*J. C. Garrison and R. Y. Chiao*

- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780198508861
- eISBN:
- 9780191708640
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198508861.003.0019
- Subject:
- Physics, Atomic, Laser, and Optical Physics

This chapter examines the evolution of an open system — the sample — with the quantum Liouville equation for the world density operator. The fundamental approximation is that the action of the sample ...
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This chapter examines the evolution of an open system — the sample — with the quantum Liouville equation for the world density operator. The fundamental approximation is that the action of the sample on the environment is negligible compared to the action of the environment on the sample. This leads to the master equation for the (reduced) sample density operator. Photons in a cavity and a two-level atom are presented as examples. The P-function representation of the sample density operator yields the Fokker-Planck equation. This is used to show the robustness of coherent states, and to describe a driven mode in a lossy cavity. The discussion next turns to quantum jumps and their experimental observation. Quantum jumps are related to the master equations by means of the Monte Carlo wavefunction algorithm, quantum trajectories, and quantum state diffusion.Less

This chapter examines the evolution of an open system — the sample — with the quantum Liouville equation for the world density operator. The fundamental approximation is that the action of the sample on the environment is negligible compared to the action of the environment on the sample. This leads to the master equation for the (reduced) sample density operator. Photons in a cavity and a two-level atom are presented as examples. The P-function representation of the sample density operator yields the Fokker-Planck equation. This is used to show the robustness of coherent states, and to describe a driven mode in a lossy cavity. The discussion next turns to quantum jumps and their experimental observation. Quantum jumps are related to the master equations by means of the Monte Carlo wavefunction algorithm, quantum trajectories, and quantum state diffusion.

*Sandip Tiwari*

- Published in print:
- 2020
- Published Online:
- November 2020
- ISBN:
- 9780198759867
- eISBN:
- 9780191820830
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198759867.003.0008
- Subject:
- Physics, Condensed Matter Physics / Materials

This chapter explores the evolution of an ensemble of electrons under stimulus, classically and quantum-mechanically. The classical Liouville description is derived, and then reformed to the quantum ...
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This chapter explores the evolution of an ensemble of electrons under stimulus, classically and quantum-mechanically. The classical Liouville description is derived, and then reformed to the quantum Liouville equation. The differences between the classical and the quantum-mechanical description are discussed, emphasizing the uncertainty-induced fuzziness in the quantum description. The Fokker-Planck equation is introduced to describe the evolution of ensembles and fluctuations in it that comprise the noise. The Liouville description makes it possible to write the Boltzmann transport equation with scattering. Limits of validity of the relaxation time approximation are discussed for the various scattering possibilities. From this description, conservation equations are derived, and drift and diffusion discussed as an approximation. Brownian motion arising in fast-and-slow events and response are related to the drift and diffusion and to the Langevin and Fokker-Planck equations as probabilistic evolution. This leads to a discussion of Markov processes and the Kolmogorov equation.Less

This chapter explores the evolution of an ensemble of electrons under stimulus, classically and quantum-mechanically. The classical Liouville description is derived, and then reformed to the quantum Liouville equation. The differences between the classical and the quantum-mechanical description are discussed, emphasizing the uncertainty-induced fuzziness in the quantum description. The Fokker-Planck equation is introduced to describe the evolution of ensembles and fluctuations in it that comprise the noise. The Liouville description makes it possible to write the Boltzmann transport equation with scattering. Limits of validity of the relaxation time approximation are discussed for the various scattering possibilities. From this description, conservation equations are derived, and drift and diffusion discussed as an approximation. Brownian motion arising in fast-and-slow events and response are related to the drift and diffusion and to the Langevin and Fokker-Planck equations as probabilistic evolution. This leads to a discussion of Markov processes and the Kolmogorov equation.

*J. C. Garrison and R. Y. Chiao*

- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780198508861
- eISBN:
- 9780191708640
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198508861.003.0003
- Subject:
- Physics, Atomic, Laser, and Optical Physics

This chapter begins with a normal-mode analysis of the classical electromagnetic field in an ideal cavity. The resulting expression for the electromagnetic energy has the same form as the energy of a ...
More

This chapter begins with a normal-mode analysis of the classical electromagnetic field in an ideal cavity. The resulting expression for the electromagnetic energy has the same form as the energy of a collection of harmonic oscillators, called radiation oscillators. This analogy is the basis for a quantization conjecture in which the classical mode amplitudes are replaced by photon creation and annihilation operators, obeying a version of the canonical commutation relations of quantum mechanics. Fock space is constructed by repeated application of creation operators to the vacuum state. Pure and mixed quantum states of light are described, respectively, by Fock-space vectors satisfying the Schrödinger equation, and density operators satisfying the quantum Liouville equation. The notions of normal ordering and vacuum fluctuations are introduced, and the latter is used to explain the Casimir effect.Less

This chapter begins with a normal-mode analysis of the classical electromagnetic field in an ideal cavity. The resulting expression for the electromagnetic energy has the same form as the energy of a collection of harmonic oscillators, called radiation oscillators. This analogy is the basis for a quantization conjecture in which the classical mode amplitudes are replaced by photon creation and annihilation operators, obeying a version of the canonical commutation relations of quantum mechanics. Fock space is constructed by repeated application of creation operators to the vacuum state. Pure and mixed quantum states of light are described, respectively, by Fock-space vectors satisfying the Schrödinger equation, and density operators satisfying the quantum Liouville equation. The notions of normal ordering and vacuum fluctuations are introduced, and the latter is used to explain the Casimir effect.