The quantized Baker's transformation

Summary

A quantum analogue of the Baker's transformation is constructed using a specially developed quantization procedure. We obtain a unitary operator acting on an N-dimensional Hilbert space, with N finite (and even), that has similar properties to the classical baker's map, and reduces to it in the classical limit, which corresponds here to N → ∞. The operator can be described as a very simple, fully explicit N×N matrix. Generalized Baker's maps are also quantized and studied. Numerical investigations confirm that this model has nontrivial features which ought to represent quantal manifestations of classical chaoticity. The quasi-energy spectrum is given by irrational eigenangles, leading to no recurrences. Most eigenfunctions look irregular, but some exhibit puzzling regular features, such as peaks at coordinate values belonging to periodic orbits of the classical Baker's map. We compare the quantal and classical time-evolutions, as applied to initially coherent quasi-classical states: the evolving states stay in close agreement for short times but seem to lose all relationship to each other beyond a critical time of the order of log₂N ∼ − logh.

INTRODUCTION

It is commonly believed that the essential features of chaotic behaviour in the classical Hamiltonian systems are basically understood [1].

The situation is rather different if one turns to the quantal transcription of classical dynamical systems. Classical chaos appears in bound systems, which implies that the quantal Hamiltonian operator has a discrete spectrum.