The Hong-Ou-Mandel Effect
The Hong-Ou-Mandel (HOM) effect is one of the most striking demonstrations of two-photon quantum interference. When two indistinguishable photons arrive simultaneously at a 50-50 beam splitter, quantum mechanics predicts something remarkable: they will never emerge in opposite output ports. Instead, they always exit together—a phenomenon called photon bunching.
What You're Seeing
The Beam Splitter Schematic
Two photons arrive at a 50-50 beam splitter from different input ports. Classically, you'd expect: 25% chance both exit at D₀, 25% both at D₁, and 50% one at each detector (coincidence). But quantum mechanics changes this dramatically when the photons are indistinguishable.
Wavepacket Visualization
The temporal wavepackets show the probability amplitude for each photon as a function of time. When Δt=0, the wavepackets perfectly overlap. The color gradient represents the amplitude profile—typically Gaussian. The overlap integral (bottom plot) quantifies how much the two wavepackets coincide.
The HOM Dip
The coincidence probability plot shows the signature "dip" at zero delay. When Δt=0 and the photons are indistinguishable, the coincidence rate drops to zero. This isn't because of energy conservation or classical interference—it's pure two-photon quantum interference. The amplitudes for "both transmitted" and "both reflected" interfere destructively with the "one transmitted, one reflected" paths.
The Physics Behind the Dip
Indistinguishability and Bosonic Statistics
Photons are bosons, meaning their joint wavefunction must be symmetric under particle exchange. At a beam splitter, there are two ways to get one photon at each detector: photon 1 transmits while photon 2 reflects, or vice versa. When the photons are indistinguishable, you cannot tell which happened—these amplitudes must be added, not probabilities.
The amplitudes for these two pathways have opposite signs (a π phase shift from the beam splitter), so they destructively interfere. The result: zero coincidence probability. The photons must bunch—they exit together at one detector or the other.
Mathematical Description
For a 50-50 beam splitter with two input modes |a⟩ and |b⟩, the transformation is:
â → (â + b̂)/√2
b̂ → (â - b̂)/√2
For an input state |1ₐ,1ᵦ⟩ (one photon in each mode), the output becomes:
|1ₐ,1ᵦ⟩ → (|2,0⟩ + |0,2⟩)/√2
The |1,1⟩ terms cancel! This is the heart of the HOM effect.
Temporal Overlap
The dip width is determined by the wavepacket duration. For Gaussian pulses of width σ, the coincidence probability as a function of delay is:
P_coinc(Δt) = 1/2 - 1/2 exp(-Δt²/2σ²)
At Δt=0, perfect overlap gives P_coinc=0. As |Δt| increases, the photons become temporally distinguishable, and the dip disappears—approaching the classical 50% coincidence rate.
Things to Try
Perfect HOM Dip: Set Δt=0 and D=0 (indistinguishable photons). The coincidence probability drops to exactly 0%. The photons always exit together—50% to D₀, 50% to D₁. This is quantum bunching in its purest form.
Scan Through the Dip: Slowly drag the delay slider from -4τ to +4τ. Watch the coincidence rate dip at zero delay. The dip width is proportional to the wavepacket duration—shorter pulses give sharper dips. This is how HOM interferometry measures temporal properties.
Distinguishability: Set Δt=0 and gradually increase D from 0 to 1. As the photons become distinguishable (different polarizations, frequencies, or paths), the dip visibility degrades. At D=1, the dip vanishes completely—you recover the classical 50% coincidence rate. Indistinguishability is essential for quantum interference.
Wavepacket Width: Change σ to make the pulses shorter or longer. Shorter pulses (small σ) give narrower HOM dips, requiring better temporal resolution to observe. Longer pulses make the dip broader and easier to measure. This trade-off is fundamental in HOM experiments.
Overlap Integral: Enable "Show overlap integral" to see the quantitative measure of wavepacket overlap. The coincidence probability is directly related to this overlap—maximum overlap (Δt=0) gives zero coincidences. The overlap function is the cross-correlation of the two wavepackets.
Compare with Classical: At large delays (|Δt| > 3σ), the photons don't overlap, and you see the classical result: 25% both at D₀, 25% both at D₁, 50% one at each. But at Δt=0, quantum mechanics says 0% coincidences. This is impossible to explain classically—it requires quantum indistinguishability.
Why This Matters
The Hong-Ou-Mandel effect is a cornerstone of quantum optics and has profound implications:
- Quantum Entanglement Generation: HOM interference is used to create entangled photon pairs in quantum communication.
- Quantum Computing: Two-photon gates in linear optical quantum computers rely on HOM-type interference.
- Quantum Metrology: The sharp HOM dip enables ultra-precise timing and distance measurements.
- Tests of Quantum Mechanics: HOM experiments rule out classical wave theories—particles don't just "clump" statistically.
- Photon Indistinguishability: The dip visibility directly measures how identical two photons are—a crucial quality metric for quantum light sources.
- Boson Behavior: This is a direct manifestation of bosonic statistics—fermions would anti-bunch instead.
First observed by Hong, Ou, and Mandel in 1987, this effect cannot be explained by any classical theory of light. It requires the full apparatus of quantum mechanics: photon indistinguishability, quantum superposition, and bosonic symmetry. Every measurement of the HOM dip reaffirms that photons are quantum particles obeying non-classical statistics.
Historical Context
C.K. Hong, Z.Y. Ou, and Leonard Mandel published their seminal paper in 1987, demonstrating the effect using parametric down-conversion to produce photon pairs. Their experiment showed a dramatic dip in coincidence counts as the delay was scanned through zero—a result with no classical counterpart. The visibility of the dip (how close it approaches zero) became a standard measure of photon quality in quantum optics. Today, HOM interference is ubiquitous in quantum technology, from quantum key distribution to linear optical quantum computing.