QUANTUM STATES AND NUMBER-PHASE UNCERTAINTY RELATIONS MEASURED BY OPTICAL HOMODYNE TOMOGRAPHY

Experiments have been performed to determine the Wigner distribution a nd the density matrix (and for pure states the wave function) of a lig ht mode, by using tomographic inversion of a set of measured probabili ty distributions for quadrature amplitudes. From these measurements th e quantum distributions of optical phase and photon number have been o btained. The measurements of quadrature-amplitude distributions for a temporal mode of the electromagnetic field are carried out using balan ced homodyne detection. We refer to this new method as optical homodyn e tomography. Given the measured density matrix, one can experimentall y infer any of the various quantum distributions of optical phase, in particular the Pegg-Barnett (or, equivalently, Shapiro-Shepard) phase distribution, the marginal Wigner distribution, and the Vogel-Schleich operational phase distribution. We have used this approach to make me asurements of the number-phase uncertainty relation for coherent-state fields. The coherent states do not attain the minimum value for the n umber-phase uncertainty product, as set by the expectation value of th e commutator of the number and phase operators; this is true theoretic ally and experimentally.