ABSTRACT

Entanglement due to the interaction of a two level atom with a laser and quantized field is investigated. The role of the nonlinearity due to these interactions is discussed. It is found that the nonlinearity changes strongly the behavior of the entanglement also the detuning parameters have important role in the structure of the measure of entanglement.

1. Introduction

Quantum information processing provides different way for manipulating information other than the classical one. This is related to entanglement, which plays an essential role in the quantum information such as quantum computing [1], teleportation [2], cryptography [3], dense coding [4] and entanglement swapping [5]. Thus intensive efforts have been done to understand theoretically and experimentally the entanglement in quantum systems. For instance, the entanglement between two qubits in an arbitrary pure state has been quantified by the concurrence [6] and Peres-Horodecki measure [7]. However, that of the mixed states is quantified by the average concurrence over all possible pure state ensemble decomposition. Additionally, the entropic relations are used in investigating the entanglement in quantum system. In this regard von Neumann entropy (NE) [8], linear entropy (LE) and the Shannon information entropy (SE) [9] have been frequently used in treating entanglement in the quantum systems. The NE [10] and the LE [11] have been applied to the JCM. It is worth mentioning that the SE involves only the diagonal elements of the density matrix and in some cases it can give information similar to that obtained from the NE and LE.

The paper is prepared in the following order: In section 2 we define the system Hamiltonian. In section 3, we derive the time evolution operator and density matrix. In section 4 we investigate the atomic inversion and idempotency defect (ID). In section 5 we summarize the main results.

2. The System Hamiltonian

The scheme, we are going to discuss exploits the passage of a single atom through a quantized cavity [12]. We wish to underline the relevance of this aspect from an experimental point of view. Preparing and controlling a single atom is certainly much easier to achieve with respect to the case when the manipulation of many atoms is required. In addition, taking into consideration the low efficiency of the atomic state detectors today used in laboratory, conditional measurement procedures involving one atom only instead of many ones, have to be preferred. The dynamics of several Hamiltonian models describing such systems is exactly treatable and, in most cases, testable in the laboratory. The point to be appreciated is indeed that, studying such systems, one has the opportunity to induce entanglement and control its evolution in a multipartite physical system. In this paper: we study a two-level atom injected into a cavity field, where the dipole-allowed transitions between the lower level and the upper levels are nonresonant with the cavity mode. Furthermore, we assume that the interaction including an arbitrary form of nonlinearity of the intensity-dependent coupling. In the rotating wave approximation (RWA), the interaction of the cavity mode with the injected atom is described by the Hamiltonian ():

(1)

where is the atom Hamiltonian, is the field Hamiltonian, is the atom-field Hamiltonian, and

is the atom-laser Hamiltonian.

The atomic transition frequency is denoted by w_o, the frequencies of the cavity and laser fields are denoted by w_c and w_L receptivity, and the laser field is assigned by the phase y_L, the operator is the annihilation (creation) operator of the cavity field and obey the commutation relation, x and r are the coupling constants associated with the cavity field and the classically described laser field respectively which are assumed real. The operators are the coherence operators for the atom, satisfying the following angular momentum commutation relations where

3. Time Evolution Operator and Density Matrix

To obtain the time evolution operator, we must eliminate the explicit time dependence of the previous Hamiltonian 1), we use the following unitary operator [13]

(2)

we redefine the Hamiltonian as :

(3)

Then the Hamiltonian 1) will be in the following form:

(4)

where and, if we consider the resonant pumping case and rearrange the Hamiltonian terms, the Hamiltonian (4) becomes.

(5)

The new atomic operators set

which obey the angular momentum commutation relations among themselves. The eigenvector of is the dressed states of the atom in the laser field alone and the operators and are the corresponding raising and lowering operators.

To associate the third and the fourth terms of we use the following unitary transformation:

(6)

where P is an atomic state dependent displacement operator of the cavity field state which is defined as :

(7)

Then the Hamiltonian 4) takes the following form:

(8)

where

(9)

We can drop which has no effect in the dynamics of the system.

This Hamiltonian describes a two level system with energy levels separated by r coupled to a single quantized mode of radiation frequency and there is an infinite sequence of probe resonance zones in the neighborhood of the Rabi sub harmonic resonances at

If we consider of the RWA, the slowly varying terms are identified and retained, in the m th resonance zone  has the zero order time dependence that is approximately canceled by the zero order time dependence of all field operators of the form, for any n.

The Hamiltonian for the resonance zone is:

(10)

In this case, we recognize as the -order multiphoton Jaynes Cumming Hamiltonian where we consider

, 

destroys m photons of frequency D_c where can be further simplified by means of another rotating wave approximation (RWA),( as discussed in detail in [12-14]).

The time evolution operator is given by:

(11)

where

(12)

and

(13)

where

.

Also,

where, , and is the Laguerre function. At the wave function of the system can be written as:

(14)

whereis the superposition state parameter

(15)

with the mean photon number, then the wave function at any time is given by

(16)

In the pure state case it is well known that the density matrix of the atom-field interaction can be written as:

(17)

In order to analyze what happens to the two level systems interacting with laser and quantized field, we trace out field variables from the state and get the reduced atomic density matrix of the system given by

(18)

where

(19)

4. Atomic Inversion and Idempotency Defect

Atomic inversion can be considered as the simplest important quantity. It is defined as the difference between the probability of finding the atom in the exited state and in the ground state, the time dependent atomic inversion in the resonance is given by:

(20)

Finally the idempotency defect as a measure of entanglement is written as:

(21)

In many cases of quantum information processing, one requires a state with high purity and large amount of entanglement. Therefore, it is necessary to consider the purity of the state and its relation with entanglement.

Here we use the idempotency defect, defined by linear entropy, as a measure of the degree of purity for a state, in analogy to what is done for the calculation of the entanglement in terms of von Neumann entropy [15] which has similar behavior in the same cases.

[In all our figures we have plotted the atomic inversion W(t) and the entanglement ID as a function of the scaled time x t for the case of two-photon processes]. In figure 1, we plot the atomic inversion and ID as a function of the scaled time. It is seen that, when we neglect the Laguerre function term, the atomic inversion shows fast Rabi oscillations, and the periodic behavior of the ID is seen. In this case the usual two-photon features are observed. Once the Laguerre term is considered the general

features of both the atomic inversion and ID are washed out, where as the time goes on the atomic inversion shows small amplitude of the oscillations and steady state of the ID (see figure 1). As the time increases further the maximum entangled state can be obtained, where At this moment the entanglement reaches its maximum value. An interesting case is seemed in figure 2, where we have considered small values of the parameter. In this case the collapse-revival phenomenon is shown. Also, the periodic oscillation occurs each for atomic inversion and the zero ID at for x = 0.1.

Now we shed the some light on the affect of the different parameters on the both of the atomic inversion and the corresponding ID. For example the effect of the detuning is considered in figure 3. In this case we see that the amplitude of the value of the detuning increased as the value of the detuning increased. The ID decreases and reaches steady state as increases further.

On the other hand, the detuning of the quantized field d, plays the opposite role, where the ID shows steady state at maximum instead of lowring its value due to the detuning of the laser field.

As d increases further (say d = 0.5) we obtain a long lived entanglement, as to show the fixed value of the ID (x = 0.5). In figure 5, we see that the collapse-revival phenomenon is clearly seen. Also, a perfect correspondence between the atomic inversion revival and the local maxima corresponds to the collapse periods. Also, we see that the collapse-revival phenomenon is clearly seen. Also, a perfect correspondence between the atomic inversion revival and the local maxima corresponds to the collapse periods. we have plotted the atomic inversion and the corresponding field Idempotency for a state of mean photon number of the coherent field as, fixed the detuning parameter and the parameter d = 0 and changed the parameter with values (0.01 and 0.1), when x = 0.01, oscillated with period equal and W(t) oscillated with same interval and has a long-lived entanglement in each interval. When x = 0.01, then has a weak oscillation and W(t) oscillated with a very small interval between maximum and minimum values .

5. Conclusions

In the present paper we show that the on idempotency defect can be used to quantify entanglement of a two level system interacting with laser and quantized field. We obtain the long living entanglement due to the idempotency defects of the field. Our results show that the nonlinearity changes strongly the behavior of the entanglement also the detuning parameters have important role in the structure of the measure of entanglement. Also, all the interaction parameters play an important role on the behavior of idempotency defect.

REFERENCES