1 Introduction

Non-commutable optical field operators make it difficult for physicists to enhance the squeezing effect. Motivated by the research work such as electron’s orbit quantization \(\oint {pdq} = n\hbar\) from Sommerfeld and Bohr and the pseudo-probability distribution function for quantum states from Wigner [5,6,7,8,9,10], this paper attempts to leverage the phase-space theory of \(q - p\), which is widely used to connect classical mechanics with quantum mechanics [1,2,3,4]. First, the method of integration within an ordered product of operator (IWOP) in quantum mechanics is employed to investigate phase space correspondence for enhanced squeezing. This approach allows for showing the theoretical representation of two independent parameters and their boundary conditions to achieve enhanced squeezing. Then, the squeezing-enhanced state is theoretically obtained by using the generalized squeezing operator \(S\left( {\lambda ,r} \right) = \exp \left[ {\frac{ - i\lambda }{2}\left( {{Q^2}{e^r} - {P^2}{e^{ - r}}} \right)} \right]\). The characteristic of phase space transformation is demonstrated as \(p \to p\cosh \lambda - q\sinh \lambda {e^r},\) \(q \to q\cosh \lambda - p{e^{ - r}}\sinh \lambda ,\) where \(\lambda ,r\) are two independent parameters, and the condition for enhanced squeezing of the quadrature operator is analyzed. Further, the integral solution of the dissipation master equation [11, 12] is utilized to analyze the attenuation of this type of squeezed field. Meanwhile, the method of IWOP [13,14,15,16,17] is employed to accomplish the task. Finally, the canonical decomposition of the enhanced squeezing operator is derived.

2 Quadrature Operator of Squeezed Light

The squeezed light [18,19,20] has non-classical properties, such as the anti-bunching effect [21,22,23], quadrature squeezing effect [24,25,26], and sub-Poissonian distribution [27,28,29], and it has been applied to optical communication [30,31,32], high-precision interferometry [33, 34] and weak signal detection[35,36,37]. The squeezed state of single-mode light [38, 39] is produced by the process of degenerate parameters. The two-photon Hamiltonian in the interaction representation can be represented as:

$$H = \frac{i\hbar }{2}\left( {{\xi^* }{a^2} - \xi {a^{\dag 2}}} \right)$$
(1)

where \(\left[ {a,{a^\dag }} \right] = 1\), \(\xi\) is the coupling constant between the classical pump light field and the incident light quantum signal in the nonlinear media. Here, \(\hbar = 1\) is set for simplicity. Let \(\xi = i\lambda\), and then the corresponding squeezing operator is

$$S\left( \xi \right) = \exp \left[ { - \frac{i\lambda }{2}\left( {{a^2} + {a^{\dag 2}}} \right)} \right]$$
(2)

The induced transformation by \(S\left( \xi \right)\) is

$${S^\dag }\left( \xi \right)aS\left( \xi \right) = a\cosh \lambda - i{a^\dag }\sinh \lambda$$
(3)
$${S^\dag }\left( \xi \right){a^\dag }S\left( \xi \right) = {a^\dag }\cosh \lambda + ia\sinh \lambda$$
(4)

Let

$${x_1} = \frac{1}{2}\left( {a + {a^\dag }} \right),{x_2} = \frac{1}{2i}\left( {a - {a^\dag }} \right)$$
(5)

Then, \({x_1} + i{x_2} = a\) can be obtained. The amplitude of the rotating light field (the angle is \(\frac{\pi }{4}\)) is introduced as:

$${Y_1} + i{Y_2} = \left( {{x_1} + i{x_2}} \right){e^{ - \frac{i\pi }{4}}} = a{e^{ - \frac{i\pi }{4}}}$$
(6)
$${Y_1} - i{Y_2} = \left( {{x_1} - i{x_2}} \right){e^{\frac{i\pi }{4}}} = {a^\dag }{e^{\frac{i\pi }{4}}}$$
(7)

It can be seen that

$${Y_1} = \frac{{a{e^{ - \frac{i\pi }{4}}} + {a^\dag }{e^{\frac{i\pi }{4}}}}}{2},{Y_2} = \frac{{a{e^{ - \frac{i\pi }{4}}} - {a^\dag }{e^{\frac{i\pi }{4}}}}}{2i}$$
(8)

\({Y_1}\) and \({Y_2}\) constitute orthogonal components

$$\left[ {{Y_1},{Y_2}} \right] = 2i$$
(9)

The squeezing operator \(S\left( \xi \right)\) transforms \(\left( {{Y_1} + i{Y_2}} \right)\) as

$$\begin{gathered} {S^\dag }\left( \xi \right)\left( {{Y_1} + i{Y_2}} \right)S\left( \xi \right) = a\cosh \lambda {e^{ - \frac{i\pi }{4}}} - {a^\dag }\sinh \lambda {e^{\frac{i\pi }{4}}} \\ = a\frac{{{e^\lambda } + {e^{ - \lambda }}}}{2}{e^{ - \frac{i\pi }{4}}} - {a^\dag }\frac{{{e^\lambda } - {e^{ - \lambda }}}}{2}{e^{\frac{i\pi }{4}}} \\ = \frac{{a{e^{ - \frac{i\pi }{4}}} + {a^\dag }{e^{\frac{i\pi }{4}}}}}{2}{e^{ - \lambda }} + i\frac{{a{e^{ - \frac{i\pi }{4}}} - {a^\dag }{e^{\frac{i\pi }{4}}}}}{2i}{e^\lambda } \\ = {Y_1}{e^{ - \lambda }} + i{Y_2}{e^\lambda } \\ \end{gathered}$$
(10)

and it changes the two orthogonal components into

$${Y_1} \to {Y_1}{e^{ - \lambda }},\;{Y_2} \to {Y_2}{e^\lambda }$$
(11)

Next, this paper finds the representation of the squeezing operator \(S\left( \xi \right)\) in the \(q - p\) phase space.

3 Representation of \(S\left( \xi \right)\) in \(q - p\) Phase Space

The Hermitian coordinate \(Q\) and momentum operator \(P\) are introduced as

$$Q = \frac{1}{\sqrt 2 }\left( {a + {a^\dag }} \right),P = \frac{1}{\sqrt 2 i}\left( {a - {a^\dag }} \right)$$
(12)

and \(\left[ {Q,P} \right] = i,\) thus

$${a^{\dag 2}} + {a^2} = {Q^2} - {P^2}$$
(13)

Therefore, \(S\left( {\xi = i\lambda } \right)\) can be represented as

$$S\left( {\xi = i\lambda } \right) = \exp \left[ { - \frac{i\lambda }{2}\left( {{Q^2} - {P^2}} \right)} \right]$$
(14)

and then we have

$${S^\dag }\left( \xi \right)QS\left( \xi \right) = \frac{1}{\sqrt 2 }\left[ {{a^\dag }\cosh \lambda + ia\sinh \lambda + a\cosh \lambda - i{a^\dag }\sinh \lambda } \right] = Q\cosh \lambda - P\sinh \lambda$$
(15)

and

$${S^\dag }\left( \xi \right)PS\left( \xi \right) = \frac{1}{\sqrt 2 i}\left[ {a\cosh \lambda - i{a^\dag }\sinh \lambda - {a^\dag }\cosh \lambda - ia\sinh \lambda } \right] = P\cosh \lambda - Q\sinh \lambda$$
(16)

The coherent state [40,41,42,43,44] \(\left| z \right\rangle\) is introduced as

$$\left| z \right\rangle = {e^{\alpha {a^\dag } - {\alpha^* }a}}\left| 0 \right\rangle = {e^{ - |z{|^2}/2 + z{a^\dag }}}\left| 0 \right\rangle$$
(17)

Let \(z = \frac{q + ip}{{\sqrt 2 }}\) and rewrite \(\left| z \right\rangle\) in canonical form \(\left\vert {p,q} \right\rangle\)

$$\left| z \right\rangle \equiv \left| {p,q} \right\rangle = {e^{i\left( {pQ - qP} \right)}}\left| 0 \right\rangle = \exp \left( { - \frac{{{p^2} + {q^2}}}{4} + \frac{q + ip}{{\sqrt 2 }}{a^\dag }} \right)\left| 0 \right\rangle$$
(18)

Then, the corresponding completeness relation is expressed as

$$\int {\int_{ - \infty }^\infty {\frac{dpdq}{{2\pi }}} } \left| {p,q} \right\rangle \left\langle {p,q} \right| = 1$$
(19)

By Eq. (15) and Eq. (16), its classical correspondence is

$$p \to {p^{\prime}} = p\cosh \lambda - q\sinh \lambda ,\;q \to {q^{\prime}} = q\cosh \lambda - p\sinh \lambda$$
(20)

Thus, the representation of \(S\left( {\xi = i\lambda } \right)\) in the \(q - p\) phase space is

$$\begin{gathered} S\left( {\xi = i\lambda } \right) = \sqrt {\cosh \lambda } \iint \frac{dpdq}{{2\pi }}\left| {p^{\prime},q^{\prime}} \right\rangle \left\langle {p,q} \right| \\ = \sqrt {\cosh \lambda } \iint \frac{dpdq}{{2\pi }}\left| {p\cosh \lambda - q\sinh \lambda ,q\cosh \lambda - p\sinh \lambda } \right\rangle \left\langle {p,q} \right| \\ \end{gathered}$$
(21)

where \(\sqrt {\cosh \lambda }\) is introduced for the unitarity of \(S\left( {\xi = i\lambda } \right)\). Based on Eq. (18) and the normally ordered form [45,46,47,48] of the vacuum state \(\left| 0 \right\rangle \left\langle 0 \right|\) (\(:\begin{array}{*{20}{c}} : \end{array}\) denotes the normal ordering)

$$\left| 0 \right\rangle \left\langle 0 \right| = :{e^{ - {a^\dag }a}}:$$
(22)

and using the IWOP method, we have

$$\begin{gathered} S\left( {\xi = i\lambda } \right) = \sqrt {\cosh \lambda } \iint \frac{dpdq}{{2\pi }}\exp \left\{ { - \frac{1}{4}{{\left( {p\cosh \lambda - q\sinh \lambda } \right)}^2} - \frac{1}{4}{{\left( {q\cosh \lambda - p\sinh \lambda } \right)}^2}} \right. \\ \left. { + \frac{{a^\dag }}{\sqrt 2 }[q\cosh \lambda - p\sinh \lambda + i\left( {p\cosh \lambda - q\sinh \lambda } \right)]} \right\}\left| 0 \right\rangle \left\langle 0 \right|\exp \left[ { - \frac{{{p^2} + {q^2}}}{4} + \frac{q - ip}{{\sqrt 2 }}a} \right] \\ = \sqrt {\cosh \lambda } \iint \frac{dpdq}{{2\pi }}:\exp \left\{ { - \frac{{{p^2} + {q^2}}}{4}\left( {1 + \mathop {\cosh }\nolimits^2 \lambda + \mathop {\sinh }\nolimits^2 \lambda } \right)} \right. \\ \left. { + \frac{p}{\sqrt 2 }(i{a^\dag }\cosh \lambda - {a^\dag }\sinh \lambda - ia + \frac{q}{\sqrt 2 }\sinh 2\lambda ) + \frac{q}{\sqrt 2 }({a^\dag }\cosh \lambda - i{a^\dag }\sinh \lambda + a) - {a^\dag }a} \right\}: \\ = \sqrt {\sec h\lambda } :\exp \left[ { - \frac{i}{2}{a^{\dag 2}}\tanh \lambda } \right]\exp \left[ {\left( {\sec h\lambda - 1} \right){a^\dag }a} \right]\exp \left[ { - \frac{i}{2}{a^2}\tanh \lambda } \right]: \\ \end{gathered}$$
(23)

Then, using

$${e^{f{a^\dag }a}} = :\exp \left[ {\left( {{e^f} - 1} \right){a^\dag }a} \right]:$$
(24)

Equation (23) can be represented as

$$S\left( {\xi = i\lambda } \right) = \exp \left[ { - \frac{i}{2}{a^{\dag 2}}\tanh \lambda } \right]\exp \left[ {\left( {{a^\dag }a + \frac{1}{2}} \right)\ln \sec h\lambda } \right]\exp \left[ { - \frac{i}{2}{a^2}\tanh \lambda } \right]$$
(25)

and the squeezed vacuum state[49,50,51,52,53] is

$$S\left( \lambda \right)\left| 0 \right\rangle = \sec {h^\frac{1}{2}}\lambda {e^{ - \frac{i}{2}{a^{\dag 2}}\tanh \lambda }}\left| 0 \right\rangle$$
(26)

Therefore, the fluctuation of the rotation amplitude of the light field in the squeezed vacuum state is

$${\left( {\Delta {Y_1}} \right)^2} = \left\langle {Y_1^2} \right\rangle - {\left\langle {Y_1} \right\rangle^2} = \frac{1}{4}{e^{ - 2\lambda }}$$
(27)
$${\left( {\Delta {Y_2}} \right)^2} = \left\langle {Y_2^2} \right\rangle - {\left\langle {Y_2} \right\rangle^2} = \frac{1}{4}{e^{2\lambda }}$$
(28)

which demonstrates standard squeezing and can be experimentally implemented in degenerate parametric amplifiers.

4 Enhanced Squeezing by Finding a Generalized Squeezing Operator

This section explores the new state that can enhance squeezing (see Eq. (47) below). For this purpose, the degenerate parametric amplifier should be improved to relate two different parameters. The squeezing-enhanced operator [54,55,56] is introduced as

$$S\left( {{\lambda_1},{\lambda_2}} \right) = \exp \left[ { - \frac{i}{2}\left( {{\lambda_1}{Q^2} - {\lambda_2}{P^2}} \right)} \right]$$
(29)

which involves two asymmetric parameters. Let \({\lambda_1} = \lambda {e^r},{\lambda_2} = \lambda {e^{ - r}}\), and we have

$$S\left( {\lambda ,r} \right) = \exp \left[ {\frac{ - i\lambda }{2}\left( {{Q^2}{e^r} - {P^2}{e^{ - r}}} \right)} \right]$$
(30)

Then, the corresponding squeezed state is derived. Using the transformation properties

$$S\left( {\lambda ,r} \right)Q{S^{ - 1}}\left( {{\lambda_1},{\lambda_2}} \right) = Q\cosh \lambda + P{e^{ - r}}\sinh \lambda$$
(31)
$$S\left( {{\lambda_1},{\lambda_2}} \right)P{S^{ - 1}}\left( {{\lambda_1},{\lambda_2}} \right) = P\cosh \lambda + Q{e^r}\sinh \lambda$$
(32)

its classical correspondence is \(p \to p\cosh \lambda - q\sinh \lambda {e^r},\) \(q \to q\cosh \lambda - p{e^{ - r}}\sinh \lambda\). Similar to Eq. (21), the squeezing-enhanced operator is proposed as

$$S\left( {\lambda ,r} \right) \sim \int {\int_{ - \infty }^\infty {dpdq} } \left| {p\cosh \lambda - q\sinh \lambda {e^r},q\cosh \lambda - p{e^{ - r}}\sinh \lambda } \right\rangle \left\langle {p,q} \right|$$
(33)

Considering that it is complex to perform this double integration in \(S\left( {\lambda ,r} \right)\), the canonical coherent state \(\left| {p,q} \right\rangle\) is expressed as

$$\left|p,q\rangle \equiv \left|\left(\begin{array}{c}q\\ p\end{array}\right)\rangle =\left|z\rangle \equiv \right.\right.\right.\left|\left(\begin{array}{c}z\\ {z}^{*}\end{array}\right)\rangle , z=\frac{q+ip}{\sqrt{2}}\right.$$
(34)

Then, the ket in Eq. (33) is expressed as

$$\begin{gathered} \left| {p\cosh \lambda - q\sinh \lambda {e^r},q\cosh \lambda - p{e^{ - r}}\sinh \lambda } \right\rangle = \left| {\left( {\begin{array}{*{20}{c}} {\cosh \lambda }&{ - {e^{ - r}}\sinh \lambda } \\ { - \sinh \lambda {e^r}}&{\cosh \lambda } \end{array}} \right)\left( {\begin{array}{*{20}{c}} q \\ p \end{array}} \right)} \right\rangle \\ \equiv \left| {\left( {\begin{array}{*{20}{c}} u&{ - v} \\ { - {v^* }}&{u^* } \end{array}} \right)\left( {\begin{array}{*{20}{c}} z \\ {z^* } \end{array}} \right)} \right\rangle \\ \end{gathered}$$
(35)

with

$$u = \cosh \lambda - i\sinh \lambda \sinh r,v = i\sinh \lambda \cosh r,|u{|^2} - |{v^2}| = 1$$
(36)

and the integration in Eq. (33) becomes

$$S\left( {\lambda ,r} \right) \to \sqrt u \int \frac{{{d^2}z}}{\pi }\left| {\left( {\begin{array}{*{20}{c}} u&{ - v} \\ { - {v^* }}&{u^* } \end{array}} \right)\left( {\begin{array}{*{20}{c}} z \\ {z^* } \end{array}} \right)} \right\rangle \left\langle {\left( {\begin{array}{*{20}{c}} z \\ {z^* } \end{array}} \right)} \right| = \sqrt u \int \frac{{{d^2}z}}{\pi }\left| {uz - v{z^* }} \right\rangle \left\langle z \right|$$
(37)

By performing this integration using the IWOP method and based on Eq. (22) and \(\left| z \right\rangle = \exp \left[ { - \frac{{|z{|^2}}}{2} + z{a^\dag }} \right]\left| 0 \right\rangle\), we have

$$\begin{gathered} S\left( {\lambda ,r} \right) = \sqrt u \int \frac{{{d^2}z}}{\pi }\left| {uz - v{z^* }} \right\rangle \left\langle z \right| \\ = \sqrt u \int \frac{{{d^2}z}}{\pi }:\exp \left[ { - |u{|^2}|z{|^2} + uz{a^\dag } + {z^* }\left( {a - v{a^\dag }} \right) + \frac{{{v^* }u}}{2}{z^2} + \frac{{v{u^* }}}{2}{z^2} - {a^\dag }a} \right]: \\ = \frac{1}{{\sqrt {u^* } }}\exp \left( { - \frac{v}{{2{u^* }}}{a^{\dag 2}}} \right):\exp \left[ {\left( {\frac{1}{{u^* }} - 1} \right){a^\dag }a} \right]:\exp \left( {\frac{{v^* }}{{2{u^* }}}{a^2}} \right) \\ = \exp \left( { - \frac{v}{{2{u^* }}}{a^{\dag 2}}} \right)\exp \left[ {\left( {{a^\dag }a + \frac{1}{2}} \right)\ln \frac{1}{{u^* }}} \right]\exp \left( {\frac{{v^* }}{{2{u^* }}}{a^2}} \right) \\ \end{gathered}$$
(38)

and the new squeezed vacuum state is

$$S\left( {\lambda ,r} \right)\left| 0 \right\rangle = \frac{1}{{\sqrt {u^* } }}\exp \left( { - \frac{v}{{2{u^* }}}{a^{\dag 2}}} \right)\left| 0 \right\rangle \equiv {\left| 0 \right\rangle_s}$$
(39)

By introducing the quadrature operators

$$\begin{gathered} {\widehat X_1} = \frac{1}{2}(Q - P) = \frac{1}{2\sqrt 2 }[a + {a^\dag } + i(a - {a^\dag })], \\ {\widehat X_2} = \frac{1}{2}(Q + P) = \frac{1}{2\sqrt 2 }[i({a^\dag } - a) + a + {a^\dag }] \\ \end{gathered}$$
(40)

with

$$[{\widehat X_1},{\widehat X_2}] = \frac{i}{2}$$
(41)

we have

$${S^{ - 1}}{\widehat X_1}S = \frac{1}{2}\left[ {Q(\cosh \lambda + \sinh \lambda {e^r}) - P(\cosh \lambda + \sinh \lambda {e^{ - r}})} \right]$$
(42)
$${S^{ - 1}}{\widehat X_2}S = \frac{1}{2}\left[ {Q(\cosh \lambda - \sinh \lambda {e^r}) + P(\cosh \lambda - \sinh \lambda {e^{ - r}})} \right]$$
(43)

Then, the following expectation values in state \({\left| 0 \right\rangle_s}\) can be calculated

$$_s\left\langle 0 \right|{\widehat X_1}{\left| 0 \right\rangle_s} = 0,{_s}\left\langle 0 \right|{\widehat X_2}{\left| 0 \right\rangle_s} = 0$$
(44)

and it follows

$$\begin{gathered} {(\Delta {\widehat X_1})^2}{ =_s}\left\langle 0 \right|\widehat X_1^2{\left| 0 \right\rangle_s} - {{(_s}\left\langle 0 \right|{\widehat X_1}{\left| 0 \right\rangle_s})^2} \\ = \frac{1}{4}(\cosh 2\lambda + 2{\sinh^2}\lambda {\sinh^2}r + \sinh 2\lambda \cosh r) \\ \end{gathered}$$
(45)

and

$$\begin{gathered} {(\Delta {\widehat X_2})^2}{ =_s}\left\langle 0 \right|\widehat X_2^2{\left| 0 \right\rangle_s} - {{(_s}\left\langle 0 \right|{\widehat X_2}{\left| 0 \right\rangle_s})^2} \\ = \frac{1}{4}(\cosh 2\lambda + 2{\sinh^2}\lambda {\sinh^2}r - \sinh 2\lambda \cosh r) \\ \end{gathered}$$
(46)

The enhancement of squeezing means

$$\;{(\Delta {\widehat X_1})^2} > \frac{1}{4}{e^{2\lambda }},{(\Delta {\widehat X_2})^2} < \frac{1}{4}{e^{ - 2\lambda }}$$
(47)

and \({(\Delta {\widehat X_1})^2} > \frac{1}{4}{e^{2\lambda }}\) indicates

$$2{\sinh^2}\lambda {\sinh^2}r + \sinh 2\lambda \cosh {\text{r}} > \sinh 2\lambda$$
(48)

When \(\lambda > 0,\) \(\sinh \lambda > 0,\) we have

$$\tanh \lambda \left( {{{\cosh }^2}r - 1} \right) > - \left( {\cosh r - 1} \right)$$
(49)

and \(\tanh \lambda \left( {1 + \cosh r} \right) > - 1\). Since \(1 - \cosh r < 0\),

$$\tanh \lambda < \frac{1}{1 + \cosh r}$$
(50)

Similarly, from the inequality \({(\Delta {\widehat X_2})^2} < \frac{1}{4}{e^{ - 2\lambda }},\) we have

$$\tanh \lambda \left( {{{\cosh }^2}r - 1} \right) < \cosh r - 1$$
(51)

which also leads to Eq. (50) and indicates that the squeezing enhancement condition is \(\tanh \lambda < \frac{1}{1 + \cosh r}\).

Is “enhanced squeezed light” can be realized by some physical mechanism? To answer this question, this paper analyzes the operator in Eq. (30), which can be expressed as

$$\begin{gathered} \begin{array}{l} S\left( {\lambda ,r} \right) = \exp \bigg[ { - \frac{i\lambda }{2}\left( {{Q^2}{e^r} - {P^2}{e^{ - r}}} \right)} \bigg] \\ \quad \; = {e^{i\left( {QP - \frac{i}{2}} \right)\sqrt r }}{e^{ - \frac{i\lambda }{2}\left( {{Q^2} - {P^2}} \right)}}{e^{ - i\left( {QP - \frac{i}{2}} \right)\sqrt r }} \ \end{array} \end{gathered}$$
(52)

where \(\exp \left[ { - \frac{i\lambda }{2}\left( {{Q^2} - {P^2}} \right)} \right]\) describes the physical mechanism of a degenerate parametric amplifier, while the operator \({e^{i\left( {QP - \frac{i}{2}} \right)\sqrt r }}\) represents the stretching or contract size of the crystal with susceptibility. Thus, using a stretched or contracted crystal may realize squeezing enhancement[57,58,59].

5 Dissipation of Enhanced Squeezed State

No system in nature can be completely isolated from its environment, the coupling between the system and the environment always generates noise, and dissipation is an irreversible process in the dynamic evolution of the system. In the framework of quantum statistical mechanics, dissipation is described by the master equation of density operator [60,61,62]. In this section, the IWOP method and coherent state representation are employed to derive the master equation of quantum amplitude attenuation and the corresponding power series solutions. Meanwhile, the attenuation law of the squeezed vacuum light field is obtained.

Since pure coherent state is the quantum state closest to the classical one, any density operator can be represented by coherent state representation

$$\rho = \int \frac{{{d^2}\alpha }}{\pi }P\left( \alpha \right)\left| \alpha \right\rangle \left\langle \alpha \right|$$
(53)

where \(P\left( \alpha \right)\) is referred to as \(P -\) representation [63,64,65]. A typical example of the evolution of light field in the amplitude attenuation channel is the amplitude attenuation of the pure coherent state density operator \(\left| \alpha \right\rangle \left\langle \alpha \right|\)

$$\left| \alpha \right\rangle \left\langle \alpha \right| \to \left| {\alpha {e^{ - \kappa t}}} \right\rangle \left\langle {\alpha {e^{ - \kappa t}}} \right|$$
(54)

where \(\kappa\) denotes the attenuation rate. The following discusses the equations that govern this evolution. By using the normal product properties and

$$\left| {\alpha {e^{ - \kappa t}}} \right\rangle \left\langle {\alpha {e^{ - \kappa t}}} \right| = :\exp \left( { - |\alpha {|^2}{e^{ - 2\kappa t}} + \alpha {e^{ - \kappa t}}{a^\dag } + {\alpha^* }{e^{ - \kappa t}}a - {a^\dag }a} \right):$$
(55)

we have

$$\begin{gathered} \frac{d}{dt}\left| {\alpha {e^{ - \kappa t}}} \right\rangle \left\langle {\alpha {e^{ - \kappa t}}} \right| = \frac{d}{dt}:\exp \left( { - |\alpha {|^2}{e^{ - 2\kappa t}} + \alpha {e^{ - \kappa t}}{a^\dag } + {\alpha^* }{e^{ - \kappa t}}a - {a^\dag }a} \right): \\ = 2\kappa |\alpha {|^2}{e^{ - 2\kappa t}}\left| {\alpha {e^{ - \kappa t}}} \right\rangle \left\langle {\alpha {e^{ - \kappa t}}} \right| - \kappa {a^\dag }\alpha {e^{ - \kappa t}}\left| {\alpha {e^{ - \kappa t}}} \right\rangle \left\langle {\alpha {e^{ - \kappa t}}} \right| - \kappa \left| {\alpha {e^{ - \kappa t}}} \right\rangle \left\langle {\alpha {e^{ - \kappa t}}} \right|{\alpha^* }{e^{ - \kappa t}}a \\ = 2\kappa a\left| {\alpha {e^{ - \kappa t}}} \right\rangle \left\langle {\alpha {e^{ - \kappa t}}} \right|{a^\dag } - \kappa {a^\dag }a\left| {\alpha {e^{ - \kappa t}}} \right\rangle \left\langle {\alpha {e^{ - \kappa t}}} \right| - \kappa \left| {\alpha {e^{ - \kappa t}}} \right\rangle \left\langle {\alpha {e^{ - \kappa t}}} \right|{a^\dag }a \\ \end{gathered}$$
(56)

Let \(\left| {\alpha {e^{ - \kappa t}}} \right\rangle \left\langle {\alpha {e^{ - \kappa t}}} \right| = \rho \left( t \right)\), and then Eq. (56) is equivalent to

$$\frac{d}{dt}\rho \left( t \right) = \kappa \left( {2a\rho {a^\dag } - \kappa {a^\dag }a\rho - \kappa \rho {a^\dag }a} \right)$$
(57)

Thus, the quantum amplitude attenuation equation is obtained. Its infinite and power series solutions are discussed below.

Considering that

$${e^{ - \kappa t{a^\dag }a}}\left| \alpha \right\rangle = {e^{ - |\alpha {|^2}/2}}{e^{ - \kappa t{a^\dag }a}}{e^{\alpha {a^\dag }}}{e^{\kappa t{a^\dag }a}}{e^{ - \kappa t{a^\dag }a}}\left| 0 \right\rangle = {e^{ - |\alpha {|^2}/2}}{e^{\alpha {e^{ - \kappa t}}{a^\dag }}}\left| 0 \right\rangle$$
(58)

we have

$$\begin{gathered} \left| {\alpha {e^{ - \kappa t}}} \right\rangle \left\langle {\alpha {e^{ - \kappa t}}} \right|:\exp \left( { - |\alpha {|^2}{e^{ - 2\kappa t}} + \alpha {e^{ - \kappa t}}{a^\dag } + {\alpha^* }{e^{ - \kappa t}}a - {a^\dag }a} \right): \\ = \sum\limits_{n = 0}^{ + \infty } \frac{{{{\left( {1 - {e^{ - 2\kappa t}}} \right)}^n}}}{n!}|\alpha {|^{2n}}{e^{ - \kappa t{a^\dag }a}}\left| \alpha \right\rangle \left\langle \alpha \right|{e^{ - \kappa t{a^\dag }a}} \\ = \sum\limits_{n = 0}^{ + \infty } \frac{{{{\left( {1 - {e^{ - 2\kappa t}}} \right)}^n}}}{n!}{e^{ - \kappa t{a^\dag }a}}{a^n}\left| \alpha \right\rangle \left\langle \alpha \right|{a^{\dag n}}{e^{ - \kappa t{a^\dag }a}} \\ \end{gathered}$$
(59)

Since any final density operator can be expressed as

$$\rho \left( t \right) = \int \frac{{{d^2}\alpha }}{\pi }P\left( {\alpha ,0} \right)\left| {\alpha {e^{ - \kappa t}}} \right\rangle \left\langle {\alpha {e^{ - \kappa t}}} \right|$$
(60)

by using Eq. (59), we have

$$\begin{gathered} \rho \left( t \right) = \int \frac{{{d^2}\alpha }}{\pi }P\left( {\alpha ,0} \right)\sum\limits_{n = 0}^{ + \infty } \frac{{{{\left( {1 - {e^{ - 2\kappa t}}} \right)}^n}}}{n!}{e^{ - \kappa t{a^\dag }a}}{a^n}\left| \alpha \right\rangle \left\langle \alpha \right|{a^{^{\dag n}}}{e^{ - \kappa t{a^\dag }a}} \\ = \sum\limits_{n = 0}^{ + \infty } \frac{{T^n}}{n!}{e^{ - \kappa t{a^\dag }a}}{a^n}{\rho_0}{a^{\dag n}}{e^{ - \kappa t{a^\dag }a}} \\ \equiv \sum\limits_{n = 0}^{ + \infty } {M_n}{\rho_0}M_n^\dag \\ \end{gathered}$$
(61)

where

$${\rho_0} = \int \frac{{{d^2}\alpha }}{\pi }P\left( {\alpha ,0} \right)\left| \alpha \right\rangle \left\langle \alpha \right|$$
(62)

and \({M_n}\) and \(T\) are defined as

$${M_n} = \frac{{T^n}}{n!}{e^{ - \kappa t{a^\dag }a}}{a^n},\;\;T = 1 - {e^{ - 2\kappa t}}$$
(63)

Now, the IWOP method is utilized to find the integral solution of the quantum attenuation equation. By using Eq. (63), Eq. (61) can be rewritten as

$$\begin{gathered} \rho \left( t \right) = \sum\limits_{n = 0}^\infty \frac{{T^n}}{n!}{e^{ - \kappa t{a^\dag }a}}\int \frac{{{d^2}\alpha }}{\pi }P(\alpha ,0){\left| \alpha \right|^{2n}}|\alpha \rangle \langle \alpha |{e^{ - \kappa t{a^\dag }a}} \\ = \int \frac{{{d^2}\alpha }}{\pi }{e^{T{{\left| \alpha \right|}^2}}}P(\alpha ,0){e^{ - \kappa t{a^\dag }a}}|\alpha \rangle \langle \alpha |{e^{ - \kappa t{a^\dag }a}} \\ \end{gathered}$$
(64)

where

$${e^{ - \kappa t{a^\dag }a}}\left| \alpha \right\rangle = {e^{ - |\alpha {|^2}/2 + \alpha {a^\dag }{e^{ - \kappa t}}}}\left| 0 \right\rangle$$
(65)

so

$$\rho \left( t \right) = \int \frac{{{d^2}\alpha }}{\pi }{e^{ - |\alpha {|^2}{e^{ - 2\kappa t}}}}P(\alpha ,0):{e^{{a^\dag }\alpha {e^{ - \kappa t}} + a{\alpha^* }{e^{ - \kappa t}} - {a^\dag }a}}:$$
(66)

The inverse relation of Eq. (62) is

$$P(\alpha ,0) = {e^{|\alpha {|^2}}}\int \frac{{{d^2}\beta }}{\pi }\left\langle { - \beta } \right|{\rho_0}\left| \beta \right\rangle {e^{|\beta {|^2} + {\beta^* }\alpha - \beta {\alpha^* }}}$$
(67)

where \(\left| \beta \right\rangle\) is also a coherent state. Substituting Eq. (67) into Eq. (66), the integral solution of the amplitude attenuation master equation is obtained:

$$\begin{array}{c}\rho(t)=\int\frac{d^2\beta}\pi\langle-\beta\vert\rho_0\vert\beta\rangle e^{\vert\beta\vert^2}\int\frac{d^2\alpha}\pi e^{-\vert\alpha\vert^2(e^{-2\kappa t}-1)}:e^{\beta^\ast\alpha-\beta\alpha^\ast+a^\dagger\alpha e^{-\kappa t}+a\alpha^\ast e^{-\kappa t}-a^\dagger a}:\\=\frac{-1}T\int\frac{d^2\beta}\pi\langle-\beta\vert\rho_0\vert\beta\rangle e^{\vert\beta\vert^2}:exp\lbrack\frac{-1}T(a^\dagger e^{-\kappa t}+\beta^\ast)(ae^{-\kappa t}-\beta)-a^\dagger a\rbrack:\\=\frac{-1}T\int\frac{d^2\beta}\pi\langle-\beta\vert\rho_0\vert\beta\rangle e^{\vert\beta\vert^2}:exp\{\frac1T\lbrack\vert\beta\vert^2+e^{-\kappa t}(\beta a^\dagger-\beta^\ast a)\rbrack-\frac{e^{-2\kappa t}}Ta^\dagger a-a^\dagger a\}:\end{array}$$
(68)

This equation has an advantage: With a given initial density operator \({\rho_0}\), when the matrix element \(\left\langle { - \beta } \right|{\rho_0}\left| \beta \right\rangle\) is calculated and Eq. (68) is integrated by the IWOP method, the final state density operator \(\rho \left( t \right)\) can be derived easily.

At this point, an approach for finding the solution to the master equation can be found in the research work of Mattos's team [66]. The method allows for obtaining analytical expressions for the time-evolved P-function of the system without needing to solve the corresponding master equation.

Now, the attenuation of the enhanced squeezed light field \({\rho_0}\) is discussed

$${\rho_0} = S\left( {\lambda ,r} \right)\left| 0 \right\rangle \left\langle 0 \right|{S^\dag }\left( {\lambda ,r} \right) = \frac{1}{{u^* }}\exp \left( { - \frac{v}{{2{u^* }}}{a^{\dag 2}}} \right)\left| 0 \right\rangle \left\langle 0 \right|\exp \left( { - \frac{{v^* }}{2u}{a^2}} \right)$$
(69)

The matrix element \(\left\langle { - \beta } \right|{\rho_0}\left| \beta \right\rangle\) is

$$\left\langle { - \beta } \right|{\rho_0}\left| \beta \right\rangle = \frac{1}{{u^* }}{e^{ - \frac{v}{{2{u^* }}}{\beta^{ * 2}} - \frac{{v^* }}{2u}{\beta^2} - |\beta {|^2}}}$$
(70)

Substituting Eq. (70) into Eq. (68) and using the integral formula

$$\int \frac{{{d^2}z}}{\pi }\exp \left( {\xi {{\left| z \right|}^2} + \xi z + \eta {z^* } + f{z^2} + g{z^{ * 2}}} \right) = \frac{1}{{\sqrt {{\xi^2} - 4fg} }}\exp \left[ {\frac{{ - {\xi^2}\eta + f{\eta^2} + g{\xi^2}}}{{{\xi^2} - 4fg}}} \right]$$
(71)

we have

$$\begin{array}{c}\rho(t)=\frac{-1}{Tu^\ast}\int\frac{d^2\beta}\pi\langle-\beta\vert exp(-\frac v{2u^\ast}a^{\dagger2})\vert0\rangle\langle0\vert exp(-\frac{v^\ast}{2u}a^2)\vert\beta\rangle e^{\vert\beta\vert^2}\\\times:\text{exp}\left\{\frac1T\left[\left|\beta\right|^2+e^{-\kappa t}\left(\beta a^\dagger-\beta^\ast a\right)\right]-\frac{e^{-2\kappa t}}Ta^\dagger a-a^\dagger a\right\}:\\=\frac{-1}{Tu^\ast}\int\frac{d^2\beta}\pi:\text{exp}\left\{\frac1T\left[\left|\beta\right|^2+e^{-\kappa t}\left(\beta a^\dagger-\beta^\ast a\right)-\frac{e^{-2\kappa t}}Ta^\dagger a-a^\dagger a\right]-\frac v{2u^\ast}\beta^{\ast2}-\frac{v^\ast}{2u}\beta^2\right\}:\\=\frac1{u^\ast\sqrt{1-T^2\frac{\vert v\vert^2}{\vert u\vert^2}}}:\text{exp}\left[\frac{e^{-2\kappa t}}2\frac{-\frac v{u^\ast}a^2-\frac{v^\ast}ua^{\dagger2}}{1-\frac{\vert v\vert^2}{\vert u\vert^2}T^2}+\left(\frac{Te^{-2\kappa t}\frac{\vert v\vert^2}{\vert u\vert^2}}{1-\frac{\vert v\vert^2}{\vert u\vert^2}T^2}-1\right)a^\dagger a\right]:\end{array}$$
(72)

here

$$\frac{{\mathop {\tanh }\nolimits^2 \lambda \mathop {\cosh }\nolimits^2 r}}{{1 + \mathop {\tanh }\nolimits^2 \lambda \mathop {\sinh }\nolimits^2 r}} = \frac{{|v{|^2}}}{{|u{|^2}}}$$
(73)

Particularly, when \(r = 0\), from Eq. (73) we have

$$u = \cosh \lambda ,v = i\sinh \lambda ,\frac{{|v{|^2}}}{{|u{|^2}}} = \mathop {\tanh }\nolimits^2 \lambda ,\;\frac{{v^* }}{{u^* }} = - i\tanh \lambda$$
(74)

and

$${\rho_0} = S\left( {\lambda ,r} \right)\left| 0 \right\rangle \left\langle 0 \right|{S^\dag }\left( {\lambda ,r} \right) = \frac{1}{{u^* }}\exp \left( { - \frac{i}{2}{a^{\dag 2}}\tanh \lambda } \right)\left| 0 \right\rangle \left\langle 0 \right|\exp \left( {\frac{i}{2}{a^2}\tanh \lambda } \right)$$
(75)

with

$$\left\langle { - \beta } \right|{\rho_0}\left| \beta \right\rangle = \sec h\lambda {e^{\frac{i}{2}\left( { - {\beta^{ * 2}} + {\beta^2}} \right)\tanh \lambda - |\beta {|^2}}}$$
(76)

In this case, Eq. (72) reduces to

$$\rho (t) = \frac{\sec h\lambda }{{\sqrt {1 - {T^2}\mathop {\tanh }\nolimits^2 \lambda } }}:\exp \left[ {\frac{{{e^{ - 2\kappa t}}i\tanh \lambda }}{2}\frac{{{a^2} - {a^{\dag 2}}}}{{1 - {T^2}\mathop {\tanh }\nolimits^2 \lambda }} + \left( {\frac{{T\mathop {\tanh }\nolimits^2 \lambda {e^{ - 2\kappa t}}}}{{1 - {T^2}\mathop {\tanh }\nolimits^2 \lambda }} - 1} \right){a^\dag }a} \right]:$$
(77)

or

$$\rho (t) = G{e^{ - i\tau {a^{\dag 2}}/2}}{e^{{a^\dag }a\ln \left( {\tau T\tanh \lambda } \right)}}{e^{i\tau {a^2}/2}}$$
(78)

where

$$G \equiv \frac{\sec h\lambda }{{\sqrt {1 - {T^2}\mathop {\tanh }\nolimits^2 \lambda } }},\;\tau = \frac{{{e^{ - 2\kappa t}}\tanh \lambda }}{{1 - {T^2}\mathop {\tanh }\nolimits^2 \lambda }}$$
(79)

It can be further proved that

$$\begin{gathered} tr\rho (t) = G\int \frac{{{d^2}z}}{\pi }\left\langle z \right|{e^{ - i\tau {a^{\dag 2}}/2}}{e^{{a^\dag }a\ln \left( {\tau T\tanh \lambda } \right)}}{e^{i\tau {a^2}/2}}\left| z \right\rangle \\ = G\int \frac{{{d^2}z}}{\pi }\exp \left[ {\left( {\tau T\tanh \lambda - 1} \right)|z{|^2} + i\tau {z^2}/2 - i\tau {z^{ * 2}}/2} \right] \\ = 1 \\ \end{gathered}$$
(80)

when \(t = 0,T = 0\), \(\rho (t)\) will go back to the initial state.

$$\begin{gathered} \rho (t = 0) = \sec h\lambda :\exp \left[ {\frac{i\tanh \lambda }{2}\left( {{a^2} - {a^{\dag 2}}} \right) - {a^\dag }a} \right]: \\ = \sec h\lambda {e^{ - \frac{i}{2}{a^{\dag 2}}\tanh \lambda }}\left| 0 \right\rangle \left\langle 0 \right|{e^{\frac{i}{2}{a^2}\tanh \lambda }} \\ \end{gathered}$$
(81)

6 Decomposition of the Enhanced Squeezing Operator

Based on Eq. (37) of the squeezing-enhancing operator

$$\begin{gathered} S\left( {\lambda ,r} \right) = \exp \left[ {\frac{ - i\lambda }{2}\left( {{Q^2}{e^r} - {P^2}{e^{ - r}}} \right)} \right] \\ = \sqrt u \int \frac{{{d^2}z}}{\pi }\left| {\left( {\begin{array}{*{20}{c}} u&{ - v} \\ { - {v^* }}&{u^* } \end{array}} \right)\left( {\begin{array}{*{20}{c}} z \\ {z^* } \end{array}} \right)} \right\rangle \left\langle {\left( {\begin{array}{*{20}{c}} z \\ {z^* } \end{array}} \right)} \right| \\ \end{gathered}$$
(82)

where

$$\begin{gathered} \left| {\left( {\begin{array}{*{20}{c}} u&{ - v} \\ { - {v^* }}&{u^* } \end{array}} \right)\left( {\begin{array}{*{20}{c}} z \\ {z^* } \end{array}} \right)} \right\rangle \equiv \left| {p\cosh \lambda - q\sinh \lambda {e^r},q\cosh \lambda - p{e^{ - r}}\sinh \lambda } \right\rangle \\ = \left| {\left( {\begin{array}{*{20}{c}} {\cosh \lambda }&{ - {e^{ - r}}\sinh \lambda } \\ { - \sinh \lambda {e^r}}&{\cosh \lambda } \end{array}} \right)\left( {\begin{array}{*{20}{c}} q \\ p \end{array}} \right)} \right\rangle \\ \end{gathered}$$
(83)

the symplectic matrix can be decomposed as

$$\left( {\begin{array}{*{20}{c}} {\cosh \lambda }&{ - {e^{ - r}}\sinh \lambda } \\ { - \sinh \lambda {e^r}}&{\cosh \lambda } \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 1&0 \\ { - \tanh \lambda {e^r}}&1 \end{array}} \right)\left( {\begin{array}{*{20}{c}} {\cosh \lambda }&0 \\ 0&{sech\lambda } \end{array}} \right)\left( {\begin{array}{*{20}{c}} 1&{ - \tanh \lambda {e^{ - r}}} \\ { - \tanh \lambda {e^r}}&1 \end{array}} \right)$$
(84)

Since the symplectic transformations constitute a symplectic group product, its quantum mechanical correspondence also supports the following decomposition

$$\begin{array}{c}exp[\frac{-i\lambda }{2}({Q}^{2}{e}^{r}-{P}^{2}{e}^{-r})]=\iint dpdq|(\begin{array}{cc}\text{cosh}\lambda & -{e}^{-r}\text{sinh}\lambda \\ -\text{sinh}\lambda {e}^{r}& \text{cosh}\lambda \end{array})(\begin{array}{c}q\\ p\end{array})\rangle \langle p,q|\\ =\iint dpdq|(\begin{array}{cc}1& 0\\ -\text{tanh}\lambda {e}^{r}& 1\end{array})(\begin{array}{c}q\\ p\end{array})\rangle \langle p,q|\\ \times \iint d{p}{^\prime}d{q}{^\prime}|(\begin{array}{cc}\text{cosh}\lambda & 0\\ 0& sech\lambda \end{array})(\begin{array}{c}{q}{^\prime}\\ {p}{^\prime}\end{array})\rangle \langle {p}{^\prime},{q}{^\prime}|\\ \times \iint d{p}{^\prime}{^\prime}d{q}{^\prime}{^\prime}|(\begin{array}{cc}1& -\text{tanh}\lambda {e}^{-r}\\ 0& 1\end{array})(\begin{array}{c}{q}{^\prime}{^\prime}\\ {p}{^\prime}{^\prime}\end{array})\rangle \langle {p}{^\prime}{^\prime},{q}{^\prime}{^\prime}|\end{array}$$
(85)

where

$$\iint {dpdq}\left| {\left( {\begin{array}{*{20}{c}} 1&0 \\ { - \tanh \lambda {e^r}}&1 \end{array}} \right)\left( {\begin{array}{*{20}{c}} q \\ p \end{array}} \right)} \right\rangle \left\langle {p,q} \right| = \exp \left[ {\frac{{ - i\tanh \lambda {e^r}}}{2}{Q^2}} \right]$$
(86)
$$\iint dp'dq'\vert(\begin{array}{cc}\text{cosh}\lambda&0\\0&sech\lambda\end{array})(\begin{array}{c}q'\\p'\end{array})\rangle\langle p',q'\vert=\text{exp}\lbrack-\frac i2(QP+PQ)\text{lncosh}\lambda\rbrack$$
(87)
$$\exp \left[ {\frac{{i\tanh \lambda {e^{ - r}}}}{2}{P^2}} \right] = \exp \left[ {\frac{{i\tanh \lambda {e^{ - r}}}}{2}{P^2}} \right]$$
(88)

Therefore, the enhanced squeezing operator is decomposed as

$$\exp \left[ {\frac{ - i\lambda }{2}\left( {{Q^2}{e^r} - {P^2}{e^{ - r}}} \right)} \right] = \exp \left[ {\frac{{ - i\tanh \lambda {e^r}}}{2}{Q^2}} \right]\exp \left[ { - \frac{i}{2}\left( {QP + PQ} \right)\ln \cosh \lambda } \right]\exp \left[ {\frac{{i\tanh \lambda {e^{ - {\text{r}}}}}}{2}{P^2}} \right]$$
(89)

One can check its correctness by evaluating

$$\exp \left[ {\frac{{i\tanh \lambda {e^{ - r}}}}{2}{P^2}} \right]Q\exp \left[ {\frac{{ - i\tanh \lambda {e^{ - r}}}}{2}{P^2}} \right] = Q + \left[ {\frac{{i\tanh \lambda {e^{ - r}}}}{2}{P^2},Q} \right] = Q + P\tanh \lambda {e^{ - r}}$$
(90)
$$\begin{gathered} \exp \left[ { - \frac{i}{2}\left( {QP + PQ} \right)\ln \cosh \lambda } \right]Q\exp \left[ {\frac{i}{2}\left( {QP + PQ} \right)\ln \cosh \lambda } \right] = Q\sec h\lambda , \\ \exp \left[ { - \frac{i}{2}\left( {QP + PQ} \right)\ln \cosh \lambda } \right]P\exp \left[ {\frac{i}{2}\left( {QP + PQ} \right)\ln \cosh \lambda } \right] = P\cosh \lambda \\ \end{gathered}$$
(91)

then

$$\begin{gathered} \exp \left[ { - \frac{i}{2}\left( {QP + PQ} \right)\ln \cosh \lambda } \right]\exp \left[ {\frac{{i\tanh \lambda {e^{ - r}}}}{2}{P^2}} \right]Q\exp \left[ {\frac{{ - i\tanh \lambda {e^{ - r}}}}{2}{P^2}} \right]\exp \left[ {\frac{i}{2}\left( {QP + PQ} \right)\ln \cosh \lambda } \right] \\ = Q\sec h\lambda + P\sinh \lambda {e^{ - r}} \\ \end{gathered}$$
(92)

Further, from

$$\exp \left[ { - \frac{{i\tanh \lambda {e^r}}}{2}{Q^2}} \right]P\exp \left[ {\frac{{i\tanh \lambda {e^{ - r}}}}{2}{P^2}} \right] = P + \tanh \lambda {e^r}Q$$
(93)

the final transform result is

$$\begin{gathered} \exp \left[ {\frac{ - i\lambda }{2}\left( {{Q^2}{e^r} - {P^2}{e^{ - r}}} \right)} \right]Q\exp \left[ {\frac{i\lambda }{2}\left( {{Q^2}{e^r} - {P^2}{e^{ - r}}} \right)} \right] \\ = Q\sec h\lambda + \left( {P + \tanh \lambda {e^r}Q} \right)\sinh \lambda {e^{ - r}} \\ = Q\cosh \lambda + P{e^{ - r}}\sinh \lambda \\ \end{gathered}$$
(94)

which agrees with Eq. (31).

7 Conclusion

To sum up, by using the IWOP method and following the quantum phase space theory, this paper analyzes the properties of the quadrature field operator of squeezed light and the representation of the squeezing operator \(S\left( \xi \right)\) in the \(q - p\) phase space. Based on this, the phase-space evolution of squeezing-enhanced light, the new squeezing state, and its attenuation can be theoretically represented. These representations give physicists more refined insights for squeezing enhancement. This paper demonstrates the approach for two parameters, and we hope our work will inspire further research that scales to multi-parameters.