Canonical Discretization. I. Discrete faces of (an)harmonic oscillator

A certain notion of canonical equivalence in quantum mechanics is proposed. It is used to relate quantal systems with discrete ones. Discrete systems canonically equivalent to the celebrated harmonic oscillator as well as the quartic and the quasi-exactly-solvable anharmonic oscillators are found. They can be viewed as a translation-covariant discretization of the (an)harmonic oscillator preserving isospectrality. The notion of the $q-$deformation of the canonical equivalence leading to a dilatation-covariant discretization preserving polynomiality of eigenfunctions is also presented.Comment: 29 pages, LaTe

Quantum register based on structured diamond waveguide with NV centers

We propose a scheme of quantum information processing with NV-centers embedded inside diamond nanostructure. Single NV-center placed in the cavity plays role of an electron spin qubit which evolution is controlled by microwave pulses. Besides, it couples to the cavity field via optical photon exchange. In their turn, neighbor cavities are coupled to each other through the photon hopping to form a bus waveguide mode. This waveguide mode overlaps with all NV-centers. Entanglement between distant centers is organized by appropriate tuning of their optical frequency relative to the waveguide frequency via electrostatic control without lasers. We describe the controlled-Z operation that is by one order of magnitude faster than in off-resonant laser-assisted schemes proposed earlier. Spectral characteristics of the one-dimensional chain of microdisks are calculated by means of numerical modeling, using the approach analogous to the tight-binding approximation in the solid-state physics. The data obtained allow to optimize the geometry of the microdisk array for the effective implementation of quantum operations.Comment: to be published in Proc. of SPI

Distinct dynamical behavior in Erd\H{o}s-R\'enyi networks, regular random networks, ring lattices, and all-to-all neuronal networks

Neuronal network dynamics depends on network structure. In this paper we study how network topology underpins the emergence of different dynamical behaviors in neuronal networks. In particular, we consider neuronal network dynamics on Erd\H{o}s-R\'enyi (ER) networks, regular random (RR) networks, ring lattices, and all-to-all networks. We solve analytically a neuronal network model with stochastic binary-state neurons in all the network topologies, except ring lattices. Given that apart from network structure, all four models are equivalent, this allows us to understand the role of network structure in neuronal network dynamics. Whilst ER and RR networks are characterized by similar phase diagrams, we find strikingly different phase diagrams in the all-to-all network. Neuronal network dynamics is not only different within certain parameter ranges, but it also undergoes different bifurcations (with a richer repertoire of bifurcations in ER and RR compared to all-to-all networks). This suggests that local heterogeneity in the ratio between excitation and inhibition plays a crucial role on emergent dynamics. Furthermore, we also observe one subtle discrepancy between ER and RR networks, namely ER networks undergo a neuronal activity jump at lower noise levels compared to RR networks, presumably due to the degree heterogeneity in ER networks that is absent in RR networks. Finally, a comparison between network oscillations in RR networks and ring lattices shows the importance of small-world properties in sustaining stable network oscillations.Comment: 9 pages, 4 figure