Abstract. The Jones-Wenzl projectors p n play a central role in quantum topology, underlying the construction of SU.2/ topological quantum field theories and quantum spin networks. We construct chain complexes P n , whose graded Euler characteristic is the "classical" projector p n in the Temperley-Lieb algebra. We show that the P n are idempotents and uniquely defined up to homotopy. Our results fit within the general framework of Khovanov's categorification of the Jones polynomial. Consequences of our construction include families of knot invariants corresponding to higher representations of U q sl.2/ and a categorification of quantum spin networks. We introduce 6j -symbols in this context.

Different types of angular magnetoresistance oscillations in quasi-one-dimensional layered materials, such as organic conductors (TMTSF)2X, are explained in terms of Aharonov-Bohm interference in interlayer electron tunneling. A two-parameter pattern of oscillations for generic orientations of a magnetic field is visualized and compared with the experimental data. Connections with angular magnetoresistance oscillations in other layered materials are discussed. AMRO [4,11,12] were formulated in terms of semiclassical electron trajectories on a cylindrical 3D Fermi surface. Then it was realized that AMRO can exist already for two layers [13,14,15], and they represent an Aharonov-Bohm interference effect in interlayer tunneling [16]. Some experimental evidence for AMRO in semiconducting bilayers has been found [17], but more systematic measurements are necessary.AMRO were also found in quasi-one-dimensional (Q1D) organic conductors with open Fermi surfaces, such as (TMTSF) 2 X [3]. These materials consist of parallel chains along the x axis, which form layers with the interlayer spacing d along the z axis and the interchain spacing b along the y axis, as shown in Fig. 1a. Origi- nally, three different AMRO were discovered in the Q1D conductors: the Lebed magic angles [18,19,20,21] for a magnetic field rotation in the (y, z) plane, the Danner, Kang, and Chaikin (DKC) oscillations in the (x, z) plane [22,23], and the third angular effect in the (x, y) plane [24,25,26]. Then Lee and Naughton [27] found combinations of all three effects for generic magnetic field rotations. It became clear that all types of AMRO in Q1D conductors have a common origin and should be explained by a single unified theory.In (TMTSF) 2 X, the in-plane tunneling amplitude between the chains, t b ∼ 250 K [22,23], is much greater than the inter-plane tunneling amplitude t c ∼ 10 K [3]. Thus, we can treat interlayer tunneling as a perturbation and study the interlayer conductivity σ c between just two layers for a tilted magnetic field B = (B x , B y , B z ), as shown in Fig. 1a. This bilayer approach [14,15] only assumes phase memory of interlayer tunneling within a decoherence time τ and does not require a well-defined momentum k z and a coherent 3D Fermi surface. It gives a simple and transparent interpretation of the most general Lee-Naughton oscillations [27] in terms of AharonovBohm interference in interlayer tunneling. The results are equivalent to other approaches based on the classical Boltzmann equation [25,26,27,28,29] and the quantum Kubo formula [30,31,32]. We calculate a contour plots of σ c as a function of two ratios B x /B z and B y /B z for models with one or several interlayer tunneling amplitudes [30,33]. This type of visualization clearly reveals agreement and disagreement between theory and experiment and allows to determine the electronic parameters of the Q1D materials. The results can be also applied to Q1D semiconducting bilayers consisting of quantum wires induced by an array parallel gates, as shown in Fig. 1a.Let us...

We report spectroscopic measurements of discrete two-level systems (TLSs) coupled to a dc SQUID phase qubit with a 16 µm 2 area Al/AlO x /Al junction. Applying microwaves in the 10 GHz to 11 GHz range, we found eight avoided level crossings with splitting sizes from 10 MHz to 200 MHz and spectroscopic lifetimes from 4 ns to 160 ns. Assuming the transitions are from the ground state of the composite system to an excited state of the qubit or an excited state of one of the TLS states, we fit the location and spectral width to get the energy levels, splitting sizes and spectroscopic coherence times of the phase qubit and TLSs. The distribution of splittings is consistent with non-interacting individual charged ions tunneling between random locations in the tunnel barrier and the distribution of lifetimes is consistent with the AlO x in the junction barrier having a frequency-independent loss tangent. To check that the charge of each TLS couples independently to the voltage across the junction, we also measured the spectrum in the 20-22 GHz range and found tilted avoided level crossings due to the second excited state of the junction and states in which both the junction and a TLS were excited.

Abstract-Rabi oscillations have been observed in many superconducting devices, and represent prototypical logic operations for quantum bits (qubits) in a quantum computer. We use a three-level multiphoton analysis to understand the behavior of the superconducting phase qubit (current-biased Josephson junction) at high microwave drive power. Analytical and numerical results for the ac Stark shift, single-photon Rabi frequency, and twophoton Rabi frequency are compared to measurements made on a dc SQUID phase qubit with Nb/AlOx/Nb tunnel junctions. Good agreement is found between theory and experiment.