# Coexistence of directed momentum current and ballistic energy diffusion in
coupled non-Hermitian kicked rotors

Jian-Zheng Li  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
Jie Liu<EMAIL_ADDRESS>Graduate School, China Academy of Engineering
Physics, Beijing 100193, China  HEDPS, Center for Applied Physics and
Technology, and College of Engineering, Peking University, Beijing 100871,
China

(August 27, 2024)

###### Abstract

We numerically investigate the quantum transport in a coupled kicked rotors
with the $\mathcal{PT}$-symmetric potential. We find that the spontaneous
$\mathcal{PT}$-symmetry breaking of wavefunctions emerges when the amplitude
of the imaginary part of the complex potential is beyond a threshold value,
which can be modulated by the coupling strength effectively. In the regime of
the $\mathcal{PT}$-symmetry breaking, the particles driven by the periodical
kicks move unidirectionally in momentum space, indicating the emergence of a
directed current. Meanwhile, with increasing the coupling strength, we find a
transition from the ballistic energy diffusion to a kind of the modified
ballistic energy diffusion where the width of the wavepacket also increases
with time in a power law. Our findings suggest that the decoherence effect
induced by the interplay between the inter-particle coupling and the non-
Hermitian driving potential is responsible for these particular transport
behaviors.

##  I Introduction

Directed transport and quantum diffusion in both real and momentum space have
attracted many interests in diverse fields of physics, ranging from condensed
matter physics Cooper19  to quantum chaos JWang20 ; LWang07 , and to
biological physics Fornes21 . It is found that a seminal phenomenon of
directed transport, namely, quantum ratchet Arzola17 , has practical
applications in the design of electron pumps Lau20 , in enhancing the
efficiency of photovoltaic cells Sogabe21  and in the construction of
molecular rotors Denisov14 . In other aspects, the quantum diffusion is
revelent for understanding the conductivity of electronics Anderson58 , the
spin transport Nardis21 ; Ljubotina19 , the energy transport Uchiyama18 , as
well as the information scrambling Moudgalya19 ; Lewis19 , thus has been a
subject of intense study in various areas of physics. A landmark of the study
on quantum diffusion is the Anderson localization (AL) of electrons in
disordered potential Anderson58 ; Lagendijk09 . Its analog in momentum space
is the dynamical localization (DL) in the quantum kicked rotor (QKR) which is
a paradigmatic model of Floquet systems Casati79 ; Fishman82 . The QKR model
with incommensurable frequency can mimic the Anderson model in 2D or 3D
disordered lattice, and is very convenient for the theoretical investigation
and experimental realization Shepelyansky83 ; Casati89 ; Lemarie08 ; Lemarie09
.

Nowadays, the QKR and its variants have been accepted as ideal systems for
exploring rich physics, for instance, Floquet-topological phase Shirley65 ;
Bukova15 ; Ho12 ; YuChen14 ; Longwen18 , dynamical phase transition Hainaut18
, and quantum walk in momentum-space lattice Dadras18 ; Dadras19 ; Xie20 .
More recently, the study of the $\cal{PT}$-symmetric extension of the kicked
rotor (PTKR) shows the spontaneous $\cal{PT}$-symmetry breaking characterized
by the emergence of the complex quasienergies of the Floquet operators
Longhi17 ; West10 . Interestingly, the $\mathcal{PT}$-symmetric kicking
potential leads to quantized acceleration of momentum current Zhao19 ;
Zhao20cpb  and quantized response of out-of-time ordered correlators Zhaowl22
in QKR model. The non-Hermiticity of Hamiltonian has nowadays been widely
accepted as a fundamental modification for the conventional quantum mechanics
Bender98 ; Ashida20 ; Moiseyev11 . It is known that open systems which
exchange particles or energy with environment can be described by non-
Hermitian Hamiltonians Ritsch13 ; Makris08 ; Li20 , for instance ultracold
atoms in dissipative optical lattice, optical wave propagation in lossy media,
electrical circuits with virtual absorption, just name a few.

The quantum transport in non-Hermitian systems has received intensive
investigations Eichelkraut13 ; Xu21 , where the fate of the directed current
(DC) and DL under the effects of inter-particle coupling is still an open
issue Borgonovi16 ; Lellouch20 ; Chicireanu21 ; Vuatelet21 ; Casati06 ; Duval
; Park03 . It is found that in a system of coupled QKRs, the non-Hermitian
driving potential can protect the DL, which otherwise is destroyed by the
inter-particle coupling in Hermitian case Huang21 . In this paper, we
investigate the effects of interaction on the quantum transport in momentum
space via a system of coupled PTKRs. Interestingly, we find the emergence of
the spontaneous $\mathcal{PT}$-symmetry breaking when the strength of the
imaginary parts of the complex kicking potential is beyond a threshold value,
which can be effectively modulated by the coupling strength. In the regime
where the $\mathcal{PT}$-symmetry is unbroken, there are two different phases
of energy diffusion, one is the DL, i.e., $\langle p_{1}^{2}\rangle\sim C$ and
another is the chaotic diffusion $\langle p_{1}^{2}\rangle\approx\gamma t$,
both of which have no the phenomenon of the DC, i.e., $\langle
p_{1}\rangle\sim 0$ (see phases I and II in Fig. 1). In the regime of
$\mathcal{PT}$-symmetry breaking, each KR moves unidirectionally in momentum
space, i.e., $\langle p_{1}\rangle=Dt$, indicating the appearance of the DC.
Interestingly, there also two different phases of energy diffusion, one is the
ballistic diffusion, i.e., $\langle p_{1}^{2}\rangle\approx D^{2}t^{2}$ (see
phase III in Fig. 1), and another is the modified ballistic diffusion (MBD),
i.e., $\langle p_{1}^{2}\rangle\approx D^{2}t^{2}+\eta t^{\alpha}$ (see phase
IV in Fig. 1). The coexistence of the DC and the MBD is a unique phenomenon in
coupled non-Hermitian systems. We numerically obtain the acceleration rate $D$
of DC and the diffusion rate $\eta$ of energy for a wide regime of system
parameters, which is helpful to guide the Floquet engineering of transport in
momentum-space lattice Keser16 ; Hainaut19 . We also numerically find that
entanglement grows with the increase of the strength of interaction. It is
reasonable to believe that the coexistence of the DC and MBD results from the
intrinsic decoherece effects in non-Hermitian chaotic systems.

The paper is organized as follows. In Sec. II, we describe the system. In Sec.
III, we show the transport behaviors in our system with emphasis on the
coexistence of the DL and MBD. A summary is presented in Sec. IV.

Figure 1: Schematic diagram for the directed current (DC) and quantum
diffusion in the parameter space $(\lambda,\varepsilon)$.

##  II Model

The Hamiltonian of the coupled PTKRs reads

$\text{H}=\text{H}_{1}+\text{H}_{2}+\text{H}_{\rm{I}}\;,$ (1)

with $\text{H}_{j}$ ($j=1,2$) of the individual particle

$\text{H}_{j}=\frac{p_{j}^{2}}{2}+V(\theta_{j})\sum^{\infty}_{n=0}\delta(t-t_{n})\;,$
(2)

and the inter-particle coupling

$\text{H}_{\text{I}}=\varepsilon\cos(\theta_{1}-\theta_{2})\sum^{\infty}_{n=0}\delta(t-t_{n}).$
(3)

Here, our consideration of the temporally delta modulation of the coupling is
just for the convenience of numerical simulation. This kind of coupling has
been widely used in previous investigations Rozenbaum17 ; Duval . In our
system, the kicking potential is $\cal{PT}$-symmetric, i.e.,
$V(\theta_{j})=K[\cos(\theta_{j})+i\lambda\sin(\theta_{j})]$ with $K$ being
the kick strength while $\lambda$ controlling the strength of imaginary part
of $V(\theta_{j})$. The $p_{j}=-i\hbar_{\rm{eff}}\partial/\partial\theta_{j}$
is the angular momentum operator and $\theta_{j}$ is the angle coordinate of
each subsystem. The $\hbar_{\rm{eff}}$ indicates the effective Planck
constant, and the parameter $\varepsilon$ is the coupling strength. The time
$t_{n}(=0,1,\ldots)$ is integer indicating the number of kicks. All variables
are properly scaled and thus in dimensionless units.

The eigenequation of $p_{j}$ has the expression
$p_{j}|\phi^{j}_{m}\rangle=p^{j}_{m}|\phi^{j}_{m}\rangle$ with
$p^{j}_{m}=m\hbar_{\rm{eff}}$ and
$\langle\theta_{j}|\phi^{j}_{m}\rangle=e^{im\theta_{j}}/\sqrt{2\pi}$. On the
basis of the product states
$|\phi^{1}_{m},\phi^{2}_{n}\rangle=|\phi^{1}_{m}\rangle\otimes|\phi^{2}_{n}\rangle$,
an arbitrary quantum state $|\psi\rangle$ can be expanded as
$|\psi\rangle=\sum_{m,n}\psi_{m,n}|\phi^{1}_{m},\phi^{2}_{n}\rangle$. One
period evolution of the quantum state from $t_{n}$ to $t_{n+1}$ is given by
$|\psi(t_{n+1})\rangle=U|\psi(t_{n})\rangle$. The Floquet operator $U$ can be
separated into two fractions,

$U=U_{\rm{}_{f}}U_{\rm{}_{K}}\;,$ (4)

where the free evolution operator of the kinetic term takes the form

$U_{\rm{}_{f}}=\exp\left(-\frac{i}{\hbar_{\rm{}_{eff}}}\sum^{2}_{j=1}\frac{p_{j}^{2}}{2}\right)\;,$
(5)

and the evolution operator of the kicking term is

$U_{\rm{}_{K}}=\exp\left\\{-\frac{i}{\hbar_{\rm{}_{eff}}}\left[\sum^{2}_{j=1}V(\theta_{j})+\varepsilon\cos(\theta_{1}-\theta_{2})\right]\right\\}\;.$
(6)

In our investigations, the initial state is set to be the product of the
ground states, i.e., $|\psi(t_{0})\rangle=|\phi^{1}_{0},\phi^{2}_{0}\rangle$.

##  III Directed current and energy diffusion in the momentum space

###  III.1 Time evolution of $\langle p_{1}\rangle$, $\langle
p_{1}^{2}\rangle$ and $\mathcal{M}_{1}$

The transport behavior of one of particles (say particle 1) in momentum space
is characterized by the momentum current $\langle
p_{1}\rangle=\rm{Tr}(\rho_{1}\emph{p}_{1})$, the energy diffusion $\langle
p^{2}_{1}\rangle=\rm{Tr}(\rho_{1}\emph{p}^{2}_{1})$, and the width of the
time-evolved state $\mathcal{M}_{1}=\langle p^{2}_{1}\rangle-(\langle
p_{1}\rangle)^{2}$. In the emergence of $\cal{PT}$-symmetry breaking, the norm
$\cal{N}$ of quantum state exponentially increases with time. To eliminate the
contribution of the norm to the observable, we normalized the quantum state to
be unity after each kick. Therefore, we define the reduced density matrix of
particle 1 as
$\rho_{1}=\frac{1}{\mathcal{N}}\rm{Tr}_{2}(|\psi\rangle\langle\psi|)$, which
is obtained by tracing out the other degree of freedom from the density matrix
of the two-particle system $\rho=|\psi\rangle\langle\psi|$.

We show the phase diagram of transport behaviors for different $\lambda$ and
$\varepsilon$ in Fig. 1. In the unbroken phase of $\cal{PT}$-symmetry, i.e.,
$\lambda<\lambda_{c}$, there are two different classes of transport behaviors,
both of which have no the DC, i.e., $\langle p_{1}\rangle\sim 0$. Class I
corresponds to DL, i.e., $\langle p_{1}^{2}\rangle\sim C$ with
$\varepsilon<\varepsilon_{g}$, and class II is for the chaotic diffusion,
i.e., $\langle p_{1}^{2}\rangle\approx\gamma t$ with
$\varepsilon>\varepsilon_{g}$. In the regime of $\cal{PT}$-symmetry breaking,
i.e., $\lambda>\lambda_{c}$, there are also two different classes. In class
III, the system displays the DC $\langle p_{1}\rangle=Dt$ Zhao19  and
ballistic diffusion $\langle p^{2}_{1}\rangle\approx D^{2}t^{2}$ Zhaowl22 .
The corresponding width of quantum state is almost a constant with time
evolution, as the wavepacket is a soliton moving unidirectionally in momentum
space Zhao20cpb . Our main finding in this work is the exotic phenomenon of
the coexistence of the DC and MBD in class IV. Specifically, the momentum
current linearly increases with time

$\langle p_{1}\rangle=Dt\;.$ (7)

The energy diffuses in a MBD of time

$\langle p^{2}_{1}\rangle\approx D^{2}t^{2}+\eta
t^{\alpha}\quad\text{with}\quad\alpha<2\;.$ (8)

Correspondingly, the width of quantum state grows in the power-law of time

$\mathcal{M}_{1}=\eta t^{\alpha}\;.$ (9)

Here, the coefficients $D$ and $\eta$, as well as the exponent $\alpha$ depend
on the system parameters.

In the regime of the unbroken phase of $\cal{PT}$-symmetry with real spectrum
of quasienergies, the transport behavior of this system has no essential
differences with that of Hermitian case. In our numerical investigations, we
choose very small $\lambda$, so that the $\cal{PT}$-symmetry breaking does not
occur with $\varepsilon=0$. Indeed, our numerical results with $\varepsilon=0$
show that there is neither momentum current, i.e., $\langle p_{1}\rangle\sim
0$ [see Fig. 2(a)], nor energy diffusion, i.e., $\langle p^{2}_{1}\rangle\sim
C$ [see Fig. 2(b)], correspondingly $\mathcal{M}_{1}\sim C$ [see Fig. 2(c)],
which is a clear evidence of the appearance of DL. In the regime of the
$\cal{PT}$-symmetry breaking, the system exhibits exotic transport behaviors.
We find that the mean value $\langle p_{1}\rangle$ increases linearly with
time, i.e., $\langle p_{1}\rangle=Dt$ [e.g., $\varepsilon=1$ in Fig. 2(a)],
which demonstrates the emergence of DC. Meanwhile, the energy diffusion
increases in a way of MBD $\langle p^{2}_{1}\rangle\approx D^{2}t^{2}+\eta
t^{\alpha}$ with $\alpha=1$ [e.g., $\varepsilon=1$ in Fig. 2(b)]. The
corresponding width of the time-evolved wavepacket grows as
$\mathcal{M}_{1}=\eta t^{\alpha}$ [e.g., $\varepsilon=1$ in Fig. 2(c)],
indicating the fact that the unbounded spreading of wavepacket occurs. Our
results, therefore, present a solid evidence of the coexistence of the DC and
MBD due to the interplay between the non-Hermitian driving and the coupling.
After extensive investigations on the energy diffusion for different
$\lambda$, we find that $\alpha$ varies with $\lambda$, which demonstrates the
influences of the non-Hermitian driven potential on the energy diffusion.

Figure 2: Time evolution of $\langle p_{1}\rangle$ (a), $\langle
p^{2}_{1}\rangle$ (b), and $\mathcal{M}_{1}$ (c) with $\varepsilon=0$
(circles), 0.2 (triangles), 1 (squares), and 5 (diamonds). Red lines in (a),
(b) and (c) indicate $\langle p_{1}\rangle=Dt$ in Eq. (7), $\langle
p^{2}_{1}\rangle\approx D^{2}t^{2}+\eta t^{\alpha}$ in Eq. (8), and
$\mathcal{M}_{1}=\eta t^{\alpha}$ in Eq. (9) with $\alpha=1$. (d) Momentum
distributions $|\psi_{1}(p)|^{2}$ at the time $t=100$ (circles) and $500$
(triangles) for $\varepsilon=5$. Red lines indicate the fitting functions of
the Gaussian form $|\psi_{1}(p)|^{2}\propto e^{-(p-p_{c})^{2}/\sigma}$. Inset:
same as in the main plot but on a logarithmic y scale. The parameters are
$K=5$, $\hbar_{\mathrm{eff}}=1$, and $\lambda=0.01$.

The probability density distributions of particle 1 in momentum space are
shown in Fig. 2(d). One can see that the momentum distribution can be well
described by the Gaussian function, i.e., $|\psi_{1}(p,t)|^{2}\propto
e^{-[p-p_{c}(t)]^{2}/\sigma(t)}$. Interestingly, the center $p_{c}(t)$ of
Gaussian wavepacket increases with time, which reveals the emergence of the DC
in momentum space. Moreover, its width $\sigma(t)$ also increases with time,
corresponding to the unbound growth of $\mathcal{M}_{1}(t)$. The appearance of
Gaussian distribution is usually regarded as a signature of the loss of
quantum coherence Gadway13 ; Casati06  which results in the exponentially-
localized quantum states, namely, a character of DL Fishman82 ; Casati89 ;
Casati79 , in momentum space. Previous investigations on Hermitian systems
have reported that the coupling induces the spreading of the Gaussian
wavepacket with time, while its center $p_{c}$ is fixed, thus no DC. Our
finding of the coexistence of the increase of both $\sigma$ and $p_{c}$ is a
new kind transport phenomenon due to the quantum decoherence effects in non-
Hermitian chaotic systems.

Figure 3: Time dependence of $\langle p_{1}\rangle$ (a), $\langle
p^{2}_{1}\rangle$ (b), and $\mathcal{M}_{1}$ (c) for $\varepsilon=5$ with
$\lambda=10^{-4}$ (squares), $10^{-3}$ (circles), $0.01$ (up triangles), $0.1$
(down triangles), $1$ (diamonds), and $5$ (pentagrams). Red lines in (a), (b)
and (c) indicate $\langle p_{1}\rangle=Dt$ in Eq. (7), $\langle
p^{2}_{1}\rangle\approx D^{2}t^{2}+\eta t^{\alpha}$ in Eq. (8), and
$\mathcal{M}_{1}=\eta t^{\alpha}$ in Eq. (9) with $\alpha<2$. (d) Momentum
distributions $|\psi_{1}(p)|^{2}$ for $t=500$ and $\varepsilon=5$ with
$\lambda=0.01$ (up triangles), $\lambda=0.1$ (down triangles), $\lambda=1$
(diamonds), and $\lambda=5$ (pentagrams). Red lines indicate the fitting
function of the Gaussian form $|\psi_{1}(p)|^{2}\propto
e^{-(p-p_{c})^{2}/\sigma}$. Other parameters are same as in Fig. 2.

We further numerically investigate the directed transport and energy diffusion
for different $\lambda$ when the coupling $\varepsilon$ is sufficiently strong
so that the $\cal{PT}$-symmetry phase breaking easily emerges for very small
$\lambda$. Figure 3(a) shows that the momentum current linearly increases with
time, i.e., $\langle p_{1}\rangle=Dt$. Meanwhile, the energy diffuses in a
kidn of MBD $\langle p^{2}_{1}\rangle\approx D^{2}t^{2}+\eta t^{\alpha}$, for
which both $\eta$ and $\alpha$ vary with $\lambda$ [see Fig. 3(b)].
Correspondingly, the width of wavepacket increases unboundedly
$\mathcal{M}_{1}=\eta t^{\alpha}$ [see Fig. 3(c)]. The momentum distributions
are shown in Fig. 3(d). One can see that for weak non-Hermitian driving [e.g.,
$\lambda\leq 1$ in Fig. 3(d)] the momentum distribution can be well described
by the Gaussian function. However, for sufficiently large $\lambda$ [e.g.,
$\lambda=5$ in Fig. 3(d)], the quantum state is clearly different from the
Gaussian wavepacket with irregular distribution in momentum space. It is
reasonable to believe that the interplay between non-Hermiticity and coupling
dramatically affects the decoherence effects, which leads to the irregular
form of the momentum distribution.

###  III.2 Growth rate of $\langle p_{1}\rangle$ and $\mathcal{M}_{1}$

Figure 4: (a) Growth rate $D$ versus $\varepsilon$ with $\lambda=10^{-4}$
(triangles), $\lambda=10^{-3}$ (circles), and $\lambda=10^{-2}$ (squares). (b)
The value of $D$ versus $\lambda$ for $\varepsilon=1$ (squares), 2 (circles),
and 5 (triangles). Red dashed line indicates the fitting function of the form
$D\propto\lambda$. (c) Dependence of the $\eta$ on $\varepsilon$ with
$\lambda=10^{-4}$ (triangles), $\lambda=10^{-3}$ (circles), and
$\lambda=10^{-2}$ (squares). Red dashed line indicates the exponential
fitting, i.e., $\eta\propto e^{\beta\varepsilon}$ with $\beta=0.1$. Other
parameters are the same as in Fig. 2.

The growth rates of the momentum current and the width of quantum state are
separately defined by $D=\langle p_{1}(t_{f})\rangle/t_{f}$ and
$\eta=\mathcal{M}_{1}(t_{f})/t_{f}^{\alpha}$. In numerical simulations, the
$t_{f}$ on the scale of hundreds of kicking periods is enough to assure the
high precision of numerical results. Figure 4(a) shows that the $D$ increases
rapidly from a very small value to saturation with increasing $\varepsilon$.
Note that the nonzero value of $D$ for $\varepsilon=0$ is due to the finite
time $t_{f}$ in numerical calculations. The saturation value of $D$ increases
with the increase of $\lambda$, which reveals that the acceleration of
momentum current is only determined by the non-Hermitian driving with no
relation to coupling. As a further step, we numerically investigate the
acceleration rate $D$ for various $\lambda$. Figure 4(b) shows that the value
of $D$ increases linearly with $\lambda$, i.e., $D\propto\lambda$, up to the
saturation. Moreover, the $D$ is almost not dependent on the variation of
$\varepsilon$ if the coupling strength is strong enough. The growth rate
$\eta$ of the $\mathcal{M}_{1}$ for a wide regime of $\varepsilon$ and
$\lambda$ is shown in Fig. 4(c). One can find that the $\eta$ exponentially
increases with $\varepsilon$, but without dependence on the variation of
$\lambda$. Therefore, the spreading of wavepacket in momentum space is mainly
determined by the inter-particle coupling. Since the MBD $\langle
p^{2}_{1}\rangle\approx D^{2}t^{2}+\eta t^{\alpha}$ has two parts, it is clear
that the first part of quadratic growth origins from the non-Hermitian
driving, while the second part, i.e., $\eta t^{\alpha}$ is dominated by the
competition between coupling and non-Hermitian driving potential. Accordingly,
this opens an opportunity for the experimental engineering of the transport
behaviors in momentum-space lattice Dadras18 ; Dadras19 ; Xie20 .

###  III.3 Spontaneous $\cal{PT}$-symmetry breaking

Figure 5: (a) Norm $\mathcal{N}$ versus time with $\lambda=0.01$ for
$\varepsilon=0$ (circles), 0.2 (triangles), 1 (squares), and 5 (diamonds). Red
solid lines indicate the fitting function of the form $\mathcal{N}\propto
e^{\gamma t}$ with $\gamma=0.0035\;(0.0051)$ for $\varepsilon=1$ (5). Green
dashed line marks $\mathcal{N}=1$. (b) The time-averaged value of norm
$\bar{\mathcal{N}}$ in the parameter space ($\lambda$, $\varepsilon$). The red
(blue) area indicates the breaking (un-breaking) phase of
$\mathcal{PT}$-symmetry. Other parameters are the same as in Fig. 2.

It is known that without interaction (i.e., $\varepsilon=0$) there is a
threshold value for the imaginary part of the kicking potential, i.e.,
$\lambda_{c}$, beyond which the system is in the regime of the
$\mathcal{PT}$-symmetry breaking phase. For convenience, the norm
$\mathcal{N}(t_{n})=\sum_{m,n}|\psi_{m,n}(t_{n})|^{2}$ Zhou18a ; Zhou18b ;
Longhi21  is applied to quantify the $\mathcal{PT}$-symmetry phase transition.
Figure 5(a) shows that, for $\varepsilon=0$, the $\mathcal{N}$ remains at
unity as time evolves, which demonstrates the maintenance of the
$\mathcal{PT}$-symmetry phase. It is interesting that, for nonzero value of
$\varepsilon$ (e.g., $\varepsilon=0.2$), the norm increases with time.
Exponential growth of norm, i.e., $\mathcal{N}(t)\propto e^{\gamma t}$, arises
for sufficiently strong coupling (e.g., $\varepsilon=1$), which is a solid
evidence of the spontaneous $\mathcal{PT}$-symmetry breaking. Therefore, the
inter-particle coupling dramatically alters the phase transition point
$\lambda_{c}$. To investigate the dependence of $\lambda_{c}$ on
$\varepsilon$, we numerically calculate the time-averaged value of norm
$\mathcal{\bar{N}}=(1/t_{M})\sum^{M}_{n=1}\mathcal{N}(t_{n})$ in the parameter
space $(\lambda,\varepsilon)$. Figure 5(b) shows that there are clearly two
different regimes, corresponding to $\mathcal{\bar{N}}>1$ and
$\mathcal{\bar{N}}=1$, respectively. Detailed observations show that the phase
transition point $\lambda_{c}$ decreases with the increase of $\varepsilon$,
which reveals the fact that the interaction is helpful to assist breaking the
$\cal{PT}$-symmetry phase. By comparison with Fig. 1, one can find that
classes III and IV of quantum transport is in the region of the breaking phase
of $\cal{PT}$-symmetry.

###  III.4 Time evolution of the Linear entropy

In the decoherence theory, the unavoidable coupling between system and
environment leads to the formation of entanglement. After tracing out the
degree of freedoms of environment, the quantum coherence in the state of
system is destroyed, resulting in a mixed state Zurek03 ; Schlosshauer04 . To
quantify entanglement, we numerically investigate the time evolution of the
linear entropy $S=1-\rm{Tr}(\rho_{1}^{2})$ Santos00 ; Buscemi07 ; Zarate11 .
Figure 6(a) shows that, for a specific $\varepsilon$ (e.g.,
$\varepsilon=0.2$), the $S$ increases linearly with time until saturates,
which demonstrates the formation of entanglement. The saturation value of $S$
increases with $\varepsilon$ up to the maximum value $S_{max}\approx 1$,
representing the growth of entanglement with coupling strength. Furthermore,
we numerically calculate the time-averaged value of the linear entropy, i.e.,
$\bar{S}=(1/t_{M})\sum^{M}_{n=1}S(t_{n})$ for a wide regime of $\varepsilon$
and $\lambda$. Figure 6(b) shows that for a specific $\lambda$ the $\bar{S}$
increases from zero to almost unity with the increase of $\varepsilon$, which
is a solid confirmation of the enhancement of entanglement by coupling.

Figure 6: (a) Linear entropy $S$ versus time for $\lambda=0.01$ with
$\varepsilon=0$ (circles), 0.05 (pentagrams), 0.1 (down triangles), 0.2 (up
triangles), 1 (squares), and 5 (diamonds). (b) Phase diagram of decoherence
quantified by the time-averaged $\bar{S}$ in the parameter space
($\lambda,\varepsilon$). Other parameters are the same as in Fig. 2.

We further investigate the time evolution of the linear entropy $S$ for a wide
regime of $\lambda$. Figure 7 shows that for small $\lambda$ (e.g.,
$\lambda\leq 0.1$) the $S$ increases very rapidly from zero to the saturation
of almost unity, demonstrating the growth of entanglement in the coupled
PTKRs. In this situation, the momentum distribution can be well described by
the Gaussian function [see Fig. 3(d)], which is a character of the onset of
the decoherence effects. Interestingly, the saturation value of $S$ decreases
with the increase of $\lambda$. For instance, the saturation of $S$ with
$\lambda=5$ fluctuates around 0.5 as time evolves. The corresponding
wavepacket differs clearly from the Gaussian function [see Fig. 3(d)], which
may imply that the quantum coherence is partially protected by non-Hermitian
driving. For sufficiently large $\lambda$ (e.g., $\lambda=10$), the $S$
remains almost at zero with time evolution, that is a clear evidence of the
disentanglement of the two particles due to the effects of the non-Hermitian
driving Huang21 ; KQHuang22 . It is reasonable to believe that the loss of
quantum coherence is dramatically affected by the interplay between coupling
and non-Hermitian driven potential. This sheds a light on the issue of the
quantum-classical transition induced by quantum decoherence effects in non-
Hermitian chaotic systems.

Figure 7: Linear entropy $S$ versus time with $\varepsilon=5$ for
$\lambda=10^{-3}$ (squares), $\lambda=0.1$ (circles), $\lambda=1$ (triangles),
$\lambda=5$ (pentagrams), and $\lambda=10$ (diamonds). Other parameters are
the same as in Fig. 2.

##  IV Conclusion and discussion

In this work, we study the transport behaviors in momentum space via a system
of coupled PTKRs. Our investigations show that the phase transition point
$\lambda_{c}$ of the spontaneous $\mathcal{PT}$-symmetry breaking can be
modulated by the coupling strength $\varepsilon$. For $\lambda<\lambda_{c}$,
the energy diffusion of each PTKR exhibits the transition from DL $\langle
p^{2}\rangle\sim C$ [see phase I in Fig. 1] to chaotic diffusion $\langle
p^{2}\rangle\approx\gamma t$ [see phase II in Fig. 1] with increasing the
coupling strength. Meanwhile, there is no the emergence of the DC in momentum
space. The DC $\langle p_{1}\rangle=Dt$ arises in the regime of
$\cal{PT}$-symmetry breaking phase (i.e., $\lambda>\lambda_{c}$). At the same
time, each PTKR exhibits the ballistic diffusion $\langle
p^{2}_{1}\rangle\approx D^{2}t^{2}$ for weak coupling [see phase III in Fig.
1], and the MBD $\langle p^{2}_{1}\rangle\approx D^{2}t^{2}+\eta t^{\alpha}$
for strong coupling [see phase IV in Fig. 1]. We numerically investigate the
time evolution of linear entropy $S$, and find that for large $\varepsilon$
the $S$ increases rapidly from zero to saturation with time evolution. The
saturation value of $S$ increases with increasing $\varepsilon$. We believe
that the decoherence effects are responsible for the appearance of the
intrinsic transport behaviors in the system of coupled PTKRs.

In recent years, the Floquet-driven systems Shirley65 ; Bukova15  with
periodical potential in time domain provide ideal platforms for investigating
novel phenomena, such as quantum thermalization Fleckenstein21 ; Geraedts16 ;
Malishava , many-body quantum chaos Rylands20 ; Zhao14 ; Lundh06 ; Wang20 ,
and topologically-protected transport Roy17 ; Fernandez19 ; Dag22 . The fate
of DL and AL under the effects of interaction has received extensive
investigations in the fields of quantum chaos and condensed matter physics. It
is shown that temporally periodical-modulated nonlinearity even induces
exponentially-fast diffusion in momentum space WLZhao19 ; WLZhao20 ;
Guarneri17 . Our finding of the coexistence of the DC and MBD in coupled PTKRs
serves as an new element of quantum transport in non-Hermitian systems.

Acknowledgements— Jian-Zheng Li is supported by the Dr. Start-up Fund of
Jiangxi University of Science and Technology (No. 205200100067), and the
Science and Technology Research Program of Jiangxi Education Department(No.
GJJ190463). Wen-Lei Zhao is supported by the National Natural Science
Foundation of China (Grant No.12065009) and Science and Technology Planning
Project of Ganzhou City (Grant No. 202101095077). Jie. Liu is supported by the
NSAF (Contract No. U1930403).

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# Anomalous internal conversion — the key to understanding the 209Bi riddle

F. F. Karpeshin
Mendeleev All-Russian Research Institute of Metrology,
Saint-Petersburg, Russia

and
M.B.Trzhaskovskaya
Petersburg Nuclear Physics Institute, Kurchatov Research Center,
Gatchina, Russia

###### Abstract

Theory of anomalous internal conversion is developed, and extended for the
description of the hyperfine splitting. Experimental data on the hyperfine
splitting in the H- and Li-like heavy ions of 209Bi are analyzed in terms of
the Bohr—Weisskopf effect. Agreement with the theory is achieved, scheduling
light on the structure of the magnetization distribution over the nuclear
volume.

##  1 Introduction

A considerable progress during the past decade was archived in investigation
of few-electron heavy ions. Specifically, this concerns study of their
electronic structure and its influence on the nuclear processes. First of all,
this is electromagnetic decay of the nuclei. Nuclear decay can be enhanced
considerably by making use of the resonance with the electronic transitions,
via irradiating with resonance field of a laser [1, 2, 3, 4]. Wonderful
experiments were performed studying the shell effects on the beta decay [5].
There is the comparative study of $\alpha$ decay in H-, He-like ions on the
urgent agenda, with respect to that in neutral atoms [6, 7, 8]. In spite of
that the influence of the electron screening on $\alpha$ decay is a very
important question, in view of many applications in astrophysics and
experiments with laser-produced plasma, it is only recently that the adequate
approach has been found [6]. It was clearly shown that the frozen-shell
approximation, which was used during half century, exaggerates the effect of
the shell at least by an order of magnitude [7]. Moreover, it gives the wrong
sign of the effect [8]. Calculations of the $\alpha$ decay rate in solids and
powerful magic fields performed within this approach provide with the basis
for the wide experimental research [9, 10].

Study of the hyperfine structure in H-like and few-electron heavy ions is of
great interest because it makes a stringent test of electronic functions.
Moreover, it can provide a test of QED. But the latter must be conducted with
proper account of classical theory of internal conversion involving anomalous
conversion. The reason is the Bohr-Weisskopf effect [11], which comprises a
contrinbution of approximately 2% to the hyperfine splitting. This effect is
due to manifestation of the finite distribution of the nuclear magnetic
currents within the nuclear volume. Its actual contribution depends on the
nuclear model [12]. We remind that the Bohr—Weisskopf effect comprises a
contribution to the lowest $1s$ and $2s$ levels in 209Bi of 2% and 2.2%,
respectively. This confines that precision with which the nuclear properties
can be calculated or predicted, in spite of that the electronic wavefunctions
are known with seemingly fantastic accuracy [13]. In order to avoid this
difficulty, a method of specific differences was proposed [14, 15, 16]. It was
suggested that by making use of the specific differences one can diminish it
in the specialy constructed linear combination of the HFS of the two atomic
levels. And the parameter $\zeta$ in the linear combination was listed up to
five digits [13, 17, 18]. However, the nuclear effects cannot be ruled out
completely [12]. Precision of the modern experiment [17] already reached such
level when discrepancy with description in terms of the specific differences
became explicitly seen. This discrepancy is noted in Refs. [17, 18], with no
explanation. In fact, such a failure of the specific difference method was
predicted in Ref. [12]. The alternative, nuclear-model independent way of
dealing with the Bohr-Weisskopf effect, was offered in [12]. It is based on
the consecutive development of the internal conversion theory, specifically,
of bound internal conversion [1, 2, 3, 4, 19]. Apriori, precision of the
proposed in HFS method of the anomalous moments should be higher than that of
the specific differences, as the model-independent method deals with the
actual nuclear moments extracted from the same experiment, whereas the
specific differences method recipe is founded on a simple conventional
Weisskopf model, see below.

In the following sections, we use the method of the anomalous moments, in
order to

1) Better realize the reasons and mechanism of the “cancellation” of the
Bohr—Weisskopf effect in the specific differences.

2) To make clear the limits on the precision of the specific differences
method.
We stress, in order to avoid misleading, that in no case we subject the
results [13, 17, 18], concerning $\zeta$ and $\Delta^{\prime}E$ values, to any
doubts, as they are obtained with electronic functions, which are probably the
best at the time being. But we remind at the same time that these values were
obtained within a simple nuclear model, and explicitly show the admissible
corridor these parameters can take values in, if realistic nuclear models are
considered. We show that the experimental data really fall within this
corridor. We draw a conclusion about the physical nature of the former
discrepancy in the $\Delta^{\prime}E$ values, which existed between [17, 18]
and [13], and thus solve seeming riddle of experiment [17, 18].

##  2 Formulae

The general expression for hyperfine splitting, allowing for the Bohr-
Weisskopf effect, was obtained in Ref. [12] as a series expansion of the
diagonal internal conversion matrix element in the limit of the transition
energy $\omega\to 0$. This expression shows explicitly that the Bohr—Weisskopf
effect is determined by the even $i$-th moments of the magnetization
distribution over the nuclear volume, which can consequently be unambiguously
retrieved from experiment. This expression was obtained using the surface
current (SC) nuclear model [20]. Though the expression contains the only model
parameter — radius of the nuclear current $R_{0}$, resulting nuclear moments
obtained by means of the analysis of the data are independent of the concrete
value of $R_{0}$. They hold in numerical variation of the parameter. This
model independency was demonstrated in Ref. [12] on the example of analysis of
data[17].

In order to separate out the Bohr-Weisskopf effect, one can set the model
radius $R_{0}=0$. This corresponds the no-penetration model in internal
conversion theory ([12] and refs. therein). The remaining terms contain
information about $R_{i}$. After the limiting transformation is performed, the
resulting expression for the hyperfine splitting (HFS) of the $i$-th level
reads as follows:

$w_{i}=\int_{0}^{\infty}g_{i}(r)f_{i}(r)dr-\frac{3}{10}c_{1}^{(i)}{R_{2}}^{2}-\frac{3}{28}c_{3}^{(i)}{R_{4}}^{4}+\ldots\,.$
(1)

Here $g_{i}(r)f_{i}(r)$ are the large an small components of the Dirac
electronic wavefunction of the $i$-th level, $i=1$ for the $1s$ level, and
$i=2$ for the $2s$ level. $c_{i}$ — the coefficients of the expansion of the
sub-integral product of $g_{i}(r)f_{i}(r)$ wave functions near the origin [11,
21, 22]. We remind that in the nuclear vicinity, the electronic wavefunction
is well reproduced by the Taylor series:

$\displaystyle g(r)=a_{0}+a_{2}r^{2}+\ldots,\qquad
f(r)=b_{1}r+b_{3}r^{3}+\ldots\qquad\text{for the $s$ states}\,,$ (2)
$\displaystyle g(r)=a_{1}r+a_{3}r^{3}+\ldots,\qquad
f(r)=b_{0}+b_{2}r^{2}+\ldots\qquad\text{for the $p_{\frac{1}{2}}$ states}\,.$
(3)

By making use of (2), (3), the electronic current in (1) can be put down as
follows:

$j(r)=g_{i}(r)f_{i}(r)=c_{1}^{(i)}r+c_{3}^{(i)}r^{3}+c_{5}^{(i)}r^{5}+\ldots\,.$
(4)

Coefficients $c_{k}^{(i)}$ are obtained by means of solving the corresponding
Dirac equation, with the help of the package of computer codes RAINE [22].
Fermi nuclear charge distribution was supposed, with typical parameters. A
conventional value of $R_{0}=1.2A^{\frac{1}{3}}$ = 7.121 fm was adopted for
the radius of the nuclear charge distribution, $A$ being the mass number of
the nucleus. The electron interaction in the case of Li-like atoms and QED
corrections in [12] were not involved in the electronic wave function, but
added separately within the frame of the perturbation theory.

The set of coupled equations (1) is solved by fit to experimental data. Baring
in mind that data [17] were obtained for the two levels, $1s$ and $2s$, we put
down explicitly two terms for description of the Bohr—Weiskoipf effect, and
neglect the remaining terms in our further consideration, in accordance with
what is said in Ref. [12]. As a result, one can find nuclear magnetization
moments $R_{2}$, $R_{4}$, under condition that the coefficients and
experimental data are correct.

Within the SC model [20], the transition current $J(R)$ is

$J(R)\,=\,D\delta(R)\,\qquad\text{--- SC nuclear model}\,.$ (5)

All the moments $R_{i}\equiv R_{0}$. In another, “opposite” nuclear model of
volume transition currents (VC) [23], assuming

$J(R)\,=\,\left\\{\begin{array}[]{l@{\hspace{2cm}}l}const,\hfil\hskip
56.9055pt&\qquad R\leq R_{0}\\\ \hfil\hskip 56.9055pt&\\\ 0\,,\hfil\hskip
56.9055pt&\qquad R\geq R_{0}\end{array}\right.\,\qquad\text{ --- VC nuclear
model}\,,$ (6)

magnetization moments ${R_{2}}^{2}=\frac{2}{3}{R_{0}}^{2}$,
${R_{4}}^{4}=\frac{1}{2}{R_{0}}^{4}$, etc., generally
${R_{i}}^{i}=\frac{4}{4+i}{R_{0}}^{i}$.

In order to make the underlying physics more transparent, first, temporarily
consider reduced set of equations (1) up to the terms with the second nuclear
moment and the first coefficients in the series expansion $c_{1}$. Then these
last terms only contain information about the Bohr—Weisskopf effect. One can
readily get rid of it in the linear combination, called specific difference:

$\Delta^{\prime}E=w^{2s}-\zeta w^{1s}$ (7)

with an evident solution for the nuclear model independent approximation,
$\zeta_{0}$, to the parameter $\zeta$:

$\zeta_{0}=\frac{c_{1}^{2s}}{c_{1}^{1s}}\,.$ (8)

In this approximation, parameter $\zeta_{0}$ can be calculated with maximal
precision with which the electronic functions are known.

In fact, this model independence is immediately broken with account of the
next, that is the second term of the series expansion in (1). Resulting
approximation $\zeta_{2}$ to the parameter $\zeta$ becomes

$\zeta_{2}=t_{2s}/t_{1s},\qquad
t_{i}=\frac{3}{10}c_{1}^{(i)}{R_{2}}^{2}+\frac{3}{28}c_{3}^{(i)}{R_{4}}^{4}\,.$
(9)

Estimation of the effect of allowance for the fourth moments can be obtained
by substitution of the coefficients found in ref. [12].

For the present purposes, it is enough to use the coefficients in the
approximate wavefunctions, calculated in Ref. [12], in order to access the
range of permissible values of $\zeta$ depending on the model used. The
coefficients are presented in Table 1. These result in the value of (8)
$\zeta_{0}$ = 0.17549, which is independent on the nuclear model and
therefore, the same for the SC and VC models. This value is listed in Table 2.
In the next approximation, involving two terms in the series (1), with
allowance for the $c_{2}$ coefficients and the $R_{4}$ moments, the parameter
$\zeta_{2}$ becomes model-dependent. Its values obtained within the SC and VC
models are also consecutively listed in Table 2. As one can see from the
presented values, account of the second terms diminishes the $\zeta_{2}$ value
as compared to $\zeta_{0}$ by approximately 3.2$\times 10^{-4}$ in the SC
model, and by 2.3$\times 10^{-4}$ — in the VC model. It is important that
within the VC model parameter $\zeta$ turns out to be by approximately
9.1$\times 10^{-5}$ higher than in the SC model. Fast divergence follows the
performed consideration.

##  3 Analysis of the experimental data for 209Bi

A little lower value of $\zeta$ = 0.16886 was reported in [15, 17, 18]. For
the specific difference in Ref. [13] a value of $\Delta_{\text{theor}}$ =
$-$61.320 meV was obtained. Experimental HFS in both $1s$ and $2s$ levels was
studied in refs. [17, 18, 24]. The results disagree with one another beyond
the error bars. We will analyze the later data [17, 18]. For the sake of
completeness, they are presented in Table 1. These were compared to the theory
[13], within the specific difference method. Substitution of these values into
Eq. (7) yields in the specific difference $\Delta^{\prime}E$ = $-$61.37 meV.

As we saw in the previous section, specifically in Table 2, with allowance for
the fourth nuclear moment, at the transition from the SC model to the VC
model, the parameter $\zeta$ increases by 9.1$\times 10^{-5}$. As the
coefficients (1) together with the parameter $\zeta$ vary regularly with
change of the wave function, there is every reason to believe that with
allowance for the higher order effects of QED, together with the electronic
interactions in the case of Li-like state, will not affect this ratio
considerably. In Ref. [13], the results are derived within the conventional
Weisskopf nuclear model. In this model, wavefunctions are assumed to be
constant within the radius $R_{0}$. The nuclear current is defined by the
matrix element
$\langle\bar{\psi}(\mathbf{r})|\mathbf{\gamma}|\psi(\mathbf{r})\rangle$, where
$\mathbf{\gamma}$ are the the three space Dirac matrices,
$\bar{\psi}(\mathbf{r})=\psi^{+}(\mathbf{r})\gamma_{0}$. In terms of the non-
relativistic wavefunctions, this matrix element leads to that of the operator
$(\mathbf{\sigma}\mathbf{p})$, where $\mathbf{p}=-i\nabla$ is the momentum
operator. Therefore, the Weisskopf model may be considered as of the surface
character. Therefore, by direct analogy with section 2, turning to the VC
model, one can anticipate a resulting $\zeta$ value of 2.6$\times 10^{-4}$
higher, which will comprise $\zeta_{\text{teor}}^{\text{VC}}$ = 0.16888.
Applying this higher value to the data [17, 18] in Table 1, we arrive at the
specific difference value of $-$61.461 meV, which value can be conditionally
considered as the upper bound, as explained above. And the value
$\zeta_{\text{W}}$ = 0.16886 therefore becomes like the lower theoretical
bound now, also the related specific difference value
$\Delta^{\prime}E_{\text{theor}}$ = $-$61.320 meV is. We see that the
discrepancy goes away. Instead, experimental data [17, 18],
$\Delta_{\text{exp}}$ = $-$61.373 meV is just between the values given by
these opposite in their physical sense models. It is closer to the SC model.
This is fairly reasonable from the simple consideration on the physical
ground. SC model [20] has a physical justification in the Pauli principle,
which suppresses nucleon motion inside the nucleus, and less suppresses it on
the nuclear surface.

Table 1: Coefficients of then series expansion of the electronic current (4), as calculated in Ref. [12], together with the experimental data concerning HFS  Electronic level |  $c_{1}$, fm-2 |  $c_{3}$, fm-4 |  $w^{\text{exp}}$, eV
---|---|---|---
$1s$ | -4.4123$\times 10^{-9}$ | 3.7255$\times 10^{-11}$ | 5.0863
$2s$ | -7.7431$\times 10^{-10}$ | 6.5494$\times 10^{-12}$ | 0.7975
Table 2: Consecutive approximations $\zeta_{0}$ and $\zeta_{2}$ to the parameter $\zeta$, calculated with the coefficients from Table 1, and the resulting volume-to-surface nuclear model ratio  Approximation to $\zeta$ | $\zeta_{SC}$ | $\zeta_{VC}$ | $\zeta_{VC}/\zeta_{SC}$
---|---|---|---
$\zeta_{0}$ | 0.175489 | 0.175489 | 1
$\zeta_{2}$ | 0.175433 | 0.175449 | 1.00010

##  4 Discussion

We applied results and the nuclear-model independent method of anomalous
nuclear moments developed in [12], deploying it in finer detail, aimed at
analysis of experimental data [17, 18]. The method is based on the set of the
coupled equations (1). From the more general approach provided by this method
we showed explicitly the limits within which the specific difference method
can be considered as independent of the nuclear model, — up to fourth decimal,
which is already lower than the experimental precision. As a result, that
model failed to explain the data, in spite of the excellent electronic
wavefunctions [13], probably the best presently. On the other hand,
experimental data [17, 18] received a reasonable explanation when the method
of anomalous nuclear moments was involved. Use of the parameter $\zeta$
calculated in Ref. [13] with high accuracy, allowed us to analyze experiment
[17, 18] on the basis of the method of the anomalous nuclear moments.
Actually, this way also offers a stringent test of QED, apriori more stringent
in comparison with the specific difference method. Moreover, model
independence of the way [12] may only be realized if both experimental data
are correct, and the electronic functions are precise. This allows one to draw
an unusual conclusion that the method can serve as a test for experimental
data themselves, not only for theory. And the main consequence in our opinion
is [12] that the method can be successfully used for the direct constructive
purpose of retrieving information on the nuclear structure from experimental
values of the Bohr-Weisskopf effect, e.g. as pursued in experiment [25], in
the spirit of the original paper [11], instead of fighting it through the
specific difference method. Such experiments just appropriate at the
contemporary stage of investigations, and can be performed on the storage-ring
facilities e.g. in Lanzhou, GSI Darmstadt.

The authors would like to acknowledge many fruitful discussions of the topic
with L.F.Vitushkin, V.M.Shabaev, I. I. Tupitsin for fruitful discussions and
helpful comments.

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