In the context of superconducting quantum computing measuments, consider the dispersive shift Hamiltonian:
$$ H/\hbar = \omega_R a^\dagger a + \frac{1}{2} (\omega_Q + \frac{2g^2}{\Delta} a^\dagger a)\sigma^z $$
where we interpret the dispersive shift as a change in the qubit frequency. What happens when we just keep adding photons into the resonator?
The first thing I would guess is that the qubit's frequency gets higher and higher, but it doesn't seem to me that it can just get higher indefinitely. So what's the limiting behavior? I once had a quantum computing guy say that the qubit stops affecting the dynamics of the system, and if you were there measuring, you would only notice a resonator.
I'm looking to connect that statement to the math. Is it that some assumption we've made to get here breaks down, RWA for example? Why can we just ignore the high-frequency qubit?
