Statistical and Squeezing Proprieties of Superposed Single-Mode Squeezed Chaotic State

In this paper we have studied the statistical and squeezing proprieties of light produced by superposition of a pair of single-mode squeezed chaotic light beams. Applying density operator of single-mode squeezed chaotic state; we obtain the anti-normal order characteristics function which enables us to find the Q function. With the resulting Q function, we calculate the photon statistics and the Quadrature squeezing for single-mode squeezed chaotic light. Moreover applying Q function of single-mode squeezed chaotic state the superposed light beams would be driven. With the resulting Q function we calculated the photon statics and the quadrature squeezing for superposed light beams. To get the maximum squeezing to be 95%, for nth = 0 and r = 1.5.


Introduction
The most important quantum states of light are chaotic state, coherent state and squeezed state. Chaotic state is one of the classical features of light with super-Poissonian photon statics. And its best example is thermal light. The coherent state is a specific superposition of number states which does not possess number of photons as well as it is known by minimum uncertainty and poissonian photon statistics. Squeezed state satisfies non-classical feature of light, with sub-poissonian photon statistics [1]- [4].
The quantum distribution of radiation is the core idea in quantum optics. This used to describe the quantum properties of light. Some of them are the P function, the Wigner function and the Q functions. The P-function is cnumber function with the anti-normal order density operator over π. And used to describe the superposition of two light beams with different states but having the same frequency. The Wigner function is the c-number function corresponding to the symmetric order density operator over π. The Q function is the most widely used one because, it is used to describe the superposition of two light beams with the same frequency but may be in the same or different states. This is described in terms of normally ordered density operator divided by π [5].

Methods
Within density operator we calculate the anti-normal order characteristic function which is used to obtain the Q function. With the help of resulting Q function we calculate the density operator for superposed single-mode squeezed Chaotic state, the mean photon number, the variance of photon number, the photon number distribution and quadrature variance for both single-mode and superposed single-mode squeezed chaotic state.
3 Single-mode squeezed chaotic state 3.1 Single-mode chaotic state Thermal light is the best example of light mode in a chaotic state, which is generated by the source in thermal equilibrium at the minimum temperature. This is not a pure state, instead it can be described by density matrix and has mean which is less than variance so, and we called as classical feature of light. It can be used to describe light of the bulb, Black-body radiation and etc. In this section we seek to determine the density operator for chaotic light in terms of the P function using Lagrangian multipliers and by maximizing entropy. The entropy of thermal light can be described as [6].

Single-mode Squeezed State
A degenerate parametric amplifier, consisting of nonlinear crystal pumped by coherent light, is a source of singlemode squeezed light. In this system a pump photon of frequency 2ω is down converted into two twines signal photons each of frequency ω as shown in Fig. (1) [6]- [10].

The Q function for single-mode squeezed chaotic state
The Squeezed chaotic state is obtained by performing squeezing operator on thermal state.Which have both classical and quantum nature rather than quantum or classical nature only because the system obeys both features simultaneously. Thus, consider the light mode initially in a chaotic state then the state vector for the squeezed chaotic light would be given by, sc = | ψ>sc <ψ| = (r)|φ>th< φ | (r) = (r)th (r).

Photon statistics
Here we wish to calculate the mean photon number; the variance of photon number and then photon number distribution of the light generated by single-mode squeezed chaotic light employing the Q function.

The mean photon number
The mean photon number in terms of the Q function is given by " # = * , , e 4 * . (10) Up on expanding the exponential function e 4 * in power series as e 4 * = R1 + S + 1 2 S + ⋯ U * α, Interims of Eqs. (4), (8) and Eq. (9) the mean photon number for single -mode squeezed chaotic state take the form " # = " # $% + 1 + 2" # $% &'"ℎ (12) Fig. (2), shows that as the mean photon number of thermal light and squeezed parameter(r) increase the mean photon number of single-mode squeezed chaotic state increases. So the use of thermal light is to increase the mean photon number of single-mode squeezed chaotic state.

The Variance of Photon Number
The variance of photon number for single-mode squeezed chaotic light in anti-normal order form is putted as, Where Z > = * , , e 4 * .

Photon number distribution
The photon number distribution for single-mode light is expressible in terms of the Q function as, \ * e \ e FGH I D * J * M | g * gh (18) Up on expanding the exponential part in a power series, we have Performing some mathematical rules we find On account of Eq. (8), (9) and Eq. (4)

Quadrature squeezing (fluctuation)
The squeezing properties of single-mode light are described by two quadrature operators defined by [13],  Eq. (4), and expanding trigonometric function in exponential form the plus and minus quadrature is rewritten as ∆ OE = 1 + 2" # $% F ∓ ‚ .
(27) Fig. (5), shows dependence of the system on its initial state. As we see from the figure while squeezing parameter increase the corresponding quadrature variance decreases. But the quadrature variance increases with mean photon number of thermal light. . (28) Where the quadrature variance for coherent state is Is the Q function for first light beam.
Furthermore, the density operator at initial time corresponds to superposed light beams can be written as, In which subscript "ss" stand for superposed light beams, and the Q function in Eq. (40), takes the form, Which represents the Q function corresponding to second light beams, Also using the fact that

Photon statistics
Here we wish to calculate the mean photon number, the variance of photon number and the photon number distribution for the superposition of two light beams employing the normally ordered density operator.

The mean photon number
The mean photon number is expressed in terms of the density operator as; " # €€ =˜ €€ .
(52) In view of Eqs. (12) and (52), we see that the mean photon number for identical superposed light beams are two times that of single-mode one.

The variance of photon number
The variance of photon number for the superposed light beams is expressible as 54 After calculating all expectation values and using Eq. (52) the variance of photon number given by, ∆" €€ = 4" # $% − 2" # $% + 6 1 + 2" # $% &'"ℎ + 4 1 + 2" # $% &'"ℎ 3 − 2 1 + 2" # $% +,&ℎ &'"ℎ 55 From Eq. (17) and (55), we see that the variance of photon number for superposed light beams is twice that of single-mode one. But the variance of photon number for the superposed light beams is greater than single-mode squeezed chaotic state. Fig. (7) Indicates both the mean photon number and variance photon number increase with squeezing parameter. But the variance of photon number is greater than the mean photon number; so the photon statics for the superposed single-mode squeezed chaotic state satisfies super-Poissonian photon statistics.

Quadrature squeezing
The quadrature variance for the superposed light beams takes the form, (61) After evaluating the expectation values of above terms the Variance for the superposed light beams can be, (65) This is the same with single-mode light beam. Fig (9), shows relation between the mean photon numbers of thermal light, squeeze parameter and quadrature squeezing. As mean photon number of thermal light increases quadrature squeezing decreases and as squeezed parameter increase, the quadrature squeezes increases.  Table (1), summarize dependence of quadrature variance, quadrature squeezing and degree of squeezing on mean photon number of thermal light. And when the mean photon number of thermal light increases, the variance and squeezing of the system increases and decrease, respectively. The quadrature squeezing is observed to be high when the initial state of the system is vacuum (no photon for convenience) which has corresponding degree of 95% for r = 1.5.
And With Q function of single-mode squeezed chaotic state we calculate the density operator for superposed light beams. The result shows the mean photon number increase with increment of the mean photon number of thermal light and squeeze parameter, but quadrature variance are decreases as both mean photon number of thermal light and squeeze parameter increase. And also the variance of photon number is greater than the mean photon number and the radiation has super-poissonian photon statics.
With density operator for superposed single-mode squeezed chaotic light and respective Q function I have calculate the mean photon number, the variance of photon number, the quadrature variance and quadrature squeezing. We have found that the mean photon number for superposed single-mode squeezed chaotic light is two times that of single-mode squeezed chaotic light with identical light beams. But the quadrature squeezing does not affected by the superposition.
The other one is that the probability of getting n-number of photon for superposed light beams are greater than single-mode squeezed chaotic state. And the variance for superposed light beams are greater than variance of single-mode one so the system still satisfies super-poissonian photon statics.