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Physics Letters A

Exact Liouvillian spectral reconstruction for a finite-temperature bosonic mode

Leonardi Hernández-Sánchez, I. Ramos-Prieto, Francisco Soto‐Eguibar, H. M. Moya-Cessa

Published Oct 21, 2026• 0 citations
Publisher

Abstract

We present an exact Liouvillian spectral reconstruction for a finite-temperature bosonic mode, including eigenvalues, physical eigenoperators, and expansion coefficients for arbitrary initial conditions. The eigenvalues are independent of the thermal occupation, whereas the physical eigenoperators retain the reservoir-temperature dependence, with the thermal stationary state corresponding to the zero Liouvillian mode. Photon-number statistics and two-time correlations connect the spectral structure with relaxation, fluctuations, and temporal coherence by revealing the Liouvillian sectors selected by different observables. The zero-temperature dynamics are recovered in the limit ̅ ̅ ̅ ̅ ̅ 𝑛 → 0 .

DOI
10.1016/j.physleta.2026.132298

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We present an exact Liouvillian spectral reconstruction for a finite-temperature bosonic mode, including eigenvalues, physical eigenoperators, and expansion coefficients for arbitrary initial conditions. The eigenvalues are independent of the thermal occupation, whereas the physical eigenoperators retain the reservoir-temperature dependence, with the thermal stationary state corresponding to the zero Liouvillian mode. Photon-number statistics and two-time correlations connect the spectral structure with relaxation, fluctuations, and temporal coherence by revealing the Liouvillian sectors selected by different observables. The zero-temperature dynamics are recovered in the limit ̅ ̅ ̅ ̅ ̅ 𝑛 → 0 .
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We develop the algebraic method based on the Lie algebra of quadratic combinations of left and right superoperators associated with matrices to study the Lindblad dynamics of multimode bosonic systems coupled a thermal bath and described by the Liouvillian superoperator that takes into account both dynamical (coherent) and environment mediated (incoherent) interactions between the modes. Our algebraic technique is applied to transform the Liouvillian into the diagonalized form by eliminating jump superoperators and solve the spectral problem. The temperature independent effective non-Hermitian Hamiltonian, H^eff\hat{H}_{eff}H^eff​, is found to govern both the diagonalized Liouvillian and the spectral properties. It is shown that the Liouvillian exceptional points are represented by the points in the parameter space where the matrix, HHH, associated with H^eff\hat{H}_{eff}H^eff​ is non-diagonalizable. We use our method to derive the low-temperature approximation for the superpropagator and to study the special case of a two mode system representing the photonic polarization modes. For this system, we describe the geometry of exceptional points in the space of frequency and relaxation vectors parameterizing the intermode couplings and, for a single-photon state, evaluate the time dependence of the speed of evolution as a function of the angles characterizing the couplings and the initial state.
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Reservoirs with the same mean relaxation rate can be indistinguishable at the level of average energy transport while producing different fluctuations. We study a bosonic mode coupled to hot and cold Poisson streams of thermal ancillas through finite beam-splitter collisions. The averaged evolution is a compound-Poisson semigroup generated by finite Gaussian event channels. Its nonequilibrium steady state is an exact mixture of thermal states governed by a random affine fixed point. Consequently, the mean-occupation dynamics and bath-resolved mean heat currents coincide with those of the matched continuous Lindblad reservoir, whereas higher correlations retain the collision strength. For equal collision transmissivities, we derive an exact superthermal bunching law controlled by the temperature contrast and collision strength. We also obtain the asymptotic rates of the first two heat cumulants for an ideal stationary event-resolved two-point-measurement record and separate local one-collision contributions from temporal correlations. Fock-space diagonalization and Monte Carlo trajectories validate the formulas and connect full resets to the weak-collision Gaussian limit.
We describe the time evolution of the quantum coherence in an open system consisting of two coupled bosonic modes embedded in a thermal reservoir. We discuss the influence of the environment in terms of the covariance matrix for initial squeezed thermal states. The coherence is quantified using the relative entropy as a measure, and its dynamics is studied in the framework of the theory of open systems based on completely positive quantum dynamical semigroups. We show that the evolution of the quantum coherence strongly depends on the initial state of the system (squeezing parameter and thermal photon numbers), the parameters characterizing the thermal reservoir (temperature and dissipation coefficient) and the intensity of the coupling between the two modes.