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Raman Transitions in Cavity QED View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Caltech Theses and Dissertations Raman Transitions in Cavity QED Thesis by David Boozer In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2005 (Defended 12 April 2005) ii c 2005 David Boozer All Rights Reserved iii This thesis is dedicated to all theses that are not dedicated to themselves. iv Acknowledgements I would first like to thank my advisor, Jeff Kimble, for giving me the opportunity to work in his research group. During my time here, I have been continually impressed by his scientific integrity as well as his ability to solve difficult problems through a combination of persistent effort and keen scientific insight. Christoph N¨agerl and Ron Legere were terrific mentors; most of what I know about the nuts and bolts of experimental work I learned from them. The cavity QED lab, which was used to perform the experiments presented in this thesis, is the product of several generations of students and postdoctoral fellows. David Vernooy and June Ye built the core of the lab and achieved the crucial first step of trapping atoms inside the cavity. Jason McKeever, Joe Buck, Alex Kuzmich, Christoph N¨agerl, and Dan Stamper-Kurn made a series of important advances, transforming the lab into a powerful tool for studying cavity QED. Andreea Boca and Russell Miller worked closely with me on the Raman project, and I owe them a special debt of gratitude. The results presented here would not have been possible without their hard work and dedication. Beyond these specific acknowledgments, I would like to thank all the members of the quantum optics group. I have always enjoyed the friendly atmosphere of the group, and it was a privilege for me to work with such an outstanding group of people. v Abstract In order to study quantum effects such as state superposition and entanglement, one would like to construct simple systems for which the damping rates are slow relative to the rate of coherent evolu- tion. One such system is strong-coupling cavity quantum electrodynamics (QED), in which a single atom is coupled to a single mode of a high finesse optical cavity. In recent years, optical trapping techniques have been applied to the cavity QED system, allowing an individual atom to remain coupled to the cavity for long periods of time. For the purpose of future cavity QED experiments, one would like to gain as much control over the trapped atom as possible; in particular, one would like to cool the center of mass motion of the atom, to measure the magnetic field at the location of the atom, and to be able to prepare the atom in a given internal state. In the first part of this thesis, I present a scheme for driving Raman transitions inside the cavity that can be used to achieve these goals. After giving a detailed theoretical treatment of the Raman scheme, I describe how it can be implemented in the lab and discuss some preliminary experimental results. In the second part of this thesis, I present a number of simple field theory models. These mod- els were developed in an attempt to understand some of the central ideas of theoretical physics by looking at how the ideas work in a highly simplified context. The hope is that by reducing the mathematical complexity of an actual theory, the underlying physical concepts can be more easily understood. Note on units: this thesis uses natural units (~ = c = 1). vi Contents Acknowledgements iv Abstract v 1 Atomic physics 1 1.1 Introduction.................................... ..... 1 1.1.1 CavityQED .................................... 1 1.1.2 Opticaltrapping ............................... ... 5 1.1.3 SomeapplicationsofcavityQED . ...... 6 1.1.4 Ramanscheme ................................... 7 1.2 Two-levelsystems ................................ ..... 9 1.2.1 Representationsofpurestates. ........ 9 1.2.2 Representationsofmixedstates. ........ 12 1.2.3 Compositesystems.............................. ... 14 1.2.4 Timeevolution ................................. .. 15 1.3 Two-levelatoms .................................. .... 17 1.3.1 Interactionpicture . ..... 17 1.3.2 Two-level atom without spontaneous decay . ......... 19 1.3.3 Two-levelatomwithspontaneousdecay . ........ 22 1.4 Opticalcavities ................................. ...... 23 1.4.1 Classicalcavity............................... .... 23 1.4.2 Quantumcavity ................................. 26 1.4.3 CavityQED .................................... 28 1.4.4 Semiclassicalapproximation. ........ 29 1.4.5 Singleatomlaser............................... ... 32 1.5 Raman transitionsin athree-levelsystem . ............ 36 1.5.1 Adiabaticelimination . ..... 36 1.5.2 FORTconfiguration .............................. .. 40 1.5.3 Ramanscheme ................................... 41 vii 1.5.4 Harmonicapproximation . .... 44 1.5.5 Numericalresults.............................. .... 46 1.5.6 Semiclassicalmodel .. .. .. .. .. .. .. .. .. .. .. .. .... 47 1.6 Many-levelatoms................................. ..... 52 1.6.1 Cesiumspectrum................................ .. 52 1.6.2 Circularpolarizationvectors . ........ 53 1.6.3 Wigner-Eckart theorem and projection theorem . ........... 54 1.6.4 Couplingtoamagneticfield . .... 56 1.6.5 Relatingreducedmatrixelements. ........ 59 1.6.6 Dipolematrixelements . .... 61 1.6.7 Atomicraisingandloweringoperators . ......... 64 1.6.8 Spontaneousdecayrate . .... 67 1.6.9 Couplingtocavitymode. .... 70 1.6.10 Couplingtoclassicalfields. ........ 72 1.7 Fardetunedlight................................. ..... 72 1.7.1 Ground state transition amplitudes . ......... 73 1.7.2 Matrix elements of J~ ................................ 77 1.7.3 FORTpotential ................................. 79 1.7.4 Ramantransitions .............................. ... 82 1.8 Experimentalapparatus . ....... 84 1.8.1 Overviewofexperiment . .... 84 1.8.2 MOTs........................................ 85 1.8.3 CavitylockingandcavityQEDprobe . ...... 85 1.8.4 FORTandRamanlasers ............................ 88 1.8.5 Photoncounters ................................ .. 90 1.8.6 Timing ....................................... 90 1.8.7 Biascoils ..................................... 91 1.9 Experimentaltechniquesandresults . ........... 92 1.9.1 Usefultransitions.. .. .. .. .. .. .. .. .. .. .. .. ..... 92 1.9.2 LoadingintotheFORT .. .. .. .. .. .. .. .. .. .. .. .. .. 93 1.9.3 Opticalpumping ................................ .. 94 1.9.4 Measuringtheatomicstate . ..... 95 1.9.5 Measuring the Raman transfer probability . .......... 96 1.9.6 Ramanspectroscopy .. .. .. .. .. .. .. .. .. .. .. .. ... 98 1.9.7 Measuring and nulling the magnetic field . ........ 101 1.9.8 Rabiflopping................................... 103 viii 1.9.9 Lasernoise .................................... 104 1.9.10 Ramancooling ................................. 110 1.9.11 Cavitytransmissionspectrum . ........ 114 1.10FORTissues ..................................... .. 115 1.10.1 FORTellipticity .. .. .. .. .. .. .. .. .. .. .. .. .. .... 115 1.10.2 ScatteringinaFORT . .. .. .. .. .. .. .. .. .. .. .. .. 117 2 Classical field theory model in 1+1 dimensions 121 2.1 Introduction.................................... ..... 121 2.2 Modelfieldtheory ................................. .... 122 2.2.1 Hamiltonianforthemodeltheory . ...... 122 2.2.2 Retardedandadvancedfields . ..... 124 2.2.3 in and out fields .................................. 130 2.2.4 Fieldenergyandmomentum . .. 132 2.2.5 Scattering.................................... 134 2.2.6 Radiatedpower................................. 137 2.2.7 Radiationreaction . .. .. .. .. .. .. .. .. .. .. .. .. .... 139 2.2.8 Scattering from a harmonically bound particle . ........... 140 2.2.9 Vectorfield .................................... 142 2.3 Extendedparticles ............................... ...... 144 2.3.1 Extendedparticleatrest . ..... 144 2.3.2 Extended particle moving at constant velocity . ............ 146 2.3.3 Extendedparticleinarbitrarymotion . ......... 146 2.4 Alternativemodelsforspace. ......... 148 2.4.1 Periodicboundaryconditions . ....... 148 2.4.2 Finite lattice with damped boundary conditions . ........... 151 2.4.3 Finite lattice with periodic boundary conditions . ............. 154 3 Quantum field theory model in 1+1 dimensions 162 3.1 Introduction.................................... ..... 162 3.2 Classicalscalarfieldonalattice. ........... 162 3.2.1 Mechanicalmodel ............................... 162 3.2.2 RelativisticLagrangian . ...... 164 3.2.3 Normalmodes ................................... 165 3.3 Fieldquantization ............................... ...... 167 3.3.1 Quantizedscalarfield . .... 167 3.3.2 Multi-modecoherentstate. ...... 168 ix 3.3.3 Singlephotonstate. .. .. .. .. .. .. .. .. .. .. .. .. .... 169 3.3.4 Second quantized Schr¨odinger equation . .......... 172 3.3.5 Singleparticlestate . ..... 177 3.4 Two-levelatomcoupledtofield . ........ 178 3.4.1 Hamiltonianforthesystem . ..... 178 3.4.2 Weisskopf-Wignerapproach . ...... 180 3.4.3 Fermi’sgoldenrule. .. .. .. .. .. .. .. .. .. .. .. .. .... 185 3.4.4 Masterequation ................................ 188 3.4.5 Propagators................................... 189 3.4.6 Scattering.................................... 194 4 Quantum field theory in 0+1 dimensions 197 4.1 Introduction.................................... ..... 197 4.2 Classicalscalarfield ............................. ....... 198 4.2.1 Hamiltonianforascalarfield . .....
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