# Dark resonance spectra of trapped ions under the influence of micromotion

Nicolás A. Nuñez Barreto<EMAIL_ADDRESS>Universidad de Buenos Aires,
Facultad de Ciencias Exactas y Naturales, Departamento de Física, Laboratorio
de Iones y Átomos Fríos, Pabellón 1, Ciudad Universitaria, 1428 Buenos Aires,
Argentina  CONICET - Universidad de Buenos Aires, Instituto de Física de
Buenos Aires (IFIBA), Pabellón 1, Ciudad Universitaria, 1428 Buenos Aires,
Argentina  Muriel Bonetto  Universidad de Buenos Aires, Facultad de Ciencias
Exactas y Naturales, Departamento de Física, Laboratorio de Iones y Átomos
Fríos, Pabellón 1, Ciudad Universitaria, 1428 Buenos Aires, Argentina  CONICET
- Universidad de Buenos Aires, Instituto de Física de Buenos Aires (IFIBA),
Pabellón 1, Ciudad Universitaria, 1428 Buenos Aires, Argentina  Marcelo A.
Luda  Universidad de Buenos Aires, Facultad de Ciencias Exactas y Naturales,
Departamento de Física, Laboratorio de Iones y Átomos Fríos, Pabellón 1,
Ciudad Universitaria, 1428 Buenos Aires, Argentina  CEILAP, CITEDEF, Buenos
Aires, 1603, Argentina  Cecilia Cormick  Instituto de Física Enrique Gaviola,
CONICET and Universidad Nacional de Córdoba, Ciudad Universitaria, X5016LAE,
Córdoba, Argentina  Christian T. Schmiegelow  Universidad de Buenos Aires,
Facultad de Ciencias Exactas y Naturales, Departamento de Física, Laboratorio
de Iones y Átomos Fríos, Pabellón 1, Ciudad Universitaria, 1428 Buenos Aires,
Argentina  CONICET - Universidad de Buenos Aires, Instituto de Física de
Buenos Aires (IFIBA), Pabellón 1, Ciudad Universitaria, 1428 Buenos Aires,
Argentina

###### Abstract

We study the influence of micromotion on the spectrum of trapped ions with a
lambda-type level scheme, leading to dark resonances due to coherent
population trapping. We work with calcium ions trapped in a ring-shaped Paul
trap, in which one can compensate excess micromotion for only one ion of the
crystal. We observe that micromotion affects the shapes of the dark resonances
and causes the appearance of “echoes” separated by intervals given by the
drive frequency. We present a theoretical model that provides good fits to the
measurements and can be used to estimate the amplitude of the micromotion
modulation of the atomic motion. We estimate an effective temperature of the
ions from the spectra and observe clear micromotion heating as well as
impaired cooling for sufficiently large excess micromotion.

Keywords: micromotion, trapped ions, atomic spectrum, dark resonance, coherent
population trapping, RF heating

##  I Introduction

Trapped-ion systems are versatile platforms which in the last decades led to
the development of quantum technologies such as quantum simulators [1, 2],
optical clocks [3, 4, 5, 6], and small-scale prototype quantum computers [7,
8]. They have also served as a platform for the study of fundamental physics
at the quantum level and for the search of new physics via precision
measurements [9, 10, 11, 12, 13].

Paul traps, and more generally radiofrequency (RF) traps, employ electric RF
fields to confine ions in vacuum [14, 15]. The dynamics of the ions within
these traps can generally be understood in terms of two contributions: macro
or secular motion, and micromotion. For appropriate trap parameters, the
oscillating RF field creates an effective static harmonic potential with a
characteristic frequency that is much slower than that of the RF field; the
movement of the ions due to this effective potential is called secular motion.
This motion can be externally controlled by cooling, coherent excitation and
preparation of quantum motional states [16]. On the other hand, the
oscillating character of the trapping field creates a driven motion at the RF
drive frequency, named micromotion. The micromotion amplitude is generally
small and, in lowest-order approximation, proportional to the displacement of
the ions with respect to the center of the trap, where the RF field vanishes.

As has been shown in the last years, it is possible to use micromotion as a
resource to enhance the speed and fidelity of entangling gates in trapped ions
[17, 18, 19]. Nevertheless, it remains generally desirable to have low
micromotion, for example to achieve efficient ion cooling [20, 21, 22], reduce
heating [23, 24], and reduce Doppler related shifts in trapped ion clocks [25,
26]. With this goals in mind, several techniques were developed that allow for
the reduction of micromotion, usually referred to as micromotion compensation
[27, 28, 29].

Micromotion-related spurious effects are particularly problematic in ion
crystals. In this case, the micromotion will affect the ions differently
depending on their locations within the trap [30]. The geometry of the trap
and the potentials used then determine the possible ion crystal shapes and the
properties of the micromotion for each ion. For instance, the linear Paul trap
is specially well-suited to work with linear ion chains since ions can be made
to lie along an axis of vanishing oscillating field. On the other hand, in
three-dimensional (3D) traps like ring traps the RF field vanishes at only one
point in space, so excess micromotion can at best be fully compensated for a
single ion.

Micromotion has important modulation effects in the atomic spectra of ions,
which are different depending on the width of the relevant spectral features.
Considering that the typical drive frequency in RF traps is usually in the MHz
regime, micromotion will be reflected either as broadening of the spectral
lines, or echoes at the drive frequency, depending on whether the width of the
spectral lines is wider or narrower than the drive frequency, respectively.
The first case is typical of dipole transitions, and its effects are commonly
used to compensate excess micromotion. The procedure consists in driving a
fluorescence transition with a detuning equal to the drive frequency and
minimizing the fluorescence when varying the voltage of several electrodes.
This reduces the micromotion, which in turn sharpens the spectrum [31]. On the
other hand, for narrow transitions like quadrupolar or stimulated Raman
transitions, the spectral lines are replicated as smaller echoes separated by
the drive frequency [32, 33].

A mixed behavior between these two regimes can be found in dark resonance
spectra, which we study in detail in this article. The atomic spectra of atoms
with lambda-type level schemes present narrow negative peaks, known as dark
resonances, when the detunings of the lasers that drive the transitions from
the lower states to the excited one are similar. This is due to coherent
population trapping (CPT) phenomena [34, 35, 36]. The widths of the dark
resonances are typically much smaller than those of dipolar transitions.

For multi-level systems, many dark resonances can emerge depending on the
polarizations of the lasers and the geometry of the optics. A typical dark
resonance spectrum is shown in Fig. 1(d), where the polarizations are such
that four dark resonances emerge. The widths are related to the product of the
two Rabi frequencies and the dephasing mechanisms at work, and reach values of
up to a few MHz. For these spectra, the micromotion will manifest as a mixture
of further broadening of the already broad main line together with echoes of
the dark resonances which appear as dips within the main line.

In this paper we present a detailed study of the effects of micromotion in
dark resonance spectra in trapped ions. We work with 40Ca+ ions confined in a
3D ring-shaped Paul trap. By displacing a single ion within the trap we
measure its dark resonance spectrum under different micromotion conditions. We
show how to retrieve the modulation factor of the ion using a theoretical
model to fit the spectra. We also estimate an effective temperature of the
ion, evidencing the presence of an additional heating mechanism due to
micromotion known as RF heating [37, 38]. Moreover, we measure the spectra of
more than one ion at the same time and retrieve the modulation factor of each
ion through a fit to the model for the full measurement.

The paper is organized as follows. In Sec. II we present the experimental
setup and the measurement algorithm as well as the compensation scheme. In
Sec. III we show how to calculate dark resonance spectra with and without
micromotion. In Sec. IV we present results for micromotion-modulated spectra
with a single ion and analyze how it alters the final temperature of the ion.
Finally, in Sec. V we present results on dark resonance spectra for two ions
and study how to model the sum of their spectra.

Figure 1:  (a) Photo of the ring trap111Photo taken by Franco Meconi,
https://www.instagram.com/terrazaalcosmos/, where the electrodes are
indicated. At the trap center, fluorescence from trapped ions can be seen. (b)
Relevant level scheme for the 40Ca+ ion with the two lasers, labelled UV and
IR, that excite each of the transitions. (c) Pulsed sequence for measuring
dark resonance spectra. An initial Doppler cooling of 100 $\mu$s is followed
by a measurement of 50 $\mu$s. After 20 repeats, a long Doppler cooling of 1
ms is performed to avoid excess heating. The whole sequence is repeated 100
times to increase the collected ion light and thus reduce statistical
uncertainties. (d) A typical dark resonance spectrum with four dark resonances
that emerge when the polarization of both lasers is $\sigma_{+}+\sigma_{-}$,
i.e., linear and perpendicular to the magnetic field that is used to obtain a
Zeeman splitting of the sublevels in each manifold.

##  II Experimental setup and compensation scheme

We perform our experiments with singly-ionized 40Ca atoms trapped in a 3D
ring-shaped Paul trap, as shown in Fig. 1(a). The ions sit at the center of
the ring-shaped electrode, set to a voltage of
$\sim~{}300~{}$V${}_{\text{pp}}$ oscillating at 22.1 MHz which is responsible
for the radial trapping. The axial confinement, on the other hand, is produced
by two endcaps, top and bottom, with $\sim-0.5~{}$V each. One extra electrode
located below the trap is used to compensate the position of the ion. The
oven, which is the source of neutral 40Ca atoms, is also used as an additional
electrode for compensation. In this work, we will vary the voltage of the top
endcap electrode to controllably add micromotion to the ions.

The calcium ion of interest has a multi-lambda type electronic level system,
schematized in Fig. 1(b). The two ground 42S1/2 states are connected through a
dipole transition near 397 nm (which we label UV for “ultraviolet”) to two
excited 42P1/2 states, with a lifetime of $\sim 7$ ns [39]. From this P level,
the ion has a probability of 0.94 to decay back to the S states, and a
probability of 0.06 to decay to one of the four metastable 32D3/2 states [40,
41]. These metastable states are connected to the excited ones through another
dipolar transition near 866 nm (labelled IR for “infrared”). A closed
fluorescence cycle, which avoids pumping to dark metastable states, is
established through the UV and IR lasers together with an additional magnetic
field [42]. The polarizations of the lasers are set to be linear and
orthogonal to the magnetic field which has a magnitude of $B\sim 4$ G. Then
both lasers have $\sigma_{+}+\sigma_{-}$ polarization in the atomic (magnetic)
eigenbasis of the ion. In such a configuration, the atomic spectrum will show
four dark resonances, as in Fig. 1(d).

We measure atomic spectra by collecting scattered ultraviolet light of the ion
for different values of IR detuning $\Delta_{\text{IR}}$ while keeping the UV
detuning $\Delta_{\text{UV}}$ fixed. To ensure that the temperature of the ion
is the same for all data points, we perform a pulsed algorithm where we
interleave cooling and detection [43, 44]. A scheme of the algorithm is
represented in Fig. 1(c). We start by performing a Doppler cooling stage for
100 $\mu$s with the UV laser red-detuned around $\Delta_{\text{UV}}=-15$ MHz
and the IR laser blue-detuned around $\Delta_{\text{IR}}=+20$ MHz. With these
cooling parameters, which we use for all measurements, we prepare the ion in a
thermal state at a temperature close to the Doppler limit for a single well-
compensated ion. After this, we switch $\Delta_{\mathrm{UV}}$ to $-25$ MHz and
$\Delta_{\mathrm{IR}}$ to the measurement value and collect the scattered
light of the ion for 50 $\mu$s. This time interval is chosen so that the
scattering does not affect significantly the temperature of the ion. We repeat
this sequence 20 times, and then perform a longer Doppler cooling stage of 1
ms to avoid excess heating of the ion and to remedy heating from other extreme
events such as background gas collisions. Each sub-loop is repeated 100 times
to reduce statistical uncertainties. Each datapoint then consists of 2000
measurements.

In order to compensate excess micromotion for a single ion we use the DC
endcaps, one compensation electrode and the atom oven, located below the trap.
For the compensation we use the following algorithm. First, we set
$\Delta_{\mathrm{IR}}=+20$ MHz and
$\Delta_{\mathrm{UV}}\sim-\Omega_{\text{RF}}$, where $\Omega_{\text{RF}}$ is
the drive frequency. Then, we manually sweep the voltage of each compensation
electrode and look for a minimum of the fluorescence. This reduces the power
in the first-order modulation sideband of the UV spectrum, hence finding the
position of the ion with minimum micromotion along the propagation direction
of the UV laser [25]. In order to do a full 3D compensation, this procedure
should be iterated with three different lasers coming from from three
different directions, ideally orthogonal between them, finding a global
minimum. For the experiments presented in this paper, we only compensate the
ion in two directions.

##  III Dark resonance spectra with micromotion

In this section we present measurements of dark resonance spectra including
micromotion effects. We then introduce a theoretical model that fits the
measurements and subsequently describe how to retrieve relevant experimental
parameters with it.

###  III.1 Experimental spectra

We measure the dark resonance spectra of a single ion for different positions
in the ion trap where it is subjected to varying degrees of micromotion. As we
vary the potential applied to the top endcap electrode, $V_{\mathrm{ec}}$, the
ion is displaced to different positions with respect to the center of the trap
following a straight line, as shown at the sketch in Fig. 2(g). When the ion
is closer to the center we expect it to experience less micromotion than when
it is further away. For each position a spectrum was measured and the
modulation factor was determined by a fit of the model. Some representative
spectra are shown in Figs. 2(a)-(f). The one with the least micromotion, when
the ion is closest to the center of the trap, corresponds to Fig. 2(c). There,
the four fundamental dark resonances can be identified. Further away from the
center, when micromotion is more pronounced, echoes of the four fundamental
resonances appear in the spectrum, as can be seen in the other subplots.

For all spectra in Fig. 2, the data are plotted along with fits to a model
which considers micromotion. In the following subsection we show how to build
such a formalism, which adds a modulation at the drive frequency
$\Omega_{\text{RF}}$ in the atomic spectra. This allows us to retrieve a
modulation factor that quantifies the amount of micromotion in the ion, as
well as the temperature of the ion.

Figure 2:  Dark resonance spectra with micromotion induced by (de)compensation
of the position of the ion by means of an endcap voltage. Subfigures (a)-(f)
depict the acquired spectra (dark-green dots) and the corresponding fitted
model (black lines) for selected voltage values. The parameter $\beta$ of the
regression model serves as a measure of micromotion modulation and is expected
to be proportional to the distance from the trap center. The ion position for
each measurement varies proportionally to the applied voltage on the endcap,
along a linear trajectory as illustrated in (g). The values of the factor
$\beta$ as a function of the applied voltage are plotted in (h) as red dots,
accompanied by the expected hyperbolic relation fitted to the data (blue
line). Residuals of the fit are presented in (i).

###  III.2 Numerical calculation of spectra including micromotion

To calculate the spectra including micromotion effects we closely follow [43]
and [45], in which similar theoretical approaches were developed. To model the
atomic dynamics of the full eight-level system interacting with the two
lasers, we resort to a master equation of the form

$\frac{d\rho}{dt}=-\frac{i}{\hbar}[H,\rho]+\mathcal{L}(\rho)\,.$ (1)

Here, $\rho$ is the density matrix of the system, $H$ is the Hamiltonian of
the eight-level system interacting with the two lasers under the rotating wave
approximation in the frame rotating with the lasers, and $\mathcal{L}$ is a
Lindblad superoperator that models the spontaneous emission from the upper
levels into the two lower manifolds, as well as finite laser linewidths and
thermal motion as described in [46].

To include micromotion in the dynamics, we consider the Doppler shift that the
ion experiences due to its periodic motion with respect to the beams. This can
be treated by adding the corresponding Doppler shift to the detuning of each
laser for the ion at rest, $\Delta_{i_{0}}$, as

$\Delta_{i}=\Delta_{i_{0}}+\vec{k}_{i}\cdot\vec{v}_{0}\cos(\Omega_{\text{RF}}t),$
(2)

where $i=\\{$IR,UV$\\}$, $\vec{k}_{i}$ is the wave vector of the corresponding
laser, $\Omega_{\text{RF}}$ is the drive frequency and
$\vec{v}_{0}\cos(\Omega_{\text{RF}}t)$ is the instantaneous velocity of the
atom, considering only the contribution due to micromotion. The magnitude of
$\vec{v}_{0}$ is proportional to the displacement of the ion from the trap
center. Motion due to thermal effects is introduced in the following section.

We define a dimensionless modulation factor $\beta$ for each beam as

$\beta_{i}\Omega_{\text{RF}}=\vec{k}_{i}\cdot\vec{v_{0}}\,,$ (3)

which quantifies the strength of micromotion effects [27]. It is important to
note that the value of $\beta_{i}$ depends on the product between the
wavevector of the corresponding beam and the driven ion motion. As in our case
both beams are co-propagating, we have that
${866}\,\text{nm}\times\beta_{\text{IR}}=397\,\text{nm}\times\beta_{\text{UV}}$.
Therefore only one of the $\beta_{i}$ needs to be specified as a free
parameter in the mathematical model. For the sake of simplicity we refer to
$\beta\equiv\beta_{\text{UV}}$ and use it as the free parameter that
quantifies the micromotion.

The explicit time-dependence of the dynamics is treated through a Floquet-like
approach, where we map the master equation to a Liouvillian equation of the
form

$\frac{d\vec{\rho}}{dt}=L(t)\vec{\rho},$ (4)

where $\vec{\rho}$ is the vectorized form of $\rho$ and $L(t)$ is a
Liouvillian superoperator in the corresponding matrix form. This can in turn
be written as

$L(t)=L_{0}+\beta\,\Omega_{\text{RF}}\,\Delta L\,\cos(\Omega_{\text{RF}}t),$
(5)

where $L_{0}$ is the Liouvillian for the system without modulation, and
$\Delta L$ is a $8^{2}\times 8^{2}$ matrix with constant elements [45].

For the trivial case where $\beta=0$, i.e. without micromotion, the steady
state can be found by inverting $L_{0}$ with a normalization constraint over
$\vec{\rho}$. We calculate the atomic spectrum by plotting, as a function of
the detuning $\Delta_{\text{IR}}$, the sum of the populations of the two
excited P states
$\rho_{{}_{\text{P}_{+1/2}\text{P}_{+1/2}}}+\rho_{{}_{\text{P}_{-1/2}\text{P}_{-1/2}}}$,
which is proportional to the fluorescence of the ion. The plot of Fig. 1(d)
was calculated in this way, and exhibits the four expected dark resonances.

If $\beta\neq 0$, the atomic populations will oscillate around a mean value
with frequency components that are multiples of $\Omega_{\text{RF}}$. This
will be reflected in the spectrum as modulation sidebands displaced from the
original dips by integer multiples of $\Omega_{\text{RF}}$. To obtain the
time-averaged value of the populations, we propose a solution for times longer
than all transient times of the system [41] with the form

$\vec{\rho}(t)=\sum_{m=-\infty}^{+\infty}\vec{\rho}_{m}e^{im\Omega_{\text{RF}}t}.$
(6)

When introducing this ansatz into Eq. (4), we get the recursive relation

$(L_{0}+in\Omega_{\text{RF}})\vec{\rho}_{n}+\frac{\beta\Omega_{\text{RF}}}{2}\Delta
L(\vec{\rho}_{n+1}+\vec{\rho}_{n-1})=0.$ (7)

We are interested in obtaining $\vec{\rho}_{0}$, i.e. the mean value of
$\vec{\rho}(t)$. To do so, we define two operators of the form

$S^{+}_{n-1}=-\left(L_{0}-in\Omega_{\text{RF}}+\frac{\beta\Omega_{\text{RF}}}{2}\,\Delta
L\,S_{n}^{+}\right)^{-1}\frac{\beta\Omega_{\text{RF}}}{2}\Delta L,$ (8)
$S^{-}_{n+1}=-\left(L_{0}-in\Omega_{\text{RF}}+\frac{\beta\Omega_{\text{RF}}}{2}\,\Delta
L\,S_{n}^{-}\right)^{-1}\frac{\beta\Omega_{\text{RF}}}{2}\Delta L.$ (9)

It is straightforward to see that $\vec{\rho}_{n+1}=S_{n}^{+}\vec{\rho}_{n}$
for $n\geq 0$ and $\vec{\rho}_{n-1}=S_{n}^{-}\vec{\rho}_{n}$ for $n\leq 0$, so
the defined operators act as raising and lowering operators respectively, in
the sense that they relate components that oscillate at neighbouring
frequencies.

In terms of these operators, we can evaluate Eq. (7) for $n=0$ and obtain

$\left(L_{0}+\frac{\beta\Omega_{\text{RF}}}{2}\,\Delta
L\,(S_{0}^{+}+S_{0}^{-})\right)\vec{\rho}_{0}=0.$ (10)

Finally, to calculate $S_{0}^{+/-}$ we truncate the sum in Eq. (6) such that
$\vec{\rho}_{\pm N_{max}}=0$. This is equivalent to saying that we will
include only up to $N_{max}-1$ modulation sidebands in the spectrum. The
appropriate value of $N_{max}$ depends on the value of $\beta$: for larger
micromotion, a higher number of sidebands will appear, and a higher $N_{max}$
will be required to properly describe the atomic spectra. One can choose
$N_{max}$ by imposing that the Bessel function $J_{N}(\beta)$ is negligible
for $N>N_{max}$. Nevertheless, setting $N_{max}\sim\beta$ is sufficient for
all practical considerations. For all fits in this work, we set $N_{max}=5$,
since $\beta$ never surpassed that value in our experiments.

###  III.3 Estimation of the modulation parameter

In Figs. 2(a)-(f) we plot, in solid line, a fit to the model for the six
measured spectra, which are shown in green dots along with their
uncertainties. The fits are in very good agreement with the measurements,
evidencing the robustness of the model to predict how micromotion affects the
atomic spectra. From them, we retrieve a value for the modulation factor
$\beta$ for each voltage, as well as the temperature $T$ of the ion as
explained in the next section.

In Fig. 2(h) we show how $\beta$ depends on the endcap voltage
$V_{\mathrm{ec}}$. There is an optimal point at $-0.55(1)~{}$V in which
$\beta$ is minimum, but not zero. This is due to extra stray fields not
compensated in directions that are orthogonal to the one of the endcaps.

Considering that the ion moves along a trajectory that misses the center of
the trap while sweeping $V_{\mathrm{ec}}$ by an amount of $y$, that its
position varies linearly with it, as it is shown in Fig. 2(g), following
$(V_{\mathrm{ec}}-V_{0})\alpha_{V}$ and the proportionality of $\beta$ with
the distance $r$ to the center ($\beta=\alpha_{\beta}\,r$), the data can be
modeled by a hyperbola function of the form

$\beta(V)=\alpha_{\beta}\,\sqrt{\alpha_{V}^{2}\,(V-V_{0})^{2}+y^{2}}\;\;\;.$
(11)

Bringing the $\alpha_{\beta}$ inside the square root, the four parameters in
the equation above can be grouped and reduced to three in the regression
model. The fitted model is plotted in a solid blue line over the experimental
data in Fig. 2(h), showing good agreement with the data. From this fit we
extract the minimum value of $\beta$ which is $\beta_{min}=0.30(5)$.
Furthermore, in Fig. 2(i) we plot the residuals of the fit, showing a random
dispersion around zero.

##  IV Micromotion and thermal effects

###  IV.1 Inclusion of thermal effects in the spectrum

We incorporate the temperature $T$ of the ion in the model following the
approximation presented in [46], where the effect is accounted for as an
additional dephasing term proportional to the square root of $T$, which is
added as an extra broadening of the laser linewidths

$\Gamma_{D}=\left|\vec{k}_{\text{IR}}-\vec{k}_{\text{UV}}\right|\sqrt{\frac{k_{B}T}{2m}}.$
(12)

One can understand this approximation the following way. The Doppler effect
generates a shift proportional to $kv$. As we are analyzing a two photon
transition, we replace $k$ for the difference
$\left|\vec{k}_{\text{UV}}-\vec{k}_{\text{IR}}\right|$. Because we are
considering a thermal state, $v$ fluctuates with a standard deviation
$\sqrt{k_{B}T/m}$. This is modelled through stochastic shifts in the frequency
of the lasers.

This method implies then two important approximations, the first by only
considering two-photon processes, and the second by approximating a Gaussian
distribution by a Lorentzian one. The $\sqrt{2}$ in the denominator of Eq.
(12) was chosen because it was found to provide the best approximation. We
note, however, that the proportionality factor leading to the best results is
generally dependent on the temperature regime of interest as well as on the
relevant linewidths. A more detailed analysis of this is beyond the scope of
this paper and will the subject of a future publication. Finally, to include
$\Gamma_{D}$ in the model, we multiply it by
$k_{i}/(k_{\text{UV}}+k_{\text{IR}})$, where $i=\\{\text{UV},\text{IR}$}, and
add it to the corresponding linewidth of the laser.

Following this procedure, the ion temperature $T$ becomes a parameter that we
can retrieve from the fits. The temperatures estimated from the spectra of
Fig. 2 are plotted as a function of the modulation factor $\beta$ in Fig. 3.
The plot shows a temperature that is approximately constant and close to the
Doppler limit for $\beta<1$. For larger micromotion, the temperature exhibits
a significant boost upon increasing $\beta$. To understand this behavior, we
now characterize the RF heating that the ion experiences and add this effect
to the equation describing the thermalization of the ion.

Figure 3:  Relationship between temperature and micromotion modulation factor
$\beta$. The data inferred from the spectra in Fig. 2 are depicted by red dots
along with uncertainty intervals (derived from the standard error of the
fitted parameters in Fig. 2). Two models have been tested to describe this
relationship: one excluding RF heating (orange dashed line) and another
incorporating it (blue straight line). Further details about each model can be
found in the main text. The estimated temperatures for low micromotion are
consistent with the Doppler limit for detuning of around -10 MHz, represented
by the gray shaded area in the graph.

###  IV.2 Thermalization in the presence of micromotion

When the ion is close to the Doppler temperature, the equation determining the
energy balance under laser cooling is

$\dot{E}=\dot{E}_{\mathrm{Dop}}+\dot{E}_{\mathrm{rec}}+\dot{E}_{\mathrm{RF}},$
(13)

where $\dot{E}_{\rm Dop}$ represents the Doppler cooling mechanism due to the
interaction of the atom with the lasers, $\dot{E}_{\rm rec}$ is the recoil
heating due to the random emission of one photon of the atom, and
$\dot{E}_{\mathrm{RF}}$ is the RF heating. In the following, we show how to
calculate the three terms separately.

####  IV.2.1 Doppler cooling

Doppler cooling works as a dissipation channel [47]. It acts on the ion
through a force of the form

$\vec{F}_{d}=\hbar\vec{k}\Gamma\rho_{ee}(\Delta,\beta),$ (14)

where $\vec{k}$ is the wave vector of the Doppler cooling laser, $\Gamma$ is
the linewidth of the cooling transition, and $\rho_{ee}$ is the excited state
population, that is modulated by the micromotion. We will approximate the
dynamics by a two-level system corresponding to the S$-$P transition which is
the most relevant for the Doppler cooling. Thus, all laser parameters in this
subsection will correspond to the UV laser.

In the low saturation limit we can express the excited state population using
Bessel functions $J_{n}$ as [20]

$\rho_{ee}(\Delta,\beta)=\left(\frac{\mu
E_{0}}{2\hbar}\right)^{2}\sum_{n}\frac{J_{n}^{2}(\beta)}{(\Delta+n\Omega_{\text{RF}})^{2}+(\Gamma/2)^{2}},$
(15)

where $\mu$ is the electric dipole moment of the transition and $E_{0}$ is the
electric field of the laser. To include the motion of the atom, we will
consider low velocities $\vec{v}$ such that
$|\vec{k}\cdot\vec{v}|\ll|\Delta|$, where $\vec{k}$ is the wave vector of the
cooling laser. We can then approximate

$\rho_{ee}(\Delta+\vec{k}\cdot\vec{v},\beta)\sim\rho_{ee}(\Delta,\beta)+\frac{d\rho_{ee}(\Delta,\beta)}{d\Delta}(\vec{k}\cdot\vec{v}).$
(16)

Considering that $\langle\vec{v}\rangle=0$ since the atom is trapped in a
harmonic potential, the cooling power turns to be

$\dot{E}_{\mathrm{Dop}}=\langle\vec{F}_{d}\cdot\vec{v}\rangle=-\hbar|\vec{k}|^{2}\Gamma\frac{d\rho_{ee}(\Delta,\beta)}{d\Delta}\langle
v_{\hat{k}}^{2}\rangle,$ (17)

where $v_{\hat{k}}$ is the velocity of the ion projected in the direction of
$\vec{k}$. Finally, assuming a thermal state we relate the mean value of the
squared of the velocity with the temperature as $\frac{1}{2}m\langle
v_{\hat{k}}^{2}\rangle=\frac{1}{2}k_{B}T$. In this form, we can write
$\dot{E}_{\mathrm{Dop}}$ in terms of the temperature of the ion.

####  IV.2.2 Recoil heating

Recoil heating is a mechanism that balances Doppler cooling leading to a non-
vanishing temperature limit. It is due to the spontaneous emission of a photon
when the atom is excited, which causes a recoil kick of the atom in a random
direction. The rate of heating is proportional to the energy of the emitted
photon, the decay rate $\Gamma$ and the emission probability linked to the
excited state population $\rho_{ee}$, resulting in

$\dot{E}_{\mathrm{rec}}=\frac{(\hbar|\vec{k}|)^{2}}{2m}\Gamma\rho_{ee}(\Delta,\beta).$
(18)

####  IV.2.3 RF heating

The RF heating mechanism is given by electric noise at frequency components of
the driving field in $\Omega_{\text{RF}}\pm\omega$, where $\omega$ is a
secular frequency of the ion. This causes a periodic ponderomotive noise force
at $\omega$ that adds extra heating. In [38], the resulting RF heating rate
was shown to scale as

$\dot{E}_{\mathrm{RF}}\sim\left[\vec{\nabla}\left(\vec{E}^{2}(\vec{r})\right)\right]^{2}\sim\beta^{2}\,.$
(19)

Here, we are using that the electric field $\vec{E}$ is approximately linear
in the ion displacement from the trap center, and that the modulation factor
$\beta$ is also proportional to this displacement. We can then express the
rate for RF heating as

$\dot{E}_{\mathrm{RF}}=C_{\mathrm{RF}}\beta^{2},$ (20)

where $C_{\mathrm{RF}}$ is a factor that depends on experimental parameters
like the amount of electric noise. Therefore, it is a quantity that must be
estimated from the measurements.

####  IV.2.4 Stationary temperature

Considering all the above contributions one can find the stationary value of
$T$ by setting $\dot{E}=0$. The expression for the temperature then becomes

$T_{0}=\,\frac{\hbar}{k_{B}}\frac{\rho_{ee}(\beta,\Delta)}{\dfrac{\partial\rho_{ee}}{\partial\Delta}(\beta,\Delta)}+\frac{C_{\mathrm{RF}}}{\left(\dfrac{\hbar|\vec{k}|^{2}\Gamma
k_{B}}{m}\right)}\,\frac{\beta^{2}}{\dfrac{\partial\rho_{ee}}{\partial\Delta}(\beta,\Delta)}.$
(21)

If we set $\beta=0$ we obtain $T_{0}=\frac{\hbar\Gamma}{2k_{B}}$, which
corresponds to the well-known Doppler limit temperature.

In Fig. 3 we show a fit to Eq. (21) in solid blue line, leaving
$C_{\mathrm{RF}}$ and $\Delta$ as fitting parameters. For comparison, we also
include in orange dashed line a fit to the model without RF heating, i.e.
setting $C_{\mathrm{RF}}=0$ and leaving $\Delta$ as the only fitting
parameter. From this last fit, we obtain $\Delta=2\pi\times-5.1(8)$ MHz.
However, this model does not properly fit the measurements for $\beta>1$. In
particular, neglecting RF heating leads to the prediction of a temperature
decrease when $\beta$ becomes larger than $\sim 2.4$, in clear contradiction
with the experiments.

We can see that the inclusion of the RF heating term captures the increasing
behavior of $T$ with increasing $\beta$, in good agreement with the
measurements. From this fit we retrieve $\Delta=2\pi\times(-11.7)(17)$ MHz,
consistent with the detuning used to Doppler cool the ion before the
measurements.

On the other hand, we obtain $C_{\mathrm{RF}}=8.4(32)\times 10^{-20}$ J/s. We
can rewrite it in terms of a phonon heating rate of one motional mode of the
ion [16]. Considering one of the radial modes of the ion, with
$\omega=2\pi\times 1~{}$MHz, the heating rate becomes

$\dot{n}_{\mathrm{RF}}=0.20(7)\times\beta^{2}~{}\mathrm{ph/s}.$ (22)

This result is in good agreement with other similar measurements of RF heating
effects [25].

##  V Multi-ion spectra

In a final phase of this work, we measure spectra for a system of two ions.
The setup used collects the light of all the trapped ions globally on a photo-
multiplier tube, not allowing to distinguish each one individually, but
allowing fast measurements with low background. The spectra are therefore
composed of the light emitted by all ions. As each ion is in a different
position in the trap, they will experience different micromotion, and
therefore contribute differently to the summed spectrum. We aimed at
extracting the modulation factor and temperature of each ion from the global
signal.

Figure 4:  Dark resonance spectrum acquired by collecting light from two
trapped ions. The data are compared with the fitted model for one (red line)
and two (dashed-black line) ions. The $\beta$ obtained in each regression are
reported in their respective colors.

In Fig. 4 we show one typical spectrum for two ions and different model fits.
We find that a two-ion model, with an independent modulation factor $\beta$
for each one, adequately fits the data, but the retrieved parameters are very
sensitive in their initial guess. Moreover, we see that a one-ion model, with
only one $\beta$, can also produce reasonable fits. Also, we studied the
behaviour of the fitted modulation factors as a function of the position of
the ion crystal in the trap, and found that the results obtained were also
very sensitive to the initialization of the parameters.

We conclude that extracting independent modulation parameters or temperatures
for multiple ions from a single signal is an ill-defined problem, with too
many free parameters. For an appropriate measurement, independent collection
of each ion’s fluorescence would be necessary. This has to be combined with
the short collection times (50 $\mu$s), which ensures that the temperatures of
the ions are marginally affected by the measurement. Such a procedure is
rather challenging with electron multiplying charge-coupled devices (EMCCDs)
or complementary metal–oxide–semiconductor (CMOS) cameras, but multi-pixel
photomultiplier tubes, avalanche photodiode detectors or photon counting
cameras could provide useful alternatives.

##  VI Conclusions

We have analyzed the effects of micromotion in dark-resonance atomic spectra.
We presented experimental results obtained with a single trapped calcium ion
for which the amplitude of the RF-driven motion was varied applying a
controllable voltage to one trap electrode. We introduced a theoretical model
that accurately describes the corresponding spectra, and used it to fit our
measurements. From these fits, we were able to retrieve the micromotion
modulation factor of each curve.

The same fits were used to estimate the temperature of the ion under different
micromotion amplitudes. Our results evidence the presence of a micromotion-
induced heating mechanism, also known as RF heating. We include this
phenomenon in the standard description of Doppler cooling and show that it
leads to good predictions of the final temperature of the ion depending on the
value of the micromotion modulation parameter.

These methods could be used to study heat transport in trapped ion systems
where excess micromotion is unavoidable due to crystal geometry, for example
2D crystals in linear traps. However, we note that a proper description of the
dynamics of several ions requires single-ion detection. Indeed, our studies of
the spectra obtained by collecting the light emitted by two ions together
indicate that global light collection is not sufficient for a good
characterization of the motional state.

## Data availability statement

The original contributions presented in the study are included in the article;
further inquiries can be directed to the corresponding author.

## Author contributions

CTS and NANB were responsible for planning the research objectives an goals.
NANB, ML and MB were responsible for carrying out the experiments and for the
analysis of the measured data. NANB developed the numerical models together
with MB and CC. The theoretical methods were supervised by CTS and CC. ML was
in charge of producing the final figures. NANB was in charge of writing the
article, with contributions from all authors who approved the submitted
version.

## Funding

NANB, MB, ML and CTS acknowledge support for grants PICT2018-03350,
PICT2019-04349 and PICT2021-I-A-01288 from ANPCyT (Argentina) and grant UBACYT
2018 Mod I - 20020170100616BA from Universidad de Buenos Aires. CC
acknowledges funding from grant PICT 2020-SERIEA-00959 from ANPCyT
(Argentina).

## Acknowledgements

We acknowledge the contribution of M. Drechsler in the building of the ion
trap and early experiments. We thank F. Schmidt-Kaler for his invaluable
support and generosity, as well as J. P. Paz for his unconditional help in the
setting up of the laboratory. Finally, we thank F. Meconi for taking the photo
of Fig. 1(a).

## Conflict of interest

The authors declare that the research was conducted in the absence of any
commercial or financial relationships that could be construed as a potential
conflict of interest.

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