Connect a temporal mode to a published experiment

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.

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.

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.

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.