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Sandbox Physics
Cavity and waveguide quantum electrodynamics

L3 · Chapter 6

Receiving a photon also means receiving its mode

Sandbox Physics ·

A receiver accepts a rhythm and a phase as well as an energy quantum. Most photons can be lost while the collected ones have excellent shape. Nearly all can arrive in a mode the receiver barely accepts. Separating these effects tells us whether to improve transmission or waveform matching.

Describe a receiver by a temporal mode

Choose a receiver envelope u(t)u(t) whose squared magnitude integrates to one within the observation window. This temporal mode specifies when amplitude is large and how phase changes. A photon in it is a coherent superposition of possible arrival times.

The unnormalized collected envelope is ξ(t)\xi(t). Its squared magnitude integrates to collection probability PcolP_{\rm col}. Multiplying by the conjugate receiver envelope and integrating instead gives the target-mode amplitude. These operations preserve different information.

∫0T∣u(t)∣2dt=1,Pcol=∫0T∣ξ(t)∣2dt,Au=∫0Tu∗(t)ξ(t) dt\int_0^T|u(t)|^2dt=1,\qquad P_{\rm col}=\int_0^T|\xi(t)|^2dt,\qquad A_u=\int_0^T u^*(t)\xi(t)\,dt
The envelopes have inverse square-root-time units and these integrals are dimensionless. The inner product retains phase cancellation.

Three numbers answer three questions

Collection probability asks how often a photon reaches the port. Conditional mode fidelity FcondF_{\rm cond} asks how closely the collected photon matches the receiver mode. Unconditional target weight WuW_u combines them, giving the target single-photon probability per attempt.

At Pcol=0P_{\rm col}=0, conditional fidelity is undefined. A high conditional value can coexist with almost no successful collection. High collection can coexist with a badly mismatched phase or timing.

This formula assumes a pure one-photon mode in the selected port. Unrecorded random phase or spectral mixing requires a density matrix. Averaging complex envelopes from different trials does not generally replace that calculation.

Fcond=∣Au∣2Pcol,Wu=∣Au∣2=PcolFcondF_{\rm cond}=\frac{|A_u|^2}{P_{\rm col}},\qquad W_u=|A_u|^2=P_{\rm col}F_{\rm cond}
This fidelity evaluates the overlap with one specified single-photon temporal mode.

Loss need not change the collected shape

An ideal frequency-independent branch collecting a fraction η\eta changes the envelope to η ξ0\sqrt\eta\,\xi_0. The overlap amplitude receives the same factor. Collection and target weight decrease, while normalized conditional fidelity stays fixed.

Worked example: collection is 0.600.60 and conditional mode fidelity is 0.900.90. Target weight per attempt is 0.540.54. Perfecting the shape with the same loss can raise this only to 0.600.60. Further gains need reduced loss or more amplitude in the receiver mode.

Equal intensity can still give a small overlap

Give one of two equal-intensity pulses a phase that rotates with time. Their intensity traces remain identical, but contributions to the inner product can cancel. A constant overall phase only rotates the final overlap arrow. A changing phase alters the mode match.

For a normalized hyperbolic-secant envelope ϕκ\phi_\kappa on the whole time axis, a temporal offset δt\delta t has the overlap below. At κδt=2\kappa\delta t=2, its amplitude is approximately 0.8510.851 and fidelity 0.7240.724. A finite record needs a fresh integral over its actual window.

ϕκ(t)=κ2sech⁡ ⁣(κt2),∫−∞∞∣ϕκ∣2dt=1\phi_\kappa(t)=\frac{\sqrt\kappa}{2}\operatorname{sech}\!\left(\frac{\kappa t}{2}\right),\qquad \int_{-\infty}^{\infty}|\phi_\kappa|^2dt=1
The hyperbolic secant is the reciprocal of the hyperbolic cosine. It gives a symmetric envelope with smoothly decaying tails.
∫−∞∞ϕκ(t)ϕκ(t−δt)dt=κδt/2sinh⁡(κδt/2)\int_{-\infty}^{\infty}\phi_\kappa(t)\phi_\kappa(t-\delta t)dt=\frac{\kappa\delta t/2}{\sinh(\kappa\delta t/2)}
The zero-offset limit is one. Timing affects overlap even when all photons still reach the same port.

A delayed pulse can miss a fixed receiver mode

Probability per scaled time∣ϕ1(x)∣2|\phi_1(x)|^2
00.1250.25-606
Scaled timex=κtx=\kappa t
  • Receiver
  • Delayed emission
Original ideal intensity curves on the whole time axis. The emitted pulse is delayed by two scaled time units. Finite-record mode overlaps are calculated separately from their complex samples.

Reading: Sandbox Physics FR11

Worked example: reverse only the second half

Choose an ideal receiver u(t)=1/Tu(t)=1/\sqrt T on 0<t<T0<t<T, zero elsewhere. Emit the same shape with collection probability P=0.8P=0.8, then reverse its phase by half a turn in the second half. Intensity remains P/TP/T throughout, so collection is still 0.8.

The inner product adds equal positive and negative contributions, cancelling exactly. Conditional fidelity and target weight become zero. Reversing the entire pulse instead leaves all contributions aligned and conditional fidelity one. The ideal example separates phase mismatch from energy loss.

ξ(t)=PT{1,0<t<T/2,−1,T/2<t<T,Au=PT(T2−T2)=0\xi(t)=\sqrt{\frac PT}\begin{cases}1,&0<t<T/2,\\-1,&T/2<t<T,\end{cases}\qquad A_u=\frac{\sqrt P}{T}\left(\frac T2-\frac T2\right)=0
The rectangular mode and instantaneous phase reversal are hand-calculation references. The interactive experiment uses smooth envelopes and continuous detuning.

Evaluate an interface from a finite record

The node experiment fixes a hyperbolic-secant receiver mode and varies emission bandwidth, collected branching, detuning and pulse centre. Its source survival probability and both output branches come from one controlled emission model and sum to one at every time.

A finite start requires survival to be normalized by its initial value. The target mode is normalized inside the same finite window. Moving a pulse can move part of its tail outside that window; check this before attributing the whole change to mode mismatch.

The finite bath also projects its right-port record onto a fixed receiver: its normalized default resonant output over the same window. This is separate from the node experiment’s sech target. After moving toward the band edge, project the new complex right-port record to see how returning light and tails affect collection, conditional fidelity and target weight.

Change one quantity at a time: branching first, then detuning, then pulse centre. The first mainly changes success probability. The other two expose the role of phase and timing for the same receiver.

Reading: Sandbox Physics FR11 · Sandbox Physics FR11

Try the prediction

Can one receiver collect a band-edge tail?

  1. Choose “Emit into the band” and record the three values under “Right-port receiving mode”. Its receiver is the fixed default resonant output.
  2. Choose “Linger near the edge”, save the complex right-port record and compare collection with conditional fidelity.
  3. Recompute target weight against the same receiver, explaining the change using phase, tails and residual system probability.

Separate loss from mode mismatch

  1. Keep bandwidth, detuning and timing fixed while reducing the collected branch. Predict all three measures.
  2. Restore the branch and add only detuning. Compare intensity, complex phase and conditional mode fidelity.
  3. Move the pulse centre and export the record. Reintegrate its complex envelope against the fixed receiver mode and assess window truncation.

Exercises and explanations

  1. Compare interfaces (Pcol,Fcond)=(0.3,1)(P_{\rm col},F_{\rm cond})=(0.3,1) and (0.8,0.5)(0.8,0.5). Which reaches the target more often per attempt?

    Read the explanation

    The second: target weights are 0.3 and 0.4. The first has better conditional shape, while the second has higher total success for this target.

  2. Does multiplying the whole envelope by minus one reduce fidelity? What if only its second half changes sign?

    Read the explanation

    A global minus sign is an overall phase and leaves squared overlap unchanged. Reversing only the second half introduces cancellation and generally lowers fidelity without changing intensity.

  3. Why is temporal-mode fidelity insufficient to claim qubit-transfer fidelity?

    Read the explanation

    Qubit transfer also depends on vacuum–one-photon coherence, relative phase, source and receiver operations, and other errors. Here we evaluate one specified single-photon projection.

Sources and calculation scope

  1. Sandbox Physics FR11, finite-band ports and collected-mode model

    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.

  2. Sandbox Physics FR11, reproducible ideal temporal-mode benchmark

    Recalculates ideal sech normalization, finite-window survival, overlap and the quarter-versus-half mean-field distinction. Parameters and calculated reference values are saved with the source; no measured points enter the benchmark.

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