Survey of the hierarchical equations of motion in tensor-train format for non-Markovian quantum dynamics

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.

Disciplines :

Physics

Author, co-author :

Mangaud, Etienne; Université Gustave Eiffel

Jaouadi AmineECE; ECE Paris

Chin Alex; Sorbonne Université

Desouter, Michèle ; Université de Liège - ULiège > Département de chimie (sciences) ; Université Paris Saclay

Language :

English

Title :

Survey of the hierarchical equations of motion in tensor-train format for non-Markovian quantum dynamics

Publication date :

2023

Journal title :

European Physical Journal. Special Topics

ISSN :

1951-6355

eISSN :

1951-6401

Publisher :

EDP Sciences, France

Volume :

232

Issue :

Pages :

1847-1869

Peer reviewed :

Peer Reviewed verified by ORBi

Funding text :

This work was performed within the French GDR 686 3575 THEMS. We want to dedicate this work to the memory of Christoph Meier and Osman Atabek. Both authors made significant contributions to the research areas described above, such as the Meier–Tannor spectral density and the introduction of the auxiliary matrices. Both have also greatly contributed to many strategies of quantum control in a wide range of processes such as molecular orientation, dynamics of excited states, isomerization, molecular cooling to cite only few. Dominik Domin is warmly acknowledged for his efficient technical support.

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since 25 September 2023

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