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Date: Wed, 22 Sep 2021 15:33:15 +0200
Subject: Expand knowledge base

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+---
+title: "Rabi oscillation"
+firstLetter: "R"
+publishDate: 2021-09-22
+categories:
+- Physics
+- Quantum mechanics
+- Optics
+
+date: 2021-09-18T00:41:43+02:00
+draft: false
+markup: pandoc
+---
+
+# Rabi oscillation
+
+In quantum mechanics, from the derivation of
+[time-dependent perturbation theory](/know/concept/time-dependent-perturbation-theory/),
+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 $\ket{n} \exp\!(-i E_n t / \hbar)$:
+
+$$\begin{aligned}
+    i \hbar \dv{c_m}{t}
+    = \sum_{n} c_n(t) \matrixel{m}{\hat{H}_1}{n} \exp\!(i \omega_{mn} t)
+\end{aligned}$$
+
+Where $\omega_{mn} \equiv (E_m \!-\! E_n) / \hbar$
+for energies $E_m$ and $E_n$.
+Note that this equation is exact,
+despite being used for deriving perturbation theory.
+Consider a two-level system where $n \in \{a, b\}$,
+in which case the above equation can be expanded to the following:
+
+$$\begin{aligned}
+    \dv{c_a}{t}
+    &= - \frac{i}{\hbar} \matrixel{a}{\hat{H}_1}{b} \exp\!(- i \omega_0 t) \: c_b - \frac{i}{\hbar} \matrixel{a}{\hat{H}_1}{a} \: c_a
+    \\
+    \dv{c_b}{t}
+    &= - \frac{i}{\hbar} \matrixel{b}{\hat{H}_1}{a} \exp\!(i \omega_0 t) \: c_a - \frac{i}{\hbar} \matrixel{b}{\hat{H}_1}{b} \: c_b
+\end{aligned}$$
+
+Where $\omega_0 \equiv \omega_{ba}$ is positive.
+We assume that $\hat{H}_1$ has odd spatial parity,
+in which case [Laporte's selection rule](/know/concept/selection-rules/)
+states that the diagonal matrix elements vanish, leaving:
+
+$$\begin{aligned}
+    \dv{c_a}{t}
+    &= - \frac{i}{\hbar} \matrixel{a}{\hat{H}_1}{b} \exp\!(- i \omega_0 t) \: c_b
+    \\
+    \dv{c_b}{t}
+    &= - \frac{i}{\hbar} \matrixel{b}{\hat{H}_1}{a} \exp\!(i \omega_0 t) \: c_a
+\end{aligned}$$
+
+We now choose $\hat{H}_1$ to be as follows,
+sinusoidally oscillating with a spatially odd $V(\vec{r})$:
+
+$$\begin{aligned}
+    \hat{H}_1(t)
+    = V \cos\!(\omega t)
+    = \frac{V}{2} \Big( \exp\!(i \omega t) + \exp\!(-i \omega t) \Big)
+\end{aligned}$$
+
+We insert this into the equations for $c_a$ and $c_b$,
+and define $V_{ab} \equiv \matrixel{a}{V}{b}$, leading us to:
+
+$$\begin{aligned}
+    \dv{c_a}{t}
+    &= - i \frac{V_{ab}}{2 \hbar} \Big( \exp\!\big(i (\omega \!-\! \omega_0) t\big) + \exp\!\big(\!-\! i (\omega \!+\! \omega_0) t\big) \Big) \: c_b
+    \\
+    \dv{c_b}{t}
+    &= - i \frac{V_{ab}}{2 \hbar} \Big( \exp\!\big(i (\omega \!+\! \omega_0) t\big) + \exp\!\big(\!-\! i (\omega \!-\! \omega_0) t\big) \Big) \: c_a
+\end{aligned}$$
+
+Here, we make the *rotating wave approximation*:
+assuming we are close to resonance $\omega \approx \omega_0$,
+we decide that $\exp\!(i (\omega \!+\! \omega_0) t)$
+oscillates so much faster than $\exp\!(i (\omega \!-\! \omega_0) t)$,
+that its effect turns out negligible
+when the system is observed over a reasonable time interval.
+
+In other words, over this reasonably-sized time interval,
+$\exp\!(i (\omega \!+\! \omega_0) t)$ averages to zero,
+while $\exp\!(i (\omega \!-\! \omega_0) t)$ does not.
+Dropping the respective terms thus leaves us with:
+
+$$\begin{aligned}
+    \dv{c_a}{t}
+    = - i \frac{V_{ab}}{2 \hbar} \exp\!\big(i (\omega \!-\! \omega_0) t \big) \: c_b
+    \qquad \quad
+    \dv{c_b}{t}
+    = - i \frac{V_{ba}}{2 \hbar} \exp\!\big(\!-\! i (\omega \!-\! \omega_0) t \big) \: c_a
+\end{aligned}$$
+
+Now we can solve this system of coupled equations exactly.
+We differentiate the first equation with respect to $t$,
+and then substitute $\dv*{c_b}{t}$ for the second equation:
+
+$$\begin{aligned}
+    \dv[2]{c_a}{t}
+    &= - i \frac{V_{ab}}{2 \hbar} \bigg( i (\omega - \omega_0) \: c_b + \dv{c_b}{t} \bigg) \exp\!\big(i (\omega \!-\! \omega_0) t \big)
+    \\
+    &= - i \frac{V_{ab}}{2 \hbar} \bigg( i (\omega - \omega_0) \: c_b 
+    - i \frac{V_{ba}}{2 \hbar} \exp\!\big(\!-\! i (\omega \!-\! \omega_0) t \big) \: c_a \bigg)
+    \exp\!\big(i (\omega \!-\! \omega_0) t \big)
+    \\
+    &= \frac{V_{ab}}{2 \hbar} (\omega - \omega_0) \exp\!\big(i (\omega \!-\! \omega_0) t \big) \: c_b - \frac{|V_{ab}|^2}{(2 \hbar)^2} c_a
+\end{aligned}$$
+
+In the first term, we recognize $\dv*{c_a}{t}$,
+which we insert to arrive at an equation for $c_a(t)$:
+
+$$\begin{aligned}
+    0
+    = \dv[2]{c_a}{t} - i (\omega - \omega_0) \dv{c_a}{t} + \frac{|V_{ab}|^2}{(2 \hbar)^2} \: c_a
+\end{aligned}$$
+
+To solve this, we make the ansatz $c_a(t) = \exp\!(\lambda t)$,
+which, upon insertion, gives us:
+
+$$\begin{aligned}
+    0
+    = \lambda^2 - i (\omega - \omega_0) \lambda + \frac{|V_{ab}|^2}{(2 \hbar)^2}
+\end{aligned}$$
+
+This quadratic equation has two complex roots $\lambda_1$ and $\lambda_2$,
+which are found to be:
+
+$$\begin{aligned}
+    \lambda_1
+    = i \frac{\omega - \omega_0 + \tilde{\Omega}}{2}
+    \qquad \quad
+    \lambda_2
+    = i \frac{\omega - \omega_0 - \tilde{\Omega}}{2}
+\end{aligned}$$
+
+Where we have defined the **generalized Rabi frequency** $\tilde{\Omega}$ to be given by:
+
+$$\begin{aligned}
+    \boxed{
+        \tilde{\Omega}
+        \equiv \sqrt{(\omega - \omega_0)^2 + \frac{|V_{ab}|^2}{\hbar^2}}
+    }
+\end{aligned}$$
+
+So that the general solution $c_a(t)$ is as follows,
+where $A$ and $B$ are arbitrary constants,
+to be determined from initial conditions (and normalization):
+
+$$\begin{aligned}
+    \boxed{
+        c_a(t)
+        = \Big( A \sin\!(\tilde{\Omega} t / 2) + B \cos\!(\tilde{\Omega} t / 2) \Big) \exp\!\big(i (\omega \!-\! \omega_0) t / 2 \big) 
+    }
+\end{aligned}$$
+
+And then the corresponding $c_b(t)$ can be found
+from the coupled equation we started at,
+or, if we only care about the probability density $|c_a|^2$,
+we can use $|c_b|^2 = 1 - |c_a|^2$.
+For example, if $A = 0$ and $B = 1$,
+we get the following probabilities
+
+$$\begin{aligned}
+    |c_a(t)|^2
+    &= \cos^2(\tilde{\Omega} t / 2)
+    = \frac{1}{2} \Big( 1 + \cos\!(\tilde{\Omega} t) \Big)
+    \\
+    |c_b(t)|^2
+    &= \sin^2(\tilde{\Omega} t / 2)
+    = \frac{1}{2} \Big( 1 - \cos\!(\tilde{\Omega} t) \Big)
+\end{aligned}$$
+
+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.
+
+The name **generalized Rabi frequency** suggests
+that there is a non-general version.
+Indeed, the **Rabi frequency** $\Omega$ is based on
+the special case of exact resonance $\omega = \omega_0$:
+
+$$\begin{aligned}
+    \Omega
+    \equiv \frac{V_{ab}}{\hbar}
+\end{aligned}$$
+
+As an example, Rabi oscillation arises
+in the [electric dipole approximation](/know/concept/electric-dipole-approximation/),
+where $\hat{H}_1$ is:
+
+$$\begin{aligned}
+    \hat{H}_1(t)
+    = - q \vec{r} \cdot \vec{E}_0 \cos(\omega t)
+\end{aligned}$$
+
+After making the rotating wave approximation,
+the resulting Rabi frequency is given by:
+
+$$\begin{aligned}
+    \Omega
+    = \frac{\vec{d} \cdot \vec{E}_0}{\hbar}
+\end{aligned}$$
+
+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
+of the electron between orbitals $\ket{a}$ and $\ket{b}$.
+
+
+
+## References
+1.  D.J. Griffiths, D.F. Schroeter,
+    *Introduction to quantum mechanics*, 3rd edition,
+    Cambridge.
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