# Floquet engineering of continuous-time quantum walks: towards the simulation
of complex and next-to-nearest neighbor couplings

Leonardo Novo  Centre for Quantum Information and Communication, École
Polytechnique de Bruxelles, Université Libre de Bruxelles, Belgium  Sofia
Ribeiro  Joint Quantum Centre (JQC) Durham-Newcastle, Department of Physics,
Durham University, United Kingdom

###### Abstract

The formalism of continuous-time quantum walks on graphs has been widely used
in the study of quantum transport of energy and information, as well as in the
development of quantum algorithms. In experimental settings, however, there is
limited control over the coupling coefficients between the different nodes of
the graph (which are usually considered to be real-valued), thereby
restricting the types of quantum walks that can be implemented. In this work,
we apply the idea of Floquet engineering in the context of continuous-time
quantum walks, i.e., we define periodically-driven Hamiltonians which can be
used to simulate the dynamics of certain target continuous-time quantum walks.
We focus on two main applications: i) simulating quantum walks that break
time-reversal symmetry due to complex coupling coefficients; ii) increasing
the connectivity of the graph by simulating the presence of next-to-nearest
neighbor couplings. Our work provides explicit simulation protocols that may
be used for directing quantum transport, engineering the dispersion relation
of one-dimensional quantum walks or investigating quantum dynamics in highly
connected structures.

##  I Introduction

Over the past decades, many efforts have been made in the study of quantum
walks [1]. Apart from describing fundamental physical processes such as the
quantum transport of mass and energy, quantum walks also provide a framework
for developing quantum state transfer protocols [2, 3, 4] or quantum
algorithms [5]. This has motivated several experimental realizations of
quantum walks, both in the discrete-time [6, 7] or continuous-time setting
[8], in a wide range of experimental platforms such as optical lattices [9,
10], ion traps[11, 12], NMR [13, 14] or photonic devices [15, 16].

In this work, we focus on continuous-time quantum walks (CTQWs), where a
quantum particle or excitation propagates in a network of coupled nodes via
continuous-time evolution. Such a general model finds widespread applications
in condensed matter physics as well as quantum information and computation.
For one-dimensional chains, this model can be used to study, for example, the
propagation of a single spin excitation in a spin chain [2], or localization
phenomena in disordered media [17]. More complicated highly connected networks
have been considered in a variety of contexts, such as in models of energy
transport in specific biological systems [18, 19, 20], in the analysis of
quantum walks on complex networks [21, 22] or in spatial search algorithms
[23, 24, 25].

In most of the applications of CTQWs, the Hamiltonian is assumed to have real-
valued parameters. This implies that time-reversal symmetry is respected,
i.e., the probability for a particle to be transferred from one node to
another at time $\displaystyle t=T$ is the same as at $\displaystyle t=-T$.
However, complex-valued coupling parameters, which appear naturally in the
description of charged particles in a magnetic field [26], can break time-
reversal symmetry, offering the possibility of biasing the direction of
propagation of the quantum walk as well as enhancing or suppressing quantum
transport between different nodes [27, 28]. In an experimental setting,
however, tuning coupling coefficients to arbitrary complex values are, in
general, not possible, especially in systems of neutral particles such as
photons or ultracold atoms. Furthermore, in physical systems the coupling
strength between different nodes typically decays with the distance, thereby
restricting the quantum walks on graphs that can be implemented experimentally
or observed in nature. One possibility to partially overcome the latter
restriction was proposed in Ref. [29] where by coupling internal degrees of
freedom of the particles (in this case, the photon polarization) with spatial
degrees of freedom it was possible to implement quantum walks in certain
highly connected graphs.

In our work, we exploit the idea of Floquet engineering [30] in the context of
CTQWs and demonstrate that this versatile technique can be used to create
complex coupling coefficients in quantum walks as well as increase the
connectivity of the graph where the quantum walk takes place. The idea behind
Floquet engineering is to design a time-periodic Hamiltonian in a specific
way, such that the dynamics is approximately described by an effective
Hamiltonian with the characteristics that we would like to simulate. This
effective description provides an accurate approximation to the system
dynamics when the period is much shorter than the typical system time-scales
[30]. This method has been applied in several contexts, such as the creation
of synthetic gauge fields [31, 32, 33], of topological insulators [34, 35], or
in order to tune the coupling coefficients of quantum annealers [36].
Discrete-time quantum walks can also be seen as periodically driven systems
and analyzed using Floquet theory [37, 38, 39].

In this article, we consider two central applications of Floquet engineering
in the context of simulating CTQWs. The first is to use this method to
simulate _chiral_ quantum walks, i.e., quantum walks breaking time-reversal
symmetry, such as the ones presented in Ref. [27]. We provide an explicit
protocol to simulate the quantum walk on a switch, where a complex coupling
parameter can be used to control the propagation direction of the walk at the
intersection between three branches. This protocol is easily generalized to
implementing a chain of triangles, where end-to-end transport can be enhanced
or suppressed by controlling the phase of the complex coupling parameters
[27]. The protocol involves fast periodic driving of on-site energies, akin to
the methods for creating synthetic gauge fields in optical lattices via
”lattice shaking” [31].

The second application is to effectively increase the connectivity of the
graph where the quantum walk takes place. Starting from a time-dependent
quantum walk on a given graph, we demonstrate that it is possible to create
effective next-to-nearest neighbor (NNN) couplings via suitable time-periodic
modulation of the couplings of the original graph. We present two illustrative
examples of this idea. In the first one, we demonstrate how to simulate NNN
couplings in a one-dimensional quantum walk by periodically modulating the
strength of nearest-neighbor (NN) couplings. Interestingly, our protocol
introduces purely imaginary NNN couplings, which leads to time-reversal
symmetry breaking and a significantly different dynamics from the usual one-
dimensional CTQW. In the second example, we consider a time-dependent quantum
walk on the star graph with $\displaystyle N+1$ nodes and show that, although
this graph has only $\displaystyle N$ edges, it can be used to simulate
quantum walks on highly connected graphs with
$\displaystyle\mathcal{O}(N^{2})$ edges. Throughout the work, we discuss the
possibility of implementing some of our proposals using integrated photonic
circuits [16, 15]. Experiments with femtosecond laser-written waveguides have
provided so far a high level of control, stability, and accuracy, making them
one of the leading experimental platforms for implementing quantum walks.

Our work is structured as follows. We start with a small introduction to
Floquet theory and the Magnus expansion in Sec. II, explaining how quantum
walks with periodically driven Hamiltonians can be described effectively by
quantum walks in a network that can be different from the original one. We
derive conditions on the modulation period which guarantee that the effective
description is a good approximation of the real dynamics. We then focus on
engineering chiral quantum walks in Sec. III from quantum walks with real
coupling coefficients and periodically modulated on-site energies, covering
different examples. In Sec. IV, we describe how NNN couplings can be
introduced in different graphs via fast periodic modulation of the coupling
coefficients. Finally, we present some concluding remarks in Sec. V.

##  II Floquet engineering of continuous-time quantum walks

We begin by considering a general model describing a quantum particle hopping
in an arbitrary network. The network can be represented by a graph
$\displaystyle G$ with $\displaystyle N$ sites corresponding to quantum states
$\displaystyle\left\\{\left|1\right\rangle,...,\left|N\right\rangle\right\\}$,
and a set of edges $\displaystyle E_{G}$, containing the coupled pairs of
sites $\displaystyle(i,j)$. We write the Hamiltonian describing the dynamics
of the system as

$\displaystyle\displaystyle
H=\sum_{i}\epsilon_{i}\left|i\right\rangle\left\langle
i\right|+\\!\\!\\!\sum_{i\neq j:(i,j)\in
E_{G}}\\!\\!\\!J_{ij}\left|i\right\rangle\left\langle j\right|,$ (1)

where the coefficients $\displaystyle J_{ij}$ represent the coupling strength
between sites $\displaystyle\left|i\right\rangle$ and
$\displaystyle\left|j\right\rangle$, with $\displaystyle J_{ij}=J^{*}_{ji}$,
whereas the real parameters $\displaystyle\epsilon_{i}$ are usually
interpreted as on-site energies. Mathematically, this Hamiltonian can be
interpreted as the adjacency matrix of a weighted graph $\displaystyle G$ with
edge weights $\displaystyle J_{ij}$ and self-loops
$\displaystyle\epsilon_{i}$.

In an experimental setting, the values of $\displaystyle J_{ij}$ are usually
real-valued tunable parameters that rapidly decay with the distance between
sites. For example, in continuous-time photonic quantum walks on waveguide
lattices, the evanescent coupling between waveguides decreases exponentially
with the distance, and usually, only nearest-neighbor couplings are considered
relevant [40]. This constraint highly restricts the kind of quantum walks that
can be realized experimentally. However, in various experimental platforms,
including photonic waveguides, the parameters describing the Hamiltonian are
highly tunable and can even be made time or space-dependent. The ability to
modulate in time the parameters describing the quantum walk can be exploited
to simulate other types of walks that, a priori, would not be possible to
implement directly due to experimental restrictions. In particular, we will
focus on the case where both the on-site potentials and couplings can be tuned
_periodically in time_ and use Floquet theory [30] to derive an effective
time-independent Hamiltonian describing the quantum walk we would like to
engineer.

###  II.1 Effective Hamiltonian and the Magnus expansion

Let us assume we can prepare a quantum system described by a time-dependent
Hamiltonian $\displaystyle H(t)$ of the form of Eq. (1) whose parameters
$\displaystyle\epsilon_{i}(t)$ and $\displaystyle J_{ij}(t)$ are periodic
functions of time with period $\displaystyle T$ such that $\displaystyle
H(t+T)=H(t)$. The time-evolution operator after a period $\displaystyle T$ is
given by

$\displaystyle\displaystyle U(T)$
$\displaystyle\displaystyle=\mathcal{T}\left\\{e^{-\text{i}\int_{t_{0}}^{t_{0}+T}dt\leavevmode\nobreak\
H(t)}\right\\}\equiv e^{-\text{i}H_{\text{eff}}T},$ (2)

where $\displaystyle\mathcal{T}$ represents the time-ordering operator. For
simplicity, we set $\displaystyle t_{0}=0$ and consider units where
$\displaystyle\hbar=1$. The stroboscopic dynamics for times that are multiples
of the period $\displaystyle T$ is fully described by the effective
Hamiltonian $\displaystyle H_{\text{eff}}$ since

$\displaystyle\displaystyle
U(mT)=U^{m}(T)=e^{-\text{i}H_{\text{eff}}\leavevmode\nobreak\ mT}$ (3)

for any positive integer $\displaystyle m$. When the frequency
$\displaystyle\Omega=2\pi/T$ is sufficiently large, the effective Hamiltonian
can be approximately computed via the Magnus expansion [41]

$\displaystyle\displaystyle
H_{\text{eff}}=\sum_{n=0}^{\infty}H_{\text{eff}}^{(n)},$ (4)

where each term $\displaystyle H_{\text{eff}}^{(n)}=\mathcal{O}(\Omega^{-n})$.
Throughout this work, we will consider at most the first two terms of this
expansion, given by

$\displaystyle\displaystyle H_{\text{eff}}^{(0)}$
$\displaystyle\displaystyle=\frac{1}{T}\int_{0}^{T}dt\,H(t)=H_{0},$ (5)
$\displaystyle\displaystyle H_{\text{eff}}^{(1)}$
$\displaystyle\displaystyle=\frac{1}{2T\text{i}}\int_{0}^{T}\int_{0}^{t_{1}}dt_{1}dt_{2}\,[H(t_{1}),H(t_{2})]$
(6)
$\displaystyle\displaystyle=\frac{1}{\Omega}\sum_{l=1}^{\infty}\frac{1}{l}\left([H_{l},H_{-l}]-[H_{l},H_{0}]+[H_{-l},H_{0}]\right),$
(7)

where we have used the expansion of $\displaystyle H(t)$ in its Fourier series
as

$\displaystyle\displaystyle
H(t)=\sum_{l\in\mathbb{Z}}H_{l}\leavevmode\nobreak\ e^{\text{i}l\Omega t}.$
(8)

The idea behind our work is to simulate a target quantum walk Hamiltonian

$\displaystyle\displaystyle
H_{\text{eff}}\approx\sum_{i}\epsilon^{\text{eff}}_{i}\left|i\right\rangle\left\langle
i\right|+\\!\\!\\!\sum_{i,j:(i,j)\in
E_{G^{\prime}}}\\!\\!\\!J^{\text{eff}}_{ij}\left|i\right\rangle\left\langle
j\right|.$ (9)

by appropriately choosing the periodic functions
$\displaystyle\epsilon_{i}(t)$ and $\displaystyle J_{ij}(t)$. In general, the
effective Hamiltonian represents a quantum walk on a graph $\displaystyle
G^{\prime}$ which might be different from the original graph $\displaystyle
G$, as we will see in Sec. IV. Moreover, the effective couplings can now be
complex numbers, even if the couplings of the original time-dependent
Hamiltonian were real. This can be used to direct the propagation of the walk
and increase transport efficiency between certain nodes as we will discuss in
Sec. III.

###  II.2 Simulation errors due to truncation of the Magnus expansion

The approximation of the effective Hamiltonian obtained by truncating the
Magnus expansion naturally leads to an error in the prediction of how the
quantum system evolves in time. Even though this error vanishes as the period
of modulation tends to zero, in reality this period will always be finite. It
is thus useful to have an understanding of how large this error can be for a
finite modulation period $\displaystyle T$ and a total evolution time
$\displaystyle T_{\text{evol.}}=mT$, where $\displaystyle m$ is some positive
integer. In this work, we consider the approximation $\displaystyle
H_{\text{eff}}\approx H_{\text{eff}}^{(0)}$ in Sec. III, whereas in Sec. IV we
consider $\displaystyle H_{\text{eff}}\approx
H_{\text{eff}}^{(0)}+H_{\text{eff}}^{(1)}$. Hence, we derive here the upper
bounds for the simulation error in these two cases. Our aim is that, by
finding this upper bound for the error, it is possible to obtain an estimate
for the period $\displaystyle T$ that guarantees that the effective picture
will provide an accurate description of the real dynamics of the system.

In order to estimate the simulation error, it is useful to consider the
expansion of the unitary $\displaystyle U(T)$ from Eq. (2) as a Dyson series

$\displaystyle\displaystyle U(T)=\sum_{n=0}^{\infty}\mathcal{D}_{n},$ (10)

where

$\displaystyle\displaystyle\mathcal{D}_{n}=\frac{(-\text{i})^{n}}{n!}\int_{0}^{T}dt_{1}\dots\int_{0}^{T}dt_{n}\mathcal{T}H(t_{1})\dots
H(t_{n}).$ (11)

We define

$\displaystyle\displaystyle H_{\text{max}}=\max_{t\in[t_{0},t_{0}+T]}||H(t)||$
(12)

and assume that $\displaystyle T$ is small enough so that $\displaystyle
H_{\text{max}}T<1$. This ensures that the norm of the higher order terms of
this series will become increasingly small. In fact, an upper bound for the
error in the truncation of the Dyson series [42] is given by

$\displaystyle\displaystyle||U(T)-\sum_{n=0}^{K}\mathcal{D}_{n}||\leq\mathcal{O}\left(\frac{(H_{\text{max}}T)^{K+1}}{(K+1)!}\right).$
(13)

A simple way to understand the error caused by truncating the Magnus expansion
is to compare the time-evolution generated by the approximate effective
Hamiltonian to the truncated Dyson series. To do so, we note that, from
equations Eqs. (5) and (6), we have

$\displaystyle\displaystyle||H_{\text{eff}}^{(0)}||\,T$
$\displaystyle\displaystyle=\mathcal{O}(H_{\text{max}}T),$ (14)
$\displaystyle\displaystyle||H_{\text{eff}}^{(1)}||\,T$
$\displaystyle\displaystyle=\mathcal{O}(H_{\text{max}}^{2}T^{2}).$ (15)

We first calculate the error obtained by using the approximation
$\displaystyle H_{\text{eff}}\approx H^{(0)}_{\text{eff}}$. Expanding the
approximate time-evolution operator in Taylor series we obtain

$\displaystyle\displaystyle
e^{-\text{i}H_{\text{eff}}^{(0)}T}\approx\mathbf{1}-\text{i}\int_{0}^{T}dt\,H(t)+\mathcal{O}(H_{\text{max}}^{2}T^{2}),$
(16)

which is equal to the truncated Dyson series to 1st order if we neglect terms
of $\displaystyle\mathcal{O}(H_{\text{max}}^{2}T^{2})$ . From Eq. (13) this
implies that the approximation of $\displaystyle H_{\text{eff}}$ by the first
term in the Magnus expression leads to an error per period $\displaystyle T$
given by

$\displaystyle\displaystyle||U(T)-e^{-\text{i}H_{\text{eff}}^{(0)}T}||=\mathcal{O}(H_{\text{max}}^{2}T^{2}).$
(17)

In order to ensure that the error is bounded after an evolution time
$\displaystyle T_{\text{evol}}=mT$ we require that

$\displaystyle\displaystyle||U(mT)-e^{-\text{i}H_{\text{eff}}^{(0)}mT}||\leq\epsilon$
(18) $\displaystyle\displaystyle\Rightarrow
T=\mathcal{O}\left(\frac{\epsilon}{T_{\text{evol}}\,H_{\text{max}}^{2}}\right).$
(19)

On the other hand, if we consider the two first terms of the Magnus expansion,
we can write the approximate time-evolution operator as

$\displaystyle\displaystyle
e^{-\text{i}(H_{\text{eff}}^{(0)}+H_{\text{eff}}^{(1)})T}$
$\displaystyle\displaystyle\approx\mathbf{1}-\text{i}(H_{\text{eff}}^{(0)}+H_{\text{eff}}^{(1)})T-\frac{1}{2}(H_{\text{eff}}^{(0)}T)^{2}+\mathcal{O}(H_{\text{max}}^{3}T^{3})$
$\displaystyle\displaystyle\approx\mathbf{1}-\text{i}\int_{0}^{T}dt\,H(t)-\int_{0}^{T}\int_{0}^{t_{1}}dt_{1}dt_{2}\,H(t_{1})H(t_{2})$
$\displaystyle\displaystyle\quad+\mathcal{O}(H_{\text{max}}^{3}T^{3}),$ (20)

which is equal to the truncated Dyson series at 2nd order, if we neglect terms
of $\displaystyle\mathcal{O}(H_{\text{max}}^{3}T^{3})$. Hence, via Eq. (13) we
obtain a bound on the error per period

$\displaystyle\displaystyle||U(T)-e^{-\text{i}(H_{\text{eff}}^{(0)}+H_{\text{eff}}^{(1)})T}||=\mathcal{O}(H_{\text{max}}^{3}T^{3}).$
(21)

Hence, the total error after a time-evolution $\displaystyle
T_{\text{evol}}=mT$ can be bounded by $\displaystyle\epsilon$, by choosing the
period $\displaystyle T$ such that

$\displaystyle\displaystyle
T=\mathcal{O}\left(\frac{\sqrt{\epsilon}}{\sqrt{T_{\text{evol}}}(H_{\text{max}})^{3/2}}\right).$
(22)

We can thus use Eqs. (19) and (22) to estimate the appropriate modulation
period $\displaystyle T$, depending on the system energy scale $\displaystyle
H_{\text{max}}$ and the desired total evolution time. Note, however, that
these upper bounds are not necessarily tight and it is possible that in
specific cases the error in Eqs. (17) and (21) is overestimated. For this
reason, we also provide throughout this work numerical simulations of the real
dynamics in order to have a more accurate comparison between the real and
effective models. Nevertheless, the bounds we provide are of independent
interest and can be particularly useful when the real dynamics of the quantum
systems involved is hard to compute numerically.

##  III Simulation of quantum walks breaking time-reversal symmetry

In Ref. [27], it was shown that complex coupling coefficients in a CTQW
quantum walk can be used to break time-reversal symmetry and control the
direction of propagation of the particle. This leads to an increase in
transport efficiency between an initial and a target node in certain graphs.
However, in an experimental setting, tunable complex coupling coefficients are
not always readily available. Here, we discuss how these effects can be
simulated via quantum walks with _real_ coupling coefficients and
_periodically modulated_ on-site potentials and briefly discuss the
possibility to implement our simulation protocol in integrated photonics
circuits, one of the leading experimental platforms for implementing photonic
quantum walks.

###  III.1 Time-reversal symmetry breaking in quantum walks

Provided that the coupling parameters of a Hamiltonian are real, the dynamics
of the system respects probability time-reversal symmetry (TRS) [43] in the
following sense: the transition probability between two states
$\displaystyle\left|i\right\rangle$ and $\displaystyle\left|j\right\rangle$
after forward time evolution

$\displaystyle\displaystyle P(t)_{i\rightarrow j}=|\left\langle
j\right|e^{-\text{i}Ht}\left|i\right\rangle|^{2}$ (23)

equals the transition probability of the time-reversed process

$\displaystyle\displaystyle P(-t)_{i\rightarrow j}$
$\displaystyle\displaystyle=|\left\langle
j\right|e^{\text{i}Ht}\left|i\right\rangle|^{2}=|\left\langle
j\right|e^{-\text{i}H^{*}t}\left|i\right\rangle|^{2},$ (24)

since $\displaystyle H=H^{*}$. Equivalently, this symmetry can be interpreted
as the absence of a directional bias in the propagation as

$\displaystyle\displaystyle P(t)_{i\rightarrow j}=P(-t)_{i\rightarrow
j}\Leftrightarrow P(t)_{i\rightarrow j}=P(t)_{j\rightarrow i}.$ (25)

Thus, a necessary condition to break probability TRS in quantum walks is to
have complex coupling coefficients. However, this alone is not sufficient. The
transition probabilities are invariant under gauge transformations
$\displaystyle\left|i\right\rangle\rightarrow
e^{\text{i}\phi_{i}}\left|i\right\rangle$, which induce a transformation of
the coupling parameters $\displaystyle J_{ij}\rightarrow
e^{\text{i}(\phi_{i}-\phi_{j})}J_{ij}$. If there is a gauge transformation
such that the transformed couplings are real, then probability TRS is still
respected.

More general conditions which ensure probability TRS are derived in Ref. [43],
wherein the authors show that for structures such as trees (which includes the
linear chain) and bipartite graphs this symmetry is always respected. For
graphs that do not fall in these categories, the presence of complex coupling
coefficients can lead to probability TRS breaking.

Throughout the work, we refer to probability TRS breaking simply as TRS
symmetry breaking. In Ref. [43], the authors also discuss the concept of
amplitude TRS breaking i.e., when $\displaystyle\left\langle
j\right|e^{\text{i}Ht}\left|i\right\rangle\neq\left\langle
j\right|e^{-\text{i}Ht}\left|i\right\rangle$. Since this does not necessarily
lead to directional bias of transition probabilities, in our work we focus
only on probability TRS breaking.

###  III.2 Simulating complex couplings via Floquet engineering

In order to show how to simulate quantum walks breaking TRS, we use a
technique known as time-asymmetric lattice shaking [31], introduced in the
context of experiments with ultracold atoms with the aim of creating synthetic
magnetic fields for neutral particles.

Let us consider the time-dependent periodic Hamiltonian

$\displaystyle\displaystyle
H(t)=\sum_{i}\beta_{i}(t)\left|i\right\rangle\left\langle
i\right|+\\!\\!\\!\sum_{i,j:(i,j)\in
E_{G}}\\!\\!\\!J_{ij}\left|i\right\rangle\left\langle j\right|,$ (26)

where the on-site terms $\displaystyle\beta_{i}(t)$ are periodic functions
with period $\displaystyle T$. To analyse the dynamics of the system it is
useful to consider the rotated basis
$\displaystyle\left|\psi^{\prime}(t)\right\rangle=V(t)\left|\psi(t)\right\rangle$,
where we define

$\displaystyle\displaystyle
V(t)=\exp\left(\text{i}\sum_{i}\mathcal{V}_{i}(t)\left|i\right\rangle\left\langle
i\right|\right)$ (27)

and

$\displaystyle\displaystyle\mathcal{V}_{i}(t)=\int_{t_{0}}^{t}dt^{\prime}\,\beta_{i}(t^{\prime})-\left\langle\int_{t_{0}}^{t}dt^{\prime}\,\beta_{i}(t^{\prime})\right\rangle_{T}.$
(28)

Here, we denote the time average of a function $\displaystyle f(t)$ over a
period $\displaystyle T$ as $\displaystyle\left\langle
f(t)\right\rangle_{T}=T^{-1}\int_{t_{0}}^{t_{0}+T}\\!\\!\\!\\!\\!f(t)\leavevmode\nobreak\
dt$. Note that the outcome probabilities of a measurement of either
$\displaystyle\left|\psi(t)\right\rangle$ or
$\displaystyle\left|\psi^{\prime}(t)\right\rangle$ in the site basis are the
same so, for simplicity, in what follows we will consider only
$\displaystyle\left|\psi^{\prime}(t)\right\rangle$.

The wavefunction $\displaystyle\left|\psi^{\prime}(t)\right\rangle$ obeys the
Schrödinger equation with Hamiltonian

$\displaystyle\displaystyle\hat{H}^{\prime}(t)$
$\displaystyle\displaystyle=V(t)\,\hat{H}(t)\,V^{\dagger}(t)+\text{i}\dot{V}(t)\,V^{\dagger}(t)$
(29) $\displaystyle\displaystyle=\sum_{\left\langle
ij\right\rangle}J_{ij}\,\exp\left[\text{i}\left(\mathcal{V}_{i}(t)-\mathcal{V}_{j}(t)\right)\right]\left|i\right\rangle\left\langle
j\right|.$ (30)

We assume the period $\displaystyle T$ is small enough, so that the effective
Hamiltonian is well approximated by the first term of the Magnus expansion

$\displaystyle\displaystyle H_{\text{eff}}\approx\left\langle
H^{\prime}(t)\right\rangle_{T}=\sum_{\left\langle
ij\right\rangle}J_{ij}^{\text{eff}}\left|i\right\rangle\left\langle j\right|,$
(31)

with the renormalized coupling terms

$\displaystyle\displaystyle
J^{\text{eff}}_{ij}=J_{ij}\left\langle\exp\left[\text{i}\mathcal{V}_{ij}(t)\right]\right\rangle_{T},$
(32)

where we have defined
$\displaystyle\mathcal{V}_{ij}=\mathcal{V}_{i}-\mathcal{V}_{j}$.

The imaginary part of the effective tunneling in Eq. (32)

$\displaystyle\displaystyle\text{Im}\left(\frac{J^{\text{eff}}_{ij}}{J_{ij}}\right)$
$\displaystyle\displaystyle=\left\langle\sin\left(\mathcal{V}_{ij}(t)\right)\right\rangle_{T}$
(33)

can only be non-zero provided that the function
$\displaystyle\mathcal{\beta}_{ij}(t)=\beta_{i}(t)-\beta_{j}(t)$ breaks two
fundamental symmetries [31]: (i) The function
$\displaystyle\mathcal{\beta}_{ij}(t)$ needs to break inversion symmetry

$\displaystyle\displaystyle\mathcal{\beta}_{ij}(t-\tau)=-\mathcal{\beta}_{ij}(-t-\tau)$
(34)

with respect to all points $\displaystyle\tau$ in time, and (ii) The function
$\displaystyle\mathcal{\beta}_{ij}(t)$ needs to break shift inversion symmetry

$\displaystyle\displaystyle\mathcal{\beta}_{ij}(t)=-\mathcal{\beta}_{ij}\left(t-T/2\right).$
(35)

The specific form of the functions $\displaystyle\beta_{i}(t)$ depends on the
effective quantum walk Hamiltonian we would like to simulate. In what follows
we present a specific protocol to simulate the simplest of the Hamiltonians
which leads to time-reversal symmetry (a triangular loop). In addition, we
show how to use this protocol to engineer some of the quantum walk
Hamiltonians considered in Ref. [27], where it is possible direct the walker
and increase the transport efficiency between certain nodes using effective
complex phases.

####  III.2.1 Triangular loop

Figure 1: (Color online) a) Scheme representing a quantum walk on a triangle
where the site-energies of nodes $\displaystyle 2$ and $\displaystyle 3$ are
modulated in time with functions $\displaystyle\beta_{2}(t)$ and
$\displaystyle\beta_{3}(t)$. b) The model represented in a) can be used to
simulate the dynamics of a quantum walk on a triangle with a complex coupling
coefficient between nodes $\displaystyle 2$ and $\displaystyle 3$ represented
by the red edge.This is the simplest quantum walk breaking time-reversal
symmetry and can be used as a building block in more complex structures in
order to direct quantum transport.

The crucial building block for some of the Hamiltonians discussed in Ref. [27]
is a triangle loop where the strength of the couplings is uniform
$\displaystyle|J_{ij}|=J$ but one of the couplings, say between nodes 2 and 3,
is complex with $\displaystyle J_{23}=Je^{\text{i}\phi}$ (see Fig. 1 b). Note
that this Hamiltonian is equivalent, up to a gauge transformation, to a
Hamiltonian where

$\displaystyle\displaystyle J_{12}=J_{23}=J_{31}=Je^{\text{i}\phi/3}.$ (36)

The protocol we consider for the simulation of this system involves a fast
time-periodic modulation of two of the on-site potentials, say
$\displaystyle\beta_{2}(t)$ and $\displaystyle\beta_{3}(t)$, while keeping
$\displaystyle\beta_{1}(t)=0$, (see Fig. 1 a) ). Namely, we consider the
Hamiltonian

$H(t)=\beta_{2}(t)\left|2\right\rangle\left\langle
2\right|+\beta_{3}(t)\left|3\right\rangle\left\langle
3\right|+\sum_{i,j:(i,j)\in E_{G}}J^{\prime}\left|i\right\rangle\left\langle
j\right|,$ (37)

where in this case the graph $\displaystyle G$ is a triangle with uniform real
couplings given by $\displaystyle J^{\prime}$.

A possible form for these functions, which leads to complex coupling
coefficients, is based on sine functions that are turned on and off for a
certain time, as proposed in Refs. [31, 44] in the context of simulating
artificial magnetic fields for ultracold atoms in triangular lattices. Here,
we consider an alternative form for the modulation based on step functions,
motivated by the possibility to implement our scheme in integrated photonic
circuits. The reason is twofold. Firstly, we show that these step-like
modulation functions create higher effective complex phases for the same
modulation amplitude, when compared to the sine-like modulation of Ref. [31]
(see more details for the results using sine-function modulation in Appendix
A). This can be important if in an experimental setting the maximum modulation
amplitude is restricted. Secondly, step-like variations of waveguides’
propagation constant have already been experimentally demonstrated in
integrated photonic circuits, as it will be discussed in Sec. III.3.

The step-like modulation functions we consider are

$\displaystyle\displaystyle\beta^{(\text{step})}_{2}(t)$
$\displaystyle\displaystyle=\begin{cases}A,\leavevmode\nobreak\ 0\leq
t\leq\frac{T}{3},\\\ \leavevmode\nobreak\ \\\ -A,\leavevmode\nobreak\
\frac{T}{3}<t\leq\frac{2T}{3}\\\ \leavevmode\nobreak\ \\\
0,\leavevmode\nobreak\ \frac{2T}{3}<t\leq T,\end{cases}$ (38)
$\displaystyle\displaystyle\beta^{(\text{step})}_{3}(t)$
$\displaystyle\displaystyle=\begin{cases}0,\leavevmode\nobreak\ 0\leq
t\leq\frac{T}{3},\\\ \leavevmode\nobreak\ \\\ -A,\leavevmode\nobreak\
\frac{T}{3}<t\leq\frac{2T}{3}\\\ \leavevmode\nobreak\ \\\
A,\leavevmode\nobreak\ \frac{2T}{3}<t\leq T.\end{cases}$ (39)

From Eq. (32), the effective couplings obtained are given by

$\displaystyle\displaystyle J^{\text{eff}}$
$\displaystyle\displaystyle=J_{12}^{\text{eff}}=J_{23}^{\text{eff}}=J_{31}^{\text{eff}}$
$\displaystyle\displaystyle=J^{\prime}e^{\text{i}\frac{AT}{9}}\left[\frac{1}{3}+\frac{2\text{i}}{AT}\left(e^{-\text{i}\frac{AT}{3}}-1\right)\right].$
(40)

Hence, this choice of the modulation functions ensures that both the absolute
value of all couplings as well as their phase is the same along the loop
$\displaystyle 1\rightarrow 2\rightarrow 3\rightarrow 1$ as in Eq. (36). The
value of the effective phase is then given by

$\displaystyle\displaystyle\phi=3\text{Arg}(J^{\text{eff}})\leavevmode\nobreak\
\leavevmode\nobreak\ \text{mod}\leavevmode\nobreak\ 2\pi.$ (41)

In Fig. 2, we compare the parameters $\displaystyle|J^{\text{eff}}|$ and
$\displaystyle\phi$, extracted from a numerical calculation of $\displaystyle
H_{\text{eff}}=\text{i}\log(U(T))/T$ and from the analytical calculation from
Eq. (40), showing a good agreement between the two for the range of parameters
shown. Note that the effective coupling $\displaystyle|J^{\text{eff}}|$ is
lower than the value of $\displaystyle J^{\prime}$. Hence, in order to achieve
a desired strength $\displaystyle J$ for the effective coupling, the value of
$\displaystyle J^{\prime}$ has to be chosen accordingly, taking into account
the modulation amplitude $\displaystyle A$ and the period $\displaystyle T$.

Figure 2: (Color online) Comparison between real (marks) and effective models
(lines) for the accumulated phase around the triangular loop
$\displaystyle\phi$ from Eq. (40) (top) and absolute value of coupling
$\displaystyle|J_{12}|$ (bottom) for two values of the period $\displaystyle
T=0.3$ (in purple) and 0.5 (in blue) as a function of the modulation amplitude
$\displaystyle A$, for step-like modulation functions.

####  III.2.2 Switch and chain of triangles

In Ref. [27], the simplest example that was constructed to illustrate the role
of complex coupling coefficients in a quantum walk was the switch represented
in Fig. 3 a). Starting from node $\displaystyle S$, the direction of the
propagation of the quantum walk can be controlled by the complex hopping phase
present in the central triangular loop. To do so, the Hamiltonian of the
system is chosen such that all coupling coefficients are real, with strength
$\displaystyle J=1$ (which sets our energy units), except for the coupling
between nodes 2 and 3, which is set to $\displaystyle
J_{23}=e^{\text{i}\phi}$. If the phase $\displaystyle\phi$ is zero, due to the
graph’s reflection symmetry, the probability of observing the particle at
either nodes $\displaystyle U$ or $\displaystyle D$ is the same. A non-zero
value of $\displaystyle\phi$ breaks this reflection symmetry (and time-
reversal symmetry) and biases the direction of propagation. It was observed in
Ref. [27] that the maximum bias is achieved for values of
$\displaystyle\phi=\pi/2$ (towards $\displaystyle U$) or
$\displaystyle\phi=3\pi/2$ (towards $\displaystyle D$).

To simulate this quantum walk, we choose a time-dependent Hamiltonian with
real parameters and periodically-modulated site-energies at nodes 2 and 3, as
represented in Fig. 3 b). We choose the step-like modulation functions from
Eqs. (38) and (39), but similar results could be obtained for the sine-like
modulations discussed in Appendix A. The modulation amplitude, as well as the
period, can be tuned following two requirements: the value of $\displaystyle
AT$ is chosen to generate the desired effective phase, following Eqs. (40) and
(41); the period $\displaystyle T$ is chosen to be short enough, so the
effective model provides an accurate description of the real dynamics. The
latter requirement can be fulfilled by using the bound from Eq. (19). However,
for this particular example, we have numerically found that this constraint is
too conservative, and higher values of $\displaystyle T$ still guarantee a
good correspondence between the dynamics of the real model and the effective
one.

The results are summarized in Fig. 4, wherein we compare the probabilities of
observing the particle at $\displaystyle D$ and $\displaystyle U$, computed
via the real (represented by the marks) and the effective model (represented
by the continuous and dashed lines). We chose the initial state of the quantum
walk to be localized at node $\displaystyle S$, the modulation period
$\displaystyle T=0.2$, and two values of modulation amplitude in order to
create an effective phase of $\displaystyle\pi/2$ in Fig. 4 a) and of
$\displaystyle 3\pi/2$ in Fig. 4 b). We observe a good agreement between the
system’s real and effective descriptions, especially for lower evolution
times. The periodic modulation of the site-energies at nodes $\displaystyle 2$
and $\displaystyle 3$ of the central triangle leads to a significant bias in
the direction of propagation of the wavefunction. For an effective phase
$\displaystyle\phi_{\text{eff}}=\pi/2$, the particle goes mostly towards
$\displaystyle U$, reaching a maximum probability $\displaystyle\sim 0.8$ at
this site, whereas for $\displaystyle\phi_{\text{eff}}=3\pi/2$ a similar bias
is achieved towards $\displaystyle D$, in agreement with Ref. [27].

Note that in order to ensure that all the couplings of $\displaystyle
H_{\text{eff}.}$ have a uniform strength $\displaystyle|J_{\text{eff}.}|=1$
(in appropriate units), the values of the coupling strength of the time-
dependent Hamiltonian $\displaystyle H(t)$ have to be chosen appropriately.
Namely, the couplings represented in green in Fig. 3 b), which represent
connections to either node $\displaystyle 2$ or $\displaystyle 3$, need to be
stronger than the rest in order to compensate for the weakening of the
effective coupling strength due to the periodic site-energy modulation of
these nodes, which is visible in Fig. 2 b). If such compensation is not done,
we observe via numerical simulations that there is a significant
backscattering of the particle at the central triangle, washing out the sharp
probability peaks at nodes $\displaystyle U$ and $\displaystyle D$ observed in
Fig. 4 a) and b), respectively.

We remark that the protocol to create complex hopping phases in a triangle
from Sec. III.2.1 can be readily used to simulate the quantum walks in the
structure represented in Fig. 5 a), also studied in Ref. [27]. This structure
forms a chain of coupled triangular loops, where in each loop one of the
coupling coefficients has a complex phase $\displaystyle\phi$ as in Fig. 1 a).
It has been shown in Ref. [27] that these complex phases can either
significantly enhance or suppress the transport efficiency between the two
ends of the chain. In order to simulate this quantum walk on-site energies
should be modulated according to the pattern shown in Fig. 5 b), where the
functions $\displaystyle\beta_{2}(t)$ and $\displaystyle\beta_{3}(t)$ are
chosen according to Sec. III.2.1.

Figure 3: (Color online) Scheme of the switch structure. a) The presence of a
complex-valued coupling (red link) in the central triangle allows the biasing
of the propagation direction of quantum walk starting at $\displaystyle S$
towards $\displaystyle U$ or towards $\displaystyle D$. This effect can be
simulated via a quantum walk with _real_ coupling coefficients and
periodically modulated site-energies at nodes 2 and 3, with modulation
functions $\displaystyle\beta_{2}(t)$ and $\displaystyle\beta_{3}(t)$ (see
Eqs. (38) and (39)). The strength of couplings inside the triangular loop as
well as those between 2 and 2’ and nodes 3 and 3’ (represented by the thicker
green edges), need to be tuned appropriately in order to avoid backscattering.
Figure 4: (Color online) Comparison between the probabilities of observing the
particle at nodes $\displaystyle U$ and $\displaystyle D$ for the quantum walk
on the switch, starting from node $\displaystyle S$ (see Fig. 3). The lines
correspond to the dynamics of effective model with a complex phase
$\displaystyle\text{Arg}(J_{23})=\phi_{\text{eff}}=\pi/2$ in (a) and
$\displaystyle\text{Arg}(J_{23})=\phi_{\text{eff}}=3\pi/2$ in (b). The marks
correspond to the stroboscopic dynamics of the real model, where the on-site
energies of nodes $\displaystyle 2$ and $\displaystyle 3$ are modulated
periodically according to the step-like functions from Eqs. (38) and (39). The
modulation period was chosen to be $\displaystyle T=0.2$ and the modulation
amplitude $\displaystyle A=63.12$ in (a) and $\displaystyle A=80.2575$ in (b)
so that the real dynamics approximately simulates the quantum walk on the
switch with uniform coupling strength and a complex hopping phase
$\displaystyle\text{Arg}(J_{23})=\pi/2$ and
$\displaystyle\text{Arg}(J_{23})=3\pi/2$, respectively. Our units are chosen
such that the effective Hamiltonian has uniform couplings
$\displaystyle|J^{\text{eff}}|=1.$ Figure 5: (Color online) Chain of coupled
triangles. a) In the effective model the complex hopping phase $\displaystyle
e^{\text{i}\phi}$ represented by the red links can be used to enhance the
transport efficiency between nodes $\displaystyle S$ and $\displaystyle E$
[27]. These complex phases can be effectively implemented by modulating the
site energies via the periodic functions $\displaystyle\beta_{1}(t)$ (which
can be chosen to be $\displaystyle\beta_{1}(t)=0$),
$\displaystyle\beta_{2}(t)$ and $\displaystyle\beta_{3}(t)$, according to the
pattern represented in panel b).

###  III.3 Experimental feasibility

Several proposals have been put forward to simulate quantum dynamics showing
time-reversal breaking for atomic ensembles [31], or coherent light [32, 35,
33]. These proposals are driven by the systems’ desirable features, such as
transport along the edge states that are robust against back-scattering.
However, quantum walks with effective complex hopping phases in the single or
few particle regimes have not yet been realized. Here, we argue that the
proposals discussed in Sec. III.2.2 could, in principle, be implemented by
coupling waveguides in photonic integrated circuits [15]. This is an
experimental platform that has extensively been used to implement and study
quantum walks [16]. The waveguide fabrication is done through a permanent
change in the index of refraction induced by nonlinear absorption phenomena
when a femtosecond laser is focused on a glass substrate. A photon can hop
from a waveguide to the neighboring one via evanescent coupling, and the
coupling strength (i.e., the $\displaystyle J_{ij}$ parameters in Eq. (26))
can be precisely controlled by tuning the distance between the waveguides.
Each waveguide represents a node of the quantum walk, and planar graphs like
those represented in Fig. 3, or Fig. 5 can be realized by exploiting laser
witting in three-dimensions [45, 46]. Furthermore, the waveguide’s propagation
constant along the photon’s propagation direction plays the role of a time-
dependent on-site potential felt by the photon (i.e., the
$\displaystyle\beta_{i}(t)$ parameters in Eq. (26)). The latter depends in a
non-trivial way on the waveguide refractive index and can be tuned by
adjusting the laser intensity in the writing process [47]. This technique has
been used to create random time-dependent step-like potentials to simulate
decoherence in photonic quantum walks [45, 46]. Hence, it is in principle
possible to implement different time-asymmetric periodic modulation of the
waveguides’ propagation constants, according to the functions proposed in Sec.
III.2.1.

Interestingly, since the waveguide’s propagation constant is also a function
of the photon wavelength, photons of different wavelengths would feel
different amplitudes of the time-dependent on-site potential. The scheme we
propose to simulate the quantum walk on the switch (see Fig. 3) thus creates a
wavelength-dependent effective complex hopping phase, which, in turn,
translates into a wavelength-dependent direction bias for the photonic quantum
walk. The proposed scheme could open the possibility to experimentally study
the effect of time-reversal symmetry breaking in photonic quantum walks in the
single or few-photon regime and exploit effective complex hopping phases to
direct photon propagation.

##  IV Simulating next-to-nearest neighbor interactions via time-periodic
Hamiltonians

In physical systems, the coupling strength between two nodes of a quantum walk
tends to decrease with the distance between them, thereby restricting the
connectivity of the walks that can be implemented. In this section, we show
that time-dependent modulation of the couplings in a quantum walk can be
exploited to selectively introduce effective couplings between NNN. In
particular, we demonstrate that the relative strength between the NN and NNN
couplings can be fully controlled. We illustrate this idea for the 1D walk and
for the quantum walk on the star graph. The NNN couplings introduced are
purely imaginary and can also be used to study time-reversal symmetry breaking
effects.

###  IV.1 Introducing effective NNN couplings on general graphs

We begin by describing a quantum walk on a graph $\displaystyle G$ with a
general time-dependent Hamiltonian given by

$\displaystyle\displaystyle
H(t)=\sum_{i,j}J_{ij}(t)\,\left|i\right\rangle\left\langle j\right|,$ (42)

where $\displaystyle J_{ij}(t+T)=J_{ij}(t)$ is time-periodic with real
parameters and $\displaystyle J_{ij}(t)=J_{ji}(t)$. We assume the on-site
terms to be zero. For simplicity, we consider the sum over all pairs
$\displaystyle(i,j)$ and assume $\displaystyle J_{ij}(t)=0$ if
$\displaystyle(i,j)$ is not an edge of the graph $\displaystyle G$. We can
expand each of the coupling parameters in the Fourier series as

$\displaystyle\displaystyle
J_{ij}(t)=J^{(0)}_{ij}+\sum_{l=1}^{\infty}J^{(l)}_{ij}\cos\left(l\Omega
t+\phi^{(l)}_{ij}\right),$ (43)

where $\displaystyle J^{(l)}_{ij}=J^{(l)}_{ji}$ and
$\displaystyle\phi^{(l)}_{ij}=\phi^{(l)}_{ji}$. In contrast to Sec. III, we
will consider the first two terms of the Magnus expansion, i.e. $\displaystyle
H_{\text{eff}}\approx H_{\text{eff}}^{(0)}+H_{\text{eff}}^{(1)}$. Then, from
Eq. (5), we find

$\displaystyle\displaystyle
H_{\text{eff}}^{(0)}=\sum_{i,j}J^{(0)}_{ij}\left|i\right\rangle\left\langle
j\right|$ (44)

as, at zero-order in the Magnus expansion, only the time-independent part of
the Fourier expansion of the Hamiltonian contributes. To calculate the 1st
order term using Eq. (7), we start by evaluating the following terms

$\displaystyle\displaystyle\left\langle
i\right|[H_{l},H_{-l}]\left|k\right\rangle$
$\displaystyle\displaystyle=\frac{\text{i}}{2}\sum_{j}J^{(l)}_{ij}J^{(l)}_{jk}\sin{(\phi^{(l)}_{ij}-\phi^{(l)}_{jk})},$
(45) $\displaystyle\displaystyle\left\langle
i\right|[H_{l},H_{0}]\left|k\right\rangle$
$\displaystyle\displaystyle=\frac{1}{2}\sum_{j}\left[J^{(l)}_{ij}J^{(0)}_{jk}e^{\text{i}\phi^{(l)}_{ij}}-J^{(0)}_{ij}J^{(l)}_{jk}e^{-\text{i}\phi^{(l)}_{jk}}\right],$
(46) $\displaystyle\displaystyle\left\langle
i\right|[H_{-l},H_{0}]\left|k\right\rangle$
$\displaystyle\displaystyle=\frac{1}{2}\sum_{j}\left[J^{(l)}_{ij}J^{(0)}_{jk}e^{-\text{i}\phi^{(l)}_{ij}}-J^{(0)}_{ij}J^{(l)}_{jk}e^{\text{i}\phi^{(l)}_{jk}}\right].$
(47)

This leads to the following expression for the matrix entries of
$\displaystyle H_{\text{eff}}^{(1)}$

$\displaystyle\displaystyle\left\langle
i\right|H_{\text{eff}}^{(1)}\left|k\right\rangle=$
$\displaystyle\displaystyle\sum_{l=1}^{\infty}\frac{\text{i}}{l\Omega}\sum_{j}\left[\frac{1}{2}J^{(l)}_{ij}J^{(l)}_{jk}\sin{(\phi^{(l)}_{ij}-\phi^{(l)}_{jk})}\right.$
$\displaystyle\displaystyle\left.-J^{(l)}_{ij}J^{(0)}_{jk}\sin{\phi^{(l)}_{ij}}+J^{(0)}_{ij}J^{(l)}_{jk}\sin{\phi^{(l)}_{jk}}\right].$
(48)

Note that $\displaystyle\left\langle
i\right|H_{\text{eff}}^{(1)}\left|i\right\rangle=0$ as
$\displaystyle\phi^{(l)}_{ij}=\phi^{(l)}_{ji}$. Eq. (48) can only be non-zero
if there is a site $\displaystyle j$ such that both $\displaystyle(i,j)$ and
$\displaystyle(j,k)$ are edges of $\displaystyle G$, i.e., if there exists at
least one path of length 2 from $\displaystyle i$ to $\displaystyle k$. These
effective NNN interactions are purely imaginary and can be used to simulate
quantum walks breaking time-reversal symmetry. Note that imaginary NNN
couplings also appear from the Magnus expansion of quantum systems with
periodically driven on-site energies [48]. However, in our case, we will see
that the ability to periodically modulate the coupling coefficients gives us
extra freedom to control in a simpler way the relative strength between NN and
NNN couplings.

###  IV.2 Simulating imaginary NNN neighbor couplings in a 1D quantum walk

Figure 6: (Color online) a) Representation of the graph corresponding to a
one-dimensional quantum walk with NN and NNN couplings. b) The dynamics of
this quantum walk can be simulated by a Hamiltonian with oscillating coupling
coefficients with two different oscillating phases, represented by the red and
green wobbly lines. The Hamiltonians corresponding to the real and effective
models are shown in Table 1.

To simplify the expression in Eq. (48), we make the following choice: apart
from the time-independent term given by $\displaystyle J^{(0)}_{ij}$, we
choose two modulation frequencies, say $\displaystyle\Omega$ and
$\displaystyle 2\Omega$, such that

$\displaystyle\displaystyle\phi^{(2)}_{ij}$
$\displaystyle\displaystyle=\phi^{(1)}_{ij}+\pi,$ (49)
$\displaystyle\displaystyle J^{(2)}_{ij}$
$\displaystyle\displaystyle=2J^{(1)}_{ij},$ (50)

while keeping $\displaystyle J^{(l)}_{ij}=0$ for $\displaystyle l>2$. This
allows us to eliminate the 2nd and 3rd term of Eq. (48) and arrive at

$\displaystyle\displaystyle\left\langle
i\right|H_{\text{eff}}^{(1)}\left|k\right\rangle=\frac{3\text{i}}{2\Omega}\sum_{j}J^{(1)}_{ij}J^{(1)}_{jk}\sin{(\phi^{(1)}_{ij}-\phi^{(1)}_{jk})}.$
(51)

Let us now consider a 1D system with a time-dependent periodic Hamiltonian
given by

$\displaystyle\displaystyle
H(t)=\sum_{j=1}^{N-1}J_{j,j+1}(t)\;(\left|j\right\rangle\left\langle
j+1\right|+\left|j+1\right\rangle\left\langle j\right|).$ (52)

As a further simplification, we assume the uniformity of the terms
$\displaystyle J^{(0)}_{j,j+1}=J^{(0)}$ and $\displaystyle
J^{(1)}_{j,j+1}=J^{(1)}$. In this case, from Eq. (51) it can be shown that the
only entries of $\displaystyle H_{\text{eff}}\approx
H_{\text{eff}}^{(0)}+H_{\text{eff}}^{(1)}$ that can be non-zero are

$\displaystyle\displaystyle\left\langle
j\right|H_{\text{eff}}\left|j+1\right\rangle$
$\displaystyle\displaystyle=J^{(0)},$ (53)
$\displaystyle\displaystyle\left\langle
j\right|H_{\text{eff}}\left|j+2\right\rangle$
$\displaystyle\displaystyle=\frac{3\text{i}}{2\Omega}(J^{(1)})^{2}\sin\left(\phi^{(1)}_{j,j+1}-\phi^{(1)}_{j+1,j+2}\right),$
(54)

as well as its complex conjugate terms. The strength of each of the NNN
couplings can be controlled individually via the phase difference
$\displaystyle\phi^{(1)}_{j,j+1}-\phi^{(1)}_{j+1,j+2}$. A uniform strength can
be achieved with the choice

$\displaystyle\displaystyle\phi^{(1)}_{j,j+1}\equiv-j\frac{\pi}{2}.$ (55)

From Eq. (54) this yields

$\displaystyle\displaystyle\left\langle
j\right|H_{\text{eff}}\left|j+2\right\rangle$
$\displaystyle\displaystyle=\frac{3\text{i}}{2\Omega}(J^{(1)})^{2}.$ (56)

These choices thus lead to a protocol to simulate imaginary NNN couplings in
1D quantum walks with uniform strength, which we summarize in Table 1. As
previously mentioned, the effective NNN coupling coefficients obtained via
this protocol are purely imaginary and, in general, cannot be made real via a
gauge transformation. This leads to observable physical consequences such as
the loss of reflection symmetry of the probability distribution of a quantum
walk starting in the middle of the 1D chain, which can be seen in Fig. 7.

More precisely, let us define a family of 1D quantum walks with NN couplings
$\displaystyle K_{1}$ and NNN couplings given by $\displaystyle K_{2}$, with
Hamiltonian

$\displaystyle\displaystyle H_{1D}(K_{1},K_{2})=$
$\displaystyle\displaystyle\sum_{j}K_{1}\left|j\right\rangle\left\langle
j+1\right|+K_{1}^{*}\left|j+1\right\rangle\left\langle j\right|)$
$\displaystyle\displaystyle+\sum_{j}K_{2}\left|j\right\rangle\left\langle
j+2\right|+K_{2}^{*}\left|j+2\right\rangle\left\langle j\right|.$ (57)

It can be seen that if either $\displaystyle K_{1}$ or $\displaystyle K_{2}$
are zero, the probability distribution after a certain time $\displaystyle t$
has a reflection symmetry around the starting node. More generally, if we
write $\displaystyle K_{1}=|K_{1}|e^{\text{i}\phi_{1}}$ and $\displaystyle
K_{2}=|K_{2}|e^{\text{i}\phi_{2}}$, it can be shown that this reflection
symmetry (as well as time-reversal symmetry) is preserved for any quantum walk
where $\displaystyle\phi_{2}-2\phi_{1}=0\leavevmode\nobreak\
(\text{mod}\leavevmode\nobreak\ 2\pi)$ 111This can be seen by performing the
gauge transformation $\displaystyle\left|j\right\rangle\rightarrow
e^{-\text{i}j\phi_{1}}\left|j\right\rangle$ in Eq. (IV.2). A possible way to
break this reflection symmetry and bias the walk is to introduce imaginary NNN
couplings. This is in contrast with the 1D discrete-time quantum walk where
even when it is possible to hop only between NN, an asymmetric propagation of
an initially localized state can be achieved either by tuning the initial coin
state or by appropriately choosing the coin-flip operator [50].

The protocol we summarize in Table 1 simulates the time evolution of a quantum
walk with Hamiltonian $\displaystyle H_{1D}(|K_{1}|,\text{i}|K_{2}|)$. In
order to achieve an evolution time $\displaystyle T_{\text{evol}}$, we can set
$\displaystyle J^{(0)}=3K_{1}T/(4\pi)$, $\displaystyle
J^{(1)}=\sqrt{|K_{2}|}$. This choice implies that the effective Hamiltonian
becomes

$H_{\text{eff}}\approx\frac{3T}{4\pi}H_{1D}(|K_{1}|,\text{i}|K_{2}|),$ (58)

which is a rescaled version of $\displaystyle
H_{1D}(|K_{1}|,\text{i}|K_{2}|)$. Hence, the time needed to simulate the time
evolution of $\displaystyle H_{1D}(|K_{1}|,\text{i}|K_{2}|)$ for time
$\displaystyle T_{\text{evol}}$ using $\displaystyle H_{\text{eff}}$ is given
by

$T_{\text{sim}}=\frac{4\pi T_{\text{evol}}}{3T}.$ (59)

Note that the period $\displaystyle T$ has to be sufficiently small to ensure
a good simulation accuracy. This implies that, in general, the simulation time
is larger than $\displaystyle T_{\text{evol}}$.

We have verified numerically that the choice $\displaystyle K_{1}=1$ and
$\displaystyle K_{2}=0.2\text{i}$ leads to a particularly asymmetric
propagation of the quantum walk. In Fig. 7, we show the probability
distribution of such a walk after a fixed time and compare it to that obtained
via our simulation protocol, observing a good agreement even for values of the
period as large as $\displaystyle T=0.5$. The dynamic of this CTQW with
imaginary NNN couplings is characterized by a sharp peak propagating to the
right, with smaller waves propagating to the left at a faster speed. This is
in clear contrast with the dynamics of the quantum walk with real NNN
couplings of the same strength ($\displaystyle K_{1}=1$, $\displaystyle
K_{2}=0.2$), also shown in Fig. 7. The latter preserves time-reversal
symmetry, and the probability distribution exhibits a reflection symmetry
around the initial position of the walk. It is thus clear that changing the
phase of $\displaystyle K_{2}$ can have a striking effect on the propagation.

We also remark that the dispersion relation of the 1D quantum walk with
Hamiltonian $\displaystyle H_{1D}(|K_{1}|,\text{i}|K_{2}|)$, assuming periodic
boundary conditions, is given by

$\displaystyle\displaystyle E_{k}=2|K_{1}|\cos{(2\pi k/N)}-2|K_{2}|\sin{(4\pi
k/N)},$ (60)

where $\displaystyle k\in\\{0,1,..,N-1\\}$ labels the momentum eigenstates.
Our protocol can thus be used to simulate the propagation of particles in 1D
systems with various engineered dispersion relations.

Figure 7: (Color online) The blue bars represent the probability distribution of a one-dimensional quantum walk on a line of 50 nodes with imaginary NNN couplings after time $\displaystyle T_{\text{evol}}=7$. The initial state was chosen to be localized in node 25, the NN coupling strength was set to $\displaystyle K_{1}=1$ (arbitrary units) and the imaginary NNN coupling to $\displaystyle K_{2}=0.2i$. The purple bars represent the probability distribution of the real dynamics generated by our simulation protocol, which uses the time-dependent periodic Hamiltonian with NN couplings from Table 1. We have used the values for the period $\displaystyle T=0.5$ and chosen the parameters $\displaystyle J^{(0)}=3K_{1}T/(4\pi)$, $\displaystyle J^{(1)}=\sqrt{|K_{2}|}$ and total evolution time of $\displaystyle T_{\text{sim}}=4\pi T_{\text{evol}}/(3T)$. The red dots represent the dynamics of the quantum walk with $\displaystyle K_{1}=1$ and real NNN coupling to $\displaystyle K_{2}=0.2$. Hamiltonian |  |  Connectivity
---|---|---
real |  $\displaystyle H(t)=\sum_{j=1}^{N-1}\left[J^{(0)}+J^{(1)}\;\left(\cos(\Omega t-j\pi)-2\cos(2\Omega t-j\pi)\right)\right]\left|j\right\rangle\left\langle j+1\right|+\text{h.c.}$ |  1D with NN couplings
effective |  $\displaystyle H_{\text{eff}}\approx\sum_{j=1}^{N-1}J^{(0)}\left|j\right\rangle\left\langle j+1\right|+\sum_{j=1}^{N-2}\frac{3\text{i}}{2\Omega}(J^{(1)})^{2}\left|j\right\rangle\left\langle j+2\right|+\text{h.c.}$ |  1D with NN + NNN couplings
real |  $\displaystyle H(t)=\sum_{j\in P}J_{0j}\cos(\Omega t)\left|0\right\rangle\left\langle j\right|+\sum_{j\in\overline{P}}J_{0j}\cos(\Omega t+\pi/2)\left|0\right\rangle\left\langle j\right|+\text{h.c.}$ |  star graph
effective |  $\displaystyle H_{\text{eff}}\approx\frac{1}{2\Omega}(J^{(1)})^{2}\sum_{j\in P}\sum_{k\in\overline{P}}\left|j\right\rangle\left\langle k\right|+\text{h.c.}$ |  complete bipartite graph
Table 1: On top, we summarize the protocol to simulate uniform imaginary NNN
couplings in a quantum walk in the linear chain from a one-dimensional quantum
walk with time-dependent NN couplings. On the bottom, we summarize the
protocol to simulate a quantum walk on a complete bipartite graph from a time-
dependent quantum walk on the star graph. The frequency $\displaystyle\Omega$
has to be large enough compared to the other energy scales for the effective
description to be accurate.

###  IV.3 Simulating a quantum walk on a complete bipartite graph

Figure 8: (Color online) a) Complete bipartite graph of 7 nodes, with
partitions $\displaystyle P=\\{1,2,4,6\\}$ and
$\displaystyle\bar{P}=\\{3,5,7\\}$. b) Schematic representation of the
protocol to simulate a quantum walk on a complete bipartite graph via time-
periodic modulation of the couplings of a Hamiltonian with a star connectivity
(all nodes coupled to a single central node). The partition $\displaystyle P$
and $\displaystyle\overline{P}$ can be chosen via two different modulations of
the oscillating phase, represented by the red and green colors.

In the previous example, time-periodic coupling modulation was used to
effectively introduce NNN couplings in 1D quantum walks, increasing the number
of edges by a factor of 2. Here, we demonstrate that if we start with a graph
with different connectivity, for instance, a star graph, it is possible to
engineer quantum walks on considerably more complex graphs, where the number
of edges can be effectively increased by a factor of $\displaystyle N$. More
precisely, we will demonstrate a protocol to simulate a quantum walk on a
complete bipartite graph of $\displaystyle N$ nodes, which can have up to
$\displaystyle\mathcal{O}(N^{2})$ edges, starting from a time-dependent
quantum walk on a graph with star connectivity of $\displaystyle N+1$ nodes.

Let the central node of a star graph in Fig. 8 b) be denoted as
$\displaystyle\left|0\right\rangle$, and consider the Hamiltonian

$\displaystyle\displaystyle H(t)$
$\displaystyle\displaystyle=\sum_{j=1}^{N}J_{0j}(t)\;(\left|0\right\rangle\left\langle
j\right|+\left|j\right\rangle\left\langle 0\right|),$ (61)

where $\displaystyle J_{0j}(t+T)=J_{0j}(t)$. In this case, we assume that only
the first frequency $\displaystyle\Omega=2\pi/T$ is present in the Fourier
expansion (see Eq. (43)). This implies that the zeroth order term of the
Magnus expansion vanishes. For simplicity, we choose a uniform value for
$\displaystyle J^{(1)}_{0j}=J^{(1)}$, but our results are valid in the more
general scenario of variable coupling strength. Using Eq. (48), we obtain that

$\displaystyle\displaystyle
H_{\text{eff}}^{(1)}=\frac{\text{i}}{2\Omega}(J^{(1)})^{2}\sum_{j=1}^{N}\sum_{k=1}^{N}\sin\left(\phi^{(1)}_{0,j}-\phi^{(1)}_{0,k}\right)\left|j\right\rangle\left\langle
k\right|.$ (62)

The resulting Hamiltonian only contains couplings between the nodes
$\displaystyle\\{1,2,\dots,N\\}$, i.e., the central node $\displaystyle 0$
gets effectively decoupled. Furthermore, the choice of the configuration of
the phases $\displaystyle\phi^{(1)}_{0,j}$ determines which nodes are
effectively connected.

The connectivity of a complete bipartite graph with uniform couplings can be
obtained as follows. Let $\displaystyle P$ be a subset of the set of all nodes
$\displaystyle\\{1,2,\dots N\\}$, and $\displaystyle\overline{P}$ its
complement. A complete bipartite graph is one where each node from subset
$\displaystyle P$ is connected to all other nodes from
$\displaystyle\overline{P}$ and vice-versa (see Fig. 8 a)). Choosing the
phases $\displaystyle\phi^{(1)}_{0,j}=\pi/2$ if $\displaystyle j\in P$, and
$\displaystyle\phi^{(1)}_{0,j}=0$ if $\displaystyle j\in\overline{P}$, we
obtain from Eq. (62) the effective Hamiltonian

$\displaystyle\displaystyle
H_{\text{eff}}^{(1)}=\frac{\text{i}}{2\Omega}(J^{(1)})^{2}\sum_{j\in
P}\sum_{k\in\overline{P}}\left[\left|j\right\rangle\left\langle
k\right|-\left|k\right\rangle\left\langle j\right|\right],$ (63)

which represents the Hamiltonian of a quantum walk on a complete bipartite
graph, with an imaginary effective coupling strength
$\displaystyle\frac{\text{i}}{2\Omega}(J^{(1)})^{2}$. However, in this case,
the fact that the coupling is imaginary has no observable physical
consequences, since this Hamiltonian can be transformed into a real
Hamiltonian via a gauge transformation. More precisely, if we define
$\displaystyle\left|k^{\prime}\right\rangle=\text{i}\left|k\right\rangle$, for
$\displaystyle k\in\overline{P}$, we can rewrite Eq. (63) as

$\displaystyle\displaystyle
H_{\text{eff}}^{(1)}=\frac{(J^{(1)})^{2}}{2\Omega}\sum_{j\in
P}\sum_{k^{\prime}\in\overline{P}}\left[\left|j\right\rangle\left\langle
k^{\prime}\right|+\left|k^{\prime}\right\rangle\left\langle j\right|\right].$
(64)

Our scheme to simulate the quantum walk on the complete bipartite graph is
summarized in Table 1. This example highlights the fact that the original
connectivity of the graph can be significantly increased by using the freedom
of periodic modulation of the couplings. Furthermore, although we have focused
on a simple choice of phases that leads to the simulation of the complete
bipartite graph’s connectivity, we believe that more complex choices of phases
in Eq. (62) can be exploited towards the simulation of quantum walks on many
other different families of graphs.

###  IV.4 Experimental feasibility

For the reasons already mentioned in Sec. III.3, an ideal experimental
platform to implement our simulation protocol for a quantum walk with
imaginary NNN couplings are integrated photonic circuits [16]. The high degree
of control of the spatial configuration of laser-written waveguides can be
exploited to create the desired form for the periodic couplings of the time-
dependent Hamiltonian discussed in Sec. IV.2 (see also Table 1). Specifically,
let us assume that the waveguides are written in $\displaystyle xz$-plane,
where the $\displaystyle z$-direction corresponds to the propagation direction
of the photon, which plays the role of time, whereas the relative distance
between two waveguides is given by $\displaystyle|x_{j}(z)-x_{j+1}(z)|$. As
the evanescent coupling between two adjacent waveguides decreases
exponentially with their relative distance, we consider an ansatz for the
coupling of the form $\displaystyle
J_{j,j+1}=\kappa\exp\left[-\gamma(|x_{j}(z)-x_{j+1}(z)|)\right]$. In Fig. 9,
we represent a set of functions $\displaystyle x_{j}(z)$ such that the
$\displaystyle z$-dependence of the couplings has the form required by the
protocol for simulating imaginary NNN couplings of Sec. IV.2, i.e., they obey
the relation

$\displaystyle\displaystyle\kappa e^{-\gamma|x_{j}(z)-x_{j+1}(z)|}$
$\displaystyle\displaystyle=J^{(0)}+J^{(1)}\left[\cos(\Omega
z-j\pi)-2\cos(2\Omega z-j\pi)\right].$ (65)

Here, we have chosen the first waveguide’s shape to be $\displaystyle
x_{1}(z)=\cos(\Omega z)$. By modulating the waveguides in this way and
provided the modulation frequency $\displaystyle\Omega$ is fast enough, it is,
in principle, possible to observe the effects of asymmetric propagation of the
quantum walk with imaginary NNN couplings in this experimental platform.

We note that one-dimensional CTQWs with real NNN interactions have been
previously implemented in arrays of photonic crystals Ref. [51]. In contrast,
our results from Sec. IV.2 allow for the implementation of quantum walks of
the form of Eq. (IV.2) with imaginary NNN couplings, which break time-reversal
symmetry. In addition, we remark that such a Hamiltonian can describe the
dynamics of charged particles on a triangular ladder with a staggered magnetic
field and is thus of interest in the context of condensed matter physics [52].
A proposal to simulate such a system, motivated by its many-body properties in
the presence of interactions, was put forward in Ref. [52] and exploits light-
induced spin-orbit coupling on optical lattices of ultracold atoms. On the
other hand, our work focuses on single particle dynamics, which is easier to
implement and control via photonic quantum walks. Furthermore, an attractive
feature of our simulation method is that it does not require the coupling of
spatial and internal degrees of freedom as in Ref. [52] and only assumes time
dependent control over the coupling strength between different nodes.

Regarding the proposal from Sec. IV.3, the main challenge is to implement a
Hamiltonian with the connectivity of a star graph. This is hard to obtain in
systems where the couplings depend on the distance between different nodes of
the quantum walk, especially for quantum walks with a large number of nodes.
However, we remark that bosonic hopping Hamiltonians with the connectivity of
a star graph can be realized with superconducting qubits by coupling different
oscillators to the same so-called “bus” mode [36]. Hence, our scheme to
simulate the quanutm walks on complete bipartite graphs with tunable
connectivity could, in principle, be realized if it is possible to modulate
the coupling coefficients at high enough frequency.

Figure 9: (Color online) Proposed set of functions $\displaystyle x_{j}(z)$
required to perform the protocol for simulating imaginary NNN coupling in a
system of coupled waveguides, where couplings decrease exponentially with the
distance. We have set $\displaystyle\gamma x_{1}(z)=\cos\left(\Omega
z\right)$, and the following $\displaystyle x_{j}(z)$ are found by solving Eq.
(65) with $\displaystyle\kappa=4$, $\displaystyle J^{(0)}=1$ and
$\displaystyle J^{(1)}=0.1$.

##  V Conclusion

Floquet theory allows us to describe the dynamics of periodically driven
systems in terms of an effective Hamiltonian. We use this idea in the context
of continuous-time quantum walks (CTQWs): by fast periodic control over
certain parameters of a quantum walk on a given graph, an effective
description emerges, which can be seen as a quantum walk on a new graph with
renormalized coupling parameters. This opens up the possibility to engineer
new types of quantum walks that could not be implemented directly due to
experimental restrictions.

We have investigated two main applications of Floquet engineering to CTQWs.
Firstly, we have discussed how fast periodic modulation of on-site energy
terms can be exploited to implement effective complex coupling coefficients.
This can be used to control the propagation direction of the walker and
improve transport efficiency in specific graphs [27]. Secondly, we have shown
how effective NNN couplings can be introduced via time-dependent control over
the coupling coefficients, leading to simulation protocols for quantum walks
on highly connected structures. The presence or absence of these NNN couplings
can be controlled through the shape of the modulation functions, and their
strength can be tuned to be as large as that of the NN couplings.

The effective NNN couplings obtained by our approach are purely imaginary and
can lead to time-reversal symmetry breaking. This effect is evident in the
quantum walk dynamics in the one-dimensional chain with imaginary NNN
couplings, which is one example we explicitly show how to simulate. Our work
may inspire experimental implementations of robust protocols achieving high
fidelity quantum state transfer such as the one introduced in Ref. [53], which
requires purely imaginary tunable NNN couplings in a Su-Schrieffer-Heeger
chain. Furthermore, it would be interesting to extend our approach to two and
three-dimensional lattices. This could lead to new ways of engineering
dispersion relations or exploiting time-reversal symmetry breaking in higher-
dimensional quantum walks, possibly leading to improvements in the performance
of state transfer protocols or search algorithms on regular lattices [23, 54].
We also leave open the question of whether _real_ NNN couplings can be
effectively implemented in quantum walks via Floquet engineering. In
principle, this could be used to simulate quadratic perturbations to the usual
quantum walk Hamiltonian [55] or quantum walks on other structures such as
complex networks [22].

We remark that after the completion of this work, the long time dynamics of
the one-dimensional quantum walk with complex NNN couplings were analyzed in
Ref. [56], extending previous work for real NNN couplings [57]. This work
shows that the propagation of the walk exhibits different velocities between
right and left-moving wavefronts and that the skewness of the distribution is
maximal for purely imaginary NNN couplings.

Overall, our work opens up new possibilities for exploiting time-dependent
control of quantum walk Hamiltonians towards directing and enhancing quantum
transport in networks as well as investigating quantum dynamics in highly
connected structures.

## Acknowledgments

We thank Shantanav Chakraborty, Hugo Terças, Michael Steel, Diego Guzman Silva
and Fulvio Flamini for discussions. SR is supported by the EPSRC grant
EP/R002061/1. LN acknowledges support from F.R.S-FNRS.

##  Appendix A Alternative form for the modulation functions

Following the work of Refs. [31, 44], we consider the set of modulation
functions

$\displaystyle\displaystyle\beta^{(\sin)}_{2}(t)$
$\displaystyle\displaystyle=\begin{cases}A\sin\left(\dfrac{3\pi
t}{T}\right),\leavevmode\nobreak\ 0\leq t\leq\frac{2T}{3},\\\
\leavevmode\nobreak\ \\\ 0,\leavevmode\nobreak\ \frac{2T}{3}<t\leq
T,\end{cases}$ (66) $\displaystyle\displaystyle\beta^{(\sin)}_{3}(t)$
$\displaystyle\displaystyle=\begin{cases}0,\leavevmode\nobreak\ 0\leq
t\leq\frac{T}{3},\\\ \leavevmode\nobreak\ \\\ A\sin\left(\dfrac{3\pi
t}{T}\right),\leavevmode\nobreak\ \frac{T}{3}<t\leq T.\end{cases}$ (67)

for the site energies of a quantum walk on the triangular loop discussed in
Sec. III.2.1. These functions are lead to the effective couplings

$\displaystyle\displaystyle J^{\text{eff}}$
$\displaystyle\displaystyle=J_{12}^{\text{eff}}=J_{23}^{\text{eff}}=J_{31}^{\text{eff}}$
$\displaystyle\displaystyle=J^{\prime}\left[\frac{2}{3}J_{0}\left(\frac{AT}{3\pi}\right)e^{\text{i}\frac{AT}{9\pi}}+\frac{1}{3}e^{\text{i}\frac{2AT}{9\pi}}\right],$
(68)

where we have used Eq. (32) and the integral relation

$\displaystyle\displaystyle\int_{0}^{\pi}d\theta e^{\text{i}x\cos(\theta)}$
$\displaystyle\displaystyle=\pi J_{0}(x),\;$

with $\displaystyle J_{0}(x)$ representing the zero-order Bessel function.

In Fig. 10, we compare the parameters $\displaystyle|J^{\text{eff}}|$ and
$\displaystyle\phi$, extracted from a numerical calculation of $\displaystyle
H_{\text{eff}}=\text{i}\log(U(T))/T$ and from the analytical calculation from
Eq. (68), showing a good agreement between the two for the range of parameters
shown. It is interesting to observe that, the step-like functions lead to
higher effective phases $\displaystyle\phi$ for the same values of
$\displaystyle A$ and $\displaystyle T$, when compared to the sine-like
modulation functions. For example, if we fix the modulation period at
$\displaystyle T=0.5$, a value of the effective phase of $\displaystyle\pi/2$
can be achieved for an amplitude $\displaystyle A\approx 26$ (in units where
$\displaystyle J^{\prime}=1$) using the step-like functions (see Fig. 2),
whereas for the sine-like modulation we would need an amplitude $\displaystyle
A\approx 35$ to obtain the same effective phase. This type of considerations
is of paramount importance for experimental realizations, where there are
natural or technological limits to the possible modulation amplitudes that can
be reached (see Subsec. III.3).

Figure 10: (Color online) Comparison between real and effective models for the
accumulated phase around the triangular loop $\displaystyle\phi$ (top) and
absolute value of coupling $\displaystyle|J^{\text{eff}}_{12}|$ (bottom) for
two values of the period $\displaystyle T=0.3$ (in purple) and 0.5 (in blue)
as a function of the modulation amplitude $\displaystyle A$, for the sine-like
modulation functions from Eq. (66) and (67). The theoretical values were
extracted from Eq. (68) and the value of $\displaystyle J^{\prime}$ was set to
1.

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