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

We investigate the consequences of introducing non-uniform initial conditions in the dynamics of spin ensembles characterized by all-to-all interactions. Specifically, our study involves the preparation of a set of semi-classical spin ensembles with varying orientations. Through this setup, we explore the influence of such non-uniform initial states on the disruption of permutational invariance. Comparing this approach to the traditional scenario of initializing with spins uniformly aligned, we find that the dynamics of the spin ensemble now spans a more expansive effective Hilbert space. This enlargement arises due to the inclusion of off-diagonal coherences between distinct total angular momentum subspaces - an aspect typically absent in conventional treatments of all-to-all spin dynamics. Conceptually, the dynamic evolution can be understood as a composite of multiple homogeneous sub-ensembles navigating through constrained subspaces. Notably, observables that are sensitive to the non-uniformity of initial conditions exhibit discernible signatures of these off-diagonal coherences. We adopt this fresh perspective to reexamine the relaxation phenomena exhibited by the Dicke model, as well as a prototypical example of a boundary time crystal. Intriguingly, ensembles initialized with inhomogeneous initial conditions can show distinctive behaviors when contrasted with canonical instances of collective dynamics. These behaviors encompass the emergence of novel gapless excitations, the manifestation of limit-cycles featuring dressed frequencies due to superradiance, instances of frequency locking or beating synchronizations, and even the introduction of ``extra'' dimensions within the dynamics. In closing, we provide a brief overview of the potential implications of our findings in the context of modern cavity quantum electrodynamics (QED) platforms.