Ergodicity of quantum eigenfunctions in classically chaotic systems

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1 Ergodicity of quantum eigenfunctions in classically chaotic systems Mar 1, 24 Alex Barnett Courant Institute work in collaboration with Peter Sarnak, Courant/Princeton p.1

2 Classical billiards Point particle in 2D domain Ω, elastic reflections off boundary Γ. Position r (x, y). Phase space (r, θ). Energy is conserved. Type of motion depends on billiard table shape: 1.9 Regular: has other conserved quantities (e.g. θ) p.2

3 Classical billiards Point particle in 2D domain Ω, elastic reflections off boundary Γ. Position r (x, y). Phase space (r, θ). Energy is conserved. Type of motion depends on billiard table shape: 1.9 Regular: has other conserved quantities (e.g. θ) Chaotic: Ω Γ start ergodic: nearly every trajectory covers phase space Hyperbolicity: exponential divergence of nearby trajectories r 1 (t) r 2 (t) e Λt, Λ = Lyapunov Also: Anosov property (all Λ > ), mixing (phase space flow) p.2

Quantum billiards quantum just means wave Membrane (drum) problem: eigenfunctions φ n (r) of laplacian φ n = E n φ n, φ n (r Γ) = φ 2 ndr = 1 Ω 1 1 1 1.5.5.5.5.5 1 1.

4 Quantum billiards quantum just means wave Membrane (drum) problem: eigenfunctions φ n (r) of laplacian φ n = E n φ n, φ n (r Γ) = φ 2 ndr = 1 Ω energy eigenvalue E wavenumber k k E 2π λ cavity modes, quantum levels quantized equivalent of classical billiards (momentum i ) p.3

7, E 1 4, k 1, 15λ across 197 s to present day, field of QUANTUM CHAOS: eigenvalues (spacings,

5 Quantum chaos What happens at higher E? Depends on classical dynamics: Regular: φ n separable Chaotic: φ n disordered MOVIE low energy: n 7, E 1 4, k 1, 15λ across 197 s to present day, field of QUANTUM CHAOS: eigenvalues (spacings, correlations, RMT... ) eigenfunctions (ergodicity, correlations, matrix els... ) dynamics (scattering, resonances, dissipation, electron physics, q. chemistry... ) p.4

6 Classical and quantum averages Choose test function A(r): classical (phase space) average A 1 A(r)dr vol(ω) Ω quantum version is Â = operator in linear space of φ n s Expectation (average) φ n, Âφ n A(r) φ n (r) 2 dr Ω }{{} dµ φn density measure p.5

7 Classical and quantum averages Choose test function A(r): classical (phase space) average A 1 A(r)dr vol(ω) Ω quantum version is Â = operator in linear space of φ n s Expectation (average) φ n, Âφ n A(r) φ n (r) 2 dr Ω }{{} dµ φn density measure Quantum ergodicity: φ n, Âφ n A as E n Does this happen? For all states n? At what rate? We test numerically for certain A, up to very high n 1 6. If true for all A equidistribution in space, dµ φn uniform p.5

8 Outline Motivation: random waves, scars Ergodicity theorems, conjectures Numerical test results Rate of equidistribution: semiclassical estimate Sketch of numerical techniques which make this possible Conclusion p.6

9 Motivation: Random plane waves Conjecture (Berry 77): statistical model of eigenfunctions φ n 1 N N j=1 a j sin(k j r + α j ) iid amplitudes a j R iid phases α j [, 2π[ Wavevectors k j, spaced uniformly in direction, k j = k. Ray analogue of classical ergodicity. Predicts equidistribution as E : deviations die like φ n, Âφ n A E 1/4 p.7

10 High-energy eigenfunction φ n k 1 3 E 1 6 n p.8

11 Random plane waves stringy structures appear due to k = const. Interesting... to the eye only? p.9

12 Motivation: Scars Heller 84 observed: often mass concentrates (localizes) on short classical unstable periodic orbits (UPOs)... Theory (Heller, Kaplan): on UPO higher classical return prob. Strong scars were thought to persist as E. (No longer!) For certain A, our φ n, Âφ n is a measure of scarring p.1

13 Quantum Ergodicity Theorem QET (Schnirelman 74, Colin de Verdière 85, Zelditch ): For ergodic systems and well-behaved A, is true for almost all φ n. lim E n φ n, Âφ n A = Could exist an exceptional set (scars?) which are not ergodic This set has to be a vanishing fraction of the total number QET makes physicists happy: Correspondence Principle all quantum & classical answers agree as λ p.11

14 Quantum Unique Ergodicity QUE conjecture (Rudnick & Sarnak 94) For every single eigenfunction, lim E n φ n, Âφ n A = All converge to unique measure: dµ φ = uniform. (no scars) cf. classical flow which has many invariant measures: each UPO QUE was in context of hyperbolic manifolds... negative curvature causes chaos Constant-curvature arithmetic case: recent analytic progress... Lindenstrauss 3: measure can t collapse on to UPO Luo & Sarnak 3: bounds on sums deviations E 1/4 p.12

15 Numerical tests Analytics only for special systems (symmetries, all Λ = 1).9 Test generic chaotic system (Λ s differ) A= Ω.4 e.g. Sinai-type billiard: concave walls Anosov A=+1 Ω A A(r) = piecewise const: fast quantum calc using boundary classical A = vol(ω A) vol(ω) quantum φ n, Âφ n = probability mass inside Ω A p.13

16 Results: Expectation values.8 quantum expectations classical mean <A>.7.6 <φ n, A φ n > quantum states.3 No exceptional values > QUE upheld E n x 1 5 mean φ n, Âφ n A. Variance slowly decreasing, but how? p.14

17 Results: Equidistribution rate Quantum variance: V A (E) 1 m N n<n+m E n E φ n, Âφ n A Hard to measure: e.g. 1% needs m indep samples! Power law: V A (E) = a E γ fitted γ =.48 ±.1 V A (E) 25 quantum states E p.15

18 Results: Power law Variance V A (E) = ae γ, found γ =.48 ±.1 Consistent with conjecture that deviations E 1/4 (i.e. γ = 1/2) Previous experiments also used piecewise-constant A(r): Aurich & Taglieber 98: negatively-curved surfaces, lowest n < 6 only Bäcker 98: billiards, n < 6, but many choices of A Can see power-law not asymptotic until n we go 1 times higher! up to level n 8 1 5, E p.16

19 Results: Distribution of deviations Histogram deviations after scaling by V A (E) : using best fit a,γ 14 counts per bin N(,1) z Consistent with Gaussian (i.e. random wave model), convincing p.17

20 Theory: Semiclassical variance estimate I Feingold & Peres 86 Signal A(t) = follow A along particle trajectory r(t) Consider autocorrelation of this signal: A(t)A(t + τ) t ergod = A()A(τ) QET φ n, Â()Â(τ)φ n p.18

21 Theory: Semiclassical variance estimate I Feingold & Peres 86 Signal A(t) = follow A along particle trajectory r(t) Consider autocorrelation of this signal: A(t)A(t + τ) t noise power spectrum C A (ω) ergod = A()A(τ) QET φ n, Â()Â(τ)φ n fourier transform band profile of matrix A nm φ n, Âφ m ω distance from diagonal Barnett et al. : verified in stadium billiard Note: the diagonal is our quantum expectation! matrix A nm : ~ C( ω) n m ω p.18

22 Theory: Semiclassical variance estimate II Estimate C A (ω) numerically: Measure power spectrum of A(t) along long trajectories LISTEN to A(t) C(ω).2.1 POWER SPECTRUM ~ C(ω=) ω Physics: C A (ω) is heating (dissipation) rate under external driving by field A. (Cohen 99: fluctuation-dissipation) p.19

23 Theory: Semiclassical variance estimate II Estimate C A (ω) numerically: Measure power spectrum of A(t) along long trajectories LISTEN to A(t) C(ω).2.1 POWER SPECTRUM ~ C(ω=) ω Physics: C A (ω) is heating (dissipation) rate under external driving by field A. (Cohen 99: fluctuation-dissipation) DC limit ω gives diagonal variance: V A (E) var(a nn ) 2 vol(ω) C A (ω = ) E 1/2, }{{} γ = 1/2 prefactor a Time-reversal symmetry: extra factor var(a nn ) 2var(A nm ) p.19

24 Results: Semiclassical estimate semiclassical (no fitted parameters) 1 3 Power law: V A (E) = a E γ fitted γ =.48 ±.1 V A (E) likelihood in parameter space (γ,a) semiclassical estimate a quantum states γ E No fitted params. Good agreement, but estimate a 5% too big cf. arithmetic surfaces where a classical a quantum provably p.2

25 Results: Offdiagonal variance.3 C (qm) A C (cl) A (ω) off diagonal (ω) classical.25 7 states, E QM errorbars.7%, classical.1% power spectrum C (qm) () diagonal A ω Most accurate test ever for real system, no fitted params. Error at ω = related to QM peak exaggeration? p.21

26 Numerical methods sketch 1 Compute eigenfunctions φ n via scaling method : (Vergini & Saraceno 94; correct explanation (QET) Barnett, Cohen & Heller ) If: find A s.t. A but dynamics gives lim ω CA (ω) = Then: matrix A nm diagonal: Eigenvectors of A {φ n }. Put into a basis, size N 1/λ E. (e.g. N 4) One dense matrix diagonalization returns O(N) cluster of φ n s O(N) 1 3 faster than boundary integral equation methods! p.22

27 Numerical methods sketch 1 Compute eigenfunctions φ n via scaling method : (Vergini & Saraceno 94; correct explanation (QET) Barnett, Cohen & Heller ) If: find A s.t. A but dynamics gives lim ω CA (ω) = Then: matrix A nm diagonal: Eigenvectors of A {φ n }. Put into a basis, size N 1/λ E. (e.g. N 4) One dense matrix diagonalization returns O(N) cluster of φ n s O(N) 1 3 faster than boundary integral equation methods! 2 New BASIS SETS of Y Bessels placed outside the domain p.22

28 Numerical methods sketch 1 Compute eigenfunctions φ n via scaling method : (Vergini & Saraceno 94; correct explanation (QET) Barnett, Cohen & Heller ) If: find A s.t. A but dynamics gives lim ω CA (ω) = Then: matrix A nm diagonal: Eigenvectors of A {φ n }. Put into a basis, size N 1/λ E. (e.g. N 4) One dense matrix diagonalization returns O(N) cluster of φ n s O(N) 1 3 faster than boundary integral equation methods! 2 New BASIS SETS of Y Bessels placed outside the domain 3 Norm formula for Helmholtz solutions (little known?): φ, φ ΩA = 1 (n φ)(r φ) φ n (r φ) ds 2k 2 Ω A Overall effort scales O(N 2 ) per state (few CPU-days total) p.22

29 Missing levels? Weyl s estimate for N(E), the # eigenvalues E n < E: N Weyl (E) = vol(ω) 4π E L 4π E + O(1) N(E) x 14 Weyl test of k sequence (jump up = missed state) N(E) N Weyl (E) N 5 N(k i ) i N Weyl (E) k E 1/2 E 1/2 not 1 state missing in sequence of 6812 states k i p.23

30 Conclusion Are quantum (laplacian) eigenfunctions spatially uniform in chaotic systems as E? Measured rate of equidistribution in generic billiard Unprecendented range in E & sample size Strong support for QUE conjecture (all scars vanish) Power law consistent with conjectured γ = 1/2 Semiclassical estimate good, not perfect Directions Prefactor a: agreement in semiclassical limit? How about for other choices of A (error varies?) Variant of QUE: off-diagonal matrix elements? Scaling method: basis sets, rigor, other shapes... p.24

Asymptotic rate of quantum ergodicity in chaotic Euclidean billiards

Asymptotic rate of quantum ergodicity in chaotic Euclidean billiards Alex H. Barnett Courant Institute, New York University, 251 Mercer St, New York, NY 10012 barnett@cims.nyu.edu July 3, 2004 Abstract