We derive and discuss three different parametrizations of the generator of a dynamical semigroup which describes the Markovian relaxation of a spin j under the influence of isotropic surroundings. The relevant parametrizations that we consider are the strengths of the polarities of the interaction, the relaxation rates of the different multipoles and the transition probabilities per unit time among the Zeeman sublevels. The results are model independent and allow us to derive a set of relations and inequalities for the transition probabilities and for the relaxation rates whose validity is not bound to any specific assumption concerning the mechanisms which govern the relaxation.
We derive and discuss three different parametrizations of the generator of a dynamical semigroup which describes the Markovian relaxation of a spin j under the influence of isotropic surroundings. The relevant parametrizations that we consider are the strengths of the polarities of the interaction, the relaxation rates of the different multipoles and the transition probabilities per unit time among the Zeeman sublevels. The results are model independent and allow us to derive a set of relations and inequalities for the transition probabilities and for the relaxation rates whose validity is not bound to any specific assumption concerning the mechanisms which govern the relaxation.
This paper is devoted to the development of a rigorous theory of angular-momentum relaxation. It is shown that by incorporating quantum-dynamical semigroups in the quantum Markovian master equation, the theory becomes mathematically sound. The expansion of the initial relaxation process in terms of primary relaxation processes of rank l offers a clearer view of the relaxation process structure. A close look at experimental procedure (the system Hilbert space structure, symmetry of the environment, and multipolarities of the system-environment interaction) gives all the information needed to simplify the structure of the relaxation equation and renders it a powerful tool for considering specific physical effects. The quantum Markovian master equation in the state multipole representation is obtained and the physical meaning of the parameters involved is found. The general approach is used for the study of a Zeeman multiplet. The relaxation rates as functions of the spectral densities of correlations (describing the environment properties) are given in an explicit form. Cross relaxation and cross correlation are taken properly into account there. The important case of spin 1 is discussed in more detail. Some useful expressions and relations among the relaxation rates are given in the Appendixes.
Abstract Relaxation rates define an important characteristics of quantum dynamics. They determine how quickly a quantum system interacting with an environment thermalizes, decoheres, and dissipates. Moreover, relaxation rates are also often directly accessible through experimental measurements. An analysis of relaxation rates of Markovian semigroups covariant w.r.t. maximally commuting subgroup of the unitary group is provided. It is shown by a direct proof that these rates satisfy the recently proved a universal constraint that each single rate is upper-bounded by the sum of all rates divided by the dimension of the Hilbert space. Furthermore, we introduce longitudinal and transversal relaxation rates for a general d -level system and derive nontrivial relations between them in the covariant case. These relations generalize the well-known inequality 2Γ T ≥ Γ L between the longitudinal relaxation rate Γ L and the transversal relaxation rate Γ T for two-level systems. Interestingly, for d ≥ 3, our generalization turns out to be much tighter than those valid for general quantum dynamical semigroups, thereby providing a potential means for the experimental verification of covariant generators.
Various aspects of a rather general treatment of proton-spin relaxation in water are discussed within the framework of quantum-dynamical semigroup theory for a four-level system coupled to a reservoir in equilibrium. In particular, the specifications of the infinitesimal generator of time evolution, either in Kossakowski or in Davies form, are worked out in detail. With the help of the Lie algebra of SU(4), the results are used to derive, under suitable simplifications, generalized Bloch equations for the static and alternating-field case. The relevant correlation functions are calculated using conventional approaches but supplemented by taking into account explicitly results from a stochastic model for formation and breaking of hydrogen bridges. A further approximate reduction of the coupled general equations to simpler ordinary Bloch equations leads to an identification of the relevant relaxation times. This approach provides a somewhat different interpretation of rotational correlation times whose numerical values are estimated over a wide temperature range.
We derive algebraic expressions for the atomic multipole moments mk,q(F) describing the steady-state polarization of an atomic ensemble with angular momentum F in a static magnetic field of arbitrary direction and general multipole relaxation rates ${\ensuremath{\Gamma}}_{q}^{(k)}$. The longitudinal moments mk,0 are given in terms of truncated continued fractions, while the transverse moments ${m}_{k,q\ensuremath{\ne}0}$, representing the ensemble coherence, are given by products of truncated continued fractions. The special case of isotropic relaxation leads to particularly simple and elegant expressions. Our results are relevant for all domains of physics that consider the evolution of a spin system interacting with vector-type perturbations.
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.
We present different characterizations of the notion of irreducibility for Quantum Markov Semigroups (QMSs) and investigate its relationship with other relevant features of the dynamics, such as primitivity, positivity improvement and relaxation; in particular, we show that irreducibility, primitivity and relaxation towards a faithful invariant density are equivalent when the semigroup admits an invariant density. Moreover, in the case of uniformly continuous QMSs, we present several useful ways of checking irreducibility in terms of the operators appearing in the generator in GKLS form. Our exposition is as much self-contained as possible, we present some well known results with elementary proofs (collecting all the relevant literature) and we derive new ones. We study both finite and infinite dimensional evolutions and we remark that many results only require the QMS to be made of Schwarz maps.