What two detector clicks reveal about photon overlap

Watch both exits

A balanced beam splitter sends one photon to either output with equal probability. Begin with two distinguishable photons, perhaps arriving far apart in time. Each chooses an output. Two of the four possibilities put them in separate outputs, so both detectors click in half the ideal trials. A pair of clicks assigned to the same trial is a coincidence.

Now align the photons in time, polarization and spatial mode. There are two alternatives for separate exits: both transmit or both reflect. A balanced lossless beam splitter gives those alternatives opposite complex amplitudes. They cancel for identical pure photons. The photons leave together, and coincidences disappear. This is Hong–Ou–Mandel interference, or HOM.

P_{\mathrm{coin}}=\frac12\quad\text{(distinguishable)},\qquad P_{\mathrm{coin}}=0\quad\text{(identical pure inputs)}

These probabilities are conditioned on exactly one photon entering each input port. Zero coincidence is an ideal limit.

Reading: C. K. Hong · P. J. Mosley et al.

A wave packet has phase as well as intensity

The frequencies in a pulse resemble notes in a chord. Their intensities say how much of each note is present. Their phases say how the oscillations line up. Equal spectral intensities can therefore assemble into different pulse shapes in time. A complete wave packet with fixed phase relationships is a mode.

Describe a normalized mode by a complex function $u(\omega)$ . A complex number is an arrow: its length gives amplitude and its angle gives phase. Multiply one mode by the complex conjugate of the other and sum over frequency to obtain an overlap amplitude. Its squared magnitude gives the overlap below. Angular frequency $\omega$ is measured in radians per second.

\int |u(\omega)|^2\,d\omega=1,\qquad M=\left|\int u^*(\omega)v(\omega)\,d\omega\right|^2

Identical modes have unit overlap. Orthogonal modes have zero overlap because their complex contributions cancel. Their intensity profiles may still overlap.

One photon can occupy an uncertain mode

Drawing one ball per trial does not guarantee the same color on every draw. Likewise, a source can emit exactly one photon each time and still vary its wave-packet mode. A mixture describes probabilities for these alternatives. A pure state uses one reproducible complex amplitude. Purity is about the state, independent of brightness.

Matching two inputs is another question. Each photon may be pure, yet one arrives too early to overlap the other. Later chapters describe each input with a density matrix and calculate their overlap. For now, separate the experimental actions: prepare each photon, then align the two.

Detector loss, dark counts and extra photon pairs also affect measured clicks. We begin with exactly one photon in each input so that each added physical effect has a clear role.

Exercises and explanations

Does a click at only one detector prove that both photons left through the same port?

Read the explanation

That inference also requires perfect detection, no background and exactly one photon at each input. Otherwise a photon may have been missed. Experiments compare coincidence rates across delays with an appropriate detection model.

Do identical spectral intensities guarantee a vanishing HOM coincidence?

Read the explanation

The phase relationships and the other degrees of freedom must also match. Equal intensities can belong to different complex amplitudes. The next chapter stores amplitudes in a matrix; chapter three changes one phase to construct an explicit counterexample.

What two detector clicks reveal about photon overlap

Watch both exits

A wave packet has phase as well as intensity

One photon can occupy an uncertain mode

Try the prediction

Bring two separated wave packets together

Exercises and explanations

Sources and calculation scope