AA2 – Materials, Light, Devices

At least for a small to medium number of sites, the semi-classical and multiscale methods can be tested against fully quantum-mechanical approaches to coupled excitons and phonons modeled by Fröhlich-Holstein type Hamiltonians with on-site and nearest-neighbor interactions only. In order to mitigate the curse of dimensionality as much as possible, representations of Hamiltonians and quantum states in terms of tensor trains (TTs, aka matrix product states) have been explored.

For the time-independent Schrödinger equation [1], we introduced an approach which directly incorporates the Wielandt deflation technique into the alternating linear scheme for the solution of eigenproblems in the TT format. This has lead to almost linear scaling of storage consumption in the number of sites, along with an only slightly faster increase of the CPU time. This technique has allowed us to directly tackle the phenomenon of mutual self-trapping, see the figure below. We were able to confirm the main results of the Davydov theory, i.e., the dependence of the wave packet width and the corresponding stabilization energy on the exciton-phonon coupling strength. In future work, our approach will allow calculations also beyond the validity regime of that theory and/or beyond the restrictions of the Fröhlich-Holstein type Hamiltonians.

We also investigated tensor-train approaches to the solution of the time-dependent Schrödinger equation [2]. One class of propagation schemes that we explored builds on splitting the Hamiltonian into two groups of interleaved nearest-neighbor interactions. In addition to the first order Lie-Trotter and the second order Strang-Marchuk splitting schemes, we also implemented a 4th order Yoshida-Neri and an 8th order Kahan-Li symplectic composition which are best when very high accuracy is required for shorter chains. Another class of propagators involves explicit, time-symmetrized Euler integrators for which we also implemented methods of 4th and 6th order, the former of which represents a good compromise between accuracy and computational effort for longer chains.

Transfer of our methodological work to applications has been guaranteed by making models, algorithms, and software freely available through the open-source WaveTrain software package for numerical simulations of chain-like quantum systems with nearest-neighbor interactions [3]. This Python package is centered around TT format representations of Hamiltonian operators and state vectors. It builds on the Python tensor train toolbox Scikit_TT, which provides efficient construction methods and storage schemes for the TT format. WaveTrain software is freely available from the GitHub platform where it will also be further developed. Moreover, it is mirrored at SourceForge, within the framework of the WavePacket project for numerical quantum dynamics.