Abstract: The interaction between two-mode radiation field and several three-level atoms is studied. The variation of the average fidelity of photon cloning with time is calculated, and the numerical calculation is simplified. As for the numerical calculation of the evolution of quantum states, it is generally calculated in the whole Hilbert space. If the spatial dimension of the system is large, the numerical calculation needs to process a large amount of data and consumes a lot of calculation time. Sometimes the numerical calculation can be simplified according to the special physical properties of the system. If the invariant subspace of each quantum state is found in the whole Hilbert space, each quantum state will always evolve in its own invariant subspace under the action of the time-evolution operator. It is thus not necessary to numerically calculate the evolution of quantum states in the whole space, but only in the subspace where the initial state is located. Through comparative analysis, it is found that when numerically calculating the cloning of photons by a few three-level atoms, the amount of data to be processed when computing in the whole space is one order of magnitude larger than that in the all subspaces.

Key words: quantum cloning, stimulated emission, cloning fidelity, invariant subspace