Algebraic and Numerical Algorithms for Quantum Evolution

Quantum evolution requires the manipulation of infinite series of products of non-commuting operators. Lie algebra techniques allow us to reduce the time-dependent operator calculus to the solution of a set of coupled differential equations for scalar functions, while automatically guaranteeing unitarity of the factorized evolution operator. We discuss the development of computer programs to implement this technique of combined factorization and application to quantum states. We use them on driven anharmonic and optomechanical oscillators to find how transition probabilities in these systems evolve in time.