(\cosh\eta) x -  (\sinh\eta) y, \quad\mbox{and}\quad
  y'=  ( \cosh\eta) y  - (\sinh\eta) x .$$ This form of squeeze is illustrated in Fig. \[sqz00\], and the expansion of this squeezed Gaussian function becomes the series given in Eq.(\[seri00\]) [@knp86; @kno79ajp]. This aspect will be discussed in detail in Sec. \[excited\].

![Transformations in the two-dimensional space. The object can be rotated, squeezed, or sheared. In all three cases, the area remains invariant. []{data-label="ctran11"}](ctran11b.eps)

![Squeeze along the $45^o$ direction, discussed most frequently in the literature.[]{data-label="sqz00"}](sqz00b.eps)

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In 1976 [@yuen76], Yuen discussed two-photon coherent states, often called squeezed states of light. This series expansion served as the starting point for two-mode squeezed states. More recently, in 2003, Giedke [*et al.*]{} [@gied03] used this formula to formulate the concept of the Gaussian entanglement.

There is another way to derive the series. For the harmonic oscillator wave functions, there are step-down and step-up operators [@dir45]. These are defined as $$a = \frac{1}{\sqrt{2}}\left(x + \frac{\partial}{\partial x} \right),
              \quad\mbox{and}\quad
  a^{\dag} = \frac{1}{\sqrt{2}} \left(x - \frac{\partial}{\partial x}\right) .$$ If they are applied to the oscillator wave function, we have $$a~\chi_n(x) = \sqrt{n}~\chi_{n-1}(x),  \quad\mbox{and}\quad
 a^{\dag}~\chi_n(x) = \sqrt{n+1}~ \chi_{n + 1}(x) .$$ Likewise, we can introduce $b$ and $b^{\dag} $ operators 