Exact Constants Of Motion Research Articles

Coherent population transfer beyond the adiabatic limit: Generalized matched pulses and higher-order trapping states

We show that the physical mechanism of population transfer in a 3-level system with a closed loop of coherent couplings (loop-STIRAP) is not equivalent to an adiabatic rotation of the dark-state of the Hamiltonian but coresponds to a rotation of a higher-order trapping state in a generalized adiabatic basis. The concept of generalized adiabatic basis sets is used as a constructive tool to design pulse sequences for stimulated Raman adiabatic passage (STIRAP) which give maximum population transfer also under conditions when the usual condition of adiabaticty is only poorly fulfilled. Under certain conditions for the pulses (generalized matched pulses) there exists a higher-order trapping state, which is an exact constant of motion and analytic solutions for the atomic dynamics can be derived.

Equivalence of the Calogero-Sutherland model to free harmonic oscillators

A similarity transformation is constructed through which a system of particles interacting with inverse-square two-body and harmonic potentials in one dimension, can be mapped identically, to a set of free harmonic oscillators. This equivalence provides a straightforward method to find the complete set of eigenfunctions, the exact constants of motion and a linear $W_{1+\infty}$ algebra associated with this model. It is also demonstrated that a large class of models with long-range interactions, both in one and higher dimensions can be made equivalent to decoupled oscillators.

The `Ermakov - Lewis' invariants for coupled linear oscillators

We consider N coupled linear oscillators with time-dependent coefficients. An exact complex amplitude-real phase decomposition of the oscillatory motion is constructed. This decomposition is further used to derive N exact constants of motion which generalize the so-called Ermakov-Lewis invariant of a single oscillator. In the Floquet problem of periodic oscillator coefficients we discuss the existence of periodic complex amplitude functions in terms of existing Floquet solutions.

Variational principle for the turbulent diffusion equation

The application of functional formalism to the description of turbulent diffusion is shown. The problem of the turbulent diffusion is formulated on the basis of a variational principle for the Phi equation in the appropriate functional space. The existence of an exact constant of motion, as a consequence of this principle, is shown.

Adiabatic charged particle motion in rapidly rotating magnetospheres

When the cE × B/B² drift velocity is large enough to exceed the particle gyro velocity, adiabatic theory becomes more complicated and less useful: more complicated because there are five drifts in addition to the E × B gradient, and line curvature drifts and three more terms in the parallel equation of motion; less useful because the second and third invariants J and Φ are no longer valid in mirror geometry due to the rapid drift across field lines that destroys any semblance of periodicity in the bounce motion. (Approximate periodicity is necessary to have an adiabatic invariant.) But the special case of a rapidly rotating rigid magnetic field, with E∥ = 0, is an exception. If the field at any time is merely a rotation of that at an earlier time, the drift and parallel equations simplify, and even better, the second invariant is again valid. The drift equation now has two rather than five additional terms—a Coriolis drift and a centrifugal drift. Just as in a slowly or nonrotating mirror system, the second invariant now provides a (rapidly rotating) drift shell to which the guiding center is confined. The particle kinetic energy in the rotating frame, minus the centrifugal potential, is an exact constant of motion in a rigid rotator. This constant is well known but does not by itself put limits on radial motion toward or away from the rotation axis, hence it does not limit particle energy changes. But the drift shell does limit and make periodic any radial excursions, and this is why previous studies of particle motion, in particular examples of rigid rotators, have shown that particles experience no steady energy gain or loss.