# Quantum Mechanics and Spectrum

**Topics:**Quantum mechanics, Quantum field theory, Quantum state

**Pages:**59 (14571 words)

**Published:**January 13, 2013

INTERNAL REPORT SERIES

HIP-1999-03

QUANTUM SIMULATIONS OF ATOM-PHOTON INTERACTIONS

MARTTI HAVUKAINEN

Helsinki Institute of Physics University of Helsinki Helsinki, Finland

Academic dissertation

To be presented, with the permission of the Faculty of Science of the University of Helsinki, for public criticism in Auditorium XII on October 9th, 1999, at 10 o’clock a.m.

Helsinki 1999

ISBN 951-45-8709-X (PDF version) Helsingin yliopiston verkkojulkaisut Helsinki 1999

Preface

This work has been done at Helsinki Institute of Physics (HIP). I am most grateful to Prof. Stig Stenholm for his guidance and giving me possibility to do this work. I am also greatly indebted to Doc. K.-A. Suominen. Part of the work was done in his group. I also thank Prof. V. Buˇek and G. Drobn´ for collaboration and for their hospitality z y during my visit in Bratislava. I want to thank my parents and little sister about everything. I also thank the members of the group of Quantum Optics who have helped me in numerous problems. Finally I thank all my friends especially in our bowling and minigolf group. They have helped me during my studies and together we have had many tight matches. I also want to thank the Center for Scientiﬁc Computing (CSC) for oﬀering their computer facilities for this work. With gratitude I acknowledge the ﬁnancial support provided by the Academy of Finland.

Helsinki, September 1999 Martti Havukainen

i

Contents

Preface Abstract List of publications Contribution by the author 1 Introduction 2 Canonical quantization of the ﬁeld 2.1 A free ﬁeld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 An interacting ﬁeld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Methods to solve the problem of interacting systems 3.1 Master equations . . . . . . . . . . . . . . . . . . . . . 3.2 Cascaded master equations . . . . . . . . . . . . . . . . 3.3 The excitation expansion . . . . . . . . . . . . . . . . . 3.4 The correlation function expansion . . . . . . . . . . . 4 The stochastic harmonic oscillator 5 The 5.1 5.2 5.3 5.4 5.5 spectrum of the radiation ﬁeld Mode spectrum . . . . . . . . . . . Fourier spectrum . . . . . . . . . . Generalizations of Fourier spectrum Physical spectrum . . . . . . . . . . Analyser atom spectrum . . . . . . i iv v vi 1 3 3 5 7 7 8 10 11 12 16 16 16 17 18 19 21 28 28 30 30 31 32 33

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6 Cavity QED simulations in 1D 7 Cavity QED simulations in 2D 7.1 The general theory . . . . . . 7.2 Various simulations . . . . . . 7.2.1 A free photon . . . . . 7.2.2 A mirror . . . . . . . . 7.2.3 A beam splitter . . . . 7.2.4 A parabolic mirror . .

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7.2.5 A mirror box . . . . . . . . . . . . . 7.2.6 Circular cavities . . . . . . . . . . . . 7.2.7 Decay of an atom in a square cavity . 7.3 Possible future simulations . . . . . . . . . . 8 Conclusion Appendix: Numerics A.1 Diﬀerent languages A.1.1 Fortran . . A.1.2 C and C++ A.1.3 Java . . . . References

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