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One complex Bogoliubov ratio reduces two-mode squeezing dynamics to universal purity and entropy formulas.

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2026-07-31 15:46 UTC pith:7OLBWMVY

load-bearing objection Solid Gaussian packaging of λ=β/α into one Riccati-plus-q pipeline; math checks out, novelty is organizational, and the cosmology demo is a phenomenological two-function model rather than true Mukhanov–Sasaki dynamics. the 2 major comments →

arxiv 2607.24404 v2 pith:7OLBWMVY submitted 2026-07-27 quant-ph gr-qchep-th

A Bogoliubov-ratio framework for quantum-information diagnostics of time-dependent two-mode Boson Hamiltonian

Shi-Cheng Liu , Lei-Hua Liu , Bichu Li , Yuebing Zhou , Hai-Qing Zhang This is my paper
classification quant-ph gr-qchep-th
keywords Bogoliubov ratiotwo-mode squeezingRiccati equationreduced-state entropycosmological perturbationsoptical parametric amplifierGaussian statesquantum information diagnostics
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.

This paper shows that for any time-dependent quadratic two-mode bosonic Hamiltonian, the full state evolution collapses to a single complex Riccati equation for the Bogoliubov ratio λ = β/α. After one partner mode is traced out, every standard reduced-state diagnostic—purity, linear entropy, Rényi-2 entropy, and von Neumann entropy—depends only on the real number q = |λ|² through fixed, model-independent expressions. Model details enter solely through the two driving functions (frequency and pair-production coupling) that steer λ; once λ is known, no further reconstruction or diagonalization of the reduced density matrix is required. The authors demonstrate the separation on primordial cosmological perturbations and on a chirped-pulse optical parametric amplifier, showing how background phase rotation, frequency softening, finite pump duration, and chirp each regulate squeezing growth and late-time mixedness. The practical payoff is a standardized diagnostic pipeline that cleanly factors driving protocols from information-theoretic metrics across a broad class of parametrically driven quadratic systems.

Core claim

For a general time-dependent quadratic two-mode Hamiltonian, the Schrödinger dynamics of a normalized paired Bogoliubov state reduce exactly to the Riccati equation dλ_k/dη = g_k(1−λ_k²)−2iω_k λ_k. After tracing out one partner, the reduced spectrum is geometric with q_k = |λ_k|², so purity and the listed entropies are the universal functions of q_k alone, without model-by-model reconstruction of the reduced density matrix from coupled squeezing parameters.

What carries the argument

The Bogoliubov ratio λ_k ≡ β_k/α_k. It packages the usual squeezing amplitude and phase into one complex variable whose modulus alone fixes the entire one-mode reduced spectrum and all derived information measures.

Load-bearing premise

The evolved state must stay inside the normalized paired Bogoliubov family with |λ| less than one, so that tracing out one mode leaves a spectrum fixed entirely by |λ| squared.

What would settle it

Solve the Riccati equation for a known quadratic drive (for example the resonant constant-coupling optical amplifier), compute purity from (1−q)/(1+q), and check whether it matches an independent numerical partial trace of the two-mode state; any systematic mismatch, or any drive that leaves the paired number structure, would refute the claim.

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

If this is right

  • Cosmological backgrounds with softer effective frequency produce larger |λ|, lower one-mode purity, and higher reduced entropies for the same pair-production drive.
  • Finite Gaussian pumps and frequency chirps delay the onset of squeezing and saturate late-time entropy at values well below the ideal resonant limit.
  • Any other Hermitian quadratic two-mode system (dynamical Casimir, driven oscillators, modified-dispersion Mukhanov–Sasaki modes) can reuse the same q-to-entropy map once its ω and g are supplied.
  • Comparisons across models reduce to comparing trajectories of a single complex function rather than rebuilding reduced density matrices case by case.

Where Pith is reading between the lines

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

  • The same pipeline could benchmark how quickly different early-universe sound-speed or mass profiles drive reduced-state entropy toward the classical limit.
  • Laboratory pulsed squeezers with programmable chirp could test the predicted delayed-onset and saturation plateaus as a direct optical analogue of the cosmological competition between phase rotation and pair creation.
  • If weak non-Gaussian corrections are later added, the first observable failure mode should be a departure of the one-mode spectrum from pure geometric form at fixed |λ|.

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 / 6 minor

Summary. The manuscript proposes organizing the quantum-information diagnostics of any time-dependent quadratic two-mode bosonic Hamiltonian, H = ω(η)(N_k+N_{−k}+1) + ig(η)(a†a† − aa), around the Bogoliubov ratio λ_k = β_k/α_k. Projecting the Schrödinger equation onto the paired number-state ansatz |ψ⟩ = √(1−|λ|²)Σ λⁿ|n,n⟩ yields a single complex Riccati equation λ′ = g(1−λ²) − 2iωλ (Eq. 31 / A12); after tracing out one mode the spectrum is geometric, p_n = (1−q)qⁿ with q = |λ|², so purity, linear entropy, Rényi-2 and von Neumann entropy are universal functions of q (Eq. 27). Two applications are presented: a 'cosmological' model with ω = √(k²+α/η²), g = −1/η, and a chirped-pulse nondegenerate OPA recovering λ = tanh τ in the resonant limit. Earlier sections embed the state in a Meixner/Krylov 'deformed' construction and reparametrize it into Bogoliubov form.

Significance. If the framing is corrected, the paper provides a useful, genuinely parameter-free organizational tool: a single complex Riccati equation replacing coupled (r,φ) evolution, plus a universal, closed-form map from |λ|² to purity and the standard entropies, with an analytic benchmark (tanh τ) and reproducible numerics (DOP853, stated parameters). The unification of cosmological and optical pair production under one diagnostic chain is pedagogically valuable, and the honesty about the |λ|<1 / Hermitian-quadratic scope in Sec. VII is welcome. The framework itself is not new physics — the Riccati equation for β/α is standard in the Bogoliubov literature — but the explicit, model-independent entropy package and the chirped-OPA application have standalone value. The cosmological application as written does not realize the Mukhanov–Sasaki dynamics it invokes, which currently limits the significance of that half of the paper.

major comments (2)
  1. [Sec. V.B, Eq. (33); Sec. VI.A] Eq. (33) treats ω_k(η)=√(k²+α/η²) and g_k(η)=−1/η as independent inputs while calling ω_k 'the shifted-time form of the standard Mukhanov–Sasaki frequency'. This is internally inconsistent: a single oscillator mode u_k with u_k''+Ω²u_k=0, rewritten via instantaneous Bogoliubov coefficients, has g fixed by the same Ω, g=Ω′/(2Ω). For Ω²=k²+α/η² this gives g=−α/[2η(k²η²+α)], which equals −1/η for no α (e.g. de Sitter α=−2 gives g=1/[η(k²η²−2)]). Hence Eq. (34) is not the Riccati equation of any Mukhanov–Sasaki mode, and the abstract/Sec. VI.A narrative ('background-induced phase rotation and frequency softening regulate squeezing growth', the ν-ordering) describes a phenomenological two-function toy model. The authors should either (a) use the consistent pair {Ω, Ω′/2Ω} and re-run Figs. 1–5, or (b) relabel Eq. (33) explicitly as a toy parametrization and tone down the cosmological claims in
  2. [Secs. III–IV, Eqs. (15)–(21)] The Meixner/Krylov machinery of Sec. III and the deformation parameters u1,u2 are not used anywhere downstream. Eqs. (17)–(21) show the 'deformed' state is exactly a paired geometric state characterized by a single complex λ with |λ|<1 — i.e. a standard two-mode squeezed vacuum with redefined (r,φ). Any state of the form (17) has the geometric spectrum (25) regardless of how λ was obtained, so the claim in Sec. I that the formulation 'incorporates the state-level deformation' is vacuous: the deformation is fully absorbed into λ and carries no independent information. The identification of u2 with ω_k and |1−u1²| with |g_k|² (Sec. III) is a coefficient-level match in a dimensionless normalization that adds no constraint or prediction. The authors should state plainly that Secs. III–IV establish only that the generalized state is a TMSV in disguise, and either streamline this material to a
minor comments (6)
  1. [Sec. V] Sec. V, paragraph after Eq. (27): 'Sections 5.2 and 5.3' should read V.B and V.C (or Secs. V B and V C).
  2. [Sec. I] Introduction: the repeated use of 'open' (e.g. 'normalized open two-mode squeezed state') is misleading given that Sec. VII correctly restricts the framework to Hermitian quadratic unitary evolution and excludes Lindblad/influence-functional dynamics. Since the reduced-state mixedness here is purely from the partial trace, suggest replacing 'open' with 'deformed' throughout the introduction, as the authors themselves clarify only in passing after Eq. (2).
  3. [Sec. VII] Sec. VII, second paragraph: 'as shown in Fig. 1, which it lowers the one-mode purity' — grammatical error; also 'Figs 2, 3, 4 and 5' should be 'Figs. 2–5'.
  4. [Figs. 2–5] Figs. 2–5: the legend entry '|λk| = 1' appears in the purity/entropy plots where the quantity plotted is not |λk|; presumably it marks a reference asymptote. Please clarify the legend or remove the entry.
  5. [Secs. III–IV; App. A] Notation: the manuscript mixes Krylov parentheses |en), |O(η)) with Dirac brackets |O(η)⟩ (Eqs. (15)–(17)); please unify. Also 'Schrodinger' in Appendix A (A3) should be 'Schrödinger'.
  6. [Sec. VI.A] Sec. VI.A, ν=2 case: ω_k² vanishes at η=−√3.75, so ω_k becomes imaginary for part of the integration window. The text notes the turning point but does not discuss whether the |λ|<1 bound and the physical interpretation of q_k are affected when the Riccati equation has an imaginary ω; one sentence of comment would help, since the conclusion flags turning-point behavior as future work.

Circularity Check

1 steps flagged

Core Riccati and geometric-spectrum chain is self-contained; only mild scaffolding self-citation for the deformed wavefunction origin, not a forced identity of the diagnostics.

specific steps
  1. self citation load bearing [Sec. III–IV, Eqs. (15)–(21); citations [30,40,77]]
    "We use the generalized two-mode wave function derived from the Meixner-polynomial construction of Ref. [40]. ... In our work [77], we have demonstrated that the wave function can be represented by the Bogoliubov transformation when the Hamiltonian is of the group structure. ... Comparing Eqs. (17) and (19) gives the central identification λ_k = β_k/α_k = −√|1−u₁²| e^{2iϕ_k} tanh r_k / (1+u₂ tanh r_k)."

    The deformed paired state and its normalization are imported from the authors’ prior Krylov/Meixner papers rather than re-derived from first principles here. However this is scaffolding only: once the state is written in Bogoliubov form, the Riccati dynamics and entropy formulas follow from the Hamiltonian and partial trace independently of those citations, so the central claim is not forced by the self-citation chain.

full rationale

The load-bearing derivation does not reduce to its inputs by construction. Appendix A starts from the paired ansatz (A1) and the quadratic Hamiltonian (A2), matches coefficients of |n,n⟩ in the Schrödinger equation, and obtains the Riccati equation λ′=g(1−λ²)−2iωλ (A12) without assuming the target purity/entropy formulas. The reduced spectrum pn=(1−q)q^n and the package (27)/(A16)–(A20) then follow from the partial trace alone. These steps are standard Gaussian/TMSV algebra and are independently checkable. Self-citations to the authors’ Krylov/Meixner and purity papers supply the prior deformed wavefunction that is then reparametrized as ordinary Bogoliubov λ; that is scaffolding for motivation and scope, not a uniqueness theorem or a fit that forces the diagnostics. The cosmological (ω,g) choice is a physics-modeling issue (phenomenological two-function drive vs strict Mukhanov–Sasaki g=Ω′/2Ω), not circularity. No fitted-input-as-prediction pattern appears. Score 2 reflects only non-load-bearing self-citation plus mild packaging of known geometric-spectrum formulas as a ‘unified framework.’

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 1 invented entities

The central claim rests on standard bosonic QM plus the domain restriction to pure paired Gaussian evolution. No new physical constants are fitted to establish the formulas; demonstration parameters only set the numerical examples. Invented content is methodological (the λ-framework label), not a new particle or force. Load-bearing background is the SU(1,1)/Bogoliubov structure and geometric reduced spectrum of two-mode squeezing.

free parameters (3)
  • cosmological ν (hence α=1/4−ν²) and comoving k = ν∈{0.5,1,1.5,2}, k=1
    Chosen by hand for the numerical survey (ν=0.5,1,1.5,2; k=1); they select backgrounds, not fit the diagnostic formulas.
  • optical pump/chirp set (γ0, σ, tc, tf, Δ0, v) and dimensionless (s, τc, κ, δ0) = γ0/2π=100 MHz, σ=2 ns, tc=8 ns, v/2π=4e16 Hz/s, κ=0.637, s=1.257
    Physical drive parameters for the chirped-pulse demo; illustrate delayed onset and saturation, not calibrated to external data to prove the framework.
  • integration window η∈[−1000,−1], λ(η_i)=0 = η_i=−1000, λ=0
    Numerical IR/UV cutoffs and vacuum initial condition for cosmology; conventional but affect final |λ| values quoted.
axioms (5)
  • domain assumption Dynamics generated by the Hermitian quadratic two-mode Hamiltonian H_k=ω_k(a†_k a_k+a†_{−k}a_{−k}+1)+i g_k(a†_k a†_{−k}−a_k a_{−k}) with real ω_k,g_k.
    Sec. II and Eq. (10); defines the entire model class.
  • domain assumption State remains in the normalized paired Bogoliubov form √(1−|λ|²) Σ λ^n |n,n⟩ with |α|²−|β|²=1 and |λ|<1.
    Secs. IV–V, Eqs. (17)–(21), (28); required for spectrum p_n=(1−q)q^n.
  • standard math Schrödinger evolution i∂_η|ψ⟩=H|ψ⟩ (ℏ=1) and partial trace over the partner mode.
    Sec. V and Appendix A; standard QM.
  • standard math Reduced diagnostics are the standard functionals of the geometric spectrum (purity, linear, Rényi-2, von Neumann).
    Eqs. (27), (A16)–(A20); textbook once p_n is geometric.
  • ad hoc to paper Meixner/Krylov deformed amplitudes with parameters u1,u2 can be reparametrized exactly into Bogoliubov λ for the Hermitian case considered.
    Secs. III–IV and prior works [40,30]; motivates the wavefunction but is not needed once pure Bogoliubov form is assumed.
invented entities (1)
  • Bogoliubov-ratio framework (λ_k as sole diagnostic state variable) no independent evidence
    purpose: Compress (r_k,φ_k) and model drives into one complex ODE whose modulus feeds universal entropy formulas.
    Methodological construct, not a new physical degree of freedom; equivalent content exists in standard Bogoliubov variables.

pith-pipeline@v1.2.0-grok45-kimik3 · 27955 in / 4107 out tokens · 78514 ms · 2026-07-31T15:46:08.722192+00:00 · methodology

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read the original abstract

We present a compact and unified framework for quantum-information diagnostics of time-dependent two-mode bosonic systems based on the Bogoliubov ratio $\lambda_k(\eta) \equiv \beta_k(\eta)/\alpha_k(\eta)$. For a general time-dependent quadratic two-mode Hamiltonian, the state dynamics is exactly reduced to a single complex Riccati equation for $\lambda_k$. Upon tracing out one partner mode, the spectrum of the one-mode reduced density matrix is determined entirely by the squared magnitude $q_k(\eta) = \vert{}\lambda_k(\eta)\vert{}^2$. Consequently, we could construct the explicit, model-independent formula for the reduced-state purity, linear entropy, R\'enyi-2 entropy, and von Neumann entropy without reconstructing and diagonalizing the reduced density matrix on a model-by-model basis using coupled squeezing parameters ($r_k, \phi_k$). We demonstrate the utility of this framework in two distinct non-stationary setups: primordial cosmological perturbations and a chirped-pulse nondegenerate optical parametric amplifier. In the cosmological context, our formulation clarifies how background-induced phase rotation and frequency softening regulate squeezing growth and state mixedness; in the optical domain, it captures the delayed onset, suppression of squeezing accumulation, and late-time entropy saturation induced by finite pump duration and frequency chirp. By cleanly factorizing model-dependent driving protocols from universal information-theoretic metrics, this framework offers an efficient, standardized diagnostic tool for a broad class of parametrically driven quadratic bosonic systems.

Figures

Figures reproduced from arXiv: 2607.24404 by Bichu Li, Hai-Qing Zhang, Lei-Hua Liu, Shi-Cheng Liu, Yuebing Zhou.

Figure 1
Figure 1. Figure 1: FIG. 1: Evolution of [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: One-mode purity for the cosmological model and the standard TMSV benchmark. The [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Linear entropy for the cosmological model and the standard TMSV benchmark. The [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: R´enyi-2 entropy for the cosmological model and the standard TMSV benchmark. The [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Von Neumann entropy of the reduced [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Evolution of [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Reduced-mode purity for the four optical controls, with [PITH_FULL_IMAGE:figures/full_fig_p024_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Linear entropy for the four optical controls, with [PITH_FULL_IMAGE:figures/full_fig_p025_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: R´enyi-2 entropy for the four optical controls, with [PITH_FULL_IMAGE:figures/full_fig_p026_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Von Neumann entropy of one optical mode for the four optical controls, with [PITH_FULL_IMAGE:figures/full_fig_p027_10.png] view at source ↗

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Reference graph

Works this paper leans on

91 extracted references · 5 linked inside Pith

  1. [1]

    10: Von Neumann entropy of one optical mode for the four optical controls, withλ opt(0) = 0 and 0≤τ≤10.053

    Derivation of the evolution equation forλ k We start from the two-mode state |ψk(η)⟩=C k(η) ∞X n=0 λn k (η)|nk, n−k⟩,|C k(η)|= p 1− |λk(η)|2.(A1) 27 0 2 4 6 8 10 τ 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 SvN, opt(τ) Gaussian, chirped Gaussian, no chirp Constant, chirped Standard TMSV FIG. 10: Von Neumann entropy of one optical mode for the four optical c...

  2. [2]

    Reduced density matrix and information-theoretic quantities The density operator associated with Eq. (A1) is ρk,−k = (1−q k) ∞X n,m=0 λn k (λ∗ k)m|nk, n−k⟩⟨mk, m−k|, q k =|λ k|2.(A13) Tracing over the−kmode and using⟨m −k|n−k⟩=δ mn gives ρk = Tr−k ρk,−k = (1−q k) ∞X n=0 qn k |nk⟩⟨nk|.(A14) The eigenvalues ofρ k are therefore pn = (1−q k)qn k , ∞X n=0 pn =...

  3. [3]

    5, the frequency and coupling are gk(η) =− 1 η , ω k(η) = r k2 + α η2 .(A21) Substituting these functions into Eq

    Application to the cosmological model For the cosmological parametrization used in Sec. 5, the frequency and coupling are gk(η) =− 1 η , ω k(η) = r k2 + α η2 .(A21) Substituting these functions into Eq. (A12) gives dλk dη =− 1 η (1−λ 2 k)−2i r k2 + α η2 λk.(A22) The initial conditionλ k(η0) = 0 specifies the initial two-mode vacuum. Once Eq. (A22) is solv...

  4. [4]

    (A30) contains the residual detuning after the bare carrier rotation has been removed

    Application to the chirped-pulse optical parametric amplifier The effective two-mode Hamiltonian in the pump-rotating frame is ˆHopt(t) = ℏ∆(t) 2 ˆa†ˆa+ˆb†ˆb+ 1 +iℏγ(t) ˆa†ˆb† −ˆaˆb .(A30) The detuning and coupling are chosen as ∆(t) = ∆0 +v(t−t c), γ(t) =γ 0 exp − (t−t c)2 2σ2 .(A31) The first term in Eq. (A30) contains the residual detuning after the ba...

  5. [5]

    L. P. Grishchuk and Yu. V. Sidorov. Squeezed quantum states of relic gravitons and primordial density fluctuations.Phys. Rev. D, 42:3413–3421, 1990

    1990

  6. [6]

    Inflation and squeezed quantum states.Phys

    Andreas Albrecht, Pedro Ferreira, Michael Joyce, and Tomislav Prokopec. Inflation and squeezed quantum states.Phys. Rev. D, 50:4807–4820, 1994. 33

    1994

  7. [7]

    Starobinsky

    David Polarski and Alexei A. Starobinsky. Semiclassicality and decoherence of cosmological perturbations.Class. Quant. Grav., 13:377–392, 1996

    1996

  8. [8]

    Emergence of classicality for primordial fluctuations: Con- cepts and analogies.Annalen Phys., 7:137–158, 1998

    Claus Kiefer and David Polarski. Emergence of classicality for primordial fluctuations: Con- cepts and analogies.Annalen Phys., 7:137–158, 1998

    1998

  9. [9]

    Starobinsky

    Claus Kiefer, Ingo Lohmar, David Polarski, and Alexei A. Starobinsky. Pointer states for primordial fluctuations in inflationary cosmology.Class. Quant. Grav., 24:1699–1718, 2007

    2007

  10. [10]

    Why do cosmological perturbations look classical to us? Adv

    Claus Kiefer and David Polarski. Why do cosmological perturbations look classical to us? Adv. Sci. Lett., 2:164–173, 2009

    2009

  11. [11]

    Cosmological Inflation and the Quantum Measurement Problem.Phys

    Jerome Martin, Vincent Vennin, and Patrick Peter. Cosmological Inflation and the Quantum Measurement Problem.Phys. Rev. D, 86:103524, 2012

    2012

  12. [12]

    Quantum Discord of Cosmic Inflation: Can we Show that CMB Anisotropies are of Quantum-Mechanical Origin?Phys

    Jerome Martin and Vincent Vennin. Quantum Discord of Cosmic Inflation: Can we Show that CMB Anisotropies are of Quantum-Mechanical Origin?Phys. Rev. D, 93(2):023505, 2016

    2016

  13. [13]

    Decoherence and entropy of primordial fluctuations

    David Campo and Renaud Parentani. Decoherence and entropy of primordial fluctuations. I: Formalism and interpretation.Phys. Rev. D, 78:065044, 2008

    2008

  14. [14]

    Rigopoulos

    Tomislav Prokopec and Gerasimos I. Rigopoulos. Decoherence from Isocurvature perturba- tions in Inflation.JCAP, 11:029, 2007

    2007

  15. [15]

    Quantum Decoherence During Inflation from Gravitational Nonlinearities

    Elliot Nelson. Quantum Decoherence During Inflation from Gravitational Nonlinearities. JCAP, 03:022, 2016

    2016

  16. [16]

    Caves and Bonny L

    Carlton M. Caves and Bonny L. Schumaker. New formalism for two-photon quantum optics

  17. [17]

    Quadrature phases and squeezed states.Phys. Rev. A, 31:3068–3092, 1985

    1985

  18. [18]

    Schumaker and Carlton M

    Bonny L. Schumaker and Carlton M. Caves. New formalism for two-photon quantum optics

  19. [19]

    Mathematical foundation and compact notation.Phys. Rev. A, 31:3093–3111, 1985

    1985

  20. [20]

    Braunstein and Peter van Loock

    Samuel L. Braunstein and Peter van Loock. Quantum information with continuous variables. Rev. Mod. Phys., 77:513–577, 2005

    2005

  21. [21]

    Cerf, Timothy C

    Christian Weedbrook, Stefano Pirandola, Ra´ ul Garc ´ ıa-Patr´ on, Nicolas J. Cerf, Timothy C. Ralph, Jeffrey H. Shapiro, and Seth Lloyd. Gaussian quantum information.Rev. Mod. Phys., 84(2):621, 2012

    2012

  22. [22]

    Entanglement in continuous variable systems: Recent advances and current perspectives.J

    Gerardo Adesso and Fabrizio Illuminati. Entanglement in continuous variable systems: Recent advances and current perspectives.J. Phys. A, 40:7821–7880, 2007

    2007

  23. [23]

    Gerardo Adesso, Sammy Ragy, and Antony R. Lee. Continuous Variable Quantum Informa- tion: Gaussian States and Beyond.Open Syst. Info. Dyn., 21(01n02):1440001, 2014. 34

    2014

  24. [24]

    R. Simon. Peres-Horodecki Separability Criterion for Continuous Variable Systems.Phys. Rev. Lett., 84:2726–2729, 2000

    2000

  25. [25]

    Giedke, J

    Lu-Ming Duan, G. Giedke, J. I. Cirac, and P. Zoller. Inseparability Criterion for Continuous Variable Systems.Phys. Rev. Lett., 84(12):2722, 2000

    2000

  26. [26]

    R. P. Feynman and F. L. Vernon, Jr. The Theory of a general quantum system interacting with a linear dissipative system.Annals Phys., 24:118–173, 1963

    1963

  27. [27]

    B. L. Hu, Juan Pablo Paz, and Yu-hong Zhang. Quantum Brownian motion in a general environment: 1. Exact master equation with nonlocal dissipation and colored noise.Phys. Rev. D, 45:2843–2861, 1992

    1992

  28. [28]

    On decoherence of cosmological perturbations and stochastic inflation

    Jan Weenink and Tomislav Prokopec. On decoherence of cosmological perturbations and stochastic inflation. 8 2011

    2011

  29. [29]

    OUP Oxford, 2002

    Heinz-Peter Breuer and Francesco Petruccione.The theory of open quantum systems. OUP Oxford, 2002

    2002

  30. [30]

    Inflationary spectra and partially decohered distribu- tions.Phys

    David Campo and Renaud Parentani. Inflationary spectra and partially decohered distribu- tions.Phys. Rev. D, 72:045015, 2005

    2005

  31. [31]

    Burgess, R

    Cliff P. Burgess, R. Holman, and D. Hoover. Decoherence of inflationary primordial fluctua- tions.Phys. Rev. D, 77:063534, 2008

    2008

  32. [32]

    Vittorio Gorini, Andrzej Kossakowski, and E. C. G. Sudarshan. Completely Positive Dynam- ical Semigroups of N Level Systems.J. Math. Phys., 17:821, 1976

    1976

  33. [33]

    On the Generators of Quantum Dynamical Semigroups.Commun

    Goran Lindblad. On the Generators of Quantum Dynamical Semigroups.Commun. Math. Phys., 48:119, 1976

    1976

  34. [34]

    Complexity of non-trivial sound speed in inflation.Phys

    Lei-Hua Liu and Ai-Chen Li. Complexity of non-trivial sound speed in inflation.Phys. Dark Univ., 37:101123, 2022

    2022

  35. [35]

    A quantum information method for early universe with non-trivial sound speed.Fortsch

    Shi-Cheng Liu, Lei-Hua Liu, Bichu Li, Hai-Qing Zhang, and Peng-Zhang He. A quantum information method for early universe with non-trivial sound speed.Fortsch. Phys., 74:e70081, 2026

    2026

  36. [36]

    Quantum- information diagnostics of cosmological perturbations with nontrivial sound speed in inflation

    Shi-Cheng Liu, Lei-Hua Liu, Bichu Li, Hai-Qing Zhang, and Peng-Zhang He. Quantum- information diagnostics of cosmological perturbations with nontrivial sound speed in inflation. 4 2026

    2026

  37. [37]

    Cosmological complexity in K- essence.Phys

    Ai-chen Li, Xin-Fei Li, Ding-fang Zeng, and Lei-Hua Liu. Cosmological complexity in K- essence.Phys. Dark Univ., 43:101422, 2024. 35

    2024

  38. [38]

    Cosmological complexity of the modified dispersion relation.Phys

    Tao Li and Lei-Hua Liu. Cosmological complexity of the modified dispersion relation.Phys. Lett. B, 854:138728, 2024

    2024

  39. [39]

    Inflationary Krylov complexity.JHEP, 04:123, 2024

    Tao Li and Lei-Hua Liu. Inflationary Krylov complexity.JHEP, 04:123, 2024

    2024

  40. [40]

    Krylov complexity of thermal state in early universe.Eur

    Tao Li and Lei-Hua Liu. Krylov complexity of thermal state in early universe.Eur. Phys. J. C, 86(3):265, 2026

    2026

  41. [41]

    Krylov Complexity in Early Universe.PTEP, 2026(2):023E04, 2026

    Ke-Hong Zhai and Lei-Hua Liu. Krylov Complexity in Early Universe.PTEP, 2026(2):023E04, 2026

    2026

  42. [42]

    Inflationary power spectrum from the Lanczos algorithm.Eur

    Ke-Hong Zhai, Lei-Hua Liu, and Hai-Qing Zhang. Inflationary power spectrum from the Lanczos algorithm.Eur. Phys. J. C, 85(10):1096, 2025

    2025

  43. [43]

    Parker, Xiangyu Cao, Alexander Avdoshkin, Thomas Scaffidi, and Ehud Altman

    Daniel E. Parker, Xiangyu Cao, Alexander Avdoshkin, Thomas Scaffidi, and Ehud Altman. A Universal Operator Growth Hypothesis.Phys. Rev. X, 9(4):041017, 2019

    2019

  44. [44]

    Krylov complexity in open quantum systems.Phys

    Chang Liu, Haifeng Tang, and Hui Zhai. Krylov complexity in open quantum systems.Phys. Rev. Res., 5(3):033085, 2023

    2023

  45. [45]

    On Krylov complexity in open systems: an approach via bi-Lanczos algorithm.JHEP, 12:066, 2023

    Aranya Bhattacharya, Pratik Nandy, Pingal Pratyush Nath, and Himanshu Sahu. On Krylov complexity in open systems: an approach via bi-Lanczos algorithm.JHEP, 12:066, 2023

    2023

  46. [46]

    Generalized CV Conjecture and Krylov Complexity in Two-Mode Hermitian Systems via Information Geometry.Annals Phys., 491:170534, 2026

    Ke-Hong Zhai, Lei-Hua Liu, and Hai-Qing Zhang. Generalized CV Conjecture and Krylov Complexity in Two-Mode Hermitian Systems via Information Geometry.Annals Phys., 491:170534, 2026

    2026

  47. [47]

    Krylov complexity and Wightman power spectrum with positive chemical potential in Schr¨ odinger field theory

    Peng-Zhang He, Lei-Hua Liu, Hai-Qing Zhang, and Qing-Quan Jiang. Krylov complexity and Wightman power spectrum with positive chemical potential in Schr¨ odinger field theory. JHEP, 02:259, 2026

    2026

  48. [48]

    Starobinsky

    Lei-Hua Liu, Tomislav Prokopec, and Alexei A. Starobinsky. Inflation in an effective gravita- tional model and asymptotic safety.Phys. Rev. D, 98(4):043505, 2018

    2018

  49. [49]

    Detecting multimode entanglement by symplectic uncertainty relations.Phys

    Alessio Serafini. Detecting multimode entanglement by symplectic uncertainty relations.Phys. Rev. Lett., 96:110402, 2006

    2006

  50. [50]

    Gaussian states and operations–a quick reference.arXiv preprint arXiv:2102.05748, 2021

    Jonatan Bohr Brask. Gaussian states and operations–a quick reference.arXiv preprint arXiv:2102.05748, 2021

    Pith/arXiv arXiv 2021

  51. [51]

    On measures of entropy and information

    Alfr´ ed R´ enyi. On measures of entropy and information. InProceedings of the fourth Berkeley symposium on mathematical statistics and probability, volume 1: contributions to the theory of statistics, volume 4, pages 547–562. University of California Press, 1961

    1961

  52. [52]

    Manfredi and M

    G. Manfredi and M. R. Feix. Entropy and Wigner Functions.Phys. Rev. E, 62:4665, 2000. 36

    2000

  53. [53]

    Cambridge university press, 2010

    Michael A Nielsen and Isaac L Chuang.Quantum computation and quantum information. Cambridge university press, 2010

    2010

  54. [54]

    Quantized fields and particle creation in expanding universes

    Leonard Parker. Quantized fields and particle creation in expanding universes. i.Physical Review, 183(5):1057, 1969

    1969

  55. [55]

    Quantized fields and particle creation in expanding universes

    Leonard Parker. Quantized fields and particle creation in expanding universes. ii.Physical Review D, 3(2):346, 1971

    1971

  56. [56]

    Quantum fields in curved space

    Nicholas David Birrell and Paul Charles William Davies. Quantum fields in curved space. 1984

    1984

  57. [57]

    Number 17

    Stephen A Fulling.Aspects of quantum field theory in curved spacetime. Number 17. Cam- bridge university press, 1989

    1989

  58. [58]

    University of Chicago press, 1994

    Robert M Wald.Quantum field theory in curved spacetime and black hole thermodynamics. University of Chicago press, 1994

    1994

  59. [59]

    Cambridge university press, 2009

    Leonard Parker and David Toms.Quantum field theory in curved spacetime: quantized fields and gravity. Cambridge university press, 2009

    2009

  60. [60]

    R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley. Observation of Squeezed States Generated by Four-Wave Mixing in an Optical Cavity.Phys. Rev. Lett., 55:2409–2412, 1985

    1985

  61. [61]

    Generating and detecting short-duration pulses of squeezed light.Physical Review A, 35(8):3586, 1987

    B Yurke, P Grangier, RE Slusher, and MJ Potasek. Generating and detecting short-duration pulses of squeezed light.Physical Review A, 35(8):3586, 1987

    1987

  62. [62]

    Pulsed squeezed light: Simultaneous squeezing of multiple modes.Physical Review A—Atomic, Molecular, and Optical Physics, 73(6):063819, 2006

    Wojciech Wasilewski, Alexander I Lvovsky, Konrad Banaszek, and Czes law Radzewicz. Pulsed squeezed light: Simultaneous squeezing of multiple modes.Physical Review A—Atomic, Molecular, and Optical Physics, 73(6):063819, 2006

    2006

  63. [63]

    Probing multimode squeezing with correlation functions

    Andreas Christ, Kaisa Laiho, Andreas Eckstein, Kati´ uscia N Cassemiro, and Christine Silber- horn. Probing multimode squeezing with correlation functions. InQuantum Information and Measurement, pages QT1B–4. Optica Publishing Group, 2012

    2012

  64. [64]

    Photon temporal modes: a complete framework for quantum information science.Physical Review X, 5(4):041017, 2015

    Benjamin Brecht, Dileep V Reddy, Christine Silberhorn, and Michael G Raymer. Photon temporal modes: a complete framework for quantum information science.Physical Review X, 5(4):041017, 2015

    2015

  65. [65]

    A. I. Lvovsky, W. Wasilewski, and K. Banaszek. Decomposing a pulsed optical parametric amplifier into independent squeezers.J. Mod. Opt., 54:721, 2007

    2007

  66. [66]

    D. B. Horoshko and M. I. Kolobov. Towards single-cycle squeezing in chirped quasi-phase- 37 matched optical parametric down-conversion.Phys. Rev. A, 88:033806, 2013

    2013

  67. [67]

    High-purity pulsed squeez- ing generation with integrated photonics.arXiv preprint arXiv:2007.07387, 2020

    Chaohan Cui, Christos N Gagatsos, Saikat Guha, and Linran Fan. High-purity pulsed squeez- ing generation with integrated photonics.arXiv preprint arXiv:2007.07387, 2020

    Pith/arXiv arXiv 2007

  68. [68]

    Gerald T. Moore. Quantum Theory of the Electromagnetic Field in a Variable-Length One- Dimensional Cavity.J. Math. Phys., 11(9):2679, 1970

    1970

  69. [69]

    V. V. Dodonov. Current status of the dynamical Casimir effect.Phys. Scripta, 82:038105, 2010

    2010

  70. [70]

    Colloquium: Stimulating uncer- tainty: Amplifying the quantum vacuum¡? format?¿ with superconducting circuits.Reviews of Modern Physics, 84(1):1–24, 2012

    PD Nation, JR Johansson, MP Blencowe, and Franco Nori. Colloquium: Stimulating uncer- tainty: Amplifying the quantum vacuum¡? format?¿ with superconducting circuits.Reviews of Modern Physics, 84(1):1–24, 2012

    2012

  71. [71]

    J. R. Johansson, G. Johansson, C. M. Wilson, and Franco Nori. Dynamical Casimir Effect in a Superconducting Coplanar Waveguide.Phys. Rev. Lett., 103:147003, 2009

    2009

  72. [72]

    C. M. Wilson, G. Johansson, A. Pourkabirian, M. Simoen, J. R. Johansson, T. Duty, F. Nori, and P. Delsing. Observation of the dynamical Casimir effect in a superconducting circuit. Nature, 479:376–379, 2011

    2011

  73. [73]

    Baker-campbell-hausdorff relations and unitarity of su (2) and su (1, 1) squeeze operators.Physical Review D, 31(8):1988, 1985

    D Rodney Truax. Baker-campbell-hausdorff relations and unitarity of su (2) and su (1, 1) squeeze operators.Physical Review D, 31(8):1988, 1985

    1988

  74. [74]

    Christoph A. Stephan. The Inverse Seesaw Mechanism in Noncommutative Geometry.Phys. Rev. D, 80:065007, 2009

    2009

  75. [75]

    A conformally coupled massive scalar field in the de sitter expanding universe with the mass term treated as a perturbation.Classical and Quantum Gravity, 26(13):135019, 2009

    Atsushi Higuchi and Lee Yen Cheong. A conformally coupled massive scalar field in the de sitter expanding universe with the mass term treated as a perturbation.Classical and Quantum Gravity, 26(13):135019, 2009

    2009

  76. [76]

    Nonclassical’states in quantum optics: asqueezed’review of the first 75 years

    VV Dodonov. Nonclassical’states in quantum optics: asqueezed’review of the first 75 years. Journal of Optics B: Quantum and Semiclassical Optics, 4(1):R1–R33, 2002

    2002

  77. [77]

    Springer Science & Business Media, 2008

    Daniel F Walls and Gerard J Milburn.Quantum optics. Springer Science & Business Media, 2008

    2008

  78. [78]

    Grishchuk, H

    L. Grishchuk, H. A. Haus, and K. Bergman. Generation of squeezed radiation from vacuum in the cosmos and the laboratory.Phys. Rev. D, 46:1440–1449, 1992

    1992

  79. [79]

    Paulina Marian and Tudor A. Marian. Squeezed states with thermal noise. 1. Photon-number statistics.Phys. Rev. A, 47:4474–4486, 1993

    1993

  80. [80]

    Springer, 1994

    VS Viswanath and Gerhard M¨ uller.The recursion method: application to many-body dynam- 38 ics. Springer, 1994

    1994

Showing first 80 references.