# Motional heating of spatially extended ion crystals

D Kalincev1, LS Dreissen1, AP Kulosa1, C-H Yeh1, HA Fürst1,2, TE
Mehlstäubler1,2 1 Physikalisch-Technische Bundesanstalt (PTB), Bundesallee
100, 38116 Braunschweig, Germany  2 Institut für Quantenoptik, Leibniz
Universität Hannover, Welfengarten 1, 30167 Hannover, Germany
<EMAIL_ADDRESS>
###### Abstract

We study heating of motional modes of a single ion and of extended ion
crystals trapped in a linear radio frequency (rf) Paul trap with a precision
of $\Delta\dot{\bar{n}}\approx 0.2$ phonons s-1. Single-ion axial and radial
heating rates are consistent and electric field noise has been stable over the
course of four years. At a secular frequency of
$\omega_{\mathrm{sec}}=2\pi\times 620$ kHz, we measure $\dot{\bar{n}}=0.56(6)$
phonons s-1 per ion for the center-of-mass (com) mode of linear chains of up
to eleven ions and observe no significant heating of the out-of-phase (oop)
modes. By displacing the ions away from the nodal line, inducing excess
micromotion, rf noise heats the com mode quadratically as a function of radial
displacement $r$ by $\dot{\bar{n}}(r)/r^{2}=0.89(4)$ phonons s-1 $\mu$m-2 per
ion, while the oop modes are protected from rf-noise induced heating in linear
chains. By changing the quality factor of the resonant rf circuit from $Q=542$
to $Q=204$, we observe an increase of rf noise by a factor of up to 3. We show
that the rf-noise induced heating of motional modes of extended crystals also
depends on the symmetry of the crystal and of the mode itself. As an example,
we consider several 2D and 3D crystal configurations. Heating rates of up to
500 ph s-1 are observed for individual modes, giving rise to a total kinetic
energy increase and thus a fractional time dilation shift of up to $-0.3\times
10^{-18}$ s-1 of the total system. In addition, we detail how the excitation
probability of the individual ions is reduced and decoherence is increased due
to the Debye-Waller effect.

precision metrology, ion Coulomb crystals, vibrational mode heating, rf noise,
multi-ion clocks

##  1 Introduction

Single-ion spectroscopy has lead to accurate optical atomic clock operation
and enabled searches for new physics with high sensitivity over the past
decades [1]. This includes the search for a variation of fundamental constants
[2, 3, 4], dark matter [5], a hypothetical fifth force [6, 7, 8, 9, 10, 11,
12] and, in general, more precise tests of Einstein’s theory of general
relativity [13, 14, 15, 16].

In many cases, the resolution is limited by the poor signal-to-noise ratio
from a single atomic absorber. To overcome this limitation, several new
approaches have been proposed, which utilize multiple ions in a so-called
Coulomb crystal, e.g. based on Ca+[17], Lu+[18] or In+[19, 20] ions trapped in
linear radio frequency (rf) Paul traps. Also, quantum simulations with trapped
ions is advancing towards 2D and 3D systems to simulate more complex
Hamiltonians [21, 22]. However, new challenges arise from extending the
crystal size and dimension. These are both of fundamental nature to trapping
ions in rf-traps, e.g. frequency shifts induced by excess micromotion (EMM) or
additional electric field gradients from neighbouring ions, and of a technical
nature, e.g. reaching the required homogeneity of spectroscopy beam
intensities, electric and magnetic fields over a larger region. Many of these
effects have been investigated in linear ion chains, where micromotion can be
well controlled and uncertainties on frequency shifts are expected to be at
the $10^{-19}$ level [20, 23]. For two- and three-dimensional crystals, where
high rf electric fields are probed, techniques for cancelling rf field shifts
have been investigated [24, 25, 26, 27].

In this article, we take a linear chip trap with well controlled
electromagnetic fields as a platform to investigate the heating of ion Coulomb
crystals in two scenarios, i.e. under influence of noise induced by static
electric fields (dc) and noise induced by rf fields. From this, we extrapolate
its impact onto 2D and 3D crystals. We measure dc heating of the centre-of-
mass (com) mode and higher modes in a linear chain of ions with an uncertainty
of $\sim$0.1 phonons s-1 and find that the dc heating along the radial
direction has been consistent over the course of four years.

By applying controlled radial EMM we can systematically amplify the coupling
of rf noise to the ion motion, leading to an enhanced heating rate of the com
mode. By filtering the rf noise by resonant circuits of different quality
factors, we deduce the power spectral density of the electric field noise.
Using the experimentally obtained parameters, we investigate heating of larger
radially extended 2D and 3D ion crystals and find that rf-noise induced
heating of modes strongly depends on the symmetry of the crystal and of the
modes themselves. We also calculate the time dilation shifts and the influence
from the Debye-Waller effect on each individual ion for spectroscopy
applications of extended crystals.

##  2 Experimental setup and methods

###  2.1 Experiment

The experiment is based on sideband thermometry of ground-state cooled 172Yb+
ions, which are stored in a linear rf Paul trap. The trap consists of four
wafers with segmented electrodes (figure 1(a) and (b)). The radial confinement
is set by an rf electric field supplied by a resonant circuit to the inner two
wafers, where additionally a combination of DC voltages may be applied to lift
the degeneracy and to rotate the principal axes of the radial confinement
($\mathrm{U_{t}}$, $\mathrm{U_{e}}$) and to compensate for stray electric
fields ($\mathrm{U_{tc}}$, $\mathrm{U_{ec}}$). The axial confinement in the
trapping segment is provided by the voltages $\mathrm{U_{t}}$ of the
neighbouring segments. The trap rf drive frequency is
$\Omega_{\mathrm{rf}}=2\pi\times 24.4$ MHz and typical secular frequencies are
$\omega_{x,y,z}=2\pi\times(600,580,205)$ kHz. The trap was designed for low
axial EMM and was proposed as a platform for scalable multi-ion clock-based
experiments with mixed species of 115In+ and 172Yb+ [19, 20, 28]. A detailed
description and characterization of the Paul trap and the laser systems can be
found in [29] and [30], respectively.

Figure 1: Experimental setup and the energy level scheme of 172Yb+.

Ion trap geometry is shown in the (a) $xy$ plane and (b) in the $zy$ plane.
The ions are trapped in a 2 mm segment of the linear Paul trap, where the
adjacent segments provide axial confinement. Additional voltages may be
applied to rotate the radial axes ($\mathrm{U_{t}}$, $\mathrm{U_{e}}$) and to
compensate stray electric fields ($\mathrm{U_{tc}}$, $\mathrm{U_{ec}}$).
(c) Reduced energy level scheme of the 172Yb+ ion, showing the relevant
transitions and decay channels for Doppler cooling (370 nm), sideband cooling
(411 nm and 1650 nm), repumping (935 nm and 639 nm) and spectroscopy (411 nm).
(d) The repump lasers are aligned along the trap axis $z$, while the
spectroscopy laser at 411 nm and one of the radial Doppler cooling lasers at
370 nm are aligned vertically along $y$. Two 370 nm lasers in the $xz$ plane
are used for micromotion measurements (H1 and H2) and for state preparation
through optical pumping (H2). The quantization axis is set by the magnetic
field ($B$), which is parallel to H2.

Figure 1(c) and (d) show the relevant part of the atomic energy level scheme
of 172Yb+ and the laser geometry, respectively. The ions are loaded from a
thermal atomic beam via two-step photo-ionization and stored by laser cooling
on the $\mathrm{{}^{2}S_{1/2}}$$\rightarrow$$\mathrm{{}^{2}P_{1/2}}$ electric
dipole transition at 370 nm. The cooling-cycle is closed using a repump laser
at 935 nm. Fluorescence from the decay of the short lived
$\mathrm{{}^{2}P_{1/2}}$-state is detected with an electron multiplying
charged-coupled device (EMCCD) camera. At the beginning of the experimental
sequence, a reference image is recorded with 2.5 ms exposure time to decide
whether the measurement cycle is valid, i.e. whether the ion is in the cooling
cycle or not. The ion is then cooled to the Doppler limit of about 0.5 mK via
a radial beam with a wavelength at 370 nm along the $y$ direction. Optical
pumping into the $\ket{\mathrm{{}^{2}S_{1/2}}},\mathrm{m_{j}}=-1/2$ Zeeman-
substate is achieved via a $\sigma^{-}$-polarized beam at 370 nm (H2 in figure
1(d)), aligned along the quantization axis set by a magnetic field of
$B=65\,\mu$T at an angle of 25∘ with respect to the trap axis $z$.

From here, the ion is cooled further to its quantum mechanical ground state
via quench-assisted resolved sideband cooling (RSC) on the electric quadrupole
transition $\mathrm{{}^{2}S_{1/2}}$$\rightarrow$$\mathrm{{}^{2}D_{5/2}}$ and
the dipole allowed $\mathrm{{}^{2}D_{5/2}}\rightarrow\mathrm{{}^{2}P_{3/2}}$.
Similar schemes have been previously reported for Ca+ ions, see e.g. [31, 32].
For a single ion, this continuous-wave (cw) sideband cooling scheme is applied
for 5 ms along the $y$ direction, the principal axis of the higher-frequency
radial mode (within 3∘), to achieve $\bar{n}\leq 0.1$ phonons (ph). Here,
$\bar{n}$ describes the mean motional state occupation of an ion in the
harmonic trap potential according to a thermal distribution. For simultaneous
sideband cooling of multiple motional modes of larger crystals, the frequency
of the 411 nm laser is tuned to the centre of the red sideband spectrum (again
for the modes along the $y$ direction). The intensity of the 1650 nm laser is
first increased to sufficiently broaden the effective linewidth and address
all motional modes simultaneously, trading final temperature for cooling time.
In the final part of the cooling sequence, the intensity is reduced to cool
the modes to $\bar{n}\leq 0.1$ ph, within typical cooling times of 5 - 10 ms.

###  2.2 Thermometry

To determine the temperature of a certain mode, we use resolved sideband
thermometry on the narrow electronic quadrupole transition [33, 34]. This
technique is based on detecting the ratio of the excitation probabilities of
the first-order red ($p_{\mathrm{r}}$) and blue ($p_{\mathrm{b}}$) motional
sidebands, from which we can determine the average motional state occupation
according to [33]

$\bar{n}=\frac{1}{p_{\mathrm{b}}/p_{\mathrm{r}}-1}.$ (1)

By varying the delay $\tau$ between the end of the sideband cooling cycle and
the sideband interrogation pulse, we determine the heating rate. To detect the
red and blue sideband amplitudes, a resonant first-order blue and red sideband
pulse is alternately applied and the average excitation probability of each
sideband is extracted from 200 measurement cycles. The uncertainty of a
temperature measurement is given by

$\Delta\bar{n}=\bar{n}^{2}\frac{p_{\mathrm{b}}}{p_{\mathrm{r}}}\sqrt{\bigg{(}\frac{\Delta
p_{\mathrm{b}}}{p_{\mathrm{b}}}\bigg{)}^{2}+\bigg{(}\frac{\Delta
p_{\mathrm{r}}}{p_{\mathrm{r}}}\bigg{)}^{2}}.$ (2)

The uncertainty of $\bar{n}$ increases at higher temperature, because it is
directly proportional to $\bar{n}^{2}$. To obtain the highest measurement
accuracy, the fixed interrogation pulse time is optimized for maximum
excitation probability on the blue sideband after ground state cooling. The
maximum delay is set such that the temperature is $\bar{n}\lesssim 1$ ph,
where this method is most sensitive when using first-order sidebands. With 200
measurement cycles, a relative accuracy of $\Delta\bar{n}/\bar{n}\approx 20$%
is reached in the range of $\bar{n}=0.1$ \- $2$. For each heating-rate
measurement, we typically determine the temperature at about ten different
values of the $\tau$. Under normal operating conditions, heating rates of
$\dot{\bar{n}}\sim 1$ ph s-1 per ion are observed, which can be determined
with an uncertainty of $\Delta\dot{\bar{n}}/\dot{\bar{n}}\approx 0.1$ ph s-1.
The required maximum delay for such a measurement is on the order of a second,
leading to a total time of around 15 minutes per heating-rate measurement.

The population in the $\ket{\mathrm{{}^{2}D}_{5/2},-5/2}$ Zeeman sublevel has
a finite probability to leave the SBC cycle via decay to the long-lived
$\mathrm{{}^{2}F_{7/2}}$ state. Since no fluorescence is observed in this
case, the ion is detected as dark, leading to a background signal of 2.7%
after sideband cooling. As a result, the temperature measurement is limited to
a minimum of $\bar{n}=0.03$ (included in the error bars). After each
measurement cycle, the population is repumped from the metastable
${}^{2}\mathrm{D}_{3/2}$, the ${}^{2}\mathrm{D}_{5/2}$ and the
${}^{2}\mathrm{F}_{7/2}$ states to the ground state as shown in figure 1(c).

##  3 Heating due to dc electric field noise

Fluctuations of the static electric field at the ion position can change the
motional state in the direction of the noise field. In a macroscopic ion trap
with an electrode-to-ion distance of $r_{0}$ = 0.71 mm, the noise field
originating from the electrodes is expected to be spatially correlated at a
crystal lengths $\Delta z<r_{0}$ [33, 35] (in our case between 10 and 100
$\mu$m). This noise leads to an energy transfer to the com mode of a chain of
ions, where all ions oscillate in phase. For this process, the heating rate
increases proportionally to the number of ions ($N$) and to the frequency-
dependent electric field noise spectral density $S_{\mathrm{E}}(\omega)$,
according to [36, 37, 38]

$\dot{\bar{n}}_{\mathrm{dc}}(\omega_{\mathrm{com}})=\frac{q^{2}}{4M\hbar\omega_{\mathrm{com}}}S_{\mathrm{E}}(\omega_{\mathrm{com}})=N\frac{\mathrm{e}^{2}}{4m_{\mathrm{Yb}}\hbar\omega_{\mathrm{com}}}S_{\mathrm{E}}(\omega_{\mathrm{com}}),$
(3)

where $\hbar$ is the reduced Planck’s constant, $\omega_{\mathrm{com}}$ is the
secular frequency of the com mode, $q=N\times e$ is the total electric charge
of the crystal, with $e$ being the elementary charge, and $M=N\times
m_{\mathrm{Yb}}$ is the total mass of the crystal, with $m_{\mathrm{Yb}}$
being the atomic mass of ytterbium.

###  3.1 Further heating mechanisms

In modes other than the com mode, the ions oscillate out of phase relative to
each other. We refer to them as out-of-phase (oop) modes and distinguish them
by the number of oscillatory nodes, e.g. the first oop mode has one node, the
second oop mode has two, and so on. Heating of the oop motional modes due to
local fluctuations of electric fields on the electrodes requires gradients of
these fields along the crystal and, therefore, is suppressed relative to that
of the com mode by at least
$\dot{\bar{n}}_{\mathrm{com}}/\dot{\bar{n}}_{\mathrm{oop}}\propto r_{0}/\Delta
z$ [33, 35], which in our system is on the order of $10$ or above.

Further sources of heating can arise from the non-linearity of the Coulomb
interaction via mode mixing [39], where phonons from two different modes can
resonantly combine to an excitation in a third mode or reverse, if energy
conservation is fulfilled. In a linear chain, this can be avoided by choosing
trapping conditions that prevent the modes from coupling, i.e. far from any
structural phase transition (e.g. 1D$\rightarrow$2D). In contrast to the oop
modes, the com mode is the only one, where the centre of mass actually moves.
An external force is required to transfer energy between com and any other
mode. Thus, coupling of the com mode to oop modes is suppressed[39].

Motional modes can also be heated, if the ion samples anharmonicities of the
trapping potential and if the resonance condition
$\sum_{\alpha=1}^{3\mathrm{N}}l_{\alpha}\omega_{\alpha}=\Omega_{\mathrm{rf}}$
is fulfilled, where $l_{\alpha}$ is a rational number related to the order of
the anharmonicity, $\omega_{\alpha}$ is the secular frequency of mode $\alpha$
and $N$ is the number of ions [40, 33]. In typical Paul trap experiments, this
source of heating is avoided by choosing the Mathieu parameters
$a_{i},q^{2}_{i}\ll 1$ ($i=x,y,z$) and by properly choosing the secular
frequencies [33].

###  3.2 Heating of a single ion

Figure 2: Measured heating rates in absence of excess micromotion

(a) Measured axial (blue circles) and radial (red circles) heating rates of a
single ion as a function of the secular frequency
$\nu_{\mathrm{sec}}=\omega_{\mathrm{sec}}/2\pi$. A power law function
$\dot{\bar{n}}(\nu_{\mathrm{sec}})=a\nu^{b}_{\mathrm{sec}}$ is fitted to the
heating rate of the radial modes (grey curve), from which $b=-1.95(25)$ is
obtained. The $1\sigma$ uncertainty of the fit is indicated by the shaded grey
area. The frequency dependent heating rate of the axial modes are consistent
with that of the radial modes, indicating an isotropic noise distribution.
(b) Heating rate per ion of the radial centre-of-mass (com) mode (red circles)
and the first out-of-phase mode (blue diamonds) is shown as a function of the
number of ions. A linear fit (grey curve) shows that there is no significant
dependence as a function of ion number within a $1\sigma$ uncertainty of the
fit (grey shaded area), therefore, we extract the weighted mean value (black
dashed line) over all 4 data points of the com modes to be
$\dot{\bar{n}}=0.56(6)$ ph s-1 per ion. The secular frequencies of the com
mode are in the range of $\nu_{\mathrm{sec}}=615$ kHz to
$\nu_{\mathrm{sec}}=635$ kHz and the corresponding first oop modes are 4 to 8
kHz lower than the com modes. For the four-ion crystal the heating rates of
the other two radial out-of-phase modes at $\nu_{\mathrm{sec}}=619$ kHz (green
square) and at $\nu_{\mathrm{sec}}=607$ kHz (purple triangle) are also
measured. No significant heating of any of these modes is observed.

To study dc electric field noise, we measure the heating rate of the radial
motional mode of a single 172Yb+ ion as a function of the secular frequency
$\nu_{\mathrm{sec}}=\omega_{\mathrm{sec}}/2\pi$ (see figure 2, red circles).
Part of the data has been published in [20]. Data at about
$\nu_{\mathrm{sec}}=600$ kHz to $\nu_{\mathrm{sec}}=630$ kHz was recorded
repeatedly and shows consistent results over the course of four years. A power
law function $\dot{\bar{n}}(\nu_{\mathrm{sec}})=a\nu_{\mathrm{sec}}^{b}$ is
fitted (grey line) to the data of the radial mode (red circles) as shown in
figure 2(a), yielding $b=-1.95(25)$, in agreement with what is found in other
traps, see e.g. [41]. By fixing the exponent to $b=-2$, we obtain a heating
rate of $\dot{\bar{n}}\left(\nu_{\mathrm{sec}}\right)=2.88(17)\times
10^{11}\mathrm{s^{-1}Hz}^{2}/\nu_{\mathrm{sec}}^{2}$, which corresponds to an
electric field noise spectral density of
$S_{\mathrm{E}}\left(\nu\right)=8.49(8)\times 10^{-9}\mathrm{(V/m)}^{2}/\nu$,
according to (3). In addition, the heating rate of the axial motional mode is
measured for secular frequencies between $\nu_{\mathrm{sec}}=200$ kHz and
$\nu_{\mathrm{sec}}=300$ kHz and is found to be consistent with this electric
field noise spectral density, as shown in figure 2(a) (blue circles).
Currently, the operating voltages of $\leq 12$ V provide a maximal axial
secular frequency of $\nu_{\mathrm{sec}}\approx 300$ kHz. The heating rate of
the axial mode as a function of secular frequency is in agreement with that of
the radial mode, indicating that the DC noise field is isotropic.

###  3.3 Heating of linear crystals

During sideband thermometry of larger ion chains, the spectroscopy laser at
411 nm addresses all the ions globally, producing entangled states when
driving motional sidebands, while detection of the electronic state is done on
one of the outer ions in the chain. If the average motional state occupation
is low, i.e. the number of phonons is smaller than the number of ions, the
probability of a red sideband excitation of one of the ions depends on the
state of the other ions. We numerically calculate a correction factor for the
extracted value of $\bar{n}$ from the detected sideband ratio, which takes the
correlation between the ions into account [36, 42, 35]. A more detailed
description is reported in A.

The heating rates of radial modes of linear crystals of two, four and eleven
ions are measured at a fixed radial confinement, see figure 2(b). The axial
potential is adapted to the crystal length for several reasons. Firstly, the
critical axial secular frequency for the 1D to 2D transition (linear-to-zigzag
[43, 44, 45]) reduces with increasing number of ions. Therefore, if the radial
secular frequency is kept constant, the axial secular frequency should be kept
low enough to prevent cross-coupling of modes. Secondly, the frequency
splitting of the radial modes is proportional to the axial secular frequency
and a reduction of the axial secular frequency, therefore, facilitates
simultaneous sideband cooling of several modes. Note, that this is limited by
the ability to sufficiently resolve individual motional modes during sideband
thermometry. Using the RSC method discussed in section 2, all radial modes of
the two-ion and four-ion crystal were cooled to $\bar{n}\leq 0.1$. For the
11-ion crystal, the detuning of the spectroscopy laser at 411 nm is set such
that the four modes with the highest secular frequency are cooled efficiently.
To exclude heating from non-linear coupling between sideband cooled modes and
Doppler cooled modes in the investigated crystals, we perform molecular
dynamics simulations as described in B. With this, we numerically verify, that
under the experimental conditions there is no significant energy flow to the
cooled modes from any other mode. The calculated energy fluctuations in the
sideband-cooled modes due to non-linear coupling are below $10^{-4}$ ph.

The heating rate of the com mode is measured to be $\dot{\bar{n}}=0.56(6)$ ph
s-1 per ion, as can be seen from the black dashed line. The data is fitted to
a linear function (grey line), verifying that the heating rate per ion is
constant as function of ion number within the uncertainty (shaded grey area).
The heating rate of the first oop mode is consistent with zero within the
uncertainty of $\sigma_{\dot{\bar{n}}}=0.2$ ph s-1 per ion for all crystal
sizes, ranging between $10\,-100\,\mathrm{\mu m}$. Heating rates of the third
and fourth oop mode in the four-ion crystal are also measured and no
significant heating is observed at the level of our resolution.

##  4 RF electric field noise coupling to secular motion

When the ion is exposed to EMM, external and technical noise at Fourier
components $\Omega_{\mathrm{rf}}\pm\omega_{\mathrm{sec}}$ couple to the ion
motion, leading to additional heating at secular frequency
$\omega_{\mathrm{sec}}$ [33, 46]. The ponderomotive trapping potential [47]

$\Phi_{\mathrm{P}}=\frac{q^{2}}{2m\Omega_{\mathrm{rf}}^{2}}\times\left<\vec{E}^{2}_{\mathrm{rf}}(x,y,z,t)\right>,$
(4)

where $m$ is the ion mass, $q$ is the ion charge, $\Omega_{\mathrm{rf}}$ is
the rf drive frequency, $\vec{E}_{\mathrm{rf}}(x,y,z,t)$ the applied electric
field, which is time averaged over a full rf period of
$T=2\pi/\Omega_{\mathrm{rf}}$. Assume a trapping electric field with a small
noise contribution $\xi$ at $\Omega_{\mathrm{rf}}\pm\omega_{\mathrm{sec}}$,

$\vec{E}_{\mathrm{rf}}(\vec{r},t)=\vec{E}_{0}(\vec{r})\left\\{\cos(\Omega_{\mathrm{rf}}t)+\xi\cos[(\Omega_{\mathrm{rf}}\,\pm\,\omega_{\mathrm{sec}})t]\right\\}.$
(5)

The time average of (4) reproduces the confining pseudopotential and,
additionally, yields a $\cos(\omega_{\mathrm{sec}}t)$ \- dependent beating
term. The gradient of the beating term provides a periodic noise force at
frequency $\omega_{\mathrm{sec}}$ acting on the ion motion

$\vec{F}=-\frac{q^{2}}{2m\Omega_{\mathrm{rf}}^{2}}\vec{\nabla}E^{2}_{\mathrm{0}}(\vec{r})\xi\cos(\omega_{\mathrm{sec}}t).$
(6)

The noise force spectral density $S_{\mathrm{F}}$ is related to the voltage
noise spectral density via
$S_{\mathrm{F}}/F^{2}=S_{\mathrm{V}}/V_{0}^{2}=S_{\mathrm{E}}/E_{0}^{2}$,
providing an extension to (3) for the rf-induced heating rate of mode $\alpha$
with secular frequency $\omega_{\alpha}$. The total heating rate is then given
by
$\dot{\bar{n}}_{\mathrm{total}}(\omega_{\alpha})=\dot{\bar{n}}_{\mathrm{dc}}(\omega_{\alpha})+\dot{\bar{n}}_{\mathrm{rf}}(\omega_{\alpha})$,
where

$\dot{\bar{n}}_{\mathrm{rf}}(\omega_{\alpha})=\frac{1}{4m\hbar\omega_{\alpha}}S_{\mathrm{F}}(\omega_{\alpha})=\frac{1}{4m\hbar\omega_{\alpha}}F^{2}_{\alpha}\frac{S_{\mathrm{V}}(\Omega_{\mathrm{rf}}\pm\omega_{\alpha})}{V_{0}^{2}}.$
(7)

Since the ion senses always both contributions, we set here
$S_{\mathrm{V}}(\Omega_{\mathrm{rf}}\pm\omega_{\alpha})=S_{\mathrm{V}}(\Omega_{\mathrm{rf}}+\omega_{\alpha})+S_{\mathrm{V}}(\Omega_{\mathrm{rf}}-\omega_{\alpha})$.
Decomposing the ion oscillations in the trapping potential into normal modes
of motion [48], the modes are described by sets of eigenfrequencies
$\omega_{\alpha}$ and normalized eigenvectors $\vec{\beta}_{\alpha}$. In this
formalism, the relative motional amplitude of ion $j$ in mode $\alpha$ is
given by the $j$-th component of the eigenvector $\vec{\beta}_{\alpha}$. In
general, a mode $\alpha$ with a normalized mode vector $\vec{\beta}_{\alpha}$
will experience a noise force
$F_{\alpha}=\vec{F}(\vec{r})\cdot\vec{\beta}_{\alpha}$ according to its
projection on the force vector. For a single ion with its mode direction
parallel to the gradient of the pseudopotential, the relation for the heating
rate is given by [46]

$\dot{\bar{n}}_{\mathrm{rf}}(\omega_{\alpha})=\frac{e_{\mathrm{0}}^{4}}{16m^{3}\Omega_{\mathrm{rf}}^{4}\hbar\omega_{\alpha}}\left[\vec{\nabla}E^{2}_{\mathrm{0}}(\vec{r})\right]^{2}\frac{S_{\mathrm{V}}(\Omega_{\mathrm{rf}}\pm\omega_{\alpha})}{V_{\mathrm{0}}^{2}}.$
(8)

Thus, measuring the heating rate as function of
$\vec{\nabla}E^{2}_{\mathrm{rf}}(\vec{r})\propto r$ provides access to a part
of the voltage noise spectrum at the ion position.

###  4.1 Heating of a single ion

Figure 3: Heating under influence of excess micromotion (EMM)

(a) Heating rate of a single-ion radial mode when exposed to radial EMM. The
inset shows the orientation of the principle axes in the trap and the
direction of the spectroscopy laser. The ion is displaced along $y$ (red
circles), along $x$ (purple circles) and diagonally in the plane (green
circles). The data is fitted (grey curves) to equation (8) and the
1$\sigma$-uncertainty is shown by the grey shade.
(b) Heating rate of the radial mode of a single ion as a function of
displacement for two different values of the quality factor ($Q$) of the
resonant circuit. Data is recorded at $Q=542$ (red data points) and $Q=204$
(blue data points) and at a secular frequency $\nu_{\mathrm{sec}}=435$ kHz
(diamonds) and $\nu_{\mathrm{sec}}=620$ kHz (circles).
(c) Extracted voltage noise spectral density $S_{\mathrm{V}}$ from the curves
shown in (b). A significant increase of $S_{\mathrm{V}}$ is observed at a
reduced $Q$ and $\nu_{\mathrm{sec}}$.

To measure the heating rate of an ion under the influence of rf noise, the
experimental sequence is modified to radially displace the ion by $r$,
inducing excess micromotion. For this, the compensation voltages
$U_{\mathrm{tc}}$ and $U_{\mathrm{ec}}$ are changed, such that the ion is
displaced radially by a few $\mu$m and stored at a specific $E_{\mathrm{rf}}$
($\propto r$). After the delay $\tau$, the ion is returned to the nodal line
and the temperature of the mode is determined in the same manner as described
in section 2.

Table 1: Relations between applied voltages $\Delta U_{\mathrm{tc}},\Delta U_{\mathrm{ec}}$, electric fields $\mathrm{E_{rf}}$ and displacement $\Delta x$, $\Delta y$ for a 172Yb+ ion at radial secular frequency of $\nu_{sec}$ = 620 kHz. Additionally, the corresponding displacing static electric field $\mathrm{E_{dc}}$ is given in columns 7 and 8. $\Delta x$ | $\Delta y$ | $\Delta U_{\mathrm{tc}}$ | $\Delta U_{\mathrm{ec}}$ | $\mathrm{E_{rf,x}}$ | $\mathrm{E_{rf,y}}$ | $\mathrm{E_{dc,x}}$ | $\mathrm{E_{dc,y}}$
---|---|---|---|---|---|---|---
[$\mu$m] | [$\mu$m] | [mV] | [mV] | [V/mm] | [V/mm] | [V/m] | [V/m]
1 | - | 33 | -165 | 1.505 | 0 | 0 | 27.0
- | 1 | -2.9 | 278 | 0 | 1.505 | 27.0 | 0
0.707 | 0.707 | 21.5 | 79.7 | 1.063 | 1.063 | 19.1 | 19.1

$E_{\mathrm{rf}}$ was measured up to $E_{\mathrm{rf}}\approx 1$ V mm-1 once as
a function of the applied compensation voltages to obtain the four relations
$\partial E_{\mathrm{rf,x}}/\partial U_{\mathrm{tc}}$, $\partial
E_{\mathrm{rf,y}}/\partial U_{\mathrm{tc}}$, $\partial
E_{\mathrm{rf,x}}/\partial U_{\mathrm{ec}}$ and $\partial
E_{\mathrm{rf,y}}/\partial U_{\mathrm{ec}}$ with uncertainties below 2%, using
photon correlation method [24, 49]. From day to day, the stray electric fields
are compensated within $\sigma_{\mathrm{E_{rf}}}=0.05$ V mm-1 and the ion is
displaced in the $xy$ plane using a combination of voltages $U_{\mathrm{tc}}$
and $U_{\mathrm{ec}}$, where we extrapolate for $E_{\mathrm{rf}}\gtrsim 1$ V
mm-1. Table 1 shows exemplary values for $\nu_{\mathrm{sec}}=620$ kHz.

The low-pass filters connected to the DC electrodes suppress frequencies
higher than 113 Hz [20], resulting in a characteristic rise and fall time of 9
ms for externally applied voltage changes. This ensures adiabatic transport of
the ions as voltages are adjusted on much longer time scales than an
oscillation period of the ion. We verify that the displacement itself does not
heat the investigated mode by repeatedly shifting the ion outwards and back
without any additional delay ($\tau=0$) and measuring the temperature.
Accounting for additional small delays in experiment control, the total
duration of the displacement sequence (back and forth) is about 28 ms.
Therefore, we extend $\tau$ by a fixed additional delay time
$\tau_{\mathrm{tech}}=40$ ms, so that the experiment remains synchronous with
the 50 Hz mains line cycle. As discussed in section 2, the highest measurement
sensitivity is reached at values $\bar{n}\lesssim 1$. Therefore, the total
$\tau_{\mathrm{total}}=\tau+\tau_{\mathrm{tech}}$ is adapted at each
displacement to account for the expected heating rate.

Figure 3(a) shows the measured heating rate of the radial mode along the $y$
direction at $\omega_{\mathrm{sec}}=2\pi\times 620$ kHz as a function of
displacement in three directions. First, the ion is displaced along $y$, so
that the measured mode is parallel to $\vec{\nabla}E^{2}_{\mathrm{rf}}$ (red
points). A quadratic increase of the heating rate is observed as a function of
the displacement. A fit to the relation $\dot{\bar{n}}(r)=Ar^{2}$ yields
$A=1.03(1)$ ph s-1 $\mu$m-2, corresponding to a voltage noise spectral density
of $S_{\mathrm{V}}(\Omega_{\mathrm{rf}}\pm\omega_{\mathrm{sec}})=0.37(1)\times
10^{-11}$ V2Hz-1 according to (8). Next, the ion is displaced diagonally in
the $xy$ plane (green points) at an angle of $\phi=45(3)^{\circ}$, where less
heating is observed as a function of displacement. The fit yields a quadratic
coefficient $A=0.45(2)$ ph s-1 $\mu$m-2. From the projection of the $y$-mode
on the radial gradient
$\left[\vec{F}_{\mathrm{y}}(\vec{r})\cdot\vec{\beta}_{\mathrm{y}}\right]^{2}\propto\cos^{2}(\phi)$,
a factor of $0.50(5)$, i.e. a coefficient $A=0.52(5)$ ph s-1 $\mu$m-2, is
expected for a diagonal displacement. Finally, when shifting the ion along
$x$, where the gradient is perpendicular to the mode vector, no additional
heating of the radial $y$-mode due to rf noise is observed (purple points). It
is worth noting, that the total energy transfer from the noise field to the
ion is unchanged, but is differently distributed over the two radial modes.

###  4.2 Impact of the quality factor of the rf circuit

As the ion motion couples to noise at
$\Omega_{\mathrm{rf}}\pm\omega_{\mathrm{sec}}$ under the influence of EMM,
this type of heating is expected to be strongly dependent on both, the quality
factor ($Q$) of the resonant rf circuit that drives the confining rf field and
on the secular frequency of the interrogated mode.

In order to study this effect, the heating rate of a single ion is measured
under the influence of EMM in two different configurations. At typical
operating conditions, where the trap drive frequency is
$\Omega_{\mathrm{rf},1}=2\pi\times 24.4$ MHz, the Lorentzian-shaped rf
resonance has a full width at half maximum (FWHM) of $45$ kHz, corresponding
to a loaded quality factor of $Q_{1}=542$ (unloaded $Q=1055$). The measurement
is repeated with a reduced quality factor of $Q_{2}=204$. Besides the
increased FWHM = $116$ kHz, the resonance frequency shifts to
$\Omega_{\mathrm{rf},2}=2\pi\times 23.7$ MHz. To retain the same secular
frequency of the ion to the initial configuration, the power of the rf source
is increased. For each value of $Q$, data is recorded at two different secular
frequencies of $\omega_{\mathrm{sec},1}=2\pi\times 620$ kHz and
$\omega_{\mathrm{sec},2}=2\pi\times 435$ kHz, as is shown in figure 3(b). As
expected, no influence of $Q$ is observed on the heating rate at compensated
micromotion, i.e. within $\sigma_{\mathrm{E_{rf}}}=0.05$ V mm-1. However, when
the ion motion couples to rf noise through EMM, a significant increase of the
heating rate is observed, as can be seen from the slope of the curves in
figure 3(b). This effect becomes larger at lower secular frequency, because
then the noise at $\Omega_{\mathrm{rf}}\pm\omega_{\mathrm{sec}}$ is amplified
more by the resonant circuit. From the fits, the voltage noise spectral
density $S_{\mathrm{V}}(\Omega_{\mathrm{rf}}\pm\omega_{\mathrm{sec}})$ is
determined for $Q_{1/2}$ and $\omega_{\mathrm{sec},1/2}$, see figure 3(c), and
yields, e.g.,
$S_{\mathrm{V}}(\Omega_{\mathrm{rf},1}\pm\omega_{\mathrm{sec},1})=0.340(12)$
V2 Hz-1. At $Q_{2}=205$, the voltage noise is amplified by a factor $1.28(9)$
for a secular frequency of $\omega_{\mathrm{sec},1}=2\pi\times 620$ kHz, while
a factor of $2.80(15)$ is observed for $\omega_{\mathrm{sec},2}=2\pi\times
435$ kHz. Using the Lorentzian relation for transmission of the helical
resonator (12), a power spectral density of the noise is determined to be
$S_{\mathrm{P}}<4\times 10^{-14}$ W Hz-1, comparable with values obtained in
other systems, see e.g. [50]. The extracted values of $S_{\mathrm{P}}$ at the
four different frequencies
$\Omega_{\mathrm{rf},1,2}\pm\omega_{\mathrm{sec},1,2}$ are significantly
different, showing that the noise is not white in this range.

###  4.3 Heating of a two-ion crystal

Figure 4: Heating rate under EMM of a spatially extended crystal

(a) Radially displaced two-ion crystal and the vectors (blue arrows) of the
com mode (left) and oop mode (right), together with the pseudopotential
gradient ($\vec{\nabla}E^{2}$) (grey arrows).
(b) Schematic representation of the investigated trap segment (top) and
measured excess micromotion (error bars are smaller than symbols) along with a
spline to guide the eye (bottom). The positions in the trap segment, where
heating-rate measurements are performed are indicated by A (centre) and B
(edge). In the trap schematic, the solid line indicates the rf nodal line and
ion positions at compensated EMM. The dashed blue line exaggerates the
deviation of the ion position from the nodal line, when they are shifted to
position B (see text).
(c) Heating rate of the radial com and oop mode when exposed to EMM in the
centre of the trap segment (A, full blue circles) and at the edge of the trap
segment (B, empty blue circles). In the latter position, heating of the oop
mode is observed due to a significant axial gradient of the radial electric
field. The single ion data set reproduces consistently the one from figure 3,
which was recorded about five months earlier.

In order to investigate the impact of the rf field gradient on extended ion
Coulomb crystals, we first consider the most simple example of a two-ion
crystal. Figure 4(a) shows an axial crystal of two ions and the mode vectors
(blue arrows) of the radial com mode and the radial oop mode, together with
the pseudopotential gradient ($\vec{\nabla}E_{\mathrm{rf}}^{2}$), which is
identical for both ions (grey arrows). The energy transfer to the com mode is
expected to be twice that of a single ion as the force on the mode is doubled.
As a result, the heating rate per ion should be equal. For the oop mode
though, the mode vectors $\vec{\beta}_{\alpha,j}$ are oriented in opposite
direction ($\vec{\beta}_{\mathrm{oop},1}=-\vec{\beta}_{\mathrm{oop},2}$)
leading to a cancellation of the noise force, i.e. the net force on the mode
is zero, and thus no heating of the oop mode should occur. In fact, this is
the case for all radial oop modes in larger strings of ions, which are
radially displaced, as pairwise cancellation of the noise force occurs. Or
more fundamentally, for each oop mode ($\alpha\neq\mathrm{com}$), the sum of
the mode vectors components $\vec{\beta}_{\alpha,j}$ in positive direction is
equal to the sum of those in the negative direction, since the centre of mass
is at rest in those modes. As the ions are placed in an identical gradient
$\vec{\nabla}E_{\mathrm{rf}}^{2}(\vec{r}_{j})=\vec{\nabla}E_{\mathrm{rf}}^{2}(r)$,
the force contributions must cancel

$F_{\alpha\neq\mathrm{com}}\propto\sum_{j=1}^{N}\vec{\nabla}E_{\mathrm{rf}}^{2}(\vec{r}_{j})\cdot\vec{\beta}_{\alpha,j}=\vec{\nabla}E_{\mathrm{rf}}^{2}(r)\cdot\sum_{j=1}^{N}\vec{\beta}_{\alpha,j}=0$
(9)

As a result, only the radial com mode of an ion chain should be heated by rf
noise.

To experimentally study the influence of rf noise on a larger crystal, the
heating rate of a linear chain of two ions at a separation of 17 $\mu$m is
measured as a function of radial displacement in the same way as described in
section 4.1. The results are shown in figure 4(c), where the red points show
the heating rate of the com mode at around $\omega_{\mathrm{sec}}=2\pi\times
620$ kHz, and the filled blue points show the heating rate of the oop mode at
around $\omega_{\mathrm{sec}}=2\pi\times 615$ kHz. The data is fitted to
$\dot{\bar{n}}=Ar^{2}$, yielding $A=0.89(4)$ ph s-1 $\mu$m-2 per ion for the
com mode, which is in agreement with that of a single ion (the single-ion fit
is given by the black dashed line) and confirms the expected scaling of the
heating rate with ion number. The grey shaded area indicates the $1\sigma$
uncertainty of the fit. As expected from (9), no significant heating of the
oop mode is observed (blue full symbols in figure 4(c)) up to a displacement
$\Delta r=$ 21 $\mu$m.

In order to induce heating by rf noise of the oop mode, we apply voltages
$\mathrm{U_{tc},U_{ec}}$ in the neighbouring segment. In this manner, the
radial component of the rf electric field as seen by the ions near the segment
edge strongly depends on the $z$ coordinate, as shown in figure 4(b), i.e. the
ions will deviate from the nodal line while shifting them along $z$. Next, the
ions were moved from the segment centre (position A in figure 4(b)) by 720
$\mu$m along the axial trapping direction towards the edge of the segment
(position B in figure 4(b)), keeping the ion separation constant. The
difference in the pseudopotential gradient at the ion positions results in a
different coupling of noise to the individual ions, preventing the noise force
from fully cancelling. This differential noise coupling becomes larger as we
displace the ions radially and leads to significant heating of the oop mode at
high values of EMM (open blue circles in figure 4(c)). With respect to the
heating rate of the com mode, $16.7$ times less heating is observed in the
case of the oop mode.

##  5 RF-noise induced heating of radially extended crystals

Based on the observed rf-noise induced heating of secular motion in few-ion
systems under the presence of rf field gradients, we extend the discussion now
by theoretical calculations of more complex 2D and 3D crystals. Here, we focus
on the heating effect due to rf noise and neglect the known heating effect due
to non-linearity of the Coulomb potential, which becomes relevant at higher
temperatures [51]. The ground state configuration of an N-ion crystal is
calculated at temperature $T=0$ K under high damping using molecular dynamic
simulations. The chosen trapping potential is approximated to the second-order
and sets of $3N$ eigenfrequencies $\omega_{\alpha}$ and mode vectors
$\vec{\beta}_{\alpha,j}$ are obtained from solving for the eigensystem.

In the case of 2D or 3D crystals, the mode structure becomes complicated and
the mode vectors of individual ions are typically no longer oriented along a
single principal axis of the trap. Therefore, the total noise force
($F_{\alpha}$) on mode $\alpha$ is obtained by summing over the forces acting
on each individual ion $j$, according to

$F_{\alpha}=\sum_{j=1}^{N}\vec{F}_{j}(\vec{r}_{j})\cdot\vec{\beta}_{\alpha,j},$
(10)

which is proportional to the projection of its normalized mode vector
$\vec{\beta}_{\alpha,j}$ on the gradient of the electric field. This modifies
equation (8) to

$\dot{\bar{n}}_{\mathrm{rf}}(\omega_{\alpha})=\frac{e_{0}^{4}}{16m^{3}\Omega_{\mathrm{rf}}^{4}\hbar\omega_{\alpha}}\left[\sum_{j=1}^{N}\vec{\nabla}E^{2}_{\mathrm{rf}}(\vec{r}_{j})\cdot\vec{\beta}_{\alpha,j}\right]^{2}\frac{S_{\mathrm{V}}(\Omega_{\mathrm{rf}}\pm\omega_{\alpha})}{V_{\mathrm{0}}^{2}},$
(11)

from which the heating rate on a mode $\alpha$ can be obtained. Since
$\vec{\nabla}E^{2}_{\mathrm{rf}}(\vec{r}_{j})$ is proportional to the absolute
ion distance $r_{j}$ from the nodal line, the square of the term is dominated
by contributions from the outermost ions at radial distance around
$r_{\mathrm{max}}$ from the nodal line. Therefore, we obtain an approximate
scaling of the heating rate
$\dot{\bar{n}}_{\mathrm{rf},\alpha}\propto\omega_{\alpha}^{-1}\times
r_{\mathrm{max}}^{2}$.

For simplicity, we assume a constant power spectral density of
$S_{\mathrm{P}}=1.5\times 10^{-14}$ W/Hz over the range of frequencies
$\Omega_{\mathrm{rf}}\pm\omega_{\alpha}$ at the input of the resonant rf
circuit. This value is in the range of the experimentally obtained values, see
section 4.1. Using a Lorentzian transfer function, the spectral voltage noise
$S_{\mathrm{V}}(\Omega_{\mathrm{rf}}\pm\omega_{\alpha})$ on the rf electrode
is calculated [50]

$S_{\mathrm{V}}(\Omega_{\mathrm{rf}}\pm\omega_{\alpha})=\frac{QL\Omega_{\mathrm{rf}}}{1+4Q^{2}(\frac{\omega_{\alpha}}{\Omega_{\mathrm{rf}}})^{2}}S_{\mathrm{P}}(\Omega_{\mathrm{rf}}\pm\omega_{\alpha}).$
(12)

The transmission is dependent on the quality factor $Q=542$ and the inductance
$L=2.5$ $\mu$H of the resonant circuit and scales approximately as
$\omega_{\alpha}^{-2}$. Using (11), we obtain
$\dot{\bar{n}}_{\mathrm{rf},\alpha}\propto\omega_{\alpha}^{-3}\times
r^{2}_{\mathrm{max}}$.

Figure 5: Heating of a radially extended crystal

(a) A radially extended two-ion crystal for
$(\omega_{\mathrm{x}},\omega_{\mathrm{y}},\omega_{\mathrm{z}})=2\pi\times(225,\,160,\,223)$
kHz. The mode vectors of the out-of-phase (oop) mode and for the centre-of-
mass (com) mode are depicted with blue arrows. The direction of the gradient
of the pseudopotential ($\vec{g}=\vec{\nabla}E_{\mathrm{rf}}^{2}$) is shown
with thick grey arrows.
(b) Normalized mode vector amplitudes $\beta^{y}$ in the $y$ direction (blue
shaded bars) and individual force contributions
$\vec{g}_{j}\cdot\vec{\beta}_{j}$ on the mode (black bars) for both the oop
mode and the com mode. The mode frequency
$\nu_{\mathrm{sec}}=\omega_{\mathrm{sec}}/2\pi$, the calculated heating rate
$\dot{\bar{n}}$ in ph s-1 and $\dot{E}$ in mK s-1 of the different modes are
given in the textboxes on the right.

###  5.1 Radially oriented linear crystals

To understand the behaviour of radially extended crystals, we first consider a
simple two-ion crystal oriented along a radial direction ($y$). Please note,
that from here on, we make the choice of the weaker confinement to be along
the $y$ direction. We assume an ideal linear ion trap, where no axial rf
fields or gradients are present, and only focus on the radial components of
the modes. Figure 5(a) depicts the calculated ion positions for a confinement
of
$(\omega_{\mathrm{x}},\omega_{\mathrm{y}},\omega_{\mathrm{z}})=2\pi\times(225,\,160,\,223)$
kHz. The blue arrows indicate the direction of mode vectors of the radial oop
mode at $\omega_{\mathrm{sec}}=2\pi\times 277$ kHz and the radial com mode at
$\omega_{\mathrm{sec}}=2\pi\times 160$ kHz. The two ions are symmetrically
placed around the nodal line, at a distance of 5.8 $\mu$m. The pseudopotential
gradient ($\vec{g}_{j}=\vec{\nabla}E_{\mathrm{rf}}^{2}(\vec{r}_{j})$),
coupling the noise to each ion, points in opposite directions with respect to
the nodal line ($\vec{g}_{1}=-\vec{g}_{2}$), as depicted by grey arrows. The
mode vectors of the oop mode also point in opposite directions and are of
equal amplitude
($\vec{\beta}_{\mathrm{oop},1}=-\vec{\beta}_{\mathrm{oop},2}$). Therefore, the
individual noise force contributions
$\vec{g}_{1}\cdot\vec{\beta}_{\mathrm{oop},1}+\vec{g}_{2}\cdot\vec{\beta}_{\mathrm{oop},2}$
add constructively. This is visualized in figure 5(b) with black bars showing
the normalized individual force contributions and blue bars showing the
normalized mode amplitudes for each mode along the $y$ direction. In this
case, the oop mode is heated by 84.5 ph s-1, corresponding to
$\dot{E}=\dot{\bar{n}}\hbar\omega/k_{B}=1.1$ mK s-1.

In contrast, the mode vectors of the com mode point in the same direction
($\vec{\beta}_{\mathrm{com},1}=\vec{\beta}_{\mathrm{com},2}$). Therefore, the
force contributions of the individual ions have opposite sign, as seen from
the black bars in the lower panel of figure 5(b), and the net force on this
mode is zero
($\vec{g}_{1}\cdot\vec{\beta}_{\mathrm{com},1}+\vec{g}_{2}\cdot\vec{\beta}_{\mathrm{com},2}=0$).
As a result, the com mode of the extended two-ion crystal is protected from
rf-noise induced heating. This perfect cancellation only holds if the ions are
symmetrically placed around the nodal line. In the case of uncompensated stray
electric fields, the net force on the com mode is finite and it is also
heated.

Figure 6: Heating of 2D extended crystals

Ion positions of a (a) 10-ion and (c) an 11-ion 2D crystal for
$(\omega_{\mathrm{x}},\omega_{\mathrm{y}},\omega_{\mathrm{z}})=2\pi\times(601,600,140)$
kHz. The respective normalized mode amplitude ($y$-component blue,
$z$-component orange) and force contribution (black bars) in the $y$ and $z$
direction are shown in (b) and (d) together with the mode frequency and the
heating rate in terms of both phonon number and energy. The 10-ion crystal is
point-symmetric with respect to the crystal center, while the 11-ion crystal
is mirror-symmetric with respect to the $y$ axis ($z=0$). Heating of radial
modes occur if the symmetry of the mode is of the same kind as the symmetry of
the crystal (see text). For each crystal, the upper panel of (b) and (d) shows
a mode with radial point symmetry while the lower panel shows a mode with
radial mirror symmetry, respectively. For both crystals the most heated mode
is shown as well as an example of a protected mode.

###  5.2 2D zig-zag crystals

The calculated ion positions of a 10- and an 11-ion zig-zag crystal for trap
frequencies
$(\omega_{\mathrm{x}},\omega_{\mathrm{y}},\omega_{\mathrm{z}})=2\pi\times(601,600,140)$
kHz are shown in figure 6(a) and (c), respectively. Due to the slight radial
anisotropy, both crystals are extended in the $yz$-plane, leading to a
coupling of the $y$ and $z$ modes.

The 10-ion crystal is axially $57$ $\mu$m long and the two central ions
(labelled as 5 and 6) are at a distance of $1.7$ $\mu$m from the nodal line.
The 10-ion crystal is point symmetric (figure 6(a)), i.e. it is symmetric with
respect to the crystal center. We can identify pairs of ions, which are
positioned opposite to each other at equal distance from the nodal line
($y_{j}=-y_{N+1-j}$ with N being the number of ions). These paired ions are
subject to opposite gradients ($\vec{g}_{j}=-\vec{g}_{N+1-j}$), e.g. the two
central ions (5,6). The radial vibrational modes can be divided in two groups,
those that have point symmetry and those that have mirror symmetry with
respect to the $y$ axis (at $z=0$). A point-symmetric mode behaves like the
oop mode of the radial two-ion crystal (section 5.1), where the mode vectors
point in opposite direction ($\beta_{j}^{y}=-\beta_{N+1-j}^{y}$) and the force
contributions add constructively, leading to heating. The most heated mode is
shown in the upper panel of figure 6(b). Assuming the rf noise
$S_{\mathrm{P}}=1.5\times 10^{-14}$ W/Hz in our system, it heats with $31.7$
ph s-1 ($0.3$ mKs-1) at a mode frequency of $\omega_{\mathrm{sec}}=2\pi\times
185$ kHz. It is the breathing-like mode in the $z$ direction and has the
largest number of nodes in the $y$ direction. In contrast, in a mirror-
symmetric mode, the mode vectors of two ions in a pair point in the same
direction and have the same mode amplitude $\beta_{j}^{y}=\beta_{N+1-j}^{y}$.
Similarly as for the com mode of the radial two-ion crystal, their force
contributions cancel and the mode is protected from this source of heating. An
example of a mirror-symmetric mode at $\omega_{\mathrm{sec}}=2\pi\times 561$
kHz is shown in the lower panel of figure 6(b).

The 11-ion zig-zag crystal shown in figure 6(c) is 60 $\mu$m long and the
central ion (6) is at 2.1 $\mu$m from the nodal line. In contrast to the
10-ion crystal, it has mirror symmetry, leading to heating of mirror-symmetric
modes, while point-symmetric modes are protected. This can be understood by
drawing the analogy with the radially displaced axial two-ion crystal (see
figure 4(a)). Again, symmetrically placed pairs of ions $y_{j}=y_{N+1-j}$ are
identified, e.g. ions (5,7) or (4,8) with $\vec{g}_{j}=\vec{g}_{N+1-j}$. But
now, if the pairs of ions show a oop-like motion
($\vec{\beta}_{\alpha,j}=-\vec{\beta}_{\alpha,N+1-j}$), pairwise cancellation
occurs and, consequently, point-symmetric modes are protected from heating
(upper panel figure 6(d)). For mirror-symmetric modes (lower panel figure
6(d)), the pairs of ions behave like a com mode
($\vec{\beta}_{\alpha,j}=\vec{\beta}_{\alpha,N+1-j}$), where the force
contributions add up constructively and rf-noise induced heating occurs. For
the 11-ion crystal, the highest heating rate is calculated to be $24.6$ ph s-1
($0.2$ mK s-1) at $\omega_{\mathrm{sec}}=2\pi\times 195$ kHz. Note, that in
the 11-ion crystal, the centre ion $j=(N+1)/2$ is unpaired. It has a finite
radial mode amplitude in mirror-symmetric modes which are heated, while it is
radially at rest in point-symmetric modes
($\beta^{x,y}_{\alpha,(N+1)/2}=-\beta^{x,y}_{\alpha,(N+1)/2}=0$), which are
protected from heating. For most of the heated modes, there are also partial
cancellations, e.g. the force adds up pairwise, but partially cancels with
respect to other ion pairs. For a full list of motional modes of the 11-ion
crystal, see C.

Figure 7: Heating of a 3D extended crystal

(a) Experimental image of a 22-ion helix crystal, with a length of 160 $\mu$m
and a radial diameter of 13 $\mu$m.
(b) Calculated ion positions in two planes ($xz$ and $yz$) for a 22-ion helix
at similar confinement as in (a) at
$(\omega_{\mathrm{x}},\omega_{\mathrm{y}},\omega_{\mathrm{z}})=2\pi\times(201,\,200,\,50)$
kHz. To guide the eye, the position of the ion in the direction perpendicular
to the plane is indicated by the size and shade of the points and the ions are
connected by a spline. The crystal is point symmetric in the $xz$ plane and
mirror symmetric in the $yz$ plane.
(c) Panels with mode vector amplitudes in the $x$ (red bars), $y$ (blue) and
$z$ (orange) direction of four exemplary modes together with force
contributions (see text). The calculated heating rates, both in phonon number
and energy, for each mode are shown together with the mode frequency in the
textbox. In the most heated modes, ions that have a large radial displacement
have a large mode amplitude. However, if the frequency of the mode is low, the
$\dot{\bar{n}}$ becomes large and the mode is also significantly heated.

###  5.3 3D crystals

As an example for a three dimensional crystal, we take a 22-ion helix [43, 44,
45] at trap frequencies
$(\omega_{\mathrm{x}},\omega_{\mathrm{y}},\omega_{\mathrm{z}})=2\pi\times(201,\,200,\,50)$
kHz. It has a crystal length of 163 $\mu$m and a maximum radial extension of 7
$\mu$m. An experimentally obtained image of such a crystal under similar
conditions is shown in figure 7(a). The calculated ion positions in two planes
are plotted in figure 7(b), where the size and the brightness of the colour of
the points indicate the position in the third dimension. The panels in figure
7(c) show the normalized mode amplitudes and the individual force
contributions in all three directions for four exemplary modes. Similar as in
section 5.2, the behaviour can be understood by looking at the symmetries of
the crystal. In the $xz$ plane the crystal is point symmetric
($x_{j}=-x_{N+1-j}$, $g^{x}_{j}=-g^{x}_{N+1-j}$) and in the $yz$ plane the
crystal is mirror-symmetric ($y_{j}=y_{N+1-j}$, $g^{y}_{j}=-g^{y}_{N+1-j}$).
Consequently, modes with point-symmetric $x$ components and mirror-symmetric
$y$ components are heated. The upper panel of figure 7(c) shows the most
heated mode at $\omega_{\mathrm{sec}}=2\pi\times 271$ kHz with a heating rate
of $\dot{\bar{n}}=508$ ph s-1 (6.6 mK s-1), where the biggest contribution
comes from a few ions with large mode amplitudes, located in the helix region.
The second panel shows a mode with much smaller and evenly distributed mode
amplitudes, both in the $x$ and $y$ direction, which is still significantly
heated (1.0 mK s-1) due to the low mode frequency and, therefore, a big change
in phonon number of $\dot{\bar{n}}=285$ ph s-1. In addition, due to the
transmission function of the resonant circuit, the noise gets amplified much
more at lower secular frequencies. The lower two panels show two modes which
are very similar, but have the opposite symmetry, both in the $x$ and $y$
direction, leading to rf-noise induced heating in one case and full
cancellation in the other.

Summing over all modes, we obtain a total heating from rf noise of this
crystal of $1.57\times 10^{-25}$ J s-1 corresponding to
$\dot{E}/k_{\mathrm{B}}=11.4$ mK s-1, of which 58% is contributed by a single
mode (the mode shown in the upper panel in figure 7(c)). Due to heating of
individual modes, the crystal is not in thermal equilibrium though. For
comparison, we estimate the heating of the crystal due to DC noise taking into
account the scaling with frequency and ion number. Here, only the three com
modes are heated by 1.5 mK s-1, 1.5 mK s-1 and 6.1 mK s-1 and the total energy
increase is $1.26\times 10^{-25}$ J s-1. The low-frequency com mode in the $z$
direction shows the largest heating rate due to the frequency scaling of the
DC electronic noise, see figure 1(a).

##  6 Implications on spectroscopy

Extended ion crystals are promising candidates for more stable optical clock
operation [17, 18, 19, 20] and for quantum simulation, which are rapidly
advancing towards higher dimensional crystals [21, 22]. While the approach of
arrays of 1D crystals, with tens of ions each, allows for a high level of
control of systematic shifts at the $10^{-19}$ level [19, 20], clock
spectroscopy in 3D crystals was proposed in [18] in view of the possible
larger number of ions, ranging up to 1000 ions in a single crystal. The latter
approach possibly will require cryogenic setups to handle the collisional
heating of large Coulomb crystals, but also will require a careful
investigation of the complex many-body dynamics of a large Coulomb crystal
with $3\times N$ motional modes (= phonons).

Here, we briefly discuss the consequences of rf-noise induced heating of ion
Coulomb crystals on spectroscopy and coherent manipulation using the 3D
crystal of 22 172Yb+ ions studied in section 5.3 as an example. For
comparison, we first estimate second-order Doppler shifts due to the simple
presence of the rf trapping field in the 3D crystal configuration. We then
calculate the second-order Doppler shifts due to heating of motional modes via
rf noise. Finally, we estimate the reduction and the fluctuation of the Rabi
frequency, for individual ions inside the Coulomb crystal due to a finite
thermal excitation of motional modes. Since the Rabi-frequency governs the
excitation probability of the ions, spatial and temporal variations will limit
the stability of a clock measurement. We consider both the residual excitation
of modes after laser cooling, and the additional heating of motional modes of
the 3D crystal during the clock interrogation time.

###  6.1 Second-order Doppler shift due to excess micromotion

The relative second-order Doppler shift (time dilation shift) due to
micromotion can be calculated from $E_{\mathrm{rf}}$ at the ion position

$\left<\frac{\Delta\nu_{\mathrm{td,j}}}{\nu_{0}}\right>=-\left(\frac{eE_{\mathrm{rf}}(\vec{r}_{j})}{2mc\Omega_{\mathrm{rf}}}\right)^{2}$
(13)

where $\nu_{0}$ is the transition frequency, $e$ the electron charge, and $c$
is the speed of light. For the 22-ion crystal, the calculated relative shifts
are all below $\Delta\nu_{\mathrm{td}}/\nu_{0}=-4.9\times 10^{-16}$. The
highest values are obtained for ions $j=11$ and $j=12$, which have the largest
radial extension in the helix region at $E_{\mathrm{rf}}=3.6$ Vmm-1 (7 $\mu$m
radial distance from nodal line). For the inner 18 ions, forming the helix,
the shift ranges between $-2\times 10^{-16}$ and $-4.9\times 10^{-16}$. The
use of specific ion species with a negative differential static scalar
polarizability $\Delta\alpha_{0}$, enables the cancellation of two rf induced
shifts. By operating at the ‘magic’ rf drive frequency, the time dilation
shift and the scalar AC Stark shift can cancel each other [24, 25, 26, 27].

###  6.2 Thermal second-order Doppler shift

In a crystal of ions at finite temperature, the thermal motion of the ions
within the vibrational modes induces a time dilation shift according to the
mean squared ion velocities $\left<v_{j}^{2}\right>$. These are obtained from
the mode amplitudes $\vec{\beta}_{\alpha,j}$ and the mode temperature
$T_{\alpha}=\bar{n}_{\alpha}\hbar\omega_{\alpha}/k_{\mathrm{B}}$

$\left<\frac{\Delta\nu}{\nu_{0}}\right>_{j}=-\frac{\left<v_{j}^{2}\right>}{2c^{2}}=-\frac{1}{2mc^{2}}\sum_{\alpha=1}^{3N}k_{\mathrm{B}}T_{\alpha}\vec{\beta}_{\alpha,j}^{2}$
(14)

For the 22-ion crystal of 172Yb+ cooled to Doppler temperature
($T_{\alpha}=T_{\mathrm{D}}$ = 0.5 mK), the motional state occupation of,
e.g., the radial and axial com mode are $\bar{n}=52$ ph and $\bar{n}=208$ ph,
respectively. Assuming that the crystal is in thermal equilibrium, the
relative time dilation shift for each of the ions is $-4\times 10^{-19}$. At a
crystal temperature of $T=5\,\mu$K, this shift reduces by two orders of
magnitude to $-0.04\times 10^{-19}$.

Due to heating by rf fields in the extended crystal, we obtain an increase in
time dilation shift for ion $j$, according to

$\frac{\partial}{\partial
t}\left<\frac{\Delta\nu}{\nu_{0}}\right>_{j}=-\frac{1}{2mc^{2}}\sum_{\alpha=1}^{3N}\dot{\bar{n}}_{\alpha}\hbar\omega_{\alpha}\vec{\beta}_{\alpha,j}^{2}.$
(15)

This increase is inhomogeneous over the 22-ion crystal due to heating of
particular modes only, and ranges between $-0.7\times 10^{-19}$ and $-3\times
10^{-19}$ per second for the individual ions, in the case of a quality factor
$Q=542$ of the resonant circuit. This increase would be higher by a factor of
2.4 for reduced quality factor of $Q=204$. Note that this dynamic shift can
not be suppressed by using a magic drive frequency.

###  6.3 Debye-Waller-effect

The thermal motion of the ions also affects the coupling between the ions and
light fields. Firstly, the motion along the propagation direction of the
interrogation laser reduces the average Rabi frequency $\bar{\Omega}_{j}$ of
the ion $j$ relative to the free atom Rabi frequency $\Omega^{0}$ [33]

$\frac{\overline{\Omega}_{j}}{\Omega^{0}}=\prod_{\alpha=1}^{3N}\exp{\left[-\eta_{\alpha,j}^{2}(\bar{n}_{\alpha}+1/2)\right]}.$
(16)

where $\eta_{\alpha,j}$ is the Lamb-Dicke-factor of mode $\alpha$ for the
$j$th ion. Secondly, it introduces a fluctuation $\sigma_{\Omega_{j}}$ of the
Rabi frequency between subsequent experiments. The relative scatter from shot
to shot (rms) can be calculated using modified zeroth-order Bessel functions
$I_{0}(x)$ [33]

$\sigma_{\Omega_{j}}=\frac{\Delta\Omega^{(rms)}_{j}}{\overline{\Omega}_{j}}\
=\sqrt{\left[\prod_{\alpha}I_{0}\left(2\eta_{\alpha,j}^{2}\sqrt{\bar{n}_{\alpha}(\bar{n}_{\alpha}+1)}\right)\right]-1}.$
(17)

For radial interrogation on the electric quadrupole transition
$\mathrm{{}^{2}S_{1/2}}$$\rightarrow$$\mathrm{{}^{2}D_{5/2}}$ at 411 nm (see
figure 1) at the Doppler temperature ($\bar{n}=52$ ph), the Rabi frequencies
for all ions are reduced to $\overline{\Omega}/\Omega^{0}\lesssim 9\,\%$. For
axial interrogation ($\bar{n}=208$ ph), they are all
$\overline{\Omega}/\Omega^{0}\lesssim 1\,\%$ and the shot-to-shot scatter is
about 100 % of the reduced Rabi frequency in both cases.

Even at ideal conditions of a crystal cooled to close to the motional ground
state, the effect remains significant. At a crystal temperature of 5 $\mu$K,
the mean occupation number of the radial com modes is $\bar{n}\lesssim 0.3$ ph
and that of the axial com mode is at $\bar{n}\approx 1$ ph. For radial
interrogation, the Rabi frequencies range between 11 %
($\sigma_{\Omega_{j}}\approx 38\,\%$) and 96 % ($\sigma_{\Omega_{j}}\approx
2\,\%$) for the most affected and the least affected ion, respectively. A more
favourable direction of interrogation is along the crystal axis. In this case,
the Rabi frequencies range between 83% and 92% of the free atom Rabi
frequency, and the shot-to-shot fluctuations are
$\sigma_{\mathrm{rms}}\lesssim 5\,\%$. Within 100 ms of rf-noise induced
heating, the distribution becomes less homogeneous, as the Rabi frequencies
reduce to between 54% and 75%, with a shot-to-shot noise between
$\sigma_{\Omega_{j}}\approx 18\,\%$ and 12 %, respectively, through coupling
between radial and axial modes. Note that in a planar 2D radial crystal, the
coupling of radial heating to axial modes can be strongly reduced[22].

##  7 Conclusion

We have experimentally studied heating effects from electric field noise on
the motion of trapped ions, originating from both dc and rf electric fields.
The previously reported low heating rate of about $0.7$ ph s-1 per ion at
$\omega_{\mathrm{sec}}=2\pi\times 620$ kHz due to noise of static electric
fields [20] was not significantly changed over a time period of four years and
could be confirmed for linear chains of up to 11 ions. The heating rate of the
first out-of-phase mode of a $100\,\mu$m-crystal was below our measurement
resolution of 0.2 ph s-1. The electric field noise spectral density
($S_{\mathrm{E}}\left(\nu\right)=8.49(8)\times 10^{-9}\mathrm{(V/m)}^{2}/\nu$)
is in the lower range of values reported for other traps[52].

We have measured heating rates in the presence of excess micromotion by
radially displacing a single ion and investigated the influence of the quality
factor Q of the resonant circuit by reducing it from $Q=542$ to $Q=204$. This
led to a significant increase of motional heating of up to a factor 2.8
depending on the secular frequency. For a two-ion crystal, heating of the com
mode is observed, while the out-of-phase mode is protected from rf-noise
induced heating, even at a large radial displacement of $\Delta r=21$ $\mu$m,
confirming the suppression of this heating mechanism in the out-of-phase modes
in linear ion chains.

Based on our experimental result and measured voltage noise spectral density,
we have calculated the motional heating effect of rf noise on radially
extended 2D and 3D crystals. The rf-noise induced heating of any mode is
largely determined by the symmetry of the crystal and of the mode. Ions which
are far away from the nodal line and have a large mode amplitude contribute
most. In a 3D 22-ion helix-shaped crystal, the energy increases by
$\dot{E}/k_{\mathrm{B}}=11.4$ mK s-1 due to rf noise dominated by a single
mode (6.6 mK s-1) and is comparable to heating of com modes induced by dc
electric field noise (1.5 mK s-1 for the radial modes and 6.1 mK s-1 for the
axial mode).

The micromotion-induced fractional second-order Doppler shift
$\Delta\nu/\nu_{0}$ is on the level of $10^{-16}$, which can be cancelled in
some specific ion species by use of a magic trap rf drive frequency [24, 25,
26, 27]. The calculated rf-noise induced heating leads to an increase of the
second-order Doppler shift of up to $-3\times 10^{-19}$ s-1, which is
inhomogeneous across the crystal and cannot be suppressed by a magic frequency
drive. By reducing the quality factor of the rf circuit from $Q=542$ to
$Q=204$, this shift increases by a factor of 2.4. Furthermore, the Debye-
Waller effect leads to a dominating noise contribution to excitation of the
atomic qubit. Therefore, the exact crystal configuration should be carefully
chosen and ground state cooling is critical. The calculated spectroscopic
properties and heating rates are highly dependent on the quality factor $Q$ of
the resonant circuit. Strong filtering of rf noise is required for precision
spectroscopy of 2D and 3D crystals, such as is proposed for multi-ion clock-
based experiments [18] and scalable quantum information processing [21, 22].

We gratefully thank J Keller, I Vybornyi, C Roos and K Hammerer for fruitful
discussions and R Nigmatullin for providing parts of the calculation code. L S
D acknowledges support from the Alexander von Humboldt foundation. This
project has been funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under Germany’s Excellence Strategy – EXC-2123
QuantumFrontiers – 390837967 (RU B06) and through Grant No. CRC 1227 (DQ-mat,
project B03). This project 17FUN07 CC4C has received funding from the EMPIR
programme co-financed by the Participating States and from the European
Union’s Horizon 2020 research and innovation programme.

##  Appendix A Temperature determination of larger ion crystals

Figure 8: Ion selection for state-detection in heating-rate measurements

The ions are numbered from left to right as seen on the EMCCD camera in the
experiment.
(a) Ion 1 (blue square) was selected for heating rate measurements of all
modes in the two-ion crystal.
(b) In the four-ion crystal, ion 1 (blue square) was selected for heating rate
measurements of the com and the first and second oop mode. For heating rate
measurements of the fourth oop mode, the second ion (green square) was
selected for better coupling.
(c) In the 11-ion crystal, ion 1 (blue square) was selected for heating rate
measurements of the com and the first oop mode.

The heating rate measurements of larger ion chains were done with global
addressing and single-ion state-detection. The ion selected for detection in
the three measured crystals are shown in figure 8. Global addressing of ions
on motional sidebands creates entangled states, which has to be taken into
account for accurate temperature determination using resolved sideband
thermometry. In this section, we elaborate on this effect and its influence on
the measurements. For simplicity, only the derivation for a two-ion crystal is
given.

The two involved electronic levels are denoted by
$\ket{\mathrm{{}^{2}S_{1/2}}}\equiv\ket{\downarrow}$ and
$\ket{\mathrm{{}^{2}D_{5/2}}}\equiv\ket{\uparrow}$. The motional state of the
centre-of-mass (com) mode at frequency $\omega_{\mathrm{c}}$ is indicated with
quantum number $n$ and that of the out-of-phase (oop) mode at
$\omega_{\mathrm{s}}$ with quantum number $k$. For simplicity, we will only
consider motional excitation of the com mode. The Lamb-Dicke parameter of the
com mode for two ions is given by $\eta_{\mathrm{c}}=\eta/\sqrt{2}$, where
$\eta=\frac{\lambda}{2\pi}\sqrt{\hbar/2m\omega_{\mathrm{c}}}$ and the Rabi
frequency of the carrier, first red- and blue sideband are $\Omega$,
$\sqrt{n}\eta_{\mathrm{c}}\Omega$ and $\sqrt{n+1}\eta_{\mathrm{c}}\Omega$,
respectively. The interaction Hamiltonian for two ions that are both addressed
with equal intensity is given by (neglecting terms of order $\eta^{2}$)

$\eqalign{H_{\mathrm{I}}&=\sum_{l=1}^{2}\frac{\hbar\Omega}{2}e^{-i(\delta
t-\phi_{l})}\ket{\downarrow_{l}}\bra{\uparrow_{l}}\Big{[}\ket{n,k}\bra{n,k}+i\eta_{\mathrm{c}}(\sqrt{n}\ket{n-1,k}\bra{n,k}e^{-i\omega_{\mathrm{c}}t}\\\
&+\sqrt{n+1}\ket{n+1,k}\bra{n,k}e^{i\omega_{\mathrm{c}}t})\Big{]}+h.c.,}$ (18)

where $l$ indicates index of the ion and $\phi_{l}$ is the phase of the laser
at ion $l$. If only resonant terms are taken into account, the Hamiltonian for
the first red- and blue sideband are given by

$\displaystyle
H_{\mathrm{r}}=i\eta_{\mathrm{c}}\sqrt{n}\frac{\hbar\Omega}{2}\ket{n,k}\bra{n-1,k}\Big{[}\ket{\downarrow_{1}}\bra{\uparrow_{1}}e^{i\phi_{1}}+\ket{\downarrow_{2}}\bra{\uparrow_{2}}e^{i\phi_{2}}\Big{]}+h.c.,$
(19) $\displaystyle
H_{\mathrm{b}}=i\eta_{\mathrm{c}}\sqrt{n+1}\frac{\hbar\Omega}{2}\ket{n,k}\bra{n+1,k}\Big{[}\ket{\downarrow_{1}}\bra{\uparrow_{1}}e^{i\phi_{1}}+\ket{\downarrow_{2}}\bra{\uparrow_{2}}e^{i\phi_{2}}\Big{]}+h.c.,$
(20)

respectively.

In the experiment, both ions are first prepared in the electronic ground state
$\ket{\downarrow\downarrow,n}$. By tuning the frequency to the first red
sideband, the ions can be in one of four possible final states
$\ket{\downarrow\downarrow,n}$, $\ket{\downarrow\uparrow,n-1}$,
$\ket{\uparrow\downarrow,n-1}$ and $\ket{\uparrow\uparrow,n-2}$ taking either
0, 1 or 2 phonons out of the system. Similarly, the four possible final states
of the blue sideband are $\ket{\downarrow\downarrow,n}$,
$\ket{\downarrow\uparrow,n+1}$, $\ket{\uparrow\downarrow,n+1}$ or
$\ket{\uparrow\uparrow,n+2}$.
The time dependent evolution of the state can be found by using the propagator
$U=Ve^{-iV^{-1}HVt/\hbar}V^{-1}$, which yields [42, 35]

$\displaystyle\eqalign{\ket{\Psi_{\mathrm{r}}}=&e^{-i(\phi_{1}+\phi_{2})}\frac{\sqrt{n(n-1)}}{2n-1}[1-\cos(g_{\mathrm{r}}\Omega
t)]\ket{\uparrow\uparrow,n-2}\\\
&+e^{-i\phi_{2}}\sqrt{\frac{n}{2(2n-1)}}\sin(g_{\mathrm{r}}\Omega
t)\ket{\uparrow\downarrow,n-1}\\\
&+e^{-i\phi_{1}}\sqrt{\frac{n}{2(2n-1)}}\sin(g_{\mathrm{r}}\Omega
t)\ket{\downarrow\uparrow,n-1}+\Big{\\{}1-\frac{n}{2n-1}[1-\cos(g_{\mathrm{r}}\Omega
t)]\Big{\\}}\ket{\downarrow\downarrow,n},}$ (24)
$\displaystyle\eqalign{\ket{\Psi_{\mathrm{b}}}=&e^{-i(\phi_{1}+\phi_{2})}\frac{\sqrt{(n+1)(n+2)}}{2n+3}[1-\cos(g_{\mathrm{b}}\Omega
t)]\ket{\uparrow\uparrow,n+2}\\\
&+e^{-i\phi_{2}}\sqrt{\frac{n+1}{2(2n+3)}}\sin(g_{\mathrm{b}}\Omega
t)\ket{\uparrow\downarrow,n+1}\\\
&+e^{-i\phi_{1}}\sqrt{\frac{n+1}{2(2n+3)}}\sin(g_{\mathrm{b}}\Omega
t)\ket{\downarrow\uparrow,n+1}+\Big{\\{}1+\frac{n+1}{2n+3}[1-\cos(g_{\mathrm{b}}\Omega
t)]\Big{\\}}\ket{\downarrow\downarrow,n},}$ (28)

where $g_{\mathrm{r}}=\eta_{\mathrm{c}}\sqrt{(2n-1)/2}$ and
$g_{\mathrm{b}}=\eta_{\mathrm{c}}\sqrt{(2n+3)/2}$.
In the experiment, only the state of one of the two ions was detected.
Therefore, the excitation probability of the red- and blue sideband are given
by
$|\braket{\uparrow,n^{\prime}}{\Psi_{\mathrm{r}}}|^{2}=\braket{\uparrow\downarrow,n-1}{\Psi_{\mathrm{r}}}^{2}+|\braket{\uparrow\uparrow,n-2}{\Psi_{\mathrm{r}}}|^{2}$
and
$|\braket{\uparrow,n^{\prime}}{\Psi_{\mathrm{b}}}|^{2}=\braket{\uparrow\downarrow,n+1}{\Psi_{\mathrm{b}}}^{2}+|\braket{\uparrow\uparrow,n+2}{\Psi_{\mathrm{b}}}|^{2}$,
respectively. Using (24) and (28) and a thermal Maxwell-Boltzmann state
distribution $p(n)$ the red and blue sideband amplitude can be written as

$\displaystyle\eqalign{p_{\mathrm{r},\uparrow}&=\sum_{n=0}^{\infty}p(n+2)\frac{n(n-1)}{(2n-1)^{2}}[1-\cos(g_{\mathrm{r}}\Omega
t/2)]^{2}\\\ &+p(n+1)\frac{n}{2(2n-1)}\sin^{2}(g_{\mathrm{r}}\Omega t)}$ (31)
$\displaystyle\eqalign{p_{\mathrm{b},\uparrow}&=\sum_{n=0}^{\infty}p(n)\frac{(n+1)(n+2)}{(2n+3)^{2}}[1-\cos(g_{\mathrm{b}}\Omega
t/2)]^{2}\\\ &+p(n)\frac{n+1}{2(2n+3)}\sin^{2}(g_{\mathrm{b}}\Omega t)}$ (34)

Figure 9: Temperature as a function of the red and blue sideband ratio for a
two-ion crystal.

(a) Mean occupation number as a function of the red- and blue side ratio
obtained from the numerical solution of the two-ion wavefunction (black curve)
at optimal pi-time. The shaded area indicates the deviation from having a 10%
different pulse time. For comparison, the curve of a single-ion (dashed line)
is shown.
(b) Difference between the two-ion result and the single-ion relation. The
shaded area indicates the deviation from having a 10% different pulse time.
The vertical grey line in (b) and (c) indicates the limit of $\bar{n}=1$ below
which we typically operate.

From (31) and (34), $\bar{n}$ as a function of sideband ratio can be obtained
numerically for a specific interrogation time $t$, as can be seen in figure
9(a). Similar as in the experiment, the interrogation time is fixed to reach
maximum excitation probability on the blue sideband directly after ground
state cooling, when a typical temperature of $\bar{n}=0.05$ is reached.
However, the determined temperature is no longer independent of the
interrogation time, as it is for a single ion. The deviation from having a 10%
longer or shorter interrogation pulse is shown by the shaded area in figure 8.
For comparison also the curve for a single-ion, as given by (1), is shown with
the dashed line in figure 9. For our operating range, $\bar{n}\leq 1$ (grey
vertical line in figure 9), the two-ion result is systematically higher than
that of a single ion. In this range, it can be well approximated by a linear
function, where the slope of the two-ion result is 35% larger than that of a
single ion. The absolute difference between the two curves is shown in figure
9(b). The largest deviation occurs at $p_{\mathrm{r}}/p_{\mathrm{b}}=0.5$ and
is $\Delta\bar{n}=0.27$ ph. The shaded area indicates the calculated value
with a $\pm 10\%$ different interrogation time.

The results show that when the state of only a single ion is detected, while
sidebands are globally driven, a systematically too low temperature is
obtained from the simple relation given by (1). In order to correct for this,
the experimental data is scaled by a factor 1.35(4). The additional
uncertainty on this factor stems from taking into account a $\pm 10\%$
uncertainty of the pulse area. Note that the correction factor of the oop mode
is nearly equal to that of the com mode, because the normalized mode amplitude
of the detected ion remains the same for both modes and the mode frequencies
are almost equal.

By extending the Hamiltonian of (18), the same calculation can be carried out
numerically to obtain correction factors for the com mode of the linear 4- and
11-ion linear chains. It yields correction factors of $2.68(4)$ and $7.05(5)$
for the com mode of 4 and 11 ions, respectively. The correction factors of all
three oop modes of the linear 4-ion crystal were calculated by taking into
account the ion-dependent Lamb-Dicke factors. The normalized mode amplitudes
of the ions in 2nd oop mode are the same as those in the com mode, leading to
a nearly equal correction factor of $2.68(4)$. Similarly, due to the choice of
ion in the experiment (see figure 8), the correction factor of $1.67(4)$ for
the 1st and 3rd oop mode were nearly equal.

For the 11-ion oop mode, no correction factor was calculated due to its
complexity. Instead, the error bar was increased to $0.2$ ph s-1, which is a
relative error of $150$%. We are confident that the heating rate of this mode
is well within this range, as the calculated correction factors for the
heating rate per ion of the other measured modes lead to a reduction of the
measured values by $20$% to $60$%. Moreover, the other data points were all
scaled down by their respective correction factor, thus the uncorrected
heating rate obtained for the first oop mode of the 11-ion crystal is an
overestimation.

##  Appendix B Influence on heating rates due to non-linear mode coupling

Heating due to non-linear coupling between the ground state (GS) cooled modes
and Doppler cooled modes can be addressed by performing higher-order
expansions of the mutual Coulomb interaction [39, 53, 54, 55] or using
molecular dynamics simulations. Since we are treating large Coulomb systems
with up to 22 ions, we decided to perform molecular dynamics simulations
involving the full Coulomb potentials as described in the following. By
determining the time evolution of ion motion, we verify that under the
experimental conditions, there is no significant energy flow to the cooled
modes from any other mode. We use the respective experimental trapping
parameters for the 2, 4 and 11-ion linear crystal configurations investigated
in section 3.3. We initially sample all mode velocities from a Maxwell-
Boltzmann distribution at around 1 mK. To simulate GS cooling, we set the
initial mode-velocities of the GS cooled modes to
$0\,\mathrm{m}\,\mathrm{s}^{-1}$. We propagate the ion crystals in time and
deduce the average kinetic energy of the normal modes every 50 ms by
integration of their Fourier spectra, using a time sample of
$T_{\mathrm{fft}}=0.46\,$ms with $n_{\mathrm{fft}}=32768$ steps. Details about
the method can be found in [56]. As the strength of the non-linear coupling
depends on the mode energies of the Doppler-cooled modes, we average over 20
simulation runs for each crystal size using different initial conditions in
each run.

Figure 10: Non-linear coupling between motional modes in an 11-ion crystal.

(a) Mode occupation numbers over time for an 11-ion crystal at $T=1$ mK (red
lines), where the four highest-frequency radial modes (modes 1 through 4)
along the $y$ direction have been set to $0\,\mathrm{m}\mathrm{s}^{-1}$
initial velocity (blue lines). For better visibility, $10^{-3}$ ph is added to
$\bar{n}$ of the ground state (GS) cooled modes to shift them upwards. Each
line represents an average over 20 simulation runs.
(b) Individual simulation run of mode temperatures is shown for three of the
Doppler-cooled modes from (a) as an example of non-linear mode coupling in the
11-ion crystal (see text).
(c) Amplitudes of energy oscillations during mode coupling. The energy of mode
18 and of mode 30 are added and multiplied by 2 (black-orange dashed line),
the energy of mode 21 is multiplied by 3 (solid purple line), according to
(35). For each curve, the mean of the curve was subtracted to center them
around zero. To indicate energy conservation, the sum of the two curves is
also shown (grey line).

For both the two-ion crystal and the four-ion crystal, where all radial modes
in the directions with the strongest confinement ($y$) are cooled, we find no
significant coupling between the GS cooled modes and Doppler cooled modes. In
the case of the 11-ion crystal, only the four highest-frequency radial modes
in $y$ direction were cooled. Figure 10(a) shows the calculated mean motional
state occupation $\bar{n}$ of all motional modes over a period of a second.
For the Doppler-cooled modes (red lines), the mean motional state occupation
ranges between $\bar{n}=12$ ph and $\bar{n}=166$ ph. For better visibility,
the plots of the four GS cooled modes (blue lines) are shifted upwards by
$10^{-3}$ ph. During 1 second, the mean motional state occupation of these
four modes varies by $\Delta\bar{n}<10^{-4}$ ph.

We see non-linear coupling between some of the Doppler cooled modes,
especially in the low frequency range. As an example, figure 10(b) shows the
calculated mode energies as function of time for a single simulation run. For
clarity, only the energy of the three coupled modes are depicted. Two of them
are radial modes at $\omega_{\mathrm{sec},18}=2\pi\times 526$ kHz (black line)
and at $\omega_{\mathrm{sec},21}=2\pi\times 485$ kHz (purple) along the weaker
confined direction $x$ and the third mode is an axial mode at
$\omega_{\mathrm{sec},30}=2\pi\times 200$ kHz (orange). The index identifies
the mode, sorted from highest (1) to lowest (33) mode frequency. Periodic
energy exchange between the three modes is observed, where energy is
transferred from mode 18 and 30, oscillating in phase with each other, to mode
21, oscillating out-of-phase. In this case, two phonons of both mode 18 and
mode 30 are converted resonantly into three phonons of mode 21 and reversely,

$2\hbar\omega_{\mathrm{sec},18}+2\hbar\omega_{\mathrm{sec},30}=3\hbar\omega_{\mathrm{sec},21}.$
(35)

This is illustrated in figure 10(c), where the curves from (b) are centred
around zero by subtracting the mean mode energy. The energies of mode 18 and
30 are summed and multiplied by 2 (black-orange dashed line) according to
(35), while the energy of mode 21 is multiplied by 3 (purple line). The sum of
the amplitudes (grey line) fluctuates around zero indicating energy
conservation. The amount of energy exchanged in this process corresponds to
two phonons at $\omega_{\mathrm{sec},21}=2\pi\times 485$ kHz, which can be
relevant, if heating rates of the weaker confined radial direction ($x$) or of
the axial modes are considered. Note that we only measured heating rates along
the $y$ direction and were not bothered by this effect. In addition, a slight
change of confinement shifts the secular frequencies away from this resonance
[39].

##  Appendix C All modes of the 11-ion crystal

Figure 11: RF-noise induced heating of a 2D 11-ion crystal.

(a) Crystal geometry of the zig-zag crystal in two directions.
(b) Mode amplitude of the individual ions in two directions and the
corresponding force contributions. The frequency of the mode and the
calculated heating rates are given both in phonon number and energy transfer
in the text box on the right.

Figure 12: RF-noise induced heating of a 2D 11-ion crystal (continued). Figure
13: RF-noise induced heating of a 2D 11-ion crystal (continued).

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