  We investigate the strong coupling regime of a linear $x$-$x$ coupled
harmonic oscillator system, by performing a direct diagonalization of the
hamiltonian. It is shown that the $x$-$x$ coupled hamiltonian can be
equivalently described by a Mach-Zehnder-type interferometer with a quadratic
unitary operation in each of its arms. We show a sharp transition of the
unitary operation from an elliptical phase rotator to an elliptical squeezer as
the coupling gets stronger, which leads to the continuous generation of
entanglement, even for a significantly thermal state, in the ultra-strong
coupled regime. It is also shown that this critical regime cannot be achieved
by a classical Hookian coupling. Finally, the effect of a finite-temperature
environment is analyzed, showing that entanglement can still be generated from
a thermal state in the ultra-strong coupled regime, but is destroyed rapidly.
