Exciton pairs coupled via the long-living phonons and their superfluorescent markers

Abstract

A system of several Wannier-Mott excitons interacting with phonons in a bulk material is considered. We show that strong exciton-phonon coupling causes the formation of a coherent two-exciton state -- the exciton pair. Unlike the biexcitons, where the photons play the role of force carrier, the exciton pair is formed via entanglement with the long-living phonon mode: (i) The essentially multi-particle theory requires excitons (cobosons composed of an electron and a hole) to satisfy the mixed Bose-Fermi statistics; (ii) This allows us to formulate a system of non-linear dynamic equations, using the multiconfiguration Hartree method applied to the Frohlich Hamiltonian. The system of equations possesses a stationary solution, which, for the case of a single exciton, describes the excitonic polaron and corresponds to the exciton pair in the two-exciton case. We also compare the fluorescent spectra of exciton polarons and exciton pairs estimated from our theory with those observed in experiments on room-temperature superfluorescence (collective emission of fluorescent light) in hybrid perovskites to give an additional insight into the superfluorescence phenomenon.

Figures (4)

• Figure 1: The Fröhlich exciton-phonon interaction amplitude in thin films plotted vs. the dimensionless wavevector $\mu_e a_0k$, where $a_0$ is the exciton Bohr radius, and $\mu_e$ is the ratio of the electron effective mass to the exciton reduced mass, $\mu_e=m_h^*/(m_e^*+m_h^*)$. The position of maximum is $\mu_e a_0k_0\sim 0.8$ and almost independent on the ratio $m_h^*/m_e^*$.
• Figure 2: (a) In the truncated model the exciton wave packet is effectively represented by two quantum states a and b (with energies $W_a$ and $W_b$ and wavevectors $\bm q_a$ and $\bm q_b$, respectively) connected by the vector $\bm k_0$ of LLPM. (b) The energy $H^{(1)}$ (orange curve, eq. \ref{['energyWa']}) and numeric solution (coloured dots) of the equations of motion (eqs. \ref{['eqsigma']}, \ref{['eqJz']}, \ref{['eqQ']} for the case depicted in Fig. a) at $\kappa=0.25$. The dots show the energy (eq. \ref{['EnergyConserva']}) and the relative population $J_a$ of the exciton-phonon subsystem during its evolution. Colours depict the time step: from black at small times to red at larger times. The system evolves from some non-stationary high-energy state towards the stationary solution corresponding to the energy minimum. The function $H^{(1)}(J_a)$, plotted at $\kappa_{ba}=0.33$ (dashed black), is given for comparison.
• Figure 3: (a) The MSEM is composed of two excitons entangled with one and the same LLPM. (b) The energy of EM $H^{(2)}$ (eq. \ref{['Esemicl4p1']}) plotted as a function of the occupations $J_a$, and $J_c$ at $\kappa_{ba}=\kappa_{dc}=0.49$ (see the scheme in Fig. a, but with unequal energies $W_a\ne W_c$, and $W_b\ne W_d$) and numeric solution of the equations of motion (the dots' colours were chosen as in Fig. \ref{['Fig2']}b). In the region "A" surrounded by blue solid curve $0<\cos\Delta\le 1$, while in the regions surrounded by green curves $-1\le\cos\Delta<0$. The stationary state solution (crosspoint of thin dashed orange lines showing the stationary values $J_{c,1}=\kappa_{dc}/4$, and $J_{a,1}=\kappa_{ba}/4$) corresponds to the energy minimum at the blue curve, where $\cos\Delta=1$ .
• Figure 4: The averaged over $\kappa_{ba}$ EP+MSEM spectrum (black curve) plotted vs the frequency $\omega/\omega_0$. For comparison the spectra of single-EP are plotted for the largest $\kappa_{ba}=-0.33$ (orange dashed) and the smallest $\kappa_{ba}=0$ (blue dashed). The maxima are located at $H^{(1\pm)}_0=\left.\left(H^{(1)}_0/ \omega_0-W_a/\omega_0\right)\right|_{\kappa_{ba}=-0.33,\,0}$, and their mean value is denoted by $H^{(1\,m)}_0=(H^{(1+)}_0+H^{(1-)}_0)/2\approx -0.22$. The superfluorescent ($\delta$-)peak is positioned at $\bar{H}_0^{(2)}=\left.H_0^{(2)}/2\omega_0\right|_{\kappa_{ba}=0.49}\sim -0.52$. In the insert: the typical solution of the kinetic equations (eqs. \ref{['dotnbi']}--\ref{['dotnex']}) for the superfluorescent peak intensity (bold orange). The dashed lines shows the asymptotic: $n_{MSEM}\sim \gamma f_1 n_0^2 t^2/2$, $n_{MSEM}\sim n_0 \gamma (\gamma +\beta+f_2) t^{-2}/2f_1\beta(\beta+\gamma-f_2)^2$, and $n_{EP}\sim n_0/(1+f_1n_0t)$ (thin red), the maximum of $n_{MSEM}\sim \gamma f_1 n_0^2 /(\beta+\gamma+f_2)(1+f_1n_0t_0)^2$ is reached at $t_0\sim\sqrt[4]{(\beta+\gamma+f_2)/f_1^2 \gamma (\beta+\gamma-f_2)^2n_0^2}$.