Categories: Laser theory, Optics, Physics, Quantum mechanics, Two-level system.

Einstein coefficients

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 E1E_1 containing N1N_1 electrons, and an excited state with energy E2E_2 containing N2N_2 electrons. The resonance ω0(E2 ⁣ ⁣E1)/\omega_0 \equiv (E_2 \!-\! E_1)/\hbar is the frequency of the photon emitted when an electron falls from E2E_2 to E1E_1.

The first Einstein coefficient is the spontaneous emission rate A21A_{21}, which gives the probability per unit time that an excited electron falls from state 2 to 1, so that N2(t)N_2(t) obeys the following equation, which is easily solved:

dN2dt=A21N2    N2(t)=N2(0)exp(t/τ)\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 τ=1/A21\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 B12B_{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 N1(t)N_1(t) obeys the following equation:

dN1dt=B12N1u(ω0)\begin{aligned} \dv{N_1}{t} = - B_{12} N_1 u(\omega_0) \end{aligned}

Where u(ω)u(\omega) is the spectral energy density of the incoming light, put here to express the fact that only photons with frequency ω0\omega_0 are absorbed.

There is one more Einstein coefficient: the stimulated emission rate B21B_{21}. An incoming photon has an associated electromagnetic field, which can encourage an excited electron to drop to the ground state, such that for A21=0A_{21} = 0:

dN2dt=B21N2u(ω0)\begin{aligned} \dv{N_2}{t} = - B_{21} N_2 u(\omega_0) \end{aligned}

These three coefficients A21A_{21}, B12B_{12} and B21B_{21} are related to each other. Suppose that the system is in equilibrium, i.e. that N1N_1 and N2N_2 are constant. We assume that the number of particles in the system is constant, implying that N1(t)=N2(t)=0N_1'(t) = - N_2'(t) = 0, so:

B12N1u(ω0)=A21N2+B21N2u(ω0)=0\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(ω0)u(\omega_0), gives following expression for the radiation:

u(ω0)=A21(N1/N2)B12B21\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, meaning that the relative probability that an electron has E2E_2 compared to E1E_1 is given by the Boltzmann distribution:

Prob(E2)Prob(E1)=N2N1=g2g1exp(ω0β)\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 g2g_2 and g1g_1 are the degeneracies of the energy levels. Inserting this back into the equation for the spectrum u(ω0)u(\omega_0) yields:

u(ω0)=A21(g1/g2)B12exp(ω0β)B21\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(ω0)u(\omega_0) represents only black-body radiation, our result must agree with Planck’s law:

u(ω0)=A21B21((g1B12/g2B21)exp(ω0β)1)=ω03π2c31exp(ω0β)1\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:

A21=ω03π2c3B21g1B12=g2B21\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 E1E_1 is not the ground state, but instead some lower excited state below E2E_2, due to the principle of 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 B21N2>B12N1B_{21} N_2 > B_{12} N_1 such that N2>(g2/g1)N1N_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 B12B_{12} and B21B_{21} from the resulting transition rate. We need to make the electric dipole approximation, in which case the perturbing Hamiltonian H^1(t)\hat{H}_1(t) is given by:

H^1(t)=qrE0cos(ωt)\begin{aligned} \hat{H}_1(t) = - q \vec{r} \cdot \vec{E}_0 \cos(\omega t) \end{aligned}

Where q=eq = -e is the electron charge, r\vec{r} is the position operator, and E0\vec{E}_0 is the amplitude of the electromagnetic wave. For simplicity, we let the amplitude be along the zz-axis:

H^1(t)=qE0zcos(ωt)\begin{aligned} \hat{H}_1(t) = - q E_0 z \cos(\omega t) \end{aligned}

This form of H^1\hat{H}_1 is a well-known case for time-dependent perturbation theory, which tells us that the transition probability from a\ket{a} to b\ket{b} is (to first order):

Pab= ⁣aH1b ⁣22sin2 ⁣((ωbaω)t/2)(ωbaω)2\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=0z = 0, then generally 1\ket{1} and 2\ket{2} will be even or odd functions of zz, meaning that 1z1=2z2=0\matrixel{1}{z}{1} = \matrixel{2}{z}{2} = 0 (see also Laporte’s selection rule), leading to:

1H12=E0d2H11=E0d1H11=2H12=0\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 dq2z1d \equiv q \matrixel{2}{z}{1} is a constant, namely the zz-component of the transition dipole moment. The chance of an upward jump (i.e. absorption) is:

P12=E02d22sin2 ⁣((ω0ω)t/2)(ω0ω)2\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 PabP_{ab} is a sinc-function, and is therefore symmetric around ωba\omega_{ba}:

P21=E02d22sin2 ⁣((ω0ω)t/2)(ω0ω)2\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 R12=P12/tR_{12} = \ipdv{P_{12}}{t}, which would give us Einstein’s absorption coefficient B12B_{12}, for this specific case of coherent monochromatic light. However, the result would not be constant in time tt, 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 uu of an electric field E0E_0 is given by:

u=12ε0E02    E02=2uε0\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 ε0\varepsilon_0 is the vacuum permittivity. Putting this in the previous result for P12P_{12} gives us:

P12=2ud2ε02sin2 ⁣((ω0ω)t/2)(ω0ω)2\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 uu turns into the spectral energy density u(ω)u(\omega):

P12=2d2ε020sin2 ⁣((ω0ω)t/2)(ω0ω)2u(ω)dω\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, despite the distinction that we are integrating over frequencies rather than states.

At sufficiently large tt, the integrand is sharply peaked at ω=ω0\omega = \omega_0 and negligible everywhere else, so we take u(ω)u(\omega) out of the integral and extend the integration limits. Then we rewrite and look up the integral, which turns out to be πt\pi t:

P12=d2ε02u(ω0)sin2 ⁣(xt)x2dx=πd2ε02u(ω0)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 R12=B12u(ω0)R_{12} = B_{12} u(\omega_0) is then calculated as follows:

R12=P12t=πd2ε02u(ω0)\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 g1=g2g_1 = g_2, the Einstein coefficients are found to be as follows for a polarized incoming light spectrum:

B21=B12=πd2ε02A21=ω03d2πε0c3\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 H^1\hat{H}_1, and define the polarization unit vector n\vec{n}:

2H^11=dE0=E0(dn)\begin{aligned} \matrixel{2}{\hat{H}_1}{1} = - \vec{d} \cdot \vec{E}_0 = - E_0 (\vec{d} \cdot \vec{n}) \end{aligned}

Where dq2r1\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 dn2|\vec{d} \cdot \vec{n}|^2. In spherical coordinates, we integrate over all directions n\vec{n} for fixed d\vec{d}, using that dn=dcos(θ)\vec{d} \cdot \vec{n} = |\vec{d}| \cos(\theta) with ddx2 ⁣+ ⁣dy2 ⁣+ ⁣dz2|\vec{d}| \equiv |d_x|^2 \!+\! |d_y|^2 \!+\! |d_z|^2:

dn2=14π0π02πd2cos2(θ)sin(θ)dφdθ\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 4π4\pi (the surface area of a unit sphere) for normalization, and θ\theta is the polar angle between n\vec{n} and d\vec{d}. Evaluating the integrals yields:

dn2=2π4πd20πcos2(θ)sin(θ)dθ=d22[ ⁣ ⁣cos3(θ)3]0π=d23\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 1/31/3, the transition rate R12R_{12} is modified to:

R12=P12t=πd23ε02u(ω0)\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:

B21=B12=πd23ε02A21=ω03d23πε0c3\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}


  1. M. Fox, Optical properties of solids, 2nd edition, Oxford.
  2. D.J. Griffiths, D.F. Schroeter, Introduction to quantum mechanics, 3rd edition, Cambridge.