Collaborations: Tsampikos Kottos (Gottingen) Antonio Mendez Bermudez (Gottingen) Holger Schanz (Gottingen)

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1 Quantum chaos and perturbation theory: from the analysis of wavefunctions to the implications on pumping, dissipation and decoherence Doron Cohen, Ben-Gurion University Collaborations: Tsampikos Kottos (Gottingen) Antonio Mendez Bermudez (Gottingen) Holger Schanz (Gottingen) Discussions: Yshai Avishai (BGU) Markus Buttiker (Geneva) Thomas Dittrich (Colombia) Shmuel Fishman (Technion) Uzy Smilansky (Weizmann) dcohen cond-mat archive $GIF, $ISF

2 The subject of this talk is Γ Γ is an energy scale (inspired by Wigner). Γ is used as a measure for the size of the perturbation. Γ is used to distingushed between prt and non-prt regimes. Regimes in the theory of quantum dynamics [1] Regimes in LDOS / wavepacket dynamics [1,2,3] Regimes in the study of fidelity [4] Regimes in the theory of driven systems [1,5] The role of Γ in quantum pumping [6,7] Regimes in the theory of quantum dissipation [8] [1] DC, PRL 1999, Annals 2000 [2] DC + Heller, PRL 2000 [3] Mentez + Kottos + Cohen 2004 [4] Jacquod + Adagdeli + Beenakker, PRL 2002 [5] DC + Kottos, PRL 2000 etc. [6] DC, PRB 2003, PRB(R) 2003 [7] DC + Kottos + Holger 2004 [8] DC + Kottos, PRE(R) 2004

3 The Hamiltonian H = H(Q, P ; x) x = control parameter The system is chaotic case [1] - LDOS / wavepacket dynamics: H 0 = H(Q, P ; x 0 ) H = H(Q, P ; x) The Hamiltonians H 0 and H are equaly chaotic. The size of the perturbation: δx = x x 0 case [2] - Driven chaotic(!) system: Linear driving: Periodic driving: x(t) = x 0 + V t x(t) = A sin(ωt) The size of the perturbation: ẋ

4 The parameter Γ - case [1] H nm = E n δ nm + δxb nm = mean level spacing b = bandwidth B nm σ for E n E m < b Assume a small constant perturbation δx Γ(δx) ( σδx ) 2 [Wigner] Γ/ is the number of levels that are mixed non-perturbatively, as implied by perturbation theory (to infinite order). Γ < 1st order perturbation theory Γ < b perturbative (Wigner/FGR) regime else non-perturbative (semiclassical?) regime

5 The parameter Γ - case [2] 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 Γ < adiabatic regime Γ < b perturbative (FGR/Kubo) regime else non-perturbative (semiclassical?) regime Wilkinson and Austin, JPA An attempt to derive / challange the Qchaos Kubo DC, PRL 1999, Annals Derivation of the Qchaos Kubo with the condition Γ < b Wilkinson, JPA Alternate (semiclassical) derivation of the condition

6 case [3] - Interaction with environment H total = H 0 (x, p) + H(Q, P ; x) After linearization (say x 0 = 0) H total = H 0 (x, p) + E + xb interaction with bath (harmonic oscillators) interaction with chaos (few freedoms environment) interaction with RMT modeled environment Assume that x performs motion with amplitude A and velocity V. Γ is related to (σa) 2 and (σv ) 2 as follows: Γ = minimum ( ) σ 2, ( hσ A V 2 ) 2/3 Γ < Born-Oppenheimer regime Γ < b F-V regime (effective bath picture) else non-perturbative (semiclassical?) regime For a two level system K = 1 16π ( Γ T ) = Kondo Parameter

7 Driven Systems Non interacting spinless electrons. Held by a potential (e.g. AB ring geometry). x 1, x 2 = shape parameters x 3 = Φ = ( h/e)φ = magnetic flux Flux Ring wire Flux dot wire Flux dot

8 Ohm law For one parameter driving by EMF I = G 33 ( x 3 ) dq = G 33 dx 3 For driving by changing another parameter I = G 31 x 1 dq = G 31 dx 1 For two parameter driving I = G 31 x 1 G 32 x 2 dq = G 31 dx 1 G 32 dx 2 Q = and in general G dx F k = j G kj ẋ j

9 What is the problem? From Kubo formula we get a formal expression for G kj. Can we trust this expression? Conditions? Quantum chaos! How to use this expression? The bare Kubo formula gives no dissipation! To define an energy scale Γ Beyond first order perturbation theory! Γ in case of isolated system is due to non-adiabaticity. Γ affects both the dissipative and the non-dissipative (geometric) part of the response.

10 From Kubo to a FD relation H = H(r, p; x 1 (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) = ī h [F k (τ), F j (0)] C kj (τ) = 1 2 ( F k (τ)f j (0) + cc ) G kj = lim ω 0 Im[χ kj (ω)] ω = 0 K kj F (τ)τdτ = g(e F ) 0 C kj E F (τ)dτ DC, PRB(R) 2003

11 BPT versus Geometric magnetism G kj = g(e F ) 0 C kj E F (τ)dτ Due to non-adiabaticity C kj E (τ) C kj E (τ) e (Γ/ h)t Γ = ( ) 2/3 ( ) hσ 2/3 ẋ 1 2 L ẋ L For x 3 = 0 one obtains G kj = B kj where B kj = 2 h n f(e n ) m( n) Im [ FnmF k mn j (E m E n ) 2 + (Γ/2) 2 ] For dot-wire system in the L limit we have Γ 0 leading to G 3j = e 2πi trace ( P A S x j S ) [BPT]

12 Simple model systems - networks dq = (1 g 0 ) e π k F dx dq = 1 e π k F dx dq = [ gt 1 g T ] [ 1 g0 g 0 ] e π k F dx Adiabatic versus non-adiabatic result DC, Kottos, Schanz, cond-mat 2004

13 Speculations... E n wire states 1/C1 1/C2 dot state E n (x(t)) dot level position time For the open system, using BPT: Q 1 g 0 Is it due to non-adiabaticity? Is it a dissipative contribution? What about strict adiabatic cycle in a closed system? Shutenko, Aleiner, Altshuler (2000): If the system were closed [and strictly adiabatic], the charge distribution after each period of the perturbation would return to the original distribution, and therefore, the pumped charge would be exactly quantized. Not correct!

14 B = (B 12, B 23, B 31 ) The B field B kj = n f(e n )B kj n B kj n = 2 h m( n) Im [ F k nmf j mn (E m E n ) 2 + (Γ/2) 2 ] This field is divergenceless (for Γ = 0) [geometric magnetism] A chain of degeneracies: ( x (0) 1, x (0) 2, Φ (0) + 2π ē ) h integer The degeneracies are like Dirac monopoles - The issue of bandwidth - The effect of screening

15 The two barrier model E n wire states 1/C1 1/C2 dot state E n (x(t)) dot level position time X 1 = 1 2 (C 1 C 2 ) = left-right bias X 2 = 1 2 (C 1 + C 2 ) = dot potential floor X2 C1,C2>0 X1 X3 X2 X1