 0.1. There are sharp peaks observed with some kind of periodicity if we

further increase χ /g (χ /g = 0.5, figure 2(d)). Note that the collapse–revival phenomenon

in hni becomes less and less prominent as χ/g increases. For a strong coupling strength

of Kerr-like nonlinearity compared to the atom–field coupling the former starts dominating

the dynamics (there is nearly decoupling of the atom and field) and there is reintroduction

of some kind of regularity in the evolution of the system which is quite apparent from the

regular spikes present in figure 2(d).

Figure 2. Plots of mean photon number hni (upper full curve), phase variance (1φ)

(dotted curve), and mean phase hφi (lower full curve) of the one-photon

with a Kerr medium as functions of gt for n = 10, 1 = 0, and various χ/g: (a) χ/g = 0, (b) χ/g = 0.01, (c) χ /g = 0.1, and (d) χ/g = 0.5. The chain line marks the value π

/3—the variance of a randomly distributed phase.

As we have noticed above for χ = 0 and 1 = 0 we get h ˆφi = 0; but when χ 6= 0 we get an intensity-dependent phase shift. In other words, it is some kind of phase–

intensity correlation. To quantify the degree of this correlation, we define the number–phase correlation function as

D = h ˆn ˆφ + ˆφ ˆni − 2h ˆnih ˆφi . (33)

Using the definitions

ˆn =

s

X

n =0

n |nihn| (34)

and

ˆφ =

s

X

m =0

θ m |θ ^m ihθ ^m |

= θ ⁰ + sπ

s + 1 + 2π s + 1

s

X

j 6=k

exp[i(j − k)θ ⁰ ] |jihk|

exp[i(j − k) 2π/(s + 1)] − 1 (35)

it is easy to show that the expectation value of the number-phase anticommutator in a state

% is

h ˆn ˆφ + ˆφ ˆni = 2



θ 0 + sπ s + 1

 h ˆni

+ 2π s + 1

s

X

j 6=k

(j + k) exp[i(j − k)θ ⁰ ]

exp[i(j − k) 2π/(s + 1)] − 1 hk|%|ji . (36) For the physically realizable states we get (s → ∞)

h ˆn ˆφ + ˆφ ˆni = 2 (θ ⁰ + π) h ˆni + i

s

X

j 6=k

k + j

k − j exp[ −i(k − j)θ ⁰ ] hk|%|ji . (37) For the so-called partial phase states [12] of which the coherent state and the squeezed state are particular examples, we can write % kj in the form

% kj = e ^{i(k −j)ϕ} % ⁰ _kj (38)

where ϕ is the phase and % ⁰ _kj is a real number. Then we get

h ˆn ˆφ + ˆφ ˆni = 2 ϕ h ˆni h ˆφi = ϕ (θ 0 = ϕ − π) (39) hence we have zero correlation (D = 0). However, for the case under consideration due to Kerr nonlinearity we get non-zero number–phase correlation. From equation (37) with equation (28) we obtain for the expectation value of the number–phase anticommutator the expression

h ˆn ˆφ + ˆφ ˆni = 4 X

k>j

Q k Q j

( −1) ^{k −j}

k − j sin χt (k ² − j ² ) sin(2β k ) sin(2β j ) sin( k t ) sin( j t ) +2 X

k>j

Q _k Q _j k + j

k − j ( −1) ^{k −j} A kj (t ) sin χt (k ² − j ² ) 

−B ^kj (t ) cos χt (k ² − j ² ) . (40)

For χ /g = 0 we get D = 0 as expected. In figure 3 we have plotted D for χ/g = 0.01 (a) and χ /g = 0.1 (b). For very small nonlinearity D starts swinging around zero level (figure 3(a)) and becomes irregular if we further increase χ /g (figure 3(b)). For the high values of χ /g there are sharp spikes and regularity in the number–phase correlation function.

Again we can explain these features with the behaviour of the (1 ˆ φ) ² with respect to χ /g.

3.2. Two-photon JCM

The phase probability distribution of the cavity field in this case assumes the following form P (θ, t ) = 1

2π



1 + 2 X

k>j

Q k Q j A kj (t ) cos (k − j) θ + χt (k + j + 1)

+ 2 X

k>j

Q k Q j B kj (t ) sin (k − j) θ + χt (k + j + 1) 



(41)

in which A kj (t ) and B kj (t ) are as defined in equation (30) with  n , sin(2β n ) and cos(2β n )

given by equation (18). In figure 4 we have plotted P (θ, t) with respect to time for ¯n = 10,

1/g = 0, and for various values of χ/g. The χ/g = 0 case (figure 4(a)) clearly describes

the time evolution of the phase probability distribution of the field of the standard two-

photon JC model. As the time evolution proceeds the single peak of the initial coherent

Figure 3. Number–phase correlation function D of the one-photon

with a Kerr medium as functions of gt for n = 10, 1 = 0, and various χ/g: (a) χ/g = 0.01 and (b) χ/g = 0.1.

Figure 4. Plots of the phase distribution P (θ ) of the two-photon

with a Kerr medium as functions of scaled time gt for n = 10, 1 = 0, and various χ/g:

(a) χ /g = 0, (b) χ/g = 0.01 and (c) χ/g = 0.05.

state splits up into two peaks which move away from each other gradually. When gt = π,

the two peaks merge into a single peak at θ = ±π. Further, when gt = 2π the two

peaks meet again, but this time at θ = 0. Thus the time evolution diagram of the phase

probability distribution of the usual two-photon JCM is very regular, it is symmetric to θ = 0

and is periodic in time. This is due to the periodic nature of the interaction phenomenon in

the two-photon JCM . With a very small increase in nonlinearity (χ /g = 0.01, figure 4(b)),

the symmetry of P (θ, t) disappears due to the intensity-dependent phase shift caused by

the Kerr medium. This becomes more pronounced at higher values of the nonlinearity

(figure 4(c), χ /g = 0.05), where the periodicity is completely lost in the time evolution of

the phase probability distribution. In this case too, we believe that the unequal diffusion

Figure 5. Plots of mean photon number hni (upper full curve), phase variance (1φ)

(dotted curve) and mean phase hφi (lower full curve) of the two-photon

with a Kerr medium as functions of gt for n = 10, 1 = 0, and various χ/g: (a) χ/g = 0, (b) χ/g = 0.01, (c) χ /g = 0.05, and (d) χ/g = 1. The chain line marks the value π

/3—the variance of a randomly distributed phase.

of two phase peaks, which becomes more prominent with the increase of nonlinearity, is responsible for the loss of periodicity in the P (θ, t).

Next we write down the Hermitian phase operator expectation value and the phase fluctuation of the cavity field for the two-photon JCM with a Kerr-like medium

h ˆφi = 2 X

k>j

Q _k Q _j ( −1) ^{k −j}

k − j {A ^kj (t ) sin[χ t (k − j)(k + j + 1)]

−B ^kj (t ) cos[χ t (k − j)(k + j + 1)]}, (42)

(1 ˆ φ) ² = π ²

3 + 4 X

k>j

Q _k Q _j ( −1) ^{k −j}

(k − j) ² {A ^kj (t ) cos[χ t (k − j)(k + j + 1)]

+B ^kj (t ) sin[χ t (k − j)(k + j + 1)]} − h ˆφi ² . (43)

These quantities along with hni have been plotted in figure 5 for ¯n = 10, 1/g = 0 and

for various values of χ /g. When χ /g = 0 (figure 5(a)) h ˆφi = 0 and (1 ˆφ) ² oscillates

periodically near π ² /3 with the period 2π which is in contrast to the one-photon JCM . hni

shows a collapse and revival phenomenon which is compact and periodic. With χ non-zero

(figure 5(b), χ /g = 0.01) the scenario for both h ˆφi and (1 ˆφ) ² changes drastically. h ˆφi

oscillates about zero but in a quite irregular manner, (1 ˆ φ) ² no longer remains periodic,

nevertheless hni does show the same behaviour as in figure 5(a). Further increase in χ

Figure 6. Number–phase correlation function D of the two-photon

with a Kerr medium as a function of gt for n = 10, 1 = 0, and various χ/g: (a) χ/g = 0.05 and (b) χ/g = 1.

(figure 5(c), χ /g = 0.05) brings yet another change in the time behaviour of h ˆφi and (1 ˆ φ) ² . For higher values of χ (figure 5(d), χ /g = 1) we observe spikes and some sort of periodicity in both these quantities. These spikes become more regular at still higher values of χ , but hni tends to become flat as there is an increased tendency of population trapping [11] at such higher values of χ . This behaviour has a direct bearing on the evolution of P (θ, t) as also explained in the case of the one-photon JCM .

Finally, we look at the expression of the number–phase correlation function D for the two-photon JCM with a Kerr medium

h ˆn ˆφ + ˆφ ˆni = 8 X

k>j

Q k Q j

( −1) ^{k −j} k − j

× sin[χt (k − j)(k + j + 1)] sin(2β ^k ) sin(2β _j ) sin( _k t ) sin( _j t ) +2 X

k>j

Q _k Q _j k + j

k − j ( −1) ^{k −j}

×{A ^kj (t ) sin[χ t (k − j)(k + j + 1)] − B ^kj (t ) cos[χ t (k − j)(k + j + 1)]} . (44) As expected, when χ = 0 we get D = 0. However, even a small nonlinearity (χ/g = 0.05, figure 6(a)) brings a drastic change in the time evolution of D. There is a large swing in the values of D about zero on either side. On further increasing the χ these swings become shorter and shorter and spikes start appearing. The number–phase correlation becomes spiky but quite regular for much higher values of χ (χ /g = 1, figure 6(b)). This is because at higher values of nonlinearity there is a competition between the processes:

atom + field interaction and nonlinearity + field interaction. As a matter of fact the latter process dominates over the former and the system goes into a kind of regular evolution, characteristic of the Kerr medium [14].

4. Conclusions

In this paper we have considered the role of a Kerr-like medium on the quantum phase

properties of the field in the one- and two-photon JC models. The results show that phase

properties get considerably modified due to the presence of Kerr-like nonlinearity. In

particular, for both one- as well as two-photon JCM a very regular/symmetric (in the case

of the one-photon JCM ) or periodic (in the case of the two-photon JCM ) behaviour of phase properties become quite irregular as the coupling strength of the Kerr medium increases.

This happens because of the competing interaction of the field between the atom and the Kerr-like medium. Since the interest in Kerr-like media is quite relevant in relation to the so-called quantum non-demolition measurements hence, our results may be useful in that context.

Acknowledgments

One of us (TsG) would like to express his gratitude for the hospitality experienced at the Max-Planck-Institut f¨ur Quantenoptik. He also would like to thank the Alexander von Humboldt Foundation for financial support. RT thanks Polish Committee for Scientific Research (KBN) for financial support under grant no 2 P03B 128 8.

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