Hong–Ou–Mandel Interference

Two photons approach a 50:50 beamsplitter, one from each side. Common sense — and classical optics — offers four outcomes: both transmit, both reflect, or one of each, in either arrangement. Coin-flipping through them, you'd expect the photons to exit from opposite sides half the time. The experiment says otherwise: if the photons are truly indistinguishable, they never exit opposite sides. Both always leave together, through one port or the other, as if they had agreed in advance. No force acts between them; the beamsplitter is the same passive glass as ever. This is the Hong–Ou–Mandel (HOM) effect — the simplest experiment in physics with no classical explanation, the first genuinely two-photon phenomenon of this module, and the raw ingredient from which every photonic entangling gate is built. It is also, as we'll see, a precision ruler: the deepest measure we have of whether two photons are truly identical.

The classical baseline: distinguishable photons

First, the boring case, to fix the stakes. Send in two photons that differ in any degree of freedom a detector could exploit — arrival time, frequency, polarisation. Each independently transmits or reflects with probability \tfrac12. The four outcomes are equally likely, and two of them (both-transmit, both-reflect) put the photons in opposite output ports. The coincidence probability — both detectors clicking in the same time window — is

P_{\text{coinc}}^{\text{dist}} \;=\; \tfrac14 + \tfrac14 \;=\; \tfrac12 .

Any theory that treats the photons as independent particles, classical or quantum, lands on ½. Keep that number in view; the quantum calculation is about to erase it.

The two-photon amplitude: a cancellation

Now let the photons be indistinguishable, and work in bra-ket notation with creation operators. Input modes a, b; output modes c, d. The 50:50 beamsplitter (with the coupler's i convention from Module 3) transforms

\hat a^\dagger \to \tfrac{1}{\sqrt2}\bigl(\hat c^\dagger + i\,\hat d^\dagger\bigr), \qquad \hat b^\dagger \to \tfrac{1}{\sqrt2}\bigl(i\,\hat c^\dagger + \hat d^\dagger\bigr).

The input state is one photon in each mode, |1,1\rangle = \hat a^\dagger \hat b^\dagger |\mathrm{vac}\rangle. Substitute and expand:

\hat a^\dagger \hat b^\dagger \;\to\; \tfrac12\bigl(\hat c^\dagger + i \hat d^\dagger\bigr)\bigl(i \hat c^\dagger + \hat d^\dagger\bigr) \;=\; \tfrac12\Bigl(i\,\hat c^{\dagger 2} + \hat c^\dagger \hat d^\dagger + i^2\,\hat d^\dagger \hat c^\dagger + i\,\hat d^{\dagger 2}\Bigr).

Because the photons are indistinguishable bosons, \hat c^\dagger and \hat d^\dagger commute — the two middle terms describe the same outcome (one photon per port) and must be added as amplitudes: \hat c^\dagger \hat d^\dagger (1 + i^2) = \hat c^\dagger \hat d^\dagger (1 - 1) = 0. The coincidence amplitude cancels exactly. What survives is

|1,1\rangle \;\longrightarrow\; \frac{i}{\sqrt2}\,\bigl(|2,0\rangle + |0,2\rangle\bigr):

a superposition of both photons in port c and both photons in port d — photon bunching. The two histories that lead to a coincidence — "both transmitted" (amplitude \tfrac12) and "both reflected" (amplitude \tfrac12 \cdot i \cdot i = -\tfrac12) — are indistinguishable even in principle, so their amplitudes add before squaring, and they add to zero. Classical probability added \tfrac14 + \tfrac14; quantum mechanics added \tfrac12 - \tfrac12.

two indistinguishable photons entering a 50:50 beamsplitter from opposite ports exit together: |1,1\rangle \to \tfrac{i}{\sqrt2}(|2,0\rangle + |0,2\rangle);
the coincidence probability drops from the classical ½ to 0 — the two coincidence histories carry amplitudes +\tfrac12 and -\tfrac12 and cancel;
the cancellation requires indistinguishability in every degree of freedom: frequency, polarisation, spatial mode and arrival time;
with partial distinguishability the coincidence rate is P_{\text{coinc}} = \tfrac12(1 - V), where 0 \le V \le 1 is the squared overlap of the two photons' wavepackets — the HOM visibility.

Notice what this is not: it is not the single-photon interference of the last lesson. No phase shifter appears anywhere, and each photon alone would exit 50:50. The interference is between two-photon histories — an effect with no single-photon, and no classical-wave, counterpart. It is also our first meeting with exchange statistics doing computational work: run the same experiment with fermions and the symmetric bunched terms are the forbidden ones — two identical fermions always exit opposite ports, a perfect anti-HOM, with the Pauli principle playing enforcer.

The HOM dip: indistinguishability made visible

The experimental signature is a curve every photonics talk shows within the first five slides. Put a variable delay \tau in one input arm and record the coincidence rate. With a large delay the photons arrive at different times — perfectly distinguishable — and the rate sits at the classical plateau \tfrac12. As \tau \to 0 the wavepackets overlap, the two histories become indistinguishable, and the rate plunges into the HOM dip. For Gaussian wavepackets of coherence time \tau_c,

P_{\text{coinc}}(\tau) \;=\; \tfrac12\Bigl(1 - V\,e^{-\tau^2/\tau_c^2}\Bigr),

with the dip bottoming out at \tfrac12(1 - V). Play with both sliders: V is the photons' intrinsic indistinguishability (the quality of your sources), while \tau_c sets the dip's width — which is why the dip doubles as a femtosecond ruler.

The dip is the field's universal quality meter. Building a photonic computer means interfering photons from different sources, so "how identical are your photons?" is the make-or-break question — and V is its answer, read straight off the dip depth. State-of-the-art quantum dots now post two-photon visibilities above 0.95; every percent below 1 becomes gate error in the KLM and fusion schemes ahead.

Worked example: reading a dip

A lab measures a coincidence rate of 500 counts/s at large delay and 75 counts/s at zero delay. The plateau corresponds to P = \tfrac12, so normalise: the dip bottom is \tfrac{75}{500}\times\tfrac12 = 0.075. Solve \tfrac12(1-V) = 0.075 to get V = 0.85: the two photons' wavepackets have 85% squared overlap. Two further readings come free. First, the timing: if the dip's half-width is \tau_c \approx 300 fs, the delay stage located zero delay to a few tens of femtoseconds — using detectors whose own timing jitter is a hundred thousand times worse. Second, the diagnosis: a visibility stuck below 1 with single photons usually means spectral mismatch or residual timing jitter between sources — the dip doesn't just score the photons, its shape versus delay is a spectrometer for what differs about them.

Chung Ki Hong, Zhe Yu Ou and Leonard Mandel published the effect in 1987 with an apparatus that, read cold, sounds impossible: they resolved time intervals of a few femtoseconds using photodetectors and coincidence electronics with nanosecond-scale resolution — six orders of magnitude too slow. The trick is that the dip's width is set by the photon wavepackets' overlap, not by detector speed: sweeping a mirror on a micrometer stage sweeps the delay through the dip, and the slow detectors merely count coincidences at each setting. Their original dip, about 100 fs wide (a mirror travel of some 30 μm), measured the length of a single photon's wavepacket directly. The paper's title — "Measurement of subpicosecond time intervals between two photons by interference" — advertises the ruler, not the revolution; the revolution was noticing that two photons can conspire at a beamsplitter at all.

The photons do not pull on each other. There is no force, no collision, no exchange of energy — linear optics guarantees each photon's operator evolves independently, exactly as the last two lessons insisted. What cancels is a coincidence amplitude: the two indistinguishable histories leading to opposite-port exits interfere destructively, so that outcome simply never occurs, and the surviving probability piles into the bunched outcomes. Three corollaries are worth engraving. First, make the photons distinguishable in any way — tag one's polarisation, delay its arrival — and the "attraction" vanishes instantly, which no real force would do. Second, nothing here violates the no-interaction rule of lesson one: HOM entangles the output modes, but it is not by itself a two-qubit gate — turning this interference into logic needs the ancillas and detectors of the next lesson. Third, classical light can fake a shallow dip: two phase-randomised classical pulses show coincidence interference with visibility at most \tfrac12. Only a dip deeper than half — coincidence below ¼ — is a certificate of quantum interference, which is why experimentalists obsess over that threshold.

Where this goes next

HOM interference is the one card linear optics deals us for multi-photon physics: amplitudes for multi-photon histories interfere even though the photons never interact. The next lesson plays that card for stakes — the Knill–Laflamme–Milburn protocol routes ancilla photons through HOM-style interference and then measures them, conjuring an effective photon–photon nonlinearity out of detection itself, and with it the entangling gate this platform was missing. Later, boson sampling will scale the same two-photon cancellation up to n photons, where the interfering amplitudes number n! and classical computers stop being able to keep score.