Optical Tomograms of Multiple-Photon-Added Gaussian States via the Intermediate State Representation Theory

Optical tomogram of a quantum state can serve as an alternative to its density matrix since it contains all the information on this state. In this paper we use the explicit expression of the intermediate state |q, f, g〉(that is, an eigenvector of the coordinate and momentum operator fQ + gP) and the Radon transform between the Wigner operator Δ(p, q) and the pure-state density operator |q, f, g〉〈q, f, g| to calculate the specific tomograms for multiple-photon-added coherent states (MPACSs), thermal states (MPATSs) and displaced thermal states (MPADTSs). Their analytical tomograms are actually the finite-summation representations related to single-variable Hermite polynomials. Besides, the numerical results indicate that the larger photon-addition number can lead to the multi-peak tomographic distributions for these three states, and the measurable tomograms of MPACSs and MPADTSs exhibit π-periodic features but MPATSs can’t.