Ergodicity of quantum eigenfunctions in classically chaotic systems

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

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:.8.7.6.5.4.3 has other conserved quantities (e.g. θ).2.1.2.4.6.8 1 p.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:.8.7.6.5.4.3 has other conserved quantities (e.g. θ).2.1.2.4.6.8 1 1 1 Chaotic:.8.6.4.2 Ω Γ start.8.6.4.2 ergodic: nearly every trajectory covers phase space.2.4.6.8 1.2.4.6.8 1 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 1.5 1 1.5 1 1.5 1 1 energy eigenvalue E wavenumber k.5.5.5.5.5 1 1.5 1 1.5 1.5 1 k E 2π λ.5.5... cavity modes, quantum levels.5 1.5 1 quantized equivalent of classical billiards (momentum i ) p.3

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

Classical and quantum averages Choose test function A(r): classical (phase space) average A 1 A(r)dr vol(ω) Ω quantum version is  = 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

Classical and quantum averages Choose test function A(r): classical (phase space) average A 1 A(r)dr vol(ω) Ω quantum version is  = 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

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

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

High-energy eigenfunction φ n k 1 3 E 1 6 n 5 1 4 p.8

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

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

Quantum Ergodicity Theorem QET (Schnirelman 74, Colin de Verdière 85, Zelditch 87... ): 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

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

Numerical tests Analytics only for special systems (symmetries, all Λ = 1).9 Test generic chaotic system (Λ s differ).8.7.6.5 A= Ω.4 e.g. Sinai-type billiard: concave walls Anosov.3.2.1 A=+1 Ω A.1.2.3.4.5.6.7.8.9 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

Results: Expectation values.8 quantum expectations classical mean <A>.7.6 <φ n, A φ n >.5.4 11615 quantum states.3 No exceptional values > QUE upheld.2.5 1 1.5 2 2.5 E n x 1 5 mean φ n, Âφ n A. Variance slowly decreasing, but how? p.14

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 2 1 4 indep samples! 2 1 3 Power law: V A (E) = a E γ fitted γ =.48 ±.1 V A (E) 25 quantum states 1 4 1 4 1 5 1 6 1 7 E p.15

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 1 4... we go 1 times higher! up to level n 8 1 5, E 1.6 1 7 p.16

Results: Distribution of deviations Histogram deviations after scaling by V A (E) : 2 18 16 using best fit a,γ 14 counts per bin 12 1 8 N(,1) 6 4 2 6 4 2 2 4 6 z Consistent with Gaussian (i.e. random wave model), convincing p.17

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

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

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=) 5 1 15 ω Physics: C A (ω) is heating (dissipation) rate under external driving by field A. (Cohen 99: fluctuation-dissipation) p.19

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=) 5 1 15 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

Results: Semiclassical estimate semiclassical (no fitted parameters) 1 3 Power law: V A (E) = a E γ fitted γ =.48 ±.1 V A (E).37.36 likelihood in parameter space (γ,a) semiclassical estimate a.35 25 quantum states.34 1 4.33.46.4825.55 γ 1 4 1 5 1 6 1 7 E No fitted params. Good agreement, but estimate a 5% too big cf. arithmetic surfaces where a classical a quantum provably p.2

Results: Offdiagonal variance.3 C (qm) A C (cl) A (ω) off diagonal (ω) classical.25 7 states, E 5 1 5 QM errorbars.7%, classical.1% power spectrum.2.15.19.185 C (qm) () diagonal A.1.18.5.175.17.165.16.2.4.6.8 1 2 3 4 5 6 7 8 ω Most accurate test ever for real system, no fitted params. Error at ω = related to QM peak exaggeration? p.21

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

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

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

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) 1 9 8 7 6 N(E) 2.464 x 14 Weyl test of k sequence (jump up = missed state) 2.463 2.462 N(E) N Weyl (E) N 5 N(k i ) i 2.461 4 3 2 1 N Weyl (E) 65 65.5 65.1 65.15 k 2.46 2.459 E 1/2 E 1/2 not 1 state missing in sequence of 6812 states 2.458 65 66 67 68 69 7 71 72 73 74 75 k i p.23

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