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Measurement of very small phase fluctuations by means of the operational approach

A. Bandilla

Published May 1, 1994• 0 citations
OpenAlex

Abstract

Recently Noh, Fourgeres and Mandel (NFM) have improved the operational approach to the quantum phase problem substantially and measured the phase dispersion of coherent light down to very small means photon numbers of the order of 10(exp -2). This has prompted many other investigations and clarified some important questions in relation to what is actually measured. Although their treatment is rather general, we confine ourselves here to the case of a strong local oscillator (LO) and reproduce their measurement scheme. Surprisingly enough, this simultaneous measurement of the sine and the cosine of the phase difference is completely equivalent to an old proposal to measure the phase after strong linear amplification realized experimentally by the Welling group. The reason for this rests on the fact, that in both cases the results are determined by the Q function of the signal. This was shown for amplification in and for the measurement after beam splitting by Lai and Haus.

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Quantum and Semiclassical Optics Journal of the European Optical Society Part B•Jun 1, 1995•36 citations
Recently Noh, Fourgeres and Mandel (NFM) have improved the operational approach to the quantum phase problem substantially and measured the phase dispersion of coherent light down to very small means photon numbers of the order of 10(exp -2). This has prompted many other investigations and clarified some important questions in relation to what is actually measured. Although their treatment is rather general, we confine ourselves here to the case of a strong local oscillator (LO) and reproduce their measurement scheme. Surprisingly enough, this simultaneous measurement of the sine and the cosine of the phase difference is completely equivalent to an old proposal to measure the phase after strong linear amplification realized experimentally by the Welling group. The reason for this rests on the fact, that in both cases the results are determined by the Q function of the signal. This was shown for amplification in and for the measurement after beam splitting by Lai and Haus.
We address the problem of strong local oscillator limit of the recently introduced operational approach to phase uncertainty measurements. In this limit the operator character of the local oscillator mode is neglected and its coherent amplitude tends to infinity. Thereby reasonable new operators for the sine and the cosine of the phase difference between an arbitrary signal field and the coherent local oscillator field are found which describe an already well-known physical situation connected with strong amplification of the signal field.
Generalizing a recent theoretical result obtained by Freyberger and Schleich [Phys. Rev. A 47, R30 (1993)], we show that the phase-measurement scheme proposed and realized by Noh, Foug\`eres, and Mandel [Phys. Rev. Lett. 67, 1426 (1991); Phys. Rev. A 45, 424 (1992)] amounts to measuring the Q function for the light under investigation, provided the reference beam (used for homodyne detection) is a very strong coherent field so that it can be described classically. The desired phase distribution follows from the Q function by averaging over the field amplitude. Since an analysis of an earlier proposal by Bandilla and Paul [Ann. Phys. (Leipzig) 23, 323 (1969)] to measure phase distributions via amplification led to just the same result, a perfect physical equivalence of those two approaches has thus been established.
We examine the problem of determining the phase difference between two optical fields, first for classical and later for quantum fields, by reference to two simple measurement schemes that yield the sine and/or cosine of the phase difference between classical fields. We show that certain difficulties exist even within the framework of semiclassical radiation theory when the field is very weak, and particularly when amplitude and phase fluctuations are correlated. We find that a clear distinction has to be made between the measured values of the sine or cosine and the values that can be inferred from a series of repeated measurements. A corresponding distinction can be made also for a quantum field, although the interpretation is not the same. The dynamical variables chosen to represent the cosine and sine that emerge from the discussion of the measurement schemes commute when the sine and cosine are obtained together, but not when the measurement yields one or the other. These sine and cosine operators have well-defined values only when there is a large dispersion of the photon number. We arrive at expressions for the moments of the measured and of the inferred sines and cosines that differ from most previous treatments. The expressions are applied to optical fields in several different quantum states. Only for the Fock state and for the so-called phase state, which was treated recently at some length by Pegg and Barnett [Phys. Rev. A 39, 1665 (1989)], do the measured and the inferred moments coincide. Our analysis of the problem of phase measurement leads to the conclusion that the appropriate dynamical variables for the measured sine and cosine depend on the measurement scheme, and that different schemes correspond to different operators.
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 echect 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.
In papers by Lynch [ Phys. Rev. A 41 2841 (1990)] and Gerry and Urbanski [ Phys. Rev. A 42 662 (1990)] it has been argued that the phase-fluctuation laser experiments of Gerhardt, Büchler and Lifkin [ Phys. Lett. 49A 119 (1974)] are in good agreement with the variance of the Pegg–Barnett phase operator for a coherent state, even for a small number of photons. We argue that this is not conclusive. In fact, we show that the variance of the phase in fact depends on the relative phase between the phase of the coherent state and the off-set phase ϕ 0 of the Pegg–Barnett phase operator. This off-set phase is replaced with the phase of a reference beam in an actual experiment and we show that several choices of such a relative phase can be fitted to the experimental data. We also discuss the Noh, Fougères and Mandel [ Phys. Rev. A 46 2840 (1992)] relative phase experiment in terms of the Pegg–Barnett phase taking post-selection conditions into account.