Quantum-Coherence-Enhanced Surface Plasmon Amplification by Stimulated Emission of Radiation

We investigate surface plasmon amplification in a silver nanoparticle coupled to an externally driven three-level gain medium and show that quantum coherence significantly enhances the generation of surface plasmons. Surface plasmon amplification by stimulated emission of radiation is achieved in the absence of population inversion on the spasing transition, which reduces the pump requirements. The coherent drive allows us to control the dynamics and holds promise for quantum control of nanoplasmonic devices.

Figure 1

Schematic of the coherence-enhanced spaser. (a) A silver nanosphere surrounded by three-level quantum emitters such as atoms, molecules, rare earth ions, or semiconductor quantum dots placed in the near field of a dipolar plasmon mode. (b) The three-level gain medium is excited by an external incoherent pump $g$ to the upper state $|3⟩$ which decays to states $|1⟩$ and $|2⟩$. The $|2⟩\to |1⟩$ transition is nearly resonant with the plasmon mode of the silver nanosphere such that the state $|2⟩$ decays by emitting SPs via energy transfer. The plasmonic oscillations of the nanosphere stimulate this emission, supplying coherent feedback for the spaser. The physical mechanisms involved in the spasing process are analogous to electromagnetically induced transparency and coherent population trapping: an external coherent drive ${\Omega }_a$ is applied to the transition $|2⟩\to |3⟩$ creating an asymmetry between absorption and stimulated emission on the spasing transition $|2⟩\to |1⟩$, enabling spasing without population inversion on the spasing transition.

Figure 2

Steady-state properties of the silver nanosphere spaser with $\hslash {\omega }_n=2.5\mathrm{eV}$ as a function of the incoherent pumping rate $g$ for three driving fields: ${\Omega }_a=0$ (red line), $4\times {10}^{12}{\mathrm{s}}^{-1}$ (blue line), and $16\times {10}^{12}{\mathrm{s}}^{-1}$ (black line). The dephasing rate ${\gamma }_{\mathrm{ph}}=0$. Note that the effect of decoherence is studied in a separate plot in Fig. 3. (a) Number $N_n$ of plasmons per spasing mode. Interestingly, ${\Omega }_a$ not only controls the spaser threshold but it also controls the number of generated plasmons. (b) Population inversion ($n_{21}$) on the spasing transition $|2⟩\leftrightarrow |1⟩$. Coherent drive allows obtaining a large enhancement of a number of plasmons in the absence of population inversion at the spasing frequency and reduces the spasing threshold.

Figure 3

Number of LSPs $N_n$ versus the incoherent pumping rate $g$ for different dephasing rates ${\gamma }_{\mathrm{ph}}=0$, 80, 160, and $240\times {10}^{12}{\mathrm{s}}^{-1}$ for ${\Omega }_a=16\times {10}^{12}{\mathrm{s}}^{-1}$. Steady-state coherence-enhanced spasing is robust against decoherence and may be observed even for large dephasing rates.

Figure 4

(a) Ratio of the imaginary part of the Floquet exponent ${\gamma }_s$ normalized by the SP linewidth ${\gamma }_n$ for three values of the pumping rate $g=4.4$, 6, and $8\times {10}^{12}{\mathrm{s}}^{-1}$ for ${\gamma }_{\mathrm{ph}}=0$. ${\gamma }_s$ corresponds to the growth rate of the plasmonic field and may be increased by the coherent drive. (b) Temporal profiles of the plasmonic field intensity for ${\Omega }_a=0$ and $24\times {10}^{12}{\mathrm{s}}^{-1}$ show that the steady state is achieved much faster for a stronger drive. This corresponds to a larger value of ${\gamma }_s$ which can be controlled by the drive as shown in (a).