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In a $m$ particle quantum system, one can have $k=1,\,2,\,\ldots,\,m$ bodyinteractions. The rank of interactions and the nature of particles (fermions orbosons) can strongly affect the dynamics of the system. To explore this indetail, we study quenched quantum dynamics in many particle systems varyingrank of interactions, both for fermionic and bosonic particles. We representthe system Hamiltonian by Fermionic Embedded Gaussian Orthogonal Ensembles(FEGOE) and Bosonic Embedded Gaussian Orthogonal Ensembles (BEGOE)respectively. We show that generating function for $q$-Hermite polynomialsdescribes the semi-circle to Gaussian transition in spectral densities ofFEGOE$(k)$ and BEGOE$(k)$ (also the Unitary variants FEGUE$(k)$ and BEGUE$(k)$)as a function of rank of interactions $k$. Importantly, numerical Fouriertransform of generating function of $q$-Hermite polynomials explains theshort-time decay of survival probability in FEGOE$(1+k)$ and BEGOE$(1+k)$. Theparameter $q$ describing these properties is related to excess parameter$\gamma_2$ and we give a formula for FEGOE$(k)$, FEGUE$(k)$, BEGOE$(k)$ andBEGUE$(k)$. We illustrate that the dynamics strongly depends on the rank ofinteractions and nature of particles and these universal features may berelevant to modeling non-equilibrium quantum systems.

Quenched many-body quantum dynamics with $k$-body interactions using $q$-Hermite polynomials