Fermi Quantum Gases

Course manual 2016/2017

Course content

Part I: Theory

1. Ultracold Fermi gases. Elastic and inelastic collisions between atomic fermions.
2. Ideal Fermi gas. Ground state and excitations. Ideal Fermi gas in an external harmonic potential.
3. Weakly interacting Fermi gases. Fermi gas with repulsive interaction between particles.
4. Landau Fermi liquid theory. Zero sound.
5. Weakly interacting Fermi gas with attractive interaction between particles. Cooper instability. BCS approach and single-particle excitations.
6. Gap equation. Collective modes. Andreev reflection.
7. Superfluidity in Fermi systems. Critical velocity. Superfluid and normal components.
8. Thermodynamics. Ginzburg-Landau equations. Vortices in Fermi gases.
9. Unconventional superfluid pairing. Pairing in a harmonic potential.
10. Strongly interacting Fermi gases. Universal thermodynamics in the unitarity regime. BCS-BEC crossover.
11. Two-dimensional Fermi gases. Interaction between particles and superfluid phase transition.
12. One-dimensional Fermi gases. Spin-charge separation. Diagram of states.

Part II: Experiment

The lecture notes can be obtained beforehand by sending a brief email to schreck at uva.nl.

Introduction: Quantum simulation

Part 1: Simulation of crystalline solids: Lattices: dispersion relation, Brillouin zone, Bloch states, Wannier states, Bloch oscillations, experimental realization; Derivation of Hubbard Hamiltonian, discussion of approximations; Superfluid to Mott-Insulator phase transition: phase diagram obtained by Gutzwiller Ansatz; Experimental observation: momentum distributions, measurement of gap, precise comparison with numerical solution; Observation of Mott shells by absorption imaging; Quantum gas microscopy: observation of superfluid to Mott-insulator phase transition.

Part 2: Magnetism: Origin of magnetism in solid state, types of exchange interaction; Observation of super-exchange interaction; Quantum dynamics of spin impurity observed with quantum gas microscope; Quantum simulation of antiferromagnetic spin chains; Changing the tunnel matrix element by shaking; Quantum simulation of frustrated classical magnetism in triangular optical lattice.

Part 3: Artificial gauge fields: Artificial gauge fields by rotation, detection of vortices; The quantum Hall effect; Artificial gauge fields and Berry phase; BEC in a uniform light-induced vector potential; Synthetic magnetic fields for ultracold neutral atoms; Optical lattice with magnetic flux; The Harper-Hofstadter Hamiltonian and the Hofstadter butterfly; Realizing the Harper-Hofstadter Hamiltonian; Spin-orbit coupling

Part 4: Fermi gases: Creation and detection; Interaction tuning: Feshbach resonances; BEC-BCS crossover: what is it? Measuring the pairing gap; The unitary Fermi gas: equation of state, second sound; Polarons.

Study materials

Literature

• R.P. Feynman, 'Statistical Mechanics'.
• K. Huang, 'Statistical Mechanics'.
• Ph. Nozieres and D. Pines, 'Theory of quantum liquids', Vol II.
• S. Stringari and L. Pitaevskii, 'Bose-Einstein condensation'.
• C.J. Pethick and H. Smith, 'Bose-Einstein condensation in dilute gases'
• J.F. Annet, 'Superconductivity, Superfluids and Condensates'.
• E. Demler, Strongly correlated systems in atomic and condensed matter physics, Lecture notes for Physics 284, Harvard University (2011)
• J.I. Cirac and P. Zoller, Goals and opportunities in quantum simulation, Nature Physics, 8, 264 (2012)
• I. Bloch, J. Dalibard, S. Nascimbéne, 'Quantum simulations with ultracold quantum gases', Nature Physics, 8, 264 (2012)

Other

• lecture notes available on blackboard.

Objectives

This course will bring students close to current research topics in ultracold quantum gases and consists of an experimental and a theory part.

The experimental part, taught by Florian Schreck (www.strontiumBEC.com), introduces quantum simulation. Understanding quantum physics is often extremely hard or even impossible. This is especially true for emergent phenomena occurring in fermionic many-body systems, such as high-temperature superconductivity. Even more powerful classical supercomputers will not help. The problem is that quantum systems life in Hilbert space, which grows exponentially with system size and already for small systems exceeds the capacity of any classical computer. The solution is to use ultracold atoms or ions, over which we have perfect control, to simulate the difficult to manipulate quantum system we are interested in. Such quantum simulation even allows us to study systems that are of high theoretical interest but do not naturally occur. This course will introduce the basic building blocks of quantum simulations with ultracold atoms and explain their theoretical and experimental basis. We will use these building blocks to understand some of the most advanced quantum simulations performed to date, including quantum simulations based on Fermi gases.

The theory part treats the phenomenon of (low-temperature) superfluidity in Fermi systems. Starting from the properties of a normal Fermi gas, we give a detailed analysis of the BCS pairing instability and reveal its many-body nature. Various possible mechanisms of pairing together with the properties of superfluid fermionic systems will be discussed. We will then address the issues of superfluid pairing in two-dimensional Fermi systems and the phenomenon of spin-charge separation in one dimension. The lectures emphasize advances in theory and the description of the remarkable experimental progress with ultracold quantum gases over the last two decades.

At the end of the course, the student is able to derive and use the theory relevant for the implementation of important quantum simulations and the theory of degenerate Fermi gases. The student will also understand how a quantum simulator is constructed.

It is recommended, but not required, to take the Bose-Einstein Condensates course (5354BOEC6Y) offered October to December by the same lecturers before taking this course. Both courses together are an excellent starting point for a master thesis in condensed matter (theory or experiment) or in one of the labs exploring ultracold atoms, ions or molecules at VU, UvA, AMOLF or ARCNL.

Teaching methods

• Lecture
• Self-study
• Supervision/feedback meeting

Lectures and seminars.

Learning activities

Attendance

The programme does not have requirements concerning attendance (OER-B).

Additional requirements for this course:

Assessment

exams: each student randomly chooses one of four possible exam exercise sheets. Each sheet contains two questions on the theory part and one question on the experimental part, very closely aligned with the lectures. The student can use the lecture notes and up to 5 hours of time to give written answers to the three questions. Whenever the student is ready, the answers are checked and discussed with the two lecturers. An oral exam part follows, in which the student is quizzed on randomly chosen parts of the course. The cut-off score is five. The retake exam has the same format.

Inspection of assessed work

The inspection of the written part takes place just before the oral exam. The discussion of the oral part takes place a few minutes after the oral exam, after the examiners have decided on the mark. Additional discussions can be provided on request by email.

Assignments

exercise sheet 1

• in-class and homework exercises, in-class solved in small groups, homework done individually, corrected by TA and discussed during werkcollege. no grades.

exercise sheet 2

• in-class and homework exercises, in-class solved in small groups, homework done individually, corrected by TA and discussed during werkcollege. no grades.

exercise sheet 3

• in-class and homework exercises, in-class solved in small groups, homework done individually, corrected by TA and discussed during werkcollege no grades.

exercise sheet 4

• in-class and homework exercises, in-class solved in small groups, homework done individually, corrected by TA and discussed during werkcollege no grades.

exercise sheet 5

• in-class and homework exercises, in-class solved in small groups, homework done individually, corrected by TA and discussed during werkcollege no grades.

exercise sheet 6

• in-class and homework exercises, in-class solved in small groups, homework done individually, corrected by TA and discussed during werkcollege. no grades.

Onderstaande opdrachten komen aan bod in deze cursus:

•    Naam opdracht 1 : beschrijving 2
•    Naam opdracht 2 : beschrijving 1
•    ....

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Course structure

Timetable