Demystifying the spectral collapse in two-photon Rabi model

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.

Spectral collapse in multiqubit two-photon Rabi model

“…This is in sharp constrast to the findings reported in Ref. 24 , which explicitly shows that the discrete eigenenergy spectrum has a one-to-one mapping with that of a particle in a "Lorentzian function" potential well, and that each eigenenergy is doubly degenerate. Particularly, Ref.…”

“…All in all, similar to Ref. 24 , the present work is capable of providing a clear and concise treatment of the incomplete spectral collapse of the multi-qubit two-photon Rabi model at the critical coupling.…”

“…That is, while the model has a discrete eigenenergy spectrum for , no normalizable eigenstate exists in the Hilbert space spanned by the photon number states for . At the critical coupling the spectral collapse is, however, found to be incomplete for the eigenenergy spectrum of the model consists of both a set of discrete energy levels and a continuous energy spectrum 24 . The number of bound states available is determined by the energy difference between the two atomic levels so that the extent of the incomplete spectral collapse can be monitored straightforwardly.…”

Deciphering the spectral collapse in two-photon Rabi model

“…e . 12 – 24 . That is, as the coupling strength increases, the spacing of the discrete eigenenergy levels of the two-photon Rabi model diminishes monotonically, leading to a continuum energy spectrum beyond the critical value.…”

“…The characteristic behaviour of the eigenstates at the critical coupling thus remains as a mystery until recently. Lo 24 has rigorously shown that at the critical coupling some discrete energy levels exist below a continuum energy spectrum, and the number of these bound states available depends upon the energy difference between the two atomic levels. In other words, incomplete spectral collapse appears at the critical coupling.…”