Quantum Trajectories and Measurements in Continuous Time - The diffusive case

Quantum trajectory theory is largely employed in theoretical quantum optics and quantum open system theory and is closely related to the conceptual formalism of quantum mechanics (quantum measurement theory). However, even research articles show that not all the features of the theory are well known or completely exploited. We wrote this monograph mainly for researchers in theoretical quantum optics and related fields with the aim of giving a self-contained and solid presentation of a part of quantum trajectory theory (the diffusive case) together with some significant applications (mainly with purposes of illustration of the theory, but which in part have been recently developed). Another aim of the monograph is to introduce to this subject post-graduate or PhD students. To help them, in the most mathematical and conceptual chapters, summaries are given to fix ideas. Moreover, as stochastic calculus is usually not in the background of the studies in physics, we added Appendix A to introduce these concepts. The book is written also for mathematicians with interests in quantum theories. Quantum trajectory theory is a piece of modern theoretical physics which needs an interplay of various mathematical subjects, such as functional analysis and probability theory (stochastic calculus), and offers to mathematicians a beautiful field for applications, giving suggestions for new mathematical developments. Appendix B presents the modern formalism of quantum mechanics and has the double role of collecting notions and results used throughout the book and of introducing to this subject peoples without a background in the axiomatic of quantum mechanics. The book is divided in two parts, one for the presentation of the theoretical structure of the theory and the other for applications. The two appendices are intended to be a primary in stochastic differential equations (Appendix A) and in quantum mechanics (measurement theory and open systems - Appendix B). Chapter 2 is devoted to the Hilbert-space formulation of the theory and it is centred on the presentation of the stochastic Schrödinger equation. Chapters 3 and 5 present the formulation in terms of statistical operators; now the key concept is that of the stochastic master equation. While the observables of the theory, represented by positive operator valued measures, have been already introduced in Chapter 2, the full connection of quantum trajectory theory with the general axiomatic structure of quantum mechanics is given in Chapter 4. Here also the moments and the spectrum of the output of the measurement are studied. Chapter 6 connects quantum trajectory theory with quantum information. Measures of information, such as mutual entropies, are introduced; they quantify the information extracted from the observed quantum system by the continuous measurement. Chapter 7 gives some ideas on how to construct concrete physical models, mainly in quantum optics, and how to use the theory developed in the first part in order to describe two types of photodetection: heterodyne and homodyne detection. A concrete model for a two level atom stimulated by a monochromatic laser is given in Chapter 8 and its heterodyne and homodyne spectra are studied in Chapter 9. Chapter 10 is devoted to the effects produced on the homodyne spectrum by feedback and control; the first part of this chapter presents an interesting scheme, due to Wiseman and Milburn, which allows to introduce feedback loops in the theory. Chapters 9 and 10 present also many physical effects typical of different quantum systems, such as squeezing, line narrowing, thermal and dephasing broadening...