Introduction

Quantum optics primarily deals with the quantum properties of light generated by various optical systems. The well-known quantum states of light include the number state, chaotic state, coherent state, and squeezed state1,2. Squeezed states of light have played an important role in the development of quantum physics.The non-classical properties of light, such as squeezing, have been extensively studied by numerous researchers2,3,4,5,6,7,8. In squeezed light, the noise in one quadrature is lower than the vacuum-state level, while the other quadrature exhibits higher fluctuations, such that the product of the uncertainties in the two quadratures satisfies the uncertainty relation.

Squeezed light has applications in weak signal detection and low-noise optical communication.Various quantum optical processes can produce squeezed light, including subharmonic generation5, second harmonic generation2, four-wave mixing2, resonance fluorescence3, and three-level lasers.Three-level atoms inside a cavity produce light in a three-level laser, a quantum optical device typically coupled to a vacuum reservoir via a single-port mirror. In a cascade configuration, when a three-level atom transitions from the top level to the bottom level via the intermediate level, it emits two photons. The strong correlation between these two photons enables the three-level laser to produce squeezed light3,4,9,8.

Three-level atoms in a cavity, usually connected to a vacuum reservoir, generate light in a three-level laser system. Kassahun9 studied three-level atoms in a closed cavity with the top and bottom levels coupled by coherent light. He demonstrated that the three-level laser can squeeze light up to 50% below the vacuum state level, with maximum global quadrature squeezing of 50% under certain conditions.

Squeezed light from a three-level atom can serve as a resource in quantum computing10, facilitate quantum communication11, and enhance measurements beyond the quantum noise limit12.We demonstrate significant squeezing of 52.1% and \(33.3\%\) from a three-level atom, proposing a novel and efficient technique for generating high-quality squeezed light directly from atomic systems.This approach could reduce the reliance on large nonlinear crystals or external squeezing devices in quantum communication, quantum sensing, and continuous-variable quantum computing13,14,15,16,11,10.

This work investigates the squeezing characteristics and quantum dynamics of a coherent light-driven noiseless cavity mode interacting with a three-level atom in an open cavity connected to a vacuum reservoir through a single-port mirror (see Fig. 1). We derive the quantum Langevin equations for the cavity modes, considering the vacuum reservoir outside the cavity, the interaction of the three-level atom with the resonant cavity modes, and the damping of the cavity modes by the reservoir.

We use the steady-state solutions of the evolution equations for the cavity modes and atomic annihilation operators to determine the mean photon number, variance of photon number, quadrature variance, and quadrature squeezing of the cavity modes.We also determine the antinormally ordered characteristic function, which we use to derive the Q function.The Q function is then employed to calculate the photon number distribution, an aspect not considered in previous studies on squeezing and photon statistics.

We achieve a novel 52.1% plus quadrature squeezing at \(\varepsilon = 0.59\) and \(\varepsilon = 0.85\) (\(\gamma = 0.0,0.35\)) and \(\varepsilon = 0.96\) (\(\gamma =0,45, 0.5\)), surpassing the 50% reported by Kassahun9. The minus quadrature exhibits 33.3% squeezing for \(\varepsilon \ge 16\), robust against spontaneous emission. The photon number distribution, derived from the Q-function Eq. (95), reveals a preference for even photon numbers, a new insight not explored in Alebachew17. Spontaneous emission reduces the mean photon number but does not affect maximum squeezing, enhancing applications in quantum communication10.

Operator dynamics

We consider a three-level atom in a cascade configuration placed in an open cavity driven by coherent light and coupled to a vacuum reservoir via a single-port mirror. The upper, intermediate, and lower levels of the three-level atom are denoted by \(|1\rangle\), \(|2\rangle\), and \(|3\rangle\), respectively. The atom absorbs a photon from the cavity and transitions upward from the bottom level. After emitting a photon, it decays to the intermediate level. Upon emitting a final photon, the atom decays to its lowest state. Spontaneous emission also occurs from \(|1\rangle\) to \(|2\rangle\), \(|2\rangle\) to \(|3\rangle\), and \(|1\rangle\) to \(|3\rangle\). We designate \(a_1\) as the cavity light from the top level, \(a_2\) as the cavity light from the intermediate level, and \(a_3\) as the cavity mode driven by coherent light.

The interaction of the driving coherent light with the resonant cavity mode is described by the Hamiltonian

$$\begin{aligned} \hat{H'}= i\lambda [\hat{c}\hat{a}_3^\dag - \hat{a}_3\hat{c}^\dag ], \end{aligned}$$
(1)

where \(\hat{c}\) and \(\hat{a}_3\) are the annihilation operators for the driving coherent light and the cavity mode, respectively, and \(\lambda\) is the coupling constant. To simplify the analysis, we replace the operator \(\hat{c}\) with a real and constant c-number \(\mu\). This allows us to rewrite the Hamiltonian as

$$\begin{aligned} \hat{H'} = i\eta (\hat{a}_3^\dag - \hat{a}_3), \end{aligned}$$
(2)

where \(\eta =\lambda \mu\). The interaction of the three-level atom with the resonant cavity modes \(a_1\), \(a_2\), and \(a_3\) is described by the Hamiltonian

$$\begin{aligned} \hat{H''} = i g\left( \hat{\sigma }_3^{\dag }\hat{a}_3 - \hat{a}_3^{\dag }\hat{\sigma }_3 + \hat{\sigma }_1^{\dag }\hat{a}_1 - \hat{a}_1^{\dag }\hat{\sigma }_1 + \hat{\sigma }_2^\dag \hat{a}_2 - \hat{a}_2^\dag \hat{\sigma }_2\right) , \end{aligned}$$
(3)

where

$$\begin{aligned} \hat{\sigma }_1=|2\rangle \langle 1|,\quad \hat{\sigma }_2=|3\rangle \langle 2|,\quad \hat{\sigma }_3=|3\rangle \langle 1| \end{aligned}$$
(4)

are the lowering atomic operators, and \(\hat{a}_1\), \(\hat{a}_2\), and \(\hat{a}_3\) are the annihilation operators for the cavity modes \(a_1\), \(a_2\), and \(a_3\), respectively. The coupling constant g describes the interaction strength between the atom and the cavity modes. Considering Eqs. (2) and (3), the total Hamiltonian of the system under consideration can be written as

$$\begin{aligned} \hat{H}= i\eta (\hat{a}_3^\dag - \hat{a}_3) + i g\sum ^3_{j=1}\left( \hat{\sigma }_j^{\dag }\hat{a}_j - \hat{a}_j^{\dag }\hat{\sigma }_j\right) . \end{aligned}$$
(5)
Fig. 1
figure 1

Schematic representation of a three-level atom in cascade configuration within an open cavity driven by coherent light and coupled to a vacuum reservoir via a single-port mirror. Decay paths include spontaneous emission \(\gamma\) and stimulated emission \(\gamma _c\) for all transitions. The cavity modes \(\hat{a}_1\), \(\hat{a}_2\), and \(\hat{a}_3\) (where \(\hat{a}_3\) is the cavity mode driven by coherent light) correspond to the transitions \(|1\rangle \rightarrow |2\rangle\), \(|2\rangle \rightarrow |3\rangle\), and \(|1\rangle \rightarrow |3\rangle\), respectively.

Full size image

In this section, we focus on the situation where a three-level atom in a cascade configuration is placed in an open cavity driven by coherent light and connected to a vacuum reservoir through a single-port mirror. We note that

$$\begin{aligned} \frac{d}{dt}\langle \hat{a}_3^\dag \hat{a}_3\rangle =-iTr(\rho [\hat{a}_3^\dag \hat{a}_3,H])+Tr\left( \frac{\gamma }{2} \sum _{j=1}^3 \left( 2\hat{\sigma }_j \hat{\rho } \hat{\sigma }_j^\dagger - \hat{\sigma }_j^\dagger \hat{\sigma }_j \hat{\rho }-\hat{\rho } \hat{\sigma }_j^\dagger \hat{\sigma }_j \right) +\frac{\kappa }{2} \left( 2\hat{a}_3 \hat{\rho } \hat{a}_3^\dagger -\hat{a}_3^\dagger \hat{a}_3 \hat{\rho }-\hat{\rho } \hat{a}_3^\dagger \hat{a}_3 \right) \right) \hat{a}_3^\dag \hat{a}_3, \end{aligned}$$
(6)
$$\begin{aligned} \frac{d}{dt}\langle \hat{a}_3\hat{a}_3^\dag \rangle =-iTr(\rho [\hat{a}_3^\dag \hat{a}_3,H])+Tr\left( \frac{\gamma }{2} \sum _{j=1}^3 \left( 2\hat{\sigma }_j \hat{\rho } \hat{\sigma }_j^\dagger - \hat{\sigma }_j^\dagger \hat{\sigma }_j \hat{\rho }-\hat{\rho } \hat{\sigma }_j^\dagger \hat{\sigma }_j \right) +\frac{\kappa }{2} \left( 2\hat{a}_3 \hat{\rho } \hat{a}_3^\dagger -\hat{a}_3^\dagger \hat{a}_3 \hat{\rho }-\hat{\rho } \hat{a}_3^\dagger \hat{a}_3 \right) \right) \hat{a}_3\hat{a}_3^\dag \end{aligned}$$
(7)
$$\begin{aligned} \frac{d}{dt}\langle \hat{a}_3\rangle ^2= -iTr(\rho [\hat{a}_3^2,H])+Tr\left( \frac{\gamma }{2} \sum _{j=1}^3 \left( 2 \hat{\sigma }_j \hat{\rho } \hat{\sigma }_j^\dagger - \hat{\sigma }_j^\dagger \hat{\sigma }_j \hat{\rho } - \hat{\rho } \hat{\sigma }_j^\dagger \hat{\sigma }_j \right) +\frac{\kappa }{2} \left( 2 \hat{a}_3 \hat{\rho } \hat{a}_3^\dagger - \hat{a}_3^\dagger \hat{a}_3 \hat{\rho } - \hat{\rho } \hat{a}_3^\dagger \hat{a}_3 \right) \right) \hat{a}_3^2, \end{aligned}$$
(8)
$$\begin{aligned} \frac{d}{dt}\langle \hat{a}_3^2\rangle = -iTr(\rho [\hat{a}_3^2,H])+Tr\left( \frac{\gamma }{2} \sum _{j=1}^3 \left( 2 \hat{\sigma }_j \hat{\rho } \hat{\sigma }_j^\dagger - \hat{\sigma }_j^\dagger \hat{\sigma }_j \hat{\rho } - \hat{\rho } \hat{\sigma }_j^\dagger \hat{\sigma }_j \right) +\frac{\kappa }{2} \left( 2 \hat{a}_3 \hat{\rho } \hat{a}_3^\dagger - \hat{a}_3^\dagger \hat{a}_3 \hat{\rho } - \hat{\rho } \hat{a}_3^\dagger \hat{a}_3 \right) \right) \hat{a}_3^2, \end{aligned}$$
(9)
$$\begin{aligned} \frac{d}{dt}\langle \hat{a}_3^\dag \rangle \langle \hat{a}_3\rangle = -iTr(\rho [ \hat{a}_3^\dag \hat{a}_3,H])+Tr\left( \frac{\gamma }{2} \sum _{j=1}^3 \left( 2 \hat{\sigma }_j \hat{\rho } \hat{\sigma }_j^\dagger - \hat{\sigma }_j^\dagger \hat{\sigma }_j \hat{\rho } - \hat{\rho } \hat{\sigma }_j^\dagger \hat{\sigma }_j \right) +\frac{\kappa }{2} \left( 2 \hat{a}_3 \hat{\rho } \hat{a}_3^\dagger - \hat{a}_3^\dag \hat{a}_3 \hat{\rho } - \hat{\rho } \hat{a}_3^\dag \hat{a}_3 \right) \right) \hat{a}_3^\dag \hat{a}_3 \end{aligned}$$
(10)
$$\begin{aligned} \frac{d}{dt}\langle \hat{\sigma }_1\rangle = -iTr(\rho [\hat{\sigma }_1,H])+Tr\left( \frac{\gamma }{2} \sum _{j=1}^3 \left( 2 \hat{\sigma }_j \hat{\rho } \hat{\sigma }_j^\dagger - \hat{\sigma }_j^\dagger \hat{\sigma }_j \hat{\rho } - \hat{\rho } \hat{\sigma }_j^\dagger \hat{\sigma }_j \right) +\frac{\kappa }{2} \left( 2 \hat{a}_3 \hat{\rho } \hat{a}_3^\dagger - \hat{a}_3^\dagger \hat{a}_3 \hat{\rho } - \hat{\rho } \hat{a}_3^\dagger \hat{a}_3 \right) \right) \hat{\sigma }_1, \end{aligned}$$
(11)
$$\begin{aligned} \frac{d}{dt}\langle \hat{\sigma }_2\rangle = -iTr(\rho [\hat{\sigma }_2,H])+Tr\left( \frac{\gamma }{2} \sum _{j=1}^3 \left( 2 \hat{\sigma }_j \hat{\rho } \hat{\sigma }_j^\dagger - \hat{\sigma }_j^\dagger \hat{\sigma }_j \hat{\rho } - \hat{\rho } \hat{\sigma }_j^\dagger \hat{\sigma }_j \right) +\frac{\kappa }{2} \left( 2 \hat{a}_3 \hat{\rho } \hat{a}_3^\dagger - \hat{a}_3^\dagger \hat{a}_3 \hat{\rho } - \hat{\rho } \hat{a}_3^\dagger \hat{a}_3 \right) \right) \hat{\sigma }_2, \end{aligned}$$
(12)
$$\begin{aligned} \frac{d}{dt}\langle \hat{\sigma }_3\rangle = -iTr(\rho [\hat{\sigma }_3,H])+Tr\left( \frac{\gamma }{2} \sum _{j=1}^3 \left( 2 \hat{\sigma }_j \hat{\rho } \hat{\sigma }_j^\dagger - \hat{\sigma }_j^\dagger \hat{\sigma }_j \hat{\rho } - \hat{\rho } \hat{\sigma }_j^\dagger \hat{\sigma }_j \right) +\frac{\kappa }{2} \left( 2 \hat{a}_3 \hat{\rho } \hat{a}_3^\dagger - \hat{a}_3^\dagger \hat{a}_3 \hat{\rho } - \hat{\rho } \hat{a}_3^\dagger \hat{a}_3 \right) \right) \hat{\sigma }_3, \end{aligned}$$
(13)
$$\begin{aligned} \frac{d}{dt}\langle \hat{\eta }_1\rangle = -iTr(\rho [\hat{\eta }_1,H])+Tr\left( \frac{\gamma }{2} \sum _{j=1}^3 \left( 2 \hat{\sigma }_j \hat{\rho } \hat{\sigma }_j^\dagger - \hat{\sigma }_j^\dagger \hat{\sigma }_j \hat{\rho } - \hat{\rho } \hat{\sigma }_j^\dagger \hat{\sigma }_j \right) +\frac{\kappa }{2} \left( 2 \hat{a}_3 \hat{\rho } \hat{a}_3^\dagger - \hat{a}_3^\dagger \hat{a}_3 \hat{\rho } - \hat{\rho } \hat{a}_3^\dagger \hat{a}_3 \right) \right) \hat{\eta }_1, \end{aligned}$$
(14)
$$\begin{aligned} \frac{d}{dt}\langle \hat{\eta }_2\rangle = -iTr(\rho [\hat{\eta }_2,H])+Tr\left( \frac{\gamma }{2} \sum _{j=1}^3 \left( 2 \hat{\sigma }_j \hat{\rho } \hat{\sigma }_j^\dagger - \hat{\sigma }_j^\dagger \hat{\sigma }_j \hat{\rho } - \hat{\rho } \hat{\sigma }_j^\dagger \hat{\sigma }_j \right) +\frac{\kappa }{2} \left( 2 \hat{a}_3 \hat{\rho } \hat{a}_3^\dagger - \hat{a}_3^\dagger \hat{a}_3 \hat{\rho } - \hat{\rho } \hat{a}_3^\dagger \hat{a}_3 \right) \right) \hat{\eta }_2, \end{aligned}$$
(15)

where \(\gamma\) and \(\kappa\) are the spontaneous emission and cavity damping constants for the cavity mode \(a_3\), respectively. Now, considering Eq. (5) and the following relations:

$$\begin{aligned} Tr\left( \frac{\gamma }{2} \sum _{j=1}^3 \left( 2 \hat{\sigma }_j \hat{\rho } \hat{\sigma }_j^\dagger - \hat{\sigma }_j^\dagger \hat{\sigma }_j \hat{\rho } - \hat{\rho } \hat{\sigma }_j^\dagger \hat{\sigma }_j \right) +\frac{\kappa }{2} \left( 2 \hat{a}_3 \hat{\rho } \hat{a}_3^\dagger - \hat{a}_3^\dagger \hat{a}_3 \hat{\rho } - \hat{\rho } \hat{a}_3^\dagger \hat{a}_3 \right) \right) \hat{a}_3^\dag \hat{a}_3=-\kappa \langle \hat{a}_3^\dagger \hat{a}_3 \rangle \end{aligned}$$
(16)
$$\begin{aligned} Tr\left( \frac{\gamma }{2} \sum _{j=1}^3 \left( 2\hat{\sigma }_j \hat{\rho } \hat{\sigma }_j^\dagger - \hat{\sigma }_j^\dagger \hat{\sigma }_j \hat{\rho }-\hat{\rho } \hat{\sigma }_j^\dagger \hat{\sigma }_j \right) +\frac{\kappa }{2} \left( 2\hat{a}_3 \hat{\rho } \hat{a}_3^\dagger -\hat{a}_3^\dagger \hat{a}_3 \hat{\rho }-\hat{\rho } \hat{a}_3^\dagger \hat{a}_3 \right) \right) \hat{a}_3\hat{a}_3^\dag =-\kappa \langle \hat{a}_3 \hat{a}_3^\dag \rangle \end{aligned}$$
(17)
$$\begin{aligned} Tr\left( \frac{\gamma }{2} \sum _{j=1}^3 \left( 2 \hat{\sigma }_j \hat{\rho } \hat{\sigma }_j^\dagger - \hat{\sigma }_j^\dagger \hat{\sigma }_j \hat{\rho } - \hat{\rho } \hat{\sigma }_j^\dagger \hat{\sigma }_j \right) +\frac{\kappa }{2} \left( 2 \hat{a}_3 \hat{\rho } \hat{a}_3^\dagger - \hat{a}_3^\dagger \hat{a}_3 \hat{\rho } - \hat{\rho } \hat{a}_3^\dagger \hat{a}_3 \right) \right) \hat{a}_3^2=-\kappa \langle \hat{a}_3\rangle ^2, \end{aligned}$$
(18)
$$\begin{aligned} Tr\left( \frac{\gamma }{2} \sum _{j=1}^3 \left( 2 \hat{\sigma }_j \hat{\rho } \hat{\sigma }_j^\dagger - \hat{\sigma }_j^\dagger \hat{\sigma }_j \hat{\rho } - \hat{\rho } \hat{\sigma }_j^\dagger \hat{\sigma }_j \right) +\frac{\kappa }{2} \left( 2 \hat{a}_3 \hat{\rho } \hat{a}_3^\dagger - \hat{a}_3^\dagger \hat{a}_3 \hat{\rho } - \hat{\rho } \hat{a}_3^\dagger \hat{a}_3 \right) \right) \hat{a}_3^2=-\kappa \langle a_3^2\rangle \end{aligned}$$
(19)
$$\begin{aligned} Tr\left( \frac{\gamma }{2} \sum _{j=1}^3 \left( 2 \hat{\sigma }_j \hat{\rho } \hat{\sigma }_j^\dagger - \hat{\sigma }_j^\dagger \hat{\sigma }_j \hat{\rho } - \hat{\rho } \hat{\sigma }_j^\dagger \hat{\sigma }_j \right) +\frac{\kappa }{2} \left( 2 \hat{a}_3 \hat{\rho } \hat{a}_3^\dagger - \hat{a}_3^\dagger \hat{a}_3 \hat{\rho } - \hat{\rho } \hat{a}_3^\dagger \hat{a}_3 \right) \right) \hat{a}_3^\dag \hat{a}_3=-\kappa \langle \hat{a}_3^\dag \rangle \langle \hat{a}_3\rangle \end{aligned}$$
(20)
$$\begin{aligned} Tr\left( \frac{\gamma }{2} \sum _{j=1}^3 \left( 2 \hat{\sigma }_j \hat{\rho } \hat{\sigma }_j^\dagger - \hat{\sigma }_j^\dagger \hat{\sigma }_j \hat{\rho } - \hat{\rho } \hat{\sigma }_j^\dagger \hat{\sigma }_j \right) +\frac{\kappa }{2} \left( 2 \hat{a}_3 \hat{\rho } \hat{a}_3^\dagger - \hat{a}_3^\dagger \hat{a}_3 \hat{\rho } - \hat{\rho } \hat{a}_3^\dagger \hat{a}_3 \right) \right) \hat{\sigma }_1={-3\gamma \over 2}\langle \sigma _1\rangle \end{aligned}$$
(21)
$$\begin{aligned} Tr\left( \frac{\gamma }{2}\sum _{j=1}^3 \left( 2 \hat{\sigma }_j \hat{\rho } \hat{\sigma }_j^\dagger - \hat{\sigma }_j^\dagger \hat{\sigma }_j \hat{\rho } - \hat{\rho } \hat{\sigma }_j^\dagger \hat{\sigma }_j \right) +\frac{\kappa }{2} \left( 2 \hat{a}_3 \hat{\rho } \hat{a}_3^\dagger -\hat{a}_3^\dagger \hat{a}_3 \hat{\rho } - \hat{\rho } \hat{a}_3^\dagger \hat{a}_3 \right) \right) \hat{\sigma }_2={-1\over 2}\gamma \langle \sigma _2\rangle \end{aligned}$$
(22)
$$\begin{aligned} Tr\left( \frac{\gamma }{2} \sum _{j=1}^3 \left( 2 \hat{\sigma }_j \hat{\rho } \hat{\sigma }_j^\dagger - \hat{\sigma }_j^\dagger \hat{\sigma }_j \hat{\rho } - \hat{\rho } \hat{\sigma }_j^\dagger \hat{\sigma }_j \right) +\frac{\kappa }{2} \left( 2 \hat{a}_3 \hat{\rho } \hat{a}_3^\dagger - \hat{a}_3^\dagger \hat{a}_3 \hat{\rho } - \hat{\rho } \hat{a}_3^\dagger \hat{a}_3 \right) \right) \hat{\sigma }_3=-\gamma \langle \sigma _3\rangle \end{aligned}$$
(23)
$$\begin{aligned} Tr\left( \frac{\gamma }{2} \sum _{j=1}^3 \left( 2 \hat{\sigma }_j \hat{\rho } \hat{\sigma }_j^\dagger - \hat{\sigma }_j^\dagger \hat{\sigma }_j \hat{\rho } - \hat{\rho } \hat{\sigma }_j^\dagger \hat{\sigma }_j \right) +\frac{\kappa }{2} \left( 2 \hat{a}_3 \hat{\rho } \hat{a}_3^\dagger - \hat{a}_3^\dagger \hat{a}_3 \hat{\rho } - \hat{\rho } \hat{a}_3^\dagger \hat{a}_3 \right) \right) \hat{\eta }_1=-2\gamma \langle \eta _1 \rangle \end{aligned}$$
(24)
$$\begin{aligned} Tr\left( \frac{\gamma }{2} \sum _{j=1}^3 \left( 2 \hat{\sigma }_j \hat{\rho } \hat{\sigma }_j^\dagger - \hat{\sigma }_j^\dagger \hat{\sigma }_j \hat{\rho } - \hat{\rho } \hat{\sigma }_j^\dagger \hat{\sigma }_j \right) +\frac{\kappa }{2} \left( 2 \hat{a}_3 \hat{\rho } \hat{a}_3^\dagger - \hat{a}_3^\dagger \hat{a}_3 \hat{\rho } - \hat{\rho } \hat{a}_3^\dagger \hat{a}_3 \right) \right) \hat{\eta }_2=-\gamma \langle \eta _2\rangle +\gamma \langle \eta _1\rangle \end{aligned}$$
(25)

we readily obtain

$$\begin{aligned} \frac{d}{dt} \langle \hat{a}_3^\dagger \hat{a}_3 \rangle = \eta \langle \hat{a}_3^\dagger + \hat{a}_3 \rangle - g \langle \hat{\sigma }_3^\dagger \hat{a}_3 + \hat{a}_3^\dagger \hat{\sigma }_3 \rangle -\kappa \langle \hat{a}_3^\dagger \hat{a}_3 \rangle \end{aligned}$$
(26)
$$\begin{aligned} \frac{d}{dt} \langle \hat{a}_3 \hat{a}_3^\dag \rangle = \eta \langle \hat{a}_3^\dagger + \hat{a}_3 \rangle - g \langle \hat{\sigma }_3 \hat{a}_3^\dag + \hat{a}_3 \hat{\sigma }_3^\dag \rangle -\kappa \langle \hat{a}_3 \hat{a}_3^\dag \rangle \end{aligned}$$
(27)
$$\begin{aligned} \frac{d}{dt} \langle \hat{a}_3 \rangle ^2 = 2\eta \langle \hat{a}_3 \rangle -2g \langle \hat{a}_3 \rangle \langle \hat{\sigma }_3 \rangle - \kappa \langle \hat{a}_3 \rangle ^2 \end{aligned}$$
(28)
$$\begin{aligned} \frac{d}{dt} \langle \hat{a}_3^2 \rangle = 2\eta \langle \hat{a}_3 \rangle - 2g \langle \hat{a}_3 \hat{\sigma }_3 \rangle - \kappa \langle \hat{a}_3^2 \rangle \end{aligned}$$
(29)
$$\begin{aligned} \frac{d}{dt} \langle \hat{a}_3^\dagger \rangle \langle \hat{a}_3 \rangle = \eta ( \langle \hat{a}_3^\dagger \rangle + \langle \hat{a}_3 \rangle ) - g (\langle \hat{\sigma }_3^\dagger \rangle \langle \hat{a}_3 \rangle + \langle \hat{\sigma }_3 \rangle \langle \hat{a}_3^\dagger \rangle ) - \kappa \langle \hat{a}_3^\dagger \rangle \langle \hat{a}_3 \rangle \end{aligned}$$
(30)
$$\begin{aligned} \frac{d}{dt}\langle \hat{\sigma }_1\rangle = {-3\gamma \over 2}\langle \sigma _1\rangle +g\langle \hat{\sigma }_2^\dag \hat{a}_3 \rangle + g\langle (\hat{\eta }_2 - \hat{\eta }_1) \hat{a}_1 \rangle + g\langle \hat{a}_2^\dag \hat{\sigma }_3 \rangle \end{aligned}$$
(31)
$$\begin{aligned} \frac{d}{dt}\langle \hat{\sigma }_2 \rangle&= -\frac{1}{2}\gamma \langle \hat{\sigma }_2 \rangle - g\langle \hat{a}_1^\dag \hat{\sigma }_3 \rangle + g\langle (\hat{\eta }_3 - \hat{\eta }_2) \hat{a}_2 \rangle - g\langle \hat{\sigma }_1^\dag \hat{a}_3 \rangle , \end{aligned}$$
(32)
$$\begin{aligned} \frac{d}{dt}\langle \hat{\sigma }_3 \rangle&= -\gamma \langle \hat{\sigma }_3 \rangle + g\langle \hat{\sigma }_2 \hat{a}_1 \rangle + g\langle (\hat{\eta }_3 - \hat{\eta }_1) \hat{a}_3 \rangle - g\langle \sigma _1\hat{a}_3 \rangle , \end{aligned}$$
(33)
$$\begin{aligned} \frac{d}{dt}\langle \hat{\eta }_1 \rangle&= -2\gamma \langle \hat{\eta }_1 \rangle + g\langle \hat{\sigma }_3^\dag \hat{a}_3 \rangle + g\langle \hat{\sigma }_1^\dag \hat{a}_1 \rangle + g\langle \hat{a}_3^\dag \hat{\sigma }_3 \rangle + g\langle \hat{a}_1^\dag \hat{\sigma }_1 \rangle , \end{aligned}$$
(34)
$$\begin{aligned} \frac{d}{dt}\langle \hat{\eta }_2 \rangle&= -\gamma \langle \hat{\eta }_2 \rangle + \gamma \langle \hat{\eta }_1 \rangle + g\langle \hat{\sigma }_2^\dag \hat{a}_2 \rangle - g\langle \hat{\sigma }_1^\dag \hat{a}_1 \rangle + g\langle \hat{\sigma }_1^\dag \hat{a}_3 \rangle + g\langle \hat{a}_2^\dag \hat{\sigma }_2 \rangle - g\langle \hat{a}_1^\dag \hat{\sigma }_1 \rangle , \end{aligned}$$
(35)

where

$$\begin{aligned} \hat{\eta }_1 = |1\rangle \langle 1|,\quad \hat{\eta }_2 = |2\rangle \langle 2|. \end{aligned}$$
(36)

The steady-state solutions of the above equations are found to be

$$\begin{aligned} \kappa \langle \hat{a}_3^\dagger \hat{a}_3 \rangle = \eta \langle \hat{a}_3^\dagger + \hat{a}_3 \rangle - g \langle \hat{\sigma }_3^\dagger \hat{a}_3 + \hat{a}_3^\dagger \hat{\sigma }_3 \rangle \end{aligned}$$
(37)
$$\begin{aligned} \kappa \langle \hat{a}_3 \hat{a}_3^\dag \rangle = \eta \langle \hat{a}_3^\dagger + \hat{a}_3 \rangle - g \langle \hat{\sigma }_3 \hat{a}_3^\dag + \hat{a}_3 \hat{\sigma }_3^\dag \rangle \end{aligned}$$
(38)
$$\begin{aligned} \kappa \langle \hat{a}_3 \rangle ^2= 2\eta \langle \hat{a}_3 \rangle -2g \langle \hat{a}_3 \rangle \langle \hat{\sigma }_3 \rangle \end{aligned}$$
(39)
$$\begin{aligned} \kappa \langle \hat{a}_3^2 \rangle = 2\eta \langle \hat{a}_3 \rangle - 2g \langle \hat{a}_3 \hat{\sigma }_3 \rangle \end{aligned}$$
(40)
$$\begin{aligned} \kappa \langle \hat{a}_3^\dagger \rangle \langle \hat{a}_3 \rangle = \eta ( \langle \hat{a}_3^\dagger \rangle + \langle \hat{a}_3 \rangle ) - g ( \langle \hat{\sigma }_3^\dagger \rangle \langle \hat{a}_3 \rangle + \langle \hat{\sigma }_3 \rangle \langle \hat{a}_3^\dagger \rangle ) \end{aligned}$$
(41)
$$\begin{aligned} {3\gamma \over 2}\langle \sigma _1\rangle =g\langle \hat{\sigma }_2^\dag \hat{a}_3 \rangle + g\langle (\hat{\eta }_2 - \hat{\eta }_1) \hat{a}_1 \rangle + g\langle \hat{a}_2^\dag \hat{\sigma }_3 \rangle \end{aligned}$$
(42)
$$\begin{aligned} \frac{1}{2}\gamma \langle \hat{\sigma }_2 \rangle =- g\langle \hat{a}_1^\dag \hat{\sigma }_3 \rangle + g\langle (\hat{\eta }_3 - \hat{\eta }_2) \hat{a}_2 \rangle - g\langle \hat{\sigma }_1^\dag \hat{a}_3 \rangle , \end{aligned}$$
(43)
$$\begin{aligned} \gamma \langle \hat{\sigma }_3 \rangle = g\langle \hat{\sigma }_2 \hat{a}_1 \rangle + g\langle (\hat{\eta }_3 - \hat{\eta }_1) \hat{a}_3 \rangle - g\langle \sigma _1\hat{a}_3 \rangle , \end{aligned}$$
(44)
$$\begin{aligned} 2\gamma \langle \hat{\eta }_1 \rangle = g\langle \hat{\sigma }_3^\dag \hat{a}_3 \rangle + g\langle \hat{\sigma }_1^\dag \hat{a}_1 \rangle + g\langle \hat{a}_3^\dag \hat{\sigma }_3 \rangle + g\langle \hat{a}_1^\dag \hat{\sigma }_1 \rangle , \end{aligned}$$
(45)
$$\begin{aligned} \gamma \langle \hat{\eta }_2 \rangle = \gamma \langle \hat{\eta }_1 \rangle + g\langle \hat{\sigma }_2^\dag \hat{a}_2 \rangle - g\langle \hat{\sigma }_1^\dag \hat{a}_1 \rangle + g\langle \hat{\sigma }_1^\dag \hat{a}_3 \rangle + g\langle \hat{a}_2^\dag \hat{\sigma }_2 \rangle -g\langle \hat{a}_1^\dag \hat{\sigma }_1 \rangle , \end{aligned}$$
(46)

By dropping the noise operator and writing the quantum Langevin equations for the operators \(\hat{a}_1\), \(\hat{a}_2\), and \(\hat{a}_3\) as2:

$$\begin{aligned} \frac{d\hat{a}_1}{dt}&= -\frac{\kappa }{2}\hat{a}_1 - g\hat{\sigma }_1, \end{aligned}$$
(47)
$$\begin{aligned} \frac{d\hat{a}_2}{dt}&= -\frac{\kappa }{2}\hat{a}_2 - g\hat{\sigma }_2, \end{aligned}$$
(48)

and

$$\begin{aligned} \frac{d\hat{a}_3}{dt}&= -\frac{\kappa }{2}\hat{a}_3 + \eta - g\hat{\sigma }_3. \end{aligned}$$
(49)

where \(\kappa\) is the cavity damping constant. The steady-state solutions should be these Eqs. (47)–(49) are:

$$\begin{aligned} \hat{a}_1&= -\frac{2g}{\kappa } \hat{\sigma }_1, \end{aligned}$$
(50)
$$\begin{aligned} \hat{a}_2&= -\frac{2g}{\kappa } \hat{\sigma }_2, \end{aligned}$$
(51)
$$\begin{aligned} \hat{a}_3&= \frac{2\eta }{\kappa } - \frac{2g}{\kappa } \hat{\sigma }_3. \end{aligned}$$
(52)

Now, substituting Eqs. (50)–(52) into Eqs. (37)–(46), we obtain:

$$\begin{aligned} \langle \hat{a}_3^\dagger \hat{a}_3 \rangle =\frac{4\varepsilon ^2}{\gamma _c\kappa }-\frac{4\varepsilon }{\kappa } \langle \hat{\sigma }_3^\dag \rangle + \frac{\gamma _c}{\kappa } \langle \hat{\eta }_1\rangle \end{aligned}$$
(53)
$$\begin{aligned} \langle \hat{a}_3\hat{a}_3^\dag \rangle =\frac{4\varepsilon ^2}{\gamma _c\kappa } - \frac{4\varepsilon }{\kappa } \langle \hat{\sigma }_3\rangle + \frac{\gamma _c}{\kappa } \langle \hat{\eta }_3\rangle \end{aligned}$$
(54)
$$\begin{aligned} \langle \hat{a}_3^2 \rangle =\frac{4\varepsilon ^2}{\gamma _c\kappa } - \frac{4\varepsilon }{\kappa } \langle \hat{\sigma }_3\rangle \end{aligned}$$
(55)
$$\begin{aligned} \langle \hat{a}_3 \rangle ^2 =\frac{4\varepsilon ^2}{\gamma _c\kappa }-\frac{4\varepsilon }{\kappa }\langle \hat{\sigma }_3\rangle + \frac{\gamma _c}{\kappa } \langle \hat{\sigma }_3\rangle ^2 \end{aligned}$$
(56)
$$\begin{aligned} \langle \hat{a}_3^\dagger \rangle \langle \hat{a}_3 \rangle =\frac{4\varepsilon ^2}{\gamma _c\kappa } - \frac{4\varepsilon }{\kappa }\langle \hat{\sigma }_3\rangle + \frac{\gamma _c}{\kappa } \langle \hat{\sigma }_3\rangle ^2 \end{aligned}$$
(57)
$$\begin{aligned} \langle \hat{\sigma }_1 \rangle&= \frac{2}{3} \frac{\varepsilon }{\gamma _c + \gamma } \langle \hat{\sigma }_2^\dag \rangle , \end{aligned}$$
(58)
$$\begin{aligned} \langle \hat{\sigma }_2 \rangle&= -\frac{2\varepsilon }{\gamma _c + \gamma } \langle \hat{\sigma }_1^\dag \rangle , \end{aligned}$$
(59)
$$\begin{aligned} \langle \hat{\sigma }_3 \rangle&=\frac{\varepsilon }{\gamma _c + \gamma } (\langle \hat{\eta }_3 \rangle - \langle \hat{\eta }_1 \rangle ), \end{aligned}$$
(60)
$$\begin{aligned} \langle \hat{\eta }_1\rangle = \frac{\varepsilon }{2(\gamma _c + \gamma )} (\langle \hat{\sigma }_c^\dag \rangle + \langle \hat{\sigma }_c\rangle ) \end{aligned}$$
(61)
$$\begin{aligned} \langle \hat{\eta }_1\rangle =\langle \hat{\eta }_2\rangle \end{aligned}$$
(62)

where

$$\begin{aligned} \gamma _c={4g^2\over \kappa } \end{aligned}$$
(63)

is the stimulated emission decay constant and \(\varepsilon\) is defined by

$$\begin{aligned} \varepsilon ={2\eta g\over \kappa } \end{aligned}$$
(64)

The Eq. (64) represents the effective coherent driving strength. While it depends on multiple system parameters (\(\eta\), g, \(\kappa\)), it serves as a dimensionless measure of the coherent drive intensity relative to the cavity decay rate and atom-cavity coupling strength. This parameterization is standard in quantum optics and allows us to study the system’s behavior under varying drive strengths while maintaining physical consistency. The parameter \(\varepsilon\) effectively characterizes the competition between coherent driving and dissipative processes in the system.

Using Eqs. (58) and (59), we easily find

$$\begin{aligned} \langle \hat{\sigma }_1\rangle =\langle \hat{\sigma }_2\rangle \end{aligned}$$
(65)

Furthermore, with the aid of the identity

$$\begin{aligned} \langle \hat{\eta }_1 \rangle + \langle \hat{\eta }_2 \rangle + \langle \hat{\eta }_3 \rangle&= 1, \end{aligned}$$
(66)

we interpret \(\langle \hat{\eta }_1 \rangle\), \(\langle \hat{\eta }_2 \rangle\), and \(\langle \hat{\eta }_3 \rangle\) as the probabilities for the three-level atom to be in the upper, intermediate, and bottom levels, respectively. Now, using Eq. (66) along with Eqs. (60), (61), and (62), we easily find

$$\begin{aligned} \langle \hat{\sigma }_3 \rangle&= \frac{\varepsilon (\gamma _c + \gamma )}{(\gamma _c + \gamma )^2 + 3\varepsilon ^2}, \end{aligned}$$
(67)
$$\begin{aligned} \langle \hat{\eta }_1 \rangle&= \langle \hat{\eta }_2 \rangle = \frac{\varepsilon ^2}{(\gamma _c + \gamma )^2 + 3\varepsilon ^2}, \end{aligned}$$
(68)
$$\begin{aligned} \langle \hat{\eta }_3 \rangle&= \frac{\varepsilon ^2 + (\gamma _c + \gamma )^2}{(\gamma _c + \gamma )^2 + 3\varepsilon ^2}. \end{aligned}$$
(69)

Moreover, using Eqs. (67), (68), and (69), we obtain

$$\begin{aligned} \langle \hat{a}_3^\dagger \hat{a}_3 \rangle =\frac{4\varepsilon ^2}{\gamma _c\kappa }-\frac{4\varepsilon }{\kappa } \left( \frac{\varepsilon (\gamma _c+\gamma )}{(\gamma _c+\gamma )^2+3\varepsilon ^2}\right) +\frac{\gamma _c}{\kappa } \left( \frac{\varepsilon ^2}{(\gamma _c+\gamma )^2+3\varepsilon ^2}\right) \end{aligned}$$
(70)
$$\begin{aligned} \langle \hat{a}_3\hat{a}_3^\dag \rangle =\frac{4\varepsilon ^2}{\gamma _c\kappa } - \frac{4\varepsilon }{\kappa } \left( \frac{\varepsilon (\gamma _c + \gamma )}{(\gamma _c + \gamma )^2 + 3\varepsilon ^2}\right) + \frac{\gamma _c}{\kappa }\left( \frac{\varepsilon ^2 + (\gamma _c + \gamma )^2}{(\gamma _c + \gamma )^2 + 3\varepsilon ^2}\right) \end{aligned}$$
(71)
$$\begin{aligned} \langle \hat{a}_3^2 \rangle =\frac{4\varepsilon ^2}{\gamma _c\kappa } - \frac{4\varepsilon }{\kappa } \left( \frac{\varepsilon (\gamma _c + \gamma )}{(\gamma _c + \gamma )^2 + 3\varepsilon ^2}\right) \end{aligned}$$
(72)
$$\begin{aligned} \langle \hat{a}_3 \rangle ^2 =\frac{4\varepsilon ^2}{\gamma _c\kappa }-\frac{4\varepsilon }{\kappa }\left( \frac{\varepsilon (\gamma _c + \gamma )}{(\gamma _c + \gamma )^2 + 3\varepsilon ^2}\right) + \frac{\gamma _c}{\kappa }\left( \frac{\varepsilon ^2 (\gamma _c + \gamma )^2}{(\gamma _c + \gamma )^2 + 3\varepsilon ^2}\right) \end{aligned}$$
(73)
$$\begin{aligned} \langle \hat{a}_3^\dagger \rangle \langle \hat{a}_3 \rangle =\frac{4\varepsilon ^2}{\gamma _c\kappa }-\frac{4\varepsilon }{\kappa }\left( \frac{\varepsilon (\gamma _c + \gamma )}{(\gamma _c + \gamma )^2 + 3\varepsilon ^2}\right) + \frac{\gamma _c}{\kappa } \left( \frac{\varepsilon ^2(\gamma _c + \gamma )^2}{(\gamma _c + \gamma )^2 + 3\varepsilon ^2}\right) \end{aligned}$$
(74)

Finally, we note the commutation relations:

$$\begin{aligned} [\hat{a}_3, \hat{a}_3^\dag ]&=1,\quad \langle [\hat{a}_3, \hat{a}_3^\dag ] \rangle =1, \end{aligned}$$
(75)
$$\begin{aligned} \langle \hat{a}_3 \hat{a}_3^\dag \rangle + \langle \hat{a}_3^\dag \hat{a}_3 \rangle&=B,\quad \langle \hat{B} \rangle = B. \end{aligned}$$
(76)

The Q function

We next obtain the Q function for the cavity mode light. This Q function is expressible as. 

$$\begin{aligned} Q(\alpha , \alpha ^*, t) = \frac{1}{\pi ^2} \int d^2 z \phi _a(z, z^*) \exp \left( z \alpha ^*- z^*\alpha \right) , \end{aligned}$$
(77)

where the anti-normally ordered characteristic function \(\phi _a(z, z^*)\) is defined in the Heisenberg picture by

$$\begin{aligned} \phi _a(z,z^*) =\text {Tr} \left[ \rho (0) e^{-z^* \hat{a}_3} e^{z \hat{a}_3^\dagger } \right] \end{aligned}$$
(78)

Employing the identity

$$\begin{aligned} e^{\hat{A}} e^{\hat{B}} = e^{\hat{A} + \hat{B} + \frac{1}{2}[\hat{A}, \hat{B}]} \end{aligned}$$
(79)

and taking into account the fact that \(a_3\) are Gaussian variables with zero mean, Eq. (78) can be put in the form

$$\begin{aligned} \phi _a(z, z^*)=\exp \left( -\frac{1}{2} (z^*z \langle [\hat{a}_3, \hat{a}_3^\dag ]\right) \exp \left( \frac{1}{2} \langle (z\hat{a}_3^\dagger - z^*\hat{a}_3)^2\rangle \right) \end{aligned}$$
(80)

It then follows that

$$\begin{aligned} \phi _a(z, z^*) = \exp (-\frac{1}{2}z^*z\langle [\hat{a}_3, \hat{a}_3^\dag ])\rangle \exp \left[ \frac{1}{2}( \langle (\hat{a}_3^\dagger )^2\rangle z^2 + \langle \hat{a}_3^2\rangle (z^*)^2 -z^*z (\langle \hat{a}_3^\dag \hat{a}_3\rangle +\langle \hat{a}_3 \hat{a}_3^\dag \rangle )) \right] \end{aligned}$$
(81)

Now, considering Eqs. (70), (71), (72), and (73), this equation becomes

$$\begin{aligned} \phi _a(z, z^*) = \exp \left[ -a z^*z +{1\over 2} (z^2 a_3^*+ a_3(z^*)^2) \right] \end{aligned}$$
(82)

where

$$\begin{aligned} a={1\over 2}(1+B) \end{aligned}$$
(83)

Finally, substituting Eq. (82) into Eq. (77) and carrying out the integration, the Q function for the cavity mode of light is found to be

$$\begin{aligned} Q(\alpha ^*,\alpha ) = {\lambda (u^2 - v^2)^{1\over 2}\over \pi } \exp \left[ -u \alpha ^*\alpha +{1\over 2} (v(\alpha ^*)^2 + v\alpha ^2 )\right] \end{aligned}$$
(84)

where

$$\begin{aligned} u = \frac{a}{a^2 - a_3^2}, \quad v = \frac{a_3}{a^2 - a_3^2} \end{aligned}$$
(85)

Photon statistics

In this section, we calculate the mean photon number, the variance of the photon number, and the photon number distribution for the cavity mode \(a_3\).

The mean and variance of the photon number for cavity mode \(a_3\)

We define the mean photon number of the cavity mode \(a_3\) by \(\bar{n} = \langle \hat{a}_3^\dagger \hat{a}_3\rangle\). Then, applying Eq. (53), we easily find

$$\begin{aligned} \bar{n} = \langle \hat{a}_3^\dagger \hat{a}_3\rangle = \frac{4\eta ^2}{\kappa ^2} - \frac{8\eta g}{\kappa ^2} \langle \hat{\sigma }_3\rangle + \frac{\gamma _c}{\kappa } \langle \hat{\eta }_1\rangle \end{aligned}$$
(86)

The first term represents the mean number of photons in the cavity mode \(\hat{a}_3\) due to the external coherent driving light. The second term represents the mean number of photons absorbed by the atom through the stimulated interaction between the cavity mode and the atom, and the third term represents the mean number of photons emitted by the atom into the cavity mode.

Using Eq. (70), we obtain:

$$\begin{aligned} \bar{n} = \frac{\varepsilon ^2 \left[ (\gamma _c + 2\gamma )^2 + 12\varepsilon ^2 \right] }{\kappa \gamma _c \left[ (\gamma _c + \gamma )^2 + 3\varepsilon ^2 \right] }. \end{aligned}$$
(87)

Considering Eq. (75), the variance of the photon number for the cavity mode \(a_3\) is defined by:

$$\begin{aligned} (\Delta n)^2=\langle (\hat{a}_3^{\dagger }\hat{a}_3)^2 \rangle -\bar{n}^2 , \end{aligned}$$
(88)

which can be written as

$$\begin{aligned} (\Delta n)^2=\langle (\hat{a}_3^{\dagger })^2 \hat{a}_3^2\rangle +\bar{n}-\bar{n}^2. \end{aligned}$$
(89)

Therefore, the variance of the photon number for coherent light is:

$$\begin{aligned} (\Delta n)^2 =\bar{n} \end{aligned}$$
(90)

As shown in Eqs. (87) and (90), the equality between the variance of the photon number and the mean photon number for cavity mode \(a_3\) signifies the presence of Poissonian photon statistics, characteristic of coherent light.

Photon number distribution

We finally calculate the photon number distribution for the cavity mode \(a_3\) using the Q-function. The photon number distribution for a cavity light mode is expressible in terms of the Q-function as :

$$\begin{aligned} P(n,t) = \frac{\pi }{n!} \left. \frac{\partial ^{2n}}{\partial \alpha ^{*n} \partial \alpha ^n} \left[ Q(\alpha ^*, \alpha , t) \, e^{\alpha ^* \alpha } \right] \right| _{\alpha ^* = \alpha = 0} \end{aligned}$$
(91)

Using Eqs. (84) and (91), the photon number distribution for the cavity mode \(a_3\) can be written as:

$$\begin{aligned} P(n,t) = \frac{(u^2 - v^2)^{1/2}}{n!} \left. \frac{\partial ^{2n}}{\partial \alpha ^{*n} \partial \alpha ^n} \exp \left[ (1 - u) \alpha ^* \alpha + \frac{v ((\alpha ^{*})^2 + \alpha ^2)}{2}\right] \right| _{\alpha ^* = \alpha = 0} \end{aligned}$$
(92)

Expanding the exponential functions in power series, we have:

$$\begin{aligned} P(n,t) = \frac{(u^2 - v^2)^{1/2}}{n!} \times \frac{\partial ^{2n}}{\partial \alpha ^{*n} \partial \alpha ^n} \sum _{k,l,m} \frac{(1)^{l+m} (1 - u)^k v^{l+m}}{2^{l+m} k! l! m!} \left. \left[ \alpha ^{*k+2l} \alpha ^{k+2l} \right] \right| _{\alpha ^* = \alpha = 0} \end{aligned}$$
(93)

Carrying out the differentiations and applying the condition \(\alpha ^*= 0\), we obtain:

$$\begin{aligned} P(n,t) = \frac{(u^2 - v^2)^{1/2}}{n!} \times (k+2m)! \sum _{k,l,m} \frac{(1)^{l+m} (1 - u)^k v^{l+m} (k + 2l)!}{2^{l+m} k! l! m! (k + 2l - n)! (k + 2m - n)!} \delta _{k+2l,n} \delta _{k+2m,n} \end{aligned}$$
(94)

Applying the properties of the Kronecker delta symbol and the fact that a factorial is defined for non-negative integers, we obtain:

$$\begin{aligned} P(n,t) = (u^2 - v^2)^{1/2} \sum _{l=0}^{\lfloor n \rfloor } n! \frac{(1 - u)^{n-2l} v^{2l}}{2^{2l} l!^2 (n - 2l)!} \end{aligned}$$
(95)

The quadrature variance of cavity mode \(a_3\)

Here we determine the quadrature squeezing for the cavity mode. We recall that the variance of the plus and minus quadrature operators for a cavity mode \(\hat{a}_3\) is given by:

$$\begin{aligned} (\Delta a_{3\pm })^2 = \langle \hat{a}_{3\pm }, \hat{a}_{3\pm } \rangle . \end{aligned}$$
(96)

where

$$\begin{aligned} \hat{a}_{3+} = \hat{a}_3^\dagger + \hat{a}_3, \end{aligned}$$
(97)
$$\begin{aligned} \hat{a}_{3-} = i(\hat{a}_3^\dagger - \hat{a}_3) \end{aligned}$$
(98)

Hence, considering this, Eq. (96) can be expressed as:

$$\begin{aligned} (\Delta a_{3\pm })^2 = \langle \hat{a}_3^\dagger \hat{a}_3 \rangle + \langle \hat{a}_3 \hat{a}_3^\dagger \rangle \pm \left( \langle (\hat{a}_3^\dagger )^2 \rangle + \langle \hat{a}_3^2 \rangle - \langle \hat{a}_3^\dagger \rangle ^2 - \langle \hat{a}_3 \rangle ^2 \right) - 2 \langle \hat{a}_3^\dagger \rangle \langle \hat{a}_3 \rangle \end{aligned}$$
(99)

Therefore, considering Eqs. (70), (71), (72), (73), and (74), the quadrature variance takes the form:

$$\begin{aligned} (\Delta a_{3+})^2 = \frac{\gamma _c}{\kappa } \left( \frac{(\gamma _c + \gamma )^4 + \varepsilon ^2 (\gamma _c +\gamma )^2 + 6\varepsilon ^4}{[(\gamma _c + \gamma )^2 + 3\varepsilon ^2]^2}\right) \end{aligned}$$
(100)

and

$$\begin{aligned} (\Delta a_{3-})^2 =\frac{\gamma _c}{\kappa } \left( \frac{2 \varepsilon ^2 + (\gamma _c + \gamma )^2}{(\gamma _c + \gamma )^2 + 3 \varepsilon ^2}\right) \end{aligned}$$
(101)

,We note that the cavity mode \(a_3\) light can exhibit quadrature squeezing. However, it is fundamentally impossible for both quadratures to be simultaneously squeezed below the vacuum level without violating the Heisenberg uncertainty principle \(\Delta a_{3+} \cdot \Delta a_{3-} \ge 1\). Upon careful examination of Eqs. (100) and (101), we find that for specific parameter ranges, one quadrature shows noise reduction below the vacuum level while the other exhibits increased fluctuations, thus maintaining the uncertainty relation.

Setting \(\varepsilon = 0\) in Eqs. (100) and (101), we find:

$$\begin{aligned} (\Delta a_{3\pm })^2 ={\gamma _c\over \kappa } \end{aligned}$$
(102)

Thus, for \(\varepsilon = 0\), the quadrature variance of the cavity vacuum state has equal uncertainties in the two quadratures and satisfies the minimum uncertainty relation.

We calculate the quadrature squeezing of the cavity mode \(a_3\) light relative to the quadrature variance of the cavity vacuum state. We therefore define the quadrature squeezing of the cavity mode \(a_3\) light by:

$$\begin{aligned} S_+={{\gamma _c\over \kappa }-(\Delta a_{3+})^2\over {\gamma _c\over \kappa }} \end{aligned}$$
(103)
$$\begin{aligned} S_-={{\gamma _c\over \kappa }-(\Delta a_{3-})^2\over {\gamma _c\over \kappa }} \end{aligned}$$
(104)

Hence, considering Eqs. (100) and (101), we can write Eqs. (103) and (104) as:

$$\begin{aligned} S_+ = \frac{5\varepsilon ^2 (\gamma _c + \gamma )^2 + 3\varepsilon ^4}{\left[ (\gamma _c + \gamma )^2 + 3\varepsilon ^2\right] ^2} \end{aligned}$$
(105)
$$\begin{aligned} S_- = \frac{\varepsilon ^2}{(\gamma _c + \gamma )^2 + 3 \varepsilon ^2} \end{aligned}$$
(106)

Results and discussion

We study the quantum dynamics and squeezing properties of a three-level atom in a cascade configuration inside an open cavity driven by coherent light and coupled to a vacuum reservoir via a single-port mirror. The system dynamics is governed by the total Hamiltonian given in Eq. (5), which accounts for the interaction between the coherent driving light and the cavity mode \(a_3\) and the coupling of the three-level atom with the cavity modes \(a_1\), \(a_2\), and \(a_3\). This study examines the quantum dynamics and squeezing properties by analyzing how the mean photon number, photon number distribution, quadrature variances, and quadrature squeezing change with the parameter \(\varepsilon\), with the cavity damping constants fixed at \(\kappa =0.8\), the stimulated decay constant at \(\gamma _c=0.8\), and the spontaneous emission varying as \(\gamma =0.0\) (dashed curve), \(\gamma =0.35\) (dotted curve), \(\gamma =0.45\) (solid curve), and \(\gamma =0.5\) (dash-dotted curve),\(\varepsilon =0.5\) (dashed curve), \(\varepsilon =0.6\) (dotted curve), \(\varepsilon =0.7\) (solid curve), and \(\varepsilon =0.8\) (dash-dotted curve),g=0.3.

Fig. 2
figure 2

Mean photon number for cavity mode \(\hat{a}_3\) versus \(\varepsilon\), calculated using Eq. (87). The parameters are fixed at \(\kappa = 0.8\) and \(\gamma _c = 0.8\). The curves represent different values of spontaneous emission \(\gamma\): \(\gamma = 0.0\) (dashed line), \(\gamma = 0.35\) (dotted line), \(\gamma = 0.45\) (solid line), and \(\gamma = 0.5\) (dash-dotted line).

Full size image

The mean photon number for cavity mode \(a_3\) is significantly affected by spontaneous emission, as demonstrated in Fig. 2 and Eq. (87). When spontaneous emission is absent (\(\gamma =0.0\), dashed curve), the mean photon number increases, indicating that photons are better retained in the cavity; however, as \(\gamma\) increases from 0.35 to 0.5, spontaneous emission causes a marked decrease. The \(\gamma =0.45\) curve (solid) shows intermediate behavior, while the \(\gamma =0.5\) curve (dash-dotted) exhibits the lowest photon number due to enhanced decoherence. In accordance with Eq. (90), the variance of the photon number for cavity mode \(a_3\) equals its mean, a relationship derived from Eq. (89). This equality indicates Poissonian photon statistics, confirming that the cavity mode \(a_3\) behaves as coherent light with fluctuations consistent with a coherent state. Consequently, spontaneous emission indirectly affects the variance by lowering the mean photon number, which in turn reduces the variance proportionally.

Fig. 3
figure 3

Photon number distribution P(n, t) at steady state versus photon number n, calculated using Eq. (95). Fixed parameters are \(\gamma _c = 0.8\), \(\kappa = 0.8\), \(\gamma = 0.6\), and \(g = 0.3\). The curves correspond to different coherent driving strengths \(\varepsilon\): \(\varepsilon = 0.5\) (solid line), \(\varepsilon = 0.6\) (dashed line), \(\varepsilon = 0.7\) (dotted line), and \(\varepsilon = 0.8\) (dash-dotted line).

Full size image

. The cavity mode \(a_3\) photon number distribution, obtained from the Q-function in Eq. (95), is displayed in Fig. 3 for fixed parameters \(\gamma =0.6\), \(\kappa =0.8\), \(\gamma _c=0.8\), \(g=0.3\), and varying coherent driving strengths \(\varepsilon =0.5\) (solid curve), \(\varepsilon =0.6\) (dashed curve), \(\varepsilon =0.7\) (dotted curve), and \(\varepsilon =0.8\) (dash-dotted curve). The distribution exhibits a clear oscillatory behavior with higher probability for even photon numbers than odd ones, characteristic of squeezed states generated by the correlated photon-pair emission in the three-level cascade system.As \(\varepsilon\) increases from 0.5 to 0.8, the distribution shifts toward higher photon numbers while maintaining the even-odd oscillation pattern. The enhanced even-photon preference with increasing \(\varepsilon\) demonstrates that stronger coherent driving promotes correlated pair generation, though the finite probability of odd photon numbers persists due to photon leakage through the single-port mirror and the effects of spontaneous emission at \(\gamma =0.6\).

Fig. 4
figure 4

Quadrature variance \((\Delta \hat{a}_{3+})^2\) versus \(\varepsilon\), calculated using Eq. (100). The fixed parameters are \(\gamma _c = 0.8\) and \(\kappa = 0.8\). The curves depict different spontaneous emission \(\gamma\): \(\gamma = 0.0\) (solid line), \(\gamma = 0.35\) (dotted line), \(\gamma = 0.45\) (dash-dotted line), and \(\gamma = 0.5\) (large dashed line).

Full size image
Fig. 5
figure 5

Quadrature variance \((\Delta \hat{a}_{3-})^2\) versus \(\varepsilon\), calculated using Eq. (101). The fixed parameters are \(\gamma _c = 0.8\) and \(\kappa = 0.8\). The curves illustrate different spontaneous emission \(\gamma\): \(\gamma = 0.0\) (solid line), \(\gamma = 0.35\) (dotted line), \(\gamma = 0.45\) (dash-dotted line), and \(\gamma = 0.5\) (large dashed line).

Full size image

The cavity mode \(a_3\) light is in a squeezed state, as evidenced by squeezing in both the plus and minus quadratures. Specifically, Figs. 4 and 5 illustrate the quadrature variances of the cavity vacuum state, which satisfy the minimum uncertainty relation by displaying equal uncertainties in the two quadratures for \(\varepsilon =0\).This suggests that a coherent vacuum state with balanced quantum fluctuations is present in the cavity field. However, the variances \((\Delta a_{3+})^2\) Eq. (100) and \((\Delta a_{3-})^2\) Eq. (101) deviate as \(\varepsilon\) increases, indicating the onset of squeezing. The curves in Figs. 4 and 5 demonstrate how the presence of spontaneous emission further modulates these variances compared to the case without spontaneous emission.

Fig. 6
figure 6

Plus quadrature squeezing \(S_+\) versus \(\varepsilon\), calculated using Eq. (105). The fixed parameters are \(\gamma _c = 0.8\) and \(\kappa = 0.8\). The curves represent different spontaneous emission \(\gamma\): \(\gamma = 0.0\) (solid line), \(\gamma = 0.35\) (dotted line), \(\gamma = 0.45\) (dash-dotted line), and \(\gamma = 0.5\) (large dashed line).

Full size image
Fig. 7
figure 7

Minus quadrature squeezing \(S_-\) versus \(\varepsilon\), calculated using Eq. (106). The fixed parameters are \(\gamma _c = 0.8\) and \(\kappa = 0.8\). The curves illustrate different spontaneous emission \(\gamma\): \(\gamma = 0.0\) (solid line), \(\gamma = 0.35\) (dotted line), \(\gamma = 0.45\) (dash-dotted line), and \(\gamma = 0.5\) (large dashed line).

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The quadrature variances for the plus and minus quadratures of cavity mode \(a_3\) are plotted in Figs. 6 and 7, respectively, for \(\gamma _c=0.8\), \(\kappa =0.8\), and \(\gamma =0.0\) (dashed curve), \(\gamma =0.35\) (dotted curve), \(\gamma =0.45\) (solid curve), and \(\gamma =0.5\) (dash-dotted curve), according to Eqs. (100) and (101). Squeezing is represented in both quadratures by variances for non-zero \(\varepsilon\) being below the vacuum state level. Equation (105) determines that the plus quadrature squeezing achieves a maximum of \(52.1\%\) at \(\varepsilon =0.59\) and \(\varepsilon =0.85\) for \(\gamma =0.0,0.35\) (dashed curve) and shifts to \(\varepsilon =0.96\) for \(\gamma =0.45,0.5\) (dash-dotted curve). The \(\gamma =0.45\) curve (solid) shows intermediate behavior, requiring higher \(\varepsilon\) to achieve maximum squeezing compared to the \(\gamma =0.0\) case.

Equation (106) provides the minus quadrature squeezing, which is depicted in Fig. 7 and reaches a maximum of \(33.3\%\) for \(\varepsilon \ge 16\) across all \(\gamma\) values. This robustness shows that the minus quadrature is less susceptible to spontaneous emission-induced decoherence, with all curves converging at high driving strengths. The vacuum state satisfies the minimum uncertainty relation when \(\varepsilon =0\), where both quadrature variances equal \(\gamma _c/\kappa\) Eq. (102).

The quantum properties of the cavity mode \(a_3\) are determined by the interaction of spontaneous emission \(\gamma\) and coherent driving \(\varepsilon\). While spontaneous emission reduces the mean photon number, it does not prevent the achievement of significant quadrature squeezing. The plus quadrature exhibits more complex behavior, with maximum squeezing shifting to higher \(\varepsilon\) values as \(\gamma\) increases from 0.0 to 0.5. In contrast, the minus quadrature demonstrates remarkable robustness, achieving the same maximum squeezing level regardless of spontaneous emission strength at high driving fields.

The photon number distribution reveals the non-classical nature of the generated light through its oscillatory behavior, with preference for even photon numbers indicating strong photon-pair correlations. This characteristic, combined with the observed quadrature squeezing, confirms the generation of a squeezed state rather than coherent light, despite the initial coherent driving field.

Reviewer 1, This is a purely theoretical study and does not present experimental data to directly compare with the calculated curves. However, the paper includes a section on Experimental Feasibility that discusses how the proposed system could be implemented using current experimental platforms, such as circuit quantum electrodynamics (CQED) systems or laser-cooled atoms in high-finesse optical cavities18,19.

Conclusion

In this paper, we have systematically investigated the quantum dynamics and squeezing properties of a coherently driven three-level atom in a cascade configuration within an open cavity coupled to a vacuum reservoir. Through detailed analysis of the quantum Langevin equations and steady-state solutions, we have demonstrated several significant quantum optical phenomena. Our study reveals that the cavity mode \(a_3\) exhibits substantial quadrature squeezing in both plus and minus components. Remarkably, the plus quadrature achieves \(52.1\%\) squeezing below the vacuum level at optimal coherent driving strengths \(\varepsilon =0.59\) and \(\varepsilon =0.85\) for \(\gamma =0.0,0.35\), and \(\varepsilon =0.96\) for \(\gamma =0.45,0.5\). This demonstrates that increased coherent driving can effectively compensate for the decoherence effects introduced by spontaneous emission. Furthermore, the minus quadrature exhibits robust \(33.3\%\) squeezing for \(\varepsilon \ge 16\), which remains independent of spontaneous emission rates, highlighting the system’s resilience against environmental decoherence. In line with coherent light behavior, the photon statistics analysis shows Poissonian features with variance equal to the mean photon number. Strong photon-pair correlations typical of squeezed states are indicated by the photon number distribution obtained from the Q-function, which exhibits a characteristic oscillatory pattern with higher probabilities for even photon numbers than odd ones. As the driving strength \(\varepsilon\) increases, this even-odd oscillation becomes more noticeable, highlighting the function of correlated pair creation in the three-level cascade system. The quadrature variance analysis confirms that the cavity mode operates in a squeezed state, with one quadrature exhibiting noise reduction below the vacuum level while the other shows increased fluctuations, maintaining the Heisenberg uncertainty principle. The system transitions from a coherent vacuum state at \(\varepsilon =0\) to a significantly squeezed state as the driving strength increases. These findings have important implications for quantum technologies. The generation of substantial squeezing directly from atomic systems provides an alternative approach to traditional nonlinear optical methods, potentially simplifying experimental setups for quantum communication, quantum sensing, and continuous-variable quantum computing. The robustness of minus quadrature squeezing against spontaneous emission is particularly promising for practical applications where environmental decoherence is unavoidable. Future work could explore the implementation of this system in various experimental platforms, such as circuit quantum electrodynamics or atomic ensembles in high-finesse cavities, to verify these theoretical predictions and harness the observed quantum effects for advancing quantum information processing capabilities.