Why Identifiability, Efficiency, and Calibration Can Come Apart

The highlighted sentence is a warning about three different kinds of “correctness” in a self-calibrating Rydberg microwave receiver. In this paper, the receiver is modeled through squared dressed-state separations. At LO setting $j$ and phase sample $k$, the idealized observation has the form

$$ y_{jk} = D+\left|L_j e^{i\phi_{jk}}+s\right|^2, \qquad s=a e^{i\theta},\qquad P=a^2, $$

or, after expanding,

$$ y_{jk} = D+L_j^2+P+2L_j\sqrt{P}\cos(\phi_{jk}-\theta). $$

Here $D=\delta^2$ is the squared detuning, $L_j$ is the LO amplitude at the atoms, $s$ is the unknown signal, $P$ is the signal power in Rabi-frequency units, and $\theta$ is the signal phase. The parent document’s central point is that one can solve parts of this inverse problem from the observations themselves, but “solving” has several meanings. A design may determine a unique noiseless solution, may have a small Cramér–Rao bound under a chosen noise model, and may still be wrong as a measurement of the physical field if the model’s scale or assumptions are wrong.

Identifiable does not mean well conditioned

Identifiability is a noiseless concept. A parameter is identifiable if two different parameter values cannot produce exactly the same ideal observations. For example, suppose two LO settings are used and the relevant moments give

$$ A_j = D+L_j^2+P, \qquad B_j = 2L_j\sqrt{P}. $$

The parent document uses the fact that, under the ideal model, the signal power can be recovered through a ratio of differences, schematically

$$ P = \frac{|H_2|^2-|H_1|^2}{4(A_2-A_1)}. $$

This formula is identifiable when the denominator is not zero. Since

$$ A_2-A_1=L_2^2-L_1^2 $$

in the ideal model, the two LO powers must be different. If $L_2^2\neq L_1^2$, the noiseless inverse can be unique.

But uniqueness is not the same as numerical stability. If $L_2^2$ is only slightly different from $L_1^2$, then $A_2-A_1$ is very small. A tiny measurement perturbation in $A_1,A_2,H_1,H_2$ is then divided by a tiny number. The answer is still unique in exact arithmetic, but the inverse map has a large derivative. This is what it means for an identifiable protocol to be poorly conditioned.

A concrete example is helpful. Suppose the two LO powers differ by only one part in a thousand. Then the numerator and denominator in the power inversion both depend on subtracting nearly equal quantities. Even if the observations are generated by a unique true signal, small spectral-fitting errors can cause a large fractional error in the recovered $P$. In this case the problem is not “unidentifiable” in the strict mathematical sense; rather, it is identifiable but fragile.

A second example occurs near zero signal amplitude. When $P=0$, the signal phase $\theta$ has no physical meaning: $s=0$ is the same complex number for every $\theta$. For very small $P$, the phase may technically be estimable under some nonzero-signal assumptions, but the sinusoidal modulation depth $2L_j\sqrt{P}$ is tiny. Noise can then dominate the phase information. Again, the practical difficulty is not captured by asking only whether the ideal noiseless map is one-to-one.

This distinction is standard in inverse problems and estimation theory: identifiability concerns uniqueness, while conditioning concerns sensitivity of the inverse map to perturbations. Singular or nearly singular Fisher-information matrices are one statistical expression of the same difficulty [Stoica and Marzetta 2001].

Statistical efficiency does not guarantee physical truth

Statistical efficiency is different again. In the parent document, efficiency is discussed through Cramér–Rao bounds. Under a specified observation model, an efficient estimator has variance close to the smallest variance allowed by that model. In the usual regular parametric setting, the Cramér–Rao bound says that the covariance of any unbiased estimator is bounded below by the inverse Fisher information matrix [Kay 1993; Van Trees et al. 2013].

But this statement is conditional on the model. If the model says that the squared-separation errors are homoscedastic Gaussian errors, then the resulting CRB describes the best possible precision under that error law. It does not prove that the spectroscopic model is the correct physical description of the experiment. It also does not prove that the conversion from observed optical spectra to microwave Rabi frequencies has the right absolute scale.

For example, imagine that the data are extremely clean and the estimator nearly attains its CRB under the assumed squared-separation model. The recovered $P$ may have very small statistical uncertainty. Yet if the actual spectral splitting has been converted into microwave field amplitude using a wrong calibration factor, the estimate can be precise and biased. In ordinary measurement language, the result has small random error but a systematic error.

A simple scale example makes this clear. Suppose the physical squared separation should be $y_{jk}$, but the extraction pipeline returns

$$ \tilde y_{jk}=c\,y_{jk} $$

with an unknown scale factor $c\neq 1$. A self-consistent inversion may then return a sharply determined signal power in the wrong units or with the wrong absolute scale. The reported confidence interval can be narrow because the noise is small, while the physical amplitude remains biased because the calibration scale is wrong. This is why the parent document separates “statistical efficiency” from “physical calibration.”

This distinction is especially important in Rydberg electrometry. Rydberg atoms provide a route to SI-traceable electric-field measurements, but actual experiments still require careful treatment of systematic effects, atomic response models, line-shape extraction, field inhomogeneity, and uncertainty budgets [Holloway et al. 2017; Schlossberger et al. 2024]. A mathematically efficient estimator cannot by itself validate those physical assumptions.

Internal consistency can still return the wrong amplitude

The final part of the highlighted sentence is the most subtle: a protocol can be internally consistent while returning the wrong signal amplitude. “Internally consistent” means that the observations fit the assumed equations well. The residual may be small, the two LO settings may agree under the fitted model, and the recovered parameters may reproduce the measured spectra. Nevertheless, a hidden model error can be observationally equivalent to a different signal.

The parent document explicitly points to one such ambiguity: an LO-power-dependent contribution to squared detuning. The ideal model assumes a common $D$ independent of LO setting:

$$ y_{jk} = D+L_j^2+P+2L_j\sqrt{P}\cos(\phi_{jk}-\theta). $$

Now suppose the real apparatus instead has an additional term proportional to LO power, so that the true observations are

$$ y_{jk}^{\mathrm{true}} = D+(1+\eta)L_j^2+P + 2L_j\sqrt{P}\cos(\phi_{jk}-\theta), $$

where $\eta$ is an unknown systematic coefficient. This could represent, abstractly, an LO-dependent contribution that the ideal model has not separated from the microwave coupling. The important point is not the exact physical mechanism, but the mathematical ambiguity.

These observations can be reinterpreted exactly by the ideal model with different effective parameters. Define

$$ P'=\frac{P}{1+\eta}, \qquad L_j'^2=(1+\eta)L_j^2, $$

assuming $1+\eta>0$. Then

$$ 2L_j'\sqrt{P'} = 2\sqrt{(1+\eta)L_j^2}\sqrt{\frac{P}{1+\eta}} = 2L_j\sqrt{P}, $$

so the sinusoidal modulation depth is unchanged. The background term can also be matched by shifting the effective detuning:

$$ D' = D+P-P'. $$

Then

$$ D'+L_j'^2+P' = D+P-P'+(1+\eta)L_j^2+P' = D+(1+\eta)L_j^2+P. $$

Thus the entire dataset can be fitted perfectly by the wrong ideal-model parameters. The residual does not reveal the problem. The protocol is internally consistent, but it returns $P'$ rather than the physical $P$. The inferred signal amplitude is therefore

$$ a'=\sqrt{P'}=\frac{a}{\sqrt{1+\eta}}, $$

not the true amplitude $a$.

This example captures the sentence’s warning in its strongest form. The failure is not caused by noise, insufficient spectra, or numerical instability. It is an exact observational ambiguity caused by a model error. More spectra of the same kind may not fix it, because the incorrect physical effect has the same observational signature as a different combination of $D$, $L_j$, and $P$.

A related example involves phase-law errors. The protocol may assume that commanded LO phases follow a law such as

$$ \phi_{jk}=\phi_{j0}+k\Delta, $$

with a shared increment $\Delta$. If the actual phases follow a different but common distorted law, the two settings may still appear mutually consistent. The recovered phase and amplitude can then be biased while the fit looks coherent. This is why the parent document emphasizes unknown phase increments and phase-reference requirements rather than treating commanded generator phases as automatically equal to phases at the atoms.

What the highlighted sentence contributes to the paper

The highlighted passage is therefore not a rhetorical aside; it states the organizing principle of the paper. The paper’s identifiability results answer questions such as: How many spectra are needed for a unique noiseless inverse? Its Fisher-information and CRB calculations answer: How much precision is lost when LO amplitudes, detuning, or phase increments are nuisance parameters under a specified noise model? Its calibration and ambiguity analysis asks a different question: Even if the inverse is unique and statistically sharp, does the model correspond to the physical field at the atoms?

The examples above show why all three questions are necessary. If the LO settings are too similar, the inversion may be identifiable but unstable. If the assumed noise model is correct but the absolute spectral calibration is wrong, an estimator may be statistically efficient but physically biased. If an LO-dependent detuning term can be absorbed into effective LO amplitudes and signal power, the protocol may fit its own equations exactly while reporting the wrong signal amplitude.

That is the central lesson: self-calibration can remove some nuisance parameters, but it cannot automatically certify every physical assumption that makes the inverse meaningful.

References

[Holloway et al. 2017] C. L. Holloway, M. T. Simons, J. A. Gordon, A. Dienstfrey, D. A. Anderson, and G. Raithel, “Electric field metrology for SI traceability: Systematic measurement uncertainties in electromagnetically induced transparency in atomic vapor,” Journal of Applied Physics, vol. 121, no. 23, Article 233106, 2017.

[Kay 1993] S. M. Kay, Fundamentals of Statistical Signal Processing, Volume I: Estimation Theory. Prentice Hall, 1993.

[Schlossberger et al. 2024] N. Schlossberger et al., “Rydberg states of alkali atoms in atomic vapour as SI-traceable field probes and communications receivers,” Nature Reviews Physics, vol. 6, no. 10, pp. 606–620, 2024.

[Stoica and Marzetta 2001] P. Stoica and T. L. Marzetta, “Parameter estimation problems with singular information matrices,” IEEE Transactions on Signal Processing, vol. 49, no. 1, pp. 87–90, 2001.

[Van Trees et al. 2013] H. L. Van Trees, K. L. Bell, and Z. Tian, Detection, Estimation, and Modulation Theory, Part I, 2nd ed. Wiley, 2013.