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Physical Review Letters

Observation of a Microscopic Cascaded Contribution to the Fifth-Order Nonlinear Susceptibility

Ksenia Dolgaleva, Heedeuk Shin, Robert W. Boyd

Published Sep 9, 2009• 49 citations

Abstract

Typically, low-order nonlinearities are much stronger than high-order nonlinearities. In this Letter, we demonstrate a procedure by which strong high-order nonlinearities can be synthesized out of low-order nonlinearities. Our procedure involves the use of the previously largely overlooked process of microscopic cascading, which results from local-field effects. We have performed an experiment that allows us to distinguish the influence of microscopic cascading from the more-well-known process of macroscopic cascading, and we find conditions under which microscopic cascading can be the dominant effect. The ability to create a large high-order nonlinear response could prove useful for applications in quantum-information science that require the detection of the simultaneous presence of N entangled photons.

DOI: 10.1103/physrevlett.103.113902

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Typically, low-order nonlinearities are much stronger than high-order nonlinearities. In this Letter, we demonstrate a procedure by which strong high-order nonlinearities can be synthesized out of low-order nonlinearities. Our procedure involves the use of the previously largely overlooked process of microscopic cascading, which results from local-field effects. We have performed an experiment that allows us to distinguish the influence of microscopic cascading from the more-well-known process of macroscopic cascading, and we find conditions under which microscopic cascading can be the dominant effect. The ability to create a large high-order nonlinear response could prove useful for applications in quantum-information science that require the detection of the simultaneous presence of N entangled photons.

Contributions to the fifth-order nonlinear optical susceptibility ${\ensuremath{\chi}}^{(5)}$ of a collection of homogeneously broadened two-level atoms that scale as ${N}^{2}{({\ensuremath{\gamma}}_{\mathrm{at}}^{(3)})}^{2}$ and ${N}^{2}{\ensuremath{\mid}{\ensuremath{\gamma}}_{\mathrm{at}}^{(3)}\ensuremath{\mid}}^{2}$, where ${\ensuremath{\gamma}}_{\mathrm{at}}^{(3)}$ is the lower-order atomic hyperpolarizability and

N

is the atomic number density, are predicted theoretically. These ``cascaded'' contributions are a consequence of local-field effects. We determine them from a fifth-order solution of the Lorentz-Maxwell-Bloch equations. They are missing from a straightforward generalization of Bloembergen's result for the local field correction to the second order nonlinearity, but are recovered by a careful application of his general approach. We find that at high atomic densities $(N>{10}^{15}\phantom{\rule{0.3em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}3})$ the value of the cascaded third-order contribution can be as large as the ``direct'' fifth-order term in the expression for the fifth-order susceptibility.

The nonlinear optical response of a system of molecules often contains contributions whereby the products of lower-order processes in two separate molecules give signals that appear on top of a genuine direct higher-order process with a single molecule. These many-body contributions are known as cascading and complicate the interpretation of multidimensional stimulated Raman and other nonlinear signals. In a quantum electrodynamic treatment, these cascading processes arise from second-order expansion in the molecular coupling to vacuum modes of the radiation field, i.e., single-photon exchange between molecules, which also gives rise to other collective effects. We predict the relative phase of the direct and cascading nonlinear signals and its dependence on the microscopic dynamics as well as the sample geometry. This phase may be used to identify experimental conditions for distinguishing the direct and cascading signals by their phase. Higher-order cascading processes involving the exchange of several photons between more than two molecules are discussed.

Tunneling-induced large fifth-order nonlinearity is theoretically demonstrated in a double-quantum-dot system. The resonant tunneling induces constructive interference for the third- and fifth-order nonlinear effects. The competition between the linearity and nonlinearity leads to a transparency window at some frequency detunings, where the fifth-order nonlinear refractive index could be increased to be more than one order of magnitude larger than that on resonance. An analytical expression shows that the resonant tunneling mainly contributes to the dramatic enhancement of the fifth-order nonlinear response.