21st-century directions in de Broglie Broglie-Bohm Bohm theory and beyond

Quantum Q many-particle y p computations p with Bohmian trajectories j
Xavier Oriols E.mail: Xavier.Oriols@uab.es
21st-century directions in
de Broglie Broglie-Bohm Bohm theory and beyond
Saturday 28th August - Saturday 4th September 2010
Vallico Sotto, Tuscany, Italy

Quantum Q many-particle y p computations p with Bohmian trajectories j
1.- Quantum Many-Particle Computations with Bohmian Trajectories:
11.1.- 1 Introduction
1.2.- Our Many-particle y p Bohmian trajectories j (MPBT) ( ) theorem
2.- Application pp to Electron Transport p in Nanoelectronic Devices:
2.1.- The electron transport problem at the nanoscale
2.2.- Our quantum Monte Carlo algorithm
33.- CConclusions l i and d Future F t workk

Quantum Q many-particle y p computations p with Bohmian trajectories j
1.- Quantum Many-Particle Computations with Bohmian Trajectories:
1 1.1.- 1 Introduction
1.1.1.- Why to use Bohmian mechanics
1.1.2.- The “many-body” y y problem p
1.1.3.- Does Bohmian mechanics provide any help?
1.2.- Our Many-particles y p Bohmian trajectories j (MPBT) ( ) theorem
1.3.- Approximation
2.- Application to Electron Transport in Nanoelectronic Devices:
3.- Conclusions and Future work

1.1.1.- Introduction: Why to use Bohmian mechanics?
Reason 1: Bohmian Thinking:
.-Looking for extensions/limitations/contradictions of ‘Orthodox’ QM.
Reason 2: Bohmian Explaining:
.- An intuitive/simpler/causal explanation of QM phenomena.
J ( r,.., r , t)
v ( r,.., r , t)
ix 1 N
ix 1 N 2
( r1,.., rN, t)
Average values
and fluctuations
( r1
,.., rN
, 0)
,.., r , t)
“Quantum equilibrium”
( r1 N
4

1.1.1.- Introduction: Why Bohmian mechanics?
Reason 3. Bohmian computing:
“Analytical” : Bohmian trajectories from the wavefunction
“Synthetical” : without explicitly computing the wavefunction
Quantum Hamilton-Jacobi equation
2
Srt
( , ) S
U( r, t) Q( r, t)
0
t
2m
Quantum potential
Qrt ( , )
Rrt ( , ) 2 m
2 2
1 Rrt ( , )
CContinuity i i equation i
( , ) 0
2
N
R ( r, t) 2 S( r, t)
R r t
t k 1
m
Unknowns:
Rrt ( , )
Srt ( , )
Synthetical examples from the Physical-Chemistry community:
With Without t trajectories: t j t i HHydrodynamic drod namicq quantum ant m equation: eq ation:BB. Kendrix
With trajectories:
Lagrangian (moving with the flow): Robert E.Wyatt
Complex Action: D.Tannor
Valid for 1, 2, 3... degrees of freedom
5

1.1.2.- Introduction: The “many-body” problem
P.A.M. Dirac, 1929
“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

1.2.- Introduction: The “many-body” problem
The many-particle (non-relativistic spinless) Schrödinger equation
( r ,.., r
, t)
( ,.., , ) · ( ,.., , )
Many particle wave function
N 2
( r1,.., rN, t)
2
i r U r
k 1 rN t r1
rN t
t k k
1 2· m
1
N
The practical solution is inaccesible for more than very few electrons
Exercise: Number of hard disks to load the (finite-difference) wavefunction
in a PC memory y for N=10 particles, p , L=50nm length g (Δx=1 ( nm) )
nº of variables (1 particle) = 50 3 = 125000 variables !!
nº of variables (Total degrees of freedom) = 50 3N = 50 30 variables !!
nº of bits = 5030 variables *16 bits/variable ̴̴
138 Terabytes !!
nº of PCs ̴̴
100000000000000000000000000000000000000 Hard disks
7

1.1.2.- Introduction: Some ‘Orthodox’ solutions available in the literature
Orthodox solutions:
Hartree-Fock
•Many-electron wave-functions having the form of antysimmetric product
of single-particle wave-functions (“orbitals”).
• Leads to an effective single-particle Schrodinger equation with a
potential determined by all others “orbitals” orbitals .
Density Functional Theory (W.Kohn)
•The The “orbitals” orbitals are solutions of single single-particle particle Schrodinger equation which
depends on the charge density rather than the “orbitals” themselves.
•There are terms (the exchange potentials) that are unknown and have to be
approximated.
pp
Quantum Monte Carlo (ex: CASINO Mike Towler)
Pseudo-Bohmian solutions:
Quantum Fluid Density Functional framework: P.K. Chataraj
Hidrodynamic quantum Monte Carlo: Ivan. P. Christov
DFT Super symmetric quantum mechanics, E. Bittner
8

Quantum Q many-particle y p computations p with Bohmian trajectories j
1.- Quantum Many-Particle Computations with Bohmian Trajectories:
1.1.- Introduction
1.2.- Our Many-particles Bohmian trajectories (MPBT) theorem
1.2.1.- The basic idea
1.2.2.- The MPT theorem
1.2.3.- Good points
1.2.4.- Bad points p
1.3.- Approximation
2.- Application to Electron Transport in Nanoelectronic Devices:
3.- Conclusions and Future work

1.2.1.- Our MPBT Theorem: the basic idea
(
x , t)
Single-particle Bohmian trajectory x a[t], 1
t
Many-particle Bohmian trajectory x a[t],
t
x 1
S( x , t)/ x
vx ( [ t], t)
1 1
x x1 1
m x1x1[ t]
v x t x t t
(
x ,..., x , t)
1 N a
a( 1[
],.., N[
], )
m x x [ t],...., x x
[ t]
v x t x t t
1
N
S( x ,...., x , t) / x
1 1
S( x [ t],.., x ,.., x [ t], t) / x
m
1 a N a
a( 1[
],.., N[
], )
( x [ t],.... x ,..., x [ t], t) ( x , t)
1
a N a a
What is the equation satisfied by this single-particle wave-function ?
N N
x x t
1 1 []
10

1.2.1.- Our MPBT Theorem: the basic idea
Any arbitrary complex “function”:
can be written in Schrödinger-like equation:
2 2
a( xa, t) a(
xa, t)
2
( x , t)
a a
i W( xa, t)· a(
xa, t)
t 2 m * xa
with a complex p potential: p
i
Wr( xa, t) iWi( xa, t)
( x , t)
2 2
a( xa, t)
a( xa, t)
2
t 2 m m*
x xa
What is the complex “potential” satisfied by this single-particle wave-function ?
a a
11

1.2.1.- Our MPBT Theorem: the basic idea
What is the complex “potential” satisfied by this single-particle wave-function ?
i
W Wr ( x xa, t t) ) iW iWi( ( x xa, t t)
)
( x , t)
1 st step
2 nd step
2 2
a( xa, t)
a( xa, t)
2
t 2 m* xa
a a
a ( x a a, t) ) ( ( x 1 1[],.,
t x a a,., x N N[],)
t t)
t t
( x ,) t (
x [],., t x ,., x [],) t t
1
2 2
a a
2
xa 1 a
2
xa
N
a b
( x [ t],. x ., x [ t], t) R( x , x [ t], t)· e
a N a b
iS ( x , x [ t], t)/
3rd step Use the quantum Hamiltonian-Jacobi Hamiltonian Jacobi equations
12

1.2.2.- Our MPBT Theorem: Good points
2 2
( xa, t)
i i U ( x , x [], [ t] t) ) G( ( x , x [], [ t] t) ) i· J ( x , x [], [ t] t) ) (
( x , t)
)
2 a xa xb t t G xa xb t t i J xa xb t t
xa t
t 2 m* xa
Good points :
[X. Oriols, Phys. Rev. Lett. 98, 066803 (2007)]
1 st An exact procedure for computing many-particle Bohmian trajectories
2 nd The rest of trajectories in the potentials have to be Bohmian trajectories
3rd The correlations are introduced into the time-dependent p ppotentials
4 th The interacting potential for “classical” correlations
5th h i l i l f l i l l i
th There is a real potential to account for “non-classical” correlations
6 th There is a imaginary potential to account for non-conserving norms
7 st “Analytical” (for 1D,2D,..TDSE) + “Synthetical” (for the rest).
13

1.2.2.- Our MPBT Theorem: Bad points
2 2
( xa, t)
i i U ( x , x [], [ t] t) ) G( ( x , x [], [ t] t) ) i· J ( x , x [], [ t] t) ) (
( x , t)
)
2 a xa xb t t G xa xb t t i J xa xb t t
xa t
t 2 m* xa
Bad points :
1 st The Bohm trajectories for the rest of particles have to be known
Ok! no problem, we will use a single-particle equation for each particles
2nd The terms G and J depends on the many-particle wave-function
N
S(
x,
t)
G ( x , xb,
t)
Ub
( xb,
t)
Kk
( x,
t)
Qk
( x,
t)
· vk
( x[
t],
t)
Ga
( xa,
xb,
t)
Ub
( xb,
t)
Kk
( x,
t)
Qk
( x,
t)
vk
( x[
t],
t)
k 1,
k a
xk
N 2
2
R ( x,
t)
R ( x,
t)
S(
x,
t)
J a(
xa,
xb,
t)
· vk
( x[
t],
t)
·
2 2 2·
R ( x,
t )
k1, ka
xk
xk
m xk
K
a
2
2 2
1 S(
x,
t)
R(
x,
t)
/ x
( x,
t)
; Qa
( x,
t)
2·
m
x
2 m
xa
2 2·
m R ( x x,
t )
¡ This is exactly the same difficulty found in the DFT (or TD-DFT) !
2
a
14

Quantum Q many-particle y p computations p with Bohmian trajectories j
1.- Quantum Many-Particle Computations with Bohmian Trajectories:
1.1.- Introduction
1.2.- Our Many-particles Bohmian trajectories (MPBT) theorem
13 1.3.- Approximation
1.3.1.- The simplest approximation
132 1.3.2.- AApplication li i to systems with i hnon-identical id i l particles i l
1.3.4.- Application to systems with ‘identical’ particles
2.- Application to Electron Transport in Nanoelectronic Devices:
3.- Conclusions and Future work

1.3.1.- Approximate methods for G and J terms: The simplest approximation
Exercise: What happens if the many-particle wave-function is separable ?
( x ,..., x , t) ( x , t)·....· ( x , t)
1 N 1 1
N N
2 2
2 a b a b a b a
2 a
( xa, t)
i
t
m x
U( x , x [],) t t G( x , x [],) t t i· J( x , x [],) t t (
x ,) t
N
dSk( xk[ t], t)
G( xa, xb[ t], t) G( xb[ t], t)
k1, ka dt
N
J( x , x [],) t t J( x [],) t t
d 2
ln R ( x [],) t t
a b b k k
k1, ka2dt a() t i· a()
t
( x t) ( x t)· e
a( xa, t) a( xa, t) e 1 ( x x1 [ t t] ], t t)· ) ... a ( x xa, t t) ).. ·
N ( x xN [ t t] ], t )
16

1.3.1.- Approximate methods for G and J terms: The simplest approximation
2 2
x t
a t
2 a b a b a b
a
t 2m
xa
( , )
i U( x , x [],) t t G( x , x [],) t t i· J( x , x [],) t t ( x ,) t
Guess: G( xa, xb(),) t t G( xa(), t xb(),) t t ...
J( x , x (),) t t J( x (), t x (),) t t ...
a b a b
a a
( x , t) ( x , t)· e
( x , t) ( x , t) e
a a a a
() t i· () t
N=2 system The global time-dependent phase does not affect the velocity
i U x x t t x t x t x t dt
2 2
t
1 ( x1, t )
J1 ( x1, t)
)
( 2 1, 2[ ], )
1( 1, ) ; 1[ ] 1[
o]
2
t 2 m x
1 |
1( x1 , t)| t x x
[] t
i i U x t x t x t x t x t ddt
2 2
t
2 ( x 2 , t ) J2( x2, t)
( [ ] ) 2( 2,
)
( 1[ ], 2, ) 2( 2 ) 2[ ] 2[
]
2 (
, ) ; [] [ o]
2
t 2 m x
2 |
2( x2 , t)| t x x
[] t
o
o
2 2
1 1
17

1.3.2.- Application to systems with non-identical particles
Example: two Coulomb interacting particles
18

Positioon
x [t] andd
x [t] (nmm)
1 2
1.3.2.- Application to systems with non-identical particles
Example: two interacting tunneling particles
40
20
0
-20
-40
40
20
0
-20
-40
40
20
(a)
(b)
(c)
x 1 [t] 1D Bohm
x 1 [t] 2D Bohm
Area=A=2.25 m 2
Area=A=3600 nm 2
2 6 2 4 2 nm
A
x=0 x=L
x
x [t] 1D Bohm
2
x [t] 2D Bohm
Collector
Emitter
x [t] 2D Bohm
2
Area A 3600 nm
Area=A=36 nm 2
x=L
x=0
Electrostatic potential
x
x 1 [t o ]
x [t] 1
~ ~
x ,t ) 1 o
Collector
Active region
Emitter
t
x 1 ,t)
0
-20
-40 40
x
x=L
x=0
x [t ] 2 o
~
x x ,t ) 2 o
~
x ,t) 2
x [t]
0 20 40 60 80 100 120 140
[ o ] x [t] 2
t
Time t (fs)
V ( x1,
x2)
U E ( x1)
U
E
( x
2
2
q
)
4
· r r
0
1
2
19

1.3.3.- Application to systems with “identical” particles
The exchange g interaction appears pp in the symmetries y of the many-particle y p
wave-function (in the configuration space) when particles are exchanged
Where is the exchange interaction ?
2 2
( xa, t)
i i U ( x , xb (),) t t G ( x , xb (),) t t i· J( ( x , xb(),) ( t), t)
(
( x ,) , t)
2 a xa xbt t G xaxbt t i J xaxbt t xat t 2 m* xa
N
S(
x,
t)
Ga
( xa,
xb,
t)
Ub
( xb,
t)
Kk
( x,
t)
Qk
( x,
t)
· vk
( x[
t],
t)
k k 1 1,
k k
a
x k
J
a
( x
a
, x , t)
b
N 2
2
R
( x,
t)
R ( x,
t)
S(
x,
t)
v
k x t t
2
· ( [ ], )
·
2 2·
R ( x , t ) k k 1 1,
k k
a
x k
x k
m
x k
The exchange interaction appears (in the real space) as a potential energy !!
20

1.3.3.- Application to systems with “identical” particles
How we include the exchange g interaction ?
Write, the fermionic/bosonic wave function as a sum of a waveffunctions
ti without ith t exchange h iinteraction. t ti
One wave-function with exchange interaction:
( x1, x2 ,.., xN, t) C· P(
xp(1) , xp(2) ,.., xp( N)
, t)
Many wave-functions without exchange interaction,
() t i· () t
pa pa
( x , t) C· ( x , t)· e
a a pa a
How can we know the phases without computing them ?
21

1.3.3.- Application to systems with “identical” particles
How can we know the phases without computing them ?
By imposing exchange conditions on the set of “no exchange” wave-functions
Example: NN=2 2 fermions
2 Conditions:
( x , x ,0) ( x ,0)· ( x ,0) ( x ,0)· ( x ,0)
1 2 1 1 2 2 1 2 2 1
1( x1, t) 11( x1, t)· w11( t)
21( x1,0)· w21( t)
( x , t ) ( x , t t)· ) w () ( t t)
( x ,0) 0)· w () ( t t)
2 2 12 2 12 22 2 22
( x [ t], t)
0
1 2
( x [ t], t)
0
2 1
22

1.3.3.- Application to systems with “identical” particles
General expression:
N !
( x , t) ( x[ t], t)·...· ( x , t)·...·
( x [ t], t)· s( p( j))
a a a, p( j) 1 1 a, p( j) a a a, p( j) N N
j j1
1
a,1
a,2
.....
aN ,
1 [] x t 2 [] x t x [] t
.....
a ,11
aN ,
a a,22
x t 2 [] x t
[]
1 []
2 2
2 2
( t)
1( x1, t)
i U( x 2 1, x2[ t],...., xN[ t], t)
1( x1, t)
t 2m
x1
Only N·N N N wave-functions wave functions are needed!
Potential
Energy (mmeV)
Source
15
12
9
6
3
N
xN t
02
4 6 8 10 12 14 16 18 20
X Position (nm)
4
8
Drain
12
16
Y Po
20
osition (nm)
23

and x [t] (nmm)
2
Position P x [t] 1
1.3.3.- Application to systems with “identical” particles
Example: two (Coulomb and Exchange) interacting particles
Only antisymmetrical wave-functions are valid
20
0
-20
-40
20
0
-20
-40
(a) Collector
Emitter
~
(b) Collector
Emitter
x
x
x 1 [t o ]
x 2 [t o ]
x 1 ,t o )
t o
~
t o
~
x 2 ,t o )
x 1 ,t o )
~
x 2 ,t o )
Area=A=2.25 m 2
~
x [t] 1 x ,t) t) 1
x 1 [t]
t
~
x 1 ,t)
x 1 [t] 1D' Bohm
1 []
2
Area=A=900 nm 2
~
x 2 ,t)
x 1 [t] 2D Bohm
x 2 [t] 1D' Bohm
x 2 [t] 2D Bohm
0 20 40 60 80 100 120 140 160 180 200 220
Time t (fs)
t
~
x 2 ,t)
x 2 (nm)
Position
20
10
0
-10 10
-20
-30
(c) Collector
What is the difference due
to the Exchange g interaction?
Emitter
x' [0]=-40.50 nm
1
x' [0]= -23.25 nm
2
x [0]=-23.25 nm
1
x [0]= -40.50 nm
2
Area=A=900 nm 2
1D' Bohm
2D Bohm
Emitterr
Collecttor
-30 -20 -10 0 10 20
PPosition iti x ( (nm) ) 1
Observable results are identical when we interchange the initial position of electrons
24

Quantum Q many-particle y p computations p with Bohmian trajectories j
1.- Quantum Many-Particle Computations with Bohmian Trajectories:
2.- Application to Electron Transport in Nanoelectronic Devices:
2.1.- Electron transport at the nanoscale
2.1.1.- The “many-body” problem
22.1.2.- 1 2 Solid Solid-state state approximations
2.1.3.- The Bohmian prediction of the current
2.2.- BITLLES simulator: our quantum Monte Carlo algorithm
33.- CConclusions l i and d Future F t workk

2.1.1.- The electron transport at the nanoscale: the “many-body” problem
Many Many-particle particle Schrödinger equation
for electron devices:
Hˆ
T
U
2
2
2
2
2
1 q
k
g
2
g
2 M
k 2 m0 g 2 M g 2 k j 4 4
0 r
( R 1 , R 2 , R 3 3,...)
, ) U C ( r k , R 1 , R 2 , R 3 3,...)
, )
k j
( r )
0 E k
E g
k
k
g
ki kinetic ti energy of f the th electrons l t
kinetic energy of the atoms
electron electron-electron electron coulomb interaction
U
kj
U '
( R
Bias
)
atom atom-atom atom coulomb interaction
electron electron-atom l t atom t coulomb l bi interaction t ti
electron electron-potential potential due to external bias
atom potential p due to external bias
26

2.1.1.- The electron transport at the nanoscale: the “many-body” problem
What iis an electron device i ?
1.- Open system
2.- Statistical system
3.- Far from equilibrium
4.- Strongly correlated system
What we measure ?
CContinuity ti it equation ti
We measure the total (conduction + displacement) current in
the ammeter ammeter, which which is identical to that in a surface of the
active region.
/
t
J
0 t
I () t I () t
c
J
D /
0
J c
S S
A D
27

Quantum Q many-particle y p computations p with Bohmian trajectories j
1.- Quantum Many-Particle Computations with Bohmian Trajectories:
2.- Application to Electron Transport in Nanoelectronic Devices:
2.1.- Electron transport at the nanoscale
2.2.- 2.2. Our quantum Monte Carlo algorithm
2.1.1.- The BITLLES simulator
2.1.2.- Numerical results for DC current
22.1.3.- 1 3 Numerical results for current fluctautions
2.1.4.- Numerical results for transient and AC currents
33.- CConclusions l i and d Future F t workk

2.2.1.- Our quantum Monte Carlo algorithm: The Bittles simulator
Quantum Monte Carlo simulation for electron transport
Monte Carlo Casino (MONACO)
N N
g h 1
I( t) lim Igh
, ( t)
Ng, Nh
N · N
g h
g g1
h1
G-distribution:
initial position of Bohmian trajectory
H-distribution:
initial energy of the wave-packet
(Stationary and ergodic system)
1 T
I lim
I ' h t g, h t dt dt'
T
T 0
29

2.2.2.- Our quantum Monte Carlo algorithm: DC current
DC current for a Resonant Tunneling g Device ( (RTD) )
In the 80’s, engineers expected that RTDs would substitute the FET transistor
Now Now, it is a very useful scenario to understand QM phenomena at the nanoscale nanoscale.
30

2.2.2.- Our quantum Monte Carlo algorithm: DC current
DC current for a Resonant Tunneling Device (RTD)
Current ()
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
1e - mean-field 1e - many-fields
Source
TTmax
Source
Drain Drain
many-fields 1
many-fields 2
many-fields 3
many-fields y 4
mean-field 1
mean-field 2
mean-field 3
mean-field 4
T=Tmax
0.0 0.2 0.4 0.6 0.8 1.0
Drain Voltage (V)
[G. Albareda et al. Phys. Rev. B 79, 075315 (2009)]
Emitter
N +
Lwell
Lx Wbarrier
N +
Lz
Ly
Collector
Electron transport beyond the standard mean-field approximation.
WWe include i l d many-particle il interaction i i effects ff on the h current.
y
z
x
31

2.2.3.- Our quantum Monte Carlo algorithm: current fluctuations
Quantum Noise:
I(t)
I () t I () t I () t I C CI ( ) S SI ( w )
Fluctuations Autocorrelation Fourier transform
Engineers do not like noise, it makes errors in the device.
Physicist enjoy noise, because it shows phenomena that are not shown in DC
Fano Factor F
SI( w0)
I
32

Cuurrent
(A)
Fano Factor F
4
3
2
1
0
1.2
1.0
2.2.3.- Our quantum Monte Carlo algorithm: current fluctuations
Effect of Coulomb correlation on current and noise [X.Oriols, APL,85, 3596 (2004)]
(a)
E EFE Emitter
x=0
L
x=L
E R
Colector
Our approach
# Particles emitter
# Particles collector
Esaki results [16]
E FC
FROZEN POTENTIALS
Our approach
Buttiker results [6]
(b) FROZEN POTENTIALS
08 0.8
0.0 0.1 0.2 0.3 0.4
Applied bias (Volts)
Cuurrent
(A)
Fanoo
Factor
6
5
3
2
(a)
Our approach
# Particles emitter
# Particles collector
4 Visual guided line
1
0
1.4
1.2
1.0
(b)
Our approach
Visual guided line
E R FROZEN
SELF-CONSISTENT POTENTIALS
E R FROZEN
SELF-CONSISTENT POTENTIALS
08 0.8
0.0 0.1 0.2 0.3 0.4
Applied bias (Volts)
[X.Oriols et al., APL, 2004]
33

2.2.4.- Our quantum Monte Carlo algorithm: transient and AC current
Time dependent (particle + displacement) current
H Jc
D
/ t
Jc D
/ t
0
D
Poisson equation
[A [A. Alarcon Alarcon, JSTM, JSTM 2009(P01051) (2009)]
Continuity equation
/ t
J 0
c
34

2.2.4.- Our quantum Monte Carlo algorithm: transient and AC current
Transit simulation: [G. Albareda et al. Phys. Rev. B, 82, 085301 (2010)]
Non-ergodic system (ensemble average):
N N
g h 1
I( t) lim Igh , ( t)
N Ng, N h h
N · N
g h
g1 h1
35

Quantum Q many-particle y p computations p with Bohmian trajectories j
1.- Quantum Many-Particle Computations with Bohmian Trajectories:
2.- Application to Electron Transport in Nanoelectronic Devices:
33.- CConclusions l i and d Future F workk

3.- Conclusions and future work
We have presented a algorithm to compute many-particle Bohmian trajectories trajectories.
1.-We have shown the existence of a single-particle Schrodinger equation
th that t computes t a many-particle ti l BBohmian h i ttrajectory. j t
2.- Its practical application needs an educated guess on the potentials.
We apply pp y the previous p algorithm g to a many-particle y p qquantum
MC simulator
for computing electron quantum transport
We are able to compute DC, AC, transients and (current and voltage) noise
with ith electron-electron l t l t correlations. l ti
Simulation time 1-2 days for a complete I-V curve (N=100 electrons)
37

3.- Future work
The BITLLES Simulator
Bohmian Interacting Transport in Electronic Structures
We have a three years project by the Ministerio de Ciencia ee Innovación to
develop the BITLLES simulator through project TEC2009-06986.
38

3.- Future work
• This book provides the first comprehensive discussion on the
practical application of Bohmian ideas in several forefront research
fields written by leading experts, with an extensive updated
bibliography.
• This book provides a didactic introduction to Bohmian mechanics
easily accessible for graduate and undergraduate students including a
thorough list of exercises and easily programmable codes.
Readership
The book is addressed to students in physics, physics chemistry, chemistry electrical
engineering, applied mathematics, nanotechnology, as well as both
theoretical and experimental researchers who seek an intuitive
understanding of the quantum world and new computational tools for
their everyday research activity.
978-981-4316-39-2
Cloth, 400 pages (approx.)
Fall 2011, US$149
How to order
You can place an order from any good bookstores or email
us at sales@panstanford.com for more information.
39

3.- Conclusions and future work
Acknowledgment g
• F.L.Traversa,
• G.Albareda,
• AAl A. Alarcón, ó
• A.Benali,
• A.Padró,
• X.Cartoixà,
• R.Rurali
This work has been partially supported by the Ministerio de Ciencia e
Innovación under Project No No. TEC2009-06986 TEC2009 06986 and by the DURSI of the
Generalitat de Catalunya under Contract No. 2009SGR783.
Thank h kyou very much hffor your attention
i
40