Quantum Information Processing with Large Nuclear Spins in GaAs Semiconductors

We propose an implementation for quantum information processing based on coherent manipulations of nuclear spins $I=3/2$ in GaAs semiconductors. We describe theoretically an NMR method which involves multiphoton transitions and which exploits the nonequidistance of nuclear spin levels due to quadrupolar splittings. Starting from known spin anisotropies we derive effective Hamiltonians in a generalized rotating frame, valid for arbitrary $I$, which allow us to describe the nonperturbative time evolution of spin states generated by magnetic rf fields. We identify an experimentally observable regime for multiphoton Rabi oscillations. In the nonlinear regime, we find Berry phase interference.

Figure 1

Multiphoton transition schemes for the coherent population of the $I_z$ eigenstates $|m⟩$ of a nuclear spin $I=3/2$. (a) Quantum computation (QC) scheme: the frequencies ${\omega }_k$ of the fields $H_k$ are red ( $\mathrm{\text{−}}·$) and blue (- -) detuned. Diagrams containing blue detunings are negligible for large quadrupolar splitting, i.e., $A\gg \hslash {\delta }_k\ge 0$. (b) Rabi oscillation (RO) scheme: the magnetic fields $H_k^{\prime }\cos ({\omega }_k^{\prime }t+{\Phi }_k^{\prime })$, $k=1,2,3$, give rise to $k$-photon RO.

Figure 2

Preparation of $|s⟩=(1/\sqrt{3})\sum _{m=-1/2}^{3/2}|m⟩$ by means of Eq. (2) for ${}^{\mathrm{71}}\mathrm{G}\mathrm{a}$ nuclei in the QC scheme, which takes about 0.2 ms for $H_1=H_2=1\mathrm{G}$, $H_3=0$, ${\delta }_1=6083{\mathrm{s}}^{-1}$, and ${\delta }_2=0$. The duration of the QC operation is $\le 1/2{\nu }_{\mathrm{R}\mathrm{a}\mathrm{b}\mathrm{i}}^{(2)}$. The analytical result is confirmed by numerics. Note that $a_{-3/2}=0$.

Figure 3

Grover algorithm calculated by means of Eq. (2) in the QC scheme (numerically confirmed), where $|s⟩=(1/\sqrt{3})\sum _{m=-1/2}^{3/2}|m⟩$ is concentrated mainly into $|-1/2⟩$ after 0.55 ms for $H_2=\hslash {\delta }_2/2g_N{\mu }_N=1\mathrm{G}$, $h_1=h_2$, $h_3=0$, ${\delta }_1=0$. The duration of the QC is $\le 1/2{\nu }_{\mathrm{R}\mathrm{a}\mathrm{b}\mathrm{i}}^{(2)}$. Note that $a_{-3/2}=0$.

Figure 4

Numerical solution for the three-photon ROs of ${}^{\mathrm{75}}\mathrm{A}\mathrm{s}$ nuclei between $|3/2⟩$ and $|-3/2⟩$ driven by $H_3^{\prime }=20\mathrm{G}$ with the RO scheme (b) in cw mode. $H_1^{\prime }=H_2^{\prime }=0$.

Figure 5

Numerical solution for the two-photon ROs of ${}^{\mathrm{75}}\mathrm{A}\mathrm{s}$ nuclei between $|3/2⟩$ and $|-1/2⟩$, driven by $H_2^{\prime }=10\mathrm{G}$ according to the RO scheme in cw mode, and $H_1^{\prime }=H_3^{\prime }=0$.

Figure 6

Berry phase oscillation. The three-photon transition probability vanishes where ${\Delta }_{\mathrm{R}\mathrm{a}\mathrm{b}\mathrm{i}}^{\prime (3)}$ is zero.