This quantity was used as a second order parameters for the DPT. It should
cross from 1 (at small $B$) to 1/2 (at large $B$) with a dip at $B^{c}$.
However, given the logarithmic scaling of $C_{2}$ with $N$ at the critical
point and the limited system size of the systems under consideration, the
experiment did not observe sharp signatures in $C_{2}$ at the critical point.
Nevertheless, measurements of the distribution of domain sizes in the chain (a
direct measurement of higher-order correlations) accessible in the experiment
allowed it to observe a sharp change of behavior at the critical point.

DPT in the Lipkin–Meshkov–Glick model. After this work a similar type of DPT
was observed in two different platforms, more specifically the one in the so
called Lipkin–Meshkov–Glick (LMG) model (Eq. (2) in Sec. II.1), which
corresponds the $\alpha=0$ limit of the system described above with an
additional longitudinal field, i.e

$\hat{H}=\chi\hat{S}^{+}\hat{S}^{-}+B\hat{S}^{x}-\delta\hat{S}^{z}$ (50)

. Here we have introduced the collective spin operators
$\hat{S}^{\alpha}=\sum_{j}\hat{\sigma}^{\alpha}_{j}/2$ and
$\hat{S}^{\pm}=\hat{S}^{x}\pm i\hat{S}^{y}$. The summation runs over the
individual spins $j=1,...,N$ and the parameter $\chi$ sets the strength of an
infinite-range exchange interaction. To connect with the Hamiltonian, Eq. (2),
note that up to $1/N$ corrections
$\chi\hat{S}^{+}\hat{S}^{-}\approx\chi({\bm{\hat{S}}}\cdot{\bm{\hat{S}}}-(\hat{S}^{z})^{2})$.
Since the first term ${\bm{\hat{S}}}\cdot{\bm{\hat{S}}}$ is a constant when
restricted to the fully symmetric spin manifold, which is the case of
interest, the Hamiltonian simplifies to
$\hat{H}\to-\chi\hat{S}_{z}^{2}+B\hat{S}_{x}-\delta\hat{S}_{z}$, which up to
an overall $\pi/2$ rotation along the $y$ axis of the Bloch sphere, coincides
with the one realized in Ref. [54] with an extra term proportional to $\delta$
which sets the longitudinal field (See Fig. 10(b)).

The observation of a DPT in the Lipkin–Meshkov–Glick model was first achieved
in a cavity-QED simulator using ensembles of $N\approx 10^{5}$-$10^{6}$ atoms
[55]. We note that in the context of cavity QED great deal of experimental
progress has been done in the observation of non-equilibrium phases
characterized by different steady states which therefore depend on parameters
such as pump or loss rates but are independent of the initial conditions [282,
283, 284, 285, 286, 287, 288, 289]. We will not further discuss these
experiments in the following, as in this review we exclusively focus on DPTs
related to the unitary time evolution.

In Ref. [55] the internal spin degree of freedom was encoded in a long lived
optical transition (linewidth $\gamma/2\pi=7.5$ kHz) between the
$\ket{\downarrow}$ [1S0 ($m_{J}=0$)] and $\ket{\uparrow}$ [3P1 ($m_{J}=0$)]
states of 88Sr atoms. The atoms were confined in a 1D optical lattice and
coupled to a single common mode of the optical cavity far detuned from the
atomic transition. In this regime the photons can be adibatically eliminated.
As a result their role is reduced to mediate infinite range elastic spin
exchange interactions between the atoms, with strength set by $\chi$. The
transverse field was engineered by pumping the cavity with a laser that
generated, at resonance, Rabi flopping at frequency $B$. The longitudinal
field was simultaneously introduced by detuning the pump laser by a frequency
$\delta$ from the atomic transition.

Figure 10: Dynamical phase transitions (DPTs) observed in ultra-cold atomic
gases: a) The first DPT was observed in a trapped ion quantum simulator of the
long range Ising model plus transverse field, with up to 53 171Yb+ ions [54].
b) A similar type of DPT, but in the collective Ising limit plus an additional
longitudinal fields (so called LMG model), was also observed in a cavity QED
simulator with with $N\approx 10^{5}$-$10^{6}$ 88Sr atoms [55], followed up by
the observation of a similar DPT in a thermal gas of $N=10^{4}-10^{5}$ bosonic
87Rb atoms atoms using a sideband transition [56]. d) Using a quantum
degenerate trapped Fermi gas of $N=10^{4}$ 40K atoms the DPT between phase I
and phase II predicted to exist in BCS superconductors was observed in Ref.
[123]. e) Using three internal levels of 23Na atoms prepared in a Bose
Eisntein Condensate (BEC) with $N\sim 10^{5}$ atoms a DPT was observed as the
ratio between collective exchange interactions and an effective quadratic
magnetic field was tuned from positive to negative [290, 291].

For this system the time-averaged collective magnetization along $\hat{z}$ in
this case
$\overline{\langle\hat{S}^{z}\rangle}\equiv\lim_{T\to\infty}(1/T)\int_{0}^{T}\langle\hat{S}^{z}(t)\rangle
dt$ serves as an ideal order parameter as discussed in Sec. II.1. This is
equivalent to Eq. (5) up to an overall unitary rotation that maps
$S^{x}\leftrightarrow S^{z}$. However, in the experiment additional
inhomogeneities and other technical imperfections damped the oscillation in
the paramagnetic phase. Under these conditions the magnetization
$\langle\hat{S}^{z}\rangle$ after $4\mu$s of time evolution served as a proxy
of the long-term time-averaged magnetization. The experiment observed the DPT
in $\langle\hat{S}^{z}\rangle$ at $\delta=0$ using $B$ as the control
parameter and demonstrated the expected scaling with atom number. The DPT was
also probed by varying the longitudinal field for a fixed value of the drive
strength $B$ and by observing a sharp change of behavior of the order
parameter at the critical point. The robustness of the DPT was demonstrated by
the symmetric response of the magnetization for interaction shift $\chi
N\leftrightarrow-\chi N$. The experiment also explored the DPT as a function
of the initial state. It tracked the variation of the critical point as the
system was initialized in a coherent spin state pointing along different
directions.

The same LMG Hamiltonian was implemented in an experiment operating with a
thermal gas of $N=10^{4}-10^{5}$ bosonic 87Rb atoms in a 3D harmonic trap
using two hyperfine states to set a spin-1/2 degree of freedom [56]. The two
internal states were coupled by laser fields tuned to one of the blue side
transitions, i.e the laser not only generated a spin flip but at the same time
increased one of the motional quantum numbers in the trap by one unit. As a
consequence the two coupled states had different motional eigenstates. Even
though the interactions in this system are contact interactions during s-wave
collisions, effective long range interactions can due to the delocalized
nature of the single particle orbitals when operating in the collisionless
regime. This is achieved when the trapping potential is much larger than the
interaction strength [292, 293]. In this regime it can be assumed, to an
excellent approximation, that each atom is fixed in a mode-space lattice with
sites set by the single-particle eigenstates of the 3D harmonic trap. The only
relevant process between two colliding atoms is to either remain in the same
internal states or to exchange them. Even when dealing with bosonic samples
restricting the Hilbert space to include only empty or singly-occupied lattice
sites is appropriate when the gas temperature is above quantum degeneracy.
Under these conditions the contact interaction term can be mapped to a
spin-$1/2$ long-range XXZ spin model where the indices $i,j$ run over the
occupied mode-space lattice sites,

$H_{\mathrm{int}}=\frac{1}{4}\sum_{ij}J_{ij}\vec{\hat{\sigma}}_{i}\cdot\vec{\hat{\sigma}}_{j}+\frac{1}{4}\sum_{ij}\chi_{ij}{\hat{\sigma}}_{i}^{z}{\hat{\sigma}}_{j}^{z}+\sum_{i}B_{i}{\hat{\sigma}}_{j}^{z}.$
(51)

The XXZ couplings depend on the different scattering lengths of the atoms and
on the overlap integral of the corresponding 3D harmonic oscillator wave
functions. The use of a blue sideband transition that generate mismatched
motional states of the coupled internal levels allowed the experiment to
generate a finite $\chi_{ij}$. Otherwise it would have been negligible for Rb
if the experiment had used a carrier transition (when laser only flips the
spin). In addition to the interaction term, the interrogating laser generated
transverse and longitudinal fields with strengths set by the Rabi frequency,
and the laser detuning from the blue sideband transition.

To observe the DPT the atoms were initialized in the $\ket{\downarrow}$ state.
For this initial state, the exchange interaction term not only becomes a
constant of motion but also locks the atoms into the fully collective spin
manifold reducing the XXZ Hamiltonian to a LMG model with coupling constants
replaced by their averaged value. Similar to the cavity experiment, instead of
direct measurements of the long-time-averaged excitation which is inevitably
limited by technical issues, the order parameter was set to be the excitation
fraction at a probe time $0.5$s. The entire phase diagram was obtained by
scanning the two-photon detuning $\delta$ for fixed transverse field $B$ and
by varying interactions using different atomic densities (See Fig. 10(c)).

We note that there is a direct connection between the DPT in the LMG model and
the phenomena of macroscopic self-trapping and Josephson tunneling observed in
coupled atomic condensates [294, 295, 296, 297] and in solid state polariton
condensates [298]. In this context the ferromagnetic and paramagnetic
dynamical phases can be related to the self-trapped and tunneling phases
respectively in the corresponding systems. Under this correspondence one can
say that self-trapping experiments done years back did observe indications of
the distinct dynamical behaviors. However, they did not provide a full
characterization of the DPTs.

DPT in Richardson-Gaudin models.  A similar mapping of the single-particle
eigenstates of the harmonic trap onto a lattice in mode space, was used in a
trapped gas of $10^{4}$ ultracold fermionic potassium atoms. In this case the
experiment observed a DPT predicted to exist in quenched s-wave
superconductors described by Richardson-Gaudin models as explained in Sec.
III. As we will discuss below this DPT has only been indirectly observed in
real superconductors (see Sec. V.1). The model that was simulated in the
experiment [123] was a collective Heisenberg model, in which the non-local
spin-spin couplings $J_{ij}$ compete with an inhomogeneous axial field set by
$h_{i}$:

$\hat{H}=\sum_{i}h_{i}\hat{S}^{z}_{i}-\sum_{i,j}J_{ij}{\vec{\hat{S}}}_{i}\cdot{\vec{\hat{S}}}_{j}.$
(52)

The inhomogeneous axial field in mode space was generated by different
harmonic confinement potentials experienced by the two internal hyperfine
levels of 40K. The strength of the inhomogeneity was tuned in two ways. Using
the polarization of one of the laser beams forming the optical trap one change
the trapping frequency in a spin dependent manner. One can also change the
temperature and therefore the distribution of the modes occupied by the atoms.
Interatomic collisions were used to generate the exchange term. Due to the
extended nature of the motional wave functions, the $\mathcal{J}_{ij}$ were
again long-ranged. The mode-dependent coupling factor ${\mathcal{J}}_{ij}$ was
controlled by tuning the s-wave scattering length of the colliding atoms using
a magnetic Feshbach resonance [299]. The expected critical behavior was
observed in the experiment as a function of the mean and thermally averaged
interaction strength $J=\left\langle\sum_{i,j}J_{ij}/N^{2}\right\rangle_{T}$
and the axial field inhomogeneity
$\widetilde{h}=\left\langle\sqrt{\sum_{i}h_{i}^{2}/N-(\sum_{i}h_{i}/N)^{2}}\right\rangle_{T}$
where the indices $i,j$ run over $N$ populated modes (See Fig. 10d).

In the “all-to-all” limit, in which coupling constants are replaced by their
mean value, $J_{ij}\to J$, the Hamiltonian becomes integrable and maps to the
Bardeen-Cooper-Schrieffer Hamiltonian for fermionic superconductors expressed
in terms of Anderson pseudospin discussed in Eq. (14) with $h_{i}\to
2\varepsilon_{i}$ and $J\to G$. Note that at the level of mean field the extra
term $({\hat{S}}^{z})\to 2{\hat{S}}^{z}\langle{\hat{S}}^{z}\rangle$ acts as a
simple collective rotation along $z$ which does not modify the dynamics since
$\langle{\hat{S}}^{z}\rangle$ is conserved. Moreover this term can be removed
by choosing an initial condition with $\langle{\hat{S}}^{z}\rangle=0$.

Using the Lax analysis explained in Session III.2, it is possible to obtain
the dynamical phases of this model, characterized by the total transverse
magnetization
$\mathcal{S}(t)=2\sqrt{\langle{\hat{S}}^{x}\rangle^{2}+\langle{\hat{S}}^{y}\rangle^{2}}/N$.
The initial conditions accessible in experiment, corresponding to a spin
polarized state pointing along the $x$ direction of the Bloch sphere is very
similar to the simple example discussed in Sec.III.3, but in 3D and with a
different dispersion. In this case there is a DPT between the so called “phase
I” below a critical coupling strength $J_{c}$ where the order parameter quick
decays to zero and “phase II”above $J_{c}$ where the order parameter
$\mathcal{S}(t)$ exhibits transient oscillations at the frequency
$2|J|\mathcal{S}(\infty)$, which slowly damp as $\mathcal{S}(t)$ reaches
$\mathcal{S}(\infty)$ (See Eq. (23)). The oscillation frequency goes to zero
at $J_{c}$ in a non-analytic manner.

The experimental work described in Ref. [123] not only fully characterized the
phase diagram but in addition determined the parameter regime where the spin
model was valid. The latter was explicitly demonstrated via a many-body echo
sequence that fully reversed the spin Hamiltonian. Terms in the Hamiltonian
such as the kinetic energy, assumed to play no role in the frozen mode
approximation, were not reversed. The experiment indeed observed full
reversibility of the dynamics in the collisionless regime where the spin model
is expected to be valid, and non-reversibility otherwise.

Although the phase diagram was characterized for the first time in Ref. [123],
the understanding that exchange interactions can stabilize coherence in a very
useful way had been demonstrated experimentally years before. The experiments
were carried out using thermal Rb atoms, which feature a very similar
Hamiltonian than the potassium fermionic gas. Deep in phase II (dense sample)
a coherence time up to 58 s owas observed, while in phase I (dilute
conditions) the coherence time decreased to at most 3 s [300, 301, 302].

Although the Toronto experiment Ref. [123] was able to observe the DPT between
phase I and phase II, by simulating a spin model, phase III has not been
observed yet.

Fermionic ultracold atomic gas experiments that can actually take advantage of
the Fermi statistics of the atoms instead of an indirect mapping to Anderson
pseudospins, in principle are an ideal platform for the observation of the
dynamical phase diagram. However, unfortunately it has remained a challenge in
cold atom experiments to reach conditions required to see BCS pairing. This is
because both thermal and quantum fluctuations play a crucial role for quenches
from the normal state for systems with a finite number $N$ of Anderson
pseudospins. Moreover, the control and manipulation of interactions in these
systems, specially for the $p$-wave case, typically require Feshbach
resonances. The latter unfortunately introduce strong three-body processes
which make the gas unstable and destroy the desired pairing processes [167,
168, 169, 113].

Future progress on cooling these systems might allow new opportunities in that
direction. An alternative option is the use of an ensemble of cold atoms
trapped in an optical cavity. In this case effective Cooper pairs can be
encoded in internal states of the atoms and attractive interactions between
the internal levels can be realized via the exchange of virtual photons in the
cavity. The control of the interaction strength via the parameters of the
optical cavity combined with the tunability of the dispersion relation of the
effective Cooper pairs via Stark shifts should allow for the near term
exploration of the full dynamical phase diagram of the BCS model, as a
function of system parameters and the prepared initial state [124]. Ref. [124]
discusses a way to prepare the simple toy model discussed in Sec.III.3 in a
cavity setup.

DPT in multi-level systems.  Beyond two-level systems, DPTs have been also
observed in spinor Bose-Einstein condensates (BEC) made of $N\sim 10^{5}$ 23Na
atoms using the three Zeeman states of the lowest hyperfine level,
$m_{F}=1,0,-1$ [290, 291](See Fig. 10 e). The experiment operated in the
regime where to an excellent approximation atoms remained frozen in their
single particle motional levels. Motional relaxation was only an issue at long
times. Because all atoms in the BEC share the same motional level, in this
limit the bosonic statistics enforces atoms to remain the in the fully
symmetric spin manifold. The Hamiltonian that describes the internal dynamics
in the experiment therefore becomes a fully collective spin model:

$H=\frac{c_{2}}{2N}{\bm{\hat{S}}}\cdot{\bm{\hat{S}}}+q\sum_{i}(\hat{S}^{z}_{i})^{2}.$
(53)

The first term encapsulates the spin dependent interactions, which are
infinite-range in mode space even though the atomic collision interaction by
itself is fully local. The second term describes a quadratic Zeeman field $q$
experience by the atoms from a real magnetic field plus a microwave dressing
field. Here $\hat{S}_{z,i}$ is the spin-1 operator for the ith particle along
the $z$ component of the Bloch sphere.

To observe the DPT, the experiment prepared the majority of the atoms in the
$m_{F}=0$ level and then monitored the number of atoms remaining in that level
$N_{0}(t)$ as a function of time for different values of the control parameter
$q$. The order parameter used in the experiment was the a quantity $A_{\rm
dip}\equiv 1-N_{0}(t_{\rm dip})$, with $N_{0}(t_{\rm dip})$ being the value of
$N_{0}(t)$ at the first dip of the spin oscillations. For positive $q>0$,
$A_{\rm dip}$ was observed to remain almost at 0. Across the DPT transition
point of $q_{c}=0$, $A_{\rm dip}$ was seen to jump to finite value which
decreased linearly with $|q|$ when $q<0$ (See Fig. 10 e). Further work in this
system was able to connect the DPT to an equilibrium phase transition not for
the ground state but for the highest energy level in a subspace with zero spin
magnetization [291].

Besides observations of DPT which happened at relatively short times, a
follow-up experiment using Rb atoms instead of Na atoms observed another
complementary type of universal non-equilibrium behavior associated with the
emergence of non-thermal fixed points [226]. This type of behavior was
observed at intermediate times satisfying two conditions: i) Times that are
long enough to allow the frozen mode approximation used to observe the DPT to
become invalid. In this case the system loses the information about the
initial conditions due to motional relaxation. ii) Times that are not so long
that lead the system to reach a quasi-stationary or an equilibrium situation.
In this intermediate regime it was observed that the system develops a
universal scaling behaviour in time and space. Physically the emergence of
non-thermal fixed points have been attributed to the transport of an emergent
collective conserved quantity towards low momentum scales. A similar behavior
has observed by another experiment in a single-component Bose gas [227, 4]. In
both of these experiments it was experimentally confirmed (by preparing
different initial conditions and obtaining the same scaling behaviour) that
the observed non-thermal scaling phenomenon involved no fine-tuning of
parameters.

It is clear from all the discussion presented above that ultra-cold atomic
system are opening fantastic opportunities to probe DPTs and even more general
non-equilibrium universal phenomena in controllable settings. While most of
the observations so far have been guided by theory in conditions where either
a mean field analysis is sufficient, or when exact calculations are possible,
soon experiments will enter regimes intractable to theory. This may lead to
the discovery of new forms of non-equilibrium phenomena not yet predicted by
theory, pushing the field into even more exciting directions.

##  VI Conclusions

In this work we reviewed DPTs occurring in the large $N$, $\mathcal{N}$, or
$d$ limit of a broad variety of models ranging from statistical mechanics
(Ising models, $\phi^{4}$ field theories) to condensed matter (Richardson-
Gaudin magnets, strongly correlated fermions). We have also discussed numerous
experimental platforms where the dynamics of these models can be realized
(Sec. V). We have illustrated those regimes where experiments are well fitted
by the theory presented in Secs. II and III, as well as the limits where
descriptions beyond collision-less regimes may instead be required to describe
experiments. Such platforms include trapped ions, cavity QED systems, and
ultracold Bose and Fermi systems. They can be employed to demonstrate a broad
variety of universality classes of DPTs in isolated quantum systems, and
possibly study in the near future operational regimes where DPTs require a
full quantum mechanical treatment and cannot be captured by an effective semi-
classical description.

There are promising avenues for the future of pre-thermal collisionless DPTs.
For instance, Rydberg simulators have been recently shown to be capable to
access long-lived athermal states beyond the conventional pre-thermal paradigm
[303, 304], referred as quantum scars [305] and defying the eigenstate
thermalization hypothesis. This could be potentially employed in the near
future to study forms of DPTs which cannot be characterized within a simple
semi-classical description. Recent implementations of models with a fragmented
Hilbert space using ultracold fermions [306] suggest similar lines of
investigation in that context.

Another exciting direction, is the understanding of the fate of the DPTs in
systems where quantum fluctuations cannot be ignored. At the theoretical
level, it will be crucial to derive new types of order parameters that can
describe the distinct dynamical behavior even when quantum effects start to
play a role. The order parameters discussed in this review will decay in the
ordered phase whenever quantum effects start to play a significant role. Can a
properly defined order parameter, genuinely characterize the underlying phases
associated with a DPT? It is likely there is an affirmative answer to this
question. Feasible candidates are observables that measure the interaction
energy of the evolving system and therefore involve at the least two-body
correlators, as illustrated by the case of the O($\mathcal{N}\to\infty$) model
in Sec. II.3. Closely connected with this question is the robustness of the
topological order in non-trivial phases in the presence of quantum effects. Is
there a redefinition of the winding number that remains nonzero even in the
presence of strong quantum fluctuations?

One of the most important goals of modern quantum science is to learn how to
control and entangle many-body systems, and how to use it as a resource in
quantum technologies, such as for the development of improved quantum sensors,
materials and technologies. A big limitation in this direction comes from the
fact that entangled states are difficult to prepare and maintain, since noise
and decoherence rapidly collapse them into classical statistical mixtures. In
the context of DPTs it is likely that the same parameter regime where
interactions stabilize a finite order parameter is also promising for a
generation of robust entanglement. In fact, recent investigations have pointed
out the generation of robust spin squeezing in phase II [307]. Understanding
the dynamics of entanglement across a DPT is a fascinating new direction [308,
309]. Is entanglement maximum at the critical point? What is the best
entanglement witness for the state across the DPT?

While all these questions are theoretically extremely challenging, recent
experimental advances in synthesizing, manipulating and detecting quantum
many-body systems are bringing quantum control into a new paradigm [310, 304].
It is likely that near term experiments will stimulate new theoretical
methods, which may provide unprecedented insight into novel classes of DPTs
where strong quantum fluctuations play a key role in shaping the dynamical
phases. We hope this review will encourage future work and investigations in
this direction.

##  VII Acknowledgements

This project has been supported by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) – Project-ID 429529648 – TRR 306 QuCoLiMa
(”Quantum Cooperativity of Light and Matter”), ERC (starting grant No.
716648), AFOSR (FA9550-18-1-0319, FA9550-19-1-027), NSF QLCI OMA–2016244, NSF
Phys-1734006, DOE National Quantum Information Science Research Centers
(Quantum Systems Accelerator), and NIST.

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