From a travelling field to a detector record

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.

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.

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.

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.

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.