Thomas E. Skinner

The Bloch equation and its variants constitute the fundamental dynamical model for arbitrary two-level systems. Many important processes, including those in more complicated systems, can be modeled and understood through the two-level approximation. It is therefore of widespread relevance, especially as it relates to understanding dissipative processes in current cutting-edge applications of quantum mechanics. Although the Bloch equation has been the subject of considerable analysis in the 70 years since its inception, there is still, perhaps surprisingly, significant work that can be done. This paper extends the scope of previous analyses. It provides a framework for more fully understanding the dynamics of dissipative two-level systems. A solution is derived that is compact, tractable, and completely general, in contrast to previous results. Any solution of the Bloch equation depends on three roots of a cubic polynomial that are crucial to the time dependence of the system. The roots are typically only sketched out qualitatively, with no indication of their dependence on the physical parameters of the problem. Degenerate roots, which modify the solutions, have been ignored altogether. Here the roots are obtained explicitly in terms of a single real-valued root that is expressed as a simple function of the system parameters. For the conventional Bloch equation, a simple graphical representation of this root is presented that makes evident the explicit time dependence of the system for each point in the parameter space. Several intuitive, visual models of system dynamics are developed. A Euclidean coordinate system is identified in which any generalized Bloch equation is separable, i.e., the sum of commuting rotation and relaxation operators. The time evolution in this frame is simply a rotation followed by relaxation at modified rates that play a role similar to the standard longitudinal and transverse rates. These rates are functions of the applied field, which provides information towards control of the dissipative process. The Bloch equation also describes a system of three coupled harmonic oscillators, providing additional perspective on dissipative systems.

The Bloch equation is the fundamental dynamical model applicable to arbitrary two-level systems. Analytical solutions to date are incomplete for a number of reasons that motivate further investigation. The solution obtained here for the propagator, which generates the time evolution of the system and embodies all the system dynamics, is compact and completely general. The parameter space that results in division by zero in previous treatments is explicitly defined and accommodated in the solution. Polynomial roots required for the solution are expressed in terms of a single real root obtained using simple functional forms. A simple graphical rendition of this root is developed that clarifies and characterizes its dependence on the physical parameters of the problem. As a result, the explicit time dependence of the system as a function of its physical parameters is immediately evident. Several intuitive models of system dynamics are also developed. In particular, the Bloch equation is separable in the proper coordinate system, written as the sum of a relaxation operator and either a null operator or a commuting rotation. The propagator thus drives either pure relaxation or relaxation followed by a rotation. The paper provides a basis for increased physical insight into the Bloch equation and its widespread applications.

Description of the transitional process from a static to a dynamic frictional regime is a fundamental problem of modern physics. Previously, we developed a model based on the well-known Frenkel-Kontorova model to describe dry macroscopic friction. Here, this model has been modified to include the effect of dissipation in derived relations between the kinematic and dynamic parameters of a transition process. The main (somewhat counterintuitive) result is a demonstration that the rupture (i.e., detachment front) velocity of the slip pulse which arises during the transition does not depend on friction. The only parameter (besides the elastic and plastic properties of the medium) controlling the rupture velocity is the shear to normal stress ratio. In contrast to the rupture velocity, the slip velocity does depend on friction. The model we have developed describes these processes over a wide range of rupture and slip velocities (up to 7 orders of magnitude) allowing, in particular, the consideration of seismic events ranging from regular earthquakes, with rupture velocities on the order of a few km/s, to slow slip events, with rupture velocities of a few km/day.

Ultra‐high‐field NMR spectroscopy requires an increased bandwidth for heteronuclear decoupling, especially in biomolecular NMR applications. Composite pulse decoupling cannot provide sufficient bandwidth at practical power levels, and adiabatic pulse decoupling with sufficient bandwidth is compromised by sideband artifacts. A novel low‐power, broadband heteronuclear decoupling pulse is presented that generates minimal, ultra‐low sidebands. The pulse was derived using optimal control theory and represents a new generation of decoupling pulses free from the constraints of periodic and cyclic sequences. In comparison to currently available state‐of‐the‐art methods this novel pulse provides greatly improved decoupling performance that satisfies the demands of high‐field biomolecular NMR spectroscopy.

Ultra‐high‐field NMR spectroscopy requires an increased bandwidth for heteronuclear decoupling, especially in biomolecular NMR applications. Composite pulse decoupling cannot provide sufficient bandwidth at practical power levels, and adiabatic pulse decoupling with sufficient bandwidth is compromised by sideband artifacts. A novel low‐power, broadband heteronuclear decoupling pulse is presented that generates minimal, ultra‐low sidebands. The pulse was derived using optimal control theory and represents a new generation of decoupling pulses free from the constraints of periodic and cyclic sequences. In comparison to currently available state‐of‐the‐art methods this novel pulse provides greatly improved decoupling performance that satisfies the demands of high‐field biomolecular NMR spectroscopy.

The dynamics of states representing arbitrary N-level quantum systems, including dissipative systems, can be modelled exactly by the dynamics of classical coupled oscillators. There is a direct one-to-one correspondence between the quantum states and the positions of the oscillators. Quantum coherence, expectation values, and measurement probabilities for system observables can therefore be realized from the corresponding classical states. The time evolution of an N-level system is represented as the rotation of a real state vector in hyperspace, as previously known for density matrix states but generalized here to Schrodinger states. A single rotor in n dimensions is then mapped directly to n oscillators in one physical dimension. The number of oscillators needed to represent N-level systems scales linearly with N for Schrodinger states, in contrast to N^2 for the density matrix formalism. Although the well-known equivalence (SU(2), SO(3) homomorphism) of 2-level quantum dynamics to a rotation in real, physical space cannot be generalized to arbitrary N-level systems, representing quantum dynamics by a system of coupled harmonic oscillators in one physical dimension is general for any N. Values for the classical coupling constants are readily obtained from the system Hamiltonian, allowing construction of classical mechanical systems that can provide visual insight into the dynamics of abstract quantum systems as well as a metric for characterizing the interface between quantum and classical mechanics.