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Hypergeometric pulses

The classic ways to drive a two-level transition — Allen–Eberly, STIRAP, Demkov–Kunike — look like separate recipes, but they are one family seen from different angles. A single two-parameter envelope, built on the Gauss hypergeometric function, contains them all: each named protocol is just a choice of $(\alpha,\beta)$. Slide between them below and watch one shape morph into another.

One envelope, two knobs

The $(\alpha,\beta)$-hypergeometric envelope multiplies a smooth $\operatorname{sech}$ base by a hypergeometric factor whose two parameters bend the shape. At $\alpha=\beta=0$ the factor is trivial and the pulse is the pure $\operatorname{sech}$ of Allen–Eberly; at $\alpha=\beta=\tfrac12$ it is the STIRAP-optimal shape; at $\alpha=1,\beta=\tfrac12$ it is Demkov–Kunike. Everything in between is a valid, smooth pulse.

$$\frac{\Omega(t)}{\Omega_0} \;=\; \operatorname{sech}(\gamma t)\;\cdot\;{}_2F_1\!\Big(\alpha,\ \beta;\ \tfrac{\alpha+\beta+1}{2};\ \tfrac{1+\tanh(\gamma t)}{2}\Big)$$

The argument $\tfrac12(1+\tanh\gamma t)$ sweeps smoothly from $0$ to $1$ as time runs, so the whole pulse is the hypergeometric function traced along that path and tapered by the $\operatorname{sech}$. The panel evaluates that ${}_2F_1$ directly — a power series, transformed near the endpoint so it stays accurate — and matches the library's reference to a few parts in ten million.

Live — the $(\alpha,\beta)$ envelope and the symmetric family
your (α,β) Allen–Eberly STIRAP
protocol: — c = (α+β+1)/2: — peak Ω/Ω₀: —

The left plot is your chosen envelope against a faint Allen–Eberly $\operatorname{sech}$ and the STIRAP shape, so you can see how $(\alpha,\beta)$ pushes the pulse taller and squarer or leaves it soft. The right plot fans out the symmetric family $\alpha=\beta$ from $0$ to $1$: as the pair grows, the hypergeometric factor lifts the middle and the pulse trades a gentle sech for a flatter, higher plateau — the shape that spends its power where the transfer needs it.

Deeper: evaluating ₂F₁ in the browser
The hypergeometric factor is a genuine special function, ${}_2F_1(\alpha,\beta;c;z)$, and near $z=1$ — the end of the pulse — its plain power series converges slowly or not at all. The panel uses Euler's transformation ${}_2F_1(a,b;c;z)=(1-z)^{c-a-b}\,{}_2F_1(c-a,c-b;c;z)$ whenever $c-a-b<0$, which moves the awkward behaviour into an explicit prefactor and leaves a series that converges cleanly. Over the whole $\alpha,\beta\in[0,1]$ range the resulting envelope agrees with the library's SciPy-backed reference to better than $3\times10^{-7}$ — the $\operatorname{sech}$ taper further suppresses any residual near the tails. The reference itself is hypergeometric_envelope in the pulse-shaping module, after Ventura Meinersen et al. (2025).

Why one family is better than three recipes

Holding many protocols in a single differentiable envelope means the shape can be tuned, not just picked from a menu: an optimiser can slide $(\alpha,\beta)$ continuously to trade transfer speed against robustness for a specific device, instead of jumping between named pulses. The ICI sequence sharpens the ends of a transfer; the hypergeometric family reshapes its whole body. Together they are the platform's vocabulary for turning a required transition into an achievable waveform.

Protocol(α, β)Character
Allen–Eberly(0, 0)pure $\operatorname{sech}$, the soft baseline
STIRAP-optimal(½, ½)symmetric, dark-state transfer
Demkov–Kunike(1, ½)level-crossing with detuning
ICI (separate panel)—bang–adiabatic–bang, time-optimal
ICI pulse shaping Pulse-shaping methods Algorithms Back to overview

Evidence boundary: the envelope is the library's exact documented formula, its Gauss ${}_2F_1$ evaluated live in your browser and checked against the SciPy-backed reference to better than $3\times10^{-7}$ across $\alpha,\beta\in[0,1]$. It is the pulse shape, not a full transfer simulation or a hardware calibration; the transfer fidelity is a separate step in the library.