Skip to main content

Self-hybridization within non-Hermitian localized plasmonic systems

Figure 1 : Left : Energy variation of two modes of silver nano-daggers as a function of the vertical arm length L. A clear anti-

In a recent paper published in Nature Physics, we show that electron energy loss spectroscopy in a scanning transmission electron microscope 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 (such as mechanics, acoustics, quantum mechanics and electromagnetism), 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. One class of non-Hermitian systems consists of open systems that span a wide range of physical situations, from gravity waves close to black holes to lasers cavities or propagating surface plasmons. In those cases, quasi-normal modes (QNMs) are specially constructed so that time-reversal symmetry breaking (in other word, the presence of 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 to exactly balance dissipation. 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 for compensating dissipation. In 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 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 nanocrosses. To this aim, we have produced a series of silver nano-crosses by electron beam lithography with shifted arms ("nano-daggers") [see figure]. The series consists in first rod of 400 nm in length, and a perpendicular one with different lengths. 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 one of the arms' length increases, 2 of the nanocrosses' 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). This behaviour are 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.