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Diffstat (limited to 'source/know/concept/rabi-oscillation')
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1 files changed, 35 insertions, 35 deletions
diff --git a/source/know/concept/rabi-oscillation/index.md b/source/know/concept/rabi-oscillation/index.md index 9077cce..07f8b25 100644 --- a/source/know/concept/rabi-oscillation/index.md +++ b/source/know/concept/rabi-oscillation/index.md @@ -12,21 +12,21 @@ layout: "concept" 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 +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 $\Ket{n} \exp(-i E_n t / \hbar)$: +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 $E_n$. +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 $n \in \{a, b\}$, +Consider a two-level system where $$n \in \{a, b\}$$, in which case the above equation can be expanded to the following: $$\begin{aligned} @@ -37,8 +37,8 @@ $$\begin{aligned} &= - \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}$$ -Where $\omega_0 \equiv \omega_{ba}$ is positive. -We assume that $\hat{H}_1$ has odd spatial parity, +Where $$\omega_0 \equiv \omega_{ba}$$ is positive. +We assume that $$\hat{H}_1$$ has odd spatial parity, in which case [Laporte's selection rule](/know/concept/selection-rules/) states that the diagonal matrix elements vanish, leaving: @@ -50,8 +50,8 @@ $$\begin{aligned} &= - \frac{i}{\hbar} \matrixel{b}{\hat{H}_1}{a} \exp(i \omega_0 t) \: c_a \end{aligned}$$ -We now choose $\hat{H}_1$ to be as follows, -sinusoidally oscillating with a spatially odd $V(\vec{r})$: +We now choose $$\hat{H}_1$$ to be as follows, +sinusoidally oscillating with a spatially odd $$V(\vec{r})$$: $$\begin{aligned} \hat{H}_1(t) @@ -59,8 +59,8 @@ $$\begin{aligned} = \frac{V}{2} \Big( \exp(i \omega t) + \exp(-i \omega t) \Big) \end{aligned}$$ -We insert this into the equations for $c_a$ and $c_b$, -and define $V_{ab} \equiv \matrixel{a}{V}{b}$, leading us to: +We insert this into the equations for $$c_a$$ and $$c_b$$, +and define $$V_{ab} \equiv \matrixel{a}{V}{b}$$, leading us to: $$\begin{aligned} \dv{c_a}{t} @@ -72,8 +72,8 @@ $$\begin{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 $\exp(i (\omega \!+\! \omega_0) t)$ +assuming we are close to resonance $$\omega \approx \omega_0$$, +we argue that $$\exp(i (\omega \!+\! \omega_0) t)$$ oscillates so fast that its effect is negligible when the system is observed over a reasonable time interval. Dropping those terms leaves us with: @@ -91,8 +91,8 @@ $$\begin{aligned} \end{aligned}$$ Now we can solve this system of coupled equations exactly. -We differentiate the first equation with respect to $t$, -and then substitute $\idv{c_b}{t}$ for the second equation: +We differentiate the first equation with respect to $$t$$, +and then substitute $$\idv{c_b}{t}$$ for the second equation: $$\begin{aligned} \dvn{2}{c_a}{t} @@ -105,15 +105,15 @@ $$\begin{aligned} &= \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}$$ -In the first term, we recognize $\idv{c_a}{t}$, -which we insert to arrive at an equation for $c_a(t)$: +In the first term, we recognize $$\idv{c_a}{t}$$, +which we insert to arrive at an equation for $$c_a(t)$$: $$\begin{aligned} 0 = \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) = \exp(\lambda t)$$, which, upon insertion, gives us: $$\begin{aligned} @@ -121,7 +121,7 @@ $$\begin{aligned} = \lambda^2 - i (\omega - \omega_0) \lambda + \frac{|V_{ab}|^2}{(2 \hbar)^2} \end{aligned}$$ -This quadratic equation has two complex roots $\lambda_1$ and $\lambda_2$, +This quadratic equation has two complex roots $$\lambda_1$$ and $$\lambda_2$$, which are found to be: $$\begin{aligned} @@ -132,7 +132,7 @@ $$\begin{aligned} = i \frac{\omega - \omega_0 - \tilde{\Omega}}{2} \end{aligned}$$ -Where we have defined the **generalized Rabi frequency** $\tilde{\Omega}$ to be given by: +Where we have defined the **generalized Rabi frequency** $$\tilde{\Omega}$$ to be given by: $$\begin{aligned} \boxed{ @@ -141,8 +141,8 @@ $$\begin{aligned} } \end{aligned}$$ -So that the general solution $c_a(t)$ is as follows, -where $A$ and $B$ are arbitrary constants, +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): $$\begin{aligned} @@ -152,11 +152,11 @@ $$\begin{aligned} } \end{aligned}$$ -And then the corresponding $c_b(t)$ can be found +And then the corresponding $$c_b(t)$$ can be found from the coupled equation we started at, -or, if we only care about the probability density $|c_a|^2$, -we can use $|c_b|^2 = 1 - |c_a|^2$. -For example, if $A = 0$ and $B = 1$, +or, if we only care about the probability density $$|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} @@ -170,15 +170,15 @@ $$\begin{aligned} \end{aligned}$$ Note that the period was halved by squaring. -This periodic "flopping" of the particle between $\Ket{a}$ and $\Ket{b}$ +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 $\omega = \omega_0$: +Indeed, the **Rabi frequency** $$\Omega$$ is based on +the special case of exact resonance $$\omega = \omega_0$$: $$\begin{aligned} \Omega @@ -187,7 +187,7 @@ $$\begin{aligned} As an example, Rabi oscillation arises in the [electric dipole approximation](/know/concept/electric-dipole-approximation/), -where $\hat{H}_1$ is: +where $$\hat{H}_1$$ is: $$\begin{aligned} \hat{H}_1(t) @@ -202,10 +202,10 @@ $$\begin{aligned} = - \frac{\vec{d} \cdot \vec{E}_0}{\hbar} \end{aligned}$$ -Where $\vec{E}_0$ is the [electric field](/know/concept/electric-field/) amplitude, -and $\vec{d} \equiv q \matrixel{b}{\vec{r}}{a}$ is the transition dipole moment -of the electron between orbitals $\Ket{a}$ and $\Ket{b}$. -Apparently, some authors define $\vec{d}$ with the opposite sign, +Where $$\vec{E}_0$$ is the [electric field](/know/concept/electric-field/) amplitude, +and $$\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. |