The third process equilibrium demands

The thin foils that stop radiant heat took Planck’s curve as given and did engineering with it. The curve that would not come down showed how Planck reached it in 1900, by a count of ways to divide up energy among oscillators in the walls of a cavity. For sixteen years, that count was all there was. It reached the right answer from a model of matter that no one believed in. Nobody could say what atoms were doing as they exchanged energy with light, and yet that exchange was what had to bring the radiation to equilibrium.

Albert Einstein, in 1916, attacked the question from the opposite side. He had just completed general relativity. He set aside any model of how atoms emit and asked only what rates they must have if they are to reach equilibrium with radiation at all. Three lines of algebra then produced Planck’s law, with no oscillators and no counting of energy elements, and in addition three results that nobody was looking for: that a photon carries momentum, that an atom’s emission is random in direction and time, and that a process of emission must exist in which a light wave causes an excited atom to emit a copy of itself. This last is known as stimulated emission, and it was an inference from a curve.

Two rates cannot balance to the right answer #

Take a gas of atoms with two energy levels, lower and upper, separated by $h\nu$, in a cavity full of radiation with energy density $\rho(\nu)$ at that frequency. In equilibrium the numbers of atoms in the levels stand in the ratio of the exponential that decides everything:

$$\frac{N_2}{N_1} = e^{-h\nu/kT}$$

for two levels with equal statistical weight. An atom in the lower level can absorb at a rate proportional to the radiation: $N_1 B\rho$. An atom in the upper level can decay on its own, whatever else is going on: $N_2 A$. Bohr had supplied absorption and emission as quantum jumps three years earlier, and Rutherford had pointed out that a jump looks like a radioactive decay, with a rate constant $A$.

Set those two rates equal, and solve for $\rho$. The result is immediate:

$$\rho = \frac{A}{B}\frac{N_2}{N_1} = \frac{A}{B}e^{-h\nu/kT}.$$

That is Wien’s law, the 1896 formula that fits the high-frequency side of the spectrum and fails at low frequency. Now the failure is informative. Nothing has been assumed except the Boltzmann factor and the existence of those rates. Two processes alone cannot give Planck’s law, and it is the balance, not a detail of atoms, that says so.

Einstein then did something that took some imagination. He added a third rate, an emission from the upper level proportional to the radiation, with a coefficient $B'$:

$$N_1 B\rho = N_2 A + N_2 B'\rho.$$

In classical physics, this makes sense: a dipole already oscillating, in a driving wave, may gain from the wave or lose to it, according to phase. An excited atom may therefore be expected to lose energy to a passing wave. Solve for $\rho$ again, with $N_2/N_1$ as before.

$$\rho = \frac{A/B'}{(B/B')e^{h\nu/kT} - 1}.$$

If this is to approach infinity as the temperature rises, so that an arbitrarily hot cavity holds an arbitrarily large radiation density, $B$ and $B'$ must be equal for equal-weight levels. With that, the expression has Planck’s form, with the $-1$ in the denominator due entirely to stimulated emission.

That $-1$ has a good claim to being the most important one in physics. It distinguishes Planck from Wien, and it is the term that makes the low-frequency side classical. Viewed from another angle, it is what makes photons bosons, since it can be rewritten as a statement that the emission rate into a mode with $n$ photons is proportional to $n + 1$: the 1 is spontaneous and the $n$ is stimulated. The outcomes identical photons refuse shows that same factor at work in a beam splitter, in a very different experiment.

The ratio that gets the mode count #

Einstein’s argument also gave the value of the ratio $A/B$. Requiring that Planck’s law hold, or requiring the classical limit at low frequency, fixes it:

$$\frac{A}{B} = \frac{8\pi h\nu^3}{c^3}.$$

The $\nu^3$ in this ratio can be read as a count: $8\pi\nu^2/c^3$ is the number of modes of the field per unit volume per unit frequency, and $h\nu$ is the energy of a photon in one of them. Spontaneous emission is emission into all the empty modes, and stimulated emission is emission into one that already has a photon. How many ways there are to vibrate performed the mode count for any wave. Since spontaneous emission is spread across all the available modes, it depends on how many there are, and the lifetime that depends on where the mirror is changes the rate just by changing that count.

Two remarkable features of this result went unnoticed for some time. The first is that the coefficients are properties of the atom alone, with no reference to any temperature, so they must apply out of equilibrium too. An excited atom in a beam of laser light has the same $A$ and $B$ as one in a furnace. So does a molecule in interstellar space, and any calculation of a stellar atmosphere relies on those two numbers. The argument for their ratio was made in equilibrium, and its result applies everywhere. The second is that nothing in the argument mentions what an atom actually is. Dirac’s quantum theory of radiation derived the coefficients from first principles in 1927, and found that the ratio is exactly what Einstein said, along with the mode-count explanation.

How much of the glow is stimulated #

The ratio of stimulated to spontaneous emission at a frequency in thermal light is $B\rho/A$, and with Planck’s law it reduces to something very simple. It equals the mean number of photons per mode:

$$\frac{B\rho}{A} = \bar{n} = \frac{1}{e^{h\nu/kT} - 1}.$$

Sunlight at green wavelengths has $h\nu/kT = 4.5$, and the ratio is about 0.011. Almost all emission from the Sun is spontaneous, which is why sunlight is not coherent over any useful distance, and why a star is not a laser. Why two lamps never interfere deals with the consequences. A warm body at 310 K glowing at 10 µm has nearly the same ratio, 0.010, since it is at about an eighteenth of the temperature and glows at about an eighteenth of the frequency; only the ratio of energies counts.

At the other extreme, consider hydrogen’s 21 cm line in the cosmic microwave background, at a temperature of 2.7 K. The photon’s energy is only about a fortieth of the thermal energy, so $\bar{n}$ is about 39, and stimulated emission exceeds spontaneous emission by that factor, in ordinary interstellar hydrogen, without any inversion. Radio astronomers use a form of the radiative transfer equation that assumes this from the outset, and the line that takes eleven million years notes how long a hydrogen atom would otherwise wait. In the radio band, stimulated emission is normal, not exotic. The laser is the unusual case for the opposite reason: it requires stimulated emission to dominate in the visible, where nature does not supply enough photons per mode, by a factor of a thousand or more.

The same ratio gives the condition for that. Stimulated emission outruns absorption only when $N_2 > N_1$, since their coefficients are equal, and that can never be true in thermal equilibrium at a positive temperature. Hotter than any temperature there is puts that condition another way, as a negative temperature, which is a more honest description than it sounds.

Why light alone cannot invert two levels #

It might seem that a sufficiently bright lamp could take care of this, by lifting most of the atoms in a gas to the upper level. The third rate forbids it.

For two levels in steady light, setting the rates equal gives an upper fraction of $B\rho/(A + 2B\rho)$. Strong light makes the fraction approach one half, because absorption and stimulated emission then dominate, and they are equal for equal numbers of atoms. Light cannot produce an inversion in a two-level system, however intense, because the light that lifts an atom works equally well in bringing it down. A gas driven very hard simply becomes transparent at the driving frequency; that is saturation, and it is also how the atoms the light stops touching get to stay as cold as they do.

The route around this was found by Nikolay Basov and Aleksandr Prokhorov in Moscow and Charles Townes at Columbia, in the early 1950s. Townes’s ammonia maser of 1954 sidestepped the problem by physically separating molecules in the two states in an inhomogeneous electric field and sending only the upper ones into the cavity. For a practical device, three levels are needed, and Basov and Prokhorov proposed this: pump the atoms from a ground level to a high level; let them fall quickly to a long-lived intermediate level; and lase from that level to the ground. Atoms pile up in the long-lived level, and it can hold more than half of them, since the pump transition is at a different frequency and the pumping light does nothing to bring them down. Theodore Maiman’s ruby laser of 1960 used exactly this scheme, with three levels and a flash lamp. Four-level systems, such as neodymium in glass, do better, since the lower lasing level is empty from the start.

Thus the third process both provides a laser’s gain and dictates the layout of levels that any laser has to have. It also dictates the noise, because every amplifier based on stimulated emission must add spontaneous emission as well; the coefficient ratio ensures that, and it reappears in the noise every amplifier must add as a quantum limit on any linear amplifier.

The cost of the cube #

The ratio $A/B \propto \nu^3$ makes the frequency a brutal factor in laser design. To get gain at some transition, excited atoms must be kept up there against their spontaneous decay rate $A$. The population needed for a given gain goes as the width of the transition over the stimulated cross-section; that cross-section is $\lambda^2 A/8\pi\Delta\nu$, and so the required inversion density scales as $\Delta\nu/(\lambda^2 A)$. Each atom has to be replaced at the rate $A$, with an energy of $h\nu$. Multiply these together: $A$ cancels, and the power per unit volume goes as $\nu \cdot \nu^2 \cdot \Delta\nu$. Doppler broadening has $\Delta\nu \propto \nu$, so the result is $\nu^4$.

Moving from a $\mathrm{CO}_2$ laser at 10.6 µm to 1 nm, in the X-ray region, raises this by about sixteen orders of magnitude. The first X-ray lasers in the 1980s, developed at Livermore, had a nuclear explosion as a pump. Lab-scale ones later used short, intense optical pulses to form hot plasmas, and they give a modest output near tens of nanometres. The bright X-ray free-electron lasers that began operating in 2009 skip atoms altogether, using a relativistic electron beam in an undulator, because the magnet period that comes out as an X-ray has no population to invert. The very idea of a laser at those wavelengths had to give up on the coefficient that Einstein found.

The same factor has a benign side at long wavelengths. Spontaneous emission scales with $\nu^3$, so masers and radio-frequency devices have essentially no spontaneous emission, and so the noise of a maser amplifier can be small. In stars the same scaling has a further consequence: an atomic line’s lifetime at 500 nm is nanoseconds, while a line at 21 cm has a lifetime of millions of years, partly because that transition is magnetic and mainly because of this cube.

Other consequences of a three-line argument #

The same paper went on to consider momentum, and it has had a separate history. An atom absorbing from a beam gets a kick of $h\nu/c$; for a gas to reach a Maxwell distribution of velocities by interacting with radiation, Einstein showed, the same recoil must occur in each spontaneous emission, in a random direction. A photon must therefore carry directed momentum. A photon with a momentum follows the experimental story of that claim, which Compton confirmed only in 1923. Einstein’s reasoning also reveals an irritating feature: there seems to be no way to predict when or in what direction a particular atom emits. He wrote that this gave him pause, and it remained a sore point in his later attacks on the quantum theory.

The factor of $n + 1$ has also been important for other fields. The heat that turns into matter concerns detailed balance in a hot gas, where stimulated terms in pair processes matter; in the early universe, radiation chemistry needs these in the same way. Induced Compton scattering, where light scatters from electrons in a stimulated process, is also the kind of thing that sets the temperature a gas takes from the light around it.

Rates without phases #

Einstein’s argument gives a relation between rates; it says nothing about their absolute size. Those have to be computed from the atom’s wave functions or measured. It also does not explain coherence. That a stimulated photon has the same phase and direction as its trigger is not a part of the rate equations, and it requires a theory of the field. Einstein implied direction and frequency in his own account, but phase was first treated fully in Dirac’s 1927 theory. A rate equation also ignores any memory: atoms driven by a coherent laser undergo Rabi oscillations instead of relaxing to a steady population, and such oscillations are not contained in the curve for saturation drawn here, although they damp down to it when the light is incoherent. The levels are assumed sharp and of equal statistical weight, and the correction for unequal weights is a simple factor, with the essential argument unchanged.

The pump curve in the last figure is a scaling and assumes Doppler broadening throughout. The actual threshold for any laser depends on its cavity, its medium and its gain length, and an X-ray laser with only a single pass through a long plasma avoids the cavity entirely. So the figure reflects a trend and not a recipe.

Still open: whether gain without inversion is more than a curiosity #

A long-established result seems to be absolute: no inversion, no gain. It turns out to hold only for incoherent systems. In the late 1980s it was shown theoretically that a three-level system with quantum coherence between two of its levels could amplify light without a population inversion in any basis. A preparing field creates a coherence that cancels absorption while leaving emission intact. That is called lasing without inversion, and such gain has been demonstrated in atomic vapours. It might in principle ease the $\nu^4$ cost by a large factor, since the pump could work at a lower frequency than the output.

Whether it can be scaled to the X-ray region, where the need is greatest, has not been established, because the coherence needed is hard to preserve with strong spontaneous decay present. A second question is whether superradiant atoms or molecules can shed their energy together at a rate proportional to $N^2$ and not $N$, without any mirrors. That has been found to occur in some atomic gases under the right conditions and is still being studied for practical uses. Each of these looks for a way around the rate equation by exploiting the phase it leaves out.

Absorption and spontaneous emission alone balance to Wien’s law, which fails at low frequency. Adding a stimulated process with an equal coefficient gives Planck’s, with its $-1$. That one inference fixes $A/B = 8\pi h\nu^3/c^3$, caps a two-level population at one half and sends the cost of a laser up as about $\nu^4$. Stimulated emission is the rule in radio and the exception in the visible, because only the number of photons per mode decides which it is.