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Waveguide QED: Multiple Qubits, Inelastic Scattering, and Non-Markovianity

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Waveguide QED: Multiple Qubits, Inelastic Scattering, and Non-Markovianity Waveguide QED: Multiple Qubits, Inelastic Scattering, and Non-Markovianity by Yao-Lung L. Fang Department of Physics Duke University Date: Approved: Harold U. Baranger, Supervisor Gleb Finkelstein Shailesh Chandrasekharan Ronen Plesser David Beratan Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Physics in the Graduate School of Duke University 2017 Abstract Waveguide QED: Multiple Qubits, Inelastic Scattering, and Non-Markovianity by Yao-Lung L. Fang Department of Physics Duke University Date: Approved: Harold U. Baranger, Supervisor Gleb Finkelstein Shailesh Chandrasekharan Ronen Plesser David Beratan An abstract of a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Physics in the Graduate School of Duke University 2017 Copyright © 2017 by Yao-Lung L. Fang All rights reserved except the rights granted by the Creative Commons Attribution-Noncommercial Licence Abstract Waveguide quantum electrodynamics (QED) studies multi-level systems (or qubits) strongly interacting with one-dimensional (1D) light fields confined in a waveguide. This rapidly growing research field attracts much attention because it provides afas- cinating platform for many-body physics, quantum nonlinear optics, as well as open quantum systems (OQS). On this platform, researchers are able to control single qubit using single photons and vice versa, based on which many potential appli- cations in quantum information processing and quantum computing are proposed. Due to the reduced dimensionality, the coupling between light and matter is greatly enhanced and so is the light interference. This, together with the strong nonlinearity provided by the qubits, results in striking quantum-optical effects at the few-photon level, which have been demonstrated experimentally in the past few years on vari- ous systems such as superconducting circuits thanks to the explosive experimental progress. While much has been known with regard to a single qubit in an infinite waveguide, it is less understood how multiple qubits would reshape the properties of 1D light. For instance, there can be effective interaction between a pair of qubits in a waveguide even in the absence of dipole-dipole interaction, which in turn causes a splitting in the power spectra and interesting bunching and anti-bunching effects for the photons. Moreover, from the OQS point of view, the systems (qubits) and its environment (photons) are highly correlated, so it is natural to question and test the validity of iv Markovian dynamics in a waveguide-QED setup. In this thesis, I consider two or more distant qubits present in a waveguide under weak driving. The role of inelastic scattering and its connection to dark states and photon correlations are emphasized. I report two-photon scattering wavefunction for multiple qubits (N > 2) in an infinite or semi-infinite waveguide. The latter has a perfect mirror at the end that reflects all of the incident photons. I find that by tuning the separation between each qubit it is possible to have stronger (anti- )correlations compared to the single-qubit case, and that the correlation can last over hundreds of the lifetime of a single qubit. The inelastic scattering is highly sensitive to the qubit-qubit (or qubit-mirror) separation and the incident frequency due to the narrow widths caused by the dark states. I also investigate the differences in scattering due to the level structure of the qubits, including two-level systems (2LS), three-level systems (3LS), and their mixture. For example, I find that a “2- 3-2” setup — two distant 2LS sandwiching a 3LS — can either rectify photons or induce correlation among elastically transmitted photons. I further propose a simple waveguide-QED system as a model system for OQS studies owing to its complex yet exactly solvable nature. To this end, a time- dependent scattering is considered, from which a dynamical map describing the system evolution can be obtained. In combination with OQS tools, I present de- tailed analysis for the non-Markovian properties of the system, and point out that the scattering setup generally gives rise to a different class of dynamical maps that are largely unexplored in conventional OQS studies. The parameters considered throughout this thesis are fully accessible with existing experimental technology, so realistic tests of my work are within reach. v To my lovely wife and son. vi Contents Abstract iv List of Figures xi Acknowledgements xv 1 Introduction 1 1.1 Motivation ................................. 1 1.2 Elastic and inelastic scattering ...................... 9 1.3 Electromagnetically induced transparency ............... 13 1.4 Shaping the quantum vacuum ...................... 17 1.5 Open quantum systems and non-Markovianity in a nutshell ..... 19 1.5.1 Dynamical map and Lindblad master equation ........ 20 1.5.2 Non-Markovianity ........................ 24 1.6 Conclusion and plan of thesis ...................... 29 2 Theoretical approaches to 1D light-matter interaction 30 2.1 Summation of interference paths .................... 31 2.2 Green’s function method ......................... 35 2.2.1 Infinite waveguide ........................ 35 2.2.2 Semi-infinite waveguide ..................... 44 2.2.3 Two-photon Transmission Probability ............. 46 2.3 Master equation-like approaches ..................... 49 vii 2.3.1 Heisenberg-Langevin equation: overview ............ 49 2.3.2 Heisenberg-Langevin equation for infinite waveguide ..... 51 2.3.3 Heisenberg-Langevin equation for semi-infinite waveguide ... 57 2.3.4 The SLH formalism: overview .................. 58 2.3.5 The SLH formalism for infinite waveguide ........... 60 2.3.6 The SLH formalism for semi-infinite waveguide ........ 65 2.4 Time-dependent Schrödinger equation ................. 68 2.4.1 Infinite waveguide: one-excitation sector ............ 68 2.4.2 Infinite waveguide: two-excitation sector ............ 69 2.4.3 Semi-infinite waveguide: one-excitation sector ......... 73 2.4.4 Semi-infinite waveguide: two-excitation sector ......... 75 2.4.5 Finite-difference time-domain .................. 79 2.5 Other approaches ............................. 85 3 Waveguide QED: Power Spectra and Correlations of Two Photons Scattered Off Multiple Distant Qubits and a Mirror 88 3.1 Introduction ................................ 89 3.2 Multiple Qubits in an Infinite Waveguide: Power Spectra ....... 91 3.3 Multiple Qubits in an Infinite Waveguide: Photon Correlations .... 98 3.4 Time Delay ................................ 108 3.5 Multiple Qubits in a Semi-Infinite Waveguide: Power Spectra .... 109 3.6 Multiple Qubits in a Semi-Infinite Waveguide: Photon Correlations . 114 3.7 Conclusion ................................. 118 4 Photon Correlations Generated by Inelastic Scattering in a One- Dimensional Waveguide Coupled to Three-Level Systems 122 4.1 Introduction ................................ 123 4.2 Model and Observables .......................... 124 viii 4.2.1 Waveguide QED model with multiple three-level systems ... 124 4.2.2 Power spectrum (resonance fluorescence), S(!) ........ 126 4.2.3 Total inelastically scattered component, F (kin) ........ 126 4.2.4 Photon-photon correlation (second-order coherence), g2(t) .. 126 4.3 Single 3LS ................................. 127 4.4 3LS plus mirror .............................. 131 4.5 Two 3LS .................................. 133 4.6 Conclusions ................................ 136 5 Nonreciprocity and correlated photons at perfect elastic transmis- sion 137 5.1 Introduction ................................ 138 5.2 The model ................................. 139 5.3 Two 2LS vs. a Λ-3LS: Similarities and differences ........... 141 5.3.1 Single-photon sector ....................... 141 5.3.2 Two-photon sector ........................ 144 5.3.3 Identical 2LS ........................... 147 5.4 Rectification Linked to Inelastic Scattering ............... 148 5.5 Recovering the fluorescence: 2-3-2 .................... 152 5.6 Loss and Relaxing the Markovian Approximation ........... 156 5.7 Conclusions ................................ 158 6 Non-Markovian Dynamics of a Qubit Due to Single-Photon Scat- tering in a Waveguide 160 6.1 Introduction ................................ 161 6.2 System ................................... 163 6.3 Dynamical map .............................. 163 6.4 Non-Linbladian Master Equation .................... 166 ix 6.5 Explicit computation of DM ....................... 167 6.6 Non-Markovianity ............................ 169 6.7 Conclusion ................................. 173 7 Conclusion and Outlook 175 A Perform double and triple integrals 180 B Stability analysis 183 C Evaluating incomplete Gamma functions on the complex plane 185 D The role of A 188 E Representative results for Green’s function calculations 190 E.1 Two identical qubits in an infinite waveguide .............. 190 E.2 Three identical qubits in an infinite waveguide ............. 191 E.3 Single qubit in a semi-infinite waveguide ................ 192 F Transmission photon intensity for two detuned 2LS 193 G Derivations for constructing the scattering dynamical map 195 G.1 Derivation of the time-dependent ME .................. 195 G.2 DM in the infinite-waveguide case .................... 198 Bibliography 200 Biography 221 x List of Figures 1.1 A sketch for the interference between incident field and scattering field in 3D free space. ............................
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