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arXiv:2009.05867 · Regular and Chaotic Dynamics

Chaos in Bohmian Quantum Mechanics: A Short Review

George Contopoulos, Athanasios C. Tzemos

Published Sep 1, 2020• 20 citations
arXivHTMLPDF

Abstract

This is a short review of the theory of chaos in Bohmian quantum mechanics based on our series of works in this field. Our first result is the development of a generic theoretical mechanism responsible for the generation of chaos in an arbitrary Bohmian system (in 2 and 3 dimensions). This mechanism allows us to explore the effect of chaos on Bohmian trajectories and study in detail (both analytically and numerically) the different kinds of Bohmian trajectories where, in general, chaos and order coexist. Finally, we explore the effect of quantum entanglement on the evolution of the Bohmian trajectories and study chaos and ergodicity in qubit systems which are of great theoretical and practical interest. We find that the chaotic trajectories are also ergodic, i. e., they give the same final distribution of their points after a long time regardless of their initial conditions. In the case of strong entanglement most trajectories are chaotic and ergodic and an arbitrary initial distribution of particles will tend to Born’s rule over the course of time. On the other hand, in the case of weak entanglement the distribution of Born’s rule is dominated by ordered trajectories and consequently an arbitrary initial configuration of particles will not tend, in general, to Born’s rule unless it is initially satisfied. Our results shed light on a fundamental problem in Bohmian mechanics, namely, whether there is a dynamical approximation of Born’s rule by an arbitrary initial distribution of Bohmian particles.

DOI
10.1134/s1560354720050056

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This is a short review of the theory of chaos in Bohmian quantum mechanics based on our series of works in this field. Our first result is the development of a generic theoretical mechanism responsible for the generation of chaos in an arbitrary Bohmian system (in 2 and 3 dimensions). This mechanism allows us to explore the effect of chaos on Bohmian trajectories and study in detail (both analytically and numerically) the different kinds of Bohmian trajectories where, in general, chaos and order coexist. Finally, we explore the effect of quantum entanglement on the evolution of the Bohmian trajectories and study chaos and ergodicity in qubit systems which are of great theoretical and practical interest. We find that the chaotic trajectories are also ergodic, i. e., they give the same final distribution of their points after a long time regardless of their initial conditions. In the case of strong entanglement most trajectories are chaotic and ergodic and an arbitrary initial distribution of particles will tend to Born’s rule over the course of time. On the other hand, in the case of weak entanglement the distribution of Born’s rule is dominated by ordered trajectories and consequently an arbitrary initial configuration of particles will not tend, in general, to Born’s rule unless it is initially satisfied. Our results shed light on a fundamental problem in Bohmian mechanics, namely, whether there is a dynamical approximation of Born’s rule by an arbitrary initial distribution of Bohmian particles.
Abstract We study in detail the onset of chaos and the probability measures formed by individual Bohmian trajectories in entangled states of two-qubit systems for various degrees of entanglement. The qubit systems consist of coherent states of 1-d harmonic oscillators with irrational frequencies. In weakly entangled states chaos is manifested through the sudden jumps of the Bohmian trajectories between successive Lissajous-like figures. These jumps are succesfully interpreted by the ‘nodal point-X-point complex’ mechanism. In strongly entangled states, the chaotic form of the Bohmian trajectories is manifested after a short time. We then study the mixing properties of ensembles of Bohmian trajectories with initial conditions satisfying Born’s rule. The trajectory points are initially distributed in two sets S 1 and S 2 with disjoint supports but they exhibit, over the course of time, abrupt mixing whenever they encounter the nodal points of the wavefunction. Then a substantial fraction of trajectory points is exchanged between S 1 and S 2 , without violating Born’s rule. Finally, we provide strong numerical indications that, in this system, the main effect of the entanglement is the establishment of ergodicity in the individual Bohmian trajectories as t → ∞ : different initial conditions result to the same limiting distribution of trajectory points.
We study order, chaos and ergodicity in the Bohmian trajectories of a 2D quantum harmonic oscillator. We first present all the possible types (chaotic, ordered) of Bohmian trajectories in wavefunctions made of superpositions of two and three energy eigenstates of the oscillator. There is no chaos in the case of two terms and in some cases of three terms. Then, we show the different geometries of nodal points in bipartite Bohmian systems of entangled qubits. Finally, we study multinodal wavefunctions and find that a large number of nodal points does not always imply the dominance of chaos. We show that, in some cases, the Born distribution is dominated by ordered trajectories, something that has a significant impact on the accessibility of Born’s rule P=|Ψ|2 by initial distributions of Bohmian particles with P0≠|Ψ0|2.
Abstract We study in detail the trajectories, ordered and chaotic, of two entangled Bohmian qubits when their initial preparation satisfies (or not) Born’s rule for various amounts of quantum entanglement. For any non zero value of entanglement ordered and chaotic trajectories coexist and the proportion of ordered trajectories increases with the decrease of the entanglement. In the extreme cases of zero and maximum entanglement we have only ordered and chaotic trajectories correspondingly. The chaotic trajectories of this model are ergodic, for any given value of entanglement, namely the limiting distribution of their points does not depend on their initial conditions. Consequently it is the ratio between ordered and chaotic trajectories which is responsible for the dynamical establishment (or not) of Born’s rule.
We study in detail the trajectories, ordered and chaotic, of two entangled Bohmian qubits when their initial preparation satisfies (or not) Born's rule for various amounts of quantum entanglement. For any non zero value of entanglement ordered and chaotic trajectories coexist and the proportion of ordered trajectories increases with the decrease of the entanglement. In the extreme cases of zero and maximum entanglement we have only ordered and chaotic trajectories correspondingly. The chaotic trajectories of this model are ergodic, for any given value of entanglement, namely the limiting distribution of their points does not depend on their initial conditions. Consequently it is the ratio between ordered and chaotic trajectories which is responsible for the dynamical establishment (or not) of Born's rule.
We study the Bohmian trajectories of a generic entangled two-qubit system, composed of coherent states of two harmonic oscillators with noncommensurable frequencies and focus on the relation between ergodicity and the dynamical approach to Born's rule for arbitrary distributions of initial conditions. We find that most Bohmian trajectories are ergodic and establish the same invariant ergodic limiting distributions of their points for any nonzero amount of entanglement. In the case of strong entanglement the distribution satisfying Born's rule is dominated by chaotic-ergodic trajectories. Therefore, P→|Ψ|^{2} for an arbitrary P_{0}. However, when the entanglement is weak the distribution satisfying Born's rule is dominated by ordered trajectories, which are not ergodic. In this case the ergodic trajectories do not, in general, lead to the distribution of Born's rule, therefore P=|Ψ|^{2} is guaranteed only if P_{0}=|Ψ_{0}|^{2}. Consequently, the existence of chaotic and ergodic Bohmian trajectories does not always lead to the dynamical establishment of Born's rule.
This paper presents a number of numerical investigations of orbits in the de Broglie–Bohm version of quantum mechanics. We first clarify how the notion of chaos should be implemented in the case of Bohmian orbits. Then, we investigate the Bohmian orbits in three different characteristic quantum systems: (a) superposition of three stationary states in the Hamiltonian of two uncoupled harmonic oscillators with incommensurable frequencies, (b) wave packets in a Hénon–Heiles-type Hamiltonian and (c) a modified two-slit experiment. In these examples, we identify regular or chaotic orbits and also orbits exhibiting a temporarily regular and then chaotic behaviour. Then, we focus on a numerical investigation of the Bohm–Vigier (Bohm and Vigier 1954 Phys. Rev. 26 208) theory, that an arbitrary initial particle distribution P should asymptotically tend to |ψ| 2 , by considering the role of chaotic mixing in causing irregularity of Madelung's flow, a necessary condition for P to tend to |ψ| 2 . We find that the degree of chaos of a particular system correlates with the speed of convergence of P to |ψ| 2 . In the case of wave-packet dynamics, our numerical data show that the time of convergence scales exponentially with the inverse of the effective perturbation from the harmonic oscillator Hamiltonian. The latter result can be viewed as a quantum analogue of Nekhoroshev's (Nekhoroshev 1977 Russ. Math. Surveys 32 1) theorem of exponential stability in classical nonlinear Hamiltonian dynamics.