Quantum kinetic models of open quantum systems

Quantum kinetic models of open quantum systems

in semiconductor theory

Chiara Manzini

Scuola Normale Superiore di Pisa

Westf¨alische Wilhelms-Universit¨at M¨unster

29 Aprile 2005

Outlook

1- Motivation

2- Quantum kinetic formulation 3- Analytical difficulties

4- Open quantum systems: two examples 5- The three well-posedness results

6- New tools and perspectives

7- Final considerations

1- Motivation

Trend in semiconductor technology: miniaturization

=⇒ Challenge: simulation tools

Features of novel devices =⇒ Quantum effects

Prototype: Resonant Tunneling Diode

Doping

Density

Density

Quantum description

QS Quantum System with d degrees of freedom:

• a pure state of QS ↔ ψ ∈ L

(R

; C),

• a physical state of QS ↔ ρ ˆ “density matrix” on L

(R

; C) , i.e. ρ(x, y) ∈ L

(R

; C) kernel

ρ(x, y) = X

j∈ N

λ

ψ

(x)ψ

(y)

| {z }

pure state

,

{ψ

}

complete orthonormal set, λ

≥ 0, P

j

λ

= 1 . The position density:

n(x) = X

λ

|ψ

|

(x) ∈ L

(R

; R

) ⇔ finite mass.

Quantum Evolution: reversible dynamics

( i

ψ

= −

∆

ψ

+ V ψ

, j ∈ N, x ∈ R

−∆V (x, t) = n(x, t) = P

j

λ

|ψ

|

(x, t) ρ(x, y, t) = X

j∈ N

λ

ψ

(x, t)ψ

(y, t) , {ψ

(t)}

⊂ L

(R

; C) E.g. evolution of an electron ensemble with d d.o.f.,

ballistic regime, mean-field approximation

Quantum Evolution: reversible dynamics

( i

ψ

= −

∆

ψ

+ V ψ

, j ∈ N, x ∈ R

−∆V (x, t) = n(x, t) = P

j

λ

|ψ

|

(x, t) w(x, v) = F

X

j∈N

λ

ψ

(x + η

2 )ψ

(x − η 2 )

Quantum Liouville equation: Wigner-Poisson

 w

+ v · ∇

w − Θ[V ]w = 0

−∆

V (x, t) = n[w](x, t) = R

w(x, v, t) dv

F

{Θ[V ]w}(x, v) := i [V (x + η/2)−V (x − η/2)]F

w(x, η)

2- Quantum Kinetic description

A physical state of QS ←→ a quasiprobability on the phase space, w : (x, v) ∈ R

7→ R =⇒

w(x, v) := F

ρ 

x + η

2 , x − η 2

 ∈ L

(R

; R )

w is the Wigner function for the physical state of QS The position density

n[w](x, t) :=

Z

w(x, v, t) dv

w ∈ L

(R

; R ) 6⇒ n[w] well-defined, non-negative

3- Analytical difficulties

• w ∈ L

(R

; R ) NOT ⇒ n[w] well-defined w(x, v, t)

0 ⇒ n[w](x, t) :=

Z

w(x, v, t) dv

0 Quantum counterpart :

{ψ

(t)}

⊂ L

(R

; C), λ

≥ 0, P

j

λ

= 1

⇒ n(t) = P

j

λ

|ψ

|

(t), kn(t)k

< ∞ , ∀ t

⇒ (mass cons.) kψ

(t)k

= const.

First a-priori estimate for Schr.-Poisson systems

3- Analytical difficulties

• w ∈ L

(R

; R ) NOT ⇒ n[w] well-defined

• e

(x, t) :=

R

|v|

w(x, v, t) dv:

w(x, v, t)

0 ⇒ e

(x, t)

0 Quantum cp. : E

(t) :=

P

j

λ

k∇ψ

(t)k

1 2

P

j

λ

k∇ψ

(t)k

+ 1

2 k∇V (t)k

| {z }

E_pot² (t)

=const.

(energy cons.) ⇒ k∇ψ

(t)k

= const.

Advantages

• w ∈ L

(R

; R ) is necessary for a physical state of QS

• L

-setting is suitable for numerical approximation (cf.[Arnold. . . 96])

• pseudo-differential operator

kΘ[V ]wk

= k [V (x + η/2)−V (x − η/2)]F

wk

< Θ[V ]w, w >

= 0 (skew-simmetry) w

+ v · ∇

w − Θ[V ]w = 0

=⇒ kw(t)k

= const.

4- Open QS: irreversible dynamics

Motivation for quantum kinetic approach Two examples:

• electron ensemble with d d.o.f. in RTD

[Frensley90]: Time-dependent boundary conditions (b.c.) for w model ideal reservoir

⇒ Boundary-value problem for Wigner-Poisson:

• inflow time-dependent b.c. for w

• mixed Dirichlet-Neumann b.c. for V

Remark: Reformulation with {ψ

}

impossible:

λ NOT const.

4- Open QS: irreversible dynamics

Motivation for quantum kinetic approach Two examples:

• electron ensemble with d d.o.f. in RTD

• electron ensemble with d d.o.f. in GaAs:

electron-phonon interaction

Semiclassical cp. : Boltzmann for semiconductors

Density-matrix form. : NOT suitable for boundary-

value problems

Open QS: irreversible dynamics

w

+ v · ∇

w − Θ[V ]w = Qw

•

Qw =

(w − M

) ⇒ [Degond,. . .03] QDD,QET

Open QS: irreversible dynamics

w

+ v · ∇

w − Θ[V ]w = Qw

•

Qw =

(w − M

) ⇒ [Degond,. . .03] QDD,QET

•

Boltzmann-like collision operator

[Demeio, . . .04] ⇒ simulation

Open QS: irreversible dynamics

w

+ v · ∇

w − Θ[V ]w = Qw

•

Qw =

(w − M

) ⇒ [Degond,. . .03] QDD,QET

•

Boltzmann-like collision operator [Demeio, . . .04] ⇒ simulation

•

scattering term for electron-phonon interaction

[Fromlet, . . .99]

Open QS: irreversible dynamics

w

+ v · ∇

w − Θ[V ]w = Qw

•

Qw =

(w − M

) ⇒ [Degond,. . .03] QDD,QET

•

Boltzmann-like collision operator [Demeio, . . .04] ⇒ simulation

•

scattering term for electron-phonon interaction [Fromlet, . . .99]

•

quantum Fokker-Planck term [Castella, . . .00]

⇒ [Jüngel,. . .05] QHD

Classical cp. :Vlasov-Fokker-Planck

Quantum Fokker-Planck term

Qw = βdiv

(vw)+σ∆

w+2γdiv

(∇

w) + α∆

w (QFP) with α, β, γ ≥ 0, σ > 0.

[Caldeira . . .83]: QS= {electrons+reservoir}

β ∼ ξ coupling, σ ∼ T temp., γ, α ∼

, ǫ ∼

Lindblad condition: α σ ≥ γ

+ β

/16

Model Term Accuracy Lindblad

Classical β div

(vw) + σ∆

w O(ǫ

) NOT ok

Frictionless σ∆

w O(ǫ) ok

Quantum (QFP) O(ǫ

) ok

5- Three well-posedness results

•

WP system with inflow, time-dependent b.c. :

•

d = 1, global-in-time well-p. [M,Barletti04]

•

d = 3, local-in-time well-p. [M05]

•

WPFP system, d = 3 [Arnold,Dhamo,M05]:

•

global-in-time well-p., NEW a-priori estimates

⇒ NEW strategy for well-p. of WP(FP)

Common feature: L

-setting, ONLY kinetic tools

WP system with inflow b.c. d = 1, 3

( {∂

+ v · ∇

− Θ[V (t) + V

(t)]} w(t) = 0, t ≥ 0,

−∆

V (x, t) = n[w](x, t) = R

w(x, v, t) dv

(x, v) ∈ Ω × R

, Ω ⊂ R

open, bounded, convex, V

≡ V

(x, t), (x, t) ∈ R

× R

 



w(s, v, t) = γ(s, v, t), (s, v) ∈ Φ

, t ≥ 0, V (x, t) = 0, x ∈ ∂Ω, t ≥ 0,

w(x, v, 0) = w

(x, v) Φ

:= 

(s, v) ∈ ∂Ω × R

| v· n(s) > 0

• Weighted space:

X

:= L

(Ω × R

; (1 + |v|

)dxdv), d = 1, 3 ⇒ k = 1, 2 cf. [Markowich,. . . 89,Arnold,. . . 96]

Prop. : d = 3, w ∈ X

⇒ kn[w]k

≤ Ckwk

• Weighted space:

X

:= L

(Ω × R

; (1 + |v|

)dxdv), d = 1, 3 ⇒ k = 1, 2 cf. [Markowich,. . . 89,Arnold,. . . 96]

Prop. : d = 3, w ∈ X

⇒ kn[w]k

≤ Ckwk

=⇒ V solution of Poisson pb. with n[w] ∈ L

(Ω) : V ∈ W

(Ω) ∩ W

(Ω), kV k

≤ Ckn[w]k

Define P : X

−→ W

(R

)

P : w ∈ X

7−→ V 7−→ P w ≡ V a.e. Ω, P w = 0 outside Σ

WP system with inflow b.c. in X ₂

 



 

w

+ v · ∇

w − Θ[P w]w = 0, t ≥ 0,

w(s, v, t) = γ(s, v, t), (s, v) ∈ Φ

, t ≥ 0, w(x, v, 0) = w

(x, v) ∈ X

• for all u ∈ X

, kΘ[U ]uk

: contains kv

Θ[U ]uk

F



v

Θ[U ]u

= ∂

{ [U (x + η/2)−U (x − η/2)]F

u }

contains ∂

, ∂

U and v

, v

u

WP system with inflow b.c. in X ₂

 



 

w

+ v · ∇

w − Θ[P w]w = 0, t ≥ 0,

w(s, v, t) = γ(s, v, t), (s, v) ∈ Φ

, t ≥ 0, w(x, v, 0) = w

(x, v) ∈ X

• for all u ∈ X

, kΘ[U ]uk

: contains kv

Θ[U ]uk

F



v

Θ[U ]u

= ∂

{ [U (x + η/2)−U (x − η/2)]F

u } Prop. : kΘ[U ]uk

≤ CkU k

kuk

⇒ kΘ[P w]wk

≤ CkP wk

kwk

≤ Ckwk

WP system with inflow b.c. in X ₂

( {∂

− T

− Θ[P w](t)} w(t) = 0, t ≥ 0, w(x, v, 0) = w

(x, v) ∈ X

w(t) ∈ D(T

) := {u ∈ X

| v · ∇

u ∈ X

, u|

= γ(t)}

Remark ∀u

, u

∈ D(T

) ⇒ u

− u

∈ D(T

)

∀ p : [0, +∞) → X

, s.t. p(t) ∈ D(T

), ∀t ≥ 0

⇒ D(T

) = p(t) + D(T

)

(cf.[Barletti00])

WP system with inflow b.c. in X ₂

( {∂

− T

− Θ[P w](t)} w(t) = 0, t ≥ 0, w(x, v, 0) = w

(x, v) ∈ X

∀ w : [0, +∞) → X

, s.t. w(t) ∈ D(T

), ∀t ≥ 0 w(t) = p(t) + u(t), with u : [0, +∞) → D(T

)

( {∂

− T

− L

(t) − Θ[P u](t)} u(t) = Q

(t) u(x, v, 0) = w

(x, v) − p(x, v, 0) ∈ D(T

) L

(t)u(t) := Θ[P p](t)u(t) + Θ[P u](t)p(t)

Q

(t) := −{∂

− T

− Θ[P p](t)} p(t)

WP system with inflow b.c. in X ₂ : local-in- t solution

( {∂

− T

− Θ[P w](t)} w(t) = 0, t ≥ 0, w(x, v, 0) = w

(x, v) ∈ X

Theorem

∀ γ s.t. ∃ p as in (1) , ∀ w

∈ D(T

), ∃t

≤ ∞ s.t.

∃! classical solution w(t), ∀ t ∈ [0, t

).

If t

< ∞, then lim

kw(t)k

= ∞.

• Ex. :

γ ∈ C

([0, ∞); L

(∂Ω × R

; (v· n(s))(1 + |v|

)dsdv))

• A priori estimates are needed for global-in-t result

WPFP system, d = 3

 



 

{∂

+ v · ∇

− Θ[V (t)]} w(t) = Qw(t)

−∆

V (x, t) = n[w](x, t) w(x, v, t = 0) = w

(x, v)

w ≡ w(x, v, t), (x, v, t) ∈ R

× [0, ∞)

Qw = βdiv

(vw) + σ∆

w + 2γdiv(∇

w ) + α∆

w Assume α σ > γ

⇒ Q uniformly elliptic in x and v.

•

Classical cp. : Vlasov-PFP with

•

non-linear term: ∇

V · ∇

w ,

•

FP term: β div

(vw) + σ∆

w.

Existing results

•

“density matrix” ρ ˆ : [Arnold, . . .03]

global-in-t solution, by conservation laws.

•

Quantum Kinetic: L

-analysis [Cañizo, . . .04]

global-in-t solution, by ρ(t) ≥ 0 ˆ , cons. laws:

w(t) ∈ L

(R

; R) ⇒ n[w](t) ∈ L

(R

) (mass cons.) ⇒ kn[w](t)k

= const.

e

[w](x, t)> −∞ ⇒ e

[w](x, t)< ∞

INSTEAD keep to L

-setting, purely kinetic analysis

• Weighted space: X

:= L

(R

; (1 + |v|

)dxdv) Prop. : w ∈ X

⇒ kn[w]k

≤ Ckwk

=⇒ V (x) := 1

4π|x| ∗ n[w](x), x ∈ R

:

• V ∈ L /

, ∀p, ∇V = − x

4π|x|

∗ n[w] ∈ L

. F

Θ[V ]z(x, η) = i (V (x + η/2) − V (x − η/2))

| {z }

=:δV (x,η)

F

z(x, η)

For η fixed, kδV (. , η)k

≤ C |η|

kn[w]k

1

• Weighted space: X

:= L

(R

; (1 + |v|

)dxdv)

• V ∈ L /

, ∀p, ∇V = − x

4π|x|

∗ n[w] ∈ L

.

• kΘ[V ]zk

≤ Ckn[w]k

k|η|

F

zk

Prop. : w, z, ∇

z ∈ X

⇒

kΘ[V ]zk

≤ C{ kzk

+ k∇

zk

} kwk

⇒ parabolic reguralization is needed

The linear equation

w

= Aw(t), t > 0, Aw := −v · ∇

w + Qw w(t = 0) = w

∈ X

,

• {e

, t ≥ 0} C

–semigroup on X

(cf. [Arnold, . . .02])

⇒ kw(t)k

≤ e

kw

k

, t ≥ 0

• w(x, v, t) = G(x, v, t)∗ w

(cf. [Sparber, . . .03]) ⇒

k∇

w(t)k

≤ B t

e

kw

k

, 0 < t ≤ T

(cf. classical cp. VPFP [Carpio98])

The non-linear equation

∀ T ∈ (0, ∞) fixed

w

= Aw(t)+(Θ[V ]w)(t), t ∈ [0, T ], w(t = 0) = w

, Y

:= {z ∈ C([0, T ]; X

) | ∇

z ∈ C((0, T ]; X

),

t

k∇

z(t)k

≤ C

∀t ∈ (0, T )}.

∀ T < T

, the map w ∈ Y

7−→ w s. t. e e

w(t) = e

w

+

Z

0

e

(Θ[ e V ]w)(s) ds, t ∈ [0, T ] is a strict contraction on some ball of Y

with

τ (kw

k

).

The non-linear equation

∀ T ∈ (0, ∞) fixed

w

= Aw(t)+(Θ[V ]w)(t), t ∈ [0, T ], w(t = 0) = w

, Y

:= {z ∈ C([0, T ]; X

) | ∇

z ∈ C((0, T ]; X

),

t

k∇

z(t)k

≤ C

∀t ∈ (0, T )}.

Existence and Uniqueness Theorem

∀ w

∈ X

, ∃ t

≤ ∞ s.t. ∃ ! mild solution w ∈ Y

, ∀ T < t

. If t

< ∞, then

lim

kw(t)k

= ∞.

Global-in- t well-posedness

A priori estimates

•

kw(t)k

≤ e

kw

k

, t ≥ 0,

•

for k|v|

w(t)k

< ∞ 0 ≤ t ≤ T,

some L

-bounds for ∇

V (t) are needed

⇒Idea: get bounds for ∇

V (t) depending on kw(t)k

Strategy: exploit dispersive effects of the free-str.

(cf. [Perthame96]) =⇒

estimate(define) ∇

V via integral equation

⇒ definition of n[w] is by-passed

Anticipation: global-in- t well-posedness WITHOUT

A priori estimate for electric field: WP case

E (x, t) = −∇V (x, t) = −

∗ n(x, t) Z

R³

v

w(x, v, t) dv = Z

R³

v

w

(x − vt, v) dv + Z

R³

v

Z

0

(Θ[V ]w)(x − vs, v, t − s) ds dv Correspondingly E into

E

(x, t) = − 1 4π

x

|x|

∗

Z

R³

v

w

(x − vt, v) dv E

(x, t) = − 1

4π

x

|x|

∗

Z Z

(Θ[V ]w)(x − vs, v, t − s) ds

A priori estimate for electric field: WP, E ₁

n

(x, t) = R R

0

(Θ[V ]w)(x − vs, v, t − s) ds dv Classical cp. : Vlasov-Poisson [Perthame96]

n

(x, t) = Z

R³

v

Z

0

E · ∇

w (x − vs, v, t − s) ds dv

= div

Z

R³

v

Z

0

s(E w)(x − vs, v, t − s) ds dv

• Θ[V ]w = F

(δV w) = F b

( W [E]· d ∇

w ) since δV (x, η) = V (x +

) − V (x −

)

= η · R

−¹₂

E(x − rη)dr =: W [E](x, η)· η

A priori estimate for electric field: WP, E ₁

n

(x, t) = R R

0

(Θ[V ]w)(x − vs, v, t − s) ds dv

• Θ[V ]w = F

(δV w) = F b

( W [E]· d ∇

w ) since δV (x, η) = V (x +

) − V (x −

)

= η · R

−¹₂

E(x − rη)dr =: W [E](x, η)· η

n

(x, t) = Z

R³

v

Z

0

F

( W [E]· d ∇

w )(x − vs, v, t − s) ds

= div

Z

R³

Z

0

s F

(W [E] w)(x − vs, v, t − s)ds b

A priori estimate for electric field: WP, E ₁

E

(x, t) = −

∗ n

(x, t) =

− div



x

|x|³

 ∗ R

0

s R

F

(W [E] w)(x − vs, v, t−s)dvds b Lemma :

RF

(W [E] w)(x−vs, v, t−s)dv b

≤ s

kE(t − s)k

x

kw(t − s)k

Quantum case: just in L

.

Classical case: L

-version.

A priori estimate for electric field: WP, E ₁

E

(x, t) = −

∗ n

(x, t) =

− div



x

|x|³

 ∗ R

0

s R

F

(W [E] w)(x − vs, v, t−s)dvds b Lemma :

RF

(W [E] w)(x−vs, v, t−s)dv b

≤ s

(kE

(t − s)k

+ kE

(t − s)k

) kw(t − s)k

Prop. : kE

(t)k

≤ C

t

with ω ∈ [0, 1)

⇒ kE

(t)k

≤ C 

T, sup

kw(s)k



t

.

NEW : w ∈ C([0, T ]; L

) 7→ E [w] ∈ L

((0, T ]; L

)

A priori estimate for electric field: WP, E ₀

E

(x, t) = − 1 4π

x

|x|

∗

Z

w

(x − vt, v) dv Ex. Strichartz for free-str. [Castella-Perthame ’96],

Z

w

(x − vt, v)dv

L⁶x^/5

≤ t

kw

k

x(L⁶v^/5)

, t > 0.

Let ∀ t ∈ (0, T ]

Z

w

(x − vt, v)dv

L^6/5_x

≤ C

t

, ω ∈ [0, 1)

(HP1)

⇒ kE

(t)k

≤ C

Z

w

(x − vt, v)dv

≤ CC

t

WPFP: global-in- t, smooth solution

w(x, v, t) = G(t, x, v)∗w

+

Z

0

G(s, x, v)∗(Θ[V ]w)(t−s)ds

E

∈ L

((0, T ]; L

) by parabolic reguralization.

Classical cp. : VPFP [Castella98]

Remark: E(t) ∈ L

is needed for bootstraping kvw(t)k

, k|v|

w(t)k

≤ C, 0 ≤ t ≤ T, ∀ T.

Theorem[Arnold,Dhamo,M05]

∀ w

∈ X

s.t. (HP1) holds, ∃!w global mild solution, w(t) ∈ C

(R

), ∀t > 0,

n(t), E (t) ∈ C

(R

), ∀t > 0.

WPFP: global-in- t, smooth solution

w(x, v, t) = G(t, x, v)∗w

+

Z

0

G(s, x, v)∗(Θ[V ]w)(t−s)ds

E

∈ L

((0, T ]; L

) by parabolic reguralization.

Remarks

• NOT used pseudo-conformal law (cf. classical cp.)

• w ∈ C([0, T ]; L

) 7→ E

[w] ∈ L

((0, T ]; L

) 7→

V

[w](x, t) := −

∗ E

[w](x, t)

⇒ V

[w] ∈ L

((0, T ]; L

) ⇒ fixed-point-map for w

WPFP: global-in- t, smooth solution, without weights

7- Final considerations

• kinetic analysis, L

-framework:

•

physically consistent,

•

suitable for real device simulation,

•

admissable parallelism with classical cp.

• by-pass the definition of the particle density Perspectives

• WPFP: hypoelliptic case, WP d = 3, all-space case

• a priori estimates in the bounded domain case

• long-time-behaviour: decay t → ∞ of E(t), n(t)