has a finite $k$, which is then favoured rather than global collapse Salasnich
_et al._ (2003); Carr and Brand (2004).

Figure 19: Roton mode in a 166Er dBEC of cigar shape (cigar along $y$, dipoles
along $z$). (a) Average momentum distribution showing the appearance of two
side peaks at finite $k_{y}$ (along the cigar elongation) after a quench to
lower $a$. (b) Extracted $k_{\rm{rot}}$ for different traps, as a function of
$1/\ell_{z}$, $\ell_{z}$ the confinement length along the dipoles. Data (dots)
are compared to a semi-analytical theory (dotted line) and numerical
simulations of the GPE (dashed line). (c) Imaginary roton gap, $\Delta$,
versus $a$, extracted from the growth rate of the momentum-peak population and
comparison to the results of the semi-analytical theory (solid line). The
inset show the same data but as a function of the time after the $a$-quench.
Chomaz  _et al._ (2018).

$\bullet$ Experimental evidence for local collapse:

In dBECs, the role played by local collapse was first noted by an extensive
theoretical analysis of the data from Ref. Koch  _et al._ (2008) by Wilson et
al. Wilson  _et al._ (2009), going beyond the Gaussian ansatz (Eqs. (54)-(57))
and solving exactly the GPE (Eq. (9)). Their predictions, showing a better
agreement with the measured stability threshold at large trap anisotropy with
$\lambda\gtrsim 3$ (Fig. 17(b)), implies the occurrence of local collapse. Yet
the instability mechanism was not experimentally resolvable and later studies
of the collapse dynamics also let modulational instabilities remain elusive,
see Sec. V.1 Metz  _et al._ (2009).

More recently, the impact of such instabilities was experimentally studied on
finite-size anisotropically-trapped dBECs of 164Dy, either in pancake traps
Kadau  _et al._ (2016) or in cigar shaped ones Ferrier-Barbut  _et al._
(2016a, b) with a tight confinement along the dipoles. Here, after quenching
$a$ down, remarkable long-lived in-situ density structures were observed. The
observed density structures have been attributed to the occurrence of a
modulational instability following the quench Kadau  _et al._ (2016). Because
of the limited system size of the original pancake-shaped geometry in Ref.
Kadau  _et al._ (2016), the instability is expected to be driven by an angular
mode with $\langle k\rangle\times R_{\rm BEC}\sim 1$ and $m>1$ Wächtler and
Santos (2016a), as also experimentally evidenced in a recent set of experiment
Schmidt  _et al._ (2021). Following their first observation, the authors of
the above-cited experimental works characterised the absence or existence of a
modulational instability as a function of the trap geometry via its signature
on the final density distribution; i.e., the observation of single or multiple
droplets long after the quench Ferrier-Barbut  _et al._ (2018b). They identify
a critical trap aspect ratio $\lambda=1.87(14)$ (compared to the dipole
orientation, $z$) above which the modulational instability exists; see also
Sec. V.3.1. The instability itself was not experimentally investigated in
these works as they focused on the long-time behaviour, which results from a
subsequent intricate non-linear dynamics loosing track of the unstable mode
driving the collapse. The authors found that these final density distributions
were surprisingly stable, which revealed an unpredicted stabilisation
mechanism. This discovery set the ground for a new paradigm of quantum fluids,
which will be discussed in the next Section V.

$\bullet$ Experimental investigation of the roton instability:

As introduced in Sec. IV.1.3, Ref. Chomaz  _et al._ (2018) first observed the
roton excitation by probing the instability dynamics of a large, cigar-shaped
dBEC of 166Er with transverse magnetisation. Chomaz et al. performed a fast
interaction quench and studied the short-time evolution of dBEC. The authors
reported the transient appearance of remarkable structure in the momentum
distribution of the dBEC with a high-amplitude central peak framed along the
cigar-long axis by two lower-amplitude symmetric side peaks, see Fig. 19(a).
Based on a Bogoliubov picture relevant for the short-time dynamics, this was
interpreted as the coherent population of finite-momentum excitation modes
thanks to its privileged dynamical softening to imaginary energies, indicative
of an unstable roton mode. By further studying the peak position and the time
evolution of its population, the authors demonstrated the characteristic
scalings $k_{\rm{rot}}\sim 1/\ell_{z}$ and $\omega_{\rm
rot}\propto(a-a^{*})^{1/2}$ for the unstable regime ($a<a^{*}$); see Fig.
19(b-c). These observations are in agreement with theory predictions based on
an analytical model as well as on GPE simulations.

###  IV.2 Dipolar quantum-degenerate Fermi gases

Figure 20: Anisotropy $\Delta=1-{\rm AR}$ (${\rm AR}$ is the cloud aspect
ratio) of the time-of flight distribution of a degenerate Fermi gas of 167Er
versus the parameter $\eta=E_{\rm dip}/E_{\rm F}$. Data (dots) are compared to
the predicted Fermi surface deformation (dashed line) as well as the
prediction accounting for the interaction effects during the TOF. From Aikawa
_et al._ (2014c).

Degenerate Fermi gases of polarised dipolar fermions constitute an interesting
system in which identical fermions directly interact. Alkali-metal polarised
fermions are usually non-interacting, as, away from FRs, the short-range
interactions can be neglected due to the Pauli exclusion principle; see III.1.
Because of universal dipolar scattering (see Sec. III.2), dipolar fermions do
interact and thus offer a unique possibility to explore physics combining the
effects of Fermi statistics and interactions. In particular, this system
offers new prospects for the creation of novel Fermi liquids Sieberer and
Baranov (2011); Lu and Shlyapnikov (2012), anisotropic superfluids Baranov
_et al._ (2002b), interlayer superfluids in optical lattices Pikovski  _et
al._ (2010); Potter  _et al._ (2010); Baranov  _et al._ (2011), or topological
$p_{x}+ip_{y}$ phases Cooper and Shlyapnikov (2009). Interesting prospects
relate to higher-orbital BCS pairing. This is permitted in dipolar DFGs
(dDFGs) thanks to the partially attractive character of the DDI. Such a high-
orbital BCS transition would induce exotic superfluid behaviour. However, it
requires a very low temperature to be observed Baranov (2008); Baranov  _et
al._ (2012), beyond current possibility; see Sec. II.

In dDFGs, the DDI competes with the Fermi energy, $E_{\rm
F}=\frac{\hbar^{2}k_{\rm F}^{2}}{2m}$. The figure of merit is given by the
ratio $nS^{2}d^{2}/E_{\rm F}$, corresponding to the ratio $\eta=E_{\rm
dip}/E_{\rm k}$ also used when considering thermal gases in Sec. IV.1.2. For
small values, $\eta$ can be rewritten, using the density scaling of the ideal
Fermi gas as a function of the characteristic dipolar length and the Fermi
momentum, $\eta=Ak_{\rm F}a_{\rm dd}$, where $A$ is a numerical coefficient.
In the homogeneous case, $A=1/\pi^{2}$. In experimental systems of magnetic
atoms, $k_{\rm F}$ is a few tens of $\mu{\rm m}^{-1}$ and $a_{\rm dd}$ up to a
few tens of nm, so $\eta$ is typically of the order of a few percent. This
small value makes the observation of many-body dipolar phenomena more involved
in a dDFG as compared to a dBEC, due to the large kinetic energy stored in the
system (similar to the thermal case).

On the other hand, the small $\eta$-value enables a semiclassical (Hartree-
Fock) treatment of the physics of the dDFG. Within the Hartree-Fock theory,
introduced in Sec. I.4.2, the DDI contributes to the total mean energy of the
system via two terms: the Hartree direct term, which is the usual MF term, and
the Fock exchange interaction, which comes from the required
antisymmetrisation of the wave-function.

$\bullet$ Fermi surface deformation:

Similar to the Bose case presented in Sec. IV.1, the first effect of the DDI
lies in a deformation of the ground state compared to the non-interacting case
(magnetostriction). For a DFG, this deformation has a different flavour than
for Bose gases, because of the predominant role played by the Fock term,
which, it should be highlighted, has a pure quantum origin: While the Hartree
term dominates the ground-state distortion in position space as in the bosonic
case, it is now the exchange term which gives a dominant contribution to its
distortion in momentum space; that is to say, a distortion of the Fermi sea
itself. This effect has been extensively studied theoretically Fregoso and
Fradkin (2009); Fregoso  _et al._ (2009); Sogo  _et al._ (2009); Miyakawa  _et
al._ (2008); Baillie and Blakie (2012); Wächtler  _et al._ (2017); Veljic  _et
al._ (2017).

The deformation of the Fermi sea under the effect of the DDI has been
experimentally observed in an Er DFG via TOF expansion Aikawa  _et al._
(2014c). The experiment was performed in a harmonic trap in which no well-
defined Fermi surface exists, but interactions do deform the Fermi sea
momentum distribution. In addition, interaction effects during the expansion
weakly break the one-to-one correspondence between momentum and TOF
distributions. Collisions during TOF increase the deformation, but still the
bulk of the TOF distribution deformation stems from the momentum distribution;
see Fig. 20.

The TOF aspect ratio of the cloud is observed to rotate with the quantisation
axis of the dipoles and its value is in agreement with theory. A linear
dependence of the deduced Fermi-surface-deformation amplitude on $\eta$ has
been verified experimentally as shown in Fig. 20. Additionally, the effect has
been seen to disappear at higher temperature $k_{\rm B}T/E_{\rm F}\gtrsim 1$.
Complementary measurements and theory have been later reported in Ref. Veljic
_et al._ (2017). Here, the Fermi-surface-deformation amplitude could be
reconstructed from the observed TOF aspect ratio for an arbitrary gas
geometry. It was also shown that the Fermi surface deformation does not always
rigidly rotate with the dipoles’ orientation but can also change in amplitude
under the effect of the trap anisotropy. Yet such an effect remained elusive
in the Er system due to the too weak DDI and moderate achievable trap
anisotropies in the DFG regime.

Observation of many-body physics in spin-polarised DFGs has been for now
limited to the above described Fermi surface deformation. However, many other
interesting aspects may arise, even without reaching more degenerate samples.
Following similar trends as for the bosonic case, one can mention the effect
of the DDI on the excitation spectrum and in combination with (highly)
anisotropic confinement Baranov  _et al._ (2012). Other directions includes
the special character of impurity physics in dDFGs, the physics of
multilayered or multitube systems, etc.

##  V Dipolar collapse and quantum stabilised states of Bose gases

In Sec. IV.1.4, we discussed the instabilities of dBECs. In the framework of
MF theory, they lead to singularities in the density ($n\to\infty$) and thus
in the energy $|E|\to\infty$ for the many-body ground-state. This unphysical
conclusion comes from the fact that the simple theory framework developed in
Sec.IV (see also Sec. I.4.1) breaks down at high density. When the density is
increased, two effects should be additionally taken into account.

  * •

First, at high density the approximation neglecting few-body collisions beyond
binary ones breaks down. At the next order, three-body collisions should be
considered. Their inelastic contribution leads in particular to atomic losses
at high density Stenger  _et al._ (1999a); Roberts  _et al._ (2000); Weber
_et al._ (2003). Thus density is not locally conserved and obeys the equation
$\partial_{t}n+\nabla\cdot(n\vec{v})=-L_{3}n^{3}$, where $L_{3}$ is the
recombination loss constant. One can effectively add a non-conservative term
in the GPE which reproduces the above equation, this term reads

$i\hbar\partial_{t}\psi|_{\rm 3b}=-i\frac{\hbar}{2}L_{3}|\psi|^{4}\psi.$ (58)

We also note that three-body interactions can additionally lead to a
conservative term and, very early on, its effect has been theoretically
considered, yielding the prediction of a liquid phase Gammal  _et al._ (2000);
Bulgac (2002). Despite the fact that this mechanism has regained recent
theoretical interest in the case of dBEC Xi and Saito (2016); Bisset and
Blakie (2015); Blakie (2016), it has remained elusive in current experimental
setups Ferrier-Barbut  _et al._ (2016b). Conservative three-body interactions
will then be neglected in the following.

  * •

Second, strong enough interactions yield corrections to the population (so-
called quantum depletion) and to the energy of the BEC mode. This can be
described within Bogoliubov theory, where one still considers only two-body
interactions and assumes a macroscopic, yet not complete, occupation of the
condensate mode (i.e. single-particle ground state). Expanding to second order
in powers of the non-condensed population (i.e. population in the single-
particle excited states) yields a quadratic Hamiltonian approximating the
many-body one, which can be diagonalised Bogoliubov (1947); Stringari and
Pitaevskii (2016). At this order, the ground state energy is given by the
zero-point energy of the elementary excitations that diagonalize the quadratic
Hamiltonian, which correspond to elementary excitations. This ground-state
energy is shifted compared to the MF energy that matches that of a pure
condensate (zero population in excited single-particle states). The correction
to the MF results are thus referred to as quantum fluctuations effects, i.e.
coming from the fluctuations of the vacuum of the excitations.

The resulting modification to the equation of state for a non-dipolar BEC was
first calculated by Lee, Huang and Yang in 1957 Lee and Yang (1957); Lee  _et
al._ (1957). The energy of a homogeneous repulsive Bose gas then reads
$E=\frac{g}{2}\frac{N_{0}^{2}}{V}(1+\frac{128}{15\sqrt{\pi}}\sqrt{n_{0}\,a^{3}})$
where $N_{0}\;(n_{0}=N_{0}/V)$ is the number (density) of atoms in the BEC
occupying a volume $V$. The first term of the sum is the usual MF energy, the
second one corresponds to the first-order beyond-MF (BMF) correction, so-
called Lee-Huang-Yang (LHY) correction. The strength of the LHY correction is
set by the _gas parameter_ $n_{0}a^{3}$.

Adding the DDI modifies the spectrum of the elementary excitations, see Sec.
IV.1.3, Eq. (52), and thus should modify their zero-point energy. This was
calculated in SCHÜTZHOLD  _et al._ (2006); Lima and Pelster (2011, 2012),
giving

$E=\frac{g}{2}\frac{N_{0}^{2}}{V}\left(1+\frac{128}{15\sqrt{\pi}}\sqrt{n_{0}\,a^{3}}\,Q_{5}(\varepsilon_{\rm{dd}})\right),$
(59)

where the function $Q_{l}(x)=\int
d\theta\sin\theta(1+x(3\cos^{2}\theta-1))^{l/2}$ results from angular
averaging of Eq. (52). Since the dispersion relation becomes imaginary at some
angles for $\varepsilon_{\rm{dd}}>1$, so does $Q_{5}$. Therefore the energy
(59) is formally defined only for $\varepsilon_{\rm{dd}}\leqslant 1$. However,
the imaginary part of $Q_{5}$ remains very low for
$\varepsilon_{\rm{dd}}\lesssim 3$. Then, one might ignore it and use Eq. (59)
for $\varepsilon_{\rm{dd}}\gtrsim 1$ without a complete breakdown of the
theory.

In addition to this energy correction, the computation of the many-body ground
state (total density $n$) within Bogoliubov theory, also gives access to the
corresponding quantum depletion density $\delta n=n-n_{0}$. For a homogeneous
dBEC, one finds $\delta
n/n_{0}=\frac{8}{3\sqrt{\pi}}\sqrt{n_{0}a^{3}}Q_{3}(\varepsilon_{\rm{dd}})$
SCHÜTZHOLD  _et al._ (2006); Lima and Pelster (2011, 2012), recovering $\delta
n/n_{0}=\frac{8}{3\sqrt{\pi}}\sqrt{n_{0}a^{3}}$ in the contact interacting
case Lee and Yang (1957); Lee  _et al._ (1957). One sees that the main
assumption behind the Bogoliubov theory, namely a dominant population of the
zero-momentum state, holds for small gas parameters $\sqrt{n_{0}a^{3}}\ll 1$.
In the dipolar case, a similar discussion on the $\varepsilon_{\rm{dd}}$-range
takes place as for the energy correction. This justifies a-posteriori the
validity regime given in Sec. IV for the MF theory, namely $na^{3}\ll 1$ and
$\varepsilon_{\rm{dd}}\leqslant 1$.

The above results hold true only for infinite, isotropic, homogeneous BECs.
The connection to experimental systems is done through two further
approximations to estimate Eq. (59). First, one might neglect the quantum
depletion assuming that all the atoms are in the BEC $n_{0}=n$. One can
therefore write an equation only for the BEC density and ignore its coupling
to other modes. This approximation is mostly valid for experimental conditions
so far where $\delta n/n_{0}$ is below a few percents. Second, one might use
the local-density approximation (LDA) for the calculation of the equation of
state. This assumes that the density varies sufficiently slowly to calculate
locally, for a given density $n$, the energy shift. The derivation of Eq. (59)
involves an integral over all momenta $k$, which is dominated by the
contribution at $k\simeq\xi^{-1}$ where $\xi$ is the healing length Stringari
and Pitaevskii (2016). Typically, $\xi$ relates to the sound velocity $c_{0}$
via $\xi=hm/c_{0}$. For dBECs, following Eq. (52) (see Sec. IV.1.3), one can
only define an ’angle dependent’ healing length and this diverges for
$\varepsilon_{\rm{dd}}=1$ along the dipole orientation. At first sight, thus,
the LDA is never applicable for $\varepsilon_{\rm{dd}}\geqslant 1$. We will
see below that the use of the LDA can still be justified for typical
experimental samples at $\varepsilon_{\rm{dd}}\gtrsim 1$, see Sec. V.2. Thus,
within both approximations, the local energy shift can be calculated, and
identically the chemical potential shift $\mu_{\rm BMF}=\frac{\partial E_{\rm
BMF}}{\partial
N}=\frac{32}{3\sqrt{\pi}}\,gn\,\sqrt{na^{3}}Q_{5}(\varepsilon_{\rm{dd}})$. To
solve the equation of motion for a non-homogeneous system, one can then extend
the GPE by this extra chemical potential Wächtler and Santos (2016a); Bisset
_et al._ (2016); Saito (2016):

$i\hbar\partial_{t}\psi|_{\rm
BMF}=\frac{32}{3\sqrt{\pi}}\,g\,a^{3/2}Q_{5}(\varepsilon_{\rm{dd}})|\psi|^{3}\psi.$
(60)

Now that we have effective terms to be added to the GPE for three-body
recombination (Eq. (58)) and BMF (Eq. (60)) effects, we can compare their
magnitudes to the MF chemical potential (see Eqs. (9)-(10)) to know which one
comes first when the density increases significantly following an instability.
To that end, one must know $a_{\rm dd}$, $a$ and $L_{3}$. $a_{\rm dd}$ is
fixed for a chemical species, see Sec. II.5. Since the collapse occurs for
$\varepsilon_{\rm{dd}}\geq 1$ with the exact value depending on the particular
trap geometry, we will fix a typical value of $\varepsilon_{\rm{dd}}=1.5$,
which further fixes $a$ to the values $a=10\,a_{0}$ (52Cr), $44\,a_{0}$
(166Er), $87\,a_{0}$ (164Dy). The parameter $L_{3}$ must be measured
experimentally. In Figure 21 one can see the absolute values of three
different contributions as a function of density141414assuming for simplicity
an elongated density distribution, with a central MF chemical potential
$\mu\simeq g\,n\,(1-\varepsilon_{\rm{dd}})$ for the three dipolar atoms 52Cr,
166Er and 164Dy. The values of $L_{3}$ are extracted from Refs. Lahaye  _et
al._ (2007); Chomaz  _et al._ (2016); Ferrier-Barbut  _et al._ (2016a). For
the above given values of $a$, set using Feshbach tuning via the FRs specified
in the refs mentioned above, we have $L_{3}\simeq 2\times 10^{-40}$ for 52Cr,
$\simeq 8\times 10^{-41}$ for 166Er, and $\simeq 5\times 10^{-41}$ for 164Dy.
Note that $L_{3}$ depends on the few-body-physics details and thus can vary
with the specific FR used to tune $a$. In Figure 21, one observes different
hierarchies of mechanisms depending on the atomic species.

Figure 21: Absolute value $|\mu|$ of the different non-linear terms of the
extended Gross-Pitaevskii equation, assuming homogeneous density, and
$\varepsilon_{\rm{dd}}=1.5$. The (attractive) mean field dominates at low
density. For 52Cr three-body losses (Eq. (58)) first become stronger, while
for 166Er and 164Dy, BMF effects (Eq. (60)) first take over.

In the case of Cr, the weakness of $a_{\rm dd}$ leads to a weak $a$ beyond
instability and thus weak MF and BMF effects. As a consequence the three-body
losses will have a much stronger effect and the BMF effect can be safely
ignored from the dynamics. In this case, the instability leads to a so-called
d-wave collapse, that we review in Sec. V.1.

For Dy and Er, the BMF can be at par with the MF effects before the three-body
losses destroy the sample. This leads to an entirely different dynamics, as
well as new ground states, stabilised beyond the MF instability. We focus on
this physics in Secs. V.3-V.4, see also Ref. Böttcher  _et al._ (2021).

###  V.1 Dipolar collapse

We focus here on the case of 52Cr where the BMF effects are negligible (see
above). A collapse occurs once the instability threshold is crossed.
Experiments have focused on crossing the instability line (Fig. 17) by
lowering $a$. Once a collapse occurs, the density increases until the three-
body losses become dominant over MF attraction (Fig. 21 top). This leads to a
strong density depletion at the density maximum. Then, the MF attraction is
reduced, down to a point where the kinetic energy, which is itself increased
by the strong localisation of the wavefunction close to the collapse centre,
becomes stronger. At that point, the remnant BEC fraction is strongly
expelled, the BEC ’explodes’. This can be seen as a reflection of the
wavefunction at the collapse centre. The explosion dynamics is thus expected
to reflect the symmetry of the collapse. Being induced by the DDI, the
collapse is not isotropic. The dipolar energy is minimised by decreasing the
gas aspect ratio $\kappa_{x,y}$ (see Eqs. (53) and (57)), and so one might
expect that the BEC collapses radially, i.e. shrinks and later explodes in the
directions transverse to the dipole orientation. An interplay with the initial
anisotropy of the cloud is also expected.

Figure 22: Collapse dynamics of a Cr BEC, (a-b) in a trap
$(\nu_{x},\nu_{y},\nu_{z})=(660,400,530)\,$Hz for which the initial BEC is
nearly isotropic, adapted from Lahaye  _et al._ (2008) and (c-d) in a trap
$(\nu_{x},\nu_{y},\nu_{z})=(650,520,400)\,$Hz for which the initial BEC is
prolate, adapted from Metz  _et al._ (2009). Dipoles are oriented along $z$.
(a) and (c), time-of-flight absorption images in the $yz$ plane (upper rows)
and corresponding simulated column density distributions (lower rows) for
different hold time after the quench of $a$ (values given above). Time of
flight is 8 ms. The initial scattering length is around 30-35 $a_{0}$ while
the final around 5-8 $a_{0}$, well below $a_{\rm crit}$. The in-trap inset
shows the initial BEC aspect ratio $\kappa_{y}$. (b-d) measured (dot) and
simulated atom number as a function of the hold time after the quench, $t_{\rm
hold}$, in unit of $\tau=\tau_{x}=1/\nu_{x}=1.5\,$ms. All simulations rely on
a GPE including three-body loss effects, i.e. Eq. (9) plus Eq. (58).

The first experiments probing the collapse dynamics of a 52Cr dBEC were
performed starting from a nearly isotropic BEC Lahaye  _et al._ (2008). The
collapse was induced by a fast quench of $a$ down to an unstable value. Imaged
after time-of-flight, the expanding cloud displays a strong anisotropy, see
Fig. 22(a). As the collapse is induced in the radial direction, the relevant
timescale for the collapse, $\tau$, is set by the largest radial trapping
frequency, $\tau=\min[1/\nu_{x},1/\nu_{y}]$. Strong atom losses are observed
to occur at the initial time of the collapse (fraction of $\tau$), before the
wavefunction is reflected away, see Fig. 22(b). A cloverleaf-like pattern is
observed in the time-of-flight images after holding $0.2$ to $0.5\tau$. This
pattern, reminiscent of the d-wave symmetry of the DDI, gave the name d-wave
collapse. The density distribution is then observed to refocus after holding
$\tau$. This is because of the presence of the harmonic trap. The observed
dynamics is very well reproduced by simulations of the GPE (Eq. (9))
additionally including the three-body loss term (Eq. (58)). The lost fraction
of condensed atoms depends on the in-trap wait time after the collapse start,
also very well described by the GPE simulation results, see Fig. 22 (a-b).

The fact that the expanding cloud remains condensed was investigated by
collapsing several independent BECs in different sites of an optical lattice
Metz  _et al._ (2009). Following the collapse, the expanding clouds
overlapped, exhibiting very clear interference fringes at long time. This
proved the local coherence of the individuals expanding clouds, and thus that
condensation was not fully destroyed by the collapse.

In anisotropic traps, not only the DDI but also the trap geometry impacts the
collapse symmetry. In this way, very different patterns of the expanding BEC
can be generated, evidencing the rich interplay between trap geometry and DDI
in dBECs. This effect has been extensively studied in Ref. Metz  _et al._
(2009), where the authors studied different cases from very prolate to very
oblate geometry. They also investigated the cross-over from a prolate geometry
to the symmetric one of Ref. Lahaye  _et al._ (2008), as shown in Fig.22(a,c).
While the two traps considered here have very similar frequencies within a
simple permutation $\nu_{y}\leftrightarrow\nu_{z}$, they display significantly
different dynamical behavior. This ultimately demonstrates the role of the
anisotropic DDI in the collapse dynamics.

###  V.2 Theoretical description of dipolar quantum stabilised states

####  V.2.1 Simple description of dipolar quantum stabilisation

The phenomenology of MF unstable, strongly dipolar BECs of Er and Dy is
fundamentally modified by the BMF effects. These effects can affect not only
the dynamics of the BEC when driven to an unstable regime but also modify its
stability itself, as it introduces a stabilisation mechanism. Indeed, the LHY
correction of Eq. (60) provides an additional conservative potential, which
acts as an effective higher-order (in three-dimension, effective 5/2-body
interaction) interaction term in the GPE and which remains repulsive within
its whole validity range 151515Provided the imaginary part of
$Q_{5}(\varepsilon_{\rm{dd}})$ can be neglected.. This higher-order repulsive
term has a density dependence of a higher power than MF (2-body interaction).
In the attractive mean field regime, it thus could be thought to stabilise a
high-density state for which BMF repulsion compensates MF attraction.

Let’s first consider the simplest case of single-peaked ground state —
standard BEC in the MF stable regime and later called "droplet" in the MF
unstable regime. Using a gaussian ansatz for its wave function $\psi$ (see
also Eqs. (54)-(57)), one can calculate the MF and BMF contributions to the
ground-state energy per particle and single-out the key ingredients that lead
to stability:

$\displaystyle\frac{E_{\rm MF}}{N}$
$\displaystyle=\frac{1}{2^{3/2}}\,\frac{g}{2}\,n_{\rm c}\,(1-f_{\rm
dip}(\kappa)\,\varepsilon_{\rm{dd}})$ (61) $\displaystyle\frac{E_{\rm
BMF}}{N}$ $\displaystyle=\left(\frac{2}{5}\right)^{3/2}\,\frac{g}{2}\,n_{\rm
c}\,\frac{128}{15\sqrt{\pi}}\,\sqrt{n_{\rm
c}a^{3}}\,Q_{5}(\varepsilon_{\rm{dd}})$ (62)

where $n_{\rm c}$ is the central density. For $f_{\rm
dip}(\kappa)\varepsilon_{\rm{dd}}>1$, $E_{\rm MF}<0$ while $E_{\rm BMF}>0$.
$E_{\rm BMF}$ has a stronger dependency in the density by a power $n_{\rm
c}^{1/2}$ compared to $E_{\rm MF}$, therefore, by increasing $n_{\rm c}$, the
ground-state energy can be minimized at a finite value, corresponding to a
finite density. This stabilisation mechanism, relying on the mere effect of
quantum fluctuation, is what we coin dipolar quantum stabilisation.

####  V.2.2 Simple description of ultra-dilute liquid state.

Remarkably, in presence of MF attraction, the single-peaked state considered
above can be stabilised even in the absence of trapping, by competing MF
attraction and the BMF repulsion, the equilibrium between these two forces
fixing the peak density $n_{\rm c}$ (MF attraction dominates at low density,
and BMF repulsion at high density).

Such a stabilisation mechanism is reminiscent of a liquid phase of matter. In
ordinary liquids, the weak attraction on large distance (e.g. of van der Waals
or covalent types) is counterbalanced by a strongly repulsive core arising
from the electromagnetic forces and the Pauli exclusion between the atoms’ (or
molecules’) electron cloud. This stabilises the liquid at a large density at
which the repulsion becomes effective $n_{\rm c}{a_{0}^{3}}\sim{1}$. The
stabilisation resulting from competing MF and BMF effects in strongly dipolar
gases can similarly be seen as resulting in a liquid state.

Considering for now such an untrapped system, and neglecting kinetic energy,
the equilibrium peak density can be estimated by imposing $\frac{\partial
E}{\partial n_{\rm c}}=0$, with $E=E_{\rm MF}+E_{\rm BMF}$. One gets

$n_{\rm c}\sim\frac{1}{a^{3}}\left(\frac{f_{\rm
dip}(\kappa)\,\varepsilon_{\rm{dd}}-1}{Q_{5}(\varepsilon_{\rm{dd}})}\right)^{2}.$
(63)

We note that Eq. (63) also provides a good estimate of the peak density in a
trapped stabilised state, as long as one can neglect contributions of the
kinetic and external trapping energies.

Interestingly, from Eq. (63) one can read-off that a stabilisation of the
density occurs before reaching the dense regime $n_{\rm c}{a^{3}}\sim{1}$,
thus forming "ultradilute" liquid states. Two main effects enable to maintain
the diluteness: The equilibrium density is reduced by the fact that there are
two competing MF interactions, resulting in an effective MF attraction of much
smaller amplitude than each of the two contributions (numerator in Eq. (63)).
Second there is an amplified BMF contribution that leads to a further
reduction of the density (denominator in Eq. (63)) 161616At
$\varepsilon_{\rm{dd}}=1.5,\Re[Q_{5}(\varepsilon_{\rm{dd}})]\simeq 3$, leading
to a reduction of the density by roughly an order of magnitude..

The relatively low stabilising density resulting from these two ingredients
protects the sample against an immediate destruction by three-body
recombination. The dipolar collapse observed with Cr is thus prevented.
Instead the BEC is stabilised via the effect of its quantum fluctuations,
resulting in a distinct phase of liquid-like properties, the clouds formed are
thus named quantum (dipolar) droplets. These droplets are denser than the MF-
stable gaseous BEC, yet much more dilute than ordinary liquid (by $\approx 8$
orders of magnitude). They offer a distinct paradigm of quantum fluid where
the BMF effects are predominant yet tractable, see e.g. Secs. V.2,V.3.4.

We note that these exact same ingredients are present in another experimental
system, namely mixtures of contact-interacting BECs with repulsive intra- and
attractive inter-species interactions Cabrera  _et al._ (2018); Cheiney  _et
al._ (2018); Semeghini  _et al._ (2018). The stabilisation mechanism of
bosonic mixtures was in fact proposed prior to the observations on dipolar
BECs by D. Petrov Petrov (2015). For more information on these systems, see
Ref. Böttcher  _et al._ (2021).

Finally, we note that, despite the fact that we have for now neglected
trapping effects in this first description, they may also play a crucial role
in the newly stabilised quantum states. In the case of dipolar atoms, this
effect is not only quantitative (energy shifts) but also qualitative. Indeed,
the interplay between trap anisotropy and DDI yields new features already in
the MF-stable BEC, as reviewed in Secs. IV.1.3,IV.1.4. These new features in
the excitation spectrum, that affect the MF instability, when combined with
quantum stabilisation, may yield new ground states. This effects will be the
focus of Sec. V.4.

####  V.2.3 Toward a quantitative theory description: extended Gross-
Pitaevskii equation.

Following the experimental observation of quantum droplets that we will more
thoroughly describe in Sec. V.3.2, several theory works Wächtler and Santos
(2016a); Bisset  _et al._ (2016); Wächtler and Santos (2016b) developed the
framework based on an extension of the GPE (eGPE) to include the first order
BMF effects. As introduced at the beginning of this chapter V, this consists
in extending Eq. (9) with the term Eq. (60) (and potentially Eq. (58)). This
perturbative approach is justified because the sample remains dilute with
$na_{s}\ll 1$. Here the BMF correction comes into play not because the gas
parameter becomes large but rather because the overall MF terms are tuned
small while both contact and dipolar interaction remains individually large.
Then the BEC quantum depletion remains limited.

We note that the theory model relies on the use of the LDA for the LHY term.
This approximation is justified if the dominant contribution to the LHY term
have momentum larger than the inverse size of the ground state. Physically,
the LHY correction, which corresponds to the zero point motion of the
elementary excitations (see above), are dominated by contributions of the
_hard_ modes, i.e. the most energetic ones. Crucially, when decreasing $a$ (as
to drive the MF-instability), the modes whose excitation occurs perpendicular
to the dipoles get softer (and lead to the instability) while the one along
the dipole get harder. The latter will then dominate the LHY correction.
Remarkably, the BEC close to instability and the quantum stabilised states get
very extended in the direction of the dipole, under the effect of
magnetostriction (see Sec. IV.1.2). Therefore, the LDA may remain a
surprisingly good approximation in this regime, and even beyond the
instability threshold.

Quantitatively, Wächtler et Santos Wächtler and Santos (2016a) show that,
using an anisotropic momentum cutoff, with an anisotropy matching the
anisotropy of the droplet itself, at 1% of the inverse of the droplet extent,
one still recovers most of the BMF effects (80%). This proves that the LDA is
a good qualitative, and even quantitative, approximation, as long as the
droplet remain elongated enough. The contribution of the long-wavelength modes
may lead to small corrections, yet the authors note that this would mainly
modify the prefactor of the LHY correction while the scaling should remain
that of Eq. (60). This domination of the LHY correction by hard modes which
makes relevant the use of the LDA even in small sized quantum-stabilised
states is also found in quantum droplets of bosonic mixtures as originally
proposed by Petrov Petrov (2015), see also Refs. Cabrera  _et al._ (2018);
Cheiney  _et al._ (2018); Semeghini  _et al._ (2018); Böttcher  _et al._
(2021).

The relevance of eGPE framework was later quantitatively studied both in
theory and experiments. Results from the eGPE and quantum Monte-Carlo (QMC)
simulations of dipoles with hard-sphere repulsion were compared in Ref. Saito
(2016) and found to be in good quantitative agreement. In experiment, the
degree of agreement was found to vary depending on the exact settings, in
particular on the gas geometry and on the density, see e.g. Refs. Chomaz  _et
al._ (2016); Schmitt  _et al._ (2016); Ferrier-Barbut  _et al._ (2018a);
Chomaz  _et al._ (2018); Petter  _et al._ (2019); Chomaz  _et al._ (2019);
Tanzi  _et al._ (2019a); Böttcher  _et al._ (2019b) and later discussions.
Typical discrepancies could be simply accounted by shifting the scattering
length value by a few, and up to a few tens, of percents. More recently, a
quantitative study compared eGPE theory, diffusive QMC simulations using
finite-range interactions, and experiments Böttcher  _et al._ (2019a). A good
agreement of the QMC results and the experiments was found, whereas the eGPE
results are systematically shifted. Boettcher et al elaborate on the eGPE
mismatch and show that it may be accounted by a more sophisticated description
of the scattering, in particular by accounting for the effects of finite
collision energy on the scattering properties, due to the finite temperature
of the samples. Such effects are known to result in an effective
renormalisation of the dipolar length Ołdziejewski and Jachymski (2016), see
also Sec. III.1. By including such corrections in the eGPE yields a better
agreement with the experimental data. Note that corrected eGPE and QMC
constitutes two complementary theories that are able to describe experimental
observations by accounting for different effects. The corrected eGPE accounts
for temperature effects on the scattering properties but only includes quantum
fluctuations at a perturbative and approximate level. QMC fully accounts for
the quantum fluctuations, yet neglects thermal effects. The two theories
account for different effects, and suggest that different sources of
corrections with respect to the standard eGPE theory may be relevant. A full
theoretical modelling, accounting for the different effects at once, and
revealing their repsective role, is yet missing.

###  V.3 Dipolar Quantum droplets

We now divide the discussion of quantum-stabilised states and their
observations in two sections. In the present section, we will discuss the
regime where the underlying ground state present no self-modulation, i.e it
has only one density peak and forms either a standard BEC (repulsive MF) or a
droplet state (attractive MF). In experiments, the droplets could there be
produced either in assemblies of independent droplets, forming then a
metastable state, or individually. We will more specifically focus on the
latter case, which allows for a detailed characterisation of the state’s
properties, in presence and in absence of external trapping. In a second part,
we will focus on the generation of self-modulated states in dipolar gases with
anisotropic confinement. We will particularly focus on the global coherence
and superfluid properties of the modulated states, and discuss the existence
of a so-called supersolid phase, where solid and superfluid orders coexists,
see Sec. V.4.

####  V.3.1 single droplet ground state in an external trap, eGPE phase
diagram

Figure 23: Eigen-state phase diagram of a dipolar BEC of $N=10000$ atoms in a
cylindrical harmonic trap of aspect ratio $\lambda=\omega_{z}/\omega_{r}$.
This diagram is obtained applying a gaussian ansatz to the energy functional
47, for a fixed value of $Na_{\rm dd}/a_{\rm ho}\simeq 79$, this corresponds
for Dy (Er) to a mean trap frequency of 80 Hz (312 Hz). The colouring shows
the ground-state density in logarithmic scale and arbitrary units.

Following the seminal work establishing a theory description of the quantum-
stabilisation mechanism, theoretical works tackle the question of the ground-
state phase diagram of a dipolar gas and in the presence of the newly
discovered stabilising term Bisset  _et al._ (2016); Wächtler and Santos
(2016b).

The presence of an anisotropic harmonic trap with aspect ratio
$\lambda=\omega_{z}/\omega_{r}$ (see also Eq. (44)) was considered, yet
cylindrical symmetry around the dipoles direction $\boldsymbol{z}$ was
assumed. Mean trap frequencies and atom numbers following the experimentally
relevant values were considered. In these works, only single-peaked ground
states were predicted (i.e. no droplet assemblies, see Sec. V.4.1). When
decreasing $a$ at fixed atom number and trap geometry, the ground state was
found to change from a low-density phase, matching a standard MF-stabilised
BEC, to a high-density phase, stabilised by the LHY term, forming a “quantum
droplet". The low-density BEC has a geometry roughly following that of the
trap, while the droplet state is not. In particular, the droplet is always
elongated in the direction of the dipoles, which is why it is sometimes called
"filament". For traps elongated along the dipoles ($\lambda\gtrsim 1$), the
ground state smoothly evolves from the low-density MF-repulsive BEC to the
high-density MF-attractive quantum-droplet phase when decreasing $a$. This
smooth crossover can be apprehended as the trap elongation along the dipole
direction enables the two states to have similar geometries (elongated along
the dipoles), so that one can continuously evolve into the other, while the
BMF term suppresses the dipolar collapse described in Sec. V.1. In the
opposite case of large $\lambda$, a discontiuous transition between the two
states is found, with an intermediate region of bistability, where the two
states form local energy minima. This bistability can be apprehended from the
following argument: In such traps, a low-density state can be stabilised even
for $\varepsilon_{\rm{dd}}>1$ as the trap asymmetry forces the dipoles to lie
side-by-side enhancing the MF DDI repulsion. This is exactly how Cr BECs have
been stabilised at low $a$, as described in Sec. IV.1.4, and the addition of
the BMF term only weakly modifies this behaviour. On the other hand, the BMF
term stabilises an other solution which is elongated along the dipoles and
where MF DDI is attractive. The bistability occurs as there is an intermediate
regime of $a$ where both solutions are local energy minima, while in the
absence of the BMF term (as relevant for Cr) in this region the MF repulsive
BEC is metastable.

The theoretical phase diagram is represented in figure 23. The exact position
of the boundaries and critical point depend on the exact experimental
parameters (number of atoms, mean trap frequency, …). The crossover and the
bi-stable region are separated at a critical aspect ratio $\lambda_{\rm c}$,
typically $\lambda_{\rm c}\approx 1.5-2$. In the remainder of this section, we
review the extensive experimental exploration of this phase diagram.

We note that the description given above does not give a full picture of the
dipolar gas phase diagram in presence of quantum stabilisation. This
description encompasses the behaviour in a low-atom-number regime. For larger
atom numbers (and/or tighter traps), distinct ground states may arise, and in
particular spontaneous density modulation may occur, or, in other words,
ground states bearing several droplets may be found. This regime, achieved
more recently in experiments, will be the topic of Sec. V.4.

####  V.3.2 First observations of droplet states: metastable droplet
assemblies and the quest for the stabilisation mechanism

The first evidence for the absence of dipolar collapse and the existence of a
stabilisation mechanism was reported by Kadau et al. Kadau  _et al._ (2016) on
164Dy, before any of the theoretical development described in Secs. V.2-V.3.1.
Starting from a pancake-shaped BEC of 164Dy atoms in a trap with aspect ratio
$\lambda\simeq 3$, $a$ was lowered down to the mean field unstable regime. In
this geometry, we note that a modulational instability is expected, see Sec.
IV.1.4. Below the instability threshold, ordered droplet ensembles were
observed in the in-situ radial density distribution of the cloud, as can be
seen in Fig. 24. Contrary to the expectations at the time, these droplets were
observed to have very long lifetimes of several hundreds of ms, simply limited
by atom loss due to three-body recombination.

As shown in the early theory works Wächtler and Santos (2016a); Bisset  _et
al._ (2016), the observation of multiple droplets arises from a crossing of
the boundary between the BEC region and the bi-stable region of the phase
diagram. In this regime, a continuum of metastable states, with different
numbers of droplets, is expected to exist. These works suggest that, following
the fast change of the interaction strength, the system ends up in one of
these metastable states, explaining the observed long lifetime of the crystal
as well as the shot-to-shot variability of the structures. Experimentally, the
system’s bistability was evidenced via the hysteresis in the appearance and
disappearance of the droplet ensembles when varying $a$ down and back up, thus
supporting the theoretical picture. As observed after a long hold time, the
arrangement of the droplets was found to result from the repulsion between the
corresponding macroscopic dipoles (see also Refs.Ferrier-Barbut  _et al._
(2016a, b)).

Figure 24: Ordered droplet ensembles observed in the MF-unstable region of the
phase diagram. The magnetic field points put of the plane. Adapted from Kadau
_et al._ (2016).

A following set of experiments Ferrier-Barbut  _et al._ (2016a) showed that
the droplets have an elongated shape along the dipoles with an axial length of
a few micrometers and a radial size estimated to half a $\mu{\rm m}$ or less
(this lies below the imaging resolution). Furthermore, matter-wave
interference fringes observed from several expanding droplets proved them to
be individually superfluid. They were observed to have extremely low expansion
velocities when released into a waveguide. In Ref. Ferrier-Barbut  _et al._
(2016a), Ferrier-Barbut et al., inspired from the work of Petrov Petrov (2015)
on BEC mixtures, proposed that BMF effects act as a stabilisation mechanism.
By comparing additional expansion and lifetime measurements to a model
following the lines given in Sec. V.2.2 (neglecting kinetic and trapping
effects), they validate this idea and invalidate a possible mechanism based on
three-body forces proposed in Refs. Xi and Saito (2016); Bisset and Blakie
(2015); Blakie (2016), see also the earlier works of Refs. Bulgac (2002);
Gammal  _et al._ (2000). In particular, the lifetime $\tau$ of the sample,
being set by the three-body loss processes, gives information on the sample’s
density via $\tau=L_{3}\langle n^{2}\rangle$ (see Eq. (58)). The estimated
density scaling of density as a function of $a$ deduced from measuring $\tau$
as a function of $a$ was found to reasonably agree with the simple density
scaling of Eq. (63), and disagree with the scaling expected from three-body
forces stabilisation.

In this setup, the bistability, yielding excited metastable states of multiple
droplets after an interaction quench, prevents the formation of a large single
droplet in the ground state. This feature has limited the study of the
properties of the liquid-like state. Shortly after, it was found out, in
particular thanks to the theory development described in Sec. V.3.1, that this
issue can be resolved by taking advantage of the "crossover region" of the
phase diagram, i.e. changing the trap geometry, as we will now discussed.

####  V.3.3 Crossover from a Bose-Einstein condensate to a large quantum
droplet.

Figure 25: Examples of properties of a 166Er dBEC in the crossover from a BEC
to a single macrodroplet reported in Chomaz  _et al._ (2016). Here $a$ is
varied in 10 ms to its final value, in abscissa. (a) mean expansion velocity
measured in 16 to 28 ms free-expansion after 5 ms holding in trap. The insets
show measured distributions in the plane transverse to the dipole orientation
for 28 ms of free-expansion, and also evidence the absence of collapse
dynamics in the low $a$-regime. (b) The axial mode of the dBEC, whose
character is illustrated in inset, is excited by transiently decreasing the
confinement frequency along the dipole, $\nu_{||}$, and the mode frequency,
$\nu_{\rm ax}$, measured by recording the time-evolution of the cloud size
along the dipole. In (a) and (b), the data (squares) are compared to
simulations from real time evolution of the eGPE including (red solid line) or
not (orange dashed line) the LHY correction (Eq. (60)). In (b), $\nu_{\rm ax}$
cannot be reliably extracted for quenches to $a\leq 56a_{0}$, nor from the
experiment (open squares) neither from the eGPE theory (open circles, thin
line).

Following the first droplet observations Kadau  _et al._ (2016); Ferrier-
Barbut  _et al._ (2016a, b) and the resulting theoretical development Bisset
_et al._ (2016); Wächtler and Santos (2016b), Chomaz et al. tested the
universality of the stabilisation effect (which relies on the sole quantum-
mechanical nature of the fluid, provided that the interactions are strong
enough), by realising the first quantum droplet of a distinct chemical
species, using 166Er Chomaz  _et al._ (2016). They also used a complementary
geometry to the previous Dy observations, using a cigar-shaped trap with
$\lambda\ll 1$, and thus they observed a smooth crossover from a BEC to a
single large droplet of Er atoms containing all the condensed atoms. Because
the created droplet is isolated, the authors could study its properties such
as its elementary excitation, and its expansion dynamics, see Fig. 25. The
authors performed a systematic comparison of the measurements with the eGPE
predictions (Eq. (9) including also Eqs.(60)-(58), see Sec. V.2) using
independently measured values of $a$ and $L_{3}$. The quantitative agreement
reached confirmed the stabilisation scenario and validated the eGPE framework
in this setting. We note again, that quantum droplets have later been observed
in mixtures of contact-interacting BECs Cabrera  _et al._ (2018); Semeghini
_et al._ (2018) following Petrov’s initial proposal, establishing the quantum-
stabilisation mechanism as universal even across different interaction types,
provided that competing interactions lead to a balance of MF and BMF
contributions.

Finally, the two regimes of bi-stability and crossover where experimentally
connected in Ferrier-Barbut  _et al._ (2018b), where similar experiments were
performed using 164Dy in traps of variable aspect ratio $\lambda$. The
formation of a single droplet resulting from lowering $a$ in the crossover was
observed for small $\lambda$ up to a critical aspect ratio $\lambda_{\rm c}$
which marked the onset of a modulational instability as in Kadau  _et al._
(2016). The experimental value of $\lambda_{\rm c}$ is in agreement with the
expected critical aspect ratio marking the separation between the crossover
and bi-stable regions.

####  V.3.4 Droplet properties and their signature in the collective modes

Following the first experimental observations, several works investigated the
specific properties of the droplet states properties.

Theoretically, it is interesting to highlight that, while the trap plays a
role in the phase diagram and the transition between the different phases (see
Sec. V.3.1), it plays a much lesser role on the properties of the single
droplet state itself, at least for large enough atom number and
$\varepsilon_{\rm{dd}}$. By studying full ground-state solution from the eGPE,
Refs. Wächtler and Santos (2016b); Bisset  _et al._ (2016); Ferrier-Barbut
_et al._ (2016b) show for instance a very weak dependence of the droplet
density on the trapping potential. At very low atom number (i.e. on the order
of a few thousand), the one-body kinetic energy is non-negligible, and its
interplay with the interaction energies leads to a dependence of the central
density on $N$. However, at high atom number, the interaction energies fully
dominate. Following the simple description of Sec. V.2.2, the density then
reaches a saturation value independent of $N$, given only by the balance of MF
and BMF energies. This marks the low compressibility typical of a liquid
phase, and which stems here from the high energy cost of increasing density
due to the BMF term. While early experiments have mostly explored the low atom
regime, first measurements showing the onset of this density saturation have
been recently reported in Ref. Böttcher  _et al._ (2021).

Besides its evidence via density saturation, the low compressibility of the
dipolar quantum droplets has been revealed in different sets of measurements,
in particular focusing on the collective modes. First investigations were
preformed by Chomaz et al., studying a particular collective mode, with
compressional character, of a quantum gas of 166Er in the BEC-droplet
crossover Chomaz  _et al._ (2016). As the scattering length is lowered through
the crossover, a steep increase in the mode mode frequency was observed,
changing by $50\%$ when varying $a$ by less than $20\%$ while the trap
frequency remained constant, see Fig. 25. This shows that the compressibility
quickly drops when going from the BEC to the droplet phase.

In addition to being weakly compressible, a dipolar quantum droplet is
anisotropic. This stems of course from the anisotropy of the DDI with respect
to the external magnetic field axis. The consequence of this external breaking
of rotational invariance is the existence of a collective mode, deemed the
scissors mode. It corresponds to a rigid-body angular oscillation around the
magnetic field direction. This scissors mode has been observed first in atomic
nuclei Iudice and Palumbo (1978); Lipparini and Stringari (1983); Enders  _et
al._ (1999) and in contact-interacting BECs in anisotropic traps Guéry-Odelin
and Stringari (1999); Maragò  _et al._ (2000). In the presence of the DDI, the
rotational symmetry is broken even in isotropic traps. This was showed to
result in a scissors mode for MF dipolar BECs in Ref. van Bijnen  _et al._
(2010). This mode is well-defined only for angular oscillations in the
$xz$-plane with an amplitude lower than $\theta_{\rm max}=(\langle
z^{2}\rangle-\langle x^{2}\rangle)/(\langle z^{2}\rangle+\langle
x^{2}\rangle)$ where $z$ is the field direction Guéry-Odelin and Stringari
(1999). It is thus very challenging to observe with dipolar MF BECs, but was
observed in dipolar quantum droplets of 164Dy thanks to their considerable
anisotropy Ferrier-Barbut  _et al._ (2018a). In conclusion, studies of the
collective modes of the liquid state are a sensitive probe for its properties
Baillie  _et al._ (2017) and will likely be pushed further.

####  V.3.5 Self-bound droplets and liquid-gas phase diagram

The possibility of creating a single large quantum droplet by tuning the trap
aspect ratio, as described in Sec. V.3.3, was key in the demonstration of the
self-bound liquid nature of this phase. A quantum droplet living in
equilibrium between the repulsive BMF and the attractive MF interactions, the
question of the necessity of the trap for the existence of this phase then
arises. Two theory papers demonstrated indeed the existence of a self-bound
state in the absence of a trap, characterised by a non-vanishing peak density
in infinite volume Wächtler and Santos (2016b); Baillie  _et al._ (2016). This
defines a liquid state, existing from the mere balance between high-density
repulsion and low-density attraction. However, this liquid is in essence
quantum, which provides it with macroscopic new properties. First, it remains
phase-coherent within a droplet. Second, the self-binding potential resulting
from MF attraction from other atoms within the droplet must counter-act
kinetic energy. As a consequence if the volume is too small, kinetic energy
prevents the existence of a bound state. This yields a phase diagram as a
function of atom number and scattering length, with a single line separating
the self-bound state from a gas at low atom number and high scattering length,
see Fig. 26. The scattering length dependence of the atom number value on the
separation line is sharp. Indeed, using scaling arguments, one can show that
the minimal atom number for the existence of the self-bound state scales as
$N_{\rm c}\sim(1-\varepsilon_{\rm{dd}}\,f_{\rm dip}(\kappa))^{-2/5}$.

Figure 26: Measured self-bound liquid to gas phase diagram of Dy atoms. The
theory curve results from simulations of the eGPE. Data are shown for both
164Dy and 162Dy. For 164Dy, the background scattering length had to be adapted
from the literature value ($92(8)\,a_{0}$, green circles) to a lower value
($69(4)\,a_{0}$, red diamonds), see also discussion in Secs. V.2 and V.3.4.
The horizontal error bars are systematic errors coming from uncertainties on
FRs positions and widths. Adapted from Refs. Böttcher  _et al._ (2019a);
Schmitt  _et al._ (2016).

First indications of a self-bound behaviour were already evidenced in the
early work Ferrier-Barbut  _et al._ (2016a) through the absence of expansion
in a waveguide and record-low expansion velocities in free space. Yet, the
self-bound character could not be observed because of the limited atom number.
Because of the large loss rate occurring in Er droplet (see Fig. 21), the
evidences of the self-bound behaviour of the macro-droplet of Ref. Chomaz  _et
al._ (2016) was partial, limited by the state lifetime. Here, only a slowing
down of the expansion dynamics was observed. An unambiguous self-bound
behaviour was then observed on 164Dy in Ref. Schmitt  _et al._ (2016). Here a
single droplet was created using an axially elongated dBEC 171717the trap has
yet a prolate shape with $\lambda=1.3$. The axial elongation of the dBEC comes
from magnetostriction effects, see Sec. IV.1.2 of 3000 Dy atoms and cruising
the crossover to the single droplet state, similarly to Ref. Chomaz  _et al._
(2016). The trap was then smoothly released while the atoms were levitated
against gravity using a magnetic-field gradient (see Fig. 27). Unexpanded
droplets were observed up to 90 ms of levitation. Using the atom loss as a
probe, the liquid-to-gas transition was mapped out in the $(a,N)$-space. This
phase-diagram was later expanded to larger atom number by using samples of
162Dy atoms Böttcher  _et al._ (2019a), see Fig. 26. The experimental points
of the two isotopes match well together and agree with eGPE predictions, at
the expenses of a substantial renormalisation of the background scattering
length of the 164Dy isotope, by a factor of 3/4. The need of such an important
change of $a_{\rm{bg}}$ rose questions about the validity of the eGPE
treatment in particular in the low-atom-number and
large-$\varepsilon_{\rm{dd}}$ regimes(see also the discussion in Sec. V.2), or
again of the methods employed to extract $a_{\rm{bg}}$ (see Sec. II.4), in
particular in the Dy case.

Figure 27: Single self-bound droplets without a trapping potential can be
produced by preshaping the droplet in an appropriate trap which is then
switched off. The droplet stays bound and floating in the chamber until
eventually the losses reduce the atom number below a critical value where no
bound states exist any more. That manifests the transition to a gaseous phase
where the atoms expand like a gas. Adapted from Ref. Schmitt  _et al._ (2016).

To conclude, the self-bound liquid-gas phase diagram, being simply due to the
quantum mechanical nature of the system, is not restricted to the dipolar
case. It may in fact be generally expected in quantum gases where two
interactions of different origin compete, and as such it has been observed in
Bose-Bose mixtures Cabrera  _et al._ (2018); Semeghini  _et al._ (2018). It is
interesting to note that in lower dimensions this phase diagram is completely
modified. This has been the topics of several theoretical works, both for
dipolar and mixtures systems, see e.g. Petrov and Astrakharchik (2016); Edler
_et al._ (2017).

###  V.4 Dipolar Supersolids

In Section V.3, we have reviewed how the quantum-stabilisation mechanism
discovered in 2016 (see Sec. V.2), may yield new ground states beyond the MF
instability. There the ground state were limited to single droplet, as
relevant for the early experiment, due to the small atom numbers and shallow
traps (independent on the trap geometry, see Sec. V.3.1). Here we review how
the same stabilisation mechanism yield to formation of self-modulated and in
particular supersolid ground state, as well as their experimental
investigations.

####  V.4.1 Preliminary works

Supersolidity is a paradoxical phase of matter in which the antithetical
properties of crystal arrangement and of superfluid flow coexist. It has been
suggested more than half a century ago as a paradigmatic manifestation of a
state in which two continuous symmetries of distinct nature are simultaneously
broken Penrose and Onsager (1956); Gross (1957); Boninsegni and Prokof’ev
(2012). Originally predicted in quantum solids with mobile bosonic vacancies
Andreev and Lifshitz (1969); Chester (1970); Leggett (1970), the search for
supersolidity has spread in many different fields of physics. Observation of
supersolidity in helium was claimed Kim and Chan (2004) but the claim was
withdrawn by the same authors a few years later Kim and Chan (2012) and the
quest for supersolidity in helium is still open Boninsegni and Prokof’ev
(2012); Balibar (2010). The possibility of supersolid states in quantum gases
were theoretically proposed long ago, see Refs. Mora  _et al._ (2007); Büchler
_et al._ (2007); Astrakharchik  _et al._ (2007); Pupillo  _et al._ (2010);
Golomedov  _et al._ (2011); Boninsegni (2012, 2013); Lu  _et al._ (2015);
Roccuzzo and Ancilotto (2019); this possibility linking back to the seminal
work from E. Gross Gross (1957) on assemblies of bosons with momentum-
dependent interactions, see also Ref. Kirzhnits and Nepomnyashchii (1971);
Pomeau and Rica (1994).

Recent experiments with BECs coupled to light fields forming lattices have
shown a variation of the original idea of supersolidity where atomic density
patterns are spontaneously formed but with a fixed and externally determined
period Léonard  _et al._ (2017); Li  _et al._ (2017c). In these experiments
two continuous symmetries are broken, one of which allowing a spatial
translational invariance. The breaking of this continuous symmetry fixes the
point of origin of the lattice but not its period, which is determined by the
external light field. More recently, a continuous $U(1)$ translational
symmetry was created in system of multimode light coupled to a BEC. The
breaking of this symmetry resulted in a supersolid with vibrations exhibiting
a Goldstone dispersion of acoustic phonon excitations Guo  _et al._ (2021). We
will now discuss the simultaneous quest for a supersolid supporting crystal
phonon excitations like a real solid in a dipolar gas.

This authentic supersolid behaviour can be expected if the simultaneous
breaking of the two symmetries arise from the intrinsic interactions between
the particles. As already highlighted by E. Gross, the relevant platforms for
observing supersolidity are quantum gases with inter-particle interactions
yielding large-momentum attraction Gross (1957); Kirzhnits and Nepomnyashchii
(1971). Practical examples are Rydberg-dressed potentials, spontaneous or
light-induced dipole-dipole interactions in confined geometries Mora  _et al._
(2007); Büchler  _et al._ (2007); Astrakharchik  _et al._ (2007); Pupillo  _et
al._ (2010); Golomedov  _et al._ (2011); Boninsegni (2012, 2013); Lu  _et al._
(2015); Roccuzzo and Ancilotto (2019). In these settings, a roton-type
excitation is induced in the superfluid’ excitation spectrum by the large-$k$
attraction (see also Sec. IV.1.3). Its full softening may indicate the
transition to a density-modulated state as the roton signals an intrinsically
favoured length scale and has been seen as a precursor of crystallisation
Kirzhnits and Nepomnyashchii (1971); Pomeau and Rica (1994). The observation
of a roton mode in dBECs confined in cigar-shaped geometries Chomaz  _et al._
(2018); Petter  _et al._ (2019) was key to point promising grounds to observe
supersolidity Roccuzzo and Ancilotto (2019), see also Sec. IV.1.3.

A recurrent hindrance to the interaction-driven formation of supersolid in
quantum gases lies in the predicted MF collapse of the gas at the roton
instability, see Secs IV.1.4,V.1 and Refs. Shlyapnikov and Pedri (2006);
Komineas and Cooper (2007). Protocols to stabilise the gases beyond its
instability thanks to the engineering of higher-order (three-body) interaction
potentials have been proposed Lu  _et al._ (2015). However, these protocols
have not been implemented in experiments so far. The discovered BMF
stabilisation mechanism provides the key ingredient for an intrinsic
stabilisation.

In 2017, theoretical works based on the eGPE Wenzel  _et al._ (2017); Baillie
and Blakie (2018) demonstrated that not only a single droplet (as described in
Sec. V.3.1) but also assemblies of multiple droplets, or in other words
density-modulated states, could constitute the ground state of dipolar quantum
gases. This was found for both pancake and cigar geometries, by simply using
tighter trap and/or larger atoms numbers than in the previous works of Ref.
Bisset  _et al._ (2016); Wächtler and Santos (2016b); Baillie  _et al._
(2017). Similar observations were also achieved via quantum Monte Carlo
simulations with large densities Macia  _et al._ (2016); Cinti and Boninsegni
(2017). Starting from the picture of a single droplet state, the physical
argument to the formation of multiple-droplet ground-states can be formulated
as follow: The liquid phase has a weak compressibility that yields a sharp
increase in energy at high density, see also Sec. V.3.4. If one compresses an
(isotropic) liquid in one or two directions, it then deforms to keep a
constant density at the small cost of increasing its surface energy. Yet for
the dipolar quantum liquid, the DDI dictates an anisotropy, and deformation
costs a high amount of (dipolar) energy. Therefore, when a dipolar quantum
droplet is compressed along its long axis, it might be energetically
favourable to split it in several droplets (of smaller radial sizes),
recovering thus an anisotropy closer to the one naturally imposed by the DDI.
Ground states can thus form spontaneous density modulation. Calculations with
periodic boundary conditions along one direction were also performed, see Ref.
Wenzel  _et al._ (2017), proving that a continuous translation symmetry can
indeed be broken (at the thermodynamic limit). These predictions indicate
possibilities for quantum-stabilised dipolar “crystallised" ground states and
in particular open the doors for quantum-stabilised dipolar supersolids.

The work of Ref. Wenzel  _et al._ (2017) also experimentally produced
spontaneously density-modulated states in cigar-shaped Dy gases that show
close features to the expected ground states yet global phase coherence was
found to be absent. The lack of phase coherence was interpreted based on a
Josephson-Junction formalism, which describes the maintenance of a phase
relation via tunnelling processes between the individual droplets, and its
preclusion via quantum or thermal fluctuations. In this experimental
realisation, the tunnelling rate between the individual droplets, or in other
words, the wave-function overlap between them, was not enough to lock the
droplets in phase. The authors of Refs. Wenzel  _et al._ (2017); Baillie and
Blakie (2018) also showed that assemblies of droplets with sizeable wave-
function overlap were theoretically possible by either increasing the trap
frequencies or the atom numbers compared to the current experimental
configurations.

####  V.4.2 First experimental evidences of supersolid behaviors

Figure 28: Examples of statistical analysis of the self-interference patterns
of dipolar quantum gases showing hallmarks of supersolidity (a,c) (or normal
solidity (b,d)), i.e., density modulations with (or without) global coherence.
(a), (b) Norm of the Fourier transform of the integrated density profiles from
the individual TOF images (grey lines) and the norm of their complex average
(thick, blue or red, line) in the supersolid case and the insulating array of
droplets case. The existence of side peaks in the norm of the complex average
evidence coherence in between density peaks (see inset). Adapted from ref.
Böttcher  _et al._ (2019b). (c),(d) Representation in the complex plane of the
values of the Fourier transform of the TOF profiles at the position of the
first side peak (see a). The limited spread of the phases $\Phi$ evidences
global phase coherence. Adapted from Ref. Ilzhöfer  _et al._ (2021).

At the end of 2018, the progressive understanding of the many key features of
Bose gases of highly magnetic atoms combined in an acute picture. These
features include the discovery of the BMF stabilisation in such gases (see
Sec. V.3.2), the observation of the roton mode and its softening in cigar-
shaped clouds (see Sec. IV.1.3-IV.1.4), the possibility of density-modulated
ground states, matching an assembly of quantum-stabilised droplets (see Sec.
V.4.1), and the need of making the droplets more extensively overlap in order
to maintain the global phase coherence in such assemblies (see Sec. V.4.1).
Building on this knowledge, a set of experiments Tanzi  _et al._ (2019a);
Böttcher  _et al._ (2019b); Chomaz  _et al._ (2019) observed hallmarks of
supersolid behaviours in cigar-shaped gases of 162Dy, 166Er, as well as 164Dy
atoms. Accompanying theoretical works Böttcher  _et al._ (2019b); Chomaz  _et
al._ (2019); Roccuzzo and Ancilotto (2019); Kora and Boninsegni (2019); Zhang
_et al._ (2019) related these observations to an underlying supersolid ground
state of the gas, see Sec. V.4.3.

The three experimental works proved two hallmarks of supersolidity in
experiments, namely the simultaneous occurrence of a spontaneously formed
density modulation and of a global phase coherence. Density modulation was
revealed by the occurrence of modulated patterns in the gas absorption images,
either in situ Böttcher  _et al._ (2019b) or in time of flight. Here the
density patterns form from the self interference of the gas via free expansion
of the in situ modulation. The global coherence was demonstrated by a
statistical analysis of multiple repetitions of this self-interference TOF
patterns. Global coherence is here marked by the stability of the interference
patterns, differentiating supersolids from an incoherent array of droplets.
Global coherence was mostly quantified through an analysis of the complex
values of the Fourier transform of the TOF density profiles, see Fig. 28. We
note that the coexistence of density modulation and phase coherence does not
prove the superfluidity of the state, which can be demonstrated only by
probing dynamic-related properties of the state, see Sec. V.4.4.

All three experiments used relatively shallow cigar-shaped traps with
transverse magnetisation, similar to Ref. Chomaz  _et al._ (2018). The
spontaneous density modulation occurred only along one axis, the long axis of
trap. The coexistence of density modulation and phase coherence was observed
in narrow ranges of scattering length values, of a few $a_{0}$ wide, and
survives a few tens Tanzi  _et al._ (2019a); Böttcher  _et al._ (2019b);
Chomaz  _et al._ (2019) up to a few hundreds of ms Chomaz  _et al._ (2019) in
these first experiments, later extended up to few seconds Sohmen  _et al._
(2021). Building on this long lifetime, Ref. Chomaz  _et al._ (2019)
additionally established a different route than standard interaction tuning
towards supersolid states, which is based on direct evaporative cooling
starting from a thermal state, see also Sohmen  _et al._ (2021).

In all three works, the states with supersolid properties could be achieved by
ramping down the scattering length starting from a stable BEC, using a slower
ramp and a finer tuning of $a$ than Ref. Chomaz  _et al._ (2018). In the
earlier works of Refs. Ferrier-Barbut  _et al._ (2016b); Wenzel  _et al._
(2017) observing droplet assemblies in cigar shaped traps, the rougher tuning
of $a$ as well as the lower initial BEC atom numbers are also thought to have
prevented the observation of supersolid properties (see also Secs. V.3.2,
V.4.1). Finally, in the seminal work of Ref. Kadau  _et al._ (2016), a
distinct pancake-shaped geometry as well as smaller initial BEC atom numbers
were used. Several following theoretical as well as experimental works
indicates that the use of a cigar-shaped geometry was crucial for more easily
achieving supersolid states in experiment. In this geometry, the MF
instability is driven by the softening of a single (doubly-degenerate) roton
mode dictating the dominant wavelength of the density fluctuations Chomaz  _et
al._ (2018); Petter  _et al._ (2019); Hertkorn  _et al._ (2021a). In contrast,
the pancake case is more complex, with several radial and angular rotons,
corresponding to different structures of density fluctuations, simultaneously
softening Hertkorn  _et al._ (2021b); Schmidt  _et al._ (2021). Furthermore,
in the cigar-shaped case, it is expected that the transition to a supersolid
state can occur continuously, with the supersolid modulation directly
connecting to the softened roton mode, see e.g. Blakie  _et al._ (2020).

####  V.4.3 eGPE phase diagram beyond the single-droplet regime: droplet
assemblies and supersolid states.

Figure 29: Phase diagrams, calculated in Chomaz  _et al._ (2019) (a) and
Böttcher  _et al._ (2019b) (e) for the relevant experiments using Er, Dy
respectively. (b-e) ((g-i)) corresponding insitu axial density profile for
$N=5\times 10^{4}$ ($N=3.5\times 10^{4}$). Besides the BEC and single-droplet
phases, identified in Fig. 26, density-modulated ID and SSP ground states are
found. The SSP is identified by a non-fully contrasted density modulation. It
is sandwiched in a narrow $a$-range in between a regular BEC and a crystal of
independent droplets (ID).

Following the observations of the roton mode population in Ref. Chomaz  _et
al._ (2018) and shortly preceding the works of Refs. Tanzi  _et al._ (2019a);
Böttcher  _et al._ (2019b); Chomaz  _et al._ (2019), Rocuzzo and Ancilotto
Roccuzzo and Ancilotto (2019) theoretically explored the phase diagram of an
Er quantum gas in an infinite cigar-shaped geometry with periodic boundary
conditions. They relied on the eGPE framework developed in the context of the
quantum droplet studies (see Sec. V.2) and calculated the ground state as a
function of $a$. In this setting, they demonstrated the existence of a
supersolid phase (SSP) in an intermediate range of $a$, separating an array of
insulating droplet (ID) at low $a$ and a regular BEC at large $a$. Both the
SSP and the ID are density modulated ground states stabilised by quantum
fluctuations, similar to the single-droplet phase described in Sec. V.3.1. The
SSP distinguishes itself by bearing a density modulation of finite contrast.
In addition, Rocuzzo and Ancilotto directly computed, via dynamical
simulation, the superfluid density of the density-modulated states, thus
rigorously establishing the connection between the occurrence of non-fully-
contrasted density modulation and non-zero superfluid fraction. This work
proved the relevance of the SSP in cigar geometries, beyond finite-size
effects, see also Ref. Blakie  _et al._ (2020).

Together with their experimental results, both Refs. Böttcher  _et al._
(2019b); Chomaz  _et al._ (2019) reported on eGPE calculations of the ground-
state phase diagrams in the finite experimental geometries, see Fig. 29(a,f).
The ground states were calculated as a function of the atom number $N$ and the
scattering length $a$ for a given trap and atomic species. At low $N$, a
transition occurs from the regular BEC to the single-droplet phase when
decreasing $a$, as described in Sec. V.3.1, see also Refs. Bisset  _et al._
(2016); Wächtler and Santos (2016b); Chomaz  _et al._ (2019). In contrast, for
large-enough atom numbers, three different regimes could be identified similar
to the infinite case results of Ref. Roccuzzo and Ancilotto (2019): When
decreasing $a$, the regular BEC state transitions first to a density-modulated
state of finite contrast, identifying a SSP, and then to a density-modulated
state of contrast almost unity, forming an ID array, see Fig. 29(b-e,g-i). We
note that the atom number for the occurrence of density-modulated states
versus single-droplet depends on the atomic species (being smaller for Dy than
for Er), and on the trap, both on its overall tightness and on its shape. The
exploration of the most favourable parameters for supersolidity as well as
achieving such a phase in different settings are interesting directions that
are currently under investigation, see e.g. Sec. V.4.5.

Based on the eGPE framework, Refs. Böttcher  _et al._ (2019b); Tanzi  _et al._
(2019a); Petter  _et al._ (2021) also reported on simulations of the real-time
evolution induced by the finite scattering-length ramp used to reach the
various BEC/SSP/ID states in the experiments. These show that in the SSP
regime, the dynamical state has strong similarities with the expected ground
state and shows limited phase fluctuations. In contrast, when decreasing the
scattering length lower, to an ID regime, the dynamical state deviates from
the ground-state expectation and show large phase fluctuations.

####  V.4.4 Toward probing the dynamical response and superfluidity of dipolar
supersolids

Besides the study of the static properties of the state, as the hallmarks
reported in the first studies (see Sec. V.4.2), a huge interest is drawn by
the special dynamical properties associated to supersolidity. Indeed, such a
dynamics intrinsically connects to the superfluid character of the supersolid
and would characterise the related rigidity of its phase. The study of the
dynamical properties covers a wide range of phenomena and concepts, spanning
from the study of the spectrum of elementary excitations, to that of transport
properties, or of the response to an external rotation for instance.
Supersolidity brings exotic characteristics to these different features.

Figure 30: Observed response frequencies for quadrupole-type excitations in
the BEC-SSP-ID phase diagram, from (a) an Er sample, taken from Natale  _et
al._ (2019), and (b) a Dy sample, from Tanzi  _et al._ (2019b). Theory
predictions from (a) Bogoliubov theory, (b) real-time simulations, based on
eGPE theory are shown. In (a) all the elementary excitations, $l$, are shown
and the colour highlight their expected response $R_{l}$ to the performed trap
excitations.

The first of these aspects to be explored in dipolar gas experiments concerns
the properties of the spectrum of elementary excitations of a supersolid at
low momenta. Here, not one but two branches of low-energy (gapless) “phonons"
modes Andreev and Lifshitz (1969); Saccani  _et al._ (2012); Macrì  _et al._
(2013); Rossotti  _et al._ (2017); Macia  _et al._ (2012); Bombin  _et al._
(2017); Roccuzzo and Ancilotto (2019) are expected, the different branches
bearing excitations of different characters. Following Goldstone’s idea
Goldstone (1961), the number of gapless branch connects to the number of
spontaneously broken symmetries – two in the case of the supersolid: one
branch holds phonons of the crystal and relate to a broken translational
symmetry, while the other branch holds phonon of the superfluid (phase
phonons) and relate to a broken gauge symmetry. The phase branch corresponds
to the branch of lower velocity, and, when evolving through the BEC-SSP-ID
phase diagram, it is observed to soften and its weight in the response to
decrease, up to vanishing in the ID case. On the contrary the crystal branch
slightly harden. These properties of the supersolid excitation spectrum were
theoretically investigated in various systems Saccani  _et al._ (2012); Macrì
_et al._ (2013); Rossotti  _et al._ (2017); Macia  _et al._ (2012); Bombin
_et al._ (2017). The excitation spectrum behaviour in dipolar supersolid were
further theoretically investigated in various works Roccuzzo and Ancilotto
(2019); Natale  _et al._ (2019); Petter  _et al._ (2021); Hertkorn  _et al._
(2019).

Several experiments show signatures of these two branches Natale  _et al._
(2019); Tanzi  _et al._ (2019b); Guo  _et al._ (2019); Hertkorn  _et al._
(2021a). Due to the finite system size, the phonon branches are here
discretised (see also Sec. IV.1.3), and the experiments probe the response of
some specific low-lying modes of the trapped dipolar supersolid states. In
Refs. Natale  _et al._ (2019); Tanzi  _et al._ (2019b), the evolution of the
BEC’s lowest-lying quadrupole mode within the BEC-ID-SSP phase diagram is
probed, see Fig. 30. In the BEC regime, the system responds at a single and
roughly constant frequency, and when reaching the SSP regime, several
frequencies are observed in the system’s response. Furthermore, these
frequencies are found to organise in two branches as a function of $a$. One
branch is softening and one is hardening when decreasing $a$. These properties
are indicative of the emergence of modes of dominant phase and crystal
characters, respectively. Ref. Tanzi  _et al._ (2019b) identifies the
different modes via their distinct signatures in the time-evolution of the
gas’s self-interference patterns. Ref. Natale  _et al._ (2019) identifies
resonant frequencies via an unbiased principal component analysis of the self-
interference patterns and observes a mixed character in the associated
principal component structures. The different modes’ characters between Refs.
Natale  _et al._ (2019); Tanzi  _et al._ (2019b) may be attributed to the
different numbers of density peaks in the underlying supersolid states.
Distinctly, Guo et al. Guo  _et al._ (2019) probe the lowest lying mode of the
trapped supersolid, which has a frequency lying below the dipole mode and is a
Goldstone mode of phase character, see Fig. 31. The frequency of the mode
itself is not probed due to its low value, instead signatures of a spontaneous
population of the low-energy excited mode are observed via a statistical
analysis of the in situ density patterns. A peculiar feature of this mode is
that the motion it induces preserves the centre of mass position thanks to
superfluid flow. This implies a correlation between the array’s displacement
and the population imbalance between the density peaks, see Fig. 31. By
repeatedly producing and imaging steady-state samples, Ref. Guo  _et al._
(2019) evidenced such correlations in the limited $a$ range where the state is
supersolid. The presence of such correlations points to the existence of the
phase Goldstone mode and implies an underlying superfluid flow. An interesting
point to note is that the emergence of this Goldstone mode in the excitation
spectrum connects to the softening of the antisymmetric roton mode from the
BEC Guo  _et al._ (2019), see also Sec. IV.1.3. In contrast, the symmetric
roton mode connects to an amplitude mode, also called Higgs mode, which
sharply hardens when moving away from the SSP-BEC transition in the SSP
Hertkorn  _et al._ (2019). More recently, a detailed study of the in situ
density fluctuations in the SSP also revealed the existence of density and
crystal phonons Hertkorn  _et al._ (2021a), see also Sec. IV.1.3.

Figure 31: Dynamics induced by the lowest lying Goldstone mode of a supersolid
(top) and observed correlation between displacement and imbalance in the in
situ density patterns of experimental realisations of Dy supersolids at
$97.6a_{0}$ (bottom). Adapted from Ref. Guo  _et al._ (2019)

Besides the study of low-lying excitations, investigation of higher energy
modes of supersolids have been recently undertaken. This includes studies of
the response to rotational excitations Tanzi  _et al._ (2021), or to large-
momentum excitations Petter  _et al._ (2021). Rotation excitations in Ref.
Tanzi  _et al._ (2021) probe a scissor-type excitation, similar to that
described in Sec. V.3.4 for droplet states. Here the observed scissor-mode
frequency decreases at the transition from BEC to modulated states. This
indicates an increase of the state’s momenta of inertia. It is however not
straightforward to relate this behaviour to the superfluid fraction of the
state. Ref. Petter  _et al._ (2021) probes the scattering of quasi-free
particles via Bragg excitations (see also Sec. IV.1.3) along the BEC-SSP-ID
phase diagram. Here a decrease in the response amplitude is observed when
crossing from BEC to modulated states. From impulse-approximation theory, this
decrease probes a reduction of the population of the zero-momentum state. The
observed decrease surpasses expectations obtained from the eGPE steady states.
This was related to coherent phase evolution, induced by the dynamical
crossing of the quantum phase transition Chomaz (2020).

Besides studies of specific excited modes, the supersolid’s dynamical
properties have also been unveiled in the time evolution induced by parameter
quenches or ramps. This includes interaction quenches Ilzhöfer  _et al._
(2021) and evaporative cooling ramps Sohmen  _et al._ (2021). Ref. Ilzhöfer
_et al._ (2021) probes the dephasing and rephasing dynamics of the gas’ global
coherence when dynamically crossing the ID-SSP transition. The observed
dynamic was understood by comparing to a simple Josephson-Junction-Array model
where the Josephson tunnelling amplitude was quenched, while dissipation is
empirically introduced, using a Langevin formalism. This model shows a good
qualitative agreement with experiments, while quantitative discrepancies are
attributed to the inherently soft nature of the supersolid’s crystal and the
effects brought in by the possible excitation of the crystal’s phonons.
Studying the formation dynamics of supersolids via direct evaporative cooling
(ramp of optical dipole traps), Ref. Sohmen  _et al._ (2021) shows that
density modulation appears before global phase coherence, but after local
phase coherence, is established. Reference Sohmen  _et al._ (2021) also
studies the subsequent decay process of the supersolids, occurring
spontaneously under the effect of three-body recombination. In this decay, in
contrast to the formation dynamics, global coherence is observed to survive
longer than density modulation, while the temperature remains roughly
constant. Furthermore, in this process, the strength of the density
modulation, for a fixed value of the number of locally coherent atoms, is
observed to depend on the temperature, being weaker when colder.

The various dynamical studies performed up to now provide valuable insights
into the supersolid state special behaviours, including direct and indirect
signatures of the existence of phase and crystal phonons (near $k=0$) and
their mixing, of superfluid flow and transport, or of coherent and incoherent
phase dynamics in the experimentally produced states. Interesting prospects
include direct measurements of the superfluid properties, and in particular of
the superfluid density of supersolids, and the creation of vortices in the
supersolid rotation, see e.g. Roccuzzo  _et al._ (2020); Ancilotto  _et al._
(2021).

####  V.4.5 Supersolids with richer crystalline patterns and two-dimensional
character

As discussed in Sec. V.4.2, the first observations of supersolidity in dipolar
gases have been conducted in elongated and relatively shallow, cigar-shaped
traps, resulting in supersolids in which (i) the breaking of the translational
invariance occurs only along one dimension, (ii) the number of density
modulations found in the finite experimental system is small (typically a
handful) and thus finite size effects might be important. Yet, dipolar
supersolid states with larger and/or more complex crystalline structures, and
in particular where crystallisation occurs in two directions of space have
attracted intense interest. Theoretically, a series of works have focused on
the phase-diagram and excitation spectra of two-dimensional supersolids in
isotropic and anisotropic traps Zhang  _et al._ (2021); Hertkorn  _et al._
(2021c); Norcia  _et al._ (2021); Bland  _et al._ (2021); Poli  _et al._
(2021), on the possibility of creating vortex excitations Roccuzzo  _et al._
(2020); Ancilotto  _et al._ (2021); Gallemí  _et al._ (2020), and on the
emergence of exotic crystalline structures Zhang  _et al._ (2021); Hertkorn
_et al._ (2021c, b). Very recently, two-dimensional supersolidity has been
observed in experiments with Dy atoms, using anisotropic traps Norcia  _et
al._ (2021). As a function of the anisotropy of the trap in the directions
perpendicular to the atomic dipoles, experiments have demonstrated evaporative
phase transitions to ground states of various supersolid patterns from a
linear chain, to a zig-zag crystalline structures Norcia  _et al._ (2021), and
finally to a hexagonal structures in circular traps Bland  _et al._ (2021). In
a related effort, two-dimensional angular roton modes, analogous to the linear
roton mode in elongated traps Santos  _et al._ (2003); Chomaz  _et al._
(2018); Petter  _et al._ (2019), has been observed in radially extended traps
Schmidt  _et al._ (2021), see also Sec. IV.1.3 for more details. By further
tuning the interplay between trapping configurations, interaction parameters,
and atom numbers, even more exotic ground states are expected to appear, such
as honeycombs, ring and labyrinth-like supersolid phases Zhang  _et al._
(2021); Hertkorn  _et al._ (2021c, b). With the considerable interest that the
discovery of dipolar supersolids have generated, we expect a quick development
of the field and a blossom of works probing the various intriguing properties
of this phase as well as exploring its possibility in different settings.

##  VI Spin physics with highly magnetic atoms

In the previous Secs. IV and V, we presented the many-body physics arising in
quantum gases of magnetic atoms fully polarised in their lowest Zeeman state,
under the influence of the elastic DDI between the aligned atomic dipoles. In
the following, we discuss the physics of ultracold gases of magnetic atoms
when the spin degrees of freedom are free and taken into account. Of course,
short-range interactions among atoms can also be spin-sensitive, see
Secs.VI.1.6,II.4: magnetism is typically driven by the interplay of spin-
dependent long-range dipolar forces and spin-dependent short-range forces.
This interplay intrinsically takes on a very different flavor if the atoms are
free to move and collide, or if the experiment is performed with atoms
confined in an optical lattice wherein the spins are localised at each node of
a 3D array. In the latter case, the system realises spin-lattice models, where
spins interact both at a distance, directly through DDI, and by short-range
interactions via second-order processes in tunnelling. These so-called super-
exchange interactions will be reviewed in Sec. VII, while the current section
focuses on the situation without a lattice potential.

After introducing a few important concepts of the spin physics with ultracold
gases, including that which is unique to highly magnetic atoms in Sec. VI.1,
we will review the experimental achievements revealing the impact of the DDI
on spinor physics and on two-component Fermi mixtures; see Secs. VI.2 and
VI.3. The latter two situations can be distinguished by whether dipolar
relaxation can be neglected, resulting in a system in which the total
magnetisation is conserved in the sample; see Sec. VI.2, or whether the
demagnetisation is a key mechanism of the physics at play, yielding gases with
free magnetisation and an intrinsic spin-orbit coupling; see Sec. VI.3.
Finally, we will discuss the engineering of spin-dependent Hamiltonians using
light fields and the unique contributions provided by magnetic atoms in this
respect; in Sec. VI.4. We will examine the engineering an artificial spin-
orbit coupling, realising a special class of artificial gauge fields, and the
engineering of spin-spin interactions that yield entangled quantum spin
states.

###  VI.1 Introduction to spinor physics

####  VI.1.1 Magnetic atoms: a large composite spin

Unlike electrons, atoms possess a composite spin that may be large compared to
electrons’ $S=1/2$. This enriches spinor physics. For alkali atoms, the total
spin arises from the coupling of the spin-1/2 electron to its orbital angular
momentum $L$ and the nuclear spin $I$. This large spin allows for the study of
a new type of quantum fluid, involving the interplay between magnetism and
superfluidity Stamper-Kurn and Ueda (2013). The study of degenerate quantum
gases with a large spin degree of freedom $s>1/2$ initially focused on bosonic
atoms, i.e., Bose spinor gases Kawaguchi and Ueda (2012). Experiments explored
the spinor physics of $F=1$ Na atoms and $F=1$ and $2$ Rb atoms Stamper-Kurn
and Ueda (2013). More recently, these studies were extended to spinor fermions
using the $F=9/2$ state of fermionic K Krauser  _et al._ (2014).

Strongly magnetic atoms such as Cr and Lns are multi-electron systems, which
can possess an even larger total spin in the ground state than alkalis. For
example, bosonic Cr atoms have an electronic spin $S=3$ in the ground state
($L=0$), while bosonic Dy and Er atoms have $L=6,\,S=2$ and $L=5,\,S=2$
respectively. Including the hyperfine structure for the Fermi isotopes, one
can reach, for example, $F=21/2$ using 161Dy atoms. Besides the richer physics
at the many-body level, such large-spin atoms have also attracted attention
for studying highly non-classical behaviours involving quantum coherence and
entanglement, either at the many, few, or even single-particle levels Fernholz
_et al._ (2008); Chalopin  _et al._ (2018a); Evrard  _et al._ (2019). These
prospects may have important implications and applications for quantum
information processing, sensing, or metrology purposes. In particular, we note
that a single magnetic atom provides the realisation of large spin states
(length $F$) in an Hilbert space of relatively moderate size ($2F+1$) and
increasing linearly with the spin size. In contrast, a spin of similar length
realised by a set of spin-$1/2$ particles lives in an Hilbert space whose size
increases exponentially with the spin length. The reduction of the Hilbert
space offered by magnetic atoms is beneficial to temper the effect of
decoherence in the system.

We note that, in atomic gases, research has also explored small-spin systems,
e.g., those involving only two spin states to form an (effective) spin-1/2
system. Such a system can be accessed either by considering a hyperfine level
of $F=1/2$ (as for fermionic Li and K O’Hara  _et al._ (2002); Bourdel  _et
al._ (2003); Regal  _et al._ (2004); Zwierlein  _et al._ (2004); Bartenstein
_et al._ (2004)) or by isolating two states of a larger-$F$ level; see e.g.
Trotzky  _et al._ (2008); Anderlini  _et al._ (2007). For magnetic atoms,
however, isolation is complicated by the spin-changing DDI; see also Sec.
III.3. Nevertheless, as we have seen in Sec. III.3.4, thanks to the quantum
statistics, some subspaces can be effectively preserved in spin mixtures of
ultracold fermions on long time scales because decay rates are strongly
suppressed Burdick  _et al._ (2015). This is in particular the case for the
subspace formed by the two lowest spin states of a fermionic magnetic atom.
This feature has been used in experiments to study effective spin-1/2 dipolar
systems under the effect of an artificial spin orbit coupling Burdick  _et
al._ (2016b) and in bulk Baier  _et al._ (2018).

####  VI.1.2 Spin-dependent one-body Hamiltonian for spinor gases

In order to study pristine spinor physics, i.e., to isolate the effects
arising solely from spin-dependent interactions, it is convenient to first
consider a spin-independent one-body Hamiltonian (typically the sum of a
kinetic term and external potential; see Eq. (9)). Using a spin-space
generated from the hyperfine level of an atomic species, the kinetic part of
the Hamiltonian is spin independent. Then one need only consider the
conservative trap confining the atoms. This potential should be independent of
the magnetic sublevels. For alkali atoms, this is typically obtained by using
optical dipole traps created by far-off-resonant focused laser beams, in which
case the AC-Stark shift is approximately independent of the magnetic sublevel
Grimm  _et al._ (2000). As discussed in Sec. II.3, the situation of magnetic
atoms differs. Both for Cr and for Lns, the large electronic spin/angular
momentum results in large vector and tensor parts of the atomic
polarisability. This yields a significant dependence of the light shifts on
the magnetic sublevel, even when using light far away of optical transitions.
Typically, this is not negligible in experiments exploring dipolar magnetism.
It is then expected that spinor phases will be driven by an interplay between
spin-dependent interactions, the Zeeman effect, and the tensor light-shift.

####  VI.1.3 Spin-dependent interactions for large-spin atoms

To study of spin physics with large-spin atoms it is important to recall that
the interaction between the atoms can depend on the spin channel $\mathcal{S}$
of the collision. Here, $\mathcal{S}$ describes the total spin of two atoms
and is preserved during the collision in the absence of a DDI, see Secs. II.4
and III.

Quantum statistics plays a major role in selecting which molecular potentials
should be taken into account. Generally speaking, regardless of the statistics
of the particles involved, $\mathcal{S}+l$ must be even, where $l$ is the
relative angular momentum; see also Sec. III.1. For short-range interactions,
$s-$wave scattering dominates at low collision energy. Under these
circumstances, only even $\mathcal{S}$ need to be considered, both for bosons
and fermions.

The different molecular potentials have identical multipole expansion
parameters at long distance where only electrostatic interactions contribute.
However, at short distance, where electronic orbitals may overlap, quantum
statistics also plays a pivotal role. This is because the molecular potentials
associated with different spin channels are often quite different. Therefore,
different spin channels $\mathcal{S}$ typically correspond to different
scattering lengths $a_{\mathcal{S}}$. This is clearly true in the case of Cr
because of the large electronic spin of the atom. This leads to seven very
different molecular potentials, as calculated in Ref. Pavlović  _et al._
(2004) and confirmed by experiments Werner  _et al._ (2005); Pasquiou  _et
al._ (2010). Scattering lengths are also (most likely) spin dependent in Lns.
Because the outermost electronic shell has zero spin (identical to the case in
Yb), all molecular potentials have a similar shape even at short distances.
However, slight differences between the molecular potentials (due to, e.g.,
orbital anisotropy of unfilled submerged $f$-shell) are most likely sufficient
to lead to spin-dependent contact interactions Kotochigova (2014). As
highlighted in Sec. II.4, scattering properties of Ln are extremely difficult
to predict, while, on the experimental side, spin-dependent scattering lengths
have remained unexplored in these atoms; see also Sec. II.4.

In gases of large-spin magnetic atoms, there is, in addition to spin-dependent
contact interactions, the DDI that must be taken into account. Dipolar
interactions are not only spin-dependent, but they also induce direct spin-
orbit coupling (SOC); see Eqs. 3-3. Indeed, when including the spin degree-of-
freedom, the DDI leads to two extra terms in the Hamiltonian. These are in
addition to the bare (yet also spin-dependent) elastic term Eq. (3) that we
have considered up till now. The first term is the so-called spin-exchange
term of Eq. (4): it changes the spins of atoms at long distances while
conserving the total longitudinal magnetisation of the pair of particles. The
second term is the relaxation term of Eq. (3). This leads to a change in
magnetisation and converts spin angular momentum into orbital angular momentum
to conserve the total angular momentum in the system, thus providing a form of
SOC. Due to this peculiar SOC, and to the long-range coupling between spins,
large-spin systems thus allow the exploration of magnetism beyond paradigms
inherited from solid-state physics in addition to beyond what could be
achieved with ultracold atomic gases up till now. Experiments can be performed
both with bosonic and fermionic isotopes (see Sec. II), greatly enhancing the
scope for new magnetic behaviour.

####  VI.1.4 Mean-field spinor physics with spin-dependent contact
interactions

Because of the spin-dependence of the contact interactions described above,
the Hamiltonian of a spinor gas depends on the spin $s$ of the interacting
particles even in the absence of a DDI. The Hamiltonian becomes increasingly
complicated for increasing $s$ Kawaguchi and Ueda (2012). It is useful to
examine the simplest case for what can be considered as a large-spin system,
i.e., one beyond the spin-1/2 case. This is the case of $s=1$ atoms. In this
case, only two spin channels exist for the short-range interactions,
$\mathcal{S}=0$ and $\mathcal{S}=2$, with the corresponding scattering lengths
$a_{\mathcal{S}=0}$ and $a_{\mathcal{S}=2}$. In the Hamiltonian, given by Eq.
(13) within the mean-field picture, the difference between the two scattering
lengths is the key ingredient for spinor physics, giving rise to a so-called
$c_{1}$-interaction term with $c_{1}\propto\left(a_{2}-a_{0}\right)$.

This spin-dependent interaction term for example results in spin-exchange
dynamics that is experimentally apparent by monitoring the evolution of the
population in the different Zeeman states as a function of time. The spin-
exchange processes correspond to the Forster-like exchange of a spin
excitation $(m_{s}=0,m_{s}=0)\rightarrow(m_{s}=-1,m_{s}=1)$ and are driven in
the mean-field picture at a rate:

$\Gamma_{\rm
exc}\propto\frac{4\pi\hbar^{2}}{m}n(a_{\mathcal{S}=2}-a_{\mathcal{S}=0}).$
(64)

Spin dynamics has been widely studied in the community for both Bose and Fermi
quantum degenerate and thermal gases Stamper-Kurn and Ueda (2013);
Schmaljohann  _et al._ (2004); Kronjäger  _et al._ (2006); Sadler  _et al._
(2006); Liu  _et al._ (2009); Jacob  _et al._ (2012); Pechkis  _et al._
(2013); Krauser  _et al._ (2012, 2014); Evrard  _et al._ (2021). More
generally, the study of excitations within a spinor condensate is expected to
be very rich, with a number of possible topological excitations observable,
such as line defects, point defects, skyrmions Choi  _et al._ (2012), and
knots Kawaguchi and Ueda (2012).

Most importantly, spin-exchange processes associated with contact interactions
conserve the total longitudinal magnetisation of the pair of particles. This
conservation stems from the isotropy of contact interactions. A very important
practical consequence is that the linear Zeeman effect is gauged-out and does
not contribute to the dynamics or to the phase diagram. Spinor phases are
rather determined by the interplay between spin-dependent contact interactions
and the quadratic Zeeman effect Stamper-Kurn and Ueda (2013), leading to
quantum phase transitions Sadler  _et al._ (2006); Liu  _et al._ (2009); Jacob
_et al._ (2012). At low magnetic fields, where the quadratic Zeeman effect may
be neglected, the spinor ground state strongly depends on interactions. For
$s=1$, when $a_{2}<a_{0}$, the BEC is ferromagnetic (favouring collisions in
the $S=2$ channel), while the predicted ground state when $a_{0}<a_{2}$ is
polar Ho (1998). On the other hand, a large positive quadratic Zeeman effect
favours the polar state. As the spin increases, the wealth of possible phases
makes spinor quantum gases a promising area of research Kawaguchi and Ueda
(2012), including for the study of non-Abelian spinors Lian  _et al._ (2012).

####  VI.1.5 Beyond-mean-field spinor physics with spin-dependent contact
interactions

One of the important features of spinor gases is that quantum correlations can
naturally occur, allowing quantum fluctuations to play an important role. For
example, spin-exchange dynamics vanish at the mean-field level when a spinor
condensate is prepared in the initial state $m_{s}=0$. Quantum fluctuations
then drive the onset of spin dynamics, with analogies to parametric
amplification Bookjans  _et al._ (2011); Klempt  _et al._ (2010); Evrard  _et
al._ (2021). Atomic quantum optical effects lead to the generation of
entanglement Lücke  _et al._ (2014), which may have important applications for
atom-based squeezing and atom interferometry Kruse  _et al._ (2016).

Quantum fluctuations can also play a key role in determining the nature of the
many-body ground state. A formal mapping to quantum optics was proposed by Law
et al. Law  _et al._ (1998), who showed that the ground state differs from
mean-field predictions Ho (1998), at least for mesoscopic systems. Depending
on the sign of the spin-dependent interactions, the ground state is strongly
degenerate (with spontaneous symmetry breaking likely), or a condensate of
spin-singlet pairs. So-called ‘fragmented BECs’ have also been studied by Ho
and Yip Ho and Yip (2000), and were very recently experimentally produced in
Ref. Evrard  _et al._ (2020).

While most of the experiments have been performed with bosonic atoms, large-
spin fermions are also a promising direction for research on large-spin
systems. First studies have been performed in Hamburg Krauser  _et al._ (2012,
2014). These large-spin Fermi systems possess increased spin fluctuations due
to the large spin Wu (2006), new SU(N) symmetries for purely nuclear spins, as
it is the case for Yb and Sr Hermele  _et al._ (2009); Tey  _et al._ (2010);
Taie  _et al._ (2010, 2012); Pagano  _et al._ (2014), BCS pairing with a non-
singlet character Ho and Yip (1999), and greater than two-particle clustering
Rapp  _et al._ (2007).

####  VI.1.6 How are dipolar interactions expected to impact spinor physics?

The impact of dipolar interactions on spinor gases crucially depends on
whether the dipolar field is larger or smaller than the external magnetic
field. For a large magnetic field, magnetisation-changing collisions merely
lead to atomic losses, and therefore, within the dipole-relaxation-limited
lifetime, the experiment is essentially performed at constant magnetisation.
That is, it evolves under terms that conserve the longitudinal magnetisation
in the dipolar Hamiltonian.

Because they are spin-dependent, the DDIs modify the properties of spinor
gases when they are not negligible compared to spin-dependent contact
interactions. To judge a priori the impact of the DDI, one needs to compare
the dipolar length $a_{\rm dd}$ to differences between scattering lengths
Lahaye  _et al._ (2009). As a consequence, dipolar effects can be prominent
even for the case of alkali metal atoms, such as Na or Rb for which spin-
dependent contact interactions are much weaker than spin-independent
interactions. For example, the long-range and anisotropic nature of the DDI
introduces a nonlocal nonlinearity that can modify spin textures, as
experimentally investigated using alkali atoms Vengalattore  _et al._ (2008);
Eto  _et al._ (2014). As we will see, dipolar interactions also impact spin
dynamics.

The modifications introduced by the DDI on spinor physics are even more
pronounced when the magnetic field is not large compared to the dipolar field.
Then, the magnetisation-changing terms in the dipolar Hamiltonian cannot be
neglected. Unlike contact interactions, the DDI, by allowing changes in
magnetisation, yields an intrinsic SOC which breaks rotational symmetry; see
Sec. VI.1.3. It thus modifies the classification of the possible spinor
phases. The exact nature of the dipolar spinor ground state phases at low
magnetic field, which may depend on the trapping geometry and other spin-
dependent forces, is still a matter of theoretical investigation; see for
example Yi  _et al._ (2004) for the $s=1$ dipolar system in the single mode
approximation and Refs. Lian  _et al._ (2012); Cui  _et al._ (2013).

Spin-orbit coupling associated with the DDI corresponds to the exchange
between orbital angular momentum and spin angular momentum. The DDI can, for
example, trigger the demagnetisation of a sample, leading to spontaneous
rotation, analogous to the Einstein-de-Haas effect Kawaguchi  _et al._
(2006a); Santos and Pfau (2006); Gawryluk  _et al._ (2007); Sun and You
(2007). Furthermore, because of magnetisation-changing processes, the dipolar
spinor gas is sensitive to magnetic field, which provides new ways to study
phase transitions driven by the interplay between spin-dependent interactions
and the linear Zeeman effect Pasquiou  _et al._ (2011b). For example, it is
expected that a ferromagnetic spinor dipolar condensate may display
spontaneous circulation in the ground state at low magnetic fields Kawaguchi
_et al._ (2006b).

In the following, we will describe the experiments that address the role the
DDI has played in spinor physics, distinguishing between the cases where
magnetisation is conserved from those where it is free to evolve.

###  VI.2 Effects of dipolar interactions on spinor physics at constant
magnetisation

If the time-scales of dipolar relaxation are large compared to the other
relevant timescales in the system (see III.3), one can study spinor physics
under the effect of the DDI but at constant magnetisation. Here, the effect of
the DDI arises from the elastic term of DDI, Eq. (3), and the spin-exchange
term, Eq. (4). The intrinsic spin-orbit coupling introduced above is neglected
for now. The spin-dependent elastic DDI term impacts the spinor physics
because of its long-range and anisotropic nature.

####  VI.2.1 Dipolar-induced spin textures in BECs

One of the prominent features of the DDI, associated with its non-trivial
momentum dependence shown in Eqs. (8), is a tendency to develop spatial
structures. This has already been discussed in the spin-polarised case in
Secs. IV.1.3, IV.1.4, and V. In the context of spinor dipolar gases, this
tendency translates into the possibility of developing various spin-textures
Kawaguchi and Ueda (2012), i.e., inhomogeneous distributions of the spin
vector.

The influence of the DDI on spin domains of Bose quantum gases was
experimentally studied both in Berkeley Vengalattore  _et al._ (2008) (for a
Rb $F=1$ BEC with ferromagnetic contact interactions) and in Tokyo Eto  _et
al._ (2014) (for a Rb $F=2$ BEC whose contact interactions disfavour
ferromagnetism). In both experiments, spin domains were created using a well-
defined magnetic field gradient. In Ref. Vengalattore  _et al._ (2008), the
spin textures decayed toward a spatially modulated structure of spin domains.
The crucial role of the DDI was demonstrated by eliminating the DDI by a
sequence of rf pulses, in which case the authors observed a suppression of the
formation of the short-range domains. In Ref. Eto  _et al._ (2014), the spin
textures observed after propagation within the magnetic field gradient are
compared to numerical simulations of the Gross-Pitaevskii equation. This
reveals that the observed spatial modulation of the longitudinal magnetisation
is due to the spin precession in the effective magnetic field produced by the
DDI. Both these results demonstrate that the DDI has considerable effect, even
on spinor condensates of weakly dipolar alkali metal atoms.

####  VI.2.2 Gapped magnon excitations of BECs

Phonon excitations have been studied in the context of scalar BECs through,
e.g., Bragg excitation spectra Stenger  _et al._ (1999b); Stamper-Kurn  _et
al._ (1999); Steinhauer  _et al._ (2002); Vogels  _et al._ (2002); Ozeri  _et
al._ (2002); Bismut  _et al._ (2012); see also Sec. IV.1.3 for the dipolar
case. Moreover, there exists, in spinor BECs, spinfull excitations that also
behave like quasiparticles. Surprisingly, these excitations remain relatively
unexplored experimentally. One of the first investigations was performed at
Berkeley on a ferromagnetic Rb BEC Marti  _et al._ (2014). Magnon excitations
were precisely studied and shown to consist of a standing-wave of spin-excited
atoms above a ferromagnetic BEC. While the Goldstone theorem predicts that a
ferromagnetic state has gapless excitations (for symmetric interactions), a
gap was measured and ascribed to the presence of a weak DDI that breaks
rotational symmetry. Perhaps related to this feature, the authors also
observed a larger effective mass than expected by mean-field and beyond-mean-
field theories involving only $s$-wave interactions. Although these effects
remain small due to the weakness of the DDI in alkali atoms, they provide
qualitative departures from the paradigms of magnetism in the presence of
symmetric short-range interactions. This constitutes but one illustration of
the possibilities dipolar spinor gases offer.

In the case of Cr, trapped magnon excitations were created by applying, on a
polarised BEC, magnetic field gradients perpendicular to the magnetic field
axis Lepoutre  _et al._ (2018b). In that experiment, the wavelength of the
excitation is not small compared to the cloud’s size. The magnons that are
created are quantised in energy, and manifest themselves as collective modes
that couple the spin and the orbital degrees of freedom. The lowest energy
mode, whose frequency is set by the zero-point energy of the BEC, consists of
a sinusoidal oscillation of the local spin around its original axis, with an
oscillation amplitude that linearly depends on the spatial coordinates. The
observations are in excellent agreement with hydrodynamic equations. In the
regime the experiment was performed, the observed spin mode has a universal
character, independent of the atomic spin and spin-dependent contact
interactions, and is therefore rather insensitive to the DDI. It was
nevertheless predicted that dipolar interactions could alter such a mode,
provided that the atomic sample has a size larger than a natural wavelength
set by the DDI Lepoutre  _et al._ (2018b).

####  VI.2.3 Out-of-equilibrium spin dynamics at constant magnetisation

Strong dipolar effects have been observed in the spin dynamics of magnetic
atoms even on time scales short compared to the dipolar relaxation processes.
Studies where performed on $S=3$ 52Cr atoms, initially polarised in the lowest
energy spin state $m_{s}=-3$, and after homogeneously quenching a BEC into a
spin-excited state. This was first performed using an engineered tensor light
shift to promote all atoms into a well-defined Zeeman state $m_{s}=-2$ de Paz
_et al._ (2016). In a latter experiment, the spin excitation was performed by
simply rotating the spins using a rf pulse Lepoutre  _et al._ (2018a).

In both cases, out-of-equilibrium spin dynamics was monitored by means of a
Stern-Gerlach procedure to measure the population of atoms in the different
Zeeman states as a function of time. It was found that spin dynamics was
driven by an interplay between spin-dependent contact interactions and DDI.
Spin-dependent contact interaction played the dominant role in the spin
dynamics because they are relatively strong in the case of 52Cr atoms.

In the case where the atomic spins are tilted compared to the magnetic field
by an rf pulse, it was nevertheless demonstrated that initial spin dynamics
were entirely triggered by the DDI. This is a consequence of the
SU(2)-symmetric nature of contact interactions, which cannot trigger dynamics
after a simple rotation of the atomic spin initially in a stretched
(ferromagnetic) state. Therefore, the onset of spin dynamics after rotation of
the spins seen in Ref. Lepoutre  _et al._ (2018a) is a purely dipolar effect.
Finally, in the specific case where the spins were tilted by $\pi/2$ compared
to the magnetic field, it was found that the spin dynamics vanishes. The mean-
field theory provides a natural explanation for this phenomenon, as the torque
associated with the inhomogeneous magnetic field carried by the atoms vanishes
when the magnetisation of each atom vanishes Kawaguchi  _et al._ (2007).

In the $\pi/2$ rotation case, spin-dynamics could be recovered by applying a
magnetic field gradient providing the necessary spin-orbit coupling to trigger
dynamics. It was then discovered that spin dynamics develops while preserving
the local spin length of the condensate Lepoutre  _et al._ (2018a). This
protection of the initial ferromagnetic character of the gas was attributed to
an energy gap provided by sufficiently large spin-dependent contact
interactions. This dynamical protection of ferromagnetism is the reason why
trapped magnon excitations could be observed in the Cr experiments and
described using the hydrodynamic equation of a ferrofluid, indicating that the
Cr BEC behaves like a genuine ferrofluid; see Section VI.2.2 and Lepoutre  _et
al._ (2018b).

It should be noted that the protection of ferromagnetism in the 52Cr
condensates is tied to the relatively small strength of the DDI compared to
spin-dependent contact interactions; it is likely that similar experiments
performed (for example with Ln atoms) in the regime where dipolar interactions
overwhelm spin-dependent contact interactions may lead to qualitatively
different behaviour, since the DDI does not preserve the spin length. As we
will see in Sec. VII, working in optical lattices is another way to
investigate a purely dipolar spin system. In this case, the collective spin
length is indeed reduced during dynamics Gabardos  _et al._ (2020).

####  VI.2.4 Effective spin-1/2 mixture of dipolar fermions

Beyond the case of BEC described above, recent work has been dedicated to the
study of spinor systems of dipolar fermions Burdick  _et al._ (2015, 2016b);
Baier  _et al._ (2018). This system is of great interest for the study of how
the DDI affects superfluid pairing and the celebrated crossover from
delocalised Cooper pairs in the BCS regime to a BEC of molecules Bloch (2008);
Baranov  _et al._ (2012). By choosing specific spin mixtures, spinor gases of
constant magnetisation can be studied over long time scales, even for highly
magnetic species of Er and Dy. This is due to the quantum statistical
suppression of dipolar relaxation Burdick  _et al._ (2015); see also Sec.
III.3.4. This feature has been recently used to realise a long-lived spin-1/2
spin-orbit-coupled degenerate dipolar Fermi gas Burdick  _et al._ (2016b).
This experiment will be discussed in Sec. VI.4.1.

A spin-1/2 degenerate Fermi mixture of strongly magnetic atoms 167Er was
created Baier  _et al._ (2018), and the collisional behaviour of this mixture
experimentally investigated. This experiment used a deep 3D optical lattice as
a tool during the preparaiton of the mixtures, which serves to inhibit
collisional losses induced by large magnetic field sweeps and the crossing of
numerous FRs: A quantum-degenerate spin-polarised sample is first loaded into
a deep 3D optical lattice at low magnetic field, such that double occupancy
are precluded and fermions are spatial separated. The magnetic field can there
be ramped up to a value where quadratic Zeeman shifts are significant,
allowing spin state preparation using using rf sweeps, and later be ramped
down to the low-field region, where the physics of the spin mixture is
studied. To do so, the 3D lattice is slowly ramped down and the Fermi mixture
loaded back into a 3D trap. As already observed for the spin-polarized case –
see Sec. II.4) – the collisional properties of the spin mixtures as a function
of the bias magnetic field amplitude, $B$, is marked by a large number of
intra- and inter-spin FRs. Baier et al. Baier  _et al._ (2018) mapped the FR
spectra for $B$ in the $[0,2]$G range via loss spectroscopy. Many narrow and
overlapping features are obserrved. A comparatively broad and isolated
interspin FR is also identified. Baier et al. used this resonance to reach the
strongly-interacting regime. Here, a large collisional stability of the
balanced spin-mixture at $T/T_{F}\approx 0.3$ is observed, in particular on
the repulsive side of the FR. This paves the way toward study of BEC-BCS
physics in such systems.

###  VI.3 Spinor physics with free magnetisation

Besides the studies at constant magnetisation described above, the DDI
introduces fundamental new features due to the presence of magnetisation-
changing collisions; see Eq. (3). In this context, the linear Zeeman effect
can impact out-of-equilibrium physics and phase diagrams. In particular,
magnetisation-changing collisions can free the total magnetisation of a gas,
so that the system now becomes sensitive to the linear Zeeman effect.
Different regimes may be reached depending on the ratio of the Larmor
precession energy to other energy scales. A rather complex phenomenology
unfolds, depending on the Larmor frequency $g_{J}\mu_{B}B/h$ compared to the
following energy scales; note that typical values are indicated on the right
column for the specific case of 52Cr atoms and $g_{J}$ is the Landé factor:

Physical process | Energy scale
---|---
Trap depth |  $U_{0}/h=500$ kHz
Thermal excitations |  $k_{B}T/h=2.5$ kHz
Fermi energy |  $\epsilon_{F}/h=2.5$ kHz
Spin-dependent |  $\Gamma_{\rm exc}/h=250$ Hz
interactions |
Trap frequency |  $\omega_{L}/2\pi=250$ Hz
DDI |  $V_{\rm dd}/h=25$ Hz

$\Gamma_{\rm exc}$ was defined in Eq. (64).

  * •

$g_{J}\mu_{B}B>U_{0}$. Dipolar relaxation leads to (spin-sensitive) losses.

  * •

$U_{0}>g_{J}\mu_{B}B$. Dipolar relaxation does not lead to losses, but
introduces heating. If $g_{J}\mu_{B}B\gg k_{B}T$, most of the atoms are
polarised in the lowest energy state at equilibrium.

  * •

$g_{J}\mu_{B}B\approx k_{B}T\gg\Gamma_{\rm exc}$. The thermal gas is
spontaneously depolarised, populating a few Zeeman sublevels at equilibrium.
For bosons, Bose-stimulation insures that the BEC remains polarised in the
lowest energy state. However, for Fermi gases, the DFG may be depolarised if
$g_{J}\mu_{B}B<\epsilon_{F}$.

  * •

$g_{J}\mu_{B}B<\Gamma_{\rm exc}$. A demagnetised spinor BEC may be produced
provided spin-dependent contact interactions do not favour a polarised BEC in
the lowest energy state.

  * •

When $g_{J}\mu_{B}B\approx\hbar\omega_{L}$, the released energy due to dipolar
relaxation corresponds to one unit of excitation in the trap. This is
favourable for measuring the change in angular momentum associated with
dipolar relaxation; this is a quantum gas analogue of the Einstein-de-Haas
effect.

  * •

Reaching $g_{J}\mu_{B}B\leq V_{dd}$ would be very interesting as the dipolar
field then overcomes the ambient magnetic field. The (inhomogeneous) dipolar
field itself sets the lowest-energy spin-textures.

####  VI.3.1 Thermodynamics of a Bose gas with free magnetisation

Thermodynamics of a Bose gas with a spin degree of freedom is derived by
including the single-particle magnetic energy in the Bose occupation factor.
For non-interacting particles,

$f_{k,m_{s}}=\frac{1}{\exp\left[\left(\epsilon_{k,m_{s}}-\mu\right)/k_{\rm
B}T\right]-1},$ (65)

where $\mu$ is the chemical potential, and $\epsilon_{k,m_{s}}$ is the single-
particle energy of the (trapped) states labeled by the index $k$, in the
Zeeman state $m_{s}$. When magnetisation is free (case of dipolar particles)
$\epsilon_{k,m_{s}}$ includes the linear Zeeman effect, whereas when
magnetisation is fixed (as for example of Rb and Na atoms), the linear Zeeman
effect is gauged out and $\epsilon_{k,m_{s}}$ includes only the quadratic
Zeeman effect.

For the generic case of a $F=1$ spinor Bose gas interacting solely via contact
interactions (and thus at constant magnetisation and without the quadratic
Zeeman effect), the phase diagram has been worked out by Isoshima et al.
Isoshima  _et al._ (2000) as a function of the temperature $T$ and the
magnetisation $M$. Two phase transitions were predicted. The first transition,
at a critical temperature $T_{1}(M)$ separates the normal phase, and a phase
where a condensate forms in the (most populated) stretched state. Below a
second critical temperature $T_{2}(M)$, all other spin components condense
simultaneously (see Fig. 32).

First investigations of spinor physics with free magnetisation were performed
using cold Cr atoms in an optical dipole trap. It was observed that the spin
degrees of freedom equilibrated to the gas mechanical temperature Pasquiou
_et al._ (2012). At high temperature, the population of the different Zeeman
states followed a distribution close to the Boltzmann distribution. The spin
temperature was equal to the mechanical temperature, and thermal equilibrium
between spin and orbital degrees of freedom was ascribed to magnetisation-
changing dipolar collisions. Below the critical temperature for BEC, a
spontaneous accumulation into the Zeeman state of lowest energy was observed.
This led to a bimodal spin distribution, arising from BEC and the spontaneous
accumulation of the atoms in the lowest-energy spin state.

By varying the temperature and the magnetic field, it was possible to map out
the corresponding critical line $T_{1}(M)$ (see Fig. 32). However, the second
predicted phase transition at $T_{2}(M)$ could not be studied. Indeed, when
magnetisation is free, Bose stimulation towards the single-particle lowest-
energy state insures that the BEC is only produced in this polarised state
Pasquiou  _et al._ (2012). The second transition was recently observed in the
case of sodium atoms with negligible DDI by Frapolli et al. Frapolli  _et al._
(2017). The authors also studied the impact of spin-dependent interactions on
the double-condensation scenario.

Figure 32: Measured and predicted phase diagram of a spin 3 Cr BEC. Three
phases are observed: (A) thermal gas; (B) polarised BEC in stretched state;
(C) spin-full (depolarised) BEC. The histograms show typical experimental
population distributions. Black triangles are the measured $T_{1}$. Solid
lines shows the predictions of $T_{1,2}$ in the non-interacting case. Dashed
lines show typical evolution predicted by theories with no interactions when
the temperature is lowered at a given magnetic field $B$, for two different
regimes depending on whether magnetisation is fixed (left curve) or free
(right curve). Adapted from Pasquiou  _et al._ (2012).

In a later study, it was possible to reach a multicomponent phase with Cr
atoms by applying rapid, forced evaporative cooling to a depolarised gas, thus
performing the experiment fast compared to the time scale for magnetisation-
changing collisions. However, the spinor BECs were very small due an interplay
between Bose condensation and spin dynamics Naylor  _et al._ (2016). This
result pointed out the difficulty of fully thermalising the spin degrees of
freedom. This is a prominent effect to be taken into account for very large
spin systems such as Dy and Er where all spin states must be saturated for a
stable multicomponent BEC to be produced.

The connection between magnetic order and superfluid order and the BEC
transition is an intriguing question that remains largely unexplored. While
these orders are intrinsically connected due to Bose stimulation, it was
predicted that strong spin-dependent interactions induce spin ordering at a
finite $T$ above the BEC transition Natu and Mueller (2011). In low
dimensions, the connection between magnetic order and superfluidity promises
to be especially interesting. In particular, in the 2D polarised case,
superfluidity does not arise from condensation but by the pairing of vortices
with opposite circulation through the Berezinskii-Kosterlitz-Thouless
mechanism. This gives rise to a topological order Berezinskii (1972);
Kosterlitz and Thouless (1973); Hadzibabic  _et al._ (2006, 2006); Hadzibabic
and Dalibard (2009). The case of a depolarised gas promises new scenarios for
superfluidity, due to the existence of topological excitations such as half-
quantum vortices that include the spin degree of freedom Mukerjee  _et al._
(2006).

We note that the case of fermions will drastically differ from the scenario
described above due to the presence of the additional energy scale provided by
the Fermi energy. One consequence is that spontaneous demagnetisation of a
noninteracting Fermi sea is expected provided the Larmor energy is smaller
than the Fermi energy (Pauli paramagnetism). How interactions modify this