  We present a first-principles theoretical analysis of the entanglement of two
superconducting qubits in spatially separated microwave cavities by a
sequential (cascaded) probe of the two cavities with a coherent mode, that
provides a full characterization of both the continuous measurement induced
dynamics and the entanglement generation. We use the SLH formalism to derive
the full quantum master equation for the coupled qubits and cavities system,
within the rotating wave and dispersive approximations, and conditioned
equations for the cavity fields. We then develop effective stochastic master
equations for the dynamics of the qubit system in both a polaronic reference
frame and a reduced representation within the laboratory frame. We compare
simulations with and analyze tradeoffs between these two representations,
including the onset of a non-Markovian regime for simulations in the reduced
representation. We provide conditions for ensuring persistence of entanglement
and show that using shaped pulses enables these conditions to be met at all
times under general experimental conditions. The resulting entanglement is
shown to be robust with respect to measurement imperfections and loss channels.
We also study the effects of qubit driving and relaxation dynamics during a
weak measurement, as a prelude to modeling measurement-based feedback control
in this cascaded system.
