Download Slides - Agenda

Bohmian solution to the measurement of many-body systems:
S
Sequential
ti l currentt in
i mesoscopic
i electron
l t
devices
d i
X.Oriols, F.L.Traversa, G.Albareda.
Universitat Autònoma de Barcelona
E.mail: [email protected]
C ll b t
Collaborators:
X C t i à A.Alarcón,
X.Cartoixà,
A Al ó A.Benali,
A B li M.Yago,
MY
TN
T.Norsen.
What is the relation between electronics and Bohmian solution to
the meas
measurement
rement of many-body
man bod ssystems
stems
Electron devices
1.000.000.000 transistors per CPU !!!
Bohmian mechanics
Bohmian solution to the measurement of many-body systems:
S
Sequential
ti l currentt in
i mesoscopic
i electron
l t
devices
d i
1.- Introduction:
2.- The use of quantum (Bohmian) trajectories
3 Numerical results with the BITLLES simulator
3.-
4.- Conclusions and Future work
Bohmian solution to the measurement of many-body systems:
S
Sequential
ti l currentt in
i mesoscopic
i electron
l t
devices
d i
1.- Introduction:
1.1.- Why we have to worry about many-body systems ?
1.2.- Why we have to worry about sequential measurement ?
2.- The use of quantum (Bohmian) trajectories
3.- Numerical results with the BITLLES simulator
4.- Conclusions and Future work
1.1.- Why we have to worry about many-body systems ?
What we measure when we measure the current ?
Amperimeter
Ammeter I(t)
S
Source
Drain
W
Wire
W
DUT
I(t) measured
in the ammeter
VP(t)
Wire
RIN
RL
Integral (surface) conservation

s


J T  r , t  ds  0
VIN(t)
I(t) computed
t d
in the DUT
Differential (local) conservation

J T ( r , t )  0
Th many
The
Which magnitude accomplish the differential (local) conservation ?
body problem
 
d  
   Jc  0
 
Continuity equation
dE ( r , t ) 
dt
   J c (r , t )  
0
dt 

Poisson (Coulomb)   

  E (r , t )   (r , t ) / 
q
equation
The total current measured at the ammeter is equal to that computed at the DUT
5
1.1.- Why we have to worry about many-body systems ?
P.A.M. Dirac, 1929
The many
body problem
“The general theory of quantum mechanics is now
almost complete. The underlying physical laws
necessary for the mathematical theory of a large part
of physics and the whole of chemistry are thus
completely known, and the difficulty is only that the
exact application of these laws leads to equations
much too complicated to be soluble.”
Max Born, 1960
”It would indeed be remarkable if Nature fortified herself
against further advances in knowledge behind the
analytical difficulties of the many-body problem
problem.”
6
Multi-time measurement and displacement current
i time-dependent
in
ti
d
d t quantum
t
transport
t
t
1.- Introduction:
1.1.- Why we have to worry about many-body systems ?
1.2.- Why we have to worry about sequential measurement ?
2.- The use of quantum (Bohmian) trajectories
3.- Numerical results with the BITLLES simulator
4.- Conclusions and Future work
1.2.- Why we have to worry about sequential measurement ?
Two “orthodox” dynamical law for describing the evolution of a quantum system:
Unitary-evolution (when no measure): Schrodinger equation
1-law:
d
i
x  (t )  H x  (t )
dt
 (t )  e  iHt /   (0)  U  (0)
Non-unitary-evolution (when measure): “Collapse”
2-law:
x  (0)  x ui (t )
 (t )  ui ui  (0)
Example: a single-time
single time (ensemble) measurement
 I   I i P ( I i )   I i  ui ui 
i
i
P ( I i )   ui ui   ui 
   Iˆ ui ui 
2
i
Iˆ ui  I i ui
  Iˆ
1   ui ui
i
For ergodic
g
system
y
Time-average
g is equal
q to ensemble average
g
The prediction of DC current can be computed without worrying about “collapse”
8
1.2.- Why we have to worry about sequential measurement ?
Sequential
S
i l
measurement
Example: noise (current fluctuations) measurement
Tootal Current (mA
A)
Engineers do not like noise, it makes errors in the device.
Physicist
Ph
i i t enjoy
j noise,
i bbecause it shows
h
phenomena
h
that
th t
are not present in DC.
I (t )
I ( t )  I ( t )  I
R ( )
S( f )
Time (ps)
fluctuations
Autocorrelation
Fourier transform
S(f)=  R( ) e -j2 f d
( )I(t+
( )
  t 2  t1 R(( ))= I(t)
R( )= I(t 2 )I(t1)   IDC 

-
2
We need the measurement of I(t2) after the measurement I(t1).
9
1.2.- Why we have to worry about multi-time measurement ?
Sequential
q
measurement
How we can compute the quantum two-times correlations ?
I(t 2 )I(t1 )  
j
 I (t
j
2
)·I i (t1 )·P  I j (t 2 ),I i (t1 ) 
i
P  I j (t 2 ),I i (t1 )    (0) ui ui Uˆ † ( ) u j u j Uˆ ( ) ui ui  (0)
How we can compute the probability ?
t1
Measuring at t1
With the modulus of the wavefunction
2-law:
 (t1 )  ui ui  (0)
 (t2 )  Uˆ ( )  (t1 )
Time
Time-evolution
1-law:
t2
Measuring
g at t2
2-law:
Final state
 (t2 )  u j u j  (t2 )
 (t2 )  u j u j Uˆ ( ) ui ui  (0)
10
Bohmian solution to the measurement of many-body systems:
S
Sequential
ti l currentt in
i mesoscopic
i electron
l t
devices
d i
1.- Introduction:
2.- The use of quantum (Bohmian) trajectories
2.1.. . Bohmian
o
mechanics
ec
cs (quantum
(qu u hydrodynamics)
yd ody
cs)
2.2.- The conditional wave function: a successful tool
2.3.- The computation of the displacement current
2.4- The computation of multi-time measurement
3.- Numerical results with the BITLLES simulator
4.- Conclusions and Future work
2.1.- Bohmian mechanics (quantum hydrodynamics)
L. De Broglie
1924
D. Bohm
J.S. Bell
1952
1960
t1
“Playing” with the many-particle Schrodinger equation....
 
  
 
 ( r1,,..,, rN , t )  N
 2 2
i
  
 rk  U ( r1,.., rN , t ) · ( r1,.., rN , t )
t
2·m
 k 1

Look for a continuity equation:
Define a velocity:


2
  ( r1 ,.., r N , t )
t

N

k 1
t2


 J k  r1 ,.., r N , t   0






2
v k ( r1,.., r N , t )  J k ( r1,.., r N , t ) /  ( r1,.., r N , t )
Define a Bohmian trajectory:


r1 [ t ]  r1 [ t 0 ] 

t
to


d t '·v k ( r1 [ t '],.., r N [ t '], t ')
The computation of many
many-particle
particle trajectories exactly reproduce the evolution of the
modulus of the wavefunction.
12
4
2.1.- Bohmian mechanics (quantum hydrodynamics)
Example: A wave packet impinging upon a double barrier structure
1.- First,
2 - Second,
2.Second
3.- Third:
 ( x, t )
J ( x, t )
v( x, t ) 
| ( x , t )|2
t
x i [t ]  xoi   v ( x i [t '], t ')·dt '
0
Main criticism against Bohmian formalism:
“…In any case, the basic reason for not paying attention to the Bohm approach
is not some sort of ideological rigidity, but much simpler…It is just that we are
all too busy with our own work to spend time on something that doesn’t seem
likely to help us make progress with our real problems”.
Steven Weinberg (private comunication with Shelly Goldstein)
13
Bohmian solution to the measurement of many-body systems:
S
Sequential
ti l currentt in
i mesoscopic
i electron
l t
devices
d i
1.- Introduction:
2.- The use of quantum (Bohmian) trajectories
2.1.. . Bohmian
o
mechanics
ec
cs (quantum
(qu u hydrodynamics)
yd ody
cs)
2.2.- The conditional wave function: a successful tool
2.3.- The computation of the displacement current
2.4- The computation of multi-time measurement
3.- Numerical results with the BITLLES simulator
4.- Conclusions and Future work
The many
body problem
2.2.- The conditional wave function: a successful tool
The “BIG” many-particle wave-function for N particles:
 ( x1 ,..., x N , t )
v1 ( x1[t ],..,
] x N [t ],
] t) 
v1 ( x1[t ],..,
] xN [t ]], t ) 
M grid points
M N configuration point
J 1 ( x1 ,..., x N , t )
 ( x1 ,..., x N , t )
2
x1  x1 [ t ],...., x N  xN [ t ]
J 1 (( x1 , x2 [t ],.., xN [t ], t )

2
 ( r1 , x2 [t ],.., xN [t ], t )
x1  x1 [ t ]
wave functions for N particles:
The “LITTLE” conditional wave-functions
 ( x1 , x2 [t ],.., x N [t ], t )  1 ( x1 , t )
M ·N
M grid points
configuration point
The computation of many-particle trajectories from one BIG or from N LITTLE
conditional wave functions are exactly identical.
What is the equation satisfied by the conditional wave-function ?
15
2.2.- The conditional wave function: a successful tool

1 ( x1 , t )   2  2



i
 

U
(
x
,
x
[
t
],
]
t
)

G
(
x
,
x
[
t
],
]
t
)

i
·
J
(
x
,
x
[
t
],
]
t
)
 1 ( x1 , t )
1
1
b
1
1
b
1
1
b
2
t
 2 m x1

Good points :
[X. Oriols, Phys. Rev. Lett. 98, 066803 (2007)]
An exact procedure for computing many-particle Bohmian trajectories
The correlations are introduced into the time-dependent potentials
4th The interacting potential from (a classical-like) Bohmian trajectories
5th There is a real potential to account for “non-classical” correlations
6th There is a imaginary potential to account for non-conserving norms
Bad points :
The terms G and J depends on the many-particle wave-function
This is exactly the same difficulty found in the DFT (or TD-DFT)
16
2.2.- The conditional wave function: a successful tool
Example: two Coulomb interacting particles



q2
1 ( r1 , t )   2
2
i
 
1 
   1 ( r1 , t )
4    0 r1  r2 [t ] 
t
 2 m *

q2
 2 ( r2 , t )   2
2
i
 
2 


t
2
m
*
4    0 r1[t ]  r2



  2 ( r2 , t )

;
;

t


J 1 ( r1 , t )
r1[t ]  r1[to ]   dt
 2
|
(
r

1 1, t ) |
to

t
J 2 ( r2 , t )


r2 [t ]  r2 [to ]   dt

|  2 ( r2 , t ) |2
to
 
r1  r1 [ t ]
 
r2  r2 [ t ]
17
2.2.- The conditional wave function: a successful tool
Example: two (Coulomb and Exchange) interacting particles
Ensemble results associated to x1 are indistinguishable from those associated to x2.
But, Bohmian trajectories associated to particle x1 or to x2 are distinguishable.

x1[to]
-40
x1,to)
~


Emitter
x1[t]

~

x1,to)
to

0
x
x1,t)
2

x2,to)
~

x1[t]

x2,,t))
-20
x2[to]
-40
Emitter
0
20
40
~


to
60
x1[t] 2D Bohm
Area=A=900 nm
~

~

x2,to)

x2[t] 1D' Bohm
0
x2,t)
100 120 140 160 180 200 220
Time t (fs)
Emitter
1D Bohm
1D'
2D Bohm
-10
-20
x2[t] 2D Bohm
B h
t
80
Collector
10
x1[t] 1D' Bohm
t
(b) Collector
20
x1,t)
2
Area=A=900 nm
-30
x'1[0]=-40.50 nm
x'2[0]= -23.25
x
23 25 nm
x1[0]=-23.25 nm
x2[0]= -40.50 nm
-30
30
-20
20
[X. Oriols, Phys. Rev. Lett. 98, 066803 (2007)]
Collector
x
-20
~

(c)
Emitter
Pos
sition x1[t] and
d x2[t] (nm)
0
~

20
2
Area=A=2.25 m
Collector
Position x2 (nm)
(a)
20
-10
10
0
10
20
Position x1 (nm)
18
2.2.- The conditional wave function: a successful tool
Is it possible to improve the estimations of G and J functions ?




1 ( x1 , t )   2  2
i
 
 U1 ( x1 , xb [t ], t )  G1 ( x1 , xb [t ], t )  i·J 1 ( x1 , xb [t ], t )  1 ( x1 , t )
2
2
m
x
t



1
Taylor expansion of the many-particle wave function:
 ( x1 , x2 , t )
1  2  ( x1 , x2 , t )
 ( x1 , x2 , t )   ( x1 , x2 , t ) x  x [ t ] 
( x2  x2 [t ]) 
( x2  x2 [t ]) 2  ....
2
2
2
x2
x2
2!
x  x [t ]
x  x [t ]
2
2
2
2
The “LITTLE” conditional wave-functions for the derivatives:
 ( x1 , x2 , t ) x  x [ t ]  1 ( x1 , t )
2
2
 n  ( x1 , x2 , t )
n


( x1 , t )
n
1
x2
x  x [t ]
2
2
A co
coupled
pled system
s stem of eq
equations
ations for the deri
derivatives
ati es conditional wave-functions
a e f nctions
1n ( x1 , t )
 2  2 1n ( x1 , t )  2 n  2
d x2 [t ] n  n   n iU ( x1 , x2 , t )
n 1
i


1 ( x1 , t )  i1 ( x1 , t )
  
1i ( x1 , t )
2
n i
t
2m
x1
2m
dt
x2
i 0  i 
x  x [t ]
2
For n=0,1,2,3,4,5.......
2
[T.Norsen, Found. Phys 40, 1858 (2010)]
19
Bohmian solution to the measurement of many-body systems:
S
Sequential
ti l currentt in
i mesoscopic
i electron
l t
devices
d i
1.- Introduction:
2.- The use of quantum (Bohmian) trajectories
2.1.. . Bohmian
o
mechanics
ec
cs (quantum
(qu u hydrodynamics)
yd ody
cs)
2.2.- The conditional wave function: a successful tool
2.3.- The computation of the displacement current
2.4- The computation of multi-time measurement
3.- Numerical results with the BITLLES simulator
4.- Conclusions and Future work
2.3.- The computation of the displacement current
The TOTAL timetime-dependent current:
current:
 


E (Er,(tr), t ) 
·ds ·ds Quantum ensemble average value
I (tI)(t )  JJcc((tt))··ds
ds 
tt
SSDD
S DS D
Ensemble over all position distributions weighted by the probability
 





2   
E ( r , t )   dr1..... drN  ( r1 ,..., rN , t ) E ( r , r1 ,..., rN , t )
Solving the N-particle Poisson equation
N
  

 
E ( r , r1 ,..., rN , t )   q ( r  ri )
i 1
Solving
equation
¿??
S l i the
th N-particle
N
ti l Schrodinger
S h di
ti


The many
2
N









 ( r1 ,..., rN , t )
i
   
2k   U ( r1 ,,...,, rN , t )   ( r1 ,,...,, rN , t )body problem
t
 k 1  2 m 

21
2.3.- The computation of the displacement current
The TOTAL timetime-dependent “Bohmian”current:
Bohmian”current:
I (t ) 



J c (t ) ·ds 
SD


 E ( r, t )
t
SD

·ds
 





2   
E ( r , t )   dr
d 1..... dr
d N  ( r1 ,..., rN , t ) E ( r , r1 ,..., rN , t )
An ensemble of Bohmian trajectories reproduces the wavefunction modulus


1
2
 ( r1 ,..., rN , t )  lim
M  M
M

j 1




 ( r1  r1 j [t ])····· ( rN  rNj [t ])
Th prediction
The
di i off AC current can be
b computedd from
f
an ensemble
bl over trajectories.
j
i
j
j 
r1 [t ] ;  1 ( r1 , t )
….
j
j 
rN [t ] ;  N ( rN , t )
N
j  j
j
 j
E ( r , r1 [t ],..., rN [t ], t )   q ( r  ri [t ])
 
1
E ( r , t )  lim
M  M
i 1
M

j 1
j  j

E ( r , r1 [t ],..., rNj [t ], t )
22
2.3.- The computation of the displacement current
The TOTAL timetime-dependent “Bohmian”current:
Bohmian”current:
I (t ) 

SD


J c (t ) ·ds 

SD

 E ( r, t )
t

·ds
[A. Alarcon, JSTM (2009)]
 



dE ( r , t ) 
   J c (r , t )  
0
dt 

Each trajectory of the ensemble fulfils the TOTAL current conservation law.
23
Bohmian solution to the measurement of many-body systems:
S
Sequential
ti l currentt in
i mesoscopic
i electron
l t
devices
d i
1.- Introduction:
2.- The use of quantum (Bohmian) trajectories
2.1.. . Bohmian
o
mechanics
ec
cs (quantum
(qu u hydrodynamics)
yd ody
cs)
2.2.- The conditional wave function: a successful tool
2.3.- The computation of the displacement current
2.4- The computation of multi-time measurement
3.- Numerical results with the BITLLES simulator
4.- Conclusions and Future work
2.4- The computation of multi-time measurement
A “simple” explanation on Bohmian measurement……
measurement……
The wavefunction:
 
  r1,,...,, rN , y, t 
The trajectories:
j j j
j
r1 [t], r2 [t], r3 [t],..., rN [t], yj[t]= I(t)
The conditional wavefunction:
  j
  r1,..., rN , y [t],
] t
The many
body problem
Time (ps)
[A.Alarcón et. al. Chapter 6, Applied Bohmian Mechanics (2012)]
25
2.1.3.- Multi-time Bohmian measuremen of the current
Is the Bohmian solution of the measurement equal to the Orthodox solution ?
Bohmian trajectory
I(t)=y[t]
Copenhagen operators
I(t)= eigenvalue of an Operator
What would happen if
I(t)
were not equal to I(t)
?
Copenhagen supporters: The measuring apparatus is wrong, because Bohmian
mechanics has to reproduce orthodox quantum mechanics.
Bohmian supporters: The operator is wrong, because
orthodox quantum mechanics has to reproduce Bohmian
mechanics.
“N ï realism
“Naïve
li about
b t operators”
t ”
[D. Durr, S. Goldstein, N. Zanghi, J. Stat. Phys., 116, 959-1055, (2004)]
26
2.4- The computation of multi-time measurement
Is the Bohmian solution of the measurement equal to the Orthodox solution ?
1  k N

ˆ
Î1 
q
p
 k kx 
Lx m  k 1
Total current operator

Mˆ I1 t    px

px

px   I1 t , I  px 
Its associated ((degenerate)
g
) pprojector=POVM
j
 f  t   Mˆ I1 t   0  t 
Final (unnormalized) state after measurement is a
total momentum eigenstate.
A.- Following a collapsed wave function
The probability of being transmitted at t2 and
reflected at t3 is zero
zero.
B.- Following a trajectory
The probability of being transmitted at t2 and
reflected at t3 is zero.
27
2.4- The computation of multi-time measurement
Partition “Noise” comparison for electron devices without (many-body) Coulomb ?
[M.Buttiker PRL1990]
SAMPLE (A)
quantization approach
1.5

An electron “created” at one contact | 1  a 1 | 0 
0.2
1.0
Fano Factor
 nT n R
F=S(0)/2q(II)
can not “emerge” at two different contacts

 

 1 | b 1  b 1  b 2  b 2 | 1  0
0.3
0.1
0.5
0.0
0.0
2.0
Buttiker results
0.3
1.5
SAMPLE (B)
mission coefficiient T(E)
Transm
2nd
Buttiker results
2.0
0.2
1.0
0.1
0.5
0.0
0.00
0.05
0.10
0.15
0.20
0.0
0.25
Applied bias V (Volts)
28
Bohmian solution to the measurement of many-body systems:
S
Sequential
ti l currentt in
i mesoscopic
i electron
l t
devices
d i
1.- Introduction:
2.- Theoretical discussion
3.- Numerical results with the BITLLES simulator
4.- Conclusions and Future work
3.2.- DC current beyond mean field
[X.Oriols, APL(2005) ]
Effect of Coulomb correlation on current and noise
6
4
(a)
Our approach
5
# Particles emitter
Esaki results [16]
2
EFE
1
ER
Colector
Emitter
# Particles emitter
x=0
# Particles collector
EFC
L
x=L
4
Visual guided line
3
2
1
FROZEN POTENTIALS
SELF-CONSISTENT POTENTIALS
0
0
1.4
Our approach
1.2
Our approach
Fano
o Factor
Buttiker results [6]
Fano F
Factor
Our approach
# Particles collector
Cu
urrent (A)
Cu
urrent (A)
3
(a)
1.0
Visual guided line
ER FROZEN
1.2
ER FROZEN
1.0
(b)
0.8
0
8
0.0
(b)
FROZEN POTENTIALS
0.1
0.2
0.3
Applied bias (Volts)
0.4
0.8
0
8
0.0
SELF-CONSISTENT POTENTIALS
0.1
0.2
0.3
0.4
Applied bias (Volts)
30
Bohmian solution to the measurement of many-body systems:
S
Sequential
ti l currentt in
i mesoscopic
i electron
l t
devices
d i
1.- Introduction:
2.- Theoretical discussion
3.- Numerical results with the BITLLES simulator
4.- Conclusions and Future work
4.- Conclusions and future work
There is a need for predictions of the dynamics properties of quantum devices…..
1.- FORMALISM: We have shown the ability of many-particle Bohmian
trajectories (extracted from conditional wave functions) to accurately and
efficiently compute time dependent transport (AC, noise, transients,…).
2.- SIMULATOR: Using these ideas, we have developed a user-friendly, handmade simulator for time-dependent electron transport, named:
Bohmian Interacting Transport for non-equiLibrium eLEctronic Structures
[G. Albareda et al. Phys. Rev. B 79, 075315 (2009)]
[G. Albareda et al. Phys. Rev. B 82, 085301(2010)]
[F. L. Traversa, et al. IEEE Trans. on Electron devices, 58(7) 2104 (2011) ]
Freely available at http:\europe.uab.es\bitlles
32
4.- Conclusions and future work
Acknowledgment
g
This work has been partially supported by
the Ministerio de Ciencia e Innovación
under Project No. TEC2009-06986 and by
the DURSI of the Generalitat de Catalunya
under
d C
Contract
t t No.
N 2009SGR783.
2009SGR783
Thank you very much for your attention
BITLLES freelyy available
at http:\europe.uab.es\bitlles
X.Oriols and J.Mompart,
A li d Bohmian
Applied
B h i M
Mechanics:
h i
From Nanoscale Systems to Cosmology, (2011).
ISBN: 978-981-4316-39-2
33