Theoretical study of plasmonic lasing in junctions with many molecules

We calculate the quantum state of the plasmon field excited by an ensemble of molecular emitters, which are driven by exchange of electrons with metallic nanoparticle electrodes. Assuming identical emitters that are coupled collectively to the plasmon mode but are otherwise subject to independent relaxation channels, we show that symmetry constraints on the total system density matrix imply a drastic reduction in the numerical complexity. For $N_{\text{m}}$ three-level molecules, we may thus represent the density matrix by a number of terms scaling as $(N_{\mathrm{m}}+8)!/(8!N_{\mathrm{m}}!)$ instead of $9^{N_{\text{m}}}$, and this allows exact simulations of up to $N_{\text{m}}=10$ molecules. Our simulations demonstrate that many emitters compensate strong plasmon damping and lead to the population of high plasmon number states as well as a narrowed emission, which indicates a transition from spontaneous emission to amplified spontaneous emission. For large $N_{\text{m}}$, our exact results are reproduced by an approximate approach based on the plasmon reduced density matrix. With this approach, we have extended the simulations to more than 100 molecules and shown that the plasmon number state population follows a Poisson-like distribution, which indicates the formation of a plasmon coherent state. This clearly indicates a transition from amplified spontaneous emission to lasing. An alternative approach based on nonlinear rate equations for the molecular state populations and the mean plasmon number also reproduce the main lasing characteristics of the system.

Figure 1

The physical system consists of molecules sandwiched between a spherical left lead supporting plasmon excitations and a right hollow lead, cf. (a). Schematic (b): the light (dark) gray areas indicate unoccupied (occupied) electron states of the electrodes; the boundaries of the areas are the Fermi energies; the dashed line indicates the Fermi energies at zero-applied voltage. Energy conservation, cf. Eq. (8), requires: for charging process (i), the energy of a neutral molecule (ground state) plus that of electrons of the left lead $E_{ng}+{\varepsilon }_{L\mathbf{k}}$ is identical to the energy of a charged molecule $E_{nf}$; for discharge processes (ii) and (iii), the energy $E_{nf}$ is equal to that of a neutral molecule (excited or ground state) plus that of the unoccupied electron states of the right lead $E_{ne}+{\varepsilon }_{R\mathbf{k}}$ or $E_{ng}+{\varepsilon }_{R\mathbf{k}}$ (shown separately for clarity); process (iv) shows energy exchange with the left lead plasmon.

Figure 2

Transition operators $\left|\alpha \right.〉\left.〈\beta \right|$ based on molecular product states $\left|\alpha \right.〉,\left|\beta \right.〉$ are mapped onto vectors $\mathbf{n}$. (a) Expansion of the molecular product states. (b) Possible combinations $(a_i,b_i)$ of quantum numbers related to individual molecules. (c) Some transition operators for seven identical molecules as one example; the group of molecules 1, 3, 6 is associated with the combination $(g,e)$, the group of molecules 2, 4 with $(f,f)$, the group of molecules 5, 7 with $(e,g)$; the transition operators formed by exchanging the molecules in one group are identical; the set of those transition operators can by classified with the number of nine combinations in (b) giving us a symmetric operator.

Figure 3

Steady-state properties of junctions with $N_{\text{m}}=1,\cdots ,10$ molecules. (Top) Population $P_{\mu }$ of excited plasmon states $\left|\mu \right.〉$ for different $N_{\text{m}}$. (Middle) Properties of emission spectra for different $N_{\text{m}}$: blue curve with squares (left ordinate axis), emission linewidth (FWHM); red curve with triangles (right ordinate axis), emission maximum. (Bottom) Population of molecular states, red upper curve $P_e$, black middle curve $P_g$, and green lower curve $P_f$; blue curve (right ordinate axis) current through the junctions. Other parameters according to Table 1.

Figure 4

Steady-state properties of junctions with many molecules. (Top) Plasmon state population for junctions with $N_{\text{m}}=10,20,40,60,80,100$ molecules. (Bottom) Blue solid curve (left ordinate axis), plasmon mean number ${\mathcal{N}}_{\text{pl}}$; red upper solid curve, current $I=-I_L=I_R$ through the junctions; red middle dotted curve, current contribution proportional to $N_{\text{m}}$, cf. Eq. (25); red lower dashed curve, current contribution proportional to ${\mathcal{N}}_{\text{pl}}$, cf. Eq. (25). Other parameters according to Table 1.

Figure 5

Second-order equal-time intensity correlation function $g_{\mathrm{pl}}^{\left(2\right)}\left(0\right)$ as a function of the charging rate $\hslash k_{g\to f}^{\left(n\right)}=\hslash {\mathrm{\Gamma }}_{Lgf}^{\left(n\right)}$ and discharging rate $\hslash k_{f\to e}^{\left(n\right)}=\hslash {\mathrm{\Gamma }}_{Rfe}^{\left(n\right)}$ for different values of the number of molecules $N_{\text{m}}$. Other parameters are given in Table 1.

Figure 6

Plasmon mean number calculated with different approaches for junctions with different numbers of molecules $N_{\text{m}}$. Black stars: calculations based on symmetric reduced density matrix equation (13). Green circles: calculations based on recursion relation, cf. Eq. (24). Blue solid line: calculations based on rate Eqs. (26), (27), (28), and (30). Other parameters according to Table 1.