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arXiv:quant-ph/9701011 · WORLD SCIENTIFIC eBooks

Homodyning as Universal Detection

Giacomo Mauro D’Ariano

Published Feb 1, 2005• 0 citations

Abstract

Homodyne tomography - i. e. homodyning while scanning the local oscillator phase - is now a well assessed method for ``measuring'' the quantum state. In this paper I will show how it can be used as a kind of universal detection, for measuring generic field operators, however at expense of some additional noise. The general class of field operators that can be measured in this way is presented, and includes also operators that are inaccessible to heterodyne detection. The noise from tomographical homodyning is compared to that from heterodyning, for those operators that can be measured in both ways. It turns out that for some operators homodyning is better than heterodyning when the mean photon number is sufficiently small. Finally, the robustness of the method to additive phase-insensitive noise is analyzed. It is shown that just half photon of thermal noise would spoil the measurement completely.

DOI: 10.1142/9789812563071_0034

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Homodyne tomography - i. e. homodyning while scanning the local oscillator phase - is now a well assessed method for ``measuring'' the quantum state. In this paper I will show how it can be used as a kind of universal detection, for measuring generic field operators, however at expense of some additional noise. The general class of field operators that can be measured in this way is presented, and includes also operators that are inaccessible to heterodyne detection. The noise from tomographical homodyning is compared to that from heterodyning, for those operators that can be measured in both ways. It turns out that for some operators homodyning is better than heterodyning when the mean photon number is sufficiently small. Finally, the robustness of the method to additive phase-insensitive noise is analyzed. It is shown that just half photon of thermal noise would spoil the measurement completely.

The double-homodyne and the heterodyne detection schemes for phase shifts between two synchronous modes of the electromagnetic field are analyzed in the framework of quantum estimation theory. The probability operator-valued measures (POM's) of the detectors are evaluated and compared with the ideal one in the limit of strong local reference oscillator. The present operational approach leads to a reasonable definition of phase measurement, whose sensitivity is actually related to the output r.m.s. noise of the photodetector. We emphasize that the simple-homodyne scheme does not correspond to a proper phase-shift measurements as it is just a zero-point detector. The sensitivity of all detection schemes are optimized at fixed energy with respect to the input state of radiation. It is shown that the optimal sensitivity can be actually achieved using suited squeezed states.

Quantum noise in optical heterodyning and homodyning, usually analyzed under the assumption that the photodetector output is conditionally Poisson,1,2 is frequently supposed to arise from local oscillator (LO) shot noise. However, a fully quantum mechanical analysis from Yuen and Shapiro3 has shown that it actually arises from the signal quantum fluctuation; in particular homodyning and heterodyning are realizations of abstract quantum measurements of the field quadratures. We analyze the effect of LO noise, both quantum and excess noise, on homodyne and heterodyne detection. For conventional input light, i.e., fields in coherent states or their random superpositions, we demonstrate how a conditionally Poisson calculation would yield the correct answer but incorrect interpretation for the detection quantum noise.

Quantum-mechanical calculations of the mean-square fluctuation spectra in optical homodyning and heterodyning are made for arbitrary input and local-oscillator quantum states. In addition to the unavoidable quantum fluctuations, it is shown that excess noise from the local oscillator always affects homodyning and, when it is broadband, also heterodyning. Both the quantum and the excess noise of the local oscillator can be eliminated by coherent subtraction of the two outputs of a 50-50 beam splitter. This result also demonstrates the fact that the basic quantum noise in homodyning and heterodyning is signal quantum fluctuation, not local-oscillator shot noise.

We use our interpretation of homodyne and heterodyne detection of squeezed states, viz., quantum correlation between photons as expressed in second-order correlation functions, to study various cases. In particular, we introduce the coherence time of the local oscillator and show its influence on the possible noise reduction. We also study the case in which the local oscillator is not much stronger than the squeezed state and analyze the influence of phase jitter. This approach could explain some experimental results and suggest improvements in detection schemes.