Quantum Experiments · I · Observations and models · Chapter 06
What makes a source of light nonclassical?
Send a light beam onto a balanced \(50{:}50\) beam splitter and place single-photon detectors \(A\) and \(B\) at the two output ports, linked to a time-to-digital coincidence counter that records pairs of clicks separated by relative time delay \(\tau\). Rather than measuring only the mean count rate (brightness), this Hanbury Brown–Twiss interferometer measures how the conditional probability of registering a click at detector \(B\) at time \(t+\tau\) depends on having registered a click at detector \(A\) at time \(t\).
An ideal single-photon Fock state \(\lvert n=1\rangle\) cannot be split into two detector clicks at \(\tau=0\): it triggers either detector \(A\) or detector \(B\), never both simultaneously. Attenuating a classical laser beam with neutral-density filters behaves differently: although absorption reduces the mean photon number \(\langle n\rangle\ll 1\), a coherent state retains a Poisson photon-number distribution in which the two-photon probability scales as \(\langle n\rangle^2/2\), so the normalized coincidence ratio at \(\tau=0\) remains unity regardless of attenuation. A dim laser is therefore not a true single-photon source.
The interactive model below compares normalized second-order correlation curves \(g^{(2)}(\tau)\) for idealized antibunched, coherent (Poissonian), and thermal (bunched) light sources. In an actual laboratory setup, finite detector timing jitter, afterpulsing, dark counts, and accidental background coincidences convolve with and offset the intrinsic source correlation function.
Visual animation · Section 6.1
Hanbury Brown–Twiss beam-splitter correlations
For a single optical mode with photon-number operator \(n\) and nonzero mean \(\langle n\rangle>0\), the zero-delay normalized second-order correlation function is defined by the normal-ordered factorial moment
\[g^{(2)}(0) = \frac{\langle n(n-1)\rangle}{\langle n\rangle^2}.\]
For an ideal one-photon state \(\lvert 1\rangle\), \(n(n-1)\lvert 1\rangle=0\), yielding \(g^{(2)}(0)=0\). More generally, an \(n\)-photon Fock state \(\lvert n\rangle\) (\(n\ge 1\)) gives \(g^{(2)}(0)=(n-1)/n\). A coherent state \(\lvert\alpha\rangle\) with Poisson statistics satisfies \(\langle n(n-1)\rangle=\langle n\rangle^2\) and thus \(g^{(2)}(0)=1\), whereas a single-mode thermal (chaotic) state obeys Bose–Einstein statistics with \(\langle n(n-1)\rangle=2\langle n\rangle^2\) and \(g^{(2)}(0)=2\).
As the delay \(\tau\) increases well beyond the source coherence or emitter reset time \(\tau_c\), events at \(t\) and \(t+\tau\) become statistically independent and \(g^{(2)}(\tau)\to 1\). Our interactive explorer illustrates three canonical delay profiles with characteristic scale \(\tau_c=1\): \(g^{(2)}(\tau)=1-e^{-|\tau|/\tau_c}\) for an antibunched single emitter, \(g^{(2)}(\tau)=1\) for a coherent laser, and \(g^{(2)}(\tau)=1+e^{-|\tau|/\tau_c}\) for bunched thermal light.
Why does observing \(g^{(2)}(0)<1\) certify nonclassical light? In semiclassical photodetection theory, where clicks are generated by a nonnegative classical stochastic intensity \(I(t)\ge 0\), the zero-delay correlation is \(g^{(2)}_{\mathrm{cl}}(0)=\langle I^2\rangle/\langle I\rangle^2\). Because the variance of any real random variable is nonnegative, \(\langle(I-\langle I\rangle)^2\rangle=\langle I^2\rangle-\langle I\rangle^2\ge 0\), every classical intensity distribution must satisfy
\[g^{(2)}_{\mathrm{cl}}(0) = \frac{\langle I^2\rangle}{\langle I\rangle^2} \ge 1, \qquad g^{(2)}_{\mathrm{cl}}(0) \ge g^{(2)}_{\mathrm{cl}}(\tau).\]
Observing \(g^{(2)}(0)<1\) (sub-Poissonian statistics, commonly benchmarked below the single-emitter threshold \(g^{(2)}(0)<1/2\)) or \(g^{(2)}(0)<g^{(2)}(\tau)\) (photon antibunching) directly violates the classical intensity inequality.
Visual animation · Section 6.2
Second-order correlation and the antibunching dip
Why split the beam into two detectors \(A\) and \(B\) instead of recording time intervals on a single avalanche photodiode? Every physical photon counter experiences a recovery dead time \(\tau_{\mathrm{dead}}\) (typically tens of nanoseconds) after firing, during which it cannot register a second photon. A single detector therefore shows an artificial zero-delay dip for \(|\tau|<\tau_{\mathrm{dead}}\) even under a classical laser beam! Cross-correlating two independent outputs of a beam splitter eliminates single-channel dead time at \(\tau=0\).
While a measured dip \(g^{(2)}(0)\ll 1\) establishes that multiphoton emissions are strongly suppressed, it does not by itself prove that successive photons are emitted into a pure polarization state or into indistinguishable spectral-temporal wave packets. Testing whether two photons from the source are quantum-mechanically indistinguishable requires the two-photon Hong–Ou–Mandel interference experiment of Chapter 16.
Credible single-photon characterization reports both the raw \(g^{(2)}(0)\) extracted from a specified coincidence time bin \(\Delta\tau\) and any background-subtracted value alongside the measured dark-count rate and timing jitter.
Visual animation · Section 6.3
Dead time, accidental subtraction, and source claims
Switch among the antibunched, coherent, and bunched source models and sweep the delay \(\tau\). Compare \(g^{(2)}(0)\) with the long-delay asymptote \(g^{(2)}(\tau)\to 1\), and verify why attenuating a coherent state leaves \(g^{(2)}(0)=1\) invariant.
Interactive model · Idealized predictions
On a small screen, swipe the diagram to inspect its components. Moving marks illustrate the model; they are not measured trajectories.
A heralded single-photon source emits vacuum \(\lvert 0\rangle\) with probability \(0.80\) and a single photon \(\lvert 1\rangle\) with probability \(0.20\), with zero probability of \(n\ge 2\) photons. Calculate \(\langle n\rangle\) and \(g^{(2)}(0)\).
The mean photon number per pulse is \(\langle n\rangle = 0\times 0.80 + 1\times 0.20 = 0.20\). Because \(n(n-1)=0\) for both \(n=0\) and \(n=1\), the factorial numerator vanishes: \(\langle n(n-1)\rangle=0\), giving \(g^{(2)}(0)=0/(0.20)^2=0\). Notice that adding vacuum loss reduces the delivery efficiency \(\langle n\rangle\) without degrading the ideal single-photon purity \(g^{(2)}(0)=0\).
What is \(g^{(2)}(0)\) for an ideal two-photon Fock state \(\lvert n=2\rangle\)?
Evaluate \(\langle n(n-1)\rangle/\langle n\rangle^2\) at \(n=2\).
For \(\lvert 2\rangle\), \(\langle n\rangle=2\) and \(\langle n(n-1)\rangle=2(2-1)=2\), so \(g^{(2)}(0)=2/2^2=1/2=0.5\). Because \(g^{(2)}(0)=1/2<1\) is nonclassical yet corresponds to two photons, certifying single-photon emission against \(n\ge 2\) Fock states requires \(g^{(2)}(0)<1/2\).
Why is a balanced beam splitter with two independent detectors used to measure \(g^{(2)}(0)\) instead of timing successive clicks on a single detector?
Consider what happens to a single detector immediately after it registers a click at \(\tau=0\).
A single detector is blind during its electronic recovery dead time \(\tau_{\mathrm{dead}}\) after a click, creating a false dip near \(\tau=0\) for any light source. Splitting the beam between two independent detectors allows simultaneous arrivals at \(\tau=0\) to be recorded.
Does measuring \(g^{(2)}(0)=0.01\) guarantee that the emitted photons have pure horizontal polarization \(\lvert H\rangle\)?
Ask whether photon-number counters resolve polarization.
No. A nonpolarizing Hanbury Brown–Twiss setup measures photon-number correlations and is insensitive to whether the polarization state is pure \(\lvert H\rangle\) or completely unpolarized; polarization tomography is required separately.
Average intensity and photon-number statistics answer distinct physical questions: sub-Poissonian correlation \(g^{(2)}(0)<1\) rules out classical intensity mixtures, while \(g^{(2)}(0)<1/2\) bounds multiphoton contamination below the two-photon Fock level.
These readings provide historical context, experimental implementations, or deeper derivations. The interactive display above is an educational model, not a reproduction of their measured data.