EU 1st term (30 ECTS) | EU 2nd term (30 ECTS) |

Electrons and phonons in nanostructures | Quantum Computing |

Quantum theory of light | Quantum Information |

Advanced Solid State Physics | Nanomagnetism ans spintronics |

Photonic quantum devices | Functional Materials |

Electronic quantum devices | Internship |

Bidimensionals Materials | |

Imaging nano-objects | |

Experimental projects (3 weeks) | |

Visit to Laboratories |

The first term (September-January) includes lectures on the fundamental concepts and tools of quantum photonics and electronics in condensed matter, high-tech analysis tools (electronic microscopy, STM, AFM…), a large panorama of quantum devices and functionalized materials and proposes a series of seminars on hot research topics.

The second term (January-June) details the different fields of research (Electronic transport, Spintronics, Quantum Photonics) and includes a research project.

The course is given by lecturers from several laboratories specializing in quantum devices and nanosystems.The training is based on a permanent interaction between students and research teams, and includes: experimental projects, guided tours of laboratories and a research project.

**First term**

**Electrons and phonons in semiconductor heterostructures (3ECTS**

Professors :

- Christophe Voisin (Prof. UPD, LPA),
- Emmanuelle Deleporte (Prof. ENS Cachan, LPQM),
- Francesca Carosella (MCF UPD, LPA)

### Program:

FUNDAMENTALS OF SOLID STATE PHYSICS

- Band structure and Bloch theorem
- Density of states
- Effective mass
- Overview of phonons

ENVELOPE FUNCTION APPROXIMATION

ELECTRON – PHONON INTERACTION: WEAK COUPLING REGIME

- Fermi golden rule
- Rabi oscillations
- Importance of energy loss in opto-electronic devices

ELECTRON – PHONON INTERACTION: STRONG COUPLING REGIME

- Polarons in quantum dots
- Energy relaxation within polaron framewor

**Quantum theory of light (3ECTS)**

Professors :

- Edouard Boulat (MCF UPD, MPQ),
- Cristiano Ciuti (Prof. UPD, MPQ)

### Program:

SEMI-CLASSICAL THEORY OF LIGHT MATTER INTERACTION

- Free particle of Spin 1/2
- Jauge invariance of Schroedinger equation ; Pauli Hamiltonian
- Semiclassical theory of light – matter interaction
- Electron-field interaction and Fermi golden rule ; transition rate

QUANTUM NATURE OF LIGHT : PHOTONS

- Fock space
- Operators : electric field, momentum, photon number
- The Casimir effect
- Special states of the electromagnetic field : coherent states, squeezed states

PHOTON EMISSION AND ABSORPTION

- Hamiltonian electron-photon; revisiting the Fermi golden rule
- Spontaneous and stimulated emission
- Natural linewidth
- Dipolar electric emission
- Diffusion of a photon from an atom

**Advanced Solid State Physics (3ECTS)**

Professors :

- Alain Sacuto (Prof. UPD, MPQ)
- Francesco Sottile (DR CNRS, LSI, Ecole Polytechnique)
- Fausto Sirotti (DR CNRS, PMC, Ecole Polytechnique)

### Program:

REMINDER OF SOLID STATE PHYSICS AND INTRODUCTION (F. Sottile)

Scope of this first introductory session is to give the outline of the course, remind few concepts of basic solid-state theory, and assess the knowledge of the students on the different topics.

- Electrons and nuclei
- Born-Oppenheimer approximation
- Bloch theorem
- spin and k-points
- magnetism (diamagnetic, paramagnetic, ferromagnetic, anti-ferromagnetic, etc.)

SUPERCONDUCTIVITY (A. Sacuto)

An introduction to Superconductivity:

- Introduction to a short story of superconductivity and its fascinating properties
- The quest of very low temperature
- The discovery of superconductivity
- The high-Tc superconductors
- Their properties with experiments performed during the lecture

The Cooper’s model :

- Bound electrons in a degenerate Fermi gaz
- The superconducting gap

A first approach to the microscopic theory of Bardeen Cooper Schrieffer (BCS)

- Description of the ground state
- The BCS Hamiltonian
- The energy of the ground state and the superconducting gap

Signatures of the superconductivity in some spectroscopy probes

- Tunnelling and ARPES
- Infrared and Raman
- NMR

ELECTRONIC STRUCTURE; THE GROUND STATE (F. Sottile)

The electronic problem is introduced. In particular the state-of-the-art approach for the ground-state, Density Functional theory is presented. The needs for certain spectroscopy, introduced here, stimulates the need for the next experimental sessions.

- Ground-state quantities (lattice parameters, phonons, Bulk modulus, phase transitions)
- The many-body problem: independent particles
- Hartree and Hartree-Fock approaches
- Koopmans’s theorem and self-interaction concerns
- Density Functional Theory (theory, approximations and examples)
- Band-structure and Density of States
- Absorption in DFT ?

PHOTOEMISSION SPECTROSCOPY (F. Sirotti)

- Energy and momentum conservation
- ARPES, XPS, Spin-resolution
- Bulk surfaces and interfaces, Cross sections,
- Experimental issues: Ultra High Vacuum, X-rays sources, Electron energy analyzers,
- Examples

GREEN’S FUNCTION THEORY I (F. Sottile)

The green’s function approach is presented, wih particular emphasis on the one-particle Green’s function, that contains the removal and addiction energies of the electrons, for a direct comparison with photoemission spectroscopy.

- The need for the Green’s function
- Spectral representation
- The self-energy
- Hedin’s equations
- The GW approximations
- Quasiparticle and satellites
- Results and examples

X-RAY ABSORPTION ELLIPSOMERY (F. Sirotti)

- Valence spectroscopy and ellipsometry
- Core electrons: XAS, XANES, EXAFS,
- Magnetic systems: Linear and circular Dichroism
- Applications

GREEN’S FUNCTION THEORY II (F. Sottile)

Absorption spectroscopy require the two-particle Green’s function, which is presented briefly here.

- The need for the two-particle Green’s function
- The Bethe-Salpeter equation
- 4 points quantities
- Results and examples

SCATTERING SPESTROSCOPIES AND TDDFT (F. Sottile and F. Sirotti)

Scattering spectroscopies are presented in the first half of the session, namely Electron Energy Loss Spectroscopy and Inelastic X-ray Scattering. This gives the occasion to introduced the concept of screening and the theoretical approach Time Dependent Density Functional Theory.

- scattering process and the inverse dielectric function
- electron energy loss
- electron microscope
- inelastic x-ray scattering
- experimental resolution: energy, momentum, space, time
- Time Dependent Density Functional Theory (theory, linear response and polarizability, approximations and applications)

**Photonic quantum devices (3ECTS)**

Professors :

- C. Sirtori (Prof. ENS, LPA)
- A. Vasanelli (Prof. UPD, LPA)

### Program:

**Quantum devices in current technologies**

- Introduction to quantum optoelectronic devices
- p-n diodes
- Tunnel diodes
- Transfer matrix and tunnelling current
- Mobility and modulation doping
- Two-dimensional electron gas
- Field effect transistors
- Bipolar transistors
- Transistor HEMT

**Charge oscillations of quantized states**

- Superlattices
- Bloch oscillations and Wannier Stark quantization
- Intersubband transitions and electron dispersion
- Oscillator strength
- Dipole charge oscillations

**Quantum cascade lasers**

- Introduction: Laser diodes vs quantum cascade lasers
- Reminder of guided optics
- Plasmon waveguides
- Rate equations
- Gain
- From near infrared to THz optoelectronics

**Quantum detectors**

- Quantum well infrared photodetectors
- Quantum cascade detectors

**Magnetic field applied to 2D structures**

- Landau quantization
- Magneto-transport
- Shubnikov-de Haas effect
- Aharonov-Bohm effect

**Light-matter coupling in microcavities**

- Two level atom in an ideal cavity: Jaynes-Cummings Hamiltonian
- Spontaneous emission in a cavity
- Strong coupling / Weak coupling regime
- Strong coupling regime in semiconductor systems: polaritons
- Microcavity polaritons: exciton – polaritons, intersubband polaritons

**Electronic quantum devices (3ECTS)**

Professors :

- Philippe Joyez (Ch SPEC, CEA Saclay)
- Philippe Lafarge (Prof. UPD, MPQ)

Program:

- Rappels de physique des solides : structures de bandes, métaux, semiconducteurs, phonons, transport diffusif…
- Seconde quantification
- Transport quantique : longueurs caractéristiques, quantification de la conductance, formule de Landauer, bruit de courant dans les conducteurs quantiques, localisation…
- Electrons en champ magnétique : niveaux de Landau, effet Hall quantique entier, fractionnaire, états de bord.
- Supraconductivité : Théorie BCS, effet Josephson, supraconductivité mésoscopique, réflexion d’Andreev.
- Transport dans les nanotubes de carbone.

**Bidimensional Materials (3ECTS)**

Professors :

- Clément Barraud (MCF UPD, MPQ)
- Jérôme Lagoute (CR CNRS, MPQ)
- Yann Gallais (Prof UPD, MPQ)

### Program:

Since the discovery of graphene with its remarkable transport and optical properties, the field of two-dimensional crystals has flourished, and many materials can now be studied down to the single atomic layers. Compared to bulk materials two dimensional materials provide highly tunable platforms for novel functionalities and exotic opto-electronic phenomena. The goal of this course is to give an overview of this vibrant field by providing some basic concepts of two-dimensional materials (device fabrication, electronic and optical properties) and then focus on a selection of recent developments in the field (van der Waals heterostructures, defect engineering, di-chalcogenides, topological insulators…).

We will first review the basics of the physical properties of graphene with an emphasis on the properties of graphene-based devices and the means to characterize them. We will then introduce the physics of other two-dimensional materials like di-chalcogenides and black phosphorus which have been discovered more recently and whose optical and electrical properties differs from graphene. The course will end by an introduction to the unusual two-dimensional electronic states that forms at the surface of topological insulators.

**The Physics of graphene and its devices (12h)**

- Introduction: graphene and its band-structure
- Transport properties of graphene devices
- Optical properties and application to opto-electronic devices
- Local spectroscopies and defect engineering
- Graphene based heterostructures and van der Waals engineering: concept and fabrication

**Beyond graphene: dichalcogenides, black phosphorus and topological insulators (12h)**

- Introduction to dichalcogenides and their band structure in the 2D limit: the case of semiconducting MoS
_{2} - Spin and valley degrees of freedom in semiconducting dichalcogenide + proximity effect
- Correlated states in metallic dichalcogenides: density wave and superconductivity
- Black-phosphorus
- Introduction to topological insulators

**‘Imaging’ nano-objects**

Professors :

- Damien Alloyeau (CR CNRS, MPQ)
- Vincent Repain (Prof UPD, MPQ)

### Program:

### SINGLE NANO-OBJECTS IMAGING: FROM NANOMETER TO SUB-ANGSTROM SCALE (8h)

Microscopes history and state-of-the-art optical microscopes

Electron microscopy

- Microscope and Image formation
- Transmission mode : high resolution imaging
- Aberrations corrector : principle and unprecedented performances
- Tridimensional imaging
- Future challenges in electron optic

Near field microscopy

- A brief history
- General principle of working
- Scanning Tunneling Microscope, Atomic Force Microscope, Scanning Near-field Optical Microscope : signal to noise and resolution
- State-of-the-art examples : from single atoms to biological proteins

### STRUCTURAL AND CHEMICAL ANALYSIS OF NANO-OBJECTS (8 h)

Structural analysis

- Structure of surfaces with near field microscopes
- X-ray diffraction and synchrotron radiation
- Electron diffraction : quantitative analysis of single nano-objects

Chemical analysis

- Probing atomic-scale chemistry by electron spectroscopy (Electron Energy Loss Spectroscopy) and photo-electron spectroscopy (Energy Dispersive X-ray analysis, X-ray Photoelectron Spectroscopy, Angle Resolved PhotoEmission Spectroscopy…)
- Indirect ‘chemical’ mapping with near field microscopes (Inelastic Electron Tunneling Spectroscopy, Chemical Force Microscopy…)

### MEASURING PHYSICAL PROPERTIES OF NANO-OBJECTS (8 h)

Electronic mapping of nano-objects

- Wavefunctions in quantum dots and complex systems with Scanning Tunneling Spectroscopy
- Plasmon imaging using EELS

Magnetic properties of nano-objects

- Holography and magnetic circular dichroism
- Magnetic Force Microscope and Spin Polarized-STM

**Experimental projects in nanosciences (6ECTS)**

Professors :

- Maria Luisa Della Rocca (MCF UPD, MPQ)
- Fabrice Raineri (MdC UPD, C2N)
- R. Braive (MdC UPD, C2N)

In this original course, students will get trained with experimental techniques used in nanosciences. During the first three weeks of the Master, students will have to make an experimental project in the nanosciences field like the elaboration and characterization of metallic nanoparticles, the optic of semiconducting laser, the electronic conduction in atomic contacts or organic materials, nanotubes physics, quantum optics…

A specific nanoscience area dedicated to teaching will be available with free of use instruments like an atomic force microscope, a scanning tunnelling microscope, a transmission electron microscope or an optic microscope. All students will also be initiated to clean room techniques during three days of practise.

**Second semester**

**Quantum Computing (3ECTS)**

Professors :

- Perola Millman (DR CNRS, MPQ)
- Hélène Perrin (DR CNRS, LPL)

Program:

**Introduction to Quantum Computing (3h)**

- Quantum complexity classes
- Quantum communication
- Universal gates
- Discrete and continuous variables
- Qubit coding

**Trapped ions for quantum computing (3h)**

- Methods for ions trapping
- Cooling of ions
- Microwave quantum logic gates

**Quantum algorithms (6h)**

- Shor algorithm
- Grover algorithm
- Presentation of the IBM qubits project

- Implementation of the Shor algorithm with trapped ions
- Superconducting qubits

**Quantum error correction (6h)**

- Quantum error correction codes
- Computing by superconducting qubits with quantum error correction codes (exp)
- Other platforms for quantum computing (Si, RMN, photons,…)

**Quantum simulation (6h)**

- Dictrete and continuous quantum simulation
- Quantum simulation platforms: quantum gases (bulk or lattice), Rydberg cold atoms in optical lattices, ions, microwaves, polaritons …

**Quantum information (3ECTS)**

Professors :

- E. Diamanti (CNRS, LIP6)
- S. Ducci (Prof. UPD,MPQ)

### Program:

### THEORETICAL QUANTUM INFORMATION

The qubit and its states

- quick review of the basic quantum formalism (kets, bras and density matrices)
- No cloning theorem and Wiesner’s unforgeable banknotes
- Quantum Key Distribution and BB84 protocol

Quantum Entanglement 1: Definition and some Properties

- Formal definition (as non separable state)
- Apparent Heisenberg inequality violation
- Link with partial trace
- Entanglement detection for pure and mixed states
- Entanglement monogamy and application to QKD
- Partial transpose and its physical meaning

Quantum Entanglement 2: Bell inequalities and Application

- Entanglement is not a limitation of quantum formalism
- Bell inequalities (mainly CHSH)
- GHZ Paradox
- Some Entanglement application
- The 4 Bell States
- Quantum Dense Coding
- Quantum Teleportation

Introduction to Quantum Computation

- Grover’s Algorithm
- Quantum Error Correcting Codes

### DEVICES FOR QUANTUM INFORMATION

Introduction: Experimental implementation of quantum information : challenges and some famous experiments.

Photon sources: Single photon sources and their characterization : Hanbury Brown and Twiss interferometry, colloidal and grown quantum dots, colored centers in diamonds,..

Entangled photon sources and their characterization : Bell inequality test, density matrix reconstruction, nonlinear dielectric crystals and fibers, quantum dots, semiconductor waveguides,…

Single photon detectors: Photomultipliers, single photons avanlanche photodiodes, supraconducting detectors

Quantum metrology: absolute detector calibration, absolute radiance measurement, polarization mode dispersion, quantum ellipsometry …

Physical implementations of quantum computation: General overview, exemple of trapped ions.

**Nanomagnetism and spintronics (3ECTS)**

Professors :

- H. Jaffres (CNRS, UMR CNRS -Thales)
- P. Seneor (Prof. UPSaclay, UMR CNRS -Thales)

Program:

The ‘NanoMagnetism and Spintronics’ course targets the physics of Magnetism, of Magnetism at the nanometer scale (NanoMagnetism) and the spin-dependant transport in magnetic Nanostructures, scientific discipline designated today as Spin Electronics. After having introduced the fundamentals of orbital and spin localized magnetism in ionic systems, the course will tackle the important notions of paramagnetic, ferromagnetic and antiferromagnetic order. An important effort will be brought on the understanding of the establishment of band-ferromagnetism of 3d transition metals taking into account atomic exchange interactions. The second part of this course will be devoted some more actual problems of spin-dependent transport in Magnetic nanostructures (magnetic multilayers, nanowires, Magnetic tunnel junctions). The concepts of spin-dependent conduction in the diffusive regime, spin diffusion length and spin accumulation will be clearly emphasized to explain Giant MagnetoResistance (GMR) and Tunnel Magnetoresistance (TMR) effects. An opening will be done on the Magneto-Coulomb effects obtained with nanoparticules dispersed between ferromagnetic reservoirs and on spin transfer effects observed on metallic nanopillars and magnetic tunnel junctions.

**Functional materials (3ECTS)**

Professors :

- S. Bierman (PR, LPMC)

This lecture is composed by invited seminars given by international researchers on topics at the interface between fundamental and applied physics/materials science (i. e. Meta-Materials, 2d Materials for Valleytronics, 2d oxide heterostructures, Nanoparticles, battery materials, ….). The lecture is completely held at Ecole Polytechnique.

**Master thesis project (mars to june):**

The final four-month Master thesis project can be conducted in one of the academic or industrial laboratory supporting the formation or in another Lab in France or abroad.

The evaluation is based on a project report and an oral presentation.