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A variance reduction technique for the stochastic Liouville–von Neumann equation

  • Konstantin Schmitz
  • Jürgen T. StockburgerEmail author
Regular Article
  • 4 Downloads
Part of the following topical collections:
  1. Non-equilibrium Dynamics: Quantum Systems and Foundations of Quantum Mechanics

Abstract

The stochastic Liouville–von Neumann equation provides an exact numerical simulation strategy for quantum systems interacting with Gaussian reservoirs [J.T. Stockburger, H. Grabert, PRL 88, 170407 (2002)]. Its scaling with the extension of the time interval covered has recently improved dramatically through time-domain projection techniques [J.T. Stockburger, EPL 115, 40010 (2016)]. Here, we present a sampling strategy which results in a significantly improved scaling with the strength of the dissipative interaction, based on reducing the non-unitary terms in sample propagation through convex optimization techniques.

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References

  1. 1.
    R. Alicki, K. Lendi, in Quantum Dynamical Semigroups and Applications, Lecture Notes in Physics (Springer, Berlin, 1987), Vol. 286 Google Scholar
  2. 2.
    E.B. Davies, Commun. Math. Phys. 39, 91 (1974) ADSCrossRefGoogle Scholar
  3. 3.
    H.-P. Breuer, F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2002), p. 625 Google Scholar
  4. 4.
    A. Levy, R. Kosloff, EPL (Europhys. Lett.) 107, 20004 (2014) ADSCrossRefGoogle Scholar
  5. 5.
    J.T. Stockburger, T. Motz, Fortschr. Phys. 65, 1600067 (2017) CrossRefGoogle Scholar
  6. 6.
    R. Alicki, D.A. Lidar, P. Zanardi, Phys. Rev. A 73, 052311 (2006) ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    R. Schmidt et al., Phys. Rev. Lett. 107, 130404 (2011) ADSCrossRefGoogle Scholar
  8. 8.
    R.P. Feynman, F.L. Vernon, Ann. Phys. (N.Y.) 24, 118 (1963) ADSCrossRefGoogle Scholar
  9. 9.
    U. Weiss, in Quantum Dissipative Systems, Series in Modern Condensed Matter Physics, 3rd edn. (World Scientific, Singapore, 2008), Vol. 13 Google Scholar
  10. 10.
    R. Egger, L. Mühlbacher, C.H. Mak, Phys. Rev. E 61, 5961 (2000) ADSCrossRefGoogle Scholar
  11. 11.
    L. Mühlbacher, J. Ankerhold, C. Escher, J. Chem. Phys. 121, 12696 (2004) ADSCrossRefGoogle Scholar
  12. 12.
    Y. Tanimura, P.G. Wolynes, Phys. Rev. A 43, 4131 (1991) ADSCrossRefGoogle Scholar
  13. 13.
    Y. Tanimura, J. Chem. Phys. 141, 044114 (2014) ADSCrossRefGoogle Scholar
  14. 14.
    J.T. Stockburger, H. Grabert, Phys. Rev. Lett. 88, 170407 (2002) ADSCrossRefGoogle Scholar
  15. 15.
    J. Cao, L.W. Ungar, G.A. Voth, J. Chem. Phys. 104, 4189 (1996) ADSCrossRefGoogle Scholar
  16. 16.
    W.T. Strunz, Phys. Lett. A 224, 25 (1996) ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    J. Shao, J. Chem. Phys. 120, 5053 (2004) ADSCrossRefGoogle Scholar
  18. 18.
    Y. Tanimura, J. Phys. Soc. Jpn. 75, 082001 (2006) CrossRefGoogle Scholar
  19. 19.
    J.T. Stockburger, EPL (Europhys. Lett.) 115, 40010 (2016) ADSCrossRefGoogle Scholar
  20. 20.
    R. Kubo, J. Phys. Soc. Jpn. 17, 1100 (1962) CrossRefGoogle Scholar
  21. 21.
    C.W. Gardiner, in Stochastic Methods: A Handbook for the Natural and Social Sciences, Springer Series in Synergetics, 4th edn. (Springer, Berlin, 2009), Vol. 13 Google Scholar
  22. 22.
    H. Imai, Y. Ohtsuki, H. Kono, Chem. Phys. 446, 134 (2015) CrossRefGoogle Scholar
  23. 23.
    J.T. Stockburger, Chem. Phys. 296, 159 (2004) CrossRefGoogle Scholar
  24. 24.
    W. Koch, F. Großmann, J.T. Stockburger, J. Ankerhold, Phys. Rev. Lett. 100, 230402 (2008) ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    M. Grant, S. Boyd, CVX: Matlab Software for Disciplined Convex Programming, version 2.1 (2014), https://doi.org/cvxr.com/cvx (accessed on December 16, 2017)
  26. 26.
    M. Grant, S. Boyd, in Recent Advances in Learning and Control, Lecture Notes in Control and Information Sciences, edited by V. Blondel, S. Boyd, H. Kimura (Springer, Berlin, 2008), pp. 95–110 Google Scholar
  27. 27.
    A.J. Leggett et al., Rev. Mod. Phys. 59, 1 (1987) ADSCrossRefGoogle Scholar
  28. 28.
    A.J. Leggett et al., Rev. Mod. Phys. 67, 725 (1995) (erratum) ADSCrossRefGoogle Scholar

Copyright information

© EDP Sciences, Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute for Complex Quantum Systems, Ulm UniversityUlmGermany

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