# Modeling energy relaxation effects with density matrix formalism

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?

Take one qubit for simplicity. It was two energy levels, call them 0 and 1. 1 is at an energy $\omega$ higher than 0. So, the natural Hamiltonian is $$H_0=\omega|1\rangle\langle 1|$$ If you want to talk about energy dissipation, you probably have to go to the Lindblad equation, with an amplitude damping term $$L=\gamma|0\rangle\langle 1|.$$ However, you're probably going to add energy back in using a unitary process, $$H_1=\Omega(|0\rangle\langle 1|+|1\rangle\langle 0|).$$ Thus, the overall evolution is $$\frac{d\rho}{dt}=-i[H_0+H_1,\rho]+L\rho L^\dagger-\frac12(L^\dagger L\rho+\rho L^\dagger L).$$ Now, if you want to make sure there's a balance of energy in and energy out, you want to evaluate the expected energy of the state, $\text{Tr}(\rho H_0)$, and make sure that's a constant, i.e. that the time derivative is 0. I haven't tried to solve it (you were asking for a starting point) but I presume that to make it work, $\Omega$ will be a function of time and of the initial state. The other interesting thing to calculate will presumably be $\text{Tr}(\rho^2)$ (or perhaps its first derivative). This will map the increasing mixedness of the state $\rho$, which can never be compensated for by the unitary action of $H_1$ (which preserves mixedness).