Scattering States in One-Dimensional Non-Hermitian Baths

Abstract

A single quantum emitter coupled to a structured non-Hermitian environment shows anomalous bound states and real-time dynamics without Hermitian counterparts, as shown in [Gong et al., Phys. Rev. Lett. 129, 223601 (2022)]. In this work, we establish a general approach for studying the scattering states of a single quantum emitter coupled to one-dimensional non-Hermitian single-band baths. We formally solve the exact eigenvalue equation for all the scattering states defined on finite periodic lattices. In the thermodynamic limit, the formal solution reduces to the celebrated Lippmann-Schwinger equation for generic baths. In this case, we find that the scattering states are no longer linear superpositions of plane waves in general, unlike those in Hermitian systems; Instead, the wave functions exhibit a large, yet finite localization length proportional to the lattice size. Furthermore, we show and discuss the cases where the Lippmann-Schwinger equation breaks down. We find the analytical solutions for the Hatano-Nelson and unidirectional next-to-nearest-neighbor baths in the thermodynamic limit.

Figures (8)

• Figure 1: Illustration for a single quantum emitter (two-level) coupled to 1D nearest-neighbor hopping NH baths with $|x' - x| = 1$. Other structured baths are also studied in this work. The wave indicates that the propagation of the photon is scattered by the emitter.
• Figure 2: The magnitudes of finite-size self-energy Eq. (\ref{['SL']}) evaluated for the HN and unidirectional NNN baths. For the same randomly selected $E = h_{k}$, the finite-size self-energies do not converge with the system size.
• Figure 3: HN baths for $u=6,\ \kappa=2,\ J=20,\ \Delta=2.14$ and $L=801$. (a) The single-excitation spectrum of the $H$ obtained from exact diagonalization is shown in blue. The black dotted line indicates the dispersion $h^{\text{NH}}_{k}$. The pink dashed vertical lines indicate all the bounds states, and the positions are analytically predicted by the relevant emitter Green's functions (see main text for the numerical values). An eigenvector is selected at random (red marker) and shown in (b) (blue), which shows a good agreement with the analytical LS wave function under PBC (see main text for the discussion). We have normalized the LS wavefunction Eq. (\ref{['eq:LS-wavefunction-HN']}) by dividing $\sqrt{L}$, such that Eq. (\ref{['eq:Normalization']}) holds.
• Figure 4: For each system sizes, the value of ${\rm Im}\tilde{k}$ changes for a fixed scattering states with an exact eigenvalue $E$. The difference between ${\rm Im}\tilde{k}$ and its leading order in $L^{-1}$ shows a $\mathcal{O}(L^{-2})$ scaling for both NN and NNN baths, as predicted by Eq. (\ref{['eq:imk']}).
• Figure 5: NNN baths for $\kappa=5, \kappa'=12, L=801, J=20, \Delta=2.14$. The single-excitation spectrum of the $H$ obtained from exact diagonalization is shown in blue. The black dotted line indicates the dispersion $h^{\text{NNN}}_{k}$. The pink dashed vertical lines indicate all the bound states. Two ED eigenvectors in $k_{1}$ and $k_{2}$ are chosen at random, indicated by the red and green markers. The pink marker indicates the self-intersecting state. The real space distributions of states indicated by the red, green, and pink markers are shown in Fig. \ref{['fig:fig3_2']}.