Zero-point energy

Abstract: Majorana zero modes are quasiparticle excitations in condensed matter systems that have been proposed as building blocks of fault-tolerant quantum computers. They are expected to exhibit non-Abelian particle statistics, in contrast to the usual statistics of fermions and bosons, enabling quantum operations to be performed by braiding isolated modes around one another. Quantum braiding operations are topologically protected insofar as these modes are pinned near zero energy, with the departure from zero expected to be exponentially small as the modes become spatially separated. Following theoretical proposals, several experiments have identified signatures of Majorana modes in nanowires with proximity-induced superconductivity and atomic chains, with small amounts of mode splitting potentially explained by hybridization of Majorana modes. Here, we use Coulomb-blockade spectroscopy in an InAs nanowire segment with epitaxial aluminium, which forms a proximity-induced superconducting Coulomb island (a 'Majorana island') that is isolated from normal-metal leads by tunnel barriers, to measure the splitting of near-zero-energy Majorana modes. We observe exponential suppression of energy splitting with increasing wire length. For short devices of a few hundred nanometres, sub-gap state energies oscillate as the magnetic field is varied, as is expected for hybridized Majorana modes. Splitting decreases by a factor of about ten for each half a micrometre of increased wire length. For devices longer than about one micrometre, transport in strong magnetic fields occurs through a zero-energy state that is energetically isolated from a continuum, yielding uniformly spaced Coulomb-blockade conductance peaks, consistent with teleportation via Majorana modes. Our results help to explain the trivial-to-topological transition in finite systems and to quantify the scaling of topological protection with end-mode separation.

Abstract: (Chapter Heading): Zero-Point Energy in Early Quantum Theory. The Electromagnetic Vacuum. Some QED Vacuum Effects. Nonrelativistic Theory of Atoms in Vacuum. Interlude: Radiation Reaction. The Vacuum in Quantum Optics. Casimirand van der Waals Forces: Prelude. Casimir and van der Waals Forces: Elaborations. The Dirac Equation. Introduction to Quantum Field Theory. Self-Energies and Renormalization. Feynman Diagrams. Appendices: Oscillator Equation and Absorption Rate. Force onan Atom in a Thermal Field. Derivation of Equation (2.28). Electric Field of Radiation Reaction. Photodetection and Normal Ordering. Transverse and Longitudinal Delta Functions. Lorentz-Invariant Measure. Index.

Abstract: Chapter 1: What is Density Functional Theory? 1.1 How To Approach This Book. 1.2 Examples of DFT in Action. 1.3 The Schrodinger Equation. 1.4 Density Functional Theory - From Wavefunctions to Electron Density. 1.5 The Exchange-Correlation Functional. 1.6 The Quantum Chemistry Tourist. 1.7 What Can't DFT Do?. 1.8 Density Functional Theory in Other Fields. 1.9 How To Approach This Book (Revisited). Chapter 2: DFT Calculations for Simple Solids. 2.1 Periodic Structures, Supercells, and Lattice Parameters. 2.2 Face Centered Cubic Materials. 2.3 Hexagonal Close Packed Materials. 2.4 Crystal Structure Prediction. 2.5 Phase Transformations. Chapter 3: Nuts and Bolts of DFT Calculations. 3.1 Reciprocal Space and k-points. 3.2 Energy Cutoffs. 3.3 Numerical Optimization. 3.4 DFT Total Energies - An Iterative Optimization Problem. 3.5 Geometry Optimization. Chapter 4: DFT Calculations for Surfaces of Solids. 4.1 Why Surfaces Are Important. 4.2 Periodic Boundary Conditions and Slab Models. 4.3 Choosing k-points for Surface Calculations. 4.4 Classification of Surfaces by Miller Indices. 4.5 Surface Relaxation. 4.6 Calculation of Surface Energies. 4.7 Symmetric and Asymmetric Slab Models. 4.8 Surface Reconstruction. 4.9 Adsorbates on Surfaces. 4.10 Effects of Surface Coverage. Chapter 5: DFT Calculations of Vibrational Frequencies. 5.1 Isolated Molecules. 5.2 Vibrations of Collections of Atoms. 5.3 Molecules on Surfaces. 5.4 Zero Point Energies. 5.5 Phonons and Delocalized Modes. Chapter 6: Calculating Rates of Chemical Processes Using Transition State Theory. 6.1 A One-Dimensional Example. 6.2 Multi-dimensional Transition State Theory. 6.3 Finding Transition States. 6.4 Finding the Right Transition State. 6.5 Connecting Individual Rates to Overall Dynamics. 6.6 Quantum Effects and Other Complications. Chapter 7: Equilibrium Phase Diagrams From Ab Initio Thermodynamics. 7.1 Stability of Bulk Metal Oxides. 7.2 Stability of Metal and Metal Oxide Surfaces. 7.3 Multiple Chemical Potentials and Coupled Chemical Potentials. Chapter 8: Electronic Structure and Magnetic Properties. 8.1 Electronic Density of States. 8.2 Local DOS and Atomic Charges. 8.3 Magnetism. Chapter 9: Ab Initio Molecular Dynamics. 9.1 Classical Molecular Dynamics. 9.2 Ab Initio Molecular Dynamics. 9.3 Applications of Ab Initio Molecular Dynamics. Chapter 10: Accuracy and Methods Beyond "Standard" Calculations. 10.1 How Accurate Are DFT Calculations? 10.2 Choosing A Functional. 10.3 Examples of Physical Accuracy. 10.4 DFT+X Methods for Improved Treatment of Electron Correlations. 10.5 Large System Sizes With Linear Scaling Methods and Classical Forcefields. 10.6 Conclusion.