Quantum symbolic dynamics St´ ephane Nonnenmacher Institut de Physique Th´ eorique, Saclay Quantum chaos: routes to RMT and beyond Banff, 26 Feb. 2008
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 .?
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.
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 , � ′ → 0 f ( ψ � ′ ) = µ ( f ) lim 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.
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 Λ t ) 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 .
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 Λ t ) 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 dµ L | 2 h → 0 � N − 1 ∀ f, | f ( ψ � ,n ) − − − − → 0 . n =1 Quantum ergodicity [ Shnirelman’74, Zelditch’87, Colin de Verdi` ere’85 ]
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 Λ t ) 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 dµ L | 2 h → 0 � N − 1 ∀ f, | f ( ψ � ,n ) − − − − → 0 . n =1 Quantum ergodicity [ Shnirelman’74, Zelditch’87, Colin de Verdi` ere’85 ] → do ALL eigenstates become equidistributed [ Rudnick-Sarnak’93 ]? Or are there exceptional sequences of eigenstates?
Some counter-examples No exceptional sequences for arithmetic eigenstates of (2D) cat maps [ Rudnick- Sarnak’00 ] and for arithmetic surfaces [ Lindenstrauss’06 ].
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- evre ] and Walsh-quantized baker’s map [ Anantharaman-N’06 ] (cf. Kelmer’s talk). DeBi` B
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- evre ] and Walsh-quantized baker’s map [ Anantharaman-N’06 ] (cf. Kelmer’s talk). DeBi` B → in general, can ANY invariant measures occur as a semiclassical measure? In particular, can one have strong scars µ sc = δ P O ?
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 · · · � − 1 � 0 � 1 · · · · · · denoting its “history”. t=0 2 ε i 1 [ ] −1 ε Φ [ ] i −2 ε Φ [ ] i t=1 t=2 At each time n , the rectangle [ � 0 · · · � 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 ) = − µ ([ � 0 · · · � n ]) log µ ([ � 0 · · · � n ]) � 0 ,...,� n measures the distribution of the weights µ ([ � 0 · · · � n ]) .
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. log J u dµ L . � • Related to localization: H KS ( δ P O ) = 0 , H KS ( µ L ) = • Affine function of µ .
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. log J u dµ L . � • Related to localization: H KS ( δ P O ) = 0 , H KS ( µ L ) = • Affine function of µ . t=0 1 2 t=1 t=2 What can be the entropy of a semiclassical measure for an Anosov system?
Semiclassical measures are at least “half-delocalized” Theorem [ Anantharaman-Koch-N’07 ]: For any quantized Anosov system, any semiclas- sical measure µ satisfies � log J u dµ − 1 H KS ( µ ) ≥ 2Λ max ( d − 1) � “full scars” are forbidden. Some of the exceptional measures saturate this lower bound. p q
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 � 0 ··· � n = U − n ˜ def = U − n P � n U · · · P � 1 UP � 0 P � 0 ··· � n is a quasi-projector on the rectangle [ � 0 · · · � n ] . t=0 1 2 t=1 t=2 Can we get some information on the distribution of the weights � P � 0 ··· � n ψ � � 2 h → 0 − − − → µ ([ � 0 · · · � n ]) ?
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 � 0 ··· � n = U − n ˜ def = U − n P � n U · · · P � 1 UP � 0 P � 0 ··· � n is a quasi-projector on the rectangle [ � 0 · · · � n ] . t=0 1 2 t=1 t=2 Can we get some information on the distribution of the weights � P � 0 ··· � n ψ � � 2 h → 0 − − − → µ ([ � 0 · · · � n ]) ? YES, provided we consider times n > T E (for which the quasi-projector interpretation breaks down).
Evolution of “adapted elementary states” To estimate P � 0 ··· � n ψ � , we decompose ψ � in a well-chosen family of states ψ Λ , and compute each P � 0 ··· � n ψ Λ separately. The Anosov dynamics is anisotropic (stable/unstable foliations). p 0.5 Λ 0 q −0.5 −0.5 0 0.5 ⇒ 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 )) }
Φ Λ η Λ η Φ( ) Through the sequence of stretching-and-cutting, the state ˜ P � 0 ··· � n ψ Λ remains a nice Lagrangian state up to large times ( n ≈ C T E for any C > 1 ).
Φ Λ η Λ η Φ( ) Through the sequence of stretching-and-cutting, the state ˜ P � 0 ··· � n ψ Λ remains a nice Lagrangian state up to large times ( n ≈ C T E for any C > 1 ). • Summing over all [ � 0 · · · � n ] we recover U n ψ Λ : no breakdown of the semiclassical evolution at T E [ Heller-Tomsovic’91 ].
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