Attosecond light matter interaction is a developing area in the field of strong laser matter inter action With attosecond spectroscopy, the motion of electrons in atoms and molecules and can be observed along with the long term effects in molecular vibration due to the driven electrons For an adequate theoretical and computational description of the strong field effects in molecules, the time dependent solution for a driven many body system is essential, in spite of the conceptual, numerical and computational difficulties involved The ( t method for quantum dynamics with general time dependent Hamiltonians in the context of laser atom, laser molecule interactions is developed 1 2 3 The evolution operator for the ( t method requires a huge storage space with a large operation count for the propagation The analytical block diagonalization of the Floquet Hamiltonian is proposed for the evolution of the time dependent Hamiltonian This combination of the split operator method with the block diagonalization technique of the chosen order is approximate the full evolution operator The number of operations are drastically reduced using new algorithm 4 Hence, only matrices of the order of the number of position basis functions need to be stored Thus, the presented algorithm is an effective tool for solving the ( t t problem for interactions with a bi chromatic laser, different symmetries of laser fields, and a single frequency laser pulse with explicit interactions of the pulse envelope