https://doi.org/10.1140/epjs/s11734-023-00919-0
Regular Article
Survey of the hierarchical equations of motion in tensor-train format for non-Markovian quantum dynamics
1
MSME, Université Gustave Eiffel, UPEC, CNRS, 77454, Marne-la-Vallée, France
2
LYRIDS, ECE-Paris, Graduate School of Engineering, 75015, Paris, France
3
Institut des Nanosciences de Paris, Sorbonne Université, CNRS, 75015, Paris, France
4
Institut de Chimie Physique UMR8000, Université Paris-Saclay, CNRS, 91405, Orsay, France
d
michele.desouter-lecomte@universite-paris-saclay.fr
Received:
5
January
2023
Accepted:
2
July
2023
Published online:
26
July
2023
This work is a pedagogical survey about the hierarchical equations of motion and their implementation with the tensor-train format. These equations are a great standard in non-perturbative non-Markovian open quantum systems. They are exact for harmonic baths in the limit of relevant truncation of the hierarchy. We recall the link with the perturbative second-order time convolution equations also known as the Bloch–Redfield equations. Some theoretical tools characterizing non-Markovian dynamics such as the non-Markovianity measures or the dynamical map are also briefly discussed in the context of HEOM simulations. The main points of the tensor-train expansion are illustrated in an example with a qubit interacting with a bath described by a Lorentzian spectral density. Finally, we give three illustrative applications in which the system–bath coupling operator is similar to that of the analytical treatment. The first example revisits a model in which population-to-coherence transfer via the bath creates a long-lasting coherence between two states. The second one is devoted to the computation of stationary absorption and emission spectra. We illustrate the link between the spectral density and the Stokes shift in situations with and without nonadiabatic interaction. Finally, we simulate an excitation transfer when the spectral density is discretized by undamped modes to illustrate a situation in which the TT formulation is more efficient than the standard one.
Copyright comment Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
© The Author(s), under exclusive licence to EDP Sciences, Springer-Verlag GmbH Germany, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
