Topological Braiding and Dynamic Probing of Phase Transitions at Temporal Interfaces in Non-Hermitian Synthetic Dimensions

Abstract

Non-Hermitian systems give rise to distinct topological phenomena, yet their manifestations at temporal interfaces characterized by abrupt changes in system parameters remain largely unex plored. Upon an abrupt alteration of the Hamiltonian in a one-dimensional non-Hermitian sys tem,the ensuring temporal interface excites both reflected and refracted wave modes. By intro ducing a chiral-symmetric Hamiltonian, this study reveals the topological effects at such temporal interfaces. We find that the reflection and refraction coefficients exhibit a topological braiding struc ture. This structure is directly determined by the difference in the topological invariants across the interface, establishing a bulk-boundary correspondence for temporal interfaces in non-Hermitian systems. Furthermore, we propose a dynamical probe that leverages the geometric similarity of eigenstates at the temporal interface to detect topological phase transitions. These findings estab lish a fundamental connection between topological braiding and nonreciprocal dynamics at temporal interfaces, providing a platform to explore phase transition detection and nonreciprocal phenomena in time-varying non-Hermitian systems.

Figures (4)

• Figure 1: (a) Coupled resonator ring. (b) Synthetic frequency lattice formed by symmetric (pink dots) and antisymmetric (blue dots) modes. (c) Geometric similarity between the state vectors of the upper and lower bands after the temporal interface and the state vector of the upper band before the temporal interface, when the topological phases on the two sides differ. The reference coupling strength is denoted as$g_0 = 1$, the nonreciprocal coupling strength $\gamma = g_0/10$. Before the time interface, $g_{1i} =g_0/2$, $g_{2i} =g_0 \pm \gamma/2$, After the time interface, $g_{1f} =2g_0$, $g_{2f} =g_0 \pm \gamma/2$ (d) Same as (c), but for the case where the topological phase remains unchanged across the temporal interface. Before the time interface, $g_{1i} =3g_0/2$, $g_{2i} =g_0 \pm \gamma/2$, After the time interface, $g_{1f} =2g_0$, $g_{2f} =g_0 \pm \gamma/2$
• Figure 2: (a) Interwoven reflection (blue curve) and refraction (red curve) coefficients when topological phases differ across the temporal interface, plotted against the toroidal direction $k$ and poloidal coordinate $r$. On the Bloch sphere, red/blue shading indicates the distribution of the state vector and $\boldsymbol{b_k}$ for the upper/lower band; poles are labeled with $|0\rangle$ and $|1\rangle$. The $\boldsymbol{b_k}$ curve crosses a pole only at the topological phase transition. Diamond symbols marking the transition points at the geometric similarity curves $Q$[see Fig. 1(c)] on the $\boldsymbol{b_k}$ trajectory. (b) Interwoven reflection (blue) and transmission (red) coefficients when topological phases are identical across the temporal interface.
• Figure 3: Higher-order topological phase. (a) Topological phase diagram with fixed intracell coupling strength $g_1=2g_0$, intercell coupling strength $g_2=g_0/2{\pm}\gamma/2$, and next-nearest-neighbor coupling strengths $g_3=-2g_0$ (triangle), $g_3=-g_0/2$ (square), $g_3=g_0$ (diamond), and $g_3=2.5g_0$ (star), corresponding to winding numbers of 1, 0, 0, and 2, respectively. (b) Schematic of the temporal interface effect. (c) The system undergoes no topological phase transition. (d) The winding number difference across the temporal interface is 1. (e) The winding number difference across the temporal interface is 2.
• Figure 4: Heatmaps of wavefunction intensity evolution across a temporal interface. Color bars represent the base-10 logarithm of the wavefunction intensity modulus squared. The horizontal and vertical axes correspond to the Brillouin zone wavevector and time, respectively. $g_2$ remains constant across the temporal interface, where $g_2=g_0\pm0.05g_0$, with next-nearest-neighbor and longer-range couplings neglected. (a) Forward transmission with $g_1$ switching from 1.2$g_0$ to 0.8$g_0$ at the temporal interface. (b) Reverse transmission, complementary to (a), with $g_1$ switching from 0.8$g_0$ to 1.2$g_0$.