Dynamical symmetries and sub-recoil laser cooling

The authors consider an n-level quantum system with Hamiltonian H and density matrix ρ in which the dynamics are generated by the Liouville equation. The n2-dimensional vector space V on which H and ρ act is spanned by a representation of the Lie algebra u(n). Dynamical symmetries are manifest in a Hamiltonian living in a subspace of V which is spanned by a subalgebra of u(n). This gives rise to certain constants of motion, the exact nature of which depends on the structure of the sub-algebra. The authors have developed a procedure to automate the generation of maximal subalgebra chains of u(n) and the detection of dynamical symmetries and their corresponding constants of motion. By way of example, they consider several configurations for sub-recoil laser cooling by means of velocity selective coherent population trapping, in which the states of the system are products of internal atomic states and external momentum states. In particular they find that for certain values of the family momentum, the Hamiltonian exhibits additional dynamical symmetry, leading to additional constants of motion, to which the authors attribute the coherent population trapping in the velocity selective dark state