Plasmon–exciton coupling in a dimer cavity revisited: effect of excitonic dipole orientation

We revisit plasmon–exciton coupling of a single emitter in a dimer cavity, featuring the analysis of how the excitonic dipole orientation influences the coupling behaviour from both the spectral and temporal aspects. Results demonstrate that the dipolar mode could be suppressed to vanish while the magnitude of the pseudomode could only be suppressed to half of the maximum value. The temporal analysis gives further evidence of this effect on the dipolar mode and pseudomode. The analysis might have potential significance on the experimental community as the excitonic dipole orientation could be precisely measured and has a rather important impact on the experiments.

P lasmon-exciton coupling is a fundamental process in quantum optics. [1][2][3] It also plays an important role in an emerging field named polaritonic chemistry. [4][5][6] Plexciton, 7) equivalently the plasmon-exciton-Polariton (PEP), 8) the hybrid state formed by plasmon-exciton strong coupling, with profound significance in nanoscale science has been a physical phenomenon extensively investigated in past decades. 9) Well-developed theoretical descriptions can be generally categorized based on how (classical or quantum) light and matter are treated although the feasibility of some quantum concepts utilized in PEP strong coupling is still worth further clarifying. 10) Meanwhile, transformation optics (TO), 11) as a convenient auxiliary tool to simplify the cavity geometry and reduce the computation scale, 12) has merged into plasmonics 13) and quantum optics 14) and helped establish quasi-analytical descriptions on the strong coupling of a single quantum emitter (QE) in a dimer cavity. [15][16][17] On the other side, quantum effects arise at short distances (around and below 10 nm) between QEs and cavity surfaces. Recently established treatment has incorporated these nonclassical effects within the framework of mesoscopic electrodynamics. 18,19) Experimentally, due to advances in nanoscale fabrication and characterization, plasmon-exciton strong coupling has been achieved from emitter ensembles up to single emitter level. Notably, Baumberg et al. have implemented plasmon-exciton strong coupling at the single emitter level 20) and later obtained multilayers of WSe 2 strongly coupled to plasmonic modes. 21) If we look into the detailed analysis of the above two experiments, both considered the orientation of excitons. Reference 20 exhibits the obvious differences in scattering spectra of single NPoM (Nano-Particle-on-Mirror) cavities inside which the transition dipole moments (TDM) are vertically and horizontally orientated. Strong coupling is observed only in the case of vertically orientated TDM. Reference 21 points out that the TDM should be perfectly aligned with the gap plasmon. These notions indicate that the orientation of the excitonic dipole is a crucial factor importantly influencing the experiments. On the other side, there have already been effective methods to predict TDM orientation experimentally. References 22 and 23 are two representative works among the huge literature on the TDM measurement.
In this paper, we aim at exploring from the theoretical aspect the effect of TDM orientation on plasmon-exciton coupling in a dimer cavity, a prototype of many other nanocavities (e.g. setting the diameter of one particle to infinity switches the cavity to the NPoM cavity). Based on the transformation optics approach, the effect of TDM orientation on the Purcell enhancement and the spectral density of the emitter-cavity system is revealed. Temporal property is observed by calculating population dynamics which straightforwardly determine whether the plasmonexciton coupling is in the weak or strong coupling regime. As an outlook, we expect transplanting the methodology and analysis to the microwave regime where successful experiments have already been implemented to excite surface cavity modes, 24) which would be a suitable platform for the experimental verification of the current study.
The sketch in Fig. 1(a) depicts the system. The dimer cavity is composed of two metallic spherical nanoparticles. e , 1 e 2 and e D denotes the permittivity of the two nanoparticles and the background medium, respectively. The geometry of the dimer is rotationally-symmetric with respect to the z-axis, with R 1 and R 2 denoting the radii of the nanospheres. The single QE is located in the gap of the dimer on the z-axis. The orientation of the QE's TDM p is characterized by angle q E and j E in spherical coordinates as denoted in Fig. 1(c). The gap size of the dimer is d. The origin of the coordinate system is set at the centre of the lower sphere. As mentioned above, the TO approach helps acquire concentric spherical geometry. 12) We skip the algebraic derivation of the geometric parameters. For this part, we suggest readers to refer Refs. 16 and 17 and the results there will be directly used in this article for the avoidance of repetition. We only write down the inversion coordinate mapping The general methodology is a celebrated matrix formulation linking the source and the scattered field. 12) In the quasistatic approximation, the electric field could be described by the scalar potential. 25) In the transformed space [ Fig. 1(b)], the scalar potential F¢ could be elegantly expanded with spherical harmonics In the original space, the TDM-induced potential is [26]. Applying the coordinate mapping leads to the source (a point dipole) potential distribution in the transformed space: To facilitate the numerical calculations, we further express ¢ R E and ¢ R 0 with geometric parameters only in the original space: Through the inverse coordinate mapping, the scattered field in the original space is retrieved as ( ( )) ( ) F ¢ = F¢ ¢ r r r .

S S
The total scalar potential can be denoted as ( ) ( ) ( ) F = F + F r r r.

S E
The spectral density ( ) w J is the quantity describing the spectrum of QE-cavity coupling strength: . 2 In this definition, ( ) r w is the density of modes of the system while ( ) w g denotes the coupling strength. In macroscopic quantum electrodynamics, ( ) w J is formally expressed with the dyadic Green function  Here we have set relative permeability to unity as the system works in the optical regime. Substituting the electric field with the scalar potential, the expression is further connected directly to the potential field evaluated at the source point: In the numerical calculations, the diameter of the nanospheres are set to = R R 1 2 = 20 nm. The background is set to vacuum. The gap between the nanospheres is set to d = 1 nm. The TDM is set to | | p = 1.5 · e nm. It should be noted that we set j = 0 E in the following for simplicity. The contour plot in Fig. 2(a) explicitly exhibits the relation between the TDM orientation and the spectral density. It can be observed that the amplitude of the lowest mode (in the dashed black rectangle) is suppressed as the orientation of TDM switches from vertical (q = 0 E ) to horizontal (q p = 2 E ) direction. It has been demonstrated with an electric field map that the lowest mode is the electric dipolar mode [see Fig. 1  It is also obvious that the so-called pseudomode, which is intrinsically the superposition of spectrally overlapping high-order modes, decreases in amplitude through this process. To observe the comparison of different TDM orientations more clearly, three cases are plotted in Fig. 3(b). The amplitude of the dipolar mode vanishes and the pseudomode decreases to half the magnitude of the vertical case when the TDM is horizontally orientated.
From the other aspect, we could get a more intuitive view from the temporal dynamics. The Wigner-Weiskopf theory links the spectral information to the temporal behaviour of the emitter-cavity system. 28) The population dynamics of the emitter-cavity system could be described well with a wellknown integro-differential equation. For a more obvious comparison without loss of generality, the gap is set to d = 2 nm. Firstly, we set the natural frequency of the QE to be close to the resonance frequency of the dipolar mode. We can observe from Fig. 3(a) that for q = 0, E strong reversible dynamics occurs which indicates the intense energy exchange between the QE and the plasmonic cavity modes in a timescale much smaller than the decaying lifetime of the QE. However, with q E increases, the reversible dynamics becomes much weaker and the oscillation frequency also decreases. However, when the natural frequency of the QE is set to be close to the resonance frequency of the pseudomode, the reversible dynamics is so strong that the system enters the strong coupling regime. The decrease of the magnitude of the pseudomode only leads to a small decrease in the Rabi oscillation frequency. The difference between Figs. 3(a) and 3(b) shows the different effect the TDM orientation has on different modes. For the dipolar mode, switching the TDM orientation could let the mode magnitude decrease and finally vanish. For the pseudomode, the magnitude could only decrease to half the value of the vertical case.  In conclusion, based on the inversion coordinate mapping, we investigated how the TDM orientation of the QE influences the plasmon-exciton coupling in a dimer cavity. It has been observed that the dipolar mode could be suppressed to vanish when the TDM switches from the vertical to the horizontal direction. But the pseudomode could only be suppressed to half of the magnitude of the vertical case. This gives rise to the different control of the population dynamics when the emitter frequency is close to the resonance frequencies of the dipolar and the pseudomode. Further investigations would possibly be to give analytical analysis which could give a deeper physical insight into the evolution process of both the dipolar mode and the pseudomode when the TDM orientation varies. ) when its natural frequency is (a) close to the resonance frequency of the dipolar mode; (b) close to the resonance frequency of the pseudomode. Obvious difference could be observed, which results from the orientation having different effects on the dipolar and pseudomode.