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+---
+title: "Einstein coefficients"
+date: 2021-07-11
+categories:
+- Physics
+- Optics
+- Quantum mechanics
+- Two-level system
+- Laser theory
+layout: "concept"
+---
+
+The **Einstein coefficients** quantify
+the emission and absorption of photons by a solid,
+and can be calculated analytically from first principles
+in several useful situations.
+
+
+## Qualitative description
+
+Suppose we have a ground state with energy $E_1$ containing $N_1$ electrons,
+and an excited state with energy $E_2$ containing $N_2$ electrons.
+The resonance $\omega_0 \equiv (E_2 \!-\! E_1)/\hbar$
+is the frequency of the photon emitted
+when an electron falls from $E_2$ to $E_1$.
+
+The first Einstein coefficient is the **spontaneous emission rate** $A_{21}$,
+which gives the probability per unit time
+that an excited electron falls from state 2 to 1,
+so that $N_2(t)$ obeys the following equation,
+which is easily solved:
+
+$$\begin{aligned}
+    \dv{N_2}{t} = - A_{21} N_2
+    \quad \implies \quad
+    N_2(t) = N_2(0) \exp(- t / \tau)
+\end{aligned}$$
+
+Where $\tau = 1 / A_{21}$ is the **natural radiative lifetime** of the excited state,
+which gives the lifetime of an excited electron,
+before it decays to the ground state.
+
+The next coefficient is the **absorption rate** $B_{12}$,
+which is the probability that an incoming photon excites an electron,
+per unit time and per unit spectral energy density
+(i.e. the rate depends on the frequency of the incoming light).
+Then $N_1(t)$ obeys the following equation:
+
+$$\begin{aligned}
+    \dv{N_1}{t} = - B_{12} N_1 u(\omega_0)
+\end{aligned}$$
+
+Where $u(\omega)$ is the spectral energy density of the incoming light,
+put here to express the fact that only photons with frequency $\omega_0$ are absorbed.
+
+There is one more Einstein coefficient: the **stimulated emission rate** $B_{21}$.
+An incoming photon has an associated electromagnetic field,
+which can encourage an excited electron to drop to the ground state,
+such that for $A_{21} = 0$:
+
+$$\begin{aligned}
+    \dv{N_2}{t} = - B_{21} N_2 u(\omega_0)
+\end{aligned}$$
+
+These three coefficients $A_{21}$, $B_{12}$ and $B_{21}$ are related to each other.
+Suppose that the system is in equilibrium,
+i.e. that $N_1$ and $N_2$ are constant.
+We assume that the number of particles in the system is constant,
+implying that $N_1'(t) = - N_2'(t) = 0$, so:
+
+$$\begin{aligned}
+    B_{12} N_1 u(\omega_0) = A_{21} N_2 + B_{21} N_2 u(\omega_0) = 0
+\end{aligned}$$
+
+Isolating this equation for $u(\omega_0)$,
+gives following expression for the radiation:
+
+$$\begin{aligned}
+    u(\omega_0)
+    = \frac{A_{21}}{(N_1 / N_2) B_{12} - B_{21}}
+\end{aligned}$$
+
+We assume that the system is in thermal equilibrium
+with its own black-body radiation, and that there is no external light.
+Then this is a [canonical ensemble](/know/concept/canonical-ensemble/),
+meaning that the relative probability that an electron has $E_2$ compared to $E_1$
+is given by the Boltzmann distribution:
+
+$$\begin{aligned}
+    \frac{\mathrm{Prob}(E_2)}{\mathrm{Prob}(E_1)}
+    = \frac{N_2}{N_1}
+    = \frac{g_2}{g_1} \exp(- \hbar \omega_0 \beta)
+\end{aligned}$$
+
+Where $g_2$ and $g_1$ are the degeneracies of the energy levels.
+Inserting this back into the equation for the spectrum $u(\omega_0)$ yields:
+
+$$\begin{aligned}
+    u(\omega_0)
+    = \frac{A_{21}}{(g_1 / g_2) B_{12} \exp(\hbar \omega_0 \beta) - B_{21}}
+\end{aligned}$$
+
+Since $u(\omega_0)$ represents only black-body radiation,
+our result must agree with [Planck's law](/know/concept/plancks-law/):
+
+$$\begin{aligned}
+    u(\omega_0)
+    = \frac{A_{21}}{B_{21} \big( (g_1 B_{12} / g_2 B_{21}) \exp(\hbar \omega_0 \beta) - 1 \big)}
+    = \frac{\hbar \omega_0^3}{\pi^2 c^3} \frac{1}{\exp(\hbar \omega_0 \beta) - 1}
+\end{aligned}$$
+
+This gives us the following two equations relating the Einstein coefficients:
+
+$$\begin{aligned}
+    \boxed{
+        A_{21} = \frac{\hbar \omega_0^3}{\pi^2 c^3} B_{21}
+        \qquad \quad
+        g_1 B_{12} = g_2 B_{21}
+    }
+\end{aligned}$$
+
+Note that this result holds even if $E_1$ is not the ground state,
+but instead some lower excited state below $E_2$,
+due to the principle of [detailed balance](/know/concept/detailed-balance/).
+Furthermore, it turns out that these relations
+also hold if the system is not in equilibrium.
+
+A notable case is **population inversion**,
+where $B_{21} N_2 > B_{12} N_1$ such that $N_2 > (g_2 / g_1) N_1$.
+This situation is mandatory for lasers, where stimulated emission must dominate,
+such that the light becomes stronger as it travels through the medium.
+
+
+## Coherent light
+
+In fact, we can analytically calculate the Einstein coefficients in some cases,
+by treating incoming light as a perturbation
+to an electron in a two-level system,
+and then finding $B_{12}$ and $B_{21}$ from the resulting transition rate.
+We need to make the [electric dipole approximation](/know/concept/electric-dipole-approximation/),
+in which case the perturbing Hamiltonian $\hat{H}_1(t)$ is given by:
+
+$$\begin{aligned}
+    \hat{H}_1(t)
+    = - q \vec{r} \cdot \vec{E}_0 \cos(\omega t)
+\end{aligned}$$
+
+Where $q = -e$ is the electron charge,
+$\vec{r}$ is the position operator,
+and $\vec{E}_0$ is the amplitude of
+the [electromagnetic wave](/know/concept/electromagnetic-wave-equation/).
+For simplicity, we let the amplitude be along the $z$-axis:
+
+$$\begin{aligned}
+    \hat{H}_1(t)
+    = - q E_0 z \cos(\omega t)
+\end{aligned}$$
+
+This form of $\hat{H}_1$ is a well-known case for
+[time-dependent perturbation theory](/know/concept/time-dependent-perturbation-theory/),
+which tells us that the transition probability from $\Ket{a}$ to $\Ket{b}$ is:
+
+$$\begin{aligned}
+    P_{ab}
+    = \frac{\big|\!\matrixel{a}{H_1}{b}\!\big|^2}{\hbar^2} \frac{\sin^2\!\big( (\omega_{ba} - \omega) t / 2 \big)}{(\omega_{ba} - \omega)^2}
+\end{aligned}$$
+
+If the nucleus is at $z = 0$,
+then generally $\Ket{1}$ and $\Ket{2}$ will be even or odd functions of $z$,
+meaning that $\matrixel{1}{z}{1} = \matrixel{2}{z}{2} = 0$
+(see also [Laporte's selection rule](/know/concept/selection-rules/)),
+leading to:
+
+$$\begin{gathered}
+    \matrixel{1}{H_1}{2} = - E_0 d^*
+    \qquad
+    \matrixel{2}{H_1}{1} = - E_0 d
+    \\
+    \matrixel{1}{H_1}{1} = \matrixel{2}{H_1}{2} = 0
+\end{gathered}$$
+
+Where $d \equiv q \matrixel{2}{z}{1}$ is a constant,
+namely the $z$-component of the **transition dipole moment**.
+The chance of an upward jump (i.e. absorption) is:
+
+$$\begin{aligned}
+    P_{12}
+    = \frac{E_0^2 |d|^2}{\hbar^2} \frac{\sin^2\!\big( (\omega_0 - \omega) t / 2 \big)}{(\omega_0 - \omega)^2}
+\end{aligned}$$
+
+Meanwhile, the transition probability for stimulated emission is as follows,
+using the fact that $P_{ab}$ is a sinc-function,
+and is therefore symmetric around $\omega_{ba}$:
+
+$$\begin{aligned}
+    P_{21}
+    = \frac{E_0^2 |d|^2}{\hbar^2} \frac{\sin^2\!\big( (\omega_0 - \omega) t / 2 \big)}{(\omega_0 - \omega)^2}
+\end{aligned}$$
+
+Surprisingly, the probabilities of absorption and stimulated emission are the same!
+In practice, however, the relative rates of these two processes depends heavily on
+the availability of electrons and holes in both states.
+
+In theory, we could calculate the transition rate $R_{12} = \ipdv{P_{12}}{t}$,
+which would give us Einstein's absorption coefficient $B_{12}$,
+for this specific case of coherent monochromatic light.
+However, the result would not be constant in time $t$,
+so is not really useful.
+
+
+## Polarized light
+
+To solve this "problem", we generalize to (incoherent) polarized polychromatic light.
+To do so, we note that the energy density $u$ of an electric field $E_0$ is given by:
+
+$$\begin{aligned}
+    u = \frac{1}{2} \varepsilon_0 E_0^2
+    \qquad \implies \qquad
+    E_0^2 = \frac{2 u}{\varepsilon_0}
+\end{aligned}$$
+
+Where $\varepsilon_0$ is the vacuum permittivity.
+Putting this in the previous result for $P_{12}$ gives us:
+
+$$\begin{aligned}
+    P_{12}
+    = \frac{2 u |d|^2}{\varepsilon_0 \hbar^2} \frac{\sin^2\!\big( (\omega_0 - \omega) t / 2 \big)}{(\omega_0 - \omega)^2}
+\end{aligned}$$
+
+For a continuous light spectrum,
+this $u$ turns into the spectral energy density $u(\omega)$:
+
+$$\begin{aligned}
+    P_{12}
+    = \frac{2 |d|^2}{\varepsilon_0 \hbar^2}
+    \int_0^\infty \frac{\sin^2\!\big( (\omega_0 - \omega) t / 2 \big)}{(\omega_0 - \omega)^2} u(\omega) \dd{\omega}
+\end{aligned}$$
+
+From here, the derivation is similar to that of
+[Fermi's golden rule](/know/concept/fermis-golden-rule/),
+despite the distinction that we are integrating over frequencies rather than states.
+
+At sufficiently large $t$, the integrand is sharply peaked at $\omega = \omega_0$
+and negligible everywhere else,
+so we take $u(\omega)$ out of the integral and extend the integration limits.
+Then we rewrite and look up the integral,
+which turns out to be $\pi t$:
+
+$$\begin{aligned}
+    P_{12}
+    = \frac{|d|^2}{\varepsilon_0 \hbar^2} u(\omega_0) \int_{-\infty}^\infty \frac{\sin^2\!\big(x t \big)}{x^2} \dd{x}
+    = \frac{\pi |d|^2}{\varepsilon_0 \hbar^2} u(\omega_0) \:t
+\end{aligned}$$
+
+From this, the transition rate $R_{12} = B_{12} u(\omega_0)$
+is then calculated as follows:
+
+$$\begin{aligned}
+    R_{12}
+    = \pdv{P_{12}}{t}
+    = \frac{\pi |d|^2}{\varepsilon_0 \hbar^2} u(\omega_0)
+\end{aligned}$$
+
+Using the relations from earlier with $g_1 = g_2$,
+the Einstein coefficients are found to be as follows
+for a polarized incoming light spectrum:
+
+$$\begin{aligned}
+    \boxed{
+        B_{21} = B_{12} = \frac{\pi |d|^2}{\varepsilon_0 \hbar^2}
+        \qquad
+        A_{21} = \frac{\omega_0^3 |d|^2}{\pi \varepsilon_0 \hbar c^3}
+    }
+\end{aligned}$$
+
+
+## Unpolarized light
+
+We can generalize the above result even further to unpolarized light.
+Let us return to the matrix elements of the perturbation $\hat{H}_1$,
+and define the polarization unit vector $\vec{n}$:
+
+$$\begin{aligned}
+    \matrixel{2}{\hat{H}_1}{1}
+    = - \vec{d} \cdot \vec{E}_0
+    = - E_0 (\vec{d} \cdot \vec{n})
+\end{aligned}$$
+
+Where $\vec{d} \equiv q \matrixel{2}{\vec{r}}{1}$ is
+the full **transition dipole moment** vector, which is usually complex.
+
+The goal is to calculate the average of $|\vec{d} \cdot \vec{n}|^2$.
+In [spherical coordinates](/know/concept/spherical-coordinates/),
+we integrate over all directions $\vec{n}$ for fixed $\vec{d}$,
+using that $\vec{d} \cdot \vec{n} = |\vec{d}| \cos(\theta)$
+with $|\vec{d}| \equiv |d_x|^2 \!+\! |d_y|^2 \!+\! |d_z|^2$:
+
+$$\begin{aligned}
+    \Expval{|\vec{d} \cdot \vec{n}|^2}
+    = \frac{1}{4 \pi} \int_0^\pi \int_0^{2 \pi} |\vec{d}|^2 \cos^2(\theta) \sin(\theta) \dd{\varphi} \dd{\theta}
+\end{aligned}$$
+
+Where we have divided by $4\pi$ (the surface area of a unit sphere) for normalization,
+and $\theta$ is the polar angle between $\vec{n}$ and $\vec{d}$.
+Evaluating the integrals yields:
+
+$$\begin{aligned}
+    \Expval{|\vec{d} \cdot \vec{n}|^2}
+    = \frac{2 \pi}{4 \pi} |\vec{d}|^2 \int_0^\pi \cos^2(\theta) \sin(\theta) \dd{\theta}
+    = \frac{|\vec{d}|^2}{2} \Big[ \!-\! \frac{\cos^3(\theta)}{3} \Big]_0^\pi
+    = \frac{|\vec{d}|^2}{3}
+\end{aligned}$$
+
+With this additional constant factor $1/3$,
+the transition rate $R_{12}$ is modified to:
+
+$$\begin{aligned}
+    R_{12}
+    = \pdv{P_{12}}{t}
+    = \frac{\pi |\vec{d}|^2}{3 \varepsilon_0 \hbar^2} u(\omega_0)
+\end{aligned}$$
+
+From which it follows that the Einstein coefficients for unpolarized light are given by:
+
+$$\begin{aligned}
+    \boxed{
+        B_{21} = B_{12} = \frac{\pi |\vec{d}|^2}{3 \varepsilon_0 \hbar^2}
+        \qquad
+        A_{21} = \frac{\omega_0^3 |\vec{d}|^2}{3 \pi \varepsilon_0 \hbar c^3}
+    }
+\end{aligned}$$
+
+
+
+## References
+1.  M. Fox,
+    *Optical properties of solids*, 2nd edition,
+    Oxford.
+2.  D.J. Griffiths, D.F. Schroeter,
+    *Introduction to quantum mechanics*, 3rd edition,
+    Cambridge.