Supercanonical transformations and time-dependent Hamiltonians

Abstract

Starting from general self-adjoint linear combinations of generators of the superalgebra a time-dependent Hamiltonian of a supersymmetric quantum mechanical system is defined by computing the supercommutator of the linear forms. The resulting Hamiltonian is given by the sum of two quadratic forms in odd, respectively even, generators of the superalgebra describing a class of systems containing bosons and fermions. Linear supercanonical transformations of wave vectors leave invariant a Heisenberg superalgebra and belong to the supergroup OSp. The equations of motion for the supercanonical transformations in the Heisenberg picture are shown to be systems of ordinary differential equations. The unitary time evolution operator is constructed using the adjoint map. For periodic Hamiltonians, it is shown that this is a procedure to obtain effective Floquet Hamiltonians. The examples show that the known superalgebras-based approaches for the nuclear shell model and the Jaynes-Cummings model are incorporated in the lowest-dimensional cases. The presented approach opens the possibility to study quantum control problems defined by linear combinations of superalgebra generators with Grassmannian coefficients.