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We investigate the consequences of introducing non-uniform initial conditionsin the dynamics of spin ensembles characterized by all-to-all interactions.Specifically, our study involves the preparation of a set of semi-classicalspin ensembles with varying orientations. Through this setup, we explore theinfluence of such non-uniform initial states on the disruption of permutationalinvariance. Comparing this approach to the traditional scenario of initializingwith spins uniformly aligned, we find that the dynamics of the spin ensemblenow spans a more expansive effective Hilbert space. This enlargement arises dueto the inclusion of off-diagonal coherences between distinct total angularmomentum subspaces - an aspect typically absent in conventional treatments ofall-to-all spin dynamics. Conceptually, the dynamic evolution can be understoodas a composite of multiple homogeneous sub-ensembles navigating throughconstrained subspaces. Notably, observables that are sensitive to thenon-uniformity of initial conditions exhibit discernible signatures of theseoff-diagonal coherences. We adopt this fresh perspective to reexamine therelaxation phenomena exhibited by the Dicke model, as well as a prototypicalexample of a boundary time crystal. Intriguingly, ensembles initialized withinhomogeneous initial conditions can show distinctive behaviors when contrastedwith canonical instances of collective dynamics. These behaviors encompass theemergence of novel gapless excitations, the manifestation of limit-cyclesfeaturing dressed frequencies due to superradiance, instances of frequencylocking or beating synchronizations, and even the introduction of ``extra''dimensions within the dynamics. In closing, we provide a brief overview of thepotential implications of our findings in the context of modern cavity quantumelectrodynamics (QED) platforms.

Dynamics of inhomogeneous spin ensembles with all-to-all interactions: breaking permutational invariance