Universality in nonadiabatic behavior of classical actions in nonlinear models of Bose-Einstein condensates

Global change in action due to trapping: How to derive it whatever the rate of variation of the dynamics

In this paper, we investigate the motion of a set of charged particles acted upon by a growing electrostatic wave in the limit when the initial wave amplitude is vanishingly small and when all the particles have the same initial action, I(0). We show, both theoretically and numerically, that when all the particles have been trapped in the wave potential, the distribution in action exhibits a very sharp peak about the smallest action. Moreover, as the wave keeps growing, the most probable action tends toward a constant, I(f), which we estimate theoretically. In particular, we show that I(f) may be calculated very accurately when the particles' motion before trapping is far from adiabatic by making use of a perturbation analysis in the wave amplitude. This fills a gap regarding the computation of the action change, which, in the past, has only been addressed for slowly varying dynamics. Moreover, when the variations of the dynamics are fast enough, we show that the Fourier components of the particles' distribution function can be calculated by connecting estimates from our perturbation analysis with those obtained by assuming that all the particles have the same constant action, I=I(f). This result is used to compute theoretically the imaginary part of the electron susceptibility of an electrostatic wave in a plasma. Moreover, using our formula for the electron susceptibility, we can extend the range in ε(a) (the parameter that quantifies the slowness of the dynamics) for our perturbative estimate of I(f)-I(0). This range can actually be pushed down to values of ε(a) allowing the use of neoadiabatic techniques to compute the jump in action. Hence, this paper shows that the action change due to trapping can be calculated theoretically, regardless of the rate of variation of the dynamics, by connecting perturbative results with neoadiabatic ones.

Rapid geometrical chaotization in slow-fast Hamiltonian systems

In this Rapid Communication we demonstrate effects of a new mechanism of adiabaticity destruction in Hamiltonian systems with a separatrix in the phase space. In contrast to the slow diffusive-like destruction typical for many systems, this new mechanism is responsible for very fast chaotization in a large phase volume. To investigate this mechanism we consider a Hamiltonian system with two degrees of freedom and with a separatrix in the phase plane of fast variables. The fast chaotization is due to an asymmetry of the separatrix and corresponding geometrical jumps of an adiabatic invariant. This system describes the motion of charged particles in a inhomogeneous electromagnetic field with a specific configuration. We show that geometrical jumps of the adiabatic invariant result in a very fast chaotization of particle motion.

Nonlinear energy transfer in classical and quantum systems

In this paper we investigate the effect of slowly-varying parameters on the energy transfer in a weakly coupled system. For definiteness, we consider a system of two nonlinear oscillators, in which the directly excited first oscillator with constant parameter is attached to the oscillator with slowly time-varying frequency. It is proved that the equations of the slow passage through resonance in this system are identical to the equations of nonlinear Landau-Zener (LZ) tunneling. Three types of dynamical behavior are distinguished, namely, quasilinear, moderately nonlinear, and strongly nonlinear ones. Quasilinear systems exhibit a gradual energy transfer from the excited to the attached oscillator, while moderately nonlinear systems are characterized by an abrupt transition from the energy localization on the excited oscillator to the localization on the attached oscillator. In strongly nonlinear systems, the transition from the energy localization to strong energy exchange between the oscillators is revealed. Explicit approximate solutions describing the transient processes in moderately and strongly nonlinear systems are suggested. Correctness of the constructed approximations is confirmed by numerical results. The results presented in this paper, in addition to providing an analytical framework for understanding the transient dynamics, suggest an approximate procedure for solving the nonlinear LZ problem with arbitrary initial conditions over a finite time-interval.

Classical analog of quasilinear Landau-Zener tunneling

In this paper we develop an analytical framework to study the effect of nonlinearity on irreversible energy transfer in a system of two weakly coupled oscillators with time-dependent parameters, with special attention to an analogy between classical energy transfer and nonadiabatic quantum tunneling. For preciseness, we suppose that a linear oscillator with constant parameters is excited by an initial impulse but a coupled quasilinear oscillator with slowly varying parameters is initially at rest. It is shown that the equations of the slow passage through resonance in this system are identical to quasilinear equations of nonadiabatic Landau-Zener tunneling. Due to revealed equivalence, a recently found analogy between irreversible energy transfer in a classical linear system and conventional linear Landau-Zener tunneling can be extended to quasilinear systems. An explicit analytical solution of the quasilinear problem is found with the help of an iteration procedure, wherein the linear solution is chosen as an initial approximation. Correctness of the constructed approximations is confirmed by numerical simulations. The results presented in this paper, in addition to providing an analytical framework for understanding the transient dynamics of coupled oscillators, suggest an approximate procedure for solving the quasilinear Landau-Zener equations with arbitrary initial conditions over a finite time interval.