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Journal of Modern Optics

Quantum Theory of Multiport Optical Homodyning

Nigel G. Walker

Published Jan 1, 1987• 132 citations
Publisher

Abstract

Multiport optical homodyning concerns the combining of a local oscillator with N − 1 signals of the same frequency in a lossless 2N-port junction or an N × N directional coupler. An operator representation is derived for quantum measurements made on these signals in the limit of a strong local oscillator. The local oscillator may otherwise be in an arbitrary quantum state.

DOI
10.1080/09500348714550131

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Multiport optical homodyning concerns the combining of a local oscillator with N − 1 signals of the same frequency in a lossless 2N-port junction or an N × N directional coupler. An operator representation is derived for quantum measurements made on these signals in the limit of a strong local oscillator. The local oscillator may otherwise be in an arbitrary quantum state.
The eight-port homodyne detection apparatus is analyzed in the framework of the operational theory of quantum measurement. For an arbitrary quantum noise leaking through the unused port of the beam splitter, the positive operator valued measure and the corresponding operational homodyne observables are derived. It is shown that such an eight-port homodyne device can be used to construct the operational quantum trigonometry of an optical field. The quantum trigonometry and the corresponding phase space Wigner functions are derived for a signal field probed by a classical local oscillator and a squeezed vacuum in the unused port.
The eight-port homodyne detection apparatus is analysed in the framework of the operational theory of quantum measurement. For an arbitrary quantum noise leaking through the unused port of the beam splitter, the positive operator valued measure and the corresponding operational homodyne observables are derived. It is shown that such an eight-port homodyne device can be used to construct the operational quantum trigonometry of an optical field. The quantum trigonometry and the corresponding phase space Wigner functions are derived for a signal field probed by a classical local oscillator and a squeezed vacuum in the unused port.
The fields of precision timekeeping and spectroscopy increasingly rely on optical-frequency-comb interferometry. However, comb-based measurements are not described by existing quantum theory because they exhibit both large mode mismatch and finite-strength local oscillators. To establish this quantum theory, we derive measurement operators for homodyne detection with arbitrary mode overlap. These operators are a combination of quadrature and intensity-like measurements, which inform a filter that maximizes the quadrature-measurement signal-to-noise ratio. Furthermore, these operators establish a foundation to extend frequency-comb interferometry to a wide range of scenarios, including metrology with nonclassical states of light.
The theory of balanced homodyne and heterodyne detection is developed for inputs in which the signal field is in an arbitrary quantum state and the local-oscillator field is in a highly excited coherent state. Exact expressions are derived for the photocount moment-generating functions in the special case of a coherent signal. For more general signals, the first two moments of the photocount probability distribution are determined. The moments are evaluated for the examples of a coherent signal with a chaotic noise component, and for squeezed light derived from a degenerate and from a non-degenerate parametric amplifier. The corresponding moments for direct detection are obtained so that comparisons can be made. The Kelley-Kleiner photocount distribution formula is adapted to balanced detection schemes. Light beams are characterized throughout by their energy fluxes, and the theory accordingly describes steady-state experiments.
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.