arising from this dynamical synthetic magnetic field including topological
effects, chiral currents, fractal energy bands, and Meissner-like states. We
then consider cavity-assisted two-photon Raman schemes to engineer dynamical
spin-orbit coupling for neutral atoms, where the interplay between the
dynamical spin-orbit coupling, contact interactions, and cavity-mediated long-
range interactions results in rich physics. Finally, we discuss nonequilibrium
topological phases induced in quantum gases via the cavity field dynamics.

###  8.1 Cavity-induced synthetic gauge potentials

####  8.1.1 Dynamical synthetic magnetic fields

Following Ref. [297], consider spin-polarized fermionic atoms trapped in a
strong external classical 2D lattice in the $x$-$y$ plane with the
superlattice structure along the $x$ direction. The optical lattice is located
inside an initially empty linear cavity and created using standing-wave lasers
with wavelength $\lambda_{\rm latt}^{(y)}$ along the $y$ direction, and with
wavelengths $\lambda_{\rm latt}^{(x)}$ and $2\lambda_{\rm latt}^{(x)}$ along
the $x$ direction (which is the cavity axis). The phase difference of the two
lasers along the $x$ direction is chosen such that to create potential offsets
$\pm\hbar\Delta_{\rm latt}$ between adjacent sites of the lattice, resulting
in an imbalanced superlattice structure along the cavity (i.e., $x$) axis.
Therefore, the external lattice is separated into a set of decoupled ladders
as depicted schematically in Figure 49(a). The atoms can tunnel along the legs
of the ladders (i.e., the $y$ direction) due to their kinetic energy, while
the tunneling along the rungs (i.e., the $x$ direction) is strongly suppressed
due to the potential offset $\hbar\Delta_{\rm latt}$. The atoms are
dispersively pumped in the transverse direction by two running-wave pump
lasers with space-dependent Rabi couplings
$\Omega_{1,2}(y)=\Omega_{01,02}e^{\pm ik_{c}y}$ and frequencies
$\omega_{p1,p2}$ such that $\omega_{p2}-\omega_{p1}=2\Delta_{\rm latt}$. The
atoms are strongly coupled to a standing-wave mode of the cavity with the bare
frequency $\omega_{c}$ such that two-photon Raman transitions involving the
pump lasers and the cavity mode are close to resonant,
$\omega_{p2}-\omega_{c}\approx\omega_{c}-\omega_{p1}\approx\Delta_{\rm latt}$;
see Figure 49(b). The double $\Lambda$ scheme ensures that the system is not
optically pumped into one leg due to the cavity photon losses. The cavity mode
is initially in the vacuum state; however, it can be populated due to the
constructive, collective Raman scattering from the pump lasers by the atoms.
This in turn restores the hopping along the rungs (i.e., the cavity axis along
$x$), imprinting a phase into the wavefunction of the atoms due to the
running-wave pump lasers.

Figure 49: Cavity-induced synthetic magnetic field for fermionic atoms. (a)
The schematic sketch of the system. The external lattice provides disjoint
ladders. (b) A single rung in one ladder, showing all the transitions induced
by the pump lasers and the cavity field. The real part of the cavity-field
amplitude $\alpha$ (c) and the chiral current $j_{c}$ (d) are shown as a
function of the flux $\varphi$ and $-U_{l}A^{2}L/J_{\|}\propto\eta_{0}^{2}$
for three representative filling $n=1/2,3/8,1/4$, from left to right. The
solid red curves mark the onset of the superradiant phase transition, above
which a synthetic magnetic field appears for the fermionic atoms. As a
consequence of the latter, a chiral current is induced in the system.
Depending on the filling $n$, the atomic ground state corresponds to either a
chiral insulator (CI) or a chiral liquid (CL). Figure adapted and reprinted
with permission from Refs. [297, 298] © 2016 by the American Physical Society.

In the tight-binding limit, one ladder of the system is described in the
rotating frame of the lasers by the Hamiltonian,

$\displaystyle\hat{H}_{\rm latt}$
$\displaystyle=-J_{\|}\sum_{\ell=1,2}\sum_{m}\left(\hat{f}_{\ell,m+1}^{\dagger}\hat{f}_{\ell,m}+\text{H.c.}\right)+\hbar\eta_{0}(\hat{a}^{\dagger}+\hat{a})\sum_{m}\left(Ae^{-im\varphi}\hat{f}_{2,m}^{\dagger}\hat{f}_{1,m}+\text{H.c.}\right)$
$\displaystyle-\hbar\delta_{c}\hat{a}^{\dagger}\hat{a},$ (95)

where $\hat{f}_{\ell,m}$ is the fermionic operator annihilating a particle at
leg $\ell$ and rung $m$, $J_{\|}$ the kinetic-energy hopping amplitude along
the legs,
$\delta_{c}=(\omega_{p1}+\omega_{p2})/2-\omega_{c}-U_{0}\sum_{\ell,m}B_{\ell,m}^{(x)}\hat{n}_{\ell,m}$
the dispersively shifted cavity detuning with respect to the pumps with
$U_{0}=\mathcal{G}_{0}^{2}/\Delta_{a}$ being the dispersive shift per atom
[where $\Delta_{a}=(\omega_{p1}+\omega_{p2})/2-\omega_{a}$ is the atomic
detuning with respect to the pumps], and
$\eta_{0}=\Omega_{01}\mathcal{G}_{0}/\Delta_{a}=\Omega_{02}\mathcal{G}_{0}/\Delta_{a}$
the balanced two-photon Raman coupling. The coefficient $B_{\ell,m}^{(x)}$ is
the same as the one defined in Equation (7.1), while the coefficient $A$ has
now the form

$\displaystyle A=\int
W_{2,m}^{*}(\mathbf{r})\cos(k_{c}x)\exp{\left[-ik_{c}\left(y-\frac{1}{2}m\lambda_{\rm
latt}^{(y)}\right)\right]}W_{1,m}(\mathbf{r})d\mathbf{r}.$ (96)

The phase $\varphi=k_{c}\lambda_{\rm latt}^{(y)}/2=\pi\lambda_{\rm
latt}^{(y)}/\lambda_{c}$ arises due to the photon-assisted tunneling. In
particular, the hopping amplitude $\propto\eta_{0}(\hat{a}^{\dagger}+\hat{a})$
along the rungs is an operator depending on the cavity field operator
$\hat{a}$, which in turn depends on atomic properties highlighting the highly
nonlinear nature of the system. In the absence of the cavity photon, the
system consists of decoupled 1D legs along the $y$ direction. The nonzero
cavity field restores the hopping along the rungs, where the atoms collect the
total phase $\varphi$ while traversing a plaquette counterclockwise:
$-m\varphi+(m+1)\varphi=\varphi$. This phase simulates an electromagnetic
vector potential through the Peierls substitution, where the vector potential
corresponds to a synthetic transverse magnetic field perpendicular to the
$x$-$y$ plane. The phase and correspondingly the synthetic magnetic field can
be tuned by changing the ratio between the wavelengths $\lambda_{\rm
latt}^{(y)}/\lambda_{c}$ of the external lattice along the $y$ axis and the
cavity.

The steady state of the system can be obtained in the thermodynamic limit
using a mean-field approach; see also Section 2.4.2. To this end, the cavity
field operator $\hat{a}$ is replaced with its steady-state expectation value
$\alpha=\eta_{0}\langle
A\hat{K}_{\bot}+A^{*}\hat{K}_{\bot}^{\dagger}\rangle/(\delta_{c}+i\kappa)$,
where
$\hat{K}_{\bot}=\sum_{m}e^{-im\varphi}\hat{f}_{2,m}^{\dagger}\hat{f}_{1,m}$ is
the directed atomic tunneling in all rungs. Substituting the steady-state
field amplitude $\alpha$ in Equation (8.1.1) results in an effective atom-only
Hamiltonian for a single ladder

$\displaystyle\hat{H}_{\rm
ladd}=-J_{\|}\sum_{\ell=1,2}\sum_{m}\left(\hat{f}_{\ell,m+1}^{\dagger}\hat{f}_{\ell,m}+\text{H.c.}\right)-\sum_{m}\left(J_{\bot}e^{-im\varphi}\hat{f}_{2,m}^{\dagger}\hat{f}_{1,m}+\text{H.c.}\right),$
(97)

with the self-consistent hopping amplitude

$\displaystyle J_{\bot}=-U_{l}A\langle
A\hat{K}_{\bot}+A^{*}\hat{K}_{\bot}^{\dagger}\rangle/2,$ (98)

along the cavity axis, where
$U_{l}=4\hbar\eta_{0}^{2}\delta_{c}/(\delta_{c}^{2}+\kappa^{2})$ [cf.
definition of $U_{l}$ in Equation (7.1)]. The self-consistency condition (98)
can be solved, for instance, graphically and has a nontrivial solution only
for $\delta_{c}<0$. Furthermore, the existence of a nontrivial solution
depends also crucially on the system’s parameters including the filling
$n=N/2L$ with $N$ being the total number of the atoms and $L$ the size of the
system (i.e., the number of the rungs), the flux $\varphi$, and the pump
strength $\eta_{0}$ ($\propto\sqrt{U_{l}}$), which in turn they determine the
energy bands and the Fermi surface (i.e., Fermi points). Therefore, for a
given parameter set including a fixed $\eta_{0}$, the transverse hopping
$J_{\bot}$ is varied continuously and correspondingly $\langle
A\hat{K}_{\bot}+A^{*}\hat{K}_{\bot}^{\dagger}\rangle$ is calculated. The
system has a nontrivial self-consistent solution if the left- and right-hand
sides of Equation (98) intersect in a nonzero value.

Recall that a nontrivial solution of the self-consistency equation implies a
nonzero cavity field amplitude
$\alpha=-J_{\bot}(\delta_{c}+i\kappa)/2\hbar\eta_{0}A\delta_{c}$. Thanks to
the restored hopping $J_{\bot}$ along the rungs of the ladder for a nontrivial
solution of Equation (98), fermionic atoms become subject to a synthetic
magnetic field which induces a chiral current
$j_{c}=\sum_{m}\langle\hat{j}_{1,m}-\hat{j}_{2,m}\rangle/(L-1)$ with
$\hat{j}_{\ell,m}=-iJ_{\|}(\hat{f}_{\ell,m+1}^{\dagger}\hat{f}_{\ell,m}-\text{H.c.})$
being the local current operator along the leg $\ell$.

Figures 49(c) and (d) show, respectively, the real part of the cavity-field
amplitude $\alpha$ and the chiral current $j_{c}$ as a function of the flux
$\varphi$ and $-U_{l}A^{2}L/J_{\|}\propto\eta_{0}^{2}$ for three
representative filling $n=1/2,3/8,1/4$ [298]. The cavity field and the chiral
current build up only above a pump-strength threshold $\eta_{0c}(\varphi)$,
indication of the appearance of the cavity-induced synthetic magnetic field
beyond the nonlinear pump-strength threshold $\eta_{0c}(\varphi)$. This is in
sharp contrast to the free-space implementation of this scheme (in square
lattices) [306, 307]. For a given filling $n$, however, the pump-strength
threshold $\eta_{0c}$ is strongly suppressed at the critical flux
$\varphi_{c}=2\pi n$ (and $2\pi-\varphi_{c}$) and the hopping is restored even
for an infinitesimal pump strength. This behavior is intimately related to the
band structure and the Fermi points. For critical fluxes $\varphi_{c}$, the
cavity field amplitude increases slowly above the zero pump strength. On the
other hand, for $\varphi<\varphi_{c}\leq\pi$ and $\varphi>2\pi-\varphi_{c}$
($\varphi_{c}<\varphi<2\pi-\varphi_{c}$) the cavity-field amplitude acquires a
nonzero value only beyond a nonzero pump-strength threshold $\eta_{0c}$,
exhibiting a first (second) order phase transition into the self-ordered
superradiant phase. Depending on the filling $n$, the atomic ground state
corresponds to a band insulator or a liquid, carrying a chiral edge current
[indicated respectively by CI and CL in Figures 49(c) and (d)]. In some
parameter regimes, there exists a second solution. This second solution
vanishes at hight pump strengths, signaling the unstable nature of it.
Furthermore, the exact diagonalization of the corresponding master equation
for a small-size system reproduces solely the first stable solution (above the
pump-strength threshold $\eta_{0c}$). The field amplitude and the chiral
current exhibit similar behavior as functions of the filling $n$ and
$U_{l}A^{2}L/J_{\|}\propto\eta_{0}^{2}$ for representative fluxes. In
particular, the cavity field and the chiral current appear again only above a
pump-strength threshold $\eta_{0c}(n)$, where the threshold approaches zero at
the critical filling $n=\varphi_{c}/2\pi$.

This cavity-based scheme to implement a synthetic magnetic field in a ladder
can be straightforwardly generalized into a square/rectangle optical lattice
[305, 303], corresponding to the quantum Hall effect for electrons. This
allows one to study the Harper-Hofstadter model with fractal energy
bands—known as the Hofstadter’s butterfly—arising from a dynamical synthetic
magnetic field. Indeed, the cavity-induced synthetic magnetic field in a
finite-size system leads to topological edge states [305]. In an infinite
system, the dynamic nature of the magnetic field induces a nontrivial
deformation of the Hofstadter butterfly in the superradiant phase due to the
delicate interplay between the collective superradiant scattering inducing the
synthetic magnetic field and the emerging underlying fractal band structure
[303]. This is in sharp contrast to the free-space implementation of this
scheme as first proposed in Ref. [308], where the energy bands show the
conventional Hofstadter-butterfly structure.

In the above discussed schemes [297, 298, 305, 303], although the cavity-
induced synthetic magnetic field emerges dynamically owing to the backaction
of the atoms, the spatial profile of the synthetic magnetic field itself is
fixed by the cavity wave number $k_{c}$. The latter can also be made dynamical
as proposed in Refs. [299, 300], providing a local dynamical coupling between
the matter and the gauge potential as in the Ginzburg-Landau theory. A variant
of the scheme discussed in detail above was considered in Ref. [299], where an
external 1D lattice with an energy offset $\hbar\Delta_{\rm latt}$ between
adjacent sites is located perpendicular (i.e., along the $y$ direction) to the
axis of a single-mode linear cavity. The suppressed hopping is restored by a
cavity-assisted “directional” hopping containing a Peierls phase, which is
proportional to the phase of the cavity field and is dynamic and must be
determined self-consistently. In an infinite lattice under certain initial
conditions, a dynamical vector gauge potential appears in the system, inducing
a net, steady-state directed atomic current. In contrast, for a finite system
the atomic current shows a transient pulse behavior due to a superradiant
pulse (similar to those observed by illuminating degenerate quantum gases in
free space) and a nonequilibrium gauge potential. A generalization of this
scheme to a two-leg ladder coupled to two cavity modes revealed even more
interesting features, including the emergence of an electromotive force due to
dynamical instabilities according to simulated Faraday’s induction law for the
neutral atoms inside the cavity [83]. In a different approach, coupling a BEC
to many nearly degenerate transverse modes of a near-confocal cavity in a
particular configuration induces a synthetic magnetic field $\mathcal{B}_{x}$,
which its spatial profile, proportional to the spatial derivative of the
transverse intensity profile of the cavity light field, can evolve freely in
response to the atomic state [300]. In particular, the cavity-induced
synthetic magnetic field is expelled from the bulk of the condensate due to
the diamagnetic response of the system, reminiscent of the Meissner effect in
superconductors; see Figure 50(d). Correspondingly, vortices in the BEC
density also disappear as shown in Figure 50(b).

Figure 50: A BEC in a synthetic magnetic field. (b) The density profile of the
BEC in a dynamic, synthetic magnetic field induced by a near-confocal
multimode cavity. There is no vortex in the bulk of the BEC (i.e., the high
density region) due the repulsion of the synthetic magnetic field from the
inside of the BEC. Vortices are moved to the edge of the BEC. (d) The spatial
profile of the cavity-induced dynamic magnetic field, affected by the atomic
back-action. The synthetic magnetic field is expelled from the bulk of the
condensate, similar to the Meissner effect in superconductors. The BEC density
(a) in a static, synthetic magnetic field (c), included for the sake of
comparison. Here $a_{0}$ is the harmonic oscillator length of the trapping
potential and $f_{0}$ is a constant specifying the overall amplitude of the
pump. Figure adapted and reprinted with permission from Ref. [300] © 2017 by
the American Physical Society.

####  8.1.2 Dynamical synthetic spin-orbit coupling

As the name suggest, the spin-orbit (SO) coupling refers to the coupling
between the spin and the (linear or angular) momentum of a particle and it is
a relativistic effect. A charged particle moving with velocity ${\mathbf{v}}$
in an electric field $\mathbf{E}$ experiences a magnetic field
$\mathbf{B}=\mathbf{E}\times{\mathbf{v}}/c^{2}$ in its own rest frame
according to the Lorentz transformation up to the first order in $(v/c)^{2}$
(with $c$ being the speed of the light). Consequently, the magnetic moment
$\hat{\boldsymbol{\mu}}$ of the particle proportional to its charge $q$ and
its spin $\hat{\mathbf{s}}$ couples to this magnetic field
$\hat{\mathcal{H}}_{\rm SO}=-\hat{\boldsymbol{\mu}}\cdot\mathbf{B}\propto
q\hat{\mathbf{s}}\cdot(\mathbf{E}\times\hat{\mathbf{p}})$, resulting in the SO
coupling. Depending on the spatial form of the electric field, different forms
of SO coupling can arise.

Such a coupling between the spin and the external degree of freedom is absent
for an atom, as atoms are charge neutral. The SO coupling can be simulated for
neutral atoms via different methods such as dressing the atoms by laser
lights, where in the simplest case two atomic (ground) pseudospin states are
needed to encode a spin-1/2 degree of freedom. As for the simulation of
synthetic dynamical magnetic fields for the atoms using cavity fields
described in Section 8.1.1, dynamical SO coupling can also be engineered for
neutral atoms based on cavity fields. We will review here a few concrete
schemes based on cavity-assisted two-photon Raman processes. The Raman
technique using (non-copropagating) plane-wave lasers has so far been the only
method to experimentally engineer an equal contribution of the Rashba and
Dresselhaus dynamical SO couplings for quantum gases inside cavities [72]—it
was also the first method to implement static SO coupling for quantum gases in
free space [309, 310]. In Raman-based schemes for inducing SO coupling, it is
required to employ at least one running-wave light field, in order to impart
spin-dependent momentum recoil to the atoms.

#####  8.1.2.1 Linear-cavity-based schemes

Consider $\Lambda$-type three level bosonic atoms, with two ground spin states
$\\{\left|\downarrow\right>,\left|\uparrow\right>\\}$ and an electronic
excited state $\left|e\right>$, inside a linear cavity as in Ref. [292]. The
magnetic quantum number $m_{\tau}$ of the atomic internal states satisfy
$m_{\downarrow}-m_{e}=0$ and $m_{\uparrow}-m_{e}=1$. A bias magnetic field
applied along the $z$ direction defines the quantization axis and produces a
Zeeman shift $\hbar\omega_{Z}$ between the spin states. The atoms are driven
in the transverse direction by a standing- and a running-wave laser with
frequencies $\omega_{p1}=\omega_{p}+\Delta\omega_{p}$ and
$\omega_{p2}=\omega_{p}$, respectively. The first (second) laser is linearly
polarized along the $z$ ($x$) directions, therefore, inducing the atomic
transition $\left|\downarrow\right>\leftrightarrow\left|e\right>$
($\left|\uparrow\right>\leftrightarrow\left|e\right>$) with the position-
dependent Rabi frequency $\Omega_{1}(\mathbf{r})=\Omega_{01}\cos(k_{c}y)$
[$\Omega_{2}(\mathbf{r})=\Omega_{02}e^{ik_{c}y}$]. Furthermore, the transition
$\left|\downarrow\right>\leftrightarrow\left|e\right>$ is also strongly
coupled into an initially empty standing-wave mode of the cavity linearly
polarized along $z$, with the coupling strength
$\mathcal{G}(\mathbf{r})=\mathcal{G}_{0}\cos(k_{c}x)$. The system is depicted
schematically in Figures 51(a) and (b). Note that the atom-pump detuning
$\Delta_{a}=\omega_{p}-\omega_{a}>0$ is chosen to be positive, i.e., blue
detuned.

Figure 51: Cavity-induced dynamical spin-orbit coupling. The schematic sketch
of the system (a) and the atom-photon coupling (b). (c) The transverse spin
$\mathbf{s}_{\bot}(\mathbf{r})=(s_{x},s_{y})$ in the lattice of vortex-
antivortex pairs (LVAP) phase shows texture with opposite windings ($\pm 1$)
at even and odd sites. (d) The mean-field phase diagram of the system in the
parameter plane of $\eta_{0}$-$\Omega_{p0}$, exhibiting checkerboard, stripe,
and LVAP phases. These phases are characterized by the cavity field amplitude
$\alpha$ and the order parameters $\\{\Theta,\Xi\\}$, shown respectively in
panels (e) and (f) for a vertical cut through the phase diagram at
$\hbar\eta_{0}=0.7E_{r}$. The LVAP phase arises due to the competition between
the static and dynamic SO coupling. Figure adapted and reprinted with
permission from Ref. [292] © 2014 by the American Physical Society.

As in the case of double-$\Lambda$ system discussed in Section 6.1, in the
large atom-pump detuning limit, $|\Omega_{01,02}/\Delta_{a}|\ll 1$ and
$|\mathcal{G}_{0}/\Delta_{a}|\ll 1$, the atomic excited state can be
adiabatically eliminated. This results in an effective Hamiltonian for the
ground spin states,

$\displaystyle\hat{H}_{\rm
SO}=\int\hat{\Psi}^{\dagger}(\mathbf{r})\hat{\mathcal{H}}_{1,\rm
eff}\hat{\Psi}(\mathbf{r})d\mathbf{r}+\frac{1}{2}\sum_{\tau,\tau^{\prime}=\downarrow,\uparrow}g_{\tau\tau^{\prime}}\int\hat{\psi}_{\tau}^{\dagger}(\mathbf{r})\hat{\psi}_{\tau^{\prime}}^{\dagger}(\mathbf{r})\hat{\psi}_{\tau^{\prime}}(\mathbf{r})\hat{\psi}_{\tau}(\mathbf{r})d\mathbf{r}-\hbar\Delta_{c}\hat{a}^{\dagger}\hat{a},$
(99)

where $\Delta_{c}\equiv\omega_{p1}-\omega_{c}$, and $\hat{a}$ and
$\hat{\Psi}=(\hat{\psi}_{\uparrow},\hat{\psi}_{\downarrow})^{\top}$ are the
photonic and two-component atomic bosonic annihilation field operators,
respectively. The single-particle atomic Hamiltonian density reads,

$\displaystyle\hat{\mathcal{H}}_{1,\rm
eff}=-\frac{\hbar^{2}}{2M}\boldsymbol{\nabla}^{2}+V_{\rm
ext}(\mathbf{r})+\hbar\begin{bmatrix}\delta&\Omega_{p}(\mathbf{r})+\Omega_{c}(\mathbf{r})\hat{a}\\\
\Omega_{p}^{*}(\mathbf{r})+\Omega_{c}^{*}(\mathbf{r})\hat{a}^{\dagger}&V(\mathbf{r})+U(\mathbf{r})\hat{a}^{\dagger}\hat{a}+\eta(\mathbf{r})(\hat{a}^{\dagger}+\hat{a})\end{bmatrix},$
(100)

with the state-independent harmonic trap $V_{\rm ext}(\mathbf{r})$, and the
classical potential
$V(\mathbf{r})=\Omega_{1}^{2}(\mathbf{r})/\Delta_{a}=V_{0}\cos^{2}(k_{c}y)$,
the cavity potential
$U(\mathbf{r})=\mathcal{G}^{2}(\mathbf{r})/\Delta_{a}=U_{0}\cos^{2}(k_{c}x)$
per photon, and the interference potential
$\eta(\mathbf{r})=\mathcal{G}(\mathbf{r})\Omega_{1}(\mathbf{r})/\Delta_{a}=\eta_{0}\cos(k_{c}x)\cos(k_{c}y)$
for the spin-$\downarrow$ component. The two spin states are coupled via the
classical
$\Omega_{p}(\mathbf{r})=\Omega_{1}(\mathbf{r})\Omega_{2}^{*}(\mathbf{r})/\Delta_{a}=\Omega_{p0}\cos(k_{c}y)e^{-ik_{c}y}$
and cavity-induced
$\Omega_{c}(\mathbf{r})=\mathcal{G}(\mathbf{r})\Omega_{2}^{*}(\mathbf{r})/\Delta_{a}=\Omega_{c0}\cos(k_{c}x)e^{-ik_{c}y}$
two-photon Raman-Rabi processes. Here it has been assumed without loss of
generality $\\{\Omega_{01},\Omega_{02},\mathcal{G}_{0}\\}\in\mathbb{R}$. The
maximum strength of the potentials and Raman-Rabi couplings are defined as
$V_{0}=\Omega_{01}^{2}/\Delta_{a}$, $U_{0}=\mathcal{G}_{0}^{2}/\Delta_{a}$,
$\eta_{0}=\mathcal{G}_{0}\Omega_{01}/\Delta_{a}$,
$\Omega_{p0}=\Omega_{01}\Omega_{02}/\Delta_{a}$,
$\Omega_{c0}=\mathcal{G}_{0}\Omega_{02}/\Delta_{a}$. Here,
$\delta\equiv\omega_{Z}+\Delta\omega_{p}+\Omega_{02}^{2}/\Delta_{a}$ is the
effective two-photon detuning and can be modified independently via the
external bias magnetic field tuning $\omega_{Z}$. Both classical and quantum
Raman-Rabi interactions $\Omega_{p}(\mathbf{r})$ and
$\Omega_{c}(\mathbf{r})\hat{a}$ induce coupling between internal and external
degrees of freedom of the atom, resulting in a dynamical SO coupling. The
interplay of this dynamical SO coupling with the classical and quantum optical
potentials leads to interesting phenomena as discussed in the following.

The mean-field approach
$\hat{\psi}_{\tau}(\mathbf{r})\to\psi_{\tau}(\mathbf{r})=\langle\hat{\psi}_{\tau}(\mathbf{r})\rangle$
and $\hat{a}\to\alpha=\langle\hat{a}\rangle$ can be used to describe the
system; see Section 2.4.1. The corresponding coupled Gross-Pitaevskii
equations for the condensate wavefunctions are solved self-consistently with
the steady-state field amplitude

$\displaystyle\alpha_{\rm
ss}=\frac{\eta_{0}\Theta+\Omega_{c0}\Xi}{\Delta_{c}-U_{0}B+i\kappa},$ (101)

where $B=\int|\psi_{\downarrow}(\mathbf{r})|^{2}\cos^{2}(k_{c}x)d\mathbf{r}$,
$\Theta=\int|\psi_{\downarrow}(\mathbf{r})|^{2}\cos(k_{c}x)\cos(k_{c}y)d\mathbf{r}$
[cf. Equation (15)], and
$\Xi=\int\psi_{\downarrow}^{*}(\mathbf{r})\psi_{\uparrow}(\mathbf{r})\cos(k_{c}x)e^{ik_{c}y}d\mathbf{r}$.
Here it has been assumed the cavity decay $\kappa$ is large so that the cavity
field follows the atomic dynamics instantly. The cavity field has two
contributions: i) from collective photon scattering from the first pump with
spin-down atoms without changing their internal state characterized by
$\Theta$, and ii) from collective photon scattering from the second pump via
the spin-flipping process quantified by $\Xi$. The first scattering process
favors a checkerboard density pattern, while the second process results in the
cavity-induced SO coupling favoring a lattice of vortex-antivortex pairs
(LVAP). Therefore, $\Theta$ and $\Xi$ can be identified as order parameters,
characterizing different phases of the system.

The mean-field phase diagram of the system in the parameter plane of
$\eta_{0}$-$\Omega_{p0}$ for a fixed $\delta$ is presented in Figure 51(d). In
order to reduce the number of the free parameters, the ratio
$\Omega_{01}/\mathcal{G}_{0}$ has been fixed at 10, implying
$\Omega_{c0}=0.1\Omega_{p0}$. In relatively small $\Omega_{p0}$ the
checkerboard phase arises beyond a critical threshold $\eta_{0c}$, similar to
the one-component checkerboard self-ordering discussed in Section 3. For
chosen immiscible two-body contact interaction strengths
$g_{\downarrow\uparrow}^{2}>g_{\downarrow\downarrow}g_{\uparrow\uparrow}$, in
this phase the density structure of the spin-$\downarrow$ component is
determined by the potentials $V(\mathbf{r})$, $U(\mathbf{r})$, and
$\eta(\mathbf{r})$, while the density structure of the spin-$\uparrow$
component is mainly fixed by the contact interactions. In moderate
$\Omega_{p0}$, the system undergoes a phase transition into the density-
modulated stripe phase due to the classical potential $V(\mathbf{r})$ and the
“static” SO coupling induced by the classical pump Raman-Rabi coupling
$\Omega_{p}(\mathbf{r})$. The cavity field amplitude is zero in this phase,
signaling that it is a normal (i.e., non-superradiant) state. Note that the
stripe phase here does not possess supersolid characteristics as the
translational symmetry along the transverse ($y$) direction is broken
explicitly by the classical Raman-Rabi coupling $\Omega_{p}(\mathbf{r})$ and
the classical potential $V(\mathbf{r})$. The relative phase
$\Delta\phi(\mathbf{r})$ of the two condensate wave functions is periodically
modulated along the $y$ direction to minimize the Raman-coupling energy; see
also Section 6.2.2 for more details. The LVAP phase is, however, favored in
the strong $\Omega_{p0}$ limit where the cavity-induced “dynamical” SO
coupling by the Raman-Rabi coupling $\alpha\Omega_{c}(\mathbf{r})$ is
comparable with the static SO coupling owing to the large number of photons in
the system. In this phase, the relative phase of the two condensate wave
functions exhibits a lattice of vortex-antivortex pairs, hence the name LVAP
of this phase. Consequently, the transverse spin
$\mathbf{s}_{\bot}(\mathbf{r})=(s_{x},s_{y})$ forms a hedgehog (quadrupole)
texture with the winding number $\pm 1$ at the even/odd sites breaking the
$\mathbf{Z}_{2}$ symmetry of the system; see Figure 51(c). The cavity field
amplitude $\alpha$ and the order parameters $\\{\Theta,\Xi\\}$ are shown in
Figures 51(e) and (f), respectively, for a vertical cut through the phase
diagram shown in Figure 51(d). Note that $|\Theta/\Xi|\gg 1$ ($|\Xi/\Theta|\gg
1$) in the checkerboard (LVAP) phase, while both order parameters as well as
the cavity field are zero in the stripe phase.

We now revisit the model described in Sections 6.2 and 6.4 for realizing
density and spin self-ordering. As it was shown, the Hamiltonian of Equation
(79) for two pure running-wave pump lasers leads to a transverse conical,
chiral spin-spiral texture. In fact, this emergent spin order is due to the
cavity-induced dynamical SO coupling in the system. This has been studied in
detail in Ref. [311], by also including the two-body contact interactions
between the atom in the model. This is the first and only cavity-based scheme
where a dynamical SO coupling has been realized for ultracold atoms so far
[72], as will be discussed in more details in Section 8.1.3. This cavity-
induced SO-coupled BEC system has also been studied in 1D along the direction
of the pumps using a finite-size density matrix renormalization group (DMRG)
algorithm in the matrix product state form [302]. By adiabatically eliminating
the cavity field and discretizing the space, the system amounts to an
interacting Bose-Hubbard ladder pierced with a cavity-induced synthetic
magnetic field (similar to the ladder model for the fermionic atoms with the
cavity-induced synthetic magnetic field discussed in Section 8.1.1). In the
continuum limit, a Meissner superfluid with a net chiral (i.e., spin) current
is dynamically stabilized in some parameter regimes.

#####  8.1.2.2 Ring-cavity-based schemes

Another approach for inducing a dynamical SO coupling in the framework of
cavity QED is to utilize ring cavities [293, 294, 295, 296]. Since the natural
modes of ring cavities are plane waves, this provides the important ingredient
of a SO coupling based on the Raman scheme. Reference [294] has considered a
scenario where non-interacting $\Lambda$-type bosonic atoms are placed inside
one arm of a ring cavity. The atoms are Raman-dressed by one running-wave
laser and one counterpropagating plane-wave mode of the ring cavity pumped
longitudinally through a cavity mirror. The single-particle energy dispersion
of the system has two branches, similar to the free-space SO-coupled atoms.
However, the energy dispersion develops a loop structure in some parameter
regimes due to the nonlinearity of the system. The nonlinearity also leads to
dynamical instabilities, which are more pronounced in the limit of low cavity-
photon number. An interacting $\Lambda$-type BEC has also been considered
inside a ring cavity, where the atoms are Raman-dressed instead by two
distinct counterpropagating plane-wave modes of the cavity pumped
longitudinally through a cavity mirror [295]. The cavity fields induce a
synthetic SO coupling as well as long-range interactions between the atoms,
where the latter stems from the dynamical nature of the photonic fields and
their backaction on the atomic fields (see Section 2.2). The strength of the
cavity-mediated interactions can be tuned readily by changing the pump
amplitudes and frequencies as well as the decay rate of the cavity modes. The
interplay between the single-particle energy dispersion on one hand, and the
two-body contact and cavity-induced global interactions on the other hand
determines the many-body ground state of the system. In a wide range of
parameters, the stripe phase is favored over the plane-wave phase owing to the
cavity-induced interactions. Notably, the stripe phase is energetically
stabilized even for condensates with attractive intraspecies and interspecies
contact interactions for sufficiently large cavity-induced interactions.
Furthermore, for a symmetric double-well energy dispersion it is also possible
by tuning the cavity-mediated interactions to obtain a stripe phase with an
arbitrary superposition of atoms in the left and right wells, in sharp
contrast to a stripe phase in a free-space symmetric double-well energy
dispersion consisting of an equal number of atoms in the left and right wells.
Both Refs. [294, 295] have ignored the unpumped degenerate counterpropagating
mode(s) corresponding to the pumped mode(s) of the ring cavities. This is a
good first approximation for strongly pumped cavities in short time scales, as
the probably of scattering a photon into a strongly pumped mode is much higher
than the probability of scattering a photon into the degenerate empty mode due
to the Bose-enhancement factor. By building on Ref. [295], the effect of the
unpumped degenerate counterpropagating modes in this model has been included
and studied in Ref. [230]; though the two-body interactions were omitted.
These unpumped modes are populated solely by the coherent backscattering of
photons from the pumped modes via the atoms and play an important role in the
dynamics of the system in long times. The system exhibits three fundamentally
different steady-state phases, as the pump strengths and pump frequencies are
varied, characterized by distinct spatial density distributions and spin
textures: a combined density and spin wave, a continuous spin-spiral texture
with a homogeneous atomic density, and a spin spiral with a modulated density.
The spin-spiral states are topologically nontrivial, as discussed in Section
6.2, and are intimately related to cavity-induced SO coupling appearing beyond
a critical pump power. The topologically trivial density-wave–spin-wave state
spontaneously breaks two continuous symmetries, the $U(1)$ freedom of the
total phase of the two condensate wave functions and a $U(1)$ screw-like
symmetry corresponding to a continuous spatial translation accompanied by
phase rotations of the unpumped cavity-field amplitudes and condensate wave
functions, a characteristic of a supersolid state as discussed in Section 4.
The transitions between different phases are either topological and first
order, or non-topological and second order. Remarkably, all the phase
transitions can be detected nondestructively at the cavity output via
monitoring the unpumped cavity modes.

####  8.1.3 Experimental realization of dynamical spin-orbit coupling

The first experiment demonstrating cavity-assisted dynamical spin-orbit
coupling made use of the experimental scheme shown in Figure 52(a) and (b)
[72]. It is a variant of the scheme presented in Section 6.4, however,
employing running-wave fields for the pumps instead of the standing-wave
fields. More precisely, it is a special case of the Hamiltonian of Equation
(79) for two pure running-wave pump lasers corresponding to $R=0$ in Section
6.2. A BEC is loaded into a single-mode cavity and transversally illuminated
with two balanced counterpropagating running-wave laser fields. The
frequencies of these laser fields are chosen such that they, in combination
with vacuum fluctuations of the cavity mode, each induce a Raman transition
between two atomic hyperfine states, labeled as $\left|\downarrow\right>$ and
$\left|\uparrow\right>$. For pump rates below threshold, the atomic spins and
positions are disordered, and the Raman scattering rates are low. Above a
critical pump strength, spin and density of the gas self-order and the cavity
mode is populated with a coherent field due to superradiance. The frequency of
the emerging cavity field is the center frequency between the two coherent
pump-field frequencies.

Figure 52: Schematic of the cavity-induced dynamical spin-orbit coupling. Two
Raman pump beams (red and blue arrows), polarized along the cavity axis,
counterpropagate through a BEC of Rb atoms (purple) inside a TEM00 cavity
mode. The cavity emission (green arrow) is detected by a single-photon
counter, and the atoms are imaged after ballistic expansion. (b) Level diagram
illustrating the cavity-assisted Raman coupling between two hyperfine levels
of the 87Rb atoms acting as the spin states. $\delta^{\prime}$ is the Raman
detuning. (c),(d) Momentum-space depiction of the emergence of SO coupling.
(c) Initially atoms are in a spin-polarized state $\left|\downarrow\right>$.
(d) If the transverse pump strength is sufficiently strong, SO coupling
emerges and the spin components are in different momentum states. The $\pm$
sign of the $\left|\uparrow\right>$ spin component indicates the
$\mathbf{Z}_{2}$ symmetry-broken phase freedom. (e),(f) Energy-momentum
dispersion relation of each spin state, transitioning from free (e) to SO-
coupled (f) dispersion bands. The coupling strength
$\hat{\Omega}_{\mathrm{SOC}}$ is proportional to $\hat{a}$ and
$\hat{a}^{\dagger}$ and therefore arises dynamically as the atoms scatter pump
photons into the cavity. The zero of the momentum has been shifted with
respect to the lab frame by $-k_{c}/2$ in the plot. Figure and caption adapted
and reprinted with permission from Ref. [72] © 2019 by the American Physical
Society.

Above threshold, SO coupling emerges in this scheme. In contrast to a
situation with standing-wave pump fields where no net momentum is transferred
to a cloud (cf. Section 6.4), during the self-organization with the running-
wave pumps momentum is imparted to the system while the atomic spins are
flipped. This can also be understood from the momentum-space representation
shown in Figure 52(c) and (d). While the cloud is spin polarized in
$\left|\downarrow\right>$ below the critical pump strength and only occupies
the zero momentum state, above threshold additional momentum states are
occupied. Since the Raman scattering process is spin selective, the spin state
$\left|\uparrow\right>$ is only populated if a net momentum along the positive
$y$ axis is transferred to the atoms. In contrast along the $x$ axis, the
accompanying momentum change averages to zero due to the standing-wave mode of
the cavity. Going into an according co-moving reference frame, opposite spin
states thus move in opposite directions, which thus realizes a SO coupling.
Since this process depends self-consistently on the evolution of the atomic
cloud and the cavity field in order to maximize the superradiant collective
photon scattering, the SO coupling has a dynamical character.

The relevant physics is captured—neglecting dispersive shifts—by a Hamiltonian
obtained from Equation (79) for two pure counterpropagating running-wave pump
lasers $\Omega_{1,2}(y)=\Omega_{\pm}e^{\pm ik_{c}y}$ (corresponding to $R=0$)
after transferring to the co-moving frame,

$\displaystyle\hat{H}=-\hbar\Delta_{c}\hat{a}^{\dagger}\hat{a}+\int\hat{\Psi}^{\dagger}(\mathbf{r})\begin{pmatrix}\frac{1}{2M}\left(\hat{\mathbf{p}}+\frac{\hbar
k_{c}}{2}\mathbf{e}_{y}\right)^{2}-\hbar\tilde{\delta}&\hbar\hat{\Omega}_{\mathrm{SOC}}\cos{k_{c}x}\\\
\hbar\hat{\Omega}_{\mathrm{SOC}}^{\dagger}\cos{k_{c}x}&\frac{1}{2M}\left(\hat{\mathbf{p}}-\frac{\hbar
k_{c}}{2}\mathbf{e}_{y}\right)^{2}\\\
\end{pmatrix}\hat{\Psi}(\mathbf{r})d\mathbf{r}\,,$ (102)

where $\hbar\tilde{\delta}$ is the effective two-level spin splitting and
$\hat{\Psi}=(\hat{\psi}_{\uparrow},\hat{\psi}_{\downarrow})^{\top}$ is a
spinor containing the atomic annihilation operators
$\hat{\psi}_{\uparrow,\downarrow}$. The strength of the two-photon Raman-Rabi
rate giving rise to the SO coupling is an operator, given by

$\displaystyle\hat{\Omega}_{\mathrm{SOC}}=\frac{\mathcal{G}(y,z)\Omega_{+}}{8\sqrt{2}\Delta_{a+}}\hat{a}^{\dagger}+\frac{\mathcal{G}(y,z)\Omega_{-}}{8\sqrt{2}\Delta_{a-}}\hat{a},$
(103)

where $\Delta_{a\pm}$ are the atomic detunings with respect to the two pump
fields. Considering the dispersion relation of both spin states in presence of
the SO coupling, one finds that the energy bands are displaced from one
another due to the opposite momentum recoils and coupled via the Raman-Rabi
term. This leads to a double-minima lowest energy-band structure and a band
gap given by $\hat{\Omega}_{\mathrm{SOC}}$ proportional to $\hat{a}^{\dagger}$
and $\hat{a}$, shown schematically in Figure 52(e) and (f).

Figure 53: Experimental realization of dynamical spin-orbit coupling. (a)
Cavity emission detected by single-photon counters (solid blue curve) and
optical power in the Raman beams (dashed black line), both as a function of
time. Steady-state SO coupling persists up to a few milliseconds. (b),(c)
Spin-resolved momentum distribution in time of flight, taken at the points
labeled in (a). All atoms are in the $\left|\downarrow\right>$ state (blue)
just below threshold (b). Above threshold (c), spin-up atoms (red) have
acquired a net momentum in the $y$ direction, as shown by the spin-colored
Bragg peaks at nonzero momenta. Also shown are second-order diffraction peaks
along the cavity direction due to the reverse Raman process. Figure adapted
and reprinted with permission from Ref. [72] © 2019 by the American Physical
Society.

The superradiant state displaying SO coupling corresponds in real space to an
atomic spinor state that exhibits a helical pattern (i.e., spin spiral) along
the pump axis according to $\left|\downarrow\right>\pm
e^{ik_{c}y}\cos(k_{c}x)\left|\uparrow\right>$. This state corresponds to the
chiral spin-spiral phase in Figure 35(b) and Figure 36(a). The experimental
observables of this state are the intra-cavity light field that rises to a
finite value above threshold and the spin-selective occupation of momentum
states according to the expectations described above; see Figure 53. The
lifetime of the superradiant state is only on the order of a few milliseconds,
most likely limited by a dephasing of the two pump beams.

###  8.2 Cavity-induced topological states

####  8.2.1 Su-Schrieffer-Heeger model and topological superradiant Peierls
insulator

The landscape of the superradiant optical lattice

$\displaystyle\hbar V_{\rm SR}(x)=\hbar
U_{0}|\alpha|^{2}\cos^{2}(k_{c}x)+2\hbar\eta_{0}(\alpha+\alpha^{*})\cos(k_{c}x),$
(104)

depends crucially on the sign of
$U_{0}\propto\Delta_{a}^{-1}=(\omega_{p}-\omega_{a})^{-1}$ as illustrated in
Figure 5. The lattice potential $\hbar V_{\rm SR}(x)$ is $\lambda_{c}$
periodic due to the interference term $\propto\cos(k_{c}x)$. For a
conventional red-detuned pump laser (and cavity field) with respect to the
relevant atomic transition $U_{0}\propto\Delta_{a}^{-1}<0$, the superradiant
lattice has a single global minimum in every unit cell of length
$\lambda_{c}$. More precisely, the unit cell contains two lattice sites with
an energy offset between them. For the blue-detuned repulsive case
$U_{0}\propto\Delta_{a}^{-1}>0$, in stark contrast $\hbar V_{\rm SR}(x)$ is
homopolar and its unit cell possesses two lattice sites with no energy offset
between them. In the deep lattice limit, this latter case maps to the Su-
Schrieffer-Heeger (SSH) model, a non-interacting 1D tight-binding fermionic
model with staggered (i.e., different intracell and intercell) hopping
amplitudes [312]. The SSH model has two distinct dimerizations: i) the
intracell hopping amplitude is larger than the intercell hopping amplitude,
ii) the intracell hopping amplitude is smaller than the intercell hopping
amplitude. In the former case, the system is a trivial insulator at half
filling. Whereas in the latter case, the Bloch bands have a nontrivial
topological structure, characterized by a non-zero bulk topological
invariant—the winding number—and topologically protected edge states.

The self-ordering of spin-polarized, low-field-seeking (i.e., blue detuned
$\Delta_{a}^{-1}>0$) fermionic atoms inside a linear cavity in 1D along the
cavity axis has been studied theoretically in Ref. [59]. The system is
described by the 1D limit of the effective Hamiltonian (2.1) with
$U_{0}\propto\Delta_{a}^{-1}>0$; see also Section 2.4.2. Note that the two-
body contact interaction is identically zero owing to the Pauli exclusion
principle and $V_{\rm ext}(x)=\hbar V(x)=0$.

At low pump strengths, the system is in the normal Fermi-gas state with no
photon inside the cavity. By increasing the pump strength, the system becomes
unstable toward the superradiant phase $\alpha\neq 0$, where the Fermi gas
crystallizes under the emergent superradiant potential $\hbar V_{\rm SR}(x)$,
Equation (104), in turn enhancing constructive photon scattering from the pump
laser into the cavity mode. Across the superradiant phase transition, the
relative phase of the cavity field with respect to the pump laser is locked at
either 0 or $\pi$ (under the assumption $\Delta_{c}\gg\kappa$), spontaneously
breaking the $\mathbf{Z}_{2}$ symmetry (i.e., the invariance under the
transformation $\alpha\to-\alpha$ and $x\to x+\lambda_{c}/2$) of the system.
That is, the system chooses spontaneously one of the two possible superradiant
lattice configurations shifted by $\lambda_{c}/2$ with respect to one another,
corresponding to the two distinct dimerizations of the SSH model.

The phase diagram of the system in the parameter space of the pump strength
$\eta_{0}$ versus the Fermi momentum $k_{F}$ at a finite temperature is shown
in Figure 54(a) [59]. The emergent lattice $\hbar V_{\rm SR}(x)$ opens up gaps
in the quadratic energy dispersion of the free Fermi gas across the
superradiant phase transition. For atomic densities around half filling
$k_{F}=k_{c}/2$, the superradiant state is an insulator, while for filling
sufficiently away from half filling the system corresponds to a superradiant
metal (SRM). The transition to the insulating state with a dimerized structure
at half filling occurs through the same mechanism underlying the Peierls
instability in 1D electron-phonon models [173]. Depending on the spontaneously
chosen dimerization via the superradiant $\mathbf{Z}_{2}$ symmetry breaking,
the superradiant insulating state is either a trivial band insulator (SRBI) or
a topological insulator (SRTI), characterized by the Zak phase. For this
system the Zak phase is $\mathbf{Z}_{2}$ quantized and serves as a topological
invariant, with the value 0 ($\pi$) corresponding to the SRBI (SRTI).
Remarkably, the value of the Zak phase coincides exactly with the relative
phase of the cavity field with respect to the pump laser, providing a
nondestructive way to read out the topological Zak phase via the leaked cavity
field. Note that the superradiant phase terminates at large pump strengths,
indicated by the gray area in the phase diagram of Figure 54(a); see also
Figure 20 for a similar behavior in self-ordering of low-field-seeking bosonic
atoms. We also note that the critical pump strength at half filling becomes
infinitesimally small at zero temperature due to the Fermi nesting [140].

Figure 54: Topological superradiant Peierls insulator in 1D spin-polarized
fermionic atoms inside a linear cavity in the blue-detuned repulsive regime
$U_{0}\propto\Delta_{a}^{-1}>0$. (a) The phase diagram of the system in the
$\eta_{0}$-$k_{F}$ parameter space at the finite temperature
$k_{B}T=0.01E_{r}$. Above the critical pump strength (the blue curve), the
Fermi gas crystallizes as a consequence of the superradiant phase transition.
The superradiant crystalline phase is either a metal (SRM) or an insulator.
Depending on the spontaneously chosen dimerization (i.e., the cavity field
phase) via the $\mathbf{Z}_{2}$ symmetry breaking process, the superradiant
insulating state is either a band insulator (SRBI) or a topological insulator
(SRTI), characterized by their Zak phases. (b) The imaginary part of the
optical polarizability $\chi_{\rm dyn}(\omega)$ for a finite-size system,
corresponding to the atomic absorption of the cavity photons. The edge states
within the energy gap open an additional absorption channel in a well-defined
frequency range (the red dashed curve). The inset shows the energy spectrum of
a finite-size system, with a pair of edge states within the superradiant-opned
energy gap. Figure adapted and reprinted with permission from Ref. [59] © 2017
by the American Physical Society.

The SRTI state possesses a pair of edge states within the superradiant-opened
energy band gap in a finite size system, another indication of the nontrivial
topology of the state, represented in the inset of Figure 54(b). These edge
states can be detected indirectly via the spectral properties of the cavity
output. Figure 54(b) shows for a finite-size system the imaginary part of the
optical polarizability $\chi_{\rm dyn}(\omega)$ [see Equation (32)],
corresponding to the atomic absorption with respect to the propagation of
cavity photons. The edge states modify the polarization function significantly
and open an additional absorption channel in a well-defined frequency range as
shown in Figure 54(b).

####  8.2.2 Spin-orbit-coupled-induced topological superradiant phase

A closely related scheme to the spinor bosonic self-ordering discussed in
Section 6 has also been considered for 1D spin-1/2 fermionic atoms in the
repulsive regime, i.e., where the pump laser and the cavity field are blue
detuned with respect to relevant atomic transitions [304]; see Figure 55(a)
and (b). The transverse pump laser and the cavity mode A couple the two ground
spin states $\\{\left|\downarrow\right>,\left|\uparrow\right>\\}$ via a
double-$\Lambda$ scheme (i.e., two independent Raman processes), while the
longitudinally pumped cavity mode B provides a background spin-independent
“classical” optical lattice potential $V_{\rm ext}(x)=V_{\rm
latt}^{(0)}\cos^{2}(k_{c}x)$ for the atoms. The system is described by a
special case of the Hamiltonian density (79),

$\displaystyle\hat{\mathcal{H}}_{1,\rm
eff}=\begin{pmatrix}\frac{p_{x}^{2}}{2M}+V_{\rm ext}(x)+\hbar
U(x)\hat{a}^{\dagger}\hat{a}+\hbar\delta&\hbar\eta(x)(\hat{a}+\hat{a}^{\dagger})\\\
\hbar\eta(x)(\hat{a}+\hat{a}^{\dagger})&\frac{p_{x}^{2}}{2M}+V_{\rm
ext}(x)+\hbar U(x)\hat{a}^{\dagger}\hat{a}\end{pmatrix},$ (105)

where $\hat{a}$ is the annihilation operator of the mode A,
$U(x)=U_{0}\cos^{2}(k_{c}x)$, and $\eta(x)=\eta_{0}\cos(k_{c}x)$, with $U_{0}$
and $\eta_{0}$ defined as before. However, note that here
$U_{0}\propto\Delta_{a}^{-1}>0$ as the pump laser and the cavity field are
assumed to be blue detuned with respect to the relevant atomic transitions.
The energy difference (i.e., the effective two-photon detuning) $\hbar\delta$
between the two spin states can be tuned via an external bias magnetic field,
where the latter also fixes the quantization axis along the $z$ direction.

Figure 55: Topological superradiant state in 1D spin-1/2 fermionic atoms
inside a linear cavity in the blue-detuned repulsive regime, $\Delta_{a}>0$.
The schematic representation of the setup (a) and the atom-photon coupling
scheme (b). The Bloch energy bands before (solid) and after (dashed) the
superradiant phase transition for $\delta<\delta_{c}$ (c) and
$\delta>\delta_{c}$ (d). For $\delta<\delta_{c}$, the superradiance-induced
gap opening in finite momenta marked by circles results in the band inversion.
(e) The phase diagram of the system at half filling and finite temperature
$k_{B}T=0.005E_{r}$ is represented in the $\eta_{0}$-$\delta$ parameter plane,
exhibiting metallic (M), insulating (I), superradiant (SR), and topological
superradiant (TSR) states. The various boundaries merge at a tetracritical
point (the purple dot). The insets show the evolution of the energy bands
(solid-dotted-dashed) across the phase boundaries indicated by arrows. Note,
in particular, that the band gap closes and re-opens across the TSR-SR phase
boundary. Figure adapted and reprinted with permission from Ref. [304] © 2015
by the American Physical Society.

At small pump strengths $\eta_{0}$, there is no constructive photon scattering
from the pump laser into the cavity mode, $\langle\hat{a}\rangle=0$.
Therefore, there is no (Raman) coupling between the two spin states and the
atoms only experience the background optical lattice potential $V_{\rm
ext}(x)$ and the Zeeman field $\hbar\delta$, with Bloch energy bands
corresponding to different spins being independent from each other. The normal
Fermi gas at half filling is either a gapless metal (M) or a gapped insulator
(I) depending on $\delta$; see Figures 55(c) and (d).

By increasing the pump strength above the critical value $\eta_{0c}(\delta)$,
the system enters the superradiant phase with $\langle\hat{a}\rangle\neq 0$.
As a consequence, the cavity-assisted space-dependent Raman processes (with
periodicity twice as the background lattice potential) induce a spin-orbit
coupling which mixes different spin Bloch bands and may open gaps in the bulk
energy spectrum within the first Brillouin zone; see Figures 55(c) and (d).
Depending on $\delta$, this interband mixing in the superradiant phase can
result in a band inversion, leading to topologically nontrivial states.

The steady-state phase diagram of the system at half filling in the
$\eta_{0}$-$\delta$ parameter plane is illustrated in Figure 55(e). In
particular, for $\delta<\delta_{c}$ the system is in the metallic state below
the superradiant threshold and a gap opens in the bulk energy spectrum across
the superradiant phase transition. Intriguingly, a pair of topological edge
modes with localized wave functions appears within the superradiant-opened
energy gap in a finite-size system. Furthermore, the topologically nontrivial
nature of this superradiant state (TSR) is confirmed by a nonzero bulk winding
number, a topological invariant counting the number of times the momentum-
space spin texture sweeps the circle $S^{1}$ (i.e., does a full $2\pi$
rotation) across the first Brillouin zone. In contrast, for
$\delta>\delta_{c}$ the system is in the insulating state below the
superradiant threshold and the trivial insulating gap persists across the
superradiant phase transition, resulting in a topologically trivial
superradiant (SR) state. However, by further increasing the pump strength the
trivial superradiant state crosses into the topological superradiant phase.
This is due to the increased interband mixing, which closes and then re-opens
the lowest band gap, resulting in a band inversion; see the inset of Figure
55(e-ii).

The topological phase transition across the SR-TSR boundary can be detected in
the abrupt change of the momentum-space density distribution of the spin-up
and spin-down components via a spin-selective time-of-flight imaging
technique. However, since the cavity field here is driven by the self-ordering
of the atomic spin, this provides a nondestructive tool to monitor the atomic
spin texture and the SR-TSR phase transition. In particular, the derivative of
the cavity photon number exhibits a pronounced peak across the SR-TSR phase
boundary.

##  9 Superradiant self-ordering with emergent discrete rotational symmetry:
quasicrystals

The preceding sections, especially Sections 3, 4, and 5, were focused on
cavity-QED systems with the discrete ($\mathbf{Z}_{2}$) and continuous [$U(1)$
and $SO(3)$] symmetries, and how the self-ordering of the atoms into
“crystalline” structures in the superradiant phase spontaneously breaks these
symmetries. We noted that these hybrid atom-cavity systems exhibit
fundamentally different steady-state phase diagrams, depending on underlying
symmetries and their _spontaneous breaking_. This is the common paradigm of
condensed-matter physics, described by Landau theory. That said, in rare
exotic situations in some condensed-matter models, global [313, 314, 315, 316]
and local gauge [317, 318, 319, 320, 321] symmetries _emerge_ in the proximity
of certain quantum phase transitions and/or in some quantum phases.

Recently, a novel many-body cavity-QED setup has been proposed in Ref. [322],
where a discrete eightfold rotational symmetry $C_{8}$ (i.e., a rotation by
45°) _emerges_ in the low-energy states across the superradiance quantum phase
transition. Since the eightfold rotational symmetry $C_{8}$ is not consistent
with any translational symmetry, this results in an emergent
“quasicrystalline” superradiant potential for the atoms in analogy to natural
quasicrystalline materials. Quasicrystals are quasiordered (i.e.,
orientationally ordered) materials with no translational symmetry, rather with
crystallographically forbidden rotational symmetries such as five-, seven-,
eightfold rotational symmetries, as discovered from their diffraction patterns
first by Shechtman et al. in 1984 [323].

###  9.1 Four crossed linear cavities

####  9.1.1 Superradiant phase transition

In the setup proposed in Ref. [322], four identical linear cavities are
arranged symmetrically in a plane with a common center such that they make a
45° angle with one another. A BEC is tightly confined in the direction
perpendicular to the plane at the intersection of these initially empty four
cavities, and is strongly coupled to an in-plane polarized mode of each cavity
$\hat{a}_{j}$. The BEC is also driven with the Rabi frequency $\Omega_{0}$ by
a spatially uniform, right circularly polarized pump laser propagating
perpendicular to the cavity-BEC plane as depicted in Figure 56(a). The
involved atomic internal states satisfy the magnetic selection rule $\Delta
m=m_{e}-m_{g}=1$. Generalizing the procedure of Section 2.1 (see also Section
5.1), in the dispersive regime of a large atom-pump detuning $\Delta_{a}$ the
system is described by the effective many-body Hamiltonian $\hat{H}_{\rm
eff}=\int\hat{\psi}^{\dagger}(\mathbf{r})\hat{\mathcal{H}}_{1,\rm
eff}\hat{\psi}(\mathbf{r})d\mathbf{r}+\hat{H}_{\rm
int}-\hbar\Delta_{c}\sum_{j}\hat{a}^{\dagger}_{j}\hat{a}_{j}$ with the
effective single-particle atomic Hamiltonian density in the rotating-frame of
the laser,

$\displaystyle\hat{\mathcal{H}}_{1,\rm
eff}=-\frac{\hbar^{2}}{2M}\boldsymbol{\nabla}^{2}+V_{\rm
ext}(\mathbf{r})+\frac{\hbar}{\Delta_{a}}\Big{|}\Omega_{0}+\sum_{j=1}^{4}\hat{a}_{j}\mathcal{G}_{j}(\mathbf{r})\Big{|}^{2},$
(106)

and the two-body contact interaction Hamiltonian $\hat{H}_{\rm
int}=g_{0}\int\hat{\psi}^{\dagger}(\mathbf{r})\hat{\psi}^{\dagger}(\mathbf{r})\hat{\psi}(\mathbf{r})\hat{\psi}(\mathbf{r})d\mathbf{r}$.
The atom-cavity couplings are given by
$\mathcal{G}_{j}(\mathbf{r})=e^{i\theta_{j}}\mathcal{G}_{0}\cos(\mathbf{k}_{j}\cdot\mathbf{r})$,
where $\mathbf{k}_{1}=k_{c}\hat{e}_{x}$, $\mathbf{k}_{3}=k_{c}\hat{e}_{y}$,
and $\mathbf{k}_{2,4}=k_{c}(\hat{e}_{x}\pm\hat{e}_{y})/\sqrt{2}$ with
$k_{c}=|\mathbf{k}_{j}|$ being the cavity (and pump laser) wave number and
$\hat{e}_{x}$ ($\hat{e}_{y}$) the unit vector along the $x$ ($y$) direction.
The phase factors $\theta_{1}=\pi/2$, $\theta_{2}=5\pi/4$, $\theta_{3}=\pi$,
and $\theta_{4}=3\pi/4$ arise due to the projection of the in-plane linear
polarizations of the cavity fields onto the right circular polarization [322].

The mean-field phase diagram of the system in the
$\\{Ng_{0}/\hbar\omega_{r}\lambda_{c}^{2},\sqrt{N}\eta_{0}/\omega_{r}\\}$
parameter space is presented in Figure 56(b), where
$\eta_{0}=\Omega_{0}\mathcal{G}_{0}/\Delta_{a}$ as before. Beyond the pump-
strength threshold $\eta_{0c}(g_{0})$, there is a quantum phase transition
from the normal homogeneous (NH) state into the superradiant phase
$\alpha_{j}=|\alpha_{j}|e^{i\gamma_{j}}\neq 0$, where not only are the
absolute values of all the field amplitudes equal
$|\alpha|\equiv|\alpha_{1}|=\cdots=|\alpha_{4}|\neq 0$, but also the phase of
each field amplitude is locked at $\gamma_{j}=\gamma_{0}-\theta_{j}$ or
$\gamma_{0}-\theta_{j}+\pi$, where $\gamma_{0}$ is a common phase introduced
due to the nonzero cavity-field decay rate $\kappa$. The nature of the phase
transition (i.e., first or second order) depends on the strength of the two-
body interaction $g_{0}$ and is indicated in the figure.

Figure 56: Self-ordering of a transversely driven BEC inside four crossed
linear cavities, leading to the _emergence_ of an eightfold ($C_{8}$)
quasicrystalline symmetry. (a) Schematic sketch of the setup. A 2D BEC is
transversely driven (in the $z$ direction) by a right circularly-polarized
laser (denoted by $\hat{\sigma}_{+}$) and is coupled dispersively to an in-
plane polarized mode of each cavity (denoted by $\hat{e}_{x,y}$ and
$\hat{\epsilon}_{\pm}$). (b) Phase diagram of the system as a function of the
strength of the two-body repulsive interaction $g_{0}$ and the pumping
strength $\eta_{0}$. Beyond the pump-strength threshold $\eta_{0c}(g_{0})$,
the system exhibits a (first- or second-order) quantum phase transition from
the normal homogeneous (NH) state into the self-ordered superradiant state,
where a quasicrystalline optical potential emerges for the atoms. Depending on
the strength of the two-body interaction, in the superradiant phase either the
condensate localizes in one or few of deepest minima of the quasicrystal
potential (denoted as SRL), or a quasicrystalline density order is stabilized
in the system (denoted as SRQC). (c) A typical emergent quasicrystalline
potential. The center of the quasicrystal is fixed via a process of the
spontaneous breaking of an approximate $\otimes_{j=1}^{4}\mathbf{Z}_{2}$
symmetry. Figure adapted and reprinted with permission from Ref. [322] © 2019
by the American Physical Society.

####  9.1.2 Emergent eightfold quasicrystalline rotational symmetry

The single-particle Hamiltonian density $\hat{\mathcal{H}}_{1,\rm eff}$,
Equation (106), possesses a symmetry denoted by $\tilde{C}_{8}$: an eightfold
rotation $C_{8}$ around the $z$ axis with the transformation $x\to
x^{\prime}=(x+y)/\sqrt{2}$ and $y\to y^{\prime}=(y-x)/\sqrt{2}$ followed by
the field transformation
$\hat{a}_{1}\to\hat{a}_{2}e^{-i(\theta_{1}-\theta_{2})}$,
$\hat{a}_{2}\to\hat{a}_{3}e^{-i(\theta_{2}-\theta_{3})}$,
$\hat{a}_{3}\to\hat{a}_{4}e^{-i(\theta_{3}-\theta_{4})}$, and
$\hat{a}_{4}\to\hat{a}_{1}e^{-i(\theta_{4}-\theta_{1})}$. Although
$\hat{\mathcal{H}}_{1,\rm eff}$ is not invariant under the sole action of the
$C_{8}$ symmetry, an eightfold rotational symmetry $C_{8}$ _emerges_ in the
low-lying energy states of the superradiant phase as a consequence of the
amplitude and phase locking of the cavity fields:
$|\alpha|\equiv|\alpha_{1}|=\cdots=|\alpha_{4}|\neq 0$, and
$\gamma_{j}-\gamma_{0}+\theta_{j}=0$ or $\pi$ [322]. This leads to the
formation of a quasicrystalline superradiant optical potential with a $C_{8}$
rotational symmetry for the atoms [see Figure 56(c)], where the location of
the $C_{8}$ rotational axis (i.e., the center of the quasicrystal) is
determined spontaneously via the phases $\gamma_{j}$ of the cavity
electromagnetic fields. The two possible choices of phase,
$\gamma_{0}-\theta_{j}$ or $\gamma_{0}-\theta_{j}+\pi$, for each field
amplitude correspond to an _approximate_ $\mathbf{Z}_{2}$ symmetry for each
cavity. The approximate nature of the $\mathbf{Z}_{2}$ symmetry is due to the
lack of a translational symmetry in the system. That is, a parity
transformation of a cavity field $\hat{a}_{j}\to-\hat{a}_{j}$ cannot be
compensated “exactly” by any translation because of incommensurate cavity
wavevectors along a given cavity axis. However, the parity transformation of
each cavity field can be “approximately” compensated by a translation, leading
to an approximate $\mathbf{Z}_{2}$ symmetry for each cavity. Therefore, at the
onset of the superradiant phase transition, the center of the emergent
quasicrystal is fixed through the spontaneous breaking of an approximate
$\otimes_{j=1}^{4}\mathbf{Z}_{2}$ symmetry. The sixteen
$\otimes_{j=1}^{4}\mathbf{Z}_{2}$-symmetry-broken states form the low-lying
energy sector of the system’s Hilbert space.

####  9.1.3 Localized state versus extended quasicrystalline state

In the weakly interacting regime, the quasicrystal potential dominates over
the two-body interactions. Consequently, the condensate localizes in one or
few of deepest minima of the quasicrystal potential since the quasicrystal
potential lacks a translation symmetry and can act as a disordered potential.
In this superradiant localized (SRL) phase the atoms scatter more photons from
the pump laser into the cavity modes [see Figure 56(b)] due to constructive
interference: the more atoms are condensed in the same potential minimum, the
more photons are scattered with the same phase. As the two-body contact
interaction is increased, it is energetically more favorable for the atoms to
occupy different global and local minima of the quasicrystal optical
potential. For sufficiently strong two-body contact interactions, the system
crosses over into an extended phase, where a quasicrystalline density order is
stabilized in the system. The momentum distribution in this superradiant
quasicrystalline (SRQC) phase exhibits a dense, fractal structure with an
eightfold rotational symmetry, a characteristic of quasicrystalline order
indicating that the two-dimensional momentum space is spanned by four
incommensurate cavity wavevectors $\\{\mathbf{k}_{1},\cdots,\mathbf{k}_{4}\\}$
[322]. We stress that the SRQC phase in this setup is a genuine many-body
state as the two-body repulsive interaction between the atoms is necessary for
its stabilization.

###  9.2 Different cavity setups and other quasicrystals

In the preceding section, we discussed a scenario where a quasicrystalline
superradiant optical potential with an eightfold rotational symmetry emerges
for ultracold atoms coupled to four-crossed linear cavities, where the atoms
in turn self-order in this emergent quasicrystalline potential. This proposal
is an extension of the ETH Zurich group’s two-crossed cavity setup (see
Section 4.1). Other superradiant quasicrystalline optical potentials with
different rotational symmetries can also be implemented by employing different
cavity setups, although they might become progressively more challenging
experimentally. However, they can provide a platform to study many fundamental
open questions concerning the formation and nature of quasicrystals. For
instance, it is still not completely clear whether quasicrystals are only
entropy-stabilized high-temperature states or can also be thermodynamically
stable at low temperatures. The conditions and nature of quasicrystal growth
are also under debate with a lack of a generally accepted model. Further
enriching the physics in cavity-based composite atom-light quasicrystals
compared to solid-state quasicrystalline materials is the interplay between
two-body contact and cavity-mediated long-range interactions, both tunable in
principle.

##  10 Superradiant self-organization without a steady state

So far we mostly considered situations, where atomic and photonic degrees of
freedom reach a steady state. In the present section we will instead discuss
some interesting quantum-gas–cavity scenarios with non-steady states. In order
to put the results presented in the following into the general context of
quantum many-body dynamics, some remarks are in order. First and foremost, a
hybrid atom-cavity setup is intrinsically an open quantum system due to the
photon leakage out of the cavity. This means that the dynamics always leads to
an attractor (which needs not to be unique), defined by the eigenvalue of the
master equation (8) with vanishing real part, i.e., with only damping. This
scenario has to be distinguished from the case of isolated systems, where the
dynamics are classified in terms of ergodic versus non-ergodic behavior. In
the context of quantum many-body systems, this characterization is still an
active field of research, with the phenomenon of many-body localization being
a prominent example [324].

Besides being dissipative, the atom-cavity systems we consider are also
driven, as external lasers in conjunction with cavity fields are used to
generate the dispersive light-matter coupling discussed throughout this
review. In this case, the long-time attractor does not necessarily need to be
a steady state, even though the Liouville operator governing the dynamics has
no explicit time dependence (in the frame rotating with the driving laser).
This situation will be considered in Section 10.1, while the case of
explicitly time-dependent Liouvillians will be discussed in Section 10.2.

Some of the non-steady states discussed below can be classified, in a broader
sense, as so-called time crystals. Such phases, which spontaneously break some
time-translation symmetry and show robust periodic oscillations in time, have
recently received considerable amount of attention, with examples from
different physical platforms [325]. We will also encounter in the following
non-steady phases which do not oscillate periodically in time but rather
exhibit irregular behavior, similar to the ergodic dynamics of isolated
systems. In the context of open quantum systems, the definition and
characterization of ergodic or chaotic behavior is still an open issue, with
very recent interesting developments [326, 327, 328, 329, 330, 331, 332, 333].
In this context, atom-cavity setups seem a very promising platform for further
investigation.

Before proceeding, let us note that in a driven-dissipative setting like the
ones considered here, even a steady attractor can be very nontrival, since it
is not in general of the thermal Gibbs type [128]. We refer here to the
discussion given in Section 2.6.

###  10.1 Non-steady-state phases with time-independent driving

In this section, we discuss two examples, where a stable, non-steady behavior
emerges without an explicit time-dependent external driving. This type of
phenomena shares strong similarities with limit cycles, quantum
synchronization, and some types of dissipative time crystals.

####  10.1.1 Atomic blue-detuned laser driving

A possible scenario through which stable non-steady attractors can emerge is
via the competition between different steady attractors. This can be achieved
in the simplest single-mode superradiant self-ordering scenario by driving the
atoms with a laser blue-detuned with respect to the atomic resonance,
$\Delta_{a}>0$, as first pointed out in Ref. [63]. Blue-detuned laser fields
lead to repulsive optical potentials but still allow for superradiant self-
ordering in some limited parameter regimes, as discussed in Section 3.3.3. To
better understand the phenomenon, instead of considering the two-dimensional
configuration of that section, we re-examine here the simpler one-dimensional
blue-detuned model introduced in Section 8.2.1, this time for bosonic atoms
instead of fermionic atoms.

The phase diagram of this 1D system is shown in Figure 57(a). The competition
which destabilizes the steady states can be immediately understood from
Equation (104), expressing the optical potential felt by the atoms in the
superradiant phase. For a laser which is blue detuned from the atomic
transition, one has $U_{0}\propto\Delta_{a}^{-1}>0$. This implies that the
$\cos^{2}(k_{c}x)$ part of the potential pushes the atoms out of the self-
ordered pattern induced by the $\cos(k_{c}x)$ part, thereby favoring a
homogenous density and thus the non-superradiant (normal) phase. At very small
photon numbers, the $\cos(k_{c}x)$ term dominates so that a steady-state
superradiant phase is stabilized. However, for sufficiently strong pumps the
$\cos^{2}(k_{c}x)$ term becomes equally important, so that the system cannot
decide between the normal and the superradiant phases. This leads to self-
sustained oscillations between these two states which are stabilized by the
cavity loss. These limit-cycle oscillations are shown in the middle panel of
Figure 57(b). By further increasing the pump strength the number of harmonics
increases by the known mechanism of period doubling, eventually crossing over
into irregular behavior exemplified in the lower panel of Figure 57(b). Within
a (semi-)classical description of the system based on coupled nonlinear
equations for the cavity coherent field and the BEC wavefunction (see Section
2.4.1) this irregular behavior can be associated with a chaotic attractor
[334]. Indeed, the dissipative, i.e., non-Hamiltonian, nature of the equations
of motion leads to the contraction of the phase-space volume to within a
finite region. Still, within the attractor region the dynamics is chaotic,
that is, it features at least one positive Lyapunov exponent.

Figure 57: Emergence of stable, non-steady-state behavior for a BEC inside a
cavity, driven transversely by a blue-detuned laser. A quasi-one-dimensional
BEC is trapped along the axis of a linear cavity and coupled to a single
standing-wave mode of it. The transverse, driving laser is blue detuned from
the atomic transition, $\Delta_{a}>0$. (a) Phase diagram of the system. While
the bare cavity detuning $\Delta_{c}$ is positive in the graph, the
dispersively-shifted detuning is still negative: $\delta_{c}<0$. The phase
diagram features the usual steady-state normal (N) and superradiant self-
organized (S-O) states, as well as non-steady-state superradiant self-
organized limit-cycle (S-O-L-C) and chaotic (S-O-Chaos) phases. (b) Time
evolution of the coherent cavity field in the three different self-organized
phases: S-O, S-O-L-C, and S-O-Chaos in order. (c) Time evolution of the BEC
density in the chaotic-like phase, where the white circles indicate the space-
time coordinates where solitons are nucleated. The parameters are
$U_{0}N=12.1\omega_{r}$ and $\kappa=10\omega_{r}$. Panel (c) also features a
finite short-range atom-atom repulsion $ng_{0}=\hbar\omega_{r}$. Figure
adapted and reprinted with permission from Ref. [63] © 2015 by the American
Physical Society.

What this classically known scenario implies for the full quantum state of the
system is an open question. Differently from the closed-system case, the study
of quantum dissipative chaos is still in its beginning and matter of active
research [327, 328, 335, 332, 333]. It is thus also an open question whether
this type of classical dynamics underlie some kind of ergodic or thermal-like
behavior of our quantum many-body system. This is a matter of future research,
for which the atom-cavity systems considered in this review article seem an
ideal platform. From the point of view of the atoms, the chaotic attractor
corresponds to a fast increase of the kinetic energy [63, 189], which reaches
a value much larger than that in the limit-cycle attractor. The chaotic
attractor is thus characterized by a more effective redistribution, i.e., a
behavior which is “more ergodic” than for the limit-cycle attractor. Still,
the increase in the kinetic energy is counteracted by the cavity loss via the
mechanism of cavity cooling, which eventually stops the heating [63].

A further interesting aspect connected with the chaotic attractor is the
phenomenon of phase slippage and related soliton nucleation in the BEC [63],
as shown in Figure 57(c). This behavior is peculiar to superfluids excited out
of equilibrium, whereby spatial gradients in the condensate phase are
converted into localized excitations carrying a phase slip. In the present
case the phase gradients are caused by the modulation of the cavity field and
have a characteristic length scale fixed by $\lambda_{c}$. On the other hand,
the characteristic size of a soliton is set by the healing length, whose value
$(2Mg_{0}n)^{-1/2}$ depends on the interatomic repulsion and is set to
$0.1\lambda_{c}$ in Figure 57(c). The main contribution to the BEC kinetic
energy can be attributed to soliton proliferation. The latter is counteracted
by cavity cooling, in a situation which resembles turbulence. This interesting
analogy should be further investigated in the future.

Figure 58: (a) Extension of the mean-field phase diagram of Figure 57(a) by
including non-mean-field atomic correlations as well as an harmonic trap for
the atoms. The dimensionless potential depth is defined as
$A=N\eta_{0}\sqrt{U_{0}/\omega_{r}}/\sqrt{(\Delta_{c}-NU_{0}B)^{2}+\kappa^{2}}$,
with the overlap $B=\int dxn(x)\cos^{2}(k_{c}x)$. Besides the normal (NP) and
self-organized superfluid (SSF) phases present in Figure 57(a), a further
mean-field-type phase appears as a self-organized dimerized superfluid (SDSF).
Moreover, beyond-mean-field phases are also stabilized: a self-organized Mott
insulator (SMI) phase [also present in the phase diagram of Figure 45(b)] and
self-organized second-order superfluid (2-SSF) phase. Additionally, non-
steady-state phases are also present: A quasi-periodic attractor (QA) related
to the limit-cycle phase of Figure 57(a) and a chaotic attractor corresponding
to the chaotic phase of Figure 57(a). (b) Pictorial representation of the
different steady-state phases. Figure adapted and reprinted with permission
from Ref. [189] © 2020 by the American Physical Society.

The Gross-Pitaevskii study of Ref. [63] has been extended to include non-mean-
field correlations between the bosonic atoms via a multiconfigurational time-
dependent Hartree method for indistinguishable particles (MCTDH-X) [189], as
well as to include the effect of harmonic trapping of the atomic cloud. The
phase diagram for the one-dimensional case is shown in Figure 58(a). A non-
mean-field, steady-state phase peculiar to this blue-detuned case is found. In
this so-called self-organized second-order superfluid phase, the phase
coherence is limited to within each double well, i.e., the unit cell of the
dimerized superradiant lattice. Within this beyond-mean-field description a
limit-cycle phase is also identified, with the only difference with respect to
the mean-field version discussed above being the quasi-periodic nature of the
oscillations, which has been attributed to the combination of the external
trap and the short-range atomic repulsion [189]. The chaotic attractor is also
present within this beyond-mean-field description, indicating that this
phenomenon is not an artifact of the (semi-)classical description. This opens
the interesting perspective for further investigating of this type of chaotic
phases in a fully quantum setting.

Figure 59: Time crystals in non-steady-state self-organized phases in the
repulsive, blue-detuned regime. (a) Temporal coherence $\mathcal{C}(t)$
(defined in the text) for the self-organized phases shown in the phase diagram
of Figure 57(a) [here the DW corresponds to the S-O phase in Figure 57(a)].
The emergence of temporal coherence in the limit-cycle (LC) phase is a
characteristic of time crystals. (b) The amplitude of the oscillations of
$\mathcal{C}(t)$ as a function of the pump strength across the three phases
(the LC phase extends roughly between 0.7 and 1.5). Here the simulations are
done for a two-dimensional BEC inside a standing-wave cavity pumped
transversally with a standing-wave laser. Adapted and reprinted with
permission from Ref. [67] © 2019 by the American Physical Society.

Further insight into the limit-cycle phase and its relation to time crystals
is provided by the analysis of the temporal coherence presented in Figure 59.
These results [67] are obtained using the truncated Wigner approximation,
which goes beyond the mean-field approximation by including the noise induced
by cavity loss [see Equation (10)], as well as the quantum noise in the
initial state. In Ref. [67], the latter has been included by occupying low-
lying BEC and cavity modes with vacuum fluctuations. Observables are then
computed by averaging over a set of stochastically-generated trajectories.
Even in the presence of these forms of noise, the temporal coherence of the
limit-cycle phase is clearly visible in Figure 59(a) as finite-amplitude
harmonic oscillations in the quantity
$\mathcal{C}(t)=\langle\alpha^{*}(t)\alpha(t_{1})\rangle$, where the averaging
is performed over an ensemble of trajectories. The emergence of temporal
coherence in the limit-cycle phase is a characteristic of time crystals. On
the contrary, the chaotic phase does not show appreciable temporal coherence.
The amplitude of the oscillations of temporal coherence is shown in Figure
59(b) as a function of the ramping of the pump strength from the self-
organized steady-state phase, across the limit-cycle phase over to the chaotic
phase.

We conclude this section by noting that the above described limit-cycle phase
has not been yet experimentally demonstrated, though experiments at the ETH
Zurich group have been carried out in the blue-detuned regime, as discussed in
Section 3.3.3. One possible explanation is the two-dimensional geometry of the
ETH Zurich experiment, which is different from the two-dimensional geometry
studied in Ref. [67] and presented in Figure 59, where the pump laser is
orthogonal to the cavity axis. Another possible explanation is the large value
of the cavity loss rate in the experiment, reducing the size of the limit-
cycle region in the phase diagram. Addressing this issue requires further
investigation.

####  10.1.2 Dissipation-induced instability in spinor BECs

Non-steady-state long-time dynamic behavior can also be induced by a
competition of coherent and dissipative couplings, as has been experimentally
demonstrated in Ref. [81] and theoretically discussed in Refs. [336, 337]. We
consider the model already introduced in Section 3.3.2.1, where a spinor BEC
with equal populations $N$ of the Zeeman states $m_{F}=+1$ and $m_{F}=-1$ is
coupled to a single standing-wave cavity mode. The linear polarization of a
standing-wave transverse pump field with lattice depth $V_{0}$ can be adjusted
with an angle $\varphi$ with respect to the $y$ axis [see Figure 18(a)], which
changes the ratio of scalar and vector polarizability of the illuminated
atoms. Generalizing the effective Dicke Hamiltonian of Equation (54) for a
single-component BEC to both Zeeman states, the many-body Hamiltonian of the
system can be expressed as

$\displaystyle\begin{split}\hat{H}=&-\hbar\delta_{c}\hat{a}^{\dagger}\hat{a}+2\hbar\omega_{r}\left(\hat{J}_{z,+}+\hat{J}_{z,-}\right)\\\
&+\frac{\hbar}{2}\left[\eta_{D}(\hat{a}^{\dagger}+\hat{a})\left(\hat{J}_{x,+}+\hat{J}_{x,-}\right)-i\eta_{S}(\hat{a}^{\dagger}-\hat{a})\left(\hat{J}_{x,+}-\hat{J}_{x,-}\right)\right].\end{split}$
(107)

The subscripts $\pm$ of the effective angular momentum operators
$\hat{J}_{x,\pm}$, $\hat{J}_{y,\pm}$, and $\hat{J}_{z,\pm}$ indicate the
Zeeman states $m_{F}=\pm 1$. If the two spin components occupy the same
checkerboard pattern, the expectation value for the density modulation
$x_{D}=(\langle\hat{J}_{x,+}\rangle+\langle\hat{J}_{x,-}\rangle)/N$ becomes
non-zero and the system self-organizes its density. If on the other hand the
two spin-components occupy the opposite checkerboard patterns, the expectation
value for the spin modulation
$x_{S}=(\langle\hat{J}_{x,+}\rangle-\langle\hat{J}_{x,-}\rangle)/N$ is non-
zero. The parameters $\eta_{D}=\eta_{s0}\cos\varphi$ and
$\eta_{S}=\eta_{v0}\sin\varphi$ [cf. Equation (54)] describe the coherent
coupling strengths to the density and spin modulated states, respectively. The
coupling with the density modulated state is mediated via the real quadrature
$(\hat{a}^{\dagger}+\hat{a})$ of the cavity field, which is thus
simultaneously occupied with a finite value of $x_{D}$. While the imaginary
quadrature $i(\hat{a}^{\dagger}-\hat{a})$ is occupied with the emergence of
the spin modulated state, $x_{S}\neq 0$. Without dissipation, this system
shows stable self-organization with either a density or a spin modulation
above a critical coupling strength, when either the eigenfrequency
$\omega_{D}$ of the density modulated state or $\omega_{S}$ of the spin
modulated state have softened to zero. Density and spin modulated states are
distinguishable via the phase of the cavity field; see Section 3.3.2.1.

Figure 60: Dissipation-induced dynamical behavior in a two-component BEC
coupled to a single mode of a linear cavity via both scalar and vector
polarizabilities. (A and B) Time evolution of the mean number of photons
(black) in the cavity and the corresponding phase of the light field modulo
$2\pi$ (yellow), while the transverse pump-lattice depth $V_{0}$ is ramped up
over time (dashed line). Panel (A) displays data for a polarization angle and
detuning where stable self-organization takes place, while panel (B) shows
data for parameters where a transition to a dissipation-induced instability
happens. Panels (C) to (E) show spectrograms of the mean intra-cavity photon
number as a function of frequency while the pump-lattice depth is increased
over time. The corresponding insets show the evolution of the state of the
system in the $x_{D}$-$x_{S}$ plane. Panel (C) displays the situation for
stable self-organization, and panel (D) for the dissipation-induced
instability with degenerate mode frequencies $\omega_{S}\approx\omega_{D}$.
The observation of a single red sideband corresponds to a linearly running
phase of the light field, or a circular motion in the $x_{D}$-$x_{S}$ plane.
Panel (E) shows the situation for non-degenerate modes, which leads to the
appearance of also a blue sideband and a corresponding elliptic motion in the
$x_{D}$-$x_{S}$ plane. Figure adapted and reprinted with permission from Ref.
[81] published in 2019 by the American Association for the Advancement of
Science.

A non-zero cavity-field dissipation rate $\kappa$ leads to a phase shift
$\phi_{\kappa}=\tan^{-1}(-\kappa/\delta_{c})$ of the light field scattered
from the pump field into the cavity mode. As a consequence, a pure density
modulation, for instance, thus leads to an occupation of both the real and the
imaginary quadratures of the cavity field. The light field in the imaginary
quadrature then drives self-organization of the spin mode. This process exerts
an effective force onto the system and triggers an instability constantly
driving it between density and spin modulation already far below the threshold
for self-organization. In the experiment, this was observed via a constantly
evolving phase of the intra-cavity light field and the associated emergence of
sidebands in the spectrum of the cavity field; see Figure 60.

Adiabatically eliminating the cavity field, linearized equations of motion for
the amplitudes $x_{D}$ and $x_{S}$ can be derived [81]:

$\displaystyle\frac{d^{2}}{dt^{2}}\begin{pmatrix}x_{D}\\\
x_{S}\end{pmatrix}=\begin{pmatrix}-\omega_{D}^{2}&-K^{2}\\\
K^{2}&-\omega_{S}^{2}\end{pmatrix}\begin{pmatrix}x_{D}\\\ x_{S}\end{pmatrix},$
(108)

where $(\omega_{D}^{2}-\omega_{S}^{2})\propto\sin\delta\phi$ with
$\delta\phi=2\tan^{-1}(-\eta_{S}/\eta_{D})-\pi/2$ captures the relative
coupling strength of the BEC to the density and the spin modulated state. The
important term is the dissipative coupling strength $K^{2}\propto
V_{0}\sin^{2}\phi_{\kappa}\cos\delta\phi$ which can be increased by increasing
the cavity-induced phase shift (via the detuning $\delta_{c}$) or by making
the two modes more degenerate (via the polarization angle $\varphi$ affecting
the ratio $\eta_{D}/\eta_{S}$). Expressing the state of the system by a vector
in a plane spanned by the two amplitudes $x_{D}$, $x_{S}$, this dissipative
coupling induces a force which is always orthogonal to the instantaneous
position vector of the system. Accordingly, solutions to the equations of
motion (108) show that the system undergoes amplified rotations with fixed
chirality in the $x_{D}$-$x_{S}$ plane. This mean-field description does not
fully cover the experimental observations. Specifically, the theoretical model
predicts an amplified motion, while in the experiment an evolution similar to
limit-cycles without increasing amplitude was observed. This behavior has been
attributed to collisional interactions of the atoms.

An analysis of the eigenspectrum reveals that the emergence of this dynamical
instability is connected to the presence of exceptional points with the
according level attraction as shown in Figure 61(b). This behavior is well
known for systems described by non-Hermitian matrices such as Equation (108).
After the instability has been induced, the two eigenmodes of the system
synchronize and oscillate at a mean frequency. It was shown that the
dissipative processes in the system generate a nonreciprocal coupling between
the two collective spins that eventually induces this instability in a certain
parameter regime, shown in Figure 61(a). Going beyond adiabatic elimination of
the cavity field, it was though shown that the instability should occur in the
long-time limit for basically all parameters due to additional anti-damping
from the cavity field fluctuations. In the bad-cavity limit that applies to
the experiment, the instability is, however, confined to a finite parameter
region [336].

Figure 61: Dissipation-induced dynamical behavior in a two-component BEC
coupled to a single mode of a linear cavity via both scalar and vector
polarizabilities. Panel (A) shows the boundary of the dissipation-induced
structural instability (DSI) regime as a function of dispersively shifted
cavity detuning $\delta_{c}$ and angle of the pump field polarization
$\varphi$. Data points indicate where the sidebands from Figure 60 become
dominant, the colored background is the result of a mean-field calculation.
Panel (B) shows the frequencies of the two eigenmodes in presence (solid
lines) and absence (dashed lines) of dissipation. Without dissipation the two
modes are the spin modulation (SM, blue) and the density modulation (DM,
green). Dissipation leads to level attraction and finally synchronization in
the instability dominated region (orange). The corresponding gains of the
amplified and damped modes are shown in black and grey, respectively. Around
the critical angle ($\varphi\approx 65\degree$), the eigenmodes become
degenerate, and the DSI extends to large absolute values of $\delta_{c}$.
Figure adapted and reprinted with permission from Ref. [81] published in 2019
by the American Association for the Advancement of Science.

An analysis studying the full quantum model of this system confirmed that the
inclusion of cavity field fluctuations renders the system unstable indeed also
in regions where the mean-field model predicts stability [337]. Using the
framework of Lindblad master equation, the eigenspectrum of the system was
studied. The interpretation was that the presence of equally spaced imaginary
parts of the eigenvalues prevents a dephasing of the dynamics and thus leads
to a long-time non-steady-state dynamics. This can be contrasted to steady-
state situations, where the eigenfrequencies are dense and incommensurate and
thus lead to dephasing as it would be the case for a closed system. An
analysis of the correlation functions of the system revealed indications of
beyond mean-field behavior such as entanglement induced by dissipation. The
described behavior can be connected to dissipative time crystals breaking a
continuous symmetry, where the description in a rotating frame is given by a
time-independent master equation [337].

###  10.2 Non-steady-state phases with time-dependent driving

Non-trivial dynamics without reaching a steady state can also be induced by
applying a time-dependent transverse pump lattice, modulated either in
amplitude or in phase; see Figure 62(a). Restricting the discussion to the
self-organization of the atomic density in the low energy sector, the
modulation results in a Dicke model with an explicitly time-dependent coupling
rate, even in the frame rotating with the natural pump frequency.

####  10.2.1 Time-modulated pump fields

First investigations of a parametrically driven Dicke model showed that
additional minima in the energy landscape of the system appear already for
infinitesimally small modulation if the modulation frequency is close to an
eigenfrequency of the undriven system [47]. For strong enough modulation, the
predicted phase diagram features metastable phases and according
nonequilibrium first-order phase transitions. Including dissipation both via
the cavity and via the atoms stabilizes the system against the instability
seen on resonance for the closed system, such that a finite modulation
strength is required to enter the non-steady-state phases [338]. This work
also investigated possible trajectories of the Bloch vector in the non-steady-
state phases. Examples of such trajectories are shown in Figure 62(b). For
non-steady states in the so-called “dynamical normal phase”, the phase of the
cavity field rotates in more or less complicated trajectories and the field
amplitude averages to zero. For the atomic degree of freedom this corresponds
to a quasi-resonant switching between the two ordered density patterns allowed
by the Z2 symmetry. Likewise, the superradiant phase displays oscillations,
however, around a mean non-zero photon number. The simulations also showed the
existence of chaotic behavior: While the normal and the self-organized phases
seem to be robust against parametric heating also in presence of the modulated
drive, the above introduced dynamical normal phase shows ergodic-like behavior
similar to the one characterizing the chaotic phases discussed in Section
10.1. Numerical simulations also taking into account atomic collisional
interactions and a trapping potential confirm the existence of the
parametrically driven phases [339].

The response of a BEC to a modulated field in the case of a cavity in the
sideband-resolved limit has been numerically studied using an open system
truncated Wigner approximation [66]. Also in this work, the pump field
amplitude was assumed to be modulated periodically. In presence of this
modulation, which effectively imprints two sidebands at the modulation
frequency onto the pump field, the system was found to either relax to a
steady state or enter dynamical states. Depending on the modulation frequency,
density waves involving different higher atomic momentum modes with multiple
recoils being transferred along the pump or the cavity direction can be
excited.

Figure 62: Parametrically driven Dicke model. (a) Sketch of the setup. A BEC
inside a dissipative single-mode cavity is driven transversely by a pump laser
with an oscillating intensity $V(t)$. The atoms are coupled to the environment
with a dissipation rate $\gamma$, and photons with a rate $\kappa$. (b)
Exemplary trajectories of the atomic order parameter on a Bloch sphere as a
function of time in the steady state for different driving parameters. Figure
adapted and reprinted with permission from Ref. [338] © 2015 by the American
Physical Society.

Another theoretical proposal for a periodically driven atom-cavity system is
based on a phase modulated rather than an amplitude modulated pump field
[340]. Such a phase modulation results in a periodically shaken optical pump
lattice which the atoms are exposed to. This configuration is expected to lead
to self-organization in specific density waves, depending on the modulation
frequency. Different to the case of an amplitude modulated pump field, here
the atoms can also order in a state with density maxima at the nodes of the
emergent superradiant optical lattice, dynamically switching between the
symmetry broken states. The induced oscillations can have a very low
frequency, set by the beating between the mode eigenfrequency and the
modulation frequency. This beating leads to a subharmonic response also at
incommensurate frequency ratios, and has been interpreted as incommensurate
time crystal.

Periodic phase modulation of the cavity-induced optical lattice can also
modify the properties of the superradiant phase transition. For instance,
periodic phase modulation of a cavity field has been predicted to yield a
first-order superradiant phase transition with a hysteresis loop, provided the
modulation frequency is chosen appropriately and the bosonic gas is
sufficiently-weakly interacting [341]. The periodic phase modulation of a
cavity field can also result in cavity-assisted long-range hoppings for
fermionic atoms in an external lattice inside the cavity, leading to
topological superradiant states [342].

####  10.2.2 Experimental realizations of time-modulated pump fields

The first experimental realization of a modulated atom-cavity system employed
the beating between two different pump-field frequencies [343]. One field is
red detuned with respect to the cavity resonance and drives the self-
organization phase transition in the system for sufficiently strong
amplitudes. The amplitude of the second pump field is increased from zero only
after self-organization has set in. The frequency of this second field is
closer to, and blue detuned with respect to the cavity resonance, such that it
counteracts atomic self-organization. Indeed, a suppression of the self-
organized density-wave state was observed with increasing amplitude of the
second pump field. This behavior was present for a wide range of modulation
frequencies. It was argued [66, 343] that this mechanism of suppressing the
self-organized density wave and enhancing atomic coherence in this setup bears
analogy with light-induced restoration of superconductivity in hight-$T_{c}$
cuprates. From dynamical point of view, this suppression of the self-
organization can be understood from (reduced) rescaling of the atom-field
coupling $\eta_{0}$ in the corresponding time-independent effective Magnus
Hamiltonian [66]. Accordingly, the self-organization threshold is pushed to
stronger atom-field couplings. Resonant features as reported in Ref. [338]
were not observed employing this scheme, except for rapid excitations if the
modulation frequency hit multiples of the vibrational frequency of the pump
lattice.

Figure 63: Dynamical self-organization of a BEC in a standing-wave cavity
driven transversely by an amplitude-modulated pump lattice. (a) The carrier of
the pump field $\omega_{p}$ is red detuned with respect to the dispersively
shifted cavity resonance at $\omega_{c,{\rm eff}}$. The amplitude modulation
leads to sidebands at frequencies $\omega_{p\pm}$. (b) The driving protocol is
shown in grey, where the pump lattice depth is first increased linearly such
that the atomic system self-organizes. Deep in the self-organized phase the
amplitude modulation is ramped up. The blue line indicates the mean intra-
cavity photon number. (c) Dynamical phase diagram showing the difference of
the occupations in a state induced by the modulation $F_{3,1}$ (a density wave
with three recoil momenta along the pump direction and one recoil momentum
along the cavity direction) and in the state induced by standard self-
organization $F_{1,1}$ (one recoil momentum along both pump and cavity
directions) as a function of modulation strength and modulation frequency.
(d-f) Absorption images of the atomic cloud after ballistic expansion for a
modulation frequency of $2\pi\times$$40\text{\,}\mathrm{kHz}$ and modulation
strength (d) $f_{0}=0.60$, (e) $f_{0}=0.47$, and (f) $f_{0}=0.10$. Figure
adapted and reprinted with permission from Ref. [344].

When an amplitude modulation was implemented experimentally by imprinting
instead two sidebands onto the pump field as suggested in Ref. [66], a
resonant feature was observed in addition to the suppression of superradiance
described above [344]. The experimental scheme and observations are displayed
in Figure 63. The pump intensity is modulated at a frequency $\omega_{d}$
according to $V(t)=V_{0}[1+f_{0}\cos(\omega_{d}t)]$, which imprints frequency
sidebands at $\omega_{p\pm}=\omega_{p}\pm\omega_{d}$. The higher frequency
sideband is close to and blue detuned with respect to the dispersively-shifted
cavity resonance. Operating in the sideband-resolved regime is important to
select specific atomic excitations [345]: In a certain range of modulation
frequencies $\omega_{d}$, a different set of momentum states is occupied than
for usual self-organization induced by an unmodulated pump field. For the
given parameters, this state corresponds to a transfer of three recoil momenta
along the pump axis and one recoil momentum along the cavity axis. Such
resonant occupation of specific momentum states, corresponding to a
nonequilibrium density wave, has also been theoretically predicted (see above)
if the modulation frequency is close to multiples of the atomic momentum-
excitation energy [66, 344]. Interestingly, the observed density wave is a
subradiant non-steady state where the cavity occupation is suppressed while
higher order momentum states are occupied. As shown in Figure 63(b), the
cavity field decays rapidly once the amplitude modulation is switched on. At
the end of the sequence absorption images are taken after ballistic expansion;
examples are displayed in Figures 63(d)-(f). Away from the resonance, momentum
peaks signaling standard self-organization are observed, while on resonance
the occupation of higher momentum states is visible.

Theoretical analysis provided together with the experiment shows that the
self-organization in this nonequilibrium density wave takes place at unstable
positions, and that the atomic pattern is expected to periodically switch
between symmetry broken states, similar to the discussion in Ref. [338]. The
theoretically predicted transition to chaotic behavior for increasing
modulation depth could not be observed experimentally due to driving-induced
heating and accordingly shorter lifetimes.

A recent experimental work [346] on time-modulated pump fields employed a
heterodyne detection scheme to gain access to the phase of the light field
leaking from the cavity. The observed period doubling of the atomic response
to the modulation was interpreted as a dissipative time crystal.

##  11 Cavity-enhanced quantum measurement in quantum gases

In the previous sections we demonstrated that ultracold quantum gases trapped
together with light within high-Q optical cavities constitute an incredibly
rich field of modern AMO physics with growing connections and application
prospects towards solid-state physics. In many respects the specific quantum
properties or quantum fluctuations of driving laser light fields and cavity
modes can be neglected and a mean-field description in terms of coherent
fields yields a valid and reliable description (see Section 2.4 for more
details). However, from a wider perspective quantum-gas cavity QED, i.e., the
combination of cavity quantum electrodynamics with ultracold quantum gas
physics, goes well beyond the mean-field picture of the coherent cavity-field
mode evolution. In its full blossom the field can address and study novel
phenomena, where the quantum statistical nature of both light and ultracold
matter play equally important roles. While the light fields create forces and
dynamical potentials with quantum properties, the atoms constitute a dynamical
refractive index with genuine quantum properties as well. This wider view has
several intriguing consequences as we will discuss in the following sections.

###  11.1 Projective measurements

Beyond the mean-field approximation, a quantum description of the
electromagnetic modes allows for dynamic field-matter entanglement beyond
classical correlations. Hence, a quantized cavity mode can still induce forces
and long-range interaction via photons even for zero average field-mode
amplitude. Consequently, this opens a completely new venue for projective
measurement-based preparation of special many-body states. In particular, a
wide class of emerging atomic states can be probabilistically prepared via the
choice of optical geometry. As generic basic examples, atom number-squeezed as
well as Schrödinger-cat states can be prepared in close analogy to cavity-
based spin squeezing [89].

Figure 64: Cavity-based non-destructive measurement scheme: Light scattered by
the quantum gas from a pump laser is collected and builds up in a cavity mode.
The field leaking through the cavity mirrors can be analyzed by various
detection schemes and lead to quantum measurement back-action on the atomic
dynamics. Figure adapted and reprinted with permission from Ref. [347] © 2009
by the American Physical Society.

As we generally deal with open systems with respect to the cavity-field
dynamics, the output light continuously provides information about the
intrinsic dynamics with minimal perturbation [347]; see Figure 64. Ultimately
it can serve as a quantum non-demolition (QND) probe of the spatial
distribution, phase diagram, or the phase-transition dynamics of the ultracold
quantum gas. As an example, a Mott-insulator state in an optical lattice
inside a cavity exhibits clearly distinct light scattering properties compared
to a superfluid state. In this way the corresponding phase transition between
the two states can be directly monitored from angle resolved scattering [84,
347] or cavity transmission measurements [348] as shown in Figure 65. The QND
detection of the Bose-glass phase [131, 349] and few-body complexes of polar
molecules [350] were suggested as well. In addition to being able to monitor
non-destructively the orbital state (i.e., Mott insulator or superfluid) of
one-component atoms in an optical lattice inside a cavity through cavity
output fields, it is also feasible to probe non-destructively the spinorial
state of multi-component atoms through cavity transmission spectrum. This
provides a novel approach for non-destructive detection of various magnetic
orders in quantum gases [351]. The QND property also allows for intriguing and
promising applications for quantum enhanced sensing as in, e.g., a supersolid-
based accelerometer in a ring cavity [90]; see Section 11.3 for more details.

Figure 65: Transmission spectra of a cavity filled with a BEC in an external
lattice. The transmission peaks directly map out the full atom-number
distribution of the lattice sites within the cavity mode as shown in Figure
64. (a) A single Lorentzian for a Mott-insulating (MI) phase reflects the non-
fluctuating atom number. Many Lorentzians for a superfluid (SF) phase indicate
atom-number fluctuations, which are imprinted on the positions of narrow
resonances in the spectrum. For a very narrow bandwidth cavity, i.e. when the
cavity decay rate $\kappa$ is smaller than the cavity light shift per atom
$U_{0}$, all cavity transmission are resolved. Here we see the example of a
superfluid of $N=30$ atoms in a lattive with $M=30$ sites, where $K=15$ sites
within the cavity are illuminated. (b) The same as in panel (a) but for a
worse cavity with $\kappa=U_{0}$, which gives a smooth broadened contour for
SF. Although the individual peaks are not resolved, the spectra for SF and MI
states are still very different. (c) Same spectra as above for SF with
$N=M=70$ ($\kappa=0.05U_{0}$) and different number of sites illuminated
$K=10,35,68$. The transmission spectra have different forms, since different
atom distribution functions correspond to different $K$. Adapted and reprinted
with permission from Ref. [348] published in 2006 by the Nature Publishing
Group.

From a fundamental physics point of view such systems allow detailed studies
of quantum measurement theory for many-body quantum systems ranging from the
quantum Zeno effect [352, 353] to quantum measurement-induced ordering [95].
Implementing weak measurement schemes on the field modes even allows
controlled steering of the atom-field quantum evolution [354], and generation
and observation of multi-partite entanglement via nonlocal cavity-enhanced
measurements [93]. The generation of squeezed light has also been suggested
[355]. As one might expect the genuine quantum nature of measurement back-
action also allows to generate dynamics beyond the mean-field level. For
fermions theoretical models show the appearance of measurement-induced
antiferromagnetic order [95] as well as the appearance of many-body
localization effects [74]. By trapping atoms inside an optical cavity, one
creates optical potentials and forces, which are not externally prescribed but
are quantum dynamical variables themselves. Ultimately, cavity QED with
quantum gases requires a self-consistent solution for light and particles,
which enriches the picture of quantum many-body states of atoms trapped in
quantum potentials. A proper quantum treatment of interaction and back-action
turns out to be particularly important to implement quantum simulations or to
exploit quantum-enhanced optimization using cavity-mediated tailored and
externally controlled interactions [28]. Interestingly, a mean-field treatment
of the fields here leads to a significantly reduced success probability
hinting for a real quantum advantage [48].

###  11.2 Measurement-induced dynamics and quantum feedback

One can make use of the continuously measured output-field properties for a
feedback on the coupled atom-field system. Applying weak measurements and
feedback to a quantum system induces phase transitions driven by fundamental
quantum fluctuations due to measurements. Non-Markovianity and nonlinearity of
feedback enables simulating special spin-bath problems and Floquet time
crystals with long-range and memory time interactions [356, 94]. As an example
we show in Figure 66 the simulation of quantum trajectories for a continuous
measurement on an ultracold gas in a lattice with continuous monitoring of the
cavity outputs [357]. The interplay between measurement projection and
nonlinear dynamics creates an intriguing dynamical pattern for the atom number
distribution on even or odd sites along the cavity axis.

Figure 66: Probability distribution of the population imbalance between odd
and even sites for single quantum trajectories (column1) and averages over 200
trajectories (column 2) for different values of the feedback gain. Column 3
shows the power spectrum averaged over 200 trajectories. In the absence of
feedback [panel (a)] the oscillations of the population between even and odd
sites are visible only in a single trajectory. For an increased feedback
strength $z$ beyond a creitical value $z>z_{c}$ [panel (b)] the imbalance
between odd and even sites soon becomes frozen for each individual quantum
trajectory . Below the critical feedback strength $z<z_{c}$ the frequency of
the oscillations can be tuned above [panel (c)] or below [panel (d)] the
frequency defined by the tunneling amplitude $J$. Again, the phase varies from
run to run and the oscillatory dynamics is visible only in a single quantum
trajectory. Adapted and reprinted with permission from Ref. [357] published in
2016 by the Optical Society.

Classical feedback derived from the output of a cavity field has been
experimentally demonstrated [358]. A BEC trapped inside a linear cavity was
brought to self-organization by applying a transverse pump field. Using the
field leaking out from the cavity as a signal, the intensity of the transverse
pump laser was regulated by an electronic circuit such that the mean intra-
cavity photon number was stabilized to a given value. While this photon number
decreases without feedback due to atom losses, applying the feedback scheme
allowed to stabilize the system over multiple seconds.

###  11.3 Quantum sensing with ultracold gases in cavities

Measurement-induced back-action and feedback are intriguing tools to study
genuine quantum physics in action and the measurement outcomes provide in-
depth information on the quantum gas dynamics [85]. Furthermore, as quantum-
gas–cavity systems are extremely sensitive to external perturbations, these
hybrid systems constitute a new class of quantum sensors [359, 360, 361, 92].
A remarkable example is a recently proposed supersolid-based gravimeter in a
ring cavity, which exhibits Heisenberg-like-scaling sensitivity with respect
to the atom number [90]. For state-of-the-art experimental parameters, the
relative sensitivity $\Delta g/g$ of such a gravimeter has been predicted to
be of the order of $10^{-10}$–$10^{-8}$ for a condensate of a half a million
atoms within a few seconds; see Figure 67. Detailed numerical studies on the
dynamical behavior of such ring-cavity systems were carried out in Ref. [362].
Recently, strong evidence for the creation of a supersolid in a ring-cavity
geometry as a prerequisite for such a sensor was observed in the experiment
[202] as discussed in detail in Section 4.3.

Figure 67: Supersolid-based gravimeter in a ring cavity. (a) Schematic sketch
of the setup. A transversely driven 1D BEC is coupled to two degenerate
counterpropagating modes $\hat{a}_{\pm}$ of a ring cavity. The cavity is
tilted with respect to the horizontal direction by an angle $\theta$ such that
the BEC experiences the gravitational potential $V_{\rm ext}(x)=Mgx\sin\theta$
(cf. Section 4.2 and Figure 27). The relative phase $\phi$ of the two cavity
modes is measured in the cavity output, in order to monitor the motion of the
BEC along the cavity axis under the gravitational force. (b) Relative
sensitivity for state-of-the-art experimental parameters as a function of
time. The red dashed curve presents the sensitivity for $\kappa=0$ (for the
sake of comparison) while the blue solid curve takes into account the
effective friction arising from photon loss $\kappa\neq 0$. Figure reprinted
with permission from Ref. [90] © 2019 by the American Physical Society.

Also the real-time monitoring of Bloch-Zener oscillations via the light field
leaking from a cavity has been theoretically discussed [363, 88, 364] and
experimentally explored [365, 104]. Such a method holds the promise to help to
increase the data acquisition speed in precision measurements of forces.
Recently, more refined modeling of the minimally invasive detection of quantum
atomic currents revealed the great potential of cavities to study the presence
and consequences of synthetic gauge fields as well as dynamical quantum phases
in particular for atomic models of quantum transport [86, 87]. First results
on real-time studies of $g^{(2)}$ correlations in the BEC-BCS crossover regime
revealed the great potential of this approach [62].

##  12 Conclusions, outlook, and open challenges

Starting from a short historical overview and an introduction of the basic
underlying physical models and mechanisms, the central goal of this review has
been to present the state of the art in the field of quantum-gas–cavity QED
(QG-CQED). As some comprehensive and nice reviews on the classical gas aspects
[114], simple cavity-lattice models [50], and the early quantum-gas
developments [96] already exist, for the sake of compactness here we
concentrated on more recent important developments and outstanding
achievements in the field. Nevertheless, the overwhelming recent growth in
theory and experiment has rendered the goal of being comprehensive, detailed
enough, and complete rather futile. Even well exceeding the initially planned
length of about 100 pages did not allow us to include everything, which would
deserve to be mentioned here in sufficient detail. Even while writing this
review article intriguing new developments continuously surfaced, we finally
had to strongly restrict ourselves and leave some of the most recent
developments for future reviews. Hence, we here have to apologize for some
important omissions. Nevertheless, we hope that on the one hand this review
gives a good and solid starting point to new researchers interested in
entering the field, and on the other hand fosters new and closer
collaborations among the existing diverse community.

Let us wrap things up shortly here: starting from early theoretical ideas and
abstract proposals somewhat in the style of Gedankenexperiments at the turn of
the century [366, 367, 368], the field of QG-CQED has enormously grown and
expanded in various directions. First aims were driven by theoretical visions
of cavity-induced ground-state cooling towards implementing a continuous BEC
and a continuous-wave atom lasers [369, 370], or creating macroscopic
superposition states of larger particle numbers [44]. Nevertheless, it soon
became clear that using a simpler implementation starting from a zero-
temperature degenerate quantum gas prepared by conventional means within an
optical cavity, one already realizes a very versatile and well controllable
test ground for many-body physics in an extreme quantum limit. Here cavity-
induced thermalization limits the available measurement time, but this process
is in many cases slow enough such that it does not hinder typical experimental
sequences. As a particular example, it proved straightforward to emulate a
wide class of optomechanical Hamiltonians at zero temperature [35, 33, 38], a
goal far out of reach of other optomechanical systems at that time [371].

Almost a decade after theoretical suggestions, a series of cutting edge
foundational experiments trapping a BEC within a cavity mode has been
implemented in Zürich [372], Tübingen [37], Berkeley [33], and Paris [34].
These allowed transforming Gedankenexperiments into real setups in
laboratories and thus sparked a great deal of interest well beyond the
ultracold-atom community, in particular, in the solid-state and quantum-
information communities. The subsequent multitude of theoretical models,
predictions, and suggested applications initiated another wave of experimental
efforts, encompassing BECs in recoil-resolved linear and ring cavities [102,
155], in multi-mode cavities [153], in crossed linear cavities [58], and in
folded ring cavities [373], as well as most recently also of degenerate Fermi
gases coupled to a linear cavity [61].

A second decisive boost to the field came from the discovery that cavity-
mediated global interactions can induce self-ordering of trapped atomic clouds
[106, 31]. Most interestingly, in the zero-temperature regime this corresponds
to a reversible quantum phase transition. In a seminal experiment, this self-
ordering phase transition was explored in real time [39]. Even more
interestingly, a direct mapping of the corresponding Hamiltonian to the Dicke
model allowed to confirm an almost 50-year-old prediction of the Dicke
superradiant phase transition [41, 39], a prediction that was controversially
disputed in literature for decades with its actual direct realization being
still disputed and sought for in solid-state platforms [157, 158].
Nevertheless, this proof-of-principle demonstration highlighted the remarkable
possibilities for QG-CQED as a versatile platform for quantum simulation of
many-body Hamiltonians with all-to-all interactions. With these possibilities
at hand a new wave of theoretical work was triggered.

With the subsequent extension and refinement of experimental apparatus,
experiments went well beyond the existing theoretical capabilities. Soon it
became clear that existing 1D mean-field models, which allow quite general
understanding of the emerging physical phenomena, could not adequately account
for all observations. This became particularly obvious for the case of 3D
intra-cavity lattices or setups with several cavity modes dynamically involved
[269, 49]. As a notable example, the clear experimental observation of
signatures of a lattice-supersolid phase required new theoretical approaches
such as dynamical mean-field theories [54], Keldysh-type approaches [79], or
multi-configuration time-dependent Hartree methods [166]. As experiments
become more complex but also more precise, these theoretical efforts strongly
grow in various directions such as, e.g., a thrust to make use of powerful
DMRG methods [133], which turns out to be rather delicate due to the
competition of local and cavity-mediated global interactions.

In a next phase theory and experiments pushed the field ahead further by
exploiting Zeeman sub-levels of atomic ground-state manifolds which added an
additional spin degree of freedom to the dynamics and paved the way for
exploring emergent magnetic orders in real time [68, 70, 71, 69]. Here the
dynamical nature of the cavity fields allowed to extend models for synthetic
gauge fields to the dynamical regime [72] previously inaccessible in free-
space configuration. New realms of physics also became accessible by
introducing repulsive optical potentials in the blue-detuned regime allowing
to emulate a nonequilibrium form of the Su-Schrieffer-Heeger model [59] as
well as entering the domain of nonlinear dynamical instabilities,
deterministic chaos, and a new arising area of time-crystal formation [63,
65]. Using complex resonator geometries even allowed to study quasicrystal
formation beyond naturally existing crystal symmetries [322].

Beyond studying fundamental aspects of quantum physics, the application
perspective of QG-CQED has been theoretically discussed already since the
early stage of the field as the extreme sensitivity of the systems to external
perturbations and the real-time observation capabilities hint for high
measurement precision and accuracy. In this context several new proposals for
acceleration, force, and magnetic-field sensors have recently been put forward
[90, 92] and are currently refined. In a parallel development the large
versatility of cavity-induced interactions, i.e., in principle almost
arbitrary forms of all-to-all couplings, allowed to envisage novel types of
quantum simulators [97, 109] or annealers [28, 99]. In some special cases it
could be shown that these implementations, which typically require a number of
laser frequencies, only require a minimal q-bit overhead so that
implementations beyond the capabilities of classical computers are within
reach of current experimental technology [48]. Recently, in an another
direction QG-CQED-based setups for active atomic clocks in the form of so-
called superradiant lasers were theoretically proposed [374, 375] and many
efforts started for their experimental implementation and technical refinement
[376, 377]. Here the cavity-induced all-to-all coupling among clock atoms is
centrally responsible for the synchronization of the atomic dipoles at very
low photon numbers in the spirit of the successful micro-maser-based clock
devices.

Let us also mention that cavity QED with quantum matter recently became
possible also in solid-state platforms [378, 379, 380, 381, 382, 383, 278,
279, 384, 281, 385, 386], potentially attracting the interest of an even
larger community. Those platforms partially differ from their ultracold-atomic
counterparts both in terms of models and parameter regimes. Nevertheless, a
large number of phenomenological features are still common, so that we believe
that this review article should be useful also for readers interested in
solid-state many-body cavity QED.

Although it is very difficult to make predictions about future directions, we
finally point out some recent promising developments and comment on a possible
road map. As a recent central advancement, first cavity-QED experiments with
fermionic atoms have recently been presented [61]. This will bring the field
even closer to emulate genuine solid-state Hamiltonians and condensed-matter
phenomena, such as momentum-space pairing in high-$T_{c}$ superconductors
[130, 277, 229]. With more and more quantum-gas species trapped within
resonators, one can also envisage multi-species configurations soon [73],
which will pave the way to molecule formation via dark-state-induced
collective photoassociation [387] and subsequently superradiant quantum
chemistry [388]. Important new possibilities will arise from more complex
resonator geometries covering two and three dimensions, as already
demonstrated in 2D [58], and by the help of higher-order transverse modes
[153]. Similarly, the whole manifold of longitudinal modes available in any
optical resonator opens the possibility to emulate inter-particle forces of
virtually arbitrary form. Here the natural frequency-periodic identical nature
of cavity and frequency-comb spectra can certainly help to reduce the
complexity of corresponding setups [29]. In addition, the combination of high-
resolution optical access with QG-CQED experiments may make new schemes
possible for the shaping of cavity-induced interactions, but also for the
simultaneous preparation of multiple atomic systems, or for the in-situ
imaging of self-ordered atomic structures. Finally, the open system physics of
non-steady states and in particular dynamic periodic solutions of many-body
systems, often called “time crystals,” has drawn strongly increased recent
attention. Here the nonlinear nature of the coupled atom-field dynamics and
real-time observation capabilities of the cavity fields together with the
excellent control over the system make QG-CQED setups a prime candidate for
corresponding experimental studies and tests [67, 346].

## Acknowledgements

We are grateful to Nishant Dogra, Peter Domokos, Francesco Ferri, Jürg
Fröhlich, Yudan Gao, Sarang Gopalakrishnan, Andreas Hemmerich, Jonathan
Keeling, Ronen Kroeze, Benjamin Lev, Brendan Marsh, Igor Mekhov, Heiko Rieger,
and Claus Zimmermann, for critical reading and valuable feedback on our
manuscript. We thank Jean-Philippe Brantut, R. Chitra, Tilman Esslinger, Petr
Karpov, Corinna Kollath, Giovanna Morigi, and Oded Zilberberg for stimulating
discussions. We also thank Elvia Colella for the assistance to create Figure
4.

## Funding

F. M. is supported by the Lise-Meitner Fellowship M2438-NBL of the Austrian
Science Fund FWF. F. M. and H. R. acknowledge the International Joint Project
No. I3964-N27 of the the Austrian Science Fund FWF and the National Agency for
Research of France ANR. T. D. acknowledges funding from the Swiss National
Science Foundation SNF: NCCR QSIT and the project “Cavity-assisted pattern
recognition” (Project No. IZBRZ2_186312). T. D. and H. R. acknowledge funding
from EU Horizon 2020: ITN grant ColOpt (Project No. 721465).

##  A1 Appendix: The most important and commonly-used symbols throughout the
paper

Symbol |  Definition
---|---
Fundamental constants
$c$ |  speed of light
$\epsilon_{0}$ |  vacuum permittivity
$h$ |  Planck’s constant, $\hbar=h/2\pi$
Properties concerning the atoms
$M$ |  atomic mass
$\mathbf{d}_{ge}$ |  matrix element of the induced atomic electric dipole moment for the transition $\left|g\right>\leftrightarrow\left|e\right>$
$\omega_{a}$ |  atomic transition frequency
$\Delta_{a}$ |  atomic detuning, relative frequency between the pump field and the atomic resonance: $\Delta_{a}=\omega_{p}-\omega_{a}$
$\gamma$ |  decay rate of the atomic dipole
$\Gamma$ |  natural linewidth of the atomic transition: $\Gamma=2\gamma$
$a^{s}_{\tau\tau^{\prime}}$ |  atomic s-wave scattering length between internal atomic states $\tau,\tau^{\prime}$
$g_{\tau\tau^{\prime}}$ |  strength of the two-body contact interactions: $g_{\tau\tau^{\prime}}=4\pi a^{s}_{\tau\tau^{\prime}}\hbar^{2}/m$
$N$ |  number of atoms
$T$ |  temperature of the atomic cloud
$V_{\rm ext}(\mathbf{r})$ |  external trapping potential
$\hat{\psi}_{\tau}(\mathbf{r},t)$ |  atomic annihilation field operator for internal state $\tau$
$\hat{b}_{j},\hat{c}_{j}(\hat{f}_{j})$ |  bosonic (fermionic) atomic annihilation operator for momentum state/lattice site $j$
$\psi_{\tau}(\mathbf{r},t)$ |  condensate wave function for internal state $\tau$
Properties concerning the pump field
$\omega_{p}$ |  frequency of the pump field
$\lambda_{p}$ |  wave length of the pump field: $\lambda_{p}\simeq\lambda_{c}$
$k_{p}$ |  wave number of the pump field: $k_{p}=2\pi/\lambda_{p}\simeq k_{c}$
$\boldsymbol{\epsilon}_{p}$ |  polarization vector of the pump field
$\omega_{r}$ |  recoil frequency: $\omega_{r}=\hbar k_{c}^{2}/2M$
$P$ |  optical power of the pump field
$\Omega_{0}$ |  maximum Rabi frequency of the pump field
$\Omega(\mathbf{r})$ |  position-dependent Rabi frequency of the pump field
$\hbar V_{0}$ |  maximum pump-lattice depth: $\hbar V_{0}=\hbar\Omega_{0}^{2}/\Delta_{a}$
$\hbar V(\mathbf{r})$ |  position-dependent pump lattice: $\hbar V(\mathbf{r})=\hbar\Omega^{2}(\mathbf{r})/\Delta_{a}$
$\eta_{0}$ |  maximum two-photon Rabi frequency: $\eta_{0}=\Omega_{0}\mathcal{G}_{0}/\Delta_{a}$
$\eta(\mathbf{r})$ |  position-dependent two-photon Rabi frequency: $\eta(\mathbf{r})=\Omega(\mathbf{r})\mathcal{G}(\mathbf{r})/\Delta_{a}$
Properties concerning the cavity
$R^{c}_{i}$ |  radius of curvature of mirror $i$
$l_{\mathrm{res}}$ |  resonator length
$g_{i}$ |  $g$-parameter of the cavity: $g_{i}=1-l_{\mathrm{res}}/R^{c}_{i}$
$w_{0}$ |  beam waist radius of the cavity mode: $w_{0}^{2}=(\lambda l_{\rm res}/\pi)\sqrt{\frac{g_{1}g_{2}(1-g_{1}g_{2})}{(g_{1}+g_{2}-2g_{1}g_{2})^{2}}}$
$\mathcal{V}$ |  mode volume of the cavity: $\mathcal{V}=\pi w_{0}^{2}l_{\mathrm{res}}/4$
$\mathcal{R}_{i},\mathcal{T}_{i},\mathcal{L}_{i}$ |  reflectivity, transmissivity, and losses of mirror $i$: $\mathcal{R}_{i}+\mathcal{T}_{i}+\mathcal{L}_{i}=1$
$\nu_{\mathrm{FSR}}$ |  free spectral range of the cavity: $\nu_{\mathrm{FSR}}=c/2l_{\mathrm{res}}$