$, which leads to the Gaussian entangled state given in Eq.(\[gau03\]). Another interesting case is for $m = 0$ while $n$ is allowed to take all integer values. This single-excitation system has applications in the covariant oscillator formalism where no time-like excitations are allowed. The Gaussian entangled state is a special case of this single-excited oscillator system.

The most general case is for nonzero integers for both $n$ and $m$. The calculation for this case is available in the literature [@knp86; @rotbart81]. Seeing no immediate physical applications of this case, we shall not reproduce this calculation in this section.

For the single-excitation system, we write the starting wave function as $$\label{wf12}
     \chi_{n}(x)\chi_{0}(y) = \left[\frac{1}{\pi~2^n n!}\right]^{1/2}
              H_{n}(x) \exp{\left\{-\left(\frac{x^2 + y^2 }{2}\right)\right\}} .$$ There are no excitations along the $y$ coordinate. In order to squeeze this function, our plan is to replace $x$ and $y$ by $x'$ and $y'$ respectively and write $\chi_{n}(x')\chi_{0}(y')$ as a series in the form $$\chi_{n}(x')\chi_{0}(y') = \sum_{k',k} A_{k',k}(n)  \chi_{k'}(x)\chi_{k}(y) .$$ Since $k' - k = n$ or $ k' = n + k$, according to the eigenvalue of the differential equation given in Eq.(\[cov01\]), we write this series as $$\chi_{n}(x')\chi_{0}(y') = \sum_{k',k} A_{k}(n)  \chi_{(k + n)}(x)\chi_{k}(y) ,$$ 