FWDD

FWDD (Finite Width Dynamical Decoupling) implements Dynamical Decoupling considering the effects of FInite Pulse Widths. All times in the tutorials use microseconds $(\mu s)$.

The code presented here is used in the paper Quantum sensing with a spin ensemble in a two-dimensional material to predict noise spectra.

Installation

You can install fwdd via pip or conda (from Conda-forge).

Via pip

pip install fwdd

Via conda (Conda-forge)

conda install -c conda-forge fwdd

Local Development & Tutorials

If you are running the tutorial notebooks locally and want to make live changes, run an editable installation:

git clone https://github.com/nonohuff/FWDD.git
cd FWDD
pip install -e .

Quick Start

Here is how you can import the core modules of fwdd in your python code:

from fwdd import filter_function as ff
from fwdd import noise_spectra as ns
from fwdd import coherence_profile as cp
from fwdd import noise_learning_fitting as nlf
from fwdd import fitting_utils as fu

Coherence-Noise Relationship

A given dynamical decoupling pulse sequence of $N$ pulses (e.g. CPMG, XY8) yields a corresponding filter function, $F_N(\omega, t)$, which is related to the coherence decay as follows:

$$ \begin{align} C_N(t) = e^{-\chi_N(t)}, \quad \chi(t) = (t/T_{2})^\beta, \ \chi_N(t) = \frac{1}{\pi} \int _{0}^{\infty} d\omega S(\omega) \frac{F_N(\omega,t)}{\omega^2} \end{align} $$

Here, $C_N(t)$ is the coherence decay as a function of time characterized by two important parameters: $T_2$ - coherence time and $\beta$ - stretch factor. $F_N(\omega,t)$ is a filer function defined by the dynamical decoupling pulse sequence, and $S(\omega)$ is the power spectral density of the underlying noise (standard units $\frac{Hz^2}{Hz}$).

Delta Function Approximation

If $F_N(\omega,t)$ is approximated as a $\delta$ function peaked at $\omega_0 =\pi N/T$, where the total experiment time, $T=2N\tau+Nt_{\pi}$, $2\tau$ being the wait time between the pulses and $t_{\pi}$ is the pulse width. Then, the integral above is trival to invert and we have:

$$ \begin{equation} S(\omega) = -\pi \frac{ln(C_N(T))}{T}. \end{equation} $$

This approximation holds true when the $\pi$-pulses are themselves much shorter compared to the delay between them and as a consequence the coherence time, $T_2$.

Finite-width Pulses

When the length of the $\pi$-pulse becomes a sizable fraction of the delay (or $T_2$), such a $\delta$ approximation cannot be made. In that case, the filter function needs to be modified to accommodate for the finite pulse duration and can be expressed as:

$$ F_{N}(\omega,T)=\left|1+(-1)^{N+1}e^{i\omega T} +2\sum_{k=1}^N(-1)^ke^{i\omega t_{k}}\cos\left(\frac{\omega t_{\pi}}{2}\right)\right|^{2} $$

Where $t_k$ is the time corresponding to the center of the $k^{th}$ pulse, and $t_{\pi}$ is the pulse width.

Tutorial Notebooks

We have included several tutorial notebooks to make adapting this code to your purposes easier. They cover how we implement noise spectral densities, $S(\omega)$, the finite-width filter function, coherence profile, $C_N(t)$, and finally, how we go about fitting noise profiles to observed $C_N(t)$ data (and how to do it for you own data/noise models!). The reccomended viewwing of these notebooks are.

noise_tutorial.ipynb -> filter_function_tutorial.ipynb -> coherence_profile_tutorial.ipynb -> noise_learning_fitting.ipynb

Conventions

• $\omega = 2 \pi f$
• $t = \frac{1}{f}$
• Time varibles assume microseconds, so tau_p = 0.024 means $0.024 \mu s$ or $24 ns$

Citation

If you use fwdd in your research, please cite our work:

@misc{biswas2025quantum,
  title = {Quantum sensing with a spin ensemble in a two-dimensional material},
  author = {Souvik Biswas, Giovanni Scuri, Noah Huffman, and et al.},
  year = {2025},
  eprint = {2509.08984},
  archivePrefix = {arXiv},
  primaryClass = {cs.LG},
  url = {https://arxiv.org/abs/2509.08984}
}