$\bf A^2$-robust superradiant phase transition in hybrid qubit-cavity optomechanics

TL;DR

The paper tackles the obstacle posed by the A^2 term to observing superradiant phase transitions in cavity QED by introducing a hybrid qubit–mechanical–cavity system with an auxiliary cavity that switches the A^2 contribution. Through dispersive elimination and a Fröhlich–Nakajima transformation, the authors engineer an effective A^2 term that can counteract the intrinsic one, enabling SPT beyond the no-go theorem via a squeezing-transformed, enhanced-coupling description. They demonstrate that the second-order phonon correlation g^{(2)}(0) acts as an order parameter distinguishing normal and superradiant phases, and show that the optomechanical coupling exponentially reduces the required coupling strength. Additionally, higher-order squeezing near the SPT point provides a strong, distinct signature of the transition, suggesting a practical route to observe SPT in realistic setups.

Abstract

The $\mathbf{A}^2$ term presents a fundamental challenge to realizing the superradiant phase transition (SPT) in cavity quantum electrodynamics. Here, we propose a hybrid quantum system enabling SPT regardless of the presence of the $\mathbf{A}^2$ term. The system consist of a qubit, a mechanical mode, and an optical cavity, where the qubit and mechanical mode constitute a quantum Rabi model, while the mechanical mode and cavity form an optomechanical system. Crucially, the auxiliary cavity introduces a switchable $\mathbf{A}^2$ term that effectively counteracts or even fully eliminates the original $\mathbf{A}^2$ effect. This allows controllable observation of SPT, diagnosed via the second-order equal-time correlation function $g^{(2)}(0)$ of phonons. Furthermore, the auxiliary cavity exponentially reduces the critical coupling strength, significantly relaxing experimental requirements. In addition, we show that phonons in the normal phase display bunching, but coherent in the superradiant phase. Interestingly, higher-order squeezing is found in both phases, with near-perfect higher-order squeezing achieved at the SPT point, establishing it as a probe for SPT behavior. Our work demonstrates that hybridizing optomechanics and cavity quantum electrodynamics provides a promising route to accessing SPT physics in the presence of the $\mathbf{A}^2$ term.

Figures (5)

• Figure 1: Schematic diagram of the proposed system, consisting of a mechanical mode $b$ coupled to both a qubit (i.e., two-level system) and a single-mode optomechanical cavity $a$. The coupling strengths are $g$ and $G$.
• Figure 2: Exponentially--enhanced effective coupling $g_s$ versus the optomechanical coupling strength $\xi/\omega_b$.
• Figure 3: Equal-time second--order correlation function $g^{(2)}(0)$ as a function of the normalized coupling ratio $\xi/\omega_b$ and the critical coupling strength $\tilde{g}_c^s$ for two different values of $\alpha$. The region with $\xi/\omega_b<1/4$ indicates $\alpha = 0$, while the region with $\xi/\omega_b>1/4$ indicates $\alpha = 1.5$. The white dashed line [$g^{(2)}(0) = 1$] marks the boundary between NP and SP.
• Figure 4: Equal-time second-order correlation function $g^{(2)}(0)$ as a function of the dimensionless coupling strength $\tilde{g}_c$ for different values of $\Omega/\omega_s$. (a) $\alpha=0$, $\xi=0$; (b) $\alpha=0$, $\xi/\omega_b=0.245$; (c) $\alpha=1.5$, $\xi=0$; and (d) $\alpha=1.5$, $\xi/\omega_b=0.26$.
• Figure 5: Higher-order quadrature fluctuations $\braket{(\Delta P)^N}$ in the ground state as a function of rescaled coupling strength $\tilde{g}_c$. (a) $\alpha=0$, $\xi=0$; (b) $\alpha=0$, $\xi/\omega_b=0.245$; (c) $\alpha=1.5$, $\xi=0$; (d) $\alpha=1.5$, $\xi/\omega_b=0.26$. Horizontal lines denote $\braket{(\Delta P)^N}$ in the coherent state.