dependence of the cavity mode functions. At the same time, since $w_{\rm
BEC}\gg\lambda_{c}$, one can set all oscillating terms to zero in the
integrated equations. The cavity fields $\alpha_{mn}$ can be adiabatically
eliminated by substituting their steady-state solution [see Equation (23) for
the single-mode case] into the equations of motion for the atomic order
parameters $\Theta_{\rm c,s}$. This then leads to effective coupled equations
of motion for the atomic order parameters,

$\displaystyle i\partial_{t}\Theta_{\tau}$
$\displaystyle=(\mu+2\omega_{r})\psi_{\rm etr}^{\rm
avg}\Theta_{\tau}+\frac{g_{0}}{2}\psi_{\rm etr}^{\rm
avg}\psi_{0}^{2}\Theta_{\tau}^{*}+\frac{\mathcal{G}_{0}^{2}\Omega_{0}^{2}N}{\Delta_{a}^{2}|\Delta_{0}|}$
$\displaystyle\times\int
d\mathbf{r}_{\perp}d\mathbf{r}_{\perp}^{\prime}\sum_{\tau^{\prime}=\rm
c,s}\tilde{\mathcal{D}}_{\tau\tau^{\prime}}^{\rm
eff}(\mathbf{r}_{\perp},\mathbf{r}_{\perp}^{\prime};X_{0})\psi_{\rm
etr}(\mathbf{r}_{\perp})\psi_{\rm
etr}(\mathbf{r}_{\perp}^{\prime})|\psi_{0}|^{2}\left(\Theta_{\tau^{\prime}}+\Theta_{\tau^{\prime}}^{*}\right),$
(69)

where $\tau,\tau^{\prime}=\\{{\rm c,s}\\}$, $g_{0}$ is the short-range atom-
atom interaction strength, $\psi_{\rm etr}^{\rm avg}=\int
d\mathbf{r}_{\perp}\psi_{\rm etr}(\mathbf{r}_{\perp})$, and

$\displaystyle\tilde{\mathcal{D}}_{\tau\tau^{\prime}}^{\rm
eff}(\mathbf{r}_{\perp},\mathbf{r}_{\perp}^{\prime};X_{0})=|\Delta_{0}|\mathrm{Re}\sum_{m,n}\frac{\Xi_{mn}(\mathbf{r}_{\perp};X_{0})\Xi_{mn}(\mathbf{r}_{\perp}^{\prime};X_{0})}{\Delta_{mn}+i\kappa}\mathcal{O}_{mn}^{\tau}(X_{0})\mathcal{O}_{mn}^{\tau^{\prime}}(X_{0}),$
(70)

is the dimensionless, effective cavity-mediated interaction potential [cf.
Equation (67)]. Here the dimensionless transverse Hermite-Gaussian functions
are defined as

$\displaystyle\Xi_{mn}(\mathbf{r}_{\perp};X_{0})=\frac{w_{0}}{w(X_{0})}H_{m}\left(\frac{\sqrt{2}y}{w(X_{0})}\right)H_{n}\left(\frac{\sqrt{2}z}{w(X_{0})}\right)e^{-\mathbf{r}_{\perp}^{2}/w^{2}(X_{0})},$
(71)

and the overlap integrals (resulting from the integration along the
longitudinal direction) as
$\mathcal{O}_{mn}^{c}(X_{0})=\cos[(m+n)\phi_{0}(X_{0})]$ and
$\mathcal{O}_{mn}^{s}(X_{0})=\sin[(m+n)\phi_{0}(X_{0})]$, with

$\displaystyle\phi_{0}(X_{0})=\frac{\pi}{4}+\arctan{\frac{\lambda_{c}X_{0}}{\pi
w_{0}^{2}}}.$ (72)

An analytical expression for the effective cavity-mediated interaction
potential can be written in the following matrix form in the order-parameter
space, specified for simplicity to an even degenerate family at perfect
confocality $\epsilon=0$ and at the cavity midplane $X_{0}=0$, neglecting
losses $\kappa=0$ [150]:

$\displaystyle\mathrm{sgn}(\Delta_{0})\tilde{\mathcal{D}}^{\rm
eff}(\mathbf{r}_{\perp},\mathbf{r}_{\perp}^{\prime};X_{0})\bigg{|}_{\epsilon=\kappa=X_{0}=0}=\mathbb{I}\tilde{\mathcal{D}}_{\rm
loc}^{\rm
eff}(\mathbf{r}_{\perp},\mathbf{r}_{\perp}^{\prime})+\sigma^{z}\tilde{\mathcal{D}}_{\rm
nonloc}^{\rm eff}(\mathbf{r}_{\perp},\mathbf{r}_{\perp}^{\prime}),$ (73)

where $\mathbb{I}$ and $\sigma^{i}$ are the identity and Pauli matrices in the
order-parameter space. The potential is split into a local part,

$\displaystyle\tilde{\mathcal{D}}_{\rm loc}^{\rm
eff}(\mathbf{r}_{\perp},\mathbf{r}_{\perp}^{\prime})\bigg{|}_{\epsilon=0}=\frac{1}{4}\left[\delta\left(\frac{\mathbf{r}_{\perp}-\mathbf{r}_{\perp}^{\prime}}{w_{0}}\right)+\delta\left(\frac{\mathbf{r}_{\perp}+\mathbf{r}_{\perp}^{\prime}}{w_{0}}\right)\right],$
(74)

and a nonlocal part,

$\displaystyle\tilde{\mathcal{D}}_{\rm nonloc}^{\rm
eff}(\mathbf{r}_{\perp},\mathbf{r}_{\perp}^{\prime})\bigg{|}_{\epsilon=\kappa=X_{0}=0}=\frac{1}{\pi}\cos\left(\frac{2\mathbf{r}_{\perp}\cdot\mathbf{r}_{\perp}^{\prime}}{w_{0}^{2}}\right).$
(75)

The former originates from the peak of size $\xi$ visible in the actual
interatomic potential in Figure 30(a). The Dirac-delta form of
$\tilde{\mathcal{D}}_{\rm loc}^{\rm
eff}(\mathbf{r}_{\perp},\mathbf{r}_{\perp}^{\prime})$ is a peculiarity of the
ideal confocal case where an infinite number of modes contributes to the
cavity-induced interatomic potential. Slightly away from confocality, one has
instead [27],

$\displaystyle\tilde{\mathcal{D}}_{\rm loc}^{\rm
eff}(\mathbf{r}_{\perp},\mathbf{r}_{\perp}^{\prime})=\frac{M^{*}}{4\pi}\sum_{s=\pm
1}K_{0}\left(\frac{\mathbf{r}_{\perp}-s\mathbf{r}_{\perp}^{\prime}}{\xi}\sqrt{1+\frac{(\mathbf{r}_{\perp}+s\mathbf{r}_{\perp}^{\prime})^{2}}{8M^{*}w_{0}^{2}}}\right),$
(76)

with $K_{0}$ being the modified Bessel function of the second kind. In both
Equations (74) and (76), there is a contribution to the local interaction
potential from the mirror image located at $-\bm{r}_{\perp}^{\prime}$. The
nonlocal term (75) instead oscillates without decay, with a period set by the
beam waist $w_{0}$. This corresponds to the residual long-range oscillations
in the interatomic interaction potential in Figure 30(a).

Let us now turn to the actual experimental characterization of the interaction
potential. As mentioned, this relies on inferring its strength from the value
of the superradiant threshold. For a BEC of small transverse size $w_{\rm
BEC}$ compared to the beam waist $w_{0}$, we see from Equation (69) that close
to the confocal point the local part of the interaction potential is much more
important than the nonlocal part in determining the threshold, which would
depend on the self-interaction $\tilde{\mathcal{D}}^{\rm
eff}(\mathbf{R}_{\perp 0},\mathbf{R}_{\perp 0};X_{0})$ around the center of
the small atomic cloud $\mathbf{R}_{0}=(X_{0},\mathbf{R}_{\perp
0})=(X_{0},Y_{0},Z_{0})$. The contribution of the nonlocal part to the self-
interaction is indeed of order one, while the contribution from the local part
is of order $M^{*}\gg 1$ near confocality. For the determination of the
superradiant threshold, the two coupled equations for the order-parameter
components $\Theta_{\rm c,s}$ can be thus considered approximately decoupled.
This allows to obtain a simple relation between the critical value
$\Omega_{0\rm c}$ of the laser strength and the strength of the local part of
the effective self-interaction potential [27]:

$\displaystyle\tilde{\mathcal{D}}_{\rm loc}^{\rm eff}(\mathbf{R}_{\perp
0},\mathbf{R}_{\perp
0})\simeq-\frac{\omega_{r}\Delta_{a}^{2}|\Delta_{0}|}{N\mathcal{G}_{0}^{2}\Omega_{0\rm
c}^{2}}.$ (77)

It is evident that the self-interaction has to be negative (i.e., the short-
range part of the interatomic interaction potential must be attractive) in
order for the superradiant threshold to be reachable. We note at this point
the different sign convention between the present review article and Ref.
[27].

The relation of Equation (77) has been used in the experiment [27] to
reconstruct the spatial dependence of the local effective interaction (76).
More precisely, due to the divergence of the Bessel function at zero argument,
Equation (77), which assumes the transverse envelope of the BEC to be
infinitely localized $\psi_{\rm
etr}\propto\delta(\mathbf{r}_{\perp}-\mathbf{R}_{\perp 0})$, cannot be
directly used. The transverse spatial average in Equation (69) needs instead
to be performed over a finite BEC size. The experimental results are presented
in Figure 31 for different cavity lengths. As clear from Equation (76), even
though the experimental data of Figure 31(b) only measure the self-interaction
as a function of the center-of-mass position $Y_{0}$, they already show the
relevant scales characterizing the local interaction potential: the width
$\xi$ of the (attractive) peak and a background value decaying on a scale much
slower than $w_{0}$. Those features of the effective potential are also found
in the microscopic interatomic interaction potential shown in Figure 30.
Measurements involving two separate BECs have also been performed, which
allowed to access the finite-distance behavior of the local part of the
effective interaction potential. This first Stanford experiment [27] did focus
only on the local part of the interaction potential. The nonlocal part was the
object of a second experiment [221], to which we turn next.

Figure 31: Experimental characterization of the cavity-mediated interatomic
potential close to and away from confocality. (a) Cavity transmission spectrum
for five different cavity lengths, from confocal $\delta l_{\rm res}=l_{\rm
res}-R^{c}=0$ to a regime similar to the one presented in Figure 29, where the
even mode families are well separated. For all the different cavity lengths in
(a), panel (b) shows the absolute value of the strength of the effective self-
interaction potential for a small-sized BEC located at different positions on
the cavity midplane: $X_{0}=0$, measured using the theoretical model of
Equation (77). The change in the width of the peak at the cavity center as a
function of the degree of confocality is appreciable. The inset shows a zoom
of the data close to the cavity center. Figure adapted and reprinted with
permission from Ref. [27] © 2018 by the American Physical Society.

While the nonlocal part does not appreciably modify the superradiant
threshold, it is crucial in determining the associated density modulation of
the BEC. This is already evident from Equation (73), where the nonlocal
interaction favors one of the two components $\Theta_{\rm c,s}$ of the order
parameter depending on the transverse position. This means that, depending on
the BEC location the superradiant longitudinal density modulation would be
either cosinusoidal or sinusoidal. In fact, as discussed in Ref. [150], away
from the cavity midplane the nonlocal part of the effective interaction is not
a real quantity anymore, which results in a coupling between the two order
parameters via an additional term in Equation (73) proportional to
$\sigma^{x}\mathrm{Im}\tilde{\mathcal{D}}_{\rm nonloc}^{\rm eff}$. This
implies that in general the superradiant density modulation along the
longitudinal direction is not either cosinusoidal or sinusoidal, but rather a
superposition of the two with a phase dependent on the position of the BEC.
The behavior of this relative phase has also been characterized experimentally
by means of the cavity output field [221]. The principle of such measurement
can be understood in the adiabatic cavity limit, which we used to derive
Equation (69). In this limit, the cavity field components $\alpha_{mn}$ are
locked to the order-parameter components $\Theta_{\rm c,s}$, i.e., the cavity
fields are fully determined by the instantaneous value of the atomic order
parameters. The total cavity field in real space is then given by a sum over
the Hermite-Gaussian modes, weighted by the coefficients $\alpha_{mn}$. The
measured cavity output corresponds to the forward-travelling component of the
total cavity field passing through one of the mirrors. Choosing the parameters
as in Equation (73), it reads [150]:

$\displaystyle\alpha^{\rm F}(\mathbf{r}_{\perp})\propto\int
d\mathbf{r}_{\perp}^{\prime}\psi_{\rm
etr}(\mathbf{r}_{\perp}^{\prime})\tilde{\mathcal{D}}_{\rm loc}^{\rm
eff}(\mathbf{r}_{\perp},\mathbf{r}_{\perp}^{\prime})+e^{-2i\text{arg}\left(\Theta_{\rm
c}+i\Theta_{\rm s}\right)}\int d\mathbf{r}_{\perp}^{\prime}\psi_{\rm
etr}(\mathbf{r}_{\perp}^{\prime})\tilde{\mathcal{D}}_{\rm nonloc}^{\rm
eff}(\mathbf{r}_{\perp},\mathbf{r}_{\perp}^{\prime}),$ (78)

where we have assumed the BEC wavefunction to be real. In the cavity output
field, the two terms in Equation (78) can be clearly separated, as shown in
Figure 32. In panels (a) and (c), for two different positions of the BEC the
light intensity is presented, showing two bright spots corresponding to the
BEC and its mirror image, together with interference fringes. In accordance
with the discussion above, the bright spots localized around the BEC and its
image originate from the local part of the effective interaction potential,
while the periodic fringes originate from the nonlocal part. Having separated
the two contributions, a measurement of the local phase of the output field,
as shown in panels (b) and (d), allows then to extract the relative phase
between the local and nonlocal parts of the interaction potential, and thus
the relative weight of the cosinusoidal and sinusoidal modulations of the
superradiant BEC density. Panel (e) shows the theoretical prediction for the
relative phase as a function of the BEC position, with the two circles marking
the values used for the experimental output of panels (b) and (d). The
measured sign flip is consistent with the theoretical prediction.

Figure 32: Experimental characterization of the superradiant density wave via
the cavity output field at the confocal point. For two different positions of
a single BEC in the transverse direction, the light intensity, (a) and (c),
and phase, (b) and (d), are shown. Panel (e) shows the theoretical prediction
(see text) for the relative phase between the fringes and the bright spot,
with circles marking the values used in panels (b) and (d). Figure adapted and
reprinted with permission from Ref. [221] © 2019 by the American Physical
Society.

We conclude this section by discussing the symmetries of the effective model
of Equation (69). In the generic case just discussed, where the nonlocal part
of the interaction potential shifts and also couples the two components
$\Theta_{\rm c,s}$ of the order parameter, one is left with the usual single-
mode scenario where the superradiant transition corresponds to the spontaneous
breaking of a discrete $\mathbf{Z}_{2}$ symmetry [see Equation (15)]. In the
present notation, this symmetry corresponds to the choice of the global shift
$\gamma=0,\pi$ in Equation (68), while $\Theta_{\rm s,c}$ are not independent
so that a superposition of fixed relative phase is chosen, as shown in Figure
32. However, under certain conditions the two components of the order
parameter can become independent and equivalent, so that a continuous $U(1)$
symmetry emerges in the system [150]. Assuming perfect confocality and
choosing the cavity midplane $X_{0}=0$, the effective interaction is given by
Equation (73). Taking the transverse BEC size $w_{\rm BEC}$ much smaller than
$\lambda_{c}$, one can approximately consider only the self-interaction
$\mathbf{r}_{\perp}=\mathbf{r}_{\perp}^{\prime}$ in Equation (69) for the
order-parameter components. By choosing the position of the BEC such that
$\tilde{\mathcal{D}}_{\rm nonloc}^{\rm eff}(\mathbf{R}_{\perp
0},\mathbf{R}_{\perp 0})=0$, the equations for $\Theta_{\rm c,s}$ are not only
decoupled but also equivalent, so that their relative weight can be chosen
arbitrarily, thus a $U(1)$ symmetry emerges. Such special positions exist also
out of the cavity midplane, and the $U(1)$ symmetry can exist approximately
also for small deviations from confocality or finite BEC size [150].

###  5.4 Superradiant quantum crystals versus self-bound crystalline droplets

Having discussed the characteristics of the cavity-mediated interaction
potential around the confocal degeneracy, let us now explore the consequences
for the many-body physics. In particular, we focus on the types of crystalline
phases and phase transitions that can appear in this case. As already observed
in Section 5.3.1, the main feature of the cavity-mediated interaction in or
around the confocal point is the presence of an attractive minimum at zero
interatomic distance [see Figure 30(a) and (b)]. This feature is quite generic
to laser-driven atoms coupled to multiple cavity modes, as the strongest
constructive light scattering occurs typically when all the atoms are located
at the same position. For a laser which is red detuned from a degenerate mode
family, the resulting potential is attractive. As indicated by Equation (77),
red detuning is necessary to observe superradiance and the accompanying
crystallization of the atoms. Moreover, for a blue-detuned laser with respect
to the cavity modes, heating of the atomic cloud might become an issue.

The consequence of an attractive interaction minimum at zero interatomic
distance for an atomic bosonic cloud is the instability of the latter towards
collapse. This collapse is counteracted by the finite temperature or the
quantum kinetic energy, as well as by the short-range atomic repulsion,
therefore leading to the formation of a stable, self-bound droplet state.
Considering that the cavity-mediated interaction generically features also an
oscillatory part, depending on the period of such oscillations compared to the
droplet size, the latter may exhibit a density modulation associated with
superradiance. The density crystallization breaks spontaneously a (typically
discrete) translation symmetry of the system, accompanied with a phase-locking
of the superradiant cavity field with respect to the pump laser. Noting
Equation (67), we stress that the oscillatory part of the interaction might
stem from the cavity modes, i.e., from $\tilde{\mathcal{D}}$, and/or from the
spatial profile of the driving laser $\Omega(\mathbf{r})$.

Figure 33: Qualitative phase diagram for a BEC of atoms interacting via a
cavity-mediated interaction potential with an attractive minimum of finite
width $\xi$ at zero distance, as is the case in confocal cavities (see Figure
30). The axes correspond to the strength of the atom-cavity coupling
$\mathcal{G}_{0}$ and the strength of the short-range interatomic repulsion
$g_{0}$. The color scale indicates the value of the droplet order parameter or
the inverse participation ratio $\int
d\mathbf{r}\rho^{2}(\mathbf{r})/\left[\int
d\mathbf{r}\rho(\mathbf{r})\right]^{2}$. Adapted from Ref. [222].

Such superradiant droplets have been predicted to appear for multimode
cavities [223, 222] and also for a related geometry involving a retroreflected
beam passing through the atomic cloud [224]. The generic zero-temperature
phase diagram, as a function of the strength of the atom-cavity coupling
$\mathcal{G}_{0}$ versus the interatomic-repulsion strength $g_{0}$, is shown
in Figure 33. As expected, a strong enough interatomic repulsion is necessary
to stabilize an extended crystalline phase against the droplet. Otherwise, the
superradiant instability of the homogeneous phase always leads to the
formation of a self-bound, crystalline droplet state through a first-order
phase transition [223, 222]. Taking further into account the static optical
lattice created by the pump laser, a strong enough repulsion $g_{0}$
suppresses number fluctuations and brings in the Mott-insulator physics, as
discussed in Section 7 for extended superradiant states. As shown in Ref.
[222], this can lead to different types of quantum liquid-like droplets. In
Section 9, we will see another instance of transition between a superradiant
localized state and a superradiant extended quasicrystalline phase due to the
competition between the atom-cavity coupling (i.e., the cavity-mediated
attractive atomic interaction) and the interatomic repulsion in a different
geometry involving few cavity modes.

The qualitative phase diagram of Figure 33 is generic to systems where the
attractive minimum of the cavity-mediated interaction at zero distance has a
finite width $\xi$, as is the case close to a confocal degeneracy (see Figure
30). The attractive minimum of size $\xi$ has been indeed observed from the
superradiant light emission using a BEC inside a confocal cavity as discussed
in the previous section. Due to the tight tweezer trap, the BEC had a
characteristic size $w_{\rm BEC}\sim 10~{}\mu\text{m}\sim\text{min}\xi\sim
w_{0}/2$ (see Figure 31). Therefore, even though the droplet size can be
appreciably smaller than $\xi$ (depending on the interatomic repulsion), the
cavity-output spatial resolution available in Refs. [27, 221] would not have
allowed to identify the droplet. However, the use of a shallower trap creating
a BEC whose transverse extension is above $\xi$ should allow to distinguish
the superradiant droplet from the extended crystalline phase.

In the phase diagram of Figure 33, the nature of the transition between the
homogeneous and the crystalline phase has not been specified. The reason is
that this depends on the actual range of the interaction and not strictly on
the width $\xi$ of the attractive minimum at zero distance. A clear example is
the confocal interaction potential of Figure 30, where the actual range of the
interaction is infinite (compared to available atom-cloud sizes) while $\xi$
can be small. As already discussed, this is a consequence of the imperfect
destructive interference due to the missing odd or even modes in a confocal
family. In this situation, the crystallization transition is of second-order
and of the mean-field type, i.e., the same as in the single-mode case (see
Section 3.1.1). However, as already mentioned above, by keeping the cloud away
from the cavity axis and eliminating the long-range oscillating background via
an additional laser, a confocal cavity can be used to engineer a genuine
short-range interaction with a scale given by $\xi$, in which case beyond-
mean-field effects would become accessible.

A further option is to approach the concentric limit. As discussed in Sections
2.8.1 and 5.1, a degenerate family in the concentric case possesses both even
and odd transverse modes, which allows for completely destructive interference
at long distances and thus a finite cavity-mediated interaction range. As
already mentioned before, the perfect concentric limit can never be achieved
as it corresponds to an unstable point, so that only a near-concentric regime
is accessible. As shown in Figure 7(c), this regime has the peculiarity that
higher-order transverse modes lie lower in frequency, which means that for a
red-detuned pump laser the highest-order transverse modes are the closest ones
to resonance and thus mostly coupled with the atoms. Therefore, the resultant
cavity-mediated interaction potential strongly depends on the mode cutoff,
i.e., on the details of the cavity mirrors beyond the simple curvature and
reflectivity. A sharp mode cutoff is certainly not a good approximation. While
proper mode-dependent loss rates $\kappa_{\nu}$ in Equation (64) can work well
in concentric cavities, where among others diffraction losses due to finite
mirror sizes need to be accurately modeled [225]. Therefore, qualitative
features of the photon-mediated interaction in near-concentric cavities will
depend on the actual cavity employed. While a large portion of the phase
diagram should in general feature a droplet phase due to the attractive
minimum at zero distance expected also in near-concentric cavities, the
properties of the transition between the homogeneous and the extended
crystalline phase have to be discussed case by case.

Figure 34: Brazovskii-type superradiant crystallization. (a) Proposed setup
involving a quasi-two-dimensional cloud of bosons trapped on a longitudinal
plane around the cavity center. (b) Nodal structure of one of the modes of the
concentric cavity (TEM12 with 5 longitudinal nodes in the picture). (c) Sketch
of the free energy landscape as a function of the superradiant order
parameter. The right column corresponds to a fluctuation-induced first-order
transition of the Brazovskii-type. Panels (a) and (b) have been adapted and
reprinted with permission from Ref. [26] published at 2009 by the Nature
Publishing Group, and panel (c) from Ref. [134] © 2010 by the American
Physical Society.

Here we limit ourselves to briefly discuss a particular example investigated
theoretically in Ref. [26]. Though considering an idealized case, it well
illustrates the promise of multimode cavities for realizing interesting many-
body physics of crystallization. The proposed setup is illustrated in Figure
34. The basic idea is that (thermal or quantum) fluctuations, which become
important for a short-enough interaction range, can modify the order of the
transition as illustrated by the free energy sketched in Figure 34(c). In
Refs. [26, 134], starting from the microscopic description of the atom-cavity
system, an effective free energy for the superradiant order parameter is
derived. We will not present the calculation here and limit ourselves to
describe the qualitative picture instead. Assuming one can approximately
realize a radial symmetry on the atomic plane for a perfectly concentric
cavity, the superradiant order parameter corresponds to a density modulation
with a wave vector whose modulus is set by the laser wavelength but its
direction is arbitrary chosen in the plane. This plane-wave-like picture
relies on an approximate momentum conservation valid in the limit of very
large clouds with respect to the wavelength, and also on excluding (via a
sharp cutoff) modes which have a more radial than longitudinal nodes. Within
this description, getting closer to the superradiant threshold the order
parameter fluctuations grow. Since they can have an arbitrary direction in the
plane, they form a quasi-continuum which impacts the critical behavior very
strongly, so that the order of the transition is changed from the second-order
mean-field type to a fluctuation-induced weak first order. This mechansim is
known as the Brazovskii mechanism [226, 227]. Besides relying on a quasi-
continuum of order-parameter components with radial symmetry, it also requires
the free energy not to contain cubic terms, so that at the mean-field level
the crystallization transition can remain of second order (which is actually
not the typical case in most materials [227]). In the specific geometry of
Figure 34(a) and within the approximations employed in Ref. [26], one can show
that the cubic terms are indeed absent from the effective free energy for the
superradiant order parameter [26, 134]. We note that the instability towards
the self-bound droplets discussed above, which corresponds to a strongly
first-order transition, is automatically excluded if the cubic terms are
absent.

##  6 Multi-component quantum gases inside cavities: density and spin self-
ordering

So far, we have mainly focused on many-body cavity QED systems where the
internal dynamics of atoms are not relevant (i.e., they evolve much faster
than the external degrees of freedom) and are omitted. That is, ultracold
atomic gases are approximated as polarizable quantum media. Nonetheless, it is
natural to wonder what will happen if the internal dynamics of the atoms also
become important. This results in a new degree of freedom—“pseudospin”—which
follows the same algebra as a genuine “spin”, and allows in turn for studying
combined density and spin self-ordering dynamics [228, 68, 71, 130, 69, 229,
230, 231].222We will just use spin instead of pseudospin to refer to internal
states and internal dynamics of the atom throughout this review paper. In
particular, the exchange of cavity photons by the atoms in this case can also
result in long-range interactions between atomic spins independent of the
temperature of the atomic cloud, similar to dipolar interactions between polar
molecules. This is in contrast to spin exchange interactions stemming from the
Hubbard contact interactions in atomic gases trapped in free-space optical
lattices, which only come into play at very low temperatures. The cavity-
mediated spin interactions can be modified and tuned experimentally, allowing
one to study different magnetic orders. Therefore, many-body cavity-QED
systems provide a promising, versatile platform to implement various spin-
model Hamiltonians and explore quantum magnetism in a controlled way.

###  6.1 Multi-component ultracold atoms coupled to cavities: Fundamentals

Spin-dependent self-organization of a spinor BEC was already discussed in
Section 3.3.2, where the population of each spin component was conserved. Here
we instead focus on more general processes, where the cavity-assisted photon
scattering drives spin-changing Raman processes and therefore the population
of individual spin components is not conserved across the superradiant phase
transition. This leads to the concept of combined, stationary density and spin
self-ordering introduced in Ref. [68], where they studied the Dicke
superradiance phase transition for a generalized atomic system with both
internal and external quantized degrees of freedom. The nonequilibrium
dynamics of a similar, related setup was also studied in Ref. [228]. The
considered setup consists of four-level ultracold atoms with the ground-state
manifold $\\{\downarrow,\uparrow\\}$ and excited states $\\{1,2\\}$ with
energies
$\\{\hbar\omega_{\downarrow}=0,\hbar\omega_{\uparrow},\hbar\omega_{1},\hbar\omega_{2}\\}$,
tightly confined into the $x$-$y$ plane containing the axis of a standing-wave
linear cavity [108, 71, 69]. The atoms are pumped in the transverse direction
using two sufficiently red-detuned, non-interfering pump lasers and also
strongly coupled to an off-resonance empty mode of the cavity. Namely, the
first (second) laser with the frequency $\omega_{p1}$ ($\omega_{p2}$) drives
the transition $\uparrow\>\leftrightarrow 1$ ($\downarrow\>\leftrightarrow 2$)
with the position-dependent Rabi rate $\Omega_{1}(\mathbf{r})$
[$\Omega_{2}(\mathbf{r})$], while the transitions $\downarrow\>\leftrightarrow
1$ and $\uparrow\>\leftrightarrow 2$ are both induced by the cavity mode with
the frequency $\omega_{c}$ and the same coupling strength
$\mathcal{G}(x)=\mathcal{G}_{0}\cos(k_{c}x)$ as depicted in Figure 35(a). For
instance, for an atomic species with Zeeman sublevels with corresponding
magnetic quantum numbers $m_{\downarrow}=m_{1}$ and
$m_{\uparrow}=m_{2}=m_{\downarrow}+1$, the polarization of the cavity field is
chosen to be linear along the quantization axis $z$, while the pump lasers
have in-plane polarizations.333Alternatively, the pump lasers
$\Omega_{1,2}(\mathbf{r})$ can be aligned in the $z$ direction with left/right
circular polarizations, where now the physics in the $x$-$z$ plane is
identical to the one presented in the text for given choices. The scheme
constitutes a double $\Lambda$ configuration with large atomic detunings
$\Delta_{1}\equiv(\omega_{p1}+\omega_{p2})/2-\omega_{1}$ and
$\Delta_{2}\equiv\omega_{p2}-\omega_{2}$, where the ground states
$\\{\downarrow,\uparrow\\}$ are the relevant spin states and form a spin-1/2.

Figure 35: Scheme and phase diagram for the cavity-induced density and spin
self-ordering. (a) Four-level bosonic atoms are coupled to a single standing-
wave mode of a linear cavity and pumped in the transverse direction by two
pump lasers. As can be seen from the inset, the atom-photon couplings comprise
a double $\Lambda$ scheme, with far off-resonant, red-detuned transitions. (b)
The phase diagram of the system in the $R$-$\eta_{0}$ plane for
$\text{Re}(\tilde{\delta}_{c})<0$ exhibits four fundamentally different
cavity-induced spin orders. Below the pump-laser strength threshold
$\eta_{0c}(R)$, the ground state of the system is a spin-polarized
ferromagnetic (FM) normal state. Increasing the pump-laser strength beyond the
threshold, the system undergoes the Dicke superradiant phase transition and
exhibits an emergent magnetic order: domain-wall $X$-antiferromagnetic
($X$-AFM), domain-wall spin-spiral, or chiral spin-spiral order depending on
the competition among different cavity-induced long-range spin-spin
interactions. The structure of the density self-ordering is different in each
phase. Figure adapted and reprinted with permission from Ref. [69] © 2019 by
the American Physical Society.

Adiabatic elimination of the atomic excited states yields an effective
Hamiltonian $\hat{H}_{\rm
eff}=\int\hat{\Psi}^{\dagger}(\mathbf{r})\hat{\mathcal{H}}_{1,\rm
eff}\hat{\Psi}(\mathbf{r})d\mathbf{r}-\hbar\Delta_{c}\hat{a}^{\dagger}\hat{a}$,
where $\Delta_{c}\equiv(\omega_{p1}+\omega_{p2})/2-\omega_{c}$, and $\hat{a}$
and $\hat{\Psi}=(\hat{\psi}_{\uparrow},\hat{\psi}_{\downarrow})^{\top}$ are
the photonic and two-component (spinor) atomic annihilation field operators,
respectively. The single-particle atomic Hamiltonian density reads [69],

$\displaystyle\hat{\mathcal{H}}_{1,\rm
eff}=\hbar\begin{pmatrix}-\frac{\hbar}{2M}\boldsymbol{\nabla}^{2}+\delta+V_{\uparrow}(\mathbf{r})+U_{\uparrow}(\mathbf{r})\hat{a}^{\dagger}\hat{a}&\eta_{1}^{*}(\mathbf{r})\hat{a}+\eta_{2}(\mathbf{r})\hat{a}^{\dagger}\\\
\eta_{1}(\mathbf{r})\hat{a}^{\dagger}+\eta_{2}^{*}(\mathbf{r})\hat{a}&-\frac{\hbar}{2M}\boldsymbol{\nabla}^{2}+V_{\downarrow}(\mathbf{r})+U_{\downarrow}(\mathbf{r})\hat{a}^{\dagger}\hat{a}\end{pmatrix},$
(79)

with the classical pump potentials
$V_{\uparrow(\downarrow)}(\mathbf{r})=|\Omega_{1(2)}(\mathbf{r})|^{2}/\Delta_{1(2)}$,
the quantum cavity potentials
$U_{\uparrow(\downarrow)}(\mathbf{r})=|\mathcal{G}(x)|^{2}/\Delta_{2(1)}$, and
the two-photon Raman-Rabi coupling strength
$\eta_{j}(\mathbf{r})=\mathcal{G}^{*}(x)\Omega_{j}(\mathbf{r})/\Delta_{j}$
with $j=1,2$. Here,
$\delta\equiv\omega_{\uparrow}-(\omega_{p2}-\omega_{p1})/2+\delta_{\rm
Zeeman}$ is the effective frequency difference between the two spin states,
where an external magnetic field $B_{\rm ext}$ has been applied to fix the
quantization axis (along $z$) and tune the effective two-photon detuning via
the Zeeman shift $\delta_{\rm Zeeman}(B_{\rm ext})$. The two-body contact
interactions are assumed to be small and are omitted. Note that for _bosonic_
atoms by restricting to an appropriate low-energy manifold including both
internal and external states, one can map the effective Hamiltonian
$\hat{H}_{\rm eff}$, corresponding to Equation (79), into the Dicke
Hamiltonian [232, 71].

From the steady-state photonic field operator,

$\displaystyle\hat{a}_{\rm
ss}=\frac{\int\left[\eta_{1}(\mathbf{r})\hat{s}_{-}(\mathbf{r})+\eta_{2}(\mathbf{r})\hat{s}_{+}(\mathbf{r})\right]d\mathbf{r}}{\Delta_{c}+i\kappa-\int\left[U_{\downarrow}(\mathbf{r})\hat{n}_{\downarrow}(\mathbf{r})+U_{\uparrow}(\mathbf{r})\hat{n}_{\uparrow}(\mathbf{r})\right]d\mathbf{r}},$
(80)

where
$\hat{n}_{\tau}(\mathbf{r})=\hat{\psi}^{\dagger}_{\tau}(\mathbf{r})\hat{\psi}_{\tau}(\mathbf{r})$
and
$\hat{\mathbf{s}}(\mathbf{r})=\hat{\Psi}^{\dagger}(\mathbf{r})\boldsymbol{\sigma}\hat{\Psi}(\mathbf{r})$
(with $\boldsymbol{\sigma}$ being the vector of Pauli matrices) are the local
density and spin operators, respectively, one can see that the cavity field is
derived by pump-photon scattering processes involving the atomic spin
operators
$\hat{s}_{+}(\mathbf{r})=\hat{s}_{-}^{\dagger}(\mathbf{r})=\hat{\psi}^{\dagger}_{\uparrow}(\mathbf{r})\hat{\psi}_{\downarrow}(\mathbf{r})$,
rather than the atomic density as in the normal self-ordering of single
component atoms; cf. for instance Equation (11). That is, the scattering of
pump-laser photons into the cavity mode is accompanied by the atomic spin
flips $\uparrow\leftrightarrow\downarrow$ and corresponding momentum kicks to
the atom depending on the pump lasers’ spatial profiles
$\Omega_{1,2}(\mathbf{r})$.

####  6.1.1 Cavity-induced long-range spin-spin interactions and effective
spin Hamiltonians

For the cavity-field dynamics evolving on a much faster time scale compared to
the internal (i.e., spin) and external (i.e., center of mass) atomic dynamics,
the former is locked to the latter ones in the sense of the Born-Oppenheimer
approximation. To this end, the steady-state photonic field operator (80) can
be adiabatically eliminated in close analogy to the single-component atoms
described in Section 2.2. This leads to an effective atom-only Hamiltonian
consisting of a (local) spin-independent non-interacting part for the center-
of-mass motion, plus a long-range interaction part for the spin degree of
freedom [69]—cf. the penultimate term in the single-component effective atom-
only Hamiltonian (2.2),

$\displaystyle\hat{H}_{\rm spin}=\iint\Big{\\{}\sum_{\beta=x,y}$
$\displaystyle J_{\rm
Heis}^{\beta}(\mathbf{r}^{\prime},\mathbf{r})\hat{s}_{\beta}(\mathbf{r}^{\prime})\hat{s}_{\beta}(\mathbf{r})+J_{\rm
DM}^{z}(\mathbf{r}^{\prime},\mathbf{r})\left[\hat{s}_{x}(\mathbf{r}^{\prime})\hat{s}_{y}(\mathbf{r})-\hat{s}_{y}(\mathbf{r}^{\prime})\hat{s}_{x}(\mathbf{r})\right]$
$\displaystyle+$ $\displaystyle J_{\rm
cc}^{xy}(\mathbf{r}^{\prime},\mathbf{r})\left[\hat{s}_{x}(\mathbf{r}^{\prime})\hat{s}_{y}(\mathbf{r})+\hat{s}_{y}(\mathbf{r}^{\prime})\hat{s}_{x}(\mathbf{r})\right]\Big{\\}}d\mathbf{r}d\mathbf{r}^{\prime}+\int
B_{z}(\mathbf{r})\hat{s}_{z}(\mathbf{r})d\mathbf{r}.$ (81)

The first term in the effective spin Hamiltonian $\hat{H}_{\rm spin}$,
Equation (6.1.1), corresponds to the $x$ and $y$ components of a Heisenberg-
type interaction
$\hat{\mathbf{s}}(\mathbf{r}^{\prime})\cdot\hat{\mathbf{s}}(\mathbf{r})$. The
second term corresponds to the $z$ component of a Dzyaloshinskii-Moriya-type
(DM) interaction
$\hat{\mathbf{s}}(\mathbf{r}^{\prime})\times\hat{\mathbf{s}}(\mathbf{r})$. The
third term is a long-range coupling between the $x$ and $y$ components of the
spins, hence referred to as the cross-component spin interaction. Finally, the
last term serves as a local magnetic bias field, fixing the spin quantization
axis. Depending on the sign of $J_{\rm Heis}^{\beta}$, the Heisenberg
interaction favors ferromagnetic (FM) or antiferromagnetic (AFM) ordering. On
the other hand, the DM and cross-component interactions favor chiral spin
states such as spin spiral and skyrmion. It is the competition among these
cavity-induced spin-spin interactions as well as the local magnetic bias field
that determines the nature of the emergent magnetic order in the superradiant
phase. The magnetic field and the spin-coupling coefficients are position
dependent and are related to the cavity mode function and the pump-field
spatial profiles as [cf. the density-density interaction strength
$\mathcal{D}(\mathbf{r}^{\prime},\mathbf{r})$ in Equation (13)]:
$B_{z}(\mathbf{r})=\hbar\delta/2+\hbar|\Omega_{1}(\mathbf{r})|^{2}/2\Delta_{1}-\hbar|\Omega_{2}(\mathbf{r})|^{2}/2\Delta_{2}$,
$J_{\rm Heis}^{x/y}=\text{Re}(c_{1})\pm\text{Re}(c_{2})$, $J_{\rm
DM}^{z}=-\text{Im}(c_{1})$, and $J_{\rm cc}^{xy}=-\text{Im}(c_{2})$ with

$\displaystyle c_{1}(\mathbf{r}^{\prime},\mathbf{r})$
$\displaystyle=2\hbar\mathcal{G}(x^{\prime})\mathcal{G}(x)\left[\frac{1}{\Delta_{1}^{2}\tilde{\delta}_{c}}\Omega_{1}(\mathbf{r}^{\prime})\Omega_{1}^{*}(\mathbf{r})+\frac{1}{\Delta_{2}^{2}\tilde{\delta}_{c}^{*}}\Omega_{2}^{*}(\mathbf{r}^{\prime})\Omega_{2}(\mathbf{r})\right],$
$\displaystyle c_{2}(\mathbf{r}^{\prime},\mathbf{r})$
$\displaystyle=\frac{2\hbar\mathcal{G}(x^{\prime})\mathcal{G}(x)}{\Delta_{1}\Delta_{2}}\left[\frac{1}{\tilde{\delta}_{c}}\Omega_{2}(\mathbf{r}^{\prime})\Omega_{1}^{*}(\mathbf{r})+\frac{1}{\tilde{\delta}_{c}^{*}}\Omega_{1}^{*}(\mathbf{r}^{\prime})\Omega_{2}(\mathbf{r})\right].$
(82)

The dispersively-shifted complex cavity detuning
$\tilde{\delta}_{c}\equiv\delta_{c}+i\kappa=\Delta_{c}-\int[U_{\downarrow}(\mathbf{r})\hat{n}_{\downarrow}(\mathbf{r})+U_{\uparrow}(\mathbf{r})\hat{n}_{\uparrow}(\mathbf{r})]d\mathbf{r}+i\kappa$
has been introduced for the shorthand. We note that the coupling coefficients
are periodic and sign changing, reminiscent of the
Ruderman–Kittel–Kasuya–Yosida interaction in some heavy-fermion materials and
metallic spin glasses [109].

One can implement other spin models by utilizing different atom-photon
coupling schemes. For instance, in a single $\Lambda$ scheme where one leg of
the transition is driven by a pump laser and the second leg is coupled to
another pump laser and a linear-cavity mode, in addition to the long-range
$XXZ$ Heisenberg interactions and the cross coupling between the $x$ and $z$
components of the spins, cavity photons also induce long-range density-density
and density-spin interactions [231]. Therefore, this cavity-QED system offers
an alternative approach for simulating $t$-$J$-$V$-$W$-like models implemented
via polar molecules in optical lattices [233]. It has also been recently
predicted that long-range, cavity-mediated interactions of the
antiferromagnetic Heisenberg type can frustrate magnetic order enough to
stabilize spin liquid phases, even in absence of geometric frustration [234].
We also note that the range of the cavity-mediated spin-spin (and density-
spin) interactions can be tuned and become finite range by utilizing multimode
cavities similar to the tunable density-density interactions in the single-
component atoms coupled to multimode cavities as discussed in Section 5.3
[109].

###  6.2 Spinor Bose gases

In this section, we focus on bosonic atoms with the effective Hamiltonian (79)
[69]. The steady state of the system can be obtained for any given laser
profiles $\Omega_{1,2}(\mathbf{r})$. An instructive and interesting scenario
is to consider the spatial profiles $\Omega_{1,2}(y)=\Omega_{01,02}(e^{\pm
ik_{c}y}+Re^{\mp ik_{c}y})/(1+R)$, where $0\leqslant R\leqslant 1$ is the
reflectivity of mirrors retroreflecting the pump lasers. $R=0$ and 1
correspond to pure running-wave and pure standing-wave pump lasers,
respectively, while $0<R<1$ yields a mixture of running and standing waves.
The amplitudes $\Omega_{01,02}$ and $\mathcal{G}_{0}$ are taken to be real,
with the balanced Raman condition
$\eta_{0}\equiv\mathcal{G}_{0}\Omega_{01}/\Delta_{1}=\mathcal{G}_{0}\Omega_{02}/\Delta_{2}$.
Once again we note that the pump lasers do not interfere with each other, and
the pump lattices $V_{\uparrow/\downarrow}(\mathbf{r})$ do not add up or
cancel one another as they act on different internal atomic spin states. The
two extreme limits of the above considered laser profiles corresponding to
$R=1$ and $0$ have been experimentally realized and will be discussed in
Sections 6.4 and 8.1.3, respectively.

The mean-field approach for spinor bosonic atoms is identical to the single-
component bosonic atoms discussed in Section 2.4.1, saving that here
$\Psi(\mathbf{r})=\langle\hat{\Psi}(\mathbf{r})\rangle=(\psi_{\uparrow},\psi_{\downarrow})^{\top}$
is a two-component condensate wave function. The mean-field phase diagram of
the system in the $R$-$\eta_{0}$ plane is shown in Figure 35(b) for
$\text{Re}(\tilde{\delta}_{c})<0$, to ensure the possibility of the
superradiance phase transition. Below the pump-laser strength threshold
$\eta_{0c}(R)$, the ground state of the system is a spin-polarized FM normal
state with no photon in the cavity. At the threshold $\eta_{0c}(R)$, the
system undergoes a Dicke superradiant phase transition into a magnetically
ordered state. The nature of the emergent magnetic order depends on the
spatial profiles of the lasers through $R$.

For pure standing-wave pumps $\Omega_{1,2}(y)=\Omega_{01,02}\cos(k_{c}y)$
corresponding to $R=1$, a spin wave is stabilized in the superradiant phase;
see Figure 36(b) which shows the projection of the normalized local spin
$\tilde{\mathbf{s}}(\mathbf{r})\equiv\mathbf{s}(\mathbf{r})/s_{n}(\mathbf{r})$,
with $\mathbf{s}(\mathbf{r})=\langle\hat{\mathbf{s}}(\mathbf{r})\rangle$ and
$s_{n}(\mathbf{r})=\sqrt{s_{x}^{2}(\mathbf{r})+s_{y}^{2}(\mathbf{r})+s_{z}^{2}(\mathbf{r})}$,
in the $\tilde{s}_{x}$-$\tilde{s}_{z}$ plane as a function of $\mathbf{r}$.
Note that different spin domains are separated by domain-wall lines. In strong
pump strengths, the optical potentials
$V_{\tau}(\mathbf{r})+U_{\tau}(\mathbf{r})|\alpha|^{2}$ (with
$\tau=\\{\downarrow,\uparrow\\}$) and the Raman coupling
$\eta_{1}^{*}(\mathbf{r})\alpha+\eta_{2}(\mathbf{r})\alpha^{*}=2\eta_{0}\text{Re}(\alpha)\cos(k_{c}x)\cos(k_{c}y)$
localize the atoms in a $\lambda_{c}/2$-periodic density pattern. Hence, the
spin wave evolves into a $\lambda_{c}$-periodic checkerboard $X$-AFM spin
lattice order in a more conventional sense.

In contrast, for pure running-wave lasers
$\Omega_{1,2}(y)=\Omega_{01,02}e^{\pm ik_{c}y}$ corresponding to $R=0$,
transverse conical, chiral spin-spiral order is favored. The spirals appear in
the $\tilde{s}_{x}$-$\tilde{s}_{y}$ plane as the spin performs a full $2\pi$
rotation in this plane over one wave length $\lambda_{c}$ along the $y$
direction; see Figure 36(a). For $0<R<1$, the system exhibits “domain-wall
spin-spiral” texture; see Figure 36(c). In this phase, although the projection
of the spin in the $\tilde{s}_{x}$-$\tilde{s}_{y}$ plane still sweeps a full
$2\pi$ angle over one wave length $\lambda_{c}$ along the $y$ direction, the
$y$ component of the spin is considerably suppressed compared to the chiral
spin-spiral order in the $R=0$ case.

Figure 36:  Typical cavity-induced chiral spin-spiral, domain-wall
$X$-antiferromagnetic ($X$-AFM), and domain-wall spin-spiral textures. (a)
Shown is a chiral spin-spiral order (a-i) and its projections in different
mutual spin planes (a-ii)-(a-iv). The spin does a full $2\pi$ rotation in the
$\tilde{s}_{x}$-$\tilde{s}_{y}$ plane along the $y$ direction over one
$\lambda_{c}$, as can be seen clearly in the projection of
$\tilde{\mathbf{s}}(\mathbf{r})$ into the $\tilde{s}_{x}$-$\tilde{s}_{y}$ spin
plane (a-ii). The projection of the normalized spin texture into the (b)
$\tilde{s}_{x}$-$\tilde{s}_{z}$ spin plane for the domain-wall $X$-AFM order
and (c) $\tilde{s}_{x}$-$\tilde{s}_{y}$ spin plane for the domain-wall spin-
spiral texture. The color code in each figure indicates the respective spin
angle, e.g., $\varphi=\tan^{-1}(\tilde{s}_{y}/\tilde{s}_{x})$ in (a-ii).
Figure adapted and reprinted with permission from Ref. [69] © 2019 by the
American Physical Society.

We note that in the case of two pure running-wave pumps
$\Omega_{1,2}(y)=\Omega_{01,02}e^{\pm ik_{c}y}$, a dynamical spin-orbit
coupling is induced for the BEC on the superradiant phase and the emergent
spin-spiral magnetic order is indeed intimately connected to this dynamical
spin-orbit coupling; this will be discussed in more detail in Section 8.1.2.
By choosing more general pump configurations $\Omega_{1,2}(\mathbf{r})$, more
exotic spin textures can be obtained.

####  6.2.1 The effective spin Hamiltonian and the spin self-ordering

The various emergent magnetic orders in the superradiant phase can be
described and accounted for in an elegant way based on cavity-mediated spin-
spin interactions and the spin Hamiltonian $\hat{H}_{\rm spin}$, Equation
(6.1.1). For the pure standing-wave pumps
$\Omega_{1,2}(y)=\Omega_{01,02}\cos(k_{c}y)$ corresponding to $R=1$, the
coefficients $c_{1}$ and $c_{2}$ are identical (i.e., $c_{1}=c_{2}$) and real.
Therefore, all the long-range spin-spin interactions vanish except the $x$
component of the Heisenberg Hamiltonian [see Equation (6.1.1) and the
paragraph below it]. The spin Hamiltonian $\hat{H}_{\rm spin}$ then reduces to
a $X$-type long-range Heisenberg model $\iint J_{\rm
Heis}^{x}(\mathbf{r}^{\prime},\mathbf{r})\hat{s}_{x}(\mathbf{r}^{\prime})\hat{s}_{x}(\mathbf{r})d\mathbf{r}d\mathbf{r}^{\prime}$
with the periodically modulated coupling strength $J_{\rm
Heis}^{x}\propto\text{Re}(\tilde{\delta}_{c})\cos(k_{c}x^{\prime})\cos(k_{c}x)\cos(k_{c}y^{\prime})\cos(k_{c}y)$.
For the red-detuned dispersively shifted cavity frequency
$\text{Re}(\tilde{\delta}_{c})<0$, the cavity-induced Heisenberg interaction,
therefore, leads to the checkerboard domain-wall $X$-AFM spin texture shown in
Figure 36(b). On the other hand for the pure running-wave pumps
$\Omega_{1,2}(y)=\Omega_{01,02}e^{\pm ik_{c}y}$ corresponding to $R=0$, the
coefficients $c_{1}$ and $c_{2}$ are both complex and different from each
other, $c_{1}\neq c_{2}$. As a consequence, all the long-range spin-spin
interactions in the effective spin Hamiltonian $\hat{H}_{\rm spin}$ are
nonzero. They all have the same
$\text{Re}(\tilde{\delta}_{c})\cos(k_{c}x^{\prime})\cos(k_{c}x)$ position
dependence along the $x$ direction, but different modulations along the $y$
direction: $J_{\rm Heis}^{x}\propto\cos(k_{c}y^{\prime})\cos(k_{c}y)$, $J_{\rm
Heis}^{y}\propto\sin(k_{c}y^{\prime})\sin(k_{c}y)$, $J_{\rm
DM}^{z}\propto-\sin k_{c}(y^{\prime}-y)$, and $J_{\rm cc}^{xy}\propto\sin
k_{c}(y^{\prime}+y)$. The DM and cross-component spin interactions stabilize
transverse, conical spin-spiral order, while the magnetic domains are favored
by the Heisenberg interactions as before; see Figure 36(a). However, these
cavity-mediated interactions are compatible with one another and do not lead
to a frustration for this chosen geometry of the pump lasers and the cavity.
The spirals appear solely in the $\tilde{s}_{x}$-$\tilde{s}_{y}$ plane as the
DM interaction has only the $z$ component and the cross-component spin
interaction only couples the $x$ and $y$ components of the spins. Note also
that the $\lambda_{c}/4$ shift of the magnetic domains in the
$\tilde{s}_{x}$-$\tilde{s}_{z}$ and $\tilde{s}_{y}$-$\tilde{s}_{z}$ planes
along the $y$ axis is accurately accounted for by the $y$ dependence of
$J_{\rm Heis}^{x/y}$ given above. For $0<R<1$, all the cavity-mediated spin-
spin interactions are again nonzero. However, the coupling coefficients
$J_{\rm Heis}^{y}$ and $\\{J_{\rm DM}^{z},J_{\rm cc}^{xy}\\}$ approach zero,
respectively, as $(1-R)^{2}$ and $1-R^{2}$ when $R\to 1$. Whereas, the
coupling coefficient $J_{\rm Heis}^{x}$ is independent of $R$ and remains the
dominant coupling coefficient. This is indeed the reason for the suppression
of the $y$ component of the spin in the domain-wall spin-spiral phase compared
to the spin-spiral state where all the spin-spin interactions are in the same
order of magnitude.

####  6.2.2 The origin of the self-ordering in the generalized Dicke model

For chosen pump lasers $\Omega_{1,2}(y)=\Omega_{01,02}(e^{\pm ik_{c}y}+Re^{\mp
ik_{c}y})/(1+R)$, the system possesses a $\mathbf{Z}_{2}\otimes U(1)$
symmetry, where $\mathbf{Z}_{2}$ is the parity symmetry
$\hat{a}\to\hat{a}^{\prime}=-\hat{a}$ and
$\mathbf{r}\to\mathbf{r}^{\prime}=\mathbf{r}+(\lambda_{c}/2)\hat{e}_{x}$, and
the continuous $U(1)$ symmetry represents the freedom of the total phase of
the two bosonic atomic fields $\hat{\psi}_{\tau}(\mathbf{r})$. The $U(1)$
symmetry associated with the freedom of the relative phase of the two atomic
fields is explicitly broken on the onset of the superradiant phase transition
by the cavity-induced position-dependent Raman coupling term with the zero
quantum average below the superradiant threshold
$\langle\eta_{1}(\mathbf{r})\hat{a}^{\dagger}+\eta_{2}^{*}(\mathbf{r})\hat{a}\rangle=0$
and with the always zero spatial average
$\int[\eta_{1}(\mathbf{r})\hat{a}^{\dagger}+\eta_{2}^{*}(\mathbf{r})\hat{a}]d\mathbf{r}=0$;
see Equation (79). As a consequence, in the self-ordered phase the relative
condensate phase varies in space to minimize the Raman coupling energy,
leading to the spin orders. For instance, for the pure standing-wave pump
lasers with $R=1$, the Raman energy is
$2\hbar\eta_{0}(\alpha+\alpha^{*})\int\sqrt{n_{\uparrow}n_{\downarrow}}\cos(k_{c}x)\cos(k_{c}y)\cos(\Delta\phi)d\mathbf{r}$,
where $n_{\tau}(\mathbf{r})=\langle\hat{n}_{\tau}(\mathbf{r})\rangle$ are the
condensate densities and $\Delta\phi(\mathbf{r})$ is the relative phase
between the two condensate wave functions. As $\cos(k_{c}x)$ and
$\cos(k_{c}y)$ change sign along the $x$ and $y$ directions, respectively, the
relative phase $\Delta\phi(\mathbf{r})$ then jumps (discontinuously) along
both $x$ and $y$ directions to minimize the energy, resulting in domain-wall
AFM spin order. On the other hand, for the pure running-wave pump lasers with
$R=0$, the Raman energy is
$2\hbar\eta_{0}(\alpha+\alpha^{*})\int\sqrt{n_{\uparrow}n_{\downarrow}}\cos(k_{c}x)\cos(k_{c}y-\Delta\phi)d\mathbf{r}$.
As before the relative phase $\Delta\phi(\mathbf{r})$ jumps (discontinuously)
along the $x$ direction to minimize the energy. However, to minimize the
energy along the $y$ direction, it changes smoothly in a screw-like manner due
to the term $\cos(k_{c}y-\Delta\phi)$, resulting in the chiral spin-spiral
texture along the $y$ direction. Owing to the explicit breaking of the $U(1)$
relative-phase symmetry by the zero quantum- (below the superradiant
threshold) and spatial-averaged Raman field, one may envisage the self-
organization and the magnetic ordering in this generalized Dicke model (with
both internal and external quantized degrees of freedom of the atoms included)
as an order-by-disorder process [68], with disorder stemming from strong
fluctuations of the cavity field around the superradiant threshold. This can
be considered reminiscent of the spontaneous ordering in a two-dimensional
classical ferromagnetic $XY$ spin model with a uniaxial random magnetic field
[235, 236, 237, 238]. The relative condensate phase is related to the spin
angle, and the position-dependent Raman field with zero quantum (below the
superradiant threshold) and spatial average plays the role of the random
magnetic field. However, one has to note that quantum fluctuations here might
not necessarily lower the free energy as in the original concept of order-by-
disorder processes [239, 240].

###  6.3 Spinful Fermi gases

####  6.3.1 The density and spin self-ordering

Closely related schemes to the model discussed in Section 6.1 have been
considered with ultracold fermionic atoms [241, 130, 229]. Reference [241] has
considered a similar double-$\Lambda$ atom-photon-coupling scheme as in the
inset of Figure 35(a) with a 2D Fermi gas perpendicular to the cavity axis
(i.e., in the $y$-$z$ plane), where the pumps are also aligned along the
cavity axis. In the mean-field limit, above a certain critical pump strength
the Fermi gas undergoes a phase transition from a normal state into a
superradiant state with partial polarization. Depending on the effective
energy separation between the two spin states, the Dicke phase transition is
either first order or second order. A variant of the scheme discussed in
Section 6.1 has been considered for a 1D Fermi gas within a ring cavity [229].
In particular, the transitions $\downarrow\>\leftrightarrow 1$ and
$\uparrow\>\leftrightarrow 2$ are coupled to a pair of degenerate,
counterpropagating modes of the ring cavity with the coupling strength
$\mathcal{G}_{\pm}(x)=\mathcal{G}_{0}e^{\pm ik_{c}x}$. Below the Dicke
superradiant phase transition, the system is in a normal mixed or polarized
Fermi-gas state depending on the effective energy separation between the two
spin states. A spin wave is stabilized in the superradiant phase, which
evolves into an $X$-AFM lattice order in the strong pump limit. Due to the
ring geometry of the cavity the system possesses a $U(1)\times\mathbf{Z}_{2}$
symmetry, where the former is spontaneously broken by the emergent density
order and the latter by the $X$-AFM spin order at the onset of the
superradiant phase transition. Furthermore, in moderate pump strengths the
self-ordered states exhibit unexpected positive momentum pair correlations
between fermions with opposite spin.

####  6.3.2 Self-organization across the BEC-BCS crossover

The recent experimental realization of a tunable Fermi gas inside an optical
cavity [61] has opened up interesting avenues of research with interacting
spinful Fermi gases. In the experiments performed at the EPF Lausanne the
scattering length governing the interactions between fermions in different
spin states can be tuned using a standard Feshbach resonance. This allows to
control the pairing between the atoms within the cavity, interpolating between
the BEC limit (small positive scattering length), where tightly-bound pairs
are well described as a condensate of localized bosons, and the BCS limit
(small negative scattering length), where weakly-bound pairs are formed.
Between these two limits the so-called unitary point is crossed, at which the
scattering length diverges. The behavior of the cavity-induced self-
organization transition across the BEC-BCS crossover has been studied
theoretically in Ref. [242]. The main finding, shown in Figure 37, is that
self-organization is maximally enhanced in the unitary regime for low
densities, in the BCS regime for moderate densities close to Fermi surface
nesting, or in the BEC regime for high densities.

Figure 37: Behavior of the density response function $\chi_{\rm stat}$ at the
cavity wave vector, for interacting fermions in the BEC-BCS crossover. The
latter is parametrized by the inter-species scattering length
$a_{\uparrow\downarrow}^{s}$. The images below the plot illustrate pictorially
how the size of the Cooper pairs changes in the BEC-BCS crossover. As
discussed below Equation (31), $\chi_{\rm stat}$ quantifies the response of
the atoms to a cavity photon and thus the tendency of the system to self-
organize into the superradiant phase. Figure adapted and reprinted with
permission from Ref. [242] © 2015 by the American Physical Society.

####  6.3.3 Cavity-induced superfluid pairing without superradiance

In the scenario just discussed in the preceding section, the pairing is
induced by the short-range atom-atom interactions, while the cavity-mediated
interactions favor the self-organization of the atomic cloud into a density
and spin pattern. Another interesting question is whether and how pairing can
be instead induced by the cavity-mediated interactions. This question has been
addressed in Ref. [130], where $\Lambda$-type fermionic atoms are coupled to a
far red-detuned transverse pump laser with the Rabi rate $\Omega(\mathbf{r})$
and a standing-wave cavity mode with strength $\mathcal{G}_{0}\cos(k_{c}x)$.
In the bad cavity limit, cavity photons mediate global-range interaction
between the atomic ground spin states,

$\displaystyle\iint\mathcal{D}(\mathbf{r},\mathbf{r^{\prime}})\hat{\psi}_{\downarrow}^{\dagger}(\mathbf{r})\hat{\psi}_{\uparrow}(\mathbf{r})\hat{\psi}_{\uparrow}^{\dagger}(\mathbf{r^{\prime}})\hat{\psi}_{\downarrow}(\mathbf{r^{\prime}})d\mathbf{r}d\mathbf{r^{\prime}},$
(83)

where
$\mathcal{D}(\mathbf{r},\mathbf{r^{\prime}})=(\mathcal{D}_{0}/\Omega_{0}^{2})\Omega(\mathbf{r})\Omega(\mathbf{r}^{\prime})\cos(k_{c}x)\cos(k_{c}x^{\prime})$
with
$\mathcal{D}_{0}=\hbar\Delta_{c}\Omega_{0}^{2}\mathcal{G}_{0}^{2}/\Delta_{a}^{2}|\tilde{\Delta}_{c}|^{2}$;
cf. Equation (13). Restricting to 1D along the cavity axis, i.e.,
$\Omega(\mathbf{r})=\Omega_{0}$, for $\mathcal{D}_{0}<0$ one finds the
equivalent of the $X$-AFM spin texture discussed above and shown in Figure
36(b). One the other hand, for $\mathcal{D}_{0}>0$ a superfluid phase of
paired fermions is favored. In Ref. [243], the possibility of generating the
more exotic Fulde-Ferrel-Larkin-Ovchinnikov type of pairing—where the fermion
pair has a net center-of-mass momentum—is discussed. As we discussed
previously, for positive $\mathcal{D}_{0}>0$ corresponding to $\Delta_{c}>0$,
superradiance is strongly suppressed. The superfluid paired phase is indeed
not of the superradiant type, i.e., coherent macroscopic occupation of the
cavity mode does not occur in any pump strength. In this case, superradiant
and pairing processes, therefore, strongly exclude each other, and in
particular for $\mathcal{D}_{0}<0$ only superradiance is possible. In Section
7.3.3 we discuss another scheme for cavity-induced pairing in an external
optical lattice inside ring cavities in the red-detuned regime,
$\Delta_{c}<0$.

###  6.4 Experimental realizations of spin self-organization

Spin self-organization of atomic systems has been experimentally explored
using schemes similar to the one discussed in Sections 6.1 and 6.2. While
first experiments were based on thermal atoms [110, 244], we here discuss a
more recent realization using a BEC [71]. Operating in the quantum degenerate
regime implies that experiments take place at effectively zero temperature,
that all atoms couple identically to light fields as they share the same
spatial wave function, and that the self-organization of the matter wave can
be clearly observed in the appearance of sharp momentum peaks. In the spin-
changing superradiant phases as discussed in Section 6.2, the atomic spins are
flipped due to the cavity-assisted Raman scattering processes [see Figure
35(a)]. Correspondingly, recoil momenta associated with the Raman processes
are imparted to the atoms. As a result, the spinor superradiant phases are
spin-decorated patterns (see Figure 36).

Figure 38: Observation of the density and spin self-organization of a spinor
BEC. (a) Experimental scheme using two hyperfine states of 87Rb as the
effective spin-1/2 system. These states are coupled via two separate cavity-
assisted Raman transitions, where each transition involves the cavity mode and
an additional coherent pump field. (b) The Raman scattering processes flip the
spin of the atoms while transferring them to a superposition of momentum
states (bottom). The system undergoes a transition from the normal phase with
no spin ordering to a spin-decorated checkerboard superradiant state breaking
a $\mathbf{Z}_{2}$ symmetry. (c) Spin-selective absorption images of the
atomic cloud after ballistic expansion in the normal (left) and the
superradiant phase (right). The images reveal that the excited momentum states
in the superradiant phase are only populated by spin flipped atoms. Figure
adapted and reprinted with permission from Ref. [71] © 2018 by the American
Physical Society.

The experimental scheme used in Ref. [71] is shown in Figure 38(a), which
corresponds to the $R=1$ case discussed in Section 6.2. Two different
hyperfine ground states of 87Rb are chosen as the effective spin-1/2 system
($\left|\uparrow\right>$, $\left|\downarrow\right>$) and coupled via two
different Raman processes, each involving a coherent pump field and the TEM00
cavity mode. The amplitudes of the Raman processes are balanced, such that an
effective Dicke model is realized. The standing-wave pump beams illuminate the
atoms from the side and are arranged such that the resulting pump lattices are
in phase at the location of the atoms. The frequency of the light field
scattered into the cavity by either Raman process is the mean frequency of the
two coherent Raman pump fields illuminating the atoms. Since the frequencies
of the pump fields and the cavity field are different, the potentials for the
densities are square lattices while the spin potential has a checkerboard
shape [see Equation (79)]. During self-organization, an according spin-
decorated checkerboard pattern forms.

Figure 38(b) shows the underlying scattering processes together with
illustrations of the normal and possible superradiant phases breaking a
$\mathbf{Z}_{2}$ symmetry. In the organized phase, coherent Raman scattering
creates a superposition of atoms in the initial zero-momentum state with the
spin state $\left|\downarrow\right>$ and of atoms in the recoiled momentum
state with the spin state $\left|\uparrow\right>$. Spin-selective absorption
imaging of the atomic cloud after ballistic expansion allows to distinguish
both the momentum states and the spin states. According images in the normal
and the superradiant phases are shown in Figure 38(c).

The experiment was repeated by utilizing the TEM10 mode instead of the TEM00
mode. The node in the field of the TEM10 mode is equivalent to a $\pi$ phase
shift in the plane perpendicular to the cavity axis. This spatial phase shift
of the cavity field leads to the creation of a spinor domain wall across the
superradiant self-organization. In the experiment, the phase of the field
leaking from the cavity was analyzed spatially by interfering it with a local
oscillator in a camera, thus realizing a spatially resolved homodyne
detection. Indeed, a $\pi$ phase shift across the nodal line was observed in
the cavity field. This nodal line acts as a domain wall in the atomic systems.
Accordingly, also the momentum distribution of the atomic cloud—derived from
absorption images after ballistic expansion—shows a node in the Bragg peaks of
the self-organized atomic cloud.

##  7 Lattice superradiance: generalized extended Hubbard models

So far for single cavity setups, we have considered scenarios where the atoms
are initially free along the cavity axis below the superradiance phase
transition. Only beyond the superradiant phase transition, photons scattered
from the pump laser into the cavity create an optical lattice along the cavity
axis for the atoms. The atoms then self-order in this emergent potential into
a Bragg density grating which maximizes the photon scattering from the pump
into the cavity. We now consider lattice scenarios where the atoms are already
initially trapped in a strong 2D “external”, static optical lattice below the
superradiant phase transition. As before, photons scattered by the atoms from
a transverse pump field into the cavity result in cavity-mediated long-range
interactions, competing directly with the kinetic energy and the local
interactions of the strongly correlated atoms [107, 52, 53, 54, 55, 245, 246,
247, 248, 249, 250, 251, 136, 137, 252, 253, 254, 255, 256, 60, 75, 257].
Here, for instance, the cavity-mediated long-range interactions can be
incommensurate with respect to the external static lattice spacing, leading to
frustration.

For ultracold atoms in a static optical lattice, the competition between the
kinetic energy and the local contact interaction leads to the quantum phase
transition between a superfluid (SF) state and a Mott-insulator (MI) phase
[258, 14]. The Mott-insulator state is favored in the strongly interacting
regime where the atoms are localized in potential minima and the coherence
between the atoms is lost. Adding the cavity-mediated long-range interactions
into this system enriches significantly the physics resulting, in addition to
the superfluid and Mott-insulator phases, in new states in the superradiant
phase including lattice-supersolid (LSS), charge-density-wave (CDW), and Bose-
glass (BG) states. The lattice supersolid is the lattice variant of the
supersolid discussed in Section 4, where here the crystalline order is
stabilized through a discrete spatial symmetry breaking (cf. Section 4). The
charge-density wave is an insulating state with a modulated periodic density.
The Bose glass is a compressible state, containing disjoint superfluid islands
in the Mott-insulator background. It is a characteristic state of disordered
Bose-Hubbard models.

###  7.1 One-component extended Bose-Hubbard model

Let us now consider a generic lattice model inside a linear cavity [52, 54,
55, 246, 247, 248, 249, 250, 251, 136, 137, 254]. Ultracold bosonic atoms are
trapped in a three dimensional external _static_ lattice, generated by three
mutually orthogonal standing-wave laser fields not interfering with each
other. The BEC is split into a stack of identical 2D pancakes along the $z$
direction by the lattice, and we focus our description into one of these
pancakes. In the $x$-$y$ plane, the atoms experience the external, static
square optical lattice

$\displaystyle V_{\rm ext}(\mathbf{r})=V_{\rm latt}^{(0)}\left[\cos^{2}(k_{\rm
latt}x)+\cos^{2}(k_{\rm latt}y)\right],$ (84)

with the lattice spacing $\lambda_{\rm latt}/2=\pi/k_{\rm latt}$ and lattice
depth $V_{\rm latt}^{(0)}$. The polarizations and frequencies of the lasers
generating this lattice potential are chosen such that the photon scattering
from these lasers into the cavity by the atoms are either not possible or
strongly suppressed. As depicted in Figure 39, the atoms are further driven in
the $y$ direction by a standing-wave pump laser with wavenumber
$k_{c}=2\pi/\lambda_{c}$ and strongly coupled to an empty mode of the cavity
with the mode function $\cos(k_{c}x)$, similar to the basic model described in
Section 2.1. The initially empty cavity mode is populated by photons scattered
off the atoms from the transverse pump field. This results in an additional,
_dynamical_ potential on top of the classical external static lattice. In the
2D $x$-$y$ plane, the system is still described by the Hamiltonian of Equation
(2.1), with $V_{\rm ext}(\mathbf{r})$ given by the lattice potential of
Equation (84). The external static lattice and the dynamical potentials can be
commensurate or incommensurate with each other, depending on the ratio
$\lambda_{c}/\lambda_{\rm latt}$. These two cases will be considered
separately in Sections 7.1.1.1 and 7.1.1.2.

Figure 39: Schematic sketch of a setup for realizing a generic lattice model
with cavity-mediated global interactions. Bosonic atoms in a 2D pancake in the
$x$-$y$ plane are trapped by a strong external, statice square lattice
generated by two standing-wave fields (indicated in blue) inside a cavity. The
cavity mode is populated via photon scattering of the pump laser (indicated in
red) off the atoms. The cavity photons mediate global, all-to-all interactions
among the atoms, described by an extended Bose-Hubbard-type Hamiltonian.
Likewise, an extended Hubbard-type Hamiltonian can be realized for fermionic
atoms. Figure adapted and reprinted with permission from Ref. [55] © 2015 by
the American Physical Society.

For a very strong external classical lattice $V_{\rm ext}(\mathbf{r})$, as a
lowest-order approximation one can ignore the modification of $V_{\rm
ext}(\mathbf{r})$ by the transverse pump and dynamical potentials and expand
the atomic field operator in the basis of Wannier functions of the lowest
Bloch band of the classical lattice as
$\hat{\psi}(\mathbf{r})=\sum_{\mathbf{j}}W_{\mathbf{j}}(\mathbf{r})\hat{b}_{\mathbf{j}}$,
where $W_{\mathbf{j}}(\mathbf{r})$ is the lowest-band Wannier function
localized at lattice site $\mathbf{r}_{\mathbf{j}}=\mathbf{j}\lambda_{\rm
latt}/2=(j_{x},j_{y})\lambda_{\rm latt}/2$ with
$\\{j_{x},j_{y}\\}\in\mathbb{Z}$ and $\hat{b}_{\mathbf{j}}$ is the
corresponding atomic annihilation operator satisfying the canonical bosonic
commutation relation. By doing so, the transverse pump and the dynamical
potentials are basically treated as perturbations; see Section 7.1.2 for a
brief discussion regarding beyond perturbative treatments. Therefore, in the
lowest-order approximation the system is described by the lattice Hamiltonian,

$\displaystyle\hat{H}_{\rm latt}$
$\displaystyle=-J\sum_{\langle\mathbf{j},\mathbf{j}^{\prime}\rangle}(\hat{b}_{\mathbf{j}}^{\dagger}\hat{b}_{\mathbf{j}^{\prime}}+\text{H.c.})+\frac{1}{2}U_{s}\sum_{\mathbf{j}}\hat{n}_{\mathbf{j}}(\hat{n}_{\mathbf{j}}-1)+\sum_{\mathbf{j}}\left(\hbar
V_{0}B_{\mathbf{j}}^{(y)}-\mu\right)\hat{n}_{\mathbf{j}}$
$\displaystyle+\hbar\sum_{\mathbf{j}}\left[U_{0}\hat{a}^{\dagger}\hat{a}B_{\mathbf{j}}^{(x)}+\eta_{0}(\hat{a}^{\dagger}+\hat{a})A_{\mathbf{j}}\right]\hat{n}_{\mathbf{j}}-\hbar\Delta_{c}\hat{a}^{\dagger}\hat{a},$
(85)

where $J$ is the hopping amplitude, $U_{s}$ the strength of the short-range
contact interaction,
$\hat{n}_{\mathbf{j}}=\hat{b}_{\mathbf{j}}^{\dagger}\hat{b}_{\mathbf{j}}$ the
local particle number operator at site $\mathbf{j}$, $\mu$ is the chemical
potential included explicitly to fix the number of the particles, and
$\\{V_{0},U_{0},\eta_{0}\\}$ are defined as before (see Section 2.1). Like
$\\{J,U_{s}\\}$, the coefficients $A_{\mathbf{j}}$ and
$B_{\mathbf{j}}^{(x/y)}$ are related to overlap integrals of the Wannier
functions (i.e., matrix elements of the spatial part of the pump and dynamic
cavity potentials):

$\displaystyle A_{\mathbf{j}}$
$\displaystyle\equiv\int|W_{\mathbf{j}}(\mathbf{r})|^{2}\cos(k_{c}x)\cos(k_{c}y)d\mathbf{r},$
$\displaystyle B_{\mathbf{j}}^{(x)}$
$\displaystyle\equiv\int|W_{\mathbf{j}}(\mathbf{r})|^{2}\cos^{2}(k_{c}x)d\mathbf{r},$
$\displaystyle B_{\mathbf{j}}^{(y)}$
$\displaystyle\equiv\int|W_{\mathbf{j}}(\mathbf{r})|^{2}\cos^{2}(k_{c}y)d\mathbf{r}.$
(86)

Note that off-site matrix elements of the potentials have been ignored here,
as these contributions are normally much smaller than on-site matrix elements,
Equation (7.1). The contribution of these higher-order corrections will be
discussed briefly in Section 7.1.2. We note that in the absence of the cavity
field [the second line in Equation (7.1)], the lattice Hamiltonian
$\hat{H}_{\rm latt}$ reduces to the common Bose-Hubbard Hamiltonian describing
the dynamics of locally interacting bosonic atoms in the static lattice
$V_{\rm ext}(\mathbf{r})$ with a site-dependent chemical potential $\mu-\hbar
V_{0}B_{\mathbf{j}}^{(y)}$ due to the transverse pump-field lattice potential
$\hbar V(\mathbf{r})$ that can have a different wave vector than the original
static lattice.

The distinction between phenomena arising due to the dynamical (semiclassical)
nature of cavity-light potentials and effects due to their quantumness was
underlined in Refs. [245, 259]. Here we focus on the large cavity decay-rate
limit, where the cavity field reaches very fast its steady state,

$\displaystyle\hat{a}_{\rm
ss}=\frac{\eta_{0}\sum_{\mathbf{j}}A_{\mathbf{j}}\hat{n}_{\mathbf{j}}}{\Delta_{c}+i\kappa-
U_{0}\sum_{\mathbf{j}}B_{\mathbf{j}}^{(x)}\hat{n}_{\mathbf{j}}}.$ (87)

The cavity field acquires a nonzero value only if the “weighted” average
density operator [cf. Equation (15)],

$\displaystyle\hat{\Theta}\equiv\frac{1}{N_{\Lambda}}\sum_{\mathbf{j}}A_{\mathbf{j}}\hat{n}_{\mathbf{j}},$
(88)

with $N_{\Lambda}$ being the total number of lattice sites, is nonzero.
Adiabatically eliminating the steady-state photonic field operator
$\hat{a}_{\rm ss}$ (87) under the assumption that $\Delta_{c},\kappa\gg
U_{0}\sum_{\mathbf{j}}B_{\mathbf{j}}^{(x)}\hat{n}_{\mathbf{j}}$ and retaining
terms up to $1/\Delta_{a}^{2}$ yields a generalized, extended Bose-Hubbard
model,

$\displaystyle\hat{H}_{\rm eBH}$
$\displaystyle=-J\sum_{\langle\mathbf{j},\mathbf{j}^{\prime}\rangle}\left(\hat{b}_{\mathbf{j}}^{\dagger}\hat{b}_{\mathbf{j}^{\prime}}+\text{H.c.}\right)+\frac{1}{2}U_{s}\sum_{\mathbf{j}}\hat{n}_{\mathbf{j}}(\hat{n}_{\mathbf{j}}-1)+\sum_{\mathbf{j}}\left(\hbar
V_{0}B_{\mathbf{j}}^{(y)}-\mu\right)\hat{n}_{\mathbf{j}}$
$\displaystyle+N_{\Lambda}U_{l}\Big{(}\frac{1}{N_{\Lambda}}\sum_{\mathbf{j}}A_{\mathbf{j}}\hat{n}_{\mathbf{j}}\Big{)}^{2},$
(89)

where the last term represents the cavity-mediated global interaction with the
strength $U_{l}=2\hbar
N_{\Lambda}\Delta_{c}\eta_{0}^{2}/(\Delta_{c}^{2}+\kappa^{2})$ [246]; cf.
Equations (2.2) and (13). A negative cavity-mediated long-range interaction
$U_{l}<0$ favors a state which maximizes the absolute value of
$\Theta=\langle\hat{\Theta}\rangle$. The parameter $\Theta$ is zero in the
superfluid and Mott-insulator states with average uniform densities. The onset
of nonzero $\Theta$ signals a quantum phase transition into other ordered
states with modulated densities, driven by the cavity-mediated interaction
$N_{\Lambda}U_{l}\hat{\Theta}^{2}$. The nonzero $\Theta$ also corresponds to
the superradiant phase with nonzero cavity field amplitude $\alpha_{\rm
ss}=\langle\hat{a}_{\rm ss}\rangle\neq 0$; see Equation (87). Therefore,
$\Theta=\langle\hat{\Theta}\rangle$ can be considered as an order parameter,
the lattice analog of the continuum order parameter introduced in Equation
(15). The nature of the states with $\Theta\neq 0$ depends crucially on the
ratio $\lambda_{c}/\lambda_{\rm latt}$, as we will see shortly in Sections
7.1.1.1 and 7.1.1.2.

The extended Bose-Hubbard Hamiltonian (7.1) can be diagonalized exactly for
small system sizes. Other numerical techniques such as quantum Monte Carlo
[249], variational Monte Carlo [260], multi-configurational time-dependent
Hartree method [255], and even exact diagonalization of an infinite system in
the hardcore limit (in 1D) [252, 253] have also been utilized to find the
ground state of this system. Due to the fully-connected, long-range nature of
cavity-mediated interactions, the mean-field approach provides, however, a
faithful description of the system in a wide range of parameters in the
thermodynamics limit which we focus on in the following section (see also
Section 2.4). That said, one has to note that only the long-range interaction
part of the Hamiltonian is treated appropriately by the mean-field
approximation. When the short-range hopping terms and/or the on-site
repulsions are also included, the dimensionality plays an important
role—implying, e.g., substantial shifts of phase boundaries and/or even
completely vanishing of some phases. Before discussing the mean-field
approach, we finally also note that such extended Bose-Hubbard models can also
be obtained in the deep superradiant phase without an external optical lattice
[53, 255] or even in longitudinally pumped cavities due to the optomechanical
interaction [107, 261, 262].

####  7.1.1 Mean-field approximation

The extended Bose-Hubbard Hamiltonian, Equation (7.1), can be studied most
readily using a mean-field method. This mean-field approach decouples the off-
site terms in the Hamiltonian $\hat{H}_{\rm eBH}$ [263]:

$\displaystyle\hat{b}_{\mathbf{j}}^{\dagger}\hat{b}_{\mathbf{j}^{\prime}}+\hat{b}_{\mathbf{j}^{\prime}}^{\dagger}\hat{b}_{\mathbf{j}}$
$\displaystyle\simeq\beta_{\mathbf{j}^{\prime}}\hat{b}_{\mathbf{j}}^{\dagger}+\beta_{\mathbf{j}^{\prime}}^{*}\hat{b}_{\mathbf{j}}+\beta_{\mathbf{j}}\hat{b}_{\mathbf{j}^{\prime}}^{\dagger}+\beta_{\mathbf{j}}^{*}\hat{b}_{\mathbf{j}^{\prime}}-\left(\beta_{\mathbf{j}}^{*}\beta_{\mathbf{j}^{\prime}}+\beta_{\mathbf{j}}\beta_{\mathbf{j}^{\prime}}^{*}\right),$
$\displaystyle\hat{\Theta}^{2}$ $\displaystyle\simeq
2\Theta\hat{\Theta}-\Theta^{2},$ (90)

where $\beta_{\mathbf{j}}=\langle\hat{b}_{\mathbf{j}}\rangle$ is the
superfluid order parameter and $\Theta=\langle\hat{\Theta}\rangle$ is the
“Bragg density” order parameter, assumed to be independent of the superfluid
order parameter. This assumption is used frequently in the literature, but its
validity has not yet been checked thoroughly.

#####  7.1.1.1 Commensurate: Superfluid, Mott-insulator, density-wave, and
lattice-supersolid states

We first consider the simplest case corresponding to $\lambda_{c}=\lambda_{\rm
latt}$. One then has $B=B_{\mathbf{j}}^{(x)}=B_{\mathbf{j}}^{(y)}$ independent
of site $\mathbf{j}$ and $A_{\mathbf{j}}=(-1)^{j_{x}+j_{y}}A$ (with
$A\equiv|A_{\mathbf{j}}|$) alternating its sign between $\ell\equiv
j_{x}+j_{y}\in$ even (e) and odd (o) sites. The Hamiltonian (7.1) can then be
recast as,

$\displaystyle\hat{H}_{\rm ceBH}$ $\displaystyle=-J\sum_{\langle
e,o\rangle}\left(\hat{b}_{e}^{\dagger}\hat{b}_{o}+\hat{b}_{o}^{\dagger}\hat{b}_{e}\right)+\frac{1}{2}U_{s}\sum_{\ell\in
e,o}\hat{n}_{\ell}(\hat{n}_{\ell}-1)-\left(\mu-\hbar
V_{0}B\right)\sum_{\ell\in e,o}\hat{n}_{\ell}$
$\displaystyle+\frac{A^{2}}{N_{\Lambda}}U_{l}\Big{(}\sum_{e}\hat{n}_{e}-\sum_{o}\hat{n}_{o}\Big{)}^{2},$
(91)

where the total number of atoms
$N=\langle\hat{N}\rangle=\sum_{j}\langle\hat{n}_{j}\rangle$ has to be
determined self-consistently. Let us first heuristically discuss the extended
Bose-Hubbard Hamiltonian (7.1.1.1) and consider in particular the effect of
the cavity-mediated long-range interactions
$N_{\Lambda}U_{l}\hat{\Theta}^{2}\propto
U_{l}(\sum_{e}\hat{n}_{e}-\sum_{o}\hat{n}_{o})^{2}$. In the absence of the
cavity-mediated interaction, $U_{l}=0$, the Hamiltonian (7.1.1.1) reduces to
the common Bose-Hubbard model, where for $J/U_{s}>J_{c}(\mu/U_{s})/U_{s}$ with
$J_{c}(\mu/U_{s})$ being the critical hopping amplitude the system is in the
superfluid state. For $J/U_{s}<J_{c}(\mu/U_{s})/U_{s}$, the system enters into
the Mott-insulator state, where the atoms are localized on lattice sites with
an equal, integer number of atoms per site and the quantum coherence between
the atoms is lost. Both states respect the symmetry between even and odd
sites. However, a negative cavity-mediated long-range interaction $U_{l}<0$
favors a population imbalance between even and odd sites, which breaks the
$\mathbf{Z}_{2}$ symmetry of even and odd sites [originating from the
independence of the long-range interaction energy
$\propto(\sum_{e}\hat{n}_{e}-\sum_{o}\hat{n}_{o})^{2}$ on the sign of the
population imbalance]. For large enough cavity-mediated interaction strength,
this results in a $\lambda_{c}$-periodic checkerboard density structure which
acts as a Bragg grating and maximizes the photon scattering from the pump
lattice to the cavity field, signaling the superradiant phase transition. The
resultant state in the superradiant phase with a population imbalance is
either a charge-density wave or a lattice supersolid, depending on if the
quantum coherence between the atoms is lost or not.

The phase diagram of the system can be obtained using the decoupling mean-
field approach, Equation (7.1.1). Application of the mean-field decoupling
(7.1.1) to the Hamiltonian $\hat{H}_{\rm ceBH}$ (7.1.1.1) leads to an
effective two-site mean-field Hamiltonian

$\displaystyle\hat{H}_{\rm ceBH}^{\rm MF}$ $\displaystyle\simeq\sum_{\ell\in
e,o}\left[-J\tilde{\beta}_{\ell}\left(\hat{b}_{\ell}^{\dagger}+\hat{b}_{\ell}-\beta_{\ell}\right)+\frac{U_{s}}{2}\hat{n}_{\ell}(\hat{n}_{\ell}-1)-\left(\mu-\hbar
V_{0}B\right)\hat{n}_{\ell}\right]$
$\displaystyle+2AU_{l}\Theta\Big{(}\sum_{e}\hat{n}_{e}-\sum_{o}\hat{n}_{o}\Big{)}-N_{\Lambda}U_{l}\Theta^{2},$
(92)

where, without loss of generality, we have assumed that the superfluid order
parameters are real $\beta_{e/o}\in\mathbb{R}$ (as there is no gauge
potential), $\tilde{\beta}_{e}=\sum_{n.n.o}\beta_{o}=4\beta_{o}$ (assuming an
isotropic situation in this 2D model each even site has four identical,
nearest-neighbor odd sites and vice versa), $\tilde{\beta}_{o}=4\beta_{e}$,
and
$\Theta=\langle\hat{\Theta}\rangle=A(\sum_{e}n_{e}-\sum_{o}n_{o})/N_{\Lambda}$
with $n_{e/o}=\langle\hat{n}_{e/o}\rangle$. The Bose-Hubbard model with the
cavity-mediated infinite-range interaction in the mean-field level,
represented by the Hamiltonian $\hat{H}_{\rm ceBH}^{\rm MF}$ in Equation
(7.1.1.1), is equivalent to the extended Bose-Hubbard Hamiltonian with the
nearest-neighbor density-density interaction
$\propto\sum_{\langle\mathbf{j},\mathbf{j}^{\prime}\rangle}\hat{n}_{\mathbf{j}}\hat{n}_{\mathbf{j}^{\prime}}$
[246]. The mean-field Hamiltonian $\hat{H}_{\rm ceBH}^{\rm MF}$ can be
diagonalized in a unit cell consisting of an even and an odd lattice site,
with the order parameters $\beta_{e/o}$ and $\Theta$ to be determined self-
consistently.

Figure 40: Phase diagram of a cavity-induced commensurate extended Bose-
Hubbard model with $\lambda_{c}=\lambda_{\rm latt}$. (a) Phase diagram as
function of rescaled parameters $\mu/U_{s}$, $zJ/U_{s}$, and
$-A^{2}U_{l}/U_{s}$ with coordination number $z$. The colorbar refers to
$zJ/U_{s}=0$ and indicates the average imbalance $\Theta$ between the number
of atoms on an even and an odd site. The transparent colors show the different
phases for $zJ/U_{s}>0$: MI (green), CDW (red), and LSS (blue). The SF phase
is not indicated, but fills the remaining space. The labels correspond to the
site populations ($n_{e}$,$n_{o}$). (b) Evolution of the different order
parameters as a function of tunneling $zJ/U_{s}$ for filling 1/2,
corresponding to a cut through the CDW(1,0) lobe at $U_{l}/U_{s}=0.3$ in panel
(a). With increasing $zJ/U_{s}$, the system evolves from CDW over LSS to SF.
Here, all order parameters change continuously across the phase transitions.
Figure and caption adapted and reprinted with permission from Ref. [246] ©
2016 by the American Physical Society.

All phases of the system can be identified using the order parameters
$\beta_{e/o}$ and $\Theta$. Zero superfluid order parameters
$\beta_{e}=\beta_{o}=0$ indicate an incompressible insulating state
$\partial\bar{n}/\partial\mu=0$, with
$\bar{n}=\sum_{j}\langle\hat{n}_{j}\rangle/N_{\Lambda}=N/N_{\Lambda}$ being
the average density. This incompressible state is a Mott insulator if
$\Theta=0$, otherwise a charge-density-wave state. On the other hand
$\beta_{e}=\beta_{o}\neq 0$ and $\Theta=0$ corresponds to a superfluid sate,
while $\beta_{e/o}\neq 0$ with $\beta_{e}\neq\beta_{o}$ and $\Theta\neq 0$
signals the existence of a so-called lattice supersolid. The phase diagram of
the system is shown in Figure 40. The system can be studied in a similar way
for other integer ratios of $\lambda_{c}/\lambda_{\rm latt}$.

#####  7.1.1.2 Incommensurate: Bose-glass phase

We now consider the case where the ratio $\lambda_{c}/\lambda_{\rm latt}$ is
not a rational number [131, 264, 56, 75]. Therefore, the external classical
lattice is not commensurate with the pump and dynamical potentials, resulting
in a non-periodic configuration. One may envisage the cavity-mediated
interaction as a disorder in this case, since the amplitude of the interaction
proportional to $A_{\mathbf{j}}$ oscillates at the cavity wavelength
$\lambda_{c}$, which is incommensurate with respect to the external lattice.
The phase diagrams of the system in 1D (along the cavity axis $x$) and 2D are
shown in Figures 41(a) and (b), obtained from quantum Monte Carlo and mean-
field simulations, respectively [131]. In addition to the superfluid and Mott-
insulator states, the system exhibits another phase with interesting
properties. It comprises a compressible state $\partial\bar{n}/\partial\mu\neq
0$ with vanishing superfluid density. The atomic density in this phase
exhibits a checkerboard density-wave pattern with an envelope at the beating
wavenumber $|k_{\rm latt}-k_{c}|$, such that the Bragg density order parameter
$\Theta=\sum_{\mathbf{j}}A_{\mathbf{j}}\langle\hat{n}_{\mathbf{j}}\rangle/N_{\Lambda}$
acquires a nonzero value. The envelope arises due to the fact that the system
tries to maximize $\Theta$ by convoluting the (checkerboard) density as
$A_{\mathbf{j}}$ oscillates at the cavity wavelength $\lambda_{c}$ and becomes
out of phase with respect to the external lattice with the incommensurate
wavelength $\lambda_{\rm latt}$. Therefore, the atoms can constructively
scatter photons from the pump laser into the cavity. This state has the
characteristic of a Bose-glass (BG) phase typical in disordered systems.

The effect of the cavity-mediated interactions is particularly evident in the
1D case, where the pump term is a constant shift in the chemical potential
[cf. Equation (7.1)]: At small hopping amplitudes the sizes of the Mott lobes
reduce and they give way for the Bose-glass states (compare it with the dashed
curves for the phase diagram without cavity-mediated interactions $U_{l}=0$).
By increasing the hopping amplitude, the Bose glass melts and gives way to the
superfluid state. Note that in 1D the Bose-glass state corresponds exactly to
the superradiant phase, where the cavity field is populated macroscopically.
However, in 2D a Bose-glass state can also arise due to the pump potential
which is also incommensurate with respect to the external optical lattice. The
color map in Figure 41(b) represents the average superfluid order parameter
$\sum_{\mathbf{j}}\beta_{\mathbf{j}}/N_{\Lambda}$ and the dashed line
delineates the superradiant phase. In the single-particle case, the
incommensurate cavity potential can lead to the Anderson-type localization of
the atom and mobility edges—energy eigenstates that separate coexisting
extended and localized states—even in the purely optomechanical regime [265,
266, 267].

Figure 41: Phase diagram of a cavity-induced incommensurate extended Bose-
Hubbard model in the $\mu$-$J$ plane for the irrational ratio of
$\lambda_{c}/\lambda_{\rm latt}=830/785$. The phase diagram of the system
includes Mott-insulator (MI), superfluid (SF), and Bose-glass (BG) states,
obtained in 1D via quantum Monte Carlo simulations (a) and in 2D using mean-
field calculations (b). In panel (b) the dotted and solid curves delineate the
regions with a small, arbitrarily chosen threshold for average superfluid
order parameter and the density fluctuations, respectively, while inside the
dashed curve the number of photons is at least 100 times larger than outside.
The region with finite compressibility and vanishing superfluid order
parameter corresponds to the Bose-glass phase. In the Bose-glass phase, the
system forms clusters with checkerboard density distribution, where the atoms
scatter photons in phase into the cavity mode. Panel (a) adapted and reprinted
with permission from Ref. [131] and panel (b) from Ref. [264] © 2013 by the
American Physical Society.

In Ref. [268], a closely related scenario to the schemes discussed in this
section has been studied: it has been assumed that non-interacting bosonic
atoms are already trapped in a 1D static, external quasiperiodic
potential—composed of two incommensurate lattices and described by the single-
band Aubry-André model—along the cavity axis and pumped by a laser in the
transverse direction. If the strength of disorder in the external Aubry-André
lattice exceeds a critical value, it leads to the Anderson localization of the
atoms. As a consequence, the atoms scatter pump photons into the cavity mode
completely in phase and constructively, driving the Dicke-superradiant
instability even in vanishingly small pump strengths. Such superradiant
instability with vanishing pump strengths has also been predicted for
fermionic atoms, but due to a completely different mechanism—Fermi surface
nesting [140].

####  7.1.2 Higher-order corrections to cavity-induced extended Bose-Hubbard
models

In deriving the lattice Hamiltonian (7.1), we have assumed that the external
classic lattice $V_{\rm ext}(\mathbf{r})$ and its lowest-band Wannier
functions are not modified by the transverse pump and dynamical potentials.
However, this is not strictly true, specially in the strong pump and the deep
superradiant regime. Therefore, one has to find _self-consistently_ the
Wannier functions of the lowest Bloch band of the total lattice potential,
$V_{\rm ext}(\mathbf{r})+\hbar V(\mathbf{r})+\hbar
U(\mathbf{r})\hat{a}^{\dagger}\hat{a}+\hbar\eta(\mathbf{r})(\hat{a}^{\dagger}+\hat{a})$.
That is, the steady-state cavity field $\hat{a}_{\rm ss}$, Equation (87),
depends on the matrix elements (7.1) calculated using the Wannier functions,
which in turn depend on the cavity field through the total lattice potential.
Correspondingly, the hopping amplitude $J$ and the strength of the short-range
contact interaction $U_{s}$ must be also determined self-consistently. This is
a genuine many-body cavity-QED nonlinear effect, where the coupled matter-
field dynamics depend crucially on the atomic density. Such a nonlinear effect
is most prominent when the dynamic cavity potential is the sole lattice
potential in the system, e.g., by longitudinally pumping a cavity [107] where
the existence of overlapping, competing Mott-insulator states as well as
bistable behavior has been predicted [261, 262].

As mentioned already, in deriving the lattice Hamiltonian (7.1) only the on-
site matrix elements of the spatial part of the pump and dynamic cavity
potentials, Equation (7.1), have been kept. Although much smaller compared to
the on-site matrix elements, retaining the off-site matrix elements of the
potentials alters the lattice Hamiltonian (7.1) crucially. Namely, after
adiabatic elimination of the steady-state cavity field $\hat{a}_{\rm ss}$
these off-site contributions modify the bare hopping amplitude $J$ and result
also in cavity-induced correlated two-particle hoppings proportional to
$[\sum_{\langle\mathbf{j},\mathbf{j}^{\prime}\rangle}(\hat{b}_{\mathbf{j}}^{\dagger}\hat{b}_{\mathbf{j}^{\prime}}+\text{H.c.})]^{2}$
in the extended Bose-Hubbard Hamiltonian. As a consequence, phase boundaries
between different phases are modified [107, 52]. In particular, in the
vicinity of phase boundaries different photon-number states are associated
with different atomic states, so that the ground state of the system in the
proximity of phase boundaries contains atomic states of different nature
correlated with different photon numbers due to photon-number fluctuations. A
scenario for enhancing the off-site matrix elements while suppressing the on-
site ones which can result in additional quantum phases was discussed in Refs.
[245, 259].

####  7.1.3 Experimental realizations of the extended Bose-Hubbard model

#####  7.1.3.1 Quantum phases from competing interactions

The experimental realization of the above described commensurate extended
Bose-Hubbard model with cavity-mediated global-range interactions (Section
7.1.1.1) requires to create a static but tunable 3D optical lattice inside an
optical cavity. The depth of this lattice controls the strength of the on-site
short-range collisional interactions between the atoms, while the cavity is
used to induce competing global-range interactions. For a sufficiently deep 3D
optical lattice, the BEC will—for small enough global-range
interactions—undergo a quantum phase transition from a superfluid phase to a
Mott-insulating phase featuring an integer number of particles with equal
occupations of even and odd sites [14]. In contrast, cavity-mediated global-
range interactions favor superradiant states with an occupation imbalance
between even and odd sites. For small enough short-range interactions, the
system will undergo a transition from the SF phase to a lattice supersolid
phase with a modulated superfluid state. If both short-range and global-range
interactions are strong, the system will enter a superradiant, insulating
charge-density wave state with a periodically modulated integer site
occupation.

Figure 42: Experimental setup to realize the commensurate extended Bose-
Hubbard model. A BEC is trapped at the center of an optical cavity and loaded
into a deep optical lattice along the $z$ axis, which slices the cloud into 2D
layers in the $x$-$y$ plane. A TEM00 mode of the cavity is pumped to create an
intra-cavity lattice along the $x$ direction, which forms together with a free
space lattice along the $y$ direction a square optical lattice of tunable
depth $V_{\rm latt}^{(0)}$. The lattice along the $y$ direction simultaneously
acts as a transverse pump with lattice depth $V_{0}=V_{\rm latt}^{(0)}$ for
the orthogonally-polarized TEM00 cavity mode inducing cavity-mediated global-
range interactions. The figure on the right displays the fundamental processes
determining the physics within a 2D layer: atoms are subject to global
interactions with strength $\propto U_{l}$, favoring a density modulation, and
to repulsive on-site interactions with strength $U_{s}$, favoring localization
of the atomic wave function on individual lattice sites. Finally, tunneling
between neighboring lattice sites with amplitude $J$ favors delocalization of
the atomic wavefunction. Figure adapted and reprinted with permission from
Ref. [49] published at 2016 by the Nature Publishing Group.

Figure 42 displays the basic experimental setup together with the relevant
atomic processes. A BEC trapped at the center of an optical cavity by external
fields is loaded into a deep optical lattice along the $z$ axis, effectively
freezing out tunneling in this direction. This slices the cloud into 2D
pancakes in the $x$-$y$ plane. These 2D layers are further exposed to a
classical optical lattice in the $x$-$y$ plane, formed by one free space
lattice along the $y$ direction and one intra-cavity optical standing wave
along the $x$ direction. By a choice of perpendicular polarizations and a
large enough relative detuning, these two fields do not interfere such that
the resulting potential is a square lattice. This corresponds to the external
optical lattice $V_{\mathrm{ext}}(\boldsymbol{r})$ introduced in Equation (84)
with adjustable lattice depth $V_{\rm latt}^{(0)}$, allowing to tune the
strength of the short-range interactions as well as the hopping amplitude. At
the same time, the standing-wave potential along the $y$ axis acts
additionally as a transverse pump lattice with lattice depth $V_{0}=V_{\rm
latt}^{(0)}$ [cf. Equation (7.1)] for another cavity mode with the similar
polarization, inducing the cavity-mediated global interactions. The resonance
frequency of the cavity is chosen with an adjustable detuning $\Delta_{c}$
with respect to the frequency of this pump field. As a result, the strength of
the cavity-mediated global-range interactions $\propto A^{2}U_{l}$ can be
tuned by changing $\Delta_{c}$ or by varying $V_{\rm latt}^{(0)}$. Most
importantly, this arrangement allows to change the ratio of short-range and
global-range interactions which can thus be brought into competition. The
physics of this system is captured in a wide range of parameters by a
Hamiltonian of the form of Equation (7.1.1.1). It is, however, important to
note that due to the twofold role of the transverse pump as the external
lattice and the pump field simultaneously, the transverse pump is not a
perturbative term anymore. This requires to calculate the Wannier functions,
$J$, $U_{s}$, and other coefficients consistently using $V_{\rm
latt}^{(0)}=V_{0}$. The fundamental processes underlying the different terms
in the Hamiltonian are visualized in Figure 42.

In order to detect different quantum phases reached by tuning the parameters,
two observables are necessary. First, the presence or absence of an intra-
cavity light field is a measure whether the atomic system has an imbalanced
occupation of the even and odd lattice sites or not [see Equations (87) and
(88)], allowing to distinguish the superfluid and Mott-insulating phases from
the lattice-supersolid and the charge-density-wave phases. Second, absorption
images of the atomic cloud after ballistic expansion are used to probe the
coherence of the gas. Observation of narrow momentum peaks and a high
condensate fraction indicate the presence of coherence typical for the
superfluid and lattice-supersolid phases. In contrast, broadened peaks and a
vanishing condensate fraction indicate the loss of coherence and can be
associated with the loss of superfluidity and the formation of an insulating
phase (i.e., the Mott-insulator and charge-density-wave phases). Combining
both observables—related to $\beta_{e/o}$ and $\Theta$ in the mean-field
description of Section 7.1.1.1—thus allows to distinguish all the possible
phases of the system.

This scheme was experimentally implemented in Ref. [49], and Figure 43
displays the measurement of both observables as a function of the lattice
depth $V_{\rm latt}^{(0)}$ and the relative cavity-pump detuning $\Delta_{c}$.
The BEC fraction vanishes for increasing lattice depth $V_{\rm latt}^{(0)}$
and a threshold for the transition from the superfluid to insulating phases
can be defined, as indicated by the white data points; see Figure 43(a) and
(b). Figure 43(c) and (d) show respectively the simultaneously measured intra-
cavity photon number and the imbalance $\Theta=\langle\hat{\Theta}\rangle$
between even and odd sites as defined in Equation (88), but assuming
completely localized atoms on even or odd sites. This imbalance is directly
calculated from the mean intra-cavity photon number using Equation (87). Also
for this observable, a threshold can be defined that indicates the transition
from a normal phase with balanced occupation of even and odd sites to a
superradiant state with an imbalanced occupation.

Figure 43: Experimental observables of the cavity-induced extended Bose-
Hubbard model as a function of the cavity-pump detuning $\Delta_{c}$ and the
lattice depth $V_{\rm latt}^{(0)}$. Panel (b) shows the BEC fraction extracted
from bimodal fits to absorption images after ballistic expansion of the atomic
cloud [examples are given in panel (a)]. Panel (d) shows the imbalance
$\Theta$ between even and odd lattice sites, calculated from the measured mean
intra-cavity photon number $n_{\rm ph}$ [example traces of $n_{\rm ph}$ are
shown in panel (c)]. The white and black data points indicate the threshold
for transitions between superfluid and insulating phases [panel (b)] and
between balanced and imbalanced site occupations [panel (d)], respectively.
Figure adapted and reprinted with permission from Ref. [49] published at 2016
by the Nature Publishing Group.

Combining the measurement results a phase diagram can be constructed, as shown
in Figure 44. All four expected phases [i.e., superfluid (SF), lattice
supersolid (LSS), Mott insulator (MI), and charge-density wave (CDW)] can be
identified. The transition line between the superfluid and the lattice
supersolid shifts to smaller values of $V_{\rm latt}^{(0)}$ when approaching
cavity resonance, as expected from self-organization without the additional
lattices (see Section 3). The transition line between the lattice supersolid
and the charge-density wave follows also this trend, indicating that the
insulating phase is entered for smaller lattice depths if a density modulated
state is favored. This has been attributed to a reduced nearest-neighbor
tunneling due to an increasing energy offset between even and odd lattice
sites in the superradiant phases. The transition between the charge-density
wave and the Mott insulator phase is especially interesting, since the system
has to transition between two different incompressible density patterns while
tunneling is strongly suppressed. An experimental exploration of this first-
order phase transition is discussed below in Section 7.1.3.2.

Figure 44: Phase diagram of the cavity-induced extended Bose-Hubbard model
constructed from the thresholds identified in Figure 43. All four
theoretically expected phases can be identified: superfluid (SF), lattice
supersolid (LSS), Mott insulator (MI), and charge-density wave (CDW). Panel
(a) shows the phase diagram as a function of the experimental parameters
$\Delta_{c}$ and $V_{\rm latt}^{(0)}$, while panel (b) shows the same phase
diagram as a function of the Hamiltonian parameters $U_{s}/J$ and
$-A^{2}U_{l}/J$, calculated from the experimental parameters. Figure adapted
and reprinted with permission from Ref. [49] published at 2016 by the Nature
Publishing Group.

The experimental setup also included a shallow harmonic potential trapping the
atomic cloud, leading to an inhomogeneous density profile. As a consequence,
it is energetically more costly for the atoms to be away from the trap center
and the insulating phases will form a wedding cake structure, also allowing
for the coexistence of phases. Furthermore, since the global character of the
cavity-mediated interactions leads to a break-down of the local-density
approximation, a direct comparison of the experimental results with
theoretically calculated phase diagrams is challenging. The role of the
trapping potential in this system has, however, been theoretically discussed
in Refs. [54, 248].

As already mentioned, even without an external optical lattice important
aspects of the competition between short-range and cavity-induced global-range
interactions can be explored in the deep superradiant phase [53, 255]. Indeed,
such a setup was the first experimental scheme to explore the competition
between short-range and global-range interactions [269]. Figure 45(a) shows
the experimental scheme of the Hamburg group where the atomic cloud was sliced
into 2D pancakes, however, no additional lattice was applied along the cavity
axis. This scheme relies on the fact that the emergent superradiant
checkerboard lattice can become sufficiently deep such that the self-organized
superfluid (SSF) phase undergoes eventually a transition into a self-organized
Mott-insulator (SMI) phase. Since there is no underlying 3D optical lattice,
this system can not enter a Mott insulating phase without imbalanced
occupations between even and odd sites.

The phase diagram can again be constructed by combining measurements of the
intra-cavity field with the interpretation of absorption images after
ballistic expansion of the atomic cloud. The resultant phases are shown in
Figure 45(b) on top of the measured mean intra-cavity photon number as a
function of pump-cavity detuning and pump lattice depth. Besides the
homogeneous superfluid phase (HSF), the self-organized superfluid and the
self-organized Mott insulator could be identified.

Figure 45: Extended Bose-Hubbard model realized in the emergent checkerboard
lattice of the superradiant phase. (a) Schematic representation of the
experimental setup. A BEC is magnetically trapped at the center of an optical
cavity (oriented vertically) and sliced into 2D layers by an external lattice.
A second lattice, not interfering with the first one, is applied in the
perpendicular direction and also acts simultaneously as a pump beam for the
cavity mode inducing global-range interactions. Photons leaking from the
cavity are recorded with a photon counter and measure the density modulation
of the atomic system. (b) The measured mean intra-cavity photon number
$n_{\mathrm{ph}}$ is shown as a function of the dispersively shifted pump-
cavity detuning $\delta_{c}$ and the pump lattice depth $V_{0}$. Combining
this measurement with absorption images taken after time of flight expansion
of the atomic cloud allows to identify the different phases of the system. The
homogeneous superfluid (HSF) phase undergoes a phase transition with
increasing pump lattice depth into a self-organized superfluid (SSF) phase and
then finally into a self-organized Mott-insulating (SMI) state. Figure adapted
and reprinted with permission from Ref. [269] © 2015 by the American Physical
Society.

#####  7.1.3.2 Metastability at the MI-CDW transition

The phase transition between the Mott-insulator state and the charge-density-
wave phase described above is a first order phase transition with an according
parameter region of metastability [see hatched region in Figure 44(a)].
Metastability, hysteresis, and a self-induced switching between the Mott
insulator and the charge-density wave have been observed experimentally using
the real-time access typical to cavity-based quantum gas experiments [270].

Figure 46(a) shows the experimental sequence for this study. After the system
is prepared in the superfluid phase at a detuning $\Delta_{c}$ corresponding
to vanishing global-range interactions, the depth $V_{\rm latt}^{(0)}$ of the
square lattice is ramped up such that the system undergoes a phase transition
to the Mott-insulator state. This constitutes the starting point for the
experiments exploring the first-order phase transition between the Mott-
insulating and charge-density-wave states. In the next step, the absolute
value of the detuning is first reduced (ramp I) and then increased again (ramp
II) over time, which brings the global-range interactions $\propto U_{l}$ into
competition with the short-range interactions $\propto U_{s}$ and back,
driving the system from the Mott insulator to the charge-density-wave state
and back. The light field leaking out of the cavity is a measure for the
spatial ordering of the atoms, captured by the imbalance $\Theta$. While
$\Theta\approx 0$ in the Mott-insulator state, the charge-density-wave phase
is characterized by a large imbalance $\Theta$. A typical single trajectory
measuring the evolution of the imbalance during the two ramps is shown in
Figure 46(b). The trajectory encloses a hysteresis loop and encompasses a
characteristic jump, indicated by the blue symbol.

Figure 46: (a) Schematic phase diagram of the system with a superfluid (SF,
gray), a lattice supersolid (LSS, blue), a Mott insulator (MI, orange), and a
charge-density wave (CDW, green) phase. The shaded region between the MI and
CDW indicates a region of hysteresis between the phases. The black arrow
illustrates the experimental sequence: The atoms are prepared in the SF phase
and the 3D optical lattice is ramped up to increase $U_{s}$, which brings the
system into the MI phase. Subsequently, a detuning ramp toward cavity
resonance is carried out which increases the strength of the global-range
interaction. (b) Non-normalized imbalance, as derived from the intra-cavity
light field. The imbalance created during ramp I is shown in orange and the
imbalance during ramp II is shown in green. Arrows indicate the ramp
directions. (c) Mean-field results from the toy model assuming unit filling.
In the presence of short- and global-range interactions, atoms placed in a
lattice potential can show metastable behavior. States (indicated by circles)
can be protected by an energy barrier and the present state of the system
depends on its history, leading to hysteresis. The Mott insulator (orange
line) and the charge-density wave (green lines) are stable (solid), metastable
(dashed), or unstable. Figure and caption adapted and reprinted with
permission from Ref. [270] published in 2018 by the United States National
Academy of Science.

A simple toy model can be used to explain the occurrence of a first order
phase transition in this system. Neglecting tunneling for simplicity in
$\hat{H}_{\rm ceBH}$ (7.1.1.1) in deep insulating phases, the system can be
characterized by the competition of short-range and global-range interactions
captured by the Hamiltonian $\hat{H}_{\rm insul}=(U_{s}/2)\sum_{\ell\in
e,o}\hat{n}_{\ell}(\hat{n}_{\ell}-1)+(A^{2}U_{l}/N_{\Lambda})\left(\sum_{e}\hat{n}_{e}-\sum_{o}\hat{n}_{o}\right)^{2}$.
Figure 46(c) shows the average ground-state energy per particle as a function
of the imbalance $\Theta$ and the ratio of the interactions $U_{l}/U_{s}$. In
the limit of small global-range interactions, a global energy minimum at zero
imbalance stabilizes the system in the Mott-insulator phase. With increasing
global-range interactions, $\mathbf{Z}_{2}$-symmetric local minima at finite
imbalance emerge. By further increasing global-range interactions, the Mott-
insulator state becomes a local minimum in the energy landscape, separated
from the charge-density-wave states by an energy barrier resulting from
competing short- and global-range interactions.

Not captured by this simple model is the observed jump in the imbalance [blue
symbol in Figure 46(b)], where the atomic system is strongly re-organized via
tunneling between neighboring sites in a dynamic process. Time-resolved
studies of this regime indicate that the process is a collective tunneling of
several thousand atoms within the expected tunneling time of a single double
well. The microscopic description of the observed dynamics relies on the
harmonic trapping potential experienced by the atoms. The atomic system
accordingly forms a wedding-cake structure with a Mott-insuolator core and a
superfluid surface. Particles in the surface possess a high mobility and can
locally organize into a density-modulated state. Local self-organization leads
to a global energy offset between even and odd sites. Depending on the
position in the trap, this offset can bring the tunneling between neighboring
sites into resonance, which leads to an avalanche dynamics and very fast
collective tunneling.

###  7.2 Two-component extended Bose-Hubbard model

As discussed in Section 6, when the internal states of atoms become relevant
[271], the exchange of cavity photons by the atoms can result in global-range
spin-spin and spin-density interactions among the atoms in addition to the
density-density interactions [69]. When the atoms are trapped in an external
lattice, the cavity-induced long-range spin interactions can then compete
directly with contact-interaction-induced short-range spin interactions in the
deep insulating phases, opening the possibility for realizing intriguing
lattice spin models including topological spin orders and frustrated spin
states.

Cavity-induced lattice spin models can be obtained by imposing an external
optical lattice potential in the spinor models discussed in Section 6. The
simplest model of these kinds is the lattice version of a transversely-driven
spinor BEC with standing-wave pump lasers
$\Omega_{1,2}(y)=\Omega_{01,02}\cos(k_{c}y)$ inside a single-mode linear
cavity. Namely, a state-independent external, static optical lattice potential
$V_{\rm ext}(\mathbf{r})$, Equation (84), is added to the effective two-
component Hamiltonian $\hat{\mathcal{H}}_{1,\rm eff}$, Equation (79).
Following exactly the same procedure as discussed in Section 7.1, for a strong
external optical lattice and after adiabatic elimination of the cavity field
one obtains up to $1/\Delta_{a}^{2}$ a two-component extended Bose-Hubbard
model,

$\displaystyle\hat{H}_{\rm 2eBH}$
$\displaystyle=-J\sum_{\tau,\langle\mathbf{j},\mathbf{j}^{\prime}\rangle}\left(\hat{b}_{\tau\mathbf{j}}^{\dagger}\hat{b}_{\tau\mathbf{j}^{\prime}}+\text{H.c.}\right)+\frac{1}{2}\sum_{\tau,\mathbf{j}}U_{s,\tau}\hat{n}_{\tau\mathbf{j}}(\hat{n}_{\tau\mathbf{j}}-1)+U_{s,\uparrow\downarrow}\sum_{\mathbf{j}}\hat{n}_{\uparrow\mathbf{j}}\hat{n}_{\downarrow\mathbf{j}}$
$\displaystyle+\sum_{\tau,\mathbf{j}}\left(\hbar
V_{0}B_{\mathbf{j}}^{(y)}-\mu\right)\hat{n}_{\tau\mathbf{j}}+N_{\Lambda}U_{l}\Big{(}\frac{1}{N_{\Lambda}}\sum_{\mathbf{j}}A_{\mathbf{j}}\hat{s}_{x,\mathbf{j}}\Big{)}^{2},$
(93)

where $\hat{b}_{\tau\mathbf{j}}$ is the bosonic annihilation operator
destroying a boson at spin state $\tau$ and site $\mathbf{j}$, and satisfies
the bosonic commutation relation
$[\hat{b}_{\tau\mathbf{j}},\hat{b}_{\tau^{\prime}\mathbf{j}^{\prime}}^{\dagger}]=\delta_{\tau,\tau^{\prime}}\delta_{\mathbf{j},\mathbf{j}^{\prime}}$.
Here we have assumed the balanced Raman condition as well as
$V_{\uparrow}(\mathbf{r})=V_{\downarrow}(\mathbf{r})=V_{0}\cos^{2}(k_{c}y)$.
The last term in the Hamiltonian $\hat{H}_{\rm 2eBH}$ is the cavity-mediated
global spin-spin interaction. It is indeed nothing but the lattice version of
the continuum spin Hamiltonian $\hat{H}_{\rm spin}$, Equation (6.1.1), in a
special case where only the $x$ component of the Heisenberg term is present
(recall that for standing-wave pump lasers other spin-spin interactions are
absent; see Section 6.2.1).

For the case of the commensurate lattice with $\lambda_{c}=\lambda_{\rm
latt}$, the global spin-spin interaction in Equation (7.2) reduces to the
imbalance of the $x$ component of the spin on even and odd sites, $\propto
U_{l}(\sum_{e}\hat{s}_{x,e}-\sum_{o}\hat{s}_{x,o})^{2}$. Therefore, for large
enough (negative) cavity-mediated interaction strength $U_{l}$, a ground state
with spin imbalance between even and odd sites
$\sum_{e}\langle\hat{s}_{x,e}\rangle\neq\sum_{o}\langle\hat{s}_{x,o}\rangle$
is favored. The ground state in this regime is either a spin-density wave or a
lattice supersolid depending the superfluid order parameter is zero or finite
[272].

More interesting lattice spin models can be obtained by considering different
pump-laser configurations as discussed for the continuum models in Section 6.
Therefore, a variety of emergent magnetic orders can be explored in the
framework of lattice cavity QED with multi-component quantum gases; see also
the next section for the fermionic variants.

###  7.3 Extended Fermi-Hubbard models

Similar to the bosonic atoms discussed in the previous section, ultracold
fermionic atoms can also be loaded into the external lattice (84) inside the
cavity, yielding extended Fermi-Hubbard-like Hamiltonians with global, all-to-
all atomic interactions [273]. The only difference with respect to the
extended Bose-Hubbard models, Equations (7.1) and (7.2), is that the only
nonzero onsite atomic interaction would be the inter-species interaction
$U_{s,\uparrow\downarrow}\sum_{\mathbf{j}}\hat{n}_{\uparrow\mathbf{j}}\hat{n}_{\downarrow\mathbf{j}}$
for two-component fermionic atoms due to the Pauli exclusion principle.

####  7.3.1 Many-body localization in the presence of cavity-mediated global
interactions

The effect of the cavity-mediated global, all-to-all commensurate interactions
$\propto U_{l}[\sum_{\ell}(-1)^{\ell}\hat{n}_{\ell}]^{2}$ on the many-body
localization of spin-polarized fermionic atoms in a 1D external, static
optical lattice with nearest-neighbor atomic interactions
$U_{n.n.}\sum_{\ell}\hat{n}_{\ell}\hat{n}_{\ell+1}$ and random onsite disorder
$\sum_{\ell}\epsilon_{\ell}\hat{n}_{\ell}$, with $\epsilon_{\ell}$ being
uncorrelated random variables uniformly distributed in $[-W,W]$, has been
explored via an exact diagonalization for a small system size. It has been
found that the many-body localization persists in the presence of the cavity-
mediated global interactions, but occurs at larger disorder values [74]; see
Figure 47(a). Similar results have also been predicted for one-component
bosonic atoms inside cavities with on-site contact and random interactions
[74], where the existence of Bose glass phases is also anticipated [274, 275].

Figure 47: Steady-state phase diagram of cavity-induced commensurate, 1D
extended Fermi-Hubbard model for spin-polarized (a) and spin-1/2 (b) fermions.
(a) In addition to the cavity-mediated global interactions, fermions
experience nearest-neighbor interactions with strength $U_{n.n.}$ and random
onsite disorder characterized by $W$. While the disorder drives the system
into the many-body localized phase (the blue region characterized by the mean
gap ratio shown as the background color map), the global interactions favor
eigenvector-thermalization-hypothesis phase (the yellow region). The
simulation has been done at half-filling for a lattice of sixteen sites. The
solid yellow line indicates the phase boundary between the two phases,
obtained via finite-size scaling analysis. (b) The cavity-mediated global spin
interaction with strength $U_{l}$ induces different magnetic orders:
ferromagnetic (FM), antiferromagnetic (AFM), and incommensurate
antiferromagnetic (IAFM) orders. The subscripts in the figure indicate the
direction of the magnetic order in the internal spin space. The magnetic
orders in the repulsive cavity-mediated interaction regime,
$U_{l}\propto\Delta_{c}>0$, are induced by quantum fluctuations of the
photonic field. Panel (a) is adapted and reprinted with permission from Ref.
[74] published in 2019 by the SciPost and panel (b) from Ref. [276] © 2018 by
the American Physical Society.

####  7.3.2 Cavity-fluctuation-induced lattice magnetic orders

The steady-state phase diagram of spin-1/2 (i.e., two-component) fermionic
atoms subject to the cavity-mediated global, commensurate spin interaction
$\propto U_{l}[\sum_{\ell}(-1)^{\ell}\hat{s}_{x,\ell}]^{2}$ in a 1D lattice at
half filling has been mapped out using a numerical density-matrix-
renormalization-group method [276]; see Figure 47(b). The phase diagram
exhibits a rich structure containing anisotropic magnetic orders: FM (AFM)
indicates a state where both $\hat{s}_{x}$ and $\hat{s}_{z}$ exhibit
ferromagnetic (antiferromagnetic) correlations, while FMz-IAFMx (AFMz-IAFMx)
is a state with a ferromagnetic (antiferromagnetic) correlation for
$\hat{s}_{z}$ and an incommensurate—with respect to the static optical
lattice— antiferromagnetic correlation for $\hat{s}_{x}$. In the absence of
the cavity-mediated global interaction, $U_{l}=0$, a repulsive onsite
interaction $U_{s,\uparrow\downarrow}>0$ already favors an AFM order at the
half filling. The attractive cavity-mediated interaction $U_{l}<0$ further
stabilizes the AFM order. In contrast, the repulsive cavity-mediated
interaction $U_{l}>0$ competes with the repulsive onsite interaction,
stabilizing the other magnetic orders in the phase diagram. Interestingly, the
region of the positive cavity-mediated interaction $U_{l}\propto\Delta_{c}>0$
(i.e., the blue pump-cavity detuning region) corresponds to a non-superradiant
regime where the coherent field amplitude is vanishingly small,
$\alpha=\langle\hat{a}\rangle\approx 0$. Nonetheless, cavity-field
fluctuations
$\langle\delta\hat{a}^{\dagger}\delta\hat{a}\rangle/\langle\hat{a}^{\dagger}\hat{a}\rangle$
is considerably large (in the order of one) in this regime, signaling that the
FM, FMz-IAFMx, and AFMz-IAFMx magnetic orders are induced by quantum
fluctuations of the photonic field.

####  7.3.3 Cavity-induced superfluid pairing in a lattice

In Section 6.3.3 we described a scenario for superfluid pairing mediated via
cavity fluctuations in the blue-detuned regime, $\Delta_{c}>0$. Here we
consider a lattice scenario for cavity-induced pairing in the red-detuned
regime $\Delta_{c}<0$ corresponding to attractive cavity-mediated interactions
$\mathcal{D}_{0}<0$. In particular, two ring cavities oriented along axes of a
2D external optical lattice (84) mediate global interactions

$\displaystyle\hat{H}_{\rm
int}=\mathcal{D}_{0}\sum_{\mathbf{k},\mathbf{k}^{\prime},\mathbf{k}_{c}}\sum_{\tau,\tau^{\prime}}\hat{f}_{\tau,\mathbf{k}\pm\mathbf{k}_{c}}^{\dagger}\hat{f}_{\tau,\mathbf{k}}\hat{f}_{\tau^{\prime},\mathbf{k}^{\prime}\mp\mathbf{k}_{c}}^{\dagger}\hat{f}_{\tau^{\prime},\mathbf{k}^{\prime}},$
(94)

among spin-1/2 fermionic atoms in the external optical lattice, where
$\mathbf{k}_{c}=k_{c}\hat{e}_{i}$ with $i=x,y$. In Ref. [273], such cavity-
mediated interactions have been considered in a regime where
$|\mathbf{k}_{c}|\sim k_{\rm latt}$, which is characterized by the strong
competition between a cavity-induced superfluid paired phase and other types
of non-paired, charge- or spin-density phases. On the other hand, in the
regime where $|\mathbf{k}_{c}|\ll k_{\rm latt}$ considered in Ref. [277], the
superfluid pairing instability dominates. Furthermore, different types of
pairing compete with one another, and can give rise to interesting exotic
topological superfluid states. This phenomenon is a consequence of an
accidental degeneracy of competing paired phases, shown in Figure 48, due to
the global range of the cavity-mediated interaction, leaving the system in a
frustrated state. The perturbative addition of local interactions splits this
degeneracy and creates topological $p+id$ wave superfluid-paired state
featuring Majorana fermions. The resulting phase diagram is shown in Figure
48(e).

Figure 48: Cavity-induced pairing in a 2D Fermi gas trapped in an external,
square optical lattice. (a)-(d) Solutions of the gap equation for the
superfluid order parameter induced via cavity-mediated interactions (94). The
initial state for the iterative solution is chosen with (a) $s$-wave, (b)
$d$-wave, and (c)-(d) $p$-wave symmetry, respectively. The cavity wave number
is set to $k_{c}=0.02\pi$ (with lattice spacing $a_{\rm latt}=1$). The other
parameters are $\mathcal{D}_{0}=-0.1J$, $U_{s,\uparrow\downarrow}=0$, and
$\mu=-0.5J$. (e) Dominant pairing symmetry as a function of the spin-dependent
chemical potential $\delta\mu$ splitting the energy of the two spin states,
and the local Feshbach interaction $U_{s,\uparrow\downarrow}$. The cavity-
mediated interaction strength is $\mathcal{D}_{0}=-0.1J$, and the chemical
potential is $\mu=-3J$. Figure adapted and reprinted with permission from Ref.
[277] © 2019 by the American Physical Society.

These results highlight the promise of cavities for inducing novel mechanisms
for pairing as well as stabilizing more exotic superconducting states. This
direction of investigation has recently received a lot of attention also in
the context of solid-state materials [278, 279, 280, 281].

##  8 Cavity-induced synthetic gauge potentials and topological states

Gauge potentials and quantum gauge theories play central roles in our
understanding of Nature. The simplest example is electromagnetism with its
Abelian vector and scaler potentials describing the coupling between charged
matter and electromagnetic fields. Quantum electrodynamics (QED), a
relativistic gauge theory describing the coupling of a relativistic charged
matter field to the Abelian electromagnetic gauge potentials, is among the
most successful and most accurate theories of physics. In the Standard Model
of elementary particles, gauge fields are the mediators of fundamental
interactions among the elementary particles. Gauge potentials are also of
great significance in condensed-matter physics [282]. In fact, they are the
essential ingredients of topological states of matter [283, 284]. In addition
to their fundamental significance, topological insulators are also of great
importance from a technological point of view: they possess conducting edge or
surface states which are topologically protected and robust against external
perturbations. These topological edge states can have exotic properties. In
the case of quantum Hall insulators, such edge states carry precisely one
quantum of conductance. They can also host anyonic excitations, quasiparticles
that do not obey Bose or Fermi statistics.

Since ultracold atoms are charge neutral, they do not couple to gauge
potentials the same way charged particles (such as electrons) do. However, a
variety of gauge potentials and minimal couplings can be engineered via the
light-matter interaction for neutral atoms, to mimic phenomena encountered not
only in condensed-matter physics but also in high-energy physics [8, 285, 9,
286, 287, 288]. The induction of such synthetic gauge potentials generally
relies on geometric phases acquired through adiabatic motion of quantum
particles with internal structures. That is, the light-matter interaction
couples parametrically the atomic internal dynamics to its external degree of
freedom (i.e., position), resulting in the appearance of a geometric vector
potential under special conditions. The laser-induced gauge potentials are
normally _static_ , in the sense that the back-action of the quantum matter on
these gauge fields is negligible. They are solely background gauge potentials,
unaffected by the matter dynamics (though they can still have an externally
imposed time dependence) and described by extra fixed terms in the Hamiltonian
of the system. For instance, the synthetic electric and magnetic fields
induced by the light-matter interaction for quantum gases need not to obey the
Maxwell’s equations. However, gauge potentials encountered in quantum gauge
theories such as QED, quantum chromodynamics, etc., and strongly correlated
many-body condensed-matter systems are normally _dynamic_. That is, the
dynamics of these gauge potentials are governed by their own Hamiltonians and
affected by the back-action of the matter. Motivated by these dynamical gauge
potentials, some proposals have been put forward to simulate dynamical gauge
potentials for quantum gases [289]. A simplest interacting gauge potential
toward full dynamical gauge potentials is perhaps a density-dependent gauge
potential [290, 291].

An alternative, natural route to implement dynamic gauge potentials for
neutral atoms is the framework of cavity QED. This hinges on the non-linear,
dynamical nature of cavity fields which can induce gauge potentials. Unlike
lasers interacting with atoms in free space, cavity fields are dynamical and
affected by the atomic back-action. Therefore, cavity-induced gauge potentials
inherit the dynamical nature of the cavity fields and respond naturally to the
atomic back-action. Furthermore, as radiation fields decay out of a cavity,
quantum-gas–cavity systems with induced gauge potentials are naturally out of
equilibrium. This leads to nonequilibrium topological phases in the framework
of cavity QED, a scenario beyond common condensed-matter and free-space
quantum-gas systems. Guided by this nonlinear coupled dynamics of matter and
radiation in cavity QED, proposals have been put forward to induce dynamical
gauge fields [292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 69, 302, 303]
and topological phases [304, 305, 59] for quantum gases in cavities, with the
experimental realization of a dynamical spin-orbit coupling [72]. Another
peculiarity of cavity-induced gauge potentials and topological states, in
addition to their nonequilibrium dynamical nature, is the appearance of them
on the onset of the superradiance phase transition. Since the threshold of the
superradiant quantum phase transition depends crucially on atomic properties,
the onset of a cavity-induced gauge potential and topological phase is also
sensitive to atomic structures and is self-consistent.

Here we will review some of the cavity-based schemes for inducing gauge
potentials for quantum gases. We start with the cavity-QED implementation of a
synthetic magnetic field corresponding to a position-dependent vector
potential, the simplest gauge potential with the $U(1)$ gauge symmetry. Our
focus shall be mainly on lattice models where the synthetic magnetic field is
induced via a cavity-assisted Peierls phase. We then study rich physics