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arXiv:2605.20019

Induced transitions in non-Hermitian spin-boson models with time-dependent boundaries

Andreas Fring, Marta Reboiro

quant-phmath-phSubmitted May 19, 2026• Announced May 20, 2026• 0 citations
v1 · May 19, 2026v2 · Aug 4, 2026
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Abstract

We study a time-dependent non-Hermitian extension of the Schütte-Da~Providência spin-boson Hamiltonian with complex couplings. A time-dependent Dyson map containing a squeezing transformation maps the model, in an admissible bounded regime, to a Hermitian Hamiltonian with real instantaneous energy spectrum. The squeezing contribution generates a dilatation term allowing the Hermitian partner to be interpreted as a fixed-domain representation of a system with moving boundaries. While the fixed-boundary Hermitian model conserves $Q=N-S_0$ and forbids transitions between sectors differing by two bosonic quanta, the boundary motion opens such channels. For closed boundary protocols with constant background parameters the first-order integrated transition amplitude vanishes, reflecting the unitary nature of constant squeezing. Nontrivial transition control arises when the non-Hermitian parameter varies during the boundary motion, changing the dressed basis and allowing boundary-induced transitions to be suppressed or enhanced by coherent interference.

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16 pages, 1 figure

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We study a time-dependent non-Hermitian generalisation of the Schütte-Da~Providência model describing a bosonic mode coupled to collective particle-hole excitations. Using a time-dependent Dyson map, we construct a Hermitian counterpart and reduce the collective fermionic sector by means of the random phase approximation (RPA). The resulting dynamics is mapped to two time-dependent harmonic-oscillator branches with instantaneous RPA frequencies W±(t)W_\pm(t)W±​(t). We determine the corresponding stability regions and compute transition probabilities between instantaneous oscillator states. In first-order instantaneous-basis perturbation theory the leading transition n→n+2n\to n+2n→n+2 is proportional to W˙j/Wj\dot W_j/W_jW˙j​/Wj​, showing that it is purely nonadiabatic and absent in the time-independent case. We compare this result with exact Lewis-Riesenfeld transition amplitudes within the RPA approximation. Numerical examples show that different components of the Dyson map provide distinct driving mechanisms: the scaling parameter modulates the effective coupling, while the squeezing parameter acts through a moving-boundary contribution. In both cases the induced collective transitions exhibit parametric-resonance peaks and sideband structures.
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We develop a comprehensive, gauge-covariant geometric framework for non-Hermitian quantum systems in the quasi-Hermitian regime, that is, the region of parameter space where the non-Hermitian Hamiltonian admits a real spectrum and a positive-definite metric operator. We build this framework by elevating the Dyson map to a central geometric object. This map is the transformation that converts a non-Hermitian Hamiltonian into an equivalent Hermitian one. From it we construct the Dyson connection and decompose it into Hermitian and anti-Hermitian parts, identified respectively as {\it stretching } and {\it rotation } components. This decomposition cleanly separates the genuine physical metric deformations from the unitary gauge redundancies. Working with manifestly gauge-covariant states, we then derive the complex non-Hermitian Berry phase and the quantum geometric tensor (QGT), and show that the non-Hermitian geometric curvature originates from the non-commutativity of the stretching components at the operator level. We further analyse the geometric singularities near an exceptional point (EP) and uncover a distinct hierarchy of divergences. For a general two-level non-Hermitian model, the quantum metric tensor (QMT) exhibits a leading-order divergence ∼∣εμ∣−2\sim |ε_μ|^{-2}∼∣εμ​∣−2, while the Berry curvature shows a weaker, subleading divergence ∼∣εμ∣−3/2\sim |ε_μ|^{-3/2}∼∣εμ​∣−3/2, with εμε_μεμ​ denoting the parameter displacement from the EP along an individual parameter axis μμμ. Finally, we examine physical realizations of this model, including the non-Hermitian Su--Schrieffer--Heeger (SSH) and Hatano--Nelson (HN) models, where exact analytical results confirm the predicted critical scaling laws and illustrate the metric-deformation-driven non-Hermitian geometries.
We provide a time-dependent Dyson map and metric for the two dimensional harmonic oscillator with a non-Hermitian ixyi xyixy coupling term. This particular time-independent model exhibits spontaneously broken PT\mathcal{PT}PT-symmetry and becomes unphysical in the broken regime, with the spectrum becoming partially complex. By introducing an explicit time-dependence into the Dyson map, we provide a time-dependent metric that renders the model consistent across the unbroken and broken regimes.
We establish a quantitative adiabatic estimate for a class of finite-dimensional non-Hermitian Schrödinger dynamics. The original non-Hermitian Hamiltonian is assumed to be diagonalizable with real spectrum and non-crossing eigenvalues. We then construct a dynamically compatible time-dependent metric operator, and its positive square root defines a Dyson map. The associated Dyson-transformed Hamiltonian is Hermitian. When this Dyson-transformed Hamiltonian satisfies the Hermitian adiabatic assumption, the standard resolvent-projection method gives an explicit adiabatic estimate in the Hermitian representation. Pulling this estimate back through the Dyson map gives an approximation in the original non-Hermitian representation. The resulting Dyson-pulled-back projections are then compared with the spectral projections of the original non-Hermitian Hamiltonian by using a contour-resolvent estimate. The final bound contains two contributions: the pulled-back Hermitian adiabatic error and the projection-comparison error. A two-level non-Hermitian model is presented to illustrate the hypotheses of the theorem and the two contributions appearing in the final estimate.