Non-Hermitian tight-binding network engineering

We suggest a simple method to engineer a tight-binding quantum network based on proper coupling to an auxiliary non-Hermitian cluster. In particular, it is shown that effective complex non-Hermitian hopping rates can be realized with only complex onsite energies in the network. Three applications of the Hamiltonian engineering method are presented: the synthesis of a nearly transparent defect in an Hermitian linear lattice; the realization of the Fano-Anderson model with complex coupling; and the synthesis of a $\mathcal{PT}$-symmetric tight-binding lattice with a bound state in the continuum.

Figure 1

(a) Schematic of a tight-binding network S coupled to an auxiliary cluster A made of $M$ sites. (b) Control of hopping amplitude between sites $|n_0{\rangle }_S$ and $|m_0{\rangle }_S$ via an auxiliary site $|{\alpha }_0{\rangle }_A$.

Figure 2

(a) Schematic of a linear tight-binding lattice S side-coupled to an auxiliary site A. (b) Equivalent lattice after elimination of the auxiliary site, with energy-dependent renormalized hopping rate ${\theta }^{\prime }$ and site potential ${\sigma }^{\prime }$, defined by Eqs. (20) and (21) given in the text. (c) Behavior of the spectral transmittance ${\left|t\left(E\right)\right|}^2$ and phase of $t\left(E\right)$ vs energy $E=2\kappa cos\left(q\right)$ for parameter values $\theta /\kappa =0.2,\sigma /\kappa =-0.8$ and for $U/\kappa =-5,\omega /\kappa =2$ (curve 1); $U/\kappa =-10,\omega /\kappa =2\sqrt{2}$ (curve 2); $U/\kappa =-20,\omega /\kappa =4$ (curve 3); and $U/\kappa =-40,\omega /\kappa =4\sqrt{2}$ (curve 4).

Figure 3

(a) Schematic of a bound state coupled to a continuum of states. When the energy of the bound state is not embedded in the continuous spectrum, for small coupling $g$ there is not a decay while the energy of the bound state is renormalized. The ghost regime of the Lee model is realized when the coupling $g$ to the continuum is imaginary ($g=-iG$). (b) Tight-binding lattice realization of the Lee model in the ghost regime. The continuum of states is given by the Bloch modes of a semi-infinite homogeneous lattice. The end of the semi-infinite lattice is attached by complex coupling $g$ to the localized state $|0\rangle$. (c) Phase diagram of the Lee model. Below the curve 2 the system is in the unbroken $\mathcal{PT}$ phase, with one bound state in region I (the so-called physical particle state) and two bound states in region II (the so-called physical particle state plus the so-called ghost). Above curve 2 the $\mathcal{PT}$ symmetry is broken. Analytic equations of curves 1 and 2 are $G=\kappa \sqrt{-2+(\sigma /\kappa )}$ and $G=\kappa \sqrt{-1+{(\sigma /2\kappa )}^2}$, respectively. (d) Implementation of the non-Hermitian (imaginary) coupling by an auxiliary site ${|1\rangle }_A$. (d) Numerically computed evolution of the occupation probability $P\left(t\right)$ as obtained from the exact Lee Hamiltonian (solid curve) and from the synthesized lattice (dashed curve). Parameter values are given in the text.

Figure 4

(a) Schematic of the $\mathcal{PT}$-symmetric tight-binding lattice, with Hermitian hopping amplitudes and imaginary potentials $\pm iU$ at the auxiliary sites ${|1\rangle }_A$ and ${|2\rangle }_A$, that admits of a bound state in the continuum. The hopping rates ${\kappa }_n$ ($n\ne 0,1$) are defined by Eq. (37) given in the text. (b) Equivalent tight-binding lattice model obtained after elimination of the two auxiliary sites. The energy-dependent hopping amplitudes ${\theta }_{0,1}$ and site potentials ${\sigma }_{0,1,2}$ are given by ${\theta }_0\left(E\right)={\omega }^2/(E+iU),{\theta }_1\left(E\right)={\omega }^2/(E-iU),{\sigma }_{-1}\left(E\right)={\omega }^2/(E+iU),{\sigma }_0\left(E\right)=2E{\omega }^2/(E^2+U^2)$, and ${\sigma }_1\left(E\right)={\omega }^2/(E-iU)$. (c) Numerically computed energy spectrum of the tight-binding lattice of Fig. 4 for $\omega /\kappa =1$ and $U/\kappa =0.4$. The lattice comprises $N=403$ sites. Eigenmodes are ordered for increasing values of eigenenergy. The two arrows in the figure highlight the existence of four bound states with energies in the gap, two above and the other two below the tight-binding energy band. (d) Numerically computed participation ratio $R$ of the lattice eigenmodes. Localized modes, corresponding to low values of $R$, are highlighted by the arrows in the figure. The central eigenmode, with energy $E_0=0$ in the middle of the allowed band, corresponds to the BIC state.