Figure 1

(Color online) Schematic diagram of experimental system. (a) Qubit level diagram showing relevant transitions and energy levels (see text). [Inset (a)] Microwave resonance transition showing Rabi line shape for square excitation pulses. Vertical axis is normalized counts. (b) Rabi oscillations for directly driven qubit transition showing ${\tau }_{\pi }\sim 185\mu \text{s}$. (c) Effective Larmor precession measured using a Ramsey interferometer. Open markers correspond to DDS detuning during free-precession time of 100 kHz; solid markers correspond to 50 kHz detuning. (d) Schematic of microwave generation system. $\text{CPL}=\text{coupler}$. (e) Schematic of system for generating microwave frequency noise. (f) Schematic of trap system showing relative positions of hardware elements. $\text{AOM}=\text{acousto-optic}$ modulator. Inset (f) Top-view optical micrograph of an ion array showing hexagonal crystal order, obtained by imaging ion fluorescence parallel to the magnetic field axis. (g) Schematic of experimental control system.Reuse & Permissions

Figure 2

(Color online) (a) Schematic pulse sequence showing relevant time scales. Capitalized letters correspond to specific times and relate to images in lower panel. Primary and final tails set as indicated in the text. For standard CPMG and UDD, primary tail has $12.5\mu \text{s}$ and final tail has $17.5\mu \text{s}$. (b) Bloch sphere representation of key experimental procedures and events in application of a dynamical decoupling pulse sequence. Curved arrow indicates rotation axis during driving or free precession. Driven rotations are always about $\widehat{X}$ and free-precession is always about $\widehat{Z}$. (c) CPMG (closed markers) and UDD (open markers) fractional pulse locations as a function of pulse number, illustrating increasing divergence between the sequences with larger $n$.Reuse & Permissions

Figure 3

(Color online) (a)–(d) Color scale images of measured qubit decoherence under ambient noise as a function of total free-precession time and pulse number, $n$, for various sequences. Color scale illustrates phase-error probability as normalized counts ($0.5=50\%$ probability of ending sequence as $|\uparrow ⟩$ or $|\downarrow ⟩$), with light colors corresponding to low qubit errors and black corresponding to full dephasing. Axis of rotation is designated in the figure. ${\pi }_Y$ rotations driven by a combination of a ${\pi }_X$ pulse and a net ${\pi }_Z$ rotation during each free-precession period. (e) and (f) Theoretical fits to qubit decoherence using the filter function as defined in Eq. (2) and the noise spectrum plotted in panel (g). (g) Measured noise spectrum showing $1/{\omega }^2$ field fluctuations $\left[S_{\beta }\left(\omega \right)\propto 1/{\omega }^4\right]$. Note strong spur at $\omega \sim 2\pi \times 153\text{Hz}$. Dashed line is a guide for the eyes representing $S_{\beta }\left(\omega \right)\propto 1/{\omega }^4$ scaling.Reuse & Permissions

Figure 4

(Color online) UDD (open markers) and CPMG (solid markers) performances under the application of an Ohmic noise spectrum with a sharp-high-$\omega$ cutoff. (a) UDD and CPMG for fixed noise power and various values of $n$. Representative error bars presented for $n=6$. Fits show good agreement with data (see text). Vertical axis corresponds to an experimental measure of $\frac{1}{2}\left[1-W\left(t\right)\right]$ as in previous figures. Data offset vertically for clarity. Inset (a) smoothed measured Ohmic noise spectrum with high-$\omega$ cutoff at $2\pi \times 500\text{Hz}$. Actual spectrum used for fitting extends from $2\pi \times 10\text{mHz}$ to $2\pi \times 1\text{MHz}$, truncated for visibility. (b) Evolution of pulse sequences with increasing noise strength, parametrized by $V_N$, showing that UDD performance relative to CPMG improves as noise intensity increases. Data offset vertically for clarity. (c) Quadratic scaling of extracted fit parameters for various noise strengths as a function of $V_N$ on a log-log plot. All fit parameters normalized to extracted fit parameter for $V_N=0.1\text{V}$. Solid line represents quadratic scaling of the normalized value of $\alpha$ with $V_N$. Data for both CPMG and UDD presented.Reuse & Permissions

Figure 5

(Color online) Robustness of $n=6$ spin echo and UDD pulse sequences to systematic rotation errors for a variety of input states, measured for free-precession time of 8 ms and ambient noise. (a) and (b) Color scale representation of qubit rotation to $|\uparrow ⟩$ as a function of microwave reference frequency (centered on $f_0$) and initial state. Dark colors indicate rotation errors. (c) Difference between UDD and echo results for above parameters. White indicates regions where UDD outperforms spin echo. (d) and (e) Color scale representation of normalized counts as a function of actual “$\pi$-pulse” rotation angle $\left(\pi =185\mu \text{s}\right)$, where ${\tau }_{\pi }$ is manipulated to mimic systematic rotation errors. (f) Difference between UDD and echo results for $\pi$-pulse rotations; white shows regions where UDD outperforms echo. Randomly located black dots correspond to laser dropouts during data collection. (g) Bloch sphere representations of several relevant starting states used in color scale images above.Reuse & Permissions