Experimental investigation of the elastic enhancement factor in a transient region between regular and chaotic dynamics

We present the results of an experimental study of the elastic enhancement factor $W$ for a microwave rectangular cavity simulating a two-dimensional quantum billiard in a transient region between regular and chaotic dynamics. The cavity was coupled to a vector network analyzer via two microwave antennas. The departure of the system from an integrable one due to the presence of antennas acting as scatterers is characterized by the parameter of chaoticity $\kappa =2.8$. The experimental results for the rectangular cavity are compared with those obtained for a microwave rough cavity simulating a chaotic quantum billiard. The experimental results were obtained for the frequency range $\nu =16\text{–}18.5$ GHz and moderate absorption strength $\gamma =5.2\text{–}7.4$. We show that the elastic enhancement factor for the rectangular cavity lies below the theoretical value $W=3$ predicted for integrable systems, and it is significantly higher than that obtained for the rough cavity. The results obtained for the microwave rough cavity are smaller than those obtained within the framework of random matrix theory, and they lie between them and those predicted within a recently introduced model of the two-channel coupling [V. V. Sokolov and O. V. Zhirov, arXiv:1411.6211 [nucl-th]].

Figure 1

(a) The rectangular microwave cavity and (b) the rough cavity, which were used for measuring the two-port scattering matrix $\widehat{S}$. The rough cavity side wall segments are marked by (1) and (2) (see text). The scattering matrix $\widehat{S}$ of the cavities was measured in the frequency window: 16–18.5 GHz. The vector network analyzer Agilent E8364B was connected to the microwave antennas, which were introduced inside the cavities [holes $A_1,A_2,A_3,A_4$, and $A_5$ in (a) and $A_1$ and $A_2$ in (b)] through the flexible microwave cables HP 85133-616 and HP 85133-617. The width of the rectangular cavity was 20 cm. The length of the cavity was changed from $L_1=41.5$ to 36.5 cm in 25 steps of 0.2 cm length. To create different realizations of the rough cavity, a metallic perturber [see panel (b)] was moved inside the cavity.

Figure 2

(a) The nearest-neighbor spacing distribution $P\left(s\right)$ for the microwave rectangular cavity coupled to the external channels via antennas (bars) obtained for the frequency range $\nu =16\text{–}17$ GHz. The experimental distribution $P\left(s\right)$ is compared to the Poisson distribution (broken line), which is characteristic for classically integrable systems, and to the theoretical prediction for GOE in RMT (full line). The inset shows the numerically reconstructed nearest-neighbor spacing distribution $P\left(s\right)$ for the chaoticity parameter $\kappa =2.8$. (b) The nearest-neighbor spacing distribution $P\left(s\right)$ for the microwave rough cavity obtained for the frequency range $\nu =6\text{–}9$ GHz (bars) is compared to the theoretical prediction for GOE (full line) and to the Poisson distribution (broken line).

Figure 3

Typical spectra of the rectangular cavity in the frequency range 16–17 GHz [panel (a)] and the rough cavity in the frequency range 8–9 GHz [panel (b)].

Figure 4

(a) The elastic enhancement factor $W$ of the two-port scattering matrix $\widehat{S}$ of the rectangular cavity coupled to the external channels via antennas simulating a quantum system with the chaoticity parameter $\kappa =2.8\pm 0.5$ (full circles). The RMT results [5, 6] are shown by empty circles. (b) The elastic enhancement factor $W$ of the two-port scattering matrix $\widehat{S}$ of the microwave rough cavity simulating a quantum chaotic system (full black rhombi). The RMT results are shown by empty rhombi. The measurements were done in the frequency window $\nu =16\text{–}18.5$ GHz. The two black broken lines $W=2$ and 3 show, respectively, the RMT limits for very strong and very low absorption. The latter limit $W=3$ is also expected for the integrable systems.