Fig. 1. Experimental setup for non-unitary quantum walks with alternating losses^[200]. A photon pair is generated in a Bell state through spontaneous parametric down-conversion. One photon serves as a trigger and can be prepared in any desired polarization state using a quarter-wave plate (QWP), a half-wave plate (HWP), and a polarizer for projection measurement. The other photon undergoes a series of processes, including controlled polarization rotations, polarization-dependent loss operations, and spatial translations. To explore the topological characteristics of the quantum walk, unmounted HWPs are strategically placed along specific paths to introduce spatially varying polarization rotations. The resulting probability distribution is recorded using avalanche photodiodes.

Fig. 2. (a) Conceptual illustration of the quantum walk dynamics^[207]. The shift operator S and coin operator R are depicted, with lattice sites represented at the intersection points of the laser beams (red). Quasiperiodic phase modulation is introduced by the phase operator P, illustrated as the dashed gray curves. (b) Detailed experimental setup using a time-multiplexed architecture. Key components include: a pulsed laser source, a neutral density (ND) filter for attenuation, HWPs for polarization control, an electro-optic modulator (EOM) for phase modulation, polarizing beam splitters (PBSs) for routing photons, and avalanche photodiodes (APDs) for detection. The two fiber loops with different lengths encode the walker’s position in the time domain, enabling the simulation of non-Hermitian quasicrystals.

Fig. 3. Integrated circuit for disordered quantum walks^[210]. (a) A schematic of a network of directional couplers is shown, implementing an eight-step one-dimensional quantum walk with static disorder. Different phase shifts are represented by distinct colors, while violet waveguides indicate accessible paths. (b) The optical path length is controlled by deforming one of the two S-bent waveguides at the output of each directional coupler, effectively functioning as a phase shifter. (c) The deformation is modeled by a nonlinear coordinate transformation dependent on a deformation coefficient. The plot compares the undeformed (solid line) and deformed (dashed line) S-bends. (d) A Mach–Zehnder structure is used as the unit cell for the directional coupler network, which is fabricated for calibrating the phase shift induced by the deformation. (e) The phase shift induced by the deformation is presented, showing the theoretical curve (solid line) calculated from the geometric deformation and experimental data (diamonds) confirming the results.

Fig. 4. A probe beam is launched into a disordered photonic lattice, which is periodic in the transverse x–y dimensions and invariant along the propagation direction (z). The photonic lattice features a triangular arrangement with a periodicity of 11.2 µm and a refractive index contrast of approximately 5.3×10−4^[217].

Fig. 5. Experimental setup for the study of exceptional-point sensors^[223]. (a) A tunable laser diode (TLD) injects probe light into a whispering-gallery-mode (WGM) microtoroid resonator via a fiber-taper waveguide. The injection direction—CW or CCW—is selected using an optical switch (OS). Fiber polarization controllers (PCs) are employed to optimize the coupling efficiency. Transmitted and reflected signals are detected at the output ports by photodetectors (PDs) and recorded through an oscilloscope (OSC). The relative positions of the resonator and nearby silica nano-tips are precisely controlled using nano-positioning stages. Additional components include an optical circulator (C). (b) Optical microscope image showing the microtoroid resonator coupled to the fiber-taper and surrounded by three silica nano-tips. (c) Scanning electron microscope (SEM) image of the microtoroid, with measured major and minor diameters of approximately 80 and 100 µm, respectively.

Fig. 6. (a) Quasi-energy spectrum of the effective Hamiltonian of non-Hermitian quantum walks under the PBC. The red lines represent the analytical results, and the blue dots represent the numerical calculation. (b) Distribution of eigenstates. The parameters selected are θ1=π/5, θ2=0, and γ=0.2.

Fig. 7. (a) Eigenenergy spectrum of non-Hermitian quantum walks under the OBC. (b) Distribution of eigenstates. For all eigenstates, the distributions are localized near the boundary. The parameters selected are the same as those under the PBC.

Fig. 8. (Upper row) Spatial probability distributions of a seven-step unitary quantum walk with γ=0 and different initial states |Φ(0)⟩. Both the time-dependent probability distribution (a), (c) and the distribution at the last step (b), (d) are shown. (Lower row) Spatial probability distributions of a seven-step non-unitary quantum walk with γ=0.2746 and different initial states |Φ(0)⟩. Both the time-dependent probability distribution (e), (g) and the distribution at the last step (f), (h) are shown^[232].

Fig. 9. (a) BZ and GBZ in the complex plane. The red solid line represents the GBZ, while the black dashed line represents the BZ. The red circle is smaller than the black circle, indicating that all bulk states are localized at the left boundary position. (b) The eigenenergy spectrum under the GBZ. (c) The analytical results of the eigenstates, which are localized at the left boundary position. The parameters selected are θ1=π/5, θ2=0, and γ=0.2.

Fig. 10. Non-Bloch topological invariants are calculated with Eq. (59). Non-Bloch topological invariants (a) ν˜0 and (b) ν˜π. The red (blue) solid line represents ν˜L (ν˜R).

Fig. 11. Detecting non-Bloch winding numbers^[233]. Topological phase diagrams under the (a) non-Bloch and (b) Bloch band theories. Different topological phases are characterized by the non-Bloch (Bloch) winding numbers νβ(ν) as functions of the coin parameters (θ1,θ2). (c) The measured average chiral displacement C¯ (blue symbols) for six-step quantum walks. (d) The measured Δn¯z (blue symbols) along the blue path for six-step quantum walks.

Fig. 12. Results of non-Bloch topological invariants. (a) Quasienergy spectrum’s absolute values for various θ1R. (b) Winding number differences between the two bulks for the π- and zero-modes (blue and red, respectively). θ1L=0.2π, θ2R=0.1π, θ2L=−0.7π, and γ=0.2 are the parameters.

Fig. 13. Experimental measurements of photon dissipations^[235]. (a), (b) Photon dissipation probabilities. (c), (d) Accumulated dissipations P(n) that correspond, respectively, to (a) and (b), with experimental (theoretical) results drawn in red (blue).

Fig. 14. Sketch of the non-Hermitian disordered SSH model^[251]. The dotted box represents the unit cell, with mj and tj(l;r) indicating Hermitian intercellular and nonreciprocal intercellular hoppings, respectively.

Fig. 15. (a) Disorder-averaged winding number ν as a function of non-Hermiticity γ and disorder strength W^[251]. f(γ)=t′γ. (b) ν as a function of γ and W for a nonreciprocal term f(γ)=t′(1−γ+γ2), with t′=t. Other parameters are t=1, t′=1.2t.

Fig. 16. Lyapunov exponent for bulk dynamics^[206]. (a), (b) Polarization-averaged growth rates λ¯(v) and (c), (d) polarization-resolved photon distribution for Hermitian quantum walks with γ=0 (first column) and non-Hermitian quantum walks with γ=0.1 (second column). The horizontal dashed lines indicate the threshold below which significant photon loss renders experimental data unreliable. The numerical findings for horizontal (vertical) polarizations are represented by the blue top (red bottom) parts for each bar in (c) and (d). The experimental probability distributions for vertical polarization (the total of the two polarizations) are represented by the white (black) dots.

Fig. 17. Measured λ¯(v) and polarization-resolved probability distributions with non-zero W^[206]. The other parameters of the quantum walks are θ1=4.3, θ2=2.175, and γ=0.1. It should be noted that the system is topologically trivial in (b) but topologically nontrivial in (a) and (c). The results are averaged over 20 disorder setups in the upper panel. The same symbols as in Fig. 16 are used here. (d)-(f) Experimental data (symbols) and numerical results (bars) corresponding to (a)-(c) in Ref. [206].

Fig. 18. Results for characterizing topology with W=0.25^[206]. (a) Phase diagram characterized by numerically evaluated biorthogonal local marker, where θ1=4.3+δθ(x) and γ=0.1. The non-Hermitian topological Anderson insulator state corresponds to the yellow region with W>0, which is labeled by NHTAI. (b) Measured chiral displacement for nine steps of quantum walks with θ2=2.175 averaged over 10 different configurations of δθ(x) randomly chosen within the range [−W,W]. The blue (red) symbol represent the result for γ=0 (γ=0.1). The blue and red dashed lines represent numerical results averaged over 2000 random-disorder configurations. The blue and red solid lines represent numerical results for 400-step quantum walks averaged over 200 disorder configurations. (c), (d) Measured probability distributions near the boundary (the vertical dashed lines) after the final step.

Fig. 19. Distinctive contours (ECs) present in the non-Hermitian Weyl semimetal model (NH WSM)^[255]. It displays both the real (green) and imaginary (red) parts of the exceptional surfaces for cases (a) γ=0.3 and (b) γ=0.5. The intersections of these surfaces correspond to the exceptional contours. It is noteworthy that, with an increase in the strength of non-Hermiticity, the size of the ECs expands accordingly. Additionally, four EPs (indicated as blue dots) form at kx=ky=0 along the kz axis.

Fig. 20. Anderson localization in the non-Hermitian quasicrystal^[261]. (a) Dynamic inverse participation ratio (dIPR, blue) and dynamic normalized inverse participation ratio (dNIPR, red) are plotted with the initial state set to |x=0⟩⊗|V⟩. Solid symbols correspond to experimental results, while hollow symbols indicate numerical simulations. (b) Numerical results for the average inverse participation ratio (IPR¯, blue) and average normalized participation ratio (NPR¯, red) are shown for a lattice size of N=1000 under the condition of γ=0.1.

Fig. 21. PT-symmetry transition in the non-Hermitian quasicrystal^[261]. (a) Imaginary components of the quasienergies of the non-Hermitian quantum walk are presented for a lattice size of N=200 with γ=0.1. The vertical dashed line at θ/π4≈1.25 indicates the PT-symmetry transition point. (b) Corrected overall probabilities P(t) are illustrated for the unbroken and broken phases of PT symmetry. Symbols represent experimental data, while solid lines indicate theoretical predictions.

Fig. 22. Conceptual illustration of light funneling^[267]. The lattice design directs light from any point toward an interface, dubbed funnel opening, where it is efficiently collected.

Fig. 23. (a) Parametric path, δ1(t)=0.09sin(±2πt/T), Ω1(t)=0.08+0.09cos(±2πt/T). Trajectories of the CW rotation (b) and the CCW rotation (c) against the eigenspectra of the non-Hermitian Hamiltonian H. The red (blue) color code indicates the eigenstate with a larger (smaller) imaginary component, and hence smaller (larger) loss. The black triangles represent the initial states, the black squares denote the final states, and the black lines illustrate the evolution trajectories.