## Nonreciprocal and chiral single-photon scattering for giant atoms

### Two-level giant atom coupled to a single waveguide

As schematically shown in Fig. 1a, we consider a two-level giant atom coupled to a waveguide at two separated points *x* = 0 and *x* = *d*. The atom-waveguide coupling coefficients are \(g{e}^{i{\theta }_{1}}\) and \(g{e}^{i{\theta }_{2}}\), respectively, with local coupling phases *θ*_{1} and *θ*_{2} for inducing some intriguing interference effects to the scattering properties as will be discussed below. With superconducting quantum devices, the local coupling phases can be introduced with Josephson-junction loops threaded by external fluxes^{69}.

Under the rotating wave approximation, the real-space Hamiltonian of the model can be written as (*ћ* = 1 hereafter)

$$H=\, {H}_{w}+{H}_{a}+{H}_{I},\\ {H}_{w}= \int\nolimits_{-\infty }^{+\infty }dx\left[{a}_{L}^{{{\dagger}} }(x)\left({\omega }_{0}+i{v}_{g}\frac{\partial }{\partial x}\right){a}_{L}(x)\right.\\ \left.+ \, {a}_{R}^{{{\dagger}} }(x)\left({\omega }_{0}-i{v}_{g}\frac{\partial }{\partial x}\right){a}_{R}(x)\right],\\ {H}_{a}= \left({\omega }_{e}-i\frac{{\gamma }_{e}}{2}\right)\left|e\right\rangle \left\langle e\right|,\\ {H}_{I}= \int\nolimits_{-\infty }^{+\infty }dx\left\{\delta (x)g{e}^{i{\theta }_{1}}\big[{a}_{R}^{{{\dagger}} }(x)+{a}_{L}^{{{\dagger}} }(x)\big]|g\rangle \left\langle e\right|\right.\\ +\left.\delta (x-d)g{e}^{i{\theta }_{2}}\left[{a}_{R}^{{{\dagger}} }(x)+{a}_{L}^{{{\dagger}} }(x)\right]|g\rangle \left\langle e\right|+\,{{\mbox{H.c.}}}\right \}.$$

(1)

Here *H*_{w} represents the free Hamiltonian of the waveguide modes with *v*_{g} being the group velocity of photons in the waveguide. *a*_{R,L} (\({a}_{R,L}^{{{\dagger}} }\)) are the bosonic annihilation (creation) operators of the right-going and left-going photons in the waveguide, respectively; *ω*_{0} is the frequency around which the dispersion relation of the waveguide mode is linearized^{1,70}. *H*_{a} is for the atom, where *ω*_{e} describes the transition frequency between the ground state \(|g\rangle \) and the excited state \(\left|e\right\rangle \); *γ*_{e} is the external atomic dissipation rate due to the non-waveguide modes in the environment. *H*_{I} describes the interactions between the atom and the waveguide, where the Dirac delta functions *δ*(*x*) and *δ*(*x* − *d*) indicate that the atom-waveguide couplings occur at *x* = 0 and *x* = *d*, respectively.

Considering that the total excitation number is conserved in rotating wave approximation, the eigenstate of the system can be expressed in the single-excitation subspace as

$$|{{\Psi }}\rangle = \int\nolimits_{-\infty }^{+\infty }dx\big[{\Phi }_{R}(x){a}_{R}^{{{\dagger}} }(x)+{\Phi }_{L}(x){a}_{L}^{{{\dagger}} }(x)\big]|0,g\rangle\\ +{u}_{e}|0,e\rangle,$$

(2)

where Φ_{R,L}(*x*) are the density of probability amplitudes of creating the right-going and left-going photons at position *x*, respectively; *u*_{e} is the excitation amplitude of the atom; \(|0,g\rangle \) denotes the vacuum state of the system. The probability amplitudes can be determined by solving the eigenequation \(H\left|{{\Psi }}\right\rangle =E\left|{{\Psi }}\right\rangle \), which leads to

$$E{\Phi }_{R}(x)=\, \left({\omega }_{0}-i{v}_{g}\frac{\partial }{\partial x}\right){\Phi }_{R}(x)\\ +g\big[{e}^{i{\theta }_{1}}\delta (x)+{e}^{i{\theta }_{2}}\delta (x-d)\big]{u}_{e},\\ E{\Phi }_{L}(x)=\, \left({\omega }_{0}+i{v}_{g}\frac{\partial }{\partial x}\right){\Phi }_{L}(x)\\ +g\big[{e}^{i{\theta }_{1}}\delta (x)+{e}^{i{\theta }_{2}}\delta (x-d)\big]{u}_{e},\\ E{u}_{e}=\, \left({\omega }_{e}-i\frac{{\gamma }_{e}}{2}\right){u}_{e}+g{e}^{-i{\theta }_{1}}\big[{\Phi }_{R}(0)+{\Phi }_{L}(0)\big]\\ +g{e}^{-i{\theta }_{2}}\big[{\Phi }_{R}(d)+{\Phi }_{L}(d)\big].$$

(3)

Assuming that a photon with the renormalized wave vector *k* satisfying the linear dispersion relation *E* = *ω*_{0} + *k**v*_{g} is incident from port 1 of the waveguide, the wave functions Φ_{R,L}(*x*) can be written as

$${\Phi }_{R}(x)=\; {e}^{ikx}\left\{\right.\Theta (-x)+A\left[\right.\Theta (x)-\Theta (x-d)\left]\right.\\ +t\Theta (x-d)\left\}\right.,\\ {\Phi }_{L}(x)=\; {e}^{-ikx}\left\{\right.r\Theta (-x)+B\left[\right.\Theta (x)-\Theta (x-d)\left]\right.\left\}\right.,$$

(4)

where Θ(*x*) is the Heaviside step function. Here, *t* and *r* denote the single-photon transmission and reflection amplitudes in the regions of *x* > *d* and *x* < 0, respectively. We define *A* and *B* as the probability amplitudes for the right-going and left-going photons between the two coupling points (0 < *x* < *d*), respectively.

Substituting Eq. (4) into Eq. (3), we obtain

$$0= -i{v}_{g}(A-1)+g{e}^{i{\theta }_{1}}{u}_{e},\\ 0= -i{v}_{g}(t-A){e}^{i\phi }+g{e}^{i{\theta }_{2}}{u}_{e},\\ 0= -i{v}_{g}(r-B)+g{e}^{i{\theta }_{1}}{u}_{e},\\ 0= -i{v}_{g}B{e}^{-i\phi }+g{e}^{i{\theta }_{2}}{u}_{e},\\ 0= \frac{g}{2}{e}^{-i{\theta }_{1}}(A+B+r+1)+\frac{g}{2}{e}^{-i{\theta }_{2}}\\ \times (A{e}^{i\phi }+B{e}^{-i\phi }+t{e}^{i\phi })-\left(\Delta +i\frac{{\gamma }_{e}}{2}\right){u}_{e}$$

(5)

with Δ = *E* − *ω*_{e} being the detuning between the incident photon and the atomic transition. In the case of near resonant couplings with *E* ~ *ω*_{e} (i.e., ∣Δ/*ω*_{e}∣ ≪ 1), the transmission and reflection amplitudes can be obtained from solving Eq. (5) as

$$ t=\frac{\Delta +i\frac{{\gamma }_{e}}{2}-2\Gamma {e}^{i\theta }\sin \phi }{\Delta +i\frac{{\gamma }_{e}}{2}+2i\Gamma (1+{e}^{i\phi }\cos \theta )},$$

(6a)

$$r=\frac{\left[\right.2i\Gamma (1+{e}^{i\phi }\cos \theta )+2\Gamma {e}^{i\theta }\sin \phi \left]\right.\left[\right.1+{e}^{i(\theta +\phi )}\left]\right.}{\left[\right.\Delta +i\frac{{\gamma }_{e}}{2}+2i\Gamma (1+{e}^{i\phi }\cos \theta )\left]\right.\left[\right.1+{e}^{i(\theta -\phi )}\left]\right.},$$

(6b)

where *θ* = *θ*_{2} − *θ*_{1} is the phase difference between the two atom-waveguide coupling channels and Γ = *g*^{2}/*v*_{g} is the rate of the atomic emission into the waveguide. Compared with the setup of a two-level small atom coupled locally to a waveguide, such giant atom shows phase-dependent effective detuning and decay rate given by \(\Delta -2\Gamma \cos \theta \sin \phi \) and \({\gamma }_{e}/2+2\Gamma (1+\cos \theta \cos \phi )\), respectively^{23}. In fact, a left-incident (right-incident) photon can propagate from *x* = 0 to *x* = *d* (from *x* = *d* to *x* = 0) via two different paths: it can either keep on propagating along the waveguide, or be absorbed at *x* = 0 (*x* = *d*) and re-emitted at *x* = *d* (*x* = 0) by the atom, as shown in Fig. 1b. For the left-incident photon, the two paths yield phase accumulations *ϕ* and *θ*, respectively, which determine the phase-dependent interference effect jointly.

For the right-incident photon, the propagation process is equivalent to that of the left-incident one yet with exchanged coupling phases, i.e., *θ*_{1} ↔ *θ*_{2}. Therefore, the transmission and reflection amplitudes for the right-incident photon are expressed as

$${t}^{\prime}=\frac{\Delta +i\frac{{\gamma }_{e}}{2}-2\Gamma {e}^{-i\theta }\sin \phi }{\Delta +i\frac{{\gamma }_{e}}{2}+2i\Gamma (1+{e}^{i\phi }\cos \theta )},$$

(7a)

$${r}^{\prime}=\frac{\left[\right.2i\Gamma (1+{e}^{i\phi }\cos \theta )+2\Gamma {e}^{-i\theta }\sin \phi \left]\right.\left[\right.1+{e}^{-i(\theta -\phi )}\left]\right.}{\left[\right.\Delta +i\frac{{\gamma }_{e}}{2}+2i\Gamma (1+{e}^{i\phi }\cos \theta )\left]\right.\left[\right.1+{e}^{-i(\theta +\phi )}\left]\right.},$$

(7b)

which are also consistent with the results obtained by rewriting the wave functions for the right-incident photon. In addition, it is worth noting that the accumulated phase of a propagating photon can be written as *ϕ* = (*k*_{0} + *k*)*d* = *ϕ*_{0} + (*E* − *ω*_{0})*τ* = *ϕ*_{0} + *τ*Δ with *ϕ*_{0} = *k*_{0}*d* and *τ* = *d*/*v*_{g} by taking *ω*_{0} = *ω*_{e} for convenience. As usual, we have discarded *k*_{0} in *H*_{w} of Eq. (1) and Φ_{L,R} of Eq. (4) without changing other equations, and will take *ϕ*_{0} = *α* to replace *ϕ*_{0} = *α* + 2*m**π* with *m* being a positive integer and 0 ≤ *α* < 2*π* in the following discussions. Hence, it is viable to work in the Markovian regime with ∣*τ*Δ∣ ~ *τ*Γ ≪ 1 ~ *ϕ*_{0} when *d* is not too large, while in the non-Markovian regime with ∣*τ*Δ∣ ~ *τ*Γ ~ 1 ~ *ϕ*_{0} when *d* is large enough. For a transmon qubit considered here, *ω*_{e} and Γ are of the order of GHz and 0.1MHz^{25,27,28}, respectively, ensuring thus the validity of rotating wave approximation mentioned before Eqs. (1) and (2).

#### Reciprocal and nonreciprocal transmissions

We first focus on the Markovian regime of *τ* ≪ 1/(2Γ + *γ*_{e}/2), where *ϕ* ≈ *ϕ*_{0} according to the Taylor expansion because this substitution gives correct Lamb shift and modified emission rate in the Markovian limit^{24,29}. In Fig. 2, we plot the transmission rates *T*_{1→2} = ∣*t*∣^{2} and \({T}_{2\to 1}=| {t}^{\prime}{| }^{2}\) as functions of the detuning Δ and the phase difference *θ* with and without external atomic dissipations. Owing to the interference between two photon paths mentioned above, the scattering behavior changes periodically with *θ*. For *γ*_{e} = 0 as shown in Fig. 2a, b, the single-photon scattering is reciprocal, i.e., *T*_{1→2} ≡ *T*_{2→1}, although the time-reversal symmetry is broken due to the nontrivial phase difference *θ* arising from the interference.

This counterintuitive phenomenon can be explained by comparing Eqs. (6a) and (7a). On one hand, the transmission amplitudes *t* and \({t}^{\prime}\) share the same denominator that is an even function of *θ*. On the other hand, the numerators of *t* and \({t}^{\prime}\) in Eqs. (6a) and (7a) can be rewritten as

$$\Delta -2\Gamma \sin \phi \cos \theta +i\left(\frac{{\gamma }_{e}}{2}\, -2\Gamma \sin \phi \sin \theta \right),\\ \Delta -2\Gamma \sin \phi \cos \theta +i\left(\frac{{\gamma }_{e}}{2}\, +2\Gamma \sin \phi \sin \theta \right).$$

(8)

Equation (8) clearly shows that nonreciprocal single-photon transmissions \((| t{| }^{2}\ne | {t}^{\prime}{| }^{2})\) can be achieved only if a finite external atomic dissipation rate is taken into account (*γ*_{e} > 0). This can be observed by the transmission spectra shown in Fig. 2c, d.

When *γ*_{e} = 0, Fig. 3a depicts the transmission rates *T*_{1→2} and *T*_{2→1} versus the detuning Δ with various *θ*. For *θ* = *π*/2, we find *T*_{1→2} = *T*_{2→1} ≡ 1 over the whole range of the detuning, implying that reflections are prevented for both directions. For *θ* = *π*, however, the transmission spectrum exhibits the Lorentzian line shape with phase-dependent Lamb shift and linewidth (decay rate)^{23}. In both cases (*θ* = *π*/2, *π*), the transmissions are reciprocal, yet the atomic excitation probabilities are different as will be discussed below. When *γ*_{e} ≠ 0, as shown in Fig. 3b and (c), the scattering becomes nonreciprocal if *θ* = *π*/2; however, with *θ* = *π*, the scatterings are still reciprocal even in the presence of the external dissipation. We also can see from the three curves corresponding to *θ* = *ϕ*_{0} = *π*/2 a non-monotonic behavior of transmission rate *T*_{1→2} (in particular, *T*_{1→2} = 0 at Δ = 0 for *γ*_{e}/Γ = 4) with the increase of external decay rate *γ*_{e} while *T*_{2→1} remains unity independent of Δ and *γ*_{e}. This can be understood by resorting to Eqs. (6a) and (7a) restricted by *θ* = *ϕ*_{0} = *π*/2, from which it is easy to find that \({T}_{2\to 1}=| {t}^{\prime}{| }^{2}\equiv 1\) while *T*_{1→2} = ∣*t*∣^{2} exhibits a vanishing (nonzero) minimum for Δ = 0 (Δ ≠ 0) at the optimal \({\gamma }_{e}=2\sqrt{{\Delta }^{2}+4{\Gamma }^{2}}\) as determined by setting ∂*T*_{1→2}/∂*γ*_{e} = 0. Physically, this is an interference result of two propagating paths. In Eq. (6a), the direct path denoted by \(-2\Gamma {e}^{i\theta }\sin \phi \) is along the waveguide from *x* = 0 to *x* = *d*; the indirect path denoted by Δ + *i**γ*_{e}/2 is via an absorption at *x* = 0 and an emission at *x* = *d*. The two paths will contribute a perfect destructive interference leading to *t* = 0 in the case of \(\sin \phi =\pm \scriptstyle \sqrt{{\gamma }_{e}^{2}\,+\,4{\Delta }^{2}}/4\Gamma \) and tan*θ* = *γ*_{e}/2Δ indicating that *T*_{1→2} can also exhibit a vanishing minimum for Δ ≠ 0 if we have \(\scriptstyle \sqrt{{\gamma }_{e}^{2}\,+\,4{\Delta }^{2}}\le 4\Gamma \). A similar analysis on Eq. (7a) shows that a perfect destructive interference leading to \({t}^{\prime}=0\) will occur in the case of \(\sin \phi =\pm \sqrt{{\gamma }_{e}^{2}+4{\Delta }^{2}}/4\Gamma \) and tan*θ* = − *γ*_{e}/2Δ due to a reversed phase difference (*θ* → − *θ*) in the direct path. Hence, it is impossible to simultaneously have *T*_{1→2} = 0 and *T*_{2→1} = 0 for a nonzero external decay (*γ*_{e} ≠ 0).

The yellow dot-dashed, red dotted, and blue dashed lines in Fig. 3d depict the contrast ratio

$$I=\frac{{T}_{2\to 1}-{T}_{1\to 2}}{{T}_{2\to 1}+{T}_{1\to 2}}$$

(9)

versus the coupling phase difference *θ* with different atomic dissipation rates. The observed phase-dependent nonreciprocal transmission can be easily understood by resorting to Eqs. (6a) and (7a). For instance, it is viable to have *t* = 0 (\({t}^{\prime}=0\)) in the case of \(\Delta =2\Gamma \sin \phi \cos \theta \) and \({\gamma }_{e}=4\Gamma \sin \phi \sin \theta \) (\({\gamma }_{e}=-4\Gamma \sin \phi \sin \theta \)). Thus, perfectly nonreciprocal transmission denoted by *I* = ± 1 can be attained by tuning the accumulated phase *ϕ* and the coupling phase difference *θ* if we have ∣Δ∣≤2Γ and *γ*_{e}≤4Γ. Note, in particular, that the conditions for attaining *I* = ± 1 will reduce to *θ* = *π*/2 + 2*n**π* and \({\gamma }_{e}=\pm 4\Gamma \sin \phi \) as well as *θ* = − *π*/2 + 2*n**π* and \({\gamma }_{e}=\mp 4\Gamma \sin \phi \) for Δ = 0, with *n* being an arbitrary integer.

Furthermore, the underlying physics of the reciprocal and nonreciprocal scatterings can be understood via examining the atomic excitation by the single photon. To this end, we define the contrast ratio *D* of the atomic excitation probabilities for two opposite propagating directions as

$$D=\frac{{|{u}_{{e}_{2\to 1}}|}^{2}-{|{u}_{{e}_{1\to 2}}|}^{2}}{{|{u}_{{e}_{2\to 1}}|}^{2}+{|{u}_{{e}_{1\to 2}}|}^{2}}$$

(10)

with

$${u}_{{e}_{1\to 2}}=\frac{t-1}{-i\frac{g}{{v}_{g}}[{e}^{i{\theta }_{1}}+{e}^{i({\theta }_{2}+\phi )}]},\\ {u}_{{e}_{2\to 1}}=\frac{{t}^{\prime}-1}{-i\frac{g}{{v}_{g}}[{e}^{i{\theta }_{2}}+{e}^{i({\theta }_{1}+\phi )}]}.$$

(11)

According to Eqs. (6a) and (7a), parameters *t* − 1 and \({t}^{\prime}-1\) have the same denominator containing *γ*_{e} but different numerators without *γ*_{e}. Furthermore, because the denominator that contains *γ*_{e} is eliminated when calculating Eq. (10), the contrast ratio *D* is independent of dissipation rate *γ*_{e}. Note that the contrast ratio *D* can be used to capture the difference of the atomic excitation probabilities for opposite directions even if the eigenstate Eq. (2) is unnormalized. It is also not difficult to find from Eq. (6a) that *t* = 1 and hence \({u}_{{e}_{1\to 2}}=0\) in the case of \(1+\cos (\phi -\theta )=0\) while from Eq. (7a) that \({t}^{\prime}=1\) and hence \({u}_{{e}_{2\to 1}}=0\) in the case of \(1+\cos (\phi +\theta )=0\). Then, with *ϕ* ∓ *θ* = (2*n* + 1)*π*, we can attain *D* = ± 1 as a measure of the optimal difference of atomic excitation probabilities for oppositely propagating photons.

We plot in Fig. 3d the contrast ratio *D* (black solid line) as a function of the phase difference *θ* with *ϕ*_{0} = *π*/2. For *θ* = *π*/2, *D* = − 1 means that the atom can only be excited by the left-incident photon, and thus the atom-waveguide interaction becomes ideally equivalent chiral^{69,71}. In this case, the right-incident photon is guided transparently because it does not interact with the atom. While in the Markovian regime, the reflections are lacking for both directions under the ideally equivalent chiral coupling^{18,46}, this is in fact not true in the non-Markovian regime as will be discussed in the “Non-Markovian regime” subsection. For *θ* = 3*π*/2, *D* = 1 corresponds to the ideally equivalent chiral case where the atom can only be excited by the right-incident photon. For other cases of *D* = 0 and 0 < ∣*D*∣ < 1, the equivalent atom-waveguide couplings are nonchiral and nonideal chiral, respectively. In fact, the nonreciprocal scatterings arise from the different dissipations into the environment, i.e., the energy loss into the environment is proportional to the dissipation rate *γ*_{e} as well as the atomic population. In addition, we demonstrate in Fig. 3e that the contrast ratio *D* is also sensitive to the propagating phase. This provides an alternative way to tune the equivalent chirality of the atom-waveguide interaction and the reciprocal/nonreciprocal scattering on demand. Note that the equivalent chiral coupling found here is not in the standard form featuring different coupling strengths^{19,43}, but is a direct result of asymmetric interference effects, between the left- and right-incident photons.

The results above can also be interpreted from the aspect of Hermitian and non-Hermitian scattering centers^{72,73,74}. In our system with *γ*_{e} = 0 (*γ*_{e} ≠ 0), the giant atom can be regarded as a Hermitian (non-Hermitian) scattering center of the Aharonov-Bohm structure supporting two spatial interference paths. For the Hermitian case, the scattering remains reciprocal; however, when introducing an imaginary potential, e.g., the external atomic dissipation, the combination of the non-Hermiticity and the broken time-reversal symmetry gives rise to nonreciprocal scatterings^{72,73}. It is noted that, as discussed in the case in Fig. 3b, c (*θ* = *π*), not all non-Hermitian scattering centers can demonstrate nonreciprocal transmissions. The exceptions include, e.g., \({{{{{\mathcal{P}}}}}}\)-, \({{{{{\mathcal{T}}}}}}\)-, or \({{{{{\mathcal{PT}}}}}}\)-symmetric scattering centers^{72,74}. In our model, although the giant atom can exhibit chiral spontaneous emission corresponding to the time-reversal symmetry breaking if *θ* ≠ *n**π*^{69}, the scatterings are still reciprocal unless the additional non-Hermiticity (such as external dissipations) are introduced. Similar equivalent asymmetric couplings are also observed in the setup of emitters coupled to photonic lattice^{75}.

The nonreciprocal scatterings can also be observed as the external decay rate is included for a two-level chiral giant atom coupled to a waveguide with asymmetric coupling strengths *g*_{L} ≠ *g*_{R} (see Supplementary Note 1). The problem lies in that it is difficult or impossible to tune the degree of chirality *η* = *g*_{R}/*g*_{L} in the full (non-periodic) range of {0, *∞*}, e.g., for a special photonic crystal waveguide (PCW) with one side shifted half the lattice constant relative to the other side^{44}. In our model, however, it is much easier and more flexible to engineer the nonreciprocal scatterings in a standard PCW by tuning *ϕ* and *θ* in the full (periodic) range of 2{*n*, (*n* + 1)}*π* via the separation of two coupling points and the magnetic fluxes threading different Josephson junctions^{69,76,77}, respectively.

#### Non-Markovian regime

With nontrivial local coupling phases, as demonstrated above, the current giant-atom model (in the Markovian regime) is able to simulate a chiral atom-waveguide system. However, one important characteristic of the giant atom is the peculiar scattering behaviors arising in the non-Markovian regime, where the propagating phase accumulation *ϕ* = *ϕ*_{0} + *τ*Δ is sensitive to the detuning Δ due to the large enough *τ* that is comparable to or larger than the lifetime of the atom^{29}. Such a detuning-dependent phase will undoubtedly result in the non-Markovian features in the transmission and reflection spectra^{25,66}. Here we just consider our system in the non-Markovian regime and demonstrate the reflection with *ϕ*_{0} = *π*/2 and *θ* = *π*/2. Note that the reflection is totally prevented in the small-atom case with an ideal chiral coupling, which has been demonstrated in ref. ^{18}.

We plot in Fig. 4 the reflection rates *R* = ∣*r*∣^{2} for the left-incident photon in the Markovian and non-Markovian regimes. The yellow solid curve shows that the reflection in the Markovian regime disappears completely. Such a reflectionless behavior occurs in the case of *D* = ± 1, independent of the external atomic dissipation. However, in the non-Markovian regime, due to the Δ-dependent propagating phase *ϕ*, the reflection revives with multiple peaks aligning periodically in the frequency domain. In addition, the maximums of the reflection peaks decrease gradually with the increasing of *γ*_{e}. The underlying physics is that, in the phase accumulation *ϕ*, the non-Markovian contribution *τ*Δ cannot be ignored relative to *ϕ*_{0}; thus, *τ*Δ and *ϕ*_{0} determine the scattering behaviors jointly. The reflection disappears at some discrete Δ points satisfying *τ*Δ = *n**π*.

Moreover, we can find from Eq. (6b) the condition

$$ {(\Delta -2\Gamma \sin \phi \cos \theta )}^{2}+{\gamma }_{e}^{2}/4+2{\gamma }_{e}\Gamma (1+\cos \theta \cos \phi )\\ +4{\Gamma }^{2}{\sin }^{2}\phi {\sin }^{2}\theta =0,$$

(12)

for achieving the perfect reflection (*R* = 1). It is easy to see that this equation has no solutions for any choices of Δ/Γ, *θ*, and *ϕ* in the case of *γ*_{e} ≠ 0. Hence, it is viable to achieve the perfect reflection only by simultaneously requiring \(\sin \phi \sin \theta =0\) and \(\Delta =2\Gamma \sin \phi \cos \theta \) in the case of *γ*_{e} = 0, which have solutions (*i*)*ϕ* = *n**π* and Δ = 0; (*i**i*)*θ* = 2*n**π* and \(\Delta =2\Gamma \sin \phi \); (*i**i**i*)*θ* = (2*n* + 1)*π* and \(\Delta =-2\Gamma \sin \phi \). The underlying physics is that constructive interference occurs between two reflection paths contributed by coupling points *x* = 0 and *x* = *d*, respectively, which becomes perfect (imperfect) in the case of *γ*_{e} = 0(*γ*_{e} ≠ 0) due to a vanishing (nonzero) possibility for losing photons via the external decay. In the inset of Fig. 4, we plot *R* against Δ/Γ for *θ* = *π*/2 and *ϕ*_{0} = *π* as an example and find that *R* = 1 at the resonance point (Δ = 0) as predicted by solution (*i*).

### Three-level giant atom coupled to double waveguides

In this section, we consider two types of three-level giant atoms to explore the possibility of realizing nonreciprocal scatterings, as well as relevant single-photon router and circulator applications, without the additional non-Hermiticity (i.e., external atomic dissipation). As shown in Fig. 5a, we propose a ∇-type giant atom coupled to two single-mode waveguides via different transitions sharing the same ground state and driven by an external field on the third transition between two excited states. Figure 5b shows instead a Δ-type giant atom whose two ground states, interacting with an external field, are further coupled to the same excited state via different waveguide modes. The two proposals can be implemented with a three-level transmon coupled to two PCWs by considering that the energy bandgaps of different PCWs don’t overlap each other. In this case, it is viable to assume that only one transition driven by the external field exhibits a frequency falling outside the bandgaps of both PCWs, while the frequencies of other two transitions fall outside the bandgaps of different PCWs as shown in Fig. 5c. It is then justified that no atomic transitions will be coupled to both PCWs as long as the incident photons are not resonant with the transition driven by an external field. Experimentally, the upper and lower edges of a PCW’s bandgap may be controlled by adding defects^{78}, adjusting the waveguide widths^{79}, and varying widths of permalloy and cobalt stripes in crystals^{80}, while the transition frequencies of a transmon can be tuned via the external flux of a magnetic coil^{11}.

As shown in Fig. 5a, the atomic transition \(|g\rangle \leftrightarrow |{e}_{1}\rangle \) of frequency \({\omega }_{{e}_{1}}\) is coupled to waveguide *W*_{a} with complex coupling coefficient \({g}_{1}{e}^{i{\theta }_{1,2}}\) at two separated points *x* = 0 and *x* = *d*_{a}, respectively; the transition \(|g\rangle \leftrightarrow |{e}_{2}\rangle \) of \({\omega }_{{e}_{2}}\) is coupled to *W*_{b} with \({g}_{2}{e}^{i{\theta }_{3,4}}\) at *x* = 0 and *x* = *d*_{b}, respectively. The excited states \(|{e}_{1,2}\rangle \) are coupled to an external coherent field of Rabi frequency Ω and initial phase *α*. The atom is initialized in the ground state \(|g\rangle \). The Hamiltonian of the ∇-type giant atom coupled to two waveguides can be written as

$${H}^{\prime}=\, {H}_{w}^{\prime}+{H}_{a}^{\prime}+{H}_{I}^{\prime},\\ {H}_{w}^{\prime}= \int\nolimits_{-\infty }^{+\infty }dx\left[{a}_{L}^{{{\dagger}} }(x)\left({\omega }_{0}+i{v}_{g}\frac{\partial }{\partial x}\right){a}_{L}(x) \right. \\ + \left. {a}_{R}^{{{\dagger}} }(x)\left({\omega }_{0}-i{v}_{g}\frac{\partial }{\partial x}\right){a}_{R}(x) \right ]\\ +\int\nolimits_{-\infty }^{+\infty }dx \left [{b}_{L}^{{{\dagger}} }(x)\left({\omega }_{0}+i{v}_{g}\frac{\partial }{\partial x}\right){b}_{L}(x)\right. \\ +\left. {b}_{R}^{{{\dagger}} }(x)\left({\omega }_{0}-i{v}_{g}\frac{\partial }{\partial x}\right){b}_{R}(x)\right],\\ {H}_{a}^{\prime}= \left({\omega }_{{e}_{1}}-i\frac{{\gamma }_{{e}_{1}}}{2}\right)|{e}_{1}\rangle \langle {e}_{1}|+ \left({\omega }_{{e}_{2}}-i\frac{{\gamma }_{{e}_{2}}}{2} \right)|{e}_{2}\rangle \langle {e}_{2}| \\ +(\Omega {e}^{i\alpha }|{e}_{1}\rangle \langle {e}_{2}|+ \, {{\mbox{H.c.}}}), \\ {H}_{I}^{\prime}= \int \nolimits_{-\infty }^{+\infty }dx \{ \delta (x){g}_{1}{e}^{i{\theta }_{1}} \Big [{a}_{R}^{{{\dagger}} }(x)+{a}_{L}^{{{\dagger}} }(x) \Big ]|g \rangle \langle {e}_{1}| \\ +\delta (x-d){g}_{1}{e}^{i{\theta }_{2}} \Big [{a}_{R}^{{{\dagger}} }(x)+{a}_{L}^{{{\dagger}} }(x) \Big ]|g \rangle \langle {e}_{1}| \\ +\delta (x){g}_{2}{e}^{i{\theta }_{3}} \Big[{b}_{R}^{{{\dagger}} }(x)+{b}_{L}^{{{\dagger}} }(x)\Big]|g\rangle \langle {e}_{2}| \\ +\delta (x-d){g}_{2}{e}^{i{\theta }_{4}}[{b}_{R}^{{{\dagger}} }(x)+{b}_{L}^{{{\dagger}} }(x)]|g \rangle \langle {e}_{2}|+ \, {{\mbox{H.c.}}} \},$$

(13)

where *a*_{R,L}/*b*_{R,L} (\({a}_{R,L}^{{{\dagger}} }/{b}_{R,L}^{{{\dagger}} }\)) annihilates (creates) right-going and left-going photons in the waveguide *W*_{a}/*W*_{b}, respectively. In the single-excitation subspace, the eigenstate of the system can be expressed as

$$\left|{{\Psi }}\right\rangle= \int\nolimits_{-\infty }^{+\infty }dx\Big[{\Phi }_{aR}(x){a}_{R}^{{{\dagger}} }(x)+{\Phi }_{aL}(x){a}_{L}^{{{\dagger}} }(x)\\ +{\Phi }_{bR}(x){b}_{R}^{{{\dagger}} }(x)+{\Phi }_{bL}(x){b}_{L}^{{{\dagger}} }(x)\Big]|0,g\rangle \\ +{u}_{{e}_{1}}|0,{e}_{1}\rangle +{u}_{{e}_{2}}|0,{e}_{2}\rangle,$$

(14)

where Φ_{aR,aL}(Φ_{bR,bL}) are the probability amplitudes of creating the right-going and left-going photons in *W*_{a}(*W*_{b}), respectively.

Assuming that a photon with renormalized wave vector *k*_{a} is emanated from port 1 of *W*_{a}, the probability amplitudes can be written as

$${\Phi }_{aR}(x)=\, {e}^{i{k}_{a}x}\left\{\right.\Theta (-x)+M\left[\right.\Theta (x)-\Theta (x-{d}_{a})\left]\right.\\ +{s}_{1\to 2}\Theta (x-{d}_{a})\left\}\right.,\\ {\Phi }_{aL}(x)=\, {e}^{-i{k}_{a}x}\left\{\right.{s}_{1\to 1}\Theta (-x)+N\left[\right.\Theta (x)-\Theta (x-{d}_{a})\left]\right.\left\}\right.,\\ {\Phi }_{bR}(x)=\, {e}^{i{k}_{b}x}\left\{\right.Q\left[\right.\Theta (x)-\Theta (x-{d}_{b})\left]\right.+{s}_{1\to 4}\Theta (x-{d}_{b})\left\}\right.,\\ {\Phi }_{bL}(x)=\, {e}^{-i{k}_{b}x}\left\{\right.{s}_{1\to 3}\Theta (-x)+W\left[\right.\Theta (x)-\Theta (x-{d}_{b})\left]\right.\left\}\right.,$$

(15)

where the wave vectors \({k}_{a}=(E^{\prime} -{\omega }_{0})/{v}_{g}\) with the eigenenergy \(E^{\prime} \) in *W*_{a} and \({k}_{b}={k}_{a}+({\omega }_{{e}_{2}}-{\omega }_{{e}_{1}})/{v}_{g}\) in *W*_{b}. When excited to state \(|{e}_{1}\rangle \) by the incident photon from port 1, the atom can either re-emit a photon with the same frequency to *W*_{a} via decaying back to state \(|g\rangle \) directly, or radiate a photon with frequency \({\omega }_{{e}_{2}}\) to *W*_{b} via first transferring from state \(|{e}_{1}\rangle \) to state \(|{e}_{2}\rangle \) due to the external driving and then decaying to state \(|g\rangle \)^{67,81}. If a photon with renormalized wave vector *k*_{b} is sent from port 4 of *W*_{b}, the probability amplitudes can be written as

$${\Phi }_{aR}(x)=\, {e}^{i{k}_{a}x}\left\{\right.{s}_{4\to 2}\Theta (x-{d}_{a})+M^{\prime} \left[\right.\Theta (x)-\Theta (x-{d}_{a})\left]\right.\left\}\right.,\\ {\Phi }_{aL}(x)=\, {e}^{-i{k}_{a}x}\left\{\right.N^{\prime} \left[\right.\Theta (x)-\Theta (x-{d}_{a})\left]\right.+{s}_{4\to 1}\Theta (-x)\left\}\right.,\\ {\Phi }_{bR}(x)=\, {e}^{i{k}_{b}x}\left\{\right.Q^{\prime} \left[\right.\Theta (x)-\Theta (x-{d}_{b})\left]\right.+{s}_{4\to 4}\Theta (x-{d}_{b})\left\}\right.,\\ {\Phi }_{bL}(x)=\, {e}^{-i{k}_{b}x}\left\{\right.\Theta (x-{d}_{b})+W^{\prime} \left[\right.\Theta (x)-\Theta (x-{d}_{b})\left]\right.\\ +{s}_{4\to 3}\Theta (-x)\left\}\right..$$

(16)

By solving the stationary Schrödinger equation, one can obtain the scattering amplitudes of ∇-type giant atom for this case.

#### Nonreciprocal scattering

For simplicity, we start by supposing *ϕ*_{b} = *θ*_{3} = *θ*_{4} = 0, i.e., the transition \(|g\rangle \leftrightarrow |{e}_{2}\rangle \) is coupled to *W*_{b} at a single point. Then, the scattering probabilities can be calculated from *S*_{1→2} = ∣*s*_{1→2}∣^{2} and *S*_{1→3(4)} = ∣*s*_{1→3(4)}∣^{2} with the scattering amplitudes given by

$${s}_{1\to 2} =\frac{{\Delta }^{\prime}+i\frac{{\gamma }_{{e}_{1}}}{2}-{\Omega }^{2}f-2{\Gamma }_{1}{e}^{i\theta }\sin {\phi }_{a}}{{\Delta }^{\prime}+i\frac{{\gamma }_{{e}_{1}}}{2}-{\Omega }^{2}f+2i{\Gamma }_{1}(1+{e}^{i{\phi }_{a}}\cos \theta )},\\ {s}_{1\to 3(4)} =\frac{{g}_{2}\Omega {e}^{-i\alpha }({s}_{1\to 2}-1)}{{g}_{1}({\Delta }^{\prime}+i{\gamma }_{{e}_{2}}/2+i{\Gamma }_{2})\left[\right.{e}^{i{\theta }_{1}}+{e}^{i({\theta }_{2}-{\phi }_{a})}\left]\right.}$$

(17)

with the detuning \({\Delta }^{\prime}=E^{\prime} -{\omega }_{{e}_{1}}\) and the atomic emission rates \({\Gamma }_{1,2}={g}_{1,2}^{2}/{v}_{g}\). As discussed above, we make the substitution *ϕ*_{a} = (*k*_{a} + *k*_{0})*d*_{a} ≃ *ϕ*_{a0} in the Markovian regime. Likewise, one can also obtain *S*_{2→1} = ∣*s*_{2→1}∣^{2} and *S*_{2→3(4)} = ∣*s*_{2→3(4)}∣^{2} with

$${s}_{2\to 1} =\frac{{\Delta }^{\prime}+i\frac{{\gamma }_{{e}_{1}}}{2}-{\Omega }^{2}f-2{\Gamma }_{1}{e}^{-i\theta }\sin {\phi }_{a}}{{\Delta }^{\prime}+i\frac{{\gamma }_{{e}_{1}}}{2}-{\Omega }^{2}f+2i{\Gamma }_{1}(1+{e}^{i{\phi }_{a}}\cos \theta )},\\ {s}_{2\to 3(4)} =\frac{{g}_{2}\Omega {e}^{-i\alpha }({s}_{2\to 1}-1)}{{g}_{1}({\Delta }^{\prime}+i{\gamma }_{{e}_{2}}/2+i{\Gamma }_{2})\left[\right.{e}^{i{\theta }_{2}}+{e}^{i({\theta }_{1}-{\phi }_{a})}\left]\right.},$$

(18)

which are achieved via exchanging *θ*_{1} and *θ*_{2} in Eq. (17). It is found that *s*_{4→1} = *s*_{2→3} and thus *S*_{4→1} = ∣*s*_{4→1}∣^{2} = *S*_{2→3}.

Compared with Eq. (6a) and Eq. (7a) of the two-level giant atom, both *s*_{1→2} and *s*_{2→1} include an additional coupling term Ω^{2}*f* with

$$f=\frac{1-i\left(\frac{{\gamma }_{{e}_{2}}}{2}+{\Gamma }_{2}\right)}{{\Delta }^{^{\prime} 2}+{\left(\frac{{\gamma }_{{e}_{2}}}{2}+{\Gamma }_{2}\right)}^{2}},$$

(19)

which describes the photon transfer from *W*_{a} to *W*_{b}. It can be seen from Eqs. (17)–(19) that, in contrast to the two-level giant-atom scheme, the transmission between ports 1 and 2 in *W*_{a} is nonreciprocal even if the external dissipations are not considered. In fact, for \({\gamma }_{{e}_{1}}={\gamma }_{{e}_{2}}=0\), the imaginary part of *f* describing the decay of \(|{e}_{2}\rangle \to |g\rangle \) into *W*_{b} plays the role of an external dissipation for the transition \(|{e}_{1}\rangle \to |g\rangle \).

It is worth noting that the scattering probabilities of the ∇-type system are independent of the phase *α* of the external coherent field Ω in spite of the closed-loop atom-level structure. This is because the ∇-type atom cannot provide the inner two-path quantum interference. For instance, when excited to state \(|{e}_{1}\rangle \) by an incident photon from port 1, the atom may be pumped to state \(|{e}_{2}\rangle \) by the external field Ω and then return to state \(|g\rangle \) after emitting a photon into *W*_{b}, which is the only path for the photon transferring from *W*_{a} to *W*_{b}. This is radically different from the Δ-type structure as will be discussed in the “Comparison with the Δ-type scheme” subsection. In fact, the photon cannot be routed from *W*_{a} to *W*_{b} in the absence of the field Ω, implying that the ∇-type three-level giant atom reduces to a two-level one. This is also consistent with the fact that *S*_{1→3(4)} = *S*_{2→3(4)} = 0 when Ω = 0.

Figure 6 shows the single-photon scattering spectra as functions of the detuning \({\Delta }^{\prime}\) and the phase difference *θ*. As discussed above, it can be seen from Fig. 6a, b that the nonreciprocal scattering can still be realized in *W*_{a} (*S*_{1→2} ≠ *S*_{2→1}) with *W*_{b} playing the role of the external thermal reservoir in the two-level scheme as analyzed above. According to the conclusion in the “Reciprocal and nonreciprocal transmissions” subsection, for *θ* ≠ *n**π* and *ϕ*_{a0} = *π*/2 + 2*n**π*, the excitation probabilities \(| {u}_{{e}_{1}}{| }^{2}\) for two opposite directions are unequal, i.e., the interaction between the atom and *W*_{a} is equivalent chiral. Then, as shown in Fig. 6c, d, the nonreciprocal scattering between ports 1 and 4 can be led to by the equivalent chiral coupling, since the scattering probability *S*_{1→4} (*S*_{4→1}) is related to the coupling between the atomic transition \(|{e}_{1}\rangle \leftrightarrow |g\rangle \) and the right-going (left-going) mode in *W*_{a}. When *θ* = *π*/2 (3*π*/2), *S*_{1→4} (*S*_{4→1}) approaches 0.5 and *S*_{4→1} (*S*_{1→4}) falls to 0. This corresponds to the ideally equivalent chiral case where the atom is only coupled to the right-going (left-going) modes effectively in *W*_{a}. When *θ* = *π*, the scatterings between ports 1 and 4 are reciprocal, similar to the results of the equivalent nonchiral case in the “Reciprocal and nonreciprocal transmissions” subsection.

#### Chiral scattering

Next, we turn to study another kind of asymmetric scattering phenomenon proposed recently called “chiral scattering”. Specifically, the transmission from port 1 to port 4 and that from port 2 to port 3 are different. Quantitatively, the chiral scattering can be evaluated by the chirality defined as^{82}

$$C=\frac{{S}_{1\to 4}-{S}_{2\to 3}}{{S}_{1\to 4}+{S}_{2\to 3}}.$$

(20)

Figure 7a shows the chirality as a sinusoidal function of the phase difference *θ*. In view of this, chiral scatterings can be observed as long as *θ* ≠ *n**π*, where the chirality *C* ≠ 0 means *S*_{1→4} ≠ *S*_{2→3}. This can be further verified by the scattering spectra as shown in Fig. 7b, c. Note that *C* = 1 (*C* = − 1) corresponding to *θ* = *π*/2(*θ* = 3*π*/2), implies that only the scattering from port 2 (1) to port 3 (4) is prevented, as shown in Fig. 7b [Fig. 7c].

The underlying physics of the chiral scattering can also be attributed to the difference between the atomic excitation probabilities for two incident directions as discussed above. The excitation probabilities \(| {u}_{{e}_{1}}{| }^{2}\) by the photon incident from port 1 and port 2 can be unequal, and thus the atom is pumped from \(|{e}_{1}\rangle \) to \(|{e}_{2}\rangle \) with unequal probabilities. This leads to different probabilities of routing photons from *W*_{a} to *W*_{b}. Furthermore, as shown in Fig. 7, the chiral scattering scheme here shows the in-situ tunability that the scattering chirality can be controlled by tuning the phase difference *θ*.

#### Targeted router and circulator

In this subsection, we would like to demonstrate how to realize a single-photon targeted router and circulator based on the asymmetric scatterings above. Specifically, one can send a single photon deterministically from port 1 to one of the other three ports on demand. Note that the router and circulator can run with very high efficiency in such a non-loss system. Here we assume the transition \(|{e}_{2}\rangle \leftrightarrow |g\rangle \) coupled to *W*_{b} at two separated points, i.e., *ϕ*_{b} ≠ 0, as shown in Fig. 5a, and define \({\theta }^{\prime}={\theta }_{4}-{\theta }_{3}\).

The mechanism of the targeted router can be understood from Fig. 8a–c showing the scattering probabilities from port 1 to other ports versus the detuning \({\Delta }^{\prime}\). When turning off the external field (Ω ≡ 0), as shown in Fig. 8a, the incident photon from port 1 cannot be routed to *W*_{b}; particularly for *θ* = *π*/2, the photon is routed to port 2 totally. Next, we turn on the external field to enable photon routing to the desired port in *W*_{b} with high efficiency. When setting \({\theta }^{\prime}=\pi /2\), as shown in Fig. 8b, a photon resonant with the transition \(|g\rangle \leftrightarrow |{e}_{1}\rangle \) can be routed from port 1 to port 4 totally. Likewise, when setting \({\theta }^{\prime}=3\pi /2\) as shown in Fig. 8c, the resonant photon can be routed to port 3 totally. In addition, both the propagating phases *ϕ*_{a0} and *ϕ*_{b0} determine the output port of photons in *W*_{b}, which is a unique feature of the giant-atom model.

It is worth noting that one ∇ -type small atom with chiral asymmetric couplings (*g*_{1L} ≠ *g*_{1R} and *g*_{2L} ≠ *g*_{2R}) to two waveguides has also been explored to realize a deterministic routing^{19}, with subscripts “1,2” referring to the first and second waveguides while “*L*, *R*” to the left- and right-moving photons, respectively. In principle, it is viable to observe any desired routing results by tailoring two degrees of chirality *η*_{1} = *g*_{1R}/*g*_{1L} and *η*_{2} = *g*_{2R}/*g*_{2L}, e.g., via the amplitude of a magnetic field applied upon a quantum dot serving as the small atom^{44}. The problem is that *η*_{1} and *η*_{2} exhibit similar changing trends and are located at a single point, hence cannot be tuned independently. In our giant-atom model, however, it is much easier to observe different routing results by tailoring *θ* and \({\theta }^{\prime}\) individually, e.g., via the magnetic fluxes threading Josephson junctions at different coupling points. Such a selective tunability of coupling phase differences can also be used to realize a perfect circulator as shown below, which is impracticable yet by tailoring *η*_{1} and *η*_{2}. Our giant-atom model bears also another nontrivial feature – the non-Markovian retardation effect, which could result in multiple peaks in the reflection (and also transmission) spectra as shown in Fig. 4, and might enable the simultaneous manipulation of more than one incident photon with different frequencies.

More interestingly, the ∇-type giant atom is also a promising candidate of realizing a single-photon circulator. When turning on the external field and setting *θ* = *π*/2 and \({\theta }^{\prime}=3\pi /2\), the two waveguides are coupled to the atom with ideally equivalent chiral couplings in opposite manners, respectively. That is to say, the atom is only coupled to the left-incident photons in *W*_{a} yet to right-incident photons in *W*_{b}. Then, as shown in Fig. 8d, one has *S*_{2→1} = *S*_{3→4} ≡ 1 over the whole frequency range and *S*_{1→3} = *S*_{4→2} = 1 around the resonance. Consequently, for a resonant photon, directional scattering along the direction 1 → 3 → 4 → 2 → 1 can be realized suggesting a high-performance single-photon circulation scheme for quantum networks^{59,60}.

#### Comparison with the Δ-type scheme

Finally, we consider a Δ-type giant-atom scheme where the ∇ -type atom in Fig. 5a is replaced by a Δ-type one in Fig. 5b and compare the single-photon scatterings of these two schemes. The Δ-type structure is constructed with an external coherent filed *ϵ**e*^{iβ} which couples the two ground states \(|{g}_{1,2}\rangle \) of a Λ-type atom that has been broadly studied to demonstrate quantum interference phenomena, such as coherent population trapping^{83} and electromagnetically induced transparency^{84}.

For the Δ-type giant-atom system, the Hamiltonians of the atom and the atom-waveguide interaction become

$${H}_{a}^{\prime}= \left({\omega }_{{g}_{2}}-i\frac{{\gamma }_{{g}_{2}}}{2}\right)|{g}_{2}\rangle \langle {g}_{2}|+\left({\omega }_{e}-i\frac{{\gamma }_{e}}{2}\right)\left|e\right\rangle \left\langle e\right|\\ +(\varepsilon {e}^{i\beta }|{g}_{1}\rangle \langle {g}_{2}|+\,{{\mbox{H.c.}}}),\\ {H}_{I}^{\prime}= \int\nolimits_{-\infty }^{+\infty }dx\left\{\right.\delta (x){g}_{1}{e}^{i{\theta }_{1}}\Big[{a}_{R}^{{{\dagger}} }(x)+{a}_{L}^{{{\dagger}} }(x)\Big]|{g}_{1}\rangle \left\langle e\right|\\ +\delta (x-d){g}_{1}{e}^{i{\theta }_{2}}\Big[{a}_{R}^{{{\dagger}} }(x)+{a}_{L}^{{{\dagger}} }(x)\Big]|{g}_{1}\rangle \left\langle e\right|\\ +\delta (x){g}_{2}{e}^{i{\theta }_{3}}\Big[{b}_{R}^{{{\dagger}} }(x)+{b}_{L}^{{{\dagger}} }(x)\Big]|{g}_{2}\rangle \left\langle e\right|\\ +\delta (x-d){g}_{2}{e}^{i{\theta }_{4}}\Big[{b}_{R}^{{{\dagger}} }(x)+{b}_{L}^{{{\dagger}} }(x)\Big]|{g}_{2}\rangle \langle e |+\,{{\mbox{H.c.}}}\,\big\}\\ $$

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The single-excitation eigenstate of the system takes the form

$$\left|{{\Psi }}\right\rangle= \int\nolimits_{-\infty }^{+\infty }dx \Big \{\Big[{\Phi }_{aR}(x){a}_{R}^{{{\dagger}} }(x)+{\Phi }_{aL}(x){a}_{L}^{{{\dagger}} }(x)\Big]|0,{g}_{1} \rangle \\ +\Big[{\Phi }_{bR}(x){b}_{R}^{{{\dagger}} }(x)+{\Phi }_{bL}(x){b}_{L}^{{{\dagger}} }(x) \Big ]|0,{g}_{2}\rangle\Big \}+{u}_{e}|0,e \rangle .$$

(22)

With the same procedure above (see Supplementary Note 2), one can obtain the scattering probabilities in this case.

Setting the atom in the ground state \(|{g}_{1}\rangle \) initially, we plot in Fig. 9 the scattering spectra of \({\tilde{S}}_{1\to 4}\) and \({\tilde{S}}_{4\to 1}\). It is worth noting that, even in the absence of the local coupling phases, i.e., \(\theta ={\theta }^{\prime}=0\), the nonreciprocal scatterings still exist. This is obviously distinct from the ∇ -type case. The nonreciprocity of the ∇ -type case stems from the equivalent chiral couplings owing to the nontrivial coupling phase difference, and is independent of the phase of the external field. For the Δ-type scheme, however, the nonreciprocity arises from the typical which-way quantum interference, i.e., the interference between the two transition paths \(|{g}_{1}\rangle \to |{g}_{2}\rangle \) and \(|{g}_{1}\rangle \to |e\rangle \to |{g}_{2}\rangle \). In this case, the optical responses are typically sensitive to the phase of the external field encoded in the closed-loop level structure^{85}. However, the main drawback to the Δ-type scheme is that one cannot switch on/off the photon transfer between the two waveguides by tuning the external field solely.

Our ∇ -type and Δ-type giant atoms will reduce to the corresponding small atoms if we set *ϕ*_{a} = *ϕ*_{b} = 0 (i.e., *d*_{a} = *d*_{b} = 0). In this case, chiral scattering disappears for both ∇-type and Δ-type small atoms due to the intrinsic symmetry of atom-waveguide interactions. On the other hand, nonreciprocal scattering still can be observed for the Δ-type small atom due to the asymmetric interference (for left- and right-incident photons) between two transitions sharing the same starting and ending states, but will not occur for the ∇-type small atom in the presence of only one accompanied transition and thus absence of any interference effects (see Supplementary Note 3).