Nonreciprocal and chiral single-photon scattering for giant atoms

Quantum optics with giant atoms has provided a new paradigm to study photon scatterings. In this work, we investigate the nontrivial single-photon scattering properties of giant atoms being an effective platform to realize nonreciprocal and chiral quantum optics. For two-level giant atoms, we identify the condition for nonreciprocal transmission: the external atomic dissipation is further required other than the breaking of time-reversal symmetry by local coupling phases. Especially, in the non-Markovian regime, unconventional revival peaks periodically appear in the reflection spectrum. To explore more interesting scattering behaviors, we extend the two-level giant-atom system to {\Delta}-type and {\nabla}-type three-level giant atoms coupled to double waveguides with different physical mechanisms to realize nonreciprocal and chiral scatterings. Our proposed giant-atom structures have potential applications of high-efficiency targeted routers that can transport single photons to any desired port deterministically and circulators that can transport single photons between four ports in a cyclic way.

In general, the atom can be viewed as a point when coupled with the waveguide due to its negligible size compared to the wavelength of waveguide modes. Nevertheless, in a recent experiment, a superconducting transmon qubit was designed to interact with surface acoustic waves via multiple coupling points whose separation distances can be much larger than the wavelength of the waves 21 . Instead, a generalized theory called "giant atom" has been developed to describe such situations 22 . Since the first theoretical study in 2014 23 , the giant-atom scheme has been broadly investigated with superconducting qubits [24][25][26][27][28] , coupled waveguide arrays 29 , and cold atoms 30 . With such nonlocal coupling schemes, a series of tempting quantum phenomena have been demonstrated, including frequency-dependent relaxation rate and Lamb shift 23,27,31 , non-exponential atomic decay 24,25 , decoherence-free interatomic interaction 27,32,33 , exotic bound states 26,34 , modified topological effects 35 , and quantum Zeno and quantum anti-Zeno effects 36 . Giant atoms have emerged as a new paradigm in quantum optics and require more comprehensive understanding in physics.
In this paper, we investigate how external atomic dissipations outside the waveguide and local coupling phases affect the single-photon scattering properties of a two-level giant atom with two atom-waveguide coupling points. By taking into account the phase difference between two coupling points, we find that the giant atom behaves like a chiral small atom in the Markovian regime but exhibits peculiar giant-atom effects in the non-Markovian regime. We physically demonstrate that the breaking of time-reversal symmetry by local coupling phases is not sufficient for realizing nonreciprocal photon scatterings. In fact, in the absence of the external atomic dissipation, the scatterings are always reciprocal even if the atomic spontaneous emission becomes chiral 68,69 . In order to realize asymmetric scattering for a giant atom without external dissipation, we propose a ∇-type giant atom coupled to two waveguides. In such way, we realize the nonreciprocal and chiral scatterings with single ∇-type atom. Targeted routing and circulation schemes can also be realized via such scatterings with proper phases. Finally, we consider a Δ-type giant atom and compare its properties with that of ∇-type one. We reveal that, the nonreciprocial scatterings stem from the quantum interference effect in the closed-loop atom-level structure for the Δ-type giant atom, but from the nontrivial coupling phase difference for the ∇-type giant atom.

Results and discussion
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 ge iθ 1 and ge iθ 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 Josephsonjunction 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) j i e h j; 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 y R;L ) are the bosonic annihilation (creation) operators of the right-going and left-going photons in the waveguide, respectively; Fig. 1 Schematic representation of the model and the photon paths. a A two-level giant atom coupled to a waveguide at x = 0 and x = d, respectively, with individual local coupling phases θ 1,2 . b Two paths of a single photon propagating from port 1 to port 2 (left) or from port 2 to port 1 (right). 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 jgi and the excited state e j i; 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 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; j0; gi denotes the vacuum state of the system. The probability amplitudes can be determined by solving the eigenequation Assuming that a photon with the renormalized wave vector k satisfying the linear dispersion relation E = ω 0 + kv g is incident from port 1 of the waveguide, the wave functions Φ R,L (x) can be written as 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 ¼ À iv g ðA À 1Þ þ ge iθ 1 u e ; 0 ¼ À iv g ðt À AÞe iϕ þ ge iθ 2 u e ; 0 ¼ À iv g ðr À BÞ þ ge iθ 1 u e ; 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 where θ = θ 2 − θ 1 is the phase difference between the two atomwaveguide 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 Δ À 2Γ cos θ sin ϕ and γ e =2 þ 2Γð1 þ cos θ cos ϕÞ, 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 leftincident 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 ð7aÞ 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 + τΔ 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 + 2mπ with m being a positive integer and 0 ≤ α < 2π in the following discussions. Hence, it is viable to work in the Markovian regime with |τΔ|~τΓ ≪ 1~ϕ 0 when d is not too large, while in the non-Markovian regime with |τΔ|~τΓ~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!1 ¼ jt 0 j 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 0 share the same denominator that is an even function of θ. On the other hand, the numerators of t and t 0 in Eqs. (6a) and (7a) can be rewritten as Equation (8) clearly shows that nonreciprocal single-photon transmissions ðjtj 2 ≠jt 0 j 2 Þ 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!1 ¼ jt 0 j 2 1 while T 1→2 = |t| 2 exhibits a vanishing (nonzero) minimum for Δ = 0 (Δ ≠ 0) at the optimal γ e ¼ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Δ 2 þ 4Γ 2 p 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 À2Γe iθ sin ϕ is along the waveguide from x = 0 to x = d; the indirect path denoted by Δ + iγ 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 Fig. 3 The influence of the coupling phase difference and accumulated phase on transmission rates and contrast ratios. Transmission rates T 1→2 and T 2→1 versus the detuning Δ with accumulated phase ϕ 0 = π/2 and different atomic dissipation rates a γ e /Γ = 0; b γ e /Γ = 4; c γ e /Γ = 20, where Γ is the atomic emission rate into the waveguide. d Contrast ratios I and D versus the coupling phase difference θ with ϕ 0 = π/2 and Δ = 0. The yellow dot-dashed, red dotted, and blue dashed lines are I with γ e /Γ = 0, γ e /Γ = 4, and γ e /Γ = 20, respectively, and the black solid one represents D independent of γ e . e Contrast ratio D versus θ and ϕ 0 with Δ = 0. Other parameters: photon propagation time τΓ = 0.01.
A similar analysis on Eq. (7a) shows that a perfect destructive interference leading to t 0 ¼ 0 will occur in the case of 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 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 0 ¼ 0) in the case of Δ ¼ 2Γ sin ϕ cos θ and γ e ¼ 4Γ sin ϕ sin θ (γ e ¼ À4Γ sin ϕ sin θ). 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 + 2nπ and γ e ¼ ± 4Γ sin ϕ as well as θ = − π/2 + 2nπ and γ e ¼ Ç4Γ sin ϕ 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 with ; According to Eqs. (6a) and (7a), parameters t − 1 and t 0 À 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!2 ¼ 0 in the case of 1 þ cosðϕ À θÞ ¼ 0 while from Eq. (7a) that t 0 ¼ 1 and hence u e 2!1 ¼ 0 in the case of 1 þ cosðϕ þ θÞ ¼ 0. Then, with ϕ ∓ θ = (2n + 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 leftincident photon, and thus the atom-waveguide interaction becomes ideally equivalent chiral 69,71 . In this case, the rightincident 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 = 3π/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 atomwaveguide 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 72-74 . 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., P-, T -, or 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 ≠ nπ 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 . Reflection rate R versus the detuning Δ with the coupling phase difference θ = π/2 and the accumulated phase ϕ 0 = π/2. The inset depicts R versus Δ with θ = π/2 and ϕ 0 = π in the case of photon propagation time τΓ = 1 and atomic dissipation rate γ e /Γ = 0, where Γ is the atomic emission rate into the waveguide.
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 + τΔ 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 leftincident 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 τΔ cannot be ignored relative to ϕ 0 ; thus, τΔ and ϕ 0 determine the scattering behaviors jointly. The reflection disappears at some discrete Δ points satisfying τΔ = nπ.
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 jgi $ je 1 i of frequency ω e 1 is coupled to waveguide W a with complex coupling coefficient g 1 e iθ 1;2 at two separated points x = 0 and x = d a , respectively; the transition jgi $ je 2 i of ω e 2 is coupled to W b with g 2 e iθ 3;4 at x = 0 and x = d b , respectively. The excited states je 1;2 i are coupled to an external coherent field of Rabi frequency Ω and initial phase α. The atom is initialized in the ground state jgi. The and e 2 are coupled to a coherent field Ωe iα . b Δ-type atom: W a (W b ) is coupled to g 1 $ e j i ( g 2 $ e j i) at two separated points. Two ground states g 1 and g 2 are coupled to a coherent field εe iβ . c Band structures of two PCWs used to implement our proposals.
Hamiltonian of the ∇-type giant atom coupled to two waveguides can be written as where a R,L /b R,L (a y R;L =b y R;L ) 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 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 Φ aR ðxÞ ¼ e ik a x fΘðÀxÞ þ M½ΘðxÞ À Θðx À d a Þ þ s 1!2 Θðx À d a Þg; Φ aL ðxÞ ¼ e Àik a x fs 1!1 ΘðÀxÞ þ N½ΘðxÞ À Θðx À d a Þg; where the wave vectors k a ¼ ðE 0 À ω 0 Þ=v g with the eigenenergy E 0 in W a and k b ¼ k a þ ðω e 2 À ω e 1 Þ=v g in W b . When excited to state je 1 i by the incident photon from port 1, the atom can either reemit a photon with the same frequency to W a via decaying back to state jgi directly, or radiate a photon with frequency ω e 2 to W b via first transferring from state je 1 i to state je 2 i due to the external driving and then decaying to state jgi 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 Φ aR ðxÞ ¼ e ik a x fs 4!2 Θðx À d a Þ þ M 0 ½ΘðxÞ À Θðx À d a Þg; Φ aL ðxÞ ¼ e Àik a x fN 0 ½ΘðxÞ À Θðx À d a Þ þ s 4!1 ΘðÀxÞg; By solving the stationary Schrödinger equation, one can obtain the scattering amplitudes of ∇-type giant atom for this case.
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 je 1 i by an incident photon from port 1, the atom may be pumped to state je 2 i by the external field Ω and then return to state jgi 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 Δ 0 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 ≠ nπ and ϕ a0 = π/2 + 2nπ, the excitation probabilities ju e 1 j 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 je 1 i $ jgi and the right-going (left-going) mode in W a . When θ = π/2 (3π/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 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 ≠ nπ, 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 = 3π/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 ju e 1 j 2 by the photon incident from port 1 and port 2 can be unequal, and thus the atom is pumped from je 1 i to je 2 i 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 insitu tunability that the scattering chirality can be controlled by tuning the phase difference θ.   Fig. 8 Targeted routing and circulating scheme with a three-level ∇-type giant-atom system. Scattering probabilities versus the detuning Δ 0 with different Rabi frequencies of the external coherent field Ω and coupling phase differences θ and θ 0 a Ω = 0, θ = π/2; b Ω = 2Γ 1 , θ = π/2, θ 0 ¼ π=2; c and d Ω = 2Γ 1 , θ = π/2, θ 0 ¼ 3π=2. Other parameters: atomic dissipation rates γ e 1 ¼ γ e 2 ¼ 0, the ratio of two atomic emission rates into different waveguides Γ 2 /Γ 1 = 1, and accumulated phases of the photon traveling in waveguides W a and W b ϕ a0 = ϕ b0 = π/2.
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 je 2 i $ jgi coupled to W b at two separated points, i.e., ϕ b ≠ 0, as shown in Fig. 5a, and define 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 Δ 0 . 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 θ 0 ¼ π=2, as shown in Fig. 8b, a photon resonant with the transition jgi $ je 1 i can be routed from port 1 to port 4 totally. Likewise, when setting θ 0 ¼ 3π=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 θ 0 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 θ 0 ¼ 3π=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 highperformance 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 ϵe iβ which couples the two ground states jg 1;2 i 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 The single-excitation eigenstate of the system takes the form 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 jg 1 i initially, we plot in Fig. 9 the scattering spectra ofS 1!4 andS 4!1 . It is worth noting that, even in the absence of the local coupling phases, i.e., θ ¼ θ 0 ¼ 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 jg 1 i ! jg 2 i and jg 1 i ! jei ! jg 2 i. In this case, the optical responses are typically sensitive to the phase of the external field encoded in the closedloop 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.  9 Nonreciprocal scattering behaviors in a three-level Δ-type giantatom system. Scattering probabilities versus the detuningΔ 0 with different phases of the external coherent filed β. Other parameters: atomic dissipation rates γ e 1 ¼ γ e 2 ¼ 0, the ratio of two atomic emission rates into different waveguides Γ 2 /Γ 1 = 1, Rabi frequency of the external coherent field ε = 30Γ 1 , and accumulated phases of the photon traveling in waveguides W a and W b ϕ a0 = ϕ b0 = π/2.
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).

Conclusions
In summary, we have investigated step-by-step the conditions of single-photon nonreciprocal and chiral scatterings in the twolevel and three-level giant-atom structures with tunable local phase on each atom-waveguide coupling. We found that the atomic excitation in the two-level giant-atom structure depends on the propagation direction of waveguide modes and can be tuned by the nontrivial coupling phase difference. In such scenario, our two-level giant atom in the Markovian regime is equivalent to a two-level small atom chirally coupled to the waveguide mode. However, it is worth noting that the realization of nonreciprocal scatterings requires the combination of the timereversal symmetry breaking induced by the local coupling phases and the non-Hermiticity induced by the external atomic dissipation due to the surrounding non-waveguide modes. Moreover, in the non-Markovian regime, the reflection spectra exhibit peculiar non-Markovian features with multiple reflection peaks that are absent in the chiral small-atom case.
For exploring more interesting asymmetric scattering properties and applications with such giant-atom structures, we have extended the two-level structure to the three-level ∇-type and Δ-type ones coupled to two waveguides via different atomic transitions. We found that, for the atomic transition coupled to one waveguide, the transition coupled to the other waveguide can serve as the external dissipation channel. Such three-level giantatom structures coupled to double waveguides enable the nonreciprocal and chiral scatterings without external dissipations. Based on this mechanism, the high-efficiency single-photon targeted router and circulator can be implemented. Finally, we explained the different physical mechanisms that lead to the nonreciprocal and chiral scatterings for the two phase-sensitive closed-loop three-level giant-atom structures. We believe that our results have promising applications in designing effective and efficient single-photon optical elements for quantum network engineering and optical communications.

Methods
In this theoretical work, the methods used are solving the stationary Schrödinger equation with the Hamiltonian and single-excitation eigenstate (as described in the main text [Eq. (3)] and Supplementary Material).

Data availability
All data are available in the main text or in the supplementary materials.

Code availability
The code used to produce the figures in this article is available from the corresponding author upon request.
Received: 1 March 2022; Accepted: 4 August 2022; Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/ licenses/by/4.0/.