# Unconventional Quantum Electrodynamics with Hofstadter-Ladder Waveguide

Xin Wang  Institute of Theoretical Physics, School of Physics, Xi’an Jiaotong
University, Xi’an 710049, People’s Republic of China  Zhao-Min Gao  Institute
of Theoretical Physics, School of Physics, Xi’an Jiaotong University, Xi’an
710049, People’s Republic of China  Jia-Qi Li  Institute of Theoretical
Physics, School of Physics, Xi’an Jiaotong University, Xi’an 710049, People’s
Republic of China  Huai-Bing Zhu  Institute of Theoretical Physics, School of
Physics, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China
Hong-Rong Li  Institute of Theoretical Physics, School of Physics, Xi’an
Jiaotong University, Xi’an 710049, People’s Republic of China

###### Abstract

We propose a novel quantum electrodynamics (QED) platform where quantum
emitters interact with a Hofstadter-ladder waveguide. We demonstrate several
intriguing phenomena stemming from the nontrivial dispersion relation and
vacuum mode properties led by the effective spin-orbit coupling. First, by
assuming emitter’s frequency to be resonant with the lower band, we find that
the spontaneous emission is chiral with most photonic field decaying
unidirectionally. Both numerical and analytical results indicate that the
Hofstadter-ladder waveguide can be engineered as a well-performed chiral
quantum bus. Second, the dynamics of emitters of giant atom form is explored
by considering their frequencies below the lower band. Due to quantum
interference, we find that both the emitter-waveguide interaction and the
amplitudes of bound states are periodically modulated by giant emitter’s size.
The periodical length depends on the positions of energy minima points induced
by the spin-orbit coupling. Last, we consider the interaction between two
giant emitters mediated by bound states, and find that their dipole-dipole
interaction vanishes (is enhanced) when maximum destructive (constructive)
interference happens.

##  I introduction

The interactions between quantum emitters and the unavoidable baths with large
degrees of freedom are the central topic of quantum optics Cohen-Tannoudji
_et al._ (1998); Clerk  _et al._ (2010). For example, in the present of a
vacuum bath, the emitter will spontaneously decay to its ground state as well
as its frequency being renormalized due to Lamb shifts Lamb and Retherford
(1947); Scully and Zubairy (1997). By shaping the size of the environment or
narrowing its spectrum bandwidth, many intriguing phenomena, such as isotropic
propagation of photons and non-Markovian evolution arise John and Quang
(1994); Lambropoulos  _et al._ (2000); Giraldi and Petruccione (2011); Lodahl
_et al._ (2015); Stewart  _et al._ (2020); Ferreira  _et al._ (2021). As
discussed in Refs. Petersen  _et al._ (2014); Bliokh and Nori (2015); Bliokh
_et al._ (2015); Lodahl  _et al._ (2017); Lang  _et al._ (2022), chiral
emission can be observed via the subwavelength confinement in nanophotonic
systems, which opens the possibilities to realize cascaded quantum networks.
Moreover, when considering an emitter coupling to the bandgaps of a bath John
and Wang (1990); Goban  _et al._ (2014); González-Tudela  _et al._ (2015);
Douglas  _et al._ (2016); Liu and Houck (2017); Chang  _et al._ (2018); Wang
_et al._ (2021a), photonic bound states (in the form of an evanescent field)
emerge Douglas  _et al._ (2015). In this scenario, between atoms there are
long-range dipole-dipole interactions by exchanging the virtual photons in the
waveguide Shahmoon and Kurizki (2013); Ying  _et al._ (2019).

In recent years, exploring quantum electrodynamics (QED) with emitters
coupling to structured lattice environments, has attracted a lot of interests
Ramos  _et al._ (2016); Calajo  _et al._ (2016); González-Tudela and Cirac
(2018); González-Tudela  _et al._ (2019); Leonforte  _et al._ (2021). Those
artificial lattice reservoirs are widely studied in condensed matter physics,
and usually have unconventional spectra, or topological properties with
nontrivial vacuum modes. In Refs. González-Tudela and Cirac (2017, 2017), by
considering emitters interacting with a 2D tight-binding lattice environment,
the authors showed that both superradiance and subradiance of collective atoms
emerge in the nonperturbative regime. The unusual chiral bound states and
directional dipole-dipole interaction were also demonstrated in a topological
waveguide QED system Bello  _et al._ (2019); Kim  _et al._ (2021). In Refs.
Roccati  _et al._ (2022); Gong  _et al._ (2022), the authors discussed how to
realize bound states and dipole-dipole interactions in non-Hermitian photonic
lattices. All these studies indicate those structured lattice reservoirs with
reduced dimensionality are versatile toolboxes for exploring novel quantum
optical phenomena, as well as the potential applications in quantum
information processing.

In artificial baths with the spin-orbit coupling, the motion and spin freedoms
of a particle are linked, and many anomalous phenomena such as spin-Hall
effect and topological insulators can be observed Zhang (2000); Murakami  _et
al._ (2003); Sinova  _et al._ (2004); Wunderlich  _et al._ (2005); Kane and
Mele (2005); Galitski and Spielman (2013); Zhou  _et al._ (2013); Wu  _et al._
(2016); Kartashov  _et al._ (2017); Livi  _et al._ (2016); Liu  _et al._
(2011). Since neither the spin nor the momentum is the well-defined quantum
number to describe the dispersion relation, the spin-orbit coupling will
produce nontrivial energy bands and photonic modes Sala  _et al._ (2015);
Salerno  _et al._ (2017). The quantum optics with emitters interacting with
baths of the spin-orbit coupling, is rarely studied. It is a simple but
interesting toy model in condensed matter physics (see Fig. 1) Creutz (2001);
Narozhny  _et al._ (2005); Jaefari and Fradkin (2012); Atala  _et al._ (2014);
Tai  _et al._ (2017); Yuan  _et al._ (2019); Guan  _et al._ (2020). As
discussed in Ref. Hügel and Paredes (2014), the ladder contains two legs which
play the roles of two freedoms in an effective spin. In the present of
synthetic gauge fields, the effective spin will be locked to momentum freedom.

In this work, we discuss QED phenomena in a setup composed by quantum emitters
and a Hofstadter-ladder waveguide. Different from previous studies based on
lattice environment with synthetic gauge fields Sánchez-Burillo  _et al._
(2020); Wang  _et al._ (2020); De Bernardis  _et al._ (2021); Dong  _et al._
(2021), here we mainly focus on unconventional QED phenomena induced by the
spin-orbit coupling. First, we assume that the emitter is of small atom form
which frequency is resonant with the lower energy band. Due to spin-momentum
locking, the emitter chirally dissipates almost all its energy into one
direction of the waveguide, which is different from the directional emission
along the edge states of a 2D topologically non-trivial lattice Mittal  _et
al._ (2014). In our study the chiral emission into the 1D Hofstadter-ladder
waveguide stems from the effective spin-orbit coupling, and does not require
any topological protection. Our proposal is possible to demonstrate chiral
quantum optics, which has been extensively studied in Refs. Calajó  _et al._
(2019); Gheeraert  _et al._ (2020); Wang  _et al._ (2022); Solano  _et al._
(2021); Kannan  _et al._ (2022). Second, the emitter is considered as giant
atom form, and couples to the waveguide at multiple sites Kockum  _et al._
(2014); Guo  _et al._ (2017); Kockum  _et al._ (2018); Kockum (2020); Kannan
_et al._ (2020); Zhao and Wang (2020); Du  _et al._ (2021); Zhang  _et al._
(2021); Wang  _et al._ (2021b); Du  _et al._ (2022). Given that emitters’
frequency is below two degenerate minima points induced by the spin-orbit
coupling, there will be bound state in which the photonic energy will be
trapped. The bound state induced by time-delay effects of giant atom has been
investigated in Ref. Guo  _et al._ (2020). In this work, we focus on another
giant atom effect, i.e., the quantum interference between different coupling
points. We find that, due to quantum interference and unconventional spectrum
of the Hofstadter-ladder waveguide, the bound state will be periodically
modulated by the giant atom’s size. The periodical length is tunable by
controlling the parameters of the Hofstadter ladder waveguide. Based on this
mechanism, we show that by tuning the interference as
constructive/destructive, the dipole-dipole interaction between two giant
emitters will be enhanced/suppressed.

##  II spectrum and spin-orbit coupling of Hofstadter-ladder waveguide

Figure 1: A two-level quantum emitter interacts with a waveguide in the form
of the photonic analog of a Hofstadter-ladder model. The ladder rungs are
hopped with strength $t^{\prime}$. The nearest neighbor sites in two legs are
coupled at rate $t$, together with a synthetic gauge phase $e^{i\phi}$
($e^{-i\phi}$) for channel A (B). The effective magnetic field through each
plaquette is $\Phi=2\phi$.

The model of the QED setup we study is depicted in Fig. 1, where a quantum
emitter interacts with an artificial one-dimensional waveguide along the $x$
direction, which behaves as a photonic analog of the Hofstadter-ladder model.
The Hofstadter ladder can be viewed as the two-leg edge of the Harper-
Hofstadter model Ozawa  _et al._ (2019), where a synthetic gauge field
$\Phi=2\phi$ is applied through each plaquette (see Fig. 1). Here we consider
it working as a 1D artificial waveguide which allows photons traveling along
it. In this situation, two legs in of the ladder waveguide serve as channel A
and B of the waveguide. For convenience, we set the length of one unit site as
$d_{0}=1$. The ladder waveguide is composed by two legs, which can be viewed
as two quantum channels for the emitter. Two sites in each rung are coupled
with strength $t^{\prime}$, which is set as $t^{\prime}=1$. By adopting a
Landau gauge along the $x$ direction, the phase connections only appear in
each leg. Therefore, the hopping amplitude between two nearest neighbor sites
is $te^{i\phi}$ ($te^{-i\phi}$) for channel A (B). Consequently, by setting
$\hbar=1$, the tight-binding Hamiltonian of the waveguide is Guan  _et al._
(2020)

$\displaystyle H_{B}$ $\displaystyle=$
$\displaystyle\sum_{x}\omega_{0}(a_{x}^{\dagger}a_{x}+b_{x}^{\dagger}b_{x})-\Big{[}t^{\prime}\sum_{x}{a_{x}^{\dagger}b_{x}}$
(1)
$\displaystyle+t\sum_{x}{\left(e^{i\phi}a_{x+1}^{\dagger}a_{x}+e^{-i\phi}b_{x+1}^{\dagger}b_{x}\right)}+\mathrm{H}.\mathrm{c.}\Big{]},$

where $a_{x},b_{x}$ ($a_{x}^{\dagger},b_{x}^{\dagger}$) are the annihilation
(creation) operators of the sites $a,b$ at position $x$, and $\omega_{0}$ is
the identical frequency of those bosonic modes. In the following we work in
the rotating frame of the constant part
$\sum_{x}\omega_{0}(a_{x}^{\dagger}a_{x}+b_{x}^{\dagger}b_{x})$.

Under the periodic boundary condition and in the momentum space with

$\displaystyle
a_{k}^{\dagger}=\frac{1}{\sqrt{N}}\sum_{x}{e^{ikx}a_{x}^{\dagger}},\quad
b_{k}^{\dagger}=\frac{1}{\sqrt{N}}\sum_{x}{e^{ikx}b_{x}^{\dagger}},$
$\displaystyle k=\frac{2\pi}{N}n,\quad n\in(-N/2,N/2],$ (2)

we can diagonalize the waveguide Hamiltonian as

$\displaystyle H_{B}$ $\displaystyle=$
$\displaystyle-2t\left[\begin{matrix}a_{k}^{\dagger}\quad b_{k}^{\dagger}\\\
\end{matrix}\right]\mathcal{H}_{B}\left[\begin{array}[]{c}a_{k}\\\ b_{k}\\\
\end{array}\right],$ (5) $\displaystyle\mathcal{H}_{B}$ $\displaystyle=$
$\displaystyle\left[\begin{matrix}g\left(k\right)+f\left(k\right)&\eta\\\
\eta&g\left(k\right)-f\left(k\right)\\\ \end{matrix}\right]$ (6)
$\displaystyle=$ $\displaystyle
g\left(k\right)I+f\left(k\right)\sigma_{z}+\eta\sigma_{x},$

where $\eta=t^{\prime}/2t$, $g\left(k\right)$ and $f\left(k\right)$ are
respectively expressed as

$g\left(k\right)=\cos\phi\cos k,\quad f\left(k\right)=\sin\phi\sin k.$ (7)

As shown in Eq. (6), the Hamiltonian $H_{B}$ is expressed in the Pauli
operators, indicating that the upper-lower leg degree of freedom behaves as an
effective spin. Due to the synthetic gauge field, $H_{B}$ contains the
effective spin-orbit coupling term $(\sin\phi\sin k)\sigma_{z}$, which will
lead to spin-momentum locking Hügel and Paredes (2014). Note that in condensed
matter physics the concept of “spin” is extensively used for models consisting
of“A” and “B” sublattices. Similarly, the spin-orbit coupling is a generalized
concept from atomic physics, which describes a two-component internal freedom
coupling to the momentum of a particle Ozawa  _et al._ (2019). For example, in
Refs. Sala  _et al._ (2015); Hafezi  _et al._ (2011), the spin-orbit coupling
and spin Hall insulators for photons have been successfully demonstrated in
experiments.

The energy spectrum can be derived by simply diagonalizing $\mathcal{H}_{B}$.
Consequently, the energy bands and eigenmodes are derived as

$\displaystyle
E_{\pm}\left(k\right)=-2t\left[g\left(k\right)\mp\sqrt{f^{2}\left(k\right)+\eta^{2}}\right],$
(8) $\displaystyle
C_{k-}^{\dagger}=\left(\cos\frac{\theta_{k}}{2}a_{k}^{\dagger},\quad\sin\frac{\theta_{k}}{2}b_{k}^{\dagger}\right),$
(9) $\displaystyle
C_{k+}^{\dagger}=\left(\sin\frac{\theta_{k}}{2}a_{k}^{\dagger},\quad-\cos\frac{\theta_{k}}{2}b_{k}^{\dagger}\right),$
(10)

where $\theta_{k}=\arctan[\eta/f(k)]$. Now we define the average spin as

$\langle\sigma_{z}\rangle_{k}=\langle a_{k}^{\dagger}a_{k}\rangle-\langle
b_{k}^{\dagger}b_{k}\rangle=\cos^{2}\frac{\theta_{k}}{2}-\sin^{2}\frac{\theta_{k}}{2}.$
(11)

Given that $\langle\sigma_{z}\rangle_{k}>0$
($\langle\sigma_{z}\rangle_{k}<0$), the mode $k$ asymmetrically distributes on
channel $A$ ($B$) with more probabilities. In Fig. 2(a), we plot
$E_{\pm}\left(k\right)$ and $\langle\sigma_{z}\rangle_{k}$ of two bands versus
$k$. It is found that, due to spin-momentum locking, the chiral current
$\langle\sigma_{z}\rangle_{k}$ of the lower band is opposite to the upper band
for certain momentum $k$.

Moreover, the spin-orbit coupling significantly modifies the dispersion
relation of the waveguide. For a free particle without spin-orbit coupling,
the energy minimum point is usually with zero momentum $k=0$ (or $k=\pm\pi$).
When spin-orbit coupling appears, the spin-up and spin-down modes minimize
their energies by carrying non-zero opposite momentum $k$ Galitski and
Spielman (2013). As depicted in Fig. 2, the photonic dispersion relation of
the ladder waveguide becomes spin-dependent, and the energy minima are
degenerate at two points with non-zero momentum $k\neq 0$. Moreover, there is
Kramers degeneracy for a pair of modes $\pm k$ due to spin-momentum
interaction, and the field distribution is mostly localized in channel A (B)
for $k>0$ ($k<0$) Hügel and Paredes (2014). Those properties allow us to
realize unconventional phenomena of quantum optics, which will be addressed in
the following discussions.

Figure 2: (a) The dispersion relations for two energy bands of a Hofstadter-
ladder waveguide. The effective spin $\langle\sigma_{z}\rangle_{k}$ describing
the population difference between channel A and B, is mapped with colors. When
discussing the chiral emission, the emitter’s frequency is assumed in the cyan
area, around which the group velocity is $v_{g}$. The detuning to the lower
(upper) band edge is denoted by $\delta_{-k0}$ ($\delta_{+k0}$). (b) The
analytical decay rates $\Gamma_{A(B)}$ [in the unit $g^{2}/(2v_{g})$, see Eq.
(23)] into channel A (B) and the chiral factor change with $k$. The cyan area
corresponds to the Markovian decay regime where both band edges and the upper
energy band do not take apparent effects. The cross points with the dashed
vertical lines correspond to the decay rates of the emitter with frequency in
(a). Parameters of the whole system: $t^{\prime}=1$, $t=2$ and $\phi=\pi/3$.

##  III chiral spontaneous emission

We first consider that the two-energy-level emitter is of small atom form,
i.e., couples to the Hofstadter-ladder waveguide at one site $x=0$ (see Fig.
1). Its frequency lies resonantly within the lower energy band. In the
rotating frame of bosonic frequency $\omega_{0}$, the system Hamiltonian is
written as

$\displaystyle H_{s}=H_{0}+H_{\mathrm{int}},$ (12) $\displaystyle
H_{0}=\frac{1}{2}\Delta_{q}\sigma_{z}+H_{B},\quad
H_{\mathrm{int}}=g\left(\sigma_{-}a_{0}^{\dagger}+\sigma_{+}a_{0}\right),$
(13)

where $\Delta_{q}=\omega_{q}-\omega_{0}$ with $\omega_{q}$ being the emitter’s
transition frequency, $\sigma_{z}=|e\rangle\langle e|-|g\rangle\langle g|$ and
$\sigma_{+}=(\sigma_{-})^{\dagger}=|e\rangle\langle g|$, with $|e\rangle$
($|g\rangle$) being the exited (ground) state of the emitter. Applying inverse
Fourier transform, one obtains
$a^{\dagger}_{0}=\sum_{k}a_{k}^{\dagger}/\sqrt{N}$. According to Eqs. (9, 10),
$a_{k}^{\dagger}$ can be decomposed as the superposition of
$C_{k\pm}^{\dagger}$. Finally, the interaction Hamiltonian is written as

$\displaystyle
H_{\mathrm{int}}\\!=\\!\\!\frac{g}{\sqrt{N}}\sum_{k}{\\!\sigma_{-}\\!\\!\left(\\!\cos\frac{\theta_{k}}{2}C_{k-}^{\dagger}\\!+\\!\sin\frac{\theta_{k}}{2}C_{k+}^{\dagger}\\!\right)\\!+\\!\mathrm{H}.\mathrm{c}.}$
(14)

As shown in Fig. 2(a), $\omega_{q}$ is set in the cyan regime, and only the
lower band $E_{-}(k)$ is resonant with the emitter. To avoid the non-Markovian
effects led by the band tops Calajo  _et al._ (2016), we require $\Delta_{q}$
far away from two band edges, i.e., $|\delta_{\pm k0}|\gg 0$. By dropping the
off-resonant terms with upper band modes $C_{k+}^{\dagger}$, the interacting
Hamiltonian is reduced as

$H_{\mathrm{int}}=\frac{g}{\sqrt{N}}\left(\sum_{k}{\cos\frac{\theta_{k}}{2}\sigma_{-}\,\,C_{k-}^{\dagger}+\mathrm{H}.\mathrm{c}.}\right).$
(15)

After substituting Eq. (9) into Eq. (15), we can divide $H_{\mathrm{int}}$
into two parts which describe interactions with channel A and B respectively:

$\displaystyle H_{\mathrm{int}}=H_{\mathrm{int},A}+H_{\mathrm{int},B},$ (16)
$\displaystyle
H_{\mathrm{int},A}=\frac{g}{\sqrt{N}}\left(\sum_{k}{\cos^{2}\frac{\theta_{k}}{2}\sigma_{-}a_{k}^{\dagger}+\mathrm{H}.\mathrm{c}.}\right),$
(17) $\displaystyle
H_{\mathrm{int},B}=\frac{g}{\sqrt{N}}\left(\sum_{k}{\cos\frac{\theta_{k}}{2}\sin\frac{\theta_{k}}{2}\sigma_{-}b_{k}^{\dagger}+\mathrm{H}.\mathrm{c}.}\right).$
(18)

From Eq. (16) and as depicted in Fig. 2(a), we find four dissipation terms by
assuming the resonant position at $\Delta_{q}=E_{-}(\pm k_{r})$, i.e., the
left/right direction of channel A (B). After applying the unitary
transformation $U_{0}(t)=\exp(-iH_{0}t)$, the interaction operator with
channel A becomes

$\sum_{k}{(\sigma_{-}a_{k}^{\dagger})}\rightarrow\frac{N}{2\pi}\int_{-\pi}^{\pi}{\left(\sigma_{-}a_{k}^{\dagger}e^{i\Delta_{k}t}\right)dk},$
(19)

where $\Delta_{k}=E_{-}\left(k\right)-\Delta_{q}$. Similar to Eq. (19), the
interaction operator with channel B can also be written in an integral form.
We consider the spontaneous decay process with an excitation initially
localized in the emitter. In the single-excitation subspace, the state of the
whole system is expressed as
$|\psi(t)\rangle=\sum_{k}[c_{ka}(t)|g,1_{ka}\rangle+c_{kb}(t)|g,1_{kb}\rangle]+c_{e}(t)|e,0\rangle$,
and the evolution of the whole system governed by $H_{\mathrm{int}}$ is
derived from the following differential equations

$\displaystyle\dot{c}_{e}(t)=-i\sum_{k}\frac{ge^{-i\Delta_{k}t}}{\sqrt{N}}\left[\cos^{2}\frac{\theta_{k}}{2}c_{ka}(t)+\frac{\sin\theta_{k}}{2}c_{kb}(t)\right],$
(20)
$\displaystyle\dot{c}_{ka}(t)=-i\frac{g^{*}}{\sqrt{N}}e^{i\Delta_{k}t}\cos^{2}\frac{\theta_{k}}{2}c_{e}(t),$
(21)
$\displaystyle\dot{c}_{kb}(t)=-i\frac{g^{*}}{\sqrt{N}}e^{i\Delta_{k}t}\frac{\sin\theta_{k}}{2}c_{e}(t).$
(22)

By substituting the internal form of Eqs. (21, 22) into Eq. (20), the
evolution of $c_{e}(t)$ is derived as

$\dot{c}_{e}\left(t\right)=-\frac{g^{2}}{2\pi}\sum_{\pm}\left[\left(\frac{\cos\theta_{\pm
kr}+1}{2}\right)^{2}+\left(\frac{\sin\theta_{\pm
kr}}{2}\right)^{2}\right]\left|\int_{0}^{\pm\pi}{dk}\int_{0}^{t}{\left[c_{e}\left(t^{\prime}\right)e^{-i\Delta_{k}\left(t-t^{\prime}\right)}\right]dt^{\prime}}\right|.$
(23)

As depicted in Fig. 2(a), we approximate the dispersion relation around $\pm
k_{r}$ to be linear, i.e.,

$\displaystyle v_{g}$ $\displaystyle=$
$\displaystyle\frac{dE_{-}\left(k\right)}{dk}\Big{|}_{k_{r}}$ (24)
$\displaystyle=$
$\displaystyle-2t\sin\left(k_{r}\right)\left(-\cos\phi+\frac{\sin^{2}\phi\cos
k_{r}}{\sqrt{f^{2}\left(k_{r}\right)+\eta^{2}}}\right),$

where $v_{g}$ is the group velocity at $k_{r}$ ($k_{r}>0$). By setting $\delta
k=k-k_{r}$, the detuning is written as $\Delta_{k}\simeq v_{g}\delta k$. In
the Born-Markovian regime, the decay rate is required to be much smaller than
the band width $\\{\delta_{+k0},\delta_{-k0}\\}$, and we can extend the
integral bound $\pm\pi$ to be infinite. Consequently, Eq. (23) is reduced as

$\dot{c}_{e}\left(t\right)=-\sum_{i=A,B}\left(\Gamma_{i+}+\Gamma_{i-}\right)c_{e}\left(t\right),$
(25)

where $\Gamma_{i\pm}$ correspond to the emission rates into the right/left
direction of channel $i$, which are derived as

$\displaystyle\Gamma_{A\pm}=\frac{g^{2}}{2v_{g}}\left(\frac{\cos\theta_{\pm
kr}+1}{2}\right)^{2},$ (26)
$\displaystyle\Gamma_{B\pm}=\frac{g^{2}}{2v_{g}}\left(\frac{\sin\theta_{\pm
kr}}{2}\right)^{2},$ (27)

which show that $\Gamma_{A(B)\pm}$ are determined by $\theta_{\pm kr}$. In
this part we only focus on the Markovian decay regime (cyan area in Fig. 2)
where both band edges and the upper energy band do not take apparent effects.
We plot $\Gamma_{A,B}$ in Fig. 2(b) [in units of $g^{2}/(2v_{g})$], and find
that the emission to channel A (B) is spatially asymmetric (symmetric), i.e.,

$\Gamma_{A+}>\Gamma_{A-},\qquad\Gamma_{B+}=\Gamma_{B-}.$

Specially, under the following condition

$\left(\frac{\cos\theta_{kr}+1}{2}\right)^{2}\gg\left\\{\left(\frac{\cos\theta_{-kr}+1}{2}\right)^{2},\left(\frac{\sin\theta_{\pm
kr}}{2}\right)^{2}\right\\},$ (28)

the emission field mostly distributes on the right side of channel A.
Therefore, the spontaneous emission field will chirally propagate along the
Hofstadter-ladder waveguide. As discussed in Sec. II, the chirality is led by
the effective spin-orbit coupling mechanism.

Figure 3: (a) The spontaneous decay of the emitter by setting $g=0.4$ and
$g=3$, respectively. The curves with (without) symbols correspond to numerical
calculations (Markovian exponential decay). (b) Given that $g=0.4$, the field
distributions along channel A and B at $t=100$. The parameters are adopted the
same with those in Fig. 2.

We assume that the coupling position is at $x=0$, and therefore, the field
intensities on the right (left) side of channel A and B are defined as

$\Phi_{A(B)\pm}=\sum_{x=0}^{\pm N/2}|c_{A(B),x}|^{2},$ (29)

where $c_{A(B),x}$ is the field amplitude of site $a(b)$ of the rung at $x$ of
the ladder. For example, if the right side of channel A is the desired
direction, both the dissipation into channel B and into the left hand side of
channel A will lead to photonic leakage. Consequently, the chiral factor
$\mathcal{C}$ is defined as Lodahl  _et al._ (2017)

$\mathcal{C}=\frac{\Phi_{A+}}{\sum_{i=A,B}\left(\Phi_{i+}+\Phi_{i+}\right)}=\frac{\Gamma_{A+}}{\sum_{i=A,B}\left(\Gamma_{i+}+\Gamma_{i-}\right)}.$
(30)

By adopting expressions of $\Gamma_{A(B)\pm}$ in Eqs. (26,27), the analytical
chiral factor is derived as

$\mathcal{C}=\frac{\left(f\left(k_{r}\right)+|f\left(k_{r}\right)|\right)^{2}}{\left(f\left(k_{r}\right)+|f\left(k_{r}\right)|\right)^{2}+2\eta^{2}},$
(31)

where we employ the relation $\tan\theta_{k}=\eta/f(k)$. In Fig. 2(b), we plot
chiral factor $\mathcal{C}$ changing with resonant wave number $k_{r}$. Given
that $-\pi<\phi<0$, $f\left(k_{r}\right)=\sin\phi\sin k_{r}<0$ (Note that we
have restricted $k_{r}$ as positive, i.e., $k_{r}>0$). In this case,
$\mathcal{C}=0$, indicating that the field hardly dissipates into the desired
channel. When $0<\phi<\pi$, the chiral factor is simplified as

$\mathcal{C}=\frac{2f^{2}\left(k_{r}\right)}{2f^{2}\left(k_{r}\right)+\eta^{2}}=\frac{1}{1+\frac{\eta^{2}}{2f^{2}\left(k_{r}\right)}}.$
(32)

Therefore, under the condition

$\frac{\eta^{2}}{2f^{2}\left(k_{r}\right)}\ll
1\longrightarrow\mathcal{C}\rightarrow 1,$ (33)

most excitation energy will dissipate into the right side of channel A. Those
discussions indicate that both the waveguide’s parameters and the resonant
position $k_{r}$ will directly determine the chiral factor.

By adopting the parameters in Fig. 2, we obtain

$\eta^{2}/[2f^{2}\left(k_{r}\right)]\simeq 0.058\rightarrow\mathcal{C}\simeq
0.944,$ (34)

which is a high chiral factor to realize cascaded quantum networks with
multiple nodes. To verify our above analysis, in the following we numerically
simulate the system’s evolution by adopting the system’s Hamiltonian in Eq.
(12). Note that the ladder’s Hamiltonian is expressed in real space [see Eq.
(1)], and the ladder length is set as $N=1000$, which is long enough to avoid
field reflection by the bounds. Figure 3(a) shows the evolution of the emitter
with $g=0.4$ and $g=3$, respectively. Given that $g=0.4$, the analytical decay
rate is calculated as $\sum_{i=A,B}\left(\Gamma_{i+}+\Gamma_{i-}\right)\simeq
0.0014$ [according to Eqs. (26, 27)], which is much smaller than the band
width $\delta_{\pm k0}$, and the Markovian approximation is valid. The
emitter’s evolution $|c_{e}(t)|^{2}$ is shown in Fig. 3(a), which decays with
time and matches well with the analytical exponential form. In Fig. 3(b), we
plot the field distributions at $t=100$ for both channel A and B, where the
photonic field mostly distributes on the right side of channel A. By adopting
Eq. (29) the numerical chirality is about $\mathcal{C}\simeq 0.943$, which is
very close to the analytical result derived in Eq. (34).

Note that all the above results for the chiral decay are derived within the
Born-Markovian approximation. When the emitter-waveguide interaction strength
is comparable to the band width $\\{\delta_{+k0},\delta_{-k0}\\}$, both modes
around $k=0$ and two energy minima points $k_{\pm\text{min}}$ (see Fig. 2 and
Fig. 5) with zero group velocity will prevent the emitter from decaying.
Partial excitation energy will be trapped around the coupling position in the
form of bound states Calajo  _et al._ (2016). By increasing the emitter-
waveguide interaction beyond the Markovian regime with $g=3$, we plot the
emitter’s evolution in Fig. 3(a), which is no longer of exponential decaying
form. After dissipating partial energy into the waveguide, the rest is trapped
within the emitter. Therefore, to work as a well-performed cascaded quantum
system, the emitter in each node should couple to the waveguide within the
Markovian regime. Next, we will discuss the behavior of bound states of both
small and giant emitters due to band edge effects.

As discussed in experimental work in Refs. Underwood  _et al._ (2012); Houck
_et al._ (2012), the bosonic modes in the Hofstadter-ladder waveguide can be
made by cavities or LC resonators, which will experience decoherence led by
the noisy environment. The decay of each site is assumed in the Lindblad form,
i.e.,

$\mathcal{L}\rho=\kappa\sum_{O=a,b}\sum_{x}O_{x}\rho
O_{x}^{{\dagger}}-\frac{1}{2}(O_{x}^{{\dagger}}O_{x}\rho-\rho
O_{x}^{{\dagger}}O_{x}),$ (35)

where $\kappa$ is the photonic decay rate of each cite. By setting $\kappa=0$
and $0.01$ respectively, we plot the corresponding field distributions along
the waveguide at $t=100$. When the photonic wavepacket propagates along the
dissipative waveguide, it will decay the energy into the environment. We find
that the amplitude of the photonic field becomes much lower than the non-
dissipative case. However, the field distribution is still chiral, which is
not affected by this local coherence.

Additionally, in experiments the waveguide’s length cannot be infinite, and
therefore, the chiral field will be reflected by the hard-wall boundary of the
waveguide after a long-time propagation. In Fig. 4(c), we plot the field
distribution at $t=180$, when the wavepacket already touches the waveguide’s
boundary. The energy flow direction is shown in the insets. We find that, due
to the hopping rates $t^{\prime}$ between two channels at the boundary, most
of the energy will be reflected into channel B. Since the photon decay of each
rate is also considered, the amplitude of the reflected wavepacket is much
lower than that in Fig. 4(a).

Figure 4: (a, b) By setting the photonic decay rate for each site as
$\kappa=0$ and $\kappa=0.01$, the field distributions along channel A and B at
$t=100$. (c) By setting $\kappa=0.01$ the field distributions after being
reflected by the hard-wall boundary of the waveguide. The energy flow
direction is shown in the inset. Other parameters are adopted the same with
those in Fig. 3.

##  IV periodical interference behavior modulated by giant emitter’s size

###  IV.1 bound state of a single giant emitter

Figure 5: (a) Around two energy minima $\pm k_{\text{min}}$, the dispersion
relation of the lower band can be fit with the quadratic relation [dashed
curves, see Eq. (50)]. To observe bound states, the emitter’s frequency is set
below the band edge, i.e., $\Delta_{q}<E_{k_{\text{min}}}$. (b) A giant
emitter interacts with the waveguide at two coupling points $x_{\pm}$. The
emitter’s size is $d_{s}=x_{+}-x_{-}$. (c) Given that
$k_{\text{min}}\simeq\pi/3$, the coupling phase relation between two coupling
points changes with $d_{s}$.

Besides emitting and absorbing real photons, different emitters can also be
coherently mediated via exchanging virtual photons in the waveguide, which
requires the emitter’s frequency to be outside the spectrum of the waveguide
bath Douglas  _et al._ (2016); Bello  _et al._ (2019); Wang  _et al._ (2021a).
Here we restrict the frequency detuning $\Delta_{q}$ below the lower bound of
$E_{-}(k)$ [see Fig. 5(a)]. Additionally, the emitter is assumed to be of
giant atom form Kockum  _et al._ (2014); Guo  _et al._ (2017); Kockum  _et
al._ (2018); Kockum (2020); Kannan  _et al._ (2020); Zhao and Wang (2020); Du
_et al._ (2021); Zhang  _et al._ (2021); Wang  _et al._ (2021b); Du  _et al._
(2022), which couples to the waveguide at two points $x_{\pm}$ of channel A
(or B), as depicted in Fig. 5(b). The separation distance is denoted as
$d_{s}=x_{+}-x_{-}$, which corresponds to the giant emitter’s size. Similar to
previous discussions, the interaction strengths with channel A and B [see Eqs.
(17,18)] are written as

$\displaystyle
g_{ka}=\frac{g}{\sqrt{N}}\cos^{2}\frac{\theta_{k}}{2}(e^{-ikx_{-}}+e^{-ikx_{+}}),$
(36) $\displaystyle
g_{kb}=\frac{g}{\sqrt{N}}\cos\frac{\theta_{k}}{2}\sin\frac{\theta_{k}}{2}(e^{-ikx_{-}}+e^{-ikx_{+}}),$
(37)

where $g$ is the interaction strength with a single point. Note that under the
condition $d_{s}=0$, two coupling positions coincide at the same site, and the
giant atom degrades as a small atom.

Similar to Eqs. (20-22), we can obtain differential equations for $c_{e}(t)$
and $c_{ka(b)}(t)$. Defining
$e^{i\Delta_{k}t}\tilde{C}_{ka\left(b\right)}\left(t\right)=c_{ka\left(b\right)}\left(t\right)$,
the evolution is derived in Laplace space with
$c_{e}(t)\rightarrow\tilde{c}_{e}(s)$ and
$C_{ka(b)}(t)\rightarrow\tilde{C}_{ka(b)}(s)$ Calajo  _et al._ (2016); Bello
_et al._ (2019); Wang and Li (2022)

$\displaystyle
s\tilde{c}_{e}\left(s\right)=-i\sum_{k}{\left[g_{ka}\tilde{C}_{ka}\left(s\right)+g_{kb}\tilde{C}_{kb}\left(s\right)\right]},$
(38) $\displaystyle
s\tilde{C}_{ki}\left(s\right)=-i\Delta_{k}\tilde{C}_{ki}\left(s\right)-ig_{ki}^{*}\tilde{c}_{e}(s),\quad
i=a,b.$ (39)

Consequently, $\tilde{C}_{ki}(s)$ is obtained as

$\tilde{C}_{ki}\left(s\right)=\frac{-ig_{ki}^{*}c_{e}(s)}{s+i\Delta_{k}},\quad
i=a,b.$ (40)

By substituting Eq. (40) into Eq. (38), $\tilde{c}_{e}(s)$ is derived as

$\displaystyle\tilde{c}_{e}\left(s\right)=\frac{1}{s+\Sigma_{e}(s)},$ (41)
$\displaystyle\Sigma_{e}(s)=\sum_{k}{\frac{|g_{ka}|^{2}+|g_{kb}|^{2}}{s+i\Delta_{k}}},$
(42)

where $\Sigma_{e}(s)$ is the self-energy. The time-dependent evolution is
recovered from the inverse Laplace transform Ramos  _et al._ (2016)

$c_{e}\left(t\right)=\frac{1}{2\pi i}\lim_{E\rightarrow\infty}\int_{\epsilon-
iE}^{\epsilon+iE}\tilde{c}_{e}\left(s\right)e^{st}ds,\quad\epsilon>0.$ (43)

Given that the waveguide is long enough to avoid reflection effects, we can
write the self-energy in the integral form by replacing $\Sigma_{k}$ with
$N/(2\pi)\int dk$. Substituting the relations

$\displaystyle\cos^{2}\frac{\theta_{k}}{2}=\frac{\frac{f\left(k\right)}{\sqrt{f^{2}\left(k\right)+\eta^{2}}}+1}{2},$
(44)
$\displaystyle\frac{\sin\theta_{k}}{2}=\frac{\frac{\eta}{\sqrt{f^{2}\left(k\right)+\eta^{2}}}}{2},$
(45)

into Eq. (42), the self-energy is expressed as

$\Sigma_{e}(s)=\frac{1}{4\pi}\int_{-\pi}^{\pi}{\frac{\left(\frac{f\left(k\right)}{\sqrt{f^{2}\left(k\right)+\eta^{2}}}+1\right)|G_{k}|^{2}}{s+i\Delta_{k}}dk},$
(46)

where $G_{k}=g(e^{-ikx_{-}}+e^{-ikx_{+}})$.

Due to the effective spin-orbit coupling, the energy minimum point is split
into two with non-zero momentum $\pm k_{\text{min}}\neq 0$. At the positions
of two dips, the group velocities are zero, which can be derived from Eq. (24)

$v_{g}=\frac{dE_{-}\left(k\right)}{dk}\Big{|}_{k=\pm k_{\text{min}}}=0.$ (47)

Therefore, their positions are derived as

$\sin\left(k_{\min}\right)=\pm\sqrt{\sin^{2}\phi-\eta^{2}\cot^{2}\phi}.$ (48)

At $\pm k_{\min}$, the second-order derivatives are non-zero, and we denote
the curvature as

$\alpha=\frac{d^{2}E_{-}\left(k\right)}{dk^{2}}\Big{|}_{k=\pm
k_{\text{min}}}.$ (49)

As depicted in Fig. 5(a), we employ the effective mass approximation, to fit
the dispersion around the band edges with quadratic relations

$E_{-}\left(k\right)=E_{\min}+\alpha\left(k\pm k_{\min}\right)^{2}.$ (50)

By substituting Eq. (50) into Eq. (46), the self-energy is calculated as

$\displaystyle\Sigma_{e}(s)$ $\displaystyle\simeq$
$\displaystyle\frac{1}{4\pi}\Bigg{\\{}\int_{-\pi}^{0}{\frac{\left(\frac{f\left(k\right)}{\sqrt{f^{2}\left(k\right)+\eta^{2}}}+1\right)|G_{k}|^{2}}{s+i\left[\Delta_{0}+\alpha\left(k+k_{\min}\right)^{2}\right]}dk}$
(51)
$\displaystyle+\int_{0}^{\pi}{\frac{\left(\frac{f\left(k\right)}{\sqrt{f^{2}\left(k\right)+\eta^{2}}}+1\right)|G_{k}|^{2}}{s+i\left[\Delta_{0}+\alpha\left(k-k_{\min}\right)^{2}\right]}dk}\Bigg{\\}},$

where $\Delta_{0}=E_{\min}-\Delta_{q}$ is the detuning to the band edge [see
Fig. 5(a)]. We assume that $\Delta_{0}$ is small, and therefore, only the
modes around $\pm k_{\min}$ contribute significantly to $\Sigma_{e}(s)$.
Consequently, we can approximate

$G_{\pm k}\simeq G_{\pm k_{\text{min}}},\quad f\left(\pm k\right)\simeq f(\pm
k_{\text{min}}),$ (52)

and extend the integral bound of Eq. (51) to be infinite. Finally, we obtain

$\Sigma_{e}(s)\simeq\frac{|G_{k_{\text{min}}}|^{2}}{2}\frac{1}{\sqrt{-\alpha\left(\Delta_{0}-is\right)}},$
(53)

where we have employed the following relations

$|G_{k_{\text{min}}}|=|G_{-k_{\text{min}}}|,\quad f(k_{\min})=-f(-k_{\min}).$
(54)

Figure 6: (a) The probabilities $|c_{e}(t)|^{2}$ of the giant emitter
remaining in its excited state change with $t$ for different $d_{s}$. In the
limit $t\rightarrow\infty$, $|c_{e}(t)|^{2}$ reaches its steady value, which
is nonzero due to the band edge effect. The corresponding field distributions
of the photonic bound states in channel A and B are shown in (b) and (c),
respectively. The emitter frequency is set below the lower band with
$\Delta_{q}=-4.2$. The coupling strength is $g=0.1$. The waveguide parameters
are the same with those in Fig. 2.

In this work, the adopted parameters of the waveguide satisfy

$\sin^{2}\phi\gg\eta^{2}\cot^{2}\phi\longrightarrow
k_{\text{min}}\simeq\phi=\pi/3.$ (55)

As shown in Fig. 5(c), the phase relation between two coupling points will
rotate counter-clockwise when increasing giant emitter’s size
($d_{s}=0,1,2,3...$). Therefore, the relative phase between $x_{\pm}$ is very
essential for the dynamics of the giant emitter. The interference between two
points is maximum destructive (constructive) given that $d_{s}=3(2N+1)$
($d_{s}=6N$). When $\Delta_{0}$ is much stronger than $G_{k_{\text{min}}}$,
most energy will be trapped in the emitter in the form of the bound state
Calajo  _et al._ (2016); González-Tudela and Cirac (2017); Wang and Li (2022).
The trapped excitation probability is determined by the pure imaginary pole
$s_{0}$ of the following transcendental equation Wang and Li (2022)

$s_{0}+\Sigma_{e}(s_{0})\simeq
s_{0}+\frac{|G_{k_{\text{min}}}|^{2}}{2}\frac{1}{\sqrt{-\alpha\left(\Delta_{0}-is_{0}\right)}}=0,$
(56)

and the steady state population of the emitter is derived via the residue
theorem Calajo  _et al._ (2016)

$\displaystyle|c_{e}(t=\infty)|^{2}=|\text{Res}(s_{0})|^{2},$ (57)
$\displaystyle\text{Res}(s_{0})=\frac{1}{1+\partial_{s}\Sigma_{e}(s)}\Big{|}_{s=s_{0}}.$
(58)

In Fig. 6(a), we plot dynamical evolutions of $|c_{e}(t)|^{2}$ for different
giant emitter’s sizes $d_{s}$. When $d_{s}=0$ (a small emitter), the effective
interacting strength $|G_{k_{\text{min}}}|$ is strongest, which is enhanced by
the constructive interference between two coupling legs. The steady state
$|c_{e}(t)|^{2}$ reaches lowest, and the emitter can effectively distribute
its energy into the waveguide. That is, the photonic bound state is mostly
localized in channel A due to the relation

$\cos^{2}\frac{\theta_{k_{\text{min}}}}{2}\gg\cos\frac{\theta_{k_{\text{min}}}}{2}\sin\frac{\theta_{k_{\text{min}}}}{2},$
(59)

which can be seen clearly by comparing Fig. 6(b) and Fig. 6(c).

When increasing the distance between two coupling points, the constructive
interference will be reversed as destructive, with $|G_{k_{\text{min}}}|$
being significantly weaken. When $d_{s}=3$, the phase difference satisfies
$k_{\text{min}}d_{s}\simeq\pi$, indicating that the interaction strength is
approximately zero, i.e., $|G_{k_{\text{min}}}|\simeq 0$ [see Fig. 5(c)]. The
giant emitter approximately decouples with the waveguide. Due to this
decoupling mechanism, the excitation trapped in the emitter reaches its
maximum, as shown in Fig. 6(a). Figure 6(b, c) show that the destructive
interference will also suppress the bound state’s amplitude significantly. We
plot the steady state population $|c_{e}(t\rightarrow\infty)|^{2}$ changing
with $d_{s}$ in Fig. 7(a), which clearly presents a periodical interference
pattern of $|c_{e}(t\rightarrow\infty)|^{2}$ modulated by the giant atom’s
size. Note that we only take $k_{\text{min}}\simeq\pi/3$ for example. Note
that The periodical length is tunable by controlling the parameters of the
Hofstadter ladder waveguide. By adopting another $k_{\text{min}}$ according
Eq. (48), different spatial interference patterns can also be observed. The
oscillating pattern in Fig. 7(a) is due to interference between different
points in the giant atom, and the peaks (dips) correspond to the positions
where the maximum constructive (destructive) interference happens. Therefore,
we can define the contrast ratio $R$ for the interference as

$R=\frac{\min\\{|c_{e}(t\rightarrow\infty,d_{s})|^{2}\\}}{\max\\{|c_{e}(t\rightarrow\infty,d_{s})|^{2}\\}}.$
(60)

For the parameters employed in Fig. 7(a),
$\min\\{|c_{e}(t\rightarrow\infty,d_{s})|^{2}\\}$
($\max\\{|c_{e}(t\rightarrow\infty,d_{s})|^{2}\\}$) is located at $d_{s}=0$
($d_{s}=\pm 3$), which corresponds to a small (giant) atom. A smaller $R$
indicates that the dynamical difference between small and giant atoms is more
apparent. In Fig. 7(b), we plot maximum/minimum
$|c_{e}(t\rightarrow\infty,d_{s})|^{2}$ and $R$ changing with the frequency
detuning $\Delta_{q}$. When shifting $\Delta_{q}$ away from the band edge, the
contributions of the band-edge modes to the bound states will be significantly
suppressed. Therefore, a larger $\Delta_{q}$ will reduce the interference
effects, which corresponds to $R\rightarrow 1$ [see Fig. 7(b)]. To observe
better interference effect in experiments, the emitter frequency can be close
to the edge of the lower energy band.

Figure 7: (a) The periodical behavior of the trapped excitation
$|c_{e}(t\rightarrow\infty)|^{2}$ changing with the size of giant emitters.
The analytical results are calculated via the residue theorem in Eq. (55). (b)
The maximum/minimum $|c_{e}(t\rightarrow\infty,d_{s})|^{2}$ and the contrast
ratio $R$ changes with the emitter frequency $\Delta_{q}$.

###  IV.2 dipole-dipole interactions

Figure 8: (a) Two giant and (b) two small emitters, which are separated with
distance $D_{q}$, interact with a common ladder waveguide. For giant emitter
$i$, the coupling points are located at different positions $x_{i}^{\pm}$. For
small emitter $i$, $x_{i}^{\pm}$ are coincided at the same site. (c) The Rabi
oscillations between two giant (small) emitters which are marked with
triangles (squares). The corresponding analytical results (curves without
symbols) are derived from Eq. (71). The separation distances between two
emitters in (a, b) are set as $D_{q}=4$. To avoid virtual photons in the
waveguide being excited with high probabilities, we adopt a weak coupling
strength $g=0.015$. The other parameters are the same with those in Fig. 6.

By exchanging virtual photons in the waveguide, between emitters there are
long-range dipole-dipole interactions which are determined by the overlap
areas between their bound states González-Tudela  _et al._ (2015); Douglas
_et al._ (2016); Bello  _et al._ (2019). As shown in Fig. 8(a, b), we now
discuss the multiple emitters interacting with the same ladder waveguide. Due
to the interference mechanism presented above, we focus on revealing the
relation between giant emitter’s size and the quantum dynamics where dipole-
dipole interactions are involved.

Both two giant emitters are assumed to couple with channel A, and the coupling
topology is of separation form Kockum  _et al._ (2018). Similar to the single
emitter case, the interaction Hamiltonian is written as

$H_{\mathrm{int},2}=\sum_{i=1,2}\sum_{k}\frac{G_{ik}}{\sqrt{N}}{\cos\frac{\theta_{k}}{2}\sigma^{-}_{i}\,\,C_{k-}^{\dagger}+\mathrm{H}.\mathrm{c}.},$
(61)

where $G_{ik}=g(e^{-ikx^{i}_{-}}+e^{-ikx^{i}_{+}})$ is the coupling strength
between emitter $i$ and the waveguide. We consider two emitters with identical
frequency $\Delta_{q}$ which is also below the lower bound of $E_{-}(k)$. To
proceed, we define the average distance between two emitters as

$D_{q}=\frac{x^{2}_{+}+x^{2}_{-}}{2}-\frac{x^{1}_{+}+x^{1}_{-}}{2}.$ (62)

The dipole-dipole interaction can be tediously derived via the standard
resolvent-operator techniques González-Tudela  _et al._ (2015); Douglas  _et
al._ (2016); Bello  _et al._ (2019); Wang  _et al._ (2021a). However, given
that the detuning $\Delta_{0}$ is large and the waveguide is only virtually
excited, $J_{0}$ corresponds to the effective coupling strength mediated by
the waveguide’s modes, which can be simply derived via the effective
Hamiltonian methods James and Jerke (2007). In the rotating frame of free
energies of both emitters and waveguide, we first write $H_{\mathrm{int},2}$
in Eq. (61) in the time-dependent form

$H_{\mathrm{int},2}(t)=\sum_{i=1,2}\sum_{k}\frac{G_{ik}}{\sqrt{N}}{\cos\frac{\theta_{k}}{2}e^{i\Delta_{k}t}\sigma^{-}_{i}\,\,C_{k-}^{\dagger}+\mathrm{H}.\mathrm{c}.},$
(63)

where $\Delta_{k}=E_{-}(k)-\Delta_{q}$. We first calculate the coupling rate
$J_{12,k}$ mediated by one mode $k$. By employing the methods in Ref. [76],
the one-mode-mediated effective Hamiltonian is derived as

$\displaystyle H_{q-q,k}\\!$ $\displaystyle=$
$\displaystyle\\!\frac{G_{1k}G^{*}_{2k}}{\Delta_{k}}\frac{\cos^{2}\frac{\theta_{k}}{2}}{N}\times$
(64)
$\displaystyle\\!\left(\sigma^{-}_{1}C_{k-}^{\dagger}\sigma^{+}_{2}C_{k-}\\!-\\!\sigma^{+}_{2}C_{k-}\sigma^{-}_{1}C^{\dagger}_{k-}\right)\\!+\\!\mathrm{H}.\mathrm{c}.$

Since the waveguide is only virtually excited and approximately in its vacuum
state, we can trace off the freedom of mode $k$ by adopting the following
approximation

$\langle C_{k-}^{\dagger}C_{k-}\rangle\simeq 0,\quad\langle
C_{k-}C_{k-}^{\dagger}\rangle\simeq 1.$ (65)

Consequently, $H_{q-q,k}$ is simplified as

$H_{q-q,k}\simeq-\frac{G_{1k}G^{*}_{2k}}{\Delta_{k}}\frac{\cos^{2}\frac{\theta_{k}}{2}}{N}\sigma^{+}_{2}\sigma^{-}_{1}+\mathrm{H.c.}$
(66)

Note that $H_{q-q,k}$ is mediated by one mode, and the total dipole-dipole
interaction should take all the modes’ contribution into account.
Consequently, the total dipole-dipole interaction Hamiltonian mediated by all
the waveguide’s modes is derived as

$H_{q-q}=\sum_{k}H_{q-q,k}=J_{12}\left(\sigma_{1}^{-}\sigma_{2}^{+}+\mathrm{H}.\mathrm{c}.\right).$
(67)

where $J_{12}$ is the total interaction strength which is written as

$\displaystyle J_{12}$ $\displaystyle=$
$\displaystyle-\sum_{k}\frac{G_{1k}G^{*}_{2k}}{\Delta_{k}}\frac{\cos^{2}\frac{\theta_{k}}{2}}{N}$
(68) $\displaystyle\simeq$
$\displaystyle-\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{G_{1k}G^{*}_{2k}\cos^{2}\frac{\theta_{k}}{2}}{\Delta_{k}}dk.$

Since the emitter frequency is below the edge of the lower energy band [see
Fig. 5(a)], the dispersion relation can be fit with the quadratic relations in
Eq. (50). Without loss of generality, we set $(x^{1}_{+}+x^{1}_{-})/2=0$ since
only relative distance matters. As depicted in Fig. 8(a), two emitters are
separated with a distance $D_{q}$. Given that $D_{q}>d_{s}$, $J_{12}$ can be
written as

$\displaystyle J_{12}$ $\displaystyle\simeq$
$\displaystyle-\frac{1}{2\pi}\Bigg{\\{}\int_{-\pi}^{0}\frac{|G_{1k}||G^{*}_{2k}|e^{ikD_{q}}\left(\frac{\cos\theta_{k}+1}{2}\right)}{\Delta_{0}+\alpha\left(k+k_{\min}\right)^{2}}dk$
(69) $\displaystyle+$
$\displaystyle\int_{0}^{\pi}\frac{|G_{1k}||G^{*}_{2k}|e^{ikD_{q}}\left(\frac{\cos\theta_{k}+1}{2}\right)}{\Delta_{0}+\alpha\left(k-k_{\min}\right)^{2}}dk\Bigg{\\}}$

As shown in Eq. (26) and depicted in Fig. 2(b), the following relation

$|\frac{\cos\theta_{-k}+1}{2}|\ll|\frac{\cos\theta_{k}+1}{2}|\simeq 1,\quad
k>0$

is valid for the parameters employed in this work. Consequently, we can
neglect the first part ($k<0$) in Eq. (69), and $J_{12}$ is derived as

$\displaystyle
J_{12}\simeq\int_{0}^{\pi}\frac{|G_{1k}||G^{*}_{2k}|e^{ikD_{q}}}{\Delta_{0}+\alpha\left(k-k_{\min}\right)^{2}}dk,$
(70)

which indicates that that dipole-dipole interactions are determined by the
overlap areas between two bound states of emitters González-Tudela  _et al._
(2015). Similar to the process obtaining the self-energy in Eqs. (51, 53), we
derive $J_{12}$ as

$J_{12}=\frac{|G_{1k\min}||G_{2k\min}^{*}|}{2\sqrt{\alpha\Delta_{0}}}e^{-|\sqrt{\frac{\Delta_{0}}{\alpha}}D_{q}|},$
(71)

which shows that $J_{12}$ exponentially decays with emitters’ separation
distance. Similar results have been obtained in Refs. González-Tudela  _et
al._ (2015); Douglas  _et al._ (2016); Wang  _et al._ (2021a).

As shown in Fig. 7(b, c), the amplitude of the bound state will periodically
change with $d_{s}$ due to the interference effect. When $d_{s}=3$, the
photonic bound state approximately disappears. Consequently, the overlapping
area is also nearly zero. By assuming the initial excitation being in emitter
1, we numerically plot the Rabi oscillations between giant emitters
($d_{s}=3$) and small emitters ($d_{s}=0$) in Fig. 8(c), respectively. We find
that, due to the destructive interference, when $d_{s}=3$, two emitters hardly
exchange energy, and decouple with each other ($J_{12}\simeq 0$). On the
contrary, two emitters will exchange excitation rapidly at a large rate due to
the constructive interference when $d_{s}$ is reduced to be zero.

##  V Conclusion and outlooks

In this work, we explore the unconventional quantum optics by considering a
Hofstadter-ladder waveguide interacting with both small and giant emitters.
Due to the effective spin-orbit coupling, both the waveguide’s vacuum modes
and spectrum show nontrivial properties. In the first part, we consider a
small emitter which frequency is resonant with one energy band. Two legs of
the ladder, which can be viewed as the freedoms in an effective spin locked
with the momemtum freedom, provide two dissipating channels. The spontaneous
emission is chiral with most photonic field decaying unidirectionally. Both
numerical and analytical results show that the Hofstadter-ladder waveguide can
work as a well-performed quantum bus of a chiral network.

In the second part, the emitters are assumed of giant atom form with
frequencies below the lower energy band. In this scenario, only the modes
around two energy minima points induced by the spin-orbit coupling contribute
significantly to the system’s dynamics, which will lead to emitter-waveguide
bound states. Since the energy minima modes carry non-zero momentum, the
coupling strengths are periodically modulated by the giant emitter’s size due
to quantum interference. Specially the giant emitter will decouple with the
waveguide when maximum destructive interference happens. This mechanism
provides a novel approach to control long-range dipole-dipole interactions
between emitters by modulating their sizes.

In this work, the giant emitter is assumed to couple with the sites of the
same channel. Other intriguing effects might be observed by considering the
coupling points on different channels. Moreover, for multiple giant emitters,
we just consider the coupling points arranged in the sepration form. In fact
there are some other distinct topologies, which are nested and braided cases
Kockum  _et al._ (2018). Exploring the coupling topology effects might also
bring novel quantum phenomena. There are already plenty of work about
realizing artificial Hofstadter ladders in various quantum systems such as
cold atoms and circuit-QED Tai  _et al._ (2017); Gu  _et al._ (2017); Guan
_et al._ (2020); Weitenberg and Simonet (2021), which can be potential
platforms to demonstrate above unconventional QED phenomena. We hope that our
work will open the possibilities of exploring novel quantum effects in
artificial spin-orbit-coupling environments.

##  VI Acknowledgments

The quantum dynamical simulations are based on open source code QuTiP
Johansson  _et al._ (2012, 2013). X.W. is supported by the National Natural
Science Foundation of China (NSFC) (Grant No. 12174303 and No. 11804270), and
China Postdoctoral Science Foundation No. 2018M631136. H.R.L. is supported by
the National Natural Science Foundation of China (NSFC) (Grant No.11774284).

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