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Nonclassical States of Light and Atomic Ensembles: Generation and New Applications Nonclassical states of light and atomic ensembles: Generation and New Applications A thesis presented by Axel Andr´e to The Department of Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of Physics Harvard University Cambridge, Massachusetts May 2005 c 2005 - Axel Andr´e All rights reserved. Thesis advisor Author Mikhail D. Lukin Axel Andr´e Nonclassical states of light and atomic ensembles: Generation and New Applications Abstract This thesis considers several novel methods for generating nonclassical states of light and atomic ensembles, and describes applications of these methods to precision measurements, generation of Fock states with controllable waveform and few-photons nonlinear optics. We study the generation of spin-squeezed states of an ensemble of N atoms, and the conditions necessary to achieve high degree of squeezing taking into account im- perfections such as decay and finite number of atoms. A specific implementation of this model based on atom-atom interactions via quantized photon exchange is pre- sented in detail. We analyze the effect of realistic noise sources for an atomic clock consisting of a local oscillator that is actively locked to a spin-squeezed (entangled) ensemble of N atoms. We show that the use of entangled states with a moderate degree of entanglement yields the maximal clock stability. We study the dynamics of Raman scattering in optically thick atomic media, and the ensuing correlations between the atomic spin coherence and the Stokes photons created via Raman generation. The theoretical model and experimental highlights Abstract iv are presented, demonstrating generation of pulses of light with controllable photon numbers, propagation direction, timing, and pulse shapes. We describe two methods to dynamically control the propagation of light in atomic media using EIT. We show that propagating light pulses can be coherently converted into stationary excitations with nonvanishing photonic components, and present high- lights of an experiment demonstrating this effect. We then show that these ideas can be further extended to localize optical pulses in all three spatial dimensions, and to dramatically enhance nonlinear interactions between weak optical pulses. Finally, we report on experimental progress towards nonlinear optical interactions in atomic me- dia confined inside hollow-core photonic crystal fibers. We describe the experimental setup used to load Rubidium atoms along with a buffer gas inside the optical fiber, and report our preliminary results towards achieving this goal. Contents TitlePage.................................... i Abstract..................................... iii TableofContents................................ v Citations to Previously Published Work . viii Acknowledgments................................ x Dedication.................................... xii 1 Introduction 1 1.1 Motivation................................. 1 1.2 Overview.................................. 5 1.3 QuantumControlofLight ........................ 8 1.3.1 Strongcouplingoflightandmatter . 10 1.3.2 Electromagnetically Induced Transparency (EIT) . .... 11 1.3.3 Dark-statepolaritons . 13 1.3.4 Quantum memory using dynamic EIT . 15 1.4 Entanglement Generation in Atomic Ensembles . ... 16 2 Spin squeezing and Heisenberg-limited spectroscopy 18 2.1 Motivation................................. 18 2.2 Overview.................................. 21 2.3 Ramsey Spectroscopy with Correlated Atoms . .. 24 2.4 Two Axis Counter-twisting Model . 30 2.5 Coherent Atom Interactions via Slow Light . ... 37 2.6 DiscussionandConclusion . 49 3 Stability of Realistic Atomic Clocks Based on Entangled Atoms 51 3.1 Introduction................................ 51 3.2 QuantumProjectionNoise . 53 3.3 Local Oscillator and Environmental Noise . ... 56 3.4 Simple Model of Frequency Control . 59 3.5 Long Term Stability of the Atomic Clock . 63 v Contents vi 3.6 Conclusions ................................ 67 4 Raman Generation of Quantum States of Light and Atoms 69 4.1 Motivation................................. 69 4.2 Introduction................................ 72 4.3 QuantumDescriptionofRamanScattering . .. 76 4.4 Three-dimensional Theory of Raman Scattering . .... 80 4.5 Dynamics: One-dimensional Model . 83 4.5.1 Transientregime ......................... 87 4.6 RetrievalofStoredAtomicExcitation. ... 92 4.6.1 Adiabaticsolution . .. .. 94 4.6.2 Fastread-out ........................... 98 4.6.3 Optimal geometry for retrieval of stored excitation . ..... 100 4.7 Four-wave Mixing: Continuous Wave Picture . ... 104 4.8 ExperimentalHighlights . 112 4.8.1 Quantum correlations and delay in multi-photon experiments . 112 4.8.2 Quantum control and generation of few-photon pulses . .... 117 5 Dynamical control of the propagation properties of light pulses 124 5.1 Introduction................................ 124 5.2 Stationary Pulses of Light: Dispersive Case . ..... 125 5.2.1 Dispersive modulation via EIT . 125 5.2.2 Dynamically controlled photonic band gap . 128 5.2.3 Trappingoflightpulses . 131 5.3 Stationary Pulses of Light: Absorptive Case . .... 135 5.3.1 BasicIdea ............................. 135 5.3.2 Spatially Modulated Transparency . 137 5.4 Stationary Pulses of Light: Experiments . ... 142 6 Nonlinear Optics with Stationary Pulses of Light 147 6.1 Introduction................................ 147 6.2 Waveguiding and Three-dimensional Localization of Light Pulses . 149 6.3 Controlled Effective Kerr Nonlinearity and Single Photon Phase Shift 154 6.4 Conclusions ................................ 158 6.5 Photonic Crystal Fiber Experiment . 158 6.5.1 Hollow-core Photonic Crystal Fibers . 160 6.5.2 ExperimentalSetup. 164 6.5.3 Current status of experiment and outlook . 167 6.5.4 Summary ............................. 168 Contents vii A Appendices to Chapter 2 170 A.1 RamseySpectroscopy. 170 A.2 Spin Squeezed States - Wigner Function Representation . ....... 176 A.3 Adiabatic Elimination of Excited State in Raman Scattering ..... 177 A.4 Adiabatic Elimination of Bright Polariton . .... 179 B Appendices to Chapter 3 184 B.1 ModelofFrequencyControlLoop . 184 B.2 Stochastic Differential Equation approach . .... 191 B.3 Long-termStability. .. .. 195 B.3.1 LinearFeedback. 195 B.3.2 NonlinearFeedback. 197 B.4 PerturbativeSolutionofSDE . 199 B.4.1 Linearfeedback . .. .. 199 B.4.2 Nonlinearfeedback . 202 B.5 GaussianStates.............................. 203 C Appendices to Chapter 4 205 C.1 Three-dimensional quantum description of resonant atom-field interac- tions.................................... 205 C.1.1 Equationsofmotionforfields . 205 C.1.2 Atom-fieldequationsofmotion . 209 D Appendices to Chapter 5 211 D.1 ShapingStationaryPulses . 211 D.2 Multi-component Spatial Coherence Grating . .... 215 D.2.1 Release of stored pulse: multi-component approach . .... 221 Bibliography 224 Citations to Previously Published Work Some of the introductory material appears in “Quantum Control of Light Using Electromagnetically Induced Trans- parency”, A. Andr´e, M. D. Eisaman, R. L. Walsworth, A. S. Zibrov, and M. D. Lukin, Journal of Physics B 38, S589 Special Issue: Einstein Year, 2005. The material of Chapter 2 has been published as “Atom Correlations and Spin Squeezing near the Heisenberg Limit: Finite- size Effect and Decoherence”, A. Andr´eand M. D. Lukin, Phys. Rev. A 65, 053819 (2002), quant-ph/0112126; and appears also partly in “Coherent Atom Interactions Mediated by Dark-state Polaritons”, A. Andr´e, L.-M. Duan, and M. D. Lukin, Phys. Rev. Lett. 88, 243602 (2002), quant-ph/0107075; Chapter 3 is based on “Stability of Atomic Clocks Based on Entangled Atoms”, A. Andr´e, A. S. Sørensen, and M. D. Lukin, Phys. Rev, Lett. 92, 230801 (2004), quant-ph/0401130 as well as “Long-term Stability of Atomic Clocks Based on Entangled Atoms”, A. Andr´e, A. S. Sørensen, and M. D. Lukin, in preparation Experimental results from Chapter 4 appear partly in “Atomic Memory for Correlated Photon States”, C. H. Van der Wal, M. D. Eisaman, A. Andr´e, R. L. Walsworth, D. F. Phillips, A. S. Zibrov, and M. D. Lukin, Science 301, 196 (2003). and in “Shaping Quantum Pulses of Light Via Coherent Atomic Memory”, M. D. Eisaman, L. Childress, A. Andr´e, F. Massou, A. S. Zibrov, and M. D. Lukin, Phys. Rev, Lett. 93, 233602 (2004), quant-ph/0406093 Contents ix The material of Chapter 5 is based partly on “Manipulating Light Pulses via Dynamically Controlled Photonic Band Gap”, A. Andr´eand M. D. Lukin, Phys. Rev, Lett. 89, 143602 (2002), quant-ph/0205072 and the experimental data presented in Chapter 5 appeared in “Stationary Pulses of Light in an Atomic Medium”, M. Bajcsy, A. S. Zibrov, and M. D. Lukin, Nature 426, 638 (2003), quant-ph/0311092 Chapter 6 appears partly as “Nonlinear Optics with Stationary Pulses of Light”, A. Andr´e, M. Bajcsy, A. S. Zibrov, and M. D. Lukin, Phys. Rev, Lett. 94, 063902 (2005), quant-ph/0410157 Electronic preprints (shown in typewriter font) are available on the Internet at the following URL: http://xxx.lanl.gov Acknowledgments First, I would like to thank my advisor Mikhail Lukin for his support and encour- agement during the completion of the work presented in this thesis. His enthusiasm in conducting research and his relentless curiosity, have been an inspiration and have kept me challenged during those years that I had the privilege to be his student. Thank you Misha for giving me the opportunity to learn from you. I would also like to thank Eric Heller for his support while I was in his group.
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