Spontaneous parametric down-conversion (SPDC)

Spontaneous parametric down-conversion is the process in which a pump photon passing through a crystal with a second-order nonlinearity splits into two photons of lower energy, conventionally called the signal and the idler. No seed beam is present; the splitting is stimulated by vacuum fluctuations, which makes SPDC the low-gain limit of an optical parametric amplifier. The two photons are created within a time shorter than their coherence time, typically picoseconds, and their energies, momenta and, with suitable design, polarizations are correlated. A 405 nm diode laser pumping a beta-barium borate (BBO) or periodically poled KTP (PPKTP) crystal is the standard laboratory source of 810 nm photon pairs. The process is weak: in bulk crystals roughly one pump photon in 10⁹ to 10¹² is converted, the higher end in long PPKTP crystals, depending on crystal length, nonlinear coefficient and the collected bandwidth.

Conservation laws and phase matching

Energy conservation fixes the sum of the photon frequencies,

$$\frac{1}{\lambda_p} = \frac{1}{\lambda_s} + \frac{1}{\lambda_i},$$

so a 405 nm pump photon of 3.061 eV yields two 810 nm photons of 1.531 eV each in the degenerate case, or any other pair that sums correctly. Phase matching, $\mathbf{k}_p = \mathbf{k}_s + \mathbf{k}_i$, decides which pairs are emitted and in which directions. In a bulk crystal the photons leave on cones around the pump axis, with each signal photon diametrically opposite its idler partner.

The polarization configuration names the type:

• Type 0: pump, signal and idler share one polarization. It requires quasi-phase-matching and uses the largest coefficient, $d_{33}$, in PPLN or PPKTP.
• Type I: signal and idler share a polarization orthogonal to the pump's.
• Type II: signal and idler have orthogonal polarizations, one ordinary and one extraordinary. With a suitable crystal cut the two cones, ordinary and extraordinary, are displaced and intersect along two lines; birefringence sets their separation.

Entanglement and heralding

Collecting light where the two type II cones intersect gives pairs for which neither photon has a definite polarization, but the two are always orthogonal: a polarization-entangled state. Two thin type I crystals with optic axes rotated by 90° produce a similar state, and Sagnac-loop sources with PPKTP give high brightness and stability. Such sources underpin entanglement-based quantum key distribution, Bell-inequality tests, and photonic quantum computing experiments.

Because the photons are created together, detecting one heralds the presence of the other. A heralded single-photon source has antibunched statistics (a heralded $g^{(2)}(0)$ well below 1), unlike a laser, which has Poissonian photon statistics. Its limitation is probabilistic emission: raising the pump power raises the probability of producing two pairs at once, which scales as the square of the single-pair probability, so heralded sources are usually operated at pair probabilities of a few percent per pulse or per coherence time.

Measuring pair rates

Pairs are detected with two single-photon avalanche diodes or superconducting nanowire detectors feeding a coincidence counter. A 10 mW pump at 405 nm carries $2.0 \times 10^{16}$ photons/s, so conversion efficiencies of $10^{-12}$ to $10^{-9}$ correspond to about $2 \times 10^4$ to $2 \times 10^7$ pairs generated per second. Detected coincidences are lower by the product of the two collection and detection efficiencies. Accidental coincidences, from uncorrelated photons that arrive within the coincidence window $\tau_w$, occur at a rate

$$R_\text{acc} \approx R_1 R_2 \tau_w .$$

With singles rates of $10^5$ s⁻¹ on each detector and a 1 ns window, $R_\text{acc}$ ≈ 10 s⁻¹. The coincidence-to-accidental ratio is the usual figure of merit and falls as pump power rises.

The spectral bandwidth of the pairs follows from the phase-matching bandwidth. For a 1 nm bandwidth at 810 nm the frequency width is about 457 GHz and the coherence time about 1 to 2 ps, which is short compared with typical detector jitter of tens of picoseconds or more, so the coincidence peak width in the histogram measures the detectors and timing electronics.

Pitfalls

Pump light leaks through at a level many orders of magnitude above the pair flux, so dichroic mirrors, long-pass filters and narrow bandpass filters are all needed before the detectors. Fluorescence from the crystal and filters adds uncorrelated background. Pump bandwidth matters: a multimode diode laser broadens the energy correlation and degrades two-photon interference. Coupling pairs into single-mode fiber requires matching the pump focus to the collection mode, which trades brightness against heralding efficiency.

Common questions

Is SPDC the same as an OPO below threshold?

The physics is the same parametric process. An optical parametric oscillator operated below threshold emits down-converted light into its cavity modes, which narrows the pair bandwidth and, observed as a continuous field, produces squeezed light.

Can photon pairs be made on a chip?

Yes. Periodically poled thin-film lithium niobate waveguides give SPDC with much higher efficiency per unit pump power than bulk crystals because of the tight mode confinement. Silicon and silicon nitride, which lack a second-order nonlinearity, produce pairs by spontaneous four-wave mixing instead.

Why is 810 nm common for SPDC pairs?

Inexpensive 405 nm diode lasers are available, and silicon single-photon avalanche diodes have high detection efficiency near 800 nm. Telecom-band pairs near 1550 nm, pumped at 775 nm, are used for fiber transmission.

References: D. C. Burnham and D. L. Weinberg, Phys. Rev. Lett. 25, 84 (1970); C. K. Hong and L. Mandel, Phys. Rev. Lett. 56, 58 (1986); P. G. Kwiat et al., Phys. Rev. Lett. 75, 4337 (1995); P. G. Kwiat et al., Phys. Rev. A 60, R773 (1999); Mandel & Wolf, Optical Coherence and Quantum Optics (Cambridge, 1995).