Preserving multilevel quantum coherence by dynamical decoupling

Quantum information processing with multilevel systems (qudits) provides additional features and applications than the two-level systems. However, qudits are more prone to dephasing and dynamical decoupling for qudits has never been experimentally demonstrated. Here, as a proof-of-principle demonstration, we experimentally apply dynamical decoupling to protect superpositions with three levels of a trapped ${}^9\mathrm{Be}^{+}$ ion from ambient noisy magnetic field, prolonging coherence by up to approximately an order of magnitude. Our demonstration, straightforwardly scalable to more levels, may open up a path toward long coherence quantum memory, metrology, and information processing with qudits.

Figure 1

Scheme of the dynamical decoupling sequence and the experimental setup. (a) Scheme of dynamical decoupling sequence. The total sequence consists of $N$ repetitions of the basic unit, which includes three pairs of $\pi$ pulses, denoted as ${\pi }_{ij}$ for transition $|i\rangle \leftrightarrow |j\rangle$, separated by duration $\tau$. During the experiment, we prepare the superposition states from $|0\rangle$ and apply the dynamical decoupling sequence with various $\tau$. After the sequence, we measure the fidelity between the final state and the prepared state by reversing the preparation sequence and detect the fidelity of $|0\rangle$. (b) The ion trap system with segmented blades. The radio frequency (RF) blades are made of whole blades, the DC blades have five segments to fine-tune the potential and ion position. The RF and microwave (MW) antenna drive the transitions around 100 MHz and 1 GHz, respectively. (c) The energy levels and transitions used for the demonstration of the three level dynamical decoupling sequence.

Figure 2

The energy levels of the ground state of ${}^9\mathrm{Be}^{+}$ and the detection fidelity of $|0\rangle , |1\rangle$, and $|2\rangle$ with precooling method. (a) The energy levels of the ground state of ${}^9\mathrm{Be}^{+}$. The qutrit is implemented with $|2,2\rangle \equiv |0\rangle , |2,1\rangle \equiv |1\rangle$, and $|1,1\rangle \equiv |2\rangle$, which together with the transitions used in the sequence are colored in blue. The gray dashed transitions show the shelving path. (b) The detection of $|0\rangle , |1\rangle$, and $|2\rangle$ with precooling method. The upper (lower) row shows the result of detection with $5\mu \mathrm{s}$ (50 ms) waiting time after the preparation with $400\mu \mathrm{s}$ detection time. The histogram shows the counts distribution of 5000 repetitions with counts $\le 8$ colored in blue and counts $>8$ colored in orange. The black curves fit the counts with Poisson distribution. The detection fidelities with $5\mu \mathrm{s}$ (50 ms) wait time are $99.2\%$ ($96.0\%$), $99.0\%$ ($97.0\%$), and $97.4\%$ ($97.0\%$) for $|0\rangle , |1\rangle$, and $|2\rangle$, respectively.

Figure 3

The fidelity (population) of the state after the MLDD sequence. $N=0$ are the data without the protection of MLDD sequence. $N=1,2,4,8$ are the results of the sequences by repeating MLDD $N$ times. Each data point is obtained from 300 trials and the error bar is estimated by binomial distribution of the raw data. The solid lines are the fit result in the framework of multilevel dynamical decoupling explained in the main text.

Figure 4

The effect of the MLDD sequence on various superposition states. The subfigures are the result of MLDD with four repetitions, where we obtain coherence time for $|s0\rangle \equiv \frac{1}{\sqrt{2}}\left(\right|0\rangle +|1\rangle )$ with 24(1) ms, $|s1\rangle \equiv \frac{1}{\sqrt{2}}\left(\right|1\rangle +|2\rangle )$ with 36(2) ms, $|s2\rangle \equiv \frac{1}{\sqrt{2}}\left(\right|0\rangle +|2\rangle )$ with 16.5(8) ms coherence time, respectively. The coherence times of $|s3\rangle \equiv \frac{1}{\sqrt{3}}\left(\right|0\rangle -|1\rangle -|2\rangle )$ and $|s4\rangle \equiv \frac{1}{\sqrt{3}}\left(\right|0\rangle -i|1\rangle +i|2\rangle )$ are set to be the same as $\frac{1}{\sqrt{3}}\left(\right|0\rangle +|1\rangle +|2\rangle )$ in the fit and we obtain a result of 35.2(9) ms.

Figure 5

The coherence time and randomized benchmarking result of the two-level system. (a) The raw coherence time for each transition among $|0\rangle , |1\rangle$, and $|2\rangle$ fitted with $a{e}^{-{\left(\frac{t}{T_2}\right)}^2}+b$. The fitted results give coherence times of 1.04(4) ms, 1.57(6) ms, and 3.1(2) ms for $|0\rangle \leftrightarrow |2\rangle , |0\rangle \leftrightarrow |1\rangle$ and $|1\rangle \leftrightarrow |2\rangle$, respectively. The coherence times correspond to the respective sensitivities of the transitions. (b) The randomized benchmarking results of each transition among $|0\rangle , |1\rangle$, and $|2\rangle$. The average operation fidelity is 99.8(1)%, 99.5(3)%, and 99.95(2)% for transitions $|0\rangle \leftrightarrow |2\rangle , |1\rangle \leftrightarrow |2\rangle$, and $|0\rangle \leftrightarrow |1\rangle$, respectively.

Figure 6

The fit result with multiple frequencies and the contribution of each frequency component. (a) The raw data (blue star) and the fit results with single frequency (orange) and multiple frequencies (green) in the three level DD experiment with $N=4$. (b) The figure shows the fidelities with only a single noise frequency component, which helps us to identify the frequency of the noise clearly.