Long-range surface-plasmon-polariton excitation at the quantum level

We provide the quantum-mechanical description of the excitation of long-range surface-plasmon polaritons (LRSPPs) on thin metallic strips. The excitation process consists of an attenuated-reflection setup, where efficient photon-to-LRSPP wave-packet transfer is shown to be achievable. For calculating the coupling, we derive the first quantization of LRSPPs in the polaritonic regime. We study quantum statistics during propagation and characterize the performance of photon-to-LRSPP quantum state transfer for single-photons, photon-number states, and photonic coherent superposition states.

Figure 1

(Color online) Single-photon excitation of LRSPPs using attenuated reflection. (a) A photon wave packet is injected into the system at a specific angle $\theta$, with a prism mediating an interaction between the photon and LRSPP modes. The minimum prism size is diffraction limited. (b) The ATR excitation geometry considered. (c) Transfer process for the photon and LRSPP mode operators.

Figure 2

(Color online) (a) Dispersion relations for the ${\omega }^{\pm }$ excitations as a function of $k$ and metal thickness $d_1$. Inset shows the curves ${\omega }^{+}\left(k\right)$ (dashed) and ${\omega }^{-}\left(k\right)$ (solid) corresponding to $d_1=20\text{nm}$ and the same curves merged at $d=100\text{nm}$ (dashed-dotted). (b) Coupling angle $\theta$ as a function of frequency and metal thickness $d_1$. Inset shows the coupling angle $\theta$ for ${\omega }^{+}\left(k\right)$ (dashed) and ${\omega }^{-}\left(k\right)$ (solid) at $d_1=20\text{nm}$ and the same curves merged at $d=100\text{nm}$ (dashed-dotted).

Figure 3

(Color online) Values of the parameters $B^{\pm }$, $P^{\pm }$, and $C^{\pm }$ corresponding to the optimized coupling $g^{\pm }\left(\omega \right)$ over the separation $d_2$.

Figure 4

(Color online) Optimization of the coupling over the prism separation $d_2$, subject to the constraints $B^{\pm }\ge 1$, $P^{\pm }\le 1$, and $C^{\pm }\ge 1$. (a) ${\omega }^{+}$ excitation; (b) ${\omega }^{-}$ excitation. Panels (c) and (d) show the values of coupling parameter $\left|g^{\pm }\right|$ (dashed line), metal thickness $d_1$ (solid), and prism separation $d_2$ (dot-dashed), after numerical optimization. (c) ${\omega }^{+}$ excitation; (d) ${\omega }^{-}$ excitation. For $d_1$, by increasing the resolution of the numerical calculation, one finds the surface plots in (a) and (b) become smoother and the points in (c) and (d) tend toward the best fit line.

Figure 5

(Color online) Phenomenological model to include the effect of damping for the LRSPP propagation [25]. (a) Array of beam splitters and bath of field excitations. [(b) and (c)] Normalized mean excitation number $⟨\tilde{m}⟩=⟨m⟩/n$ as a function of the frequency and the distance traveled from the injection point. The optimal profiles obtained in Fig. 4 are used for the (b) ${\omega }^{+}$ excitation and (c) ${\omega }^{-}$ excitation (c). A detector efficiency of $\mu =0.65$ is used [8].

Figure 6

(Color online) The entropy $S_V$ as a function of frequency and distance traveled for fixed values of cat-state amplitude $\alpha$. (a) and (c) correspond to the ${\omega }^{+}$ LRSPP excitations with $\alpha =2$ and 5, respectively. (b) and (d) correspond to the ${\omega }^{-}$ excitations with $\alpha =2$ and 5, respectively.