Non-Hermitian interferometer: Unidirectional amplification without distortion

A non-Hermitian interferometer can realize asymmetric transmission in the presence of imaginary potential and magnetic flux. Here, we propose a non-Hermitian dimer with an unequal hopping rate by an interferometerlike cluster in the framework of a tight-binding model. The intriguing features of this design are the wave-vector independence and unidirectionality of scattering, which amplify the wave packet without distortion and absorb the incoherent wave without reflection. The absorption relates to the system spectral singularities. The dynamical behaviors of the spectral singularities are also investigated analytically and numerically.

Figure 1

Schematic illustration of the associated effect of imaginary potential and magnetic flux. (a) Transmission of imaginary potential. (b) Wave vector under magnetic flux. (c),(d) A non-Hermitian ring with the non-Hermiticity arising from the imaginary potentials. The magnetic flux breaks left-right symmetry. Initially, a stationary wave packet (blue) with vector $k_0=0$ is located at the (c) left or (d) right, respectively. In both cases, the central momentum of the wave packet can be shifted to $\pi /2$ by a well-prepared flux. It makes two wave packets face two different situations. (c) The wave packet moves up and encounters the potential $2i$. It acquires an infinite transmission. (d) The wave packet moves down and is scattered by the potential $-2i$ and gets a finite transmission.

Figure 2

Sketch of the system considered in this work and schematic illustration of the equivalent Hamiltonian with asymmetric transmission. The black bonds denote Hermitian hopping, while the blue arrowed bond denotes the hopping with asymmetric amplitudes. (a) The non-Hermitian scattering center embedded in a discrete waveguide. It consists of Hermitian hopping, magnetic flux, and imaginary potentials. The coexistence of flux and imaginary potentials breaks the symmetry of transmission. (b) The equivalent system of (a), in which the scattering center is simply an asymmetric dimer with unequal hopping strength $(\mu ,\nu )$.

Figure 3

The profiles of the Dirac norm for the time evolution of initial Gaussian wave packets with $w=0.15$ and several typical $k_0$ under the systems with parameters $\delta =-1.25$ and $\gamma =0.75$ but different fluxes. (a) $k_0=\pi /2$ and $\varphi =\pi /4$. The blue line represents the time evolution at several typical instants. As a comparison, the red circle shows the time evolution of the same initial state under the system with parameters $\delta =-1$ and $\gamma =0$, which corresponds to a uniform chain. It shows a perfect reflectionless amplification with $\nu =2$ . (b)–(d) show the time evolution at instant $t=70$ for different initial states with (b) $k_0=\pi /3$, (c) $k_0=\pi /2.5$, and (d) $k_0=\pi /2$, respectively. Here, $\varphi$ are deviated from $\pi /4$ in units of $d=\pi /100$. It shows that this amplifier is immune to the deviation of $\varphi$ for the signal around $k_0=\pi /2$.

Figure 4

The profiles of the Dirac norm for evolved states of four types of initial states: (a) the state defined by $\left|\alpha \right.〉+i\nu \left|\beta \right.〉$, (b) the state defined by $\left|\alpha \right.〉-i\nu \left|\beta \right.〉$, (c) a Gaussian wave packet with $w=0.15$ and $k_0=\pi /2$, and (d) the state defined by Eq. (27) with $w=0.15$ and $k_0=\pi /2$. The time evolutions are governed by the Hamiltonian with $\delta =0.75$ and $\gamma =1.25$ for all figures, which correspond to $\mu =-2.0$ and $\nu =0.5$, satisfying the spectral singularity condition $\mu \nu =-1.0$.

Figure 5

Schematic illustration of a non-Hermitian configuration to demonstrate the perfect absorption for the incoherent state. It is the same system as Fig. 2 with the open boundary condition at the left side. (a) Initially, a mixed state is located in the region $[-N_0,-1]$. (b) The incoherent perfect absorption is achieved if the total Dirac probability in the whole system vanishes at relaxation time $\tau$.

Figure 6

The plots of $P\left(t\right)$ for the initial state in Eq. (42), obtained from the time evolution under the configuration illustrated in Fig. 5 with $N_0=20$. The time is dimensionless and in units of 1. The inset parameters $(\delta ,\gamma )$ correspond to $\nu =0.5$, 0.4, and 0.1, respectively.