Energy Variance in Decoherence

We study the effect of the initial-state energy variance to the short-time behavior of the Loschmidt echo (LE) in a purely dephasing model. We find that the short-time LE behaves as a Gaussian function with the width determined by the initial-state energy variance of the interaction Hamiltonian, while it is a quartic decaying function with the width determined by the initial-state energy variance of the commutator between the interaction Hamiltonian and the environmental Hamiltonian when the initial state is an eigenstate of the interaction Hamiltonian. Furthermore, the Gaussian envelope in the temporal evolution of LE in strong coupling regime is determined by the inband variance. We will also verify the above conclusion in the XY spin model (as environment).