Realistic Measurement of Phase

The experimental schemes for measuring quantum-mechanical phase properties of light suggested and partly also realized thus far, namely (i) amplification, with the help of a quantum amplifier, of the microscopic field before phase measurement, (ii) heterodyning the field with a strong local oscillator, and (iii) performing two separate homodyne measurements on the field after beam splitting, are compared from a theoretical point of view. They share the common feature that undesired noise enters the experimental setup, which makes the measurement fuzzy. It will be pointed out that all three schemes amount to measuring the Q function of the original field, and hence are fully equivalent. Since the Q function can be interpreted as a smoothed Wigner function, one may associate with the introduced noise a smoothing process in which intrinsically quantum-mechanical features displayed by finer details of the Wigner function - especially by the occurrence of negative values - are lost. As a consequence, the measured phase distribution will be broader than the "true" one based on the concept of a quantum-mechanical phase operator. In realistic experiments, the nonunit detection efficiency further deteriorates the measuring results. It will be shown that also this effect can be properly described by an (additional) smoothing process leading to a certain s-parametrized quasiprobability distribution, with a parameter s that is connected with the detection efficiency in a simple way, as the distribution that is actually measurable.