I know there are measures that can be taken to mitigate the effects of dephasing (I'm referring here to Dynamic Decoupling and the other ideas it led to). I find it surprising that there is no equivalent procedure to mitigate the effects of energy leakage. In principle, it seems like if you add energy to the system at the rate it leaves (due to interactions with the measurement equipment, etc) you could keep a qubit at the same energy level indefinitely. That is, $T_1 \to \infty$.
I know that just have the qubit at the correct energy probably isn't enough to ensure that it is in a pure state, so I want to try to formulate this idea using the density matrix formalism and try to see if there is any clever way to add the energy in such a way to keep the qubit in an initial .
I can't get started, so I'm asking for help. How would you model a system at constant energy, with energy leaving and entering at the same rate?