More Electron Traps — Question 8

Question 8

Tunneling between identical traps produces a level splitting and coherent transfer

Two localized orthonormal states $|L\rangle,|R\rangle$ are modeled by $H=E_0I-J(|L\rangle\langle R|+|R\rangle\langle L|)$, with $J>0$. An electron starts in $|L\rangle$. Use $\hbar=6.582\times10^{-16}\,\mathrm{eV\,s}$ and $J=.00200\,\mathrm{eV}$.

1. Find the two energy eigenstates and their splitting.

2. Derive the right-trap probability and find the first complete-transfer time.

Question 8 - Solution

Tunneling between identical traps produces a level splitting and coherent transfer

(a) Method and calculation

$$|+\rangle=(|L\rangle+|R\rangle)/\sqrt2,\ E_+=E_0-J;\quad|-\rangle=(|L\rangle-|R\rangle)/\sqrt2,\ E_-=E_0+J.$$$$\boxed{\Delta E=2J=.00400\,\mathrm{eV}}.$$

(b) Interpretation and further calculation

$$|\Psi(t)\rangle=e^{-iE_0t/\hbar}[\cos(Jt/\hbar)|L\rangle+i\sin(Jt/\hbar)|R\rangle],\quad\boxed{P_R=\sin^2(Jt/\hbar)}.$$$$\boxed{t_{\mathrm{transfer}}=\pi\hbar/(2J)=5.16949\times10^{-13}\,\mathrm s}.$$

Physical check

Coherent transfer results from the relative phase of the split eigenstates. Environmental dephasing would require a different dynamical model.