# Complex decoherence-free interactions between giant atoms

Lei Du  Center for Theoretical Physics and School of Science, Hainan
University, Haikou 570228, China  Center for Quantum Sciences and School of
Physics, Northeast Normal University, Changchun 130024, China  Lingzhen Guo
Department of Applied Physics, Nanjing University of Science and Technology,
Nanjing 210094, China  Max Planck Institute for the Science of Light, 91058,
Erlangen, Germany  Yong Li  Center for Theoretical Physics and School of
Science, Hainan University, Haikou 570228, China  Synergetic Innovation Center
for Quantum Effects and Applications, Hunan Normal University, Changsha
410081, China

###### Abstract

Giant atoms provide a promising platform for engineering decoherence-free
interactions which is a major task in modern quantum technologies. Here we
study systematically how to implement complex decoherence-free interactions
among giant atoms resorting to periodic coupling modulations and suitable
arrangements of coupling points. We demonstrate that the phase of the
modulation, which is tunable in experiments, can be encoded into the
decoherence-free interactions, and thus enables phase-dependent dynamics when
the giant atoms constitute an effective closed loop. Moreover, we consider the
influence of non-Markovian retardation effect arising from large separations
of the coupling points and study its dependence on the modulation parameters.

##  I Introduction

Giant atoms fiveyear  have become a powerful quantum optical paradigm, which
breaks up a longstanding wisdom that atoms are usually modeled as single
points based on the electric-dipole approximation. Specifically, giant atoms
can be understood as quantum emitters that are coupled to a (propagating)
bosonic field at multiple separate points. As the separation distances between
different coupling points are comparable to the wavelength of bosonic field,
giant atoms feature a peculiar self-interference effect leading to a series of
unprecedented quantum optical phenomena, including frequency-dependent Lamb
shift and relaxation rate LambAFK ; GLZ2017 , unconventional bound states
oscillate ; WXchiral1 ; ZhaoWbound ; VegaPRA ; YuanGA ; TopoCheng ; 2DtopoGA ;
oscillate2 , advanced single-photon scatterings DLlambda ; DLprr ; JiaGA1 ;
JiaGA2 ; YinScattering ; ZhaoWScattering ; CYTcp ; ZhuScattering , non-
Markovian decay dynamics nonexp ; LonghiGA ; DLretard ; LvGA , and chiral
light-matter interactions AFKchiral ; WXchiral2 ; DLsyn , to name a few. Even
more strikingly, by engineering the geometrical arrangements of the coupling
points, a set of giant atoms can be made fully dissipationless but featuring
field-mediated coherent interactions NoriGA ; braided ; FCdeco . This
phenomenon realizes the so-called decoherence-free interaction (DFI) that has
potential important applications in quantum technologies, e.g., engineering
large-scale quantum networks. Although DFIs can also be realized in discrete
photonic lattices by tuning the atomic frequencies within the photonic band
gaps disDFI1 ; disDFI2 ; disDFI3 , this kind of interactions, however, is
typically of short range and only operates within certain bandwidths since
they are mediated by overlapped atom-field bound states.

It is known that electrons can acquire path-dependent phases when traveling in
a magnetic field AB1959 , while photons and phonons are immune to physical
magnetic fields due to their charge neutrality. Given this fact, many efforts
have been made to create synthetic magnetic fields for bosonic systems syn1 ;
syn2 ; syn3 ; syn4 ; syn5 ; syn6 ; syn7 ; syn8 ; syn9 ; Roushan ; JinPRL ;
JiaST . While most of these seminal works have concentrated on systems where
the targets (e.g., atoms and resonators) are spatially close and non-Markovian
retardation effects are typically ignored, very little is known about the
effect of synthetic magnetic fields in large-scale quantum networks featuring
field-mediated long-range interactions. Moreover, it is natural to ask if the
DFIs between giant atoms, which are the result of a virtual-photon process,
can be endowed with synthetic magnetism.

In this paper, we demonstrate how to realize complex DFIs between _detuned_
giant atoms. By modulating the atom-field couplings (or the atomic transition
frequencies) properly, the phase of the modulation can be encoded into the
DFI. Such a complex DFI is tunable _in situ_ and leads to observable phase-
dependent effects when the effective Hamiltonian of the giant atoms has a
closed-loop form. We find that the non-Markovian retardation effect, which is
intrinsic to giant-atom systems, only introduces finite dissipation to the
atoms without affecting their dynamics qualitatively. This detrimental effect
can be mitigated with a smaller modulation frequency, yet an extremely slow
modulation can smear the effect of the synthetic magnetic field due to the
contribution of anti-rotating-wave terms.

##  II Model and equations

Figure 1: Schematics of model architectures. (a) Two-level giant atoms $A$ and
$B$ are coupled to each other via a time-dependent decoherence-free
interaction. (b) A third atom $C$ is coupled directly to $A$ and $B$ to form a
closed-loop atomic trimer. (c) and (d) Protected all-to-all couplings for
atoms $A$, $B$, and $C$ resorting to (c) two different waveguides and (d) a
single waveguide. Atoms $B$ and $C$ are assumed to be resonant with each other
and detuned from atom $A$. The coupling points are equally spaced in all
panels.

We start by considering a pair of two-level giant atoms (labeled as atoms $A$
and $B$, respectively), each of which is coupled to the one-dimensional
waveguide at two coupling points. As shown in Fig. 1(a), the atom-waveguide
coupling points are arranged in a braided manner that allows for a DFI between
the two giant atoms NoriGA ; braided : under certain conditions, both atoms do
not dissipate into the waveguide yet there is a field-mediated coherent
coupling between them. For simplicity, we assume that the coupling points are
equally spaced by distance $d$ (DFIs are allowed even if the coupling points
are not equally spaced). In contrast to the previous standard model where the
atom-waveguide coupling strengths are constant NoriGA ; braided , here we
assume that the coupling strength $g(t)$ of atom $A$ is time dependent and the
strength $g_{0}$ of atom $B$ is constant (for each atom the coupling strength
is assumed to be real and identical at the two coupling points). In circuit
quantum electrodynamics, such time-dependent couplings can be implemented by
using a superconducting quantum interference device with tunable inductance to
mediate the atom-waveguide interaction and modulate its inductance via a bias
current WXchiral2 ; moduscheme . Moreover, we assume that there is a small
detuning $\Delta$ between the transition frequencies of the two atoms. This
detuning is crucial for realizing the synthetic magnetic field as will be
shown below. With the assumptions above, the Hamiltonian of the giant-atom
dimer can be written as (hereafter $\hbar=1$)

$\displaystyle H$ $\displaystyle=$ $\displaystyle
H_{\text{a}}+H_{\text{w}}+H_{\text{int}},$ (1) $\displaystyle H_{\text{a}}$
$\displaystyle=$
$\displaystyle\omega_{0}\sigma_{A}^{+}\sigma_{A}^{-}+(\omega_{0}+\Delta)\sigma_{B}^{+}\sigma_{B}^{-},$
(2) $\displaystyle H_{\text{w}}$ $\displaystyle=$ $\displaystyle\int
dk\omega_{k}a_{k}^{{\dagger}}a_{k},$ (3) $\displaystyle H_{\text{int}}$
$\displaystyle=$ $\displaystyle\int
dk\left[g(t)\left(1+e^{2ikd}\right)\sigma_{A}^{+}a_{k}\right.$ (4)
$\displaystyle\left.+g_{0}\left(e^{ikd}+e^{3ikd}\right)\sigma_{B}^{+}a_{k}+\text{H.c.}\right],$

where $\omega_{0}$ is the transition frequency of atom $A$; $\sigma_{A}^{+}$
and $\sigma_{B}^{+}$ ($\sigma_{A}^{-}$ and $\sigma_{B}^{-}$) are the raising
(lowering) operators of atoms $A$ and $B$, respectively; $\omega_{k}$ is the
frequency of the waveguide field, which can be either linearly dependent on
the amplitude of wave vector $k$ or linearizable around the frequency
$\omega_{0}$ (with the corresponding wave vector $k_{0}$). Having in mind that
the total excitation number is conserved [due to the rotating-wave
approximation used in Eq. (4)], the state of the model in the single-
excitation subspace can be written as

$\begin{split}|\psi(t)\rangle&=\int
dkc_{k}(t)a_{k}^{{\dagger}}e^{-i\omega_{k}t}|G\rangle+\left[u_{A}(t)\sigma_{A}^{+}\right.\\\
&\left.\quad\,+u_{B}(t)\sigma_{B}^{+}\right]e^{-i\omega_{0}t}|G\rangle,\end{split}$
(5)

where $c_{k}$ is the probability amplitude of creating a photon with wave
vector $k$ in the waveguide; $u_{A}$ and $u_{B}$ are the excitation amplitudes
of atoms $A$ and $B$, respectively; $|G\rangle$ denotes that the atoms are in
the ground states and there is no photon in the waveguide. Solving the
Schrödinger equation with Eqs. (1)-(5), one has

$\displaystyle\dot{u}_{A}(t)$ $\displaystyle=$ $\displaystyle-i\int
dkg(t)\left(1+e^{2ikd}\right)c_{k}(t)e^{-i(\omega_{k}-\omega_{0})t},$ (6)
$\displaystyle\dot{u}_{B}(t)$ $\displaystyle=$ $\displaystyle-i\Delta
u_{B}(t)-i\int dkg_{0}\left(e^{ikd}+e^{3ikd}\right)$ (7) $\displaystyle\times
c_{k}(t)e^{-i(\omega_{k}-\omega_{0})t},$ $\displaystyle\dot{c}_{k}(t)$
$\displaystyle=$
$\displaystyle-i\left[g(t)\left(1+e^{-2ikd}\right)u_{A}(t)\right.$ (8)
$\displaystyle\left.+g_{0}\left(e^{-ikd}+e^{-3ikd}\right)u_{B}(t)\right]e^{i(\omega_{k}-\omega_{0})t}.$

By substituting the formal solution of the field amplitude (assuming that the
waveguide is initially in the vacuum state)

$\begin{split}c_{k}(t)&=-i\int_{0}^{t}dt^{\prime}\left[g(t^{\prime})\left(1+e^{-2ikd}\right)u_{A}(t^{\prime})\right.\\\
&\left.\quad\,+g_{0}\left(e^{-ikd}+e^{-3ikd}\right)u_{B}(t^{\prime})\right]e^{i(\omega_{k}-\omega_{0})t^{\prime}}\end{split}$
(9)

into Eqs. (6) and (7), one can obtain the following time-delayed dynamical
equations (see Appendix A for more details):

$\displaystyle\dot{u}_{A}(t)$ $\displaystyle=$ $\displaystyle-\frac{2\pi
g(t)}{v_{g}}\left[2g(t)u_{A}(t)+2g(t-2\tau)D_{A,2}(t)\right.$ (10)
$\displaystyle\left.+3g_{0}D_{B,1}(t)+g_{0}D_{B,3}(t)\right],$
$\displaystyle\dot{u}_{B}(t)$ $\displaystyle=$ $\displaystyle-i\Delta
u_{B}(t)-\frac{2\pi g_{0}}{v_{g}}\left[2g_{0}u_{B}(t)+2g_{0}D_{B,2}(t)\right.$
(11) $\displaystyle\left.3g(t-\tau)D_{A,1}(t)+g(t-3\tau)D_{A,3}(t)\right],$

where $D_{j,l}(t)=\text{exp}(il\phi)u_{j}(t-l\tau)\Theta(t-l\tau)$
($j=A,\,B,\,...$ and $l=1,\,2,\,3$), with $\phi=k_{0}d$ and $\tau=d/v_{g}$
being the phase accumulation and the propagation time (time delay) of a photon
traveling between adjacent coupling points, respectively; $\Theta(x)$ is the
Heaviside step function.

Equations (10) and (11) describe the non-Markovian dynamics of the two giant
atoms, revealing that the non-Markovian retardation effect depends on not only
the coupling strength $g(t)$ _at this moment_ but also its values $g(t-l\tau)$
_at earlier moments_. Such a feature arises from the multiple time delays
among these coupling points. In Sec. V, we will also demonstrate this non-
Markovian feature in a number of extended models as shown in Figs. 1(b)-1(d),
where an additional atom $C$ is coupled to $A$ and $B$ directly or in a
decoherence-free manner via the waveguides. Before doing this, we would like
to demonstrate how to implement complex DFIs in the giant-atom dimer discussed
above.

##  III DFI in the Markovian regime

The multiple retardations in Eqs. (10) and (11) make the dynamics of the
giant-atom dimer a bit complicated. However, if $\tau$ is negligible compared
to all the other characteristic time scales footnote , i.e., in the Markovian
limit, Eqs. (10) and (11) can be simplified to

$\displaystyle\dot{u}_{A}(t)$ $\displaystyle=$ $\displaystyle-\frac{4\pi
g(t)^{2}}{v_{g}}\left(1+e^{2i\phi}\right)u_{A}(t)$ (12)
$\displaystyle-\frac{2\pi
g(t)g_{0}}{v_{g}}\left(3e^{i\phi}+e^{3i\phi}\right)u_{B}(t),$
$\displaystyle\dot{u}_{B}(t)$ $\displaystyle=$ $\displaystyle-i\Delta
u_{B}(t)-\frac{4\pi g_{0}^{2}}{v_{g}}\left(1+e^{2i\phi}\right)u_{B}(t)$ (13)
$\displaystyle-\frac{2\pi
g(t)g_{0}}{v_{g}}\left(3e^{i\phi}+e^{3i\phi}\right)u_{A}(t).$

Clearly, both atoms are dissipationless and their effective interaction is
purely coherent when $\phi=(m+1/2)\pi$ ($m$ is an arbitrary integer). Now we
consider cosine-type time-dependent couplings for atom $A$, i.e.,

$g(t)=\Delta_{g}\cos{(\Omega t+\theta)}$ (14)

with $\Delta_{g}$, $\Omega$, and $\theta$ being the amplitude, frequency, and
initial phase of the modulation, respectively. If
$\Omega=\Delta\gg|2\pi\Delta_{g}g_{0}/v_{g}|$ and using the transformation
$u_{B}(t)\rightarrow u_{B}(t)\text{exp}(-i\Delta t)$, Eqs. (12) and (13)
become

$\displaystyle\dot{u}_{A}(t)$ $\displaystyle\simeq$ $\displaystyle-
iG_{m}e^{i\theta}u_{B}(t),$ (15) $\displaystyle\dot{u}_{B}(t)$
$\displaystyle\simeq$ $\displaystyle-iG_{m}e^{-i\theta}u_{A}(t),$ (16)

where $\phi=(m+1/2)\pi$ has been assumed and
$G_{m}=(-1)^{m}2\pi\Delta_{g}g_{0}/v_{g}$. One can see from Eqs. (15) and (16)
that the modulation phase $\theta$ is encoded into the DFI (with effective
strength $G_{m}$), mimicking a synthetic magnetic flux for photons
transferring between $A$ and $B$. Although the coupling phase $\theta$ can be
gauged away for such a two-atom model (thus it has no particular interest in
this case), it can significantly affect the dynamics of the system when a
third atom is introduced to form a closed-loop trimer DLretard ; Roushan ;
arxivClerk ; WXNJP , as will be shown in Sec. V.

Although the above analysis is only applicable in the single-excitation
subspace, the decoherence-free nature of our model can also be illustrated by
resorting to the theory of effective Hamiltonian FCdeco ; GAcollision ;
James2007 . As shown in Appendix B, in the Markovian regime, the effective
Hamiltonian of the giant-atom dimer can be given by

$H_{\text{eff,dim}}=G_{m}e^{-i\theta}\sigma_{B}^{+}\sigma_{A}^{-}+\text{H.c.},$
(17)

which shows a complex DFI between atoms $A$ and $B$. Moreover, we have checked
that the average interaction between the giant atoms and the waveguide field
vanishes (thus the atoms are dissipationless) in this case.

Before proceeding, we briefly discuss the influence of the non-Markovian
retardation effect on the result above. It is clear from Eqs. (10) and (11)
that the retardation effect arising from the non-negligible time delay $\tau$
may smear the DFI (such that the atoms are not perfectly dissipationless) and
makes the dynamics much more complicated. To mitigate this detrimental effect,
one can either consider a small enough $\tau$, or assume
$\text{mod}(\Omega\tau,\pi)=0$ (a large enough $\Omega$) such that complete
atomic decay can be prevented DLprr2 .

##  IV Dynamics with effective decoherence-free interactions

Figure 2: Dynamics of atomic excitation probabilities $P_{A}(t)$ and
$P_{B}(t)$ in the atomic dimer [Fig. 1(a)] with different values of (a)
$\Omega$ and (b) $\chi$. We assume $\chi=1$ in panel (a) and
$\Omega/\Gamma_{0}=10\pi$ in panel (b). Moreover, we set $\Delta=\Omega$ for
all lines, except for the case “without modulation” in panel (a): $\Omega=0$
and $\Delta/\Gamma_{0}=10\pi$ in this case. Other parameters are
$\Gamma_{0}=2\pi g_{0}^{2}/v_{g}$, $\phi=\pi/2$, $\theta=0$,
$\tau\Gamma_{0}=0.001$, and $|\psi(t=0)\rangle=\sigma_{A}^{+}|G\rangle$.

In this section we would like to verify the above analysis by numerically
solving the time-delayed dynamical equations (10) and (11) with appropriate
parameters. For clarity, we use $\Gamma_{0}=2\pi g_{0}^{2}/v_{g}$ (which is
the radiative decay rate of atom $B$ at each coupling point) as the unit of
energies, and define $P_{A}(t)=|u_{A}(t)|^{2}$ and $P_{B}(t)=|u_{B}(t)|^{2}$
as the excitation probabilities of atoms $A$ and $B$, respectively. Moreover,
we introduce a dimensionless parameter $\chi=\Delta_{g}/g_{0}$ so that the
time-dependent coefficients in Eqs. (10) and (11) [e.g., $2\pi
g(t)g_{0}/v_{g}$] can be expressed with $\Gamma_{0}$ and $\chi$. Since we
focus on the DFI of the giant atoms, hereafter we will always assume
$\phi=\pi/2$ (i.e., $m=0$) and $\tau\Gamma_{0}\ll 1$.

Figure 3: Dynamics of atomic excitation probabilities $P_{A}(t)$, $P_{B}(t)$,
and $P_{C}(t)$ in the atomic trimer [Fig. 1(b)] with different values of
$\theta$. The lower plots illustrate the excitation transfer in the trimer,
corresponding to panels (a)-(c), respectively. Other parameters are
$\Gamma_{0}=2\pi g_{0}^{2}/v_{g}$, $\phi=\pi/2$,
$\Delta/\Gamma_{0}=\Omega/\Gamma_{0}=10\pi$, $\chi=1$, $\tau\Gamma_{0}=0.001$,
and $|\psi(t=0)\rangle=\sigma_{A}^{+}|G\rangle$.

Figure 2(a) shows the time evolutions of $P_{A}(t)$ and $P_{B}(t)$ with the
initial state $|\psi(t=0)\rangle=\sigma_{A}^{+}|G\rangle$ (atom $A$ is
initially excited) and with different values of modulation frequency $\Omega$.
As discussed above, $\Omega=\Delta\gg|G_{m}|$ is required to justify the
rotating-wave approximation [i.e., dropping high-frequency terms as in Eqs.
(15) and (16)]. Indeed, we find that the two atoms exhibit a nearly
decoherence-free excitation exchange (Rabi-like oscillation) when $\Omega$ is
large enough (see, e.g., the orange line with circles and the green line with
stars), while the dynamics deviate markedly from this typical form when
$\Omega$ is small (see, e.g., the blue solid and red dashed lines). The Rabi-
like line shapes exhibit additional tiny oscillations (thus we refer to them
as “Rabi-like”) due to the cosine-type coupling modulations. Note that the
interatomic interaction almost disappears and atom $A$ exhibits a long-lived
population in the absence of modulations (in this case we assume $\Omega=0$
and $\Delta/\Gamma_{0}=10\pi$ instead). This is intuitive since the two atoms
have very different transition frequencies. From this point of view, the
coupling modulation allows for protected interactions between _detuned_ giant
atoms, which is significant on its own.

We also plot in Fig. 2(b) the time evolutions of the atomic excitation
probabilities with different values of $\chi$. It shows that the Rabi-like
oscillation becomes faster for larger $\chi$ (i.e., larger $\Delta_{g}$),
since the effective coupling strength $G_{m}$ between the two atoms is
proportional to $\Delta_{g}$. This thus provides an _in situ_ tunable scheme
for manipulating the interactions between remote quantum emitters.

##  V Directional excitation circulation

Figure 4: (a, b) Dynamics of atomic excitation probabilities $P_{A}(t)$,
$P_{B}(t)$, and $P_{C}(t)$ in the atomic trimer [Fig. 1(c)] with (a) $\chi=2$
and (b) $\chi=1$. (c) Dynamics of total atomic excitation probability
$P_{\text{tot}}(t)$ in the atomic trimer [Fig. 1(c)] with different values of
$\chi$. Other parameters are $\Gamma_{0}=2\pi g_{0}^{2}/v_{g}$, $\phi=\pi/2$,
$\Delta/\Gamma_{0}=\Omega/\Gamma_{0}=10\pi$, $\theta=\pi/2$,
$\tau\Gamma_{0}=0.001$, and $|\psi(t=0)\rangle=\sigma_{A}^{+}|G\rangle$.

As discussed in Sec. III, the effective coupling phase $\theta$ of the giant-
atom dimer has no actual physical meaning since it can always be gauged away
(indeed, such a coupling phase is sensitive to the choice of the initial
time). In view of this, we consider an additional two-level atom (labeled as
atom $C$, described by the ladder operators $\sigma_{C}^{\pm}$ and excitation
amplitude $u_{C}$) coupled directly to $A$ and $B$, forming a closed-loop
trimer as shown in Fig. 1(b). To be specific, we assume: (i) atom $C$ is
resonant with atom $B$ (thus it is detuned from atom $A$ by $\Delta$); (ii)
atom $C$ is coupled to atom $A$ with a time-dependent coupling strength
$\lambda(t)=2G_{0}\cos{(\Omega t)}$ and to atom $B$ with a constant coupling
strength $G_{0}$ ($G_{0}\coloneqq G_{m=0}=\chi\Gamma_{0}$). Considering all
these assumptions, the Hamiltonian describing atom $C$ and its interaction
with the other atoms can be written as

$\begin{split}H_{\text{add}}&=(\omega_{0}+\Delta)\sigma_{C}^{+}\sigma_{C}^{-}+\left[\lambda(t)\sigma_{A}^{+}\sigma_{C}^{-}\right.\\\
&\left.\quad\,+G_{0}\sigma_{B}^{+}\sigma_{C}^{-}+\text{H.c.}\right].\end{split}$
(18)

Combined with Eqs. (1)-(5), the dynamical equations of the trimer can be
immediately obtained as

$\displaystyle\dot{u}_{A}(t)$ $\displaystyle=$ $\displaystyle-\frac{4\pi
g^{2}(t)}{v_{g}}u_{A}(t)-\frac{4\pi
g(t)g(t-2\tau)}{v_{g}}D_{A,2}(t)-\frac{2\pi
g(t)g_{0}}{v_{g}}\left[3D_{B,1}(t)+D_{B,3}(t)\right]-i\lambda(t)u_{C}(t),$
(19) $\displaystyle\dot{u}_{B}(t)$ $\displaystyle=$ $\displaystyle-i\Delta
u_{B}(t)-\frac{4\pi
g_{0}^{2}}{v_{g}}\left[u_{B}(t)+D_{B,2}(t)\right]-\frac{2\pi
g_{0}}{v_{g}}\left[3g(t-\tau)D_{A,1}(t)+g(t-3\tau)D_{A,3}(t)\right]-iG_{0}u_{C}(t),$
(20) $\displaystyle\dot{u}_{C}(t)$ $\displaystyle=$ $\displaystyle-i\Delta
u_{C}(t)-i\left[\lambda(t)u_{A}(t)+G_{0}u_{B}(t)\right].$ (21)

If $\phi=\pi/2$, $\Omega=\Delta$, $\tau\rightarrow 0$, and
$g(t)=\Delta_{g}\cos{(\Omega t+\theta)}$, Eqs. (19)-(21) can be simplified to

$\displaystyle\dot{u}_{A}(t)$ $\displaystyle\simeq$ $\displaystyle-
iG_{0}\left[e^{i\theta}u_{B}(t)+u_{C}(t)\right],$ (22)
$\displaystyle\dot{u}_{B}(t)$ $\displaystyle\simeq$ $\displaystyle-
iG_{0}\left[e^{-i\theta}u_{A}(t)+u_{C}(t)\right],$ (23)
$\displaystyle\dot{u}_{C}(t)$ $\displaystyle\simeq$ $\displaystyle-
iG_{0}\left[u_{A}(t)+u_{B}(t)\right].$ (24)

In this way, the excitation can acquire a gauge-invariant phase $\theta$ when
it travels along the closed loop made of the three atoms. Such a phase
simulates the synthetic magnetic flux threading the closed loop (which is
typically defined as $\oint\vec{A}\cdot d\vec{r}$, with $\vec{A}$ the
effective vector potential and the integral performed over the closed path
syn3 ; syn4 ; syn5 ) and thus leads to phase-dependent dynamics as will be
shown below.

Figure 3 shows the dynamics governed by Eqs. (19)-(21) [we define
$P_{C}(t)=|u_{C}(t)|^{2}$ as the excitation probability of atom $C$], with the
initial state $|\psi(t=0)\rangle=\sigma_{A}^{+}|G\rangle$ and different values
of $\theta$. It shows that phase $\theta$ plays a key role in this case. In
particular, as shown in Figs. 3(a) and 3(c), directional excitation
circulation Roushan  can be observed if $\text{mod}(\theta,\pi)=\pi/2$, with
the circulation direction determined by the sign of $\theta$. According to
Eqs. (22)-(24), the excitation transfer should be symmetric if
$\text{mod}(\theta,\pi)=0$. However, as shown in Fig. 3(b), there is a minor
difference between the time evolutions of $P_{B}(t)$ and $P_{C}(t)$, which we
conclude arises from the finite retardation effect between atoms $A$ and $B$.
We have checked that such a difference tends to vanish as $\tau$ decreases
gradually.

Note that the direct interactions between $C$ and the other atoms impose some
limitations on the architecture of the model. For example, atoms $A$ and $B$
have to be spatially close in order to interact directly with atom $C$. In
view of this, we would like to extend the above trimer to a purely giant-atom
version, where all the three atoms interact with each other via waveguide-
mediated DFIs. As shown in Fig. 1(c), atoms $A$ and $C$ exhibit a DFI through
the upper waveguide, while the DFIs between them and atom $B$ are mediated by
the lower waveguide. In particular, atoms $B$ and $C$ are coupled to the
waveguides with identical and constant strength $g_{0}$, whereas atom $A$ is
coupled to the lower and upper waveguides with different time-dependent
coupling strengths $g(t)$ and $g^{\prime}(t)$, respectively (for each
waveguide the two couplings of $A$ are identical). For simplicity, we still
assume that the coupling points are equally spaced by distance $d$ [in the
lower waveguide, atoms $A$ and $C$ share a common coupling point as shown in
Fig. 1(c)].

The time-delayed dynamical equations of this extended model are given in
Appendix C [see Eqs. (35)-(37)], which, under certain conditions, show a
protected all-to-all interaction (i.e., all the atoms are coupled to each
other via DFIs). One may argue that the protected all-to-all interaction can
also be realized by using only one waveguide as shown in Fig. 1(d) NoriGA .
However, we do not concentrate on this model since the global coupling phase
(i.e., the total synthetic magnetic flux threading the closed loop) is always
zero in this case (see Appendix C for more details). Hereafter, we assume
$g(t)=\Delta_{g}\cos{(\Omega t+\theta)}$ and
$g^{\prime}(t)=\Delta_{g}\cos{(\Omega t)}$ for the model in Fig. 1(c) so that
$\theta$ plays the role of the global coupling phase.

We plot in Figs. 4(a) and 4(b) the time evolutions of the atomic excitation
probabilities in such a giant-atom trimer with $\theta=\pi/2$. When $\chi=2$,
the excitation “hops” directionally in sequence of $A\rightarrow B\rightarrow
C\rightarrow A$, similar to that in Fig. 3(a), yet the damping of the total
atomic excitation probability $P_{\text{tot}}(t)=P_{A}(t)+P_{B}(t)+P_{C}(t)$
is enhanced due to the stronger retardation effect in this model. From Figs.
4(a) and 4(b) one can find that the effective coupling strength between $A$
and the other atoms (which determines the transfer efficiency and the period
of the circulation) can be controlled by tuning $\chi$ (i.e., tuning the
modulation amplitude $\Delta_{g}$). Moreover, as shown in Fig. 4(c),
$P_{\text{tot}}(t)$ shows a slower damping for smaller $\chi$, since the
effective decay rate of atom $A$ [described by the first two terms on the
right side of Eq. (35)] decreases gradually as $\Delta_{g}$ goes to zero. This
can also be seen by comparing the dynamics in Figs. 4(a) and 4(b).

Figure 5: Dynamics of atomic excitation probabilities $P_{A}(t)$, $P_{B}(t)$,
and $P_{C}(t)$ in the atomic trimer [Fig. 1(c)] with (a)
$\Omega/\Gamma_{0}=10\pi$, (b) $\Omega/\Gamma_{0}=5\pi$, (c)
$\Omega/\Gamma_{0}=3\pi$, and (d) $\Omega/\Gamma_{0}=\pi$. All panels in this
figure share the same legend. Other parameters are $\Gamma_{0}=2\pi
g_{0}^{2}/v_{g}$, $\Delta=\Omega$, $\phi=\pi/2$, $\chi=2$, $\theta=\pi/2$,
$\tau\Gamma_{0}=0.01$, and $|\psi(t=0)\rangle=\sigma_{A}^{+}|G\rangle$.

Finally, we would like to demonstrate the influence of a stronger retardation
effect on the present results and discuss how to mitigate this effect to some
extent by tuning the modulation parameters. For relatively large $\tau$, as
shown in Fig. 5(a), the atomic excitation probabilities become strongly damped
and fall to zero rapidly, although the directional excitation circulation can
still be observed. Such a rapid damping, however, can be weakened by using a
smaller modulation frequency as shown in Figs. 5(b) and 5(c) ($\Delta=\Omega$
is always satisfied). This phenomenon can be understood again from the
effective decay rate of atom $A$: as shown in Eq. (35), atom $A$ can be
finally dissipationless if $g(t)=g(t-2\tau)$ and
$g^{\prime}(t)=g^{\prime}(t-2\tau)$ [i.e., $\text{mod}(\Omega\tau,\pi)=0$],
while its effective decay increases with $\Omega$ if
$0<\text{mod}(\Omega\tau,\pi)\ll\pi$. However, decreasing the value of
$\Omega$ also smears the directional excitation circulation since the anti-
rotating-wave terms (i.e., the high-frequency oscillating terms in the
effective Hamiltonian and the dynamical equations) come into play eventually.
As shown in Fig. 5(d), the excitation transfer becomes ruleless when $\Omega$
is small enough. In other words, there is a tradeoff between the retardation-
induced dissipation and the effect of synthetic magnetic field in this case.

##  VI Frequency-modulation schemes

In principle, the synthetic magnetic field can also be created by modulating
the transition frequencies of the giant atoms. For example, recalling the
giant-atom dimer in Fig. 1(a), one can assume constant and uniform coupling
strengths for both atoms and a time-dependent transition frequency for atom
$B$ (thus the detuning between $A$ and $B$ is time dependent). Then a complex
DFI between the two atoms can be realized under certain conditions, as shown
in Appendix D. However, the coupling-modulation scheme shows two major
advantages over the frequency-modulation one arxivClerk : (i) the requirements
for the rotating-wave approximation to be valid are less severe in the
coupling-modulation scheme; (ii) for the frequency-modulation scheme, there
are many sidebands that cannot be neglected in many cases (especially when
multiple frequency modulations are considered or a relatively faster
modulation is employed), which may smear the DFI. Therefore we concentrate on
the coupling-modulation scheme in this paper.

##  VII Conclusions and outlooks

In summary, we have demonstrated how to create a synthetic magnetic field for
the effective decoherence-free Hamiltonian of giant atoms resorting to
periodic coupling modulations and suitable arrangements of atom-waveguide
coupling points. With our scheme one can not only realize DFIs between detuned
giant atoms, but also observe phase-dependent dynamics in closed-loop chains
of giant atoms. Moreover, we have considered the non-Markovian retardation
effect and studied its influence on the atomic dynamics. The retardation
effect does not alter the phase dependence of the dynamics qualitatively, and
its resulting dissipation can be controlled via the modulation parameters
within a certain range.

The results in this paper can be applied to many applications and further
investigations. For example, our scheme highlights a way towards quantum
simulations of many-body systems that are subject to various gauge fields and
towards engineering more high-fidelity quantum gates NoriGA ; braided . It is
also possible to generate fractional quantum Hall states of light by simply
increasing the size of our models (e.g., implementing two-dimensional square
or quasi-one-dimensional ladder lattices of giant atoms with tailored
couplings) Roushan . Although in this paper we have concentrated on models
made up of superconducting qubits and microwave transmission lines, our
proposal is general and can be immediately extended to other possible setups,
such as quantum emitters coupled to real-space or synthetic discrete lattices.
Moreover, the synthetic gauge field offers the opportunity of implementing
richer topological phases based on the effective spin Hamiltonians of giant
atoms WXchiral1 .

## Acknowledgments

We would like to thank F. Ciccarello, Y. Zhang and Y.-T. Chen for helpful
discussions. This work was supported by the National Natural Science
Foundation of China (under Grants No. 12274107 and No. 12074030).

##  Appendix A Time-delayed dynamical equations of the giant-atom dimer

In this Appendix we demonstrate in detail how to derive the time-delayed
dynamical equations (10) and (11) of the giant-atom dimer. By substituting Eq.
(9) into Eqs. (6) and (7), we have

$\displaystyle\dot{u}_{A}(t)$ $\displaystyle=$
$\displaystyle-\int_{0}^{t}dt^{\prime}\int_{-\infty}^{+\infty}dke^{-i(\omega_{k}-\omega_{0})(t-t^{\prime})}\left\\{2g(t)g(t^{\prime})[1+\cos{(2kd)}]u_{A}(t^{\prime})\right.$
(25)
$\displaystyle\left.+g(t)g_{0}\left(e^{ikd}+2e^{-ikd}+e^{-3ikd}\right)u_{B}(t^{\prime})\right\\},$
$\displaystyle\dot{u}_{B}(t)$ $\displaystyle=$ $\displaystyle-i\Delta
u_{B}(t)-\int_{0}^{t}dt^{\prime}\int_{-\infty}^{+\infty}dke^{-i(\omega_{k}-\omega_{0})(t-t^{\prime})}\left\\{2g_{0}^{2}[1+\cos{(2kd)}]u_{B}(t^{\prime})\right.$
(26)
$\displaystyle\left.+g(t^{\prime})g_{0}\left(2e^{ikd}+e^{-ikd}+e^{3ikd}\right)u_{A}(t^{\prime})\right\\}.$

If we change the integration variable as
$\int_{-\infty}^{+\infty}dkf(k)\rightarrow\int_{0}^{+\infty}d\omega_{k}[f(k)+f(-k)]/v_{g}$
and write the dispersion relation of the waveguide as
$\omega_{k}=\omega_{0}+\nu_{k}=\omega_{0}+(k-k_{0})v_{g}$ JTShen2005 ;
JTShen2009 , with $k_{0}$ the wave vector corresponding to frequency
$\omega_{0}$ and $v_{g}$ the group velocity of the emitted photon, Eqs. (25)
and (26) become

$\displaystyle\dot{u}_{A}(t)$ $\displaystyle=$
$\displaystyle-\frac{1}{v_{g}}\int_{0}^{t}dt^{\prime}\int_{-\infty}^{+\infty}d\nu_{k}e^{-i\nu_{k}(t-t^{\prime})}\left\\{4g(t)g(t^{\prime})[1+\cos{(2kd)}]u_{A}(t^{\prime})\right.$
(27)
$\displaystyle\left.+g(t)g_{0}\left(3e^{ikd}+3e^{-ikd}+e^{3ikd}+e^{-3ikd}\right)u_{B}(t^{\prime})\right\\},$
$\displaystyle=$
$\displaystyle-\frac{2\pi}{v_{g}}\int_{0}^{t}dt^{\prime}\left\\{2g(t)g(t^{\prime})\left[2\delta(t-t^{\prime})+e^{2i\phi}\delta(t-t^{\prime}-2\tau)\right]u_{A}(t^{\prime})\right.$
$\displaystyle\left.+g(t)g_{0}\left[3e^{i\phi}\delta(t-t^{\prime}-\tau)+e^{3i\phi}\delta(t-t^{\prime}-3\tau)\right]u_{B}(t^{\prime})\right\\},$
$\displaystyle\dot{u}_{B}(t)$ $\displaystyle=$ $\displaystyle-i\Delta
u_{B}(t)-\frac{1}{v_{g}}\int_{0}^{t}dt^{\prime}\int_{-\infty}^{+\infty}d\nu_{k}e^{-i\nu_{k}(t-t^{\prime})}\left\\{4g_{0}^{2}[1+\cos{(2kd)}]u_{B}(t^{\prime})\right.$
(28)
$\displaystyle\left.+g(t^{\prime})g_{0}\left(3e^{ikd}+3e^{-ikd}+e^{3ikd}+e^{-3ikd}\right)u_{A}(t^{\prime})\right\\}$
$\displaystyle=$ $\displaystyle-i\Delta
u_{B}(t)-\frac{2\pi}{v_{g}}\int_{0}^{t}dt^{\prime}\left\\{2g_{0}^{2}\left[2\delta(t-t^{\prime})+e^{2i\phi}\delta(t-t^{\prime}-2\tau)\right]u_{B}(t^{\prime})\right.$
$\displaystyle\left.+g(t^{\prime})g_{0}\left[3e^{i\phi}\delta(t-t^{\prime}-\tau)+e^{3i\phi}\delta(t-t^{\prime}-3\tau)\right]u_{A}(t^{\prime})\right\\},$

where $\phi=k_{0}d$ and $\tau=d/v_{g}$. In the last steps of Eqs. (27) and
(28), we have omitted the time-advanced terms containing
$\delta(t-t^{\prime}+l\tau)$ ($l=1,2,3$) since they do not contribute to the
integral $\int_{0}^{t}(\cdots)dt^{\prime}$. Finally, one can obtain the time-
delayed dynamical equations (10) and (11) by using the sifting property $\int
dtf(t)\delta(t-t^{\prime})=f(t^{\prime})$ of $\delta$ functions.

##  Appendix B Effective Hamiltonian

In this Appendix we would like to demonstrate the decoherence-free mechanism
of the giant-atom dimer in Fig. 1(a) by deriving its effective Hamiltonian. We
first consider a more general situation where a set of two-level giant atoms
are coupled to a common waveguide with arbitrary arrangements of coupling
points. Similar to the models studied in this paper, one of the atoms (with
transition frequency $\omega_{0}$; labeled as atom $A$) is detuned from the
others by $\Delta$ and is coupled to the waveguide with time-dependent
coupling strength $g(t)$ at two coupling points, while the other giant atoms
have the same transition frequency ($\omega_{0}+\Delta$) and are coupled to
the waveguide with constant coupling strength $g_{0}$. In this case, the
Hamiltonian in the interaction picture can be written as

$\begin{split}V(t)&=\int
dk\Big{[}g(t)\left(e^{-ikx_{A1}}+e^{-ikx_{A2}}\right)\sigma_{A}^{-}a_{k}^{{\dagger}}e^{i\Delta_{k}t}+g_{0}\sum_{j,l}e^{-ikx_{jl}}\sigma_{j}^{-}a_{k}^{{\dagger}}e^{i(\Delta_{k}-\Delta)t}+\text{H.c.}\Big{]}\\\
&=\int
dk\Big{\\{}g(t)\left[e^{-i\varphi_{A1}}e^{i\Delta_{k}(t-\tau_{A1})}+e^{-i\varphi_{A2}}e^{i\Delta_{k}(t-\tau_{A2})}\right]\sigma_{A}^{-}a_{k}^{{\dagger}}\\\
&\quad\,+g_{0}\sum_{j,l}e^{-i\varphi_{jl}}e^{i\Delta_{k}(t-\tau_{jl})}e^{-i\Delta
t}\sigma_{j}^{-}a_{k}^{{\dagger}}+\text{H.c.}\Big{\\}},\end{split}$ (29)

where $\Delta_{k}=\omega_{k}-\omega_{0}$. $x_{jl}$ is the position of the
$l$th coupling point of atom $j$, with which we define
$\tau_{jl}=x_{jl}/v_{g}$ and $\varphi_{jl}=k_{0}x_{jl}$. In the second step of
Eq. (29) we have used the linearized dispersion relation
$\omega_{k}=\omega_{0}+(k-k_{0})v_{g}$. If we consider a discrete time axis
$t_{n}=nT$ with the time interval $T$ short enough compared with the
characteristic time of interaction, the average interaction can be defined as
FCdeco ; GAcollision

$\bar{V}=\frac{1}{T}\int_{t_{n-1}}^{t_{n}}dsV(s),$ (30)

and the effective Hamiltonian of the giant atoms can be given by

$H_{\text{eff}}=\frac{-i}{2T}\int_{t_{n-1}}^{t_{n}}ds\int_{t_{n-1}}^{s}ds^{\prime}[V(s),V(s^{\prime})].$
(31)

To realize decoherence-free Hamiltonians, it is necessary to fulfill the
condition $\bar{V}=0$. Now if we consider the giant-atom dimer in Fig. 1(a)
with cosine-type time-dependent couplings $g(t)=\Delta_{g}\cos{(\Delta
t+\theta)}$ for atom $A$ and perform the transformation $a_{k}\rightarrow
a_{k}\text{exp}(-i\Delta t)$, Eq. (29) becomes

$\begin{split}V(t)&=\int
dk\Big{\\{}\frac{\Delta_{g}}{2}\left[e^{i\Delta_{k}t}+e^{-2i\phi}e^{i\Delta_{k}(t-2\tau)}\right]\left(e^{-i\theta}+e^{2i\Delta
t}e^{i\theta}\right)\sigma_{A}^{-}a_{k}^{{\dagger}}\\\
&\quad\,+g_{0}\left[e^{-i\phi}e^{i\Delta_{k}(t-\tau)}+e^{-3i\phi}e^{i\Delta_{k}(t-3\tau)}\right]\sigma_{B}^{-}a_{k}^{{\dagger}}+\text{H.c.}\Big{\\}},\end{split}$
(32)

where we have assumed
$\\{x_{A1},\,x_{B1},\,x_{A2},\,x_{B2}\\}=\\{0,\,d,\,2d,\,3d\\}$,
$\phi=k_{0}d$, and $\tau=d/v_{g}$ as defined in the main text. Substituting
Eq. (32) into Eq. (31) we can obtain the effective Hamiltonian of the giant-
atom dimer, i.e.,

$\begin{split}H_{\text{eff,dim}}&\simeq\frac{-i}{2T}\frac{2\pi\Delta_{g}g_{0}}{v_{g}}\int_{t_{n-1}}^{t_{n}}ds\int_{t_{n-1}}^{s}ds^{\prime}\Big{\\{}\Big{[}2e^{i\phi}[\delta(s^{\prime}-s+\tau)-\delta(s-s^{\prime}+\tau)]\\\
&\quad\,+e^{3i\phi}[\delta(s^{\prime}-s+3\tau)-\delta(s-s^{\prime}+3\tau)]+e^{-i\phi}[\delta(s^{\prime}-s-\tau)-\delta(s-s^{\prime}-\tau)]\Big{]}e^{-i\theta}\sigma_{B}^{+}\sigma_{A}^{-}+\text{H.c.}\Big{\\}}\\\
&=\frac{-i\pi\Delta_{g}g_{0}}{v_{g}}\left[(2e^{i\phi}+e^{3i\phi}-e^{-i\phi})e^{-i\theta}\sigma_{B}^{+}\sigma_{A}^{-}+\text{H.c.}\right],\end{split}$
(33)

where we have assumed that all the time delays $l\tau$ are negligible compared
to $T$ (Markovian regime) and have dropped the high-frequency oscillating
terms containing $\text{exp}(2i\Delta t)$. When $\phi=(m+1/2)\pi$, the
effective Hamiltonian becomes

$H_{\text{eff,dim}}=G_{m}e^{-i\theta}\sigma_{B}^{+}\sigma_{A}^{-}+\text{H.c.}$
(34)

with $G_{m}=(-1)^{m}2\pi\Delta_{g}g_{0}/v_{g}$, which shows a complex DFI
between atoms $A$ and $B$. Moreover, one can see from Eqs. (30) and (32) that
the average interaction between the giant atoms and the waveguide field
vanishes (i.e., $\bar{V}=0$) in this case.

##  Appendix C Dynamical equations of the models in Figs. 1(c) and 1(d)

For the giant-atom trimer in Fig. 1(c), the time-delayed dynamical equations
of the atomic excitation amplitudes can be immediately given by

$\displaystyle\dot{u}_{A}(t)$ $\displaystyle=$
$\displaystyle-\frac{4\pi[g^{2}(t)+g^{\prime
2}(t)]}{v_{g}}u_{A}(t)-\frac{4\pi[g(t)g(t-2\tau)+g^{\prime}(t)g^{\prime}(t-2\tau)]}{v_{g}}D_{A,2}(t)-\frac{2\pi
g(t)g_{0}}{v_{g}}\left[3D_{B,1}(t)+D_{B,3}(t)\right]$ (35)
$\displaystyle-\frac{2\pi
g(t)g_{0}}{v_{g}}\left[u_{C}(t)+2D_{C,2}(t)+D_{C,4}(t)\right]-\frac{2\pi
g^{\prime}(t)g_{0}}{v_{g}}\left[3D_{C,1}(t)+D_{C,3}(t)\right],$
$\displaystyle\dot{u}_{B}(t)$ $\displaystyle=$ $\displaystyle-i\Delta
u_{B}(t)-\frac{4\pi
g_{0}^{2}}{v_{g}}\left[u_{B}(t)+D_{B,2}(t)\right]-\frac{6\pi
g(t-\tau)g_{0}}{v_{g}}D_{A,1}(t)$ (36) $\displaystyle-\frac{2\pi
g(t-3\tau)g_{0}}{v_{g}}D_{A,3}(t)-\frac{2\pi
g_{0}^{2}}{v_{g}}\left[3D_{C,1}(t)+D_{C,3}(t)\right],$
$\displaystyle\dot{u}_{C}(t)$ $\displaystyle=$ $\displaystyle-i\Delta
u_{C}(t)-\frac{8\pi
g_{0}^{2}}{v_{g}}\left[u_{C}(t)+D_{C,2}(t)\right]-\frac{2\pi
g(t)g_{0}}{v_{g}}u_{A}(t)-\frac{4\pi
g(t-2\tau)g_{0}}{v_{g}}D_{A,2}(t)-\frac{2\pi
g(t-4\tau)g_{0}}{v_{g}}D_{A,4}(t)$ (37) $\displaystyle-\frac{6\pi
g^{\prime}(t-\tau)g_{0}}{v_{g}}D_{A,1}(t)-\frac{2\pi
g^{\prime}(t-3\tau)g_{0}}{v_{g}}D_{A,3}(t)-\frac{2\pi
g_{0}^{2}}{v_{g}}\left[3D_{B,1}(t)+D_{B,3}(t)\right].$

which can be simplified to

$\displaystyle\dot{u}_{A}(t)$ $\displaystyle=$ $\displaystyle-i\frac{4\pi
g(t)g_{0}}{v_{g}}u_{B}(t)-i\frac{4\pi g^{\prime}(t)g_{0}}{v_{g}}u_{C}(t),$
(38) $\displaystyle\dot{u}_{B}(t)$ $\displaystyle=$ $\displaystyle-i\Delta
u_{B}(t)-i\frac{4\pi g(t)g_{0}}{v_{g}}u_{A}(t)-i\frac{4\pi
g_{0}^{2}}{v_{g}}u_{C}(t),$ (39) $\displaystyle\dot{u}_{C}(t)$
$\displaystyle=$ $\displaystyle-i\Delta u_{C}(t)-i\frac{4\pi
g^{\prime}(t)g_{0}}{v_{g}}u_{A}(t)-i\frac{4\pi g_{0}^{2}}{v_{g}}u_{B}(t),$
(40)

if $\phi=\pi/2$ and $\tau\rightarrow 0$. By assuming cosine-type time-
dependent couplings $g(t)=\Delta_{g}\cos{(\Omega t+\theta)}$ and
$g^{\prime}(t)=\Delta_{g}\cos{(\Omega t)}$ for atom $A$ with
$\Omega\equiv\Delta$, one finally has

$\displaystyle\dot{u}_{A}(t)$ $\displaystyle\simeq$ $\displaystyle-
iG_{0}e^{i\theta}u_{B}(t)-iG_{0}u_{C}(t),$ (41) $\displaystyle\dot{u}_{B}(t)$
$\displaystyle\simeq$ $\displaystyle-
iG_{0}e^{-i\theta}u_{A}(t)-2i\Gamma_{0}u_{C}(t),$ (42)
$\displaystyle\dot{u}_{C}(t)$ $\displaystyle\simeq$ $\displaystyle-
iG_{0}u_{A}(t)-2i\Gamma_{0}u_{B}(t),$ (43)

which shows a protected all-to-all interaction with synthetic magnetic flux
$\theta$. Having in mind that $G_{0}=\chi\Gamma_{0}$, directional excitation
circulation can be expected if $\chi=2$ and $\text{mod}(\theta,\pi)=\pi/2$.

As mentioned in the main text, the protected all-to-all interaction among
atoms $A$, $B$, and $C$ can also be implemented by using only one waveguide,
provided that the coupling points of the three atoms are arranged according to
the configuration in Fig. 1(d). In this case, we assume that the coupling
points are equally spaced by $d^{\prime}$ such that the phase accumulation
(propagation time) of the field between adjacent coupling points becomes
$\phi^{\prime}=k_{0}d^{\prime}$ ($\tau^{\prime}=d^{\prime}/v_{g}$). Again,
atoms $B$ and $C$ are coupled to the waveguide with identical and constant
strength $g_{0}$, while atom $A$ interacts with the waveguide with time-
dependent strength $g(t)$ at each coupling point. After some algebra, the
dynamical equations of the model can be obtained as

$\displaystyle\dot{u}_{A}(t)$ $\displaystyle=$ $\displaystyle-\frac{2\pi
g(t)}{v_{g}}\left\\{2g(t)u_{A}(t)+2g(t-3\tau^{\prime})D^{\prime}_{A,3}(t)+g_{0}\left[2D^{\prime}_{B,1}(t)+D^{\prime}_{B,2}(t)+D^{\prime}_{B,4}(t)\right.\right.$
(44)
$\displaystyle\left.\left.+D^{\prime}_{C,1}(t)+2D^{\prime}_{C,2}(t)+D^{\prime}_{C,5}(t)\right]\right\\},$
$\displaystyle\dot{u}_{B}(t)$ $\displaystyle=$ $\displaystyle-i\Delta
u_{B}(t)-\frac{2\pi
g_{0}}{v_{g}}\left\\{2g_{0}u_{B}(t)+2g_{0}D^{\prime}_{B,3}(t)+2g(t-\tau^{\prime})D^{\prime}_{A,1}(t)+g(t-2\tau^{\prime})D^{\prime}_{A,2}(t)\right.$
(45)
$\displaystyle\left.+g(t-4\tau^{\prime})D^{\prime}_{A,4}(t)+g_{0}\left[2D^{\prime}_{C,1}(t)+D^{\prime}_{C,2}(t)+D^{\prime}_{C,4}(t)\right]\right\\},$
$\displaystyle\dot{u}_{C}(t)$ $\displaystyle=$ $\displaystyle-i\Delta
u_{C}(t)-\frac{2\pi
g_{0}}{v_{g}}\left\\{2g_{0}u_{C}(t)+2g_{0}D^{\prime}_{C,3}(t)+g(t-\tau^{\prime})D^{\prime}_{A,1}(t)+2g(t-2\tau^{\prime})D^{\prime}_{A,2}(t)\right.$
(46)
$\displaystyle\left.+g(t-5\tau^{\prime})D^{\prime}_{A,5}(t)+g_{0}\left[2D^{\prime}_{B,1}(t)+D^{\prime}_{B,2}(t)+D^{\prime}_{B,4}(t)\right]\right\\},$

where
$D^{\prime}_{j,l}(t)=\text{exp}(il\phi^{\prime})u_{j}(t-l\tau^{\prime})\Theta(t-l\tau^{\prime})$.
When $\phi^{\prime}=(2m+1/3)\pi$ and $\tau\rightarrow 0$, the above three
equations become

$\displaystyle\dot{u}_{A}(t)$ $\displaystyle=$ $\displaystyle-i\frac{4\pi
g(t)g_{0}}{v_{g}}\sin{\left(\frac{\pi}{3}\right)}u_{B}(t)-i\frac{4\pi
g(t)g_{0}}{v_{g}}\sin{\left(\frac{\pi}{3}\right)}u_{C}(t),$ (47)
$\displaystyle\dot{u}_{B}(t)$ $\displaystyle=$ $\displaystyle-i\Delta
u_{B}(t)-i\frac{4\pi
g(t)g_{0}}{v_{g}}\sin{\left(\frac{\pi}{3}\right)}u_{A}(t)-i\frac{4\pi
g_{0}^{2}}{v_{g}}\sin{\left(\frac{\pi}{3}\right)}u_{C}(t),$ (48)
$\displaystyle\dot{u}_{C}(t)$ $\displaystyle=$ $\displaystyle-i\Delta
u_{C}(t)-i\frac{4\pi
g(t)g_{0}}{v_{g}}\sin{\left(\frac{\pi}{3}\right)}u_{A}(t)-i\frac{4\pi
g_{0}^{2}}{v_{g}}\sin{\left(\frac{\pi}{3}\right)}u_{B}(t),$ (49)

which are identical with Eqs. (38)-(40), except for the modified effective
coupling strengths. By assuming $g(t)=\Delta_{g}\cos(\Omega t+\theta)$ and
performing the transformation $u_{B,C}(t)\rightarrow
u_{B,C}(t)\text{exp}(-i\Delta t)$, Eqs. (47)-(49) become

$\displaystyle\dot{u}_{A}(t)$ $\displaystyle\simeq$ $\displaystyle-
iG^{\prime}e^{i\theta}u_{B}(t)-iG^{\prime}e^{i\theta}u_{C}(t),$ (50)
$\displaystyle\dot{u}_{B}(t)$ $\displaystyle\simeq$ $\displaystyle-
iG^{\prime}e^{-i\theta}u_{A}(t)-2i\Gamma^{\prime}u_{C}(t),$ (51)
$\displaystyle\dot{u}_{C}(t)$ $\displaystyle\simeq$ $\displaystyle-
iG^{\prime}e^{-i\theta}u_{A}(t)-2i\Gamma^{\prime}u_{B}(t),$ (52)

where $G^{\prime}=G_{0}\sin{(\pi/3)}=2\pi\Delta_{g}g_{0}\sin{(\pi/3)}/v_{g}$
and $\Gamma^{\prime}=\Gamma_{0}\sin{(\pi/3)}=2\pi
g_{0}^{2}\sin{(\pi/3)}/v_{g}$. Clearly, the effective coupling phase can
always be gauged away via the transformation $u_{A}(t)\rightarrow
u_{A}(t)\text{exp}(i\theta)$. Therefore, phase-dependent dynamics cannot be
observed in this case.

##  Appendix D Frequency-modulation scheme

In this Appendix, we consider that atoms $A$ and $B$ (recalling the giant-atom
dimer) are coupled to the waveguide in the braided manner, yet with constant
and uniform couplings (coupling strength $g_{0}$) instead. While the
transition frequency $\omega_{0}$ of atom $A$ is assumed to be constant, we
consider a frequency modulation for atom $B$ such that there is a small time-
dependent detuning $\Delta_{0}+\Delta(t)$ between the two atoms. In this case,
the Hamiltonian of the model can be written as

$\displaystyle H^{\prime}$ $\displaystyle=$ $\displaystyle
H_{\text{a}}^{\prime}+H_{\text{w}}+H_{\text{int}}^{\prime},$ (53)
$\displaystyle H_{\text{a}}^{\prime}$ $\displaystyle=$
$\displaystyle\omega_{0}\sigma_{A}^{+}\sigma_{A}^{-}+[\omega_{0}+\Delta_{0}+\Delta(t)]\sigma_{B}^{+}\sigma_{B}^{-},$
(54) $\displaystyle H_{\text{int}}^{\prime}$ $\displaystyle=$
$\displaystyle\int
dkg_{0}\left[\left(1+e^{2ikd}\right)\sigma_{A}^{+}a_{k}\right.$ (55)
$\displaystyle\left.+\left(e^{ikd}+e^{3ikd}\right)\sigma_{B}^{+}a_{k}+\text{H.c.}\right],$

where $H_{\text{w}}$ is identical with that in Eq. (3). With the single-
excitation state of the system given in Eq. (5) and a similar calculation
procedure as shown in Sec. II, one can obtain the dynamical equations of the
atomic excitation amplitudes as

$\displaystyle\dot{u}_{A}(t)$ $\displaystyle=$
$\displaystyle-\Gamma_{0}\left[2u_{A}(t)+2D_{A,2}(t)+3D_{B,1}(t)\right.$ (56)
$\displaystyle\left.+D_{B,3}(t)\right],$ $\displaystyle\dot{u}_{B}(t)$
$\displaystyle=$
$\displaystyle-i[\Delta_{0}+\Delta(t)]u_{B}(t)-\Gamma_{0}\left[2u_{B}(t)\right.$
(57) $\displaystyle\left.+2D_{B,2}(t)+3D_{A,1}(t)+D_{A,3}(t)\right],$

where $\Gamma_{0}=2\pi g_{0}^{2}/v_{g}$ and
$D_{j,l}(t)=\text{exp}(il\phi)u_{j}(t-l\tau)\Theta(t-l\tau)$ as defined in the
main text. Once again, in the Markovian regime with negligible time delays and
if $\phi=\pi/2$, the above two equations can be simplified to

$\displaystyle\dot{u}_{A}(t)$ $\displaystyle=$
$\displaystyle-2i\Gamma_{0}u_{B}(t),$ (58) $\displaystyle\dot{u}_{B}(t)$
$\displaystyle=$
$\displaystyle-i[\Delta_{0}+\Delta(t)]u_{B}(t)-2i\Gamma_{0}u_{A}(t).$ (59)

Now we consider a cosine-type modulation
$\Delta(t)=\Delta_{g}^{\prime}\cos{(\Omega^{\prime}t+\theta^{\prime})}$ (where
$\Delta_{g}^{\prime}$, $\Omega^{\prime}$, and $\theta^{\prime}$ are the
amplitude, frequency, and initial phase of the modulation, respectively) for
the detuning and perform a transformation

$u_{B}(t)\rightarrow
u_{B}(t)e^{-i\Delta_{0}t}e^{-i\eta\sin{(\Omega^{\prime}t+\theta^{\prime})}}$
(60)

with $\eta=\Delta_{g}^{\prime}/\Omega^{\prime}$, Eqs. (58) and (59) become

$\displaystyle\dot{u}_{A}(t)$ $\displaystyle=$
$\displaystyle-2i\Gamma_{0}u_{B}(t)e^{-i\Delta_{0}t}e^{-i\eta\sin{(\Omega^{\prime}t+\theta^{\prime})}},$
(61) $\displaystyle\dot{u}_{B}(t)$ $\displaystyle=$
$\displaystyle-2i\Gamma_{0}u_{A}(t)e^{i\Delta_{0}t}e^{i\eta\sin{(\Omega^{\prime}t+\theta^{\prime})}}.$
(62)

Assuming $\Omega^{\prime}=\Delta_{0}\gg 2\Gamma_{0}$ and using the Jacobi-
Anger expansion

$e^{-iz\sin{x}}=\sum_{q=-\infty}^{+\infty}J_{q}(z)e^{-iqx},$ (63)

where $J_{q}(z)$ is the Bessel function of the first kind, one finally has

$\displaystyle\dot{u}_{A}(t)$ $\displaystyle\simeq$
$\displaystyle-2i\Gamma_{0}J_{-1}(\eta)u_{B}(t)e^{i\theta^{\prime}},$ (64)
$\displaystyle\dot{u}_{B}(t)$ $\displaystyle\simeq$
$\displaystyle-2i\Gamma_{0}J_{-1}(\eta)u_{A}(t)e^{-i\theta^{\prime}}.$ (65)

Clearly, a complex DFI between atoms $A$ and $B$ can also be created in this
case.

## References

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