The lifetime that depends on where the mirror is

The clock read by counting atoms relied on what makes radiocarbon work at all: a nucleus’s decay is unaffected by anything around it. Boil it, freeze it, put it under pressure, and the half-life of carbon-14 stays the same to any measurable level. The half-life that chemistry can change found a rare exception, where electron capture depends on what the atom’s own electrons are doing. These two facts have given the idea of a lifetime a sense of being the most private thing an object possesses. A nucleus with no clock added the statistical picture, in which each atom has a fixed chance of decaying per second, with nothing to be done about it.

Spontaneous emission from an excited atom appears to be in that family. It obeys the same exponential law, and it is usually treated in the same words. An excited sodium atom has a lifetime of about 16 nanoseconds, and that number is in tables. It would seem to be a property of sodium.

It is not. The number in the tables is the lifetime of sodium in free space, and an atom’s surroundings can raise it, lower it or bring it to zero. A mirror placed near an atom will do this, without touching it and without acting on it with any force that matters. The reason is in what spontaneous emission is, and it has nothing mysterious about it, since it can be drawn with an oscillating dipole and some geometry.

What an atom emits into #

Quantum mechanics gives the rate of a transition from Fermi’s golden rule: it is proportional to the square of the coupling between the initial and final states, times the number of final states available at the right energy. For an excited atom, the initial state is “atom up, no photon”. The final states are “atom down, one photon in some mode of the electromagnetic field”. The coupling depends on the atom’s dipole moment and on the strength of that mode’s field at the atom’s position.

Both of the last two things belong to the room. What free space offers is a continuum of modes in every direction, with a count that gives the $\omega^3$ factor the third process equilibrium demands derives by another route. Put a reflecting surface near the atom and these modes are replaced by standing waves with nodes and antinodes. An atom sitting at a node of a mode cannot emit into it, and an atom sitting at an antinode couples to it more strongly than to a free mode. The count of final states at the atom’s position is thus a geometrical quantity. It is called the local density of states, and the lifetime is its reciprocal, up to constants.

That is a quantum statement. For the geometry at hand, the same result follows exactly from a classical calculation, which is much easier to picture. A dipole oscillating near a mirror interacts with its own reflected field, and the rate at which it loses energy to that field goes up or down with the relative phase. Fortunately the ratio of the emission rate near the mirror to that in free space is the same in both treatments. The solution that is thrown away discussed this in another setting, in the sense that an emitter’s rate is a measure of how its field is received.

A mirror at the distance of the wavelength #

A perfect plane mirror at a distance $d$ from a dipole acts as if there were an image dipole at $-d$. A charge’s image has the opposite sign, which sets the direction of the image dipole. A dipole across the mirror has an image pointing the same way, and one along the mirror has an image pointing the opposite way.

The rate relative to free space depends only on $x = 2kd = 4\pi d/\lambda$, and it has a compact closed form:

$$\frac{\Gamma_\perp}{\Gamma_0} = 1 - 3\left(\frac{\cos x}{x^2} - \frac{\sin x}{x^3}\right), \qquad \frac{\Gamma_\parallel}{\Gamma_0} = 1 - \frac32\left(\frac{\sin x}{x} + \frac{\cos x}{x^2} - \frac{\sin x}{x^3}\right).$$

The two limits at the surface are easy to check against intuition. As $d$ goes to zero, the dipole across the mirror has twice its original rate, and the dipole along it has none at all. Further out, each oscillates about the free rate with a period of half a wavelength in distance, because the round trip to the mirror gains a whole wavelength of phase each time, and the swings die away as $1/x$. The along-mirror rate overshoots to about 1.35 times the free value a third of a wavelength out, and its next peak, near 0.85 of a wavelength, reaches only about 1.14.

The picture at the surface explains the factor of two, which might have been expected to be four. A dipole across the mirror and its in-step image add their fields, doubling the amplitude and quadrupling the intensity everywhere in the half-space. But only the half-space exists, so the power per unit time from a dipole of given strength is four times the free value over half the solid angle, which is twice. The dipole along the mirror has an image reversed and a tiny distance away, so the two together make up a quadrupole, much less effective at radiating, and the emission vanishes in proportion to $d^2$. This is the same thing that happens to a radio antenna: a vertical antenna over conducting ground works well, and a horizontal antenna close to ground barely radiates, as anyone who has set up a low dipole has found. Whether a charge on a table glows is a reminder of how readily such a dipole radiates in the first place.

The pattern a decay rate adds up #

That the rate changes at all might suggest that something is pulling on the atom. The calculation shows there is no need for any such force, and it can be done again a different way. Treat the radiation as a far-field interference pattern between the dipole’s direct wave and its reflection. The total power that leaves is the integral of that pattern over the half-space.

For a dipole across the mirror, the pattern is $\sin^2\theta$, the ordinary doughnut, multiplied by $2 + 2\cos(2kd\cos\theta)$, the fringe that when two waves meet deals with for any two sources. At a tenth of a wavelength the two waves are close to in step everywhere, and the pattern is a slightly wider doughnut. At a quarter of a wavelength a null appears straight out from the mirror. At half a wavelength the light leaves in lobes. Integrating these patterns over all directions gives 1.85, 1.30 and 0.92 at those distances, matching the expression above to within half a per cent. That comparison was made independently, by numerical integration, and it is an exact check on the physical statement: each photon goes somewhere, and the rate at which they go is the total of the angular distribution, so a pattern with more interference nulls has less rate.

Energy conservation thus has one account and one only. An atom beside a mirror does not store up its emission for later. It emits into an interference pattern, and its total rate is what the pattern permits. A radiating dipole’s interaction with its own reflection is what makes this happen at the level of the dipole itself, in the classical picture: the reflected field arrives back at the source with a certain phase and either drives it, or is driven by it, to give the rate.

An experiment with layers of soap #

The first person to see this in an experiment was Karl Drexhage, working in Marburg in the late 1960s, and the difficulty he had to overcome was setting a distance. A wavelength of visible light is half a micrometre, so to see one oscillation he needed to place an emitter at chosen distances from a mirror, in steps of a few tens of nanometres, and do so reliably.

His spacer was a monolayer of fatty acid. Irving Langmuir and Katharine Blodgett had shown decades earlier that a film of fatty acid one molecule thick, floating on water, can be transferred layer by layer to a solid plate by dipping. Each layer is about 2.6 nm thick, and the thickness of a stack is exactly its number of layers times that. Drexhage built up such stacks on a silver mirror and laid on them a single layer holding europium ions, which glow red at 612 nm with a conveniently long lifetime of about a millisecond, easy to measure with the electronics of that time.

The lifetime rose and fell as a function of the number of layers, along the curve in the figure, and the angular distribution of the emitted light changed with it. This was the first clear demonstration that the surroundings can alter spontaneous emission in the weak-coupling regime, and with an effect that was large and easy to read. Its theory, due to Kuhn and to Chance, Prock and Silbey in the following years, is close to what is drawn here, with corrections for the real silver film.

The simulation in the figure does not reproduce his data, only the same trend. It draws 4,000 decay times from an exponential at each distance and takes their mean; the dots show how well a modest number of decays measures a lifetime. This matters because a decay curve is the only observable, and a few thousand photons at each thickness were enough to fix it. Any real experiment at this scale must also watch for the energy lost non-radiatively to the metal itself, which grows very steeply as the emitter approaches within about twenty nanometres. That near-field quenching takes over from everything drawn here, and the figure, with its perfect mirror, does not include it. The rising trend to the left of the figure is thus cut short in a real film.

A gap the photon does not fit in #

A single mirror changes a rate by a limited factor. Two parallel mirrors can make it vanish.

Between two perfect mirrors with a gap $L$, only certain standing waves fit. A mode polarised along the mirrors needs a whole number of half-wavelengths across the gap, at least one, so a wave with a free-space wavelength $\lambda$ cannot exist in a gap smaller than $\lambda/2$ with that polarisation. This is the same reason a wave cannot travel in a waveguide below its cut-off, which the pipe that will not carry a low note describes in acoustics. An atom in that gap with its dipole along the mirrors has no mode into which to emit. There is no photon final state. Its decay rate is then exactly zero, and the formula obtained by summing over the allowed modes bears this out:

$$\frac{\Gamma}{\Gamma_0} = \frac{3\lambda}{4L}\sum_{m\ \text{odd},\ m < 2L/\lambda}\left(1 + \frac{m^2\lambda^2}{4L^2}\right).$$

As the gap opens to half a wavelength, the first mode appears, and the rate jumps from zero to about three times the free value, since the atom’s whole emission must go into few modes in a small volume. Beyond that the rate is a sawtooth, falling as each mode’s weight spreads and rising with each new one, and it settles to 1 as the gap grows. Computed at forty wavelengths it is within a per cent or two of the free value.

Daniel Kleppner argued in 1981 that this inhibition ought to be observable, and in 1985 Randall Hulet, Eric Hilfer and Kleppner did it. They used caesium atoms excited to a very high Rydberg state, which the atom the size of a bacterium describes, because such an atom emits at a long wavelength of 3.4 µm and a mirror gap of 2.3 µm is practical to make. They sent the atoms through a gap with a field aligning their dipoles, and measured how many survived over the flight. The atoms lasted over twenty times their free lifetime. When a small field rotated the dipole out of the plane, so that it was across the mirrors and had a mode available, the decay came back at once.

That experiment is the cleanest statement of the argument. An emission can be turned off by what is not there. A vacuum is a thing with modes, and an atom’s emission is a transition into them. The attraction that needs no charge drew attention to one consequence of those same modes between two plates, which is a force. Here the consequence is a lifetime.

Purcell’s factor: a mode instead of a mirror #

The reverse is also possible, and it is more useful. An atom in a small, resonant cavity may be forced to emit much faster than it would in free space. The solution that is thrown away quoted the result of Edward Purcell, in a short abstract of 1946, in which he noted that nuclear spin relaxation is hopelessly slow in free space but ought to be fast in a resonant circuit: $F = 3Q\lambda^3/4\pi^2 V$. A high quality factor $Q$ and a small volume $V$ both increase the local density of states at the resonant frequency. The result has a clear physical meaning: the atom’s linewidth is much narrower than a free mode’s width in a cavity with large $Q$, and all of the modal density is concentrated into a narrow peak where the atom can use it.

That enhancement is the basis of bright single-photon sources for quantum communications and of faster light-emitting diodes. A semiconductor quantum dot, an artificial atom a few nanometres across, in a micropillar cavity a micrometre wide can be made to emit ten or more times faster than in bulk, and it delivers its photon to one chosen mode with most of the efficiency, an aim that free-space emission, spreading in all directions, can never achieve. The principle is one that the mode that lives in the mistake provides in another form: a structure with a designed defect can produce a very localised mode of light with a very small volume.

If the coupling is much stronger than either the emission rate or the cavity’s loss, there is no longer a decay to enhance. The atom and the cavity exchange a single excitation back and forth, an oscillation first seen in 1992 by H. Jeff Kimble’s group as a splitting of the resonance. Spontaneous emission has stopped being spontaneous, and it has also stopped being irreversible. The exponential law from which all this began is then the limit in which the emitted photon goes away and never returns; the exponential that is only true in the middle shows that even in free space the law is only an approximation.

Why a nucleus does not care #

This does not contradict the constancy that radiocarbon relies on, and the reason is one of scale. A carbon-14 nucleus emits an electron and an antineutrino, and those particles have short wavelengths of order a picometre and pass straight through anything in the surroundings. No mirror exists for them at the distance of an atom. Neither does an atomic nucleus’s gamma-ray emission care much, for the wavelength involved is smaller than any cavity that can be built, and mirrors for gamma rays are almost nonexistent.

The relevant scale is the wavelength of what is emitted, compared with the distance to the first structure. Visible light at half a micrometre is comparable with engineering lengths and molecules stacked on a surface, and radio waves far more so. It is only in that regime that the surroundings can have an effect. That is why it took until the 1960s to find this effect, by which time people were already used to assuming that an atomic lifetime is intrinsic: in any ordinary situation the nearest surface is thousands of wavelengths away, where the oscillation in the first figure has faded to a tiny fraction of a per cent. A sodium lamp’s emission is in that regime. So the lifetime that appears in tables is correct, and is a statement about an atom a long way from anything.

A perfect mirror and a point dipole #

The curves use a perfect mirror, a point dipole and a two-level emitter. Real metals absorb, and at optical frequencies silver’s reflection has a phase different from the image dipole’s sign rule, so the oscillation is displaced and weakened. A molecule rather than an atom has its emission spread across a band of wavelengths, over which the oscillation partly washes out. Dielectric media change the wavelength and bring in their own guided modes. The image-dipole picture applies only in the far zone and for small emitters.

The model is also a weak-coupling one throughout: it treats the atom’s decay as exponential with a modified rate, and this fails if the emitted photon can return in the coupling’s own time. Between two mirrors, the formula’s sharp zero depends on perfect mirrors and an exactly aligned dipole, and a real gap with imperfect metal would have a small non-zero rate from evanescent coupling. No part of the discussion here concerns whether an excited atom’s levels shift near a surface, which they do, as a separate Casimir–Polder effect, small at these distances.

Still open: how far a lifetime can be engineered #

A photonic crystal, a periodic structure with a band gap for light, is a three-dimensional version of the two-mirror gap: there can be no mode at all, in any polarisation or direction, within its forbidden band. Eli Yablonovitch proposed in 1987 that this would inhibit spontaneous emission entirely, and that a great deal of the waste in a laser might be eliminated by it. Inhibition by factors of ten or so has been seen for quantum dots in such crystals, along with the difficulty that real emitters sit near surfaces and have other, non-radiative paths for decay, which no crystal can inhibit.

At the other extreme, plasmonic structures concentrate light in volumes far below a cubic wavelength, with predicted rate enhancements of many thousands. Their usefulness is limited by metal’s absorption, which turns an enhanced emission into a heat loss. How to reach very large enhancement while still getting the photon out in a useful mode is a practical matter under active investigation.

A lifetime in a table is the lifetime of an atom with nothing near it. Put a mirror a fraction of a wavelength away and its rate swings between 0 and 2 according to the distance and the dipole’s direction, by an amount matching the integral of the interference pattern it radiates. Put two mirrors less than half a wavelength apart and a dipole along them has no photon state to go to, and does not decay.