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The rapidly developing field of optomechanics aims at the combined control of optical and mechanical modes. In cold atoms, the spontaneous emergence of spatial structures due to optomechanical back-action has been observed in one dimension in optical cavities or highly anisotropic samples. Extensions to higher dimensions that aim to exploit multimode configurations have been suggested theoretically. Here, we describe a simple experiment with many spatial degrees of freedom, in which two continuous symmetries-rotation and translation in the plane orthogonal to a pump beam axis-are spontaneously broken. We observe the simultaneous long-range spatial structuring (with hexagonal symmetry) of the density of a cold atomic cloud and of the pump optical field, with adjustable length scale. Being based on coherent phenomena (diffraction and the dipole force), this scheme can potentially be extended to quantum degenerate gases. [ABSTRACT FROM AUTHOR]

We study non-equilibrium spatial self-organization in cold atomic gases, where long-range spatial order spontaneously emerges from fluctuations in the plane transverse to the propagation axis of a single optical beam. The self-organization process can be interpreted as a synchronization transition in a fully connected network of fictitious oscillators, and described in terms of the Kuramoto model. [ABSTRACT FROM AUTHOR]

We investigate transverse symmetry-breaking instabilities emerging from the optomechanical coupling between light and the translational degrees of freedom of a collisionless, damping-free gas of cold, two-level atoms. We develop a kinetic theory that can also be mapped on to the case of an electron plasma under ponderomotive forces. A general criterion for the existence and spatial scale of transverse instabilities is identified; in particular, we demonstrate that monotonically decreasing velocity distribution functions are always unstable. [ABSTRACT FROM AUTHOR]

Theoretical analysis of the optomechanics of degenerate bosonic atoms with a single feedback mirror shows that self-structuring occurs only above an input threshold that is quantum mechanical in origin. This threshold also implies a lower limit to the size (period) of patterns that can be produced in a condensate for a given pump intensity. These thresholds are interpreted as due to the quantum rigidity of Bose-Einstein condensates, which has no classical counterpart. Above the threshold, the condensate self-organizes into an ordered supersolid state with a spatial period self-selected by optical diffraction. [ABSTRACT FROM AUTHOR]