1. Fast transients in mesoscopic systems Molecular Bridge in Transient Regime B. Velický, Charles University and Acad. Sci. of CR, Praha A. Kalvová, Acad. Sci. of CR, Praha V. Špička, Acad. Sci. of CR, Praha PNGF4 Glasgow, 17 – 21 August, 2009

2. Fast transients in mesoscopic systems Molecular Bridge in Transient Regime B. Velický, Charles University and Acad. Sci. of CR, Praha A. Kalvová, Acad. Sci. of CR, Praha V. Špička, Acad. Sci. of CR, Praha PNGF4 Glasgow, 17 – 21 August, 2009

3. Introductory note 3 We study the JWM model canonical by now for study of transient processes using NGF: Generally known advantages – rich physical content & easy analytic study More recently, several groups used this model to treat transients starting at a finite initial time: This paper contributes to the topic of finite time initial conditions. We confine ourselves to the non-interacting limit, which permits to obtain basic results in a closed analytic form. We use this advantage to look into the contrast between correlated and uncorrelated initial conditions in detail.

4. NGF for a finite initial time Reduction for a non-interacting system fancy letters – entire system 4

5. Molecular bridge: NGF for a transient For a non-interacting bridge, the NGF for a transient starting at t I from an arbitrary initial state can be obtained directly by a simple quantum mechanical calculation which can be given a field-theoretic appearance and interpretation.

6. Bridge model Projectors bound island states decay lead states 6 RIGHT LEAD JRJR LEFT LEAD ISLANDJLJL coupling Hamiltonian free Hamiltonian Hamiltonian block structure

7. NGF for the bound (island) states 7 Bound states decay into continua of leads states... genuine GF behavior without interactions RIGHT LEADJRJR LEFT LEADISLANDJLJL Bound state Green’s functions

8. Analogy with interacting systems 8 Bound states decay into continua of leads states... genuine GF behavior without interactions Interacting systemsNon-interacting bridge one-electron excitationsisland bound states multi-electron excitationslead one-electron decay states many-body interactionsisland-lead coupling

9. Dyson equations for propagators by Löwdin (Hilbert space) partitioning 9 Free propagators are block diagonal  partitioning expressions Dyson equations for the island state propagators result This coincides, of course, with the standard JWM result... propagators in the non-interacting case do not depend on statistics... on the initial state. - - - and the same for G A Dyson equations for global propagators are unitary

10. Dyson equation for particle correlation function by Löwdin (Hilbert space) partitioning 10 Particle correlation function is sensitive to initial conditions:

11. Dyson equation for particle correlation function by Löwdin (Hilbert space) partitioning 11 Dyson equation with initial conditions results Particle correlation function is sensitive to initial conditions:

12. Dyson equation for particle correlation function 12

13. Dyson equation for particle correlation function

15. Analogy with the interacting systems 15 Bound states decay into continua of leads states... genuine GF behavior without interactions Interacting systemsNon-interacting bridge one-electron excitationsisland bound states multi-electron excitationslead one-electron decay states many-body interactionsisland-lead coupling

16. Analogy with the interacting systems continued: role of initial state 16 Bound states decay into continua of leads states... genuine GF behavior without interactions Interacting systemsNon-interacting bridge one-electron excitationsisland bound states multi-electron excitationslead one-electron decay states many-body interactionsisland-lead coupling many-body statistical operatorglobal one-electron density matrix correlated initial stateisland and leads coupled uncorrelated initial stateisland and leads uncoupled

17. Uncorrelated initial condition is defined by Initial distribution is block-diagonal  the island and lead states are only coupled dynamically, through, not by the initial condition. As expected, The Dyson equation reduces to which is equivalent with the standard Keldysh form of the initial condition Uncorrelated initial condition exploring the analogy 17

18. Uncorrelated initial condition is defined by Initial distribution is block-diagonal  the island and lead states are only coupled dynamically, through, not by the initial condition. As expected, The Dyson equation reduces to which is equivalent with the standard Keldysh form of the initial condition Uncorrelated initial condition exploring the analogy 18

19. Self-energy does not simplify for the uncorrelated initial condition Stronger condition implies independent of initial time somewhat like the fluctuation-dissipation structure... towards the KB Ansatz identical with the JWM form of self-energy Uncorrelated initial condition self-energy independent of initial time 19

20. Initial states created by switch-on processes A “physical” initial state is prepared at t = t I by a switch-on process antecedent to our transient. The initial conditions at this instant are fully captured by the NGF for the joint process {preparation & transient}. The transient NGF is extracted by a projection on times future with respect to t I (time partitioning).

21. Steps to solve the equations in a direct fashion 21

22. Steps to solve the equations in a direct fashion FUTUREPAST 22

23. Steps to solve the equations in a direct fashion pulse envelope equilibrium observation period transient process uncorrelated initial state correlated equil. state FUTUREPAST Switch-on transient process with Keldysh initial condition 23

24. Steps to solve the equations in a direct fashion pulse envelope equilibrium observation period preparation Switch-on process with Keldysh init. cond. and preparation stage uncorrelated initial state correlated NE state pulse envelope equilibrium observation period transient process Switch-on transient process with Keldysh initial condition uncorrelated initial state correlated equil. state transient process FUTUREPAST 24

25. Dyson equation for particle correlation function TRANSIENTHOST PROCESS 25

26. Dyson equation for particle correlation function TRANSIENTHOST PROCESS different integration ranges !!! 26

27. Dyson equation for particle correlation function TRANSIENTHOST PROCESS... the purple components of have to add to the integrand of the right hand integral to compensate for the reduced integration range as compared with the left hand integral... TASK: express the future-future block of in terms of the past-past and past-future blocks of the host GF and self-energies PARTITIONING-IN-TIME METHOD different integration ranges !!! 27

28. Time partitioning for Switch-on states: RESULT uncorrelated IC correlated IC 28

29. Time partitioning and decay of correlations Typical contribution to G < : 29

30. Time partitioning and decay of correlations Typical contribution to G < : propagation in the future propagation in the past 30

31. Time partitioning and decay of correlations Typical contribution to G < : propagation in the future propagation in the past self-energies link the past and the future 31

32. Time partitioning and decay of correlations Typical contribution to G < : propagation in the future propagation in the past self-energies link the past and the future If a finite time for the decay of correlations exists... the self-energies are concentrated to a strip around the equal time diagonal of a width, the depth of interpenetration of the past and the future around t I is the same (Bogolyubov principle). 32

33. Various approaches to correlated initial conditions Two complementary techniques dealing with correlated initial conditions in current use are compared :  those using characteristics of the initial state at t I Here... the direct method  those using the NGF along an extended Schwinger-Keldysh loop Here... the time partitioning

34. Correlated initial conditions have more recently been attacked along two complementary lines:  the diachronous techniques: the finite-time Keldysh loop is extended, commonly by an imaginary stretch, the NGF determined along the extended contour starting at an uncorrelated state; either this is the result, or the finite-loop NGF is deduced by a contraction Fujita  Hall  Danielewicz  …  Wagner  Morozov&Röpke …  the synchronous techniques: the correlated initial state represented by a chain of correlation functions at a single -- initial time instant and suitably terminated Klimontovich  Kremp  …  Bonitz&Semkat … Two approaches to the correlated initial conditions 34

35. Diachronous vs. synchronous for our bridge model Use of the Keldysh switch-on states with a subsequent time-partitioning is a variant of diachronous methods Direct solution by Hilbert space partitioning permits in this special case an exact explicit result by the synchronous approach With the two solutions at hand, we were able to derive one from the other verifying their equivalence and visualizing the underlying complementarity of both views. Basic idea 35

36. Diachronous vs. synchronous for our bridge model Use of the Keldysh switch-on states with a subsequent time-partitioning is a variant of diachronous methods Direct solution by Hilbert space partitioning permits in this special case an exact explicit result by the synchronous approach With the two solutions at hand, we were able to derive one from the other verifying their equivalence and visualizing the underlying complementarity of both views. Basic idea The rest is an algebra. 36

37. Molecular bridge with time-dependent coupling One source of transient behavior in the bridge structure are fast changes in the coupling strengths of both junctions. We contrast different sudden transitions between a coupled and an uncoupled state of a junction, some reducing to the uncorrelated initial state, other manifest- ing the correlated behavior.

38. Time dependent coupling 38 Changes in the coupling strength of the junctions are interesting can be very fast – extreme of possibility of ensuing transients exotic – represent a time dependent “interaction” strength in our analogy partitioning in time is technically well suited Bridge Hamiltonian further specialized All time dependence in the  amplitudes Processes start as switch-on at, the couplings switched on adiabatically in part (preparation), then vary arbitrarily starting from (transient – observation)

39. Expressions for self-energy of the switch-on processes 39 All components of the self-energy have only regular parts (... ) identical with usual JWM expressions: These “host process” self-energies will serve as input for generating the transient self-energies by partitioning-in-time

40. Definition of two coupling transients Asymptotic stationary processes serving as building blocks in time partitioning Two transient processes excited by sudden turning on of lead-island coupling T R A N S I E N T

41. Correlation function for U process 41

42. Correlation function for U process 42 Uncorrelated initial condition

43. Correlation function for C process 43

44. Correlation corredtions for C process 44 In the correlation correction the past soaks up into the future