  We study the asymptotic emergent dynamics of two models that can be thought
of as extensions of the well known Schr\"odinger-Lohe model for quantum
synchronization. More precisely, the interaction strength between different
oscillators is determined by intrinsic parameters, following Cucker-Smale
communication protocol. Unlike the original Schr\"odinger-Lohe system, where
the interaction strength was assumed to be uniform, in the cases under our
consideration the total mass of each quantum oscillator is allowed to vary in
time. A striking consequence of this property is that these extended models
yield configurations exhibiting phase, but not space, synchronization. The
results are mainly based on the analysis of the ODE systems arising from the
correlations, control over the well known Cucker-Smale dynamics, and the
dynamics satisfied by the quantum order parameter.
