Long-time storage of entangled logical states in decoherence-free subspaces

Abstract

The maintenance of quantum entanglement lays the elementary building block of quantum information processing, requiring an integration of long coherence time, sufficient storage capacity, and high-fidelity entangling gates. Here we encode two-qubit entangled states into the decoherence-free subspaces (DFS) of four ions in a cryogenic trap. By crosstalk-free sympathetic cooling under dual-type encoding and multi-state detection which discards the collision-induced leakage error, we achieve a storage lifetime of about one hour for the entangled logical states. We further study the second-order DFS and show its advantage in suppressing the spatially nonuniform noise over the first-order DFS. Our work paves the way for applications of DFS quantum memories in quantum computing, quantum network and precision measurement.

Figures (6)

• Figure 1: Experimental Scheme. (a) Six ${}^{171}\mathrm{Yb}^+$ ions form a one-dimensional chain, with two edge ions (blue) for sympathetic cooling and four central ions encoding two logical qubits (green and orange) in decoherence-free subspaces (DFS). (b) Experimental sequence. We initialize the ions in $|0_S\rangle$, and use focused $411\,$nm laser beams and a global microwave to prepare the entangled states between the four memory ions in the DFS. The memory ions are then transferred to the $F$-type for long-time storage, while the coolant ions are not affected by shelving into an ancilla level. After a verification step which checks the correct qubit types, the $F$-type memory qubits are stored for time $T$ with a microwave spin echo inserted in the middle, while the $S$-type coolant ions continuously provide sympathetic laser cooling. Finally, the memory ions undergo analysis pulses and multi-state detection to measure the storage fidelity and the leakage error. (c) Relevant level structure of the ${}^{171}\mathrm{Yb}^+$ ion. Dual-type qubits are defined by the clock states of $S_{1/2}$ ($|0_S\rangle$ and $|1_S\rangle$) and $F_{7/2}$ ($|0_F\rangle$ and $|1_F\rangle$) levels, with their coherent conversion achieved by global bichromatic $411\,$nm and $3432\,$nm laser beams. The population on the $F$-type qubits may slowly leak to the nearby Zeeman levels due to the collision of the ions with the background $H_2$ gas molecules. We employ global $370\,$nm laser for laser cooling, optical pumping and fluorescence detection, $976\,$nm repump laser to clear the $D_{5/2}$ states, and focused $411\,$nm laser beams for single-qubit and two-qubit gates.
• Figure 2: Entanglement of DFS-encoded qubits. (a) Quantum circuit to prepare the entangled state $|\psi^{+}_L\rangle\equiv\frac{1}{\sqrt{2}}(|0_L 1_L\rangle + |1_L 0_L\rangle)=\frac{1}{\sqrt{2}}(|1001\rangle+|0110\rangle)$. $R_X$ and $R_Y$ rotations are achieved by global microwave pulses and hence also act undesirably on the coolant ions, which will be compensated by the single-qubit $R_Z(3\pi/2)$ rotations indicated by the blue boxes. The $R_{ZZ}(\pm\pi/2)\equiv e^{\mp i (\pi/4)Z\otimes Z}$ and $R_Z$ gates are realized by focused $411\,$nm laser beams. (b) The fidelity of the GHZ-like entangled state is decomposed into three terms $F=\frac{1}{4}(2\langle O_{1}\rangle + \langle O_{2}\rangle - \langle O_{3}\rangle)$ which are measured separately by the three circuits. (c) Fidelity of the prepared $|\psi^{+}_L\rangle$ state without conversion to $F$-type or storage. A fidelity $F=95.3(5)\%$ is obtained.
• Figure 3: Storage fidelity $F$ versus storage time $T$ for two DFS-encoded Bell states $|\psi^{+}_L\rangle=\frac{1}{\sqrt{2}}(|1001\rangle+|0110\rangle)$ (red circles) and $|\phi^{+}_L\rangle=\frac{1}{\sqrt{2}}(|1010\rangle+|0101\rangle)$ (blue diamonds) after discarding the leakage events. The solid lines represent exponential fitting results, with the shaded regions showing $68\%$ confidence intervals.
• Figure 4: Parity oscillation for the first- and the second-order DFS-encoded states on magnetic-field-sensitive levels. (a) A first-order DFS state $|+_{L1}\rangle=\frac{1}{\sqrt{2}}(|1^\prime0\rangle+|01^\prime\rangle)$ encoded on two central ions. The parity dynamics are measured for two $350$-ms time windows starting from $T=0$ (left panel) and $T=6\,$s (right panel), respectively. (b) A second-order DFS state $|+_{L2}\rangle\equiv|\psi^{+}_L\rangle=\frac{1}{\sqrt{2}}(|1^\prime001^\prime\rangle+|01^\prime1^\prime0\rangle)$ encoded on four central ions.
• Figure 5: (a) Experimental sequence for multi-state detection of $F_{7/2}$ levels. (b) Lookup table for the measurement outcome ("D" for dark and "B" for bright). (c) Measured survival probability for the four-ion state versus storage time.