# Towards a zero magnetic field environment for ultracold atoms experiments

C. Rogora1 R. Cominotti1 C. Baroni1 D. Andreoni1 G. Lamporesi1,2
<EMAIL_ADDRESS>A. Zenesini1,2 G. Ferrari1,2 1Pitaevskii BEC
Center, CNR-INO and Dipartimento di Fisica, Università di Trento, 38123
Trento, Italy  2Trento Institute for Fundamental Physics and Applications,
INFN, 38123 Trento, Italy

###### Abstract

The minimization of the magnetic field plays a crucial role in ultracold gas
research. For instance, the contact interaction dominates all the other energy
scales in the zero magnetic field limit, giving rise to novel quantum phases
of matter. However, lowering magnetic fields well below the mG level is often
challenging in ultracold gas experiments. In this article, we apply Landau-
Zener spectroscopy to characterize and reduce the magnetic field on an
ultracold gas of sodium atoms to a few tens of $\mu$G. The lowest magnetic
field achieved here opens to observing novel phases of matter with ultracold
spinor Bose gases.

##  I INTRODUCTION

Controlling magnetic fields is critical in many contexts involving fundamental
and applied physics experiments and quantum technologies. Often, the
performance of a measurement critically depends on the stability against
fluctuations of the background magnetic field, as it happens in electron
microscopy experiments [1, 2, 3], atom interferometry [4], nuclear magnetic
resonance [5], atomic clock experiments [6], and ultracold gases experiments
involving coherently-coupled BEC mixtures [7, 8, 9]. In other contexts, such
as zero-field nuclear magnetic resonance [10, 11], the constraint relates to
the magnitude of the magnetic field, which generally should be minimized and,
depending on the type of measurement, needs to be kept below the threshold
values. Historically, the measurement of relatively small magnetic fields is
achieved using SQUIDs in a cryogenic environment [12], exploiting atomic
coherence in room temperature gas cells [13, 14, 15], or atomic spin-alignment
[16], with applications in diverse fields, including biomedical imaging [17,
18], metal detection [19, 20], and material characterization [21]. Nowadays, a
wide variety of experimental platforms and techniques are available, as
described, for instance, in Ref. [22]. In the context of magnetometry, cold
gases have been consistently employed as magnetic field sensors [23, 24], in
various cases with micrometric scale spatial resolution [25, 26, 27, 28].

Finding a way to reduce the magnetic field, control it with high accuracy, and
be able to measure such small values would pave the way for the investigation
of new physical phenomena, such as the zero-magnetic-field physics of spinor
condensates, i.e. condensates with a vector order parameter. Spinor
condensates can develop different configurations, and their ground state
depends on the strength of the interactions between the internal states and on
the strength of the externally applied magnetic field, which removes the
degeneracy between different spin states [29, 30]. The magnetic field is
typically large enough that the corresponding magnetic energy splitting
between spin states dominates over the spin-dependent interaction energy. A
regime where the two energies are comparable has been studied with dipolar
chromium gases in the presence of magnetic fields of a few tenths of a mG,
thanks to the large magnetic moment of chromium atoms [31].

For alkali atoms, the spin-dependent interaction is essentially dominated by
the contact interaction, given the negligible contribution of the long-range
magnetic dipole coupling [32].

This further reduces the threshold magnetic field below which interesting and
unobserved spinor phases are expected [29, 30, 33]. The strength of the
magnetic field below which these novel phases appear is known, in the case of
the zero spatial mode approximation [33], to be on the order of a few hundreds
of $\mu$G. However, this threshold is expected at even lower magnetic fields
in the case of spatially extended condensates. Here, the experimental
challenge relies on the difficulty of controlling the magnetic field stability
at the $\mu$G level and reaching and maintaining such small magnetic field
values over the whole extension of the atomic sample both during a single
experimental sequence and within different runs.

This work presents an experimental technique for minimizing the magnetic field
at the 10-$\mu$G level. Since it is impossible to integrate an external device
in the vacuum chamber of ultracold atom experiments to characterize and
minimize the field, developing an independent technique for the magnetic field
characterization using the trapped atoms as sensors becomes necessary. In
particular, we describe an application of Landau-Zener (LZ) spectroscopy [34]
over an atomic gas of sodium. Reaching such a low field is possible thanks to
the presence of a magnetic shield [35], which demonstrated its efficiency in
stabilizing the field in several previous works [9, 8, 36, 37]. The results
presented in the following open the way to studying the zero-magnetic-field
ground state of an $F$=1 system.

In Sec. II, we describe the experimental platform. Section III contains the
theoretical framework and experimental protocols. In Sec. IV we discuss the
results, while in Sec. V we report concluding remarks and outlooks.

##  II Experimental platform

The experimental platform consists of a bosonic gas of 23Na atoms, trapped in
the elongated optical trap potential generated by a single far-detuned
infrared (1064 nm) laser beam, and therefore equal for all spin components. A
thermal sample is loaded in the optical trap and presents a Gaussian spatial
density distribution in all three directions. In the optical trap
($\omega_{\mathrm{x}}/2\pi\sim 10$ Hz, $\omega_{\mathrm{y,z}}/2\pi\sim 1$
kHz,), the cloud has an elongated shape with a 1:100 aspect ratio, having the
long axis along $x$ and the short axes along $y$ and $z$.

Figure 1: Schematics of the core setup. (a) The innermost layer of the
magnetic shield (dark gray and transparent) contains a 3D-printed plastic
support structure for the coils (red), the coils themselves (blue), and the
glass cell (light gray). (b) $yz$ section of the apparatus.

The sample is prepared inside a magnetic shield, which guarantees the
stability of the magnetic field at the $\text{\,}\mu\mathrm{G}$ level [38,
35]. Pairs of magnetic coils inside the innermost layer of the magnetic shield
allow for the application of controllable magnetic fields in each of the three
spatial directions. Figure 1 schematically represents the apparatus and shows
the glass cell, coils, and magnetic shield. The applied magnetic field’s
stability is ensured using high-stability current supplies [Stanford research
system (SRS) LDC501 for the longitudinal field and Delta Elektronica ES 015-10
with 10:1 current dividers for transverse fields]. Ramping from positive to
negative values of the longitudinal field within the same experimental run is
achieved using two unipolar current supplies with opposite orientations
arranged in parallel. A master SRS is set to the steady drive of 100 mA
($\sim$ 245 mG), while a second SRS introduces a tunable current in the
opposite direction.

##  III Measurement scheme

Spectroscopic methods are typical solutions to measure a magnetic field. They
consist of interrogating an atomic two-level system with constant radiation at
different frequencies for a given time and recording the resulting energy
spectrum. In such measurements, the Fourier broadening does not represent a
limitation at high magnetic fields when the Zeeman splitting ($\sim 700$ kHz/G
for atomic 23Na) is much larger than the resonance linewidth. Conversely, when
approaching the $\mu$G regime, resolving the Zeeman structure and exerting
suitable control on the polarization of the coupling field becomes
challenging.

An alternative method to characterize the magnetic field around the null value
relies on ramps of the magnetic field amplitude and directions, taking
advantage of the adiabatic/diabatic dynamics of the atomic spin rotation. At
low magnetic field, i.e., when the Larmor frequency is of the order or lower
than the velocity of rotation of the magnetic field direction, LZ theory [39,
40] applies, resulting in a powerful tool for the magnetic field
characterization.

We developed two protocols based on LZ sweeps on the $z$ (taken as a
quantization axis) component of the field. The first one aims to minimize the
magnetic field in the transverse $xy$ plane and is implemented by ramping (or
sweeping) the $z$ field component from positive to negative finite values. The
second protocol involves a ramp with a variable endpoint to find the minimum
field along $z$.

In the following, we will use the magnetic field nomenclature presented in
Tab. 1.

$B$ | Magnetic field modulus
---|---
$B_{i}$ ($i=x,y,z$) | Actual magnetic field components
$B_{\perp}$ | Transverse field
$B_{z}$ | Longitudinal field
$B_{i,\mathrm{coils}}$ | Field induced by the $i$ coil
$B_{i,\mathrm{ramp}}$ | Starting $B_{i,\mathrm{coils}}$ of the LZ ramps
$B_{i,0}$ | Optimal $B_{i,\mathrm{coils}}$ from fit
$B_{z,c}$ | Field set during condensation
$B_{z,\mathrm{fin}}$ | Final value of longitudinal field protocol
Table 1: Magnetic field nomenclature

###  III.1 Energy levels and Landau-Zener theory

The hyperfine ground state of sodium has a total angular momentum $F=1$ with
three magnetic sublevels $m_{F}=0,\pm 1$ defined as the projection of $F$
along the $z$-axis. The energy of the three states can be estimated with high
accuracy using the Breit-Rabi formula [41] once the magnetic field value is
known, and, in the small field limit, a linear dependence on the field (first
order Zeeman regime) is expected. Let us consider $z$ as the quantization axis
and the magnetic field with a finite transverse component
$B_{\perp}=(B_{\mathrm{x}}^{2}+B_{\mathrm{y}}^{2})^{1/2}$. If $B_{\mathrm{z}}$
is linearly ramped with a slope $\dot{B_{\mathrm{z}}}$, from large positive to
large negative values passing across zero, the energy of the states as a
function of time is the one shown in Fig. 2. At $B_{\mathrm{z}}=0$, the
magnetic field is equal to $B_{\perp}$, which introduces an avoided crossing
between the states.

The time-dependent Hamiltonian of the system for the wavefunction
$\\{\psi_{-1},\psi_{0},\psi_{+1}\\}$ reads

$H(t)=g_{F}\mu_{B}\begin{pmatrix}\alpha t&\Delta&0\\\ \Delta&0&\Delta\\\
0&\Delta&-\alpha t\\\ \end{pmatrix},$ (1)

where $g_{F}=-1/2$ is the Landé factor, $\mu_{B}$ is the Bohr magneton, and
$\Delta=B_{\perp}/\sqrt{2}$ acts as a coupling between the states and defines
the gap in the energy levels shown in fig:fig1eq:Eq. (1)fig:fig1fig:Fig.
1fig:fig1tab:Table 1fig:fig1appendix:Appendix 1fig:fig1sec:Section 1. Here
$t=0$ is defined as the instant when $B_{z}=0$. The parameter $\alpha$
originates from the time derivative of the Larmor frequency at the energy
levels crossing

$\alpha=\left||B(t)|\frac{\partial}{\partial
t}\left(\arctan\frac{\dot{B}_{\mathrm{z}}t}{|B(t)|}\right)\right|_{t=0}=|\dot{B}_{\mathrm{z}}|.$
(2)

Figure 2: Transverse field and LZ ramps. For each row, the left panels show a
schematic view of the transverse magnetic field components ($B_{\perp}$
decreases from top to bottom). Central panels show the energy of the three
states $m_{F}=0,\pm 1$ as a function of time for the corresponding
$B_{\perp}$. The right panels show the relative populations of the atoms in
the three states at the end of the ramp, obtained experimentally and measured
through Stern-Gerlach imaging. For the three different values of $B_{\perp}$
we obtain full adiabatic transfer (upper panel, large $B_{\perp}$), partial
transfer (central panel), and complete diabatic transfer (lower panel, low
$B_{\perp}$).

While the atoms are initially prepared in the $m_{F}=-1$ at large (as compared
to $B_{\perp}$), positive values of $B_{\mathrm{z}}$, ramping $B_{\mathrm{z}}$
in time may induce the transfer to different $m_{F}$ states according to the
analytical model discussed in [34]. In the following, we apply the results of
[42] as we measure the $m_{F}$ population distribution at long times after the
inversion of $B_{\mathrm{z}}$:

$\displaystyle P_{1}$ $\displaystyle=p^{2},$ (3) $\displaystyle P_{0}$
$\displaystyle=2p(1-p),$ (4) $\displaystyle P_{-1}$ $\displaystyle=(1-p)^{2},$
(5)

with

$p=e^{-2\pi|g_{F}|\mu_{B}\frac{\Delta^{2}}{\hbar\alpha}}$ (6)

being the LZ transfer probability. From Eq. (3) and Eq. (6), one finds that a
complete diabatic transfer of the population from the $m_{F}=-1$ to the
$m_{F}=+1$ state takes place when $p\approx 1$ (large ramp speed $\alpha$ or
small gap $\Delta$), while adiabaticity is preserved for small $\alpha$ and
large $\Delta$. This results from implementing the adiabatic condition to the
spin dynamics in a rotating magnetic field. In other words, adiabaticity is
fulfilled when the variation of the magnetic field direction is smaller than
the Larmor frequency, i.e.,
${|\dot{B_{\mathrm{z}}}}/{B}|\ll|{g_{F}\mu_{B}B}/{\hbar}|$. For instance, the
probability of the transition to $m_{F}=1$ has a Gaussian distribution with
RMS width $\frac{\sigma_{B}}{\sqrt{2}}$ where
$\sigma_{B}=(\dot{B}_{\mathrm{z}}\hbar/(2\pi|g_{F}|\mu_{B}))^{1/2}$ is defined
from the LZ transfer probability and it is the width used in the following.

###  III.2 Experimental protocols

Figure 3: Transverse field minimization protocol. $B_{\mathrm{z}}$ is shown as
a function of time. Scheme A (red line) consists in a ramp from
$B_{\mathrm{z,ramp}}$ to -$B_{\mathrm{z,ramp}}$ performed on a thermal cloud,
then $B_{\mathrm{z}}$ is brought down to $-B_{\mathrm{z,c}}$ in 50 ms before
the evaporation in the optical trap. Scheme B (blue line) starts with the
condensation at a finite value of the field $B_{\mathrm{z,c}}$, then after the
evaporation, the field is ramped down to $B_{\mathrm{z,ramp}}$ in 50 ms. Here
the condensate experiences the field ramp across zero, from
$B_{\mathrm{z,ramp}}$ to $-B_{\mathrm{z,ramp}}$. In both cases, the dotted
line corresponds to the field crossing the minimum value with a tunable ramp
duration. The shaded gray area denotes the time during which the sample is
evaporated to obtain a BEC. Figure 4: Transverse field determination. (a)
Schematic representation of the iterative measurement process; see main text.
(b) Relative populations of the three Zeeman levels $m_{F}=-1,0,+1$ (from top
to bottom) for decreasing ramp velocities $\alpha$ (from left to right). All
panels show the experimental data (dots) and the fitted function (solid line)
plotted as a function of the transverse field in the $x$ directions. For each
panel, the upper axis shows the field’s value generated by the coils, while
the lower axis is shifted according to the fit result. The speed of the ramp
is specified in the orange box as well as the value of the compensation field
in the $y$ direction used for each scan. Note the offset of the compensating
field in the first column, whose data were acquired with a few weeks delay
with respect to the other two data sets.

####  III.2.1 Transverse field minimization

The goal is to find the optimal current values for each coil to compensate for
the residual transverse magnetic field (which is not screened by the magnetic
shield or due to the shield’s permanent magnetization).

The protocol to minimize the transverse field amplitude $B_{\perp}$ starts
with a thermal atomic sample by setting the values of the currents in the
coils for the transverse directions, generating the fields
$B_{\mathrm{x,coils}}$ and $B_{\mathrm{y,coils}}$. In the following, we
discuss the two experimental schemes depicted in fig:fig3eq:Eq.
(3)fig:fig3fig:Fig. 3fig:fig3tab:Table 3fig:fig3appendix:Appendix
3fig:fig3sec:Section 3.

Scheme A (red line) consists in changing $B_{\mathrm{z}}$ from
$B_{\mathrm{z,ramp}}$ to $-B_{\mathrm{z,ramp}}$ with a linear ramp of variable
duration $\Delta t^{*}$. After this first ramp, $B_{\mathrm{z}}$ is reduced to
$-B_{\mathrm{z,c}}\approx-130$ mG $\ll-B_{\mathrm{z,ramp}}$ in 50 ms. Then,
the sample is evaporatively cooled to Bose condensation by reducing the dipole
trapping beam intensity. To image the spin state, after switching off the
trapping potential, a vertical magnetic field gradient of 8 G/cm is applied to
separate the three states in a Stern-Gerlach-like scheme. A simultaneous
absorption image of the three states is made after a time-of-flight of 18 ms.
Bose condensing the sample before imaging favors the spatial spin resolution
of the Stern-Gerlach imaging, given the relatively small amount of the applied
magnetic field gradient and ballistic expansion time.

To reduce the sensitivity to magnetic field inhomogeneities, in Scheme B (blue
line) the order between LZ ramp and evaporation is reversed. The atomic sample
is first cooled below the critical temperature for condensation at
$B_{\mathrm{z,c}}\approx$ 130 mG, and then the field is decreased to
$B_{\mathrm{z,ramp}}$ to apply the LZ ramp from $B_{\mathrm{z,ramp}}$ to
$-B_{\mathrm{z,ramp}}$. By doing so, the spatial extent of the condensed
atomic sample is considerably smaller during the LZ ramp, suppressing the
contributions from the magnetic field’s spatial inhomogeneities.

The ramp duration $\Delta t^{*}$ should be much longer than the coils time
constant $\tau=L/R\simeq$ 0.5 ms, where $L$ and $R$ are the inductance and the
resistance of the coils, respectively, and much shorter than the sample
lifetime in the trap ($\tau_{\mathrm{BEC}}\simeq$ 1 s, $\tau_{\mathrm{th}}\gg$
1 s). The value $B_{\mathrm{z,ramp}}$ was chosen depending on the ramp
duration, from a maximum of 1.5 mG to a minimum of 750 $\mu$G, with
$|\dot{B_{\mathrm{z}}}|$ ranging from a maximum of 3 G/s to a minimum of 3.75
mG/s.

The values $B_{\mathrm{x,0}}$ and $B_{\mathrm{y,0}}$ that best compensate for
the residual transverse field are the ones giving the maximal transfer to the
$m_{F}=+1$ state. They are found through an iterative procedure, as sketched
in fig:fig4eq:Eq. (4)fig:fig4fig:Fig. 4fig:fig4tab:Table
4fig:fig4appendix:Appendix 4fig:fig4sec:Section 4a. The first iteration is a
scan in $B_{\mathrm{x,coils}}$ setting $|\dot{B}_{\mathrm{z}}|=300$ mG/s and
$B_{\mathrm{y,coils}}=0$. We obtain the value $B_{\mathrm{x,0}}$ for which we
observe maximum transfer in $m_{F}=+1$. Then, we set
$B_{\mathrm{x,coils}}=B_{\mathrm{x,0}}$ and we perform a scan in
$B_{\mathrm{y,coils}}$ obtaining $B_{\mathrm{y,0}}$ for which we have the
maximal transfer of the population in the state $m_{F}=+1$. We repeat this
procedure by slowing down the ramp in $B_{\mathrm{z}}$ at each iteration. In
this way, the $P_{+1}$ distribution shrinks, increasing our sensitivity to
determine the field value that best compensates for the residual one. The
first iteration starts finding the value $B_{\mathrm{x,0}}$ with a scan in
$B_{\mathrm{x,coils}}$ at fixed $B_{\mathrm{y,coils}}=0$, with a given ramp
speed $|\dot{B}_{\mathrm{z}}|=300$ mG/s. Then, at fixed
$B_{\mathrm{x,coils}}=B_{\mathrm{x,0}}$ we perform a scan in
$B_{\mathrm{y,coils}}$ obtaining $B_{\mathrm{y,0}}$.

Figure 5: Longitudinal field protocol. Representation of the field in $z$ and
$y$ directions. For the field in the longitudinal direction, the protocol is
analogous to Scheme B with the only difference that the speed of the ramp is
fixed to be $\dot{B}_{\mathrm{z}}=15$mG/s while the final value of the ramp
$B_{\mathrm{z,fin}}$ is varied. The field in the $y$-direction is fixed to the
optimal value $B_{\mathrm{y,0}}$ previously determined, it is adiabatically
ramped to a finite value and then back o $B_{\mathrm{y,0}}$ for the imaging
procedure. The field along $x$ is not shown in the figure, as it is kept at a
constant value $B_{\mathrm{x,0}}$ for the whole duration of the experiment.

Figure 4b presents the three hyperfine relative populations, $n_{-1}$,
$n_{0}$, $n_{+1}$, for three subsequent scans of the magnetic field
$B_{\mathrm{x,coils}}$, with decreasing value of $|\dot{B}_{\mathrm{z}}|$ and
$B_{\mathrm{y,coils}}$ set to the value determined in the previous iteration
step. The experimental data were fitted to Eq. (3-5) from which we extract the
center $B_{\mathrm{x,0}}$, the maximum transfer $P_{1,\mathrm{max}}$, and the
width $\sigma_{B}$ of the transfer peak. At first, the center of the observed
structures corresponds to $B_{\mathrm{x}}=0$ and this allows us to determine
$B_{\mathrm{x,coils}}$ to compensate any residual field along $x$. The maximum
in the transferred population $P_{1,\mathrm{max}}$ depends on the residual
field in $y$,

$B_{y}=\sqrt{-\frac{\hbar\dot{B}_{\mathrm{z}}}{\pi\mu_{B}}\ln(P_{1,\mathrm{max}})},$
(7)

which is minimized after each $B_{\mathrm{x}}$ scan (scan in
$B_{\mathrm{y,coils}}$ are not shown in figure). As expected, slower ramps
lead to smaller $\sigma_{B}$ (as highlighted by the white windows, which
always mark a region of 200 $\mu$G), and consequently to increase the
precision at which $B_{\mathrm{x}}=0$ (and $B_{\mathrm{y}}$) is determined.

####  III.2.2 Longitudinal field

Figure 6: Example of the longitudinal field characterization. Each panel shows
the relative population for the three states as a function of
$B_{\mathrm{z,fin}}$. The dots are experimental data obtained averaging
different experimental realizations, with standard deviation smaller than the
marker size, while dotted line is the result of the sigmoidal fit. The value
$B_{\mathrm{z,0}}$ compensating for the residual field is pointed by the red
star.

The protocol presented in the previous section allows for minimizing the
transverse magnetic field. The best-obtained values are then used as
transverse fields setting $B_{\mathrm{x,coils}}=B_{\mathrm{x,0}}$ and
$B_{\mathrm{y,coils}}=B_{\mathrm{y,0}}$ for the characterization of the
residual longitudinal field $B_{\mathrm{z}}$.

Here, we introduce the procedure to characterize the residual longitudinal
field $B_{\mathrm{z}}$. The gas is first evaporated to obtain a BEC at
$B_{\mathrm{z,c}}=130$ mG, then $B_{\mathrm{z,coils}}$ is ramped from a
positive value $B_{\mathrm{z,ramp}}$ down to a variable one
$B_{\mathrm{z,fin}}$ with constant ramp $\dot{B}_{\mathrm{z}}=-15$ mG/s. As
the Stern-Gerlach imaging is implemented at positive $B_{\mathrm{z}}$, we
constrain the diabatic spin dynamics to the decreasing ramp on
$B_{\mathrm{z,coils}}$ by raising the transverse field to a finite value after
the end of the $B_{\mathrm{z,coils}}$ ramp, see Fig. 5. If
$B_{\mathrm{z,fin}}>0$, the LZ transfer does not take place since the ramp is
interrupted before the zero crossing on $B_{\mathrm{z,coils}}$ hence leaving
the atoms in $m_{F}=-1$. If $B_{\mathrm{z,coils}}<0$, on the other hand, the
transfer to $m_{F}=+1$ takes place. An example of such a scan is given in Fig.
6 where the relative populations in the three states are represented as a
function of the value $B_{\mathrm{z,fin}}$. We verified that using a slower
ramp does not affect the experimental result.

##  IV RESULTS

From the fits of the experimental data, as the examples shown in Fig. 4, it is
possible to extract both the width of the transfer peak $\sigma_{B}$ and, from
the contrast of the LZ population transfer, the residual transverse magnetic
field $B_{\perp}$. Figure 7a shows the measured width $\sigma_{B}$ of the
transfer peak as a function of $|\dot{B}_{\mathrm{z}}|$ for both protocols
(red symbols for A and blue for B). Each point was obtained by averaging the
widths from different experimental runs with the same ramp speed. The
experimental values of the width are consistent with $\sigma_{B}$ predicted
from the LZ theory (dashed line in the figure). We attribute the general
increase in the experimental values to effects not included in the LZ
discussed so far.

Figure 7:  a) Width $\sigma_{B}$ of LZ transfer probability as a function of
the ramp speed. The dashed black line is the theoretical prediction from LZ
theory; the symbols are the experimental results using scheme A (red symbols)
or scheme B (blue symbols). Empty (filled) symbols are obtained without (with)
compensation for the residual gradient, respectively. The error bars have been
estimated considering the fluctuations of different datasets. The two insets
are examples of the population in the state $m_{F}=+1$ obtained at a ramp
velocity of 3.75 mG/s with (left inset) and without (right inset) the
compensation of the gradient. The blue dashed line is the contribution of the
residual magnetic field gradient to the peak width. b) Transverse residual
magnetic field $B_{\perp}$ extracted from the population fits as a function of
the inverse of the ramp velocities. Note that the error bars for the data
obtained compensating the magnetic field inhomogeneities are too small to be
seen in the plot.

For instance, comparing the measurements taken on a thermal sample (empty red
symbols in Fig. 7) to the ones taken on the smaller Bose condensed sample
(empty blue symbols ), the flattening trend of $\sigma_{B}$ observed with the
thermal sample around $30\,\mu$G can be attributed to magnetic field
inhomogeneities, i.e. magnetic field gradients along the elongated direction
of the atomic sample $y$. Here, the residual magnetic field difference between
opposite regions gives rise to an additional contribution
$\sigma_{\mathrm{grad}}$ in the determination of $\sigma_{B}$, which we expect
to sum in quadrature with $\sigma_{\mathrm{LZ}}$ predicted by the LZ theory:

$\sigma_{B}=\sqrt{\sigma^{2}_{\mathrm{LZ}}+\sigma_{\mathrm{grad}}^{2}}.$ (8)

We determine the optimal compensation for this gradient by employing a
dedicated set of coils in anti-Helmholtz configuration, which is driven to
reduce $\sigma_{B}$ further (inset in Fig. 7a, and filled dots in 7a and b).
From eq:eq7eq:Eq. (8)eq:eq7fig:Fig. 8eq:eq7tab:Table 8eq:eq7appendix:Appendix
8eq:eq7sec:Section 8 and the comparison between data with and without
compensation obtained with protocol B, we estimate the non-compensated
gradient contribution at $\sigma_{\mathrm{grad}}=22\pm 3\,\mu$G (horizontal
dashed blue line in Fig. 7a). This is consistent with the estimation from
cloud size and magnetic field gradient applied to minimize $\sigma_{B}$. We
notice that $\sigma_{B}$ acquired in schemes A and B are comparable upon
magnetic gradient compensation.

In fig:fig7eq:Eq. (7)fig:fig7fig:Fig. 7fig:fig7tab:Table
7fig:fig7appendix:Appendix 7fig:fig7sec:Section 7b, we plot $B_{\perp}$
obtained by fitting Eq. (7) on the experimental data both for measurements
without (empty symbols) and with (filled triangles) the gradient compensation.
The smallest width observed allows us to determine the minimal residual
transverse field at $14\pm 2\,\mu$G, a value that could be limited by the
noise of the current supplies driving the compensation coils. Also, it is
worth mentioning that the condition for the residual field compensation has
generally been stable for several weeks. Still, we did observe jumps in the
compensation field at the level of hundreds of $\mu$G (a few events over six
months of measurements), which we could not clearly attribute to technical
circumstances.

The longitudinal field was minimized by following the protocol explained in
Sec.III.2.2. $B_{\mathrm{z,0}}$ is chosen as the center of the sigmoidal fit
of the data shown in Fig. 6, $B_{\mathrm{z,0}}=-323\pm 10\,\mu$G.

By combining the minimal transverse and longitudinal field results, we
estimate we can reach a minimal field modulus of $18\pm 5\,\mu$G, which
complies with the conditions for observing a nematic phase in a 23Na BEC.

##  V CONCLUSIONS AND OUTLOOK

In this paper, we present a technique to characterize and compensate magnetic
fields at the level of 10 $\mu$G in experiments with ultracold atomic gases.
The method is based on monitoring the Zeeman populations in diabatic atomic
spin rotation. The simulations based on the LZ dynamics for a three-level
system reproduced the experimental data well. These results pave the way to
studying the unexplored scenario of condensation in zero magnetic fields in
extended spinor gases, when spin interactions may become dominant over all
other contributions of the Hamiltonian. For instance, the ground state of F=1
BEC characterized by antiferromagnetic interaction is expected to develop an
order parameter with a nematic character, which has not been reported on
superfluid matter so far.

## Acknowledgements

We acknowledge funding from Provincia Autonoma di Trento, from the European
Union’s Horizon 2020 research and innovation Programme through the STAQS
project of QuantERA II (Grant Agreements No. 101017733 and No. 101017733) and
from the European Union - Next Generation EU through PNRR MUR project
PE0000023-NQSTI. This work was supported by Q@TN, the joint lab between the
University of Trento, FBK - Fondazione Bruno Kessler, INFN - National
Institute for Nuclear Physics and CNR - National Research Council.

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