Quantum Kinetic Theory Michael Bonitz Quantum Kinetic ...bonitz/talks/08/balescu_mb2.pdf · Quantum Kinetic Theory Michael Bonitz Outline Outline R. Binder et al. PRB 45, 1107-1115

Quantum KineticTheory

Michael Bonitz

OutlineQuantum Kinetic Theory for Artificial Atoms

Michael Bonitz and Karsten Balzer

Institut fur Theoretische Physik und Astrophysik

Christian-Albrechts-Universitat zu Kiel, Germany

Solvay Workshop for Radu Balescu, Brussels, March 6 2008

Michael Bonitz (CAU Kiel) Quantum Kinetic Theory 1 / 46


Michael Bonitz

Outline

Outline

R. Binder et al. PRB 45, 1107-1115 (1992)

Lenard-Balescu collision integral, Phys. of Fluids 3, 52 (1960)

dynamically screened Coulomb potential

W (q, ω) =V (q)

1 − V (q)P(q, ω)= V (q) ǫ−1(q, ω)

unscreened potential V (q) =4πe2

Vq2

P(q, ω) = limδ→02X

α,k

fα(k) − fα(|q + k|)

ǫα(k) − ǫα(|q + k|) + ~ω + iδ



Michael Bonitz

Outline

Outline

Outline

IntroductionArtificial atomsFemtosecond relaxation processesLimitations of Boltzmann-type kinetic equations

Basics of Nonequilibrium Green’s FunctionsSecond quantizationDefinition of Nonequilibrium Green’s functions (NEGFs)Equations of motion for the NEGF

Numerical Results. ApplicationsHomogenous Coulomb systems. PlasmasDynamics of charged particles in a trap

Conclusions



Michael Bonitz

Introduction

Artificial atoms

fs-Dynamics

Boltzmann equation

NonequilibriumGreen’s Functions

2nd Quantization

Definition andproperties of NEGFs

KKB Equations

NumericalApplications

Homogeneousplasmas

Charged particles ina trap

Conclusions

Introduction Artificial atoms

Outline




Conclusions



Michael Bonitz

Introduction

Artificial atoms

fs-Dynamics

Boltzmann equation


2nd Quantization


KKB Equations


Homogeneousplasmas


Conclusions

Introduction Artificial atoms

Artificial Atoms

Electrons in artificial atoms, R.C. Ashoori, Nature 379, 413-419 (1996)

gated transport experiments

Simulation: N = 2, 3, . . . , 6 fermions in a 2D parabolical trap.



Michael Bonitz

Introduction

Artificial atoms

fs-Dynamics

Boltzmann equation


2nd Quantization


KKB Equations


Homogeneousplasmas


Conclusions

Introduction fs-Dynamics

Outline




Conclusions



Michael Bonitz

Introduction

Artificial atoms

fs-Dynamics

Boltzmann equation


2nd Quantization


KKB Equations


Homogeneousplasmas


Conclusions

Introduction fs-Dynamics

Femtosecond relaxation processes

Femtosecond laser excitation of matter coherent semiconductor optics time resolved relaxation dynamics of solids fs dynamics of atoms and molecules

Interaction of high intensity lasers, free electron lasers with matter strong nonlinear excitation correlation effects on fs and sub-fs time scale

Need: Nonequilibrium many-body theory⇒ selfconsistent treatment of correlations, quantum and spin effectsNeccesary to go beyond standard Boltzmann kinetic equations



Michael Bonitz

Introduction

Artificial atoms

fs-Dynamics

Boltzmann equation


2nd Quantization


KKB Equations


Homogeneousplasmas


Conclusions

Introduction Boltzmann equation

Outline




Conclusions



Michael Bonitz

Introduction

Artificial atoms

fs-Dynamics

Boltzmann equation


2nd Quantization


KKB Equations


Homogeneousplasmas


Conclusions


R. Binder et al. PRB 45, 1107-1115 (1992)

screened Coulomb potential

W (q, ω) =V (q)

1 − V (q)P(q, ω)= V (q) ǫ−1(q, ω)

unscreened potential V (q) =4πe2

Vq2

P(q, ω) = limδ→02X

α,k

fα(k) − fα(|q + k|)

ǫα(k) − ǫα(|q + k|) + ~ω + iδ



Michael Bonitz

Introduction

Artificial atoms

fs-Dynamics

Boltzmann equation


2nd Quantization


KKB Equations


Homogeneousplasmas


Conclusions


Limitations of Boltzmann-type kinetic equations

∂

∂t+

∂E

∂p

∂

∂R−

∂E

∂R

∂

∂p

ff

f (p, R, t) = I (p, R, t) (1)

I (p1, t) =2

~

Z

dp2

(2π~)3dp1

(2π~)3dp2

(2π~)3

˛

˛

˛

˛

V (p1 − p1)

ǫRPA(p1 − p1, E(p1) − E(p1))

˛

˛

˛

˛

2

×(2π~)3δ(p1 + p2 − p1 − p2) · 2πδ(E1 + E2 − E1 − E2)

×˘

f1 f2(1 ± f1)(1 ± f2) − f1f2(1 ± f1)(1 ± f2)¯

|t

with quasiparticle energy Ei = E(pi ), Ei = E(pi ), fi = f (pi ), fi = f (pi )

Example: QuantumLenard-Balescu collision integral(Coulomb scattering)

|p1>

|p1’>

|p2>

|p2’>

Properties of Equation (1):

1. Conservation of kinetic energy,ddt〈E〉(t) = 0

2. Equilibrium solution: f (p, t) →Bose/Fermi/Maxwell distribution→ thermodynamic and transportproperties of ideal gas

3. Eq. (1) limited to times largerthan correlation time, t ≫ τcorr

Properties (1)–(3) in conflict with present goal ⇒ Generalizations necessary



Michael Bonitz

Introduction

Artificial atoms

fs-Dynamics

Boltzmann equation


2nd Quantization


KKB Equations


Homogeneousplasmas


Conclusions

Nonequilibrium Green’s Functions 2nd Quantization

Outline




Conclusions



Michael Bonitz

Introduction

Artificial atoms

fs-Dynamics

Boltzmann equation


2nd Quantization


KKB Equations


Homogeneousplasmas


Conclusions


Hamiltonian in second quantization

Hamiltonian for interacting system

H(t) =

Z

dq Ψ†(q) h0(q, t) Ψ(q) +

1

2

ZZ

dq d q Ψ†(q) Ψ†(q) hint(q, q) Ψ(q) Ψ(q)

e.g. h0(x, t) = (p − e A(x, t))2 + Φ(x, t) , hint(x, x) = e2

|x−x|

Field operators Ψ, Ψ(†) withcommutation relation for bosons (-),anti-commutation for fermions (+)

h

Ψ(†)(q), Ψ(†)(q)i

∓= 0

h

Ψ(q), Ψ†(q)i

∓= δ(q − q)

⇒ Theory has “built in” spinstatistics

Symmetry/anti-symmetry ofN−particle states exactly guaranteed



Michael Bonitz

Introduction

Artificial atoms

fs-Dynamics

Boltzmann equation


2nd Quantization


KKB Equations


Homogeneousplasmas


Conclusions


Macroscopic Observables

Equilibrium ensemble average. Density operator

〈O〉 = Tr(ρ O), ρ =1

Zexp (−[β H − µ N ])

Nonequilibrium expectation values. Switch on perturbing field at t = t0〈O〉 → 〈O〉 (t), use Heisenberg operators

〈OH(t)〉 =Tr

“

eβµNU(t0 − iβ, t0) U(t0, t) O(t) U(t, t0)”

Tr(eβµNU(t0 − iβ, t0))

with evolution operator (~ = 1)

U(t, t0) = exp

−i

Z

t

t0

dt H(t)

!

e−βH ≡ U(t0 − iβ, t0)

Time evolution runs alongKeldysh-contour C [L.V. Keldysh,Sov. Phys. JETP, 20, 1018

(1965)]

-iΒ

t0+

-

È

t

t’

C = t ∈ C|Re t ∈ [t0,∞],Im t ∈ [t0,−β]



Michael Bonitz

Introduction

Artificial atoms

fs-Dynamics

Boltzmann equation


2nd Quantization


KKB Equations


Homogeneousplasmas


Conclusions

Nonequilibrium Green’s Functions Definition and properties of NEGFs

Outline




Conclusions



Michael Bonitz

Introduction

Artificial atoms

fs-Dynamics

Boltzmann equation


2nd Quantization


KKB Equations


Homogeneousplasmas


Conclusions


Definition of Nonequilibrium Green’s functions

replace 〈O〉 →D

Ψ† ΨE

: One-particle Green function

G(xt, x′t′) = ±

1

i

D

TC

“

Ψ†(xt) Ψ(x′t′)”E

time ordering along C

t, t′ belong to one of 3 branches, G is 3 × 3matrix

Keldysh Green functions (matrixelements on C), denote 1 ≡ r1, t1, sz1

-iΒ

t0+

-

È

t

t’

correlation functions

g<(1, 1′) = ±

1

i

D

Ψ†(1′)Ψ(1)E

g>(1, 1′) =

1

i

D

Ψ(1)Ψ†(1′)E

retarded/advanced functions

gR/A(1, 1′) =

±Θ[±(t1 − t′1)]n

g> − g

<o

real branch - imaginary branchcomponents

g⌈(1, 1′), g

⌉(1, 1′)

imaginary branch component:Matsubara (equilibrium) GF

gM

(1, 1′)



Michael Bonitz

Introduction

Artificial atoms

fs-Dynamics

Boltzmann equation


2nd Quantization


KKB Equations


Homogeneousplasmas


Conclusions


Information contained in the Nonequilibrium Green’s functions

Physical content: Time-dependent macroscopic observables

D

O(t)E

=

Z

dxh

o(x′t)D

Ψ†(xt) Ψ(x′t)Ei

x=x′

= ∓ i

Z

dxh

o(x′t) g

<(xt, x′t)i

x=x′

Particle density

〈n(x, t)〉 = 〈n(1)〉 = ∓ i g<(1, 1)

Density matrix

F (x1, x′1 , t) = ∓ i g

<(1, 1′)˛

˛

t1=t′1

Current density

D

j(1)E

= ∓ i

»„

∇1

2i−

∇1′

2i+ A(1)

«

g<(1, 1′)

–

1′=1

Interaction energy, also follows from 1-particle function, [Baym/Kadanoff]

〈V (t)〉 = ± iV

4

Z

dp

(2π~)3

(

(i ∂t − i ∂t′ ) −p2

m

)

g<(p, t, t

′)|t=t′



Michael Bonitz

Introduction

Artificial atoms

fs-Dynamics

Boltzmann equation


2nd Quantization


KKB Equations


Homogeneousplasmas


Conclusions


Wigner distribution

f (p, R, T ) = ± i

Z

dω g<(RT , pω)

CoM and relative variables

T = (t1 + t2)/2 , t = t1 − t2

R = (x1 + x2)/2 , r = x1 − x2

Single particle spectrum (Spectral function)

a(ω;R, p, T ) = i

Z

dt eiωth

g> − g

<i

„

T +t

2, T −

t

2, R, p

«

Nonequilibrium Density of states, ρ(ω) = Tr(a(ω; R, p, T ))

Example: Electrons in harmonicoscillator potential V

Ε0 Ε1 Ε2 Ε3 Ε4

Ω

ΡHΩL

ideal

corr.

VHxL

02

46

810

t10

2

4

6

8

10

t2

-1-0.5

00.5

1

Im g<(t1,t2)



Michael Bonitz

Introduction

Artificial atoms

fs-Dynamics

Boltzmann equation


2nd Quantization


KKB Equations


Homogeneousplasmas


Conclusions

Nonequilibrium Green’s Functions KKB Equations

Outline




Conclusions



Michael Bonitz

Introduction

Artificial atoms

fs-Dynamics

Boltzmann equation


2nd Quantization


KKB Equations


Homogeneousplasmas


Conclusions


Equations of motion for Keldysh Green function G

Heisenberg’s equation of motion:

i ∂tΨ(†)(q, t) =

h

Ψ(†)(q, t), H(t)i

−

Result: Martin-Schwinger Hierarchy for one, two ... s-particle Green functions

(i∂t + h(1)) G(1, 1′) = δc (1 − 1′) ± i

Z

Cd2 h

int(1 − 2) G(12, 12+) & adjoint

Formal decoupling of hierarchyintroducing selfenergy Σ

± i

Z

Cd2 h

int(1 − 2) G(12, 12+)

=

Z

Cd2 Σ(1, 2) G(2, 1

′)

Conserving approximations

Σ[G ](1, 2) =δΦ[G ]

δG(2, 1)

[G. Baym, Phys. Rev., 127, 1391(1962)]

Example for Σ: 1st and 2nd order diagrams ⇒ Hartree-Fock + Second Born.



Michael Bonitz

Introduction

Artificial atoms

fs-Dynamics

Boltzmann equation


2nd Quantization


KKB Equations


Homogeneousplasmas


Conclusions


Keldysh-Kadanoff-Baym equations for matrix components∗ of G

(i∂t − h(1)) g≷(1, 2) =

h

ΣR g≷ + Σ≷ g

A + Σ⌉ ⋆ g⌈i

(1, 2)

(−i∂t′ − h(2)) g≷(1, 2) =

h

gR Σ≷ + g

≷ ΣA + g⌉ ⋆ Σ⌈

i

(1, 2)

(i∂t − h(1)) g⌉(1, 2) =

h

ΣR g⌉ + Σ⌉ ⋆ g

Mi

(1, 2)

(−i∂t′ − h(2)) g⌈(1, 2) =

h

g⌈ ΣA + g

M ⋆ Σ⌈i

(1, 2)

(−∂τ − h(1)) gM (1, 2) = iδ(τ − τ ′) +

h

ΣM ⋆ gMi

(1, 2)

(∂τ′ − h(2)) gM

(1, 2) = iδ(τ − τ′) +h

ΣM

⋆ gMi

(1, 2)

∗ matrix algebra: DuBois, Langreth, ..., Short notation:

[A B](12) =

∞Z

t0

dt dx A(1, x t) B(x t, 2) , [A ⋆ B](12) = −i

βZ

0

d τ dx A(1, x τ) B(x τ , 2)

Equilibrium initial state: g(t0, t0)

defined by precomputed gM

Nonequilibrium initial state: g(t0, t0)

given by f (t0), initial pair correlations

c12(t0), additional selfenergy ΣIC

[Danielewicz, Kremp/Semkat/Bonitz]



Michael Bonitz

Introduction

Artificial atoms

fs-Dynamics

Boltzmann equation


2nd Quantization


KKB Equations


Homogeneousplasmas


Conclusions

Numerical Applications

Numerical solution of Keldysh-Kadanoff-Baym Equations.

Full two-time solutions: Danielewicz, Schafer, Kohler/Kwong,Bonitz/Semkat, Haug, Jahnke, van Leeuwen ...

-Β

0

Time

propagation

iterative procedure

t

t′

τgM(τ)

g≷(t, t′)

g⌈(τ, t′)

g⌉(t, τ)

τ ∈ [−β, β]

Equilibrium initial conditions

g≷(0, 0) = i gM (0±)

g⌉(0,−iτ ) = i gM (−τ )

g⌈(−iτ, 0) = i gM (τ )

Numerical scheme

1. Choose proper basis set

2. Solve equilibrium problem:Dyson equation → gM (τ )

3. Time-propagation: SolveKeldysh-Kadanoff-Baymequations

→ g≷(t, t′), g⌉/⌈(t, t′)



Michael Bonitz

Introduction

Artificial atoms

fs-Dynamics

Boltzmann equation


2nd Quantization


KKB Equations


Homogeneousplasmas


Conclusions

Numerical Applications Homogeneous plasmas

Outline




Conclusions



Michael Bonitz

Introduction

Artificial atoms

fs-Dynamics

Boltzmann equation


2nd Quantization


KKB Equations


Homogeneousplasmas


Conclusions


Sub-femtosecond energy relaxation in dense plasma

Dense hydrogen plasma, T = 10, 000K , n = 1021cm−3, k = 0.6/aB

Solution of KB equations conserves total energy H(t) = T (t) + U(t) = H(0)

Initial state uncorrelated (zerocorrelation energy U)

Correlations build up → increase of |U|→ Increase of kinetic energy T .

1.5

2.0

0.0 0.5 1.0 1.5 2.0 2.5

Zeit T[fs]

-0.5

0.0

Ene

rgie

[10-5

Ryd

a B-3

]

T (t) and U(t) saturate at correlation

time t ≈ τcor ∼ ω−1pl

Uncorrelated vs. over-correlated initialstate

0.0 0.4 0.8 1.2 1.6

Time T[fs]

-1

0

1

2

Ene

rgy[

10-5

Ryd

/aB

3 ]

Preparing system in over-correlatedinitial state leads to cooling.

Semkat, Kremp, Bonitz, Phys. Rev. E 59, 1557 (1999)Bonitz/Semkat, Introduction to Computational Methods for Many Body Systems,

Rinton Press, Princeton 2006



Michael Bonitz

Introduction

Artificial atoms

fs-Dynamics

Boltzmann equation


2nd Quantization


KKB Equations


Homogeneousplasmas


Conclusions


Selfconsistent plasmon spectrum of the correlated electron gas

Solution of KKB equations for short monochromaticexcitation, U(t) = U0(t) cos q0r

Periodic density fluctuation, withLandau plus correlation damping

-9 0 9 18 27

Time [Ry-1

]

-0.15

-0.1

-0.05

0.0

0.05

0.1

n(q

,t)

[r0-3

]

Excitation U0(t)HF + second BornHF2

HF

Fourier transform yieldsdynamic structure factor S(q, ω)

0.0 0.2 0.4 0.6 0.8 1.0

[Ry]

0.0

0.4

0.8

1.2

1.6

S(q,

)[R

y-1]

HF + second BornHF2

HF

• Conservation properties of KBE guarantee exact sum rule preservation ofplasmon spectrum S(q, ω)• Simple approximations for selfenergy (such as 2nd Born approximation) givehigh-level correlation effects, including vertex corrections, in S

[Kwong/Bonitz, PRL 84, 1768 (2000)]



Michael Bonitz

Introduction

Artificial atoms

fs-Dynamics

Boltzmann equation


2nd Quantization


KKB Equations


Homogeneousplasmas


Conclusions

Numerical Applications Charged particles in a trap

Outline




Conclusions



Michael Bonitz

Introduction

Artificial atoms

fs-Dynamics

Boltzmann equation


2nd Quantization


KKB Equations


Homogeneousplasmas


Conclusions


Hamiltonian and system parameters. 1D model

H =

Z

dx Ψ†(x) h0(x , t) Ψ(x) +1

2

Z

dx1 dx2 Ψ†(x1) Ψ†(x2)w(x1, x2) Ψ(x2) Ψ(x1)

h0(x , t) = −~2∇2

x

2m+

m

2Ω2 x2 + Vext(x , t) , w(x1, x2) = λ

x0 ~Ωp

(x1 − x2)2 + a2

dipole field

Vext(x , t) = −e E(t) x

E(t) = a0 e−

(t−t0)2

2 t1 cos(ω t)

ω, a0: dimensionless laserfrequency and amplitude

Coupling parameter λ, inversetemperature β:

λ = e2

x0 ~Ω, x2

0 = ~

mΩ; β = ~Ω/kT

Expand Green functions inoscillator basis, g → gkl

-Β

0

Time

propagation

iterative procedure

t

t′

τgM

kl (τ)

g≷kl

(t, t′)

g⌈kl

(τ, t′)

g⌉kl

(t, τ)

τ ∈ [−β, β]



Michael Bonitz

Introduction

Artificial atoms

fs-Dynamics

Boltzmann equation


2nd Quantization


KKB Equations


Homogeneousplasmas


Conclusions


Equilibrium density n(x) and energy distribution f (ǫk − µ)

N = 6 fermions (1D trap)



Michael Bonitz

Introduction

Artificial atoms

fs-Dynamics

Boltzmann equation


2nd Quantization


KKB Equations


Homogeneousplasmas


Conclusions


Energy evolution after laser excitation



Michael Bonitz

Introduction

Artificial atoms

fs-Dynamics

Boltzmann equation


2nd Quantization


KKB Equations


Homogeneousplasmas


Conclusions


Occupation number dynamics for off-resonant laser excitation



Michael Bonitz

Introduction

Artificial atoms

fs-Dynamics

Boltzmann equation


2nd Quantization


KKB Equations


Homogeneousplasmas


Conclusions


Occupation number dynamics for near-resonant laser excitation



Michael Bonitz

Introduction

Artificial atoms

fs-Dynamics

Boltzmann equation


2nd Quantization


KKB Equations


Homogeneousplasmas


Conclusions


Evolution of Green function of ground state

02

46

810

0

2

46

810

-1

0

1

24

68

10

tt

t′

Im g<00

along diagonal: weak change of ground state population,across: renormalized ground state energy (w.r. to µ)



Michael Bonitz

Introduction

Artificial atoms

fs-Dynamics

Boltzmann equation


2nd Quantization


KKB Equations


Homogeneousplasmas


Conclusions


Evolution of Green function of level 3

02

46

810

0

2

46

810

-1

0

1

24

68

10

tt

t′

Im g<33

along diagonal: reduction of population of level 3



Michael Bonitz

Introduction

Artificial atoms

fs-Dynamics

Boltzmann equation


2nd Quantization


KKB Equations


Homogeneousplasmas


Conclusions


Evolution of Green function of level 5

02

46

810

0

2

46

810

-1

0

1

24

68

10

tt

t′

Im g<55

Build up of population of level 5 (along diagonal) andof correlated spectrum (across)



Michael Bonitz

Introduction

Artificial atoms

fs-Dynamics

Boltzmann equation


2nd Quantization


KKB Equations


Homogeneousplasmas


Conclusions

Conclusions

Conclusion & Outlook

fs and sub-fs radiation sources ⇒ ultrafast dynamical response of matter

Ultrafast dynamics of Coulomb correlations. Correlation build up ⇒ requiresnon-Markovian kinetic treatment

Real-time nonequilibrium Green functions (Keldysh/Kadanoff-Baym) selfconsistent non-perturbative treatment of strong fields total energy conservation, exact spin statistics

Numerical applications: propagation of NEGF in two-time plain semiconductors, dense plasmas, electron gas dynamics of trapped electrons, excitons, Bose condensates atoms, molecules in strong fields ⇒ GW approximation

for larger/complex systems: nonequilibrium combinations with DFT, QMC

Text books, reviews:• Introduction to Computational Methods in Many Body Physics,M. Bonitz and D. Semkat (eds.), 2006• M. Bonitz Quantum Kinetic Theory, 1998• Progress in Nonequilibrium Green Functions I, II, III 2000, 2003, 2006

Announcement: Interdisciplinary conference Progress in Nonequilibrium GreenFunctions IV, August 17-22 2009, Glasgow, Scotland

Web page: www.theo-physik.uni-kiel.de/∼bonitz



Michael Bonitz

Introduction

Artificial atoms

fs-Dynamics

Boltzmann equation


2nd Quantization


KKB Equations


Homogeneousplasmas


Conclusions

Conclusions

Basis representation of Nonequilibrium Green’s Functions.

With an arbitrary but orthonormal set φk (x) :

g≷

(xt, x′t′) =

X

kl

φ∗k (x) φl (x) g

≷kl

(t, t′) , Ψ

(†)(x, t) =

X

k

φ(∗)k

(x) a(†)k

(t)

Equations of motion (KKBE) → matrix equations of exactly same structure. Butspatial degrees of freedom separated.

Dyson equation for Matsubara (equilibrium) Green’s function.

h

−∂τ − H0i

gM (τ ) = δ(τ ) +

Z β

0

d τ Σ[gM ](τ − τ) gM(τ )

Real-time stepping: Kadanoff-Baym equations.

i ∂t g>(t, t′) = hHF+Ext(t) g>(t, t

′) + I>1 (t, t′)

−i ∂t g<(t′, t) = g<(t′, t) hHF+Ext(t) + I<2 (t′, t)

i ∂t g⌉(t, −iτ ) = hHF+Ext(t) g⌉(t,−iτ ) + I⌉(t,−iτ )

−i ∂t g⌈(−iτ, t) = g⌈(−iτ, t) hHF+Ext(t) + I⌈(−iτ, t)


Documents

Quantum Kinetic Theory Michael Bonitz Quantum Kinetic ...bonitz/talks/08/balescu_mb2.pdf · Quantum Kinetic Theory Michael Bonitz Outline Outline R. Binder et al. PRB 45, 1107-1115