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Strongly driven transmon as an incoherent noise source

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Pith's one-line read A strongly driven transmon whose ground state sits in a chaotic layer acts, at short times, as an incoherent noise source to a coupled two-level system, with exponential relaxation matching an infinite-temperature bath.

desk verdict Genuinely new chaos-to-noise mechanism for transmon crosstalk, but the advertised analytical 1/T1 formula is off by a factor of 64. read the letter →

arxiv 2505.24549 v1 pith:UAFRVGW4 submitted 2025-05-30 quant-ph

Linda Greggio , Rémi Robin , Mazyar Mirrahimi , Alexandru Petrescu This is my paper
classification quant-ph
keywords transmonquantumchaosdynamicallocalizationreflectedBrownianmotionnoisespectroscopyFloquetmodesinfinite-temperaturebathsuperconductingcircuits
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

The paper tries to establish that the chaotic dynamics created by strong transmon drives can propagate to neighboring circuit elements: a transmon prepared with its ground state inside the chaotic layer behaves, at short times, as an incoherent noise source toward a coupled spectator two-level system. The authors compare a full quantum simulation with two semiclassical descriptions, one in which the transmon charge follows a chaotic classical pendulum and one in which it follows a reflected Brownian motion with diffusion constant $D = \lambda^2/\xi_d$. In the fast-resonance-crossing regime the Brownian description reproduces the quantum relaxation dynamics, with the TLS decaying exponentially and with equal excitation and relaxation rates ($\gamma_\downarrow \approx \gamma_\uparrow$), the signature of an infinite-temperature bath. The paper also identifies where the analogy fails: at long times the finite chaotic Hilbert space saturates the TLS population at a plateau, and at stronger drives dynamical localization suppresses relaxation below the classical prediction.

What carries the argument

The central object is the reflected Brownian motion (RBM), a diffusive random process confined to the interval $[-\bar{p},\bar{p}]$ by reflecting boundaries, which replaces the transmon's charge variable in the semiclassical limit. Starting from the driven-pendulum Hamiltonian obtained at small effective Planck constant, the paper uses the fast-resonance-crossing approximation to reduce the stroboscopic dynamics to the standard map with $k = 2\sqrt{\pi\lambda}/\sqrt{\xi_d}$; successive kicks have effectively random phases, so the momentum diffuses with $D = \lambda^2/\xi_d$, and the bounded chaotic layer is modeled by the reflecting diffusion. This RBM supplies the classical noise $\tilde{p}_t$ that drives the TLS in Eq. (18), and its power spectral density, computed from the Fokker-Planck eigenfunctions of the reflecting diffusion, enters the golden-rule rate Eq. (19), giving $\gamma_\downarrow = \gamma_\uparrow$. For the long-time plateau the paper instead uses the Floquet modes of the coupled Hamiltonian, whose random-phase overlaps with the chaotic layer yield the estimate $z_{\mathrm{ss}} = \frac{1}{2N_{\mathrm{ch}}}\sum_\alpha |z_\alpha|^2 = \frac{1}{2}\mathrm{Var}_\alpha(z_\alpha)$.

What would settle it

Measure the excitation and relaxation rates of a spectator qubit as a function of its transition frequency and of the transmon drive amplitude inside the chaotic window, and compare with the golden-rule estimate Eq. (19); unequal up and down rates, or a rate that departs from the diffusive prediction away from drive resonances, would break the infinite-temperature Brownian picture, while a long-time plateau that changes with the initial qubit state would confirm the finite-bath mechanism.

Watch

Extended reading notes

Core claim

The central discovery is that a coherently driven transmon can act as a source of incoherent noise to another circuit element coupled to it, provided the transmon's state lies in a chaotic layer of its classical phase space. Concretely, the paper shows numerically that when the transmon is initialized in its ground state, which strong driving spreads over the chaotic region, the population of a transversally coupled TLS decays exponentially toward a maximally mixed state with $\gamma_\downarrow \approx \gamma_\uparrow$, as if the TLS were attached to a bath at infinite temperature. The short-time quantum dynamics is matched by a semiclassical model in which the transmon's charge variable follows a chaotic classical pendulum, and, for relaxation only, by a reflected Brownian motion on $[-\bar{p},\bar{p}]$ with diffusion constant $D = \lambda^2/\xi_d$. The paper derives from the power spectral density of that Brownian process a golden-rule estimate for the relaxation rate, Eq. (19). On longer times the analogy breaks because the chaotic layer is a finite-dimensional subspace of the transmon Hilbert space: the TLS population saturates at a plateau set by the number of chaotic Floquet modes and the initial TLS state, and at sufficiently strong drives dynamical localization makes the induced relaxation weaker than the classical and diffusive predictions.

Load-bearing premise

The derivation rests on replacing the transmon's actual noisy effect on the neighbor qubit with a prescribed classical random signal, with no reaction from the qubit back onto the transmon and no memory of the transmon's quantum level structure.

Editorial extensions

If this is right

  • Energy relaxation of a spectator TLS in the fast-crossing regime is exponential with equal excitation and relaxation rates, so the driven transmon acts as an infinite-temperature bath for relaxation.
  • The golden-rule formula Eq. (19) gives an analytic estimate of the spectator's $1/T_1$ directly from transmon parameters, drive amplitude, and the chaotic-layer boundary.
  • The reflected Brownian motion is not a valid model for pure dephasing, because the fast-crossing approximation misses the low-frequency noise that controls fourth-order dephasing; the chaotic classical pendulum is needed there.
  • At long times the TLS population does not fully depolarize: it saturates at a plateau whose height depends on the number of chaotic Floquet modes and on the initial TLS state.
  • Beyond a localization threshold, dynamical localization suppresses the transmon's noise, so the induced relaxation is weaker than the classical and diffusive models predict.

Reading between the lines

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

  • If multiple spectator qubits couple to the same chaotic transmon, the reflected-Brownian-motion picture predicts cross-correlated noise between them; measuring that correlation would extend the single-spectator result to multi-qubit crosstalk.
  • The plateau formula implies the effective bath size is tunable: decreasing the effective Planck constant increases the number of chaotic Floquet modes and pushes the steady state toward the infinite-temperature result, so the plateau height could serve as a probe of how classical the transmon noise is.
  • In bosonic-error-correction schemes where transmon ancillas are strongly driven for tomography, this mechanism would predict spurious excitation and relaxation of the stored bosonic mode even without extrinsic losses, with the analytic rate giving an order-of-magnitude estimate.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The paper considers a strongly driven transmon as an incoherent noise source for a capacitively coupled spectator TLS. Three models are studied: the full quantum model, the classical driven-pendulum limit, and, in the fast-crossing regime, a reflected Brownian motion (RBM) for the charge variable. The central claims are that (i) at short times the driven transmon induces exponential TLS relaxation with equal up/down rates, as if coupled to an infinite-temperature bath, with an analytical Fermi Golden Rule estimate in Eq. (19); (ii) the RBM captures this relaxation when the drive is strong but before dynamical localization; (iii) on longer times the finite-dimensional chaotic Floquet subspace leads to a plateau in TLS polarization, with an analytical estimate in Eq. (31); and (iv) dynamical localization suppresses the relaxation at very strong drives. The conclusions are supported by extensive numerical comparison of the three models over drive strengths and TLS frequencies.

Significance. If the quantitative estimates are corrected, the qualitative mechanism is significant for circuit-QED crosstalk: a transmon ancilla driven into the chaotic regime can act as an effective thermal/infinite-temperature bath for neighboring circuit elements, with implications for multi-transmon processors and bosonic codes. The paper's strengths include deriving D and p-bar from the classical pendulum and standard-map theory rather than fitting them to TLS data, providing an explicit Floquet-plateau formula, and honestly stating the validity limitations (RBM fails for dephasing; finite chaotic subspace yields a plateau; localization at strong drive). The main quantitative weak point is the FGR formula in Eq. (19), which is inconsistent with the RBM spectral density by a large factor; this must be corrected before the quantitative claims can be accepted.

major comments (2)
  1. [§III A, Eq. (19), Appendix B 3] The printed FGR formula is not consistent with the RBM spectral density. From the correlation function in Eq. (B24) with a = π²D/(8p̄²), the correct spectral density, keeping the n=1 and n=3 modes, is Spp(ω) = (8D/π²)/(ω² + π⁴D²/(64p̄⁴)) + (8D/(9π²))/(ω² + 81π⁴D²/(64p̄⁴)). Eq. (19), after clearing denominators, has the ω̃q² coefficient 64 times smaller in both terms and a second-term zero-frequency coefficient that is a factor 81 too large relative to the correct transform of Eq. (B24). In the high-frequency limit Eq. (19) overestimates γ↓=γ↑ by a factor 64 relative to the printed Eq. (B25) and by a factor 576 relative to the correct transform of Eq. (B24). Because Eq. (19) is the paper's analytical estimate for 1/T1 and is compared with numerics in Fig. 5, this discrepancy is load-bearing; the formula and the comparison must be corrected.
  2. [Appendix B 3, Eq. (B25)] The coefficient 18a in the second term of Eq. (B25) is inconsistent with Eq. (B24). Since the n=3 term in Eq. (B24) is (1/81) exp(-9a|τ|), its Fourier transform contributes (2a/9)/(ω²+(9a)²), not 18a/(ω²+(9a)²). This 81-fold error propagates to Eq. (19) and should be corrected independently of the comparison in Fig. 5.
minor comments (2)
  1. [§III A, before Fig. 5] The text states that Eq. (19) is shown 'at ξd = 1.5', while the Fig. 5 caption and panel label indicate ξd = 2.5; please reconcile which parameter set is plotted.
  2. [§III A, summary paragraph] The phrase 'reflected Bronwnian motion' should read 'reflected Brownian motion'.

Circularity Check

0 steps flagged · score 0.0 of 10

No load-bearing circularity; the TLS relaxation rates and plateau are derived from an explicit stochastic model and independently checked against full quantum simulation.

full rationale

The paper's central claim is that a strongly driven transmon can act as an incoherent noise source for a coupled TLS. This is supported by a self-contained derivation chain. The diffusion constant D=λ²/ξd is obtained from the variance of the momentum jump in the fast-crossing approximation, Eq. (B9)-(B10), not fitted to TLS data. The chaotic-layer boundary p̄ is estimated from Chirikov resonance-overlap theory in Appendix B 2, again without reference to the TLS relaxation. The noise spectral density Spp(ω) in Eq. (B25) is computed from the exact eigenfunctions of the reflected Brownian motion Fokker-Planck operator, Eqs. (B16)-(B25). Equation (19) is then a Fermi Golden Rule evaluation using that spectral density; no parameter is calibrated against the TLS population dynamics, and the comparison with the full quantum model in Figs. 4-5 is an independent numerical check rather than an input to the derivation. The long-time plateau formula Eq. (31) is derived from the Floquet expansion and the random-phase approximation Eq. (27), then compared to numerics, so it is not imposed. Self-citations, including [6] and [21], are used as background for chaos-assisted transitions and are not load-bearing for the new stochastic-noise or plateau results. The apparent discrepancy between Eq. (19) and Eq. (B25) noted in the skeptic analysis is an internal arithmetic/correctness issue, not a circularity, since the quoted formula is not a fitted input masquerading as a prediction. No step was found where a claimed prediction is equivalent to its inputs by construction.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

No parameter is fitted to the TLS relaxation data. The stochastic model parameters D and p̄ are derived from the classical pendulum and standard-map theory, although p̄ still requires a numerical or Chirikov-based estimate. The listed analysis thresholds (IPR cut, fit window) are the main hand-chosen numbers. The key axioms are Floquet and FGR background plus the physical assumptions of weak backaction, random chaotic phases, and closed unitary dynamics.

free parameters (3)
  • p̄ (boundary of chaotic layer) = numerically extracted from Chirikov resonance overlap; no closed form
    Sets the RBM domain and enters the FGR formula Eq. (19); the authors note in Sec. II D and Appendix B 2 that only a coarse analytic estimate exists.
  • IPR cutoff for selecting chaotic Floquet modes = 0.3
    Ad hoc threshold used in Sec. III B when comparing the plateau estimate Eq. (31); no sensitivity analysis is reported.
  • Rate extraction window = first 200 drive periods
    Used for linear-regression extraction of γ↑ and γ↓ in Fig. 5; no error bars or window-dependence are shown.
assumptions (7)
  • standard math Floquet theorem: a T-periodic Hamiltonian has a complete basis of T-periodic modes with quasienergies (Eq. 21).
    Used for the long-time plateau derivation in Sec. III B; not proved in the paper.
  • standard math Fermi's Golden Rule for weak coupling: rate γ = g² S(ωq) for a TLS driven by classical noise.
    Used to obtain Eq. (19) from the RBM spectral density in Appendix B 3.
  • domain assumption The charge-driven transmon Hamiltonian Eq. (1), with readout resonator fluctuations neglected, describes the device.
    Starting model in Sec. II A; neglects resonator quantum fluctuations and external dissipation.
  • domain assumption Resonance-crossing phases in the fast-crossing regime are IID uniform random variables, following Chirikov and Shepelyansky [27].
    Basis for the diffusion approximation and D=λ²/ξd in Appendix B 1.
  • domain assumption Chaotic Floquet modes have random overlap signs and uniform magnitude |d_α|≈1/√Nch (Eq. 27).
    Needed for the plateau formula Eq. (31); validated only indirectly through numerical agreement.
  • domain assumption The TLS is a weakly coupled classical noise sensor with no backaction on the transmon (Eq. 18).
    Underlies the semiclassical comparison; breaks near resonances where the drive dresses the TLS (Fig. 5).
  • domain assumption The system is closed and unitary; no external bath is present, so all TLS relaxation is induced by the transmon.
    The paper deliberately omits environmental dissipation in Sec. II A; long-time recurrence follows from the finite-dimensional Hilbert space.

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Pith. "Pith review of Strongly driven transmon as an incoherent noise source." pith.science (2026). https://pith.science/paper/UAFRVGW4

@misc{pith2026250524549,
  author       = {Pith},
  title        = {Pith review of: Strongly driven transmon as an incoherent noise source},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UAFRVGW4}},
  note         = {Machine review of arXiv:2505.24549}
}
read the original abstract

Under strong drives, which are becoming necessary for fast high-fidelity operations, transmons can be structurally unstable. Due to chaotic effects, the computational manifold is no longer well separated from the remainder of the spectrum, which correlates with enhanced offset-charge sensitivity and destructive effects in readout. We show here that these detrimental effects can further propagate to other degrees of freedom, for example to neighboring qubits in a multi-qubit system. Specifically, a coherently driven transmon can act as a source of incoherent noise to another circuit element coupled to it. By using a full quantum model and a semiclassical analysis, we perform the noise spectroscopy of the driven transmon coupled to a spectator two-level system (TLS), and we show that, in a certain limit, the interaction with the driven transmon can be modeled as a stochastic diffusive process driving the TLS.

Figures

Figures reproduced from arXiv: 2505.24549 by the authors.

Figure 1
Figure 1. FIG. 1. A lumped-element representation of a periodically [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (a) Poincar´e section of an undriven pendulum, overlaid on the Husimi function of the transmon ground state at [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Asymptotic time-averaged standard deviation of mo [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: We consider ℏeff = 0.16, and three drive strengths ξd = 1.5, 2.5 and 4.5 (as in [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 4
Figure 4. Figure 4: FIG. 4. Short-time TLS relaxation dynamics when initial [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 6
Figure 6. Figure 6: FIG. 6. (a) Same conventions as in of Fig. 4(b) for a longer [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. (a)-(c) Upper envelope of the damped oscillations [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Estimate for the boundary of the chaotic region (thick [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]
Figure 10
Figure 10. Figure 10: shows |Rαβk| 2 versus the transition frequency ∆αβk, for different values of ℏeff. We overlay plots for a sweep of the offset charge ng in [0, 0.5]. As we decrease ℏeff, the spectrum, and hence the distribution of |Rαβk| 2 , becomes denser since the density of states …

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