Global and local billiard level dynamics

Michael Barth · Ulrich Kuhl · Hans-Jürgen Stöckmann

Motivation

Conductance through a system is related to the sensitivity of eigenvalues on the change of an external parameter.
[E. Akkermans and G. Montambaux (1992)]

Rescaling of eigenvalues E_n and parameter X,

leads to universal behavior (Δ: mean level spacing), independent of the kind of perturbation.
[B.D. Simons and B.L. Altshuler (1993)]

These predictions are tested with two types of perturbations:

length variation
displacement of one impurity

Experiment

There is a one-to-one correspondence between the stationary Schrödinger equation and the Helmholtz equation.

Measurement of eigenfrequency spectra of billiard-shaped microwave cavities
==> Experimental approach to quantum chaos.

Level dynamics measurements:

variation of a single length in a Sinai billiard
=> field distributions are changed globally.
Length X: 460..480 mm (step: 0.1 mm), Width: 200 mm, Radius: 70 mm

Typical example for a global level dynamics:
variation of the position of one disk in a rectangular billiard with randomly distributed disks
=> field distributions are changed locally.
Length: 340 mm, Width: 240 mm, 19 fixed disks,
1 movable disk at position X (step: 1mm)

Typical example for a local level dynamics:

Global level dynamics

velocity distribution	velocity autocorrelation function

The blue curve corresponds to the expected Gaussian behavior, the red one to a modified Bessel function (see below).	The blue curve corresponds to the universal function of Simons and Altshuler. Calculated by E.R. Mucciolo: Simulation of 500×500 Gaussian random matrices of the form H(x) = H₁ cos(x) + H₂ sin(x).

Local level dynamics

Whether the level dynamics is global or local depends on δ=kD (wavenumber: k = 2 π ν / c, diameter of the disk: D).

a) 0.35 < δ < 0.65 (D=5 mm, 3.4GHz < ν < 6GHz)
b) 1.4 < δ < 2.6 (D=20 mm, 3.4GHz < ν < 6GHz)
c) 5.1 < δ < 5.9 (D=20 mm, 12.5GHz < ν < 14.5GHz)

velocity distributions:

transition from modified Bessel to Gaussian behavior with increasing δ

velocity autocorrelation functions:

oscillations with wavelengths depending on delta

Interpretation

Movable disk can be interpreted as a perturber probing the field in the resonator.

In two-dimensional billiards a metallic perturber leads to a negative frequency shift proportional to the field intensity at the position X of the perturber.

==> The eigenvalue velocity reads

where α is a constant depending on the geometry of the perturber.

Under the assumption that chaotic wavefunctions can be described by a random superposition of plane waves [M.V. Berry (1977)],
ψ and ∇ψ are uncorrelated, and Gaussian distributed [V.N. Prigodin et al. (1995)].

This leads to the new velocity distribution

with β = A / (2 α k), A is billiard area.
The red curves correspond to this modified Bessel function (also called MacDonald function).

With 1 / β² = 〈 ( dE_n / dX )² 〉 and Δ=4π/A (mean level spacing),
the rescaled parameter x is obtained as

Result:
x is not universal for local level dynamics!
One should use xbar = kX instead.

The autocorrelation function is obtained as:

If the autocorrelation function is plotted as a function of the rescaled parameter xbar, a new universal behavior is found.

Literature:

E. Akkermans and G. Montambaux, Phys. Rev. Lett. 68, 642 (1992)
B.D. Simons and B.L. Altshuler, Phys. Rev. B 48, 5422 (1993)
M.V. Berry, J. Phys. A 10, 2083 (1977)
V.N. Prigodin et al., Phys. Rev. Lett. 75, 2392 (1995)
M. Barth, U. Kuhl, H.-J. Stöckmann, Phys. Rev. Lett. 82, 2026 (1999)
M. Barth, U. Kuhl, H.-J. Stöckmann, Annalen der Physik (Leipzig) 8, 733 (1999)
M. Barth, Dissertation (2001)