Finite-dimensional Lie algebras in bosonic quantum dynamics: The single-mode case

We study, classify, and explore the mathematical properties of finite-dimensional Lie algebras occurring in the quantum dynamics of single-mode and self-interacting bosonic systems. These Lie algebras are contained in the real skew-hermitian Weyl algebra hat{A}₁, defined as the real subalgebra of the Weyl algebra A₁ consisting of all skew-hermitian polynomials. A central aspect of our analysis is the choice of basis for hat{A}₁, which is composed of skew-symmetric combinations of two elements of the Weyl algebra called monomials, namely strings of creation and annihilation operators combined with their hermitian conjugate. Motivated by the quest for analytical solutions in quantum optimal control and dynamics, we aim at answering the following three fundamental questions: (i) What are the finite-dimensional Lie subalgebras in hat{A}₁ generated by monomials alone? (ii) What are the finite-dimensional Lie subalgebras in hat{A}₁ that contain the free Hamiltonian? (iii) What are the non-abelian and finite-dimensional Lie subalgebras that can be faithfully realized in hat{A}₁? We answer the first question by providing all possible realizations of all finite-dimensional non-abelian Lie algebras that are generated by monomials alone. We answer the second question by proving that any non-abelian and finite-dimensional subalgebra of hat{A}₁ that contains a free Hamiltonian term must be a subalgebra of the Schrödinger algebra. We partially answer the third question by classifying all nilpotent and non-solvable Lie algebras that can be realized in hat{A}₁, and comment on the remaining cases. Finally, we also discuss the implications of our results for quantum control theory. Our work constitutes an important stepping stone to understanding quantum dynamics of bosonic systems in full generality.