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Macroscopic Schr\"{o}dinger-cat states of nonequilibrium electrons induced by cat-state optical driving and projective measurements on the light field

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Pith's one-line read The paper shows that measuring the light field, not just irradiating matter with it, can create macroscopic Schrödinger-cat states in many-electron systems, even in the thermodynamic limit.

desk verdict Measurement postselection really does convert bright cat-state light into macroscopic electronic cats—the exact Tavis-Cummings numerics say so—but the thermodynamic-limit claim outruns the proof and leans on an unproven XFA uniqueness. read the letter →

arxiv 2508.11769 v2 pith:FIO5YBU4 submitted 2025-08-15 quant-ph cond-mat.mes-hallphysics.optics

Shohei Imai This is my paper · ORCID
classification quant-phcond-mat.mes-hallphysics.optics
keywords macroscopicSchrödinger-catstatequantumlightprojectivemeasurementphoton-numberparityquadraturehomodynedetectionFisherinformationTavis–Cummingsmodelexternal-fieldapproximation
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 projective measurements on the light field, not just irradiation, are what create macroscopic quantum superpositions in many-electron systems driven by quantum light. For $N$ independent two-level electrons driven by a large-amplitude even Schrödinger-cat light state, tracing out the light leaves a classical mixture: the coherence terms shrink as $\langle -\alpha_0|\alpha_0\rangle = e^{-2|\alpha_0|^2}$. The paper shows that projecting the light onto photon-number parity or onto a selected quadrature converts that mixture into a genuine superposition of the two Rabi-oscillating electron states, with quantum Fisher information $F_Q=O(N^2)$. If true, this provides a practical route to matter cat states using standard light-matter coupling plus projective light measurement.

What carries the argument

The external-field approximation (XFA): the initial photonic state $|\chi(0)\rangle$ is written as $\int d^2\alpha\, f(\alpha)|\alpha\rangle$ and, neglecting electronic backaction, the bosonic operator $\hat a$ is replaced by the c-number $\alpha(t)=\alpha e^{-i\omega t}$. This yields the approximate total wave function $|\Psi_{\mathrm{XFA}}(t)\rangle = \int d^2\alpha\, f(\alpha)\,|\psi_\alpha(t)\rangle_e |\alpha(t)\rangle_p$, so each coherent component of the light carries its own classically driven electron state. For an even cat state, this reduces to $|\psi_{+\alpha_0}\rangle|\alpha_0\rangle + |\psi_{-\alpha_0}\rangle|-\alpha_0\rangle$, and photon-number-parity or quadrature projectors t

What would settle it

Take the even cat state with $N=32$, $\alpha_0=30$, and $\gamma\alpha_0=0.1$. Compute the parity-postselected QFI density exactly from the Tavis–Cummings model and also from the two XFA expansions (18) and (53) as $\gamma$ decreases to 0.01 and below. If the two XFA results do not converge to the same curve (and to the exact numerics) as $\gamma\to0$, the thermodynamic-limit prediction is representation-dependent and the central claim is not unique; alternatively, if an experiment with photon-number-resolving detection fails to see $F_Q/N$ approach $N$ during the first Rabi cycle, the ROC-stat

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Extended reading notes

Core claim

The central claim is that photon-number-parity or quadrature projective measurements can restore a macroscopic cat state in nonequilibrium electrons driven by cat-state light, even when the light amplitude is large enough that the unmeasured state would be a classical mixture. The mechanism is captured by an external-field approximation (XFA) in which the initial light state is expanded in coherent states and each coherent component $\alpha$ drives the electrons classically, giving the approximate total state $|\Psi\rangle_{\mathrm{ep}} \sim |\psi_+\rangle|\alpha_0\rangle + |\psi_-\rangle|-\alpha_0\rangle$. Applying the even-parity projector $P_+$ yields the postselected electronic state $|\

Load-bearing premise

The argument rests on the external-field approximation being unique in the thermodynamic limit: however you write the initial light state as a sum of coherent states, the same effective electron state must emerge as $\gamma\to0$ with $\gamma\sqrt{\langle \hat{n}\rangle}$ fixed; the paper verifies this for two different expansions numerically but leaves a full proof for future work.

Editorial extensions

If this is right

  • Macroscopic electronic cat states (GHZ-type superpositions such as $|\leftarrow\cdots\leftarrow\rangle + |\rightarrow\cdots\rightarrow\rangle$) become preparable in ensembles of independent two-level emitters without direct inter-particle interactions, using only light-matter coupling and a projective light measurement.
  • The same protocol works with photon-number parity or quadrature detection, and with both cat and kitten states of light, giving two experimentally distinct routes to the same kind of matter cat state.
  • The postselected QFI density reaches its $N$-qubit maximum $F_Q/N\approx N$, so the produced state is certified as genuinely multipartite entangled and as a macroscopic quantum superposition by the $F_Q = O(N^2)$ criterion.
  • In the few-photon regime, parity-conditional dynamics remove the sudden birth-and-death kinks seen without postselection, letting the QFI rise quickly — an advantage for systems with finite decoherence times.
  • The validity of the XFA picture extends beyond the rotating-wave approximation: including counter-rotating terms only adds $2\omega$ modulations to the QFI and does not change the ROC-state description.

Reading between the lines

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

  • If the representation-independence question is settled, the same XFA-plus-measurement recipe should work for arbitrary superpositions of coherent states, not only symmetric cat or kitten states; any two distinguishable coherent branches with a projector satisfying the equal-overlap condition would yield a conditional matter superposition.
  • The $\Delta x \cdot |\alpha_0|$ scaling suggests a practical resource trade-off: a given entanglement depth demands quadrature resolution that improves linearly with the light amplitude, which is exactly the same precision regime needed to certify the nonclassicality of the incident light itself.
  • Because the XFA total state remains an entangled superposition at all times, the parity measurement need not be performed early; delaying the projective readout should still herald the ROC state, so an experiment could store the light-matter entanglement until the detector is ready.
  • The paper's ROC/ROK states are expressed in terms of the analytical Tavis–Cummings solution, so the same formulas give quantitative predictions for the time-resolved QFI, making the transient measurement-induced cat state testable in current attosecond and high-harmonic setups.
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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 / 5 minor

Summary. The paper studies N independent two-level electrons coupled to a single cavity mode initially prepared in Schrödinger-cat or -kitten states of light. It develops an external-field approximation (XFA) in which the light is represented by a coherent-state expansion and the electronic system evolves under c-number fields, yielding an approximate total wave function (Eq. 14). For even cat-state light, the XFA gives a superposition of two classically driven electronic states correlated with the two coherent components (Eq. 19). Without postselection, the reduced electronic state is a classical mixture in the large-amplitude regime; with photon-number-parity or quadrature projective measurements, the postselected electronic state becomes a Rabi-oscillation cat (ROC) or kitten (ROK) state, respectively, with QFI density reaching N, i.e., F_Q = O(N^2). These predictions are benchmarked against exact Tavis–Cummings and Dicke-model simulations for N=8 and N=32, with good agreement in the first Rabi cycles and improved agreement as γ decreases. The paper also analyzes the effect of finite measurement resolution and checks the rotating-wave approximation.

Significance. If the central claim holds, the paper establishes a conceptually new route to preparing macroscopic superposition states of matter: standard light–matter interaction followed by projective measurement of the light, without engineered particle–particle interactions. The main result—parity or quadrature postselection converts the otherwise classical mixture produced by large-amplitude cat light into a genuinely multipartite entangled electronic state—is physically interesting and supported by independent exact numerics that do not rely on the XFA. The paper is also strong in explicit diagnostics: it uses QFI with entanglement-depth bounds, spin Wigner functions, and analytic ROC/ROK states, and it quantifies the resolution requirement for homodyne detection. The XFA is transparently presented and the authors clearly identify its main limitation, the representation dependence of the coherent-state expansion, with numerical evidence for convergence. The remaining issue is that the abstract's 'even in the thermodynamic limit' statement depends on an unproven uniqueness property of the XFA, which the text explicitly defers to future work.

major comments (2)
  1. [Sec. V C, Eqs. (16), (52)–(54)] The central claim 'even in the thermodynamic limit' rests on the XFA state (14) being well defined in the limit (16), but the coherent-state expansion (6) is overcomplete. The ROC state (52) is obtained from the delta-function representation (18); the identity-inserted representation (53) gives a different XFA state (54) at finite γ. The text states that a full proof that any coherent-state expansion yields the same XFA state in the limit (16) is left for future work. The numerics in Fig. 8 support convergence for N=8 at γ|α0|=0.1, but they do not establish uniqueness. Without such a proof, the thermodynamic-limit prediction is not unique, and the abstract's strongest statement is not demonstrated. Please either prove convergence for a specified class of admissible f(α) or reformulate the claim as a prediction of the delta-function representation supported by numerical convergence.
  2. [Sec. V A, Figs. 4–6] The claimed Heisenberg scaling F_Q = O(N^2) is shown numerically only for N=32 at finite γ=0.01 and α0=30, i.e., γ√<n>=0.3. This is the largest N reported, and no finite-size scaling of the postselected QFI is given. Since the ROC state (52) is an analytic function of N through Eq. (48), an explicit evaluation of F_Q(N)/N for larger N (or a plot for N up to, say, 10^3) would directly substantiate the O(N^2) claim and would also make the γ→0 extrapolation more transparent. As it stands, 'macroscopic' is evidenced by a single system size and a single finite value of γ.
minor comments (5)
  1. [Sec. IV B] Typo: 'developes' should be 'develops'.
  2. [Eqs. (17) and (19)] The symbol N is used both for the number of electrons and for the normalization constant of the cat state. Please use a distinct script (e.g., N_ψ or 𝒩) to avoid confusion.
  3. [Sec. V C, Eq. (54)] In the text, the state |˜Ψ_XFA(t)⟩ is sometimes written with subscript 'e', but it is a total (electron+photon) state. The subscript should be 'ep' for consistency with Eq. (14).
  4. [Sec. II, Eq. (11)] The XFA derivation truncates photonic fluctuations at zeroth order, α_q=0, with no explicit error bound. The exact numerical benchmarks mitigate this concern, but the formal status of the approximation would benefit from a short statement of the conditions under which the Born approximation is controlled.
  5. [Acknowledgments] Typo: 'The author thank' should be 'The author thanks'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central ROC/ROK predictions are independently checked against exact Tavis–Cummings simulations, and the acknowledged XFA representation-dependence gap is a rigor concern, not an input–output equivalence.

full rationale

The paper's derivation chain is not circular. The XFA state (14) is constructed from the coherent-state expansion (6) and a zeroth-order Born treatment (12); the ROC state (52) then follows by applying the photon-number-parity projector to that approximate state. Crucially, the central claim is verified by exact Tavis–Cummings simulations that do not use the XFA (blue curves in Figs. 4, 5, and 8). No parameter is fitted to the target result: gamma, alpha0, and N are stated control parameters, and the measurement settings follow from the projector criterion (28) rather than from the measured outcome. The only self-citation (Ref. [23], the author's earlier density-matrix formulation) is used as background/contrast and is not load-bearing. The paper explicitly flags one gap: Sec. V C shows that the XFA total-system wave function is representation-dependent at finite gamma and states, 'A full proof that any coherent-state expansion yields the same XFA state in the thermodynamic limit (16) is left for future work.' That is an unproven uniqueness assumption and a correctness/rigor concern, but it is not circular: the same section provides numerical evidence of convergence as gamma decreases (Fig. 8), and the main predictions are checked against independent exact simulation. Therefore no circular step can be exhibited under the required standard.

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

The central claim rests on the XFA approximation, whose core assumption (negligible backaction) is physically motivated and validated at finite parameters, but whose representation-independence in the thermodynamic limit is asserted and only partially verified. No free parameters are fitted to data; the listed values are control parameters of the simulations.

free parameters (3)
  • electron-photon coupling gamma = 0.01 (also 0.05 and 0.1 in convergence tests)
    Set small to satisfy the XFA validity criterion (15); varied systematically in Fig. 8. Not fitted to data.
  • cat-state amplitude alpha0 = 1, 2, 4, 10, 30, and sqrt(N/2)
    Chosen to realize small, intermediate, and large photon-number regimes; the intermediate value sqrt(N/2) follows from energy conservation, not from fitting.
  • quadrature measurement parameters (x, phi) = x=0, phi(t)=pi/2 - omega t
    Chosen to satisfy the optimal projector criterion (28), making the two coherent-state components have equal quadrature overlap. This is a design choice, not a fit.
assumptions (5)
  • standard math Coherent states form an overcomplete basis, so any photonic state can be expanded as |chi(0)> = ∫ d²α f(α)|α> (Eq. 6).
    Standard property of coherent states, used to represent the photonic initial state and to derive the XFA state (14).
  • domain assumption Electron-photon backaction can be neglected (zeroth-order Born approximation, αq=0), with validity criterion ⟨n⟩ >> N_exc (Eq. 15).
    This is the physical basis of the XFA; it is not proven rigorously but justified by small γ and verified numerically for N=8 in Fig. 8.
  • ad hoc to paper In the thermodynamic limit γ→0 with γ√⟨n⟩ constant (Eq. 16), the XFA total wave function becomes independent of the coherent-state representation.
    The paper asserts this convergence and verifies it numerically for two specific expansions, but explicitly leaves a full proof for future work (Sec. V C).
  • domain assumption The rotating-wave approximation is valid for the parameter regime considered.
    Used to define the Tavis-Cummings Hamiltonian (23); checked in Sec. VII A by comparing with the full Dicke model, where only small 4π-period modulations appear.
  • standard math Total spin J=N/2 is conserved, so the electronic Hilbert space can be restricted to Dicke states |J,m>.
    Follows from the collective coupling structure of the Dicke/Tavis-Cummings Hamiltonian; used in the numerics and in the analytical solution (48).

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Cite this review

Pith. "Pith review of Macroscopic Schr\"{o}dinger-cat states of nonequilibrium electrons induced by cat-state optical driving and projective measurements on the light field." pith.science (2026). https://pith.science/paper/FIO5YBU4

@misc{pith2026250811769,
  author       = {Pith},
  title        = {Pith review of: Macroscopic Schr\"odinger-cat states of nonequilibrium electrons induced by cat-state optical driving and projective measurements on the light field},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FIO5YBU4}},
  note         = {Machine review of arXiv:2508.11769}
}
abstract

We show that projective measurements on quantum light can induce macroscopic cat states in many-electron systems driven by such light. Here we investigate the quantum dynamics of $N$ independent two-level electrons interacting with Schr\"{o}dinger-cat or -kitten states of light. Without measurement, a macroscopic cat state of the electrons appears only in an ultrashort time window. In contrast, we demonstrate that photon-number parity or quadrature projective measurements can restore a macroscopic cat state in nonequilibrium electrons, even in the thermodynamic limit. These dynamics are captured by an external-field approximation, in which the electronic system evolves into a Rabi-oscillation cat state. Our results highlight the need for precise quantum measurement techniques for light to control macroscopic quantum states of matter driven by quantum light.

Figures

Figures reproduced from arXiv: 2508.11769 by the authors.

Figure 1
Figure 1. FIG. 1. Schematic of measurement-assisted generation of macro [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Many-body electronic dynamics driven by the even cat-state light without postselection. (a)–(f) Time evolution of the QFI density [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. FWHM [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Time evolution of the QFI density [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Comparison of QFI dynamics for the XFA’s ROC state. [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Spin Wigner functions [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Effect of coherent-state overcompleteness on the XFA [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Dependence of the maximum QFI density [PITH_FULL_IMAGE:figures/full_fig_p012_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Effect of the RWA on the QFI dynamics under even [PITH_FULL_IMAGE:figures/full_fig_p013_11.png]

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