= Solution
<spontaneous parametric down-conversion> uses a second-order nonlinear polarization to convert pump <energy> into correlated signal and idler <photons>. With a classical undepleted pump, a single-mode interaction-picture model is
$$
H_I=i\hbar(\kappa a_s^\dagger a_i^\dagger-\kappa^*a_sa_i).
$$
The creation term acts on the signal/idler vacuum, so no classical signal seed is required. The pump supplies the emitted <energy>; vacuum fluctuations are not an extra <energy> source. In the small-gain limit the <SPDC pair state> is
$$
|\Psi\rangle=|0,0\rangle+\kappa t|1,1\rangle+O(|\kappa t|^2),
$$
with normalization corrections understood at second order. More generally it is a <two-mode squeezed vacuum> with correlated pair numbers. A classical calculation with both generated fields exactly zero would miss the spontaneous start.
Efficient generation requires $\omega_p=\omega_s+\omega_i$ and approximate wave-vector matching $\mathbf k_p=\mathbf k_s+\mathbf k_i$. <birefringence> or a reciprocal grating vector in quasi-phase matching can compensate dispersion. In a continuum-mode treatment the pair amplitude includes
$$
\Phi(\omega_s,\omega_i)\propto\alpha_p(\omega_s+\omega_i)
L\operatorname{sinc}\left(\frac{\Delta kL}{2}\right)e^{i\Delta kL/2}.
$$
The pump spectrum constrains the sum <frequency>, while <phase matching> constrains the allowed <frequency> combinations and emission directions. <type-I phase matching> produces daughters in the same polarization class; <type-II phase matching> produces orthogonal ordinary/extraordinary classes. Correlation of pair numbers or orthogonal polarizations does not alone guarantee a polarization-entangled state in two chosen output paths.
One application is a <heralded single-photon source>: detect one partner and condition measurements on the other. Low gain suppresses multiple-pair contamination, whose probability begins at fourth order in the gain. Efficient collection and low detector noise matter. Spectral entanglement with an unresolved herald can leave the partner mixed; engineering or filtering the joint spectrum improves purity at a possible cost in brightness. Heralding does not make a <photon> appear in every pump pulse.
A second application is entangled-pair generation for <quantum communication> and tests of <Bell inequalities>. If the assignments $H_sV_i$ and $V_sH_i$ are indistinguishable except for polarization, their amplitudes can form $(|H_sV_i\rangle+e^{i\varphi}|V_sH_i\rangle)/\sqrt2$. Walk-off, group delay or <frequency> labels can reveal which assignment occurred and destroy polarization interference. Their compensation is part of preparing the state, not merely an improvement in count rate.
A primary example pumped a 3-mm beta-barium-borate crystal with a $351.1\,\mathrm{nm}$ argon-ion <laser>. Collecting the two intersections of its type-II emission cones selected paired output directions. Half-wave rotation and compensating birefringent crystals reduced temporal walk-off, and polarization analysers with avalanche photodiodes measured coincidence fringes. Varying analyser settings tested the correlations; accidentals, mode distinguishability and detector/collection assumptions limited interpretation. The reported visibility exceeded 97 percent and supported a Bell violation under the stated auxiliary assumptions, rather than a loophole-free claim.<New High-Intensity Source of Polarization-Entangled Photon Pairs by Paul G. Kwiat and others>[pp. 4337–4340, Figs. 1–2, compensation, detectors and coincidence correlations]{ref}
\Image[/past-exam-of-the-nonlinear-optics-and-quantum-states-of-light-course-of-the-university-of-cambridge/2015/paper-1-spdc.png]
{title=Original down-conversion and heralding schematic with an illustrative joint-frequency selection model}
Other uses include preparing interfering <photons> for quantum-information experiments, correlation-based imaging, and pair-based detector calibration. All rely on specified spatial and spectral modes and coincidence statistics. A pump-blocked background run, delayed-coincidence estimate, gain dependence and polarization/temporal-overlap scans distinguish correlated pairs from fluorescence, detector dark counts and accidental coincidences. The useful resource is the controlled joint <quantum state>, not simply two bright output beams.
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