Self-Calibrating Rydberg Microwave Reception: Identifiability, Information Loss, and Unknown-Phase-Increment Designs

Mujirin, Nomi Lae, and Hyundong Shin (Kyung Hee University; corresponding author: Hyundong Shin)

Abstract

Coherent Rydberg microwave sensing requires distinguishing an incident signal from an imperfectly characterized local oscillator (LO). This paper studies separation-based reception when the LO amplitudes, common microwave detuning, and phase increments are unknown. Starting from a two-setting moment inversion, we characterize identification requirements and the information loss caused by self-calibration. For uniform phase sampling with homoscedastic squared-separation noise, the signal phase is orthogonal to the LO-amplitude and detuning nuisance parameters, whereas the amplitude penalty depends on the LO-power ratio and the calibrated benchmark. We then establish an eight-spectrum protocol with a shared unknown phase increment and an unknown inter-setting phase offset: five measurements identify the increment at one LO setting, and three additional measurements identify the second setting. Within the class of two consecutive equal-step bursts, eight spectra are necessary and sufficient for identification at every phase and offset. Independent unknown increments require five measurements at each setting for the same guarantee. We also identify observational ambiguities that cannot be detected by a small fitting residual, including LO-power-dependent squared detuning and common phase-law errors. These analytical results separate identifiability, statistical efficiency, and physical calibration, and provide design principles for quasi-static channel sounding and phase-referenced microwave sensing.

Index Terms: Rydberg atomic receiver, microwave sensing, self-calibration, identifiability, Cramér–Rao bound, phase-shifting interferometry, channel sounding.

Introduction

Rydberg atoms provide a microwave-field readout through optical interrogation of highly excited atomic states. Their large transition dipole moments and spectroscopically characterized energy levels make them useful for microwave electrometry and quantum-enabled reception [1, 2, 3, 4]. Demonstrations of amplitude-modulated communication, atomic superheterodyne reception, and phase-sensitive sensing have established that an atomic sensor can provide more than a measurement of incident field strength [5, 6, 7, 8]. Related demonstrations of radio-over-fiber, frequency-shift-keyed, phase-modulated, and embedded-antenna reception [19, 20, 21, 22, 23], recent communication-oriented surveys [24, 25], and sensitivity approaching the standard quantum limit [26] further motivate a careful treatment of reference calibration.

A phase-sensitive receiver combines the signal with a reference field. In an LO-assisted atomic receiver, the interference between the signal and the LO changes the microwave coupling experienced by the atoms. The resulting optical response depends on the relative microwave phase. This mechanism enables coherent measurements, but also introduces nuisance parameters: the LO amplitude at the atoms, the microwave detuning, and the actual phase evolution of the LO.

An externally calibrated LO is not always available. Cable losses, spatial field nonuniformity, vapor-cell interactions, and instrument drift can make the field at the atoms differ from the field inferred from the generator output. Moreover, accurate phase commands at the source do not automatically imply accurate relative phases during a sequence of spectral acquisitions. The distinction between generator settings and the field experienced by the atoms is therefore central to receiver calibration.

Our earlier conference work introduced a two-setting, six-spectrum inversion that eliminates the unknown LO amplitudes and a common squared detuning from an ideal dressed-state separation model [9]. The present paper treats that inversion as a starting point rather than a new result. We ask three further questions:

1. How many spectra are required for identification, both with known phases and with unknown phase increments?

2. What information is lost when the LO amplitudes, detuning, and phase increments are treated as nuisance parameters?

3. Which calibration errors can be detected from the observations, and which are observationally indistinguishable from a different signal?

These questions require separating three concepts that are sometimes conflated. Identifiability concerns uniqueness from noiseless observations. Statistical efficiency concerns uncertainty under a specified observation model. Physical calibration concerns the validity and absolute scale of that model. A protocol can be identifiable but poorly conditioned, statistically efficient but physically biased, or internally consistent while returning the wrong signal amplitude.

Relation to Phase-Shifting Interferometry

At a fixed LO amplitude, the squared-separation model is a sampled sinusoid with an unknown background and modulation depth. Its phase-recovery component is therefore closely related to phase-shifting interferometry [10, 11, 12]. The sinusoidal recurrence used below is classical; we do not claim a new recurrence or a new general-purpose phase-shifting algorithm. Phase-shift-error analyses and unknown-step or randomly stepped algorithms [27, 28, 29, 30, 31, 32, 33] address the one-setting sinusoid, not the coupled two-setting inversion studied here.

The additional structure in the atomic problem is that the background and modulation depth depend jointly on the unknown signal and LO amplitudes: $$A_j=D+L_j^2+P,\qquad B_j=2L_j\sqrt{P}.$$ A second LO setting allows this structure to be inverted. The acquisition-count results in this paper concern that coupled physical model, including an unknown phase offset between the two settings.

Contributions

The main contributions are as follows.

1. Identification and acquisition complexity: We distinguish generic identification from identification at every signal phase. For a shared unknown phase increment and an unknown inter-setting offset, we establish a five-plus-three protocol and prove its eight-spectrum minimality within the class of two consecutive equal-step bursts. We also characterize the exceptional set of the shared-increment four-plus-four design and the ten-spectrum requirement for independent increments.

2. Information loss under explicit benchmarks: We derive closed-form Cramér–Rao bounds (CRBs) under a homoscedastic squared-separation model and distinguish same-data calibration penalties from comparisons that change the acquisition budget. We give a separate Fisher-information formulation for Gaussian errors in extracted separations.

3. Limits of internal calibration: We characterize exact invariances and observational ambiguities. In particular, an LO-power-dependent contribution to squared detuning can be indistinguishable from a different signal amplitude. Common phase-law errors can also preserve inter-setting phase consistency while biasing the recovered signal.

4. Receiver-design implications: We formulate physically qualified LO-design rules, acquisition-budget comparisons, and phase-reference requirements for channel sounding and angle-of-arrival sensing.

This is a theoretical study. It does not claim experimental validation or numerical validation of a particular vapor-cell spectral model. The implementation and validation requirements needed to connect the analytical model to an apparatus are stated explicitly. The figures are noiseless inversions, analytic-Jacobian Cramér–Rao evaluations, and exact-invariance checks of the analytical model, generated by the accompanying code; they are not vapor-cell simulations.

Notation

All microwave frequencies and Rabi frequencies are cyclic frequencies unless otherwise stated. For a complex number $z$, $z^*$ denotes conjugation, $|z|$ its modulus, and $\arg z$ its phase modulo $2\pi$. The operators $\mathop{\mathrm{Re}}$ and $\mathop{\mathrm{Im}}$ denote real and imaginary parts. The signal is $$s=a e^{\mathrm{i}\theta},\qquad a>0,\qquad P=a^2.$$ The squared microwave detuning is $D=\delta^2$. The transpose and Hermitian transpose are denoted by $(\cdot)^\mathsf{T}$ and $(\cdot)^\mathsf{H}$, respectively.

Measurement Model and Scope

Ideal Dressed-State Separation

Consider an isolated microwave transition between two Rydberg states, optically interrogated through an EIT ladder [13, 14]. The signal and the LO couple the same transition through the same polarization component. Their complex cyclic Rabi frequencies are $s$ and $L_j e^{\mathrm{i}\phi_{jk}}$, respectively, where $L_j>0$ is the LO amplitude at setting $j$.

For a monochromatic signal and LO at the same microwave frequency, the total microwave coupling is $$\Omega_{jk}=L_j e^{\mathrm{i}\phi_{jk}}+s.$$ The ideal two-state dressed separation is $$\nu_{jk}=\sqrt{\delta^2+|\Omega_{jk}|^2}.$$ Consequently, $$\begin{aligned} y_{jk} &\equiv \nu_{jk}^2 \nonumber\\ &=D+L_j^2+P+2L_j a\cos(\theta-\phi_{jk}). \end{aligned}$$

For an open ladder, phases of individual atomic coupling matrix elements can be removed by redefining state phases. The relative phase between the signal and the LO nevertheless remains observable because these two fields add before determining the total microwave coupling.

If $d$ is the relevant transition dipole matrix element and $E_s$ is the peak signal-field amplitude in the selected polarization, $$a=\frac{dE_s}{h}, \qquad E_s=\frac{h}{d}a. $$

Assumptions

Unless stated otherwise, the identification results assume:

1. The selected microwave transition is sufficiently isolated for (3) to describe the extracted dressed-state separation.

2. The signal amplitude and the relevant phase trajectory remain consistent with the model throughout a measurement block.

3. The LO amplitude is constant within each setting, and $L_1\ne L_2$.

4. The squared detuning $D$ is common to both settings.

5. The frequency-axis scale and the conversion from the optical readout to $\nu_{jk}$ are specified.

Known-phase protocols additionally require known LO phases relative to a common origin. Unknown-increment protocols replace this requirement by an equal-increment model, a known rotation sense, and the restriction $0<\beta<\pi$.

The interior parameter domain used for regular statistical analysis has $$D>0,\quad a>0,\quad L_j>0,\quad L_1\ne L_2.$$ Noiseless inversion can include $D=0$, but ordinary interior CRB and residual-distribution arguments need not remain valid at that boundary.

Extracted Separation Versus Optical Peak Distance

Equation (3) describes an ideal dressed-state separation. It is not an exact identity for the distance between arbitrarily selected optical peaks.

For a scanned-probe, counter-propagating ladder configuration, a commonly used Doppler conversion relates the optical peak distance to a microwave splitting through a factor involving the probe and coupling wavelengths [1, 2]. Its applicability depends on the spectroscopic regime. Finite linewidth, imperfect peak resolution, optical saturation, Doppler averaging, and detuning-dependent contrast can alter the relation between optical extrema and dressed-state eigenvalues.

Accordingly, $\nu_{jk}$ should be regarded as either:

1. a separation extracted using a validated conversion model; or

2. a physical parameter obtained from an appropriate full-spectrum fit.

Self-calibration of $L_j$ and $D$ does not validate this extraction step or remove its absolute scale uncertainty.

Phase Reference

The recovered $\theta$ is a phase relative to the LO phase at a specified reference acquisition. It is not an absolute electromagnetic phase.

A common phase origin must therefore be preserved or tracked across measurement blocks when the output is used for coherent channel estimation. Similarly, multiple sensors need a known relative LO reference before their phases can be interpreted as a propagation-phase difference.

Known-Phase Identification

Moment Inversion

Let each setting use the three phases $$\phi_k=\frac{2\pi k}{3},\qquad k=0,1,2.$$ Define $$A_j=\frac{1}{3}\sum_{k=0}^{2}y_{jk}, \qquad H_j=\frac{2}{3}\sum_{k=0}^{2}y_{jk}e^{\mathrm{i}\phi_k}. $$ The root-of-unity identities give $$A_j=D+L_j^2+P,\qquad H_j=2L_js. $$

Theorem 1 (Two-setting moment inversion [9]). Suppose (3) holds, $a>0$, $L_j>0$, and $L_1\ne L_2$. Then $$\begin{aligned} P&=\frac{|H_2|^2-|H_1|^2}{4(A_2-A_1)}, \\ \theta&=\arg H_1=\arg H_2, \\ L_j^2&=\frac{|H_j|^2}{4P}, \\ D&=A_j-L_j^2-P. \end{aligned}$$ The signal $s$, the positive LO amplitudes, and $|\delta|$ are uniquely determined. The sign of $\delta$ is not identified by separations alone.

Proof. Let $\Lambda=L_2^2-L_1^2$. Equation (6) gives $$A_2-A_1=\Lambda,\qquad |H_2|^2-|H_1|^2=4P\Lambda.$$ Since $\Lambda\ne0$, their ratio yields $P$. Positivity of $L_j$ gives the phase, and the remaining expressions follow directly. ◻

The inversion is an algebraic elimination of the LO amplitudes and common squared detuning. It does not imply that the recovered $D$ is numerically well conditioned: (10) subtracts quantities that can be much larger than $D$.

Single-Setting Ambiguity

Proposition 2 (Feasible signal powers from one setting). For a single setting with $H\ne0$, the phase is identified as $\theta=\arg H$, while the feasible signal powers satisfy $$P\in[P_-,P_+], \qquad P_\pm=\frac{A\pm\sqrt{A^2-|H|^2}}{2}. $$ For each feasible $P$, $$L^2=\frac{|H|^2}{4P}, \qquad D=A-P-L^2\ge0.$$

Proof. The condition $D\ge0$ is equivalent to $$P^2-AP+\frac{|H|^2}{4}\le0,$$ which yields (11). ◻

The interval is generally nondegenerate. At a special boundary point it can collapse. Thus a second setting is needed for generic amplitude identification, not for every exceptional parameter configuration.

Generic and Phase-Uniform Identification

Definition 3 (Phase-uniform identification). A design is phase-uniformly identifying if its noiseless observations uniquely determine the specified parameters for every admissible signal phase and, when applicable, every admissible inter-setting phase offset.

The definition concerns uniqueness, not numerical conditioning. A phase-uniformly identifying design can still be sensitive to noise near other degeneracies.

Proposition 4 (Known-phase acquisition counts). Consider two positive, unknown LO amplitudes and a common known phase reference.

1. Generic regular identification of $(D,L_1,L_2,a,\theta)$ requires at least five scalar observations.

2. Three distinct known phases at each setting suffice for phase-uniform identification.

3. Among nonadaptive two-setting phase schedules, six observations are necessary for phase-uniform identification of the full parameter vector.

Proof. The first statement follows from the five-dimensional interior parameterization.

For the second statement, three distinct phases give a full-rank regression with columns $1$, $\cos\phi$, and $\sin\phi$. They therefore determine the background and complex harmonic at each setting, after which Theorem 1 applies.

For the third statement, any setting supplies at most three independent noiseless sinusoidal parameters. A five-observation design with a chance of full identification must therefore allocate observations as three-plus-two, up to relabeling. Let the two phases at the second setting be $\phi_a,\phi_b$. At $$\theta=\frac{\phi_a+\phi_b}{2}\pmod{\pi},$$ the two cosine factors are equal. They then supply only one constraint on the second-setting background and modulation amplitude. The first setting determines its own background, modulation amplitude, and phase, but a continuum remains in the second-setting background and modulation amplitude. At an interior physical point, the inverse map (7)–(10) maps nearby members of this continuum to feasible physical parameters. Hence full identification fails at that phase. ◻

Uniform spacing is not the only identifying six-spectrum schedule. Its special advantage is orthogonality and phase-independent moment covariance under homoscedastic squared-separation noise.

LO-Off and Adaptive Alternatives

If the first LO setting is known to be zero, one obtains $$y_{\mathrm{off}}=D+P.$$ A three-phase cycle at a nonzero LO then gives $$P=\frac{|H_2|^2}{4(A_2-y_{\mathrm{off}})}. $$ This is a four-observation design with four unknown parameters. Its practical limitation is that the LO-off spectrum may not provide a resolvable separation for a weak signal.

A noiseless adaptive five-observation design is also possible. A first three-phase cycle identifies $\theta$. Two observations at the second setting with phases $\theta$ and $\theta+\pi$ then give its background and modulation amplitude.

With noisy data, the second-stage phases depend on a noisy first-stage estimate. The resulting sequential experiment must be analyzed as such. A CRB evaluated at phases aligned with the true $\theta$ is an oracle benchmark, not the performance of the implemented adaptive procedure.

Unknown Phase Increments

Shared-Increment Model

Suppose the two LO settings are acquired as consecutive equal-step bursts: $$\phi_{1k}=k\beta, \qquad \phi_{2k}=\chi+k\beta, $$ where $0<\beta<\pi$ and $\chi$ are unknown.

The first acquisition at setting 1 defines the phase origin. The observations become $$y_{jk}=A_j+B_j\cos(\psi_j-k\beta), $$ where $$B_j=2L_ja,\qquad \psi_1=\theta,\qquad \psi_2=\theta-\chi.$$

There are seven real unknowns: $$\boldsymbol{q}=(D,L_1,L_2,a,\theta,\beta,\chi)^\mathsf{T}. $$

The assumption of a shared unknown increment is weaker than knowledge of the phase steps, but stronger than arbitrary unknown phase errors. In particular, it does not cover sample-dependent phase nonlinearity.

Five Samples Identify One Unknown-Step Sinusoid

Lemma 5 (Five-sample sinusoidal identification). Let $$y_k=A+B\cos(\psi-k\beta),$$ where $B>0$ and $0<\beta<\pi$. Five consecutive samples uniquely determine $A$, $B$, $\psi$ modulo $2\pi$, and $\beta$.

Proof. The samples satisfy $$y_{k+1}+y_{k-1}=2cy_k+g, \qquad c=\cos\beta,\quad g=2(1-c)A. $$ For five samples, the three equations indexed by $k=1,2,3$ form a three-by-two linear system for $(c,g)$.

Its two columns are linearly dependent only if $y_1=y_2=y_3$. Equality of the first pair requires $$\psi=\frac{3\beta}{2}\pmod{\pi},$$ while equality of the second pair requires $$\psi=\frac{5\beta}{2}\pmod{\pi}.$$ Together they imply $\beta=0\pmod{\pi}$, which is excluded. Thus the recurrence identifies $c$ and $g$, and $$\beta=\arccos c,\qquad A=\frac{g}{2(1-c)}. $$

Finally, $$y_k-A=p\cos(k\beta)+q\sin(k\beta)$$ has full column rank for $0<\beta<\pi$, giving $$B=\sqrt{p^2+q^2}, \qquad \psi=\mathop{\mathrm{atan2}}(q,p).$$ ◻

The recurrence is related to classical self-calibrating phase-shifting and sinusoidal parameter estimation [11, 12, 15]. The result is used here to establish the acquisition requirements of the coupled two-setting physical model.

Eight-Spectrum Shared-Increment Protocol

Theorem 6 (Five-plus-three identification). Under (14), assume $B_j>0$, $0<\beta<\pi$, and $L_1\ne L_2$. Five consecutive samples at setting 1 and three consecutive samples at setting 2 identify $\boldsymbol{q}$ in (16), modulo phase wrapping. The sign of $\delta$ remains unidentified.

Proof. Lemma 5 determines $(A_1,B_1,\psi_1,\beta)$.

With $\beta$ known, the three samples at setting 2 satisfy $$\begin{bmatrix} 1&1&0\\ 1&\cos\beta&\sin\beta\\ 1&\cos2\beta&\sin2\beta \end{bmatrix} \begin{bmatrix} A_2\\p_2\\q_2 \end{bmatrix} = \begin{bmatrix} y_{20}\\y_{21}\\y_{22} \end{bmatrix}. $$ The determinant of the matrix is $$2\sin\beta(1-\cos\beta)>0.$$ Hence $A_2,p_2,q_2$ are uniquely determined. Set $$B_2=\sqrt{p_2^2+q_2^2}, \qquad \psi_2=\mathop{\mathrm{atan2}}(q_2,p_2).$$ The physical parameters follow from $$\begin{aligned} P&=\frac{B_2^2-B_1^2}{4(A_2-A_1)}, \\ L_j^2&=\frac{B_j^2}{4P}, \qquad D=A_1-L_1^2-P, \\ \theta&=\psi_1, \qquad \chi=\psi_1-\psi_2\pmod{2\pi}. \end{aligned}$$ ◻

The protocol uses eight spectra, rather than requiring five spectra at both settings. The additional two spectra in a five-plus-five design provide redundancy and may improve conditioning, but they are not necessary for identification when the increment is shared.

Minimality Within Consecutive Equal-Step Bursts

Theorem 7 (Restricted eight-spectrum minimality). Consider two positive, unknown LO amplitudes, an unknown common increment $0<\beta<\pi$, and an unknown inter-setting offset. Within the class of two consecutive equal-step bursts, eight spectra are necessary and sufficient for identification of the full interior parameter vector at every signal phase and offset.

Proof. Sufficiency follows from Theorem 6.

The parameter vector has seven real components, so fewer than seven observations cannot provide generic regular identification.

At least three observations are needed at each setting. Indeed, even if the first setting and the shared increment were fully characterized, the other setting would generally retain three independent sinusoidal parameters: its background, modulation amplitude, and phase. The physical inverse (23)–(25) does not remove this local freedom at an interior point.

A seven-observation design must therefore use four-plus-three samples, up to relabeling. For the four-sample setting, consider $$\psi=\frac{3\beta}{2}\pmod{\pi}.$$ The samples then have the symmetric form $$(y_0,y_1,y_2,y_3)=(u,v,v,u). $$ For any sufficiently close $\beta'$ in $(0,\pi)$, choose $\psi'=3\beta'/2$ with the appropriate addition of $\pi$. The two distinct values in (26) determine new $A'$ and $B'$ because $$\cos(3\beta'/2)-\cos(\beta'/2) =-2\sin\beta'\sin(\beta'/2)\ne0.$$ The other setting’s three samples can be fitted for each nearby $\beta'$ by (21).

At an interior point, positivity of $B_j$, $P$, $D$, and $L_j$ and the nonzero LO gap are preserved under sufficiently small perturbations. Thus a continuum of physical parameter vectors produces the same seven observations. Identification fails at the symmetric phase. ◻

The theorem is not a universal lower bound over every possible atomic measurement protocol. It applies to the specified two-setting separation model and consecutive equal-step sampling structure. Additional observables, calibrated parameters, adaptive sampling, or external references can change the acquisition requirement.

The Shared-Increment Four-Plus-Four Design

Proposition 8 (Exceptional set of four-plus-four sampling). For four samples at each setting with a shared increment, the parameters are identified whenever at least one setting satisfies $y_{j1}\ne y_{j2}$.

Both settings are simultaneously singular only when $$\psi_1=\frac{3\beta}{2}\pmod{\pi}, \qquad \psi_2=\frac{3\beta}{2}\pmod{\pi}. $$ Consequently, for a fixed offset $\chi\not\equiv0\pmod{\pi}$, the four-plus-four design identifies the parameters at every signal phase. It does not provide the same guarantee over arbitrary unknown offsets.

Proof. Subtracting the two recurrence equations available from four samples gives $$2c(y_1-y_2)=y_0-y_3+y_2-y_1.$$ If $y_1\ne y_2$, this identifies $c=\cos\beta$, after which three samples suffice to fit each setting.

For $0<\beta<\pi$, the condition $y_1=y_2$ is equivalent to $\psi=3\beta/2\pmod{\pi}$. Simultaneous singularity therefore requires (27), which implies $\chi=\psi_1-\psi_2=0\pmod{\pi}$. ◻

Independent Unknown Increments

Suppose instead that $$\phi_{1k}=k\beta_1,\qquad \phi_{2k}=\chi+k\beta_2,$$ where $0<\beta_j<\pi$ are independent unknowns.

Corollary 9 (Independent-increment acquisition count). Five samples at each setting suffice for phase-uniform identification with independent unknown increments. Within the class of two consecutive equal-step bursts, five samples per setting are also necessary for identification at every phase and offset.

Proof. Sufficiency follows by applying Lemma 5 separately at each setting and then using (23)–(25).

A setting with at most three samples cannot identify its four sinusoidal parameters even generically. A setting with four samples has the continuum ambiguity (26). Because the increments are independent, observations at the other setting cannot determine that increment. The physical inverse maps a sufficiently small ambiguity at an interior point to a feasible physical ambiguity. Hence each setting requires at least five samples for a phase-uniform guarantee. ◻

Identification guarantees for two-setting protocols

Identification Does Not Imply Uniform Conditioning

The inverse in Lemma 5 becomes sensitive as $\beta$ approaches $0$ or $\pi$. Similarly, the physical inversion becomes sensitive as $L_2^2-L_1^2$ approaches zero or as the signal amplitude approaches zero.

For the five-plus-three design, the noiseless inverse is smooth locally at every interior admissible point, after choosing local phase coordinates. Therefore the measurement Jacobian has full column rank. On a compact parameter set bounded away from the foregoing degeneracies, its smallest singular value, under a fixed parameter scaling, has a positive lower bound. This is a conditioning statement on a restricted set, not a claim of noise immunity over the entire parameter domain. Figure 1 illustrates this for the example of Section XI with the corrected recurrence: the noiseless inversion is exact to numerical precision over most of $(0,\pi)$ and degrades toward the endpoints.

Noise Models and Estimation

Gaussian Separation Errors

A common high-SNR approximation for an extracted separation is $$\widehat\nu_{jk}=\nu_{jk}+n_{jk}, \qquad n_{jk}\sim\mathcal N(0,\sigma_\nu^2), $$ independently across observations.

The approximation concerns the output of a spectral extraction procedure. It is not a fundamental atomic-noise model. In an apparatus, the variance can depend on linewidth, contrast, optical power, microwave coupling, and acquisition time.

Let $\widehat y=\widehat\nu^2$. Then $$\begin{aligned} \mathbb{E}[\widehat y]&=y+\sigma_\nu^2, \\ \mathop{\mathrm{Var}}(\widehat y)&=4\sigma_\nu^2y+2\sigma_\nu^4. \end{aligned}$$ An unbiased squared-separation estimate is $$\widehat y_{\mathrm{db}} =\widehat\nu^2-\sigma_\nu^2. $$

For $K$ uniform phases, define $\widehat H_j$ using the factor $2/K$. An unbiased estimator of $\mathop{\mathrm{Var}}(\widehat y_{jk})$ under (30) is $$\widehat v_{jk} =4\sigma_\nu^2\widehat\nu_{jk}^2-2\sigma_\nu^4.$$ The harmonic-power moment can therefore be corrected as $$\widehat{|H_j|^2}_{\mathrm{db}} = |\widehat H_j|^2 -\frac{4}{K^2}\sum_k\widehat v_{jk}. $$

Correcting these moments does not make the ratio (7) unbiased. Moreover, unbiased moment corrections can yield negative intermediate values. Such outcomes require a feasibility flag or a specified constrained estimator, rather than silent clipping.

Homoscedastic Squared-Separation Model

For analytical insight, consider the separate surrogate model $$\widehat y_{jk}=y_{jk}+\epsilon_{jk}, \qquad \epsilon_{jk}\sim\mathcal N(0,\sigma_j^2), $$ independently, with variance constant within setting $j$.

This model is not equivalent to (30). It provides exact Fisher-information expressions for a useful idealized observation model.

For $K\ge3$ uniform phases, $$\begin{aligned} \mathop{\mathrm{Var}}(\widehat A_j)&=\frac{\sigma_j^2}{K},\\ \mathop{\mathrm{Var}}(\mathop{\mathrm{Re}}\widehat H_j) &=\mathop{\mathrm{Var}}(\mathop{\mathrm{Im}}\widehat H_j) =\frac{2\sigma_j^2}{K}. \end{aligned}$$ All covariances among these three components vanish.

The local inverse-variance phase combination is $$\widehat\theta = \arg\left( \sum_{j=1}^{2} \frac{|\widehat H_j|\widehat H_j}{\sigma_j^2} \right). $$ The factors $1/\sigma_j^2$ cannot be omitted unless the within-setting variances are equal.

Likelihood Estimation

Under independent Gaussian separation errors with specified standard deviations $\sigma_{jk}$, a constrained likelihood estimator minimizes $$\widehat{\boldsymbol{q}} \in \arg\min_{\boldsymbol{q}\in\mathcal Q} \sum_{j,k} \frac{ \left[\widehat\nu_{jk} -\nu_{jk}(\boldsymbol{q})\right]^2 }{\sigma_{jk}^2}, $$ where $\mathcal Q$ enforces the physical constraints and the appropriate known- or unknown-phase model.

For unknown increments, the recurrence solution is an initializer rather than a likelihood estimator. In noisy data, both the response and the regressor in (17) contain errors, so ordinary least squares on that recurrence is an errors-in-variables procedure.

Reliable implementation should:

1. use the algebraic inversion when feasible;

2. employ observation-based multistart initialization when it is not;

3. retain optimizer convergence and boundary flags;

4. report initialization and fitting failures;

5. never use the true simulation parameters as a fallback initialization.

Local Versus Global Error Measures

The denominator in (7) can approach zero under noise. The unconstrained ratio can therefore develop large outliers, and local variance expressions do not by themselves establish finite global moments.

A numerical evaluation should report separately:

1. the probability of an invalid or ill-conditioned algebraic reconstruction;

2. the conditional error given a valid reconstruction;

3. the unconditional performance of a fully specified operational estimator, including its fallback rule.

An RMSE computed after discarding invalid estimates is a conditional RMSE and must be labeled accordingly.

Information Limits and Calibration Cost

Closed-Form Bounds Under Homoscedastic Noise

Theorem 10 (Local information limits). Assume (36), $K\ge3$ uniform phases at each setting, and an interior physical parameter point. Let $\Lambda=L_2^2-L_1^2$. When $D,L_1,L_2$ are unknown, $$\begin{aligned} \mathop{\mathrm{CRB}}_{\mathrm{sc}}(P) &= \frac{ P\left[ \sigma_1^2(2L_1^2+P) +\sigma_2^2(2L_2^2+P) \right] }{ K\Lambda^2 }, \\ \mathop{\mathrm{CRB}}_{\mathrm{sc}}(\theta) &= \frac{1}{ 2KP\left( L_1^2/\sigma_1^2+L_2^2/\sigma_2^2 \right) }. \end{aligned}$$ The first-order covariance of the moment inversion and (39) attains these bounds.

Proof. A derivation is given in Appendix A. ◻

The theorem is a Fisher-information statement for (36). Attainment by the algebraic estimator is local and high-SNR; it is not an exact finite-noise variance identity for a ratio estimator.

A Same-Data Calibrated Benchmark

Suppose the LO amplitudes are known but $D$ is unknown, and the calibrated receiver uses the same $2K$ observations.

Corollary 11 (Same-data calibration penalty). Under the assumptions of Theorem 10, $$\begin{aligned} \mathop{\mathrm{CRB}}_{\mathrm{cal}}(P) &= \frac{2P}{ K\left( L_1^2/\sigma_1^2+L_2^2/\sigma_2^2 \right) }, \\ \mathop{\mathrm{CRB}}_{\mathrm{cal}}(\theta) &=\mathop{\mathrm{CRB}}_{\mathrm{sc}}(\theta). \end{aligned}$$ For $\sigma_1^2=\sigma_2^2=\sigma_y^2$, define $$\rho=\frac{L_1^2}{L_2^2}, \qquad \epsilon=\frac{P}{L_2^2}.$$ Then $$\frac{\mathop{\mathrm{CRB}}_{\mathrm{sc}}(P)} {\mathop{\mathrm{CRB}}_{\mathrm{cal}}(P)} = \frac{(1+\rho)(1+\rho+\epsilon)} {(1-\rho)^2}. $$

Proof. When $L_j$ are known, the means $A_j=D+L_j^2+P$ cannot separate changes in $P$ from changes in the unknown $D$. After eliminating $D$, the means provide no additional information about $P$.

The modulation amplitudes satisfy $B_j=2L_j\sqrt P$, with local variance $2\sigma_j^2/K$. Their combined information about $P$ is $$\sum_j \frac{(L_j/\sqrt P)^2}{2\sigma_j^2/K} = \frac{K}{2P} \sum_j\frac{L_j^2}{\sigma_j^2},$$ which yields (43).

Uniform phase sampling makes the tangential phase direction orthogonal to the background and radial directions, giving (44). Taking the ratio for equal variances gives (45). ◻

For amplitude $a=\sqrt P$, $$\mathop{\mathrm{CRB}}(a)=\frac{\mathop{\mathrm{CRB}}(P)}{4P}.$$ Thus the variance penalty is unchanged, while the standard-deviation penalty is its square root.

Equation (44) is not a universal statement about atomic receivers. It relies on uniform sampling and the homoscedastic squared-separation model.

Different Benchmarks Give Different Ratios

If the calibrated receiver uses only the second setting, its modulation-based bound is $$\mathop{\mathrm{CRB}}_{\mathrm{cal},2}(P) =\frac{2P\sigma_2^2}{KL_2^2}.$$ For equal variances, $$\frac{\mathop{\mathrm{CRB}}_{\mathrm{sc}}(P)} {\mathop{\mathrm{CRB}}_{\mathrm{cal},2}(P)} = \frac{1+\rho+\epsilon}{(1-\rho)^2}. $$

Equations (45) and (48) answer different questions. The latter compares different observation sets and must not be presented as a same-data information loss.

A receiver that also knows $D$ has a further advantage: the phase averages then contain information about $P$. That benchmark must be labeled separately.

Fisher Information for Gaussian Separation Errors

Let $$t_{jk}=\theta-\phi_{jk}.$$ For the known-phase parameterization $$\boldsymbol{x}=(D,L_1,L_2,a,\theta)^\mathsf{T},$$ the nonzero derivatives of $y_{jk}$ are $$\begin{aligned} \frac{\partial y_{jk}}{\partial D}&=1,\\ \frac{\partial y_{jk}}{\partial L_j} &=2L_j+2a\cos t_{jk},\\ \frac{\partial y_{jk}}{\partial a} &=2a+2L_j\cos t_{jk},\\ \frac{\partial y_{jk}}{\partial\theta} &=-2L_ja\sin t_{jk}. \end{aligned}$$

For the shared-increment model, $$\begin{aligned} \frac{\partial y_{jk}}{\partial\beta} &=2L_ja k\sin t_{jk},\\ \frac{\partial y_{jk}}{\partial\chi} &= \begin{cases} 0,&j=1,\\ 2L_2a\sin t_{2k},&j=2. \end{cases} \end{aligned}$$ Since $\nu_{jk}=\sqrt{y_{jk}}$, $$\frac{\partial\nu_{jk}}{\partial\boldsymbol{q}} = \frac{1}{2\nu_{jk}} \frac{\partial y_{jk}}{\partial\boldsymbol{q}}.$$

For specified independent Gaussian variances, $$\mathbf{F} = \sum_{j,k} \frac{1}{4\sigma_{jk}^2y_{jk}} \left( \frac{\partial y_{jk}}{\partial\boldsymbol{q}} \right) \left( \frac{\partial y_{jk}}{\partial\boldsymbol{q}} \right)^\mathsf{T}. $$

If the observation covariance itself depends on unknown parameters, its derivatives contribute additional Fisher information. Equation (56) treats the specified variances as fixed.

Unlike (36), the weights in (56) depend on phase through $y_{jk}$. Consequently, exact phase orthogonality and phase-independent bounds do not follow from uniform sampling alone.

The Cost of Unknown Phase Increments

Partition the parameters into quantities of interest $\boldsymbol{\eta}$ and nuisance parameters $\boldsymbol{\zeta}$: $$\mathbf{F}= \begin{bmatrix} \mathbf{F}_{\eta\eta}&\mathbf{F}_{\eta\zeta}\\ \mathbf{F}_{\zeta\eta}&\mathbf{F}_{\zeta\zeta} \end{bmatrix}.$$ At a regular point, the efficient information is $$\mathbf{F}_{\mathrm{eff}} = \mathbf{F}_{\eta\eta} - \mathbf{F}_{\eta\zeta} \mathbf{F}_{\zeta\zeta}^{-1} \mathbf{F}_{\zeta\eta}. $$

The fact that $A_j$ and $B_j$ do not explicitly contain $\beta$ does not make their estimates orthogonal to the estimate of $\beta$. Unknown increments and offsets can therefore reduce both amplitude and phase information.

A comparison should separately evaluate:

1. known increment and known offset;

2. known increment and unknown offset;

3. shared unknown increment and unknown offset;

4. independent unknown increments and unknown offset.

No universal numerical amplitude or phase penalty follows from the identification theorem. Figure 2 nevertheless evaluates the Gaussian-separation bound (56) for one parameter point as a numerical instance of these comparisons. It is an example, not a universal penalty; the benchmark definitions and acquisition counts differ across curves and must be kept in mind when reading it.

Near rank loss, the singular values of the whitened Jacobian should be examined rather than directly inverting a poorly conditioned normal matrix. A singular full information matrix does not imply that every parameter of interest is unidentifiable [16].

Heteroscedastic Delta-Method Expression

For the three-plus-three inversion, let $t_1=-1$ and $t_2=1$. At noiseless moments, $$\frac{\partial P}{\partial y_{jk}} = \frac{t_j}{3\Lambda} \left[ \mathop{\mathrm{Re}}\!\left(H_j^*e^{\mathrm{i}\phi_k}\right)-P \right]. $$ Under (30), the corresponding local variance is $$\mathop{\mathrm{Var}}(\widehat P) \approx \sum_{j,k} \left( \frac{\partial P}{\partial y_{jk}} \right)^2 \left( 4\sigma_\nu^2y_{jk}+2\sigma_\nu^4 \right). $$

This retains the phase-dependent squared-separation variances. It is preferable to substituting a cycle-averaged variance into a homoscedastic expression when evaluating the Gaussian-separation model.

LO and Acquisition Design

A Qualified Corner Rule

Under equal, LO-independent squared-separation variance, (41) becomes $$\mathop{\mathrm{CRB}}_{\mathrm{sc}}(P) = \frac{2P\sigma_y^2}{K} \frac{L_1^2+L_2^2+P}{(L_2^2-L_1^2)^2}. $$

Proposition 12 (LO separation under fixed squared-noise variance). Assume $0<L_1<L_2$, fixed $P$, fixed $K$, and LO-independent equal squared-separation noise variance. Then (60) increases with $L_1^2$ and decreases with $L_2^2$.

Under independent box constraints on the two LO powers, the optimum is therefore the smallest admissible $L_1$ and the largest admissible $L_2$.

Proof. Set $x=L_1^2$, $z=L_2^2$, and $$f(x,z)=\frac{x+z+P}{(z-x)^2}.$$ Then $$\begin{aligned} \frac{\partial f}{\partial x} &=\frac{x+3z+2P}{(z-x)^3}>0,\\ \frac{\partial f}{\partial z} &=-\frac{3x+z+2P}{(z-x)^3}<0. \end{aligned}$$ ◻

The proposition does not establish a universal physical optimum. Its assumptions exclude LO-dependent linewidth, contrast, fitting variance, and model error.

Resolution and Model-Validity Constraints

The coupling experienced by the atoms is $$|\Omega|=|Le^{\mathrm{i}\phi}+s|,$$ not $L$ alone. For $L\ge a$, $$\min_\phi |Le^{\mathrm{i}\phi}+s|=L-a.$$ Thus a phase-uniform coupling threshold $|\Omega|\ge\Omega_{\min}$ can be enforced conservatively by $$L\ge a_{\max}+\Omega_{\min}, $$ if a valid upper bound $a_{\max}$ is available.

Even (64) is only a coupling constraint. Actual peak resolvability also depends on detuning, optical parameters, and contrast. An apparatus-specific resolution criterion should therefore be determined from a validated spectral model or measured data.

The strong LO is likewise limited by off-resonant coupling, power-dependent shifts, spatial nonuniformity, and departure from the isolated-transition model. The appropriate practical design minimizes a criterion that includes variance, model-mismatch bias, and failure probability.

Fixed-Acquisition-Budget Comparisons

Comparing protocols at equal per-spectrum uncertainty is useful for understanding information structure, but it does not compare equal total resources.

Let $$T_{\mathrm{block}} = \sum_{j,k} \left(T_{jk}^{\mathrm{scan}}+T_{jk}^{\mathrm{settle}}\right) + T^{\mathrm{switch}}. $$ Under a white-noise-limited approximation, $$\sigma_{\nu,jk}^2 = \frac{c_{jk}}{T_{jk}^{\mathrm{scan}}}, $$ where $c_{jk}$ can depend on spectral shape.

A fair protocol comparison should fix $T_{\mathrm{block}}$ and account for settling and switching. The eight-spectrum protocol need not have lower error than a ten-spectrum protocol at equal per-spectrum variance, while the ordering can change under a fixed total time.

A general design objective is $$\min_{\mathcal D} \ \sup_{\boldsymbol{q}\in\mathcal Q_0} \operatorname{tr}\!\left[ \mathbf{W}\, \mathbf{C}_{\eta}(\boldsymbol{q};\mathcal D) \right], $$ subject to acquisition-time, resolution, and model-validity constraints. Here $\mathcal D$ specifies the LO settings, sample counts, timing, and phase-increment policy, and $\mathbf{W}$ weights the parameters relevant to the application.

Robustness and Undetectable Calibration Errors

Setting-Dependent Squared Detuning

Proposition 13 (Bias under unequal squared detunings). Suppose the true squared detunings are $D_1,D_2$ and the data otherwise satisfy the known-phase model. Let $$\Delta D=D_2-D_1,\qquad \Lambda=L_2^2-L_1^2.$$ Applying the common-detuning inversion gives $$\widehat P = P\frac{\Lambda}{\Lambda+\Delta D}, $$ whenever the denominator is nonzero. The individual harmonic phases remain equal to $\theta$.

Proof. The harmonic powers remain $|H_j|^2=4L_j^2P$, while $$A_2-A_1=\Lambda+\Delta D.$$ Substitution into (7) gives the result. ◻

If the reconstructed parameters are physically feasible, these observations are fitted exactly by a common-detuning model with a different signal amplitude. An internal least-squares residual therefore need not detect the fault.

An Observational Ambiguity for Any Number of Settings

Theorem 14 (Affine squared-detuning ambiguity). Suppose, for any number of LO settings, $$D_j=D_0+\eta L_j^2, \qquad 1+\eta>0. $$ Define $$P'=\frac{P}{1+\eta}, \qquad L_j'=\sqrt{1+\eta}\,L_j, \qquad D'=D_0+P-P'. $$ If the true and transformed parameters are physically admissible, the observations are identical to those of a common-detuning model with parameters $(D',L_j',P',\theta)$.

Proof. Under (69), $$A_j=D_0+P+(1+\eta)L_j^2, \qquad B_j=2L_j\sqrt P.$$ The transformed parameters satisfy $$D'+P'+(L_j')^2=A_j, \qquad 2L_j'\sqrt{P'}=B_j.$$ Thus the entire sinusoidal observation at every setting is unchanged. ◻

Equation (69) is a specified phenomenological law for squared detuning. It is not asserted to be an exact description of every power-dependent atomic shift. It nevertheless establishes a precise nonidentifiability within a plausible enlarged model class.

An external constraint is needed to distinguish the two parameterizations. Possible sources include an independently characterized shift law, another atomic transition, or spectral observables beyond separation.

For three or more settings, the common-detuning model implies $$B_j^2=4P A_j-4P(D+P). $$ Departure from this line can reveal some forms of model mismatch. The ambiguity in Theorem 14 preserves the line, so additional LO settings do not universally resolve calibration ambiguity. Figure 3 verifies this numerically for three settings.

Common Affine Distortion of Squared Separation

Proposition 15 (Affine readout invariance). Suppose all squared separations undergo the same transformation $$\widetilde y_{jk}=b+\gamma y_{jk}, \qquad \gamma>0. $$ Then the inversion returns $$\widetilde P=\gamma P,\qquad \widetilde L_j^2=\gamma L_j^2,\qquad \widetilde D=b+\gamma D, $$ and leaves phase unchanged.

Proof. Uniform phase moments satisfy $$\widetilde A_j=b+\gamma A_j,\qquad \widetilde H_j=\gamma H_j.$$ Substitution into the inversion gives (73). ◻

A common additive error in squared separation therefore leaves the signal amplitude unchanged while biasing the inferred squared detuning. A common multiplicative scale error does not cancel.

This distinction is relevant when evaluating peak-extraction errors. An empirical approximation $\widetilde y=b+\gamma y+r$ should be assessed by measuring the nonaffine residual $r$, rather than assuming that any spectral distortion is harmless.

Inter-Setting Phase Offset

If the actual phases at setting 2 are $\phi_k+\chi$, then $$H_2=2L_2s e^{-\mathrm{i}\chi}.$$ The amplitude inversion remains unchanged because it uses $|H_2|$, but the second-setting phase is $\theta-\chi$.

A naive equal-noise combined phase estimate satisfies, in the noiseless limit, $$\widehat\theta = \theta+ \arg\!\left(L_1^2+L_2^2e^{-\mathrm{i}\chi}\right). $$ Thus amplitude invariance under an offset does not imply invariance of the combined phase estimate.

The unknown-offset models in Section IV explicitly estimate $\chi$ instead of combining inconsistent phase references.

Common Phase-Law Errors Can Evade Phase Consistency

Suppose the receiver processes three samples using nominal phases $\phi_k=2\pi k/3$, but the actual phases are $\alpha_k$, identical across the two LO settings. Define $$\begin{aligned} F&=\frac{1}{3}\sum_{k=0}^{2}e^{\mathrm{i}\alpha_k},\\ C_-&=\frac{1}{3}\sum_{k=0}^{2} e^{\mathrm{i}(\phi_k-\alpha_k)},\\ C_+&=\frac{1}{3}\sum_{k=0}^{2} e^{\mathrm{i}(\phi_k+\alpha_k)},\\ z&=sC_-+s^*C_+. \end{aligned}$$

Proposition 16 (Common phase-law distortion). Processing the observations with nominal phases gives $$\begin{aligned} \widetilde A_j &=D+L_j^2+P+2L_j\mathop{\mathrm{Re}}(s^*F), \\ \widetilde H_j&=2L_jz. \end{aligned}$$ If the inversion is defined, its signal-power output is $$\widetilde P = \frac{|z|^2}{ 1+ 2\mathop{\mathrm{Re}}(s^*F)/(L_1+L_2) }, $$ and its harmonic phase is $\arg z$. Nevertheless, $\widetilde H_2/\widetilde H_1=L_2/L_1$ whenever $z\ne0$.

Proof. Write the true observations as $$y_{jk} = D+L_j^2+P+ L_js e^{-\mathrm{i}\alpha_k} + L_js^*e^{\mathrm{i}\alpha_k}.$$ Averaging and applying the nominal first-harmonic transform gives (80) and (81). Their differences satisfy $$\widetilde A_2-\widetilde A_1 = (L_2-L_1) \left[L_1+L_2+2\mathop{\mathrm{Re}}(s^*F)\right],$$ and $$|\widetilde H_2|^2-|\widetilde H_1|^2 = 4(L_2^2-L_1^2)|z|^2.$$ The inversion yields (82). ◻

A phase-shifter scale error corresponds to $\alpha_k=(1+\varepsilon)\phi_k$. It is therefore a special case of Proposition 16.

The proposition shows that inter-setting phase agreement is not a general test of phase-step accuracy. If the reconstructed physical parameters remain feasible, a common phase-law error can preserve the exact six-spectrum model while changing the recovered signal.

An unknown-increment protocol corrects the error only when the actual phases satisfy its equal-increment model. It does not correct arbitrary $\alpha_k$.

Consistency Tests and Their Limits

For a model with $M$ independent Gaussian observations and $p$ locally identifiable interior parameters, define $$T= \min_{\boldsymbol{q}\in\mathcal Q} \sum_{j,k} \frac{ [\widehat\nu_{jk}-\nu_{jk}(\boldsymbol{q})]^2 }{\sigma_{jk}^2}. $$ Under a correct model and in a regular high-SNR regime, linearization gives $$T\ \dot{\sim}\ \chi^2_{M-p}. $$

Examples include:

• known-phase three-plus-three: $M-p=6-5=1$;

• shared-increment five-plus-three: $M-p=8-7=1$;

• shared-increment five-plus-five: $M-p=10-7=3$;

• independent-increment five-plus-five: $M-p=10-8=2$.

These degrees of freedom apply at regular points. Active constraints, weak signals, coincident LO settings, singular phase configurations, or inaccurate noise variances can invalidate the approximation.

A residual test evaluates compatibility with the assumed observation manifold. It does not establish that the fitted physical parameters are correct. In particular, the following can escape detection:

1. setting-dependent detuning that maps to another feasible common-detuning parameter vector;

2. the affine squared-detuning ambiguity of Theorem 14;

3. common affine readout errors;

4. common phase-law errors described by Proposition 16.

Conversely, a large residual indicates incompatibility but does not uniquely identify its physical cause. Possible causes include phase drift, unequal increments, signal evolution, spectral-extraction failure, and violation of the isolated transition model.

A receiver should therefore report both:

1. an internal consistency statistic; and

2. the external calibration assumptions that the statistic cannot test.

Phase-Referenced Communication and Sensing

Pilot-Based Channel Estimation

Let a known pilot $x$ produce a local microwave signal $$s=g h x, $$ where $h$ is the channel coefficient and $g$ collects known field-to-Rabi and normalization factors. Then $$\widehat h=\frac{\widehat s}{g x}.$$

For small amplitude and phase errors, $$\widehat s-s \approx e^{\mathrm{i}\theta} \left[(\widehat a-a)+\mathrm{i}a(\widehat\theta-\theta)\right].$$ Consequently, $$\frac{\mathbb{E}|\widehat s-s|^2}{a^2} \approx \frac{\mathbb{E}(\widehat a-a)^2}{a^2} + \mathbb{E}(\widehat\theta-\theta)^2. $$ The right-hand side includes both variance and squared bias. An internally consistent calibration error can therefore dominate channel-estimation NMSE even when the estimator approaches its model-based CRB.

The phase in (85) must be referenced consistently across pilot blocks. Estimating an unknown within-block increment does not independently establish the LO phase relation between different blocks.

Angle-of-Arrival Estimation

For two sensors with spacing $d_{\mathrm{s}}$ and a plane wave of wavelength $\lambda$, a calibrated differential phase satisfies $$\Delta\varphi = 2\pi\frac{d_{\mathrm{s}}}{\lambda}\sin\alpha. $$ Let the two phase errors have variances $V_1,V_2$ and covariance $C_{12}$. Local propagation gives $$\mathop{\mathrm{Var}}(\widehat\alpha) \approx \frac{ V_1+V_2-2C_{12} }{ \left[ 2\pi(d_{\mathrm{s}}/\lambda)\cos\alpha \right]^2 }. $$

Independent phase-error assumptions are not automatic when the sensors share an LO. Common reference noise can cancel, while unequal reference paths can introduce a systematic differential phase.

Equation (89) is a local bound. Phase wrapping and spatial aliasing remain relevant; increasing the spacing beyond $\lambda/2$ can introduce angular ambiguities without additional information.

Stationarity and Acquisition Rate

The protocol estimates a quasi-static complex field over $T_{\mathrm{block}}$. Its usefulness therefore depends on the channel and source remaining sufficiently stable during that interval.

A nominal complex-sample rate of $1/T_{\mathrm{block}}$ is an acquisition rate, not a receiver communication bandwidth. The protocol is naturally suited to pilot tones, channel sounding, calibration signals, and slowly varying field maps.

If a free-running signal–LO frequency difference $\Delta\omega$ generates the phase increment, the phase changes during a spectral scan as well as between scans. The instantaneous-spectrum approximation requires $$|\Delta\omega|T_{\mathrm{scan}}\ll1. $$ A substantial phase increment between acquisitions is compatible with (90) only if the timing or measurement mechanism supports the required separation of time scales. Otherwise a time-dependent optical response model is necessary.

Analytical Example and Reproducibility Requirements

An Exact Algebraic Example

Consider $$s=0.6+0.8\mathrm{i}\ \,\mathrm{MHz},\quad L_1=4\ \,\mathrm{MHz},\quad L_2=7\ \,\mathrm{MHz},\quad \delta=2\ \,\mathrm{MHz}.$$ Then $P=1\ \mathrm{MHz}^2$ and $D=4\ \mathrm{MHz}^2$. For the uniform three-phase design, the squared separations are $$\begin{aligned} \boldsymbol{y}_1 &= \left( 25.8,\, 18.6+3.2\sqrt3,\, 18.6-3.2\sqrt3 \right)^\mathsf{T},\\ \boldsymbol{y}_2 &= \left( 62.4,\, 49.8+5.6\sqrt3,\, 49.8-5.6\sqrt3 \right)^\mathsf{T}, \end{aligned}$$ in $\mathrm{MHz}^2$.

The moments are $$A_1=21,\quad A_2=54,\quad H_1=8s,\quad H_2=14s.$$ Hence $$\widehat P = \frac{196-64}{4(54-21)} =1,$$ and the remaining parameters follow exactly.

For the homoscedastic squared-separation model, $$\rho=\frac{16}{49},\qquad \epsilon=\frac{1}{49}.$$ The same-data variance penalty is $$\frac{\mathop{\mathrm{CRB}}_{\mathrm{sc}}(P)} {\mathop{\mathrm{CRB}}_{\mathrm{cal}}(P)} = \frac{130}{33},$$ whereas comparison against the calibrated second cycle alone gives $$\frac{\mathop{\mathrm{CRB}}_{\mathrm{sc}}(P)} {\mathop{\mathrm{CRB}}_{\mathrm{cal},2}(P)} = \frac{98}{33}.$$ These exact ratios illustrate why benchmark definitions matter. They are not performance claims for a vapor-cell experiment.

Required Software Tests

A reproducible implementation should include at least the following tests:

1. Exact recovery of the known-phase moment inversion over randomly generated admissible noiseless parameters.

2. Exact recovery of a five-sample unknown-step sinusoid, including phases near the four-sample symmetric configuration.

3. Exact recovery of the shared-increment five-plus-three protocol over phase and offset sweeps.

4. Explicit demonstration of the four-plus-three ambiguity and the four-plus-four exceptional set.

5. Agreement between analytic Jacobians and finite-difference derivatives away from boundaries.

6. Verification of the transformations in Theorem 14 and Propositions 15 and 16.

7. Correct handling of negative moment corrections, near-zero LO gaps, optimizer failure, and active physical constraints.

The recurrence regression must use rows $(2y_k,1)$ for the unknowns $(c,g)$ in (17). Using a second column equal to two while retaining $A=g/[2(1-c)]$ introduces a factor-of-two error in the recovered background. The accompanying implementation uses the corrected rows, passes tests 2, 3 and 5 above, and reproduces Figs. 1–3 with no random numbers and no truth-based initialization.

Statistical Evaluation

Monte Carlo comparisons should specify:

• the observation domain in which noise is generated;

• the calibrated parameters in each benchmark;

• whether the acquisition count or total time is fixed;

• initialization and fallback rules;

• invalid-estimate and optimizer-failure rates;

• bias, conditional error, and operational error;

• Monte Carlo uncertainty on reported performance.

The principal design comparison should include known-phase three-plus-three, shared-increment five-plus-three, shared-increment five-plus-five, and independent-increment five-plus-five protocols.

Near singular configurations, reporting only a sampled grid maximum is insufficient. The analysis should combine analytic exceptional sets with targeted conditioning tests.

Physical Spectral Validation

Before applying the model to raw optical peak distances, a spectral validation should compare:

1. ideal dressed-state separations;

2. separations extracted from simulated or measured optical extrema;

3. parameters obtained from full-spectrum fitting.

For a Doppler-averaged master-equation implementation, the following convergence checks are essential:

• probe-detuning grid refinement;

• velocity-domain enlargement and quadrature refinement;

• trace, Hermiticity, positivity, and steady-state residual checks for the density matrix;

• stability of peak identification under scan-window and interpolation changes;

• explicit detection of unresolved or low-contrast doublets.

The optical and microwave frequency conventions, decay-rate normalization, velocity distribution, and effective dephasing model must be stated. Selecting the two deepest local minima without contrast or tracking checks is not a sufficient general peak-identification rule.

A useful diagnostic is to fit the extraction relation to $$\widetilde y=b+\gamma y+r.$$ The affine part has the exact consequences in Proposition 15; the nonaffine residual must be propagated through the inversion or included in the likelihood model.

Discussion

What Is Self-Calibrated

The ideal model permits joint identification of the signal, the two LO amplitudes, and the magnitude of a common detuning. The unknown-increment protocols additionally identify specified phase-evolution parameters.

The procedure does not independently calibrate:

• the dipole matrix element;

• the absolute frequency-axis scale;

• polarization projection;

• spatial field averaging;

• the validity of the isolated-transition approximation;

• the optical separation-extraction model.

These limitations are consistent with the broader uncertainty budget of atomic electrometry [17, 4] The term self-calibrating is therefore more precise than calibration-free.

What Is Established About Phase Control

The five-plus-three protocol does not require the numerical value of the phase increment or the inter-setting phase offset. It does require the increment to be constant and shared.

The five-plus-five independent-increment protocol removes the shared-increment assumption but still requires a constant increment within each setting. Neither protocol covers arbitrary unknown phase sequences.

The restriction $0<\beta<\pi$ resolves sampled-sinusoid direction and aliasing ambiguities. Without an appropriate restriction or external information about rotation direction, phase reconstruction can remain ambiguous under conjugation.

Classical Estimation Limits Versus Quantum Limits

The CRBs in this paper are classical bounds for specified extracted-observation models [34, 35]. They do not establish a quantum Fisher-information limit or an improvement over the quantum-limited sensitivity of an atomic receiver.

Atomic projection noise, optical shot noise, transit noise, and technical fluctuations determine the achievable observation covariance and bandwidth [18]. Connecting those physical noise mechanisms to the extraction model is necessary for an apparatus-level sensitivity claim.

Implications for Experimental Design

The results suggest three complementary experimental checks:

1. vary the LO settings while keeping the signal fixed and examine whether the reconstructed signal remains invariant within its uncertainty;

2. compare separation-based reconstruction with a validated full-spectrum or independent field reference;

3. deliberately introduce known phase and detuning perturbations to distinguish detectable mismatch from observationally equivalent bias.

An unchanged residual under a perturbation is not necessarily evidence of robustness. It can instead reveal an ambiguity of the observation model.

Conclusion

This paper has characterized identification and local estimation limits for a two-setting Rydberg microwave receiver with unknown LO amplitudes and detuning. The analysis separates noiseless uniqueness, statistical information, and physical calibration.

For known phases, three observations at each setting provide phase-uniform identification, while uniform spacing additionally orthogonalizes the moment errors under homoscedastic squared-separation noise. Under that model, treating the LO amplitudes as unknown does not reduce phase information, but does incur an amplitude penalty whose value depends on the calibrated benchmark.

For a shared unknown phase increment and an unknown inter-setting offset, five observations at one setting and three at the other suffice. Eight observations are minimal for phase-uniform identification within the specified class of consecutive equal-step bursts. Independent increments require five observations per setting for the same guarantee.

Finally, exact observational ambiguities show why internal consistency cannot by itself establish correct amplitude or phase calibration. LO-power-dependent squared detuning and common phase-law errors can produce biased yet internally consistent reconstructions. Reliable implementation therefore requires both model-based estimation and external validation of the spectral and phase-reference assumptions.

These results provide a theoretical basis for self-calibrating atomic receivers used in quasi-static channel sounding, pilot-based estimation, and phase-referenced microwave sensing.

Derivation of the Homoscedastic Information Bounds

Let $$B_j=|H_j|=2L_j\sqrt P.$$ Under uniform phase sampling and (36), $$\mathop{\mathrm{Var}}(\widehat A_j)=\frac{\sigma_j^2}{K}.$$ Resolve the harmonic perturbation into radial and tangential components: $$\widehat H_j-H_j = e^{\mathrm{i}\theta}(r_j+\mathrm{i}t_j).$$ Then $$\mathop{\mathrm{Var}}(r_j)=\mathop{\mathrm{Var}}(t_j)=\frac{2\sigma_j^2}{K},$$ and $r_j,t_j,\widehat A_j-A_j$ are mutually uncorrelated. Because the underlying errors are Gaussian, they are also independent.

The physical map $$(D,L_1,L_2,P) \longleftrightarrow (A_1,A_2,B_1,B_2)$$ has the smooth inverse (23)–(24) when $P>0$ and $\Lambda\ne0$.

Define $$N=B_2^2-B_1^2=4P\Lambda.$$ The first-order perturbation of $P=N/(4\Lambda)$ is $$dP= \frac{B_2\,dB_2-B_1\,dB_1}{2\Lambda} - \frac{P}{\Lambda}(dA_2-dA_1). $$ Since the radial and mean errors are independent, $$\begin{aligned} \mathop{\mathrm{Var}}(dP) &= \frac{ B_1^2(2\sigma_1^2/K) + B_2^2(2\sigma_2^2/K) }{ 4\Lambda^2 } \nonumber\\ &\quad+ \frac{P^2}{\Lambda^2} \left( \frac{\sigma_1^2}{K} + \frac{\sigma_2^2}{K} \right) \nonumber\\ &= \frac{ P[ \sigma_1^2(2L_1^2+P) + \sigma_2^2(2L_2^2+P) ] }{ K\Lambda^2 }. \end{aligned}$$

The sinusoidal mean lies in the span of the background and first-harmonic regressors. Under Gaussian homoscedastic noise, the corresponding least-squares moments contain all parameter information; orthogonal residual components contain no information about the mean parameters. Transforming their Fisher information through the smooth physical inverse gives the same expression, establishing (41).

For the phase, $$d\theta_j=\frac{t_j}{B_j},$$ so $$\mathop{\mathrm{Var}}(d\theta_j) = \frac{2\sigma_j^2/K}{4L_j^2P} = \frac{\sigma_j^2}{2KL_j^2P}.$$ The two settings are independent. Summing their phase information yields $$\mathop{\mathrm{CRB}}(\theta) = \left( \sum_{j=1}^{2} \frac{2KL_j^2P}{\sigma_j^2} \right)^{-1},$$ which is (42).

The background and radial directions are orthogonal to the tangential phase direction. Thus eliminating amplitude and detuning nuisance parameters leaves the phase information unchanged under this observation model.

Gradient of the Six-Spectrum Power Inversion

For three uniform phases, $$\frac{\partial A_j}{\partial y_{jk}}=\frac13, \qquad \frac{\partial H_j}{\partial y_{jk}} =\frac23e^{\mathrm{i}\phi_k}.$$ Hence $$\frac{\partial |H_j|^2}{\partial y_{jk}} = \frac43 \mathop{\mathrm{Re}}(H_j^*e^{\mathrm{i}\phi_k}).$$

Let $$N=|H_2|^2-|H_1|^2,\qquad \Lambda=A_2-A_1,\qquad P=\frac{N}{4\Lambda}.$$ For $t_1=-1,t_2=1$, $$\frac{\partial N}{\partial y_{jk}} = t_j\frac43\mathop{\mathrm{Re}}(H_j^*e^{\mathrm{i}\phi_k}), \qquad \frac{\partial\Lambda}{\partial y_{jk}} =\frac{t_j}{3}.$$ Applying the quotient rule gives $$\frac{\partial P}{\partial y_{jk}} = \frac{t_j}{3\Lambda} \left[ \mathop{\mathrm{Re}}(H_j^*e^{\mathrm{i}\phi_k})-P \right],$$ which proves (58).

The expression is evaluated at the noiseless parameter point. Moment-debiasing corrections modify higher-order terms and do not turn the resulting delta-method expression into an exact finite-noise variance.

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