Quantum entanglement driven by electron-vibrational mode coupling

In this work, we show how electron-vibrational mode coupling can be used to drive maximally entangled states. The physical system consists of two pairs of quantum dots, each pair with a single electron able to tunnel between the dots, thus encoding a qubit. The electron-vibrational mode coupling, described by the strength parameter $g$, yields to an effective electron-electron interaction, responsible for the formation of a maximally entangled quantum state. Both the formation time and the relative phase of the aforementioned state are strongly dependent on $g$. The quantum dynamics exhibits a nonmonotonic behavior as $g$ increases, an effect explained through an effective Hamiltonian that takes into account high-order transition processes.

Figure 1

Illustration of our system of interest: Quantum dots 1 and 3 (2 and 4) are coupled with the vibrational (bosonic) mode VM1 (VM2) with frequency ${\omega }_{1\left(2\right)}$. Additionally, the dots 1 (3) and 2 (4) are coupled by tunneling with real parameter ${\mathrm{\Delta }}_{12\left(34\right)}$, where ${\varepsilon }_i$ is the electronic level of the $n\mathrm{th}$ dot ($n=1,...,4$). If two quantum dots share a single electron, the system can encode a qubit, as shown by dashed lines.

Figure 2

Spectrum and entanglement degree as functions of ${\delta }_1$, of the first (solid black line), the second (dashed black line), the third (solid brown line), and the fourth (dashed brown line) eigenstate of the general Hamiltonian, given by Eq. (1). (a) The energies and (b) the concurrence are shown, considering ${\delta }_2=0$, and (c) the energies and (d) the concurrence for ${\delta }_2=0.1\omega$. Anticrossings are indicated by arrows and identified with letters (see main text for details). Physical parameters are $g=0.1\omega$ and ${\mathrm{\Delta }}_1={\mathrm{\Delta }}_2=5\times {10}^{-3}\omega$.

Figure 3

Stacked bar graph showing the populations of states of the computational basis for the eigenstates at the anticrossing positions of Fig. 2: (a) first order and (b) second order. Colors and patterns for the main contributors (states with $m=l=0$) are described on the right side of the panel. Solid gray is used for the sum of the populations of other states. Physical parameters are $g=0.1\omega$ and ${\mathrm{\Delta }}_1={\mathrm{\Delta }}_2=5\times {10}^{-3}\omega$.

Figure 4

Dynamics of the entanglement degree of the electrons as a function of $\theta$ for different values of $g$, considering the physical parameters of the anticrossing ${\mathrm{C}}^{\prime }$: $\mathrm{\Delta }={\mathrm{\Delta }}_1={\mathrm{\Delta }}_2=5\times {10}^{-3}\omega$ and $\delta ={\delta }_1={\delta }_2=0.1\omega$. Here, the numerical calculation of $C$ (black dots) and the analytic expression $C_{2\mathrm{ls}}\left(t\right)$ as defined in Eq. (23) (brown line) are plotted considering (a) $g=0.05\omega$, (b) $g=0.10\omega$, ($\mathrm{c})g=0.15\omega$, (d) $g=0.22\omega$, (e) $g=0.40\omega$, and (f) $g=0.50\omega$.

Figure 5

Ratio $r=\mathrm{\Omega }/{\mathrm{\Omega }}_0$ as a function of the electron-vibrational mode coupling $g$, as given by Eq. (20), considering the physical parameters of the anticrossing ${\mathrm{C}}^{\prime }$: $\mathrm{\Delta }={\mathrm{\Delta }}_1={\mathrm{\Delta }}_2=5\times {10}^{-3}\omega$ and $\delta ={\delta }_1={\delta }_2=0.1\omega$.

Figure 6

Evolution of the fidelity $\mathcal{F}$, considering the target state $|\mathrm{\Psi }\left({\phi }_{\mathrm{tar}}\right)\rangle$ for two choices of relative phase: ${\phi }_{\mathrm{tar}}=-\pi /2$ (brown dots) and ${\phi }_{\mathrm{tar}}=\pi /2$ (black triangles). The physical parameters correspond to those considered in Fig. 4 with (a) $g=0.1\omega$ and (b) $g=0.4\omega$. Lines show the evolution of ${\mathcal{F}}_{2\mathrm{ls}}\left(\theta \right)$ for the same physical conditions considering ${\phi }_{\mathrm{tar}}=-\pi /2$ (solid black line) and ${\phi }_{\mathrm{tar}}=\pi /2$ (dotted black line).

Figure 7

Evolution of the concurrence and the fidelity, assuming two different values of charge dephasing rate ${\mathrm{\Gamma }}_{\mathrm{deph}}$, calculated by solving a Lindblad master equation, with $\omega =20$ meV, $\mathrm{\Delta }=5\times {10}^{-3}\omega =0.1$ meV, and ${\delta }_1={\delta }_2=0.1\omega =2$ meV, with $g=0.15\omega =3$ meV corresponding to the case illustrated in Fig. 4. (a) Concurrence $C$ (black dots) for ${\mathrm{\Gamma }}_{\mathrm{deph}}=1\times {10}^{-4}\omega$; (b) fidelity considering the target state $|\mathrm{\Psi }\left({\phi }_{\mathrm{tar}}\right)\rangle$ for two choices of relative phase, ${\phi }_{\mathrm{tar}}=-\pi /2$ (brown dots) and ${\phi }_{\mathrm{tar}}=\pi /2$ (black triangles), considering the same value ${\mathrm{\Gamma }}_{\mathrm{deph}}$ in (a); (c) concurrence $C$ (black dots) for ${\mathrm{\Gamma }}_{\mathrm{deph}}=2\times {10}^{-4}\omega$; (d) fidelity, also for ${\mathrm{\Gamma }}_{\mathrm{deph}}=2\times {10}^{-4}\omega$, using the same scheme of colors and symbols as (b). In all panels, we illustrate the behavior when the dephasing is neglected (brown solid line).