Quantum phase transition in the anisotropic Rabi model induced by parametric amplification

TL;DR

This work analyzes the quantum phase transition in the anisotropic Rabi model, revealing a microscopic competition among three fundamental patterns that drive the superradiant transition in the classical oscillator limit. It introduces a parametric amplification scheme in a driven Jaynes-Cummings system, enabling a squeezed-frame simulation of the anisotropic Rabi model under experimentally accessible conditions. The study shows that at the critical point the excitation energies vanish, the ground-state energy is continuous while its second derivative is discontinuous, and the photon number diverges in the superradiant phase, demonstrating a clear QPT. Overall, it provides both a microscopic mechanism for QPT in anisotropic QRM and a practical route to realize and tune this transition via squeezing.

Abstract

In this manuscript, we analyze the mechanism of the superradiant phase transition in the anisotropic Rabi model under the classical oscillator limit using the pattern picture. By expanding the anisotropic Rabi model Hamiltonian in operator space, we obtained three patterns, and we find that the phase transition arises from the competition between patterns. The difficulty in achieving the classical oscillator limit motivates our investigation into the quantum phase transition within a parametrically-driven Jaynes-Cummings model. This parametrically-driven Jaynes-Cummings model can reproduce the dynamics of a ultrastrong-coupling anisotropic Rabi model in a squeezed-light frame. According to the eigenenergies and eigenstates of the normal and superradiant phases of this equivalent anisotropic Rabi model, we find that the excitation energy of the normal phase and the superradiant phase vanishes at the critical point. The photon number becomes infinite beyond the critical point. These results indicate that the system undergoes a superradiant phase transition at the critical point.

Figures (11)

• Figure 1: Marks of the patterns obtained by diagonalizing the Hamiltonian $H_{\mathrm{AN}}$.
• Figure 2: First four pattern energy levels as a function of $k/k_c$. The insets show an enlarged region of the $k/k_c$ ranging from 0.9 to 1.1. We set $\Omega=100\omega_0$ and $\xi_1=0.1\omega_0$.
• Figure 3: Comparison between the relevant physical quantities obtained from patterns and those obtained from ED. (a) Summation of patterns energy levels (lines) and their comparison with the results obtained directly by ED (diamond). (b) Comparison of the summations of the patterns’ photon number (lines) $\langle a^\dagger a\rangle$ to those obtained by ED (diamond). The parameters are the same as Fig. \ref{['F2']}.
• Figure 4: (a1),(b1) Sum of patterns of ground state energy level (black solid line) and corresponding pattern components (red, yellow, and blue solid lines) as as functions of coupling strength $\xi_1/\xi_{1c}$ when $k=0.9$ and $k=1$. (a2),(b2) Second-order derivatives of the corresponding energy levels (black solid line) and corresponding pattern components (red, yellow, and blue solid lines).
• Figure 5: (a1), (b1) Sum of patterns of ground state energy level (black solid line) and corresponding pattern components (red, yellow, and blue solid lines) as as functions of coupling strength $\xi_1/\xi_{1c}$ when $k=0.5$ and $k=1.5$. (a2), (b2) Second-order derivatives of the corresponding energy levels (black solid line) and corresponding pattern components (red, yellow, and blue solid lines).