Quantum Optics

Roy Glauber supplied the framework in 1963. The question "is this light coherent?" had meant "does it produce fringes?", a statement about the first-order correlation of the field. Glauber showed that a photodetector does not measure the field; it measures normally ordered correlations of the quantised field, and there is a whole hierarchy of them. Light can be first-order coherent and second-order anything.

The second-order quantity, written $g^{(2)}(0)$, is the probability of detecting two photons at once divided by what it would be if detections were independent. It takes three characteristic values. For thermal light — a star, a filament, any hot body — the field amplitude itself fluctuates, bright moments deliver pairs, and $g^{(2)}(0) = 2$: the Hanbury Brown–Twiss bunching. For a laser well above threshold, the amplitude is steady and photon arrivals are Poisson, giving $g^{(2)}(0) = 1$. And for a single atom, $g^{(2)}(0) = 0$, because having just emitted, it has nothing left to emit until it is re-excited.

That third case is the one classical physics cannot reach. Any classical field, however exotic, has $g^{(2)}(0) \ge 1$. Jeff Kimble, Mario Dagenais and Leonard Mandel measured $g^{(2)}(0) < 1$ from sodium atoms in 1977, and in doing so produced the first light that demonstrably required quantisation of the field, rather than merely of the matter absorbing it.

Start with the scale. A 1 mW beam at 550 nm carries photons of energy

$$E = \frac{hc}{\lambda} = \frac{(6.626 \times 10^{-34})(3.00 \times 10^{8})}{550 \times 10^{-9}} = 3.6 \times 10^{-19} \text{ J},$$

so the flux is $10^{-3} / 3.6 \times 10^{-19} \approx 2.8 \times 10^{15}$ photons per second. Individual photons are not a scarce resource in ordinary light; the issue is their arrival pattern.

For a coherent state of mean photon number $\mu$, the number arriving in a given interval is Poisson:

$$P(n) = e^{-\mu}\frac{\mu^{n}}{n!}.$$

Now consider quantum key distribution. The BB84 protocol is secure because an eavesdropper cannot copy a single photon without disturbing it. If a pulse contains two identical photons, she can take one and let the other through, learning a bit with no disturbance at all — the photon-number-splitting attack. Real systems mostly use an attenuated laser rather than a true single-photon source, so what matters is how often a non-empty pulse contains more than one photon.

Attenuate to $\mu = 0.1$ photons per pulse:

$$P(0) = e^{-0.1} = 0.9048, \quad P(1) = 0.0905, \quad P(2) = 0.00452, \quad P({\ge}2) = 0.00468.$$

Of the pulses that contain anything at all, the fraction carrying two or more is

$$\frac{0.00468}{1 - 0.9048} = \frac{0.00468}{0.0952} = 4.9\%.$$

One pulse in twenty of the useful ones is leaky. Lower $\mu$ to improve that and you lose signal proportionally: at $\mu = 0.01$ the multi-photon fraction falls to about 0.5%, but 99% of pulses are empty, so the key rate falls by a factor of ten. The trade-off is set entirely by Poisson statistics, and it is why decoy-state protocols — which estimate the eavesdropper's advantage by varying $\mu$ — had to be invented, and why a deterministic single-photon source with $g^{(2)}(0) \approx 0$ would be worth so much.

The same arithmetic run backwards is the shot-noise limit. Counting $N$ photons gives a relative precision of $1/\sqrt{N}$, so an interferometer using $10^{20}$ photons per second measures a phase to about $10^{-10}$ radians — unless the light is squeezed, in which case the uncertainty is redistributed and the phase can be measured better at the cost of the amplitude being measured worse. That is the trick running inside LIGO since 2019.

The last part of the subject turns the relationship around: instead of using matter to make interesting light, use light to control matter. An atom absorbing a photon takes its momentum, $h/\lambda$, which at sodium's 589 nm is $1.1 \times 10^{-27}$ kg·m/s. Divided by the mass of a sodium atom, $3.8 \times 10^{-26}$ kg, that is a velocity change of 2.9 cm/s per photon. Do it ten thousand times a second with a laser tuned slightly below resonance, so that only atoms moving towards the beam are Doppler-shifted into resonance, and the atoms are slowed. Six beams make a viscous medium for atoms, and Steven Chu, William Phillips and Claude Cohen-Tannoudji brought sodium to microkelvin temperatures this way — colder, by then, than anything else known. The Bose–Einstein condensate of 1995 was made from laser-cooled atoms, and the optical lattice clock followed.

Serge Haroche took the complementary route: trap the photon and send atoms past it. Microwave photons between superconducting mirrors of extraordinary quality survive more than a tenth of a second, during which the photon travels some 40,000 kilometres between reflections. A Rydberg atom crossing the cavity acquires a phase shift that depends on whether a photon is there, without absorbing it, so the same photon can be interrogated hundreds of times and its eventual disappearance watched as a quantum jump.

The techniques here feed two directions. Engineering light's statistics and entanglement for computation and communication is quantum information. Making light intense, short and coherent enough to be an industrial and scientific instrument is lasers and photonics, and almost everything in this chapter was done with a laser in the first place.