Simulation of a T2 echo procedure on a simulated qubit with relaxation and dephasing

[Daniel Benvenutti]

I am simulating the typical characterization procedures of a qubit on qutip like ramsey and rabi oscillations and the T2 echo.

The problem is, given a T2 and T1 (from which I get a T\phi) parameters for the dephasing and relaxation collapse operators on my simulation I am getting a T2echo from the fit in a T2 echo procedure that is considerably smaller than the initial T2 parameter I used as input. Using a smaller step improves it a bit up until the step is around 2pi over twice the frequency of the qubit resonance.

Lowering the qubit resonance frequency and the T2 seem to both improve the rapport T2echo/T2.

But I can't really find the cause or at least how to compensate this problem in the simulation.

You can check my notebook for more details on my simulation at https://github.com/Danielgb23/superconducting_qubit_3/blob/main/Daniel-simulation-qubit.ipynb

[Abdul Aziz Kazi.]

  Use a time step that resolves the qubit oscillations, not just the decay envelope:

If you're simulating the qubit in the lab frame (with a Hamiltonian like H=ωq​σz​/2), then the state vector rotates rapidly, which requires high time resolution.

• Decoherence happens in the rotating frame, but if your simulation is done in the lab frame, the echo envelope is modulated by fast qubit rotation, leading to under-sampling errors.

• Shift your simulation to the rotating frame, especially during T₂ and echo experiments.

• Redefine the Hamiltonian so that in the rotating frame it becomes something like:

  Hrot​=Δσz​/2+Hdrive​

  where Δ=ωq​−ωdrive​≈0 during free evolution

If your simulated data is noisy or undersampled, a simple exponential fit might not extract T2,echo​ correctly.

• Fit only the envelope of the oscillations.

• Use an analytic model for echo decay, or filter out high-frequency oscillations before fitting.