OBE (Optical Bloch Equation) Python Tools

OBE Tools is a collection of functions written in Python that provide solution methods for the optical Bloch equations (OBEs). Alongside the functions available in the OBE_Tools.py module are two Jupyer notebooks containing more information on how the functions are set up (Functions.ipynb) and examples of how they can be used to model atom-light systems (Examples.ipynb). For more information please see the publication Simple Python tools for modelling few-level atom-light interactions [1].

To run the Jupyter notebooks in an interactive browser window, click here:

Prerequisites

• Numpy
• Scipy (constants, stats, linalg sub-packages)
• Sympy (requires mpmath)
• Matplotlib (for plotting in the Examples notebook)

Background

The optical Bloch equations (OBEs) arise from the Master equation which describes the time dependence of the density matrix through

$$\frac{\partial \hat{\rho}}{\partial t} = -\frac{i}{\hbar}\left[\hat{H}, \hat{\rho}\right] + \hat{\mathcal{L}}(\hat{\rho})$$

where $\hat{H}$ is the Hamiltonian of the atom-light system, $\hat{\rho}$ is the density matrix and $\hat{\mathcal{L}}$ describes the decay and dephasing in the system.

For systems with fewer than 5 levels, analytic solutions can be found, but these require using approximations and so are only valid within certain parameter ranges. The functions in the OBE_Tools module allows the OBEs to be set up and solved numerically for any parameter regime. It uses Sympy to derive the OBEs from an interaction Hamiltonian and decay/dephasing operator, then expresses these as matrix equations and uses a singular value decomposition (SVD) to solve them. It also provides functions for the analytic weak-probe solutions.

Notes

• $\hbar = 1$
• Parameters are in units of $2\pi,\rm{MHz}$ (the $2\pi$ is omitted from plots for clarity)
• Examples are purely illustrative and do not represent any particular physical experiment
• The functions are in no way optimised for speed, they are intended to be as transparent as possible to aid insight.
• Variable names have been chosen to align with the notation used in the paper. Parameters (and hence variables) referring to a field coupling two levels $i$ and $j$ with have the subscript $ij$, whereas parameters that are relevant to a single atomic level $n$ will have the single subscript $n$. Variable names with upper/lower-case letters denote upper/lower-case Greek symbols respectively, for example Gamma denotes $\Gamma$ while gamma denotes $\gamma$.

The code makes a number of assumptions:

• That the number of atomic levels is equal to 1 plus the number of fields. This is consistent with a ladder system in which each level is coupled to the ones above and below it (except for the top and bottom levels).
• That each level only decays to the level below it, and there is no decay out of the bottom level.
• That laser linewidths do not matter (unless they are explicitly specified).

References

[1] Lucy A. Downes, Simple Python tools for modelling few-level atom-light interactions J. Phys. B: At. Mol. Opt. Phys. 56 223001 (2023) https://iopscience.iop.org/article/10.1088/1361-6455/acee3a