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arXiv:2007.04664 · Journal of Physics A Mathematical and Theoretical

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

Ching Kwan Chan

Published Jul 8, 2020• 14 citations
arXivHTMLPDF

Abstract

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 the 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 is known as incomplete spectral collapse. From the eigenvalue equation in real space, a pair of second order differential equations, which are similarly to the Schrödinger equation, are derived at the collapse point. These differential equations provide an explanation to the existence of isolated bound states below E = − ω 2 with the presence of the spin slitting ω 0 and a better numerical method to generate those bound states.

DOI
10.1088/1751-8121/aba3e0

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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 the 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 is known as incomplete spectral collapse. From the eigenvalue equation in real space, a pair of second order differential equations, which are similarly to the Schrödinger equation, are derived at the collapse point. These differential equations provide an explanation to the existence of isolated bound states below E = − ω 2 with the presence of the spin slitting ω 0 and a better numerical method to generate those bound states.
In this communication, based upon a squeezed-state trial wave function, we have performed a simple variational study of the spectral collapse in the anisotropic two-photon Rabi model. Our analysis indicates that the light-matter interaction and the spin-flipping (together with the anisotropy) effectively constitute two competing impacts upon the radiation mode. Whilst the former tries to decrease the radiation mode frequency, the latter may counteract or reinforce it. The light-matter interaction appears to dominate the frequency modulation as its coupling strengths go beyond the critical values, leading to the emergence of the spectral collapse. However, at the critical couplings the dominance of the light-matter interaction is not complete, and incomplete spectral collapse appears. Accordingly, at the critical couplings the eigenenergy spectrum comprises both a set of discrete energy levels and a continuous energy spectrum. The discrete eigenenergy spectrum can be derived via a simple one-to-one mapping to the bound state problem of a particle of variable effective mass in a finite potential well, and the number of bound states available is determined by the energy difference between the two atomic levels. Each of these eigenenergies has a twofold degeneracy corresponding to the spin degree of freedom.
Spectral collapse, the transition from a discrete to a continuous spectrum, is a characteristic in quantum Rabi models. We explore this phenomenon in the two-photon quantum Rabi model using optical phase space, and we find that, in the so-called degenerate qubit regime, the collapse is similar to the transition from a harmonic to an inverted oscillator with the free-particle potential as a critical transition point. In the degenerate qubit regime, we construct Dirac-normalizable eigenfunctions with well-defined parity for the model. In the general model, we use parity to diagonalize the system in the qubit basis and numerically find that the qubit frequency does not change the critical point where spectral collapse occurs. We numerically confirm the existence of an exceptional state at the critical coupling, and we argue its analytic provenance from both a Born-Oppenheimer and a variational approximation.
In this communication, based upon a squeezed-state trial wave function, we have performed a simple variational study of the spectral collapse of the two-photon Rabi model. Our analysis indicates that the light-matter interaction and the spin-flipping effectively constitute two competing impacts upon the radiation mode. Whilst the former tries to decrease the radiation mode frequency, the latter may counteract or reinforce it, contingent upon the state of the atomic system. The light-matter interaction appears to dominate the frequency modulation as its coupling strength goes beyond the critical value, leading to the emergence of the spectral collapse. However, at the critical coupling the dominance of the light-matter interaction is not complete, and incomplete spectral collapse appears. The extent of incomplete spectral collapse is found to depend upon the energy difference between the two atomic levels as well.
Abstract The two-photon quantum Rabi model is analyzed using the Z 4 -symmetry and su ( 1 , 1 ) -algebra. We derive the G -function whose zeros give the exact eigenvalues of the Hamiltonian. The singularity structure of the G -function allows to draw conclusions about the distribution of these eigenvalues along the real axis and we derive the spectral collapse phenomenon at critical coupling g c , found numerically before. The spectrum at g c consists of a discrete and a continuous part: the ground state is always separated from the continuum by a finite excitation gap, ruling out a quantum phase transition in the usual sense. For large qubit splitting, also other low lying states split off from the continuum. However, perturbation theory predicts the vanishing of the gap to all orders, demonstrating its non-perturbative nature. We corroborate this result with a variational calculation for the ground state.
The anisotropic two-photon quantum Rabi model is studied using the Bogoliubov operator approach. Doubly degenerate exceptional states are identified using analytical methods. By adjusting the position of the last exceptional point belonging to two adjacent energy levels, we derive a condition for the absence of the discrete spectrum at the critical coupling where ``spectral collapse'' occurs. In this special case, the spectrum becomes fully continuous above a threshold energy, with no bound states existing, whereas the ground state remains gapped in the general case. More interestingly, we rigorously find that a finite number of bound states exist between two anisotropy-dependent critical atomic frequencies, with infinitely many bound states beyond this frequency regime. In this manner, all issues in the two-photon quantum Rabi model are resolved.
We employ a polaron picture to investigate the properties of the two-photon quantum Rabi model (QRM), which describes a two-level or spin-half system coupled with a single bosonic mode by a two-photon process. In the polaron picture, the coupling in the two-photon process leads to spin-related asymmetry so that the original single bosonic mode splits into two separated frequency modes for the opposite spins, which correspond to two bare polarons. Importantly, the tunneling causes these two bare polarons to exchange their components with each other, thus leading to additional induced polarons. According to this picture, the variational ground-state wave function of the two-photon QRM can be correctly constructed, with the ground-state energy and other physical observables in good agreement with the exact numerics in all the coupling regimes. Furthermore, generalization to multiple induced polarons involving higher orders in the tunneling effect provides a systematic way to yield a rapid convergence in accuracy even around the difficult spectral collapse point. In addition, the polaron picture provides a distinctive understanding of an incomplete spectral collapse behavior that is about the existence of discrete energy levels apart from the collapsed spectrum at the spectral collapse point. This work illustrates that the polaron picture is helpful to capture the key physics in this nonlinear light-matter interaction model and indicates that this method can be applicable to more complicated QRM-related models.