Case Study 2: Hong-Ou-Mandel — Proving Photons Are Quantum

The Physical Situation

In 1987, three physicists at the University of Rochester — Chung Ki Hong, Zhe Yu Ou, and Leonard Mandel — performed an experiment so elegant that it has become the gold standard for testing photon indistinguishability. They sent two photons into a 50:50 beam splitter, one in each input port, and measured the rate of coincident detections at the two outputs.

Classical intuition predicts a coincidence rate of 50%: each photon independently goes to either output with equal probability, so the probability of one photon in each output is $2 \times (1/2)(1/2) = 1/2$. But Hong, Ou, and Mandel measured a coincidence rate of nearly zero when the two photons arrived simultaneously. The photons always exited the same port — either both in port 3 or both in port 4. This is the Hong-Ou-Mandel (HOM) effect, and it has no classical explanation.

Setting Up the Experiment

The Photon Source: Spontaneous Parametric Down-Conversion

Hong, Ou, and Mandel used spontaneous parametric down-conversion (SPDC) to generate photon pairs. In SPDC, a "pump" photon (typically UV) enters a nonlinear crystal (e.g., KDP, BBO, or PPKTP) and spontaneously splits into two lower-energy photons called the "signal" and "idler." Energy and momentum conservation require:

$$\omega_p = \omega_s + \omega_i, \qquad \mathbf{k}_p = \mathbf{k}_s + \mathbf{k}_i$$

These phase-matching conditions determine the frequencies and directions of the emitted photon pairs. For type-I SPDC, both photons have the same polarization; for type-II, they have orthogonal polarizations.

The key quantum property is that the two photons are produced simultaneously (within the coherence time of the pump, typically $\sim 1$ ps) and in a correlated quantum state. For the HOM experiment, the signal and idler photons are collected into single-mode fibers and directed to the two input ports of a beam splitter.

The Beam Splitter and Detectors

The beam splitter is a standard 50:50 dielectric cube or plate. At each output, a single-photon detector (originally a photomultiplier tube, now typically an avalanche photodiode or superconducting nanowire detector) records arrival times. A coincidence counter registers events where both detectors fire within a narrow time window (typically $\sim 1$ ns).

The Crucial Variable: Time Delay

One photon's path length can be adjusted by a translation stage, introducing a controllable time delay $\tau$ between the two photons' arrivals at the beam splitter. This is the experiment's key control parameter.

The Analysis

Quantum Calculation

The input state is $|1\rangle_1|1\rangle_2 = \hat{a}_1^\dagger\hat{a}_2^\dagger|0,0\rangle$. The 50:50 beam splitter transforms the creation operators:

$$\hat{a}_1^\dagger \to \frac{1}{\sqrt{2}}(\hat{a}_3^\dagger - i\hat{a}_4^\dagger), \qquad \hat{a}_2^\dagger \to \frac{1}{\sqrt{2}}(-i\hat{a}_3^\dagger + \hat{a}_4^\dagger)$$

The output state is:

$$\hat{a}_1^\dagger\hat{a}_2^\dagger|0,0\rangle \to \frac{1}{2}(\hat{a}_3^\dagger - i\hat{a}_4^\dagger)(-i\hat{a}_3^\dagger + \hat{a}_4^\dagger)|0,0\rangle$$

Expanding:

$$= \frac{1}{2}\left[-i(\hat{a}_3^\dagger)^2 + \hat{a}_3^\dagger\hat{a}_4^\dagger - \hat{a}_4^\dagger\hat{a}_3^\dagger - i(\hat{a}_4^\dagger)^2\right]|0,0\rangle$$

The cross terms cancel because $[\hat{a}_3^\dagger, \hat{a}_4^\dagger] = 0$ (creation operators for different modes commute):

$$\hat{a}_3^\dagger\hat{a}_4^\dagger - \hat{a}_4^\dagger\hat{a}_3^\dagger = 0$$

This leaves:

$$= \frac{-i}{2}\left[(\hat{a}_3^\dagger)^2 + (\hat{a}_4^\dagger)^2\right]|0,0\rangle = \frac{-i}{\sqrt{2}}\left(|2,0\rangle + |0,2\rangle\right)$$

The $|1,1\rangle$ component is identically zero. Both photons exit the same port.

Physical Interpretation: Two-Path Interference

There are four "histories" for two photons at a beam splitter:

The two histories leading to $|1,1\rangle$ have amplitudes $+1/2$ and $-1/2$ — they cancel perfectly. This cancellation requires the photons to be identical (indistinguishable). If they differ in any property — frequency, polarization, arrival time, spatial mode — the histories become distinguishable, the amplitudes do not interfere, and coincidences reappear.

The HOM Dip

When a time delay $\tau$ is introduced between the two photons, the coincidence probability becomes:

$$P_{\text{coinc}}(\tau) = \frac{1}{2}\left[1 - \exp\!\left(-\frac{\tau^2}{2\sigma_\tau^2}\right)\right]$$

where $\sigma_\tau$ is related to the spectral bandwidth of the photons: $\sigma_\tau \sim 1/\Delta\omega$.

• At $\tau = 0$: Perfect temporal overlap, full destructive interference, $P_{\text{coinc}} = 0$.
• At $|\tau| \gg \sigma_\tau$: No temporal overlap, photons distinguishable by arrival time, $P_{\text{coinc}} = 1/2$ (classical value).

The plot of $P_{\text{coinc}}(\tau)$ shows a characteristic dip centered at $\tau = 0$ — the HOM dip. The width of the dip is the coherence time of the photons, and its depth (or visibility) measures the degree of photon indistinguishability:

$$\mathcal{V} = \frac{P_{\text{coinc}}(\infty) - P_{\text{coinc}}(0)}{P_{\text{coinc}}(\infty)}$$

For perfectly indistinguishable photons, $\mathcal{V} = 1$. For completely distinguishable photons, $\mathcal{V} = 0$.

Experimental Results

The original 1987 experiment achieved a visibility of $\mathcal{V} \approx 0.85$, limited by the spectral and spatial mode purity of the SPDC source. Modern experiments with engineered photon sources routinely achieve $\mathcal{V} > 0.99$:

The steady improvement in HOM visibility over three decades reflects advances in photon source engineering — a critical technology driver for photonic quantum computing.

Why It Matters: Boson Statistics Made Visible

The HOM effect is the most direct experimental manifestation of the bosonic nature of photons. The cancellation occurs because creation operators commute: $\hat{a}_3^\dagger\hat{a}_4^\dagger = \hat{a}_4^\dagger\hat{a}_3^\dagger$. For fermions, the anti-commutation relation $\hat{c}_3^\dagger\hat{c}_4^\dagger = -\hat{c}_4^\dagger\hat{c}_3^\dagger$ gives the opposite sign, and the calculation shows that fermions always emerge in different ports (one in each output). This is the Pauli exclusion principle in action — fermionic antibunching.

The HOM experiment thus provides a clean binary test: does a given pair of particles exhibit bunching (bosons), antibunching (fermions), or no correlation (classical)?

Applications of the HOM Effect

1. Quantum Source Characterization

The HOM visibility is the standard metric for photon source quality. Any source intended for quantum information applications — SPDC, quantum dots, color centers, atomic ensembles — must demonstrate high HOM visibility to be useful. A visibility $\mathcal{V} > 0.9$ is typically required; $\mathcal{V} > 0.99$ is needed for fault-tolerant photonic quantum computing.

2. Quantum State Teleportation

The Bell-state measurement at the heart of quantum teleportation (Ch 25) relies on two-photon interference at a beam splitter — essentially an HOM effect. The quality of teleportation fidelity is directly linked to HOM visibility.

3. Linear Optical Quantum Computing

The KLM scheme and its successors (fusion-based quantum computing) use beam splitter interference between photons as the fundamental entangling operation. Every two-qubit gate in a photonic quantum computer depends on HOM-quality interference.

4. Quantum Metrology

The $N00N$ state $\frac{1}{\sqrt{2}}(|N,0\rangle + |0,N\rangle)$ — a generalization of the HOM output for $N = 2$ — enables phase measurements with Heisenberg-limited sensitivity $\Delta\phi \sim 1/N$ instead of the shot-noise limit $\Delta\phi \sim 1/\sqrt{N}$. This has applications in optical lithography and biological imaging.

5. Quantum Network Verification

When two distant nodes of a quantum network attempt to share entanglement, they can verify success by performing a HOM experiment on photons from the two nodes. High visibility confirms that the network is faithfully transmitting quantum states.

Connection to Main Chapter

This case study brings together the chapter's central themes:

1. Photon as QHO excitation — the beam splitter transformation is a linear transformation of oscillator operators, directly following from Section 27.2.

2. Fock states vs. coherent states — the HOM effect requires Fock state inputs ($|1,1\rangle$). Coherent state inputs produce no HOM dip.

3. Indistinguishability — the cancellation requires the two photons to be identical in every degree of freedom. Any distinguishing information destroys the interference.

4. Quantum computing connection — HOM-quality photon interference is the enabling operation for photonic quantum computing (Section 27.9).

Further Questions

1. Fermion anti-bunching. Repeat the HOM calculation for fermions, replacing the bosonic commutation relation $[\hat{a}_3^\dagger, \hat{a}_4^\dagger] = 0$ with the anti-commutation relation $\{\hat{c}_3^\dagger, \hat{c}_4^\dagger\} = 0$. Show that the coincidence probability is 100% — fermions always exit in different ports.

2. Three-photon HOM. What happens when three identical photons enter a 3-port beam splitter (a "tritter")? Compute the output state $|1,1,1\rangle_{\text{in}}$ and identify which output patterns are suppressed.

3. Mixed-state input. Suppose one photon is in a pure state $|1\rangle$ and the other is described by a thermal density matrix $\hat{\rho} = \bar{n}/(1+\bar{n})^2 |1\rangle\langle 1| + \ldots$. How does the HOM visibility depend on $\bar{n}$?

4. Timing precision. If the photon coherence time is $\tau_c = 200$ fs, what mechanical translation precision (in nanometers) is needed to achieve 99% HOM visibility? (Hint: $\Delta x = c\tau/2$ for a folded path.)

5. Multiphoton interference. Boson sampling (Problem 27.28) exploits multiphoton HOM-like interference in an $M$-mode network. Explain why the computational complexity grows exponentially with $N$ while the physical resources grow only polynomially.