Superradiant phase transitions (SPTs) are important for understanding light-matter interactions at the quantum level, and play a central role in criticality-enhanced quantum sensing. So far, SPTs have been observed in driven-dissipative systems, but the emergent light fields did not show any nonclassical characteristic due to the presence of strong dissipation. Here we report an experimental demonstration of the SPT featuring the emergence of a highly nonclassical photonic field, realized with a resonator coupled to a superconducting qubit, implementing the quantum Rabi model. We fully characterize the light-matter state by Wigner matrix tomography. The measured matrix elements exhibit quantum interference intrinsic of a photonic mesoscopic superposition, and reveal light-matter entanglement.

Quantum sensing improves the accuracy of measurements of relevant parameters by exploiting the unique properties of quantum systems. The divergent susceptibility of physical systems near a critical point for quantum phase transition enables criticality-enhanced quantum sensing. The quantum Rabi model (QRM), composed of a single qubit coupled to a single bosonic field, represents a good candidate for realizing such critical enhancement for its simplicity, but it is experimentally challenging to achieve the ultrastrong qubit-field coupling required to realize the critical phenomena. In this work, we explore an alternative to construct the analog of the QRM for the sensing, exploiting the criticality appearing in the Jaynes-Cummings (JC) model whose bosonic field is parametrically driven, not necessitating the ultrastrong coupling condition, thus to some extent relaxing the requirement for practical implementation.

Quantum phase transition and quench dynamics in the Jaynes-Cummings model with squeezed light are investigated. We find a special unitary transformation that removes the nonintegrable squeezing interaction when the qubit frequency far outweighs the field frequency. The eigenenergies and the eigenstates of the normal and superradiant phases are derived analytically. We demonstrate that the system exhibits a second-order quantum phase transition at a phase-boundary-induced nonlinearly by squeezed light. This phase boundary requires neither an ultrastrong coupling regime nor multiqubits, which lessens the difficulty of the experiment. The entanglement entropy is very close to its maximum value as the squeezing strength increases in the superradiant phase. In quench dynamics, the nonlinear relation between the residual energy and the squeezing strength is obtained analytically.

How to analytically deal with the general entanglement dynamics of separate Jaynes–Cummings nodes with continuous-variable fields is still an open question, and few analytical approaches can be used to solve their general entanglement dynamics. Entanglement dynamics between two separate Jaynes–Cummings nodes are examined in this article. Both vacuum state and coherent state in the initial fields are considered through the numerical and analytical methods. The gap between two nonidentical qubit-field coupling strengths shifts the revival period and changes the revival amplitude of two-qubit entanglement. For vacuum-state fields, the maximal entanglement is fully revived after a gap-dependence period, within which the entanglement nonsmoothly decreases to zero and partly recovers without exhibiting sudden death phenomenon. For strong coherent-state fields, the two-qubit entanglement decays exponentially as the evolution time increases, exhibiting sudden death phenomenon, and the increasing gap accelerates the revival period and amplitude decay of the entanglement, where the numerical and analytical results have an excellent coincidence.

We study the quantum phase transition in the Dicke model beyond the thermodynamic limit. With the Kibble–Zurek mechanism and adiabatic dynamics, we find that the residual energy is inversely proportional to the number of qubits, indicating that more qubits can obtain more energies from the oscillator as the number of qubits increases. Finally, we put forward a promising experiment device to realize this system.