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Maple Transactions

Critical points and trajectories of the Bohmian quantum flow

Athanasios C. Tzemos, G. Contopoulos

Published Aug 28, 2023• 0 citations
Publisher

Abstract

In the present work we study the critical points of the Bohmian quantum flow, namely the nodal point and its associated X-point, which are responsible for the generation of chaos in Bohmian trajectories. In the first part of the paper we find an analytical equation for the position of the X-point in a planar 2-d Bohmian system with a single nodal point and test its accuracy numerically. We then calculate its asymptotic curves and comment on the way they affect the evolution of the nearby Bohmian trajectories. In the second part we present our first results on the position of the X-point and its asymptotic curves in a 3d partially integrable system, where the Bohmian trajectories evolve on spherical surfaces.

DOI
10.5206/mt.v3i2.15546

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In the present work we study the critical points of the Bohmian quantum flow, namely the nodal point and its associated X-point, which are responsible for the generation of chaos in Bohmian trajectories. In the first part of the paper we find an analytical equation for the position of the X-point in a planar 2-d Bohmian system with a single nodal point and test its accuracy numerically. We then calculate its asymptotic curves and comment on the way they affect the evolution of the nearby Bohmian trajectories. In the second part we present our first results on the position of the X-point and its asymptotic curves in a 3d partially integrable system, where the Bohmian trajectories evolve on spherical surfaces.
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.
Abstract Chaos in Bohmian quantum mechanics is an open field of research. In general, most of the 3D Bohmian trajectories are free to wander around the 3D space. However, there are cases where the evolution of the trajectories is dictated by exact or approximate integrals of motion. A first case corresponds to partial integrability, where the trajectories (ordered and chaotic) evolve on certain integral surfaces. A second case corresponds to ordered trajectories. In this paper we extend our previous work in 3D Bohmian chaos by using both forms of integrability and discuss their physical implications.
We study in detail the critical points of Bohmian flow, both in the inertial frame of reference (Y-points) and in the frames centered at the moving nodal points of the guiding wavefunction (X-points), and analyze their role in the onset of chaos in a system of two entangled qubits. We find the distances between these critical points and a moving Bohmian particle at varying levels of entanglement, with particular emphasis on the times at which chaos arises. Then, we find why some trajectories are ordered, without any chaos. Finally, we examine numerically how the Lyapunov Characteristic Number (LCN) depends on the degree of quantum entanglement. Our results indicate that increasing entanglement reduces the convergence time of the finite-time LCN of the chaotic trajectories toward its final positive value.
We consider the Bohmian trajectories in a 2-d quantum harmonic oscillator with non commensurable frequencies whose wavefunction is of the form Ψ=aΨm1,n1(x,y)+bΨm2,n2(x,y)+cΨm3,n3(x,y)Ψ=aΨ_{m_1,n_1}(x,y)+bΨ_{m_2,n_2}(x,y)+cΨ_{m_3,n_3}(x,y)Ψ=aΨm1​,n1​​(x,y)+bΨm2​,n2​​(x,y)+cΨm3​,n3​​(x,y). We first find the trajectories of the nodal points for different combinations of the quantum numbers m,nm,nm,n. Then we study, in detail, a case with relatively large quantum numbers and two equal m′sm'sm′s. We find %We find first the nodal points where Ψ=0Ψ=0Ψ=0. The nodes can be found analytically only if mmm and nnn are small. If two m′sm'sm′s (or two n′sn'sn′s are equal we can find explicitly the nodal points , which are of two types (1) fixed nodes independent of time and (2) moving nodes which from time to time collide with the fixed nodes and at particular times they go to infinity. Finally, we study the trajectories of quantum particles close to the nodal points and observe, for the first time, how chaos is generated in a complex system with multiple nodes scattered on the configuration space.
Abstract In the present paper we study the Bohmian dynamics of a partially integrable 3d Bohmian system whose trajectories evolve on spherical surfaces. By use of spherical coordinates (R, ϕ, θ) we study the behaviour of unstable fixed points that generate chaos on the (ϕ, θ) plane and discuss the differences between them and those of planar 2d systems. Finally, we show for the first time that the chaotic trajectories of this system are ergodic although the number of its nodes is very small (two). Thus we can observe ergodicity not only in multinodal 2d systems but also in partially integrable systems with few nodes due to their curved geometry.