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At the heart of quantum technology development is the control of quantumsystems at the level of individual quanta. Mathematically, this is realisedthrough the study of Hamiltonians and the use of methods to solve the dynamicsof quantum systems in various regimes. Here, we present a pedagogicalintroduction to solving the dynamics of quantum systems by the use of a Liealgebra decoupling theorem. As background, we include an overview of Lie groupsand Lie algebras aimed at a general physicist audience. We then prove thetheorem and apply it to three well-known examples of linear and quadraticHamiltonian that frequently appear in quantum optics and related fields. Theresult is a set of differential equations that describe the most Gaussiandynamics for all linear and quadratic single-mode Hamiltonian with generictime-dependent interaction terms. We also discuss the use of the decouplingtheorem beyond quadratic Hamiltonians and for solving open-system dynamics.

Solving quantum dynamics with a Lie algebra decoupling method