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Continuous phase transitions occur in a wide range of physical systems, andprovide a context for the study of non-equilibrium dynamics and the formationof topological defects. The Kibble-Zurek (KZ) mechanism predicts the scaling ofthe resulting density of defects as a function of the quench rate through acritical point, and this can provide an estimate of the critical exponents of aphase transition. In this work we extend our previous study of themiscible-immiscible phase transition of a binary Bose-Einstein condensate (BEC)composed of two hyperfine states in which the spin dynamics are confined to onedimension [J. Sabbatini et al., Phys. Rev. Lett. 107, 230402 (2011)]. Thetransition is engineered by controlling a Hamiltonian quench of the couplingamplitude of the two hyperfine states, and results in the formation of a randompattern of spatial domains. Using the numerical truncated Wigner phase spacemethod, we show that in a ring BEC the number of domains formed in the phasetransitions scales as predicted by the KZ theory. We also consider the sameexperiment performed with a harmonically trapped BEC, and investigate how thedensity inhomogeneity modifies the dynamics of the phase transition and the KZscaling law for the number of domains. We then make use of the symmetry betweeninhomogeneous phase transitions in anisotropic systems, and an inhomogeneousquench in a homogeneous system, to engineer coupling quenches that allow us toquantify several aspects of inhomogeneous phase transitions. In particular, wequantify the effect of causality in the propagation of the phase transitionfront on the resulting formation of domain walls, and find indications that thedensity of defects is determined during the impulse to adiabatic transitionafter the crossing of the critical point.

Causality and defect formation in the dynamics of an engineered quantum phase transition in a coupled binary Bose-Einstein condensate