Global and local billiard level dynamics
Michael Barth · Ulrich Kuhl · Hans-Jürgen Stöckmann
MotivationConductance 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 En 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:
ExperimentThere 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:
Global level dynamics
Local level dynamicsWhether 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)
InterpretationMovable 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 = 〈 ( dEn / dX )² 〉 and Δ=4π/A (mean level spacing),
the rescaled parameter x is obtained as
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.