# Quantum transport in nonlinear Rudner-Levitov models

Lei Du  Beijing Computational Science Research Center, Beijing 100193, China
Jin-Hui Wu  Center for Quantum Sciences and School of Physics, Northeast
Normal University, Changchun 130024, China  M. Artoni<EMAIL_ADDRESS>Department of Engineering and Information Technology and INO-CNR Sensor Lab,
Brescia University, 25133 Brescia, Italy  G. C. La Rocca<EMAIL_ADDRESS>NEST,
Scuola Normale Superiore, 56126 Pisa, Italy

###### Abstract

Quantum transport in a class of nonlinear extensions of the Rudner-Levitov
model is numerically studied in this paper. We show that the quantization of
the mean displacement, which embodies the quantum coherence and the
topological characteristics of the model, is markedly modified by
nonlinearities. Peculiar effects such as a “trivial-nontrivial” transition and
unidirectional long-range quantum transport are observed. These phenomena can
be understood on the basis of the dynamic behavior of the effective hopping
terms, which are time and position dependent, containing contributions of both
the linear and nonlinear couplings.d nonlinear couplings.

##  I Introduction

The Rudner-Levitov (RL) model RLmodel , which can be viewed as a simple non-
Hermitian extension of the Su-Schrieffer-Heeger (SSH) model SSH , has been
predicted to support quantized quantum transport with topological protection
RLmodel ; RLarxiv  has been verified experimentally in a lattice of optical
waveguides RL2015 . This important feature not only shows the possibility of
accessing nontrivial topological phases in physical systems where dissipations
are intrinsic and unavoidable, but also provides a direct way to detect the
topology of such systems by monitoring their bulk properties. Specifically,
for a dimerized ladder of discrete quantum states, with one sublattice lossy
and the other one neutral, the mean displacement of a particle which is
initialized at a neutral site shows well quantized behaviors, depending on the
ratio between the intra- and inter-cell coupling constants.

Recently, the intersection of nonlinear optics and topological photonics has
unlocked a series of peculiar effects that have no counterparts in condensed
matter topologies NonTopoRev , such as magnet-free nonreciprocity MFnr1 ;
MFnr2 , active tunability tuna1 ; tuna2 ; tuna3 , self-induced topological
transitions Hadad ; Hadad2 ; Ezawa , topological solitons tsoliton1 ;
tsoliton2 ; tsoliton3 ; tsoliton4 ; chong2021 , and topological frequency
conversions FC1 ; FC2 ; FC3 . Most recently, Xia et al. have considered a
nonlinear PT-symmetric optical waveguide lattice to investigate the mutual
interplay between topology, PT symmetry, and nonlinearity XiaScience ,
demonstrating that global effects, i.e., topological and PT transitions, can
be actively controlled by the local nonlinearity. The RL model has already
been generalized by including on-site Kerr-type nonlinearities, which shows
that the quantized quantum transport is severely affected by the nonlinearity
induced decoherence effects Rapedius  and that the nonlinear effect becomes
pronounced in the PT-broken region RLSR . However, the study of the influence
of nonlinearity on the quantum transport in the RL model is still at the
initial stage. In particular, work on how nonlinearities affect the hopping
terms, rather than the on-site energies, is still missing; the latter in fact
may affect quantum transport in the RL model precisely because its topological
properties are dictated by the ratio between the intra- and inter-cell
coupling constants.

In this paper, we consider specific extensions of the RL model with different
types of nonlinear couplings and numerically study the quantum transport
therein. As it happens for the case of on-site nonlinearities Rapedius ; RLSR
, our results show that the nonlinear coupling generates departures from the
typical quantized behavior of the particle transport. Our hopping
nonlinearities turn out to be responsible for new effects, such as (i.) a
“trivial-nontrivial” transition, by which the mean displacement will always
take the value of $0$ or $1$ regardless of the linear coupling imbalance, and
(ii.) a long-range transport, by which the particle can move unidirectionally
across many unit cells before its decay. These phenomena can be understood on
the basis of the dynamic behavior of the effective time and position dependent
hopping terms containing contributions of both the linear and nonlinear
couplings. Our findings are relevant to experimental platforms, such as arrays
of coupled optical waveguides and cavities, but may also be adapted to
acoustic or electric circuit resonators as well as superconducting quantum
circuits Kono ; Smirn ; Xue .

##  II The Rudner-Levitov model

Figure 1: (Color online) Schematic illustration of the Ruder-Levitov model,
where $\mu$ ($\nu$) and $\gamma_{a}$ are the intracell (intercell) coupling
constant and the decay rate of the lossy sites, respectively.

We start by briefly reviewing the original RL model RLmodel ; RLarxiv ; RL2015
, where each unit cell of the one-dimensional dimerized lattice contains a
lossy site $A$ and a neutral site $B$, as shown in Fig. 1. The intra- and
inter-cell coupling constants are denoted by $\mu$ and $\nu$, respectively,
while the decay rate of each lossy site is denoted by $\gamma_{a}$. In the
lossless limit $\gamma_{a}\rightarrow 0$, such a linear bipartite lattice is
equivalent to the SSH model SSH  which shows a topological transition
depending on the ratio $\mu/\nu$ zak ; asboth . It has been shown that the
nonvanishing loss in the RL model plays the key role for observing quantized
quantum transport. The real-space Hamiltonian of the RL model is given by
($\hbar=1$ hereafter)

$\begin{split}H_{0}&=\sum_{m}\Big{[}-i\gamma_{a}|m,a\rangle\langle
m,a|-(\nu|m,a\rangle\langle m-1,b|\\\ &\quad\,-\mu|m,a\rangle\langle
m,b|+\text{H.c.})\Big{]},\end{split}$ (1)

where $m\in[-N,N]$ labels the unit cells of the lattice; $|m,a\rangle$ and
$|m,b\rangle$ denote the Wannier states of sites $A$ and $B$ in the $m$th unit
cell, respectively.

Setting $|\psi(t)\rangle=\sum_{m}[a_{m}(t)|m,a\rangle+b_{m}(t)|m,b\rangle]$ as
a general eigenstate of the system, with $a_{m}$ ($b_{m}$) the field amplitude
of $A$ ($B$) in the $m$th unit cell, it is straightforward to attain the
following dynamic equations

$\begin{split}\frac{\partial a_{m}}{\partial t}&=-\gamma_{a}a_{m}+i\nu
b_{m-1}+i\mu b_{m},\\\ \frac{\partial b_{m}}{\partial t}&=i\mu a_{m}+i\nu
a_{m+1}.\end{split}$ (2)

Hereafter in this paper, we assume $\mu=0.5-\delta g$ and $\nu=0.5+\delta g$
with $\delta g$ denoting the coupling imbalance ($|\delta g|\leq 0.5$). Then,
one can numerically calculate the mean displacement of a particle before its
decay, i.e.,

$\langle\Delta
m\rangle=\sum_{m}m\int_{0}^{\infty}2\gamma_{a}|a_{m}(t)|^{2}dt.$ (3)

This is equivalent to the winding number of the relative phase between
components of the Bloch wave function [i.e., the eigenstate of the Fourier
transformation of $H(m)$] and is thus topologically protected RLmodel ;
RLarxiv ; zak ; asboth . Specifically, if one initializes a single particle at
a neutral site, the mean displacement $\langle\Delta m\rangle$ takes the value
of $0$ for $\nu<\mu$ and takes the value of $1$ for $\nu>\mu$, which precisely
predicts the topological phase transition of this model.

Such an archetype model has been well investigated both theoretically RLmodel
; RLarxiv  and experimentally RL2015 , and has also been extended in many
directions Rapedius ; RLSR ; LiExtend ; LieuExtend ; LonghiExtend ; JinExtend
; YuceExtend ; RLdl ; SAP ; WuExtend ; BergExtend . In the following, we
generalize the RL model to a few instances with nonlinear hopping terms and
study the quantum transport therein.

##  III Nonlinearity induced trivial-nontrivial transition

We anticipate that all our nonlinear extensions have symmetric nonlinear
couplings so that the norm of a quantum state still evolves according to
$d\langle\psi|\psi\rangle/dt=-\sum_{m}2\gamma_{a}|a_{m}|^{2}$ as in the linear
RL model (see Appendix A); this validates in turn the definition of the mean
displacement in Eq. (3). In the absence of losses, this assures the
conservation of the standard norm of a quantum state; other nonlinear models,
such as those related to the Ablowitz-Ladik system NDDE ; ALpra ; renormAL ,
do not have this property and are outside the scope of this paper.

Figure 2: (Color online) Mean displacement $\langle\Delta m\rangle$ of the
particle versus coupling imbalance $\delta g$ for (a) model $A$ and (b) model
$B$, with initial state $|\psi(t=0)\rangle=|m=0,b\rangle$, integral time
$\gamma_{a}T=50$, and different values of nonlinear coefficient $U$. (c)
$\langle\Delta m\rangle$ versus $\delta g$ for model $A$ with
$|\psi(t=0)\rangle=|m=0,b\rangle$, $U=0$, and different values of integral
time $T$. (d) $\langle\Delta m\rangle$ versus $\delta g$ for model $A$, with
identical parameters to those in (a) except for $\mu\rightarrow-\mu$ and
$\nu\rightarrow-\nu$. Other parameters are $\mu=0.5-\delta g$, $\nu=0.5+\delta
g$ and$\gamma_{a}=2$.

Model A. First, consider the case in which the intercell coupling terms
contain a Kerr-type nonlinear part that depends on the field amplitudes of
both lossy and neutral sites. The relevant dynamics follows the nonlinear
Schrödinger equations

$\begin{split}\frac{\partial a_{m}}{\partial t}&=-\gamma_{a}a_{m}+i\mu
b_{m}+i(\nu-\xi_{m})b_{m-1},\\\ \frac{\partial b_{m}}{\partial t}&=i\mu
a_{m}+i(\nu-\xi_{m+1})a_{m+1}.\end{split}$ (4)

where $\xi_{m}=U(|a_{m}|^{2}+|b_{m-1}|^{2})$ corresponds to the nonlinear part
of the intercell coupling between the $m$th and $(m-1)$th unit cells, the
strength of which is described by the nonlinear coefficient $U$. Eq. (4)
embeds a non-Hermitian extension of the model in Ref. Hadad  and can be
employed to describe arrays of coupled optical and acoustic cavities, as well
as circuit resonators.

We plot in Fig. 2(a) the mean displacement $\langle\Delta m\rangle$ as a
function of the coupling imbalance $\delta g$ for model $A$ with the initial
state $|\psi(t=0)\rangle=|m=0,b\rangle$. In the absence of nonlinearities
($U=0$), the mean displacement of the particle before its decay is quantized
with two integer values $0$ and $1$: the particle initially localized at the
central neutral site will hop to the lossy site in the right ($m=1$) unit cell
and then decays from the lattice if $\nu>\mu$, or it will hop to the left
lossy site within the initial ($m=0$) unit cell and then decays if $\nu<\mu$.
Note that the transition from $\langle\Delta m\rangle=0$ to $\langle\Delta
m\rangle=1$ here is not perfectly quantized due to the finite integration
time. As shown in Fig. 2(c), the quantization behavior becomes more and more
ideal with the increase of the integration time $T$. As the nonlinear
coefficient increases from zero, the mean displacement deviates from the
quantized behavior gradually. In particular, the original “topological
trivial” region (where $\langle\Delta m\rangle=0$ in the linear case RLmodel ;
RLarxiv ) almost disappears upon increasing $U$, with the values of
$\langle\Delta m\rangle$ approaching unity over the whole parametric range. In
other words, the particle always hops to the right unit cell for large enough
$U$, regardless of the relative values of $\mu$ and $\nu$.

Model B. We consider next the case in which the intracell coupling is instead
modified by a Kerr-type nonlinearity. The dynamics can now be described by,

$\begin{split}\frac{\partial a_{m}}{\partial
t}&=-\gamma_{a}a_{m}+i(\mu-\zeta_{m})b_{m}+i\nu b_{m-1},\\\ \frac{\partial
b_{m}}{\partial t}&=i(\mu-\zeta_{m})a_{m}+i\nu a_{m+1},\end{split}$ (5)

with $\zeta_{m}=U(|a_{m}|^{2}+|b_{m}|^{2})$ and the corresponding numerical
results are plotted in Fig. 2(b). It can be seen that the mean displacement
for model $B$ shows exactly reverse behaviors compared with those for model
$A$: upon increasing $U$, the originally “nontrivial” region (where
$\langle\Delta m\rangle=1$ in the linear case) almost disappears. In this
case, the particle always hops from a neutral site to the left lossy one
within the same unit cell.

Figure 3: (Color online) Dynamic evolutions of displacement $\Delta m(t)$
(upper) and effective coupling contrast $Z_{m}$ (lower) for model $A$, with
(a) $U=0.5$, (b) $U=3$, and (c) $U=5$. (d) Dynamic evolutions of effective
coupling contrast $Z_{1}$ with different values of $U$. The initial state is
assumed as $|\psi(t=0)\rangle=|m=0,b\rangle$. Other parameters are $\delta
g=-0.4$ for (a)-(c), $\delta g=-0.1$ for (d), and $\gamma_{a}=2$.

The results above can be understood from the effective inter- and intra-cell
couplings which contain the contributions of the nonlinear parts. As an
example, we define for model $A$ the effective coupling contrast
$Z_{m}=|\nu_{\text{eff},m}|-|\mu|$ with $\nu_{\text{eff},m}=\nu-\xi_{m}$ the
effective intercell coupling constant, and plot in Fig. 3 the dynamic
evolutions of $Z_{m}$ with different values of $U$. Moreover, we also plot the
evolutions of the time-dependent particle displacement $\Delta
m(t)=\sum_{m}m\int_{0}^{t}2\gamma_{a}|a_{m}(t^{\prime})|^{2}dt^{\prime}$
Rapedius  showing how it reaches its final value. In the absence of
nonlinearities, the coupling contrast $Z_{m}$ reduces to $Z=|\nu|-|\mu|$,
which is position independent and can be used to predict the transition of
$\langle\Delta m\rangle$, i.e., $\langle\Delta m\rangle=1$ if $Z>0$ and
$\langle\Delta m\rangle=0$ if $Z<0$. In the presence of nonlinearities,
however, $Z_{m}$ becomes dependent on $m$. It can be seen from Figs. 3(a)-3(c)
that for $\delta g=-0.4$ ($\nu=0.1$), the value of $Z_{m=1}$ increases with
$U$ from negative to positive monotonously. On the other hand, for $\delta
g=-0.1$ ($\nu=0.4$) as shown in Fig. 3(d), with the increase of $U$, the value
of $Z_{m=1}$ first diminishes until $Z_{m=1}=-|\mu|$, then it turns to
increase with $U$ gradually from negative to positive. This is also why the
the mean displacement shows a non-monotonic behavior as $U$ increases in this
case. In Appendix B, we provide a detailed explanation for the physical
meaning of the positive and negative fractional values of $\langle\Delta
m\rangle$. As a consequence, the topology in the “trivial” region can be
modified by the nonlinearities and thus one can always observe nontrivial
quantum transport for large enough $U$. Although the region where the contrast
$Z_{m}$ is modified by the nonlinearity disappears with time, its duration is
long enough for the particle displacement to reach the final values [see the
figures in the upper row of Figs. 3(a)-3(c)]. Intuitively, the influence of
the nonlinearities disappears after the particle spreads out and decays from
the lattice via the lossy sites. Similarly, the “nontrivial-to-trivial”
transition of model $B$ can also be understood from the effective coupling
contrast.

In Eqs. (4) and (5), we have assumed that the linear and nonlinear parts of
the coupling terms tend to cancel each other, i.e., the signs of $\mu$ and
$\nu$ are opposite to that of $U$. Such an assumption is responsible for the
non-monotonic behavior of $\langle\Delta m\rangle$ as $U$ changes. For
comparison, we reverse the signs of $\mu$ and $\nu$ (i.e.,
$\mu\rightarrow-\mu$ and $\nu\rightarrow-\nu$) in Eq. (4) and plot in Fig.
2(d) $\langle\Delta m\rangle$ versus $\delta g$ in this case. It can be seen
that the values of $\langle\Delta m\rangle$ in the originally trivial region
increase monotonously with $U$ to approach unity over the whole parametric
range. This is different from the non-monotonic behavior in Fig. 2(a). Such a
result is due to the fact that the effective coupling contrast $Z_{m=1}$
increases with $U$ monotonously in this case.

At the end of this section, we would like to point out that the “trivial-
nontrivial” transition can also be observed even if the nonlinear coupling
terms only depend on the field amplitudes of the neutral sites. To prove this,
we consider in Appendix C such a case and demonstrate the behaviors of the
quantum transport therein.

##  IV Nonlinearity induced unidirectional long-range displacement

Figure 4: (Color online) Mean displacement $\langle\Delta m\rangle$ of the
particle versus coupling imbalance $\delta g$ for [(a) and (b)] model $C$ and
(c) model $D$, with initial state $|\psi(t=0)\rangle=|m=0,b\rangle$ and
different values of nonlinear coefficient $U$. We assume $\gamma_{a}=0.2$ for
(a) and (c), while $\gamma_{a}=2$ for (b).

Model {C. & D.} Finally, we consider in this section the case in which the
nonlinear coupling constants are only dependent on the field amplitudes of the
lossy sites. The relevant dynamic equations take the form (Model C.),

$\begin{split}\frac{\partial a_{m}}{\partial t}&=-\gamma_{a}a_{m}+i\mu
b_{m}+i(\nu-\chi_{m})b_{m-1},\\\ \frac{\partial b_{m}}{\partial t}&=i\mu
a_{m}+i(\nu-\chi_{m+1})a_{m+1}\end{split}$ (6)

with $\chi_{m}=U|a_{m}|^{2}$ describing the nonlinear part of the intercell
couplings in this case. Similarly, when the intracell couplings are modified
by the field amplitudes of the lossy sites in a nonlinear manner one has for
the dynamic equations (Model $D.$),

$\begin{split}\frac{\partial a_{m}}{\partial
t}&=-\gamma_{a}a_{m}+i(\mu-\chi_{m})b_{m}+i\nu b_{m-1},\\\ \frac{\partial
b_{m}}{\partial t}&=i(\mu-\chi_{m})a_{m}+i\nu a_{m+1}.\end{split}$ (7)

As in the last section, we first examine the mean displacement $\langle\Delta
m\rangle$ of the particle versus the coupling imbalance $\delta g$ for both
models $C$ and $D$, as shown in Figs. 4(a)-4(c). Interestingly, we find that
long-range particle displacements, i.e., $\langle\Delta m\rangle<0$ or
$\langle\Delta m\rangle>1$, are allowed within these two extensions of the RL
model. As shown in Fig. 4(a), for instance, the value of $\langle\Delta
m\rangle$ can be smaller than $-1$ for large $U$, implying that the particle
initially located at a neutral site can hop on average several times to the
left before its decay if the nonlinearities are strong enough. Note that a
relatively small $\gamma_{a}$ ($\gamma_{a}=0.2$) is assumed in this case in
order to observe the such unidirectional long-range displacements, while for
large $\gamma_{a}$ the mean displacement becomes less sensitive to $U$, as
shown in Fig. 4(b). Moreover, model $D$ displays reverse behaviors of the mean
displacement, i.e., the particle will move to the right with $\langle\Delta
m\rangle>1$ for large enough $U$.

Figure 5: (Color online) Dynamic evolutions of displacement $\Delta m(t)$
(upper) and effective coupling contrast $Z_{m}$ (lower) for model $C$, with
(a) $U=0.5$, (b) $U=3$, and (c) $U=5$. The initial state is assumed as
$|\psi(t=0)\rangle=|m=0,b\rangle$. Other parameters are $\delta g=-0.4$ and
$\gamma_{a}=0.2$.

The underlying physics of the above results can be qualitatively understood as
follows. For positive $\delta g$ the intercell coupling is stronger than the
intracell one, thus the particle initially localized at a neutral site will
hop to the right lossy site and eventually decay. In this case, the value of
the mean displacement cannot be larger than $1$, because the particle will not
keep on moving due to the weak intracell coupling. In contrast, for negative
$\delta g$, the particle will hop to the left lossy site within the initial
unit cell (i.e., the $0$th unit cell in Fig. 4). For model $C$, the effective
intercell coupling between the $0$th and $-1$th unit cells is enhanced after
the particle has moved to the site $|m=0,a\rangle$. If the nonlinearity is
strong enough, the particle can keep on moving towards the left and thereby
the mean displacement should increase with $U$. Of course, owing to the
dissipation at each lossy site, the mean displacement is still finite since
the effective intercell coupling will no longer be stronger than the adjacent
intracell coupling after the particle has moved across several lossy sites.
This is also why the long-range particle displacement tends to vanish if the
decay rate $\gamma_{a}$ is large enough. The reverse behaviors in model $D$
can be interpreted in a similar way.

Similar to the last section, we plot in Figs. 5(a)-5(c) the dynamic evolutions
of the time-dependent displacement $\Delta m(t)$ as well as the effective
coupling contrast $Z_{m}=|\nu-\chi_{m}|-|\mu|$ of model $C$ with different
values of $U$. Once again, one can find that the values of $Z_{m}$ in a
certain region increase with $U$. Unlikely in Fig. 3, however, where only the
nearest-neighbor intercell coupling (the coupling between the $0$th and the
$1$th unit cells) is markedly modified by the nonlinearities, here model $C$
exhibits a larger modification region that extends toward the left side of the
model. This again reflects the origin of the long-range displacements. For all
cases in Fig. 5, the particle displacement reaches its final value before the
modification region disappears.

Figure 6: (Color online) Dynamic evolutions of initial state
$|\psi(t=0)\rangle=|m=0,b\rangle$ for model $C$ with (a) $U=5$ and $\delta
g=-0.4$, (b) $U=3$ and $\delta g=-0.4$, (c) $U=0.5$ and $\delta g=-0.4$, and
(d) $U=0.5$ and $\delta g=0$. In all cases, we take $\gamma_{a}=0.2$.

The results in Fig. 5 can also be verified by the time evolutions of the
initial state $|\psi(t=0)\rangle=|m=0,b\rangle$. As shown in Figs. 6(a)-6(c),
the particle transport tends to be unidirectional as the nonlinearities
strengthen. Note that for $\delta g=0$, the lattice is homogeneous at the
initial time, such that the particle transport should be almost symmetric,
insensitive to the value of $U$. As an example, Fig. 6(d) shows the dynamic
evolution of the initial state for $\delta g=0$ and $U=0.5$, which is similar
to the symmetric diffusion in a one-dimensional homogeneous lattice. In Fig.
6(c), however, the excitation is confined within a narrow region around $m=0$
because the effective intercell couplings are extremely weak (both the linear
and nonlinear parts are weak) in this case.

##  V Conclusions

In summary, we have extended the Rudner-Levitov (RL) model allowing for
nonlinear hopping and assessed how nonlinearities affect the mean displacement
in quantum transport. At variance with on-site nonlinearities, dynamic
modifications of the ratio between the effective intra- and inter-cell
couplings directly affect the behavior of quantum transport related to the
topological characteristics of the RL model. Our results encompass different
extensions of the RL model, which are interpreted qualitatively in terms of
the local and time-dependent effective coupling contrast function $Z_{m}(t)$.
There is one specific extension (See Sect. IV) whereby for strong
nonlinearities and low losses unidirectional long-range quantum transport may
emerge. The present work is relevant to experimental platforms that hinge on
tight-binding Hamiltonians, including non-Hermitian Hamiltonians and those
describing nonlinear effects in optics, in acoustics, and in electronics.

## Acknowledgments

This work is supported by the National Natural Science Foundation of China
(No. U1930402 and No. 12074061), the Cooperative Program by the Italian
Ministry of Foreign Affairs and International Cooperation (MAECI) through
Italy-China International Cooperation (PGR00960), and the “Laboratori
Congiunti” 2019 Program of National Research Council (CNR) of Italy.

##  Appendix A Norm evolution with symmetric couplings

It has been shown that the norm of a quantum state should evolve according to

$\frac{d}{dt}\langle\psi|\psi\rangle=-\sum_{m}2\gamma_{a}|a_{m}|^{2}$ (8)

to guarantee the validity of the definition of the mean displacement in Eq.
(3) RLmodel . In this appendix, we would like to prove that Eq. (8) still
holds for nonlinear RL models with symmetric couplings. For example, according
to Eq. (4) in the main text, the norm of a quantum state in model $A$ should
follows

$\begin{split}\frac{d\langle\psi|\psi\rangle}{dt}&=(\frac{d}{dt}\langle\psi|)|\psi\rangle+\langle\psi|(\frac{d}{dt}|\psi\rangle)\\\
&=\sum_{m}\Big{[}(\frac{da_{m}^{*}}{dt})a_{m}+(\frac{db_{m}^{*}}{dt})b_{m}\\\
&\quad\,+a_{m}^{*}(\frac{da_{m}}{dt})+b_{m}^{*}(\frac{db_{m}}{dt})\Big{]}\\\
&=\sum_{m}\Big{[}-\gamma_{a}|a_{m}|^{2}-ig_{r}b_{m}^{*}a_{m}-i(g_{l}-\xi_{m})b_{m-1}^{*}a_{m}\\\
&\quad\,+\gamma_{a}|a_{m}|^{2}+ig_{r}b_{m}a_{m}^{*}+i(g_{l}-\xi_{m})b_{m-1}a_{m}^{*}\\\
&\quad\,-ig_{r}a_{m}^{*}b_{m}-i(g_{l}-\xi_{m+1})a_{m+1}^{*}b_{m}\\\
&\quad\,+ig_{r}a_{m}b_{m}^{*}+i(g_{l}-\xi_{m+1})a_{m+1}b_{m}^{*}\Big{]}\\\
&=-\sum_{m}2\gamma_{a}|a_{m}|^{2}.\end{split}$ (9)

Similarly, one can verify that models B, C, and D also respect the norm
evolution in Eq. (8) due to the symmetric couplings.

On the other hand, the RL model could also be extended to include asymmetric
couplings. As an example, one could consider a model described by the dynamic
equations

$\begin{split}\frac{\partial a_{m}}{\partial
t}&=-\gamma_{a}a_{m}+i(\mu-\chi_{a,m})b_{m}+i(\nu-\chi_{a,m})b_{m-1},\\\
\frac{\partial b_{m}}{\partial
t}&=i(\mu-\chi_{b,m})a_{m}+i(\nu-\chi_{b,m})a_{m+1}\end{split}$ (10)

with $\chi_{\alpha,m}=U|\alpha_{m}|^{2}$ ($\alpha=a,\,b$), which might be
viewed as a simple dimerization of the Ablowitz-Ladik equation NDDE ; ALpra ;
renormAL . For such a model, however, the evolution of the norm cannot be
simply described by a sum over local decays from each lossy site and the
definition of the mean displacement given in Eq. (3) cannot be applied.

##  Appendix B Coherent vs incoherent transport

It has been shown that the mean displacement $\langle\Delta m\rangle$ displays
continuous dependence on the relative values of $\nu$ and $\mu$ in the case of
incoherent hopping and in the absence of nonlinearities RLmodel . Moreover,
similar incoherent transport behavior has been obtained with on-site
nonlinearities due to the interaction-induced decoherence effect Rapedius .
The underlying physics of such results is quite useful for understanding the
positive and negative fractional values of $\langle\Delta m\rangle$ in Fig. 2.

For incoherent hopping, the original RL model exhibits continuous variation of
$\langle\Delta m\rangle$ from $0$ to $1$ and thus can also take fractional
values. This arises from the competition between the probabilities of the
particle hopping from the initial site to all other sites. The particle
initially located at site $|m=0,b\rangle$ can hop to either $|m=0,a\rangle$ or
$|m=1,a\rangle$, corresponding to $\langle\Delta m\rangle=0$ and
$\langle\Delta m\rangle=1$, respectively. However, the contributions of all
subsequent hoppings should cancel out in this case (the hopping probabilities
from $|m=0,b\rangle$ to $|m=-1,b\rangle$ and to $|m=+1,b\rangle$ are the
same). In view of this, the value of $\langle\Delta m\rangle$ is determined by
the probability of hopping to $A_{1}$ initially, which is given by
$\langle\Delta m\rangle=\nu^{2}/(\nu^{2}+\mu^{2})$ RLmodel ; Rapedius .

For models $A$ and $B$ here considered, the fractional values of
$\langle\Delta m\rangle$ can be understood in a similar way. However, the most
significant difference compared with the linear model above is that the
contributions of long-range hopping processes no longer exactly cancel out due
to the $m$-dependent effective couplings, i.e., the effective couplings on the
left and right sides of the initial site are no longer symmetric. In
particular, also negative fractional values of $\langle\Delta m\rangle$ are
allowed, if the left-hand effective couplings are stronger than the right-hand
ones.

##  Appendix C Nonlinear couplings with only neutral-site contributions

Figure 7: (Color online) (a) Mean displacement $\langle\Delta m\rangle$ of the
particle versus coupling imbalance $\delta g$ for model $E$ with different
values of $U$. (b) Comparison of $\langle\Delta m\rangle$ versus $\delta g$
for models $A$ and $E$ with relatively strong nonlinearities. The initial
state is assumed as $|\psi(t=0)\rangle=|m=0,b\rangle$. Other parameters are
$\mu=0.5-\delta g$, $\nu=0.5+\delta g$ and $\gamma_{a}=2$.

In this appendix, we consider an extended RL model (model $E$) where the
nonlinear intercell couplings only depend on the neutral-site field
amplitudes, described by the dynamic equations

$\begin{split}\frac{\partial a_{m}}{\partial t}&=-\gamma_{a}a_{m}+i\mu
b_{m}+i(\nu-\eta_{m-1})b_{m-1},\\\ \frac{\partial b_{m}}{\partial t}&=i\mu
a_{m}+i(\nu-\eta_{m})a_{m+1}\end{split}$ (11)

with $\eta_{m}=U|b_{m}|^{2}$ in this case.

Due to the fact that the particle is initially located at a neutral site with
$|\psi(t=0)\rangle=|m=0,b\rangle$, the nearest-neighbor intercell coupling can
still be markedly modified at the initial time by the local field amplitude
$b_{0}$, similar to the case of model $A$. In view of this, Fig. 7(a) shows
qualitatively the same behaviors as those in Fig. 2(a): as $U$ increases, the
curve of $\langle\Delta m\rangle$ versus $\delta g$ tends to be flat, with the
values approaching unity over the whole parametric range. In other words, the
“trivial-nontrivial” transition in model $A$ can still be achieved even if the
nonlinear intercell couplings only depend on the neutral-site field
amplitudes. As shown in Fig. 7(b), however, the effective intercell couplings
are a bit weaker due to the absence of the contributions of the lossy sites,
such that the values of $\langle\Delta m\rangle$ for model $E$ are smaller
than those for model $A$, especially in the region of $\delta g<0$. To achieve
similar modification effects, stronger nonlinearities are required in this
case (see for instance the red dashed and green dotted lines).

## References

  * (1) M. S. Rudner and L. S. Levitov, Topological transition in a non-Hermitian quantum walk, Phys. Rev. Lett. 102, 065703 (2009).
  * (2) W. P. Su, J. R. Schrieffer, and A. J. Heeger, Solitons in polyacetylene, Phys. Rev. Lett. 42 1698-1701 (1979).
  * (3) M. S. Rudner, M. Levin, and L. S. Levitov, Survival, decay, and topological protection in non-Hermitian quantum transport, arXiv:1605.07652.
  * (4) J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, Observation of a Topological Transition in the Bulk of a Non-Hermitian System, Phys. Rev. Lett. 115, 040402 (2015).
  * (5) D. Smirnova, D. Leykam, Y. Chong, and Y. Kivshar, Nonlinear topological photonics, Appl. Phys. Rev. 7, 021306 (2020).
  * (6) X. Zhou, Y. Wang, D. Leykam, and Y. D. Chong, Optical isolation with nonlinear topological photonics, New J. Phys. 19, 095002 (2017).
  * (7) W. Chen, D. Leykam, Y. Chong, and L. Yang, Nonreciprocity in synthetic photonic materials with nonlinearity, MRS Bull. 43, 443 (2018).
  * (8) D. Leykam, S. Mittal, M. Hafezi, and Y. D. Chong, Reconfigurable Topological Phases in Next-Nearest-Neighbor Coupled Resonator Lattices, Phys. Rev. Lett. 121, 023901 (2018).
  * (9) A. Amo, When quantum optics meets topology, Science 359, 638 (2018).
  * (10) D. A. Dobrykh, A. V. Yulin, A. P. Slobozhanyuk, A. N. Poddubny, and Y. S. Kivshar, Nonlinear Control of Electromagnetic Topological Edge States, Phys. Rev. Lett. 121, 163901 (2018).
  * (11) Y. Hadad, A. B. Khanikaev, and A. Alù, Self-induced topological transitions and edge states supported by nonlinear staggered potentials, Phys. Rev. B 93, 155112 (2016).
  * (12) Y. Hadad, J. C. Soric, A. B. Khanikaev, and A. Alù, Self-induced topological protection in nonlinear circuit arrays, Nat. Electronics 1, 178 (2018).
  * (13) M. Ezawa, Nonlinearity-induced transition in the nonlinear Su-Schrieffer-Heeger model and a nonlinear higher-order topological system, Phys. Rev. B 104, 235420 (2021).
  * (14) Y. Lumer, M. C. Rechtsman, Y. Plotnik, and M. Segev, Instability of bosonic topological edge states in the presence of interactions, Phys. Rev. A 94, 021801 (2016).
  * (15) D. Solnyshkov, O. Bleu, B. Teklu, and G. Malpuech, Chirality of topological gap solitons in bosonic dimer chains, Phys. Rev. Lett. 118, 023901 (2017).
  * (16) D. A. Smirnova, L. A. Smirnov, D. Leykam, and Y. S. Kivshar, Topological edge states and gap solitons in the nonlinear Dirac model, Laser Photonics Rev. 13, 1900223 (2019).
  * (17) W. Zhang, X. Chen, Y. V. Kartashov, V. V. Konotop, and F. Ye, Coupling of edge states and topological Bragg solitons, Phys. Rev. Lett. 123, 254103 (2019).
  * (18) L.-J. Lang, S.-L. Zhu, and Y. D. Chong, Non-Hermitian topological end breathers, Phys. Rev. B 104, L020303 (2021).
  * (19) S. Kruk, A. Poddubny, D. Smirnova, L. Wang, A. Slobozhanyuk, A. Shorokhov, I. Kravchenko, B. Luther-Davies, and Y. Kivshar, Nonlinear light generation in topological nanostructures, Nat. Nanotechnol. 14, 126 (2019).
  * (20) Y. Wang, L.-J. Lang, C. H. Lee, B. Zhang, and Y. D. Chong, Topologically enhanced harmonic generation in a nonlinear transmission line metamaterial, Nat. Commun. 10, 1102 (2019).
  * (21) D. Smirnova, S. Kruk, D. Leykam, E. Melik-Gaykazyan, D.-Y. Choi, and Y. Kivshar, Third-harmonic generation in photonic topological metasurfaces, Phys. Rev. Lett. 123, 103901 (2019).
  * (22) S. Xia, D. Kaltsas, D. Song, I. Komis, J. Xu, A. Szameit, H. Buljan, K. G. Makris, Z. Chen, Nonlinear tuning of PT symmetry and non-Hermitian topological states, Science 372, 72 (2021).
  * (23) K. Rapedius and H. J. Korsch, Interaction-induced decoherence in non-Hermitian quantum walks of ultracold bosons, Phys. Rev. A 86, 025601 (2012).
  * (24) A. K. Harter, A. Saxena, and Y. N. Joglekar, Fragile aspects of topological transition in lossy and parity-time symmetric quantum walks, Sci. Rep. 8, 12065 (2018).
  * (25) V. V. Konotop, J. Yang, and D. A. Zezyulin, Nonlinear waves in PT-symmetric systems, Rev. Mod. Phys. 88, 035002 (2016).
  * (26) D. Smirnova, D. Leykam, Y. Chong, and Y. Kivshar, Nonlinear topological photonics, Appl. Phys. Rev. 7, 021306 (2020).
  * (27) Z.-Y. Xue and Y. Hu, Topological Photonics on Superconducting Quantum Circuits with Parametric Couplings, Adv. Quant. Techn. 4, 2100017 (2021).
  * (28) P. Delplace, D. Ullmo, and G. Montambaux, Zak phase and the existence of edge states in graphene, Phys. Rev. B 84, 195452 (2011).
  * (29) J. K. Asboth, L. Oroszlany, and A. Palyi, A short course on topological insulators, Lecture Notes in Physics, Vol. 919 (Springer International Publishing, Cham, 2016).
  * (30) L.-H. Li, Z.-H. Xu, and S. Chen, Topological phases of generalized Su-Schrieffer-Heeger models, Phys. Rev. B 89, 085111 (2014).
  * (31) S. Lieu, Topological phases in the non-Hermitian Su-Schrieffer-Heeger model, Phys. Rev. B 97, 045106 (2018).
  * (32) S. Longhi, Probing one-dimensional topological phases in waveguide lattices with broken chiral symmetry, Opt. Lett. 43, 4639 (2018).
  * (33) L. Jin and Z. Song, Bulk-boundary correspondence in a non-Hermitian system in one dimension with chiral inversion symmetry, Phys. Rev. B 99, 081103(R) (2019).
  * (34) C. Yuce, Spontaneous topological pumping in non-Hermitian systems, Phys. Rev. A 99, 032109 (2019).
  * (35) L. Du, J.-H. Wu, M. Artoni, and G. C. La Rocca, Fractional quantum transport and staggered topological transition in a lossy trimerized lattice, Phys. Rev. A 100, 052102 (2019).
  * (36) L. Du, J.-H. Wu, M. Artoni, and G. C. La Rocca, Phase-dependent topological interface state and spatial adiabatic passage in a generalized Su-Schrieffer-Heeger model, Phys. Rev. A 100, 012112 (2019).
  * (37) H. C. Wu, L. Jin, and Z. Song, Topology of an anti-parity-time symmetric non-Hermitian Su-Schrieffer-Heeger model, Phys. Rev. B 103, 235110 (2021).
  * (38) E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Exceptional topology of non-Hermitian systems, Rev. Mod. Phys. 93, 015005 (2021).
  * (39) M. J. Ablowitz and J. F. Ladik, Nonlinear differential–difference equations and Fourier analysis, J. Math. Phys. 17, 1011 (1976).
  * (40) M. Salerno, Quantum deformations of the discrete nonlinear Schrödinger equation, Phys. Rev. A 46, 6856 (1992).
  * (41) M. Salerno, A new method to solve the quantum Ablowitz-Ladik system, Phys. Lett. A 162, 381 (1992).


* Mondéjar _et al._ (2010) Mondéjar, J., J. H. Piclum, and A. Czarnecki, 2010, Phys. Rev. A 81, 062511.
  * Mount _et al._ (2010) Mount, B. J., M. Redshaw, and E. G. Myers, 2010, Phys. Rev. A 82, 042513.
  * Nagy _et al._ (2006) Nagy, S., T. Fritioff, M. Björkhage, I. Bergström, and R. Schuch, 2006, Europhys. Lett. 74(3), 404.
  * Nakamura _et al._ (2010) Nakamura, K., K. Hagiwara, K. Hikasa, H. Murayama, M. Tanabashi, T. Watari, C. Amsler, M. Antonelli, D. M. Asner, H. Baer, H. R. Band, R. M. Barnett, _et al._ , 2010, J. Phys. G 37, 075021.
  * Neronov and Aleksandrov (2011) Neronov, Y. I., and V. S. Aleksandrov, 2011, Pis’ma. Zh. Eksp. Teor. Fiz. 94(6), 452, [JETP Lett. 94(6), 418 (2011)].
  * Neronov and Karshenboim (2003) Neronov, Y. I., and S. G. Karshenboim, 2003, Phys. Lett. A 318(1-2), 126.
  * Nevado and Pineda (2008) Nevado, D., and A. Pineda, 2008, Phys. Rev. C 77, 035202.
  * Newton _et al._ (1979) Newton, G., D. A. Andrews, and P. J. Unsworth, 1979, Philos. Trans. R. Soc. London, Ser. A 290(1373), 373\.
  * Odom _et al._ (2006) Odom, B., D. Hanneke, B. D’Urso, and G. Gabrielse, 2006, Phys. Rev. Lett. 97, 030801.
  * Olver _et al._ (2010) Olver, F. W. J., D. W. Lozier, R. F. Boisvert, and C. W. Clark, 2010, _NIST Handbook of Mathematical Functions_ (National Institute of Standards and Technology and Cambridge University Press, Gaithersburg and Cambridge).
  * Pachucki (1993a) Pachucki, K., 1993a, Phys. Rev. A 48(4), 2609.
  * Pachucki (1993b) Pachucki, K., 1993b, Phys. Rev. A 48(1), 120.
  * Pachucki (1994) Pachucki, K., 1994, Phys. Rev. Lett. 72(20), 3154.
  * Pachucki (1995) Pachucki, K., 1995, Phys. Rev. A 52(2), 1079.
  * Pachucki (2001) Pachucki, K., 2001, Phys. Rev. A 63, 042503.
  * Pachucki (2006) Pachucki, K., 2006, Phys. Rev. Lett. 97, 013002.
  * Pachucki (2008) Pachucki, K., 2008, Phys. Rev. A 78, 012504.
  * Pachucki _et al._ (2005a) Pachucki, K., A. Czarnecki, U. D. Jentschura, and V. A. Yerokhin, 2005a, Phys. Rev. A 72, 022108.
  * Pachucki and Grotch (1995) Pachucki, K., and H. Grotch, 1995, Phys. Rev. A 51(3), 1854.
  * Pachucki and Jentschura (2003) Pachucki, K., and U. D. Jentschura, 2003, Phys. Rev. Lett. 91, 113005.
  * Pachucki _et al._ (2004) Pachucki, K., U. D. Jentschura, and V. A. Yerokhin, 2004, Phys. Rev. Lett. 93, 150401, erratum: see Pachucki _et al._ (2005b).
  * Pachucki _et al._ (2005b) Pachucki, K., U. D. Jentschura, and V. A. Yerokhin, 2005b, Phys. Rev. Lett. 94, 229902.
  * Pachucki and Karshenboim (1995) Pachucki, K., and S. G. Karshenboim, 1995, J. Phys. B 28, L221.
  * Pachucki and Karshenboim (1999) Pachucki, K., and S. G. Karshenboim, 1999, Phys. Rev. A 60(4), 2792.
  * Pachucki and Sapirstein (2010) Pachucki, K., and J. Sapirstein, 2010, in _Lepton Dipole Moments_ , edited by B. L. Roberts and W. J. Marciano (Advanced Series on Directions in High Energy Physics, Vol. 20, World Scientific, Singapore), chapter 7, pp. 219–272.
  * Pachucki and Yerokhin (2009) Pachucki, K., and V. A. Yerokhin, 2009, Phys. Rev. A 79, 062516, erratum: see Pachucki and Yerokhin (2010b).
  * Pachucki and Yerokhin (2010a) Pachucki, K., and V. A. Yerokhin, 2010a, Phys. Rev. Lett. 104, 070403.
  * Pachucki and Yerokhin (2010b) Pachucki, K., and V. A. Yerokhin, 2010b, Phys. Rev. A 81, 039903.
  * Pachucki and Yerokhin (2011a) Pachucki, K., and V. A. Yerokhin, 2011a, Can. J. Phys. 89(1), 95.
  * Pachucki and Yerokhin (2011b) Pachucki, K., and V. A. Yerokhin, 2011b, J. Phys. Conf. Ser. 264, 012007.
  * Park _et al._ (1999) Park, P. G., C. H. Choi, C. S. Kim, V. Y. Shifrin, and V. N. Khorev, 1999, J. Korean Phys. Soc. 34(4), 327.
  * Parks and Faller (2010) Parks, H. V., and J. E. Faller, 2010, Phys. Rev. Lett. 105, 110801.
  * Parthey _et al._ (2011) Parthey, C. G., A. Matveev, J. Alnis, B. Bernhardt, A. Beyer, R. Holzwarth, A. Maistrou, R. Pohl, K. Predehl, T. Udem, T. Wilken, N. Kolachevsky, _et al._ , 2011, Phys. Rev. Lett. 107, 203001.
  * Parthey _et al._ (2010) Parthey, C. G., A. Matveev, J. Alnis, R. Pohl, T. Udem, U. D. Jentschura, N. Kolachevsky, and T. W. Hänsch, 2010, Phys. Rev. Lett. 104, 233001.
  * Penin (2009) Penin, A. A., 2009, Phys. Rev. B 79, 113303, erratum: see Penin (2010).
  * Penin (2010) Penin, A. A., 2010, Phys. Rev. B 81, 089902.
  * Peres (2010) Peres, N. M. R., 2010, Rev. Mod. Phys. 82(3), 2673.
  * Petley and Donaldson (1984) Petley, B. W., and R. W. Donaldson, 1984, Metrologia 20(3), 81.
  * Phillips _et al._ (1977) Phillips, W. D., W. E. Cooke, and D. Kleppner, 1977, Metrologia 13(4), 179.
  * Phillips _et al._ (1984) Phillips, W. D., D. Kleppner, and F. G. Walther, 1984, private communication.
  * Pitre (2011) Pitre, L., 2011, private communication.
  * Pitre _et al._ (2009) Pitre, L., C. Guianvarc’h, F. Sparasci, A. Guillou, D. Truong, Y. Hermier, and M. E. Himbert, 2009, C. R. Physique 10(9), 835.
  * Pitre _et al._ (2011) Pitre, L., F. Sparasci, D. Truong, A. Guillou, L. Risegari, and M. E. Himbert, 2011, Int. J. Thermophys. 32(9), 1825.
  * de Podesta (2011) de Podesta, M., 2011, private communication.
  * de Podesta _et al._ (2010) de Podesta, M., E. F. May, J. B. Mehl, L. Pitre, R. M. Gavioso, G. Benedetto, P. A. Giuliano Albo, D. Truong, and D. Flack, 2010, Metrologia 47(5), 588.
  * Pohl _et al._ (2011) Pohl, R., F. D. Amaro, A. Antognini, F. Biraben, J. M. R. Cardoso, D. S. Covita, A. Dax, S. Dhawan, L. M. P. Fernandes, A. Giesen, T. Graf, T. W. Hänsch, _et al._ , 2011, J. Phys. Conf. Ser. 264, 012008.
  * Pohl _et al._ (2010) Pohl, R., A. Antognini, F. Nez, F. D. Amaro, F. Biraben, J. M. R. Cardoso, D. S. Covita, A. Dax, S. Dhawan, L. M. P. Fernandes, A. Giesen, T. Graf, _et al._ , 2010, Nature 466(7303), 213.
  * Prades _et al._ (2010) Prades, J., E. de Rafael, and A. Vainshtein, 2010, in _Lepton Dipole Moments_ , edited by B. L. Roberts and W. J. Marciano (Advanced Series on Directions in High Energy Physics, Vol. 20, World Scientific, Singapore), chapter 9, pp. 303–317.
  * Puchalski _et al._ (2011) Puchalski, M., U. D. Jentschura, and P. J. Mohr, 2011, Phys. Rev. A 83, 042508.
  * Quinn (1989) Quinn, T. J., 1989, Metrologia 26(1), 69.
  * Quinn (2001) Quinn, T. J., 2001, Metrologia 38(1), 89.
  * Quinn _et al._ (2001) Quinn, T. J., C. C. Speake, S. J. Richman, R. S. Davis, and A. Picard, 2001, Phys. Rev. Lett. 87, 111101.
  * Rainville _et al._ (2005) Rainville, S., J. K. Thompson, E. G. Myers, J. M. Brown, M. S. Dewey, E. G. Kessler Jr, R. D. Deslattes, H. G. Börner, M. Jentschel, P. Mutti, and D. E. Pritchard, 2005, Nature 438(7071), 1096\.
  * Ramsey (1990) Ramsey, N. F., 1990, in _Quantum Electrodynamics_ , edited by T. Kinoshita (World Scientific, Singapore), chapter 13, pp. 673–695.
  * Redshaw _et al._ (2008) Redshaw, M., J. McDaniel, and E. G. Myers, 2008, Phys. Rev. Lett. 100, 093002.
  * Roberts (2009) Roberts, B. L., 2009, private communication.
  * Robinson (2012) Robinson, I. A., 2012, Metrologia 49(1), 113.
  * Robinson and Kibble (2007) Robinson, I. A., and B. P. Kibble, 2007, Metrologia 44(6), 427.
  * Ron _et al._ (2011) Ron, G., X. Zhan, J. Glister, B. Lee, K. Allada, W. Armstrong, J. Arrington, A. Beck, F. Benmokhtar, B. L. Berman, W. Boeglin, E. Brash, _et al._ , 2011, Phys. Rev. C 84, 055204.
  * Rudziński _et al._ (2009) Rudziński, A., M. Puchalski, and K. Pachucki, 2009, J. Chem. Phys. 130, 244102.
  * Sapirstein (2010) Sapirstein, J., 2010, J. Phys. B 43, 074015.
  * Sapirstein _et al._ (1984) Sapirstein, J. R., E. A. Terray, and D. R. Yennie, 1984, Phys. Rev. D 29(10), 2290.
  * Sapirstein and Yennie (1990) Sapirstein, J. R., and D. R. Yennie, 1990, in _Quantum Electrodynamics_ , edited by T. Kinoshita (World Scientific, Singapore), chapter 12, pp. 560–672.
  * Schlamminger _et al._ (2006) Schlamminger, S., E. Holzschuh, W. Kündig, F. Nolting, R. E. Pixley, J. Schurr, and U. Straumann, 2006, Phys. Rev. D 74, 082001.
  * Schmidt _et al._ (2007) Schmidt, J. W., R. M. Gavioso, E. F. May, and M. R. Moldover, 2007, Phys. Rev. Lett. 98, 254504.
  * Schwob _et al._ (1999) Schwob, C., L. Jozefowski, B. de Beauvoir, L. Hilico, F. Nez, L. Julien, F. Biraben, O. Acef, and A. Clairon, 1999, Phys. Rev. Lett. 82(25), 4960, erratum: see Schwob _et al._ (2001).
  * Schwob _et al._ (2001) Schwob, C., L. Jozefowski, B. de Beauvoir, L. Hilico, F. Nez, L. Julien, F. Biraben, O. Acef, and A. Clairon, 2001, Phys. Rev. Lett. 86(18), 4193.
  * Shabaev _et al._ (1998) Shabaev, V. M., A. N. Artemyev, T. Beier, and G. Soff, 1998, J. Phys. B 31(8), L337.
  * Shifrin _et al._ (1998a) Shifrin, V. Y., V. N. Khorev, P. G. Park, C. H. Choi, and C. S. Kim, 1998a, Izmer. Tekh. 1998(4), 68.
  * Shifrin _et al._ (1998b) Shifrin, V. Y., P. G. Park, V. N. Khorev, C. H. Choi, and C. S. Kim, 1998b, IEEE Trans. Instrum. Meas. 47(3), 638\.
  * Shifrin _et al._ (1999) Shifrin, V. Y., P. G. Park, V. N. Khorev, C. H. Choi, and S. Lee, 1999, IEEE Trans. Instrum. Meas. 48(2), 196.
  * Shiner and Dixson (1995) Shiner, D. L., and R. Dixson, 1995, IEEE Trans. Instrum. Meas. 44(2), 518.
  * Sick (2003) Sick, I., 2003, Phys. Lett. B 576(1-2), 62.
  * Sick (2007) Sick, I., 2007, Can. J. Phys. 85(5), 409.
  * Sick (2008) Sick, I., 2008, in _Precision Physics of Simple Atoms and Molecules_ , edited by S. G. Karshenboim (Springer, Lect. Notes Phys. 745, Berlin, Heidelberg), pp. 57–77.
  * Sick (2011) Sick, I., 2011, Few-Body Syst. 50(1-4), 367.
  * Sick (2012) Sick, I., 2012, to be published, Prog. Part. Nuc. Phys.
  * Sienknecht and Funck (1985) Sienknecht, V., and T. Funck, 1985, IEEE Trans. Instrum. Meas. IM-34(2), 195.
  * Sienknecht and Funck (1986) Sienknecht, V., and T. Funck, 1986, Metrologia 22(3), 209.
  * Small _et al._ (1997) Small, G. W., B. W. Ricketts, P. C. Coogan, B. J. Pritchard, and M. M. R. Sovierzoski, 1997, Metrologia 34(3), 241.
  * Smiciklas and Shiner (2010) Smiciklas, M., and D. Shiner, 2010, Phys. Rev. Lett. 105, 123001.
  * Solders _et al._ (2008) Solders, A., I. Bergström, S. Nagy, M. Suhonen, and R. Schuch, 2008, Phys. Rev. A 78, 012514.
  * Steiner _et al._ (2007) Steiner, R. L., E. R. Williams, R. Liu, and D. B. Newell, 2007, IEEE Trans. Instrum. Meas. 56(2), 592.
  * Stock (2011) Stock, M., 2011, Phil. Trans. R. Soc. A 369(1953), 3936\.
  * Stöckinger (2010) Stöckinger, D., 2010, in _Lepton Dipole Moments_ , edited by B. L. Roberts and W. J. Marciano (Advanced Series on Directions in High Energy Physics, Vol. 20, World Scientific, Singapore), chapter 12, pp. 393–438.
  * Sturm _et al._ (2011) Sturm, S., A. Wagner, B. Schabinger, J. Zatorski, Z. Harman, W. Quint, G. Werth, C. H. Keitel, and K. Blaum, 2011, Phys. Rev. Lett. 107, 023002.
  * Suhonen _et al._ (2007) Suhonen, M., I. Bergström, T. Fritioff, S. Nagy, A. Solders, and R. Schuch, 2007, J. Instrum. 2, P06003.
  * Sutton _et al._ (2010) Sutton, G., R. Underwood, L. Pitre, M. de Podesta, and S. Valkiers, 2010, Int. J. Thermophys. 31(7), 1310.
  * Taylor (2011) Taylor, B. N., 2011, J. Res. Natl. Inst. Stand. Technol. 116(6), 797.
  * Taylor and Mohr (2001) Taylor, B. N., and P. J. Mohr, 2001, IEEE Trans. Instrum. Meas. 50(2), 563.
  * Tomaselli _et al._ (2002) Tomaselli, M., T. Kühl, W. Nörtershäuser, S. Borneis, A. Dax, D. Marx, H. Wang, and S. Fritzsche, 2002, Phys. Rev. A 65, 022502.
  * Trapon _et al._ (2003) Trapon, G., O. Thévenot, J. C. Lacueille, and W. Poirier, 2003, Metrologia 40(4), 159.
  * Trapon _et al._ (2001) Trapon, G., O. Thévenot, J.-C. Lacueille, W. Poirier, H. Fhima, and G. Genevès, 2001, IEEE Trans. Instrum. Meas. 50(2), 572.
  * Tu _et al._ (2010) Tu, L.-C., Q. Li, Q.-L. Wang, C.-G. Shao, S.-Q. Yang, L.-X. Liu, Q. Liu, and J. Luo, 2010, Phys. Rev. D 82, 022001\.
  * Uzan (2011) Uzan, J.-P., 2011, Living Rev. Relativity 14(2), 1.
  * Valkiers _et al._ (2011) Valkiers, S., G. Mana, K. Fujii, and P. Becker, 2011, Metrologia 48(2), S26.
  * Valkiers _et al._ (2010) Valkiers, S., D. Vendelbo, M. Berglund, and M. de Podesta, 2010, Int. J. Mass Spectrom. 291(1-2), 41.
  * Van Dyck (2010) Van Dyck, R. S., Jr., 2010, private communication.
  * Van Dyck _et al._ (2006) Van Dyck, R. S., Jr., D. B. Pinegar, S. V. Liew, and S. L. Zafonte, 2006, Int. J. Mass Spectrom. 251(2-3), 231.
  * Van Dyck _et al._ (1987) Van Dyck, R. S., Jr., P. B. Schwinberg, and H. G. Dehmelt, 1987, Phys. Rev. Lett. 59(1), 26.
  * Verdú (2006) Verdú, J., 2006, private communication.
  * Wapstra _et al._ (2003) Wapstra, A. H., G. Audi, and C. Thibault, 2003, Nucl. Phys. A 729(1), 129.
  * Weis and von Klitzing (2011) Weis, J., and K. von Klitzing, 2011, Phil. Trans. R. Soc. A 369(1953), 3954\.
  * Weitz _et al._ (1995) Weitz, M., A. Huber, F. Schmidt-Kaler, D. Leibfried, W. Vassen, C. Zimmermann, K. Pachucki, T. W. Hänsch, L. Julien, and F. Biraben, 1995, Phys. Rev. A 52(4), 2664.
  * Werth (2003) Werth, G., 2003, private communication.
  * Wichmann and Kroll (1956) Wichmann, E. H., and N. M. Kroll, 1956, Phys. Rev. 101(2), 843.
  * Wicht _et al._ (2002) Wicht, A., J. M. Hensley, E. Sarajlic, and S. Chu, 2002, Phys. Scr. T102, 82\.
  * Williams _et al._ (1989) Williams, E. R., G. R. Jones, Jr., S. Ye, R. Liu, H. Sasaki, P. T. Olsen, W. D. Phillips, and H. P. Layer, 1989, IEEE Trans. Instrum. Meas. 38(2), 233.
  * Williams _et al._ (1998) Williams, E. R., R. L. Steiner, D. B. Newell, and P. T. Olsen, 1998, Phys. Rev. Lett. 81(12), 2404.
  * Winkler _et al._ (1972) Winkler, P. F., D. Kleppner, T. Myint, and F. G. Walther, 1972, Phys. Rev. A 5(1), 83.
  * Wood and Solve (2009) Wood, B. M., and S. Solve, 2009, Metrologia 46(6), R13.
  * Wundt and Jentschura (2008) Wundt, B. J., and U. D. Jentschura, 2008, Phys. Lett. B 659(3), 571.
  * Yerokhin (2000) Yerokhin, V. A., 2000, Phys. Rev. A 62, 012508.
  * Yerokhin (2010) Yerokhin, V. A., 2010, Eur. Phys. J. D 58(1), 57.
  * Yerokhin (2011) Yerokhin, V. A., 2011, Phys. Rev. A 83, 012507.
  * Yerokhin _et al._ (2005a) Yerokhin, V. A., A. N. Artemyev, V. M. Shabaev, and G. Plunien, 2005a, Phys. Rev. A 72, 052510.
  * Yerokhin _et al._ (2002) Yerokhin, V. A., P. Indelicato, and V. M. Shabaev, 2002, Phys. Rev. Lett. 89, 143001.
  * Yerokhin _et al._ (2003) Yerokhin, V. A., P. Indelicato, and V. M. Shabaev, 2003, Phys. Rev. Lett. 91, 073001.
  * Yerokhin _et al._ (2005b) Yerokhin, V. A., P. Indelicato, and V. M. Shabaev, 2005b, Phys. Rev. A 71, 040101.
  * Yerokhin _et al._ (2005c) Yerokhin, V. A., P. Indelicato, and V. M. Shabaev, 2005c, Zh. Eksp. Teor. Fiz. 128(2), 322, [JETP 101(2), 280 (2005)].
  * Yerokhin _et al._ (2007) Yerokhin, V. A., P. Indelicato, and V. M. Shabaev, 2007, Can. J. Phys. 85(5), 521.
  * Yerokhin _et al._ (2008) Yerokhin, V. A., P. Indelicato, and V. M. Shabaev, 2008, Phys. Rev. A 77, 062510.
  * Yerokhin and Jentschura (2008) Yerokhin, V. A., and U. D. Jentschura, 2008, Phys. Rev. Lett. 100, 163001.
  * Yerokhin and Jentschura (2010) Yerokhin, V. A., and U. D. Jentschura, 2010, Phys. Rev. A 81, 012502.
  * Yi _et al._ (2012) Yi, H., T. Li, X. Wang, F. Zhang, W. Huo, and W. Yi, 2012, Metrologia 49(1), 62.
  * Zelevinsky _et al._ (2005) Zelevinsky, T., D. Farkas, and G. Gabrielse, 2005, Phys. Rev. Lett. 95, 203001.
  * Zhan _et al._ (2011) Zhan, Z., K. Allada, D. S. Armstrong, J. Arrington, W. Bertozzi, W. Boeglin, J.-P. Chen, K. Chirapatpimol, S. Choi, E. Chudakov, E. Cisbani, P. Decowski, _et al._ , 2011, Phys. Lett. B 705(1-2), 59.
  * Zhang (2011) Zhang, J. T., 2011, private communication.
  * Zhang _et al._ (2011) Zhang, J. T., H. Lin, X. J. Feng, J. P. Sun, K. A. Gillis, M. R. Moldover, and Y. Y. Duan, 2011, Int. J. Thermophys. 32(7-8), 1297.
  * Zhang _et al._ (1995) Zhang, Z., X. Wang, D. Wang, X. Li, Q. He, and Y. Ruan, 1995, Acta Metrol. Sin. 16(1), 1.