We analyze the angular dependence of the purity-loss rate in a restricted class of open quantum systems: a qubit coupled to a thermal bosonic bath through a commuting interaction operator. Within the Born–Markov approximation, we derive the dissipative generator in closed form andestablish four results. First, the restricted induced trace norm of the generator (on the Hermitian traceless subspace) isindependent of the coupling angle, ‖ℒ(𝐷)𝜃 ‖1→1,ℬ0 = 4𝜋𝜂𝑘𝐵 𝑇,and we give a closed-form proof of this fact based on the rotational structure of the coupling family, together with an independent numerical cross-check.Second, the initial purity-loss rate of a pure reference state depends on the coupling angle through the geometric quantity Γ(𝜃) = |⟨𝜓|𝐴(𝜃)|𝜓⟩|2, which for the qubit family 𝐴(𝜃) = cos 𝜃 𝜎𝑥 + sin 𝜃 𝜎𝑧 takes the exact form Γ(𝜃) = (𝑒𝑥 cos 𝜃 + 𝑒𝑧 sin 𝜃)2. Third, the local curvature of Γ(𝜃) at its maximum defines a curvature-derived angular scale 𝜎geo = 1/√2, universal within this coupling family and independent of every physical parameter of the model; the two forms agree to within ∼ 5% at |𝜃 − 𝜇| = 𝜎geo (Appendix E). Fourth, we quantify explicitly the algebraic (non-exponential) decay of the Markov tail integral for the Ohmic–Lorentzian bath considered here, and we show that a slowly time-dependent coupling angle, 𝜃(𝑡), admits a well-defined adiabatic-Markovian generator; we then prove that the resulting purity-loss functional is not proportional, in general, to the trajectory-integrated Frobenius-norm witness of Ref. [1], identifying the specific obstructions (coherent-precession contamination and a linear-versus-quadratic mismatch in Bloch-vector language) responsible, and we pose as an open problem the narrower question of whether any restricted or reformulated version of the witness admits a well-defined relation to the present results. These results provide an exact microscopic benchmark for trajectory-dependent effective decoherence models: the angular dependence originates from the geometry of the system–environment coupling and the evolving state, not from an intrinsic angular preference of the bath, while a full reduction of the phenomenological model of Ref. [1] to this microscopic setting remains open.
We analyze the angular dependence of the purity-loss rate in a restricted class of open quantum systems: a qubit coupled to a thermal bosonic bath through a commuting interaction operator. Within the Born–Markov approximation, we derive the dissipative generator in closed form andestablish four results. First, the restricted induced trace norm of the generator (on the Hermitian traceless subspace) isindependent of the coupling angle, ‖ℒ(𝐷)𝜃 ‖1→1,ℬ0 = 4𝜋𝜂𝑘𝐵 𝑇,and we give a closed-form proof of this fact based on the rotational structure of the coupling family, together with an independent numerical cross-check.Second, the initial purity-loss rate of a pure reference state depends on the coupling angle through the geometric quantity Γ(𝜃) = |⟨𝜓|𝐴(𝜃)|𝜓⟩|2, which for the qubit family 𝐴(𝜃) = cos 𝜃 𝜎𝑥 + sin 𝜃 𝜎𝑧 takes the exact form Γ(𝜃) = (𝑒𝑥 cos 𝜃 + 𝑒𝑧 sin 𝜃)2. Third, the local curvature of Γ(𝜃) at its maximum defines a curvature-derived angular scale 𝜎geo = 1/√2, universal within this coupling family and independent of every physical parameter of the model; the two forms agree to within ∼ 5% at |𝜃 − 𝜇| = 𝜎geo (Appendix E). Fourth, we quantify explicitly the algebraic (non-exponential) decay of the Markov tail integral for the Ohmic–Lorentzian bath considered here, and we show that a slowly time-dependent coupling angle, 𝜃(𝑡), admits a well-defined adiabatic-Markovian generator; we then prove that the resulting purity-loss functional is not proportional, in general, to the trajectory-integrated Frobenius-norm witness of Ref. [1], identifying the specific obstructions (coherent-precession contamination and a linear-versus-quadratic mismatch in Bloch-vector language) responsible, and we pose as an open problem the narrower question of whether any restricted or reformulated version of the witness admits a well-defined relation to the present results. These results provide an exact microscopic benchmark for trajectory-dependent effective decoherence models: the angular dependence originates from the geometry of the system–environment coupling and the evolving state, not from an intrinsic angular preference of the bath, while a full reduction of the phenomenological model of Ref. [1] to this microscopic setting remains open.
We explore how the dissipative geometric phase evolves within the spin-boson model, focusing specifically on coupling to an Ohmic bath in the weakly coherent regime. To determine the non-unitary time evolution of the system's reduced density matrix, we employ the non-interacting blip approximation (NIBA). For the localized pure initial state used throughout, we derive a compact Bloch-sphere expression showing that the mixed-state geometric phase is a weighted azimuthal winding of the dissipative trajectory. We then map geometric-phase accumulation across various system-bath coupling strengths, temperatures, static biases, and bath cutoff frequencies. Benchmarking representative results against the numerically exact time-evolving matrix product operator (TEMPO) technique shows that NIBA reproduces the population dynamics almost indistinguishably. It also captures the geometric phase quantitatively and qualitatively, although TEMPO reveals a clearer trend in the stationary coherence. Our results show that quantum dissipation suppresses the geometric phase through two complementary mechanisms: thermal noise reduces the state's purity, while static bias localizes the dynamics and reduces the accessible geometric area.
The theory of open quantum systems represents one of the major developments of quantum mechanics over the last 50 years. Its central object of study is constituted by quantum systems that cannot be regarded as fully isolated, but instead interact with additional degrees of freedom, giving rise to irreversible phenomena such as dissipation and decoherence. The tools provided by this framework are nowadays of paramount importance across many active areas of research, including quantum computation, thermodynamics, metrology and biology. In this thesis, we investigate three main topics within the field. The first contribution concerns the microscopic derivation of the evolution equations, socalled quantum master equations, for open systems which are also subject to external driving fields. Standard techniques developed for time-independent Hamiltonians encounter several difficulties in the time-dependent case, largely restricting their applicability to specific instances, such as periodic or adiabatic drivings. In contrast, a core result of the derivation presented in the thesis is that, when the environment is sufficiently Markovian (essentially, memoryless), the non-adiabatic contributions to the driving field can be neglected in a controlled fashion, yielding an accurate master equation with time-dependent generator, capable of correctly reproducing the open dynamics even for strong drivings. This result is further supported by benchmarks against numerically exact simulations. The second contribution addresses the long-time behavior of general types of (Markovian) quantum master equations, including those of the form introduced above. While this problem has been extensively analyzed in the time-independent setting, virtually no results are known for time-dependent generators. In particular, we establish the foundations of a general theory of relaxation, that is, convergence towards an asymptotic trajectory in the long-time limit, independently of the initial state. The main results include a theorem guaranteeing sufficient conditions for relaxation and the derivation of a rate function governing the convergence to the asymptotic trajectory. The analysis is further extended to a simple non-Markovian case. Moreover, we treat in detail the specific master equation derived in the first part, connecting the aforementioned relaxation property directly to the underlying microscopic dynamics. The third contribution investigates chirality-induced spin selectivity (CISS) effects from the perspective of open quantum systems. Although the majority of theoretical studies aiming to model CISS are based on a coherent description of the spin dynamics, such effects have been observed also for electrons subject to incoherent transport across molecular networks. In this regard, we provide a rigorous ab initio derivation of the microscopic Hamiltonian describing the electronic transport, incorporating electron spin effects and contributions stemming from the nuclear vibrations. Within this result, we critically revise a master equation approach aiming to explain the origin of CISS effects observed in photosynthetic complexes. Taken together, these contributions offer fresh insights and original results on the interplay between driving fields and dissipation, relaxation phenomena, and electronic spin transport in the presence of incoherent dynamics. More broadly, these results provide new theoretical tools with several potential applications in quantum technologies, from modeling of control pulses in noisy environments to quantum reservoir engineering, and contribute to a deeper understanding of the role of quantum effects in biological systems.
Complementarity relations constrain the distribution of coherence, predictability, and openness in quantum systems. Here we show that, in open quantum systems, these local constraints acquire a geometric interpretation through quasistatic transport. For a driven dissipative qubit, the complementarity variables define cylindrical coordinates on the Bloch sphere, while openness appears geometrically as a radial deficit associated with reduction from a larger Hilbert space. Quasistatic driving induces a work connection on the resulting steady-state manifold whose curvature determines the cyclic response. Hamiltonian-aligned dissipation produces an exact work connection and vanishing cyclic work, whereas fixed pointer-basis dissipation generates non-integrable transport, finite curvature, and holonomic response. The resulting curvature admits a phase-resolved representation on the triality manifold and develops perturbatively with pointer--Hamiltonian mismatch. In the weak-mismatch limit, the curvature is governed by a competition between coherence-preserving and pure-dephasing channels, producing symmetry-related positive- and negative-curvature sectors. These results establish a direct connection between complementarity, dissipation, and geometric thermodynamic response, and show that cyclic quasistatic work provides an operational probe of nonequilibrium quantum geometry.
Nonassociative deformations of phase-space structures arise naturally in the presence of magnetic charge, where the Jacobi identity for momentum components fails and the corresponding Moyal product becomes nonassociative. While such structures are well understood at the level of single-particle kinematics, their implications for open-system quantum dynamics remain largely unexplored. Here we derive a Born-Markov master equation for a system coupled to a bath when the underlying operator product is weakly nonassociative. The deformation enters through associators appearing in the second-order kernel, while pairwise operator products and dissipators retain their standard form. The resulting correction is dispersive and modifies the Liouville-von Neumann part of the generator without introducing additional dissipative channels. We then embed this structure into a two-qubit transverse-field Ising model using a Stratonovich-Weyl representation and an Ising-aligned twisted Poisson structure. In the zero-temperature limit, the nonassociative terms produce a nonlinear correction in which the instantaneous population imbalance of each qubit feeds back into the dynamics as a state-dependent longitudinal field. Numerical simulations in the weak-coupling regime, where the Born-Markov derivation is quantitatively controlled, show that increasing the nonassociativity parameter suppresses steady-state entanglement by up to 59%, reduces purity, and increases entropy, while leaving the relaxation timescale set by the dissipative rates unchanged. These results demonstrate that weak nonassociativity manifests as a coherent, population-dependent deformation of open-system dynamics rather than an additional dissipative mechanism.
We develop a Lie-Jordan geometric formulation of finite-dimensional open quantum dynamics, building on the algebraic framework introduced in our previous work (arXiv:2606.26477). The Hilbert-Schmidt operator space is endowed with an orthonormal Hermitian basis, in which the commutator and anticommutator are encoded by the structure tensors Cμνλ and Bμνλ. Within this formulation, the von Neumann and Gorini-Kossakowski-Lindblad-Sudarshan equations admit a direct component representation. Our central result is the identification of a basis-independent universal trilinear dissipative map, D(X,Y)Z=XZY−21{YX,Z}, whose components define a universal operator-space tensor depending only on the Lie-Jordan structure tensors. The physical dissipator is obtained by contracting this tensor with the expansion coefficients of the Lindblad operators and the Kossakowski matrix, thereby separating the universal algebraic structure from the model-dependent physical information. We further show that the combinations (B+C) and (B−C) generate internal left and right transports, allowing the elementary dissipative map to be expressed as a left-right bimodule action corrected by an ordered Jordan contribution. The universal map satisfies basis-independent trace and Hermitian-conjugation identities, from which trace preservation, Hermiticity preservation of the complete dissipator, and the reality of its component representation in a Hermitian Hilbert-Schmidt basis follow. We also derive its Hilbert-Schmidt adjoint and illustrate the formalism for a qubit with pure dephasing and amplitude-damping channels. This construction provides a tensorial and affine-geometric interpretation of the universal superoperator structure derived in arXiv:2606.26477 while keeping the algebraic and model-dependent sectors explicitly separated.
We propose that the quantum-jump (decay) rate of an open quantum system can depend on the relative phase δ = θ − φ between the system coherence and a reference oscillation, taking the form Γ(δ) = Γ₀ exp(α cos δ) ≈ Γ₀[1 + α cos δ]. The resulting distribution of jump phases is a von Mises distribution whose first Fourier coefficient is M₁ = I₁(α)/I₀(α) ≈ α/2. Standard Lindblad theory employs a constant rate γ₀; this work replaces it with a phase-dependent rate inside the Lindblad dissipator. The operational definitions are explicit: θ = arg ρ_eg is the phase of the system coherence in a fixed reference frame, and φ is the phase of a classical reference oscillation (a drive tone or a phase-lockable bath mode). The relative phase δ is therefore defined only when such a reference exists — pure Fock-state (photon or phonon number) jumps carry no phase information and cannot test the hypothesis. A two-dimensional metric on the torus (θ, φ), g = [[1, −cos δ], [−cos δ, 1]], is offered solely as geometric motivation for the cos δ form. Its Hessian is singular at δ = nπ, but this singularity does not by itself imply an enhanced decay rate, and it cannot select the sign of α. The step from "Legendre failure" to "rate modulation" is an explicit postulate, not a derivation. This work does not claim to solve the measurement problem. A reproducible injection–recovery test with null controls (4000 trajectories per replicate, seed 2026) recovers injected values of α to within statistical error. For the linear law, the estimator α̂ = 2⟨cos δ⟩ is exactly unbiased with standard error √(2/N); for the exponential (von Mises) law, a maximum-likelihood estimator is obtained by solving M₁ = I₁(α)/I₀(α). Under the null (α = 0), the maximum |M₁| observed across 1500 independent replicates is 0.046, with 95th and 99th percentiles of 0.027 and 0.035. Power analysis indicates that a 3σ detection requires N ≈ 2(3/α)² jumps and a 5σ detection requires N ≈ 2(5/α)² jumps. These simulations validate only the statistical estimator: because α was injected by construction, they are not evidence for the physical hypothesis. The manuscript states the platform requirements for a real test (a driven qubit or oscillator with an explicit classical reference phase, measurable system phase from continuous I/Q records, and trajectory-resolved jump detection), the pre-registration requirements, the null controls, and the rival explanations that must be excluded (phase-dependent detection efficiency, ordinary drive-phase dependence, population-dependent rates, and intrinsic heterodyne detector phase response). No experimental result is claimed. The link between the singular metric and the rate modulation is postulated rather than derived; the phases θ and φ must be tied to a concrete Hamiltonian and reference in each physical system; and the sign of α remains a free parameter of the hypothesis. Keywords: open quantum systems; quantum jumps; phase-dependent decay; von Mises distribution; Lindblad equation; quantum trajectories; injection–recovery; null controls; estimator validation