Michael Barth · Young-Hee Kim · Ulrich Kuhl · Hans-Jürgen Stöckmann
One-to-one correspondence between Schrödinger equation and Helmholtz equation in flat (quasi two-dimensional) microwave resonators.
- electric field density Ez and wave function ψ
- Poynting vector S and probability current density j; for details see page vortex.
Field and current patternTransport ⇒ complex wave functions ψ=ψR+iψI
a) open system ('quantum dot billiard'):
b) closed billiard with ferrite insert ('Robnik billiard'):
Berry's conjectureFor chaotic wave functions, ψ(r) may be written as random superposition of plane waves [Berry 1977]
Directions and amplitueds are random variables.
It follows: ψ and all components of ∇ψ are uncorrelated and Gaussian distributed!There are numerous tests of Berry's conjecture in numerics and experiment for distribution and spatial autocorrelations of ψ.Also local level dynamics can be explained quantitatively by this conjecture, see page level dynamics
.For currents etc. there exist numerical studies by Berggren and coworkers
; in the following: experimental tests!
Current distributions for the Robnik billiard (top: x and y component, bottom: linear and logarithmic plot of absolute value).Here σ2=〈jx2〉=〈jy2〉=0.5 〈j2〉 = k2〈ψR2〉〈ψI2
Distribution of vorticity for Robnik billiard (left) and quantum dot billiard (right).Here λ2=〈ω2〉 = 0.5 k4〈ψR2〉〈ψI2
Vortex correlationsAt every vortex is ψ=0, i.e. ψR=0 and ψI
=0.Pair correlation function g(kr), studied by Berry/Dennis (2000) and Saichev et al. (2001): Probability of two vortices in distance r at wave number k
(weighted with their vorticities).
Vortex pair correlation for Robnik billiard with ferrite ring.
(in cooperation with M. Vraničar, M. Robnik)Levy et al 1990
: Observation of (magnetic) persistent currents in mesoscopic copper rings due to flux quantization. In microwave billiards, inserts of ferritic material break time reversal symmetry and act like magnetic fields in electron billiards.
'Persistent current' around ferrite insert (left), and phase of the current in grey scale (right). The vortex around the ferrite gives rise to a phase shift of 4π.
- M.V. Berry, J. Phys. A 10, 2083 (1977)
- L.P. Lévy, G. Dolan, J. Dunsmuir, H. Bouchiat,Phys. Rev. Lett. 64, 2074 (1990)
- M.V. Berry, M.R. Dennis, Proc. R. Soc. Lond. A 456, 2059 (2000)
- A.I. Saichev, K.-F. Berggren, A.F. Sadreev, Phys. Rev. E 64, 036222 (2001)
- A.I. Saichev, H. Ishio, A.F. Sadreev, K.-F. Berggren, J. Phys. A 35, L87 (2002)
- M. Barth, Dissertation (2001)
- M. Barth, H.-J. Stöckmann, Phys. Rev. E 65 066208 (2002)
- M. Vraničar, M. Barth, G. Veble, M. Robnik, H.-J. Stöckmann, J. Phys. A 35 4929 (2002)
- Y.-H. Kim, M. Barth, U. Kuhl, H.-J. Stöckmann, Prog. Theor. Phys. Suppl. 150, 105 (2003)