Present address: ] Safran Reosc, Sain-Pierre-du-Perray 91280, France

Present address: ] ColdQuanta, Oxford Centre for Innovation, OX1 1BY, United
Kingdom

Present address: ] Korea Research Institute of Standards and Science, Daejeon
34113, South Korea

# Low-field Feshbach resonances and three-body losses
in a fermionic quantum gas of 161Dy

E. Soave  Institut für Experimentalphysik, Universität Innsbruck, Austria  V.
Corre  [  Institut für Experimentalphysik, Universität Innsbruck, Austria  C.
Ravensbergen  [  Institut für Experimentalphysik, Universität Innsbruck,
Austria  J. H. Han  [  Institut für Experimentalphysik, Universität Innsbruck,
Austria  M. Kreyer  Institut für Experimentalphysik, Universität Innsbruck,
Austria  E. Kirilov  Institut für Experimentalphysik, Universität Innsbruck,
Austria  Institut für Quantenoptik und Quanteninformation (IQOQI),
Österreichische Akademie der Wissenschaften, Innsbruck, Austria  R. Grimm
Institut für Experimentalphysik, Universität Innsbruck, Austria  Institut für
Quantenoptik und Quanteninformation (IQOQI), Österreichische Akademie der
Wissenschaften, Innsbruck, Austria

###### Abstract

We report on high-resolution Feshbach spectroscopy on a degenerate, spin-
polarized Fermi gas of 161Dy atoms, measuring three-body recombination losses
at low magnetic field. For field strength up to 1 G, we identify as much as 44
resonance features and observe plateaus of very low losses. For four selected
typical resonances, we study the dependence of the three-body recombination
rate coefficient on the magnetic resonance detuning and on the temperature. We
observe a strong suppression of losses with decreasing temperature already for
small detunings from resonance. The characterization of complex behavior of
three-body losses of fermionic 161Dy is important for future applications of
this peculiar species in research on atomic quantum gases.

††preprint: APS/123-QED

##  I Introduction

Over the past decade, the exotic interactions of submerged-shell lanthanide
atoms have tremendously boosted experimental research on ultracold quantum
gases [1]. The exciting properties of such atoms result from long-range
anisotropic interactions in combination with tunability of the contact
interaction via magnetically controlled Feshbach resonances [2]. Prominent
examples for novel states of matter created in the laboratory are quantum
ferrofluids of Dy [3] and supersolids realized with both Dy and Er [4, 5, 6].
Progress has also been made with quantum-gas mixtures of different lanthanide
atoms (Dy-Er) [7, 8] and mixtures of lanthanide and alkali-metal atoms (Dy-K)
[9, 10], with a wide potential for future experiments on exotic states of
quantum matter.

For interaction control, magnetic lanthanide atoms offer a rich spectrum of
Feshbach resonances [11, 12, 13, 14], much denser as compared to alkali-metal
atoms. This experimentally well-established fact is a consequence of
anisotropy stemming both from the strong magnetic dipole-dipole interaction
and from the van-der-Waals interaction for electronic ground states with non-
zero orbital angular momentum [15, 16]. The anisotropic interaction leads to a
strong mixing of different partial waves. If hyperfine structure is present,
such as for the fermionic isotopes 161Dy and 167Er, the Feshbach spectrum is
even more complex, and the blessing of tunability may turn into a curse of
omnipresent three-body recombination losses.

The experiments performed with 161Dy in our laboratory are motivated by the
prospect to realize novel superfluid states in mass-imbalanced fermion
mixtures [17, 18, 19]. In a Fermi-Fermi mixture of 161Dy and 40K atoms, we
have recently demonstrated hydrodynamic behavior as a manifestation of strong
interactions, realized on top of an interspecies Feshbach resonance [10].
Further experiments are in progress on the formation of bosonic Feshbach
molecules, paired fermionic many-body states, and collective behavior of the
strongly interacting mixture. In all these experiments, the Dy-Dy intraspecies
Feshbach resonances and accompanying three-body losses represent a
complication and appropriate strategies have to be developed to minimize
unwanted effects.

Feshbach resonances in spin-polarized fermionic quantum gases result from
scattering in odd partial waves. Accordingly, $p$-wave resonances have been
observed in early experimental work [20, 21, 22] and studied theoretically
[23, 24]. More recent experiments [25, 26] have provided deeper insights into
the scaling laws and universal properties of three-body recombination losses
near $p$-wave resonances. Our present situation of 161Dy, however, is more
complex because of the strong coupling between different partial waves and the
possible interaction between different closely spaced or overlapping
resonances, which makes a theoretical description very challenging.
Experiments are needed to find out in how far our resonances in 161Dy behave
in a similar way.

In this article, we report on the experimental investigation of the ultradense
Feshbach spectrum of 161Dy at low magnetic field strength (up to about 1 G)
with high resolution ($\sim$1 mG). To minimize the effect of finite collision
energies, i.e. broadening effects and the influence of higher partial waves,
we work in the deeply quantum-degenerate regime at rather low values of the
Fermi energy down to a few $100$ nK. In Sec. II, we present the Feshbach loss
spectrum, exhibiting nearly 50 loss features in a 1 G wide range. We also
identify plateaus of very low losses, which can be used for efficient
evaporative cooling. In Sec. III, we then present case studies on four typical
resonances, where we report on the dependence of the three-body rate
coefficient on the magnetic detuning and the temperature of the sample. Our
measurements show that even very small detunings from resonance of a few mG
are sufficient to enter a regime where losses are strongly suppressed with
decreasing temperature.

##  II Low-field Feshbach Spectrum

###  II.1 Sample preparation

All our experiments begin with the production of a degenerate Fermi gas of
161Dy atoms. We follow the procedures described in detail in Ref. [9]: After
capturing the atoms in a magneto-optical trap (MOT) operated at the 626-nm
intercombination line [27], the sample is transferred into a crossed-beam
optical dipole trap (ODT), which uses near-infrared light at a wavelength of
$1064\text{\,}\mathrm{nm}$. Here forced evaporative cooling is performed by
ramping down the trapping potential. Under optimized conditions, we obtain a
sample of up to $N=$1.5\text{\times}{10}^{5}\text{\,}$$ atoms in a nearly
harmonic trap (geometrically averaged trap frequency
$\bar{\omega}=2\pi\times$150\text{\,}\mathrm{Hz}$$) at a temperature of
$T=$80\text{\,}\mathrm{nK}$$. With a Fermi temperature of
$T_{F}=\hbar\bar{\omega}(6N)^{1/3}/k_{B}=$695\text{\,}\mathrm{nK}$$, this
corresponds to deeply degenerate conditions with $T/T_{F}=0.12$ and a peak
number density of
$\hat{n}=$1.6\text{\times}{10}^{14}\text{\,}{\mathrm{cm}}^{-3}\text{/}$$ in
the center of the trap. Our sets of measurements are taken over typically many
hours (sometimes even a few days), where long-term drifts may reduce the
maximum atom number provided by roughly a factor of two. In a last preparation
stage, the ODT is modified by replacing one of the laser beams (horizontally
propagating) with a beam of larger waist. This modification provides us with
more flexibility to vary the trap frequency $\bar{\omega}$ and, in particular,
it allows us to realize very shallow traps to work at lower atomic number
densities. For each experiment, the trap is chosen in a way to avoid residual
evaporation. The particular conditions for each set of measurements are listed
in App. A.

The cloud is fully spin polarized in the lowest hyperfine Zeeman sub-level
$\ket{F,m_{F}}=\ket{21/2,-21/2}$ as a result of optical pumping during the MOT
stage [28] and subsequent rapid dipolar relaxation of residual population in
higher spin states in the ODT [29]. For the fully spin-polarized sample,
inelastic two-body losses are suppressed already at very low magnetic field
values. The minimization of three-body losses, essential for efficient
evaporative cooling, depends very sensitively on the particular magnetic field
applied. Our evaporation sequence is performed at a magnetic bias field of
$230\text{\,}\mathrm{mG}$, which we found to work slightly better than at
$430\text{\,}\mathrm{mG}$, as applied in Ref. [9].

###  II.2 Loss Scan

Figure 1: Low-field Feshbach spectrum of a degenerate sample of spin-polarized
161Dy. The magnetic field is varied in steps of $1.6\text{\,}\mathrm{mG}$. The
error bars show the sample standard deviation of three individual measurements
at the same magnetic field. We observe about 44 loss features, which we
attribute to Feshbach resonances. The shaded areas correspond to the four
resonances that are investigated in detail in Sec. III.

.

We study the low-field Feshbach spectrum by measuring atom losses for a
variable magnetic field strength [2] in the range between 0 and
$1\text{\,}\mathrm{G}$. After preparation of the sample in a very shallow ODT
(for experimental parameters see App. A), we ramp the magnetic field from the
evaporation field to the variable target one in $20\text{\,}\mathrm{ms}$. The
low trap frequency of $\bar{\omega}=2\pi\times$100\text{\,}\mathrm{Hz}$$ is
chosen to minimize losses induced by the magnetic field ramp. We hold the
cloud for $7\text{\,}\mathrm{s}$, and then release it from the ODT. An
absorption image is taken after a time of flight of $10\text{\,}\mathrm{ms}$.

The magnetic-field stability is essential for resolving narrow loss features.
Using radio-frequency spectroscopy111We investigate the magnetic-field
stability by performing radio-frequency spectroscopy on 40K. The possibility
to work with potassium in the same setup follows from the fact that our
apparatus is designed for mixture experiments [9, 10]. we identified a 50-Hz
ripple in the ambient magnetic field as the main source of noise, with a peak-
to-peak value of $1.7\text{\,}\mathrm{mG}$. Other noise sources, such as noise
in the current of our coils, stay well below an estimated rms level of
$1\text{\,}\mathrm{mG}$.

In Fig. 1 we plot the remaining atom number as a function of the magnetic
field. We count $\simeq 44$ loss features, which we assign to Feshbach
resonances. On resonance the three-body recombination rate is greatly enhanced
and leads to more than a factor of 10 reduction in atom number. At these
positions we also observe substantial heating (not shown). The resonances seem
to mostly gather in groups, with flat, typically tens of
$\text{\,}\mathrm{mG}$ wide, plateaus between them. Within these plateaus,
losses are rather weak and stay within a few percent even for the long hold
time of $7\text{\,}\mathrm{s}$ applied.

The recorded Feshbach spectrum resembles previous observations in submerged-
shell lanthanide atoms (Er [11, 14], Dy [12, 13], Tm [31]), which are known to
exhibit a dense and very complex resonance spectrum. In the cases of the
fermionic isotopes 161Dy and 167Er, where hyperfine structure is present, the
resonance density is extremely high. While for 167Er a density of about 25
resonances per gauss has been reported in the range between 0 and
$4.5\text{\,}\mathrm{G}$ [11], previous work on 161Dy has revealed between
about 10 resonances per gauss in a range between 0 and $6\text{\,}\mathrm{G}$
[12] and up to about 100 resonances in a 250-mG wide range near
$34\text{\,}\mathrm{G}$ [13]. With our 44 resonances in a range between 0 and
$1\text{\,}\mathrm{G}$, we apparently resolve more resonances than in Ref.
[12], which we attribute to our higher magnetic field resolution. We believe
that a further improved magnetic field stability to well below
$2\text{\,}\mathrm{mG}$ would reveal even more resonances and a substructure
of some of our observed features. The complex spectrum of resonances may be
further analyzed using statistical methods [14, 31], which is beyond the scope
of the present work.

##  III Case Studies of Selected Resonances

We now perform a systematic investigation of the $K_{3}$ coefficient as a
function of the magnetic-field and the temperature for selected resonances. In
Sec. III.1 we first show how, from atom number decay measurements, we obtain
the value of the three-body recombination coefficient $K_{3}$. In Secs. III.2
and III.3 we investigate the dependence of $K_{3}$ on the magnetic field and
the initial temperature, respectively.

###  III.1 Three-body decay curves and loss-rate coefficients

In the absence of two-body losses, the evolution of the number of trapped
atoms $N(t)$ can be modeled based on the differential equation

$\dot{N}(t)=-\Gamma_{v}N(t)-3K_{3}\int d^{3}r\,n^{3}(\mathbb{r},t),$ (1)

where $\Gamma_{v}$ is the one-body loss rate from collisions with rest-gas
particles, and $n(\mathbb{r},t)$ represents the number density distribution of
the cloud. The quantity $K_{3}$ denotes the three-body event rate coefficient,
which for a single atomic species is related to the commonly used three-body
loss rate coefficient by $L_{3}=3K_{3}$. Note that, according to our
phenomenological definition, the $K_{3}$ coefficient represents a thermal
average over the distribution of collision energies in the sample, and does
not represent the coefficient for a specific collision energy as used in
theoretical work [32]. For our experiments we estimate a rest-gas limited
lifetime as long as $1/\Gamma_{v}\simeq$60\text{\,}\mathrm{s}$$. Given such a
low one-body loss rate, $\Gamma_{v}$ can be neglected in the analysis of near-
resonance decay curves, while it is relevant for cases on the long-lived
plateaus.

Figure 2: Typical decay curves. The measurements have been performed (a) on
resonance at $B=$678\text{\,}\mathrm{mG}$$, (b) for small detuning at
$B=$989\text{\,}\mathrm{mG}$$, and (c) on a minimum-loss plateau at
$B=$225\text{\,}\mathrm{mG}$$. The solid lines show fits by the heuristic
model introduced in App. B.

In Fig. 2 we show three typical decay curves, on resonance (a), near a
resonance (b) and far away from any resonance (c). The sample is held in the
ODT at a fixed magnetic field. After a variable hold time the cloud is
released and the number of remaining atoms is measured by time-of-flight
imaging. To analyse the decay curves and to extract values for $K_{3}$, we
apply a heuristic model (for details see App. B) to quantify the initial slope
$\dot{N}(0)$. From the initial decay rate $1/\tau=-\dot{N}(0)/N(0)$ and
knowledge of the experimental parameters at $t=0$, we then calculate the
resulting values for $K_{3}$. This approach, which focuses on the initial
decay, avoids complications by the heating of the sample during the decay.
Depending on the experimental conditions under consideration, decay times can
vary from a few ms to many seconds. As an example, the measurement reported in
Fig. 2(a) was carried out under typical experimental conditions (see App. A)
very close to the center of the 679-mG resonance, with initially about
$1.2\text{\times}{10}^{5}\text{\,}$ atoms. Our fit yields an initial decay
time $\tau=$22(6)\text{\,}\mathrm{ms}$$, and for the three-body rate
coefficient we obtain
$K_{3}=$4(1)\text{\times}{10}^{-26}\text{\,}{\mathrm{cm}}^{6}\text{/}\mathrm{s}$$.
The same measurement, performed few mG detuned from the resonance at
$995\text{\,}\mathrm{mG}$ and reported in Fig. 2(b), already shows a
significant longer decay time ($\tau=$0.50(9)\text{\,}\mathrm{s}$$). We
calculate a three-body recombination coefficient value
$K_{3}=$7(2)\text{\times}{10}^{-28}\text{\,}{\mathrm{cm}}^{6}\text{/}\mathrm{s}$$,
two orders of magnitude lower than on resonance.

The measurement in Fig. 2(c) is carried out at a magnetic field of
$225\text{\,}\mathrm{mG}$, on a minimum-loss plateau, and reveals a very long
lifetime. To observe the effect of three-body losses we worked in tightly
compressed trap with $\bar{\omega}=2\pi\times$380\text{\,}\mathrm{Hz}$$,
leading to a peak-density of
$\hat{n}_{0}\approx$6\text{\times}{10}^{14}\text{\,}{\mathrm{cm}}^{-3}\text{/}$$,
which is exceptionally high for a degenerate Fermi gas. We measure an initial
decay time $\tau=$12(3)\text{\,}\mathrm{s}$$, from which a value
$K_{3}=$5(3)\text{\times}{10}^{-32}\text{\,}{\mathrm{cm}}^{6}\text{/}\mathrm{s}$$
is obtained. This value is extraordinary low, which is highlighted by
comparison with 87Rb as a widely used bosonic species, where the $K_{3}$
coefficient has been measured to be
$\sim$6\text{\times}{10}^{-30}\text{\,}{\mathrm{cm}}^{6}\text{/}\mathrm{s}$$
[33, 34]. Such an extremely weak three-body decay, together with the sizeable
elastic scattering cross section from dipolar collisions [35], explains why
Fermi gases of submerged-shell lanthanide atoms facilitate highly efficient
evaporative cooling [36, 9].

###  III.2 Dependence on magnetic-field detuning

Figure 3: $K_{3}$ coefficient as a function of the magnetic-field, measured
for the four selected resonances. Note that in (a)-(c) the full range covers
12 mG, whereas in (d) it is three times wider. Each data point is obtained
from an individual decay curve as discussed in Sec. III.1. The dashed lines
indicate the lower limit to the measurable $K_{3}$ value, imposed by one-body
losses.

In this Section, we discuss selected loss features as typical examples for the
many resonances observed in our Feshbach spectrum. We focus on three resonant
features that lead to relatively strong losses in the measured Feshbach
spectrum (near $358\text{\,}\mathrm{mG}$, $677\text{\,}\mathrm{mG}$ and
$999\text{\,}\mathrm{mG}$, see loss scan in Fig. 1). For reference, we also
investigate a weaker loss resonance (near $170\text{\,}\mathrm{mG}$), which
appears to be well isolated from other resonances. We consider their line
shapes and widths by presenting measurements on the $K_{3}$ values as a
function of the magnetic detuning from resonance. Here we work in the deeply
degenerate regime, with $T/T_{F}\simeq 0.2$ at a low
$T_{F}\simeq$400\text{\,}\mathrm{nK}$$ (for details see App. A), which
minimizes line broadening stemming from the finite kinetic energy [14, 37].
Our results are displayed in Fig. 3(a-d). The measured values for $K_{3}$ vary
over more than four orders of magnitude. Maximum values are found to exceed
${10}^{-25}\text{\,}{\mathrm{cm}}^{6}\text{/}\mathrm{s}$. The presence of weak
one-body losses (see Sec. III.1) imposes a lower limit for the measurable
$K_{3}$ value, which is in the range of a few
${10}^{-30}\text{\,}{\mathrm{cm}}^{6}\text{/}\mathrm{s}$. This lower limit is
indicated by the dashed horizontal lines in Fig. 3.

Figure 3(a) shows the resonance near $170\text{\,}\mathrm{mG}$, which is the
weakest of the four selected features. We observe a full width of about
$3.5\text{\,}\mathrm{mG}$ 222We define the width as the full magnetic field
range where the $K_{3}$ value exceeds the geometric average between its
maximum and minimum. This corresponds to the full width at half maximum on a
logarithmic scale.. The line shape is essentially symmetric, which may first
appear surprising in view of the expected asymmetric line shapes of Feshbach
resonances in higher partial waves, which usually show a sharp edge on the
lower side (marking the resonance position) along with a tail on the upper
side [37, 39, 40, 41]. We assume that the shape of the weak feature is
dominated by the magnetic-field fluctuations in our experimental setup (see
Sec. II.2), which may affect the observed loss features in a range of a few
mG. The fluctuations will smear out any narrower feature and mask the true
resonance line shape (see discussion on broadening effects in App. C). This
interpretation is supported by the fact that we never observe any narrower
feature. We therefore believe that the observed behavior of narrower
resonances, such as the 170-mG feature, is dominated by magnetic-field
fluctuations.

In Fig. 3(b) we show a double feature of two resonances, separated by about
$4\text{\,}\mathrm{mG}$. While the weaker feature near
$355\text{\,}\mathrm{mG}$ closely resembles the one in Fig. 2(a), the stronger
feature near $359\text{\,}\mathrm{mG}$ shows a peak value for $K_{3}$
exceeding ${10}^{-25}\text{\,}{\mathrm{cm}}^{6}\text{/}\mathrm{s}$, which is
an order of magnitude higher. The stronger feature also shows indications of
the tail expected on the upper side for such resonances. The resonance appears
to be wide enough that its true structure is not fully masked by the magnetic-
field fluctuations. Figure 3(c) shows a feature near
$677\text{\,}\mathrm{mG}$, which in the Feshbach scan in Fig. 1, appeared to
be a single, relatively strong resonance. A closer investigation, however,
reveals a shoulder on the lower side, which is likely to be caused by another
weak overlapping resonance. On the upper side, the $K_{3}$ coefficient falls
off in a way resembling the expected tail. Figure 3(d) finally displays our
strongest observed loss feature; note the three times wider magnetic-field
range. We see a double feature separated by about $10\text{\,}\mathrm{mG}$.
The line shapes of the two resonances correspond to the expectation of a
sharper edge on the lower side and a tail on the upper side. Here, at least
for these broader features, magnetic-field fluctuations do not have a
substantial effect on the line shape.

###  III.3 Temperature dependence

We now turn our attention to the dependence of the $K_{3}$ coefficient on the
temperature of the cloud, for different magnetic detunings from the resonance
center. We vary the temperature of the cloud by interrupting the evaporation
sequence in a controlled way and by adiabatically varying the final trap
frequency. For these measurements, decay is observed in a 160-Hz and a 400-Hz
trap, for lower and higher temperatures, respectively. The $K_{3}$ coefficient
is obtained according to Eq.(5) or Eq.(4), depending on the initial $T/T_{F}$
of the sample. We introduce the effective temperature $\tilde{T}$, such that
the mean energy per particle is $3k_{B}\tilde{T}$. This definition takes into
account that, for a degenerate Fermi gas, the relative momentum and thus the
collision energy stay finite even at $T=0$. In the limit of a thermal gas,
$\tilde{T}=T$, while for a degenerate one we have
$\tilde{T}=T\,\mathrm{Li}_{4}(-\zeta)/\mathrm{Li}_{3}(-\zeta)$ [42], where
$\zeta$ is the fugacity and $\mathrm{Li}_{i}$ is the polylogarithm of order
$i$.

Figure 4 reports the behavior of the $K_{3}$ value as a function of
$\tilde{T}$ for three different magnetic detunings relative to the center of
the 358-mG resonance.

We first discuss the $K_{3}$ behavior in the high energy regime
($\tilde{T}\gtrsim$3\text{\,}\mathrm{\SIUnitSymbolMicro K}$$). Here the three
curves decrease in a similar manner, showing no dependence on the magnetic
field. Such a behavior reflects the unitarity limit for the $K_{3}$
coefficient, which was predicted and observed in several resonantly
interacting systems, fermionic and bosonic ones (e.g. [43, 44, 45, 46, 25]). A
non-degenerate atomic system enters the unitarity-limited regime when the
thermal de Broglie wavelength $\lambda_{\rm{dB}}=\hbar\sqrt{2\pi/(mk_{B}T)}$
becomes comparable to a characteristic length associated with the resonance at
a given magnetic detuning 333The length scale that characterizes the
interaction is the scattering length $a$ for an s-wave resonance and by
$\sqrt{|V_{p}|k_{\rm{eff}}}$ in the case of p-wave Feshbach resonances. Here
$V_{p}$ and $k_{\rm{eff}}$ are the scattering volume and the effective range,
respectively.. We attribute to the competition between those two length scales
the fact that the larger the detuning, the higher is the temperature at which
the $K_{3}$ value enters the unitary regime. In this regime the $K_{3}$ value
is expected to scale as $T^{-2}$ [23]:

$K_{3}=\zeta\frac{12\sqrt{3}\pi^{2}\hbar^{5}}{m^{3}(k_{B}T)^{2}}.$ (2)

The prefactor $\zeta$ relates to the efficiency of three colliding atoms
forming a dimer and a free atom, and is believed to be a non-universal (i.e.
species-dependent) quantity. A fit to our data, considering only the points
with $\tilde{T}>$5\text{\,}\mathrm{\SIUnitSymbolMicro K}$$, yields a value
$\zeta=0.022(2)$. In Ref. [26] the authors extracted a value $\zeta=0.09$ for
6Li. Those two results are about an order of magnitude below what has been
observed for Bose gases, where values of $\zeta=0.9,\,0.3$, and 0.24 have been
derived for 7Li [43], 39K [44], and 164Dy [45], respectively.

We now discuss the temperature-dependence of $K_{3}$ far below the unitarity-
limited regime. Figure 4 demonstrates that even a very small magnetic
resonance detuning of a few mG can have a dramatic effect on the low-
temperature behavior. The data taken at $\delta=6.7$ mG (typical uncertainty
0.2 mG) show a reduction of $K_{3}$ from a maximum value of the order of
$10^{-27}$ cm6/s at $\tilde{T}=3.5\,\mu$K to a minimum of about $10^{-30}$
cm6/s at $\tilde{T}\lesssim 200\,$nK. Note that the minimum value that we can
observe is limited by one-body decay (dashed line), so that the true
suppression will be even larger. A very similar behavior is observed closer to
resonance at $\delta=2.6$ mG. Here a maximum $K_{3}$ value of $\sim 5\times
10^{-27}$ cm6/s is found at $\tilde{T}$ of the order of $1\,\mu$K, which is
reduced by three orders of magnitude at our lowest temperature
$\tilde{T}\approx 200\,$nK. The main effect of the smaller detuning appears to
be a shift of the qualitatively similar behavior to lower temperatures.

These observations on the low-temperature behavior can be compared with recent
experimental work studying three-body recombination on $p$-wave Feshbach
resonances in 6Li [25, 26]. For the limit of very low collision energies
$E_{\rm coll}$, a threshold law $K_{3}\propto E_{\rm coll}^{2}$, as originally
predicted in Ref. [23], has been observed in Ref. [25] for a thermal
($T/T_{F}>1$) Fermi gas, where $K_{3}\propto T^{2}$. This observation of
threshold-law behavior required a relatively large resonance detuning. In our
case, with rather small detunings, the threshold-law regime would require
extremely low collision energies. This regime, however, remains inaccessible
in our present experiments because of two limitations: The Fermi energy gives
a lower limit to the collision energy ($\tilde{T}=T_{F}/4$ at $T=0$), and one-
body losses do not allow us to measure $K_{3}$ values below $\sim 10^{-30}$
cm6/s. However, beyond the threshold-law regime, we observe the same steep
increase with temperature as seen in Ref. [25] for relatively large magnetic
detunings. The breakdown of the threshold law has been interpreted [25] in
terms of the effective range of the resonance.

The case very close to resonance (data points for $\delta=-1\,$mG in Fig. 4)
reveals a different behavior. Here we do not observe any loss suppression with
decreasing temperature. The $K_{3}$ value appears to level off at about
$10^{-26}$ cm6/s. This, however, does not rule out the possibility of loss
suppression at values of $\tilde{T}$ that are even lower than what we can
realize experimentally in the deeply degenerate situation. The single data
point shown for $\delta=0$ at $\tilde{T}\approx 75$ nK corresponds to the loss
maximum in Fig. 3(b). This measurement highlights that three-body losses can
be very strong on top of the resonance, suggesting no suppression at low
temperatures. A similar on-resonance behavior has been observed in Refs. [26,
21], but in contrast to our work these experiments were limited to the non-
degenerate case.

In the narrow detuning range of $|\delta|\lesssim 1$ mG, the interpretation of
our present results is impeded by the sensitivity of the experiments to
magnetic field noise (see Sec. II.2 and App. C). The on-resonance behavior of
three-body recombination at ultralow collision energies, which has also been
subject to recent theoretical investigations [48], thus remains a topic for
future research.

Figure 4: Temperature dependence of $K_{3}$, for various detunings (typical
uncertainty 0.2 mG) relative to the center of the 358-mG resonance. For
$\tilde{T}\lesssim$1\text{\,}\mathrm{\SIUnitSymbolMicro K}$$ the samples have
initial $T/T_{F}<1$. The decay curves have been measured in a
$\bar{\omega}/2\pi=$160-Hz (circles) and 400-Hz trap (squares). As a
reference, we plot the $K_{3}$ resonant peak value, corresponding to the
maximum in Fig. 3(b). The dot-dashed line shown for
$T>$5\text{\,}\mathrm{\SIUnitSymbolMicro K}$$ indicates the $T^{-2}$
dependence, according to Eq. (2), with $\zeta=0.022$. The dashed line
indicates the lower limit to the measurable $K_{3}$ value, imposed by one-body
losses.

##  IV Conclusion

In summary, we have carried out Feshbach spectroscopy on an optically trapped
spin-polarized degenerate Fermi gas of 161Dy atoms by measuring three-body
recombination losses. We have focused on the range of low magnetic fields up
to 1 G, scanned with a high resolution on the order of 1 mG. The ultradense
loss spectrum revealed a stunning complexity with 44 resolved loss features,
some of them showing up in groups and other ones appearing as isolated
individual features. We also observed low-loss plateaus, which are typically a
few 10 mG wide and which are free of resonances. Here very low three-body
losses facilitate highly efficient evaporative cooling.

We have studied selected resonance features in more detail by measuring the
three-body recombination rate coefficient $K_{3}$ upon variation of the
magnetic resonance detuning and the temperature. In general, the observed
behavior shows strong similarities with recent observations on $p$-wave
Feshbach resonances [25, 26]. At higher temperatures (above a few $\mu$K) we
observed the unitarity limitation of resonant three-body losses. At low
temperatures in the nanokelvin range, we observed a strong suppression of
losses with decreasing temperature, provided a small detuning of just a few mG
is applied. Right on top of the resonance, however, three-body losses remain
very strong even at the lowest temperatures we can realize.

Our work shows that in experiments employing fermionic 161Dy gases special
attention must be payed to choosing and controlling the magnetic field in a
way to avoid detrimental effects of three-body recombination losses. For our
specific applications targeting at strongly interacting fermion mixtures of
161Dy and 40K [10], those magnetic-field regions are of particular interest
where one can combine near-resonant interspecies Dy-K interaction with low-
loss regions of Dy.

###### Acknowledgements.

We acknowledge support by the Austrian Science Fund (FWF) within Projects No.
P32153-N36 and P34104-N, and within the Doktoratskolleg ALM (W1259-N27). We
further acknowledge a Marie Sklodowska Curie fellowship awarded to J.H.H. by
the European Union (project SIMIS, Grant Agreement No. 894429). We thank the
members of the ultracold atom groups in Innsbruck for many stimulating
discussions and for sharing technological know how.

##  Appendix A Summary of the initial experimental conditions

In Table 1, we report the experimental conditions under which the measurements
reported in Figs. 1, 2, 3 and 5 have been carried out. For the measurement in
Fig. 4, where the value of $K_{3}$ as a function of the $\tilde{T}$ is
reported, the different temperatures have been achieved by interrupting the
evaporation in a controlled way. This unavoidably has led to initial
experimental conditions which vary over a wide range. The initial atom number
ranges from $5\text{\times}{10}^{4}\text{\,}$ to
$1\text{\times}{10}^{6}\text{\,}$. The coldest samples have $T/T_{F}\simeq
0.18$ and peak densities
$\hat{n}\simeq~{}$1\text{\times}{10}^{14}\text{\,}{\mathrm{cm}}^{-3}\text{/}$$.

Table 1:  Experimental parameters for the different sets of measurements: geometrically averaged trap frequency $\bar{\omega}$, atom number $N$, temperature $T$, Fermi temperature $T_{F}$, reduced temperature $T/T_{F}$, the function $\beta(x)$ according to Eq. (6), and the peak density $\hat{n}$. All quantities represent the initial values after preparation (at $t=0$). Figure | $\bar{\omega}/2\pi($\text{\,}\mathrm{Hz}$)$ | $N$ | $T~{}($\text{\,}\mathrm{nK}$)$ | $T_{F}~{}($\text{\,}\mathrm{nK}$)$ | $T/T_{F}$ | $\beta(T/T_{F})$ | $\hat{n}~{}($\text{\,}{\mathrm{cm}}^{-3}\text{/}$)$
---|---|---|---|---|---|---|---
Fig. 1 | 100 | $5\text{\times}{10}^{4}\text{\,}$ | 70 | 320 | 0.21 | 0.616 | $4.8\text{\times}{10}^{13}\text{\,}$
Fig. 2(a) | 58 | $1.22(4)\text{\times}{10}^{5}\text{\,}$ | 29(3) | 250 | 0.12 | 0.84 | $3.1\text{\times}{10}^{13}\text{\,}$
Fig. 2(b) | 120 | $4.3(1)\text{\times}{10}^{4}\text{\,}$ | 83(3) | 367 | 0.23 | 0.59 | $5.7\text{\times}{10}^{13}\text{\,}$
Fig. 2(c) | 380 | $1.53(3)\text{\times}{10}^{5}\text{\,}$ | 358(8) | $1.77\text{\times}{10}^{3}\text{\,}$ | 0.20 | 0.63 | $6\text{\times}{10}^{14}\text{\,}$
Fig. 3(a) | 88 |  $\simeq$ $1.2\text{\times}{10}^{5}\text{\,}$ | 64 | 379 | 0.17 | 0.72 | $6\text{\times}{10}^{13}\text{\,}$
Fig. 3(b) | 118 |  $\simeq$ $4\text{\times}{10}^{4}\text{\,}$ | 77 | 352 | 0.22 | 0.59 | $5\text{\times}{10}^{13}\text{\,}$
Fig. 3(c) | 118 | 4 - $6\text{\times}{10}^{4}\text{\,}$ | 77 - 88 | 352 - 403 | 0.22 | 0.59 | 5 - $6\text{\times}{10}^{13}\text{\,}$
Fig. 3(d) | 118 |  $\simeq$ $5\text{\times}{10}^{4}\text{\,}$ | 106 | 379 | 0.28 | 0.46 | $5.2\text{\times}{10}^{13}\text{\,}$
Fig. 5(a) green stars | 118 |  $\simeq$ $4\text{\times}{10}^{4}\text{\,}$ | 77 | 352 | 0.22 | 0.59 | $5\text{\times}{10}^{13}\text{\,}$
Fig. 5(a,b) blue diamonds | 157 | 4.5 - $6\text{\times}{10}^{4}\text{\,}$ | 86 - 126 | 487 - 536 | 0.16 - 0.26 | 0.50 - 0.74 | 0.78 - $1\text{\times}{10}^{14}\text{\,}$
Fig. 5(b) red squares | 157 | 4 - $5.5\text{\times}{10}^{4}\text{\,}$ | 88 - 187 | 468 - 520 | 0.17 - 0.4 | 0.26 - 0.71 | 0.6 - $1\text{\times}{10}^{14}\text{\,}$

##  Appendix B Extraction of the loss-rate coefficient

Here we summarize our method to extract values for the three-body rate
coefficient $K_{3}$ from the decay curves. Basically the same procedures have
been applied in Ref. [10].

The particles that are more likely to collide and leave the trap are the ones
in the center of the trap, with highest density and lowest potential energy.
Therefore losses are accompanied by heating of the cloud, which is known as
antievaporation heating in the thermal case [49] or hole heating in the case
of a degenerate Fermi gas [50]. As a consequence the shape of the density
distribution $n(\mathbb{r},t)$ changes with time. Taking a time-dependent
temperature $T(t)$ into account, Eq. (1) leads to a set of coupled
differential equations (see for instance [49]). We circumvent this
complication by focusing on the initial decay rate $1/\tau=-\dot{N}(0)/N(0)$,
where $N(0)$ and $\dot{N}(0)$ represent the atomic number and its time
derivative, respectively, both at $t=0$. For the initial decay and thus the
decay time $\tau$ only the initial number density distribution
$n_{0}(\mathbb{r})=n(\mathbb{r},t=0)$ is relevant.

Neglecting one-body losses and considering only the initial part of the decay,
Eq.(1) leads to

$K_{3}=-\frac{N(0)}{3\tau}\left(\int
d^{3}r\,n_{0}^{3}(\mathbb{r})\right)^{-1}.$ (3)

For the limit of a thermal (Gaussian) distribution [49], the integration
results in

$K_{3}=\sqrt{3}\frac{T_{0}^{3}}{\tau N_{0}^{2}}\left(\frac{2\pi
k_{B}}{m\bar{\omega}^{2}}\right)^{3}.$ (4)

Here $\hbar$ is the reduced Planck constant, $k_{B}$ is the Boltzmann
constant, $m$ is the atomic mass, and $T_{0}$ is the initial temperature of
the sample. For the number density distribution of a degenerate Fermi gas, we
find

$K_{3}=\frac{3\pi^{4}}{4}\frac{1}{\tau
N_{0}}\left(\frac{\hbar}{m\bar{\omega}}\right)^{3}\frac{1}{\beta(T_{0}/T_{F})}.$
(5)

The function $\beta(T/T_{F})$ is defined as the three-body integral of a
finite-temperature Fermi gas normalized to the zero-temperature case:

$\beta(T/T_{F})=\frac{\int d^{3}\mathbb{r}\,n^{3}(\mathbb{r})}{\int
d^{3}\mathbb{r}\,n^{3}_{\textrm{TF}}(\mathbb{r})},$ (6)

where $n(\mathbb{r})$ describes the density profiles of non-interacting
fermion systems at finite temperature, and $n_{\textrm{TF}}(\mathbb{r})$
refers to the Thomas-Fermi profile at $T=0$. By numerical integration, we find
that the function can be well approximated numerically for $x=T/T_{F}\lesssim
1$ by $\beta\left(x\right)\simeq(1+12.75x^{2}+31.05x^{4}-8.46x^{6})^{-1}$.

In order to obtain the initial decay rate, we fit the decay curve with

$N(t)=\frac{N_{0}}{\sqrt[\alpha-1]{1+(\alpha-1)t/\tau}},$ (7)

which is the solution of the differential equation
$\dot{N}/N_{0}=-\tau^{-1}\,(N/N_{0})^{\alpha}$ for decay by few-body process
of order $\alpha$. This heuristic model allows us to access the initial time
decay $\tau$ without making an assumption on the true order of the loss
process. The fit parameter $\alpha$ absorbs the order $n$ of the recombination
process together with effects from heating. The initial atom number $N_{0}$ is
also derived from the fit, whereas the initial temperature $T_{0}$ is measured
separately. The value of $K_{3}$ is finally obtained from Eq. (4) or Eq. (5).

##  Appendix C Broadening Effects

Figure 5:  Effect of magnetic field broadening. The $K_{3}$ value is plotted
versus the magnetic field, in the region around $358\text{\,}\mathrm{mG}$. (a)
Effect of the magnetic field gradient used for levitating Dy. (b) Effect of a
sinusoidal magnetic field modulation, with a frequency of
$600\text{\,}\mathrm{Hz}$ and an peak-to-peak amplitude of 8 mG.

When dealing with a Feshbach spectrum dense of narrow resonances, it is
important to understand and possibly eliminate potential broadening effects.
In our system, we identify two sources of broadening: magnetic levitation and
magnetic field noise. In experiments where a decrease of the trapping
frequencies leads to a trapping potential not deep enough to hold atoms
against gravity, magnetic field levitation is often used to cancel (or reduce)
the gravitational sag [51]. However, the presence of a magnetic field gradient
introduces an inhomogeneity of the magnetic field along the vertical extent of
the cloud. Assuming full levitation for dysprosium
($\partial_{z}B=$2.83\text{\,}\mathrm{G}\text{/}\mathrm{cm}$$) and a typical
Thomas-Fermi radius $\sim\\!\\!$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$$,
our atomic sample is subjected to a magnetic-field variation of
$\sim\\!$6\text{\,}\mathrm{mG}$$ over the trap volume. In Fig. 5(a) we
demonstrate the effect of levitation broadening on the $K_{3}$ coefficient. We
work at the 358-mG resonance. If no gradient is applied we can resolve a
double-peak structure. The two features have a full width of about
$2\text{\,}\mathrm{mG}$, with peak values of
$6(3)\text{\times}{10}^{-27}\text{\,}\mathrm{a}$nd
$1.5(4)\text{\times}{10}^{-25}\text{\,}{\mathrm{cm}}^{6}\text{/}\mathrm{s}$,
respectively. The presence of the magnetic field gradient reduces the
resolution and the two peaks almost merge into a single feature 7.7-mG wide.
We can still distinguish two local maxima, whose values are reduced with
respect to the no-levitation case. In view of this broadening effect, we
decided to carry out all measurements reported in the main text without
magnetic levitation. To achieve low enough trap frequencies in the shallow
trap where the atoms are transferred at the end of the evaporation, we employ
the second ODT stage with an increased waist of the horizontal trapping beam,
as mentioned in Sec. II.1. Such a trap geometry allows us to reach low
trapping frequencies without too much sacrificing the trap depth.

In another set of measurements, we investigate the effect of magnetic field
fluctuations. We artificially introduce noise into the system by adding a
sinusoidal magnetic field modulation to the bias field. We chose a modulation
frequency of $600\text{\,}\mathrm{Hz}$, i.e. faster than the trap frequencies,
but still slow enough to avoid technical complications. We then measure decay
curves for different modulation strengths and magnetic field detunings. In
Fig. 5(b) we report the obtained $K_{3}$ values versus magnetic field
detunings for 10-mG peak to peak modulation. As a reference, we plot the
$K_{3}$ profile measured in the absence of artificial noise. The modulation
results in a broadening of the feature and consequent loss of field
resolution. The peak value decreases and the curve flattens off.

Regardless of the source of broadening, a magnetic-field inhomogeneity (in
time and space) has an averaging effect on the $K_{3}$ coefficient, and leads
to a broadening and weakening of the narrow loss features characterizing the
dysprosium Feshbach spectrum.

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