Quantum States and Number-Phase Uncertainty Relations Measured by Optical Homodyne Tomography

Experiments have been performed to determine the Wigner distribution and the density matrix (and for pure states the wave function) of a light mode, by using tomographic inversion of a set of measured probability distributions for quadrature amplitudes. From these measurements the quantum distributions of optical phase and photon number have been obtained. The measurements of quadrature-amplitude distributions for a temporal mode of the electromagnetic field are carried out using balanced homodyne detection. We refer to this new method as optical homodyne tomography. Given the measured density matrix, one can experimentally 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 measurements of the number-phase uncertainty relation for coherent-state fields. The coherent states do not attain the minimum value for the number-phase uncertainty product, as set by the expectation value of the commutator of the number and phase operators; this is true theoretically and experimentally.