General Scheme for the Construction of a Protected Qubit Subspace

We present a new robust decoupling scheme suitable for levels with either half-integer or integer angular momentum states. Through continuous dynamical decoupling techniques, we create a protected qubit subspace, utilizing a multistate qubit construction. Remarkably, the multistate system can also be composed of multiple substates within a single level. Our scheme can be realized with state-of-the-art experimental setups and thus has immediate applications for quantum information science. While the scheme is general and relevant for a multitude of solid-state and atomic systems, we analyze its performance for the case composed of trapped ions. Explicitly, we show how single qubit gates and an ensemble coupling to a cavity mode can be implemented efficiently. The scheme predicts a coherence time of $\sim 1\mathrm{s}$, as compared to typically a few milliseconds for the bare states.

Figure 1

Level structure of the calcium ion, ${}^{40}\mathrm{Ca}^{+}$. The $D_{3/2}$ subspace, which has a lifetime of $\sim 1\mathrm{s}$, serves as the protected subspace. The $S_{1/2}-P_{1/2}$ transitions and the $D_{3/2}-P_{1/2}$ transitions are used in the initialization and construction of the protected subspace.

Figure 2

Realization of dark states. (a) The black (red) driving fields result in the first (second) dark state. The driving fields of each dark state also operate on the subspace of the other dark state (dashed lines), resulting in small energy shifts. The detunings are given by $\Delta 2=-\Delta 1=\Omega +4B/5$, where $\Omega$ is the energy gap between the two $P_{1/2}$ states, introduced by the $S_{1/2}-P_{1/2}$ coupling. Blue arrow shows optical pumping to the first dark state, $|D_1⟩$. (b) Level structure in the dark states basis. The dark states $|D_1⟩$ and $|D_2⟩$ form the protected subspace.

Figure 3

Realization of (i) a single qubit gate (blue) (ii) coupling to a cavity mode (green).