Non-Hermitian Continuous-Variable Systems Overview

• Non-Hermitian continuous-variable systems are quantum models governed by complex Hamiltonians that exhibit exceptional points, mobility edges, and the non-Hermitian skin effect.
• They enable advanced quantum control and simulation by leveraging gauge potentials, nontrivial boundary conditions, and robust state transfer protocols.
• Applications span photonics, ultracold atoms, and topological materials, offering novel insights into open-system dynamics and quantum metrology.

Non-Hermitian continuous-variable systems comprise quantum or wave-mechanical models whose governing equations—typically of Schrödinger or related form—feature non-Hermitian operators acting on fields defined over continuous spatial (and/or momentum) degrees of freedom. These models exhibit phenomena fundamentally absent in Hermitian counterparts, such as exceptional points, the non-Hermitian skin effect, nontrivial spectral topology, mobility edges, and anomalous localization. The field spans fundamental mathematical constructs, quantum simulation platforms, photonic systems, and engineered open-system dynamics, with direct relevance to quantum control, metrology, and topological materials.

1. Mathematical Formulation and Canonical Structures

The generic framework for non-Hermitian continuous-variable systems starts from linear operators of the form

$\hat{h} = -\frac{\hbar^2}{2m}\nabla^2 + V(x)$

where $V(x)\in\mathbb{C}$ may encode spatially varying gain, loss, or complex coupling. Canonical quantization associates right and left eigenfields $\psi(x), \bar\phi(x)$, leading to a Hamiltonian functional

$$H[q,p] = \int_{\mathbb{R}^d} d^dx\, \bar\phi(x)\,\hat{h}\,\psi(x)$$

where $q(x)\equiv\psi(x)$, $p(x)\equiv i\hbar\,\bar\phi(x)$, and Poisson brackets are defined accordingly. Dynamics obey generalized Hamilton’s equations and possess conserved biorthogonal charge densities $Q = \int \bar\phi(x)\psi(x)$, per Noether’s theorem. Adiabatic invariants generalize the standard occupation-number conservation via biorthogonal expansions, supporting persistent action integrals under slow parameter variation (Zhang, 2023).

Non-Hermitian boundary terms may originate from integration by parts, even in closed geometries, and manifest as quantized fluxes of generalized current densities $J_g[B]$, leading to both conventional (Bohr-type) and topological (QHE-type) quantization of observables. In multiply-connected spaces, emergent non-Hermitian contributions correspond to topological anomalies, affecting modern polarization, orbital magnetization theory, and bulk geometric phases (Moulopoulos, 2018).

2. Localization, Mobility Edges, and Spectral Topology

Continuous non-Hermitian quasiperiodic systems—and in particular, those governed by

$$H_g = \frac{1}{2m}\left[ -i\hbar\frac{d}{dx} + i\,g\right]^2 + V(x)$$

with incommensurate bichromatic potentials—exhibit both Anderson-type localization and genuine mobility edges (ME), where the real spectrum exhibits a transition between localized and extended states. The localization regime is identified using biorthogonal inverse participation ratios (IPR) and fractal dimensions, distinguishing regimes of delocalization (IPR $\to 0$) from localization (finite IPR, small $\tau$). Critical amplitude $\lambda_c$ for the quasiperiodic potential marks the onset of localization, and the mobility edge $E_c$ lies inside real spectral gaps. For fixed $g/E_r$, numerics confirm mobility edge intervals, e.g., $3.2 \lesssim \lambda \lesssim 6.0$ and $7.0 \lesssim \lambda \lesssim 10.5$ for $g/E_r=2$.

Under periodic boundary conditions (PBC), the spectrum forms open curves or closed loops in the complex energy plane, with point-gap topology quantified by winding numbers $\omega(E_B)$ around base points. Open arcs associated with extended states carry quantized non-zero winding numbers ($|\omega|=1$), denoting nontrivial topological phases (Jiang et al., 14 Aug 2024).

3. Non-Hermitian Skin Effect and Generalized Band Theory

The non-Hermitian skin effect (NHSE) represents the anomalous accumulation of bulk eigenstates at domain boundaries under open boundary conditions (OBC). In continuous models, all bulk states share a universal localization length $\xi$ independent of boundary detail, set by the imaginary part of the complex Bloch wave number $k$ in the generalized Brillouin zone (GBZ). The GBZ arises by extending $k$ to the complex plane for OBC, forming a closed contour $|\beta|=r$ where solutions decay exponentially into the bulk.

Bulk–edge correspondence is recovered via Zak phases defined on the GBZ; a $2\pi$ jump in the GBZ-Zak phase as the sample termination is varied signals the presence of topological edge modes. In photonic crystals with non-Hermitian parameters, all transverse-electric (TE) modes for a fixed transverse wave vector $k_y$ exhibit equal localization length and pile up at a single boundary (Yokomizo et al., 2021). Theoretical mapping to lattice models confirms that boundary fluxes of generalized currents underlie NHSE and exceptional-point physics (Moulopoulos, 2018, Yokomizo et al., 2021).

4. Exceptional Points, Dynamical Singularities, and Control

Non-Hermitian continuous-variable systems feature both Hamiltonian and Liouvillian exceptional points (EPs). In periodic or confining potentials augmented by spatially varying imaginary terms, discrete Fourier mappings yield effective Stark-ladder Hamiltonians: $$H_{\rm eff} = J\sum (c_{j+1}^\dagger c_j+ \mathrm{h.c.}) + iF\sum (j-\tfrac{N+1}{2}) c_j^\dagger c_j$$ whose spectral discriminant $\Delta$ dictates the coalescence of eigenvalues and eigenvectors at EPs. Two-state ($\mathrm{EP}_2$) and higher-order ($\mathrm{EP}_n$) coalescences yield Jordan block structures with time evolution governed by Puiseux scaling—probabilities grow as $P(t)\propto t^{2(n-1)}$ near EP$_n$. Such dynamical singularities can be harnessed for mode-selective amplification, sensing, and transport (Wang et al., 9 Nov 2024).

In driven-dissipative Kerr-cat qubits (parametrically stabilized oscillator subspaces), dissipation implements bidirectional quantum jumps, generating third-order Liouvillian exceptional points (LEP3s) when three eigenvalues and eigenvectors coalesce. Topological characterization via winding of resultant vectors in parameter space identifies the unique continuous-variable phenomenon: LEP3s cannot occur in standard two-level systems, as their jump operators are unidirectional (Han et al., 23 Nov 2025).

5. Control Theory for Non-Hermitian Continuous Bosonic Networks

Universal quantum control over non-Hermitian bosonic systems proceeds via gauge-potential formalism in instantaneous ancillary frames. For $N$-mode quadratic Hamiltonians,

$H(t) = \vec{a}^\dagger H^a(t) \vec{a}$

ancilla rotations $\mathcal{M}(t)$ enable decomposition into stationary frames supplemented by a geometric gauge potential $\mathcal{A}(t)$. An upper triangularization constraint on the rotated coefficient matrix guarantees decoupling of two nonadiabatic Heisenberg passages, facilitating perfect state transfer or nonreciprocal absorption. This passage-design is independent of spectrum, exceptional points, or $\mathcal{PT}$-symmetry; norm conservation is restored at protocol end without artificial normalization. Explicit control waveforms realize robust quantum state manipulation in cavity-magnonic systems and other multimode bosonic networks (Jin et al., 4 Dec 2025).

6. Novel Bound States, Device Applications, and Quantum Simulation Platforms

Non-Hermitian continuous-variable Hamiltonians with imaginary momentum and Landau-type vector potentials support true continua of Gaussian-localized bound states (continuum Landau modes, CLMs) filling the complex energy plane. The eigenenergy condition is subverted: $\tau>0$ yields normalizable modes for all $E\in\mathbb{C}$ and real momentum labels, violating the canonical Hermitian principle of discretized bound-state energies. Lattice analogs replicate CLMs and enable applications such as rainbow traps (spatial response tied to input frequency) and wave funnels (energy concentration at boundaries without nonreciprocal hopping). These concepts are implementable in photonics, topo-electrical circuits, and fiber networks (Wang et al., 2022).

Experimentally, non-Hermitian continuous-variable physics is accessible via ultracold atoms in optical potentials with controlled dissipation, cavity/circuit QED platforms (e.g., Kerr-cat qubits), metamaterials, and photonic crystals. Engineering complex potentials, coupling networks, and spatial loss enables in-situ paper of NHSE, topological transitions, and exceptional-point physics (Jiang et al., 14 Aug 2024, Wang et al., 9 Nov 2024, Yan et al., 6 Nov 2025, Han et al., 23 Nov 2025).

7. Criticality, Universality, and Outlook

A universal critical exponent $\nu\simeq 1/3$ governs the localization transition in continuous non-Hermitian quasiperiodic models, distinguishing them from lattice analogs (e.g., $\nu\sim0.5$ in AAH-type models). Experimental realization of criticality, point-gap topology, and mobility edges in genuinely continuous systems marks a phase diagram with regimes of purely extended, mixed, and localized states.

Broader implications include newly accessible universality classes, distinct critical scaling, breakdown of self-duality, and fertile ground for studies of many-body localization, anomalous transport, and topological pumping in quantum continua. Control over non-Hermitianity via engineered dissipation and drive parameters transforms error mechanisms into resources for quantum information and metrology, leveraging enhanced sensitivity near EPs and topologically robust state transfer protocols (Yan et al., 6 Nov 2025, Jin et al., 4 Dec 2025).

Collectively, non-Hermitian continuous-variable systems constitute a paradigm that merges foundational mathematical structures with frontiers in quantum technology, enabling comprehensive investigation of open-system dynamics, spectral topology, quantum control, and device functionalities in infinite-dimensional Hilbert spaces.