<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>A</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> -robust superradiant phase transition in hybrid qubit-cavity optomechanics

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 consists 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 are 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.