Exploring the role of quantum fluctuations and chaos in dynamical phase transitions of the Dicke model

Investigations of dynamical phase transitions (DPTs), which are signaled by nonanalytic behavior in a time-averaged order parameter after a quench, have been primarily motivated by well-established connections to the classical dynamics of integrable collective spin models. We theoretically and experimentally study the role of quantum fluctuations, and their interplay with chaos, by examining a series of DPTs in the paradigmatic Dicke model describing a coupling between a single collective spin and a bosonic oscillator. We study the dynamics of quantum fluctuations using observables such as squeezing and contrast results from initial conditions that are distinguished by the presence or absence of a well-defined classical limit. For the latter, we further investigate the distinct dynamics between integrable and chaotic regimes. We compare our theoretical predictions with experimental data obtained in a trapped-ion quantum simulator of the Dicke model: An ensemble of qubits is encoded in the internal states of a 2D array of Be ions in a Penning trap, while the center-of-mass collective vibrational motion of the array encodes the bosonic mode that is coupled to the qubits via a spin-dependent optical dipole force. Experimental tunability of the parameters of the engineered Dicke model combined with controllably prepared different initial states enable us to map out the predicted dynamical phase diagram and examine the growth and survival of quantum fluctuations even in the presence of decoherence.