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Completely Mixing Quantum Open Systems and Quantum Fractals


Ph. Blanchard, A. Jadczyk and R. Olkiewicz

Summary: Departing from classical concepts of ergodic theory, formulated in terms of probability densities, measures describing the chaotic behavior and the loss of information in quantum open systems are proposed. As application we discuss the chaotic outcomes of continuous measurement processes in the EEQT framework. Simultaneous measurement of four noncommuting spin components is shown to lead to a chaotic jump on quantum spin sphere and to generate specific fractal images - nonlinear ifs (iterated function system). The model is purely theoretical at this stage, and experimental confirmation of the chaotic behavior of measuring instruments during simultaneous continuous measurement of several noncommuting quantum observables would constitute a quantitative verification of Event Enhanced Quantum Theory.

Keywords:entropy,chaos,quantum,quantum measurement,continuous measurement,event enhanced quantum theory,eeqt,lyapunov,mixing,noncommuting,zeno,zeno effect,markov,classical,spin,tetrahedron,jump,iterated function system,ifs,nonlinear ifs,fractal

Computer simulations of a continuous simultaneous monitoring of four (distributed at the edges of a regular tetrahedron, with all four coupling constants equal to the same parameter value alpha) noncommuting spin projections produced the following quantum state trajectories:

Note: interesting pictures culminate around alpha=0.6. For smaller alpha points are distributed evenly on the sphere. For alpha approaching 1 they tent to concentrate at the four veritces. Selfsimilarity is clearly seen for alpha=0.7

Fig. 0. Quantum state trajectory for alpha=0.1, 10000000 points.

Fig. 1. Quantum state trajectory for alpha=0.2, 10000000 points.

Fig. 1a. Quantum state trajectory for alpha=0.3, 10000000 points.

Fig. 2. Quantum state trajectory for alpha=0.4, 10000000 points.

Fig. 2a. Quantum state trajectory for alpha=0.5, 10000000 points.

Fig. 3. Quantum state trajectory for alpha=0.6, 10000000 points.

Fig. 4a. Quantum state trajectory for alpha=0.7, 1000000000 points.

Fig. 4b. Quantum state trajectory for alpha=0.7, zoom=2x, 1000000000 points.

Fig. 4c. Quantum state trajectory for alpha=0.7, zoom=4x, 1000000000 points.

Fig. 4d. Quantum state trajectory for alpha=0.7, zoom=16x, 1000000000 points.

Fig. 5. Quantum state trajectory for alpha=0.8, 10000000 points.

Fig. 5. Quantum state trajectory for alpha=0.9, 10000000 points.

Fig. 6. Animation through different coupling constants

New paper: "How Events Come Into Being: EEQT, Particle Tracks, Quantum Chaos, and Tunneling Time." This paper has graphics displaying iterated Markov operator .


Last modified on: December 12, 1999.