Chiral transmission by an open evolution trajectory in a non-Hermitian system

Exceptional points (EPs), at which two or more eigenvalues and eigenstates of a resonant system coalesce, are associated with non-Hermitian Hamiltonians with gain and/or loss elements. Dynamic encircling of EPs has received significant interest in recent years, as it has been shown to lead to highly nontrivial phenomena, such as chiral transmission in which the final state of the system depends on the encircling handedness. Previously, chiral transmission for a pair of eigenmodes has been realized by establishing a closed dynamical trajectory in parity-time- (PT-) or anti-PT-symmetric systems. Although chiral transmission of symmetry-broken modes, more accessible in practical photonic integrated circuits, has been realized by establishing a closed trajectory encircling EPs in anti-PT-symmetric systems, the demonstrated transmission efficiency is very low due to path-dependent losses. Here, we demonstrate chiral dynamics in a coupled waveguide system that does not require a closed trajectory. Specifically, we explore an open trajectory linking two infinite points having the same asymptotic eigenmodes (not modes in PT- and anti-PT-symmetric systems), demonstrating that this platform enables high-efficiency chiral transmission, with each eigenmode localized in a single waveguide. This concept is experimentally implemented in a coupled silicon waveguide system at telecommunication wavelengths. Our work provides a new evolution strategy for chiral dynamics with superior performance, laying the foundation for the development of practical chiral-transmission devices.


Supplementary Note 1: Derivation of evolution equation
In sections 1 and 3, 0 , the coupled mode equation can be written as da i a i a dz da i a i a dz where 1 a and 2 a are the amplitude of the system state, 1 β and 2 β are the propagation constants, and κ is the coupling coefficient.Eq.S1 can be rewritten as λ are real numbers).Consequently, the Hamiltonian satisfying these eigenstates and eigenvalues simultaneously is ( ) ' 2 ' 2 can be rewritten as ( ) The formula of coupling length is ( ) , with wave number 0 k , effective refractive indices, even n and odd n for even and odd modes, associated with ( ) . In principle, we can simultaneously introduce loss to even and odd modes to generate a non-zero value of ' γ .In our practical design of the coupled waveguide system, we only exert loss to odd mode by attaching the adiabatic couplers.
Comparing Eq.S2 and S4, the evolution equation for the entire system can be written as with the definition of  k k .Therefore, we can write the total power at port I at the right side Note that here the summation of power is an approximation under the condition that one term is much larger than another term (an accurate approach should be the summation of field i b instead of the power 2 i b , and this can be done by the transfer matrix approach).In the expression of 11 T , under the condition that 1 γ <<  , the first term dominates, and therefore Similarly, the value of other elements of  can be derived 1 and that is: By writing ( ) ( ) In our design, 1 3 K K K ≈ ≡ , therefore we simplify Eq. (S7) as , Because of reciprocity, the transmission matrix T ′ for excitation from right side is 11 12 11 21 The dependence of β and κ on W ∆ and d is depicted in Fig. S3.β and κ used in the trajectory can be selected accordingly.The calculation of γ is based on Beer-Lambert-Bouguer law [1].With the assumption of the complex refractive index of the optical absorbing medium, ' '' , the optical intensity in the medium is  When ζ is big enough, the attenuation constant in each section remains constant.
According to Eq. S7, the optical intensity at   The two coupled waveguides have a larger gap separation at the terminal points, associated with a smaller κ .Because of the larger width of Waveguide II than that of Waveguide I in the first section, and is reversed in the third section, Section 1 and Section 3 associate with a negative and positive β , respectively [Fig.S6(a)].
Consequently, the cross sections for the left and right sides correspond to the infinite points ( , 0) −∞ and ( , 0) +∞ , respectively.In the interval between ( , 0) −∞ and A, or B and ( , 0) +∞ , the gap separation is varied, associated with a continuously-varied κ  In order to further demonstrate the chiral response as indicated by Fig. 3  β κ γ κ = , the system state can be expressed as ( ) ( ) ( ) − , and ε is a small number, since  1 is triggered as the adiabatic evolution is not strictly fulfilled.
The dominant eigenmode is  2 and the other eigenmode  1 is infinitesimal at A. In the interval from A to B with 0 β = , the system maintains the two eigenstates as The system state at B can be written as , according to Eq. ( 5) in the main text, associated with ( ) approaches zero, and is smaller than ε , i.e., NAJ occurs.In this situation, the system state at B is dominated by In the last interval denoted by B end z z − in Fig. S7, the system keeps its dominate state as  1 , which finally evolves into For the backward process, the initial system state is The system state at B can be expressed as 1 is always dominant from A z to 0 z , and finally evolves into In the whole process from end z to 0 z , the system state is always dominated by  1 and its associated eigenvalue has ( ) Im 0 E = , the conversion efficiency between ( ) X z is close to 100%.Meanwhile, the triggered  2 in the interval from end z to B z is nearly dissipated due to the selective coupling loss.
Using the above-mentioned method, we can obtain the dynamical evolution process as [ ] For all different input states and evolution directions, the conversion efficiency evolving from ( ) process is maximum.This conversion efficiency corresponds to the chiral transmission efficiency of the output port ΙΙ at the left of the device when light inputs from the port Ι at the right,  21 ′ , demonstrating the chiral response (Fig. 4f in the main text).We have noted in some previous studies on chiral transmission based on EP-encircling strategies, the chiral transmission efficiency is selected as the transmittance for all the different input states and encircling directions are maximum 2,3 .In our work, we have used the same definition of chiral transmission efficiency, which is consistent with the proposed open evolution trajectory.It is also worth pointing out that high-efficiency chiral transmission efficiency is guaranteed in principle as the system loss merely occurs during the NAJ process, superior over the previous schemes encircling an EP in (anti-) PT-symmetric systems with path-dependent loss.In addition, it can be seen from Figs.
S7(a), S7(c), S7(e) and S7(g) that, the system evolving between A and B will not cause additional crosstalk for the output state.The crosstalk of the output state merely comes from the end of the evolution process, i.e., B to ( , 0) +∞ in the forward path, and A to ( , 0) −∞ in the backward path.
out of the silicon waveguide back into the fiber.The decoupled light will be collected by the optical power meter (PMSII-A) and a spectrometer (YOKOGAWA AQ6370C).We can only directly record the transmission power with the power meter and spectrometer.But this does not affect how we can extract the transmission efficiencies at different ports.The optical power reaching the control waveguides is much smaller than the light source as light has to go by PBS, polarization controller and optical fiber.
During this process, the major loss comes from the coupling loss between the fiber and the GC prior to the control waveguides.In the measurement, we have adjusted the coupling angle between the fiber and grating to maximize the optical power coupled into the control waveguides.This optimal coupling angle is determined if the optical power reaching the power meter is maximum, and is used for the fabricated sample ' exp( ),

1 FIG
FIG. S1 | System states evolving on the Riemann surfaces.(a, b) Forward and (c, d) backward

Figure 1 X 2 X 1 T 1 k 3 k
Figure S1 shows the dynamic evolution paths of the Hamiltonian for forward (Figs.S1(a) and S1(b)) and backward evolution [Figs.S1(c) and S1(d)].As for forward evolution [Figs.S1(a) and S1(b)], the initial state [0,1] T is located on the red sheet of FIG. S3 | Hamiltonian parameters.(a, b) β and κ versus the width difference, the attenuation constant with angular frequency ω and velocity of light in vacuum c , z is propagation distance.As a result, 4 α γ = , and γ is accessible according to the distributions of ( ) I z along z direction in two middle waveguides.

1 .
FIG. S5 | Structural parameters of the coupled waveguides.(a) The top view of coupled silicon waveguides.(b-g) The width of the first middle waveguide, 1 W , the gap distances of two middle waveguides and two side waveguides, d , 1 c d and 2 c d , the widths of two side waveguides, 1 c W

FigureW
Figure S5(a) schematically shows the structural configuration of double-coupled

FIG. S6 |
FIG. S6 | Hamiltonian parameters used in the path.(a-c) Dependence of β (a), κ (b), andγ (c) on the spatial coordinate z at 1500, 1550, and 1600 nm. (d) Evolution path at 1500, 1550, and 1600 nm.A and B are the points of division in the evolution.

[
in the main text, the Hamiltonian parameters at 1550 nm are selected to calculate the dynamics of such an unclosed path.The total propagation distance is assumed to be 243 Z m µ = .For the forwrd process, the Hamitonian starts at an infinie point, Figs.S7(a) and S7(b)].When the Hamiltonian evolves slowly to A, The distance between A and B has been set sufficiently long to incur high loss on [ ]

0, 1 T
is used in the initial state.The final state is [ ] 1, 0 T [Figs.S7(e) and S7(f)] for the forward process, and [ ] 0,1 T for the backward process owing to the occurance of NAJ [Figs.S7(g) and S7(h)].The output state is always dominted by [ ] 1, 0 T for the forward process and [ ] 0,1 T for the backward process, regardless of the initial input state.

Figures
Figures S8(b)-S8(d) show the measured SEM images of the fabricated chip samples.

P
consisting of DCWs and the control device without DCWs to ensure that the identical power is injected into the control waveguides.Subsequently, the transmittance at different ports can be obtained by comparing the loss differences between the fabricated sample consisting of DCWs and the control device without DCWs.Specifically, we record the output power b ij P , c P , and d P in Fig. S8(b), S8(c) and S8(d), respectively, where b ij P represents the output power of the output port i when light inputs from the port j ( , 1, 2, 3 and 4 i j = ).The input power, in P , is defined as the remaining optical power in the fiber after light source passes through the PBS and polarization controller.The fabricated sample consisting of DCWs and the control device without DCWs in Figs.S8(b)-S8(d) have the same GC, which incurs the identical loss on the input power.The bending radii in Figs.S8(b) and S8(d) are the same and sufficiently large, associated with near-zero loss on the input power.As a result, c P is equal to d P .The loss coefficient from one GC is assumed to be GC α .DCW ij α represents the loss coefficient from the DCWs, when light inputs from the port j and outputs from the port i ( , must be satisfied.We can thus establish three equations associated with Figs.measured transmittance in Eq. (S14) accurately reflects the transmission characteristics of the DCWs.The measured power at different ports with the optical spectrometer is shown in Figs.S9.The measured transmittance spectra shown in Fig. 4 of the main text and supplementary Fig.

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FIG. S9 | The measured optical power at different ports.(a) The power through two GCs in Fig. S8(c).(b) The power through two GCs with two bending waveguides in Fig. S8(d).(c, d) The power in Fig. S8(b) as the DCWs is involved.