The Hong–Ou–Mandel (HOM) effect, named after Chung Ki Hong, Zhe Yu Ou and Leonard Mandel, is a two-photon quantum interference phenomenon occurring when two indistinguishable photons are sent simultaneously on the two input ports of a 50:50 beamsplitter (BS). The photon distribution at the output shows that the two photons always emerge together from the same output port of the beam splitter. This behavior, referred to as photon bunching, contradicts the classical prediction, where two independent particles would have a 50% probability of exiting through different ports. The HOM effect suppresses this splitting outcome, leaving only the bunching behavior.
Mathematical description:
Formally, when two indistinguishable photons enter each input port of a balanced beam splitter, the input state is given by
$$ |\psi_{\text{in}}\rangle = \hat{a}_1^{\dagger}\hat{a}_2^{\dagger}|0\rangle, $$
where \(\hat{a}\) is the annihilation operator that satisfies the commutation relation \([\hat{a},\hat{a}^\dagger] = 1.\)
After the beam splitter transformation,
$$ \begin{pmatrix} \hat{b}_1 \\ \hat{b}_2 \end{pmatrix} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} \begin{pmatrix} \hat{a}_1 \\ \hat{a}_2 \end{pmatrix}, $$
the output state becomes $$|\psi_{\text{out}}\rangle = \frac{1}{2}\left(\hat{b}_1^{\dagger 2} – \hat{b}_2^{\dagger 2}\right)|0\rangle= \frac{1}{\sqrt{2}}\left(|2,0\rangle – |0,2\rangle\right), $$
which shows that both photons exit together in the same output mode, with no coincidence terms of the form \( \hat{b}_1^{\dagger}\hat{b}_2^{\dagger} .\) In the general case where the photons are not perfectly identical, the coincidence probability depends on the overlap of their temporal, spectral, and polarization wavefunctions \(\langle \psi_1 | \psi_2 \rangle\). It is given by
$$ P_c = \frac{1}{2}\left(1 – \left|\langle \psi_1 | \psi_2 \rangle \right|^2\right). $$
For perfectly indistinguishable photons \( |\langle \psi_1 | \psi_2 \rangle| = 1\), the coincidence probability \(P_c \) drops to zero, as expected from complete destructive interference. Conversely, for fully distinguishable photons \(|\langle \psi_1 | \psi_2 \rangle| = 0\), the coincidence probability reduces to the classical value \(P_c = \tfrac{1}{2}\). In this regime, interference is absent and the beamsplitter simply produces the classical \(50/50\) distribution of output events.
Experimental measurement:
The experimental protocol of the HOM effect relies on the observation of the states \(|1,1\rangle\) via coincidence measurements, which indicate whether one photon appears in each output port. The coincidence rate is measured by introducing a variable temporal delay between the photons using a delay line. By scanning this delay, one records how the coincidence rate changes as the photons arrive at the beamsplitter at different times. When the photons reach the beamsplitter simultaneously, the coincidence rate drops to a minimum known as the HOM dip, revealing the point of strongest two-photon interference.
Frequently asked questions:
- What is the HOM effect used for? The HOM effect is employed to quantify photon indistinguishability, generate photonic entanglement, investigate fundamental bosonic statistics, among other applications.
- Can the effect be generalized to more photons? Yes, multi-photon generalizations exist, leading to complex interference patterns, e.g., in boson sampling experiments.
- How is partial distinguishability measured experimentally? Partial distinguishability is measured by scanning the delay between photons and measuring the coincidence rate, one can extract the overlap wavefunctions. The minimum coincidence rate at zero delay (HOM dip) provides a direct measure of photon indistinguishability.