Chaotic Bohmian trajectories for stationary states

Cesa, Alexandre; Martin, John P.; Struyve, Ward

doi:10.1088/1751-8113/49/39/395301

Cited by 18 publications

(23 citation statements)

References 53 publications

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“…The mechanism of generation of chaos is thought to be well understood in 2-d quantum systems [1,2,13,14,15,16,17,18,19,20,21]. However, only few works exist in the 3-d or higher dimensional cases, due to the far higher complexity of the problem [16,22,23,24].…”

Section: Introductionmentioning

confidence: 99%

Origin of chaos in 3-d Bohmian trajectories

2016

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We study the 3-d Bohmian trajectories of a quantum system of three harmonic oscillators. We focus on the mechanism responsible for the generation of chaotic trajectories. We demonstrate the existence of a 3-d analogue of the mechanism found in earlier studies of 2-d systems [1,2], based on moving 2-d 'nodal point -X-point complexes'. In the 3-d case, we observe a foliation of nodal point -X-point complexes, forming a '3-d structure of nodal and X-points'. Chaos is generated when the Bohmian trajectories are scattered at one or more close encounters with such a structure.

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Section: Introductionmentioning

confidence: 99%

Origin of chaos in 3-d Bohmian trajectories

2016

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“…( 28) is nothing else than the linear entropy of its reduced density matrix. However in this case it is trivial to calculate the V N E of the reduced density matrix for both states (10) and (11), the so called entanglement entropy EE:…”

Section: Entanglementmentioning

confidence: 99%

“…Having as a goal the exploration of chaos in the presence of QE (see also [23,24,25]), in a previous paper [26] we gave the basic characteristics of the Bohmian trajectories in an entangled two-qubit system, where the basic qubit states in one dimension are realized as properly engineered coherent states of the unperturbed quantum harmonic oscillator. We chose to work with this system, since it is quite well understood from a quantum mechanical standpoint and is easily constructed and controlled in the laboratory.…”

Section: Introductionmentioning

confidence: 99%

Bohmian trajectories in an entangled two-qubit system

2019

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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.

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“…On the other hand, the existence of trajectories in BQM allows us to define and study quantum chaotic behaviour by applying all the techniques of classical dynamical systems. Thus chaos in Bohmian Dynamics has attracted a lot of interest in the last decades [20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39].…”

Section: Introductionmentioning

confidence: 99%

Chaos in Bohmian Quantum Mechanics: A short review

Contopoulos,

Tzemos

2020

Preprint

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This is a short review in 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 tends 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.

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Chaotic Bohmian trajectories for stationary states

Cited by 18 publications

References 53 publications

Origin of chaos in 3-d Bohmian trajectories

Origin of chaos in 3-d Bohmian trajectories

Bohmian trajectories in an entangled two-qubit system

Chaos in Bohmian Quantum Mechanics: A short review

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