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Coupled Quantum Dots as Two-Level Systems: A Variational Monte Carlo Approach

J. Kim, D.V. Melnikov, J.-P. Leburton

Published Nov 4, 2021• 0 citations
Publisher

Abstract

Recent works indicate that coupled quantum dots (QD) are promising nanostructures for realizing quantum gates that are the basic elements of a quantum computer. In QDs, fundamental quantum logic operations can be achieved in a quantum gate by controlling the entanglement between two electron spins by means of external electric and magnetic fields. This chapter describes the computational approach used to solve the many-body Schrodinger equation. It provides results of the simulations for both the strongly coupled and weakly coupled QDs and analyzes the important differences between the two cases. The utilization of simple trial wavefunctions in VMC have lead to good agreement with the ED method, when electrons are decoupled, which validates two-level approaches such as the Heitler-London approximation in weakly coupled QDs.

DOI
10.1201/9781003148494-32

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Recent works indicate that coupled quantum dots (QD) are promising nanostructures for realizing quantum gates that are the basic elements of a quantum computer. In QDs, fundamental quantum logic operations can be achieved in a quantum gate by controlling the entanglement between two electron spins by means of external electric and magnetic fields. This chapter describes the computational approach used to solve the many-body Schrodinger equation. It provides results of the simulations for both the strongly coupled and weakly coupled QDs and analyzes the important differences between the two cases. The utilization of simple trial wavefunctions in VMC have lead to good agreement with the ED method, when electrons are decoupled, which validates two-level approaches such as the Heitler-London approximation in weakly coupled QDs.
The properties of an electron pair confined in a coupled quantum dot (QD) system are currently being intensively investigated both experimentally and theoretically. Indeed the manipulation of the two-electron spin states is paramount for the realization of controllable entanglement so as to produce the fundamental quantum gate operations in a solid state quantum computer. The works usually concentrate on the properties of a two-electron system without going into detailed analysis of the contributions from various single-particle states or the effects produced by the constituent single QDs. Zeeman splitting is neglected as it affects only the magnitude of the exchange energy but not the single-particle state mixing. In summary, after numerically diagonalizing the two-electron Schrodinger equation in the model confinement potential of both single and laterally coupled double quantum dots, we computed the contributions from the various single-particle states to the final two-particle wave function.
We consider a quantum-gate mechanism based on electron spins in coupled semiconductor quantum dots. Such gates provide a general source of spin entanglement and can be used for quantum computers. We determine the exchange coupling JJJ in the effective Heisenberg model as a function of magnetic (B)(B)(B) and electric fields, and of the interdot distance aaa within the Heitler-London approximation of molecular physics. This result is refined by using sp\mathrm{sp}sp hybridization, and by the Hund-Mulliken molecular-orbit approach, which leads to an extended Hubbard description for the two-dot system that shows a remarkable dependence on BBB and aaa due to the long-range Coulomb interaction. We find that the exchange JJJ changes sign at a finite field (leading to a pronounced jump in the magnetization) and then decays exponentially. The magnetization and the spin susceptibilities of the coupled dots are calculated. We show that the dephasing due to nuclear spins in GaAs can be strongly suppressed by dynamical nuclear-spin polarization and/or by magnetic fields.
We use a variational Monte Carlo algorithm to solve the electronic structure of two-dimensional semiconductor quantum dots in external magnetic field. We present accurate many-body wave functions for the system in various magnetic field regimes. We show the importance of symmetry, and demonstrate how it can be used to simplify the variational wave functions. We present in detail the algorithm for efficient wave function optimization. We also present a Monte Carlo-based diagonalization technique to solve the quantum dot problem in the strong magnetic field limit where the system is of a multiconfiguration nature. PACS numbers: 71.10.-w, 73.21.La, 02.70.Ss 1.
We report on our recent progress in applying semiconductor quantum dots for spin-based quantum computation, as proposed by Loss and DiVincenzo (1998 Phys. Rev. A 57 120). For the purpose of single-electron spin resonance, we study different types of single quantum dot devices that are designed for the generation of a local ac magnetic field in the vicinity of the dot. We observe photon-assisted tunnelling as well as pumping due to the ac voltage induced by the ac current driven through a wire in the vicinity of the dot, but no evidence for ESR so far. Analogue concepts for a double quantum dot and the hydrogen molecule are discussed in detail. Our experimental results in laterally coupled vertical double quantum dot device show that the Heitler–London model forms a good approximation of the two-electron wavefunction. The exchange coupling constant J is estimated. The relevance of this system for two-qubit gates, in particular the SWAP operation, is discussed. Density functional calculations reveal the importance of the gate electrode geometry in lateral quantum dots for the tunability of J in realistic two-qubit gates.
Exchange energies and charge densities for few electrons in electrically tunable quantum dot (QD) systems are investigated by using the variational Monte Carlo (VMC) method in the presence of magnetic fields. For N = 2 electrons in double QD, the singlet Coulomb energy decreases in magnetic fields as a consequence of magnetic localization in both strongly and weakly coupled QDs. However, the triplet Coulomb energy reaches a maximum value at intermediate magnetic fields before decreasing. We observe good agreement between the results for VMC with s-p orbital trial wavefunction and exact diagonalization (ED) method in the exchange energies. For N = 2 electrons in triple quantum dot (TQD), we observe a discontinuity in the J-derivative with detuning voltage ( dJdVT ) in triangular TQDs at B = 0 T as crossing of the eigenenergy levels leads to abrupt spatial symmetry change of the triplet densities (density rotation) and relocalization. For B ≠ 0 T, the angular momentum provided to the electrons quenches this effect. The density rotation is absent in collinear TQD for all magnetic fields as the lowest excited state remains the px state and as such, no change in symmetry is possible. By varying the triangular TQD configuration, we show the discontinuity in ( dJdVT ) persists for top angle between 20° and 70°. For N = 3 electrons in symmetric triangular TQD, the monotonicity of the exchange curve from B = 0 to 4 T remains intact for decoupled QDs but not for coupled QDs. Finally, additional energy for the triangular TQD system is computed for up to N = 3 electrons.
Semiconductor quantum dots (QDs) are one of the leading candidates for realizable qubits, as well as for many other advances in quantum computing and quantum communication. The spatial overlapping of wavefunctions describing each single electron in these nanoscale devices results in tunable spatial entanglement. In this article, we explore the case of two electrons in two-dimensional double quantum dot systems. We compute the two-particle wavefunction through a variational method combined with Hermite finite elements and study the spatial entanglement of electrons. We show that symmetry in the geometry of the double quantum dots plays a role in obtaining optimal entanglement, while a broken symmetry can lead to additional resonances in entanglement that are associated with the crossings of states. We also show that one can finely tune the level of spatial entanglement by altering the geometry of the quantum dots or by applying external fields, which corresponds to an “entanglement spectroscopy.” Finally, we study how impurities in the potential profile of the QDs affect the level of entanglement.