Int J Theor PhysDOI 10.1007/s10773-013-1697-x

Effects of Cavity-Field Statistics on AtomicEntanglement in Two-Mode Jaynes-Cummings Model

Sudha Singh · Amrita

Received: 13 February 2013 / Accepted: 10 June 2013© Springer Science+Business Media New York 2013

Abstract We study the entanglement properties of a pair of two-level Rydberg atoms pass-ing one after another into a lossless cavity with two modes. The initial joint state of twosuccessive atoms that enter the cavity is unentangled. Interactions mediated by the cavityfield results in the final two-atom mixed entangled type state. The entanglement of forma-tion of the joint two-atom state as a function of the Rabi angle gt is calculated for Fock statefield, coherent field and thermal field respectively inside the cavity. We present a compara-tive study of two-atom entanglement corresponding to the different field statistics.

Keywords Two-mode Jaynes-Cummings (J-C) model · Micromaser · Fock state ·Coherent state · Thermal state

1 Introduction

Entanglement is one of the striking features of quantum mechanics that distinguishes quan-tum information theory from a classical one. It plays a key role in quantum information,quantum computation and quantum cryptography [1, 2]. The simplest scheme to investigatethe atom- field entanglement is the Jaynes-Cummings Model (JCM) [3] that describes theinteraction of a two-level atom with a single mode quantized radiation field. The model isexactly solvable in the framework of the rotating wave approximation and has been experi-mentally realized [4, 5]. Two-photon processes in atomic systems are important in QuantumOptics due to the high degree of correlation between the emitted photons. With the success-ful operation of a single mode two-photon maser in a high Q cavity [6] the discussion onthe generalized Two-photon Jaynes-Cummings Models have acquired added importance.

An important generalization of the model is the non-degenerate Two-mode Two-photonJaynes Cummings Model. In a non-degenerate two mode process a two-level Rydberg atom

S. Singh (�) · AmritaUniversity Department of Physics, Ranchi University, Ranchi 834008, Jharkhand State, Indiae-mail: sudhhasingh@gmail.com

S. Singhe-mail: vasudha_rnc1@rediffmail.com

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interacts with two different modes of a quantum electromagnetic field in a high Q cavity viaa non-degenerate two-photon transition.

In the present work we study the dynamical generation of entanglement between twotwo-level atoms mediated by various cavity fields through the above mentioned two-photonnon-degenerate process. Quantum entanglement between two atoms interacting with thecavity field in the above manner is of interest as it may produce stronger interaction due tothe correlation of the two photons that are involved in atomic transitions.

Since the atoms do not interact directly with each other, the properties of the radiationfield encountered produce considerable influence on the nature of atomic entanglement. Inthis paper we study the effect of different field statistics on the magnitude of two-atomentanglement.

In the last few years a number of entanglement measures for bipartite states has beenintroduced and analyzed [7–12]. Since the joint state of the two atoms emanating from thecavity is not a pure state, we use the well known measure appropriate for the mixed states,i.e. the entanglement of formation [8, 11, 12] to quantify the entanglement. We investigatethe effect of the statistics of different radiation fields on the quantitative dynamics of atomicentanglement.

The paper is organized as follows. In the following section, we describe the essentialfeatures of the model and obtain an expression for the system wavefunction. In Sect. 3 thedynamics of entanglement and the effect of different field statistics has been discussed. Inthis section we study the entanglement mediated by the Fock state field, the coherent fieldand the thermal field. A summary of our results is presented in the final section.

2 Description of the Model

We consider a model consisting of a lossless cavity with two modes ω1 and ω2 throughwhich two two-level Rydberg atoms pass one after another. The first atom interacts withthe cavity field as a result of which the cavity field statistics is changed. Since the cavityis lossless, the changes remain in an unaltered state until the second atom enters the cavity.Thus, the second atom interacts with the field with the changes made by its interaction withthe first atom. It has been shown earlier [13–15] that the two atoms get entangled in theprocess even though they do not interact directly, nor do they interact with the cavity field atthe same time as it happens in the Dicke system.

The non-degenerate two-photon Jaynes-Cummings model is an effective two level atominteracting with two different modes ω1 and ω2 of the field. The Hamiltonian for the modelin the rotating wave approximation is written as [16, 17].

H = �ω0

2σ3 + �ω1a

†1 a1 + �ω2a

†2 a2 + �g(B− + B+) (1)

where B− and B+ is given by

B− = a†1 a

†2 σ−; B+ = a1a2σ+ (2)

Here ωo is the transition frequency and g is the atom-field coupling constant, σ3 is the in-version operator and σ+, σ− are the Pauli raising and lowering operators respectively. a

†1, a1

and a†2, a2 are the creation & annihilation operators for mode 1 & 2 respectively. The σ ’s

and the a’s obey the following commutation relations.

[σ3, σ±] = ±2σ±, [σ+, σ−] = σ3,[a, a†

] = 1 (3)

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We consider the atoms in their upper states at the start of their individual journey throughthe cavity. Hence, the initial condition for the system when the first atom enters the field is

∣∣ψ(t = 0)

⟩ =∞∑

n1=0

∞∑

n2=0

Cn1Cn2 |a1, n1, n2〉 (4)

where |n1〉 and |n2〉 represent the cavity photon number states with an initial distributionPni

= |Cni|2 (i = 1,2). a1 represents the upper state of the first atom. At resonance (ω1 +

ω2) ≈ ω0 and the time evolution of the atom-field system wavefunction can be written as

∣∣ψ(t)⟩ = e−iH t

∣∣ψ(t = 0)⟩

(5)

We expand e−iH t and operate each term on the initial state |ψ(t = 0)〉. At resonance(ω1 + ω2) ≈ ω0, all the terms can be summed up giving us the wavefunction |ψ1(t)〉 in asimple form [18].

∣∣ψ1(t)

⟩ = cos tg[√

(n1 + 1)√

(n2 + 1)]|a1, n1, n2〉

− i sin tg[√

(n1 + 1)√

(n2 + 1)]|b1, n1 + 1, n2 + 1〉 (6)

where t is now the duration of the atom-field interaction.The second atom enters the cavity at a time t at which the first atom has already left the

cavity. Since Q = ∞, the cavity field statistics remain unchanged in the duration betweenthe first atom leaving the cavity and the second atom entering the cavity. Thus, the initialcondition for the second atom can be written as |ψ2(t = 0)〉 = |a2〉|ψ1(τ )〉 so that we get

∣∣ψ2(t = 0)

⟩ = cos tg√

(n1 + 1)√

(n2 + 1)|a1, a2, n1, n2〉− i sin tg

√(n1 + 1)

√(n2 + 1)|b1, a2, n1 + 1, n2 + 1〉 (7)

The state vector representing the two atoms and the cavity for t > 2τ is given by

∣∣ψ(t)

⟩a−a−f

= γ1|a1, a2, n1, n2〉 + γ2|a1, b2, n1 + 1, n2 + 1〉 + γ3|b1, a2, n1 + 1, n2 + 1〉+ γ4|b1, b2, n1 + 2, n2 + 2〉 (8)

Here a2 and b2 are the upper and lower states respectively of the second atom. The γ ′sare given by

γ1 = cos2 tg√

(n1 + 1)√

(n2 + 1)

γ2 = {cos tg

√(n1 + 1)

√(n2 + 1)

}{−i sin tg√

(n1 + 1)√

(n2 + 1)}

γ3 = {cos tg

√(n1 + 2)

√(n2 + 2)

}{−i sin tg√

(n1 + 1)√

(n2 + 1)}

γ4 = {−i sin tg√

(n1 + 2)√

(n2 + 2)}{−i sin tg

√(n1 + 1)

√(n2 + 1)

}

(9)

Here the second atom also interacts with the cavity field for the same time duration t .This can be arranged by a velocity selector as used in micro-maser devices. In derivingEqs. (6) and (7) we have used the following general properties of the combined atom and

Int J Theor Phys

field operators:

B+|a,n1, n2〉 = 0

B−|b,n1, n2〉 = 0

B+|b,n1, n2〉 = √n1

√n2|a,n1 − 1, n2 − 1〉

B−|a,n1, n2〉 = √(n1 + 1)

√(n2 + 1)|b,n1 + 1, n2 + 1〉

B+B−|a,n1, n2〉 = (n1 + 1)(n2 + 1)|b,n1 + 1, n2 + 1〉B−B+|a,n1, n2〉 = 0

(10)

where a and b are again the atomic upper and lower levels respectively.

3 Atom–Atom Entanglement and Effect of Field Statistics

The density operator of the two atoms and the field is given by

ρatom−atom−f ield = ∣∣ψ(t)

⟩⟨ψ(t)

∣∣ (11)

where |ψ(t)〉 is given by Eq. (8). The atom–atom density operator is given by

ρatom−atom = T rf ieldρatom−atom−f ield (12)

Since the joint state of two atoms emanating from the cavity is not a pure state, theentanglement of the two-atom system can be quantified by the concurrence as proposed byWootters [11, 12]. This has been widely used to study bipartite entanglement.

The concurrence of the system is given by

C(ρ) = max{0,√

λ1 − √λ2 − √

λ3 − √λ4} (13)

where λ are the four eigenvalues of the non-Hermitian matrix

R = ρ(σy ⊗ σy)ρ•(σy ⊗ σy) (14)

arranged in a decreasing order and ρ is the (4 × 4) density matrix of the two atom system.Entanglement can be quantified by another function, called the entanglement of formationEf (ρ), monotone of C. It can be defined as

Ef (ρ) = h

[1 + √

1 − C2(ρ)

2

](15)

where h(x) = −x log2 x − (1 − x) log2(1 − x).The cavity field statistics is expected to have considerable influence on the atom-atom

entanglement [13–15]. In the following, we study the influence of the cavity field obeyingvarious statistics.

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3.1 Cavity Field in a Fock State

The reduced mixed density state of two-atoms after tracing over the field is given by (wedisplay the nonvanishing terms only)

ρa−a = γ 21 |a1a2〉〈a1a2| + γ 2

2 |a1b2〉〈a1b2| + γ2γ∗3 |b1a2〉〈a1b2|

+ γ ∗2 γ3|a1b2〉〈b1a2| + γ 2

3 |b1a2〉〈b1a2| + γ 24 |b1b2〉〈b1b2| (16)

When the cavity field is in a photon number state |n1, n2〉, the four eigenvalues of R inEq. (14) is given by

λ1 = 4|γ2|2|γ3|2λ2 = λ3 = |γ1|2|γ4|2 (17)

λ4 = 0

where γ ′s are given by Eq. (9). Hence the concurrence of the two atom bipartite system isgiven by

C = max{0, sin 2tg(

√n1 + 1

√n2 + 1) sin tg(

√n1 + 1

√n2 + 1 −√

n1 + 2√

n2 + 2)}

(18)

In Fig. 1 we plot the entanglement of formation Ef which is a monotone of concurrencefor

n1 = n2 = 0, n1 = n2 = 2 and n1 = n2 = 5.

It is observed that there is a finite entanglement induced by the vacuum field. This is dueto the fact that the first atom in excited state interacts with the vacuum cavity field and inthis process disturbs the field. So the cavity field is no longer in the vacuum state when thesecond atom enters. This makes the atoms entangled.

For n1, n2 = 0, the argument of the sine functions increase with increasing n. Hence, dur-ing these oscillations, the concurrence get non-zero more frequently. These characteristicsfor different values of n1, n2 are shown in Fig. 1.

The displayed characteristics are qualitatively similar to those obtained earlier in case ofatom-atom entanglement in the single mode Intensity Dependent Jaynes-Cummings Model[15]. As the photon number n1, n2 increases the entanglement of formation Ef increases.

Earlier it has been observed that the simultaneous interaction of two excited atoms withFock state field never results in two-atom entanglement [19]. On the other hand here, in thetwo-mode two-photon JC dynamics modelling the micromaser we always observe two-atomentanglement mediated by the Fock state cavity field.

3.2 Cavity Field in a Coherent State

The effect of field statistics on the atom–atom entanglement has been studied in the JaynesCummings Model and Intensity Dependent Jaynes Cummings Model and it has been noticedthat the atom-atom entanglement is sensitive to the field statistics [13–15].

In the present paper we study the effect of different field statistics of the two modes onthe atom-atom entanglement. We consider two kinds of statistical field, namely coherentfield and thermal field.

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Fig. 1 Evolution of atomicentanglement for both the atomsinitially in the excited state andthe field in a number state.(a1) n1 = n2 = 0(a2) n1 = n2 = 2(a3) n1 = n2 = 5

When we consider both the modes with frequency ω1 and ω2 to be in an initial coherentstate, the complete wavefunction for the model is written as

∣∣ψ(t)

⟩a−a−f

=∞∑

n1=0

∞∑

n2=0

Cn1Cn2

[γ1|a1, a2, n1, n2〉 + γ2|a1, b2, n1 + 1, n2 + 1〉

+ γ3|b1, a2, n1 + 1, n2 + 1〉 + γ4|b1, b2, n1 + 2, n2 + 2〉] (19)

where

∣∣Cni

(αi)∣∣2 = Pni

(ni) = ∣∣〈ni |αi〉

∣∣2 = exp(−ni )

nni

i

ni ! (i = 1,2) (20)

Pn1(n1) and Pn2(n2) represents the coherent field probability distribution functions for pho-ton numbers in the Poisson statistics. ni = |α|2 is the initial average photon number of thefield in the i-th mode. The coherent states |α〉 are given by.

|α〉 = exp

(−1

2|α|2

) ∞∑

n=0

αn

√n! |n〉 (21)

Since we are interested in calculating the entanglement of the joint two-atom state afterthe atoms emerge from the cavity, we consider the reduced density state ρ(t)a−a of the two

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atoms obtained after taking trace over the field variables.

ρ(t)a−a =∞∑

n1=0

∞∑

n2=0

[〈n1, n2|∣∣ψ(t)a−a−f

⟩⟨ψ(t)a−a−f

∣∣]|n1, n2〉 (22)

Making use of the above expression we obtain the matrix elements of ρatom−atom as:

ρ11 =∞∑

n1=0

∞∑

n2=0

|Cn1 |2|Cn2 |2 cos4 tg√

n1 + 1√

n2 + 1 (23a)

ρ12 =∞∑

n1=0

∞∑

n2=0

Cn1+1Cn2+1

(cos2 tg

√n1 + 2

√n2 + 2

)C∗

n1C∗

n2

× (i sin tg√

n1 + 1√

n2 + 1)(cos tg√

n1 + 1√

n2 + 1) (23b)

ρ13 =∞∑

n1=0

∞∑

n2=0

Cn1+1Cn2+1

(cos2 tg

√n1 + 2

√n2 + 2

)C∗

n1C∗

n2

× (i sin tg√

n1 + 1√

n2 + 1)(cos tg√

n1 + 2√

n2 + 2) (23c)

ρ14 =∞∑

n1=0

∞∑

n2=0

Cn1+2Cn2+2(cos2 tg

√n1 + 3

√n2 + 3

)C∗

n1C∗

n2

× (i sin tg√

n1 + 1√

n2 + 1)(i sin tg√

n1 + 2√

n2 + 2) (23d)

ρ21 = −ρ12 (23e)

ρ22 =∞∑

n1=0

∞∑

n2=0

|Cn1 |2|Cn2 |2(cos2 tg

√n1 + 1

√n2 + 1

)(sin2 tg

√n1 + 1

√n2 + 1

)(23f)

ρ23 =∞∑

n1=0

∞∑

n2=0

|Cn1 |2|Cn2 |2(cos tg√

n1 + 1√

n2 + 1)(sin2 tg

√n1 + 1

√n2 + 1

)

× (cos tg√

n1 + 2√

n2 + 2) (23g)

ρ24 =∞∑

n1=0

∞∑

n2=0

Cn1+1Cn2+1(cos tg√

n1 + 2√

n2 + 2)(−i sin tg√

n1 + 2√

n2 + 2)C∗n1

C∗n2

× (i sin tg√

n1 + 1√

n2 + 1)(i sin tg√

n1 + 2√

n2 + 2) (23h)

ρ31 = −ρ13 (23i)

ρ32 = ρ23 (23j)

ρ33 =∞∑

n1=0

∞∑

n2=0

|Cn1 |2|Cn2 |2(cos2 tg

√n1 + 2

√n2 + 2

)(sin2 tg

√n1 + 1

√n2 + 1

)(23k)

ρ34 =∞∑

n1=0

∞∑

n2=0

Cn1+1Cn2+1(−i sin tg√

n1 + 2√

n2 + 2)(cos tg√

n1 + 3√

n2 + 3)C∗n1

C∗n2

× (i sin tg√

n1 + 1√

n2 + 1)(i sin tg√

n1 + 2√

n2 + 2) (23l)

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Fig. 2 Evolution of atomicentanglement for both the atomsinitially in the excited state andthe field in Two-mode coherentstate with various photonnumbers. (a1) 〈n1〉 = 〈n2〉 = 10(a2) 〈n1〉 = 〈n2〉 = 20 (a3)〈n1〉 = 〈n2〉 = 40

ρ41 = ρ14 (23m)

ρ42 = −ρ24 (23n)

ρ43 = −ρ34 (23o)

ρ44 =∞∑

n1=0

∞∑

n2=0

|Cn1 |2|Cn2 |2(sin2 tg

√n1 + 1

√n2 + 1

)(sin2 tg

√n1 + 2

√n2 + 2

)(23p)

For a detailed analysis of atom-atom entanglement, Ef (ρ) in Eq. (15) needs to be eval-uated numerically. The entanglement of formation Ef is computed separately for low andhigh photon numbers. In Fig. 2 we plot Ef versus the Rabi angle gt for 〈n1〉 = 〈n2〉 = 10,〈n1〉 = 〈n2〉 = 20, 〈n1〉 = 〈n2〉 = 40. It is observed that the two atoms disentangle from eachother periodically and then get entangled again.

For gt = mπ or (m + 1)π/2(m = 0,1,2, . . .) all the four eigenvalues are zero so thereis no entanglement between the two atoms. It is observed that the maximum entanglementoccurs for gt = (2m + 1) π

4 . The periodic disentanglement is also known as collapse andrevival of two-qubit entanglement in the literature. This property in fact follows from thecollapse and revivals in population of atomic levels taking place for two-level atoms in acavity as seen in many previous works. During the time the atoms are disentangled, theentanglement gets transferred to the composite system comprising of one of the two atomsand the cavity field. In fact at times when concurrence becomes zero, the atom is in its purestates.

Here we have used equal average photon numbers for both the modes. It is very wellillustrated in the curves obtained that the magnitude of entanglement of the two atoms is seen

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Fig. 3 Evolution of atomic entanglement for both the atoms initially in the excited state and the fieldin Two-mode coherent state with various photon numbers (a) 〈n1〉 = 〈n2〉 = 0.5 (b) 〈n1〉 = 〈n2〉 = 0.1(c) 〈n1〉 = 〈n2〉 = 0.05, (d) 〈n1〉 = 〈n2〉 = 0.01

to increase with the average cavity photon numbers of the modes as seen in case of singlemode intensity dependent Jaynes-Cummings Model [15]. This is in contrast to the case ofstandard JC Model [20] and the Tavis-Cummings (TC) Model [21] where the magnitude ofatomic entanglement falls with increase of cavity photon number.

In Figs. 3(a), 3(b), 3(c) and 3(d) we plot the entanglement of formation Ef versus theRabi angle gt for n1 = n2 = 0.5,0.1,0.05 and 0.01 respectively. We notice that for low val-ues of n the plot shows a different nature and the peak value of Ef increases with decreaseof cavity photon number n = |α|2. Poisson distributions are uniquely characterized by theirmean value n. The distribution peaks at n and gets broader as n increases. Hence, we obtainsharp narrow peaks for small values of n.

The peaks of the entanglement of formation are reflective of the photon statistics that aretypical in micromaser dynamics [22, 23].

It is observed that for low photon numbers n1, n2, Ef falls off sharply as n1, n2 in-creases. Peaks of the coherent distribution function shifts to left as the average photon num-ber decreases in contrast to the case of thermal distribution function that always peaks atn ≈ 0.

Hence for small photon numbers the evolution of Ef for the coherent distribution func-tion is similar to the case when a thermal field is inside the cavity [24].

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Fig. 4 Time Evolution of atomic entanglement in thermal fields with various photon numbers(a) 〈n1〉 = 〈n2〉 = 0.5, (b) 〈n1〉 = 〈n2〉 = 0.1 (c) 〈n1〉 = 〈n2〉 = 0.05 (d) 〈n1〉 = 〈n2〉 = 0.01

3.3 Cavity Field in a Two Mode Thermal State

The thermal field is the most easily available radiation field and its influence on the entangle-ment is of much interest. The field at thermal equilibrium obeying Bose-Einstein statisticshas an average photon number at temperature T K given by 〈n〉 = (e

�ωkT − 1)−1. The photon

statistics is governed by the distribution Pn given by

Pn = 〈n〉n(1 + 〈n〉)n+1

(24)

For a thermal field distribution function, the joint two-atom-cavity state is obtained bysumming over all n for both the modes. Using the above procedure we compute the en-tanglement of formation numerically. In Figs. 4(a), 4(b), 4(c) and 4(d) we plot the entan-glement of formation of the two atoms interacting with the two mode thermal field forn1, n2 = 0 · 5,0 · 1,0.05 and 0 · 01 respectively. It is observed that the magnitudes of theatomic entanglement increases with decrease of cavity photon number.

A comparison of Figs. 3 and 4 reveals that the evolution of Ef for a two-mode thermalfield is similar to the case for the two-mode coherent field inside the cavity for the small val-ues of mean number of photons in the two modes. Quantum effects which are predominantprimarily when the photon number is low, helps to increase the peak value of Ef .

Earlier, it has been shown that [25] two identical dipole-dipole coupled atoms passingsimultaneously through the cavity can be entangled through non-degenerate two-photon in-teraction with two-mode thermal field with the exception of the case when both atoms are

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Fig. 5 Dynamics ofentanglement for atoms in a qubitfield state; (a) solid line:atom-atom, (b) dotted line:second atom-field (c) dashedline: first atom-field

excited. In contrast, here we observe that for the two-mode thermal cavity field, entangle-ment between two atoms in the excited state passing successively can be generated as aconsequence of Two-mode Jaynes-Cummings dynamics.

3.4 Cavity Field in a Qubit State

If we take the first atom and the second atom as the first system and second system respec-tively, the third system involved in the interaction is the cavity field. During the time theatoms are disentangled, the entanglement gets transferred to the composite system compris-ing of one of the two atoms and the cavity field. In this composite system, the atom is aqubit and the radiation field is described by a many particle system (photons). To get aninsight into the transfer of entanglement, we consider the radiation field with frequency ω2

in a vacuum state that is, n2 = 0 and the radiation field with frequency ω1 in a two state fieldi.e. |ψ〉f = ∑

i ci |n1〉 where ci = 0 for n1 = 0 and 1 only. The cavity field of frequency ω1

is in a qubit state viz.

|ψ〉f = 1√2

(|0〉 + |1〉) (25)

The micromaser cavity field [26] can generate a mixture of photonic states close to aqubit state. In such a situation, we have three mutually interacting qubits. We numericallyfind the eigenvalues defining three possible bipartite entanglements. In Fig. 5 we plot the en-tanglement of formation against gt for (a) atom-atom (represented by solid line) (b) secondatom-field (represented by dotted line) and (c) first atom-field (represented by dashed line).

The dynamics of entanglement among the three bipartite systems has been displayed inFig. 5 where the characteristics of transfer of entanglement can be observed roughly. It isobserved that the maximum of one of the entanglements does not exactly correspond to theminimum of the other and vice versa. Also all the three entanglements are zero at gt = mπ .This is due to the way we have treated the field statistics. Such a state has been assumedonly to get a rough understanding of the transfer of entanglement which is otherwise notpossible since the field is a function of n number states in general. The situation may furtherbe modified if the effect of the field reservoir is taken into account.

4 Summary and Conclusion

In this work, we consider a micromaser set-up where a couple of atoms traverse the cav-ity one after another so that there is no overlap between the passing atoms. The generation

Int J Theor Phys

of entanglement for such a set up has already been investigated for the standard Jaynes-Cummings model [13, 14] and intensity dependent Jaynes-Cummings Model [15]. Here westudy the generated entanglement when the atom interacts with the cavity field through a nondegenerate two-photon process. We consider the two mode cavity fields to be in the Fockstate, coherent state and thermal state respectively. For the initial two-mode coherent state,we observe periodic entanglement and disentanglement. This property in fact follows fromthe collapse and revival in population of atomic levels (population inversion) taking placefor two-level atoms in a cavity interacting with two modes of the radiation field. Since theentanglement between the atoms is mediated by the cavity fields in the Two-mode Jaynes-Cummings Model such an effect is also seen in the entanglement of two atoms. Two distinctpatterns of entanglement are seen to emerge for the cases corresponding to low and highaverage cavity photon numbers respectively. In the former case the quantum nature of ra-diation field plays a prominent role in enhancing atomic entanglement with the decrease ofn. The situation is reverse for high n case where the increase of n leads to a slight increaseof Ef .

For small values of mean number of photons, it is observed that, the evolution of Ef

for the initial two-mode Thermal field is similar to the case for the two-mode coherent fieldinside the cavity.

Acknowledgements The first author (Sudha Singh) wishes to acknowledge the support from the UniversityGrants Commission (UGC, New Delhi, India) in the form of a Major Research Project (F.No. 37-327/2009(SR)).

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