Relativistic quantum mechanics. Path Integrals. WKB Method. Coherent states and particle creation by a classical source. Landau Levels. Berry’s Phase. Introduction to quantum open systems: entanglement, superoperators, master equations.

Relativistic quantum mechanics and quantum field theory. Scalar fields. Photons and the electromagnetic field. Electrons and the Dirac field. Perturbation theory and Feynman diagrams. Basics of loop diagrams.

Ideal quantum gases. Many body systems and second quantization. Interacting quantum gases. The degenerate electron gas. Spontaneous symmetry breaking: superconductivity and superfluidity. Introduction to the renormalization group.

Each compulsory course involves 4 hours per week of lectures, equivalent to 5 ECTS.

OPTIONAL COURSES:

Five optional courses must be taken during the second quarter (January-March). Each course involves 3 hours per week of lectures, equivalent to 4 ECTS. The list of optional courses actually offered will depend on the number and interests of the students and may vary from year to year. Here is a list of courses that could potentially be offered.

Concepts on General Relativity and Quantum Field Theory. Quantum field theory in curved space times. Quantum mechanics in the early universe. Inflation. Gravitational waves. Perturbations in cosmology.

Branches of quantum information: Communications, metrology, simulation, computation. The entanglement advantage. Entanglement and interferometry. Quantum random number generator. Different architectures for quantum information. Gates for quantum computation.

The second quantum revolution. Quantum metrology: The atomic clock and other ultraprecise devices. Quantum teleportation. Quantum cryptography and communications. Gates for quantum computation.

Density Functional Theory (theory and practice). Several topics in condensed matter physics, such as electron-phonon coupling and superconductivity, electronic and magnetic linear response: magnons and plasmons, topological insulators.

COLD MATTER PHYSICS (Not offered this year)

Introduction to ultracold atom physics. Atom-light interaction. Laser cooling and trapping. Bose-Einstein condensation. Gross-Pitaevskii theory. Bogoliubov theory. Optical lattices and tight binding models.

Introduction: classical atom-field interactions. Quantum state: Pictures, Wigner distribution, P and Q functions. Semiclassical theory: Rate equations, Two-level system, Three-level systems. Quantum theory of light-matter interactions: Quantization of the EM field, cavity QED and Jaynes-Cummings model, spontaneous emission.

Semiconductor nanostructures: quantum wells, quantum wires, and quantum dots. Quantum Hall effect in two-dimensional electron systems. Graphene as an ideally two-dimensional material. Quantum transport in one-dimensional systems. Quantum dots as spin-based qubits. Spin-orbit coupling and spin manipulation by electric field.

CALENDAR

You can find the calendar and the schedule for the classes in this semester in the following links.

## Courses

## COMPULSORY COURSES:

You can find a syllabus of each course by following the link in their titles.

During the first quarter (October-December) the student must take the following courses:

## ADVANCED QUANTUM MECHANICS

Relativistic quantum mechanics. Path Integrals. WKB Method. Coherent states and particle creation by a classical source. Landau Levels. Berry’s Phase. Introduction to quantum open systems: entanglement, superoperators, master equations.

## QUANTUM FIELD THEORY

Relativistic quantum mechanics and quantum field theory. Scalar fields. Photons and the electromagnetic field. Electrons and the Dirac field. Perturbation theory and Feynman diagrams. Basics of loop diagrams.

## QUANTUM OPTICS AND INFORMATION

Field quantization. Coherent states. Atom-field interactions: semiclassical and fully quantum; Jaynes-Cummings model. Quantum coherence functions. Beam splitters and interferometers. Nonclassical light. Dissipative interactions and decoherence. Trapped ions. Introduction to quantum information.

## QUANTUM STATISTICAL MECHANICS AND CONDENSED MATTER

Ideal quantum gases. Many body systems and second quantization. Interacting quantum gases. The degenerate electron gas. Spontaneous symmetry breaking: superconductivity and superfluidity. Introduction to the renormalization group.

Each compulsory course involves 4 hours per week of lectures, equivalent to 5 ECTS.

## OPTIONAL COURSES:

Five optional courses must be taken during the second quarter (January-March). Each course involves 3 hours per week of lectures, equivalent to 4 ECTS. The list of optional courses actually offered will depend on the number and interests of the students and may vary from year to year. Here is a list of courses that could potentially be offered.

## FIELDS AND PARTICLES

Topics selected among the following: Electromagnetic and weak interactions, Quantum Chromo Dynamics, Unification, Higgs particle, Elementary particles, Supersymmetry, Primordial Nucleosynthesis, Topological defects, Cosmic Rays.

## QUANTUM ASPECTS OF COSMOLOGY AND ASTROPHYSICS

Concepts on General Relativity and Quantum Field Theory. Quantum field theory in curved space times. Quantum mechanics in the early universe. Inflation. Gravitational waves. Perturbations in cosmology.

## QUANTUM INFORMATION

Branches of quantum information: Communications, metrology, simulation, computation. The entanglement advantage. Entanglement and interferometry. Quantum random number generator. Different architectures for quantum information. Gates for quantum computation.

## QUANTUM TECHNOLOGIES

The second quantum revolution. Quantum metrology: The atomic clock and other ultraprecise devices. Quantum teleportation. Quantum cryptography and communications. Gates for quantum computation.

## SUPERSTRINGS AND SUPERSYMMETRY (Not offered this year)

Supersymmetry. Supergravity. Superparticles. Superstrings. Super-P-branes, D-branes. ADS/CFT correspondence and applications. Other recent developments.

## TOPICS IN FUNDAMENTAL PHYSICS

Density Functional Theory (theory and practice). Several topics in condensed matter physics, such as electron-phonon coupling and superconductivity, electronic and magnetic linear response: magnons and plasmons, topological insulators.

## COLD MATTER PHYSICS (Not offered this year)

Introduction to ultracold atom physics. Atom-light interaction. Laser cooling and trapping. Bose-Einstein condensation. Gross-Pitaevskii theory. Bogoliubov theory. Optical lattices and tight binding models.

## ADVANCED QUANTUM OPTICS

Introduction: classical atom-field interactions. Quantum state: Pictures, Wigner distribution, P and Q functions. Semiclassical theory: Rate equations, Two-level system, Three-level systems. Quantum theory of light-matter interactions: Quantization of the EM field, cavity QED and Jaynes-Cummings model, spontaneous emission.

## MATHEMATICAL TOOLS

Differential Geometry, Lie Groups, Fiber Bundles and Yang-Mills theory, Functional Analysis.

## SEMICONDUCTOR PHYSICS, TRANSPORT AND SPINTRONICS

Semiconductor nanostructures: quantum wells, quantum wires, and quantum dots. Quantum Hall effect in two-dimensional electron systems. Graphene as an ideally two-dimensional material. Quantum transport in one-dimensional systems. Quantum dots as spin-based qubits. Spin-orbit coupling and spin manipulation by electric field.

## CALENDAR

You can find the calendar and the schedule for the classes in this semester in the following links.