Communications in Mathematical Physics

, Volume 320, Issue 1, pp 37–71 | Cite as

Wave Decoherence for the Random Schrödinger Equation with Long-Range Correlations

Article

Abstract

In this paper, we study the loss of coherence of a wave propagating according to the Schrödinger equation with a time-dependent random potential. The random potential is assumed to have slowly decaying correlations. The main tool to analyze the decoherence phenomena is a properly rescaled Wigner transform of the solution of the random Schrödinger equation. We exhibit anomalous wave decoherence effects at different propagation scales.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adler R.J., Taylor J.: Random fields and geometry. Springer, New York (2007)MATHGoogle Scholar
  2. 2.
    Bal G.: On the self-averaging of wave energy in random media. SIAM Multiscale Mo. Sim. 2, 398–420 (2004)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Bal G., Papanicolaou G., Ryzhik L.: Radiative transport limit for the random Schrödinger equation. Nonlinearity 15, 513–529 (2002)MathSciNetADSMATHCrossRefGoogle Scholar
  4. 4.
    Bal G., Papanicolaou G., Ryzhik L.: Self-averaging in time reversal for the parabolic wave equation. Stoch. Dyn. 2, 507–532 (2002)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Bal G., Komorowski T., Ryzhik L.: Self-averaging of the Wigner transform in random media. Commun. Math. Phys. 242, 81–135 (2003)MathSciNetADSMATHGoogle Scholar
  6. 6.
    Bal G., Komorowski T., Ryzhik L.: Kinetic limits for waves in a random medium. Kin. Rela. Mod. 3, 529–644 (2010)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Bal G., Komorowski T., Ryzhik L.: Asymptotics of the phase of the solutions of the random Schrödinger equation. Arch. Rat. Mech. Anal. 100, 613–664 (2011)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Billingsley, P.: Convergence of probability measure. 2nd ed., New York: Wiley InterScience, 1999Google Scholar
  9. 9.
    Blomgren P., Papanicolaou G., Zhao H.: Super-resolution in time-reversal acoustics. J. Acoust. Soc. Am. 111, 230–248 (2002)ADSCrossRefGoogle Scholar
  10. 10.
    Caffarelli L., Chan C.H., Vasseur A.: Regularity theory for nonlinear integral operators. J. Amer. Math. Soc. 24, 849–869 (2011)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Da Prato, G., Zabczyk, J.: Stochastic equations in infinite dimensions. Encyclopedia of Mathematics and its applications 44, Cambridge: Cambridge University Press, 1992Google Scholar
  12. 12.
    Dolan S., Bean C., Riollet B.: The broad-band fractal nature of heterogeneity in the upper crust from petrophysical logs. Geophys. J. Int. 132, 489–507 (1998)Google Scholar
  13. 13.
    Erdös L., Salmhofer M., Yau H.-T.: Quantum diffusion of the random Schrödinger evolution in the scaling limit I. The non-recollision diagrams. Acta Math. 200, 211–277 (2008)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Erdös L., Salmhofer M., Yau H.-T.: Quantum diffusion of the random Schrödinger evolution in the scaling limit II. The recollision diagrams. Commun. Math. Phys. 271, 1–53 (2007)ADSMATHCrossRefGoogle Scholar
  15. 15.
    Erdös L., Yau T.H.: Linear Boltzmann equation as the weak coupling limit of a random Schrödinger Equation. Comm. Pure Appl. Math. 53, 667–735 (2000)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Fannjiang A.: Self-Averaging Scaling Limits for Random Parabolic Waves. Arch. Rat. Mech. Anal. 175, 343–387 (2005)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Fannjiang A.: White-Noise and Geometrical Optics Limits of Wigner-Moyal Equation for Wave Beams in Turbulent Media. Commun. Math. Phys. 254, 289–322 (2005)MathSciNetADSMATHCrossRefGoogle Scholar
  18. 18.
    Fannjiang A., Solna K.: Propagation and Time-reversal of Wave Beams in Atmospheric Turbulence. SIAM Multiscale Mod. Sim. 3, 522–558 (2005)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Fouque J.-P.: La convergence en loi pour les processus à à valeur dans un espace nucléaire. Ann. Inst. Henri Poincaré 20, 225–245 (1984)MathSciNetMATHGoogle Scholar
  20. 20.
    Garnier J., Papanicolaou G.: Passive sensor imaging using cross correlations of noisy signals in a scattering medium. SIAM J. Imag. Sci. 2, 396–437 (2009)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Garnier J., Sølna K.: Pulse propagation in random media with long-range correlation. SIAM Multiscale Mod. Sim. 7, 1302–1324 (2009)MATHCrossRefGoogle Scholar
  22. 22.
    Gérard P., Markowich P., Mauser N., Poupaud F.: Homogenization limits and Wigner transforms. Comm. Pure Appl. Math. 50, 323–380 (1997)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Gomez C.: Radiative Transport Limit for the Random Schrödinger Equation with Long-Range Correlations. J. Math. Pures. Appl. 98(3), 295–327 (2012)MathSciNetMATHGoogle Scholar
  24. 24.
    Ho T., Landau L., Wilkins A.: On the weak coupling limit for a Fermi gas in a random potential. Rev. Math. Phys. 5, 209–298 (1993)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic. Calculus New York, Springer, 2nd edition 1991Google Scholar
  26. 26.
    Komorowski T., Ryzhik L.: On asymptotics of a tracer advected in a locally self-similar correlated flow. Asymp. Anal. 53, 159–187 (2007)MathSciNetMATHGoogle Scholar
  27. 27.
    Kushner, H.-J.: Approximation and weak convergence methods for random processes. Cambridge, MA: MIT Press, 1984Google Scholar
  28. 28.
    Lions P.L., Paul T.: Sur les mesures de Wigner. Rev. Mat. Iberoamer. 9, 553–618 (1993)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Lukkarinen J., Spohn H.: Kinetic limit for wave propagation in a random medium. Arch. Rat. Mech. Anal. 183, 93–162 (2007)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Marty R., Sølna K.: Acoustic waves in long range random media. SIAM J. Appl. Math. 69, 1065–1083 (2009)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Marty R., Solna K.: A general framework for waves in random media with long-range correlations. Ann. Appl. Probab. 21, 115–139 (2011)MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Mitoma I.: On the sample continuity of \({\mathcal{S}'}\) -processes. J. Math. Soc. Japan 35, 629–636 (1983)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Papanicolaou G., Weinryb S.: A functional limit theorem for waves reflected by a random medium. Appl. Math. Optim. 30, 307–334 (1994)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Picard J.: On the existence of smooth densities for jump processes. Probab. Theory Rel. Fields 105, 481–511 (1996)MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Rozanov, Y.A.: Stationary random processes. San Francisco, CA: Holden-Day, Inc., 1967Google Scholar
  36. 36.
    Sato, K.: Lévy processes and infinitely divisible distributions. New York: Cambridge University Press, 1999Google Scholar
  37. 37.
    Sidi C., Dalaudier F.: Turbulence in the stratified atmosphere: Recent theoretical developments and experimental results. Adv. in Space Res. 10, 25–36 (1990)ADSCrossRefGoogle Scholar
  38. 38.
    Sølna K.: Acoustic pulse spreading in a random fractal. SIAM J. Appl. Math. 63, 1764–1788 (2003)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Spohn H.: Derivation of the transport equation for electrons moving through random impurities. J. Stat. Phys. 17, 385–412 (1977)MathSciNetADSMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of MathematicsStanford UniversityCaliforniaUSA
  2. 2.Laboratoire d’Analyse, Topologie, Probabilités, UMR 7353Aix-Marseille UniversityMarseilleCedex 7France

Personalised recommendations