Firstly, the motion equations of the giant atom limited in the single-photon subspace can be studied by using the similar method introduced in Ref [29]. The total state of a Λ-type giant atom and the microwave field in the waveguide can be described by | Φ t 〉 = C a t | a , 0 〉 + ∑ α = 1,2 ∫ d ω k C b α k t | b , 1 k 〉 + ∑ α = 1,2 ∫ d ω q C c α q t | c , 1 q 〉 , (5) where |0⟩ represents the vacuum state of the microwave filed in the SQUID-chain waveguide. The integral terms describe the state of a single photon propagating in the waveguide towards right α = 1 or left α = 2, with the giant atom in the ground state |b⟩ (or substable state |c⟩). Using the Schrödinger equation iℏ∂/∂t|Φ(t)⟩ = H|Φ(t)⟩, we obtain the differential equations for coefficients: C _a (t) being the probability amplitude of the giant atom in exicted state |a⟩, C _b1(2)k and C _c1(2)q denoting the photon spectrums in the transmission line with the giant atom staying in the ground state |b⟩ and the substable state |c⟩, respectively. The time evolution of C _a (t) is then turned to d C a t d t = − i ω a C a t − γ 1 C a t +     Θ t − T e − i ω b T C a t − T − γ 2 C a t + Θ t − T e − i ω c T C a t − T − i γ 1 v g 2 e − i ω b t C b 1 − L / 2 − v g t , 0 + C b 1 L / 2 − v g t , 0 − i γ 1 v g 2 e − i ω b t C b 2 − L / 2 + v g t , 0 + C b 2 L / 2 + v g t , 0 − i γ 2 v g 2 e − i ω c t C c 1 − L / 2 − v g t , 0 + C c 1 L / 2 − v g t , 0 − i γ 2 v g 2 e − i ω c t C c 2 − L / 2 + v g t , 0 + C c 2 L / 2 + v g t , 0 . (6)

The first term denotes free evolution. The following two terms show the atom relaxing process with γ _1,2 and feedback effect with C _a (t − T), where T = L/v _g is the delay time. Here, Θ(•) is the Heaviside step function showing the time-delay feedback between the coupling points. We introduced the notion C b 1 ( 2 ) ( x , t ) = 1 2 π v g ∫ d ω k e ± i ω k x / v g C b 1 ( 2 ) k ( t ) for describing photon distribution and time evolution. Then, the remaining four terms imply that the incident photon in both directions can excite the giant atom.

C _a (t) can be analytically solved and finally has the spectrum form expansion (details in Appendix A) C a t = 1 2 π i ∫ ϵ − i ∞ ϵ + i ∞ C a s e s t d s = 1 2 π i ∫ d s C a 0 e s t s + i ω a + γ 1 D 1 + γ 2 D 2 − i γ 1 4 π ∫ d s C b s 0 e s T / 2 + e − s T / 2 e s t e − i ω b t s + i ω a + γ 1 D 1 + γ 2 D 2 − i γ 2 4 π ∫ d s C c s 0 e s T / 2 + e − s T / 2 e s t e − i ω c t s + i ω a + γ 1 D 1 + γ 2 D 2 , (7) where D 1 ( 2 ) = 1 + e − i ω b ( c ) T e − s T is the additional phase of the decay rate γ ₁₍₂₎ due to the time-delay feedback effect. Here, we introduce the notations C _bs (0) = C _b1s (0) + C _b2s (0) and C _cs (0) = C _c1s (0) + C _c1s (0), denoting the initial photon of frequency ω _s in the waveguide incident from both direction.

To study scattering properties of the giant atom, we set a right-propagating photon incident into the waveguide, whose frequency is centered at ω _k (C _b1k 0 ≠ 0), with the giant atom initially being in the ground state. The dynamical evolution equation of the giant atom is then derived from Eq. 7, C a t = − i γ 1 4 π ∫ d s C b 1 s 0 e s T / 2 + e − s T / 2 e s t e − i ω b t s + i ω a + γ 1 D 1 + γ 2 D 2 . (8)

In the long-time limit, the emission spectrum of outgoing photons can be obtained (details in Appendix B): C b 1 k ∞ = e − i ω k t C b 1 k 0 1 − γ 1 1 + cos ω k T i ω a − ω k + γ 1 1 + e i ω k T + γ 2 1 + e i ω q T , (9) C b 2 k ∞ = e − i ω k t C b 1 k 0 − γ 1 1 + cos ω k T i ω a − ω k + γ 1 1 + e i ω k T + γ 2 1 + e i ω q T , (10) C c 1 q ∞ = e − i ω k t C b 1 k 0 − 2 γ 1 γ 2 cos ω q T / 2 cos ω k T / 2 i ω a − ω k + γ 1 1 + e i ω k T + γ 2 1 + e i ω q T , (11) where the ground state energy is set to zero ω _b = 0 with variable substitution ω _k = ω _q + ω _c . Then, we obtain the transmission rate T _k , the reflection rate R _k , and the forward conversion rate T _q , T k = | C b 1 k ∞ C b 1 k 0 | 2 = ω k − ω a − γ 1 ⁡ sin ω k T − γ 2 ⁡ sin ω q T 2 + γ 2 2 1 + cos ω q T 2 ω k − ω a − γ 1 ⁡ sin ω k T − γ 2 ⁡ sin ω q T 2 + γ 1 1 + cos ω k T + γ 2 1 + cos ω q T 2 , (12) R k = | C b 2 k ∞ C b 1 k 0 | 2 = γ 1 2 1 + cos ω k T 2 ω k − ω a − γ 1 ⁡ sin ω k T − γ 2 ⁡ sin ω q T 2 + γ 1 1 + cos ω k T + γ 2 1 + cos ω q T 2 , (13) T q = | C c 1 q ∞ C b 1 k 0 | 2 = 4 γ 1 γ 2 ⁡ cos 2 ω k T / 2 cos 2 ω q T / 2 ω k − ω a − γ 1 ⁡ sin ω k T − γ 2 ⁡ sin ω q T 2 + γ 1 1 + cos ω k T + γ 2 1 + cos ω q T 2 . (14)

In contrast to small atoms, Eqs 12–14 imply that the giant atom has the corrected transition frequency Ω _a = ω _a + γ ₁ sin(ω _k T) + γ ₂ sin(ω _q T) and decay rate Γ₁₍₂₎ = γ ₁₍₂₎(1 + cos(ω _k(q) T)) [43]. From Eq. 13, the conditions for total reflection, R _k = 1, can be obtained directly ω a − ω k + γ 1 ⁡ sin ω k T + γ 2 ⁡ sin ω q T = 0 , γ 2 1 + cos ω q T = 0 . (15)

There are two ways to make the second condition satisfied. One is γ ₂ = 0, which means the transition path |a⟩ → |c⟩ directly turned off. The other one is ω _q T = 2 (n + 1)π, which infers that the decay channel of |a⟩ → |c⟩ is suppressed by the destructive interference between coupling points [35]. Either of these two cases can cause that the Λ configuration atom could be reduced to a simple two-level atom. However, the first solution is trivial and will not be considered in our case. Then the two conditions in Eq. 15 can be rewritten as ω _k − ω _a = γ ₁ sin (ω _c T) and ω _q T = 2 (n + 1)π. Here, we introduce the detuning, Δ = ω _k − ω _a , of the incident photon from the |a⟩ − |b⟩ transition. The total reflection conditions are written as Δ = γ ₁ sin (ω _c T) and Δ = ω _c − ω _a + 2 (n + 1)π/T.

While the total transmission, T _k = 1 and R _k = T _q = 0, happens in the case of ω _k T = (2n + 1)π, no matter with ω _c T. Because of the nondirectional decay process of the excited atom, the forward nonelastic scattering photon strength is equal to the backward one. Thus, it is reasonable to set the frequency conversion rate to be twice the forward conversion rate, T _c = 2T _q . The fundamental source affecting the conversion rate is the ratio of the decay rates, i.e., γ ₁/γ ₂. For simplicity, we set γ ₁/γ ₂ = 1 and use γ regardless of the subscript in the context.

In Figure 2, we plot the scattering rates as the function of the scaled detuning Δ/γ and the scaled substable level frequency ω _c /γ, for ω _a T = 2000π, where ω _a /γ = 10,000 and T = 0.2π/γ. Figure 2A,B depict the reflection rates for the Markovian (γT = 0.2π) and non-Markovian (γT = 3π) regimes, respectively. Compared with Figure 2A,B shows two different features. First, there are additional small peaks at two sides of the primary maximum for the certain ω _c . Second, the frequency band of the total reflection is broadened. The giant atom can provide a wide spectrum window for the strong reflection in the non-Markovian regime. Figure 2C,D represent the single-photon transmittance and the frequency conversion rate, respectively, of the giant atom in the Markovian regime.

Based on the above investigation, it is clear that the single-photon scattering effect of the Λ-type giant atom is intensely affected by the two crucial phases, ω _k T and ω _c T. These phases are dependent on the atom’s energy level structure and the coupling points separation distance. Therefore, it is intuitive to control the spatial length between the coupling points for coherently tuning the phase. However, the spatial length is fixed once the circuit sample is fabricated. Here, we introduce the SQUID-chain transmission line, in which the photon’s phase velocity is flux-dependent. Although the spatial distance of two points in the waveguide is determined, changes in the photon’s phase velocity will change the cost time T for photons traveling between two fixed points, equivalent to the effective length and accumulated phase. Consequently, the effect will affect the dynamical evolution of this Λ-type giant atom as well as the scattering properties.

In addition to infinite length condition L → ∞, if the superconducting transmission line also satisfies following conditions [32]: d 0 ≪ λ k ≪ L , k ≪ 1 d 0 C g C J , (16) the dispersion relation Eq. 2 can be reduced to ω k 0 ≃ k d 0 L J C g = k v J , (17) i.e., the phase velocity is d 0 / L J C g . The effective inductance of the SQUID is controlled by the magnetic field through it. Then the photon’s phase velocity v can be modulated by controlling the field through the SQUID loops v = d 0 L J C g = d 0 cos π Φ / Φ 0 L 0 C g . (18)

The change in phase velocity leads to the change of the traveling time of photons propagating between two points, so does the accumulating phase. We set the separation distance of the two coupling points as d = nd ₀, i.e., n SQUIDs constituting the waveguide between the coupling points. Then, the traveling time of a photon is T = d v = n L 0 C g cos π Φ / Φ 0 . (19)

By modulating the photon’s traveling time between two coupling points, we can explore some interesting applications that will be discussed in Section 4.

The linear dispersion relation (17) of the SQUID-chain waveguide is valid over a wide frequency domain ω _k < 10 GHz, when the circuit parameters are taken as [32]; [41,40]: L ₀ = 0.2 nH, C _g = 0.4 fF, C _J = 90 fF, d ₀ = 1 μm. The transition frequencies of the circuit Λ-type atom can be set to fall within the frequency domain. For example, we can take these parameters: ω _ac /2π = 6.4828 GHz, ω _ab /2π ≈ 6.48452 GHz under a driving field with the frequency ω _d /2π = 5.6466 GHz and the coupling strength Ω _d /2π = 1.46 MHz [37], where ω _ab(c) = ω _a − ω _b(c) is the transition frequency between |a⟩ and |b(c)⟩. The original traveling time now is T ₀ ≈ 10^–9 s without the magnetic field. With normal reliable decay rates γ (∼MHz) for the capacitive coupling, we can realize γT changing from the regime of γT ≪ 1 to γT ≈ 1 theoretically. In other words, the whole system can transit to the non-Markovian regime from the Markovian regime gradually. Thus, taking advantage of this setup, we can study novel physical phenomena in both Markovian and Non-Markovian regimes. It also allows fine-grained operations.

For the sake of intuitively seeing the tunable scattering effects, we substitute Eq. 19 into Eq. 12–14 and obtain the relationship between the scattering rates and the flux bias as: T k = ω k − Ω a 2 + Γ 2 2 ω k − Ω a 2 + Γ 1 + Γ 2 2 , (20) R k = Γ 1 2 ω k − Ω a 2 + Γ 1 + Γ 2 2 , (21) T q = Γ 1 Γ 2 ω k − Ω a 2 + Γ 1 + Γ 2 2 , (22) where the corrected transition frequency Ω a = ω a + γ 1 ⁡ sin ( n ω k L 0 C g cos ( π Φ / Φ 0 ) ) + γ 2 ⁡ sin ( n ω q L 0 C g cos ( π Φ / Φ 0 ) ) and decay rate Γ 1 ( 2 ) = γ 1 ( 2 ) [ 1 + cos ( n ω k ( q ) L 0 C g cos ( π Φ / Φ 0 ) ) ] are the functions of the flux bias. Then we plot the atom’s scattering effect to characterize their relationship.

First, we demonstrate the giant atom’s scattering characteristics R _k , T _k and T _c as the function of the external scaled magnetic flux Φ/Φ₀ and the scaled detuning from the excited state frequency Δ/γ in Figures 3A–C. The decay rates and the energy level structure of the giant atom are set to γ ₁ = γ ₂ = ω _c /10 = ω _a /10000. The original traveling time of the photon is set as T ₀ = 0.001/γ. Figures 3A–C clearly show that the scattering rate is a single value around zero flux. While the flux is increasing, the frequency band of the rates is broadened, as discussed in Section 3.2. Continue to increase magnetic flux approaching 0.5Φ₀, the scattering rates begin violently oscillating, which implies γT > 1 in the non-Markovian regime. Then, the detailed scattering rates of the system near the white dashed line in Figures 3A–C are shown in Figures 3D–F. The shape of scattering rates spreading out like a water wave demonstrates their periodicity, which originates from the cosine function of the magnetic flux. The flux Φ _a along the white dashed line satisfies T a = 1 / γ = T 0 / cos ( π Φ a / Φ 0 ) , while the blue dashed line corresponds to Φ = 0. Therefore, the large region between the two white dashed lines is the parameter space of the Markovian regime, and the remaining edge areas are the parameter space of the non-Markovian regime. From the perspective of Figure 3, the adjustable ranges of the reflection rate and the transmission rate are 0, −,1. The total reflection case only happends in the non-Markovian regime. The frequency conversion rate varies from 0, −,0.5 as the flux changes, which is limited by the ratio of the decay rates.

Figures 4A–C show the scattering rates R _k , T _k and T _c along the cuts of dashed lines in Figures 3A–C versus the normalized detuning Δ/γ. Here, the red curve describes the scattering rates for γT = 1 (the cut of white dashed lines), and the blue dashed curve describes the scattering rates for γT = 0.001 (the cut of blue dashed lines). The reflection rate and the transmission rate are significantly affected by the traveling time T with the maximum reflection rate increasing and the minimum transmission rate decreasing, respectively, as shown in Figures 4A,B . Apart from the main peak around the zero detuning, some small peaks arise on two sides of the resonance point. In contrast, the peaks of the frequency conversion rates for two cases only exhibit a slight shift, as shown in Figure 4C.

Based on the facts presented in Figures 3, 4, we find that the maximum reflection rate in the non-Markovian regime is much bigger than the corresponding rate in the Markovian regime with a narrower peak width. Giant atoms prefer to intensely interact with the photon modes with certain frequencies because these photon modes were selected out in a “cavity” formed by the two coupling points of the artificial giant atom, which acts as resonant mirrors [44].

Distinguishing scattering properties in both Markovian and non-Markovian regimes can serve future quantum technologies. Excitingly, all of these are now implemented by using one setup, in which we only need to change the external magnetic field through the SQUID loops. In addition, our circuit allows observing the transition between these two regimes by only modulating the external field gradually from one to the other.

The tunable traveling time in our Λ-type giant atom can also be used to store quantum information. Because the corrected decay rates Γ _i depend on the phases, we can find an unusual phase ω _c T, with which Γ₂ can reach the maximum value as well as Γ₁ at its minimum value. These tunable decay rates enable one to realize state transition between |b⟩ and |c⟩ in our system. For example, |b⟩ → |a⟩ → |c⟩ transitions can be achieved by switching off the decay Γ₁, when the atom is excited by an incident ω _k photon.

Recalling Eq. 7 with the initial values, C _a (0) = 1, C _bs (0), and C _cs (0) = 0, we obtain the atomic excitation C a t = 1 2 π i ∫ d s e s t s + i ω a + γ 1 D 1 + γ 2 D 2 . (23)

The emission spectrum of outgoing photons in the long-time limit is C b α k ∞ = e − i ω k t γ 1 / π cos ω k T / 2 i ω a − ω k + γ 1 1 + e i ω k T + γ 2 1 + e i ω q T , (24) C c α q ∞ = e − i ω q t γ 2 / π cos ω q T / 2 i ω a − ω q + γ 1 1 + e i ω k T + γ 2 1 + e i ω q T . (25)

When the phases satisfy the conditions, ω _k T ₁ = ω _ab T ₁ = (2n + 1)π and ω _q T ₁ = ω _ac T ₁ = 2nπ, the resonant photon with frequency ω _ab [C _bαk (∞)] vanishes corresponding to extreme depression of the decay path |a⟩ → |b⟩, while the conversion photon with frequency ω _ac [C _cαq (∞)] reaches the maximum corresponding to the enhancement of the decay from |a⟩ → |c⟩.

Using Residue theorem, we can derive C a ( t ) = L − 1 [ C a ( s ) ] = ∑ n R e s [ C a ( s ) , s n ] e s n t , where s _n is the root of the equation [29,45]. s + i ω a + γ 1 1 + e − i ω b T e − s T + γ 2 1 + e − i ω c T e − s T = 0 . (26)

Interestingly, when s _n is a purely imaginary number s _n = iω _s , there will be a mode s _n without decay which can survive in the-long time limit. Therefore, we have the stationary solution C a t → ∞ = e − i ω s t 1 − γ 1 e − i ω b T + γ 2 e − i ω c T T e − i ω s T , (27) which yields a dark state decoupled with the open transmission line.

If the giant atom is initially prepared in the excited state and the transition frequencies between levels satisfy ω _ac = ω _ab = (2n + 1)π/T, a dark-state mode, s _n = −iω _a , C a ( t → ∞ ) = e − i ω a t 1 + ( γ 1 + γ 2 ) T , exists. The fields in the transmission line have the form as C b x , t = γ 1 2 v g − 1 n e − i ω a t 2 ⁡ cos ω a b x v g × Θ − x + L / 2 Θ x + L / 2 , (28) C c x , t = γ 2 2 v g − 1 n e − i ω a t 2 ⁡ cos ω a c x v g × Θ − x + L / 2 Θ x + L / 2 , (29) which are referring to continuum bound states [31,32]; [46,47]. Unlike the bound state protected by the energy gap [48-55], it derives from the destructive interference among the coupling points of the giant atom. Particularly, it can be created and relieved after introducing the adjustable traveling time. This method of preserving quantum information in an atom-waveguide bound state could be widely applied in quantum storage.

In this part, we briefly derive the photon distruibution and time evolution of the single-photon process. From the single-excitation wave function Eq. 5, we can substitute the state formula into the Schröndinger equation and obtain the dynamical equation for the giant atom d C a t d t = − i ω a C a t − i ∑ α = 1,2 ∫ ℏ γ 1 4 π C b α k t e − i c α ω k T / 2 + e i c α ω k T / 2 d ω k − i ∑ α = 1,2 ∫ ℏ γ 2 4 π C c α q t e − i c α ω q T / 2 + e i c α ω q T / 2 d ω q . (41)

The propogating photon fields in the transmission line can be written as d C b α k t d t = − i ω b + ω k C b α k t − i γ 1 4 π C a t e − i c α ω k T / 2 + e i c α ω k T / 2 , d C c α q t d t = − i ω c + ω q C c α q t − i γ 2 4 π C a t e − i c α ω q T / 2 + e i c α ω q T / 2 . (42)

By integrating above equations, we can obtain Eq. 6. Then taking Laplace transform C a ( s ) = ∫ 0 ∞ C a ( t ) e − s t d t and introducing the coupling strength V 1 ( 2 ) = γ 1 ( 2 ) v g / 2 , the differential equation can be simplified to s C a s − C a 0 = − i ω a C a s − D 1 C a s − D 2 C a s − i V 1 2 π v g e − i ω b t C b 1 s 0 + C b 2 s 0 e s T / 2 + e − s T / 2 − i V 2 2 π v g e − i ω c t C c 1 s 0 + C c 2 s 0 e s T / 2 + e − s T / 2 . (43)

Here we set the right-propagating incident photon field C _b1k (0) ≠ 0, which frequency band is around ω _ab . With the giant atom initially being in the ground state, the probability amplitude of the exctied state is given by C a t = − i γ 1 4 π ∫ d s C b 1 s 0 e s T / 2 + e − s T / 2 e s t e − i ω b t s + i ω a + γ 1 D 1 + γ 2 D 2 . (44)

From Eq. 42, we obtain the solutions of the forward-scattered field C b 1 k t = e − i ω b + ω k t C b 1 k 0 − γ 1 4 π e − i ω k T / 2 + e i ω k T / 2 × ∫ d s C b 1 s 0 e s T / 2 + e − s T / 2 e s t ′ e − i ω b t ′ s + i ω a + γ 1 D 1 + γ 2 D 2 × ∫ 0 t d t ′ e i ω b + ω k t ′ , (45) C c 1 q t = e − i ω c + ω q t − γ 1 γ 2 4 π 2 e − i ω q T / 2 + e i ω q T / 2 × ∫ d s C b 1 s 0 e s T / 2 + e − s T / 2 e s t ′ e − i ω b t ′ s + i ω a + γ 1 D 1 + γ 2 D 2 × ∫ 0 t d t ′ e i ω c + ω q t ′ , (46) and the backward-scattered fields C b 2 k t = e − i ω b + ω k t − γ 1 4 π e − i ω k T / 2 + e i ω k T / 2 × ∫ d s C b 1 s 0 e s T / 2 + e − s T / 2 e s t ′ e − i ω b t ′ s + i ω a + γ 1 D 1 + γ 2 D 2 × ∫ 0 t d t ′ e i ω b + ω k t ′ , (47) where D 1 ( 2 ) = 1 + e − i ω b ( c ) T e − s T is the phase of the decay rate sourced from the coupling points distance, defined in Sec.3.1 of the main text. After taking the long-time limit, we can finally solve the above integrations and obtain the scattering rates R _k , T _k and T _q in Eqs. 12–14.