  We introduce jumptime unraveling as a distinct description of open quantum
systems. As our starting point, we consider quantum jump trajectories, which
emerge, physically, from continuous quantum measurements, or, formally, from
the unraveling of Markovian quantum master equations. If the stochastically
evolving quantum trajectories are ensemble-averaged at specific times, the
resulting quantum states are solutions to the associated quantum master
equation. We demonstrate that quantum trajectories can also be
ensemble-averaged at specific jump counts. The resulting jumptime-averaged
quantum states are then solutions to a discrete, deterministic evolution
equation, with time replaced by the jump count. This jumptime evolution
represents a trace-preserving quantum dynamical map if and only if the
associated open system does not exhibit dark states. In the presence of dark
states, on the other hand, jumptime-averaged states may decay into the dark
states and the jumptime evolution may eventually terminate. Jumptime-averaged
quantum states and the associated jumptime evolution are operationally
accessible in continuous measurement schemes, when quantum jumps are detected
and used to trigger the readout measurements. We illustrate the jumptime
evolution with the examples of a two-level system undergoing relaxation or
dephasing, a damped harmonic oscillator, and a free particle exposed to
collisional decoherence.
