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diff --git a/source/know/concept/multi-photon-absorption/index.md b/source/know/concept/multi-photon-absorption/index.md index af433ac..5dd9887 100644 --- a/source/know/concept/multi-photon-absorption/index.md +++ b/source/know/concept/multi-photon-absorption/index.md @@ -11,10 +11,10 @@ categories: layout: "concept" --- -Consider a quantum system where there are many eigenstates $\Ket{n}$, +Consider a quantum system where there are many eigenstates $$\Ket{n}$$, e.g. atomic orbitals, for an electron to occupy. Suppose an [electromagnetic wave](/know/concept/electromagnetic-wave-equation/) -passes by, such that its Hamiltonian gets perturbed by $\hat{H}_1$, given in the +passes by, such that its Hamiltonian gets perturbed by $$\hat{H}_1$$, given in the [electric dipole approximation](/know/concept/electric-dipole-approximation/) by: $$\begin{aligned} @@ -23,19 +23,19 @@ $$\begin{aligned} \approx -\vu{p} \cdot \vb{E} e^{-i \omega t} \end{aligned}$$ -Where $\vb{E}$ is the [electric field](/know/concept/electric-field/) amplitude, -and $\vu{p} \equiv q \vu{x}$ is the transition dipole moment operator. +Where $$\vb{E}$$ is the [electric field](/know/concept/electric-field/) amplitude, +and $$\vu{p} \equiv q \vu{x}$$ is the transition dipole moment operator. Here, we have made the [rotating wave approximation](/know/concept/rotating-wave-approximation/) -to neglect the $e^{i \omega t}$ term, +to neglect the $$e^{i \omega t}$$ term, because it turns out to be irrelevant in this discussion. -We call the ground state $\Ket{0}$, +We call the ground state $$\Ket{0}$$, but other than that, the other states need *not* be sorted by energy. However, we demand that the following holds -for all even-numbered states $\Ket{e}$ and $\Ket{e'}$, -and for all odd-numbered ($u$neven) states $\Ket{u}$ and $\Ket{u'}$: +for all even-numbered states $$\Ket{e}$$ and $$\Ket{e'}$$, +and for all odd-numbered ($$u$$neven) states $$\Ket{u}$$ and $$\Ket{u'}$$: $$\begin{aligned} \matrixel{e}{\hat{H}_1}{e'} = \matrixel{u}{\hat{H}_1}{u'} = 0 @@ -46,7 +46,7 @@ $$\begin{aligned} This is justified for atomic orbitals thanks to [Laporte's selection rule](/know/concept/selection-rules/). Therefore, [time-dependent perturbation theory](/know/concept/time-dependent-perturbation-theory/) -says that the $N$th-order coefficient corrections are: +says that the $$N$$th-order coefficient corrections are: $$\begin{aligned} c_e^{(N)}(t) @@ -56,8 +56,8 @@ $$\begin{aligned} &= -\frac{i}{\hbar} \sum_{e}^{\mathrm{even}} \int_0^t \matrixel{u}{\hat{H}_1(\tau)}{e} \: c_e^{(N-1)}(\tau) \: e^{i \omega_{ue} \tau} \dd{\tau} \end{aligned}$$ -Where $\omega_{eu} = (E_e \!-\! E_u) / \hbar$. -For simplicity, the electron starts in the lowest-energy state $\Ket{0}$: +Where $$\omega_{eu} = (E_e \!-\! E_u) / \hbar$$. +For simplicity, the electron starts in the lowest-energy state $$\Ket{0}$$: $$\begin{aligned} c_0^{(0)} = 1 @@ -65,8 +65,8 @@ $$\begin{aligned} c_u^{(0)} = c_{e \neq 0}^{(0)} = 0 \end{aligned}$$ -Finally, we prove the following useful relation for large $t$, -involving a [Dirac delta function](/know/concept/dirac-delta-function/) $\delta$: +Finally, we prove the following useful relation for large $$t$$, +involving a [Dirac delta function](/know/concept/dirac-delta-function/) $$\delta$$: $$\begin{aligned} \lim_{t \to \infty} \bigg| \frac{e^{i x t} - 1}{x} \bigg|^2 @@ -86,7 +86,7 @@ $$\begin{aligned} = i e^{i x t / 2} \int_{-t/2}^{t/2} e^{i x \tau} \dd{\tau} \end{aligned}$$ -By taking the limit $t \to \infty$, +By taking the limit $$t \to \infty$$, it can be turned into a nascent Dirac delta function: $$\begin{aligned} @@ -102,7 +102,7 @@ $$\begin{aligned} = 4 \pi^2 \delta^2(x) \end{aligned}$$ -However, a squared delta function $\delta^2$ is not ideal, +However, a squared delta function $$\delta^2$$ is not ideal, so we take a step back: $$\begin{aligned} @@ -111,7 +111,7 @@ $$\begin{aligned} = \delta(x) \lim_{t \to \infty} \frac{t}{2 \pi} \end{aligned}$$ -Where we have set $x = 0$ according to the first delta function. +Where we have set $$x = 0$$ according to the first delta function. This gives the target: $$\begin{aligned} @@ -119,6 +119,7 @@ $$\begin{aligned} = 4 \pi^2 \delta^2(x) = 2 \pi \: \delta(x) \: t \end{aligned}$$ + </div> </div> @@ -127,7 +128,7 @@ $$\begin{aligned} To warm up, we start at first-order perturbation theory. Thanks to our choice of initial condition, -nothing at all happens to any of the even-numbered states $\Ket{e}$: +nothing at all happens to any of the even-numbered states $$\Ket{e}$$: $$\begin{aligned} c_e^{(1)}(t) @@ -135,8 +136,8 @@ $$\begin{aligned} = 0 \end{aligned}$$ -While the odd-numbered states $\Ket{u}$ have a nonzero correction $c_u^{(1)}$, -where $\vb{p}_{u0} = \matrixel{u}{\vu{p}}{0}$: +While the odd-numbered states $$\Ket{u}$$ have a nonzero correction $$c_u^{(1)}$$, +where $$\vb{p}_{u0} = \matrixel{u}{\vu{p}}{0}$$: $$\begin{aligned} c_u^{(1)}(t) @@ -157,10 +158,10 @@ $$\begin{aligned} } \end{aligned}$$ -Since $\big| c_u^{(1)}(t) \big|^2$ is the probability -of finding the electron in $\Ket{u}$, -its transition rate $R_u^{(1)}(t)$ is as follows, -averaged since the beginning $t = 0$: +Since $$\big| c_u^{(1)}(t) \big|^2$$ is the probability +of finding the electron in $$\Ket{u}$$, +its transition rate $$R_u^{(1)}(t)$$ is as follows, +averaged since the beginning $$t = 0$$: $$\begin{aligned} R_u^{(1)}(t) @@ -169,7 +170,7 @@ $$\begin{aligned} \cdot \bigg| \frac{e^{i (\omega_{u0} - \omega) t} - 1}{\omega_{u0} - \omega} \bigg|^2 \end{aligned}$$ -For large $t \to \infty$, we can use the formula we proved earlier +For large $$t \to \infty$$, we can use the formula we proved earlier to get [Fermi's golden rule](/know/concept/fermis-golden-rule/): $$\begin{aligned} @@ -180,17 +181,17 @@ $$\begin{aligned} \end{aligned}$$ This well-known formula represents **one-photon absorption**: -it peaks at $\omega_{u0} = \omega$, i.e. when one photon $\hbar \omega$ -has the exact energy of the transition $\hbar \omega_{u0}$. -Note that this transition is only possible when $\matrixel{u}{\vu{p}}{0} \neq 0$, -i.e. for any odd-numbered final state $\Ket{u}$. +it peaks at $$\omega_{u0} = \omega$$, i.e. when one photon $$\hbar \omega$$ +has the exact energy of the transition $$\hbar \omega_{u0}$$. +Note that this transition is only possible when $$\matrixel{u}{\vu{p}}{0} \neq 0$$, +i.e. for any odd-numbered final state $$\Ket{u}$$. ## Two-photon absorption Next, we go to second-order perturbation theory. Based on the previous result, this time -all odd-numbered states $\Ket{u}$ are unaffected: +all odd-numbered states $$\Ket{u}$$ are unaffected: $$\begin{aligned} c_u^{(2)}(t) @@ -198,8 +199,8 @@ $$\begin{aligned} = 0 \end{aligned}$$ -While the even-numbered states $\Ket{e}$ have the following correction, -using $\omega_{eu} \!+\! \omega_{u0} = \omega_{e0}$: +While the even-numbered states $$\Ket{e}$$ have the following correction, +using $$\omega_{eu} \!+\! \omega_{u0} = \omega_{e0}$$: $$\begin{aligned} c_e^{(2)}(t) @@ -213,7 +214,7 @@ $$\begin{aligned} - \frac{e^{i (\omega_{eu} - \omega) \tau}}{i (\omega_{eu} - \omega)} \bigg]_0^t \end{aligned}$$ -The second term represents one-photon absorption between $\Ket{u}$ and $\Ket{e}$. +The second term represents one-photon absorption between $$\Ket{u}$$ and $$\Ket{e}$$. We do not care about that, so we drop it, leaving only the first term: $$\begin{aligned} @@ -224,7 +225,7 @@ $$\begin{aligned} } \end{aligned}$$ -As before, we can define a rate $R_e^{(2)}(t)$ +As before, we can define a rate $$R_e^{(2)}(t)$$ for all transitions represented by this term: $$\begin{aligned} @@ -234,7 +235,7 @@ $$\begin{aligned} \cdot \bigg| \frac{e^{i (\omega_{e0} - 2 \omega) t} - 1}{\omega_{e0} - 2 \omega} \bigg|^2 \end{aligned}$$ -Which for $t \to \infty$ takes a similar form to Fermi's golden rule, +Which for $$t \to \infty$$ takes a similar form to Fermi's golden rule, using the formula we proved: $$\begin{aligned} @@ -245,19 +246,19 @@ $$\begin{aligned} } \end{aligned}$$ -This represents **two-photon absorption**, since it peaks at $\omega_{e0} = 2 \omega$: -two identical photons $\hbar \omega$ are absorbed simultaneously -to bridge the energy gap $\hbar \omega_{e0}$. -Surprisingly, such a transition can only occur when $\matrixel{e}{\vu{p}}{0} = 0$, -i.e. for any even-numbered final state $\Ket{e}$. -Notice that the rate is proportional to $|\vb{E}|^4$, +This represents **two-photon absorption**, since it peaks at $$\omega_{e0} = 2 \omega$$: +two identical photons $$\hbar \omega$$ are absorbed simultaneously +to bridge the energy gap $$\hbar \omega_{e0}$$. +Surprisingly, such a transition can only occur when $$\matrixel{e}{\vu{p}}{0} = 0$$, +i.e. for any even-numbered final state $$\Ket{e}$$. +Notice that the rate is proportional to $$|\vb{E}|^4$$, so this effect is only noticeable at high light intensities. ## Three-photon absorption For third-order perturbation theory, -all even-numbered states $\Ket{e}$ are unchanged: +all even-numbered states $$\Ket{e}$$ are unchanged: $$\begin{aligned} c_e^{(3)}(t) @@ -265,7 +266,7 @@ $$\begin{aligned} = 0 \end{aligned}$$ -And the odd-numbered states $\Ket{u}$ get the following third-order corrections: +And the odd-numbered states $$\Ket{u}$$ get the following third-order corrections: $$\begin{aligned} c_u^{(3)}(t) @@ -294,7 +295,7 @@ $$\begin{aligned} } \end{aligned}$$ -The resulting transition rate $R_u^{(3)}(t)$ +The resulting transition rate $$R_u^{(3)}(t)$$ is found to have the following familiar form: $$\begin{aligned} @@ -317,31 +318,31 @@ $$\begin{aligned} } \end{aligned}$$ -This represents **three-photon absorption**, since it peaks at $\omega_{u0} = 3 \omega$: -three identical photons $\hbar \omega$ are absorbed simultaneously -to bridge the energy gap $\hbar \omega_{u0}$. +This represents **three-photon absorption**, since it peaks at $$\omega_{u0} = 3 \omega$$: +three identical photons $$\hbar \omega$$ are absorbed simultaneously +to bridge the energy gap $$\hbar \omega_{u0}$$. This process is similar to one-photon absorption, -in the sense that it can only occur if $\matrixel{u}{\vu{p}}{0} \neq 0$. -The rate is proportional to $|\vb{E}|^6$, +in the sense that it can only occur if $$\matrixel{u}{\vu{p}}{0} \neq 0$$. +The rate is proportional to $$|\vb{E}|^6$$, so this effect only appears at extremely high light intensities. ## N-photon absorption A pattern has appeared in these calculations: -in $N$th-order perturbation theory, -we get a term representing $N$-photon absorption, -with a transition rate proportional to $|\vb{E}|^{2N}$. +in $$N$$th-order perturbation theory, +we get a term representing $$N$$-photon absorption, +with a transition rate proportional to $$|\vb{E}|^{2N}$$. Indeed, we can derive infinitely many formulas in this way, although the results become increasingly unrealistic -due to the dependence on $\vb{E}$. +due to the dependence on $$\vb{E}$$. -If $N$ is odd, only odd-numbered destinations $\Ket{u}$ are allowed -(assuming the electron starts in the ground state $\Ket{0}$), -and if $N$ is even, only even-numbered destinations $\Ket{e}$. +If $$N$$ is odd, only odd-numbered destinations $$\Ket{u}$$ are allowed +(assuming the electron starts in the ground state $$\Ket{0}$$), +and if $$N$$ is even, only even-numbered destinations $$\Ket{e}$$. Note that nothing has been said about the energies of these states -(other than $\Ket{0}$ being the minimum); -everything is determined by the matrix elements $\matrixel{f}{\vu{p}}{i}$. +(other than $$\Ket{0}$$ being the minimum); +everything is determined by the matrix elements $$\matrixel{f}{\vu{p}}{i}$$. |