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arXiv:quant-ph/0701078 · Journal of Physics B Atomic Molecular and Optical Physics

Input–output relations for multiport ring cavities

Matteo G. A. Paris

Published Feb 1, 2007• 4 citations
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Abstract

Quantum input-output relations for a generic $n$-port ring cavity are obtained by modeling the ring as a cascade of $n$ interlinked beam splitters. Cavity response to a beam impinging on one port is studied as a function of the beam-splitter reflectivities and the internal phase-shifts. Interferometric sensitivity and stability are analyzed as a function of the number of ports.

DOI
10.1088/0953-4075/40/4/f03

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•Apr 19, 2016•0 citations
Quantum input-output relations for a generic nnn-port ring cavity are obtained by modeling the ring as a cascade of nnn interlinked beam splitters. Cavity response to a beam impinging on one port is studied as a function of the beam-splitter reflectivities and the internal phase-shifts. Interferometric sensitivity and stability are analyzed as a function of the number of ports.
The second-order coherence of optical fields at output ports of a beam splitter is investigated. The properties of the second-order coherence are not only strongly dependent on the input states but also quite involved with the nature of the beam splitter. When parameters of both the input states and the beam splitter are chosen appropriately, the fields at the output ports will exhibit quantum behaviour. Between the two output modes, there stands mode–mode correlation which can even be anticorrelation. Some nonclassical properties, such as the photon antibunching effect of the input fields, may be destroyed by the beam splitter.
The present article is concerned with the study of quantum electrodynamics (QED) in interacting Fock space and a generalised approach of modelling quantum feedback systems of the optical QED cavities. The dynamical behaviour and the state space representation in stochastic field of the optical QED system provide the basis for developing quantum feedback QED control system. This study is also motivated by the design of quantum networks of any number of optical QED baths connected in parallel. Nyquist stability of the feedback control system in the interacting Fock space using two-mode beam splitter device at the input and a single-mode fractional beam splitter device at the output of the cavity has been analysed. The degree of stability of the quantum feedback control system of the cavity QED is studied by measuring gain margin and phase margin of the system in terms of parameters of the beam splitter.
Here we introduce interferometric devices by combining optical feedback (cavities) with unbiased multiports, which, unlike traditional beam dividers, allow light to reflect back out of the port from which it originated. By replacing the traditional, directionally biased beam-splitter in a Michelson interferometer with an unbiased multiport, the functional dependence of the scattering amplitudes changes. As a result, the derivative of transmittance with respect to an external phase perturbation can be made substantially large. This significantly enhances the resolution of phase measurement and allows the phase response curves to be altered in real time by tuning an externally controllable phase shift.
In this paper, we present a very simple approach to input-output relations in optical cavities, limiting ourselves to one- and two-photon states of the field. After field quantization, we derive the nonunitary transformation between inside and output annihilation and creation operators. Then we express the most general two-photon state generated by inside creation operators, through base states generated by output creation operators. After renormalization of coefficients of the inside two-photon state, we calculate the outside photon-number probability distribution in a general case; then we treat with some detail the single-mode and symmetrical cavity case. We found that the statistics of emitted photons may exhibit either quantum or classical behavior, depending both on source properties and on cavity characteristics.
An obvious question is, how to relate the internal cavity mode to externallymeasured fields. Normally, themeasuring instruments are external to the cavity. In this case, the input-output formalism is use- ful. Strictly speaking, the input-output results are approximations, and can be obtained from considering the boundary conditions at lossy cavity mirrors. In this chapter, we consider the input-output formalism for a single-sided cavity in which the cavity mode is coupled to an external multi-mode environment through a partially transparent broadband mirror.