``Dynamical driving'' means driving a system by another dynamical system, in contrast to driving by an external force (which acts on the system but is not acted upon by the system). The model consists of a localized spin driven by a polarized beam of spins. A remarkably complex variety of behaviors is possible as a result of the competition between the aligning effect of interaction with the beam and the precession due to the external magnetic field. The state of the localized spin can evolve from pure to mixed, or from mixed to pure, and its motion is characterized by an attractor. The final state of the driven spin is usually independent of its initial state, so the model provides an example of quantum-state preparation. These effects cannot be produced by a prescribed external force.
``Dynamical driving'' means driving a system by another dynamical system, in contrast to driving by an external force (which acts on the system but is not acted upon by the system). The model consists of a localized spin driven by a polarized beam of spins. A remarkably complex variety of behaviors is possible as a result of the competition between the aligning effect of interaction with the beam and the precession due to the external magnetic field. The state of the localized spin can evolve from pure to mixed, or from mixed to pure, and its motion is characterized by an attractor. The final state of the driven spin is usually independent of its initial state, so the model provides an example of quantum-state preparation. These effects cannot be produced by a prescribed external force.
We study the dynamics of a particle in continuous time and space, the displacement of which is governed by an internal degree of freedom (spin). In one definite limit, the so-called quantum random walk is recovered but, although quite simple, the model possesses a rich variety of dynamics and goes far beyond this problem. Generally speaking, our framework can describe the motion of an electron in a magnetic sea near the Fermi level when linearization of the dispersion law is possible, coupled to a transverse magnetic field. Quite unexpected behaviours are obtained. In particular, we find that when the initial wave packet is fully localized in space, the J z angular momentum component is frozen; this is an interesting example of an observable which, although it is not a constant of motion, has a constant expectation value. For a non-completely localized wave packet, the effect still occurs although less pronounced, and the spin keeps for ever memory of its initial state. Generally speaking, as time goes on, the spatial density profile looks rather complex, as a consequence of the competition between drift and precession, and displays various shapes according to the ratio between the Larmor period and the characteristic time of flight. The density profile gradually changes from a multimodal quickly moving distribution when the scattering rate is small, to a unimodal standing but flattening distribution in the opposite case.
I will discuss the spin dynamics of two magnetic atoms trapped in an external trap. Assuming that initially the atoms are in their orbital ground state I will discuss in details all possible elementary processes of spin flipping. Dipole-dipole interactions couple spin degrees of freedom of the two particles to their quantized orbital motion. Because of conservation of the total angular momentum if a magnetization of the system changes the system acquires an orbital angular momentum and atoms circulate around the center of the trap. I show that spin dynamics driven by dipolar interactions is possible only when the energies of the initial and the final sates are equal. In a case of weak magnetic moments of atoms this condition can be fulfilled utilizing a resonant external magnetic field, which tunes energies of involved states via the linear Zeeman effect. I will illustrate the resonant dynamics comparing the theoretical predictions to some results of experiments on Chromium atoms.
We study the dynamics of a particle in continuous time and space, the displacement of which is governed by an internal degree of freedom (spin). In one definite limit, the so-called quantum random walk is recovered but, although quite simple, the model possesses a rich variety of dynamics and goes far beyond this problem. Generally speaking, our framework can describe the motion of an electron in a magnetic sea near the Fermi level when linearisation of the dispersion law is possible, coupled to a transverse magnetic field. Quite unexpected behaviours are obtained. In particular, we find that when the initial wave packet is fully localized in space, the Jz angular momentum component is frozen; this is an interesting example of an observable which, although it is not a constant of motion, has a constant expectation value. For a non-completely localized wave packet, the effect still occurs although less pronounced, and the spin keeps for ever memory of its initial state. Generally speaking, as time goes on, the spatial density profile looks rather complex, as a consequence of the competition between drift and precession, and displays various shapes according to the ratio between the Larmor period and the characteristic time of flight. The density profile gradually changes from a multimodal quickly moving distribution when the scatttering rate is small, to a unimodal standing but flattening distribution in the opposite cas case.
A classical spin, dynamically driven by a polarized beam, is shown to exhibit a complex variety of behaviors, incuding stable fixed points, bifurcations, limit cycles, and chaos. In the classical (large-spin) limit of the corresponding quantum model, the probability distributions for the spin components converge to their classical counterparts, even though the classical motions are qualitatively different for different ranges of parameters. However, the nature of the quantum-to-classical limit has some unexpected features, which are discussed.
Classical and quantum models of a localized spin driven by a polarized-spin beam have been shown to exhibit a rich variety of dynamical behaviors. In the classical model these include stable limit points, limit cycles, and chaos; however, in the quantum model the density matrix of the localized spin generally approaches a steady state. This paradox is resolved by considering the autocorrelation functions of the spin components, which reveal the expected quantum-classical correspondence. The noncommuting of the limits j\ensuremath{\rightarrow}\ensuremath{\infty} and t\ensuremath{\rightarrow}\ensuremath{\infty} is explained in this model by the limiting behavior of certain eigenvalue distributions, and the significance of these limits is discussed. The study also highlights the unresolved problem of interpreting the complex correlation functions of noncommuting operators.
Two classical subjects in statistical physics are kinetic theory and phase transitions. The latter has been traditionally studied in the equilibrium framework, where the goal is to characterize the ordered phases that arise below the critical temperature in systems with short-range interactions. A more recent development has been to investigate how this order is formed dynamically . In the present and following chapter, we focus on this dynamics. Phenomenology of coarsening The theory of phase transitions was originally motivated by the goal of understanding ferromagnetism. The Ising model played a central role in this effort and provided a useful framework for investigating dynamics. The basic entity in the Ising model is a spin variable that can take two possible values, s = ±1, at each site of a lattice. A local ferromagnetic interaction between spins promotes their alignment, while thermal noise tends to randomize their orientations. The outcome of this competition is a disordered state for sufficiently high temperature, while below a critical temperature the tendency for alignment prevails and an ordered state arises in which the order parameter – the average magnetization – is non-zero. Suppose that we start with an Ising model in an equilibrium disordered phase and lower the temperature. To understand how the system evolves toward the final state, we must endow this model with a dynamics and we also need to specify the quenching procedure. Quenching is usually implemented as follows: • Start at a high initial temperature T i > T c , where spins are disordered; here T c is the critical temperature. • Instantaneously cool the system to a lower temperature T f .