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Chinese Physics B

New light fields based on integration theory within the Weyl ordering product of operators

Ke Zhang, Lanlan Li, Da-wei Guo, Hong-Yi Fan

Published Jul 5, 2022• 4 citations
Publisher

Abstract

The development of quantum optics theory based on the method of integration within an ordered product of operators (IWOP) has greatly stimulated the study of quantum states in the light field, especially non-Gaussian states with various non-classical properties. In this paper, the two-mode squeezing operator is derived with integral theory within the Weyl ordering product of operators using a combinatorial field in which one mode is a chaotic field and the other mode is a vacuum field. The density operator of the new light field, its entanglement property and photon number distribution are analyzed. We also note that tracing a three-mode pure state can yield this new light field. These methods represent a theoretical approach to investigating new density operators of light fields.

DOI
10.1088/1674-1056/ac7e34

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The development of quantum optics theory based on the method of integration within an ordered product of operators (IWOP) has greatly stimulated the study of quantum states in the light field, especially non-Gaussian states with various non-classical properties. In this paper, the two-mode squeezing operator is derived with integral theory within the Weyl ordering product of operators using a combinatorial field in which one mode is a chaotic field and the other mode is a vacuum field. The density operator of the new light field, its entanglement property and photon number distribution are analyzed. We also note that tracing a three-mode pure state can yield this new light field. These methods represent a theoretical approach to investigating new density operators of light fields.
For the density operator(mixed state) describing squeezed chaotic light(SCL) we search for its thermal vacuum state(a pure state) in the real-fictitious space. Using the method of integration within ordered product(IWOP) of operators we find that it is a kind of one- and two-mode combinatorial squeezed state. Its application in evaluating the quantum fluctuation of photon number reveals: the stronger the squeezing is, the larger a fluctuation appears. The second-order degree of coherence of SCL is also deduced which shows that SCL is classic. The new thermal vacuum state also helps to derive the Wigner function of SCL.
For the density operator (mixed state) describing squeezed chaotic light (SCL) we search for its thermal vacuum state (a pure state) in the real-fictitious space. Using the method of integration within ordered product (IWOP) of operators we find that it is a kind of one- and two-mode combinatorial squeezed state. Its application in evaluating the quantum fluctuation of photon number reveals: the stronger the squeezing is, the larger a fluctuation appears. The second-order degree of coherence of SCL is also deduced which shows that SCL is classic. The new thermal vacuum state also helps to derive the Wigner function of SCL.
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.
We propose a new two-mode thermo- and squeezing-mixed optical field, described by the new density operator ρ = ( 1 − e f − | g | 2 ) e g a † b † e f a † a | 0 ⟩ b b ⟨ 0 | e g ∗ a b , where | 0 ⟩ b b ⟨ 0 | is the b -mode vacuum, e f a † a represents the thermo-field, and e g a † b † indicates squeezing. The photon statistics for ρ is studied by virtue of the method of integration within ordered product (IWOP) of operators. Such a field can be generated when a two-mode squeezed state passes through a one-mode dissipation channel.
We present a general unified approach for arranging quantum operators of optical fields into ordered products (normal ordering, antinormal ordering, Weyl ordering (or symmetric ordering)) by fashioning Dirac's symbolic method and representation theory. We propose the technique of integration within an ordered product (IWOP) of operators to realize our goal. The IWOP makes Dirac's representation theory and the symbolic method more transparent and consequently more easily understood. The beauty of Dirac's symbolic method is further revealed. Various applications of the IWOP technique, such as in developing the entangled state representation theory, nonlinear coherent state theory, Wigner function theory, etc, are presented.
By the newly developed technique of integration within an ordered product (IWOP) of operators, we explore quantum optical version of classical optical transformations such as optical Fresnel transform, Hankel transform, fractional Fourier transform, Wigner transform, wavelet transform and Fresnel-Hadmard combinatorial transform etc. In this way one may gain benefit for developing classical optics theory from the research in quantum optics, or vice-versa. We can not only find some new quantum mechanical unitary operators which correspond to the known optical transformations, deriving a new theorem for calculating quantum tomogram of density operators, but also can reveal some new classical optical transformations. We derive GFO's normal product form and its canonical coherent state representation and find that GFO is the loyal representation of symplectic group multiplication rule. We show that GFT is just the transformation matrix element of GFO in the coordinate representation such that two successive GFTs is still a GFT. The ABCD rule of the Gaussian beam propagation is directly demonstrated in the context of quantum optics. Especially, the introduction of quantum mechanical entangled state representations opens up a new area to finding new classical optical transformations. The complex wavelet transform and the condition of mother wavelet are studied in the context of quantum optics too. Throughout our discussions, the coherent state, the entangled state representation of the two-mode squeezing operators and the IWOP technique are fully used. All these confirms Dirac's assertion: " ... for a quantum dynamic system that has a classical analogue, unitary transformation in the quantum theory is the analogue of contact transformation in the classical theory".