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Physical Review A

Considerations on Hund's first rule in a planar two-electron quantum dot

Sebastian Schröter, Harald Friedrich, Javier Madroñero

Published Apr 12, 2013• 17 citations
Publisher

Abstract

We give a detailed analysis of the applicability of Hund's first rule for harmonic planar two-electron quantum dots by means of entanglement witnesses. We find that for purely harmonic confinement there is only one pair of singlet and triplet states for which it can be applied. We also discuss the origin and validity for this case and extend the discussion to a quartic confining potential, a hard-wall potential, and a combination of harmonic confinement with quartic perturbation. A generalized rule, the alternating rule, is found to be applicable and valid for vanishing angular momentum states in all these cases. Furthermore, we are able to clarify the role of entanglement in general harmonic two-electron models for vanishing interaction strength. This behavior can be attributed to the special separability properties of these models.

DOI
10.1103/physreva.87.042507

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We give a detailed analysis of the applicability of Hund's first rule for harmonic planar two-electron quantum dots by means of entanglement witnesses. We find that for purely harmonic confinement there is only one pair of singlet and triplet states for which it can be applied. We also discuss the origin and validity for this case and extend the discussion to a quartic confining potential, a hard-wall potential, and a combination of harmonic confinement with quartic perturbation. A generalized rule, the alternating rule, is found to be applicable and valid for vanishing angular momentum states in all these cases. Furthermore, we are able to clarify the role of entanglement in general harmonic two-electron models for vanishing interaction strength. This behavior can be attributed to the special separability properties of these models.
Abstract We investigate the influence of quartic anharmonic confinement on spatial entanglement in two-electron quantum dots. By transforming the Hamiltonian into center-of-mass and relative coordinates, we develop a computational framework combining variational methods, Numerov integration, and Monte Carlo evaluation of linear entropy. We demonstrate that anharmonicity reduces the equilibrium inter-electronic separation, leading to enhanced Coulomb correlations and increased entanglement. In the weakly anharmonic regime, the linear entropy exhibits an approximately linear dependence on the anharmonicity parameter. The results are validated through convergence analysis and comparison with known harmonic-limit results. These findings establish anharmonic confinement as a controllable mechanism for tuning entanglement in semiconductor quantum dots, with potential applications in quantum information processing.
We study the validity of Hund's first rule for the spin multiplicity in circular molecules---made of real or artificial atoms such as quantum dots---by considering a perturbative approach in the Coulomb interaction in the extended Hubbard model with both on-site and long-range interactions. In this approximation, we show that an anti-Hund rule always defines the ground state in a molecule with 4N4N4N atoms at half-filling. In all other cases (i.e., number of atoms not a multiple of four, or a 4N4N4N molecule away from half-filling) both the singlet and the triplet outcomes are possible, as determined primarily by the total number of electrons in the system. In some instances, the Hund rule is always obeyed and the triplet ground state is realized mathematically for any values of the on-site and long-range interactions, while for other filling situations the singlet is also possible but only if the long-range interactions exceed a certain threshold, relatively to the on-site interaction.
We consider a system of two purely capacitively coupled singlet-triplet qubits and numerically simulate the energy structure of four electrons in two double quantum dots with a large potential barrier between them. We calculate the interqubit coupling strength using an extended Hund-Mulliken approach which includes excited orbitals in addition to the lowest-energy orbital for each quantum dot. We show the coupling strength as a function of the qubit separation as well as plotting it against the detunings of the two double quantum dots and show that the general qualitative features of our results can be captured by a potential-independent toy model of the system.
ABSTRACT Accurate estimation of structural properties is essential for understanding the behavior of atomic systems, particularly in confined environments where they display unique physical phenomena. Amongst these, the correlation energy plays a pivotal role in shaping the overall dynamics of the system. In this study, we investigate the influence of external confinement on both correlation energy and Collins' conjecture by evaluating different quantum information and quantum entanglement entropies. For this purpose, we model a two‐electron quantum dot using an isotropic harmonic potential to explore these effects. Various geometrical properties are estimated to showcase the acceptability and viability of the present method. Additionally, we investigate the impact of harmonic confinement on the quantum similarity measures e.g. quantum similarity index and quantum dissimilarity between He(1) and He(1 and 1), as well as between He(1) and He(1). These results for both ground and singly excited states of two‐electron system are reported here for first time.
Excitonic phases described by the quantum condensation of electron-hole pairs (or excitons) are studied in the two-orbital Hubbard model with the Hund's rule coupling J . Using the variational cluster approximation, we calculate the ground-state energy and order parameters, and show that the Hund's rule coupling always stabilizes the excitonic spin-density- wave state and destabilizes the excitonic charge-density-wave state. The pair coherence lengths calculated in the spin-singlet and spin-triplet states indicate that only the spin-triplet excitons are paired more tightly with increasing J . The single-particle spectrum and density of states are also calculated to clarify the characters of these excitonic density-wave states.
We calculate the spatial entanglement between two electrons trapped in a nanostructure for a broad class of confinement potentials, including single and double quantum dots, and core-shell quantum dot structures. By using a parametrized confinement potential, we are able to switch from one structure to the others with continuity and to analyze how the entanglement is influenced by the changes in the confinement geometry. We calculate the many-body wave function by ``exact'' diagonalization of the time-independent Schr\"odinger equation. We discuss the relationship between the entanglement and specific cuts of the wave function, and show that the wave function at a single highly symmetric point could be a good indicator for the entanglement content of the system. We analyze the counterintuitive relationship between spatial entanglement and Coulomb interaction, which connects maxima (minima) of the first to minima (maxima) of the latter. We introduce a potential quantum phase transition which relates quantum states characterized by different spatial topology. Finally we show that by varying shape, range, and strength of the confinement potential, it is possible to induce strong and rapid variations in the entanglement between the two electrons. This property may be used to tailor nanostructures according to the level of entanglement required by a specific application.