Mean-field methods are a very powerful tool for investigating weakly interacting many-body systems in many branches of physics. In particular, they describe with excellent accuracy trapped Bose-Einstein condensates. A generic, but difficult question concerns the relation between the symmetry properties of the true many-body state and its mean-field approximation. Here, we address this question by considering, theoretically, vortex nucleation in a rotating Bose-Einstein condensate. A slow sweep of the rotation frequency changes the state of the system from being at rest to the one containing one vortex. Within the mean-field framework, the jump in symmetry occurs through a turbulent phase around a certain critical frequency. The exact many-body ground state at the critical frequency exhibits strong correlations and entanglement. We believe that this constitutes a paradigm example of symmetry breaking in -or change of the order parameter of -quantum many-body systems in the course of adiabatic evolution.PACS numbers: 03.75. Hh, 03.75.Kk, 67.40.Vs In classical physics, examples of the usefulness of meanfield theory go back to the "molecular field theory" of magnetism [1]. In the classical world, symmetry changes (or breaking) are driven by thermal fluctuations, and in the standard Landau-Ginsburg scenario are associated with increase of classical correlations. In quantum physics, the paradigm example of applicability of the mean field concerns a weakly interacting quantum Bose gas and Bose-Einstein condensation [2]. The mean-field description of the gas assumes that its ground state Ψ is approximated by a product state Ψ( r 1 , . . . , r N ) = ψ( r 1 ) . . . ψ( r N ), of essencially uncorrelated particles forming a superfluid Bose-Einstein condensate with order parameter ψ.Of particular interest for quantum gases are quantum phase transitions and symmetry changes/breaking driven by quantum fluctuations. A celebrated example is the superfluid to Mott-insulator transition of bosons in an optical lattice [3].Another example yet to be explored experimentally is the case of a fast rotating gas, when the number of vortices is similar to the number of particles, or equivalently angular momentum L ∼ N 2 [4]. The ground state of the system is then a strongly correlated quantum liquid such as the Laughlin state, analogous to those emerging in quantum Hall physics [5]. Here, we consider another situation, dealing with the case of a relatively slowly rotating gas at the threshold of the nucleation of the first vortex. We show that owing to the symmetries of the system, the many-body state at nucleation is strongly correlated and characterize its properties.The symmetry change/breaking that results from vortex nucleation has drawn a lot of attention since the discovery of superfluids [6]. For quantum gases, atoms are usually confined in an isotropic harmonic trap and experience an extra quadratic potential rotating at angular frequency Ω (for a review see ref.7). From a theoretical point of view, the vortex nucleation can be...

Two-dimentional systems of trapped samples of few cold bosonic atoms submitted to strong rotation around the perpendicular axis may be realized in optical lattices and microtraps. We investigate theoretically the evolution of ground state structures of such systems as the rotational frequency ⍀ increases. Various kinds of ordered structures are observed. In some cases, hidden interference patterns exhibit themselves only in the pair correlation function; in some other cases explicit broken-symmetry structures appear that modulate the density. For N Ͻ 10 atoms, the standard scenario, valid for large sytems is absent, and is only gradually recovered as N increases. On the one hand, the Laughlin state in the strong rotational regime contains ordered structures much more similar to a Wigner molecule than to a fermionic quantum liquid. On the other hand, in the weak rotational regime, the possibility to obtain equilibrium states, whose density reveals an array of vortices, is restricted to the vicinity of some critical values of the rotational frequency ⍀.

We study strongly correlated ground and excited states of rotating quasi-2D Fermi gases constituted of a small number of dipole-dipole interacting particles with dipole moments polarized perpendicular to the plane of motion. As the number of atoms grows, the system enters an intermediate regime, where ground states are subject to a competition between distinct bulk-edge configurations. This effect obscures their description in terms of composite fermions and leads to the appearance of novel quasihole ground states. In the presence of dipolar interactions, the principal Laughlin state at filling 1=3 exhibits a substantial energy gap for neutral (total angular momentum conserving) excitations and is well-described as an incompressible Fermi liquid. Instead, at lower fillings, the ground state structure favors crystalline order.

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Using exact diagonalization for a small system of cold bosonic atoms, we analyze the emergence of strongly correlated states in the presence of an artificial magnetic field. This gauge field is generated by a laser beam that couples two internal atomic states, and it is related to Berry's geometrical phase that emerges when an atom follows adiabatically one of the two eigenstates of the atom-laser coupling. Our approach allows us to go beyond the adiabatic approximation, and to characterize the generalized Laughlin wave functions that appear in the strong magnetic field limit.