Violation of Leggett-Garg type inequalities in a driven two level atom interacting with a squeezed thermal reservoir

Arun M. Jayannavar; Javid Naikoo; Subhashish Banerjee

REVIEW 3 major objections 4 minor 42 references

Violation of Leggett-Garg type inequalities in a driven two level atom interacting with a squeezed thermal reservoir

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A driven two-level atom can violate Leggett-Garg type inequalities in the underdamped regime, approaching the quantum bound 3/2 as the spontaneous emission rate vanishes.

desk verdict A useful but sloppy application of Leggett-Garg-type inequalities to a squeezed thermal bath; the qualitative results are plausible, but the central equations contain typos and an unverified expression that need sorting out before the plots can be trusted. read the letter →

arxiv 1908.04054 v1 pith:H4FK4XLZ submitted 2019-08-12 quant-ph

Javid Naikoo , Subhashish Banerjee , Arun M. Jayannavar This is my paper · ORCID

classification quant-ph

keywords Leggett-Gargtypeinequalitiesdriventwo-levelatomsqueezedthermalreservoirunderdampedregimeweakmeasurementsquantumcoherenceMarkovianopensystem

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 establishes a concrete open-system test bed for Leggett-Garg type inequalities (LGtIs): a single two-level atom driven by a resonant field and placed in a squeezed thermal reservoir. The paper shows that the LG parameters K+ and K−, which stay below 1 for classical macrorealistic theories, exceed 1 in the underdamped regime and saturate at the quantum bound 3/2 when the drive dominates spontaneous emission. Temperature and reservoir squeezing both erode the violation, while stronger driving and ideal projective measurements enhance it. These are experimentally tunable predictions, so the result matters as a practical quantum-coherence witness for driven open qubits.

What carries the argument

The machinery is the LGtI correlation function $C(t_i,t_j)$ built from the survival probability of the ground state: for the dichotomic observable $\hat M=|g\rangle\langle g|-|e\rangle\langle e|$, $C(t_0,t)=2p_g(t)-1$, leading to $K_{\pm}=\pm 2F(t)-F(2t)\mp 1$ with $F(t)$ given by a damped-oscillation expression. The behavior is governed by the effective frequency $\mu_s=\sqrt{\Omega^2-(\gamma_s/4)^2}$: real $\mu_s$ (underdamped) yields oscillatory correlations that overshoot the classical bound, imaginary $\mu_s$ (overdamped) yields monotone decay with no violation. In the strong-driving limit the expression reduces to $K_{\pm}\approx\pm 2\cos(\Omega t)-\cos(2\Omega t)$, which is the explicit source of the violation.

What would settle it

Measure $K_+$ and $K_-$ on a strongly driven two-level system in a squeezed thermal bath (e.g., a superconducting qubit or trapped ion). If $K_{\pm}$ never exceeds 1 even as the spontaneous emission rate $\gamma_0\to 0$ while the Rabi drive is kept large, or if violations appear when $\Omega<\gamma_s/4$, the central prediction fails. A direct quantitative check: in the underdamped regime the data should follow $K_{\pm}\approx\pm 2\cos(\Omega t)-\cos(2\Omega t)$, with the quantum bound $3/2$ reached only in the projective-measurement limit $\xi=1$.

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Extended reading notes

Core claim

The paper's central claim is that the LG parameters $K_{\pm}=\pm 2C(t_0,t)-C(t_0,2t)$ exceed the classical upper bound 1 precisely when the driven atom is in the underdamped regime, i.e., when $\Omega>\gamma_s/4$ with $\gamma_s=\gamma+2\gamma_0 M$; in the limit $R=\gamma_0/\Omega\to 0$ they reach the quantum bound $3/2$. The violation is controlled by the single dimensionless ratio $R$ and by the reservoir parameters $\gamma$ and $M$ that enter through the squeezed thermal bath. Temperature and squeezing monotonically reduce the violation and shorten the time interval over which it lasts, while stronger driving favors it. The paper also claims that the violation is maximal for ideal projective measurements and scales as $\xi^2$ for weak measurements.

Load-bearing premise

Everything rests on treating the squeezed thermal reservoir as memoryless (Markovian) and on stationary conditional probabilities; if the reservoir has memory or its statistics vary in time, the tested inequalities no longer apply.

Editorial extensions

If this is right

Crossing the threshold $\Omega=\gamma_s/4$ switches LGtI violation on: above it $K_{\pm}$ exceed 1, below it they do not.
As $R=\gamma_0/\Omega\to 0$ in the strong-driving limit, $K_{\pm}$ reach the quantum bound $3/2$; finite $R$ degrades the violation.
Raising temperature (lowering $\beta$) shortens the time window over which $K_{\pm}>1$ and reduces its peak value.
Increasing reservoir squeezing $s$ suppresses the violation and can push the system into the overdamped regime.
Weak measurements reduce the LG parameters by the factor $\xi^2$, so ideal projective measurements give the largest violation.

Reading between the lines

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

The $\xi^2$ scaling suggests a metrological use: sweeping measurement strength $\xi$ could map the decoherence rate, since the measured $K_{\pm}$ versus $\xi$ curve carries the same information as the correlation decay.
The condition $\Omega=\gamma_s/4$ defines a tunable quantum-classical crossover whose location depends on temperature and squeezing; one could use the same device to study how reservoir engineering shifts the boundary.
Because $K_+$ and $K_-$ are complementary in the strong-driving limit, a combined witness (e.g., the maximum of the two) would detect violation over a wider time window than either alone; the paper plots them separately.

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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

The genuinely new thing here is the combination: LGtIs (the stationarity-based variant of Leggett-Garg) applied to a driven two-level atom in a squeezed thermal reservoir, including weak-measurement effects. That specific map—where violations survive in the underdamped regime, how temperature and squeezing suppress them, and how projective measurements maximize them—is not in the cited literature. The paper does this with a standard Bloch-vector master-equation treatment, and the qualitative physics is exactly what you'd expect: coherence oscillations in the underdamped regime produce violations, noise and squeezing degrade them. The strong-driving limit correctly reduces to the known unitary Rabi result. Credit where it's due: the model setup is clear, the figures tell a coherent story, and the weak-measurement extension (K± → ξ²K±) is a nice, simple addition.

The soft spots are real. The stress-test note is right: Eq. (24) has a spurious ∓1. From Eq. (23), C(t0,t) = 2p_g(t) − 1, so K± = ±2C(t) − C(2t) = ±2F(t) − F(2t) if you define F = C. The extra ∓1 makes the printed formula inconsistent with Eq. (27), which is exact in the strong-driving limit. This is not a purely cosmetic typo: Eq. (25), the F(t) expression that generates all the violation plots, is quoted without derivation, and with the inconsistency you cannot verify that the figures follow from the equations as printed. Eq. (20) also has a sign error—it gives p_g(0) = 1.5 for the declared initial ground state. These are exactly the kind of errors that make a referee nervous, because they undermine confidence in the central computation even if the underlying physics is sound.

That said, I want to be fair: the errors look fixable, and nothing here suggests the qualitative conclusions are wrong. The underdamped condition Ω > γ_s/4 is derived from the eigenvalues, and the strong-driving limit reduces to a known result, which is a good sanity check. But the missing derivation of Eq. (25) is a genuine gap, not a minor omission; a referee should ask for it in full. The Markovian squeezed-reservoir assumption is a limitation worth stating, but it's a standard and reasonable starting point.

Who is this for? People designing macrorealism or temporal-correlation experiments in dissipative driven systems, and open-systems folks who want a worked example of LGtIs in a non-standard bath. It's not groundbreaking, but it's a useful contribution once the algebra is cleaned up. I'd send it to peer review, but with a request for the derivation of F(t), corrected Eq. (24) and Eq. (20), and ideally a short calculation or notebook showing the figures. For my own work, I wouldn't cite it in its current form—not because the idea is bad, but because I'd want to verify the equations first.

Referee Report

3 major / 4 minor

Summary. This manuscript studies Leggett-Garg-type inequalities (LGtIs) for a resonantly driven two-level atom coupled to a squeezed thermal reservoir. The authors write a Markovian Lindblad master equation, solve the resulting Bloch equations analytically, and construct the two-time correlation C(t0,t) from the ground-state survival probability. Their central claim, stated in Sec. IV, is that the LG parameters K± defined in Eq. (3) exceed the classical bound 1 in the underdamped regime Ω > γs/4 and approach the quantum bound 3/2 as R = γ0/Ω → 0. They further report that thermal effects and reservoir squeezing reduce the violation, that stronger driving favors violation, and that ideal projective measurements give the maximum violation. All numerical results in Figures 3–7 are based on the closed form F(t) in Eq. (25) and the strong-driving reduction in Eq. (27).

Significance. If the central derivation were correct, the paper would provide a simple, explicitly solvable open-quantum-system model in which LGtI violations appear in experimentally relevant parameters, with transparent dependence on temperature, squeezing, driving strength, and measurement strength. The calculation is self-contained: no free parameters are fitted, the master equation is standard, and the underdamped/overdamped boundary is given in closed form. The qualitative statement that temperature and squeezing suppress violations is physically plausible. However, the manuscript currently contains an algebraic inconsistency in the formula for K±, and the closed-form expression on which all figures rest is quoted without derivation. These issues affect every reported violation, so the central claim is not yet established.

major comments (3)

[Sec. III, Eq. (24)] The formula K± = ±2F(t) − F(2t) ∓ 1 is inconsistent with the definition of K± in Eq. (3) and the expression for C(t0,t) in Eq. (23). Since Eq. (23) gives C(t0,t) = 2p_g(t) − 1, substitution into Eq. (3) yields K± = ±2C(t0,t) − C(t0,2t), i.e., K± = ±2F(t) − F(2t) when F(t) is identified with C(t0,t). The extra '∓1' term is absent. This is not a minor typo: in the strong-driving limit it changes the maximum of K+ from 3/2, as claimed in Eq. (27), to 1/2, so the reported violation and the approach to the quantum bound are not supported. The authors should correct Eq. (24) and recompute all results, or give an explicit reconciliation of Eq. (24) with Eq. (27).
[Sec. II, Eq. (20)] The survival probability is written as p_g(t) = 1 − (⟨Σ3(t)⟩ + ⟨σ3⟩s)/2. With the initial condition in Eq. (17), ⟨Σ3(0)⟩ = −1 − ⟨σ3⟩s, which gives p_g(0) = 3/2, an unphysical result. The correct expression is p_g(t) = (1 − (⟨Σ3(t)⟩ + ⟨σ3⟩s))/2. Since Eq. (23) defines C(t0,t) = 2p_g(t) − 1, this error propagates into F(t) and all of the K± plots. The authors must re-derive the correlation function and the subsequent formulas with the corrected p_g.
[Sec. III, Eqs. (25)–(27)] The closed form F(t) in Eq. (25) is quoted without derivation, and the strong-driving reduction to F(t) ∝ cos(Ωt) and hence Eq. (27) is asserted without showing the limit or the algebra. Because Eq. (25) is the sole basis for Figures 3–7, the paper should provide the explicit derivation starting from Eq. (19) and the corrected Eq. (20), or a machine-checkable supplement, confirming the coefficient formulas in Eq. (26). In particular, the stated approximations A ≈ Ω^{-3}, B ≈ C ≈ Ω^3, and D ≈ Ω^2 must be verified against the exact coefficients, since the claimed approach to the quantum bound 3/2 depends on them.

minor comments (4)

[Throughout] There are several typographical errors, e.g., 'inequlalities' in Sec. I, 'paramter' and 'coherence paramter' in Sec. IV, and 'the two time correlation function becomes' in the weak-measurement paragraph. A careful proofread is recommended.
[Sec. II, Eq. (21)] The expression for the off-diagonal element appears to double-count the stationary contribution: as written, ⟨σ+(t)⟩ = (⟨σ1(t)⟩ + i⟨σ2(t)⟩)/2 + ⟨σ+⟩s includes ⟨σ+⟩s twice if ⟨σ1⟩ and ⟨σ2⟩ are the full Bloch components. This equation is not used in the final results, but it should be corrected to avoid confusion, e.g., by writing it in terms of the deviation ⟨Σ⟩.
[Sec. II, Eq. (8)] The paper would benefit from a brief statement of the validity regime of the Markovian Lindblad master equation for a squeezed reservoir with finite bandwidth. The stationarity assumption underlying the LGtI form is tied to Markovian dynamics, so a comment on this limitation would strengthen the experimental discussion.
[Sec. III, Eq. (29)] The weak-measurement result K±|weak = ξ² K± is stated without showing the intermediate algebra. The derivation is short and should be included, since the section is otherwise self-contained.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the LGtI calculation is a self-contained application of a standard Lindblad master equation with free physical parameters, and the self-citations are not load-bearing.

full rationale

The paper's derivation chain is not circular. The input parameters (Ω, γ0, nth, s, θ) are physical reservoir and driving parameters; none of them are fitted to the LG parameter K± or to the violation plots. The master equation (7) is the standard Markovian Lindblad form for a driven two-level atom in a squeezed thermal reservoir, and the solution (19) and survival probability (20) are obtained by direct solution of the Bloch equations. The LGtI criterion (3) is an independent inequality based on stationarity, and the correlation function is computed from the dynamical map in Eq. (23). The subsequent evaluation of K± in Eq. (24) and the strong-driving reduction (27), while possibly underived or algebraically questionable, is not an instance of a fitted input being renamed a prediction, nor does it make the target result an input by construction. The self-citations [16–19] are cited as earlier studies of LGIs in other contexts; they are not used to justify the central formulas, to supply a uniqueness theorem, or to smuggle in an ansatz. Likewise, the weak-measurement scaling C_weak = ξ²C follows from the weak-projector definition in Eq. (28); even if one doubts its correctness, it is a stated consequence of the model rather than a circular equivalence. The possible inconsistency between Eq. (24) and Eq. (27) flagged in the reader's take is a correctness or verification issue, not a circularity issue: it does not reduce the prediction to its inputs. Therefore no circular step is exhibited, and the honest finding is a circularity score of 0.

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

The central claim depends only on physical inputs from the model (Ω, γ0, n_th, s) and the standard formalism of open quantum systems and LGtI. No fitted parameters or invented entities are introduced. The main extra assumptions are the Markovian master equation and the stationarity condition needed for the LGtI form.

assumptions (4)

domain assumption The reduced dynamics of the driven atom is Markovian and described by the Lindblad master equation (7).
This master equation, with the squeezed-reservoir parameters n and M from Eq. (8), is the starting point for all results. It assumes a broadband squeezed thermal reservoir, dipole and rotating-wave approximations, and θ=0.
domain assumption The two-time correlation function is evaluated with ideal projective measurements and the map E_{tj←ti}.
Equation (2) and the projection formula (23) define the correlation function; the subsequent weak-measurement generalization (28)-(29) is an extension.
domain assumption Stationarity of the conditional probabilities, required for the LGtI form K± ≤ 1, holds for this Markovian system prepared in a definite state.
The paper states this condition is satisfied in the model considered, in the paragraph after Eq. (3).
standard math The standard quantum postulates (Born rule, collapse) underpin the measurement statistics.
Used implicitly throughout the derivation of correlation functions.

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Cite this review

Pith. "Pith review of Violation of Leggett-Garg type inequalities in a driven two level atom interacting with a squeezed thermal reservoir." pith.science (2026). https://pith.science/paper/H4FK4XLZ

@misc{pith2026190804054,
  author       = {Pith},
  title        = {Pith review of: Violation of Leggett-Garg type inequalities in a driven two level atom interacting with a squeezed thermal reservoir},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H4FK4XLZ}},
  note         = {Machine review of arXiv:1908.04054}
}

read the original abstract

The violation of Leggett-Garg type inequalities (LGtIs) is studied on a two level atom, driven by an external field in the presence of a squeezed thermal reservoir. The violations are observed in the underdamped regime where the spontaneous transition rate is much smaller compared to the Rabi frequency. Increase in thermal effects is found to decrease the extent of violation as well as the time over which the violation lasts. With increase in the value squeezing parameter the extent of violation of LGtIs is seen to reduce. The violation of LGtIs is favored by increase in the driving frequency. Further, the interplay of the degree of violation and strength of the measurements is studied. It is found that the maximum violation occurs for ideal projective measurements.

Figures

Figures reproduced from arXiv: 1908.04054 by the authors.

**Figure 2.** FIG. 2: (color online). Probability of finding the atom in [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3: (color online). Evolution of the LG parameters [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: FIG. 4: (color online). Complementary behavior of LG pa [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

**Figure 5.** Figure 5: FIG. 5: (color online). Temperature dependence of LG pa [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

**Figure 7.** Figure 7: FIG. 7: (color online). Variation LG parameter [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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Schr¨ odinger, Naturwissenschaften23, 823 (1935)

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