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Quantum multibaker maps: extreme quantum regime.

Daniel K Wójcik1, J R Dorfman

  • 1Institute for Physical Science and Technology, University of Maryland, College Park 20742, USA. danek@ipst.umd.edu

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|October 9, 2002
PubMed
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We introduce quantum multibaker maps (QMB), models for quantum random walks. Uniform QMB models show extended eigenstates and allow analytic solutions, unlike random QMB models with localized states.

Area of Science:

  • Quantum mechanics
  • Statistical physics
  • Condensed matter theory

Background:

  • Classical multibaker models describe chaotic dynamics.
  • Quantum mechanical random walks are crucial for quantum computing and transport phenomena.
  • Weyl quantization provides a method to translate classical systems into quantum ones.

Purpose of the Study:

  • Introduce and analyze quantum multibaker maps (QMB), a family of models for quantum mechanical, one-dimensional random walks.
  • Investigate two classes of QMB: uniform and random, based on phase properties.
  • Explore the deep quantum regime of uniform QMB and present initial findings for random QMB.

Main Methods:

  • Weyl quantization of classical multibaker models.
  • Analysis of eigenstates for uniform and random QMB (extended vs. localized).

Related Experiment Videos

  • Derivation of analytic solutions for time-dependent states and steady-state properties in uniform QMB under specific conditions (large Planck's constant, periodic/open chains).

    • Main Results:

    • Uniform QMB models exhibit extended eigenstates, while random QMB models display localized eigenstates.
    • Analytic solutions for uniform QMB are achievable for large Planck's constant, including time-dependent states and steady-state relaxation.
    • Numerical results confirm the derived analytical findings for the uniform QMB.

    Conclusions:

    • Quantum multibaker maps offer a tractable framework for studying quantum random walks.
    • The distinction between uniform (extended states) and random (localized states) QMB is significant.
    • Analytical tractability in the deep quantum regime of uniform QMB facilitates understanding of quantum transport and relaxation phenomena.