FR11 · A photon leaves a finite band

Structured Bath and Output Ports

How does the surrounding waveguide change the photon that reaches a port? The waveguide supports only a finite range of frequencies. Moving an emitter toward its band edge changes how the excitation spreads, so the output pulse and the excitation left near the atom change together.

Release one atomic excitation into a finite waveguide. Follow the emitted field to two output ports, then move the atomic frequency toward the band edge and watch the emission slow down.

Atomic population now $.0400$

Both ports so far $.350$

Waveguide population now $.580$

Intrinsic loss so far $.0300$

Model assumptions

The rotating-wave model contains one excitation, a nearest-neighbour chain and a central two-level emitter. Time is measured in inverse reference hopping; frequency uses the reference hopping. Chain sites have unit spacing. Endpoint ports and intrinsic emitter loss are Markov reservoirs. Endpoint mismatch can reflect the field; these finite-chain returns are not a prescribed time-delay feedback equation. The scene shows probability density at schematic scale. Exported complex output amplitudes and integrated losses come from the same evolution; no detector click sequence is sampled.

The pulse that reaches an output port

Cyan: right-port flux. Violet: left-port flux. Amber: right-port intensity after a causal timing response. The response acts on intensity and is separate from a quantum-mode projection.

Output flux and finite timing response

|\xi_R(t)|^2,\ |\xi_L(t)|^2

Excitation in the atom, bath and output

P_e,\ P_{\rm bath},\ P_{\rm out}

Right-port receiving mode

The receiver keeps the normalized complex right-port output of the default resonant bath, over the same recording window. Change the atomic frequency or bandwidth to compare collection with temporal-mode matching.

Right-port probability $P_R$: $0.439$
Conditional mode fidelity $F_R$: $1.00$
Target-mode probability $W_R=P_RF_R$: $0.439$

The overlap uses saved complex amplitudes before the detector timing response. Loss and a mismatch to this fixed mode contribute separately to the target-mode probability.

Read the saved record

Output channels are Markov continua attached to the finite chain. The detector response time is 0.6 in reference time units. It broadens a probability-flux record; no stochastic counting data are generated.

Start with the essentials

Focus question: How does the surrounding waveguide change the photon that reaches a port?
One-sentence intuition: Atomic population and emitted output must be followed together through the finite observation window.

Core mathematical model

A finite propagation band

H/J_{\rm ref}=-j\sum_n(a_{n+1}^\dagger a_n+\mathrm{h.c.})+\Delta\sigma^+\sigma^-+g(a_m^\dagger\sigma^-+\mathrm{h.c.}),\quad\omega(k)=-2j\cos k

The emitter couples at the central site. The band extends over four times the hopping frequency; its group velocity vanishes at either edge.

Save what leaves the system

\xi_L=\sqrt{\kappa_{\rm p}}c_1,\quad\xi_R=\sqrt{\kappa_{\rm p}}c_N,\qquad P_e+\sum_n|c_n|^2+P_L+P_R+P_{\rm loss}=1

The port amplitudes, populations and accumulated losses all follow from the same single-excitation state.

Common difficulties

A delayed peak has a cause

Typical misconceptionEvery revival proves intrinsic delayed feedback.

Better mental modelAn endpoint can reflect the field back through the chain. Compare port coupling and travel time before attributing a return to the emitter’s environment.

Run the experiment

01
Move toward the band edge
Compare Emit into the band and Linger near the edge. Predict whether the atom empties sooner.
What to observe: Near the edge, the output waveform and the remaining population change together.
02
Identify an endpoint return
Choose An imperfect output port and follow the field back along the chain.
What to observe: The endpoint sends part of the field back. This return is tied to the finite chain geometry.
03
Move the atom closer
Drag the glowing atom toward the waveguide.
What to observe: The coupling and emission history update continuously. The vertical spacing is a schematic control for coupling.

Structured Bath and Output Ports

The pulse that reaches an output port

The environment shapes the emitted photon

Core mathematical model

A finite propagation band

Save what leaves the system

Common difficulties

A delayed peak has a cause

Run the experiment

Move toward the band edge

Identify an endpoint return

Move the atom closer

Related experiments and articles

From one excitation to photon pairs and collected modes