Quantum Thermodynamics of Open Systems Coupled to Structured Reservoirs

Abstract

Standard weak-coupling master equations assume a flat, memoryless reservoir spectral density, yet many
solid-state and molecular platforms couple a quantum system to reservoirs with pronounced structure, sharp
phonon bands, cavity-filtered photon baths, that violate this assumption and can qualitatively change the entropy
produced during relaxation. We simulate a two-level system coupled to a bosonic reservoir with a Lorentzian
spectral density of variable width, using the hierarchical equations of motion (HEOM) to obtain numerically exact
dynamics, and compare entropy production and heat-current statistics against a standard Lindblad treatment across
54 reservoir configurations spanning coupling strengths from 0.02 to 0.4 (in units of the qubit gap) and spectral
widths from 0.05 to 5.0 (in units of the reservoir peak frequency). For narrow, strongly structured reservoirs (width
below 0.3), the HEOM-derived entropy production rate fell below the Lindblad prediction by 34-61% during the first
oscillation period and displayed transient negative intervals, indicating backflow of entropy from reservoir to system,
in 22 of 54 configurations. Total entropy produced over a fixed evolution window of 20 qubit periods converged to
within 6% of the Markovian estimate only once spectral width exceeded roughly 1.5, defining an operational
non-Markovianity threshold. A non-Markovianity measure based on trace-distance revivals correlated with the
fractional entropy-production deficit at r = 0.81 (p < 0.001, n = 54), and the peak heat-current backflow amplitude
scaled approximately as the inverse square root of spectral width across the structured regime, reaching 2.3 times
the corresponding Markovian peak current at the narrowest width tested. Coupling strength modulated the
magnitude but not the qualitative onset of these effects: at weak coupling (0.02-0.05) the Lindblad approximation
remained accurate to within 8% even for narrow reservoirs, while at strong coupling (0.2-0.4) deviations exceeded
50% across nearly the full width range tested. These results quantify when reservoir structure must be treated
non-perturbatively to obtain reliable entropy-production estimates and provide a practical width-versus-coupling
map for deciding whether a Markovian approximation suffices in a given open quantum system.