A thermodynamically consistent Markovian master equation beyond the secular approximation

Markovian master equations provide a versatile tool for describing open quantum systems when memory effects of the environment may be neglected. As these equations are of an approximate nature, they often do not respect the laws of thermodynamics when no secular approximation is performed in their derivation. Here we introduce a Markovian master equation that is thermodynamically consistent and provides an accurate description whenever memory effects can be neglected. The thermodynamic consistency is obtained through a rescaled Hamiltonian for the thermodynamic bookkeeping, exploiting the fact that a Markovian description implies a limited resolution for heat. Our results enable a thermodynamically consistent description of a variety of systems where the secular approximation breaks down.

Local master equations bypass the secular approximation

Master equations are a vital tool to model heat flow through nanoscale thermodynamic systems. Most practical devices are made up of interacting sub-system, and are often modelled using either local master equations (LMEs) or global master equations (GMEs). While the limiting cases in which either the LME or the GME breaks down are well understood, there exists a 'grey area' in which both equations capture steady-state heat currents reliably, but predict very different transient heat flows. In such cases, which one should we trust? Here, we show that, when it comes to dynamics, the local approach can be more reliable than the global one for weakly interacting open quantum systems. This is due to the fact that the secular approximation, which underpins the GME, can destroy key dynamical features. To illustrate this, we consider a minimal transport setup and show that its LME displays exceptional points (EPs). These singularities have been observed in a superconducting-circuit realisation of the model \cite{partanen2019exceptional}. However, in stark contrast to experimental evidence, no EPs appear within the global approach. We then show that the EPs are a feature built into the Redfield equation, which is more accurate than the LME and the GME. Finally, we show that the local approach emerges as the weak-interaction limit of the Redfield equation, and that it entirely avoids the secular approximation.

Thermodynamics of the Coarse-Graining Master Equation

We study the coarse-graining approach to derive a generator for the evolution of an open quantum system over a finite time interval. The approach does not require a secular approximation but nevertheless generally leads to a Lindblad–Gorini–Kossakowski–Sudarshan generator. By combining the formalism with full counting statistics, we can demonstrate a consistent thermodynamic framework, once the switching work required for the coupling and decoupling with the reservoir is included. Particularly, we can write the second law in standard form, with the only difference that heat currents must be defined with respect to the reservoir. We exemplify our findings with simple but pedagogical examples.

Dark matter spin–spin interaction through the pseudo-scalar vacuum field

We suggest that the pseudo-scalar vacuum (PSV) field in the dark matter (DM) sector of the Universe may be as important as the electromagnetic vacuum field in the baryonic sector. In particular, the spin–spin interaction between the DM fermions, mediated by PSV, may represent the strongest interaction between the DM fermions due to the absence of the electric charge and the magnetic dipole moment. Based on this assumption, we consider the influence of the spin–spin interaction, mediated by PSV, on the spin precession of the DM fermions (e.g. neutralino). In the secular approximation, we obtain the exact expression describing the frequency of the precession and estimate the decoherence rate.

Observing collisions beyond the secular approximation limit

Quantum coherence plays an essential role in diverse natural phenomena and technological applications. The unavoidable coupling of the quantum system to an uncontrolled environment incurs dissipation that is often described using the secular approximation. Here we probe the limit of this approximation in the rotational relaxation of molecules due to thermal collisions by using the laser-kicked molecular rotor as a model system. Specifically, rotational coherences in N2O gas (diluted in He) are created by two successive nonresonant short and intense laser pulses and probed by studying the change of amplitude of the rotational alignment echo with the gas density. By interrogating the system at the early stage of its collisional relaxation, we observe a significant variation of the dissipative influence of collisions with the time of appearance of the echo, featuring a decoherence process that is well reproduced by the nonsecular quantum master equation for modeling molecular collisions.

Potential softening and eccentricity dynamics in razor-thin, nearly Keplerian discs

ABSTRACTIn many astrophysical problems involving discs (gaseous or particulate) orbiting a dominant central mass, gravitational potential of the disc plays an important dynamical role. Its impact on the motion of external objects, as well as on the dynamics of the disc itself, can usually be studied using secular approximation. This is often done using softened gravity to avoid singularities arising in calculation of the orbit-averaged potential – disturbing function – of a razor-thin disc using classical Laplace–Lagrange theory. We explore the performance of several softening formalisms proposed in the literature in reproducing the correct eccentricity dynamics in the disc potential. We identify softening models that, in the limit of zero softening, give results converging to the expected behaviour exactly, approximately or not converging at all. We also develop a general framework for computing secular disturbing function given an arbitrary softening prescription for a rather general form of the interaction potential. Our results demonstrate that numerical treatments of the secular disc dynamics, representing the disc as a collection of N gravitationally interacting annuli, are rather demanding: for a given value of the (dimensionless) softening parameter, ς ≪ 1, accurate representation of eccentricity dynamics requires N ∼ Cς−χ ≫ 1, with C ∼ O(10), 1.5 ≲ χ ≲ 2. In discs with sharp edges a very small value of the softening parameter ς (≲ 10−3) is required to correctly reproduce eccentricity dynamics near the disc boundaries; this finding is relevant for modelling planetary rings.