  We study the vortex nucleation dynamics in inhomogeneous atomic Bose gases
quenched into a superfluid phase and investigate the dependence of the
Kibble-Zurek (KZ) scaling exponent on the underlying trap configuration. For
samples in a number of different inhomogeneous traps, we observe the
characteristic power-law scaling of the vortex number with the thermal quench
rate, as well as an enhanced vortex suppression in the outer regions with lower
particle density, in agreement with the causality effect as encapsulated in the
inhomogeneous Kibble-Zurek mechanism (IKZM). However, the measured KZ scaling
exponents show significant differences from the theoretical estimates, and
furthermore their trends as a function of the underlying trap configuration
deviate from the IKZM prediction. We also investigate the early-time coarsening
effect using a two-step quench protocol as proposed in a recent study and show
that the interpretation of the measurement results without including the
causality effect might be misleading. This paper provides a comprehensive study
of vortex formation dynamics in quenched Bose gases confined in inhomogeneous
trapping potentials and calls for a refined theoretical framework for
quantitative understanding of the phase transition and defect formation
processes in such inhomogeneous systems.
