  In this work we present an effective Hamiltonian description of the quantum
dynamics of a generalized Lambda system undergoing adiabatic evolution. We
assume the system to be initialized in the dark subspace and show that its
holonomic evolution can be viewed as a conventional Hamiltonian dynamics in an
appropriately chosen extended Hilbert space. In contrast to the existing
approaches, our method does not require the calculation of the non-Abelian
Berry connection and can be applied without any parametrization of the dark
subspace, which becomes a challenging problem with increasing system size.
