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

Universal Journal of Physics Research, 2022, 1, 277 www.scipublications.org/journal/index.php/ujpr DOI: 10.31586/ujpr.2022.277 Article Quantum Properties of Coherently Driven Three-Level Atom Coupled to Vacuum Reservoir Bessie Mengesha * Department of Physics, Jimma University, P.O.Box: 378, Jimma, Ethiopia *Correspondence: Bessie Mengesha ([email protected]) Abstract: 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. Keywords: Atom; Photon Statistics; Vacuum Reservoir; Squeezing; Quadrature Variance 1. Introduction How to cite this paper: Mengesha, B. (2022). Quantum Properties of Coherently Driven Three-Level Atom Coupled to Vacuum Reservoir. Universal Journal of Physics Research, 1(2), 48–62. Retrieved from https://www.scipublications.com/jou rnal/index.php/ujpr/article/view/277 Received: April 20, 2022 Accepted: October 21, 2022 Published: October 23, 2022 Copyright: © 2022 by the author. Submitted for possible open access publication under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses /by/4.0/). 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 |𝑏⟩ and eventually to level |𝑐⟩. 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]. 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]. DOI: https://doi.org/10.31586/ujpr.2022.277 Universal Journal of Physics Research Bessie Mengesha 2 of 15 Figure 1. The model The Hamiltonian describes how coherent light couples the top and bottom levels of a three-level atom [5, 6], ̂ 1= iΩ [σ H ̂†c − σ ̂c ], (1) 2 where is a lowering operator and ̂c = |c⟩⟨a|, σ (2) Ω = 2ελ. (3) Here, 𝜆 is the coupling constant between the driving coherent light and a three-level atom, and 𝜀 is the amplitude of the driving coherent light, which is considered to be real and constant. The interaction of light modes a and b with the atom at resonance is represented by the Hamiltonian [2, 7, 8] where ̂ 2 = ig(σ H ̂†a σ ̂ − ↠σ ̂a + σ ̂†b b̂ − b̂ † σ ̂b ), (4) ̂a = |b⟩⟨a|, σ (5) ̂b = |c⟩⟨b|, σ (6) and are lowering atomic operators, g is the coupling constant between the atom and cavity mode a or b, and â and b̂ are the annihilation operators for light modes a and b. Thus, the total Hamiltonian is given by ̂ = iΩ (σ H ̂†c − σ ̂c ) + ig(σ ̂†a â − ↠σ ̂a + +σ ̂†b b̂ − b̂ † σ ̂b ). 2 (7) The master equation for a three-level atom interacting with a two-mode vacuum reservoir has the form [5] ̂ dρ dt γ ̂ , ρ̂] + γ (2σ ̂b ), ̂b ρ̂ − ρ̂σ ̂†b σ ̂†b σ = −i[H ̂a ρ̂σ ̂†a − σ ̂†a σ ̂a ρ̂ − ρ̂σ ̂†a σ ̂a ) + (2σ ̂b ρ̂σ ̂†b − σ 2 2 (8) where γ is the spontaneous emission decay constant. Hence with the aid of equation (2) the master equation can be put in the form dρ̂ γ γ ̂b ) ̂b ρ̂ − ρ̂σ ̂†b σ ̂†b σ = (2σ ̂a ρ̂σ ̂†a − σ ̂†a σ ̂a ρ̂ − ρ̂σ ̂†a σ ̂a ) + (2σ ̂b ρ̂σ ̂†b − σ dt 2 2 Bessie Mengesha 3 of 15 Ω γ γ ̂†b − η̂b ρ̂ − ρ̂η̂b ) + (2σ + (σ ̂†c ρ̂ − σ ̂c ρ̂ − ρ̂σ ̂†c + ρ̂σ ̂c ) + (2σ ̂†b ρ̂σ ̂†a ρ̂σ ̂†a − η̂a ρ̂ − ρ̂η̂a ). 2 2 2 (9) When the noise operators associated with the vacuum reservoir are set in normal order and the noise forces have no effect on the dynamics of the cavity mode operators, the equations of motion for the operators a and b assume the form [1, 8, 9] dâ κ ̂ ], = − â − i[a, ̂H ̂ db κ ̂H ̂ ]. = − b̂ − i[b, dt and dt (10) 2 (11) 2 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 b̂ turn out to be dâ = − â − gσ ̂a , ̂ db κ = − b̂ − gσ ̂b . dt and dt κ (12) 2 (13) 2 Upon adding equations (12) and (13), we get dĉ dt where, κ = − ĉ − g(σ ̂a + σ ̂b ), (14) 2 ĉ = â + b̂, (15) is the annihilation operator for the superposition of light modes a and b. By employing the relation [8, 10] d dt we btain ̂ ), ̂ 〉 = Tr (dρ̂ A 〈A (16) dt γ γ d Ω 〈σ ̂†b ) + Tr(−ρ̂σ ̂a ) + Tr(−ρ̂̂ σa ) ̂a 〉 = gTr[ρ̂η̂b â − ρ̂âη̂a + ρ̂b̂ † σ ̂c ] + Tr(ρ̂σ 2 2 dt 2 = g[〈η̂a â〉 − 〈η̂a η̂a 〉 + 〈b̂ † σ ̂c 〉] + Ω † 〈σ ̂a 〉. ̂ 〉 − γ〈σ 2 b With the same procedure one can obtain the following d dt d dt Ω γ 〈σ ̂c 〉] − 〈σ ̂b 〉 = g[〈η̂c b̂〉 − 〈η̂b b̂〉 − 〈↠σ ̂†a 〉 − 〈σ ̂b 〉, 2 2 γ Ω 〈σ ̂b â〉 − 〈σ ̂a b̂〉] − [〈η̂c 〉 − 〈η̂a 〉] − 〈σ ̂c 〉 = g[〈σ ̂c 〉, 2 2 (17) (18) (19) Bessie Mengesha 4 of 15 d 〈η̂a 〉 = g[〈σ ̂†c 〉] − γ〈η̂a 〉, ̂c 〉 + 〈σ ̂†a â〉 + 〈â†a σ ̂a 〉] + [〈σ (20) d 〈η̂b 〉 = g[〈σ ̂†a â〉 − 〈↠σ ̂a 〉] + γ[〈η̂a 〉 − 〈η̂b 〉], ̂†b â〉 + 〈b̂ † σ ̂b 〉 − 〈σ (21) d 〈η̂c 〉 = −g[〈σ ̂†b b̂〉 + 〈b̂ † σ ̂b 〉] − [〈σ ̂†c 〉] + γ〈η̂b 〉, ̂c 〉 + 〈σ dt dt where dt Ω 2 Ω 2 (22) η̂a = |a⟩⟨a|, (23) η̂c = |c⟩⟨c|. (25) η̂b = |b⟩⟨b|, (24) Equations (17)-(22) are nonlinear differential equations. Now, by applying the largetime approximation [11], the solutions of equations (12) and (13) becomes â = − and 2g κ ̂a , σ (26) 2g 𝑏̂ = − σ ̂b . (27) κ At steady state, these would obviously be exact relationships. When equations. (26), (27), and their adjoints are introduced, one gets d dt 〈σ ̂a 〉 = g [− 2g κ 〈η̂b σ ̂a 〉 + 2g κ 〈η̂a σ ̂a 〉 − 2g κ Ω 〈σ ̂c 〉] + 〈σ ̂a 〉. ̂†b σ ̂†b 〉 − γ〈σ 2 (28) Considering equations (2), (5), (6) and their adjoints, one obtains 〈η̂b σ ̂a 〉, ̂a 〉 = 〈|b〉〈b|b〉〈a|〉 = 〈|b〉〈a|〉 = 〈σ 〈η̂a σ ̂a 〉 = 0, 〈σ ̂a 〉. ̂c 〉 = 〈|b〉〈c|c〉〈a|〉 = 〈|b〉〈a|〉 = 〈σ ̂†b σ (29) (30) (31) Substitution of equations (29)-(31) into (28) gives d dt Ω 〈σ ̂a 〉 + 〈σ ̂a 〉 = −(γ + γc )〈σ ̂†b 〉. 2 (32) Similarly, the equations of evolution of the atomic operators' expectation values take the form 1 Ω (33) 1 Ω (34) d 〈σ ̂b 〉 − 〈σ ̂b 〉 = − (γ + γc )〈σ ̂†a 〉, d 〈σ ̂c 〉 + [〈η̂c 〉 − 〈η̂a 〉], ̂c 〉 = − (γ + γc )〈σ d 〈η̂a 〉 = −(γ + γc )〈η̂c 〉 + [〈σ ̂†c 〉], ̂c 〉 + 〈σ dt dt dt 2 2 2 2 Ω 2 (35) Bessie Mengesha 5 of 15 d 〈η̂b 〉 = −(γ + γc )〈η̂b 〉 + (γ + γc )〈η̂a 〉, d 〈η̂c 〉 = (γ + γc )〈η̂b 〉 − [〈σ ̂†c 〉 + 〈σ ̂c 〉]. dt With dt Ω 2 γc = 4g2 (36) (37) , (38) η̂a + η̂b + η̂c = Î. (39) 〈η̂a 〉 + 〈η̂b 〉 + 〈η̂c 〉 = 1, (40) κ is the stimulated emission decay constant. The completeness relation has the form [12] Then, we see that [13, 14] where 〈η̂a 〉 is the probability to find the atom in the top level, 〈η̂b 〉 is the probability to find the atom in intermediate level, and 〈η̂c 〉 is the probability to find the atom in the bottom level. The steady state solutions of equations (32)-(37) are found to be Ω 〈σ ̂a 〉 = (γ+γ ) 〈σ ̂†b 〉, c Ω 〈σ ̂†a 〉, ̂b 〉 = − (γ+γ ) 〈σ Ω c 〈σ ̂c 〉 = (γ+γ ) (〈η̂c 〉 − 〈η̂a 〉), 〈η̂a 〉 = c Ω 2(γ+γc ) 〈η̂b 〉 = 〈η̂a 〉. (〈σ ̂†c 〉 + 〈σ ̂c 〉), (41) (42) (43) (44) (45) Furthermore, with the aid of equation. (40), one readily obtains 〈η̂c 〉 = 1 − 〈η̂a 〉 − 〈η̂b 〉. In view of equations (45), equation (46) has the form 〈𝜂̂ 𝑐 〉 = 1 − 2〈𝜂̂ 𝑎 〉. (46) (47) Now, on account of equation (47), equation (43) can be expressed as Ω 〈σ ̂c 〉 = (γ+γ ) [1 − 3〈η̂a 〉]. (48) 〈𝜎̂𝑐† 〉 = 〈𝜎̂𝑐 〉. (49) 〈𝜎̂𝑎 〉 = 〈𝜎̂𝑏 〉 = 0. (50) c With the aid of equation (48), one can observe that Also, from equations (41), (42), and (43), one can readily obtains Ω 〈σ 〈η̂a 〉 = ̂c 〉. (γ+γ ) c (51) Bessie Mengesha 6 of 15 By substituting equation (48) into (51) yields 〈η̂a 〉 = (γ+γ Ω2 . (52) . (53) . (54) . (55) 2 2 c ) +3Ω Moreover, on account of equation (45), one can obtain 〈η̂b 〉 = (γ+γ Now, by Substituting (52) in (47), we have Ω2 2 2 c ) +3Ω Ω2 +(γ+γc )2 〈η̂c 〉 = (γ+γ 2 2 c ) +3Ω Finally, on account of equation (52), equation (48) takes the form Ω(γ+γc ) 〈σ ̂c 〉 = (γ+γ 2 2 c ) +3Ω 3. Photon Statistics The mean photon number for the cavity light modes a and b is given by [15] 𝑛̅𝑎 = 〈𝑎̂† 𝑎̂〉 = 〈𝑎̂𝑎̂† 〉 = 𝛾𝑐 𝜅 〈𝜂̂ 𝑎 〉. (56) On account of equations (26) and (52), equation (56) can be written as 𝑛̅𝑎 = 𝛾𝑐 n̅a = γc 𝛺2 [ 𝜅 (𝛾+𝛾𝑐 )2 +3𝛺2 ]. (57) For non-spontaneous case (𝛾 = 0), the mean photon number of light mode a has the form κ ( Ω2 2 γ2 c +3Ω ). (58) In addition, for 𝛺 ≫ 𝛾𝑐 , equation (58) becomes 𝑛̅𝑎 = 𝛾𝑐 3𝜅 . (59) The mean photon number of light mode b is determined using the same procedure as n̅b = 〈b̂ † b̂〉 = γc κ 〈η̂b 〉 = γc κ [(γ+γ For non-spontaneous case, equation (60) takes the form n̅b = γc κ ( Ω2 2 γ2 c +3Ω ). In addition, for Ω ≫ γc , equation (61) reduces to 𝑛̅𝑏 = 𝛾𝑐 3𝜅 . Ω2 2 2 c ) +3Ω ]. (60) (61) (62) 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 Bessie Mengesha 7 of 15 n̅c = 〈ĉ † ĉ〉. (63) The mean photon number has the form when using the steady state solution of equation (14) and its adjoint n̅c = γc n̅c = γc n̅c = γc n̅c = 2γc κ (〈η̂a 〉 + 〈η̂b 〉). (64) Substituting equations (57) and (58) in (64) for the steady state solution of (14) yields κ 2Ω2 ((γ+γ 2 2 c ) +3Ω ). (65) Now, the mean photon number in the non-spontaneous scenario is in the form For Ω ≫ γc , equation (66) reduces to κ ( 3κ 2Ω2 2 γ2 c +3Ω ). (66) . (67) Furthermore, the variance of the photon number is expressible as [3, 5] (Δn)2 = 〈n̅2 〉 − 〈n̅〉2 . (68) (Δna )2 = 〈↠â〉〈â↠〉 + 〈â†2 〉〈â2 〉. (69) On account of equation (56), the variance of the photon number for light mode a is described as Upon use of equations (26) and (50), one obtains Moreover, 〈â〉 = − 2g κ 〈σ ̂a 〉 ≡ 0. 〈â2 〉 = 〈(− In view of equation (5), one readily obtains 〈â2 〉 = γc κ 2g κ (70) γc ̂a )〉 = σ κ 〈σ ̂2 〉. (71) 〈(|b⟩⟨a|)2 〉 = 0. (72) Equations (69), and (71) are used to calculate the variance of the photon number for light mode a as (𝛥𝑛𝑎 )2 = 〈𝑎̂† 𝑎̂〉〈𝑎̂𝑎̂† 〉. (73) On account of equations (52), (53), and (56), equation (73) becomes γ 2 (Δna )2 = ( c ) [ ((γ+γ κ Ω4 2 2 2 c ) +3Ω ) ]. (74) Furthermore, for non-spontaneous case, the photon number variance has the form 𝛾 For Ω ≫ γc , 2 (𝛥𝑛𝑎 )2 = ( 𝑐 ) [ 𝜅 𝛺4 2 (𝛾𝑐2 +3𝛺2 ) ]. (75) Bessie Mengesha 8 of 15 γ 2 (Δna )2 = ( c ) . 3κ (76) With the same procedure one can obtain the variance of the photon number for light mode b as γ 2 Ω4 +Ω2 (γ+γc )2 (Δnb )2 = ( c ) [ ((γ+γ κ 2 2 2 c ) +3Ω ) For non-spontaneous case, equation (75) takes the form γ 2 Ω2 (Ω2 +γ2 )2 c γ 2 (Δnb )2 = ( c ) [ For Ω ≫ γc , equation (75) reduces to κ 2 (γ2 c +3Ω ) 2 ]. ]. (78) (Δnb )2 = ( c ) = (Δnb )2 = n̅2a ≡ n̅2b , 3κ (77) (79) which represents the normally-ordered variance of the photon number for the chaotic light. Furthermore, equation (79) indicates that (𝛥𝑛𝑎 )2 > 𝑛̅𝑎 and (𝛥𝑛𝑏 )2 > 𝑛̅𝑏 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 γ 2 4Ω4 +3Ω2 (γ+γ )2 c (Δnc )2 = ( c ) [ ((γ+γ κ 2 2 2 c ) +3Ω ) For non-spontaneous case, equation (80) has the form γ 2 4Ω4 +3Ω2 γ2 c (Δnc )2 = ( c ) [ κ 2 2 (γ2 c +3Ω ) Additionally, Ω ≫ γc , equation (81) reduces to (Δnc )2 = ( ]. ]. 2γc 2 3κ ) ≡ n̅2c , (80) (81) (82) which represents the normally-ordered variance of the photon number for chaotic light. Furthermore, inspection of equation (82) indicates that (Δnc )2 > n̅2c and hence the photon statistics of the two-mode light is super-poissionian. 4. Quadrature Squeezing and The mean Photon number The squeezing properties of light mode a are described by the two quadrature operators [16-18] and â+ = 𝑎̂+ + 𝑎̂, (83) 𝑎̂− = 𝑖(𝑎̂+ − 𝑎̂). (84) In view of equations (83) and (84), the commutation relation becomes [𝑎̂− , 𝑎̂+ ] = 2𝑖 𝛾𝑐 𝜅 (𝜂̂ 𝑎 − 𝜂̂ 𝑏 ). (85) The uncertainty relation for the two Hermitian operators 𝐴̂ and 𝐵̂ satisfies the commutation relation 𝑖𝐶̂ , which is described as [5] Bessie Mengesha 9 of 15 1 𝛥𝐴𝛥𝐵 ≥ |〈𝐶̂ 〉|. (86) 2 Upon use of equation (86), one can readily obtains 1 ∆𝑎+ ∆𝑎− ≥ |〈[𝑎̂− , 𝑎̂+ ]〉| ≥ |〈𝑎̂+ 𝑎̂〉 − 〈𝑎̂𝑎̂+ 〉|. 2 (87) On account of equation (57) along with (75), one obtains ∆𝑎+ ∆𝑎− ≥ 0. (88) (∆a+ )2 = 〈â2+ 〉 − 〈â+ 〉2 , (89) (∆a− )2 = 〈â2− 〉 − 〈â− 〉2 . (90) Next the variance of the plus and minus quadrature operators becomes [17] and In consideration of equation (84), equation (87) can be expressed in terms of the raising and lowering operators as (∆a± )2 = 〈â↠〉 + 〈↠â〉 ± 〈â〉2 ± 〈↠〉2 ∓ 〈â2 〉 ∓ 〈â†2 〉 ∓ 2〈â〉〈↠〉. (91) In view of equations (70) and (72), equation (91) reduces to (∆𝑎± )2 = 〈𝑎̂† 𝑎̂〉 + 〈𝑎̂𝑎̂† 〉. (92) Now, by using equations (56) and (64), one obtains (∆a± )2 = γc (∆a± )2 = γc (∆a± )2 = γc (∆𝑎± )2 = 2𝛾𝑐 κ [〈η̂a 〉 + 〈η̂b 〉]. (93) On substituting equations (52) and (53), the quadrature variance for the light mode a becomes κ 2Ω2 [(γ 2 2 c +γ) +3Ω [(γ 2 2 c ) +3Ω ]. For non-spontaneous case (𝛾 = 0), the quadrature variance has the form κ 2Ω2 In addition, for 𝛺 ≫ 𝛾𝑐 , equation (94) reduces to 3𝜅 . ]. (94) (95) (96) In view of equation (63), the quadrature variance of light mode a can be written in terms of the mean photon number as (∆𝑎± )2 = 2𝑛̅𝑎 , (97) 𝑛̅𝑐 = (𝛥𝑎± )2 . (98) which is the normally-ordered quadrature variance for chaotic light. In the absence and presence of spontaneous emission the mean photon number of the two-mode light is the same as with the quadrature variance of light mode a. This can be written as In the same procedure the quadrature variance of light mode b can be obtained as Bessie Mengesha 10 of 15 (∆b± )2 = For 𝛾 = 0, equation (99) reduces to (∆b± )2 = γc 2Ω2 +(γc +γ)2 [(γ κ 2 2 c +γ) +3Ω γc 2Ω2 +γc 2 κ [ γc 2 +3Ω2 ]. ]. (100) And for Ω ≫ γc , Hence, equation (100) merely becomes (∆𝑏± )2 = 2𝛾𝑐 3𝜅 (99) . (101) Thus, the quadrature variance of light mode b is written in terms of the mean photon number as (∆𝑎𝑏± )2 = 2𝑛̅𝑏 , (102) (103) and 𝑐̂+ = 𝑐̂ † + 𝑐̂ , (104) where, ĉ− = i(ĉ † − ĉ). ĉ = â + b̂. (105) which is the normally-ordered quadrature variance for chaotic light. The squeezing properties of the two-mode cavity light can be described as With the aid of equations (103) and (104), the commutation relation is found to be [ĉ− , ĉ+ ] = 2i γc κ (η̂a − η̂c ). (106) The quadrature operators’ uncertainty relation for two-mode cavity light is expressed as [10, 11] 1 Δc+ Δc− ≥ |〈[ĉ− , ĉ+ ]〉|. (107) 2 Now, in view of equation (106), one can re-write equation (107) as Δc+ Δc− ≥ γc Δc+ Δc− ≥ γc Δc+ Δc− ≥ γc Δc+ Δc− ≥ γc κ |〈η̂a 〉 − 〈η̂c 〉|. (108) By employing equations (53) and (54), equation (108) can be written as κ |(γ (γc +γ)2 2 2 c +γ) +3Ω |. In the absence of spontaneous emission (𝛾 = 0), it becomes | γ2 c 2 κ γ2 c +3Ω |. In the absence of deriving coherent light (Ω = 0) κ , (109) (110) (111) which is the uncertainty relation for vacuum state. The variance of the plus and minus quadrature operators of the two-mode cavity light are defined as [19] Bessie Mengesha 11 of 15 and (Δc+ )2 = 〈ĉ+2 〉 − 〈ĉ− 〉2 , (112) (Δc− )2 = 〈ĉ−2 〉 − 〈ĉ− 〉2 . (113) On account of equations (105), (112) and (113), the plus and minus quadrature variance for the creation and annihilation operators can be written as (∆c± )2 = 〈ĉĉ † 〉 + 〈ĉ † ĉ〉 ± 〈ĉ〉2 ± 〈ĉ † 〉2 ∓ 〈ĉ 2 〉 ∓ 〈ĉ †2 〉 ∓ 2〈ĉ〉〈ĉ † 〉. (114) Now, using the steady state solution of equation (14) along with (50), one can get 〈𝑐̂ 〉 = 0. (115) In view of equation (115), the quadrature variance becomes (∆𝑐± )2 = 〈𝑐̂ 𝑐̂ † 〉 + 〈𝑐̂ † 𝑐̂ 〉 ± 〈𝑐̂ 〉2 ± 〈𝑐̂ † 〉2 . (116) Thus, with the aid of equation (68) along with (111), equation (116) becomes (∆𝑐± )2 = 𝛾𝑐 𝜅 [〈𝜂̂ 𝑎 〉 + 2〈𝜂̂ 𝑏 〉 ± 〈η̂𝑐 〉 ± 〈𝜎̂𝑐 〉 ± 〈𝜎̂𝑐† 〉]. (117) By substituting equations (64)-(57) in (117), one obtains (∆𝑐± )2 = 𝛾𝑐 4𝛺2 +(𝛾𝑐 +𝛾)2 ±2𝛺(𝛾𝑐 +𝛾) 𝜅 [ (𝛾𝑐 +𝛾)2 +3𝛺2 ]. In the absence of spontaneous emission (𝛾 = 0), equation (118) turns to (∆𝑐± )2 = 𝛾𝑐 4𝛺2 +𝛾𝑐2 ±2𝛺𝛾𝑐 𝜅 [ 𝛾𝑐2 +3𝛺2 ]. Besides, for 𝛺 ≫ 𝛾𝑐 , equation (119) will have the form (∆𝑐± )2 = 4𝛾𝑐 3𝜅 . (118) (119) (120) In view of equation (71), equation (120) can be expressed as (∆𝑐± )2 = 2𝑛̅𝑐 , (121) n̅c = 0, (122) where this represents the normally-ordered quadrature variance for chaotic light. For Ω = 0, equations (69), (110), and (119) become Δc+ Δc− ≥ (∆c± )2 = γc κ γc κ . , (123) (124) The mean photon number and quadrature variance of a two-mode vacuum condition are represented by equations (122), (123), and (124). 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= 2 (∆c− )2 υ −(Δc− ) (∆c− )2 υ . Equations (109) and (120) are used to obtain (125) Bessie Mengesha 12 of 15 S=1−[ 4Ω2 +(γc +γ)2 −2Ω(γc +γ) (γc +γ)2 +3Ω2 For 𝛾 = 0, the above expression reduces to S= 2Ωγc −Ω2 2 γ2 c +3Ω ≡ 2η−η2 3η2 +1 ]≡ with η = 2Ω(γc +γ)−Ω2 (γc +γ)2 +3Ω2 Ω γc . . (126) (127) 5. Physical Analysis 1. Plots mean photon number (As shown in the Figure 2 below) Figure 2. Plots of eqs (57), (58) versus Ω for γc=0.5, κ=0.8, γ=0 (red curve) and γ=0.1 (blue curve). 2. Plots of the variance of the photon number for light mode a (As shown in the Figure 3 below) Figure 3. Plots of Eqs. (74) And (75) versus Ω for γc=0.5, κ=0.8, γ=0 (red curve) and γ=0.1 (blue curve). Bessie Mengesha 13 of 15 3. Plots of the variance of the photon number for light mode b (As shown in the Figure 4 below) Figure 4. Plots of Eqs. (77) And (78) versus Ω for γc=0.5, κ=0.8, γ=0 (red curve) and γ=0.1 (blue curve). 4. Plots of quadrature variance of two modes light (As shown in the Figure 5 below) Figure 5. Plots of Eqs. (118) and (119) versus Ω for γc=0.5, γ=0 & κ=0.8 (red color) and γ=0.1 (blue color). Bessie Mengesha 14 of 15 5. Plots of quadrature squeezing (As shown in the Figure 6 below) Figure 6. Plots of Eq. (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 𝛾 = 0 . The plots, however, overlap at that moment 𝛺 = 0.55 in Figure 3. This demonstrates that when the variation of the light's photon number is greater for 𝛾 = 0 mode b than for 𝛾 = 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 𝛺 < 0.24 and is anticipated to be bigger in the absence of spontaneous emission when 𝛺 > 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. 6. 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 Bessie Mengesha 15 of 15 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. Acknowledgments: I would like to thank the anonymous reviewers of the paper for their useful comments. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] B. Daniel and K. Fesseha, "Opt. Commun," pp. 151, 384, 1998. A. Eschmann and M. D. Reid, Phys. Rev A, Vols. 49, 2881, 1994. A. Eschmann and M. D. Reid., Phys. Rev. A 49, 2881, 1994. N. Ansari, pp. A 48, 4686, 1993. K.Fesseha, Opt.Commun.156, 145, 1998. M.O. Scully, K. Wodkiewicz, M.S. Zubairy, J. Bergou, N. Lu, and J. Meyerter Veh, Phys. Rev. Lett, pp. 60, 1832, 1988. Fesseha Kassahun, Opt. Commun., vol. 1357, p. 284, 2011. K.Fesseha, "Opt.Commun," 1998, pp. 156,145. G. Asmelash, "PhD dissertation," Addis Ababa University, 2017. N. Lu, F.X. Zhao and J. Bergou., Phys. Rev. A39, 5189, 1989. G. New, Introduction to Nonlinear Optics, New York: Cambridge University Press, 2011. F. Kassahun, "Fundamentals of Quantum Optics," Lulu Press Inc., 2010. F. Kassahun, "arXiv:1105.1438v3," quant-ph, 2012. M.O. Scully, K. Wodkiewicz, M.S. Zubairy, J. Bergou, N. Lu, and J. Meyerter Veh., Phys. Rev. Lett. 60, 1832, 1988. F. Kassahun., "Fundamentals of Quantum Optics," Lulu Press Inc., North Car-, 2008. F. Kassahun, Basic Quantum Mechanics, Elizabeth printing press, 2017. C.A. Blockely and D.F.Walls, Phys. Rev., pp. A 43, 5049, 1991. Eyob Alebachew and K. Fesseha, Opt. Commun, pp. 265, 314, 2006. J. Anwar andM.S. Zubairy, Phys. Rev. A 49,481, 1994. A. Sizmann, R. J. Borowicz, G. Wagner, and G. Lcuchs., Opt. Commun. 80, 138, 1990.