--- title: "Connect a temporal mode to a published experiment" description: "Read a fixed Kurpiers photon-interface paper, reproduce its chosen ideal envelope and distinguish squared mean field from photon flux." canonical_url: "https://sandboxphysics.com/stories/waveguide-qed/kurpiers-photon-interface/" language: "en" page_type: "story" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/stories/waveguide-qed/kurpiers-photon-interface/" last_modified: "2026-10-28" --- [Cavity and waveguide quantum electrodynamics](https://sandboxphysics.com/stories/waveguide-qed/) L4 · Chapter 7 # Connect a temporal mode to a published experiment Sandbox Physics · 2026-10-28 A curve shaped like a light pulse need not be a single-photon count rate. Its interpretation depends on the prepared state and measured operator. We reproduce one fully specified link: normalization and time scale of a hyperbolic-secant envelope, together with its mean-field observable. ## Before you begin Temporal-mode overlap from Chapter 6 and model scope from Chapter 3. Distinguish the squared mean of a quantity from its mean square. We fix a paper version and derive its observable mapping; apparatus control and process tomography remain further reading. [Review the preceding chapter](https://sandboxphysics.com/stories/waveguide-qed/collected-temporal-modes/) - Derive the different factors for squared mean field and flux from a declared state. - Specify a bounded reproduction with parameters, sources and repeatable numerical checks. ## Fix the version and observable The source version is Kurpiers et al., arXiv:1712.08593v1, submitted 22 December 2017, associated with the 2018 Nature paper in volume 558. Figure and appendix numbers here refer to this preprint. We select Figure 2c,f and Appendix C. A cavity-assisted process emits a symmetric microwave photon; heterodyne detection reads the prepared state’s mean field. The caption gives effective bandwidths of $10.4\,\mathrm{MHz}$ and $10.6\,\mathrm{MHz}$. This reproduction explicitly selects the first scale. The reproduced objects are the ideal envelope and its observable relation. Process tomography, measured transfer efficiency and apparatus loss need additional models and data. All displayed curves are original calculations; no experimental points, source artwork or raw measurements are redistributed. Reading: Kurpiers et al. ## One photon mode gives distinct observables Let $|1_\phi\rangle$ be a single photon with normalized envelope $\phi(t)$. Applying the output annihilation operator $b(t)$ gives vacuum times $\phi(t)$. The mean field of a pure one-photon state is zero because vacuum and the original state are orthogonal. An equal coherent superposition with vacuum is $|\chi\rangle=(|0\rangle+|1_\phi\rangle)/\sqrt2$. Expanding its bra and ket leaves one nonzero term in the mean field, with coefficient one half. Squaring gives one quarter. Flux takes the operator product before averaging. Only the one-photon branch contributes, with probability one half. These observables share a temporal shape but have different normalization factors. $$ \langle b(t)\rangle_\chi=\frac{\phi(t)}2,\qquad |\langle b(t)\rangle_\chi|^2=\frac{|\phi(t)|^2}{4} $$ Squared mean field retains vacuum–one-photon coherence when the preparation phase is fixed. $$ \langle b^\dagger(t)b(t)\rangle_\chi=\frac{|\phi(t)|^2}{2},\qquad \langle b^\dagger(t)b(t)\rangle_{1_\phi}=|\phi(t)|^2 $$ These are flux in the superposition and flux in a pure one-photon state, respectively. Label them separately from squared mean field. ## Calculate a physical pulse width The declared envelope is $\phi(t)=\sqrt\kappa\,\operatorname{sech}(\kappa t/2)/2$. Substitute $x=\kappa t$. Using the derivative of the hyperbolic tangent, its integrated probability is exactly one. At $\kappa/(2\pi)=10.4\,\mathrm{MHz}$, one dimensionless time unit is $1/\kappa\simeq15.30\,\mathrm{ns}$. Half the peak intensity requires a hyperbolic cosine of square-root two, giving intensity full width at half maximum $53.95\,\mathrm{ns}$. This is a calculated width of the selected ideal envelope. $$ \int_{-\infty}^{\infty}\frac14\operatorname{sech}^2(x/2)\,dx=1 $$ Changing variables changes both the horizontal scale and the probability density. $$ T_{\rm FWHM}=\frac{4\operatorname{arcosh}(\sqrt2)}{\kappa} $$ This width refers to intensity. A half-maximum amplitude criterion gives a different width. A delayed pulse can miss a fixed receiver mode Probability per scaled time $|\phi_1(x)|^2$ Scaled time $x=\kappa t$ - 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 ## Construct a controlled source for the envelope An ideal Markov source can be designed backwards from its survival probability. Flux is minus its time derivative. Dividing flux by survival gives the required instantaneous release rate. Starting infinitely early, a decreasing logistic survival $S(t)$ gives release rate $\lambda(t)$ and amplitude $\sqrt{\lambda S}$ with the desired shape. A finite numerical start uses $S(t)/S(0)$ and retains the residual source population at the window end. This source evaluates an interface envelope and its mode projection. Reproducing the experimental control pulse would additionally require the emitter–transfer-cavity dynamics, frequency shifts and apparatus parameters. $$ S(t)=\frac{1}{1+e^{\kappa(t-t_c)}},\qquad \lambda(t)=-\frac{\dot S}{S}=\kappa[1-S(t)],\qquad \lambda S=\frac\kappa4\operatorname{sech}^2\!\left(\frac{\kappa(t-t_c)}2\right) $$ Survival, flux and release rate come from one function, permitting probability checks at every time. Reading: Sandbox Physics FR11 ## Use the limited calculation as a research reference Compare numerical overlap with the analytic limit at zero loss, matched timing and zero detuning. Change one condition and predict whether collection or conditional fidelity will change. A saved complex record lets a new receiver mode be tested without regenerating emission. More detailed interfaces can add receiver-cavity dynamics, drift, filtering or phase noise. Each addition should retain a limit that returns to the known ideal envelope. This turns a small reproducible calculation into a useful reference for a larger model. ## Try the prediction ### [Recalculate the selected ideal observable](https://sandboxphysics.com/labs/quantum-node-photon-collection/) 1. Save a complex envelope and confirm its version, window and normalization. 2. Convert time units using 15.30 nanoseconds and compare intensity width with the analytic value. 3. Construct pure one-photon flux, superposition flux and squared superposition mean field, explaining each normalization factor. ## Exercises and explanations 1. Does zero mean field of a pure photon mean there can be no detection? Read the explanation No. Counting depends on the mean field-operator product and gives nonzero flux. Mean amplitude instead depends on coherence between different photon-number branches. 2. What is integrated squared mean field for the equal vacuum superposition, and how does it compare with flux? Read the explanation It is one quarter for a normalized mode. Mean photon number in that superposition is one half, so the two differ by a factor of two. Pure single-photon flux integrates to one. 3. Does reproducing the ideal width establish the reported qubit-transfer fidelity? Read the explanation No. Width checks one temporal scale. Transfer also depends on reception, loss, decoherence and operations, and requires comparison with the relevant process-tomography data. [Previous: Receiving a photon also means receiving its mode](https://sandboxphysics.com/stories/waveguide-qed/collected-temporal-modes/) [Next: Turn a scattering picture into a reproducible study](https://sandboxphysics.com/stories/waveguide-qed/reproducible-waveguide-study/) ## Sources and calculation scope 1. [Kurpiers et al., Deterministic quantum state transfer and remote entanglement, Nature 558, 264–267 (2018)](https://arxiv.org/abs/1712.08593v1) Frozen preprint v1, 22 December 2017, Figure 2c,f and Appendix C. We recalculate the ideal sech envelope and the vacuum–one-photon mean-field mapping at the declared 10.4 MHz scale. No experimental data or source artwork is redistributed; transfer efficiency and process tomography are outside the reproduced observable. 2. [Sandbox Physics FR11, reproducible ideal temporal-mode benchmark](https://github.com/yangzichao/sandboxphysics/blob/main/scripts/frontier/fr11/reproduce-node-benchmark.mjs) 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.