This paper studies the evolution of non-Gaussian phase fluctuations in stochastic and quantum systems. We construct an observable based on second and fourth cumulants and show that it approaches a constant value under broad physical conditions.The main result is the emergence of a universal invariant:Y(t) = σ⁴(t) / (κ(t) · t)We show that this quantity converges to a constant λ under short-memory, finite-cumulant dynamics. This connects stochastic processes, decoherence physics, and renormalization flow into a unified structure. In many physical systems, noise is commonly approximated as Gaussian. This assumption simplifies analysis but hides higher-order statistical structure.However, in real quantum systems (e.g., superconducting qubits, optical interferometers, spin environments), deviations from Gaussianity are observed due to:environmental memory effectsrare event contributionsinteraction-induced correlationsThe question addressed here is:Does non-Gaussianity decay in a universal way under time evolution?To answer this, we define a measurable invariant based on cumulants.
This paper studies the evolution of non-Gaussian phase fluctuations in stochastic and quantum systems. We construct an observable based on second and fourth cumulants and show that it approaches a constant value under broad physical conditions.The main result is the emergence of a universal invariant:Y(t) = σ⁴(t) / (κ(t) · t)We show that this quantity converges to a constant λ under short-memory, finite-cumulant dynamics. This connects stochastic processes, decoherence physics, and renormalization flow into a unified structure. In many physical systems, noise is commonly approximated as Gaussian. This assumption simplifies analysis but hides higher-order statistical structure.However, in real quantum systems (e.g., superconducting qubits, optical interferometers, spin environments), deviations from Gaussianity are observed due to:environmental memory effectsrare event contributionsinteraction-induced correlationsThe question addressed here is:Does non-Gaussianity decay in a universal way under time evolution?To answer this, we define a measurable invariant based on cumulants.
This paper studies the evolution of non-Gaussian phase fluctuations in stochastic and quantum systems. We construct an observable based on second and fourth cumulants and show that it approaches a constant value under broad physical conditions.The main result is the emergence of a universal invariant:Y(t) = σ⁴(t) / (κ(t) · t)We show that this quantity converges to a constant λ under short-memory, finite-cumulant dynamics. This connects stochastic processes, decoherence physics, and renormalization flow into a unified structure. In many physical systems, noise is commonly approximated as Gaussian. This assumption simplifies analysis but hides higher-order statistical structure.However, in real quantum systems (e.g., superconducting qubits, optical interferometers, spin environments), deviations from Gaussianity are observed due to:environmental memory effectsrare event contributionsinteraction-induced correlationsThe question addressed here is:Does non-Gaussianity decay in a universal way under time evolution?To answer this, we define a measurable invariant based on cumulants.
Abstract Understanding universal behavior of far-from-equilibrium transport dynamics at a quantum many-body level is a longstanding challenge. A full characterization of universal dynamics of nonlocal correlation functions still remains largely unknown. In this letter, we uncover universal scaling laws of higher-order cumulants in one-dimensional isotropic Heisenberg model, revealing anomalous behaviors of nonequilibrium dynamics exclusively accessible in higher-order correlation functions. Using numerical simulations and full counting statistics, we determine the power laws of both the spin polarization transfer and contrast cumulants for different kinds of helix and domain-wall initial states. Building on such physical states, we unify the scaling behavior of the higher-order cumulants as two types of dynamics: anomalous diffusive and superdiffusive. For the former, these higher cumulants show a deviation from Gaussian statistics, with the scaling exponents being identical for the first four orders. For the latter, however, we observe a breakdown of KPZ universality, with the exponents of the third and fourth orders differing significantly from those of the first two. Our results are also agreeable with recent experimental observations, advancing understanding of far-from-equilibrium transport phenomena.
This work investigates the structural origin of non‑Gaussian quantum noise by introducing two universal degrees of freedom: spread and contraction. These two components appear broadly in quantum optics, measurement theory, and statistical physics, and they characterize how a distribution acquires smoothness, nonlocality, sharpness, or irreversibility. Using a minimal 1+1 dimensional dynamical model, we show that the competition between spread and contraction determines the shape of the distribution intensity p(x). When the ratio between the two crosses a critical value, the system undergoes a continuous yet abrupt transition through three regimes: Gaussian → non‑Gaussian → two‑component. The model reproduces key signatures of non‑Gaussianity—such as changes in kurtosis, skewness, and the emergence of two‑component spectra—without introducing additional parameters. This framework provides a unified structural perspective on phenomena including the disappearance of interference fringes, sudden contraction in weak measurements, and non‑Gaussian noise observed in optical state tomography and decoherence experiments. The spread–contraction plane serves as a general classification map for understanding how non‑Gaussian noise arises from the balance between smoothing and localization mechanisms, independent of any specific physical theory.
This preprint introduces the Kaan Invariant, a compact unifying structure underlying decoherence, dissipation, and evolution in open quantum systems. Across a wide range of physical platforms — including superconducting qubits, NV centers, trapped ions, and Lindbladian models — the characteristic decoherence time tau_dec is empirically observed to scale inversely with an effective environmental coupling. This note formalises that pattern through a dimensionless relation: tau_dec * ||G_env|| ≈ c where G_env is the noise/environment generator norm and c is a constant of order unity. The paper shows how this form unifies quasi-static dephasing, Markovian master equations, the Caldeira–Leggett model, spin–boson dynamics, filter-function noise analysis, dynamical decoupling limits, and geometric quantum speed limits. The goal is conceptual clarity: many disparate results share the same underlying time–generator structure, and the invariant provides a single organising principle connecting coherence times, control strengths, and viability conditions in quantum technologies. This document forms Part I of the Kaan Invariant series.
This record contains the manuscript “Canonical Invariants, Quantum Noise, and Memory in Gaussian Open Systems.” We develop an invariant-based formulation of Gaussian open-system dynamics for a time-dependent quantum harmonic oscillator subject to dissipation and noise. The construction combines canonical invariants with stochastic and retarded-mode descriptions, allowing Markovian thermal relaxation and finite-memory non-Markovian dynamics to be treated within a common framework. A central result is that, away from equilibrium, the instantaneous energy is not determined by the entropy alone. For local relaxation, the energy admits a differential state representation of the form[E=E!\left(S,\dot S;\frac{\omega(t)}{\nu_B(t)},\gamma_{\rm eff}(t)\right),]so that the entropy rate (\dot S) carries essential nonequilibrium information. Finite-memory dynamics enlarges the required state data while preserving this basic structure. The formulation also makes explicit the consistency relation between dissipation and the asymmetric quantum-noise spectrum, connects the Markovian limit with thermal GKLS dynamics, and provides a finite-dimensional localization of colored-noise memory kernels. The framework is intended as a basis for studying driven Gaussian open systems and for possible extensions to quantum-field and electromagnetic settings. This version has been submitted to Physical Review Research.
Gaussian models provide an excellent effective description of many quantum many-body systems ranging from condensed-matter systems1,2 all the way to neutron stars3. Gaussian states are common at equilibrium when the interactions are weak. Recently it was proposed that they can also emerge dynamically from a non-Gaussian initial state evolving under non-interacting dynamics4,5,6,7,8,9,10,11. Here we present the experimental observation of such a dynamical emergence of Gaussian correlations in a quantum many-body system. This non-equilibrium evolution is triggered by abruptly switching off the effective interaction between the observed collective degrees of freedom, while leaving the interactions between the microscopic constituents unchanged. Starting from highly non-Gaussian correlations, consistent with the sine–Gordon model12, we observe a Gaussian state emerging over time as revealed by the decay of the fourth- and sixth-order connected correlations in the quantum field. A description of this dynamics requires a novel mechanism for the emergence of Gaussian correlations, which is relevant for a wide class of quantum many-body systems. In our closed system with non-interacting effective degrees of freedom, we do not expect full thermalization13,14,15,16,17,18,19. This memory of the initial state is confirmed by observing recurrences20 of non-Gaussian correlations.