For the Davydov Hamiltonian several special cases are known which can be solved analytically. Starting from these cases we show that the initial state for a simulation using Davydov's \ensuremath{\Vert}
D1〉 approximation has to be constructed from a given set of initial lattice displacements and momenta in the form of a coherent state with its amplitudes independent of the lattices site, corresponding to Davydov's \ensuremath{\Vert}
D2〉 approximation. The site dependences in the \ensuremath{\Vert}
D1〉 ansatz evolve from this initial state exclusively via the equations of motion. Starting the \ensuremath{\Vert}
D1〉 simulation from an ansatz with site-dependent coherent-state amplitudes leads to an evolution which is different from the analytical solutions for the special cases. Thus also in applications of the \ensuremath{\Vert}
D1〉 ansatz to polyacetylene \ensuremath{\Vert}
D2〉-type initial states always have to be used in contrast to our previous suggestion [W. F\"orner, J. Phys. Condens. Matter 6, 9105 (1994)]. Further we expand the known exact solutions in Taylor series in time and compare expectation values in different orders with the exact results.We find that for an approximation up to third order in time (for the wave function) norm, and total energy, as well as displacements and momenta are reasonably correct for a time up to \ensuremath{\approxeq}0.12--0.14 ps, depending somewhat on the coupling strengh for the transportless case. We performed long-time simulations using the \ensuremath{\Vert}
D1〉 approximation where we computed expectation values of the relevant operators with the state (H^/J)\ensuremath{\Vert}
D1〉 and the deviation \ensuremath{\Vert}\ensuremath{\delta}〉 from the exact solution over long times, namely 10 ns. We found that in the very long-time scale the \ensuremath{\Vert}
D1〉 ansatz is very close to an exact solution. Further we report results from an investigation of the very short-time behavior of the \ensuremath{\Vert}
D1〉 state compared with that of an expansion of the exact solution in powers of time t. Within a time of roughly 0.10--0.15 ps the second- and third-order corrections turned out to be not very important. This is due to the fact that our first-order state contains already some terms of the expansion, summed up to infinite order. We found good agreement of the results obtained with our expansion and those from the corresponding \ensuremath{\Vert}
D1〉 simulations within the time of about 0.10 ps. Altogether we have shown that the \ensuremath{\Vert}
D1〉 state, although of approximative nature, is very close to an exact solution of the Davydov model on time scales from some fs up to ns. \textcopyright{} 1996 The American Physical Society.
Abstract A recently developed quantum Monte Carlo approach to the Holstein model with one electron [Phys. Rev. B 69, 024301 (2004)] is extended in several aspects. In addition to a straight‐forward generalization to higher dimensions, a checkerboard breakup for the hopping part of the Hamiltonian is introduced, which allows significantly faster simulations. Moreover, results are extrapolated towards the limit of continuous imaginary time to remove the error due to the Trotter–Suzuki approximation. The performance and accuracy of the current approach are compared to existing quantum Monte Carlo methods. To demonstrate the applicability of our method for almost arbitrary parameters, we study the small‐polaron crossover in one to three dimensions by calculating the electronic kinetic energy. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
Abstract We propose a new version of the Davydov Ansatz for the approximate solution of the time-dependent Schrödinger equation (TDSE) and apply it to a spin 1/2 system coupled to a single bosonic mode. In complexity the new Ansatz is between the well-studied D1 and D2 Ansätze and is therefore termed D1.5 Ansatz. It is tailored to specific but still rather general initial conditions and is free of numerical inconsistencies that plague the D1 Ansatz. Finally, a generalization to nontrivial multiplicity in the oscillator state space is given that leads, in principle, to an exact solution of the TDSE for arbitrary system parameters. By employing Bargmann coherent states, the equations of motion could be made explicit, which is a big advantage, computationally.
This Perspective presents an overview of the development of the hierarchy of Davydov's Ansätze and a few of their applications in many-body problems in computational chemical physics. Davydov's solitons originated in the investigation of vibrational energy transport in proteins in the 1970s. Momentum-space projection of these solitary waves turned up to be accurate variational ground-state wave functions for the extended Holstein molecular crystal model, lending unambiguous evidence to the absence of formal quantum phase transitions in Holstein systems. The multiple Davydov Ansätze have been proposed, with increasing Ansatz multiplicity, as incremental improvements of their single-Ansatz parents. For a given Hamiltonian, the time-dependent variational formalism is utilized to extract accurate dynamic and spectroscopic properties using Davydov's Ansätze as its trial states. A quantity proven to disappear for large multiplicities, the Ansatz relative deviation is introduced to quantify how closely the Schrödinger equation is obeyed. Three finite-temperature extensions to the time-dependent variation scheme are elaborated, i.e., the Monte Carlo importance sampling, the method of thermofield dynamics, and the method of displaced number states. To demonstrate the versatility of the methodology, this Perspective provides applications of Davydov's Ansätze to the generalized Holstein Hamiltonian, variants of the spin-boson model, and systems of cavity-assisted singlet fission, where accurate dynamic and spectroscopic properties of the many-body systems are given by the Davydov trial states.
Developed originally for the Holstein polaron, the Davydov D1 ansatz is an efficient, yet extremely accurate trial state for time-dependent variation of the spin-boson model [N. Wu, L. Duan, X. Li, and Y. Zhao, J. Chem. Phys. 138, 084111 (2013)]. In this work, the Dirac-Frenkel time-dependent variational procedure utilizing the Davydov D1 ansatz is implemented to study entanglement dynamics of two qubits under the influence of two independent baths. The Ohmic spectral density is used without the Born-Markov approximation or the rotating-wave approximation. In the strong coupling regime finite-time disentanglement is always found to exist, while at the intermediate coupling regime, the entanglement dynamics calculated by Davydov D1 ansatz displays oscillatory behavior in addition to entanglement disappearance and revival.
Abstract A method for propagating multi‐dimensional wave packets in time using a product ansatz for the wave function is presented. Other than in the time‐dependent Hartree approach which is based on a variational scheme, the current method is based on the stochastic Ito calculus. The correlations between the different one‐dimensional wave functions are taken into account via stochastic processes such that on average the original Schrödinger equation is reproduced. Numerical tests of the proposed scheme have been performed in two and four dimensions. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)