Collective Emission Interface Limits

Almost complete photon collection does not guarantee faithful quantum-state transfer.

A symmetric ensemble stores an unknown quantum state and releases its excitations as light. A receiver, configured before the input is known, retains one oscillator. The question is when the collective spin can be treated as a harmonic oscillator for this transfer—not merely for the average number of photons it delivers.

Read next	Purpose
Scientific account · Model and objective	Enter the connected account through its definitions and proof dependencies
Common-mode boundary · Claim and evidence map	Check the exact theorem, finite bounds and supporting consequences
Learning route · 30 milestones · Source bridge	Build understanding from one educational anchor and local worked examples
Physical scope · References	Check receiver resources, realization limits and attribution
Verification guide	Reproduce the scientific checks and interpret the evidence
LLM guide	Identify relevant questions and authoritative files for further reading

Model and transfer task

Task	What one receiving pulse must preserve
Photon collection	Nearly all the mean photon number
Canonical quantum-state transfer	Every number amplitude and coherence in the declared input space, including correlations with an external reference

The source has $N$ emitters and supports excitation numbers $0,\ldots,M$, with $M\leq N$. The intended map uses the same normalized temporal mode $f$ for every input:

$$\sum_{m=0}^{M}c_m\lvert D_N^m\rangle \longmapsto \sum_{m=0}^{M}c_m\lvert m\rangle_f.$$

One spatial output channel can contain many temporal modes. Capturing the beam is not therefore the same operation as storing its state in one oscillator.

The common-mode boundary

Let $\mathcal F_{N,M}$ be the optimized worst-input squared entanglement fidelity for complete ideal collective decay and a predetermined, photon-number-preserving linear receiver with vacuum auxiliaries. The optimization allows every common waveform.

$$\boxed{ \frac{M}{N^{2/3}}\longrightarrow c>0 \quad\Longrightarrow\quad \mathcal F_{N,M}\longrightarrow e^{-c^3/192} }$$

The fidelity tends to one for $M=o(N^{2/3})$ and to zero when $M/N^{2/3}\to\infty$. In contrast, one fixed waveform collects a mean photon fraction tending to one throughout $M=o(N)$. Taking $M=\lfloor N^{3/4}\rfloor$ gives

$$\begin{aligned} M/N &\longrightarrow 0,\\\ \text{mean collected photon fraction} &\longrightarrow 1,\\\ \text{optimal worst-input fidelity} &\longrightarrow 0. \end{aligned}$$

This is a uniform channel limitation, not a statement that every input fails. The boundary proof supplies the construction, all-waveform converse and finite bounds. The photon-collection proof establishes the larger domain for the mean collected fraction.

Why the two criteria differ

The collective ladder decays at $\gamma k(N-k+1)$, rather than the oscillator rate $\gamma Nk$. Each known subextensive photon number can be matched to an excellent individual pulse. But different numbers prefer slightly different pulses.

A pulse difference of order $M/N$ produces a one-photon mismatch of order $(M/N)^2$. Requiring all photons in a many-photon component to occupy the receiving mode amplifies this into the scale $M^3/N^2$. The matched-pulse argument controls the correlated emitted field before making that product-pulse comparison; independence is not assumed.

Even one logical qubit can witness the critical limit: use the populated numbers $\lfloor M/4\rfloor$ and $M$. Their physical excitation numbers grow. Vacuum plus one populated number is a different, successful comparator. See the two-sector witness.

One tutorial, then this result

The selected learning anchor is:

A. H. Kiilerich and K. Mølmer, Quantum interactions with pulses of radiation,
Physical Review A 102, 023717 (2020).
Author tutorial, arXiv:2003.04573 · Published article

Its virtual input/output cavities give the physical language of the interface. The learning guide maps its sections to this repository. The source-to-project bridge supplies the additional Dicke-ladder, fidelity and uniform-bound steps locally; no second external tutorial is required. Worked emission, mode-mismatch and coherence examples lead into the proof; loss and finite capture supply the physical qualifications.

The 30-milestone map has two phases. M01–M12 assemble the pulse, source and channel framework; M13–M30 connect the controlled field approximation, pulse geometry, optimality proof and physical consequences into the whole result. Each milestone states a central question, understanding target, dependencies, conceptual connection and source pointers. It is a shared map of the science, independent of a reader's prior knowledge or learning history.

Boundaries and prior work

The ideal source is symmetric, has known $N$, and emits completely into one vacuum Markov channel. The receiver retains one oscillator after fixed passive processing. The input is an arbitrary state in the symmetric excitation code, including its correlations with a reference. The full emitted field retains that information; the theorem quantifies transfer into the specified single receiving oscillator. The physical analysis gives the loss, duration and coupling requirements for capture.

The loss-competition result also identifies when prescribed mode-independent transmission loss masks the common-mode limitation. Its pure-loss channel is distinct from independent atomic decay and from mode-dependent filtering. Finite capture is quantified on the same excitation code, with coupling and duration allowed to scale with that code.

The Dicke cascade, conventional-pulse cubic mismatch, number-dependent temporal modes, mean-occupation optimization and pulse-capture formalism have direct predecessors. The comparison and background account separate those ingredients from the optimized uniform limit. The Tziperman comparison examines its overlapping collective source and selected-state transfer construction alongside the uniform common-code guarantee. The interpretation connects the consequences, while the verification guide locates their executable checks.

Evidence and reproduction

python -m pip install -r requirements.txt
python tools/check_presentation.py --self-test
python verify.py --integrity-only
python verify.py --artifacts-dir verification-artifacts --require-reference
python tools/check_loss_competition.py --output loss-competition-report.json

The eight scientific suites cover 39 groups and 612 cases. The scientific scripts and saved results are preserved; generated outputs are compared without rewriting references. The reproduction policy distinguishes exact bytes, reviewed numerical agreement and passing assertions. The presentation check is separate from the scientific test count. The 50 supplementary loss-competition checks are counted separately. The verification guide explains the commands, expected outputs and the distinction between finite calculations and analytical proof.

Purpose and contact

This repository serves as a record of the work and a guide for the author’s self-directed learning. For discussion or potential collaboration, please contact Ruge Lin at gogoko699@gmail.com.

The LLM guide describes relevant research questions, search terms and the authoritative reading order for automated assistants and other readers. Code is available under the MIT license.