Quantum chaotic scattering

Transcription

1 Quantum chaotic scattering S. Nonnenmacher (Saclay) + M. Zworski, M. Rubin Toulouse, 7/4/26 A resonant state for a dielectric cavity, computed by Ch.Schmit.

2 Outline Quantum chaotic scattering, resonances Semiclassics: fractal Weyl law, resonance gap. Link with classical trapped set the quantum baker s map as a toy model for chaotic scattering fractal Weyl law, resonance gap and eigenstates distribution a toy-of-the-toy solvable model: the Walsh-quantized baker s map conductance fluctuations for the Walsh-baker cavity

3 Scattering quantum systems V(x,y) x y In a scattering Hamiltonian system (e.g. in 2D) Left: free motion in an open cavity Right: particle scattering on a potential V (q) supported on a bounded region the classical flow at energy E > is unbounded. The corresponding quantum Hamiltonian H = 2 2 spectrum on R +. + V (q) has a purely continuous

4 Quantum resonances E γ h z j The resolvent (z H ) may be continued meromorphically from Im z > to Im z < ; its poles {z j } are the resonances of H. Each z j = E j iγ j /2 is associated with a metastable (non-normalizable) state of H, with lifetime τ j = /Γ j = long-living state if Γ j = O( ). 2 Questions in the semiclassical régime: Distribution of long-living resonances at E j E, Γ j = O( ). Description of the metastable states associated with these resonances.

5 Distribution of resonances - Trapped set Main semiclassical idea: the distribution of resonances near E depends on the dynamics near the set K E of trapped trajectories at energy E. We consider chaotic systems, for which K E is a hyperbolic repeller, with a fractal geometry. It has a fractal (Hausdorff/box) dimension dim(k E ) = 2µ E + (µ E < d )

6 Counting resonances - Fractal Weyl law How numerous are the long-living resonances when? Conjecture: in the limit, the density of resonances near E is given by a fractal Weyl law γ >, C E (γ), # { z j E < γ } = ( C E (γ) + o() ) µ E Remind that dim(k E ) = 2µ E +. E γ h n(h)~? Let us first check that this Weyl law is compatible with simple scattering situations.

7 Case of tunnelling resonances V(x,y) x y K E of positive volume: resonances are very close from being real (Im z j = O(e c/ )), and from eigenvalues of a closed system (tunnelling). The usual Weyl law predicts n( ) (d ). E γh Im z~ c/h e

8 Opposite case: very open systems Ex. 2: empty trapped set: resonances are very deep ( Im z j 2/3 n( ) = ). Ex 3: the trapped set = a single unstable orbit The resonances form a quasi-lattice (Bohr-Sommerfeld + inverted harm. osc.) n( ). E γh Im z< h λ/2

9 Chaotic scattering hbar=.7. gamma E Fractal hyperbolic repeller: proofs of the upper bound O( µ E) [Sjöstrand,Zworski,Guillopé]. Numerics: difficult to check the Weyl law [Lin-Sridhar-Zworski]

10 Resonance gap for filamentary repellers [Ikawa 88, Burq 93]: if n convex obstacles in R d are far enough from e.o. = gap in the semiclassical resonance spectrum: g > s.t. any resonance with Re z j ( ) /2 satisfies Im z j ( ) g. Equivalently, the quantum lifetimes τ j ( ) (2g). gh E γh [Gaspard-Rice 89] (3 disks in R 2 ): If the dimension µ E < /2 then there is a gap g = P E (/2). The topological pressure P E (s) involves complexity and hyperbolicity of the flow on K E : P E (s) = ( sup h KS (ν) s ν M(K E ) ) log J + (ρ) dν(ρ). [N.-Zworski 7]: proof of this gap for smooth scattering flows on R d.

11 The open baker s map Start with a canonical chaotic map on T 2, the baker s map B with 3 components of widths (b, b, b 2 ). B b b b 2 Now, open the system by piercing a hole at {b < q < b 2 }, through which particles escape to infinity: this is the open baker s map B. This open map mimics the Poincaré section of a chaotic scattering flow, or the boundary map of an open chaotic cavity. S

12 Trapped set for the open baker (/3,*,/3) p q Left: forward trapped set Γ = C [, ]. C is a simple Cantor set with dim(c) = µ given by b µ + bµ 2 = Center: backward trapped set Γ + = [, ] C. Right: the trapped set K = C C (hyperbolic repeller) of dimension 2µ.

13 Weyl quantization of the open baker For (2π ) = N N, the closed baker can be quantized into a unitary matrix [Balazs-Voros,Saraceno] on H N = Span{ q j, j =,..., N }: B N = F N F b N F b N F b2 N, (F N ) j j = e 2iπj j /N N discrete FT. To open the system, we kill the states supported in the hole [Saraceno-Vallejos] B N = B N (Id Π hole ) = F N F b N F b2 N. The quantized open kicked rotor has been used to model the ionization of a chaotic atom [Borgonovi-Guarneri-Shepelyansky], but in the diffusive regime (limit of long dwell time). A different version of the open kicked rotor, more similar to our open baker, has been studied by [Beenakker-Tajic-Tworzyd lo] with applications to transport through quantum dots.

14 Matrices of the (open) quantum bakers Matrices of the open baker B (/3,, /3) quantized à la Weyl (left) and à la Walsh (right).

15 Dictionary: Hamiltonian system quantum map N N = (2π ) exp { it (H ) }, t > (B N ) n, n =,, e itz/, z resonance of H λ n, λ eigenvalue of B N z { z E γ } λ { λ > r > } dim(k E ) = 2µ E + dim(k) = 2µ

16 Fractal Weyl law for the open quantum baker Conjecture: For fixed r >, C(r) such that, when N, Remember µ = dim(k) 2 <. Heuristics: # {λ Spec(B N ) : λ r} = ( C(r) + o() ) N µ. Count the states supported in the -neighbourhood of the forward trapped set Γ. States supported outside this neighbourhood are killed by some BN n. short-lived eigenstates, with eigenvalues. They span [Schomerus-Tworzyd lo] observed a fractal Weyl laws for the open kicked rotator in a chaotic régime (no precise analysis of the exponent). They predict a universal form of the profile function C(r) (ensemble of truncated random unitary matrices [Fyodorov-Sommers,Sommers-Zyczkowski]).

17 Spectra of B N for baker (/3,, /3).8 be8 be be be

18 Check of the fractal Weyl law (asymm. baker) To avoid arithmetical problems, we use an asymmetric baker (/32,, 2/3) Baker /32-2/3 Unscaled resonance counting Baker /32-2/3 Rescaled resonance counting n(n,r) 4 2 N=728 N=234 N=288 N=3456 N=432 N=468 N=584 N=576 n(n,r)/n^d N=728 N=234 N=288 N=3456 N=432 N=468 N=584 N= r r Left: counting function n(n, r) = # {λ Spec(B N ) : λ r} Left: function rescaled by the theoretical factor N µ (here µ =.493). Universality of the profile function?

19 Phase space structure of the resonant eigenstates Understand the phase space structure of the long-living (right) eigenstates ψ N of B N (resp. the resonant eigenstates of H ). Consider their Husimi functions (probability densities) H ψn (x) = N x ψ N 2, x T 2. As already noticed in [Casati-Maspero-Shepelyansky 99], the Husimi function is essentially supported on a fractal subset, namely the past trapped set Γ +. Semiclassical limit measures From any sequence {H ψn } N, extract a limit probability measure ν. Theorem. [N.-Rubin 6] If ν does not charge the discontinuity sets of B invariant, supported on Γ +. and B, then ν is conditionally A T 2, [B ν](a) def = ν(b (A)) = e γ ν ν(a) (γ ν = escape rate of ν). The eigenvalues of {ψ N } must asymptotically have the same radius: λ N 2 N e γ ν

20 A few eigenstates of B N There remain many questions: are all limit measures conditionally invariant? (pb. of discontinuities) which CIM ν can be limit measures? Continuous spectral density = many limit measures. For a given radius < r < r, is there a favored limit measure? ( Quantum ergodicity ) a unique limit measure? ( Quantum unique ergodicity ) [Keating-Novaes-Sieber- Prado 6] for r = e γ cl/2, is the natural measure a limit measure?

21 Toy model for the quantum baker s map (/3,, /3) To obtain analytical results, we replace B N by a skeleton matrix T N. T 9 = 3 ω ω 2 ω 2 ω ω ω 2 ω 2 ω ω ω 2 ω 2 ω, ω = e 2πi/3. The unitary version T N was introduced as a toy quantization of the baker, in the context of quantum information [Schack-Caves,Saraceno]. For N = 3 k, T N can be obtained by replacing in the definition of B N the discrete FT F N by the Walsh transform W N. It is then a solvable model.

22 Fractal Weyl law for the toy model (N = 3 k ).8.6 k= k= k= k= The spectrum of T N is known analytically. It is highly degenerate. In the semiclassical limit N = 3 k, it satisfies the and the fractal Weyl law, with a singular profile function C(r) = Θ(r r).

23 Eigenstates of T N in Walsh-Husimi representation We define a Walsh-Husimi density. Any limit measure ν is a CIM. Left/right: extremal eigenvalues λ max /λ min. Quantum Unique Ergodicity near these radii [Keating-N.-Novaes-Sieber]. Center: exemple of Bernoulli (purely fractal) limit measure in the bulk The natural measure does not appear naturally as a limit measure.

24 Transport through a chaotic cavity L L 2 U n π (Uπ Int U π Left: closed cavity. Right: open cavity, connected with 2 leads. ) n L L 2 ~ B

25 Conductance and quantum shot noise Using the Walsh-quantization of the 4-baker (Walsh-baker cavity), we compute both the conductance and the noise power ( in the semiclassical ) limit. Expressed from the r t scattering matrix of the cavity, S(ϑ) = 2 : t 2 r 22 conductance g tr(t 2t 2 ) (Landauer) noise power P tr { t 2t 2 (t 2t 2 ) 2} (Büttiker) In our case, the N/4 N/4 transmission matrix t 2 (ϑ) is given by t 2 (ϑ) = π 2 e inϑ BN (π Int BN ) n π (ϑ = quasi-energy). n For N = 4 k, there are N/4 injected channels, among which (N/4) /2 nonclassical channels carry the quantum noise, while the remaining ones are classically scattered. We compute in the limit N = 4 k : the average conductance = 2 the average noise power per nonclassical channel = 8. Remark: if S nonclass is a random matrix, the average noise power = 8! [Jalabert-Pichard-Beenakker,Baranger-Mello]

26 Final remarks open quantum maps form an interesting testing ground for scattering systems (easy numerics) Weyl-quantized open baker: check of the fractal Weyl law. Universality of the profile function? Walsh-quantized baker: first rigorous case of fractal Weyl law Two different interpretations of T N : Walsh-quantized open baker. Weyl quantization of a multivalued (ray-splitting) map. resonant eigenstates: variety of fractal limit measures better study the transmission spectrum for the Walsh-baker cavity

27 Spectra of B N for baker (/5, 3/5, /5).8 gap 4.8 gap gap gap P (/2) = log ( 2 5 ) < : gap at r.894.

28 7 6 5 N=2*3^k, r=.5 N random, r=.5 N=27x3^k, r=.5 N=27x3^k, r=.5 N=27x3^k, r=.3 theoretical slope Even eigenvalues of the open 3-baker Counting eigenvalues in logarithmic scale log(n(n,r)) log(n)

29 Free motion on a co-compact quotient of H. The grey area is the union of 2 fundamental domains representing the quotient H/Γ (Γ a Schottky group), a hyperbolic surface of infinite area, with 3 hyperbolic leads (right). Here the resonances of 2 LB are simply related with the zeroes of the Selberg zeta function.