Spectral kissing and its dynamical consequences in the squeezed Kerr-nonlinear oscillator

Transmon qubits are the predominant element in circuit-based quantum information processing due to their controllability and ease of engineering implementation. But more than qubits, transmons are multilevel nonlinear oscillators that can be employed in the discovery of new fundamental physics. Here, they are explored as simulators of excited state quantum phase transitions (ESQPTs), which are generalizations of quantum phase transitions to excited states. We show that the coalescence of pairs of adjacent energy levels (spectral kissing) recently observed with a squeezed Kerr oscillator [arXiv:2209.03934] is an ESQPT precursor. The classical limit of this system explains the origin of the quantum critical point and its consequences for the quantum dynamics, which includes both the fast scrambling of quantum information, characterized by the exponential growth of out-oftime-ordered correlators, and the slow evolution of the survival probability at initial times, caused by the localization of the energy eigenstates at the vicinity of the ESQPT. These signatures of ESQPT in the spectrum and in the quantum dynamics are simultaneously within reach for current superconducting circuits experiments.