REVIEW ARTICLE: Wavefunctions of two-mode states in entangled-state representation

Two-mode entangled-state wavefunctions and with complex variables z and y are introduced and investigated. They are direct analogues of the wavefunctions and in the single-mode case or are linearly related to each other by two-dimensional Fourier transformation. The states and are eigenstates of two pairs of commuting operators (Z,Z†) (or (Q+,P−)) and (Y,Y†) (or (P+,Q−)) to eigenvalues (z,z*) and (y,y*), respectively, and are normalized by means of the two-dimensional delta function. The entangled states and are represented as two limiting cases of two-mode squeezed coherent states. Different representations of these states, in particular, the analogue of the Agarwal–Simon representation of squeezed vacuum states, are derived and the properties of these states are discussed. The transition from the single-mode to the two-mode case is made using the common property of the squeezing operators to belong to different realizations of the abstract SU(1,1) group. The Wigner quasiprobability in representation of the states and is discussed and is explicitly calculated for two-mode squeezed vacuum (and squeezed coherent) states.