The Jaynes-Cummings model has been studied for six decades, yet its Nakajima-Zwanzig projected Liouvillian QLQ hides an unexpected structure: although manifestly non-Hermitian, its spectrum is purely real across all numerically accessible parameters. We resolve this anomaly by constructing a positive-definite metric eta > 0 such that (QLQ)^dag eta = eta (QLQ), proving that the generator of quantum memory is pseudo-Hermitian. The effect is genuine, not a block-diagonal artefact: the Delta N = 0 and Delta N = +/- 2 sectors remain strongly non-Hermitian, yet the global spectrum is protected by eta. Building on Ng and Rabani's identification of exceptional-point physics in QLQ, we show that the Jaynes-Cummings point sits inside a symmetry-protected pseudo-Hermitian phase. Under continuous deformation toward the full Rabi model this protection breaks and re-enters, producing a U(1) -> complex -> Z_2 phase structure bracketed by exceptional-point boundaries. The result supplies a structural reason for Hardy-space analyticity of the memory kernel in the canonical quantum-optical model and identifies a phase transition between oscillatory and genuinely dissipative quantum memory.
We propose a reliable scheme to realize the ultrastrong Jaynes-Cummings (JC) model by simultaneously modulating the resonance frequencies of the two-level system and the bosonic mode in the ultrastrong quantum Rabi model. We find that in both the high- and low-frequency modulation regimes, the counter-rotating terms can be completely suppressed without reducing the coupling strength of the rotating-wave terms, and hence the ultrastrong JC Hamiltonian is achieved. The ultrastrong JC interaction can not only be used to implement ultrafast quantum operations such as ultrafast quantum state transfer and quantum logic gates but also will open up a new route to the demonstration of quantum phase transition associated with the JC Hamiltonian across the deep-strong-coupling point. Some discussions on the experimental implementation of this scheme with circuit-QED systems are presented.
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)/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.
We present the analytical solution of the Tavis-Cummings (TC) model for more than one qubit inhomogeneously coupled to a single mode radiation field beyond the rotating-wave approximation (RWA). The significant advantage of the displaced oscillator basis enables us to apply the same truncation techniques adopted in the single qubit Jaynes-Cummings (JC) model to the multiple qubits system. The derived analytical spectrum match perfectly the exact diagonalization numerical solutions of the inhomogeneous TC model in the parameter regime where the qubits transition frequencies are far off-resonance with the field frequency and the interaction strengths reach the ultra-strong coupling regime. The two-qubit TC model is quasi-exactly solvable because part of the spectra can be determined exactly in the homogeneous coupling case with two identical qubits or with symmetric(asymmetric) detuning. By means of the fidelity of quantum states we identify several nontrivial level crossing points in the same parity subspace, which implies that homogeneous coupled two-qubit TC model with
ω1=ω2 or
ω1±ω2=2ωc is integrable. We further explore the time evolution of the qubit's population inversion and the entanglement behavior taking two qubits as an example. The analytical methods provide unexpectedly accurate results in describing the dynamics of the qubit in the present experimentally accessible coupling regime, showing that the collapse-revival phenomena emerge, survive, and are finally destroyed when the coupling strength increases beyond the ultra-strong coupling regime. The suggested procedure applies readily to the multiple qubits system such as the GHZ state entanglement evolution and quantum entanglement between a single photon and superconducting qubits of particular experiment interest.
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+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
z 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) with real spectrum, the biorthogonal Gibbs functional
ωbi(A)=Zbi−1∑ne−βEn⟨φn∣A∣ψn⟩ satisfies
ωbi(A†A)≥0 for all
A if and only if
H is quasi-Hermitian. The proof reconstructs the metric
η directly from the eigenprojectors of
ωbi via the Riesz representation theorem, yielding a metric-free criterion for quasi-Hermiticity. Under the quasi-Hermitian hypothesis, we prove that the
η-Gibbs state
ωη(A)=Zη−1Tr[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, 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
η-Gibbs state in finite dimensions, while the construction of a compatible
C∗-norm and the proof of
σ-weak continuity remain open.