Shoresh Shafei

The influence of geometry and topology of quantum graphs on their nonlinear-optical properties

We analyze the nonlinear optics of quasi one-dimensional quantum graphs and manipulate their topology and geometry to generate for the first time nonlinearities in a simple system approaching the fundamental limits of the first and second hyperpolarizabilities. Changes in geometry result in smooth variations of the nonlinearities. Topological changes between geometrically-similar systems cause profound changes in the nonlinear susceptibilities that include a discontinuity due to abrupt changes in the boundary conditions. This work may inform the design of new molecules or nano- scale structures for nonlinear optics and hints at the same universal behavior for quantum graph models in nonlinear optics that is observed in other systems.

THE EFFECT OF EXTREME CONFINEMENT ON THE NONLINEAR-OPTICAL RESPONSE OF QUANTUM WIRES

This work focuses on understanding the nonlinear-optical response of a 1-D quantum wire embedded in 2-D space when quantum-size effects in the transverse direction are minimized using an extremely weighted delta function potential. Our aim is to establish the fundamental basis for understanding the effect of geometry on the nonlinear-optical response of quantum loops that are formed into a network of quantum wires. It is shown that in the limit of full confinement, the sum rules are obeyed when the transverse infinite-energy continuum states are included. While the continuum states associated with the transverse wavefunction do not contribute to the nonlinear optical response, they are essential to preserving the validity of the sum rules. This work is a building block for future studies of nonlinear-optical enhancement of quantum graphs (which include loops and bent wires) based on their geometry. These properties are important in quantum mechanical modeling of any response function of quantum-confined systems, including the nonlinear-optical response of any system in which there is confinement in at least one dimension, such as nanowires, which provide confinement in two dimensions.

Geometry-controlled nonlinear optical response of quantum graphs

We study for the first time the effect of the geometry of quantum wire networks on their nonlinear optical properties and show that for some geometries, the first hyperpolarizability is largely enhanced and the second hyperpolarizability is always negative or zero. We use a one-electron model with tight transverse confinement. In the limit of infinite transverse confinement, the transverse wavefunctions drop out of the hyperpolarizabilities, but their residual effects are essential to include in the sum rules. The effects of geometry are manifested in the projections of the transition moments of each wire segment onto the 2D lab frame. Numerical optimization of the geometry of a loop leads to hyperpolarizabilities that rival the best chromophores. We suggest that a combination of geometry and quantum-confinement effects can lead to systems with ultralarge nonlinear response.

Paradox of the many-state catastrophe of fundamental limits and the three-state conjecture

The calculation of the fundamental limits of nonlinear susceptibilities posits that when a quantum system has a nonlinear response at the fundamental limit, only three energy eigenstates contribute to the first and second hyperpolarizability. This is called the three-level ansatz and is the only unproven assumption in the theory of fundamental limits. All calculations that are based on direct solution of the Schrodinger equation yield intrinsic hyperpolarizabilities less than 0.709 and intrinsic second hyperpolarizabilities less than 0.6. In this work, we show that relaxing the three-level ansatz and allowing an arbitrary number of states to contribute leads to divergence of the optimized intrinsic hyperpolarizability in the limit of an infinite number of states - what we call the many-state catastrophe. This is not surprising given that the divergent systems are most likely not derivable from the Schrodinger equation, yet obey the sum rules. The sums rules are the second ingredient in limit theory, and apply also to systems with more general Hamiltonians. These exotic Hamiltonians may not model any real systems found in nature. Indeed, a class of transition moments and energies that come form the sum rules do not have a corresponding Hamiltonian that is expressible in differential form. In this work, we show that the three-level ansatz acts as a constraint that excludes many of the nonphysical Hamiltonians and prevents the intrinsic hyperpolarizability from diverging. We argue that this implies that the true fundamental limit is smaller than previously calculated. Since the three-level ansatz does not lead to the largest possible nonlinear response, contrary to its assertion, we propose the intriguing possibility that the three-level ansatz is true for any system that obeys the Schrodinger equation, yet this assertion may be unprovable.

Nonlinear optics of quantum graphs

Quantum graphs are graphical networks comprised of edges supporting Hamiltonian dynamics and vertices conserving probability flux. Lateral confinement of particle motion on every edge results in a quasi one-dimensional quantum-confined system for which nonlinear optical effects may be calculated. Our ongoing research program is the first to investigate the nonlinear optical properties of quantum graphs. We seek to discover configurations with intrinsic first and second hyperpolarizabilities approaching their respective fundamental limits, to explore the NLO variation with the geometry and topology of the graphs, and to develop scaling laws for more complex graphs with self-similar properties. This paper describes a new methodology for calculating the hyperpolarizabilities of a class of graphs comprised of sequentially-connected edges. Such graphs include closed-loop topologies and their geometrically-similar but topologically-different open loop cousins, as well as other bent wire graphs and their combinations.

Optimum topology of quasi-one-dimensional nonlinear optical quantum systems

In this paper, 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.

Topological optimization of nonlinear optical quantum wire networks

Spatially extended molecular structures, modeled as quantum graphs with one-dimensional electron dynamics, exhibit optical responses that can approach the fundamental limits. We present the results of a comprehensive study of the topological dependence of the nonlinearities of quantum graphs and show exactly how the first and second hyperpolarizability of a graph depend upon its topological class and how the hyperpolarizability tensors vary with graph geometry. We show how graphs with star motifs share universal scaling behavior near the maximum nonlinear responses and articulate design rules for quantum-confined, quasi-one dimensional systems that may be realized using molecular elements and nanowires.

Applying universal scaling laws to identify the best molecular design paradigms for third-order nonlinear optics

We apply scaling and the theory of the fundamental limits of the second-order molecular susceptibility to identify material classes with ultralarge nonlinear-optical response. Size effects are removed by normalizing all nonlinearities to get intrinsic values so that the scaling behavior of a series of molecular homologues can be determined. Several new figures of merit are proposed that quantify the desirable properties for molecules that can be designed by adding a sequence of repeat units, and used in the assessment of the data. Three molecular classes are found. They are characterized by sub-scaling, nominal scaling, or super-scaling. Super-scaling homologues most efficiently take advantage of increased size. We apply our approach to data currently available in the literature to identify the best super-scaling molecular paradigms with the aim of identifying desirable traits of new materials.

Potential energy optimization and Monte Carlo simulations of the first hyperpolarizability: a comparative study

We contrast two numerical approaches that are used to optimize the intrinsic hyperpolarizability: potential optimization and sum-rule-constrained Monte Carlo simulations. Our aim is to resolve inconsistencies between the two. We show that while the first method accurately reflects the properties of real physical systems, the second requires exotic hamiltonians that obey sum rules but may not represent a physical reality. Under certain extreme conditions, the sum-rule-constrained approach leads to systems that may not be representable by any Schrodinger Equation in differential equation form.

Using geometry to enhance the nonlinear response of quantum confined systems

We study the effect of geometry on the nonlinear response of a network of quantum wires that form loops. Exploiting the fact that a loop’s transition moment matrix and energies are exactly solvable for each wire segment, they can be pieced together to determine a loop’s properties. A Monte Carlo method is used to sample the configuration space of all possible geometries to determine the shape that optimizes the intrinsic hyperpolarizability. We suggest that a combination of wire geometry and confinement effects can lead to artificial systems with ultra-large nonlinear response, which can be potentially made using known nanofabrication techniques.