Dark Bell states in tunnel-coupled spin qubits

We investigate the dynamical purification of maximally entangled electron states by transport through coupled quantum dots. Under resonant ac driving and coherent tunneling, even-parity Bell states perform Rabi oscillations that decouple from the environment, leading to a dark state. The two electrons remain spatially separated, one in each quantum dot. We propose configurations where this effect will prove as antiresonances in transport spectroscopy experiments.

Figure 1

Dynamical generation of entanglement. (a) In a spin blockade double quantum dot, charge flows through states of two electrons with opposite spins, $\left\{\right|\uparrow ,\downarrow \rangle ,|\downarrow ,\uparrow \rangle \}$, coupled by intradot tunneling $\tau$, to the doubly occupied singlet $|S_R\rangle$, from where an electron tunnels to the drain lead. (b) The occupation of the even-parity subspace blocks the current, forming a mixed state ${\widehat{\rho }}_{SB}$ with statistical weights $p_{\sigma }$. Coherent spin rotations by an ac magnetic field, $B_x$, open the system to transport again. For a particular frequency $\Omega$, an even-parity eigenstate of the magnetic field becomes a dark state $|{\psi }_d\rangle$. (c) The system evolves towards a pure steady state composed of Bell superpositions. Coherence of the steady state is apparent in the undamped oscillations of the off-diagonal elements.

Figure 2

Current as a function of the driving frequency $\omega$. The collective spin rotation results in a nontrivial line shape not centered at the ESR conditions (at $\hslash \omega ={\Delta }_L^z,{\Delta }_R^z$, vertical dashed lines) and a vanishing current (dark state) at $\hslash \Omega =({\Delta }_L^z+{\Delta }_R^z)/2$. Cases for different driving amplitudes $B_x$ are vertically offset. Parameters (for all figures, except where indicated): $\hslash \Gamma ={10}^{-2}$ meV, $\tau =20\Gamma$, $\gamma =0.4$ meV/T, $B_z=0.1$ T, and $a=0.8$. Interdot tunneling is resonant: $\delta \varepsilon =0$. The inset shows the shift of the antiresonance frequency ${\omega }_m$ with the amplitude of the ac field. The solid line is a quadratic fit.

Figure 3

A pure and coherent steady state. (a) Purity of the steady state as a function of the driving frequency. At the antiresonance frequency $\Omega$ a pure state is formed. Far from resonance, the purity tends to 1/2, consistently with a spin blockaded density matrix ${\widehat{\rho }}_{SB}$. Here, $B_x=B_z/5$. (b) Mean and variance of a state tomography at $\omega =\Omega$. Only two-electron states are represented for clarity. Finite off-diagonal elements reveal a coherent superposition. Their explicit time dependence [cf. Eq. (5)] will be averaged out in the mean of any measurement, but will affect its variance.

Figure 4

Transport spectroscopy for different dc magnetic fields, with $B_x=B_z^0=0.1$T. (a) For finite detuning ($\delta \varepsilon =2$ meV), a double peak structure coinciding with the individual ESR conditions ($\hslash \omega ={\Delta }_l$, marked by color dots) is consistent with previous experimental observations. (b) Resonant interdot tunneling ($\delta \varepsilon =0$) enhances the visibility of the collective features. The dark antiresonance is robust upon increasing the magnetic field. At low $B_z$ two peaks are visible which do not correspond to individual ESR. The latter appear as humps for larger magnetic fields. An offset is introduced for clarity. Inset: Frequency splitting of the current maxima, $\Delta {\omega }_M$. In orange, the expected behavior for the individual ESR.