Quantum symbolic dynamics

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1 Quantum symbolic dynamics Stéphane Nonnenmacher Institut de Physique Théorique, Saclay Quantum chaos: routes to RMT and beyond Banff, 26 Feb. 2008

2 What do we know about chaotic eigenstates? Hamiltonian H(q, p), such that the dynamics on Σ E is chaotic. H = Op (H) has discrete spectrum (E,n, ψ,n ) near the energy E. Laplace operator on a chaotic cavity, or on a surface of negative curvature. What do the eigenstates ψ look like in the semiclassical limit 1.?

3 Chaotic quantum maps chaotic map Φ on a compact phase space propagators U N (Φ), N 1 (advantage: easy numerics, some models are partially solvable ). Husimi densities of some eigenstates.

4 Phase space localization Interesting to study the localization of ψ in both position and momentum: phase space description. Ex: for any bounded test function (observable) f(q, p), study the matrix elements f(ψ ) = ψ, Op (f) ψ = dqdp f(q, p) ρ ψ (q, p) Depending on the quantization, the function ρ ψ can be the Wigner function, the Husimi function. Def: from any sequence (ψ ) 0, one can always extract a subsequence (ψ ) such that for any f, lim f(ψ ) = µ(f) 0 µ is a measure on phase space, called the semiclassical measure of the sequence (ψ ). µ takes the macroscopic features of ρ ψ into account. Fine details (e.g. oscillations, correlations, nodal lines) have disappeared.

5 Quantum-classical correspondence For the eigenstates of H, µ is supported on Σ E. Let us use the flow Φ t generated by H, and call U t = e ith / the quantum propagator. Egorov s theorem: for any observable f, U t Op (f) U t = Op (f Φ t ) + O( e Λt ) Idem for a quantum map U = U N (Φ). Breaks down at the Ehrenfest time T E = log /Λ (cf. R.Whitney s talk). the semiclassical measure µ is thus invariant through the classical dynamics.

6 Quantum-classical correspondence For the eigenstates of H, µ is supported on Σ E. Let us use the flow Φ t generated by H, and call U t = e ith / the quantum propagator. Egorov s theorem: for any observable f, U t Op (f) U t = Op (f Φ t ) + O( e Λt ) Idem for a quantum map U = U N (Φ). Breaks down at the Ehrenfest time T E = log /Λ (cf. R.Whitney s talk). the semiclassical measure µ is thus invariant through the classical dynamics. If Φ is ergodic w.r.to the Liouville measure, one can show (again, using Egorov) that almost all eigenstates ψ,n become equidistributed when : N f, N 1 f(ψ,n ) f dµ L 2 h 0 0. n=1 Quantum ergodicity [Shnirelman 74, Zelditch 87, Colin de Verdière 85]

7 Quantum-classical correspondence For the eigenstates of H, µ is supported on Σ E. Let us use the flow Φ t generated by H, and call U t = e ith / the quantum propagator. Egorov s theorem: for any observable f, U t Op (f) U t = Op (f Φ t ) + O( e Λt ) Idem for a quantum map U = U N (Φ). Breaks down at the Ehrenfest time T E = log /Λ (cf. R.Whitney s talk). the semiclassical measure µ is thus invariant through the classical dynamics. If Φ is ergodic w.r.to the Liouville measure, one can show (again, using Egorov) that almost all eigenstates ψ,n become equidistributed when : f, N 1 N f(ψ,n ) f dµ L 2 h 0 0. n=1 Quantum ergodicity [Shnirelman 74, Zelditch 87, Colin de Verdière 85] do ALL eigenstates become equidistributed [Rudnick-Sarnak 93]? exceptional sequences of eigenstates? Or are there

8 Some counter-examples No exceptional sequences for arithmetic eigenstates of (2D) cat maps [Rudnick- Sarnak 00] and for arithmetic surfaces [Lindenstrauss 06].

9 Some counter-examples No exceptional sequences for arithmetic eigenstates of (2D) cat maps [Rudnick- Sarnak 00] and for arithmetic surfaces [Lindenstrauss 06]. explicit exceptional semiclassical measures for the quantum cat map [Faure-N- DeBièvre] and Walsh-quantized baker s map [Anantharaman-N 06] (cf. Kelmer s talk). B

explicit exceptional semiclassical measures for the quantum cat map [Faure-N- DeBièvre] and Walsh-quantized

10 Some counter-examples No exceptional sequences for arithmetic eigenstates of (2D) cat maps [Rudnick- Sarnak 00] and for arithmetic surfaces [Lindenstrauss 06]. explicit exceptional semiclassical measures for the quantum cat map [Faure-N- DeBièvre] and Walsh-quantized baker s map [Anantharaman-N 06] (cf. Kelmer s talk). B in general, can ANY invariant measures occur as a semiclassical measure? In particular, can one have strong scars µ sc = δ P O?

11 Symbolic dynamics of the classical flow To classify Φ-invariant measures, one may use a phase space partition; each trajectory will be represented by a symbolic sequence ɛ 1 ɛ 0 ɛ 1 denoting its history. 1 t=0 2 [ ε i ] Φ 1 [ ε ] i Φ 2 [ ] εi t=2 t=1 At each time n, the rectangle [ɛ 0 ɛ n ] contains all points sharing the same history between times 0 and n (ex: [121]). Let µ be an invariant proba. measure. The time-n entropy H n (µ) = ɛ 0,...,ɛ n µ([ɛ 0 ɛ n ]) log µ([ɛ 0 ɛ n ]) measures the distribution of the weights µ([ɛ 0 ɛ n ]).

12 KS entropy of semiclassical measures The Kolmogorov-Sinai entropy H KS (µ) = lim n n 1 H n (µ) represents the information complexity of µ w.r.to the flow. Related to localization: H KS (δ P O ) = 0, H KS (µ L ) = log J u dµ L. Affine function of µ.

13 KS entropy of semiclassical measures The Kolmogorov-Sinai entropy H KS (µ) = lim n n 1 H n (µ) represents the information complexity of µ w.r.to the flow. Related to localization: H KS (δ P O ) = 0, H KS (µ L ) = log J u dµ L. Affine function of µ. t=0 1 2 t=2 t=1 What can be the entropy of a semiclassical measure for an Anosov system?

14 Semiclassical measures are at least half-delocalized Theorem [Anantharaman-Koch-N 07]: For any quantized Anosov system, any semiclassical measure µ satisfies H KS (µ) log J u dµ 1 2 Λ max(d 1) full scars are forbidden. bound. Some of the exceptional measures saturate this lower p q

15 Quantum partition of unity Using quasi-projectors P j = Op (χ j ) on the components of the partition, we construct a quantum partition of unity Id = J j=1 P j. Egorov thm for n < T E the operator P ɛ0 ɛ n = U n Pɛ0 ɛ n def = U n P ɛn U P ɛ1 UP ɛ0 is a quasi-projector on the rectangle [ɛ 0 ɛ n ]. t=0 1 2 t=2 t=1 Can we get some information on the distribution of the weights P ɛ0 ɛ n ψ 2 h 0 µ([ɛ 0 ɛ n ])?

16 Quantum partition of unity Using quasi-projectors P j = Op (χ j ) on the components of the partition, we construct a quantum partition of unity Id = J j=1 P j. Egorov thm for n < T E the operator P ɛ0 ɛ n = U n Pɛ0 ɛ n def = U n P ɛn U P ɛ1 UP ɛ0 is a quasi-projector on the rectangle [ɛ 0 ɛ n ]. t=0 1 2 t=2 t=1 Can we get some information on the distribution of the weights P ɛ0 ɛ n ψ 2 h 0 µ([ɛ 0 ɛ n ])? YES, provided we consider times n > T E (for which the quasi-projector interpretation breaks down).

17 Evolution of adapted elementary states To estimate P ɛ0 ɛ n ψ, we decompose ψ in a well-chosen family of states ψ Λ, and compute each P ɛ0 ɛ n ψ Λ separately. The Anosov dynamics is anisotropic (stable/unstable foliations). 0.5 p Λ 0 q use states adapted to these foliations We consider Lagrangian states associated with Lagrangian manifolds close to the unstable foliation: ψ Λ (q) = a(q) e is Λ(q)/ is localized on Λ = {(q, p = S Λ (q))}

18 Λ η Φ Φ( Λ η ) Through the sequence of stretching-and-cutting, the state P ɛ0 ɛ n ψ Λ remains a nice Lagrangian state up to large times (n C T E for any C > 1).

19 Λ η Φ Φ( Λ η ) Through the sequence of stretching-and-cutting, the state P ɛ0 ɛ n ψ Λ remains a nice Lagrangian state up to large times (n C T E for any C > 1). Summing over all [ɛ 0 ɛ n ] we recover U n ψ Λ : no breakdown of the semiclassical evolution at T E [Heller-Tomsovic 91].

20 Λ η Φ Φ( Λ η ) Through the sequence of stretching-and-cutting, the state P ɛ0 ɛ n ψ Λ remains a nice Lagrangian state up to large times (n C T E for any C > 1). Summing over all [ɛ 0 ɛ n ] we recover U n ψ Λ : no breakdown of the semiclassical evolution at T E [Heller-Tomsovic 91]. The amplitude of Pɛ0 ɛ n ψ Λ is governed by the unstable Jacobian along the path ɛ 0 ɛ t : P ɛ0 ɛ n ψ Λ J n u (ɛ 0 ɛ n ) 1/2 e Λn/2 From the decomposition ψ = 1/h η=1 c η ψ Λη, one obtains the bound [Anantharaman 06] P ɛ0 ɛ n h 1/2 J n u (ɛ 0 ɛ n ) 1/2 h 1/2 e Λn/2. This hyperbolic estimate is nontrivial for times t > T E (no more a quasi-projector).

21 What to do with this estimate? [Anantharaman-N 06 07]: rewrite the operator at time 2n as U n P ɛ0 ɛ 2n = P ɛn+1 ɛ 2n U n P ɛ0 ɛ n This can be seen as a block matrix element of the unitary propagator U n, expressed in the block-basis {P ɛ0 ɛ n }. Setting n = T E, the hyperbolic estimate at time 2n states that these block matrix elements are all 1/2. An entropic uncertainty principle then implies that the entropy constructed from the weights P ɛ0 ɛ n ψ 2 satisfies H n (ψ ) log 1/2 = n Λ 2. From this bound at n = T E, one uses subadditivity and Egorov to get a similar bound at finite time n, and then the bound for H KS (µ).

22 Where else to use this hyperbolic estimate? semiclassical resonance spectra of chaotic scattering systems. 0 E c E E+c gh Discrete model: open quantum baker B 8 subunitary propagator B N = U N Π N.

23 Gap in the resonance spectrum Use a quantum partition outside the hole: Π N = j P j. (B N ) n = Pɛ0 ɛ n 1 ɛ 0 ɛ n 1 = (B N ) n P ɛ0 ɛ n 1 ɛ 0 ɛ n 1 ɛ 0 ɛ n 1 h 1/2 J n u (ɛ 0 ɛ n 1 ) 1/2 For n T E, the RHS is approximately given by the topological pressure P( log J u /2) associated with the classical trapped set: (B N ) n exp ( n P( log J u /2) ) If P( log J u /2) < 0 ( thin trapped set ), this gives an upper bound on the quantum lifetimes [Ikawa 88,Gaspard-Rice 89,N-Zworski 07].

24 Perspectives To obtain nontrivial information on eigenstates, it was crucial to analyze the dynamics beyond the Ehrenfest time. The partition allows to control the evolution of Lagrangian states (also wavepackets) [Heller-Tomsovic 91,Schubert 08] The decomposition into P ɛ0 ɛ n could also be useful to: analyze the phase space structure of resonant states [Keating-Novaes-Prado- Sieber 06, N-Rubin 06] expand the validity of the Gutzwiller trace formula to times n T E [Faure 06] show that (some) resonances are close to the zeros of the Gutzwiller-Voros Zeta function analyze transport through chaotic cavities

Quantum decay rates in chaotic scattering

Quantum decay rates in chaotic scattering S. Nonnenmacher (Saclay) + M. Zworski (Berkeley) National AMS Meeting, New Orleans, January 2007 A resonant state for the partially open stadium billiard, computed