(Fabry-)pyrot is an open-source python package for the physics of one-dimensional (1D) Fabry-Pérot cavities containing two-level atoms interacting with the light-field.
Currently, it implements the following features:
- Transfer matrix formalism (also known as Parratt's formalism) to calculate cavity scattering.
- Linear dispersion theory to compute linear scattering (i.e. in the weak driving limit) when two-level atoms are present inside the cavity.
- Calculation of the cavity Green's function via a recursive algorithm.
- Via the Green's function, one can set up Markovian Master equations for the atom ensemble at weak light-matter coupling.
Note that this software and its algorithms are mainly designed with the goal of transparent physics and to illustrate theoretical concepts. It is not designed for realistic practical applications or numerical efficiency.
It is recommended to first create a virtual environment.
pyrot is available on pip and can be installed using
pip install pyrotA detailed documentation is currently not available and will be added at a
later time. The current main documentation is given in form of explanatory jupyter notebooks,
which can be found in demo/.
As a short documentation on the underlying algorithms, mathematical details and physics background is given in the following resources:
- Parratt's formalism is particularly known from x-ray scattering on thin films. It essentially solves Maxwell's equations for a stack of layers with a given refractive index. The formalism works recursively by adding up all the paths between the layer interfaces, whose response is encoded in their Fresnel coefficients. [see https://doi.org/10.1103/PhysRev.95.359]
- The transfer matrix formalism can be considered a rewriting of Parratt's formalism, which packs the recursively algorithm into a Matrix multiplication. In addition, it allows for polarization. However, this package deals with normal incidence only (for now) and polarization is ignored (for now). [see e.g. https://en.wikipedia.org/wiki/Transfer-matrix_method_(optics) and references therein]
-
Linear dispersion theory is a method to encode the response of level schemes (such as atoms) into a
frequency dependent refractive index. This simplification is possible at weak excitation, where the
level schemes behave like classical oscillators, such that the response is linear in the
excitation field. It relies on the approximation
$\langle\hat{a}(t)\hat{\sigma}^-(t)\rangle\approx-\langle\hat{a}(t)\rangle$ or similar formulations. [see e.g. https://doi.org/10.1103/PhysRevLett.64.2499, https://doi.org/10.1103/PhysRevA.93.012120, https://doi.org/10.1103/PhysRevX.10.011008, https://doi.org/10.1103/PhysRevResearch.2.023396] - The classical electromagnetic Green's tensor is defined by the equation
$$[\nabla\times\nabla\times - \frac{\omega^2}{c^2} \varepsilon(\mathbf{r}, \omega)] \mathbf{G}(\mathbf{r}, \mathbf{r}', \omega) = \delta(\mathbf{r} - \mathbf{r}') ,,$$ Here, we consider the 1D special case, which can be regarded the normal incidence component of a layer stack in Fourier space and is available analytically via a recursive algorithm [https://doi.org/10.1103/PhysRevA.51.2545]. - The Green's function can e.g. be used to set up Markovian Master equations for the atom ensemble in the weak coupling limit. [see https://arxiv.org/abs/0902.3586]
For a summary of these methods and their connection see also https://doi.org/10.11588/heidok.00030671.
The package was released together with https://arxiv.org/abs/2107.11775 and is used therein. If you use pyrot in your research, please cite this preprint or the corresponding journal article once available.
pyrot further builds on techniques developed in https://doi.org/10.1103/PhysRevX.10.011008, https://doi.org/10.1103/PhysRevResearch.2.023396 and summarized in https://doi.org/10.11588/heidok.00030671. Please consider citing these papers if you find them useful.