Encircling an exceptional point

We calculate analytically the geometric phases that the eigenvectors of a parametric dissipative two-state system described by a complex symmetric Hamiltonian pick up when an exceptional point (EP) is encircled. An EP is a parameter setting where the two eigenvalues and the corresponding eigenvectors of the Hamiltonian coalesce. We show that it can be encircled on a path along which the eigenvectors remain approximately real and discuss a microwave cavity experiment, where such an encircling of an EP was realized. Since the wave functions remain approximately real, they could be reconstructed from the nodal lines of the recorded spatial intensity distributions of the electric fields inside the resonator. We measured the geometric phases that occur when an EP is encircled four times and thus confirmed that for our system an EP is a branch point of fourth order.

Figure 1

A path in the complex $B$ plane [cf. Eq. (6) in the main text] surrounding an EP situated at $B^{\text{EP}}=-i$. Along this path the eigenfunctions of $H$ remain approximately real.

Figure 2

A photograph of the opened microwave billiard employed for the observation of EPs. A circular copper cavity is divided into two semicircular parts. The two parts are variably coupled by a slit of width $s$. One of the semicircular cavities can be perturbed by adjusting the position $\delta$ of a Teflon stub inside the resonator.

Figure 3

Development of the reconstructed electrical field distributions of two modes of the resonator shown in Fig. 2 while an EP is encircled. The initial states, i.e., the “start” configurations, are labeled as $\mid 1⟩$ and $\mid 2⟩$ in agreement with the definition (37). Their field distributions can be reconstructed from the recorded nodal line patterns for all settings $\left(s,\delta \right)$.

Figure 4

Development of the basis states $\mid 1⟩$ and $\mid 2⟩$ during four consecutive turns around an EP. For each loop, symbolized by $$$\circlearrowleft$, the development from the initial to the final states has been tracked according to Fig. 3. Four turns are needed in order to restore the original situation.