Fast Dynamical Decoupling of the Mølmer-Sørensen Entangling Gate

Engineering entanglement between quantum systems often involves coupling through a bosonic mediator, which should be disentangled from the systems at the operation’s end. The quality of such an operation is generally limited by environmental and control noise. One of the prime techniques for suppressing noise is by dynamical decoupling, where one actively applies pulses at a rate that is faster than the typical time scale of the noise. However, for boson-mediated gates, current dynamical decoupling schemes require executing the pulses only when the boson and the quantum systems are disentangled. This restriction implies an increase of the gate time by a factor of $\sqrt{N}$, with $N$ being the number of pulses applied. Here we propose and realize a method that enables dynamical decoupling in a boson-mediated system where the pulses can be applied while spin-boson entanglement persists, resulting in an increase in time that is at most a factor of $\pi /2$, independently of the number of pulses applied. We experimentally demonstrate the robustness of our entangling gate with fast dynamical decoupling to ${\sigma }_z$ noise using ions in a Paul trap.

Figure 1

Trajectory in phase space of a spin with initial state $|\uparrow ,\alpha ⟩$, where $\alpha$ is the coherent state of the trap (marked as a bullet), due to the MS unitary [Eq. (2)]. (a) Applying a $\pi$ pulse when the spin and the boson are disentangled, at times $t_n=2\pi n/ϵ$, causes the spin to change between the blue and orange trajectories. (b) Applying a $\pi$ pulse at time $t=\pi /ϵ$ causes the spin to change from the blue to the orange trajectory, resulting in an effective $4\tilde{\mathrm{\Omega }}/ϵ$ displacement at time $t=2\pi /ϵ$.

Figure 2

Phase space trajectory of a spin under the MS Hamiltonian [Eq. (1)] dynamics and multiple DD pulses, where $\alpha$ is the coherent state of the trap. (a) Utilizing DD pulses for enlarging the area in phase space. A flower-shaped area is generated by applying $N$ DD pulses with a specific time separation. The area enclosed by the flower shape, which circumscribes a polygon (orange dashed line) and flower petals (blue line), is proportional to the accumulated geometric phase. At the limit of many pulses $N\gg 1$, the polygon can be approximated as a circle, and the petals’ area contribution nulls. (b) Every two $\pi$ pulses separated by $\mathrm{\Delta }t\approx \pi /ϵ$ give rise to a spin-dependent displacement (orange dashed line) $2\tilde{\mathrm{\Omega }}/ϵ$, where the path effective velocity is $2\tilde{\mathrm{\Omega }}/\pi$. In comparison to the regular strong coupling entangling gate where the path effective velocity is $\tilde{\mathrm{\Omega }}$ [Eq. (1)], we find a factor of $\pi /2$ in the gate durations.

Figure 3

An experimental and numerical comparison of ${\sigma }_z$ robustness for three Mølmer-Sørensen entangling gate protocols: the fast DD scheme, as proposed in this Letter (blue), the slow DD scheme (green), and the slow DD scheme without DD pulses (red). Circles represent measurements connected by a solid line as a guide to the eye. Simulation results are presented in dashed lines. The number of pulses is denoted as $N$. The comparison is shown for differing pulse sequences enumerated by the number of DD arms. Two ${}^{88}{\mathrm{Sr}}^{+}$ ions were entangled according to the appropriate protocol using a 674 nm laser and their final state measured via state-selective fluorescence. ${\sigma }_z$ noise was implemented by detuning all driving lasers from their resonance frequencies. The fidelity with the specific fully entangled state, calculated from measurement results, is shown. Confidence intervals of 95% are under $\pm 0.03$ and are not plotted. Numerical simulations were done for the MS Hamiltonian with the appropriate pulse sequence and a detuning term, alternating the dynamics between the MS Hamiltonian [Eq. (1)] with detuning $(\mathrm{\Delta }/2)\sum _i{\sigma }_{z,i}$ and the detuned $\pi$ pulse. Graphs (a)–(d) show the fidelity for a specific state. Graph (e) shows the fidelity for an entangled state up to an arbitrary phase; it shows that high-quality entangled states are achieved even at large detunings, albeit at a phase that differs from the zero-detuning case. Graph (f) compares the fast MS gate with $N=0$ (red) and MS with $N=1$ (green), the most popular schemes, vs the flower scheme with $N=10$ (blue). The ten-pulse flower DD scheme offers a significantly more robust response with only a small overhead in time.