# All-optical measurement of magnetic fields for quantum gas experiments

Suthep Pomjaksilp  Department of Physics and Research Center OPTIMAS,
Rheinland-Pfälzische Technische Universität Kaiserslautern-Landau, 67663
Kaiserslautern, Germany  0009-0004-1162-801X Sven Schmidt  Department of
Physics and Research Center OPTIMAS, Rheinland-Pfälzische Technische
Universität Kaiserslautern-Landau, 67663 Kaiserslautern, Germany
0009-0000-5626-2630 Aaron Thielmann  Department of Physics and Research Center
OPTIMAS, Rheinland-Pfälzische Technische Universität Kaiserslautern-Landau,
67663 Kaiserslautern, Germany  0009-0000-0799-9927 Thomas Niederprüm
Department of Physics and Research Center OPTIMAS, Rheinland-Pfälzische
Technische Universität Kaiserslautern-Landau, 67663 Kaiserslautern, Germany
0000-0001-8336-4667 Herwig Ott  Department of Physics and Research Center
OPTIMAS, Rheinland-Pfälzische Technische Universität Kaiserslautern-Landau,
67663 Kaiserslautern, Germany  Corresponding author<EMAIL_ADDRESS>0000-0002-3155-2719

We present an all-optical method to measure and compensate for residual
magnetic fields present in a cloud of ultracold atoms trapped in an optical
dipole trap. Our approach leverages the increased loss from the trapped atomic
sample through electromagnetically induced absorption. Modulating the
excitation laser provides coherent sidebands, resulting in $\Lambda$-type
pump-probe scheme. Scanning an additional magnetic offset field leads to pairs
of sub-natural linewidth resonances, whose positions encode the magnetic field
in all three spatial directions. Our measurement scheme is readily implemented
in a typical quantum gas experiments and has no particular hardware
requirements.

## Introduction

Magnetic field measurement and calibration is a cornerstone in quantum gas
experiments. It allows for precision control of the atomic Zeeman levels,
needed in a large variety of applications, like Raman transitions, spinor
gases [1], spin dynamics and entanglement [2], Feshbach resonances [3], or
metrology [4, 5]. Traditional methods such as fluxgate magnetometers [6, 7],
or high-end methods such as NV centers [8] or SQUID sensors [9, 10] can only
measure the field outside the vacuum chamber and microwave equipment, which
allows for the precise in-situ measurement of magnetic fields is not always
available. Atomic magnetometers [11, 12, 13] use magneto-optical effects to
measure the spin precession in a magnetic field. They are close to the
application discussed here, however, they are often optimized for vapor cell
applications. In a quantum gas experiment, atomic magnetometry is
straightforward to implement since the necessary components, such as lasers,
atom detection and magnetic field management are readily available. It is
therefore possible to adapt the core principles of atomic magnetometers to the
requirements and constraints of ultracold atom experiments.

Here, we demonstrate an all-optical technique to measure a magnetic field
vector in an ultracold atom experiment without prior calibration. Our method
is easily applicable in most quantum gas and optical tweezer setups. The
method relies on electromagnetically induced absorption (EIA) [14, 15] on the
optical cooling transition in combination with fast chopping of the excitation
laser. The latter creates coherent sidebands at a well defined frequency.
Spectroscopy is performed by scanning an additional magnetic field in all
three spatial dimensions. Thereby, the carrier and the sidebands can be
brought into two-photon resonance with neighboring Zeeman sublevels in a
$\Lambda$ type setting. The resonance condition is signaled by an increased
absorption due to EIA and corresponding peaks in the spectra. The measurement
signal can be any experimentally accessible observable, which depends on the
optical absorption, e.g. the fluorescence or the sample lifetime. Our method
does not require any microwave field and a single laser beam is, in principle,
sufficient, to measure the field.

Figure 1:  We probe the atomic sample with two counter-propagating beams
(dashed black arrows) having circular polarization of opposite handedness.
Three pairs of coils in Helmholtz configuration provide a constant magnetic
offset field $\vec{B}=(B_{x},B_{y},B_{z})$. The probing beams are not aligned
to any of the coil axis. The field $|\vec{B}|$ splits the Zeeman sates of the
involved $5\text{S}_{1/2}\text{F}=2$ and $5\text{P}_{3/2}\text{F'}=3$ levels.

In the following, we first introduce the optical tweezer setup, which we use
in our experiment. We then introduce the general measurement scheme, the
underlying theory and discuss the obtained spectra. We close with a discussion
of possible applications, extensions and further developments.

## Experimental Setup

To demonstrate the method, we use mesoscopic atomic ensembles of rubidium
atoms trapped in optical tweezers. In brief, we load ^87Rb atoms from the
background gas into a 6-axis magneto-optical-trap (MOT), located in a ultra-
high vacuum glass cell. The optical tweezers (waist
$w_{0}\approx$2\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ at
$1064\text{\,}\mathrm{nm}$) are directly loaded from the MOT. We typically
trap 10-20 atoms in a single optical tweezer. An $\text{NA}=$0.4$$ objective,
which is used for the creation of the optical tweezers, is also used to
collect fluorescence light from the atomic sample both on the D1 and D2 line.
Our setup has an overall photon detection efficiency of approximately
$2\text{\,}\mathrm{\char 37\relax}$. The fluorescence imaging is essentially
background free ($<0.01$ photons/second/pixel). For the spectroscopy, we use a
separate set of co-axial counter-propagating beams, which are resonant to the
D2 cooling transition $\text{F}=$2$\rightarrow\text{F'}=$3$$ of ^87Rb. Both
beams are switched by an acousto-optical modulator (AOM). See Fig. 1 for a
sketch of the laser beam geometry and the relevant level scheme.

(a)

(b)

Figure 2:  (a) Alternating sequence of trapping (top) and probing (bottom) the
sample. Modulating the probe beam with a modulation frequency $\nu$ and a duty
cycle $d=0.5$ leads to the formation of AM sidebands. Using a smaller duty
cycle for the optical dipole trap $d=$0.4$$ omits trap-induced lightshifts
during probing. (b) Measured frequency spectrum of the probing beam for
$\nu=$500\text{\,}\mathrm{kHz}$$ and $d=$0.5$$ revealing sidebands with
frequency offsets $(2n+1)\nu\;\forall n\in\mathbb{N}_{0}$. $S$ is normalized
to the spectral power of the carrier.

For the spectroscopy, we chop the probing beams and the optical tweezers with
a frequency of $\nu=$500\text{\,}\mathrm{kHz}$$ and a phase shift of $\pi$
with respect to each other (see Fig. 2(a)). A duty-cycle of
$d=\tau_{\mathrm{on}}\nu=$0.5$$ for the probe beams and $0.4$ for the optical
tweezer ensures that the probe beams interact with free atoms without any
residual trapping light and resulting lightshifts present. The modulation of
the probing beams leads to the formation of coherent sidebands at frequencies
$\pm(2n+1)\nu\;\forall n\in\mathbb{N}_{0}$ as shown in Fig. 2(b). We typically
apply $250$ pulses of the probe beams, before we measure the fluorescence of
the remaining atoms on the D1 line. As the probe beams are resonant to the
atomic transition, photon scattering leads to a heating of the atomic sample
and a loss of atoms. We model the time evolution of the atom number
$\overline{N}(\tau)$ with an exponential decay

$\overline{N}(\tau)=\overline{N_{0}}\exp(-\gamma\tau)$ (1)

where $\overline{N}$ denotes the mean number of trapped atoms and $\gamma$ the
loss rate. Fig. 3 shows typical decay curves of the atom number. The setup is
completed by three orthogonal pairs of coils in Helmholtz configuration around
the science cell, which can be used to apply external homogeneous magnetic
offset fields during the probing sequence.

Figure 3:  Mean D1 fluorescence signal count plotted against the probing
duration $\tau$. The dashed lines shown an exponential fit to the data. The
three curves corresponds to the three vertical lines in Fig. 5(a).

## Electromagnetically Induced Absorption via Optical Sidebands

The atomic sample is initially prepared in the $\text{F}=$2$$ hyperfine ground
state for which the Zeeman levels split according to $\Delta
E=m_{F}g_{F}\mu_{\text{B}}|\vec{B}|$ with $g_{F}=1/2$ in a constant magnetic
field $\vec{B}=(B_{x},B_{y},B_{z})$. For typical ambient fields, this
splitting is well below the linewidth of the optical probing transition and
direct spectroscopic measurement of the magnetic field is impossible. However,
in three-level systems interferences between excitation pathways can suppress
the admixture of the short-lived excited state and thus allow for a
spectroscopic resolution below the linewidth of the optical transition.

Figure 4:  Schematics of the $\Lambda$ scheme used in the experiment (top).
Exemplarily, the carrier of the probe beam drives the $\pi$ transition to the
upper hyperfine manifold whereas the sidebands couple to a neighboring $m_{F}$
state with $\sigma^{\pm}$ transitions. Additional EIA excitation pathways with
the same initial and final state are possible but not shown for clarity. The
bottom half shows the expected absorption spectrum. The left peak depicts the
situation where the external field $B_{x}$ brings the $\Delta m_{F}=\pm 1$
level in resonance with the $\pm\nu$ sideband according to Eq. 2. On the
right, the $\Delta m_{F}=\pm 1$ level is in resonance with the $\mp\nu$
sideband. Both lead to an enhanced absorption via EIA. The center peak only
occurs if the orthogonal field components $B_{\perp}$ vanish such that all
Zeeman levels are degenerate.

Here we use EIA in a $\Lambda$ scheme as depicted in Fig. 4, where the pump
and probe lasers are created by the carrier and the sidebands of the
amplitude-modulated excitation laser. Since the atoms are illuminated by two
counter-propagating beams with circular polarization every magnetic transition
between the Zeeman levels in the upper F’=3 and the lower F=2 manifold is
possible as long as the quantization axis is along one of the three coil axes.
While the two-photon transition in principle allows for ($\sigma^{\pm}$,
$\sigma^{\mp}$) transitions with $\Delta m_{F}=\pm 2$ and ($\pi$,
$\sigma^{\pm}$) transitions with $\Delta m_{F}=\pm 1$, a summation over the
relevant Clebsch-Gordon coefficients reveals that the latter is up to a factor
of eight stronger than the former. Therefore, it is possible to observe
increased photon absorption due to EIA if the two-photon detuning vanishes,
$\delta=\frac{g_{F}\mu_{\text{B}}}{h}-\nu=0$, i.e. if the magnetic energy
splitting between adjacent Zeeman states matches the modulation frequency
creating the sidebands

$h\nu=\pm g_{F}\mu_{\mathrm{B}}|\vec{B}|$ (2)

As depicted in Fig. 4, different pathways generated by the sidebands come into
resonance depending on the magnetic splitting of the Zeeman levels. Since EIA
also stays effective for small one-photon detunings
($\Delta<\Gamma_{\mathrm{5P}}$), the magnetic splitting of the excited state
Zeeman levels can be neglected and only becomes relevant if a detailed
understanding of the observed line strength is necessary. The increased
absorption of photons at $\delta=0$ directly leads to an increased heating-
induced loss rate $\gamma$ from the trap and comprises an easily accessible
measurement signal that allows to characterize the external magnetic field.

For the magnetic field, we consider a system of 3 pairs of coils which create
a magnetic field with mutually perpendicular components

$\displaystyle B_{n}$ $\displaystyle=\alpha_{n}I_{n}$ (3)

at the position of the atoms. Here $I_{n}$ is the current through the
respective coil pair and $\alpha_{n}$ is the system specific conversion factor
between current and magnetic field. We further introduce an unknown offset
field
$\vec{B}^{\mathrm{res}}=(B_{x}^{\mathrm{res}},B_{y}^{\mathrm{res}},B_{z}^{\mathrm{res}})$.
For convenience, we express the offset field with help of the corresponding
current in the coils $B_{n}^{\mathrm{res}}=\alpha_{n}I_{n}^{0}$. The modulus
of the magnetic field is then given by

$|\vec{B}|=\sqrt{\alpha_{x}^{2}(I_{x}^{2}+I_{x}^{0})^{2}+\alpha_{y}^{2}(I_{y}^{2}+I_{y}^{0})^{2}+\alpha_{z}^{2}(I_{z}^{2}+I_{z}^{0})^{2}}$
(4)

Combining Eq. 2 and Eq. 4, we obtain the following relation for the appearance
of the EIA peaks

$\displaystyle\left(\frac{h\nu}{g_{F}\mu_{\text{B}}}\right)^{2}$
$\displaystyle=\alpha_{x}^{2}(I_{x}-I_{x}^{0})^{2}+\alpha_{y}^{2}(I_{y}-I_{y}^{0})^{2}+\alpha_{z}^{2}(I_{z}-I_{z}^{0})^{2}$
(5)

To experimentally probe these field-induced EIA resonances, we create a
varying magnetic field in one of the Cartesian directions, e.g. the
x-direction, while setting $I_{y}=I_{z}=0$. For each applied magnetic field,
we measure the atom loss from the sample as a function of the excitation pulse
duration and extract the decay constant $\gamma$ (Fig. 3). Plotting the decay
rates as a function of the applied magnetic field reveals the spectrum shown
in Fig. 5(a). Except for an overall slope the measured spectrum shows the
expected peak structure from Fig. 4. The observed peaks correspond to the
$(\pi,\sigma^{\pm})$ transitions, while the strongly suppressed
$(\sigma^{\pm},\sigma^{\mp})$ resonances remain hidden in the noise floor of
the measurement. The central peak from Fig. 4 only appears if also the
orthogonal field components $B_{y}=B_{z}\approx 0$ approximately vanish. This
is only the case if the offset field is small in these two directions or
already compensated. Still, the symmetry of the spectrum allows to identify
the current required to compensate the offset field, i.e. $I_{x}=-I_{x}^{0}$
in the x direction, by measuring the center between the two first side peaks.

(a)

(b)

(c)

Figure 5:  (a) Measured sample loss rate $\gamma$ while scanning the magnetic
field in x-direction and keeping the other respective coil currents fixed. We
determine the positions of the peaks by fitting the spectrum with a multi
Lorentzian model (gray line). The dotted red line indicates $I_{x}^{0}$
whereas the solid red lines mark the peak positions used to determine $I_{x}$.
The sample decays for $+$, $-$ and $\ast$ are shown in Fig. 3. (b), (c) The
remaining measurements for the field directions y and z.

Compensating the field in the x-direction and conducting an analogue
measurement for the y-direction yields the peak structure shown in Fig. 5(b).
After compensating the field in the y-direction, the spectrum for the
z-direction shown in Fig. 5(c) differs in that a central peak in between the
pair of symmetric EIA resonances emerges. This feature corresponds to a
situation where the external fields are zeroed and all Zeeman sublevels are
degenerate, as previously observed in [16, 17, 14]. In this situation, the
carrier alone is able to drive all allowed transitions between
$\text{F}=2\rightarrow\text{F'}=3$, greatly increasing the sample absorption
and loss rate. Note that this feature is only visible in the measurement in
z-direction since we deliberately compensated the field in the orthogonal
directions by setting $I_{x}=-I_{x}^{0}$ and $I_{y}=-I_{y}^{0}$ through the
values extracted from the measurements in x- and y-direction.

With the information on the offset currents $I_{n}^{0}$, we can apply Eq. 5,
for the peaks $I_{n}^{\pm}$ appearing in the three measurements and obtain a
system of three linear equation which we then solve for the conversion
coefficients $\alpha_{x}$, $\alpha_{y}$ and $\alpha_{z}$. Finally, we
determine the residual magnetic fields in the experiment as

$|B_{n}^{\text{res}}|=|\alpha_{n}I_{n}^{0}|$ (6)

from the centers of the peak structures.

Table 1:  Measured values of the current to field conversion factor $\alpha$ and residual magnetic field $B^{\text{res}}$ for each coil pair. For comparison of the conversion factors, we list the conversion factors $\alpha_{\mathrm{sim}}$ simulated for our coil geometry. To compare the measured residual magnetic field, we list the earth magnetic field $B^{\mathrm{earth}}$ at the position of our lab according to the world magnetic model [18]. The uncertainties are calculated from the covariance of the fitted peaks. position  axis |  $|\alpha|$ $($\mathrm{mG}\text{\,}{\mathrm{A}}^{-1}$)$ |  $|\alpha_{\text{sim}}|$ $($\mathrm{mG}\text{\,}{\mathrm{A}}^{-1}$)$ |  $|B^{\text{res}}|$ $($\mathrm{mG}$$) |  $|B^{\text{earth}}|$ $($\mathrm{mG}$$)
---|---|---|---|---
x | $562(17)$ | $680$ | $498(16)$ | $208$
y | $530(17)$ | $580$ | $91(3)$ | $6$
z | $10\,061(17)$ | $9060$ | $488(5)$ | $482$

As an example, we show the conversion factors $\alpha_{n}$ and residual fields
$B_{n}^{\text{res}}$ of our setup in Tab. Tab. 1. While the residual field in
the z-direction is given by the earths magnetic field, the deviation in the x-
and y-direction can be qualitatively explained by a nearby ion-getter pump
located $0.5\text{\,}\mathrm{m}$ beside the science chamber that creates a
magnetic field mainly in the horizontal plane. For comparison of the extracted
conversion coefficients, we calculated the conversion factors
$\alpha_{\text{sim}}$ by numerically integrating Biot-Savart’s law for our
coil geometry. The measured values show an overall good agreement with the
simulated values and deviate by less than $20\text{\,}\mathrm{\char
37\relax}$. The mismatch can be attributed to the position uncertainties,
small remanent magnetic fields of the setup and imperfections in the coil
manufacturing.

## Discussion

This work demonstrates a technique to measure magnetic fields in a quantum gas
experiment without prior calibration of the coil system. The possibility to
determine the conversion factors $B_{n}=\alpha_{n}I_{n}$ on the fly
distinguishes our method from magnetometers based on the Hanle effect while
providing a comparable measurement accuracy of
$\approx\pm$8\text{\,}\mathrm{mG}$$ [13, 12]. While competing methods, which
work without prior coil calibration, for example microwave spectroscopy, show
a higher accuracy, they typically need additional equipment. In contrast, our
approach needs in essence only resonant laser light and beam chopping.
Preliminary investigations show that the symmetric structure of EIA peaks in
Fig. 5(a) can even be produced with only one modulated beam. Our method is
also flexible with respect to the detection scheme. In principle, any signal
that is dependent on the photon scattering rate, such as the sample loss rate,
the fluorescence or the transparency of the sample can serve as an observable.
The simplicity and the independence of a prior calibration makes the technique
readily available in a typical quantum gas experiments. Finally, since it is
intrinsically Doppler-free in a single beam, our scheme might also find
applications in vapor cell magnetometers, using a single probe beam while
directly monitoring its transmitted intensity. The measurement speed is then
only limited by the time it takes to perform the magnetic field sweeps.

##  1 Data availability

The data that support the plots within this paper and other findings of this
study are available from the corresponding author upon request.

##  2 Code availability

The code that support the simulations within this paper and other findings of
this study are available from the corresponding author upon request.

##  3 Acknowledgements

We acknowledge financial support by the DFG within the collaborative research
center TR185 OSCAR (number 277625399).

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##  4 Author Contribution

S.P., S.S. and A.T. performed the experiments. S.P. and S.S. analysed the
data. S.P. prepared the manuscript. H.O. and T.N. supervised the experiment.
All authors contributed to the data interpretation and manuscript preparation.

##  5 Competing financial interests

The authors declare no competing financial interests.