fast transients in mesoscopic systems molecular bridge in transient regime b. velický, charles...
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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
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
Bound states decay into continua of leads states ... genuine GF behavior without interactions
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
Dyson equation for particle correlation function
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
Dyson equation for particle correlation function
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
Bound states decay into continua of leads states ... genuine GF behavior without interactions
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
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 systems Non-interacting bridge
one-electron excitations island bound states
multi-electron excitations lead one-electron decay states
many-body interactions island-lead coupling
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
Steps to solve the equations in a direct fashion
FUTUREPAST
22
Steps to solve the equations in a direct fashion
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
Steps to solve the equations in a direct fashion
t
t Ut
pulse envelope
Itequilibrium
observation period
preparation
Switch-on process with Keldysh init. cond. and preparation stage
uncorrelated initial state
correlatedNE state
t
t Ut
pulse envelope
Itequilibrium
observation period
transient process
Switch-on transient process with Keldysh initial condition
uncorrelated initial state
correlated equil. state
transient process
FUTUREPAST
24
Dyson equation for particle correlation function
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
Dyson equation for particle correlation function
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
different integration ranges !!!
26
Dyson equation for particle correlation function
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
TRANSIENT HOST 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
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
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
propagation in the future
propagation in the past
30
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
propagation in the future
propagation in the pastself-energies link the past and the future
31
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
propagation in the future
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
G t t G t t t G t t t t 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
G t t G t t t G t t t t 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
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