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arXiv:physics/0611196 · New Journal of Physics

Time-dependent quantum Monte Carlo: preparation of the ground state

Ivan P. Christov

Published Mar 22, 2007• 66 citations
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Abstract

We study one-dimensional (1D) and 2D helium atoms using a new time-dependent quantum Monte Carlo (TDQMC) method. The TDQMC method employs random walkers, with a separate complex-valued guiding wave attached to each walker. The ground state is calculated by a self-consistent solution of complex-time Schrödinger equations for the guiding waves and of equations for the velocity fields of the walkers. Our results show that the many-body wavefunction and the ground state energy of the model atoms are very close to those predicted by the standard diffusion quantum Monte Carlo method. The obtained ground state can further be used to examine correlated time-dependent processes which include, for example, interaction of atoms and molecules with external electromagnetic fields.

DOI
10.1088/1367-2630/9/3/070

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Here we study the dynamics of many-body quantum systems using the time-dependent quantum Monte Carlo method where the evolution is described by ensembles of particles and guide waves. The exponential time scaling inherent to the quantum many-body problem is reduced to polynomial-time computation by solving concurrently a set of coupled Schrodinger equations for the guide waves in physical space and a set of first-order equations for the Monte Carlo walkers. We use effective potentials to account for the local and nonlocal quantum correlations in time-varying fields, where for fermionic states an exchange "hole" is introduced explicitly through screened Coulomb potentials. The walker distributions for the ground states of para- and ortho-helium reproduce well the statistical properties, such as the electron-pair density function, of the real atoms. Our predictions for the dipole response and the ionization of an atom exposed to strong ultrashort optical pulse are in good agreement with the exact results.
Abstract By using stochastic ensembles of walkers in physical and in one-body Hilbert spaces the recently proposed time-dependent quantum Monte Carlo (TDQMC) method offers the unique capability to calculate one-body density matrices at fully correlated level, without referencing the many-body quantum state. Here TDQMC is applied to study entanglement of simple systems such as Moshinsky atom (oscillator potentials) and atoms with Coulomb potentials in one spatial dimension. Our findings indicate that the dynamic entanglement of atoms exposed to powerful ultrashort laser pulse can be easily manipulated by introducing an appropriate phase modulation where the negative chirp enhances the entanglement while the positive one suppresses it. These findings can be used to explore the correlation properties of different constituents of complex quantum systems subjected to appropriately shaped laser radiation.
Abstract In this paper we apply the time-dependent quantum Monte Carlo (TDQMC) method to explore a modified single- and double-slit diffraction of matter waves. By using a simplified model of two electrons prepared in the ground state of an atom (molecule) and then suddenly released we are able to calculate the diffraction patterns in one spatial dimension in close correspondence with the numerically exact results. Through the Coulomb repulsion the one electron serves as an environment for the other, thus introducing decoherence in the quantum state which is easily quantified. It is demonstrated that the set of single particle wave functions yield by TDQMC can be used to directly construct a density matrix for that particle without tracing out the other constituents from the density matrix of the whole system. In this way it is possible to build explicitly time-dependent density matrices for different components of a complex quantum system straight within the TDQMC algorithm which may widen our understanding of quantum dynamics in many-body systems.
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The standard method of quantum Monte Carlo for the solution of the Schrödinger equation in configuration space can be described quite generally as devising a random walk that generates-at least asymptotically-populations of random walkers whose probability density is proportional to the wave function of the system being studied. While, in principle, the energy eigenvalue of the Hamiltonian can be calculated with high accuracy, estimators of operators that do not commute the Hamiltonian cannot. Bilinear quantum Monte Carlo (BQMC) is an alternative in which the square of the wave function is sampled in a somewhat indirect way. More specifically, one uses a pair of walkers at positions x and y and introduces stochastic dynamics to sample phi(i)(x)t(x,y)phi(j)(y), where phi(i)(x) and phi(j)(y) are eigenfunctions of (possibly different) Hamiltonians, and t(x,y) is a kernel that correlates positions x and y. Using different Hamiltonians permits the accurate computation of small energy differences. We review the conceptual basis of BQMC, discuss qualitatively and analytically the problem of the fluctuations in the branching, and present partial solutions to that problem. Finally we exhibit numerical results for some model systems including harmonic oscillators and the hydrogen and helium atoms. Further research will be necessary to make this a practical and generally applicable scheme.