First Experimental Evidence for Quantum Echoes in Scattering Systems

A self-pulsing effect termed quantum echoes has been observed in experiments with an open superconducting and a normal conducting microwave billiard whose geometry provides soft chaos, i.e., a mixed phase space portrait with a large stable island. For such systems a periodic response to an incoming pulse has been predicted. Its period has been associated with the degree of development of a horseshoe describing the topology of the classical dynamics. The experiments confirm this picture and reveal the topological information.

Figure 1

The open billiard used in this work possesses a stable orbit (dotted line) and two unstable periodic orbits (dashed lines) at the necks. Between the unstable orbits, a wave packet can get trapped. The circles indicate the positions of the antennas ( labeled by numbers from left to right) used in the microwave experiment.

Figure 2

A Poincaré section of the open billiard consists of a stable island and a surrounding chaotic layer with two hyperbolic points. Each point in the section corresponds to the angle of reflection $\varphi$ and arc length $l$ of a point of reflection at the lower boundary of the billiard. The rectangle enclosing the rhs unstable fixed point shows a heteroclinic structure.

Figure 3

Spectrum obtained from a measurement of the transmission parameter $|S_{12}|$ between antennas 1 and 2 in the superconducting resonator. The arrows mark four narrow resonances. More of those are detected with other antenna combinations. All sit on top of a continuous spectrum.

Figure 4

Time spectra resulting from a FFT of the spectrum in Fig. 3 (upper part) and from the same measurement at room temperature (lower part). The arrows indicate the early visible echoes. The additional large peaks are caused by standing waves on the cables from the rf source to the resonator and from the resonator to the network analyzer. The different offset times in the upper and the lower part are due to different cable lengths. The inset shows echoes at very late times. They could be detected only in the superconducting case.

Figure 5

Comparison of transmission measurements between antennas on different sides (upper part) and on the same side (lower part) of the resonator. A shift of the peaks by about half a period is visible as expected from the classical model. Note the shortening of the echo periods $T$ with time.