Course contents
Chapters in Breuer & Petruccione are referred to as BP1, BP2, etc.
Lectures 1-6 contain a compressed version of the planned Master level course, sufficient to following the PhD course proper. Lectures 7-18 contain the actual PhD level course.
Lecture 1: Overview of the course contents. Repetition of the basics of quantum mechanics. Pure states as rays in Hilbert space. Observables and Hermitian operators. Schrödinger picture, Heisenberg picture and interaction picture. Schrödinger equation. Time evolution of pure states and unitary operators. Mixed states and density matrices. The von Neumann equation describing the time evolution of a density matrix. BP2.
Lecture 2: Basics of quantum information processing. Entanglement. Von Neumann entropy as a measure of entanglement. Einstein-Podolski-Rosen experiment as an example of quantum correlations than can be given a classical interpretation (local hidden variables). Bell theorem as an example of quantum correlations that cannot be given a classical interpretation (no local hidden variables). The Local operations and classical communication (LOCC) paradigm. The quantum no cloning theorem (without proof).
Lecture 3: Basics of open quantum system dynamics. The time evolution of a reduced density matrix. Kraus operators. Number of parameters describing a general open quantum system evolution. The case of quantum Markov processes. Lindblad equation and Lindblad operators. BP3 and BP4.
Lecture 4: Decoherence and dissipation. Feynman-Vernon influence functionals. Derivation of the Feynman-Vernon influence functional using path integrals (without evaluating the actual integrals). Kernels in the Feynman-Vernon functionals are operator correlation functions in the environment. Derivation of Lindblad equation as a memory-less limit of the Feynman-Vernon approach.
Lecture 5: Practical and theoretical limits to the Feynman-Vernon approach. Overview of the alternatives if the open system itself is a harmonic oscillator. If that is not the case nor a Lindblad equation limit is appropriate, the open quantum system dynamics has to be solved numerically.
Lecture 6: The Jaynes-Cumming and the spin-boson models. Quantum memory effects. A survey of simulation techniques for open quantum systems with memory. The complexity of open quantum system dynamics in even simple set-ups at low enough temperature. BP7 & BP10.
Lecture 7: Evolution of open quantum systems as PDEs and as integro-differential equations.
Stochastic simulation methods. The quantum jump method. Stochastic Schrödinger equations.
BP7.
Lecture 8. The hierachical equations of motion method (HEOM). [extra material]
Lecture 9. Driven harmonic oscillators. Comparison between different numerical methods. BP7.
Lecture 10. Applications to quantum optics systems I. Continuous measurements in quantum electrodynamics. The microscopic Hamiltonian. Incomplete measurements. BP8.
Lecture 11. Applications to quantum optics systems II. Dark states. An atom evolving in interaction with the quantum electrodynamic field as an environment. BP8.
Lecture 12. Application to quantum optics III. Strong field interaction and the Floquet picture. BP8.
Lecture 13. Relativistic quantum theory on the formal level. Schwinger-Tomonaga equation. States as functionals of spacelike hypersurfaces. Foliations of space-time. The measurement of local observables. Relativistic state reduction. BP11.
Lecture 14.
EPR correlations. Non-local measurements and causality. Entangled quantum probes. Quantum state verification. Quantum non-demolition verification of non-local states. BP11.
Lecture 15.
Quantum teleportation. Teleportation and Bell-state measurement. A survey of xperimental realization and implementations. BP11 and additional material.
Lecture 16.
Density matrix theory for quantum electro-dynamics. Field equations and correlation functions. The influence functional (Feynman-Vernon functional). BP12.
Lecture 17.
Vacuum-to-vacuum amplitudes. Decoherence by the emission of brehmstrahlung. The decoherence functional. Evolution of the decoherence functional for a quantum test body interacting with the quantized electromagnetic field. BP12.
Lecture 18.
Decoherence of many-particle states. Limits to quatum information processing from the interactions with photons. Bp12.