On Nori's Fundamental Group Scheme

We determine the quotient category which is the representation category of the kernel of the homomorphism from Nori's fundamental group scheme to its \'etale and local parts. Pierre Deligne pointed out an error in the first version of this article. We profoundly thank him, in particular for sending us his enlightning example reproduced in Remark 2.4 2).Comment: 29 page

Quantum interference from sums over closed paths for electrons on a three-dimensional lattice in a magnetic field: total energy, magnetic moment, and orbital susceptibility

We study quantum interference effects due to electron motion on a three-dimensional cubic lattice in a continuously-tunable magnetic field of arbitrary orientation and magnitude. These effects arise from the interference between magnetic phase factors associated with different electron closed paths. The sums of these phase factors, called lattice path-integrals, are ``many-loop" generalizations of the standard ``one-loop" Aharonov-Bohm-type argument. Our lattice path integral calculation enables us to obtain various important physical quantities through several different methods. The spirit of our approach follows Feynman's programme: to derive physical quantities in terms of ``sums over paths". From these lattice path-integrals we compute analytically, for several lengths of the electron path, the half-filled Fermi-sea ground-state energy of noninteracting spinless electrons in a cubic lattice. Our results are valid for any strength of the applied magnetic field in any direction. We also study in detail two experimentally important quantities: the magnetic moment and orbital susceptibility at half-filling, as well as the zero-field susceptibility as a function of the Fermi energy.Comment: 14 pages, RevTe

Inverse Landau-Zener-Stuckelberg problem for qubit-resonator systems

We consider theoretically a superconducting qubit - nanomechanical resonator (NR) system, which was realized by LaHaye et al. [Nature 459, 960 (2009)]. First, we study the problem where the state of the strongly driven qubit is probed through the frequency shift of the low-frequency NR. In the case where the coupling is capacitive, the measured quantity can be related to the so-called quantum capacitance. Our theoretical results agree with the experimentally observed result that, under resonant driving, the frequency shift repeatedly changes sign. We then formulate and solve the inverse Landau-Zener-Stuckelberg problem, where we assume the driven qubit's state to be known (i.e. measured by some other device) and aim to find the parameters of the qubit's Hamiltonian. In particular, for our system the qubit's bias is defined by the NR's displacement. This may provide a tool for monitoring of the NR's position.Comment: 10 pages, 7 figure

Unified single-photon and single-electron counting statistics: from cavity-QED to electron transport

A key ingredient of cavity quantum-electrodynamics (QED) is the coupling between the discrete energy levels of an atom and photons in a single-mode cavity. The addition of periodic ultra-short laser pulses allows one to use such a system as a source of single photons; a vital ingredient in quantum information and optical computing schemes. Here, we analyze and ``time-adjust'' the photon-counting statistics of such a single-photon source, and show that the photon statistics can be described by a simple `transport-like' non-equilibrium model. We then show that there is a one-to-one correspondence of this model to that of non-equilibrium transport of electrons through a double quantum dot nanostructure. Then we prove that the statistics of the tunnelling electrons is equivalent to the statistics of the emitted photons. This represents a unification of the fields of photon counting statistics and electron transport statistics. This correspondence empowers us to adapt several tools previously used for detecting quantum behavior in electron transport systems (e.g., super-Poissonian shot noise, and an extension of the Leggett-Garg inequality) to single-photon-source experiments.Comment: 8 pages, 3 figure

Enhancing the conductance of a two-electron nanomechanical oscillator

We consider electron transport through a mobile island (i.e., a nanomechanical oscillator) which can accommodate one or two excess electrons and show that, in contrast to immobile islands, the Coulomb blockade peaks, associated with the first and second electrons entering the island, have different functional dependences on the nano-oscillator parameters when the island coupling to its leads is asymmetric. In particular, the conductance for the second electron (i.e., when the island is already charged) is greatly enhanced in comparison to the conductance of the first electron in the presence of an external electric field. We also analyze the temperature dependence of the two conduction peaks and show that these exhibit different functional behaviors.Comment: 16 pages, 5 figure

Quantum super-cavity with atomic mirrors

We study single-photon transport in an array of coupled microcavities where two two-level atomic systems are embedded in two separate cavities of the array. We find that a single-photon can be totally reflected by a single two-level system. However, two separate two-level systems can also create, between them, single-photon quasi-bound states. Therefore, a single two-level system in the cavity array can act as a mirror while a different type of cavity can be formed by using two two-level systems, acting as tunable "mirrors", inside two separate cavities in the array. In analogy with superlattices in solid state, we call this new "cavity inside a coupled-cavity array" a super-cavity. This supercavity is the quantum analog of Fabry-Perot interferometers. Moreover, we show that the physical properties of this quantum super-cavity can be adjusted by changing the frequencies of these two-level systems.Comment: 13 pages, 9 figure