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arXiv:2107.03834 · Entropy

Spatial Entanglement of Fermions in One-Dimensional Quantum Dots

Ivan P. Christov

Published Jul 7, 2021• 6 citations
arXivHTMLPDF

Abstract

The time-dependent quantum Monte Carlo method for fermions is introduced and applied in the calculation of the entanglement of electrons in one-dimensional quantum dots with several spin-polarized and spin-compensated electron configurations. The rich statistics of wave functions provided by this method allow one to build reduced density matrices for each electron, and to quantify the spatial entanglement using measures such as quantum entropy by treating the electrons as identical or distinguishable particles. Our results indicate that the spatial entanglement in parallel-spin configurations is rather small, and is determined mostly by the spatial quantum nonlocality introduced by the ground state. By contrast, in the spin-compensated case, the outermost opposite-spin electrons interact like bosons, which prevails their entanglement, while the inner-shell electrons remain largely at their Hartree-Fock geometry. Our findings are in close correspondence with the numerically exact results, wherever such comparison is possible.

DOI
10.3390/e23070868

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The time-dependent quantum Monte Carlo method for fermions is introduced and applied in the calculation of the entanglement of electrons in one-dimensional quantum dots with several spin-polarized and spin-compensated electron configurations. The rich statistics of wave functions provided by this method allow one to build reduced density matrices for each electron, and to quantify the spatial entanglement using measures such as quantum entropy by treating the electrons as identical or distinguishable particles. Our results indicate that the spatial entanglement in parallel-spin configurations is rather small, and is determined mostly by the spatial quantum nonlocality introduced by the ground state. By contrast, in the spin-compensated case, the outermost opposite-spin electrons interact like bosons, which prevails their entanglement, while the inner-shell electrons remain largely at their Hartree-Fock geometry. Our findings are in close correspondence with the numerically exact results, wherever such comparison is possible.
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.
Given a specific interacting quantum Hamiltonian in a general spatial dimension, can one access its entanglement properties, such as the entanglement entropy corresponding to the ground state wave function? Even though progress has been made in addressing this question for interacting bosons and quantum spins, as yet there exist no corresponding methods for interacting fermions. Here we show that the entanglement structure of interacting fermionic Hamiltonians has a particularly simple form-the interacting reduced density matrix can be written as a sum of operators that describe free fermions. This decomposition allows one to calculate the Renyi entropies for Hamiltonians which can be simulated via determinantal quantum Monte Carlo calculations, while employing the efficient techniques hitherto available only for free fermions. The method presented works for the ground state, as well as for the thermally averaged reduced density matrix.
We model a core-shell quantum dot using a one dimensional system of two interacting electrons with a potential that can vary from one corresponding to a dot with comparable width of core and shell to one where the core well width becomes very small, but is always non-zero, while the shell becomes wide. We calculate the entanglement due to the spatial degrees of freedom and also the site or local entanglement found in a partition of the potential accessible to measurement. We quantify both using the linear entropy of the reduced density matrix. A sharp variation of the entanglement had been seen in earlier work for the transformation from a one dimensional model core-shell dot to a double dot [1]. We investigate whether the spatial entanglement and energy derivatives display any discontinuity as the core well width becomes small and if the two-electron state is always bound to the core well.
We perform the quantitative evaluation of the entanglement dynamics in scattering events between two indistinguishable electrons interacting via the Coulomb potential in one- and two-dimensional semiconductor nanostructures. We apply a criterion based on the von Neumann entropy and the Schmidt decomposition of the global state vector suitable for systems of identical particles. From the time-dependent numerical solution of the two-particle wave function of the scattering carriers we compute their entanglement evolution for different spin configurations: two electrons with the same spin, with different spin, and singlet and triplet spin states. The procedure allows us to evaluate the mechanisms that govern entanglement creation and their connection with the characteristic physical parameters and initial conditions of the system. The cases in which the evolution of entanglement is similar to the one obtained for distinguishable particles are discussed.
Quantum dot structures confine electrons in a small region of space. Some properties of semiconductor quantum dots, such as the discrete energy levels and shell filling effects visible in addition spectra, have analogies to those of atoms and indeed dots are sometimes referred to as 'artificial atoms'. However, atoms and dots show some fundamental differences due to electron correlations. For real atoms, the kinetic energy of electrons dominates over their mutual Coulomb repulsion energy and for this reason the independent electron approximation works well. For quantum dots the confining potential may be shallower than that of real atoms leading to lower electron densities and a dominance of mutual Coulomb repulsion over kinetic energy. In this strongly correlated regime the independent electron picture leads to qualitatively incorrect results. This thesis concentrates on few-electron quantum dots in the strongly correlated regime both for quasi-one-dimensional and two-dimensional dots in a square confining potential. In this so-called 'Wigner' regime the ground-state electronic charge density is localised near positions of classical electrostatic minima and the interacting electronic spectrum consists of well separated spin multiplets. In the strongly correlated regime the structure of low-energy multiplets is explained by mapping onto lattice models with extended-Hubbard and Heisenberg effective Hamiltonians. The parameters for these effective models are calculated within a Hartree approximation and are shown to reproduce well the exact results obtained by numerical diagonalisation of the full interacting Hamiltonian. Comparison is made between square dots and quantum rings with full rotational symmetry. In the very low-density regime, direct diagonalisation becomes impractical due to excessive computer time for convergence. In this regime a numerical renormalisation group method is applied to one-dimensional dots, enabling effective spin-interactions to be pr
Confined geometries such as semiconductor quantum dots are promising candidates for fabricating quantum computing devices. When several quantum dots are in proximity, spatial correlation between electrons in the system becomes significant. In this paper, we employ a variational formulation for calculating accurate two-electron wavefunctions in configuration space, irrespective of potential geometry. To evaluate the Coulomb integrals with high accuracy, a numerical integration method using multiple Gauss quadratures is used. With this approach, we investigate the confinement of two electrons in double quantum dots, and evaluate the spatial entanglement. We investigate the dependence of spatial entanglement on various geometrical parameters. We derive the two-particle wavefunctions in the asymptotic limit of the separation distance between quantum dots and obtain universal saturation values for the spatial entanglement. Resonances in the entanglement values due to avoided level crossings of states are observed. We also demonstrate the formation of electron clusters and show that the entanglement value is a good indicator for the formation of such clusters. Further, we show that a precise tuning of the entanglement values is feasible with applied external electric fields.