Dynamics of a narrow-band exciton coupled with optical phonons: A time-convolutionless master-equation approach

A time-convolutionless master equation is established for studying the dynamics of a narrow-band exciton coupled with optical phonons. Within the nonadiabatic weak-coupling limit, the diagonal hypothesis works quite well so that the exciton-phonon dynamics is mainly governed by the so-called time-dependent dephasing function. It has been shown that the dephasing function tends to zero by exhibiting damped oscillations that characterize a series of dephasing-rephasing mechanisms. Indeed, the correlation time of the exciton-phonon interaction is defined as the time needed to the exciton to cover a few lattice sites. Therefore this correlation time is sufficiently long so that the system dynamics remains sensitive to the coherent nature of the lattice vibrations. Because the phonon memory recurs periodically, the exciton experiences a series of dephasing-rephasing processes. Although each rephasing does not exactly compensate the previous dephasing, the coherence survives. Consequently, the exciton keeps its wavelike nature and a coherent energy transfer occurs according to an effective hopping constant smaller than the bare hopping constant.

Figure 1

Criteria to check the diagonally dominant behavior of the relaxation operator ${\mathcal{J}}_0\left(t\right)$ (see the text) for $T=300\text{K}$, ${\Omega }_0=60{\text{cm}}^{-1}$, ${\Delta }_0=3{\text{cm}}^{-1}$, and $\Phi =6{\text{cm}}^{-1}$.

Figure 2

Criteria to check the diagonally dominant behavior of the evolution operator ${\mathcal{U}}_0\left(t\right)$ (see the text) for $T=300\text{K}$, ${\Omega }_0=60{\text{cm}}^{-1}$, ${\Delta }_0=3{\text{cm}}^{-1}$, and $\Phi =6{\text{cm}}^{-1}$.

Figure 3

Time evolution of the dephasing function ${\Gamma }_1^{\ast }\left(t\right)$ for $T=300\text{K}$, ${\Omega }_0=60{\text{cm}}^{-1}$, ${\Delta }_0=3{\text{cm}}^{-1}$ and (a) $\Phi =6{\text{cm}}^{-1}$, (b) $\Phi =8{\text{cm}}^{-1}$, (c) $\Phi =10{\text{cm}}^{-1}$, (d) $\Phi =12{\text{cm}}^{-1}$, and (e) $\Phi =14{\text{cm}}^{-1}$.

Figure 4

Time evolution of the decoherence function ${\varphi }_1\left(t\right)$ for $T=300\text{K}$, ${\Omega }_0=60{\text{cm}}^{-1}$, ${\Delta }_0=3{\text{cm}}^{-1}$ and (a) $\Phi =6{\text{cm}}^{-1}$, (b) $\Phi =8{\text{cm}}^{-1}$, (c) $\Phi =10{\text{cm}}^{-1}$, (d) $\Phi =12{\text{cm}}^{-1}$, and (e) $\Phi =14{\text{cm}}^{-1}$.

Figure 5

Decoherence factor $\text{exp}\left(-{\varphi }_1\left(\infty \right)\right)$ vs $B$ for $T=300\text{K}$ and ${\Omega }_0=60{\text{cm}}^{-1}$. ${\Delta }_0=1{\text{cm}}^{-1}$ (circles), ${\Delta }_0=3{\text{cm}}^{-1}$ (squares) and ${\Delta }_0=5{\text{cm}}^{-1}$ (triangles). Full lines represent the corresponding analytical expressions (see Sec. 5).

Figure 6

Time evolution of the $\eta \left(t\right)$ for $T=300\text{K}$, ${\Omega }_0=60{\text{cm}}^{-1}$, ${\Delta }_0=3{\text{cm}}^{-1}$ and (a) $\Phi =6{\text{cm}}^{-1}$, (b) $\Phi =8{\text{cm}}^{-1}$, (c) $\Phi =10{\text{cm}}^{-1}$, (d) $\Phi =12{\text{cm}}^{-1}$, and (e) $\Phi =14{\text{cm}}^{-1}$.

Figure 7

Time evolution of the diffusion coefficients (a) $D_B\left(t\right)$, (b) $D_C\left(t\right)$ and (c) $D\left(t\right)$ for $T=300\text{K}$, ${\Omega }_0=60{\text{cm}}^{-1}$, and $\Phi =6{\text{cm}}^{-1}$. ${\Delta }_0=1{\text{cm}}^{-1}$ (full lines), ${\Delta }_0=2{\text{cm}}^{-1}$(long dashed lines), ${\Delta }_0=3{\text{cm}}^{-1}$ (short dashed lines) and ${\Delta }_0=4{\text{cm}}^{-1}$ (dotted lines).

Figure 8

Effective hopping constant $\widehat{\Phi }$ vs $B$ for $T=300\text{K}$ and ${\Omega }_0=60{\text{cm}}^{-1}$. ${\Delta }_0=1{\text{cm}}^{-1}$ (circles), ${\Delta }_0=3{\text{cm}}^{-1}$ (squares) and ${\Delta }_0=5{\text{cm}}^{-1}$ (triangles). Full lines represent the corresponding analytical expressions (see Sec. 5.).

Figure 9

Fourier transform of the transfer function (open circles). The full line represents the approximate expression given by Eq. (28).