#### OBJECTIVES

The Quantum Devices program is a training course for students having validated the first year of a master’s degree.

Its objective is to provide a very high level of theoretical and experimental training on several types of quantum phenomena with particular emphasis on devices exploiting quantum principles.

This largely transversal field touches on several disciplines, theoretical advancements are accompanied by progress in material science and by the realization of new and unique experimental techniques. These advancements have had important consequences in fundamental physics. Today we are able to observe and manipulate single atoms or conceive complex quantum devices at the nanometer scale: semiconductor sources and detectors, molecular transistor, superconducting circuits for quantum information, hard disks based on giant magnetoresistance, transistor made from two-dimensional materials….

Part of the training is dedicated to quantum technologies, with lectures on quantum computing, quantum information and communication. Forming high-level students on these « hot » topics is a current challenge perfectly fitting the scientific initiatives federating quantum communities nowadays, such as the* European Flagship on Quantum Technologies* and the French *Plan Quantiques*. Furthermore, studying original electronic and photonics devices and exploring electronic and structural properties of new materials (such as 2D Materials, nanoparticles,…) represent essential aspects of the formation enriching the scientific background of a young physicist expert in the quantum properties of matter.

After the master’s degree, many of our students begin a PhD thesis in a research laboratory (public or private) with a grant from the Ministry of Education or from other forms of financial support (CIFRE, DGA, Ile de France, CNano,…)

Thanks to this versatile training, both theoretical and applied, students will be able to quickly integrate both a public research organization (after a PhD thesis) or an industrial Research and Development group. Various industrial laboratories are directly associated with this training (Thales, ONERA, CEA, etc.).

#### CALENDAR

#### OUTLINE OF THE COURSES

The training includes modules introducing the fundamental concepts and tools of photonics and quantum electronics in condensed matter, state-of-the-art analysis instruments (electron microscopy, STM, AFM, etc.), and a broad panorama of quantum devices and low dimensional materials. Lectures dedicated to quantum technologies are offered dealing with quantum communication and computation. Specialized courses cover also the domains of spintronics and functional materials. An original experimental training is realized at the beginning of the academic year with the experimental projects in nanosciences. Students receive a state of the art training in nanofabrication and nanocharacterization, thanks to the access to the clean-room of the Université de Paris and the dedicated nanoscience teaching platform equipped with top-level facilities.

Throughout the year, students can participate to seminars organized by the university on topical research themes given by researchers from public and/or industrial laboratories.

The formation is also based on the permanent interaction between students and research teams in the field of quantum devices. The permanent interaction with research activities in the domain of quantum devices is further enriched by guided tours of laboratories and the end-of-year internship.

The program is completely though in English.

#### TRAINING COURSE

### Electrons and phonons in nanostructures (3ECTS)

*Teachers: *

*Christophe Voisin (PR UP, LPENS)*

* Emmanuelle Deleporte (PR ENS Cachan, LPQM)*

*Francesca Carosella (MCF UP, LPENS)*

**Part 1**

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 framework

**Part 2**

Optical absorption in a bulk material

Direct absorption, indirect absorption, selection rules

Excitons

Optical absorption in a quantum well

Interband and intraband transitions

Type I and type II quantum wells, superlattice

Excitonic effects

Optical emission in bulk materials and quantum wells

Einstein coefficients

Luminescence

Different kinds of experience: electroluminescence, photoluminescence, excitation spectroscopy, time-resolved photoluminescence

Effect of an external electric field on heterostructure electronic states and optical properties

Effect of an external magnetic field on heterostructure electronic states and optical properties

Examples of problem class:

Density of states and energy states calculation in various kind of heterostructures

Determination of electrons lifetime in presence of phonons

Calculation of absorption coefficient in a bulk material

Optical absorption in a quantum well

Landau levels and magnetoabsorption

### Quantum Theory of light (3ECTS)

*Teachers : *

*Edouard Boulat (MCF UP, LMPQ)*

* Loic Lanco (MCF UP, C2N)*

**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)

*Teachers : *

*A. Sacuto (PR, LMPQ)*

* F. Sirotti (DR CNRS, LPMC École Polytechnique)*

* F. Sottile (DR CNRS, LSI École Polytechnique)*

*Reminder of Solid State Physics and Introduction to the course:*

Electrons and nuclei

Born-Oppenheimer approximation

Bloch theorem

Spin and k-points

Magnetism (diamagnetic, paramagnetic, ferromagnetic, anti-ferromagnetic, etc.)

*Superconductivity*:

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*:

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*:

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 Functions theory I*:

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 and Ellipsometry*:

Valence spectroscopy and ellipsometry

Core electrons: XAS, XANES, EXAFS,

Magnetic systems: Linear and circular Dichroism

Applications

*Green’s Functions theory II*:

Need for the two-particle Green’s function

Bethe-Salpeter equation

4 points quantities

Results and examples

*Scattering spectroscopies and TDDFT:*

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

### Electronic Quantum Devices (3ECTS)

*Teachers :*

*P. Joyez (DR CEA Saclay, Lab SPEC)*

*P. Lafarge (PR UP, Lab MPQ)*

Basics of Solid State Physics : band structure, metals, semiconductors, phonons, balistic and diffusive electronic transport,…

Second quantization

Quantum transport : characteristic lenght scales, conductance quantum, Landauer formula, current noise in quantum conductors, localization, …

Electrons in magnetic field : Landau levels, integer and fractionary quantum Hall effect, edge states, …

Superconductivity : BCS theory, Josephson effect, mesoscopic superconductivity, Andreev reflexions.

Electronic transport in carbon nanotubes.

### 2D Materials (3ECTS)

*Teachers : *

*Y. Gallais (Prof UP, LMPQ)*

*J. Lagoute (CR CNRS, LMPQ)*

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 single atomic layers. Compared to bulk materials, two-dimensional materials provide highly adjustable platforms for new functionality, which can be the source of exotic optoelectronic phenomena. The objective of this course is to give an overview of this highly dynamic research field by providing some basic concepts of two-dimensional materials (device fabrication, electronic and optical properties) and by focusing on a selection of recent developments in the field (van der Waals heterostructures, defects engineering, transition metal dichalcogenides, topological insulators, etc.).

We will first review the physical properties of graphene with an emphasis on the properties of graphene-based devices and the ways to characterize them. We will then introduce the physics of other two-dimensional materials such as transition metal dichalcogenides and black phosphorus, which have been discovered more recently and whose optical and electrical properties differ from graphene. The course will end with an introduction to the unusual two-dimensional electronic states formed on the surface of topological insulators.

**The physics of graphene and its devices**

Introduction: graphene and its band structure

Transport properties of graphene devices

Optical properties and application to optoelectronic devices

Local spectroscopies and defects engineering

Graphene-based heterostructures and van der Waals engineering: concept and manufacturing

*Beyond graphene: transition metal dichalcogenides (TMDs), black phosphorus (BP) and topological insulators (TI)*

Introduction to transition metal dichalcogenides and their band structure in the 2D limit: the case of semiconductor MoS2

Degrees of freedom of spin and valley in semiconductor dichalcogenide and proximity effect

Correlated states in transition metal dichalcogenides: density wave and superconductivity

Black-phosphorus

Introduction to topological isolators

### Experimental projects in nanosciences (6ECTS)

*Teachers : *

*ML Della Rocca (MCF UP, LMPQ)*

*F. Raineri (MCF UP, C2N)*

*R. Braive (MCF UP, C2N)*

In this original course, students will get trained with experimental techniques used in nanosciences. During the first three weeks of the formation, students will realize in complete autonomy an experimental project in the field of nanosciences, on hot-topics such as electronic transport or optical properties of graphene and carbon nanotubes, molecular electronics, nanoplasmonics, photonic crystals, organic electronics, quantum transport in tunnel diodes,…

A specific nanoscience platform equipped with advanced facilities (AFM – atomic force microscopes and STM- tunneling effect microscopes, TEM – transmission electron microscope, SEM – scanning electron microscope, spectrometers, cryogenics, electronic transport measurements, etc.) will be available with free use of these instruments. All students will also be initiated to clean room techniques and activity by practicing the realization of their own device.

### Quantum Computing (3ECTS)

*Teachers : *

*P. Milman (DR CNRS, LMPQ)*

*B. Laburthe (DR CNRS, LPL)*

THEORY

– From classical to quantum computing

What’s « to calculate » ?

Hilbert’s problem

Russell paradox

Gödel’s theorem and Turing’s halting problem

The Church-Turing thesis’

Complexity classes

Quantum gates and universality

Clifford, non-Clifford and the Gottesman-Knill theorem

– Why is it so hard to build a quantum computer ?

Decoherence of a Schrödinger cat : simple model

Neumark’s extension and partial trace

Decoherence of a qubit : simple model. The master equation

The stabilizer formalism

Error correction

Teleportation and measurement based quantum computing

– The Grover algorithm

The oracle

The algorithm (and discussion)

Geometrical implementation

The N=4 case

The Deutsch-Jozsa algorithm

– Quantum simulation

Feynman’s idea

The Trotter expansion

Complexity

Analogic simulations

EXPERIMENTS

– From transistors to quantum dots

What’s inside a computer?

From transistors to « quantum transistors“

The orthodox model of quantum dot

Tunneling from a reservoir to an island

Single-electron transistors

Quantum dots qbits

Q-bit manipulation and 2 qbit gate

Quantum dots and beyond

– Superconducting circuits

Basics of electrodynamics

Go quantum (LC quantum circuit)

The Josephson junction

Hamiltonian of the superconducting qbit

How to tune a qbit

Coupling the qbit to electromagnetic radiation

How to measure the qbit

Two qbit interaction

– Ions from quantum computing to quantum simulation

Quantum computing with ion: experimental facts

Light atom/ion interactions

Manipulation of a qbit by laser light

Manipulation of qbit-S

Collective mode

The Sorensen-Molmer gate

Quantum computing and simulation with transverse modes

– Quantum simulation with cold atoms

Nice assets of cold atoms for quantum many-body physics

How to produce « zero entropy » gases

Interactions between ground-state cold atoms

Two-qbit gate

Optical lattices and the quantum microscope approach

Rydberg atoms in optical tweezers

(Beyond gates and quantum magnetism)

### Quantum Communication (3ECTS)

*Teachers :*

* E. Diamanti (DR CNRS, LIP6)*

*S. Ducci (PR UP, LMPQ)*

Quantum Communication constitutes one of the pillars of the field of quantum information and encapsulates a vast array of technologies that range from laboratory experiments, to real-world implementations and to commercial reality. Its applications can have a profound impact in cybersecurity and in communication practices in next-generation network infrastructures. Photonics plays a central role in this field, as it is based on techniques from classical, nonlinear and quantum optics, and light-matter interactions.

This course covers the different aspects of this rapidly evolving field: from theoretical concepts, to the development of integrated sources and detectors of quantum states of light, circuits for their manipulation, and then to major protocols such teleportation and quantum key distribution, and to their implementation within fiber and satellite-based quantum networks.

The lectures are highly interactive, with students presenting recent scientific papers during the sessions, and include a ‘live’ experimental demonstration on the generation of Bell states and their analysis.

**Theoretical concept and protocol implementation**

Introduction to quantum information theory concepts. Entanglement and Bell inequalities

Applications of entanglement: quantum teleportation and entanglement swapping

Theory and implementation of quantum key distribution

Quantum networks with fiber-optic and satellite links

**Photonics devices for quantum communication**

Photon statistics; photon antibunching (Handbury-Brown and Twiss setup).

Established technologies for single photon detection; implementation of integrated single photon sources (requirements, design and experimental evaluation of their performances)

Physical processes generating two-photon entangled states and experimental evaluation of entanglement level

Implementation of integrated sources of entangled states and quantum photonic circuits

**Experiment:**

Bell’s inequality violations and density matrix reconstruction with a Quantum Entanglement Demonstrator

### Nanomagnetism and spintronics (3ECTS)

*Teachers : *

*H. Jaffres (PR École Plytechnique, **UMR CNRS -Thales)*

*P. Seneor (PR Paris Saclay, UMR CNRS -Thales)*

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.

### Functionals Materials (3ECTS)

*Teacher : *

*S. Biermann (PR École Polytechnique, LPMC)*

This course is at the interface between applications and fundamental science (industrial and academic research). It takes place completely at the Ecole Polytechnique.

It consists of a series of half-day seminars held by guest researchers at the cutting edge of themes at the interface between fundamental and applied physics and materials science (material design, meta-materials, 2D materials for valleytronics, 2D oxide heterostructures,…).

An introductory course is initially given by Prof. Biermann with updating of the knowledge necessary to follow the seminars.

### Photonics Quantum Devices (3ECTS)

*Teachers : *

*A. Vasanelli (PR, LPENS)*

* C. Sirtori (PR ENS, LPENS)*

*Basics of semiconductor physics:*

Electrons in solids: wavefunctions, band structures, effective mass

Statistics of semiconductors: Fermi-Dirac, semi-classical approximation, free-carrier density

Semiconductor doping: donors and acceptors, temperature regimes

Optical absorption: matrix element and absorption coefficient in direct-bandgap semiconductors, joint density of states, phonons and absorption in indirect-bandgap semiconductors

Non-radiative recombination

*Basics of semiconductor devices:*

Transport in semiconductors: diffusion and conductivity, Drude and Boltzmann

Quasi-neutral approximation: rate equations in doped semiconductors, minority-carrier evolution, application to photocarrier injection and surface recombination

p-n junctions: space charge and band profile, I-V characteristics and Shockley approximation, quasi Fermi levels

Photovoltaic detectors

*When electric fields come into play:*

Perturbation of electronic states: enveloppe function approximation, Franz-Keldysh effect

Application to heterostructures: quantum wells, intersubband transitions, QWIPs

Modulators: Quantum Confined Stark effect, QCSE vs. FK, designs

Introduction to non-linear optics: coupled-wave equations, slowly-varying-amplitude approximation, second-order processes and wave-vector mismatch

Second-order non-linear optics in semiconductors: susceptibility enhancement, phase-matching schemes

*Light emission in semiconductors:*

Radiative recombination and photoluminescence spectrum

Light-Emitting Diodes: carrier lifetime, internal quantum yield, light extraction

Stimulated emission: absorption, optical gain and Bernard-Duraffourg inversion condition

Double-heterostructure laser: electron and photon confinement, threshold, processing

Quantum-well laser: separate confinement, interband absorption and gain in quantum wells, threshold, comparison with DH, structures

Introduction to quantum-cascade laser: unipolar scheme, active part, superlattices and injector design

*From optoelectronics to photonic devices:*

Distributed-feedback lasers: principle, mode coupling, DFB operation

Vertical-cavity surface-emitting lasers: principle, Bragg mirrors, cavity design, electrical injection

Introduction to photonic crystals: DBR as 1D photonic crystals, modes and band structures, 2D and 3D generalisation, application to integrated optics, analogy with electron states and limits

Application to light extraction: emission from a cavity, light extraction and refractive-index engineering

### Nano-objets at the atomic scale (3ECTS)

*Teachers :*

* D. Alloyeau (CR CNRS, LMPQ)*

*Vincent Repain (PR UP, LMPQ)*

*H. Amara (CR ONERA)*

*Electronic, magnetic and optical properties down to the molecular scale:*

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

Diffraction principle, optical resolution

Beyond diffraction

*Near field microscopy:*

A brief history

General principle of working

Scanning Tunneling Microscope and Atomic Force Microscope: signal to noise and resolution

*Electronic properties:*

Local Density of States

Quantized levels and wavefunctions mapping

Superconductivity at the nanoscale

*Magnetic properties:*

Local Tunnel Magneto-Resistance

Single atom magnetism, superparamagnetism and non-collinear magnetism

*Optical properties*:

Optical Luminescence from a nanometer scale junction

Tip Enhanced Raman Scattering

*Structure-related properties of nanomaterials:*

The atomic structure of nanomaterials: a key to understand and optimize their properties

Revealing the atomic structure and the electronic properties of nanomaterials with a transmission electron microscope

Image and diffraction

Phase-contrast microscopy at the atomic scale (high-resolution TEM)

Electron and X-ray spectroscopies

Plasmon mapping at the nanoscale

*Studying the dynamics of nanomaterials in realistic environments:*

In situ electron microscopy and X-ray scattering methods

Nucleation and growth phenomena

Life cycle of nanomaterials in biological media

*Modlisation of structural and electronic properties of nanomaterials:*

Different approaches at atomic scale

DFT calculations

Tight-binding formalism (diagonalization scheme, order N method, Green function, second moment approximation …)

Empirical potentials (Lennard Jones, EAM, MEAM, Brenner, Tersoff, …)

Different types of atomic calculations (static, Molecular Dynamics, Monte Carlo, energy landscape exploration methods, …)

*Electronic properties of nano-objects:*

Carbon nanomaterials : nanotube, graphene

Green functions formalism

Carbon nanotubes : imaging molecular orbitals

Doped Graphene : DFT vs Tight-binding

*Structural properties of nano-objects:*

Thermodynamic of nanoalloys (driving forces : size, surface energy, ordering tendency, …) : empirical and semi-empirical approaches

Growth mechanisms (nanorod, carbon nanotube, graphene)

### Internship (from March to June) (18 ECTS)

The 4-month end-of-study internship can be carried out in one of the academic or industrial laboratories supporting the Master or in other laboratories in France or abroad. The final assessment is carried out on an internship report and an oral presentation.