Phase Shift Experiments Identifying Kramers Doublets in a Chaotic Superconducting Microwave Billiard of Threefold Symmetry

The spectral properties of a two-dimensional microwave billiard showing threefold symmetry have been studied with a new experimental technique. This method is based on the behavior of the eigenmodes under variation of a phase shift between two input channels, which strongly depends on the symmetries of the eigenfunctions. Thereby a complete set of 108 Kramers doublets has been identified by a simple and purely experimental method. This set clearly shows Gaussian unitary ensemble statistics, although the system is time-reversal invariant.

Figure 1

Schematic view of the experimental setup. The chaotic microwave cavity (left side) possesses threefold symmetry. The microwaves provided by port 1 (P1) of the network analyzer are split into two signals before they are coupled into the cavity by two different channels. The phase shift $\Delta \Phi$ between these two channels can be varied by a phase shifter. Power is coupled out by a third antenna and led back to the network analyzer, where the transmission to port 2 (P2) is measured.

Figure 2

Part of a transmission spectrum measured at $T=4.2\mathrm{K}$. The maxima of the resonance line mark the resonance frequencies. One clearly observes a signal-to-noise ratio of about 4 orders of magnitude (40 dB). Singlets and doublets are marked by S and D, respectively.

Figure 3

Resonance line for a typical singlet. The shape of the line depends only weakly on the phase shift varied. Nevertheless, there is a change in amplitude of more than 1 order of magnitude, i.e., a difference of more than 10 dB in the logarithmic scale.

Figure 4

Resonance line for a typical doublet. The shape of the line strongly depends on the phase shift varied. Note that this is a semilogarithmic plot, which compresses the changes in amplitude.

Figure 5

Long range correlations for the singlets and doublets: in the upper half the Dyson-Mehta statistics obtained in the framework of our previous work is shown. As one can see there are distinct deviations from GOE for the singlets. Also there is no saturation behavior for singlets and doublets. In the lower part the Dyson-Mehta statistics for the results of the present work are plotted. The statistics show for both singlets and doublets saturation, although the singlets still deviate slightly from GOE for small interval lengths $L$.