Quantum-Squeezing-Induced Algebraic Non-Hermitian Skin Effects and Ultra Spectral Sensitivity

TL;DR

This paper addresses the emergence of algebraic non-Hermitian skin effects in higher-dimensional Hermitian systems by introducing a 2D bosonic lattice with on-site and off-site squeezing implemented within a Bogoliubov–de Gennes framework. It demonstrates quasi-long-range, power-law localization of bosonic quasiparticle excitations and reveals ultra spectral sensitivity to pairs of infinitesimal impurities and long-range hopping impurities, explained via a Green’s function analysis that highlights nonlocal bound-state formation. The key contributions include establishing algebraic NHSE in a Hermitian bosonic setting, showing robustness to lattice geometry, and deriving a nonperturbative spectral response mechanism, with potential realizations in superconducting circuits, photonic lattices, and optomechanical arrays. The findings offer a foundation for exploiting bosonic squeezing to realize and harness higher-dimensional non-Hermitian phenomena for quantum sensing and amplification before coupling to external reservoirs.

Abstract

The well-established non-Bloch band theory predicts exponential localization of skin-mode eigenstates in one-dimensional (1D) non-Hermitian systems. Recent studies, however, have uncovered anomalous algebraic localization in higher dimensions. Here, we extend these ideas to Hermitian bosonic quadratic Hamiltonians incorporating quantum squeezing, offering a genuine quantum framework to explore non-Hermitian phenomena without external reservoirs. We construct a two-dimensional (2D) bosonic lattice model with two-mode squeezing and study its spectral properties of bosonic excitation within the Bogoliubov-de Gennes (BdG) formalism. We demonstrate an algebraic non-Hermitian skin effect (NHSE), characterized by quasi-long-range power-law localization of complex eigenstates. The system shows ultra spectral sensitivity to double infinitesimal on-site and long-range hopping impurities, while remaining insensitive to single impurities. Analytical treatment via the Green's function reveals that this sensitivity originates from the divergence of the nonlocal Green's function associated with the formation of nonlocal bound states between impurities. Our study establishes a framework for realizing novel higher-dimensional non-Hermitian physics in Hermitian bosonic platforms such as superconducting circuits, photonic lattices, and optomechanical arrays, with the demonstrated ultraspectral sensitivity enabling quantum sensing and amplification via bosonic squeezing.

Figures (6)

• Figure 1: Schematic of a two-dimensional quadratic bosonic Hermitian lattice system. $J_x$, $J_y$, and $J_{xy}$ represent single-particle hopping amplitudes along the $x$ direction, $y$ direction, and anti-diagonal directions, respectively. $\Delta_0$ characterizes on-site quantum squeezing, and $\Delta_x$ accounts for off-site quantum squeezing between adjacent sites along the $x$ direction.
• Figure 2: (a,e) Complex eigenenergies (orange dots) of quasiparticle excitations computed from $\mathcal{M}_\text{B}$ under the square geometry with OBCs for (a) $(J_x,J_y,J_{xy},\Delta_0,\Delta_x)=(1i,1,3i,-1,2i)$, and (e) $(J_x,J_y,J_{xy},\Delta_0,\Delta_x)=(0,1i,4i,3,2)$. Gray regions mark the PBC spectra. The corresponding eigenenergy-resolved fractal dimensions $\mathcal{D}[\psi_i]$ are shown in (b,f). (c,g) Probability densities $\abs{\Psi_i(\mathbf{r})}^2 = \abs{\psi_{p,i}(\mathbf{r})}^2 + \abs{\psi_{h,i}(\mathbf{r})}^2$ for the eigenstates with $E_1=-0.97+2.43i$ from (a) and $E_2=2.09+9.23i$ from (e). (d,h) Layer-resolved densities $\mathcal{P}_1(x)$ and $\mathcal{P}_2(x)$ for $E_1$ and $E_2$, shown on a logarithmic scale (black dots). Red and blue curves represent exponential and power-law fits, respectively. (i,k) $\mathcal{D}[\psi_i]$ under oblique-square geometries with varying oblique angles for $(J_x,J_y,J_{xy},\Delta_0,\Delta_x)=(1i,1,3i,-1,2i)$. (j,l) Corresponding probability densities $\abs{\Psi_i(\mathbf{r})}^2$ for eigenstates with $E_3=-0.18+3.18i$ from (i), and $E_4=-0.97+2.77i$ from (k).
• Figure 3: Complex eigenenergies of quasiparticle excitations (a,e) under PBCs along both the $x$ and $y$ directions, (b–d) and (f–h) under OBC along the $x$ direction and PBC along the $y$ direction. Panels (c,g) correspond to a single impurity with sufficiently weak onsite potential $V_1=0.01$ ($V_2=0$) coupled to the left edge, while panels (d,h) correspond to two distant impurities with sufficiently weak onsite potentials $V_1=0.01$ and $V_2=0.01$, each coupled to the left and right edges, respectively, as indicated by green dots in the top schematic plot. In (d), the red arrow marks states that disappear when two impurities are present, whereas in (d,h), the black arrows denote states induced solely by the two impurities. The parameters are $(J_x,J_y,J_{xy},\Delta_0,\Delta_x)=(1i,1,3i,-1,2i)$ for (a–d) and $(J_x,J_y,J_{xy},\Delta_0,\Delta_x)=(0,1i,4i,3,2)$ for (e–h). The lattice size is $L_x\times L_y=50\times 50$, with impurities positioned at $\mathbf{r}_1=(1,1)$ and $\mathbf{r}_2=(L_x,L_y/2)$.
• Figure 4: Complex quasiparticle eigenenergies in the presence of a long-range hopping impurity under OBC along the $x$ direction and PBC along the $y$ direction. Panels (a) and (b) correspond to different hopping ranges: (a) $(x_1,y_1)=(1,1)$, $(x_2,y_2)=(3L_x/5, L_y/2)$ with $t_\textrm{p}=0.01$, and (b) $(x_1,y_1)=(1,1)$, $(x_2,y_2)=(L_x, L_y/2)$ with $t_\textrm{p}=0.005$. The arrows indicate the new states arising from the long-range hopping impurity. The other parameters are $(J_x,J_y,J_{xy},\Delta_0,\Delta_x) = (0,1i,4i,3,2)$ with $L_x \times L_y = 50 \time 50$.
• Figure 5: Two roots $\beta_{1,2}^+(k_y)$ and $\beta_{1,2}^-(k_y)$ of characteristic equations $E-E_+(\beta_x,k_y)=0$ and $E-E_-(\beta_x,k_y)=0$ with $\beta_x = e^{i k_x + \mu_x}$, where $E$ is outside the eigenspectrum of $\mathcal{M}_{p/m}$ with (a1-a4) $E = 0.85+7.59i$ indicated by the orange circle in Fig. \ref{['Fig3']}(h), and (b1-b4) $E = -2.12-7.63i$ indicated by the red circle in Fig. \ref{['Fig3']}(h). The red and blue dots represent the maximum values $\mu_\text{max,1}^\pm = \max_{k_y\in[0,2\pi]} \ln [\beta_1^\pm(k_y)]$, and minimum values of $\mu_\text{min,2}^\pm = \min_{k_y\in[0,2\pi]}\ln [\beta_2^\pm(k_y)]$.