Geometric Landau-Zener-Stückelberg-Majorana interferometry for hybrid spin registers

Hybrid systems of electron and nuclear spins are a fundamental building block for scalable quantum information processing. Here we propose a fast and high-fidelity control scheme based on geometric Landau-Zener-Stückelberg-Majorana interferometry, using a nitrogen-vacancy center in diamond as an example. When this hybrid spin multilevel system is driven through avoided crossings, geometric phases accumulate in each spin subspace and manifest as interference patterns in the dynamics of the whole system. We systematically study the effects of various decoherence mechanisms and fluctuations of control parameters on this geometric phase process. We also consider the application of quantum memory, where electron spins are used as a processing qubit and nuclear spins are used as a storage qubit. This scheme can achieve a transfer fidelity of 0.985 and a total storage fidelity of 0.96. These results pave the way for geometric coherent manipulations on a variety of hybrid quantum platforms.

Figure 1

Multiple energy levels versus external magnetic field $B_z$. The electron and nuclear spin states are labeled as $|m_S,m_I\rangle$. Here the constant magnetic field has $B_x=B_y=2$ G. The inset shows only the electron spin states.

Figure 2

(a) External magnetic field applied to the full system. Here the amplitudes of the external field are $\mathrm{\Delta }=2$ G, $\epsilon =6$ G, and the solid angle is $\varphi =\frac{\pi }{2}$. (b) Effective magnetic fields in each spin subspace. (c) Evolution paths along the instantaneous eigenstates in a spin subspace.

Figure 3

Population of the state $|-1,1\rangle$ as a function of solid angle $\varphi$ and total sweeping time $T$. Here the control parameters are $\mathrm{\Delta }=2$ G and $\epsilon =6$ G. The dotted lines are calculated according to the analytical condition of interference.

Figure 4

Population of the state $|-1,1\rangle$ as a function of evolution time $t$ for (a) a destructive and (b) a constructive interference process. The solid line is the evolution without decoherence and the dash-dotted line is the evolution with decoherence. Here the control parameters of the external magnetic field are amplitude $\mathrm{\Delta }=2$ G, detuning $\epsilon =6$ G, and solid angle (a) $\varphi =4.77$ and (b) $\varphi =1.63$.

Figure 5

Transfer fidelity as a function of (a) electron spin relaxation rate ${\mathrm{\Gamma }}_{1e}$, (b) electron spin dephasing rate ${\mathrm{\Gamma }}_{2e}$, and (c) nuclear spin dephasing rate ${\mathrm{\Gamma }}_{2n}$. Here we consider a destructive interference process and use the control parameters of the external magnetic field, i.e., amplitude $\mathrm{\Delta }=2$ G, detuning $\epsilon =6$ G, and solid angle $\varphi =4.77$.

Figure 6

Transfer fidelity as a function of deviations of control pulse parameters for different cases (a) ${\sigma }_{\epsilon }=0$ G, (b) ${\sigma }_{\mathrm{\Delta }}=0$ G, (c) ${\sigma }_{\epsilon }=0.1$ G, and (d) ${\sigma }_{\mathrm{\Delta }}=0.1$ G. Here we consider a destructive interference process and use the control parameters of the external magnetic field, i.e., amplitude $\mathrm{\Delta }=2$ G, detuning $\epsilon =6$ G, and solid angle $\varphi =4.77$.

Figure 7

(a) Population of the electron spin state $|m_s=0\rangle$ vs storage time $t_m$. (b) Quantum state tomography of the quantum memory: density matrices for initial ${\sigma }_X, {\sigma }_Y$, and ${\sigma }_Z$ states (top row) and corresponding recovered states (bottom row). Here the control parameters are $\mathrm{\Delta }=0.1$ G, $\epsilon =6$ G, $\varphi =\frac{\pi }{2}$, transfer time $t_{\text{trans}}=240\mathrm{ns}$, and storage time $t_m=20\mu \mathrm{s}$.

Figure 8

(a) Nonadiabatic error as a function of driving field time for different amplitude values. (b) Nonadiabatic error as a function of driving field time for different detuning values. Here we use the control parameters of the external magnetic field: (a) detuning $\epsilon =6$ G and (b) amplitude $\mathrm{\Delta }=2$ G

Figure 9

Simulation of quantum process tomography for the quantum memory in the basis of $(I,{\sigma }_X,{\sigma }_Y,{\sigma }_Z)$. In the process matrix, the $[I,I]$ component is dominant, indicating the high fidelity of this memory process. Here the control parameters are amplitudes $\mathrm{\Delta }=0.1$ G, $\epsilon =6$ G, solid angle $\varphi =\frac{\pi }{2}$, transfer time $t_{\text{trans}}=240\mathrm{ns}$, and storage time $t_m=20\mu \mathrm{s}$.