By Christophe Salomon, Georgy V. Shlyapnikov, Leticia F. Cugliandolo

This publication gathers the lecture notes of classes given on the 2010 summer season college in theoretical physics in Les Houches, France, consultation XCIV. Written in a pedagogical sort, this quantity illustrates how the sector of quantum gases has flourished on the interface among atomic physics and quantum optics, condensed topic physics, nuclear and high-energy physics, non-linear physics and quantum details.

The physics of correlated atoms in optical lattices is roofed from either theoretical and experimental views, together with the Bose and Fermi Hubbard versions, and the outline of the Mott transition. Few-body physics with chilly atoms has made awesome development and designated recommendations for 3-body and 4-body difficulties were bought. The striking collisional balance of weakly certain molecules is on the center of the stories of molecular BEC regimes in Fermi gases. Entanglement in quantum many-body platforms is brought and is a key factor for quantum info processing. swiftly rotating quantum gases and optically brought about gauge fields determine a striking reference to the fractional quantum corridor impression for electrons in semiconductors. Dipolar quantum gases with lengthy diversity and anisotropic interplay result in new quantum degenerate regimes in atoms with huge magnetic moments, or electrically aligned polar molecules. Experiments with ultracold fermions exhibit how quantum gases function ''quantum simulators'' of complicated condensed subject structures via measurements of the equation of country. equally, the new remark of Anderson localization of topic waves in a disordered optical capability makes a fruitful hyperlink with the behaviour of electrons in disordered platforms.

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**Extra info for Many-Body Physics with Ultracold Gases: Lecture Notes of the Les Houches Summer School: Volume 94, July 2010**

**Sample text**

We thus see that the combination of kinetic energy, Pauli principle, and interaction leads to a remarkable exchange term between the spins, which looks very much like the dipolar one that would exist for the direct magnetic exchange between magnetic moments. However, there are also remarkable diﬀerences. This exchange, nicknamed superexchange, is responsible for many of the magnetic properties of the strongly interacting quantum systems in solids (Auerbach 1998). Some noteworthy points are as follows: 1.

2011)). The Bose–Hubbard model 23 Interpreting such data is, of course, not easy, since one must deal with a full timedependent Hamiltonian, which is diﬃcult to deal with analytically and numerically. Analytically, it is possible to use a linear response to study the eﬀects of the shaking (Iucci et al. 2006). The results depend crucially on whether the lattice is weak or strong. We will concentrate here on the case of the strong lattice and refer the reader to the literature for the other limit.

1 25 Peculiarities of one dimension Before we embark on the one-dimensional world, let us brieﬂy recall some of the points of the typical solution for a bosonic system in higher dimensions. 3 for a high (meaning d > 1) dimensional system, one can expect that there is a well-deﬁned superﬂuid order. 54) where ρ(x ) is the density of particles at point x and θ(x ) the phase of the wavefunction at the same point. The presence of superﬂuid order implies that we can use ρ(x ) → ρ0 and θ(x ) acquires a ﬁnite expectation value θ(x ) → θ0 , so that the wavefunction has a coherent phase through the whole sample.