International Journal of Theoretical Physics

, Volume 51, Issue 11, pp 3536–3550 | Cite as

Analysis Technique for Exceptional Points in Open Quantum Systems and QPT Analogy for the Appearance of Irreversibility

  • Savannah Garmon
  • Ingrid Rotter
  • Naomichi Hatano
  • Dvira Segal
Article

Abstract

We propose an analysis technique for the exceptional points (EPs) occurring in the discrete spectrum of open quantum systems (OQS), using a semi-infinite chain coupled to an endpoint impurity as a prototype. We outline our method to locate the EPs in OQS, further obtaining an eigenvalue expansion in the vicinity of the EPs that gives rise to characteristic exponents. We also report the precise number of EPs occurring in an OQS with a continuum described by a quadratic dispersion curve. In particular, the number of EPs occurring in a bare discrete Hamiltonian of dimension nD is given by nD(nD−1); if this discrete Hamiltonian is then coupled to continuum (or continua) to form an OQS, the interaction with the continuum generally produces an enlarged discrete solution space that includes a greater number of EPs, specifically \(2^{n_{\mathrm{C}}} (n_{\mathrm{C}} + n_{\mathrm{D}} ) [ 2^{n_{\mathrm{C}}} (n_{\mathrm{C}} + n_{\mathrm{D}} ) - 1] \), in which nC is the number of (non-degenerate) continua to which the discrete sector is attached. Finally, we offer a heuristic quantum phase transition analogy for the emergence of the resonance (giving rise to irreversibility via exponential decay) in which the decay width plays the role of the order parameter; the associated critical exponent is then determined by the above eigenvalue expansion.

Keywords

Exceptional points Open quantum systems Resonant state Time irreversibility Exponential decay Phase transition 

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Savannah Garmon
    • 1
  • Ingrid Rotter
    • 2
  • Naomichi Hatano
    • 3
  • Dvira Segal
    • 1
  1. 1.Chemical Physics Theory Group, Department of Chemistry and Center for Quantum Information and Quantum ControlUniversity of TorontoTorontoCanada
  2. 2.Max-Planck-Institut für Physik Komplexer SystemeDresdenGermany
  3. 3.Institute of Industrial ScienceUniversity of TokyoTokyoJapan

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