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.