Expedited Noise Spectroscopy of Transmon Qubits

Bhavesh Gupta; Nicolas Gheeraert; Prabha Mandayam; Siddharth Dhomkar; Udit Kandpal; Vismay Joshi

REVIEW 3 major objections 5 minor 68 references

Expedited Noise Spectroscopy of Transmon Qubits

T0 review · 3 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read A convolutional neural network, trained only on synthetic data, can infer a transmon qubit's dephasing noise spectrum from a single CPMG-32 coherence decay curve in near real time.

desk verdict A clean synthetic demonstration of CNN-based noise spectroscopy, but the experimental validation on IBM qubits is only a round-trip consistency check, so the claim of accurate real-device spectra is not yet supported. read the letter →

arxiv 2502.00679 v2 pith:EMCH3OAM submitted 2025-02-02 quant-ph physics.app-ph

Bhavesh Gupta , Vismay Joshi , Udit Kandpal , Prabha Mandayam , Nicolas Gheeraert , Siddharth Dhomkar This is my paper · ORCID

classification quant-phphysics.app-ph PACS 03.67.-a03.65.Yz85.25.-j07.05.Mh

keywords noisespectroscopytransmonqubitsdynamicaldecouplingconvolutionalneuralnetworkdecoherencefunctional1/fCPMGsequencetime-dependent

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that a convolutional neural network, pre-trained entirely on synthetic noise spectra and their corresponding coherence decays, can take a single CPMG-32 coherence-decay measurement from a transmon qubit and output the dephasing noise spectrum S(ω) in near real time. The claim matters because conventional noise spectroscopy requires long, repeated measurements and deconvolution of an ill-posed integral equation, and because noise on real devices drifts faster than those slow protocols can track. If correct, the method gives a fast, hardware-agnostic way to benchmark qubits, monitor time-varying noise, and tailor dynamical decoupling sequences to the actual noise of each qubit. The authors validate the workflow on superconducting qubits aboard a 127-qubit processor, reconstructing measured decay curves from predicted spectra with errors dominated by the experimental noise level.

What carries the argument

The load-bearing objects are the decoherence functional $\chi(t)$ and the filter-function formalism of Eq. (3), which ties a measurable coherence curve $C(t)$ to the noise power spectral density $S(\omega)$. The paper's inversion engine is a convolutional autoencoder trained on pairs $(C(t), S(\omega))$ generated by convolving synthetic three-regime spectra with a CPMG-32 filter function and adding Gaussian noise; the network takes a single coherence curve as input and outputs the spectrum directly, replacing iterative deconvolution. A secondary piece is the use of the predicted $S(\omega)$ as an objective-function input: substituting $S(\omega)$ and a candidate dynamical-decoupling sequence's filter function into Eq. (3) lets a classical optimizer time the pulses to minimize $\chi(t)$.

What would settle it

Take a qubit whose dephasing is dominated by a narrow discrete two-level-system peak (for instance, by tuning a strong defect into resonance with the qubit), measure C(t) with CPMG-32, feed it to the network, and independently determine S(ω) with a conventional multi-sequence noise-spectroscopy protocol; a predicted spectrum that misses the narrow peak while still reproducing C(t) through Eq. (3) would show that the training-family constraint, not the data, controls the answer.

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Extended reading notes

Core claim

The central discovery is that inverting the decoherence functional can be learned rather than solved. Given the relation $\chi(t)=-\ln C(t)=\int_0^\infty \frac{d\omega}{\pi}\frac{S(\omega)F(\omega t)}{\omega^2}$, where $F(\omega t)$ is the filter function of a CPMG-32 pulse sequence, the authors generated tens of thousands of synthetic spectra $S(\omega)$ from a three-regime model (white noise at low frequency, $1/\omega$ noise at intermediate frequency, $k/(k^2+\omega^2)$ noise at high frequency), computed the corresponding $C(t)$ curves, added Gaussian noise, and trained an autoencoder-style convolutional network to predict $S(\omega)$ from a single noisy $C(t)$. On held-out synthetic data the prediction error is about 3.6% mean absolute error; on real qubits the spectra are predominantly white-noise dominated with a $1/\omega$ crossover near 1 MHz, and the decay curves reconstructed from predicted spectra match the measured ones. The method also resolves time-dependent changes: ten consecutive T1/T2 measurement pairs on one qubit yield noise spectra that track abrupt jumps, which the authors interpret as quasi-instantaneous noise profiling.

Load-bearing premise

The load-bearing premise is that a real qubit's dephasing noise is made only of white noise at low frequencies, 1/f noise at intermediate frequencies, and Lorentzian noise at high frequencies, with no other spectral features; if a measured qubit has noise outside this family, the network's spectrum will be biased and the paper's consistency check cannot catch that bias.

Editorial extensions

If this is right

One CPMG-32 coherence decay suffices to infer a transmon's dephasing noise spectrum, cutting the measurement burden by the factor implied by the 81-fold shot-count range tested.
Noise spectroscopy can be repeated often enough to catch drift: consecutive runs about ten minutes apart on one qubit show T1, T2, and the extracted S(ω) changing abruptly and in a non-correlated way.
Predicted spectra can drive bespoke dynamical decoupling; on the white-noise-dominated qubits tested the optimized sequences gave no visible improvement, which confirms the spectra rather than improving coherence.
Because the network is trained on synthetic data with no device-specific calibration, the same trained model can be applied to other qubit hardware without retraining on experimental data.
A network trained at an intermediate noise level (3% added Gaussian noise) stays accurate when test noise varies from 1% to 9%, so experimenters can reduce measurement time without precisely matching the training signal-to-noise ratio.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper: if real dephasing noise contains spectral features outside the white + 1/ω + Lorentzian training family, such as discrete two-level-system peaks or strongly non-Gaussian fluctuations, the network will return a biased spectrum; the paper's consistency check (reconstructing C(t) from predicted S(ω) via the same Eq. (3)) cannot detect that bias because the inversion is constrained t
Beyond the paper: retraining the same architecture on a broader distribution, including Lorentzian peaks at arbitrary frequencies and non-Gaussian noise models, would extend the method to devices where localized defects dominate dephasing; the paper's robustness heatmap suggests the dominant error would then be how well the training family spans reality.
Beyond the paper: near-instant inference opens a control-loop possibility not demonstrated here, namely feeding a freshly extracted spectrum back into dynamical-decoupling optimization in real time so that error suppression adapts as the noise drifts, provided measurement and inference were integrated with low-latency control hardware.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

This is a direct extension of the authors' own CNN-based noise spectroscopy work (refs [32,33]) to IBM transmon qubits. The genuinely new parts: application to a multi-qubit cloud device, time-dependent noise tracking over ten consecutive runs, and design of bespoke dynamical decoupling sequences from the predicted spectra. The paper also honestly reports that the customized sequences did not improve coherence because the noise is white-noise dominated, which is a point in its favor.

The synthetic benchmark is clean. Mean absolute error of 3.6% on the test set, and a robustness analysis showing the network handles mismatched noise levels. The network architecture and training-data generation are described well enough to reproduce.

The soft spot is the one the stress-test note identifies: the experimental validation is a closed loop. The measured coherence curve goes into the network; the predicted S(ω) is fed back into Eq. (3) with the same CPMG-32 filter; the resulting curve is compared to the same measured curve. That verifies internal consistency within the generator used for training, not that the true spectrum was recovered. The inverse problem is ill-posed, as the paper admits, so many in-family spectra can produce nearly identical CPMG-32 decays. Any real spectral structure outside the training family — TLS peaks, non-Gaussian statistics, non-Lorentzian bumps — would go undetected while the round-trip check still passes. The 3.6% synthetic error measures invertibility on the training distribution, not accuracy on real hardware.

Minor issues: the match between predicted and measured coherence curves is called an 'authentication,' which overstates what it is. The time-dependent part is one qubit, ten runs, no external reference. The data and code are not actually linked; the data availability statement says 'available from the authors,' which is not sufficient for a reproducibility-minded referee.

Who gets value: people working on ML-based quantum characterization and experimentalists wanting fast, hardware-agnostic qubit noise screening. The paper deserves a serious referee, because the synthetic result is solid and the application is timely. But the referee should require an independent benchmark against a standard QNS protocol on the same qubits, plus error bars on the extracted spectra.

My take: the load-bearing claim — that a single CPMG-32 curve yields accurate noise spectra on real transmon hardware — is not yet supported. Worth engaging with, but as a method in need of validation, not as a demonstrated result.

Referee Report

3 major / 5 minor

Summary. The paper presents a machine-learning method for extracting the dephasing noise spectrum S(ω) of a transmon qubit from a single CPMG-32 coherence decay C(t). A convolutional neural network is trained on synthetic (S(ω), C(t)) pairs generated via the filter-function relation Eq. (3), with spectra restricted to a smoothed combination of white, 1/ω, and Lorentzian components. The authors report 3.6% synthetic test error, robustness to injected Gaussian noise, and apply the method to IBM Osaka qubits, using the extracted spectra to design optimized dynamical decoupling sequences and to track time-dependent noise. The core technical pipeline is clear, but the experimental validation is a round-trip through the same forward model used for training.

Significance. If the accuracy claim were established, this would be a practically useful fast benchmarking tool for transmon dephasing, and the combination with time-resolved spectroscopy is attractive. The synthetic experiments are clean and the robustness analysis is a genuine strength; code availability is a further plus. The main limitation is that the only experimental evidence consists of reconstructing C(t) from the predicted S(ω) with Eq. (3), the very model used to generate the training data, so the central claim of accurate spectra on real devices is not yet supported. The problem is identifiable and potentially fixable with an independent QNS comparison or an out-of-distribution test.

major comments (3)

[Section IV.C, Eq. (3)] The validation on IBM Osaka is a closed loop. The predicted noise spectra in Fig. 3(c) are evaluated by inserting them into Eq. (3) to reconstruct coherence curves and comparing those reconstructions with the same C(t) that was fed to the network (Fig. 3(d)-(f)). Because Eq. (3) is also the forward model used to generate the synthetic training data, this procedure demonstrates only that the predicted spectrum lies in the preimage of the observed C(t) under the training generator. The paper itself states in Section II that the deconvolution problem is ill-posed; many spectra can yield nearly identical CPMG-32 decays. To support the claim of accurate spectra on real hardware, the authors should compare their predicted S(ω) with an independent QNS measurement, or at least show that the round-trip check fails for plausible out-of-family spectra.
[Section III, step 1] The CNN is trained exclusively on spectra of the stitched form: white at low frequencies, 1/ω at intermediate frequencies, and Lorentzian k/(k^2+ω^2) at high frequencies, with smoothed cutoffs. The paper then claims that the extracted spectra are accurate for IBM qubits. That conclusion requires the real dephasing noise to be in this parametric family (and stationary, Gaussian, and z-axis coupled). No test is reported with spectra outside the family, such as discrete TLS peaks, non-Lorentzian bumps, or non-Gaussian fluctuations, so the network's behavior on such noise is unknown and the consistency check cannot detect a bias. Please add an out-of-distribution test or otherwise justify the prior. In addition, the low-frequency white / intermediate 1/ω ordering is not the standard transmon picture (usually 1/f at low frequencies), so the choice of spectral shapes should be justified physically.
[Section III, step 3, Eq. (5)] Eq. (5) is presented as the extraction of the pure-dephasing coherence from T1 and T2 data. With the stated definitions, P0(T2) is the probability of |0> in a T2 experiment and P1(T1) is the probability of |1> in a T1 experiment; thus P0(T2) = (1 + C_meas(t))/2 (up to readout calibration) and P1(T1) = e^{-Γ1 t}. The identity C(t) = P0(T2)/sqrt(P1(T1)) = e^{-Γ2 t}/sqrt(e^{-Γ1 t}) then does not hold unless P0(T2) is the bare coherence signal, not a probability. Please clarify the normalization and calibration of P0(T2) and correct Eq. (5), since this processing is the direct input to the network.

minor comments (5)

[Section IV.A] The quoted 3.6% mean absolute error is the error in the coherence curves reconstructed from the predicted spectra, not the error in S(ω); please state this explicitly and, where possible, report a spectral-domain error metric.
[Section IV.E] The text says the extracted noise spectra are presented in Fig. 5(c), but Fig. 5(c) is the Tφ heatmap; the spectra appear in Fig. 5(d).
[Figs. 1 and 4] The axis labels contain the typo 'Frequancy'; the introduction also contains 'substatially' instead of 'substantially'.
[Data Availability / Code Availability] The code availability statement refers to a public repository without giving its URL; please add the repository identifier so the claim is verifiable.
[Appendix B, Table II] The network is described with Conv2D layers although the input is one-dimensional; using 1D convolution terminology (or explaining the 2D convention) would avoid confusion for readers.

Circularity Check

2 steps flagged · score 5.0 of 10

IBM validation of extracted S(ω) is a closed loop through the same forward model (Eq. 3) used to generate training data; no external ground-truth spectrum is measured.

fitted input called prediction [Section IV.C, Fig. 3(e,f); training data generation in Section III, step 1]
"The model accurately predicted the noise spectrum for each qubit, as evidenced by a minimal prediction error in the coherence functions reconstructed from the extracted noise spectra; two example curves are shown in Fig. 3(e) and (f)."

The 'reconstructed' coherence functions are obtained by inserting the network's extracted S(ω) into Eq. (3), the same forward model used in Section III, step 1 to generate every training pair (S(ω), C(t)). The experimental check therefore verifies only that the predicted spectrum is one of the many spectra that map back to the input coherence curve under the generator; it does not compare S(ω) to any independently measured noise spectrum. Because the inverse map from a single C(t) to S(ω) is ill-posed, a spectrum outside the training family could also reproduce C(t), and the round-trip error would not detect it. Calling this 'accurate' prediction is thus a consistency check on the training distribution, not validation of the real noise spectrum.
self definitional [Section IV.C, paragraph on IBM results; Section III, step 1]
"The analysis revealed that the noise spectrum predominantly exhibited white noise characteristics, with a 1 /ω-type profile emerging around 1 MHz, confirming the model’s effectiveness in identifying dominant noise features."

The network's output space was defined in Section III, step 1 to be exactly the stitched family: 'white noise dominates at relatively low frequencies, 1 /ω-type noise becomes prominent at intermediate frequencies, and k/(k2 + ω2)-type noise takes over at relatively high frequencies.' Any output of the trained network is therefore guaranteed, by construction, to contain white and 1/ω components; observing these features in the extracted IBM spectra is a property of the parameterization, not independent evidence that the true spectra have been identified. The 'confirming' claim is self-definitional rather than an external test.

full rationale

The central training procedure is self-contained: synthetic (S, C) pairs are generated from an assumed parametric family via Eq. (3), and the synthetic test set (3.6% MAE) demonstrates that the CNN can invert the forward model within that family. This part is not circular. The circularity enters when the paper claims experimental validation on IBM qubits. In Fig. 3(e,f), the agreement is between the measured coherence input and a coherence curve re-computed from the predicted S using Eq. (3), the same equation used to create the training labels. Since the forward map is many-to-one, matching C(t) does not certify the true S(ω); it only certifies consistency with the generator. Similarly, the observation that extracted IBM spectra show white and 1/ω features is forced by the training-family ansatz, which is adopted as the only allowed output shape. The self-citations [32,33] are not load-bearing here because the network is retrained on new synthetic data and the functional ansatz is taken from an external review. Overall, the real-device claim of 'accurate noise spectra' is partially circular: it reduces to a closed-loop consistency check, while the genuinely non-circular content is the in-family synthetic invertibility result.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the standard filter-function formalism (Eq. 3), the Gaussian-stationary-z-axis dephasing model, the approximate factorization in Eq. (5), and the assumption that real transmon noise falls within the three-component synthetic family. No new physical entities are introduced. The main burden is the ad hoc training-distribution assumption.

free parameters (5)

White noise amplitude (A_white)
Sampled over a range to generate synthetic training spectra (Section III, step 1); the CNN's output distribution is constrained by this range.
1/f noise amplitude (A_1f)
Sampled over a range for the intermediate-frequency component of S(omega).
High-frequency Lorentzian amplitude and width (k)
Sampled over a range for the k/(k^2 + omega^2) component at high frequencies.
Frequency cutoffs and smoothing parameters
Chosen to stitch the three noise regimes into a smooth S(omega); these shape the training set.
CNN hyperparameters (layers, filters, pooling, dropout)
Hand-chosen architecture (Table II, Appendix B); reasonable but not optimized, and differs from the exact prior architecture in refs [32,33].

assumptions (4)

domain assumption Noise is stationary, Gaussian, and couples exclusively along the qubit z-axis (pure dephasing).
Stated in Section III. Required for the filter-function integral (Eq. 3) to apply and for the CNN input to be the pure-dephasing coherence curve.
standard math The decoherence functional relation chi(t) = integral d(omega)/pi S(omega) F(omega t)/omega^2 holds for the CPMG sequence.
Eq. (3), from Cywinski et al. [44]; this is the standard filter-function formalism for dephasing under dynamical decoupling.
domain assumption The measured T2 coherence decay factorizes so that C(t) = P0(T2)/sqrt(P1(T1)) isolates pure dephasing.
Eq. (5); this assumes the off-diagonal decay is e^{-Gamma1/2 t} e^{-chi(t)} and ignores gate errors, detuning, and measurement artifacts.
ad hoc to paper Real qubit noise spectra lie within the three-component functional family (white + 1/f + Lorentzian) used to generate training data.
Section III, step 1; if the real S(omega) has features outside this family, the CNN predictions will be biased. The paper checks that experimental curves fall within training bounds but does not verify the spectral family itself.

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Cite this review

Pith. "Pith review of Expedited Noise Spectroscopy of Transmon Qubits." pith.science (2026). https://pith.science/paper/EMCH3OAM

@misc{pith2026250200679,
  author       = {Pith},
  title        = {Pith review of: Expedited Noise Spectroscopy of Transmon Qubits},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EMCH3OAM}},
  note         = {Machine review of arXiv:2502.00679}
}

read the original abstract

There has been tremendous progress in the physical realization of quantum computing hardware in recent times, bringing us closer than ever before to realizing the promise of quantum computing. However, noise continues to pose a crucial challenge when it comes to scaling up present day quantum processors. While decoherence limits the qubits ability to store information for long periods in the presence of uncontrollable noise sources, the erroneous implementation of control methods for state preparation and measurements leads to faulty implementations of quantum circuits. Conventional noise spectroscopy protocols can characterize and model environmental noise but are usually resource intensive and lengthy. Moreover, the underlying noise can vary in nature over time, making noise profile extraction futile as this new information cannot be harnessed to improve quantum error correction or dynamical decoupling protocols. In this work, we address this challenge using a machine learning-based methodology to quickly extract noise spectra of multiple qubits and demonstrate a possible noise mitigation strategy. The procedure involves implementing undemanding dynamical decoupling sequences to record coherence decays of the investigated qubits and then predict the underlying noise spectra with the help of a convolution neural network pre-trained on a synthetic dataset. While our protocol is virtually hardware-agnostic, we validate its effectiveness using superconducting qubits available on the IBM Quantum platform. We further use these rapidly obtained, yet accurate, noise spectra to design bespoke dynamic decoupling sequences and perform time-dependent noise spectroscopy.

Figures

Figures reproduced from arXiv: 2502.00679 by the authors.

**Figure 1.** FIG. 1. (a) Randomly chosen 5 test input decoherence curves as functions of evolution time; the dashed lines denote the curves [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

**Figure 2.** FIG. 2. The heatmap displaying percentage mode values asso [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3. (a), (b),and (d) display [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: FIG. 4. The first and second columns correspond to two different qubits from IBM-Osaka. (a), (e) Decoherence curves [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: FIG. 5. (a)—(c) Heatmaps displaying time-dependent [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 7.** Figure 7: FIG. 7. % Coherence Improvement for a randomly chosen [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

**Figure 6.** Figure 6: FIG. 6. Distribution of fitted [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]

**Figure 8.** Figure 8: FIG. 8. % error in the predicted [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Training Data Generation : Based on the previ- ous work [39], we assumed that the noise spectra S(ω) follow some complex yet well-defined func- tional form. Particularly, we considered that white noise dominates at relatively low frequencies, 1 /ω- type noise becomes prominent at intermediate fre- quencies, and k/(k2 + ω2)-type noise takes over at relativ...

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A detailed description is provided in the SI

Network Construction and Training : Using the numerically generated noise spectral densities and their associated coherence curves C(t), we trained a convolution neural network [33] to identify the noise spectral density S(ω) based on a single deco- herence curve C(t) provided at the network input. A detailed description is provided in the SI

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After determining the opti- mal amplitude for the square-shaped π-pulse, we performed sequential T1 and T2 decay experiments using a CPMG-32 pulse protocol

Experimental Data Acquisition : For experimen- tal validation, we performed measurements on IBM’s superconducting quantum processors using the Qiskit package. After determining the opti- mal amplitude for the square-shaped π-pulse, we performed sequential T1 and T2 decay experiments using a CPMG-32 pulse protocol. The choice of CPMG-32 sequence over conve...

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [1]

Training Data Generation : Based on the previ- ous work [39], we assumed that the noise spectra S(ω) follow some complex yet well-defined func- tional form. Particularly, we considered that white noise dominates at relatively low frequencies, 1 /ω- type noise becomes prominent at intermediate fre- quencies, and k/(k2 + ω2)-type noise takes over at relativ...

work page

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A detailed description is provided in the SI

Network Construction and Training : Using the numerically generated noise spectral densities and their associated coherence curves C(t), we trained a convolution neural network [33] to identify the noise spectral density S(ω) based on a single deco- herence curve C(t) provided at the network input. A detailed description is provided in the SI

work page

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After determining the opti- mal amplitude for the square-shaped π-pulse, we performed sequential T1 and T2 decay experiments using a CPMG-32 pulse protocol

Experimental Data Acquisition : For experimen- tal validation, we performed measurements on IBM’s superconducting quantum processors using the Qiskit package. After determining the opti- mal amplitude for the square-shaped π-pulse, we performed sequential T1 and T2 decay experiments using a CPMG-32 pulse protocol. The choice of CPMG-32 sequence over conve...

work page

[4] [4]

Noise Spectra Prediction: The noise spectrum pre- diction is then obtained almost instantaneously, by providing the acquired experimental data as an in- put to our trained neural network

work page

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By substitut- ing the extracted noise spectrum S(ω) into Eq.(3), one can effectively obtain an objective function that seeks to minimize χ(t)

Optimization of DD Pulses : The optimization be- gins by defining the filter function F (ωt) corre- sponding to a given DD sequence. By substitut- ing the extracted noise spectrum S(ω) into Eq.(3), one can effectively obtain an objective function that seeks to minimize χ(t). The goal is to find the optimal timing of pulses that reduces the overlap between...

work page

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3% noise, is extremely resilient to the changes in signal-to-noise ratio

First, the neural network trained with intermedi- ate levels of experimental noise, e.g. 3% noise, is extremely resilient to the changes in signal-to-noise ratio. This signifies that the experimenter can re- duce the measurement time significantly without a substantial reduction in prediction accuracy. More- over, this also means that the experimenter nee...

work page

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Secondly, classical optimizers are extremely sensi- tive to initial guesses and frequently get trapped at local minima. On the contrary, it is known 6 Frequency (MHz) → → Evolution time, τ (μs)Evolution time, τ (μs)Evolution time, τ (μs) Evolution time, τ (μs) Evolution time, τ (μs) S(ω) (arbitrary units) FIG. 3. (a), (b),and (d) display T1, T2 and Tϕ dat...

work page