Entangled photons

Entangled photons are pairs (or larger groups) of photons described by one joint quantum state that cannot be factored into a state for each photon. Neither photon of a polarization-entangled pair has a definite polarization on its own, yet measurements on the two are correlated more strongly than any local classical model allows. The standard laboratory source is spontaneous parametric down-conversion (SPDC): a 405 nm pump in a nonlinear crystal produces pairs near 810 nm, and good sources give Bell-test values of the CHSH parameter close to the quantum limit of 2√2 ≈ 2.83, where any classical explanation is capped at 2.

Bell states

For polarization qubits written in the horizontal (H) and vertical (V) basis, the four maximally entangled two-photon states are the Bell states:

$$|\Phi^{\pm}\rangle = \tfrac{1}{\sqrt{2}}\big(|HH\rangle \pm |VV\rangle\big)$$ $$|\Psi^{\pm}\rangle = \tfrac{1}{\sqrt{2}}\big(|HV\rangle \pm |VH\rangle\big)$$

For $|\Phi^{+}\rangle$ measured with polarizers at angles $a$ and $b$, the correlation coefficient is $E(a,b) = \cos[2(a-b)]$, the same in every linear polarization basis. A product state such as $|HH\rangle$ shows perfect correlation in the H/V basis and none in the diagonal basis; an entanglement test therefore checks correlations in at least two bases.

Generation

Kwiat and co-workers (1995) produced polarization entanglement with type-II SPDC in a BBO crystal: the ordinary and extraordinary emission cones intersect along two lines, and a photon collected at an intersection may be either H or V, with its partner always orthogonal, giving a $|\Psi\rangle$-type state after compensation of walk-off. A second design (Kwiat et al., 1999) stacks two thin type-I crystals with optic axes rotated by 90°; a diagonally polarized pump creates $|HH\rangle$ in one crystal and $|VV\rangle$ in the other, and when the two processes are indistinguishable the output is $|\Phi\rangle$. Periodically poled crystals in a Sagnac loop give brighter, more stable sources. Time-bin entanglement survives fiber transmission better than polarization.

The CHSH test

Bell tests measure the correlation $E$ for two analyzer settings on each side, $a, a'$ and $b, b'$, and combine them:

$$\begin{aligned} S = \big|\,&E(a,b) - E(a,b') \\ &+ E(a',b) + E(a',b')\,\big| \end{aligned}$$

Any local hidden-variable model gives $S \le 2$. With $|\Phi^{+}\rangle$ and analyzer angles $a = 0°$, $a' = 45°$, $b = 22.5°$, $b' = 67.5°$, each correlation has magnitude $\cos 45° = 0.707$, and

$$S = 4 \times 0.707 = 2\sqrt{2} \approx 2.83,$$

the largest value quantum mechanics permits (the Tsirelson bound). In practice the measured two-photon fringe visibility $V$ scales the result to $S = 2\sqrt{2}\,V$, so a violation requires $V > 1/\sqrt{2} \approx 0.71$. Detection uses single-photon avalanche diodes or superconducting nanowire detectors with a coincidence window of a few nanoseconds. The 2022 Nobel Prize in Physics went to Alain Aspect, John Clauser and Anton Zeilinger for experiments with entangled photons establishing Bell violations and pioneering quantum information science.

Hong-Ou-Mandel interference

When two indistinguishable photons enter the two input ports of a 50:50 beam splitter at the same time, the amplitudes for "both transmitted" and "both reflected" cancel, and the photons always leave through the same port. Scanning the delay of one photon makes the coincidence rate between the two outputs fall to a dip, the Hong-Ou-Mandel (HOM) dip. Its width is set by the single-photon coherence time, the time counterpart of the coherence length, roughly

$$\tau_c \approx \frac{\lambda^2}{c\,\Delta\lambda}.$$

For 1 nm bandwidth at 810 nm this gives $\tau_c \approx 2.2$ ps, a path difference of about 0.66 mm in air; the exact width depends on the spectral shape of the filters. The dip visibility measures indistinguishability: 1 for photons identical in frequency, timing, polarization and spatial mode. Unlike classical interference, the effect is insensitive to the phase between the photons, which makes it the standard test of single-photon sources.

Applications

Entanglement-based quantum key distribution (Ekert, 1991) derives a key from correlated measurements and uses the Bell violation itself as a security check. Quantum teleportation transfers a photon's polarization state using a shared entangled pair and a Bell-state measurement (Bouwmeester et al., 1997). Photonic quantum computing builds larger entangled states, such as cluster states, from pairs and HOM-type interference. Entangled pairs also serve in quantum imaging and in interferometric measurements with photon pairs.

Pitfalls

Correlations in one basis alone do not prove entanglement; a classical mixture of $|HH\rangle$ and $|VV\rangle$ produces them. Multi-pair emission and dark counts add accidental coincidences that lower the visibility, so the mean pair number is kept to a few percent per pump pulse or coincidence window. Distinguishability between the two emission paths turns entanglement into a mixture, and fiber birefringence rotates polarization states unless compensated.

Common questions

How are entangled photons made?

Most often by SPDC in a nonlinear crystal or by spontaneous four-wave mixing in a waveguide or fiber. Semiconductor quantum dots can also emit entangled pairs through a biexciton cascade.

Can entangled photons transmit information faster than light?

No. Each photon's individual measurement outcomes are random, and the correlations become visible only when the two records are compared over a classical channel.

What does a CHSH value of 2.83 mean?

It equals $2\sqrt{2}$, the maximum quantum correlation for two qubits. Measured values between 2 and 2.83 violate the classical bound; reaching the maximum requires a pure Bell state and perfect visibility.

References: J. F. Clauser, M. A. Horne, A. Shimony, R. A. Holt, Phys. Rev. Lett. 23, 880 (1969); C. K. Hong, Z. Y. Ou, L. Mandel, Phys. Rev. Lett. 59, 2044 (1987); A. K. Ekert, Phys. Rev. Lett. 67, 661 (1991); P. G. Kwiat et al., Phys. Rev. Lett. 75, 4337 (1995); P. G. Kwiat et al., Phys. Rev. A 60, R773 (1999); D. Bouwmeester et al., Nature 390, 575 (1997); L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).