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Quantum trajectories and measurements in continuous time. The diffusive case. (English) Zbl 1182.81001

Lecture Notes in Physics 782. Berlin: Springer (ISBN 978-3-642-01297-6/hbk; 978-3-642-01298-3/ebook). xiv, 325 p. (2009).
This quite interesting book is devoted to the theory of continuous measurements in quantum mechanics. Indeed continuous measurements on quantum systems are common experimental practice, e.g. in quantum optics, even if the topic is often ignored in standard textbook quantum mechanics. This monograph addresses the subject in a mathematically rigorous way, still paying great attention to the physical meaning and the interpretation of the results. Many telling remarks indeed help the reader in a proper understanding of the key passages and results, together with their physical meaning. In this spirit the chapters end with very useful brief summaries, which collect the basic mathematical quantities introduced in the chapter, as well as the main results obtained. These summaries together with a strict logical presentation and a detailed outline make the presentation very clear, despite the wealth of results provided in the book, also allowing for a selective reading. The presentation is aimed at both mathematicians and physicists, keeping together mathematical rigor and physical motivation. A background in both probability theory and quantum mechanics is certainly of help to the reader. To this aim two appendixes providing a concise presentation of relevant results in both topics can be found at the end of the monograph.
The book is mainly concerned with the approach to continuous measurements based on classical stochastic differential equations and the notion of a posteriori state. It further shows the equivalence of this approach to the operational one relying on operator-valued measures and instruments, thus explicitly expressing the continuous observations within the general formulation of quantum mechanics. The approach to continuous measurement based on quantum stochastic calculus and quantum stochastic differential equations is not considered in detail and has been reviewed in a previous monograph [A. Barchielli, Continual measurements in quantum mechanics and quantum stochastic calculus. Open quantum systems III. Recent developments. Lecture notes of the summer school, Grenoble, France, June 16th – July 4th 2003. Berlin: Springer. Lecture Notes in Mathematics 1882, 205–292 (2006; Zbl 1142.81014)]. The presentation is devoted to finite dimensional systems and to the diffusive case in which the driving noises are Wiener processes.
The volume is divided into two main parts. The first part introduces the general theory, while the second part addresses concrete physical applications, using the two-level atom as a reference system.
In the first part Chapter 2 considers the Hilbert-space formulation of the theory, of interest also for dynamical reduction models. Both linear and nonlinear stochastic differential equations are considered, showing that the linear stochastic differential equation is a sound starting point for the theory of continuous observations. The physical interpretation of the obtained results is given, stressing how this interpretation more naturally arises in the linear case. Advantages and disadvantages of both approaches are considered. In Chapter 3 the formulation of the theory in terms of statistical operators is given, so that also incomplete measurements can be considered; the connection with quantum master equations and quantum dynamical semigroup is also clarified. In Chapter 4 the connection with the operational approach and with the general formulation of quantum mechanics is thoroughly discussed, also introducing the notion of characteristic operator, kind of Fourier transform of the instrument, which allows to obtain explicit formulae for the moments of the measurement output. In Chapter 5 it is shown that the nonlinear stochastic differential equation can also equivalently be taken as a solid starting point for the whole theory. In Chapter 6 quantum trajectory theory is related to quantum information theory, showing how to quantify the information extracted through continuous observation from the quantum system. Exploiting the fact that an instrument can be seen as a channel from quantum to quantum/classical states important inequalities characterizing the information gain are derived in a general setting.
The second part starts with Chapter 7, where hints are given, based on results coming from the approach through quantum stochastic calculus, on how to choice the operators appearing in the abstract theory, thus determining the physical model. In Chapter 8 the case of a two-level atom stimulated by a laser is considered in full detail, providing in particular the equations for the reduced dynamics and the explicit expression of the stochastic differential equations describing the continuous measurement. In Chapter 9 the analytic expressions of the heterodyne and homodyne spectra for the two-level atom are obtained and analyzed. Chapter 10 finally considers the issue of feedback and control of quantum systems, discussing in particular the Markovian feedback scheme of Wiseman and Milburn and applying it to the two-level atom.
The presentation ends with two very well written and useful appendixes, providing a brief and self-contained introduction to stochastic differential equations as well as the general formulation of quantum mechanics. Besides collecting the relevant notions used in the book, these two appendixes are meant to help the physicists or the mathematicians which are not confident with the formalism or the physical interpretation, respectively.

MSC:

81-02 Research exposition (monographs, survey articles) pertaining to quantum theory
81P15 Quantum measurement theory, state operations, state preparations
81S25 Quantum stochastic calculus
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
81V80 Quantum optics
81S22 Open systems, reduced dynamics, master equations, decoherence
81Q93 Quantum control

Citations:

Zbl 1142.81014
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