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arXiv:2212.13430 · Symmetry

Quantum Fluctuations in the Small Fabry–Perot Interferometer

Игорь Евгеньевич Проценко, Alexander V. Uskov

Published Jan 27, 2023• 3 citations

Abstract

Spectra of the small Fabry–Perot interferometer (FPI) of the size of the order of the wavelength, with the main mode excited by a quantum field from a nano–LED or a laser, are investigated. The input field is detuned from the FPI mode with only a few photons. We formulate the convenient model for the FPI interacting with a quantum field, and provide novel explicit expressions for the field and the photon number fluctuation spectra inside and outside the FPI, with clearly identified contributions of the quantum and the classical noise. As a result, we found the spectra structures are quite different for the field, the photon number fluctuations inside the FPI, for the transmitted and the reflected fields and note asymmetries in spectra. The quantum noise is colored (or white) inside (or outside) the FPI, which explains differences in spectra. As another novel result, we calculate the second-order time auto–correlation functions for the FPI field; they oscillate and are negative under certain conditions. Results will help the study, design, manufacture, and use of the small elements of quantum optical integrated circuits, such as delay lines or optical transistors.

DOI: 10.3390/sym15020346

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Spectra of the small Fabry–Perot interferometer (FPI) of the size of the order of the wavelength, with the main mode excited by a quantum field from a nano–LED or a laser, are investigated. The input field is detuned from the FPI mode with only a few photons. We formulate the convenient model for the FPI interacting with a quantum field, and provide novel explicit expressions for the field and the photon number fluctuation spectra inside and outside the FPI, with clearly identified contributions of the quantum and the classical noise. As a result, we found the spectra structures are quite different for the field, the photon number fluctuations inside the FPI, for the transmitted and the reflected fields and note asymmetries in spectra. The quantum noise is colored (or white) inside (or outside) the FPI, which explains differences in spectra. As another novel result, we calculate the second-order time auto–correlation functions for the FPI field; they oscillate and are negative under certain conditions. Results will help the study, design, manufacture, and use of the small elements of quantum optical integrated circuits, such as delay lines or optical transistors.

We theoretically investigated the bistability in a small Fabry-Perot interferometer (FPI) with the optical wavelength size cavity, the nonlinear Kerr medium, and only a few photons, on average, excited by the external quantum field. Analytical expressions for the stationary mean photon number, the bistability domain, the field, and the photon number fluctuation spectra are obtained. Multiple stationary states of the FPI cavity field with different spectra are possible at realistic conditions, for example, in the FPI with the photonic crystal cavity and the semiconductor-doped glass nonlinear medium.

We calculate the intensity fluctuations of the light transmitted by a Fabry-P\'erot interferometer when a broadband laser is incident upon it. The laser field is assumed to have a Lorentzian spectrum and to be broadened by random amplitude fluctuations (real or complex) or random frequency fluctuations. In each case we obtain closed-form expressions for the intensity autocorrelation function, from which we can obtain the magnitude of the intensity fluctuations, as well as the power spectrum. These results indicate a strong dependence of the transmitted intensity fluctuations on the higher-order statistical properties of the incident field. Agreement with similar results concerning atomic fluorescence fluctuations is very good.

We have measured the power spectrum of the intensity fluctuations of light transmitted by a Fabry-P\'erot interferometer when the input field is the real Gaussian field. The real Gaussian field is a field characterized by real, random (Gaussian) amplitude fluctuations. The bandwidth of the real Gaussian field was varied, taking on values less than that of the interferometer, as well as greater. Comparisons of the measured spectra with calculated spectra are quite satisfactory. Of special interest is a feature in the spectra centered at the laser-interferometer detuning frequency.

Results from a study of quantum noise and its macroscopic manifestations in stimulated Raman scattering are presented.Shot to shot fluctuations in quantum noise initiated Stokes spectra are considered.The Stokes spectra were recorded experimentally using a Fabry-Perot interferometer.For comparison, numerically generated Stokes spectra were produced using the semi-classical model of stimulated Raman scattering with a numerically generated "vacuum" input field.The amount of structure on each spectrum was quantified by calculating the spectrum's standard deviation from its mean frequency.It was found that ensembles with the noisy spectra removed had narrower distributions of mean frequencies than ensembles that included noisy spectra.

A full quantum theory of the Fabry-Perot interferometer is presented. The quantum limits on the determination of small changes in the cavity length are considered, studying the case in which the interferometer is simultaneously excited by coherent and squeezed-vacuum inputs. The contributions to the limiting resolution from photon noise and radiation pressure uncertainty are evaluated in a unified calculation by means of a unitary transformation which relates the input modes to the output modes. These results are compared with the corresponding results for the Michelson interferometer.

The amplitude and phase noise spectra of the internal and external fields of a negative resistance oscillator are derived using electrical circuit theory. It is shown that the difference between internal and external fluctuations becomes important in the quantum domain (h\omega > kT) where external noise sources are not negligible as they often are in the classical domain (h\omega < kT). Using established methods to describe quantum fluctuations in electrical circuits, we obtain results in full agreement with the more exact but less general direct quantum mechanical analysis of a Fabry-Perot laser in Part I.