Introduction – Ehrenfest Time in Scattering Problems Robert S Whitney

Institut Laue Langevin, Grenoble, France.

Banff, February 2008 Introduction – Ehrenfest time in scattering problems Robert S Whitney

CLOSED RMT

1 10 system size, L wavelength,λ RMT ‘CLASSICAL’ OPEN τ << ττ >> τ E DDE

Collaborations & Discussions: P. Jacquod, C. Petitjean, P. Brouwer, E. Sukhorukov When is quantum = classical ?

QUANTUMEhrenfest theorem CLASSICAL Liouville theorem d ρˆ = −i¯h[Hˆ, ρˆ] d f(r, p)=−{H,f(r, p)} dt dt phase−space commutator Moyal bracket Poisson bracket Wigner function

minimally dispersed W wavepacket 1/2 1/2 L deBoglie L,W L λ hyperbolic stretch non−hyperbolic stretch Quantum = Classical & fold Quantum = Classical Ehrenfest time (log time)

Ehrenfest Time, τE: time during which 1/2 1/2 L,W = L λ deBoglie quantum classical

τ =2Λ−1 ln[L/λ ] hyperbolic stretch non−hyperbolic stretch E deBroglie & fold Quantum = Classical 2 Quantum = Classical with L=L, (W /L) WAVEPACKET growth ∝ exp[−Λt]

CLOSED SYSTEM – classical limit TIMESCALES: Form factor GUE − classical limit −1 ♣ (Level-spacing) ∝ (L/λdeBroglie) GOE ♣ ∝ L/λ OPEN SYSTEM Ehrenfest time ln[ deBroglie] Ehrenfest time "All" particles escape ♣ Classical scales: beforeh/∆ t

dwell time = λdeBroglie-independent ‘CLASSICAL’ Brouwer−Rahav−Tian A short and inaccurate history of the Ehrenfest time! LAST CENTURY

Ehrenfest: Ehrenfest theorem. = Liouville theorem Z. Physik 45 455 (1927)

??? Larkin-Ovchinnikov: superconducting ﬂuct. in metals Sov. Phys. JETP 28 1200 (1969)

Berman-Zaslavsky: stochasticity Physica A 91, 450 (1978), Phys. Rep. 80, 157 (1981)

Aleiner-Larkin weak-loc. in transport & level-statistics PRB 54, 14423 (1996); 55, R1243 (1997).

THIS CENTURY Semiclassics ... see other talks this week.

♣ Cut-off on loops Sieber, M¨uller-Braun-Heusler-Haake

♣ Off-diagonal (Noise, weak-loc., UCFs, etc) suppressed (or not) Rahav-Brouwer & Jacquod-Whitney

Weak localization and back−scattering Conductance fluctuations

W L W W 1/2 1/2 L λdeBroglie W W

Fano factor of shot noise not diag. approx W W W W Open systems – scattering matrix

cf. scattering from atom or node in quantum graph rt S = Scattering matrix, t r

t†t → Landauer-B¨uttiker: eigenvalues of all transport properties ♣ ∝ T Electrical conductance n n ♣ quantum noise ∝ n Tn(1 − Tn) & higher moments

Quantum noise : measures determinism Bands in classical phase-space Silvestrov-Goorden-Beenakker (2003) Tworzydlo-Tajic-Schomerus-Beenakker (2003) Whitney-Jacquod (2005) Open systems: Paths in families (bands)

cf. integrable systems

exp[−Λ t ] t ~ number bounces

Wirtz−Tang−Burgdorfer PRB (1999)

Paths associated with 4 band corners dynamics relative to black path Phase-space basis : orthogonal basis of wavepackets exp[−Λ t ] Cover: bands with area > 2π¯h ⇒ t<τ 1/2 paths with E ~λdeBroglie

unchanged under Basis states Fourier transform

  ♠ 00··· 00  0 ♠ 0 ··· 00    . .  ⇐=  00.. . .  CLASSICAL block: diagonal t =  . . .   .  | eigenvalues | = 0,1  . ♠ 00  00··· 0 ♦♦ ⇐= QUANTUM block: not diagonal 00··· 0 ♦♦ semiclassics shows non-RMT.

† Eigenvalues of t t =0, 1 ⇒ Tn(1−Tn)=0 ⇒ Noise-less Modes Conclusion Open systems: Ehrenfest time controls cross-over to NEW universal regime (non-RMT)asλdeBroglie → 0.

Open questions [A] Ehrenfest time relevant for closed systems? • need classical timescale; i.e. inter-billiard transfer time [B] Re-summing loop expansion for ﬁnite Ehrenfest time?

−1 [C] Negative Ehrenfest time τE =Λ ln[L/λdeBroglie] ?? ... for noise L = W 2/L → 0 for W L.

[D] Ehrenfest time in graphs? or • require correlated actions? Typical Ehrenfest times Zurek (2003)

QUANTUM CLASSICAL

L∼λ deBroglie Quantum dot Double pendulum −1 τE = Λ ln[1] ∼ 0 Marcus Group, Harvard (2003) www.chaoticpendulums.com 3 L∼10 λdeBroglie −1 3 τE = Λ ln[10 ] ∼ 7τ0 34 L∼10 λdeBroglie −1 34 τE = Λ ln[10 ] ∼ 80τ0 Schrodinger cat state after 2 minutes !! Decoherence??