Quantum theory of plasmon polaritons in chains of metallic nanoparticles: From near- to far-field coupling regime

We develop a quantum theory of plasmon polaritons in chains of metallic nanoparticles, describing both near- and far-field interparticle distances, by including plasmon–photon umklapp processes. Taking into account the retardation effects of the long-range dipole–dipole interaction between the nanoparticles, which are induced by the coupling of the plasmonic degrees of freedom to the photonic continuum, we reveal the polaritonic nature of the normal modes of the system. We compute the dispersion relation and radiative linewidth, as well as the group velocities of the eigenmodes, and compare our numerical results to classical electrodynamic calculations within the point-dipole approximation. Interestingly, the group velocities of the polaritonic excitations present an almost periodic sign change and are found to be highly tunable by modifying the spacing between the nanoparticles. We show that, away from the intersection of the plasmonic eigenfrequencies with the free photon dispersion, an analytical perturbative treatment of the light-matter interaction is in excellent agreement with our fully retarded numerical calculations. We further study quantitatively the hybridization of light and matter excitations through an analysis of Hopfield's coefficients. Finally, we consider the limit of infinitely spaced nanoparticles and discuss some recent results on single nanoparticles that can be found in the literature.

Figure 1

Sketch of a linear chain of identical spherical metallic nanoparticles of radius $a$, separated by a center-to-center distance $d$, arranged along the $z$ direction.

Figure 2

(a), (b) Band structure and (c), (d) radiative decay rates in units of the bare frequency ${\omega }_0$ and as a function of the reduced wave number $qd$ in half of the Brillouin zone. The left (right) panels represent longitudinal (transverse) polarizations. The red curves correspond to the results obtained within our fully retarded quantum formalism while the green ones are the results obtained within the classical model presented in Appendix pp1. The blue dashed lines represent the second-order perturbation theory and Fermi's golden rule analytical results. In the upper panels, we also display the quasistatic band structure Eq. (21) by solid black lines and the first ($l=0$) light line, $c\left|q\right|$, as a dotted black line. The center-to-center distance is $d=3a$ and $k_{0}a=0.3$.

Figure 3

Same quantities as shown in Fig. 2, except for the black dotted line in (a) and (b), which is now the light line corresponding to the photonic band index $l=1$, i.e., $c|q-2\pi /d|$. In the figure, $k_{0}a=0.3$ and $d=13a$.

Figure 4

Same quantities as shown in Figs. 2 and 3, except for the black dotted line in panels (a) and (b), which is now the light line corresponding to the photonic band index $l=-1$, i.e., $c|q+2\pi /d|$. In the figure, $k_{0}a=0.3$ and $d=23a$.

Figure 5

Group velocities $v_q^{\sigma }$ in units of the speed of light in vacuum $c$, for the (a) longitudinal and (b) transverse polarizations as a function of the reduced wave number $qd$ in half of the first Brillouin zone. The parameters used in the figure are the same as in Fig. 2. The inset shows the longitudinal group velocity in units of $c$ corresponding to the reduced wave number $qd=\pi /2$ as a function of the ratio $d/a$, computed using the analytical perturbative results.

Figure 6

Modulus squared of the plasmonic Hopfield coefficients $W_q^{\sigma }$ (orange solid line) and sum of all the moduli squared of the photonic ones $Y_{\mathit{\kappa }q}^{l,\sigma ,{\widehat{\lambda }}_{\mathit{\kappa }q}^l}$ (purple solid line), as a function of the reduced wave number $qd$ in half of the Brillouin zone. The vertical dashed line represents the mode for which the polaritonic dispersion crosses a light line. The upper and lower panels show, respectively, the longitudinal and transverse polarizations, while the three columns are associated, respectively, to center-to-center distances $d=3a, d=13a$, and $d=23a$. In the figure, $k_{0}a=0.3$.

Figure 7

Single nanoparticle radiative shift ${\delta }_0$ (upper panel) and radiative decay rate ${\gamma }_0$ (lower panel) in units of the single nanoparticle resonance frequency ${\omega }_0$ and as a function of the reduced nanoparticle radius $k_{0}a$. The green solid line represents the result obtained from the classical model, as presented in Appendix pp1, while the thick grey solid line (almost perfectly overlapped by both the fully retarded quantum results in red and the perturbative results in dashed blue) represents the single nanoparticle results of Eqs. (33) and (34). In the upper panel, the orange line represents the result for the radiative shift presented in Ref. [46], and the grey dotted line is a guide for the eye.

Figure 8

Radiative frequency shifts in units of the bare frequency ${\omega }_0$ as a function of the reduced nanoparticle radius $k_{0}a$. The same approach as for the chain and two-dimensional systems has been used, i.e., without making use of Bethe's mass renormalization. The grey dotted line is a guide for the eye. The interparticle distance is $d=3a$.

Figure 9

Ratio $\mathrm{\Delta }{\tilde{\omega }}^z/\mathrm{\Delta }{\tilde{\omega }}^{x,y}$ of longitudinal and transverse frequency splittings in a nanoparticle dimer as a function of the (reduced) interparticle distance $k_{0}d$, for increasing (reduced) nanoparticle radius $k_{0}a$, from 0.1 (green line), 0.15 (yellow line), to 0.2 (red line). The grey dotted line shows the bare ratio $\mathrm{\Delta }{\omega }^z/\mathrm{\Delta }{\omega }^{x,y}$ in the absence of the coupling with the photonic environment. The UV frequency cutoff is chosen as ${\omega }_{\mathrm{c}}=c/a$.