  We study signatures of quantum chaos in dynamics of Rydberg dressed bosonic
atoms held in a one dimensional triple-well potential. Long-range
nearest-neighbor and next-nearest-neighbor interactions, induced by laser
dressing atoms to strongly interacting Rydberg states, affect drastically mean
field and quantum many-body dynamics. By analyzing the mean field dynamics,
classical chaos regions with positive and large Lyapunov exponents are
identified as a function of the potential well tilting and dressed
interactions. In the quantum regime, it is found that level statistics of the
eigen-energies gains a Wigner-Dyson distribution when the Lyapunov exponents
are large, giving rise to signatures of strong quantum chaos. We find that both
the time averaged entanglement entropy and survival probability of the initial
state have distinctively large values in the quantum chaos regime. We further
show that population variances could be used as an indicator of the emergence
of quantum chaos. This might provide a way to directly probe quantum chaotic
dynamics through analyzing population dynamics in individual potential wells.
