Single-Photon Sources and Detectors

Single-photon hardware is the rate-limiting resource in photonic quantum technology. Quantum-dot sources in microcavities routinely deliver $g^{(2)}(0)<0.01$, Hong–Ou–Mandel indistinguishability above $0.95$, and brightness high enough that boson-sampling machines such as the Jiuzhang experiments interfere tens of photons — a regime classical computers struggle to simulate. Multiplexed heralded SPDC sources push in parallel, switching whichever probabilistic pair-generation attempt succeeded into a common output. On the detector side, waveguide-integrated SNSPD arrays now exceed $98\%$ system efficiency with jitter below $3$ ps, co-fabricated with the circuits they read out. The stakes are concrete: satellite and fiber quantum-key-distribution links have their secure distance fixed by detector dark counts and source $g^{(2)}(0)$, and metropolitan networks are stitched together by entanglement-swapping nodes whose heralding rate is indistinguishability times detection efficiency. The open frontier is a source that is at once deterministic, indistinguishable, and telecom-wavelength — the missing piece for a scalable photonic quantum internet.

Picture a perfect single-photon source as a turnstile: every push of the button lets exactly one person through — never zero, never a clump. A faint laser is instead a leaky gate that lets on average one person pass per push, but sometimes none and occasionally two at once. That clumping is the multi-photon error, and the instrument that measures it is the intensity correlator $g^{(2)}(\tau)$: split the beam onto two detectors (a Hanbury Brown–Twiss setup) and histogram the delay $\tau$ between clicks. A single quantum emitter must be re-excited after each emission, so two photons never arrive together — the histogram dips to zero at $\tau=0$. Thermal light does the opposite, bunching into a peak of two; a laser shows no correlation. The first figure is this fingerprint, with the three defining values $g^{(2)}(0)=0,\,1,\,2$.

The other quantity a photonic engineer lives by is indistinguishability: two photons interfere usefully only if they are copies of one another in every degree of freedom. The test is the Hong–Ou–Mandel effect — send one photon into each port of a balanced beam splitter, and if they are truly identical they always leave together, so the coincidence rate collapses. The detector side has its own hard reality, mapped by the second figure. A "click" detector is a smoke alarm: it reports something arrived but not how much. The technologies differ on two axes — how often they fire when a photon is present (efficiency) and how precisely they timestamp it (jitter), with dark counts, operating temperature, and number resolution as the fine print. Superconducting nanowires occupy the sweet spot that makes modern quantum networking possible.

The master quantity is the normally ordered second-order coherence: the joint probability of detecting a photon at time $t$ and another at $t+\tau$, normalized so uncorrelated light gives one. It is normally ordered because a photodetector absorbs photons, so annihilation operators act first.

where:

• $\hat a,\hat a^{\dagger}$ — photon annihilation/creation operators of the detected mode
• $\tau$ — delay between the two photodetection events
• $\langle\cdot\rangle$ — quantum and ensemble expectation in the field state

Reading it: numerator is the coincidence rate, denominator the product of singles rates; $g^{(2)}(0)<1$ is the strictly nonclassical regime no classical field can reach. At zero delay on a Fock state the correlator collapses to one number, because $\hat a|n\rangle=\sqrt n\,|n-1\rangle$ makes the double-lowering factor $n(n-1)$ while the mean is $n$.

where:

• $n$ — photon number of the Fock state $|n\rangle$
• $n(n-1)$ — value of the normally ordered numerator, vanishing for $n=0,1$

Reading it: a single photon ($n=1$) gives exactly $0$ — perfect antibunching — climbing toward the classical $1$ only as $n\to\infty$. Equivalently, using $\hat a^{\dagger}\hat a^{\dagger}\hat a\hat a=\hat n^2-\hat n$, $g^{(2)}(0)=1+(\langle(\Delta\hat n)^2\rangle-\langle\hat n\rangle)/\langle\hat n\rangle^2$, so sub-Poissonian light (variance below the mean) is exactly the nonclassical regime $g^{(2)}(0)<1$ — no positive Glauber–Sudarshan $P$ function. A single photon meeting a balanced beam splitter is delocalized, not divided: applying $\hat b^{\dagger}\to(\hat c^{\dagger}-\hat d^{\dagger})/\sqrt2$ to $|0\rangle_a|1\rangle_b$ gives $(|1\rangle_c-|1\rangle_d)/\sqrt2$, yet a number measurement always finds the whole photon in one output. Send two identical photons into opposite ports and Hong–Ou–Mandel interference cancels the cross terms, $\hat a^{\dagger}\hat b^{\dagger}\to\tfrac12(\hat c^{\dagger 2}-\hat d^{\dagger 2})|0\rangle$, so both photons always leave together and the coincidence rate vanishes — its depth measures indistinguishability.

where:

• $C_{\text{HOM}}$ — coincidence rate for the two photons made identical
• $C_{\text{dist}}$ — coincidence rate when they are made distinguishable (e.g. crossed polarization)
• $|\psi_i\rangle$ — single-photon wavepacket of each photon

Reading it: $\mathcal{V}=1$ for perfectly identical photons and $0$ for fully distinguishable ones — the single number deciding whether photons interfere usefully in a quantum circuit. A cavity around a quantum-dot emitter reshapes its emission through the Purcell effect, speeding spontaneous emission by a factor set by the cavity quality factor and mode volume.

where:

• $Q$ — cavity quality factor; $V$ — mode volume; $\lambda/n$ — emission wavelength in the medium
• $F_P$ — enhancement of spontaneous emission, $\Gamma_{\text{cav}}=F_P\,\Gamma_0$; $\beta$ — fraction funneled into the useful mode

Reading it: a large $F_P$ shortens the lifetime so the photon escapes before dephasing scrambles its wavepacket (raising $\mathcal{V}$) and funnels emission into one mode (raising brightness through $\beta$). Most detectors cannot count: they fire on "one or more" photons and are described by a two-element POVM, not a Fock projector.

where:

• $\eta$ — single-photon detection (quantum) efficiency
• $n$ — number of incident photons; $\Pi_{\text{click}},\Pi_{\text{no}}$ — the "click" and "no-click" POVM elements summing to $\mathbb{1}$

Reading it: for $0<\eta<1$ the no-click operator has eigenvalues strictly between $0$ and $1$, so it is not a projector — only the POVM formalism captures a real detector. Number resolution needs a transition-edge sensor or many multiplexed click detectors. Finally, a detector's noise is set by its dark-count contribution and its noise-equivalent power, the optical power giving unit signal-to-noise in a 1 Hz band.

where:

• $R_{\text{dark}}$ — false-click rate; $R_{\text{signal}}$ — true detection rate
• $h\nu$ — photon energy; NEP — noise-equivalent power in W Hz$^{-1/2}$

Reading it: low dark counts and high efficiency both drive the NEP down, which is why SNSPDs, with $R_{\text{dark}}$ of a few counts per second, extend quantum links to hundreds of kilometres.

The second-order coherence at zero delay classifies light across the whole span from deeply quantum to classical — the map every measurement is read against.

1 · Why an attenuated laser is not a single-photon source (fully worked)

A coherent pulse is attenuated to a mean photon number $\bar n=0.1$. Find the probabilities of zero, one, and two-or-more photons, the multi-photon fraction conditioned on a detection, and confirm $g^{(2)}(0)$.

Step 1. A coherent state is Poissonian, $P(k)=e^{-\bar n}\bar n^{k}/k!$. With $\bar n=0.1$ and $e^{-0.1}=0.904837$: $P(0)=0.904837$, $P(1)=\mathbf{0.090484}$, $P(2)=0.004524$. At least one photon: $P(\ge 1)=1-P(0)=\mathbf{0.095163}$.

Step 2. The probability of multi-photon emission is $P(\ge 2)=1-P(0)-P(1)=1-0.904837-0.090484=\mathbf{0.004679}$. Conditioned on a detection, the multi-photon fraction is $P(\ge 2)/P(\ge 1)=0.004679/0.095163=\mathbf{0.0492}$, i.e. about $4.9\%$ — matching the rule of thumb $\bar n/2=0.05$.

Step 3. The leak does not vanish with attenuation: halving $\bar n$ halves the rate but the conditional multi-photon fraction only falls to $\bar n/2$, still nonzero, and $g^{(2)}(0)=1$ for any coherent state regardless of $\bar n$. A weak laser has $g^{(2)}(0)=1$ and a ~5% multi-photon leak at $\bar n=0.1$ — not a single-photon source, which is exactly why decoy-state protocols exist.

2 · Purcell factor, brightness, and lifetime of a quantum dot (fully worked)

An InAs quantum dot emitting at $\lambda=925$ nm sits in a GaAs micropillar ($n=3.5$) with quality factor $Q=1.1\times 10^{4}$ and mode volume $V=1.2\ \mu\text{m}^3$. Find the Purcell factor, the collection fraction $\beta$, and the lifetime shortening.

Step 1. The emission wavelength in the medium is $\lambda/n=925/3.5=264.3$ nm $=0.2643\ \mu$m, so $(\lambda/n)^3=0.01846\ \mu\text{m}^3$. The prefactor is $3/(4\pi^{2})=0.07599$.

Step 2. Assemble the Purcell factor: $F_P=0.07599\times 0.01846\times(1.1\times 10^{4}/1.2)=\mathbf{12.9}$. The radiative rate is enhanced almost thirteenfold, so the lifetime shortens by the same factor — a $1$ ns free-space lifetime becomes about $\mathbf{78\ ps}$ in the cavity.

Step 3. The fraction funneled into the cavity mode is $\beta=F_P/(F_P+1)=12.9/13.9=\mathbf{0.93}$ — the ceiling on brightness from mode matching alone — while the faster emission outruns phonon dephasing, pushing indistinguishability toward $\mathcal{V}>0.95$. $F_P\approx 13$ gives $\beta\approx 0.93$ and a ~13× lifetime cut — why cavity quantum dots are the brightest deterministic sources.

3 · Detector efficiency and noise from measured rates

A heralded SPDC source gives idler singles $R_s=8.0\times 10^{5}$ s$^{-1}$ and signal–idler coincidences $R_{\text{coinc}}=2.4\times 10^{5}$ s$^{-1}$. Find the signal-arm Klyshko efficiency. Then, for an SNSPD at $\lambda=1550$ nm with $\eta=0.90$ and $R_{\text{dark}}=10$ Hz, find the noise-equivalent power.

Step 1. The Klyshko efficiency of the signal arm is the coincidence-to-idler-singles ratio: $\eta_s=R_{\text{coinc}}/R_s=2.4\times 10^{5}/8.0\times 10^{5}=\mathbf{0.30}$. Thirty percent of heralded photons survive collection, coupling, and detection — a realistic figure that caps the usable rate of any circuit fed by this source.

Step 2. The telecom photon energy is $h\nu=hc/\lambda=(6.626\times 10^{-34})(2.998\times 10^{8})/(1.55\times 10^{-6})=1.28\times 10^{-19}$ J $\approx 0.80$ eV. The noise-equivalent power is $\text{NEP}=(h\nu/\eta)\sqrt{2R_{\text{dark}}}=(1.28\times 10^{-19}/0.90)\sqrt{20}=1.42\times 10^{-19}\times 4.47=\mathbf{6.4\times 10^{-19}\ W\,Hz^{-1/2}}$.

Step 3. This sub-attowatt NEP is why SNSPDs read out signals attenuated across hundreds of kilometres of fiber; an InGaAs APD with $\eta\approx 0.25$ and far higher dark counts would fare orders of magnitude worse. Klyshko $\eta_s=0.30$; SNSPD NEP $\approx 6.4\times 10^{-19}$ W Hz$^{-1/2}$.

Single-photon physics is where the quantum theory of light became a set of measurable numbers. The person who supplied the language — coherent states, the hierarchy of correlation functions, the line between classical and nonclassical light — is Roy Glauber, and this article is written in his notation.

Texts. Nielsen & Chuang, Quantum Computation and Quantum Information, §7.4 (optical implementations) · Preskill, Lecture Notes on Quantum Computation, Ch. 9 (quantum optics and measurement) · Loudon, The Quantum Theory of Light, 3rd ed., Ch. 3, 6 (photon statistics, $g^{(2)}$) · Walls & Milburn, Quantum Optics, Ch. 3, 5, 10 (coherence functions, antibunching, resonance fluorescence) · Wilde, Quantum Information Theory, Ch. 4 (POVMs) · Watrous, The Theory of Quantum Information, Ch. 2 (measurements as channels). The key results in this article are: $g^{(2)}(0)=1-1/n$ for Fock states (from the ladder algebra); $g^{(2)}=2$ for thermal light and $1$ for coherent light; the click POVM $\Pi_{\text{no}}=\sum_n(1-\eta)^n|n\rangle\langle n|$ and its failure of idempotence; and the Klyshko ratio $R_{\text{coinc}}/R_s=\eta_i$. The worked numbers are: $\bar n=0.1$ giving $P(\ge2)/P(\ge1)=0.049$; Purcell $F_P\approx 12.9$, $\beta\approx 0.93$; and NEP $\approx 6.4\times 10^{-19}$ W Hz$^{-1/2}$. The Purcell formula is the standard cavity-QED expression. Device figures (SNSPD $\eta>90\%$, picosecond jitter, sub-$10^2$ Hz dark counts; TES number resolution) are current standard ranges, not fixed constants, and should be checked against up-to-date reviews.

Prerequisites. Photonic Qubits · DiVincenzo Criteria. Leads to. Linear-Optical Quantum Computing · Circuit Quantum Electrodynamics.

Ask the tutor: Ask me why $g^{(2)}(0)<1$ is impossible for any classical light field, or to work a full quantum-key-distribution error budget from a source's $g^{(2)}(0)$ and a detector's dark-count rate.