Generalized Kibble-Zurek mechanism for defects formation in trapped ions

The Kibble-Zurek (KZ) mechanism has played a fundamental role in defect formation with universal scaling laws in nonequilibrium phase transitions. However, this theory may not accurately predict the scaling laws in inhomogeneous systems and slow quenching processes. Here, we present a generalized KZ mechanism for the defect formation in trapped ions with the freeze-out condition gt̂ = b_0τ (t̂) , where g is a universal quenching velocity function and b 0 is a constant. We derived a differential equation φ ( x, t ) to account for the frozen correlation length of a kink in an inhomogeneous system and demonstrated a smooth crossover from a fast quenching process to a slow quenching process, which agrees well with the experiments performed by Ulm et al. [Nat. Commun. 4 , 2290 (2013)] and Pyka et al. [Nat. Commun. 4 , 2291 (2013)]. Furthermore, we confirmed our theoretical model using molecular dynamics simulation by solving the stochastic differential equation, showing excellent agreement with the results from the differential equation. Our theory provides a general theoretical framework for studying KZ physics in inhomogeneous systems, which has applications in other nonequilibrium platforms studied experimentally.