Microwave fidelity studies by varying antenna coupling

The fidelity decay in a microwave billiard is considered, where the coupling to an attached antenna is varied. The resulting quantity, coupling fidelity, is experimentally studied for three different terminators of the varied antenna: a hard-wall reflection, an open wall reflection, and a $50\Omega$ load, corresponding to a totally open channel. The model description in terms of an effective Hamiltonian with a complex coupling constant is given. Quantitative agreement is found with the theory obtained from a modified VWZ approach [J. J. M. Verbaarschot et al., Phys. Rep. 129, 367 (1985)].

Figure 1

(Color online) Geometry of the chaotic Sinai billiard, length $l=472\text{mm}$, width $w=200\text{mm}$ and a quarter-circle of radius $r=70\text{mm}$ where an antenna with different terminations may be introduced at position $c$. $a$ denotes the measuring antenna. The additional elements were inserted to reduce the influence of bouncing balls.

Figure 2

(Color online) Real part $f_R\left(t\right)$ and imaginary part $f_I\left(t\right)$ of the fidelity amplitude for three types of perturbation: ${\lambda }_{50\Omega }$ (black); ${\lambda }_{\text{hw}}$ (orange, light gray); ${\lambda }_{\text{oe}}$ (green, dark gray) in frequency range 8.0–8.5 GHz. The time is given in units of the Heisenberg time $t_H=2\pi \hslash /\Delta$, where $\Delta$ is the mean level spacing. Solid lines show the experimental results. The theoretical curves are dotted for experimental parameter (available only for the $50\Omega$ load case) and dashed for fitting parameter. The corresponding parameter and the transmission coefficient for the measuring antenna $a$ are listed in Table . The black dotted curve is nearly indistinguishable from the dashed one.

Figure 3

(Color online) Fidelity decay ${\left|f\left(t\right)\right|}^2$ for three types of perturbation: ${\lambda }_{50\Omega }$ (black); ${\lambda }_{\text{hw}}$ (orange, light gray); ${\lambda }_{\text{oe}}$ (green, dark gray) in three different frequency ranges: (i) 7.2–7.7 GHz; (ii) 8.0–8.5 GHz; (iii) 8.7–9.2 GHz. Solid lines show the experimental results and the theoretical curves are dotted for experimental parameter (available only for the $50\Omega$ load case) and dashed for fitting parameter. The corresponding parameter and the transmission coefficient for the measuring antenna $a$ are listed in Table .

Figure 4

(Color online) Geometry of the billiard with attached wave guide.

Figure 5

(Color online) Experimental coupling fidelity ${\left|f\left(t\right)\right|}^2$ (solid lines) and theoretical results for the experimental parameter ${\lambda }_W$ (dotted line) and the fit parameter ${\lambda }_{\text{fit}}$ (dashed line) for two openings $d=6.5\text{mm}$ with ${\lambda }_W=0.05$ and ${\lambda }_{\text{fit}}=-0.18i$ (black), and $d=11.2\text{mm}$ with ${\lambda }_W=0.52$ and ${\lambda }_{\text{fit}}=-0.55i$ (orange, light gray). The frequency window of the Fourier transform of the measured $S_{\text{aa}}\left(\nu \right)$ was 13 to 14 GHz and the transmission coefficient for antenna $a$ was $T_a=0.95$.