Pith. sign in

REVIEW 2 major objections 5 minor 47 references

Quantum geometry survives vacuum breakdown via operator frames

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · glm-5.2

2026-07-09 22:00 UTC pith:5U4TBGTP

load-bearing objection Operator-frame QGT stays analytic across Stokes lines where frame vacua go non-normalizable — a real construction, with a gauge-choice soft spot the 2 major comments →

arxiv 2607.06994 v1 pith:5U4TBGTP submitted 2026-07-08 quant-ph

Operator-frame geometry of non-compact quantum systems with frame-vacuum phase transitions

Satoshi Tanaka , Gonzalo Ordonez , Masatoshi Kubota , Hiroto Nakano , Kazuki Kanki This is my paper
classification quant-ph
keywords quantum geometric tensorBerry connectionoperator frameBogoliubov transformationrigged Hilbert spaceStokes linesvacuum instabilityEhresmann connection
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper claims that quantum geometry does not break down when a bosonic vacuum becomes unstable. Standard quantum geometry (Berry connection, quantum metric, Berry curvature) is defined through inner products of normalizable quantum states. When a Bogoliubov transformation makes the vacuum non-normalizable, those definitions fail. The authors shift the geometric construction from the space of states to the space of canonical operator frames — ordered pairs of annihilation and creation operators satisfying the canonical commutation relation. They build a principal bundle over a parameter space of such frames, equip it with an Ehresmann connection that preserves the commutation relations under parallel transport, and derive a quantum geometric tensor (QGT) from the infinitesimal displacement of the transported operators. The resulting QGT is an analytic function of the complex squeezing parameter β and stays finite across Stokes lines where the frame vacuum norm diverges. In the stable regime the construction reduces to the ordinary Berry connection. In the unstable regime the vacuum is reinterpreted as a generalized vector in a rigged Hilbert space, while the operator-space QGT remains well-defined. A dissipative coupling to environmental modes provides a physical mechanism that promotes the real squeezing parameter to a complex one, enabling continuous trajectories that connect stable and unstable domains while bypassing exceptional points.

Core claim

The QGT defined on the operator-frame bundle is analytic in the complex squeezing parameter β and does not singularize at the Stokes lines v = ±π/4 + nπ where the frame vacuum becomes non-normalizable. The quantum metric components g_θθ = sinh²(2β), g_ββ = 1 and the Berry curvature F_θβ = 2 sinh(2β) are all analytic functions of β. This means the frame-vacuum phase transition is a representational instability — the Hilbert-space description breaks down but the underlying operator-space geometry does not.

What carries the argument

The central object is the operator-frame principal bundle: the total space is the set of all canonical operator frames (annihilation-creation pairs satisfying [φ̂, φ̂*] = 1), the structure group is C× (complex rescaling φ̂ → λφ̂, φ̂* → λ⁻¹φ̂*), and the base manifold is the quotient by this gauge action. An Ehresmann connection on this bundle defines horizontal transport of operator frames. A gauge-fixing condition [D_μ φ̂_e, φ̂*_e] = 0 selects the Berry connection A_θ = −cosh(2β), A_β = 0. The QGT is then Q_μν = [D*_μ φ̂*_e, D_ν φ̂_e], which can be rewritten as Q_μν = [∂_μ φ̂*_e, ∂_ν φ̂_e] − [∂_μ φ̂*_e, φ̂_e][∂_ν φ̂_e, φ̂*_e], mirroring the conventional state-space QGT formula but operating纯

Load-bearing premise

The gauge-fixing condition [D_μ φ̂_e, φ̂*_e] = 0 is imposed to select the Berry connection, and the paper does not independently prove this is the unique gauge yielding the standard Berry connection in the stable regime. An alternative gauge choice could produce a different connection and QGT.

What would settle it

If an alternative gauge-fixing condition on the operator-frame bundle produced a QGT that does not reduce to the standard Berry connection in the stable regime, or that fails to be analytic across Stokes lines, the central claim of a unique well-defined operator-space QGT would be undermined.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Quantum geometric quantities (metric, curvature) can probe phase transitions in non-Hermitian or dissipative bosonic systems where the Hilbert-space description fails, extending geometric diagnostics beyond the stable regime.
  • The operator-frame geometry provides a Hamiltonian-independent background structure: the QGT is defined before any specific dynamics is chosen, suggesting quantum geometry is more fundamental than previously framed.
  • Dissipation-induced complexification of the squeezing parameter offers an experimentally accessible route to observe continuous QGT evolution across Stokes lines in systems like the dissipative dynamical Casimir effect or parametric amplification.
  • The rigged Hilbert space framework for unstable frame vacua may apply to other non-compact quantum systems with vacuum instability, such as Unruh-type observer-dependent vacua or curved-spacetime quantum field theory.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the operator-frame QGT is truly Hamiltonian-independent, one could precompute geometric phase diagrams for entire families of bosonic models without solving each Hamiltonian individually, treating geometry as a kinematic substrate.
  • The Z₄ monodromy structure on the complex β-plane suggests a topological invariant (winding number around exceptional points) that may constrain which stable and unstable domains can be connected by smooth paths, independent of the specific dissipation model.
  • The consistency between operator-space QGT and state-space QGT (differing only by excitation-number factors) hints that the operator-frame geometry could serve as a universal skeleton from which state-dependent geometries for different representations are derived by algebraic projection.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. This manuscript develops a geometric formulation of quantum geometry for non-compact bosonic systems in regimes where vacuum instability renders the relevant quantum states non-normalizable, causing conventional state-space quantum geometry (Berry connection, curvature, quantum metric) to break down. The authors formulate quantum geometry at the level of canonical operator frames, allowing for complexified Bogoliubov transformations that lift the requirement that creation operators be Hermitian conjugates of annihilation operators. The space of canonical operator frames is endowed with a principal C*-bundle structure over a parameter space, equipped with an Ehresmann connection that defines parallel transport while preserving the canonical commutation relations. The resulting operator-space quantum geometric tensor (QGT) is shown to be analytic in the complex squeezing parameter and remains well-defined across frame-vacuum phase transition boundaries (Stokes lines), even as the frame vacuum itself becomes non-normalizable and must be described within the rigged Hilbert space (RHS) framework. A physical realization using dissipative coupling to environmental modes is presented to illustrate continuous paths connecting stable and unstable regimes.

Significance. The paper addresses a genuine and important gap in quantum geometry: the breakdown of the standard Berry connection and QGT when vacuum states become non-normalizable at phase transitions. The construction of an operator-frame bundle geometry that remains analytic across Stokes lines is a novel and potentially impactful contribution. The mathematical apparatus is carefully laid out: the preservation of the canonical commutation relations under parallel transport is rigorously proven (Appendix C), the intertwining relation between the complex structure and covariant derivative is derived (Appendix D), and the QGT formulas (Eqs. 72-73) are shown to be analytic in the complex squeezing parameter. The connection to rigged Hilbert spaces for the unstable domain is physically motivated and well-executed. The physical example in Section VI provides a concrete dissipative mechanism for realizing trajectories that cross Stokes lines, which strengthens the paper's claims regarding the physical observability of the analytic continuation.

major comments (2)
  1. The central analyticity claim rests on the gauge-fixing condition [D_mu phi_e, phi_e*] = 0 (Eq. 66), which is introduced as a 'natural choice' to select the Berry connection (Eq. 69). However, the paper does not independently justify the uniqueness of this condition or demonstrate that the analyticity of the QGT (Eqs. 72-73) is gauge-invariant. Since the entire analyticity claim across Stokes lines depends on the specific connection determined by Eq. 66, the authors should clarify whether alternative gauge-fixing conditions would yield the same analytic QGT, or at minimum explain why this specific gauge is physically singled out in the unstable domain where the standard Hilbert-space inner product is unavailable. This is a load-bearing assumption that needs explicit justification.
  2. In Section VI, the physical realizability of the frame-vacuum phase transition relies on the phenomenological approximation sigma(z;k) approx kappa + i*gamma (Eq. 97), where kappa and gamma are free parameters. While this is a standard weak-coupling approximation, the claim that the QGT 'evolves smoothly across Stokes lines' along the trajectories in Figures 1 and 3 is contingent on this approximation. The authors should discuss the robustness of the smooth QGT evolution to higher-order corrections in the self-energy sigma(z;k), or clarify the regime of validity more precisely.
minor comments (5)
  1. In Eq. (80), the norm of the frame vacuum is given as exp(Im*eta) / sqrt(cos(2v)). It would help the reader to explicitly state that this expression is valid only in the stable region cos(2v) > 0, and that the divergence as v approaches pi/4 is a square-root singularity, which is mentioned later in the text but not directly at the equation.
  2. The notation for the dual and antidual spaces in Eq. (85) uses Phi' and Phi^x, but the superscript 'x' is non-standard. Consider clarifying whether this denotes the antidual space or a specific topological dual.
  3. Figure 1 is referenced before it is fully explained. The 'exceptional points' (black dots) are mentioned in the caption but their characterization is deferred to the next section. A forward reference or brief inline definition would improve readability.
  4. In Eq. (99), the expression for beta involves a logarithm of a ratio. The branch cut structure of this logarithm should be briefly discussed in relation to the Riemann sheet structure shown in Figure 2, to make the connection between the complex K_3 plane and the beta-plane more explicit.
  5. The paper uses both 'Bogoliubov-Valatin' (abstract) and 'Bogoliubov' (Section V) transformations. Consider standardizing the terminology.

Circularity Check

0 steps flagged

No significant circularity: the QGT is derived from intrinsic operator-frame geometry without fitting to data or reducing to its own inputs by construction.

full rationale

The paper's central claim—that the operator-space QGT (Eqs. 72-73) is analytic in the complex squeezing parameter β and remains well-defined across Stokes lines—is derived from the Ehresmann connection on the operator-frame principal bundle, the gauge-fixing condition (Eq. 66), and the Kähler structure on the operator vector space. The QGT components g_μν = sinh²(2β) and F_μν = 2 sinh(2β) are explicit analytic functions of β, and their analyticity across Stokes lines is a direct mathematical consequence of these formulas, not an assumption smuggled in. The reduction to the standard Berry connection in the stable regime is verified algebraically (the gauge-fixing condition recovers A_μ = i[∂_μ φ̂*_e, φ̂_e]). The state-space QGT in Appendix E is shown to be consistent with the operator-space QGT up to excitation-number factors, providing an independent cross-check rather than a circular restatement. The physical example in Sec. VI uses the Feshbach projection method (Appendix F) with a phenomenological approximation σ(z;k) ≈ κ + iγ, but this is used to construct continuous trajectories in parameter space, not to fit or 'predict' the QGT itself. The gauge-fixing condition (Eq. 66) is a choice rather than a uniquely derived necessity, and the paper does not prove that analyticity is gauge-invariant across all possible gauges—but this is a question of uniqueness and robustness, not circularity. The derivation chain does not reduce to its own inputs by construction, and no self-citation chain is load-bearing for the central result. The one minor self-citation (Ref. [41], Tanaka and Kanki) concerns the Feshbach projection framework and exceptional points, but it supports the physical realization section rather than the core geometric construction. Score 1 reflects this minor self-citation with no circularity in the main derivation.

Axiom & Free-Parameter Ledger

2 free parameters · 4 axioms · 2 invented entities

The paper introduces two main invented entities (the operator-frame bundle and the frame vacuum) and relies on one ad hoc gauge choice and one phenomenological approximation. The mathematical framework is internally consistent but lacks independent experimental or computational verification.

free parameters (2)
  • κ (dissipation parameter)
    Introduced phenomenologically in Sec. VI as part of the approximation σ(z;k) ≈ κ + iγ. No specific value is given; it is a free parameter controlling the imaginary part of K_3.
  • γ (dissipation parameter)
    Introduced alongside κ in the phenomenological approximation of the self-energy. Controls the non-Hermitian part of the effective frame generator. No specific value or derivation provided.
axioms (4)
  • ad hoc to paper Gauge-fixing condition [D_μ φ̂_e, φ̂*_e] = 0
    Imposed in Eq. (66) to select the Berry connection. While it yields the desired reduction to the standard Berry connection in the stable regime, its uniqueness or physical necessity is not independently established.
  • domain assumption Rigged Hilbert space framework for unstable vacua
    The paper assumes that non-normalizable frame vacua in the unstable domain can be consistently treated as generalized vectors in a rigged Hilbert space (Eq. 84-85), with bi-orthonormal pairing. This is a standard framework from the cited literature but is a structural assumption for the state-space QGT.
  • ad hoc to paper Phenomenological approximation σ(z;k) ≈ κ + iγ
    The replacement of the z-dependent self-energy kernel with constant complex parameters (Sec. VI) is stated as 'a standard approximation in the weak-coupling regime' but the regime of validity is not quantified.
  • standard math Canonical commutation relations preserved under complex symplectic transformations
    The construction relies on S^T Ω S = Ω holding for complex symplectic matrices, which is a standard mathematical fact.
invented entities (2)
  • Operator-frame bundle (principal C×-bundle over parameter space) no independent evidence
    purpose: Provides the geometric substrate on which the Ehresmann connection and QGT are defined, replacing the Hilbert-space line bundle.
    This is a new mathematical construction introduced by the paper. Its consistency is verified internally (CCR preservation, reduction to Berry connection), but no independent external benchmark or experimental prediction is provided that would falsify it.
  • Frame vacuum (right and left vacua in rigged Hilbert space) no independent evidence
    purpose: Defines the vacuum state associated with each operator frame, generalizing the bare vacuum to unstable regimes.
    The frame vacuum is defined algebraically by φ̂_ξ |φ_{ξ,0}⟩ = 0. Its interpretation as a generalized vector in the RHS is consistent but not independently verified against physical observables.

pith-pipeline@v1.1.0-glm · 28731 in / 3092 out tokens · 233469 ms · 2026-07-09T22:00:44.778362+00:00 · methodology

0 comments
read the original abstract

We formulate the geometric structure of non-compact bosonic quantum systems in regimes where vacuum instability renders the relevant quantum states non-normalizable, causing conventional state-space quantum geometry -- described by the Berry connection, curvature, and quantum metric -- to become ill-defined. To overcome this breakdown, we develop a formulation of quantum geometry at the level of canonical operator frames, allowing for complexified Bogoliubov-Valatin transformations that lift the requirement that creation operators be Hermitian conjugates of annihilation operators. Canonical operator frames are defined as choices of bosonic creation and annihilation operators realizing the canonical commutation relations. A natural equivalence relation among such frames generalizes the phase ambiguity of quantum states and determines a parameter space that analytically extends the stable-regime parameter space. The space of canonical operator frames forms a principal bundle over parameter space -- the operator-frame bundle -- equipped with a natural Ehresmann connection that defines parallel transport while preserving the canonical commutation relations. In the stable regime, this construction reduces to the Berry connection, while more generally it yields a well-defined operator-space quantum geometric tensor (QGT) that remains valid across vacuum instabilities. Using the framework of rigged Hilbert spaces, we define a notion of quantum frame vacuum and obtain a consistent state-space QGT. Focusing on a single bosonic mode, we demonstrate analyticity of the QGT across the quantum frame-vacuum phase transition and present the corresponding phase diagram on the complexified squeezing-parameter plane. We further introduce a realistic physical setting that allows continuous paths connecting stable and unstable regimes, along which the QGT evolves smoothly across Stokes lines.

Figures

Figures reproduced from arXiv: 2607.06994 by Gonzalo Ordonez, Hiroto Nakano, Kazuki Kanki, Masatoshi Kubota, Satoshi Tanaka.

Figure 1
Figure 1. Figure 1: FIG. 1. Stable-unstable domain phase diagram in the com [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 1
Figure 1. Figure 1: Fig.1. The Stokes lines shown as red lines and appearing [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (Color online) Continuous trajectories in the com [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

47 extracted references · 47 canonical work pages · 1 internal anchor

  1. [1]

    von Neumann,Mathematical Foundations of Quantum Mechanics(Princeton University Press, Princeton, 1955)

    J. von Neumann,Mathematical Foundations of Quantum Mechanics(Princeton University Press, Princeton, 1955)

    work page 1955

  2. [2]

    M. V. Berry, Proc. R. Soc. Lond. A392, 45 (1984). https://doi.org/10.1098/rspa.1984.0023

    work page doi:10.1098/rspa.1984.0023 1984

  3. [3]

    Burdick, and Benjamin L

    Y. Aharonov and J. Anandan, Phys. Rev. Lett.58, 1593 (1987).https://doi.org/10.1103/PhysRevLett. 58.1593

    work page doi:10.1103/physrevlett 1987

  4. [4]

    Simon, Phys

    B. Simon, Phys. Rev. Lett.51, 2167 (1983).https:// doi.org/10.1103/PhysRevLett.51.2167

    work page doi:10.1103/physrevlett.51.2167 1983

  5. [5]

    Wilczek and A

    F. Wilczek and A. Shapere,Geometric Phases in Physics, Vol. 5 of Advanced Series in Mathematical Physics (World Scientific, Singapore, 1989)

    work page 1989

  6. [6]

    T¨ orm¨ a, Phys

    P. T¨ orm¨ a, Phys. Rev. Lett.131, 240001 (2023).https: //doi.org/10.1103/PhysRevLett.131.240001

    work page doi:10.1103/physrevlett.131.240001 2023

  7. [7]

    Ozawa and B

    T. Ozawa and B. Mera, Phys. Rev. B104, 045103 (2021), https://doi.org/10.1103/PhysRevB.104.045103

    work page doi:10.1103/physrevb.104.045103 2021

  8. [8]

    Mera and T

    B. Mera and T. Ozawa, Phys. Rev. B104, 045104 (2021). https://doi.org/10.1103/PhysRevB.104.045104

    work page doi:10.1103/physrevb.104.045104 2021

  9. [9]

    A. C. Carollo and J. K. Pachos, Phys. Rev. Lett.95, 157203 (2005).https://doi.org/10.1103/ PhysRevLett.95.157203

    work page 2005

  10. [10]

    Carollo, D

    A. Carollo, D. Valenti, and B. Spagnolo, Phys. Rep. 838, 1 (2020),https://doi.org/10.1016/j.physrep. 2019.11.002

    work page doi:10.1016/j.physrep 2020

  11. [11]

    Hatsuda, S

    M. Cozzini, P. Giorda, and P. Zanardi, Phys. Rev. B75, 014439 (2007),https://doi.org/10.1103/PhysRevB. 75.014439

    work page doi:10.1103/physrevb 2007

  12. [12]

    Zanardi, P

    P. Zanardi, P. Giorda, and M. Cozzini, Phys. Rev. Lett.99, 100603 (2007).https://doi.org/10.1103/ PhysRevLett.99.100603

    work page 2007

  13. [13]

    I. M. Gelfand and N. Y. Vilenkin,Generalized Functions, Vol. 4: Applications of Harmonic Analysis(Academic Press, New York, 1964)

    work page 1964

  14. [14]

    Petrosky, I

    T. Petrosky, I. Prigogine, and S. Tasaki, Physica A173, 175 (1991).https://doi.org/10.1016/0378-4371(91) 90257-D

    work page doi:10.1016/0378-4371(91 1991

  15. [15]

    Petrosky and I

    T. Petrosky and I. Prigogine, Adv. Chem. Phys.99, 1 (1997)

    work page 1997

  16. [16]

    A. R. Bohm, R. Scurek, and S. Wikramasekara, Rev. Mex. Fis.45, 16 (1999).https://doi.org/10.48550/ arXiv.nucl-th/9902076

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  17. [17]

    Petrosky, G

    T. Petrosky, G. Ordonez, and I. Prigogine, Phys. Rev. A62, 042106 (2000).https://doi.org/10.1103/ PhysRevA.62.042106

    work page 2000

  18. [18]

    Ordonez, T

    G. Ordonez, T. Petrosky, and I. Prigogine, Phys. Rev. A63, 023201 (2001).https://doi.org/10.1103/ PhysRevA.63.052106

    work page 2001

  19. [19]

    Chru´ sci´ nski, J

    D. Chru´ sci´ nski, J. Math. Phys.44, 3718 (2003).https: //doi.org/10.1063/1.1599074

    work page doi:10.1063/1.1599074 2003

  20. [20]

    Chru´ sci´ nski, J

    D. Chru´ sci´ nski, J. Math. Phys.45, 841 (2004).https: 18 //doi.org/10.1063/1.1644751

    work page doi:10.1063/1.1644751 2004

  21. [21]

    de la Madrid, Eur

    R. de la Madrid, Eur. J. Phys.26, 287 (2005).https: //doi.org/10.1088/0143-0807/26/2/008

    work page doi:10.1088/0143-0807/26/2/008 2005

  22. [22]

    T. W. B. Kibble, Commun. Math. Phys.65, 189 (1979). https://doi.org/10.1007/BF01225149

    work page doi:10.1007/bf01225149 1979

  23. [23]

    Ashtekar and T

    A. Ashtekar and T. A. Schilling, Geometrical formulation of quantum mechanics, inOn Einstein’s Path, edited by A. Harvey (Springer, New York, 1999), pp. 23–65, https://doi.org/10.1007/978-1-4612-1422-9_3

    work page doi:10.1007/978-1-4612-1422-9_3 1999

  24. [24]

    Dutta, N

    Arvind, B. Dutta, N. Mukunda, and R. Simon, Pramana 45, 471 (1995).https://doi.org/10.1007/BF02848172

    work page doi:10.1007/bf02848172 1995

  25. [25]

    S. L. Braunstein, Phys. Rev. A71,71.055801 (2005), https://doi.org/10.1103/PhysRevA.71.055801

    work page doi:10.1103/physreva.71.055801 2005

  26. [26]

    Kobayashi and K

    S. Kobayashi and K. Nomizu,Foundations of Differential Geometry, Vol. 1(Wiley, New York, 1996)

    work page 1996

  27. [27]

    Morita,Geometry of Differential Forms(American Mathematical Society, Providence, 2001)

    S. Morita,Geometry of Differential Forms(American Mathematical Society, Providence, 2001)

    work page 2001

  28. [28]

    Nakahara,Geometry, Topology and Physics(Institute of Physics Publishing, Bristol, 2003)

    M. Nakahara,Geometry, Topology and Physics(Institute of Physics Publishing, Bristol, 2003)

    work page 2003

  29. [29]

    I. G. Avramidi, Rev. Math. Phys.11, 947 (1999).https: //doi.org/10.1142/S0129055X99000295

    work page doi:10.1142/s0129055x99000295 1999

  30. [30]

    J. P. Provost and G. Vallee, Commun. Math. Phys.76, 289301 (1980).https://doi.org/10.1007/BF02193559

    work page doi:10.1007/bf02193559 1980

  31. [31]

    Zhanget al., Opt

    H.-L. Zhanget al., Opt. Express32, 22566 (2024). https://doi.org/10.1364/OE.517716

    work page doi:10.1364/oe.517716 2024

  32. [32]

    Kolodrubetz, V

    M. Kolodrubetz, V. Gritsev, and A. Polkovnikov, Phys. Rev. B88, 064304 (2013).https://doi.org/10.1103/ PhysRevB.88.064304

    work page 2013

  33. [33]

    C. M. Bender and S. A. Orszag,Advanced Mathemati- cal Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory(Springer, New York, 2013)

    work page 2013

  34. [34]

    Barnett and P

    S. Barnett and P. M. Radmore,Methods in Theoreti- cal Quantum Optics(Oxford University Press, Oxford, 2002)

    work page 2002

  35. [35]

    C. M. Bender and A. Turbiner, Phys. Lett. A173, 442 (1993).https://doi.org/10.1016/0375-9601(93) 90153-Q

    work page doi:10.1016/0375-9601(93 1993

  36. [36]

    Garrison and E

    J. Garrison and E. Wright, Phys. Lett. A128, 177 (1988).https://doi.org/10.1016/0375-9601(88) 90905-X

    work page doi:10.1016/0375-9601(88 1988

  37. [37]

    Milburn and D

    G. Milburn and D. F. Walls, Opt. Commun.39, 401 (1981).https://doi.org/10.1016/0030-4018(81) 90232-7

    work page doi:10.1016/0030-4018(81 1981

  38. [38]

    C. W. Gardiner and C. M. Savage, Opt. Commun.50, 173 (1984).https://doi.org/10.1016/0030-4018(84) 90342-0

    work page doi:10.1016/0030-4018(84 1984

  39. [39]

    Dimer, B

    F. Dimer, B. Estienne, A. S. Parkins, and H. J. Carmichael, Phys. Rev. A75, 013804 (2007).https: //doi.org/10.1103/PhysRevA.75.013804

    work page doi:10.1103/physreva.75.013804 2007

  40. [40]

    Rom´ an-Ancheyta, I

    R. Rom´ an-Ancheyta, I. Ramos-Prieto, A. Perez-Leija, K. Busch, and R. D. J. Le´ on-Montiel, Phys. Rev. A96(2017).https://doi.org/10.1103/PhysRevA.96. 032501

    work page doi:10.1103/physreva.96 2017

  41. [41]

    Tanaka and K

    S. Tanaka and K. Kanki, Prog. Theor. Exp. Phys. 2020, 12A107 (2020).https://doi.org/10.1093/ptep/ ptaa129

    work page doi:10.1093/ptep/ 2020

  42. [42]

    Dodonov, Physics2, 67 (2020).https://doi.org/ 10.3390/physics2010007

    V. Dodonov, Physics2, 67 (2020).https://doi.org/ 10.3390/physics2010007

    work page doi:10.3390/physics2010007 2020

  43. [43]

    Feshbach, Ann

    H. Feshbach, Ann. Phys.5, 357 (1958).https://doi. org/10.1016/0003-4916(58)90007-1

    work page doi:10.1016/0003-4916(58)90007-1 1958

  44. [44]

    W. G. Unruh, Phys. Rev. D14, 870 (1976).https: //doi.org/10.1103/PhysRevD.14.870

    work page doi:10.1103/physrevd.14.870 1976

  45. [45]

    P. D. Nation, J. R. Johansson, M. P. Blencowe, and F. Nori, Rev. Mod. Phys.84, 1 (2012).https://doi. org/10.1103/RevModPhys.84.1

    work page doi:10.1103/revmodphys.84.1 2012

  46. [46]

    Capolupo and G

    A. Capolupo and G. Vitiello, Phys. Rev. D88(2013). https://doi.org/10.1103/PhysRevD.88.024027

    work page doi:10.1103/physrevd.88.024027 2013

  47. [47]

    Chru´ sci´ nski and A

    D. Chru´ sci´ nski and A. Kossakowski, Phys. Rev. Lett.111050402 (2013).https://doi.org/10.1103/ PhysRevLett.111.050402

    work page 2013