Two-dimensional ion crystals in radio-frequency traps for quantum simulation

The computational difficulty of solving fully quantum many-body spin problems is a significant obstacle to understanding the behavior of strongly correlated quantum matter. Experimental ion trap quantum simulation is a promising approach for studying these lattice spin models, but has so far been limited to one-dimensional systems. This work argues that such quantum simulation techniques are extendable to a two-dimensional (2D) ion crystal confined in a radio-frequency (rf) trap. Using appropriately chosen parameters, driven ion motion due to the rf fields can be made small and will not limit the types of quantum spin models that can be experimentally encoded. The rf-driven motion is calculated to modestly reduce the stability region of a 2D crystal and must be considered when designing the 2D trap. The system will be scalable to 100+ quantum particles, far beyond the realm of classical intractability, while maintaining the traditional ion trap strengths of individual-ion control, long quantum coherence times, and site-resolved projective spin measurements.

Figure 1

Equilibrium ion positions in the proposed rf Paul trap for (a) 20 ions and (b) 100 ions. Results were obtained with a molecular-dynamics simulation that included the effects of micromotion. The insets show the micromotion directions and amplitudes, which are small compared to the interion spacing.

Figure 2

(a) Calculated spectrum of axial mode frequencies (including micromotion effects), along with several mode eigenfunctions, for a 100-ion crystal using the parameters in Sec. 2. The highest-frequency mode is the center-of-mass motion (no spatial variation), while lower-frequency modes vary on shorter and shorter length scales. (b) Radial micromotion induces a frequency shift in the axial normal modes, compared to the no-micromotion case. The center-of-mass mode is left unchanged, while the lower-frequency modes are shifted downwards by progressively larger amounts.

Figure 3

Stability region for a 2D planar ion crystal, using the trap parameters of Sec. 2 and ${\omega }_r=0.5$ MHz. Red points show the calculated boundary in the absence of micromotion, in agreement with Eq. (4) (red dashed line). When micromotion effects are included, the entire region below the blue points becomes unstable. This stability boundary is also found to exhibit weak power-law scaling with the number of ions (blue dashed line).

Figure 4

The normal modes determine the spin-spin couplings $J_{ij}$ [Eq. (9)], which are shown for an edge spin (a) and a central spin (b) in a 100-ion array using the trap parameters detailed in the text. For these parameters, the couplings fall off algebraically with distance as $\sim 1/r^{\alpha }$, with $\alpha$ tunable between zero and three. Panel (c) shows this power-law decay for the edge spin chosen in (a).

Figure 5

By using a focused laser beam to shelve specific ions in uncoupled spin states, one can realize a variety of different lattice geometries. Four examples are depicted here: (a) Kagomé, (b) honeycomb, (c) rectangular, and (d) spin ladder. Blue circles are participating ions, empty circles are “hidden” ions, and black lines indicate the nearest-neighbor spin-spin couplings on each participating lattice site.