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Instabilities and Chaos in Quantum Optics: Semi-Classical Theory

E R Pike frs, Sarben Sarkar

Published Dec 7, 1995• 0 citations
Publisher

Abstract

Abstract The theoretical description of radiation − matter interactions in terms of master equations can be approximated by a semi-classical theory when quantum fluctuations are small. This approach, the exact definition of which will emerge later, has been enormously influential in understanding instabilities and chaos in quantum optics. Although much of the theory (and experiment) has been done only in the last few years, there were already indications in the early experiments on ruby lasers of ‘spiky’ behaviour. Instabilities are often due to competition between different modes which are coupled together non-linearly. Consequently we will first describe qualitatively the modes that are important when we deal with optical systems in Fabry-Perot or ring cavities.

DOI
10.1093/oso/9780198520320.003.0008

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Abstract The theoretical description of radiation − matter interactions in terms of master equations can be approximated by a semi-classical theory when quantum fluctuations are small. This approach, the exact definition of which will emerge later, has been enormously influential in understanding instabilities and chaos in quantum optics. Although much of the theory (and experiment) has been done only in the last few years, there were already indications in the early experiments on ruby lasers of ‘spiky’ behaviour. Instabilities are often due to competition between different modes which are coupled together non-linearly. Consequently we will first describe qualitatively the modes that are important when we deal with optical systems in Fabry-Perot or ring cavities.
Quantum-statistical theories attracted considerable attention in the early history of optical bistability.(1) More recently, there has been less activity in this area; perhaps due partially to the lack of experimental work, and partially to the increasing interest in instabilities and chaos. It has now been demonstrated, however, that experiments which come very close to realizing the theoretical ideal—two-level homogeneously broadened atoms in a ring cavity— can be performed with quantitative precision. (2) Moreover, the rising importance of optical chaos is reason for directing attention towards, rather than away from, quantum-statistical studies, since there questions concerning quantum chaos might be addressed.
Abstract This book is unique in covering phenomena in photon- matter interactions in a unified way over a range of many orders in energy. The quantum field theoretic approach to the fully relativistic theory of quantum electrodynamics (QED) is presented together with the non- relativistic theory in both confined and unconfined geometries. The predictions of QED have been verified to a greater accuracy than any other physical theory. Moreover QED is a paradigm for other gauge theories and is presented in such a way that the generalisation to other gauge theories is natural. Gauge and Poincare symmetry properties and the non-existence of a photon wave function are thoroughly discussed. Starting from the Dirac equation the non-relativistic interaction of the electron with the electromagnetic field is derived as an effective Hamiltonian of multipole expansions. Much of quantum optics is based on the lowest order dipole approximation. From this point on the treatment of fully relativistic QED and quantum optics is done in parallel. Applications of perturbation theory such as Compton and Moller scattering and the theory photdetection are given. After the impressive successes of QED, the limitation of the theory and the necessity of electroweak theory and quantumchromodynamics are discussed. The remaining chapters are devoted to quantum optics inside cavaties. Various approaches to open systems such as master equations are discussed within the context of active systems (e.g. the laser) and passive systems. Semi- classical approximarions are shown to imply a rich non- linear dynamics including chaos for certain parameter regimes. The effect of fluctuations on such non-linear dynamics is also studied. The final chapter is devoted to highly non- classical states of the light field such as photon number, squeezed and two photon entangled states. The latter are studied for the important system of parametric down conversion and the localisation properties of photons are characterised in terms of asympotic tails in photodetection probabilities as a function of time delay. The range of the book has wider benefits. Workers in quantum optics will gain a deeper understanding of the foundations of their subject and field theorists will see concrete examples of open systems, which are beginning to impinge on fundamental theories.
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The instabilities in homogeneously broadened lasers, which we described in the previous chapter, have a paradigmatic importance because they arise in models which allow one to perform the linear-stability analysis in a simple analytic form. The treatment of the corresponding instabilities in the case of in homogeneously broadened lasers is substantially more complex in general. However, it is very interesting because, as we shall see, the instabilities arise in parametric domains that are much more accessible from the experimental viewpoint. A related issue is that of the possibility of multimodal emission in a laser. It is common knowledge that in order to attain multimode emission it is necessary to have a sort of in homogeneity in the gain medium, because in a completely homogeneous configuration all modes interact with all atoms, and the arising competition among modes implies that only one of them survives. There are basically two main kinds of in homogeneity, the first is of spatial nature and the other is of spectral nature. The first kind is realized in Fabry–Perot lasers, because different cavity modes interact with atoms at different spatial positions in the cavity thanks to the standing-wave configuration of the field and to the consequent phenomenon of spatial hole- burning (see Section 14.2 and Fig. 14.1). This circumstance is capable of weakening the mode–mode competition and allowing multimode operation. In unidirectional ring lasers, on the other hand, there is an “inhomogeneous” configuration in the case of inhomogeneous broadening, because different groups of atoms have different transition frequencies and therefore interact with different modes thanks to the spectral hole-burning (see Section 15.2 and Fig. 15.3). Again, this weakens the mode–mode competition and allows multimode operation. However, these are qualitative considerations, and the only reliable and quantitative method to determine the possibility of multimodal operation is the linear-stability analysis of the single-mode stationary solutions.
Mesoscopic devices, with system sizes in the range of several to several dozens wavelengths, represent paradigmatic model systems for the observation of quantum chaotic behaviour based on semiclassical concepts. Those electronic and photonic billiard cavities are small enough for interference effects not to be ignored. Nonetheless, the classical ray or particle tracing picture can often provide a substantial understanding of the dynamics of the system along the lines of classical-quantum, or ray-wave correspondence. This well-established principle turns out to be particularly useful when applied not only in real space, but by extending it to phase space such that both location and momentum information can contribute to a deeper and more comprehensive understanding of the dynamical behaviour.