Ion-trap measurements of electric-field noise near surfaces

Electric-field noise near surfaces is a common problem in diverse areas of physics and a limiting factor for many precision measurements. There are multiple mechanisms by which such noise is generated, many of which are poorly understood. Laser-cooled, trapped ions provide one of the most sensitive systems to probe electric-field noise at MHz frequencies and over a distance range $30-3000\mu \mathrm{m}$ from a surface. Over recent years numerous experiments have reported spectral densities of electric-field noise inferred from ion heating-rate measurements and several different theoretical explanations for the observed noise characteristics have been proposed. This paper provides an extensive summary and critical review of electric-field noise measurements in ion traps and compares these experimental findings with known and conjectured mechanisms for the origin of this noise. This reveals that the presence of multiple noise sources, as well as the different scalings added by geometrical considerations, complicates the interpretation of these results. It is thus the purpose of this review to assess which conclusions can be reasonably drawn from the existing data, and which important questions are still open. In so doing it provides a framework for future investigations of surface-noise processes.

Figure 1

Various possible trap geometries. (a) The most basic geometry, of which all others are a variation, comprises perfectly hyperbolic electrodes, capable of creating a perfectly quadrupolar field. (b) In generating a three-dimensional structure, a rotationally symmetric geometry can create a pseudopotential which is confining in three dimensions. This is the basic design of a ring trap. (c) Alternatively, a translationally symmetric geometry can create a pseudopotential which is confining in two dimensions. This is the basis of the design for a linear trap, where confinement in the third dimension is provided by the addition of electrodes held at a positive dc voltage. These basic designs can be deformed to depart significantly from the idealized hyperbolae, where (d), (e) are topologically ring traps and (f), (g) are topologically linear traps. (h) The axial electrodes of linear traps can also be segmented to form multiple trapping positions in a single device. See main text for further discussion.

Figure 2

Heating-rate measurements by sideband-resolved methods. (a) Excitations of the ion to states with different phonon numbers have phonon-number-dependent coupling strengths. In the Lamb-Dicke limit, transitions of $|\mathrm{\Delta }n|>1$ can be neglected. (b) Rabi flops driven on the red sideband (open circles) and blue sideband (filled circles). The solid lines are fits to the data from Eqs. (27) and (28). (c) From these fits the populations $P_n$ can be calculated. (d) Rather than driving Rabi flops, the mean phonon number $\overline{n}$ can be calculated from the relative heights of the red and blue sidebands. (b), (c) Adapted from [85]. (d) Adapted from [34].

Figure 3

Frequency exponent $\alpha$ as a function of the ion motional frequency ${\omega }_{\mathrm{t}}$ in room-temperature traps. The lateral extent of the bars gives the frequency range over which the noise was measured. The vertical extent of the bars indicates the uncertainty in $\alpha$. Data are taken from the relevant references in Table 1. For points marked with ${}^{†}$, the uncertainty in $\alpha$ was not stated in the original paper and has been estimated from the published data. For points marked with ${}^{*}$, neither the value of $\alpha$ nor its uncertainty were stated in the original paper, and they have been estimated from the published data. For experiments in which resonance peaks were observed [points [47] [171] and [57]b 52] the regions over which the resonances occurred are indicated by a dotted line. Selected data are discussed in detail in the text and are highlighted here.

Figure 4

Frequency exponent $\alpha$ as a function of temperature for a gold-on-quartz surface trap. The values for the data points and error bars are taken from Fig. 4 of [123]. The dotted line shows the theoretically expected dependence from the model proposed [123, drawing on 66] with the dashed line showing temperature-independent $1/f$ scaling.

Figure 5

Heating rate $\dot{\overline{n}}$ as a function of trap secular frequency ${\omega }_{\mathrm{m}}$ ($\equiv {\omega }_{\mathrm{t}}$), for ring traps of two different sizes. The small trap (called 3a by Turchette et al., and [7]c in this review) had $d=125\mu \mathrm{m}$. The large trap (called 3b, or [7]d) had $d=280\mu \mathrm{m}$. The dashed lines show scaling of $\dot{\overline{n}}\propto {\omega }^{-1}$ (i.e., $\alpha =0$). Assuming this frequency scaling, the data imply a distance scaling of $\beta =3.8(6)$. From [206].

Figure 6

Spectral density of electric-field noise $S_E$ for different species of trapped ions. Data are from [57] (${}^{111}{\mathrm{Cd}}^{+}$), [206] (${}^9{\mathrm{Be}}^{+}$ I and III), [179] (${}^9{\mathrm{Be}}^{+}$ II), [62] (${}^{198}{\mathrm{Hg}}^{+}$), [176] (${}^{40}{\mathrm{Ca}}^{+}$), and [60] (${}^{137}{\mathrm{Ba}}^{+}$). The dashed line shows the $1/d^4$ scaling predicted by a simple patch-potential model. From [57].

Figure 7

Scaling of heating rates in a needle trap. Axial heating rate $\dot{\overline{n}}$ as a function of the ion-electrode separation $z_0$ ($\equiv d$). The dashed line, to which the measured data fit well, has a gradient of $-3.5$, corresponding to $\beta =3.5$. From [58].

Figure 8

Spectral density of electric-field noise $S_E$ as a function of the distance $d$ from the ion to the nearest electrode for traps operated nominally at room temperature. Data points are taken from the relevant references in Table 1. On the right, the ordinate scale is given as the equivalent heating rate of a ${}^{40}{\mathrm{Ca}}^{+}$ ion with a motional frequency of ${\omega }_{\mathrm{t}}=2\pi \times 1\mathrm{MHz}$. The shaded regions indicate an envelope scaling with $d^{-4}$. The dotted lines indicate an envelope scaling with $d^{-2}$. See Sec. 3b for discussion, including the uses and significant limitations of plotting such data on a single graph.

Figure 9

Measured values of the temperature exponent $\gamma$ for four traps ([123]). Traps [22]a,b,d were measured once at cryogenic temperatures. For trap [22]c five measurements were made. Following measurement i the trap was cycled from 7 to 130 K, without breaking vacuum. Following measurement ii the trap was cycled from 7 to 340 K without breaking vacuum. Following measurements iii and iv the trap was brought to room temperature, cleaned with lab solvents in air, and then recooled.

Figure 10

Spectral density of electric-field noise $S_E$ as a function of temperature $T$ for a single trap ([22]c) following different treatments. $S_E(T)$ was inferred from ion-heating rates taken between 7 and 100 K ([123]). The published fit parameters ([123], Table 1) have been extrapolated here to room temperature (dotted lines).

Figure 11

Spectral density of electric-field noise $S_E$ as a function of temperature $T$ for traps made of different materials. The levels of heating are similar in the three traps, despite the different trap-electrode materials. The dotted lines are fits to power laws in temperature to all data. Above 70 K the scaling exponent is $\gamma =2.13(5)$. Below 70 K it is 0.54(4). From [47].

Figure 12

Spectral density of electric-field noise $S_E$ as a function of the distance $d$ from the ion to the nearest electrode, for traps at cryogenic temperatures (all nominally around 6 K). Data points are taken from the relevant references in Table 1. The gray points are the same room-temperature data as shown in Fig. 8, for comparison. On the right, the ordinate scale is the equivalent heating rate of a ${}^{40}{\mathrm{Ca}}^{+}$ ion with a motional frequency of ${\omega }_{\mathrm{t}}=2\pi \times 1\mathrm{MHz}$. The white area ($40\mu \mathrm{m}\le d\le 230\mu \mathrm{m}$) shows the size range over which cryogenic measurements have been made. The solid black line marks ${\tilde{S}}_E$ for the cryogenic traps. The solid gray line marks ${\tilde{S}}_E$ for the room-temperature traps over the same size range.

Figure 13

Distance scaling of Johnson noise for a needle trap. (a) Electrode geometry considered for the analytic result given in Eq. (46). (b) Electrode geometry used for the numerical calculation of the electric field at the center of a needle trap. The taper angle is 4° and a conducting ground sleeve, 3 mm inside diameter, is recessed 2.3 mm from the tips. This geometry was chosen to approximate the experiment of [58]. (c) The dipole efficiency factor $\kappa$ defined in Eq. (44) plotted as a function of the tip separation.

Figure 14

Schematic representation of a second-order $RC$ low-pass filter. Capacitor ${\mathrm{C}}_2$ lumps together the capacitance of the trap with that of the filter capacitors.

Figure 15

Resonator circuit. (a) Schematic representation of a tank resonator which uses a perfect transformer to match the resonator’s resistance on resonance $R_{\mathrm{L}}$ to that of the source impedance $R_{\mathrm{S}}$ of the voltage source ${\mathrm{V}}_{\mathrm{r}\mathrm{f}}$. (b) To simplify the analysis, the voltage source and matching transformer can equivalently be considered a perfectly impedance matched voltage source.

Figure 16

Length scales considered in the patch-potential model. The ion is trapped at a distance $d$ above an electrode of characteristic dimension $R_{\mathrm{el}}$. Patch fields on the electrode produce local potential fluctuations ${\mathrm{\Phi }}_{\mathrm{p}}(\mathit{r},t)$, which are correlated over a length scale $r_{\mathrm{c}}\approx R_{\mathrm{p}}$ set by the typical radius of the patches $R_{\mathrm{p}}$.

Figure 17

Electric-field noise from patch potentials. (a) Distance dependence of the electric-field noise $S_E$ for an ion trapped between two infinite parallel plates (inset) assuming an exponential (solid line) and steplike (dashed line) cutoff for the spatial voltage correlation function $C_V({\mathit{r}}_1,{\mathit{r}}_2,\omega )$. (b) The local distance-scaling coefficient $\beta (d)$ defined in Eq. (73) for an ion trapped above a sphere of radius $R_{\mathrm{el}}$ (inset). The solid lines represent the limits for pointlike patches (PP) for which $r_{\mathrm{c}}\to 0$, and infinite patches (IP) for which $r_{\mathrm{c}}\to \infty$. The dotted lines show the results for intermediate values of $r_{\mathrm{c}}/R_{\mathrm{el}}$ as indicated in the plot. Adapted from [134].

Figure 18

(a) General setup considered in Secs. 5b, 5c, 5d for the analysis of the electric-field noise generated from microscopic processes above a planar electrode. The noise processes are modeled by a distribution of pointlike dipoles $\mathit{\mu }({\mathit{r}}_i,t)$ located within a small layer above the metal surface. (b) Double-well potential representing the energy of an atomic or electronic two-state system. $\mathrm{\Delta }$ is the “classical” energy difference between the two wells and ${\mathrm{\Delta }}_0$ is the tunnel coupling.

Figure 19

Temperature-dependent behavior of nonuniformly distributed TLFs. (a) Temperature dependence of the averaged dipole-fluctuation spectrum ${\overline{S}}_{\mu }$ (in arbitrary units) produced by an ensemble of TLFs with a Lorentzian distribution of activation energies centered around $V_0=0.3\mathrm{eV}$ and with a width $\mathrm{\Delta }V=0.15\mathrm{eV}$. (b) The temperature dependence for the same parameters of the local scaling exponent $\alpha (\omega ,T)$.

Figure 20

Sketch of the typical shape of the adatom-surface potential $U(z)$ approximated by the model potential given in Eq. (95) for $w=5.8$. The dashed lines indicate the energies $\hslash {\nu }_n$ of the bound vibrational states with wave functions indicated by the thin solid lines. From [181].

Figure 21

Dipole-fluctuation spectrum $S_{\mu }$ of a single adatom. (a) Dependence of the full spectrum on the rescaled frequency $\omega /{\mathrm{\Gamma }}_0$ for three different temperatures. The dashed lines show the corresponding spectrum under a two-level approximation. (b) Temperature dependence of $S_{\mu }(\omega \to 0)$ for the full multilevel model (solid line) and a two-level system (dashed line).

Figure 22

Adatom diffusion model. (a) Setup considered for the evaluation of electric-field noise generated by diffusing dipoles on a planar electrode. Within this model a needlelike electrode can be approximately described by considering only the electric field from dipoles within a small electrode area $A_{\mathrm{el}}$. (b) Dependence of the normalized spectral function ${\mathcal{I}}_{\tilde{R}}^{\eta }(\tilde{\omega })$ on the scaled frequency $\tilde{\omega }=\omega /{\omega }_d$, where ${\omega }_d=\mathcal{D}/d^2$. The dashed lines indicate the analytic limits given in Eqs. (110), (111), (113), (114), (116), and (117).

Figure 23

Overview of traps for which the heating mechanisms are known. For comparison, the heating rate for the first use of “anomalous heating” in the literature is shown as point [2] ([145]). Experimental data points are taken from the relevant references in Table 1. Points [A] and [B] are calculated values from [128]. The gray points (noise sources unknown) are the same as those in Fig. 8. For orientation, the gray shaded regions and dotted lines are the same as those shown in Fig. 8 and indicate slopes of $-4$ and $-2$, respectively. Despite the experiments being limited by many different sources of noise, the reported values of $S_E$ in the various experiments are broadly similar. Consequently, it is difficult to directly infer a particular noise source simply from the absolute level of noise.

Figure 24

Distance-scaling exponent $\beta$ as a function of dimensionless distance $d/r_{\text{prolate}}$ for both axial and radial modes above a spheroidal needle of radius $r_{\text{prolate}}$ and half length $100r_{\text{prolate}}$. The solid lines are for the limit of a single patch covering the entire electrode. The dotted lines correspond to intermediate patch sizes of angular extent 0.4 and 0.04. The bar shows the approximate range of ion-electrode distances and the possible values of $\beta$ ($1\sigma$) measured in this experiment of [58]. Adapted from [134].

Figure 25

Filters used by [85]. The network initially consisted of an $RC$ filter outside vacuum (cutoff at 1 kHz) and a second $RC$ filter inside vacuum (cutoff at 1 MHz). A resistor was subsequently added (shown in gray) to reduce the cutoff of the second stage to 370 kHz.

Figure 26

Only two analytical solutions to the Laplace equation are capable of trapping: (a) toroidal solutions and (b) bispherical solutions. Toroidal solutions are already well known as ring traps, although bispherical solutions would be simpler to scale in a controllable way. This could be achieved in a manner similar to the work of [58], by (c) mounting the spheres on a pair of translation stages. The vacuum chamber provides a far-off ground.

Figure 27

Effect of noise correlations in multi-ion traps. The ratio between the spectral noise densities $S_E^{(1)}$ and $S_E^{(0)}$ evaluated at equal frequencies plotted as a function of the ion-surface distance for different correlation lengths $r_{\mathrm{c}}$. For this plot an exponential cutoff for the noise correlations on the surface has been assumed.