Module 4 ended with a quietly astonishing theorem: a mesh of Mach–Zehnder interferometers implements any unitary matrix on its optical modes. We read that classically — bright laser light in, matrix product out. This module walks the same hardware through a different door. Turn the laser down. Keep turning. When exactly one photon is in the mesh, the field amplitudes we have been multiplying become probability amplitudes, the mesh becomes a quantum circuit, and the chip you learned to program in Module 4 becomes a quantum computer — or at least, the easy half of one.

You have already met the headline trade-offs in photonic qubits: room-temperature coherence, the natural flying qubit, and the crippling absence of photon–photon interactions. This lesson is the photonics-side deep dive that page could not afford: what a dual-rail qubit is in the language of optical modes, why the weakness is structural rather than an engineering shortfall, and — the part every architecture stands or falls on — where single photons actually come from.

The qubit, written in modes

A photonic chip does not naturally speak the language of qubits; it speaks modes — waveguides, polarisations, time slots — and occupation numbers: how many photons each mode holds. Write |n_1, n_2\rangle for "n_1 photons in mode 1, n_2 in mode 2". The dual-rail encoding takes two waveguides and defines the logical basis as one photon, two possible homes:

where \hat a_j^\dagger creates a photon in waveguide j. A general qubit state is a single photon smeared coherently over both rails,

which is exactly the "one input port lit, with complex amplitude" picture from the classical mesh — except the amplitudes now obey the Born rule instead of a power meter. The polarisation encoding is the same mathematics in a rotated wardrobe: |H\rangle and |V\rangle are two orthogonal modes sharing one spatial path. The two encodings interconvert with a single passive component — a polarising beamsplitter sends |H\rangle down one waveguide and |V\rangle down another, turning a polarisation qubit into a dual-rail qubit and back. On-chip work overwhelmingly uses dual-rail (waveguides preserve path far better than polarisation); free-space and fibre experiments often prefer polarisation or time-bin.

Note the bookkeeping cost, because it recurs all module: one dual-rail qubit consumes two optical modes, so n qubits need 2n waveguides before a single ancilla is added. Mode count, not photon count, is what a chip designer pays for in floor space.

The ledger: what photons buy, and the clause in the fine print

The strengths are genuine and worth restating precisely. A photon at telecom wavelength has energy \hbar\omega \approx 0.8\ \mathrm{eV}, some thirty times the room-temperature thermal energy k_BT \approx 0.025\ \mathrm{eV} — the environment is simply too cold to excite or dephase it, which is why photonic coherence needs no cryostat. And a photon moves: the same qubit that computes on the chip will happily travel kilometres of fibre, making photons the only serious candidate for networking quantum processors together.

Now the clause. Every element of Module 4's toolkit — couplers, phase shifters, whole meshes — acts on the mode operators linearly:

Apply that to a product of creation operators and each photon is rotated by the same U, independently, no matter how many photons share the mesh. Nowhere in a linear-optical circuit does one photon's presence alter another photon's evolution — the transformation has no term that couples them. But a two-qubit entangling gate is defined by such conditioning: CNOT must flip the target only when the control photon is there. So the shortfall is not an engineering gap you close with a better foundry; it is a structural absence in linear optics. The honest fixes are a strong optical nonlinearity at the single-photon level — the Kerr effect, which Module 7 showed is many orders of magnitude too weak — or the measurement-based trickery this module builds toward.

Where do single photons come from?

Here is the question that embarrasses the field at parties: the qubit is "a single photon", so — where is the photon gun? A laser will not do. An attenuated laser pulse is a coherent state, which has a Poisson-distributed photon number: dial the mean down to \bar n = 0.1 and 90% of your pulses contain no photon at all, while about 1 pulse in 200 still contains two — and a two-photon pulse in a machine whose logic assumes one is a silent error. No amount of attenuation fixes the shape of the distribution. You need light that is antibunched — one photon, then a gap — which no classical source produces.

SPDC: photons born in pairs

The workhorse for four decades is spontaneous parametric down-conversion (SPDC): pump a crystal with a mild optical nonlinearity and, rarely, one pump photon splits into two daughter photons, momentum- and energy-matched. The magic is the word pairs: detect one daughter and you know — without touching it — that its twin exists and is on its way. That announcement is called heralding, and it converts a hopelessly random process into a usable, if unscheduled, single-photon source.

The price is written in the statistics. Per pulse, the pair number of an SPDC mode is thermally distributed with mean \mu:

Pump harder and heralds come faster — but the double-pair contamination, the events that fake a herald while sneaking an extra photon into your circuit, grows as P(n\!\ge\!2)/P(1) = \mu. The chart below shows the squeeze: usable single-pair events peak and then fall, while multi-pair junk keeps climbing. Real experiments run at \mu \sim 0.01\text{–}0.1 and simply accept that the source fires usefully only a few percent of the time.

Quantum dots: an artificial atom on demand

The modern challenger is the quantum dot — a nanoscale semiconductor island that behaves as a single artificial atom. Excite it with a laser pulse and it can emit exactly one photon per pulse: a genuinely deterministic gun, with demonstrated end-to-end efficiencies past 50% and multi-photon rates a hundred times below SPDC's. Its weakness is the mirror image of SPDC's: each dot is grown, not printed, so no two are identical — and (as the next two lessons make vivid) photonic computing needs photons from different sources to be perfectly indistinguishable. Tuning many dots into agreement, or multiplexing many heralded SPDC sources into a pseudo-deterministic one, is a live engineering race with no declared winner.

Where this goes next

We have a qubit — one photon, two modes — and a certificate of its arrival. The next lesson delivers the good news in full: every single-qubit gate is a Mach–Zehnder interferometer you already know how to build, and Module 4's SU(2) algebra becomes quantum mechanics without changing a single symbol. After that, the module confronts the fine-print clause head-on: Hong–Ou–Mandel interference, the one genuinely two-photon effect linear optics does allow — and the KLM protocol that leverages it into a computer.