diff options
Diffstat (limited to 'source/know/concept/rabi-oscillation/index.md')
| -rw-r--r-- | source/know/concept/rabi-oscillation/index.md | 41 |
1 files changed, 21 insertions, 20 deletions
diff --git a/source/know/concept/rabi-oscillation/index.md b/source/know/concept/rabi-oscillation/index.md index 07f8b25..2fcdea8 100644 --- a/source/know/concept/rabi-oscillation/index.md +++ b/source/know/concept/rabi-oscillation/index.md @@ -15,11 +15,11 @@ In quantum mechanics, from the derivation of we know that a time-dependent term $$\hat{H}_1$$ in the Hamiltonian affects the state as follows, where $$c_n(t)$$ are the coefficients of the linear combination -of basis states $$\Ket{n} \exp(-i E_n t / \hbar)$$: +of basis states $$\Ket{n} e^{-i E_n t / \hbar}$$: $$\begin{aligned} i \hbar \dv{c_m}{t} - = \sum_{n} c_n(t) \matrixel{m}{\hat{H}_1}{n} \exp(i \omega_{mn} t) + = \sum_{n} c_n(t) \matrixel{m}{\hat{H}_1}{n} e^{i \omega_{mn} t} \end{aligned}$$ Where $$\omega_{mn} \equiv (E_m \!-\! E_n) / \hbar$$ @@ -31,10 +31,10 @@ in which case the above equation can be expanded to the following: $$\begin{aligned} \dv{c_a}{t} - &= - \frac{i}{\hbar} \matrixel{a}{\hat{H}_1}{b} \exp(- i \omega_0 t) \: c_b - \frac{i}{\hbar} \matrixel{a}{\hat{H}_1}{a} \: c_a + &= - \frac{i}{\hbar} \matrixel{a}{\hat{H}_1}{b} e^{-i \omega_0 t} \: c_b - \frac{i}{\hbar} \matrixel{a}{\hat{H}_1}{a} c_a \\ \dv{c_b}{t} - &= - \frac{i}{\hbar} \matrixel{b}{\hat{H}_1}{a} \exp(i \omega_0 t) \: c_a - \frac{i}{\hbar} \matrixel{b}{\hat{H}_1}{b} \: c_b + &= - \frac{i}{\hbar} \matrixel{b}{\hat{H}_1}{a} e^{i \omega_0 t} \: c_a - \frac{i}{\hbar} \matrixel{b}{\hat{H}_1}{b} c_b \end{aligned}$$ Where $$\omega_0 \equiv \omega_{ba}$$ is positive. @@ -44,10 +44,10 @@ states that the diagonal matrix elements vanish, leaving: $$\begin{aligned} \dv{c_a}{t} - &= - \frac{i}{\hbar} \matrixel{a}{\hat{H}_1}{b} \exp(- i \omega_0 t) \: c_b + &= - \frac{i}{\hbar} \matrixel{a}{\hat{H}_1}{b} e^{-i \omega_0 t} \: c_b \\ \dv{c_b}{t} - &= - \frac{i}{\hbar} \matrixel{b}{\hat{H}_1}{a} \exp(i \omega_0 t) \: c_a + &= - \frac{i}{\hbar} \matrixel{b}{\hat{H}_1}{a} e^{i \omega_0 t} \: c_a \end{aligned}$$ We now choose $$\hat{H}_1$$ to be as follows, @@ -56,7 +56,7 @@ sinusoidally oscillating with a spatially odd $$V(\vec{r})$$: $$\begin{aligned} \hat{H}_1(t) = V \cos(\omega t) - = \frac{V}{2} \Big( \exp(i \omega t) + \exp(-i \omega t) \Big) + = \frac{V}{2} \Big( e^{i \omega t} + e^{-i \omega t} \Big) \end{aligned}$$ We insert this into the equations for $$c_a$$ and $$c_b$$, @@ -64,16 +64,16 @@ and define $$V_{ab} \equiv \matrixel{a}{V}{b}$$, leading us to: $$\begin{aligned} \dv{c_a}{t} - &= - i \frac{V_{ab}}{2 \hbar} \Big( \exp\!\big(i (\omega \!-\! \omega_0) t\big) + \exp\!\big(\!-\! i (\omega \!+\! \omega_0) t\big) \Big) \: c_b + &= - i \frac{V_{ab}}{2 \hbar} \Big( e^{i (\omega - \omega_0) t} + e^{-i (\omega + \omega_0) t} \Big) \: c_b \\ \dv{c_b}{t} - &= - i \frac{V_{ab}}{2 \hbar} \Big( \exp\!\big(i (\omega \!+\! \omega_0) t\big) + \exp\!\big(\!-\! i (\omega \!-\! \omega_0) t\big) \Big) \: c_a + &= - i \frac{V_{ab}}{2 \hbar} \Big( e^{i (\omega + \omega_0) t} + e^{-i (\omega - \omega_0) t} \Big) \: c_a \end{aligned}$$ Here, we make the [rotating wave approximation](/know/concept/rotating-wave-approximation/): assuming we are close to resonance $$\omega \approx \omega_0$$, -we argue that $$\exp(i (\omega \!+\! \omega_0) t)$$ +we argue that $$e^{i (\omega + \omega_0) t}$$ oscillates so fast that its effect is negligible when the system is observed over a reasonable time interval. Dropping those terms leaves us with: @@ -82,10 +82,10 @@ $$\begin{aligned} \boxed{ \begin{aligned} \dv{c_a}{t} - &= - i \frac{V_{ab}}{2 \hbar} \exp\!\big(i (\omega \!-\! \omega_0) t \big) \: c_b + &= - i \frac{V_{ab}}{2 \hbar} \: e^{i (\omega - \omega_0) t} \: c_b \\ \dv{c_b}{t} - &= - i \frac{V_{ba}}{2 \hbar} \exp\!\big(\!-\! i (\omega \!-\! \omega_0) t \big) \: c_a + &= - i \frac{V_{ba}}{2 \hbar} \: e^{-i (\omega - \omega_0) t} \: c_a \end{aligned} } \end{aligned}$$ @@ -96,13 +96,12 @@ and then substitute $$\idv{c_b}{t}$$ for the second equation: $$\begin{aligned} \dvn{2}{c_a}{t} - &= - i \frac{V_{ab}}{2 \hbar} \bigg( i (\omega - \omega_0) \: c_b + \dv{c_b}{t} \bigg) \exp\!\big(i (\omega \!-\! \omega_0) t \big) + &= - i \frac{V_{ab}}{2 \hbar} \bigg( i (\omega - \omega_0) \: c_b + \dv{c_b}{t} \bigg) e^{i (\omega - \omega_0) t} \\ - &= - i \frac{V_{ab}}{2 \hbar} \bigg( i (\omega - \omega_0) \: c_b - - i \frac{V_{ba}}{2 \hbar} \exp\!\big(\!-\! i (\omega \!-\! \omega_0) t \big) \: c_a \bigg) - \exp\!\big(i (\omega \!-\! \omega_0) t \big) + &= - i \frac{V_{ab}}{2 \hbar} \bigg( i (\omega - \omega_0) \: c_b + - i \frac{V_{ba}}{2 \hbar} \: e^{-i (\omega - \omega_0) t} \: c_a \bigg) e^{i (\omega - \omega_0) t} \\ - &= \frac{V_{ab}}{2 \hbar} (\omega - \omega_0) \exp\!\big(i (\omega \!-\! \omega_0) t \big) \: c_b - \frac{|V_{ab}|^2}{(2 \hbar)^2} c_a + &= \frac{V_{ab}}{2 \hbar} (\omega - \omega_0) \: e^{i (\omega - \omega_0) t} \: c_b - \frac{|V_{ab}|^2}{(2 \hbar)^2} \: c_a \end{aligned}$$ In the first term, we recognize $$\idv{c_a}{t}$$, @@ -113,7 +112,7 @@ $$\begin{aligned} = \dvn{2}{c_a}{t} - i (\omega - \omega_0) \dv{c_a}{t} + \frac{|V_{ab}|^2}{(2 \hbar)^2} \: c_a \end{aligned}$$ -To solve this, we make the ansatz $$c_a(t) = \exp(\lambda t)$$, +To solve this, we make the ansatz $$c_a(t) = e^{\lambda t}$$, which, upon insertion, gives us: $$\begin{aligned} @@ -148,7 +147,7 @@ to be determined from initial conditions (and normalization): $$\begin{aligned} \boxed{ c_a(t) - = \Big( A \sin(\tilde{\Omega} t / 2) + B \cos(\tilde{\Omega} t / 2) \Big) \exp\!\big(i (\omega \!-\! \omega_0) t / 2 \big) + = \Big( A \sin(\tilde{\Omega} t / 2) + B \cos(\tilde{\Omega} t / 2) \Big) e^{i (\omega - \omega_0) t / 2} } \end{aligned}$$ @@ -173,7 +172,7 @@ Note that the period was halved by squaring. This periodic "flopping" of the particle between $$\Ket{a}$$ and $$\Ket{b}$$ is known as **Rabi oscillation**, **Rabi flopping** or the **Rabi cycle**. This is a more accurate treatment -of the flopping found from first-order perturbation theory. +of the flopping found from first-order perturbation theory in textbooks. The name **generalized Rabi frequency** suggests that there is a non-general version. @@ -185,6 +184,8 @@ $$\begin{aligned} \equiv \frac{V_{ba}}{\hbar} \end{aligned}$$ +Some authors use $$|V_{ba}|$$ instead, +but not doing that lets us use $$\Omega$$ as a nice abbreviation. As an example, Rabi oscillation arises in the [electric dipole approximation](/know/concept/electric-dipole-approximation/), where $$\hat{H}_1$$ is: |