Experimental investigations of chaos-assisted tunneling in a microwave annular billiard

We present detailed investigations of the experimental signatures of chaos-assisted tunneling in the two-dimensional annular billiard, as already summarized in Phys. Rev. Lett. 84, 867 (2000). We have performed analog experiments with two-dimensional, electromagnetic resonators allowing for a direct simulation of the corresponding quantum system. Spectra from a superconducting cavity with a high-frequency resolution are combined with electromagnetic intensity distributions of high spatial resolution experimentally determined using a normal conducting twin cavity. Thereby all eigenmodes were obtained with properly identified quantum numbers. Besides distributions of quasidoublet splittings, which serve as fundamental observables for the tunneling between whispering gallery types of modes, we also focus on the distributions of resonance widths of the doublets. These directly reflect the role of lifetime of certain modes in the tunneling process. Here, as theoretically expected, the class of so-called beach modes is found to play a particular role in mediating between regular and chaotic states to enhance the tunneling strength. This behavior is found in the spectrum and also in the structure of the wave functions.

Figure 1

(Color online) Geometry of the two-dimensional annular billiard. The eccentricity $\delta$ characterizes the center displacement of the two circles.

Figure 2

The family of annular billiards with $r+\delta =0.75$ is very suitable to study chaos-assisted tunneling. The shown trajectories (arrows) are located right within the so-called whispering gallery region of the billiard.

Figure 3

Definition of the Birkhoff coordinates for the construction of adequate Poincaré surfaces of section of the classical phase space.

Figure 4

Poincaré surfaces of section for the eccentric configurations $\delta =0.05$, 0.10, 0.15, and 0.20 together with some examples of the generating trajectories in the configuration space. Here, the upper two trajectories are whispering gallery orbits (neutrally stable), while “spo” and “upo” label stable and unstable periodic orbits, respectively.

Figure 5

Universal stability behavior for the shortest periodic orbits in case of $r>\delta$. For details see Sec. 2B.

Figure 6

(Color online) Squared wave functions of the concentric system with $r=0.75$ according to Eq. (6). On the left side the first singlet (with rotational symmetry) is given; the middle part shows the first doublet (with even and odd parity, respectively). The right-hand part shows a higher excited doublet for better understanding of the quantum numbers $(n,m)$.

Figure 7

Eigenfrequencies vs quantized angular momenta according to Eq. (8) for the concentric system with $r=0.75$. The dashed line at $S=0.75$ marks the borderline (the “beach”) between the chaotic sea and the whispering gallery region in the classical phase space of the present setup. The solid bars denote those important spectral regions, where quantum states pass the borderline and thus couple the otherwise distinct phase-space regions.

Figure 8

Modular microwave resonator of the niobium annular billiard with an outer circle of radius $R=125\mathrm{mm}$. The lid on the right-hand side has two antennas for the excitation of the electromagnetic fields.

Figure 9

Excerpt from the transmission spectrum of the superconducting resonator in the concentric configuration $\delta =0.0$. The dashed lines mark the theoretical positions of the eigenvalues calculated from Eq. (5) for all the states from (0∣1) to (10∣1).

Figure 10

Excerpt from the transmission spectrum for the superconducting resonator (lid $A$) as a function of the eccentricity $\delta$. Besides a large number of singlets exactly one quasidoublet is observed. The circles mark and enlarge this doublet. They show a systematic motion towards smaller frequencies with increasing $\delta$. The frequency axis has been stretched by the factor of 10 in the magnifications.

Figure 11

The quasidoublet taken from Fig. 10, measured with strongly (lid $A$) and weakly (lid $B$) perturbing antennas. The frequency axis is stretched by the factor of 10 in the magnifying circles.

Figure 12

Photograph of the experimental setup, which was used to measure electromagnetic intensity distributions. At the top, the open billiard in concentric configuration is seen. Below this, the positioning unit for the perturbing magnetic body, properly aligned to the resonator, is shown. The guiding magnet is located directly below the bottom plate of the billiard on a movable sledge, which is positioned by two stepper motors.

Figure 13

(Color online) Principle of control of the experimental mapping of electromagnetic field distributions in the annular billiard. The PC sets the position of the perturbing body and sends commands to the network analyzer. The measured frequency shift $\partial f$ is recorded as a function of the position $(\rho ,\phi )$.

Figure 14

(Color online) Theoretically predicted quasidoublet splittings in the vicinity of the beach of the chaotic sea at $S=0.75$ (dashed vertical line) as a function of the quantized angular momentum $S=n∕k$. The calculated data points for three of the investigated billiard configurations ($\delta =0.10$, 0.15, and 0.20) are marked by different symbols and are interconnected. In addition, all the different quasidoublets (10∣1)–(30∣1) are sorted by grey scales. Here, related data points of a given state $(n\mid 1)$ fall one upon the other and are denoted by one grey scale level. The dashed horizontal line at $\mid \Delta f∕f\mid ={10}^{-6}$ shows the experimental limit of resolution.

Figure 15

(Color online) Comparison between the normal conducting $\left(300\mathrm{K}\right)$ and the superconducting $\left(4.2\mathrm{K}\right)$ measurement and identification via experimentally obtained field distributions [quantum numbers $(n\mid m)$ including parity] for the configuration $\delta =0.20$.

Figure 16

(Color online) Comparison between the warm and the cold measurements at very high frequencies. For notation see Fig. 15. The position of the whispering gallery mode (44∣1) is nearly invariant under variations in $\delta$ (dashed line for $\delta =0$).

Figure 17

(Color online) Comparison between experimentally measured (left side) and numerically simulated (right side) states for different angular momentum quantum numbers $n$ of the concentric system. Tiny perturbations of the eccentricity lead to the development of a tilted line of symmetry.

Figure 18

(Color online) An example of a scar, found for the configuration $\delta =0.20$ in the frequency range around $20\mathrm{GHz}$—i.e., at the upper end of the spectrum. On the left-hand side the corresponding unstable periodic orbit of the classical system is shown for comparison.

Figure 19

Level scheme of identified resonances, experimentally taken using the superconducting resonator with lid $A$ in the spectral range around $10\mathrm{GHz}$. The dashed lines indicate the wandering of quasidoublets of the family $m=1$ as a function of $\delta$.

Figure 20

Sequel of Fig. 19.

Figure 21

(Color online) Mode (10∣1) under variations of the eccentricity $\delta$. At the top the drastically changing field distributions can be seen, below the corresponding frequency shift.

Figure 22

(Color online) A spectral window for modes between (27∣1) and (30∣1), where the frequencies as well as the field distributions are nearly invariant under variations in $\delta$.

Figure 23

(Color online) Numerical fit of the resonance shape given by Eq. (10) to the weakly excited quasidoublets ${(29\mid 1)}_{\pm }$ of the configuration ($r=0.65$, $\delta =0.10$). In this case the resolution of the measurement was increased to $250\mathrm{Hz}$, instead of $10\mathrm{kHz}$. The relative splitting of the doublet is $\mid \Delta f∕f\mid \approx 3\times {10}^{-6}$.

Figure 24

(Color online) For the eccentricity $\delta =0.20$ observed splittings of quasidoublets taken from the window $8.75–10.50\mathrm{GHz}$ as a function of the quantized angular momentum $S$. The family $m=2$ indicates the transition (dashed curve) between chaotic (grey) and regular modes (black), respectively.

Figure 25

(Color online) Quasidoublet splittings of the family $m=1$ (black) with angular momentum quantum numbers $n=1–30$ in the case of weakly perturbing antennas (lid $B$) again for $\delta =0.20$. For comparison also the modes with $m\ne 1$ (grey) from Fig. 24 (lower part) are included. Obviously, a local maximum of the family $m=1$ occurs at $S=0.75$.

Figure 26

(Color online) Splittings of the family $m=1$ in the zoomed-in window around $S=0.75$. The figure shows a comparison between the warm and the cold measurement, where in both cases lid $A$ was used. Different symbols denote the varying eccentricity $\delta$. Diverse grey scales have been taken for a better separation of the quasidoublets, whereby neighboring data points, all of which belonging to the same state $(n\mid 1)$, are given in the same grey scale. The splittings of the warm measurement $\left(300\mathrm{K}\right)$ fall in the range of the resolution limit for normal conducting resonator $(1∕Q⪆{10}^{-4})$. This is also the reason for some missing data points. As a consequence of the strong perturbation of the antennas, also the points of the cold measurement $\left(4.2\mathrm{K}\right)$ lie on this level (i.e., about two orders of magnitude above their resolution limit), being nearly invariant under a variation in $\delta$.

Figure 27

(Color online) Comparison of the splittings in the superconducting state $\left(4.2\mathrm{K}\right)$ using strongly (lid $A$) and weakly (lid $B$) perturbing antennas, respectively. In the latter case only three of five eccentricities were accessible. We have used the same notation as in Fig. 26. In the lower subfigure all the modes between (10∣1) and (30∣1) are displayed. While the upper figure shows nearly constant, perturbation-induced splittings, the lower illustration reproduces the already found maximum structure of Fig. 25 in the range around the beach at $S=0.75$. As already indicated in the theoretical prediction (see Fig. 14), also here the data points show a systematical rise in the chaotic sea $(S<0.75)$ and a fluctuating fall in the whispering gallery region $(S>0.75)$.

Figure 28

(Color online) Loaded quality factors determined from the states shown in Fig. 26 at 300 and $4.2\mathrm{K}$, respectively, using strongly perturbing antennas (lid $A$). Normally, for each quasidoublet two quality factors could be reduced (separately for the even and the odd state). Both subfigures show on average nearly constant quality factors as a function of the quantized angular momentum with an apparent bifurcational structure towards larger values of $S$, which is slightly present already in the normal conducting case and clearly observable in the superconducting state. The reason for this strong asymmetry in the excitation of both states of a given quasidoublet is basically composed of three effects: first, the more and more increasing localization of field distributions at higher angular momenta, and second, the nodal line structure of these fields, which is strongly correlated with the angular momentum quantum number $n$, and, third, the spatially fixed coupling to these fields.

Figure 29

(Color online) Comparison of loaded quality factors in the superconducting state $\left(4.2\mathrm{K}\right)$ using strongly (lid $A$) and weakly (lid $B$) perturbing antennas, respectively. Again, per quasidoublet there are two quality factors given (which are nearly identical in the lower subfigure). In comparison to the already explained bifurcational structure for strongly perturbing antennas (upper subfigure), the lower diagram shows quality factors, nearly exponentially increasing with $S$ ($\log Q_L$ linearly depends on $S$) up to the expected limit of ${10}^6$. This nicely reflects the systematical disapproach between the antenna (located at $S=0.625$) and the balance point of the field distribution (approximately at $S=n∕k$). In addition, it might be deduced from the very symmetrical excitation of both states of a given quasidoublet that indeed there is only very little impact on the mode structure by the weak coupling to the fields.

Figure 30

(Color online) Frequency dependence of quality factors in the case of family $m=1$ (black) and $m\ne 1$ (grey), respectively, for the eccentricity $\delta =0.20$. In the lower subfigure all states (0∣1)–(30∣1) are considered. Here, the dashed circle marks a clearly observable minimum in the beach region.

Figure 31

(Color online) Polar Fourier transformation of the modes (10∣1) from the chaotic sea, (18∣1) from the beach region, and (30∣1) from the whispering gallery region for the configuration $\delta =0.20$. Regular field distributions feature more and more sharply localized Fourier components beyond the beach (dashed lines).