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**Quantum noise. A handbook of Markovian and non-Markovian quantum stochastic methods with applications to quantum optics.
3rd ed.**
*(English)*
Zbl 1072.81002

Springer Series in Synergetics. Berlin: Springer (ISBN 3-540-22301-0/hbk). xxii, 449 p. (2004).

The book, a milestone in its field, addresses by means of a wealth of applications the thorough exposition of various methods used in the description of quantum systems interacting with some other infinite-dimensional quantum system, often seen or described as noise, so called quantum stochastic methods. The main emphasis is still on applications in quantum optics, even though this third edition further stresses that such methods are actually relevant for many other areas of research. Indeed the only addition with respect to the previous edition consists in a last Chapter with a commented bibliography on recent work in both quantum optics and condensed matter physics where the introduced methods have shown to be relevant, suggesting that both research areas could be considered as subfields of a more general quantum statistics field.

The presentation is finalized to applications, rather than to a precise explanation of the mathematical context or a detailed justification of the introduced approximations, and builds heavily on the material presented by C. W. Gardiner in his book [Handbook of stochastic methods for physics, chemistry and natural sciences (1990; Zbl 0713.60076)], where classical stochastic methods are introduced and dealt with in the same spirit. This applied viewpoint and the most concise style can sometimes hinder the understanding for a reader not familiar with applications, only willing to focuse on the methods. The reader who wants to better understand the mathematical framework is not helped by the final bibliography, but he can refer to work by [A. S. Holevo, Probabilistic and statistical aspects of quantum theory (North Holland, Amsterdam) (1982; Zbl 0497.46053)] as well as the more recent [Statistical structure of quantum theory (Lecture Notes in Physics, Springer, Berlin) (2001; Zbl 0999.81001)], where related literature can be found. A rigorous treatment of applications of quantum stochastic calculus to quantum optics can be found in the recent monograph by [A. Barchielli, Continual measurements in quantum mechanics and quantum stochastic calculus (Lecture Notes in Mathematics, Springer, Berlin) (2005; to appear)], whose work on the subject has given the basis for some of the Sections of the book. A further relevant related publication, focusing on the application of stochastic methods to quantum open systems, is the book by [H.-P. Breuer and F. Petruccione, The theory of open quantum systems (Oxford University Press, Oxford) (2002; Zbl 1053.81001)].

Chapter 3 introduces the theory of quantum Langevin equations and quantum noise; Chapter 5 deals with quantum Markov processes described in terms of quantum master-equations and quantum stochastic differential equations, further developed in Chapter 11, where stochastic SchrĂ¶dinger equations are also dealt with, discussing their connection to quantum measurement theory and the relevance for computer simulations. Chapter 2 reviews in a crude way the modern formalism of quantum mechanics, relying on statistical operator, effects and operations, allowing for the introduction of quantum measurement continuous in time, an issue fully developed in Chapter 8 on photon counting. Chapter 4 gives a thorough review of phase-space methods, widely used throughout the book. The other chapters mainly deal with applications, though in this publication theory and applications are always strongly intertwined.

The presentation is finalized to applications, rather than to a precise explanation of the mathematical context or a detailed justification of the introduced approximations, and builds heavily on the material presented by C. W. Gardiner in his book [Handbook of stochastic methods for physics, chemistry and natural sciences (1990; Zbl 0713.60076)], where classical stochastic methods are introduced and dealt with in the same spirit. This applied viewpoint and the most concise style can sometimes hinder the understanding for a reader not familiar with applications, only willing to focuse on the methods. The reader who wants to better understand the mathematical framework is not helped by the final bibliography, but he can refer to work by [A. S. Holevo, Probabilistic and statistical aspects of quantum theory (North Holland, Amsterdam) (1982; Zbl 0497.46053)] as well as the more recent [Statistical structure of quantum theory (Lecture Notes in Physics, Springer, Berlin) (2001; Zbl 0999.81001)], where related literature can be found. A rigorous treatment of applications of quantum stochastic calculus to quantum optics can be found in the recent monograph by [A. Barchielli, Continual measurements in quantum mechanics and quantum stochastic calculus (Lecture Notes in Mathematics, Springer, Berlin) (2005; to appear)], whose work on the subject has given the basis for some of the Sections of the book. A further relevant related publication, focusing on the application of stochastic methods to quantum open systems, is the book by [H.-P. Breuer and F. Petruccione, The theory of open quantum systems (Oxford University Press, Oxford) (2002; Zbl 1053.81001)].

Chapter 3 introduces the theory of quantum Langevin equations and quantum noise; Chapter 5 deals with quantum Markov processes described in terms of quantum master-equations and quantum stochastic differential equations, further developed in Chapter 11, where stochastic SchrĂ¶dinger equations are also dealt with, discussing their connection to quantum measurement theory and the relevance for computer simulations. Chapter 2 reviews in a crude way the modern formalism of quantum mechanics, relying on statistical operator, effects and operations, allowing for the introduction of quantum measurement continuous in time, an issue fully developed in Chapter 8 on photon counting. Chapter 4 gives a thorough review of phase-space methods, widely used throughout the book. The other chapters mainly deal with applications, though in this publication theory and applications are always strongly intertwined.

Reviewer: Bassano Vacchini (Milano)

### MSC:

81-02 | Research exposition (monographs, survey articles) pertaining to quantum theory |

81V80 | Quantum optics |

81S25 | Quantum stochastic calculus |

81S30 | Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics |

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

82B31 | Stochastic methods applied to problems in equilibrium statistical mechanics |