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Communications in Mathematical Physics

, Volume 267, Issue 1, pp 65–92 | Cite as

Quantum States on Harmonic Lattices

  • Norbert SchuchEmail author
  • J. Ignacio Cirac
  • Michael M. Wolf
Article

Abstract

We investigate bosonic Gaussian quantum states on an infinite cubic lattice in arbitrary spatial dimensions. We derive general properties of such states as ground states of quadratic Hamiltonians for both critical and non-critical cases. Tight analytic relations between the decay of the interaction and the correlation functions are proven and the dependence of the correlation length on band gap and effective mass is derived. We show that properties of critical ground states depend on the gap of the point-symmetrized rather than on that of the original Hamiltonian. For critical systems with polynomially decaying interactions logarithmic deviations from polynomially decaying correlation functions are found.

Keywords

Correlation Length Gaussian State Hamiltonian Matrix Polynomial Decay Correlation Decay 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Plenio M.B., Eisert J., Dreissig J., Cramer M. (2005) Entropy, entanglement, and area: analytical results for harmonic lattice systems. Phys. Rev. Lett. 94, 060503CrossRefADSMathSciNetGoogle Scholar
  2. 2.
    Cramer M., Eisert J., Plenio M.B., Dreissig J. (2005) An entanglement-area law for general bosonic harmonic lattice systems. Phys. Rev. A 73, 012309CrossRefADSGoogle Scholar
  3. 3.
    Wolf M.M. (2005) Violation of the Entropic area law for fermions. Phys. Rev. Lett. 96, 010404CrossRefGoogle Scholar
  4. 4.
    Audenaert K., Eisert J., Plenio M.B., Werner R.F. (2002) Entanglement properties of the harmonic chain. Phys. Rev. A 66, 042327CrossRefADSGoogle Scholar
  5. 5.
    Botero A., Reznik B. (2004) Spatial structures and localization of vacuum entanglement in the linear harmonic chain. Phys. Rev. A 70, 052329CrossRefADSGoogle Scholar
  6. 6.
    Asoudeh M., Karimipour V. (2005) Entanglement of bosonic modes in symmetric graphs. Phys. Rev. A 72, 0332339CrossRefADSGoogle Scholar
  7. 7.
    Plenio M.B., Semiao F.L. (2005) High efficiency transfer of quantum information and multi-particle entanglement generation in translation invariant quantum chains. New J. Phys. 7,73CrossRefADSGoogle Scholar
  8. 8.
    Plenio M.B., Hartley J., Eisert J. (2004) Dynamics and manipulation of entanglement in coupled harmonic systems with many degrees of freedom. New J. Phys. 6, 36CrossRefADSGoogle Scholar
  9. 9.
    Eisert J., Plenio M.B., Bose S., Hartley J. (2004) Towards mechanical entanglement in nano-electromechanical devices. Phys. Rev. Lett. 93, 190402CrossRefADSGoogle Scholar
  10. 10.
    Wolf M.M., Verstraete F., Cirac J.I. (2003) Entanglement and frustration in ordered systems. Int. J. Quant. Inf. 1, 465zbMATHCrossRefGoogle Scholar
  11. 11.
    Wolf M.M., Verstraete F., Cirac J.I. (2004) Entanglement frustration for gaussian states on symmetric graphs. Phys. Rev. Lett. 92, 087903CrossRefADSGoogle Scholar
  12. 12.
    Nachtergaele B., Sims R. (2006) Lieb-robinson bounds and the exponential clustering theorem. Commun. Math. Phys. 265, 119zbMATHCrossRefADSMathSciNetGoogle Scholar
  13. 13.
    Hastings M.B., Koma T. Spectral gap and exponential decay of correlations. http://arxiv.org/list/ math-ph/0507008, 2005Google Scholar
  14. 14.
    Cramer M., Eisert J. (2006) Correlations and spectral gap in harmonic quantum systems on generic lattices. New J. Phys. 8, 71CrossRefADSMathSciNetGoogle Scholar
  15. 15.
    Auerbach A. (1994) Interacting electrons and quantum magnetism. Springer Verlag, New YorkGoogle Scholar
  16. 16.
    James D.F.V. (1998) Quantum dynamics of cold trapped ions, with application to quantum computation. Appl. Phys. B 66, 181CrossRefADSGoogle Scholar
  17. 17.
    Williamson J. (1936) Amer. J. Math. 58, 141zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Birkl G., Kassner S., Walther H. (1992) Multiple-shell structures of laser-cooled Mg ions in a quadrupole storage ring. Nature 357, 310CrossRefADSGoogle Scholar
  19. 19.
    Dubin D.H.E. (1993) Theory of structural phase transitions in a Coulomb crystal. Phys. Rev. Lett. 71: 2753CrossRefADSGoogle Scholar
  20. 20.
    Enzer D.G., Schauer M.M., Gomez J.J., Gulley M.S., et al. (2000) Observation of power-law scaling for phase transitions in linear trapped ion crystals. Phys. Rev. Lett. 85: 2466CrossRefADSGoogle Scholar
  21. 21.
    Mitchell T.B., Bollinger J.J., Dubin D.H.E., Huang X.-P., Itano W.M., Baugham R.H. (1998) Direct observations of structural phase transitions in planar crystallized ion plasmas. Science 282: 1290CrossRefADSGoogle Scholar
  22. 22.
    Manuceau J., Verbeure A. (1968) Quasi-free states of the C.C.R.–Algebra and Bogoliubov transformations. Commun. Math. Phys. 9, 293zbMATHCrossRefADSMathSciNetGoogle Scholar
  23. 23.
    Holevo A.S. (1971) Quasi-free states on the C*-algebra of CCR. Theor. Math. Phys. 6, 1CrossRefGoogle Scholar
  24. 24.
    Arvind, Dutta B., Mukunda N., Simon R. (1995) The real symplectic groups in quantum mechanics and optics. Pramana 45, 471CrossRefADSGoogle Scholar
  25. 25.
    Wolf M.M., Giedke G., Krüger O., Werner R.F., Cirac J.I. (2004) Gaussian entanglement of formation. Phys. Rev. A 69, 052320CrossRefADSGoogle Scholar
  26. 26.
    Benzi M., Golub G.H. (1999). BIT Numerical Mathematics 39: 417zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Bleistein N., Handelsman R.A. (1986) Asymptotic expansions of integrals. Dover Publication, New YorkGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • Norbert Schuch
    • 1
    Email author
  • J. Ignacio Cirac
    • 1
  • Michael M. Wolf
    • 1
  1. 1.Max-Planck-Institut für QuantenoptikGarchingGermany

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