osc}(E)$ by $\rho^{\rm osc}(E;N)$, Eq. (83). Encounters between different
(periodic) mean field modes take on the role of encounters between classical
(periodic) orbits. This implies the interpretation that interfering periodic
mean-field solutions of Eqs. (86) with quasi-degenerate actions $S_{\rm
pm}(E)$, Eq. (88), lead to the emergence of universal MB spectral
fluctuations, in close correspondence with the reasoning in the single-
particle case. This includes in particular the same RMT-type expressions for
the spectral MB two-point correlator $R(\Delta E;N)\sim\langle\rho^{\rm
osc}(E;N)\rho^{\rm osc}(E+\Delta E;N)\rangle$ and its associated spectral form
factor888Note that $R(\Delta E;N)$ contains interesting new parametric
correlations with regard to (changes in) particle number and or interaction
strength..

These conclusions drawn from the semiclassical MB encounter formalism coincide
both with long known results from nuclear shell models and data [6, 7],
embedded Gaussian ensembles [72], including those restricted to two-body
interactions, as well as with recent results showing random-matrix behaviour
of the spectral form factor for a periodically kicked Ising spin-1/2 model
without a semiclassical limit (to leading orders in $t/t_{H}$ [200, 201]) and
for a Floquet spin model [202]. Moreover, Wigner-Dyson-type level statistics
have recently been numerically shown with appropriate parameters for a
discrete Bose-Hubbard system [203, 85] and the SYK-model [68] in the large
$N$-limit. They help close an important missing theoretical link in the sense
that the k-body restricted interaction ensembles, i.e. embedded random matrix
ensembles, have resisted analytic proofs of their asymptotic behaviors [204,
205, 206] unlike the original classical random matrix ensembles. Semiclassical
results for spectral determinants [107, 31, 207, 59, 111, 108] in terms of
pseudo-orbits carry over to the MB case, i.e. the semiclassical finding [108]
that pseudo- mean field paths with durations $t>t_{H}$ necessarily must
involve multiple partial traversals that do not contribute to the MB spectrum.

It is worth, highlighting again the relevance of the scrambling time scale
${t_{\mathrm{E}}}$, Eq. (90), and the role of encounters for entangling mean
field modes: The semiclassical MB propagator and the trace formula, as sums
over mean field paths, contain genuine MB interference thereby giving rise to
MB correlations. The encounter calculus is involving ergodic averages
distilling out of all these paths, otherwise mostly random interference terms,
that prevail for an observable after energy or spatial averages. Each
encounter diagram, such as those shown in Fig. 6, represents all interferences
resulting from certain types of coupled mean field trajectory configurations
with quasi-degenerate actions. If we think of entanglement as coupling between
different product states, as mean field solutions are, then encounters
generate entanglement that resists (energy) averaging. After starting to
propagate a wave packet along a separable initial (periodic) mean field path,
it will split at an encounter, acting like a rail switch and entangling the
mean field paths that come close to each other in the encounter region in Fock
phase space. The time scale of this entanglement process is given by
${t_{\mathrm{E}}}$. Whereas the relevance of encounters for entanglement
arises naturally, developing tools to measure the degree of entanglement
created through encounter structures remains open. This would also distingish
encounter-mediated entanglememt growth from some sort of entanglement captured
through TWA approaches introduced in Sec. 3.1.2. Quantum unitaries acting as
interconnects in quantum networks can be viewed as mimicking certain encounter
structures of a quantum chaotic dynamical system. For such random unitary
dynamics entanglement growth has been measured [208].

To conclude, MB semiclassical methods developed for systems with underlying
chaotic dynamics can provide a direct theoretical derivation of universality
in the spectral statistics of large-$N$ MB systems, and the applicability of
RMT more generally. In the regime of exponentially unstable mean field chaos
in the classical limit, the local fluctuations of MB spectra comply with RMT
predictions. Though beyond the objective of this contribution, keep in mind
that MB semiclassical methods accomplish more than this. They apply to
individual systems and system specific non-statistical quantities as well. In
particular, if some system displayed unexpected properties from an RMT
perspective, it would still be expected that the semiclassical theory would
capture that behavior. This is especially true of system specific MB dynamics
before, up to, and just beyond the Ehrenfest time scale, a time scale that has
collapsed to zero in RMT.

###  3.3 Out-of-time-order-correlators and commutators

####  3.3.1 Concept and heuristic semiclassical reasoning

Based on correlations between periodic mean field modes and by invoking
ergodicity and classical hyperbolicity, it was just discussed how spectral
RMT-type universality can be semiclassically explained in the MB context. For
the GOE spectral form factor beyond Ehrenfest timescales, terms based on MB
interference and organized through a hierarchy of encounters of mean field
modes, provide higher-order ${\hbar_{\mathrm{eff}}}$-corrections to Berry’s
diagonal contribution. This section presents another prime example for ergodic
MB interference, so-called out-of-time-order correlators [87, 40, 41]

$F(t)=\langle\Psi|\hat{W}^{\dagger}(t)\ \hat{V}^{\dagger}(0)\hat{W}(t)\
\hat{V}(0)|\Psi\rangle\,;$ (91)

see Fig. 10, and their closely related relatives, out-of-time-order
commutators (OTOCs) [41]

$C(t)=2-{\rm
Im}(F(t))=\langle\Psi|\left[\hat{W}(t),\hat{V}(0)\right]^{\dagger}\left[\hat{W}(t),\hat{V}(0)\right]|\Psi\rangle\,.$
(92)

Both, $F(t)$ and $C(t)$ comprise two forward and two backward propagations by
means of the Heisenberg operator
$\hat{W}(t)=\exp{(-i\hat{H}t/\hbar)}\hat{W}(0)\exp{(i\hat{H}t/\hbar)}$.

Consider a Bose-Hubbard system in which the local measurement of an atom at a
given site perturbs (locally) the MB system. The squared commutator $C(t)$ of
such a suitable (local) operator $\hat{W}(t)$ with another (local) operator
$\hat{V}(0)$ measures the temporal growth of $\hat{W}$, including its
increasing complexity at another site at a certain distance. Hence the initial
local (quantum) information is spread and scrambled across the huge Hilbert
space of the interacting MB system with its vast number of degrees of freedom
[209], making OTOCs the measures of choice for quantifying growing complexity
and instability of MB systems, thereby with relevance for quantum computing
[210]. Although OTOCs require rewinding time and their implementation is
experimentally challenging, already several measurement protocols exist [211,
199, 212, 213, 214]. For a recent comprehensive tutorial on OTOCs; see [215]).

Figure 10:  Scheme of an out-of-time-order correlator — $F(t)$ can be viewed
as the overlap between two MB states arising from the operation of $V(0)$ and
$W(t)$ on $|\Psi\rangle$ in different time order, i.e., $F(t)$ contains four
non-time ordered operators.

OTOCs represent one of the most direct quantum analogues of classical measures
for hyperbolic instability of chaotic dynamics. For the single particle case
invoking a heuristic classical-to-quantum correspondence for small $\hbar$ and
replacing the commutator (92) for pre-Ehrenfest times by Poisson brackets
$\\{\cdot,\cdot\\}$ generates a leading-order Moyal expansion, e.g.
$\hat{W}=\hat{p}_{i}$ and $\hat{V}=\hat{q}_{j}$ [87, 41],

$C(t)\longrightarrow|\rmi\hbar|^{2}\left\\{p_{i},q_{j}(t)\right\\}^{\\!2}\\!\simeq\\!\hbar^{2}\\!\left(\frac{\partial
q_{j}(t)}{\partial q_{i}}\\!\right)^{\\!2}\propto\hbar^{2}\rme^{2\lambda
t}\,.$ (93)

The leading off-diagonal monodromy matrix element $\partial q_{j}(t)/\partial
q_{i}$ is replaced by an exponential growth determined by the classical
single-particle Lyapunov exponent $\lambda_{SP}$. This close quantum-to-
classical correspondence for quantum chaotic single-particle dynamics is
intriguing, and thus there is also the quest for establishing a corresponding
MB quantum-to-classical correspondence, i.e., a MB version of such a quantum
butterfly effect. This, in particular, includes an interpretation of the
quantum mechanical growth rate of OTOCs for MB systems with a classical limit.
Most notably their growth is bounded in time and OTOCs saturate due to a MB
interference mechanism setting in at the Ehrenfest time, i.e. scrambling time.
It gives rise to an interference term that is of the same order as the
corresponding diagonal contribution [86]. Such distinct features at
${t_{\mathrm{E}}}$ render OTOC evolution a hallmark of Ehrenfest phenomena.

####  3.3.2 Pre- and post-Ehrenfest times: exponential growth and saturation

To derive the OTOC pre-Ehrenfest growth and to illustrate these genuine MB
interferences consider again Bose-Hubbard systems describing $N$ interacting
bosons with creation (annihilation) operators $\hat{b}^{\dagger}$ ($\hat{b}$)
at sites $i\\!=\\!1,\ldots,L$. Evaluate the OTOC (92) for the position and
momentum quadrature operators, Eq. (41),

$\hat{q}_{i}=(\hat{b}_{i}\\!+\\!\hat{b}_{i}^{\dagger})/\sqrt{2N}\quad,\quad\hat{p}_{i}=(\hat{b}_{i}\\!-\\!\hat{b}_{i}^{\dagger})/(\sqrt{2N}\rmi)\,,$
(94)

which are related to the occupation operators $\hat{n}_{i}$ through
$(\hat{q}_{i}^{2}\\!+\\!\hat{p}^{2}_{i})/2={\hbar_{\mathrm{eff}}}(\hat{n}_{i}\\!+\\!1/2)$.
The OTOC, Eq. (92), reads

$C(t)\\!=\\!\Braket{\Psi}{\\!\left[\hat{p}_{i},\hat{U}^{\dagger}(t)\hat{q}_{j}\hat{U}(t)\right]\\!\left[\hat{U}^{\dagger}(t)\hat{q}_{j}\hat{U}(t),\hat{p}_{i}\right]\\!}{\Psi}$
(95)

in terms of the MB time evolution operator
$\hat{U}(t)=\exp(-\rmi\hat{H}t/\hbar)$. In Eq. (95) consider an initial
coherent state $\Ket{\Psi}$ localized in both quadratures.

The semiclassical derivation is based on approximating $\hat{U}(t)$ by its
asymptotic form for small ${\hbar_{\mathrm{eff}}}$, the MB version [189, 84],
Eq. (75), of the van Vleck-Gutzwiller propagator. The corresponding sum runs
over all mean-field solutions $\gamma$ of the equations of motion
$\rmi\hbar\partial\Phi/\partial t=\partial
H_{\mathrm{cl}}/\partial\Phi^{\ast}$ of the classical Hamilton function (50)
that denotes the mean-field limit of $\hat{H}$ for
${\hbar_{\mathrm{eff}}}=1/N\rightarrow 0$:

$H_{\mathrm{cl}}\left(\mathbf{q},\mathbf{p}\right)=\sum_{i,j}h_{ij}\Phi_{i}^{*}\Phi_{j}+\sum_{i,j,i^{\prime},j^{\prime}}V_{iji^{\prime}j^{\prime}}\Phi_{i}^{*}\Phi_{i^{\prime}}\Phi_{j}^{*}\Phi_{j^{\prime}}\,.$
(96)

In the coherent sum over mean-field paths in Eq. (3.1.1) the phases are given
by classical actions
$R_{\gamma}({\mathbf{q}}^{(\textrm{f})},{\mathbf{q}}^{(\textrm{i})};t)\\!=\\!\int_{0}^{t}\rmd
t[\mathbf{p}_{\gamma}(t)\cdot\mathbf{q}_{\gamma}(t)-H^{\mathrm{cl}}\left(\mathbf{q}_{\gamma}(t),\mathbf{p}_{\gamma}(t)\right)/\hbar]$
along $\gamma$, and the weights $A_{\gamma}$ reflect their classical
(in)stability.

In order to make a connection to RMT-type universality assume that the mean-
field limit exhibits uniformly hyperbolic, chaotic dynamics with the same
Lyapunov exponent $\lambda$ at any phase space point. Evaluating Eq. (95) in
position quadrature representation, inserting unit operators, and using Eq.
(3.1.1) for the propagator $K$ gives a general semiclassical MB representation
of the OTOC. To leading order in $\hbar_{\rm eff}$, the derivatives
$\hat{p}_{i}=-\rmi{\hbar_{\mathrm{eff}}}\partial/\partial q_{i}$ only act on
the phases $R_{\gamma}$ in $K$. Employing the relations
${\mathbf{p}}^{(\textrm{i})}_{\gamma}=-\partial
R_{\gamma}/\partial{\mathbf{q}}^{(\textrm{i})}$ generates for the OTOC [86],
Eq. (95),

$\displaystyle C(t)\simeq$
$\displaystyle\\!\\!\int\\!\rmd^{n}q_{1}^{\prime}\\!\int\\!\rmd^{n}q_{2}\\!\int\\!\rmd^{n}q_{3}^{\prime}\\!\int\\!\rmd^{n}q_{4}\\!\int\rmd^{n}q_{5}^{\prime}\Psi^{*}\\!\left(\mathbf{q}_{1}^{\prime}\right)\Psi\\!\left(\mathbf{q}_{5}^{\prime}\right)$
(97)
$\displaystyle\quad\times\\!\\!\\!\\!\\!\sum_{\alpha^{\prime}:\mathbf{q}_{1}^{\prime}{\rightarrow}\mathbf{q}_{2}}\quad\sum_{\alpha:\mathbf{q}_{3}^{\prime}{\rightarrow}\mathbf{q}_{2}}\\!\\!\\!\\!A_{\alpha^{\prime}}^{*}A_{\alpha}\rme^{(\rmi/{\hbar_{\mathrm{eff}}})\left(\\!-R_{\alpha^{\prime}}+R_{\alpha}\right)}\left({p}^{(\textrm{i})}_{\alpha^{\prime},i}\\!-\\!{p}^{(\textrm{i})}_{\alpha,i}\right){q}^{(\textrm{f})}_{2,j}$
$\displaystyle\quad\times\\!\\!\\!\\!\\!\sum_{\beta^{\prime}:\mathbf{q}_{3}^{\prime}{\rightarrow}\mathbf{q}_{4}}\quad\sum_{\beta:\mathbf{q}_{5}^{\prime}{\rightarrow}\mathbf{q}_{4}}\\!\\!\\!\\!A_{\beta^{\prime}}^{*}A_{\beta}\rme^{(\rmi/{\hbar_{\mathrm{eff}}})\left(\\!-R_{\beta^{\prime}}+R_{\beta}\right)}\\!\left({p}^{(\textrm{i})}_{\beta,i}\\!-\\!{p}^{(\textrm{i})}_{\beta^{\prime},i}\right){q}^{(\textrm{f})}_{4,j}\,.$

The four time evolution operators in Eq. (95) have been converted into
fourfold sums over mean-field trajectories of time $t$ linking the various
initial and final position quadratures. In the semiclassical expression, Eq.
(97), the operators $\hat{p}_{i}$ and $\hat{q}_{j}$ are replaced by their
classical counterparts ${p}^{(\textrm{i})}_{\gamma,i}$ and
${q}^{(\textrm{f})}_{\gamma,j}$. The commutators translate into differences of
initial momenta of trajectories not restricted to start at nearby positions.

The geometric connections amongst the trajectories quadruples involved are
sketched in Fig. 11. Panel (a) shows an arbitrary orbit quadruple and (b) the
corresponding diagram. Black and orange arrows refer to contributions to $K$
and $K^{\ast}$, respectively, i.e. forward and backward propagation in time.
The grey shaded spots mimic the initial state $|\Psi\rangle$.

|  |
---|---|---
(a) |  | (b)
|  |
(c) |  | (d)
|  |
(e) |  | (f)
Figure 11:  Configurations of interfering mean-field paths that contribute to
the OTOC $C(t)$, Eq. (97). (a) arbitrary trajectory quadruple and (b)
corresponding general diagram denoting forward and backward propagation along
black and orange mean-field paths. (c)-(f): Relevant configurations
contributing predominantly to $C(t)$: The trajectory quadruples reside (c)
inside an encounter (marked by dashed box), form a ”two-leg”-diagram with an
encounter (d) at the beginning or (e) at the end, or (f) build a ”four-
leg”-diagram with the encounter in between. (From Ref. [86].)

In the semiclassical limit
$R_{\gamma}({\mathbf{q}}^{(\textrm{f})},{\mathbf{q}}^{(\textrm{i})};t)\\!\gg\\!{\hbar_{\mathrm{eff}}}$.
Hence, the corresponding phase factors in Eq. (97) are highly oscillatory with
respect to initial and final positions. Thus, contributions from arbitrary
trajectory quadruples are usually suppressed whereas correlated trajectory
quadruples with action differences such that
$R_{\alpha}\\!-\\!R_{\alpha^{\prime}}\\!+\\!R_{\beta}\\!-\\!R_{\beta^{\prime}}\simeq
O({\hbar_{\mathrm{eff}}})$ are not averaged out and contribute dominantly to
$C(t)$. For post-Ehrenfest times these are quadruples where for most of the
propagation the four paths are pairwise nearly identical except for encounter
regions where trajectory pairs approach each other, evolve in a correlated
manner, and exchange their partners. The encounter calculus applies in the
high-dimensional phase space associated with the MB Fock space.

To leading order in ${\hbar_{\mathrm{eff}}}$, the relevant quadruples for
OTOCs involve a single encounter. These can be subdivided into four classes
depicted in Fig. 11(c)-(f). Diagram (c) represents a bundle of four
trajectories staying in close vicinity to each other throughout time $t$, i.e.
forming an encounter marked by the dashed box. This diagram turns out to be
dominant for times $t<{t_{\mathrm{E}}}$, Eq. (90), the time scale for
traversing an encounter region, if the associated action differences are of
order ${\hbar_{\mathrm{eff}}}$. Due to exponential divergences in chaotic
phase space the dynamics merges beyond the encounter boundary into
uncorrelated time evolution of the individual trajectories. However, the
symplectic Hamiltonian structure implies that the exponential separation along
unstable phase space manifolds is complemented by motion near stable
manifolds. This enables the formation of pairs of exponentially close
trajectories [30], e.g. paths $\alpha^{\prime}$ and $\alpha$ or $\beta$ and
$\beta^{\prime}$ in Figs. 11(d,f). This mechanism becomes quantum mechanically
relevant for times beyond ${t_{\mathrm{E}}}$; see the discussion in Sec. 2.3.
Here it is crucial for understanding post-Ehrenfest OTOC saturation. Panels
(c, d) display diagrams with an encounter at the beginning or end of two such
trajectory pairs. The diagrams in (f) are characterized by uncorrelated motion
of four trajectory pairs before and after the encounter.

The evaluation of Eq. (97) requires a thorough consideration of the dynamics
in and outside the encounter regions. Inside an encounter, Fig. 11(c), the
hyperbolic dynamics essentially follows a common mean-field solution:
linearization in the vicinity of one reference path allows for expressing
contributions from the remaining three trajectories. The detailed evaluation
of the diagrams (d-f) in Fig. 11 is given in Ref. [86]. It involves the
calculation of corresponding encounter integrals based on phase space averages
invoking ergodicity. Diagrams similar to class (f) have been earlier
considered in the context of shot noise [123, 124, 125] and observables based
on quantum chaotic single-particle [166] and MB [57] scattering. However, the
evaluation of such encounter integrals for OTOCs requires a generalization to
high-dimensional MB phase spaces. The occurence of operators (positions and
momenta) in the case of OTOCs demand a generalization of the encounter
calculus and special treatment, depending on whether the initial or final
position quadratures are inside an encounter.

Using the amplitudes $A_{\gamma}$ in Eq. (97) to convert integrals over final
positions into initial momenta, the OTOC contribution from each diagram is
conveniently represented as an ergodic phase-space average

$C(t)\simeq\int\rmd^{n}q^{\prime}\int\rmd^{n}p^{\prime}W(\mathbf{q}^{\prime},\mathbf{p^{\prime}})I(\mathbf{q}^{\prime},\mathbf{p}^{\prime};t)\,.$
(98)

Here,

$W(\mathbf{q}^{\prime},\mathbf{p^{\prime}})\\!=\\!\int\rmd^{n}y/(2\pi)^{n}\Psi^{*}\\!\left(\mathbf{q}^{\prime}\\!+\\!\mathbf{y}/2\right)\Psi\left(\mathbf{q}^{\prime}\\!-\\!\mathbf{y}/2\right)\exp[(\rmi)\mathbf{y}\mathbf{p}^{\prime}]$
(99)

is the Wigner function [190] and
$I(\mathbf{q}^{\prime},\mathbf{p}^{\prime},t)$ comprises all encounter
intgerals. The detailed evaluation of the encounter integrals $I$ represented
by the different diagrams in Fig. 11, and thereby $C(t)$, yields the following
results for pre- and post-Ehrenfest time evolution [86]: From diagram (c) it
follows for $\lambda^{-1}<t<{t_{\mathrm{E}}}$ upon ergodic averaging in the
semiclassical limit

$I(\mathbf{q}^{\prime},\mathbf{p^{\prime}};t)\simeq F_{<}(t)\quad;\quad
F_{<}(t)\approx
e^{2\lambda(t-{t_{\mathrm{E}}})}\Theta({t_{\mathrm{E}}}-t)={\hbar_{\mathrm{eff}}}^{2}e^{2\lambda
t}\Theta({t_{\mathrm{E}}}-t)\,.$ (100)

Diagram (d) turns out to be negligible, diagrams (e, f) together yield for
$t>{t_{\mathrm{E}}}$

$I(\mathbf{q}^{\prime},\mathbf{p^{\prime}};t)\simeq
F_{>}(t)\Braket{\left(p_{i}^{\prime}-p_{i}\right)^{2}}(\Braket{q_{j}^{2}}-\Braket{q_{j}}^{2})\quad;\quad
F_{>}(t)=\Theta(t-{t_{\mathrm{E}}})\,.$ (101)

Here,

$\Braket{f}=\frac{1}{\Sigma(E)}\int\rmd^{n}q\int\rmd^{n}pf(\mathbf{q},\mathbf{p})\delta\\!\left(E\\!-\\!\mathcal{H}^{\rm
cl}\left(\mathbf{p},\mathbf{q}\right)\right)$ (102)

is the ergodic average with $\Sigma(E)$ the phase space volume of the energy
shell at energy $E$.

Figure 12:  Universal contribution to the time evolution of out-of-time-order
commutators — Exponential increase according to $F_{<}(t)$, Eq. (100), before
and according to $F_{>}(t)$, Eq. (101), after the Ehrenfest time
${t_{\mathrm{E}}}=(1/\lambda)\log N$ marked by the vertical dashed line.
Insets depict diagrams (c), (f) and (e) from Fig. 11 representing interfering
mean-field trajectories. (From Ref. [86].)

The time-dependences of the universal functions $F_{<}$ and $F_{>}$ are
sketched in Fig. 12. For $t<{t_{\mathrm{E}}}$ the semiclassical evaluation for
MB systems confirms the heuristic result, Eq. (93). The careful treatment of
the encounter dynamics, diagram (c), provides a natural cut-off (exponential
suppression) at ${t_{\mathrm{E}}}$, absent in Eq. (93). It results from the
mechanism that the initial phase space area enabling four trajectories to stay
close to each other is exponentially shrinking for $t>{t_{\mathrm{E}}}$. The
fact that for $t<{t_{\mathrm{E}}}$ all four mean-field solutions essentially
follow in the linearizable vicinity of a common one, see diagramm (c),
indicates that the initial exponential increase of an OTOC of a chaotic MB
system can be considered as a property of unstable mean-field dynamics that
would also be captured by a truncated Wigner approach.

Figure 13:  Out-of-time-order commutator of a Bose-Hubbard (BH) system —
Numerically exact calculation of Eq. (92) for a BH system with four sites and
$N=40$ particles. The system is initially described by a coherent state
localized near a hyperbolic fixed point of the classical mean-field dynamics.
For the choice of parameters $J/NU\simeq\pi/2$ in Eq. (96) with hopping
$h_{i,j}=J(\delta_{i,j+1}+\delta_{i+1,j})$ and local interactions
$V_{i,j,i^{\prime},j^{\prime}}=U\delta_{i,j}\delta_{i^{\prime},j^{\prime}}\delta_{i,j}$,
the corresponding stability exponent is given by $(J/\hbar)\lambda$. For times
$1<\lambda t<\log(N)$ (time in units of the typical hopping time between sites
$\sim\hbar/J$) a clear exponential growth due to this local hyperbolicity can
be observed. (courtesy of Mathias Steinhuber).

On the contrary, the term $F_{>}(t)$ in Eq, (101) is suppressed for
$t\\!<\\!{t_{\mathrm{E}}}$, but is indeed responsible for OTOC saturation.
After the scrambling time $t>{t_{\mathrm{E}}}$ genuine MB interference sets in
captured by encounter diagrams such as in panel (f). This diagram represents
successive forward and backward dynamics swapping back and forth along
different encounter-coupled mean-field trajectories. This involves correlated
quantum MB dynamics and the temporal build up of entanglement between mean-
field modes. This mechanism is evidently in a regime where mean-field
approaches fail [216]. Thus, genuine MB interference is the quantum mechanism
behind the commonly observed saturation of OTOCs at the scrambling or
Ehrenfest time.

Note that the expression, Eq. (101), for the OTOC contains variances of
classical quantities, e.g. the variance of the $j$-th final position
quadrature that determine the OTOC saturation level. Here different types of
classical MB dynamics at post-scrambling times, e.g. diffusive versus chaotic
evolution, may lead to a different time-evolution of these classical
variances. As shown in [86], diffusive dynamics implies a linear increase with
time, whereas a calculation assuming ergodic dynamics yields $C(t)\approx
2/L^{2}$ for $t\gg{t_{\mathrm{E}}}$ with $L$ the number of sites of a Bose-
Hubbard system (for $|\Psi\rangle$ being either a localized wave packet or
extended chaotic MB state) corresponding to the flat plateau in Fig. 12.

Figure 13 shows the OTOC Eq. (92) with $\hat{V}=\hat{n}_{0}$ and
$\hat{W}=\hat{n}_{1}$ denoting occupation operators for adjacent sites
obtained quantum mechanically for a 4-site BH system with $N=40$ particles.
These numerics confirm the semiclassical predictions. Up to ${t_{\mathrm{E}}}$
the OTOC increases exponentially with slope $2\lambda$ where $\lambda$ agrees
with the Lyapunov exponent of the (here locally) unstable MB mean-field
dynamics of the specfifc BH system. At $t\simeq{t_{\mathrm{E}}}$ saturation
sets in.

The present semiclassical analysis of MB OTOCs in the large-$N$ limit, the
vertical limit in Fig. 2, can be readily generalized to systems of $N$
particles in $d$ spatial dimensions in the complementary limit of small
$\hbar$, in particular to the quantum chaotic single-particle case. Invoking
the corresponding Gutzwiller propagator, Eq. (3), in $n=d\times N$ dimensions
the exponential increase of the OTOC $C_{N}(t)$ in Eq. (100) is then governed
by the leading Lyapunov exponent $\lambda_{N}$ of the corresponding classical
$N$-particle system. Saturation sets in at the corresponding Ehrenfest time
$(1/\lambda_{N})\log S/\hbar$ with $S$ a typical classical action. For a
chaotic phase space average the usual semiclassical limit the saturation value
$C(t)\approx\hbar^{2}N/d$ results [86]. For $N=1$ this short-time growth with
$\lambda_{1}$ has also been independently semiclassically derived in [217,
218]. The exponential increase and saturation of such single-particle OTOCs
was considered in detail in numerical case studies of the kicked rotor [219]
and quantum maps [220].

####  3.3.3 Extensions

There are two further interesting extensions of semiclassical results to
summarize for MB OTOCs. First it is worth considering how the time dependence
of the OTOC changes for open MB quantum systems, specifically for $N$-particle
systems where each particle has a large but finite average dwell time
${t_{\mathrm{D}}}>{t_{\mathrm{E}}}$ to stay in the system. The corresponding
classical decay can be incorporated into the encounter integrals $I(t)$ in Eq.
(98) by means of terms $\sim\exp{(-t/{t_{\mathrm{D}}})}$. Most notably, their
individual consideration in the encounter diagrams 11(c) and (d-f) for
$t<{t_{\mathrm{E}}}$ and $t>{t_{\mathrm{E}}}$, respectively, yields, to
leading order, different decay rates [221]

$\displaystyle F_{<}(t)$ $\displaystyle\sim e^{(2\lambda
t-t/{t_{\mathrm{D}}})}\Theta({t_{\mathrm{E}}}-t)\ ,$ (103) $\displaystyle
F_{>}(t)$ $\displaystyle\sim\ e^{(-2t/{t_{\mathrm{D}}})}\
\Theta(t-{t_{\mathrm{E}}})\,,$ (104)

as depicted in Fig. 14(a). The non-trivial, doubled decay rate
$2/{t_{\mathrm{D}}}$ in the regime of MB interference arises from the
structure of the corresponding encounter diagrams (d-f) containing two
independent “legs” (trajectory pairs) of length $\sim t$ that can lead to
particle decay, compared to correlated dynamics centered around one path of
length $t$ inside the encounter, diagram (c). Its experimental observation
would clearly indicate this subtle and possibly unexpected aspect of MB
interfernce.

Second, it is of interest to consider $k$-th order generalizations of the
usual OTOC [222], i.e. $k$-OTOCs

$C_{k}(t)=\langle\Psi|\left[\hat{p}_{i}(0),\hat{q}_{j}(t)\right]^{k}|\Psi\rangle\,.$
(105)

Note that this definition does not contain absolute values as in the
definition, Eq. (92), of the usual OTOC, i.e. $C(t)\neq C_{2}(t)$.
Generalizing the semiclassical encounter calculus to the case of a $k$-OTOC,
$k-1$ encounters can be placed into a trajectory structure comprising $2k$
paths. Careful evaluation [221] of the leading-order encounter diagrams
suggests a stepwise structure for $C_{k}(t)$ with stepsize ${t_{\mathrm{E}}}$
as visualized in Fig. 14(b).

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---|---|---
(a) |  | (b)
Figure 14:  Generalized OTOCs — (a) Semiclassical prediction for the OTOC
$C(t)$ of an open chaotic MB quantum system with decay time
${t_{\mathrm{D}}}>{t_{\mathrm{E}}}$. For $t<{t_{\mathrm{E}}}$, the classical
regime governed by a dominant mean-field path, the exponent is diminished by
the rate $1/{t_{\mathrm{D}}}$, while in the regime of genuine MB quantum
interference ($t>{t_{\mathrm{E}}}$) the OTOC exhibits an exponential decay
with twice that rate, $-2/{t_{\mathrm{D}}}$. (b) Sketch of the semiclassical
prediction for the $k$-OTOC $C_{k}(t)$, Eq. (105). Characteristic steps are
expected at multiples of the scrambling time ${t_{\mathrm{E}}}$. (From Ref.
[221] with permission.)

Returning to the usual OTOCs, ergodic quantum MB systems with a chaotic
classical limit $\hbar\rightarrow 0$ or ${\hbar_{\mathrm{eff}}}\rightarrow 0$
show an OTOC growth with exponent $2\lambda_{N}$ or $2\lambda$, respectively.
However, an exponentially increasing OTOC does not necessarily imply chaotic
dynamics, i.e. OTOCs are not necessarily indicative of chaos. Important
exceptions are large-$N$ MB systems (near criticality) where their quantum
dynamics are accompanied by unstable fixed points (separatrices) in the
associated MB mean-field dynamics that can even be integrable. The local fixed
point instability $\lambda_{s}$ also leads to an exponential increase
$C(t)\sim e^{2\lambda_{s}t}$ up to times ${t_{\mathrm{E}}}$ [223, 224]. Thus
scrambling does not necessarily imply chaos. In such systems the quantum
critical states may be viewed as residing close to separatrices that have much
in common with encounters. However, due to the integrable classical limit, the
fast initial scrambling is followed subsequently by oscillatory behavior
between reentrant localization and delocalization of information in Hilbert
space [223].

For such quantum critical systems the semiclassical leading-order
$1/N$-expansion has been refined providing $e^{2\lambda t}/N$ (and not $1/N$)
as a renormalized parameter that non-perturbatively rules the quantum-
classical transition [225]. For scrambling in many-body hyperbolic systems
this provides formal grounds for a conjectured and numerically observed multi-
exponential form of OTOCs for the SYK-model [226].

Apart from quantum chaotic and critical large-$N$ systems, the imprints on
OTOCs of non-ergodic dynamics, for example, a mixed (regular-chaotic) phase
space in the classical limit, have also been recently considered [227], but
are not in the focus of this review.

##  4 Perspectives

During the past twenty years considerable progress has been made and various
breakthroughs have been achieved in laying the foundations of a semiclassical
theory that successfully provides the understanding of quantum chaotic
universality, or more precisely, of universal (spectral) features of SP and MB
quantum systems exhibiting chaotic classical limits. However, as is generally
accepted, this RMT-type universality applies to much broader classes of
quantum systems, including those without a strict classical limit, such as
quantum graphs and networks, MB quantum circuits and spin chains (see Sec.
3.2.2 and [228] for a recent review) that often may be represented by ensemble
theories, i.e. theory of disordered systems and RMT. Why such quantum chaotic
systems in a wider sense exhibit the same universal features as quantum
chaotic systems in a narrow sense, i.e. those with a classical limit,
partially remains a mystery.

Leaving aside such conceptual questions, the now existing theoretical
framework, reviewed here, opens various interesting perspectives and
challenges. Although semiclassical theory applies to the dynamics of
individual systems, enabling a complete understanding of system specific
properties and deviations from universal behaviors (at least theoretically),
we conclude this review with perspectives of particular relevance to quantum
universality:

  * (i)

Variety of classical limits – in both complementary asymptotic limits
considered, the quantum-classical transition is singular. The limiting, but
non-vanishing, values of $\hbar/S\ll 1$ and ${\hbar_{\mathrm{eff}}}=1/N\ll 1$
imply ever shorter wave lengths and extensive quantum interference. Classical
and mean field physics arise for $\hbar/S\equiv 0$ and
${\hbar_{\mathrm{eff}}}\equiv 0$, respectively. Such singular asymptotic
behavior is generally indicative of fascinating physical phenomena. The
semiclassical theory presented provides the leading-orders (in $\hbar$ and
${\hbar_{\mathrm{eff}}}$) of quantum mechanical contributions: the former
dealing with quantum wave interference and the latter with genuine MB quantum
interference. These two quite distinct limits, sketched in Fig. 2, represent
two avenues of asymptotic analysis. It may be interesting to consider other
limiting procedures where both $\hbar$ and ${\hbar_{\mathrm{eff}}}$ come into
play in concert. Indeed, short wavelength approximations (semiclassical
methods) are applied universally across classical field theories, e.g. optics,
acoustics, gravity waves, seismic waves, etc… in which the limit leads to ray
equations underlying the motion of classical waves. In turn these rays
themselves can be chaotic. Beginning with quantum fields, it is possible to
imagine a kind of “asymptotics of asymptotics” in which there are chaotic rays
underlying the classical field solutions underlying semiclassically the
quantum field solutions. This is as opposed to just the single limit
${\hbar_{\mathrm{eff}}}\ll 1$ considered here generating some kind of
nonlinear, unstable classical field. Does the asymptotics of asymptotics
chaotic limit lead to a distinct kind of MB quantum chaos from that mainly
considered in this text? On another front, following the “diagonal” in Fig. 2,
i.e., sending $\hbar$ and ${\hbar_{\mathrm{eff}}}$ to zero simultaneously
appears particularly appealing and challenging, since there is no reason to
expect the two limits to commute and could lead in another unique direction.
There is a great deal yet to uncover.

  * (ii)

The limit of dilute local Hilbert spaces – The construction of the
semiclassical propagator relies on two key facts: the existence of a classical
limit (defined through extremizing an action identified in turn from the exact
path integral) and a semiclassical regime (identified as the small parameter
$\hbar_{\rm eff}\to 0$ scaling the action). In situations where either of
these two ingredients is not obvious or explicit, the semiclassical program
relies on further assumptions. Three very important situations where such
problems appear and await for further progress are semiclassical analysis of
fermionic Fock space, the related case of spin-$1/2$ chains, and the extension
to fields in the continuum. In systems described by (continuous or discrete)
fermionic degrees of freedom, the natural path integral based on Grassman-
valued fields leads to Grassman-valued actions [176] where the stationary
phase approximation cannot be defined in any sensible manner. This problem
reflects in turn the difficulty in identifying an $\hbar_{\rm eff}$ due to the
fundamentally quantum character of the Pauli principle, a problem shared by
spin systems with low spin with their fundamentally discrete natural basis
states (in fact, fermionic and spin-$1/2$ degrees of freedom are rigorously
mapped by means of the Jordan-Wigner transformation [229]). Progress in this
direction can be achieved by forcing a description in terms of bosonic
(commuting) classical fields as in [187, 188, 82] for both fermionic and spin
systems. The subsequent semiclassical program can be formally defined, and in
some situations provides extremely accurate results for delicate MB
interference effects as shown in [182, 187], but so far it lacks rigorous
support. A similar violation of the large local Hilbert space assumption also
occurs in bosonic systems if one considers the continuum limit. Since the
number of sites tends to infinity, any finite number of particles will get
effectively diluted thus breaking the fundamental assumption of large
occupations. Identifying a proper semiclassical regime for the propagator of
bosonic fields in the continuum holds the key for a very promising program, as
several important results are known for such non-linear classical field
equations. There are the existence of chaos [230, 231], a precise definition
of classical mean-field integrability by means of the inverse scattering
method, and the corresponding semiclassical quantization based on solitons as
building classical blocks [232]. Extending this approach into the chaotic
regime remains a fundamental and fascinating open problem.

  * (iii)

Many-body scarring and deviations from equilibration – ergodicity is commonly
related to the equidistribution of eigenfunctions linked to a chaotic phase
space energy shell leading to generic equilibration as predicted by the
eigenstate thermalization hypothesis [233, 234]. There are various properties
and mechanisms that lead to different degrees of ergodicity breaking and, as a
result, to the possibility of hindered relaxation. Corresponding settings
possibly include: (i) MB systems without a classical limit, which are subject
to additional constraints [235, 236], leading to disconnected sub-Hilbert
spaces with reduced dimensions; (ii) MB systems with a classical limit that
exhibits a non-ergodic limit, e.g. mixed phase space structures of co-existing
chaotic and regular regions. As an obvious case, quantum states residing on
tori associated with locally integrable phase space regions are long-lived and
typically decay or MB equilibrate on (exponentially) long time scales.

However, there is a hierarchy of weak ergodicity breaking mechanisms reflected
in deviations from equilibration. In this context, two examples from SP
physics are dynamical localization due to partial transport barriers leading
to additional time scales whose effects are to localize eigenfunctions [237]
and the concept of scars, discovered and introduced by Heller [238] for low-
dimensional quantum-chaotic systems. Both represent prime examples of weak
ergodicity breaking. For the latter, a quantum eigenstate that is
semiclassically anchored on or close to an unstable periodic orbit [238] is
scarred, if the period of the orbit is short and its Lyapunov exponent weak
enough, such that a single-particle wave packet launched along this orbit
shows distinct recurrences after one period. This can be cast into a rough
criterion for scar formation [238]. This concept naturally requires a
classical limit, and is intriguing because it indicates deviations from
ergodicity for a fully chaotic system that globally shows eigenstate
thermalization, and must be differentiated from the aforementioned quantum
states localized in regular regions. Very recently, scarring in Heller’s
original sense could be demonstrated for a MB Bose-Hubbard system with a high-
dimensional associated classical phase space, including the corresponding
generalization of the scar criterion [239].

Earlier, MB ”scars”, reflected in persistent oscillations of local
observables, were observed in Rydberg-based quantum simulators [235], as well
as in corresponding numerical simulations [39, 240]. They were found in spin-
chain type MB Hamiltonians that do not possess a natural classical limit. It
remains to be understood how such a “MB scarring” of this type can be related
to the semiclassical scar mechanism associated with periodic orbits.
Furthermore, for systems with a semiclassical limit, it has to be explored,
whether true MB scars prevail in the thermodynamic limit of large site or
particle number. This could be probed employing ultracold bosonic atoms in
optical lattices as quantum simulators.

  * (iv)

Entropies, entanglement and encounters – The use of quantum information
concepts in the framework of MB systems has lead to deep insights into the
mechanisms of equilibration, thermalization and the role of quantum coherence
[34, 241, 242, 243]. In this context, RMT has rather successfully been applied
to various universal aspects of particular information measures [244, 202]. It
has been of further utility for studying weakly connected bipartite chaotic
systems as well [245, 246, 247, 248]. In contrast, the use of many-body
semiclassical methods face severe technical difficulties and only limited work
has been accomplished. The origin of the problem is the extreme nonlinear way
the propagator appears when calculating such measures, typically involving
functions with dependencies such as $\log KK^{*}$, lacking the usual structure
of multiple sums over paths where further analysis based on action
correlations is possible. Only the so-called purity (a linearized version of
the entanglement entropy) has allowed for a semiclassical study in first-
quantized systems as carried out in [249, 250]. Properties of entanglement of
two (non)interacting particles in the quantum chaotic Chirikov standard map
were very recently numerically considered in Ref. [251].

  * (v)

Dual-unitary dynamics – In Refs. [63, 53] a correspondence between unitary
propagation in time and a non-unitary evolution in particle number in terms of
an operator dual to the time evolution operator was established. The case
where the dual operator is also unitary, referred to as “self-dual” [200], is
of special interest: corresponding (disordered) spin chains have been shown to
exhibit RMT behavior [201], whereas deviations from self-duality can lead to
many-body localization [252]. Correspondingly, so-called dual-unitary circuits
consist of networks of special unitary gates leading to interrelated unitary
evolution in space and time directions [253, 254]; for a recent review see
[255]. The locality of the gates, together with dual-unitary propagation
implies correlations to be possible only along a “ray” $|x|=|t|$ in “space
time”. Such dual-unitary circuits are special and particularly appealing as
they represent a class of exactly solvable quantum systems that exhibit RMT
behavior. It would be interesting to devise dual-unitary quantum dynamical
systems with a semiclassical limit. This could open up the possibility to
extend and apply the semiclassical tools outlined in this review to a
conceptually simple case that still carries quantum chaotic features.

  * (vi)

Quantum chaos meets quantum gravity – During the last two decades, quantum
chaos concepts have entered the realm of research towards possible
quantization of gravitational degrees of freedom. Early suggestions pointing
to the chaotic character of black holes [209] were finally made precise
through the study of scrambling of (quantum) information and the proposal of
black holes as systems where such scrambling is maximal, see [41] and
references therein. This connection between toy models of quantum gravity
where, unlike the realistic scenario describing the universe, a full solution
is available and quantum chaos was made precise in the cornerstone paper
[256]. There, the authors showed the dual relation, by means of the
equivalence of correlation functions, between quantized Jackiw-Teitelboim
gravity, a solvable model of gravity coupled with a dilaton field in 1+1
dimensions, and a suitably double-scaled theory of random matrices. This
finding has triggered lots of attempts towards understanding the origins of
this duality with prospective links to supersymmetry and semiclassical
analysis in quantum chaos [257].

Just these few speculations illustrate the richness of possibilities from
following various semiclassical paths to interesting future theoretical
challenges in MB quantum chaos.

##  5 Acknowledgments

This review is dedicated to the memory of our dear colleagues, Fritz Haake and
Petr Braun. We are very thankful to both of them for many many inspiring
conversations about semiclassical physics, quantum chaos and beyond science
during the last 30 years and numerous encounters at various places all over
the world. Our close connections started with the Symposium Quantum Aspects of
Nonlinear Systems, organized by Fritz Haake and Robert Graham in 1990, in
retrospective playing a role for quantum chaos similar to the role of the 1911
Solvay conference for quantum physics. K.R. is further indebted to Petr Braun
for various conversations also about Rydberg atomic physics, and for his kind
hospitality during a longer stay in St. Petersburg in 1990.

Part of the results on MB semiclassics presented here arose out of two PhD
theses [188, 258] conducted in Regensburg. Hence we particularly thank Th.
Engl and J. Rammensee for their important work and for many related
conversations. We further thank P. Schlagheck and D. Ullmo, as knowlegable
discussion partners on advanced topics of many-body semiclassical methods over
many years. We would also like to thank numerous colleagues for conversations
on topics that entered the many-body part of this review, including A.
Altland, A. Buchleitner, R. Dubertrand, B. Geiger, T. Guhr, B. Gutkin, Q.
Hummel, R. Jalabert, D. Vierl, D. Waltner and D. Wisniacki.

We acknowledge financial support from the Deutsche Forschungsgemeinschaft
(German Research Foundation) through Projects Ri681/14-1, through Ri681/15-1
within the Reinhart-Koselleck Programme, as well as funding through Vielberth
Foundation in Regensburg.

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