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arXiv:1609.03298 · Journal of the Optical Society of America B

Particle–wave dichotomy in quantum Monte Carlo: unlocking the quantum correlations

Ivan P. Christov

Published Aug 4, 2017• 9 citations
arXivHTMLPDF

Abstract

Here, a dichotomy of particles and waves is employed in a quantum Monte Carlo calculation of interacting electrons. Through the creation and propagation of concurrent stochastic ensembles of walkers in physical space and in Hilbert space one can correctly predict the ground state and the real-time evolution of a single electron interacting with larger quantum system. It is shown that such walker ensembles can be constructed straightforwardly through a stochastic sampling (windowing) applied to the mean-field approximation. Our calculations reveal that the ground state and the real-time evolution of the probability distributions and the decoherence due to the Coulomb interaction in presence of strong ultrashort laser pulse can be accounted for correctly by calculating the density matrix of the electron, without referencing to the quantum many-body state of the whole system.

DOI
10.1364/josab.34.001817

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Here, a dichotomy of particles and waves is employed in a quantum Monte Carlo calculation of interacting electrons. Through the creation and propagation of concurrent stochastic ensembles of walkers in physical space and in Hilbert space one can correctly predict the ground state and the real-time evolution of a single electron interacting with larger quantum system. It is shown that such walker ensembles can be constructed straightforwardly through a stochastic sampling (windowing) applied to the mean-field approximation. Our calculations reveal that the ground state and the real-time evolution of the probability distributions and the decoherence due to the Coulomb interaction in presence of strong ultrashort laser pulse can be accounted for correctly by calculating the density matrix of the electron, without referencing to the quantum many-body state of the whole system.
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We examine the relation between the recently proposed time-dependent quantum Monte Carlo (TDQMC) method and the principles of stochastic quantization. In both TDQMC and stochastic quantization, particle motion obeys stochastic guidance equations to preserve quantum equilibrium. In this way the probability density of the Monte Carlo particles corresponds to the modulus square of the many-body wave function at all times. However, in TDQMC, the motion of particles and guide waves occurs in physical space unlike in stochastic quantization where it occurs in configuration space. Hence, the practical calculation of time evolution of many-body fully correlated quantum systems becomes feasible within the TDQMC methodology. We illustrate the TDQMC technique by calculating the symmetric and antisymmetric ground state of a model one-dimensional helium atom, and the time evolution of the dipole moment when the atom is irradiated by a strong ultrashort laser pulse.
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.
The time-dependent quantum Monte Carlo method represents a many-electron state by an ensemble of replicas, in which each electron is described by a cloud of walkers which samples its density in physical space, one walker and one guide wave per replica, and it replaces the Hartree potential by a conditional interaction built from the walker positions. The sampling is in physical space rather than in configuration space, which keeps the cost polynomial. Here we derive that conditional interaction. Treating the walker as a localization of its electron to a finite resolution, we show that Bayes' theorem and a single empirical substitution yield the Nadaraya-Watson form which the method has used heuristically. The nonlocality length then becomes the width of the conditioning, and the pair and mean-field limits follow from one construction. Because the conditioning removes the gauge freedom which eliminates the orthonormality multipliers in Hartree-Fock, we enforce orthonormality by a term derived from the constraint. We generalize the formulation to spinors, where the Pauli suppression at coincidence becomes graded by the local spin alignment, and it reduces through the collinear and mean-field limits to the known two-particle spin equations. For two electrons in a double quantum dot in an inhomogeneous magnetic field, compared with the exact solution, the spinor equations reproduce the non-collinear spin geometry which collinear calculations cannot represent and recover the larger part of the entanglement missing from a single determinant. In the crossover between the exchange and the Zeeman regimes, where the singlet-triplet coherence sets the spin geometry, the replicas must carry that coherence themselves, as in the spin-coupled pair of valence-bond theory.
Direct sampling of multi-dimensional systems with quantum Monte Carlo methods allows exact account of many-body effects or particle correlations. The most straightforward approach to solve the Schrödinger equation, Diffusion Monte Carlo, has been used in several benchmark cases for other methods to pursue. Its robustness is based on direct sampling of a positive probability density for diffusion in imaginary time. It has been argued that the corresponding real time diffusion can not be realised, because the corresponding oscillating complex valued distribution can not be used to drive diffusion. Here, we demonstrate that this can be done by turning the distribution piecewise positive and normalizable, and also, by using four types of walkers. This study is a proof of concept demonstration using the well-known and transparent case: one-dimensional harmonic oscillator. Furthermore, we show that our novel method can be used to find not only the ground state but also excited states and even the time evolution of a given wave function. Considering fermionic systems, this method may turn out to be feasible for finding the wave function nodes for other approaches.
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.