Quantum transport in non-Hermitian impurity arrays

We study the formation of band gap bound states induced by a non-Hermitian impurity embedded in a Hermitian system. We show that a pair of bound states emerges inside the band gap when a parity-time $\left(\mathcal{PT}\right)$ imaginary potential is added in strongly coupled bilayer lattices and the bound states become strongly localized when the system approaches the exceptional point. As a direct consequence of such $\mathcal{PT}$ impurity-induced bound states, an impurity array can be constructed and protected by energy gap. The effective Hamiltonian of the impurity array is non-Hermitian Su-Schrieffer-Heeger type and hosts Dirac probability-preserving dynamics. We demonstrate the conclusion by numerical simulations for the quantum transport of wave packet in right-angle bends waveguide and $Y$-beam splitter. Our finding provides an alternative way to fabricate quantum device by non-Hermitian impurity.

Figure 1

(a) Schematic illustration of a two-layer tight-binding lattice with non-Hermitian imaginary tunneling. (b) Band structures for $H_0$ and $H$. In the absence of the interlayer tunneling, a gap in order of $2\mathrm{\Delta }$ opens. When a single tunneling $i{T}_j$ switches on, two isolated energy levels can be achieved around the midgap.

Figure 2

(a) Schematic illustration of a bichain lattice with opposite chemical potentials $\pm \mathrm{\Delta }$ and a single non-Hermitian tunneling $iT$. The intra- and interlayer hopping strengths are $\kappa$ and $iT$, respectively. (b) Profile of one of the two bound states for single imaginary tunneling $iT$. The ordinates of blue (solid) and red (dash) lines are real and imaginary numbers, which represent the wave function of the top and bottom chains, respectively. The parameters are $\mathrm{\Delta }=5,\kappa =1$, and $T=4$. Panels (c1) and (c2) are profiles of two of the four bound states for double imaginary tunneling $iT$. The parameters are $\mathrm{\Delta }=5,\kappa =1$, and $T=4$. Panels (d1)–(d3) are profiles of three of the ten bound states induced by multi(five) impurities with parameters $\mathrm{\Delta }=5,\kappa =1$, and $T=4$. The results are obtained by numerical diagonalization and the wave functions are Dirac normalized.

Figure 3

(a) Schematic illustration of a bilayer square lattice model with $\kappa$ (gray line) and $\mathrm{\Delta }$ (black line) are intra- and interlayer hopping strengths; the non-Hermitian imaginary potentials $\pm i\gamma$ are indicated by the blue $\left(i\gamma \right)$ and yellow $\left(-i\gamma \right)$ dots, and the white dots indicate zero on-site potentials. (b) The waveguide path [red dash lines in (a)] forms a non-Hermitian ladder model.

Figure 4

(a) Schematic illustration of the waveguide path, which is obtained by taking the nonzero non-Hermitian imaginary potentials $\pm i\gamma$ along the path. (b)–(e) Numerical simulations of the dynamics in waveguide. The initial state is a bilayer Gaussian wave packet. Probability distribution at $t=0,t=T_t/2$, and $t=T_t$ are shown in (b), (c), and (d), respectively. (e) The trace of the wave packet. Parameters for the system are $t=1,\mathrm{\Delta }=15$, and $\gamma =11$; the size of the system is $100\times 60\times 2$. Parameters for the initial state Eq. (21) are $\alpha =0.4,N_{\mathrm{c}}=10$, and $k_{\mathrm{c}}=-\pi /2$. The total duration of the simulation is $T_t=42J^{-1}$, where $J$ is the scale of the Hamiltonian and we take $J=1$.

Figure 5

(a) Schematic illustration of the beam splitter waveguide. In order to get high transmission rate, we take $\kappa \to \kappa /\sqrt{2}$ for the hopping (labeled by green arrows) connecting the joint. (b)–(e) Numerical simulations of the dynamics in waveguide. The initial state is a bilayer Gaussian wave packet. Probability distribution at $t=0,t=T_t/2$, and $t=T_t$ are shown in (b), (c), and (d), respectively. (e) The trace of the wave packet. Parameters for the system are $t=1,\mathrm{\Delta }=15$, and $\gamma =11$; the size of the system is $100\times 60\times 2$. Parameters for the initial state Eq. (21) are $\alpha =0.4,N_{\mathrm{c}}=10$, and $k_{\mathrm{c}}=-\pi /2$. The total duration of the simulation is $T_t=32J^{-1}$, where $J$ is the scale of the Hamiltonian and we take $J=1$.

Figure 6

Spectra of (a) ladder model and (b) SSH model with the core matrix Eq. (A39) and Eq. (A40) for different $\gamma$. The Parameters are $\kappa =1,\mathrm{\Delta }=5$, and $\gamma =0,2.5,3$.