Analytical Solutions to Asymmetric Two-Photon Rabi Model

TL;DR

The work addresses a generalized Rabi model that includes a biphoton term and an asymmetry term within the Segal-Bargmann framework, which yields a fourth-order differential equation for the wavefunction. The authors apply a unitary rotation to obtain the Hamiltonian H and then use a Bethe-ansatz reduction, aided by a φ1(z)=exp(α z^2) ansatz and the condition q4=0, to derive a quartic equation in α with Λ=λ/ω and BAEs for the zeros, from which the spectra are obtained as E_n. They present closed-form energies E_n and two parameter constraints, with explicit n=0 and n=1 results and Bethe roots z_i, and discuss normalizability in Bargmann space. The results establish (quasi-)exact solvability for multiphoton and asymmetric Rabi generalizations and provide a framework for extending to anisotropic, multi-mode, and few-qubit variants relevant to quantum optics and related technologies.

Abstract

Within the Segal-Bargmann representation, a generalized Rabi model is considered that includes both two-photon and asymmetric terms. It is shown that, through a suitable transformation, nearly exact solutions can be obtained using the Bethe ansatz approach. Applying this approach to the meromorphic structure of the resulting differential equation, solutions in exact analytical form of the fourth-order problem are presented for both an arbitrary state and for the restriction between the parameters.

Figures (5)

• Figure 1: Real and imaginary components of $\alpha_i$, $i=1,...,4$ as a function of $\Lambda=\frac{\lambda}{\omega}$, in the range $|\alpha_i| < \frac{1}{2}$.
• Figure 2: The ground state energy $E_0$\ref{['E0']}, indicated by the surfaces, and the allowed values of the parameter \ref{['del0']}, indicated by the red curves embedded in the surfaces, as functions of $\epsilon$ and $\lambda$ for fixed values of $\Delta$ and $\omega$. Here, $\alpha_2(\Lambda)$ is taken from \ref{['alpha1234']} within the allowed region $|\alpha_2| < \frac{1}{2}$.
• Figure 3: Allowed values for the parameters $\Delta$ and $\epsilon$ by \ref{['del0']} when $n=0$, with $\omega=1$ and different values of $\lambda$ and $\alpha$, and taking $\alpha_2(\Lambda)$ from \ref{['alpha1234']} within the region $|\alpha_2| < \frac{1}{2}$.
• Figure 4: The first excited state energy $E_1$\ref{['E1']}, indicated by the surfaces, and the allowed values of the parameter \ref{['del1']}, indicated by the red curves embedded in the surfaces, as functions of $\epsilon$ and $\lambda$ for fixed values of $\Delta$ and $\omega$. Again, $\alpha_2(\Lambda)$ is considered from \ref{['alpha1234']} within the allowed region $|\alpha_2| < \frac{1}{2}$.
• Figure 5: Allowed values for the parameters $\Delta$ and $\epsilon$ by \ref{['del1']} when $n=1$, with $\omega=1$ and different values of $\lambda$ and $\alpha$, and taking $\alpha_2(\Lambda)$ from \ref{['alpha1234']} within the region $|\alpha_2| < \frac{1}{2}$.