Toulouse 2012

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RELAXATION AND ENERGYDISSIPATION IN TDCDFT

Roberto DAgostaUPV/EHU and Ikerbasque

San Sebastian

[email protected]

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RELAXATION IN QUANTUM MECHANICS

Common approach (effective non unitary time evolution, e.g.Caldeira-Leggett)

Time Dependent Density Functional Theory approach(Unitary Time Evolution)

How to introduce

a time arrow in aHamiltonianframework?

Answer: Thefunctional dependson the velocity eld

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The Kohn-Sham Hamiltonian for TDCDFT

The Vignale-Kohn exchange-correlation eld

In principle, exact time evolution of particle and current densities

H KS =

1

2m Xi [ pi

+ eA

(r, t

) + eA xc

(n, j

)]

2

+ Z V (r, t ) + V ALDAxc (r, t ) n (r, t )dr

edA

xc

dt =

xcn

v = jn

xc,ij = i v j + j v i 2

3 ij v + ij v

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THE KOHN-SHAM ENERGY FUNCTIONAL

dE

dt 0

t

dE dt

= Z j(r , t ) E xc(r , t ) dr e E xc (r , t ) = dA xcdt E [n ,

~

j] = h KS |

H KS | KS i

Z V hxc(r , t )n(r , t ) dr + E ALDAhxc [n]

We assume vanishing velocity at theboundaries

is NOT the total energy but gives a time arrow E

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Numerical results

~

> 0 , t < 0 ~

= 0 , t > 0

-L/2 -L/2

L/2

L/2

1D code no memory.For similar results in the case of

systems with memory look atH. Wijewardane and C. Ullrich, Phys. Rev. Lett. 95 , 086401 (2005)

Quantum Well

E i

E (t ) E 0 e t

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z

y

x

1D Quantum well in the z direction

2D electron gas in the (x,y) plane

Energy transfer from the motion in the z direction to the 2D electron gas

Energy transfer

By using standard time-dependent perturbation theory one can test theapproximations for the xc kernel.

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TIME EVOLUTION

Energy

conservationdE

dt + T (t )

dS

dt = 0Z

0

dE

dt + T (t )

dS

dt dt = 0

quasi-static

T (t ) = T i + s 2

[ E i E (t )]

dS

dt =

dS

dT

dT

dt = C V (T )

dT

dt

S (t ) = p 2 [ E i E (t )]

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CONCLUSION - 1

TDCDFT is able to capture the relaxation of a many-particle system to a statistical equilibrium

The corresponding time evolution is unitary and doesnot contain ad hoc parameters

At least in a simple case we are able to interpret the

Kohn-Sham energy as the maximum work one canextract from the system

TDCDFT offers a natural framework where to study the dynamical timeevolution of a many-particle system

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OPEN QUANTUM SYSTEMS

Van Kampen, Stochastic Processes in Physics and Chemistry

bath l(t )

The bath induces stochasticity and:

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bath l(t )


is macroscopic

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bath l(t )


is macroscopic

is unbiased

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bath l(t )


l (t ) = 0

l (t )l (t ) = (t t )

V =

V (t ) is assigned

no feedback is macroscopic

is unbiased

has no memory

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Open Quantum Systems

bath l(t )

The system is in a mixed state and:

is out of equilibrium

is interacting

has memory

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Open Quantum Systems

bath l(t )

two formalisms

density matrixstochastic

Schrdinger eq.

The system is in a mixed state and:

is out of equilibrium

is interacting

has memory

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STOCHASTIC

SCHROEDINGER EQ.

t (r, t ) = i H (r, t ) 1

2V V (r, t ) + l(t )V (r, t )

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STOCHASTIC

SCHROEDINGER EQ.

t (r, t ) = i H (r, t ) 1

2V V (r, t ) + l(t )V (r, t )

dissipation uctuation

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STOCHASTIC

SCHROEDINGER EQ.

t (r, t ) = i H (r, t ) 1

2V V (r, t ) + l(t )V (r, t )

dissipation uctuation

(t ) =i

p i | i i |

...

1 | 1 2 | 2 M | M

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BOSE GAS IN A TRAP

Gross - Pitaevskii equation

i t (x, t ) = 12m

d2

dx 2 +

12

m 20 x2 + gn (x, t ) (x, t )

Davis, et al. PRL (2005),

Anderson, et al. Science (2005)

13

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1-D CONFINED BOSE GAS

V

0 1 1 1 . . .

0 0 0 0 . . ....

......

......

0 0 0 0 . . .

R. DAgosta and M. DiVentra PRB (2008)

.

..

E

E

E

E

Em 0

14

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V

0 1 1 1 . . .

0 0 0 0 . . ....

......

......

0 0 0 0 . . .


.

..

E

E

E

E

Em 0

No interaction,g=0

14

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0 5 10 15 200

0.2

0.4

0.6

0.8

1single run10 runs50 runs100 runsdensity matrix

p 0

t! 0

0 5 10 15 200

0.04

0.08

t! 0

|p 0 -p0 |/p 0dm sse sse


V

0 1 1 1 . . .

0 0 0 0 . . ....

......

......

0 0 0 0 . . .


.

..

E

E

E

E

Em 0

No interaction,g=0

14

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V

0 1 1 1 . . .

0 0 0 0 . . ....

......

......

0 0 0 0 . . .


.

..

E

E

E

E

Em 0

No interaction,g=0

!!"#!"$!"%!"&!"'

!"#$"

()

* ,

!

!

!"#

!"$

!"%

!"#%

-)

* ,

!

.#' .#! .' ! ' #! #'!

!"#

!"$

,/, !

!"#"

0)

* ,

!

14

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Strong interaction

! "! #! $! %! &! '!!

!("

!(#

!($

!(%

!(&

!('

) * * + , -

. / 0 1 , 3 0

4 -

4 / 5 / . 6

, !

, "

. !

, #

1-D CONFINED BOSE GASR. DAgosta and M. Di VentraPRB (2008)

15

g/ 0 = 5

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Strong interaction

! "! #! $! %! &! '!!

!("

!(#

!($

!(%

!(&

!('

) * * + , -

. / 0 1 , 3 0

4 -

4 / 5 / . 6

, !

, "

. !

, #

1-D CONFINED BOSE GASR. DAgosta and M. Di VentraPRB (2008)

!" !# $ # "$

$%$&

$%'

$%'&

$%#

()*+,-.)/+

0,*1-. ! 3451/

676$

8 6

$

15

g/ 0 = 5

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SPIN CHAIN

0 50 100 150 200 250

time

-0.0015

-0.001

-0.0005

0

0.0005

0.001

0.0015

e n e r g y c u r r e n

t

j 2,

3 j 1,

2

L > R

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EQUATION OF MOTION

average over the stochastic eld

M .Di Ventra and R. DAgosta PRL (2007)

j (r, t )

t=

F (r, t )

m+ G (r, t ) +

1

mF ext (r, t )

R. DAgosta and M. Di Ventra PRB (2008)

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EQUATION OF MOTION


F (r, t ) = i = j

(r r i ) j U (r i r j ) + m (r, t )

internal forces


j (r, t )

t=

F (r, t )

m+ G (r, t ) +

1

mF ext (r, t )


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EQUATION OF MOTION


G (r, t ) =

V

j (r, t )

V

1

2 j (r, t )

V

V

1

2V

V

j (r, t )bath induced forces


j (r, t )

t=

F (r, t )

m+ G (r, t ) +

1

mF ext (r, t )


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EQUATION OF MOTION


F ext (r, t ) = n(r, t )

m t A ext (r, t )

j (r, t )m

[ A ext (r, t )]

external forces: driving elds


j (r, t )

t=

F (r, t )

m+ G (r, t ) +

1

mF ext (r, t )


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STOCHASTIC TD-CDFTGiven l(t ), V , (r, 0)

M. Di Ventra and R. DAgosta, PRL (2007)

A ext r, tR. DAgosta and M. Di Ventra, PRB (2008)

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A SURPRISING RESULT

The theorem of STCDFT does admit for the system to beclosed, i.e.we can set for the KS system

= 0

AKS (r, t ) = Aext (r, t ) + A xc [ j, 0 , V = 0]

Yuen-Zhou et al., PCCP 2009

Relaxation and dissipation in TDCDFT are described by a KSvector potential in a closed system!

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THANKS!

Toulouse 2012

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