Quantum Holonomies in (2+1)-Dimensional Gravity

We describe an approach to the quantization of (2+1)--dimensional gravity with topology R x T^2 and negative cosmological constant, which uses two quantum holonomy matrices satisfying a q--commutation relation. Solutions of diagonal and upper--triangular form are constructed, which in the latter case exhibit additional, non--trivial internal relations for each holonomy matrix. This leads to the notion of quantum matrix pairs. These are pairs of matrices with non-commuting entries, which have the same pattern of internal relations, q-commute with each other under matrix multiplication, and are such that products of powers of the matrices obey the same pattern of internal relations as the original pair. This has implications for the classical moduli space, described by ordered pairs of commuting SL(2,R) matrices modulo simultaneous conjugation by SL(2,R) matrices.Comment: 5 pages, to appear in the proceedings of 10th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Gravitation and Relativistic Field Theories (MG X MMIII), Rio de Janeiro, Brazil, 20-26 Jul 200

Single Atom Imaging with an sCMOS camera

Single atom imaging requires discrimination of weak photon count events above background and has typically been performed using either EMCCD cameras, photomultiplier tubes or single photon counting modules. sCMOS provides a cost effective and highly scalable alternative to other single atom imaging technologies, offering fast readout and larger sensor dimensions. We demonstrate single atom resolved imaging of two site-addressable single atom traps separated by 10~$\mu$m using an sCMOS camera, offering a competitive signal-to-noise ratio at intermediate count rates to allow high fidelity readout discrimination (error $<10^{-6}$) and sub-$\mu$m spatial resolution for applications in quantum technologies.Comment: 4 pages, 4 figure

Quantum Geometry and Quantum Gravity

The purpose of this contribution is to give an introduction to quantum geometry and loop quantum gravity for a wide audience of both physicists and mathematicians. From a physical point of view the emphasis will be on conceptual issues concerning the relationship of the formalism with other more traditional approaches inspired in the treatment of the fundamental interactions in the standard model. Mathematically I will pay special attention to functional analytic issues, the construction of the relevant Hilbert spaces and the definition and properties of geometric operators: areas and volumes.Comment: To appear in the AIP Conference Proceedings of the XVI International Fall Workshop on Geometry and Physics, Lisbon - Portugal, 5-8 September 200

Experimental demonstration of high-fidelity entanglement via Rydberg blockade

The strong dipole interactions of Rydberg atoms are ideal candidates to facilitate interactions between neutral atoms to generate entanglement for quantum information processing. This offers the potential to scale to large atom arrays through well established techniques for neutral atoms, overcoming limitations of other architectures for quantum information processing. This thesis presents the design and development of an experiment for quantum information processing using Rydberg atoms, concluding with the deterministic preparation of two caesium atomic qubits in a maximumly entangled Bell state.The experiment presented achieves low error readout of two single atoms held in optical tweezers using new imaging technology as an alternative to what is typically used in the field, offering a cost effective solution whilst maintaining high shot to shot retention as is necessary for qubit operations. Qubit manipulations are demonstrated with fast two-photon rotations between the hyperfine ground states and the 69S1/2 Rydberg state. Due to the cold single atom temperatures achieved, T ≈ 10 μK, the ground-Rydberg dephasing times measured through Ramsey spectroscopy find coherence times around twice that of previously reported experiments, over an order of magnitude greater than the gate time.;Demonstration of Rydberg blockade between two atoms with a separation of 6 μm is shown with an almost compete suppression to the doubly excited state and observation of a √2-enhancement of coupling to an entangled symmetric |W〉 state. Finally the |W〉 state is mapped to the ground state qubit levels to create a maximally entangled Bell state achieving a loss-corrected fidelity of Ƒpairs = 0:81 ± 0:05. This result represents the highest corrected ground state neutral atom entanglement fidelity via Rydberg blockade and is equal to that achieved via Rydberg dressing . The limitation of this Bell state preparation is primarily due to laser phase noise as found in other experiments and is verified through the long coherence times measured in this thesis. Generation of entanglement in the magnetically insensitive hyperfine states of caesium allows long coherence times to be achieved with Ramsey spectroscopy used to measured transverse dephasing times of T*₂ = 10± 1 ms and T'2 = 150 ± 20 ms, offering an excellent platform for quantum computation.The strong dipole interactions of Rydberg atoms are ideal candidates to facilitate interactions between neutral atoms to generate entanglement for quantum information processing. This offers the potential to scale to large atom arrays through well established techniques for neutral atoms, overcoming limitations of other architectures for quantum information processing. This thesis presents the design and development of an experiment for quantum information processing using Rydberg atoms, concluding with the deterministic preparation of two caesium atomic qubits in a maximumly entangled Bell state.The experiment presented achieves low error readout of two single atoms held in optical tweezers using new imaging technology as an alternative to what is typically used in the field, offering a cost effective solution whilst maintaining high shot to shot retention as is necessary for qubit operations. Qubit manipulations are demonstrated with fast two-photon rotations between the hyperfine ground states and the 69S1/2 Rydberg state. Due to the cold single atom temperatures achieved, T ≈ 10 μK, the ground-Rydberg dephasing times measured through Ramsey spectroscopy find coherence times around twice that of previously reported experiments, over an order of magnitude greater than the gate time.;Demonstration of Rydberg blockade between two atoms with a separation of 6 μm is shown with an almost compete suppression to the doubly excited state and observation of a √2-enhancement of coupling to an entangled symmetric |W〉 state. Finally the |W〉 state is mapped to the ground state qubit levels to create a maximally entangled Bell state achieving a loss-corrected fidelity of Ƒpairs = 0:81 ± 0:05. This result represents the highest corrected ground state neutral atom entanglement fidelity via Rydberg blockade and is equal to that achieved via Rydberg dressing . The limitation of this Bell state preparation is primarily due to laser phase noise as found in other experiments and is verified through the long coherence times measured in this thesis. Generation of entanglement in the magnetically insensitive hyperfine states of caesium allows long coherence times to be achieved with Ramsey spectroscopy used to measured transverse dephasing times of T*₂ = 10± 1 ms and T'2 = 150 ± 20 ms, offering an excellent platform for quantum computation

Parasupersymmetric Quantum Mechanics of Order 3 and a Generalized Witten Index

In this paper we generalize the Rubakov-Spiridonov parasupersymmetry algebra to the order 3 case. We also generalize the notion of the Witten index, and we provide a class of models satisfying our parasupersymmetry algebra. Finally, we show that there is a correspondence between the Hamiltonian and the index in our class of models

QUANTUM HOLONOMIES AND THE HEISENBERG GROUP

Quantum holonomies of closed paths on the torus $T^2$ are interpreted as elements of the Heisenberg group $H_1$. Group composition in $H_1$ corresponds to path concatenation and the group commutator is a deformation of the relator of the fundamental group $\pi_1$ of $T^2$, making explicit the signed area phases between quantum holonomies of homotopic paths. Inner automorphisms of $H_1$ adjust these signed areas, and the discrete symplectic transformations of $H_1$ generate the modular group of $T^2$.Comment: 8 pages, 3 figure