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Effective Hamiltonian for an off-resonantly driven qubit-cavity system

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Pith's one-line read By deferring the rotating-wave approximation until after a displacement transformation, this paper derives an effective Hamiltonian that reproduces measured Stark shifts and driven interactions in a qubit-cavity system where the early-RWA…

desk verdict Late RWA is a genuine fix for the ac Stark shift, but the missing benchmark against the full Hamiltonian keeps this from being a closed result. read the letter →

arxiv 2509.03375 v1 pith:EITP37HP submitted 2025-09-03 quant-ph

Martin Jirlow , Kunal Helambe , Axel M. Eriksson , Simone Gasparinetti , Tahereh Abad This is my paper · ORCID
classification quant-ph
keywords effectiveHamiltonianoff-resonantdrivingrotating-waveapproximationdisplacementtransformationcircuitQEDacStarkshifttwo-modesqueezingbeamsplitting
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

Off-resonant multi-tone driving of a qubit-cavity system is hard to model: the standard recipe of moving to a displaced frame and then immediately applying the rotating-wave approximation predicts, for example, the wrong ac Stark shift of the qubit. This paper shows that the order of those two steps matters. Deferring the rotating-wave approximation until after the displacement keeps counter-rotating drive components that, together with anharmonic terms, produce slow, resonant contributions at small detunings, and the authors write down the resulting effective Hamiltonian explicitly. Tested against circuit-QED measurements, the model reproduces the measured Stark shifts to about one to four percent, the sign change of the shift across resonance, and the interference patterns of two-mode squeezing and beam-splitting gates.

What carries the argument

The machine that carries the argument is the time-dependent displacement transformation $U(t)=D_q[\xi_q(t)]D_c[\xi_c(t)]$, with displacement amplitudes split into co-rotating and counter-rotating parts, $\xi_i(t)=\xi_{i,1}(t)+\xi_{i,2}(t)e^{2i\omega_i t}$. The counter-rotating pieces are the ones the early-RWA approach discards; keeping them lets terms such as $b^{\dagger 2}b$ rotate slowly at small drive detunings and contribute to the effective dynamics. The amplitudes are fixed by requiring the linear drive terms to cancel, with decay rates $\kappa_q,\kappa_c$ included so that the Lindblad dissipators remain invariant. Expanding the quartic Josephson term in the displaced frame and then applying the RWA produces Eqs. (9)--(12).

What would settle it

Tune a drive tone in the same qubit-cavity setup so that a discarded superharmonic process such as $b^{\dagger 3}$ becomes resonant, then compare the measured transition rate with the prediction of Eqs. (9)--(12); the omitted term would make the effective Hamiltonian miss the resonance. A milder check is a Stark-shift-versus-detuning scan pushed well beyond $\Delta_q\approx -\alpha$, where the "small detunings" assumption should eventually break down.

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Extended reading notes

Core claim

The central claim is that the dynamics of a dispersively coupled qubit-cavity system under arbitrary off-resonant multi-tone drives is captured by the effective Hamiltonian of Eqs. (9)--(12). The derivation keeps the full cosine form of the drives, applies a displacement transformation whose amplitudes include both co-rotating ($\xi_{i,1}$) and counter-rotating ($\xi_{i,2}$) parts, and only at the final step applies the rotating-wave approximation. The resulting Hamiltonian contains a diagonal part with drive-induced frequency shifts $\delta_q$ and $\delta_c$, an interaction part $H_1$ describing displaced anharmonic and dispersive terms, and a counter-rotating correction $H_2$. The paper validates this construction by showing that its spectrum matches experimental ac Stark shifts, including the sign change near resonance and the avoided-crossing feature near $\Delta_q\approx -\alpha$, and that its dynamics reproduce the measured populations in two-mode squeezing and beam-splitting protocols.

Load-bearing premise

The load-bearing premise is that every term discarded by the final rotating-wave step really rotates fast enough to average out, in particular superharmonic processes such as $b^{\dagger 3}$; the paper assumes the drive detunings are small enough to suppress these but gives no quantitative boundary for that regime.

Editorial extensions

If this is right

  • The closed-form shifts $\delta_q=-2\alpha|\xi_{q,1}|^2-\chi|\xi_{c,1}|^2$ and $\delta_c=-2K_c|\xi_{c,1}|^2-\chi|\xi_{q,1}|^2$ make cumulative Stark effects from complicated drive spectra predictable before running a master-equation simulation.
  • Choosing opposite detunings, $\Delta_c=-\Delta_q=\Delta$, makes the two-photon term $b^\dagger a^\dagger$ resonant, enabling two-mode squeezing; choosing matched detunings, $\Delta_c=\Delta_q$, makes the exchange term $b a^\dagger$ resonant, enabling beam-splitting.
  • Since the derivation only uses the quartic expansion of the Josephson cosine, the same effective-Hamiltonian construction transfers to any anharmonic oscillator, not just the transmon.
  • The model's agreement with the measured sign change across resonance means that Stark-shift-based calibration can use $H_{\rm eff}$ to place operating points away from the avoided crossing near $\Delta_q\approx -\alpha$.
  • The truncation that drops superharmonic terms such as $b^{\dagger 3}$ is a stated boundary: transitions like $|0f\rangle\leftrightarrow|1g\rangle$, which require such terms, are not covered by the present version.

Reading between the lines

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

  • An implicit design rule, not stated in the paper, is to check any rapidly rotating term against every counter-rotating drive component before discarding it; applying that check systematically to sixth-order Josephson terms is the natural route to superharmonic transitions.
  • In platforms with weaker anharmonicity or stronger drives, the $H_2$ counter-rotating correction could grow relative to $H_1$, so the Late RWA ordering should matter in optomechanical and trapped-ion settings, not only in circuit QED.
  • A sharper validation than the ground-state Stark shift would be to measure the Fock-resolved dispersive shift under multi-tone drive; the model predicts explicit photon-number dependence through the $b^\dagger b a^\dagger a$ and $b^\dagger a^\dagger a$ terms, and a mismatch at higher $n$ would localize where the small-detuning assumption breaks.
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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

3 major / 5 minor

Summary. This manuscript derives an effective Hamiltonian for a qubit-cavity system under multi-tone off-resonant driving. The authors introduce a 'Late RWA' approach in which the rotating-wave approximation is applied only after a displacement transformation in the rotating frame, thereby retaining counter-rotating drive contributions. The resulting Hamiltonian (Eqs. (9)-(12)) contains diagonal Stark shifts, two-photon interactions (two-mode squeezing and beam-splitting), and counter-rotating corrections. The model is validated against experimental data from a previous paper by the same group, showing quantitative agreement for ac Stark shifts versus drive amplitude and detuning, two-mode squeezing chevrons, and beam-splitting dynamics. The paper claims that this model captures the near-resonant regime where the conventional 'Early RWA' approach fails.

Significance. If the central claims hold, the Late RWA effective Hamiltonian provides a practical and broadly applicable tool for simulating and designing driven bosonic interactions in circuit QED and related platforms. The key conceptual contribution is showing that counter-rotating drive terms can become resonant after displacement and that deferring the RWA captures near-resonant corrections missed by conventional displaced-frame treatments. The Stark-shift validation is a notable strength: for a single qubit drive, the model reproduces the measured shift to about 1.3% error without fitted parameters, and it captures the sign change across resonance and the avoided crossing at Δ ≈ -α. The explicit closed-form formulas make the model easily implementable, although no simulation code is shipped. The paper also usefully identifies the limits of the Early RWA method, which is widely used in the field.

major comments (3)
  1. [Supplement, after Eq. (S29); main text Eq. (9)] The final RWA step is not quantitatively justified. The only validity condition given is that 'drive detunings are small enough to neglect superharmonic processes, such as three-photon transitions (b†3) and similar terms', but no error bound or numerical benchmark against the full Hamiltonian without this final RWA is provided. The tested regime includes detunings up to about 300 MHz, which is not small compared with the qubit anharmonicity α ≈ 230 MHz, and the avoided crossing near Δ ≈ -α is precisely where a truncated RWA can miss resonant level repulsion. Since the central claim is quantitative accuracy and general applicability to arbitrary multi-tone driving, the authors should either derive a bound on neglected terms or provide a comparison of Heff dynamics with an exact simulation of the full Hamiltonian (or the displaced Hamiltonian before the final RWA) over the parameter ranges used in Figs. 1-4. The outlook statement acknowledging superharmonic processes as future work confirms that this is an open issue rather than a closed result.
  2. [Supplement, around Eq. (S30)] The derivation of the effective Hamiltonian is not shown in sufficient detail. The main text states that 'after performing the RWA, obtain' Eqs. (9)-(12), and the supplement says 'Expanding the quartic interaction ... we obtain' without displaying the intermediate algebra. Since Eqs. (9)-(12) are the main result, readers cannot verify the derivation or check which terms were kept and which were discarded. Please include the displaced expansion of the quartic term, the explicit identification of rotating vs. counter-rotating terms, and the selection rules used in the final RWA, or provide a symbolic-verification script (e.g., a Mathematica or Python notebook) as supplemental material.
  3. [Main text, Fig. 4 and surrounding text] The beam-splitting simulation uses a fitted 'experimental correction' νcorr ≈ -5.02 MHz that shifts the cavity drive frequency. This is a free parameter not predicted by the model. While the use is disclosed, it means the beam-splitting data do not provide a parameter-free validation of Heff. Please quantify how sensitive the agreement in Fig. 4 is to νcorr and clarify which of the presented validations (Stark shifts, squeezing, beam-splitting) are fully parameter-free. The abstract's claim of 'quantitatively reproduces experimentally measured ac Stark shifts and captures key interactions' is supported by the Stark-shift data, but the beam-splitting demonstration would be more convincing with a sensitivity analysis or with νcorr predicted from the model.
minor comments (5)
  1. [Supplement, after Eq. (S29)] The shorthand notation is defined incorrectly: the text says 'ξ_i,2(t) = ∑_n ξ_i,1(t)' but the second sum should be over ξ_i,2(t). Please correct the typo.
  2. [Fig. 2 caption and axis labels] The horizontal axis label 'Detuning q [MHz]' should be 'Δ_q [MHz]' to match the text, and the y-axis label should be 'Stark shift [MHz]' with the units. The current label is ambiguous.
  3. [Main text, Eq. (3) and surrounding text] The sentence 'Avoiding the conventional approximation a†e^{-iωt}+h.c., we retain the full cosine form' is confusing: the conventional approximation is typically to drop the counter-rotating term, not to use the specific complex-exponential notation. Please rephrase to clarify what the 'full cosine form' means and how Eq. (3) relates to it.
  4. [Main text, Fig. 4 caption] The symbol νcorr is introduced in the caption but is not defined in the main text. Please define it explicitly when the beam-splitting drives are first discussed.
  5. [Main text, paragraph after Eq. (11)] The statement 'In [30], only the cavities are displaced, which does not capture the effects of off-resonance qubit driving' is too brief. Provide one or two sentences explaining the qualitative difference between displacing the cavity only and displacing both modes.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity: Heff is derived from the stated Hamiltonian, not fitted; only a minor self-benchmark and one explicit experimental offset appear in the validation.

full rationale

The effective Hamiltonian in Eqs. (9)-(12) is obtained by a concrete unitary displacement transformation of the original Hamiltonian Eq. (1), with the displacement amplitudes chosen to cancel the drive terms (Eqs. (6)-(8) and Supplement Eqs. (S20)-(S29)); the ac Stark shifts δq and δc in Eq. (10) are derived coefficients, not fitted parameters. The validation against ac Stark shifts and two-mode squeezing uses parameters from the authors' prior experimental work [14], but the model is not tuned to those data. The one adjustable quantity, νcorr ≈ −5.02 MHz in the beam-splitting comparison, is explicitly labeled an experimental correction that centers the pattern, and the chevron/Rabi structure is still a prediction; this is a calibration offset rather than a renamed prediction. The Supplement's RWA assumption that drive detunings are small enough to neglect superharmonic processes such as b†3 is an acknowledged approximation with no quantitative error bound and no direct benchmark against the full undisplaced Hamiltonian; that is a correctness or validity risk, not circularity. The derivation is therefore self-contained at the equation level, with only a minor self-benchmark caveat.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central derivation uses standard transmon approximations (dispersive regime, fourth-order cosine expansion) and a domain assumption about the validity of the final RWA. Only one free parameter, the beam-splitting frequency offset nu_corr, is introduced in the validation, not in the Hamiltonian itself. No new physical entities are postulated.

free parameters (1)
  • nu_corr = -5.02 MHz
    Used in the beam-splitting simulation (Fig. 4) as an 'experimental correction' to center the pattern; it is not derived from the effective Hamiltonian and is effectively adjusted to match the experimental data.
assumptions (4)
  • domain assumption The system is in the dispersive regime, |omega_q - omega_c| >> g, allowing use of the normal-mode basis.
    Invoked in the supplement around Eqs. (S2)-(S3) to eliminate the capacitive coupling and arrive at the Hamiltonian in Eq. (2).
  • standard math The Josephson cosine potential is truncated at fourth order, neglecting higher-order terms.
    Used in Eq. (S3) to obtain the quartic Hamiltonian (2); this is the standard weakly-anharmonic transmon approximation.
  • domain assumption Drive detunings are small enough to avoid superharmonic processes such as b-dagger^3 + h.c., so the final RWA can discard them.
    Stated in the main text before Eq. (9) and in the supplement: 'To apply the RWA, we assume drive detunings are small enough to neglect superharmonic processes...' This defines the regime of validity but is not quantitatively bounded.
  • standard math The displacement transformation with decay rates kappa_i leaves the Lindblad dissipators invariant.
    Stated in the main text after Eq. (8) and in the supplement Eq. (S17); this is a standard consistency condition for displaced Lindblad equations.

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Cite this review

Pith. "Pith review of Effective Hamiltonian for an off-resonantly driven qubit-cavity system." pith.science (2026). https://pith.science/paper/EITP37HP

@misc{pith2026250903375,
  author       = {Pith},
  title        = {Pith review of: Effective Hamiltonian for an off-resonantly driven qubit-cavity system},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EITP37HP}},
  note         = {Machine review of arXiv:2509.03375}
}
read the original abstract

Accurate modeling of driven light-matter interactions is essential for quantum technologies, where natural and synthetic atoms are used to store and process quantum information, mediate interactions between bosonic modes, and enable nonlinear operations. In systems subject to multi-tone drives, however, the theoretical description becomes challenging and existing models cannot quantitatively reproduce the experimental data. Here, we derive an effective Hamiltonian that retains slowly rotating terms, providing a general framework for accurately describing driven dynamics across platforms. As a concrete application, we validate the theory in circuit QED, where it quantitatively reproduces experimentally measured ac Stark shifts and captures key interactions such as two-mode squeezing and beam-splitting. Our results establish a broadly applicable tool to engineer driven interactions in quantum information processing platforms.

Figures

Figures reproduced from arXiv: 2509.03375 by the authors.

Figure 1
Figure 1. Ac Stark shift of the qubit frequency under (a) a [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Qubit population under the two-mode squeezing [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Qubit population under the beam splitting tran [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

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