Classical Weyl correspondences of some two-mode quantum operators and their application

Jian-ming Du, Jian-guo Ma, Gang Ren

Weifeng Wu, Hong-Yi Fan

Habib Rebei

Ke Zhang, H. C. Fan

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.

Chuanmei Xie, Hong-Yi Fan

Based on the Fan—Hu's formalism, i.e., the tomogram of two-mode quantum states can be considered as the module square of the states' wave function in the intermediate representation, which is just the eigenvector of the Fresnel quadrature phase, we derive a new theorem for calculating the quantum tomogram of two-mode density operators, i.e., the tomogram of a two-mode density operator is equal to the marginal integration of the classical Weyl correspondence function of F 2 † ρ F 2 , where F 2 is the two-mode Fresnel operator. An application of the theorem in evaluating the tomogram of an optical chaotic field is also presented.

展德会 ZHAN De-hui, 卢道明 Lu Daoming, 范洪义 FAN Hong-yi