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arXiv:1511.09094 · Physical Review A

Magnetic alteration of entanglement in two-electron quantum dots

N. S. Simonović, R. G. Nazmitdinov

Published Nov 24, 2015• 17 citations
arXivHTMLPDF

Abstract

Quantum entanglement is analyzed thoroughly in the case of the ground and lowest states of two-electron axially symmetric quantum dots under a perpendicular magnetic field. The individual-particle and the center-of-mass representations are used to study the entanglement variation at the transition from interacting to noninteracting particle regimes. The mechanism of symmetry breaking due to the interaction, which results in the states with symmetries related to the latter representation only being entangled even at the vanishing interaction, is discussed. The analytical expression for the entanglement measure based on the linear entropy is derived in the limit of noninteracting electrons. It reproduces remarkably well the numerical results for the lowest states with the magnetic quantum number $M\ensuremath{\ge}2$ in the interacting regime. It is found that the entanglement of the ground state is a discontinuous function of the field strength. A method to estimate the entanglement of the ground state, characterized by the quantum number $M$, with the aid of the magnetic-field dependence of the addition energy is proposed.

DOI
10.1103/physreva.92.052332

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Quantum entanglement is analyzed thoroughly in the case of the ground and lowest states of two-electron axially symmetric quantum dots under a perpendicular magnetic field. The individual-particle and the center-of-mass representations are used to study the entanglement variation at the transition from interacting to noninteracting particle regimes. The mechanism of symmetry breaking due to the interaction, which results in the states with symmetries related to the latter representation only being entangled even at the vanishing interaction, is discussed. The analytical expression for the entanglement measure based on the linear entropy is derived in the limit of noninteracting electrons. It reproduces remarkably well the numerical results for the lowest states with the magnetic quantum number M\ensuremath≥2M\ensuremath{\ge}2M\ensuremath≥2 in the interacting regime. It is found that the entanglement of the ground state is a discontinuous function of the field strength. A method to estimate the entanglement of the ground state, characterized by the quantum number MMM, with the aid of the magnetic-field dependence of the addition energy is proposed.
We consider an exactly solvable two-dimensional model of a two-electron parabolic quantum dot (QD) under the perpendicular magnetic field. We perform a detailed study of the degree of the orbital entanglement of the ground and first excited singlet state depending on the shape of QD, the strength of the effective interaction and the magnetic field. It was found that a degree of entanglement increases with the growth of interaction between electrons. The magnetic field destroys the entanglement of electrons. However, the entanglement in deformed QDs is more stable to the effect of the magnetic field.
The nature of the ground state of a few-electron system in a parabolic twodimensional (2D) quantum dot (QD) is theoretically investigated on the basis of a variational many-body approach within a second quantized formulation. The resulting analytical expressions lead to various phase transitions: maximum density droplet, Wigner phase, etc., in a broad range of magnetic fields. It is observed that in the case of relatively strong QD confinement and weak magnetic fields, a rotationally symmetric spin-singlet state is the ground state of the system. However, in a strong magnetic field and for the same QD confinement, a broken-symmetry spinsinglet state appears to be energetically favoured over the symmetric spin-singlet state. The method developed is generalized for the investigation of possible novel phases for electron-hole interacting systems.
We explore the correlations and entanglements of exact-diagonalized few-electron wave functions in a quantum dot in magnetic fields without the Zeeman splitting. With the increase of the field, the lowest states with different spins gradually form a narrow band and the electronic states undergo a transition from liquids to rotating Wigner molecules which are accompanied by different characters of charge correlations. For both the liquid and crystal states, the spin conditional probability densities show magnetic couplings between the particles which depend on the particle numbers, the total spins and the angular momenta of the states. The von Neumann entropies show the spin-dependent entanglements between electrons. The regular magnetic-coupling oscillations and converging entanglement entropies emerge in the rotating Wigner molecular states.
The electron-electron correlations in a many-electron (Ne=1,2,…,5)({N}_{e}=1,2,\dots{},5)(Ne​=1,2,…,5) quantum dot confined by a parabolic potential is investigated in the presence of a single magnetic ion and a perpendicular magnetic field. We obtained the energy spectrum and calculated the addition energy which exhibits cusps as a function of the magnetic field. The vortex properties of the many-particle wave function of the ground state are studied and for large magnetic fields are related to composite fermions. The position of the impurity influences strongly the spin-pair-correlation function when the external field is large. In a small applied magnetic field, the spin-exchange energy together with the Zeeman terms leads to a ferromagnetic-antiferromagnetic (FM-AFM) transition. When the magnetic ion is shifted away from the center of the quantum dot, a remarkable re-entrant AFM-FM-AFM transition is found as a function of the strength of the Coulomb interaction. Thermodynamic quantities such as the heat capacity, the magnetization, and the susceptibility are also studied. The cusps in the energy levels show up as peaks in the heat capacity and the susceptibility.
We consider a simple model for two-electron quantum dots (QDs) in the perpendicular magnetic field. With the aid of this model we study the degree of the orbital entanglement as a function of the magnetic field strength and the QD shape. We found that the circular QD exhibits no entanglement in the magnetic field. Although the entanglement increases with the increase of the QD deformation at a fixed value of the magnetic field, it approaches the constant value at the fixed deformation with the increase of the magnetic field.
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.