## An Open Systems Approach to Quantum Optics: Lectures Presented at the Université Libre de Bruxelles, October 28 to November 4, 1991, Volume 18This volume contains ten lectures presented in the series ULB Lectures in Nonlinear Optics at the Universite Libre de Bruxelles during the period October 28 to November 4, 1991. A large part of the first six lectures is taken from material prepared for a book of somewhat larger scope which will be published,by Springer under the title Quantum Statistical Methods in Quantum Optics. The principal reason for the early publication of the present volume concerns the material contained in the last four lectures. Here I have put together, in a more or less systematic way, some ideas about the use of stochastic wavefunctions in the theory of open quantum optical systems. These ideas were developed with the help of two of my students, Murray Wolinsky and Liguang Tian, over a period of approximately two years. They are built on a foundation laid down in a paper written with Surendra Singh, Reeta Vyas, and Perry Rice on waiting-time distributions and wavefunction collapse in resonance fluorescence [Phys. Rev. A, 39, 1200 (1989)]. The ULB lecture notes contain my first serious atte~pt to give a complete account of the ideas and their potential applications. I am grateful to Professor Paul Mandel who, through his invitation to give the lectures, stimulated me to organize something useful out of work that may, otherwise, have waited considerably longer to be brought together. |

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### Contents

Introduction | 1 |

Lecture 1 Master Equations and Sources I | 5 |

12 Master equations | 6 |

13 Master equation for a cavity mode driven by thermal light | 9 |

14 The cavity output field | 13 |

15 Correlations between the free field and the source field | 16 |

Lecture 2 Master Equations and Sources II | 22 |

22 Master equation for a twostate atom in thermal equilibrium | 24 |

the spectrum of squeezing | 100 |

63 Vacuum fluctuations | 103 |

64 Squeezing spectra for the degenerate parametric oscillator | 107 |

65 Photoelectron counting for the degenerate parametric oscillator | 110 |

Lecture 7 Quantum Trajectories I | 113 |

71 Exclusive and nonexclusive photoelectron counting probabilities | 114 |

72 The distribution of waiting times | 116 |

73 Quantum trajectories from the photoelectron counting distribution | 117 |

23 Phase destroying processes | 28 |

24 The radiated field | 33 |

resonance fluorescence lasers parametric oscillators | 35 |

Lecture 3 Standard Methods of Analysis I | 39 |

the quantum regression theorem | 41 |

33 Optical spectra | 46 |

34 The HanburyBrownTwiss effect | 52 |

35 Photon antibunching | 53 |

Lecture 4 Standard Methods of Analysis II | 58 |

42 FokkerPlanck equation for a cavity mode driven by thermal light | 64 |

43 Stochastic differential equations | 67 |

44 Linearization and the system size expansion | 68 |

45 The degenerate parametric oscillator | 73 |

Lecture 5 Photoelectric Detection I | 78 |

52 Photoelectron counting for a general classical field | 80 |

53 Moments of the counting distribution | 82 |

54 The waitingtime distribution | 86 |

55 Photoelectron counting for quantized fields | 88 |

Lecture 6 Photoelectric Detection II | 93 |

74 Unravelling the master equation for the source | 121 |

75 Stochastic wavefunctions | 122 |

Lecture 8 Quantum Trajectories II | 126 |

82 Resonance fluorescence | 130 |

83 Cavity mode driven by thermal light | 134 |

84 The degenerate parametric oscillator | 136 |

85 Complementary unravellings | 138 |

Lecture 9 Quantum Trajectories III | 140 |

92 Homodyne detection | 143 |

93 Nonclassical photoelectron correlations | 146 |

94 Stochastic Schrodinger equation for the degenerate parametric oscillator | 148 |

95 Nonlocality | 152 |

Lecture 10 Quantum Trajectories IV | 155 |

cavity QED | 160 |

103 Spontaneous dressedstate polarization | 162 |

104 Semiclassical analysis | 164 |

105 Quantum stability phase switching and Schrodinger cats | 166 |

Postscript | 174 |

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### Common terms and phrases

Bloch sphere calculate classical stochastic coherent conditioned mean photon correlation function defined degenerate parametric oscillator density operator described detector driven by thermal emitted equations of motion evolution example exclusive probability densities field amplitude Fock Fokker-Planck equation free field frequency given H. J. Carmichael Hamiltonian homodyne detection integral interaction interval laser lectures master equation mean photon number nonlinear normal obtained operator averages operator master equation optical bistability optical cavity output field pc(t photocurrent photoelectric detection photoelectron counting distribution photoelectron emissions photoemissive source photon emissions photon flux Phys picture quadrature phase quadrature phase amplitudes quantized field quantum fluctuations quantum mechanics quantum optics quantum trajectory approach quantum-classical correspondence radiated random resonance fluorescence right-hand side Schrodinger equation Sect semiclassical shot noise solution source field source master equation spectrum of squeezing squeezed light statistical stochastic differential equation superoperator theory thermal light tion two-state atom unravelling vacuum variance waiting-time distribution