Quantum Fokker-Planck models: kinetic & operator theory approaches

Transcription

1 TECHNISCHE UNIVERSITÄT WIEN Quantum Fokker-Planck models: kinetic & operator theory approaches Anton ARNOLD with F. Fagnola (Milan), L. Neumann (Vienna) TU Vienna Institute for Analysis and Scientific Computing Victoria, July 2009 Anton ARNOLD (TU Vienna) Quantum Fokker-Planck models 1/19

2 application: electron transport in nano-semiconductors resonant tunneling diode for high frequency oscillators: 5nm each total length 60nm {}}{ GaAs Al Ga Ga As As Al Ga As GaAs } {{ }} {{ }} {{ } D>0 D=0 D>0 D(x) 0... concentration of donor ions, doping profile goal: numerical simulation of electron transport potential barrier for electrons resonant tunneling: V hetero -0.3eV x Anton ARNOLD (TU Vienna) Quantum Fokker-Planck models 2/19

3 simulation model: Wigner functions, 1-particle approxim. Wigner Fokker-Planck equ. (augmented Caldeira-Leggett model) evolution for Wigner function w(x, v, t) R: w t + v x w +Θ[V ]w = Qw, x, v R d, t > 0 w(x, v, t =0)=w 0 (x, v) Qw = D pp Δ v w +2γ div v (vw) + D qq Δ x w +2D pq div x ( v w) class. diffusion friction quantum diffusion Fokker-Planck term Q models interaction of electrons with phonon heat bath diffusive effects, open quantum system Anton ARNOLD (TU Vienna) Quantum Fokker-Planck models 3/19

4 simulation model: Wigner functions, 1-particle approxim. Wigner Fokker-Planck equ. (augmented Caldeira-Leggett model) evolution for Wigner function w(x, v, t) R: w t + v x w +Θ[V ]w = Qw, x, v R d, t > 0 w(x, v, t =0)=w 0 (x, v) Qw = D pp Δ v w +2γ div v (vw) + D qq Δ x w +2D pq div x ( v w) class. diffusion friction quantum diffusion Fokker-Planck term Q models interaction of electrons with phonon heat bath diffusive effects, open quantum system V (x, t)... electrostatic potential: Θ[V ]w(x, v) =i(2π) d 2 [V (x + η R d 2 ) V (x η )]ŵ(x,η)eiη v dη 2 n(x, t) = R w(x, v, t) dv 0... particle density d Anton ARNOLD (TU Vienna) Quantum Fokker-Planck models 3/19

5 nonlinear mean-field model: selfconsistent Hartree potential V (x, t): ΔV = n(x, t) D(x) = w(x, v, t)dv D(x) R d for RTD: d =1;inflow boundary conditions for w at x =0,x = L (due to characteristics of free transport equation): v BC o L x BC Anton ARNOLD (TU Vienna) Quantum Fokker-Planck models 4/19

6 Refs: let x R d, V = x 2 2 θ[v ]= x v w. Sparber-Carrillo-Dolbeault-Markovich [SCDM 04]:! normalized steady state w ; Gaussian (explicit comp.) w(t) t w exponentially (entropy method) (bounded) perturb. of V : difficult on Wigner level work in progress Anton ARNOLD (TU Vienna) Quantum Fokker-Planck models 5/19

7 Outline 1 density matrix, Lindblad equation 2 evolution: global solution 3 main results 4 steady state proof 5 t convergence Anton ARNOLD (TU Vienna) Quantum Fokker-Planck models 6/19

8 1) density matrix formulation Wigner-Weyl transformation: w(x, v, t) = 1 (2π) R ρ ( x + η d d 2, x η 2, t) e iη v dη w R ρ(x, y) =ρ(y, x) n(x, t) = ρ(x, x, t) 0... particle density Anton ARNOLD (TU Vienna) Quantum Fokker-Planck models 7/19

9 1) density matrix formulation Wigner-Weyl transformation: w(x, v, t) = 1 (2π) R ρ ( x + η d d 2, x η 2, t) e iη v dη w R ρ(x, y) =ρ(y, x) n(x, t) = ρ(x, x, t) 0... particle density density matrix operator on L 2 (R d ): (ϱf )(x) = ρ(x, y)f (y)dy R d... self-adjoint physical quantum states: ϱ 0, ϱ J 1 (L 2 (R d )), tr ϱ =1... positive trace class operator Anton ARNOLD (TU Vienna) Quantum Fokker-Planck models 7/19

10 evolution by Lindblad equation: t ϱ = i [H,ϱ] + l=1,2 Hamiltonian L l ϱl l 1 2 (L l L l ϱ + ϱl l L l ) =: L dissipative (ϱ) generator H = 1 2 xx + ω2 2 x2 + V (x) i γ 2 {x, x }... (adjusted) Hamiltonian subquadrat. confinement perturb. pot. potential Anton ARNOLD (TU Vienna) Quantum Fokker-Planck models 8/19

11 evolution by Lindblad equation: t ϱ = i [H,ϱ] + l=1,2 Hamiltonian L l ϱl l 1 2 (L l L l ϱ + ϱl l L l ) =: L dissipative (ϱ) generator H = 1 2 xx + ω2 2 x2 + V (x) i γ 2 {x, x }... (adjusted) Hamiltonian subquadrat. confinement perturb. pot. potential WFP representable as Lindblad equ. if: (set D pq =0here) Lindblad cond: Δ:=D pp D qq γ2 4 0 (diffusion dominates friction) hence: ϱ 0 0 ϱ(t) 0 t 0 iγ L 1 = p 2 Dpp i x + 2 Dpp q, L 2 = 2 Δ p... Lindblad op. 2 Dpp x linearly independent for Δ > 0 span (L 1, L 2 )=span(p, q) Anton ARNOLD (TU Vienna) Quantum Fokker-Planck models 8/19

12 2) global in time solution for Quantum Fokker-Planck Theorem ([AA-Sparber 04]: quadratic pot. + V L ) 1 lin.qfp;letϱ 0 J 1 (L 2 (R d ))! global, sol. ϱ C([0, ); J 1 ), ϱ(t) 0, tr ϱ(t) =tr ϱ 0 2 QFP Poisson ( ΔV (t) =n(t) =ρ(x, x, t), in R 3 ) let ϱ 0 E ( i.e. ϱ 0 J 1, E kin := 1 2 tr (Δϱ 0) < )! global, trace preserving solution ϱ C([0, ); E) Proof (by semigroup theory). construction of linear evolution semigroup in J 1, E (for V =0) nonlinearity is locally Lipschitz + a-priori estimates in E Anton ARNOLD (TU Vienna) Quantum Fokker-Planck models 9/19

13 construction of linear evolution semigroup in J 1, E-details: (for V =0) dissipative open quantum system (linear V given): { ddt ϱ(t) = L (ϱ) := i[h,ϱ]+a(ϱ), t > 0 ϱ(t =0) =ϱ 0 A(ϱ)... dissipative / Lindblad terms [E. Davies 77]: a linear C 0 -semigroup on J 1 ( minimal solution constructed by iteration) possible problems: semigroup not unique D(L ) too small not conservative: tr (ϱ(t)) tr ϱ 0 need to prove: D(L ) is big enough Anton ARNOLD (TU Vienna) Quantum Fokker-Planck models 10/19

14 Lemma ([AA-Carrillo-Dhamo 02], [AA-Sparber 04]) Let operator P = p 2 (x, i ) be a quadratic polynomial, D(P) :=C 0 (Rd ) L 2 (R d ). Proof. for f D( P): P is the maximum extension of P, i.e. D( P) ={f L 2 Pf L 2 } f n (x) := χ n (x) (f ϕ n )(x) n f -cutoff -mollifier C 0 C 0 in graph norm. P Anton ARNOLD (TU Vienna) Quantum Fokker-Planck models 11/19

15 application/limitation of lemma: Example 1: P = Δ x 2, D(P) =C 0 (Rd ) P is essentially selfadjoint on C 0 (Rd ) Example 2: P = 2 x x 4 not essentially selfadjoint on C 0 (Rd ) [Reed-Simon] lemma can t be extended to all P = p 4 (x, i ) Anton ARNOLD (TU Vienna) Quantum Fokker-Planck models 12/19

16 prove: D(L ) is big enough Lemma ([AA-Sparber, CMP 04]) Let generator L (ϱ) be quadratic in x and x (QFP, e.g.). L D is the maximum extension in J 1 D J 1... dense subset with C0 -kernels Proof. for ϱ D(Lmax) ={ϱ J 1 L (ϱ) J 1 }: n D ϑ n ϱ in graph norm. L [ ] θ n (x, y) := χ n (x) ϕn (x) x ρ(x, y) y ϕ n (y) χn (y) -cutoff -mollifier C 0 Theorem QFP: C 0 -semigroup e L t of Davies is unique & trace preserving C 0 Anton ARNOLD (TU Vienna) Quantum Fokker-Planck models 13/19

17 3) main results Theorem ([AA-Fagnola-Neumann 08]) Let V (x) c (1 + x 2 ) α 2 for some 0 α<1 (subquadratic) 1! global, trace preserving solution ϱ = ϱ(t) of QFP 2 normal steady state of Quantum Fokker-Planck (QFP) 3 if Δ=D pp D pp γ2 4 > 0:! normal steady state ϱ, ϱ is faithful (i.e. 0 / σ(ϱ ); rank ϱ = ), ϱ(t) t ϱ in J 1 (L 2 ). Anton ARNOLD (TU Vienna) Quantum Fokker-Planck models 14/19

18 4) existence of steady state 2 results (by compactness): [Fagnola Rebolledo 2001]: notation: T t... QM semigroup on ϱ J 1 ; t 0(Schrödinger picture) T t... dual QMS on (observables in) B(L 2 ) (Heisenberg picture) Y s.a. r := YE r + re r E r... spectral proj. for Y on (, r] R cut-off op. Theorem ([Fagnola Rebolledo 2001]) Let T be QMS. assume: s.a. X 0, Y b R (Y with finite dim. spectral proj. for bounded intervals) with t u, T s (Y r) u ds u, Xu t, r > 0, u D(X ) 0 B(L 2 ) normalized steady state for T, T. Anton ARNOLD (TU Vienna) Quantum Fokker-Planck models 15/19

19 condition in Th: t 0 u, T s (Y r)u ds u, Xu (1) Proof. from (1) with Y r (b + r) E r + r 1 : ɛ>0 t(ɛ) > 0, r(ɛ) > 0: 1 1 t t 0 tr ( ) T s ( u u ) E r(ɛ) ds 1 ɛ, t > t(ɛ) (= Prohorov cond.) pure state E r(ɛ)... compact support in spectrum of Y, finite rank projector i.e. ϱ(t) := 1 t t 0 T s ( u u )ds, t > t(ɛ) is tight (2) 2 J 1 - weakly convergent subsequence ϱ(t n ), t n i.e. tr ( ϱ(t n ) A) tr (ϱ A), A B(L 2 ) 3 such weak limits of (2) are normalized steady states of T. Anton ARNOLD (TU Vienna) Quantum Fokker-Planck models 16/19

20 existence of steady state simplified condition: Theorem ([Fagnola Rebolledo 2001]) Let s.a. X 0, Y b : + technical assumption (on domains) normalized steady state Proof. (3) (1) u, L(X ) u u, Yu u D (3) Anton ARNOLD (TU Vienna) Quantum Fokker-Planck models 17/19

21 existence of steady state simplified condition: Theorem ([Fagnola Rebolledo 2001]) Let s.a. X 0, Y b : + technical assumption (on domains) normalized steady state Proof. (3) (1) u, L(X ) u u, Yu u D (3) application to Quantum Fokker-Planck [AA-Fagnola-Neumann]: choose: X := rp 2 +(pq + qp)+(ω 2 r +2γ)q 2 0 ( for r > 1 ) 2γ Y := C (p 2 + q 2 ) C >0 L(X ) involves subquadratic perturbations (from V (x)) pv (q) can be compensated by Y Anton ARNOLD (TU Vienna) Quantum Fokker-Planck models 17/19

22 steady state ϱ is faithful [AA-F-N] (i.e. 0 / σ(ϱ )): let Δ > 0 L 1, L 2 linearly independent QM semigroup T on B(L 2 )isirreducible, i.e. proper invariant subspaces of evolution (for any ϱ 0 ) ϱ has full rank, i.e. faithful Theorem ([Frigerio, 1977]) Let ϱ be faithful, then: T is irreducible normal steady state ϱ is unique Anton ARNOLD (TU Vienna) Quantum Fokker-Planck models 18/19

23 steady state ϱ is faithful [AA-F-N] (i.e. 0 / σ(ϱ )): let Δ > 0 L 1, L 2 linearly independent QM semigroup T on B(L 2 )isirreducible, i.e. proper invariant subspaces of evolution (for any ϱ 0 ) ϱ has full rank, i.e. faithful Theorem ([Frigerio, 1977]) Let ϱ be faithful, then: T is irreducible normal steady state ϱ is unique Proof-Idea. let χ be a subspace of L 2 with L l (χ) χ. span (L 1, L 2 )=span(p, q) =span (a, a ) creation/annihilation proveable: e t 2 (p2 +q 2 1) χ χ t 0 χ = span (e j, j J N) e j... eigenfct. of N := 1 2 (p2 + q 2 1) a(χ) χ, a (χ) χ J = N. Anton ARNOLD (TU Vienna) Quantum Fokker-Planck models 18/19

24 5) t convergence Theorem ([Fagnola Rebolledo 98]) Let T be QMS; let faithful normalized steady state ϱ, let commutant {L l, L l ; l =1, 2} = {L l, L l, H} ( {steady states } ) + techn. assumpt. (on domains) ϱ 0 J 1, tr ϱ 0 =1:ϱ(t) ϱ in J 1 (L 2 ) application to Quantum Fokker-Planck [AA-Fagnola-Neumann]: Δ:=D pp D qq γ2 4 > 0, L 1 = iγ 2 Dpp p + 2 D pp q, L 2 = 2 Δ 2 Dpp p {L l, L l } ={p, q} = {c 1 c C} {L l, L ( trivial) l } = {L l, L l, H} convergence of quantum Fokker-Planck solution to ϱ Anton ARNOLD (TU Vienna) Quantum Fokker-Planck models 19/19