The Redfield equation, derived microscopically to second order in the system-bath coupling under Born-Markov assumptions, is accurate at weak coupling but can violate complete positivity, producing unphysical states. Farina and Giovannetti (Phys. Rev. A 100, 012107 (2019)) showed that the partial secular approximation (PSA), which retains slowly rotating nonsecular terms by coarse-graining over a finite time interval, restores complete positivity once the interval exceeds sufficient thresholds, with a necessary-and-sufficient condition known only for a single-mode dipole coupling. The problem is to delineate the validity of the PSA: (i) are the sufficient coarse-graining thresholds tight for multi-gap, structured bath spectral densities, or do substantially smaller admissible intervals exist; (ii) how do the non-Gibbsian steady states predicted by the PSA deviate from the Gibbs state as coupling strength and coarse-graining time vary, and to which order beyond the weak-coupling regime is the equation valid; (iii) what are explicit positivity criteria for interacting multipartite systems with degenerate or nearly degenerate Bohr frequencies, where local and global master-equation descriptions compete; (iv) does the finite-interval positivity recovery survive relaxing Born-Markov assumptions via higher-order or non-Markovian corrections.

Background

An open quantum system is a finite-dimensional system coupled to a macroscopic environment (bath), whose reduced state is to be described by a closed evolution equation, a quantum master equation. Expanding the microscopic dynamics perturbatively to second order in the system-bath coupling while assuming weak system-environment correlations (the Born approximation) and fast decay of bath memory (the Mark approximation) yields the time-honored Redfield equation. It is accurate at very weak coupling but, in general, its generator is not of Gorini-Kossakowski-Sudarshan-Lindblad (GKSL, or Lindblad) form, the class of Markovian generators that generate completely positive dynamical maps. Complete positivity (CP) is the requirement that the evolution map remain positive even when the system is entangled with an ancilla (an auxiliary system treated as part of a larger composite); its violation by the Redfield equation can yield unphysical predictions such as negative populations. The standard cure is the secular approximation (SA): in the eigenbasis of the system Hamiltonian, whose energy differences are called Bohr frequencies, one drops all terms oscillating at a nonzero difference of Bohr frequencies; this forces the matrix of dissipative coefficients (the Kossakowski matrix) to be block-diagonal and positive semidefinite. The SA guarantees CP and, for a single thermal bath, relaxation to the Gibbs state (the canonical equilibrium state at the bath temperature), but it destroys coherences between nearly degenerate levels and misdescribes weakly interacting multipartite systems. For such systems one distinguishes a global master equation (dissipator built from the eigenoperators of the full interacting Hamiltonian) from a local one (dissipators built from the bare subsystem Hamiltonians), and their comparison hinges on how the secular approximation is implemented.

The partial secular approximation (PSA) is an intermediate cure with a long history in coarse-graining approaches (e.g. Schaller and Brandes, 2008): instead of discarding all nonsecular terms, one averages the second-order dynamics over a finite coarse-graining time interval, which suppresses term pairs whose Bohr frequencies differ by more than about the inverse interval while retaining slowly rotating coherences. Farina and Giovannetti (2019) applied the coarse graining only after the full Born-Markov reduction and proved that the resulting master equation is CP once all blocks of the interval-dependent Kossakowski matrix, whose entries involve sinc factors (sinc x=sin⁡x/x\mathrm{sinc}\,x=\sin x/x) mixing pairs of Bohr frequencies, are positive semidefinite. Here the bath enters through its spectral density, the frequency-dependent function weighting the density of bath modes by their coupling strength; multi-gap system spectra and strongly structured spectral densities (with sharp resonances or cutoffs) make these sinc-mixed blocks genuinely non-diagonal. They derived sufficient lower thresholds on the coarse-graining time: a general bound, a more compact but weaker bound, and a provably tighter but more involved bound, all expressed through minimal eigenvalues of the secular blocks, operator norms of the coupling matrices, and minimal separations between distinct Bohr frequencies. For a two-level system coupled by a dipole to a thermal bath they obtained a necessary and sufficient condition, and comparing it with the general bounds shows the latter are conservative. In their examples (a qubit or a harmonic oscillator coupled to a bosonic or fermionic bath) a finite coarse-graining time also induces a bath-driven Hamiltonian renormalization (a Lamb shift of the kind familiar from quantum electrodynamics) that need not commute with the dissipator, and, for the oscillator, a steady state that deviates from the Gibbs state. What the paper does not provide, and what later work has not closed, is a validity theory of the PSA: tightness of the thresholds, the scaling of the non-Gibbsian steady-state deviation, explicit criteria for degenerate spectra, and behavior beyond the Born-Markov assumptions.

Problem Statement

Establish a validity theory for the partial secular approximation (PSA) as a complete-positivity recovery scheme for the microscopically derived Redfield equation, in the framework where a finite coarse-graining time Δt\Delta t is applied after the Born-Markov reduction (Farina and Giovannetti, 2019). The overarching objective — a validity theory for the PSA — is delineated by four open strands. Complete closure of the objective requires all four strands.

(i) Threshold tightness. Determine whether the sufficient coarse-graining thresholds Δtc(1)\Delta t_c^{(1)}, Δtc(2)\Delta t_c^{(2)}, Δtc(0)\Delta t_c^{(0)} are tight (optimal up to constants) for general multi-gap and strongly structured bath spectral densities, or whether substantially smaller admissible Δt\Delta t exist for specific models; in particular, prove necessity of a finite threshold, or construct explicit models whose true complete-positivity threshold is parametrically smaller than every known sufficient bound, in model classes beyond the single-mode dipole case where the necessary-and-sufficient condition is already known. Any one of these directions — tightness up to constants, necessity of a finite threshold, or an explicit model with a parametrically smaller true threshold — resolves this strand on its own; the branches not taken remain open.

(ii) Steady-state structure. Quantitatively characterize how the PSA steady state deviates from the Gibbs state of the bare system Hamiltonian as a function of the system-bath coupling strength and of Δt\Delta t, and establish the order in the coupling up to which the PSA equation and its steady state remain accurate beyond the weak-coupling regime in which they were derived.

(iii) Degenerate spectra. Provide explicit complete-positivity criteria and Δt\Delta t bounds for interacting multipartite systems whose spectra contain exactly or nearly degenerate Bohr frequencies, the regime where the PSA is needed and where local and global master-equation descriptions compete.

(iv) Beyond Born-Markov. Determine whether the finite-Δt\Delta t complete-positivity recovery survives relaxing the underlying approximations: whether master equations incorporating higher-order (in the coupling) corrections or non-Markovian memory admit analogous coarse-graining-based positivity recoveries, and if so, with what thresholds.

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