Langevin Approach to Quantum Optics with Molecules

We investigate the interaction between light and molecular systems modeled as quantum emitters coupled to a multitude of vibrational modes via a Holstein-type interaction. We follow a quantum Langevin equations approach that allows for analytical derivations of absorption and fluorescence profiles of molecules driven by classical fields or coupled to quantized optical modes. We retrieve analytical expressions for the modification of the radiative emission branching ratio in the Purcell regime and for the asymmetric cavity transmission associated with dissipative cross talk between upper and lower polaritons in the strong coupling regime. We also characterize the Förster resonance energy transfer process between donor-acceptor molecules mediated by the vacuum or by a cavity mode.

Figure 1

(a) The equilibrium mismatch $R_{\mathrm{ge}}^{(k)}$ between ground and excited electronic potential landscapes along a nuclear coordinate leads to the standard Franck-Condon physics with distinct optimal absorption (blue arrow) and emission (red arrow) lines. (b) Molecular box consisting of an electronic transition operator $\sigma$ coupled to $n$ vibrations $b_{\mathrm{k}}$ with frequencies, linewidths and strengths dictated by the spectral density function $F(\omega )$. (c) Absorption and emission profiles for a toy-model molecule with three vibrational modes (spectral density shown in inset). Solid lines show numerical master equation results, dashed lines are analytical results. (d) Cavity transmission $|{\mathrm{t}}_{\mathrm{c}}{|}^2$ on resonance ${\omega }_{\mathrm{c}}={\omega }_{\mathrm{e}}$ showing polariton asymmetry for $\lambda =0.3$. Full lines are numerical simulations with $\mathrm{\Gamma }=1.5\kappa$, $g=2\kappa$, $\nu =4\kappa$ while dotted lines are fitted Lorentzians with the analytically derived ${\kappa }_{\mathrm{UL}}$.

Figure 2

(a) Unidirectional near-field FRET processes showing donor excitation (blue) followed by energy exchanges (green) and acceptor emission (red). (b) Donor emission and acceptor absorption spectra with overlap in green region. Zero phonon lines are indicated by $A_{00}$ and $D_{00}$. (c) Solution of the rate equations for time dynamics of the D-A system with (full lines) and without coupling (dashed lines) in a pump-probe scenario. The tangents (dotted lines) show that the independent radiative emission of the donor at rate $2{\gamma }_D$ is modified by energy transfer rate ${\kappa }_{\mathrm{ET}}$, while the initial population increase of the acceptor is governed by the same quantity ${\kappa }_{\mathrm{ET}}$. Agreement with numerical simulations is indicated by the dashed-dotted curves with parameters ${\gamma }_A={\gamma }_D=\gamma$, $\mathrm{\Omega }(r)=15\gamma$, $\mathrm{\Delta }=500\gamma$, ${\nu }_A={\nu }_D=250\gamma$, ${\mathrm{\Gamma }}_A={\mathrm{\Gamma }}_D=30\gamma$, ${\lambda }_D=0.6$, and ${\lambda }_A=0.4$.