Master 2 courses in Physics - International Centre for Fundamental Physics track

2nd year Master - 1st Semester - 6 ECTS - English Level: B2 (no test required)

Location: Sorbonne Université

Brief Description

The Advanced Biophysics Course is a lecture course that covers modern concepts in experimental and theoretical physics of living systems, in the broadest sense. We will present theoretical ideas and illustrate them with experiments that show how these ideas are relevant for real systems, at the same time presenting modern experimental techniques. The topics covered by the course will include : cytoskeleton physics, tissue mechanics, non-equilibrium molecular motors, gene regulation, evolution, neuroscience, among others. The concepts covered will include landscape theory, non-equilibrium phenomena, nonlinear physics, stochastic processes, information theory applied to living systems, among others.

Contact

enseignement@phys.ens.fr

2nd year Master - 1st Semester - 6 ECTS - English Level: B2 (no test required)

Location: Sorbonne Université

Brief Description

A lot of conceptual progress in physics requires more and more precise and quantitative experiments used by theoreticians to produce new models or used to (in)validate models. This course will present some of the most recent and advanced in experimental physics used in soft matter and biological physics and minimal ingredients for their data analysis.

Prerequisites

Good knowledge in Optics, Statistical physics and basic notions in biology.

Contact

enseignement@phys.ens.fr

2nd year Master - 1st Semester - 6 ECTS - English Level: B2 (no test required)

Location: Université Paris Cité

Brief Description

The aim of the course is to present in details the advanced theoretical tools in quantum mechanics with an emphasis on degenerate many-body systems and scattering theory.

Prerequisites

Basic knowledge on Quantum Mechanics and Statistical Physics.
Elements on complex analysis: analytic functions and residue theorem.

Contact

enseignement@phys.ens.fr

2nd year Master - 1st Semester - 6 ECTS - English Level: B2 (no test required)

Brief Description

This course covers advanced topics in Statistical Physics. It assumes a very good knowledge of the Statistical Physics concepts and methods taught in standard lectures at the M1 level.

Prerequisites

We strongly suggest the students who do not have a strong background in Statistical Physics to profit from the summer break and acquire it. Any standard Statistical Physics textbook should be helpful.
We have also prepared an exercise sheet Math Support with definitions and problems. It focuses on the mathematical tools needed before the beginning of our lectures and exercise sessions. We strongly suggest the students to work on these problems as a preparation for our course. We will not explicitly correct these exercises but we will assume that the students are already proficient at using these methods.

Contact

enseignement@phys.ens.fr

2nd year Master - 1st Semester - 6 ECTS - English Level: B2 (no test required)

Brief Description

Part 1 : Out-of-equilibrium statistical physics
Part 2 : Introduction to phase transitions and critical phenomena

Prerequisites

- Equilibrium statistical physics (ensemble theory)
- Some concepts in probability : moments, variance and cumulants, generating function, central limit theorem

Contact

enseignement@phys.ens.fr

2nd year Master - 1st Semester - 6 ECTS - English Level: B2 (no test required)

Location: ENS-PSL

Brief Description

Statistical physics is witnessing a revolution : understanding the dynamics of a very large number of interactive degrees of freedom, which has been from the beginning the main aim of statistical physics, has become now a central problem in many fields such as physics, biology, computer science, just to cite a few. Now more than ever, statistical physics is for both its methods and its applications a very powerful discipline with a very broad range of theoretical methods and ramifications in many branches of science. The aim of this series of lectures is facing the students with this very rich state of the art: on one hand by teaching fundamental notions and methods of statistical physics and at the same time by presenting its modern applications in physics and beyond.

Prerequisites

- Equilibrium statistical physics: statistical ensembles, thermodynamics, basic notions of phase transitions. Students should have a good knowledge of statistical ensembles (canonical, microcanonical and grand-canonical) and their use to compute thermodynamic properties of equilibrium systems. Basic notions and examples of phase transitions are also required.
- Mathematical tools : basic probability theory, operator formalism. A good working knowledge of basic probability theory will be important, in particular probability measures, conditioned probability, averages and moments.
- Students should be familiar with Fourier transforms and Gaussian integrals.

Contact

enseignement@phys.ens.fr

2nd year Master - 1st Semester - 6 ECTS - English Level: B2 (no test required)

Location: Université Paris Cité

Brief Description

Computational physics plays a central role in all fields of physics, from classical statistical physics, soft matter problems, and hard-condensed matter. Our goal is to cover the very basic concepts underlying computer simulations in classical and quantum problems, and connect these ideas to relevant contemporary research problems in various fields of physics. In the TD’s you will also learn how to set, perform and analyse simple computer simulations by yourself. We will use Python, but no previous knowledge of this programming language is needed.

Contact

enseignement@phys.ens.fr

2nd year Master - 1st Semester - 6 ECTS - English Level: B2 (no test required)

Location: Sorbonne Université

Brief Description

Classical and phenomenological approaches to light-matter coupling. Harmonically bound electron, Einstein’s coefficients. Semi-classical approach. Atom-field interaction Hamiltonian. Field quantization. Eigenmodes of the classical field, normal variables, field energy, momentum and angular momentum. Quantum field coupled with quantum matter. Atom-field interaction. Spontaneous emission.

Prerequisites

Quantic physics course mandatory. The course assumes an excellent command of basic quantum physics : Dirac formalism, operators, measurement, Hamiltonian dynamics, spin-1/2, elementary atomic physics (hydrogen), harmonic oscillator. A working knowledge of density operator is also required, as well as a good knowledge of classical electromagnetism: Maxwell equations, propagation equations, limiting conditions, plane waves, spherical waves, dipole radiation.

Contact

enseignement@phys.ens.fr

2nd year Master - 1st Semester - 6 ECTS - English Level: B2 (no test required)

Location: Sorbonne Université

Brief Description

This course gives an introduction to a choice of topics in condensed matter theory. Starting from the basic notions of Bloch theory known from the introductory classes at Bachelor or 1st year master level, but extending them to include topological properties of matter, we make first steps in Green’s function theory to describe excitations in interacting quantum systems, and explore collective phenomena and emergent properties of matter and their theoretical description.

Contact

enseignement@phys.ens.fr

2nd year Master - 1st Semester - 6 ECTS - English Level: B2 (no test required)

Brief Description

The goal of this course is to introduce the main methods in electronic structure theory, which is at the heart of our present capability of understanding, predicting, and engineering materials properties based on accurate in-silico solutions of the many-body Schrödinger equation for electrons and their coupling with the lattice/structural degrees of freedom. We will follow a teaching/learning approach that combines detailed explanations of the underlying formalisms and approximations to solve the quantum many-body electron-ion problem and the introduction to their actual implementations in working computer codes.

Contact

enseignement@phys.ens.fr

2nd year Master - 1st Semester - 6 ECTS - English Level: B2 (no test required)

Location: Sorbonne Université

Brief Description

The main goal of this course is to present the different regimes of electronic transport in conductors and how quantum mechanical effects affect their resistance or conductance.

Contact

enseignement@phys.ens.fr

2nd year Master - 1st Semester - 6 ECTS - English Level: B2 (no test required)

Location: Sorbonne Université

Brief Description

The main goals of this course are to cover the fundamentals of the electronic properties of solids, and to provide the conceptual basis of selected modern experimental techniques used to investigate such properties. The course will be closely connected with the formalisms and topics introduced in the “Theory of Condensed Matter” lectures. We will illustrate the essential concepts and experimental tools with a few representative research subjects of current interest: strongly correlated materials, high-temperature superconductivity, lowdimensional electronic systems, topological systems, etc. In addition, practical sessions in the lab or at Synchrotron SOLEIL will provide hands-on experience on the themes and materials discussed in the lectures.

Contact

enseignement@phys.ens.fr

2nd year Master - 1st Semester - 6 ECTS - English Level: B2 (no test required)

Location: Université Paris Cité

Brief Description

This course aims at presenting a selection of advanced topics in gravitational classical dynamics. It starts with a reminder of the basics of general relativity and Riemannian geometry. The equations of motion are displayed, and Schwarzschild and Kerr solutions obtained. The black-hole properties of the latter are then discussed with special emphasis on their thermodynamic interpretation. Follows the general study of the energy-momentum tensor accompanied with the analysis of the conserved charges of the gravitational field. The last part of the course is devoted to Cartan’s formalism and exterior calculus. These are the building blocks for cosmological applications and primer tools for gravitational instantons, useful for the quantum theory.

Prerequisites

Special relativity and basics of general relativity: elements of differential geometry, Schwarzschild solution, tests of Einstein’s theory of gravity. This can be found in any modern textbook.

Contact

enseignement@phys.ens.fr

2nd year Master - 1st Semester - 6 ECTS - English Level: B2 (no test required)

Location: Sorbonne Université

Brief Description

The goal of this course is to better understand how interfaces instabilities can affect morphogenesis. We will address the following topics : from interfacial instabilities to form ; reaction-diffusion and population dynamics ; mechanical instabilities and buckling ; cellular organization, from liquid foams to tissues.

Prerequisites

Basic hydrodynamics.

Contact

enseignement@phys.ens.fr

2nd year Master - 1st Semester - 6 ECTS - English Level: B2 (no test required)

Location: Université Paris Cité

Brief Description

The theory of groups and their representations is a central and transverse subject that studies symmetries as they appear in mathematics and in the sciences in general, especially in physics. The aim of this course is to provide a mathematically sound introduction to the classical concepts and tools of Lie theory. The course covers Lie groups and algebras (of finite dimension) and their representations, with many examples.

Prerequisites

There are no specific prerequisites, except for standard linear and general algebra.

Contact

enseignement@phys.ens.fr

2nd year Master - 1st Semester - 6 ECTS - English Level: B2 (no test required)

Location: Sorbonne Université

Brief Description

The main goal of this course is to cover the physics of light-matter interaction in the context of quantum devices, and materials at the nanoscale. This UE features both theoretical aspects in lectures and tutorials - possibly based on the analysis and discussion of recent research papers - and experimental projects (12h) on research grade experiments at the end of the semester.

Contact

enseignement@phys.ens.fr

2nd year Master - 1st Semester - 6 ECTS - English Level: B2 (no test required)

Location: Université Paris Cité

Brief Description

Modern physics is characterized by an increasing complexity of systems under investigation, in domains as diverse as condensed matter, astrophysics, biophysics, etc. Due to the growing availability of experimental data, data-driven modelling is emerging as a powerful way to model those systems. The objective of the course is to provide the theoretical concepts and practical tools necessary to understand and to use these approaches.

Prerequisites

Basic level in statistical physics. The program language that we use is Python 3, but no previous experience in programming is required.

Contact

enseignement@phys.ens.fr

2nd year Master - 1st Semester - 6 ECTS - English Level: B2 (no test required)

Location: Sorbonne Université

Brief Description

Objectives: the lecture will present a broad overview of fluid mechanics at all scales, from bacterias to stars. After a review of hydrodynamics fundamentals (conservation laws and boundary conditions), the lecture will stroll through two main themes, Low Reynolds number hydrodynamics and inertia-dominated flows. A particular focus on thin films will allow for a description of flows as diverse as the relaxation of micron-thick viscous polymer melts up to planet-sized atmospherical flows.

Contact

enseignement@phys.ens.fr

2nd year Master - 1st Semester - 6 ECTS - English Level: B2 (no test required)

Location: Université Paris Cité

Brief Description

Progress in experimental quantum physics has transformed thought experiments into reality, so that an exciting new question can now be asked : How can we harness the "strange" features of quantum mechanics - such as nonlocality, entanglement, and quantum measurement - in new applications ? In this new field, broadly called “quantum technologies”, new ideas and concepts are being put forward. In all of these subjects, the relation between quantum physics and information plays a profound role. The course will tackle fundamental concepts as well as examples. In each lecture, discussion of experiments from various fields (trapped ions, ultracold atoms, superconducting circuits,) will complement the theoretical description.

Prerequisites

Good knowledge of basic quantum physics

Contact

enseignement@phys.ens.fr

2nd year Master - 1st Semester - 3 ECTS - English Level: B2 (no test required)

Location: Sorbonne Université

Brief Description

The course provides an introduction to some of the methods of quantum field theory necessary for the study of condensed matter and ultracold atomic gases. The main focus will be on the coherent-state functional integral and its applications to systems of interacting bosons and fermions: superfluidity and superconductivity, Fermi-liquid theory, antiferrromagnetism and Mott transition in the Hubbard model, superfluid—Mott-insulator transition in the Bose-Hubbard model, etc.

Contact

enseignement@phys.ens.fr

2nd year Master - 1st Semester - 6 ECTS - English Level: B2 (no test required)

Location: Sorbonne Université

Brief Description

In this lecture we will introduce the fundamental principles of Soft Matter Physics and treat some chosen examples, showing how:
- Van de Waals forces and electrostatic interactions are at the origin of intermolecular interactions, dictating the behavior of colloidal suspensions.
- The entropic elasticity of polymers chains determines their conformation at rest and under external flow and is reflected in their non-linear macroscopic rheology.
- In liquid crystals aligning torques create order in the microscopic director field giving rise to rise to a wealth of fascinating properties as structural color or defect dynamics dictating biological cell growth.
Our general approach will be to relate the microscopic properties or interactions to the resulting macroscopic behavior.

Contact

enseignement@phys.ens.fr

2nd year Master - 1st Semester - 6 ECTS - English Level: B2 (no test required)

Location: Université Paris Cité

Brief Description

This course is about how to describe complex systems using ideas of the renormalization group (‘coarse-graining’) and statistical field theory. Taking lattice models of magnetism as a starting point, we will explore a few milestones of 20th century many-body physics, such as Wilson-Fisher fixed point and Kosterlitz-Thouless transition. .

Prerequisites

• Basic concepts from statistical mechanics such as partition function, phase, transition, symmetry breaking.
• Familiarity with the idea of a functional (field) integral.
• Gaussian integration.

Contact

enseignement@phys.ens.fr

2nd year Master - 2nd Semester - 3 ECTS - English Level: B2 (no test required)

Location: ENS-PSL

Brief Description

The lectures offer a statistical-physics perspective on active matter, which encompasses systems whose fundamental constituents dissipate energy to exert forces on the environment. This out-of-equilibrium microscopic drive endows active systems with properties unmatched in passive ones. From molecular motors to bacteria and animals, active agents are found at all scales in nature. Over the past twenty years, physicists and chemists have also engineered synthetic active systems in the lab, by motorizing particles whose sizes range from nanometers to centimeters, hence paving the way towards the engineering of active materials.
The lectures will rely on the modern tools of statistical mechanics, from stochastic calculus to field theoretical methods, using both theoretical models and experimental systems to illustrate the rich physics of active matter.

Prerequisites

- Equilibrium statistical mechanics
- Stochastic calculus

Contact

enseignement@phys.ens.fr

2nd year Master - 2nd Semester - 3 ECTS - English Level: B2 (no test required)

Brief Description

The main goal of the course is to study the light-matter interaction at the fundamental level where one two-level system interacts with a single mode of the electromagnetic field.
The lecture will first present the fundamental concept of cavity quantum electrodynamics (Jaynes–Cummings model, resonant and dispersive interaction, Schrödinger cat states of light) and then moves to the more recent developments of circuit QED.

Prerequisites

Basic knowledge on Quantum Mechanics and Light/Field interaction

Contact

enseignement@phys.ens.fr

2nd year Master - 2nd Semester - 3 ECTS - English Level: B2 (no test required)

Location: Sorbonne Université

Brief Description

This course will explore how living systems make use of circuits and networks to process information from neuroscience to synthetic biology.

Prerequisites

Python

Contact

enseignement@phys.ens.fr

2nd year Master - 2nd Semester - 3 ECTS - English Level: B2 (no test required)

Location: Sorbonne Université

Brief Description

This course is an introduction to Conformal Field Theory (CFT), in the context of twodimensional (and quantum 1d) critical phenomena. The goal of this course is to provide a solid exposition of the CFT formalism in two dimensions, as well as a good basis to develop the intuition for its application to Statistical Mechanics and one-dimensional critical systems.

Prerequisites

Elementary complex analysis, basics of Statistical Mechanics, some notions of quantum field theory.

Contact

enseignement@phys.ens.fr

2nd year Master - 2nd Semester - 3 ECTS - English Level: B2 (no test required)

Brief Description

The purpose of these lectures is to provide the basis on modern theoretical cosmology. First, we’ll discuss cosmological solutions stemming from General Relativity, the observational and theoretical grounds upon which are built the so-called cosmological pillars, namely the fact of expansion, the big-bang model itself, and finally, the thermal history of the Universe. We then study large scale structure formation from a theoretical standpoint : cosmological perturbation theory and their quantum vacuum fluctuation origin during inflation. This should lead to a global perspective over the contemporary cosmological model ; we will also emphasise the limitations and extensions to currently ongoing researches. Observational aspects will not be covered in details, but we will pinpoint the relationship between the predictions and the real Universe, as revealed through measurements on primordial nucleosynthesis, the cosmic microwave background, the power spectrum of large scale structure, Hubble diagram and so on.

Contact

enseignement@phys.ens.fr

2nd year Master - 2nd Semester - 3 ECTS - English Level: B2 (no test required)

Brief Description

The goal of these lectures is to define mathematical gauge theory and relate it to gauge theory (from physics).
To achieve this, we will build a mathematical background: a lot of differential geometry, some algebraic topology, a bit of Riemannian geometry, and an ounce of category theory. We will also see how to use these effectively in physics.

Prerequisites

A tiny bit of differential calculus is required (you will need to recall how to compute gradients, divergences and curls).

Contact

enseignement@phys.ens.fr

2nd year Master - 2nd Semester - 3 ECTS - English Level: B2 (no test required)

Brief Description

In this course, we will cover the basics of ecology, evolution, and epidemiology, with the lens and tools of physics. We will show how concepts from nonlinear physics, stochastic processes, complex networks, random matrix theory and disordered systems can be leveraged to understand the complex dynamics of biological populations. Whenever relevant, we will discuss experimental approaches and results to illustrate the covered topics, from epidemiological data and genomics to experiments of laboratory evolution of yeast and bacteria.

Prerequisites

Stochastic processes; equilibrium statistical mechanics.

Contact

enseignement@phys.ens.fr

2nd year Master - 2nd Semester - 3 ECTS - English Level: B2 (no test required)

Location: ENS-PSL

Brief Description

 The main goal of this course is to provide an advanced view of the optical response of quantum materials. We will start from the notions acquired in the mandatory condensed matter theory course of the first semester and develop the modelling of the electronic excitation spectrum in particular for a framework adapted to materials with strongly correlated electrons. We will then provide a unified method to determine the optical conductivity and Raman responses. We will apply these concepts to concrete examples from present frontier research like high-Tc superconductors and graphene.

Contact

enseignement@phys.ens.fr

2nd year Master - 2nd Semester - 3 ECTS - English Level: B2 (no test required)

Location: ENS-PSL

Brief Description

This is an introductory course presenting the basics of the AdS/CFT duality. On the way, it will present some interesting physical concepts which are worth being acquainted wth, independently of anti-de Sitter or string theory, such as for example aspects of: confromal filed theories in d>2, the renormalization group, non-abelian gauge theories,  the large-N limit, black hole thermodynamics, semiclassical gravity.

Prerequisites

Good knowledge of (perturbative) QFT and General Relativity. No previous knowledge of string theory is required.

Contact

enseignement@phys.ens.fr

2nd year Master - 2nd Semester - 3 ECTS - English Level: B2 (no test required)

Location: Sorbonne Université

Brief Description

The aim of this lecture is to provide a description of quantum transport in disordered systems, with an emphasis on important phenomena like weak localization, Anderson localization and the Anderson metal-insulator transition. During the lecture, a number of important theoretical tools needed to describe quantum particle scattering in the presence of spatial disorder will be introduced in a pedagogical fashion, such as the Green's function technique, diagrammatic approaches to weak localization and transfer matrices. The lectures will be also illustrated by experimental examples and tutorials, especially taken from the physics of quantum gases and  condensed matter.

Contact

enseignement@phys.ens.fr

2nd year Master - 2nd Semester - 3 ECTS - English Level: B2 (no test required)

Location: ENS-PSL

Brief Description

The objective of this course is to cover the background required to understand one major system in future quantumbased technologies in the solid state, namely localized spins of atoms embedded in a solid state matrix. It will also provide key notions on central theories in solid-state physics (theory of group representations, defect formation, crystal field theory, and spin-relaxation in crystals). These concepts will be illustrated by experimental tools and by a presentation of the currently employed material platforms.

Prerequisites

Quantum mechanics M1 level.

Contact

enseignement@phys.ens.fr

2nd year Master - 2nd Semester - 3 ECTS - English Level: B2 (no test required)

Location: ENS-PSL

Brief Description

Statistical machine learning is a growing discipline at the intersection of computer science and applied mathematics (probability / statistics, optimization, etc.) and which increasingly plays an important role in many other scientific disciplines.
This course will cover supervised and unsupervised learning, as well as deep learning. The first part of the course on statistical machine learning will be focused on the analysis of data in high dimension, as well as the efficiency of algorithms to process the large amount of data encountered in multiple application areas. The second part of the course will present the fundamental principles and methods of recent deep learning techniques and their links with theoretical physics.

Prerequisites

- Proficiency in Python
- Basic Calculus, Linear Algebra
- Basic Probability and Statistics

Contact

enseignement@phys.ens.fr

2nd year Master - 2nd Semester - 3 ECTS - English Level: B2 (no test required)

Location: ENS-PSL

Brief Description

The numerical resolution of fluid dynamics equations is becoming increasingly important in many aspects of scientific research. In this course, we will develop and analyze the methods used to solve the partial differential equations relevant to fluid dynamics (elliptic, parabolic and hyperbolic). Emphasis will be placed on algorithms and their convergence properties as well as applications to a wide variety of fluid dynamics problems.

Prerequisites

An elementary knowledge of fluid dynamics will be assumed.

Contact

enseignement@phys.ens.fr

2nd year Master - 2nd Semester - 3 ECTS - English Level: B2 (no test required)

Location: ENS-PSL

Brief Description

Dark Matter Journey from Particle Physics to Modern Cosmology. For more than a century now, the mystery of missing mass (or dark matter) has been the subject of debates, and motivates experiments in the field of supra-colliders, satellites and underground laboratories. While its existence is now hardly questionable, paradoxically, its nature is totally unknown. In these lectures, I propose to revisit a history of the Universe, from its primordial state to the direct and indirect detection of dark matter. The structure will be chronological, and at each step, I will show the influence of this mysterious particle and how we can detect it. This module perfectly correlated with the cosmology and particle physics modules can be seen as a concrete application of the latter.

Contact

enseignement@phys.ens.fr

2nd year Master - 2nd Semester - 3 ECTS - English Level: B2 (no test required)

Location: ENS-PSL

Brief Description

The aim of this course is to introduce the most important topics needed to understand how we can test the Standard Model, then why and how we must go beyond it. It is aimed not just at future practitioners, but at anyone wanting to study High Energy Physics.

Prerequisites

The only prerequisite is QFT 1, but it should be taken alongside QFT 2.

Contact

enseignement@phys.ens.fr

2nd year Master - 2nd Semester - 3 ECTS - English Level: B2 (no test required)

Brief Description

This course is a research oriented introduction to a rapidly expanding theme in condensed matter. We present several aspects of this contemporary field including fundamental questions, materials, techniques and applications.
Six basic lessons cover the physics of graphene, 2D chalcogenides, cuprates and oxide interfaces with regard to quantum Hall and topological effects, 2D superconductivity, BKT transitions and applications. These are followed by three or four in-depth seminars on chosen topics.

Contact

enseignement@phys.ens.fr

2nd year Master - 2nd Semester - 3 ECTS - English Level: B2 (no test required)

Location: Sorbonne Université

Brief Description

The development of animals, starting from a single cell to produce a fully formed organism, is a fascinating process. Its study is currently advancing at a rapid pace thanks to combined experimental and theoretical progress, with yet many fundamental questions remaining to be understood. This course will address the fundamental theoretical concepts underlying the self-organization of multicellular systems, from gene regulation to the mechanics of active biological materials. The course will be based on various concepts from theoretical physics: dynamical systems, soft and active matter, the mechanics of continuous media, numerical modeling, etc.

Prerequisites

No prerequisite in physics but general knowledge in statistical physics, dynamical systems and continuous mechanics will help. Basic mathematic tools are essential (integration by parts, linear algebra, partial differential equations...) and Python programming skills will be required for the TDs.

Contact

enseignement@phys.ens.fr

2nd year Master - 2nd Semester - 3 ECTS - English Level: B2 (no test required)

Location: ENS-PSL

Brief Description

The goal of this course is to introduce the main concepts and challenges of quantum computing, a new set of technologies and techniques that promise to solve hard computational problems. We will review both the theoretical premises and promises of ideal quantum computers, and the practical challenges that one faces when one actually runs quantum programs on today’s physical quantum computers: what is the role of decoherence? How does one deal with it in practical quantum algorithms? The course will also introduce the key concepts of quantum error correction, which could be used when sufficient qubit counts and qualities are reached.

Prerequisites

Standard linear algebra. Quantum mechanics.

Contact

enseignement@phys.ens.fr

2nd year Master - 2nd Semester - 3 ECTS - English Level: B2 (no test required)

Brief Description

This is the follow-up couse to the quantum field theory class of the first semester. Topics we will cover include non-abelian gauge symmetry, spontaneous symmetry breaking, and the Higgs mechanism, all needed to understand the inner workings of the Standard Model, which we shall discuss in some detail.

Prerequisites

Quantum Field Theory I ; bases of Feynman diagrams calculation diagrams; bases of representations of Lie groups

Contact

enseignement@phys.ens.fr

2nd year Master - 2nd Semester - 3 ECTS - English Level: B2 (no test required)

Brief Description

In this course we will use the general framework of quantum phase estimation and of metrologically useful entanglement, which will be established (or recalled, for some of you) in the beginning of the course.
We will then apply these notions to real quantum sensors using photons and atoms and highlight some of the latest results in this rapidly evolving field.

Prerequisites

Basic quantum mechanics and notions of quantum optics and atomic physics, such as the Bloch sphere formalism.

Contact

enseignement@phys.ens.fr

2nd year Master - 2nd Semester - 3 ECTS - English Level: B2 (no test required)

Location: ENS-PSL

Brief Description

Recent years have seen enormous experimental progress in preparing, controlling and probing quantum systems in various regimes far from thermal equilibrium. Examples include systems as ultra-cold atomic quantum gases under time-dependent perturbations, driven non-linear cavity QED systems or strongly correlated electrons in solid-state materials under ultra-fast optical excitations. These developments have brought fresh new interest around a set of fundamental questions concerning quantum systems and their dynamical behaviour, at the interface between quantum statistical physics, condensed matter, atomic physics and quantum optics. The aim of this course is to introduce the students to the main concepts and motivations behind this active field of research.

Prerequisites

Only basic knowledge of quantum many-body physics will be needed.

Contact

enseignement@phys.ens.fr

2nd year Master - 2nd Semester - 3 ECTS - English Level: B2 (no test required)

Location: ENS-PSL

Brief Description

This course is an introduction to geometric random critical phenomena and their description by algebraic and probabilistic techniques and by quantum field theories. The main topics of study are conformally invariant extended objects, such as percolation clusters, walls in spin systems and self-evolving walks. We wish to describe these objects as precisely as possible, through critical exponents and correlation functions. A substantial part of the work takes place in two dimensions, but some techniques can also be extended to higher dimensions.

Prerequisites

Elementary statistical physics. An introduction to field theories would be beneficial.

Contact

enseignement@phys.ens.fr

2nd year Master - 2nd Semester - 3 ECTS - English Level: B2 (no test required)

Location: ENS-PSL

Brief Description

In recent years, new concepts related to the coupling of quantum many-body systems to tailored reservoirs are providing elegant waysto modify the electronic, optical and mechanical properties of quantum materials in condensed matter, atomic, optical and quantum circuit platforms. Moreover, reservoir-based approaches are emerging in the context of machine learning and quantum information. This M2 course and its tutorials will provide in a pedagogical way the fundamental concepts for the reservoir-control of quantum systems and show the most recent developments, applications and experimental realizations. The interdisciplinary and self-contained character of this class should resonate with students belonging to the tracks of Condensed Matter, Quantum Physics, and Theoretical Physics. 

Contact

enseignement@phys.ens.fr

2nd year Master - 2nd Semester - 3 ECTS - English Level: B2 (no test required)

Location: Sorbonne Université

Brief Description

Starting from the basics of elasticity, we will put emphasis on the relevant length and time scales specific to soft matter. Using examples primarily inspired from Nature, we will illustrate the relationship between some ingenious textures and structures and their mechanical performances. Our journey will confront us with those phenomena (adhesion, friction, fracture, shape conformation and morphology changes) living bodies have managed to deal with in order to perform their fundamental tasks (locomotion, nutrition, reproduction…).

Prerequisites

No prerequisites. The course will start from basics of the theory of elasticity.

Contact

enseignement@phys.ens.fr

2nd year Master - 2nd Semester - 3 ECTS - English Level: B2 (no test required)

Location: ENS-PSL

Brief Description

In this course, you will learn tools and ideas developed by statistical physics to deal with "complex systems". These tools can be used in different contexts, including economics and social sciences where the modelling of collective phenomena, crises, panics, and discontinuities, is more necessary than ever.

Prerequisites

Taste for modelling, probabilities and statistics.

Contact

enseignement@phys.ens.fr

2nd year Master - 2nd Semester - 3 ECTS - English Level: B2 (no test required)

Location: ENS-PSL

Brief Description

This course deals with systems in which the presence of impurities or amorphous structures (in other words, of disorder) influences radically the physics, generating novel phenomena. These phenomena involve the properties of the system at equilibrium (freezing and glass transitions), as well as their dynamical evolution out-of-equilibrium (pinning, avalanches), giving rise to ergodicity breaking both in absence and in presence of quantum fluctuations (classical metastability, quantum localization). We discuss the main statistical physics models that are able to capture the phenomenology of these systems, as well as the powerful theoretical tools (replica theory, large deviations, random matrix theory, strong-disorder expansions) that have been developed to characterize quantitatively their physics, which nowadays have a huge impact in a variety of fields that go well-beyond statistical physics (computer science, probability, condensed matter, theoretical biology).

Prerequisites

The course is mainly aimed at students with a background in physics.  However students in chemestry or mathematics are more than welcome. A basic knowledge of statistical physics or of probability/statics is useful.

Contact

enseignement@phys.ens.fr

2nd year Master - 2nd Semester - 3 ECTS - English Level: B2 (no test required)

Brief Description

The goal of this course is to introduce string theory. String Theory is the best developed contender for a complete theory of quantum gravity. This course serves a first introduction to this vast subject, studying in detail the theory in the first quantized formalism, culminating with a discussion of scattering amplitudes.

Prerequisites

Both Quantum Field Theory, General Relativity will help in the understanding of the course, although for the most part it is self contained.

Contact

enseignement@phys.ens.fr

2nd year Master - 2nd Semester - 3 ECTS - English Level: B2 (no test required)

Location: Université Paris Cité

Brief Description

The main goal of this course is to provide an introduction to the subject of topological phenomena in condensed-matter. After an introduction to the mathematical tools, we will focus on electronic models and discuss how topology plays a major role in shaping their physical properties. We will then move to the quantum Hall effect and discuss its spectacular phenomenology, deeply rooted into topological notions, from robust edge states to fractionalized excitations, such as Majorana fermions.

Contact

enseignement@phys.ens.fr

2nd year Master - 2nd Semester - 3 ECTS - English Level: B2 (no test required)

Brief Description

Turbulent flows are present all around us and are crucial in fields such as aeronautics, industry, meteorology, astrophysics, climate. We know their equation of motions (the Navier-Stokes equations). Yet, several theoretical and practical obstacles preclude the building of a complete theory, and impose to resort to modern and original tools of mathematics and physics, such as weak formalism, non-equilibrium physics, wavelets or multi-fractals. In this class, we will first explain the various difficulties associated with turbulence theory, and describe several tools and approaches that enable to draw a modern picture of turbulence. For this purpose, we will navigate between theory and practice, using data from real experiments or numerical simulations to put into practice what we have learned.

Prerequisites

Basic notion of Fluid Mechanics and probability theory are required.

Contact

enseignement@phys.ens.fr

2nd year Master - 2nd Semester - 3 ECTS - English Level: B2 (no test required)

Location: Sorbonne Université

Brief Description

In this course we discuss a few examples of new electronic phenomena emerging due to strong electron confinement in the matter: ultimate superconductors made of single atomic layers or nano-wires, topological insulators - materials which own conducting channels at their edges/surfaces, as well as hybrid nano-devices containing these exotic materials and phases. Theoretical aspects will be introduced and illustrated by the results of experimental studies by Scanning Tunneling Microscopy and Spectroscopy on a local scale, and by measurements of coherent electron transport and magnetism at mesoscopic scales.

Contact

enseignement@phys.ens.fr

2nd year Master - 2nd Semester - 3 ECTS - English Level: B2 (no test required)

Location: ENS-PSL

Brief Description

In these lectures, we present the key tools and concepts of cold-atom physics: laser cooling and trapping, measurement methods, the quantum statistics of ultracold gases. We also introduce a selection of modern research themes, such as quantum simulation with strongly interacting quantum gases.

Prerequisites

Advanced quantum mechanics. Atoms and Photons.

Contact

enseignement@phys.ens.fr