IFE states under time dependent Hamiltonians

An interaction-free evolution (IFE) of a quantum system is an evolution which is not influenced by a certain part of the Hamiltonian which is addressed as the interaction term [1]. In other words, the dynamics generated by the ‘unperturbed’ Hamiltonian H0 is essentially the same as the evolution generated by the total Hamiltonian which is the sum of H0 and the interaction term HI: H = H0 + HI. This notion, which has been introduced in Ref. [1], is somehow related to the concept of decoherence-free subspaces (DFS) [2–4]. In spite of such connection, it should be stressed that the two concepts are still different in many aspects. Generally speaking, the notion of IFE can be relevant to composite systems with different dimensions (like a small system and its environment) or with similar dimensions (for example two interacting qubits), but it can even concern different degrees of freedom of the same particle (for example atomic and vibrational degrees of freedom of a trapped ion). One can even talk about IFE states in connection with the action of a classical field on a quantum system, for example a spin under the action of a magnetic field. Subradiance [5], in its original formulation, is surely a very famous phenomenon which can be thought of as an IFE involving a matter system (several atoms) and the vacuum electromagnetic field. Here we study the non trivial extension of IFE states which applies to those cases wherein the system is governed by a time-dependent Hamiltonian. The interest in such a kind of problem is related to several aspects. On the one hand, generally speaking the resolution of dynamical problems with time-dependent Hamiltonians is a tough job due to the highly nontrivial structure of the corresponding solution