--- title: "From a travelling field to a detector record" description: "Separate port amplitudes, photon flux and detector response, and determine when a spatial pair map can become an arrival-time distribution." canonical_url: "https://sandboxphysics.com/stories/waveguide-qed/output-records/" language: "en" page_type: "story" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/stories/waveguide-qed/output-records/" last_modified: "2026-10-28" --- [Cavity and waveguide quantum electrodynamics](https://sandboxphysics.com/stories/waveguide-qed/) L3 · Chapter 5 # From a travelling field to a detector record Sandbox Physics · 2026-10-28 A detector acts on an output port and has its own temporal response. Calculate the escaping field first, then apply the stated measurement model. Sampling intracavity photon number at successive frames describes repeated system measurements, not automatically a click trajectory of one emission. ## Before you begin Chapter 2 on input–output relations and Chapter 4 on two-excitation states. You should understand integration and probability density. Port flux, normal ordering and response convolution are introduced here. [Review the preceding chapter](https://sandboxphysics.com/stories/waveguide-qed/two-photon-saturation/) - Calculate cumulative output probability from saved complex amplitudes. - Explain how detector response changes a pulse or correlation without changing the underlying system evolution. ## Define where the field leaves The structured-bath experiment keeps 35 resonators, a central emitter and a memoryless output at either end. Chain amplitudes $c_j(t)$ describe light still inside the bath. Multiplying endpoint amplitudes by the square root of the port rate gives the travelling one-photon envelopes. This model has a finite-band system with open ports. Internal propagation and returns are explicit; the outer ports use the Markov approximation. The emitter also has a separate loss channel. An endpoint-mismatch return changes with port coupling and has a different geometry from a giant atom with two coupling points. $$ \xi_L(t)=\sqrt{\kappa_p}\,c_0(t),\qquad \xi_R(t)=\sqrt{\kappa_p}\,c_{34}(t),\qquad P_r(T)=\int_0^T|\xi_r(t)|^2\,dt $$ Envelopes have inverse square-root-time units. The distinct output channels have additive cumulative probabilities. Reading: Sandbox Physics FR11 ## Retain what has not yet escaped Start with one emitter excitation. At time $T$, it may remain in the atom, remain in the chain, have left through either port or have entered intrinsic loss. These exclusive probabilities must sum to one. Worked example: suppose 0.1 remains in the system, the two ports contain 0.8 cumulatively and intrinsic loss is 0.1. Collection probability is 0.8. Assigning the remaining system population to output would count emission before it occurs. All three contributions can change when the window is extended. $$ P_e(T)+\sum_j|c_j(T)|^2+P_L(T)+P_R(T)+P_{\rm loss}(T)=1 $$ Every term comes from the same run. Intrinsic loss integrates atomic excitation probability times its loss rate. ## A detector spreads a sharp arrival in time Ideal one-photon flux is $I(t)=|\xi(t)|^2$. A causal response $h(t)$ spreads arrival times into later readings. The recorded mean flux is a convolution: earlier fluxes are summed with response-dependent weights. The experiment uses first-order response time $\tau_d=0.6$ in model units. It smooths the rising edge, delays the peak and adds a tail. A normalized response preserves total weight over an infinite window; an unfinished tail must be retained separately in a finite window. This intensity processing does not create a new quantum state with preserved phase. An optical filter acts first on field amplitudes and needs its lost ports included. Readout blur is a different physical operation. $$ I_{\rm det}(t)=\int_0^t\frac{e^{-(t-s)/\tau_d}}{\tau_d}I(s)\,ds $$ This is a declared synthetic detector response, not an instrument calibration. Mode projection uses the saved raw complex output. ## When position can be converted into arrival time The map $|\Phi_{jk}|^2$ describes two positions at the same time. Converting positions into detection times requires separated outgoing light, known velocities and negligible dispersion over the relevant band. Divide travel distances by velocity and include the Jacobian of the change of variables. Near a band edge, different frequencies travel at very different speeds. A straight relabelling of a position axis is then inadequate. Calculate a two-time output-field correlation at a defined port, or explicitly include the detector and output modes. A normally ordered field correlation describes joint detection. In a stationary experiment it is normalized by mean fluxes to compare with independent events. A finite two-excitation pulse does not supply a constant steady-state flux. $$ g^{(2)}(t,t')=\frac{\langle b^\dagger(t)b^\dagger(t')b(t')b(t)\rangle}{\langle b^\dagger(t)b(t)\rangle\langle b^\dagger(t')b(t')\rangle} $$ This normalization is undefined when a denominator vanishes. A transmitted coherent input also interferes with emitter radiation. Reading: Sheremet et al. ## Save enough information to repeat the processing An output record needs a model version, parameters, time units, sampling times and the real and imaginary components of the envelope. Intensities suffice to recalculate collection and readout blur, but not a phase-sensitive temporal-mode fidelity. The last chapter independently benchmarks side fluorescence of a weakly driven Markov two-level atom using a master equation and detector response. That stationary calculation complements the finite-waveguide pair map; each retains its own assumptions. ## Try the prediction ### [Recalculate flux from one saved output record](https://sandboxphysics.com/labs/structured-bath-output-ports/) 1. Save the default complex-port record. Check the total of atom, chain, both outputs and loss probabilities. 2. Compare raw and detector-broadened curves. Predict what extra tail an extended window would recover. 3. Change port coupling and save a second run. Use its spatial and temporal evidence to distinguish endpoint mismatch from a bandwidth effect. ## Exercises and explanations 1. Do two identical intensity traces necessarily have equal temporal-mode fidelity? Read the explanation No. Their complex phases can differ. Mode overlap adds amplitudes before squaring, and an intensity-only record loses that information. 2. Why can a normalized detector response reduce counts in a finite window? Read the explanation Part of its tail lies beyond the window. Extend it or retain the unfinished response weight to test total normalization. 3. Can an intracavity one-photon measurement be turned directly into an output click? Read the explanation No. Intracavity measurement and output detection are different operations and can change the state differently. A click model needs a specified output detection operator and conditional evolution. [Previous: Why two photons cannot always be scattered separately](https://sandboxphysics.com/stories/waveguide-qed/two-photon-saturation/) [Next: Receiving a photon also means receiving its mode](https://sandboxphysics.com/stories/waveguide-qed/collected-temporal-modes/) ## Sources and calculation scope 1. [Sheremet et al., Waveguide quantum electrodynamics, Rev. Mod. Phys. 95, 015002 (2023)](https://doi.org/10.1103/RevModPhys.95.015002) The theory overview and Appendix A place single-excitation propagation and input–output relations in their model context. Our two-amplitude examples and schematic figures are original teaching calculations. 2. [Sandbox Physics FR11, finite-band ports and collected-mode model](https://github.com/yangzichao/sandboxphysics/blob/main/docs/frontier/fr11/ports-and-node-model.md) The finite bath keeps its field amplitudes and output losses. Its detector response is a declared synthetic filter. The separate controlled source retains complex output samples for receiver-mode projection.