We have measured the probability distribution of the phase difference Phi(2) - Phi(1) between two signal fields produced by parametric downconversion from two cascaded nonlinear crystals for several different values of the degree of mutual coherence. The experimental results are compared with theory, based on our operational approach to phase operators for quantum fields, and good agreement is obtained. These results are of interest because they constitute what we believe is the first experimental test of our phase theory for nonclassical states of light.
Although the operational approach to the quantum-phase problem developed earlier by us [Phys. Rev. A 45, 424 (1992)] has now been confirmed experimentally for several different states of the electromagnetic field, so far no measurements have been reported with partially coherent light beams. This situation is remedied in the present work, where we report the results of measurements on light obeying thermal statistics with a variable degree of mutual coherence. Once again, we find agreement between experiment and our theoretical approach.
The problem of identifying the dynamic variable representing the phase difference between quantized electromagnetic fields is approached via the measurement process. By comparison with classical optics, certain operators corresponding to the measured sine and cosine of the phase difference are identified, but different measurement schemes lead naturally to different operators. First and second moments of the sine and cosine operators are calculated for a two-mode field in a coherent state. The theory is then tested in a photon counting experiment, for mean detected photon numbers varying from 30 to 10-2 which extends the range of previous measurements 300- fold. Good agreement is found between theory and experiment.
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 distribntion.We have used this approach to make measurements of the number-plase 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.
The measurement of a phase difference between two optical modes in an interferometer has several important applications in metrology. This measurement can be useful in several contexts, such as in the detection of gravitational waves. The quantum nature of light imposes a limit on how accurately a phase difference can be measured, this limit being known as the Heisenberg limit (HL). On the other hand, photonic quantum metrology studies alternatives to reach the HL through entangled or squeezed quantum states. According to the theory of parameter estimation, the minimum uncertainty possible in the estimation of a phase
θ considering classical light scales with $1/\sqrt{\Bar{N}}$, where $\Bar{N}$ is the average number of photons. This dependence is known as the standard quantum limit. However, when considering quantum light sources, there is a significant improvement in the estimation of the phase parameter, as its minimum uncertainty can scale with $1/{\Bar{N}}$, known as HL. In this thesis, starting from the relative phase probability distribution P(
ϕ) introduced by Luis and S\'anchez-Soto in 1996, we obtained this distribution for several quantum states of light useful in photonic quantum metrology. We show that, within the numerical precision of our calculations, the Fisher information obtained via the relative phase distribution is equal to the quantum Fisher information for the considered states, such that the average differences between these quantities were on the order of
0,1%. Our results indicate that the relative phase distribution can be used to predict the minimum possible uncertainty in the phase detection process in quantum metrology, since this uncertainty depends on the quantum Fisher information, at least for pure states.