Preparation of cluster states with trapped electrons on a liquid helium surface

We present a scheme for the preparation of one-dimensional （1D） and two-dimensional （2D） cluster states with electrons trapped on a liquid helium surface and driven by a classical laser beam. The two lowest levels of the vertical motion of the electron act as a two-level system, and the quantized vibration of the electron along one of the parallel directions （the x direction） serves as the bosonic mode. The degrees of freedom of the vertical and parallel motions of the trapped electron can be coupled together by a classical laser field. With the proper frequency of the laser field, the cluster states can be realized.