Decoherence Cartography with Transportable Matter and Antimatter Qubits as Probes of Beyond the Standard Model Sectors

1. Introduction

We propose to use a transported localized charged particle qubit as a null probe of environment-conditioned decoherence. The central object is not a single coherence time measured at one site, but a site-indexed family of decoherence functionals inferred after conditioning on measured nuisance channels. In this sense, the proposal resembles a field-mapping problem more than a conventional single-parameter lifetime measurement. The word cartography is therefore meant literally: the output is a map on the space of environments, not merely a yes/no statement about whether transport is possible.

The purpose of the present manuscript is to formulate a concept that is conservative experimentally but theory-driven in structure. The framework applies in principle to matter and antimatter implementations alike, including electron, proton, and antiproton platforms, provided coherent control, transport, and nuisance-channel monitoring are available. Our primary empirical anchor is nevertheless the antiproton case, because recent demonstrations of coherent spectroscopy with a single antiproton spin and physical transport of trapped antiprotons make that platform the most timely and conceptually distinctive realization of the general framework [1,2,3]. The formalism developed below is therefore species-general, while the antiproton implementation serves as the flagship example for two reasons. First, it already combines coherent single-particle control with transport. Second, as the antimatter partner of the proton, it provides access to matter–antimatter differential observables and to model classes that are not sharply testable with matter-only sensors.

What has changed recently is not the conceptual importance of the antiproton, but the operational status of the antiproton as a controllable quantum object. The BASE collaboration has reported coherent spectroscopy of a single antiproton spin, including the first observation of Rabi oscillations and a spin-coherence time of order $50\mathrm{s}$ [1]. In parallel, CERN’s BASE-STEP program has demonstrated physical transport of trapped antiprotons in a portable cryogenic Penning-trap system, with the explicit aim of moving antiprotons out of the magnetically noisy Antimatter Factory environment and into quieter laboratories for higher-precision measurements [2,4,5,6]. These two advances together create a new possibility: one can now ask theory-level questions about a portable antiproton qubit rather than only about a stationary one. BASE-STEP was not originally developed for decoherence cartography, but to improve precision CPT tests by relocating trapped antiprotons from the magnetically noisy accelerator environment to quieter offline laboratories. The present article proposes and theoretically investigates a program of an additional use of the same transport capability: environment-dependent decoherence mapping with localized matter and antimatter qubits.

The essential distinction from ordinary transport qualification should be emphasized. Engineering qualification asks whether an apparatus survives relocation and resumes operation. By contrast, matter and antimatter decoherence cartography asks whether the reduced density matrix of the same class of prepared charged particle qubit evolves identically across environments once known noise channels are accounted for. The conservative null hypothesis is that it does. In that case the main result is a validated transport and environment decoherence model for portable metrology of different particle species. If it does not, then the residual must be characterized as either hidden technical structure or genuine location-conditioned excess decoherence. The strength of the proposal is that both outcomes are physically informative. The BASE-STEP apparatus was developed as a novel autonomous, open, and transportable Penning-trap system. Its primary experimental purpose is to enable a new measurement paradigm in which trapped antiprotons are transported to magnetically quiet offline laboratories, including facilities under development at Heinrich Heine University Düsseldorf and planned at CERN. The present proposal builds on this capability, but asks a different question: once transportable trapped antiprotons are available, can the same platform be used not only for improved precision spectroscopy, but also as a controlled probe of environment-conditioned decoherence?

This paper is written deliberately at the level of formalism and inference strategy. We do not advocate a specific beyond Standard Model (BSM) explanation, nor do we assume that an exotic signal is likely. Instead, we ask what theoretical structure is required to make a future experiment interpretable once transport-enabled electron, proton and antiproton coherence measurements become available. In that spirit, we develop a set of illustrative dictionaries that map representative microscopic scenarios—including millicharged, Lorentz-violating, ultralight bosonic fields, and collapse-like noise—onto the residual observables of the cartography framework. These dictionaries are not intended as an exhaustive list of targets, but as concrete examples of how a measured residual could be translated into model-dependent parameters. The recent demonstration of coherent spectroscopy with a single antiproton spin and the successful transport of trapped antiprotons provide the empirical anchor for posing this question in a quantitative way [1,2].

More broadly, the cartography framework is not limited to the explicit dictionary examples developed below. Any BSM or cosmologically motivated sectors that induce a residual phase shift, residual dephasing, or excess motional-noise channel for the localized charged-particle qubit can be embedded in the same formalism. In particular, the framework is naturally applicable to ultralight dark matter backgrounds [7,8,9] that generate oscillatory spin-precession or frequency-shift signals [10,11], to transient topological defects that would appear as correlated site- and revisit-dependent excursions [12,13], to scalar or dilaton-like sectors that induce oscillations or drifts of fundamental constants [14,15], and to exotic spin-dependent interactions mediated by light bosons [16]. Ultralight vector fields, including kinetically mixed dark photons and related light-vector backgrounds, provide a complementary coherent-field target [17,18].

The formalism developed below is primarily intended for stationary-site cartography. The trapping platform, together with a particle reservoir, is relocated between environment classes. After transport, the apparatus is stabilized, the relevant nuisance channels are characterized, and fresh qubit states are prepared, interrogated, and read out locally. The repeated preparation and measurement cycle is therefore performed separately at each site. The relevant coherence requirement is that each locally prepared qubit state remain coherent over the interrogation time, not over the full transport duration. Transporting an already prepared coherent superposition between sites would define a distinct and more demanding coherent-transport limit of the same general framework, but it is not assumed in what follows. Possible site-to-site changes in preparation fidelity, state of motion, trap tuning, and readout response are not interpreted as residual decoherence by themselves; they enter the calibration and nuisance model and must be monitored or marginalized before assigning any nonzero a residual decoherence observable to an environment-conditioned channel.

The paper is organized as follows. Section 2 introduces the operational observables of decoherence cartography: the coherence signal, accumulated phase, survival probability, and residual cartography observable. Section 3 develops the effective dynamical description, showing how residual perturbations enter the qubit evolution through a stochastic frequency shift, its spectral density, and the associated decoherence functional. Section 4 classifies the effective residual structures that may appear in the data, including broadband colored spectra, narrow-band harmonic signals, and burst-like or transient processes. Section 5 then formulates the comparative structure of the cartography program in terms of graphs whose nodes are site or environment classes and whose edges represent pairwise comparisons or transport links. Section 6 explains how the operational observables are converted into falsifiable statistical statements about residual decoherence, including null, site-like, edge-like, and transient alternatives. Section 7 introduces microscopic dictionaries that map representative BSM and cosmology-motivated sectors onto effective residual phase or decoherence signals. Order of magnitude benchmark estimates for the residual sensitivity of the protocol are given in Section 8. Section 9 specifies what physical conclusions can, and cannot, be drawn from each class of residual structure. Section 10 presents worked hypothetical cartography scenarios illustrating how the observables, graph-based inference structure, and microscopic dictionaries would be used in practice. Conclusions and outlook are given in Section 11.

2. Operational Definition of Localized Qubit Decoherence Cartography

Consider a single charged particle of species q confined in a transportable cryogenic trap apparatus, where we allow for a species label

$$q\in \{e^{-},e^{+},p,\overline{p}\},$$

(1)

with $e^{-}$, $e^{+}$, p, and $\overline{p}$ denoting electron, positron, proton, and antiproton implementations, respectively 1. In the antiproton case this is realized by the portable Penning-trap systems developed for BASE-STEP and related CERN transport studies [4,5], while analogous proton or electron implementations would differ only in platform-specific control architecture. The logical qubit is encoded in the spin basis,

$$|{\psi }_0\rangle ={\displaystyle \frac{1}{\sqrt{2}}}\left(|\uparrow \rangle +{\mathrm{e}}^{\mathrm{i}{\varphi }_0}|\downarrow \rangle \right),$$

(2)

prepared and interrogated using the species-appropriate coherent-spectroscopy protocol. In the antiproton implementation, this protocol is anchored by the recent demonstration of coherent single-antiproton spin spectroscopy by BASE [1] 2. Here $|\uparrow \rangle$ and $|\downarrow \rangle$ denote the two spin eigenstates of the trapped charged particle along the quantization axis set by the trap magnetic field, and ${\varphi }_0$ is the controllable initial relative phase of the prepared superposition. Equation (2) therefore represents the simplest phase-sensitive pure state of the localized charged particle qubit: equal populations in the two logical basis states, together with a well-defined coherence stored in the off-diagonal element of the density matrix. Throughout the experiment, the particle remains confined within the trapping apparatus. The proposal therefore concerns a transported localized stored qubit, rather than a flying qubit propagating through a quantum communication channel 3.

Let $\mathsf{e}\in \mathcal{E}$ label an environment class or physical site, $\mathsf{s}\in \mathcal{S}$ label a control sequence, and $\mathsf{r}\in \mathbb{N}$ label repeated visits to the same site. By an environment class or site we mean either a specific laboratory location or, more abstractly, a class of conditions sharing a common noise environment, such as an accelerator hall, shielded surface laboratory, or underground site. By a control sequence we mean the pulse pattern used to prepare, evolve, and interrogate the qubit, for example Ramsey, spin echo, or multipulse echo protocols [31,32,33,34]. The label $\mathsf{r}$ is included because the same site may be revisited multiple times, and reproducibility under revisits is part of the operational definition of a genuine cartographic signal. Finally, T denotes the total interrogation time, that is, the elapsed duration between state preparation and final readout during which the qubit accumulates phase and decoherence.

For each triple $(\mathsf{e},\mathsf{s},\mathsf{r})$ we define a prepared-state quantum channel

$${\rho }_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)={\mathcal{M}}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right){\rho }_0,$$

(3)

where ${\rho }_0=|{\psi }_0\rangle \langle {\psi }_0|$ is the initially prepared density operator and ${\mathcal{M}}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)$ is the completely positive trace-nonincreasing map that represents the effective evolution of the qubit during interrogation time T [35]. In other words, ${\mathcal{M}}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)$ is the reduced dynamical map obtained after tracing over all uncontrolled degrees of freedom, including technical noise sources, trap-motion couplings, and any residual environment-conditioned channel. The map is taken to be trace nonincreasing rather than strictly trace preserving because particle loss, failed readout, or shot rejection may remove runs from the valid ensemble. The experimentally relevant observables are encoded in the off-diagonal matrix element of the reduced density operator,

$${\rho }_{\uparrow \downarrow }^{(\mathsf{e},\mathsf{s},\mathsf{r})}\left(T\right)=〈\uparrow |{\rho }_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)|\downarrow 〉={\rho }_{\uparrow \downarrow }^{(\mathsf{e},\mathsf{s},\mathsf{r})}\left(0\right)P_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)W_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)e^{-i{\mathrm{\Phi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)},$$

(4)

where $W_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)$ is the coherence signal, ${\mathrm{\Phi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)$ is the accumulated phase, ${\rho }_{e,s,r}\left(T\right)$ is the unnormalized reduce state before conditioning on survival, and $P_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)$ is the survival probability. The factor $P_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)$ is included to distinguish particle loss or invalid shots from genuine decoherence of the surviving qubit ensemble.

The theory object of interest is not the full channel ${\mathcal{M}}_{\mathsf{e},\mathsf{s},\mathsf{r}}$ itself, but the part of the decoherence that remains after subtracting the contribution predicted from ordinary, independently diagnosed noise sources. In the dephasing-dominated regime, the coherence signal of a qubit is naturally written in the form

$$W\left(T\right)\sim e^{-\chi \left(T\right)},$$

(5)

or equivalently the off-diagonal density-matrix element decays as

$${\rho }_{\uparrow \downarrow }\left(T\right)={\rho }_{\uparrow \downarrow }\left(0\right)e^{-i\mathrm{\Phi }\left(T\right)}{e}^{-\chi \left(T\right)},$$

(6)

so that $-\ln W\left(T\right)$ is the natural additive decoherence functional in standard open-system and filter-function descriptions [33,34,35]. This motivates the definition

$${\mathrm{\Xi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)\equiv -\ln W_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)-{\widehat{\chi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{nuis}}\left(T\right),$$

(7)

where ${\widehat{\chi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{nuis}}\left(T\right)$ is the decoherence predicted from independently monitored technical channels. The quantity ${\mathrm{\Xi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)$ is therefore an estimator of any additional environment-conditioned contribution beyond the nuisance model. In the conservative null model one has

$${\mathrm{\Xi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)=0\mathrm{for}\mathrm{all}(\mathsf{e},\mathsf{s},\mathsf{r},T),$$

(8)

up to statistical and modeling uncertainty. Equation (8) is not an independent dynamical law, but the operational null hypothesis of the cartography program: once the monitored nuisance contribution has been subtracted, no residual site-conditioned decoherence 4. Cartography is then the problem of determining whether $\mathrm{\Xi }$ is consistent with zero everywhere, or whether it exhibits a reproducible dependence on site, revisit, or control sequence that would require an additional residual channel.

Two features are essential. First, the map is intrinsically differential: no single site is privileged, and the robust objects are pairwise or networked comparisons among sites. A residual is scientifically interesting only if it survives such comparisons rather than being tied to an arbitrary reference choice. Second, the map is sequence dependent: if a residual exists, its scaling under changes of filter function provides immediate information about whether it behaves like low-frequency dephasing, bursty noise, nonstationary drift, or an effectively white channel [33,34,36]. This sequence dependence is crucial because it turns the program from a search for a single anomalous decay rate into an inference problem over effective noise classes. A residual that remains stable under revisits and site permutations, while transforming predictably under changes of control sequence, can be assigned a much sharper interpretation than an undifferentiated excess-loss signal. In this sense, matter and antimatter decoherence cartography is not merely a measurement of coherence times at different locations; it is a structured framework for deciding whether any observed site dependence is compatible with known nuisance channels or instead requires an additional environment-conditioned contribution.

At each experimental shot, the localized particle is prepared, or reinitialized, in the target qubit state before the interrogation begins. The physical particle need not be replaced after every measurement; rather, the same trapped particle may be repeatedly reset, coherently manipulated, and measured, as demonstrated in single-particle Penning-trap and trapped-ion platforms [1,37]. Thus ${\rho }_{e,s,r}\left(T\right)$ should be understood as the ensemble state reconstructed from repeated local preparation and measurement cycles, not as a single wave function that remains coherent during transport. The target state is common to all sites, although the experimentally realized initial density operator may exhibit small site- and revisit-dependent preparation errors.

The preparation and control fields are assumed to be strong compared with the weak residual frequency perturbations being investigated. For a semiclassical near-resonant drive, and within the rotating-wave approximation, the effective Hamiltonian in the frame rotating with the applied drive may be written as [31,38,39,40,41]

$$H_{\mathrm{prep}}^{(\mathsf{e},\mathsf{r})}\left(t\right)={\displaystyle \frac{\hslash }{2}}\left\{{\mathrm{\Omega }}_{\mathrm{R}}^{(\mathsf{e},\mathsf{r})}\left(t\right)\left[\cos {\varphi }_{\mathrm{d}}\left(t\right){\sigma }_x+\sin {\varphi }_{\mathrm{d}}\left(t\right){\sigma }_y\right]+{\mathrm{\Delta }}_{\mathsf{e},\mathsf{r}}\left(t\right){\sigma }_z\right\}.$$

(9)

Here ${\mathrm{\Omega }}_{\mathrm{R}}^{(\mathsf{e},\mathsf{r})}\left(t\right)$ is the controlled Rabi frequency, ${\varphi }_{\mathrm{d}}\left(t\right)$ is the phase of the applied drive, and ${\mathrm{\Delta }}_{e,r}\left(t\right)={\omega }_d\left(t\right)-{\omega }_q^{(e,r)}\left(t\right)$ is the instantaneous detuning between the drive and the qubit transition frequency, including any weak environment-dependent frequency shift present during preparation. The operators ${\sigma }_x$, ${\sigma }_y$, and ${\sigma }_z$ are the Pauli matrices acting in the qubit basis $\{|\uparrow \rangle ,|\downarrow \rangle \}$ 5. The transverse control term proportional to $\cos {\varphi }_{\mathrm{d}}{\sigma }_x+\sin {\varphi }_{\mathrm{d}}{\sigma }_y$ generates coherent rotations about an axis in the equatorial plane of the Bloch sphere, with the rotation axis determined by the drive phase ${\varphi }_{\mathrm{d}}$. The longitudinal term proportional to ${\sigma }_z$ describes the detuning of the applied drive from the qubit transition frequency. The special choice ${\varphi }_{\mathrm{d}}=0$ corresponds to rotations about the x axis.

Reliable state preparation requires

$$\left|{\mathrm{\Omega }}_{\mathrm{R}}^{(\mathsf{e},\mathsf{r})}\left(t\right)\right|\gg \left|{\mathrm{\Delta }}_{\mathsf{e},\mathsf{r}}\left(t\right)\right|$$

(10)

during the active part of the control pulse. The calibration of amplitude and off-resonance errors in single-qubit control is a standard component of coherent-control protocols [41].

For a simple nominally resonant pulse, the residual detuning tilts the effective rotation axis away from the intended transverse direction. The resulting state-vector error amplitude is parametrically of order

$${ϵ}_{\mathrm{prep}}^{(\mathsf{e},\mathsf{r})}\sim \left|{\displaystyle \frac{{\mathrm{\Delta }}_{\mathsf{e},\mathsf{r}}}{{\mathrm{\Omega }}_{\mathrm{R}}^{(\mathsf{e},\mathsf{r})}}}\right|,$$

(11)

while the corresponding preparation infidelity begins at quadratic order,

$$1-{\mathcal{F}}_{\mathrm{prep}}^{(\mathsf{e},\mathsf{r})}\sim {\left({\displaystyle \frac{{\mathrm{\Delta }}_{\mathsf{e},\mathsf{r}}}{{\mathrm{\Omega }}_{\mathrm{R}}^{(\mathsf{e},\mathsf{r})}}}\right)}^2,$$

(12)

where ${\mathcal{F}}_{\mathrm{prep}}^{(\mathsf{e},\mathsf{r})}\equiv 〈{\psi }_0^{\mathrm{ref}}\left|{\rho }_0^{(\mathsf{e},\mathsf{r})}\right|{\psi }_0^{\mathrm{ref}}〉$ denotes the fidelity of the experimentally prepared state ${\rho }_0^{(\mathsf{e},\mathsf{r})}$ with respect to the target pure state $|{\psi }_0^{\mathrm{ref}}\rangle$. This scaling follows directly from the detuned Rabi solution for a driven two-level system and underlies standard treatments of off-resonance single-qubit control errors [41,42].

Consequently, a weak residual perturbation may have a negligible effect on the short preparation pulse while still generating a measurable phase during the much longer local interrogation interval,

$${\mathrm{\Phi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{x}}\left(T\right)={\int }_0^{T}dt{y}_{\mathsf{s}}\left(t\right)\delta {\omega }_{\mathrm{x}}^{(\mathsf{e},\mathsf{r})}\left(t\right).$$

(13)

Here (t=0) denotes the beginning of the local interrogation interval after state preparation, not the beginning of transport. $\delta {\omega }_{\mathrm{x}}^{(\mathsf{e},\mathsf{r})}\left(t\right)$ denotes a weak residual shift of the qubit frequency not captured by the calibrated preparation model or by the monitored nuisance channels. The function $y_{\mathsf{s}}\left(t\right)$ is the switching function associated with the control sequence $\mathsf{s}$; it specifies the sign and temporal weight with which a frequency perturbation contributes to the accumulated phase. For example, $y_{\mathrm{R}}\left(t\right)=1$ during a Ramsey interrogation, whereas for a Hahn-echo sequence it changes sign at the refocusing $\pi$ pulse. These quantities are introduced here only to distinguish the short preparation interval from the subsequent sensing interval; their full dynamical definitions are given in Section 3.

This separation between short control pulses and sequence-filtered phase accumulation during interrogation is the basis of standard filter-function and noise-spectroscopy methods [33,34]. The cartographic signal therefore arises primarily during the controlled free-evolution or sensing interval, rather than during the state-preparation pulse.

To prevent site-dependent preparation and readout changes from being misidentified as residual decoherence, the cartography protocol is defined with respect to a fixed target input state, ${\rho }_0$, in Eq. (3), which is taken to be independent of the site and revisit. Any dependence on $\mathsf{e}$ or $\mathsf{r}$ is therefore not attributed to the intended initial state, but may arise from imperfections in its local experimental preparation.

The state actually produced at a given site and revisit may be represented as

$${\rho }_{0,\mathrm{prep}}^{(\mathsf{e},\mathsf{r})}={\mathcal{P}}_{\mathsf{e},\mathsf{r}}\left[{\rho }_0\right],$$

(14)

where ${\mathcal{P}}_{\mathsf{e},\mathsf{r}}$ is the experimentally calibrated preparation map. In the ideal limit, ${\mathcal{P}}_{\mathsf{e},\mathsf{r}}=\mathcal{I}$, and hence ${\rho }_{0,\mathrm{prep}}^{(\mathsf{e},\mathsf{r})}={\rho }_0$ for every site and revisit. Site-dependent deviations of ${\rho }_{0,\mathrm{prep}}^{(\mathsf{e},\mathsf{r})}$ from ${\rho }_0$ are therefore interpreted as preparation errors rather than as intended changes of the cartography protocol.

When preparation imperfections are retained explicitly, the experimentally realized evolution is

$${\rho }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\exp }\left(T\right)={\mathrm{\Lambda }}_{\mathsf{e},\mathsf{s},\mathsf{r};T}\left[{\rho }_{0,\mathrm{prep}}^{(\mathsf{e},\mathsf{r})}\right]={\mathrm{\Lambda }}_{\mathsf{e},\mathsf{s},\mathsf{r};T}\left[{\mathcal{P}}_{\mathsf{e},\mathsf{r}}\left[{\rho }_0\right]\right].$$

(15)

Here ${\mathrm{\Lambda }}_{\mathsf{e},\mathsf{s},\mathsf{r};T}$ denotes the effective reduced evolution map during the interrogation interval T. It incorporates the controlled qubit dynamics, coupling to the local environment, and the resulting sequence-dependent phase accumulation and decoherence. Its dependence on $\mathsf{e}$, $\mathsf{s}$, and $\mathsf{r}$ allows the experimentally realized evolution to vary between sites, control protocols, and repeated visits, while the target input state ${\rho }_0$ remains fixed. This decomposition separates the site-independent target state from local preparation errors and from the subsequent interrogation dynamics.

The coherence signal should accordingly be normalized to the locally measured initial coherence,

$$W_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)={\displaystyle \frac{\left|{\rho }_{\uparrow \downarrow }^{(\mathsf{e},\mathsf{s},\mathsf{r})}\left(T\right)\right|}{P_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)\left|{\rho }_{0,\mathrm{prep};\uparrow \downarrow }^{(\mathsf{e},\mathsf{r})}\right|}},$$

(16)

where ${\rho }_{0,\mathrm{prep};\uparrow \downarrow }^{(\mathsf{e},\mathsf{r})}=\langle \uparrow |{\rho }_{0,\mathrm{prep}}^{(\mathsf{e},\mathsf{r})}|\downarrow \rangle$ is the off-diagonal element of the locally prepared initial state. This normalization removes the leading effect of site-dependent variations in the initial coherence from the inferred decay function 6.

The local preparation fidelity may be quantified by

$${\mathcal{F}}_{\mathrm{prep}}^{(\mathsf{e},\mathsf{r})}=\langle {\psi }_0|{\rho }_{0,\mathrm{prep}}^{(\mathsf{e},\mathsf{r})}|{\psi }_0\rangle ,$$

(17)

which measures the overlap between the experimentally prepared state and the fixed target state. Short-interrogation reference measurements, interleaved calibration shots, and self-consistent tomography methods can be used to constrain preparation and readout errors [43,44]. Any residual uncertainty in these calibrations must be propagated into the nuisance covariance and should not be assigned directly to the environment-conditioned residual ${\mathrm{\Xi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)$.

Thus Section 2 supplies both the measurement layer and the local-calibration layer of the cartography framework. The quantities $W_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)$, ${\mathrm{\Phi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)$, $P_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)$, and ${\mathrm{\Xi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)$ are defined operationally, while site-dependent state-preparation and readout effects are incorporated into the calibrated nuisance model and its covariance. The next layer is dynamical: one must specify how monitored nuisance channels and possible residual perturbations enter the qubit Hamiltonian, how they propagate through the control sequence, and how they generate measurable phase shifts or decoherence.

3. Effective Dynamics and Residual Frequency Noise

In this section we explain how the operational observables introduced in Section 2 arise from the underlying qubit dynamics. The key object is the stochastic frequency shift $\delta \omega \left(t\right)$ experienced by the localized qubit during interrogation. Monitored components of $\delta \omega \left(t\right)$ determine the nuisance decoherence model ${\widehat{\chi }}^{\mathrm{nuis}}$, while any contribution not accounted for by that model enters the residual inferred through $\mathrm{\Xi }$. The purpose of this section is therefore to provide the bridge between the measured coherence signal and the frequency-domain noise description used later for residual classification and microscopic dictionaries.

We model the system as a localized qubit encoded in the spin degree of freedom of a single trapped charged particle and prepared and interrogated using coherent spectroscopy. The physical setting is a transportable or relocatable charged-particle trapping platform. For antiprotons and protons, this is naturally realized by the portable cryogenic Penning-trap architecture developed in the BASE-STEP program [2,4,5,6,45]. For electrons, the closest high-precision analogue is the cryogenic single-electron Penning-trap platform used in quantum-cyclotron spectroscopy, although a directly comparable transportable implementation has not yet been established [19,20]. Positrons provide the corresponding antimatter-lepton extension: their accumulation, confinement, and manipulation in Penning–Malmberg traps are well established, although an autonomous transportable single-positron spin-qubit platform suitable for the present cartographic protocol remains a future development [46,47].

The relevant degrees of freedom of the complete transported apparatus are vastly more numerous than those of the qubit itself. Besides the particle spin, they include axial and cyclotron motion, electrode and magnetic-field fluctuations, cryogenic and mechanical perturbations, control-pulse imperfections, timing jitter, and possible additional environment-conditioned couplings. Since decoherence cartography concerns the reduced evolution of the spin qubit rather than a microscopic description of the complete apparatus, it is natural to adopt the language of open quantum systems and work with an effective reduced description [34,35]. The controlled two-level subsystem is retained explicitly, while uncontrolled or only partially monitored degrees of freedom are represented through effective couplings and noise channels.

At this level of description, the qubit dynamics may be represented schematically by

$$H\left(t\right)={\displaystyle \frac{\hslash }{2}}{\omega }_s\left(t\right){\sigma }_z+H_{\mathrm{ctrl}}\left(t\right)+H_{\mathrm{aux}}\left(t\right),$$

(18)

where ${\omega }_s\left(t\right)$ is the instantaneous spin splitting of the charged particle qubit along the trap quantization axis, $H_{\mathrm{ctrl}}\left(t\right)$ denotes the externally applied control Hamiltonian used for state preparation, coherent manipulation, and readout, and $H_{\mathrm{aux}}\left(t\right)$ collects additional couplings not included in the ideal control model. Here ${\sigma }_z$ is the same Pauli operator used in Eq. (9), acting in the qubit basis $\{|\uparrow \rangle ,|\downarrow \rangle \}$. The term $H_{\mathrm{ctrl}}\left(t\right)$ represents the intended pulse sequence—for example Ramsey [31] or Hahn-echo [32] interrogation—whereas $H_{\mathrm{aux}}\left(t\right)$ represents the effective influence of motional couplings, magnetic-field fluctuations, mechanical vibration, thermal drift, timing errors, and other environment-conditioned perturbations.

The advantage of Eq. (18) is that it separates the dynamics into three conceptually distinct ingredients: coherent precession of the qubit, deliberate interrogation through the control sequence, and uncontrolled or only partially monitored perturbations. The first two specify the intended experimental evolution, whereas the last term represents the influence of the environment and apparatus. After tracing over uncontrolled degrees of freedom associated with $H_{\mathrm{aux}}\left(t\right)$, one obtains the reduced dynamical map of the qubit and, in the dephasing-dominated regime considered here, the filter-function description used below [33,34]. Equation (18) should therefore not be interpreted as a microscopic Hamiltonian of the complete transported apparatus, but as an effective starting point for inferring how site-dependent environments modify the coherence of a localized charged-particle qubit.

For the purposes of cartography, the dominant uncontrolled effect is assumed to be dephasing through fluctuations of the spin splitting. We therefore decompose

$${\omega }_s\left(t\right)={\overline{\omega }}_s\left(t\right)+\delta \omega \left(t\right),$$

(19)

where ${\overline{\omega }}_s\left(t\right)$ denotes the reference splitting, including deterministic drift or deliberately applied control-frame contributions, and $\delta \omega \left(t\right)$ contains stochastic or effectively uncontrolled frequency fluctuations about that reference value. Thus, ${\overline{\omega }}_s\left(t\right)$ describes the precession incorporated in the reference model, whereas $\delta \omega \left(t\right)$ describes the frequency noise experienced by the qubit.

For concreteness, the monitored and residual contributions may be written as

$$\begin{array}{cc}\hfill \delta \omega \left(t\right)=& \delta {\omega }_B\left(t\right)+\delta {\omega }_{\mathrm{vib}}\left(t\right)+\delta {\omega }_{\mathrm{th}}\left(t\right)+\delta {\omega }_{\mathrm{clk}}\left(t\right)+\delta {\omega }_{\mathrm{x}}\left(t\right)\hfill \\ \hfill =& \sum _{a\in \mathcal{A}}{g}_a{\xi }_a\left(t\right)+x\left(t\right),\hfill \end{array}$$

(20)

where $\mathcal{A}=\{B,\mathrm{vib},\mathrm{th},\mathrm{clk}\}$ labels the monitored magnetic, vibrational, thermal, and timing/control channels. The variables ${\xi }_a\left(t\right)$ represent the independently monitored environmental quantities in their native units, while $g_a$ are the calibrated transfer coefficients converting them into effective qubit frequency fluctuations, $\delta {\omega }_a=g_a{\xi }_a$. The residual $x\left(t\right)\equiv \delta {\omega }_{\mathrm{x}}\left(t\right)$ denotes any frequency perturbation not included in the monitored nuisance model.

Equation (20) is an effective bookkeeping decomposition and does not imply that the monitored channels are statistically independent. Magnetic fluctuations, trap motion, temperature, and control electronics may be correlated, and such correlations must be retained when constructing the nuisance spectrum. This point becomes particularly important when the cartographic residual is obtained by subtracting a predicted nuisance decoherence from the measured coherence loss.

More generally, the transfer from the monitored environmental variables to the qubit frequency can be frequency dependent. We therefore introduce the N-component monitored-noise vector

$$\xi \left(\omega \right)=\left(\begin{array}{c}{\xi }_1\left(\omega \right)\\ ⋮\\ {\xi }_N\left(\omega \right)\end{array}\right),$$

(21)

where $N=\left|\mathcal{A}\right|$ is the number of monitored nuisance channels included in the analysis, and the corresponding calibrated transfer vector $\mathit{G}\left(\omega \right)$, such that

$$\delta {\omega }_{\mathrm{nuis}}\left(\omega \right)={\mathit{G}}^{†}\left(\omega \right)\xi \left(\omega \right).$$

(22)

The components $G_a\left(\omega \right)$ are understood as causal linear-response transfer functions. The constants $g_a$ in Eq. (20) correspond to a flat-band approximation in which $G_a\left(\omega \right)$ is approximately real and frequency independent over the frequency range to which the chosen control sequence is appreciably sensitive. The time-local form in Eq. (20) is recovered in this limit with $G_a\left(\omega \right)$ approximated by the real, frequency independent constants $g_a$ .

The control sequence determines how the frequency fluctuations are sampled during the interrogation interval. Introducing the switching function $y_{\mathsf{s}}\left(t\right)$, the stochastic phase accumulated under sequence $\mathsf{s}$ is

$${\mathrm{\Phi }}_{\mathsf{s}}\left(T\right)={\int }_0^{T}dt{y}_{\mathsf{s}}\left(t\right)\delta \omega \left(t\right).$$

(23)

For an ideal Ramsey interrogation,

$$y_{\mathrm{R}}\left(t\right)=1,0<t<T,$$

(24)

whereas for an ideal Hahn-echo sequence,

$$y_{\mathrm{E}}\left(t\right)=\left\{\begin{array}{cc}+1,\hfill & 0<t<T/2,\hfill \\ -1,\hfill & T/2<t<T.\hfill \end{array}\right.$$

(25)

In the Hahn-echo sequence, the sign reversal of $y_{\mathrm{E}}\left(t\right)$ occurs at the central $\pi$-pulse 7. More general multipulse sequences are described by switching functions that alternate sign at each applied $\pi$-pulse.

The corresponding Fourier-domain switching amplitude is

$$Y_{\mathsf{s}}(\omega ,T)\equiv {\int }_0^{T}dt{y}_{\mathsf{s}}\left(t\right)e^{i\omega t},$$

(26)

and throughout this work we define the dimensionless filter function as

$$F_{\mathsf{s}}\left(\omega T\right)\equiv {\omega }^2{\left|Y_{\mathsf{s}}(\omega ,T)\right|}^2.$$

(27)

With this normalization, the Ramsey and Hahn-echo filter functions are [31,32,33,34]

$$F_{\mathrm{R}}\left(u\right)=4{\sin }^2\left({\displaystyle \frac{u}{2}}\right),F_{\mathrm{E}}\left(u\right)=16{\sin }^4\left({\displaystyle \frac{u}{4}}\right),u=\omega T.$$

(28)

The subscripts $\mathrm{R}$ and $\mathrm{E}$ label the Ramsey and Hahn-echo interrogation sequences, respectively. The sequence dependence of Eq. (28) implies that even a stationary site-conditioned environment can generate different measured coherence functions under different control protocols [33,34,36].

For dephasing-dominated evolution with negligible spin flips during the interrogation window, it is useful to separate the reproducible frequency shift from its stochastic fluctuations,

$$\delta \omega \left(t\right)=\delta {\omega }_{\mathrm{coh}}\left(t\right)+\delta {\omega }_{\mathrm{stoch}}\left(t\right),$$

(29)

where $\delta {\omega }_{\mathrm{coh}}\left(t\right)$ denotes the coherent or reproducible component and $\langle \delta {\omega }_{\mathrm{stoch}}\left(t\right)\rangle =0$. The corresponding accumulated phases are

$${\mathrm{\Phi }}_{\mathsf{s}}^{\mathrm{coh}}\left(T\right)={\int }_0^{T}dt{y}_{\mathsf{s}}\left(t\right)\delta {\omega }_{\mathrm{coh}}\left(t\right),{\mathrm{\Phi }}_{\mathsf{s}}^{\mathrm{stoch}}\left(T\right)={\int }_0^{T}dt{y}_{\mathsf{s}}\left(t\right)\delta {\omega }_{\mathrm{stoch}}\left(t\right).$$

(30)

The normalized off-diagonal density-matrix element then evolves in the general form

$${\rho }_{\uparrow \downarrow }\left(T\right)={\rho }_{\uparrow \downarrow }\left(0\right)e^{-i{\mathrm{\Phi }}_{\mathsf{s}}^{\mathrm{coh}}\left(T\right)}{\mathcal{C}}_{\mathsf{s}}\left(T\right),$$

(31)

where

$${\mathcal{C}}_{\mathsf{s}}\left(T\right)\equiv 〈\exp \left[-i{\mathrm{\Phi }}_{\mathsf{s}}^{\mathrm{stoch}}\left(T\right)\right]〉$$

(32)

is the coherence factor associated with the stochastic frequency fluctuations [33,34]. Equivalently, ${\mathcal{C}}_{\mathsf{s}}\left(T\right)$ is the characteristic function of the random accumulated phase ${\mathrm{\Phi }}_{\mathsf{s}}^{\mathrm{stoch}}\left(T\right)$, obtained by evaluating the characteristic functional of the stochastic process $\delta {\omega }_{\mathrm{stoch}}\left(t\right)$ on the switching function $y_{\mathsf{s}}\left(t\right)$.

The measurable coherence magnitude is therefore

$$W_{\mathsf{s}}\left(T\right)=\left|{\mathcal{C}}_{\mathsf{s}}\left(T\right)\right|.$$

(33)

Equation (31) refers to the normalized qubit state conditioned on particle survival; the survival factor $P_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)$ introduced in Section 2 is treated separately.

For stationary Gaussian classical frequency noise, or more generally when the cumulant expansion can be truncated consistently at second order, the coherence factor reduces to

$${\mathcal{C}}_{\mathsf{s}}\left(T\right)=\exp \left[-{\chi }_{\mathsf{s}}\left(T\right)\right],$$

(34)

and hence

$$W_{\mathsf{s}}\left(T\right)=\exp \left[-{\chi }_{\mathsf{s}}\left(T\right)\right],$$

(35)

with

$${\chi }_{\mathsf{s}}\left(T\right)={\displaystyle \frac{1}{2}}\mathrm{Var}\left[{\mathrm{\Phi }}_{\mathsf{s}}^{\mathrm{stoch}}\left(T\right)\right].$$

(36)

The filter-function expressions for ${\chi }_{\mathsf{s}}\left(T\right)$ given below are understood in this Gaussian or second-cumulant sense. For genuinely non-Gaussian residual channels, higher cumulants need not vanish and the coherence must instead be obtained from the full characteristic function in Eq. (32). Such cases include, for example, burst-like or discrete phase-kick processes and are treated explicitly in Section 4.

To retain correlations among the monitored environmental variables, we introduce their two-sided cross-spectral matrix,

$${\left[{\mathit{S}}_{\xi }^{\left(2\right)}\left(\omega \right)\right]}_{ab}\equiv S_{ab}^{\left(2\right)}\left(\omega \right)={\int }_{-\infty }^{+\infty }d\tau e^{i\omega \tau }〈{\xi }_a(t+\tau ){\xi }_b\left(t\right)〉.$$

(37)

For real stationary stochastic processes,

$$S_{ab}^{\left(2\right)}(-\omega )={\left[S_{ba}^{\left(2\right)}\left(\omega \right)\right]}^{*},$$

(38)

so that the spectral matrix is Hermitian at each frequency. Its diagonal elements are the individual auto-spectra, whereas the off-diagonal elements encode correlations among distinct monitored channels.

The complete monitored nuisance spectrum is then

$$S_{\mathrm{nuis}}^{\left(2\right)}\left(\omega \right)={\mathit{G}}^{†}\left(\omega \right){\mathit{S}}_{\xi }^{\left(2\right)}\left(\omega \right)\mathit{G}\left(\omega \right).$$

(39)

For real, frequency-independent transfer coefficients $g_a$, this reduces to

$$S_{\mathrm{nuis}}^{\left(2\right)}\left(\omega \right)=\sum _a{g}_a^2{S}_{aa}^{\left(2\right)}\left(\omega \right)+2\sum _{a<b}{g}_a{g}_b\mathrm{Re}{S}_{ab}^{\left(2\right)}\left(\omega \right).$$

(40)

Thus, correlations among monitored channels can either enhance or suppress the nuisance contribution relative to the result obtained by adding their individual auto-spectra. They therefore cannot in general be neglected or absorbed into a positive residual auto-spectrum.

To describe possible correlations between the monitored nuisance sector and the residual channel $x\left(t\right)$, we further define

$${\left[{\mathit{S}}_{\xi x}^{\left(2\right)}\left(\omega \right)\right]}_a\equiv S_{ax}^{\left(2\right)}\left(\omega \right)={\int }_{-\infty }^{+\infty }d\tau e^{i\omega \tau }〈{\xi }_a(t+\tau )x\left(t\right)〉,$$

(41)

and

$$S_{xx}^{\left(2\right)}\left(\omega \right)={\int }_{-\infty }^{+\infty }d\tau e^{i\omega \tau }〈x(t+\tau )x\left(t\right)〉.$$

(42)

The total two-sided frequency-noise spectrum is therefore

$$\begin{array}{cc}\hfill S_{\delta \omega }^{\left(2\right)}\left(\omega \right)=& S_{\mathrm{nuis}}^{\left(2\right)}\left(\omega \right)+S_{xx}^{\left(2\right)}\left(\omega \right)\hfill \\ \hfill & +2\mathrm{Re}\left[{\mathit{G}}^{†}\left(\omega \right){\mathit{S}}_{\xi x}^{\left(2\right)}\left(\omega \right)\right].\hfill \end{array}$$

(43)

Equation (43) is the correlated-noise generalization of the simple additive spectral decomposition. In particular, the last term shows explicitly that an unresolved correlation between the monitored and residual sectors cannot in general be represented by an independent positive residual power spectrum.

With the filter normalization in Eq. (27), the decoherence functional corresponding to the two-sided spectral convention is

$${\chi }_{\mathsf{s}}\left(T\right)={\displaystyle \frac{1}{4\pi }}{\int }_{-\infty }^{+\infty }d\omega S_{\delta \omega }^{\left(2\right)}\left(\omega \right){\displaystyle \frac{F_{\mathsf{s}}\left(\omega T\right)}{{\omega }^2}}.$$

(44)

For broadband stochastic noise it is often convenient to use a one-sided spectrum. We define

$$S_{\delta \omega }^{\left(1\right)}\left(\omega \right)\equiv 2S_{\delta \omega }^{\left(2\right)}\left(\omega \right),\omega >0,$$

(45)

and analogously for the nuisance, residual, and cross-spectral quantities. For real scalar stationary noise, Eq. (44) is then equivalent to

$${\chi }_{\mathsf{s}}\left(T\right)={\displaystyle \frac{1}{4\pi }}{\int }_0^{+\infty }d\omega S_{\delta \omega }^{\left(1\right)}\left(\omega \right){\displaystyle \frac{F_{\mathsf{s}}\left(\omega T\right)}{{\omega }^2}}.$$

(46)

Equations (44) and (46) are therefore identical descriptions written with consistently normalized two-sided and one-sided spectra. In the dictionary sections below, broadband residual spectra may be written using the one-sided convention, whereas explicit pairs of spectral lines such as $\delta (\omega -{\omega }_0)+\delta (\omega +{\omega }_0)$ are most naturally written using the two-sided convention.

The monitored nuisance decoherence functional is consequently

$${\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{nuis}}\left(T\right)={\displaystyle \frac{1}{4\pi }}{\int }_{-\infty }^{+\infty }d\omega S_{\mathrm{nuis}}^{(2;\mathsf{e},\mathsf{r})}\left(\omega \right){\displaystyle \frac{F_{\mathsf{s}}\left(\omega T\right)}{{\omega }^2}},$$

(47)

or, equivalently,

$${\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{nuis}}\left(T\right)={\displaystyle \frac{1}{4\pi }}{\int }_0^{+\infty }d\omega S_{\mathrm{nuis}}^{(1;\mathsf{e},\mathsf{r})}\left(\omega \right){\displaystyle \frac{F_{\mathsf{s}}\left(\omega T\right)}{{\omega }^2}}.$$

(48)

For diagnostic purposes, the contribution associated with an individual auto-spectrum may still be defined as

$${\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\left(a\right)}\left(T\right)={\displaystyle \frac{1}{4\pi }}{\int }_0^{+\infty }d\omega {\left|G_a\left(\omega \right)\right|}^2{S}_{aa}^{(1;\mathsf{e},\mathsf{r})}\left(\omega \right){\displaystyle \frac{F_{\mathsf{s}}\left(\omega T\right)}{{\omega }^2}},$$

(49)

but these diagonal contributions do not in general exhaust the nuisance decoherence. The cross-spectral contributions are

$${\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\left(ab\right)}\left(T\right)={\displaystyle \frac{1}{2\pi }}{\int }_0^{+\infty }d\omega \mathrm{Re}\left[G_a^{*}\left(\omega \right)S_{ab}^{(1;\mathsf{e},\mathsf{r})}\left(\omega \right)G_b\left(\omega \right)\right]{\displaystyle \frac{F_{\mathsf{s}}\left(\omega T\right)}{{\omega }^2}},a<b.$$

(50)

Hence,

$${\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{nuis}}\left(T\right)=\sum _a{\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\left(a\right)}\left(T\right)+\sum _{a<b}{\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\left(ab\right)}\left(T\right).$$

(51)

The second sum may contain contributions of either sign, although the complete physical nuisance spectrum $S_{\mathrm{nuis}}$ and its associated total dephasing functional remain non-negative.

In the notation of Section 2, the experimentally inferred nuisance model is therefore

$${\widehat{\chi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{nuis}}\left(T\right)\equiv \widehat{{\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{nuis}}\left(T\right)},$$

(52)

where the hat emphasizes that the quantity is reconstructed from measured auto- and cross-spectra together with calibrated transfer functions. The uncertainties of the measured spectral matrix and of $\mathit{G}\left(\omega \right)$ must be propagated into the nuisance covariance. In particular, statistically significant off-diagonal elements of ${\mathit{S}}_{\xi }$ should not be discarded.

The total decoherence functional obtained from Eq. (43) can be written as

$${\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)={\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{nuis}}\left(T\right)+{\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{x}}\left(T\right)+{\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{cross}}\left(T\right),$$

(53)

where

$${\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{x}}\left(T\right)={\displaystyle \frac{1}{4\pi }}{\int }_0^{+\infty }d\omega S_{xx}^{(1;\mathsf{e},\mathsf{r})}\left(\omega \right){\displaystyle \frac{F_{\mathsf{s}}\left(\omega T\right)}{{\omega }^2}},$$

(54)

and

$${\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{cross}}\left(T\right)={\displaystyle \frac{1}{2\pi }}{\int }_0^{+\infty }d\omega \mathrm{Re}\left[{\mathit{G}}^{†}\left(\omega \right){\mathit{S}}_{\xi x}^{(1;\mathsf{e},\mathsf{r})}\left(\omega \right)\right]{\displaystyle \frac{F_{\mathsf{s}}\left(\omega T\right)}{{\omega }^2}}.$$

(55)

The cross contribution is not required to be positive.

The cartography observable introduced in Section 2 is

$${\mathrm{\Xi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)=-\ln W_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)-{\widehat{\chi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{nuis}}\left(T\right).$$

(56)

Its interpretation depends on the statistical relation between the monitored nuisance sector and the residual channel.

If the residual is second-order uncorrelated with every monitored nuisance variable,

$$S_{ax}^{\left(2\right)}\left(\omega \right)=0,a\in \mathcal{A},$$

(57)

then

$$S_{\delta \omega }^{\left(2\right)}\left(\omega \right)=S_{\mathrm{nuis}}^{\left(2\right)}\left(\omega \right)+S_{xx}^{\left(2\right)}\left(\omega \right),$$

(58)

and

$${\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)={\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{nuis}}\left(T\right)+{\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{x}}\left(T\right).$$

(59)

In the idealized limit of an unbiased nuisance model,

$${\mathrm{\Xi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)\simeq {\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{x}}\left(T\right)\ge 0,$$

(60)

up to statistical uncertainty, calibration uncertainty, and limitations of the Gaussian or second-cumulant approximation. Under this stronger interpretation, the conservative null hypothesis is

$$S_{xx}^{(\mathsf{e},\mathsf{r})}\left(\omega \right)=0,{\mathrm{\Xi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)\simeq 0.$$

(61)

If Eq. (57) is not satisfied, however, $\mathrm{\Xi }$ is not the decoherence functional of an independent positive residual spectrum. Instead,

$${\mathrm{\Xi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)\simeq {\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{x}}\left(T\right)+{\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{cross}}\left(T\right),$$

(62)

and the second term may be positive or negative. In this more general case, $\mathrm{\Xi }$ should be interpreted conservatively as a signed discrepancy functional between the measured dephasing and the complete monitored nuisance prediction. A statistically significant negative value of $\mathrm{\Xi }$ is therefore not intrinsically inconsistent; it may signal anticorrelation between unresolved dynamics and the monitored nuisance sector, or a deficiency in the calibrated nuisance model.

This distinction is important for the logic of decoherence cartography. Correlations among monitored environmental channels belong to the nuisance model itself and are retained through the full cross-spectral matrix ${\mathit{S}}_{\xi }$. The additional condition Eq. (57) is required only if the residual $\mathrm{\Xi }$ is to be interpreted as the dephasing produced by a statistically independent stochastic channel. The more conservative cartographic analysis does not require this assumption: it tests whether a reproducible discrepancy remains after the complete monitored nuisance contribution, including measured cross correlations, has been propagated through the filter function.

As emphasized in Eq. (29), an additional frequency perturbation may contain both coherent and stochastic components. The coherent component $\delta {\omega }_{\mathrm{coh}}\left(t\right)$ contributes to the reproducible accumulated phase ${\mathrm{\Phi }}_{\mathsf{s}}^{\mathrm{coh}}\left(T\right)$, whereas the stochastic component $\delta {\omega }_{\mathrm{stoch}}\left(t\right)$ contributes to the coherence magnitude through its characteristic function and, in the Gaussian or second-cumulant limit, through the spectral decoherence functional ${\chi }_{\mathsf{s}}\left(T\right)$. This distinction is particularly useful for the dictionaries developed below: coherent Lorentz-violating or ultralight-field signals may appear primarily as reproducible phase or narrow-band frequency modulations, whereas broadband stochastic or burst-like sectors naturally produce coherence loss.

A general qubit–environment interaction may also contain transverse terms proportional to ${\sigma }_x$ and ${\sigma }_y$, which can induce real transitions between the qubit energy levels [35,36,48,49]. We denote by ${\mathrm{\Gamma }}_{\uparrow }\equiv {\mathrm{\Gamma }}_{g\to e}$ the excitation rate from the lower-energy state g to the upper-energy state e, and by ${\mathrm{\Gamma }}_{\downarrow }\equiv {\mathrm{\Gamma }}_{e\to g}$ the corresponding relaxation rate. In the approximately Markovian regime, $T_1^{-1}={\mathrm{\Gamma }}_{\uparrow }+{\mathrm{\Gamma }}_{\downarrow }$, and population relaxation contributes the additional coherence factor $\exp [-({\mathrm{\Gamma }}_{\uparrow }+{\mathrm{\Gamma }}_{\downarrow })T/2]$. Such spin transitions do not reduce the particle-survival probability, since they preserve the trapped particle and the trace of the qubit density matrix. The pure dephasing approximation therefore requires $({\mathrm{\Gamma }}_{\uparrow }+{\mathrm{\Gamma }}_{\downarrow })T\ll 1$, or, if this condition is not satisfied, an independently measured relaxation contribution must be included in ${\widehat{\chi }}^{\mathrm{nuis}}$. A sequence-independent residual should in particular be checked against unmodeled spin relaxation, since a Hahn echo can refocus slow longitudinal frequency noise but cannot reverse an incoherent spin transition.

The formalism developed here is species independent at the level of the reduced dynamical map, switching-function description, coherence functional, and cartography observable. Species dependence enters through the transition frequency, control Hamiltonian, transfer functions $\mathit{G}$, monitored spectral matrix ${\mathit{S}}_{\xi }$, and the microscopic residual dictionary. In this sense, electron, positron, proton, and antiproton realizations instantiate the same cartographic framework with different calibration and response layers.

The scientific roles of the four platforms are nevertheless distinct. Electron qubits offer strong magnetic response, mature single-particle spectroscopy, and comparatively accessible coherent spin manipulation. Positron qubits provide the corresponding antimatter-lepton platform and, when compared with electrons, enable charge-sign and matter–antimatter differential tests in the lepton sector. Proton qubits furnish the closest matter analogue of the antiproton implementation and share a closely related trapped-particle metrology architecture. Antiproton qubits are distinctive because comparison with protons provides a baryon–antibaryon differential observable, opening access to antimatter-sensitive, $CPT$-odd, and baryon-dependent sectors that cannot be isolated as sharply in matter-only implementations. Taken together, the $e^{-}/e^{+}$ and $p/\overline{p}$ pairs make it possible to test whether a residual is associated primarily with the lepton sector, baryon sector, charge sign, or more general particle–antiparticle character.

The quantities introduced in this section constitute the dynamical ingredients of decoherence cartography. The stochastic shift $\delta \omega \left(t\right)$ specifies the perturbation at the effective Hamiltonian level, the cross-spectral matrix ${\mathit{S}}_{\xi }^{(\mathsf{e},\mathsf{r})}\left(\omega \right)$ describes the monitored environment and its internal correlations, and ${\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)$ determines the resulting coherence loss under a chosen control sequence. The cartographic residual ${\mathrm{\Xi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)$ is therefore the experimentally inferred remainder after the complete monitored nuisance contribution has been propagated through the filter-function response. The next step is to ask what generic temporal and spectral forms such a remainder may exhibit, which motivates the toy residual classes introduced in Section 4.

4. Toy Model: Residual Site-Dependent Spectra and Sequence Discrimination

In this section we classify the possible residual structures that may appear in ${\mathrm{\Xi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)$ before assigning them to any particular microscopic model. This separation is important: the same cartographic observable may be generated by ordinary unmodeled noise, by a nonstandard particle physics sector, or by a cosmology-motivated background. The role of the toy models is therefore to describe the effective residual phenomenology—broadband colored spectra, narrow-band harmonic signals, and burst-like or transient processes—that later dictionary sections map onto concrete microscopic sectors.

To make the cartography program operational, it is useful to exhibit a minimal toy model in which the residual environment-conditioned contribution can be inferred from transport data in a falsifiable way. We therefore write the measured coherence of a localized qubit at site a, for control sequence $\mathsf{s}$ and interrogation time T, as

$$W_{a,\mathsf{s}}\left(T\right)=\exp \left[-{\chi }_{a,\mathsf{s}}^{\mathrm{pred}}\left(T\right)-{\chi }_{a,\mathsf{s}}^{\mathrm{res}}\left(T\right)\right],$$

(63)

where ${\chi }_{a,\mathsf{s}}^{\mathrm{pred}}$ is the nuisance contribution predicted from independently monitored magnetic, vibrational, thermal, timing, and control channels, while ${\chi }_{a,\mathsf{s}}^{\mathrm{res}}$ denotes the residual environment-conditioned functional to be inferred.

A minimal residual-noise model is a site-dependent low-frequency spectrum,

$$S_a^{\mathrm{res}}\left(\omega \right)=A_a{\omega }^{-\alpha }\mathrm{\Theta }(\omega -{\omega }_{\mathrm{IR}})\mathrm{\Theta }({\omega }_{\mathrm{UV}}-\omega ),0<\alpha <2,$$

(64)

with site-dependent amplitude $A_a$, common spectral index $\alpha$, and infrared/ultraviolet cutoffs ${\omega }_{\mathrm{IR}}$ and ${\omega }_{\mathrm{UV}}$. The interval $0<\alpha <2$ is meant as a phenomenological range for colored $1/f^{\alpha }$-type noise, as commonly used in qubit-noise modeling [50,51]. Because the cutoffs in Eq. (64) are kept explicit, the residual-noise integral below is finite for the phenomenological range considered here.

Within the standard filter-function description [33,34], the residual decoherence functional is

$${\chi }_{a,\mathsf{s}}^{\mathrm{res}}\left(T\right)={\displaystyle \frac{1}{4\pi }}{\int }_0^{\infty }d\omega S_a^{\mathrm{res}}\left(\omega \right){\displaystyle \frac{F_{\mathsf{s}}\left(\omega T\right)}{{\omega }^2}},$$

(65)

where $F_{\mathsf{s}}$ is the filter function of the control sequence, as in Eq. (28). If the interrogation time lies in an intermediate regime in which the cutoffs do not dominate Eq. (65), one may attempt to write the scale-free form

$${\chi }_{a,\mathsf{s}}^{\mathrm{res}}\left(T\right)\simeq K_{\mathsf{s}}\left(\alpha \right)A_a{T}^{1+\alpha },$$

(66)

where

$$K_{\mathsf{s}}\left(\alpha \right)={\displaystyle \frac{1}{4\pi }}{\int }_0^{\infty }du{\displaystyle \frac{F_{\mathsf{s}}\left(u\right)}{u^{2+\alpha }}}$$

(67)

is a sequence-dependent coefficient.

The validity of Eq. (66), however, is more restrictive than the cutoff-regulated model in Eq. (64). It requires the scale-free coefficient $K_{\mathsf{s}}\left(\alpha \right)$ to converge. For the Ramsey filter,

$$F_{\mathrm{R}}\left(u\right)=4{\sin }^2\left({\displaystyle \frac{u}{2}}\right)\sim u^2(u\to 0),$$

(68)

so the integrand in Eq. (67) behaves as $u^{-\alpha }$ in the infrared. The Ramsey scale-free coefficient is therefore convergent only for $-1<\alpha <1$. By contrast, for Hahn echo,

$$F_{\mathrm{E}}\left(u\right)=16{\sin }^4\left({\displaystyle \frac{u}{4}}\right)\sim {\displaystyle \frac{u^4}{16}}(u\to 0),$$

(69)

so the integrand behaves as $u^{2-\alpha }$ in the infrared. The Hahn-echo scale-free coefficient is therefore convergent for the wider range $-1<\alpha <3$. In both cases the ultraviolet condition is $\alpha >-1$, since the filter functions are bounded at large u.

Thus the phenomenological range $0<\alpha <2$ is compatible with the cutoff-regulated spectrum in Eq. (64), and it is also compatible with the scale-free Hahn-echo approximation. For Ramsey, however, the interval $1\le \alpha <2$ is infrared sensitive and cannot be summarized by a universal cutoff-independent coefficient $K_{\mathrm{R}}\left(\alpha \right)$. In that regime the explicit cutoff dependence on ${\omega }_{\mathrm{IR}}$ must be retained.

Equation (66) shows that a residual cartographic signal is not characterized by a single site-dependent decay rate. It contains three separable pieces of information: the site-dependent amplitude $A_a$, the interrogation-time scaling $T^{1+\alpha }$, and the sequence-dependent coefficient $K_{\mathsf{s}}\left(\alpha \right)$. The first identifies where the residual is largest, the second constrains the spectral character of the residual channel, and the third tests that interpretation by changing the filter function of the measurement. Thus measurements at multiple sites, times, and control sequences can distinguish a genuine residual spectral family from an arbitrary excess decay. This is precisely the logic exploited in filter-function based noise spectroscopy and sequence discrimination [34,36].

It is convenient to define a differential site observable

$${\mathrm{\Xi }}_{ab}^{\left(\mathsf{s}\right)}\left(T\right)\equiv -\ln {\displaystyle \frac{W_{a,\mathsf{s}}\left(T\right)}{W_{b,\mathsf{s}}\left(T\right)}}-\left[{\chi }_{a,\mathsf{s}}^{\mathrm{pred}}\left(T\right)-{\chi }_{b,\mathsf{s}}^{\mathrm{pred}}\left(T\right)\right],$$

(70)

which removes the nuisance model to first order and isolates the residual site difference. In the power-law toy model of Eq. (64), motivated by the broad phenomenology of low-frequency and $1/f^{\alpha }$-type noise, [51] one finds

$${\mathrm{\Xi }}_{ab}^{\left(\mathsf{s}\right)}\left(T\right)\simeq K_{\mathsf{s}}\left(\alpha \right)\left(A_a-A_b\right)T^{1+\alpha }.$$

(71)

Several immediate consequences follow. First, if a common spectral family describes all sites, then all pairwise observables ${\mathrm{\Xi }}_{ab}^{\left(\mathsf{s}\right)}$ must exhibit the same time exponent $1+\alpha$. Second, the sequence ratio

$${\mathcal{R}}_{ab}\left(T\right)\equiv {\displaystyle \frac{{\mathrm{\Xi }}_{ab}^{\left(\mathrm{E}\right)}\left(T\right)}{{\mathrm{\Xi }}_{ab}^{\left(\mathrm{R}\right)}\left(T\right)}}\simeq {\displaystyle \frac{K_{\mathrm{E}}\left(\alpha \right)}{K_{\mathrm{R}}\left(\alpha \right)}}$$

(72)

is independent of the site pair and, in the scaling regime, independent of T. Thus a stable, pair-independent ${\mathcal{R}}_{ab}$ is a nontrivial diagnostic of a common residual spectral family, whereas strong pair dependence or drift with T falsifies the single-family model.

As a contrasting toy model, consider a rare-burst environment in which the localized qubit experiences stochastic phase kicks at site a according to a Poisson process of rate ${\lambda }_a$. This is a compound-Poisson phase-noise model, a standard construction in stochastic-process descriptions of randomly occurring kicks [52]. Let each burst produce a sequence-filtered phase kick $\delta {\varphi }_{\mathsf{s}}$, drawn from a site- and sequence-dependent distribution with characteristic function

$${\mathcal{F}}_{a,\mathsf{s}}\left(k\right)\equiv {〈e^{ik\delta {\varphi }_{\mathsf{s}}}〉}_{a,\mathsf{s}}.$$

(73)

If $N_T$ burst events occur during the interrogation interval, the total burst-induced phase is

$${\mathrm{\Phi }}_{a,\mathsf{s}}^{\mathrm{burst}}\left(T\right)={\displaystyle \sum _{j=1}^{N_T}}\delta {\varphi }_{\mathsf{s},j},$$

(74)

where $\delta {\varphi }_{\mathsf{s},j}$ is the phase kick produced by the j-th burst after filtering by the control sequence $\mathsf{s}$. The random variable $N_T$ is Poisson distributed with mean ${\lambda }_{a}T$, while the individual kicks $\delta {\varphi }_{\mathsf{s},j}$ are drawn independently from the distribution entering Eq. (73).

The burst-only ensemble-averaged coherence factor is then

$${\mathcal{B}}_{a,\mathsf{s}}\left(T\right)\equiv 〈e^{-i{\mathrm{\Phi }}_{a,\mathsf{s}}^{\mathrm{burst}}\left(T\right)}〉=\exp \left[{\lambda }_{a}T\left({\mathcal{F}}_{a,\mathsf{s}}(-1)-1\right)\right],$$

(75)

which is the characteristic function of the compound-Poisson accumulated phase and is the same type of object that appears in stochastic line-shape theory [52,53] 8. Including the predicted nuisance decoherence, the full off-diagonal coherence factor is

$${\mathcal{C}}_{a,\mathsf{s}}\left(T\right)\simeq \exp \left[-{\chi }_{a,\mathsf{s}}^{\mathrm{pred}}\left(T\right)+{\lambda }_{a}T\left({\mathcal{F}}_{a,\mathsf{s}}(-1)-1\right)\right],$$

(76)

provided the bursts are short on the timescale of the control sequence. This quantity is generally complex and therefore should not itself be identified with the real coherence signal. The measured coherence magnitude is instead

$$W_{a,\mathsf{s}}^{\mathrm{burst}}\left(T\right)=\left|{\mathcal{C}}_{a,\mathsf{s}}\left(T\right)\right|\simeq \exp \left[-{\chi }_{a,\mathsf{s}}^{\mathrm{pred}}\left(T\right)-{\lambda }_{a}T\left(1-\mathrm{Re}{\mathcal{F}}_{a,\mathsf{s}}(-1)\right)\right].$$

(77)

The imaginary part of ${\mathcal{F}}_{a,\mathsf{s}}(-1)$ contributes instead to an effective burst-induced phase shift,

$${\mathrm{\Phi }}_{a,\mathsf{s}}^{\mathrm{burst},\mathrm{eff}}\left(T\right)=-{\lambda }_{a}T\mathrm{Im}{\mathcal{F}}_{a,\mathsf{s}}(-1).$$

(78)

Thus the burst model contributes both to coherence loss and, in general, to the accumulated phase. For a symmetric phase-kick distribution, $\mathrm{Im}{\mathcal{F}}_{a,\mathsf{s}}(-1)=0$, and the burst contribution is purely dephasing.

This kind of description is closely related to standard stochastic line-shape theory and to models of low-frequency telegraph or burst-like qubit noise [51,53,54]. In this case, the residual edge observable becomes

$${\mathrm{\Xi }}_{ab}^{\left(\mathsf{s}\right)}\left(T\right)\simeq \left[{\lambda }_a\left(1-\mathrm{Re}{\mathcal{F}}_{a,\mathsf{s}}(-1)\right)-{\lambda }_b\left(1-\mathrm{Re}{\mathcal{F}}_{b,\mathsf{s}}(-1)\right)\right]T,$$

(79)

after subtraction of the predicted nuisance contribution. If the phase-kick distributions are symmetric and only weakly sequence dependent, this expression reduces to an approximately linear growth in T with little dependence on the choice of control sequence. The power-law and burst models are therefore distinguishable at the level of cartographic observables: the former predicts a nontrivial sequence-dependent scaling exponent, whereas the latter predicts nearly sequence-independent linear growth.

The toy models introduced in this section are not intended as realistic microscopic theories, but as an effective low-energy classification of the residual structures accessible to decoherence cartography. They provide the intermediate layer between the general operational formalism and the specific BSM and cosmology-motivated dictionaries developed below. Broadband colored residuals may describe unmodeled low-frequency noise, collapse-like diffusion, or related excess-noise sectors; narrow-band harmonic residuals provide the natural effective description of coherent ultralight fields and Lorentz-violating backgrounds; and burst-like residuals characterize scattering processes or transient phenomena such as topological defects [10,12,55,56,57].

These residual classes are neither mutually exclusive nor intended as a classification of ultraviolet theories. Rather, they constitute an experimentally accessible taxonomy of the structures that a transported localized qubit can measure, including coherent phase shifts, harmonic modulations, broadband dephasing spectra, burst processes, and matter–antimatter differential responses. A nonzero residual becomes physically interpretable only if it is reproducible under revisits and site permutations, exhibits a consistent sequence dependence, and can be mapped onto one or more of these effective structures. Conversely, the cartographic null hypothesis corresponds to ${\mathrm{\Xi }}_{ab}^{\left(\mathsf{s}\right)}\left(T\right)\simeq 0$, within uncertainty, for all tested site pairs, sequences, and revisits after nuisance subtraction. The dictionary sections below perform the subsequent mapping from these observable residual structures to representative microscopic sectors.

5. Graph Structure, Differential Observables, and Identifiability

The cartography program is intrinsically comparative. A single measurement of the coherence of a localized qubit at one site can reveal an excess decay relative to a nuisance model, but by itself it cannot determine whether that excess is site specific, calibration dependent, sequence dependent, or caused by an unmodeled global drift. For this reason, the natural object is not an isolated site measurement but a graph of site comparisons.

Let

$$\mathcal{G}=(\mathcal{E},\mathcal{L})$$

(80)

be a graph whose nodes $\mathcal{E}$ are laboratory sites or environment classes and whose edges $\mathcal{L}$ are experimentally realized comparisons between pairs of sites [58,59]. An edge $\mathcal{\ell }=(a,b)\in \mathcal{L}$ means that the same species of localized qubit, control sequence, and analysis protocol have been used to compare site a with site b. The graph need not be complete. In practice, the first generation of experiments may involve only a small connected graph, for example an accelerator hall, a nearby quiet laboratory, and one shielded or underground location.

For a fixed control sequence $\mathsf{s}$, revisit index $\mathsf{r}$, and interrogation time T, the node-level residual observable is

$${\mathrm{\Xi }}_{a,\mathsf{s},\mathsf{r}}\left(T\right)=-\ln W_{a,\mathsf{s},\mathsf{r}}\left(T\right)-{\widehat{\chi }}_{a,\mathsf{s},\mathsf{r}}^{\mathrm{nuis}}\left(T\right),$$

(81)

where $a\in \mathcal{E}$. A direct comparison between two sites is then encoded in the edge observable

$$\mathrm{\Delta }{\mathrm{\Xi }}_{ab,\mathsf{s},\mathsf{r}}\left(T\right)\equiv {\mathrm{\Xi }}_{a,\mathsf{s},\mathsf{r}}\left(T\right)-{\mathrm{\Xi }}_{b,\mathsf{s},\mathsf{r}}\left(T\right).$$

(82)

Equivalently, using Eq. (81), this can be written directly in terms of measured coherence signals and nuisance estimates as

$$\mathrm{\Delta }{\mathrm{\Xi }}_{ab,\mathsf{s},\mathsf{r}}\left(T\right)=-\ln {\displaystyle \frac{W_{a,\mathsf{s},\mathsf{r}}\left(T\right)}{W_{b,\mathsf{s},\mathsf{r}}\left(T\right)}}-\left[{\widehat{\chi }}_{a,\mathsf{s},\mathsf{r}}^{\mathrm{nuis}}\left(T\right)-{\widehat{\chi }}_{b,\mathsf{s},\mathsf{r}}^{\mathrm{nuis}}\left(T\right)\right].$$

(83)

This form makes clear why pairwise comparisons are useful: any nuisance contribution that is common to both sites, or any global normalization error that enters the two measurements in the same way, cancels to first order, as in relative measurement estimation on graphs [60].

The conservative null model is

$${\mathrm{\Xi }}_{a,\mathsf{s},\mathsf{r}}\left(T\right)=0\mathrm{for}\mathrm{all}(a,\mathsf{s},\mathsf{r},T),$$

(84)

and therefore

$$\mathrm{\Delta }{\mathrm{\Xi }}_{ab,\mathsf{s},\mathsf{r}}\left(T\right)=0\mathrm{for}\mathrm{all}((a,b),\mathsf{s},\mathsf{r},T).$$

(85)

A nonzero edge observable does not by itself identify new physics. It only states that the two sites cannot both be explained by the same nuisance-subtracted null model. The task of graph inference is to determine whether the collection of edge observables is consistent with reproducible node-level residuals, edge-specific transport systematics, or statistical fluctuations.

A useful first model is the node-potential model,

$$\mathrm{\Delta }{\mathrm{\Xi }}_{ab,\mathsf{s},\mathsf{r}}\left(T\right)={\mathrm{\Theta }}_{a,\mathsf{s}}\left(T\right)-{\mathrm{\Theta }}_{b,\mathsf{s}}\left(T\right)+{ϵ}_{ab,\mathsf{s},\mathsf{r}}\left(T\right),$$

(86)

where ${\mathrm{\Theta }}_{a,\mathsf{s}}\left(T\right)$ is the reproducible residual associated with site a and sequence $\mathsf{s}$, while ${ϵ}_{ab,\mathsf{s},\mathsf{r}}\left(T\right)$ denotes statistical uncertainty and revisit-dependent modeling error. In this model, a true cartographic signal is a scalar field over the graph nodes. The absolute offset of all ${\mathrm{\Theta }}_{a,\mathsf{s}}$ is unobservable from pairwise differences alone, so one must fix a gauge, for example

$$\sum _{a\in \mathcal{E}}{\mathrm{\Theta }}_{a,\mathsf{s}}\left(T\right)=0,$$

(87)

or choose one reference site $a_0$ and set

$${\mathrm{\Theta }}_{a_0,\mathsf{s}}\left(T\right)=0.$$

(88)

Only differences of node residuals are physically meaningful unless an absolute external calibration of $\mathrm{\Xi }$ is available.

For a connected graph, the node-potential model can be written compactly using the incidence matrix [58,60]. Assign an arbitrary orientation to each edge. If the oriented edge ℓ connects two sites as $\mathcal{\ell }:b\to a$, then b is called the tail of ℓ and a is called the head of ℓ 9. The incidence matrix $B_{\mathcal{\ell }c}$ is then defined by

$$B_{\mathcal{\ell }c}=\left\{\begin{array}{cc}+1,\hfill & c=a\mathrm{if}\mathcal{\ell }:b\to a,\hfill \\ -1,\hfill & c=b\mathrm{if}\mathcal{\ell }:b\to a,\hfill \\ 0,\hfill & c\notin \{a,b\}.\hfill \end{array}\right.$$

(89)

Let ${\mathbf{d}}_{\mathsf{s}}\left(T\right)$ be the vector of measured edge observables and ${\mathbf{\Theta }}_{\mathsf{s}}\left(T\right)$ the vector of node residuals. Then

$${\mathbf{d}}_{\mathsf{s}}\left(T\right)=B{\mathbf{\Theta }}_{\mathsf{s}}\left(T\right)+{ϵ}_{\mathsf{s}}\left(T\right).$$

(90)

If the edge-noise covariance is $C_{\mathsf{s}}\left(T\right)$, the weighted least-squares estimator [60,61] of the node residuals is obtained by minimizing

$${\chi }_{\mathrm{graph}}^2={\left[{\mathbf{d}}_{\mathsf{s}}\left(T\right)-B{\mathbf{\Theta }}_{\mathsf{s}}\left(T\right)\right]}^{\mathrm{T}}{C}_{\mathsf{s}}^{-1}\left(T\right)\left[{\mathbf{d}}_{\mathsf{s}}\left(T\right)-B{\mathbf{\Theta }}_{\mathsf{s}}\left(T\right)\right],$$

(91)

subject to the gauge condition in Eq. (87) or Eq. (88). This provides a simple inference procedure for deciding whether the observed residuals are compatible with a reproducible site map.

The graph structure also provides an important internal consistency test. If the residuals are purely node based, then the sum of edge observables around any closed cycle $\mathcal{C}\subset \mathcal{G}$ must vanish:

$$\sum _{\mathcal{\ell }\in \mathcal{C}}{s}_{\mathcal{\ell },\mathcal{C}}\mathrm{\Delta }{\mathrm{\Xi }}_{\mathcal{\ell },\mathsf{s},\mathsf{r}}\left(T\right)=0,$$

(92)

where $s_{\mathcal{\ell },\mathcal{C}}=\pm 1$ records whether the orientation of edge ℓ agrees with the orientation of the cycle. A statistically significant violation of Eq. (92) indicates that the data cannot be represented by site potentials alone. Such a violation would point instead to edge-dependent transport effects, path-dependent systematics, nonstationary drift, or a residual that depends on the history of relocation rather than only on the endpoint site.

To account for this possibility, one may extend Eq. (86) to

$$\mathrm{\Delta }{\mathrm{\Xi }}_{ab,\mathsf{s},\mathsf{r}}\left(T\right)={\mathrm{\Theta }}_{a,\mathsf{s}}\left(T\right)-{\mathrm{\Theta }}_{b,\mathsf{s}}\left(T\right)+{\mathrm{\Lambda }}_{ab,\mathsf{s}}\left(T\right)+{ϵ}_{ab,\mathsf{s},\mathsf{r}}\left(T\right),$$

(93)

where ${\mathrm{\Lambda }}_{ab,\mathsf{s}}\left(T\right)$ is an edge-specific residual. A nonzero ${\mathrm{\Theta }}_{a,\mathsf{s}}$ represents a site-conditioned effect, whereas a nonzero ${\mathrm{\Lambda }}_{ab,\mathsf{s}}$ represents a comparison- or path-conditioned effect. This distinction is crucial. A site-conditioned residual could be associated with local overburden, shielding, magnetic environment, or a local exotic background. A path-conditioned residual could instead arise from transport vibration, route-dependent magnetic exposure, acceleration history, or other relocation-specific effects.

Revisits provide the next identifiability layer. For a fixed site a, sequence $\mathsf{s}$, and interrogation time T, the residuals measured on different visits should satisfy

$${\mathrm{\Xi }}_{a,\mathsf{s},{\mathsf{r}}_1}\left(T\right)-{\mathrm{\Xi }}_{a,\mathsf{s},{\mathsf{r}}_2}\left(T\right)\simeq 0$$

(94)

within the uncertainty of the nuisance model, if the residual is genuinely site conditioned and stationary. Failure of Eq. (94) does not necessarily falsify the cartography program; rather, it changes the interpretation. It may indicate a transient background, a seasonal or sidereal modulation, a slow instrumental drift, or an unmodeled dependence on transport history.

Control sequence dependence gives an additional discriminator. A residual that appears under Ramsey control but is suppressed under echo control is naturally interpreted as low-frequency dephasing. A residual that survives echo and multipulse sequences may instead indicate high-frequency noise, burst-like events, or pulse-independent loss mechanisms. Therefore one should treat the graph as sequence resolved:

$$\mathcal{G}⟶{\mathcal{G}}_{\mathsf{s},T},$$

(95)

where each choice of $\mathsf{s}$ and T defines a different weighted graph over the same set of physical sites [62]. The comparison of maps across $\mathsf{s}$ and T is part of the signal classification, not merely a robustness check.

The graph formalism also clarifies what is meant by discovery and falsification in this proposal. A null result is not simply the absence of visible excess decoherence at one site. It is the stronger statement that

$${\mathbf{d}}_{\mathsf{s}}\left(T\right)\simeq \mathbf{0}$$

(96)

for all measured edges, sequences, revisits, and interrogation times after nuisance subtraction. A site-like non-null result is a reproducible node map ${\mathrm{\Theta }}_{\mathsf{s}}\left(T\right)$ that satisfies cycle consistency and revisit stability. An edge-like non-null result is a reproducible failure of cycle consistency or a stable nonzero ${\mathrm{\Lambda }}_{ab,\mathsf{s}}\left(T\right)$. A transient non-null result is a violation of revisit stability with timing correlations across sites. These cases correspond to different physical interpretations and should not be conflated.

In this sense, the graph layer is the statistical and logical backbone of decoherence cartography. The operational observables define what is measured, the dynamics define how residual noise enters, and the toy models define possible residual classes. The graph formalism determines whether the measured residuals form a reproducible map over sites, a transport-path effect over edges, or a time-dependent transient pattern over revisits. Only after this classification is performed is it meaningful to apply the microscopic dictionaries for millicharged sectors, Lorentz-violating backgrounds, collapse-like noise, ultralight fields, or other BSM and cosmology-motivated sectors.

6. Inference Strategy and Decision Criteria

The falsifiability of decoherence cartography rests on treating the residual observables as a constrained statistical data set rather than as isolated excess-decay measurements. In this section we discuss how to convert the operational observables defined above into a falsifiable statement about residual decoherence. The primitive data are the measured coherence signals, phases, and survival probabilities obtained for a set of sites, control sequences, revisits, and interrogation times. For the decoherence cartography problem, the relevant derived data are the residual observables, presented by Eq. (56) and, for differential comparisons, expressed by Eq. (82). The central statistical question is whether these quantities are consistent with zero within the uncertainty of the nuisance model, or whether they require a reproducible residual component.

We collect all measured residuals into a data vector

$$\mathbf{x}=\left\{{\mathrm{\Xi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)\right\},$$

(97)

or, in the graph formulation, into an edge-data vector

$$\mathbf{d}=\left\{\mathrm{\Delta }{\mathrm{\Xi }}_{ab,\mathsf{s},\mathsf{r}}\left(T\right)\right\}.$$

(98)

The covariance of these data has two conceptually distinct contributions,

$$C=C_{\mathrm{stat}}+C_{\mathrm{nuis}},$$

(99)

where $C_{\mathrm{stat}}$ denotes statistical readout uncertainty and $C_{\mathrm{nuis}}$ denotes uncertainty propagated from the monitored nuisance channels. The latter includes imperfect knowledge of transfer coefficients, finite precision in measured magnetic or vibrational spectra, thermal and timing diagnostics, and any uncertainty in the filter-function reconstruction of ${\widehat{\chi }}^{\mathrm{nuis}}$.

The conservative null model is

$${\mathcal{H}}_0:{\mathrm{\Xi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)=0\mathrm{for}\mathrm{all}\mathrm{measured}(\mathsf{e},\mathsf{s},\mathsf{r},T).$$

(100)

Equivalently, in the graph representation,

$${\mathcal{H}}_0:\mathrm{\Delta }{\mathrm{\Xi }}_{ab,\mathsf{s},\mathsf{r}}\left(T\right)=0\mathrm{for}\mathrm{all}\mathrm{measured}\mathrm{edges}.$$

(101)

A simple null statistic is therefore

$${\chi }_0^2={\mathbf{x}}^{\mathrm{T}}{C}^{-1}\mathbf{x},$$

(102)

or, for purely differential graph data,

$${\chi }_{0,\mathrm{graph}}^2={\mathbf{d}}^{\mathrm{T}}{C}_{\mathrm{edge}}^{-1}\mathbf{d}.$$

(103)

Here $C_{\mathrm{edge}}$ denotes the covariance matrix of the edge-data vector $\mathbf{d}=\left\{\mathrm{\Delta }{\mathrm{\Xi }}_{ab,\mathsf{s},\mathsf{r}}\left(T\right)\right\}$ (see Eq. (91))

$${\left(C_{\mathrm{edge}}\right)}_{\mathcal{\ell }{\mathcal{\ell }}^{\prime }}=\mathrm{Cov}\left[\mathrm{\Delta }{\mathrm{\Xi }}_{\mathcal{\ell },\mathsf{s},\mathsf{r}}\left(T\right),\mathrm{\Delta }{\mathrm{\Xi }}_{{\mathcal{\ell }}^{\prime },{\mathsf{s}}^{\prime },{\mathsf{r}}^{\prime }}\left(T^{\prime }\right)\right].$$

(104)

Its diagonal entries contain the variances of the pairwise residual observables, while its off-diagonal entries encode correlations between different edge comparisons arising from shared sites, common calibration parameters, common nuisance-channel uncertainties, or repeated use of the same transported qubit platform. A failure of the null model means that the measured residuals cannot be explained by monitored nuisance channels alone. Such a failure is not by itself a discovery claim; it is the beginning of residual classification.

A first non-null alternative is a site-conditioned residual,

$${\mathcal{H}}_{\mathrm{site}}:{\mathrm{\Xi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)={\mathrm{\Theta }}_{\mathsf{e},\mathsf{s}}\left(T\right)+{ϵ}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right),$$

(105)

where ${\mathrm{\Theta }}_{\mathsf{e},\mathsf{s}}\left(T\right)$ is a reproducible site-dependent contribution and $ϵ$ denotes statistical and revisit-dependent modeling error. In the graph language this becomes

$$\mathrm{\Delta }{\mathrm{\Xi }}_{ab,\mathsf{s},\mathsf{r}}\left(T\right)={\mathrm{\Theta }}_{a,\mathsf{s}}\left(T\right)-{\mathrm{\Theta }}_{b,\mathsf{s}}\left(T\right)+{ϵ}_{ab,\mathsf{s},\mathsf{r}}\left(T\right).$$

(106)

This model is appropriate when the residual is associated with a local environment, for example overburden, shielding, local backgrounds, or a site-dependent exotic field.

A second non-null alternative is an edge- or path-conditioned residual, where “edge” refers to the graph edges $\mathcal{\ell }=(a,b)\in \mathcal{L}$ introduced in Section 5. These edges represent experimentally realized pairwise comparisons or transport links between sites. In this case one writes

$${\mathcal{H}}_{\mathrm{edge}}:\mathrm{\Delta }{\mathrm{\Xi }}_{ab,\mathsf{s},\mathsf{r}}\left(T\right)={\mathrm{\Theta }}_{a,\mathsf{s}}\left(T\right)-{\mathrm{\Theta }}_{b,\mathsf{s}}\left(T\right)+{\mathrm{\Lambda }}_{ab,\mathsf{s}}\left(T\right)+{ϵ}_{ab,\mathsf{s},\mathsf{r}}\left(T\right),$$

(107)

where ${\mathrm{\Lambda }}_{ab,\mathsf{s}}\left(T\right)$ is a residual associated with the edge $(a,b)$, rather than with either endpoint site alone. This model is required if cycle consistency fails, or if the residual depends on relocation history rather than only on endpoint sites.

A third alternative is a transient or revisit-dependent residual,

$${\mathcal{H}}_{\mathrm{tr}}:{\mathrm{\Xi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)={\mathrm{\Theta }}_{\mathsf{e},\mathsf{s}}\left(T\right)+{{\rm Y}}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)+{ϵ}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right),$$

(108)

where ${{\rm Y}}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)$ is not stable under revisits. This class includes time-localized backgrounds, seasonal or sidereal modulations, transient topological defects, and unmodeled nonstationary technical effects.

The inference problem is therefore not binary. One should not simply ask whether $\mathrm{\Xi }\ne 0$. Instead, one asks which of the following statements is best supported:

$$\begin{array}{cccc}\hfill & {\mathcal{H}}_0:\hfill & \hfill & \mathrm{no}\mathrm{residual}\mathrm{after}\mathrm{nuisance}\mathrm{subtraction},\hfill \\ \hfill & {\mathcal{H}}_{\mathrm{site}}:\hfill & \hfill & \mathrm{a}\mathrm{reproducible}\mathrm{site}\text{-}\mathrm{conditioned}\mathrm{residual},\hfill \\ \hfill & {\mathcal{H}}_{\mathrm{edge}}:\hfill & \hfill & \mathrm{a}\mathrm{reproducible}\mathrm{path}\text{-}\mathrm{or}\mathrm{comparison}\text{-}\mathrm{conditioned}\mathrm{residual},\hfill \\ \hfill & {\mathcal{H}}_{\mathrm{tr}}:\hfill & \hfill & \mathrm{a}\mathrm{time}\text{-}\mathrm{dependent}\mathrm{or}\mathrm{transient}\mathrm{residual}.\hfill \end{array}$$

(109)

These alternatives correspond to physically distinct interpretations and should be tested separately.

Model comparison may be performed with likelihood ratios, information criteria, or Bayesian evidence. For a Gaussian residual model with parameter vector $\mathit{\theta }$, the likelihood takes the form

$$\mathcal{L}\left(\mathit{\theta }\right)\propto \exp \left[-{\displaystyle \frac{1}{2}}{\left(\mathbf{x}-\mathit{\mu }\left(\mathit{\theta }\right)\right)}^{\mathrm{T}}{C}^{-1}\left(\mathbf{x}-\mathit{\mu }\left(\mathit{\theta }\right)\right)\right],$$

(110)

where $\mathit{\mu }\left(\mathit{\theta }\right)$ is the predicted residual pattern for the model under consideration. For the null model $\mathit{\mu }=0$. For a site model, $\mu$ is determined by node residuals ${\mathrm{\Theta }}_{\mathsf{e},\mathsf{s}}\left(T\right)$. For a dictionary model, $\mathit{\theta }$ may instead denote microscopic or effective parameters such as a harmonic amplitude, a burst rate, or a diffusion coefficient.

The minimum requirement for interpretability is reproducibility under changes that should not affect the proposed residual. A site-like residual must be stable under revisits and satisfy graph-cycle consistency. A path-like residual must be reproducible for the same edge or transport history. A transient residual must show timing correlations across sites or revisits consistent with the hypothesized event class. A sequence-dependent residual must transform predictably under changes of filter function. Without these checks, an apparent excess in $\mathrm{\Xi }$ should be treated as an unmodeled nuisance contribution rather than as evidence for a new sector.

This leads to the following decision criteria:

1.

If ${\mathrm{\Xi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)$ and $\mathrm{\Delta }{\mathrm{\Xi }}_{ab,\mathsf{s},\mathsf{r}}\left(T\right)$ are consistent with zero for all measured sites, sequences, revisits, and interrogation times, the cartography map is null.

2.

If a nonzero residual is stable under revisits and satisfies cycle consistency, it may be interpreted as a site-conditioned map.

3.

If a nonzero residual violates cycle consistency but is reproducible for particular transport paths or pairwise comparisons, it should be interpreted as an edge- or path-conditioned effect.

4.

If a residual is not revisit-stable but shows correlated timing across sites, it should be treated as a transient candidate.

5.

If a residual has the time and sequence scaling predicted by one of the toy residual classes, it can be passed to the corresponding microscopic dictionary for model-dependent interpretation.

The outcome of the inference procedure is therefore not a single number, but a classified residual structure: null, site-like, edge-like, transient, or dictionary-compatible. This classification is essential before attaching any BSM or cosmological interpretation to the data.

7. Dictionaries from Microscopic Sectors to Cartography Observables

The final layer of the framework assigns microscopic interpretations to the residual classes of Section 4. A dictionary is a map from model parameters to an effective residual signal,

$$\left\{\mathrm{microscopic}\mathrm{parameters}\right\}⟶\delta {\omega }_{\mathrm{x}}\left(t\right)\mathrm{or}{S}_{\mathrm{x}}\left(\omega \right)⟶{\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{x}}\left(T\right)⟶{\mathrm{\Xi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right).$$

(111)

The dictionaries are not discovery claims and do not assume that any exotic signal is likely. They specify what would have to be true, at the level of effective qubit dynamics, for a residual cartographic signal to be interpreted as evidence for a particular class of BSM or cosmology-motivated physics.

The proposal is valuable only if it can fail sharply. We therefore phrase the primary decision criteria as follows.

(i) Transport-only feasibility test. If the stored particle or particle reservoir cannot survive controlled relocation, or if reproducible coherent preparation, interrogation, and readout cannot be re-established after stabilization at the new site, then stationary-site cartography is precluded at the current level of technological maturity. This is a genuine physical boundary on portable charged particle qubit quantum control.

(ii) Nuisance closure test. If all site dependence is absorbed by the monitored nuisance model, then the null hypothesis survives and the result is a predictive environment-conditioned decoherence model.

(iii) Residual-structure test. A claim of irreducible site-conditioned decoherence requires nonzero residuals that survive repeat visits, site order permutations, and control sequence changes, with uncertainties propagated through the nuisance model.

(iv) Graph-consistency test. If the measured edge residuals violate the cycle-closure condition in Eq. (92) beyond the propagated uncertainty, they cannot be represented consistently by a site-dependent scalar map. Such a failure instead indicates unresolved transport-history dependence, temporal drift, edge-specific systematics, or deficiencies in the nuisance or covariance model.

These decision criteria are deliberately stricter than the detection of an isolated anomaly. They are designed to ensure that a claimed residual is structurally rich enough to support physical interpretation.

7.1. Dictionary Example: Millicharged-Sector Contribution to the Residual Channel

As a first concrete example of a nonstandard residual coupling, consider a background of millicharged particles $\mathcal{Q}$ of mass $m_{\mathcal{Q}}$ carrying an electric charge much smaller than the elementary charge. Such particles arise naturally in BSM scenarios containing an additional Abelian gauge sector. In the simplest realization, kinetic mixing between the visible electromagnetic field and a hidden $U{\left(1\right)}_X$ gauge field causes matter charged under $U{\left(1\right)}_X$ to acquire a small effective electromagnetic charge

$$q_{\mathcal{Q}}={ϵ}_{\mathcal{Q}}e,\left|{ϵ}_{\mathcal{Q}}\right|\ll 1,$$

(112)

where e is the magnitude of the electron charge [63,64]. Millicharged particles are therefore a representative example of feebly coupled hidden-sector matter, rather than an ad hoc violation of electric charge quantization. If stable or sufficiently long lived, they may contribute to the dark-matter density, form a subdominant interacting dark-matter component, or occur as part of a more general dark-radiation sector.

The allowed parameter space is strongly constrained but remains dependent on the microscopic realization. Stellar energy loss arguments, Big Bang nucleosynthesis, the effective number of relativistic species, cosmic microwave background (CMB) anisotropies, structure formation, and terrestrial or accelerator searches probe different combinations of $m_{\mathcal{Q}}$, ${ϵ}_{\mathcal{Q}}$, the cosmological abundance, and the properties of the hidden mediator [65,66,67,68,69]. In particular, cosmological bounds typically require an electromagnetically coupled component to be either sufficiently weakly interacting or only a subdominant fraction of the total dark matter, although additional hidden-sector interactions can modify this conclusion [70]. Any cartographic constraint must therefore be stated conditionally in terms of the assumed abundance, velocity distribution, mediator structure, and local environmental population.

Complementary analyses [71,72] show that terrestrial thermalization and electrostatic environments can substantially alter the local population of millicharged particles, underscoring the need to specify the relevant environmental assumptions when interpreting a cartographic residual.

Millicharged particles are especially relevant to trapped-particle cartography because their scattering from a confined charged particle can produce individual energy-deposition events or an excess heating rate [57]. In the present coherence-based formulation, the same microscopic interaction may instead appear as stochastic phase kicks, burst-like dephasing, or a site-dependent residual rate if the local millicharged-particle density, velocity distribution, shielding, or thermalization environment differs between measurement sites. The purpose of the dictionary below is not to identify every residual with millicharged matter, but to map the microscopic quantities onto the observable residual channel and thereby permit comparison with existing laboratory, astrophysical, and cosmological bounds.

Ion-trap searches are naturally sensitive to such particles because scattering can appear either as individual impulsive events or as an excess motional-heating rate [5,6,57]. In the present framework, the millicharged sector is treated as one possible microscopic realization of the residual channel $\delta {\omega }_{\mathrm{x}}\left(t\right)$ introduced in Eq. (20).

For a site $\mathsf{e}$ and revisit $\mathsf{r}$, let $n_{\mathcal{Q}}^{(\mathsf{e},\mathsf{r})}$ denote the local millicharged-particle number density and $f_{\mathsf{e},\mathsf{r}}\left(\mathbf{v}\right)$ the corresponding velocity distribution. A coarse-grained scattering rate may then be written as

$${\mathrm{\Gamma }}_{\mathsf{e},\mathsf{r}}^{\mathrm{m}\mathcal{Q}}=n_{\mathcal{Q}}^{(\mathsf{e},\mathsf{r})}\int d^{3}v{f}_{\mathsf{e},\mathsf{r}}\left(\mathbf{v}\right)v{\sigma }_{\mathrm{eff}}(v;m_{\mathcal{Q}},\epsilon ),$$

(113)

where ${\sigma }_{\mathrm{eff}}$ is the trap-relevant effective cross section. The resulting collisions induce stochastic perturbations of the trapped charged particle motion and thereby generate an additional contribution to the measured decoherence of the localized qubit.

In an event-based description, the millicharged contribution is incorporated as an additive term in the residual decoherence functional,

$${\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{x}}\left(T\right)\supset {\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{m}\mathcal{Q}}\left(T\right)\simeq {\mathrm{\Gamma }}_{\mathsf{e},\mathsf{r}}^{\mathrm{m}\mathcal{Q}}{\eta }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{m}\mathcal{Q}}T,$$

(114)

where ${\eta }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{m}\mathcal{Q}}$ is an effective single-event dephasing factor encoding how strongly one scattering event perturbs the qubit under control sequence $\mathsf{s}$. The measured coherence signal then becomes

$$W_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)=\exp \left[-{\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{nuis}}\left(T\right)-{\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{m}\mathcal{Q}}\left(T\right)\right],$$

(115)

and the corresponding cartographic observable is

$${\mathrm{\Xi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)\supset {\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{m}\mathcal{Q}}\left(T\right).$$

(116)

Thus a site-dependent millicharged population maps directly onto a site-dependent residual signal.

In a complementary Gaussian-limit description, many weak kicks may instead be represented by an additional site-conditioned frequency-noise spectrum,

$$S_{\delta \omega }^{(\mathsf{e},\mathsf{r})}\left(\omega \right)=\sum _{a\in \mathcal{A}}{g}_a^2{S}_a^{(\mathsf{e},\mathsf{r})}\left(\omega \right)+S_{\mathrm{m}\mathcal{Q}}^{(\mathsf{e},\mathsf{r})}\left(\omega \right),$$

(117)

with $\mathcal{A}=\{B,\mathrm{vib},\mathrm{th},\mathrm{clk}\}$ as in Eq. (20). The millicharged contribution to the residual decoherence functional is then

$${\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{m}\mathcal{Q}}\left(T\right)={\displaystyle \frac{1}{\pi }}{\int }_0^{\infty }d\omega S_{\mathrm{m}\mathcal{Q}}^{(\mathsf{e},\mathsf{r})}\left(\omega \right){\displaystyle \frac{F_{\mathsf{s}}\left(\omega T\right)}{{\omega }^2}},$$

(118)

so that

$${\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{x}}\left(T\right)={\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{m}\mathcal{Q}}\left(T\right)+{\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{other}}\left(T\right)$$

(119)

if other nonstandard residual channels are also present.

In this notation, the dictionary from microscopic model to cartographic observable is

$$\{m_{\mathcal{Q}},\epsilon ,n_{\mathcal{Q}}^{(\mathsf{e},\mathsf{r})},f_{\mathsf{e},\mathsf{r}}\}⟶{\mathrm{\Gamma }}_{\mathsf{e},\mathsf{r}}^{\mathrm{m}\mathcal{Q}}\mathrm{or}{S}_{\mathrm{m}\mathcal{Q}}^{(\mathsf{e},\mathsf{r})}\left(\omega \right)⟶{\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{m}\mathcal{Q}}\left(T\right)⟶{\mathrm{\Xi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right).$$

The role of transport is therefore explicit: by moving the same localized antiproton qubit among sites with different overburden or electromagnetic environment, one tests whether the inferred residual channel exhibits the site dependence expected for a thermalized or attenuated millicharged population [5,6,57].

Complementary information on the millicharged sector is provided by direct searches at high-energy colliders, where the observable is their production in particle collisions rather than their interaction with an assumed local or cosmological population. A dedicated LHC experiment exploiting the small but nonzero ionization deposited by particles carrying a tiny electric charge was proposed in Ref. [73]. This concept was subsequently developed into the milliQan experiment at the LHC. Using proton–proton collision data collected during Run 3 with the completed milliQan bar detector, the collaboration has obtained stringent direct constraints on the millicharge as a function of the particle mass [74]. These searches are complementary to the astrophysical, cosmological, cartographic, and other low-energy laboratory probes discussed above: collider experiments test the direct production of millicharged states, whereas decoherence cartography would be sensitive to their local interaction with a trapped quantum sensor and therefore depends on the terrestrial abundance and environmental distribution of the relevant sector.

7.2. Dictionary Example: Lorentz-Violating Background Contribution to the Residual Channel

As a second example of a nonstandard residual coupling, consider a Lorentz-violating background in the sense of the BSM, an observer-covariant effective field theory obtained by supplementing the Standard Model (SM) and, in its gravitational formulation, general relativity with coordinate-independent contractions of conventional fields and Lorentz-violating background coefficients [75,76,77]. The minimal BSM contains the power-counting-renormalizable operators of mass dimension $d\le 4$, whereas the nonminimal BSM includes operators of higher mass dimension. General fermion operators of arbitrary mass dimension and the corresponding observable coefficient combinations have been classified in Ref. [78]. Experimental constraints and coefficient conventions across the matter, photon, neutrino, and gravity sectors are compiled and regularly updated in the Data Tables for Lorentz and $CPT$ violation [79].

Penning traps provide particularly clean probes of the fermion sector of the BSM. Lorentz- and $CPT$-violating operators perturb the energy levels of a confined charged fermion and thereby generate shifts in its cyclotron and anomaly frequencies. Depending on the coefficient combination, these shifts may contain constant offsets, sidereal or annual modulations, orientation dependence, boost-suppressed contributions, and differences between particles and antiparticles. Early BSM analyses of electron–positron and proton–antiproton Penning-trap observables were developed in Refs. [80,81]. They were subsequently extended to nonminimal fermion operators of mass dimension up to six, to charge-to-mass-ratio comparisons, and to effects at linear order in the laboratory boost [55,56,82,83]. These results provide the microscopic basis for the effective phase and harmonic-modulation dictionary constructed below.

In the present framework, such a background is treated as one possible microscopic realization of the residual channel $\delta {\omega }_{\mathrm{x}}\left(t\right)$ introduced in Eq. (20). For a site $\mathsf{e}$, revisit $\mathsf{r}$, and species q, the Lorentz-violating contribution to the localized qubit may be represented as an additive frequency shift

$$\delta {\omega }_{\mathrm{LV}}^{(\mathsf{e},\mathsf{r};q)}\left(t\right)=C_0^{(\mathsf{e},\mathsf{r};q)}+{\displaystyle \sum _{n=1}^2}\left[C_n^{(\mathsf{e},\mathsf{r};q)}\cos \left(n{\omega }_{\oplus }t\right)+S_n^{(\mathsf{e},\mathsf{r};q)}\sin \left(n{\omega }_{\oplus }t\right)\right]+\delta {\omega }_{\mathrm{boost}}^{(\mathsf{e},\mathsf{r};q)}\left(t\right).$$

(120)

where ${\omega }_{\oplus }$ is the sidereal frequency and $\delta {\omega }_{\mathrm{boost}}^{(\mathsf{e},\mathsf{r};q)}\left(t\right)$ denotes boost-suppressed terms. The coefficients $C_0^{(\mathsf{e},\mathsf{r};q)}$, $C_n^{(\mathsf{e},\mathsf{r};q)}$, and $S_n^{(\mathsf{e},\mathsf{r};q)}$ are determined by the relevant BSM coefficients together with the local laboratory geometry, including colatitude, magnetic field orientation, and the relation between the trap frame and the Sun-centered frame [55,56]. These coefficients depend, in general, on the particle species q, since different fermion sectors probe different combinations of BSM coefficients and have different experimental response factors. Accordingly, the induced frequency perturbation $\delta {\omega }_{\mathrm{LV}}^{(\mathsf{e},\mathsf{r};q)}$ is also species dependent. For notational simplicity, the label q will be left implicit in the remainder of this subsection unless species comparisons are being discussed explicitly.

If the phase is resolved shot by shot, the primary Lorentz-violating observable is a residual phase contribution,

$${\mathrm{\Phi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{LV}}\left(T\right)={\int }_0^{T}dt{y}_{\mathsf{s}}\left(t\right)\delta {\omega }_{\mathrm{LV}}^{(\mathsf{e},\mathsf{r})}\left(t\right),$$

(121)

where $y_{\mathsf{s}}\left(t\right)$ is the switching function of the control sequence. In this case, the dictionary is most naturally one from microscopic BSM coefficients to a site- and sequence-dependent phase signal.

If, however, the experiment averages over unresolved sidereal phase, revisit time, or orientation drift, then the same deterministic background induces an effective residual decoherence functional,

$${\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{LV}}\left(T\right)={\displaystyle \frac{1}{2}}{\mathrm{Var}}_{\mathsf{r}}\left[{\mathrm{\Phi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{LV}}\left(T\right)\right],$$

(122)

which contributes to the measured coherence signal as

$$W_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)=\exp \left[-{\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{nuis}}\left(T\right)-{\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{LV}}\left(T\right)\right].$$

(123)

In the notation of the present manuscript,

$${\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{x}}\left(T\right)\supset {\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{LV}}\left(T\right),{\mathrm{\Xi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)\supset {\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{LV}}\left(T\right).$$

(124)

Equivalently, after coarse graining over unresolved sidereal phase, revisit time, or slow orientation drift, the oscillatory part of the Lorentz-violating contribution may be represented by an effective two-sided residual spectrum. Since the frequency shift in Eq. (120) is real, the spectral lines at positive and negative frequencies must appear with equal weights. Defining

$${\mathcal{A}}_n^{(\mathsf{e},\mathsf{r})}\equiv {\left[C_n^{(\mathsf{e},\mathsf{r})}\right]}^2+{\left[S_n^{(\mathsf{e},\mathsf{r})}\right]}^2,n=1,2,$$

(125)

the oscillatory Lorentz-violating contribution may be written as

$$S_{\mathrm{LV},\mathrm{osc}}^{(\mathsf{e},\mathsf{r})}\left(\omega \right)={\displaystyle \frac{\pi }{2}}{\displaystyle \sum _{n=1}^2}{\mathcal{A}}_n^{(\mathsf{e},\mathsf{r})}\left[\delta (\omega -n{\omega }_{\oplus })+\delta (\omega +n{\omega }_{\oplus })\right]+S_{\mathrm{boost}}^{(\mathsf{e},\mathsf{r})}\left(\omega \right).$$

(126)

The corresponding decoherence functional is then evaluated using the two-sided filter-function convention,

$${\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{LV},\mathrm{osc}}\left(T\right)={\displaystyle \frac{1}{4\pi }}{\int }_{-\infty }^{\infty }d\omega S_{\mathrm{LV},\mathrm{osc}}^{(\mathsf{e},\mathsf{r})}\left(\omega \right){\displaystyle \frac{F_{\mathsf{s}}\left(\omega T\right)}{{\omega }^2}}.$$

(127)

For the idealized monochromatic sidereal harmonics in Eq. (126), this gives

$${\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{LV},\mathrm{osc}}\left(T\right)={\displaystyle \frac{1}{4}}{\displaystyle \sum _{n=1}^2}{\mathcal{A}}_n^{(\mathsf{e},\mathsf{r})}{\displaystyle \frac{F_{\mathsf{s}}\left(n{\omega }_{\oplus }T\right)}{{\left(n{\omega }_{\oplus }\right)}^2}}+{\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{boost}}\left(T\right),$$

(128)

where ${\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{boost}}$ denotes the contribution obtained by inserting $S_{\mathrm{boost}}^{(\mathsf{e},\mathsf{r})}$ into Eq. (127). This form makes explicit that changes of control sequence probe the sidereal and boost harmonics with different filter-function weights.

The static term $C_0^{(\mathsf{e},\mathsf{r})}$ in Eq. (120) is treated separately from the oscillatory residual spectrum because, when stable over repeated interrogations, it represents a zero-frequency coherent frequency offset and therefore produces deterministic phase accumulation rather than decoherence. If the nominally static offset varies randomly between repeated interrogations while remaining approximately constant during each interrogation, it instead acts as quasi-static frequency noise and contributes to ensemble dephasing. It is treated directly through the phase integral in Eq. (121). If the static offset varies between revisits or orientations and is not phase-resolved, its dephasing contribution is obtained from the variance form

$${\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{LV},0}\left(T\right)={\displaystyle \frac{1}{2}}{\mathrm{Var}}_{\mathsf{r}}\left[C_0^{(\mathsf{e},\mathsf{r})}{\int }_0^{T}dt{y}_{\mathsf{s}}\left(t\right)\right],$$

(129)

rather than from a zero-frequency spectral line. Thus the spectral representation in Eq. (126) is intended for the oscillatory sidereal and boost components, while the static contribution is handled as a deterministic or slowly varying phase offset. Equivalently, a random quasi-static offset may be represented by a zero-frequency spectral contribution.

In this notation, the dictionary from microscopic model to cartographic observable is

$$\begin{array}{c}\hfill \left\{\mathrm{BSM}\mathrm{coefficients}\mathrm{in}\mathrm{the}\mathrm{Sun}\text{-}\mathrm{centered}\mathrm{frame}\right\}⟶\delta {\omega }_{\mathrm{LV}}^{(\mathsf{e},\mathsf{r})}\left(t\right)⟶{\mathrm{\Phi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{LV}}\left(T\right)\\ \hfill \mathrm{or}{\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{LV}}\left(T\right)⟶{\mathrm{\Xi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right).\end{array}$$

(130)

The role of transport is then immediate: by moving the same localized antiproton qubit among sites with different colatitude, magnetic-field orientation, and transport history, one changes the map from Sun-centered BSM coefficients to the laboratory signal and thereby tests whether the inferred residual follows the harmonic structure expected for a Lorentz-violating background [55,56].

7.3. Dictionary Example: Coherent Ultralight-Field Contribution

Ultralight bosonic fields constitute a broad and well motivated class of dark matter candidates. Representative examples include the QCD axion and axion-like particles, as well as dilaton- or modulus-like scalar fields arising in BSMs and in ultraviolet theories [7,8,9]. Such fields can be produced nonthermally in the early Universe and, because of their large occupation numbers, may behave locally as coherent or quasicoherent classical fields rather than as dilute fluxes of individually resolved particles. Their phenomenology depends on the field mass, its contribution to the local dark matter density, its coherence time and length, and the operator through which it couples to SM particles.

For decoherence cartography, the important point is that ultralight bosonic fields naturally generate residuals that are coherent in time, narrow-band in frequency, and potentially correlated across sites. Axionlike or pseudoscalar fields can produce oscillatory spin-precession signals and effective pseudo-magnetic frequency shifts, as in CASPEr-type searches and related spin-precession proposals [10,11,84]. Scalar or dilatonlike fields can instead modulate particle masses, effective couplings, and fundamental constants, thereby producing oscillatory shifts in atomic or spin-transition frequencies and clock-like observables [14,15,85]. In a distinct transient regime, ultralight fields may form topological defect or domain wall configurations that generate correlated excursions across spatially separated sensors [12,13]. Thus, at the level relevant for the present framework, the microscopic field may enter a localized qubit as a coherent phase shift, a narrow-band frequency modulation, or a transient excursion. The purpose of the dictionary below is therefore to map the microscopic field parameters and couplings onto the experimentally accessible quantities $\delta {\omega }_{\mathrm{UL}}^{(\mathsf{e},\mathsf{r};q)}\left(t\right)$, ${\mathrm{\Phi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{UL}}\left(T\right)$, and ${\mathrm{\Xi }}_{ab}^{(\mathsf{s};q)}\left(T\right)$.

The standard low-energy description of ultralight bosonic dark matter is that the highly occupied field behaves, over laboratory timescales shorter than its coherence time, as a classical coherently oscillating background [9,14]. Locally, this may be represented by the leading plane-wave component

$$\phi (\mathbf{x},t)={\phi }_0^{\left(\mathsf{e}\right)}\cos \left[{\omega }_{\phi }t-{\mathbf{k}}_{\phi }·{\mathbf{x}}_{\mathsf{e}}+{\alpha }_{\mathsf{e}}\right],$$

(131)

where ${\omega }_{\phi }\simeq m_{\phi }$ in units with $\hslash =c=1$, ${\mathbf{k}}_{\phi }\simeq m_{\phi }\mathbf{v}$ encodes the nonrelativistic field velocity, and ${\alpha }_{\mathsf{e}}$ is the effective local phase at site $\mathsf{e}$. If the field accounts for an effective local energy density ${\rho }_{\phi }^{\left(\mathsf{e}\right)}$, then parametrically ${\phi }_0^{\left(\mathsf{e}\right)}\simeq \sqrt{2{\rho }_{\phi }^{\left(\mathsf{e}\right)}}/m_{\phi }$, up to convention-dependent normalization factors. The site dependence in Eq. (131) is written explicitly because stationary-site cartography compares local interrogations performed after the apparatus has stabilized at different environment classes.

At the level of the qubit Hamiltonian, the leading effect of such a field can be represented as an additional residual frequency shift,

$$\delta {\omega }_{\mathrm{UL}}^{(\mathsf{e};q)}\left(t\right)=A_{\mathrm{UL}}^{(\mathsf{e};q)}\cos \left[{\omega }_{\phi }t+{\alpha }_{\mathsf{e}}\right],$$

(132)

where q labels the qubit species Eq. (1). The amplitude $A_{\mathrm{UL}}^{(\mathsf{e};q)}$ contains the relevant microscopic coupling, the field amplitude, and the species-dependent response coefficient. For example, a spin-coupled axionlike background contributes as an effective oscillatory pseudo-magnetic field, whereas a scalar or dilatonlike field may enter through a small oscillation of parameters that determine the spin splitting.

Under a control sequence $\mathsf{s}$, the corresponding residual phase is

$${\mathrm{\Phi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{UL}}\left(T\right)={\int }_0^{T}dt{y}_{\mathsf{s}}\left(t\right)\delta {\omega }_{\mathrm{UL}}^{(\mathsf{e};q)}\left(t\right),$$

(133)

where $y_{\mathsf{s}}\left(t\right)$ is the sequence-dependent switching function. If the ultralight field phase is stable over the experimental record, the primary observable is a coherent, narrow-band contribution to the accumulated phase. In that case the signature appears most directly in

$${\mathrm{\Phi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)={\mathrm{\Phi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{nuis}}\left(T\right)+{\mathrm{\Phi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{UL}}\left(T\right)+{ϵ}_{\mathrm{\Phi },\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right).$$

(134)

If, instead, the phase of the ultralight field is unresolved, averaged over many coherence times, or randomized between revisits, the same underlying oscillatory perturbation contributes to dephasing. In that limit one may write

$${\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{UL}}\left(T\right)={\displaystyle \frac{1}{2}}\mathrm{Var}\left[{\mathrm{\Phi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{UL}}\left(T\right)\right],$$

(135)

so that the residual cartography observable becomes

$${\mathrm{\Xi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{UL}}\left(T\right)\simeq {\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{UL}}\left(T\right).$$

(136)

Equivalently, in the frequency-domain language of Section 3, the coherent oscillation corresponds to a narrow-band residual spectrum. Since Eq. (132) describes a real-valued frequency shift, its spectral representation contains lines at both positive and negative frequencies. It is therefore most convenient to use the two-sided spectral-density convention,

$${\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{UL}}\left(T\right)={\displaystyle \frac{1}{4\pi }}{\int }_{-\infty }^{+\infty }d\omega S_{\mathrm{UL}}^{(\mathsf{e};q)}\left(\omega \right){\displaystyle \frac{F_{\mathsf{s}}\left(\omega T\right)}{{\omega }^2}},$$

(137)

where $F_{\mathsf{s}}\left(\omega T\right)$ is even in $\omega$. In this convention, the idealized monochromatic ultralight-field contribution is

$$S_{\mathrm{UL}}^{(\mathsf{e};q)}\left(\omega \right)={\displaystyle \frac{\pi }{2}}{\left[A_{\mathrm{UL}}^{(\mathsf{e};q)}\right]}^2\left[\delta (\omega -{\omega }_{\phi })+\delta (\omega +{\omega }_{\phi })\right],$$

(138)

up to the finite linewidth set by the field coherence time and the finite measurement record. Substituting Eq. (138) into Eq. (137) gives

$${\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{UL}}\left(T\right)={\displaystyle \frac{1}{4}}{\left[A_{\mathrm{UL}}^{(\mathsf{e};q)}\right]}^2{\displaystyle \frac{F_{\mathsf{s}}\left({\omega }_{\phi }T\right)}{{\omega }_{\phi }^2}}.$$

(139)

This form makes explicit that different pulse sequences probe the ultralight-field frequency through their filter functions. If one instead uses a one-sided convention with integration over $0<\omega <\infty$, the positive-frequency spectral line alone should be used with the corresponding one-sided normalization.

The dictionary from microscopic parameters to cartography observables can therefore be summarized as

$$\left\{m_{\phi },g_{\phi q},{\rho }_{\phi }^{\left(\mathsf{e}\right)},{\alpha }_{\mathsf{e}},{\mathbf{k}}_{\phi }\right\}⟶A_{\mathrm{UL}}^{(\mathsf{e};q)}⟶\delta {\omega }_{\mathrm{UL}}^{(\mathsf{e};q)}\left(t\right)⟶\left\{{\mathrm{\Phi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{UL}}\left(T\right),{\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{UL}}\left(T\right)\right\}⟶{\mathrm{\Xi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right),$$

(140)

where $g_{\phi q}$ denotes the relevant effective coupling to species q, and ${\rho }_{\phi }^{\left(\mathsf{e}\right)}$ denotes the effective local field energy density or field amplitude at site $\mathsf{e}$.

Several qualitative signatures distinguish this dictionary from the millicharged-scattering dictionary. First, the residual is expected to be narrow-band or quasi-periodic rather than a purely rate-like excess decoherence. Second, it can appear as a coherent phase signal before it appears as dephasing. Third, its response is strongly sequence dependent through $F_{\mathsf{s}}\left({\omega }_{\phi }T\right)$ or the corresponding phase integral. Fourth, if the field is coherent over the spatial scale of the experiment, the phases ${\alpha }_{\mathsf{e}}$ measured at different sites should be correlated rather than arbitrary.

In stationary-site cartography, the most direct test is therefore not a search for a monotonic excess rate, but a search for a reproducible frequency component whose amplitude and phase are consistent across revisits and whose sequence dependence follows the expected filter-function response. A residual of this type would be classified as ultralight-field compatible only if the same ${\omega }_{\phi }$ and a common set of microscopic parameters can account for the measured pattern across sites, species, sequences, and interrogation times.

Transient ultralight-field configurations, such as domain-wall or defect-like backgrounds, form a related but distinct limit. Instead of the harmonic signal in Eq. (132), one may write

$$\delta {\omega }_{\mathrm{UL},\mathrm{tr}}^{(\mathsf{e};q)}\left(t\right)=A_{\mathrm{tr}}^{\left(q\right)}h\left[t-t_{\mathsf{e}}^{\mathrm{cross}}\right],$$

(141)

where $h\left(t\right)$ is the temporal profile of the transient crossing and $t_{\mathsf{e}}^{\mathrm{cross}}$ is the site-dependent crossing time. Such a signal would not appear as a stationary site amplitude, but as a correlated transient pattern across revisits or across separated sites. In the graph language, this corresponds to a time-ordered residual map rather than a static node potential. Its defining signature is not a constant ${\mathrm{\Theta }}_{\mathsf{e},\mathsf{s}}\left(T\right)$, but a correlated excursion with a propagation pattern compatible with a common transient background.

A residual compatible with the ultralight-field dictionary should not, however, be identified solely by the observation of a narrow-band modulation or a transient excursion. A common-field interpretation requires the inferred signal to reproduce, where applicable, a consistent frequency or temporal profile, phase evolution, interrogation-sequence response, spatial correlation, and species dependence across repeated measurements. These cross-checks provide a means of distinguishing a coherent ultralight-field contribution from local technical noise, site-specific systematics, or other residual processes.

Spatially distributed quantum sensors provide useful experimental precedents for this type of search. Spatially and temporally separated atomic clocks and cavity-stabilized lasers have been used to probe temporal and spatial variations associated with oscillating scalar dark matter [86]. At larger scales, existing and developing long-baseline atom-interferometer programs, including AION [87], MAGIS-100 [88], AEDGE [89] and AICE [90] concepts, provide complementary approaches to searches for ultralight bosonic dark matter and other new physics. Although clock, cavity, and atom-interferometric experiments differ substantially from the transportable trapped-particle qubit considered here, they illustrate the broader principle that controlled quantum phase evolution, spatial comparison, and correlated measurements provide sensitivity to coherent ultralight fields. In the present approach, relocation of the same trapped-particle platform adds a complementary possibility: site dependence can be tested while retaining a common sensor architecture and explicitly comparing different control sequences, revisits, and particle species.

Ultralight vector fields, including kinetically mixed dark photons or other light vector backgrounds [17,18], provide a complementary realization of the coherent-field dictionary. Unlike scalar or pseudoscalar backgrounds, a vector field carries a polarization direction, so the laboratory response can depend on the orientation of the trap axis and can acquire characteristic modulation from Earth’s rotation. Depending on the microscopic coupling, the field may act as an oscillatory effective electric or magnetic perturbation, producing motional excitation, a frequency shift, or—after calibrated spin–motion transduction—a residual spin-phase signal. Transport between sites with different trap orientations therefore provides an additional discriminator through the combined frequency, polarization, and orientation dependence of the inferred residual.

7.4. Dictionary Example: Collapse-Like Contribution to the Residual Channel

As a final example of a nonstandard residual coupling, consider a collapse-like modification of quantum dynamics, such as continuous spontaneous localization (CSL) or a related gravity-inspired localization model [91,92,93,94]. In the present framework, such a modification is treated as one possible microscopic realization of the residual channel $\delta {\omega }_{\mathrm{x}}\left(t\right)$ introduced in Eq. (20).

For a localized trapped particle, collapse-like noise is most naturally represented not as a direct spin coupling but as a stochastic perturbation of the motional state. At the coarse-grained level, this may be encoded either as a momentum-diffusion coefficient $D_{\mathsf{e},\mathsf{r}}^{\mathrm{col}}$ or, equivalently, as an additional site-conditioned motional-noise spectrum. The corresponding microscopic parameters are model dependent; for example, in CSL they are usually expressed in terms of a collapse rate ${\lambda }_{\mathrm{CSL}}$ and localization length $r_C$ [91,92].

In an event- or diffusion-based description, the collapse-induced contribution to the residual decoherence functional may be written as

$${\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{x}}\left(T\right)\supset {\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{col}}\left(T\right)\simeq {\zeta }_{\mathsf{e},\mathsf{s},\mathsf{r}}{D}_{\mathsf{e},\mathsf{r}}^{\mathrm{col}}T,$$

(142)

where ${\zeta }_{\mathsf{e},\mathsf{s},\mathsf{r}}$ is an effective transfer coefficient that quantifies how motional diffusion is converted into dephasing of the localized charged particle qubit under sequence $\mathsf{s}$. The measured coherence signal then takes the form

$$W_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)=\exp \left[-{\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{nuis}}\left(T\right)-{\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{col}}\left(T\right)\right],$$

(143)

so that

$${\mathrm{\Xi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)\supset {\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{col}}\left(T\right).$$

(144)

In a spectral description, the same physics is represented by an additional residual contribution to the effective frequency-noise spectrum,

$$S_{\delta \omega }^{(\mathsf{e},\mathsf{r})}\left(\omega \right)=\sum _{a\in \mathcal{A}}{g}_a^2{S}_a^{(\mathsf{e},\mathsf{r})}\left(\omega \right)+S_{\mathrm{col}}^{(\mathsf{e},\mathsf{r})}\left(\omega \right),$$

(145)

with $\mathcal{A}=\{B,\mathrm{vib},\mathrm{th},\mathrm{clk}\}$ as in Eq. (43). The collapse-like contribution to the residual decoherence functional is then

$${\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{col}}\left(T\right)={\displaystyle \frac{1}{\pi }}{\int }_0^{\infty }d\omega S_{\mathrm{col}}^{(\mathsf{e},\mathsf{r})}\left(\omega \right){\displaystyle \frac{F_{\mathsf{s}}\left(\omega T\right)}{{\omega }^2}}.$$

(146)

In this notation,

$${\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{x}}\left(T\right)={\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{col}}\left(T\right)+{\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{other}}\left(T\right)$$

(147)

if additional residual channels are present.

The corresponding dictionary from microscopic model to cartographic observable is therefore

$$\begin{array}{c}\hfill \{{\lambda }_{\mathrm{CSL}},r_C,\mathrm{or}\mathrm{analogous}\mathrm{collapse}\mathrm{parameters}\}⟶D_{\mathsf{e},\mathsf{r}}^{\mathrm{col}}\\ \hfill \mathrm{or}{S}_{\mathrm{col}}^{(\mathsf{e},\mathsf{r})}\left(\omega \right)⟶{\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{col}}\left(T\right)⟶{\mathrm{\Xi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right).\end{array}$$

(148)

This example is particularly natural in the cartography setting because collapse-like models are often tested through excess noise or heating, and because underground or shielded environments can alter the relative importance of ordinary backgrounds that might otherwise mimic such a residual channel [5,6,93,94].

8. Benchmark Estimates

We now give order of magnitude benchmark estimates for the residual sensitivity of the cartography protocol. These estimates are not exclusion limits. They are intended only to translate the abstract observable ${\mathrm{\Xi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)$ into the characteristic amplitudes, rates, or frequency shifts that different microscopic dictionaries would have to generate in order to be visible.

The benchmark should be formulated in a species-dependent way. Let $T_{*}^{(q,\mathsf{s})}$ denote the local interrogation time used for species q of Eq. (1) under sequence $\mathsf{s}$. In stationary-site cartography, $T_{*}^{(q,\mathsf{s})}$ is constrained by the coherence of the locally prepared qubit state and by the readout protocol, not by the time required to transport the apparatus between sites. Thus short coherence platforms can still participate in cartography, provided the interrogation is performed locally after the apparatus has stabilized.

For each species and sequence we define the residual uncertainty

$${\sigma }_{\mathrm{\Xi }}^{(q,\mathsf{s})}\left(T_{*}\right)\equiv {\left[\mathrm{Var}\left({\mathrm{\Xi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\left(q\right)}\left(T_{*}\right)\right)\right]}^{1/2},$$

(149)

including both statistical uncertainty and the uncertainty of the nuisance subtraction. A residual contribution is detectable at the order of magnitude level if

$$\left|{\chi }_{\mathsf{e},\mathsf{s}}^{\mathrm{x},\left(\mathrm{q}\right)}\left(T_{*}\right)\right|\gtrsim n_{\sigma }{\sigma }_{\mathrm{\Xi }}^{(q,\mathsf{s})}\left(T_{*}\right),$$

(150)

where $n_{\sigma }$ specifies the desired significance threshold. For rate-like residuals it is convenient to introduce the effective residual-rate sensitivity

$${\gamma }_{\mathrm{\Xi }}^{(q,\mathsf{s})}\left(T_{*}\right)\equiv {\displaystyle \frac{{\sigma }_{\mathrm{\Xi }}^{(q,\mathsf{s})}\left(T_{*}\right)}{T_{*}^{(q,\mathsf{s})}}}.$$

(151)

The longer the local interrogation time and the smaller the residual uncertainty, the smaller the excess rate that can be resolved.

The benchmark times $T_{*}^{(q,\mathsf{s})}$ should not be read as universal coherence times for all platforms. Their empirical status differs substantially between species. For antiprotons, coherent spectroscopy of a single spin has demonstrated coherence times greater than $50\mathrm{s}$, providing the most direct heavy-antimatter benchmark for the present proposal [1]. For protons, single-spin flips, double Penning trap spin-state analysis, and high precision magnetic moment spectroscopy have been demonstrated, but a directly analogous published Ramsey-style single-proton spin-qubit coherence time is not yet available [21,22,95]. For free electrons in Penning traps, quantum-cyclotron spectroscopy and cavity-controlled single-particle measurements are mature, but again they do not currently provide a directly comparable transportable free-electron spin-cartography coherence benchmark [19,20,96]. A useful related benchmark is supplied by a bound-electron spin qubit in a single ⁹${\mathrm{Be}}^{+}$ ion in a microfabricated Penning trap, for which Ramsey coherence of $1.9\left(2\right)\mathrm{ms}$ and Uhrig-decoupled coherence times up to $8.0\left(7\right)\mathrm{ms}$ have been reported [37]. For positrons, trapping and electron–positron comparison techniques are well established, but a dedicated positron spin-qubit cartography coherence time has not yet been demonstrated [46,47]. Thus, except for the antiproton benchmark, $T_{*}^{(q,\mathsf{s})}$ should be treated as a platform-dependent experimental input rather than as a fixed theoretical number.

For the antiproton platform, a natural first benchmark is set by the demonstrated coherent single-spin spectroscopy time scale, of order

$$T_{*}^{\left(\overline{p}\right)}\sim 50\mathrm{s},$$

(152)

with the understanding that this is an empirical hardware scale rather than a fundamental limit [1]. For proton, electron, and positron platforms, the appropriate $T_{*}^{(q,\mathsf{s})}$ should be treated as a platform-specific calibration parameter. In particular, storage or transport time should not be identified with coherence time: the relevant quantity for stationary-site cartography is the duration over which a locally prepared superposition remains phase sensitive under the chosen interrogation sequence.

As a numerical illustration, if

$${\sigma }_{\mathrm{\Xi }}^{(q,\mathsf{s})}\left(T_{*}\right)\approx {10}^{-2},$$

(153)

then the corresponding rate sensitivity is

$${\gamma }_{\mathrm{\Xi }}^{(q,\mathsf{s})}\left(T_{*}\right)=2\times {10}^{-4}{\mathrm{s}}^{-1}\left({\displaystyle \frac{50\mathrm{s}}{T_{*}^{(q,\mathsf{s})}}}\right)\left({\displaystyle \frac{{\sigma }_{\mathrm{\Xi }}^{(q,\mathsf{s})}}{{10}^{-2}}}\right).$$

(154)

For a millisecond-scale platform with $T_{*}\approx 10\mathrm{ms}$, the same residual uncertainty would correspond instead to

$${\gamma }_{\mathrm{\Xi }}^{(q,\mathsf{s})}\simeq 1{\mathrm{s}}^{-1}\left({\displaystyle \frac{{\sigma }_{\mathrm{\Xi }}^{(q,\mathsf{s})}}{{10}^{-2}}}\right)\left({\displaystyle \frac{10\mathrm{ms}}{T_{*}^{(q,\mathsf{s})}}}\right).$$

(155)

Equation (155) should be interpreted with care. A short local interrogation time weakens sensitivity to weak rate-like residuals because ${\gamma }_{\mathrm{\Xi }}={\sigma }_{\mathrm{\Xi }}/T_{*}$ increases as $T_{*}$ decreases. Short-coherence platforms are therefore not optimal for very small quasi-static excess-decoherence rates. They may nevertheless be useful in three regimes. First, if the residual contrast between environment classes is large, then ${\chi }_{\mathsf{e}}^{\mathrm{x}}\left(T_{*}\right)$ can exceed ${\sigma }_{\mathrm{\Xi }}\left(T_{*}\right)$ even for small $T_{*}$. Second, for fast oscillatory signals, the relevant figure of merit is not only $T_{*}$, but the overlap of the signal with the control sequence response; suitably chosen Ramsey, echo, or multipulse sequences can target frequency components within the accessible interrogation window. Third, repeated local operation can reduce the statistical part of ${\sigma }_{\mathrm{\Xi }}$ approximately as $N^{-1/2}$, provided the nuisance model is stable and systematic uncertainties do not dominate. Thus long-coherence heavy-particle platforms are advantageous for weak rate-like residuals, while short-coherence platforms are best viewed as high-repetition, local-contrast probes rather than as optimal sensors for weak, slowly accumulating rate-like residuals.

For the millicharged-particle dictionary, the residual is naturally rate-like. Writing

$${\chi }_{\mathsf{e},\mathsf{s}}^{\mathrm{m}\mathcal{Q},\left(\mathrm{q}\right)}\left(T\right)={\mathrm{\Gamma }}_{\mathsf{e}}^{\mathrm{m}\mathcal{Q},\left(\mathrm{q}\right)}{\eta }_{\mathsf{s}}^{\left(q\right)}\left(T\right)T,$$

(156)

where ${\mathrm{\Gamma }}_{\mathsf{e}}^{\mathrm{m}\mathcal{Q},\left(\mathrm{q}\right)}$ is the effective scattering or heating-event rate and ${\eta }_{\mathsf{s}}^{\left(q\right)}\left(T\right)$ is the sequence- and platform-dependent coherence-loss efficiency, the benchmark condition becomes

$${\mathrm{\Gamma }}_{\mathsf{e}}^{\mathrm{m}\mathcal{Q},\left(\mathrm{q}\right)}{\eta }_{\mathsf{s}}^{\left(q\right)}\left(T_{*}\right)\gtrsim n_{\sigma }{\gamma }_{\mathrm{\Xi }}^{(q,\mathsf{s})}\left(T_{*}\right).$$

(157)

Thus the same microscopic millicharged flux can produce different cartographic visibility for different species, because the trap response, charge sign, mass, motional frequencies, and readout efficiency all enter the effective factor ${\eta }_{\mathsf{s}}^{\left(q\right)}$.

For the ultralight-field dictionary, the residual is generally not rate-like. A coherent scalar, pseudoscalar, or dark-photon-like background instead produces an oscillatory frequency perturbation,

$$\delta {\omega }_{\mathrm{UL}}^{(\mathsf{e};q)}\left(t\right)=A_{\mathrm{UL}}^{(\mathsf{e};q)}\cos \left({\omega }_{\phi }t+{\alpha }_{\mathsf{e}}\right),$$

(158)

where $A_{\mathrm{UL}}^{(\mathsf{e};q)}$ is the species-dependent response amplitude. The corresponding phase response under sequence $\mathsf{s}$ is

$${\mathrm{\Phi }}_{\mathsf{e},\mathsf{s}}^{\mathrm{UL},\left(\mathrm{q}\right)}\left(T\right)=A_{\mathrm{UL}}^{(\mathsf{e};q)}{\mathcal{I}}_{\mathsf{s}}({\omega }_{\phi },T,{\alpha }_{\mathsf{e}}),$$

(159)

with

$${\mathcal{I}}_{\mathsf{s}}({\omega }_{\phi },T,\alpha )\equiv {\int }_0^{T}dt{y}_{\mathsf{s}}\left(t\right)\cos \left({\omega }_{\phi }t+\alpha \right).$$

(160)

It is also useful to introduce the corresponding complex sequence response for which

$${\mathcal{I}}_{\mathsf{s}}(\omega ,T,\alpha )=\mathrm{Re}\left[e^{i\alpha }{\tilde{\mathcal{I}}}_{\mathsf{s}}(\omega ,T)\right],$$

(161)

where ${\tilde{\mathcal{I}}}_{\mathsf{s}}(\omega ,T)\equiv Y_{\mathsf{s}}(\omega ,T)$ is defined in Eq. (26).

If the phase of the ultralight field is stable over the measurement record, the relevant benchmark is a phase sensitivity,

$$A_{\mathrm{UL}}^{(\mathsf{e};q)}\gtrsim {\displaystyle \frac{n_{\sigma }{\sigma }_{\mathrm{\Phi }}^{(q,\mathsf{s})}\left(T_{*}\right)}{\left|{\mathcal{I}}_{\mathsf{s}}({\omega }_{\phi },T_{*},{\alpha }_{\mathsf{e}})\right|}},$$

(162)

where ${\sigma }_{\mathrm{\Phi }}^{(q,\mathsf{s})}$ is the uncertainty of the residual phase after nuisance subtraction.

If instead the field phase is randomized between statistically independent measurements or coherence intervals, averaging over the phase produces

$${\mathcal{C}}_{\mathsf{e},\mathsf{s}}^{\mathrm{UL},\left(\mathrm{q}\right)}\left(T\right)=J_0\left(A_{\mathrm{UL}}^{(\mathsf{e};q)}\left|{\tilde{\mathcal{I}}}_{\mathsf{s}}({\omega }_{\phi },T)\right|\right),$$

(163)

where $J_0$ denotes the Bessel function of the first kind of order zero. In the weak-signal regime, $A_{\mathrm{UL}}\left|{\tilde{\mathcal{I}}}_{\mathsf{s}}\right|\ll 1$, this gives

$${\chi }_{\mathsf{e},\mathsf{s}}^{\mathrm{UL},\left(\mathrm{q}\right)}\left(T\right)\simeq {\displaystyle \frac{1}{4}}{\left[A_{\mathrm{UL}}^{(\mathsf{e};q)}\right]}^2{\left|{\tilde{\mathcal{I}}}_{\mathsf{s}}({\omega }_{\phi },T)\right|}^2,$$

(164)

and hence

$$A_{\mathrm{UL}}^{(\mathsf{e};q)}\gtrsim {\displaystyle \frac{2\sqrt{n_{\sigma }{\sigma }_{\mathrm{\Xi }}^{(q,\mathsf{s})}\left(T_{*}\right)}}{\left|{\tilde{\mathcal{I}}}_{\mathsf{s}}({\omega }_{\phi },T_{*})\right|}}.$$

(165)

This estimate shows explicitly why ultralight-field cartography is controlled by frequency matching between ${\omega }_{\phi }$ and the sequence response, rather than only by the total interrogation time.

For scalar or dilaton-like ultralight fields, the amplitude in Eq. (158) may be written schematically as

$$A_{\mathrm{sc}}^{(\mathsf{e};q)}={\omega }_0^{\left(q\right)}\sum _i{K}_i^{\left(q\right)}{d}_i{\phi }_0^{\left(\mathsf{e}\right)},$$

(166)

where ${\omega }_0^{\left(q\right)}$ is the reference spin or transition frequency, $d_i$ are effective scalar couplings to SM parameters, $K_i^{\left(q\right)}$ are species- and transition-dependent sensitivity coefficients, and ${\phi }_0^{\left(\mathsf{e}\right)}$ is the local scalar field amplitude [9,14,15] and normalization is chosen such that $d_i{\phi }_0^{\left(\mathsf{e}\right)}$ is dimensionless. Equation (166) is deliberately schematic: the purpose is to show how a scalar field model enters the cartography observable, not to provide a platform-specific atomic-structure calculation.

For Lorentz-violating or CPT-violating dictionaries, the residual often appears as a deterministic modulation of the spin splitting, for example at sidereal or annual sideband frequencies. The benchmark is therefore similar to the coherent ultralight field case:

$$\delta {\omega }_{\mathrm{LV}}^{(\mathsf{e};q)}\left(t\right)=A_{\mathrm{LV}}^{(\mathsf{e};q)}\cos \left({\omega }_{\mathrm{LV}}t+{\alpha }_{\mathsf{e}}\right),$$

(167)

with detectability condition

$$A_{\mathrm{LV}}^{(\mathsf{e};q)}\gtrsim {\displaystyle \frac{n_{\sigma }{\sigma }_{\mathrm{\Phi }}^{(q,\mathsf{s})}\left(T_{*}\right)}{\left|{\mathcal{I}}_{\mathsf{s}}({\omega }_{\mathrm{LV}},T_{*},{\alpha }_{\mathsf{e}})\right|}}$$

(168)

for phase resolved measurements, or the analogue of Eq. (165) if the modulation is treated as dephasing. Matter–antimatter comparisons are especially important here because the response amplitude can differ between $q=p$ and $q=\overline{p}$, or between $q=e^{-}$ and $q=e^{+}$, depending on the CPT properties of the underlying coefficient. If instead the modulation is effectively phase randomized or unresolved between statistically independent measurements, the corresponding weak-signal dephasing benchmark follows from Eq. (165) with $A_{\mathrm{UL}}\to A_{\mathrm{LV}}$ and ${\omega }_{\phi }\to {\omega }_{\mathrm{LV}}$.

For collapse-like or diffusion-like residuals, the benchmark is again rate-like. If

$${\chi }_{\mathsf{e},\mathsf{s}}^{\mathrm{col},\left(\mathrm{q}\right)}\left(T\right)={\zeta }_{\mathsf{s}}^{\left(q\right)}{D}_{\mathrm{col}}^{\left(q\right)}T,$$

(169)

where $D_{\mathrm{col}}^{\left(q\right)}$ is the effective diffusion strength and ${\zeta }_{\mathsf{s}}^{\left(q\right)}$ is the platform response factor, then the detectability condition is

$${\zeta }_{\mathsf{s}}^{\left(q\right)}{D}_{\mathrm{col}}^{\left(q\right)}\gtrsim n_{\sigma }{\gamma }_{\mathrm{\Xi }}^{(q,\mathsf{s})}\left(T_{*}\right).$$

(170)

This has the same mathematical structure as the millicharged-scattering benchmark, although the microscopic interpretation is different.

The species dependence can be summarized as follows. Electron and positron platforms may have strong magnetic response and are natural for lepton-sector or charge-sign comparisons, but their useful cartographic range depends on the locally achievable interrogation time. Proton platforms provide the closest matter analogue of the antiproton architecture and are essential for baryonic matter-reference measurements. Antiproton platforms are the most distinctive for antimatter-sensitive, CPT-odd, and baryon–antibaryon differential dictionaries. In all cases, the benchmark sensitivity is controlled by three quantities:

$$\left\{T_{*}^{(q,\mathsf{s})},{\sigma }_{\mathrm{\Xi }}^{(q,\mathsf{s})}\left(T_{*}\right),\mathrm{dictionary}\mathrm{response}\mathrm{factor}\right\}.$$

(171)

The main lesson is that no single benchmark time or sensitivity characterizes the entire program. The different benchmark scalings are summarized in Table 1. Rate-like dictionaries benefit directly from long interrogation times through ${\gamma }_{\mathrm{\Xi }}\propto 1/T_{*}$. Coherent ultralight-field and Lorentz-violation dictionaries depend instead on the phase response integral ${\mathcal{I}}_{\mathsf{s}}$, and can be enhanced by choosing sequences matched to the target frequency. Species comparisons add a further layer of discrimination: a residual common to all q is naturally interpreted as species-generic, a residual confined to $e^{-}/e^{+}$ is lepton-sector-like, a residual confined to $p/\overline{p}$ is baryon-sector-like, and a residual changing under $q\to \overline{q}$ is a candidate matter–antimatter differential effect.

9. Interpretation of Null and Non-Null Cartographic Outcomes

The inference procedure of Section 6 classifies the residual structure as null, site-like, edge-like, transient, or dictionary-compatible. The purpose of the interpretation is to specify what physical conclusions can and cannot be drawn from each class. This distinction is essential because decoherence cartography is a null-test framework. A nonzero residual is not automatically evidence for new physics; it is first evidence for an inadequacy of the monitored nuisance model.

A fully null result corresponds to

$${\mathrm{\Xi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)\simeq 0$$

(172)

for all measured sites, control sequences, revisits, and interrogation times, within statistical and modeling uncertainty. Such an outcome would not be a failure of the program. It would establish that, at the achieved sensitivity, the coherence of the transported localized qubit is fully accounted for by monitored magnetic, vibrational, thermal, timing, and control channels. The result would then become a quantitative bound on residual environment-conditioned decoherence:

$$\left|{\chi }_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{x}}\left(T\right)\right|\lesssim {\sigma }_{\mathrm{\Xi },\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right),$$

(173)

where ${\sigma }_{\mathrm{\Xi },\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)$ is the uncertainty on the nuisance-subtracted cartographic observable. Through the dictionaries developed above, this bound can be translated into model-dependent constraints on millicharged scattering rates, Lorentz-violating frequency shifts, collapse-like diffusion constants, or other effective residual channels.

A site-like non-null result is one for which the data are well described by a reproducible node map,

$${\mathrm{\Xi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)={\mathrm{\Theta }}_{\mathsf{e},\mathsf{s}}\left(T\right)+{ϵ}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right),$$

(174)

with ${\mathrm{\Theta }}_{\mathsf{e},\mathsf{s}}\left(T\right)$ stable under revisits and consistent under graph-cycle tests. Such an outcome would indicate that the residual is associated with the endpoint environment rather than with the transport path alone. Physically, this could arise from unmodeled local technical conditions, local overburden, shielding, ambient radiation, site-dependent electromagnetic backgrounds, or an environment-dependent new-physics channel. The first interpretation to test is always the conservative one: incomplete nuisance modeling. Only after known environmental variables fail to explain the node pattern should a microscopic dictionary be invoked.

An edge-like non-null result is instead characterized by a reproducible path- or comparison-conditioned term,

$$\mathrm{\Delta }{\mathrm{\Xi }}_{ab,\mathsf{s},\mathsf{r}}\left(T\right)={\mathrm{\Theta }}_{a,\mathsf{s}}\left(T\right)-{\mathrm{\Theta }}_{b,\mathsf{s}}\left(T\right)+{\mathrm{\Lambda }}_{ab,\mathsf{s}}\left(T\right)+{ϵ}_{ab,\mathsf{s},\mathsf{r}}\left(T\right),$$

(175)

with ${\mathrm{\Lambda }}_{ab,\mathsf{s}}\left(T\right)$ stable for a given comparison or transport history. This class should not be interpreted as a site map. It points to transport-specific effects such as route-dependent magnetic exposure, vibration spectra, acceleration history, thermal cycling, or calibration changes between endpoint configurations. A genuinely exotic path-dependent effect is logically possible, but the interpretation burden is higher than for a node-like residual because transport itself introduces many ordinary edge-specific systematics.

A transient non-null result is one in which the residual is not stable under revisits but exhibits timing structure,

$${\mathrm{\Xi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)={\mathrm{\Theta }}_{\mathsf{e},\mathsf{s}}\left(T\right)+{{\rm Y}}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)+{ϵ}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right),$$

(176)

where ${{\rm Y}}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)$ is localized in time or correlated across sites. This class is the natural interpretation channel for transient exotic backgrounds, such as topological defect crossings, but it is also the channel most vulnerable to ordinary nonstationary noise. A transient candidate would therefore require timing correlations, environmental vetoes, and consistency across independent qubit species or independent sensors before being assigned a cosmological interpretation.

The interpretation also depends on control sequence scaling. A residual that appears in Ramsey measurements but is suppressed by Hahn echo or multipulse control is naturally associated with low-frequency dephasing. A residual that survives echo protocols may indicate higher-frequency noise, impulsive events, loss-like mechanisms, or phase accumulation that is not refocused by the chosen sequence. This sequence response can be summarized schematically as

$${\mathrm{\Xi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)⟶\left\{{\mathrm{\Xi }}_{\mathsf{e},\mathrm{R},\mathsf{r}}\left(T\right),{\mathrm{\Xi }}_{\mathsf{e},\mathrm{E},\mathsf{r}}\left(T\right),{\mathrm{\Xi }}_{\mathsf{e},\mathrm{MP},\mathsf{r}}\left(T\right)\right\},$$

(177)

where $\mathrm{R}$, $\mathrm{E}$, and $\mathrm{MP}$ denote Ramsey, echo, and multipulse protocols. The pattern across this sequence vector is part of the physical interpretation. It determines whether a residual is compatible with a broadband spectrum, a narrow-band harmonic signal, a burst process, or a loss-like channel.

A dictionary-compatible result is stronger than a generic non-null residual. It requires not only

$${\mathrm{\Xi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)\ne 0,$$

(178)

but also a match between the residual pattern and the predicted structure of a microscopic sector. For example, a Lorentz-violating interpretation requires the expected sidereal, boost, or orientation dependence; an ultralight-field interpretation requires a coherent oscillatory component with the appropriate frequency and phase behavior; a millicharged-sector interpretation requires consistency with a scattering or heating channel and with site dependence such as overburden; and a collapse-like interpretation requires compatibility with excess diffusion or heating not attributable to ordinary backgrounds. Thus the dictionaries are not labels assigned after a residual is found. They are predictive maps from model parameters to the residual structures that must be observed.

It is useful to separate three levels of claim:

$$\begin{array}{cc}\hfill \mathrm{Level}\mathrm{I}\text{:}& \mathrm{a}\mathrm{nuisance}\text{-}\mathrm{subtracted}\mathrm{residual}\mathrm{is}\mathrm{detected},\hfill \\ \hfill \mathrm{Level}\mathrm{II}\text{:}& \mathrm{the}\mathrm{residual}\mathrm{has}\mathrm{a}\mathrm{stable}\mathrm{site},\mathrm{edge},\mathrm{transient},\mathrm{or}\mathrm{sequence}\mathrm{structure},\hfill \\ \hfill \mathrm{Level}\mathrm{III}\text{:}& \mathrm{the}\mathrm{structure}\mathrm{matches}\mathrm{a}\mathrm{specific}\mathrm{microscopic}\mathrm{dictionary}.\hfill \end{array}$$

(179)

Only Level III supports a model dependent BSM or cosmological interpretation. Level I alone merely indicates that the nuisance model is incomplete. Level II establishes that the residual is organized enough to be physically classified, but not necessarily that it is exotic.

For matter and antimatter implementations, interpretation should also distinguish species-generic and species-differential residuals. A residual observed in electron, proton, and antiproton qubits with the same effective scaling is most naturally interpreted as a generic environmental or detector-platform effect. A residual that appears with different sign, amplitude, or harmonic structure in proton and antiproton implementations is more suggestive of an antimatter sensitive or CPT-odd sector, provided ordinary charge-sign and magnetic-moment systematics are excluded. The relevant comparison may be written schematically as

$${\mathrm{\Delta }}_{\mathrm{C}}{\mathrm{\Xi }}_{\mathsf{s},\mathsf{r}}\left(T\right)\equiv {\mathrm{\Xi }}_{\overline{p},\mathsf{s},\mathsf{r}}\left(T\right)-\mathcal{C}\left[{\mathrm{\Xi }}_{p,\mathsf{s},\mathsf{r}}\left(T\right)\right],$$

(180)

where $\mathcal{C}$ denotes the calibration map that accounts for known proton–antiproton differences in charge, magnetic moment sign, trap configuration, and control response. A nonzero ${\mathrm{\Delta }}_{\mathrm{C}}\mathrm{\Xi }$ is meaningful only after this calibration map is validated.

The strongest outcome of the program would be a residual that is simultaneously nuisance-subtracted, revisit-stable, graph-consistent, sequence-classifiable, and dictionary-compatible. Conversely, the most conservative outcome is a null map that improves bounds on residual channels. Both outcomes are scientifically useful. The first would identify a structured anomaly requiring targeted follow-up; the second would establish that transported localized qubits can serve as calibrated null instruments for BSM and cosmology-motivated sectors.

The guiding principle is therefore conservative interpretability. Decoherence cartography does not claim that any excess decoherence is exotic. It provides a hierarchy of tests that determine whether an excess is reproducible, where it lives in the graph, how it transforms under control sequences, whether it is species generic or antimatter differential, and only then whether it can be mapped onto a microscopic dictionary.

10. Worked Cartography Scenarios

The preceding sections define the cartography observable, the graph-based inference structure, and the dictionaries that map microscopic sectors onto residual phase or decoherence signals. We now illustrate how these ingredients would be used in concrete hypothetical cartography programs. The examples below are not discovery claims and do not assume that any nonstandard sector is present. Their purpose is to show how a residual map would be tested for dictionary compatibility after the conservative null hypothesis has failed.

We consider two representative cases. The first is a millicharged sector compatible residual, which is expected to appear primarily as a rate-like excess decoherence or heating channel. The second is an ultralight scalar compatible residual, which is expected to appear primarily as a narrow-band phase or frequency residual, with decoherence arising only when the field phase is unresolved or averaged. These two cases are deliberately chosen because they stress different parts of the formalism: the millicharged example tests node-like rate residuals in ${\mathrm{\Xi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)$, whereas the ultralight scalar example tests coherent sequence-dependent structure in the residual phase $R_{\mathrm{\Phi },\mathsf{e},\mathsf{s},\mathsf{r}}\left(T\right)$.

In both examples, the conclusion is organized in the same hierarchy. A Level-I conclusion means only that a residual has been detected after nuisance subtraction. A Level-II conclusion means that the residual is cartographically structured, for example reproducible under revisits and compatible with a node-like or edge-like graph pattern. A Level-III conclusion means that the structured residual is compatible with a specific microscopic dictionary. Only this final level permits a statement of dictionary compatibility, and even then the appropriate interpretation is compatibility rather than discovery.

10.1. Worked Cartography Scenario: a Millicharged-Compatible Residual

We now illustrate how the interpretation framework operates for a residual compatible with the millicharged-sector dictionary of Section 7.1. This example is hypothetical. Its purpose is not to claim that such a signal is expected, but to show what a residual cartographic map would have to look like in order to be described as millicharged-compatible.

The essential distinction from the ultralight-field scenario is that a millicharged sector contribution is naturally rate-like. In the dictionary of Section 7.1, the microscopic parameters of the millicharged sector are mapped onto an effective scattering, heating, or motional-noise contribution, which then generates a residual decoherence functional. The explicit microscopic mapping from the millicharged particle mass, charge, local density, velocity distribution, and trap response to ${\chi }_{\mathsf{e},\mathsf{s}}^{\mathrm{m}\mathcal{Q},\left(\mathrm{q}\right)}\left(T\right)$ as described in Section 7.1, where q labels the charged particle species as in Eq. (1).

Consider stationary-site cartography over a small set of environment classes,

$$\mathcal{E}=\{A,B,C\},$$

(181)

where A may denote the production or accelerator-adjacent environment, B a quieter laboratory, and C a shielded or underground environment. At each site the apparatus is stabilized before the qubit is prepared, interrogated, and read out locally. The same control sequences, interrogation times, and nuisance diagnostics are used across sites and revisits.

The measured residual observable is

$${\mathrm{\Xi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\left(q\right)}\left(T\right)=-\ln W_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\left(q\right)}\left(T\right)-{\widehat{\chi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{nuis},\left(\mathrm{q}\right)}\left(T\right),$$

(182)

where ${\widehat{\chi }}^{\mathrm{nuis}}$ is constructed from independently monitored magnetic, vibrational, thermal, and control-reference noise. In a millicharged-compatible interpretation, this residual should be fit by

$${\mathrm{\Xi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\left(q\right)}\left(T\right)={\chi }_{\mathsf{e},\mathsf{s}}^{\mathrm{m}\mathcal{Q},\left(\mathrm{q}\right)}\left(T\right)+{ϵ}_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\left(q\right)}\left(T\right),$$

(183)

where ${ϵ}_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\left(q\right)}$ contains statistical uncertainty and residual nuisance model uncertainty. Equation (183) is the operational version of the millicharged dictionary: the microscopic sector is not inferred directly from a single excess decay, but through its ability to reproduce the residual map across sites, sequences, revisits, species, and interrogation times.

The first inference step is the conservative null test,

$${\mathcal{H}}_0:{\mathrm{\Xi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\left(q\right)}\left(T\right)=0\mathrm{for}\mathrm{all}(\mathsf{e},\mathsf{s},\mathsf{r},q,T),$$

(184)

within the covariance assigned to the nuisance model. Failure of this null hypothesis does not imply millicharged particles. It only establishes that the monitored nuisance channels do not fully account for the observed coherence loss.

The second step is to test whether the residual has the graph structure expected for a local scattering or heating channel. In the simplest millicharged-compatible interpretation, the residual is node-like:

$${\mathrm{\Theta }}_{\mathsf{e},\mathsf{s}}^{\mathrm{m}\mathcal{Q},\left(\mathrm{q}\right)}\left(T\right)={\chi }_{\mathsf{e},\mathsf{s}}^{\mathrm{m}\mathcal{Q},\left(\mathrm{q}\right)}\left(T\right).$$

(185)

The corresponding edge observable is

$$\mathrm{\Delta }{\mathrm{\Xi }}_{ab,\mathsf{s},\mathsf{r}}^{\left(q\right)}\left(T\right)={\mathrm{\Theta }}_{a,\mathsf{s}}^{\mathrm{m}\mathcal{Q},\left(\mathrm{q}\right)}\left(T\right)-{\mathrm{\Theta }}_{b,\mathsf{s}}^{\mathrm{m}\mathcal{Q},\left(\mathrm{q}\right)}\left(T\right)+{ϵ}_{ab,\mathsf{s},\mathsf{r}}^{\left(q\right)}\left(T\right),$$

(186)

where $(a,b)\in \mathcal{L}$ is an edge of the site graph. A millicharged-compatible residual should therefore be reproducible under revisits to the same environment class and should be explainable primarily by site amplitudes, rather than by arbitrary path-dependent edge terms. If the data require a large independent ${\mathrm{\Lambda }}_{ab,\mathsf{s}}\left(T\right)$, the interpretation as a local millicharged scattering or heating effect is weakened unless the path dependence can be independently associated with a physical change in the relevant particle density or shielding environment.

The third step is to test the dependence on interrogation time and control sequence. A millicharge sector residual is expected to behave as an excess decoherence or heating channel, rather than as a coherent phase modulation. Using the rate-like form introduced in Section 7.1, one expects schematically

$${\mathrm{\Xi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\left(q\right)}\left(T\right)\simeq {\mathrm{\Gamma }}_{\mathsf{e}}^{\mathrm{m}\mathcal{Q},\left(\mathrm{q}\right)}{\eta }_{\mathsf{s}}^{\left(q\right)}\left(T\right)T,$$

(187)

up to statistical and nuisance model uncertainty. The first dictionary-compatible signature is therefore an approximately linear dependence on interrogation time,

$${\displaystyle \frac{\partial {\mathrm{\Xi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\left(q\right)}\left(T\right)}{\partial T}}\simeq {\mathrm{\Gamma }}_{\mathsf{e}}^{\mathrm{m}\mathcal{Q},\left(\mathrm{q}\right)}{\eta }_{\mathsf{s}}^{\left(q\right)}\left(T\right)+{\mathrm{\Gamma }}_{\mathsf{e}}^{\mathrm{m}\mathcal{Q},\left(\mathrm{q}\right)}T{\displaystyle \frac{\partial {\eta }_{\mathsf{s}}^{\left(q\right)}\left(T\right)}{\partial T}}.$$

(188)

If ${\eta }_{\mathsf{s}}^{\left(q\right)}\left(T\right)$ is slowly varying over the accessible range of T, this reduces to an approximately constant excess-decoherence rate. A residual dominated instead by sidereal, harmonic, or phase-coherent structure would be more naturally associated with the Lorentz-violation or ultralight-field dictionaries.

The fourth step is to test species dependence. The same microscopic millicharged parameters must generate a consistent set of residual amplitudes across the available species Eq. (1), after accounting for charge sign, mass, trap frequencies, motional response, and readout efficiency. Electron and positron platforms would probe lepton sector and charge sign response, while proton and antiproton platforms would probe baryonic matter–antimatter response. A residual that is common to all species after calibration may indicate a species-generic environmental channel, whereas a residual with a nontrivial dependence on q would require a corresponding dependence in the millicharged sector response model.

The microscopic interpretation is then tested by asking whether a common parameter set can account for the full residual map. Schematically, the requirement is

$$\left\{{\mathrm{\Xi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\left(q\right)}\left(T\right)\right\}\mathrm{is}\mathrm{compatible}\mathrm{with}\left\{{\chi }_{\mathsf{e},\mathsf{s}}^{\mathrm{mQ},\left(\mathrm{q}\right)}\left(T\right)\right\}$$

(189)

for a single millicharged sector model, together with site dependent densities, shielding factors, or environmental contrasts allowed by the dictionary. If no common microscopic parameter region can reproduce the residuals across sites, sequences, revisits, and species, then the residual may remain real but should not be described as millicharged-compatible.

As in Section 9, the conclusion should be stated in levels. A Level-I conclusion is only that a residual decoherence channel is present:

$${\mathrm{\Xi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\left(q\right)}\left(T\right)\ne 0.$$

(190)

A Level-II conclusion is that the residual is cartographically structured, for example node-like and reproducible under revisits:

$${\mathrm{\Xi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\left(q\right)}\left(T\right)\simeq {\mathrm{\Theta }}_{\mathsf{e},\mathsf{s}}^{\mathrm{mQ},\left(\mathrm{q}\right)}\left(T\right).$$

(191)

A Level-III conclusion is that the structured residual is compatible with the millicharged sector dictionary:

$${\mathrm{\Theta }}_{\mathsf{e},\mathsf{s}}^{\mathrm{mQ},\left(\mathrm{q}\right)}\left(T\right)\simeq {\chi }_{\mathsf{e},\mathsf{s}}^{\mathrm{mQ},\left(\mathrm{q}\right)}\left(T\right)$$

(192)

for a common microscopic parameter set and calibrated species response. Only this final level would justify describing the observation as millicharged-compatible, and even then the correct statement would be dictionary compatibility rather than discovery.

This worked example also clarifies how the millicharged interpretation can fail. It is disfavored if the residual is not positive, is not reproducible under revisits, is dominated by coherent phase modulation rather than excess decoherence, requires arbitrary edge terms, lacks the effective rate-like behavior specified by the dictionary, or cannot be generated by a common microscopic parameter region across sites and species. These failure modes are essential because they make the millicharged sector interpretation falsifiable rather than merely descriptive.

10.2. Worked Cartography Scenario: an Ultralight Scalar-Compatible Residual

We now illustrate how the interpretation framework operates for a residual compatible with the ultralight scalar field dictionary of Section 7.3. This example is hypothetical. Its purpose is not to claim that such a field is present, but to show what a residual cartographic signal would have to look like in order to be described as scalar-compatible.

The essential difference from the millicharged-particle scenario is that an ultralight scalar field does not primarily generate a Poissonian sequence of scattering or heating events. Instead, a coherently oscillating scalar field can produce an oscillatory shift of the effective constants entering the qubit frequency. Such a mechanism is the basis of scalar or dilatonlike dark matter searches with clock and frequency-comparison observables [9,14,15]. In the cartography language, the primary signature is therefore a narrow-band residual in the phase or spin-frequency channel, with dephasing appearing only when the scalar phase is unresolved, averaged, or fluctuating over the measurement record. We therefore use the dictionary map of Eq. (140), with the scalar-induced frequency perturbation represented by Eq. (132). The primary observable is the residual phase generated by the sequence-dependent integral in Eq. (133). Only if the scalar phase is unresolved, randomized, or averaged over many coherence intervals does the same perturbation appear primarily as a dephasing contribution, as in Eqs. (135) and (136).

Consider stationary-site cartography over the small set of environment classes introduced in Section 10.1 and defined in Eq. (181). For phase-resolved operation, the measured phase is written as

$${\mathrm{\Phi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\left(q\right)}\left(T\right)={\widehat{\mathrm{\Phi }}}_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{nuis},\left(\mathrm{q}\right)}\left(T\right)+{\mathrm{\Phi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{sc},\left(\mathrm{q}\right)}\left(T\right)+{ϵ}_{\mathrm{\Phi },\mathsf{e},\mathsf{s},\mathsf{r}}^{\left(q\right)}\left(T\right),$$

(193)

where ${\widehat{\mathrm{\Phi }}}^{\mathrm{nuis}}$ is the phase predicted from monitored conventional sources, ${\mathrm{\Phi }}^{\mathrm{sc}}$ is the scalar-compatible contribution described by the ultralight-field dictionary, and ${ϵ}_{\mathrm{\Phi }}$ contains statistical and residual modeling uncertainty. The phase residual is therefore

$$R_{\mathrm{\Phi },\mathsf{e},\mathsf{s},\mathsf{r}}^{\left(q\right)}\left(T\right)\equiv {\mathrm{\Phi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\left(q\right)}\left(T\right)-{\widehat{\mathrm{\Phi }}}_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{nuis},\left(\mathrm{q}\right)}\left(T\right).$$

(194)

In a scalar-compatible interpretation, $R_{\mathrm{\Phi }}$ should contain a narrow-band component at a common frequency ${\omega }_{\phi }$, with sequence dependence governed by the response integral defined in Eq. (160) or, equivalently, by the phase response of Eq. (133).

If the scalar phase is not resolved, the same underlying frequency perturbation contributes to decoherence. In that regime the relevant residual is not $R_{\mathrm{\Phi }}$ but

$${\mathrm{\Xi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\left(q\right)}\left(T\right)=-\ln W_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\left(q\right)}\left(T\right)-{\widehat{\chi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\mathrm{nuis},\left(\mathrm{q}\right)}\left(T\right),$$

(195)

and scalar compatibility is tested using the dephasing form of Eqs. (135)–(136). Thus the same microscopic dictionary can appear either as a phase residual or as a coherence-loss residual, depending on the stability of the scalar phase during the experimental record.

The cartographic interpretation proceeds in stages. First, the data must fail the conservative null hypothesis,

$${\mathcal{H}}_0:R_{\mathrm{\Phi },\mathsf{e},\mathsf{s},\mathsf{r}}^{\left(q\right)}\left(T\right)=0,{\mathrm{\Xi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\left(q\right)}\left(T\right)=0,$$

(196)

within the covariance assigned to the nuisance model. Failure of this null hypothesis does not imply an ultralight scalar field; it only shows that the monitored conventional model is incomplete.

Second, the residual must be narrow-band and reproducible. A scalar-compatible signal should be fit by

$$R_{\mathrm{\Phi },\mathsf{e},\mathsf{s},\mathsf{r}}^{\left(q\right)}\left(T\right)=A_{\mathrm{sc}}^{(\mathsf{e};q)}{\mathcal{I}}_{\mathsf{s}}\left({\omega }_{\phi },T,{\alpha }_{\mathsf{e}}\right)+{ϵ}_{\mathrm{\Phi },\mathsf{e},\mathsf{s},\mathsf{r}}^{\left(q\right)}\left(T\right),$$

(197)

where ${\mathcal{I}}_{\mathsf{s}}$ is the sequence response integral and the same ${\omega }_{\phi }$ must describe all relevant sites, revisits, sequences, and species. A residual whose fitted frequency changes arbitrarily between revisits is not compatible with a coherent ultralight scalar.

Third, the residual must show the correct sequence dependence. Changing from Ramsey to echo or multipulse control changes the response integral ${\mathcal{I}}_{\mathsf{s}}$. Therefore a scalar-compatible residual should change in the corresponding calculable way. A signal that is insensitive to sequences with very different filter responses is more naturally interpreted as an unmodeled offset, drift, or nuisance systematic.

Fourth, the graph structure should be consistent with a field-like node residual. In the simplest scalar interpretation, the site dependence enters through a local amplitude and phase, so one may write

$${\mathrm{\Theta }}_{\mathsf{e},\mathsf{s}}^{\mathrm{sc},\left(\mathrm{q}\right)}\left(T\right)=A_{\mathrm{sc}}^{(\mathsf{e};q)}{\mathcal{I}}_{\mathsf{s}}\left({\omega }_{\phi },T,{\alpha }_{\mathsf{e}}\right).$$

(198)

The corresponding edge observable is

$$\mathrm{\Delta }{R}_{\mathrm{\Phi },ab,\mathsf{s},\mathsf{r}}^{\left(q\right)}\left(T\right)={\mathrm{\Theta }}_{a,\mathsf{s}}^{\mathrm{sc},\left(\mathrm{q}\right)}\left(T\right)-{\mathrm{\Theta }}_{b,\mathsf{s}}^{\mathrm{sc},\left(\mathrm{q}\right)}\left(T\right)+{ϵ}_{\mathrm{\Phi },ab,\mathsf{s},\mathsf{r}}^{\left(q\right)}\left(T\right).$$

(199)

If arbitrary edge or path terms ${\mathrm{\Lambda }}_{ab,\mathsf{s}}\left(T\right)$ are required, the interpretation as a local scalar-field node residual is weakened unless the edge dependence can be independently associated with a physical scalar-field or environmental modulation along the route.

Fifth, the species dependence must be consistent with the scalar-field dictionary. For a scalar or dilatonlike field that modulates effective constants, the species dependence is encoded in the response amplitudes $A_{\mathrm{sc}}^{(\mathsf{e};q)}$, or equivalently in the sensitivity coefficients appearing in Eq. (166). In particular,

$${\displaystyle \frac{A_{\mathrm{sc}}^{(\mathsf{e};q_1)}}{A_{\mathrm{sc}}^{(\mathsf{e};q_2)}}}\simeq {\displaystyle \frac{{\omega }_0^{\left(q_1\right)}{\sum }_i{K}_i^{\left(q_1\right)}{d}_i}{{\omega }_0^{\left(q_2\right)}{\sum }_i{K}_i^{\left(q_2\right)}{d}_i}},$$

(200)

up to calibration and nuisance uncertainty. A scalar-compatible interpretation requires a common set of scalar parameters to describe the data across species q (Eq. (1)), after calibration of charge sign, magnetic moment, and reference frequency. A residual that changes sign between matter and antimatter species in a way not implied by these calibrated response functions would be more naturally associated with a CPT-odd or charge-sign dependent dictionary than with a simple scalar-field one.

The result should again be stated in levels. A Level-I conclusion is only that a residual phase or decoherence signal is present,

$$R_{\mathrm{\Phi },\mathsf{e},\mathsf{s},\mathsf{r}}^{\left(q\right)}\left(T\right)\ne 0\mathrm{or}{\mathrm{\Xi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\left(q\right)}\left(T\right)\ne 0.$$

(201)

A Level-II conclusion is that the residual is structured: narrow-band, reproducible under revisits, and consistent with node-like graph behavior,

$$R_{\mathrm{\Phi },\mathsf{e},\mathsf{s},\mathsf{r}}^{\left(q\right)}\left(T\right)\simeq {\mathrm{\Theta }}_{\mathsf{e},\mathsf{s}}^{\mathrm{sc},\left(\mathrm{q}\right)}\left(T\right).$$

(202)

A Level-III conclusion would be that the same frequency ${\omega }_{\phi }$, the same scalar couplings $d_i$, and a consistent set of site amplitudes and phases explain the residuals across sites, species, sequences, and interrogation times:

$${\mathrm{\Theta }}_{\mathsf{e},\mathsf{s}}^{\mathrm{sc},\left(\mathrm{q}\right)}\left(T\right)\simeq {\omega }_0^{\left(q\right)}\left(\sum _i{K}_i^{\left(q\right)}{d}_i\right){\phi }_0^{\left(\mathsf{e}\right)}{\mathcal{I}}_{\mathsf{s}}\left({\omega }_{\phi },T,{\alpha }_{\mathsf{e}}\right).$$

(203)

Only this final level would justify describing the data as ultralight-scalar compatible.

This example also clarifies how the scalar interpretation can fail. It is disfavored if the residual is purely rate-like with no narrow-band phase structure, if the fitted frequency is not reproducible, if the response does not follow the control sequence, if arbitrary edge terms are required, if the species dependence cannot be described by a common scalar response model, or if a matter–antimatter sign reversal is observed after calibration. These failure modes are essential because they make the scalar-field interpretation falsifiable rather than merely descriptive.

11. Conclusions and Outlook

We have developed a species-general framework for transport-enabled decoherence cartography with localized charged-particle qubits. At the level of the reduced qubit dynamics, filter-function description, graph-based comparison, and statistical inference, the formalism does not depend on a specific particle species. Its experimental realization, however, is strongly platform dependent: the achievable interrogation time, control fidelity, transport architecture, nuisance monitoring, and microscopic response coefficients must be established separately for each implementation.

The primary operational mode considered in this work is stationary-site cartography. In this mode, the trapping platform, together with a particle reservoir is relocated between environment classes and allowed to stabilize before the qubit is prepared, interrogated, and read out locally. Transport enables the comparison by exposing the same measurement platform to distinct nuisance spectra and possible environment dependent residual channels. This formalism does not require a previously prepared superposition to remain coherent throughout the journey.

Coherent-transport cartography, in which a state is prepared before relocation and read out only after transport, constitutes a more demanding special case. Its feasibility is constrained by the requirement that the state remain coherent for at least the transport duration. By contrast, in stationary-site cartography the relevant requirement is only that the locally prepared state remain coherent throughout the interrogation window. Consequently, even platforms with comparatively short coherence times may participate in facility-scale or longer-baseline cartography, provided that local preparation, interrogation, and readout can be repeated reproducibly after relocation.

The scale of a first implementation is constrained by present hardware. Coherent spectroscopy of a single trapped antiproton has demonstrated Rabi oscillations with an effective spin-coherence time of order $50\mathrm{s}$ [1]. In addition, the BASE collaboration has demonstrated the physical transport of trapped antiprotons in the portable cryogenic BASE-STEP system [2,4,5]. These achievements make a transport-enabled antiproton program experimentally concrete, but they do not yet imply a continent-scale sensor network or multi-hour coherent transport. A realistic first cartography program would instead compare a small number of carefully characterized environment classes, such as the accelerator-adjacent production environment, a nearby magnetically quieter laboratory, and, eventually, a shielded or underground measurement site.

From the theoretical perspective, such a restricted first stage is already sufficient. Decoherence cartography does not require the largest possible geographical baseline. It requires measurable contrast between environment classes, independent monitoring of the dominant nuisance channels, and reproducible revisits. The immediate objective is therefore to maximize environmental contrast and repeatability, rather than distance alone. A particularly informative initial protocol would consist of an $A\to B\to A$ sequence: establish a reference data set at site A, relocate the platform to site B, and then return to A. Such a cycle directly tests whether an apparent site difference is reproducible and whether the original response is recovered after transport.

The broader BASE program, including transportable magnet concepts and future off-site precision stations, points toward increasingly flexible relocation and measurement infrastructure [6]. Related progress in sympathetic cooling of a single proton through coupling to laser-cooled ⁹${\mathrm{Be}}^{+}$ ions provides a credible path toward improved motional preparation and deeper quantum control of heavy trapped baryons [23]. These developments are relevant not only for precision spectroscopy, but also for reducing the technical-noise floor against which cartographic residuals would be inferred.

For any species q, a viable cartography platform must provide a localized and sufficiently long-lived stored-particle qubit; reproducible coherent preparation, interrogation, and readout; controlled, or at least quantitatively characterized, relocation between environment classes; and sufficiently complete monitoring of nuisance channels to construct ${\widehat{\chi }}^{\mathrm{nuis}}$ together with its covariance. The last requirement is particularly important. A long coherence time alone does not make a platform suitable for cartography if changes in magnetic noise, trap motion, control calibration, state preparation, or readout cannot be propagated consistently into the nuisance model.

The antiproton platform currently provides the primary empirical motivation. It combines coherent single-spin spectroscopy with demonstrated transport in a portable cryogenic Penning-trap system [1,2]. Its distinctive scientific role does not lie in a generic enhancement of sensitivity, but in providing access to an antimatter baryon and hence to matter-antimatter differential observables, CPT-odd effects, and antimatter sensitive sectors.

The proton platform is the closest matter analogue. Loss-free proton relocation using BASE-STEP has been demonstrated, while single-proton spin-state detection and high-precision magnetic-moment spectroscopy are mature [21,22,45,95]. Proton measurements can therefore serve both as a development platform for the transport and inference protocol and as the natural matter reference for antiproton cartography.

Single-electron Penning-trap spectroscopy provides exceptionally mature single-particle frequency metrology and strong magnetic response [19,20,96]. A directly analogous transportable single-electron spin-cartography system has not yet been demonstrated, however. Electron implementations should consequently be regarded as promising experimental platform for the species-general formalism, particularly for lepton-sector effects and residuals with relatively rapid temporal variation, although the required transport architecture and coherence benchmarks remain to be established.

The positron supplies the antimatter-lepton counterpart of the electron. Single electrons and positrons have historically been isolated under closely matched Penning-trap conditions for high-precision comparisons of their magnetic moments [46], and positron accumulation, cooling, compression, and transfer between trap regions are well-developed elements of antihydrogen experiments [47,97]. These capabilities are not yet equivalent to an autonomous, transportable single-positron spin-qubit platform with coherent local interrogation and cartography-grade nuisance monitoring. Positrons should therefore be regarded as a well-motivated future extension that would distinguish lepton-sector, charge-sign, and antimatter-dependent residuals from effects specific to baryons.

Taken together, the four-species set provides a useful hierarchy of comparisons. Electron–positron measurements probe matter–antimatter differences in the lepton sector, while proton–antiproton measurements probe the corresponding baryonic sector. Cross-sector comparisons can further test whether a residual scales with mass, magnetic moment, electric charge, baryon number, or matter-antimatter character. Such comparisons are meaningful only after the species-dependent transfer functions, control responses, and nuisance models have been calibrated.

The central output of the proposed program is not an isolated excess-decay measurement, but a structured residual map together with its covariance. The conservative null result is ${\mathrm{\Xi }}_{\mathsf{e},\mathsf{s},\mathsf{r}}^{\left(q\right)}\left(T\right)=0$ for all tested sites, sequences, revisits, interrogation times, and species, within statistical and nuisance model uncertainties. Such a null result would still be scientifically useful: it would bound environment-conditioned decoherence channels and constrain the microscopic sectors represented by the dictionaries developed above.

A nonzero residual would require a hierarchy of increasingly restrictive tests. It would first have to remain after accounting for local state-preparation and readout effects and after propagating the calibrated nuisance-channel contributions. It would have to exhibit reproducible dependence on site, transport edge, control sequence, time, or particle species. Only after passing these tests could it be compared with a microscopic dictionary.

Even at the final stage, compatibility with a millicharged-particle, Lorentz-violating, ultralight-field, or collapse-like dictionary would not by itself constitute a discovery claim. Competing technical and microscopic explanations would have to be tested through additional sequences, revisits, orientations, species comparisons, and independent experiments.

The principal experimental challenge is consequently not transport alone, but the transfer of calibration across sites. State-preparation and measurement errors, changes in trap eigenfrequencies, magnetic shielding, mechanical coupling, thermal conditions, and control electronics can all produce apparent site dependence. The credibility of the cartography program will therefore depend on interleaved reference measurements, common calibration protocols, blind or held-out nuisance tests, and explicit propagation of modeling uncertainties into the residual covariance.

The framework presented here is deliberately conservative. It neither assumes that an exotic residual exists nor privileges a particular microscopic interpretation. Its purpose is to provide an operational and falsifiable framework for future transport-enabled experiments: to define what is measured, separate locally calibrated nuisance contributions from residual dynamics, compare environments through a graph structure, and test whether any surviving pattern is reproducible and compatible with a microscopic dictionary. Recent progress in coherent antiproton spectroscopy and portable trapped-particle transport makes this question timely, while the extension to proton, electron, and positron platforms demonstrates that the underlying cartographic strategy is more general than any single experimental realization.