Scattering experiments with microwave billiards at an exceptional point under broken time-reversal invariance

Scattering experiments with microwave cavities were performed and the effects of broken time-reversal invariance (TRI), induced by means of a magnetized ferrite placed inside the cavity, on an isolated doublet of nearly degenerate resonances were investigated. All elements of the effective Hamiltonian of this two-level system were extracted. As a function of two experimental parameters, the doublet and the associated eigenvectors could be tuned to coalesce at a so-called exceptional point (EP). The behavior of the eigenvalues and eigenvectors when encircling the EP in parameter space was studied, including the geometric amplitude that builds up in the case of broken TRI. A one-dimensional subspace of parameters was found where the differences of the eigenvalues are either real or purely imaginary. There, the Hamiltonians were found to be $\mathcal{P}\mathcal{T}$ invariant under the combined operation of parity ($\mathcal{P}$) and time reversal ($\mathcal{T}$) in a generalized sense. The EP is the point of transition between both regions. There a spontaneous breaking of $\mathcal{P}\mathcal{T}$ occurs.

Figure 1

Scheme of the experimental setup (not to scale) to study the properties of the ferrite. The antennas 1 and 2 are connected to the vector network analyzer. The inner circle is a copper disk introduced into the resonator to realize singlet resonances.

Figure 2

Photograph of the microwave billiard. The top plate has been removed. In the measurements it was screwed tightly to the middle plate, which had a hole with the shape of the resonator, and the bottom plate through the displayed holes. The ferrite is marked by an arrow. Rings of solder are visible along the boundary of the resonator and the inner disk. They ensured good electrical contact between the top and bottom plates of the resonator.

Figure 3

Sectional drawing of the setup used to magnetize the ferrite. The ferrite was placed inside the resonator and above and below it were NdFeB magnets outside the cavity. Each one was held in place by a screw thread mechanism allowing us to vary the distance between the magnets and thus the field strength at the ferrite.

Figure 4

The singlet at $2.84$ GHz in the annular billiard. The complex functions $S_{12}\left(f\right)$ (open circles) and $S_{21}\left(f\right)$ (solid circles) have been measured at $B=119.3$ mT. For clarity only every $14\mathrm{th}$ data point is shown. The two complex functions coincide up to a deviation due to errors of $5\times {10}^{-3}$ for the real and imaginary parts. Thus reciprocity holds within this error.

Figure 5

The doublet at $2.696$ GHz in the circular billard with the magnetic field $B=36.0$ mT. The upper panel shows $|S_{11}|$ (solid line) and $|S_{22}|$ (dashed line), the lower panel $|S_{12}|$ (solid line) and $|S_{21}|$ (dashed line). Violation of reciprocity is clearly visible.

Figure 6

Field patterns of the fourth doublet. The small triangles symbolize strength and direction of the magnetic fields. The gray shades represent the electric field strength. The darker the color the stronger is the electric field. The upper modes are found at $3.18$ GHz, the lower ones at $3.20$ GHz. The arrows point at the ferrite. In part A of the figure violation of reciprocity is not observed because in the upper mode the magnetic field vanishes at its position. In part B the ferrite has been shifted 20 mm towards the center of the circle, i.e., to a position where the field is nonzero in both modes.

Figure 7

Comparison of the fitted matrix elements $|S_{12}|$ (solid line) and $|S_{21}|$ (dashed line) to the experimental data points (circles) also shown in the lower panel of Fig. 5. Fitting of expression (1) to the data reproduces the data points within errors of $\approx$$5\times {10}^{-4}$ for both real and imaginary parts. For clarity only every 10th experimental point is shown.

Figure 8

The antisymmetric part $H_{12}^A$ of $\mathcal{H}$. The data have been taken at the third doublet of Table I at a mean frequency of $\overline{f}=2.914$ GHz in the circular billiard. The maximum in $|H_{12}^A|$ and the decrease of $\mathrm{Arg}\left(H_{12}^A\right)$ by the amount of $\pi$ display the ferromagnetic resonance. The error bars indicate the variance of the results obtained in five independent experiments [116].

Figure 9

The upper part sketches the resonator used in Secs. 6, 7, 8, 9. Two parameters can be set from outside, the opening $s$ between the two approximate semicircles and the position $\delta$ of the Teflon piece with respect to the center of the cavity. As in Sec. 2 there are two antennas 1 and 2 reaching into the cavity. The ferrite is denoted by F. The parameter $s$ actually denotes the vertical position of the gate as shown in the lower part of the figure. The broken line indicates the groove in the bottom plate.

Figure 10

The eigenvalues $f_j-i{\Gamma }_j/2$ of $\mathcal{H}$ plotted as functions of $\delta$ at $s=s_{\mathrm{EP}}=1.66$ mm and $B=53$ mT. The eigenvalues cross at ${\delta }_{\mathrm{EP}}=41.25$ mm. There $f_{\mathrm{EP}}=2.728$ GHz and ${\Gamma }_{\mathrm{EP}}=2.08$ MHz.

Figure 11

Modulus and phase of the ratio ${\nu }_j=\left|{\nu }_j\right|exp\left(i{\Phi }_j\right)$ of the components of the left eigenvectors ${\vec{l}}_j$, where $j=1,2$, at $s=s_{\mathrm{EP}}=1.66$ mm and $B=53$ mT. The eigenvectors coalesce at ${\delta }_{\mathrm{EP}}=41.25$ mm. There the TRI-breaking phase $\tau$ can be read off as the deviation of ${\Phi }_{1,2}$ from $\pi /2$. The present figure relies on the same data as Fig. 10.

Figure 12

The relative phase (dots with error bars) of the components of the left eigenvector at the EP given as a function of the magnetic field $B$ that activates the ferrite. For $B=0$ the phase equals $\pi /2$. Thus an earlier result [30] is recovered. The model (15) for the TRI-breaking coefficient $H_{12}^A$ yields the solid line, see text. The shaded vertical bar indicates the range of $B$ where the center of the ferromagnetic resonance is expected.

Figure 13

Modulus square of the Fourier transform of $S_{ba}\left(f\right)$. Data taken from Ref. [32] (solid line) are compared to the Fourier transform of Eq. (34) (dashed line). The sharp peak at $t=30\pm 5$ns indicates the time needed for the signal to travel through the coaxial cables connecting the VNA with the antennas. At about $t=1.8\mu \mathrm{s}$ the noise level is reached. Between these instants of time the temporal behavior is well described by the function $t^{2}exp\left(2\mathrm{Im}{E}_{\mathrm{EP}}t\right)$—in agreement with the second-order pole entering Eq. (34). Compare with Fig. 3 of Ref. [32], where only a fraction of the available data points had been plotted.

Figure 14

Differences of the real and the imaginary parts of the complex eigenvalues in the notation of Eq. (35). The data have been taken at $B=53$ mT. The darker the color the smaller the respective difference. It is vanishingly small at the darkest colors. The differences of the real parts, shown in blue, are small to the left of the EP, those of the imaginary parts, shown in red, to the right. In the regions of white colors both differences are large and beyond the scale of the color code. The dotted curve is the double loop around the EP discussed in the text.

Figure 15

The TRI-violating phase $\tau \left(t\right)$ for $B=53$ mT with $t$ varied along the dotted double loop shown in Fig. 14. At the end of either loop $\tau$ returned to its initial value. Counting the points of measured Hamiltonians along the path yields the “time” scale $t$ with $t_1=25,t_2=150$.

Figure 16

Geometric phases $\gamma \left(t\right)$ gathered when the EP was experimentally encircled twice. Panel (a) displays $\gamma \left(t\right)$ along the dotted double loop marked in Fig. 14. The green triangle (upward) marks the initial point at $t=0$. Moving counterclockwise along the dotted line, the red diamond was reached at the end $t_1=25$ of the inner loop. The blue triangle (downward) completes the outer loop at $t_2=150$. Compare Fig. 15. The points, where the direction of $\gamma \left(t\right)$ switches, occurred at the extreme values of $\tau \left(t\right)$. At the end of the path we found $\gamma \left(t_2\right)\ne \gamma \left(0\right)$. Panel (b) shows $\gamma \left(t\right)$ when the EP is encircled twice along the outer loop of Fig. 14. Then we found $\gamma \left(t_2\right)=\gamma \left(0\right)$, i.e., the end point coincided with the initial point.

Figure 17

Geometric phase $\gamma \left(t\right)$ gathered when the EP is encircled 4 times by following twice the double loop shown in Fig. 14. The green triangle (upward) marks $\gamma \left(0\right)$. With increasing $t$ the geometric phase follows the black dots counterclockwise. At the end of the first double loop (blue triangle downward) it continues along the red ones. It ends at the blue diamond.

Figure 18

Differences of the complex eigenvalues of the effective Hamiltonians in a neighborhood of the EP, compare Fig. 14. The three panels show results for the magnetization of the ferrite with $B=0,38,61$ mT.