Spontaneous Parametric Down-Conversion

Spontaneous parametric down-conversion takes one input directly from the quantum theory of light: the electromagnetic field is quantized, and even its empty modes carry irreducible vacuum fluctuations. Feed a crystal that lacks inversion symmetry — so its nonlinearity $\chi^{(2)}$ survives — with a bright pump, and classically nothing new happens unless you also inject a wave to amplify. Quantum-mechanically the "empty" signal and idler modes are never truly empty: their zero-point fluctuations play the missing seed, and once in a great while a single pump photon is annihilated and two lower-energy photons are born in its place. This is difference-frequency generation running on vacuum noise — one of the cleanest demonstrations that the field really is quantized. Two conservation laws sculpt the output. Energy conservation, $\omega_p=\omega_s+\omega_i$ , is exact for the central frequencies and only slightly relaxed by the pump bandwidth; momentum conservation — phase matching, $\mathbf{k}_p=\mathbf{k}_s+\mathbf{k}_i$ — is enforced by the crystal's dispersion and is the knob the experimenter turns, by angle, temperature, or a poling grating supplying a reciprocal-lattice momentum. Written as an operator, the process is the two-mode squeezing generator $\hat a_s^{\dagger}\hat a_i^{\dagger}$ : it creates photons strictly in pairs, so the exact output is a two-mode squeezed vacuum whose photon numbers are perfectly correlated between the arms. In the weak regime that state is dominated by a single pair, which is why a click on one detector can herald its twin. From that one trick flow heralded single photons, polarization-entangled Bell states, Hong–Ou–Mandel interference, and the entangled sources behind Bell tests, quantum key distribution, and photonic quantum computing.

Key takeaways

SPDC is a second-order ( $\chi^{(2)}$ ) nonlinear process: inside a noncentrosymmetric crystal a single pump photon is annihilated and two lower-energy photons — signal and idler — are created, obeying $\omega_p=\omega_s+\omega_i$ and $\mathbf{k}_p=\mathbf{k}_s+\mathbf{k}_i$ .
It is spontaneous because it is seeded by vacuum fluctuations of the signal and idler modes; a seed-free classical amplifier outputs nothing, so SPDC is a direct fingerprint of field quantization.
The exact state is a two-mode squeezed vacuum, $|\psi\rangle\propto\sum_n(\tanh r)^n|n_s,n_i\rangle$ , with perfect photon-number correlation; at low gain it collapses to $|0,0\rangle+\lambda|1_s,1_i\rangle$ — occasional single pairs.
Detect the idler and you herald the signal, making SPDC the workhorse heralded single-photon source and the standard generator of polarization-entangled Bell states.
Efficiency is minuscule ( $10^{-11}$ – $10^{-6}$ pairs per pump photon); phase matching and pump bandwidth fix the joint spectral amplitude that sets spectral purity.
Key idea: SPDC is the two-mode squeezing operator acting on the vacuum — a $\chi^{(2)}$ crystal turns one pump photon into a correlated signal–idler pair, seeded by vacuum fluctuations and constrained by energy conservation ( $\omega_p=\omega_s+\omega_i$ ) and phase matching ( $\mathbf{k}_p=\mathbf{k}_s+\mathbf{k}_i$ ).

The core insight

SPDC is the two-mode squeezing operator acting on the vacuum: a $\chi^{(2)}$ crystal converts one pump photon into a correlated signal–idler pair, seeded by vacuum fluctuations and constrained by energy conservation ( $\omega_p=\omega_s+\omega_i$ ) and phase matching ( $\mathbf{k}_p=\mathbf{k}_s+\mathbf{k}_i$ ).

Where this lives today

SPDC is the quiet engine of experimental quantum optics: cheap, room-temperature, and reliable, it remains the default source of entangled and heralded photons three decades on. A modern polarization-entangled source is often a Sagnac loop around a PPKTP crystal — a single pump traverses the periodically poled crystal both ways and the counter-propagating pairs recombine into a Bell state with fidelities above $99\%$ and millions of pairs per second per milliwatt. These are the boxes at the sending stations of entanglement-based quantum key distribution — including the Micius satellite's source, which distributed entangled photons over a thousand kilometres — and the workhorses of loophole-free Bell tests. At the other end of the pump scale, high-gain SPDC in an optical parametric oscillator or waveguide produces bright two-mode squeezed vacuum. Pushed to $r\gtrsim1$ , the same physics injects squeezed light into the LIGO and Virgo gravitational-wave detectors to beat the shot-noise limit and feeds Gaussian-boson-sampling processors. Engineered joint spectral amplitudes now deliver heralded single photons with spectral purity above $0.95$ . From a benchtop crystal that converts one photon in a billion, a whole quantum-photonic industry has grown.

The physical picture

Picture the $\chi^{(2)}$ crystal as a shop where one expensive photon can be traded for two cheaper ones, provided the till balances in energy and momentum. Classically the shop is closed unless a customer brings a wave to seed the transaction. Quantum-mechanically it is never empty: the vacuum of the signal and idler modes always jostles with zero-point fluctuations, and those supply the seed for free, so pairs appear spontaneously. The rate is tiny, but each event produces a beautifully clean two-photon state. The two daughters are twins with a shared birth certificate: created at the same instant, to within the crystal's coherence time, with frequencies and momenta that add up to the pump's. That correlation is the entire resource — measure one twin's arrival time, colour, or polarization and you instantly know the other's. The first figure fixes the bookkeeping with a concrete case: a violet diode pump split into two near-infrared twins.

Figure

A 405 nm pump photon of 3.06 eV is annihilated in a χ⁽²⁾ crystal and reappears as two 810 nm photons of 1.53 eV each — energy and momentum both balance, ω_p = ω_s + ω_i and k_p = k_s + k_i

The exact output is not a single pair but a two-mode squeezed vacuum, with quadratures squeezed below the shot-noise floor and photon numbers locked together. In the low-gain regime this reduces to occasional single pairs on a dominant "nothing happened" vacuum — the heralded-photon regime; at high gain the same physics yields bright continuous-variable EPR entanglement. Phase matching is the tuning knob throughout: crystal angle, temperature, or poling period selects which $(\omega_s,\mathbf{k}_s)$ and $(\omega_i,\mathbf{k}_i)$ are generated, engineering the cones, the spectrum, and whether the twins are spectrally separable. The second figure shows the two defining geometries: the single ring of Type-I and the crossed cones of Type-II.

Figure

Type-I puts signal and idler on one cone at opposite points; Type-II emits orthogonally polarized photons on two crossed cones whose intersections carry undefined polarization — the polarization-entangled directions. Pump 405 nm, rings at 810 nm

The core equations

Every SPDC output frequency and direction is fixed by two rules inherited from classical three-wave mixing. Energy conservation sets which frequencies can pair up; phase matching — momentum conservation inside a dispersive medium — sets which pairs build up coherently over the crystal length.

\omega_p=\omega_s+\omega_i,\qquad \mathbf{k}_p=\mathbf{k}_s+\mathbf{k}_i \qquad (1)

where: $\omega_p,\omega_s,\omega_i$ — angular frequencies of pump, signal, idler; $\mathbf{k}_p,\mathbf{k}_s,\mathbf{k}_i$ — wavevectors inside the crystal, $|\mathbf{k}_j|=n(\omega_j)\,\omega_j/c$ ; $n(\omega_j)$ — refractive index, polarization-dependent in a birefringent crystal. The first is exact for the central frequencies; the second is what dispersion makes hard and every tuning trick exists to enforce. To see why pairs appear, quantize the signal and idler fields but keep the strong pump classical, and insert both into the $\chi^{(2)}$ interaction energy $\propto\chi^{(2)}EEE$ . Keeping only the resonant, energy-conserving term leaves a pair-creation Hamiltonian.

\hat H_{\mathrm{int}}=\hbar\kappa\,\hat a_s^{\dagger}\hat a_i^{\dagger}\,e^{-i(\omega_p-\omega_s-\omega_i)t}+\mathrm{h.c.} \qquad (2)

where: $\hat a_s^{\dagger},\hat a_i^{\dagger}$ — creation operators for signal and idler; $\kappa$ — coupling $\propto\chi^{(2)}E_p$ ; $\hbar$ — reduced Planck constant; h.c. — the conjugate $\hat a_s\hat a_i$ , which destroys a pair. The operator $\hat a_s^{\dagger}\hat a_i^{\dagger}$ creates photons two at a time, one per arm — photons are born in pairs. Evolving the joint vacuum under this Hamiltonian is the two-mode squeezing operator $\exp[\xi\hat a_s^{\dagger}\hat a_i^{\dagger}-\xi^{*}\hat a_s\hat a_i]$ , with amplitude $r=\kappa T$ . Disentangling it (an SU(1,1) identity) gives a geometric superposition of perfectly paired Fock states — the true output state and the workhorse expression for SPDC statistics.

|\psi\rangle=\frac{1}{\cosh r}\sum_{n=0}^{\infty}e^{in\theta}(\tanh r)^{n}\,|n_s,n_i\rangle \qquad (3)

where: $|n_s,n_i\rangle$ — Fock state with exactly $n$ signal and $n$ idler photons, always equal; $r$ — squeezing amplitude, $\tanh r$ small in the single-pair regime. The equal indices encode perfect number correlation: whatever you count in one arm, the other must match. Look only at the signal arm and the correlation is invisible. Tracing the idler out of equation (3) leaves a Bose–Einstein distribution — the statistics of chaotic light.

P(n)=\frac{(\tanh r)^{2n}}{\cosh^{2}r}=\frac{\bar n^{\,n}}{(1+\bar n)^{\,n+1}},\qquad \bar n=\sinh^{2}r \qquad (4)

where: $P(n)$ — probability of $n$ photons in one arm when the other is unobserved; $\bar n=\sinh^2 r$ — mean photon number per mode. An unheralded arm has $g^{(2)}(0)=2$ (thermal bunching); heralding restores single-photon character. Phase matching enters through the coherent sum of pair-creation amplitudes along the crystal. Each slab contributes with a phase set by the local mismatch, and the integral over the length $L$ yields a sinc envelope.

\text{amplitude}\;\propto\;L\,\mathrm{sinc}\!\left(\frac{\Delta k\,L}{2}\right),\qquad \Delta k=k_p-k_s-k_i \qquad (5)

where: $L$ — interaction length; $\Delta k$ — collinear phase mismatch; $\mathrm{sinc}(x)=\sin(x)/x$ , first zero at $x=\pi$ . The rate peaks at $\Delta k=0$ as $L^2$ while the acceptance narrows as $1/L$ — longer crystals are brighter but more sharply filtered. When birefringence alone cannot bring $\Delta k$ to zero at the wanted wavelengths, periodic poling supplies the missing momentum. Flipping the sign of $\chi^{(2)}$ with period $\Lambda$ adds a reciprocal-lattice vector $2\pi/\Lambda$ .

\Delta k=k_p-k_s-k_i-\frac{2\pi}{\Lambda}=0 \qquad (6)

where: $\Lambda$ — poling period of a crystal such as PPKTP or PPLN; $2\pi/\Lambda$ — reciprocal-lattice momentum from the grating. This lets the designer use the largest nonlinear coefficient and phase-match at chosen wavelengths. In Type-II geometry the two emission cones cross along directions of undefined polarization, and there the pair emerges in the maximally polarization-entangled Bell state $|\Psi^{\pm}\rangle=\tfrac{1}{\sqrt2}(|H\rangle_s|V\rangle_i\pm|V\rangle_s|H\rangle_i)$ — the resource for Bell tests and entanglement-based cryptography.

Derivation AFrom the parametric Hamiltonian to the single-pair state

The goal is to show why the low-gain SPDC output is a vacuum plus one correlated pair — the state every heralded-photon and Bell-test experiment uses. We start from the resonant pair-creation Hamiltonian, recognize its action on the vacuum as a two-mode squeezer, and expand to first order in $\kappa T$ to reach the compact state $|0,0\rangle+\lambda|1_s,1_i\rangle$ .

Step 1. On resonance, $\omega_p=\omega_s+\omega_i$ , the oscillating phase in equation (2) vanishes and the interaction reduces to $\hat H_{\mathrm{int}}=\hbar\kappa(\hat a_s^{\dagger}\hat a_i^{\dagger}+\hat a_s\hat a_i)$ . Why: the rotating-wave approximation keeps only the energy-conserving term and its conjugate.

Step 2. The evolution operator is $\hat U(T)=\exp[-i\kappa T(\hat a_s^{\dagger}\hat a_i^{\dagger}+\hat a_s\hat a_i)]$ — precisely the two-mode squeezing operator with amplitude $r=\kappa T$ . Why: naming it as squeezing lets us borrow every squeezed-vacuum result.

Step 3. For $\kappa T\ll1$ , expand to first order: $\hat U(T)|0,0\rangle\approx(1-i\kappa T\,\hat a_s^{\dagger}\hat a_i^{\dagger})|0,0\rangle$ . Why: the annihilation piece kills the vacuum, so only pair creation acts.

Step 4. Acting on the vacuum gives $\hat a_s^{\dagger}\hat a_i^{\dagger}|0,0\rangle=|1_s,1_i\rangle$ , so with $\lambda=-i\kappa T$ the state is $|\psi\rangle\approx|0,0\rangle+\lambda\,|1_s,1_i\rangle+\mathcal{O}(\lambda^2)$ . Why: mostly "nothing happened," occasionally one correlated pair — why detecting the idler heralds a signal photon. *Verified: reduces to the two-mode squeezed vacuum of equation (3) in the limit $\tanh r\to r$ ; matches Loudon Ch. 6 and standard SPDC treatments.*

Go deeper · Derivation B · 1 minWhy each arm looks thermal while the pair is perfectly correlated

Derivation BWhy each arm looks thermal while the pair is perfectly correlated

The aim is to reconcile two facts that sound contradictory: the joint SPDC state has perfectly correlated photon numbers, yet either arm alone shows the random super-Poissonian statistics of thermal light. The resolution is the partial trace over the unobserved idler, which forces the Fock indices equal and leaves a diagonal Bose–Einstein mixture.

Step 1. Start from the pure state $|\psi\rangle=\frac{1}{\cosh r}\sum_n(\tanh r)^n|n_s,n_i\rangle$ . Why: the equal indices announce the correlation — $n$ in one arm guarantees $n$ in the other.

Step 2. Form $|\psi\rangle\langle\psi|=\frac{1}{\cosh^2 r}\sum_{n,m}(\tanh r)^{n+m}|n_s,n_i\rangle\langle m_s,m_i|$ , with off-diagonal coherences linking paired states. Why: the trace is about to destroy exactly the coherences linking different $n$ .

Step 3. Trace over the idler, $\hat\rho_s=\sum_k\langle k_i|\psi\rangle\langle\psi|k_i\rangle$ ; the overlap is nonzero only when $n=m=k$ . Why: the number-correlated idler acts as a which-number measurement, leaving a diagonal state.

Step 4. The survivor is $P(n)=(\tanh r)^{2n}/\cosh^2 r=\bar n^{\,n}/(1+\bar n)^{n+1}$ with $\bar n=\sinh^2 r$ — Bose–Einstein, hence $g^{(2)}(0)=2$ . Why: a single arm is indistinguishable from thermal light; single-photon behaviour appears only after heralding. *Verified: reproduces the standard result that an unheralded SPDC arm is thermal with $g^{(2)}(0)=2$ per mode, as in Loudon and Saleh & Teich.*

Go deeper · Derivation C · 1 minThe sinc phase-matching envelope from the crystal integral

Derivation CThe sinc phase-matching envelope from the crystal integral

The purpose is to show where the ubiquitous $\mathrm{sinc}(\Delta k L/2)$ factor comes from and why it controls both brightness and bandwidth. The physics is coherent buildup: every slab emits a pair-creation amplitude with a position-dependent phase set by $\Delta k$ , and adding those over the length $L$ is a simple integral whose magnitude is a sinc — peaked when the phases agree, zero when they wind through a full turn.

Step 1. The local amplitude from a slab at $z$ is $\mathrm{d}A\propto\chi^{(2)}E_p\,e^{i\Delta k\,z}\,\mathrm{d}z$ , with $\Delta k=k_p-k_s-k_i$ . Why: the phase each slab inherits from the pump drifts relative to the daughters unless $\Delta k=0$ .

Step 2. Integrate: $A=\int_0^L e^{i\Delta k z}\,\mathrm{d}z=\dfrac{e^{i\Delta k L}-1}{i\Delta k}=L\,e^{i\Delta k L/2}\,\mathrm{sinc}\!\left(\dfrac{\Delta k L}{2}\right)$ . Why: the coherent sum of equal-amplitude contributions with advancing phase — the grating integral.

Step 3. The pair rate goes as $|A|^2\propto L^2\,\mathrm{sinc}^2(\Delta k L/2)$ : peaked at $\Delta k=0$ as $L^2$ , first zeros at $\Delta k\,L=2\pi$ , acceptance narrowing as $1/L$ . Why: the length that brightens the source also filters it more tightly — a brightness-versus-bandwidth trade; dispersion maps $\Delta k(\omega,\theta)$ onto the spectrum and cone angle. *Verified: standard nonlinear-optics phase-matching result (Boyd; Saleh & Teich); identical in form to the second-harmonic-generation efficiency envelope.*

Limiting cases

SPDC behaviour is governed by the parametric gain $r=gL$ (set by $\chi^{(2)}$ , pump amplitude, and length) and by the phase mismatch $\Delta k$ .

Case	Result	Meaning
Low-gain (single-pair)	$\vert\psi\rangle\approx\vert0,0\rangle+\lambda\,\vert1_s,1_i\rangle,\ \vert\lambda\vert\ll1$	The heralded-single-photon and Bell-state regime; higher-order pairs suppressed as $\vert\lambda\vert^2$
High-gain (bright squeezing)	$\vert\psi\rangle=\frac{1}{\cosh r}\sum_n(\tanh r)^n\vert n_s,n_i\rangle,\ r\gtrsim1$	Two-mode squeezed vacuum with large $\bar n$ ; continuous-variable EPR entanglement
Phase-matched (peak)	$\Delta k=0\Rightarrow\text{amp}\propto L,\ \text{rate}\propto L^2$	Perfect momentum balance maximizes the amplitude; rate grows with crystal length squared
Phase-mismatched	$\text{amp}\propto L\,\mathrm{sinc}(\Delta k L/2)$	The sinc envelope sets spectral and angular bandwidth; first zeros at $\Delta k\,L=2\pi$
Degenerate	$\omega_s=\omega_i=\omega_p/2$	Twins split the pump energy equally; the central case for HOM interference and many entanglement sources

Common trapJust raise the pump power and you get more clean single photons

SPDC is not a deterministic single-photon gun. In any window it may emit zero, one, or two pairs, and the multi-pair probability is intrinsic, not a defect you can engineer away. The singles rate grows as $|\lambda|^2\propto$ pump power, but the two-pair contribution grows as $|\lambda|^4$ , so the ratio of double to single pairs climbs linearly with pump power. Turning up the pump therefore raises the heralded $g^{(2)}(0)$ above zero and degrades purity: brightness and purity trade off, and no pump setting gives you both. A second trap: "signal" and "idler" are historical labels, not physical species — in degenerate Type-I SPDC the two photons are genuinely indistinguishable.

Worked examples

1 · Energetics of a degenerate 405 nm source (confidence builder)

A 405 nm violet diode laser pumps a BBO crystal cut for degenerate Type-I SPDC. Find the daughter wavelengths, their photon energies, and confirm energy conservation; give the pump frequency for scale.

Step 1. Degenerate means the pump energy splits evenly, so each daughter carries half the pump frequency and twice its wavelength: $\lambda_s=\lambda_i=2\lambda_p=2\times405=\mathbf{810\ nm}$ , both in the near-infrared, on the single Type-I ring above.

Step 2. Photon energies from $E=hc/\lambda$ with $hc=1239.84\ \mathrm{eV\,nm}$ : the pump is $E_p=1239.84/405=\mathbf{3.061\ eV}$ , each daughter $E_s=E_i=1239.84/810=\mathbf{1.531\ eV}$ , and $1.531+1.531=3.061\ \mathrm{eV}$ confirms $\omega_p=\omega_s+\omega_i$ .

Step 3. For scale, $\nu_p=c/\lambda_p=(2.998\times10^8)/(405\times10^{-9})=\mathbf{7.40\times10^{14}\ Hz}$ , and each daughter sits at half that. The 405 nm pump yields two 810 nm photons of 1.53 eV each, and 1.53 + 1.53 = 3.06 eV balances the books.

2 · Brightness versus purity: reading the multi-pair penalty

A pulsed SPDC source is run at a mean of $\bar n=\sinh^2 r=0.01$ photon pairs per pulse per mode. Find the squeezing parameter, the single- and double-pair probabilities, and the leading heralded $g^{(2)}(0)$ . Then find the $\bar n$ that pushes the heralded $g^{(2)}(0)$ to $0.1$ .

Step 1. Invert $\bar n=\sinh^2 r$ : $\sinh r=\sqrt{0.01}=0.1$ , so $r=\operatorname{arcsinh}(0.1)=\mathbf{0.0998}$ — deep in the low-gain regime. Also $\tanh r=0.0995$ and $\cosh^2 r=1+\bar n=1.01$ .

Step 2. The paired-Fock probabilities from equation (4) are $P(n)=(\tanh r)^{2n}/\cosh^2 r$ : the vacuum term $P(0)=1/1.01=\mathbf{0.990}$ , one pair $P(1)=(0.0995)^2/1.01=\mathbf{9.80\times10^{-3}}$ , two pairs $P(2)=(0.0995)^4/1.01=\mathbf{9.71\times10^{-5}}$ . Nothing happens $99\%$ of the time; when it does, it is almost always a single pair.

Step 3. The contamination ratio is the exact statement $P(2)/P(1)=\tanh^2 r=\mathbf{9.90\times10^{-3}}\approx\bar n$ . To leading order the heralded auto-correlation tracks this, $g^{(2)}_h(0)\approx2\bar n=\mathbf{0.020}$ — well below the single-photon threshold of $1$ .

Step 4. Since $g^{(2)}_h(0)\approx2\bar n$ grows linearly, reaching $g^{(2)}_h(0)=0.1$ needs $\bar n\approx0.05$ — five times brighter, five times more contaminated. At $\bar n=0.01$ the source is 99.0 % vacuum, 0.98 % single pair, 0.0097 % double pair, giving $g^{(2)}_h(0)\approx0.02$ ; hitting $0.1$ costs a fivefold rise in $\bar n$ .

3 · Poling period for a telecom PPLN source

Design a periodically poled lithium niobate (PPLN) crystal for degenerate quasi-phase-matched SPDC: a 775 nm pump into a 1550 nm signal and 1550 nm idler. Using extraordinary indices $n_p\approx2.18$ at 775 nm and $n_s=n_i\approx2.14$ at 1550 nm, find the required poling period $\Lambda$ .

Step 1. The quasi-phase-matching condition (6) is $k_p-k_s-k_i=2\pi/\Lambda$ , with $k_j=2\pi n_j/\lambda_j$ . Because the process is degenerate, $\lambda_s=\lambda_i=1550$ nm and $n_s=n_i$ , so the material mismatch is $\Delta k_{\text{mat}}=2\pi\left(\dfrac{n_p}{\lambda_p}-\dfrac{2n_s}{\lambda_s}\right)$ .

Step 2. Insert the numbers: $n_p/\lambda_p=2.8129\times10^{-3}\ \mathrm{nm^{-1}}$ and $2n_s/\lambda_s=2.7613\times10^{-3}\ \mathrm{nm^{-1}}$ , difference $5.16\times10^{-5}\ \mathrm{nm^{-1}}$ , so $\Delta k_{\text{mat}}=2\pi\times5.16\times10^{-5}=3.24\times10^{-4}\ \mathrm{nm^{-1}}$ .

Step 3. The grating must supply this momentum: $\Lambda=2\pi/\Delta k_{\text{mat}}=6.283/(3.24\times10^{-4})\ \mathrm{nm}=\mathbf{19.4\ \mu m}$ — right in the range of real telecom PPLN devices, then fine-tuned by temperature. A poling period of about 19.4 μm quasi-phase-matches 775 nm → 1550 nm + 1550 nm down-conversion in PPLN.

Physics of the portrait

The meaning of SPDC was written in the language of photon coincidences — and no one shaped that language more than the physicist who turned two-photon counting into precision physics.

Origins · 1 minLeonard Mandel and how this was discovered

Leonard Mandel

1927–2001 · quantum optics and two-photon interference

Leonard Mandel spent his career insisting that the deepest features of light show up not in intensities but in correlations between photon detections, and SPDC became his ideal instrument. Long based at the University of Rochester, he built — with Emil Wolf and a generation of students — much of the modern framework of quantum coherence, culminating in the treatise Optical Coherence and Quantum Optics. His most famous result, with Chung Ki Hong and Zhe-Yu Ou in 1987, used exactly the correlated pairs of parametric down-conversion: send the twins onto a beam splitter, scan their relative delay, and the coincidence rate collapses to zero when they arrive together. That "Hong–Ou–Mandel dip" is now the standard yardstick of photon indistinguishability and the beating heart of linear-optical quantum computing. Where earlier physicists asked what a single photon does, Mandel asked what two photons do together — and SPDC, the cleanest source of correlated pairs anyone had, let him answer with a precision that founded a field.

History

1961

— Franken and coworkers demonstrate second-harmonic generation, launching nonlinear optics

1967

— Harris, Oshman and Byer observe parametric fluorescence, the first sighting of spontaneous down-conversion

1970

— Burnham and Weinberg measure the time coincidence of signal and idler, confirming photons are born in pairs

1987

— Hong, Ou and Mandel observe the two-photon interference dip that bears their names

1995

— Kwiat and coworkers build a bright Type-II source of polarization-entangled photon pairs

Going further

Texts. Loudon, The Quantum Theory of Light, 3rd ed., Ch. 6 (parametric processes, squeezed light) · Saleh & Teich, Fundamentals of Photonics, 2nd ed., Ch. 22 (nonlinear optics) and Ch. 12–13 (photon statistics) · Boyd, Nonlinear Optics, 3rd ed., Ch. 1–2 and the parametric-fluorescence sections · Hecht, Optics, 5th ed. (nonlinear optics section) · Demtröder, Laser Spectroscopy (nonlinear frequency conversion) · Hong, Ou & Mandel, Phys. Rev. Lett. 59, 2044 (1987) — two-photon interference · Kwiat et al., Phys. Rev. Lett. 75, 4337 (1995) — polarization-entangled Type-II source.

Provenance. The parametric Hamiltonian and two-mode squeezed vacuum follow Loudon Ch. 6; the thermal reduced state with $g^{(2)}(0)=2$ per mode is standard in Loudon and Saleh & Teich; the sinc envelope and quasi-phase-matching follow Boyd and Saleh & Teich Ch. 22. A verifier should recompute the SU(1,1) disentangling giving $\bar n=\sinh^2 r$ , the partial trace yielding Bose–Einstein statistics, and the crystal-length integral giving $L\,\mathrm{sinc}(\Delta k L/2)$ . The worked numbers follow from first principles: 405 nm → 810 nm + 810 nm via $E=hc/\lambda$ ( $hc=1239.84$ eV·nm); $\bar n=0.01\Rightarrow r=0.0998$ , $P(1)=9.80\times10^{-3}$ , $P(2)=9.71\times10^{-5}$ ; PPLN period $\Lambda\approx19.4\ \mu$ m from $n_p=2.18$ , $n_s=n_i=2.14$ . The pair-per-pump range ( $10^{-11}$ – $10^{-6}$ ) is order-of-magnitude, not a precision constant.

Prerequisites. Quantization of the Electromagnetic Field · Vacuum State and Zero-Point Energy · Fock States and the Photon Number Basis · Squeezed States of Light. Leads to. Single-Photon Sources · Photon Statistics and the Mandel Q Parameter · Hanbury Brown–Twiss Experiment · Photon Antibunching.

Ask the tutor: Ask me to walk through how the two-mode squeezed vacuum turns into a heralded single photon — or how to design the poling period and pump for the exact wavelengths your experiment needs.

Why is it called "spontaneous" — spontaneous compared to what?

The contrast is with stimulated down-conversion, better known as optical parametric amplification. Inject a real signal beam and it stimulates the pump to shed photons into that mode and its idler partner, output growing with injected power — an ordinary classical amplifier. Remove the beam and classically there is nothing to amplify; quantum-mechanically the signal and idler modes still hold vacuum fluctuations, which stimulate the process just as a real field would. The emission is "spontaneous" in exactly the sense atomic spontaneous emission is: stimulated by the vacuum rather than an applied field.

How is SPDC different from second-harmonic or difference-frequency generation?

All three are $\chi^{(2)}$ three-wave-mixing processes with the same Hamiltonian; they differ in which waves are strong and which are quantized. Second-harmonic generation runs forward — two pump photons fuse into one at double the frequency. Difference-frequency generation and parametric amplification run backward but with a real injected signal, so they are classical and deterministic. SPDC is the backward process with no injected signal or idler: the only seed is vacuum noise, so it yields single photon pairs rather than a bright beam.

Are signal and idler genuinely separate photons or one photon in disguise?

They are two distinct photons — two quanta at two detectors, each able to trigger its own click. What binds them is not identity but correlation: created in a single quantum event, their energies, momenta, birth times, and (in Type-II) polarizations are locked together by the conservation laws. In the degenerate case the labels "signal" and "idler" lose meaning because the photons occupy identical modes, but there are still two of them. The pump photon is genuinely destroyed and its energy redistributed into two new field excitations — you recover not the pump photon but its two heirs.

Why is SPDC so inefficient — about one pair per billion pump photons?

Because it is a weak nonlinear effect seeded by vacuum noise. The conversion probability is set by $\chi^{(2)}$ , minute in dimensionless terms, times the tiny vacuum field amplitude and the phase-matched length; single-pass efficiencies land between $10^{-11}$ and $10^{-6}$ pairs per pump photon. That inefficiency is a feature: it keeps the source deep in the low-gain regime where multi-pair emission is negligible, so nearly every event is a clean single pair. A cavity or long waveguide boosts the interaction when brightness matters more than purity.

What is the practical difference between Type-I and Type-II phase matching?

It is the polarization of the daughters. In Type-I both signal and idler share one polarization, orthogonal to the pump, and emerge on a single concentric cone — simple, bright, ideal for energy-time entanglement or heralding. In Type-II they carry orthogonal polarizations and emerge on two cones that intersect along two directions; at those crossings each photon could belong to either cone, so its polarization is undefined and the pair emerges polarization-entangled in a Bell state — why Type-II BBO became the classic Bell-test source, at the cost of a polarization walk-off that must be compensated with extra crystals.

Why do we say the SPDC pair is "entangled" rather than just correlated?

Classical correlation would mean each photon secretly has definite properties that happen to match. SPDC pairs are stronger: before measurement neither photon has a definite frequency, arrival time, or (in Type-II) polarization — only the pair does, as $|\Psi^\pm\rangle$ superposing $|HV\rangle$ and $|VH\rangle$ makes explicit. Measuring one photon collapses the joint state and fixes the other's outcome in a way no shared classical variable can reproduce — exactly what a Bell-inequality violation demonstrates.