diff options
Diffstat (limited to 'content/know/concept/rabi-oscillation')
| -rw-r--r-- | content/know/concept/rabi-oscillation/index.pdc | 24 |
1 files changed, 15 insertions, 9 deletions
diff --git a/content/know/concept/rabi-oscillation/index.pdc b/content/know/concept/rabi-oscillation/index.pdc index cf393a4..a488de0 100644 --- a/content/know/concept/rabi-oscillation/index.pdc +++ b/content/know/concept/rabi-oscillation/index.pdc @@ -87,11 +87,15 @@ while $\exp\!(i (\omega \!-\! \omega_0) t)$ does not. Dropping the respective terms thus leaves us with: $$\begin{aligned} - \dv{c_a}{t} - = - i \frac{V_{ab}}{2 \hbar} \exp\!\big(i (\omega \!-\! \omega_0) t \big) \: c_b - \qquad \quad - \dv{c_b}{t} - = - i \frac{V_{ba}}{2 \hbar} \exp\!\big(\!-\! i (\omega \!-\! \omega_0) t \big) \: c_a + \boxed{ + \begin{aligned} + \dv{c_a}{t} + &= - i \frac{V_{ab}}{2 \hbar} \exp\!\big(i (\omega \!-\! \omega_0) t \big) \: c_b + \\ + \dv{c_b}{t} + &= - i \frac{V_{ba}}{2 \hbar} \exp\!\big(\!-\! i (\omega \!-\! \omega_0) t \big) \: c_a + \end{aligned} + } \end{aligned}$$ Now we can solve this system of coupled equations exactly. @@ -186,7 +190,7 @@ the special case of exact resonance $\omega = \omega_0$: $$\begin{aligned} \Omega - \equiv \frac{V_{ab}}{\hbar} + \equiv \frac{V_{ba}}{\hbar} \end{aligned}$$ As an example, Rabi oscillation arises @@ -195,7 +199,7 @@ where $\hat{H}_1$ is: $$\begin{aligned} \hat{H}_1(t) - = - q \vec{r} \cdot \vec{E}_0 \cos(\omega t) + = - q \vec{r} \cdot \vec{E}_0 \cos\!(\omega t) \end{aligned}$$ After making the rotating wave approximation, @@ -203,12 +207,14 @@ the resulting Rabi frequency is given by: $$\begin{aligned} \Omega - = \frac{\vec{d} \cdot \vec{E}_0}{\hbar} + = - \frac{\vec{d} \cdot \vec{E}_0}{\hbar} \end{aligned}$$ Where $\vec{E}_0$ is the [electric field](/know/concept/electric-field/) amplitude, -and $\vec{d} \equiv q \matrixel{a}{\vec{r}}{b}$ is the transition dipole moment +and $\vec{d} \equiv q \matrixel{b}{\vec{r}}{a}$ is the transition dipole moment of the electron between orbitals $\ket{a}$ and $\ket{b}$. +Apparently, some authors define $\vec{d}$ with the opposite sign, +thereby departing from its classical interpretation. |