Erratum: Chiral state conversion without encircling an exceptional point [Phys. Rev. A 96, 052129 (2017)]

Figure 1

Two different cw parameter cycles are shown in panels (a) and (d) along with the ensuing behavior of $\chi$ in the corresponding panels (b),(c) and (e),(f) in each row. The loop in panel (a) lies away from the EP (EP is shown as a cross) with $(g_0,\rho )=(0.82,0.1)$. In the one shown in panel (d), the contour includes the EP with $(g_0,\rho )=(0.95,0.1)$. The terminal points, where the two eigenvectors $|{\psi }_{1,2}\rangle$ are found, are marked by a yellow circle and the arrow shows the direction of encirclement. In panels (b) and (e), the resulting variation in $\chi$ at all times is shown when the rate of cycling is relatively large, i.e., $\gamma =0.5$. Plots on the left (shown in red) depict the case when the system is excited with $|{\psi }_1\rangle$ and those on the right (shown in blue) provide results for excitations with $|{\psi }_2\rangle$. In these plots, solid (dashed) lines represent real (imaginary) parts of $\chi$. As mentioned in the text, for this cw cycle, the state expected at the output is $|{\psi }_1\rangle$ that corresponds to $\chi \to e^{i\theta }$. The real (imaginary) part of this expected result is shown as a filled (empty) circle at $t=2\pi {\gamma }^{-1}$. In the upper panels, these two circles lie very close to each other. In panels (c) and (f), the rate of cycling is reduced to $\gamma =0.1$ and both excitations end up at the correct location even for the non-EP enclosing case, panel (c). Although mode conversion is not robust in panel (b)—consider the plot on the right—results for the EP-inclusive loop show robust state conversion not only when the encirclement is slow [in panel (f)], but also when it is fast, $\gamma =0.5$, as in panel (e).

Figure 2

Assuming adiabatic conditions (small values of $\gamma$), the shaded area shows the range of values of the loop center $g_0$ and radius $\rho$, for which chiral mode conversion can take place even without enclosing an EP. The red line is the curve obtained from Eq. (1) and the black line depicts the boundary where the loop starts touching the EP located at $(g_0+\rho =1)$.