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Physical Review Letters

Exact Linearization of the Radiation-Damped Spin System

David E. Rourke, Matthew P. Augustine

Published Feb 21, 2000• 14 citations
Publisher

Abstract

Nonlinear evolution of the Landau-Lifshitz type can be exactly linearized. Special cases include the radiation-damped spin system and the superradiant system in the semiclassical regime, in the presence of time-varying driving fields. For these, the resultant linear system is simply that of a spin 1 / 2 particle, with the radiation damping rate, or superradiant characteristic time, manifested as an imaginary addition to the spin's resonance frequency. Consequently, methods from inverse scattering theory can be used to design driving fields. The behavior of these systems under stochastic excitation can be determined exactly.

DOI
10.1103/physrevlett.84.1685

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Nonlinear evolution of the Landau-Lifshitz type can be exactly linearized. Special cases include the radiation-damped spin system and the superradiant system in the semiclassical regime, in the presence of time-varying driving fields. For these, the resultant linear system is simply that of a spin 1 / 2 particle, with the radiation damping rate, or superradiant characteristic time, manifested as an imaginary addition to the spin's resonance frequency. Consequently, methods from inverse scattering theory can be used to design driving fields. The behavior of these systems under stochastic excitation can be determined exactly.
The transient behavior of a two-level spin system coupled to an electric circuit is investigated by using the equations of Bloembergen and Pound. The equations are solved, in the limit where the circuit ringing time is very short compared with all other characteristic times, for two cases: 1) the spin-lattice and spin-spin relaxation times both infinite, with an externally applied driving field, and 2) the spin-lattice relaxation time infinite but the spin-spin relaxation time finite, in the absence of an external field. In case 1), it is shown that the motion of an initially inverted magnetization under the action of an applied signal consists roughly of two stages: in the first stage, the effect of radiation damping is unimportant and the motion of the system is determined principally by the applied signal via the ordinary Bloch equations, whereas in the second stage, the motion is essentially the same as if the applied signal had been turned off and only radiation damping were present. In case 2), it is shown that a delayed peak in the emitted radiation should be observed under certain conditions. The delayed peak condition is identical with that derived by Bloom. Curves are presented showing the peak power and the time at which the delayed peak occurs as functions of the relevant parameters. In connection with the ordinary maser behavior of a two-level spin system, it is shown that for values of the parameters typical of steady-state maser amplification, the effects of radiation damping should be unimportant. Finally, systems are examined for which the radiation damping time is much shorter than all other characteristic times (super-regenerative systems). It is indicated how such systems might be operated as one-shot multivibrators or as linear amplifiers. For the latter type of operation, an expression for the gain is derived which is found to be similar to that encountered in ordinary circuit theory.
We consider the time-optimal control of an ensemble of uncoupled spin 1/2 particles in the presence of relaxation and radiation damping effects, whose dynamics is governed by nonlinear equations generalizing the standard linear Bloch equations. For a single spin, the optimal control strategy can be fully characterized analytically. However, in order to take into account the inhomogeneity of the static magnetic field, an ensemble of isochromats at different frequencies must be considered. For this case, numerically optimized pulse sequences are computed and the dynamics under the corresponding optimal field is experimentally demonstrated using nuclear magnetic resonance techniques.
When a two-state quantum system, capable of making radiative transitions, is driven by a high-frequency field, there occur damping effects due to the reaction of the sample to its own radiated field. This damping, which becomes important for high-Q rf structures and for samples of large dc susceptibilities, is here analyzed classically for a simple prototype system—a magnetic spin system obeying the Bloch equations. These equations are augmented to include the radiation-damping torques and are then solved for the four cases of usual interest. Firstly, for steady-state slow passage, radiation damping broadens and lowers the absorption and dispersion signals. Secondly, for adiabatic fast passage, damping causes the dispersion signal to dip at the line center and causes the otherwise absent absorption signal to become finite. However, if the radiation damping field becomes larger than the rf driving field a complete adiabatic inversion is shown to be no longer possible. Thirdly, the conditions for the occurrence of delayed power peaks of the free-precession signal are discussed. Fourthly, for the case of the driven nonsteady state, the stimulated emission may, for large radiation damping, become quite large if the system is driven from an initially ``negative temperature'' condition.
The problem of the evolution in time of a system of two-level atoms that are coupled through their electromagnetic fields, is studied and solved in the framework of semiclassical radiation theory. The atoms may be in any initial states and the radiation reaction is fully taken into account. Both superradiance and time-dependent frequency shift or chirping effects are found. The assumption that all the atoms see the same field is shown to place severe constraints on the evolution of the system, in the form of constants of the motion. These constraints are most easily pictured in terms of the Bloch-vector representation of the atomic system, and they lead to explicit solutions for the time development of each atom. It is shown that the evolution of the complex system is describable by means of a collective super Bloch vector, whose behavior is similar to the behavior of the Bloch vector for a single isolated atom. The constraints on the motion also imply that the system cannot radiate all its energy coherently; some of it remains trapped, to be dissipated ultimately by incoherent processes. Some curves are presented to illustrate the behavior of the system in special cases.
Abstract Radiation damping of an accelerated particle coupled with the field is a classical problem. Still there are difficulties. Time symmetry is broken. So the radiation damping belongs to a class of phenomena that includes transport properties. In a series of our recent articles we proposed a radically new approach based on the extension of the dynamics of integrable systems. There, we considered a simple model: a harmonic oscillator interacting with a field. For integrable systems, it is well known that there exists a unitary transformation U. However, in the radiation damping we have resonances between the action of a charged particle and the actions of the field modes. This makes the system an example of Poincaré nonintegrable systems. We extended the unitary operator to a nonunitary operator Λ. This changes the dynamical description of radiation damping. Once we know the Hamiltonian, we can write the Hamilton equations. But we have the opportunity to go to new descriptions. The invertible Λ transformation gives many new aspects, which are hidden in the initial description. For example, we showed that there are fluctuations in the emitted field. In the current article we apply the Λ transformation to describe the evolution of the emitted field from the charged particle and show that the fluctuations lead to a Langevin equation for the field modes. As a consequence, we have a stochastic wave equation that is an extension of Maxwell–Lorentz equation for classical matter–field interacting systems. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004
The Hamiltonian of a non-relativistic free charged particle interacting with one quantised mode of the electromagnetic field is diagonalised exactly in dipole approximation. The diagonalisation is simpler for a circularly polarised mode, but is is shown that the case of a linearly polarised mode can be reduced to the circularly polarised one. Exact stationary and time-dependent states of this Hamiltonian are then calculated. Based on these results, the rederivation of the nonlinear inverse and direct bremsstrahlung's cross section is presented and the connection with earlier (semiclassical) approaches is also established. Photon number distribution functions for different initial conditions are also determined in analytical form.