# Giant Atoms in Synthetic Frequency Dimensions

Lei Du  Beijing Computational Science Research Center, Beijing 100193, China
Yan Zhang  Center for Quantum Sciences and School of Physics, Northeast Normal
University, Changchun 130024, China  Jin-Hui Wu  Center for Quantum Sciences
and School of Physics, Northeast Normal University, Changchun 130024, China
Anton Frisk Kockum  Department of Microtechnology and Nanoscience (MC2),
Chalmers University of Technology, 412 96 Gothenburg, Sweden  Yong Li
<EMAIL_ADDRESS>Beijing Computational Science Research Center, Beijing
100193, China

###### Abstract

Giant atoms with nonlocal couplings to real-space waveguides have attracted
extensive attention due to their unique interference effects. Here, we propose
a feasible scheme for constructing giant atoms in synthetic frequency
dimensions with, e.g., a dynamically modulated microring resonator and a
tailored multi-level atom. Both analytical and numerical calculations show
good agreement between this scheme and real-space two-level giant atoms. Our
scheme is proven to have potential applications for manipulating the frequency
of light, such as excitation confinement and release in the frequency domain.
More importantly, time-reversal symmetry of the model can be readily broken by
tuning the relative phase between the external fields applied on the atom,
enabling implementations of chiral quantum optics in frequency dimensions.

Giant atoms, which support nonlocal couplings with the surrounding environment
(waveguides), have attracted growing research interest in the past few years
due to various intriguing phenomena arising from them fiveyears . In general,
giant atoms can be achieved by coupling (artificial) atoms to propagating
fields whose wavelengths are much smaller than the atomic sizes (e.g., surface
acoustic waves) transmon1 ; transmon2 ; transmonA1 ; transmonA2 ; transmonA3 ;
transmonA4 ; transmonA5 ; transmon3 ; transmonA6 , or by coupling atoms to
meandering waveguides at separated points Lamb ; braided ; EngineerCW . With
such nonlocal interactions, one should naturally consider phase accumulations
of photons between different atom-waveguide coupling points, which lead to a
series of striking phenomena that are absent for small atoms, such as
frequency-dependent Lamb shifts and relaxation rates Lamb , decoherence-free
interaction between atoms through the waveguide GANori ; braided ; DFmechanism
; AFKchiral , and oscillating bound states oscillate ; osci2 , to name a few.

On the other hand, the concept of synthetic dimensions has been recently
proposed and extensively explored in a variety of physical systems such as
photonic structures synOL ; synOptica ; synOptRev ; synPRB ; LiNbO3PRL ;
LiNbO3PRB ; multiphoton , cold atoms syncold1 ; syncold2 ; syncold3 , and
superconducting circuits synsc1 ; synsc2 ; synsc3 ; Wilson2021 ; synSAW . With
synthetic dimensions, it is possible to explore richer physical effects with
fewer geometric dimensions. A simple way to create synthetic dimensions in
photonic systems is to actively couple modes at different frequencies via
dynamical modulations synOL ; synOptica ; synOptRev ; synPRB ; LiNbO3PRL ;
LiNbO3PRB . Moreover, one can also create synthetic dimensions based on other
internal degrees of freedom of photons such as momentum SLdwwang ; SLOptica
and orbital angular momentum OAM1 ; OAM2 ; OAM3 . The construction of
synthetic dimensions not only enables significant reduction of physical
resources, but also provide possibilities for manipulating the relevant
degrees of freedom synOptRev .

In this Letter, we demonstrate how to implement giant atoms in synthetic
frequency dimensions, where the one-dimensional (1D) frequency lattice acting
as a discrete waveguide can be readily achieved with a dynamically modulated
microring resonator synOL ; synOptica ; synOptRev ; synPRB . We consider a
cyclic four-level atom, where the two lower transitions couple to different
sites of the frequency lattice and the two upper transitions are driven by two
external fields. The atom is assumed to exhibit two two-photon resonant
transitions, such that it can be effectively described as a two-level giant
atom that couples nonlocally with the lattice. We reveal that such an
effective model can not only support some typical effects of real-space giant
atoms such as long-lived populations, but also shows the potential for
manipulating the frequency of light.

As shown in Fig. 1(a), we consider a microring resonator which is modulated by
an electro-optic modulator (EOM). For the finite frequency range where the
group velocity dispersion can be neglected, the resonant frequencies of the
ring (labeled by $m=0,\pm 1,\pm 2,...$) are approximately given by
$\omega_{m}=\omega_{0}+m\Omega_{s}$, where $\omega_{0}$ is the central
frequency of the considered range; $\Omega_{s}=2\pi v_{g}/L$ is the free
spectral range, with $v_{g}$ and $L$ the group velocity at $\omega_{0}$ and
the circumference of the ring, respectively. Due to the dynamic modulation
introduced by the EOM, adjacent resonant modes are effectively coupled and
thus a synthetic 1D frequency lattice is formed synOL ; synOptica ; synOptRev
; synPRB . For weak modulations, the Hamiltonian of the synthetic lattice can
be written as (see Sec. I of SM  for more details; $\hbar=1$ throughout this
Letter)

$H_{r}=\sum_{m}[\omega_{m}a_{m}^{{\dagger}}a_{m}+2J\cos{(\Omega
t)}(a_{m}^{{\dagger}}a_{m+1}+\text{H.c.})],$ (1)

which in the interaction picture becomes

$H_{r}^{\prime}=\sum_{m}J(a_{m}^{{\dagger}}a_{m+1}+\text{H.c.})$ (2)

if the modulation frequency $\Omega=\Omega_{s}$ (we focus on this case
hereafter). Here $a_{m}$ ($a_{m}^{{\dagger}}$) is the annihilation (creation)
operator of the $m$th resonant mode of the ring; $J=\alpha/2T_{R}$ is the
effective coupling strength between adjacent modes, with $\alpha$ the
modulation amplitude and $T_{R}$ the round-trip propagation time of the ring;
H.c. denotes the Hermitian conjugate.

Figure 1: (a) Schematic illustration of the model under consideration. The
ring resonator is modulated by an electro-optic modulator. A four-level atom
interacts with two resonant modes of the ring and two external fields, which
form two two-photon resonant transitions between $|g\rangle$ and $|e\rangle$.
(b) Effective two-level giant atom after adiabatic elimination. (c) Ladder-
type implementation scheme of the effective giant atom.

In addition, we consider a four-level atom with a ground state $|g\rangle$,
two middle states $|f_{1}\rangle$ and $|f_{2}\rangle$, and an excited state
$|e\rangle$. We select $|g\rangle$ as the reference so that the energies of
$|f_{1}\rangle$, $|f_{2}\rangle$, and $|e\rangle$ are denoted by
$\omega_{f_{1}}$, $\omega_{f_{2}}$, and $\omega_{e}$, respectively. The two
lower transitions $|g\rangle\leftrightarrow|f_{1}\rangle$ and
$|g\rangle\leftrightarrow|f_{2}\rangle$ are coupled with modes $a_{0}$ and
$a_{N}$, respectively, while the upper transitions
$|f_{1}\rangle\leftrightarrow|e\rangle$ and
$|f_{2}\rangle\leftrightarrow|e\rangle$ are driven by two external fields of
amplitudes (frequencies) $\eta_{1}$ and $\eta_{2}$ ($\omega_{d,1}$ and
$\omega_{d,2}$), respectively. In the interaction picture and with the
rotating-wave approximation, the total Hamiltonian of the system can be
written as (see Sec. II of SM  for more details)

$\begin{split}H&=H_{r}^{\prime}+\Big{(}g_{1}a_{0}^{{\dagger}}|g\rangle\langle
f_{1}|e^{i\Delta_{1}t}+g_{2}a_{N}^{{\dagger}}|g\rangle\langle
f_{2}|e^{i\Delta_{2}t}\\\ &\quad+\eta_{1}e^{i\theta}|e\rangle\langle
f_{1}|e^{i\Delta_{d,1}t}+\eta_{2}|e\rangle\langle
f_{2}|e^{i\Delta_{d,2}t}+\text{H.c.}\Big{)},\end{split}$ (3)

where $g_{1}$ ($g_{2}$) is the coupling strength between mode $a_{0}$
($a_{N}$) and the transition $|g\rangle\leftrightarrow|f_{1}\rangle$
($|g\rangle\leftrightarrow|f_{2}\rangle$), and is assumed to be real hereafter
for simplicity; $\theta$ is the relative phase between the external fields;
$\Delta_{1}=\omega_{0}-\omega_{f,1}$ ($\Delta_{2}=\omega_{N}-\omega_{f,2}$) is
the detuning between mode $a_{0}$ ($a_{N}$) and the transition
$|g\rangle\leftrightarrow|f_{1}\rangle$
($|g\rangle\leftrightarrow|f_{2}\rangle$);
$\Delta_{d,1}=\omega_{e}-\omega_{f,1}-\omega_{d,1}$
($\Delta_{d,2}=\omega_{e}-\omega_{f,2}-\omega_{d,2}$) is the detuning between
the transition $|f_{1}\rangle\leftrightarrow|e\rangle$
($|f_{2}\rangle\leftrightarrow|e\rangle$) and the external field $\eta_{1}$
($\eta_{2}$). In the case of $\Delta_{d,1(2)}=\Delta_{1(2)}$, where two two-
photon resonant transitions between $|g\rangle$ and $|e\rangle$ are
constructed Stark1993 ; AdiaElim ; YanDong , Eq. (3) is equivalent to the
time-independent Hamiltonian

$\begin{split}H^{\prime}&=H_{r}^{\prime}-\Delta_{1}|f_{1}\rangle\langle
f_{1}|-\Delta_{2}|f_{2}\rangle\langle f_{2}|\\\
&\quad+\Big{(}g_{1}a_{1}^{{\dagger}}|g\rangle\langle
f_{1}|+g_{2}a_{N}^{{\dagger}}|g\rangle\langle f_{2}|\\\
&\quad+\eta_{1}e^{i\theta}|e\rangle\langle f_{1}|+\eta_{2}|e\rangle\langle
f_{2}|+\text{H.c.}\Big{)}.\end{split}$ (4)

Note that we only have considered resonant modes propagating along one
direction of the ring. Generally speaking, counterpropagating modes should
also be taken into account due to the randomness of the atomic spontaneous
emission. However, chiral atom-waveguide interactions can be achieved, for
example, with the spin-momentum locking effect chiralAW ; SMlocking ;
chiralZoller .

Assuming the single-excitation states of the system as

$|\psi(t)\rangle=\sum_{m}u_{m}(t)a_{m}^{{\dagger}}|0,g\rangle+\sum_{\beta=f_{1},f_{2},e}u_{\beta}(t)|0,\beta\rangle$
(5)

and solving the Schrödinger equation, we have

$\begin{split}i\dot{u}_{e}&=\eta_{1}e^{i\theta}u_{f_{1}}+\eta_{2}u_{f_{2}},\\\
i\dot{u}_{f_{1}}&=-\Delta_{1}u_{f_{1}}+g_{1}u_{0}+\eta_{1}e^{-i\theta}u_{e},\\\
i\dot{u}_{f_{2}}&=-\Delta_{2}u_{f_{2}}+g_{2}u_{N}+\eta_{2}u_{e},\\\
i\dot{u}_{m}&=J(u_{m+1}+u_{m-1})+g_{1}u_{f_{1}}\delta_{m,0}\\\
&\quad+g_{2}u_{f_{2}}\delta_{m,N},\end{split}$ (6)

where $u_{m}$ and $u_{\beta}$ are the probability amplitudes of exciting the
$m$th mode and the state $|\beta\rangle$, respectively.

Considering the regime of
$\\{|\Delta_{1}|,\,|\Delta_{2}|\\}\gg\\{g_{1},\,g_{2},\,\eta_{1},\,\eta_{2}\\}$,
where $|f_{1}\rangle$ and $|f_{2}\rangle$ can be adiabatically eliminated if
they are initially unpopulated Stark1993 ; AdiaElim ; YanDong , Eq. (6) can be
approximately simplified to

$\begin{split}i\dot{u}_{e}&=\Delta_{e,e}u_{e}+g_{e,1}e^{i\theta}u_{0}+g_{e,2}u_{N},\\\
i\dot{u}_{m}&=J(u_{m+1}+u_{m-1})+(g_{e,1}e^{-i\theta}u_{e}\\\
&\quad+\Delta_{e,1}u_{0})\delta_{m,0}+(g_{e,2}u_{e}+\Delta_{e,2}u_{N})\delta_{m,N},\end{split}$
(7)

where
$\Delta_{e,e}=(\eta_{1}^{2}\Delta_{2}+\eta_{2}^{2}\Delta_{1})/\Delta_{1}\Delta_{2}$
and $\Delta_{e,1}=g_{1}^{2}/\Delta_{1}$ ($\Delta_{e,2}=g_{2}^{2}/\Delta_{2}$)
are the effective frequency shifts of $|e\rangle$ and $a_{0}$ ($a_{N}$),
respectively; $g_{e,1}=g_{1}\eta_{1}/\Delta_{1}$
($g_{e,2}=g_{2}\eta_{2}/\Delta_{2}$) is the effective coupling strength
between $a_{0}$ ($a_{N}$) and the transition
$|g\rangle\leftrightarrow|e\rangle$ due to the corresponding two-photon
transition. For simplicity, we assume $\Delta_{1}\equiv\Delta_{2}=\Delta$ and
$\eta_{1}\equiv\eta_{2}=\eta$ hereafter. In the case of $\theta=0$ and
$\\{\Delta_{e,e},\Delta_{e,1},\Delta_{e,2}\\}\rightarrow 0$, Eq. (7) is
similar to the dynamic equation of a real-space two-level giant atom coupled
to a $1$D lattice at two separated sites LonghiGA , which reads

$\begin{split}i\dot{w}_{e}&=\lambda_{1}w_{0}+\lambda_{2}w_{N},\\\
i\dot{w}_{m}&=J(w_{m+1}+w_{m-1})+w_{e}(\lambda_{1}\delta_{m,0}+\lambda_{2}\delta_{m,N}).\end{split}$
(8)

Here $w_{m}$ and $w_{e}$ are the probability amplitudes of exciting the $m$th
site of the lattice and the atom, respectively$;\lambda_{1}$ ($\lambda_{2}$)
is the atom-waveguide coupling strength at the $0$th ($N$th) site. In view of
this, our model is equivalent to a two-level giant atom in the frequency
dimension, as shown in Fig. 1(b).

Figure 2: Dynamic evolutions of (a) $P_{e}(t)$ and $P_{e}^{\prime}(t)$ with
$\Delta/J=60$ and different values of $N$, (b) $P_{e}(t)$ with $N=2$ and
different values of $\Delta$. In panel (a), we assume
$\lambda_{1}=\lambda_{2}=0.1J$ for the real-space case. Other parameters are
$g_{1}=g_{2}=3J$, $\eta=2J$, $\theta=0$, and $|\psi(t=0)\rangle=|0,e\rangle$.

We first verify the validity of the effective giant atom in the familiar case
of $\theta=0$ by numerically solving Eq. (6) with appropriate parameters and
comparing the results with those of the real-space analogue [obtained with Eq.
(8)]. In practice, the frequency dimension typically extends over a limited
number of modes where the group velocity dispersion can be neglected. In view
of this, we consider $25$ lattice sites (resonant modes) in total, which is
experimentally accessible OAM3 , and assume that $N$ is much smaller than the
lattice length to avoid undesired boundary effects. Figure 2(a) shows the
dynamic evolutions of the populations $P_{e}(t)=|u_{e}(t)|^{2}$ and
$P_{e}^{\prime}(t)=|w_{e}(t)|^{2}$ with the initial state
$|\psi(t=0)\rangle=|0,e\rangle$ and with different values of $N$. Clearly, the
evolutions of $P_{e}(t)$ and $P_{e}^{\prime}(t)$ can be well fitted if
$|\Delta|\gg\\{g_{1},\,g_{2},\,\eta_{1},\,\eta_{2}\\}$ and
$\lambda_{1(2)}=g_{e,1(2)}$. In particular, long-lived population can be
observed if $N=2$ and $g_{e,1}=g_{e,2}$, similar to a decoherence-free giant
atom in real space Lamb ; LonghiGA . Physically, this is because the two atom-
waveguide coupling channels interfere destructively in this case. Moreover, we
see that the decay rate of $P_{e}(t)$ depends on $N$, which embodies the
phase-dependent spontaneous emissions of real-space giant atoms.

We also demonstrate in Fig. 2(b) the dependence of the dynamics on detuning
$\Delta$. The long-lived population becomes more and more ideal as $\Delta$
increases. Physically, this is due to two reasons: on one hand, the adiabatic
elimination method becomes more accurate as $\Delta$ increases; on the other
hand, a larger $\Delta$ leads to longer coherence time $t_{c}$ of the emitted
photons ($t_{c}\propto 1/(g_{e,1}^{2}+g_{e,2}^{2})$ Solano ) and thus the
propagation time $\tau$ between the two coupling points ($\tau=N/2J$ LonghiGA
) tends to be negligible compared with $t_{c}$. Consequently, the non-
Markovian retardation effect can be ignored and the giant atom becomes almost
decoherence-free with large enough $\Delta$ LonghiGA ; retardDu .

In Fig. 2, we have considered the regime of $J\gg\\{g_{e,1},g_{e,2}\\}$, where
the effective giant atom can be almost decoherence-free. In such a regime,
however, the giant atom can hardly affect the dynamic diffusion of a single-
mode excitation along the frequency lattice due to the weak atom-waveguide
couplings, as shown in Fig. S1(a) SM . In view of this, we now consider the
regime where the effective coupling strengths $g_{e,1}$ and $g_{e,2}$ are
comparable with $J$. Nevertheless, the effective frequency shifts
$\Delta_{e,1}$ and $\Delta_{e,2}$ can also be considerable in this case such
that the excitations residing between sites $a_{0}$ and $a_{N}$ are always
strongly confined therein regardless of the giant-atom structure (i.e., the
value of $N$; see Sec. III of SM  for more details). Therefore, we introduce
opposite frequency shifts for modes $a_{0}$ and $a_{N}$ by coupling them with
two extra auxiliary modes $b_{1}$ and $b_{2}$, respectively, i.e.,

$\begin{split}i\dot{u}_{m}&=J(u_{m+1}+u_{m-1})+g_{1}(u_{f_{1}}+u_{b_{1}})\delta_{m,0}\\\
&\quad+g_{2}(u_{f_{2}}+u_{b_{2}})\delta_{m,N},\\\ i\dot{u}_{b_{1(2)}}&=\Delta
u_{b_{1(2)}}+g_{1(2)}u_{0},\end{split}$ (9)

where $u_{b_{1,2}}$ are the probability amplitudes of exciting $b_{1}$ and
$b_{2}$, respectively. The dynamic evolutions of $u_{e}$, $u_{f_{1}}$, and
$u_{f_{2}}$ are identical with those in Eq. (6).

Figure 3: Dynamic evolutions of $P_{m}(t)$ with (a) $N=2$, (b) $N=4$, and
$P_{m}^{\prime}(t)$ with (c) $N=2$, (d) $N=4$. Dynamic excitation release
[i.e., $P_{m}(t)$] (e) and profiles at specific moments (f) with $N=2$ and
$\eta(t)=5J[1-\Theta(Jt-3)]$. We assume $g_{1}=g_{2}=60J$, $\eta=5J$ [except
for (e) and (f)], $\Delta=200J$ for calculating $P_{m}(t)$ and
$\lambda_{1}=\lambda_{2}=1.5J$ for calculating $P_{m}^{\prime}(t)$. The
initial state is assumed to be
$|\psi(t=0)\rangle=a_{1}^{{\dagger}}|0,g\rangle$.

We plot in Figs. 3(a) and 3(b) the dynamic evolutions of the modal excitation
probabilities $P_{m}(t)=|u_{m}(t)|^{2}$ in the presence of the auxiliary modes
[solving Eq. (9) numerically] with $N=2$ and $N=4$, respectively, and
demonstrate in Figs. 3(c) and 3(d) the dynamic evolutions of
$P_{m}^{\prime}(t)=|w_{m}(t)|^{2}$ of the real-space case for comparison (the
initial state is assumed to be
$|\psi(t=0)\rangle=a_{1}^{{\dagger}}|0,g\rangle$ in this case). We find that
the effective giant atom exhibits essentially the same evolutions as those of
the real-space analogue for both $N=2$ and $N=4$. Different from the case
shown in Fig. S2 SM , where excitation confinement always can be observed
(regardless of $N$) due to the large effective frequency shifts, here the
single-mode excitation can be strongly confined only if $N=2$ whereas it
diffuses along the frequency lattice in a more free manner if $N=4$. We point
out that the confined excitation can be dynamically released by turning off
the external fields. As shown in Figs. 3(e) and 3(f), the giant atom
disappears effectively at $Jt=3$ with the assumption $\eta(t>3)=0$, such that
the excitation is released and diffuses freely along the frequency lattice.

Physically, the phenomena above can be understood from an effective “giant-
small atom model” synEIT , where a lattice (waveguide) is coupled
simultaneously with a small atom at $m=M$ and with a giant atom at $m=0$ and
$m=N$. In fact, the dynamics of our model with
$|\psi(t=0)\rangle=a_{M}^{{\dagger}}|0,g\rangle$ can be effectively described
by that of the giant-small atom model with an initially excited small atom
located at $m=M$. When $\theta=0$ and $0<M<N$, the giant atom is effectively
coupled to the small one with symmetric coupling strength
$g_{GS}\propto(e^{ik_{0}M}+e^{ik_{0}D})$ in the Markovian limit, where $D=N-M$
and $k_{0}=-\pi/2$ (see Sec. IV of SM  for more details). For $M=D=1$ in Figs.
3(a) and 3(c), the excitation oscillation between the small atom (or say, mode
$a_{1}$ of the ring) and the giant one is dominated by the nonzero coupling
strength $g_{GS}$. Note that $M=D=1$ is the only choice achieving nonzero
$g_{GS}$ as long as $N=2$ and $\theta=0$ SM . For $M=1,D=3$ in Figs. 3(b) and
3(d), however, the giant atom is decoupled from the small one due to
$g_{GS}=0$. In this case, the excitation should diffuse freely as in a bare
lattice. However, considering the nonnegligible retardation effects and the
finite effective frequency shift $\Delta_{e,e}$ in this case, the giant atom
can still be weakly excited and thus the excitation diffusion is modified as
shown in Figs. 3(b) and 3(d).

Figure 4: (a) Dynamic evolutions of $P_{m}(t)$ and corresponding profile
$P_{m}$ at $Jt=5$ with initial state $|\psi(t=0)\rangle=|0,e\rangle$. (b)
Dynamic evolutions of $P_{e}$ with different initial states. (c) Profiles of
$P_{m}$ at $Jt=5$ with different initial states and different values of
$\theta$. We assume $g_{1}=g_{2}=3J$, $\eta=2J$, $\Delta=60J$, $N=3$, and
$\theta=\pi/2$ in panels (a) and (b), while $g_{1}=g_{2}=40J$, $\eta=5J$,
$\Delta=200J$, and $N=3$ in panel (c).

Up to now, we have considered the case $\theta=0$, which simulates in the
synthetic frequency dimension a conventional two-level giant-atom. Now we turn
to consider the case $\theta\neq 0$, which can be readily achieved by tuning
the phase difference between the external fields. In this case, the time-
reversal symmetry of the Hamiltonian Eq. (4) is broken such that the
spontaneous emission of the giant atom should be chiral. In particular, the
giant atom exhibits almost unidirectional emission for $\theta=\pi/2$ and
$N=3$, as shown in Fig. 4(a). This phenomenon can be understood again from the
giant-small atom models for $M<0<N$ and $0<N<M$ (see Sec. IV of SM  for more
details), where the small atom is located at the left and right sides of the
giant atom, respectively. For $\theta=\pi/2$ and $M<0<N$ ($0<N<M$), the
effective coupling strength from the giant atom to the small one,
$g_{G\rightarrow S}\propto(e^{i(k_{0}|M|-\pi/2)}+e^{ik_{0}D})$
[$g_{G\rightarrow S}\propto(e^{i(k_{0}M-\pi/2)}+e^{ik_{0}|D|})$], is always
zero (nonzero) in the Markovian limit, implying that the resonant modes on the
left (right) side of the giant atom cannot (can) be excited. We point out that
the weak excitations of the left-side modes here arise from the finite-size
effect of the lattice. The chiral profile tends to become more ideal with the
increase of the evolution time (and also with the increase of the lattice
length to avoid boundary effects), as shown in Fig. S2 SM . On the other hand,
such a giant atom can hardly be excited by photons coming from the right side,
as shown in Fig. 4(b). This is because the effective coupling strength from a
right-side small atom to the giant one, $g_{S\rightarrow
G}\propto(e^{i(k_{0}M+\pi/2)}+e^{ik_{0}|D|})$ is always zero in the case of
$M>3$. This thus provides the possibility of realizing cascaded interactions
cascade1 ; cascade2 ; cascade3  in the frequency dimension, with which the
excitation of the present giant atom can be transferred to another such one
but not vice versa, as shown in Fig. S3 SM .

Moreover, we demonstrate that chiral diffusion of modal excitations can also
be achieved based on the mechanism above. As shown in Fig. 4(c), for $N=3$,
the modal excitation profile becomes quite asymmetric with respect to the
initially excited mode if $\theta=\pi/2$ and
$|\psi(t=0)\rangle=a_{2}^{{\dagger}}|0,g\rangle$, or if $\theta=-\pi/2$ and
$|\psi(t=0)\rangle=a_{1}^{{\dagger}}|0,g\rangle$. The two choices of
parameters lead to reversed chiral profiles. This result arises from the fact
that a portion of the excitation is transferred to the giant atom and
reemitted in the chiral manner shown in Fig. 4(a), and then it interferes with
another portion of the excitation that has diffused along the lattice. We have
checked that the chiral diffusion is most obvious when
$J\approx\\{g_{e,1},g_{e,2}\\}$ and disappears gradually when deviating from
this condition. Note that the giant atom can hardly be excited (and thus the
chiral modal profile almost disappears) if one only reverses $\theta$ but
keeps the initial state unchanged [see the green line with squares in Fig.
4(c) and Sec. IV of SM  for more details].

Finally, we would like to point out that the effective giant atom in the
frequency dimension can also be implemented with a Ladder-type three-level
atom, which is coupled to different sites of the synthetic lattice via
multiple two-photon resonant transitions, as shown in Fig. 1(c). The
feasibility and advantages of such a scheme are discussed in detail in Sec. V
of SM  using effective Hamiltonian theory EH2007 . Considering the
implementations of synthetic dimensions based on superconducting circuits
synsc1 ; synsc2 ; synsc3 ; Wilson2021 , this also shows the possibility of
constructing effective giant atoms in the microwave regime.

In summary, we propose in this Letter a feasible scheme for constructing giant
atoms in synthetic frequency dimensions. Compared with real-space giant atoms,
our scheme has several advantages: (i) The present model shows much smaller
scale for practical on-chip integration and is more resource-efficient than
the real-space analogue, which often requires large numbers of optical
elements. (ii) Higher-dimensional giant atoms HDgiant  can be simulated with
systems of lower geometric dimensions, which are easier to construct
experimentally. In particular, the zero-dimensional model here greatly reduces
the difficulty of introducing other physical effects, such as nonlinearity
that may bring richer exotic phenomena to giant atoms toposyn ; ZHWprl . (iii)
The present scheme supports some useful applications for manipulating the
frequency of light, such as excitation confinement and release, as well as
chiral atom-waveguide interactions in frequency dimensions. The scheme in this
Letter can also be extended to include other synthetic dimensions OAM3 , which
paves a promising way for manipulating various internal degrees of freedom of
photons.

###### Acknowledgements.

We acknowledge helpful discussions with G. C. La Rocca and Peng Zhang. YL is
supported by the National Natural Science Foundation of China (under Grants
No. 12074030 and No. U1930402). AFK acknowledges support from the Swedish
Research Council (grant number 2019-03696), and from the Knut and Alice
Wallenberg Foundation through the Wallenberg Centre for Quantum Technology
(WACQT). JHW acknowledges support from the National Natural Science Foundation
of China (under Grant No. 12074061).

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Supplemental Material for “Giant Atoms in Synthetic Frequency Dimensions”

Lei Du1, Yan Zhang2, Jin-Hui Wu2, Anton Frisk Kockum3, and Yong Li1,∗

1 Beijing Computational Science Research Center, Beijing 100193, China
2 Center for Quantum Sciences and School of Physics,
Northeast Normal University, Changchun 130024, China
3 Department of Microtechnology and Nanoscience (MC2),
Chalmers University of Technology, 412 96 Gothenburg, Sweden

##  I I. Implementations of synthetic frequency lattices

In this section, we briefly review some physical implementations of the
synthetic frequency lattice considered in the main text. For a ring resonator
with zero group velocity dispersion in a specific frequency range, the
resonant modes of the ring can be assumed to be equally spaced in frequency
synOL ; synOptica ; synPRB ; synOptRev . The frequency of the $m$th resonant
mode (we use $m=0,\pm 1,\pm 2,...$ to label the resonant modes and select the
central mode of frequency $\omega_{0}$ as the $0$th mode) is then given by

$\omega_{m}=\omega_{0}+m\Omega_{s},$ (S1)

where $\Omega_{s}=2\pi v_{g}/L$ is the free spectral range, with $v_{g}$ and
$L$ the group velocity at $\omega_{0}$ and the circumference of the ring,
respectively. In view of this, the electric field in the ring can be expanded
as walls

$E(t,r_{\bot},z)=\sum_{m}a_{m}(t,z)\varepsilon_{m}(r_{\bot})e^{-i\omega_{m}t},$
(S2)

where $z$ and $r_{\bot}$ correspond to the propagation direction along the
ring and the direction perpendicular to $z$, respectively;
$\varepsilon_{m}(r_{\bot})$ is the modal profile of the ring; $a_{m}(t,z)$ is
the modal amplitude of the $m$th mode, which satisfies the periodic boundary
condition $a_{m}(t,z+L)=a_{m}(t,z)$ due to the geometry of the ring. If the
ring resonator is dynamically modulated by an electric-optic modulator (EOM)
located at $z=z_{0}$, the electric field acquires a time-dependent phase
whenever it passes through the EOM, i.e.,

$E(t^{+},r_{\bot},z_{0})=E(t^{-},r_{\bot},z_{0})e^{i\alpha\cos{(\Omega t)}},$
(S3)

where $\alpha$ ($\Omega$) is the modulation amplitude (frequency) of the EOM
and $t^{\pm}=t+0^{\pm}$. Substituting Eq. (S3) into Eq. (S2) and using the
Jacobi-Anger expansion, one can obtain

$a_{m}(t^{+},z_{0})=J_{0}(\alpha)a_{m}(t^{-},z_{0})+\sum_{l=1}^{\infty}i^{l}J_{l}(\alpha)\Big{[}a_{m+l}(t^{-},z_{0})e^{-il\delta_{s}t^{-}}+a_{m-l}(t^{-},z_{0})e^{il\delta_{s}t^{-}}\Big{]},$
(S4)

where $\delta_{s}=\Omega_{s}-\Omega$ and $J_{l}$ is the Bessel function of the
$l$th order. For weak modulations (i.e., $\alpha\rightarrow 0$), one can only
keep the terms $l=\pm 1$ and approximately has $J_{0}(\alpha)\approx 1$ and
$J_{1}(\alpha)\approx\alpha/2$. In this way, we have

$\frac{\partial a_{m}}{\partial
t}=iJ(a_{m+1}e^{-i\delta_{s}t}+a_{m-1}e^{i\delta_{s}t}),$ (S5)

where $J=\alpha/2T_{R}$ with $T_{R}$ the round-trip propagation time of the
ring. Clearly, Eq. (S5) describes similar dynamics as those of one-dimensional
tight-binding models, implying that a one-dimensional synthetic frequency
lattice is created. Such dynamics can also be described by an equivalent
Hamiltonian synOL ; synOptica ; synPRB  ($\hbar=1$ throughout this
Supplemental Material)

$H_{r}=\sum_{m}[\omega_{m}a_{m}^{{\dagger}}a_{m}+2J\cos{(\Omega
t)}(a_{m}^{{\dagger}}a_{m+1}+\text{H.c.})],$ (S6)

which in the interaction picture becomes

$H_{r}^{\prime}=\sum_{m}J(a_{m}^{{\dagger}}a_{m+1}e^{-i\delta_{s}t}+\text{H.c.})$
(S7)

by taking the rotating wave approximation
($\delta_{s}=\Omega_{s}-\Omega\approx 0$).

In addition to the ring-resonator scheme discussed above, such a synthetic
frequency lattice can also be achieved by dynamically modulating the
refractive index of a waveguide in terms of $n(z,t)=n_{0}+\Delta n\cos{(\Omega
t-qz+\phi)}$ LiNbO3PRL ; LiNbO3PRB , where $n_{0}$ is the static background
refractive index, $\Delta n$, $\Omega$, $q$, and $\phi$ are the amplitude,
frequency, wave vector, and initial phase of the modulation, respectively.
With the phase-matching condition $\Omega/q=c/n_{0}$, one can obtain formally
the same dynamic equation as that in Eq. (S5).

Besides photonic systems, superconducting quantum circuits have also shown the
potential in creating synthetic dimensions, for both microwave photons synsc3
; Wilson2021  and surface acoustic waves synSAW . In this case, a three-level
$\Delta$-type structure is sufficient to simulate the effective giant atom of
our interest, without the need of one more intermediate state as that in Fig.
1(a) in the main text. To be specific, a single-photon transition
$|g\rangle\leftrightarrow|e\rangle$ (resonantly coupled to the $0$th site of
the lattice with coupling strength $g_{1}$) and a two-photon resonant
transition $|g\rangle\leftrightarrow|f\rangle\leftrightarrow|e\rangle$ (one of
the two transitions is coupled to the $N$th site off-resonantly with coupling
strength $g_{2}$ while the other one is driven by an external field) coexist
in this case coexist . For artificial atoms, $\Delta$-type structure can be
readily achieved with qubits operated away from their optimal points, and
$g_{1}$ can be engineered much smaller than $g_{2}$ ReviewGu  (in this way,
the effective coupling strength between $|g\rangle$ and $|e\rangle$ induced by
the two-photon process can be comparable with $g_{1}$). Moreover, the
effective giant atom can also be simulated with a Ladder-type three-level
structure, as shown later in Sec. V, which is even simpler to construct than a
$\Delta$-type one. However, the $\Delta$-type scheme here has the advantage
that only one external field is needed.

##  II II. Total Hamiltonian in the interaction picture

According to the main text, the total Hamiltonian of the system can be written
as

$H=H_{a}+H_{r}+H_{\textrm{int}},$ (S8)

where $H_{a}=\omega_{f,1}|f_{1}\rangle\langle
f_{1}|+\omega_{f,2}|f_{2}\rangle\langle f_{2}|+\omega_{e}|e\rangle\langle e|$
is the free Hamiltonian of the atom, with $\omega_{f,1}$, $\omega_{f,2}$, and
$\omega_{e}$ the frequencies of the states $|f_{1}\rangle$, $|f_{2}\rangle$,
and $|e\rangle$ with respect to the state $|g\rangle$, respectively; $H_{r}$
is the Hamiltonian of the modulated ring resonator, which is given in Eq.
(S6)$;H_{\textrm{int}}$ is the Hamiltonian that describes the interaction of
the atom with the ring resonator and the external fields, which reads

$H_{\textrm{int}}=g_{1}a_{0}^{{\dagger}}|g\rangle\langle
f_{1}|+g_{2}a_{N}^{{\dagger}}|g\rangle\langle
f_{2}|+\eta_{1}e^{-i\omega_{d,1}t}e^{i\theta}|e\rangle\langle
f_{1}|+\eta_{2}e^{-i\omega_{d,2}t}|e\rangle\langle f_{2}|+\text{H.c.},$ (S9)

where $g_{1}$ ($g_{2}$) is the coupling strength between mode $a_{0}$
($a_{N}$) and the transition $|g\rangle\leftrightarrow|f_{1}\rangle$
($|g\rangle\leftrightarrow|f_{2}\rangle$); $\theta$ is the relative phase
between the two external fields. In the interaction picture with respect to
$H_{0}=H_{a}+\sum_{m}\omega_{m}a_{m}^{{\dagger}}a_{m}$ and using the rotating-
wave approximation, Eq. (S8) can be transformed into

$\begin{split}H^{\prime}&=\sum_{m}J(a_{m}^{{\dagger}}a_{m+1}+\text{H.c.})+\Big{(}g_{1}a_{0}^{{\dagger}}|g\rangle\langle
f_{1}|e^{i\Delta_{1}t}+g_{2}a_{N}^{{\dagger}}|g\rangle\langle
f_{2}|e^{i\Delta_{2}t}\\\ &\quad\,+\eta_{1}e^{i\theta}|e\rangle\langle
f_{1}|e^{i\Delta_{d,1}t}+\eta_{2}|e\rangle\langle
f_{2}|e^{i\Delta_{d,2}t}+\text{H.c.}\Big{)}\end{split}$ (S10)

in the case of $\delta_{s}=0$, where $\Delta_{1}=\omega_{0}-\omega_{f,1}$
($\Delta_{2}=\omega_{N}-\omega_{f,2}$) is the detuning between mode $a_{0}$
($a_{N}$) and the transition $|g\rangle\leftrightarrow|f_{1}\rangle$
($|g\rangle\leftrightarrow|f_{2}\rangle$);
$\Delta_{d,1}=\omega_{e}-\omega_{f,1}-\omega_{d,1}$
($\Delta_{d,2}=\omega_{e}-\omega_{f,2}-\omega_{d,2}$) is the detuning between
the transition $|f_{1}\rangle\leftrightarrow|e\rangle$
($|f_{2}\rangle\leftrightarrow|e\rangle$) and the external field $\eta_{1}$
($\eta_{2}$). Clearly, the Hamiltonian in Eq. (S10) is exactly the time-
independent one in Eq. (4) in the main text.

##  III III. Excitation confinement without auxiliary modes

Figure S1: Dynamic evolutions of the modal excitation probability $P_{m}(t)$
with (a) $J\gg\\{g_{e,1},g_{e,2}\\}$ and (b)-(d) $J\sim\\{g_{e,1},g_{e,2}\\}$.
In panel (a) we assume $g_{1}=g_{2}=3J$, $\eta=2J$, $\Delta=60J$, and $N=4$,
while in panels (b)-(d) we assume $g_{1}=g_{2}=40J$, $\eta=5J$, $\Delta=200J$,
and $N=4,3,2$, respectively. The initial state is assumed to be
$|\psi(t=0)\rangle=a_{1}^{{\dagger}}|0,g\rangle$.

As discussed in the main text, the effective giant atom can hardly affect the
propagation of a single-mode excitation along the synthetic frequency lattice
in the regime of $J\gg\\{g_{e,1},g_{e,2}\\}$. Indeed, as shown in Fig. S1(a),
the dynamic evolution of the initial state
$|\psi(t=0)\rangle=a_{1}^{{\dagger}}|0,g\rangle$ behaves as in a bare lattice
due to the weak atom-waveguide couplings. In view of this, we consider the
regime where the effective coupling strengths $g_{e,1}$ and $g_{e,2}$ are
comparable with $J$. As shown in Figs. S1(b)-S1(d), the two atom-waveguide
coupling points located at $m=0$ and $m=N$ act as two cavity mirrors between
which the excitation is reflected back and forth. The oscillation region is
fully determined by the separation $N$ of the two coupling points. In
particular, the excitation is almost confined at the initial mode $a_{1}$ for
$N=2$ due to the narrow region between the two “mirrors”.

It is worth mentioning, however, that the confinements above are actually
caused by the large effective frequency shifts of modes $a_{0}$ and $a_{N}$,
i.e., $\Delta_{e,1}=g_{1}^{2}/\Delta$ and $\Delta_{e,2}=g_{2}^{2}/\Delta$,
which serve as two $\delta$-like potentials to confine the excitations between
them. In order to remove the potentials and thereby focus on the giant-atom
effect, we introduce opposite frequency shifts for modes $a_{0}$ and $a_{N}$
by coupling them with two extra auxiliary modes $b_{1}$ and $b_{2}$,
respectively, as shown in Eq. (9) in the main text.

##  IV IV. Effective giant-small atom model

In this section, we provide physical interpretations for the results in Figs.
3 and 4 in the main text with the help of an effective “giant-small atom
model”. Considering a one-dimensional discrete waveguide (lattice) that is
coupled with a two-level giant atom (at sites $m=0$ and $m=N$) and a two-level
small one (at site $m=M$) simultaneously, the Hamiltonian of the system can be
written as

$H=\omega_{0}(b^{{\dagger}}b+c^{{\dagger}}c)+\Big{[}J\sum_{m}a_{m}^{{\dagger}}a_{m+1}+gb^{{\dagger}}(a_{0}e^{i\theta}+a_{N})+\xi
c^{{\dagger}}a_{M}+\text{H.c.}\Big{]},$ (S11)

where $\omega_{0}$ is the transition frequency of the two atoms; $a_{m}$ and
$b$ ($c$) are the annihilation operator of the $m$th lattice site and the
lowering operator of the giant (small) atom, respectively; $g$ ($\xi$) is the
coupling strength between the lattice and the giant (small) atom; $\theta$ is
the relative phase between the two coupling channels of the giant atom. By
transforming the lattice operators into the Bloch representation, i.e.,

$a_{k}=\frac{1}{\sqrt{2\pi}}\sum_{m}a_{m}e^{-ikm},$ (S12)

with $-\pi\leqslant k<\pi$ the Bloch wave vector, Eq. (S11) can be rewritten
as

$H^{\prime}=\omega_{0}(b^{{\dagger}}b+c^{{\dagger}}c)+\int
dk\omega_{k}a_{k}^{{\dagger}}a_{k}+\int
dk[(G_{k,1}b^{{\dagger}}+G_{k,2}c^{{\dagger}})a_{k}+\text{H.c.}],$ (S13)

where $\omega_{k}=2J\cos{k}$ is the dispersion relation of the lattice;
$G_{k,1}=g[\textrm{exp}(i\theta)+\textrm{exp}(ikN)]/\sqrt{2\pi}$ and
$G_{k,2}=\xi\text{exp}(ikM)/\sqrt{2\pi}$ are the spectral coupling functions.
In the single-excitation subspace, the wave function of the system is given by

$|\psi(t)\rangle=\int
dku_{k}(t)e^{-i\omega_{k}t}a_{k}^{{\dagger}}|V\rangle+\sum_{\beta=b,c}u_{\beta}(t)e^{-i\omega_{0}t}\beta^{{\dagger}}|V\rangle,$
(S14)

where $u_{k}$ and $u_{b}$ ($u_{c}$) are the probability amplitudes of creating
a photon with wave vector $k$ in the lattice and of exciting the giant (small)
atom to the excited state, respectively; $|V\rangle$ is the vacuum state of
the whole system. Solving the Schrödinger equation, we have

$\begin{split}\frac{du_{b}}{dt}=&-i\int
dkG_{k,1}u_{k}e^{-i(\omega_{k}-\omega_{0})t},\\\ \frac{du_{c}}{dt}=&-i\int
dkG_{k,2}u_{k}e^{-i(\omega_{k}-\omega_{0})t},\\\
\frac{du_{k}}{dt}=&-i(G_{k,1}^{*}u_{b}+G_{k,2}^{*}u_{c})e^{i(\omega_{k}-\omega_{0})t}.\end{split}$
(S15)

The formal solution of $u_{k}$ is written as

$u_{k}(t)=-i\int_{t_{0}}^{t}dt^{\prime}e^{i(\omega_{k}-\omega_{0})t^{\prime}}[G_{k,1}^{*}u_{b}(t^{\prime})+G_{k,2}^{*}u_{c}(t^{\prime})],$
(S16)

where $t_{0}<t$ is the initial time and $u_{k}(t_{0})=0$ has been assumed due
to the initially vacuum lattice. Substituting Eq. (S16) into the dynamic
equations of $u_{b}$ and $u_{c}$ in Eq. (S15), we have

$\begin{split}\frac{du_{b}}{dt}&=-\frac{1}{2\pi}\int_{t_{0}}^{t}dt^{\prime}e^{i\omega_{0}(t-t^{\prime})}\int_{-\infty}^{+\infty}dk\Big{\\{}2g^{2}[1+\cos{(\theta-
kN)}]u_{b}(t^{\prime})\\\ &\quad\,+g\xi[e^{i(\theta-
kM)}+e^{ikD}]u_{c}(t^{\prime})\Big{\\}}e^{-i\omega_{k}(t-t^{\prime})},\\\
\frac{du_{c}}{dt}&=-\frac{1}{2\pi}\int_{t_{0}}^{t}dt^{\prime}e^{i\omega_{0}(t-t^{\prime})}\int_{-\infty}^{+\infty}dk\Big{\\{}\xi^{2}u_{c}(t^{\prime})\\\
&\quad\,+g\xi[e^{-i(\theta-
kM)}+e^{-ikD}]u_{b}(t^{\prime})\Big{\\}}e^{-i\omega_{k}(t-t^{\prime})}\end{split}$
(S17)

with $D=N-M$. By changing the integration variable with $\int
dk\rightarrow\int d\omega_{k}/v_{g}$, we further have

$\begin{split}\frac{du_{b}}{dt}&=-\frac{1}{2\pi}\int_{t_{0}}^{t}dt^{\prime}e^{i\omega_{0}(t-t^{\prime})}\int_{0}^{+\infty}\frac{d\omega_{k}}{v_{g}}\Big{\\{}4g^{2}(1+\cos{\theta}\cos{kN})u_{b}(t^{\prime})\\\
&\quad\,+2g\xi(e^{i\theta}\cos{kM}+\cos{kD})u_{c}(t^{\prime})\Big{\\}}e^{-i\omega_{k}(t-t^{\prime})},\\\
\frac{du_{c}}{dt}&=-\frac{1}{2\pi}\int_{t_{0}}^{t}dt^{\prime}e^{i\omega_{0}(t-t^{\prime})}\int_{0}^{+\infty}\frac{d\omega_{k}}{v_{g}}\Big{\\{}2\xi^{2}u_{c}(t^{\prime})\\\
&\quad\,+2g\xi(e^{-i\theta}\cos{kM}+\cos{kD})u_{b}(t^{\prime})\Big{\\}}e^{-i\omega_{k}(t-t^{\prime})},\end{split}$
(S18)

where $v_{g}=-2J\sin{k}$ is the group velocity of photons in the lattice.
According to the Weisskopf-Wigner theory walls , the intensity of the emission
spectrum is concentrated around the atomic transition frequency $\omega_{0}$.
Moreover, $v_{g}$ varies quite slowly and thus can be treated as a constant if
$\omega_{0}$ is far away from the cut-off frequency of the lattice (i.e., if
$\omega_{0}\rightarrow 0$). In this case, one has
$\omega_{k}\simeq\omega_{0}+\nu=\omega_{0}+(k-k_{0})v_{0}$ with
$\omega_{0}=2J\cos{k_{0}}$ and $v_{0}=-2J\sin{k_{0}}$, which lead to

$\begin{split}\frac{du_{b}}{dt}&=-\int_{-\infty}^{+\infty}d\nu\int_{t_{0}}^{t}dt^{\prime}e^{-i\nu(t-t^{\prime})}\Big{\\{}\frac{\gamma_{b}}{\pi}\Big{[}2+\cos{\theta}\Big{(}e^{i(k_{0}N+\nu\frac{N}{v_{0}})}+e^{-i(k_{0}N+\nu\frac{N}{v_{0}})}\Big{)}\Big{]}u_{b}(t^{\prime})\\\
&\quad\,+\frac{\gamma_{bc}}{2\pi}\Big{[}e^{i\theta}e^{i(k_{0}M+\nu\frac{M}{v_{0}})}+e^{i\theta}e^{-i(k_{0}M+\nu\frac{M}{v_{0}})}+e^{i(k_{0}D+\nu\frac{D}{v_{0}})}+e^{-i(k_{0}D+\nu\frac{D}{v_{0}})}\Big{]}u_{c}(t^{\prime})\Big{\\}},\\\
\frac{du_{c}}{dt}&=-\int_{-\infty}^{+\infty}d\nu\int_{t_{0}}^{t}dt^{\prime}e^{-i\nu(t-t^{\prime})}\Big{\\{}\frac{\gamma_{c}}{\pi}u_{c}(t^{\prime})+\frac{\gamma_{bc}}{2\pi}\Big{[}e^{-i\theta}e^{i(k_{0}M+\nu\frac{M}{v_{0}})}+e^{-i\theta}e^{-i(k_{0}M+\nu\frac{M}{v_{0}})}\\\
&\quad\,+e^{i(k_{0}D+\nu\frac{D}{v_{0}})}+e^{-i(k_{0}D+\nu\frac{D}{v_{0}})}\Big{]}u_{b}(t^{\prime})\Big{\\}},\end{split}$
(S19)

where $\gamma_{b}=g^{2}/v_{0}$, $\gamma_{c}=\xi^{2}/v_{0}$, and
$\gamma_{bc}=g\xi/v_{0}$. With the definition of the delta function $\int d\nu
e^{-i\nu t}=2\pi\delta(t)$ and assuming $0<M<N$ (the small atom is located
between the two atom-waveguide coupling points of the giant one), we finally
have

$\begin{split}\frac{du_{b}(t)}{dt}=&-2\gamma_{b}\Big{[}u_{b}(t)+\cos{\theta}e^{ik_{0}N}u_{b}\Big{(}t-\frac{N}{v_{0}}\Big{)}\Theta\Big{(}t-\frac{N}{v_{0}}\Big{)}\Big{]}-\gamma_{bc}\Big{[}e^{i(k_{0}M+\theta)}u_{c}\Big{(}t-\frac{M}{v_{0}}\Big{)}\Theta\Big{(}t-\frac{M}{v_{0}}\Big{)}\\\
&+e^{ik_{0}D}u_{c}\Big{(}t-\frac{D}{v_{0}}\Big{)}\Theta\Big{(}t-\frac{D}{v_{0}}\Big{)}\Big{]},\\\
\frac{du_{c}(t)}{dt}=&-\gamma_{c}u_{c}(t)-\gamma_{bc}\Big{[}e^{i(k_{0}M-\theta)}u_{b}\Big{(}t-\frac{M}{v_{0}}\Big{)}\Theta\Big{(}t-\frac{M}{v_{0}}\Big{)}+e^{ik_{0}D}u_{b}\Big{(}t-\frac{D}{v_{0}}\Big{)}\Theta\Big{(}t-\frac{D}{v_{0}}\Big{)}\Big{]},\end{split}$
(S20)

where $\Theta(x)$ is the Heaviside step function. In the Markovian limit where
retardation effects can be neglected
($\\{\frac{N}{v_{0}},\frac{M}{v_{0}},\frac{D}{v_{0}}\\}\ll\\{\frac{1}{2\gamma_{b}},\frac{1}{\gamma_{c}}\\}$),
Eq. (S20) can be further simplified to

$\begin{split}\frac{du_{b}(t)}{dt}=&-2\gamma_{b}(1+\cos{\theta}e^{ik_{0}N})u_{b}(t)-\gamma_{bc}[e^{i(k_{0}M+\theta)}+e^{ik_{0}D}]u_{c}(t),\\\
\frac{du_{c}(t)}{dt}=&-\gamma_{c}u_{c}(t)-\gamma_{bc}[e^{i(k_{0}M-\theta)}+e^{ik_{0}D}]u_{b}(t).\end{split}$
(S21)

Similarly, one has

$\begin{split}\frac{du_{b}(t)}{dt}=&-2\gamma_{b}(1+\cos{\theta}e^{ik_{0}N})u_{b}(t)-\gamma_{bc}[e^{i(k_{0}|M|+\theta)}+e^{ik_{0}D}]u_{c}(t),\\\
\frac{du_{c}(t)}{dt}=&-\gamma_{c}u_{c}(t)-\gamma_{bc}[e^{i(k_{0}|M|-\theta)}+e^{ik_{0}D}]u_{b}(t)\end{split}$
(S22)

if $M<0<N$ (the small atom is located at the left side of the giant one), and

$\begin{split}\frac{du_{b}(t)}{dt}=&-2\gamma_{b}(1+\cos{\theta}e^{ik_{0}N})u_{b}(t)-\gamma_{bc}[e^{i(k_{0}M+\theta)}+e^{ik_{0}|D|}]u_{c}(t),\\\
\frac{du_{c}(t)}{dt}=&-\gamma_{c}u_{c}(t)-\gamma_{bc}[e^{i(k_{0}M-\theta)}+e^{ik_{0}|D|}]u_{b}(t)\end{split}$
(S23)

if $0<N<M$ (the small atom is located at the right side of the giant one).
Note that for the case of $\omega_{0}\rightarrow 0$ (i.e.,
$\Delta_{e,e}\rightarrow 0$) considered in this Letter, one has
$k_{0}\approx-\pi/2$ LonghiGA ; Longhiretard .

Figure S2: Dynamic evolutions of the modal excitation probability $P_{m}(t)$
with (a) $m_{\textrm{tot}}=25$, (b) $m_{\textrm{tot}}=65$, and (c)
$m_{\textrm{tot}}=105$. The lower $2$D plots in each panel are the
corresponding modal excitation profiles $P_{m}$ at the final time $Jt=5$,
$Jt=15$, and $Jt=25$, respectively. Other parameters are assumed as
$g_{1}/J=g_{2}/J=3$, $\eta/J=2$, $\Delta/J=60$, $\theta=\pi/2$, $N=3$ and the
initial state is assumed to be $|\psi(t=0)\rangle=|0,e\rangle$.

The results in Fig. 3 in the main text can be well understood from Eq. (S21):
the dynamics of the our model in Eq. (S10) with the initial state
$|\psi(t=0)\rangle=a_{M}^{{\dagger}}|0,g\rangle$ can be effectively described
by the giant-small atom model with an initially excited small atom located at
$m=M$. For the case of $\theta=0$ in Fig. 3, the effective coupling strength
between the giant and small atoms $g_{GS}\propto-(e^{ik_{0}M}+e^{ik_{0}D})$ is
reciprocal. When $M=1$ and $D=1$ [Figs. 3(a) and 3(c)], the excitation can
oscillate between the two atoms (or say, between mode $a_{1}$ and the state
$|e\rangle$) due to the nonzero effective coupling strength and thus it is
confined in the frequency dimension in an oscillating manner. Note that this
is the only choice to achieve nonzero effective coupling as long as $N=2$ and
$\theta=0$, i.e., $g_{GS}$ is always zero for other values of $M$ and $D$, as
can be seen from Eqs. (S21)-(S23). When $M=1$ and $D=3$ [Figs. 3(b) and 3(d)],
the giant atom decouples from the small one (mode $a_{1}$) due to $g_{GS}=0$,
such that the excitation diffuses as in a bare lattice. The results in Figs.
3(b) and 3(d) deviate slightly from the bare-lattice case due to the
nonnegligible non-Markovian retardation effect in the case of
$J\approx\\{g_{e,1},g_{e,2}\\}$.

Equations (S22) and (S23) reveal that if $\theta\neq n\pi$ ($n$ is an
arbitrary integer), the effective couplings between the two atoms become
different for $M<0<N$ and $0<N<M$ (the effective coupling strength depends on
which side of the giant atom the small one is located at). In particular, for
$\theta=\pi/2$ and $N=3$, the effective coupling strength from the giant atom
to the small one $g_{G\rightarrow
S}\propto[e^{i(k_{0}|M|-\theta)}+e^{ik_{0}D}]$ is always zero as long as
$M<0$, implying that the lattice sites (resonant modes) on the left side of
the giant atom cannot be excited in the Markovian limit. We would like to
point out that the weak excitations of the left-side modes in Fig. 4(a) in the
main text arise from the finite-size effect of the lattice. As shown in Figs.
S2(a)-S2(c), the chiral modal excitation profile tends to become more ideal
(i.e., the weak excitations of the left-side modes tend to vanish) with the
increase of the evolution time. To avoid undesired boundary effects, we also
extend the length of the frequency lattice accordingly [the total number of
the lattice sites is assumed to be $m_{\textrm{tot}}=25,65,105$ for Figs.
S2(a)-S2(c), respectively], which is experimentally available with synthetic
lattices of other internal degrees of freedom such as orbital angular
momentum. Note that such a chiral spontaneous emission can be reversed by
tuning $\theta$ from $\pi/2$ to $-\pi/2$.

On the other hand, as mentioned in the main text, such a giant atom can hardly
be excited by photons coming from its right side (or say, by an excited small
atom located at its right side). This can be understood again from Eq. (S23),
which shows that the effective coupling strength from a right-side small atom
to the giant one $g_{S\rightarrow
G}\propto[e^{i(k_{0}M+\theta)}+e^{ik_{0}|D|}]$ is always zero as long as
$\theta=\pi/2$ and $N=3$. This thus provides the possibility of realizing
cascaded interactions between multiple such giant atoms, i.e., the interatomic
interactions can be directional based on this mechanism. As an example, we
consider here that another giant atom (labeled as “atom $B$”) is coupled to
the frequency lattice at sites $m=4$ and $m=7$, while the present one (labeled
as “atom $A$”) is coupled to the lattice at $m=0$ and $m=3$. The two giant
atoms are assumed to be identical except for the positions of the coupling
points. We plot in Fig. S3 the dynamic evolutions of the atomic populations
$P_{e,A}(t)$ of atom $A$ and $P_{e,B}(t)$ of atom $B$ with two initial states
$|\psi(t=0)\rangle=|0,e,g\rangle$ and $|\psi(t=0)\rangle=|0,g,e\rangle$. As
predicted above, in the case of $\theta=\pi/2$, the excitation of atom $A$ can
be partially transferred to atom $B$ [Fig. S3(a)], whereas atom $A$ can hardly
be excited if atom $B$ is initially excited [Fig. S3(b)]. This indicates that
one can implement cascaded systems in the synthetic frequency dimension.

Figure S3: Dynamic evolutions of the atomic populations $P_{e,A}(t)$ and
$P_{e,B}(t)$ with the initial states (a) $|\psi(t=0)\rangle=|0,e,g\rangle$ and
(b) $|\psi(t=0)\rangle=|0,g,e\rangle$. Other parameters are assumed as
$g_{1}=g_{2}=10J$, $\eta=4J$, $\Delta=100J$, $\theta=\pi/2$, and $N=3$.

Finally, one can see from Eq. (S21) that the effective coupling strength from
the small atom to the giant one $g_{S\rightarrow
G}\propto[e^{i(k_{0}M+\theta)}+e^{ik_{0}D}]=0$ if $M=2$, $D=1$, and
$\theta=-\pi/2$, or if $M=1$, $D=2$, and $\theta=\pi/2$. In these cases, the
giant atom cannot be excited and thus the chiral modal excitation profile
almost disappears, as shown in Fig. 4(c) in the main text (see the green line
with squares). On the contrary, $g_{S\rightarrow G}$ reaches its maximum if
$M=1$, $D=2$, and $\theta=-\pi/2$, or if $M=2$, $D=1$, and $\theta=\pi/2$. The
excited giant atom exhibits reversed chiral spontaneous emission for
$\theta=\pi/2$ and $\theta-\pi/2$, as discussed above, and thus yields
reversed modal excitation profiles as shown in Fig. 4(c) in the main text (see
the blue line with dots and the red line with circles).

##  V V. Effective Hamiltonian of the Ladder-type scheme

In this section, we would like to demonstrate that the effective giant atom of
our interest also can be implemented by exploiting a Ladder-type three-level
atom coupled with two sites of the synthetic frequency lattice via two two-
photon resonant transitions. For example, as shown in Fig. 1(c) in the main
text, the lower transition $|g\rangle\leftrightarrow|f\rangle$ of the Ladder-
type atom is coupled off-resonantly to the $0$th and $N$th lattice sites with
coupling strengths (detunings) $g_{1}$ and $g_{2}$ ($\Delta_{1}$ and
$\Delta_{2}$), respectively. Meanwhile, the upper transition
$|f\rangle\leftrightarrow|e\rangle$ is driven by two external fields of
amplitudes $\eta_{1}$ and $\eta_{2}$ in order to construct two two-photon
resonant transitions. In the interaction picture and with the rotating-wave
approximation, the total Hamiltonian can be written as
$H_{\text{tot}}=H_{r}^{\prime}+H_{L}(t)$ with

$H_{L}(t)=g_{1}a_{0}^{{\dagger}}|g\rangle\langle
f|e^{i\Delta_{1}t}+g_{2}a_{N}^{{\dagger}}|g\rangle\langle
f|e^{i\Delta_{2}t}+\eta_{1}e^{i\theta}|e\rangle\langle
f|e^{i\Delta_{1}t}+\eta_{2}|e\rangle\langle f|e^{i\Delta_{2}t}+\text{H.c.}.$
(S24)

Now we exploit effective Hamiltonian theory EH2007  to remove the time
dependence of $H_{L}(t)$. According to this theory, if the time-dependent
Hamiltonian has the form of

$H(t)=\sum_{m}h_{m}^{{\dagger}}e^{i\Delta_{m}t}+\text{H.c.},$ (S25)

then the effective Hamiltonian can be given by

$\begin{split}H_{\text{eff}}&=-iH(t)\int^{t}_{0}dt^{\prime}H(t^{\prime})\\\
&=\sum_{m,n}\frac{-1}{\Delta_{n}}[h_{m}^{{\dagger}}h_{n}^{{\dagger}}e^{i(\Delta_{m}+\Delta_{n})t}+h_{m}h_{n}^{{\dagger}}e^{-i(\Delta_{m}-\Delta_{n})t}+h_{m}^{{\dagger}}h_{n}e^{i(\Delta_{m}-\Delta_{n})t}+h_{m}h_{n}e^{-i(\Delta_{m}+\Delta_{n})t}],\end{split}$
(S26)

where we have discarded the high-frequency terms containing $\text{exp}(\pm
i\Delta_{m}t)$ in Eq. (S26). For $H_{L}(t)$ given in Eq. (S24), we have

$h_{1}^{{\dagger}}=g_{1}a_{0}^{{\dagger}}|g\rangle\langle
f|,\,\,\,h_{2}^{{\dagger}}=g_{2}a_{N}^{{\dagger}}|g\rangle\langle
f|,\,\,\,h_{3}^{{\dagger}}=\eta_{1}e^{i\theta}|e\rangle\langle
f|,\,\,\,h_{4}^{{\dagger}}=\eta_{2}|e\rangle\langle f|.$ (S27)

Substituting Eq. (S27) into Eq. (S26), the effective Hamiltonian can be
derived as

$H_{\text{eff}}=\sum_{m=n}...+\sum_{m\neq n}...$ (S28)

with

$\begin{split}\sum_{m=n}...&=\frac{g_{1}^{2}}{\Delta_{1}}[n_{0}|g\rangle\langle
g|-(n_{0}+1)|f\rangle\langle
f|]+\frac{g_{2}^{2}}{\Delta_{2}}[n_{N}|g\rangle\langle
g|-(n_{N}+1)|f\rangle\langle f|]\\\
&\quad\,+\frac{\eta_{1}^{2}}{\Delta_{1}}(|e\rangle\langle e|-|f\rangle\langle
f|)+\frac{\eta_{2}^{2}}{\Delta_{2}}(|e\rangle\langle e|-|f\rangle\langle
f|)\end{split}$ (S29)

and

$\begin{split}\sum_{m\neq
n}...&=\frac{g_{1}g_{2}}{\Delta_{1}}a_{N}^{{\dagger}}a_{0}|g\rangle\langle
g|e^{i(\Delta_{2}-\Delta_{1})t}-\frac{g_{1}g_{2}}{\Delta_{1}}a_{N}a_{0}^{{\dagger}}|f\rangle\langle
f|e^{i(\Delta_{1}-\Delta_{2})t}+\frac{g_{1}\eta_{1}}{\Delta_{1}}e^{i\theta}a_{0}|e\rangle\langle
g|+\frac{g_{1}\eta_{2}}{\Delta_{1}}a_{0}|e\rangle\langle
g|e^{i(\Delta_{2}-\Delta_{1})t}\\\
&\quad\,+\frac{g_{1}g_{2}}{\Delta_{2}}a_{0}^{{\dagger}}a_{N}|g\rangle\langle
g|e^{i(\Delta_{1}-\Delta_{2})t}-\frac{g_{1}g_{2}}{\Delta_{2}}a_{0}a_{N}^{{\dagger}}|f\rangle\langle
f|e^{i(\Delta_{2}-\Delta_{1})t}+\frac{g_{2}\eta_{1}}{\Delta_{2}}e^{i\theta}a_{N}|e\rangle\langle
g|e^{i(\Delta_{1}-\Delta_{2})t}+\frac{g_{2}\eta_{2}}{\Delta_{2}}a_{N}|e\rangle\langle
g|\\\
&\quad\,+\frac{g_{1}\eta_{1}}{\Delta_{1}}e^{-i\theta}a_{0}^{{\dagger}}|g\rangle\langle
e|+\frac{g_{2}\eta_{1}}{\Delta_{1}}e^{-i\theta}a_{N}^{{\dagger}}|g\rangle\langle
e|e^{i(\Delta_{2}-\Delta_{1})t}+\frac{\eta_{1}\eta_{2}}{\Delta_{1}}e^{-i\theta}|e\rangle\langle
e|e^{i(\Delta_{2}-\Delta_{1})t}-\frac{\eta_{1}\eta_{2}}{\Delta_{1}}e^{i\theta}|f\rangle\langle
f|e^{i(\Delta_{1}-\Delta_{2})t}\\\
&\quad\,+\frac{g_{1}\eta_{2}}{\Delta_{2}}a_{0}^{{\dagger}}|g\rangle\langle
e|e^{i(\Delta_{1}-\Delta_{2})t}+\frac{g_{2}\eta_{2}}{\Delta_{2}}a_{N}^{{\dagger}}|g\rangle\langle
e|+\frac{\eta_{1}\eta_{2}}{\Delta_{2}}e^{i\theta}|e\rangle\langle
e|e^{i(\Delta_{1}-\Delta_{2})t}-\frac{\eta_{1}\eta_{2}}{\Delta_{2}}e^{-i\theta}|f\rangle\langle
f|e^{i(\Delta_{2}-\Delta_{1})t},\end{split}$ (S30)

where $n_{0}=a_{0}^{{\dagger}}a_{0}$ and $n_{N}=a_{N}^{{\dagger}}a_{N}$ are
the photon number operators of the $0$th and $N$th lattice sites,
respectively. Considering that $\Delta_{2}-\Delta_{1}=N\Omega_{s}\gg 0$ in our
model ($\Omega_{s}\gg\\{g_{1},g_{2},\eta_{1},\eta_{2}\\}$), Eqs. (S29) and
(S30) can be further simplified to

$\sum_{m=n}...=(\frac{\eta_{1}^{2}}{\Delta_{1}}+\frac{\eta_{2}^{2}}{\Delta_{2}})|e\rangle\langle
e|+(\frac{g_{1}^{2}}{\Delta_{1}}n_{0}+\frac{g_{2}^{2}}{\Delta_{2}}n_{N})|g\rangle\langle
g|$ (S31)

and

$\sum_{m\neq
n}...=\frac{g_{1}\eta_{1}}{\Delta_{1}}e^{i\theta}a_{0}|e\rangle\langle
g|+\frac{g_{2}\eta_{2}}{\Delta_{2}}a_{N}|e\rangle\langle
g|+\frac{g_{1}\eta_{1}}{\Delta_{1}}e^{-i\theta}a_{0}^{{\dagger}}|g\rangle\langle
e|+\frac{g_{2}\eta_{2}}{\Delta_{2}}a_{N}^{{\dagger}}|g\rangle\langle e|,$
(S32)

respectively, where we have neglected the high-frequency terms containing
$\text{exp}[\pm i(\Delta_{2}-\Delta_{1})t]$ in Eq. (S32). Moreover, we have
also discarded the Stark shift terms of the state $|f\rangle$ in Eq. (S31)
since $|f\rangle$ is decoupled from the other states in the effective
Hamiltonian. It is clear from Eqs. (S31) and (S32) that such a Ladder-type
scheme with two two-photon resonant transitions can also mimic a two-level
giant atom with two coupling points in the synthetic frequency dimension. In
particular, the two-photon processes induced Stark shifts [e.g.,
$\eta_{1}^{2}/\Delta_{1}+\eta_{2}^{2}/\Delta_{2}$ and
$g_{1}^{2}n_{0}/\Delta_{1}+g_{2}^{2}n_{N}/\Delta_{2}$ in Eq. (S31)] can be
removed if $\Delta_{1}=-\Delta_{2}$, $|g_{1}|=|g_{2}|$, and
$|\eta_{1}|=|\eta_{2}|$, such that the giant-atom effects become more ideal.
Finally, we would like to point out that the Ladder-type scheme can be
extended to include more two-photon processes such that an effective giant
atom with multiple atom-waveguide coupling points can be simulated.

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