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Quantum State Engineering of Light with Continuous-wave Optical Parametric Oscillators
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On Quantum Ergodicity for Higher Dimensional Cat Maps.

Pär Kurlberg1, Alina Ostafe2, Zeev Rudnick3

  • 1Department of Mathematics, Royal Institute of Technology, 100 44 Stockholm, Sweden.

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PubMed
Summary
This summary is machine-generated.

We show that eigenfunctions of higher dimensional cat maps become uniformly distributed for most quantum parameters. This extends previous results for two-dimensional systems, offering new insights into quantum chaos.

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Area of Science:

  • Quantum chaos
  • Mathematical physics
  • Number theory

Background:

  • Cat maps are a model for quantum chaos, defined by linear symplectic maps.
  • Eigenfunction localization is a key phenomenon in quantum chaotic systems.
  • Previous work established uniform distribution for 2D cat maps.

Purpose of the Study:

  • To investigate eigenfunction localization in higher dimensional cat maps.
  • To extend the understanding of quantum chaos models beyond two dimensions.
  • To prove uniform distribution of eigenfunctions for a density one sequence of quantum parameters.

Main Methods:

  • Utilizing tools from additive combinatorics, including Bourgain's bound for Mordell sums.
  • Analyzing the tensor product structures specific to higher dimensional cat maps.
  • Developing new mathematical techniques to handle the increased complexity of higher dimensions.

Main Results:

  • Demonstrated uniform distribution of eigenfunctions for a density one sequence of integers N.
  • Showed that this result holds for higher dimensional cat maps (g > 1).
  • The methods employed differ significantly from those used for the 2D case.

Conclusions:

  • The study successfully extends results on eigenfunction distribution to higher dimensions.
  • New mathematical tools, particularly from additive combinatorics, are crucial for higher dimensional analysis.
  • This work deepens the understanding of quantum chaos in more complex systems.