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Completely Mixing Quantum Open Systems and Quantum
Fractals
by
##

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 .

Homepage: http://www.cassiopaea.org/quantum_future/homepage.htm

Last modified on: December 12, 1999.
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