$\text{textttSymdyn}$: an automated algebraic solution for high-order quantum systems

Abstract

Many significant quantum physical systems are characterized by Hamiltonians expressible as a linear combination of time-independent generators of a closed Lie algebra, $\hat{H}(t)={\sum }_{l=1}^L{\eta }_l(t){\hat{g}}_l$. The Wei-Norman method provides a framework for determining the coefficients of the corresponding time evolution operator in its factorized representation, $\hat{U}(t)={\prod }_{l=1}^L{e}^{{\Lambda }_l(t){\hat{g}}_l}$. This work introduces $\text{Symdyn}$, a Python library that automates the application of this method. The library efficiently computes similarity transformations and the nonlinear differential equations intrinsic to derive Baker-Campbell-Hausdorff-like relations and the time evolution of high-order quantum systems ($L\ge 6$). We demonstrate its robustness by deriving the time evolution operator for a system of two time-dependent coupled harmonic oscillators. Additionally, we specialize the library to the Lie group $\text{SU}(N)$, showing its versatility with $\text{SU}(2)$, $\text{SU}(3)$ and $\text{SU}(4)$ examples, relevant to quantum computing.