# Dynamical transition of quantum scrambling in a non-Hermitian Floquet
synthetic system

Liang Huo  School of Information Engineering, Jiangxi University of Science
and Technology, Ganzhou 341000, China  Han Ke  School of Science, Jiangxi
University of Science and Technology, Ganzhou 341000, China  Wen-Lei Zhao
<EMAIL_ADDRESS>School of Science, Jiangxi University of Science and
Technology, Ganzhou 341000, China

###### Abstract

We investigate the dynamics of quantum scrambling, characterized by the out-
of-time ordered correlators (OTOCs), in a non-Hermitian quantum kicked rotor
subjected to quasi-periodical modulation in kicking potential. Quasi-periodic
modulation with incommensurate frequencies creates a high-dimensional
synthetic space, where two different phases of quantum scrambling emerge: the
freezing phase characterized by the rapid increase of OTOCs towards
saturation, and the chaotic scrambling featured by the linear growth of OTOCs
with time. We find the dynamical transition from the freezing phase to the
chaotic scrambling phase, which is assisted by increasing the real part of the
kicking potential along with a zero value of its imaginary part. The opposite
transition occurs with the increase in the imaginary part of the kicking
potential, demonstrating the suppression of quantum scrambling by non-
Hermiticity. The underlying mechanism is uncovered by the extension of the
Floquet theory. Possible applications in the field of quantum information are
discussed.

##  I Introduction

Understanding the connection between scrambling and chaos is fundamentally
important in different fields of physics, such as quantum information and
quantum chaos Dowling23prl ; TXu20prl ; ritzsch22pre ; JWang21pre . It is
found that the out-of-time ordered correlators (OTOCs) can be exploited as an
effective tool to detect the quantum information scrambling, quantum chaos and
quantum entanglement LewisSwan19 ; XDHu23 . The underlying chaos with
exponential instability generally gives rise to the fast scrambling of quantum
information in many-body systems Belyansky20prl ; JKim21prb , which is
quantified by the exponential growth of OTOCs with the rate denoted by quantum
Lyapunov exponent Maldacena16 ; Rozenbaum17 . Interestingly, genuine quantum
chaos induced by many-body interaction ZQi23  and Floquet-engineered nonlinear
interaction WLZhao21prb  even induces the fastest scramblers, being
characterized by the superexponential growth of OTOCs in the LMG model ZQi23
and in the Gross-Pitaevskii map model WLZhao21prb . The rich physics
represented by OTOCs’ dynamics enlightens our understanding on the
nonequilibrium dynamics of closed quantum many-body systems, such as quantum
thermalization Murthy19prl , ergodicity Rozenbaum19prb  and topological phase
transitions Bin23prb , which is of interest for fundamental physics from both
theoretical Kukuljan17prb  and experimental viewpoints Chaudhury09 .

Floquet engineering technique has now been widely employed to create versatile
synthetic systems which have potential advantages to mimic the electronic
transport in a nanotube threaded by an artificial gauge field Hainaut18nc ,
the Anderson metal-insulator transition in the high dimensional space Lopez13
, as well as the topological fractional pumping of cold atoms Taddia17prl ,
just to name a few. It is more interesting that the complex potentials are
experimentally achievable by using a decay channel of cold atoms to excited
state in the standing laser field Chudesnikov91 ; Keller97  and by taking
account to the loss feature of optical medium Berry04 ; Berry08 ; Yin13 . This
non-Hermitian extension of conventional systems induces the flourishing
advancement of novel platforms, such as non-Hermitian photonic crystal, non-
Hermitian sensors Budich20 , and non-Hermitian electrical circuits Stegmaier21
, which have been widely used to explore the fundamental concepts of topology,
degeneracies, and adiabaticity WYWang22pra ; XWang23njp  in non-Hermitian
physics. For $\cal{PT}$-symmetric systems, the dynamics of OTOCs can diagnose
Yang-Lee edge singularity Zhai20prb , exhibit the quantized response to
external potential Wlzhao22prr , and display the scaling law at the transition
to the spontaneous $\cal{PT}$-symmetry breaking Wlzhao23pra , which sheds
light on the fruitful physics in Floquet synthetic systems.

In this context, we investigate the dynamics of OTOCs in a non-Hermitian
kicked rotor (NQKR) model for which the complex kicking potential is modulated
quasi-periodically in time. In this Floquet synthetic system, we observe a
rapid increase of the OTOCs towards saturation during time evolution, when the
real part of the kicking potential is small and its the imaginary part is
zero. This demonstrates the freezing phase of quantum scrambling for long-term
evolution behaviors. In this Hermitian case, a dynamical transition for the
OTOCs from the freezing behavior to power law growth emerges as the real part
of kicking potential increases, indicating the chaos assisted scrambling.
Interestingly, the opposite transition from power-law behavior to freezing
behavior occurs by increasing the imaginary part of the kicking potential. We
further obtain the phase diagram of the quantum scrambling by utilizing a
well-trained long short-term memory network (LSTM). By extending the Floquet
theory, we predict that a quantum state will finally evolve to a quasi-
eigenstate of the instantaneous Floquet operator with the large most value of
the imaginary part of quasi-energy. This is verified by our numerical results
of the fidelity between time evolved quantum state and quasi-eigenstates. The
feature of exponential localization of quasi-eigenstates determines the
freezing of quantum scrambling of the NQKR model.

The paper is organized as follows. In Sec. II, we describe our model and show
the dynamics of OTOCs. We discuss the mechanism of the freezing of quantum
scrambling in Sec. III. Conclusion and discussions presented in Sec. IV.

##  II Model and main results

The dimensionless Hamiltonian of the QPKR model reads

${\rm H}=\frac{{p}^{2}}{2}+V_{K}(\theta,t)\sum_{n}\delta(t-t_{n})\>,$ (1)

with the time-dependent kicking potential

$V_{K}(\theta,t)=(K+i\lambda)[1+\eta\cos(\omega_{1}t)\cos(\omega_{2}t)]\cos(\theta)\;,$
(2)

where $p=-i\hbar_{\mathrm{eff}}\partial/\partial\theta$ is the angular
momentum operator, $\theta$ is the angle coordinate, satisfying the
commutation relation $[\theta,p]=i\hbar_{\mathrm{eff}}$ with
$\hbar_{\mathrm{eff}}$ effective Planck constant. Here, the parameters $K$ and
$\lambda$ control the strength of the real and imaginary parts of the kicking
potential, respectively. The time $t_{n}$ is integer, i.e., $t_{n}=1,2\ldots$,
indicating the kicking number. The $\omega_{j}$ ($j=1,2$) indicates the
modulation frequency on the kicking strength, with the modulation strength
denoted by $\eta$. When $\omega_{1}$ and $\omega_{2}$ are incommensurate with
each other, and both of them are incommensurate with $2\pi$, this quasi-
periodical extension of the QKR model mimics a three-dimensional QKR Casati89
; Shepelyansky83 ; Lopez13 ; Hainaut2018 . We consider the case with
$\omega_{1}=2\pi/\kappa$ and $\omega_{2}=2\pi/\kappa^{2}$, where
$\kappa=1.3247\ldots$ is the real root of the cubic equation Shepelyansky14 .

The eigenequation of angular momentum operator is $p|n\rangle=p_{n}|n\rangle$
with eigenvalue $p_{n}=n\hbar_{\mathrm{eff}}$ and eigenstate
$\langle\theta|n\rangle=e^{in\theta}/\sqrt{2\pi}$. With this complete basis,
an arbitrary state can be expanded as
$|\psi\rangle=\sum_{n}\psi_{n}|n\rangle$. The advancements of the delta-
kicking systems is that the Floquet operator can be split into two components,
namely $U(t)=U_{f}U_{K}(t)$, with the free evolution operator
$U_{f}=\exp(-ip^{2}/2\hbar_{\mathrm{eff}})$ and the kicking term
$U_{K}(t)=\exp[-iV_{K}(\theta,t)/2\hbar_{\mathrm{eff}}]$. The time evolution
of a quantum state from $t_{n}$ to $t_{n+1}$ is governed by
$|\psi(t_{n+1})\rangle=U(t_{n})|\psi(t_{n})\rangle$.

The OTOCs are defined as $C(t)=-\langle[A(t),B]^{2}\rangle$, where both
$A(t)=U^{\dagger}(t)AU(t)$ and $B$ are operators evaluated in Heisenberg
picture, and $\langle\cdot\rangle=\langle\psi(t_{0})|\cdot|\psi(t_{0})\rangle$
denotes the average over an initial state. Since the thermal state of Floquet
systems are not well defined, there is no need to perform the average over the
ensemble of thermal state in evaluating the $C$ DAlessio14 ; Zhao21 ; Zhao22 .
We consider the case that $A$ is the translation operator $A=e^{-i\epsilon p}$
and $B$ is the projection operator on the initial state
$B=|\psi(t_{0})\rangle\langle\psi(t_{0})|$. Straightforward derivation yields
the relation $C(t)=\mathcal{N}^{2}(t)-|\langle\psi(t)|e^{i\epsilon
p}|\psi(t)\rangle|^{2}$ with the norm
$\mathcal{N}(t)=\langle\psi(t)|\psi(t)\rangle$. The non-unitary evolution of
non-Hermitian Hamiltonian leads to the exponential growth of the norm, i.e.,
$\mathcal{N}(t)=e^{\gamma t}$. In order to reduce the impact of the norm to
OTOCs, we introduce the rescaled OTOCs as
$C(t)=1-|\langle\psi(t)|e^{-i\epsilon
p}|\psi(t)\rangle|^{2}/\mathcal{N}^{2}(t)$. In condition that $\epsilon\ll 1$,
we can derive the approximate equivalence

$C(t)\approx\epsilon^{2}[\langle p^{2}(t)\rangle-\left(\langle
p(t)\rangle\right)^{2}]\;,$ (3)

where we used the approximation $e^{-i\epsilon p}\approx 1-i\epsilon p$
Zhao23arx15 ; Zhao23arx62 . This relation clearly demonstrates the underlying
connection between quantum scrambling and energy diffusion Lewis19 ; Meier19 ,
which has practical applications in enhancing the precision of quantum
measurement ZYLi2022 ; Garttner18 . In numerical simulations, we choose the
ground state of the angular momentum operator as initial state, i.e.,
$|\psi(t_{0})\rangle=1/\sqrt{2\pi}$.

We numerically investigate the time evolution of $C$ for a wide regime of $K$.
Figure 1(a) shows that $C$ increases during short time duration, and
eventually fluctuates around a saturation level for small $K$. The saturation
level increases with the increase of $K$. This demonstrates the suppress of
quantum scrambling by the dynamical localization of quantum diffusion. In
fact, we have uncovered the power-law dependence of a different OTOCs on the
kicking strength in the dynamical localization regime Zhao23pra201 .
Interestingly, the OTOCs linearly increases with time $C(t)=Dt$ when $K$ is
sufficiently large (e.g., $K=7$ and 10), indicating the emergence of chaotic
scrambling. Corresponding to the freezing of scrambling, the wavepackets are
exponentially localized in momentum space, i.e.,
$|\psi(p)|^{2}\propto\exp(-|p|/\xi)$ with a constant $\xi$, which is a
signature of the dynamical localization [see Fig. 1(b)]. For the linear-law of
scrambling, the momentum distribution can be well described by the Gaussian
function $|\psi(p)|^{2}\propto\exp(-p^{2}/\sigma)$, which characterizes the
chaotic diffusion.

Figure 1: (a) Time evolution of $C$ for $\lambda=0$ with $K=1$ (black line), 4
(green line), 5 (pink line), 7 (blue line), and 10 (cyan line). Red-dashed
line indicates our theoretical prediction $C\approx(\epsilon K)^{2}t/2$. (b)
Momentum distributions at the time $t=1000$ for $K=4$ (squares) and 10
(circles). Red solid (green dashed) line indicates the Gaussian function
(exponentially-localized function), i.e.,
$|\psi(p)|^{2}\propto\exp(-p^{2}/\sigma)$
($|\psi(p)|^{2}\propto\exp(-|p|/\xi)$). (c) Phase diagram of the quantum
scrambling in the parameter space $(\eta,K)$. One can see clearly a phase
boundary. Here, the value of $\rho$ indicates the probability of a parameter
value $(\eta,K)$ being within regime of the freezing of scrambling ($\rho=0$)
or the chaotic scrambling ($\rho=1$). (d) The growth rate $D$ versus $K$.
Solid line denotes our theoretical prediction ($D\approx K^{2}/2$). The
parameters are $\eta=0.75$, $\epsilon=10^{-5}$ and
$\hbar_{\mathrm{eff}}=2.89$.

It is now widely accepted that the machine learning method can be used to
determine the feature of the time sequence of observables by using finite time
evolution of these observables HZhao21 . Previously, we have use a well-
trained LSTM network to successfully acquire the phase diagram of the
spontaneous $\cal{PT}$-symmetry breaking in a QKR model Zhao23pra201 . Here,
we use the LSTM network to explore the phase diagram of the quantum scrambling
in parameter space $(\eta,K)$. For a specific value of $(\eta,K)$, it generate
a probability $\rho$ for the finite time sequence of the $C(t)$, which is
featured by the asymptotical saturation ($\rho=0$) or the linear growth
($\rho=1$). Figure 1(c) demonstrates a clear phase boundary $\eta_{c}$
separating the phase of the freezing of scrambling ($\rho=0$) and the chaotic
scrambling phase ($\rho=1$). The critical value $\eta_{c}$ monotonically
decreases with the increase of $K$. That is to say, the larger $K$ makes it
easier for the transition from the freezing scrambling phase to chaotic
scrambling occurs. In fact, the parameter $K$ controls the hopping distance in
momentum-space lattice Lopez13 . Therefore, strong kick strength is helpful
for enhancing the quantum transport behavior. Since the $C$ is proportional to
the variance of wavepackets in momentum space, the phase diagram of $C$ in the
parameter in space $(\eta,K)$ is essentially same as that of the mean energy
Lopez13 ; Mano21 .

Since both $\omega_{1}$ and $\omega_{2}$ are irrational numbers in the time
dependent kick strength, this kind of temporal modulation induces effectively
random noises ($\eta\ll 1$) to kick strength in time domain. It is well-known
that random noises in kick strength can destroy DL and lead to classically
normal diffusion in the system, i.e., $\langle p^{2}\rangle\approx K^{2}t/2$
Lemarie10JMO . Substituting this relation into Eq. (3) yields
$C(t)\approx(\epsilon K)^{2}t/2$, where we use the condition $\langle
p\rangle=0$ as the quantum state is symmetric in momentum space. Therefore,
the scrambling rate $D=dC/dt$ equals to $(\epsilon K)^{2}/2$, which is
validated by our numerical results in Fig. 1(d).

Figure 2: (a) Dependence of $C$ on time with $K=10$ for $\lambda=0$ (black
line), 1.5 (green line), 2 (cyan line), and 5 (blue line). Red-dashed line
indicates the linear growth $C\approx(\epsilon K)^{2}t/2$. (b) Momentum
distributions at the time $t=1000$ for $\lambda=2$ (squares) and 5 (circles).
Red solid lines denote the exponentially-localized function
$|\psi(p)|^{2}\propto\exp(-|p|/\xi)$ with $\xi\approx 14$ and $4.5$ for
$\lambda=2$ and 5, respectively. (c) Phase diagram of the quantum scrambling
in the parameter space $(\lambda,K)$. Other parameters are the same as in Fig.
1(d).

In order to reveal the effect of the non-Hermitian driven potential on the
quantum scrambling dynamics, we numerically investigate the time dependence of
$C$ for different $\lambda$. Figure 2(a) shows that for $\lambda=1.5$ the
growth of the $C$ is reduced compared with the $C$ for $\lambda=0$, which
demonstrates the suppress of the scrambling by non-Hermiticity. For
sufficiently large $\lambda$ (e.g., $\lambda=2$ and $5$), the $C$ exhibits the
asymptotical saturation after long time evolution, moreover the saturation
level decreases with the increase of $\lambda$, indicating the total freezing
of quantum scrambling for $\lambda\gg 1$. Correspondingly, the wavepackets are
exponentially localized in momentum space,i.e.,
$|\psi(p)|^{2}\propto\exp(-|p|/\xi)$ with localization length $\xi$ decreasing
with the increase of $\lambda$ [see Fig. 2(a) for $\lambda=5$], which reveals
the mechanism of the suppress of quantum scrambling by dynamical localization.
The decrease of $\xi$ with the increase of the $\lambda$ also reflects the
enhancement of the dynamical localization by the non-Hermitian driven
potential, whose mechanism has been uncovered by our previous investigations
Wlzhao23sym ; Wywang22 ; KQHuang21pra . We further use the LSTM network to
investigate the dynamical transition of the $C$ in the parameter space
$(\lambda,K)$. Figure 2(c) shows that there is a clear boundary quantified by
$\lambda_{c}$ between the freezing of scrambling ($\lambda>\lambda_{c}$) and
the chaotic scrambling phases ($\lambda<\lambda_{c}$). This phase diagram
broadens our understanding on the quantum scrambling in non-Hermitian Floquet
systems.

##  III Mechanism of the freezing of scrambling in non-Hermitian regime

A character of the non-unitary evolution of non-Hermitian systems is the
growth of the norm with time. We numerically investigate the time evolution of
the norm $\mathcal{N}$ for different $\lambda$. Figure 3(a) shows that, for
$\lambda$ smaller than a threshold value $\lambda_{c}$, the value of
$\mathcal{N}$ equals almost to unity with time evolution corresponding to the
real quasienergy. For $\lambda>\lambda_{c}$, the $\mathcal{N}$ increases
exponentially with time, i.e., $\mathcal{N}=e^{\mu t}$, for which the growth
rate $\mu$ linearly increases with the increase of $\lambda$ [see the inset in
Fig. 3(a)]. For periodically-driven systems, the Floquet theory governs the
eigenequation
$U|\varphi_{\varepsilon}\rangle=e^{-i\varepsilon}|\varphi_{\varepsilon}\rangle$.
In analogy with this equation, we define the eigenequation of the
instantaneous Floquet operator as
$U(t+1,t)|\varphi_{\varepsilon}(t)\rangle=e^{-i\varepsilon(t)}|\varphi_{\varepsilon}(t)\rangle$.
It is reasonable to believe that the quasienergies are complex
$\varepsilon=\varepsilon_{r}+i\varepsilon_{i}$ for non-unitary Floquet
operator. To confirm this issue, we numerically investigate the time evolution
of the average value of the imaginary part of the quasienergy
$|\bar{\varepsilon}_{i}(t)|=\sum_{j=1}^{N}|\varepsilon_{i}^{j}(t)|/M$ where
$M$ is the dimension of the Floquet operator matrix. Figure 3(b) shows that
the value of $|\bar{\varepsilon}_{i}|$ is virtually zero for small $\lambda$,
corresponding to $\mathcal{N}=1$ in Fig. 3(a). For sufficiently large
$\lambda$ (e.g., $\lambda=2$), the $|\bar{\varepsilon}_{i}(t)|$ oscillates
around a saturation level, which appearantly increases with $\lambda$. The
appearance of complex quasienergies leads to the exponential growth of the
norm. We further investigate the long-time average value of the norm
$\bar{\mathcal{N}}$ for a wide range of $\lambda$. Figure 3(c) shows that, for
a specific $\hbar_{\mathrm{eff}}$, the $\bar{\mathcal{N}}$ remains at unity
for $\lambda$ smaller than a threshold value $\lambda_{c}$, and increases
rapdily with the increase of $\lambda$ for $\lambda>\lambda_{c}$. The critical
value $\lambda_{c}$ increases with the increase of $\hbar_{\mathrm{eff}}$.

Figure 3: (a) Norm $\mathcal{N}$ versus time for $\lambda=0$ (squares), 0.3
(circles), 0.5 (up triangles), 1 (diamonds), 1.5 (pentagrams), and 2 (down
triangles). Solid lines indicate the exponential increase, i.e.,
$\mathcal{N}=e^{\mu t}$. Dashed line denotes $\mathcal{N}=1$. Inset: The $\mu$
versus $\lambda$. Solid line indicates the linear growth $\mu\propto\lambda$.
(b) Time dependence of the average value of the imaginary parts of the quasi-
energies $|\bar{\varepsilon}_{i}|$ for $\lambda$ = 0 (diamonds), 1.5
(triangles), 2 (circles), 5 (squares). The parameters are $K=10$ and
$\hbar_{\mathrm{eff}}=2.89$. (c) Average value $\bar{\mathcal{N}}$ versus
$\lambda$ for $K=10$ with $\hbar_{\mathrm{eff}}=0.5$ (squares), 1.5 (circles),
and 2.89 (triangles). Arrow marks the critical value $\lambda_{c}$. Solid line
denotes $\mathcal{N}=1$. Other parameters are the same as in Fig. 1(d).

We have found both theoretically and numerically that the quantum state of
non-Hermitian Floquet systems will evolved to the quasi-eigenstate whose
imaginary part of the quasi-energie is maximum. Thus, the feature of this
state determines the dynamical behaviors of the system Wlzhao23sym ; Wywang22
; KQHuang21pra ; KQHuang21jpcm . As a futher step, we numerically investigate
the fidelity between the time-evolved quantum state and the quasieigenstates,
i.e., $\mathcal{F}(t)=|\langle\psi(t)|\varphi_{\varepsilon}(t)\rangle$ for
$\lambda\gg\lambda_{c}$. Figure (4)(a) shows that for $\lambda=5$, the
$\mathcal{F}$ at the time $t=200$ almost equals to unity at the maximum
$\varepsilon_{i}=2.45398$. In addition, we compare the probability density
distribution between quantum state $|\psi(t=200)\rangle$ and quasieigenstate
$|\varphi_{\varepsilon}(t=200)\rangle$ with $\varepsilon_{i}=2.45398$ in Fig.
(4)(a). One can find that the two states almost completely overlap with each
other, both of which are exponentially localized in momentum space. It is
apparent that the exponential localization of quasieigenstates suppresses the
quantum scrambling.

Figure 4: (a) Fidelity $\mathcal{F}$ at the time $t=200$ versus the imaginary
part of the quasi-energy $\varepsilon_{i}$ for $K=10$ and $\lambda=5$. Red
diamond indicates $\mathcal{F}\approx 1$ for $\varepsilon_{i}=2.45398$. (b)
Comparison of the probability density distribution between the state
$|\psi(t=200)\rangle$ and the quasi-eigenstate
$|\varphi_{\varepsilon}(t=200)\rangle$ with $\varepsilon_{i}=2.45398$. Solid
line indicates the exponentially-localized distribution of
$|\psi(p)|^{2}\propto\exp(-|p|/\xi)$ with $\xi\approx 3.4$. Other parameters
are the same as in Fig. 1(d).

##  IV Conclusion and discussions

In the present work, we investigate the dynamics of quantum scrambling in high
dimensional synthetic space created by quasi-periodical modulation on a NQKR
model. In Hermitian case (i.e., $\lambda=0$), we find a dynamical transition
of quantum scrambling from freezing phase to the chaotic scrambling of linear
growth of OTOCs by increasing the real part $K$ of the kicking strength. By
increasing the imaginary part of the kicking potential, the system
demonstrates the dynamical transition from chaotic scrambling to freezing
behavior, demonstrating the suppress of the scrambling by non-Hermiticity. We
obtain the phase diagram of the quantum scrambling with a kind of machine
learning method, i.e., the LSTM. In order to reveal the dynamical transtion of
the quantum scrambling, we extend the Floquet theory to our system, and find
the emergence of complex quasi-energies of the instantaneous Floquet operator.
The fidelity between time evolved quantum state with quasi-eigenstates reveals
that the quantum state finally evolves to the quasi-eigenstate with large most
value of the imaginary quasi-energy. Its localized property dominates the
freezing of quantum scrambling. Our findings serve as new elements of the
quantum scrambling in synthetic systems generated by Floquet engineering.

## ACKNOWLEDGMENTS

This work is supported by the National Natural Science Foundation of China
(Grant No. 12065009 and 12365002), the Science and Technology Planning Project
of Jiangxi province (Grant No. 20224ACB201006, 20224BAB201023, and
20212BAB204024).

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