Pseudo-Hermiticity of the Nakajima-Zwanzig Projected Liouvillian in the Jaynes-Cummings Model

The Nakajima-Zwanzig projected Liouvillian QLQ, the generator of the exact memory kernel in open quantum dynamics, is manifestly non-Hermitian yet has been reported to possess a purely real spectrum in the Jaynes-Cummings model -- an anomaly unexplained since observation. We resolve this anomaly by showing that QLQ is pseudo-Hermitian in the Mostafazadeh sense: a positive-definite metric eta>0 exists such that (QLQ)^dag eta = eta (QLQ), forcing the spectrum to be real. The pseudo-Hermiticity is genuine: the Delta N = 0 and Delta N = +/-2 sectors are individually non-Hermitian (residuals 1.70 and 5.06, respectively), yet the global spectrum is protected by eta. The metric survives the bath-truncation limit (N_max = 3--20, matrix dimension up to 1764 x 1764) with intertwining residual <10^{-11}. A continuous deformation to the full Rabi model reveals a re-entrant pseudo-Hermitian phase with two exceptional-point boundaries, in which the metric condition number diverges. The result supplies a structural reason for Hardy-space analyticity of the memory kernel in the canonical quantum-optical model.

Source: arXiv:2604.25801v1 - http://arxiv.org/abs/2604.25801v1 PDF: https://arxiv.org/pdf/2604.25801v1 Original Link: http://arxiv.org/abs/2604.25801v1