Fast encirclement of an exceptional point for highly efficient and compact chiral mode converters

Exceptional points (EPs) are degeneracies at which two or more eigenvalues and eigenstates of a physical system coalesce. Dynamically encircling EPs by varying the parameters of a non-Hermitian system enables chiral mode switching, that is, the final state of the system upon a closed loop in parameter space depends on the encircling handedness. In conventional schemes, the parametric evolution during the encircling process has to be sufficiently slow to ensure adiabaticity. Here, we show that fast parametric evolution along the parameter space boundary of the system Hamiltonian can relax this constraint. The proposed scheme enables highly efficient transmission and more compact footprint for asymmetric mode converters. We experimentally demonstrate these principles in a 57 μm-long double-coupled silicon waveguide system, enabling chiral mode switching with near-unity transmission efficiency at 1550 nm. This demonstration paves the way towards high-efficiency and highly integrated chiral mode switching for a wide range of practical applications.

I have a few comment that the authors may include in the revised manuscript, as delineated below.
(1) The explanation for the key idea is provided in the second paragraph on page 5 but it is incomplete yet. In this paragraph, high evolution speed <i>V</i> = |∂s/∂<i>t</i>| is understood well on the basis of the characteristics |<i>E</i><sub>2</sub>-<i>E</i><sub>1</sub>| → ∞ and <i>ξ<sub>m</sub></i> ≈ 0 along the parametric space boundary. However, total length <i>L</i> of the encircling-EP loop along the parametric space boundary tends to infinity and thereby total evolution time, or device footprint length equivalently, is not necessarily small even with extremely large <i>V</i>. In a complete explanation, the authors may provide a theoretical argument verifying that the adiabatic condition allows <i>V</i> scaling with <i>L</i> or higher, and the total evolution time converges to a certain finite value which can be significantly smaller than that for conventional loops in the moderate size regime.
(2) Another important aspect of the proposed approach is high modal purity but related explanation is not included. Modal purity of a dynamic state is associated with non-adiabatic transition amplitude [ref. 27 in the manuscript] with a factor |<i>E</i><sub>2</sub>-<i>E</i><sub>1</sub>|<sup>-1</sup> which tends to 0 along the parametric space boundary. It should be good to include an appropriate explanation on this basis. In addition, the authors may include some comment on the modal purity towards the end of a process over which the parametric condition returns to the initial condition from the parametric space boundary. Note that the final state purity must be determined for this stage, not for steering on the parametric space boundary.
(3) In the definition of factor <i>ξ<sub>m</sub></i>, the additional index <i>n</i> is dummy as there is a constraint <i>n</i> ≠ <i>m</i> out of two indices 1 and 2 but <i>n</i> seems to be an independent index in its appearance in the definition. It is good to state this point or use an alternative indexrepresentation scheme in the definition in order to avoid potential confusion. What about using an indexing scheme like "<i>m</i>′" (instead of "<i>n</i>") and say <i>m</i> and <i>m</i>′ are binary exclusive indices?
(4) Simple typo: Eq. (3) in line 5 on page 5 should be Eq. (2). 1 Response to the Reviewers' Comments on the manuscript NCOMMS-21-34305 entitled "Fast 1 Encirclement of an Exceptional Point for Highly Efficient and Compact Chiral Mode Converters" 2 submitted to Nature Communications 3 We thank the editor for handling our manuscript and the reviewers for careful and constructive 4 opinions. We feel truly encouraged by the shared comments from three reviewers that our work 5 presents interesting chiral mode converters with high efficiency and compact footprint by fast 6 encirclement of an exceptional point. We are also very grateful for the minor criticisms from the 7 reviewers, which have motivated us to significantly improve our manuscript. Below, please find the 8 point-by-point response in details. We hope our efforts could be appreciated by the reviewers and 9 firmly believe that our novel findings in this majorly revised manuscript can meet the stringent 10 requirements and attract general interest of broad readerships of Nature Communications.   should not be lower than 96%, which is also the simulated transmittance value for the 57-µm -long 56 asymmetric mode converter in the experiment. There are a number of combinations of ( (1) L , (2) L ,

57
(3) L ) to construct the asymmetric mode converter. We have shown in Fig. S11a

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If the device length is further reduced to 10 µm, we obtain a rather different and inferior

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"The physics behind the Hamiltonian hopping is that the eigenstates of the system Hamiltonian 100 converge as its parameters approach infinity, and the hopping is accessible by transitioning between 101 these states." on page 3, paragraph 1 in the main text.
represents the transmission efficiency of TEu mode that 133 outputs from right (left) port when TEv mode inputs from left (right) port in each section (Fig. S10b).

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(1) In Fig. 5a, only the two thin lines (marked by "waveguides") are the actual waveguides, right? The 198 two wider and bright stripes appear as waveguides on the first look.

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The transmittance of the entire device can be written as