## Principle of adiabaticity control in encircling exceptional points

When the Hamiltonian \(H(\tau )\) slowly changes over time \(\tau\) in a Hermitian system, the system remains in the same eigenstate throughout the evolution, based on the adiabatic theorem40–42. However, this theorem is not applicable for evolutions in non-Hermitian systems. For an *N*-order non-Hermitian Hamiltonian system, with \(\left| {{\varphi _m}} \right\rangle\) and \(\left| {{\varphi _n}} \right\rangle\) being two arbitrary eigenstates (or called right eigenstates), \(\left| {{\varphi _m}} \right\rangle\) being the initial eigenstate and \(\left| {{\varphi _n}} \right\rangle\) the eigenstate with lowest loss (\(m,\;n \in [0,N - 1]\)), \(\left| {{\varphi _n}} \right\rangle\) can be excited during the evolution, since adiabaticity is not stringently fulfilled, and \(\left| {{\varphi _n}} \right\rangle\) becomes the final output state due to a nontrivial NAT31,39. The adiabaticity parameter is defined as \({\xi _{m,n}}=\left| {{{(\hbar \left\langle {{{{\tilde {\varphi }}_m}}} \mathrel{\left | {\vphantom {{{{\tilde {\varphi }}_m}} {{{\dot {\varphi }}_n}}}} \right. \kern-0pt} {{{{\dot {\varphi }}_n}}} \right\rangle )} \mathord{\left/ {\vphantom {{(\hbar \left\langle {{{{\tilde {\varphi }}_m}}} \mathrel{\left | {\vphantom {{{{\tilde {\varphi }}_m}} {{{\dot {\varphi }}_n}}}} \right. \kern-0pt} {{{{\dot {\varphi }}_n}}} \right\rangle )} {({E_m} - {E_n})}}} \right. \kern-0pt} {({E_m} - {E_n})}}} \right|\), where \(\hbar\) is the reduced Planck constant, \(\left\langle {{{\tilde {\varphi }}_m}} \right|\) is the left eigenstate with \({H^\dag }\left| {{{\tilde {\varphi }}_m}} \right\rangle ={E_m}^{ * }\left| {{{\tilde {\varphi }}_m}} \right\rangle\), \(\left| {{{\dot {\varphi }}_n}} \right\rangle ={{\partial \left| {{\varphi _n}} \right\rangle } \mathord{\left/ {\vphantom {{\partial \left| {{\varphi _n}} \right\rangle } {\partial \tau }}} \right. \kern-0pt} {\partial \tau }}\), \({E_m}\) and \({E_n}\) are the corresponding eigenvalues for \(\left| {{\varphi _m}} \right\rangle\) and \(\left| {{\varphi _n}} \right\rangle\), respectively, and the dot above the function denotes the time derivative. To prevent the excitation of \(\left| {{\varphi _n}} \right\rangle\) during the evolution, \({\xi _{m,n}}\) should be zero. The output can be mixed with multiple eigenstates having simultaneously the lowest loss, if the non-adiabaticity degree is not purposely controlled39.

The eigenstate at \(\tau\), \(\left| {{\varphi _n}(\tau )} \right\rangle\), can be expressed as the superposition of all eigenstates at the initial time \({\tau _0}\), \(\left| {{\varphi _n}(\tau )} \right\rangle =\sum\nolimits_{{j=0}}^{{j=N - 1}} {{c_{j,n}}(\tau )\left| {{\varphi _j}({\tau _0})} \right\rangle }\), with \({c_{j,n}}(\tau )\) being the expansion coefficients. The numerator of \({\xi _{m,n}}\) can be then expressed as

$$\begin{gathered} \left\langle {{{{\tilde {\varphi }}_m}({\tau _0})}} \mathrel{\left | {\vphantom {{{{\tilde {\varphi }}_m}({\tau _0})} {{{\dot {\varphi }}_n}({\tau _0})}}} \right. \kern-0pt} {{{{\dot {\varphi }}_n}({\tau _0})}} \right\rangle =\left\langle {{{\tilde {\varphi }}_m}({\tau _0})} \right|\mathop {\lim }\limits_{{\Delta \tau \to 0}} \frac{{\left| {{\varphi _n}({\tau _0}+\Delta \tau )} \right\rangle - \left| {{\varphi _n}({\tau _0})} \right\rangle }}{{\Delta \tau }} \\ =\mathop {\lim }\limits_{{\Delta \tau \to 0}} \frac{{\sum\nolimits_{{j=0}}^{{j=N - 1}} {{c_{j,n}}({\tau _0}+\Delta \tau )} \left\langle {{{{\tilde {\varphi }}_m}({\tau _0})}} \mathrel{\left | {\vphantom {{{{\tilde {\varphi }}_m}({\tau _0})} {{\varphi _j}({\tau _0})}}} \right. \kern-0pt} {{{\varphi _j}({\tau _0})}} \right\rangle - \left\langle {{{{\tilde {\varphi }}_m}({\tau _0})}} \mathrel{\left | {\vphantom {{{{\tilde {\varphi }}_m}({\tau _0})} {{\varphi _n}({\tau _0})}}} \right. \kern-0pt} {{{\varphi _n}({\tau _0})}} \right\rangle }}{{\Delta \tau }} \\ \end{gathered}$$

1

Because the left and right eigenstates of are bi-orthogonal, i.e., \(\left\langle {{{{\tilde {\varphi }}_m}}} \mathrel{\left | {\vphantom {{{{\tilde {\varphi }}_m}} {{\varphi _m}}}} \right. \kern-0pt} {{{\varphi _m}}} \right\rangle =1\) and \(\left\langle {{{{\tilde {\varphi }}_m}}} \mathrel{\left | {\vphantom {{{{\tilde {\varphi }}_m}} {{\varphi _n}}}} \right. \kern-0pt} {{{\varphi _n}}} \right\rangle =0\) for \(m \ne n\)43. Eq. (1) can be simplified as \(\left\langle {{{{\tilde {\varphi }}_m}({\tau _0})}} \mathrel{\left | {\vphantom {{{{\tilde {\varphi }}_m}({\tau _0})} {{{\dot {\varphi }}_n}({\tau _0})}}} \right. \kern-0pt} {{{{\dot {\varphi }}_n}({\tau _0})}} \right\rangle ={\dot {c}_{m,n}}({\tau _0})\). We can let \({c_{m,n}}({\tau _0})=0\) to yield \({\xi _{m,n}}=0\). By using \(P(\tau )=[\begin{array}{*{20}{c}} {\left| {{\varphi _0}(\tau )} \right\rangle }&{\left| {{\varphi _1}(\tau )} \right\rangle }& \cdots &{\left| {{\varphi _{N - 1}}(\tau )} \right\rangle } \end{array}]\) to represent all eigenstates of \(H(\tau )\), the transformation relation of eigenstates at \(\tau\) and \({\tau _0}\) can be described as \(P(\tau )=P({\tau _0})C\), where is the transformation matrix. If the eigenstate space can be divided into direct sum subspaces as \(P={P_1} \oplus {P_2} \oplus \cdots \oplus {P_K}\) (\({P_k}\) is an invariant subspace under the transformation,\(k=1,2, \ldots ,K\)), the transformation relation can be rewritten as

$$\left[ {\begin{array}{*{20}{c}} {{P_1}(\tau )}&{{P_2}(\tau )}& \cdots &{{P_K}(\tau )} \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} {{P_1}({\tau _0})}&{{P_2}({\tau _0})}& \cdots &{{P_K}({\tau _0})} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{C_1}}&0&{}&{} \\ 0&{{C_2}}&{}&{} \\ {}&{}& \ddots &0 \\ {}&{}&0&{{C_K}} \end{array}} \right]$$

2

with \({P_k}(\tau )={P_k}({\tau _0}){C_k}\). Therefore, any two eigenstates belonging to different \({P_k}\) satisfy \({\xi _{m,n}}=0\), i.e., an initial eigenstate in one subspace does not trigger any other eigenstate in a different subspace during the evolution.

Interestingly, a Hamiltonian with a double-symmetric matrix form can group all eigenstates into two subspaces. The double-symmetric form indicates the \(N \times N\) Hamiltonian is symmetric, with respect to its main diagonal and skew diagonal elements, simultaneously. The eigenstates in one subspace are symmetric, and the eigenstates in the other subspace are anti-symmetric. The eigenstates in different subspaces cannot excite each other during the evolution (see Supplementary Note 1 for a detailed demonstration). A four-coupled optical waveguide system can then be designed to support a double-symmetric Hamiltonian with

$$H=\left[ {\begin{array}{*{20}{c}} {\beta - i\gamma }&\kappa &{}&{} \\ \kappa &{ - \beta }&\kappa &{} \\ {}&\kappa &{ - \beta }&\kappa \\ {}&{}&\kappa &{\beta - i\gamma } \end{array}} \right]$$

3

where \(\beta\), \(\gamma\) and \(\kappa\) represent the detuning, relative loss rate, and coupling strength, respectively. The eigenvalue spectrum of forms two sets of self-intersecting RSs, RS1 and RS2, in the parameter space \(({\beta \mathord{\left/ {\vphantom {\beta \kappa }} \right. \kern-0pt} \kappa },{\gamma \mathord{\left/ {\vphantom {\gamma \kappa }} \right. \kern-0pt} \kappa })\), as schematically presented in Fig. 1. Two eigenstates are located in the upper surfaces, with an EP at \((0.5,2)\), and the other two eigenstates are located in the lower surfaces, with an EP at \(( - 0.5,2)\). Figure 1 shows the dynamic trajectory of the Hamiltonian for clockwise (CW) (Fig. 1a) and anti-clockwise (ACW) loops (Fig. 1b) around the two EPs.

The state evolution obeys a Schrödinger-type equation, \({{i\partial } \mathord{\left/ {\vphantom {{i\partial } {\partial \tau }}} \right. \kern-0pt} {\partial \tau }}\left| \varphi \right\rangle =H\left| \varphi \right\rangle\). At the starting point \((0,0)\) of the trajectory, the four eigenstates from top to bottom are \(\left| {{\varphi _0}} \right\rangle ={[a,b,b,a]^T}\), \(\left| {{\varphi _1}} \right\rangle ={[b,a, - a, - b]^T}\), \(\left| {{\varphi _2}} \right\rangle ={[b, - a, - a,b]^T}\), and \(\left| {{\varphi _3}} \right\rangle ={[a, - b,b, - a]^T}\) (\(a={{\sqrt {5 - \sqrt 5 } } \mathord{\left/ {\vphantom {{\sqrt {5 - \sqrt 5 } } {2\sqrt 5 }}} \right. \kern-0pt} {2\sqrt 5 }}\)and \(b={{\sqrt {5+\sqrt 5 } } \mathord{\left/ {\vphantom {{\sqrt {5+\sqrt 5 } } {2\sqrt 5 }}} \right. \kern-0pt} {2\sqrt 5 }}\)), respectively, in which \(\left| {{\varphi _0}} \right\rangle\), \(\left| {{\varphi _2}} \right\rangle\) are symmetric, and \(\left| {{\varphi _1}} \right\rangle\), \(\left| {{\varphi _3}} \right\rangle\) are anti-symmetric. An eigenstate with eigenvalue with larger imaginary part suffers lower loss.

For CW encircling, \(\left| {{\varphi _0}} \right\rangle\) (\(\left| {{\varphi _1}} \right\rangle\)) evolves along the yellow (green) line on the red surface with low loss. The system ends at \(\left| {{\varphi _2}} \right\rangle\) (\(\left| {{\varphi _3}} \right\rangle\)), caused by the self-intersecting property of the RS. The two evolution trajectories are located on two different RSs, as indicated by the right panels in Fig. 1a. The two evolution processes do not interfere with each other, as the symmetric and anti-symmetric eigenstates belong to different subspaces. For the ACW encircling direction, as shown in Fig. 1b, \(\left| {{\varphi _0}} \right\rangle\) (\(\left| {{\varphi _1}} \right\rangle\)) initially evolves on the RS with high loss and switches to the eigenstates with low loss on the upper (lower) red surfaces due to a NAT. Two eigenstates in each subspace are associated with one RS. Consequently, the system state will jump to the red RS1, rather than to the red RS2, when \(\left| {{\varphi _0}} \right\rangle\) is injected. Meanwhile, the system will jump to the red RS2 when \(\left| {{\varphi _1}} \right\rangle\) is injected. In other words, the adopted grouping method ensures that only one eigenstate with low loss is dominant after the NAT process. Finally, \(\left| {{\varphi _0}} \right\rangle\) and \(\left| {{\varphi _1}} \right\rangle\) return to their initial state. Overall, the system can independently enable chiral transmission for the two eigenstate groups. For the encircling trajectories on RS1 (RS2), the output state is locked to \(\left| {{\varphi _2}} \right\rangle\) (\(\left| {{\varphi _3}} \right\rangle\)) and \(\left| {{\varphi _0}} \right\rangle\) (\(\left| {{\varphi _1}} \right\rangle\)) for CW and ACW handedness, respectively (see Supplementary Note 2 for the evolution paths with the injected states being of \(\left| {{\varphi _2}} \right\rangle\) and \(\left| {{\varphi _3}} \right\rangle\)) For CW and ACW evolution, we have numerically calculated the dynamic evolution along the encircling trajectories presented in Fig. 1, verifying chiral switching for each group of eigenstates (see Supplementary Note 3).

## Experimental verification of multi-state chiral switching

We map our theory onto a realistic geometry consisting of four-coupled optical waveguides, associated with a double-symmetric Hamiltonian, as schematically shown in Fig. 2a. The following design strategies have been employed to map the Hamiltonian parameters in Eq. (3) onto the device structure. First, we used sufficiently large gap separation between adjacent waveguides to introduce weak coupling, guaranteeing a symmetric Hamiltonian along the main diagonal. Then, the four waveguides are symmetrically positioned with respect to the center (denoted by green plane), making sure that the Hamiltonian is symmetric along the anti-diagonal. Finally, the gap separations between adjacent waveguides are kept the same.

For the first and fourth waveguides, a metallic chromium strip of 20-nm thickness was used to introduce a position-dependent absorption loss. The chromium strip is tapered on both sides, with the purpose of preventing reflections caused by abrupt structural change. Based on coupled-mode theory, the relationship between the structural parameters and the Hamiltonian parameters can be then established44. The Hamiltonian parameters \(\beta\) and \(\gamma\) are proportional to \({w_1} - {w_2}\) and \({w_{\text{m}}}\), respectively, and \(\kappa\) is inversely proportional to 45. The detailed relationship can be found in Fig. S3 of Supplementary Note 4. The evolution time \(\tau\) is mapped to the propagation distance , and the CW and ACW loops correspond to light propagation along the positive and negative directions with respect to the axis, respectively. The EP-encircling evolution trajectory is schematically shown in Fig. 2b. At the two ports of the device - positions A and D in Fig. 2a - all four waveguides have the same width \({w_1}={w_2}\). The four-coupled waveguides support four eigenmodes, i.e., TE0, TE1, TE2, and TE3 modes, corresponding to the four eigenstates \(\left| {{\varphi _0}} \right\rangle\), \(\left| {{\varphi _1}} \right\rangle\), \(\left| {{\varphi _2}} \right\rangle\), and \(\left| {{\varphi _3}} \right\rangle\). These four modes are divided into two groups, i.e., the even modes TE0 and TE2, and the old modes TE1 and TE3. The four coupled waveguides for chiral mode switching are linked to bus waveguides using two branches on both ends, which are used for our experimental demonstration.

Figures 3a-3d show the electric field evolution during propagation in a four-coupled silicon waveguides system, simulated with the Finite-Difference Time-Domain (FDTD) method (see Supplementary Note 5 for the detailed geometrical parameters of the four-coupled silicon waveguides). When the TE0 mode enters from the left port, it propagates to the two middle waveguides from A to B, as shown in the first row of Fig. 3a. The mode evolution from B and C experiences almost no energy loss, and finally evolves to TE2 at D. For the TE0 mode excited from the right port, as shown in the second row of Fig. 3a, the system evolves to the side waveguides from D to C, but quickly attenuates from C and B, due to absorption in the Cr layer. A portion of the energy leaks due to non-adiabatic evolution into the middle waveguides. This branch of mode evolution suffers no loss, and hence it becomes the dominant mode, associated with a NAT, and finally the energy exits at A as TE0. As a summary, TE2 and TE0 modes are produced when TE2 enters from the left and right ports, respectively (Fig. 3b). In other words, the output mode is locked to TE2 (TE0) for the left-side (right-side) input, regardless of TE0 or TE2 mode injection. The other group of eigenmodes, i.e., TE1 and TE3 modes, show a similar evolution process as TE0 and TE2 modes (Figs. 3c and 3d). The output mode is locked to TE3 (TE1) for the left-side (right-side) input, regardless of a TE1 or TE3 mode injection. Benefitting from selectively governing the non-adiabaticity degree, during the entire evolution one group of eigenmodes, i.e., TE0 and TE2 modes, will not excite the other group of eigenmodes, i.e., TE1 and TE3 modes, and vice versa.

A scanning electron microscope (SEM) image of the four-coupled silicon waveguides in one of the fabricated samples is shown in Fig. 4a (See Method and Supplementary Note 6 for the fabrication details). Figures 4b-4e show the measured transmittance for different mode inputs within 1540–1560 nm. The measurement details can be found in Supplementary Note 7. The output is dominated by TE2 (TE0) mode when TE0 (Fig. 4b) and TE2 (Fig. 4c) modes excite the left (right) port, and is dominated by TE3 (TE1) mode when TE1 (Fig. 4d) and TE3 (Fig. 4e) modes input from left (right) port. These measured results clearly demonstrate chiral switching for two groups of optical eigenmodes, i.e., TE2 and TE0 modes, TE1 and TE3 modes. It should be noted that, due to fabrication errors and interference in the measurement setup, all other modes are also excited. To make sure that all other modes are negligible compared to the expected output modes, AB and CD sections in Fig. 2(a) are shortened to reduce the evolution adiabaticity in designing the final device, guaranteeing that the expected modes are dominantly excited. As a result, the expected modes have significantly larger amplitudes than the modes excited by fabrication imperfections.