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Survival of Classical and Quantum Particles in the Presence of Traps

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Abstract

We present a detailed comparison of the motion of a classical and of a quantum particle in the presence of trapping sites, within the framework of continuous-time classical and quantum random walk. The main emphasis is on the qualitative differences in the temporal behavior of the survival probabilities of both kinds of particles. As a general rule, static traps are far less efficient to absorb quantum particles than classical ones. Several lattice geometries are successively considered: an infinite chain with a single trap, a finite ring with a single trap, a finite ring with several traps, and an infinite chain and a higher-dimensional lattice with a random distribution of traps with a given density. For the latter disordered systems, the classical and the quantum survival probabilities obey a stretched exponential asymptotic decay, albeit with different exponents. These results confirm earlier predictions, and the corresponding amplitudes are evaluated. In the one-dimensional geometry of the infinite chain, we obtain a full analytical prediction for the amplitude of the quantum problem, including its dependence on the trap density and strength.

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Notes

  1. The notation \(\ell \mid N\) means that \(\ell \) is a divisor of \(N\), i.e., that \(N\) is a multiple of \(\ell \).

  2. We keep with the long tradition in mathematical physics [15, 16] of using the word sphere, even though the word ball would be more appropriate to describe the volume delimited by a sphere.

References

  1. Montroll, E.W., Weiss, G.H.: Random walks on lattices. II. J. Math. Phys. 6, 167 (1965)

    Article  ADS  MathSciNet  Google Scholar 

  2. Balagurov, B.Y., Vaks, V.G.: Random walk of a particle in a lattice with traps. J.E.T.P. 38, 968 (1974)

    Google Scholar 

  3. Donsker, M., Varadhan, S.R.S.: Asymptotics for the Wiener sausage. Commun. Pure Appl. Math. 28, 525 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  4. Donsker, M., Varadhan, S.R.S.: On the number of distinct sites visited by a random walk. Commun. Pure Appl. Math. 32, 721 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  5. Grassberger, P., Procaccia, I.: The long time properties of diffusion in a medium with static traps. J. Chem. Phys. 77, 6281 (1982)

    Article  ADS  Google Scholar 

  6. Toussaint, D., Wilczek, F.: Particle–antiparticle annihilation in diffusive motion. J. Chem. Phys. 78, 2642 (1983)

    Article  ADS  Google Scholar 

  7. Klafter, J., Zumofen, G., Blumen, A.: Long-time properties of trapping on fractals. J. Phys. (Paris) 45, L49 (1984)

    Article  Google Scholar 

  8. Bramson, M., Lebowitz, J.L.: Asymptotic behavior of densities in diffusion-dominated annihilation reactions. Phys. Rev. Lett. 61, 2397 (1988)

    Article  ADS  Google Scholar 

  9. Bramson, M., Lebowitz, J.L.: Asymptotic behavior of densities in diffusion-dominated annihilation reactions (erratum). Phys. Rev. Lett. 62, 694 (1989)

    Article  ADS  Google Scholar 

  10. Ben-Naim, E., Redner, S., Weiss, G.H.: Partial absorption and ‘virtual’ traps. J. Stat. Phys. 71, 75 (1993)

    Article  ADS  MATH  Google Scholar 

  11. Haus, J.W., Kehr, K.W.: Diffusion in regular and disordered lattices. Phys. Rep. 150, 263 (1987)

    Article  ADS  Google Scholar 

  12. Havlin, S., Ben-Avraham, D.: Diffusion in disordered media. Adv. Phys. 36, 695 (1987)

    Article  ADS  Google Scholar 

  13. Krapivsky, P.L., Redner, S., Ben-Naim, E.: A Kinetic View of Statistical Physics. Cambridge University Press, Cambridge (2010)

    Book  MATH  Google Scholar 

  14. Lifshitz, I.M.: The energy spectrum of disordered systems. Adv. Phys. 13, 483 (1964)

    Article  ADS  Google Scholar 

  15. Lifshitz, I.M., Gredeskul, S.A., Pastur, L.A.: Introduction to the Theory of Disordered Systems. Wiley, New York (1988)

    Google Scholar 

  16. Pastur, L., Figotin, A.: Spectra of Random and Almost-Periodic Operators. Springer, Berlin (1992)

    Book  MATH  Google Scholar 

  17. Rosenstock, H.B.: Random walks on lattices with traps. J. Math. Phys. 11, 487 (1970)

    Article  ADS  Google Scholar 

  18. Barkema, G.T., Biswas, P., van Beijeren, H.: Diffusion with random distribution of static traps. Phys. Rev. Lett. 87, 170601 (2001)

    Article  ADS  Google Scholar 

  19. Friedberg, R., Luttinger, J.M.: Density of electronic energy levels in disordered systems. Phys. Rev. B 12, 4460 (1975)

    Article  ADS  MathSciNet  Google Scholar 

  20. Cardy, J.L.: Electron localisation in disordered systems and classical solutions in Ginzburg–Landau field theory. J. Phys. C 11, L321 (1978)

    Article  ADS  Google Scholar 

  21. Luttinger, J.M., Tao, R.: Electronic density of levels in a disordered system. Ann. Phys. 145, 185 (1983)

    Article  ADS  Google Scholar 

  22. Lubensky, T.C.: Fluctuations in random walks with random traps. Phys. Rev. A 30, 2657 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  23. Nieuwenhuizen, T.M., Luck, J.M.: Singularities in spectra of disordered systems: an instanton approach for arbitrary dimension and randomness. Europhys. Lett. 9, 407 (1989)

    Article  ADS  Google Scholar 

  24. Nieuwenhuizen, T.M.: Trapping and Lifshitz tails in random media, self-attracting polymers, and the number of distinct sites visited: a renormalized instanton approach in three dimensions. Phys. Rev. Lett. 62, 357 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  25. Zee, A., Affleck, I.: Hopping between random locations: spectrum and instanton. J. Phys. Cond. Matt. 12, 8863 (2000)

    Article  ADS  Google Scholar 

  26. Parris, P.E.: One-dimensional trapping kinetics at zero temperature. Phys. Rev. Lett. 62, 1392 (1989)

    Article  ADS  Google Scholar 

  27. Parris, P.E.: One-dimensional quantum transport in the presence of traps. Phys. Rev. B 40, 4928 (1989)

    Article  ADS  Google Scholar 

  28. Edwards, J.W., Parris, P.E.: Quantum transport in the presence of random traps. Phys. Rev. B 40, 8045 (1989)

    Article  ADS  Google Scholar 

  29. Parris, P.E.: Quantum and stochastic aspects of low-temperature trapping and reaction dynamics. J. Stat. Phys. 65, 1161 (1991)

    Article  ADS  Google Scholar 

  30. Pearlstein, R.M.: Impurity quenching of molecular excitons. I. Kinetic comparison of Förster–Dexter and slowly quenched Frenkel excitons in linear chains. J. Chem. Phys. 56, 2431 (1972)

    Article  ADS  Google Scholar 

  31. Hemenger, R.P., Lakatos-Lindenberg, K., Pearlstein, R.M.: Impurity quenching of molecular excitons. III. Partially coherent excitons in linear chains. J. Chem. Phys. 60, 3271 (1974)

    Article  ADS  Google Scholar 

  32. Huber, D.L.: Decay of the \( \overrightarrow{{\rm k}} = 0\) exciton mode at a finite trap concentration. Phys. Rev. B 24, 1083 (1981)

    Article  ADS  Google Scholar 

  33. Benk, H., Silbey, R.: Exciton-phonon scattering and exciton trapping in one-dimensional exciton systems. J. Chem. Phys. 79, 3487 (1983)

    Article  ADS  Google Scholar 

  34. Aharonov, Y., Davidovich, L., Zagury, N.: Quantum random walks. Phys. Rev. A 48, 1687 (1993)

    Article  ADS  Google Scholar 

  35. Farhi, E., Gutmann, S.: Quantum computation and decision trees. Phys. Rev. A 58, 915 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  36. Childs, A.M., Goldstone, J.: Spatial search by quantum walk. Phys. Rev. A 70, 022314 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  37. Childs, A.M.: Universal computation by quantum walk. Phys. Rev. Lett. 102, 180501 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  38. Childs, A.M.: On the relationship between continuous- and discrete-time quantum walk. Commun. Math. Phys. 294, 581 (2010)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  39. Kempe, J.: Quantum random walks: an introductory overview. Contemp. Phys. 44, 307 (2003)

    Article  ADS  Google Scholar 

  40. Ambainis, A.: Quantum walks and their algorithmic applications. Int. J. Quant. Inf. 1, 507 (2003)

    Article  MATH  Google Scholar 

  41. Perets, H.B., Lahini, Y., Pozzi, F., Sorel, M., Morandotti, R., Silberberg, Y.: Realization of quantum walks with negligible decoherence in waveguide lattices. Phys. Rev. Lett. 100, 170506 (2008)

    Article  ADS  Google Scholar 

  42. Peruzzo, A., Lobino, M., Matthews, J.C.F., Matsuda, N., Politi, A., Poulios, K., Zhou, X.Q., Lahini, Y., Ismail, N., Wörhoff, K., Bromberg, Y., Silberberg, Y., Thompson, M.G., OBrien, J.L.: Quantum walks of correlated photons. Science 329, 1500 (2010)

    Article  ADS  Google Scholar 

  43. Konno, N.: Quantum random walks in one dimension. Quant. Inf. Process. 1, 345 (2002)

    Article  MathSciNet  Google Scholar 

  44. Mackay, T.D., Bartlett, S.D., Stephenson, L.T., Sanders, B.C.: Quantum walks in higher dimensions. J. Phys. A 35, 2745 (2002)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  45. Tregenna, B., Flanagan, W., Maile, R., Kendon, V.: Controlling discrete quantum walks: coins and initial states. New J. Phys. 5, 83 (2003)

    Article  ADS  Google Scholar 

  46. de Toro Arias, S., Luck, J.M.: Anomalous dynamical scaling and bifractality in the one-dimensional Anderson model. J. Phys. A 31, 7699 (1998)

    Article  ADS  MATH  Google Scholar 

  47. ben-Avraham, D., Bollt, E.M.: One-dimensional continuous-time quantum walks. Quant. Inf. Process. 3, 295 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  48. Akkermans, E., Comtet, A., Desbois, J., Montambaux, G., Texier, C.: Spectral determinant on quantum graphs. Ann. Phys. 284, 10 (2000)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  49. Strauch, F.W.: Connecting the discrete- and continuous-time quantum walks. Phys. Rev. A 74, 030301 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  50. Venegas-Andraca, S.E.: Quantum walks: a comprehensive review. Quant. Inf. Process. 11, 1015 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  51. Haroche, S., Raimond, J.M.: Exploring the Quantum: Atoms, Cavities, and Photons. Oxford University Press, New York (2006)

    Book  Google Scholar 

  52. Konno, N., Namiki, T., Soshi, T., Sudbury, A.: Absorption problems for quantum walks in one dimension. J. Phys. A 36, 241 (2003)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  53. Mülken, O., Blumen, A., Amthor, T., Giese, C., Reetz-Lamour, M., Weidenmüller, M.: Survival probabilities in coherent exciton transfer with trapping. Phys. Rev. Lett. 99, 090601 (2007)

    Article  Google Scholar 

  54. Mülken, O., Pernice, V., Blumen, A.: Slow excitation trapping in quantum transport with long-range interactions. Phys. Rev. E 78, 021115 (2008)

    Article  ADS  Google Scholar 

  55. Agliari, E., Mülken, O., Blumen, A.: Continuous-time quantum walks and trapping. Int. J. Bif. Chaos 20, 271 (2010)

    Article  MATH  Google Scholar 

  56. Agliari, E.: Trapping of continuous-time quantum walks on Erdös-Rényi graphs. Phys. A 390, 1853 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  57. Mülken, O., Blumen, A.: Continuous-time quantum walks: models for coherent transport on complex networks. Phys. Rep. 502, 37 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  58. Gönülol, M., Aydiner, E., Shikano, Y., Müstecaplioglu, Ö.E.: Survival probability in a one-dimensional quantum walk on a trapped lattice. New J. Phys. 13, 033037 (2011)

    Article  Google Scholar 

  59. Ampadu, C., Gönülol, M., Aydiner, E.: Survival probability of the quantum walk with phase parameters on the two-dimensional trapped lattice. J. Quant. Inf. Sc. 3, 51 (2013)

    Article  Google Scholar 

  60. Anishchenko, A., Blumen, A., Mülken, O.: Geometrical aspects of quantum walks on random two-dimensional structures. Phys. Rev. E 88, 062126 (2013)

    Article  ADS  Google Scholar 

  61. Nieuwenhuizen, T.M., Luck, J.M.: Lifshitz singularities in random harmonic chains: periodic amplitudes near the band edge and near special frequencies. J. Stat. Phys. 48, 393 (1987)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  62. Luck, J.M.: Systèmes désordonnés unidimensionnels. Collection Aléa, Saclay (1992)

    Google Scholar 

  63. Selstø, S.: Formulae for partial widths derived from the Lindblad equation. Phys. Rev. A 85, 062518 (2012)

    Article  ADS  Google Scholar 

  64. Selstø, S., Kvaal, S.: Absorbing boundary conditions for dynamical many-body quantum systems. J. Phys. B 43, 065004 (2010)

    Article  ADS  Google Scholar 

  65. Diaz-Torres, A.: Absence of decoherence in the complex-potential approach to nuclear scattering. Phys. Rev. C 81, 041603(R) (2010)

    Article  ADS  Google Scholar 

  66. Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products. Academic, New York (1965)

    Google Scholar 

  67. Holz, D.E., Orland, H., Zee, A.: On the remarkable spectrum of a non-Hermitian random matrix model. J. Phys. A. 36, 3385 (2003)

    Article  ADS  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Physics, Boston University, Boston, MA , 02215, USA

    P. L. Krapivsky

  2. Institut de Physique Théorique, CEA Saclay and CNRS URA 2306, 91191 , Gif-sur-Yvette Cedex, France

    J. M. Luck & K. Mallick

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  1. P. L. Krapivsky
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  2. J. M. Luck
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  3. K. Mallick
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Correspondence to J. M. Luck.

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Krapivsky, P.L., Luck, J.M. & Mallick, K. Survival of Classical and Quantum Particles in the Presence of Traps. J Stat Phys 154, 1430–1460 (2014). https://doi.org/10.1007/s10955-014-0936-8

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