Identification of dynamical Lie algebras for finite-level quantum control systems

The problem of identifying the dynamical Lie algebras of finite-level quantum systems subject to external control is considered, with special emphasis on systems that are not completely controllable. In particular, it is shown that the dynamical Lie algebra for an N-level system with symmetrically coupled transitions, such as a system with equally spaced energy levels and uniform transition dipole moments, is a subalgebra of so(N) if N = 2l + 1, and a subalgebra of sp(l) if N = 2l. General criteria for obtaining either so(2l + 1) or sp(l) are established.