Degenerate eigensubspace in a triangle-level system and its geometric quantum control

The non-Abelian geometric phases appear only in a degenerate subspace and the most familiar model with such a subspace is the $N$-pod system. Here we propose an alternative system to realize a non-Abelian gauge structure. We demonstrate that a three-level system with a triangular form can have two degenerate eigenstates which can induce non-Abelian geometric phases, and notably they are the lowest eigenstates of the system. As an application of the degenerate subspace, we propose an experimentally feasible scheme to realize a universal set of quantum gates based on the adiabatic non-Abelian geometric phases. To shorten the evolution time constrained by the adiabatic condition, the shortcut to adiabaticity is adopted to speed up the adiabatic geometric quantum gates. The combination of the intrinsic geometric characteristic and shortcut to adiabaticity makes the quantum operations fast and robust against certain control errors.

Figure 1

(a) Coupling scheme of three-level triangle-type system. Rabi frequencies ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_2$ couple the ground states $|1\rangle$ and $|2\rangle$ with the excited state $|3\rangle$ respectively and ${\mathrm{\Omega }}_3$ couples the ground states $|1\rangle$ and $|2\rangle$. A special condition is applied to the single-photon detuning $\mathrm{\Delta }$ to achieve degenerate eigenvalues, as described in the text. (b) Eigenvalues $E$ (in the unit of ${\mathrm{\Omega }}_0$) of the three-level triangle-type system. Two dressed states ${\lambda }_1$ and ${\lambda }_2$ are always degenerated in the parameters space ($\theta ,\phi$ in the unit of $\pi$) when the parameters satisfy the degenerate conditions listed in the text.

Figure 2

Two physical models to realize the three-level triangle-type Hamiltonian. (a) Cold atomic system of ${}^{87}\mathrm{Rb}$. Zeeman sublevels with a quantum number difference $\mathrm{\Delta }{m}_F=\pm 1$ can be coupled by radio frequencies or the two-photon Raman transition (${\sigma }^{+}-\pi$), while the sublevels with a quantum number difference $\mathrm{\Delta }{m}_F=\pm 2$ may be coupled by the ${\sigma }^{+}-{\sigma }^{-}$ transition. (b) Nitrogen-vacancy center. State $|0,\uparrow (\downarrow )\rangle [|1,\uparrow (\downarrow )\rangle ]$ can be coupled to $|a,\uparrow (\downarrow )\rangle$ through a microwave field, while $|0,\uparrow (\downarrow )\rangle$ and $|1,\uparrow (\downarrow )\rangle$ can be coupled by magnetic coupling between the spin and a nanomechanical oscillator.

Figure 3

(a) Pulse shapes of $\mathrm{\Omega }$ and ${\mathrm{\Omega }}_3$ and the corresponding single-photon detuning $\mathrm{\Delta }$ to realize a not gate. $\mathrm{\Delta }$ is divided by a factor of 10 for the convenience of contrast. (b) Population dynamics under the driving of the pulses in Fig. 3, showing the fidelities $[\langle \psi (t\left)\right|k\rangle ]$ of states $|1\rangle ,|2\rangle$, and $|3\rangle .$ Dephasing and decoherence have not been included.

Figure 4

Modified pulse shape ${\mathrm{\Omega }}^{\prime }$ according to the shortcut to adiabaticity and the original pulse $\mathrm{\Omega }$ . A slightly larger Rabi frequency can speed up the transfer by 20 times.

Figure 5

(a) Fidelity of the not gates versus the variation of Rabi frequency. The green solid line represents the not gate realized by non-Abelian GPs and shortcut to adiabaticity, while the blue dotted-dashed line represents the one of nonadiabatic non-Abelian GPs. The red-dashed line shows the fidelity if one keeps revising ${\mathrm{\Omega }}^{\prime }$ when $\mathrm{\Omega }$ changes. (b) Fidelity of the not gate versus the amplitude of the random noise. (c) Fidelity of the proposed not gate versus the violation of degeneration condition by making ${\mathrm{\Omega }}_1\ne {\mathrm{\Omega }}_2$. (d) Fidelity of thenot gate achieved by three different ways versus the frequency detuning shift. Green solid line: four-level tripod system; blue dotted-dashed line: nonadiabatic non-Abelian GPs; black dotted line: non-Abelian GPs with shortcut to adiabaticity. The red dashed line represents the variation of amplitude of frequency detuning. The inset shows the fidelity of the not gate in a smaller region.

Figure 6

(a) Qubit-state population dynamics under decoherence with initial state $|1\rangle$. (b) Fidelity of not gate as a function of the decay rates in the NV centers system.