Chaos and quantum scars in Bose-Josephson junction coupled to a bosonic mode

We consider a model describing Bose-Josephson junction (BJJ) coupled to a single bosonic mode exhibiting quantum phase transition (QPT). Onset of chaos above QPT is observed from semiclassical dynamics as well from spectral statistics. Based on entanglement entropy we analyze the ergodic behavior of eigenstates with increasing energy density which also reveals the influence of dynamical steady state known as $\pi$-mode on it. We identify the imprint of unstable $\pi$-oscillation as many body quantum scar (MBQS), which leads to the deviation from ergodicity and quantify the degree of scarring. Persistence of phase coherence in nonequilibrium dynamics of such initial state corresponding to the $\pi$-mode is an observable signature of MBQS which has relevance in experiments on BJJ.