Fig. 1. Non-Hermitian LZ tunneling in synthetic temporal lattices. (a) Schematic diagram of two fiber loops connected by a variable optical coupler (VOC) for yielding synthetic temporal lattice. The phase modulations of ±ϕ(m) are introduced in the short and long loops through phase modulators (PMs). The intensity modulator (IM) is utilized to introduce loss with the loss rate γ. (b) Complex eigenvalues (quasi energies) in the space spanned by Bloch momentum and loss rate. Here, sin2(β)=0.05. Note the spectral phase transition at γ=γc. (c) Complex band structures for γ=0.1 and 0.6. (d) Numerical results of pulse intensity (total intensity of the short and long loops) evolutions under a driving electric field E=π/20 for γ=0, 0.1, and 0.6. The initial eigenmode is prepared at Q0=−π/2 and θ+=π/2.

Fig. 2. Method of the measurement. (a) Band structures under the modulation of gauge potentials for the truncating time step at mt=10 and mt=12. The red (blue) arrows indicate the group velocities of the upper (lower) band during evolution. The gray arrow indicates that the band structure undergoes a horizontal displacement in the presence of effective vector potential. The dotted gray curves depict the band structure after modulation as a reference. (b) Temporal evolution of effective gauge potential and electric field for the truncation time mt=10 and mt=12. Here Δϕ=(mt−10)E. (c), (d) Simulated intensity profiles in the long loop for mt=12 with Δϕ=0 (c) and Δϕ=π/10 (d). Here, E=π/20, sin2(β)=0.05, and γ=0.1.

Fig. 3. Experimental interference patterns of the long loop. (a), (b) Measured interference patterns for the truncation time mt=10 (a) and mt=12 (b). An abrupt change of gauge potential Δϕ=π/10 is applied in the latter case. Here, E=π/20, sin2(β)=0.05, and γ=0.1 are set in experiment. (c), (d) Measured field intensities (blue dots) at each time step are shown for mt=10 (c) and mt=12 (d).

Fig. 4. Measured dynamics of non-Hermitian LZ tunneling. (a)–(c) Measured band occupancies in adiabatic basis for γ=0, 0.1, and 0.6, respectively. (d)–(f) Same as (a)–(c) but for diabatic basis. The solid and dashed curves denote theoretical results and the experimental data are plotted in dots and circles. (g) Tunneling probabilities as a function of the loss rate in different bases. Influence of loss rate on (h) |c+|2 in adiabatic basis and (i) |cl|2 in diabatic basis. All have E=π/20 and sin2(β)=0.05.

Fig. 5. Non-orthogonality of eigenmodes. (a), (b) Total band occupancies in adiabatic (a) and diabatic (b) bases. (c) Overlap of the eigenmodes |⟨φ+|φ−⟩| in dependence on step m and loss rate γ. (d) Deviation of the total band occupancies in different bases. The solid curves represent the theoretical results, and the experimental data are plotted in dots.