Introduction: noninteracting atomic gases; Bose-Einstein distribution; condensation transition at low temperatures; cooling and trapping of atomic gases. Quantum scattering of two atoms: scattering length; description of atom-atom interactions via pseudopotential.
Gross-Pitaevskii equation for the condensate wave function: dynamics of the condensate; free expansion; solitons. Microscopic description of a Bose gas: Bogoliubov transformation; elementary excitations; rotating condensates; synthetic magnetic fields for cold atoms; interference and correlations in condensates.
Optical lattices: dimensionality of lattices; energy scales; bands. Bose-Hubbard model. Superfluid to Mottinsulator transition. Cold atoms in low dimensional systems: Berezinskii-Kosterlitz-Thouless ( BKT) transition in two-dimensions, correlation functions and limit of strong interactions in one-dimensional gases.
Introduction to Monte Carlo methods, pseudorandom numbers, random walk, error assessment. Variational Monte Carlo. Trial wave functions suitable for simulations of atoms, molecules, clusters, fluids and solids and their optimisation. Diffusion Monte Carlo. Fermions and excited states - sign problem. Determining expectation values of observables. Finite-temperature calculations: path integral quantum Monte Carlo. All methods will be accompanied by examples in the field of atomic and molecular physics and condensed matter physics
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- Pethick, C. J., and H. Smith, 2008, Bose-Einstein Condensation in Dilute Gases (Cambridge University Press, Cambridge, UK)
- Pitaevskii, L., and S. Stringari, 2003, Bose-Einstein Condensation (Oxford University Press, UK)
- Cohen-Tannoudji, C., and D. Guery-Odelin, 2011, Advances In Atomic Physics: An Overview (World Scientific Publishing Company, Singapore)
- Lewenstein, M., A. Sanpera, and V. Ahufinger, 2012, Ultracold Atoms in Optical Lattices (Oxford University Press, Oxford, UK).
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