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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
Quantum KineticTheory
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 (CAU Kiel) Quantum Kinetic Theory 2 / 46
Quantum KineticTheory
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 (CAU Kiel) Quantum Kinetic Theory 3 / 46
Quantum KineticTheory
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
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 (CAU Kiel) Quantum Kinetic Theory 4 / 46
Quantum KineticTheory
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
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 (CAU Kiel) Quantum Kinetic Theory 5 / 46
Quantum KineticTheory
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 fs-Dynamics
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 (CAU Kiel) Quantum Kinetic Theory 6 / 46
Quantum KineticTheory
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 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 (CAU Kiel) Quantum Kinetic Theory 7 / 46
Quantum KineticTheory
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 Boltzmann equation
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 (CAU Kiel) Quantum Kinetic Theory 8 / 46
Quantum KineticTheory
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 Boltzmann equation
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 (CAU Kiel) Quantum Kinetic Theory 9 / 46
Quantum KineticTheory
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 Boltzmann equation
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)
ע
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 (CAU Kiel) Quantum Kinetic Theory 10 / 46
Quantum KineticTheory
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
Nonequilibrium Green’s Functions 2nd Quantization
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 (CAU Kiel) Quantum Kinetic Theory 11 / 46
Quantum KineticTheory
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
Nonequilibrium Green’s Functions 2nd Quantization
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 (CAU Kiel) Quantum Kinetic Theory 12 / 46
Quantum KineticTheory
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
Nonequilibrium Green’s Functions 2nd Quantization
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 (CAU Kiel) Quantum Kinetic Theory 13 / 46
Quantum KineticTheory
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
Nonequilibrium Green’s Functions Definition and properties of NEGFs
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 (CAU Kiel) Quantum Kinetic Theory 14 / 46
Quantum KineticTheory
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
Nonequilibrium Green’s Functions Definition and properties of NEGFs
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 (CAU Kiel) Quantum Kinetic Theory 15 / 46
Quantum KineticTheory
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
Nonequilibrium Green’s Functions Definition and properties of NEGFs
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 (CAU Kiel) Quantum Kinetic Theory 16 / 46
Quantum KineticTheory
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
Nonequilibrium Green’s Functions Definition and properties of NEGFs
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 (CAU Kiel) Quantum Kinetic Theory 17 / 46
Quantum KineticTheory
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
Nonequilibrium Green’s Functions KKB Equations
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 (CAU Kiel) Quantum Kinetic Theory 18 / 46
Quantum KineticTheory
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
Nonequilibrium Green’s Functions KKB Equations
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 (CAU Kiel) Quantum Kinetic Theory 19 / 46
Quantum KineticTheory
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
Nonequilibrium Green’s Functions KKB Equations
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 (CAU Kiel) Quantum Kinetic Theory 20 / 46
Quantum KineticTheory
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
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 (CAU Kiel) Quantum Kinetic Theory 21 / 46
Quantum KineticTheory
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
Numerical Applications Homogeneous plasmas
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 (CAU Kiel) Quantum Kinetic Theory 22 / 46
Quantum KineticTheory
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
Numerical Applications Homogeneous plasmas
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 (CAU Kiel) Quantum Kinetic Theory 23 / 46
Quantum KineticTheory
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
Numerical Applications Homogeneous plasmas
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 (CAU Kiel) Quantum Kinetic Theory 24 / 46
Quantum KineticTheory
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
Numerical Applications Charged particles in a trap
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 (CAU Kiel) Quantum Kinetic Theory 25 / 46
Quantum KineticTheory
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
Numerical Applications Charged particles in a trap
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 (CAU Kiel) Quantum Kinetic Theory 26 / 46
Quantum KineticTheory
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
Numerical Applications Charged particles in a trap
Equilibrium density n(x) and energy distribution f (ǫk − µ)
N = 6 fermions (1D trap)
Michael Bonitz (CAU Kiel) Quantum Kinetic Theory 27 / 46
Quantum KineticTheory
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
Numerical Applications Charged particles in a trap
Energy evolution after laser excitation
Michael Bonitz (CAU Kiel) Quantum Kinetic Theory 28 / 46
Quantum KineticTheory
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
Numerical Applications Charged particles in a trap
Occupation number dynamics for off-resonant laser excitation
Michael Bonitz (CAU Kiel) Quantum Kinetic Theory 29 / 46
Quantum KineticTheory
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
Numerical Applications Charged particles in a trap
Occupation number dynamics for near-resonant laser excitation
Michael Bonitz (CAU Kiel) Quantum Kinetic Theory 30 / 46
Quantum KineticTheory
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
Numerical Applications Charged particles in a trap
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 (CAU Kiel) Quantum Kinetic Theory 31 / 46
Quantum KineticTheory
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
Numerical Applications Charged particles in a trap
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 (CAU Kiel) Quantum Kinetic Theory 32 / 46
Quantum KineticTheory
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
Numerical Applications Charged particles in a trap
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 (CAU Kiel) Quantum Kinetic Theory 33 / 46
Quantum KineticTheory
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
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 (CAU Kiel) Quantum Kinetic Theory 34 / 46
Quantum KineticTheory
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
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)
Michael Bonitz (CAU Kiel) Quantum Kinetic Theory 35 / 46
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