The effects of decoherence on Fermi's golden rule

Abstract

Fermi's golden rule which describes the transition rates between two electronic levels under external stimulations is used ubiquitously in different fields of physics. The original Fermi's golden rule was derived from perturbative time-dependent Schrödinger's equation without the direct contribution by decoherence effect. However, as a result of recent developments of quantum computing and ultra fast carrier dynamics, the decoherence becomes a prominent topic in fundamental research.Here, by using the non-adiabatic molecular dynamics which goes beyond the time-dependent Schrödinger's equation by introducing decoherence, we study the effect of decoherence on Fermi's golden rule for the fixed basis and the adiabatic basis, respectively. We find that when the decoherence time becomes short, there is a significant deviation from the Fermi's golden rule for both bases. By using monolayer $\mathrm{WS_2}$ as an example, we investigate the decoherence effect in the carrier transitions induced by the electron-phonon coupling with first-principle method.

Figures (4)

• Figure 1: State $\left| 2 \right\rangle$ occupation at $t=\infty$ for various frequencies ($\omega$) of the external perturbation but with fixed state energy different ($\hbar\omega_{12}=0.03$ eV). The initial state is a fully occupied state on $\left| 1 \right\rangle$. Comparisons of FGR and the numerical method calculation with (a) the fixed-basis and with (b) the adiabatic basis.
• Figure 2: State $\left| 2 \right\rangle$ occupation at $t=\infty$ for the various gap energies $\hbar\omega_{12}$ with a fixed frequency ($\hbar\omega=0.03$ eV) of the external perturbation. The initial state is a fully occupied state on $\left| 1 \right\rangle$.
• Figure 3: The NAMD-simulated excited carrier of the real system and the TLS. Two phonons are considered, whose phonon energies are 0.046 and 0.020 eV. The corresponding EPC strength are 0.007 and 0.016 eV. (a) Tensiled structure ($1\%$ tensile): energy gap is 0.056 eV. (b) Stress-free structure: energy gap is 0.067 eV. (c) Compressed structure ($0.4\%$ compression): energy gap is 0.12 eV. The initial state is a fully occupied state on VBM-2.
• Figure 4: State $\left| 2 \right\rangle$ occupation at $t=\infty$ of the two-phonon TLS model: phonon energies $\hbar \omega_{1}$ = 0.047 eV, $\hbar \omega_{2}$ = 0.020 eV; EPC strengths $V_1$ = 0.00001 eV, $V_2$ = 0.0001 eV; energy gap $\left| \Delta \right|$ = 0.056 eV. Black: the numerical result of the P-matrix NAMD method. Red: analytical result based on SI equ.28. Blue: analytical result of the first phonon ($\hbar \omega_{1}$, $V_1$). Gold: analytical result of the second phonon ($\hbar \omega_{2}$, $V_2$). Green: analytical result of the interference terms.