Collaborations: Tsampikos Kottos (Gottingen) Chen Sarig (BGU)

Transcription

1 Quantum Dissipation due to the Interaction with Chaos Doron Cohen, Ben-Gurion University Collaborations: Tsampikos Kottos (Gottingen) Chen Sarig (BGU) Discussions: Massimiliano Esposito (Brussels) Pierre Gaspard (Brussels) Leonid Pastur (Kharkov) dcohen LANL cond-mat archive $GIF, $ISF

2 System - Environment H total = H 0 (x, p) + H(Q, P ; x) x dissipation fluctuations Q driving source "slow" DoF "system" driven system "fast" DoF "environment"

3 Interaction with bath: ZCL/DLD models V x V H total = H 0 (x, p) + H(Q, P ; x) H 0 (x, p) = H ZCL = α H DLD = α 2M p2 ( Pα 2 2m α + 2 m αω 2 αq α 2 ( Pα 2 2m α + 2 m αω 2 αq α 2 ) ) x c α Q α α c α Q α u(x x α ) α

4 Brownian Motion modeling V ZCL model V DLD model (disorder) (Q,P) V piston model: interaction with chaos

5 Interaction with chaos x (t) A V (Q,P) x(t) (Q,P) x B(t) H total = H 0 (x, p) + H(Q, P ; x) H 0 (x, p) = 2M p2 H(Q, P ; x) = 2m P 2 + U(Q; x) [dephasing in this model: DC, PRE 2002]

6 2 Spin interacting with chaos H total = H 0 + H(Q, P ; x) x = vσ 3 = position in the double well H 0 = ( hω/2)σ The Hamiltonian of a nearby chaotic system: H(Q, P ; x) = 2 (P 2 +P Q 2 +Q 2 2) + (+x) Q 2 Q Q Q Nuclear physics application: boundary may have either of two shapes

7 The Hamiltonian matrix for interaction with chaos H = 2 (P 2 +P Q 2 +Q 2 2) + (+x) Q 2 Q 2 2 H = E + xb It can be argued that B nm is a banded matrix. bandwidth = b = h/τ cl H total = 2 hωσ + E + vσ 3 B H total = [ E+vB Ω/2 Ω/2 E vb ]

8 Simulations Ψ(t = 0) = 2 ( + ) ψ (E) basis: ν n ρ ν,ν (t) = n Ψ ν,n (t) Ψ ν,n(t) [ + M3 M im 2 ] ρ(t) = 2 ( + M σ) 2 M + im 2 M 3 S(t) = (2 trace(ρ(t) 2 ) ) = M M

9 Pedagogical remark: Given a basis ν for the representation of the spin, the wavefunction Ψ can always be written as Ψ = ν ν ψ (ν) where the unnormalized wavefunction ψ (ν) is called the relative state of the environment. With this notation the elements of the reduced probability matrix are: ρ ν,ν = ψ (ν) ψ (ν ) Hence the overlap of the relative states determines the purity of the spin state. In particular orthogonality of the relative states implies a maximally mixed spin state.

10 Numerical observations v< S(t) t v>0.6 2DW 0.02<v< RMT v< v> (vt) 2 RMT 0.02<v< v 2 t 2.8 < E < 3.2 T.3 d T < v < 0.3 τ cl h = 0.03 b

11 H = α The Hamiltonian matrix for interaction with bath ( P 2 α 2m α + 2 m αω 2 αq 2 α ) x α c α Q α The states of the bath are: n = {n α } = n, n 2, n 3,... E n = α ω α n α We can write: H = E + xb B nm is non-zero only for one-photon excitations. For such excitations E m E n = hω α Consequently B nm is a sparse banded matrix. bandwidth = b = hω c

12 Fluctuations H = H(Q, P ; x) F(t) = H x (Q(t), P (t); x) The (asymmetrized) correlation function: C(τ) = F(τ)F(0) The power spectrum of the fluctuations (assuming preparation in the nth state): C(ω) = m F mn 2 2πδ ( ω E m E n h ) Conclusion: The bandprofile of the matrix F nm is related to the power spectrum C(ω)

13 Numerical example H = 2 (P 2 +P Q 2 +Q 2 2) + (+x) Q 2 Q 2 2 F = Q 2 Q C(ω) classical h=0.030 h= H = E + xb B = { F nm } E 3 τ cl 4.3 h 2 b h/τ cl b / h

14 H = α The bandprofile for bath ( P 2 α 2m α + 2 m αω 2 αq 2 α ) x α c α Q α F = α c α Q α = α c α ( h 2m α ω α ) /2 (a α + a α) For preparaion in state n C(ω) = c 2 α n α ± Q α n α 2 2πδ(ω ω α ) α C(ω) = α ± π hc 2 [ ] α (+n α )δ(ω ω α ) + n α δ(ω + ω α ) m α ω α For canonical preparation C T (ω) = 2J(ω) e = J(ω) βω sinh(ω/(2t )) eω/(2t ) where we define J(ω) = π 2 α c 2 α m α ω α δ(ω ω α ) = ηωe ω/ω c (with anti-symmetric continuation)

15 Two notions of temperature T EQLB = d de ln(g(e)) equilibrium T FLCT = 2 d dω ln( C(ω)) ω 0 dissipation which means that [ g(e + ω) g(e) exp [ C(ω) C(ω 0) exp ] ω T EQLB ] ω 2T FLCT For interaction with chaos C E (ω) = B mn 2 2πδ(ω (E m E n )) m = g(e+ω) σ(e+ω E) 2 T FLCT 2 T EQLB

16 The FD relation For canonical preparation µ = friction coef = 2T C(ω 0) For interaction with microcanonical chaos d [ µ = g(e) CE (ω 0) ] = 2g(E) de = d dω C E (ω) ω 0 = 2T FLCT CE (ω 0) From now on we characterize the fluctuations by the symmetrized correlation function, and regard the temperature as an independent parameter. ) ) ( hω ( hω C(ω) = 2πσ 2 δ(ω) + 2π hσ2 R G G() = semiclassical envelope (bandprofile) b R() = lower cutoff function (level repulsion) b = h τ cl = hω c = bandwidth

17 The parameters of the theory H total = H 0 + E + xb parameter h d b h T d T ɛ Γ significance environment mean level spacing environment bandwidth environment temperature environment heat capacity system energy strength of coupling d T = ( ) dt = heat capacity d de Assuming that x performs motion with amplitude A and velocity V, then Γ is related to (σa) 2 and (σv ) 2. Γ = minimum ( ) σ 2, ( hσ A V 2 ) 2/3

18 The parameter Γ H nm = E n δ nm + xb nm = mean level spacing b = bandwidth B nm σ for E n E m < b Assume a small constant perturbation x = δx Γ(δx) ( σδx ) 2 Γ/ is the number of levels that are mixed non-perturbatively, as implied by perturbation theory (to infinite order). Re-write the Hamiltonian in the adiabatic basis: H nm = E n δ nm + ẋ i hb nm E n E m Assume a slow variation ẋ Γ(ẋ) ( hσẋ 2 ) 2/3

19 Parameters: The thermodynamic Limit T, d T, h d, b h, ɛ, Γ Mathematical definition of the Thermodynamic Limit: d (infinitely many degrees of freedom). Physical definition of the Thermodynamic Limit: ɛ/d T T (having well defined temperature). In case of two level (spin) system: Ω d T T In case of d 0 dimensional system: d 0 d T

20 Parameters: The High Temperature Limit T, d T, h d, b h, ɛ, Γ Mathematical definition of the high temperature limit: T (vanishing friction effect in this limit) Note: high temperature does not imply a lot of noise! Physical definition of the high temperature limit: hω/t for the physically relvevnat frequencies Sufficient condition: T b In case of the Spin-Boson model: T Ω, Γ K = 6π ( Γ T ) = Kondo Parameter

21 The Semiclassical Limit Parameters: T, d T, h d, b h, ɛ, Γ Mathematical definition of the semiclassical limit: h 0 (scaled planck constant ). Physical definition of the semiclassical limit: Γ > b (the non-perturbative regime) This should be contrasted with: Γ b (Fermi golden rule regime) Note that the adiabatic / standard-prt regime is : Γ < (Born-Oppenheimer regime)

22 Feynman Vernon formalism K(x t, x t x 0, x 0) = x,x F [x, x ] e i(a[x] A[x ]) F [x A, x B ] = ψ (E) U[xB ] U[x A ] ψ (E) = U[x B ] ψ (E) U[xA ] ψ (E) U[x] = T exp ( ī h (E + x(t)b) ) Given a driving scheme we define P(t) = F [x A, x B ] 2

23 FV picture and dephasing Using the FV expression [ P(t) = exp t t ] h 2 C(t t )(x A (t ) x B (t )) 2 dt dt 0 0 with x = ±v and the symmetrized C(ω) = 2πσ 2 δ(ω) + 2π hσ2 R ( hω ) G ( hω b ) then we get: P(t) = exp( 4C(0)(vt/ h) 2 ) P(t) = exp( 2γt) requires Γ b P(t) = exp( (σvt/ h) 2 ) with: ( γ = 2 v h ) 2 C(ω 0) 2 which is identified as the /T rate Γ h

24 The adiabatic picture d dt ψ = ī H(x(t)) ψ h ψ = n a n (t) n(x(t)) da n dt = ī h E na n + ī h m j ẋ j A j nma m A j nm(x) = i h n(x) ( A j i h H nm(x) = E m E n x j m(x) x j ) nm for n m da n dt = ī h (E n ẋa n )a n ī h m W nm a m W nm j ẋ j A j nm for n m, else zero

25 Linear Response Theory (Kubo) H = H(r, p; x (t), x 2 (t), x 3 (t)) F k = H x k Generalized Ohm law: F k = j G kj ẋ j K kj (τ) = (i/ h) [F k (τ), F j (0)] C kj (τ) = 2 ( F k (τ)f j (0) + cc) G kj = lim ω 0 Im[χ kj (ω)] ω = 0 K kj E (τ)τdτ = g(e) d de g(e) 0 C kj E (τ)dτ

26 The Born-Oppenheimer picture H total = 2M j p 2 j + H(Q, P ; x) basis: x, n(x) = x n(x) Ψ = n,x Ψ n (x) x, n(x) x, n(x) H x 0, m(x 0 ) = δ(x x 0 ) δ nm E n (x) x, n(x) p j x 0, m(x 0 ) = = ( i j δ(x x 0 )) n(x) m(x 0 ) = i j δ(x x 0 )δ nm δ(x x 0 )A j nm(x) hence: p j i j A j nm(x). H total = 2M j (p j A j (x)) 2 + E(x) H interaction j ẋ j A j (x)