Abstract
We have investigated the eigenenergy spectrum of the two-photon Rabi model at the critical coupling, particularly the special feature “spectral collapse”, by means of an elementary quantum mechanics approach. The eigenenergy spectrum is found to consist of both a set of discrete energy levels and a continuous energy spectrum. Each of these eigenenergies has a two-fold degeneracy corresponding to the spin degree of freedom. The discrete eigenenergy spectrum has a one-to-one mapping with that of a particle in a “Lorentzian function” potential well, and the continuous energy spectrum can be derived from the scattering problem associated with a potential barrier. The number of bound states available at the critical coupling is determined by the energy difference between the two atomic levels so that the extent of the “spectral collapse” can be monitored in a straightforward manner.
Highlights
We have investigated the eigenenergy spectrum of the two-photon Rabi model at the critical coupling, the special feature “spectral collapse”, by means of an elementary quantum mechanics approach
The discrete eigenenergy spectrum has a one-to-one mapping with that of a particle in a “Lorentzian function” potential well whose eigenspectrum can be determined by an elementary quantum mechanics approach and the continuous energy spectrum can be derived from the scattering problem associated with a potential barrier
The disappearance of spin eigenvalues in Eq (22) implies that each eigenstate is doubly degenerate. In this communication we have shown that at the critical coupling ǫc the eigenenergy spectrum of the two-photon Rabi model consists of both a set of discrete energy levels and a continuous energy spectrum, and that each of these eigenenergies has a two-fold degeneracy corresponding to the spin degree of freedom
Summary
We have investigated the eigenenergy spectrum of the two-photon Rabi model at the critical coupling, the special feature “spectral collapse”, by means of an elementary quantum mechanics approach. In 1998, via exact numerical diagonalization, Ng et al.[27] first demonstrated that while the quantum two-photon Rabi model has a discrete eigenenergy spectrum for ǫ < ǫc , no normalizable eigenstate exists in the Hilbert space spanned by the photon number states for ǫ > ǫc .
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