Abstract:
We develop a theory of quantum fluctuations and squeezing in a three-dimensional Bose-Einstein condensate atom interferometer with nonlinear losses. We use stochastic equations in a truncated Wigner representation to treat quantum noise. Our approach includes the multi-mode spatial evolution of spinor components and describes the many-body dynamics of a mesoscopic quantum system.

Abstract:
We review some aspects of three-dimensional quantum gravity with emphasis in the `CFT -> Geometry' map that follows from the Brown-Henneaux conformal algebra. The general solution to the classical equations of motion with anti-de Sitter boundary conditions is displayed. This solution is parametrized by two functions which become Virasoro operators after quantisation. A map from the space of states to the space of classical solutions is exhibited. Some recent proposals to understand the Bekenstein-Hawking entropy are reviewed in this context. The origin of the boundary degrees of freedom arising in 2+1 gravity is analysed in detail using a Hamiltonian Chern-Simons formalism.

Abstract:
We consider the coupling between massive and spinning particles and three dimensional gravity. This allows us to construct geometric operators (distances between particles) as Dirac observables. We quantize the system a la loop quantum gravity: we give a description of the kinematical Hilbert space and construct the associated spin-foam model. We construct the physical disctance operator and consider its quantization.

Abstract:
We study a three dimensional non-commutative space emerging in the context of three dimensional Euclidean quantum gravity. Our starting point is the assumption that the isometry group is deformed to the Drinfeld double D(SU(2)). We generalize to the deformed case the construction of the flat Euclidean space as the quotient of its isometry group ISU(2) by SU(2). We show that the algebra of functions becomes the non-commutative algebra of SU(2) distributions endowed with the convolution product. This construction gives the action of ISU(2) on the algebra and allows the determination of plane waves and coordinate functions. In particular, we show that: (i) plane waves have bounded momenta; (ii) to a given momentum are associated several SU(2) elements leading to an effective description of an element in the algebra in terms of several physical scalar fields; (iii) their product leads to a deformed addition rule of momenta consistent with the bound on the spectrum. We generalize to the non-commutative setting the local action for a scalar field. Finally, we obtain, using harmonic analysis, another useful description of the algebra as the direct sum of the algebra of matrices. The algebra of matrices inherits the action of ISU(2): rotations leave the order of the matrices invariant whereas translations change the order in a way we explicitly determine.

Abstract:
Geometric phase (GP) independent of energy and time rely only on the geometry of state space. It has been argued to have potential fault tolerance and plays an important role in quantum information and quantum computation. We present the first experiment for producing and measuring an Abelian geometric phase shift in a three-level system by using NMR interferometry. In contrast to existing experiments, based on the geometry of $S^2$, our experiment concerns the geometric phase with the geometry of $SU(3)/U(2)$. Two interacting qubits have been used to provide such a three-dimensional Hilbert space.

Abstract:
We describe some results concerning the phase space of 3-dimensional Einstein gravity when space is a torus and with negative cosmological constant. The approach uses the holonomy matrices of flat SL(2,R) connections on the torus to parametrise the geometry. After quantization, these matrices acquire non-commuting entries, in such a way that they satisfy q-commutation relations and exhibit interesting geometrical properties. In particular they lead to a quantization of the Goldman bracket.

Abstract:
All existing experimental results are currently interpreted using classical geometry. However, there are theoretical reasons to suspect that at a deeper level, geometry emerges as an approximate macroscopic behavior of a quantum system at the Planck scale. If directions in emergent quantum geometry do not commute, new quantum-geometrical degrees of freedom can produce detectable macroscopic deviations from classicality: spatially coherent, transverse position indeterminacy between any pair of world lines, with a displacement amplitude much larger than the Planck length. Positions of separate bodies are entangled with each other, and undergo quantum-geometrical fluctuations that are not describable as metric fluctuations or gravitational waves. These fluctuations can either be cleanly identified or ruled out using interferometers. A Planck-precision test of the classical coherence of space-time on a laboratory scale is now underway at Fermilab.

Abstract:
We generalize the noncommutative relations obeyed by the guiding centers in the two-dimensional quantum Hall effect to those obeyed by the projected position operators in three-dimensional (3D) topological band insulators. The noncommutativity in 3D space is tied to the integral over the 3D Brillouin zone of a Chern-Simons invariant in momentum-space. We provide an example of a model on the cubic lattice for which the chiral symmetry guarantees a macroscopic number of zero-energy modes that form a perfectly flat band. This lattice model realizes a chiral 3D noncommutative geometry. Finally, we find conditions on the density-density structure factors that lead to a gapped 3D fractional chiral topological insulator within Feynman's single-mode approximation.

Abstract:
New quantum degrees of freedom of space-time, originating at the Planck scale, could create a coherent indeterminacy and noise in the transverse position of massive bodies on macroscopic scales. An experiment is under development at Fermilab designed to detect or rule out a transverse position noise with Planck spectral density, using correlated signals from an adjacent pair of Michelson interferometers. A detection would open an experimental window on quantum space-time.

Abstract:
The subject of this talk was the review of our study of three ($2+1$) dimensional Quantum Chromodynamics. In our previous works, we showed the existence of a phase where parity is unbroken and the flavor group $U(2n)$ is broken to a subgroup $U(n)\times U(n)$. We derived the low energy effective action for the theory and showed that it has solitonic excitations with Fermi statistic, to be identified with the three dimensional ``baryon''. Finally, we studied the current algebra for this effective action and we found a co-homologically non trivial generalization of Kac-Moody algebras to three dimensions.