--- title: "When an environment sends light back" description: "Separate finite bandwidth, giant-atom propagation and boundary reflections, then explain a localized photon cloud near a band edge." canonical_url: "https://sandboxphysics.com/stories/waveguide-qed/bandwidth-and-return/" language: "en" page_type: "story" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/stories/waveguide-qed/bandwidth-and-return/" last_modified: "2026-10-28" --- [Cavity and waveguide quantum electrodynamics](https://sandboxphysics.com/stories/waveguide-qed/) L2 · Chapter 3 # When an environment sends light back Sandbox Physics · 2026-10-28 An atom population can rise because emitted light has returned, or because the atom still exchanges excitation with a nearby localized field. The spatial path and travel time distinguish these cases. Finite bandwidth, separated coupling points and a distant chain boundary leave different signatures. ## Before you begin Chapter 2, memory kernels and exponential decay. You should know sine, cosine and derivatives as slopes. Dispersion, group velocity and band edges are introduced here. A bound state means a joint eigenstate that remains localized. [Review the preceding chapter](https://sandboxphysics.com/stories/waveguide-qed/input-output-markov/) - Estimate propagation times from a resonator-chain dispersion. - Use spatial profiles, chain length and coupling separation to distinguish return mechanisms. ## A resonator chain transmits a finite frequency band Represent a waveguide by resonators spaced by $a$, with nearest-neighbour hopping $J>0$. Set their common frequency to zero. A plane wave accumulates phase $ka$ between neighbouring sites. Substitution into the discrete propagation equation gives a finite band. No extended chain mode has a frequency outside $[-2J,2J]$. The slope of frequency against wave number is group velocity, the velocity of a narrow wavepacket centre. It is largest at band centre and vanishes at the edges. A broad pulse contains different velocities and changes shape. $$ \omega(k)=-2J\cos(ka),\qquad v_g(k)=2Ja\sin(ka),\qquad |v_g|\le 2Ja $$ Frequency has inverse-time units and wave number has inverse-length units. Browser chain models often choose hopping and spacing as their units. Reading: Sandbox Physics ## Why light can remain beside an atom in a gap Outside the propagating band, an atom–chain eigenstate can carry a photon tail that decays away from the atom. That tail cannot carry all its probability to infinity as a propagating wave. This localized mixture of atom and light is a bound state. Write $E$ for physical energy divided by the reduced Planck constant, with the same inverse-time units as $J$. Outside the band, let the distant amplitude fall as $r^{|j|}$. Substituting it into the source-free chain equation gives the relation below. Localization requires $|r|<1$. The atomic coupling selects energies that also satisfy the central matching condition. A bare excited atom is usually a superposition of extended and bound eigenstates. The extended contribution leaves; the bound contribution stays nearby. Atomic population alone measures only one component. A finite spatial window can also contain travelling light that has not yet left. $$ E=-J(r+r^{-1}),\qquad \ell=\frac{a}{-\log|r|} $$ A long localization length makes chain size and observation time more consequential. Compare a bright local cloud with an eigenstate calculation or long-time behaviour. Reading: Sandbox Physics ## Two coupling points turn a distance into a delay A giant atom connects the same quantum system to two separated points. Light emitted at one point can reach the other and drive the same atom again. This path survives when the ends of the waveguide are moved arbitrarily far away. Near band centre, take separation $d=32a$ and units $J=a=1$. Group velocity is two, giving a first travel-time estimate of $16$. This estimate describes narrow band-centre components. Finite bandwidth spreads the return in time, so every small peak cannot be assigned the same exact delay. Changing the coupling-point separation changes this return time and its phase. Moving only a distant endpoint should not change the first traversal between coupling points. That gives a useful numerical diagnostic. $$ \tau_{\rm between}\simeq\frac{d}{|v_g(k_0)|} $$ This is a one-way journey between coupling points. A trip to a boundary and back has a different length. Two return paths have different lengths - Between two connections - To the endpoint and back Original geometry schematic; distances are not to scale. Move a coupling point to change the cyan path. Move the endpoint to change the amber round trip. Reading: Sandbox Physics ## How a boundary reflection can mimic memory An atom a distance $L$ from a hard boundary receives reflected light after roughly $2L/|v_g|$. Lengthening the chain moves that echo later. The existing single-excitation model has 401 sites, about 200 spacings from centre to endpoint. With $J=a=1$, the fastest round-trip estimate is about 200 time units. Its default run ends at time 64, before the main wavefront completes that trip. A lattice has small propagation tails, so endpoint population is still checked. A clean absorbing boundary requires a reflection check alongside its norm loss. Reading: Sandbox Physics ## Diagnose a returning pulse Worked example: an atom population rises again near time sixteen. Lengthening the chain by half leaves this time nearly unchanged; halving the separation of the coupling points advances it. That supports propagation between the coupling points. A stationary cloud outside the band instead calls for a bound-state projection analysis. Bandwidth determines environmental correlation times. A second coupling point or mirror supplies a particular return path. A finite endpoint may add an unintended echo. A model can contain all three, so identify their contributions before asking which returning field can be collected. ## Try the prediction ### [Change a path and find the first return](https://sandboxphysics.com/labs/giant-atom-waveguide-echo/) 1. Follow the wave between the default coupling points and predict what shortening their separation will do. 2. Compare “Hear the echo” with “Shorten the delay”, estimating the time from the first traversal. 3. Choose “One connection only”. Explain which path is removed and why finite bandwidth remains. ### [Separate a localized cloud from departing light](https://sandboxphysics.com/labs/bandgap-bound-photon-cloud/) 1. At the default gap frequency, identify the outward wave and the cloud remaining nearby. 2. Move into the band and then toward its edge. Compare cloud size and propagation speed. 3. Use the infinite-chain bound-state reference to discuss the remaining atom population and explain why nearby light at finite time is not the total bound-state weight. ## Exercises and explanations 1. Near band centre with $J=a=1$, separation increases from 24 to 48. What happens to travel time? Read the explanation Group velocity is approximately two, so time increases from twelve to twenty-four. A pulse near a band edge has a broader velocity distribution and needs a more careful estimate. 2. A late peak moves when chain length changes but barely responds to coupling-point separation. What would you check first? Read the explanation Check endpoint reflection, endpoint population and the shortest round-trip time. Repeat with a longer chain or independently validated absorbing boundary. [Previous: How to keep track of light after it leaves](https://sandboxphysics.com/stories/waveguide-qed/input-output-markov/) [Next: Why two photons cannot always be scattered separately](https://sandboxphysics.com/stories/waveguide-qed/two-photon-saturation/) ## Sources and calculation scope 1. [Sandbox Physics, single-excitation finite-band waveguide model and bound-state references](https://github.com/yangzichao/sandboxphysics/blob/main/docs/quantum/waveguide-memory.md) Nearest-neighbour chain dispersion, infinite-chain bound-state reference and finite-window definitions. These are model calculations with declared units, not measured photonic-crystal data. 2. [Sandbox Physics, existing giant-atom propagation model](https://github.com/yangzichao/sandboxphysics/tree/main/src/scripts/physics-lab/experiments/waveguide-memory/model) The existing model uses 401 sites, maximum time 64 and explicit propagation between two connections to the same emitter. Default separation is 32 and the compact-state preset uses 30. Boundary monitoring and analytic references are retained.