Bound states of two-photon Rabi model at the collapse point

Abstract

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 ω 0. However, many numerical studies have pointed out that with the presence of ω 0 , some low-level isolated states exist while other high energy states collapse to E = − ω 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 = − ω 2 with the presence of the spin slitting ω 0 and better numerical method to generate those bound states.

Spectral determinant of the two-photon quantum Rabi model

“…There are now two different parts of the spectrum, E > −1 and E < −1. The threshold energy is E 0 = −1, in accord with [17,18]. The four Stokes sectors are given in (13).…”

“…However, exactly this behavior is seen in all numerical evaluations of the spectrum, simply because the Hilbert space used in these calculations has finite dimension. The continuous part of the spectrum at ω c covers the interval [E 0 , ∞[ with E 0 = −ω c /2 = −1 which follows from a direct treatment of this case [17,18]. The critical point ω c = 2 allows for a mapping of the eigenvalue problem to a 1D Schrödinger equation in an energy-dependent potential.…”

Spectral collapse in multiqubit two-photon Rabi model

“…Particularly, Ref. 34 shows no sign of the two-fold degeneracy. Perhaps it may be due to some flaws in the compicated numerical calculations or the entire formulation.…”

“…(20), (22) and (46) into the time-independent Schrödinger equation of the bound state problem with a particle of unit mass moving in a finite potential well. Finally, we would like to thank an anonymous referee for bringing a recent paper on the bound states of the two-photon Rabi model at the critical coupling to our attention 34 . The paper investigates the discrete eigenenergy levels via solving a time-independent Schrödinger equation with an energy-dependent effective potential, whose solutions are found by self-consistent iterative numerical calculations.…”

Mathematical Models with Nonlocal Initial Conditions: An Exemplification from Quantum Mechanics

“…For example, as mentioned earlier, they arise naturally in inverse scattering problems [242], where the presence of measured scatted data nonlocal boundary conditions are typical, and advanced data-based methodologies, such as deep learning schemes [215], may be required. Other substantial sources are provided by open driven systems, and in particular driven quantum systems [50,158,[243][244][245], including those for Floquet engineering [246], quantum information control with quantum computing applications [247], as well as various nonlinear problems such as those described by Rabi's models, providing, among other things, new perspectives on the entanglement via the von Neumann entropy [248,249]. It is envisaged that in dealing with such systems, communication complexity will play a progressively growing role.…”