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Journal of the Optical Society of America B

Quantum noise in spectral filtering of light

L. Knöll, W. Vogel, D.‐G. Welsch

Published Oct 1, 1986• 66 citations
Publisher

Abstract

The quantum-theoretical description of the action of a spectral filter is given. It is shown that spectral filtering of light is unavoidably connected with quantum noise, which prevents the spectrometer from performing a Fourier analysis even under ideal conditions. The quantum noise is important in the study of spectral properties of nonclassical light fields, such as spectral squeezing in single-atom resonance fluorescence.

DOI
10.1364/josab.3.001315

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The quantum-theoretical description of the action of a spectral filter is given. It is shown that spectral filtering of light is unavoidably connected with quantum noise, which prevents the spectrometer from performing a Fourier analysis even under ideal conditions. The quantum noise is important in the study of spectral properties of nonclassical light fields, such as spectral squeezing in single-atom resonance fluorescence.
The problem of spectral filtering of quantized light fields is studied, based on the recently developed quantum-optical theory of the action of passive, lossless optical systems [L. Kn\"oll, W. Vogel, and D.-G. Welsch, Phys. Rev. A 36, 3803 (1987)]. Expressions for the operator of the electric field strength of the light and the normally and time-ordered field-correlation functions are derived for the case of a Fabry-P\'erot interferometer being present. Various kinds of field decomposition that are usually considered in classical optics are studied. The results are compared with the Fourier approach to spectral properties of light. It is shown that, dependent on the experimental scheme used, new quantum effects appear, which may prevent the observation of the Fourier structure of the light as predicted from classical optics. Quantitatively this is demonstrated for the example of spectral squeezing in resonance fluorescence, where significant discrepancies between the measured and the full Fourier spectrum are found.
Based on the measurement of quantum correlation functions, the quantum statistical properties of spectral measurements are studied for broadband radiation fields. The spectral filtering of light before its detection is compared with the direct detection followed by the spectral analysis of the recorded photocurrents. As an example, the squeezing spectra of the atomic resonance fluorescence are studied for both types of filtering procedures. The conditions for which the detection of the nonclassical signatures of the radiation is possible are analyzed. For the considered example, photocurrent filtering appears to be the superior option to detect nonclassicality due to the vacuum-noise effects in the optical filtering.
We study the two-time correlation function of spectrally filtered light from a quantum mechanical light source. The bunching and anti-bunching properties of the photo-statistics of a field display the quantum nature of the source. We study whether these properties are affected by the presence of filters that are modelled by dielectric slabs. We quantize the radiation field in the presence of the filters and express the correlation function in terms of source operators. As a quantum source we take the resonance fluorescence of a driven two-level atom. We find that the spectral correlations depend on the linewidth of the filters. When the pass time of the filter is much longer than the time scale on which the source evolves, the presence of the filter can conceal the quantum nature of the source. However, quantum mechanical correlations between opposite side bands survive even for very narrow filters.
A theoretical investigation of spectrally filtered broadband light fields is undertaken. More specifically, the balanced homodyne correlation measurement scheme is extended using spectral filters. The technique provides a way to spectrally resolve correlation functions, making it possible to measure the squeezing spectrum. Two-level resonance fluorescence is included as a concrete example of this, providing some predictions concerning the parameters of the setup for potential experiments. Furthermore, spectral filtering is compared to electronically filtering the current generated in the photodetectors. This is of interest since filtering the current is a classical operation, whereas having optical filters in the path of the radiation is a quantum one.
The problems concerning the influence of spectral filters on the quantum properties of light have recently attracted great attention in connection with quantum cryptography and quantum data transmission. In this paper, we consider the influence of a spectral filter on the second-order coherence function of a field of a resonator mode and a two-level atom in the framework of the Jaynes-Cummings model. Since the Heisenberg equations for the operators of the field of the resonator mode and the atom can be solved exactly, it is possible to obtain exact analytical Fourier transformation of the dynamics of operators of the resonator mode and two-level atom. We demonstrate that the second-order coherence function of the resonator mode and the two-level atom is equal to zero for all possible frequencies in the spectrum of operator oscillations. We find the interbeam second-order coherence function between different frequencies of the Fourier spectrum and show that in the limit of a large number of quanta, it can take the values in the range from zero to two. Thus, non-classical correlations are formed between certain frequencies in the Fourier spectrum of emitted light. We demonstrate that in the limit of a large number of quanta in the resonator mode, when the filter sums up the frequencies near the resonator eigenfrequency, the second-order coherence function of the field of the resonator mode is not affected by the interaction with the two-level atom.