Phase-Space Evolution of Squeezing-Enhanced Light and Its Attenuation

As physicists continue to find approaches to achieve quantum squeezing enhancement, a key challenge is to identify the specific parameters of the squeezing operator and their boundary conditions. This paper employs the method of integration within an ordered product of operator (IWOP) in quantum mechanics to solve this problem for two independent parameters. First, the \(q - p\) phase space correspondence for enhanced squeezing is investigated, where \(q,p\) represent the coordinate and momentum, respectively. Then, the squeezing-enhanced state is theoretically obtained by finding the generalized squeezing operator \(S\left( {\lambda ,r} \right) = \exp \left( { - \frac{v}{{2{u^* }}}{a^{\dag 2}}} \right)\exp [\left( {{a^\dag }a + \frac{1}{2}} \right)\ln \frac{1}{{u^* }}]\exp \left( {\frac{{v^* }}{{2{u^* }}}{a^2}} \right)\). It is demonstrated that the characteristic of the phase space transformation is \(p \to p\cosh \lambda - q\sinh \lambda {e^r},\) \(q \to q\cosh \lambda - p{e^{ - r}}\sinh \lambda ,\) \(u = \cosh \lambda - i\sinh \lambda \sinh r\). Here, \(\lambda ,r\) are two independent parameters, and they must satisfy \(\tanh \lambda \left( {{{\cosh }^2}r - 1} \right) < \cosh r - 1\) to achieve enhanced squeezing of the quadrature operator. Further, the integral solution of the dissipation master equation is adopted to analyze the attenuation of this type of squeezed field. Meanwhile, the paper presents the canonical decomposition of the enhanced squeezing operator \(S\left( {\lambda ,r} \right)\). Our research provides physicists with more refined insights to enhance the squeezing effect with more precision.