Fit and analyze scattering parameter data from resonators.
The package can fit raw VNA data from which the delay has not been removed:
The same data and fit, normalized to the device plane.
Data from a resonator can be fit and analyzed in a few lines of code. For example, to fit a resonator in the shunt-coupled configuration, print data about the fit, then plot the data, fit, and response at resonance in the complex plane, do the following:
from resonator import shunt, see
frequency, s21 = get_the_resonator_data()
lsf = shunt.LinearShuntFitter(frequency=frequency, data=s21)
print(lsf.result.fit_report())
fig, axes = see.real_and_imaginary(resonator=lsf)where frequency is a numpy array of frequencies corresponding to the complex s21 data array.
The scattering parameter models are parameterized in terms of resonator inverse quality factors, which are called losses in the code.
Thus, the fit report above will show parameters called internal_loss and coupling_loss, which are the inverses of the corresponding quality factors.
See below for discussion of this choice.
The fitter object (r above) makes the best-fit parameters as well as quality factors, energy decay rates, and the standard errors of all of these available for attribute access.
Try print(dir(r)) to see a list of the available attributes.
For example, r.Q_i is the internal quality factor, and r.coupling_energy_decay_rate_error is the standard error of the coupling energy decay rate.
This package was developed to fit data from superconducting microwave resonators, which are measured at low temperature in cryostats.
In these systems, measured signals through the microwave lines includes effects such as loss, amplifier gain, phase shifts, and cable delay.
This package can usually fit data from resonators with reasonably high quality factors even when the system gain and phase have not been characterized.
It does this by assuming that the measured scattering parameter is the product of an ideal resonator model, which is what one would measure in an imaginary on-chip microwave measurement, and the transmission of everything else in the system.
The system response is called the background, and the resonator response is called the foreground, so the models are of the form
model = background * foreground.
The user can choose between background models with various free parameters, such as a cable delay or a slope in the transmission magnitude.
Models with more free parameters can be used when the background has more complex structure, while models with fewer free parameters will produce lower error bars if they can adequately describe the data.
The modules reflection.py, shunt.py, and transmission.py contain classes to fit data from resonators in the following coupling configurations: shunt-coupled (signal transmitted past resonator), reflection (signal reflected from resonator), and transmission (signal transmitted through resonator).
The module background.py contains models for everything except for the resonator.
The module see.py contains functions to plot resonator data and fits using matplotlib.
The examples folder contains Jupyter notebooks with detailed examples of fitting.
The fitting is done using lmfit, a fitting package that is built on routines in scipy.optimize but allows for more control and flexibility.
Check out the repository from GitHub:
/directory/for/code$ git clone https://github.com/danielflanigan/resonator.gitI recommend installing the package in editable mode:
/directory/for/code$ pip install -e resonatorInstead of moving the code to the site-packages directory, this command creates a link there that tells Python where to find the code. This makes it easier to pull updates.
The package requirements are lmfit >= 0.9.3, numpy, and matplotlib for the see.py plotting functions.
The code should run in Python 2.7 or 3.6+.
Models for resonators are typically written either in terms of quality factors (e.g. Q_internal, Q_external) or in terms of energy decay rates that are equal to the resonance angular frequency divided by a quality factor (e.g. kappa_external = omega_r / Q_external)).
The resonator models in this package use inverse quality factors, which are called "losses" in the code. These have the expected definition:
loss_x = 1 / Q_x = P_x / (omega_r * E),
where P_x is the power lost to channel x, omega_r is the resonance angular frequency, and E is the total energy stored in the resonator. For calculations, these are more useful than quality factors because energy losses to independent channels simply add. For example, if Q_i is the internal quality factor and Q_c is the coupling (or "external") quality factor, then internal_loss = 1 / Q_i, coupling_loss = 1 / Q_c, and thus total_loss = internal_loss + coupling_loss, the inverse of the total (or "resonator" or "loaded") quality factor. Inverse quality factors are preferred here over the energy decay rates because they are dimensionless.
In order to make this choice transparent to users, the ResonatorFitter class (and thus all of its subclasses) has properties that calculate the quality factors and energy decay rates as well as their standard errors.