Dynamics of a one-dimensional Holstein polaron with Davydov Ans¨ atze Jin Sun, Bin Luo, and Yang Zhao ∗ School of Materials Science and Engineering, Nanyang Technological University, Singapore 639798 (Dated: November 14, 2018) Following the Dirac-Frenkel time-dependent variational principle, dynamics of a one-dimensional Holstein polaron is probed by employing the Davydov D 2 Ansatz with two sets of variational pa- rameters, one for each constituting particle in the exciton-phonon system, and a simplified variant of the Davydov D 1 Ansatz, also known as the ˜ D Ansatz, with an additional set of phonon displace- ment parameters. A close examination of variational outputs from the two trial states reveals fine details of the polaron structure and intricacies of dynamic exciton-phonon interactions. Superradi- ance coherence sizes, speeds of exciton-induced phonon wave packets, linear optical absorption, and polaron energy compositions are also included in the study. PACS numbers: An electron in an insulating crystal induces a local lat- tice distortion around itself as it is excited by a photon. The electron together with the locally distorted lattice around it can be viewed as a quasiparticle which is also known as a polaron. Relaxation dynamics of photoex- cited entities such as polarons in liquids and solids has recently received much interest thanks to the advent of the ultrafast laser spectroscopy $^{1–3}$. It is now commonly accepted that dephasing and relaxation time scales in condensed matter are approximately picoseconds to tens of picoseconds. Emerging technological capabilities to control femtosecond pulse durations and down-to-one- hertz bandwidth resolutions provide novel probes on vi- brational dynamics and excitation relaxation. For ex- ample, progress in femtosecond spectroscopic techniques has made it possible to observe a coherent phonon wave packet oscillating along an adiabatic potential surface associated with a self-trapped exciton in a crystal with strong exciton-phonon interactions $^{4}$. Ultrafast events occur on femtosecond to picosecond timescales, and studies of ultrafast relaxation dynamics are a strategic research domain from both fundamental and technological points of view. It is the aim of an ultrafast optical experiment to provide information on the details of temporal evolution on a femtosecond scale, which, in turn, offers insights into fundamental processes governing the dynamics. Developments in ultrafast laser physics and technologies now allow studies of nonequi- librium carrier/exciton dynamics that is previously inac- cessible to traditional linear optical spectroscopy. How- ever, theoretical studies of polaron dynamics has not re- ceived much-deserved attention due to inherent difficul- ties in obtaining reliable solutions. Previously, a time- dependent form of the Merrifield-type polaron wave func- tion with zero crystal momentum has been employed to yield an approximative solution to the Schr¨ odinger equa- tion that governs the ultrafast relaxation process of a one- dimensional molecular chain $^{5–8}$. Results show that tem- poral changes of the exciton coherence size and related energy relaxation strongly depend on the exciton transfer integral, the exciton-phonon coupling strength, and the phonon bandwidth. The applicability of the Merrifield wave function, however, is restricted to the narrow-band regime where the electronic coupling between neighbor- ing molecules is sufficiently weak leaving the electron- phonon coupling at a dominant role. In addition, in the presence of off-diagonal electron-phonon interactions, the Merrifield Ansatz is shown to fail $^{9}$. Beyond the Merrifield Ansatz, there exist several trial wave functions of increasing sophistication to describe the polaron state in a translationally invariant man- ner such as the Toyozawa Ansatz $^{10}$, the Global-Local (GL) Ansatz formulated by Zhao and coworkers in the early 90s $^{11,12}$, and a delocalized form of the Davydov D$_{1}$ Ansatz that has been constructed very recently $^{13}$. By using these Ans¨ atze, we have previously investigated the ground state polaron energy band and the self-trapping phenomenon of a static Holstein polaron. Far superior results have been obtained compared with the Merri- field Ansatz $^{9,12,14}$. Closely related to those polaron trial states are the Davydov Ans¨ atze which originated from the theory of “Davydov soliton.” Seeking to explain stor- age and transport of biological energy in protein struc- tures, Davydov proposed in 1973 that quantum units of peptide vibrational energy might become “self-localized” through interactions with lattice phonons $^{15,16}$. Follow- ing his suggestion, many related studies have been car- ried out on the “Davydov soliton,” an essentially one- dimensional object that maintains dynamic integrity by balancing the effects of nonlinearity against those of dis- persion. The original Davydov Ans¨ atze include two forms of varying sophistication, namely, the D$_{1}$ 17–23 and D$_{2}$ Ans¨ atze, with the latter being a special case of the for- mer. In this work, we employ the Davydov D$_{2}$ Ansatz and a localized form factor of the GL Ansatz, previously known as the ˜ D Ansatz, to study time evolution of the Holstein polaron following the Dirac-Frenkel time-dependent vari- ∗ Electronic address: YZhao@ntu.edu.sg arXiv:1001.3198v2 [cond-mat.mtrl-sci] 22 Mar 2010 I. INTRODUCTION