Contemplating on Brush Numbers of Mycielski Jaco Graphs,
$\mu(J_{n}(1)),n\in\mathbb{N}$

(Johan Kok, Susanth C, Sunny Joseph Kalayathankal)111Affiliation of author:
Johan Kok (Tshwane Metropolitan Police Department), City of Tshwane, Republic
of South Africa
e-mail<EMAIL_ADDRESS>

Susanth C (Department of Mathematics, Vidya Academy of Science and
Technology), Thalakkottukara, Thrissur-680501, Republic of India
e-mail<EMAIL_ADDRESS>

Sunny Joseph Kalayathankal (Department of Mathematics, Kuriakose Elias
College), Mannanam, Kottayam- 686561, Kerala, Republic of India
e-mail<EMAIL_ADDRESS>
###### Abstract

The concept of the brush number $b_{r}(G)$ was introduced for a simple
connected undirected graph $G$. The concept will be applied to the Mycielski
Jaco graph $\mu(J_{n}(1)),n\in\mathbb{N},$ in respect of an _optimal
orientation_ of $J_{n}(1)$ associated with $b_{r}(J_{n}(1))$.

Further to the above the concept of a _brush centre_ of a simple connected
graph will be introduced. Because brushes themselves may be technology of
kind, the technology in real worl application will normally be the subject of
maitenance or calibration or virus vetting or alike. Finding a _brush centre_
of a graph will allow for well located maintenance centres of the brushes
prior to a next cycle of cleaning.

Keywords: Brush number, Jaco graph, Mycielski Jaco graph, Brush centre

AMS Classification Numbers: 05C07, 05C12, 05C20, 05C38, 05C70

##  1 Introduction

For a general reference to notation and concepts of graph theory see [1]. For
ease of self-containess we shall briefly introduce the concepts of _brush
numbers_ and _Mycielskian graphs_.

###  1.1 The brush number of a simple connected graph $G$

The concept of the brush number $b_{r}(G)$ of a simple connected graph $G$ was
introduced by McKeil [5] and Messinger et. al. [7]. The problem is initially
set that all edges of a simple connected undirected graph $G$ is _dirty_. A
finite number of brushes, $\beta_{G}(v)\geq 0$ is allocated to each vertex
$v\in V(G).$ Sequentially any vertex which has $\beta_{G}(v)\geq d(v)$ brushes
allocated may clean the vertex $v$ and send exactly one brush along a dirty
edge and in doing so allocate an additional brush to the corresponding
adjavent vertex (neighbour). The reduced graph $G^{\prime}=G-vu_{\forall vu\in
E(G),\beta_{G}(v)\geq d(v)}$ is considered for the next iterative cleaning
step. Note that a neighbour of vertex $v$ in $G$ say vertex $u$, now have
$\beta_{G^{\prime}}(u)=\beta_{G}(u)+1.$

Clearly for any simple connected undirected graph $G$ the first step of
cleaning can begin if and only if at least one vertex $v$ is allocated,
$\beta_{G}(v)=d(v)$ brushes. The minimum number of brushes that is required to
allow the first step of cleaning to begin is, $\beta_{G}(u)=d(u)=\delta(G).$
Note that these conditions do not guarantee that the graph will be cleaned.
The conditions merely assure at least the first step of cleaning.

If a simple connected graph $G$ is orientated to become a directed graph,
brushes may only clean along an out-arc from a vertex. Cleaning may initiate
from a vertex $v$ if and only if $\beta_{G}(v)\geq d^{+}(v)$ and $d^{-}(v)=0.$
The order in which vertices sequentially initiate cleaning is called the
_cleaning sequence_ in respect of the orientation $\alpha_{i}$. The minimum
number of brushes to be allocated to clean a graph for a given orientation
$\alpha_{i}(G)$ is denoted $b_{r}^{\alpha_{i}}$. If an orientation
$\alpha_{i}$ renders cleaning of the graph undoable we define
$b_{r}^{\alpha_{i}}=\infty.$ An orientation $\alpha_{i}$ for which
$b_{r}^{\alpha_{i}}$ is a minimum over all possible orientations is called
_optimal_.

Now, since the graph $G$ having $\epsilon(G)$ edges can have $2^{\epsilon(G)}$
orientations, the _optimal orientation_ is not necessary unique. Let the set
$\mathbb{A}=\\{\alpha_{i}|\emph{ $\alpha_{i}$ an orientation of G}\\}.$

######  Lemma 1.1.

For a simple connected directed graph $G$, we have that:

$b_{r}(G)=min_{\emph{over all $\alpha_{i}\in\mathbb{A}$}}(\sum_{v\in
V(G)}max\\{0,d^{+}(v)-d^{-}(v)\\})=min_{\forall\alpha_{i}}b_{r}^{\alpha_{i}}.$

###### Proof.

See [9]. ∎

Although we mainly deal with simple connected graphs it is easy to see that
for set of simple connected graphs $\\{G_{1},G_{2},G_{3},...,G_{n}\\}$ we have
that, $b_{r}(\cup_{\forall i}G_{i})=\sum\limits_{i=1}^{n}b_{r}(G_{i}).$

###  1.2 Mycielskian graph $\mu(G)$ of a graph, $G$

Mycielski [8] introduced an interesting graph transformation in 1955. The
transformation can be described as follows:

(1) Consider any simple connected graph $G$ on $n\geq 2$ vertices labelled
$v_{1},v_{2},v_{3},...,v_{n}$.
(2) Consider the extended vertex set
$V(G)\cup\\{x_{1},x_{2},x_{3},...,x_{n}\\}$ and add the edges
$\\{v_{i}x_{j},v_{j}x_{i}|$ iff $v_{i}v_{j}\in E(G)\\}.$
(3) Add one more vertex $w$ together with the edges $\\{wx_{i}|\forall i\\}.$

The transformed graph _(Mycielskian graph of G or Mycielski G)_ denoted
$\mu(G),$ is the simple connected graph with
$V(\mu(G))=V(G)\cup\\{x_{1},x_{2},x_{3},...,x_{n}\\}\cup\\{w\\}$ and
$E(\mu(G))=E(G)\cup\\{v_{i}x_{j},v_{j}x_{i}|$ iff $v_{i}v_{j}\in
E(G)\\}\cup\\{wx_{i}|\forall i\\}.$

##  2 Brush Numbers of Mycielski Jaco Graphs, $\mu(J_{n}(1)),n\in\mathbb{N}$

The infinite Jaco graph (_order 1_) was introduced in $[2],$ and defined by
$V(J_{\infty}(1))=\\{v_{i}|i\in\mathbb{N}\\}$,
$E(J_{\infty}(1))\subseteq\\{(v_{i},v_{j})|i,j\in\mathbb{N},i<j\\}$ and
$(v_{i},v_{j})\in E(J_{\infty}(1))$ if and only if $2i-d^{-}(v_{i})\geq j.$

The graph has four fundamental properties which are;
$V(J_{\infty}(1))=\\{v_{i}|i\in\mathbb{N}\\}$ and, if $v_{j}$ is the head of
an edge (arc) then the tail is always a vertex $v_{i},i<j$ and, if $v_{k},$
for smallest $k\in\mathbb{N}$ is a tail vertex then all vertices
$v_{\ell},k<\ell<j$ are tails of arcs to $v_{j}$ and finally, the degree of
vertex $k$ is $d(v_{k})=k.$ The family of finite directed graphs are those
limited to $n\in\mathbb{N}$ vertices by lobbing off all vertices (and edges
arcing to vertices) $v_{t},t>n.$ Hence, trivially we have $d(v_{i})\leq i$ for
$i\in\mathbb{N}.$

For ease of reference we repeat a few definitions found in [2].

######  Definition 2.1.

The infinite Jaco Graph $J_{\infty}(1)$ is defined by
$V(J_{\infty}(1))=\\{v_{i}|i\in\mathbb{N}\\}$,
$E(J_{\infty}(1))\subseteq\\{(v_{i},v_{j})|i,j\in\mathbb{N},i<j\\}$ and
$(v_{i},v_{j})\in E(J_{\infty}(1))$ if and only if $2i-d^{-}(v_{i})\geq j.$

######  Definition 2.2.

The family of finite Jaco Graphs are defined by $\\{J_{n}(1)\subseteq
J_{\infty}(1)|n\in\mathbb{N}\\}.$ A member of the family is referred to as the
Jaco Graph, $J_{n}(1).$

######  Definition 2.3.

The set of vertices attaining degree $\Delta(J_{n}(1))$ is called the Jaconian
vertices of the Jaco Graph $J_{n}(1),$ and denoted, $\mathbb{J}(J_{n}(1))$ or,
$\mathbb{J}_{n}(1)$ for brevity.

######  Definition 2.4.

The lowest numbered (indiced) Jaconian vertex is called the prime Jaconian
vertex of a Jaco Graph.

######  Definition 2.5.

If $v_{i}$ is the prime Jaconian vertex of a Jaco Graph $J_{n}(1)$, the
complete subgraph on vertices $v_{i+1},v_{i+2},\cdots,v_{n}$ is called the
Hope subgraph of a Jaco Graph and denoted, $\mathbb{H}(J_{n}(1))$ or,
$\mathbb{H}_{n}(1)$ for brevity.

Note that the Fisher Algorithm determines $d^{+}(v_{i})$ on the assumption
that the Jaco Graph is always sufficiently large, so at least $J_{n}(1),n\geq
i+d^{+}(v_{i}).$ So technically the value $d^{+}(v_{i})$ determined by the
Fisher Algorithm is, $d^{+}(v_{i})=d_{J_{\infty}}^{+}(v_{i}).$ For a smaller
graph the degree of vertex $v_{i}$ is given by
$d_{J_{n}(1)}(v_{i})=d^{-}(v_{i})+(n-i).$ In $[2]$ Bettina’s theorem describes
an arguably, closed formula to determine $d^{+}(v_{i})$. Since
$d^{-}(v_{i})=n-d^{+}(v_{i})$ it is then easy to determine
$d_{J_{n}(1)}(v_{i})$ in a smaller graph $J_{n}(1),n<i+d^{+}(v_{i}).$ It is
important to note that Definition 2.2 read together with Definition 2.1,
prescribes a well-defined orientation of the underlying Jaco graph. So we have
one defined orientation of the $2^{\epsilon(J_{n}(1))}$ possible orientations.
In [4] the following theorem is proven.

######  Theorem 2.1.

For the finite Jaco Graph $J_{n}(1),n\in\mathbb{N},$ with prime Jaconian
vertex $v_{i}$ we have that:

$b_{r}(J_{n}(1))=\sum\limits_{j=1}^{i}(d^{+}(v_{j})-d^{-}(v_{j}))+\sum\limits_{j=i+1}^{n}max\\{0,(n-j)-d^{-}(v_{j})\\}.$

In general we have that if $\beta_{G^{\prime}}(v)$ at a particular cleaning
step has $\beta_{G^{\prime}}(v)>d_{G^{\prime}}(v)$, exactly
$\beta_{G^{\prime}}(v)-d_{G^{\prime}}(v)$ brushes are left redundant and can
clean along new edges linked to vertex $v$ if such are added through
transformation of the graph $G$. The latter observation allows for an adaption
of Theorem 2.1 to obtain the _brush number_ of $\mu(J_{n}(1))n\geq 3$. Note
that $\mu(J_{1}(1))\simeq K_{1}\cup P_{2},$ hence a disconnected graph. Easy
to see that $\mu(J_{2}(1))\simeq C_{5}$ hence $b_{r}(\mu(J_{2}(1))=2.$

######  Theorem 2.2.

For the Jaco graph $J_{n}(1),n\in\mathbb{N},n\geq 2$ the brush number of the
Mycielski Jaco graph is given by:

$b_{r}(\mu(J_{n}(1))=2\sum\limits_{i=1}^{n}d^{+}_{J_{n}(1)}(v_{i}).$

###### Proof.

Consider the Jaco graph, $J_{3}(1)$. From [4] it follows that
$b_{r}(J_{3}(1))=1$ with brush allocations
$\beta_{J_{3}(1)}(v_{1})=1,\beta_{J_{3}(1)}(v_{2})=0,\beta_{J_{3}(1)}=0.$ In
the graph $\mu(J_{3}(1))$ we add the set of vertices
$\\{x_{1},x_{2},x_{3}\\}\cup\\{w\\}$ and we add the edges
$\\{v_{1}x_{2},v_{2}x_{1},v_{2}x_{3},v_{3}x_{2}\\}\cup\\{wx_{1},wx_{2},wx_{3}\\}.$

Clearly $v_{1}$ has degree, $d_{\mu(J_{3}(1))}(v_{1})=2.$ So besides the
normal cleaning sequence within $J_{3}(1)$, vertex $v_{1}$ must either
dispatch a second brush along edge $v_{1}x_{2}$ or await a brush dispatched
from vertex $x_{2}$. In the latter case a brush will be left redundant at
vertex $v_{1}$. So without loss of generality and to ensure optimality,
allocate a second brush to $v_{1}$. At this stage we have the _first brush
number term_ of $\mu(J_{3}(1)),$ $2d^{+}_{J_{3}(1)}(v_{1})=2.$

On initiating the cleaning process one brush will be dispatched to $v_{2}$
along the arc $(v_{1},v_{2})$ (edge $v_{1}v_{2}$). In the next step of
cleaning the brush from vertex $v_{1}$ now at $v_{2}$ can be dispatched to
$v_{3}$ provided that two more brushes are initially allocated to $v_{2}$ to
begin with. But in $\mu(J_{3}(1))$ we have the additional edges
$v_{2}x_{1},v_{2}x_{3},$ so two additional brushes must be allocated to vertex
$v_{2}$. So we have the _second brush number term_ of $\mu(J_{3}(1))$,
$2d^{+}_{J_{3}(1)}(v_{2})=2.$

In the second step of cleaning the brush, initially dispatched from $v_{1}$ to
$v_{2}$ can be dispatched to vertex $v_{3}$. The third step of cleaning may
proceed with no further allocation to $v_{3}$. Hence $v_{3}$ requires _zero_
additional brushes. Now we have the _third and final brush number term_ of
$\mu(J_{3}(1))$, $2d^{+}_{J_{3}(1)}(v_{3})=0.$ Hence we have the result:

$b_{r}(\mu(J_{3}(1)))=2\sum\limits_{i=2}^{3}d^{+}_{J_{3}(1)}(v_{i}).$

We settled the result through induction. Assume the the result holds for
$J_{k}(1)$, with prime Joconian vertex $v_{j}$. Thus we assume that:

$b_{r}(\mu(J_{k}(1)))=2\sum\limits_{i=2}^{k}d^{+}_{J_{k}(1)}(v_{i}),$ holds.

Now consider the Jaco graph $J_{k+1}(1)$ to begin with. This extension adds
the vertex $v_{k+1}$ and the set of arcs,
$\\{(v_{i+1},v_{k+1}),(v_{i+2},v_{k+1}),(v_{i+3},v_{k+1}),...,(v_{n},v_{k+1})\\}$
to $J_{k}(1)$ to obtain $J_{k+1}(1).$

Each vertex $v_{j},i+1\leq j\leq k$ receives two additional arcs namely
$(v_{j},v_{k+1})$ and $(v_{j},x_{k+1})$ in the transformed Mycielski Jaco
graph. The minimum additional brushes per such vertex is thus two. We have
that the respective brush number terms of $\mu(J_{k+1}(1))$ are
$(d^{+}_{J_{k}}(v_{j})+2)=2d^{+}_{J_{k+1}(1)}(v_{j}).$ This implies we have
the partial brush number:

$2\sum\limits_{i=1}^{k}d^{+}_{J_{k+1}(1)}(v_{i}).$

With regards to the vertex $v_{k+1}$ we have,
$d_{J_{k+1}(1)}(v_{k+1})=d^{-}(v_{k+1})$ stemming from the $k-j$ vertices,
$v_{j+1},v_{j+2},v_{j+3},...,v_{k}.$ Hence, $k-i$ brushes will be allocated to
vertex $v_{k+1}$ through the iterative cleaning process. In the Mycielski Jaco
graph the additional edges
$v_{k+1}x_{k},v_{k+1}x_{k-1},v_{k+1}x_{k-2},...,v_{k+1}x_{j+1}$, (_exactly
$k-j$ edges_) have been added to vertex $v_{k+1}$ so no additional brushes are
needed. It means that the _final brush number term_ is,
$2d^{+}_{J_{k+1}(1)}(v_{k+1})=2.0=0.$ Since all the brush number terms of
$\mu(J_{k+1}(1))$ have now been determined at the absolute minimum we have the
result, $b_{r}(\mu(J_{k+1}(1))=2\sum\limits_{i=1}^{k+1}d^{+}(v_{i}).$

Through induction we settle the result that:

$b_{r}(\mu(J_{n}(1))=2\sum\limits_{i=1}^{n}d^{+}_{J_{n}(1)}(v_{i}),\forall
J_{n}(1),n\in\mathbb{N},n\geq 3.$ ∎

##  3 Brush Centre of a Graph

From Theorem 2.1 and Lemma 1.1 the brush allocations can easily be determined
for Jaco graphs. See [4]. For example, $J_{9}(1)$ requires the minimum brush
allocations,
$\beta_{J_{9}(1)}(v_{1})=1,\beta_{J_{9}(1)}(v_{2})=0,\beta_{J_{9}(1)}(v_{3})=1,\beta_{J_{9}(1)}(v_{4})=2,\beta_{J_{9}(1)}(v_{5})=1,\beta_{J_{9}(1)}(v_{6})=1,\beta_{J_{9}(1)}(v_{7})=0,\beta_{J_{9}(1)}(v_{8})=0,\beta_{J_{9}(1)}(v_{9})=0.$
We note that the allocations of $\beta(v_{i})>0$ are located at vertices
$v_{1},v_{3},v_{4},v_{5},v_{6}$. The _end_ allocation itself is a minimum
allocation associated with an optimal orientation.

So far cleaning was restricted to a brush transversing a dirty edge only once.
If the latter restriction is relaxed to, after the first complete cleaning
sequence a brush may transverse an edge for a second time for another complete
reversed cleaning sequence, the initial allocation of brushes or a deviation
thereof can be obtained.This observation leads to the concept of a _brush
centre_. The question is, what is the minimium set of vertices,
$\mathbb{B}_{r}(G)\subseteq V(G)$ (_primary condition_) to allocate the
$b_{r}(G)$ brushes to, to ensure cleaning of graph $G$ and on return _(second
cleaning)_ the brushes are clustered as centrally as possible for
_maintenance_ (_secondary condition is the min(max(distance between vertices
of the brush centre))_). Finding a _brush centre_ of a graph will allow for
well located maintenance centres of the brushes prior to a next cycle of
cleaning. Because brushes themselves may be technology of kind, the technology
in real worl application will normally be the subject of maitenance or
calibration or virus vetting or alike.

It is easy to see that for the path $P_{n}$ the allocation of one brush to
either $\\{v_{1}\\}$ or $\\{v_{n}\\}$ is a minimum set and clustered
absolutely centrally so both represent a _brush centre_. So $P_{n}$ has two
possible _brush centres_. Similarly, the allocation of two brushes to any set
$\mathbb{B}_{r}(C_{n})=\\{v_{i}\\},v_{i}\in V(C_{n})$ of the cycle cycle
$C_{n}$ represents a _brush centre_. So $C_{n}$ has $n$ possible _brush
centres_. Hence the _brush centre_ of a graph $G$ is not necessary unique.
However, for the star $K_{1,n}$ the _brush centre_ is indeed unique, namely,
$\mathbb{B}_{r}(K_{1,n})=\\{v_{1}\\}_{\emph{($v_{1}$ central)}}$, with
$\beta_{K_{1,n}}(v_{1})_{\emph{($v_{1}$ central)}}=n.$

###  3.1 Brush centre of the Mycielski Jaco Graph,
$\mu(J_{n}(1)),n\in\mathbb{N}$

Let us immediately jump paths and consider $J_{5}(1)$. In the defined Jaco
graph $J_{5}(1)$ the brush number is $b_{r}(J_{5}(1))=2$ [4], with the brush
allocation
$\beta_{J_{5}(1)}(v_{1})=1,\beta_{J_{5}(1)}(v_{2})=0,\beta_{J_{5}(1)}(v_{3})=1,\beta_{J_{5}(1)}(v_{4})=0,\beta_{J_{5}(1)}(v_{5})=0.$
We note that after the first cleaning sequence both brushes are allocated to
the vertex $v_{5}$. The latter allocation of brushes with an appropriate re-
orientation of $J_{5}(1)$ also clean the Jaco graph. On a second cleaning
sequence the brushes can _park_ at $v_{5}$ for _maintenance_. Clearly the set
$\\{v_{5}\\}$ with $\beta_{J_{5}(1)}(v_{5})=2$ is a (_the_) brush centre.

######  Theorem 3.1.

Consider the initial minimal brush allocation of $b_{r}(J_{n}(1))$ brushes to
the finite Jaco graph, $J_{n}(1),n\in\mathbb{N}$, [4]. The location of the
brushes at the end of the cleaning sequence represents a brush centre of
$J_{n}(1),n\in\mathbb{N}.$

###### Proof.

For the paths $J_{1}(1),J_{2}(1),J_{3}(1),J_{4}(1)$ the result is obvious. As
observed in the introduction the set $\mathbb{B}_{r}(J_{5}(1))=\\{v_{5}\\}$
with $\beta_{J_{5}(1)}(v_{5})=2$ is a (_the_) brush centre of $J_{5}(1)$.

We prove the result through induction. Assume the result holds for $J_{k}(1)$
which has the prime Jaconian vertex $v_{i}$. So our assumption implies that
after the first cleaning sequence the brushes are clustered amongst vertices
in the vertex set
$\mathbb{B}_{r}\subseteq\\{v_{i+1},,v_{i+2},v_{i+3},..,v_{k}\\}$ in compliance
with both the _primary condition_ and the _secondary condition_ namely, the
min(max(distance between vertices of the brush centre)). Note that
$min(max(d_{v,u\in\mathbb{B}_{r}}(v,u)))=1$ because
$\mathbb{B}_{r}(J_{k}(1))\subseteq\mathbb{H}(J_{k}(1)).$

Now consider the Jaco graph $J_{k+1}(1)$ to begin with. This extension adds
the vertex $v_{k+1}$ and the set of edges,
$\\{v_{j+1}v_{k+1},v_{j+2}v_{k+1},v_{j+3}v_{k+1},...,v_{n}v_{k+1}\\}$ to
$J_{k}(1)$ to obtain $J_{k+1}(1).$

Each vertex $v_{j},i+1\leq j\leq k$ receives one additional edges namely
$v_{j}v_{k+1}.$ In addition the prime Jaconian vertex may, or may not change
to $v_{i+1}$. Exactly $k-i$ new edges were added in the extension from
$J_{k}(1))$ to $J_{k+1}(1)$. Since $b_{r}(J_{k+1}(1))\geq b_{r}(J_{k}(1))$
additional brushes might be needed at some of the vertices
$v_{i+1},v_{i+2},...,v_{k_{1}}$ if $v_{i}$ remains the prime Jaconian vertex
of $J_{k+1}(1).$ Else, additional brushes might be needed at some of the
vertices $v_{i+2},...,v_{k_{1}}$. Note that the vertex $v_{k}$ will not
require additional brushes. Since the minimum additional brushes to be
allocated is always possible (_primary condition_) and
$\mathbb{B}_{r}(J_{k+1}(1))\subseteq\mathbb{H}(J_{k+1})$ the
$min(max(d_{v,u\in\mathbb{B}_{r}(J_{k+1}(1))}(v,u)))=1$ (_secondary
condition_), the result follows for $J_{k+1}(1).$

Hence the result is settled for all Jaco graphs, $J_{n}(1),n\in\mathbb{N}.$ ∎

_Open access:_ This paper is distributed under the terms of the Creative
Commons Attribution License which permits any use, distribution and
reproduction in any medium, provided the original author(s) and the source are
credited.

References (Limited)

$[1]$ Bondy, J.A., Murty, U.S.R., _Graph Theory with Applications,_ Macmillan
Press, London, (1976).
$[2]$ Kok, J., Fisher, P., Wilkens, B., Mabula, M., Mukungunugwa, V.,
_Characteristics of Finite Jaco Graphs, $J_{n}(1),n\in\mathbb{N}$_, arXiv:
1404.0484v1 [math.CO], 2 April 2014.
$[3]$ Kok, J., Fisher, P., Wilkens, B., Mabula, M., Mukungunugwa, V.,
_Characteristics of Jaco Graphs, $J_{\infty}(a),a\in\mathbb{N}$_, arXiv:
1404.1714v1 [math.CO], 7 April 2014.
$[4]$ Kok, J., _A note on the Brush Number of Jaco Graphs,
$J_{n}(1),n\in\mathbb{N}$_, arXiv: 1412.5733v1 [math.CO], 18 December 2014.
$[5]$ McKeil, S., _Chip firing cleaning process_ , M.Sc. Thesis, Dalhousie
University, (2007).
$[6]$ Messinger, M. E., _Methods of decontaminating a network_ , Ph.D. Thesis,
Dalhousie University, (2008).
$[7]$ Messinger, M.E., Nowakowski, R.J., Pralat, P., _Cleaning a network with
brushes_. Theoretical Computer Science, Vol 399, (2008), 191-205.
$[8]$ Mycielski, J., _Sur le coloriages des graphes_ , Colloq. Maths. Vol 3,
(1955), 161-162.
$[9]$ Ta Sheng Tan,_The Brush Number of the Two-Dimensional Torus,_ arXiv:
1012.4634v1 [math.CO], 21 December 2010.


MIT-CTP 5593

# Floquet engineering with spatially non-uniform driving fields

Stella T. Schindler1 Hanan Herzig Sheinfux2<EMAIL_ADDRESS>1Center for Theoretical Physics, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA
2Physics Department, Bar-Ilan Univeresity, Ramat Gan, 5290002 Israel

###### Abstract

We generalize the scope of Floquet engineering to include spatially-dependent
modulations of an optical system. As an application, we show that we can
transform large classes of Hamiltonians into one another by driving them in a
time-periodic but spatially non-uniform manner. We propose several
experimental realizations in 1D optical lattices, including freeing disordered
lattices from Anderson localization, as well as effectively disconnecting all
their sites. These techniques straightforwardly extend to more complex classes
of systems.

Driving a system periodically can generate new out-of-equilibrium phases,
which can exhibit quite different properties from the original system. Driven
phases are ubiquitous and extensively studied in physics, from Kapitza’s
pendulum in mechanics [1] to Thouless pumping in quantum physics [2, 3]. An
important milestone for periodic driving was the discovery of Dynamic
Localization (DL) in atomic physics in 1986 [4]. Naively, one might expect
that applying a periodic force to a wavepacket would cause it to accelerate or
scatter. However, as the name suggests, in DL one can drive a lattice
containing a single electron in a way that suppresses the coupling between
lattice sites, making them behave as if fully disconnected. This
counterintuitive influence of DL has been observed in manifold experiments and
is now understood to be a general wave phenomenon [5, 6, 7, 8, 9, 10, 11, 12,
13, 14, 15, 16, 17, 18].

In the last two decades, there has been a surge of interest in so-called
Floquet engineering, a considerable extension of DL. Floquet engineering
encompasses a range of techniques to modify a wave system’s behavior: not just
arresting expansion (as is the case with DL), but also the application of
artificial gauge fields and the imbuing of a system with topological
properties. Experimentally, Floquet driving has been realized in a range of
systems, from condensed matter [19, 20, 21, 22, 23, 24], to cold atoms [25,
26, 27, 28, 29, 30, 31, 32], to photonics [33, 34, 35, 36, 37, 38, 39, 40,
41]. Notably, the demonstration of Floquet topological insulators in curved
waveguide arrays had a progenitive role in topological photonics [42, 43, 44,
45, 46, 47, 48, 49].

The beauty of Floquet engineering lies in its simplicity: driving a field
periodically gives us new degrees of control over a system, allowing us to
generate novel behaviors. At the same time, conventional Floquet engineering
is also a rather blunt instrument, with only a couple parameters (driving
strength, frequency, and in some cases polarization) giving us limited
leverage to induce a desired behavior in a prefabricated physical platform.
The exploitation of spatial degrees of freedom has only been explored in
limited contexts [21, 50, 51, 52, 53].

Here, we open up spatial degrees of freedom in Floquet engineering more
broadly, giving us greater control over systems. Specifically, we show that
with properly tailored driving fields, we can make a large class of 1D
nearest-neighbor tight-binding Hamiltonians behave as any other, to leading
order in perturbation theory. This allows us to carry out applications
previously thought impossible, such as making a lattice effectively free of
disorder by undoing Anderson Localization (AL) [54, 55, 56, 57, 58, 59, 60,
61, 62, 63, 64] and turning the system to a diffusive transport state. We also
show that applying a spatially dependent driving can do the opposite: inducing
AL in a regular lattice.

In this paper, we focus on the concrete example of using curvature to induce a
driving field in paraxial waveguide arrays, and we provide a recipe for
achieving desired phenomena. Spatially non-uniform Floquet engineering
techniques are, however, mathematically general and thus broadly applicable
across other physical platforms, driving field types, and phenomena.

Curved waveguide setup. Consider an array of paraxial waveguides with low
refractive index contrast. In the slowly-varying envelope (paraxial)
approximation, this system obeys a Schrödinger-like equation
$i\partial_{z}\psi(\vec{x},z)={\cal{H}}\psi=\Big{(}\frac{i}{2k_{0}}\nabla^{2}-\frac{k_{0}}{r_{0}}[n(\vec{r})-n_{0}]\Big{)}\psi.$
Here, ${\cal{H}}$ is the Hamiltonian, $\psi$ is the wavefunction, the
propagation axis $z$ plays the role of time, $n(\vec{r})$ is the local
distribution of the refractive index, $n_{0}$ is the homogeneous waveguide
cladding index, and $k_{0}=2\pi n_{0}/\lambda$ is the wavenumber. In this
approximation, the array behaves like a tight-binding model.

To induce a driving field in a waveguide array, we curve the waveguides along
zig-zagged paths in the $x$-$z$ plane, as sketched in Fig. 1. Each waveguide
has a periodically undulating path and width, altering the refractive index in
a manner mathematically equivalent to driving a quantum system with a
$z$-periodic field $F(z)$. We give $F(z)$ period $T$ and frequency
$\omega=2\pi/T$.

We can carry out a Magnus (perturbative) expansion of
${\cal{H}}(z)={\cal{H}}_{0}+F(z)$ to find a $z$-independent effective
description ${\cal{H}}_{\rm eff}$ of system behavior:

$\displaystyle T{\cal{H}}_{\rm
eff}\\!=\\!\\!\int_{0}^{T}\\!\\!\\!\\!dz_{1}{\cal{H}}(z_{1})\\!+\\!\frac{1}{2}\\!\int_{0}^{T}\\!\\!\\!\\!\\!\\!dz_{1}\\!\\!\int_{0}^{z_{1}}\\!\\!\\!\\!\\!\\!dz_{2}[{\cal{H}}(z_{1}),{\cal{H}}(z_{2})]\\!+\\!...$
(1)

The first term is ${\cal{O}}(\omega^{0})$ and describes the dispersion that
would occur were driving force averaged out. The second term, ${\cal{H}}_{1}$,
is an ${\cal{O}}(\omega^{-1})$ correction. (In DL,
${\cal{H}}_{0}+{\cal{H}}_{1}$ is often treated as a single term.) Note that
${\cal{H}}_{0}$ may depend on the driving field and differ from the undriven
case.

At frequencies of experimental interest, it is often sufficient to truncate
the expansion at the second term. In this case, finding an appropriate driving
field $F(t)$ that makes ${\cal{H}}_{0}$ behave as ${\cal{H}}_{\rm eff}$ is
equivalent to solving an inverse commutator problem. The essence of this paper
lies in eq. (1) and in our ability to approximately solve the inverse
commutator problem in a broad range of cases.

Figure 1: System schematic, side and top views. Set of waveguides evenly
spaced in the $x$-direction but randomly space along the $y$-axis. Waveguides
take periodic trajectories in space and have periodic thickness modulations.
Shape is exaggerated for purpose of illustration.

Reversing Anderson localization. Let us consider a simple case: nearest-
neighbor (nn) coupled waveguides, described by a tridiagonal matrix. The
waveguides oscillate in unison, so their distances are $z$-independent. We
make the waveguide trajectories locally parabolic, inducing an artificial
gauge field on the wavepackets and causing the nn couplings $\gamma$ to rotate
phase as $\gamma_{k}\exp(\pm i\omega z)$. We vary the on-site potential and
waveguide thickness as $V$ and $\epsilon\cos(\omega z)$, alternating sign
$\pm$ after each period. The propagation of light through this system is
governed by the $z$-dependent Hamiltonian

$\displaystyle{\cal{H}}(z)=\gamma e^{i\omega z}+\gamma^{\dagger}e^{-i\omega
z}+V+\epsilon\cos(\omega z)$ (2)

We take our waveguides to be uniformly spaced in the $x$-direction and
randomly spaced in $y$, so that the distances (and thus couplings) between
each waveguide pair are random. For our calculations, we take the couplings
$\gamma_{k}\in[0.2,1]$ in eq. (2) from a uniform random distribution.

In Fig. 2a, we show how waves propagate through the array described by eq. (2)
when the waveguides are straight (zero driving field). These waveguides are
disordered, so we expect Anderson Localization to bring transverse transport
in the array to a halt [65]. In Fig. 2a, we see that a localized beam incident
on the array does indeed remain localized.

In Fig. 2b, we consider propagation through the waveguide when acted upon by a
simple, spatially-uniform AC driving field. As expected, an AC field generally
causes a wavepacket to diffuse and eventually arrests its propagation, except
in extreme cases, like when the field strength dwarfs the disorder.

In Fig. 2c, we construct the curvature (driving field) that counteracts the
disorder in the waveguides, making the system behave as an effectively
uniform, ordered array. To do so, we calculate appropriate curvatures (choices
of $V$ and $\epsilon$) that create the desired effective Hamiltonian using
numerical optimization. Notably, we can make a choice of $\epsilon$ that is
pure diagonal.

The tailored driving field (curvature) removes disorder up to
${\cal{O}}(\omega^{-2})$ corrections, which are too small to induce
localization. In many cases, it is possible to reduce the effect of
${\cal{O}}(\omega^{-2})$ corrections without explicitly working at next order
in the expansion of $H_{\rm eff}$, by numerically optimizing the choice of $V$
and $\epsilon$. All results here are obtained for relatively low $\omega=6$.
Because the magnitude of ${\cal{H}}_{\rm eff}$ is proportional to
$\omega^{-1}$, the dynamics of the driven system are comparable in speed to
the dynamics of the undriven system, a clear advantage for experiments.

Applying an arbitrary non-periodic driving force (not shown) results in
diffusive behavior and possibly even faster-than-ballistic anomalous diffusive
transport [66] and should not be confused with the disorder-free case.

Inducing dynamic localization. DL is believed to generally fail for
inhomogeneous lattices, though a notable exception is Glauber-Fock lattices
[67]. It is clear, however, from our explorations of eqs. (1) and (2) that an
appropriately tailored driving field could induce DL in such a lattice, giving
us the powerful ability to sever specific links or even cut out part of a
lattice. For example, we could create an artificial sharp edge in a
topological system like in Refs. [27, 25] and expect to see edge modes there.
This technique can extend to complicated, highly-nonuniform, and even
multidimensional lattices, which all obey a similar operator form to eq. (1).

Other physical platforms. When extending our driving scheme beyond curved
waveguides, we must strike a delicate balance between challenges in the
mathematical calculation and physical implementation. The nature of the Magnus
expansion places restrictions on the Hamiltonians and driving fields.

Figure 2: Wave propagation in nearest-neighbor coupled waveguides with and
without driving, described in eq. (2). (A) An undriven, disordered lattice
with random nearest-neighbor couplings. The beam width remains limited, as
expected for Anderson localization. (B) The same lattice driven by an AC
electric field with an amplitude designed to induce DL for $\omega=6$. (C) The
same lattice, driven by an appropriately tailored time-periodic spatially
nonuniform field. The nature of the driving field was determined to make the
lattice behave as an effectively disorder-free lattice with uniform couplings,
using numerical optimization. The beam exhibits discrete diffraction, as
expected.

For example, to solve eq. (1) at ${\cal{O}}(\omega^{-1})$, we generally must
solve an inverse commutator problem of the form ${\cal{H}}_{\rm
eff}=[{\cal{H}}_{0},F]$. If ${\cal{H}}_{\rm eff}$ and ${\cal{H}}_{0}$ are both
Hermitian, then $F$ must be skew-Hermitian. Likewise, if ${\cal{H}}_{\rm eff}$
and $F$ are Hermitian, then ${\cal{H}}_{0}$ is skew-Hermitian. Skew-Hermitian
waveguide schemes or driving fields may be difficult to fabricate. This does
not, however, render Magnus expansion methods impracticable. Indeed, our
curved waveguide setup satisfied the skew-Hermitian criterion, and other
potential implementations exist.

Alternatively, one could circumvent the problem of skew-Hermiticity by
choosing a driving field $F$ such that the ${\cal{O}}(\omega^{-1})$ term
$[{\cal{H}}_{0},F]$ vanishes and the leading term in the Magnus expansion is
an ${\cal{O}}(\omega^{-2})$ integral over the double-commutator
$[F,[{\cal{H}}_{0},F]]$. In this case, ${\cal{H}}_{\rm eff}$, ${\cal{H}}_{0}$,
and $F$ can be Hermitian simultaneously. However, even if ${\cal{H}}_{\rm
eff}$ and ${\cal{H}}_{0}$ are tridiagonal, $F$ may have nonzero terms on its
second off-diagonal (or vice versa). This has its own set of challenges for
experimental realization.

Additionally, one may simply use numerical optimization to determine a
spatially-nonuniform driving field that produces as close to the desired
behaviors as possible, while simultaneously fulfilling physical criteria.

Outlook. We open up Floquet engineering to a breadth of spacetime-dependent
driving fields, giving us greater control over wave systems. By exploiting
spatial degrees of freedom, we showed that we could both induce dynamic
localization in non-uniform lattices and undo Anderson localization by giving
photonic waveguides appropriate curvature. The technique is more broadly
applicable, and we foresee its use in a range of atomic, optical, and
condensed matter systems. In the future, perhaps this technique can help in
fabricating classes of Hamiltonians that currently are difficult or
implausible to realize, such as imaginary gauge fields, Hamiltonians with only
next-nearest-neighbor coupling, or non-Abelian Hamiltonians.

Acknowledgments. We thank Mordechai Segev and Yaakov Lumer for initial work on
this project many years past [68], and S.T.S. is grateful to Moti for hosting
her as a summer research student during that time. We thank Momchil Minkov for
interesting discussions. S.T.S. was partially supported by the U.S. National
Science Foundation through a Graduate Research Fellowship under Grant No.
1745302.

## References

  * Kapitza [1951] P. L. Kapitza, Uspekhi Fizicheskikh Nauk 44, 7 (1951).
  * Thouless [1983] D. J. Thouless, Phys. Rev. B 27, 6083 (1983).
  * Niu and Thouless [1984] Q. Niu and D. J. Thouless, Journal of Physics A: Mathematical and General 17, 2453 (1984).
  * Dunlap and Kenkre [1986] D. H. Dunlap and V. M. Kenkre, Phys. Rev. B 34, 3625 (1986).
  * Grossmann  _et al._ [1991] F. Grossmann, T. Dittrich, P. Jung, and P. Hänggi, Phys. Rev. Lett. 67, 516 (1991).
  * Madison  _et al._ [1998] K. W. Madison, M. C. Fischer, R. B. Diener, Q. Niu, and M. G. Raizen, Phys. Rev. Lett. 81, 5093 (1998).
  * Grifoni and Hänggi [1998] M. Grifoni and P. Hänggi, Physics Reports 304, 229 (1998).
  * Lignier  _et al._ [2005] H. Lignier, J. Chabé, D. Delande, J. C. Garreau, and P. Szriftgiser, Phys. Rev. Lett. 95, 234101 (2005).
  * Eckardt  _et al._ [2005] A. Eckardt, C. Weiss, and M. Holthaus, Phys. Rev. Lett. 95, 260404 (2005).
  * Longhi  _et al._ [2006] S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, Phys. Rev. Lett. 96, 243901 (2006).
  * Lignier  _et al._ [2007] H. Lignier, C. Sias, D. Ciampini, Y. Singh, A. Zenesini, O. Morsch, and E. Arimondo, Phys. Rev. Lett. 99, 220403 (2007).
  * Kierig  _et al._ [2008] E. Kierig, U. Schnorrberger, A. Schietinger, J. Tomkovic, and M. K. Oberthaler, Phys. Rev. Lett. 100, 190405 (2008).
  * Eckardt  _et al._ [2009] A. Eckardt, M. Holthaus, H. Lignier, A. Zenesini, D. Ciampini, O. Morsch, and E. Arimondo, Phys. Rev. A 79, 013611 (2009).
  * Szameit  _et al._ [2009] A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, and Y. S. Kivshar, Nature Physics 5, 271 (2009).
  * Szameit  _et al._ [2010] A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, S. Longhi, and Y. S. Kivshar, Phys. Rev. Lett. 104, 223903 (2010).
  * Struck  _et al._ [2012] J. Struck, C. Ölschläger, M. Weinberg, P. Hauke, J. Simonet, A. Eckardt, M. Lewenstein, K. Sengstock, and P. Windpassinger, Phys. Rev. Lett. 108, 225304 (2012).
  * Struck  _et al._ [2013] J. Struck, M. Weinberg, C. Ölschläger, P. Windpassinger, J. Simonet, K. Sengstock, R. Höppner, P. Hauke, A. Eckardt, M. Lewenstein, and L. Mathey, Nature Physics 9, 738–743 (2013).
  * Yuan and Fan [2015] L. Yuan and S. Fan, Phys. Rev. Lett. 114, 243901 (2015).
  * Oka and Aoki [2009] T. Oka and H. Aoki, Phys. Rev. B 79, 081406 (2009).
  * Kitagawa  _et al._ [2011] T. Kitagawa, T. Oka, A. Brataas, L. Fu, and E. Demler, Phys. Rev. B 84, 235108 (2011).
  * Lindner  _et al._ [2011] N. H. Lindner, G. Refael, and V. Galitski, Nature Physics 7, 490 (2011).
  * Wang  _et al._ [2013] Y. H. Wang, H. Steinberg, P. Jarillo-Herrero, and N. Gedik, Science 342, 453 (2013).
  * Shan  _et al._ [2021] J.-Y. Shan, M. Ye, H. Chu, S. Lee, J.-G. Park, L. Balents, and D. Hsieh, Nature 600, 235 (2021).
  * Zhou  _et al._ [2023] S. Zhou, C. Bao, B. Fan, H. Zhou, Q. Gao, H. Zhong, T. Lin, H. Liu, P. Yu, P. Tang, S. Meng, W. Duan, and S. Zhou, Nature 614, 75 (2023).
  * Aidelsburger  _et al._ [2013] M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch, Phys. Rev. Lett. 111, 185301 (2013).
  * Miyake  _et al._ [2013] H. Miyake, G. A. Siviloglou, C. J. Kennedy, W. C. Burton, and W. Ketterle, Phys. Rev. Lett. 111, 185302 (2013).
  * Jotzu  _et al._ [2014] G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, and T. Esslinger, Nature 515, 237 (2014).
  * Rahav  _et al._ [2003] S. Rahav, I. Gilary, and S. Fishman, Phys. Rev. A 68, 013820 (2003).
  * Goldman and Dalibard [2014] N. Goldman and J. Dalibard, Phys. Rev. X 4, 031027 (2014).
  * Abanin  _et al._ [2015] D. A. Abanin, W. De Roeck, and F. m. c. Huveneers, Phys. Rev. Lett. 115, 256803 (2015).
  * Lee [2016] T. E. Lee, Phys. Rev. A 94, 040701 (2016).
  * Choi  _et al._ [2020] J. Choi, H. Zhou, H. S. Knowles, R. Landig, S. Choi, and M. D. Lukin, Phys. Rev. X 10, 031002 (2020).
  * Lenz  _et al._ [1999] G. Lenz, I. Talanina, and C. M. de Sterke, Phys. Rev. Lett. 83, 963 (1999).
  * Longhi  _et al._ [2007] S. Longhi, G. Della Valle, M. Ornigotti, and P. Laporta, Phys. Rev. B 76, 201101 (2007).
  * Joushaghani  _et al._ [2009] A. Joushaghani, R. Iyer, J. K. S. Poon, J. S. Aitchison, C. M. de Sterke, J. Wan, and M. M. Dignam, Phys. Rev. Lett. 103, 143903 (2009).
  * Garanovich  _et al._ [2012] I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, Physics Reports 518, 1–79 (2012).
  * Plotnik  _et al._ [2016] Y. Plotnik, M. A. Bandres, S. Stützer, Y. Lumer, M. C. Rechtsman, A. Szameit, and M. Segev, Phys. Rev. B 94, 020301 (2016).
  * Minkov and Savona [2016] M. Minkov and V. Savona, Optica 3, 200 (2016).
  * Sounas and Alù [2017] D. L. Sounas and A. Alù, Nature Photonics 11, 774 (2017).
  * Huidobro  _et al._ [2019] P. A. Huidobro, E. Galiffi, S. Guenneau, R. V. Craster, and J. B. Pendry, Proceedings of the National Academy of Sciences 116, 24943 (2019).
  * Yin  _et al._ [2022] S. Yin, E. Galiffi, and A. Alù, eLight 2, 10.1186/s43593-022-00015-1 (2022).
  * Fang  _et al._ [2012a] K. Fang, Z. Yu, and S. Fan, Phys. Rev. Lett. 108, 153901 (2012a).
  * Fang  _et al._ [2012b] K. Fang, Z. Yu, and S. Fan, Nature Photonics 6, 782 (2012b).
  * Rudner  _et al._ [2013] M. S. Rudner, N. H. Lindner, E. Berg, and M. Levin, Phys. Rev. X 3, 031005 (2013).
  * Rechtsman  _et al._ [2013] M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, Nature 496, 196 (2013).
  * Tzuang  _et al._ [2014] L. D. Tzuang, K. Fang, P. Nussenzveig, S. Fan, and M. Lipson, Nature Photonics 8, 701 (2014).
  * Ozawa  _et al._ [2016] T. Ozawa, H. M. Price, N. Goldman, O. Zilberberg, and I. Carusotto, Phys. Rev. A 93, 043827 (2016).
  * Yuan  _et al._ [2016] L. Yuan, Y. Shi, and S. Fan, Opt. Lett. 41, 741 (2016).
  * Lustig  _et al._ [2019] E. Lustig, S. Weimann, Y. Plotnik, Y. Lumer, M. A. Bandres, A. Szameit, and M. Segev, Nature 567, 356–360 (2019).
  * Katan and Podolsky [2013] Y. T. Katan and D. Podolsky, Phys. Rev. Lett. 110, 016802 (2013).
  * Morina  _et al._ [2018] S. Morina, K. Dini, I. V. Iorsh, and I. A. Shelykh, ACS Photonics 5, 1171 (2018).
  * Sharabi  _et al._ [2022] Y. Sharabi, A. Dikopoltsev, E. Lustig, Y. Lumer, and M. Segev, Optica 9, 585 (2022).
  * Peng [2022] Y. Peng, Phys. Rev. Lett. 128, 186802 (2022).
  * Anderson [1958] P. W. Anderson, Phys. Rev. 109, 1492 (1958).
  * De Raedt  _et al._ [1989] H. De Raedt, A. Lagendijk, and P. de Vries, Phys. Rev. Lett. 62, 47 (1989).
  * Wiersma  _et al._ [1997] D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Nature 390, 671 (1997).
  * Schwartz  _et al._ [2007] T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
  * Sanchez-Palencia  _et al._ [2007] L. Sanchez-Palencia, D. Clément, P. Lugan, P. Bouyer, G. V. Shlyapnikov, and A. Aspect, Phys. Rev. Lett. 98, 210401 (2007).
  * Hu  _et al._ [2008] H. Hu, A. Strybulevych, J. H. Page, S. E. Skipetrov, and B. A. van Tiggelen, Nature Physics 4, 945 (2008).
  * Lahini  _et al._ [2008] Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 100, 013906 (2008).
  * Lagendijk  _et al._ [2009] A. Lagendijk, B. v. Tiggelen, and D. S. Wiersma, Physics Today 62, 24 (2009), https://doi.org/10.1063/1.3206091 .
  * Martin  _et al._ [2011] L. Martin, G. D. Giuseppe, A. Perez-Leija, R. Keil, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, A. F. Abouraddy, D. N. Christodoulides, and B. E. A. Saleh, Opt. Express 19, 13636 (2011).
  * Kondov  _et al._ [2011] S. S. Kondov, W. R. McGehee, J. J. Zirbel, and B. DeMarco, Science 334, 66 (2011), https://www.science.org/doi/pdf/10.1126/science.1209019 .
  * Karbasi  _et al._ [2014] S. Karbasi, R. J. Frazier, K. W. Koch, T. Hawkins, J. Ballato, and A. Mafi, Nature Communications 5, 3362 (2014).
  * Segev  _et al._ [2013] M. Segev, Y. Silberberg, and D. N. Christodoulides, Nature Photonics 7, 197 (2013).
  * Levi  _et al._ [2012] L. Levi, Y. Krivolapov, S. Fishman, and M. Segev, Nature Physics 8, 912 (2012).
  * Longhi and Szameit [2012] S. Longhi and A. Szameit, Journal of Physics: Condensed Matter 25, 035603 (2012).
  * Sheinfux  _et al._ [2017] H. H. Sheinfux, S. Schindler, Y. Lumer, and M. Segev, in  _2017 Conference on Lasers and Electro-Optics (CLEO)_ (2017) pp. 1–2.
  * Bender and Boettcher [1998] C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998), arXiv:physics/9712001 .
  * Ruschhaupt  _et al._ [2005] A. Ruschhaupt, F. Delgado, and J. G. Muga, Journal of Physics A: Mathematical and General 38, L171 (2005).
  * El-Ganainy  _et al._ [2007] R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, Opt. Lett. 32, 2632 (2007).
  * Guo  _et al._ [2009] A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).
  * Rüter  _et al._ [2010] C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nature Physics 6, 192 (2010).

##  Appendix A General reverse-engineering techniques

The field of Floquet engineering aims to perturb time-independent systems with
a time-periodic driving field to induce a desired behavior. Perturbation
theory allows us to extract key characteristics of a physical model in an
approximate but systematically improvable manner, oftentimes with far greater
ease than trying to find an exact solution. Perturbative expansions are a
powerful tool for Floquet engineering, as the period of the driving field’s
temporal oscillations serves as a natural expansion parameter.

###  A.1 Magnus expansion

We illustrate the perturbative Magnus expansion with a simple (1+1)D
differential equation. Consider wave propagation in a simple system described
by a Schrödinger-like equation of the form

$\frac{d}{dz}\psi(\vec{x},z)={\cal{H}}(\vec{x},z)\psi(\vec{x},z)$ (3)

where ${\cal{H}}$ is an arbitrary Hamiltonian operator into which we have
absorbed all constants, and $z$ is the direction of propagation. From here
onwards, we suppress indices/arguments $\vec{x}$ related to the transverse
direction(s) for clarity and concision. Given some initial state $\psi(z=0)$,
we can construct a solution of eq. (3) at some later position $z$:

$\psi(z)=e^{U(z)}\psi(0).$ (4)

where for $z$-independent Hamiltonians ${\cal{H}}(z)={\cal{H}}_{0}$, we have
the simple expression $U(z)=z{\cal{H}}_{0}$, but for $z$-dependent
Hamiltonians ${\cal{H}}(z)$, we have

$\displaystyle U(z)=\int dz\,{\cal{H}}(z)\,.$ (5)

We may perturbatively expand the $z$-evolution operator $U(z)$ in terms of
${\cal{H}}(z)$ using a perturbative Magnus expansion:

$\displaystyle U(z)$
$\displaystyle\approx\int_{0}^{z}dz_{1}\,{\cal{H}}(z_{1})+\frac{1}{2}\int_{0}^{z}dz_{1}\int_{0}^{z_{1}}dz_{2}\,\left[{\cal{H}}(z_{1}),{\cal{H}}(z_{2})\right]$
$\displaystyle\qquad+\frac{1}{6}\int_{0}^{z}dz_{1}\int_{0}^{z_{1}}dz_{2}\int_{0}^{z_{2}}dz_{3}\Big{\\{}\big{[}{\cal{H}}(z_{1}),\left[{\cal{H}}(z_{2}),{\cal{H}}(z_{3})\right]\big{]}+\big{[}{\cal{H}}(z_{3}),\left[{\cal{H}}(z_{2}),{\cal{H}}(z_{1})\right]\big{]}\Big{\\}}+...$
(6)

where the $n$th-order term consists of a set of commutators nested $(n-1)$
times.

###  A.2 Floquet effective Hamiltonian

Let us apply the Magnus expansion eq. (A.1) to eq. (3). Consider a discrete,
$z$-independent matrix Hamiltonian ${\cal{H}}_{0}$. We wish to find a
$z$-periodic, spatially-nonuniform field $F(z)$ that we can drive
${\cal{H}}_{0}$ with, that makes it behave like some new $z$-independent
effective Hamiltonian ${\cal{H}}_{\rm eff}$. That is, we try to find an $F$
such that:

${\cal{H}}(\vec{x},z)={\cal{H}}_{0}(\vec{x})+F(\vec{x},z)\overset{?}{\approx}{\cal{H}}_{\rm
eff}(\vec{x}).$ (7)

We again suppress transverse indices $\vec{x}$ from here onwards. We consider
a simple illustrative driving field:

$F(z)=F_{0}\sin(\omega z),$ (8)

though our results straightforwardly extend. Here, $F_{0}$ is a
$z$-independent, spatially non-uniform operator that we will solve for. Using
eq. (A.1) evaluated over one period of oscillation $T=2\pi/\omega$, we have
that

$\displaystyle\frac{2\pi}{\omega}{\cal{H}}_{\rm
eff}\approx\frac{2\pi}{\omega}{\cal{H}}_{0}+\frac{2\pi}{\omega^{2}}[{\cal{H}}_{0},F_{0}]-\frac{3\pi}{2\omega^{3}}[F_{0},[{\cal{H}}_{0},F_{0}]]+{\cal{O}}(\omega^{-4})$
(9)

Finding an appropriate driving field $F(z)$ that makes ${\cal{H}}_{0}$ behave
as some new effective Hamiltonian ${\cal{H}}_{\rm eff}$ at
${\cal{O}}(\omega^{-1})$ accuracy thus amounts to solving an inverse
commutator problem:

$\displaystyle{\cal{H}}_{\rm
eff}-{\cal{H}}_{0}=\frac{1}{\omega}[{\cal{H}}_{0},F_{0}]\,,$ (10)

where the goal is to calculate $F_{0}$ in terms of ${\cal{H}}_{0}$ and
${\cal{H}}_{\rm eff}$. Note that while there are many choices of $z$-dependent
driving schemes, they typically all reduce to an inverse commutator problem
resembling eq. (10).

###  A.3 Inverse commutator problem

Consider a generic inverse commutator problem:

$A=[B,C]$ (11)

where $A$ and $B$ are known operators, and we wish to solve for $C$. For the
lattice waveguide Hamiltonians considered in this paper, we take $A$ and $B$
as $n\times n$ Hermitian matrices with real entries. For more general physical
applications, $A$ and $B$ may take on more complicated forms.

While our focus in this paper is pragmatic computations for practical
applications, we begin by pointing out the straightforward underlying
mathematics. By nature, eq. (11) defines the elements of a Lie algebra. In the
case of interest here, real $n\times n$ matrices, the relevant algebra is the
general linear Lie algebra on the real numbers,
$\mathfrak{gl}_{n}(\mathbb{R})$. The set of possible choices for $A$ is called
the commutator subgroup of the algebra. The commutator subgroup of
$\mathfrak{gl}_{n}(\mathbb{R})$ is the special linear Lie algebra
$\mathfrak{sl}_{n}(\mathbb{R})$. Properties of Lie algebras are well known,
and place restrictions on the form of exactly (rather than approximately)
solvable inverse commutator problems.

The physical systems of interest for this article (matrices $A$ and $B$) are
typically described by sparse Hermitian matrices with real nonzero entries
only along the main and first few off-diagonals. This typically implies that
matrix $C$ will likewise be sparse. Let us see what additional information we
can glean about $A$, $B$, and $C$. First, we verify that $A$ must indeed be
traceless (a property of $\mathfrak{sl}_{n}(\mathbb{R})$) by taking the trace
of both sides of eq. (11):

$\displaystyle\text{Tr}(A)=\text{Tr}[B,C]=\text{Tr}(BC)-\text{Tr}(CB)=0.$ (12)

where in the second-to-last step we have used the fact that the trace is
cyclic. Next, let us examine the physical constraint of Hermiticity of $A$ and
$B$, which restricts the form of $C$:

$\displaystyle
A^{\dagger}=[B,C]^{\dagger}=C^{\dagger}B^{\dagger}-B^{\dagger}C^{\dagger}=C^{\dagger}B-BC^{\dagger}=[B,-C^{\dagger}]\,.$
(13)

From $A=A^{\dagger}$ and $A$ real, we need $C$ to be skew-Hermitian, i.e.,
$C=-C^{\dagger}$. Skew-Hermiticity and reality of $C$ implies that its
diagonal vanishes:

$\displaystyle C_{ii}=0$ (14)

for all $i$. To solve for components $C_{ij}$ with $i\neq j$, we first label
the eigenvalues and eigenvectors of $B$ as $B\vec{v}_{i}=E_{i}\vec{v}_{i}$. We
arrange the eigenvectors into a matrix $V=(\vec{v}_{i},\,...,\,\vec{v}_{n})$.
Multiplying eq. (11) on both sides by $V$ and $V^{-1}=V^{\dagger}$ and
examining the $(i,j)$, we have that

$\displaystyle(V^{-1}AV)_{ij}=\vec{v}_{i}^{\,T}[B,C]\vec{v}_{j}=(E_{i}-E_{j})(V^{-1}CV)_{ij}\equiv(E_{i}-E_{j})\tilde{C}_{ij}\,.$
(15)

Thus, for $i\neq j$, we have that

$\displaystyle\tilde{C}_{ij}=(VCV^{-1})_{ij}=\frac{\vec{v}_{i}A\vec{v}_{j}^{\,T}}{E_{i}-E_{j}}\,.$
(16)

We also note that this is only consistent with $C$ being real skew-Hermitian
if and only if

$\displaystyle\vec{v}_{i}^{\,T}A\vec{v}_{i}\stackrel{{\scriptstyle?}}{{=}}0,$
(17)

which is in general not true. However, in practice, this usually is not a
major concern, given that we are working only to lowest order in a
perturbative expansion.

Now that we know the entries of $\tilde{C}$ from eq. (16), it is
straightforward to solve for $C$ by simply multiplying on the left by $V^{-1}$
and on the right by $V$:

$\displaystyle C=V\tilde{C}V^{-1}\,.$ (18)

We emphasize again that Floquet engineering schemes can be extended to more
general classes of operators, such as non-Hermitian systems [69, 70, 71, 72,
73], but we do not examine other cases in depth here.

##  Appendix B Tight-binding model derivation

In this section, we review the tight-binding model equivalent of a waveguide
lattice. Next, we generalize it to the case of waveguides that have
oscillating widths and different trajectories. To the best of our knowledge,
the latter derivation has not explicitly been written down elsewhere in the
literature; it may thus be of interest to the reader.

It is useful to first analyze the case of a single waveguide, which we take as
centered about the curve $\vec{r}=[x(z),y(z),z]$ with shape function
$u(\vec{r})$. Under the slowly-varying envelope (paraxial) approximation,
light propagating through this waveguide is described by the paraxial wave
equation

$i\frac{\partial}{\partial
z}\psi(\vec{r})=-\frac{1}{2k_{0}}\left(\frac{\partial^{2}}{\partial
x^{2}}+\frac{\partial^{2}}{\partial
y^{2}}\right)\psi(\vec{r})-\frac{k_{0}}{n_{0}}u(\vec{r})\psi(\vec{r}).$ (19)

This equation is isomorphic (mathematically equivalent) to the Schrödinger
equation, albeit with the propagation direction $z$ of waves in optics playing
the traditional role of time $t$ in quantum mechanics. The wavenumber is
$k_{0}=2\pi n_{0}/\lambda$, where $n_{0}$ is the refractive index of the bulk
material and $\lambda$ is the wavelength of the light. The shape function can
be related to the total refractive index as $u(\vec{r})=n(\vec{r})-n_{0}$.

We let the waveguide shape evolve slowly and remain localized around
$\vec{r}$, so that it is a simple single-mode waveguide. We denote the local
phase accumulation speed of the propagating mode as
$\beta(z)=\beta_{R}(z)+i\beta_{I}(z)$, where $\beta_{I}$ is a $z$-dependent
loss term arising from coupling to continuum modes. If $u(\vec{r})$ changes
adiabatically (i.e. slowly relative to $\beta_{R}$), then this rate of loss is
small with $\beta_{I}\ll\beta_{R}$. This statement describes many physical
systems. For example, an array of low-contrast waveguides created by laser-
writing techniques has strongly paraxial waveguides, and $\beta_{I}$
contributes to a uniform decay of light in all waveguides within the array. We
remark here that even small decay can have a major practical effect in certain
cases, like long arrays.

Next, let us consider a set of $n$ single-mode waveguides labeled
$k=1,2,...,n$ centered about the curves $\vec{r}_{k}=[x_{k}(z),y_{k}(z),z]$.
Here, $z$ is the longitudinal direction along which waves propagate, and
$x_{k}(z)$ and $y_{k}(z)$ are functions describing how the waveguides curve
through space in the $x$-$y$ plane, i.e., transverse to the propagation
direction $z$. The width of each of these waveguides varies in space, and we
describe the radius of a waveguide in the $x$-$y$ plane at each point along
its trajectory in the $z$-direction by the shape function
$u_{k}(\vec{r})=U(\vec{r_{k}})\delta(\vec{r}-\vec{r}_{k})$. We now can write
the behavior of the full array as

$\displaystyle u(\vec{r})=\sum_{j=1}^{n}u_{j}(\vec{r})$ (20)

In the case discussed in this article, we choose the waveguides to be equally
spaced in the $x$-direction (i.e. $x_{1}=x_{0}$, $x_{2}=2x_{0}$,
$x_{3}=3x_{0}$, … ) with the same periodically curved trajectory $X(z)$, i.e.,

$\displaystyle x_{k}(z)=kx_{0}+X(z)\,.$ (21)

We distribute the waveguides at random spacings relative to one another along
the $y$-direction. We write the local mode profile of the wave as
$\phi_{k}(x,y)\equiv\phi\big{(}x_{k}(z),y\big{)}$ and the complex phase
accumulation rate at the $k$th waveguide as $\beta_{k}(z)$. Note that for all
$k=1,...,n$, we have that

$\displaystyle\left[-\frac{1}{2k_{0}}\left(\frac{\partial^{2}}{\partial
x^{2}}+\frac{\partial^{2}}{\partial y^{2}}\right)+\sum_{j\neq
k}u_{j}(\vec{r})\right]\phi_{k}=\beta_{k}\phi_{k}\,.$ (22)

Let us write a wavepacket as the sum

$\displaystyle\psi(\vec{r})=\sum_{k=1}^{n}a_{k}(z)\phi_{k}(x,y)\,,$ (23)

where $a_{k}(z)$ are a set of coefficients. Plugging eq. (23) into eq. (19),
we have that

$\displaystyle i\sum_{k=1}^{n}\left[\frac{\partial a_{k}}{\partial
z}+a_{k}\frac{\partial x_{k}}{\partial z}\frac{\partial}{\partial
x}\right]\phi_{k}=-\sum_{k=1}^{n}\left[\frac{1}{2k_{0}}a_{k}\left(\frac{\partial^{2}}{\partial
x^{2}}+\frac{\partial^{2}}{\partial
y^{2}}\right)+\frac{k_{0}}{n_{0}}a_{k}u(\vec{r})\right]\phi_{k}$ (24)

We can rearrange terms to find

$\displaystyle i\sum_{k=1}^{n}\frac{\partial a_{k}}{\partial z}\phi_{k}$
$\displaystyle=-\sum_{k=1}^{n}\left\\{\frac{1}{2k_{0}}\left[\left(\frac{\partial}{\partial
x}+ik_{0}\frac{\partial x_{k}}{\partial
z}\right)^{2}+\frac{\partial^{2}}{\partial
y^{2}}\right]+\frac{k_{0}}{2}\left(\frac{\partial x_{k}}{\partial
z}\right)^{2}+\frac{k_{0}}{n_{0}}u(\vec{r})\right\\}a_{k}\phi_{k}\,.$ (25)

Next, we make the transform $\phi_{k}\to\exp\big{[}-ik_{0}\frac{\partial
x_{k}}{\partial z}\left(x-x_{k}\right)\big{]}\phi_{k}$, giving us

$\displaystyle i\sum_{k=1}^{n}e^{-ik_{0}\frac{\partial x_{k}}{\partial
z}\left(x-x_{k}\right)}\frac{\partial a_{k}}{\partial z}\phi_{k}$
$\displaystyle=-\sum_{k=1}^{n}e^{-ik_{0}\frac{\partial x_{k}}{\partial
z}\left(x-x_{k}\right)}\left[\frac{1}{2k_{0}}\left(\frac{\partial^{2}}{\partial
x^{2}}+\frac{\partial^{2}}{\partial
y^{2}}\right)+\frac{k_{0}}{2}\left(\frac{\partial x_{k}}{\partial
z}\right)^{2}+\frac{k_{0}}{n_{0}}u(\vec{r})\right]a_{k}\phi_{k}\,.$ (26)

Then, we plug eq. (22) into eq. (26), giving us

$\displaystyle i\sum_{k=1}^{n}e^{-ik_{0}\frac{\partial x_{k}}{\partial
z}\left(x-x_{k}\right)}\frac{\partial a_{k}}{\partial z}\phi_{k}$
$\displaystyle=-\sum_{k=1}^{n}e^{-ik_{0}\frac{\partial x_{k}}{\partial
z}\left(x-x_{k}\right)}\left[\beta_{k}+\frac{k_{0}}{2}\left(\frac{\partial
x_{k}}{\partial
z}\right)^{2}+\left(\frac{k_{0}}{n_{0}}-1\right)u(\vec{r})+u_{k}(\vec{r})\right]a_{k}\phi_{k}$
(27)

Next, we multiply both sides by
$\sum_{j}\phi_{j}^{*}\exp\big{[}ik_{0}\frac{\partial x_{j}}{\partial
z}\left(x-x_{j}\right)\big{]}$. Integrating over $x$, we get

$\displaystyle-iS_{jk}\frac{\partial a_{k}}{\partial
z}=\epsilon_{j}S_{jk}a_{k}+\Gamma_{jk}a_{k}+\Delta_{k}a_{k}\,,$ (28)

where we have defined

$\displaystyle S_{jk}$
$\displaystyle=\exp\left[ik_{0}\left(\frac{dx_{k}}{dz}x_{k}-\frac{dx_{j}}{dz}x_{j}\right)\right]\int
dx\,\phi_{j}^{*}\phi_{k}\exp\left[ik_{0}x\left(\frac{\partial x_{k}}{\partial
z}-\frac{\partial x_{j}}{\partial z}\right)\right]$
$\displaystyle\epsilon_{k}$
$\displaystyle=\beta_{k}+\frac{k_{0}}{2}\left(\frac{\partial x_{k}}{\partial
z}\right)^{2}$ $\displaystyle\Gamma_{jk}$
$\displaystyle=\left(\frac{k_{0}}{n_{0}}-1\right)\exp\left[ik_{0}\left(\frac{dx_{k}}{dz}x_{k}-\frac{dx_{j}}{dz}x_{j}\right)\right]\int
dx\,\phi_{j}^{*}u(\vec{r})\phi_{k}\exp\left[ik_{0}x\left(\frac{\partial
x_{k}}{\partial z}-\frac{\partial x_{j}}{\partial z}\right)\right]$
$\displaystyle\Delta_{k}$ $\displaystyle=\int
dx\,u_{k}(\vec{r})|\phi_{k}|^{2}\,.$ (29)

Here, we make the crucial assumption that the modes are tightly-bound matrices
and the $S_{jk}$ matrix is purely real and mostly diagonal:
$S_{jj}/S_{jk}\approx\delta_{jk}$. The $\Gamma_{ij}$ terms are more nuanced
and may lead to complicated behavior.

Typically, we can make significant simplifications to eq. (28) if $U$ is a
narrow potential well that changes on a short distance scale relative to
$\left(k_{0}\frac{dx}{dz}\right)^{-1}$; using eq. (21), we write

$\Gamma_{jk}\propto\exp\left[ik_{0}(x_{k}-x_{j})\frac{dX}{dz}\right]\int
dx\,\phi_{j}^{*}U_{k}\phi_{k}\,.$ (30)

Using the nearest-neighbor approximation, we have that

$\int dx\,\phi_{j}^{*}U_{k}\phi_{k}\propto\delta_{j+1,k}+\delta_{j-1,k}\,.$
(31)

We thus obtain a standard tight-binding model:

$i\frac{da_{k}}{dz}=\epsilon_{k}a_{k}+c_{k,k+1}e^{i\theta_{k}}a_{k+1}+c_{k,k-1}e^{-i\theta_{k}}a_{k-1}$
(32)

where we have written that $\theta_{k}=k_{0}(x_{k}-x_{j})\frac{dX}{dz}$ and
$c_{k,k+1}=c_{k,k-1}=|\Gamma_{jk}|/S_{kk}$.

##  Appendix C Curved waveguide example

Suppose that we have generated the system

${\cal{H}}=\gamma e^{i\omega z}+\gamma^{\dagger}e^{-i\omega
z}+V+\epsilon\cos(\omega z),$ (33)

where $\epsilon$ represents the driving field, $V$ is an on-site potential
(diagonal matrix), and $\gamma$ represents couplings between sites. Let us
find the lowest-order effective Hamiltonian that describes the behavior of
this system. Applying the Magnus expansion in eq. (A.1) to eq. (33) and
averaging over one period, we have

$\displaystyle{\cal{H}}_{\rm
eff}=V+\frac{1}{\omega}\Big{(}2[\gamma,\gamma^{\dagger}]-[\epsilon,\gamma-\gamma^{\dagger}]+2[V,\gamma-\gamma^{\dagger}]\Big{)}+{\cal{O}}(\omega^{-2})$
(34)

As an example, consider the case of nearest-neighbor couplings, i.e. $\gamma$
is a Hermitian off-diagonal matrix $\gamma_{ij}=\gamma_{i}\delta_{i+1,j}$.
Then the terms $V$ and $[\gamma,\gamma^{\dagger}]$ are both diagonal matrices.
The other two commutators in eq. (34) have nonzero elements only on their
first off-diagonals.

###  C.1 Engineering desired behaviors

We now apply the gauged driving scheme to eq. (33). We start with undriven
(straight) waveguides and apply a $z$-dependent driving field (curvature) to
produce a $z$-independent effective Hamiltonian ${\cal{H}}_{\rm eff}$

${\cal{H}}(z)=V+\gamma e^{i\omega z}+\gamma^{\dagger}e^{-i\omega
z}+\epsilon\cos(\omega z)\qquad\approx\qquad{\cal{H}}_{\rm eff}=V_{\rm
eff}+\gamma_{\rm eff}+\gamma^{\dagger}_{\rm eff}.$ (35)

Here we take ${\cal{H}}_{\rm eff}$ to be a nearest-neighbor (Hermitian
tridiagonal) Hamiltonians for simplicity. The matrices $V$ and $V_{\rm eff}$
are diagonal and represent an on-site potential. The $\gamma$-matrices are
off-diagonal matrices representing nearest-neighbor couplings. We take
$\gamma_{ij}=\gamma_{i}\delta_{i+1,j}$. Let us now solve for $\epsilon$.
Recall that the Schrödinger equation carries an extra factor of $-i$, which
both lies in ${\cal{H}}_{\rm eff}$ and propagates throughout the Magnus
expansion. Equating the Hamiltonian we wish to create with the Magnus
expansion of eq. (33), we have that

$\displaystyle V_{\rm eff}+\gamma_{\rm eff}+\gamma_{\rm
eff}^{\dagger}=V+\frac{1}{\omega}\Big{(}2[\gamma,\gamma^{\dagger}]-[\epsilon,\gamma-\gamma^{\dagger}]+2[V,\gamma-\gamma^{\dagger}]\Big{)}$
(36)

Let us rearrange these terms:

$\displaystyle[\epsilon,\gamma-\gamma^{\dagger}]=\Big{(}\omega
V+2[\gamma,\gamma^{\dagger}]-\omega V_{\rm
eff}\Big{)}+\Big{(}2[V,\gamma-\gamma^{\dagger}]-\omega\gamma_{\rm
eff}-\omega\gamma_{\rm eff}^{\dagger}\Big{)}$ (37)

On the right-hand side (RHS), the first parenthetical set of terms are purely
diagonal, and the second set of terms are purely off-diagonal. It is
straightforward to check that the RHS is purely Hermitian. Likewise, on the
LHS, $\gamma-\gamma^{\dagger}$ is skew-Hermitian. Thus, $\epsilon$ is
Hermitian, i.e. $\epsilon_{ij}=\epsilon_{ji}$. Let us examine the terms of the
$\epsilon$-commutator:

$\displaystyle[\epsilon,\gamma-\gamma^{\dagger}]_{ik}$
$\displaystyle=\gamma_{k-1}\epsilon_{i,k-1}-\gamma_{k}\epsilon_{i,k+1}-\gamma_{i}\epsilon_{i+1,k}+\gamma_{i-1}\epsilon_{i-1,k}$
(38)

This commutator is also tridiagonal. We can examine each diagonal $k=i,i\pm 1$
separately:

$\displaystyle[\epsilon,\gamma-\gamma^{\dagger}]_{i,i+1}$
$\displaystyle=\gamma_{i}(\epsilon_{i,i}-\epsilon_{i+1,i+1})$
$\displaystyle[\epsilon,\gamma-\gamma^{\dagger}]_{ii}$
$\displaystyle=2\gamma_{i-1}\epsilon_{i,i-1}\,,$ (39)

Combining eq. (C.1) with eq. (37), we find that

$\displaystyle\epsilon_{i-1,i}=\epsilon_{i,i-1}$
$\displaystyle=\frac{1}{2\gamma_{i-1}}\Big{(}\omega
V+2[\gamma,\gamma^{\dagger}]-\omega V_{\rm eff}\Big{)}_{ii}$
$\displaystyle\epsilon_{i,i}-\epsilon_{i+1,i+1}$
$\displaystyle=\frac{1}{\gamma_{i}}\Big{(}2[V,\gamma-\gamma^{\dagger}]-\omega\gamma_{\rm
eff}\Big{)}_{i,i+1}$ (40)

The latter equation allows us to construct the diagonal terms recursively:

$\displaystyle\epsilon_{ii}$
$\displaystyle=\sum_{k=1}^{i}\frac{1}{\gamma_{n-i}}\Big{(}2[V,\gamma-\gamma^{\dagger}]-\omega\gamma_{\rm
eff}\Big{)}_{n-i,n-i+1}\,,$ (41)

with $\epsilon_{nn}=0$. These results are accurate to
${\cal{O}}(\omega^{-3})$. If we use large driving frequencies $\omega$, all
terms of ${\cal{H}}_{\rm eff}$ are small, and thus the dynamics are slow and
of small bandwidth. We remark that using this scheme, we can also alter the
on-site potential $V$ to change the properties of $\epsilon$.

###  C.2 Example parameters

Suppose we have an original undriven nearest-neighbor-coupled system described
by

$\displaystyle{\cal{H}}_{0}=\gamma+\gamma^{\dagger}\,,$ (42)

where $\gamma_{ij}=\gamma_{i}\delta_{i+1,j}$, with couplings
$\gamma_{i}\in[0.2,1]$ chosen at random. We wish to use driving to produce the
Hamiltonian:

$\displaystyle{\cal{H}}_{\rm eff}=\gamma_{\rm eff}+\gamma_{\rm
eff}^{\dagger}\,,$ (43)

where the coefficients of the matrices are uniform, $\gamma^{\rm
eff}_{ij}=\Gamma\delta_{i,j+1}$.

Example 1. If we drive eq. (42) with a field $\epsilon\cos(\omega z)$ as in
eq. (35), then we can identify:

$\displaystyle\epsilon_{i-1,i}$
$\displaystyle=\epsilon_{i,i-1}=\frac{1}{\gamma_{i-1}}(\gamma_{i}^{2}-\gamma_{i-1}^{2})$
$\displaystyle\epsilon_{ii}$
$\displaystyle=-\omega\Gamma\sum_{k=1}^{i}\frac{1}{\gamma_{n-i}}\,.$ (44)

Example 2. We could also choose to work with both an on-site potential and
driving field $V+\epsilon\cos(\omega z)$. In this case, let us try to
eliminate the off-diagonal terms in $\epsilon$ by fixing $V$ properly. This
means:

$\displaystyle V_{ij}$
$\displaystyle=-\frac{2}{\omega}[\gamma,\gamma^{\dagger}]_{ij}=-\frac{2}{\omega}(\gamma_{i}^{2}-\gamma_{i-1}^{2})\delta_{ij}$
$\displaystyle\epsilon_{i,i\pm 1}$ $\displaystyle=0$
$\displaystyle\epsilon_{ii}$
$\displaystyle=\sum_{k=1}^{i}\frac{1}{\gamma_{n-i}}\Big{(}-\frac{4}{\omega}\Big{[}[\gamma,\gamma^{\dagger}],\gamma-\gamma^{\dagger}\Big{]}-\omega\Gamma\Big{)}_{n-i,n-i+1}=-\omega\Gamma\sum_{k=1}^{i}\frac{1}{\gamma_{n-i}}\,.$
(45)

###  C.3 Realization in a physical system

We now examine how to physically construct a Hamiltonian like eq. (33) using
curved waveguides. Because we are creating a periodic potential, we need
waveguides with a shape that is periodic along the $z$-direction. We let the
phase accumulation rate change as $\beta_{i}(z)=\beta_{0}+\beta_{i}\cos(\omega
z)$. There are several ways to fabricate such a system. First, we could
modulate the depth of the waveguide. This would also change the magnitude of
$\beta_{i}$, but this change is often small relative to $\beta_{0}$ and thus
would have a negligible effect on system dynamics. A second option is to
change the shape of the potential asymmetrically so that the area of the
waveguide (and thus the value of $\beta$) would change significantly, whereas
the coupling coefficient in a particular direction would not. It is most
practicable to carry out a full calculation, accounting for and correcting for
the effects of a $z$-dependent coupling.

To induce a rotating phase, we oscillate the waveguide trajectories in unison,
as described in eq. (21). One possible trajectory we could take is
$x_{k}=kx_{0}+\frac{\alpha}{2}z^{2}$, where $x_{0}$ is the separation between
waveguides in the $x$-direction, similar to that of [10]. However, this
trajectory rapidly diverges, which is a hindrance to experiment. So, instead
we choose the convergent, scalloped trajectory

$\displaystyle
x_{k}(z)=\begin{cases}kx_{0}+\frac{\alpha}{2}\left(z-\tau\left\lfloor\frac{z}{\tau}\right\rfloor\right)^{2}&\qquad
0<(z\,\,{\rm mod}\,\,2\tau)<\tau\\\
kx_{0}+\frac{\alpha}{2}\left[1-\left(z-\tau\left\lfloor\frac{z}{\tau}\right\rfloor\right)^{2}\right]&\qquad\tau<(z\,\,{\rm
mod}\,\,2\tau)<2\tau\end{cases}$ (46)

Here $\lfloor...\rfloor$ indicates the floor operation. The scalloping is
$2\tau$-periodic and has kinks at every $\tau$; however, mild experimental
smoothening should not be detrimental to the working principles behind this
Floquet engineering scheme.

Because eq. (46) is locally parabolic, it gives rise to linear accumulation of
phase

$\theta_{k}=k_{0}(x_{k+1}-x_{k})\frac{dx_{k}}{dz}=\pm k_{0}x_{0}\alpha z.$
(47)

If we set $\alpha=\omega/(k_{0}x_{0})$, we obtain the desired phase rotation
of the coupling coefficient. Recall, however, that the waveguides are not
uniformly distributed in the $y$-direction, which results in an overall
nonuniform coupling coefficient

$\displaystyle\gamma_{i}={\rm
exp}\left[-\kappa\sqrt{x_{0}^{2}-(y_{i+1}-y_{i})^{2}}\right]$ (48)

with $\kappa$ the modal decay distance. If we take $y_{i}\in(0,x_{0}]$, then
$\gamma_{i}\in[e^{-\kappa x_{0}},1]$.

As a consequence, we have an experimentally-realizable setup that is similar
but not identical to eq. (33):

$\displaystyle{\cal{H}}=\begin{cases}\gamma e^{i\omega
t}+\gamma^{\dagger}e^{-i\omega t}+V+\epsilon\cos(\omega t)&\qquad 0<(z\,\,{\rm
mod}\,\,2\tau)<\tau\\\ \gamma e^{i\omega t}+\gamma^{\dagger}e^{-i\omega
t}+V-\epsilon\cos(\omega t)&\qquad\tau<(z\,\,{\rm
mod}\,\,2\tau)<2\tau\end{cases}$ (49)

Here, just as in the main text, we assume that the sign of $\epsilon$ changes
every half-cycle $\tau$. This again would create a discontinuity in an
experiment.