Controlling long ion strings for quantum simulation and precision measurements

Scaling a trapped-ion based quantum simulator to a large number of ions creates a fully controllable quantum system that becomes inaccessible to numerical methods. When highly anisotropic trapping potentials are used to confine the ions in the form of a long linear string, several challenges have to be overcome to achieve high-fidelity coherent control of a quantum system extending over hundreds of micrometers. In this paper, we describe a setup for carrying out many-ion quantum simulations including single-ion coherent control that we use for demonstrating entanglement in 50-ion strings. Furthermore, we present a set of experimental techniques probing ion qubits by Ramsey and Carr-Purcell-Meiboom-Gill pulse sequences that enable detection (and compensation) of power-line-synchronous magnetic-field variations, measurement of path-length fluctuations, and detection of the wavefronts of elliptical laser beams coupling to the ion string.

Figure 1

(a) Magnetization dynamics under a $XY$ Hamiltonian for a 51-ion chain initially prepared in a Néel state $|\downarrow \uparrow \downarrow \uparrow \cdots \rangle$ (Sec. 5). (b) Log negativity measured for connected pairs after $T=3\mathrm{ms}$ of time evolution.

Figure 2

(a) Ion string in a linear Paul trap. A string of up to 51 ions is cooled, manipulated, and detected with laser beams impinging from various angles. Two counterpropagating beams creating a polarization gradient along the axis of the ion string are directed through holes in the ion trap's endcaps. (b) Partial level scheme of $\stackrel{+}{{}^{40}\mathrm{Ca}}$. The qubit is encoded in the states $4^2{S}_{1/2}(m=+1/2)$ and $3^2{D}_{5/2}(m=+5/2)$ (gray dots).

Figure 3

Normalized heating rate vs axial trap frequency, normalized by the number of ions $N$. Open circles show data measured with a single ion (red circles), 28 ions (blue square), and 50 ions (green diamond). The line is a fit of an expression $dn/dt\propto \frac{1}{{\omega }^{\alpha }}$ to the experimental data, where we estimate $\alpha =1.9\left(2\right)$.

Figure 4

Survival probability of the ion crystal against melting by collisions vs time for 25-ion and 51-ion strings.

Figure 5

(a) Pulse timing of a CPMG pulse sequence containing $N_p\pi$ pulses and total duration $\tau$. (b) Magnitude of a filter function with $N_p=2$ and $\tau =20\mathrm{ms}$. (c) Mean excitation after a CPMG sequence ($N_p=2$, $\tau =20$ ms) probing 50-Hz noise before (red circles) and after (blue squares) field compensation. Solid curves are fits of Eq. (4) to the data. (d) Ramsey measurement with $\tau =4.5\mathrm{ms}$ probe time.

Figure 6

Wavefront detection using a CPMG sequence. The detected signal is produced by scanning the wait time between the pulses of the CPMG sequence. (a) Elliptical astigmatic beam. (b) Elliptical nonastigmatic beam. (c) The tilted and curved wavefronts of the astigmatic beam of the old beam shaping optics give rise to peaks under a CPMG sequence ($N_p=20$ pulses, axial trapping frequency ${\omega }_z=2\pi \times 112\mathrm{kHz}$, ion string temperature $T=4.6\mathrm{mK}$). (d) After the implementation of new beam shaping optics that deliver straight wavefronts, the peaks vanish.

Figure 7

Autocorrelations from a single-ion experiment. Plotted are both exponential (blue dashed line) and Gaussian (green solid line) fits to the data (black points). Insets: Normal $Q-Q$ plots for each fit, along with their associated $R^2$ values. It can be seen that the Gaussian decay better fits the data, indicating that the decay in correlations is predominantly affected by slow phase noise.

Figure 8

Single-ion addressing of a 51-ion string illustrated for a few selected ions. The ions were initialized in the $X$ basis, then addressed by a laser pulse inducing an ac-Stark shift with the pulse length set to realize a $\pi$ rotation, and measured in the $X$ basis. For ion 51 the double-frequency component of the AOD leads to an excitation of ion 1 and 2.

Figure 9

Ions prepared in different initial states: (a) ion 26 addressed, (b) Néel state with odd ions addressed, (c) Néel state with even ions addressed, and (d) ions that have a prime number as index addressed (ions numbered from right to left).

Figure 10

Cross-talk in a 51-ion string, measured as the ratio of the Rabi frequency of a nonaddressed and of the addressed ion. This measurement was repeated for every fifth ion in the ion string.

Figure 11

Addressing flop quality after PG cooling for 1 ms (blue points) without PG cooling (red squares) and with an extra 20 ms waiting time after PG cooling (black diamonds) of a 51-ion string. Here we show only addressing flops for the 26th ion. Solid lines are fits to the experimental data.

Figure 12

Log negativities measured for adjacent ion triplets after $T=3\mathrm{ms}$ of entangling operation in a 51-ion chain [for the corresponding plot showing log negativities of pairs, please see Fig. 1].

Figure 13

Excitation after a CPMG-10 sequence. The example of a numerical simulation for a Fock state $|\downarrow ,n=50\rangle$ was calculated for the trapping frequency $\omega =2\pi$, Rabi frequency $\mathrm{\Omega }$, $\mathrm{\Delta }=0$, and $\eta =0.01$. The variation of the ratio $\mathrm{\Omega }/\omega$ shows the emergence of the intermediate peaks for small Rabi frequencies, respectively, large $\pi$ times $t_{\pi }=\pi /\mathrm{\Omega }$. The wait time denotes the time difference between the start of two consecutive $\pi$ pulses.

Figure 14

Comparison of numerical simulation and experiment. (a) The numerical simulation describes the full measured CPMG signal (ion 1), while the semiclassical model captures only the main peaks. (b), (c) Detailed scans of the first and second peak show the different peak heights of the main and the intermediate peaks. The parameters were the same as for Fig. 6 ($N_p=20$ pulses, axial trapping frequency ${\omega }_z=2\pi \times 112\mathrm{kHz}$, ion string temperature $T=4.6\mathrm{mK}$).