Quantum Physics

Noncommutative geometry is a more general formulation of geometry that does not require coordinates to commute. As such it unifies quantum theory and geometry and should appear in any effective theory of quantum gravity. In this general talk we present quantum groups as a microcosm of this unification in the same way that Lie groups are a microcosm of usual geometry, and give a flavour of some of the deeper insights they provide. One of them is the ability to interchange the roles of quantum theory and gravity by `arrow reversal'. Another is that noncommutative spaces typically carry a canonical 1-parameter evolution or intrinsic time created from the fundamental conflict between noncommuting coordinates and differential calculus. In physical terms one could say that quantising space typically has an anomaly for the spatial translation group and this forces the system to evolve. We give an example where we derive Schroedinger's equation in this way.

In this talk I'll discuss recent joint work with Raymond Laflamme, David Poulin and Maia Lesosky in which a unified approach to quantum error correction is presented, called "operator quantum error correction". This scheme relies on a generalized notion of noiseless subsystems and includes the known techniques for the error correction of quantum operations --i.e., the standard model, the method of decoherence-free subspaces, and the noiseless subsystem method--as special cases. Correctable codes in this approach take the form of operator algebras and operator semigroups. The condition from the standard model is shown to be necessary for all of the known methods of error correction, and we'll see this as part of a discussion on conditions that characterize correctability for the general case.

Superconducting circuits based on Josephson junctions are promising candidates for the implementation of solid-state qubits. In most of the recent experiments on these circuits, the qubits are controlled by a classical field containing a large number of photons. The possibility of coherently coupling these systems to a single photon has been recently suggested, opening the possibility to study analogs of quantum optics in condensed matter systems. I will review one of these proposals based on a superconducting charge qubit fabricated inside a high quality transmission line resonator and will describe its recent experimental realization. When the qubit is brought into resonance with the resonator, vacuum Rabi splitting is observed indicating that the regime of strong coupling has been reached. When the qubit is detuned from the cavity, I will explain how quantum non-demolition measurement can be realized. I will discuss how the measurement process can be quantitatively understood in this regime allowing us to explore the effect of measurement back-action on the qubit and to extract, for the first time in superconducting qubits, large visibility in Rabi oscillations.