Nodal points and the transition from ordered to chaotic Bohmian trajectories

Abstract

We explore the transition from order to chaos for the Bohmian trajectories of a simple quantum system corresponding to the superposition of three stationary states in a 2D harmonic well with incommensurable frequencies. We study in particular the role of nodal points in the transition to chaos. Our main findings are (a) a proof of the existence of bounded domains in configuration space which are devoid of nodal points, (b) an analytical construction of formal series representing regular orbits in the central domain as well as a numerical investigation of its limits of applicability, (c) a detailed exploration of the phase-space structure near the nodal point. In this exploration we use an adiabatic approximation and we draw the flow chart in a moving frame of reference centered at the nodal point. We demonstrate the existence of a saddle point (called X-point) in the vicinity of the nodal point which plays a key role in the manifestation of exponential sensitivity of the orbits. One of the invariant manifolds of the X-point continues as a spiral terminating at the nodal point. We find cases of Hopf bifurcation at the nodal point and explore the associated phase space structure of the nodal point-X-point complex. We finally demonstrate the mechanism by which this complex generates chaos. Numerical examples of this mechanism are given for particular chaotic orbits, and a comparison is made with previous related works in the literature.

Applied Bohmian mechanics

“…In a series of works published later on by Wisniacki and coworkers [282,283] it was shown that the origin of Bohmian chaos is in the evolution in time of those vortices. An analogous conclusion was also found by Efthymiopoulos and coworkers [197,[284][285][286][287], who showed that chaos is due to the presence of moving quantum vortices forming nodal point-X-point complexes. In particular, in reference [197] a theoretical analysis of the dependence of Lyapunov exponents of Bohmian trajectories on the size and speed of the quantum vortices is presented, which explains their earlier numerical findings [284,285].…”

Ordered and chaotic Bohmian trajectories

“…These series are similar to Lindstedt series with fixed frequencies, or to the 'third integral' series of galactic dynamics (Contopoulos 1960). In Efthymiopoulos et al (2007) we demonstrated the consistency of the construction. Namely, contrary to the case of classical Lindstedt series, no secular terms appear in the quantum-mechanical series as a result of keeping the frequencies constant.…”

“…In the model (12) (Efthymiopoulos et al 2007), we find that the flow lines in the moving frame of reference (u, v) are spirals terminating at the nodal point. Furthermore, one spiral is always connected to one of the invariant manifolds of a simply hyperbolic point (called 'X-point') that exists in the vicinity of the nodal point.…”

“…Further details on this problem in the example of two harmonic oscillators are given in Efthymiopoulos et al (2007).…”

“…In recent years several authors (Dürr et al 1992;Faisal and Schwengelbeck 1995;Parmenter and Valentine 1995;de Polavieja 1996;Dewdney and Malik 1996;Iacomelli and Pettini 1996;Frisk 1997;Konkel and Makowski 1998;Wu and Sprung 1999;Makowski et al 2000;Cushing 2000;Falsaperla and Fonte 2003;de Sales and Florencio 2003;Wisniacki and Pujals 2005;Valentini and Westman 2005;Efthymiopoulos and Contopoulos 2006;Efthymiopoulos et al 2007) presented a consistent approach to the problem of chaos in quantum-mechanical systems that does not rely on the classical properties of the same systems. This is the Bohmian (or 'pilot wave') approach (de Broglie 1926;Bohm 1952a,b, etc), that considers deterministic quantum orbits.…”