Illustrated Physics
physics, drawn
Quantum

The flicker that is half wave and half particle

In 1909, sixteen years before quantum mechanics, Einstein asked how much the energy in a small corner of a hot oven should fluctuate, and answered with nothing but thermodynamics and Planck's law. The answer has two terms. One is exactly what a spray of independent particles would give; the other is exactly what randomly overlapping waves would give. Neither theory of light could produce the sum, and the sum is what the oven does. The two terms are equal where each mode of the radiation holds one photon on average — a dividing line that runs through every hot thing, from the microwave sky to the Sun.
17 min read 5 figures The same equation againWho is measuring

Assumes: Light arrives in lumps, and brightness only changes how many · The gas that nobody counted

In 1905 Albert Einstein proposed that light, in some respects, behaves as if it were made of independent quanta of energy hνh\nu. His argument did not come from the photoelectric effect, which he used only as a supporting example. It came from the entropy of radiation: Wien’s law for the spectrum of a hot body, at high frequencies, implies that the entropy of radiation changes with its volume in exactly the way the entropy of a gas of independent particles does. Light at high frequency, he concluded, is thermodynamically a gas.

The argument was a strange one, because Wien’s law was known by then to be wrong at low frequencies, where Planck’s law and the classical Rayleigh–Jeans law agree with experiment. Four years later Einstein took the whole of Planck’s law and asked a sharper question of it. Not what the radiation’s entropy is, but how much its energy flickers. The answer, presented at a meeting in Salzburg in 1909, was the first place where the waves and the particles of light appeared in the same equation, added together.

How much a heat bath must flicker

Any part of a system in thermal equilibrium exchanges energy with the rest, and its energy fluctuates. Thermodynamics fixes how much, without any model of what the energy is carried by. A subsystem whose mean energy ⟨E⟩\langle E\rangle depends on temperature has a mean-square fluctuation

⟨ΔE2⟩=kT2 ∂⟨E⟩∂T.\langle\Delta E^2\rangle = kT^2\,\frac{\partial\langle E\rangle}{\partial T}.

The relation follows from the Boltzmann distribution in a few lines. If the subsystem occupies its states with probabilities proportional to e−E/kTe^{-E/kT}, the mean energy is a ratio of two sums, and differentiating it with respect to temperature produces the mean of E2E^2 minus the square of the mean of EE, divided by kT2kT^2. Nothing is assumed about what the states are or how they are spaced, and the content of the result is that a system whose energy is very sensitive to temperature must also wander a lot in energy at fixed temperature. The jiggle that proved atoms was Einstein’s other use of fluctuations in the same years: the random motion of a pollen grain, whose size measured the number of molecules hitting it. Fluctuations count the carriers of energy, because a few large carriers flicker more than many small ones.

So take a volume VV inside a cavity of black-body radiation, and the radiation in it within a narrow range of frequencies. Its mean energy is given by Planck’s law, and the formula gives its fluctuation directly. Counted per mode of the radiation — per independent standing-wave pattern the volume can hold, which the gas that nobody counted used as the unit of a photon gas — with nˉ=1/(ehν/kT−1)\bar n = 1/(e^{h\nu/kT} - 1) the mean number of quanta in the mode, the result is

⟨Δn2⟩=nˉ+nˉ2.\langle\Delta n^2\rangle = \bar n + \bar n^2.

Two terms, two theories

Einstein recognised both terms at once, because each is the answer a different theory of light would give.

The first, nˉ\bar n, is the variance of a count of independent things. If quanta arrived in the volume at random and independently, like molecules of a dilute gas or raindrops on a tile, the number present would follow a Poisson distribution, whose variance equals its mean. That term is what a gas of particles of energy hνh\nu would fluctuate by, and nothing more.

The second, nˉ2\bar n^2, is the fluctuation of waves. A volume filled with radiation from many independent sources, with random phases, contains a field that is the sum of a great many random contributions. Why two lamps never interfere followed the consequence: such a sum is a random field with Gaussian statistics, and the intensity of a Gaussian field — the square of its amplitude — fluctuates by as much as its own mean. The energy in one mode of a random wave field flickers with a standard deviation equal to its average, so its variance is the square of the mean. That is the term the Rayleigh–Jeans picture of the cavity, as an assembly of classical waves, would give.

Two kinds of flicker in one heat bath. The variance of the number of photons in one mode of black-body radiation, against hν/kT, both on logarithmic axes, split into its two parts: n̄, the variance a shower of independent particles would have, and n̄², the variance of the intensity of randomly phased waves; their sum is what thermodynamics demands of Planck's law. Where the photons are rare, at hν much more than kT, the particle term wins; where they are crowded, the wave term wins; they are equal at hν/kT = ln 2 = 0.693, where each mode holds one photon on average. At hν = 5kT the wave term is 0.68 per cent of the particle term; at hν = kT/10 the particle term is 10.5 per cent of the wave term.
Fig. 1 The variance of the photon number in one mode of black-body radiation against hν/kTh\nu/kT, both on logarithmic axes, split into the particle term nˉ\bar n and the wave term nˉ2\bar n^2; dashed, their sum. The two are equal at hν/kT=ln⁡2=0.693h\nu/kT = \ln 2 = 0.693, where each mode holds one photon on average. At hν=5kTh\nu = 5kT the wave term is 0.68 per cent of the particle term; at hν=kT/10h\nu = kT/10 the particle term is 10.5 per cent of the wave term.

Planck’s law requires both, added. Where a mode is nearly always empty — at frequencies well above kT/hkT/h — the particle term dominates, and the radiation fluctuates like a dilute gas. Where a mode holds many quanta — at frequencies well below — the wave term dominates, and the radiation fluctuates like a classical field. The two are equal where the mean occupation is exactly one, at hν=kTln⁡2h\nu = kT\ln 2. Einstein drew the conclusion explicitly: the next theory of light would have to be a kind of fusion of the wave theory and the emission theory, and neither could be discarded.

Einstein gave a second argument in the same paper, which reaches the same two terms by mechanics rather than by counting energy. Imagine a thin mirror free to move in a cavity full of radiation, reflecting some frequencies and passing others. Radiation pressure on its two faces pushes it about at random, and the drag of radiation on a moving mirror — it meets the light ahead of it slightly blue-shifted and the light behind slightly red-shifted, so it is pushed back harder from the front — slows it down. In equilibrium the mirror must end up with the same mean kinetic energy, 12kT\tfrac12 kT per direction, as any other body in the cavity, and that fixes how large the random pushes must be. Calculated from Planck’s law, the mean square of the momentum the radiation delivers again comes out as a sum: one term that is what independent particles of momentum hν/ch\nu/c would deliver, bouncing off the mirror one at a time, and one term that is what the interference of random waves would deliver. A mirror in a box of light jiggles as if it were hit by both at once.

What each old law knew

The same thermodynamic formula can be applied to the two laws that preceded Planck’s, and each yields only half of his answer.

Each old law knew one half of the answer. The energy fluctuation per mode that thermodynamics derives from three radiation laws — Wien's, which Einstein had shown described light as independent quanta; Rayleigh and Jeans's, which follows from classical waves; and Planck's — each divided by Planck's, against hν/kT on a logarithmic axis. Wien's law gets the fluctuation right where photons are rare and misses the wave part where they are crowded: at hν = kT/10 it gives 0.009 of the correct value. Rayleigh–Jeans gets it right where photons are crowded and is wrong where they are rare: at hν = 5kT it gives 5.9 times the correct value, because classical waves keep fluctuating by their own size however few photons the mode really holds. Only Planck's law, which contains both, fluctuates correctly at every frequency.
Fig. 2 The energy fluctuation per mode implied by Wien’s law, by the Rayleigh–Jeans law and by Planck’s law, each divided by Planck’s, against hν/kTh\nu/kT. Wien’s law is right where photons are rare and gives 0.009 of the correct value at hν=kT/10h\nu = kT/10; Rayleigh–Jeans is right where they are crowded and gives 5.9 times the correct value at hν=5kTh\nu = 5kT.

Wien’s law, nˉ=e−hν/kT\bar n = e^{-h\nu/kT}, gives exactly the particle term and nothing else. That is the thermodynamic content of Einstein’s 1905 argument restated: the law that made light look like a gas fluctuates like a gas. The Rayleigh–Jeans law, which gives each mode a mean energy kTkT as the curve that would not come down found it must for classical waves, gives exactly the wave term, a fluctuation of (kT)2(kT)^2 per mode. Each law is right in its own region of frequency, and each is wrong in the other region in a characteristic way: Wien’s misses the wave fluctuations where the photons are crowded, and Rayleigh–Jeans’s goes on fluctuating as if every mode were full where the modes are nearly empty. Planck’s formula, invented as an interpolation between the two, interpolates their fluctuations too, by adding them.

There is something unusual about an argument that settles a question about the nature of light by differentiating a formula for its spectrum. It works because thermodynamics constrains what any theory can do, and the spectrum carries more information than its shape: the way it changes with temperature encodes the size and independence of whatever carries the energy. The spectrum alone, measured carefully enough, told physicists in 1909 that light was neither of the things they knew how to describe.

The count in a single mode

The two terms can be made more concrete by asking not only for the variance but for the full distribution — the probability of finding exactly kk quanta in one mode. For thermal radiation it is geometric: P(k)=nˉk/(1+nˉ)k+1P(k) = \bar n^k/(1 + \bar n)^{k+1}, falling by the same factor with each additional quantum, which is what the Boltzmann factor gives when the energy levels of a mode are equally spaced. Its variance is nˉ+nˉ2\bar n + \bar n^2, and set beside the Poisson distribution with the same mean it shows the two terms as shapes.

How many photons one mode holds. The probability of finding k photons in a single mode of thermal light (bars) and in light whose photons arrive independently with the same mean, the Poisson distribution (dots), on a logarithmic axis, for a mean of 0.2 photons — a mode well above kT in energy — and a mean of 5, well below it. At a mean of 0.2 the two are nearly the same: thermal light is 83.3 per cent empty against 81.9 for Poisson, and its variance is 0.24 against 0.20. At a mean of 5 they are utterly different: thermal light is most often empty — 17 per cent of the time — and falls off geometrically, with a variance of 30, while independent arrivals cluster near 5 with a variance of 5. The extra spread is the wave term, n̄².
Fig. 3 The probability of finding kk photons in one mode of thermal light (bars) and for independent arrivals with the same mean, the Poisson distribution (dots), on a logarithmic axis, for means of 0.2 and 5. At 0.2 the two nearly agree: 83.3 against 81.9 per cent empty, variance 0.24 against 0.20. At 5 thermal light is most often empty — 17 per cent of the time — with a variance of 30, while independent arrivals cluster near 5.

At a mean of 0.2 the geometric and Poisson distributions are nearly indistinguishable: a mode that is mostly empty, and occasionally holds one quantum, looks the same whether the quanta are independent or not. At a mean of 5 they are completely different. The single most likely number of quanta in a thermal mode is zero, however full the mode is on average; large numbers are far more likely than for independent arrivals; and the distribution has no peak near its mean at all. Thermal light in a well-filled mode arrives in bunches, separated by lulls — the behaviour of a randomly fluctuating wave, whose intensity spends much of its time near zero and occasionally swells to several times its mean.

That bunching is measurable. The correlation that survives what the phase does not followed Robert Hanbury Brown and Richard Twiss’s measurement of it in 1956: two detectors watching the same thermal source within its coherence time and area record coincidences twice as often as independent arrivals would give, which is the wave term made visible as a correlation. With a laser, whose light is not thermal, the distribution is Poisson and the extra coincidences vanish, as Fortunato Arecchi showed in 1965 by measuring the full count distribution of laser light and of laser light scattered through a moving ground-glass disc, which turns it thermal.

Why detectors usually see particles

If the wave term dominates wherever photons are numerous, ordinary light from a lamp ought to flicker enormously. It does not, because a detector does not collect one mode. It collects every mode that falls within its area, its range of directions and its bandwidth, for as long as it integrates — and the modes fluctuate independently.

Why a detector usually sees only particles. The variance of a photon count divided by its mean (the Fano factor; one for independent arrivals) against the number of modes the detector collects at once, on logarithmic axes, for thermal light delivering on average 1, 100 and 10,000 photons per count. With M modes sharing N photons, the variance is N(1 + N/M). The wave term, N²/M, is diluted by every mode added: 10,000 photons gathered from a single mode fluctuate 10,001 times as widely as independent particles would, but gathered from a million modes only 1.01 times. A detector watching a lamp collects billions of modes in each sample and sees shot noise; only a detector smaller than the light's coherence area and faster than its coherence time, collecting about one mode, sees the waves' extra flicker.
Fig. 4 The variance of a photon count divided by its mean against the number of modes collected at once, for thermal light delivering on average 1, 100 and 10,000 photons per count; with MM modes sharing NN photons the variance is N(1+N/M)N(1 + N/M). Ten thousand photons from a single mode fluctuate 10,001 times as widely as independent particles; from a million modes, 1.01 times.

Summed over MM independent modes sharing NN photons, the particle terms add to NN and the wave terms to N2/MN^2/M, so the wave term is diluted by the number of modes. A photodiode watching a lamp for a microsecond collects something like 101210^{12} modes, holding a vanishingly small fraction of a photon each, and sees pure shot noise: the particle term. Only a detector smaller than the light’s coherence area and faster than its coherence time — collecting about one mode — sees the waves’ extra flicker. That is why Hanbury Brown and Twiss needed huge mirrors, fast electronics and long averaging to see the bunching of starlight, which holds about 10−310^{-3} photons per mode.

The radio astronomer lives in the opposite regime. A radio receiver at a few gigahertz, looking at a source at a few tens of kelvin, sees modes holding several quanta each, and its noise is the Gaussian noise of waves, with the photons’ discreteness invisible. Its sensitivity is governed by the radiometer equation, in which the noise falls as the square root of the bandwidth times the integration time — the number of independent modes collected — which is the wave term’s dilution by MM written in the language of receivers. The same electronic noise appears in a warm resistor: half a kT in a piece of wire found its voltage noise to be the classical 4kTR4kTR per unit bandwidth, which is the Rayleigh–Jeans end of exactly the same formula and acquires a particle correction only at frequencies approaching kT/hkT/h, about six terahertz at room temperature.

Where the crossover lies

The line hν=kTln⁡2h\nu = kT\ln 2 runs through every hot object, and where it falls in frequency says which kind of fluctuation that object’s light has in which part of the spectrum.

Where thermal light turns from waves into particles. The frequency at which the wave and particle terms of thermal radiation's fluctuations are equal, ν = kT ln 2/h, against temperature, both on logarithmic axes, with four sources marked; shaded, the radio-and-microwave, infrared and visible bands. Below the line the radiation's flicker is mostly wave-like, above it mostly particle-like. For the cosmic background the crossover is at 39 GHz, a quarter of the frequency at which the background is brightest; for a room, 4.3 THz in the far infrared; for the Sun, 83 THz, at a wavelength of 3.6 µm. Visible light from any hot body is far above its crossover and flickers like particles, while the radio noise of a warm resistor is far below and flickers like waves.
Fig. 5 The frequency at which the wave and particle terms are equal, kTln⁡2/hkT\ln 2/h, against temperature, both on logarithmic axes, with four sources marked; shaded, the radio-and-microwave, infrared and visible bands. For the cosmic background the crossover is at 39 GHz, for a room 4.3 THz, for the Sun 83 THz, at a wavelength of 3.6 µm.

The cosmic microwave background, at 2.7 kelvin, crosses over at 39 gigahertz: below that its fluctuations are wave-like, above it particle-like, and the instruments that measure it use both kinds of detector, coherent amplifiers at low frequencies and bolometers that count energy at high ones. Room-temperature radiation crosses over in the far infrared. The Sun’s crosses over at 3.6 micrometres, so that all of its visible light, and most of its energy, flickers like a gas of particles. Visible light from any thermal source is far above its crossover, which is why the photon picture is so natural for it, and why the wave term of sunlight — real, and measurable with enough care — went unnoticed until Hanbury Brown and Twiss went looking for it.

The first quantum field

Einstein’s formula waited sixteen years for a theory that produced it. In the 1925 paper by Max Born, Werner Heisenberg and Pascual Jordan that set out matrix mechanics, a section written by Jordan treated the modes of a vibrating string as quantum oscillators and calculated the energy fluctuations in a segment of it. Both terms came out: the nˉ\bar n from the oscillators’ discrete levels and the nˉ2\bar n^2 from the interference of their waves. It was the first calculation in what became quantum field theory, and its point was Einstein’s: a quantised wave field is not a compromise between waves and particles but a single object whose fluctuations contain both, with no need to say which it is in between measurements.

The counting that underlies the geometric distribution was supplied in 1924 by Satyendra Nath Bose, who derived Planck’s law by counting the ways to distribute indistinguishable quanta among modes, and whose counting is the reason the quanta bunch. The outcomes identical photons refuse found the same indistinguishability making two photons meeting at a beamsplitter always leave together. The wave term is, in quantum language, the extra probability that indistinguishable bosons have of being found in the same mode.

What the formula does not decide

The fluctuation formula is often presented as a proof that light is made of photons, and it is not quite that. The wave term needs no photons at all. The particle term needs a quantum somewhere, but not necessarily in the light: a detector made of atoms, absorbing energy in units of hνh\nu from a perfectly classical wave, produces counts with Poisson statistics, and the semiclassical theory of photodetection reproduces both terms of the count distribution of thermal light with a classical field. What no classical field can produce is a variance smaller than the particle term — the antibunching that light arrives in lumps described, seen first in 1977 from a single atom’s fluorescence — and the squeezed light that the noise pushed below the floor uses to beat the shot-noise limit. Einstein’s formula showed that the classical picture of the radiation field was incomplete; the experiments that showed the field itself had to be quantised came seventy years later.

The figures assume radiation in thermal equilibrium, a mode-by-mode description that holds for a volume large compared with the wavelength, and detectors that collect whole modes. A small volume, comparable to the wavelength, does not contain a well-defined set of modes, and its fluctuations depend on its shape; Einstein’s formula is for a volume large enough to have a smooth density of modes in the chosen band. The domain of the drawings is black-body radiation at hν/kTh\nu/kT from a hundredth to sixteen, and sources from 1 to 30,000 kelvin.

Still open: whether gravity’s waves flicker the same way

If gravitational waves are carried by quanta, gravitons, then a gravitational-wave background in thermal equilibrium would fluctuate by the same two terms, and so would a detector’s response to it. The wave term has been the working assumption of every gravitational-wave measurement: the signals are classical waves, and their noise is classical. Whether any measurement could see the particle term — the discreteness of gravitons in a detector’s noise — has been examined in recent years, with the conclusion that for every realistic source the occupation numbers are so enormous that the particle term is buried under the wave term by tens of orders of magnitude, exactly as it is for a radio receiver. The question of whether gravity is quantised in the sense light is, in the sense that Einstein’s formula would detect, remains open, and the formula itself says why it is hard to answer.

The arithmetic Einstein did in 1909 is still the clearest statement of what light is. The energy in one mode of thermal radiation fluctuates by nˉ+nˉ2\bar n + \bar n^2, a particle term from Wien’s law and a wave term from Rayleigh–Jeans’s, equal where each mode holds one photon at hν = kT ln 2 — 39 GHz for the cosmic background, 83 THz for the Sun — so that one mode holding five photons has a variance of 30, while a detector collecting a million modes sees the wave term diluted to almost nothing. The radiation in a hot oven is not sometimes waves and sometimes particles; its flicker is both, added, in a proportion its temperature sets.

Part 6 of 6

This essay is one argument about Photon. The others:

Four states of light, each with the same area of noise. The field of a single mode of light drawn as a point in a plane whose two axes are its two quadratures — the parts of the wave in step with a reference and a quarter-cycle out of step. The distance from the centre is the amplitude and the angle is the phase. Each state is a cloud of 400 sampled measurements with its two-standard-deviation outline, in units where the vacuum's noise is one in every direction. The vacuum is a round cloud at the centre. Steady laser light is the same round cloud moved away from the centre. Light squeezed by 4 dB is an ellipse of the same area: narrower than the vacuum by a factor of 0.63 in one direction and wider by 1.58 in the other. Pointed along the direction from the centre it is quiet in amplitude; pointed across it, quiet in phase. The sampled spreads were checked against each state's widths. The noise pushed below the floor Part 5 — A perfectly steady laser beam still flickers, by an amount set by the vacuum itself, and for most of the twentieth century that flicker was treated as the floor of any optical measurement. It is a floor only for one shape of noise. Light can be made quieter than the vacuum in one property by being made louder in another — and the price, the fragility and the use of that trade are all visible in how the noise is shaped, which is why the world's gravitational-wave detectors now run on it.

Where this goes next

Photocurrent against applied voltage, at two intensities. Photocurrent against retarding voltage for caesium lit at 12×10¹⁴ hertz, at relative intensities of 1 and 2. Both curves reach zero at the same stopping voltage of 2.82 volts, because the brighter light delivers more photons and not more energetic ones. The saturation currents are in the ratio of the intensities. The slope between the stopping voltage and zero assumes the emitted electrons' energies are spread uniformly below the maximum, which they are not; the crossing point does not depend on that assumption. Light arrives in lumps, and brightness only changes how many Shine dim blue light on a metal and electrons come out. Shine intense red light and none do, however long the wait. The frequency decides whether anything happens; the intensity decides only how much. The same source measured by amplitude and by intensity. The degree of coherence of a 47 milliarcsecond disc at 550 nm, and its square, against the separation of two apertures. A Michelson interferometer measures the upper curve, because fringe contrast is |γ|. Correlating the intensities at the two apertures instead measures the lower one, because the excess correlation of two thermal beams is |γ|² — the same information about the source, since one curve determines the other, and reaching zero at the same baseline of 2.94 m. What is lost is the phase of γ, which the intensity correlation never sees; what is bought is that a path error of many wavelengths does not matter, because the quantity being correlated is a slow fluctuation of brightness rather than a wave. The half-coherence baselines differ — 1.71 m against 1.25 m — which is the practical statement that the squared curve is the steeper one to measure against. The correlation that survives what the phase does not Two telescopes can measure a star's diameter by interfering the light, which requires holding two paths equal to a fraction of a wavelength through an atmosphere that will not hold still. Or they can throw the phase away entirely and correlate the brightness fluctuations, which needs the paths equal to a few metres and works. Energy, pressure and entropy of a gas nobody counted. The energy density, pressure and entropy density of blackbody radiation against temperature, on logarithmic axes, together with the pressure a monatomic gas of the same energy density would have. Every curve is a power of the temperature — the fourth for energy and pressure, the third for entropy — because the only length in the problem is the thermal wavelength and the only energy is kT. The pressure is exactly a third of the energy density, where an ordinary gas's is two thirds, a factor of 2: a photon carries momentum E/c and a slow molecule carries √(2mE), and that difference is the whole of it. Some values: at room temperature the radiation pressure is 1.86e-6 pascals, which is a ten thousand millionth of an atmosphere; at 1e+7 kelvin it is 2.52e+12, which is where radiation rather than matter holds a star up. The gas that nobody counted A box of gas holds however many molecules were put in it. A box of radiation holds however many photons the temperature says, because the walls make and destroy them until the free energy is least — and one dropped assumption changes every result. The pressure becomes a third of the energy density instead of two thirds, the entropy goes as the cube of the temperature, and the adiabatic index comes out at exactly four thirds.

The objects named here

The third axis, after the field and the reading path: the things themselves, and every essay that touches each one.

Blackbody radiationCoherenceFluctuationsModePhotonPlanck lawShot noiseThermal light