Deterministic preparation of Dicke states of donor nuclear spins in silicon by cooperative pumping

For donor nuclear spins in silicon, we show how to deterministically prepare various symmetric and asymmetric Dicke states which span a complete basis of the many-body Hilbert space. The state preparation is realized by cooperative pumping of nuclear spins by coupled donor electrons, and the required controls are in situ to the prototype Kane proposal for quantum computation. This scheme only requires a subgigahertz donor exchange coupling, which can be readily achieved without atomically precise donor placement, and hence it offers a practical way to prepare multipartite entanglement of spins in silicon with current technology. All desired Dicke states appear as the steady state under various pumping scenarios, and therefore the preparation is robust and does not require accurate temporal controls. Numerical simulations with realistic parameters show that Dicke states of 10–20 qubits can be prepared with high fidelity in the presence of decoherence and unwanted dynamics.

Figure 1

(a) Donor spins in silicon controlled by patterned electrodes. Pumping of nuclear spins is realized through ac voltage control of the A gates. (b) Energy level scheme for the exchange-coupled electrons in magnetic field along $z$ direction. The transition between the ground state $|g\rangle$ and a nondegenerate one-spin-excitation state $|e_0\rangle$ is used. (c), (d) Pumping of nuclear spins by ac hyperfine coupling ${\widehat{H}}_c$ (double-head arrows), assisted by the electron spin decay channel from $|e_0\rangle$ to $|g\rangle$ (wavy lines). The central frequency of ${\widehat{H}}_c$ is tuned in resonance with the energy separation between $|g\rangle |J,M+1\rangle$ and $|e_0\rangle |J^{\prime },M\rangle$, which is a quantity dependent on $M$. (c) When the ac voltage control uses the first scenario for phases (see text), population is deterministically transferred from the collective nuclear state $|J,M+1\rangle$ to $|J,M\rangle$. (d) When the ac voltage control uses the second scenario for phases (see text), population is transferred from $|J,M+1\rangle$ to $|J^{\prime },M\rangle$, where $J^{\prime }-J=0$ or $\pm 1$.

Figure 2

Preparation of the symmetric Dicke states of 20 nuclear spin qubits initially on the fully polarized state. For the numerical simulation in (a), (b), the amplitude $\Omega$ of the ac control is held as a constant while the central frequency $\omega$ steps down after a finite interval. Symmetric Dicke states $|J=10,M\rangle$ with different $M$ are obtained sequentially with a probability $\ge$$99.8\%$. In (c), (d), the control sequence $\Omega \left(t\right)$ is optimized for speed, and each Dicke state is obtained with a probability $\ge$$99.5\%$. $\Gamma /2\pi =60$ kHz.

Figure 3

Preparation of asymmetric Dicke state $|J=7,M=7\rangle$ in the subspace $(j_A=4,j_B=6)$ for 20 nuclear spin qubits initially on the fully polarized state. (a) Population flows by the cooperative pumping. (b), (c) Numerical simulation of the state preparation. (b) Populations on the various Dicke states: $|J=10,M=10\rangle$ (dashed blue); $|J=9,M=9\rangle$ (dotted green); $|J=8,M=8\rangle$ (dotted red). The solid curve gives the population on the target state $|J=7,M=7\rangle$ (or its time reversal state in the shaded interval), and its final population is $99.5\%$. (c) The magnitude $\Omega$ and central frequency $\omega$ for the ac tuning of hyperfine interaction. In the unshaded (shaded) interval, the first (second) scenario for the phases is applied (see text). The vertical arrows indicate timing of global $\pi$ flip of all nuclear spins. $\Gamma /2\pi =60$ kHz.

Figure 4

Preparation of Dicke states $|J,M,\{j_{l,k}\}\rangle$ which span a complete basis for the nuclear spin qubits. (a) The quantum numbers $j_{l,k}$ are the collective spin of subsets of qubits. (b) Concatenated preparation process. In step $l$, Dicke states are prepared in parallel in each $2^l$-qubit subset with the collective spin being the specified value $j_{l,k}$. The quantum numbers $j_{l<l_0,k}$ are all conserved in step $l_0$. Red solid (black dashed) lines indicate the donor electron exchange “on” (“off”) when nuclear spins are pumped.