The quantum and statistical properties of light generated by an external classical field in a correlated emission laser with a parametric amplifier and coupled to a squeezed vacuum reservoir are investigated using the combination of the master and stochastic differential equations. First, the solutions of the cavity-mode variables and correlation properties of noise forces associated to the normal ordering are obtained. Next, applying the resulting solutions, the mean photon number of the separate cavity modes and their crosscorrelation, smallest eigenvalue of the symplectic matrix, mean photon number, intensity difference fluctuation, photon number variance, and intensity correlation are derived for the cavity-mode radiation. The entanglement produced is studied employing the logarithmic negativity criterion. It is found that pumping atoms from the lower energy state to excited state, introducing the nonlinear crystal into the cavity and coupling the system to a biased noise fluctuation, generate a bright and strong squeezing and entanglement with enhanced statistical properties although the atoms are initially in the ground state.

A search for quantum systems that would generate a strongly correlated two photons is an active area of theoretical and experimental investigations [

Moreover, several authors have demonstrated that introducing a nonlinear crystal into a correlated emission laser amplifies the nonclassical properties of radiation [

However, the squeezing, entanglement, and mean photon number of the cavity radiation have been found to be insignificant for the case in which all the atoms are initially prepared in the bottom level [

In this work, the squeezing, entanglement, and photon statistics of a correlated emission laser coupled to a two-mode squeezed vacuum reservoir and containing the parametric amplifier are studied. The motivation of this work is that the parametric amplifier and squeezed vacuum reservoir could enhance the nonclassical properties of interest. Moreover, on the basis of the existing practical challenges in preparing the atoms initially in an arbitrary atomic coherence and due to the sensitivity of the quantum features to the inevitable effect decoherence from an external environment, it is supposed that the quantum system under consideration can be one of the interesting schemes in generating a significantly enhanced quantum and statistical features. In order to carry out our analyses, the master equation is derived by applying the linear and adiabatic approximations in the good cavity limit following the standard method presented in [

The three-level cascade atoms, which are initially prepared in a bottom level, are injected into the cavity at a constant rate and removed from the laser cavity after they spontaneously decay to the external environment. The top and bottom levels are linked by the classical driving radiation similar to the case of lasing without population inversion. Three-level atoms interact with resonant cavity modes and a nondegenerate parametric amplifier (NLC) in the laser cavity. Here, the cavity light is coupled to the squeezed vacuum environment. As it can be seen in Figure

Schematic representation of a correlated emission laser with a nonlinear crystal (NLC) and coupled to a two-mode squeezed vacuum reservoir.

In the nondegenerate three-level laser, a pump mode of frequency,

On the contrary, the interaction of a three-level atom with a two-mode cavity light can be described in the interaction picture with the electric dipole and rotating-wave approximations as

Moreover, the interaction Hamiltonian operator describing the coupling of the dipole forbidden transition is describable in the interaction picture as

On the basis of equations (

Moreover, in this work, the three-level cascade atoms are assumed to be initially prepared in the bottom (lower energy) level which corresponds to the absence of coherence among the noninteracting atoms. These atoms are injected into a cavity at constant rate

The result presented in equation (

In this section, we apply equation (

We see that the cavity-mode operators in equations (

The solutions of the cavity-modes variables, following the procedure outlined in [

Now, supposing the initial states of the cavity modes to be in a vacuum state, in view of the fact that the noise force at some time

The result in equation (

In this section, we proceed to study the squeezing of the radiation produced by the proposed scheme. The intracavity squeezing of a two-mode cavity radiation can be studied by the phase and amplitude quadrature operators constructed from the separate cavity-mode operators,

The uncertainty relation of the two noncommuting operators can be described in the form

Using equations (

Now, employing equations (

The result in equation (

The threshold condition for the system under consideration is described as

Next, the explicit dependence of the squeezing on the parametric amplifier, and squeeze parameter is investigated. Since the squeezing occurs in

It is shown in Figure

Plots of the minus quadrature variance (equation (

Plots of the minus quadrature variance (equation (

In this section, the entanglement of the generated radiation applying the logarithmic negativity, which is introduced in [

It is not difficult to find the extended covariance matrix in terms of the

Next, on account of equations (

We now investigate the variation of the entanglement of the cavity radiation with the amplitude of the parametric amplifier and squeeze parameter. To this end, we plot

It is not difficult to note in Figure

Plots of

Moreover, we plot in Figure

Plots of

In this section, we investigate the intensity of the cavity light. In order to get insight about the intensity of the generated light and its relation with the other nonclassical properties, it is worthwhile to study the mean number of photon of the two-mode cavity radiation. In terms of the annihilation operator of the cavity radiation, the mean photon number of the cavity light can be defined as

Equation (

Equation (

We next investigate the variation of the mean photon number of the system, equation (

The parametric amplifier and squeeze parameter enhance the mean photon number of the cavity light, as reflected in Figures

Plots of the steady state mean photon number (equation (

Plots of the steady state mean photon number (equation (

Here, we study the variance in the photon number for the cavity light. The variance of the photon number is one of the features categorizing a light source exhibiting quantum and classical properties [

Equation (

It is realized that

In this section, we analyze the photon number correlation of the cavity radiation. Nonclassical features of the radiation are usually studied via the amplification of the correlation in the quadrature operators. In many instances, experimental realization of the theoretical prediction in this line has been found to be a formidable task due to the complication involved in the homodyne and phase measurements. Thus, an alternative approach for studying the quantum features of the cavity radiation which involves simultaneous photon count measurement is required. The equal time photon number correlation for light mode

In view of the fact that the separate cavity-mode variables are

This equation describes equal time steady state photon number correlation of a two-mode cavity radiation. In the following, we investigate the explicit dependence of the steady state photon number correlation on the amplitude of the parametric amplifier and squeeze parameter.

Figure

Plots of the steady state photon number correlation (equation (

Plots of the steady state photon number correlation (equation (

In this section, the intensity difference fluctuation of the cavity radiation is studied. This study is based on the assumption that there is a difference between the mean photon numbers of the two radiations due to the disparity of the absorption emission mechanism among the involved atomic energy levels. The intensity difference fluctuation allows us to investigate how the difference between the mean photon numbers of the two radiations deviates from each other. Therefore, the fluctuation of the intensity difference can be defined as

We then investigate the explicit dependence of the intensity difference fluctuation on the amplitude of the parametric amplifier and squeeze parameter.

It is not subtle to see from Figures

Plots of the steady state intensity difference fluctuation (equation (

Plots of the steady state intensity difference fluctuation (equation (

In this paper, the quantum statistical properties of the two-mode cavity light, which is generated by a correlated emission laser with a nondegenerate parametric amplifier and coupled to the squeezed vacuum reservoir, have been investigated. First, the master equation in the good cavity limit, linear, and adiabatic approximation schemes has been determined. Applying the stochastic master equation, the equations of evolution of the first and second moments of the cavity-mode variables have been derived. With the aid of these equations, the quadrature fluctuations, smallest eigenvalue of the symplectic matrix, mean photon number, photon number fluctuation, photon number correlation, and intensity difference fluctuation of the cavity radiation are obtained. Then, the quantum statistical properties of the cavity light have been analyzed varying the parameters involved in the system.

A robust squeezed and entangled light of

Besides, a quite robust intensity of the cavity radiation has also been demonstrated upon manipulating the parameters that has enhanced the squeezing and entanglement in the proposed scheme. The photon number correlation, similar to the squeezing and entanglement, has correctly demonstrated the effect of the huge external environment that dominates the small cavity system. The correlation between the photon numbers tends to be minimum in regions where the squeezing and entanglement are maximum due to the fact that the mean photon number of the separate cavity modes is inversely related to the photon number correlation. Furthermore, the intensity difference fluctuation increases with the amplitude of the parametric amplifier and squeeze parameter, and it is almost vanished in the absence of these parameters. The system also falls to the category of super-Poissonian photon statistics for all cases while exhibiting the various nonclassical properties of the generated radiation. In general, it can be concluded that a correlated emission laser can be a promising optical device which produces a strong and bright squeezed and entangled cavity light with a rich varieties of statistical properties.

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.