Hyperfine interactions and quantum information processing in quantum dots A dissertation presented by Jacob Mason Taylor 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 August 2006 c 2006 - Jacob Mason Taylor All rights reserved. Thesis advisor Author Mikhail D. Lukin Jacob Mason Taylor Hyperfine interactions and quantum information processing in quantum dots Abstract This thesis explores the feasibility of using electron spins in semiconductor quan- tum dots as fundamental building blocks for quantum information processing. We start from a general perspective, evaluating the possible limits to operation of such a spin-based system. We show that the coherence properties of electron spins are limited by their inter- action with lattice nuclear spins. We then consider approaches that take advantage of the long coherence and relaxation times of the lattice nuclear spins to mitigate their effect on the electron spin, and show how they could be used as a resource for long-term quantum memory in the solid-state. Next, we develop techniques for experimentally probing electron spin–nuclear spin interactions in quantum dots, and demonstrate the controlled creation and measurement of entangled electron spin states. A new paradigm for quantum information processing in the presence of nuclear spins emerges—one that uses two electron spins to form a single quantum bit. We demonstrate theoretically and experimentally the potential for long-lived quantum memory using this system system, and find improved, specialized approaches to quantum communication and quantum computing using pairs of electron spins. Finally, we consider methods for scaling the system to large numbers of quantum bits such that techniques for robust computation, including fault-tolerant computation, might be realized. Contents TitlePage........................................ i Abstract......................................... iii TableofContents.................................... iv ListofFigures ..................................... vii ListofTables...................................... ix CitationstoPreviouslyPublishedWork . ...... x Acknowledgments.................................... xi Dedication........................................ xiv 1 Introduction 1 1.1 Background.................................... 1 1.2 Overview ..................................... 2 1.3 Hyperfine interactions in a single quantum dot . ....... 4 2 Dephasing of quantum bits by a quasi-static mesoscopic environment 11 2.1 Thephysicalsystem ............................... 14 2.2 Freeevolution................................... 22 2.3 Driven-evolution ................................ 27 2.4 Application to experimental systems . ...... 37 2.5 Methods to mitigate bath effects . 42 2.6 Conclusions .................................... 47 3 Long-lived quantum memory in a mesoscopic environment 49 3.1 Nuclearensemblememory . 50 3.2 Preparation and control of a mesoscopic environment . .......... 60 4 Quantum information processing using localized ensembles of nuclear spins 71 4.1 Qubitsandquantumoperations. 73 4.2 Errormechanisms ................................ 82 4.3 Materialsandestimates . 89 4.4 Conclusions .................................... 90 iv Contents v 5 Quantum measurement of a mesoscopic spin ensemble 93 5.1 Phaseestimation ................................. 95 5.2 Themeasurementscheme . .. .. .. .. .. .. .. .. 97 5.3 Performanceofthescheme. 102 5.4 Errors and Fluctuations in Az .......................... 103 5.5 Example: Estimating collective nuclear spin in a quantumdot........ 107 5.6 Conclusions .................................... 111 6 Relaxation, dephasing, and quantum control of electron spins in double quantum dots 113 6.1 Electronspinsinadoublequantumdot . 115 6.2 Nuclear-spin-mediated relaxation in double dots . .......... 130 6.3 Quantum control of two electron spin states . 138 6.4 Exchange gates and echo techniques . 159 6.5 Conclusions .................................... 167 7 Electronic pumping of nuclear spin polarization in a double quantum dot 171 7.1 Experimentalsetup............................... 173 7.2 Electron spin-nuclear spin interactions . ......... 176 7.3 Nuclear spin polarization . 179 7.4 Theory of polarization in the weak polarization limit . ........... 183 7.5 Conclusions .................................... 185 8 Robust non-local entanglement generation in quantum dots 187 8.1 A single electron quantum repeater . 190 8.2 Limits to the single-electron scheme . 202 8.3 Twoelectronencoding ............................. 206 8.4 Conclusions .................................... 215 9 Cavity quantum electrodynamics with semiconductor double-dot molecules on a chip 217 9.1 EffectiveHamiltonian ............................. 219 9.2 Operatingpoint.................................. 221 9.3 Generalizedapproach ............................. 223 9.4 Noiseanddecoherence ............................. 223 9.5 Quantumcontrol ................................. 225 9.6 Conclusions .................................... 228 10 Fault-tolerant architecture for quantum computation using electrically controlled semiconductor spins 229 10.1 Protected qubits and quantum gates . 231 10.2 Errorsinquantumoperations . 236 10.3 Fault tolerance and error thresholds . ....... 240 10.4Methods...................................... 244 vi Contents 10.5Outlook ...................................... 246 A Details for Chapters 1: breakdown of the quasi-static approximation 249 B Details for Chapter 2: spin bath-related integrals and models 255 B.1 Integrals involved in driven evolution . ....... 255 B.2 Comparison with superconducting qubits: model . ........ 257 C Details for Chapter 6: inelastic decay and charge-based dephasing 259 C.1 Adiabatic elimination for nuclear-spin-mediated inelastic decay . 259 C.2 Dephasingpowerspectra . 261 D Reprint of Ref. [96], “Triplet-singlet spin relaxation via nuclei in a double quantum dot” 265 E Details for Chapter 7: spin diffusion via nuclear dipole-dipole interactions 271 F Details for Chapter 8: electron spin-based quantum repeaters and ms = 0 measurement methods 275 F.1 Measurement-based preparation . 275 F.2 Purification with ms = 0 measurement for single electron proposal . 277 F.3 Adiabaticpreparation . 281 Bibliography 285 List of Figures 1.1 Electron spin in a quantum dot interacting with lattice nuclear spins . 5 2.1 Heirarcichal system–bath–environment coupling . ........... 17 2.2 Free-inductiondecay .. .. .. .. .. .. .. .. 24 2.3 Low-powerlineshape .............................. 30 2.4 Rabioscillations ................................ 34 2.5 Rabi oscillations with decorrelation of the bath . .......... 37 2.6 Comparison of theory to superconducting qubit experiments......... 41 3.1 Storage of electron spin states in an ensemble of nuclear spins........ 52 3.2 Transfer characteristics for different nuclear polarizations . 58 3.3 Dark state polarization and entropy . ..... 63 3.4 Dark state interaction parameters . ...... 69 3.5 Quantum memory fidelity with dark states . 70 4.1 Quantum dot based approaches for nuclear ensemble-based computation . 73 4.2 Level structure for nuclear ensemble qubits . ........ 80 4.3 A network of small scale quantum computing devices . ........ 81 4.4 Error for nuclear ensemble computation . ...... 90 5.1 Illustration of the first two steps of the measurement procedure ....... 98 5.2 Estimation in the presence of preparation and measurement error . 104 6.1 Experimentalsetup............................... 116 6.2 Charge and orbital states for a double dot . 119 6.3 Exact solution: charge and orbital states . ........ 121 6.4 The far detuned and charge transition regimes . ....... 122 6.5 A double quantum dot in the (1,1) configuration . 124 6.6 Level structure: far detuned and charge transition cases ........... 125 6.7 Cycleandmeasuringexchange . 128 6.8 Energy level structure as a function of detuning . ......... 132 6.9 Field dependence of spin-blockade . 136 6.10 Chargesignalofspinblockade. 137 vii viii List of Figures 6.11 Rapid- and slow-adiabatic passage . 140 6.12 Pulsesequences................................. 144 6.13 Measuring T2∗ ................................... 147 6.14 Exchangeoscillations. 153 6.15 Dephasing in the far detuned and charge transition regimes ......... 155 6.16 Decay of exchange oscillations . 158 6.17 Expectedechosignal . 162 6.18 Singlet-tripletspinecho . 164 7.1 Schematic of the experimental setup . 174 7.2 Time and energy dependence of the singlet–T+ resonance . 178 7.3 The singlet–T+ crossing ............................. 180 7.4 Polarization: simulation and experiment . ........ 182 8.1 Scheme for entanglement generation and purification withfourspins . 189 8.2 Generating and measuring entanglement in a double quantumdot ..... 192 8.3 Purificationcurveforsinglespins . 198 8.4 Measurement scheme for purification . 209 8.5 Gate network for entanglement generation . ....... 210 8.6 Gatenetworkforpurification . 213 9.1 Two double quantum dots interacting via a transmission line resonator . 219 9.2 Coupling between a double quantum dot and a resonator . ....... 224 9.3 Dephasing and double dot–resonator coupling . ........ 226 10.1 Architecture for quantum computation . ....... 232 10.2Logicalqubit ................................... 233 10.3 Long-rangequbittransport . 237 10.4 Teleportation-based gates . 243 F.1 Projective measurement with gradients . ......