Classication of Squeezing Transformations and Squeezed States

Coordinate wave functions for the one-mode squeezed states produced by the quantum analog of the general linear transformation in phase space are calculated. The probability density [\ensuremath{\Vert}\ensuremath{\psi}(q)${\ensuremath{\Vert}}^{2}$] for these states is Gaussian with center predicted by the classical transformation. The quantum image (which includes the traditional two-mode squeeze operator) of a three-parameter symplectic map in two-mode phase space equally generates squeezed states having Gaussian \ensuremath{\Vert}\ensuremath{\psi}( ${q}_{1}$ , ${q}_{1}$ )${\ensuremath{\Vert}}^{2}$. The center of the two-mode Gaussian is again predicted by the classical mapping.Coordinate wave functions for the one-mode squeezed states produced by the quantum analog of the general linear transformation in phase space are calculated. The probability density [\ensuremath{\Vert}\ensuremath{\psi}(q)${\ensuremath{\Vert}}^{2}$] for these states is Gaussian with center predicted by the classical transformation. The quantum image (which includes the traditional two-mode squeeze operator) of a three-parameter symplectic map in two-mode phase space equally generates squeezed states having Gaussian \ensuremath{\Vert}\ensuremath{\psi}(${q}_{1}$,${q}_{1}$)${\ensuremath{\Vert}}^{2}$. The center of the two-mode Gaussian is again predicted by the classical mapping.

Physical Review AOct 1, 198919 citations

An Introduction to Squeezed States

John R. Klauder

Springer eBooksJan 1, 19861 citation

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

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.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.

PhotonicsApr 9, 20260 citations

Squeezed States: Basic Principles

P. L. Knight, Vladimír Bužek

Springer series on atomic, optical, and plasma physicsJan 1, 200414 citations

A note on squeezed states

I. A. Pedrosa, B. Baseia

Jan 1, 19860 citations

Linear canonical transformations of coherent and squeezed states in the Wigner phase space

D. Han, Y. S. Kim, Marilyn E. Noz

It is shown that classical linear canonical transformations are possible in the Wigner phase space. Coherent and squeezed states are shown to be linear canonical transforms of the ground-state harmonic oscillator. It is therefore possible to evaluate the Wigner functions for coherent and squeezed states from that for the harmonic oscillator. Since the group of linear canonical transformations has a subgroup whose algebraic property is the same as that of the (2+1)-dimensional Lorentz group, it may be possible to test certain properties of the Lorentz group using optical devices. A possible experiment to measure the Wigner rotation angle is discussed.It is shown that classical linear canonical transformations are possible in the Wigner phase space. Coherent and squeezed states are shown to be linear canonical transforms of the ground-state harmonic oscillator. It is therefore possible to evaluate the Wigner functions for coherent and squeezed states from that for the harmonic oscillator. Since the group of linear canonical transformations has a subgroup whose algebraic property is the same as that of the (2+1)-dimensional Lorentz group, it may be possible to test certain properties of the Lorentz group using optical devices. A possible experiment to measure the Wigner rotation angle is discussed.

Physical Review AFeb 1, 198882 citations

{q}_{1}

{q}_{1}

)${\ensuremath{\Vert}}^{2}$. The center of the two-mode Gaussian is again predicted by the classical mapping.