Advertisement

The European Physical Journal Special Topics

, Volume 224, Issue 11, pp 2127–2171 | Cite as

The dissipative Bose-Hubbard model

Methods and examples
  • G. Kordas
  • D. Witthaut
  • P. Buonsante
  • A. Vezzani
  • R. Burioni
  • A. I. Karanikas
  • S. Wimberger
Review
Part of the following topical collections:
  1. The Dissipative Bose-Hubbard Model - Methods and Examples

Abstract

Open many-body quantum systems have attracted renewed interest in the context of quantum information science and quantum transport with biological clusters and ultracold atomic gases. The physical relevance in many-particle bosonic systems lies in the realization of counter-intuitive transport phenomena and the stochastic preparation of highly stable and entangled many-body states due to engineered dissipation. We review a variety of approaches to describe an open system of interacting ultracold bosons which can be modeled by a tight-binding Hubbard approximation. Going along with the presentation of theoretical and numerical techniques, we present a series of results in diverse setups, based on a master equation description of the dissipative dynamics of ultracold bosons in a one-dimensional lattice. Next to by now standard numerical methods such as the exact unravelling of the master equation by quantum jumps for small systems and beyond mean-field expansions for larger ones, we present a coherent-state path integral formalism based on Feynman-Vernon theory applied to a many-body context.

Keywords

European Physical Journal Special Topic Master Equation Phase Noise Wigner Function Dark Soliton 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    E.A. Cornell, C.E. Wieman, Rev. Mod. Phys. 74, 875 (2002)ADSCrossRefGoogle Scholar
  2. 2.
    W. Ketterle, Rev. Mod. Phys. 74, 1131 (2002)ADSCrossRefGoogle Scholar
  3. 3.
    M. Albiez, R. Gati, J. Fölling, S. Hunsmann, M. Cristiani, M.K. Oberthaler, Phys. Rev. Lett. 95, 010402 (2005)ADSCrossRefGoogle Scholar
  4. 4.
    C. Gross, T. Zibold, E. Nicklas, J. Estève, M.K. Oberthaler, Nature 464, 1165 (2010)ADSCrossRefGoogle Scholar
  5. 5.
    W.S. Bakr, J.I. Gillen, A. Peng, S. Foelling, M. Greiner, Nature 462, 74 (2009)ADSCrossRefGoogle Scholar
  6. 6.
    M. Miranda, R. Inoue, Y. Okuyama, A. Nakamoto, M. Kozuma, Phys. Rev. A 91, 063414 (2015)ADSCrossRefGoogle Scholar
  7. 7.
    J.F. Sherson, C. Weitenberg, M. Endres, M. Cheneau, I. Bloch, S. Kuhr, Nature 467, 68 (2010)ADSCrossRefGoogle Scholar
  8. 8.
    C. Weitenberg, M. Enders, J.F. Sherson, M. Cheneau, P. Schauß, T. Fukuhara, I. Bloch, S. Kuhr, Nature 471, 319 (2011)ADSCrossRefGoogle Scholar
  9. 9.
    R. Labouvie, B. Santra, S. Heun, S. Wimberger, H. Ott, Phys. Rev. Lett. 115, 05060 (2015)CrossRefGoogle Scholar
  10. 10.
    T. Gerike, P. Würtz, D. Reitz, T. Langen, H. Ott, Nature Phys. 4, 949 (2008)ADSCrossRefGoogle Scholar
  11. 11.
    J. Hubbard, Proceedings of the Royal Society of London 276, 238 (1963)Google Scholar
  12. 12.
    R. Grimm, M. Weidemüller, Y.B. Ovchinnikov, Adv. Atom. Mol. Opt. Phys. 42, 95 (2000)ADSCrossRefGoogle Scholar
  13. 13.
    O. Morsch, M. Oberthaler, Rev. Mod. Phys. 78, 179 (2006)ADSCrossRefGoogle Scholar
  14. 14.
    I. Bloch, J. Dalibard, W. Zwerger, Rev. Mod. Phys. 80, 885 (2008)ADSCrossRefGoogle Scholar
  15. 15.
    M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen(De), U. Sen, Adv. Phys. 56, 243 (2007)ADSCrossRefGoogle Scholar
  16. 16.
    T. Langen, et al., Science 348, 207 (2015)MathSciNetADSCrossRefGoogle Scholar
  17. 17.
    S. Braun, et al., PNAS 112, 3641 (2015)Google Scholar
  18. 18.
    J.P. Brantut, J. Meineke, D. Stadler, S. Krinner, T. Esslinger, Science 337, 1069 (2012)ADSCrossRefGoogle Scholar
  19. 19.
    S. Eckel, J.G. Lee, F. Jendrzejewski, N. Murray, C.W. Clark, C.J. Lobb, W.D. Phillips, M. Edwards, G.K. Campbell, Nature 506, 200 (2014)ADSCrossRefGoogle Scholar
  20. 20.
    I. Bloch, J. Dalibard, S. Nascimbene, Nat. Phys. 8, 267 (2012)CrossRefGoogle Scholar
  21. 21.
    M.P.A. Fisher, P.B. Weichman, G. Grinstein, D.S. Fisher, Phys. Rev. B 40, 546 (1989)ADSCrossRefGoogle Scholar
  22. 22.
    D. Jaksch, C. Bruder, J.I. Cirac, C.W. Gardiner, P. Zoller, Phys. Rev. Lett. 81, 3108 (1998)ADSCrossRefGoogle Scholar
  23. 23.
    D. Jaksch, P. Zoller, Ann. Phys. 315, 52 (2005)zbMATHADSCrossRefGoogle Scholar
  24. 24.
    A. Tomadin, S. Diehl, M.D. Lukin, P. Rabl, P. Zoller, Phys. Rev. A 86, 033821 (2012)ADSCrossRefGoogle Scholar
  25. 25.
    C. Bruder, R. Fazio, G. Schön, Ann. Phys. 14, 566 (2005)zbMATHCrossRefGoogle Scholar
  26. 26.
    H.J. Carmichael, An Open Systems Approach to Quantum Optics, Springer Lecture Notes in Physics (Springer-Verlag, Berlin, 1993)Google Scholar
  27. 27.
    C.W. Gardiner, P. Zoller, Quantum Noise (Springer Series in Synergetics, Berlin Heidelberg New York, 2004)Google Scholar
  28. 28.
    C.W. Gardiner, Stochastic Methods: A Handbook for the Natural Social Sciences (Springer Series in Synergetics, Berlin Heidelberg New York, 2009)Google Scholar
  29. 29.
    D.F. Walls, G.J. Milburn, Quantum Optics (Springer-Verlag, Heidelberg, 2010)Google Scholar
  30. 30.
    A.J. Daley, Adv. Phys. 63, 77 (2014)ADSCrossRefGoogle Scholar
  31. 31.
    N. Syassen, D.M. Bauer, M. Lettner, T. Volz, D. Dietze, J.J. García-Ripoll, J.I. Cirac, G. Rempe, S. Dürr, Science 320, 1329 (2008)ADSCrossRefGoogle Scholar
  32. 32.
    D. Witthaut, F. Trimborn, S. Wimberger, Phys. Rev. Lett. 101, 200402 (2008)ADSCrossRefGoogle Scholar
  33. 33.
    S. Diehl, A. Micheli, A. Kantian, B. Kraus, H.P. Büchler, P. Zoller, Nat. Phys. 4, 971 (2008)CrossRefGoogle Scholar
  34. 34.
    B. Kraus, H.P. Büchler, S. Diehl, A. Kantian, A. Micheli, P. Zoller, Phys. Rev. A 78, 042307 (2008)ADSCrossRefGoogle Scholar
  35. 35.
    F. Verstraete, M.M. Wolf, J.I. Cirac, Nat. Phys. 5, 633 (2009)CrossRefGoogle Scholar
  36. 36.
    E. Arimondo, G. Oriolls, Lett. Nuovo Cim. 17, 333 (1969)ADSCrossRefGoogle Scholar
  37. 37.
    G.J. Yang, O. Zobay, P. Meystre, Phys. Rev. A 59, 4012 (1999)ADSCrossRefGoogle Scholar
  38. 38.
    G.D. Scholes, T. Mirkovic, D.B. Turner, F. Fassioli, A. Buchleitner, Energy Environ. Sci. 5, 9374 (2012)CrossRefGoogle Scholar
  39. 39.
    A.W. Chin, J. Prior, R. Rosenbach, F. Caycedo-Soler, F. Huelga, M.B. Plenio, Nat. Phys. 9, 113 (2013)CrossRefGoogle Scholar
  40. 40.
    C. Sabín, J. Kohlrus, D.E. Bruschi, I. Fuentes [arXiv:1505.01302] (2015)
  41. 41.
    J.M. Gerton, C.A. Sackett, B.J. Frew, R.G. Hulet, Phys. Rev. A 59, 1514 (1999)ADSCrossRefGoogle Scholar
  42. 42.
    T. Kraemer, M. Mark, P. Waldburger, J.G. Danzl, C. Chin, B. Engeser, A.D. Lange, K. Pilch, A. Jaakkola, H.-C. Nägerl, R. Grimm, Nature 440, 315 (2006)ADSCrossRefGoogle Scholar
  43. 43.
    U. Eismann, L. Khaykovich, S. Laurent, I. Ferrier-Barbut, B.S. Rem, A.T. Grier, M. Delahaye, F. Chevy, C. Salomon, L.-C. Ha, C. Chin [arXiv:1505.04523] (2015)
  44. 44.
    T. Weber, J. Herbig, M. Mark, H.-C. Nägerl, R. Grimm, Phys. Rev. Lett. 91, 123201 (2003)ADSCrossRefGoogle Scholar
  45. 45.
    T. Gerike, A scanning electron microscope for ultracold quantum gases, Ph.D. thesis, Johannes Gutenberg-Universität Mainz, 2010Google Scholar
  46. 46.
    B. Santra, H. Ott [arXiv:1504.00277] (2015)
  47. 47.
    G. Barontini, R. Labouvie, F. Stubenrauch, A. Vogler, V. Guarrera, H. Ott, Phys. Rev. Lett. 110, 035302 (2013)ADSCrossRefGoogle Scholar
  48. 48.
    M. Greiner, O. Mandel, T. Esslinger, T.W. Hänsch, I. Bloch, Nature 415, 39 (2002)ADSCrossRefGoogle Scholar
  49. 49.
    M.J. Hartmann, F.G.S.L. Brandão, M.B. Plenio, New J. Phys. 10, 033011 (2008)ADSCrossRefGoogle Scholar
  50. 50.
    M. Leib, M.J. Hartmann, New J. Phys. 12, 093031 (2010)ADSCrossRefGoogle Scholar
  51. 51.
    U. Naether, F. Quijandría, J.J. García-Ripoll, D. Zueco, Phys. Rev. A 91, 033823 (2015)ADSCrossRefGoogle Scholar
  52. 52.
    H.-P. Breuer, F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, 2006)Google Scholar
  53. 53.
    G. Kordas, S. Wimberger, D. Witthaut, EPL 100, 30007 (2012)ADSCrossRefGoogle Scholar
  54. 54.
    G. Kordas, S. Wimberger, D. Witthaut, Phys. Rev. A 87, 043618 (2013)ADSCrossRefGoogle Scholar
  55. 55.
    J. Anglin, Phys. Rev. Lett. 79, 6 (1997)ADSCrossRefGoogle Scholar
  56. 56.
    F. Trimborn, D. Witthaut, S. Wimberger, J. Phys. B 41, 171001 (2008)ADSCrossRefGoogle Scholar
  57. 57.
    D. Witthaut, F. Trimborn, S. Wimberger, Phys. Rev. A 79, 033621 (2009)ADSCrossRefGoogle Scholar
  58. 58.
    D. Witthaut, F. Trimborn, H. Hennig, G. Kordas, T. Geisel, S. Wimberger, Phys. Rev. A 83, 063608 (2011)ADSCrossRefGoogle Scholar
  59. 59.
    H. Pichler, A.J. Daley, P. Zoller, Phys. Rev. A 82, 063605 (2010)ADSCrossRefGoogle Scholar
  60. 60.
    D. Poletti, J.-S. Bernier, A. Georges, C. Kollath, Phys. Rev. Lett. 109, 045302 (2012)ADSCrossRefGoogle Scholar
  61. 61.
    D. Poletti, P. Barmettler, A. Georges, K. Kollath, Phys. Rev. Lett. 111, 195301 (2013)ADSCrossRefGoogle Scholar
  62. 62.
    B. Sciolla, D. Poletti, K. Kollath, Phys. Rev. Lett. 114, 170401 (2015)ADSCrossRefGoogle Scholar
  63. 63.
    A.J. Daley, J.M. Taylor, S. Diehl, M. Baranov, P. Zoller, Phys. Rev. Lett. 102, 179902 (2009)ADSCrossRefGoogle Scholar
  64. 64.
    A.J. Moerdijk, B.J. Verhaar, Phys. Rev. A 53, 19R (1996)ADSCrossRefGoogle Scholar
  65. 65.
    Y. Li, Y. Castin, A. Sinatra, Phys. Rev. Lett. 100, 210401 (2008)ADSCrossRefGoogle Scholar
  66. 66.
    J.J. García-Ripoll, S. Dürr, N. Syassen, D.M. Bauer, M. Lettner, G. Rempe, J.I. Cirac, New J. Phys. 11, 013053 (2009)ADSCrossRefGoogle Scholar
  67. 67.
    M. Kiffner, M. Hartmann, New J. Phys. 13, 053027 (2011)ADSCrossRefGoogle Scholar
  68. 68.
    S. Diehl, A. Tomadin, A. Micheli, R. Fazio, P. Zoller, Phys. Rev. Lett. 105, 015702 (2010)ADSCrossRefGoogle Scholar
  69. 69.
    G. Watanabe, H. Mäkelä, Phys. Rev. A 85, 023604 (2012)ADSCrossRefGoogle Scholar
  70. 70.
    R.C.F. Caballar, S. Diehl, H. Mäkel, M. Oberthaler, G. Watanabe, Phys. Rev. A 89, 013620 (2014)ADSCrossRefGoogle Scholar
  71. 71.
    R. Franzosi, R. Livi, G.-L. Oppo, A. Politi, Nonlinearity 24, R89 (2011)zbMATHMathSciNetADSCrossRefGoogle Scholar
  72. 72.
    R. Franzosi, R. Livi, G.-L. Oppo, J. Phys. B 40, 1195 (2007)ADSCrossRefGoogle Scholar
  73. 73.
    G.S. Ng, H. Henning, R. Fleischmann, T. Kottos, T. Geisel, New J. Phys. 11, 073045 (2009)ADSCrossRefGoogle Scholar
  74. 74.
    H. Hennig, R. Fleischmann, Phys. Rev. A 87, 033605 (2013)ADSCrossRefGoogle Scholar
  75. 75.
    E.M. Graefe, H.J. Korsch, A.E. Niederle, Phys. Rev. Lett. 101, 150408 (2008)ADSCrossRefGoogle Scholar
  76. 76.
    V.S. Shchesnovich, D.S. Mogilevtsev, Phys. Rev. A 82, 043621 (2010)ADSCrossRefGoogle Scholar
  77. 77.
    F. Trimborn, D. Witthaut, H. Hennig, G. Kordas, T. Geisel, S. Wimberger, Eur. Phys. J. D 63, 63 (2011)ADSCrossRefGoogle Scholar
  78. 78.
    S. Wimberger, C.A. Parra-Murillo, G. Kordas, J. Phys.: Conf. Ser. 442, 012029 (2013)ADSGoogle Scholar
  79. 79.
    R. Livi, R. Franzosi, G.-L. Oppo, Phys. Rev. Lett. 97, 060401 (2006)ADSCrossRefGoogle Scholar
  80. 80.
    P. Barmettler, C. Kollath, Phys. Rev. A 84, 041606(R) (2011)ADSCrossRefGoogle Scholar
  81. 81.
    T. Pudlik, H. Hennig, D. Witthaut, D.K. Campbell, Phys. Rev. A 88, 063606 (2013)ADSCrossRefGoogle Scholar
  82. 82.
    K.V. Kepesidis, M.J. Hartmann, Phys. Rev. A 85, 063620 (2012)ADSCrossRefGoogle Scholar
  83. 83.
    V. Penna, Phys. Rev. E 87, 052909 (2013)ADSCrossRefGoogle Scholar
  84. 84.
    V.S. Shchesnovich, Phys. Rev. A 85, 053616 (2012)ADSCrossRefGoogle Scholar
  85. 85.
    V.S. Shchesnovich, V.V. Konotop, EPL 99, 60005 (2012)ADSCrossRefGoogle Scholar
  86. 86.
    C. Ates, B. Olmos, W. Li, I. Lesanovsky, Phys. Rev. Lett. 109, 233003 (2012)ADSCrossRefGoogle Scholar
  87. 87.
    B. Cui, S.C. Hou, W. Wang, X.X. Yi, J. Phys. B 47, 215303 (2014)ADSCrossRefGoogle Scholar
  88. 88.
    J. Dalibard, Y. Castin, K. Mølmer, Phys. Rev. Lett. 68, 580 (1992)ADSCrossRefGoogle Scholar
  89. 89.
    R. Dum, P. Zoller, H. Ritsch, Phys. Rev. A 45, 4879 (1992)ADSCrossRefGoogle Scholar
  90. 90.
    M.B. Plenio, P.L. Knight, Rev. Mod. Phys. 70, 101 (1998)ADSCrossRefGoogle Scholar
  91. 91.
    D.K. Campbell, S. Flach, Y.S. Kivshar, Phys. Today 467, 57 (2004)Google Scholar
  92. 92.
    S. Flach, A.V. Gorbach, Phys. Rep. 467, 1 (2008)ADSCrossRefGoogle Scholar
  93. 93.
    H. Henning, J. Dorignac, D.K. Campbell, Phys. Rev. A 82, 053604 (2010)ADSCrossRefGoogle Scholar
  94. 94.
    P.G. Kevrekidis, The Discrete Nonlinear Schrödinger Equation (Springer Tracts in Modern Physics, Springer-Verlag Berlin Heidelberg, 2009)Google Scholar
  95. 95.
    P.G. Kevrekidis, D.J. Frantzeskakis, Mod. Phys. Lett. B 18, 173 (2004)ADSCrossRefGoogle Scholar
  96. 96.
    V.A. Brazhnyi, V.V. Konotop, Mod. Phys. Lett. B 18, 627 (2004)ADSCrossRefGoogle Scholar
  97. 97.
    P. Buonsante, R. Burioni, D. Cassi, V. Penna, Vezzani, Phys. Rev. B 70, 224510 (2004)ADSCrossRefGoogle Scholar
  98. 98.
    L.E. Reichl, A Modern Course in Statistical Physics (J. Wiley and Sons, New York, 2004)Google Scholar
  99. 99.
    V.A. Brazhnyi, V.V. Konotop, V.M. Pérez-García, H. Ott, Phys. Rev. Lett. 102, 144101 (2009)ADSCrossRefGoogle Scholar
  100. 100.
    M. Kreibich, J. Main, H. Cartarius, G. Wunner, Phys. Rev. A 87, 051601(R) (2014)ADSCrossRefGoogle Scholar
  101. 101.
    D. Dast, D. Haag, H. Cartarius, G. Wunner, Phys. Rev. A 90, 052120 (2014)ADSCrossRefGoogle Scholar
  102. 102.
    E.-M. Graefe, H.J. Korsch, A.E. Niederle, Phys. Rev. A 82, 013629 (2010)ADSCrossRefGoogle Scholar
  103. 103.
    E.M. Graefe, U. Günther, H.J. Korsch, A.E. Niedele, J. Phys. A 41, 255206 (2008)MathSciNetADSCrossRefGoogle Scholar
  104. 104.
    V.V. Konotop, D.E. Pelinovsky, D.A. Zezyulin, EPL 100, 56006 (2012)ADSCrossRefGoogle Scholar
  105. 105.
    D.E. Pelinovsky, P.G. Kevrekidis, D.J. Frantzeskakis, EPL 101, 11002 (2013)ADSCrossRefGoogle Scholar
  106. 106.
    K. Li, P.G. Kevrekidis, D.J. Frantzeskakis, C.E. Rüter, D. Kip, J. Phys. A 46, 375304 (2013)MathSciNetCrossRefGoogle Scholar
  107. 107.
    A.S. Rodrigues, K. Li, V. Achilleos, P.G. Kevrekidis, D.J. Frantzeskakis, C.M. Bender, Rom. Rep. Phys. 65, 5 (2013)Google Scholar
  108. 108.
    D. Dast, D. Haag, H. Cartarius, G. Wunner, Phys. Rev. A 90, 052120 (2014)ADSCrossRefGoogle Scholar
  109. 109.
    R. Benzi, A. Sutera, A. Vulpiani, J. Phys. A 14, L453 (1981)MathSciNetADSCrossRefGoogle Scholar
  110. 110.
    L. Gammaitoni, P. Hänggi, P. Jung, F. Marchesoni, Rev. Mod. Phys. 70, 223 (1998)ADSCrossRefGoogle Scholar
  111. 111.
    T. Wellens, V. Shatokhin, A. Buchleitner, Rep. Prog. Phys. 67, 45 (2004)ADSCrossRefGoogle Scholar
  112. 112.
    A. Vardi, J.R. Anglin, Phys. Rev. Lett. 86, 568 (2001)ADSCrossRefGoogle Scholar
  113. 113.
    J.R. Anglin, A. Vardi, Phys. Rev. A 64, 013605 (2001)ADSCrossRefGoogle Scholar
  114. 114.
    I. Tikhonenkov, J.R. Anglin, A. Vardi, Phys. Rev. A 75, 013613 (2007)ADSCrossRefGoogle Scholar
  115. 115.
    A.J. Leggett, Rev. Mod. Phys. 73, 307 (2001)ADSCrossRefGoogle Scholar
  116. 116.
    B. Wu, Q. Niu, New J. Phys. 5, 104 (2003)ADSCrossRefGoogle Scholar
  117. 117.
    L.D. Carr, J. Brand, S. Burger, A. Sanpera, Phys. Rev. A 63, 051601R (2001)ADSCrossRefGoogle Scholar
  118. 118.
    B. Wu, J. Liu, Q. Niu, Phys. Rev. Lett. 88, 034101 (2002)ADSCrossRefGoogle Scholar
  119. 119.
    D.J. Frantzeskakis, J. Phys. A 43, 213001 (2010)MathSciNetADSCrossRefGoogle Scholar
  120. 120.
    S. Burger, K. Bongs, S. Dettmer, W. Ertmer, K. Sengstock, Phys. Rev. Lett. 83, 5198 (1999)ADSCrossRefGoogle Scholar
  121. 121.
    J. Denschlag, J.E. Simsarian, D.L. Feder, C.W. Clark, L.A. Collins, J. Cubizolles, L. Deng, E.W. Hagley, K. Helmerson, W.P. Reinhardt, S.L. Rolston, B.I. Schneider, W.D. Phillips, Science 287, 97 (2000)ADSCrossRefGoogle Scholar
  122. 122.
    D. Jaksch, V. Venturi, J.I. Cirac, C.J. Williams, P. Zoller, Phys. Rev. Lett. 89, 040402 (2002)ADSCrossRefGoogle Scholar
  123. 123.
    P. Buonsante, V. Penna, J. Phys. A 41, 175301 (2008)MathSciNetADSCrossRefGoogle Scholar
  124. 124.
    W. Krauth, M. Caffarel, J.P. Bouchaud, Phys. Rev. B 45, 3137 (1992)ADSCrossRefGoogle Scholar
  125. 125.
    K. Sheshadri, H.R. Krishnamurthy, R. Pandit, T.V. Ramakrishnan, EPL 22, 257 (1993)ADSCrossRefGoogle Scholar
  126. 126.
    D. van Oosten, P. van der Straten, H.T.C. Stoof, Phys. Rev. A 63, 053601 (2001)ADSCrossRefGoogle Scholar
  127. 127.
    J. Zakrzewski, Phys. Rev. A 71, 043601 (2005)ADSCrossRefGoogle Scholar
  128. 128.
    P. Buonsante, V. Penna, A. Vezzani, P.B. Blakie, Phys. Rev. A 76, 011602 (2007)ADSCrossRefGoogle Scholar
  129. 129.
    P. Buonsante, F. Massel, V. Penna, A. Vezzani, Phys. Rev. A 79, 013623 (2009)ADSCrossRefGoogle Scholar
  130. 130.
    A.E. Niederle, H. Rieger, New J. Phys. 15, 075029 (2013)ADSCrossRefGoogle Scholar
  131. 131.
    K. Sheshadri, H.R. Krishnamurthy, R. Pandit, T.V. Ramakrishnan, Phys. Rev. Lett. 75, 4075 (1995)ADSCrossRefGoogle Scholar
  132. 132.
    M. Snoek, W. Hofstetter, Phys. Rev. A 76, 051603 (2007)ADSCrossRefGoogle Scholar
  133. 133.
    M. Endres, T. Fukuhara, D. Pekker, M. Cheneau, P. Schauß, C. Gross, E. Demler, S. Kuhr, I. Bloch, Nature 487, 454 (2012)ADSCrossRefGoogle Scholar
  134. 134.
    A. Rapp, Phys. Rev. A 87, 043611 (2013)ADSCrossRefGoogle Scholar
  135. 135.
    P. Buonsante, L. Pezzè, A. Smerzi, Phys. Rev. A 91, 031601 (2015)ADSCrossRefGoogle Scholar
  136. 136.
    Pižorn, Phys. Rev. A 88, 043635 (2013)ADSCrossRefGoogle Scholar
  137. 137.
    I. Vidanovic, D. Cocks, W. Hofstetter, Phys. Rev. A 89, 053614 (2014)ADSCrossRefGoogle Scholar
  138. 138.
    P. Buonsante, A. Vezzani, Phys. Rev. Lett. 98, 110601 (2007)ADSCrossRefGoogle Scholar
  139. 139.
    A.O. Caldeira, A.J. Leggett, Ann. Phys. 149, 374 (1983)ADSCrossRefGoogle Scholar
  140. 140.
    J.I. Cirac, A. Schenzle, P. Zoller, EPL 27, 123 (1994)ADSCrossRefGoogle Scholar
  141. 141.
    D.A. Zezyulin, V.V. Konotop, G. Barontini, H. Ott, Phys. Rev. Lett. 109, 020405 (2012)ADSCrossRefGoogle Scholar
  142. 142.
    V.S. Shchesnovich, V.V. Konotop, Phys. Rev. A 81, 053611 (2010)ADSCrossRefGoogle Scholar
  143. 143.
    M.J. Werner, P.D. Drummond, J. Comput. Phys. 132, 312 (1997)zbMATHMathSciNetADSCrossRefGoogle Scholar
  144. 144.
    A. Sinatra, C. Lobo, Y. Castin, J. Phys. B 35, 3599 (2002)ADSCrossRefGoogle Scholar
  145. 145.
    A. Polkovnikov, Ann. Phys. 325, 1790 (2010)zbMATHMathSciNetADSCrossRefGoogle Scholar
  146. 146.
    B. Berg, L.I. Plimak, A. Polkovnikov, M.K. Olsen, M. Fleischhauer, W.P. Schleich, Phys. Rev. A 80, 033624 (2009)ADSCrossRefGoogle Scholar
  147. 147.
    F. Trimborn, D. Witthaut, H.J. Korsch, Phys. Rev. A 77, 043631 (2008)ADSCrossRefGoogle Scholar
  148. 148.
    F. Trimborn, D. Witthaut, H.J. Korsch, Phys. Rev. A 79, 013608 (2009)ADSCrossRefGoogle Scholar
  149. 149.
    A. Polkovnikov, Phys. Rev. A 68, 053604 (2003)ADSCrossRefGoogle Scholar
  150. 150.
    J. Dujardin, T. Engl, J. Urbina, P. Schlagheck, Ann. Phys. (submitted) (preprint) [arXiv:1502.02890]
  151. 151.
    R.A. Pepino, J. Cooper, D. Meiser, D.Z. Anderson, M.J. Holland, Phys. Rev. A 82, 013640 (2010)ADSCrossRefGoogle Scholar
  152. 152.
    D.B. Gutman, Y. Gefen, A.D. Mirlin, Phys. Rev. B 85, 125102 (2012)ADSCrossRefGoogle Scholar
  153. 153.
    M. Bruderer, W. Belzig, Phys. Rev. A 85, 013623 (2012)ADSCrossRefGoogle Scholar
  154. 154.
    L.H. Kristinsdóttir et al., Phys. Rev. Lett. 110, 085303 (2013)ADSCrossRefGoogle Scholar
  155. 155.
    T. Prosen, Phys. Rev. Lett. 112, 030603 (2014)ADSCrossRefGoogle Scholar
  156. 156.
    C. Nietner, G. Schaller, T. Brandes, Phys. Rev. A 89, 013605 (2014)ADSCrossRefGoogle Scholar
  157. 157.
    F.G. Marcos, G. Platero, C. Nietner, G. Schaller, T. Brandes, Phys. Rev. A 90, 033614 (2014)ADSCrossRefGoogle Scholar
  158. 158.
    G. Kordas, D. Witthaut, S. Wimberger, Ann. Phys., doi: 10.1002/andp.201400189 (2015)Google Scholar
  159. 159.
    A. Ivanov, G. Kordas, A. Komnik, S. Wimberger, Eur. Phys. J.B 86, 345 (2013)ADSCrossRefGoogle Scholar
  160. 160.
    M. Naraschewski, R.J. Glauber, Phys. Rev. A 59, 4595 (1999)ADSCrossRefGoogle Scholar
  161. 161.
    P. Schlagheck, F. Malet, J.C. Cremon, S.M. Reimann, New J. Phys. 12, 065020 (2010)ADSCrossRefGoogle Scholar
  162. 162.
    M.D. Lukin, M. Fleischhauer, R. Cote, L.M. Duan, D. Jaksch, J.I. Cirac, P. Zoller, Phys. Rev. Lett. 87, 037901 (2001)ADSCrossRefGoogle Scholar
  163. 163.
    G.-L. Ingold, Y.V. Nazarov, in Single Charge Tunneling, edited by H. Grabert, M.H. Devoret (Plenum, New York, London, 1992)Google Scholar
  164. 164.
    R.P. Feynman, Rev. Mod. Phys. 20, 367 (1948)MathSciNetADSCrossRefGoogle Scholar
  165. 165.
    H. Kleinert, Path integrals in quantum mechanics, statistics, polymer physics financial markets (World Scientific Publishing Co, 2006)Google Scholar
  166. 166.
    M. Baranger, M.A.M. de Aguiar, F. Keck, H.J. Korsch, B. Schellhaaß, J. Phys. A 34, 7227 (2001)zbMATHMathSciNetADSCrossRefGoogle Scholar
  167. 167.
    A.L. Xavier, M.A.M. de Aguiar, Phys. Rev. A 54, 1808 (1996)MathSciNetADSCrossRefGoogle Scholar
  168. 168.
    M. Stone, K.-S. Park, A. Garg, J. Math. Phys. (NY) 41, 8025 (2000)zbMATHMathSciNetADSCrossRefGoogle Scholar
  169. 169.
    E.A. Kochetov, J. Phys. A 31, 4473 (1998)zbMATHMathSciNetADSCrossRefGoogle Scholar
  170. 170.
    J.H. Wilson, V. Galitski, Phys. Rev. Lett. 106, 110401 (2011)MathSciNetADSCrossRefGoogle Scholar
  171. 171.
    G. Kordas, S. Mistakidis, A.I. Karanikas, Phys. Rev. A 90, 032104 (2014)ADSCrossRefGoogle Scholar
  172. 172.
    R.P. Feynman, F.L. Vernon, Ann. Phys. 24, 118 (1963)MathSciNetADSCrossRefGoogle Scholar
  173. 173.
    A.O. Caldeira, A.J. Leggett, Physica A 121, 587 (1983)zbMATHMathSciNetADSCrossRefGoogle Scholar
  174. 174.
    K.K. Rajagopal, Physica A 429, 231 (2015)ADSCrossRefGoogle Scholar
  175. 175.
    J.-H. An, W.-M. Zhang, Phys. Rev. A 76, 042127 (2007)ADSCrossRefGoogle Scholar
  176. 176.
    M.W.Y. Tu, W.M. Zhang, Phys. Rev. B 78, 235311 (2008)ADSCrossRefGoogle Scholar
  177. 177.
    W.-M. Zhang, P.-Y. Lo, H.-N. Xiong, M.W.-Y. Tu, F. Nori, Phys. Rev. Lett. 109, 170402 (2012)ADSCrossRefGoogle Scholar
  178. 178.
    T. Hartmann, J. Michl, C. Petitjean, T. Wellens, J.-D. Urbina, K. Richter, P. Schlagheck, Ann. Phys. (Amsterdam) 327, 1998 (2012)zbMATHADSCrossRefGoogle Scholar
  179. 179.
    T. Engl, J. Dujardin, A. Argüelles, P. Schlagheck, K. Richter, J. Urbina, Phys. Rev. Lett. 112, 140403 (2014)ADSCrossRefGoogle Scholar
  180. 180.
    L. Simon, W.T. Strunz, Phys. Rev. A 89, 052112 (2014)ADSCrossRefGoogle Scholar
  181. 181.
    R. Dubertrand, S. Müller, conference talk, MPIPKS Dresden (2015)Google Scholar
  182. 182.
    A.R. Kolovsky, A. Buchleitner, EPL 68, 632 (2004)ADSCrossRefGoogle Scholar
  183. 183.
    A. Tomadin, R. Mannella, S. Wimberger, Phys. Rev. Lett. 98, 130402 (2007)ADSCrossRefGoogle Scholar
  184. 184.
    C.A. Parra-Murillo, J. Madroñero, S. Wimberger, Phys. Rev. A 89, 053610 (2014)ADSCrossRefGoogle Scholar
  185. 185.
    M. Lubasch, Quantum Chaos, Entanglement in the Bose-Hubbard Model, Diploma thesis, Heidelberg University, 2009Google Scholar
  186. 186.
    J. Jing, X. Zhao, J.Q. You, W.T. Strunz, T. Yu, Phys. Rev. A 88, 052122 (2013)ADSCrossRefGoogle Scholar
  187. 187.
    C. Sias, A. Zenesini, H. Lignier, S. Wimberger, D. Ciampini, O. Morsch, E. Arimondo, Phys. Rev. Lett. 98, 120403 (2007)ADSCrossRefGoogle Scholar
  188. 188.
    A.U.J. Lode, A.I. Streltsov, K. Sakmann, O.E. Alon, L.S. Cederbaum, PNAS 109, 13521 (2012)ADSCrossRefGoogle Scholar
  189. 189.
    A.S. Sørensen, L.-M. Duan, J.I. Cirac, P. Zoller, Nature 409, 63 (2001)ADSCrossRefGoogle Scholar
  190. 190.
    R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, Rev. Mod. Phys. 81, 865 (2009)zbMATHMathSciNetADSCrossRefGoogle Scholar

Copyright information

© EDP Sciences and Springer 2015

Authors and Affiliations

  • G. Kordas
    • 1
  • D. Witthaut
    • 2
    • 3
  • P. Buonsante
    • 4
  • A. Vezzani
    • 5
    • 6
  • R. Burioni
    • 6
    • 7
  • A. I. Karanikas
    • 1
  • S. Wimberger
    • 6
    • 7
  1. 1.Physics Department, Nuclear & Particle Physics SectionUniversity of AthensAthensGreece
  2. 2.Institute for Energy and Climate Research (IEK-STE)Forschungszentrum JülichJülichGermany
  3. 3.Institute for Theoretical PhysicsUniversity of CologneKölnGermany
  4. 4.QSTAR, INO-CNR and LENSFirenzeItaly
  5. 5.S3, CNR Istituto di NanoscienzeModenaItaly
  6. 6.Dipartimento di Fisica e Scienze della TerraUniversità di ParmaParmaItaly
  7. 7.Sezione di Milano Bicocca, Gruppo Collegato di ParmaINFNParmaItaly

Personalised recommendations