FR11 · Two photons meet one atom

Two-Photon Waveguide Scattering

What changes when a second photon reaches an atom before the first has left? A two-level atom can hold one excitation. Two overlapping photons therefore scatter together differently from two independent photons. Their joint position probability shows the change even when the average density looks similar.

Send two wavepackets past a single atom. Watch their shared probability spread, and compare the scattered pair with two independently scattered photons.

Atom excited $0.506$

Photons on the right $0.399$

Both excitations in the field $0.494$

Pair-probability difference $0.123$

Choose how the pulses meet

One atom, two incoming photons

Pulse-center separation

d/a

0

Atom–waveguide coupling

g/J

0.800

Atom detuning

\Delta/J

0

Drag the gold input handle to separate the pulses. The dashed curve shows their linear reference.

What the picture represents

Cyan height is the mean photon number at a lattice site; the atom glows with its excitation probability. The waveguide is drawn schematically. The gold handle sets input separation and does not follow a photon trajectory.

$a=J=\hbar=1,\quad\omega(k)=-2\cos k$

The 81-site chain has open ends, finite bandwidth and no absorbing layer. Exactly two total excitations are retained. No drive, dephasing or unguided loss is included. Playback spans 22 hopping-time units; reduced motion holds a representative interaction frame.

Focus the scene and use the left/right keys to change separation.

Model assumptions

A finite nearest-neighbor waveguide and a two-level emitter evolve with exactly two total excitations. The reference uses a harmonic emitter with the same single-excitation response. Maps show equal-time field-position probabilities from the simulated quantum state. Open chain ends can reflect light; the boundary diagnostic reports population near them. This lattice has finite bandwidth and is not a calibrated optical device.

Simulated quantum state

Where can the pair be together?

Joint field-position probability

Change caused by saturation

Both axes locate a field photon at the same simulation time. Cyan means more probability than the linear reference; coral means less. The difference map uses the joint map’s probability scale.

Normalization and finite-chain checks

\langle\Psi|\Psi\rangle=1

The maps integrate to the probability that both excitations are in the waveguide. Values are not renormalized after the atom becomes excited. Pair-probability difference is half the summed absolute difference before binning; it is a field-pair diagnostic. Boundary occupation is the fraction of excitations in the outer four sites at each end.

Nysteen et al. (2015), two-photon wavefunction dynamics · Continue the waveguide QED course

Start with the essentials

Focus question: How does atomic saturation change the outgoing pair?
One-sentence intuition: The linear reference inherits every one-photon reflection and delay. A difference in the pair map therefore reveals the additional effect of restricting the emitter to one excitation.

Core mathematical model

Finite-band waveguide Hamiltonian

\begin{aligned}H/J={}&-\sum_j(a_{j+1}^{\dagger}a_j+\mathrm{h.c.})\\&+(\Delta/J)\sigma^+\sigma^-\\&+(g/J)(a_c^{\dagger}\sigma^-+\sigma^+a_c)\end{aligned}

A field quantum hops between adjacent sites. The emitter at the central site exchanges excitations with the field. Hopping energy fixes the time unit; the lattice spacing fixes distance.

Two total excitations

\begin{aligned}|\Psi\rangle={}&\frac{1}{\sqrt2}\sum_{ij}\Phi_{ij}a_i^{\dagger}a_j^{\dagger}|g,0\rangle\\&+\sum_j e_j a_j^{\dagger}|e,0\rangle,\quad\Phi_{ij}=\Phi_{ji}\end{aligned}

The first term has two field photons and a ground-state atom. The second has one field photon and an excited atom. The position map uses the squared magnitude of the symmetric field amplitude, summed into pairs of neighboring sites.

A common single-photon reference

\Phi_{ij}^{\rm lin}=\frac{u_i v_j+v_i u_j}{\sqrt{2(1+|\langle u|v\rangle|^2)}}

Each incoming mode evolves in the same one-excitation Hamiltonian. Their overlap stays in the normalization, including when the pulses coincide. The harmonic reference can also store two excitations in its emitter mode.

Common difficulties

A density is an average

Typical misconceptionTwo bright bumps trace two individual photon paths.

Better mental modelThe waveguide shows an expected photon number at each site. The joint map gives the probabilities for two positions in repeated measurements.

Position and detection time differ

Typical misconceptionThe joint map is already a measured coincidence histogram.

Better mental modelThe map samples the simulated field at one time. A collected signal also depends on the selected output port, propagation and detector response.

Run the experiment

01
Watch the shared encounter
Use Arrive together and watch the atom brighten. Compare the filled density with the dashed linear reference, then inspect the difference map below.
What to observe: A single-excitation response is insufficient to predict the two-photon probability. Cyan marks an increase relative to the linear reference; coral marks a decrease.
02
Separate the pulses
Drag the gold pulse handle to the left, or choose Give the atom time. Replay the full encounter.
What to observe: With more time between arrivals, the atom can release an excitation before the other pulse arrives. Finite pulse tails and the chosen coupling set how close the pair comes to independent scattering.
03
Keep the output state
Pause after the pulses leave the center. Save the run and compare the joint map, norm error and boundary occupation.
What to observe: Every displayed map and density comes from the same state. The saved arrays are equal-time probabilities; a detector timing histogram requires an additional detection model.

Two-Photon Waveguide Scattering

Where can the pair be together?

When two photons share an atom

Core mathematical model

Finite-band waveguide Hamiltonian

Two total excitations

A common single-photon reference

Common difficulties

A density is an average

Position and detection time differ

Run the experiment

Watch the shared encounter

Separate the pulses

Keep the output state

Related experiments and articles

From one excitation to photon pairs and collected modes