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.
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.
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.
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.
Quantum-confined systems, such as quantum wires, have attracted attention because their nonlinear optical (NLO) properties are enhanced due to confinement effects. In the present work, we have developed a method to study what other factors may affect the NLO properties of such systems. For this purpose, we develop a model system that eliminates confinement effect contributions to the NLO properties. We test the validity of our model by verifying that the sum rules, which are a direct consequence of quantum mechanics, are obeyed. This allows us to focus on how the NLO response of a quantum wire depends on parameters such as the geometry and topology of quantum loops, which are built from networks of quantum wires.
KEYWORDS: Molecules, Complex systems, Monte Carlo methods, Nonlinear optics, Nonlinear response, Physics, Systems modeling, Electrons, Oscillators, Chemical species
In the present work we use the Monte Carlo method to study the nonlinear optical response classified by energy
spacing of the system, aiming to understand certain unresolved questions including the gap between experimental
values of the off-resonant hyperpolarizabilities of molecules and the fundamental limit. The results
suggest an explanation for the origin of the factor of 20-30 gap between the best molecules and the fundamental
limits and also confirm the validity of three-level ansatz, which states that when the first and second
hyperpolarizabilities of a quantum system are at the limit, only three states contribute to the nonlinear response.
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