Beam splitters and Hong–Ou–Mandel interference
A beam splitter (BS) is the basic interferometric element of quantum optics. Classically it divides power into reflected and transmitted parts, $R+T=1$. Quantum mechanically it is a unitary transformation on the two input modes. For a symmetric 50:50 BS (one common convention):
Unitarity is required for photon-number conservation and for the commutation relations of the output modes.
| Input | Output state | Detection statistics |
|---|---|---|
| $|1,0\rangle$ (one photon in a) | $\tfrac1{\sqrt2}(|1,0\rangle+|0,1\rangle)$ | 50:50, never both detectors: antibunching, $g^{(2)}(0)=0$ |
| $|2,0\rangle$ (two photons in a) | $\tfrac12|2,0\rangle+\tfrac1{\sqrt2}|1,1\rangle+\tfrac12|0,2\rangle$ | Probabilities 1/4, 1/2, 1/4 |
| $|1,1\rangle$ (one photon per port) | $\tfrac1{\sqrt2}(|2,0\rangle-|0,2\rangle)$ | HOM effect: no coincidences, photons “bunch” |
| Coherent $|\alpha\rangle$ in a, vacuum in b | $|\alpha/\sqrt2\rangle\otimes|\alpha/\sqrt2\rangle$ | Poisson at each output; classical split |
The $|1,1\rangle\to$ bunching result (Hong–Ou–Mandel 1987) is pure two-photon interference: it requires the photons to be indistinguishable in frequency, polarization, spatial mode and arrival time. Scanning the relative delay gives a dip in coincidence counts with visibility $V=(C_{\max}-C_{\min})/C_{\max}$. Classical light cannot exceed $V=1/2$, so higher visibility is a quantum signature. HOM visibility is the figure of merit for MDI-QKD relays and entanglement-swapping nodes.
Bell-state measurement (BSM)
Overlap two photons on a BS and detect the outputs with polarization resolution: the Bell states $|\Psi^\pm\rangle$ give distinct signatures while $|\Phi^\pm\rangle$ look the same. With linear optics alone the BSM succeeds with probability at most 50%; ancilla photons or other resources raise it. BSMs power teleportation, entanglement swapping and MDI-QKD.
Beyond optics: frequency transduction
A red-detuned frequency conversion process has the same Hamiltonian structure as a beam splitter, $H\propto g(a^\dagger b+ab^\dagger)$. Thus the BS formalism also models converters that connect telecom photons to memories at other wavelengths. This is essential for networking heterogeneous quantum devices.
Photon sources
- Weak coherent pulses. Attenuated lasers, Poisson statistics with $\mu\approx0.1$–$0.5$. Cheap and fast; need decoy states.
- Spontaneous parametric down-conversion (SPDC). A pump photon splits into a signal–idler pair, conserving energy $\omega_p=\omega_s+\omega_i$. Used for entangled pairs (polarization, time-bin, energy-time) and heralded single photons. Multi-pair emission at high pump power limits purity.
- Solid-state emitters. Quantum dots, colour centres in diamond (NV, SiV), providing near-on-demand single photons and spin–photon entanglement.
- Atoms and ions. Trapped ions and neutral-atom ensembles: photon-matter entanglement for repeaters.
Single-photon detectors
Key parameters: detection efficiency, dark-count rate, dead time, timing jitter, afterpulsing and photon-number resolution.
| Detector | Typical use | Notes |
|---|---|---|
| Si SPAD | Free-space, ~700–900 nm | High efficiency, room-temperature-ish cooling |
| InGaAs/InP SPAD | Fibre, 1310 / 1550 nm | Gated operation, moderate efficiency, afterpulsing; compact and affordable |
| Superconducting nanowire (SNSPD) | Long distance, TF-QKD, MDI | Efficiency above 90%, very low dark counts, ~ps jitter; needs cryogenics (about 1–3 K) |
| Balanced homodyne | CV-QKD | Measures quadratures; no single-photon clicks |
Detectors are a prime target of attacks (blinding, time-shift) and the reason MDI-QKD was introduced.
Channels: fibre and free space
Fibre. Standard fibre loses about 0.2 dB/km at 1550 nm (ultra-low-loss fibre about 0.16). Transmittance $\eta=10^{-\alpha L/10}$: 100 km is 20 dB ($\eta=0.01$), 1000 km is 200 dB. Impairments: Rayleigh and Raman noise when sharing fibre with classical traffic (dozens of dB isolation required), polarization drift (favouring phase/time-bin encoding), chromatic dispersion and temperature-induced phase drift.
Free space. Total transmittance factorises, $\eta=\eta_{\rm geo}\eta_{\rm atm}\eta_{\rm opt}$ (geometric spreading, atmosphere, optics). Challenges: turbulence, pointing and tracking, beam divergence and background light (daylight operation is hard). Satellite downlinks outperform uplinks because turbulence affects the beam only near the receiver. Free-space links are essential for remote regions, ship/air/satellite platforms, and global reach.
Modulation, QRNG and integrated photonics
- Encoding. Polarization (free space), phase and time-bin (fibre), frequency, orbital angular momentum (high-dimensional), quadratures (CV).
- Quantum random number generators. Photon path (single photon on a 50:50 BS), arrival-time, vacuum-fluctuation (homodyne) QRNGs. Raw bits are biased, so Toeplitz hashing extracts uniform bits using the leftover hash lemma. Source-device-independent and device-independent QRNGs certify randomness with weaker trust.
- Integrated photonics. Silicon, silicon-nitride, indium-phosphide and lithium-niobate chips integrate sources, modulators and interferometers, enabling compact, mass-producible QKD transmitters and receivers.
Quantum memories and transduction
Memories store photonic qubits while a link is established elsewhere, essential for repeaters. Platforms: rare-earth-doped crystals, warm and cold atomic ensembles, NV/SiV centres, trapped ions and neutral atoms. Metrics: storage time, efficiency, fidelity, multimode capacity and telecom-compatible interface. Transducers convert between telecom light and memory wavelengths (or microwaves for superconducting qubits).