I want to simulate the evolution of a transmon qubit driven by a classical voltage using qutip. The problem is that the qubit (and the drive) have a frequency of around 5GHz and I typically run simulations for a few hundreds of microseconds, and hence the number of steps needed in the simulation is quite large and the simulation quite slow (you need several integration steps per oscillation -> the simulation takes more time for higher frequencies).
I therefore considered doing my simulation in the rotating frame, since the relative frequency between the drive and the qubit is small. I however have some collapse operators in my model (to take into account the relaxation and dephasing of the qubit) and I couldn’t find a reference on how to convert collapse operators into the rotating frame. Does any of you have a good reference where I could find this?
The Hamiltonian of my Transmon qubit and the drive Hamiltonians are (using that $\hbar = 1$):
$$
H_q = \omega_q b^{\dagger}b + \frac{\alpha}{2}b^{\dagger}b^{\dagger}b b \qquad H_{drive} = g (b^{\dagger} + b) V(t).
$$
And in the qubit’s rotating frame we therefore have (after one page of derivation that I can share if you are interested)
$$
H_{q\ rot} = \frac{\alpha}{2}b^{\dagger}b^{\dagger}b b \qquad V_{rot}(t) = V(t) e^{-i\omega_q t} \qquad H_{drive\ rot} = g \left( (b^{\dagger} + b) Re(V_{rot}(t)) + i(b -b^{\dagger} ) Im(V_{rot}(t)) \right).
$$
And the collapse operators in the lab frame are:
$$
C_{relax} = \sqrt{\frac{1}{T_1}}b \qquad C_{pure\ dephasing} = \sqrt{\frac{1}{T_2}-\frac{1}{2}\frac{1}{T_1}}b^{\dagger}b.
$$
I tried naively to keep the same collapse operators in the rotating frame, but my simulation results show that this doesn’t work. You can see in this first plot the result of a Ramsey experiment simulation without using the rotating frame, which takes an eternity to compute but gives out the expected value for the dephasing time.
I ran the same simulation using the Hamiltonian in the rotating frame, by using the same expression for the collapse operators as in the lab frame. It was super fast to get the result, and the qubit’s physics is pretty similar, but as you can see in this second plot, there is no effect of the collapse operators (we should observe some dephasing of the qubit), which strongly suggests that they must look differently in the rotating frame.
Does anyone have a suggestion on how these collapse operators should look like in the rotating frame or at least some recipe on how to transform them?
