Dressed quantum graphs with optical nonlinearities approaching the fundamental limit

We dress bare quantum graphs with finite delta function potentials and calculate optical nonlinearities that are found to match the fundamental limits set by potential optimization. We show that structures whose first hyperpolarizability is near the maximum are well described by only three states, the so-called three-level Ansatz, while structures with the largest second hyperpolarizability require four states. We analyze a very large set of configurations for graphs with quasi-quadratic energy spectra and show how they exhibit better response than bare graphs through exquisite optimization of the shape of the eigenfunctions enabled by the existence of the finite potentials. We also discover an exception to the universal scaling properties of the three-level model parameters and trace it to the observation that a greater number of levels are required to satisfy the sum rules even when the three-level Ansatz is satisfied and the first hyperpolarizability is at its maximum value, as specified by potential optimization. This exception in the universal scaling properties of nonlinear optical structures at the limit is traced to the discontinuity in the gradient of the eigenfunctions at the location of the delta potential. This is the first time that dressed quantum graphs have been devised and solved for their nonlinear response, and it is the first analytical model of a confined dynamic system with a simple potential energy that achieves the fundamental limits

Dalgarno-Lewis perturbation theory for nonlinear optics

We apply the quadrature-based perturbation method of Dalgarno and Lewis to the evaluation of the nonlinear optical response of quantum systems. This general operator method for perturbation theory allows us to derive exact expressions for the first three electronic polarizabilities which require only a good estimate of the ground state wave function, makes no explicit reference to the underlying potential, and avoids complexities arising from excited state degeneracies. We apply this method to simple examples in 1D quantum mechanics for illustration, exploring the sensitivity of this method to variational solutions as well as poor numerical sampling. Finally, to the best of our knowledge, we extend the Dalgarno-Lewis method for for the first time to time-harmonic perturbations, allowing dispersion characteristics to be determined from the unperturbed ground state wave function alone

Fundamental limits on the electro-optic device figure of merit

Device figures of merit are commonly employed to assess bulk material properties for a particular device class, yet these properties ultimately originate in the linear and nonlinear susceptibilities of the material which are not independent of each other. In this work, we calculate the electro-optic device figure of merit based on the half-wave voltage and linear loss, which is important for phase modulators and serves as the simplest example of the approach. This figure of merit is then related back to the microscopic properties in the context of a dye-doped polymer, and its fundamental limits are obtained to provide a target. Surprisingly, the largest figure of merit is not always associated with a large nonlinear-optical response, the quantity that is most often the focus of optimization. An important lesson to materials design is that the figure of merit alone should be optimized. The best device materials can have low nonlinearity provided that the loss is low; or, near resonance high loss may be desirable because it is accompanied by resonantly-enhanced, ultra-large nonlinear response so device lengths are short. Our work shows which frequency range of operation is most promising for optimizing the material figure of merit for electro-optic devices.Comment: Higher resolution figures available in final publicatio

Scaling and universality in nonlinear optical quantum graphs containing star motifs

Quantum graphs have recently emerged as models of nonlinear optical, quantum confined systems with exquisite topological sensitivity and the potential for predicting structures with an intrinsic, off-resonance response approaching the fundamental limit. Loop topologies have modest responses, while bent wires have larger responses, even when the bent wire and loop geometries are identical. Topological enhancement of the nonlinear response of quantum graphs is even greater for star graphs, for which the first hyperpolarizability can exceed half the fundamental limit. In this paper, we investigate the nonlinear optical properties of quantum graphs with the star vertex topology, introduce motifs and develop new methods for computing the spectra of composite graphs. We show that this class of graphs consistently produces intrinsic optical nonlinearities near the limits predicted by potential optimization. All graphs of this type have universal behavior for the scaling of their spectra and transition moments as the nonlinearities approach the fundamental limit

Optimum topology of quasi-one dimensional nonlinear optical quantum systems

We determine the optimum topology of quasi-one dimensional nonlinear optical structures using generalized quantum graph models. Quantum graphs are relational graphs endowed with a metric and a multiparticle Hamiltonian acting on the edges, and have a long application history in aromatic compounds, mesoscopic and artificial materials, and quantum chaos. Quantum graphs have recently emerged as models of quasi-one dimensional electron motion for simulating quantum-confined nonlinear optical systems. This paper derives the nonlinear optical properties of quantum graphs containing the basic star vertex and compares their responses across topological and geometrical classes. We show that such graphs have exactly the right topological properties to generate energy spectra required to achieve large, intrinsic optical nonlinearities. The graphs have the exquisite geometrical sensitivity required to tune wave function overlap in a way that optimizes the transition moments. We show that this class of graphs consistently produces intrinsic optical nonlinearities near the fundamental limits. We discuss the application of the models to the prediction and development of new nonlinear optical structures

A PMU Scheduling Scheme for Transmission of Synchrophasor Data in Electric Power Systems

With the proposition to install a large number of phasor measurement units (PMUs) in the future power grid, it is essential to provide robust communications infrastructure for phasor data across the network. We make progress in this direction by devising a simple time division multiplexing scheme for transmitting phasor data from the PMUs to a central server: Time is divided into frames and the PMUs take turns to transmit to the control center within the time frame. The main contribution of this work is a scheduling policy based on which PMU transmissions are ordered during a time frame. The scheduling scheme is independent of the approach taken to solve the PMU placement problem, and unlike strategies devised for conventional communications, it is intended for the power network since it is fully governed by the measure of electrical connectedness between buses in the grid. To quantify the performance of the scheduling scheme, we couple it with a fault detection algorithm used to detect changes in the susceptance parameters in the grid. Results demonstrate that scheduling the PMU transmissions leads to an improved performance of the fault detection scheme compared to PMUs transmitting at random.Comment: 9 pages, 6 figures; an extra figure included in the published version. appears in IEEE Transactions on Smart Grid, Special Issue on Cyber Physical Systems and Security for Smart Grid, 201

Optimization of eigenstates and spectra for quasi-linear nonlinear optical systems

Quasi-one-dimensional quantum structures with spectra scaling faster than the square of the eigenmode number (superscaling) can generate intrinsic, off-resonant optical nonlinearities near the fundamental physical limits, independent of the details of the potential energy along the structure. The scaling of spectra is determined by the topology of the structure, while the magnitudes of the transition moments are set by the geometry of the structure. This paper presents a comprehensive study of the geometrical optimization of superscaling quasi-one-dimensional structures and provides heuristics for designing molecules to maximize intrinsic response. A main result is that designers of conjugated structures should attach short side groups at least a third of the way along the bridge, not near its end as is conventionally done. A second result is that once a side group is properly placed, additional side groups do not further enhance the response

Energy cascade and scaling in supersonic isothermal turbulence

Supersonic turbulence plays an important role in a number of extreme astrophysical and terrestrial environments, yet its understanding remains rudimentary. We use data from a three-dimensional simulation of supersonic isothermal turbulence to reconstruct an exact fourth-order relation derived analytically from the Navier-Stokes equations (Galtier and Banerjee, Phys. Rev. Lett., vol. 107, 2011, p. 134501). Our analysis supports a Kolmogorov-like inertial energy cascade in supersonic turbulence previously discussed on a phenomenological level. We show that two compressible analogues of the four-fifths law exist describing fifth- and fourth-order correlations, but only the fourth-order relation remains `universal' in a wide range of Mach numbers from incompressible to highly compressible regimes. A new approximate relation valid in the strongly supersonic regime is derived and verified. We also briefly discuss the origin of bottleneck bumps in simulations of compressible turbulence.Comment: Accepted to JFM Rapids, 11 pages, 6 figure

General solution to nonlinear optical quantum graphs using Dalgarno-Lewis summation techniques

We develop an algorithm to apply the Dalgarno-Lewis (DL) perturbation theory to quantum graphs with multiple, connected edges. We use it to calculate the nonlinear optical hyperpolarizability tensors for graphs and show that it replicates the sum over states computations, but executes ten to fifty times faster. DL requires only knowledge of the ground state of the graph, eliminating the requirement to determine all possible degeneracies of a complex network. The algorithm is general and may be applied to any quantum graph

An Electrical Structure-Based Approach to PMU Placement in the Electric Power Grid

The phasor measurement unit (PMU) placement problem is revisited by taking into account a stronger characterization of the electrical connectedness between various buses in the grid. To facilitate this study, the placement problem is approached from the perspective of the \emph{electrical structure} which, unlike previous work on PMU placement, accounts for the sensitivity between power injections and nodal phase angle differences between various buses in the power network. The problem is formulated as a binary integer program with the objective to minimize the number of PMUs for complete network observability in the absence of zero injection measurements. The implication of the proposed approach on static state estimation and fault detection algorithms incorporating PMU measurements is analyzed. Results show a significant improvement in the performance of estimation and detection schemes by employing the electrical structure-based PMU placement compared to its topological counterpart. In light of recent advances in the electrical structure of the grid, our study provides a more realistic perspective of PMU placement in the electric power grid.Comment: 8 pages, submitted to IEEE Transactions on Smart Grid. arXiv admin note: text overlap with arXiv:1309.130