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arXiv:quant-ph/0611003

A chaotic dynamical reduction model for the quantum mechanical state vector

Henrik Brusheim-Johansson, Johan Hansson

Published Oct 31, 2006• 2 citations
arXivHTMLPDF

Abstract

A new model is proposed for the purpose of modelling the ``wave function collapse'' of a two-state quantum system. The collapse to a classical state is driven by a nonlinear evolution equation with an extreme sensitivity to absolute phase. It is hypothesized that the phase, or part of it, is displaying chaotic behaviour. This chaotic behaviour can then be responsible for the apparent indeterminacy we are experiencing for a single quantum system. Through this randomness, the statistical ``ensemble'' behaviour, due to Born, to describe a single quantum system, is no longer needed.

DOI
10.48550/arxiv.quant-ph/0611003

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A new model is proposed for the purpose of modelling the wave function collapse'' of a two-state quantum system. The collapse to a classical state is driven by a nonlinear evolution equation with an extreme sensitivity to absolute phase. It is hypothesized that the phase, or part of it, is displaying chaotic behaviour. This chaotic behaviour can then be responsible for the apparent indeterminacy we are experiencing for a single quantum system. Through this randomness, the statistical ensemble'' behaviour, due to Born, to describe a single quantum system, is no longer needed.
A non-local toy-model is proposed for the purpose of modelling the ``wave function collapse'' of a two-state quantum system. The collapse is driven by a nonlinear evolution equation with an extreme sensitivity to absolute phase. It is hypothesized that the phase, or a part of it, is displaying chaotic behaviour. This chaotic behaviour can then be responsible for the indeterminacy we are experiencing for a single quantum system. Through this randomness, we no longer need the statistical ``ensemble'' behaviour to describe a single quantum system. A brief introduction to the ``measurement problem'' is also given.
Nonlinearity in Quantum Mechanics may have extrinsic or intrinsic origins and is a liable route to a chaotic behaviour that can be of difficult observations. In this paper, we propose two forms of nonlinear Hamiltonian, which explicitly depend upon the phase of the wave function and produce chaotic behaviour. To speed up the slow manifestation of chaotic effects, a resonant laser field assisting the time evolution of the systems causes cumulative effects that might be revealed, at least in principle. The nonlinear Schrödinger equation is solved within the two-state approximation; the solution displays features with characteristics similar to those found in chaotic Classical Mechanics: sensitivity on the initial state, dense power spectrum, irregular filling of the Poincaré map and exponential separation of the trajectories of the Bloch vector σ in the Bloch sphere.
Two major deviations from causality in the existing formulations of quantum mechanics, related respectively to quantum chaos and indeterminate wave reduction, are eliminated within the new, universal concept of dynamic complexity. The analysis involves a new paradigm for description of a system with interaction, the principle of dynamic multivaluedness (redundance), and the ensuing concept of the fundamental dynamic uncertainty. It is shown that both the wave reduction and truly unpredictable (chaotic) behaviour in quantum systems can be completely and causally understood as a higher sublevel of the same dynamic complexity that provides the causally complete picture of the unified wave-particle duality and relativity at its lowest level (quant-ph/9902015,16). The presentation is divided into five parts. The first three parts deal with intrinsic randomness in Hamiltonian (isolated) quantum systems as the basic case of dynamical chaos. In the last two parts a causal solution to the problem of quantum indeterminacy and wave reduction is proposed. Part I introduces the method of effective dynamical functions as a generalisation of the optical potential formalism. The method provides a legal transformation of the Schroedinger equation revealing the hidden multivaluedness of interaction process, i. e. its self-consistent, dynamical splitting into many equally real, but mutually incompatible branches, called 'realisations'. Each realisation incorporates the usual "complete" set of eigenfunctions and eigenvalues for the entire problem. The method is presented in detail for the Hamiltonian system with periodic (not small) perturbation, both in its time-independent and time-dependent versions.
Using the key properties of chaos, i.e. ergodicity and exponential instability, as a resource to control classical dynamics has a long and considerable history. However, in the context of controlling "chaotic" quantum unitary dynamics, the situation is far more tenuous. The classical concepts of exponential sensitivity to trajectory initial conditions and ergodicity do not directly translate into quantum unitary evolution. Nevertheless properties inherent to quantum chaos can take on those roles: i) the dynamical sensitivity to weak perturbations, measured by the fidelity decay, serves a similar purpose as the classical sensitivity to initial conditions; and ii) paired with the fact that quantum chaotic systems are conjectured to be statistically described by random matrix theory, implies a method to translate the ergodic feature into the control of quantum dynamics. With those two properties, it can be argued that quantum chaotic dynamical systems, in principle, allow for full controllability beyond a characteristic time that scales only logarithmically with system size and ℏ−1\hbar^{-1}ℏ−1. In the spirit of classical targeting, it implies that it is possible to fine tune the immense quantum interference with weak perturbations and steer the system from any initial state into any desired target state, subject to constraints imposed by conserved quantities. In contrast, integrable dynamics possess neither ergodicity nor exponential instability, and thus the weak perturbations apparently must break the integrability for control purposes. The main ideas are illustrated with the quantum kicked rotor. The production of revivals, cat-like entangled states, and the transition from any random state to any other random state is possible as demonstrated.
The measurement problem in quantum mechanics represents a disconnect between the Schr\"{o}dinger equation's unitary evolution and the non-unitary collapse of the wave function. This work resolves this issue by modeling measurement as a deterministic, thermodynamically driven process. We modify the Liouville-von Neumann equation by introducing a dual mechanism: a dephasing term that eliminates quantum coherence and a state-resolution dynamic that amplifies one outcome to certainty. An infinitesimal environmental perturbation breaks the initial symmetry, determining which outcome emerges. Numerical simulations of a qubit in superposition demonstrate that (1) coherence is conserved during unitary evolution, (2) a rapid phase transition occurs upon measurement, and (3) Born Rule statistics emerge from ensemble averaging over random environmental fluctuations. We conclude that quantum collapse need not be a separate postulate but can be modeled as a unified, deterministic dynamical law with testable experimental predictions.
We consider the quantum-to-classical transition for macroscopic systems coupled to their environments. By applying Born's Rule, we are led to a particular set of quantum trajectories, or an unravelling, that describes the state of the system from the frame of reference of the subsystem. The unravelling involves a branch dependent Schmidt decomposition of the total state vector. The state in the subsystem frame, the conditioned state, is described by a Poisson process that involves a non-linear deterministic effective Schrodinger equation interspersed with quantum jumps into orthogonal states. We then consider a system whose classical analogue is a generic chaotic system. Although the state spreads out exponentially over phase space, the state in the frame of the subsystem localizes onto a narrow wave packet that follows the classical trajectory due to Ehrenfest's Theorem. Quantum jumps occur with a rate that is the order of the effective Lyapunov exponent of the classical chaotic system and imply that the wave packet undergoes random kicks described by the classical Langevin equation of Brownian motion. The implication of the analysis is that this theory can explain in detail how classical mechanics arises from quantum mechanics by using only unitary evolution and Born's Rule applied to a subsystem.