  Quantum phase transitions universally exist in the ground and excited states
of quantum many-body systems, and they have a close relationship with the
nonequilibrium dynamical phase transitions, which however are challenging to
identify. In the system of spin-1 Bose-Einstein condensates, though dynamical
phase transitions with correspondence to equilibrium phase transitions in the
ground state and uppermost excited state have been probed, those taken place in
intermediate excited states remain untouched in experiments thus far. Here we
unravel that both the ground and excited-state quantum phase transitions in
spinor condensates can be diagnosed with dynamical phase transitions. A
connection between equilibrium phase transitions and nonequilibrium behaviors
of the system is disclosed in terms of the quantum Fisher information. We also
demonstrate that near the critical points parameter estimation beyond standard
quantum limit can be implemented. This work not only advances the exploration
of excited-state quantum phase transitions via a scheme that can immediately be
applied to a broad class of few-mode quantum systems, but also provides new
perspective on the relationship between quantum criticality and quantum
enhanced sensing.
