Numerical modeling of laser cooling in molecules: From simple diatomics to polyatomics and radioactive species

TL;DR

The paper addresses the challenge of predicting laser cooling dynamics in molecules with extensive internal structure and introduces MoleCool, a Python toolbox that unifies rate-equation and optical Bloch equation approaches for multi-level molecular systems. It demonstrates the framework through educational and species-specific examples (BaF, YbOH, RaF), covering optical cycling, EIT, slowing with frequency chirps, and bichromatic forces while incorporating full hyperfine and rotational structure. The results show how to quantify population dynamics, photon scattering, and optical forces in realistic molecules, enabling detailed trajectory calculations and optimization of cooling schemes. The work provides a practical, extensible tool for both teaching and advancing molecular laser cooling experiments, with potential integration with Bayesian optimization and broader applications beyond diatomic species.

Abstract

Optical Bloch equations and rate equations serve as powerful tools to model light-matter interactions from textbook-like two-level atoms to the complex internal dynamics of molecules. A particular challenge in this context is posed by molecular laser cooling, where many dozens or hundreds of levels need to be taken into account for a comprehensive modeling. Here, we present MoleCool, a numerically efficient Python toolbox to implement and solve the corresponding differential equation systems. We illustrate both the capabilities of the toolbox and some of the intricacies of molecular laser cooling by educational examples, which range from simple Rabi oscillations to spontaneous and coherent cooling schemes for various currently studied or considered molecular species. This includes, in particular, a comprehensive modeling of laser cooling dynamics with full hyperfine structure resolution in radioactive radium monofluoride (RaF), as well as studies of other complex species such as barium monofluoride (BaF) and ytterbium monohydroxide (YbOH).

Figures (10)

• Figure 1: Core classes, their relationships, important methods, properties, and attributes of the Python package MoleCool. This object-oriented simulation toolbox enables simulating internal dynamics of multi-level systems interacting with a laser light and magnetic field via the rate equations and the OBEs. Class composition and inheritance is denoted by arrows with diamonds and open tips, respectively. Methods are indicated by parenthesis.
• Figure 2: Rabi oscilations in a simple two-level system for three different detunings $\Delta$. The damping of the oscillation amplitude is a result of the finite lifetime of the excited state. On resonance ($\Delta = 0$), the excited-state population approaches its maximum steady-state value of $n^{\text{e}} = 1/2$. For finite detuning, the oscillation frequency increases while the average scattering rate $\Gamma n^{\text{e}}$ decreases.
• Figure 3: Real (dispersion) and imaginary part (absorption) of the density matrix element (susceptibility) in comparison for a simple 1+1 two-level system and a 2+1 three-level system demonstrating the effect of electromagnetically-induced transparency. The pump laser in the 2+1 system is assumed to be on resonance with its pump transition and only the probe laser is detuned.
• Figure 4: Steady-state scattering rate $R_\text{sc}$ as a function of Larmor precession frequency $\omega_L$ for several Rabi frequencies $\Omega$ in a $3+1$ system with excited state lifetime $\Gamma$. In such a type-II transition, the usual saturation effect that manifests in $R_\text{sc}$ for high intensities (or, equivalently, high $\Omega$) disappears. For each Rabi frequency, $R_\text{sc}$ peaks at a distinct magnetic field strength (or, equivalently, $\omega_L$), but then strongly decays. This behavior is typical for a system with dark state remixing.
• Figure 5: (a) Example $12+4$ level system. Specific parameters are shown for BaF molecules, but the structure of the rotationally-closed $X^{2}\Sigma^{+}$($N=1$) $\rightarrow$$A{}^{2}\Pi_{1/2}$($J=1/2^+$) transition is the same for all other common monohalides considered for laser cooling. (b) Steady-state scattering rate $R_\text{sc}$ as a function of magnetic-field strength $B$ for several cooling-laser intensities $I_\text{tot}$ in this level system. As in Fig. \ref{['fig:Fig4_3+1system_B-field']}, the usual saturation of $R_\text{sc}$ at high intensities disappears for certain magnetic field strengths. In addition, sharp dips in the scattering rate emerge. These dips are reminiscent of EIT, occurring when the Larmor frequency matches the laser detuning from a power-broadened neighboring transition.