Einstein Coefficients

Einstein coefficients package three weak-field radiative processes between a lower level $1$ and an upper level $2$:

$$\begin{aligned} R_{1\rightarrow2} &= B_{12}^{(\omega)} u_\omega(\omega_0), \\ R_{2\rightarrow1} &= A_{21} + B_{21}^{(\omega)} u_\omega(\omega_0). \end{aligned}$$

Here

$$\hbar\omega_0=E_2-E_1>0.$$

The three terms describe:

• absorption, proportional to $B_{12}^{(\omega)}$;
• stimulated emission, proportional to $B_{21}^{(\omega)}$;
• spontaneous emission, proportional to $A_{21}$.

The superscript $(\omega)$ is not decorative. It declares that $u_\omega(\omega)$ is the total isotropic radiation energy density per unit angular frequency, summed over propagation directions and polarizations. A different spectral variable changes the numerical $B$ coefficient.

The $A$ coefficient is a rate. A $B$ coefficient is not: it becomes a rate only after multiplication by the matching spectral energy density. Neither coefficient is a finite-time Born-rule probability.

Required background. Transition rates distinguishes rates from finite-time probabilities; oscillator strengths defines the E1 line strengths converted below.

The model treats weak incoherent coupling to a free-space radiation reservoir, with uniformly populated substates in each degenerate level. The levels are sharp enough to evaluate the radiation density at $\omega_0$. Coherent driving, anisotropic substate preparations, and structured reservoirs require their corresponding field and density-matrix descriptions.

A and B Coefficients

Spontaneous-emission coefficient A

The coefficient $A_{21}$ is the spontaneous transition rate per system prepared in level $2$ for the specified downward radiative channel:

$$dP_{2\rightarrow1} =p_2(t)A_{21}\,dt, \qquad [A_{21}]=\mathrm{s}^{-1}.$$

Here $p_2(t)$ is the upper-level population probability and $dt$ is a coarse-grained rate interval. Thus $A_{21}$ is conditional on still occupying level 2; it is not the derivative of an exact transition probability at zero time.

For a population $N_2$, the expected number of photons emitted into that channel per unit time is

$$\dot N_{\gamma,21} = N_2A_{21}$$

when reabsorption, stimulated processes, and detector effects are excluded. The corresponding emitted power is

$$\mathcal P_{21} = N_2\hbar\omega_0A_{21}.$$

Atomic databases often call $A_{21}$ a transition probability, but its units show that it is a probability rate, not a dimensionless probability.

Radiative lifetime and branching

If upper level $2$ can decay radiatively to several lower levels $\ell$, its total radiative width in inverse-time units is

$$\Gamma_{\mathrm{rad},2} = \sum_\ell A_{2\ell}.$$

The radiative lifetime is

$$\tau_{\mathrm{rad},2} = \frac{1}{\Gamma_{\mathrm{rad},2}}.$$

The radiative branching fraction into channel $2\rightarrow1$ is

$$b_{21}^{(\mathrm{rad})} = \frac{A_{21}} {\sum_\ell A_{2\ell}}.$$

If nonradiative, collisional, predissociative, or quenching channels are present,

$$\Gamma_{\mathrm{tot},2} = \Gamma_{\mathrm{rad},2} + \Gamma_{\mathrm{nr},2} + \Gamma_{\mathrm{coll},2} + \cdots,$$

and

$$\tau_{\mathrm{obs},2} = \frac{1}{\Gamma_{\mathrm{tot},2}}.$$

The radiative quantum yield is

$$\Phi_{\mathrm{rad}} = \frac{\Gamma_{\mathrm{rad},2}} {\Gamma_{\mathrm{tot},2}}.$$

Thus one $A$ coefficient, a radiative lifetime, an observed lifetime, and a fluorescence quantum yield are four different quantities.

Stimulated coefficients B

The coefficient $B_{12}^{(\omega)}$ converts spectral energy density into an absorption rate:

$$R_{1\rightarrow2}^{(\mathrm{abs})} = B_{12}^{(\omega)} u_\omega(\omega_0).$$

Similarly,

$$R_{2\rightarrow1}^{(\mathrm{stim})} = B_{21}^{(\omega)} u_\omega(\omega_0).$$

In SI units,

$$\left[ u_\omega \right] = \mathrm{J\,m^{-3}} \left(\mathrm{rad\,s^{-1}}\right)^{-1},$$

so, treating radians as dimensionless,

$$\left[ B^{(\omega)} \right] = \mathrm{m^3\,J^{-1}\,s^{-2}}.$$

Only the products $B_{12}^{(\omega)}u_\omega$ and $B_{21}^{(\omega)}u_\omega$ have dimensions of inverse time.

Finite line profiles

For a normalized absorption profile,

$$\int_0^\infty \phi_{12}^{(\omega)}(\omega)\,d\omega =1,$$

the weak-field rate is more generally

$$R_{1\rightarrow2}^{(\mathrm{abs})} = B_{12}^{(\omega)} \int_0^\infty u_\omega(\omega) \phi_{12}^{(\omega)}(\omega)\,d\omega.$$

If $u_\omega$ varies negligibly across the line, this reduces to

$$R_{1\rightarrow2}^{(\mathrm{abs})} \simeq B_{12}^{(\omega)} u_\omega(\omega_0).$$

For spontaneous emission, a normalized profile can distribute the total coefficient:

$$\begin{aligned} \frac{dA_{21}}{d\omega} &= A_{21} \phi_{21}^{(\omega)}(\omega), \\ \int_0^\infty \frac{dA_{21}}{d\omega}\,d\omega &= A_{21}. \end{aligned}$$

The integrated coefficient and the spectral line shape are separate inputs. For broad vibronic bands, absorption and emission profiles need not be identical, and the sharp-line detailed-balance derivation below should not be applied point by point without additional assumptions.

Ordinary frequency versus angular frequency

Let $u_\nu(\nu)$ be energy density per unit ordinary frequency. Conservation of energy in a spectral interval requires

$$u_\nu(\nu)\,d\nu = u_\omega(\omega)\,d\omega.$$

Because

$$\omega=2\pi\nu, \qquad d\omega=2\pi\,d\nu,$$

the densities satisfy

$$u_\nu(\nu) = 2\pi u_\omega(\omega).$$

The same physical rate can be written

$$B_{ij}^{(\nu)}u_\nu = B_{ij}^{(\omega)}u_\omega,$$

so

$$B_{ij}^{(\nu)} = \frac{1}{2\pi} B_{ij}^{(\omega)}.$$

An unlabeled $B_{ij}$ is therefore incomplete. The factors of $2\pi$ cannot be repaired by dimensional analysis because the two $B$ conventions have the same SI dimensions when radians are treated as dimensionless.

Energy density is not beam intensity

Einstein’s thermal argument uses isotropic spectral energy density. For isotropic specific intensity $I_\nu$,

$$u_\nu = \frac{4\pi}{c} I_\nu.$$

A collimated polarized laser beam is not isotropic. Its irradiance, polarization, propagation direction, and coherence must be converted to the field quantity appropriate to the transition. In coherent spectroscopy, a Rabi frequency or optical Bloch description is usually clearer than inserting beam intensity into $B u$ as though it were an isotropic blackbody density.

Detailed Balance

Thermal level populations

At temperature $T$, thermal equilibrium gives

$$\frac{N_2}{N_1} = \frac{g_2}{g_1} \exp\left( -\frac{\hbar\omega_0}{k_{\mathrm B}T} \right).$$

Define

$$\beta = \frac{1}{k_{\mathrm B}T}.$$

The ratio can then be written

$$\frac{N_2}{N_1} = \frac{g_2}{g_1} e^{-\beta\hbar\omega_0}.$$

The degeneracies matter because thermal equilibrium populates individual microstates according to Boltzmann weights. A level with more substates has a larger total population at the same energy per substate.

Balance of upward and downward events

At equilibrium, the total upward transition count equals the total downward count:

$$N_1B_{12}^{(\omega)}u_\omega = N_2 \left[ A_{21} + B_{21}^{(\omega)}u_\omega \right].$$

Solving for the equilibrium radiation density gives

$$u_\omega = \frac{A_{21}} { \dfrac{N_1}{N_2} B_{12}^{(\omega)} - B_{21}^{(\omega)} }.$$

Using the Boltzmann population ratio,

$$u_\omega = \frac{A_{21}} { \dfrac{g_1}{g_2} B_{12}^{(\omega)} e^{\beta\hbar\omega_0} - B_{21}^{(\omega)} }.$$

Einstein’s three-process rate model for two sharp levels in an isotropic radiation field. Thermal equilibrium fixes the degeneracy relation between the two stimulated coefficients and the ratio of spontaneous to stimulated emission coefficients.

Matching Planck’s law

The blackbody energy density per unit angular frequency is

$$u_\omega(\omega,T) = \frac{\hbar\omega^3} {\pi^2c^3} \frac{1} {e^{\beta\hbar\omega}-1}.$$

For the rate-balance expression to have this form at every temperature, the coefficients must obey

$$g_1B_{12}^{(\omega)} = g_2B_{21}^{(\omega)}$$

and

$$\frac{A_{21}} {B_{21}^{(\omega)}} = \frac{\hbar\omega_0^3} {\pi^2c^3}.$$

In ordinary-frequency notation, the same relations are

$$g_1B_{12}^{(\nu)} = g_2B_{21}^{(\nu)}$$

and

$$\frac{A_{21}} {B_{21}^{(\nu)}} = \frac{8\pi h\nu_0^3} {c^3}.$$

The two $A/B$ formulas are physically equivalent. Their different numerical prefactors follow entirely from the definitions of spectral density and $B$.

What equilibrium does and does not determine

Thermodynamic balance determines:

• the ratio of absorption to stimulated-emission coefficients after degeneracies are included;
• the ratio of spontaneous to stimulated-emission coefficients after the radiation-density convention is fixed.

It does not determine the absolute magnitude of $A_{21}$ or $B_{21}^{(\omega)}$. That magnitude depends on the transition matrix element. Modern quantum electrodynamics supplies it microscopically.

The first relation is sometimes written $B_{12}=B_{21}$. That is correct for two nondegenerate states or for matched state-to-state coefficients. For level-averaged coefficients,

$$B_{12}^{(\omega)} = \frac{g_2}{g_1} B_{21}^{(\omega)}.$$

Ignoring this distinction is a common source of factors $2J+1$.

Microscopic reversibility

For resolved substates and a specified photon mode, absorption and emission use adjoint pieces of the interaction and have equal material matrix-element magnitudes. For circular polarization this involves the conjugate field component; spherical tensor components obey $(D_q)^\dagger=(-1)^qD_{-q}$. Level-averaged absorption sums over final substates while stimulated emission averages over a different initial manifold. Counting those substates produces

$$g_1B_{12}=g_2B_{21}.$$

The degeneracy relation is therefore consistent with microscopic reversibility; it does not assert that the two level-averaged numbers must be equal.

Blackbody Radiation Connection

Photon occupation form

The thermal mean occupation of a photon mode is

$$\overline n(\omega,T) = \frac{1} {e^{\beta\hbar\omega}-1}.$$

Planck’s angular-frequency energy density can be factored as

$$u_\omega(\omega,T) = u_0^{(\omega)}(\omega) \overline n(\omega,T),$$

where

$$u_0^{(\omega)}(\omega) = \frac{\hbar\omega^3} {\pi^2c^3}.$$

This is a convenient free-space mode-density scale, not the divergent zero-point energy density of the electromagnetic vacuum. At the transition frequency it is precisely the ratio

$$u_0^{(\omega)}(\omega_0) = \frac{A_{21}} {B_{21}^{(\omega)}}.$$

It converts the spontaneous rate to the stimulated coefficient.

Thermal upward and downward rates

Using the Einstein relations,

$$B_{21}^{(\omega)} u_\omega(\omega_0,T) = A_{21} \overline n(\omega_0,T).$$

Thus the downward rate per system in level $2$ is

$$R_{2\rightarrow1}^{(\mathrm{th})} = A_{21} \left[ \overline n(\omega_0,T)+1 \right].$$

The upward rate per system in level $1$ is

$$R_{1\rightarrow2}^{(\mathrm{th})} = \frac{g_2}{g_1} A_{21} \overline n(\omega_0,T).$$

For equal degeneracies, these reduce to the familiar quantum-optical expressions proportional to $\overline n$ and $\overline n+1$.

The identity

$$\overline n+1 = e^{\beta\hbar\omega_0} \overline n$$

ensures that the Boltzmann population ratio makes total upward and downward event counts equal.

Low- and high-temperature limits

When

$$\hbar\omega_0\gg k_{\mathrm B}T,$$

the occupation is

$$\overline n \simeq e^{-\beta\hbar\omega_0}\ll1.$$

Stimulated processes from a thermal field are then negligible compared with spontaneous emission. This is typical for visible transitions at room temperature.

When

$$\hbar\omega_0\ll k_{\mathrm B}T,$$

the occupation is

$$\overline n \simeq \frac{k_{\mathrm B}T} {\hbar\omega_0}\gg1.$$

Stimulated emission and absorption dominate over the extra spontaneous $+1$. Microwave and radio-frequency transitions can therefore be strongly affected by thermal photons even at ordinary laboratory temperatures.

The stimulated and spontaneous downward contributions are equal when

$$\overline n=1,$$

or

$$T = \frac{\hbar\omega_0} {k_{\mathrm B}\ln 2}.$$

Historical logic

Einstein introduced the three-process rate structure in 1916–1917 and used thermal equilibrium with the known radiation law to constrain the coefficients. Stimulated emission emerged as a necessary process: absorption and spontaneous emission alone cannot reproduce Planck’s denominator.

The argument was extraordinarily predictive, but it was not yet a modern quantized-field calculation of $A_{21}$. Dirac’s 1927 radiation theory and later quantum electrodynamics supplied a microscopic account of emission and absorption amplitudes. The historical development should not be compressed into the claim that equilibrium thermodynamics alone calculated every coefficient.

Relation to Transition Dipole Moments

Nondegenerate E1 transition

For a nondegenerate electric-dipole transition in vacuum, define

$$\mathbf d_{21} = \langle2| \widehat{\mathbf D} |1\rangle.$$

The free-space spontaneous-emission coefficient is

$$A_{21} = \frac{\omega_0^3} {3\pi\epsilon_0\hbar c^3} \left| \mathbf d_{21} \right|^2.$$

Combining this with the angular-frequency Einstein relation gives

$$B_{21}^{(\omega)} = \frac{\pi} {3\epsilon_0\hbar^2} \left| \mathbf d_{21} \right|^2.$$

For two nondegenerate states,

$$B_{12}^{(\omega)} = B_{21}^{(\omega)}.$$

In ordinary-frequency notation,

$$B_{21}^{(\nu)} = \frac{1}{6\epsilon_0\hbar^2} \left| \mathbf d_{21} \right|^2.$$

These equations display the convention dependence directly:

$$B_{21}^{(\omega)} = 2\pi B_{21}^{(\nu)}.$$

Degenerate angular-momentum levels

For an E1 transition between levels with reduced line strength

$$S_{21} = \left| \left\langle \gamma_2J_2 \left\| \widehat D^{(1)} \right\| \gamma_1J_1 \right\rangle \right|^2,$$

the level-averaged coefficients are

$$\begin{aligned} A_{21} &= \frac{\omega_0^3} {3\pi\epsilon_0\hbar c^3g_2} S_{21}, \\ B_{21}^{(\omega)} &= \frac{\pi} {3\epsilon_0\hbar^2g_2} S_{21}, \\ B_{12}^{(\omega)} &= \frac{\pi} {3\epsilon_0\hbar^2g_1} S_{21}. \end{aligned}$$

The first coefficient averages over upper substates; the absorption coefficient averages over lower substates. The common reduced line strength makes the degeneracy relation manifest.

Relation to oscillator strength

For an E1 absorption oscillator strength $f_{12}$,

$$A_{21} = \frac{e^2\omega_0^2} {2\pi\epsilon_0m_ec^3} \frac{g_1}{g_2} f_{12}.$$

The stimulated coefficients are

$$\begin{aligned} B_{12}^{(\omega)} &= \frac{\pi e^2} {2\epsilon_0m_e\hbar\omega_0} f_{12}, \\ B_{21}^{(\omega)} &= \frac{g_1}{g_2} B_{12}^{(\omega)}. \end{aligned}$$

The line-strength and averaging conventions are explained in Oscillator Strengths.

Frequency scaling

For a fixed transition dipole,

$$A_{21}\propto\omega_0^3, \qquad B_{21}^{(\omega)} \propto \left|\mathbf d_{21}\right|^2.$$

The spontaneous rate carries the free-space photon mode-density factor $\omega_0^2$ and one additional frequency from field normalization. The angular-frequency $B$ coefficient does not carry that $\omega_0^3$ factor. The relation

$$\frac{A_{21}} {B_{21}^{(\omega)}} \propto \omega_0^3$$

therefore reflects the available free-space radiation modes, not a temperature dependence of either intrinsic coefficient.

Beyond electric dipole and free space

Einstein $A$ coefficients are also tabulated for M1, E2, and higher-multipole transitions, but their microscopic line-strength formulas differ from the E1 expressions above. The equilibrium $A/B$ relation for a sharp one-photon transition still relies on the declared radiation-mode convention.

In a cavity, waveguide, dielectric, near a surface, or in another structured photonic environment, spontaneous emission depends on the local electromagnetic density of states and field mode functions. The free-space formula can be enhanced, suppressed, or made anisotropic. One should then use an environment-specific mode sum or electromagnetic Green tensor rather than silently reusing the vacuum $A_{21}$. See Spontaneous Emission for the Green-tensor and Purcell-factor preview.

Link to Lasers and Spontaneous Emission

Net stimulated gain

In a radiation field, the net stimulated downward event rate per volume is

$$\begin{aligned} \mathcal R_{\mathrm{stim,net}} &= N_2B_{21}^{(\omega)}u_\omega - N_1B_{12}^{(\omega)}u_\omega \\ &= B_{21}^{(\omega)}u_\omega \left( N_2 - \frac{g_2}{g_1}N_1 \right). \end{aligned}$$

Net gain requires

$$\frac{N_2}{g_2} > \frac{N_1}{g_1}.$$

This is population inversion per substate. The simpler condition $N_2>N_1$ is equivalent to this criterion when $g_1=g_2$. For unequal degeneracies it can be too weak or unnecessarily restrictive.

A thermal two-level ensemble cannot satisfy this inequality because

$$\frac{N_2/g_2}{N_1/g_1} = e^{-\beta\hbar\omega_0} <1.$$

A nonequilibrium pump is required. Direct bidirectional Einstein pumping of a closed two-level system approaches equal populations per substate from below; additional levels or a selective injection/removal process can maintain inversion. A coherent $\pi$ pulse can transiently invert a two-level system, which is a different dynamical problem.

Role of spontaneous emission

For independently excited emitters without a prepared optical phase, spontaneous emission depletes the upper state and supplies fluorescence and quantum noise. Its directions and polarizations follow the transition and environment; they are not universally equiprobable.

Stimulated emission increases the occupation of an already populated field mode, with its frequency, polarization, and spatial mode function. In a classical coherent-field limit it provides gain with a phase relation to that field. A number-state photon has no definite optical phase, so the process should not be pictured as copying an arbitrary photon’s unknown quantum state.

The Einstein model describes population transfer and small-signal gain. Laser phase dynamics require a Maxwell–Bloch, master-equation, or other quantum-optical treatment.

When rate equations fail

The Einstein equations assume that optical coherence can be neglected and that transition events can be described by approximately constant rates. They become inadequate when:

• a coherent pulse drives resolved Rabi oscillations;
• the field dresses the levels or creates coherences that cannot be eliminated on the population timescale;
• coherence between degenerate substates matters;
• one cavity mode exchanges excitations reversibly with the emitter;
• reservoir memory or strong light–matter coupling invalidates Markovian decay;
• interference among multiple pathways changes the transition amplitude.

Saturation alone does not prove rate equations invalid: population equations can describe saturation by incoherent pumping, and suitable adiabatic elimination can yield saturated steady-state rates. Resolved coherent transients require more information.

The appropriate coherent two-level treatment is developed in Optical Bloch Equations.

Spontaneous emission is not classical driving

A prescribed classical field can produce absorption and stimulated emission, but it does not derive spontaneous emission from an initially empty field. In the quantized-field description, downward coupling to a photon mode has the matrix-element factor

$$\sqrt{n+1},$$

while absorption has

$$\sqrt n.$$

The surviving unity at $n=0$, summed over vacuum modes, produces the spontaneous rate. The full derivation and free-space dipole radiation pattern are derived in Transition Rates in Light–Matter Interaction.

Exercises

Exercise 1: Convert B between spectral conventions

Show that

$$B_{21}^{(\nu)} = \frac{B_{21}^{(\omega)}}{2\pi}$$

when $u_\nu$ and $u_\omega$ describe the same radiation field. Verify that the stimulated rate is unchanged.

Solution

The spectral energy in a physical interval is invariant:

$$u_\nu\,d\nu = u_\omega\,d\omega.$$

Since $d\omega=2\pi\,d\nu$,

$$u_\nu=2\pi u_\omega.$$

Rate invariance requires

$$B_{21}^{(\nu)}u_\nu = B_{21}^{(\omega)}u_\omega.$$

Substitution gives

$$B_{21}^{(\nu)} (2\pi u_\omega) = B_{21}^{(\omega)}u_\omega,$$

and therefore

$$B_{21}^{(\nu)} = \frac{B_{21}^{(\omega)}}{2\pi}.$$

The coefficient and density change inversely, leaving their product unchanged.

Exercise 2: Can bidirectional pumping invert two levels?

Let $g_1=g_2=1$, let the stimulated rate in each direction be $R$, and let the spontaneous decay rate be $A$. With $p_1+p_2=1$, find the stationary upper population and its limit as $R/A\to\infty$.

Solution

The population equation is

$$\dot p_2=R(1-p_2)-(R+A)p_2 =R-(2R+A)p_2.$$

Hence $p_2^{\mathrm{ss}}=R/(2R+A)<1/2$ for $A>0$. It approaches $1/2$ at large pump rate, giving saturation without inversion. For fixed $R$, deviations relax as $\exp[-(2R+A)t]$. An independently prescribed one-way pump would change the equation and its physical assumptions.

Exercise 3: Degenerate levels

An E1 line connects

$$J_1=\frac{1}{2}, \qquad J_2=\frac{3}{2}.$$

Find $B_{12}^{(\omega)}/B_{21}^{(\omega)}$ for level-averaged coefficients.

Solution

The statistical weights are

$$\begin{aligned} g_1&=2J_1+1=2, \\ g_2&=2J_2+1=4. \end{aligned}$$

Detailed balance gives

$$g_1B_{12}^{(\omega)} = g_2B_{21}^{(\omega)}.$$

Therefore

$$\frac{B_{12}^{(\omega)}} {B_{21}^{(\omega)}} = \frac{g_2}{g_1} =2.$$

This factor does not violate microscopic reversibility. It arises because the absorption coefficient averages over two lower substates while the stimulated-emission coefficient averages over four upper substates.

Exercise 4: Thermal photons at optical and microwave frequencies

Evaluate the thermal occupation

$$\overline n = \frac{1}{e^{h\nu/(k_{\mathrm B}T)}-1}$$

at $T=300\ \mathrm K$ for:

1. $\nu=5.00\times10^{14}\ \mathrm{Hz}$;
2. $\nu=1.00\times10^{10}\ \mathrm{Hz}$.

Interpret the ratio of stimulated to spontaneous downward rates.

Solution

For the optical transition,

$$\frac{h\nu}{k_{\mathrm B}T} \simeq 79.99,$$

so

$$\overline n_{\mathrm{opt}} \simeq 1.83\times10^{-35}.$$

Thermal stimulated emission is negligible compared with spontaneous emission.

For the microwave transition,

$$\frac{h\nu}{k_{\mathrm B}T} \simeq 1.600\times10^{-3},$$

so

$$\overline n_{\mathrm{mw}} \simeq 6.25\times10^2.$$

Because

$$\frac{R_{\mathrm{stim}}}{R_{\mathrm{sp}}} = \overline n,$$

thermal stimulated emission dominates for the microwave transition. The same thermal field also drives strong absorption.

Exercise 5: Lifetime, branching, and quantum yield

An upper level has two radiative channels,

$$\begin{aligned} A_{21}&=2.0\times10^7\ \mathrm{s}^{-1}, \\ A_{20}&=6.0\times10^7\ \mathrm{s}^{-1}, \end{aligned}$$

and a nonradiative decay rate

$$\Gamma_{\mathrm{nr}} = 2.0\times10^7\ \mathrm{s}^{-1}.$$

Find the radiative lifetime, observed lifetime, radiative quantum yield, and the total-event branching probabilities.

Solution

The radiative rate is

$$\Gamma_{\mathrm{rad}} = A_{21}+A_{20} = 8.0\times10^7\ \mathrm{s}^{-1},$$

so

$$\tau_{\mathrm{rad}} = 12.5\ \mathrm{ns}.$$

The total rate is

$$\Gamma_{\mathrm{tot}} = 1.0\times10^8\ \mathrm{s}^{-1},$$

giving

$$\tau_{\mathrm{obs}} = 10.0\ \mathrm{ns}.$$

The radiative quantum yield is

$$\Phi_{\mathrm{rad}} = \frac{8.0\times10^7}{1.0\times10^8} =0.80.$$

The probabilities that a decay event uses the three channels are

$$\begin{aligned} p_{21}&=0.20, \\ p_{20}&=0.60, \\ p_{\mathrm{nr}}&=0.20. \end{aligned}$$

Conditioned on a radiative event, the two radiative branching fractions are $0.25$ and $0.75$.

Exercise 6: A coefficient from a transition dipole

A nondegenerate E1 transition in vacuum has

$$\lambda=500\ \mathrm{nm}, \qquad \left|\mathbf d_{21}\right| = ea_0.$$

Estimate $A_{21}$ and the single-channel radiative lifetime. Use $ea_0=8.478\times10^{-30}\ \mathrm{C\,m}$.

Solution

The angular frequency is

$$\omega_0 = \frac{2\pi c}{\lambda} \simeq 3.767\times10^{15}\ \mathrm{s}^{-1}.$$

Then

$$A_{21} = \frac{\omega_0^3|\mathbf d_{21}|^2} {3\pi\epsilon_0\hbar c^3} \simeq 1.62\times10^7\ \mathrm{s}^{-1}.$$

If this is the only decay channel,

$$\tau_{\mathrm{rad}} = \frac{1}{A_{21}} \simeq 6.17\times10^{-8}\ \mathrm s = 61.7\ \mathrm{ns}.$$

Exercise 7: Test the gain condition

A transition has $g_1=2$ and $g_2=6$. Determine whether each population pair provides net stimulated gain:

$$\begin{array}{c|cc} &N_1&N_2\\ \hline \mathrm{A} &3.0\times10^{15} &5.0\times10^{15} \\ \mathrm{B} &3.0\times10^{15} &1.2\times10^{16} \end{array}$$

Solution

The inversion criterion is

$$\frac{N_2}{g_2} > \frac{N_1}{g_1},$$

or equivalently

$$N_2 > \frac{g_2}{g_1}N_1 = 3N_1.$$

For case A, gain would require $N_2>9.0\times10^{15}$, but $N_2=5.0\times10^{15}$. The medium absorbs.

For case B, $N_2=1.2\times10^{16}>9.0\times10^{15}$. The stimulated contribution gives net gain.

The Einstein-coefficient reference collects these relations, and the line-shape reference connects population and coherence rates to measured spectral widths.

References

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