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.

Field quantization. Coherent states. Emission and absorption of radiation by atoms. Dissipative interactions and decoherence. Experiments in cavity QED and with trapped ions.

Identical particles. Second quantization. Ideal Gases. The mean field approximation. The electron gas in perturbation theory. Phase transitions. The Landau-Ginzburg Hamiltonian. Continuos symmetry and Goldstone modes. Fluctuations. The scaling hypothesis. The renormalization group. Perturbative RG. Phenomenology of superconductivity. Electrons in metals. BCS theory. The Josephson effect. Inhomogeneous superconductors.

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.

Classical field theory. Symmetries and conservation laws. Gauge invariance. Global and local symmetries. Spontaneous symmetry breaking. Higgs mechanism. Electroweak theory. Topological defects. Early universe. Particle standard model. Beyond the standard model.

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

Branches of QI: Communications, metrology, simulation, computation. Entanglement and interferometry. Quantum random number generator. Different architectures for QI. Quantum Computation: Quantum Circuit Model, Di Vincenzo’s criteria, Universal quantum computation, Physical realizations based on NMR, Trapped Ion etc., Quantum Error Correction. Quantum Algorithms and quantum simulation.

Bosonic string model vs quantum field theory. Supersymmetry, Superspace, Super-Yang-Mills, Supergravity. Extra dimensions. Superstring, Superparticles, p-branes, Dp-branes. Dualities. M-theory. AdS/CFT correspondence, String Landscape, ‘Swampland conjectures’ and other recent developments.

## 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. Emission and absorption of radiation by atoms. Dissipative interactions and decoherence. Experiments in cavity QED and with trapped ions.

## QUANTUM STATISTICAL MECHANICS AND CONDENSED MATTER

Identical particles. Second quantization. Ideal Gases. The mean field approximation. The electron gas in perturbation theory. Phase transitions. The Landau-Ginzburg Hamiltonian. Continuos symmetry and Goldstone modes. Fluctuations. The scaling hypothesis. The renormalization group. Perturbative RG. Phenomenology of superconductivity. Electrons in metals. BCS theory. The Josephson effect. Inhomogeneous superconductors.

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

Classical field theory. Symmetries and conservation laws. Gauge invariance. Global and local symmetries. Spontaneous symmetry breaking. Higgs mechanism. Electroweak theory. Topological defects. Early universe. Particle standard model. Beyond the standard model.

## 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. Cosmic Inflation. Gravitational waves. Perturbations in cosmology.

## QUANTUM INFORMATION

Branches of QI: Communications, metrology, simulation, computation. Entanglement and interferometry. Quantum random number generator. Different architectures for QI. Quantum Computation: Quantum Circuit Model, Di Vincenzo’s criteria, Universal quantum computation, Physical realizations based on NMR, Trapped Ion etc., Quantum Error Correction. Quantum Algorithms and quantum simulation.

## QUANTUM TECHNOLOGIES

Ultra-cold atoms in optical lattices. Superconducting quantum technologies, NV centers, Trapped Ions.

## SUPERSTRINGS AND SUPERSYMMETRY (Not offered this year)

Bosonic string model vs quantum field theory. Supersymmetry, Superspace, Super-Yang-Mills, Supergravity. Extra dimensions. Superstring, Superparticles, p-branes, Dp-branes. Dualities. M-theory. AdS/CFT correspondence, String Landscape, ‘Swampland conjectures’ and other recent developments.

## TOPICS IN FUNDAMENTAL PHYSICS

Theoretical Formalism: Introduction to Density Functional Theory (DFT). Computer based simulations: Hands of DFT.

## 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

Quantum Coherence Functions. Nonclassical light. Applications of entanglement. Introduction to light-matter interaction. Advanced mathematical tools. Quantum communication. Quantum computation with photonics. Quantum sensing.

## MATHEMATICAL TOOLS

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

## SEMICONDUCTOR PHYSICS, TRANSPORT AND SPINTRONICS

Charge transport. Spintronics. Topology of solids.

## CALENDAR

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