Splitting a Photon: How One Photon Becomes an Entangled Pair

One photon in, two photons out

Shine a blue laser through a small transparent crystal — beta-barium borate (BBO), KTP, or lithium niobate are the usual choices — and almost nothing happens: the beam passes straight through. But once in roughly 10⁷ to 10¹¹ pump photons, a single photon is annihilated and two longer-wavelength photons are created in its place. By convention the two daughters are called the signal and idler. In the symmetric degenerate case each carries exactly half the pump energy, so a 405 nm blue photon becomes two 810 nm near-infrared photons.

The output is not a beam but a set of faint, rainbow-tinted cones of light surrounding the pump direction. The two photons of a given pair are born within a few femtoseconds of each other and leave the crystal entangled — their polarizations, colors, and directions are correlated more tightly than any classical pair could ever be. That correlation, not the splitting itself, is what makes SPDC the backbone of experimental quantum optics.

Two conservation laws, and why a vacuum won't do

Every down-conversion event obeys two conservation laws exactly:

Energy: ℏω_p = ℏω_s + ℏω_i — the pump frequency equals the sum of the daughter frequencies.
Momentum (phase matching): k_p = k_s + k_i — the photons' wavevectors inside the crystal must add up.

You cannot do this in empty space. Two things forbid it. First, photons in vacuum simply don't couple to one another — there is no three-photon interaction, so nothing drives the split. Second, even inside an ordinary transparent material, chromatic dispersion sabotages the momentum equation: the refractive index rises with frequency, so n(ω_p) > n(ω_p/2) and the pump's momentum is too large to balance. A birefringent, non-centrosymmetric crystal solves both problems at once. Its asymmetric bound electrons supply the nonlinear coupling, and by launching the pump on the crystal's fast (extraordinary) axis you can tune the angle until n_e(ω_p, θ) = n_o(ω_p/2) — the momenta finally match. The pair-creation amplitude scales as χ⁽²⁾ · L · sinc(ΔkL/2), where Δk = k_p − k_s − k_i is the phase mismatch and L the crystal length; phase matching just means driving Δk → 0.

The causal chain, step by step

1. A pump photon enters the crystal and its oscillating electric field drives the bound electrons of the lattice.
2. In a non-centrosymmetric medium the electrons respond nonlinearly: P = ε₀(χ⁽¹⁾E + χ⁽²⁾E² + …). That second-order χ⁽²⁾E² term is a three-wave mixer — it couples the pump to two other optical modes.
3. Those two modes, the future signal and idler, start out empty. The process is triggered by the ever-present vacuum fluctuations of the electromagnetic field — which is exactly why it is called spontaneous.
4. The crystal annihilates one pump photon and creates one signal and one idler photon.
5. Only splits that satisfy both energy and momentum conservation add up in phase over the crystal length; all others interfere destructively. This is why the pairs emerge on sharply defined cones rather than in every direction.
6. The pair leaves entangled. Detecting the idler tells you a signal photon exists and where to look — the basis of "heralded" single-photon sources.

Where the entanglement comes from

Entanglement arises because, at the moment of the split, nature genuinely does not decide which allowed outcome occurred — yet whatever happens to one photon is perfectly mirrored in the other.

Energy–time entanglement is present in every pair. The pump energy can divide in infinitely many ways, so the two-photon state is a superposition over all of them: |ψ⟩ = ∫ dω f(ω) |ω⟩_s |ω_p − ω⟩_i. Neither photon has a definite color, but their colors always sum to the pump. This is why the cones are rainbow-colored — a redder signal is paired with a bluer idler at a matching angle.

Polarization entanglement comes from crystal cut. In Type-I phase matching both daughters share one polarization and emerge on concentric rings. In Type-II they take orthogonal polarizations, forming two cones (ordinary and extraordinary) that cross at two points. Right at those crossings you cannot tell which photon is horizontal and which is vertical, producing the Bell state |ψ⟩ = (|H⟩_s|V⟩_i + |V⟩_s|H⟩_i)/√2 — the polarization-entangled source demonstrated by Kwiat and colleagues in 1995.

A worked example: 405 nm into twin 810 nm photons

Take a cheap 405 nm violet diode laser as the pump. Its photon energy is E_p = 1240 eV·nm ÷ 405 nm ≈ 3.06 eV. In the degenerate case each daughter is 810 nm, carrying 1240 ÷ 810 ≈ 1.53 eV — and 1.53 + 1.53 = 3.06 eV, so the books balance exactly. The daughters are near-infrared, at precisely twice the pump wavelength.

Push a few milliwatts through a 1–5 mm BBO crystal and you collect on the order of 10⁵–10⁶ pairs per second into optical fibers, at a conversion efficiency near 10⁻⁹. The two photons of a pair are created essentially simultaneously — within sub-picosecond of each other — so a coincidence-counting window of about 1 ns cleanly separates true pairs from accidental background. And the split need not be symmetric: dividing 3.06 eV into, say, 1.2 eV + 1.86 eV is equally allowed and emerges at its own cone angle, which is exactly why the down-converted light fans out into colored rings rather than a single line.

History and where it shows up

SPDC was predicted in the late 1960s, most influentially by David Klyshko in Moscow, who called it parametric scattering or parametric fluorescence. It was first observed by David Burnham and Donald Weinberg in 1970, who measured the tell-tale simultaneity of the emitted pairs (Phys. Rev. Lett. 25, 84). Klyshko's insight that the two photons are born together also gave metrology an elegant gift: an absolute way to calibrate a single-photon detector's efficiency using the pair itself, with no reference lamp.

The field's turning point was the 1995 Type-II polarization-entanglement source of Kwiat, Mattle, Weinfurter, Zeilinger, Sergienko, and Shih, which used a 351 nm argon-ion pump to make 702 nm entangled pairs — far brighter and more directional than the older atomic-cascade sources used in the 1980s Bell tests. That workhorse enabled quantum teleportation (1997), Ekert-style quantum key distribution, quantum imaging, and — carried into orbit — the Micius satellite's distribution of entanglement over 1200 km in 2017. Earlier SPDC sources had already delivered the Hong–Ou–Mandel effect (1987), the two-photon interference dip at the heart of linear-optics quantum computing. A common misconception to retire: the pump photon is not sliced in two like a physical object. It is annihilated, and two brand-new photons are created; the crystal merely enforces that their energies and momenta add back up to the original.

Spontaneous down-conversion versus its stimulated and inverse χ⁽²⁾ cousins — same nonlinearity, very different regimes.

Property	SPDC (spontaneous)	OPA (stimulated down-conversion)	SHG (up-conversion)
Energy flow	1 pump → signal + idler (splits)	pump + seed → amplified signal + idler	2 fundamental → 1 harmonic (merges)
What you inject	Pump beam only	Pump + a real seed beam	Fundamental beam
Seeded by	Vacuum fluctuations	The injected signal	The input field itself
Output character	Entangled photon pairs (quantum)	Bright amplified beams / squeezed light	Coherent classical beam
Typical efficiency	≈ 10⁻⁷–10⁻¹¹ per pump photon	Gain can be large (>10)	Up to tens of %
Signature use	Entangled pairs, heralded single photons, Bell tests	Tunable ultrafast lasers, squeezed states	Green laser pointers (1064 nm → 532 nm)