# AT1: Bose-Einstein Condensation in Ultra Cold Gases

Posted October 9th, 2012 by mpags

1. Bose and Fermi quantum statistics. One body density matrix, momentum distribution and physical observables. Off Diagonal Long Range order. Thermodynamics of ideal Bose gas. Bose-Einstein condensation and off-diagonal long range order. Influence of harmonic trapping potential on BEC of ideal gas, effect of density of states.

2. Interatomic interactions. Low energy collisions and scattering length. Mean field description of interactions. Order parameter and Gross-Pitaevskii equation. Role of the phase and irrotational hydrodynamics. Nonuniform condensates: Thomas Fermi regime and small amplitude oscillations. Persistent currents and Quantised vortices.

3. Bogoliubov theory of weakly interacting bosons: excitation spectrum and quantum fluctuations. Quantum hydrodynamics and Beliaev decay of phonons.

4. Landau theory of superfluidity. Elementary excitations and Landau criterion of superfuid flow. Two-fluid hydrodynamics: normal and superfluid components. First and second sound. Rotation of superfluids.

5. Phase coherence of condensates. Absence of phase coherence in low dimensions (Mermin-Wagner theorem). Interference between two condensates. Josephson effect and its quantisation. Bose-Hubbard Hamiltonianof bosons in optical lattices.

## Module title: Bose-Einstein Condensation in Ultra Cold Gases (AT1)

### Module convenor: Dr. Dimitri Gangardt (Birmingham)

### Module aims:

After its theoretical prediction by Sateyndra N. Bose and Albert Einstein in 1925 the phenomenon of Bose -Einstein condensation (BEC) and related phenomenon of Superfluidity was crucial for understanding behaviour of various physical systems, from liquid helium to neutron stars. Its direct experimental observation with ultra cold alkali gases in 1995 leads to impressive growth of experimental and theoretical studies due to unprecedental level of tunability and control achieved with these systems. In this lecture course I will discuss theoretical concepts of BEC and long range order, elementary excitations, quantised vortices and their relation to phenomenon of superfluidity and macroscopic phase coherence of condensates. Where possible I will discuss experimental observation of these effects. I will also present recent developments of the field and discuss effects of strong correlations between ultracold atoms arising in low dimensional systems and optical lattices. My presentation will be based on the standard theoretical tools of many body physics, such as first and second quantisation as well as path integrals.### Learning objectives:

To understand and use in their own research the concepts outlined in the Syllabus (below). Completion of derivation presented in lectures will serve as exercises. A short individual interview with each student during Mid-term MPAGS Workshop will be used to assess students' progress.### Syllabus

0. Short historical introduction. Review of recent experiments with ultracold atoms.1. Bose and Fermi quantum statistics. One body density matrix, momentum distribution and physical observables. Off Diagonal Long Range order. Thermodynamics of ideal Bose gas. Bose-Einstein condensation and off-diagonal long range order. Influence of harmonic trapping potential on BEC of ideal gas, effect of density of states.

2. Interatomic interactions. Low energy collisions and scattering length. Mean field description of interactions. Order parameter and Gross-Pitaevskii equation. Role of the phase and irrotational hydrodynamics. Nonuniform condensates: Thomas Fermi regime and small amplitude oscillations. Persistent currents and Quantised vortices.

3. Bogoliubov theory of weakly interacting bosons: excitation spectrum and quantum fluctuations. Quantum hydrodynamics and Beliaev decay of phonons.

4. Landau theory of superfluidity. Elementary excitations and Landau criterion of superfuid flow. Two-fluid hydrodynamics: normal and superfluid components. First and second sound. Rotation of superfluids.

5. Phase coherence of condensates. Absence of phase coherence in low dimensions (Mermin-Wagner theorem). Interference between two condensates. Josephson effect and its quantisation. Bose-Hubbard Hamiltonianof bosons in optical lattices.

### Bibliography

The course will be based on the book "Bose Einstein Condensation" by Lev Pitaevskii and Sandro Stringari. Although the material in lectures is intended to be self contained, a basic knowledge of Quantum Mechanics and Statistical Physics will be required. The students are encouraged to use the corresponding Landau&Lifshits volumes (III,V and IX). Additional sources (such as articles and reviews) will be provided during the course.Duration:

2hsAcademic year:

2012-2013Starts:

08/10/2012 - 15:00