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The low-temperature crystalline phase of barium borate, used as a nonlinear optical crystal. A properly phase-matched and compensated source can use it for spontaneous parametric down-conversion; the choice of collected modes and compensation determines the resulting polarization entanglement.

An experimental optical physicist who coauthored a 1986 generation of optical squeezed vacuum. A cavity enhanced spontaneous parametric down-conversion, while balanced homodyne detection measured reduced optical quadrature noise. The comparison with attenuation models linked a directly observed noise reduction to the source's underlying quantum field.^[ref]

An idler photon is the partner of a signal photon in a specified nonlinear optical process. In spontaneous parametric down-conversion, one pump photon supplies the energy of the pair, $\hslash {\omega }_p=\hslash {\omega }_s+\hslash {\omega }_i$.

The idler can be measured to herald a signal photon or to condition a correlation measurement. It is not a physically expendable photon: losses in its channel reduce heralding rates and can bias an uncorrected coincidence sample. When the two output modes have the same frequency, the signal/idler distinction comes from their assigned spatial, polarization or experimental roles rather than from energy ordering.

An experimental optical physicist who coauthored a 1986 demonstration of optical squeezed vacuum. Degenerate spontaneous parametric down-conversion inside a cavity produced reduced fluctuations of an optical quadrature, measured with balanced homodyne detection. The result made a noise level below the vacuum reference a directly measured optical property.^[ref]

N. Boeuf and collaborators developed calculations of non-collinear phase matching in uniaxial and biaxial nonlinear crystals. The NIST-hosted 1999 draft extended calculations beyond cases where interacting waves all propagate along one line.^[ref]

For spontaneous parametric down-conversion, the pump, signal and idler satisfy ${\omega }_p={\omega }_s+{\omega }_i$ and ${\mathbf{k}}_p={\mathbf{k}}_s+{\mathbf{k}}_i$. The second condition is a vector relation, so crystal anisotropy and propagation angles jointly constrain possible outputs. Solving only the frequency equation does not establish an efficient optical interaction.

A crystal used for its nonlinear optics response, allowing processes such as spontaneous parametric down-conversion when the appropriate phase matching is satisfied.

In quantum dense coding, a shared Bell pair lets Alice send two classical bits by transmitting one qubit. Use photon polarization as the computational basis, $|0⟩=|H⟩$, $|1⟩=|V⟩$, and define the Bell statesAlice and Bob initially share $|{\varphi }^{+}{⟩}_{AB}$. Alice encodes a two-bit message $(m,n)$ by applying the Pauli operator $X^n{Z}^m$ to her qubit:The last minus sign is a global phase. Alice transmits her photon to Bob, who now holds both qubits and uses a Bell-basis measurement to identify the message. The four Bell states are orthogonal, so an ideal Bell state analyzer decodes them without error. The resource cost is one previously shared Bell pair and one transmitted qubit per two classical bits; the shared quantum entanglement is consumed. Before transmission Bob's reduced density matrix is $I/2$ for every message, consistent with the no-communication theorem.

The BBO crystal supplies the polarization-entangled photon pair. An ultraviolet laser pumps this nonlinear optical crystal; spontaneous parametric down-conversion produces signal and idler photons with ${\omega }_p={\omega }_s+{\omega }_i$ and phase matching ${\mathbf{k}}_p={\mathbf{k}}_s+{\mathbf{k}}_i$. In a noncollinear type-II phase matching arrangement, collect the intersections of the two emission cones. The alternatives $|H{⟩}_A|V{⟩}_B$ and $|V{⟩}_A|H{⟩}_B$ must be indistinguishable in arrival time, spectrum and spatial mode. Compensation of birefringence and adjustment of the relative quantum phase then give $(|HV⟩+|VH⟩)/\sqrt{2}$; a local Pauli X gate converts it to $|{\varphi }^{+}⟩$. Half-wave plates and other waveplates implement Alice's Pauli gates. The BBO is the pair source; the message is encoded subsequently by the local optical elements.^[ref]

A basic passive linear optics Bell state analyzer with no extra resources identifies only two of the four polarization Bell states unambiguously. Thus the ideal two-bit capacity presupposes complete Bell-basis measurement; that basic optical implementation alone does not attain it deterministically.

The beta barium borate crystal supplies spontaneous parametric down-conversion: a weak ultraviolet pump can produce a pair of lower-frequency photons, with their frequencies and wavevectors selected by conservation of energy and phase matching. A suitably aligned and compensated source makes the two polarization alternatives coherent, giving the stated Bell state. The crystal alone does not guarantee that every collected pair has that state.

The two polarizing beam splitters route horizontal polarization and vertical polarization to different ports. Take $H$ from arm $a$ to the trigger $T$ and $V$ to arm $c$. The ordinary beam splitter in arm $b$ is a lossless, polarization-independent 50:50 splitter, with the unused input in the vacuum state. It splits probability amplitudes, rather than measuring which output a photon takes. The half-wave plate changes the relative phase of its principal polarizations by $\pi$; orient it so that a $V$ photon in $c$ becomes $(H+V)/\sqrt{2}$ in $c^{\prime }$, up to a common phase. Finally, photodetectors absorb arriving photons and supply electrical clicks. They need not resolve photon number for the ideal four-photon, fourfold-coincidence calculation: a click at every detector already requires one photon at each.

Choose the relative wave phases so that the final polarizing beam splitter sends $H_{c^{\prime }}$ to $H_2$, $V_{c^{\prime }}$ to $V_1$, $H_d$ to $H_1$ and $V_d$ to $V_2$. Such phase conventions must be used consistently: an uncompensated relative phase would give a phase-shifted GHZ state.

One implementation uses type-II phase matching for spontaneous parametric down-conversion in a nonlinear birefringent crystal pumped by a coherent laser. A pump photon produces two lower-frequency photons of orthogonal polarizations, subject to energy and momentum phase matching. Collect two spatial modes in which the alternatives “horizontal photon in Alice's mode and vertical photon in Bob's mode” and the reversed assignment are indistinguishable. Compensation crystals, filters and matched optical paths erase polarization-dependent timing and spectral information. The amplitudes then add coherently to give a state proportional to $|HV⟩+e^{i\phi }|VH⟩$; a birefringent phase element tunes $\phi$.

The essential requirement is coherent indistinguishability of the two assignments. Producing two orthogonally polarized photons with information revealing which assignment occurred would instead give a mixture without the required polarization entanglement.

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 isThe 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 iswith 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 includesThe 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 $H_s{V}_i$ and $V_s{H}_i$ are indistinguishable except for polarization, their amplitudes can form $(|H_s{V}_i⟩+e^{i\phi }|V_s{H}_i⟩)/\sqrt{2}$. 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.^[ref]

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.

An experimental optical physicist who coauthored a bright source of polarization-entangled photon pairs in 1995. Noncollinear type-II spontaneous parametric down-conversion generated the pairs, while compensation of distinguishability allowed strong polarization entanglement. A source of this kind supplies repeatable inputs for Bell tests and experiments in quantum information.^[ref]

In 2015, Marissa Giustina, Anton Zeilinger and collaborators combined spontaneous parametric down-conversion with rapid random analyzer choices and efficient transition-edge sensors. Local records included settings and detector signals; their Bell inequality used detected and undetected trials, avoiding fair sampling. Locally fixed time slots prevented coincidence-based trial selection. Calibrated optical and electrical delays checked spacelike separation of each choice from the remote readout. The violation addressed detection and locality together. Measurement independence still requires assumptions about the hidden state's origin; the experiment cannot exclude every possible common cause of source and settings.^[ref]

In 1998, Jian-Wei Pan, Dik Bouwmeester, Harald Weinfurter and Anton Zeilinger created two photon pairs by two passages of a pulsed pump through a crystal using spontaneous parametric down-conversion. Interfering the inner photons at a nonpolarizing beam splitter implemented a partial Bell-basis measurement. Narrow optical filters and spatial filtering removed distinguishing information; changing the outer polarizer bases checked the conditional correlations. Fourfold detections selected the entanglement swapping events. The reported interference supported conditional quantum entanglement, but the paper reserved a statistically significant Bell inequality violation for improved visibility; it was not a demonstration closing the detection loophole.^[ref]

The ideal calculation makes the distinction between conditional and unconditional quantum states explicit. Start with spin singlet states $|{\psi }^{-}{⟩}_{12}|{\psi }^{-}{⟩}_{34}$, where $|{\psi }^{-}⟩=(|HV⟩-|VH⟩)/\sqrt{2}$ and $H,V$ denote orthogonal photon polarizations. Taking the inner product with $⟨{\psi }^{-}{|}_{23}$ gives $-|{\psi }^{-}{⟩}_{14}/2$: expanding the four terms leaves precisely the two antisymmetric outer terms. That Bell-basis measurement outcome has probability $1/4$ and leaves the outer pair in a spin singlet state. Ignoring the inner outcome instead leaves the outer density matrix $I_4/4$, a maximally mixed state. A conditional preparation flag is useful for later Bell tests only with the timing and setting conditions in setting-independent heralding preserves Bell locality.

A signal photon is the photon assigned to a designated signal mode in a photon-pair process such as spontaneous parametric down-conversion. Its partner is called the idler photon. The labels identify modes or experimental roles; they do not universally mean the higher-energy and lower-energy photon.

For a narrowband pump, pair production obeys ${\omega }_p={\omega }_s+{\omega }_i$. Spatial and polarization correlations also depend on the phase matching arrangement. Detecting the idler can herald the presence of the signal, but heralding efficiency depends on collection and loss. At degeneracy, signal and idler can have equal frequencies while remaining distinguishable by propagation direction or another mode label.

In spontaneous parametric down-conversion, a strong pump drives a second-order nonlinear optical susceptibility, producing paired output photons in initially unseeded modes. The two outputs are conventionally called the signal photon and idler photon. They need not have equal frequencies. For a monochromatic pump, conservation of energy gives

Efficient bulk generation also requires phase matching: the nonlinear source and emitted waves must remain in phase. In the ideal plane-wave geometry this is ${\mathbf{k}}_p={\mathbf{k}}_s+{\mathbf{k}}_i$. Different refractive indices and propagation directions determine which pairs satisfy it. A finite interaction length supplies a mismatch bandwidth rather than an infinitely sharp wavevector constraint. ^[ref]

At weak gain, the SPDC pair state contains a large vacuum component and a small pair amplitude. Detecting one output conditions the other, producing a heralded single-photon source. This is probabilistic: stronger pumping also increases unwanted multiple-pair events. Experiments collect two output modes with optical filters and single-photon detectors, comparing coincident detections with background counts. ^[ref]