Quantum Hall effects

It is explained how plateaux are seen in the Hall conductance of two dimensional electron gases, at cryogenic temperatures, when the magnetic field is scanned from zero to ~10T. On a Hall plateau σ‎xy = ne 2/h, where n is integral, while the longitudinal conductance vanishes. This is the integral quantum Hall effect. Free electrons in such devices are shown to occupy quantized Landau levels, analogous to classical cyclotron orbits. The stability of the IQHE is shown to be associated with a mobility gap rather than an energy gap. The analysis showing the topological origin of the IQHE is reproduced. Next the fractional QHE is described: Laughlin’s explanation in terms of an IQHE of quasiparticles is presented. In the absence of any magnetic field, the quantum spin Hall effect is observed, and described here. Time reversal invariance and Kramer pairs are seen to be underlying requirements. It’s topological origin is outlined.

Review of basic quantum physics

Basic experimental evidence is sketched: the black body radiation spectrum, the photoeffect, Compton scattering and electron diffraction; the Bohr model of the atom. Quantum mechanics is reviewed using the Copenhagen interpretation: eigenstates, observables, hermitian operators and expectation values are explained. Wave-particle duality, Schrödinger’s equation, and expressions for particle density and current are described. The uncertainty principle, the collapse of the wavefunction, Schrödinger’s cat and the no-cloning theorem are discussed. Dirac delta functions and the usage of wavepackets are explained. An introduction to state vectors in Hilbert space and the bra-ket notation is given. Abstracts of special relativity and Lorentz invariants follow. Minimal electromagnetic coupling and the gauge transformations are explained.

Particle physics II

Quantum chromodynamics the quantum gauge theory of strong interactions is presented: SU(3) being the (colour) symmetry group. The colour content of strongly interacting particles is described. Gluons, the field particles, carry colour so that they mutually interact – unlike photons. Renormalization leads to the coupling strength declining at large four momentum transfer squared q 2 and to binding of quarks in hadrons at small q 2. The cutoff in the range of the strong interaction is shown to be due to this low q 2 behaviour, despite the gluon being massless. In high energy interactions, say proton-proton collisions, the initial process is a hard (high q 2) parton+parton to parton+parton process. After which the partons undergo softer interactions leading finally to emergent hardrons. Experiments at DESY probing proton structure with electrons are described. An account of electroweak unification completes the book. The weak interaction symmetry group is SUL(2), L specifying handedness. This makes the electroweak symmetry U(1)⊗SUL(2). The weak force carriers, W± and Z0, are massive, which is at odds with the massless carriers required by quantum gauge theories. How the BEH mechanism resolves this problem is described. It involves spontaneous symmetry breaking of the vacuum with scalar fields. The outcome are massive gauge field particles to match the W± and Z0 trio, a massless photon, and a scalar field with a massive particle, the Higgs boson. The experimental programmes that discovered the vector bosons in 1983 and the Higgs in 2012 are described, including features of generic detectors. Finally puzzles revealed by our current understanding are outlined.

Superfluid 4He

The superfluid transition of 4He at 2.17K to He-II and the inference of an underlying condensate are introduced. The fountain effect is interpreted. Andronikashvili’s experiment and the determination of superfluid fraction versus temperature are discussed. Sound and second sound are described. Relationships between the condensate and superfluid fractions, and to off diagonal long-range order (ODLRO) are deduced. The revelation of topological quantization of circulation by Vinen’s experiment is recounted. Spontaneous symmetry breaking by the condensate’s phase coherence is explained. Excitations and their dispersion relations described with Landau’s interpretation, including the explanation of the critical velocity of superflow. Vortices, their interpretation in terms of quantized circulation, and their visualization are described.

Symmetry and topology

Space-time symmetries, conservation laws and Nöther’s theorem are discussed. The Poincaré group, generators and Casimir invariants are outlined. Local charge conservation and the corresponding U(1) charge symmetry underlying electromagnetism are presented, showing the roles of minimal electromagnetic coupling and gauge transformations. Experimental demonstrations of the Aharonov–Bohm effect are described and the topological interpretation is recounted. How the Aharonov–Casher effect survives in the classical world is mentioned. Berry’s revelation of geometric phase is presented. The Bitter–Dubbers experiment confirming this analysis is presented. Some comments are given on a Hilbert space with a simple topology.

Transitions

A derivation of Fermi’s golden rule is given: this is the interface into which matrix elements from theory can be slotted to provide a prediction testable by experiment. The example of the prediction of the 2p→1s decay in hydrogen is worked through in detail. Selection rules, spectral line shapes (Breit–Wigner and Gaussian) and broadening processes are explained. The formula for the experimental cross-section in terms of the matrix element is produced. The Born approximation is presented and applied to Rutherford scattering. Then the decay rate for allowed β‎-decays is calculated in Fermi’s model and fitted to the observed rates. Low energy s-wave scattering is analysed in terms of phase shift and scattering length. The example of cold alkali metal atom scattering (≤10−6eV) is treated in preparation for use later with gaseous Bose–Einstein condensates. Ramsauer–Townsend effect explained.

Particle physics I

Particle families (quarks and leptons), their properties and their interactions are introduced. The exchange mechanism and the Yukawa potential are discussed. Natural units are explained. The cross-section for e − + e + → μ‎− + μ‎+ is calculated using a first order Feynman diagram. Comparison with data reveals the existence of the Z0-boson and makes a link between electroweak processes. Higher orders diagrams give divergences and their removal by renormalization is described. Neutrino properties are outlined and the determination of the number of light neutrinos related. The weak interaction is discussed: parity and charge parity are seen to be maximally violated in W-boson exchange, but the product is approximately conserved. Handedness is pursued in an appendix using Dirac spinors. The neutrino mass and weak eigenstates differ and this leads to oscillations between weak eigenstates in flight. Measurements of the neutrino flux from the sun revealing this behaviour are described. Weak and strong eigenstates of quarks also differ by a unitary transformation, the CKM matrix. This difference leads to oscillations of certain neutral mesons from particle to antiparticle. This behaviour is explored for neutral K-mesons and for B0 d mesons. CP violation is observed, which is required for the survival of matter in the universe.

Gaseous Bose–Einstein condensates

The (gaseous) BECs are introduced: clouds of 106−8 alkali metal atoms, usually 87Rb or 23Na, below ~1 μ‎K. The laser cooling and magnetic trapping are described including the evaporation step needed to reach the conditions for condensation. The magnetooptical and Ioffe–Pritchard traps are described. Imaging methods, both destructive and non-destructive are described. Evidence of condensation is presented; and of interference between separated clouds, thus confirming the coherence of the condensates. The measurement of the condensate fraction is recounted. The Gross–Pitaevskii analysis of condensate properties is given in an appendix. How Bragg spectroscopy is used to obtain the dispersion relation for excitations is detailed. Finally the BEC/BCS crossover is introduced and the role therein of Feshbach resonances.

Cavity quantum physics

The model of a cavity-enclosed 2-state atom with transition frequency near resonant with a cavity mode is introduced. For conditions where their coupling dominates the Jaynes–Cummings model is described. Rabi flopping of energy between atom’s excited state and the cavity mode is recounted. Hybrid states and the AC Stark effect are discussed. Experiments with Rydberg atoms revealing the quantum nature of the cavity-atom state are discussed. Then mechanisms for trapping ions are outlined and the use of a single mercury ion as the pendulum of an optical clock is described. This relies on shelving to make non-demolition measurements on the ion. Then the measurement of (g-2) for the electron using an electron in a Penning trap is related. The quantity of interest, is the difference between the cyclotron and spin precession frequencies: its measurement by a different non-demolition technique is detailed. Finally the Purcell effect is presented, by which the lifetime of an atomic state in a cavity can be shortened or lengthened.

Entanglement

The distiction between classical product states and quantum entangled states is disclosed with examples. Spontaneous parametric down conversion as a source of entangled photons is described. The action of a perfect beam splitter is analysed using creation and annihilation operators. The HOM interferometer is described. Its use in demonstrating the indistinguishability of photons and in measuring bandwidth of sources at the level of femtoseconds is recounted. Two particle entanglement is analysed using the Bloch sphere representation showing how the full knowledge of the entangled state does not fix the state of the individual particles. The four Bell states, eigenstates of two particle entanglement, are introduced. Teleportation of a photon state using entangled photons is described, and an experiment to entangle the quantum states of atoms at space-like separation outlined.