Critical Spectrum and Quantum Criticality in the Two-Photon Rabi-Stark Model

Abstract

We investigate the spectral properties and quantum criticality of the two-photon Rabi-Stark model. Using the exact solution of this model, we rigorously derive a condition for complete spectral collapse, where all bound states vanish. In this case, the energy gap closes at a critical coupling, signaling a continuous quantum phase transition. The corresponding gap exponent differs from those in both the one-photon Rabi-Stark model and the quantum Rabi model, suggesting a distinct universality class. While in the general case, an infinite number of discrete bound states exist when spectral collapse occur and the energy gap remains open. By mapping to an inverse square potential well, these bound levels approach the threshold energy exponentially. Our results offer new insights into novel spectral phenomena in nonlinear quantum Rabi models, with potential implications for experimental realizations in circuit QED and trapped ion systems.

Figures (6)

• Figure 1: Energy spectra at $\Delta = 0.50$ and $U = 0.10$ as functions of $g$ for $q= 1/4$ (left panel) and $q = 3/4$ (right panel). The black solid lines represent eigenvalues obtained through ED, while the dashed lines denote the pole lines $E_n^{(q, \mathrm{pole})}$. Blue circles indicate the zeros of $G_+^{(q)}$, while red ones represent zeros of $G_-^{(q)}$. A spectral collapse occurs at $g_c=\sqrt{1-U^2}/2 \simeq 0.4975$.
• Figure 2: The scaled energy spectra $E^{\prime} = [\gamma^{2} (E+U\Delta/2) + 1/2] /(2\beta) - q$ for $\Delta=0.2$ and $q=1/4$ at $U=0.40$ (left), $U=0.20$ (middle), and $U=0.10$ (right). The horizontal axis is logarithmically scaled, given by $x=-\log _{10}(1-g/g_{c})$. Blue (red) solid lines indicate energy levels with even (odd) parity, while black dashed lines represent the pole lines. The open circles denote the doubly degenerate points.
• Figure 3: The left panel presents the scaled energy spectra $E^{\prime} = [\gamma^{2} (E+U\Delta/2) + 1/2] /(2\beta) - q$, while the right panel displays the special $G$ function $G_{\pm}^{(q)}(m, g)$ associated with the $m = 1$ pole line. The horizontal axis is logarithmically scaled as $x = -\log_{10}(1 - g / g_c)$. Blue (red) solid lines represent even (odd) parity, and black dashed lines indicate pole positions. The open circles mark the positions of the special nondegenerate points. The parameters are $\Delta = 5.00$, $U = 0.25$, and $q = 1/4$.
• Figure 4: $\ln (\kappa_n^2 / \kappa_0^2)$ for the first several bound states at the critical coupling, $g = g_c$, $U = 0.25$, and $q = 1/4$, for $\Delta = 5.00$ (left) and $\Delta = 10.00$ (right). The open blue circles represent all data obtained via numerical ED, while the red filled circles denote the data used for fitting, which approaches the threshold energy $E_n^{(c)}$. The red line indicates a linear fit.
• Figure 5: Logarithmically scaled energy gap under the condition $\Delta = \Delta_c$ for $U=0.20$ (left) and $U = 0.50$ (right). The horizontal axis is logarithmically scaled as $x=-\mathrm{log}_{10} (1-g/g_c)$; Open circles represent results obtained via numerical ED, while the red line denotes a linear fit.