Influence of geometry and topology of quantum graphs on their nonlinear optical properties

Lytel, R.; Shafei, Shoresh; Smith, Julian H.; Kuzyk, Mark G.

doi:10.1103/physreva.87.043824

Cited by 25 publications

(63 citation statements)

References 49 publications

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“…2 shows the full ground state wavefunction ψ 0 (x, y) for the one-prong graph, where it is seen that the electron wavefunction along the main direction has a kink caused by the presence of the prong. We note that computation of quantum graphs requires solutions in both x and y directions [12] but the x direction is the dominant contributor to the response for this graph. Fig.…”

Section: Figmentioning

confidence: 99%

Phase disruption as a new design paradigm for optimizing the nonlinear-optical response

2015

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The intrinsic optical nonlinearities of linear structures, including conjugated chain polymers and nanowires, are shown to be dramatically enhanced by the judicious placement of a charge diverting path sufficiently short to create a large phase disruption in the dominant eigenfunctions along the main path of probability current. Phase disruption is proposed as a new general principle for the design of molecules, nanowires and any quasi-1D quantum system with large intrinsic response and does not require charge donor-acceptors at the ends. © The design and realization of ultrafast nonlinear optical materials with large responses to optical frequency electric fields remains an active field of pure and applied research [1][2][3][4]. To date, no general design rules for obtaining large nonlinearities from any structure have been articulated, and the response of materials remains well below that allowed by quantum physics. In this letter, we propose a general principle that may explain why modern molecules fall short of their potential and use it to demonstrate simple structures with nonlinear responses approaching the physical limits.The off-resonance electronic nonlinear optical response of a molecule is completely determined by its energy spectrum and transition moments. Normalized to its maximum value, the sum over states expression for the intrinsic first hyperpolarizability along the x-axis may be written as [5] where the sum is over all states, x nℓ is the n, ℓ element of the position matrix withx = x − x 00 , E n0 = E n − E 0 is the difference in energy of eigenstates n and 0, N is the number of electrons and m their mass. We include as many as 100 energy eigenstates to calculate β, but systems with large β are always well approximated using only three states. For brevity, we focus on the first hyperpolarizability, though our results also hold for the second hyperpolarizability, γ xxxx . For the remainder of this letter, all hyperpolarizabilities are divided by their maximum values and represent intrinsic tensor properties. It is evident from Eqn.(1) that a system optimized for nonlinear optics will necessarily have eigenfunctions with a large degree of overlap as well as a large change of dipole moment between contributing levels. This is an essential trade-off to achieve a large response. It is generally recognized that the hyperpolarizabilities of most molecules fall at least 30 times short of the limits. [5,6]. Monte Carlo simulations discovered the optimum energy spectrum for such molecules and provided a strong indicator of the origin of the gap [7]. Optimization of the effective potential energy an electron experiences across the main direction of a quasi-linear molecule showed that the hyperpolarizabilities may be increased by tuning a few parameters [8], by modulation of conjugation along a chain [9], or by donor-acceptor substitution and insertion of spacers [10]. However, there is no general rule about how to construct the ideal potential energy profile across a molecule to maximize the nonlinea...

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Section: Figmentioning

confidence: 99%

Phase disruption as a new design paradigm for optimizing the nonlinear-optical response

2015

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“…For the remainder of this work all calculations of the hyperpolarizability will be normalized to this maximum and therefore represent the intrinsic value β int = β/β max , which is invariant under a global length scale change. Through extensive potential optimization [10][11][12] it has become clear that there exists an apparent limit to the hyperpolarizability of real systems which is 0.7089β max , while molecules engineered for nonlinear-optical applications are often a factor of 30 below the fundamental limit. However, by sampling random transition moments and energy spectra constrained only by the sum-rules, Shafei, et al, [6] showed that the fundamental limit is achievable in principal only by energy spectra which scale as n 2 or faster.…”

Section: B Characteristics Of the Optimum Hyperpolarizabilitymentioning

confidence: 99%

Hybrid quantum systems for enhanced nonlinear optical susceptibilities

Sullivan¹,

Mossman²,

Kuzyk³

2016

J. Opt. Soc. Am. B

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Significant effort has been expended in the search for materials with ultra-fast nonlinear-optical susceptibilities, but most fall far below the fundamental limits. This work applies a theoretical materials development program that has identified a promising new hybrid made of a nanorod and a molecule. This system uses the electrostatic dipole moment of the molecule to break the symmetry of the metallic nanostructure that shifts the energy spectrum to make it optimal for a nonlinearoptical response near the fundamental limit. The structural parameters are varied to determine the ideal configuration, providing guidelines for making the best structures.

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“…This paper is presented as follows. Section 2 begins with a concise summary of the standard method we developed for calculating the hyperpolarizability tensors of a large ensemble of topologically equivalent graphs whose geometry is varied in a Monte Carlo algorithm 34,46,35,36 . We then invoke the motif method 36 to determine the characteristic function of the compressed delta atom in Section 3, as preparation for all subsequent work.…”

Section: Rljnopmmentioning

confidence: 99%

“…We have used a general Monte Carlo method that explores a very large configuration space to discover structures with optimum nonlinearities, which has been extensively reviewed in our prior publications for undressed graphs 34,46,35,36 . We start by randomly selecting vertices in 2D space, connecting them with metric edges representing a desired topology (eg, a loop or star), endowing them with single-electron dynamics, and then solving for the exact eigenstates and spectra of the entire graph by first computing the edge states that are eigenfunctions of the Hamiltonian on each edge with the same spectrum as all other edges and then using an appropriate union operation on edge states to create an eigenstate of a Hilbert space that is a direct sum of those on each edge 34 :…”

Section: Calculation Of Optical Nonlinearities Of Quantum Graphsmentioning

confidence: 99%

Dressed Quantum Graphs With Optical Nonlinearities Approaching the Fundamental Limit

Lytel

Kuzyk

2013

J. Nonlinear Optic. Phys. Mat.

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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.

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Influence of geometry and topology of quantum graphs on their nonlinear optical properties

Cited by 25 publications

References 49 publications

Phase disruption as a new design paradigm for optimizing the nonlinear-optical response

Phase disruption as a new design paradigm for optimizing the nonlinear-optical response

Hybrid quantum systems for enhanced nonlinear optical susceptibilities

Dressed Quantum Graphs With Optical Nonlinearities Approaching the Fundamental Limit

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