_{1}

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In this paper, we present the analytical solution for the model that describes the interaction between a three-level atom and two systems of N-two level atoms. The effects of the quantum numbers and the coupling parameters between spins on the Pancharatnam phase and the atomic inversion, for some special cases of the initial states, are investigated. The comparison between the two effects shows that the analytic results are well consistent.

The use of statistical mechanics is fundamental to the concepts of quantum optics: Light is described in terms of field operators for creation and annihilation of photons [

In addition, based on Dicke’s superradiance, Eyob A. Sete et al. studied the collective spontaneous emission from an ensemble of N identical two-level atoms prepared by absorption of a single photon―a.k.a. single photon Dicke superradiance [

In the present communication we are concerned with the type of atom-atom (spin-spin) interaction, the interaction between a three-level atom and two systems of N-two level atoms. The time evolution of dynamical systems has attracted considerable attention over the past several decades because of its various applications. An important aspect in this regard is the quantum phase associated with the evolution of these states in certain circumstances [

In recent years much attention has paid to the quantum phases [

The Pancharatnam phase or most commonly Berry phase is a phase difference acquired over the course of a cycle when a system is subjected to cyclic adiabatic processes, which results from the geometrical properties of the parameter space of the Hamiltonian. The Pancharatnam phase is very important in the propagation of a light beam where its polarization state is changing periodically [

This paper is organized as follows: in section 2, we will describe the Hamiltonian of the system of interest, and obtain the explicit analytical solution of the model describing the interaction between a three-level atom and two systems of N-two level atoms. The case discussed in this paper is considered to be the generalization of the atom-atom interaction, and most of the previous papers which handled this interaction are, for the most part, considered to be a special case of our case. In section 3, different cases are studied to demonstrate the effects due to both the quantum numbers m_{1}, m_{2} and the coupling parameters between spins λ_{1}, λ_{2} on the atomic inversion _{1}, m_{2} and the coupling parameters between spins λ_{1}, λ_{2} on the Pancharatnam phase

The Hamiltonian of our model describes the interaction between a three-level atom coupled to two systems of N-two level atoms. In this case the Hamiltonian of the whole system can be written in the form:

where,

where

while

and

with the operator

We define

Let

From Schrödinger equation

we get from Equations (1), (15)

where,

where

Define

by substituting from Equation (24) in Equations (17)-(19) we get the following equations:

from

So, we can write the Equations (25)-(27) as the following:

We solve the Equations (29)-(31) analytically, we get:

where,

similarly

and

So from Equation (24) we get

As applications to the solution of our case, a three-level atom coupled to two systems of N-two level atoms, we calculate the atomic inversion, the Pancharatnam phase and the correlation functions.

The atomic population inversion

In Figures 2(a)-(c), we consider (Δ = 0, λ_{1} = λ_{2} = 1 and j_{1} = 30, j_{2} = 20). We investigate the effect of the

quantum numbers m_{1}, m_{2} on the atomic inversion. In

inversion (m_{1} = m_{2} = 1) has regular and periodic oscillations. It starts from its maximum value, _{1}, m_{2} increase (m_{1} = m_{2} = 18), the phase of periodic oscillations gradually decreases, until it reaches zero (m_{1} = m_{2} = 20) (straight line), but the minimum value increases until it equalizes the maximum value (

(straight line). In

and periodic oscillations. It starts from its maximum value, _{1}, m_{2} increases, the number of periodic oscillations and the phase of periodic oscillations gradually decrease, but the minimum value increases remarkably. In Figure

2(c), the initial state is

It starts from its maximum value, _{1} = m_{2} = 1) has small oscillations in the middle of the apexes of the regular and periodic oscillations. When the quantum numbers m_{1}, m_{2} increase, the number of periodic oscillations decreases and the small oscillations disappear gradually. In _{1} = m_{2} = 1, j_{1} = 30, j_{2} = 20 and the initial state is_{1}, λ_{2} on the

atomic inversion. The atomic inversion has regular and periodic oscillations. It starts from its maximum value, _{1}, λ_{2} decrease and the number of periodic oscillations gradually decreases.

The total phase both dynamic and geometric phase parts for an arbitrary quantum evolution of a system from a state at t = 0 to a final state at time t. Without invoking the fact that any initial state vector

In _{1} = λ_{2} = 1 and j_{1} = 30, j_{2} = 20 and the initial state is_{1}, m_{2} on the Pancharatnam phase. It has regular and periodic straight lines following the shape of the letter N. It starts from zero then decreases until it reaches its minimum value, P = −3, then it increases until it reaches its maximum value, P = 3. We note that there is a regular repeat in the behavior of the Pancharatnam phase. We observe, when the quantum numbers m_{1}, m_{2} increase, each period of the Pancharatnam phase has widen on the time axe and the overall number of periods decreases. In _{1} = m_{2} = 1, j_{1} = 30, j_{2} = 20 and the initial state is_{1}, λ_{2} on the Pancharatnam phase. It has regular and periodic straight lines following the shape of the letter N. It starts from zero then decreases until it reaches its minimum value, P = −3, then it increases until it reaches its maximum value, P = 3. We note that there is a regular repeat in the behavior of the Pancharatnam phase. We observe, when the coupling parameters between spins λ_{1}, λ_{2} decrease, each period of the Pancharatnam phase has widen on the time axe and the overall number of periods decreases remarkably. Finally, comparing the change of the Pancharatnam phase in _{1}, m_{2} is larger than the effect of the coupling parameters between spins λ_{1}, λ_{2} in respect to the overall number of periods.

In this section, we discuss the behavior of the second-order correlation function where the examination of the correlation function is usually used to discuss the correlated or uncorrelated behavior from which we can distinguish between classical and nonclassical behavior. The normalized second-order correlation function is defined by [

To discuss the behavior of the correlation function, we have to calculate the expectation value of the quantity

We know that

So

we get from Equations (15) (41)

where

In _{1} = λ_{2} = 1,

gate the effect of the quantum numbers m_{1}, m_{2} on the correlation functions

2 respectively. When

functions

the coupling parameters between spins λ_{1}, λ_{2} on the correlation functions _{1}, λ_{2} increase the number of periodic oscillations increases.

In this paper, we analytically solved the model that described the interaction between a three-level atom coupled to two systems of N-two level atoms. We calculated the atomic inversion and the Pancharatnam phase for some special cases of the initial states_{1}, λ_{2}. The atomic inversion has regular and periodic oscillations. We observe, at the initial state_{1}, m_{2} increase, the phase of periodic oscillations gradually decreases, until the phase of periodic oscillations reaches zero (straight line), but the minimum value increases until it equalizes the maximum value

oscillations when the quantum numbers m_{1}, m_{2} increase, the number of periodic oscillations decreases and the small oscillations disappear gradually. We observe that there is constant interval at the maximum value. It increases remarkably when the coupling parameters between spins λ_{1}, λ_{2} decrease and the number of periodic oscillations gradually decreases. The Pancharatnam phase has regular and periodic straight lines following the shape of the letter N. We note that there is a regular repeat in the behavior of the Pancharatnam phase. When the quantum numbers m_{1}, m_{2} increase and the coupling parameters between spins λ_{1}, λ_{2} decrease, each period of the Pancharatnam phase widens on the time axe and the overall number of periods decreases. Comparing the change of the Pancharatnam phase in _{1}, m_{2} is larger than the effect of the coupling parameters between spins λ_{1}, λ_{2} in respect to the overall number of periods. Finally, we discuss the second-order correlation function where the examination of the second-order correlation function leads to better understanding for the nonclassical behavior of the system. When the quantum numbers m_{1}, m_{2} increase the correlated behavior shows remarkably.

The model presented in this paper can further be applied to two-two level atom or two qubits where in this case j = 1. This can make contribution to more understanding and possible applications in the field of quantum optics as well as solid-state physics. In addition, the atom-atom (i.e. spin-spin) interaction is a promising candidate for implementing the quantum computer, which accordingly can be connected vitally to the demonstration of spin dynamics in semiconductor structures [

I would like to express my deep gratitude to Professors Abdel-Shafy F. Obada, Mohamed M. A. Ahmed, Department of Mathematics, Faculty of Science, Al-Azhar University and Professor Mahmoud Abdel-Aty, Zewail City of Science and Technology, Giza, Egypt, for their support, care, their useful suggestions, useful discussion and for their continuous help and guidance.

D. A. M. Abo-Kahla, (2016) The Pancharatnam Phase of a Three-Level Atom Coupled to Two Systems of N-Two Level Atoms. Journal of Quantum Information Science,06,44-55. doi: 10.4236/jqis.2016.61006