Classical Correspondence of Squeezing Operators and the Extension of Bohr’s Correspondence Principle

Abstract

Bohr’s correspondence principle acts as a link between quantum physics and classical physics theory, while squeezed light, as a special nonclassical quantum state in quantum physics, achieves precision measurements and gravitational wave detection by minimizing quantum noise in one quadrature component of the optical field. Consequently, determining whether the classical counterpart of the squeezing operator reflects classical spatial scaling transformations is of significant theoretical importance. This paper establishes a universal integral formula that transforms any operator into its Weyl ordering form using the method of integration within the ordered product of operators, combined with the coherent state representation and integration theory within Weyl ordering. By transforming both single-mode and two-mode squeezing operators into their corresponding Weyl ordering forms, their classical counterpart functions are derived. This elucidates the classical correspondence of the squeezed light field density operator and demonstrates that this correspondence fundamentally represents a classical scaling transformation. As a practical application of the classical counterpart of the single-mode squeezing operator, the photon number distribution characteristics in a single-mode squeezed light field are obtained, confirming its noise-squeezing effect. This study not only deepens the theoretical implications of Bohr’s correspondence principle from the perspective of “transformation correspondence” but also introduces novel insights into the establishment of the mathematical foundations of quantum optics and quantum statistical theory.

1. Introduction

The Bohr correspondence principle serves as an important resource for effectively bridging the theoretical systems of quantum mechanics and classical physics. It reveals their inherent unity under limiting conditions and lays a crucial foundation for the study of quantum–classical transition problems. Historically, German physicists Franck and Hertz carried out extensive experiments to investigate collisions between electrons and atoms. They noted the quantized characteristics of electron energy loss, which confirmed the correctness of Bohr’s stationary state hypothesis. Nonetheless, the concept of stationary states fundamentally contradicts traditional physics theories. In his examination of the atomic model, Bohr recognized that as the quantum numbers gradually increase, the results of quantum theory align more closely with those of classical theory. Based on this observation, he formulated the famous Bohr Correspondence Principle. According to this principle, at the microscopic scale (such as inside atoms), the behavior of a system must be depicted by quantum theory. However, as quantum numbers become large or Planck’s constant tends to zero, quantum laws should gradually converge with classical physics. This establishes a seamless and continuous decoherence spectrum from microscopic quantum states to macroscopic classical states, illustrating the inseparable relationship between quantum mechanics and classical mechanics. Bohr’s correspondence principle defines the boundary where quantum theory must converge with classical physics. Within the “quantum-to-classical transition” theoretical framework, as a representative non-classical light field in quantum optics and quantum information science, squeezed light [1,2,3,4,5] has been widely applied in frontier areas such as continuous-variable quantum information processing, high-precision interferometry, and the preparation of non-classical quantum light fields. The squeezing property of quantum noise in squeezed light reflects a profound difference between quantum and classical physics. Within the framework of classical physics, noise cannot surpass the standard quantum limit; however, squeezed light achieves noise suppression beyond the classical limit on a specific orthogonal component through asymmetric reconstruction of quantum fluctuations in a pair of conjugate variables (e.g., amplitude and phase) of the light field. This important property makes squeezed light an ideal candidate for testing Bohr’s correspondence principle: in the macroscopic limit (i.e., a large photon number or the weak squeezing limit), squeezed light degenerates into a classical coherent state, consistent with Bohr’s correspondence principle. In contrast, under strong squeezing or few-photon conditions, its non-classicality becomes increasingly prominent and must be systematically described by quantum theory. Thus, the study of squeezed light not only verifies the general validity of Bohr’s correspondence principle but also provides a controllable theoretical foundation for exploring the quantum–classical boundary within the framework of quantum–classical transition.

The density operator $\rho$ for single-mode squeezed light in a quantum optical field can be represented as:

$$\rho ={\mathrm{sech}}^{1/2}\lambda e^{{\displaystyle \frac{\tanh \lambda }{2}}{a}^{†2}}\left|0\right.〉\left.〈0\right|e^{{\displaystyle \frac{\tanh \lambda }{2}}{a}^2}.$$

(1)

where $\lambda$ is the squeezing parameter, $[a,a^{†}]=1$, and $\left|0\right.〉〈\left.0\right|$ denotes the squeezed vacuum state. Critically important yet often neglected questions arise: What exactly is the classical counterpart of the density operator $\rho$ for squeezed light fields as a non-classical resource? Does the phase-space scaling transformation associated with this density operator suggest the existence of corresponding geometric scaling in classical physical space? Furthermore, how can squeezed states of light converge to classical behavior in the macroscopic limit, thereby validating Bohr’s correspondence principle? This paper aims to determine whether the transformation represented by the squeezing operator corresponds to a scaling transformation function in classical physics, thereby examining whether it is feasible to extend Bohr’s correspondence principle to incorporate the deeper connotation of a mutual correspondence between classical and quantum transformations. This study employs the integration within an ordered product (IWOP) technique [6,7,8,9,10,11], the coherent state representation [12,13,14,15,16,17,18] and Weyl ordering [19,20,21,22,23,24] theory to devise a quantization scheme for analyzing quantum-classical correspondence. By introducing Weyl ordering symbols, a universal integral formula is established that allows for the transformation of any quantum mechanical operator into its Weyl-ordered form. On this basis, and by integrating the Weyl-ordered forms of both single- and two-mode vacuum states with the Weyl–Wigner quantization scheme of the two-mode Wigner operator, this study systematically and analytically solves the classical phase-space functions corresponding to the density operators of both single- and two-mode squeezed light fields. This research explicitly provides a precise classical counterpart for squeezed light fields and confirms that Bohr’s correspondence principle can indeed incorporate the correspondence between classical and quantum transformations. Therefore, it realizes a significant deepening and expansion of the theoretical connotation of Bohr’s correspondence principle.

2. Weyl Ordering Quantization Scheme

De Broglie’s insight into wave-particle duality transformed the description of objects in physics from deterministic values of a particle’s location to quantum states that articulate the probabilities of a system existing in various conditions. Physicists have progressed beyond the mere documentation of classical labels such as position and momentum and have begun to systematically explore the definition, evolution, and measurement of this quantum state of information, which encapsulates all potential information and correlations. To align with the significant implications of quantum states, we need to elevate the directly readable numerical values in classical mechanics (such as position and energy) into operators that can act upon quantum states. Only by manipulating quantum states through these operators can all observable information concerning the system’s state and behavior be extracted. The transformation from a classical phase-space function to a quantum operator is essentially a mapping problem involving degrees of freedom. Given the non-commutative relationship between coordinates and momenta, various quantization ordering rules emerge. As a result, the quantization of classical functions inevitably faces the challenge of operator ordering. Among the various quantization schemes, Weyl quantization is commonly utilized as an operator ordering method. Its essence lies in ensuring, by definition, an optimal symmetric correspondence between quantum operators and classical physical quantities. By symmetrically handling coordinates and momenta in classical phase-space functions, it guarantees that the quantized operators remain formally unchanged under coordinate transformations, thereby preventing non-physical artificial asymmetries introduced by arbitrary ordering choices.

The primary aim of the Weyl quantization scheme is to transform a classical exponential function $e^{iqv+ipw}$ into its corresponding operator $e^{iQv+iPw}$,

$$e^{iqv+ipw}\to e^{iQv+iPw},$$

(2)

where $v$ and $w$ are real parameters, $p$ and $q$ represent the momentum and the coordinate, $P$ and $Q$ represent the momentum operator and the coordinate operator, respectively, in the quantum phase-space theory system. This transformation process indicates a specific operator ordering rule, for which the symbol $\begin{array}{c}:\\ :\end{array}\begin{array}{c}\end{array}\begin{array}{c}:\\ :\end{array}$ is introduced to represent Weyl ordering. The operator $e^{iQv+iPw}$ quantized via Weyl ordering can be represented as:

$$e^{iQv+iPw}=\begin{array}{c}:\\ :\end{array}\begin{array}{c}{e}^{iQv+iPw}\end{array}\begin{array}{c}:\\ :\end{array},$$

(3)

Since

$$Q={\displaystyle \frac{a+a^{†}}{\sqrt{2}}},\begin{array}{c}\end{array}P={\displaystyle \frac{a-a^{†}}{\sqrt{2}i}},$$

(4)

where $a$ and $a^{†}$ are the annihilation and creation operators, respectively. The formula $e^{i{\xi }^{\ast }{a}^{†}+i\xi a}=\begin{array}{c}:\\ :\end{array}\begin{array}{c}{e}^{i{\xi }^{\ast }{a}^{†}+i\xi a}\end{array}\begin{array}{c}:\\ :\end{array}$ also reveals Weyl ordering quantization. Just as creation and annihilation operators within the normal ordering [25,26,27,28,29,30] symbol $:\begin{array}{c}\end{array}:$ can be exchanged based on their product power series relation, coordinate operators $Q$ and momentum operators $P$ can similarly be exchanged within the Weyl ordering symbol $\begin{array}{c}:\\ :\end{array}\begin{array}{c}\end{array}\begin{array}{c}:\\ :\end{array}$. The Weyl quantization scheme for classical functions $e^{iqv+ipw}$ in phase space can be implemented via the integral transformation in Equation (5):

$$e^{iQv+iPw}=\begin{array}{c}:\\ :\end{array}\begin{array}{c}{e}^{iQv+iPw}\end{array}\begin{array}{c}:\\ :\end{array}={\displaystyle \int {\displaystyle {\int }_{-\infty }^{\infty }dpdq{e}^{iqv+ipw}\mathrm{\Delta }\left(p,q\right)}},$$

(5)

The integral kernel operator $\mathrm{\Delta }\left(p,q\right)$ is composed of Gaussian integrals using the completeness of the coordinate representation and the momentum representation. By comparing Equations (3) and (5), we have

$$\begin{array}{l}\mathrm{\Delta }\left(p,q\right)={\displaystyle \frac{1}{\pi }}:\exp \left[-2i\left(q-Q\right)\left(p-P\right)\right]:\\ ={\displaystyle \frac{1}{\pi }}\begin{array}{c}:\\ :\end{array}\exp \left[-2i\left(q-Q\right)\left(p-P\right)\right]\begin{array}{c}:\\ :\end{array}\\ ={\displaystyle \iint \frac{dudv}{4{\pi }^2}}\begin{array}{c}:\\ :\end{array}\exp \left[-{\displaystyle \frac{iuv}{2}}+i\left(q-Q\right)u+i\left(p-P\right)v\right]\begin{array}{c}:\\ :\end{array}\\ ={\displaystyle \iint \frac{dudv}{4{\pi }^2}\begin{array}{c}:\\ :\end{array}}{e}^{i\left(p-P\right)v}{e}^{i\left(q-Q\right)u}\begin{array}{c}:\\ :\end{array}\\ =\begin{array}{c}:\\ :\end{array}\delta \left(p-P\right)\delta \left(q-Q\right)\begin{array}{c}:\\ :\end{array}\end{array}$$

(6)

From Equations (5) and (6), it is evident that the integral kernel $\mathrm{\Delta }\left(p,q\right)$ corresponds exactly to the Weyl ordering Dirac $\delta -$ function, which relates the classical function to its quantum counterpart through the integral transformation $e^{iQv+iPw}={\displaystyle \int {\displaystyle {\int }_{-\infty }^{\infty }dpdq{e}^{iqv+ipw}\mathrm{\Delta }\left(p,q\right)}}$.

Since the integral kernel operator $\mathrm{\Delta }\left(p,q\right)={\displaystyle {\int }_{-\infty }^{\infty }\frac{du}{2\pi }}{e}^{ipu}\left|q+{\displaystyle \frac{u}{2}}\right.〉\left.〈q-{\displaystyle \frac{u}{2}}\right|$, integrating over $p$ and $q$ separately, then applying the completeness of the coordinate representation, we arrive at the completeness relation for $\mathrm{\Delta }\left(p,q\right)$:

$${\displaystyle \int {\displaystyle {\int }_{-\infty }^{\infty }dqdp\mathrm{\Delta }\left(p,q\right)}}={\displaystyle \int {\displaystyle {\int }_{-\infty }^{\infty }dqdp\begin{array}{c}:\\ :\end{array}\begin{array}{c}\delta \left(q-Q\right)\delta \left(p-P\right)\end{array}\begin{array}{c}:\\ :\end{array}}}=1.$$

(7)

In 1932, Weyl introduced a quantization scheme that maps classical phase-space functions to quantum operators through an integral transform, whose integral kernel is the Wigner operator $\mathrm{\Delta }\left(p,q\right)$.

$$F\left(P,Q\right)={\displaystyle \iint dpdq\mathrm{\Delta }\left(p,q\right)}f\left(p,q\right),$$

(8)

Here, the classical function $f\left(p,q\right)$ serves as the classical counterpart of the operator $F\left(P,Q\right)$, a correspondence now known as the Weyl–Wigner [31,32,33] quantization scheme. Therefore, any operator $F\left(P,Q\right)$ in quantum mechanics can be expanded through the Wigner operator as in Equation (8).

Substituting $\mathrm{\Delta }\left(p,q\right)=\begin{array}{c}:\\ :\end{array}\begin{array}{c}\delta \left(q-Q\right)\delta \left(p-P\right)\end{array}\begin{array}{c}:\\ :\end{array}$ obtained from Equation (6) into Equation (8), we obtain

$$\begin{array}{l}F\left(P,Q\right)={\displaystyle \int {\displaystyle {\int }_{-\infty }^{\infty }dpdqf\left(p,q\right)}\mathrm{\Delta }\left(p,q\right)}\\ ={\displaystyle \int {\displaystyle {\int }_{-\infty }^{\infty }dpdqf\left(p,q\right)\begin{array}{c}:\\ :\end{array}\delta \left(p-P\right)\begin{array}{c}\delta \left(q-Q\right)\end{array}\begin{array}{c}:\\ :\end{array}}}\\ =\begin{array}{c}:\\ :\end{array}{\displaystyle \int {\displaystyle {\int }_{-\infty }^{\infty }dpdqf\left(p,q\right)\delta \left(p-P\right)\begin{array}{c}\delta \left(q-Q\right)\end{array}\begin{array}{c}:\\ :\end{array}}},\end{array}$$

(9)

Here, $f\left(p,q\right)$ is an ordinary classical function (not involving $Q$ and $P$), and therefore it can be moved in and out of the Weyl ordering symbol without restriction. Upon integrating over $p$ and $q$ in the Dirac delta function, and applying the sifting property of the Dirac delta function, we obtain:

$${\displaystyle \int {\displaystyle {\int }_{-\infty }^{\infty }dpdqf\left(p,q\right)\delta \left(p-P\right)\begin{array}{c}\delta \left(q-Q\right)\end{array}}}=f\left(P,Q\right),$$

(10)

so

$$F\left(P,Q\right)=\begin{array}{c}:\\ :\end{array}\begin{array}{c}f\left(P,Q\right)\end{array}\begin{array}{c}:\\ :\end{array}.$$

(11)

Equation (10) indicates that the classical counterpart of any Weyl ordering operator $\begin{array}{c}:\\ :\end{array}\begin{array}{c}f\left(P,Q\right)\end{array}\begin{array}{c}:\\ :\end{array}$ can be derived directly through the substitution $P\to p,Q\to q$.

From Equation (6), then by using the IWOP method, the Baker–Hausdorff formula [34,35,36], the coordinate projection operator $\left|q\right.〉\left.〈q\right|={\displaystyle \frac{1}{\sqrt{\pi }}}:e^{-{\left(q-Q\right)}^2}:$, and the momentum projection operator $\left|p\right.〉\left.〈p\right|={\displaystyle \frac{1}{\sqrt{\pi }}}:e^{-{\left(p-P\right)}^2}:$, we also have

$$\begin{array}{l}\mathrm{\Delta }\left(p,q\right)={\displaystyle \frac{1}{\pi }}:e^{-{\left(q-Q\right)}^2-{\left(p-P\right)}^2}:\\ ={\displaystyle \int {\displaystyle {\int }_{-\infty }^{\infty }\frac{dudv}{4{\pi }^2}}}:\begin{array}{c}\exp \left[i\left(q-Q\right)u+i\left(p-P\right)v\right]\end{array}:\\ ={\displaystyle \int {\displaystyle {\int }_{-\infty }^{\infty }\frac{dudv}{4{\pi }^2}}}\begin{array}{c}:\\ :\end{array}\begin{array}{c}\exp \left[i\left(q-Q\right)u+i\left(p-P\right)v\right]\end{array}\begin{array}{c}:\\ :\end{array}\\ ={\displaystyle \int {\displaystyle {\int }_{-\infty }^{\infty }\frac{dudv}{4{\pi }^2}}}\exp \left[i\left(q-Q\right)u+i\left(p-P\right)v\right],\end{array}$$

(12)

Let $\alpha ={\displaystyle \frac{q+ip}{\sqrt{2}}}$ and using Equation (4), since $a$ and $a^{†}$ are commutative within $\begin{array}{c}:\\ :\end{array}\begin{array}{c}\end{array}\begin{array}{c}:\\ :\end{array}$, the operator integral kernel $\mathrm{\Delta }\left(p,q\right)$ can be rewritten as:

$$H\left(a,a^{†}\right)=2{\displaystyle \int d^2\alpha }h\left(\alpha ,{\alpha }^{\ast }\right)\mathrm{\Delta }\left(\alpha \right);\hspace{1em}\mathrm{\Delta }\left(\alpha \right)=\mathrm{\Delta }\left(p,q\right),$$

(13)

where $H\left(a,a^{†}\right)$ is an arbitrary quantum operator. Therefore, we have

$${\displaystyle \frac{1}{2}}\begin{array}{c}:\\ :\end{array}\begin{array}{c}\delta \left(a^{†}-{\alpha }^{\ast }\right)\delta \left(a-\alpha \right)\end{array}\begin{array}{c}:\\ :\end{array}=\mathrm{\Delta }\left(\alpha \right).$$

(14)

Moreover, by applying the Baker–Hausdorff formula, $F\left(P,Q\right)$ in Equation (11) can be transformed into the following normal ordering form:

$$\exp \left[i\left(q-Q\right)u+i\left(p-P\right)v\right]=:\exp \left[i\left(q-Q\right)u+i\left(p-P\right)v-{\displaystyle \frac{u^2+v^2}{4}}\right]:,$$

(15)

so

$$\begin{array}{l}\mathrm{\Delta }\left(\alpha \right)={\displaystyle \int {\displaystyle {\int }_{-\infty }^{\infty }\frac{dudv}{4{\pi }^2}}}\exp \left[i\left(q-Q\right)u+i\left(p-P\right)v\right]\\ ={\displaystyle \frac{1}{\pi }}:e^{-{\left(q-Q\right)}^2-{\left(p-P\right)}^2}:\\ ={\displaystyle \frac{1}{\pi }}:e^{-2({\alpha }^{\ast }-a^{†})(\alpha -a)}:.\end{array}$$

(16)

Equation (16) represents the normal ordering form of the operator integral kernel $\mathrm{\Delta }\left(\alpha \right)$.

3. The Formula for Converting Any Operator into Weyl Ordering

The main role of Weyl ordering is to establish a one-to-one correspondence between quantum operators and classical phase-space functions, thereby ensuring a natural correspondence between quantum observables and classical physical quantities. It serves as a bridge for conversion among different ordering schemes, such as normal ordering and anti-normal ordering, and is widely applied in quantum phase-space theory, preparation and identification of nonclassical states, quantum optical field evolution, and continuous-variable quantum information processing. In this section, we will comprehensively employ the coherent state representation, the normal ordering, and the integration method within Weyl ordering to systematically derive a general formula for transforming any quantum mechanical operator $H\left(a,a^{†}\right)$ into its Weyl ordering form. Using this formula as a direct theoretical tool, the classical counterpart function of the squeezed light field density operator can be subsequently solved, providing a universal method for investigating the classical phase-space representation of other quantum states. First, using Equation (8), we can derive the matrix element relation of the operator $H\left(a,a^{†}\right)$ in the coherent state representation as

$$\left.〈-\beta \right|H\left(a,a^{†}\right)\left|\beta \right.〉=2{\displaystyle \int }{d}^2\alpha \left.〈-\beta \right|\mathrm{\Delta }\left(\alpha ,{\alpha }^{\ast }\right)\left|\beta \right.〉h\left(\alpha ,{\alpha }^{\ast }\right),$$

(17)

where $h\left(\alpha ,{\alpha }^{\ast }\right)$ denotes the classical counterpart of the operator $H\left(a,a^{†}\right)$, and $\left|\beta \right.〉$ represents the coherent state

$$\left|\beta \right.〉=\exp \left(-{\displaystyle \frac{|\beta {|}^2}{2}}+\beta a^{†}\right)\left|0\right.〉,$$

(18)

and the inner product of coherent state $\left|\beta \right.〉$ is

$$\left.〈-\beta \right|\beta 〉=\exp \left(-2|\beta {|}^2\right).$$

(19)

Substituting the normal product ordering Equation (16) for the operator integral kernel $\mathrm{\Delta }\left(\alpha \right)$ into Equation (17) yields

$$\begin{array}{l}\left.〈-\beta \right|H\left(a,a^{†}\right)\left|\beta \right.〉=2{\displaystyle \int }{d}^2\alpha h\left(\alpha ,{\alpha }^{\ast }\right)\left.〈-\beta \right|{\displaystyle \frac{1}{\pi }}:e^{-2(a^{†}-{\alpha }^{\ast })(a-\alpha )}:\left|\beta \right.〉\\ =2{\displaystyle \int \frac{d^2\alpha }{\pi }h\left(\alpha ,{\alpha }^{\ast }\right)\exp \left(2\beta {\alpha }^{\ast }-2\alpha {\beta }^{\ast }-2|\alpha {|}^2\right)}\begin{array}{c}\end{array},\end{array}$$

(20)

Notably, $2\beta {\alpha }^{\ast }-2\alpha {\beta }^{\ast }$ in the exponential term on the right-hand side of Equation (20) is a purely imaginary number. Hence, ${\displaystyle \frac{1}{2\pi }}\left.〈-\beta \right|H\left(a,a^{†}\right)\left|\beta \right.〉$ can be regarded as the Fourier transform of $h\left(\alpha ,{\alpha }^{\ast }\right)e^{-2|\alpha {|}^2}$

$$h\left(\alpha ,{\alpha }^{\ast }\right)=2{\displaystyle \int \frac{d^2\beta }{\pi }\left.〈-\beta \right|H\left(a,a^{†}\right)\left|\beta \right.〉\exp \left(2|\alpha {|}^2-2\beta {\alpha }^{\ast }+2\alpha {\beta }^{\ast }\right)}\begin{array}{c}\end{array},$$

(21)

where $h\left(\alpha ,{\alpha }^{\ast }\right)$ is an arbitrary function. Following the Weyl ordering correspondence rule, it can be further derived that

$$\begin{array}{l}H\left(a,a^{†}\right)=2{\displaystyle \int d^2}\alpha h\left(\alpha ,{\alpha }^{\ast }\right)\mathrm{\Delta }\left(\alpha \right)\\ =2{\displaystyle \int d^2}\alpha 2{\displaystyle \int \frac{d^2\beta }{\pi }}\left.〈-\beta \right|H\left(a,a^{†}\right)\left|\beta \right.〉\exp \left(2|\alpha {|}^2-2\beta a^{†}+2a{\beta }^{\ast }\right)\\ \times {\displaystyle \frac{1}{2}}\delta \left(a^{†}-{\alpha }^{\ast }\right)\delta \left(a-\alpha \right)\\ =\begin{array}{c}2{\displaystyle \int \frac{d^2\beta }{\pi }\left.〈-\beta \right|H\left(a,a^{†}\right)\left|\beta \right.〉\exp \left(2|\alpha {|}^2-2\beta a^{†}+2a{\beta }^{\ast }\right)}\end{array}\begin{array}{c}\end{array}.\end{array}$$

(22)

Equation (22) serves as the general integral formula for transforming any quantum operator into Weyl ordering form. From Equation (22), it is evident that once the normal ordering form of operator $H\left(a,a^{†}\right)$ is solved, the matrix element $\left.〈-\beta \right|H\left(a,a^{†}\right)\left|\beta \right.〉$ in the coherent state representation can be further evaluated. Subsequently, using Equation (22), its corresponding Weyl ordering form can be derived. For example,

$$\begin{array}{l}\left|0\right.〉\left.〈0\right|=\begin{array}{c}:\\ :\end{array}2{\displaystyle \int \frac{d^2\beta }{\pi }\left.〈-\beta \right|0〉〈0\left|\beta \right.〉\exp \left(2|\alpha {|}^2-2\beta a^{†}+2a^{†}a\right)}\begin{array}{c}:\\ :\end{array}\\ =\begin{array}{c}:\\ :\end{array}\begin{array}{c}2{\displaystyle \int \frac{d^2\beta }{\pi }\exp \left(-|\beta {|}^2+2|\alpha {|}^2-2\beta a^{†}+2a^{†}a\right)}\end{array}\begin{array}{c}:\\ :\end{array}\\ =2\begin{array}{c}:\\ :\end{array}\begin{array}{c}{e}^{-2a^{†}a}\end{array}\begin{array}{c}:\\ :\end{array}\begin{array}{c}\end{array}.\end{array}$$

(23)

Equation (23) presents the Weyl ordering form of the vacuum projection operator $\left|0\right.〉\left.〈0\right|$.

4. Weyl Ordering Form of the Single-Mode Squeezing Operator

By utilizing the integral transformation relationship established in Section 3 for transforming any quantum operator into Weyl ordering form, the single-mode squeezed light density operator $\rho$ can be transformed into its corresponding Weyl ordering form:

$$\begin{array}{l}\rho =\mathrm{sech}\lambda e^{{\displaystyle \frac{\tanh \lambda }{2}}{a}^{†2}}\left|0\right.〉\left.〈0\right|e^{{\displaystyle \frac{\tanh \lambda }{2}}{a}^2}\\ =2\mathrm{sech}\lambda {\displaystyle \int \frac{d^2\gamma }{\pi }\begin{array}{c}:\\ :\end{array}\left.〈-\gamma \right|e^{\frac{\tanh \lambda }{2}{a}^{†2}}\left|0\right.〉\left.〈0\right|e^{\frac{\tanh \lambda }{2}{a}^2}\left|\gamma \right.〉\exp \left[2\left({\gamma }^{\ast }a-\gamma a^{†}+a^{†}a\right)\right]}\begin{array}{c}:\\ :\end{array}\\ =2\mathrm{sech}\lambda {\displaystyle \int \frac{d^2\gamma }{\pi }\begin{array}{c}:\\ :\end{array}\begin{array}{c}\exp \left[-|\gamma {|}^2+\frac{\tanh \lambda }{2}\left({\gamma }^{\ast 2}+{\gamma }^2\right)+2\left({\gamma }^{\ast }a-\gamma a^{†}+a^{†}a\right)\right]\begin{array}{c}:\\ :\end{array}\end{array}}\\ =2\begin{array}{c}:\\ :\end{array}\begin{array}{c}\exp \left[{{\displaystyle \cosh }}^2\lambda (-4a^{†}a+2\tanh \lambda \left(a^{†2}+a^2\right)+2a^{†}a\right]\end{array}\begin{array}{c}:\\ :\end{array}\\ =2\begin{array}{c}:\\ :\end{array}\begin{array}{c}\exp \left[\left(a^{†2}+a^2\right)\sinh 2\lambda -2a^{†}a\cosh 2\lambda \right]\end{array}\begin{array}{c}:\\ :\end{array}\begin{array}{c}\end{array},\end{array}$$

(24)

Equation (24) mathematically illustrates the classical phase-space structure corresponding to this quantum state. It offers an essential analytical expression for validating its correspondence with classical scale transformations and deepening the understanding of Bohr’s correspondence principle, laying the theoretical groundwork for subsequent analysis of quantum-classical transformation correspondences.

Then, using $P,Q$ as defined in Equation (4), Equation (24) can be rewritten as

$$\begin{array}{l}\rho =\mathrm{sech}\lambda e^{{\displaystyle \frac{\tanh \lambda }{2}}{a}^{†2}}\left|0\right.〉\left.〈0\right|e^{{\displaystyle \frac{\tanh \lambda }{2}}{a}^2}\\ =2\begin{array}{c}:\\ :\end{array}\begin{array}{c}\exp \left\{{\displaystyle \frac{1}{2}}\left[{\left(Q-iP\right)}^2+{\left(Q+iP\right)}^2\right]\sinh 2\lambda -\left(Q^2+P^2\right)\cosh 2\lambda \right\}\end{array}\begin{array}{c}:\\ :\end{array}\\ =2\begin{array}{c}:\\ :\end{array}\exp \left\{\left(Q^2-P^2\right)\sinh 2\lambda -\left(Q^2+P^2\right)\cosh 2\lambda \right\}\begin{array}{c}:\\ :\end{array}\\ =2\begin{array}{c}:\\ :\end{array}\exp \left[-Q^2{e}^{-2\lambda }-P^2{e}^{2\lambda }\right]\begin{array}{c}:\\ :\end{array}\begin{array}{c}\end{array},\end{array}$$

(25)

This is the Weyl ordering form of the classical single-mode squeezing operator $\rho$. Thus, its Weyl correspondence is given by

$$2\exp \left[-q^2{e}^{-2\lambda }-p^2{e}^{2\lambda }\right],$$

(26)

That is, the following relationship exists:

$$\mathrm{sech}\lambda e^{{\displaystyle \frac{\tanh \lambda }{2}}{a}^{†2}}\left|0\right.〉\left.〈0\right|e^{{\displaystyle \frac{\tanh \lambda }{2}}{a}^2}\to 2\exp \left[-q^2{e}^{-2\lambda }-p^2{e}^{2\lambda }\right].$$

(27)

From Equations (26) and (27), it is evident that the squeezing mechanism corresponds to the coordinate-space scale transformation $e^{-2\lambda }$ and the momentum-space inverse scale transformation $e^{2\lambda }$.

5. Weyl Ordering Form of the Two-Mode Squeezing Operator

In this section, the Weyl ordering form of the two-mode squeezing operator will be systematically studied. The two-mode squeezed light field acts as an essential resource in quantum information processing (such as continuous-variable quantum entanglement and quantum teleportation). The classical correspondence of its density operator is crucial for comprehending the connection between its macroscopic quantum behavior and its classical limit. The density operator ${\rho }_2$ of two-mode squeezed light is

$${\rho }_2={\mathrm{sech}}^2\lambda e^{a^{†}{b}^{†}\tanh \lambda }\left|00\right.〉\left.〈00\right|e^{ab\tanh \lambda },$$

(28)

Since the classical Weyl counterpart function of the Weyl ordering operator $\begin{array}{c}:\\ :\end{array}\begin{array}{c}h(a^{†},a;b^{†},b)\end{array}\begin{array}{c}:\\ :\end{array}$ in phase space $(\alpha ,{\alpha }^{\ast };\beta ,{\beta }^{\ast })$ is $h(\alpha ,{\alpha }^{\ast };\beta ,{\beta }^{\ast })$, there is

$$\begin{array}{c}:\\ :\end{array}\begin{array}{c}h(a^{†},a;b^{†},b)\end{array}\begin{array}{c}:\\ :\end{array}=4{\displaystyle \int d^2\alpha }{\displaystyle \int d^2\beta }\mathrm{\Delta }(\alpha ,{\alpha }^{\ast };\beta ,{\beta }^{\ast })h(\alpha ,{\alpha }^{\ast };\beta ,{\beta }^{\ast })\begin{array}{c}\end{array},$$

(29)

and, using the two-mode Wigner operator $\mathrm{\Delta }(\alpha ,{\alpha }^{\ast };\beta ,{\beta }^{\ast })$, we have

$$\mathrm{\Delta }(\alpha ,{\alpha }^{\ast };\beta ,{\beta }^{\ast }){\displaystyle \frac{1}{{\pi }^2}}:\exp \left[-2\left(a^{†}-{\alpha }^{\ast }\right)\left(a-\alpha \right)-2\left(b^{†}-{\beta }^{\ast }\right)\left(b-\beta \right)\right]:\begin{array}{c}\end{array},$$

(30)

The normal product ordering form of the density operator ${\rho }_2$ for two-mode squeezed light can be derived as follows:

$$\begin{array}{l}{\rho }_2={\mathrm{sech}}^2\lambda {\displaystyle \int \frac{d^2{\gamma }_1{d}^2{\gamma }_2}{{\pi }^2}}\left.〈-{\gamma }_1,-{\gamma }_2\right|e^{a^{†}{b}^{†}\tanh \lambda }\left|00\right.〉\left.〈00\right|e^{ab\tanh \lambda }\left|{\gamma }_1,{\gamma }_2\right.〉\\ \times \begin{array}{c}:\\ :\end{array}\begin{array}{c}\exp \left[2\left({\gamma }_1^{\ast }a-{\gamma }_1{a}^{†}+a^{†}a\right)+2\left({\gamma }_2^{\ast }b-{\gamma }_2{b}^{†}+b^{†}b\right)\right]\end{array}\begin{array}{c}:\\ :\end{array}\\ ={\mathrm{sech}}^2\lambda {\displaystyle \int \frac{d^2{\gamma }_1{d}^2{\gamma }_2}{{\pi }^2}}\\ \times \begin{array}{c}:\\ :\end{array}\begin{array}{c}\exp \left\{\left({\gamma }_1^{\ast }{\gamma }_2^{\ast }+{\gamma }_1{\gamma }_2\right)\tanh \lambda -|{\gamma }_1{|}^2-|{\gamma }_2{|}^2+2\left({\gamma }_1^{\ast }a-{\gamma }_1{a}^{†}+a^{†}a\right)+2\left({\gamma }_2^{\ast }b-{\gamma }_2{b}^{†}+b^{†}b\right)\right\}\end{array}\begin{array}{c}:\\ :\end{array}\\ =\begin{array}{c}:\\ :\end{array}\exp \left\{2\left(a^{†}{b}^{†}+ab\right)\sinh 2\lambda -2\left(b^{†}b+a^{†}a\right)\cosh 2\lambda \right\}\begin{array}{c}:\\ :\end{array}\begin{array}{c}\end{array}.\end{array}$$

(31)

Since

$$\left(Q_1-i{P}_1\right)\left(Q_2+i{P}_2\right)+\left(Q_1+i{P}_1\right)\left(Q_2-i{P}_2\right)=2\left(Q_1{Q}_2+P_1{P}_2\right),$$

(32)

$$\left(b^{†}b+a^{†}a\right)=\left(Q_1^2+P_1^2\right)+\left(Q_2^2+P_2^2\right),$$

(33)

Here, $Q_1$ and $Q_2$ are the two-mode coordinate operators, and $P_1$ and $P_2$ are the two-mode momentum operators. We have

$${\rho }_2=\begin{array}{c}:\\ :\end{array}\begin{array}{c}\exp \left\{4\left(q_1{q}_2+p_1{p}_2\right)\left({\displaystyle \frac{e^{2\lambda }+e^{-2\lambda }}{2}}\right)-\left[\left(q_1^2+p_1^2\right)+\left(q_2^2+p_2^2\right)\right]\left({\displaystyle \frac{e^{2\lambda }-e^{-2\lambda }}{2}}\right)\right\}\end{array}\begin{array}{c}:\\ :\end{array},$$

(34)

and the corresponding classical relationship is

$$\begin{array}{l}4\left(q_1{q}_2+p_1{p}_2\right)\left({\displaystyle \frac{e^{2\lambda }+e^{-2\lambda }}{2}}\right)-\left[\left(q_1^2+p_1^2\right)+\left(q_2^2+p_2^2\right)\right]\left({\displaystyle \frac{e^{2\lambda }-e^{-2\lambda }}{2}}\right)\\ ={\left[\left(q_1-q_2\right)e^{\lambda }\right]}^2+{\left[\left(p_1-p_2\right)e^{-\lambda }\right]}^2.\end{array}$$

(35)

where $q_1$ and $q_2$ are the two-mode coordinate, and $p_1$ and $p_2$ are the two-mode momentum. Equation (35) articulates mathematically that the physical mechanism of two-mode squeezing is a squeezed transformation $e^{2\lambda }$ in coordinate space $\left(q_1-q_2\right)$ and an inverse squeezed transformation $e^{-2\lambda }$ in the conjugate momentum space $\left(p_1-p_2\right)$. This transformation rigorously maintains the invariance of phase-space volume and serves as a concrete example of classical canonical transformations in the quantum domain. It offers evidence for Bohr’s correspondence principle by demonstrating how a multi-particle quantum-correlated system corresponds to a classically coupled transformation.

6. Applications of the Classical Correspondence of Single-Mode Squeezing

By directly applying the classical correspondence of the single-mode squeezing operator, we can determine the photon number distribution in a single-mode squeezed light field. Its quantum operator expression is given by:

$$\mathrm{Tr}\left(\mathrm{sech}\lambda e^{{\displaystyle \frac{\tanh \lambda }{2}}{a}^{†2}}\left|0\right.〉\left.〈0\right|e^{{\displaystyle \frac{\tanh \lambda }{2}}{a}^2}{a}^{†}a\right),$$

(36)

Since the classical correspondence of photon number operator $a^{†}a$ is $\left(q_1^2+p_1^2\right)/2$, and given the following relationship:

$${\displaystyle \int {\displaystyle {\int }_{-\infty }^{\infty }\frac{d{p}_{1}d{q}_1}{\pi }}}\exp \left[-q_1^2{e}^{-2\lambda }-p_1^2{e}^{2\lambda }\right]=1,$$

(37)

The solution of Equation (35) can be changed to solving

$$\begin{array}{l}2{\displaystyle \int {\displaystyle {\int }_{-\infty }^{\infty }\frac{d{p}_{1}d{q}_1}{\pi }}}{\displaystyle \frac{\left(q_1^2+p_1^2\right)}{2}}\exp \left[-q_1^2{e}^{-2\lambda }-p_1^2{e}^{2\lambda }\right]\\ ={\displaystyle {\int }_{-\infty }^{\infty }\frac{d{p}_1}{\sqrt{\pi }}}{e}^{-p_1^2{e}^{2\lambda }}{\displaystyle {\int }_{-\infty }^{\infty }\frac{d{q}_1}{\sqrt{\pi }}}{q}_1^2{e}^{-q_1^2{e}^{-2\lambda }}+{\displaystyle {\int }_{-\infty }^{\infty }\frac{d{q}_1}{\sqrt{\pi }}}{e}^{-q_1^2{e}^{-2\lambda }}{\displaystyle {\int }_{-\infty }^{\infty }\frac{d{p}_1}{\sqrt{\pi }}{p}_1^2}{e}^{-p_1^2{e}^{2\lambda }}\\ =e^{-\lambda }{\displaystyle {\int }_{-\infty }^{\infty }\frac{d{q}_1}{\sqrt{\pi }}}{q}_1^2{e}^{-q_1^2{e}^{-2\lambda }}+e^{\lambda }{\displaystyle {\int }_{-\infty }^{\infty }\frac{d{p}_1}{\sqrt{\pi }}{p}_1^2}{e}^{-p_1^2{e}^{2\lambda }}\\ ={\displaystyle \frac{1}{2}}\left(e^{2\lambda }+e^{-2\lambda }\right)\\ =\cosh 2\lambda .\end{array}$$

(38)

Since $\lambda$ is the squeezing parameter, Equation (38) clearly illustrates the function of squeezing. Through our calculations, we have not only shown how to derive quantum statistical information (the photon number distribution) from the classical counterpart function, but more importantly, we have also confirmed the correctness and practical relevance of the classical counterpart function. That is, the complete information of the quantum state is fully encapsulated in its classical phase-space function. This further reinforces the fundamental argument of the “correspondence between quantum transformations and classical transformations”.

7. Conclusions

By using the integration method within ordered operators, the coherent state representation, and integration theory within Weyl ordering, this paper systematically derives the classical phase-space functions corresponding to the density operators of single-mode and two-mode squeezed light fields. This addresses the critical issue of “whether the squeezing operator possesses a classical correspondence function that reflects scaling transformations.” The findings indicate that the quantum unitary transformation realized by the squeezing operator directly corresponds to a canonical scaling transformation at the classical level. This result verifies that Bohr’s correspondence principle can be extended to incorporate the deeper connotation of “mutual correspondence between classical transformations and quantum transformations.” The research theoretically elucidates that any classical canonical transformation can be represented by a corresponding unitary operator in quantum mechanics. This “correspondence of transformations” can be considered a vital component of Bohr’s correspondence principle. This conclusion not only significantly expands the theoretical connotation of Bohr’s correspondence principle by introducing the new dimension of “transformation correspondence” but also establishes a rigorous mapping between quantum operators and classical functions, offering new analytical tools and conceptual frameworks for laying the mathematical foundations of quantum optics and quantum statistics.