For a Schrödinger cat state |α⟩+|−α⟩ evolving under thermal Lindblad dynamics in an open bosonic system, we derive a closed-form expression for the time-evolved density matrix ρ(t) via the Glauber-Sudarshan P-representation, reproducing numerical Lindblad evolution at fidelity above 99% across |α|∈[2,3] and β∈[0.5,8]. On this trajectory we compute and compare two quantitatively independent observables. The first is the Wigner negativity W_neg(t), a coherence-amplitude indicator with smooth exponential decay analytic via the Joos-Zeh rate γ_coh = 2γ|α|²(2n̄+1). The second is the integrated autocorrelation time τ_int(ρ(t)), a dynamical-memory indicator extracted via the quantum regression theorem on non-stationary ρ(t) through Liouvillian eigendecomposition and spectral integration. Both observables decay from cat-specific values toward thermal stationary values along the trajectory, but with qualitatively different functional forms and parameter dependence. Across a 16-case parameter scan in (α,β), the half-life ratio tτ/tW has mean 6.03 and coefficient of variation 41%, confirming that τ_int carries independent dynamical-memory information beyond what coherence amplitude captures. The observable τ_int is the trajectory extension of the Wasserstein-Fisher ratio at the center of the stationary dissipation-curvature theorems, computed here in the non-stationary regime via quantum regression. Supplementary topological, information-geometric, and informational indicators (cubical persistence H1 count, dissipation-curvature ratio, purity) are tracked and discussed but do not carry primary contributions of the paper.
For a Schrödinger cat state |α⟩+|−α⟩ evolving under thermal Lindblad dynamics in an open bosonic system, we derive a closed-form expression for the time-evolved density matrix ρ(t) via the Glauber-Sudarshan P-representation, reproducing numerical Lindblad evolution at fidelity above 99% across |α|∈[2,3] and β∈[0.5,8]. On this trajectory we compute and compare two quantitatively independent observables. The first is the Wigner negativity W_neg(t), a coherence-amplitude indicator with smooth exponential decay analytic via the Joos-Zeh rate γ_coh = 2γ|α|²(2n̄+1). The second is the integrated autocorrelation time τ_int(ρ(t)), a dynamical-memory indicator extracted via the quantum regression theorem on non-stationary ρ(t) through Liouvillian eigendecomposition and spectral integration. Both observables decay from cat-specific values toward thermal stationary values along the trajectory, but with qualitatively different functional forms and parameter dependence. Across a 16-case parameter scan in (α,β), the half-life ratio tτ/tW has mean 6.03 and coefficient of variation 41%, confirming that τ_int carries independent dynamical-memory information beyond what coherence amplitude captures. The observable τ_int is the trajectory extension of the Wasserstein-Fisher ratio at the center of the stationary dissipation-curvature theorems, computed here in the non-stationary regime via quantum regression. Supplementary topological, information-geometric, and informational indicators (cubical persistence H1 count, dissipation-curvature ratio, purity) are tracked and discussed but do not carry primary contributions of the paper.
Abstract Understanding thermalization and irreversibility in open quantum systems requires char-acterizing the competition between coherent unitary dynamics and dissipative environmen-tal interactions. In this work, we present a systematic numerical study of thermalization dynamics in a two-qubit open quantum system governed by a completely positive and trace-preserving Lindblad master equation. The coherent evolution rate is controlled by a real scaling parameter α multiplying the Hamiltonian commutator, while decoherence is indepen-dently controlled by a dephasing rate γ . Although such a scaling of coherent dynamics can be absorbed into a rescaling of time in isolated systems, it becomes physically meaningful in open systems where decoherence introduces an absolute timescale. We investigate how the relative strength of coherent evolution and decoherence influences observable dynamics, purity decay, entropy production, and thermalization times. Using robust local and two-site observables, we show that varying α at fixed γ produces distinct transient trajectories, even though all evolutions converge to the same steady state. We further define an operational thermalization time τ ( γ, α ) based on a purity threshold and map the resulting thermalization-time surface across a broad parameter range. Increasing α generally accelerates oscillatory dynamics and modifies the approach to equilibrium, leading to measurable shifts in thermalization times without altering asymptotic states. These findings provide a clear and physically consistent characterization of how Hamil-tonian strength and decoherence rate jointly determine thermalization behavior in open quantum systems. The framework offers a practical parameterization for numerical mod-eling and experimental studies of coherent–dissipative competition in quantum information processing, quantum control, and open-system dynamics.
The Schrödinger equation describes a quantum system that does not interact with anything outside itself. The participation measure evolves unitarily; the norm is preserved; coherent superpositions are maintained indefinitely. This is a useful idealization but it is not what real quantum systems do. Every real quantum system is in contact with its environment — thermal photons, lattice vibrations, stray fields, gas‑phase collisions, observation by experimenters. Coherent superpositions decay; populations relax toward thermal equilibrium; entanglement with the environment proliferates. The closed‑system Schrödinger equation cannot describe these phenomena because, by construction, it has no environment. In 1976, Lindblad — and independently Gorini, Kossakowski, and Sudarshan — proved that the most general Markovian dynamics for a quantum density operator that preserves Hermiticity, trace, and complete positivity has a specific functional form: ∂ₜ ρ̂ = −(i/ℏ)[Ĥ, ρ̂] + ∑_α ( L̂_α ρ̂ L̂_α† − ½{ L̂_α† L̂_α , ρ̂ } ) The first term is the von Neumann equation. The sum is the “dissipator” — jump operators L̂_α mediating environmental coupling. This is the master equation of decoherence theory: qubits coupled to reservoirs, atoms in optical cavities, mesoscopic conductors with electron‑phonon coupling, open‑system everything. The standard derivation traces out the environment and applies the Born–Markov approximation. It works empirically but leaves open: why is Markov valid? Why is Born valid? What is the substrate‑level meaning of the jump operators? This walkthrough shows how the Event Density framework forces the Lindblad form at substrate level rather than deriving it from approximations. The density‑operator description is forced by environmental phase‑independence, derived from substrate primitives, rather than postulated as the natural object after partial‑tracing. Between commitment events, the density operator evolves unitarily — U3 in density‑operator form. The jump operators are the substrate‑level representation of Primitive 11 commitment events: discrete, channel‑selective updates triggered when environmental coupling drives the chain’s individuation. The Markov property is forced by P11’s locality (commitment events do not carry long‑range memory). The anti‑commutator normalization is forced by P04 (bandwidth conservation) plus trace preservation. Complete positivity is automatic from the projector structure of commitment events. The structural payoff: decoherence is not approximate physics derived from a Born–Markov simplification of a more fundamental closed‑system theory. It is what U3 plus substrate primitives produces directly when the chain is non‑isolated. The Lindblad form is forced rather than approximate. Complete positivity, trace preservation, and the Markovian property — usually justified by appeals to physical reasonableness — are forced consequences of bandwidth conservation, environmental phase‑independence, and commitment‑event locality respectively.
Using the path integral representation of the density matrix propagator of quantum Brownian motion, we derive its asymptotic form for times greater than the so-called localization time $(\ensuremath{\Elzxh}/\ensuremath{\gamma}{kT)}^{1/2},$ where \ensuremath{\gamma} is the dissipation and T the temperature of the thermal environment. The localization time is typically greater than the decoherence time, but much shorter than the relaxation time ${\ensuremath{\gamma}}^{\ensuremath{-}1}.$ We use this result to show that the reduced density operator rapidly evolves into a state which is approximately diagonal in a set of generalized coherent states. We thus reproduce, using a completely different method, a result we previously obtained using the quantum state diffusion picture [Phys. Rev. D 52, 7294 (1995)]. We also go beyond this earlier result, in that we derive an explicit expression for the weighting of each phase space localized state in the approximately diagonal density matrix, as a function of the initial state. For sufficiently long times it is equal to the Wigner function, and we confirm that the Wigner function is positive for times greater than the localization time (multiplied by a number of order 1).
Part VII of the Temporal Information Dynamics (TID) series develops the full quantum-information layer of the Scalar Temporal Field Ontology (STFO), where time is a physical scalar field τ(x) and information evolves along its integral curves. This work extends Parts I–VI by deriving the complete quantum dynamics implied by τ-geometry: the τ-modified Schrödinger equation governing intrinsic temporal evolution, the temporal decoherence kernel generated by fluctuations δτ, a covariant τ-Lindblad master equation for open quantum systems, directional entanglement flow induced by ∇μτ, τ-dependent deformations of the quantum Fisher metric, modified quantum-channel capacities, coherence lengths, and error exponents, τ-corrected propagators, Green’s functions, and canonical commutators, and a fully geometric τ-weighted path-integral formulation. Across all levels—Hamiltonian, thermodynamic, geometric, and quantum—information flow becomes a direct probe of τ-gradients and temporal curvature. The resulting framework is testable in atom interferometry, optical and microwave clock networks, spin-echo experiments, and quantum computing hardware. This paper completes the quantum-information pillar of STFO and unifies classical information flow, thermodynamic irreversibility, geometric curvature, and quantum coherence into one continuous, falsifiable theory of temporal dynamics.
The Lindblad equation is commonly used to model dissipation and decoherence in open quantum systems. Yet many of the properties of Lindblad dynamics are still little understood. This thesis explores various aspects of the Lindblad equation and its connections with non-Hermitian quantum theories. The evolution speed of Lindblad dynamics is investigated and an explicit expression for the speed is derived. The radial component, connected to the rate at which the purity of the state changes, is shown to be determined by the modified skew information of the Lindblad operators. The semiclassical time evolution of Gaussian states under Lindblad dynamics is then considered and a new type of phase-space dynamics is derived for the centre of a Gaussian Wigner function. By viewing the phase-space Lindblad equation as a Schroedinger equation with a non-Hermitian Hamiltonian, a further set of semiclassical equations are derived. These can describe the interference terms in Wigner functions. Further, a PT-symmetric two-level system driven through two consecutive exceptional points at finite speed is considered and an analytical expression for the probability of a non-adiabatic transition is derived. It is shown that the transition through the exceptional points can be experimentally addressed in a PT-symmetric lattice using Bloch oscillations. Finally, a quantum implementation of this PT-symmetric system using ultracold atoms in an optical lattice with single-particle losses is investigated.
Based on open quantum systems and non-equilibrium field dynamics, a unified physical theory connecting cosmic evolution and microscopic quantum decoherence is established. Two core axioms are proposed: the non-equilibrium field difference axiom provides the external rigid condition for the emergence of ordered structures, and the quantum intrinsic upward chemotaxis axiom describes the internal driving force for systems to evolve toward higher information integration and complexity. Using an ultra-low temperature global field and a local ordered microenvironment as an analog model, the underlying isomorphic logic of cosmic and life evolution is revealed. The theory is embedded into the open quantum system framework, and the intrinsic chemotaxis correction is introduced into the standard Lindblad master equation to construct a full quantum master equation at the density matrix level. The time-dependent effective decoherence rate Γₑբբ(t)=Γ−βαe⁻ᵅᵗ is rigorously derived, and the analytical solution of coherence decay C(t)=exp(−Γt+β(1−e⁻ᵅᵗ)) is obtained. The theory predicts that quantum coherence exhibits non-exponential stretched decay under ultra-low temperature and low noise conditions, which is falsifiably different from the mainstream pure exponential decay. The physical definitions and theoretical constraints of parameters β and α are clarified, and a semi-quantitative magnitude estimate is given (optimistic upper bound β≲10⁻⁶, to be determined experimentally). Four groups of controlled experiments are designed based on platforms such as superconducting qubits and ion traps, and four quantitative test criteria are proposed to evaluate the experimental feasibility. This theory defines the second law of thermodynamics as an effective law for macroscopic isolated systems, establishes intrinsic chemotaxis as the core driving force for directed evolution, and forms a complete logical closed-loop from unified physical framework and cosmic evolution model to microscopic experimental predictions.