The Hierarchical Equation Of Motion And Its Application To Study The Entanglement And Spectroscopy

The hierarchical equations of motion (HEOM) is a very effective dynamics method which isnon-Markovian and non-perturbative and can be used in quantum dissipative systems. This methodcan be applied to study quantum entanglement, the energy transfer process in photosynthesis,spectroscopy and many other fields. In this paper, we use the HEOM method to study the evolutionof entanglement and two-dimensional (2D) electronic spectroscopy in small molecular systems.In Chapter1, we review some fundamentals of quantum dissipation theory and spectroscopy,including the reduced density matrix of the system, correlation function, entanglement of discretesystems and third-order nonlinear response function.In Chapter2, we describe the HEOM method and introduce some forms of HEOM, such as asingle-mode bath, multi-modes baths, modified scaled, high-temperature approximation andrandom forms.In Chapter3, for any form of response function can be expressed as a polynomial through theTaylor expansion, we study the random HEOM method of which response function is polynomialbased on the exponential. We compared with the two methods and found they are in agreementwith each other well in some certain conditions. For the case of the number of exponential term islarge, it can reduce the computation if we change the exponential into polynomial form.In Chapter4, we investigate the entanglement dynamics of two coupled chromophore pairsembedded in the protein-solvent environment by HEOM method. It is shown that the evolutions ofthe entanglement described by the measures of negativity and concurrence are in agreement witheach other. They decay to zero in a short time and the sudden death and sudden birth can beobserved in the process of the evolutions of the entanglement.In Chapter5, we use a three-level Frenkel exciton model investigation the2D electronicspectroscopy by the methods of the cumulant expansion and hierarchical equation of motion.Although there is slight different between the values of peaks, the shape of the spectroscopyobtained with the two methods are in good agreement with each other. We observe that the degreeof the elongation is dwindling with the increase of waiting time from the spectroscopy.In the last chapter, we give some conclusions and some expectations of the HEOM method.