Linear Response and Collective Modes

Suppose a system described by an interacting hamiltonian, \(H=H_0+V\) , perturbed by an external force acting at \(t>0\) : \({\tilde{H}}=H-f(t) A=H+\theta (t) V(t),\) where A is any bosonic operator, such as the electron density or spin density. where A is any bosonic operator, such as the electron density or spin density. We would like to find the temporal evolution of the average value of A, A(t) in the Heisenberg representation under the effect of the external time-dependent field. Hence, how do we obtain \(\langle A_H(t) \rangle \) ?. We start by expressing \( A_H(t)\) in the interaction representation: \(A_H(t)= U^\dagger (t,0) A_I(t) U(t,0),\) where \(U(t,t')\) is the solution to the EOM: \(i{\partial U(t,t') \over \partial t} = V_I(t) U(t,t')\) which is valid even if the hamiltonian in the Schrödinger picture is explicitly time-dependent.