Chebyshev hierarchical equations of motion for systems with arbitrary spectral densities and temperatures

Abstract

The time evolution in open quantum systems, such as a molecular aggregate in contact with a thermal bath, still poses a complex and challenging problem. The influence of the thermal noise can be treated using a plethora of schemes, several of which decompose the corresponding correlation functions in terms of weighted sums of exponential functions. One such scheme is based on the hierarchical equations of motion (HEOM), which is built using only certain forms of bath correlation functions. In the case where the environment is described by a complex spectral density or is at a very low temperature, approaches utilizing the exponential decomposition become very inefficient. Here we utilize an alternative decomposition scheme for the bath correlation function based on Chebyshev polynomials and Bessel functions to derive a hierarchical equations of motion approach up to an arbitrary order in the environmental coupling. These hierarchical equations are similar in structure to the popular exponential HEOM scheme, but are formulated using the derivatives of the Bessel functions. The proposed scheme is tested up to the fourth order in perturbation theory for a two-level system and compared to benchmark calculations for the case of zero-temperature quantum Ohmic and super-Ohmic noise. Furthermore, the benefits and shortcomings of the present Chebyshev-based hierarchical equations are discussed.

Time-dependent atomistic simulations of the CP29 light-harvesting complex

“…We note in passing that work is in progress trying, e.g., by our group, to derive density matrix approaches, which can handle numerically obtained spectral densities. 90 For certain parameter regimes, however, the Ehrenfest (NISE) wave packet approach has been shown to yield the same results as accurate density matrix calculations properly representing the dephasing but not the relaxation. 77 In the wave packet-based scheme, the time-dependent Schrödinger equation needs to be solved for the time-dependent system Hamiltonian,…”

Statistical distributions of the tuning and coupling collective modes at a conical intersection using the hierarchical equations of motion

“…Different expansions have been proposed 77 among which the Tannor-Meier one leading to Eqs.16 and 17 90,92 , the Padé approximation 97 , and the Fano spectrum decomposition scheme 98 . A direct decomposition of the bath correlation function based on Chebyshev polynomials and Bessel functions has also been proposed leading to the so called C-HEOM scheme 99 .…”

Funneling dynamics in a phenylacetylene trimer: Coherent excitation of donor excitonic states and their superposition

“…The efficiency of the algorithm relies on the expansion of the correlation functions C j (t) as a sum of n ncor,j complex exponential functions associated to pseudo decay modes. Different expansions have been proposed 71 among which the Tannor-Meier scheme 96,97 , the Padé approximation 98 , the Fano spectrum decomposition 99 and an expansion on Chebyshev polynomials and Bessel functions 100 . According to the Tannor-Meier method, the correlation function is written as :…”