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Transients in Quantum Transport:

SEMINÁŘ TEORETICKÉ HO ODD. FZ Ú SLOVANKA 14. ÚNORA 2006. Transients in Quantum Transport:. B. Velický, Charles University and Acad. Sci. of CR, Praha A. Kalvová, Acad. Sci. of CR, Praha V. Špička , Acad. Sci. of CR, Praha. SEMINÁŘ TEORETICKÉ HO ODD. FZ Ú SLOVANKA 14. ÚNORA 2006.

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Transients in Quantum Transport:

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  1. SEMINÁŘ TEORETICKÉHO ODD. FZÚ SLOVANKA 14. ÚNORA 2006 Transients in Quantum Transport: B. Velický, Charles University and Acad. Sci. of CR, Praha A. Kalvová, Acad. Sci. of CR, Praha V. Špička, Acad. Sci. of CR, Praha

  2. SEMINÁŘ TEORETICKÉHO ODD. FZÚ SLOVANKA 14. ÚNORA 2006 Transients in Quantum Transport:I. Semi-Group Property of Propagators and the Gauge Invariance of the 1st Kind B. Velický, Charles University and Acad. Sci. of CR, Praha A. Kalvová, Acad. Sci. of CR, Praha V. Špička, Acad. Sci. of CR, Praha

  3. SEMINÁŘ TEORETICKÉHO ODD. FZÚ SLOVANKA 21. ÚNORA 2006 Transients in Quantum Transport:II. Correlated Initial Condition for Restart Process by Time Partitioning A. Kalvová, Acad. Sci. of CR, Praha B. Velický, Charles University and Acad. Sci. of CR, Praha V. Špička, Acad. Sci. of CR, Praha

  4. SEMINÁŘ TEORETICKÉHO ODD. FZÚ SLOVANKA 21. ÚNORA 2006 Transients in Quantum Transport:II. Correlated Initial Condition for Restart Process by Time Partitioning Progress in Non-Equilibrium Green’s Function III, Kiel Aug 22, 2005 Topical Problems in Statistical Physics, TU Chemnitz, Nov 30, 2005

  5. Prologue Transients in Quantum TransportII ...

  6. (Non-linear) quantum transport non-equilibrium problem Many-body system Initial state External disturbance many-body Hamiltonian many-body density matrix additive operator Transients in Quantum TransportII ...

  7. (Non-linear) quantum transport non-equilibrium problem Many-body system Initial state External disturbance Response many-body Hamiltonian many-body density matrix additive operator one-particle density matrix Transients in Quantum TransportII ...

  8. a closed equation for (Non-linear) quantum transport non-equilibrium problem Many-body system Initial state External disturbance Response many-body Hamiltonian many-body density matrix additive operator one-particle density matrix Quantum Transport Equation generalized collision term Transients in Quantum TransportII ...

  9. a closed equation for (Non-linear) quantum transport non-equilibrium problem Many-body system Initial state External disturbance Response many-body Hamiltonian many-body density matrix additive operator one-particle density matrix Quantum Transport Equation interaction term • existence, construction of • incorporation of the many-particle initial condition QUESTIONS Transients in Quantum TransportII ...

  10. DIRECT use a NGF solver This talk: orthodox study of quantum transport using NGF TWO PATHS INDIRECT use NGF to construct a Quantum Transport Equation Transients in Quantum TransportII ...

  11. Lecture on NGF This talk: orthodox study of quantum transport using NGF DIRECT TWO PATHS use a NGF solver INDIRECT use NGF to construct a Quantum Transport Equation Transients in Quantum TransportII ...

  12. This talk: orthodox study of quantum transport using NGF DIRECT TWO PATHS use a NGF solver Lecture on NGF…continuation Transients in Quantum TransportII ...

  13. This talk: orthodox study of quantum transport using NGF Real timeNGF choices DIRECT TWO PATHS use a NGF solver Lecture on NGF…continuation Transients in Quantum TransportII ...

  14. This talk: orthodox study of quantum transport using NGF DIRECT TWO PATHS use a NGF solver INDIRECT use NGF to construct a Quantum Transport Equation

  15. Standard approach based on GKBA Real timeNGF our choice Transients in Quantum TransportII ...

  16. Standard approach based on GKBA Real timeNGF our choice Keldysh IC:simple initial state permits to concentrate on the other issues DYSON EQUATIONS Transients in Quantum TransportII ...

  17. Standard approach based on GKBA Real timeNGF our choice  GKBE Transients in Quantum TransportII ...

  18. Standard approach based on GKBA Real timeNGF our choice  GKBE  Specific physical approximation -- self-consistent form Transients in Quantum TransportII ...

  19. Standard approach based on GKBA Real timeNGF our choice  GKBE  Specific physical approximation -- self-consistent form  Elimination of by an Ansatz widely used: KBA (for steady transport), GKBA (transients, optics) Transients in Quantum TransportII ...

  20. Standard approach based on GKBA Real timeNGF our choice  GKBE  Specific physical approximation -- self-consistent form  Elimination of by an Ansatz GKBA Resulting Quantum Transport Equation Transients in Quantum TransportII ...

  21. Standard approach based on GKBA Real timeNGF our choice  GKBE  Specific physical approximation -- self-consistent form  Elimination of by an Ansatz GKBA Resulting Quantum Transport Equation • Famous examples: • Levinson eq. • (hot electrons) • Optical quantum • Bloch eq. • (optical transients) Transients in Quantum TransportII ...

  22. Act I • reconstruction Transients in Quantum TransportII ...

  23. Exact formulation -- Reconstruction Problem GENERAL QUESTION: conditions under which a many-body interacting system can be described in terms of its single-time single-particle characteristics Transients in Quantum TransportII ...

  24. Exact formulation -- Reconstruction Problem GENERAL QUESTION: conditions under which a many-body interacting system can be described in terms of its single-time single-particle characteristics Reminiscences: BBGKY, Hohenberg-Kohn Theorem Transients in Quantum TransportII ...

  25. Exact formulation -- Reconstruction Problem GENERAL QUESTION: conditions under which a many-body interacting system can be described in terms of its single-time single-particle characteristics Reminiscences: BBGKY, Hohenberg-Kohn Theorem Here: time evolution of the system Transients in Quantum TransportII ...

  26. … in fact: express , a double-time correlation function, by its time diagonal  Eliminate by an Ansatz GKBA Exact formulation -- Reconstruction Problem New look on the NGF procedure: Any Ansatz is but an approximate solution… ¿Does an answer exist, exact at least in principle? Transients in Quantum TransportII ...

  27. Reconstruction Problem – Historical Overview INVERSION SCHEMES Transients in Quantum TransportII ...

  28. Reconstruction Problem – Historical Overview INVERSION SCHEMES Transients in Quantum TransportII ...

  29. Parallels G E N E R A L S C H E M E LABEL Bogolyubov Postulate/Conjecture: typical systems are controlled by a hierarchy of times separating the initial, kinetic, and hydrodynamic stages. A closed transport equation holds for Transients in Quantum TransportII ...

  30. Parallels G E N E R A L S C H E M E LABEL Bogolyubov Postulate/Conjecture: typical systems are controlled by a hierarchy of times separating the initial, kinetic, and hydrodynamic stages. A closed transport equation holds for Transients in Quantum TransportII ...

  31. Parallels G E N E R A L S C H E M E LABEL TDDFT Runge – Gross Theorem: Let be local. Then, for a fixed initial state , the functional relation is bijective and can be inverted. NOTES: U must be sufficiently smooth. no enters the theorem. This is an existence theorem, systematic implementation based on the use of the closed time path generating functional. Transients in Quantum TransportII ...

  32. Parallels G E N E R A L S C H E M E LABEL TDDFT Runge – Gross Theorem: Let be local. Then, for a fixed initial state , the functional relation is bijective and can be inverted. NOTES: U must be sufficiently smooth. no enters the theorem. This is an existence theorem, systematic implementation based on the use of the closed time path generating functional. Transients in Quantum TransportII ...

  33. Parallels G E N E R A L S C H E M E LABEL Schwinger Closed Time Contour Generating Functional (Schwinger): Used to invert the relation EXAMPLES OF USE: Fukuda et al. … Inversion technique based on Legendre transformation  Quantum kinetic eq. Leuwen et al. … TDDFT context Transients in Quantum TransportII ...

  34. Parallels G E N E R A L S C H E M E LABEL Schwinger Closed Time Contour Generating Functional (Schwinger): Used to invert the relation EXAMPLES OF USE: Fukuda et al. … Inversion technique based on Legendre transformation  Quantum kinetic eq. Leuwen et al. … TDDFT context Transients in Quantum TransportII ...

  35. Parallels: Lessons for the Reconstruction Problem G E N E R A L S C H E M E • „Bogolyubov“: importance of the time hierarchy REQUIREMENTCharacteristic times should emerge in a constructive manner during the reconstruction procedure. • „TDDFT“ :analogue of the Runge - Gross Theorem REQUIREMENTConsider a general non-local disturbance U in order to obtain the full 1-DM  as its dual. • „Schwinger“: explicit reconstruction procedure REQUIREMENTA general operational method for the reconstruction (rather than inversion in the narrow sense). Its success in a particular case becomes the proof of the Reconstruction theorem at the same time. LABEL NGF Reconstruction Theorem

  36. Reconstruction Problem – Summary INVERSION SCHEMES Transients in Quantum TransportII ...

  37. Reconstruction Problem – Summary INVERSION SCHEMES Transients in Quantum TransportII ...

  38. Reconstruction theorem:Reconstruction equations Keldysh IC:simple initial state permits to concentrate on the other issues DYSON EQUATIONS Two well known “reconstruction equations” easily follow: RECONSTRUCTION EQUATIONS LSV, Vinogradov … application! Transients in Quantum TransportII ...

  39. Reconstruction theorem:Reconstruction equations Keldysh IC:simple initial state permits to concentrate on the other issues DYSON EQUATIONS Two well known “reconstruction equations” easily follow: RECONSTRUCTION EQUATIONS • Source terms … the Ansatz • For t=t' … tautology  … input Transients in Quantum TransportII ...

  40. GKB EQ. DYSON EQ. RECONSTRUCTION EQ. Reconstruction theorem:Coupled equations Transients in Quantum TransportII ...

  41. Reconstruction theorem: operational description NGF RECONSTRUCTION THEOREM determination of the full NGF restructured as a DUAL PROCESS quantum transport equation   reconstruction equations Dyson eq. Transients in Quantum TransportII ...

  42. Reconstruction theorem: formal statement NGF RECONSTRUCTION THEOREM determination of the full NGF restructured as a DUAL PROCESS quantum transport equation   reconstruction equations Dyson eq. "THEOREM" The one-particle density matrix and the full NGF of a process are in a bijective relationship, Transients in Quantum TransportII ...

  43. Act II • reconstruction • and initial conditions • NGF view Transients in Quantum TransportII ...

  44. General initial state For an arbitrary initial state at start from the NGF Problem of determination of G extensively studied Fujita  Hall  Danielewicz  … Wagner  Morozov&Röpke … KlimontovichKremp … Bonitz&Semkat , Morawetz … Take over the relevant result for : The self-energy depends on the initial state (initial correlations) has singular components Transients in Quantum TransportII ...

  45. General initial state: Structure of Structure of Transients in Quantum TransportII ...

  46. General initial state: Structure of Structure of continuous time variable t > t0 singular time variable fixed at t = t0 Transients in Quantum TransportII ...

  47. General initial state: Structure of Structure of Danielewicz notation continuous time variable t > t0 singular time variable fixed at t = t0 Transients in Quantum TransportII ...

  48. General initial state: Structure of Structure of Danielewicz notation continuous time variable t > t0 singular time variable fixed at t = t0 Transients in Quantum TransportII ...

  49. GKB EQ. DYSON EQ. RECONSTRUCTION EQ. General initial state: A try at the reconstruction DANIELEWICZ CORRECTION

  50. GKB EQ. DYSON EQ. RECONSTRUCTION EQ. General initial state: A try at the reconstruction

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