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Input–Output Theory

Z. Ficek, Mohamed Ridza Wahiddin

Published Apr 19, 2016• 0 citations
Publisher

Abstract

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.

DOI
10.1201/b16875-25

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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.
Within the framework of exact quantization of the electromagnetic field in dispersing and absorbing media the input-output problem of a high-QQQ cavity is studied, with special emphasis on the absorption losses in the coupling mirror. As expected, the cavity modes are found to obey quantum Langevin equations, which could be also obtained from quantum noise theories, by appropriately coupling the cavity modes to dissipative systems, including the effect of the mirror-assisted absorption losses. On the contrary, the operator input-output relations obtained in this way would be incomplete in general, as the exact calculation shows. On the basis of the operator input-output relations the problem of extracting the quantum state of an initially excited cavity mode is studied and input-output relations for the sss-parametrized phase-space function are derived, with special emphasis on the relation between the Wigner functions of the quantum states of the outgoing field and the cavity field.
With the exception of Chapter 1, up to this point we have considered cavityless optical systems. With this chapter, instead, we start focussing on the case that the atomic medium is located in an optical cavity. We describe in detail the case of a unidirectional ring resonator, the boundary condition which characterizes it and the relations between its output fields and the intracavity field (Sections 8.1–8.3). We discuss the transmission of an empty cavity (Section 8.4) and of a cavity that contains a linear medium (Section 8.5). Optical cavities An optical cavity or optical resonator is formed by at least two mirrors, appropriately arranged in such a way that the radiation beam follows a closed path. The beam splitters allow the radiation to escape from the cavity (output) and allow one to inject radiation into the cavity (input). In order to confine the radiation transversally with respect to the direction of propagation, the mirrors are usually spherical (see Fig. 1.2). The case of cavities with spherical mirrors will be described in Part III of this volume. On the other hand, an analytical treatment of the problems is possible only within the plane-wave approximation which we will use in the remainder of Part I and in Part II. In accord with this, we assume here that the mirrors are planar. The two most common types of optical resonator are the Fabry–Perot cavity and the ring cavity . The planar Fabry–Perot cavity consists in two beam splitters in a parallel configuration as shown in Fig. 8.1. Within the cavity there are two counterpropagating fields, E F propagating in the forward direction and E B propagating in the backward direction with respect to the input field E I . Because we have derived the Maxwell–Bloch equations for the unidirectional configuration, in this and the following chapters we focus on the case of a ring cavity, which allows unidirectionality. The Fabry–Perot case will be considered in Chapter 14.
In this chapter, a one-dimensional optical cavity with output coupling is con-sidered. The optical cavity has transmission loss at one or both of the end surfaces. The classical, natural cavity mode is defined, and decaying or growing mode functions are derived using the cavity boundary conditions. A series of resonant modes appears. But these modes are not orthogonal to each other and are not suitable for quantum-mechanical analysis of the optical field inside or outside of the cavity. Hypothetical boundaries are added at infinity in order to obtain orthogonal wave mode functions that satisfy the cavity and infinity boundary conditions. These new mode functions are suitable for field quantization, where each mode is quantized separately and the electric field of an optical wave is made up of contributions from each mode. Some results of quantization are described in the next chapter. Chapter 3 deals with the usual quasimode model: a perfect cavity with distributed internal loss or with a fictitious loss reservoir.
Setting off from the classic input–output formalism, we develop a theoretical framework to characterize the Gaussian quantum channels relating the initial correlations of an open bosonic system to those of properly identified output modes. We then proceed to apply our formalism to the case of quantum harmonic oscillators, such as the motional degrees of freedom of trapped ions or nanomechanical oscillators, interacting with travelling electromagnetic modes through cavity fields and subject to external white noise. We thus determine the degree of squeezing that can be transferred from an intra-cavity oscillator to light and show that the intra-cavity squeezing can be transformed into distributed optical entanglement if one can access both output fields of a two-sided cavity.
The input-output formalism is the basis to study the response of an optical cavity to the external stimulations. The existing theories usually handle cavity systems with only one internal mode. However, there is growing interest in more complex systems, especially the hybrid cavity-matter systems, which contain at least two internal modes, one or more from both the optical cavity and the matter. Here we propose a graphical loop theory to calculate and visualize the reflection and transmission spectrum of such a multimode cavity, resembling the role of Feynman diagrams in the quantum field theory. This loop theory gives a unified picture to interpret the experimental observations of a hybrid magnet-light system and is extremely easy to apply to arbitrary complicated problems without any calculations.