^{1}

^{2}

^{3}

^{1}

^{2}

^{3}

We investigate theoretically the generation of squeezed states in spontaneous and stimulated six-wave mixing process quantum mechanically. It has been found that squeezing occurs in field amplitude, amplitude-squared, amplitude-cubed, and fourth power of field amplitude of fundamental mode in the process. It is found to be dependent on coupling parameter “g” (characteristics of higher-order susceptibility tensor) and phase values of the field amplitude under short-time approximation. Six-wave mixing is a process which involves absorption of three pump photons and emission of two probe photons of the same frequency and a signal photon of different frequency. It is shown that squeezing is greater in a stimulated interaction than the corresponding squeezing in spontaneous process. The degree of squeezing depends upon the photon number in first and higher orders of field amplitude. We study the statistical behaviour of quantum field in the fundamental mode and found it to be sub-Poissonian in nature. The signal-to-noise ratio has been studied in different orders. It is found that signal-to-noise ratio is higher in lower orders. This study when supplemented with experimental observations offers possibility of improving performance of many optical devices and optical communication networks.

Over the past three decades, particular attention has been focused on theoretical investigations and experimental observations in generation of squeezed light, for improving the performance of many optical devices and optical communication networks. The concept of squeezed light is concerned with reduction of quantum fluctuations in one of the quadrature, at the expense of increased fluctuations in the other quadrature. In general, the two important nonclassical effects, squeezing and antibunching (or Sub-Poissonian photon statistics), are not interrelated; that is, some states exist that exhibit the first but not the second and vice versa. However, squeezing can be detected using simple photon counting in higher-order sub-Poissonian statistics.

A lot of work has appeared in the literature on the theoretical and experimental investigations on generation of squeezed states of electromagnetic field. Mandel [

Recently, Giri and Gupta [

Squeezing is a purely quantum mechanical phenomenon which cannot be explained on the basis of classical physics. The coherent states do not exhibit nonclassical effects, but a superposition of coherent states can exhibit normal squeezing, higher-order squeezing, and sub-Poissonian photon statistics. A coherent state changes to a superposition of coherent states when it interacts with a non linear medium. Squeezed states of an electromagnetic field are the states with reduced noise below the vacuum limit in one of the canonical conjugate quadratures. Normal squeezing is defined in terms of the operators

The model considers the process involving absorption of three pump photons of frequency

Six-wave interaction model.

The Hamiltonian for this process is as follows (

The Heisenberg equation of motion for mode

The right-hand side of the expression (

In parallel to the spontaneous interaction, the stimulated emission is caused due to the coupling of the atom to the other states of the field. Therefore, the study of squeezing in stimulated interaction in six-wave mixing process requires initial quantum state as a product of coherent states for modes 1, 2 and vacuum state for 3, that is,

Using (

Using (

For fourth-order squeezing, amplitude is expressed as

The right-hand sides of (

Using (

Signal-to-noise ratio is defined as ratio of the magnitude of the signal to the magnitude of the noise. With the approximations

Using (

The results show the presence of squeezing in field amplitude, amplitude-squared, amplitude-cubed, and fourth-order field amplitude of fundamental mode in six-wave mixing process. To study squeezing, we denote the right-hand sides of relations (

Dependence of first-order squeezing (a)

Dependence of amplitude-squared squeezing (a)

Dependence of amplitude-cubed squeezing (a)

Dependence of fourth-order field amplitude squeezing (a)

A comparison between results of spontaneous and stimulated processes shows the occurrence of multiplication factor

Signal-to-noise ratio for different order squeezing.

Figures

It has also been found that the fundamental mode of field amplitude shows sub-Poissonian behavior as shown in relation (