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Theoretical Framework in the Stationary Regime

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Quench Dynamics in Interacting and Superconducting Nanojunctions

Part of the book series: Springer Theses ((Springer Theses))

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Abstract

In this chapter the main theoretical tools and models used along the thesis are introduced. We will focus first on the non-interacting situation, providing a brief overview about the non-equilibrium Green function formalism. We will also discuss the minimal models including electron-electron, electron-phonon interaction and superconducting correlations at the nanoscale. Some specific methods for treating interactions are discussed in the stationary regime. The final part of the chapter is devoted to the full counting statistics analysis in both the interacting and non-interacting situations.

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References

  1. Langreth DC (1966) Friedel sum rule for Anderson’s model of localized impurity states. Phys Rev 150:516

    Article  ADS  Google Scholar 

  2. Keldysh LV (1964) Diagram technique for nonequilibrium processes. Zh Eksp Teor Fiz 47:1515 [Sov Phys JETP 20:1018 (1965)]

    Google Scholar 

  3. Meir Y, Wingreen NS (1992) Landauer formula for the current through an interacting electron region. Phys Rev Lett 68:2512

    Article  ADS  Google Scholar 

  4. Bulla R, Costi TA, Pruschke T (2008) Numerical renormalization group method for quantum impurity systems. Rev Mod Phys 80:395

    Article  ADS  Google Scholar 

  5. Gull E, Millis AJ, Lichtenstein AI, Rubtsov AN, Troyer M, Werner P (2011) Continuous-time monte Carlo methods for quantum impurity models. Rev Mod Phys 83:349

    Article  ADS  Google Scholar 

  6. Haussmann R (1999) Self-consistent quantum-field theory and bosonization for strongly correlated electron systems. Lecture notes in physics monographs. Springer, Berlin

    MATH  Google Scholar 

  7. Kozik E, Ferrero M, Georges A (2015) Nonexistence of the Luttinger-Ward functional and misleading convergence of skeleton diagrammatic series for Hubbard-like models. Phys Rev Lett 114:156402

    Article  ADS  Google Scholar 

  8. Anderson PW (1961) Localized magnetic states in metals. Phys Rev 124:41

    Article  ADS  MathSciNet  Google Scholar 

  9. Hewson A (1997) The Kondo problem to heavy fermions. Cambridge studies in magnetism. Cambridge University Press, Cambridge

    Google Scholar 

  10. Wiegmann PB (1980) Towards an exact solution of the Anderson model. Phys Lett 80A:163

    Article  ADS  MathSciNet  Google Scholar 

  11. Kawakami N, Okiji A (1981) Exact expression of the ground-state energy for the symmetric Anderson model. Phys Lett A 86:483

    Article  ADS  MathSciNet  Google Scholar 

  12. Andrei N, Furuya K, Lowenstein JH (1983) Solution of the Kondo problem. Rev Mod Phys 55:331

    Article  ADS  MathSciNet  Google Scholar 

  13. MartĂ­n-Rodero A, Flores F, Baldo M, Pucci R (1982) A new solution to the Anderson-Newns hamiltonian of chemisorption. Solid State Commun 44:911

    Article  ADS  Google Scholar 

  14. White JA (1992) Self-consistent Green functions for the Anderson impurity model. Phys Rev B 45:1100

    Article  ADS  Google Scholar 

  15. MartĂ­n-Rodero A, Louis E, Flores F, Tejedor C (1986) Interpolative solution for the periodic Anderson model of mixed-valence compounds. Phys Rev B 33:1814

    Article  ADS  Google Scholar 

  16. Yeyati AL, MartĂ­n-Rodero A, Flores F (1993) Electron correlation resonances in the transport through a single quantum level. Phys Rev Lett 71:2991

    Article  ADS  Google Scholar 

  17. Kajueter H, Kotliar G (1996) New iterative perturbation scheme for lattice models with arbitrary filling. Phys Rev Lett 77:131

    Article  ADS  Google Scholar 

  18. Anders FB (2008) A numerical renormalization group approach to non-equilibrium Green functions for quantum impurity models. J Phys: Condens Matter 20:195216

    ADS  Google Scholar 

  19. Fujii T, Ueda K (2003) Perturbative approach to the nonequilibrium Kondo effect in a quantum dot. Phys Rev B 68:155310

    Article  ADS  Google Scholar 

  20. AgraĂŻt N, Untiedt C, Rubio-Bollinger G, Vieira S (2002) Onset of energy dissipation in ballistic atomic wires. Phys Rev Lett 88:216803

    Article  ADS  Google Scholar 

  21. de la Vega L, MartĂ­n-Rodero A, AgraĂŻt N, Yeyati AL (2006) Universal features of electron-phonon interactions in atomic wires. Phys Rev B 73:075428

    Article  ADS  Google Scholar 

  22. Leturcq R, Stampfer C, Inderbitzin K, Durrer L, Hierold C, Mariani E, Schultz MG, von Oppen F, Ensslin K (2009) Franck-Condon blockade in suspended carbon nanotube quantum dots. Nat Phys 5:327 EP

    Article  ADS  Google Scholar 

  23. Park H, Park J, Lim AKL, Anderson EH, Alivisatos AP, McEuen PL (2000) Nanomechanical oscillations in a single-C60 transistor. Nature 407:57 EP

    Article  ADS  Google Scholar 

  24. Holstein T (1959) Studies of polaron motion: part i. the molecular-crystal model. Ann Phys 8:325

    Article  ADS  MATH  Google Scholar 

  25. Mitra A, Aleiner I, Millis AJ (2004) Phonon effects in molecular transistors: quantal and classical treatment. Phys Rev B 69:245302

    Article  ADS  Google Scholar 

  26. Riwar R-P, Schmidt TL (2009) Transient dynamics of a molecular quantum dot with a vibrational degree of freedom. Phys Rev B 80:125109

    Article  ADS  Google Scholar 

  27. Urban DF, Avriller R, Levy Yeyati A (2010) Nonlinear effects of phonon fluctuations on transport through nanoscale junctions. Phys Rev B 82:121414

    Google Scholar 

  28. Utsumi Y, Entin-Wohlman O, Ueda A, Aharony A (2013) Full-counting statistics for molecular junctions: fluctuation theorem and singularities. Phys Rev B 87:115407

    Article  ADS  Google Scholar 

  29. Murakami Y, Werner P, Tsuji N, Aoki H (2015) Interaction quench in the Holstein model: thermalization crossover from electron- to phonon-dominated relaxation. Phys Rev B 91:045128

    Article  ADS  Google Scholar 

  30. Lang IG, Firsov YA (1962) Kinetic theory of semiconductors with low mobility. Sov Phys JETP 16:1301

    ADS  MATH  Google Scholar 

  31. Maier S, Schmidt TL, Komnik A (2011) Charge transfer statistics of a molecular quantum dot with strong electron-phonon interaction. Phys Rev B 83:085401

    Article  ADS  Google Scholar 

  32. Flensberg K (2003) Tunneling broadening of vibrational sidebands in molecular transistors. Phys Rev B 68:205323

    Article  ADS  Google Scholar 

  33. Seoane Souto R, Yeyati AL, Martín-Rodero A, Monreal RC (2014) Dressed tunneling approximation for electronic transport through molecular transistors. Phys Rev B 89:085412

    Google Scholar 

  34. Dong B, Ding GH, Lei XL (2013) Full counting statistics of a single-molecule quantum dot. Phys Rev B 88:075414

    Article  ADS  Google Scholar 

  35. Monreal RC, MartĂ­n-Rodero A (2009) Equation of motion approach to the Anderson-Holstein hamiltonian. Phys Rev B 79:115140

    Article  ADS  Google Scholar 

  36. Blonder GE, Tinkham M, Klapwijk TM (1982) Transition from metallic to tunneling regimes in superconducting microconstrictions: excess current, charge imbalance, and supercurrent conversion. Phys Rev B 25:4515

    Article  ADS  Google Scholar 

  37. Beenakker CWJ, van Houten H (1991) Josephson current through a superconducting quantum point contact shorter than the coherence length. Phys Rev Lett 66:3056

    Article  ADS  Google Scholar 

  38. Hurd M, Datta S, Bagwell PF (1996) Current-voltage relation for asymmetric ballistic superconducting junctions. Phys Rev B 54:6557

    Article  ADS  Google Scholar 

  39. Cuevas JC, MartĂ­n-Rodero A, Yeyati AL (1996) Hamiltonian approach to the transport properties of superconducting quantum point contacts. Phys Rev B 54:7366

    Article  ADS  Google Scholar 

  40. Scheer E, Joyez P, Esteve D, Urbina C, Devoret MH (1997) Conduction channel transmissions of atomic-size aluminum contacts. Phys Rev Lett 78:3535

    Article  ADS  Google Scholar 

  41. Scheer E, Belzig W, Naveh Y, Devoret MH, Esteve D, Urbina C (2001) Proximity effect and multiple Andreev reflections in gold atomic contacts. Phys Rev Lett 86:284

    Article  ADS  Google Scholar 

  42. Yeyati AL, Cuevas JC, López-Dávalos A, Martín-Rodero A (1997) Resonant tunneling through a small quantum dot coupled to superconducting leads. Phys Rev B 55:R6137

    Article  ADS  Google Scholar 

  43. De Franceschi S, Kouwenhoven L, Schönenberger C, Wernsdorfer W (2010) Hybrid superconductor-quantum dot devices. Nat Nanotechnol 5:703 EP. Review Article

    Google Scholar 

  44. MartĂ­n-Rodero A, Yeyati AL (2011) Josephson and Andreev transport through quantum dots. Adv Phys 60:899

    Article  ADS  Google Scholar 

  45. Cuevas JC (1999) Electronic transport in normal and superconducting nanocontacts. Doctoral thesis

    Google Scholar 

  46. Levitov LS, Lesovik GB (1993) Charge distribution in quantum shot noise. JETP Lett 58:230

    ADS  Google Scholar 

  47. Levitov LS, Lee H, Lesovik GB (1996) Electron counting statistics and coherent states of electric current. J Math Phys 37:4845

    Article  ADS  MathSciNet  MATH  Google Scholar 

  48. Nazarov YV (1999) Universalities of weak localization. Ann Phys (Leipzig) 8:507

    Google Scholar 

  49. Bagrets DA, Nazarov YV (2003) Full counting statistics of charge transfer in Coulomb blockade systems. Phys Rev B 67:085316

    Article  ADS  Google Scholar 

  50. Gogolin AO, Komnik A (2006) Towards full counting statistics for the Anderson impurity model. Phys Rev B 73:195301

    Article  ADS  Google Scholar 

  51. Fujisawa T, Hayashi T, Tomita R, Hirayama Y (2006) Bidirectional counting of single electrons. Science 312:1634

    Article  ADS  Google Scholar 

  52. Gustavsson S, Leturcq R, SimoviÄŤ B, Schleser R, Ihn T, Studerus P, Ensslin K, Driscoll DC, Gossard AC (2006) Counting statistics of single electron transport in a quantum dot. Phys Rev Lett 96:076605

    Article  ADS  Google Scholar 

  53. Gustavsson S, Leturcq R, SimoviÄŤ B, Schleser R, Studerus P, Ihn T, Ensslin K, Driscoll DC, Gossard AC (2006) Counting statistics and super-Poissonian noise in a quantum dot: Time-resolved measurements of electron transport. Phys Rev B 74:195305

    Article  ADS  Google Scholar 

  54. Sukhorukov EV, Jordan AN, Gustavsson S, Leturcq R, Ihn T, Ensslin K (2007) Conditional statistics of electron transport in interacting nanoscale conductors. Nat Phys 3:243 EP

    Article  ADS  Google Scholar 

  55. Fricke C, Hohls F, Wegscheider W, Haug RJ (2007) Bimodal counting statistics in single-electron tunneling through a quantum dot. Phys Rev B 76:155307

    Article  ADS  Google Scholar 

  56. Flindt C, Fricke C, Hohls F, NovotnĂ˝ T, NetoÄŤnĂ˝ K, Brandes T, Haug RJ (2009) Universal oscillations in counting statistics. Proc Natl Acad Sci 106:10116

    Article  ADS  Google Scholar 

  57. Wagner T, Strasberg P, Bayer JC, Rugeramigabo EP, Brandes T, Haug RJ (2016) Strong suppression of shot noise in a feedback-controlled single-electron transistor. Nat Nanotechnol 12:218 EP

    Article  ADS  Google Scholar 

  58. ar**v:1910.08420

  59. Agarwalla BK, Jiang J-H, Segal D (2015) Full counting statistics of vibrationally assisted electronic conduction: transport and fluctuations of thermoelectric efficiency. Phys Rev B 92:245418

    Article  ADS  Google Scholar 

  60. Yu Z, Tang G-M, Wang J (2016) Full-counting statistics of transient energy current in mesoscopic systems. Phys Rev B 93:195419

    Article  ADS  Google Scholar 

  61. Tang G, Yu Z, Wang J (2017) Full-counting statistics of energy transport of molecular junctions in the polaronic regime. New J Phys 19:083007

    Article  Google Scholar 

  62. Levitov LS, Reznikov M (2004) Counting statistics of tunneling current. Phys Rev B 70:115305

    Article  ADS  Google Scholar 

  63. Kamenev A (2011) Field theory of non-equilibrium systems. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  64. Beenakker CWJ, Schonenberger C (2003) Quantum shot noise. Phys Today 56:37

    Article  ADS  Google Scholar 

  65. Avriller R, Souto RS, MartĂ­n-Rodero A, Yeyati AL (2019) Buildup of vibron-mediated electron correlations in molecular junctions. Phys Rev B 99:121403

    Article  ADS  Google Scholar 

  66. Flindt C, NovotnĂ˝ T, Braggio A, Sassetti M, Jauho A-P (2008) Counting statistics of non-Markovian quantum stochastic processes. Phys Rev Lett 100:150601

    Article  ADS  Google Scholar 

  67. Golubev DS, Marthaler M, Utsumi Y, Schön G (2010) Statistics of voltage fluctuations in resistively shunted Josephson junctions. Phys Rev B 81:184516

    Article  ADS  Google Scholar 

  68. Fricke C, Hohls F, Sethubalasubramanian N, Fricke L, Haug RJ (2010) High-order cumulants in the counting statistics of asymmetric quantum dots. Appl Phys Lett 96:202103

    Article  ADS  Google Scholar 

  69. Beenakker CWJ, Schomerus H (2004) Antibunched photons emitted by a quantum point contact out of equilibrium. Phys Rev Lett 93:096801

    Article  ADS  Google Scholar 

  70. Kambly D, Flindt C, BĂĽttiker M (2011) Factorial cumulants reveal interactions in counting statistics. Phys Rev B 83:075432

    Article  ADS  Google Scholar 

  71. Stegmann P, Sothmann B, Hucht A, König J (2015) Detection of interactions via generalized factorial cumulants in systems in and out of equilibrium. Phys Rev B 92:155413

    Article  ADS  Google Scholar 

  72. Stegmann P, König J (2017) Inverse counting statistics based on generalized factorial cumulants. New J Phys 19:023018

    Article  Google Scholar 

  73. Brandner K, Maisi VF, Pekola JP, Garrahan JP, Flindt C (2017) Experimental determination of dynamical Lee-Yang zeros. Phys Rev Lett 118:180601

    Article  ADS  Google Scholar 

  74. Abanov AG, Ivanov DA (2008) Allowed charge transfers between coherent conductors driven by a time-dependent scatterer. Phys Rev Lett 100:086602

    Article  ADS  Google Scholar 

  75. Abanov AG, Ivanov DA (2009) Factorization of quantum charge transport for noninteracting fermions. Phys Rev B 79:205315

    Article  ADS  Google Scholar 

  76. Hickey JM, Flindt C, Garrahan JP (2013) Trajectory phase transitions and dynamical Lee-Yang zeros of the Glauber-Ising chain. Phys Rev E 88:012119

    Article  ADS  Google Scholar 

  77. Hedges LO, Jack RL, Garrahan JP, Chandler D (2009) Dynamic order-disorder in atomistic models of structural glass formers. Science 323:1309

    Article  ADS  Google Scholar 

  78. Garrahan JP, Armour AD, Lesanovsky I (2011) Quantum trajectory phase transitions in the micromaser. Phys Rev E 84:021115

    Article  ADS  Google Scholar 

  79. Hickey JM, Flindt C, Garrahan JP (2014) Intermittency and dynamical Lee-Yang zeros of open quantum systems. Phys Rev E 90:062128

    Article  ADS  Google Scholar 

  80. Yang CN, Lee TD (1952) Statistical theory of equations of state and phase transitions. i. theory of condensation. Phys Rev 87:404

    Article  ADS  MathSciNet  MATH  Google Scholar 

  81. Lee TD, Yang CN (1952) Statistical theory of equations of state and phase transitions. ii. lattice gas and ising model. Phys Rev 87:410

    Article  ADS  MathSciNet  MATH  Google Scholar 

  82. Blythe RA, Evans MR (2002) Lee-Yang zeros and phase transitions in nonequilibrium steady states. Phys Rev Lett 89:080601

    Article  ADS  MathSciNet  MATH  Google Scholar 

  83. Blythe RA (2006) An introduction to phase transitions in stochastic dynamical systems. J Phys: Conf Ser 40:1

    ADS  MathSciNet  Google Scholar 

  84. Flindt C, Garrahan JP (2013) Trajectory phase transitions, Lee-Yang zeros, and high-order cumulants in full counting statistics. Phys Rev Lett 110:050601

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. Division of Solid State Physics and NanoLund, Lund University, Lund, Sweden

    Dr. Rubén Seoane Souto

  2. Center for Quantum Devices and Station Q Copenhagen, Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark

    Dr. Rubén Seoane Souto

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  1. Dr. Rubén Seoane Souto
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Correspondence to Rubén Seoane Souto .

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Seoane Souto, R. (2020). Theoretical Framework in the Stationary Regime. In: Quench Dynamics in Interacting and Superconducting Nanojunctions. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-36595-0_2

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