  Keeping a quantum system in a given instantaneous eigenstate is a control
problem with numerous applications, e.g., in quantum information processing.
The problem is even more challenging in the setting of open quantum systems,
where environment-mediated transitions introduce additional decoherence
channels. Adiabatic passage is a well established solution, but requires a
sufficiently slow evolution time that is dictated by the adiabatic theorem.
Here we develop a systematic projection theory formulation for the
transitionless evolution of general open quantum systems described by
time-local master equations. We derive a time-convolutionless dynamical
equation for the target instantaneous eigenstate of a given time-dependent
Hamiltonian. A transitionless dynamics then arises in terms of a competition
between the average Hamiltonian gap and the decoherence rate, which implies
optimal adiabaticity timescales. We show how eigenstate tracking can be
accomplished via control pulses, without explicitly incorporating
counter-diabatic driving, thus offering an alternative route to shortcuts to
adiabaticity. We examine rectangular pulses, chaotic signals, and white noise,
and find that, remarkably, the effectiveness of eigenstate tracking hardly
depends on the details of the control functions. In all cases the control
protocol keeps the system in the desired instantaneous eigenstate throughout
the entire evolution, along an accelerated adiabatic path.
