Selective Interactions in the Quantum Rabi Model

E. Solano; I. Arrazola; J. Casanova; L. Cong; L. Lamata; S. Felicetti

REVIEW 2 major objections 5 minor 55 references

Selective Interactions in the Quantum Rabi Model

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Adding a Stark term to the quantum Rabi model makes odd-order k-photon transitions selective, with resonance frequencies controlled by the boson number; the paper derives this from perturbation theory and shows a trapped ion can simulate…

desk verdict The selective k-photon mechanism is convincingly demonstrated for k=1 and k=3 with an honest, well-validated trapped-ion proposal; the general odd-k formula is an unproven ansatz and should be treated as such. read the letter →

arxiv 1908.07358 v2 pith:IUG5ILNO submitted 2019-08-20 quant-ph

L. Cong , S. Felicetti , J. Casanova , L. Lamata , E. Solano , I. Arrazola This is my paper · ORCID

classification quant-ph

keywords quantumRabimodelRabi–Starkselectiveinteractionsmulti-photontransitionsultrastrongcouplingtime-dependentperturbationtheorytrapped-ionsimulationFockstates

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 quantum Rabi model with a Stark coupling term—the Rabi–Stark model—produces selective $k$-photon interactions in the strong and ultrastrong coupling regimes. The central claim is that the rotating and counter-rotating parts of the linear Rabi coupling jointly generate odd-order multi-photon transitions, and that the Stark term makes each transition resonant for a chosen boson Fock state while neighboring Fock states stay out of resonance. The authors derive this from time-dependent perturbation theory, obtaining an effective Hamiltonian whose $k$-photon strength scales as $(g/\omega)^k$ and whose resonance frequency depends on the Fock index through the Stark shift. They check the prediction numerically for three- and five-photon transitions and propose a single trapped ion as a simulator of the full model. If correct, this turns the Rabi–Stark model into a way to address one Fock state at a time, with direct use for bosonic state preparation and reconstruction.

What carries the argument

The central machinery is the interaction-picture Hamiltonian $H_I(t)=\sum_n\Omega_n(\sigma_+ e^{i\delta^+_n t}+\sigma_- e^{i\delta^-_n t})|n+1\rangle\langle n|+\mathrm{H.c.}$, with $\Omega_n=g\sqrt{n+1}$ and Stark-shifted detunings $\delta^\pm_n=\omega\pm[\omega_0+\gamma(2n+1)]$. The nonzero $\gamma$ makes these detunings Fock-state-dependent, which is what converts a degenerate multiphoton ladder into a selective one. A Dyson-series expansion of this Hamiltonian produces, at odd order $k$, the effective $k$-photon Hamiltonian of Eq. (5) with amplitudes $\Omega^{(k)}_{n\pm}\propto (g/\omega)^k$ and resonance frequencies $\delta^{(k)}_{n\pm}=(k-1)\omega+\delta^{\pm}_{n+(k-1)/2}$; even-order terms are either diagonal (inducing a small Stark-like shift of the resonance) or average away under the rotating-wave approximation by parity symmetry. This effective-Hamiltonian construction is what carries the argument, because it converts the dynamical phenomenon of selective $k$-photon exchange into a concrete resonance condition and a calculable interaction strength.

What would settle it

Scan $\omega_0$ in a numerical simulation of the full Rabi–Stark Hamiltonian at $g/\omega=0.1$, $\gamma/\omega=0.9$, with initial state $|g,7\rangle$ and evolution time $t=\pi/(2\Omega^{(5)}_{2-})$. If the population of $|e,2\rangle$ does not peak near $\omega_0/\omega=-3.227$ while $|g,6\rangle$ and $|e,1\rangle$ stay out of resonance, the effective five-photon Hamiltonian is not capturing the dynamics. The paper itself reports that at $g/\omega\approx 0.3$ the simple Jaynes–Cummings-type transfer degrades and population leaks to $|g,N_0\pm1\rangle$, so observing that leakage at modestly larger coupling would mark the regime where the perturbative selective-interaction claim breaks down.

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Extended reading notes

Core claim

Adding a diagonal Stark term $\gamma a^\dagger a\sigma_z$ to the quantum Rabi model changes the physical content of the model: for odd $k$, the combined rotating and counter-rotating terms generate effective $k$-photon Jaynes–Cummings/anti-Jaynes–Cummings interactions of the form $H_I^{(k)}(t)=\sum_n(\Omega^{(k)}_{n+}e^{i\delta^{(k)}_{n+}t}\sigma_+ + \Omega^{(k)}_{n-}e^{i\delta^{(k)}_{n-}t}\sigma_-)|n+k\rangle\langle n|+\mathrm{H.c.}$, with $\Omega^{(k)}_{n\pm}\propto (g/\omega)^k$ and with resonance condition $\delta^{(k)}_{n\pm}=(k-1)\omega+\delta^{\pm}_{n+(k-1)/2}=0$, where the index-dependent piece contains $\omega_0+\gamma(2(n+(k-1)/2)+1)$. Consequently a specified initial Fock state $|g,N_0\rangle$ or $|e,N_0\rangle$ can be brought into resonant $k$-photon exchange with $|e,N_0+k\rangle$ or $|g,N_0+k\rangle$ while other Fock states remain far from resonance. Even-$k$ transitions are absent by parity symmetry, and second-order terms act as a Stark-like energy shift that moves the resonance slightly away from its naive value. Numerical evolution of the full Hamiltonian confirms selective three-photon and five-photon population transfer in the strong and ultrastrong regimes, and the trapped-ion derivation shows how this model can be realized with one ion and three laser drivings.

Load-bearing premise

The load-bearing premise is that the expansion in successive interactions, keeping only low orders and discarding fast-oscillating terms, stays accurate at the ultrastrong couplings studied here, so the analytic resonance condition locates the true multiphoton peak.

Editorial extensions

If this is right

For each odd $k$, the Rabi–Stark model has selective $k$-photon resonances whose frequencies are shifted by the Stark term, so a chosen Fock state can be addressed without disturbing its neighbors.
The one-photon case inherits the same selectivity: setting $\omega-\omega_0=\gamma(2N_0+1)$ makes only the doublet $\{|e,N_0\rangle,|g,N_0+1\rangle\}$ resonant, recovering earlier selective Jaynes–Cummings physics.
The strength of $k$-photon processes decreases as $(g/\omega)^k$, so higher-order transitions require longer times; the numerical results show three-photon exchange at $g/\omega=0.1$ and five-photon exchange with partial transfer at the same coupling.
Even-$k$ multiphoton transitions are suppressed by parity, so the observable multiphoton spectrum of the Rabi–Stark model is organized by odd orders.
A single trapped ion driven by two sideband fields plus a carrier field reproduces the Rabi–Stark Hamiltonian with independently tunable $\omega_0$, $g$, and $\gamma$, allowing the selective interactions to be observed in the strong and ultrastrong coupling regimes.

Reading between the lines

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

If the same selectivity survives with a nonperturbatively corrected resonance condition, the largest practical payoff would be Fock-state-dependent operations: one could prepare, rotate, or read out a specific boson number state by choosing $\gamma$ and $\omega_0$, something the paper motivates through state preparation but does not develop into gates.
The reported five-photon peaks deviate from the second-order analytic values by roughly $0.1$–$0.2\,\omega$, suggesting the perturbative formulas are a starting point; a resummed or exact-spectrum version of the resonance condition would be needed to predict peak positions as $g/\omega$ grows past $0.1$.
A clean experimental test suggested by the parity argument is to scan for even-$k$ resonances: observing a two- or four-photon transition at finite $g$ would contradict the parity-based cancellation, while observing only odd orders would support the effective-Hamiltonian picture.
Since the trapped-ion implementation can choose $\gamma$ of either sign and can reach $g/\omega$ beyond $0.1$, it could also probe where the selective Jaynes–Cummings-type description breaks down, for instance by checking the paper's own expectation that at $g/\omega\approx 0.3$ population leaks to neighboring Fock states.

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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

Quick summary: the central phenomenon—Stark-induced selective multi-photon transitions in the Rabi-Stark model, with Fock-state-dependent resonance frequencies—is convincingly demonstrated for the one- and three-photon cases, and the trapped-ion implementation is carefully validated. I would send this to referees. But the general odd-k formula is asserted, not derived, and that distinction matters here.

The strengths first. The k=3 effective Hamiltonian is genuinely derived in Appendix A via Dyson series, and the agreement with exact numerics in Fig. 2 is real. Note that the agreement only works after including the second-order Stark shifts, by moving to an interaction picture with respect to Eq. (3); that is an honest, checkable piece of perturbation theory. The one-photon selectivity in Fig. 1 cleanly shows the Stark mechanism. The trapped-ion proposal is a cut above the usual 'in principle' simulator: Appendix C includes second-order corrections from the carrier interaction, and the numerics compare Eq. (10) against the full ion Hamiltonian with realistic parameters. Dissipative effects are addressed in Appendix B, including the honest statement that a five-photon process at g/omega = 0.1 needs kappa/omega < 1e-5. The citations to prior multiphoton-resonance work and to earlier selective interactions are appropriate; Stark-induced selectivity of multi-photon transitions is genuinely new, even if the perturbative machinery is standard.

The soft spots, in proportion. First, Eqs. (5)-(7) for general odd k are asserted 'following the same procedure' with no derivation. This is not a formality: at k=3 the naive resonance condition delta(3)=0 does not match the numerics, and the paper has to add a second-order level-shift correction. The general formula contains no such correction. Second, the five-photon validation uses the uncorrected resonance condition, and the observed peaks are systematically ~4% off, the deviation growing with N0. The paper calls them 'close,' which is fair, but it does not test whether the k=3-style correction closes the gap. Until that is done, the (g/omega)^k scaling and the resonance condition for arbitrary odd k are plausible extrapolations, not established results. The paper itself concedes that analytic resonance frequencies for higher orders 'rapidly become challenging,' which reads as an honest admission that Eqs. (5)-(7) are conjectural.

Who this is for: readers working on Rabi-Stark models, USC dynamics, and trapped-ion simulation will get real value from the k=3 mechanics and the ion implementation. The general-k claim should not be cited as proven. Recommendation: send to peer review, and have the referee press on the status of Eqs. (5)-(7). With either a derivation or an explicit narrowing of the claim, this becomes a solid PRA-type paper.

Referee Report

2 major / 5 minor

Summary. The manuscript studies the quantum Rabi model with a Stark coupling term (the Rabi-Stark model), H = (ω0/2)σz + ω a†a + γ a†a σz + g(σ+ + σ−)(a + a†), and claims that the interplay of rotating and counter-rotating terms produces selective k-photon interactions whose resonance frequencies depend on the bosonic Fock state through the Stark shift. The authors derive an interaction-picture Hamiltonian, obtain second- and third-order effective Hamiltonians via time-dependent perturbation theory, and assert a general odd-k formula (Eqs. (5)–(7)). They validate the three-photon dynamics against numerical integration of the full Hamiltonian after including a second-order Stark-shift correction, and report numerical five-photon resonances. The second half proposes a trapped-ion implementation using three laser drivings, derives an effective Rabi-Stark Hamiltonian that includes higher-order carrier corrections, and compares the full ion Hamiltonian with the effective model for one- and three-photon processes.

Significance. The selective multiphoton interactions and the trapped-ion simulation, if fully established, would be a useful contribution to quantum control and quantum simulation in the strong and ultrastrong coupling regimes. The paper has clear strengths: the Dyson-series derivations for the second- and third-order Hamiltonians are explicit and detailed; the numerical validations are performed on the full Hamiltonian (1) with no fitted parameters; the trapped-ion mapping in Appendix C keeps important second-order corrections; and dissipative effects are analyzed in Appendix B. The main weakness is that the general odd-k analytical claim, which is the central result, is asserted rather than derived and is incomplete without the second-order resonance shifts that are essential already at k=3; the five-photon numerical check shows a systematic resonance shift that Eqs. (5)–(7) do not predict.

major comments (2)

[Section II.B, Eqs. (5)–(7) and Fig. 3] The general odd-k formulas are not supported by the evidence presented. For k=3, the raw resonance condition δ^(3)_5+ = 0 does not locate the resonance in Fig. 2(a); agreement requires shifting to δ̃^(3)_5+ = δ^(3)_5+ + Δ^e_8 − Δ^g_5 from the second-order Hamiltonian (3). Yet Eqs. (5)–(7) contain no analogous shift, and no derivation of the corrected condition for general k is given. For k=5, the predicted qubit frequencies ω0^c/ω = 5 − γ(2N0 + 5) give −3.1, −4.9, and −6.7, while the numerical peaks in Fig. 3 are −3.227, −5.072, and −6.918, a systematic 3–4% shift that grows with N0. This shows that either Eq. (5) is not the complete effective description, or higher-order corrections are required. The claim that Eqs. (5)–(7) describe selective k-photon interactions for arbitrary odd k is therefore not established. The authors should derive the general corrected resonance condition, or explicitly restrict the analytical claim to k=3 and present Fig. 3 as numerical evidence only.
[Section II.B, Eqs. (5)–(7) and Fig. 3] The paper does not quantify the accuracy or selectivity of the effective Hamiltonian at k=5. In Fig. 3 the population transfer is partial, and the authors themselves note that the remaining population goes to states |g,N0+1⟩ and |g,N0−1⟩. No comparison between the full dynamics of Eq. (1) and the effective Hamiltonian (5) at k=5 is provided, and no fidelity or leakage estimate is given. Since the central claim is selectivity, it is necessary to show quantitatively how well Eq. (5) reproduces the full dynamics and over what parameter range, rather than only locating approximate resonance peaks.

minor comments (5)

[Eq. (6)] The displayed summation expression for δ^(k)_n± is malformed: it contains the undefined symbol δ±_k and the equality to (k−1)ω + δ±_{n+(k−1)/2} does not follow from the sum as printed. The simplified version is understandable, but the sum should be rewritten correctly.
[Eq. (7)] The product in Ω^(k)_n± is written as s=1,3,... without an explicit upper limit or step size; it should specify s=1,3,...,k−2, and the notation δ^(s)_n± should be defined.
[Section II.B, even-k statement] The statement that even-k transitions 'will average out as a consequence of the RWA' is misleading: the absence of even-k transitions follows from the exact parity symmetry of Hamiltonian (1), as the authors note later. The parity argument should be stated explicitly.
[Appendix C, Eq. (59)] After the RWA, the term involving a†a should be σx rather than σ+, since Eq. (60) and the basis {|+⟩,|−⟩} require σx. Please check this notation.
[Fig. 2(a) caption] The caption says the dark curve represents 'lower values of log10|δ̃|', but since the log diverges away from zero, the curve should be described as the locus of minima (or approximate zeros) of δ̃.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the effective k-photon Hamiltonians are derived from the model and checked against independent numerical integration of the full Rabi-Stark Hamiltonian.

full rationale

The paper's central derivation is self-contained. The interaction-picture Hamiltonian in Eq. (2) follows exactly from the Rabi-Stark Hamiltonian Eq. (1), and the second-order Hamiltonian Eq. (3), the third-order Hamiltonian Eq. (4), and the trapped-ion mapping Eq. (10) are derived explicitly in Appendices A and C. The three-photon resonance shift is not fitted: it is obtained as a higher-order correction from the same Dyson-series expansion, namely moving to an interaction picture with respect to Eq. (3) and using the derived Lamb-shift-like terms Δe_n and Δg_n. The general odd-k formulas in Eqs. (5)-(7) are presented as an extrapolation 'following the same procedure' rather than as a circular identity, and the paper does not claim that the uncorrected resonance condition δ(k)=0 exactly gives the five-photon peaks; instead, it reports approximate values and explicitly acknowledges that the numerically observed peaks deviate from the approximate analytic values, with no fitted parameter adjusted to force agreement. The five-photon peaks are located by numerical integration of the full Hamiltonian Eq. (1), which is an independent check, not a restatement of the perturbative input. Self-citations in the manuscript (e.g., trapped-ion QRM implementations and deep-strong-coupling dynamics) are used as background or as standard techniques and are not load-bearing for the new derivation. The possible concern that the general odd-k formula is asserted without a full derivation, or that higher-order corrections are omitted, is a correctness or rigor issue, not circularity. Consequently, no step reduces by construction to its own input, and the appropriate circularity score is 0.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new particles or entities. The model parameters (γ, g, ω0, ω) are physical inputs, not fitted. The free parameters listed are only the specific values chosen for numerical demonstrations and do not feed back into the derivation.

free parameters (3)

γ/ω = -0.4, -0.25, 0.9
Stark coupling ratio, chosen for numerical demonstration; not fitted to external data.
g/ω = 0.02, 0.05, 0.1
Rabi coupling ratio, chosen to lie in the strong and ultrastrong coupling regimes; not fitted.
N0 = 2, 3, 4, 5
Selected Fock state for resonance; chosen by hand to demonstrate selectivity.

assumptions (5)

domain assumption Rotating-wave approximation: terms oscillating at δ±_n or 2(ω±γ) average out over the relevant timescales.
Used throughout the perturbative derivation, e.g., in deriving Eq. (3) from Eq. (29) and Eq. (4) from Eqs. (37)-(38).
domain assumption Time-dependent perturbation theory (Dyson series) truncated at finite order k accurately describes the dynamics for g/ω < 1.
Central to the derivation of the effective k-photon Hamiltonian (Eqs. 3-7), especially the claim that higher-order terms generate selective interactions.
standard math Parity conservation in the Rabi-Stark model forbids transitions between |e,n> and |g,n+k> for even k.
Invoked in Section II.B to justify why even-order k-photon terms are averaged out.
domain assumption Lamb-Dicke approximation (η√<n> ≪ 1) and vibrational RWA for the trapped-ion implementation.
Used to derive Eq. (9) from Eq. (8).
domain assumption In the trapped-ion derivation, 1/(ν±δ_j) ≈ 1/ν for the second-order terms.
Used in Appendix C to combine HB(t) terms; assumes the detunings δ_j are much smaller than the trap frequency ν.

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Pith. "Pith review of Selective Interactions in the Quantum Rabi Model." pith.science (2026). https://pith.science/paper/IUG5ILNO

@misc{pith2026190807358,
  author       = {Pith},
  title        = {Pith review of: Selective Interactions in the Quantum Rabi Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IUG5ILNO}},
  note         = {Machine review of arXiv:1908.07358}
}

abstract

We demonstrate the emergence of selective $k$-photon interactions in the strong and ultrastrong coupling regimes of the quantum Rabi model with a Stark coupling term. In particular, we show that the interplay between the rotating and counter-rotating terms produces multi-photon interactions whose resonance frequencies depend, due to the Stark term, on the state of the bosonic mode. We develop an analytical framework to explain these $k$-photon interactions by using time-dependent perturbation theory. Finally, we propose a method to achieve the quantum simulation of the quantum Rabi model with a Stark term by using the internal and vibrational degrees of freedom of a trapped ion, and demonstrate its performance with numerical simulations considering realistic physical parameters.

Figures

Figures reproduced from arXiv: 1908.07358 by the authors.

**Figure 3.** FIG. 3. Five-photon selective interactions of the Rabi-Stark model. [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 4.** FIG. 4. Selective one-photon and three-photon interactions with a [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

Figures from the paper (1 more)

**Figure 5.** Figure 5: FIG. 5. Three-photon interactions with dissipative e [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

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