Entangling Remote Nuclear Spins Linked by a Chromophore

Molecular nanostructures may constitute the fabric of future quantum technologies, if their degrees of freedom can be fully harnessed. Ideally one might use nuclear spins as low-decoherence qubits and optical excitations for fast controllable interactions. Here, we present a method for entangling two nuclear spins through their mutual coupling to a transient optically excited electron spin, and investigate its feasibility through density-functional theory and experiments on a test molecule. From our calculations we identify the specific molecular properties that permit high entangling power gates under simple optical and microwave pulses; synthesis of such molecules is possible with established techniques.

Figure 1

(a) A triplet state of an electron and hole couples two nuclear spin qubits. The triplet can be $|T_{+}⟩=|{\uparrow }_e{\uparrow }_h⟩$, $|T_0⟩=\frac{1}{\sqrt{2}}(|{\downarrow }_e{\uparrow }_h⟩+|{\uparrow }_e{\downarrow }_h⟩)$, or $|T_{-}⟩=|{\downarrow }_e{\downarrow }_h⟩$. (b) An example molecular embodiment of such a model system is bis-diethyl malonate fullerene. (c) Entangling power of the time evolution operators $U_{+}$ (solid curve) and $U_0$ (dashed curve); here $a_{+}\approx 32a_0$. (d) Maximal attainable entangling power $m_{-}$ for $U_{-}$ as the symmetry of the two nuclear spins is reduced. ${\Delta }_1=\frac{|{\omega }_{n^{\prime }}-{\omega }_n|}{{\omega }_n}$; ${\Delta }_2=\frac{|A^{\prime }-A|}{A}$; $A=2.5\mathrm{MHz}$; $D=-296\mathrm{MHz}$; ${\omega }_e=9.6\mathrm{GHz}$; ${\omega }_n=3.7\mathrm{MHz}$.

Figure 2

(a) Decay traces and simulations of the flash delay (FD) and inversion recovery (IR) experiments at 20 K. Upper inset: DEMF molecule. Lower inset: electron spin echo field sweep of DEMF at 20 K. (b) Mims ENDOR simulations and experimental data acquired at 20 K at fields A–F from (a). The largest peak, around 3.9 MHz, was found to correlate to the ${}^{13}\mathrm{C}\mathrm{\text{−}}{T}_0$ interaction, while the weaker peak, at 2.2 MHz, related to ${}^2\mathrm{H}\mathrm{\text{−}}{T}_0$ interaction from the solvent (toluene-$d8$). Also observable in the ENDOR spectrum is the field dependent component of the hyperfine coupling from the ${}^{13}\mathrm{C}\mathrm{\text{−}}{T}_{\pm }$ triplet levels.

Figure 3

Spin density in the lowest triplet excited state of mono- and bis-diethyl malonate fullerenes. The chosen charge isosurface has an isovalue of $0.07\mathrm{\text{electron}}/{\AA }^3$ and the overlaid color indicates the magnitude of the spin density. The spin density is mostly gathered around the fullerene cage. Images rendered using GABEDIT [26].

Figure 4

Entanglement of formation [27] of the nuclear spins after applying our scheme and when the excitation has decayed as a function of the HF strength for different initial polarization of the excitation. The nuclear spins for both curves are assumed to be initially in the state $|\downarrow \uparrow ⟩$; parameters as in Fig. 1. For a fully polarized electronic excitation, the simple protocol as described on the first page is applied (solid); lifetimes as below. (Dashed) Here the initial triplet populations based on our experimental findings (see text) are $p_{-}=0.49$, $p_0=0.02$, $p_{+}=0.49$ with associated lifetimes ${\tau }_{-}=0.57\mathrm{ms}$, ${\tau }_0=0.02\mathrm{ms}$, ${\tau }_{+}=0.57\mathrm{ms}$. In this case, the protocol is applied twice in succession, first for the population in $|T_{+}⟩$ and then for the population in $|T_{-}⟩$.