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Circuit Quantum Electrodynamics with Electrons on Helium

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Circuit Quantum Electrodynamics with Electrons on Helium Circuit Quantum Electrodynamics with Electrons on Helium A Dissertation Presented to the Faculty of the Graduate School of Yale University in Candidacy for the Degree of Doctor of Philosophy by Andreas Arnold Fragner Dissertation Director: Professor Robert J. Schoelkopf December 2013 c 2013 by Andreas Arnold Fragner All rights reserved. ii Abstract Circuit Quantum Electrodynamics with Electrons on Helium Andreas Arnold Fragner 2013 This thesis describes the theory, design and implementation of a circuit quantum electro- dynamics (QED) architecture with electrons floating above the surface of superfluid helium. Such a system represents a solid-state, electrical circuit analog of atomic cavity QED in which the cavity is realized in the form of a superconducting coplanar waveguide resonator and trapped electrons on helium act as the atomic component. As a consequence of the large elec- tric dipole moment of electrons confined in sub-μm size traps, both their lateral motional and spin degrees of freedom are predicted to reach the strong coupling regime of cavity QED, with estimated motional Rabi frequencies of g/2π ∼ 20 MHz and coherence times exceed- ing 15 μs for motion and tens of milliseconds for spin. The feasibility of the architecture is demonstrated through a number of foundational experiments. First, it is shown how copla- nar waveguide resonators can be used as high-precision superfluid helium meters, allowing us to resolve film thicknesses ranging from 30 nm to 20 μm and to distinguish between van- der-Waals, capillary action and bulk films in micro-channel geometries. Taking advantage of the capacitive coupling to submerged electrodes and the differential voltage induced as a result of electron motion driven at a few hundred kHz, we realize the analog of a field-effect transistor on helium at milli-Kelvin temperatures on a superconducting chip and use it to measure and control the density of surface electrons. Finally, the trapping and detection of an electron ensemble in a DC-biased superconducting resonator is reported. The presence of electrons in the resonator mode volume manifests itself as trap-voltage dependent frequency shifts of up to ∼ 10 cavity linewidths and increases in cavity loss of up to ∼ 45 %. iii Contents List of Figures viii List of Tables xii Acknowledgements xiii 1. Introduction 1 1.1. Circuit Quantum Electrodynamics ......................... 2 1.2. Electrons on Superfluid Helium ........................... 5 1.3. Thesis Overview ................................... 6 2. Electrons on Superfluid Helium 10 2.1. Quantized Vertical Motion .............................. 11 2.1.1. Rydberg Surface States ............................ 11 2.1.2. Stark Shift and External Fields ....................... 15 2.1.3. Quantum Information Processing With Vertical States .......... 17 2.2. Many-Electron States on Helium .......................... 20 2.2.1. Hamiltonian and Phase Diagram ...................... 20 2.2.2. Two-Dimensional Electron Gas and Coulomb Liquid .......... 24 2.2.3. Wigner Crystallization ............................ 25 2.3. Superfluid Helium and Quantum Liquids ..................... 28 2.3.1. Thermodynamic Properties ......................... 29 2.3.2. Transport Properties & Thin Film Dynamics ............... 30 2.3.3. Charged Helium Films and Hydrodynamic Instability ......... 31 2.3.4. Capillary Action and Micro-Channel Geometries ............ 32 2.3.5. Ripplons and Elementary Surface Excitations ............... 34 iv Contents 2.3.6. Alternatives to Superfluid Helium ..................... 36 3. Circuit Quantum Electrodynamics with Electrons on Helium 38 3.1. Cavity Quantum Electrodynamics ......................... 39 3.1.1. Resonant Strong Coupling Limit ...................... 44 3.1.2. Dispersive Limit ............................... 47 3.2. Superconducting Coplanar Waveguide Cavities ................. 48 3.2.1. Terminated Transmission Lines ....................... 49 3.2.2. Inductively- and Capacitively-Coupled LCR Oscillators ........ 51 3.2.3. Inductively- and Capacitively-Coupled Transmission Line Resonators 55 3.2.4. Circuit Quantization ............................. 58 3.2.5. Coplanar Waveguide Geometry ...................... 59 3.3. Quantum Dots on Superfluid Helium ....................... 62 3.3.1. Lateral Electrostatic Traps .......................... 63 3.3.2. Quartic Anharmonic Oscillator Model ................... 65 3.3.3. Numerical Methods and Trap Simulations ................ 70 3.3.4. Comparison to Transmon Qubits ...................... 72 3.4. Circuit QED: Single Electron-Cavity Coupling .................. 73 3.5. Spin-Motion Coupling ................................ 76 3.6. Decoherence Mechanisms .............................. 79 3.6.1. Decoherence Primer ............................. 80 3.6.2. Radiative Decay and Spontaneous Emission ............... 82 3.6.3. Decay via Superfluid Excitations ...................... 85 3.6.4. Dephasing due to Voltage Fluctuations .................. 88 3.6.5. Ripplon-Induced Dephasing ........................ 90 3.6.6. Classical Helium Level Fluctuations .................... 91 3.6.7. Summary of Motional Decoherence Rates ................. 92 3.6.8. Spin Decoherence ............................... 93 3.7. Trapped Many Electron States ............................ 94 3.7.1. One-Dimensional Electron Chains in Parabolic Traps .......... 95 3.7.2. Electron Chain-Cavity Coupling ...................... 100 3.7.3. Helium Curvature Effects .......................... 101 v Contents 4. Experimental Setup and Device Fabrication 105 4.1. Measurement Setup .................................. 105 4.1.1. Setup Overview ................................ 106 4.1.2. Capillary Lines and Helium Supply System ................ 110 4.1.3. Hermetically-Sealed Sample Cells ..................... 113 4.1.4. Low-Energy Cryogenic Electron Sources ................. 113 4.2. RF, Microwave and Audio-Frequency Signal Processing ............. 117 4.2.1. Phase-Sensitive Detection and Lock-in Amplification .......... 118 4.2.2. Heterodyne and Homodyne Detection ................... 120 4.3. Nano- and Microfabrication of Superconducting Devices ............ 121 5. Superfluid Helium on Coplanar Waveguide Cavities 127 5.1. Superconducting Resonators as Helium-Level Meters .............. 128 5.1.1. Helium-Induced Frequency Shifts ..................... 128 5.1.2. Analytic Approximations: Thick-Film Limit ............... 130 5.1.3. Numerical Simulations: Thin-Film Limit ................. 131 5.2. Fill Dynamics and Level-Meter Measurements .................. 133 5.3. Helium-Level Tuning ................................. 141 5.3.1. Electromechanical Force on Helium Film Surface ............ 141 5.3.2. Level Tuning in a DC-biased Center Pin Resonator ........... 143 5.3.3. Voltage Offsets ................................ 145 6. On-Chip Detection of a Two-Dimensional Electron Gas on Helium 147 6.1. Sommer-Tanner Method ............................... 148 6.1.1. Geometry and Measurement Principle ................... 148 6.1.2. Lumped Element Circuit Model ...................... 150 6.1.3. Transmission Line Mapping ......................... 153 6.2. Device & Measurement Setup ............................ 155 6.2.1. Inductively-Coupled Cavity Helium Meter ................ 155 6.2.2. Sommer-Tanner Configuration ....................... 155 6.3. Field-Effect Transistor on Superfluid Helium ................... 157 6.4. Density Measurements ................................ 161 vi Contents 7. Trapping Electrons in a Superconducting Resonator 165 7.1. Device and Simulations ............................... 166 7.1.1. DC-biased Center Pin Resonators ..................... 166 7.1.2. Many-Electron Cavity Coupling Mechanisms .............. 170 7.2. Electron-Induced Frequency and Q Shifts ..................... 177 7.3. Loss and Hysteresis Measurements ......................... 182 8. Conclusion & Outlook 185 Appendix 190 A. Electron-Field Interactions 191 B. One-Dimensional N Electron Chains 193 C. Image Charge Effects 198 Bibliography I vii List of Figures 1.1. Comparison of different candidate systems for the implementation of a hybrid circuit QED architecture. ............................... 3 2.1. Single electron above the surface of superfluid helium: Illustration, vertical wave functions and energy levels. ......................... 11 2.2. Stark-shift spectroscopy measurements by Grimes et al. ............. 15 2.3. Stark-shifted vertical electron binding potentials and transition frequencies. 16 2.4. Sketch of a two-dimensional many-electron system floating on superfluid he- lium. .......................................... 21 2.5. Parametrized phase diagram for two-dimensional electrons on helium. .... 24 2.6. Phase diagram of 4He near the lambda transition line. .............. 28 2.7. Superfluid thin film formation: Van-Waals wall coating and capillary action filling of micro-channel arrays. ........................... 31 3.1. Schematic layout of a cavity QED system ..................... 40 3.2. Energy level diagram of the Jaynes-Cummings Hamiltonian .......... 45 3.3. Lumped element LCR oscillator circuits with various load impedances .... 52 3.4. External quality factors of coupled transmission lines at ω0/2π =5GHz and Z0 = RL =50Ω. .................................... 57 3.5. Schematic cross-section and top view of coplanar waveguide geometry .... 60 3.6. Schematic top view of coplanar waveguide resonators and their voltage dis- tributions for the lowest two modes. ........................ 61 3.7. Optical microscope
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