algorithm[ALG.]

††thanks: Current address: Institute for Biomedical Physics, Medical
University of Innsbruck, Austria<EMAIL_ADDRESS>
# A ship-in-a-bottle quantum gas microscope for magnetic mixtures

Maximilian Sohmen *  — Institute for Quantum Optics & Quantum Information,
Austrian Academy of Sciences, 6020 Innsbruck, Austria  Institute for
Experimental Physics, Leopold-Franzens-Universität, 6020 Innsbruck, Austria
Manfred J. Mark *  — Institute for Quantum Optics & Quantum Information,
Austrian Academy of Sciences, 6020 Innsbruck, Austria  Institute for
Experimental Physics, Leopold-Franzens-Universität, 6020 Innsbruck, Austria
Markus Greiner  Department of Physics, Harvard University, Cambridge,
Massachusetts 02138, USA  Francesca Ferlaino *  — Institute for Quantum Optics
& Quantum Information, Austrian Academy of Sciences, 6020 Innsbruck, Austria
Institute for Experimental Physics, Leopold-Franzens-Universität, 6020
Innsbruck, Austria

###### Abstract

Quantum gas microscopes are versatile and powerful tools for fundamental
science as well as promising candidates for enticing applications such as in
quantum simulation or quantum computation. Here we present a quantum gas
microscopy setup for experiments with highly magnetic atoms of the lanthanoid
elements erbium and dysprosium. Our setup features a non-magnetic, non-
conducting, large-working-distance, high-numerical-aperture, in-vacuum
microscope objective, mounted inside a glue-free quartz glass cell. The quartz
glass cell is enclosed by a compact multi-shell ferromagnetic shield that
passively suppresses external magnetic field noise by a factor of more than a
thousand. Our setup will enable direct manipulation and probing of the rich
quantum many-body physics of dipolar atoms in optical lattices, and bears the
potential to put exciting theory proposals – including exotic magnetic phases
and quantum phase transitions – to an experimental test.

2D

    two-dimensional
AC

    alternating current
AISI

    American Iron and Steel Institute
AOD

    acousto-optic deflector
AOM

    acousto-optic modulator
AR

    aspect ratio
BDG

    Bogoliubov–de Gennes
BEC

    Bose–Einstein condensate
BKT

    Berezinskii–Kosterlitz–Thouless
CAD

    computer-aided design
CCD

    charge-coupled device
CNC

    computer numerical control
CF

    con-flat
DC

    direct current
DDI

    dipole-dipole interaction
DMD

    digital micromirror device
DOI

    digital object identifier
EFL

    effective focal length
EMCCD

    electron-multiplying charge-coupled device
EOM

    electro-optic modulator
FEM

    finite-element method
FOV

    field of view
FWHM

    full width at half maximum
GP

    Gross–Pitaevskii
GPE

    Gross–Pitaevskii equation
IC

    infinite conjugation
IR

    infrared
LHY

    Lee–Huang–Yang
LDA

    local density approximation
LRO

    long-range order
MF

    mean field
MOPA

    master-oscillator power amplifier
MOT

    magneto-optical trap
MW

    microwave
NA

    numerical aperture
NEG

    non-evaporable getter
OD

    optical depth
ODLRO

    off-diagonal long-range order
ODT

    optical dipole trap
OPD

    optical path difference
OTF

    optical transfer function
PCA

    principal-component analysis
PSF

    point-spread function
QF

    quantum fluctuations
QFT

    quantum field theory
RF

    radio frequency
RMS

    root mean square
RWA

    rotating-wave approximation
SCMOS

    scientific complementary metal-oxide semiconductor
SIL

    solid-immersion lens
SNR

    signal”=to”=noise ratio
SNOM

    scanning near-field optical microscopy
TDL

    thermodynamic limit
TOF

    time of flight
UHV

    ultra-high vacuum
UV

    ultraviolet
USAF

    United States Air Force
WD

    working distance

## Introduction

###  .1 Background

Understanding and control of the interplay between light and matter at the
microscopic level has enabled revolutionary insights in fundamental physics
and plays an increasing role for applications [1, 2, 3, 4, 5]. A pivotal
advantage of such quantum optics approaches is the ability to custom-tailor
quasi-pure model systems, which can be prepared, manipulated and probed with
high fidelity while being near-perfectly isolated from environmental sources
of noise. Quantum gas microscopes, in particular, enable the manipulation and
imaging of individual, ultracold particles (typically: neutral atoms) pinned
in a two-dimensional (2D) optical lattice potential, allowing to study quantum
many-body physics on the single-particle level [6, 7, 8]. Owing to the
superior signal”=to”=noise ratio, site-resolved detection is usually based on
fluorescence imaging. To counteract recoil-heating of the pinned atoms by the
scattered fluorescence photons (which could cause them to hop to nearby
lattice sites), it is typically crucial to combine the imaging procedure with
an in-trap optical cooling scheme.

The first quantum gas microscopes used bosonic alkali atoms (87Rb) and went
into operation in 2009/10 [9, 10]. They enabled studies of quantum phase
transitions and the associated Higgs mode [11], particle correlations [12],
quantum dynamics [13, 14] and other, similarly fundamental phenomena [6, 8].
About five years later, two groups demonstrated microscopes for the bosonic
lanthanoid 174Yb [15, 16], whose complex electronic level structure promises
opportunities for realising new quantum information protocols. Microscopy of
alkali fermions – in contrast to the alkali boson 87Rb – initially proved
challenging, since the small hyperfine splitting and low mass required the
refinement of optical cooling procedures [17, 6]. The coming into operation of
five different fermion microscopes in 2015 – using either 6Li [18, 19] or 40K
[20, 21, 22] – marked the beginning of a long series of important results for
Fermi systems, such as the direct observation of band [19] and (fermionic)
Mott insulators [21, 23], antiferromagnetic ordering [24, 25], and many more
[6, 8].

The vast majority of experiments with quantum gas microscopes has so far
concentrated on atoms interacting via a short-range contact interaction. In
such systems, atoms do not directly experience an energy shift depending on
whether a neighbouring lattice site is occupied or not – neighbour
interactions are only introduced via a (weak) second-order tunnelling process,
the so-called super-exchange interaction [26]. If, in contrast to purely
contact-interacting systems, an additional interaction of long-range character
– i.e., with an associated length scale similar to or larger than the lattice
spacing – is present in a system, profound differences and new physics are to
be expected [27, 28].

Currently, three different platforms are the main candidates for realising
experiments with long-range interactions on optical lattices, and each of them
has individual advantages and drawbacks. First, coupling optical-lattice
ground-state atoms to Rydberg states induces a shift of the electronic energy
levels, effectively dressing the atoms with a Rydberg character [29, 30, 31].
This results in an effective interaction potential between the atoms, whose
strength and range can be controlled, e.g., via the detuning and intensity of
the dressing laser, or the specific Rydberg state used. However, Rydberg
admixtures have a limited lifetime and are highly susceptible to environmental
stray electric fields [29]. Second, one can exploit the electric dipole-dipole
interaction (DDI) between ground-state polar molecules [32, 33, 34]. Polar
molecules, too, feature strong, long-range interactions which are – over a
certain range – tunable via an external electric field, but this comes at the
price of rather demanding molecule preparation schemes (many steps for, e.g.,
association, cooling, collisional shielding, and others) as well as
substantial particle losses due to complex, reactive collision processes [35].
Third, one can exploit the anisotropic and long-range magnetic DDI between
atoms with a strong, permanent magnetic dipole moment such as chromium [36],
dysprosium [37], or erbium [38]. Despite their lower interaction strength
compared to Rydberg atoms and polar molecules, experimental realisations
benefit tremendously from straightforward cooling and preparation schemes as
well as long lifetime of samples of ultracold magnetic atoms.

Some direct effects of magnetic DDI on lattice physics have already been
studied using conventional time-of-flight techniques [39, 40, 41, 42].
Ultimately, however, it would be very desirable to perform experiments with
dipolar atoms on a lattice in a quantum gas microscope, where many phenomena
are much more directly accessible.

###  .2 Key features of erbium and dysprosium

Table 1: Selected optical transitions of erbium and dysprosium. $\sim\Gamma/2\pi$ |  $\lambda_{\text{{Er}}}/$nm |  $\lambda_{\text{{Dy}}}/$nm | usage example | refs
---|---|---|---|---
$30$ MHz |  $401$ |  $421$ |  Zeeman slowing, fluorescence imaging |  [43, 44]
$100$ kHz |  $583$ |  $626$ |  narrow-line MOT |  [43, 44]
kHz |  $741$ |  $841$ |  narrow-line MOT |  [45, 46]
Hz |  $1299$ |  $1001$ |  spin manipulation |  [47, 48]

All quantum gas microscopes are tailored specifically to the needs of their
atomic species. This is necessitated by the demanding properties such a setup
has to offer, for example high optical resolution or dedicated imaging and
cooling schemes. Hence, let us first summarise some of the decisive features
of erbium and dysprosium that distinguish them, e.g., from the alkali metals
which predominate in quantum gas microscopy today.

Magnetic moment. Most importantly, of course, erbium and dysprosium feature
large permanent magnetic dipole moments of $7\,\mu_{\text{B}}$ and
$10\,\mu_{\text{B}}$, respectively, where $\mu_{\text{B}}$ is the Bohr
magneton (cf. rubidium: $1\,\mu_{\text{B}}$).

Isotope options. Both erbium and dysprosium posses a variety of bosonic as
well as fermionic isotopes, which can be cooled and captured efficiently in a
parallel magneto-optical trap [49] and of which several combinations have been
brought to mixture degeneracy [50].

Feshbach resonances. The erbium and dysprosium isotopes offer dense Feshbach
spectra with a comfortable number of broad resonances at easily accessible
field strengths, favourable for contact interaction tuning [51, 52, 53] or
molecule formation [54]. Also interspecies Feshbach spectra of several isotope
combinations provide conveniently broad resonances [55].

Electronic energy spectra. Erbium and dysprosium possess a large number of
electronic spectral lines (see, e.g., Fig. 1 in Ref. [56]), with a great
variety of different widths, from broad to extremely narrow (see Table 1).
From these transitions the most suitable ones can be picked for the desired
task (such as cooling, imaging, shelving, etc.). Note that the broadest
transitions of erbium and dysprosium are in the blue part of the visible
spectrum, hence yield a high resolution according to the Abbe limit.

Large mass. Erbium and dysprosium are comparatively heavy elements (depending
on the isotope, between 161 and 170 atomic mass units). Therefore, recoil
velocities are exceptionally low for all optical transitions (in particular,
compared to the light alkali elements).

Large spin manifold. One of the reasons for the large magnetic moments of
erbium and dysprosium is the large angular momentum in the electronic ground
states, with $J=6$ ($J=8$) for the bosonic isotopes of erbium (dysprosium),
and $F=19/2$ ($F=21/2$) for fermionic erbium (dysprosium). The corresponding
Zeeman and hyperfine states can be used to emulate spin Hamiltonians or to
implement synthetic dimensions [57, 58].

These features of erbium and dysprosium give opportunity for new techniques
that we would like to try with our quantum gas microscope.

Free-space imaging. It has been proposed [59] that – similar to free-space
imaging of a single atom released from an optical tweezer [60, 61] – for
erbium and dysprosium atoms on an optical lattice the combination of a broad
electronic transition (i.e., high photon scattering rate) and large atomic
mass (i.e., slow diffusion) might enable fast imaging protocols that work
without optical cooling and pinning lattice. A first experimental
demonstration of such a free-space imaging of a lattice gas has been reported
very recently [62]. The ability to reliably reconstruct site occupations using
such a fast, free-space imaging protocol can greatly simplify the detection
scheme and offer a way to avoid the parity-projection problem almost all
current quantum gas microscopes are facing [8]. In addition, it might prove
useful for imaging of bulk systems with high resolution.

Spin manipulation and detection. The combination of large ground-state spin
space and narrow optical transitions makes erbium and dysprosium ideal
candidates for high-fidelity spin-state control [48, 47]. This can be used to
reliably prepare and probe specific states in the quantum gas microscope, to
dynamically drive phase transitions [63], and to perform spin-selective
imaging and shelving [64].

DDI-mediated lattices. In a doubly-dipolar mixture (species A and B – in our
case: one erbium, one dysprosium) it is possible employ an optical lattice
whose wavelength $\lambda_{l}$ is tuned to a value where the polarisability
vanishes for species A, $\alpha_{\text{A}}(\lambda_{l})\approx 0$, but not for
species B.111More precisely, $\Im\\{\alpha_{\text{B}}(\lambda_{l})\\}\approx
0$, so negligible absorption [65], but
$\Re\\{\alpha_{\text{B}}(\lambda_{l})\\}>0$. Whereas this lattice will be
invisible for species A, it can be used to pin species B. Since, however, both
species interact via DDI, species A will see an effective periodic potential
stemming from the pinned species B. As the pinned atoms can vibrate, the DDI-
mediated periodic potential itself supports phononic excitations – in contrast
to a conventional optical lattice, which is infinitely stiff. Following a
different approach, very recently an optical lattice supporting phonons was
realised using a confocal optical resonator [66], shedding first light on the
rich physics of elasticity in quantum solids.

###  .3 Potential research directions

There exists a wealth of promising research proposals that could be followed
using a quantum gas microscope for dipolar atoms such as erbium or dysprosium,
and yet further ones for a combination of two dipolar species. For instance,
in lattice systems, the additional presence of the DDI requires an extension
of the standard Bose- and Fermi-Hubbard models and lattice spin models. This
gives rise to novel, qualitatively different quantum phases and transport
dynamics. Here we list some examples of potential research directions that
could in our opinion be particularly interesting to follow.

Exotic ground states. The long-range and anisotropic nature of the DDI
dramatically changes the behaviour of bosonic as well as fermionic atoms in
optical lattices. Dipolar bosons on a square lattice, for example, are
expected to possess many-body ground-states resembling a charge-density wave.
At half-filling, this can take the form of checkerboard or stripe
configurations, and the phase diagram might host supersolid regions [67, 27,
28]. From theory it is expected that changing the dipole orientation drives
direct transitions between these phases; in close proximity of the transition,
the existence of metastable emulsion phases has been predicted [68]. For spin-
polarised dipolar fermions on a 2D lattice, in contrast, changing the dipole
orientation is expected to give rise to a topological phase transition linked
to the deformation of the Fermi surface, the so-called Lifshitz transition
[69]. Taking the spin degree of freedom into account, erbium as well as
dysprosium allow to implement a large number of different models, including
spin-$1/2$ and spin-$1$ systems, with up to 20 (22) available spin states for
fermionic erbium (dysprosium). A recent proposal suggests to use a 1D chain of
spin-$1/2$ dipolar fermions to form a bond”=order”=wave phase, which is
strongly correlated and topologically protected [70]. Other theory works have
shown that also topological flat bands [71] or fractional Chern insulators
[72] might be realisable. In principle, the whole class of systems described
by spin Hamiltonians – like quantum magnetism models, spin liquids, and
frustration – could be implemented using erbium and/or dysprosium atoms [73,
74].

Non-equilibrium dynamics. Beyond the preparation and detection of exotic
ground states, a quantum gas microscope is also well suited to study system
dynamics. For example, it is expected that the DDI leads to cluster formation,
where two or more atoms on neighbouring lattice sites are effectively bound
together. These clusters can move around, but their speed is non-trivially
dependent on the lattice geometry and system parameters [75, 76].
Interestingly, such clusters can become fully localised without a need for
disorder. Another phenomenon in reach is Levy flights – spin excitations that
move through a sparse, randomly occupied lattice [77]. As a final example, it
could be interesting to study the long-term dynamics of spin states within the
Heisenberg XXZ model, where a peculiar difference in the equilibrium
distribution is expected between scenarios with half”=integer- compared to
integer-spin manifolds [78].

Evidently, this is only a subjective choice of promising research directions.
Many more already exist or might become apparent once a dipolar quantum gas
microscope is in operation. In the following, we describe the quantum gas
microscope that we have designed as an addition to our erbium-dysprosium
quantum mixture experiment. The microscope is housed in a separate ultra-high
vacuum (UHV) section and has been assembled, evacuated, baked, tested, and
finally attached to the existing apparatus through a mechanical UHV gate
valve.

##  I Microscope objective

Figure 1: Types of quantum gas microscopes. (a) All optics outside vacuum chamber, (b) SIL close to sample, (c) optics partly in vacuum, (d) entire optics in vacuum.  Table 2: Design types of quantum gas microscopes. type | pro | contra | refs
---|---|---|---
(a) |  easy alignment
long WD |  limited NA |  [22, 10, 19, 16]
(b) |  high NA
mech. rigid |  short WD
$\lambda_{l}$ fixed by coating |  [21, 15, 18]
(c) |  moderate NA
moderate WD |  difficult alignment
relative vibrations |  [79, 80]
(d) |  high NA
easy alignment
long WD |  UHV risks |  [81, 82, 83, 84]

The most important component of the Er–Dy quantum gas microscope is its
imaging objective. In the following, we will first discuss design options and
motivate our choice. Next, we detail our design’s optical and mechanical
properties, and finally present the measured performance.

Figure 2: Calculated key characteristics of our microscope objective at the
erbium and dysprosium imaging wavelength. (a) On-axis PSF vs radial coordinate
$\rho$ (in object-space units); the value at $\rho=0$ equals the Strehl ratio.
(b) RMS wavefront error vs lateral field (left) and vs focus position (right).
Dashed horizontal lines give the diffraction limit. (c) Tangential fan of
optical path difference (OPD) compared to chief ray vs normalised pupil
coordinate $P_{y}$. Solid (dashed) lines correspond to zero
($40\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$) lateral field along $y$.
(d) OTF; the solid black line gives the diffraction limit. Magenta (cyan)
corresponds to to zero ($40\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$)
lateral field along $y$. Solid (dashed) lines correspond to tangential
(sagittal) direction. Insets: A zoom-in for intermediate spatial frequencies.

###  I.1 Initial options and considerations

Quantum gas microscopes are highly complex machines, often operating at the
edge of what is technologically possible. Therefore, the optical design
typically needs to be carefully tailored to the experimental needs. For the
Er–Dy experiment, we identified the following key requirements for the imaging
system:

  1. 1.

a high numerical aperture (NA) $>0.8$ to be able to (i) resolve lattices of
small spacing ($\leq$532\text{\,}\mathrm{nm}\text{/}$$) and for (ii) a high
fluorescence photon collection rate;

  2. 2.

a large working distance on the order of millimetres to give optical access
from the side, to grant full freedom of (transverse) lattice laser wavelengths
$\lambda_{l}$, and to avoid possible effects of close surfaces on the dipolar
atoms;

  3. 3.

use of non-magnetic and non-conducting materials to avoid magnetisation and
eddy currents;

  4. 4.

near achromaticity at the imaging wavelengths of erbium and dysprosium.

When a sample inside a UHV environment is imaged, the vacuum window
necessarily becomes part of the light path and needs to be considered in the
optical layout. The aberrations introduced by a plane-parallel glass plate are
mainly spherical and chromatic and scale with the thickness of the plate [85].
Different quantum gas experiments have found different solutions for dealing
with these aberrations, which can be broadly grouped in four categories as
sketched in Fig. 1. Table 2 gives a brief overview over the most important
properties and limitations of these approaches.

(a) Optics outside vacuum. This straight-forward option usually features a
long working distance (WD) of some to tens of millimetres which, for
reasonable optics diameters, limits the NA to below $\sim 0.7$. The usual
approach is to use a custom objective carefully corrected for a window of some
millimetres thickness [10, 19, 16]. However, the use of a commercial objective
in combination with a very thin ($<1\,$mm) and – due to bending under vacuum –
small-diameter window has also been demonstrated [22].

(b) Solid-immersion lens (SIL). Lenses in shape of a truncated sphere close to
the sample increase the NA [86]. Hemispherical SILs offer an enhancement
factor equal to their refractive index, $n$, and can be mounted in vacuum [9]
or be optically contacted to the glass window [21, 15, 18]. Notably, a window
that is part of the SIL does not add aberrations. SILs with Weierstraß
truncation offer an even higher NA enhancement by $n^{2}$ [87, 88, 84].
However, SILs have a very small (typically micrometre-scale) working distance,
necessitating sophisticated transport strategies to bring the atoms into
focus. Additionally, most often horizontal as well as vertical lattice beams
have to be reflected off the front surface, posing many constraints for the
optical coating, whose performance can quickly become limiting.

(c) Optics partly in vacuum. A first lens inside vacuum can reduce the
marginal ray angles and hence aberrations by the following window. Imaging
systems of this type have been demonstrated with moderate NAs around $0.5$
[79, 80]. Their disadvantage is that the relative alignment of in- and ex-
vacuum components is absolutely crucial and that great care must be taken to
prevent mechanical vibrations or long-term drifts.

(d) All optics in vacuum. In-vacuum objectives are conceptually simple and
allow for both a millimetre-level working distance and high NA. If the
objective is at infinite conjugation, the glass window does not add
aberrations. Despite these advantages, this design is less frequently
encountered in cold-atoms experiments, since the objective must be UHV-
compatible and the UHV chamber must be large. Nevertheless, objectives of this
kind have successfully been demonstrated in metal chambers [81, 83], in a
glued glass cell [82] and, very recently, in a quartz glass cell similar to
ours [84] – a parallel development for a neighbouring setup in Innsbruck.

For our objective, option (a) was rejected since it can hardly deliver the
targeted NA and resolution. The SIL design (b) – due to its short WD and
coating limitations – seemed incompatible with the number of different
wavelengths that need to be directed onto (multiple lattices, cooling, spin
manipulation, etc.) and collected from (broad-angle fluorescence) the erbium
and dysprosium atoms. Option (c) seemed risky since, at the level of optical
resolution aimed for, tolerances on alignment imperfections are small, and
drifts or relative vibrations could have proven fatal for the optical
performance. We hence opted for an in”=vacuum objective (d), which – in
combination with a glue-free glass cell – reminded us of the challenge to
build a ship in a bottle.

###  I.2 The Er–Dy objective design

Table 3: Important design values of the Er–Dy objective.

quantity | unit | design value
---|---|---
eff. focal length |  mm | 20.0
total length |  mm | 70.0
chrom. focal shift |  $\mathrm{\SIUnitSymbolMicro m}\text{/}$ | 0.39
depth of focus |  $\mathrm{\SIUnitSymbolMicro m}\text{/}$ | $\pm 0.25$
working f-numbera |  | 0.56
object-space NA |  | 0.89
|  |  $401\text{\,}\mathrm{nm}\text{/}$ |  $421\text{\,}\mathrm{nm}\text{/}$
object-space Airy radius |  $\mathrm{\SIUnitSymbolMicro m}\text{/}$ |  0.27 |  0.29
wavefront errora peak-valley |  $\lambda$ |  0.098 |  0.11
wavefront errora RMS |  $\lambda$ |  0.032 |  0.021
Strehl ratiob |  |  $0.96$ |  $0.98$
Ø diff.-lim. FOV |  $\mathrm{\SIUnitSymbolMicro m}\text{/}$ |  $160$ |  $180$

  * a

Evaluated over full pupil.

  * b

On axis (zero field).

Figure 3: In-vacuum optics. (a) The microscope objective with the miniature
lattice mirror (inset). The red dashed arrow indicates a vertical lattice
beam. (b) The objective mount with crown mirrors. Round mirrors serve to
reflect light onto the atoms and back (red dotted arrows), the rectangular
mirror protects the objective from the divergent transport beam (green dashed
arrows).

In this section, we present the high-NA in-vacuum objective for microscopy of
erbium and dysprosium atoms which we have developed together with a
manufacturer222Special Optics, Inc., NJ/USA. It was designed for achromatic
performance on the broad, blue imaging transitions of erbium and dysprosium at
$401$ and $421\text{\,}\mathrm{nm}\text{/}$ wavelength, respectively, as well
as for $633\text{\,}\mathrm{nm}\text{/}$, which is the alignment wavelength of
the manufacturer and, by coincidence, close to the red dysprosium transition
at $626\text{\,}\mathrm{nm}\text{/}$. The objective contains five singlet
lenses of different glasses with a sufficient transmission for blue light, and
no cemented elements to avoid outgassing. The objective tube and all lens
retaining mechanics are fully fabricated from machinable ceramics, see Fig. 3
(a). All volumes inside the housing and in between lenses are vented through
borings to avoid virtual leaks under UHV. The objective’s optical design
values are summarised in Table 3; moreover, some important calculated
characteristics are plotted in Fig. 2.

A custom miniature dielectric mirror333Optics Technology, Inc., NY/USA of
$1.5\text{\,}\mathrm{mm}\text{/}$ diameter has been glued to the objective
front lens using a high-performance UHV-compatible adhesive444Optocast 3415,
Electronic Materials, Inc., CO/USA, see Fig. 3 a. This miniature mirror blocks
only $\sim$3\text{\,}\mathrm{\char 37\relax}\text{/}$$ of the solid angle
covered by the objective lens ($\text{\acs{NA}}=0.89$), so hardly affects the
number of collected photons, and will serve to reflect off the vertical
lattice beams ($\lambda_{l}=$1550\text{\,}\mathrm{nm}\text{/}$$). This fixes
the lattice position relative to the objective and will help to reduce drifts
and vibrations, thus facilitate keeping the atoms in focus.

The objective is mounted in a home-built ceramics part, shown in Fig. 3 (b).
It features three titanium flat springs around its perimeter and sits on three
ruby balls to define the objective position within the glass cell (cf. Section
II.1). The individual pieces of the mount are assembled using vented titanium
screws and beryllium-copper disc springs which take up mechanical stress, such
as upon temperature changes during bake-out.

The top rim of the mount bears an arrangement of custom solid”=quartz”=glass
mirrors555Optico AG, Switzerland with UV-enhanced aluminium coating, which we
colloquially refer to as the ‘crown mirrors’. Eight crown mirrors (elliptic in
Fig. 3 b) are in staggered alignment with the side windows of the glass cell
and will serve to reflect laser beams onto the atoms, entering and exiting
through the large top window. One crown mirror (rectangular in Fig. 3 b)
protects the objective from being hit by the divergent, high-power optical-
transport beam (when its focus is away from the glass cell). We chose the
angle of the crown mirrors such that normal incidence on the top window (and,
thus, any standing-wave backreflection) is avoided.

We highlight that our microscope cell and interior are exclusively built from
materials which are non-magnetic and – besides small parts like titanium
screws and beryllium-copper disc springs – non-conducting. We thus maximally
avoid magnetisation effects and eddy currents that could deteriorate the
magnetic environment close to our magnetic atoms as well as limit field
switching times.

###  I.3 Optical performance

The performance of the microscope objective was tested by imaging the tip of a
scanning near-field optical microscopy (SNOM) fibre666MF001, Tipsnano OÜ,
Estonia onto a CCD camera with $50\times$ magnification [89, 88]. The SNOM
fibre tip has a nominal aperture of 50 to 100 nm and therefore can be used as
a good approximation of a point source.

Figure 4: (a) Chromatic focal shift. Normalised intensity vs on-axis position
($z$), for for $401\text{\,}\mathrm{nm}\text{/}$ (blue) and
$421\text{\,}\mathrm{nm}\text{/}$ (red). Solid lines are Gaussian fits to the
maxima; dashed lines represent the centres, grey shadings the standard
deviations. The distance between the maxima is around
$2.4\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$. (b, c) Optical
resolution. Images of a SNOM fibre tip (respective left) at
$401\text{\,}\mathrm{nm}\text{/}$ (b) and $421\text{\,}\mathrm{nm}\text{/}$
(c). Distances $(x,y)$ are in object-plane units. The azimuthally averaged
intensity profiles (respective right) are plotted vs the radial coordinate
($\rho$). The red lines are Gaussian fits.

At constant magnification, the peak intensity in a power-normalised image is
proportional to the Strehl ratio [90]. Figure 4 (a) shows the peak normalised
intensity when moving the SNOM fibre tip along the optical axis using a piezo
actuator. We observe a clear maximum - corresponding to the respective focus
position - for each of the two investigated wavelengths ($401$ and
$421\text{\,}\mathrm{nm}\text{/}$). The axial distance between these maxima,
the chromatic focal shift, is $\sim$2.4\text{\,}\mathrm{\SIUnitSymbolMicro
m}\text{/}$$. The objective is therefore close to, but not fully achromatic,
as initially targeted. According to the manufacturer, this is most probably
due to insufficient accuracy of their prior knowledge of refractive indices of
the lens glasses (they had to be extrapolated down to 401 nm wavelength, where
the dispersion is steep). For imaging of only one species per time, this does
not pose a problem, since the shift can be corrected by re-adjusting the
camera position. To be able to image both species in a single experimental
run, the chromatic shift needs to be corrected for. Possible experimental
solutions include (i) using a dichroic mirror to separate the beam paths and
image the two species on separate cameras, or to image both species shortly
after each other in combination with either (ii) an adaptive optical element,
such as a fast focus-tunable lens [91], (iii) an imaging lens on a fast
translation stage to dynamically adjust the focus position, or by (iv)
dynamically shifting the vertical lattice position between the respective
focal planes.

Figure 4 (b) shows images close to the foci [i.e., around the maxima in Fig. 4
(a)] at 401 and 421 nm, respectively, as well as azimuthally averaged and
fitted spot profiles. These measurements give an upper bound on the optical
resolution $d_{0}$ according to the Rayleigh criterion of

$d_{0}=\begin{cases}$0.29(1)\text{\,}\frac{\mathrm{\SIUnitSymbolMicro
m}}{}$\quad&\text{at}\quad\lambda=$401\text{\,}\mathrm{nm}\text{/}$,\\\
$0.30(1)\text{\,}\frac{\mathrm{\SIUnitSymbolMicro
m}}{}$\quad&\text{at}\quad\lambda=$421\text{\,}\mathrm{nm}\text{/}$,\end{cases}$

close to the values predicted by our simulations (Table 3).

The fluorescence from the atoms is collimated by the microscope objective
(focal length $f=$20\text{\,}\mathrm{mm}\text{/}$$), passes the vacuum window,
and needs to be re-focused (focal length $f^{\prime}$) on a camera chip to
form an image. For Nyquist sampling, a sufficient image magnification $M$ is
needed. Aiming for, e.g., a sampling of five pixels per lattice site with a
sensor pixel size of $d_{\text{px}}=$16.5\text{\,}\mathrm{\SIUnitSymbolMicro
m}\text{/}$$ (a typical value for EMCCD cameras), a lattice constant of
$0.266\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$, we would require
$M\gtrsim 310$ and $f^{\prime}\gtrsim$6.2\text{\,}\mathrm{m}$$. For larger
lattice spacings or smaller pixel size these numbers are smaller, but still
likely on the order of metres. Such long light paths would naturally suffer
from stability problems caused by mechanical vibrations or air flow. Therefore
we used numerical methods to design a telefocus system which consists solely
of stock lenses, has a large effective focal length
($6.2\text{\,}\mathrm{m}\text{/}$) but a small physical length
($1.1\text{\,}\mathrm{m}\text{/}$) and is fully achromatic at $401$ and
$421\text{\,}\mathrm{nm}\text{/}$.

##  II Vacuum integration

Experiments with ultracold atoms have to be conducted under UHV on the order
of $10^{-11}$ millibar to isolate the samples from the environment and permit
sufficiently long lifetimes. The basic design of the our microscope UHV setup
consists of a quartz glass cell – housing the objective – attached to a
stainless-steel777$316$LN and $316$L (AISI classification) for low relative
magnetic permeability. tube cross which connects to vacuum instrumentation and
the main experiment.

###  II.1 Quartz glass cell

The microscope objective is mounted inside a quartz glass cell (Fig. 5). The
quartz glass cell (compared to, e.g., a metal vacuum chamber) brings three
main advantages: First, superb vacuum quality, second, it is inherently non-
magnetic and non-conducting, third, it offers maximum optical access. Our
custom-manufactured888Precision Glassblowing of Colorado, Inc., CO/USA. glass
cell consists of a hollow, octagonal quartz glass corpus with one $3$” window
attached to the top and seven $1$” windows attached to side borings using a
glass-frit bonding technique. Fused silica (i.e., high-purity synthetic quartz
glass) was our window material of choice due to its low light absorption down
to below $400\text{\,}\mathrm{nm}\text{/}$, and small thermal lensing effects
even at high light intensity [92]. On the inside, our windows feature an
extremely broadband (reflectivity $<0.5$ % over several hundred nanometres)
and broad-angle ($0\text{\,}\mathrm{\SIUnitSymbolDegree}\text{/}$ to
$>$45\text{\,}\mathrm{\SIUnitSymbolDegree}\text{/}$$) gradient-index
antireflection nanostructure coating.999RAR.L$2$, Tel Aztec LLC, MA/USA On the
outside, the windows are uncoated; this combination offers maximum flexibility
with respect to the wavelengths that can be transmitted into or out of the
cell while still allowing external cleaning of the cell from dust, etc.

Figure 5: The microscope cell. Manufactured from quartz glass (body) and fused
silica (windows). Polished faces on left and bottom tube are for indium
sealing. The viewports feature a broadband, broad-angle, gradient-index
nanostructure coating on the inside. For scale, the hole distance in the
breadboard below is $25\text{\,}\mathrm{mm}\text{/}$.

One small- and one large-diameter quartz glass tube with polished flat end
lips (Fig. 5) are attached to the bottom and to one of the side borings of the
quartz glass corpus, respectively. The bottom tube allows to insert the
microscope objective and its mount (Fig. 3) into the quartz glass cell. The
side tube connects the cell to the existing UHV apparatus and forms the single
support of the cell against gravity (Fig. 7), whereby we avoid static
overdetermination which could lead to stress peaks and breaking.

###  II.2 Mounting concept

Figure 6: _Vacuum connection from main chamber _(left)_ via steel cross
_(centre)_ to quartz glass cell _(right)_._ The optical transport axis for the
atoms is indicated in green. The magnetic shielding around the quartz glass
cell is also shown.

During the microscope assembly, the objective in its mount, sitting on three
ruby balls101010Edmund Optics, Inc., NJ/USA on the bottom fused-silica window,
was carefully inserted into the quartz glass cell from below using a scissor
jack, until a glass”=to”=glass indium seal could be formed (see below). The
three titanium flat springs around the objective mount define the horizontal
position of the objective along the axis of the bottom tube [Fig. 3 (b)]. To
connect the quartz glass side tube to the tube cross, we engineered a small,
steel flat”=to”=knife-edge adapter piece. In a first, critical step, the flat
face of this adapter piece (cf. Fig. 3 in Ref. [84]) was connected to the
quartz glass cell using indium sealing. In a second, uncritical step, the
knife-edge face of the adapter was connected to the steel cross (Fig. 6) using
a standard copper gasket and con-flat (CF) steel flange. To form our indium
seals in a controlled fashion and to protect them afterwards, we designed
clamping rings that could be assembled around the quartz glass cell tube lips.
These clamps were in-house machined from a high-performance polymer.111111PAS-
PEEK GF$30$, glass-fibre reinforced polyether ether ketone, Faigle GmbH,
Austria Screw holes arranged circularly around the clamps allowed to press the
respective sealing surface together and squash the indium O-ring in between.
Conical beryllium-copper disc”=springs, placed head”=to”=head under each
screw, facilitated the loading of the clamps with even forces. The horizontal
arms of the custom steel cross connect the quartz glass cell to the
experimental main chamber via a UHV gate valve121212VAT Vakuumventile AG,
Switzerland and form part of the transport distance for our atomic samples.
The bottom vertical arm connects to a non-evaporable getter (NEG)
element131313Capacitorr Z$200$, SAES Getters S.p.A., Italy, whereas the top
arm connects to (i) a combined NEG/ion pump module141414Nextorr D$200$, SAES
Getters S.p.A., Italy, (ii) an ionisation gauge151515Tungsten-filament Bayard-
Alpert gauge, Agilent Technologies, Inc., CA/USA, and (iii) an all-metal angle
valve12 for attachment of external mechanical pumps. All metal vacuum pieces
were electropolished and vacuum-glowed161616Reuter Technologie GmbH, Germany.
prior to assembly for reduction of H2 outgassing as well as for reduction of
magnetic permeability.

For microscope experiments, quantum gas samples produced in the Er–Dy
apparatus main chamber are loaded into a single-beam optical dipole trap (ODT,
$532$ nm, around $15$ W). The ODT focus is then shifted through the vacuum
connection into the microscope cell by translation of a relay lens on an air-
bearing linear stage (see Fig. 6).

###  II.3 Indium sealing

Forming a glass”=to”=metal connection for UHV applications is not trivial. For
example, braze-alloy seals – as used by the majority of commercial
manufacturers – require that the thermal expansion coefficients of the metal
and the glass do not differ too strongly; otherwise temperature changes (which
are not completely avoidable during the production process or bake-out) could
lead to cracking of the glass.

Therefore, e.g., connecting stainless steel to quartz glass as in our case
would require to form a gradient-index transition,171717Larson Electronic
Glass LLC, CA/USA; private communication. i.e., to use several different
glasses between the two end materials to gradually match the expansion
coefficients. However, gradient-index transitions are typically long ($10$ to
$20\text{\,}\mathrm{cm}\text{/}$) and mechanically weak.

In our design process, two concerns disfavoured a gradient-index transition.
First, the prolongation of the atom transport distance; second, the risk of
breaking when the cell and microscope are supported solely by a long, fragile
gradient-index transition. We therefore decided to produce a direct
quartz”=glass”=to”=steel connection via indium sealing [93, 94, 95]. We also
used this indium sealing procedure to form a glass”=to”=glass connection
between our glass cell bottom tube and the bottom window once the objective
was inserted. For us, indium sealing offered two main advantages. First,
compared to adhesive glues, indium yields superior vacuum quality. Second,
compared to, e.g., heat-diffusion bonding, it can be formed at room
temperature – which is crucial since our objective may not be heated to above
$90\text{\,}\mathrm{\SIUnitSymbolCelsius}\text{/}$.181818Special Optics, Inc.,
NJ/USA; private communication.

Indium is a soft metal with a low melting point (about
$156\text{\,}\mathrm{\SIUnitSymbolCelsius}\text{/}$ [96]), low permeability,
and low outgassing rates [97]. Exposed to air it is covered by a thin,
passivating oxide layer. When mechanically deformed, however, like when
pressed onto a glass surface, this oxide layer breaks and fresh, reactive
metal is exposed. Such activated indium wets and reacts with glass, forming a
tight seal. In the following, we outline some cornerstones of our indium
sealing procedure.

Indium material. We used round indium wire191919Ø
$0.05\text{''}\approx$1.3\text{\,}\mathrm{mm}\text{/}$$, In
$99.995\text{\,}\mathrm{\char 37\relax}\text{/}$, Indium Corporation of
America Co., MD/USA, activated in hydrochloric acid ($37\text{\,}\mathrm{\char
37\relax}\text{/}$) for around $1\text{\,}\mathrm{min}\text{/}$ straight
before use, and connected two freshly cut, angled ends to an O-ring.

Surface finish. Our metal contact surfaces were lathed (not milled), with the
stroke direction concentric with the indium O-ring; our quartz glass contact
surfaces were polished (optically clear, not matt) and remained uncoated in
the sealing area.

Surface cleaning. Polluted surfaces underwent usual UHV cleaning: (i) cleansed
in water and detergent, (ii) rinsed with distilled water, (iii) blow dried,
(iv) acetone-cleaned in ultrasonic bath, (v) rinsed with fresh acetone, (vi)
air-dried.

Seal formation. We gently and evenly squashed each indium O-ring between the
two respective, clean contact surfaces by alternate tightening of screws
around the clamping ring (visible on right and bottom of Fig. 7); a feeler
gauge can help to monitor this process. We tightened the screws up to a few Nm
torque, accompanied by careful visual inspection. After tightening of the
screws, the indium seals typically had a thickness of around
$0.3\text{\,}\mathrm{mm}\text{/}$. Thereafter we performed helium leak tests;
small leaks may be closed by (i) stronger squashing, (ii) warming up of the
seal region, or (iii) simply leaving the indium flowing for some hours [98].

###  II.4 Baking and attachment

Figure 7: The quartz glass cell with objective under UHV after attachment to
the experiment.

When no more helium leaks were detected, our microscope assembly (still
detached from the main experiment) underwent a gentle bake”=out. For this, the
quartz glass cell and the connection to the steel cross were enclosed in a
clean metal container with resistive heating pads on the inside and insulation
foam mats on the outside. The less sensitive parts of the steel cross were
wrapped in layers of aluminium foil, heating wire, and insulating foam mats.
Several thermocouples were used for temperature monitoring and to avoid
gradients larger than a few degrees celsius over the whole assembly. We
linearly increased the temperature by a few degrees celsius per hour up to
$90\text{\,}\mathrm{\SIUnitSymbolCelsius}\text{/}$, then kept this temperature
while monitoring the vacuum on a residual gas analyser.202020Stanford Research
Systems, Inc., CA/USA After about two weeks, all relevant gas traces (in
particular, H2O) had fallen by several orders of magnitude and levelled off.
We then linearly ramped down to room temperature by a few degrees celsius per
hour. Note that the disk springs in our clamping rings serve a double purpose
during the bake”=out: first, they can take up force peaks when parts expand;
second, they maintain the force when the indium softens and flows. After the
bake”=out, clamping screws were re-tightened to about the same torque as
before; later, no more tightening was necessary.

Figure 8: Magnetic field coil system for the microscope chamber. The fields
are calculated by direct integration of the Biot–Savart law for $1$ A of
current, respectively. Note that in this figure we label the respective local
coil symmetry axis by $z$. Slow vertical bias coils (left column), fast
vertical bias or RF coils (middle column) and horizontal coils (right column).
First row: Coil geometry. Second row: Field on axis (note that we plot the
offset with respect to the atom position). Third row: Field gradient on axis.
Fourth row: Field curvature on axis.

After the bake”=out was completed, the microscope assembly was flooded with
dry argon gas, moved on its breadboard to our experiment table, and connected
to the Er–Dy main chamber through a CF flange. After evacuation, no further
baking was required. When the vacuum level in the microscope section had
dropped to a level of around $10^{-11}$ millibar, the UHV gate valve between
microscope section and Er–Dy main chamber was opened; lifetimes of quantum gas
samples measured in the main chamber remained unaffected by this. Figure 7
shows a photograph of the quartz glass cell with objective, after attachment
to the Er–Dy apparatus.

##  III Magnetic Environment

In all ultracold-atom experiments the ability to set the magnetic field in a
precise manner is absolutely essential, to define quantisation axes, to
control level splittings, or to tune s-wave interactions at Feshbach
resonances. For magnetic atoms such as erbium and dysprosium, where magnitude
as well as direction of the magnetic field are of decisive importance for the
stability and behaviour of a sample, this is even more the case.

In the following, we describe our coil system for shaping magnetic fields in
the quartz glass cell (Section III.1), as well as a ferromagnetic enclosure we
have designed to shield the atoms from external magnetic fields and noise
(Section III.2).

###  III.1 Microscope coils

Table 4: Coil specifications for the microscope chamber.

| vert. slow | vert. fast | horz.
---|---|---|---
$r_{0}$ (mm) |  $59$ |  $64$ |  $22.5$
coil separation (mm) |  $54$ |  $78$ |  $98$
windings |  56 |  4 |  16
centre field ($\mathrm{G}\text{/}\mathrm{A}$) |  $8.0$ |  $0.5$ |  $0.6$
inductancea ($\mathrm{F}\text{/}$) |  $3{\times}10^{-5}$ |  $2{\times}10^{-6}$ |  $3{\times}10^{-6}$
resistanceb ($\mathrm{m\SIUnitSymbolOhm}\text{/}$) |  700 |  55 |  80

  * a

Analytical approximation for ideal Helmholtz pairs.

  * b

For $1$-mm2 copper wire.

The trade-off in the design process of the microscope cell coils was between
achieving a maximum flexibility in the shaping and switching of magnetic
fields, blocking a minimum of optical access, as well as constraints posed by
our magnetic shield (spatially and in terms of material magnetisation).

Our coil system consists of four pairs of coils which are held in place by a
mechanical support structure. The individual pieces of the support structure
are CNC-milled from a high-performance polymer,212121PAS-PEEK GF$30$, glass-
fibre reinforced polyether ether ketone, Faigle GmbH, Austria assembled around
the quartz glass cell, and mounted directly to the cell flange. Two pairs of
coils (slow vs fast) are along the vertical (i.e., gravity) direction ($z$),
close to Helmholtz configuration; see Fig. 8. In contrast to the slow (high-
field) pair of vertical bias coils, the fast vertical coil pair has only few
windings and is intended for fast jumps in magnetic fields and for generation
of RF radiation. Two identical, mutually orthogonal horizontal coil pairs are
arranged along the diagonals of the transport direction (cf. Fig. 8) and allow
to set the field in the $(x,y)$-direction. All coils except the two bottom
vertical coils could be wound prior to assembly.

Even though FEM simulations (Section III.3) indicate that the magnetisation
threshold of the innermost layer of our magnetic shielding will not be reached
up to fields corresponding to more than a hundred Gauss in the cell centre, in
order to avoid magnetisation effects it will be advisable to restrict the
fields at the atom location to the few- or low tens”=of”=Gauss level. Table 4
summarises the characteristics of the coils, Fig. 8 shows the parts of the
coil design as well as the corresponding calculated fields, gradients and
curvatures.

###  III.2 Magnetic shielding guidelines

Figure 9: FEM simulation of shielding efficiency. (a–c) Vector fields (magenta
arrows) of $\vec{B}$ when an external homogeneous magnetic fields is applied
along the one of the three spatial directions, respectively. $\vec{e}_{x}$ is
along the transport axis, $\vec{e}_{z}$ is along the cylinder axis. The
strength of magnetic flux inside the metal is colour-coded, increasing from
blue to red. (d–f) Calculated magnetic flux density along the $x$-, $y$-, and
$z$-axis, plotted for the same external fields as in the top row (note that
axes pass through shield openings).

For very field-sensitive measurements it becomes necessary to protect the
atomic sample from magnetic stray fields in the environment, such as the earth
magnetic field or fields created by electric instrumentation. The protection
strategy depends on the noise frequency.

Low-frequency (tens of Hz down to DC) noise can be reduced either by active
compensation222222Since no probe for feedback can be put locally into the
vacuum chamber, this is often realised using feed-forward. or by enclosing the
experimental chamber in a passive shielding. There are two types of passive
magnetic shields, (i) superconducting shields at cryogenic temperature, and
(ii) soft-ferromagnetic shields. Since ferromagnetic shields work at room
temperature, they are much easier to integrate into quantum gas experiments
[99, 100, 101]. The working principle of ferromagnetic shields is flux
shunting, i.e., the shield has a high relative permeability and thus ‘guides’
the field lines around the protected volume. Among the most commonly used soft
ferromagnetic materials are Mu-metal and Supra-50. Of these two, Mu-metal has
the higher relative permeability and Supra-50 has the higher saturation flux
density (see Table 5).

For fast oscillating fields (tens of Hz and higher), eddy current cancellation
(‘skin effect’) becomes the dominant shielding process. This effect is the
strongest for good conductors such as copper, but in practice also
ferromagnetic DC shields typically provide a sufficient AC shielding [102].
The crucial task is therefore to find a good shielding for slowly-varying
fields. For the Er–Dy experiment, we designed a passive multilayer
ferromagnetic shield which will be surrounded by an additional, active field
stabilisation system.

Table 5: Relative permeability, $\mu_{r}$, and saturation flux density, $B_{s}$, for the Er–Dy shield materials [103]. material | $\mu_{r}$ |  $B_{s}$ (G)
---|---|---
Mu-metal |  $4.7\times 10^{5}$ |  $0.75\times 10^{4}$
Supra-50 |  $2\times 10^{5}$ |  $1.5\times 10^{4}$
Figure 10: FEM simulation of microscope coil effects. Left column (a, c) shows
$B$ on the innermost shield layer (colour code); small magenta arrows indicate
flux direction. Right column (b, d) shows the flux density $B$ along spatial
axes (note that all axes pass through shield openings). Top row (a, b) is for
$10\text{\,}\mathrm{A}\text{/}$ of current in the slow vertical coil pair
(along $\vec{e}_{z}$). Bottom row (c, d) is for
$10\text{\,}\mathrm{A}\text{/}$ in one of the (diagonal) horizontal coil pairs
(direction $\vec{e}_{+}$, where
$\vec{e}_{\pm}=(\vec{e}_{x}\pm\vec{e}_{y})/\sqrt{2}$). In (d), the blue,
green, and red line correspond to the flux density $B$ along the coil axis
($\vec{e}_{+}$), along the perpendicular axis ($\vec{e}_{-}$), and along $z$,
respectively.

The performance of a magnetic shield is characterised by the shielding factor

$S=B^{\prime}/B,$ (1)

where $B$ ($B^{\prime}$) is the field at the centre point in presence
(absence) of the shield. Estimated requirements for our future microscope
experiments suggested to target a shielding factor $S\sim 10^{3}$. Basic
analytical estimates (see Appendix) can already provide important guidelines
for designing a shield, some of which we summarise below:

Geometry. The best shielding performance is obtained for a spherical shell,
followed by a cylinder (intermediate) and box (inferior) [102]. Shields that
‘approximate a sphere’ (i.e., have a similar characteristic length along all
three spatial directions) show better performance.

Size. At fixed wall thickness, smaller shields are superior (Eq. 5).

Multilayer shields. Nesting multiple shields with thin walls is better than a
single shield with a thick wall – ‘it helps to shield the shielding’ (Eq. 8).

Discontinuities. Avoid discontinuities (cuts, improper welds, etc.) to ensure
unhindered guiding of magnetic flux. If discontinuities are unavoidable (e.g.,
for an assembly of multiple parts), the parts should have sufficient overlap
and mechanical contact.

Holes. Avoid openings – they lead to flux leakage. If an opening is
unavoidable, it can help to add a collar (exponential vs cubic suppression;
Eqs. 3–4).

###  III.3 Microscope shield design

Our guidelines discussed above led us to favour a compact multilayer
cylindrical shield. In particular, ‘compact’ means that the shield fits only
the cell and magnetic coils, such that, e.g., all optomechanical components
need to be placed outside. In our design process, we first drafted a prototype
design for a four-shell shielding that complied with all spatial constraints
posed by our experiment. Each shell consists of a bottom and a top half,
allowing to assemble the shield around the microscope cell. Second, we meshed
the detailed prototype CAD model – including holes for optical access, cables,
screws, etc. – and numerically analysed it using a finite-element method
(FEM). The simulations allowed us to study and refine our design with respect
to two major factors: passive shielding performance and magnetic saturation
caused by the microscope coils inside.

Shielding efficiency. To study of the shielding efficiency, we placed our
model in an external, static, homogeneous magnetic field pointing along the
three spatial directions.232323Note that for a more conservative simulation of
the shielding performance it could be helpful to artificially add small air
gaps between shield pieces. In this way one takes into account that – due to
manufacturing tolerances – pieces are typically not perfectly flush, which
reduces the flux guiding. In our case, the two most important improvements
upon this study were (i) to maximally downsize the hole for the vacuum
connection and to give it a collar, and (ii) to conically reduce the hole
diameters for the side windows from outside to inside (at constant NA).
Shielding efficiency simulation data for the revised design is shown in Fig.
9.

Avoiding saturation. We simulated the effects of the microscope coils inside
our magnetic shielding. In our case, the two most important improvements
following this study were (iii) to change the material of the innermost shell
from Mu-metal to Supra-50 and, wherever needed, (iv) rounding of edges as well
as adjustment of hole patterns and hole diameters to reduce local flux
focusing. Some results of the saturation analysis for the revised design are
shown in Fig. 9.

Figure 11: Four-shell ferromagnetic shield. (a) Bottom (left) and top half
(right, flipped upside-down) of the magnetic shield. (b) The fully assembled
shield. The collar on the side increases the shielding efficiency over the
opening for the vacuum connection along the transport direction.

Our revised shield design was manufactured by Magnetic Shields Ltd, UK (Fig.
11). All shells are held in position by nylon spacers. After fabrication, the
shells underwent a heat treatment ($4$ hours at
$1150\text{\,}\mathrm{\SIUnitSymbolCelsius}\text{/}$) for magnetic annealing.
The experimentally measured shielding factors are $>10^{3}$ in both axial and
transverse direction, meeting our initial target.

## Summary & Outlook

To conclude, we have designed and constructed a quantum gas microscopy
apparatus for the highly magnetic lanthanoid elements erbium and dysprosium.
Our microscope objective is non-magnetic, non-conducting, and features a high
NA as well as a millimetre-scale working distance. The objective is mounted in
vacuum, inside a glue-free, nanotexture-coated quartz glass cell. This
combination offers a unique flexibility in terms of laser wavelengths that can
be delivered onto the atoms, from $<380$ nm to over $1700$ nm.

We have developed and tested procedures and mechanical part designs for the
formation of compact glass-glass and glass-metal (in particular: quartz glass
to stainless steel) UHV seals using indium wire, which might inspire new
routes in UHV apparatus design.

In terms of magnetic field control, we have designed, simulated and
manufactured a versatile coil system which allows an accurate tuning of
magnetic field in all spatial directions but impacts the optical access as
little as possible. Further, we have designed and optimised a compact, four-
layer ferromagnetic magnetic shield which can be assembled around quartz glass
cell and coils. The shield suppresses external magnetic field noise by a
factor of $>10^{3}$ in all spatial directions and will facilitate conducting
high-precision measurements.

Work towards getting our setup ready for dipolar lattice experiments is
currently underway. Important next steps include (i) the implementation of
optical lattices, (ii) lattice loading from the transport ODT, (iii)
implementation of the fluorescence excitation and detection systems as well as
(iv) the development of – possibly lattice- and/or cooling-free – imaging
procedures. Once fully operational, we believe that our microscope will make
important contributions to the understanding of the complex physics of dipolar
quantum many-body systems.

## Conflict of Interest Statement

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

## Author Contributions

This work is based on a chapter of MS’s dissertation, presented to the
University of Innsbruck in July 2021 [104]. MS designed, simulated, assembled,
and tested the microscope objective, UHV setup, and magnetic shield, with help
from all team members. MG offered practical advice and general expertise
during the design process of the microscope. MJM and FF lead the Er–Dy
experiment and supervised the development of the microscope apparatus. MS
wrote the manuscript with valuable input from all other authors.

## Funding

This study received support from the European Research Council through the
Advanced Grant DyMETEr (No. 101054500) and the QuantERA grant MAQS by the
Austrian Science Fund (FWF, No. I4391-N). M.S. acknowledges funding from the
FWF via DK-ALM (No. W1259-N27).

## Acknowledgements

We would like to thank all past and present members of the Er-Dy team,
especially Philipp Ilzhöfer, Gianmaria Durastante, Arno Trautmann, Claudia
Politi, Matthew A. Norcia, Lauritz Klaus, and Eva Casotti. We thank our
group’s Erbium, Tweezer, and Theory team, as well as the whole Innsbruck
Ultracold Atoms community. We are particularly indebted to Emil Kirilov,
Innsbruck, for valuable discussions during the development process of indium
handling and sealing procedures. We thank Aaron Krahn, Anne Hébert, Gregory A.
Phelps, Sepehr Ebadi, and Susannah Dickerson from the Harvard Erbium team for
fruitful exchange. MS is particularly grateful for an insightful research stay
at Harvard. We acknowledge Dimitrios Trypogeorgos, INO-CNR and Università di
Trento, for helpful discussions during the early design process of our
magnetic shielding. We would further like to thank the IQOQI mechanical
workshop team for their expert advice and manufacturing of many pieces
necessary for the construction of our microscope setup.

## Data Availability Statement

The data sets presented in this work are willingly available upon reasonable
request.

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## Appendix

### Analytic estimates for ferromagnetic shields

In general, the shielding efficiency $S$ (Eq. 1) of a ferromagnetic shield
depends on (i) the relative permeability $\mu_{r}$ of the shield material,
(ii) the geometry of the shield, and (iii) the effect of potential holes.

#### Shield geometries

Geometrically, the best shielding performance would be obtained for a
spherical shell, however, for manufacturing reasons, the majority of magnetic
shields has a cylinder (intermediate) or box form (inferior) [102].

#### Effect of holes

Flat surface. Consider an infinite plane with with surface normal along
$\vec{e}_{x}$. Then a field $B$ entering through an opening in the plane will
drop as [102]

$B(x)\approx B(0)\,x^{-3}.$ (2)

Tube. Consider a tube along $\vec{e}_{x}$ with diameter $D$. Then, magnetic
fields entering entering through one end of the tube will decrease as [102]

$\displaystyle B(x)$ $\displaystyle\approx B(0)\,\text{e}^{-\beta
x/D}\quad\text{with}$ (3) $\displaystyle\beta$
$\displaystyle\approx\begin{cases}7&\text{for}\>\vec{B}\perp\vec{e}_{x}\\\
4.5&\text{for}\>\vec{B}\parallel\vec{e}_{x}.\end{cases}$ (4)

I.e., shielding is better for transverse ($\perp$) than for axial
($\parallel$) fields.

#### Cylindrical shields

Let the shield be a hollow cylinder with $\mu_{r}\gg 1$, diameter $D$, length
$L$, and shell thickness $d\ll D,L$.

We first consider the case $L\geq D$ and a homogeneous, transverse field. The
flux entering through the ends is exponentially suppressed, hence, the
residual field on the inside is strongly dominated by field “spilling” through
the walls (cf. Fig. 1 in Ref. [105], e.g.). Therefore, such a cylinder can
well be approximated by an infinite cylinder, for which the shielding
efficiency has the analytic solution [106]

$S^{\perp}=\frac{\mu_{r}d}{D}.$ (5)

For axial fields, except for pathologic cases [105], the performance is
improved when end caps are added to the cylinder. If they are taken into
account, the axial shielding efficiency for a cylinder of size ratio
$\alpha=L/D>1$ reads

$\displaystyle S^{\parallel}$ $\displaystyle=\frac{4\eta
S^{\perp}+1}{1+1/(2\alpha)}\qquad\text{with}$ (6) $\displaystyle\eta$
$\displaystyle=\frac{1}{\alpha^{2}-1}\left(\frac{\alpha}{\sqrt{\alpha^{2}-1}}\ln{(\alpha+\sqrt{\alpha^{2}-1})}-1\right),$
(7)

a geometry-dependent demagnetisation factor [107, 106]. For short cylinders
($\alpha\gtrsim 1$), the axial and transverse shielding efficiency are similar
in magnitude, $S^{\perp}\gtrsim S^{||}$. For long cylinders ($\alpha\gg 1$),
however, the axial shielding loses effect ($S^{||}\rightarrow 1$) due to field
spilling through the walls. This demonstrates the advantage of cylindrical
shields with $L\approx D$.

#### Multi-layer shields

The effect of a shield can be enhanced by enclosing it in a bigger shield.
Following a magnetic circuit approach [108, 102], the efficiency of a nesting
of $N$ individual shields can be worked out analytically, with the dominant
term

$S\approx
S_{N}\prod_{i=1}^{N-1}S_{i}\left(1-\left(\frac{X_{i}}{X_{i+1}}\right)^{k}\right).$
(8)

Here, $X\in\\{D,L\\}$ is the characteristic length scale of layer $i$, and we
count the layers from inside to outside ($X_{i}<X_{i+1}$).
$S\in\\{S^{\perp},S^{\parallel}\\}$ is the shielding factor in the respective
direction and – for cylindrical shields – given by Eqs 5–6. The scaling
exponent $k$ is geometry-dependent; in good approximation $k=3$ for a
spherical shield, and $k=2$ and $k=1$ for a cylindrical shield in transverse
and axial direction, respectively [102].

  *[DDI]: dipole-dipole interaction
  *[UHV]: ultra-high vacuum
  *[SIL]: solid-immersion lens
  *[WD]: working distance
  *[NA]: numerical aperture
  *[PSF]: point-spread function
  *[OPD]: optical path difference
  *[FOV]: field of view
  *[UV]: ultraviolet
  *[SNOM]: scanning near-field optical microscopy
  *[CF]: con-flat
  *[NEG]: non-evaporable getter
  *[RF]: radio frequency
  *[CNC]: computer numerical control
  *[FEM]: finite-element method
  *[DC]: direct current
  *[AC]: alternating current