*Heinz-Peter Breuer and Francesco Petruccione*

- Published in print:
- 2007
- Published Online:
- January 2010
- ISBN:
- 9780199213900
- eISBN:
- 9780191706349
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199213900.003.01
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter contains a survey of classical probability theory and stochastic processes. It starts with a description of the fundamental concepts of probability space and Kolmogorov axioms. These ...
More

This chapter contains a survey of classical probability theory and stochastic processes. It starts with a description of the fundamental concepts of probability space and Kolmogorov axioms. These concepts are then used to define random variables and stochastic processes. The mathematical formulation of the special class of Markov processes through classical master equations is given, including deterministic processes (Liouville equation), jump processes (Pauli master equation), and diffusion processes (Fokker–Planck equation). Special stochastic processes which play an important role in the developments of the following chapters, such as piecewise deterministic processes and Lévy processes, are described in detail together with their basic physical properties and various mathematical formulations in terms of master equations, path integral representation, and stochastic differential equations.Less

This chapter contains a survey of classical probability theory and stochastic processes. It starts with a description of the fundamental concepts of probability space and Kolmogorov axioms. These concepts are then used to define random variables and stochastic processes. The mathematical formulation of the special class of Markov processes through classical master equations is given, including deterministic processes (Liouville equation), jump processes (Pauli master equation), and diffusion processes (Fokker–Planck equation). Special stochastic processes which play an important role in the developments of the following chapters, such as piecewise deterministic processes and Lévy processes, are described in detail together with their basic physical properties and various mathematical formulations in terms of master equations, path integral representation, and stochastic differential equations.

*Heinz-Peter Breuer and Francesco Petruccione*

- Published in print:
- 2007
- Published Online:
- January 2010
- ISBN:
- 9780199213900
- eISBN:
- 9780191706349
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199213900.003.08
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter deals with a presentation of continuous measurement theory on the basis of the microscopic equations of quantum electrodynamics. The emphasis lies on the derivation of the quantum ...
More

This chapter deals with a presentation of continuous measurement theory on the basis of the microscopic equations of quantum electrodynamics. The emphasis lies on the derivation of the quantum operations and the corresponding stochastic processes for various detection schemes directly from the Hamiltonian, which describes the interaction of the matter degrees of freedom with the quantized electromagnetic field. The chapter treats many examples and applications to atomic physics and quantum optics, such as dark state resonances and laser cooling of atoms. This example illustrates the interplay between incoherent processes and quantum interference effects which leads to the emergence of Lévy-type distributions for the atomic waiting time. Finally, dissipative phenomena in the dynamics of open quantum systems in strong driving fields are studied, for which appropriate master equations and stochastic wave function methods can be derived by employing a representation in terms of Floquet states.Less

This chapter deals with a presentation of continuous measurement theory on the basis of the microscopic equations of quantum electrodynamics. The emphasis lies on the derivation of the quantum operations and the corresponding stochastic processes for various detection schemes directly from the Hamiltonian, which describes the interaction of the matter degrees of freedom with the quantized electromagnetic field. The chapter treats many examples and applications to atomic physics and quantum optics, such as dark state resonances and laser cooling of atoms. This example illustrates the interplay between incoherent processes and quantum interference effects which leads to the emergence of Lévy-type distributions for the atomic waiting time. Finally, dissipative phenomena in the dynamics of open quantum systems in strong driving fields are studied, for which appropriate master equations and stochastic wave function methods can be derived by employing a representation in terms of Floquet states.