Confined non-Hermitian skin effect in a semi-infinite Fock-state lattice

TL;DR

Addresses how inhomogeneous, $\sqrt{n}$-scaled coupling affects the non-Hermitian skin effect in a semi-infinite Fock-state lattice. An exact analytical solution is obtained by a similarity transformation that maps the effective non-Hermitian SSH Hamiltonian to a Jaynes–Cummings model, yielding real eigenvalues and a biorthogonal eigenbasis. The findings reveal a confined non-Hermitian skin effect where bulk eigenmodes localize within a finite range dictated by the inhomogeneous profile, and nonreciprocity enforces a one-sided dynamical trajectory. The work enables tunable skin effects through engineered coupling profiles in synthetic dimensions and proposes a trapped-ion platform for experimental realization.

Abstract

In this paper, we investigate the non-Hermitian skin effect in a semi-infinite Fock-state lattice, where the inherent coupling scales as \sqrt{n}. By analytically solving a non-uniform, non-reciprocal SSH model, we demonstrate that the intrinsic inhomogeneous coupling, in combination with nonreciprocity, fundamentally modifies the conventional skin effect. Instead of accumulating at the physical boundary, all eigenmodes become compressed and skewed within a finite spatial range determined by the inhomogeneous profile-a phenomenon we term the confined non-Hermitian skin effect. Consequently, the evolution of the probability distribution on the lattice starting from a single site is doubly confined: it is spatially bounded to a finite range by the inhomogeneous coupling, and further restricted to a one-sided trajectory at the edge of this range by the non-reciprocity. Moreover, a feasible experimental scheme based on a single trapped ion is also proposed. This work reveals how engineered coupling profiles in synthetic dimensions can reshape non-Hermitian properties and enable new protocols for quantum state manipulation.

Figures (6)

• Figure 1: Schematic of the nonreciprocal, semi-infinite SSH model. (a) Decomposition of the system into two degrees of freedom: an internal three-level subsystem and external harmonic motion. (b) Coupling scheme among the energy levels. The coupling strengths $J_1$, $J_2$, and $J_3$ are all real. All couplings are resonant except for $J_2$, which is red-detuned by a phonon frequency $v$. The auxiliary level $|f\rangle$ decays to other levels (not shown) at a rate $2\gamma$. (c) Coupling structure between lattice sites, where each triangle $\{ |g,n\rangle,\ |e,n\rangle,\ |f,n\rangle \}$ represents a unit cell.
• Figure 2: Probability distribution of the normalized right eigenvector with respect to the unit-cell index. Subfigures (a)-(d) respectively display the distributions for the states $|\psi_0^R\rangle$, $|\psi_{0,\pm}^R\rangle$, $|\psi_{10,\pm}^R\rangle$ and $|\psi_{50,\pm}^R\rangle$, each compared across the three parameter sets: ${J_3=0, \phi=0}$ (blue), ${J_3=3, \phi=\pi/2}$ (green), and ${J_3=3, \phi=-\pi/2}$ (purple). The remaining parameters are fixed at $J_1=1.5$ , $J_2=1$ and $\gamma=50$.
• Figure 3: Comparison of the mean position $\langle \hat{n} \rangle$ versus eigenenergy $E$ between uniformly and inhomogeneously coupled nonreciprocal SSH models. Here, $\hat{n} = a^\dagger a$ is the phonon number operator, corresponding to the unit-cell index in the lattice representation. Panels (a,b) present the uniformly coupled model (removing the $\sqrt{n+1}$ coefficient in Eq. (\ref{['ham_effnew']}), with total cell number $N = 100$). Panels (c,d) show the semi-infinitely long inhomogeneously coupled model (displaying the zero mode and its nearby 198 modes). Parameters: $J_2 = 1$ and $\gamma=50$ are fixed. Panels (a,c): $J_1 = 0.6$; Panels (b,d): $J_1 = 1.5$. Other settings: $J_3 = 0$ (red open triangles), $J_3 = 3$, $\phi = \pi/2$ (blue open diamonds), $J_3 = 3$, $\phi = -\pi/2$ (green open circles).
• Figure 4: Comparison of the IPR versus eigenenergy $E$ between uniformly and inhomogeneously coupled nonreciprocal SSH models. Panels (a)-(d) use the same parameter sets and share the same marker scheme as the corresponding panels in Fig. \ref{['figure3']}.
• Figure 5: Normalized time evolution in the inhomogeneous SSH model from the initial state $|g,40\rangle$. Parameters: $J_2 = 1$ and $\gamma = 50$ are fixed. The left and right columns correspond to $J_1 = 0.6$ and $J_1 = 1.5$, respectively. From top to bottom, the three rows represent: $J_3 = 0$ (panels (a,b)), $J_3 = 3$, $\phi = \pi/2$ (panels (c,d)), and $J_3 = 3$, $\phi = -\pi/2$ (panels (e,f)).