01SPQPF

A.A. 2023/24

Course Language

Inglese

Course degree

Master of science-level of the Bologna process in Physics Of Complex Systems (Fisica Dei Sistemi Complessi) - Torino/Trieste/Parigi

Course structure

Teaching | Hours |
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Teachers

Teacher | Status | SSD | h.Les | h.Ex | h.Lab | h.Tut | Years teaching |
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Teaching assistant

Context

SSD | CFU | Activities | Area context |
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FIS/02 FIS/03 |
3 3 |
C - Affini o integrative B - Caratterizzanti |
Attività formative affini o integrative Discipline matematiche, fisiche e informatiche |

2022/23

Out of equilibrium physics is a fascinating and multi-disciplinary topic that finds applications in various contexts, where standard equilibrium approach cannot be applied.
The course is meant to provide an introduction to some general theoretical methods and techniques to approach out of equilibrium problems. A particular focus will be devoted to quantum systems, with examples and a hands-on approach to solve problems in condensed matter physics and to discuss their experimental realization.

Out of equilibrium physics is a fascinating and multi-disciplinary topic that finds applications in various contexts, where standard equilibrium approach cannot be applied.
The course is meant to provide an introduction to some general theoretical methods and techniques to approach out of equilibrium problems. A particular focus will be devoted to quantum systems, with examples and a hands-on approach to solve problems in condensed matter physics and to discuss their experimental realization.

After this course the student is expected to learn some general theoretical methods and techniques to approach problems of out of equilibrium physics. Furthermore, the student will be able to apply these methods to some problems of condensed matter physics, and to explicitly compute quantities that are experimentally measurable.

After this course the student is expected to learn some general theoretical methods and techniques to approach problems of out of equilibrium physics. Furthermore, the student will be able to apply these methods to some problems of condensed matter physics, and to explicitly compute quantities that are experimentally measurable.

Quantum Mechanics and Equilibrium Statistical Mechanics

Quantum Mechanics and Equilibrium Statistical Mechanics

Part 1: The Classical Boltzmann Equation
-Short review of motion of a classical particle under a stochastic force, the phenomenological Drude model for transport.
-Construction of the Boltzmann equation, hypothesis, the collision integral, and properties.
-Example: The Boltzmann Equation in a random potential. The diffusion limit and Brownian motion
Part 2: Towards out of equilibrium physics of quantum systems
-Quantum ingredients in the Semi-Classical Boltzmann Equation
-The relaxation-time approximation.
-Linear response theory, Fluctuation-dissipation theorem. Onsager relations and its applications
-Applications to Condensed Matter: derivation of Ohm’s law, scattering processes of electrons in solids, charge and heat transport
Part 3: Quantum approaches to out of Equilibrium Quantum Systems
A) The coherent limit.
Introduction to quantum mesoscopic physics. Scattering matrix formalism, the Landauer-Büttiker approach to compute the electron current in mesoscopic systems. The quantum of resistance. Examples of quantum interference phenomena in nanosystems (Aharonov-Bohm effect, Weak localization).
Optional: The noise of electrons. Josephson effect and mesoscopic effects with superconductors
B) Effects of dissipation and decoherence.
-Phenomenological approach: the T1-T2 model, applications to a two-level system exposed to a time-dependent perturbation
Depending on student interest:
-Microscopic approach 1: The Density matrix formalism, applications to a two-level system coupled to a thermal bath. Markov limit, reduced density matrix, the Quantum Optical Master Equation.
-Microscopic approach 2: The Keldysh-Schwinger formalism and non-equilibrium Green functions. Applications to quantum transport in nanodevices: the Meir-Wingreen formula for conductance, the Coulomb blockade effect.

PART 1: Classical and semi-classical approaches
-The early approaches to out of equilibrium phenomena. The phenomenological Drude model for transport.
-Classical particle under a stochastic force. The Langevin Equation.
-Microscopic modelling of a system-bath coupling. The damping kernel and the bath spectral function.
-The Boltzmann equation, hypotheses, the collision integral, and properties.
-Toward the quantum world: Quantum ingredients in the semi-classical Boltzmann Equation
-The relaxation-time approximation;
-.Out of Equilibrium distribution in the classical Linear response theory;
-Applications of the Boltzmann Equation to Condensed Matter: scattering processes of electrons in solids.
PART 2: Out of Equilibrium Quantum Systems
A) General Aspects
-The density matrix and its properties. Pure and mixed states. The Liouville von-Neumann Equation. Example: a two-level system
-Quantum pictures: Schrödinger, Heisenberg and Interaction pictures, time-ordering and anti-time ordering of quantum operators
-Quantum Linear response theory.
-The fluctuation-dissipation theorem. Kramers-Kronig relations
-Quantum systems coupled to a bath: The Caldeira-Leggett model
-Decoherence effects. The spin-boson model. The reduced density matrix.
-Phenomenological approach to dissipation and decoherence: the T1-T2 model, applications to a two-level system
[if time is enough: The Lindblad formalism for density matrix or the Keldysh-Schwinger formalism]
B) The coherent limit
-Quantum effects in Nanoscience
-Introduction to quantum mesoscopic physics.
-Scattering matrix formalism, the Landauer-Büttiker formula for transport in mesoscopic systems.

51 ore: Lezioni in Aula
9 ore: Esercitazioni in Aula

The course consists in 34 Theory lectures (=51 hours) + 6 Exercise classes (=9 hours). Concerning the Exercise classes, a few days before the class the exercise texts will be handed out to the students, who are supposed to cooperate to find the solution. During the Exercise classes, students are asked to show how they approached the problem, in an interactive way with the teacher and their colleagues. The attendance to the Exercise classes will be registered, in view of the final exam.
(in the case the Exercise classes must take place in online mode because of the pandemics, the students are required to use a tablet or a display with an electronic pencil, or at least a physical whiteboard with a thick pencil)

R. Zwanzig, Nonequilibrium Statistical Mechanics (Oxford Univ Press)
L. E. Reichl, A modern course in Stitistical Physics (John Wiley & Sons)
J. Rammer, Quantum Field Theory of nonequilibrium states (Cambridge Univ. Press)
A. Kamenev, Field Theory of Nonequilibrium systems (Cambridge Univ. Press)
H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics and Financial markets (World Scientific)
Haug-Jauho, Quantum kinetics in Transport and optics of semiconductors (Springer)

R. Zwanzig, Nonequilibrium Statistical Mechanics (Oxford Univ Press)
L. E. Reichl, A modern course in Statistical Physics (John Wiley & Sons)
J. Rammer, Quantum Field Theory of nonequilibrium states (Cambridge Univ. Press)
H.P. Breuer and F. Petruccione, The theory of open quantum systems (Oxford Univ. Press)
U. Weiss, Quantum Dissipative Systems (World Scientific)

...
The exam is based on an oral test at the blackboard, with questions on 2-3 topics of the material discussed during the course. The first topic can be chosen by the student. The knowledge of the out of equilibrium physics methods is tested by asking the student to illustrate the key concepts, to derive proofs of the main results, and to apply the techniques to some examples and applications discussed during the course.

Gli studenti e le studentesse con disabilità o con Disturbi Specifici di Apprendimento (DSA), oltre alla segnalazione tramite procedura informatizzata, sono invitati a comunicare anche direttamente al/la docente titolare dell'insegnamento, con un preavviso non inferiore ad una settimana dall'avvio della sessione d'esame, gli strumenti compensativi concordati con l'Unità Special Needs, al fine di permettere al/la docente la declinazione più idonea in riferimento alla specifica tipologia di esame.

The exam consists in an oral test at the blackboard, with questions about 2-3 topics chosen from a list of topics. The list will be provided at the end of the course and covers the material discussed during the course. The first topic may be chosen by the student. The other topics are chosen by the teacher. The knowledge of the out of equilibrium physics methods is tested by asking the student to illustrate the key concepts, to derive proofs of the main results, and to apply the techniques to some examples and applications discussed during the course. Students who have not attended regularly the exercise classes will be asked to additionally carry out one of the exercises. Students are not supposed to "reject the mark".

In addition to the message sent by the online system, students with disabilities or Specific Learning Disorders (SLD) are invited to directly inform the professor in charge of the course about the special arrangements for the exam that have been agreed with the Special Needs Unit. The professor has to be informed at least one week before the beginning of the examination session in order to provide students with the most suitable arrangements for each specific type of exam.

© Politecnico di Torino

Corso Duca degli Abruzzi, 24 - 10129 Torino, ITALY

Corso Duca degli Abruzzi, 24 - 10129 Torino, ITALY