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arXiv:2604.25801

A symmetry-protected pseudo-Hermitian phase of quantum memory-kernel generators

Kejun Liu

quant-phmath-phSubmitted Apr 28, 2026• Announced Apr 29, 2026• 0 citations• Revised May 11, 2026 (v3)
v1 · Apr 28, 2026v2 · Apr 30, 2026v3 · May 11, 2026
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Abstract

The Jaynes-Cummings (JC) model, introduced in 1963 and central to cavity quantum electrodynamics, describes a two-level system coupled to a single bosonic mode under the rotating-wave approximation. When the mode is projected out via the Nakajima-Zwanzig (NZ) formalism, the memory-kernel generator QLQ is manifestly non-Hermitian -- yet we prove analytically that its spectrum is strictly real at every coupling and every finite truncation, for both vacuum and thermal baths. For the vacuum bath the characteristic polynomial factorises completely; the nonzero eigenvalues reproduce the JC dressed-state ladder for n >= 2, while the lowest mode is suppressed by exactly sqrt(2) relative to the bare-Hamiltonian prediction. For any thermal Gibbs state, the squared generator reduces to an asymmetric rank-one perturbation symmetrised by a closed-form diagonal metric, with eigenvalues guaranteed real by Cauchy interlacing. We construct a positive-definite metric eta_osc intertwining QLQ with its adjoint on the nonzero spectral subspace, proving hidden pseudo-Hermiticity. A classification theorem extends these results to the full U(1)-conserving single-photon-exchange class with arbitrary complex couplings. The phase boundary under counter-rotating deformation is mapped analytically at resonance and numerically across the coupling-truncation plane, revealing a weak-coupling protected wedge, re-entrant real-spectrum bubbles, and N-universal plateaus organised by a three-family band catalog with closed-form level spacings. The full phase structure is proved well-defined in the N -> infinity thermodynamic limit. The sqrt(2) suppression provides a platform-independent experimental falsification target spanning seven orders of magnitude in coupling strength.

Comments
14 pages, 4 figures, 2 tables. Supplementary Information included (30 pages, 3 figures, 5 tables). 44 pages total. Replaces arXiv:2604.25801v2

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The Jaynes-Cummings (JC) model, introduced in 1963 and central to cavity quantum electrodynamics, describes a two-level system coupled to a single bosonic mode under the rotating-wave approximation. When the mode is projected out via the Nakajima-Zwanzig (NZ) formalism, the memory-kernel generator QLQ is manifestly non-Hermitian -- yet we prove analytically that its spectrum is strictly real at every coupling and every finite truncation, for both vacuum and thermal baths. For the vacuum bath the characteristic polynomial factorises completely; the nonzero eigenvalues reproduce the JC dressed-state ladder for n >= 2, while the lowest mode is suppressed by exactly sqrt(2) relative to the bare-Hamiltonian prediction. For any thermal Gibbs state, the squared generator reduces to an asymmetric rank-one perturbation symmetrised by a closed-form diagonal metric, with eigenvalues guaranteed real by Cauchy interlacing. We construct a positive-definite metric eta_osc intertwining QLQ with its adjoint on the nonzero spectral subspace, proving hidden pseudo-Hermiticity. A classification theorem extends these results to the full U(1)-conserving single-photon-exchange class with arbitrary complex couplings. The phase boundary under counter-rotating deformation is mapped analytically at resonance and numerically across the coupling-truncation plane, revealing a weak-coupling protected wedge, re-entrant real-spectrum bubbles, and N-universal plateaus organised by a three-family band catalog with closed-form level spacings. The full phase structure is proved well-defined in the N -> infinity thermodynamic limit. The sqrt(2) suppression provides a platform-independent experimental falsification target spanning seven orders of magnitude in coupling strength.
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We study a time-dependent non-Hermitian extension of the Schütte-Da Providência spin-boson Hamiltonian with complex couplings. A time-dependent Dyson map relates the model to a Hermitian counterpart and induces a positive physical metric. Separating the positive and unitary parts of the map, we prove a no-go result for a natural Gaussian number-squeeze-number class: for a nonvanishing linear spin-boson interaction, Hermiticity and bounded invertibility force the entire bosonic part of the Dyson map to be unitary. The squeezing parameter therefore selects a time-dependent Hermitian frame rather than contributing to the metric. The squeezed and non-squeezed Hamiltonians are related exactly by a time-dependent unitary transformation. The conserved quantity formed from the boson number and spin projection is replaced by a transported dynamical invariant, so a closed squeezing-frame protocol cannot generate transitions between distinct invariant sectors. We then introduce an independently driven single-mode cavity with quadratic term iχ(t)(b†2−b2)/2iχ(t)(b^{\dagger2}-b^2)/2iχ(t)(b†2−b2)/2. This physical drive breaks the corresponding continuous symmetry while preserving parity and couples dressed sectors differing by two bosonic quanta. The non-Hermitian asymmetry parameter, which also determines the bounded metric, tunes the effective Hermitian coupling, transition strengths and resonance conditions. Direct numerical propagation confirms the distinction between passive frame-induced mixing and genuine cavity transitions, the asymmetry-controlled resonance shift, and the validity of the first-order transition formula in the weak-driving regime.
The theory of non-Hermitian systems and the theory of quantum deformations have attracted a great deal of attention in the past decades. In general, non-Hermitian Hamiltonians are constructed by an ad hoc manner. Here, we study the (2+12+12+1) Dirac oscillator and show that in the context of the $\ensuremath{\kappa}$-deformed Poincar\'e-Hopf algebra its Hamiltonian is non-Hermitian but has real eigenvalues. The non-Hermiticity stems from the $\ensuremath{\kappa}$-deformed algebra. From the mapping in Bermudez et al., Phys. Rev. A 76, 041801(R) (2007), we propose the $\ensuremath{\kappa}$-Jaynes-Cummings and $\ensuremath{\kappa}$-anti-Jaynes-Cummings models, which describe an interaction between a two-level system with a quantized mode of an optical cavity in the $\ensuremath{\kappa}$-deformed context. We find that the $\ensuremath{\kappa}$ deformation modifies the Zitterbewegung frequencies and the collapses and revivals of quantum oscillations. In particular, the total angular momentum in the zzz direction is not conserved anymore, as a direct consequence of the deformation.
It is shown that for a given Hermitian Hamiltonian possessing supersymmetry, there is always a non-Hermitian Jaynes–Cummings-type Hamiltonian (JCTH) admitting entirely real spectra. The parent supersymmetric Hamiltonian and the corresponding non-Hermitian JCTH are simultaneously diagonalizable. The exact eigenstates of these non-Hermitian Hamiltonians are constructed algebraically for certain shape-invariant potentials, including a non-Hermitian version of the standard Jaynes–Cummings (JC) model for which the parent supersymmetric Hamiltonian is the superoscillator. It is also shown that a non-Hermitian version of the several physically motivated generalizations of the JC model admits entirely real spectra. The positive-definite metric operator in the Hilbert space is constructed explicitly along with the introduction of a new inner product structure, so that the eigenstates form a complete set of orthonormal vectors and the time evolution is unitary.
We investigate the extension of the Kubo--Martin--Schwinger (KMS) thermal equilibrium condition to bounded non-Hermitian Hamiltonians with real spectra and biorthogonal eigensystems, providing a unified framework through three complementary constructions: a complete KMS theorem under quasi-Hermiticity, a biorthogonal KMS-type identity whose positivity characterises quasi-Hermiticity, and a quantum-detailed-balance condition for the associated open-system dynamics. Our main result is a thermodynamic characterisation of quasi-Hermiticity: for any diagonalisable H∈Md(C)H\in M_d(\mathbb C)H∈Md​(C) with real spectrum, the biorthogonal Gibbs functional ωbi(A)=Zbi−1∑ne−βEn⟨φn∣A∣ψn⟩ω_{\rm bi}(A)=Z_{\rm bi}^{-1}\sum_n e^{-βE_n}\langleφ_n|A|ψ_n\rangleωbi​(A)=Zbi−1​∑n​e−βEn​⟨φn​∣A∣ψn​⟩ satisfies ωbi(A†A)≥0ω_{\rm bi}(A^\dagger A)\ge0ωbi​(A†A)≥0 for all AAA if and only if HHH is quasi-Hermitian. The proof reconstructs the metric ηηη directly from the eigenprojectors of ωbiω_{\rm bi}ωbi​ via the Riesz representation theorem, yielding a metric-free criterion for quasi-Hermiticity. Under the quasi-Hermitian hypothesis, we prove that the ηηη-Gibbs state ωη(A)=Zη−1Tr[ηe−βHA]ω_η(A)=Z_η^{-1}{\rm Tr}[ηe^{-βH}A]ωη​(A)=Zη−1​Tr[ηe−βHA] satisfies the full analytic KMS condition using the Hadamard three-line theorem and Bari's theorem on Riesz bases. The transported state generally differs from the Gibbs state of the isospectral Hermitian partner whenever [η,h]≠0[η,h]\neq0[η,h]=0, so the KMS property cannot be obtained by similarity transformation alone. Finally, within the Haag--Hugenholtz--Winnink programme, we establish the Tomita--Takesaki modular structure of the ηηη-Gibbs state in finite dimensions, while the construction of a compatible C∗C^*C∗-norm and the proof of σσσ-weak continuity remain open.