  This paper presents a proof of the existence of novel bound states of the
two-photon quantum Rabi model at the collapse point. The two-photon Rabi model
is interesting not only for its important role on non-linear light-matter
interaction, but also for the exhibition of many-energy-levels degenerating
process called the "spectral collapse". The squeezing property of the
two-photon annihilation and creation operators is the origin for this
phenomenon which is well studied without the energy-slitting term $\omega_0$.
However, many numerical studies have pointed out that with the presence of
$\omega_0$ , some low-level isolated states exist while other high energy
states collapse to $E=-\frac{\omega}{2}$, which known as incomplete spectral
collapse. From the eigenvalue equation in real space, pair of second order
differential equations, which are similarly to the Schrodinger equation, are
derived at the collapse point. These differential equations provide explanation
to the existence of isolated bound states below $E=-\frac{\omega}{2}$ with the
presence of the spin slitting $\omega_0$ and better numerical method to
generate those bound states.

Quantum Physics