Logarithmic Spectral Phase Deformations: Observable Signatures and the Limits of Spontaneous Dephasing

Abstract:We study a unitary deformation of the relation between a positive self-adjoint reference Hamiltonian $H$ and the generator of phase evolution, \begin{equation*} F_\beta(H) = H + \beta H \log(H/E_*). \end{equation*} Since $F_\beta(H)$ is a real function of $H$, it has the same eigenvectors, and it preserves unitarity, purity and the Born rule. It does not cause decoherence in the sense of a loss of purity. Its effect is a nonlinear distortion of the spectrum. Changing the reference scale $E_*$ adds a term linear in $H$, which cannot be distinguished from a rescaling of the spectrum. As a result, a single transition frequency does not determine $\beta$, and neither does the static dephasing of an ensemble; a finite-width Ramsey model makes this explicit. The part of the deformation that does not depend on $E_*$ is contained in second and higher spectral differences, which sample the curvature of the spectral map. For a nominally harmonic ladder $E_n=\omega(n+\delta)$, the adjacent-transition anharmonicity is $A_n/\omega=\beta c_n(\delta)$, with $c_n\sim(n+\delta)^{-1}$. This decreasing template can be distinguished in a joint fit from ideal Kerr and specified polynomial nonlinearities using sufficiently many resolved levels, subject to platform-specific modeling of other effects. We describe a spectroscopy protocol, estimate the precision of a joint fit, and show a synthetic example. The result depends on the choice of energy origin, which second differences do not remove, and it applies only to a single system during free evolution, because the deformation is not additive over subsystems.

14 pages, 1 figure. Unitary logarithmic spectral deformation; analysis of reference-scale redundancy, energy-origin dependence, divided-difference observables, static ensemble dephasing, and multilevel spectroscopy

arXiv:2512.09236 [quant-ph]

(or arXiv:2512.09236v5 [quant-ph] for this version)