Abstract
Intense efforts have been made in recent years to realize nonlinear optical interactions at the single-photon level. Much of this work has focused on achieving strong third-order nonlinearities, such as by using single atoms or other quantum emitters, while the possibility of achieving strong second-order nonlinearities remains unexplored. Here, we describe a novel technique to realize such nonlinearities using graphene, exploiting the strong per-photon fields associated with tightly confined graphene plasmons in combination with spatially nonlocal nonlinear optical interactions. We show that in properly designed graphene nanostructures, these conditions enable extremely strong internal down-conversion between a single quantized plasmon and an entangled plasmon pair, or the reverse process of second harmonic generation. A separate issue is how such strong internal nonlinearities can be observed, given the nominally weak coupling between these plasmon resonances and free-space radiative fields. On one hand, by using the collective coupling to radiation of nanostructure arrays, we show that the internal nonlinearities can manifest themselves as efficient frequency conversion of radiative fields at extremely low input powers. On the other hand, the development of techniques to efficiently couple to single nanostructures would allow these nonlinear processes to occur at the level of single input photons.
Highlights
Second-order nonlinear conductivity of grapheneGraphene has attracted tremendous interest due to its ability to support tightly confined, electrostatically tunable surface plasmons (SPs) [17,18,19,20,21,22,23,24]
We first study the implications of such nonlinearities in a finite-size nanostructure, obtaining a general scaling law for the nonlinearity as a function of the linear dimension of the structure and the doping
We show that efficient coupling would enable second harmonic generation (SHG) or down conversion (DC) with inputs at the single-photon level, and predict a set of experimental signatures in the output fields that would verify that strong quantum nonlinear interactions are occurring between graphene plasmons
Summary
Graphene has attracted tremendous interest due to its ability to support tightly confined, electrostatically tunable SPs [17,18,19,20,21,22,23,24]. Solving in the Fourier domain, the perturbed distribution function f (1) can be inserted into equation (2) to find the resulting current This yields a linear relation between the current and field, J(k, ω) = σ(1)(ω)E(k, ω), where the proportionality constant is the familiar Drude conductivity [18, 19]. If the nonlinear response is spatially local, J(2)(2ω, r) = σ(2)(ω)E (ω, r), spatial inversion symmetry implies that −J = σ(2)(−E), which enforces that σ(2) = 0 This argument breaks down if the conductivity is nonlocal [33], for example if σ (ω, q) ∝ q, such that the current depends on the electric field gradient, J(2) = σ(2)(ω)E∂rE. Below we show that k knl ∼ qp kF ≲ 1 emerges as the relevant quantity to characterize the strength of nonlocal nonlinearities in graphene After these considerations, we calculate the second-order conductivity using the procedure explained above. This result can be converted into a relation between the electrostatic potential and the induced charge, which reproduces previously obtained results for the nonlinear polarizability [27]
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