Multi-ion sensing of dipolar noise sources in ion traps

Trapped-ion quantum platforms are subject to “anomalous” heating due to interactions with electric-field noise sources of nature not yet completely known. There is ample experimental evidence that this noise originates at the surfaces of the trap electrodes, and models assuming fluctuating pointlike dipoles are consistent with observations, but the exact microscopic mechanisms behind anomalous heating remain undetermined. Here we show how a two-ion probe displays a transition in its dissipation properties, enabling experimental access to the mean orientation of the dipoles and the spatial extent of dipole-dipole correlations. This information can be used to test the validity of candidate microscopic models, which predict correlation lengths spanning several orders of magnitude. Furthermore, we propose an experiment to measure these effects with currently available traps and techniques.

Figure 1

Ratio $S_{\mathrm{cross}}/S_{\mathrm{self}}$ for the different motional degrees of freedom of a two-ion system: $x$ (black, bottom), $y$ (blue, middle), and $z$ (orange, top). The dipoles are assumed to be uniformly distributed and pointing along $y$. The electrode area included in this simulation is a square of side $20d$, which is enough to avoid finite-size effects. Inset: Sketch of a surface-electrode Paul trap with segmented electrodes, similar to Ref. [28]. The ion-ion distance is $l$ and the ion-electrode distance is $d$, which is usually similar to the width of the central (horizontal) electrode.

Figure 2

Ratio of cross- to self-noise for uncorrelated dipoles homogeneously covering the surface of (rf driven) electrodes in the segmented planar trap of Ref. [28] (two rf electrodes of length $L_x$, one of them at positive $z$ with width $L_z\simeq L_x/10$, the other starting at negative $z=-L_z$ of width $2L_z$; see Fig. 1). We set $d\simeq L_z$, and plot (top) the noise experienced by the axial motion ($x$) and (bottom) by radial motion along the $y$ (solid) and $z$ (dashed) axes. Note that the dashed blue line and the solid orange line overlap in this plot.

Figure 3

Experimental sequence. The four radial normal modes of motion of a two-ion system are first cooled close to the ground state [33]. After being exposed to interactions with the environment, the ions are combined in a common potential well. There the motional states of all four modes are read by coupling to the internal electronic states [34], giving access to the desired normal modes' heating rates ${\mathrm{\Gamma }}_{\pm }$.

Figure 4

Top: Spatial dependence of cross-noise for $x$ motion and dipoles oriented along $y$. Bottom: Same plot for monopolar sources. Depending on the ratio $l/d=2,1,0.5$ (black, blue, red) the shape encloses a negative, null, or positive area, from which the noise crossover behavior emerges.

Figure 5

Noise ratios $S_{\mathrm{cross}}/S_{\mathrm{self}}$ for the $x$ motion of two ions above two different geometries, with homogeneous distribution of dipoles pointing normal to the surface. Top: Square electrode $L_x=L_z$, when the rest of the $y=0$ plane is dielectric (or empty). Bottom: A stylus trap as in Ref. [15] with an inner disk electrode of radius $R$ and an outer ring from $3R$ to $5R$; further apart there are four disks, but for the distances considered here they do not affect the ions; the spaces between electrodes act as empty spaces.

Figure 6

Noise felt by a single ion in its motion along $x$ when dipoles are pointing in different directions, with an electrode of infinite size arranged as in Fig. 1. The noise has been normalized by the usual scaling $d^{-4}$ for clarity. The electrode is of infinite size. Noise along the trap axis is clearly dominant.

Figure 7

Noise felt by one ion oscillating along $x$ when dipoles in the surface are pointing along $y$, with spatial dipole-dipole correlation distance $\xi$. The typical scaling $d^{-4}$ is broken for ion-electrode distances $d<\xi$, where it becomes $d^{-1}$. Distance is given in arbitrary units.

Figure 8

Noise ratios $S_{\mathrm{cross}}/S_{\mathrm{self}}$ for two ions oscillating along $x$ (top), $y$ (middle), and $z$ (bottom), when dipoles in the surface are pointing along $x,y,z$ (left, middle, right). Influence of spatial correlations is noticeable for $\xi >0.1L_z$.

Figure 9

Cross-noise (arbitrary units) along $x$ motion for a chain of ten strongly coupled ions when dipoles in the surface are pointing along $x$, as a function of the inter-ion distance $l\in [{10}^{-5}{L}_z,Lz]$. The electrode has dimensions $L_x=10L_z$ and the ions are above the electrode at $d=L_z$. The black line is the mode with the longest wavelength.