fast transients in mesoscopic systems molecular bridge in transient regime b. velický, charles...

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

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.

NGF for a finite initial time

Reduction for a non-interacting system

fancy letters – entire system 4

†

Iaverage arbitrary: non-equilibrium, ove with cr many-body orrelations .. initial sta .te

causal Green function on the Keldysh contour

(1,1') i (1) (1') G CT

0 A W

R

L

G GG

G

G

I I I

I I I

i ( , ') ( ')

i ( , ') ( ' )

( , ') i ( , ) ( , ')

i ( , ) ( , ')

R

A

R A

t t t t

t t t t

t t t t t t

t t t t

G S

G S

G S SP

G GP†

I I I I

one-particle Hamiltonian

one-particle evolution operator

initial one-particle density matrix

i ( , ') ( ) ( , ') ( , )

( , ') ( , ) ( ', )

t t t t t t t t

x x x t x t

S H S S 1

P

Molecular bridge: NGF for a transient

For a non-interacting bridge, the NGF for a transient starting at tI 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.

Bridge model

Projectors

bound island states decay lead states

6

RIGHT LEADJRLEFT LEAD ISLANDJL

L R

HH H

0L 0B 0R

0

H H H

H

op

L R

, 0

,b r

b b b b r r

P Q PQ

P Q Q Q

1

coupling Hamiltonian

free Hamiltonian

0( ) ( ) ( )t t t H H H

Hamiltonian block structure

0

0 0 0

( ) ( ) ( )t t t

H H H

H PH P QH Q

H PH Q QH P

NGF for the bound (island) states

7

Bound states decay into continua of leads states ... genuine GF behavior without interactions

RIGHT LEADJRLEFT LEAD ISLANDJL

L R r

b b r r

Q P Q

Bound state Green’s functions

,

,

,

R R R R R R

A A A A A A

b b b b G G b b

b b b b G G b b

b b b b G G b b

G PG P G P G

G PG P G P G

G PG P G P G

Analogy with interacting systems

8


Interacting systems Non-interacting bridge

one-electron excitations island bound states

multi-electron excitations lead one-electron decay states

many-body interactions island-lead coupling

Dyson equations for propagators by Löwdin (Hilbert space) partitioning

9

Free propagators are block diagonal partitioning expressions

0 0,R R R R R Rb b G b G b QG QG QH G Q H QG Q

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 GA

0 0 0 0

0

,R R R R R R R R R R

R R

G G G G G G G G

b b

H QG QH

Dyson equations for global propagators are unitary

0 0 0 0,R R R R R R R R G G G H G G G G H G

Dyson equation for particle correlation function by Löwdin (Hilbert space) partitioning

10

I I I

I

( , ') ( , ')

i ( , ) ( , ')

i

R A

R A

G t t b t t b

b t t t t b

G b b

G

G GP

G P Q P Q GP

Particle correlation function is sensitive to initial conditions:

Dyson equation for particle correlation function by Löwdin (Hilbert space) partitioning

11

Dyson equation with initial conditions results

I I I

I

( , ') ( , ')

i ( , ) ( , ')

i

R A

R A

G t t b t t b

b t t t t b

G b b

G

G GP

G P Q P Q GP by partitioning by partitioning

0 I 0... four

( ) i ( ) terms

R R A AG b b b b G H QG Q QG QHP

Particle correlation function is sensitive to initial conditions:

Dyson equation for particle correlation function

12

0 I 0( ) i ( )R R A AG G b b b b G H QG Q QG QHP

R AG G G

I I I

I I

I I

i ( ) ( ) ( ' )

( , ) ( ' )

( , ') ( )

( , ')

t t t t t

t t t t

t t t t

t t

I I I

0 I I I

I 0 I I

0 I I 0 I

i ( ) ( ' )

i ( , ) ( ' )

i ( , ) ( )

i ( , ) ( , )

R

A

R A

b b t t t t

b t t b t t

b t t b t t

b t t t t b

PH QG QP

QG QHPH QG Q QG QHP


0 I 0( ) ( )R R A AG G b b b b G H QG Q QG QHP

I I I

I I

I I

i ( ) ( ) ( ' )

( , ) ( ' )

( , ') ( )

( , ')

t t t t t

t t t t

t t t t

t t

I

projected one-electron distrib

depend on off

ution

depends on unlike t

-diagonal blocks of

one-electron distrib

hat of

ution

JWM

Meaning of self-energy for the non-interacting bridge

t

R AG G G

uncorrelated initial condi

attributes of initial co

tion

regular self-

rrelatio

ener

ns

famous and

gy

Structure of self-energy well known in the field theory

:

I I I

0 I I I

I 0 I I

0 I I 0 I

i ( ) ( ' )

i ( , ) ( ' )

i ( , ) ( )

i ( , ) ( , )

R

A

R A

b b t t t t

b t t b t t

b t t b t t

b t t t t b

PH QG QP

QG QHPH QG Q QG QHP


0 I 0( ) ( )R R A AG G b b b b G H QG Q QG QHP

I I I

I I

I I

i ( ) ( ) ( ' )

( , ) ( ' )

( , ') ( )

( , ')

t t t t t

t t t t

t t t t

t t

I I I

0 I I I

I

I 0 I

I

I

0

i ( ) ( ' )

i ( , ) ( ' )

i ( , ) ( )

i ( , ; )

R

A

b b t t t t

b t t b t t

b t t b

t

t t

b t t b

PH QG QP

QG QHPH QG QH

uncorrelated initial condi

attributes of initial co

tion

regular self-

rrelatio

ener

ns

famous and

gy

Structure of self-energy well known in the field theory

:

I

projected one-electron distrib

depend on off

ution

depends on unlike t

-diagonal blocks of

one-electron distrib

hat of

ution

JWM

Meaning of self-energy for the non-interacting bridge

t

R AG G G

Analogy with the interacting systems

15






Analogy with the interacting systems continued:role of initial state

16






many-body statistical operator global one-electron density matrix

correlated initial state island and leads coupled

uncorrelated initial state island and leads uncoupled

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 conditionexploring the analogy

I I0, 0b b Q QP P

H

uncorrelated limit

0

0

"transport"initialtransient term

I

i

R A R AG G G G G

1 1

0 0 I 0 0

0 KELDYSH FORM

(1 ) (1 )

R R R A A A R A

R R A A R A

G G G G G G G G G

G G G G G G

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 conditionexploring the analogy

I I0, 0b b Q QP P

H

uncorrelated limit

0

0

"transport"initialtransient term

I

i

R A R AG G G G G

1 1

0 0 I 0 0

0 KELDYSH FORM

i

(1 ) (1 )

R R R A A A R A

R R A A R A

G G G G G G G G G

G G G G G G

18

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 conditionself-energy independent of initial time

I I0, 0b b Q QP P

I 0 II I( , ; ) i ( , ; )t t b t t bt t H QG QH

0 I I( ), 0,t t t

H P,

I I0 ( , ), 0,R A t t t t

G P

I

I

0 I I 0

I

I I

0 0

0 I

drops out

spectral density

( , ; ) i ( , ) ( , )

i ( , ) ( , )

( , )

( )

R A

R A

t t b t t t t b

b t t t t b

b t t b

t

t

H QG Q QG QHPH Q G G QHPH QA QHP

19

Initial states created by switch-on processes

A “physical” initial state is prepared at t=tI 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 tI (time partitioning).

Steps to solve the equations in a direct fashion

21


FUTUREPAST

22


t

t Ut

pulse envelope

Itequilibrium

observation period

transient process

uncorrelated initial state

correlated equil. state

FUTUREPAST

Switch-on transient process with Keldysh initial condition

23


t

t Ut

pulse envelope

Itequilibrium

observation period

preparation

Switch-on process with Keldysh init. cond. and preparation stage


correlatedNE state

t

t Ut

pulse envelope

Itequilibrium

observation period

transient process

Switch-on transient process with Keldysh initial condition


correlated equil. state

transient process

FUTUREPAST

24


R AG G G R AG G G

I I

( , ) d dt t

R

t

A

tG t t u vG G

( , ) d d

t tR AG t t u vG G

TRANSIENT HOST PROCESS

25


R AG G G R AG G G

I I

( , ) d dt t

R

t

A

tG t t u vG G

( , ) d d

t tR AG t t u vG G


different integration ranges !!!

26


R AG G G R AG G G

I I

( , ) d dt t

R

t

A

tG t t u vG G

( , ) d d

t tR AG t t u vG G

I I I

II

I I

i ( ) ( ) ( ' )

(( , )

( , '

' )

( )

( )

)

, '

t t t t t

t t

t t

t

t t

t t

t

( , ') ( , ')t tt t


. . . 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

Time partitioning for Switch-on states: RESULT

I

I

I

II I I I

I I I

I

I

( , ) i ( , ) ( , ) ,

( , ) d ( i) d ( , ) ( , ) (

(

, ) ( , ) ( , )

d ( , ) ( i) d ( , ) ( , ) ( ,

)

( )

,

}

{

{

R A

ttR R A A

t

tR A R

G t t G t t G t t t t t t

G t t u t G t t t u G t t t u G u t

vG t v t v t G t t v

t

t u

I

I I

I

I

I) ( , ) ( , )

d ( ,d ( , ){ }

( , )

( (, ) )) ,

}t

A

t

t tR A

t t

t G t t G t t

v uG t v

v

G u t

t

v uv u

I I

( , ) ( , ) d d { }t t

R R R A A Av u v u t t G G G

uncorrelated IC

correlated IC

28

Time partitioning and decay of correlations

I I

I I

( , ) ( , )

d d d d

( , )

t tt tRR R A

t tR

G t t t u

v u t t G G G

v t

Typical contribution to G< :

I

I

,t v t t

t t u t

29


I I

I I

( , ) ( , )

d d d d

( , )

t tt tRR R A

t tR

G t t t u

v u t t G G G

v t


I

I

,t v t t

t t u t

propagation in the future

propagation in the past

30


I I

I I

( , ) ( , )

d d d d

( , )

t tt tRR R A

t tR

G t t t u

v u t t G G G

v t


I

I

,t v t t

t t u t


propagation in the pastself-energies link the past and the future

31


I I

I I

( , ) ( , )

d d d d

( , )

t tt tRR R A

t tR

G t t t u

v u t t G G G

v t


I

I

,t v t t

t t u t


propagation in the pastself-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 tI is the same

(Bogolyubov principle).

32

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 tI Here ... the direct method those using the NGF along an extended Schwinger-Keldysh loop Here ... the time partitioning

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

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

I 0

I

I I

I

( , ) i

( , ) (

( ,

, )

) ,R

R A

t t b t

t

b

t t

t

t

H QG Q

G

P

G P

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

The rest is an algebra.

I 0 I I

I I

( , ) i ( , ) ,

( , ) ( , )

R

R A

t t b t t b

t t t t

H QG QP

G GP

36

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.

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)

0

R R L L

( ) ( )

( ) ( ) ( )

t t

t t t

H H H

H V V

t

It

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

t

0

L L L R R R

L L 0 L

R R 0 R

( , ) ( ) ( )

( ) ( , ) ( ) ( ) ( , ) ( )

( , )

( , ) , , ,

X X

X X

X X

X X

t t b t t b

t t t t t t t t

t t b b

t t b b X R A

H QG QH

V QG QV

V QG QV

R R L L( ) ( ) ( )t t t H V V

It t It tL R

L I R I

L R I

process ( ) ( )

ˆ ˆ( ) ( )

ˆ ˆ ( )

t t GF

t t t t G

t t G

U

C

U

C

It t It tL

L I R I

R

L

L R

process

ˆ

( ) ( )

0 0

ˆ 0

ˆ ˆ

ˆ( ) ( )t t t

t t F

G

t

G

G

G

C

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

Correlation function for U process

41

0

P A S TX X XG G G U

, ,

F U T U R ER A R AG G U

L L L

R R R

( , ) ( ) ( )

( ) ( )

X X

X

t t t t

t t

P F or F P

0X

U

0

P A S TX X XG G G U

, ,

F U T U R ER A R AG G U

L L L

R R R

( , ) ( ) ( )

( ) ( )

X X

X

t t t t

t t

P F or F P

0X

U

Correlation function for U process

42

I

I I

I

I I I I I

correl. corrections

( , ) i ( , ) ( )

d d ( , )

( , ) ,

d d ( , ) ( , ) ( , )

{ } ( , )

R A

t tR A

t t

t tR A

t t

G t t G t t t G t t t t t t

v uG t v

v uG t v

v u G u t

G u t

U

Uncorrelated initial condition

I I

I I

I I I I I

correl. corrections

( , ) i ( , ) ( ) ( , ) ,

d d ( , ) ( , ) ( , )

d d ( , ){ } ( , )

R A

t tR A

t t

t tR A

t t


v uG t v v u G u t

v uG t v G u t

C

P A S T

G G C

, ,

F U T U R ER A R AG G C

L L L

R R R

( , ) ( ) ( )

( ) ( )

X X

X

t t t t

t t

P F or F

0

PX X

C

Correlation function for C process

43

I I

I I

I I I I I

correl. corrections

( , ) i ( , ) ( ) ( , ) ,

d d ( , ) ( , ) ( , )

d d ( , ){ } ( , )

R A

t tR A

t t

t tR A

t t


v uG t v v u G u t

v uG t v G u t

C

Correlation corredtions for C process

44

A R

R A

R R R A A A

G G

G G

G G G

In the correlation correction the past soaks up into the future

fast transients in mesoscopic systems molecular bridge in transient regime b. velický, charles...

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