\right) ,$$ generates the same shear in the $p~q$ space.

Entangled Excited States {#excited}
========================

In Sec. \[2dim\], we discussed the entangled ground state, and noted that the entangled state of Eq.(\[seri00\]) is a series expansion of the squeezed Gaussian function. In this section, we are interested in what happens when we squeeze an excited oscillator state starting from $$\label{wf10}
\chi_{n}(x)\chi_{m}(y) .$$ In order to entangle this state, we should replace $x$ and $y$ respectively by $x'$ and $y'$ given in Eq.(\[trans22\]).

The question is how the oscillator wave function is squeezed after this operation. Let us note first that the wave function of Eq.(\[wf10\]) satisfies the equation $$\label{cov01}
\frac{1}{2}\left\{\left(x^2 - \frac{\partial^2}{\partial x^2}\right)
 - \left(y^2 - \frac{\partial^2}{\partial y^2}\right)\right\}
 \chi_n(x) \chi_m(y) = (n - m) \chi_n(x) \chi_m(y) .$$ This equation is invariant under the squeeze transformation of Eq.(\[trans22\]), and thus the eigenvalue $(n - m)$ remains invariant. Unlike the usual two-oscillator system, the $x$ component and the $y$ component have opposite signs. This is the reason why the overall equation is squeeze-invariant [@kno79jmp; @kn05job; @fkr71].

We then have to write this squeezed oscillator in the series form of Eq.(\[seri22\]). The most interesting case is of course for $m = n =0 