Multi-ion sensing of dipolar noise sources in ion traps

TL;DR

This paper introduces a method using two-ion probes to analyze electric-field noise sources in ion traps, providing insights into the microscopic mechanisms behind anomalous heating.

Contribution

It presents a novel approach to determine dipole orientations and correlations, enabling testing of microscopic models of surface noise in ion traps.

Findings

Demonstrates a transition in dissipation properties of two-ion probes

Provides a way to measure dipole orientation and correlation length

Proposes an experimentally feasible method with current technology

Abstract

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 point-like 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 mag- nitude. Furthermore, we propose an experiment to measure these effects with currently-available traps and techniques.

Tables

Table 1. Table 1: Truth table indicating the presence (✓) or absence (✗) of noise crossover for radial motion along the y 𝑦 y and z 𝑧 z axes for a given dipole orientation. This information together with the results of heating-rate measurements of the radial normal modes allow for discrimination of surface-dipole orientations.

Crossover y-motion	Crossover z-motion	Dipole orientation
✓	✓	${\mu }_x$
✓	✗	${\mu }_y$
✗	✗	${\mu }_z$

Equations

$${\Gamma }_{\pm }=\frac{e^2}{4m\hslash {\Omega }_{\pm }}{S}_{\pm },$$

$$S_{\pm }=2{\int }_{-\infty }^{\infty }\text{d}\tau e^{-\text{i}{\Omega }_{\pm }\tau }⟨E_x^{(\pm )}(\tau )E_x^{(\pm )}(0)⟩$$

$$s_{i,j}(\tau )=\sum _{l,k}\frac{⟨{\mu }_l(t){\mu }_k(0)⟩}{(4\pi {ϵ}_0{)}^2}{g}_x({\vec{r}}_i,{\vec{r}}_l)g_x({\vec{r}}_j,{\vec{r}}_k),$$

$$g_x({\vec{r}}_1,{\vec{r}}_i)=\frac{d(x_i-x_1)}{((x_i-x_1{)}^2+z_i^2+d^2{)}^{5/2}},$$

$$H_I$$

$$\sqrt{\frac{\hslash }{2m{\Omega }_j}}(A_j{e}^{-i{\Omega }_{j}t}+A_j^{†}{e}^{i{\Omega }_{j}t})$$

$${\Gamma }_{0\to 1}^{(j)}:={\Gamma }_j=\frac{e^2}{4m\hslash {\Omega }_j}{S}_{\text{E}}({\Omega }_j),$$

$$S_{\text{E}}({\Omega }_j)=2{\int }_{-\infty }^{\infty }d\tau e^{-i{\Omega }_j\tau }⟨{\tilde{E}}_x^{(j)}(\tau ){\tilde{E}}_x^{(j)}(0)⟩.$$

$$⟨{\tilde{E}}_x^{(j)}(\tau ){\tilde{E}}_x^{(j)}(0)⟩=\sum _{k,l}{f}_{j,k}{f}_{j,l}⟨E_x({\vec{r}}_k,\tau )E_x({\vec{r}}_l)⟩.$$

$$f=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}1& 1\\ 1& -1\end{array}\right)$$

$$⟨{\tilde{E}}_x^{\pm }(\tau ){\tilde{E}}_x^{\pm }(0)⟩=\frac{1}{2}\left[⟨E_x({\vec{r}}_1,\tau )E_x({\vec{r}}_1,0)⟩+⟨E_x({\vec{r}}_2,\tau )E_x({\vec{r}}_2,0)⟩\pm ⟨E_x({\vec{r}}_1,\tau )E_x({\vec{r}}_2,0)⟩\pm ⟨E_x({\vec{r}}_2,\tau )E_x({\vec{r}}_1,0)⟩\right],$$

$$D(\rho )=\sum _{\omega }\sum _{\alpha ,\beta }{\gamma }_{\alpha ,\beta }(\omega )\left(A_{\beta }\rho A_{\alpha }^{†}-\frac{1}{2}\{A_{\alpha }^{†}{A}_{\beta },\rho \}\right)$$

$$⟨{\tilde{E}}_x^{+}(\tau ){\tilde{E}}_x^{-}(0)⟩=\frac{1}{2}\left[⟨E_x({\vec{r}}_1,\tau )E_x({\vec{r}}_1,0)⟩-⟨E_x({\vec{r}}_2,\tau )E_x({\vec{r}}_2,0)⟩+⟨E_x({\vec{r}}_2,\tau )E_x({\vec{r}}_1,0)⟩-⟨E_x({\vec{r}}_1,\tau )E_x({\vec{r}}_2,0)⟩\right].$$

$$D(\rho )=\sum _{\omega }\sum _{\alpha ,\beta }{\gamma }_{\alpha ,\beta }(\omega )\left(a_{\beta }\rho a_{\alpha }^{†}-\frac{1}{2}\{a_{\alpha }^{†}{a}_{\beta },\rho \}\right)$$

$${\gamma }_{\alpha ,\beta }(\omega )=\mathcal{F}\left(⟨E_x({\vec{r}}_{\alpha },t)E_x({\vec{r}}_{\beta })⟩\right)$$

$$S_{\mathrm{cross}}$$

$$S_{\mathrm{self}}$$

$$\varphi =\frac{1}{4\pi {ϵ}_0}\frac{\vec{\mu }\cdot \vec{R}}{|\vec{R}{|}^3}$$

$$g_x(\vec{r})$$

$$g_y(\vec{r})$$

$$g_z(\vec{r})$$

$$\frac{(x_d-l/2)(x_d+l/2)}{(d^2+z_d^2+(x_d-l/2{)}^2{)}^{5/2}(d^2+z_d^2+(x_d+l/2{)}^2{)}^{5/2}}.$$

$$\frac{(x_d-l/2)(x_d+l/2)}{(d^2+z_d^2+(x_d-l/2{)}^2{)}^{3/2}(d^2+z_d^2+(x_d+l/2{)}^2{)}^{3/2}},$$

$$S$$

$$\sum _{i,j}{g}_x({\vec{r}}_i)g_x({\vec{r}}_j)f(|{\vec{r}}_i-{\vec{r}}_j|).$$

$$g_x(\vec{r})=\frac{3dx}{(d^2+x^2+z^2{)}^{5/2}}$$

$$g_x(\vec{r})\sim 3x/d^4.$$

Full text

Trapped atomic ions constitute prominent candidates for deployable technologies exploiting the unintuitive properties of quantum mechanics 09Home ; 17Linke . A number of scalable architectures have been proposed 02Kielpinski ; 00Cirac ; 17Lekitsch , but technical constraints limit the current computational power of high-fidelity trapped-ion quantum machines to less than ten qubits 16Debnath . One key aspect towards the most notorious scalable schemes is trap miniaturization. This eases scalability and allows for faster quantum operations on the computational space (internal electronic states) 03Leibfried ; 11Ospelkaus as well as the quantum bus (ions’ motion) 12Bowler ; 12Walther . However, trapped-ion experiments suffer from motional heating due to interactions with noise sources of origin not yet completely known 00Turchette ; blatt ; wineland . The measured effects of this so-called ‘anomalous’ heating scale strongly with the inverse of the ion-electrode distance, posing a major obstacle to trap miniaturization.

Systematic experimental studies suggest that the origin of anomalous heating is due to contaminants on the surfaces of trap electrodes blatt . In 100-fold the NIST ion-storage group treated electrode surfaces with ion bombardment. The 100-fold reduction in the observed heating rates points at adsorbates as probable culprits for the noise. But recent studies show that this is not the whole picture wineland2017 , suggesting that only electrode surfaces subject to radio-frequency drives (as required for ion trapping) are accountable for the heating. This result has a profound impact on the search for possible microscopic models since, to our knowledge, all previous studies considered thermally-driven processes (see blatt for a review of proposed microscopic models). In particular, the diffusion of adsorbates diffusion1 ; diffusion2 ; diffusion3 ; diffusion4 is consistent with the most advanced surface-science experiments realized to date on a trap setup Sadeghpour , appearing to be a plausible mechanism for anomalous heating.

Up until now studies of the noise origin have focused exclusively on its scaling for a single trapped ion. This is usually assumed as $S_{E}~{}\sim\omega^{-\alpha}d^{-\beta}T^{\gamma}$ with $d$ the ion’s distance to the electrode, $\omega$ its motional frequency and $T$ the trap-electrode’s temperature. For different microscopic models a strong distance scaling $\beta>3$ is predicted and is consistent with experiments blatt ; wineland , whereas frequency and temperature dependencies vary for different models. Here we propose a new way of measuring noise which can give finer details on its microscopic origin.

We show that for a trap holding two or more ions a noise crossover effect must take place: there is a point where the heating rates of the center-of-mass (COM) and relative motions equate. This previously unknown crossover occurs when ions are separated by comparable distances from each other ($l$) and from the trap electrodes ($d$, see fig. 1). We will show that its characteristics depend not only on the spatial extent of correlations of dipole-dipole fluctuations $\xi$ (or the size of patches), but also on the average orientation of the dipoles. This phenomenon is in stark contrast to that found for an homogeneous lattice environment FerSciReps , where the properties of the propagator (anisotropy, resonant manifold, etc.) determine the crossover characteristics.

We derive estimates of the correlation lengths $\xi$ of dipole-dipole fluctuations for different microscopic models in the literature and show that the one-ion noise-level departs from its typical $d^{-4}$ scaling blatt when $d\sim\xi$. Likewise the noise crossover for two ions is shifted to higher ion-ion distances $l$ in this regime. Further, the absence/presence of noise crossover for different motional degrees of freedom uniquely determines the mean orientation of dipoles. As a side result, we show that the effect of dipole orientations can also be observed with a single ion. This might explain recent results at NIST, where noise levels were measured to be highest for ion motion parallel to the surface projection of the sputter beam at a well-defined angle PrivateHite . Finally, we propose a realistic experiment to measure and characterize the noise crossover. This can be carried out with current state-of-the-art Paul traps and techniques.

I Electric field noise

Every dipole source on the surface of an electrode represents a noisy source of electric potential at the ion position: $\phi(\vec{r})=\vec{\mu}\cdot\vec{r}/|\vec{r}|^{3}$, with $\vec{\mu}$ the dipole moment of the ion and $\vec{r}$ the position of the dipole relative to the ion. Heating rates for two ions along a given motional degree of freedom, say $x$, depend on the correlators $\langle E_{x}(\vec{r}_{i},\tau)E_{x}(\vec{r}_{j},0)\rangle$, with $E_{x}(\vec{r}_{i},\tau)$ the total electric field component along $x$ at time $\tau$ at the ion’s position $\vec{r}_{i}$. We derive (appendix A) the master equation for two ions in Lindblad form, exhibiting the desired heating rates.

Let us consider two coupled identical ions aligned along the $x$-axis (see Fig. 1), separated by $l$ from each other and $d$ from the trap electrodes. We will assume in what follows that all relevant modes of motion have been cooled close to their respective ground states. It is convenient to move to a normal-mode picture $x_{\pm}=(x_{1}\pm x_{2})/\sqrt{2}$, where $x_{1,2}$ are the ions’ positions with respect to a lab reference frame. Here, the $(+)$ mode corresponds to the center-of-mass motion (COM) and the $(-)$ mode to relative motion (stretch mode), with eigenfrequencies $\Omega_{\pm}$. The heating rates of the normal modes are given by

[TABLE]

with $e$ and $m$ the ions’ charge and mass respectively, $\hbar$ the reduced Planck constant, $\Omega_{j}$ the normal-mode frequencies, and

[TABLE]

the electric-field fluctuations’ spectral densities. The fields acting on the normal modes are a linear combination of the fields seen by the individual ions: $\tilde{E}_{x}^{(\pm)}=[E_{x}(\vec{r}_{1})\pm E_{x}(\vec{r}_{2})]/\sqrt{2}$. Defining $s_{i,j}(\tau):=\langle E_{x}^{(i)}(\tau)E_{x}^{(j)}(0)\rangle$, we can write $S_{\pm}=\int_{-\infty}^{\infty}\text{d}\tau e^{-\text{i}\Omega_{\pm}\tau}[s_{1,1}+s_{2,2}\pm(s_{1,2}+s_{2,1})]$. Although environmental noise can lead to coupling between both normal modes if the ions are weakly coupled breuer , a sufficiently homogeneous electrode guarantees that this coupling is negligible (see discussion in appendix A), leading to independent decay channels for the normal modes. For the two ions, this translates into a self-noise for each ion ($S_{1,1}$, $S_{2,2}$) and a cross-noise ($S_{1,2}$, $S_{2,1}$). The cross-noise governs the transition from common bath (CB) to separate baths (SB) breuer , two emblematic dissipation scenarios in open quantum systems. The former dissipates only the coordinate $x_{+}$ and leaves $x_{-}$ unaffected, whereas the latter yields equal-rate dissipation for both. Regrouping $S_{\rm{self}}=(S_{1,1}+S_{2,2})/2$ and $S_{\rm{cross}}=(S_{1,2}+S_{2,1})/2$, normal modes $(\pm)$ dissipate as $S_{\pm}=S_{\rm self}\pm S_{\rm cross}$. SB occurs (no frozen mode) when $S_{\rm{cross}}=0$, and CB (frozen $x_{-}$) when $S_{\rm{cross}}=S_{\rm{self}}$. We will later show that the counterintuitive ‘anti-common’ bath case (aCB) where $S_{\rm{cross}}=-S_{\rm{self}}$ (frozen $x_{+}$) is also possible.

The $x$ component of the electric field at a position $\vec{r}$ is given by $E_{x}(\vec{r},t)=-\partial_{x}\phi(\vec{r})=\sum_{i}(1/4\pi\epsilon_{0})\mu_{i}(t)g_{x}(\vec{r},\vec{r}_{i})$, where $g_{n}(\vec{r},\vec{r}_{i})$ are geometric functions (appendix B) which depend on the orientation of dipoles, and $\mu_{i}\hat{=}|\vec{\mu}_{i}|$. Thus, the cross- and self-noise are given by expressions

[TABLE]

where $\left<\mu_{l}(t)\mu_{k}(0)\right>$ is a correlation function between dipoles $l$ and $k$. This dipole-dipole correlator features separated temporal and spatial terms for proposed microscopic models blatt , so we can approximate it by $\left<\mu_{l}(t)\mu_{k}(0)\right>\simeq s_{\mu}(t)f(\vec{r}_{l},\vec{r}_{k})$ footnote . Here, $f$ is a spatial correlation profile (it can be a phononic correlation decay in the electrode, a domain function for dipoles in the same patch, etc.) and the approximation is valid if time-fluctuations are similar across the whole surface. Thus, after Fourier integration ($\mathcal{F}$) we will have two main ingredients: the dipole fluctuation spectrum at the eigen-frequencies, $S_{\mu}(\Omega_{\pm})\hat{=}\mathcal{F}[s_{\mu}](\Omega_{\pm})$, and geometric contributions (from $f$ and $g$). For ions in separate wells the Coulomb coupling is small compared to the eigen-frequencies and $S_{\mu}(\Omega_{+})\simeq S_{\mu}(\Omega_{-})$. The focus of our work will therefore be on the geometric part.

II Noise crossover

We start by considering the noise characteristics of an electrode containing dipoles which point normal to the surface. This is a typical assumption even in cases where microscopic details are calculated to a large extent Sadeghpour . In order to give a clear picture of the origin of the crossover, let us consider the simplest case: all dipoles are pointing normal to the surface $\vec{\mu}_{i}=\mu_{i}\hat{u}_{y}\ \forall i$, they are uncorrelated, and we focus on the motional mode along $x$ as a function of $l$. In this case, the cross-noise is proportional to the sum $\sum_{i}g_{x}(\vec{r}_{1},\vec{r}_{i})g_{x}(\vec{r}_{2},\vec{r}_{i})$. In the simplest scenario we can assume the dipoles to be almost homogeneously distributed on the surface and replace the sum by an integral. Noting that (see e.g. haffner )

[TABLE]

one can see that the integral with respect to $z$ is always finite and positive. However, the integral with respect to the dipole coordinate $x$ has an M shape, meaning that for particular parameters the area enclosed by this shape will vanish (appendix C). The crossover originates precisely due to the fact that for a given combination $\{d,l\}$ the cross-noise integral will be zero and the cross-noise changes sign. In figure 1 we see the ratio $S_{\rm{cross}}/S_{\rm{self}}$ for all three degrees of freedom. When $d\gg l$ we have $+1$ (CB), and around $d=l$ we have 0 (SB). When $d<l$ we have negative values, which means that the stretch mode dissipates at a rate $S_{-}=S_{\rm{self}}-S_{\rm{cross}}$ higher than $S_{+}=S_{\rm{self}}+S_{\rm{cross}}$. The crossover is absent for $z$-motion (in agreement with figure 27 of blatt ), however for $x$- and $y$-motion it is present, leading to an aCB regime.

One could wonder whether a pure aCB regime is at all possible. We show in appendix D that for a stylus-trap configuration 09Maiwald one can reach $S_{\rm{cross}}/S_{\rm{self}}$ ratios approaching -1. However, in such a trap at least one of the ions will be necessarily driven by micromotion 98Wineland2 , which could make the aCB regime hard to observe.

III Dipole orientation

Before dealing with possible spatial correlations among dipoles, let us consider what happens when dipoles are not normal to the surface; to our knowledge this has never been considered before. Since the effective dipole of the ion is given by its displacement from the radio-frequency (RF) null and along its motion, different dipole orientations can cause different noise levels along different directions. These can vary by a factor of 6 depending on dipole orientation (see appendix E and fig. 6), would be detectable with a one-ion probe, and might be behind recent observations with a single ion at NIST PrivateHite . After treating the electrode surfaces with ion-beam sputtering, they observed that two orthogonal motional degrees of freedom were subject to different noise levels, suggesting a preferential orientation of surface dipoles which could be caused by the generation of Gold nanochannels on the treated surfaces.

In order to sense anomalous heating with two ions, it is convenient to have both ions at RF null zones. This can be achieved in segmented, linear Paul traps (fig. 1), where trap heights are typically $d\simeq L_{z}$, with $L_{z}$ the width of the central axial electrode. The ions can be placed at arbitrary positions along the extent of the linear section $L_{x}$, allowing for a tunable inter-ion separation $l$. In the coming analysis we will use as a reference the setup in wineNAT2011 . Note that this does not compromise the generality of our results. We will further assume that only RF electrodes are sources of noise, corresponding to the upper and lower (long) electrodes in figure 1.

Different dipole orientations result in different dependencies on $l$ of the self- and cross-noise terms (fig. 2). Therefore, experimental measurements of the $S_{\rm{cross}}/S_{\rm{self}}$ ratio can reveal the mean orientation of the dipole fluctuators. Note that the increased sensitivity of the radial modes as compared to the axial motion render the former as most suitable for this analysis. Table 1 can be used to gain qualitative insight about the mean orientation of the dipoles. For arbitrary orientations the resulting curves lie between those plotted for the three principal axes, but the structure of crossovers in the table is still valid. For example, any orientation $\mu_{r}$ will keep the $\{$✓, ✗$\}$ signature for any direction $r\neq z$ in the $yz$-plane, even if the crossover for $y$-motion is less steep note1 .

IV Spatial dipole-dipole correlations

Different microscopic models of dipolar fluctuations result in different two-point spatio-temporal correlation functions $\langle\mu_{l}(t)\mu_{k}(0)\rangle\simeq s_{\mu}(t)f(\vec{r}_{l},\vec{r}_{k})$, so it is natural to wonder whether we can measure its spatial dependence $f$ with our scheme, and thus falsify given models. For one trapped ion we find that the consequence of spatial correlations is the breakdown of the typical $d^{-4}$ scaling for the spectral noise density when $d\lesssim\xi$, tending towards $d^{-1}$. A simple mean-field argument (appendix F) explains the saturation and value of this scaling. For two trapped ions, spatial correlations translate into a shift of the crossover point to higher $l/L_{z}$, except for the case of $y$-motion with $\mu_{x}$ pointing dipoles, where the crossover can even disappear. Importantly, however, these effects are also appreciable only for $\xi\sim L_{z}$ (appendix G).

We derive next the size $\xi$ of correlations for several proposed models and discuss the possibility to probe them. For patch models the dipoles are electronic cloud deformations at the surface due to different crystallographic orientations of domains in the electrode metal, so the function $f$ satisfies $f=1$ whenever two dipoles lie on the same domain and [math] otherwise. Patch sizes in the range $[$10\text{,}\mathrm{n}\mathrm{m}$,$10\text{,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$]$ have been measured patches , so ions at distances of tens of microns could potentially feel effects of big enough patches. Another proposed model is based on adatoms (or molecules) stuck to the electrode surface with their induced dipole fluctuating through phononic thermal noise safavi . In such model the dipoles of two adatoms would be spatially correlated through a phonon manifold resonant with their motional bound states (\approx$$300\text{\,}\mathrm{G}\mathrm{H}\mathrm{z} for Neon on Gold) safavi . Taking the dispersion relation of Gold (\approx$$5\text{\,}\mathrm{T}\mathrm{H}\mathrm{z} at $\lambda\approx$21\text{,}\mathrm{p}\mathrm{m}, [gold ](#bib.bib32)), such frequency would correspond to wavelengths $\sim$1\text{\,}\mathrm{n}\mathrm{m}. This indicates that correlations would decay at distances $\xi$ on the order of nano-meters, possibly of $\sim$100\text{,}\mathrm{n}\mathrm{m} for heavier adsorbed molecules, still far away from the scale of tens of microns. If we assume that dipole fluctuations are RF-driven [wineland2017 ](#bib.bib15), then a drive at $\sim100\text{,}\mathrm{M}\mathrm{H}\mathrm{z}$would correspond to wavelengths of$\sim1\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$ (provided a viable mechanism relating RF and phonon excitations exists). For the model of adatoms diffusing on the surface, the spatial scale for dipole-dipole correlations is \sqrt{D/\Omega_{\pm}}$, with$D$the diffusion constant. Considering a range$D\in[10^{-14},10^{-10}]\text{\,}\mathrm{m}^{2}\mathrm{H}\mathrm{z}$$, and frequencies of order MHz, we obtain \xi\lesssim$1\text{,}\mathrm{n}\mathrm{m}$$.

We can thus conclude that within the reach of current distance scales in trap setups, it will be hard to observe the predicted effects of correlations, unless patch noise is the correct origin of anomalous heating.

V Experimental proposal

In what follows we present an experimental routine designed to measure the noise crossover from radiative sources on the surfaces of ion-trap electrodes (Fig. 3). With two ions, a straightforward approach is to measure the heating rate $\Gamma_{\pm}$ of the different normal modes as a function of the distance $l$ between the ions, while keeping the ion-electrode separation $d$ constant. From equation (1) we obtain the noise spectral densities $S_{+}$ and $S_{-}$, and from them we calculate $S_{\rm self}=(S_{+}+S_{-})/2$ and $S_{\rm cross}=(S_{+}-S_{-})/2$. As explained earlier, for the pure CB case the noise ratio $S_{\rm cross}/S_{\rm self}=1$, and only the COM mode will heat up; for pure aCB $S_{\rm cross}/S_{\rm self}=-1$, only the relative motion heats up; for SB $S_{\rm cross}/S_{\rm self}=0$, they will both get excited according to the spectral noise-density present at the modes’ frequencies.

Such an experiment can be carried out in a linear trap, where radial modes are orthogonal to the trap axis. In our scheme, the heating rates of the radial COM and rocking modes are measured for different ion separations. The plots in Fig. 2 show that an interesting range for the ratio $l/d$ goes from 0.5 to 10 - throughout this range the noise ratio varies strongly in all the cases we have simulated. For an ion-trap height $d=$50\text{,}\mathrm{\SIUnitSymbolMicro}\mathrm{m} we should therefore be able to vary $l$ from 25 to $500\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$. With realistic experimental parameters the coupling rates between the motional modes of two ions spaced by $500\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ cannot be expected to exceed $\sim$1\text{\,}\mathrm{H}\mathrm{z} 14Krauth , making it impossible to spectrally resolve the normal modes at mega-hertz frequencies. One way of overcoming this limitation is to let the ions heat up while separated and then bring them together to the same potential well to determine the normal-mode states, as in Fig. 3. Ion-chain splitting and recombination operations have been successfully carried out with negligible effects on the radial degrees of freedom 16Kaufmann . When the ions crystallize in the same trap, the normal modes are easily resolvable 98Wineland2 .

In order to resolve the relevant features in figure 2 it suffices with an absolute uncertainty of the $S_{\rm cross}/S_{\rm self}$ ratio $\delta S_{\text{ratio}}\lesssim 0.1$. Assuming a relative uncertainty $\delta\Gamma_{+}/\Gamma_{+}=\delta\Gamma_{-}/\Gamma_{-}=\epsilon$ on the determination of the normal-mode heating-rates, as well as no correlations between $\delta\Gamma_{+}$ and $\delta\Gamma_{-}$, we find that $\delta S_{\text{ratio}}=\epsilon\sqrt{1+S_{\text{ratio}}^{2}}\frac{\sqrt{\Gamma_{+}^{2}+\Gamma_{-}^{2}}}{\Gamma_{+}+\Gamma_{-}}\leq\epsilon\sqrt{2}$. With uncertainties $\epsilon\lesssim 5\%$, which are experimentally feasible, we expect the crossovers predicted in table 1 to be clearly resolvable.

As mentioned before, directionality effects have been observed after treating electrode surfaces with ion bombardment with a well-defined angle, and might be indicative of the microscopic origin of the noise sources PrivateHite . Our scheme is sensitive to such anisotropy, which would lead to measurable differences in the spectral noise densities for the different radial axes.

VI Conclusion and outlook

Microscopic models for anomalous heating are based on dipolar sources with different characteristics: frequency scalings have been a major concern to distinguish among them, but insufficient attention has been paid to their geometric characteristics. Here we find that the normal-mode heating-rates of two ions experience a previously unknown crossover that can be used to distinguish mean dipole orientations and dipole-dipole correlations (or patch sizes). We provide estimates of the latter for well known models in the literature and show their effect for one- and two-ions configurations. We also propose an experiment which is feasible with current state-of-the-art setups. The idea of exploring spatial characteristics of the noise with more than one ion is left as a new tool for future investigations, and has interesting consequences. For example, in a coupled chain of $N$ ions (appendix H) the most noise-resistant normal modes are odd ones, and should be the ones used as quantum information buffers. Further, the availability of many normal modes could potentially give finer details on geometric features of dipole arrangements and correlations.

Acknowledgements.

JA thanks Dustin Hite for discussions. This work has been supported by the EU through the H2020 Project QuProCS (Grant Agreement 641277), by MINECO/AEI/FEDER through projects NoMaQ FIS2014-60343-P, QuStruct FIS2015-66860-P and EPheQuCS FIS2016-78010-P. The research is partly based upon work supported by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via the U.S. Army Research Office grant W911NF-16-1-0070. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the ODNI, IARPA, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the view of the U.S. Army Research Office.

Appendix A Lindblad equation for 2 and N ions

Expanding the interaction Hamiltonian $q\phi(\vec{R})$ as in Appendix B around $\vec{r}=\{0,d,0\}$ (with the ion quantum fluctuations around that position $\delta\vec{r}$) yields $H=\vec{\delta r}\cdot\vec{\nabla}\phi(\vec{r})=-\delta\vec{r}\cdot\vec{E}(\vec{R})$. Let us concentrate for simplicity on the interaction (and resultant noise) along $x$ and drop the $\delta$; thus the interaction energy for two ions is $H_{I}=-x_{1}E_{x}(\vec{r}_{1})-x_{2}E_{x}(\vec{r}_{2})$, and similarly for $N$ ions we have $\sum_{i=1}^{N}x_{i}E_{x}(\vec{r}_{i})$. We assume that in general the ions are coupled by direct Coulomb interaction, so they will form a set of $N$ normal modes $Q_{i}=\sum_{j}f_{i,j}x_{j}$ with eigenfrequencies $\Omega_{i}$. Any perturbative noise calculation that we do must be referred to that eigenset (see breuer ). Let us rewrite the interaction Hamiltonian

[TABLE]

with the new ‘electric noises’ $\tilde{E}_{x}^{(j)}=\sum_{i=1}^{N}f_{j,i}E_{x}(\vec{r}_{i})$, and where we have used the fact that the transformation matrix $f$ to normal modes is orthogonal (the inverse is its transpose). In the interaction picture the system variables $Q_{j}$ rotate as

[TABLE]

and we can already calculate heating rates in two ways: either we use the usual argument of obtaining the probability to jump from $|0\rangle$ to $|1\rangle$ in the Fock basis of a given eigenmode, or we obtain a master equation for the set of eigenmodes. The first one assumes ground state cooling, while the second is generic.

A first approach which (naively) applies single heating rates (see e.g. a derivation in appendix A of blatt ) to each eigenmode yields

[TABLE]

with

[TABLE]

The correlator can be expanded

[TABLE]

In the case of two ions the matrix $f$ is

[TABLE]

or simply, $\tilde{E}_{x}^{(\pm)}=[E_{x}(\vec{r}_{1})\pm E_{x}(\vec{r}_{2})]/\sqrt{2}$, with $+/-$ corresponding to center of mass/stretch modes, and also to $Q_{1/2}$, as intuition tells. Finally, the noise kernels suffered by center of mass and stretch modes are

[TABLE]

which can seen to consist of a self-damping part (first two terms) and a cross-damping part (last two terms).

A generically correct second approach, is to derive the full dissipator in the Lindblad equation breuer

[TABLE]

with $\omega$ spanning $\Omega_{\pm}$ and $A_{\alpha}=A_{+},A_{-},A_{+}^{\dagger},A_{-}^{\dagger}$ are the COM and stretch modes’ ladder operators. The basic difference with the previous approach is that, in addition to the (correctly predicted) heating rates $\Gamma_{j}$ written above, there appear now cooling rates too, but also cross-heating and cross-cooling rates which are neglected in the previous approach. The appearance of cooling is obvious from the time-symmetry of the evolution, but we need not care about it since we are interested in an experimental routine where we cool the normal modes at the beginning of each experimental run.

Interestingly, and sometimes overlooked in the literature, there are cross-terms in the dissipator which couple the normal modes, with kernels of the type

[TABLE]

Typically these terms can be neglected because the normal modes have different frequencies and thus these terms rotate fast [$\Omega_{+}-\Omega_{-}\gg\gamma(\Omega_{\pm})$]. However, when the coupling to the environment is strong enough, or the normal modes frequencies small enough (our case here because ions are heated when they are far apart and feel almost no Coulomb coupling), these terms become important [$\Omega_{+}-\Omega_{-}\sim\gamma(\Omega_{\pm})$].

Luckily enough, for a sufficiently homogeneous sample the first two terms will cancel out (through similar noise conditions in the two ions positions) and also the last two terms. If this is fulfilled, the Lindblad dissipator separates into two independent dissipation channels, one for each normal mode. Thus finally, the first approach seems to be sufficient if we consider ground state cooled normal modes, and that the electrode is more or less homogeneously (though random microscopically) populated by adsorbed atoms.

Uncoupled ions.- In the proposed experimental implementation we cool the motion of two ions separated by $l$, where they can be only very weakly coupled. If we consider the ion-ion coupling to be negligible [$\Omega_{+}-\Omega_{-}\ll\gamma(\Omega_{\pm})$], the Lindblad equation breuer is

[TABLE]

with $\alpha,\beta=a_{1},a_{2},a_{1}^{\dagger},a_{2}^{\dagger}$ the usual creation-annihilation operators for the axial ions motion. This dissipator can be diagonalized in the basis $a_{\pm}=(a_{1}\pm a_{2})/2$, if we have $\gamma_{1,1}=\gamma_{2,2}\hat{=}\gamma_{\textrm{self}}$. It yields independent dissipation for the C.O.M. and stretch modes, as was expected by symmetry (i.e. as before but the Lindbladian does not couple center of mass and stretch modes). Respectively they dissipate with $\gamma_{\pm}=\gamma_{\textrm{self}}\pm\gamma_{1,2}$. These coefficients are again the Fourier transform of the time-correlation functions

[TABLE]

Summary.- All this discussion was intended to show all the pitfalls that exist when considering the generic problem of a coupled two-body dissipative system: there is a regime where cross-coupling between normal modes exists. This regime however is not of significance if we assume enough homogeneity of the noise sources, which leads to independent heating rates for the normal modes.

We thus arrive to an intuitive picture: whenever we have that ‘cross’$\sim\left<E_{x}(\vec{r}_{1},t)E_{x}(\vec{r}_{2},0)\right>$ is similar to ‘self’$\sim\left<E_{x}(\vec{r}_{1},t)E_{x}(\vec{r}_{1},0)\right>=\left<E_{x}(\vec{r}_{2},t)E_{x}(\vec{r}_{2},0)\right>$, we will have what is normally called a common bath or a spatially-correlated environment, and the stretch mode will not dissipate. Note also that the sign of the cross term is very important: if it is positive it will induce higher dissipation for the COM, while when negative the stretch will suffer more.

Hence, we call for short (in analogy with common-use nomenclature)

[TABLE]

and compare their magnitudes and relative sign in the main text.

Appendix B Dipole geometric functions

The spatial functions describing the interaction of one dipole $\vec{\mu}$ with the ion motion in a given axis are given here. Noting that the ion is at $\vec{r}=\{x,y,z\}$ (fluctuating close to the point $\{0,d,0\}$), the dipole is at $\vec{r}_{d}=\{x_{d},0,z_{d}\}$, the distance is defined as $\vec{R}=\vec{r}-\vec{r}_{d}$ and the electric potential between both is

[TABLE]

we can easily obtain the total electric field in any direction $\vec{E}=-\vec{\nabla}\phi$. To obtain the noise felt by the ion in one of its eigenmotions, axial or radial, we need to calculate the corresponding component of that electric field; we will also write down the expressions when assuming that the dipoles are pointing only along a given direction. We define the dipole functions as $g_{n}(\vec{r})=-(4\pi\epsilon_{0}/|\vec{\mu}|)\partial_{n}\phi(\vec{r})$, after expanding the potential around $\vec{r}\simeq\{0,d,0\}$:

Noise along x motion:

[TABLE]

Noise along y motion:

[TABLE]

Noise along z motion:

[TABLE]

In the case of 2 ions, their positions will now be $\vec{r}_{1}=\{-l/2,d,0\}$ and $\vec{r}_{2}=\{l/2,d,0\}$ and we can use the former expressions by substituting $x_{d}\to x_{d}\pm l/2$ respectively.

Appendix C Origin of cross-noise vanishing

We have argued that the cross-noise vanishes for some ion motions and dipoles orientations. Let us take for example motion along $x$ and dipoles pointing normal to the electrode ($\vec{\mu}=\mu\hat{u}_{y}$). Considering for the moment a collection of uncorrelated, homogeneously distributed dipoles, we have that the cross-noise of two ions sitting at $\{-l/2,d,0\}$ and $\{l/2,d,0\}$ is proportional to

[TABLE]

Integrating this function along $z_{d}$ yields always a finite value, however the behavior along $x_{d}$ is like an M, as seen in figure 4 (top). The area enclosed by this curve can be positive, negative or zero depending on the ratio $d/l$, with the change of sign occurring near $d=l$. For other directions of motion and dipole orientations, using the functions $g_{n}(\vec{r})$ given in Appendix B, it is easy to deduce the properties that we have summed up in figure 2 and Table I.

What about monopolar charges? In that case, the electric potential $\phi\propto 1/|\vec{R}|$ is even (instead of odd) under reflection $\vec{R}\to-\vec{R}$. Still in this case, we have a crossover, see figure 4 (bottom). Take as before $x$-motion and dipoles along $y$, the cross-noise is here proportional to

[TABLE]

which is the same as above only that instead of $5/2$ we have $3/2$ exponents. The difference in crossover is that for monopoles it occurs at $l/d\simeq 2$.

Appendix D Pure anti-common bath regime

It is intuitive that when two coupled units are close to each other and far away from a noisy environment they will experience a common bath (CB), and this is what has been predicted in the main text. But, what about the opposite, is there a regime where the stretch mode is the only dissipative degree of freedom? In the main text figures we have seen that the minimum $S_{\rm cross}/S_{\rm self}$ was around $-0.4$. We investigate this in figure 5 for two geometries: a square finite electrode, and a stylus trap as in wineland2017 . For the square electrode a value of almost -1 is reached around $\{d,l\}=\{1.2L_{x},1.4L_{x}\}$. A value lower than $-0.9$ is reached $\{d,l\}\simeq\{L_{x},L_{x}\}$. For the stylus trap a lowest value of $\simeq-0.95$ is reached at reasonable distances $\{d,l\}=\{1.5R,2R\}$, although the absence of a common RF null for two ions in this configuration might require active stabilization of the RF-drive amplitude, whose noise could otherwise mask anomalous heating.

Appendix E 1-ion noise level for different dipole orientations

Dipole-dipole interactions, as that caused by a surface dipole and the displaced charged ion from its equilibrium position, have a preferred direction. It is hence to be expected that different orientations of dipole sources will yield different noise levels even for a 1-ion configuration. Next we plot the simplest situation in which a uniform planar infinite electrode is filled with dipoles oriented along the 3 possible directions: an ion’s motion along $x$ feels highest noise levels when dipoles are along $\hat{u}_{y}$, while for in-plane dipole orientations, $\hat{u}_{x}$ produces a noise 4 times higher than $\hat{u}_{z}$. The asymmetry between the 3 directions is a direct consequence of the dipole dipole interaction form $3(\vec{m}_{1}\cdot\hat{r})(\vec{m}_{2}\cdot\hat{r})-\vec{m}_{1}\cdot\vec{m}_{2}$, with $\vec{m}_{1}$ the ion displacement [along $\hat{u}_{x}$], $\hat{r}$ the ion-dipole unit vector [along $\hat{u}_{y}$] and $\vec{m}_{2}$ the dipole moments on the surface.

In a similar spirit, Schindler and coworkers Harmut2 showed (although assuming dipoles perpendicular to the electrode) that one can use the asymmetry in noise levels to distinguish between technical noise and anomalous heating sources. Also, the idea of 1-ion noise sensing for studying stray fields is used in FSK .

Appendix F Mean field dipole

The blurring out into a mean-field dipole in the presence of spatial dipole-dipole correlations can be understood as follows. We can rewrite the noise spectral density seen by 1 ion as the Fourier transform of

[TABLE]

where $\vec{r}_{i}$ is the distance between dipole $i$ and the ion. As explained in the main text, the spatial-temporal separability of $\left<\mu_{i}(t)\mu_{k}(0)\right>$ allows us to approximate it by $\left<\mu_{i}(t)\mu_{i}(0)\right>f(\vec{r}_{i},\vec{r}_{j})\simeq S_{\mu}(t)f(\vec{r}_{i},\vec{r}_{j})$ where we have assumed that temporal dipole fluctuations are similar for different spatial regions of the electrode, something rather reasonable. Hence, we can write

[TABLE]

Further, we can assume that the spatial dipole-dipole correlation profile $f(\cdot)$ is sufficiently translational invariant in the regimes of interest (e.g. ions do not approach too close the borders of electrodes) and thus only depends on the absolute distance between dipoles: $f(\vec{r}_{i},\vec{r}_{j})\simeq f(|\vec{r}_{i}-\vec{r}_{j}|)$. We can sum up the situation by writing the spatial dependence of the spectral noise as

[TABLE]

If the function $f$ decays significantly for distances greater than $\xi$ as e.g. if $f(\vec{r}_{i},\vec{r}_{j})=e^{-|\vec{r}_{i}-\vec{r}_{j}|/\xi}$, we can define an effective, or ‘mean field’, function $\tilde{g}_{x}(\vec{r}_{i};\xi):=\sum_{j}g_{x}(\vec{r}_{j})f(|\vec{r}_{i}-\vec{r}_{j}|)$ centered at $\vec{r}_{i}$ and averaged over a size $\xi$. The noise sum then becomes $\sum_{i}g_{x}(\vec{r}_{i})\tilde{g}_{x}(\vec{r}_{i};\xi)$, to be compared with the case of uncorrelated dipoles $\sum_{i}g_{x}(\vec{r}_{i})^{2}$. Now recall that the $g$ functions are meaningful only in an area of order $\mathcal{A}\sim\mathcal{O}(d)$ around the ion position, so contributions $g_{x}$ of dipoles far from this spot can be neglected. This allows us to compare the region $\mathcal{A}_{d}$ of ion-dipoles influence, to the region size $\mathcal{A}_{\xi}$ defining the averages for the mean-field dipoles. We thus have the following situations:

• •

$\xi\ll d$ ($\mathcal{A}_{d}\gg\mathcal{A}_{\xi}$): Mean-field dipole is seen as an effectively individual dipole and so $\tilde{g}_{x}(\vec{r}_{i};\xi)\to g_{x}(\vec{r}_{i})$. This is the uncorrelated dipoles case, which is known to scale as $d^{-4}$.

• •

$\xi\geq d$ : When $\xi$ increases, we reach a point where the averaging region $\mathcal{A}_{\xi}$ becomes bigger than $\mathcal{A}_{d}$, but every dipole lying out of it has a negligible influence on the ion. This means that making $\xi$ bigger yet will not modify the value of the mean-dipole function, which at this $\xi$ saturates.

This argument explains why the scaling drops from $d^{-4}$ when $\xi\sim d$ is reached, however it does not explain the new scaling $d^{-1}$ for $d<\xi$. We can, however give an argument why the exponent is less than $4$:

When $\mathcal{A}_{\xi}\geq\mathcal{A}_{d}$ the sum for the mean-field dipole includes the full region of influence see by the ion. If we do the brutal simplification $f(|\vec{r}_{i}-\vec{r}_{j}|)\simeq 1$ [$\forall i,j\in\mathcal{A}_{d}]$, $\sum_{i,j}g_{x}(\vec{r}_{i})g_{x}(\vec{r}_{j})f(|\vec{r}_{i}-\vec{r}_{j}|)\to\sum_{i\in\mathcal{A}}g_{x}(\vec{r}_{i})\sum_{j\in\mathcal{A}}g_{x}(\vec{r}_{j})=[\sum_{i\in\mathcal{A}}g_{x}(\vec{r}_{i})]^{2}$ this sum, for dipoles pointing normal to the surface, would yield simply [math] (a constant, i.e. $d^{0}$). However if instead of taking $f(|\vec{r}_{i}-\vec{r}_{j}|)\to 1$, we use a linear expansion of the exponential function $f\sim 1-|\vec{r}_{i}-\vec{r}_{j}|/\xi$ the sum yields a scaling $d^{-1}$ as is observed in the next Appendix and is consistent with correlated patch models guidoni . It is not obvious that in general the exact resulting exponent should be $-1$, but it seems intuitive that the strength of the scaling is tamed.

Further, for the specific case of next appendix [ion motion along $x$ and dipoles pointing along $y$] we can approximate the dipoles integration as follows: let us take the dipole geometric function from appendix B, which reads

[TABLE]

and approximate its biggest contribution which occurs for $\{x,d\}\ll d$ [this approximation usually gives a good estimate of scalings with respect to $d$],

[TABLE]

From now on we will drop numerical factors since we are interested only in the scalings. The noise felt by 1 ion is then

[TABLE]

Transforming into integrals we have

[TABLE]

If dipoles are uncorrelated, $f(\vec{r}_{i},\vec{r}_{j})=\delta(\vec{r}_{i},\vec{r}_{j})$, and we have $d^{-8}\int dx\int dzx^{2}$, which gives (recall that the important contribution from dipoles comes from an area of size $\sim d$) $d^{-4}$ as expected.

For the case of very big correlation length $\xi$, we can expand the exponential

[TABLE]

The part with the 1 gives 0 by symmetry, and the rest can be integrated by performing the change of variables $Q_{\pm}=(x\pm x^{\prime})/\sqrt{2}$ and $Z_{\pm}=(z\pm z^{\prime})/\sqrt{2}$, so

[TABLE]

A final change of variables for the area enclosed by $Z_{-}$ and $Q_{-}$ (of order $d$) to polar $Q_{-}=r\cos\phi$, $Z_{-}=r\sin\phi$, gives

[TABLE]

which after simple integration yields $d^{-1}$. This highlights the power of approximating the dipole functions in this way.

Appendix G Modified noise due to correlated dipoles

Here we study what is the effect of a finite extent $\xi$ of spatial dipole-dipole correlations on the characteristics of the noise crossover. For simplicity we take a profile function

[TABLE]

although for phononic-induced dipole vibrations of adatoms we should use the more realistic sinc($|\vec{r}_{i}-\vec{r}_{j}|/\xi)$, and for diffusion of adatoms the Kelvin function ${\rm Ker}_{0}(|\vec{r}_{i}-\vec{r}_{j}|/\xi)$. In the case of one ion the noise follows, as is well known in the literature blatt , a $d^{-4}$ behavior. However, for correlated dipoles, this scaling is modified for $d<\xi$, becoming $d^{-1}$ as can be seen in figure 7.

We plot in figure 8 the crossover equivalents of figure 2 with correlated dipoles, taking the electrode configuration described in its caption.

Appendix H Chains of coupled ions

We briefly investigate what happens in a configuration where 10 ions are forming a Coulomb crystal. When they are strongly coupled we expect that normal modes of the chain can divide into sinusoidal waves with 2 kinds of parities: even or odd (under reflection in the center of the chain). It is intuitively expected that even modes will coupled strongly to a CB type of bath, while odd ones will do so for aCB types of baths. This is precisely what we observe in figure 9: when ions get closer and closer to each other [$d=L_{z}$, $l\ll L_{z}$) they begin to see a pure CB configuration, and the noise suffered by odd modes vanishes while for even modes it stays high. Further notice that when $l\simeq L_{z}$, i.e. when the ion chain almost occupies the full length $L_{x}$ of the electrode, the mode with longest wavelength (black line) sees the least noise. This is caused by ions being at the point nearest to the aCB regime, and thus the even symmetry of this mode makes it most isolated to noise. These features are of importance for the use of normal modes as buses for quantum information and for quantum simulations where they provide effective spin-spin interactions among different ions porras .