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Self-hybridization within non-Hermitian localized surface plasmons

Electron energy loss spectroscopy reveals the possibility for two eigenmodes from the same nanoparticle to hybridize – a physical effect that cannot be observed in day-to-day (Hermitian) linear physics.

In any situation described by a Hermitian equation, the usual approach in linear physics is to apply the concept of eigenmodes. Examples are endless : the vibrations of a guitar string are best understood as a superposition of the string eigenmodes and the properties of an atom can be simply deduced from its orbitals’ properties. It is thus tempting to adapt this concept to systems in which eigenmodes are harder to define, namely for non-Hermitian systems. Non-Hermitian systems span a wide range of physical situations, such as gravity waves close to black holes, lasers cavities, or propagating surface plasmons. In those cases, quasi-normal modes (QNMs) are specially constructed so that time-reversal symmetry breaking (in other words, energy dissipation) does not prevent the establishment of a complete basis, especially when parity–time symmetry is preserved. QNMs are described by a bi-orthogonal rather than orthogonal basis. Bi-orthogonality has a few famous and exciting consequences, including the existence of ‘exceptional points’ where both the energy and wavefunctions coalesce. Exceptional points are usually associated with the apparition of non-trivial physical effects, such as asymmetric mode switching. Such effects have only very recently been studied experimentally because manipulating QNMs in open systems requires an exact balance of dissipation.

Left : Energy variation of two modes of silver nano-daggers as a function of the vertical arm length L. A clear anti-crossing is revealed. Right : EELS maps of the modes much before the crossing point (σ2 and σ3) and close to it (σ-and σ+). The strong hybridization can be clearly observed.

We develop a totally different approach where non-Hermiticity is not related to time-invariance breaking but to special spatial symmetry breakings, therefore avoiding the burden of compensating dissipation. With this aim, we introduce localized surface plasmons (LSP) as a new platform to investigate non-Hermitian physics. LSPs are resonant electronic excitations at the surface of a metallic nano-object. Their energy, which appears for noble metals mostly in the visible range, and spatial variations are highly dependent on the size and shape of their corresponding nano-objects. LSPs have many applications, from sensing to cancer therapy, most of them related to the fact that they can concentrate the electromagnetic energy at the nanoscale in regions called "hot spots". We demonstrate theoretically and experimentally that the manifestation of the non-Hermiticity is much simpler to observe and manipulate in these systems. As a clear demonstration, we introduce the concept of self-hybridization, a counter-intuitive phenomenon that cannot be observed in regular Hermitian systems. Imagine the s and p orbitals of the same atom that could hybridize without external field or symmetry breaking, or a guitar string on which a fundamental vibration and its harmonics would couple – it simply does not make any sense. This is however what we predict theoretically and observe by fast electron beam spectroscopy for harmonic plasmon modes in silver nano-daggers (see figure). The nano-daggers consist in a 400 nm long rod, with an off-center perpendicular rod of different length. We used electron energy loss spectroscopy (EELS) in a scanning transmission electron microscope (STEM) to map the spatial and spectral variations of the plasmonic signal. As shown in the figure, as the arm length increases, two of the nano-daggers modes exhibit a typical anti-crossing behavior. The anti-crossing is a strong indication of hybridization of the two modes, further evidenced by the drastic change of symmetry of the modes before and at the crossing point (see figure, right). This behaviour is confirmed by extensive simulations and analytical approaches.

The concepts we bring and their theoretical and experimental demonstration are quite novel. Indeed, this is a rare demonstration of a physical effect driven by a real (non-complex) non-Hermiticity. Also, it is worth noting the impressive developments in the design and synthesis or fabrication of plasmonic nano objects, where virtually any shape, size and composition of nanoparticles can be created. Therefore, since the burden of dissipation compensation is lifted with LSPs, they are a new and much easier playground for investigating non-Hermiticity experimentally, and therefore this should impact a wide community of physicists. Finally, a consequence of our finding is the demonstration of a robust way of manipulating and observing strong coupling physics in plasmonics systems. One of the main interests of plasmons are their associated hot spots. The self-hybridization offers the possibility of designing new types of hot spots. Beyond the physicists, the whole transdisciplinary field of surface plasmons may be impacted.


Self-hybridization within non-Hermitian localized plasmonic systems
H. Lourenço-Martins, P. Das, R. Weil, L. H. G. Tizei and M. Kociak
Nature Physics (2018)


Mathieu Kociak