  The out-of-time-ordered correlators (OTOC) have been established as a
fundamental concept for quantifying quantum information scrambling and
diagnosing quantum chaotic behavior. Recently, it was theoretically proposed
that the OTOC can be used as an order parameter to dynamically detect both
equilibrium quantum phase transitions (EQPTs) and dynamical quantum phase
transitions (DQPTs) in one-dimensional many-body systems. Here we report the
first experimental observation of EQPTs and DQPTs in a quantum spin chain via
quench dynamics of OTOC on a nuclear magnetic resonance quantum simulator. We
observe that the quench dynamics of both the order parameter and the two-body
correlation function cannot detect the DQPTs, but the OTOC can unambiguously
detect the DQPTs. Moreover, we demonstrate that the long-time average value of
the OTOC in quantum quench signals the equilibrium quantum critical point and
ordered quantum phases, thus one can measure the EQPTs from the non-equilibrium
quantum quench dynamics. Our experiment paves a way for experimentally
investigating DQPTs through OTOCs and for studying the EQPTs through the
non-equilibrium quantum quench dynamics with quantum simulators.
