Reply to "Comment on `Winding around non-Hermitian singularities' by Zhong et al., Nat. Commun. 9, 4808 (2018)"

TL;DR

This paper defends a previously proposed method for analyzing exceptional points in non-Hermitian systems against recent criticism, demonstrating that their approach yields correct results when properly applied.

Contribution

The authors clarify and validate their method for encircling multiple exceptional points, countering claims of errors and misconceptions in recent commentaries.

Findings

Their method correctly analyzes exceptional points when properly applied.

Counter-arguments show the criticized approach remains valid.

The timing of publications indicates independent development.

Abstract

A comment has been recently posted on the arXiv (arXiv:1902.07504) that discuss our recent work on encircling multiple exceptional points in Nat. Commun. 9, 4808 (2018). In that comment, the authors claim that our approach is prone to errors. In discussing their findings, they also indicate that a method presented by their team in Phys. Rev. A 98, 023818 (2018) (presumed to give correct results in all situations) was published prior to our work. First, we would like to note that both their work and ours were posted on the arXiv within a week from each other. Second and more importantly, as we will show below, their analysis and conclusion concerning the validity of our approach as presented in the comment article (arXiv:1902.07504) are not correct. As we will demonstrate, a proper application of our method does indeed provide the correct results.

Equations

$$\begin{array}{rl}H& =\left[\begin{array}{ccc}1& z& 0\\ z& -1& 0\\ 0& 0& 2z\end{array}\right],\end{array}$$

$$\begin{array}{rl}{M}_1& =\left[\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right],M_2=\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right],M_3=\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right].\end{array}$$

Full text

Reply to “Comment on ‘Winding around non-Hermitian singularities’ by

Zhong et al., Nat. Commun. 9, 4808 (2018)”

Q. Zhong1

qizhong@mtu.edu

  

M. Khajavikhan2

  

D.N. Christodoulides2

  

R. El-Ganainy1,3

ganainy@mtu.edu

1 Department of Physics, Michigan Technological University, Houghton, Michigan, 49931, USA

2 College of Optics $\&$ Photonics-CREOL, University of Central Florida, Orlando, Florida, 32816, USA

3 Center for Quantum Phenomena, Michigan Technological University, Houghton, Michigan, 49931, USA

A comment has been recently posted on the arXiv Pap et al. (2019) that discuss our recent work on encircling multiple exceptional points Zhong et al. (2018). In that comment, the authors claim that our approach is prone to errors. In discussing their findings, they also indicate that a method presented by their team Pap et al. (2018) (presumed to give correct results in all situations) was published prior to our work. First, we would like to note that both their work and ours were posted on the arXiv within a week from each other. Second and more importantly, as we will show below, their analysis and conclusion concerning the validity of our approach as presented in the comment article Pap et al. (2019) are not correct. As we will demonstrate, a proper application of our method does indeed provide the correct results.

Let us first recall the example studied in Pap et al. (2019):

[TABLE]

which has the eigenvalues $\pm\sqrt{1+z^{2}}$ and $2z$. In our approach, one would first pick a sorting scheme based on some chosen criterion. For example, we can sort the eigenvalues based on their magnitude, real part or imaginary part. After that, the eigenvalues must be sorted at every point in the complex plane according to the chosen method. This will naturally lead to a set of branch lines that separate the different solutions based on the sorting scheme. One can then associate a permutation matrix with each line to describe the transition between the different solution branches. We refer the interested reader to Zhong et al. (2018) for the detailed description of that procedure. Let us now apply this approach here. In order to make our point, we follow Pap et al. (2019) and sort the eigenvalues based on their real part. This results in the coloring scheme for the Riemann surfaces shown in Fig. 1(a). By projecting this on the complex domain, we find that we have two branch points but four branch lines (see Fig. 1(b)). In contrast, the analysis in Pap et al. (2019) identifies only two branch lines which is inconsistent with the sorting scheme. Back to Fig. 1(b), we note that the additional two lines (described by $M_{2}$ and $M_{3}$) are rather unusual branches since they do not end at branch points. This a direct outcome of the artificial nature of the example given in Pap et al. (2019) which does not arise any realistic physical situations (it amounts for analyzing two separate, uncoupled experiment by grouping their data together). Next, by inspecting the connectivity of the sheets across these branch lines, we can identify the following permutation matrices:

[TABLE]

We can now use the above matrices to study the stroboscopic evolution of the eigenstates around any arbitrary loop. For illustration purpose, we consider the two loops shown in Fig. 1(b). The starting point for the larger loop is denoted by $z_{1}$. The loop crosses the lines associated with the matrices $M_{1,3,2}$ in that order. Thus after one cycle, the exchange relation of the states is given by the matrix product $M_{2}M_{3}M_{1}(s_{1},s_{2},s_{3})^{\mathsf{T}}=(s_{2},s_{1},s_{3})^{\mathsf{T}}$. In simple terms, this means that state $s_{1}$ associated with the solution $\sqrt{1+z^{2}}$ will swap with state $s_{2}$ associated with $-\sqrt{1+z^{2}}$, as expected. On the other hand, state $s_{3}$ which is associated with the solution $2z$ remains on the same sheet, also as expected. Moving to the smaller loop with the starting point $z_{2}$, we note that that it crosses only the line associated with $M_{1}$ which results in $M_{1}(s_{1},s_{2},s_{3})^{\mathsf{T}}=(s_{3},s_{2},s_{1})^{\mathsf{T}}$. These are the expected results since at point $z_{2}$, the states $s_{1,2,3}$ belong to the solutions $\sqrt{1+z^{2}}$, $2z$, $-\sqrt{1+z^{2}}$, respectively (because $\text{Re}[\sqrt{1+z_{2}^{2}}]>\text{Re}[2z_{2}]>\text{Re}[-\sqrt{1+z_{2}^{2}}]$).

Interestingly, the authors in Pap et al. (2019), make a comment also about the efficiency of various approaches. It is not clear to us how can one compare the efficiency of various methods without having a rigorous mathematical definition for the term ‘efficiency’.

In summary, we have addressed the comments raised in Pap et al. (2019) and have shown that its conclusion is wrong.To illustrate this, we used the same example put forward in Pap et al. (2019) and demonstrated that our method provides indeed the correct results.