# Resonator punchout#

With the resonator spectroscopy at high power we found the bare resonator frequency (also called high power frequency), but this is not the frequency that will be used to perform measurements. We need to find the frequency, where the resonator is actually coupled to the qubit. To do that, we first have to find the correct amplitude of the readout pulse. This experiment can be initially performed with a VNA, to check that everything is working and to obtain a rough estimation of the parameters to be used by the control devices. We do again a resonator spectroscopy experiment, measuring at different frequencies, but this time scanning in a narrow frequency span and also for different pulse amplitudes. We expect the resonator frequency to have a strong dependency on the amplitude: in particular we expect it to be fixed at first (in the high power regime) then undergo a transition phase and then be fixed again at a different frequency because of the coupling with the qubit.

This is and example of a runcard for a resonator punchout:

```
platform: <platform_name>
qubits: [0]
actions:
- id: resonator punchout
operation: resonator_punchout
parameters:
freq_width: 40_000_000
freq_step: 500_000
amplitude: 0.03
min_amp_factor: 0.1
max_amp_factor: 2.4
step_amp_factor: 0.3
nshots: 2048
relaxation_time: 5000
```

Eventually, we want to have a plot like the one below:

This experiment is the first one where we are actually “seeing” the qubit and it’s extremely important also to check that the qubit is working properly. During characterization, various experimental problems can happen and can lead experimentalists to believe that the qubit is no longer working: this experiment gives us an easy way to check it.

Compared to a low power resonator spectroscopy, with punchout we can see the shift in the resonator frequency and thus ensure that we are in the low power regime.

Moreover, from this experiment we can actually already have an estimate of the qubit frequency using [3, 4]:

Where \(\Delta = \omega_{rl} - \omega_q\).

\(\omega_{rh}\) is the resonator frequency in the high power regime, and \(\omega_{rl}\) is the resonator frequency at low power. So, if we know, maybe from design specifications, the expected value of \(g\), we can have an estimate of the qubit frequency. If, as in most cases, we don’t have information on \(g\), then we still can infer if the qubit frequency is higher or lower than the resonator frequency.

To obtain a clear plot, this experiment usually requires multiple tries, to choose “good parameters”. In particular:

the amplitude range and step must be chosen carefully because punchout can lead to extremely long experiments (that we would like to avoid), but large steps will inevitably confuse the plot;

the frequency usually shifts very little, but depending on \(g\) and \(\Delta\), so it is difficult to set the frequency span and step;

if the scan in amplitude is linear, it will be difficult to have a clear view of all the three regimes at the same time. If it is possible, it may be worth to do a logarithmic scan.

Note also that, in literature, this plot is often presented as a scan in the attenuation of the readout line. The effect is the same as of changing the amplitude (although it is by default a logarithmic scan), but the plot is reversed: at the top (high attenuation) we will see the low power regime and at the bottom (low attenuation) the high power one.

Note that in the plots measured amplitudes are normalized for every readout amplitude and we actually have very high amplitudes in high power, and much lower amplitude elsewhere. From these plots we can extract few things:

the pulse amplitude that we will use for the next experiments for the readout pulse;

the peak frequency (ground state frequency).

We should try to use the amplitude that maximizes the signal to noise (S/N) ratio for the resonator, so in general the highest amplitude at low power. However, we still need to be sure to not enter the transition regime, something that could lead to very noisy experiments and eventually prevent us to reach any sensible result. To be sure that, at the given pulse amplitude, we are not in the transition regime, we can repeat the standard resonator spectroscopy with a finer scan, checking if the peak is Lorentzian. A resonator spectroscopy is also useful to better check the resonator frequency at low power. We can also write down the maximum value of the peak. Since we are not interacting directly with the qubit, we are effectively measuring the amplitude of the ground state and, from now on we will not change the resonator frequency, so we expect to always measure this amplitude for the zero state. Here it is not needed to do a precise measurement, but it is nevertheless very useful to have an approximate value, that can later be used to check, in other experiments, that we are not exciting the qubit by error (if we see a change in amplitude, then maybe the qubit state has changed).

It is also possible to run the punchout experiment with attenuation:

This is and example of a runcard for a resonator punchout with attenuation:

```
platform: <platform_name>
qubits: [0]
actions:
- id: resonator_punchout_attenuation
operation: resonator_punchout_attenuation
parameters:
freq_width: 10_000_000
freq_step: 500_000
max_att: 60
min_att: 4
nshots: 1000
step_att: 4
nshots: 2048
```

References

Alexandre Blais, Ren-Shou Huang, Andreas Wallraff, S. M. Girvin, and R. J. Schoelkopf. Cavity quantum electrodynamics for superconducting electrical circuits: an architecture for quantum computation. *Physical Review A*, jun 2004. URL: https://doi.org/10.1103%2Fphysreva.69.062320, doi:10.1103/physreva.69.062320.

Yvonne Y. Gao, M. Adriaan Rol, Steven Touzard, and Chen Wang. A practical guide for building superconducting quantum devices. 2021. arXiv:2106.06173.

Amy Greene. *Calibration and Utilization of High-Fidelity Two-Qubit Operations*. PhD thesis, Massachusetts Institute of Technology, 2008.

Xiu Gu, Anton Frisk Kockum, Adam Miranowicz, Yu-xi Liu, and Franco Nori. Microwave photonics with superconducting quantum circuits. *Physics Reports*, 718-719:1–102, November 2017. doi:10.1016/j.physrep.2017.10.002.

Andreas Wallraff, David I Schuster, Alexandre Blais, Luigi Frunzio, R-S Huang, Johannes Majer, Sameer Kumar, Steven M Girvin, and Robert J Schoelkopf. Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics. *Nature*, 431(7005):162–167, 2004. doi:https://doi.org/10.1038/nature02851.