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We describe how to characterize dynamical phase transitions in open quantumsystems from a purely dynamical perspective, namely, through the statisticalbehavior of quantum jump trajectories. This approach goes beyond consideringonly properties of the steady state. While in small quantum systems dynamicaltransitions can only occur trivially at limiting values of the controllingparameters, in many-body systems they arise as collective phenomena and withinthis perspective they are reminiscent of thermodynamic phase transitions. Weillustrate this in open models of increasing complexity: a three-level system,a dissipative version of the quantum Ising model, and the micromaser. In theseexamples dynamical transitions are accompanied by clear changes in staticbehavior. This is however not always the case, and in general dynamical phasebehavior needs to be uncovered by observables which are strictly dynamical,e.g. dynamical counting fields. We demonstrate this via the example of a classof models of dissipative quantum glasses, whose dynamics can vary widelydespite having identical (and trivial) stationary states.

Characterization of dynamical phase transitions in quantum jump trajectories beyond the properties of the stationary state