Quantum Trajectories and Measurements in Continuous Time

The Diffusive Case

  • Alberto Barchielli
  • Matteo Gregoratti

Part of the Lecture Notes in Physics book series (LNP, volume 782)

Table of contents

  1. Front Matter
    Pages i-xiii
  2. Alberto Barchielli, Matteo Gregoratti
    Pages 1-7
  3. General Theory

    1. Front Matter
      Pages 9-9
    2. Alberto Barchielli, Matteo Gregoratti
      Pages 11-49
    3. Alberto Barchielli, Matteo Gregoratti
      Pages 51-75
    4. Alberto Barchielli, Matteo Gregoratti
      Pages 77-110
    5. Alberto Barchielli, Matteo Gregoratti
      Pages 111-123
    6. Alberto Barchielli, Matteo Gregoratti
      Pages 125-142
  4. Physical Applications

    1. Front Matter
      Pages 143-143
    2. Alberto Barchielli, Matteo Gregoratti
      Pages 145-150
    3. Alberto Barchielli, Matteo Gregoratti
      Pages 151-182
    4. Alberto Barchielli, Matteo Gregoratti
      Pages 183-220
    5. Alberto Barchielli, Matteo Gregoratti
      Pages 221-262
  5. Back Matter
    Pages 1-61

About this book


This course-based monograph introduces the reader to the theory of continuous measurements in quantum mechanics and provides some benchmark applications.

The approach chosen, quantum trajectory theory, is based on the stochastic Schrödinger and master equations, which determine the evolution of the a-posteriori state of a continuously observed quantum system and give the distribution of the measurement output. The present introduction is restricted to finite-dimensional quantum systems and diffusive outputs. Two appendices introduce the tools of probability theory and quantum measurement theory which are needed for the theoretical developments in the first part of the book.

First, the basic equations of quantum trajectory theory are introduced, with all their mathematical properties, starting from the existence and uniqueness of their solutions. This makes the text also suitable for other applications of the same stochastic differential equations in different fields such as simulations of master equations or dynamical reduction theories.

In the next step the equivalence between the stochastic approach and the theory of continuous measurements is demonstrated.

To conclude the theoretical exposition, the properties of the output of the continuous measurement are analyzed in detail. This is a stochastic process with its own distribution, and the reader will learn how to compute physical quantities such as its moments and its spectrum. In particular this last concept is introduced with clear and explicit reference to the measurement process.

The two-level atom is used as the basic prototype to illustrate the theory in a concrete application. Quantum phenomena appearing in the spectrum of the fluorescence light, such as Mollow’s triplet structure, squeezing of the fluorescence light, and the linewidth narrowing, are presented.

Last but not least, the theory of quantum continuous measurements is the natural starting point to develop a feedback control theory in continuous time for quantum systems. The two-level atom is again used to introduce and study an example of feedback based on the observed output.


Quantum mechanics quantum control and feedback quantum measurements in continuous time quantum system quantum trajectories spectra stochastic Schrödinger equation stochastic master equation

Authors and affiliations

  • Alberto Barchielli
    • 1
  • Matteo Gregoratti
    • 2
  1. 1.Dipto. MatematicaPolitecnico di MilanoMilanoItaly
  2. 2.Dipto. MatematicaPolitecnico di MilanoMilanoItaly

Bibliographic information

  • DOI
  • Copyright Information Springer-Verlag Berlin Heidelberg 2009
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Physics and Astronomy
  • Print ISBN 978-3-642-01297-6
  • Online ISBN 978-3-642-01298-3
  • Series Print ISSN 0075-8450
  • Series Online ISSN 1616-6361
  • Buy this book on publisher's site