Chiral state conversion without encircling an exceptional point

Dynamically varying system parameters along a path enclosing an exceptional point is known to lead to chiral mode conversion. But is it necessary to include this non-Hermitian degeneracy inside the contour for this process to take place? We show that a sufficiently slow variation of parameters, even away from the system's exceptional point, can also lead to a robust asymmetric state exchange. To study this process, we consider a prototypical two-level non-Hermitian Hamiltonian with a constant coupling between elements. Closed-form solutions are obtained when the amplification or attenuation coefficients in this arrangement are varied in conjunction with the resonance detuning along a circular contour. Using asymptotic expansions, this input-independent mode conversion is theoretically proven to take place irrespective of whether the exceptional point is enclosed upon encirclement. Our results significantly broaden the range of parameter space required for the experimental realization of such chiral mode conversion processes.

Figure 1

Two different c.w. 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 far away from the EP (EP is shown as a cross) with $(g_0,\rho )=(0.3,0.3)$. In the one shown in panel (d), the contour includes the EP with $(g_0,\rho )=(0.8,0.3)$. 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., $\gamma =0.1$. 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 c.w. 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 panels (c) and (f), the rate of cycling is reduced to $\gamma =0.01$ 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, $\gamma =0.1$, as in panel (e).

Figure 2

Dependence of the mode conversion efficiency $\mathcal{C}$ on the corresponding variables used (a) $\gamma$, (b) $g_0$, and (c) $\rho$. In panel (a), the radius of the parameter circle is kept constant at $\rho =0.3$ and $\mathcal{C}$ is plotted vs $\gamma$ for various values of $g_0$, listed in blue beside each curve. The inset highlights the region of small $\gamma$' s for a range of values of $g_0$ (linearly spaced between $0.3\text{and}0.4$). Mode conversion $\mathcal{C}$ improves as the loop center $g_0$ moves closer to the EP, as shown in panel (b), again with having $\rho =0.3$. Different curves correspond to different values of $\gamma$ that are listed in red. For these same values of $\gamma$, $\mathcal{C}$ is plotted for different radii $\rho$ when the loop center is at $g_0=0.6$, as shown in panel (c). Note that in all cases, thinner lines have been used for each successive curve.