16.1 Hong–Ou–Mandel coincidence suppression at a beam splitter
16.1.1 Two-photon input modes and output coincidence counting
In a single-photon interferometer (Chapters 6 and 7), one photon enters a 50:50 beam splitter and interferes with itself across two paths. In the 1987 Hong–Ou–Mandel (HOM) experiment, two independent photons enter a lossless 50:50 beam splitter simultaneously—one in input port \(a\) and one in input port \(b\)—while single-photon detectors monitor output ports \(c\) and \(d\) through a coincidence circuit. A relative optical delay stage \(\Delta\tau\) in one input arm scans the arrival-time difference between the two wavepackets without creating any stable optical phase lock between the two input paths.
16.1.2 Temporal, spectral, polarization, and spatial mode overlap
When the path delay \(|\Delta\tau|\) far exceeds the photon coherence time \(\tau_c\), the two photons arrive at distinguishable times and behave as independent particles: each photon is transmitted or reflected with probability \(1/2\), yielding a coincidence probability \(P_{cd}(\infty) = 1/2\). As \(\Delta\tau \to 0\), if the two photons are also matched in polarization, center frequency, bandwidth, and transverse spatial mode, the coincidence rate drops sharply toward zero—forming the Hong–Ou–Mandel dip.
16.1.3 Destructive interference of two-photon coincidence pathways
A coincidence click between output ports \(c\) and \(d\) can occur in two ways: both photons are transmitted (\(T\text{–}T\)), or both photons are reflected (\(R\text{–}R\)). When the two photons are completely indistinguishable in all internal and spatiotemporal degrees of freedom, the final state \(|1_c, 1_d\rangle\) produced by \(T\text{–}T\) is physically identical to the final state produced by \(R\text{–}R\). Because a lossless symmetric beam splitter imparts a relative phase factor of \(i\) per reflection (\(r = i/\sqrt{2}\), \(t = 1/\sqrt{2}\)), the \(T\text{–}T\) amplitude is \(t^2 = +1/2\) while the \(R\text{–}R\) amplitude is \(r^2 = (i/\sqrt{2})^2 = -1/2\). The two coincidence pathways interfere destructively and cancel, forcing both photons to exit bunched together in the same output port.
Visual animation · Section 16.1
Hong–Ou–Mandel coincidence suppression at a beam splitter
Interactive visual model for 16.1 Hong–Ou–Mandel coincidence suppression at a beam splitter (Two-photon input modes and output coincidence counting · Temporal, spectral, polarization, and spatial mode overlap · Destructive interference of two-photon coincidence pathways). Use Animate or Step to follow the physical state evolution and quantitative predictions.
16.2 Bosonic mode algebra and the overlap dip
16.2.1 Unitary transformation of bosonic creation operators
For a lossless beam splitter with power transmittance \(T\) and reflectance \(R = 1 - T\), the input creation operators \(\hat{a}^\dagger, \hat{b}^\dagger\) map to the output creation operators \(\hat{c}^\dagger, \hat{d}^\dagger\) via
When the two photons occupy identical internal modes (\([\hat{c}, \hat{c}^\dagger] = 1\)), the input state \(|1_a, 1_b\rangle = \hat{a}^\dagger\hat{b}^\dagger|0\rangle\) evolves to
At a balanced 50:50 beam splitter (\(T = R = 1/2\)), the coincidence term \((T - R)|1_c, 1_d\rangle\) vanishes identically, leaving the bunched two-photon NOON state \(i(|2_c, 0_d\rangle + |0_c, 2_d\rangle)/\sqrt{2}\).
16.2.2 Coincidence probability as a function of mode inner product
When the two input photons carry normalized internal wavefunctions \(|\psi_a\rangle\) and \(|\psi_b(\Delta\tau)\rangle\) with mode overlap \(J(\Delta\tau) = |\langle\psi_a|\psi_b(\Delta\tau)\rangle|^2 \in [0,1]\), only the indistinguishable projection \(J(\Delta\tau)\) undergoes two-photon interference while the orthogonal fraction \(1 - J(\Delta\tau)\) scatters independently. For general beam-splitter transmittance \(T\) and reflectance \(R = 1 - T\), the coincidence probability is
For a balanced 50:50 splitter (\(T = R = 1/2\)) and Gaussian wavepackets with overlap \(J(\Delta\tau) = V_{\mathrm{mode}}\exp(-\Delta\tau^2/\tau_c^2)\), this simplifies to
16.2.3 Distinguishing two-photon bunching from polarization entanglement
Although the output state \((|2_c, 0_d\rangle + |0_c, 2_d\rangle)/\sqrt{2}\) is a mode-entangled superposition in photon number between spatial arms \(c\) and \(d\), the standard HOM dip does not require the two input photons to be entangled prior to the beam splitter—it works equally well for two completely unentangled single photons emitted by separate quantum dots or trapped ions, provided their wavepackets are indistinguishable (\(J \to 1\)). Conversely, if two photons are prepared in the antisymmetric polarization singlet state \(|\Psi^-\rangle = (|H_a V_b\rangle - |V_a H_b\rangle)/\sqrt{2}\), spatial exchange symmetry flips sign and produces a coincidence peak (\(P_{cd}(0) = 1\)) instead of a dip.
Visual animation · Section 16.2
Bosonic mode algebra and the overlap dip
Interactive visual model for 16.2 Bosonic mode algebra and the overlap dip (Unitary transformation of bosonic creation operators · Coincidence probability as a function of mode inner product · Distinguishing two-photon bunching from polarization entanglement). Use Animate or Step to follow the physical state evolution and quantitative predictions.
16.3 Baseline normalization and entanglement witnesses
16.3.1 Calibrating large-delay baselines and multiphoton contamination
The operational HOM visibility is defined relative to the distinguishable large-delay baseline \(C(\infty)\) as
For perfectly balanced splitting (\(T = R = 1/2\)) and pure single-photon inputs, \(V_{\mathrm{HOM}} = J(0)\) directly measures the mode indistinguishability. In practice, two systematic effects reduce \(V_{\mathrm{HOM}}\) below \(1\): beam-splitter asymmetry (\(T \neq R\)), which caps the dip visibility at \(V_{\max} = 2TR/(T^2 + R^2)\), and multi-pair emission (\(g^{(2)}(0) > 0\)) from spontaneous parametric down-conversion or attenuated lasers, which injects accidental coincidences at \(\Delta\tau = 0\). Classical phase-randomized coherent pulses cannot exceed \(V_{\mathrm{HOM}} = 0.50\), so observing \(V_{\mathrm{HOM}} > 0.50\) without accidental subtraction certifies nonclassical two-photon interference.
16.3.2 Why mode indistinguishability and entanglement require distinct tests
Observing \(V_{\mathrm{HOM}} \approx 1\) proves that two photons share nearly identical spectral-temporal-polarization modes and obey bosonic exchange symmetry, which is the prerequisite for linear-optical two-qubit gates and Bell-state analyzers. However, a single HOM dip does not by itself certify nonlocal Bell entanglement between spatially separated particles; verifying nonlocal correlations requires switched multi-setting analyzers on separated arms (Chapter 17).
16.3.3 Bridging two-photon interference to correlation inequalities
By combining a polarizing beam splitter or a balanced beam splitter with polarization rotators in the output ports, two-photon HOM interference projects an incoming pair onto Bell states. That projection links the mode-overlap physics of this chapter directly to the EPR–Bell correlation tests of Chapter 17 and the quantum teleportation circuit of Chapter 19.
Visual animation · Section 16.3
Baseline normalization and entanglement witnesses
Interactive visual model for 16.3 Baseline normalization and entanglement witnesses (Calibrating large-delay baselines and multiphoton contamination · Why mode indistinguishability and entanglement require distinct tests · Bridging two-photon interference to correlation inequalities). Use Animate or Step to follow the physical state evolution and quantitative predictions.
Explore the Apparatus
Scan the relative arrival-time delay \(\Delta\tau\), the internal mode overlap \(V_{\mathrm{mode}}\) (via relative polarization angle or spectral mismatch), and the beam-splitter transmittance \(T\). Compare the coincidence probability \(P_{cd}(\Delta\tau)\) against the bunched two-photon probabilities \(P_{2c,0d}\) and \(P_{0c,2d}\) at the center of the dip.
Interactive model · Idealized predictions
Overlap two photons
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Read the values as a table
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Worked Check
Problem: Two single photons with mode overlap \(J(0) = 0.92\) interfere at a slightly asymmetric beam splitter with transmittance \(T = 0.55\) and reflectance \(R = 0.45\). Calculate the large-delay coincidence probability \(P_{cd}(\infty)\), the zero-delay coincidence probability \(P_{cd}(0)\), and the resulting HOM dip visibility \(V_{\mathrm{HOM}}\).
Solution:
1. At large delay (\(\Delta\tau \gg \tau_c\), \(J(\infty) = 0\)), the two photons scatter independently:
When two indistinguishable photons arrive simultaneously at opposite ports of a balanced 50:50 beam splitter, the reflection–reflection and transmission–transmission coincidence amplitudes interfere with opposite signs (\(\frac{1}{2} - \frac{1}{2} = 0\)).
Scanning the arrival delay \(\Delta\tau\) traces out the Hong–Ou–Mandel (HOM) coincidence dip, where visibility exceeding \(50\%\) certifies nonclassical two-photon interference.
Polarization-entangled Bell pairs produce phase-sensitive two-photon fringes \(\cos^2(\alpha - \beta)\) across arbitrary analyzer settings while keeping each individual photon’s marginal statistics completely unpolarized.
Practice Problems
Problem 16.1: Polarization rotation between input photons
Two otherwise identical single photons enter a balanced 50:50 beam splitter (\(T = R = 1/2\)) at \(\Delta\tau = 0\) with linear polarizations rotated by angle \(\theta = 30^\circ\) relative to each other. What are the mode overlap \(J(0)\), the coincidence probability \(P_{cd}(0)\), and the dip visibility \(V_{\mathrm{HOM}}\)?
Hint
For linear polarizations separated by angle \(\theta\), the inner product squared is \(J(0) = \cos^2\theta\).
Solution
The polarization overlap is \(J(0) = \cos^2(30^\circ) = 3/4 = 0.75\). At a 50:50 splitter, \(V_{\mathrm{HOM}} = J(0) = 0.75\) and \(P_{cd}(0) = \frac{1}{2}(1 - 0.75) = 0.125\).
Problem 16.2: Inferring coherence time from dip width
A Gaussian-envelope HOM scan \(C(\Delta\tau) = C(\infty)[1 - V\exp(-\Delta\tau^2/\tau_c^2)]\) exhibits a full width at half minimum (FWHM) of \(\Delta\tau_{\mathrm{FWHM}} = 1.665\text{ ps}\). What is the \(1/e\) coherence time \(\tau_c\) and the corresponding spatial wavepacket length \(L_c = c\tau_c\) in free space?
Hint
For \(\exp(-\Delta\tau^2/\tau_c^2) = 1/2\) at \(\Delta\tau = \Delta\tau_{\mathrm{FWHM}}/2\), note that \(\Delta\tau_{\mathrm{FWHM}} = 2\sqrt{\ln 2}\,\tau_c \approx 1.665\,\tau_c\).
Solution
Since \(2\sqrt{\ln 2} \approx 1.665\), the coherence time is \(\tau_c = 1.00\text{ ps}\), corresponding to a free-space wavepacket length \(L_c = (3.00\times 10^8\text{ m/s})(1.00\times 10^{-12}\text{ s}) = 0.300\text{ mm} = 300\,\mu\text{m}\).
Problem 16.3: Maximum HOM visibility for an unbalanced beam splitter
If two input photons are 100% indistinguishable (\(J(0) = 1\)) but the beam splitter has \(T = 0.60\) and \(R = 0.40\), what is the highest possible HOM visibility \(V_{\mathrm{HOM}}\)?
Hint
Evaluate \(V_{\max} = 2TR/(T^2 + R^2)\).
Solution
Substituting \(T = 0.60\) and \(R = 0.40\) gives \(V_{\max} = 2(0.60)(0.40)/(0.36 + 0.16) = 0.48/0.52 = 12/13 \approx 0.923\). Even with ideal single photons, beam-splitter imbalance leaves a residual coincidence fraction \(P_{cd}(0) = (T - R)^2 = 0.04\).
Key Takeaways
When two indistinguishable single photons enter separate ports of a balanced 50:50 beam splitter, the transmitted–transmitted (\(t^2 = +1/2\)) and reflected–reflected (\(r^2 = -1/2\)) coincidence amplitudes cancel destructively.
The output coincidence probability obeys \(P_{cd}(\Delta\tau) = T^2 + R^2 - 2TR\,J(\Delta\tau)\), where \(J(\Delta\tau) = |\langle\psi_a|\psi_b(\Delta\tau)\rangle|^2\) is the joint internal and temporal mode overlap.
Observing \(V_{\mathrm{HOM}} > 0.50\) surpasses the classical coherent-pulse ceiling and quantifies single-photon indistinguishability.
Two-photon bunching is a consequence of bosonic exchange symmetry and mode overlap; certifying nonlocal entanglement requires separate multi-setting correlation tests.