Program on « QUANTUM DYNAMICS OUT OF EQUILIBRIUM » Institut Henri Poincaré 12 November - 14 December 2007 Pierre GASPARD Center for Nonlinear Phenomena and Complex Systems, Université Libre de Bruxelles, Brussels, Belgium 1)

SEMICLASSICS AND PERIODIC-ORBIT QUANTIZATION OF CHAOTIC SCATTERING

2)

SLOWING DOWN OF QUANTUM DECAYS IN CLASSICALLY CHAOTIC SCATTERING

3)

DECAY OF QUANTUM STATISTICAL MIXTURES IN CLASSICALLY CHAOTIC SCATTERING

4)

NONEQUILIBRIUM TRANSIENTS AND TRANSPORT IN LARGE QUANTUM SYSTEMS

DECAY OF QUANTUM STATISTICAL MIXTURES IN CLASSICALLY CHAOTIC SCATTERING Pierre GASPARD Brussels, Belgium D. Alonso, Tenerife F. Barra, Chili I. Burghardt, Paris S. A. Rice, Chicago • QUANTUM DYNAMICS OF STATISTICAL MIXTURES • SEMICLASSICS • POLLICOTT-RUELLE RESONANCES

PURE STATES VERSUS STATISTICAL MIXTURES CLASSICAL Newtonian or Hamiltonian eqs. Hamiltonian function PURE STATES:

 ∂H q˙ = + ∂p  p˙ = − ∂H  ∂q probability density

€

€ STATISTICAL MIXTURES:

€ €

p = p(q,p)

QUANTUM Schrödinger eq. Hamiltonian operator

1 ˆ ∂t ψ = H ψ ih density matrix

ρˆ = ∑ ψ j P j ψ j j

∂t p = {H, p} ≡ Lˆ p € Liouville eq. Liouvillian operator €

∂t ρˆ =

1 ˆ ˆ [ H, ρˆ ] ≡ Lˆ ρˆ ih

Landau-von Neumann eq. Liouvillian superoperator

PHOTODISSOCIATION photoabsorption cross-section (Heller 1978) +∞ ω − βhω ˆ (0) ⋅ D ˆ (t) κω = (1− e ) ∫ dt e−iωt D 6ε0 hc −∞

β

dipole-dipole autocorrelation function:

€

zero temperature

ˆ (0) ⋅ D ˆ (t) D

∞

ˆ ⋅ e iHˆ t / h D ˆϕ = e−iE 0 t / h ϕ 0 D 0

ˆϕ D 0

non-zero temperature

€

ˆ (0) ⋅ D ˆ (t) D

β

ˆ ⋅ e iHˆ t / h D ˆ e−iHˆ t / h = tr ρˆ β D

€

€ ϕ0

€

TIME AUTOCORRELATION FUNCTIONS Time autocorrelation functions are important in linear response theory and the theory of transport properties. cf. Heller formula for photoabsorption and Kubo formula for electric conductivity:

1 +∞ σ (ω ) = ∫ dt e−iωt 2 −∞

β

∫ dλ

Jˆ (0)Jˆ (t + ihλ)

0

ˆ ˆ Aˆ (0) Aˆ (t) = tr ρˆ Aˆ e iHt/h Aˆ e−iHt/h = ∑ c mn exp(iω mn t)

Bohr frequencies:

ω mn ≡

m,n

Em − En h

decomposition onto the energy shells: €

ˆA(0) Aˆ (t) = tr p( Hˆ ) Aˆ e iHˆ t/h Aˆ e−iHˆ t/h

€

= =

iHˆ t/h ˆ −iHˆ t/h ˆ ˆ dE p(E) tr δ (E − H ) A e Ae ∫

∫ dE p(E) C

E

€

t Heisenberg ≈ h n av (E)

(t)

iHˆ t/h ˆ −iHˆ t/h ˆ ˆ CE (t) ≡ tr δ (E − H ) A e A e

€

€

periodic-orbit expansion?

€

almost-periodic oscillations (quantum beats)

SEMICLASSICS OF TIME AUTOCORRELATION FUNCTIONS time autocorrelation function on an energy shell:

iHˆ t/h ˆ −iHˆ t/h ˆ ˆ CE (t) ≡ tr δ (E − H ) A e A e

using the method of Gutzwiller trace formula:

dqdp − f +1 δ E − H A (0)A (t) + O(h ) ( ) ∫ (2πh) f cl cl cl €  Sp π  cos r − r µ p  ∞ 2  1  h 0 + ∑∑ A ( τ )A ( τ + t) d τ + O(h ) ∫ cl cl 1/ 2 πh p r=1 det (M r − I) p

CE (t) =

p

classical time evolution of observables:

Acl (t) = exp(− Lˆ cl t) Acl (q,p)

€ with the classical Liouvillian operator: for each periodic orbit:

€

reduced action:

Lˆ cl ≡ {H, }Poisson Sp =

∫ p ⋅ dq

linearized Poincaré map: €

€

Maslov index:

Mp €

µp

SEMICLASSICS OF QUANTUM SURVIVAL PROBABILITY quantum survival probability: pure state:

2

ˆ

ˆ

∫ ψ(r,t) dr = ψ (t) χ D (rˆ ) ψ (t) = tr χ D (rˆ ) e−iHt/h ρˆ 0 e +iHt/h

P(t) =

D

χ D (r) statistical mixture:

€

indicator function of the domain D of the configuration space

P(t) = ∑ Pi P j ψ i (0) ψ j (t)

2

ˆ

ˆ

= tr ρˆ 0 e−iHt/h ρˆ 0 e +iHt/h

i, j

Semiclassical € approximation:

dqdp − Lˆcl t − f +1 ρ e χ + O(h ) ∫ (2πh) f 0,cl cl  Sp π  cos r − r µ p  ∞ 2  1  h − Lˆcl t 0 + dE ρ e χ d τ + O(h ) ∑ ∑ ∫ ∫ 0,cl cl 1/ 2 πh p p r=1 det (M rp − I)

P(t) =

€

classical Liouvillian operator: for each€ periodic orbit:

Lˆ cl ≡ {H, }Poisson

reduced action:

Sp =

∫ p ⋅ dq

linearized Poincaré map:

€

Mp

Maslov index:

µp

POLLICOTT-RUELLE CLASSICAL LIOUVILLIAN RESONANCES Pollicott-Ruelle resonances: eigenvalues of classical Liouvillian operator: Lˆ cl = {H cl ,

Lˆ cl f α ) = sα fα )

(gα Lˆ cl = sα (gα

leading Pollicott-Ruelle resonance: classical escape rate

€

(on each energy shell)

s0 =€−γ cl

quasi-classical term € of the survival probability:

P(t) ≈ =

∫

dqdp − Lˆcl t − f +1 ρ e χ + O(h ) 0,cl cl € f (2πh)

∫ dE ∑ (χ

cl

f α ,E ) exp(sα ,E t) ( gα ,E ρ 0,cl ) + O(h− f +1 )

α

≈

∫ dE a(E) exp(−γ

cl,E

t)

quasi-classical term of the time autocorrelation function on an energy shell:

€

CE (t) ∝ exp(−γ cl,E t)

}

CLASSICAL TRACE FORMULA Cvitanovic & Eckhardt (1991)

∫ dX δ(X − Φ X) = Tr t E

∞

E

ˆ

∫ dt e−st TrE e Lcl t = TrE 0

e

Lˆcl t

∞

= ∑ ∑ Tp p r=1

δ (t − rTp ) det (M rp − I)

1 d = ln Z cl (s;E) s − Lˆ cl ds

€ m +1

 exp[−sTp (E)]  Z cl (s;E) = ∏ ∏ 1−  m p m= 0   Λ p (E) Λ p (E)  linearized Poincaré map: M p period: ∞

classical Zeta function:

€

for each periodic orbit:

Λp

instability eigenvalue:

€

Pollicott-Ruelle resonances:

€

€

Λ p = exp( λ p Tp ) > 1

€ Z cl (sα ;E) = 0

€

Tp

det€(M p − ΛI) = 0 €

€

leading Pollicott-Ruelle resonance: classical escape rate

f =2

s0 = −γ cl

SPECTRAL CORRELATIONS S-matrix element:

Sij (E)

transition probability (cross-section):

σ ij (E) = Sij (E)

spectral autocorrelation function: C˜ E (ε) ≡ €

€

C˜ E (ε) =

2

σ ij (E − ε /2)σ ij (E + ε /2)

E

+∞ iε t / h C (t) e dt ∫ E −∞

€

CE (t) ∝ exp(−γ cl,E t )

€

C˜ E (ε) ∝

Blümel & Smilansky (1988):

1 ε 2 + (hγ cl,E ) 2

Lorentzian of width given by the classical escape rate

cf. regime of Ericson fluctuations at (relatively) high energies in neutron-nuclei scattering Agam (2000):

€ Pollicott-Ruelle resonances:

sα,E = −γ α ,E ± iωα ,E

CE (t) ≈ ∑ aα exp(−γ α,E t ) cos ωα,E t α

C˜ E (ε) ≈ ∑ α

a˜α

(ε − hωα,E )

2

+ (hγ α ,E )

2

+ (ε → −ε)

POLLICOTT-RUELLE CLASSICAL LIOUVILLIAN RESONANCES quasi-monoenergetic excitation: decay controled by Hamiltonian resonance

P(t) ∝ exp(−Γr t /h)

r

hγ cl /2

Hˆ ϕ r = E rϕ r = (εr − iΓr /2)ϕ r

€ €

broad excitation: early decay controled

€ ∝ exp(−γ cl t)

by classical Liouvillian resonance, so-called Pollicott-Ruelle resonance, here equal to the classical escape rate

P(t) ∝ exp(−γ cl t) Lˆ cl f ) = γ cl f )

∝ exp(−Γr t /h) €

€

CLASSICAL FOUR-DISK SCATTERER

classical survival probability

€

QUANTUM FINGERPRINTS OF CLASSICAL POLLICOTT-RUELLE RESONANCES W. Lu, M. Rose, K. Pance, & S. Sridhar, Phys. Rev. Lett. 82 (1999) 5233 K. Pance, W. Lu, & S. Sridhar, Phys. Rev. Lett. 85 (2000) 2737 classical escape rate

wave numbers k, κ

C˜ (κ ) ∝

1 κ 2 + γ˜ cl 2

2 2 C˜ (κ ) ≡ S21 (k − κ /2) S21 (k + κ /2)

€ Pollicott-Ruelle resonances: positions = exp.: filled squares th.: open circles (bars = widths)

quantum escape rate

k

QUANTUM GRAPHS Kottos & Smilansky (1997) 1D Schrödinger equation on each bond b:

closed graphs

d  2  2 + k ψ b (x) = 0  dx  2

S-matrix at each vertex i:

ψ aout (i) =

∑σ

i ab

ψ bin (i)

b

€

open graphs

Quantization condition:

Z(k) = det [I − R(k)] = 0 €

€ €

€

Dab (k) = δab e

ikl a

R(k) = T⋅ D(k)

Tab = σ

s a ab

δasbr

Zeta function:

 exp ikl − iπµ / 2  ( ) p p   Z(k) = ∏ 1− € 1/ 2  Λp p   In the corresponding classical dynamics, the particle is moving at constant velocity along the bonds and is scattered with transition probabilities at each vertex. The classical dynamics is thus random with a finite KS entropy because of ray splitting.

SCATTERING AND DECAY IN QUANTUM GRAPHS classically periodic repeller

i σ ab = (2/ν i ) − δab

€

classically chaotic repeller

SCATTERING AND DECAY IN EXTENDED QUANTUM GRAPHS η1 = 0.1 η2 = ( 5 −1) /2 la = 0.5 lb = 2

isin η cos η  σ =   cosη isin η

topological pressure €

€

quantum survival probability

UNIMOLECULAR DISSOCIATION RATES: STATISTICAL THEORY RRKM reaction rate:

ν (E) k (E) ≈ ≈ γ cl (E) 2πh n av (E)

number of open channels at energy E=Re E:

ν (E)

density of energy levels at E=Re E in the quasi-bounded domain, also the density of resonances at E=Re E:

€

n av (E)

interpreted as the leading Pollicott-Ruelle resonance, i.e., as the classical escape rate

€

€ weakly open potential:

strongly open potential:

CONCLUSIONS LIOUVILLIAN RESONANCES In the quasiclassical limit where many Hamiltonian resonances are excited, the early decay of the survival probability is controled by the classical escape rate which is the leading Pollicott-Ruelle resonance. The Pollicott-Ruelle resonances are the eigenvalues of the classical Liouvillian operator at complex frequencies:

E r − E r' srr' = ih

→ sα

h →0

In many-body quantum systems, these Liouvillian resonances are the eigenvalues of the master equation for the reduced density matrix in the long-time limit and € to obtain the reaction rates. they can be used