Role of hyperfine interaction for cavity-mediated coupling between spin qubits

We consider two qubits interacting by means of an optical cavity, where each qubit is represented by a single-electron spin confined to a quantum dot. It is known that electron spins in III-V semiconductor quantum dots are affected by the decoherence due to the hyperfine interaction with nuclear spins. Here we show that the interaction between two qubits is influenced by the Overhauser field as well. Starting from an unpolarized nuclear ensemble, we investigate the dependance of the fidelities for two-qubit gates on the Overhauser field. We include the hyperfine interaction pertubatively to second order in our analytical results and to arbitrary precision numerically.

Figure 1

Cavity mode mediated spin-spin interaction. If the difference of the detunings $\Delta$ for two quantum dots coincides, then they can exchange virtual cavity photons of energy ${\omega }_c$. Here ${\omega }_c$ and ${\omega }_L^i$ are the frequencies of the cavity mode and the laser acting on quantum dot $i$, and $\Delta$ denotes the difference between the detunings, i.e., the detuning from the two-photon resonance on a given quantum dot.

Figure 2

Energy level scheme for a quantum dot filled with a single electron and coupled to a cavity mode. The Zeeman-split single-electron states can be excited to a trion (negatively charged exciton) state $|X^{-}\rangle$ by coupling to the cavity or laser field. Both cavity and laser field frequencies are detuned by ${\Delta }_c$ and ${\Delta }_L$ from resonance, and the combined system is detuned from its two-photon resonance by $\Delta ={\Delta }_c-{\Delta }_L$. The Overhauser shift caused by the hyperfine coupling to the nuclear spins leads to a fluctuating detuning, thus reducing the fidelity of the optically generated quantum gates.

Figure 3

Effect of the Overhauser fields $h_i$ and $h_j$ on the time evolution of the two-electron spin state: change of the interaction phase and precessing of the rotation axes in the $x$-$y$ plane with $(h_i-h_j)t.$

Figure 4

Fidelity $F$ of the unitary time evolution operator $U$ generated by the XY interaction in the presence of nuclear spins as a function of the interaction phase $\phi \left(0\right)\propto t$, where $t$ denotes the interaction time. Both lines show the analytical result calculated in second-order perturbation theory in the nuclear field. The fidelity can be increased by tuning the parameters $\Delta$ and ${\Delta }_L$. For smaller $\Delta$ and ${\Delta }_L$, the fidelity increases due to faster gate operation. The other parameters are held fixed and are given in the text.

Figure 5

(a) The average fidelity $F$ of the unitary time evolution operator $U$ for $\phi \left(0\right)=\pi /4$ calculated in second order of the nuclear field as function of the laser detuning ${\Delta }_L$ and two-photon detuning $\Delta$. An adjustment of these parameters can lead to an increase of the fidelity or to its reduction. However, the choice of the optimal parameters is limited by the conditions of the applied formalism. Further details are given in the text. (b) The ratio between two contributing mechanism to reduction of the fidelity ${\mathrm{D}}^2(\Delta ,{\Delta }_L)·{\tilde{g}}_{ij}^2(\Delta ,{\Delta }_L)$. It increases for smaller detunings and consequently for improved fidelities.

Figure 6

The analytical and numerical results for fidelity of $n$ subsequent CNOT operations in the presence of nuclear spins for different detunings. The analytically calculated values are represented by circles and the numerically calculated ones by stars.

Figure 7

Numerically calculated average fidelity of the time evolution operator for large $\phi \left(0\right)$. (a) The comparison between analytical and numerical calculations for ${\Delta }_L$ $=$ 4.5 meV and $\Delta$ $=$ 0.3 meV. The analytically obtained fidelity diverges for large $\phi \left(0\right)$ while the numerical fidelity saturates to 0.5. (b) Comparison between two numerically calculated fidelities for the gate $U\left(\phi \right)$ with different detunings.