Quantum Properties of Coherently Driven Three-Level Atom Coupled to Vacuum Reservoir

: A three-level laser with an open cavity and a two-mode vacuum reservoir is explored for its quantum properties. Our investigation begins with a normalized order of the noise operators associated with the vacuum reservoir. The master equation and linear operators' equations of motion are used to determine the equations of evolution of the atomic operators' expectation values. The equation of motion answers are then used to calculate the mean photon number, photon number variance, and quadrature variance for single–mode cavity light and two–mode cavity light. As a result, for γ=0, the quadrature variance of light mode a is greater than the mean photon number for two-mode cavity light. As a result, for the two-mode cavity light, the maximum quadrature squeezing is 43.42 percent.


Introduction
A quantum optical system in which light is created by three-level atoms inside a cavity coupled to a vacuum reservoir is known as a three-level laser. A source of coherent or chaotic light emitted by an atom inside a cavity coupled to a vacuum reservoir is known as a three-level atom [1]. A three-level atom's top, intermediate, and bottom levels are designated by |a⟩ κ , |b⟩ κ , and |c⟩ κ , respectively. Due to stimulated or spontaneous emission, a three-level atom in the top level may decay to level | ⟩ and eventually to level | ⟩.
The purpose of this paper is to investigate the squeezing and statistical properties of light produced by a coherently driven three-level atom in an open cavity connected to a two-mode vacuum reservoir through a single-port mirror. We also determined equations of evolution of the atomic operators' expectation values using the master equation and large-time approximation. The mean photon number, photon number variance, and quadrature variances of single-mode cavity light beams were calculated using the derived solutions. We calculated the mean photon number, photon number variance, and quadrature squeezing of the two-mode cavity light using the same approaches. We perform our calculations by conventionally grouping the noise operators connected with the vacuum reservoir [1,2].

Dynamics of Linear Operators
As shown in the Figure 1, the atoms' top, intermediate, and bottom levels are indicated by |a⟩ κ , |b⟩ κ , and |c⟩ κ , respectively. When an atom transitions from level a to level b and from level b to level c, two photons with the same or different frequencies are emitted, with direct transitions between levels a and c being completely prohibited. When an atom transitions from the top to the intermediate level, light mode a is emitted, whereas light mode b is emitted when the atom transitions from the intermediate to the bottom level [3,4]. The Hamiltonian describes how coherent light couples the top and bottom levels of a three-level atom [5,6], and Here, κ, is the cavity damping constant and considered to be the same for cavity modes a and b. Then in view of equation (7), equations of motion for the operators â and b turn out to be Upon adding equations (12) and (13), we get where, is the annihilation operator for the superposition of light modes a and b. By employing the relation [8,10] With the same procedure one can obtain the following where η a = |a⟩⟨a|, Equations (17)-(22) are nonlinear differential equations. Now, by applying the largetime approximation [11], the solutions of equations (12) and (13) becomes Considering equations (2), (5), (6) and their adjoints, one obtains 〈η b σ a 〉 = 〈|b〉〈b|b〉〈a|〉 = 〈|b〉〈a|〉 = 〈σ a 〉, 〈η a σ a 〉 = 0, Substitution of equations (29)-(31) into (28) gives With is the stimulated emission decay constant. The completeness relation has the form [12] η a + η b + η c = Î.
Then, we see that [13,14] 〈η a 〉 + 〈η b 〉 + 〈η c 〉 = 1, where 〈η a 〉 is the probability to find the atom in the top level, 〈η b 〉 is the probability to find the atom in intermediate level, and 〈η c 〉 is the probability to find the atom in the bottom level. The steady state solutions of equations (32)-(37) are found to be Furthermore, with the aid of equation. (40), one readily obtains In view of equations (45), equation (46) has the form Now, on account of equation (47), equation (43) can be expressed as With the aid of equation (48), one can observe that Also, from equations (41), (42), and (43), one can readily obtains By substituting equation (48) into (51) yields Moreover, on account of equation (45), one can obtain Now, by Substituting (52) in (47), we have Finally, on account of equation (52), equation (48) takes the form
On account of equations (26) and (52), equation (56) can be written as For non-spontaneous case ( = 0), the mean photon number of light mode a has the form In addition, for ≫ , equation (58) becomes The mean photon number of light mode b is determined using the same procedure as For non-spontaneous case, equation (60) takes the form In addition, for Ω ≫ γ c , equation (61) reduces to The mean photon number for light modes a and b is the same in both spontaneous and non-spontaneous scenarios, as shown above. The mean photon number for two-mode cavity light can then be expressed as follows n ̅ c = 〈ĉ † ĉ〉. (63) The mean photon number has the form when using the steady state solution of equation (14) and its adjoint Substituting equations (57) and (58) in (64) for the steady state solution of (14) yields Now, the mean photon number in the non-spontaneous scenario is in the form For Ω ≫ γ c , equation (66) reduces to Furthermore, the variance of the photon number is expressible as [3,5] (Δn) 2 = 〈n ̅ 2 〉 − 〈n ̅〉 2 .
In view of equation (5), one readily obtains Equations (69) Furthermore, for non-spontaneous case, the photon number variance has the form For Ω ≫ γ c , (76) With the same procedure one can obtain the variance of the photon number for light mode b as For non-spontaneous case, equation (75) takes the form For Ω ≫ γ c , equation (75) reduces to which represents the normally-ordered variance of the photon number for the chaotic light. Furthermore, equation (79) indicates that ( ) 2 > ̅ and ( ) 2 > ̅ and hence the photon statistics of each light-mode is super-poissonian.
With the same approach one can readily obtain the variance of the photon number for superposed light modes a and b as which represents the normally-ordered variance of the photon number for chaotic light. Furthermore, inspection of equation (82) indicates that (Δn c ) 2 > n ̅ c 2 and hence the photon statistics of the two-mode light is super-poissionian.
The quadrature squeezing of two-mode cavity light in relation to the quadrature variance of the two-mode cavity vacuum state can be determined using the formula [20] S =  125) and (126) versus Ω for γc=0.5, γ=0 (red color) and γ=0.1 (blue color).
According to Figs. 1 and 2, light mode a's mean photon number and photon number variance are higher than those for = 0 . The plots, however, overlap at that moment = 0.55 in Figure 3. This demonstrates that when the variation of the light's photon number is greater for = 0 mode b than for = 0.1, and vice versa.
The plot from Fig.4 clearly demonstrates that the quadrature variance of the twomode light is less in the absence of spontaneous emission when < 0.24 and is anticipated to be bigger in the absence of spontaneous emission when > 0.24 for the quadrature variance of two light modes. Finally, we discovered from Figure 5 that the maximum quadrature squeezing for both light modes is 43.42 percent and that the plots intersect at the spot.

Conclusion
A coherently driven three-level atom with an open cavity coupled to a two-mode vacuum reservoir by a single port mirror has its quantum features thoroughly examined. The master equation was used to find the steady-state solutions of the equations of motion for linear operators and the equation of evolution of the expectation values of atomic operators with stable solutions. Using steady state solutions of the equations of motion for linear operators and equations of evolution of the expectation values, we estimated the mean photon number, the photon number variance, and the quadrature variance for single-mode cavity light beams as well as two-mode light beams. We also calculated quadrature squeezing for the two mode-lights. The mean photon number, the variance of the photon number for light mode a, the variance of the photon number for the two-mode cavity light, and the quadrature variance of light mode a for γ = 0 is greater than for γ = 0.1. From the plots of variance of the photon number of light mode b cross each other at the point Ω=0.55. This shows that when Ω<0.55 the variance of the photon number for γ=0 is greater than for γ=0.1 and vice versa. From the calculation the quadrature variance of light mode b for γ=0 is less than for γ=0.1. The quadrature variance of the two-mode cavity light is less in the absence of spontaneous emission when Ω<0.24 and grater in the absence of luminescence when Ω>0.24. The plots of quadrature squeezing cross each other at the point Ω=0.24. When Ω<0.24, the quadrature squeezing for γ=0 is greater than that for γ=0.1 and vice versa. Finally, it was found that the maximum quadrature squeezing of the twomode cavity light is 43.42% for both γ=0 and γ=0.1 below the vacuum-state level.
Funding: This research received no external funding.