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The complex symmetric Lanczos algorithm (LA) has proven to be a very efficient means of calculating magnetic resonance line shapes and spectral densities associated with Fokker–Planck forms. However, the relative importance of the various components of the basis set in an accurate representation of the spectrum and the proper number of recursive steps are not easily assessed in practice using the Lanczos algorithm. A systematic and objective procedure for the determination of optimal basis sets and number of recursive steps is developed using a generalization of the conjugate gradient method (CGM) appropriate for the type of complex symmetric matrices occuring in these problems. The relative importance of the individual basis vectors is determined by using the CGM to obtain the ‘‘solution vector’’ from the set of algebraic equations defining the spectrum. This is done at several values of the sweep variable (e.g., the frequency or the magnetic field). The maximum (over these values of sweep variable) for each component of the solution vector is taken to be a measure of the overall importance of the corresponding basis vector in the complete spectrum. Using this method signficant basis set truncation is conveniently possible. The number of recursive steps needed for an accurate representation of the spectrum is easily obtained by monitoring the residual in the approximate solution vector at the center of the spectrum and by recognizing the close relationship between the LA and the CGM. It is this relationship that enables construction of the Lanczos tridiagonal matrix with the CGM which can either be used to calculate the cw ESR spectrum directly or else the eigenvalues. The information obtained from the CGM can be used to ‘‘turbocharge’’ the LA by taking advantage of the nearly optimal basis set and number of recursive steps. Significant savings in computation time are possible, and relative savings are greatest for the most difficult problems. This is illustrated with a variety of examples of slow-motional cw ESR spectra and of the new two-dimensional electron-spin-echo technique. In keeping with the greater sensitivity of the latter technique to motional dynamics, it is consistently found to require significantly larger optimal basis sets and number of recursive steps for an accurate representation. One of the most challenging problems for both types of spectroscopy is the case of macroscopically oriented samples where the macroscopic director is tilted at an angle relative to the applied static magnetic field, since this removes much of the symmetry in the problem. This case is found to yield to very significant truncation of basis sets, and a new symmetry-based decoupling of certain basis vectors was found in this study for the particular example of a 90° tilt angle.

Through an investigation of the dynamics of solitons in three-level atoms, we demonstrate the possibility of optical pulse control and shaping in coherently driven media. It is also shown that solitons generated in three-level atoms, in contrast to two-level atoms, can propagate at the speed of light.[ S0031-9007(97) PACS numbers: 42.81. Dp, 31.15.Ar, 42.65.Tg The development of new techniques for pulse shaping and control is central to generating tailored pulses for communications, study of ultrafast processes, and preparation of atoms and molecules in desired quantum states. In this Letter we investigate solitons in three-level atoms and, in addition to revealing the striking differences from two-level atom results, we also demonstrate that a coherently driven resonant medium can be utilized for pulse shaping and control. Driven atom dynamics continue to receive tremendous attention, and our work represents a qualitatively different way of exploiting these dynamics. We focus on solitons due to their special fundamental properties [1] and their many applications [2][3][4][5]. The principal results are that in a nonabsorbing, resonant L system [inset of Fig. 1(a)] (i) a weak field of arbitrary profile at the Stokes transition is parametrically amplified into the replica of a soliton at the pump transition (cloning), (ii) the degree of overlap between the input pump and Stokes pulses permits a control over the temporal location of the Stokes soliton (dragging), as well as its amplitude and phase, and (iii) the cloned soliton, which has a different frequency from the pump soliton, travels at the speed of light, c, and hence is a steady state pulse [i.e., dependent only on pulse-local coordinate t͑t 2 z͞c͒, and not on z ͑z͒]. This is an unusual property of the cloned pulses generated in a three-level system since solitons generated in two-level atoms always propagate with a speed less than c, and so depend on both t and z .We begin by referring to the inset of Fig. 1(a) where a soliton, with Rabi frequency V p ͑z, t͒, is applied at the pump ͑j1͘ $ j3͒͘ transition. In a two-level system, this pulse would propagate unchanged, as known from selfinduced transparency (SIT) [1]. Now we apply a weak field with arbitrary profile, of Rabi frequency V s ͑t, z͒, at the Stokes ͑j1͘ $ j2͒͘ transition. Note that we are studying a nonabsorbing medium, and so the physical problem and the associated results are different from other work on absorbing media [6-8], as we discuss later. The results are also distinct from simultons [9]. The governing equations are the coupled Schroedinger-Maxwell equations in the slowly varying envelope approximation, which are given by [6][7][8] ≠ic 1 ͞≠t 2͑1͞2͒V p c 3 2 ͑1͞2͒V s c 2 , (1a)where c i ͑i 1, 2, 3͒ are the probability amplitudes of the atomic levels, m p ͑m s ͒ is the propagation constant for the pump (Stokes) pulse with dimensions of frequency/ length, and D 1 ͑D 2 ͒ is the detuning of the pump (Stokes) pulse from its transition. In Eq.(1), all quantities are made dimensionless by usin...