Magnetic alteration of entanglement in two-electron quantum dots

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 $M\ge 2$ 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.

Figure 1

Lowest energy levels (in $\hslash {\omega }_0$ units) of an axially symmetric two-electron QD with ${\omega }_z\gg {\omega }_0$ for (a) noninteracting electrons ($\lambda =0$) and (b) $\lambda =2$ (in $\hslash {\omega }_0{\ell }_0$ units). In order to get a better resolution the energy is defined with respect to the $E_0^{\left(0\right)}$ level. The Zeeman splitting is taken into account: The effective Landé factor $g^{*}=-0.44$. Closed circles mark the $M=2$ and $M_S=0$ levels at the field strength ${\omega }_L/{\omega }_0=1.65$. At this value the lower level $E_{<}$ corresponds to the ground-state energy (thick gray line).

Figure 2

(a) Full range of (real) values of the coefficients $c_1$ and $c_2$ in the superposition (24) that represents the lowest symmetric orbital states ${\psi }^{(+)}$ with $M=2$ at $\lambda =0$ (green circular line) and in the limit $\lambda \to 0$ (red $\odot$'s). The values, marked by (blue) open circles, correspond to the uncorrelated basis states $\pm u_1^{(+)}$ and $\pm u_2^{(+)}$. (b) Lowest ${\psi }^{(+)}$ states with $M=2$ and the related eigenenergies (in $\hslash {\omega }_0$ units) are shown in the $\theta -E_{\mathrm{tot}}$ diagram, where $\theta =arctan(c_2/c_1)$, for $\lambda =0$ (green line) and $\lambda =2$ (red closed circles) (in $\hslash {\omega }_0{\ell }_0$ units). The full $\theta$ domain $\theta \in (-\pi ,\pi )$, which corresponds to $\lambda =0$ (degenerate level $E_2^{\left(0\right)}$), is reduced to two discrete values $\theta \approx \pm \pi /4$ as soon as the interaction is switched on. The eigenenergies are calculated for ${\omega }_L/{\omega }_0=1.65$ [at $\lambda =2$ the state at $\theta \approx -\pi /4$ is the ground state; see Fig. 1]. (c) The $\theta -E_{\mathrm{tot}}$ diagram (the energy is in $\hslash {\omega }_0$ units) showing the transition of the lowest (orbital symmetric) states at $\lambda =2$ to the states ${\psi }_{\gtrless }^{(+)}$ [see Eq. (30)], obtained by reducing gradually the interaction. The states remain entangled at any $\lambda \ne 0$ ($\theta \approx \pm \pi /4$), even in the limit $\lambda \to 0$ (then $\theta =\pm \pi /4$).

Figure 3

Entanglement measure $\mathcal{E}$ of the lowest states with different $m$ as a function of the electron-electron interaction strength (the parameter $\lambda$ in $\hslash {\omega }_0{\ell }_0$ units) for (a) ${\omega }_L=0$ and (b) ${\omega }_L/{\omega }_0=2.5$. The dashed, dash-dotted, and solid lines correspond to the (3D) QDs with ${\omega }_z/{\omega }_0=2$ and ${\omega }_z/{\omega }_0=5$ and to the 2D model of QDs, respectively. The red and blue values correspond to the symmetric and antisymmetric orbital states [i.e., even and odd ${\psi }_{\mathrm{rel}}\left({\mathbf{r}}_{12}\right)$ functions], respectively.

Figure 4

Entanglement of the lowest state with $m=0$ of axially symmetric two-electron QDs with $\lambda =2$ (in $\hslash {\omega }_0{\ell }_0$ units) and different ratios ${\omega }_z/{\omega }_0$ as a function of the parameter ${\omega }_L/{\omega }_0$. The closed circles denote the values of ${\omega }_L/{\omega }_0$ when the dots with the given ratios ${\omega }_z/{\omega }_0$ become spherically symmetric.

Figure 5

Measure of entanglement $\mathcal{E}$ of the lowest states ($n_{\text{c.m.}}=n_z^{\left(\text{c.m.}\right)}=m_{\text{c.m.}}=0, n=n_z=0$) for various $m$ in the limit of noninteracting electrons.

Figure 6

(a) Lowest energy levels (thin red and blue lines) and the ground-state energy (thick lines) of axially symmetric two-electron QDs with $\lambda =2$ (in $\hslash {\omega }_0{\ell }_0$ units) for ${\omega }_z/{\omega }_0\gg 1$. In order to get better resolution, the energies are defined with respect to the $E_0^{\left(0\right)}$ level (all energies are defined in $\hslash {\omega }_0$ units). The results for $g^{*}=0$ and $-0.44$ are represented by dotted and solid lines, respectively. The numbers in parentheses are the values of quantum numbers $(m,S)$ that characterize the corresponding states. (b) Entanglement of the states corresponding to the levels shown in (a). Thick dotted and solid (orange) lines represent the entanglement of the ground state at $g^{*}=0$ and $-0.44$, respectively. Dash-dotted lines show the entanglement measure of the lowest levels with $M=0$ and 1 obtained within the first-order approximation [see Eqs. (58) and (63)].

Figure 7

Entanglement of the lowest states with different $m$ (thin red and blue lines) of axially symmetric two-electron QDs with $\lambda =2$ (in $\hslash {\omega }_0{\ell }_0$ units), $g^{*}=-0.44$, and two values of the ratio ${\omega }_z/{\omega }_0$ equal 2 (dashed lines) and 5 (solid lines). The numbers in parentheses are the values of quantum numbers $(m,S)$ that characterize the corresponding states. Entanglement of the ground state for these two cases is represented by the thick green line (dashed and solid, respectively).