Realizing quantum optics in structured environments with giant atoms

To go beyond quantum optics in free-space setups, atom-light interfaces with structured photonic environments are often employed to realize unconventional quantum electrodynamics (QED) phenomena. However, when employed as quantum buses, those long-distance nanostructures are limited by fabrication disorders. In this work, we alternatively propose to realize structured lightmatter interactions by engineering multiple coupling points of hybrid giant atom-conventionalenvironments without any periodic structure. We present a generic optimization method to obtain the real-space coupling sequence for multiple coupling points. We report a broadband chiral emission in a very wide frequency regime, with no analog in other quantum setups. Moreover, we show that the QED phenomena in the band gap environment, such as fractional atomic decay and dipole-dipole interactions mediated by a bound state, can be observed in our setup. Numerical results indicate that our proposal is robust against fabrication disorders of the coupling sequence. Our work opens up a new route for realizing unconventional light-matter interactions.

In this work, we show that structured light-matter interactions can be alternatively realized by spatially designing a giant atom's coupling sequence with a conventional photonic waveguide without any periodic structure.We present a generic optimization method to obtain real-space coupling sequences for any target momentum-space interaction.As examples, we show that both broadband chiral emission and band gaps effects can be realized by considering tens of coupling points in a conventional one-dimensional (1D) waveguide.Numerical results indicate that our proposal is robust against fabrication disorders in the coupling sequences, and can avoid localization and decoherence of photons appearing in long-distance nanostructures.Optimizing coupling sequence.-Thegeneric Hamiltonian of a quantum emitter interacting with a bosonic bath can be written as (setting = 1) where ∆ k = ω k − ω q , with ω q being the atomic transition frequency.Assuming a giant atom interacting with the waveguide at multiple points X = {x 1 , ..., x N } (see Fig. 1), the k-space interaction is thus written as G k = xi g (x i ) e −ikxi , with g(x i ) = A(x i )e iθ(xi) being the interaction strength at x i (see Fig. 1).For small-atom setups, G k is approximately a constant due to the point-like coupling between the emitter and waveguide.Therefore, the structural engineering of the photonic waveguide's dispersion relation ∆ k plays an important role in achieving exotic quantum dynamics in previous studies [70][71][72][73][74].In contrast, for our proposal, the bosonic environment is no longer designed, and a conventional waveguide is used.It has a linearized dispersion within the photonic bandwidth to which the giant atom significantly couples, i.e., ω k = c|k| with c being the group velocity.
Unlike previous setups using identical giant atomphoton interacting strength at each coupling point and equal distances between coupling points [24][25][26], our device relies on the optimal design of the coupling sequence.An intuitive method for realizing the desired G k is to find the real-space function g(x i ) via inverse Fourier transformation (iFT) [8].However, too many coupling points are required for some target structured interactions (bounded by the Nyquist-Shannon sampling theorem), which is challenging for experimental realizations.Moreover, in most conventional QED setups with linear couplers, g(x i ) are of the same sign, i.e., θ(x i ) ≡ 0 for each coupling points.However, g(x i ), obtained via iFT, always alter theirs signs, i.e., requiring additional π phase differences (see Sec.I in Ref. [69]).If nonlinear QED elements are employed (i.e., a tunable Josephson coupler in circuit-QED), the additional local phase θ(x i ) is possible to be encoded at x i via time-dependent modulations [75][76][77][78][79][80][81].Note that the nonlinear elements will add more overheads in experiments compared with the linear coupling elements.More problems about iFT methods are discussed in detail in Ref [69]).
A more optimized solution to the above problems is to consider the unequal contribution of the modes with different unbalanced weights.We introduce a generic optimization algorithm to find the desired coupling sequence.We now illustrate this algorithm by taking the optimized g(x i ) for realizing broadband chiral emission (see the next section for details) as an example.We optimize the target k-space interaction G I k with a wide asymmetric band-gap (with a width k d = 1) centered at k = −1.5, as shown in Fig. 2(a).We achieve [82][83][84], indicating that chiral emission of photons can be observed [85][86][87][88].In this case, an additional local phase θ(x i ) should be encoded via nonlinear QED elements.
To achieve the optimal sets {x i , A(x i ), θ(x i )}, we choose the constraint conditions as: where λ 0 = 2π/k 0 is the wavelength at the center of the asymmetric band gap.In condition 1, η sets the lower bound of the distance between neighbour points due to fabrication limitations.Taking the circuit QED system as an example, the minimal coupling distance in giant atoms should be much larger than the size of the coupling capacitance (or inductance).In condition 2, L 0 is the average size of a decaying photonic wavepacket from a single coupling point.This restriction guarantees a Markovian process by neglecting retardation effects [11].Condition 3 sets the maximum number of coupling points.Note that the constraint conditions stated in Eq. ( 2) can be different, depending on problems studied and experimental setups employed [69].
In addition to constraint conditions in Eq. ( 2), we define an objective function which should be minimized during the optimization process which can evaluate the likelihood between the target coupling G I k and the optimized result G k in each step.The weight function w(k) is introduced to reduce the required coupling points and improve the efficiency of the algorithm.We now consider the realization of broadband chiral emission [see Fig. 2(a) and next section for details].To achieve chiral emission, the leftpropagating modes should be decoupled from the giant atoms, a most important feature for G k .Therefore, the weight function w(k) inside the asymmetric gap is set to be much larger than outside the gap [69].Searching the optimal sets {x i , A(x i ), θ(x i )} is now converted as a convex optimization problem by minimizing C m under the restrictions in Eq. ( 2) [69].
Broadband chiral emission-We proceed to study the broadband chiral emission by designing g(x i ) based on the proposed optimized procedure.Contrasting previous studies on chiral quantum optics targeting only on a single frequency [57,59], our proposal can route directionally photons in a broadband frequency regime with a large bandwidth of the chiral channel.The amplitudes A(x i ) and phases θ(x i ) of the coupling sequence are depicted in Fig. 1.The total coupling points are N = 10.The giant atom's size L = x N − x 1 < 2λ 0 .It is much smaller than the width of the photonic wavepacket emitted from any single point at x i .Therefore, the time-delay effect can be neglected [69].As depicted in Fig. 2(a), inside the asymmetric gap, the optimized G k is approximately zero, and matches the target coupling function.
The chiral factor β ± can be derived by employing the Weisskopf-Wigner theory [69] where k r = ω q /c, G ±kr are the coupling strengths at the resonant positions, and +(−) corresponds to the right (left) propagating mode.The asymmetric coupling with G kr G −kr indicates a right chiral emission.Moreover, the asymmetric regime is very broad [see Fig. 2(b)], indicating a broadband chiral emission.When ω q varies in a wide frequency regime, the chiral factor always approaches β + 1.Such broadband chiral behavior has not yet reported in other quantum setups.For example, the chiral bandwidth of nanophotonic structures is equal to the Lorentzian transmission width of the emitter, which is much narrower than that in our proposal [63].
For the experimental realization of our setup, the fabrication errors can perturb the optimized coupling sequence.To include this disorder effect, we add random perturbations to the coupling strength as g( . The random offsets are sampled from Gaussian distributions with amplitude (phase) disorder width σ α A(x i ) (σ φ ).We plot the disorder averaged k-space coupling for different {σ A , σ φ }, as shown in Fig. 2(a).The asymmetric band gap is lifted due to disorders.The evolution shows that the chiral factor is approximately one in a very wide frequency range for disorder strengths {σ A = 0.02, σ φ = 0.02π} [see Fig. 2(b)].Even with stronger disorders, i.e., {σ A = 0.1, σ φ = 0.1π}, the chirality remains above 0.85, indicating that the broadband chiral emission realized in our proposal is robust to fabrication errors in the coupling sequence.
Bound states.-Giventhat the QED setup is constructed via conventional linear elements, the local phase θ(x i ) ≡ 0, and G * k = G −k is valid.In this We will show that our setup shares the same QED phenomena as conventional light-matter hybrid structures with photonic band gaps.We mainly focus on the fractional decay and bound state of the setup.In the single-excitation subspace, considering an initial excitation in the giant atom, the time-dependent state vector of the hybrid system is |ψ(t) = k c k (t)|g, 1 k + c e (t)|e, 0 .The evolutions of the atomic population |c e (t)| 2 are shown in Fig. 3(b) for different ω q .There, |c e (t)| 2 shows fractional decay with most energy being trapped inside the atom when ω q is in the band gap.

The trapped population is approximately |c
Once ω q is shifted far away from the gap area, |c e (t)| 2 can exponentially decay to zero.Moreover, there exists a static bound state with its wavefunction localized around the atom.As derived in Ref.
[69], the real-space distribution for the photonic part of the bound state is In Fig. 3(c), we plot the field distribution by solving the system evolution to t = 500, which is well described by the stable bound state obtained in Eq. ( 5).All the above phenomena are very similar to those observed in setups with band-gap environments [50][51][52][53][54].The counterintuitive phenomenon is that there is no stable bound state if a small atom is coupled to the conventional waveguide.While, for giant atoms coupled to the waveguide, fields emitted from different coupling points interfere with each other [see Eq. ( 5)], which results in a time-independent φ b (x).Note that, in our discussion, the propagating time inside the giant atom is negligible [69].Since the waveguide supports only modes with nonzero group velocity, the wavepacket outside the coupling regime cannot be reflected by any point and will propagate away.Therefore, φ b (x) exactly lies within the coupling regime.This is different from its counterpart in structured environments.The structured environment supports modes with zero group velocity [51,54,73], φ b (x) can spread far away from the coupling point.
To include disorder effects, we sample the error δg(x i ) randomly from a Gaussian distribution centered around zero with width σ A g(x i ).We plot the disorder averaged excitation being trapped inside the atom versus ω q (t = 300), as shown in Fig. 3(d).The averaged coupling inside the gap becomes non-zero due to the random noise [69].Therefore, the protection from the band gap is destroyed, and |c e (t)| 2 decays.The decoherence rate increases with growing σ A , as shown in Fig. 3(d).However, for σ A < 0.1, the evolution is only slightly affected by disorders.Detailed discussions about disorder effects are presented in Ref. [69].
Dipole-dipole interactions.-Wenow include two giant atoms coupled to the same waveguide with the optimal coupling sequence, and explore the atom-atom interaction.By tracing out the photonic degree of freedom [69], the Hamiltonian of the effective dipoledipole interaction is ), with: where L w → ∞ is the waveguide's length adopted in the numerical simulations.A long waveguide is utilized to avoid photons being reflected by the boundary.In Eq. ( 6), we assume that the coupling sequence of atom b is the same with a, translated a distance d s to a. Figure 5(a) depicts J AB versus d s [Eq.( 6)], which matches well the numerical dynamical evolutions [obtained from the two atoms' Rabi oscillating frequency 2J AB , see Fig. 5(b)].When ω q lies in the band gap, due to the protection of the band gap effects, both collective and individual decays are zero.and the dipole-dipole exchange is free of decoherence.Equation ( 6) also indicates that the dipole-dipole interaction is proportional to the overlap area between the two bound states (see Fig. 4).Since each bound state's distribution area coincide with the coupling regime, J AB is nonzero only when two atoms' coupling regimes overlap with each other.The dipole-dipole interaction vanishes when d s is larger than the coupling distance L (L = 17λ 0 ), as depicted in Fig. 4(b).We also consider two coupling sequences both experiencing independent disorders, the average Rabi oscillations are shown in Fig. 5(c).For σ A 0.02, the decay of the exchange process is not apparent, and the two atoms can coherently exchange excitations with a high fidelity.
Conclusion.-In this work, we explore the possibilities to realize quantum optics in structured photonic environments with giant atoms.We show that most phenomena can be reproduced by designing the couplings between giant atoms and conventional environments without any nanostructure.We first introduce a generic method to find the optimized coupling sequences for arbitrary structured light-matter interaction.Given that a position-dependent phase is added to each coupling point, the giant atom can chirally emission photons in a very wide frequency regime, which has no analog in other quantum setups.We also show that the quantum effects in a band-gap environment (such as atomic fractional decay, static bound state and non-dissipative dipoledipole interactions) can all be observed.Numerical results indicate that all the above QED phenomena can be observed even in the present of fabrication disorder in coupling sequences.Our proposed methods are very general and can also realize other types of structured environments, e.g., with multiple band gaps or a narrow spectrum bandwidth.Other quantum effects in those artificial environments, such as non-Markovian dynamics or multi-photon processes, can also be revisited [89][90][91][92], and new quantum effects might be observed.This supplementary material includes the following: In Sec.I, we first discuss the limitations of the inverse Fourier transformation (iFT) method for obtaining the real-space coupling of a target k-space interaction.Then we introduce the numerical optimization method under physical constraints.In Sec.II, we focus on how to realize the band-gap effect with the optimized coupling sequence.Unconventional phenomena (such as trapped bound state, fractional decay, and dipole-dipole interaction) are investigated.By employing numerical simulation methods, we discuss how disorder in the coupling sequences affect our proposal.In Sec.III, by adding synthetic phases at coupling points, we obtained the optimized coupling sequence for broadband chiral emission.The disorder effects of coupling amplitudes and phases are addressed.We now present a generic optimization algorithm to find the desired coupling sequence.In the main text, we start our discussions with the example of realizing broadband chiral emission, which require optimizing the local phase sequence θ(x i ).In experiments, the additional phase θ(x i ) at x i can be realized by time-dependent modulating the nonlinear coupling elements.However, in this supplementary material, we will show that there is no local phase and all coupling strengths g(x i ) are of the same sign, given that the QED setup is mediated by conventional linear elements.In this case, the corresponding algorithm is more complex and needs much more time due to the lack of phase freedom.Therefore, we take the band-gap environment as an example, to introduce our proposed optimization method.

A. Analytical method and its problems
We assume giant atoms interacting with a 1D waveguide, which has a linearized dispersion within the photonic bandwidth to which the giant atom significantly couples, i.e., ω k = c|k| with c being the phase (group) velocity.As discussed in the main text, the following k-space coupling function equivalently describes an atom interacting with a band-gap environment That is, the gap's width is k d and is centred at ±k 0 .The coupling constant is denoted by G 0 .For convenience the ultraviolet cut-off frequency is set at ck max , which should be large enough to approximate the regime k 0 + k d 2 < |k| < k max as an infinite bandwidth environment.The inverse Fourier transform (iFT) of G I k is derived as which is a continuous function in real space.In experiments, giant atoms usually couple at multiple discretized positions on a waveguide.Therefore, we assume that the coupling function g I (x) is sampled by the following function where W (x) is a window function with a width 2L, and P (x) is the sample sequence composed by δ-function series which are equally spaced with distance X T .To avoid retardation effects, the total coupling length 2L (i.e., the giant atom's size) should be much smaller than the size of the decaying photonic wavepacket [S1].All those physical restraints will be addressed in later discussions.
According to the convolution theorem, the Fourier transformation of S(x) is written as where * represents the convolution of two functions.Equation (S4) indicates that the width of the δ-functions δ(k − 2πn X T ) in k-space are broadened as ∼ 2π/L.To resolve the narrow band gap, the following relation should be satisfied Additionally, to avoid spectrum aliasing effect, the sample distance is bounded by the Nyquist-Shannon sampling theorem 2π Consequently, the coupling number of the giant atom is bounded by We now consider the target k-space coupling function with k max = 3, k 0 = 1.5 and k d = 0.1 [see Fig. S1(a)].According to Eq. (S7), the minimum coupling number is calculated as N = 120.In Fig. S1(d), we plot the sampled real-space coupling sequence by setting N = 300.The corresponding k-space coupling function is shown in Fig. S1(a), and the amplitude is mapped with color in Fig. S1(b).Note that λ 0 = 2π/k 0 is the wavelength of the central mode in the gap, and is employed as the unit length in this work.To mimic the band gap, the most important feature of G k is the vanishing of the coupling strength around k 0 .The enlarged plot of this regime is depicted in Fig. S1(c), which shows that the remnant coupling is still about 0.02G 0 .In principle one can keep increasing both N and L to suppress the non-zero coupling.However, much more coupling points are need, which is very challenging in experiments.Moreover, given that L is comparable to the wavepacket decaying from a single point, the propagating time cannot be neglected.
We also note that the coupling strengths g(x i ) which are sampled from Eq. (S2), alter their signs [see Fig. S1(d)].That is, g(x i ) needs an additional π-phase difference, which leads to another problem when implementing the coupling sequence in experiments.We now consider the circuit-QED as an example, where giant atoms are mostly discussed.As depicted in Fig. S2, a transmon (working as a giant atom) is capacitively coupled to a 1D waveguide at multiple points.As discussed in Ref. [S2-S4], the interaction Hamiltonian of the whole setup is written as where C J is the Josephson capacitance of the transmon, C g (x i ) is the coupling capacitance at point ), and C t is the total capacitance of the waveguide.In Eq. (S9) we replace ω k → ω q for the zero-point fluctuations of the voltage operators because only modes around ω q contribute significantly to the dynamics.Under this condition, the local coupling strength g(x i ) is proportional to the coupling capacitance C g (x i ), and the coupling sequence in Fig. S1(d) can be directly encoded into C g (x i ) [see Fig. S2].We notice that the coupling signs of g(x i ) are fixed because {C g (x i ), C Σ } ≥ 0. That is, there are no additional π-phase differences between different coupling points, and the discretized coupling obtained via the iFT method cannot be implemented with a linear coupling capacitance (or inductance).The additional local phase θ(x i ) can be encoded at x i via the time-dependent modulating of the nonlinear QED elements, which however will add more overheads in the experiments [S5-S11], In conclusion, to realize a structured environment with a giant atom, the analytical iFT method has the following problems: 1) Too many coupling points might be needed, which is challenging for the experimental realization.
2) The remnant non-zero coupling in the band gaps is still high.
3) The coupling strengths alter their signs, which is unfeasible with the linear coupling elements used in the experiments.

B. Optimization method
To solve the above problems, we consider the coupling function inside and outside the gap separately.The coupling strength should be exactly zero inside the gap area, which is the most important feature for a band-gap environment.Outside the band gap area, even if the interaction varies with k slightly (of the same order), the dynamics, such as trapped bound state and non-exponential decay led by band gaps, can still be observed.For the modes far away from the gap, strong coupling strengths do not affect the system's evolution due to large detuning relations.Therefore, the constraint requiring the k-space coupling strength outside the band gap to be identical, is too strong.All these indicate that the desired k-space coupling can be obtained even when the real-space coupling strengths are of the same sign.Moreover, relaxing the conditions by allowing G k to vary with k can also reduce the required number of coupling points.Now we convert realizing the target k-space coupling function as an optimization problem, which can be solved numerically.The constraint conditions for this problems are summarized as follows: Condition 1 restricts that all real-space coupling strengths are of the same sign, which avoids the coupling sign problem in QED setups with linear couplers.Condition 2 sets the lower bound of the distance between two neighbor points.The reason for this restriction is that the coupling is mediated via physical elements with finite sizes (for example, capacitances or inductances in circuit-QED).Due to fabrication limitations and to avoid crosstalk, two neighbouring points cannot be too close to each other.
In condition 3, L i is the size of a decaying photonic wavepacket from a small atom which just couples to the waveguide at a single point g(x i ), and L0 is the average size of all the decaying wavepackets.This restriction guarantees that the re-absorption and re-emission of photons due to time retardation can be neglected.
Condition 4 sets the maximum coupling number, which is much smaller than that bounded by Eq. ( S7).Considering a real-space sequence g (x i ) satisfying condition (1-4), its k-space coupling function is denoted as G (k), which is obtained from Eq. (S9).To find the optimized real-space sequence, we define an objective function which can quantify the difference between the obtained G (k) and the target coupling function.In Eq. (S11) we introduce a weight function w(k) to control the similarities for modes in different regimes.For simplicity, we assume w(k) to be   I.The real-space coupling sequence obtained via the optimized method proposed in this work.The corresponding k-space coupling is shown in Fig. S3(a).
Since the likehood between G (k) and G I (k) in the band gap regime is much more important, we set w 1 w 0 in Eq. (S11).The optimization process minimizes C m by searching the possible functions g(x i ) satisfying the constraint condition in Eq. (S10).
In this work we set L 17λ 0 , η = 0.1λ 0 , N max = 30, and w 1 = 60w 0 , and the obtained coupling sequence g(x i ) (of the same sign) is listed in Table I.The corresponding k-space coupling is shown in Fig. S3(a), and the enlarged plot around the band gap is in Fig. S3(b).We find that the number of points is reduced as N = 28, and the remnant non-zero coupling in gap area is decreased below 10 −4 G 0 , which is much weaker than those in Fig. S1(c).Specially, G k varies slightly with k for the modes outside the band gap, and the dc part (around k 0) will strongly couple to the giant atom because the g(x i )'s signs are the same [see Fig. S3(a)].To demonstrate the band-gap effect, the atomic frequency is usually set around ck 0 , and therefore, the interaction with those low-frequency components is negligible due to large detuning effects, which can be verified from the numerical discussion in next section.

A. Fractional decay and bound states
We show that most QED phenomena in band-gap environments can be reproduced with our proposal.In numerical simulations, the mode number in the regime −k max < k < k max is discretized with an interval δk = 10 −3 , which is equal to considering a waveguide with length L w = 2π/δk 6.28 × 10 3 .Such a long waveguide guarantees the propagating wavepacket never touches the boundary.We first estimate the size of the decaying wavepacket for a single coupling point.According to the Weisskopf-Wigner theory, given that a small atom coupling at point x i , the decay rate and the corresponding wavepacket size are respectively derived as The coupling constant is set as G 0 = 0.002 in our discussion.By employing the coupling sequence in Table I, the maximum and average sizes of the wavepacket are respectively calculated as max{L i } 2×10 2 λ 0 and L0 8×10 4 λ 0 , which are both much larger than the giant atom's size L. Therefore, we can neglect the time retardation effects.Assuming a single excitation initially trapped inside the giant atom, the system's state at time t is expanded as |ψ(t) = k c k (t)|g, 1 k + c e (t)|e, 0 .The dynamical evolution is numerically solved in this single-excitation subspace by discretizing the waveguide's modes in k space.A similar method can be found in Ref. [S12].We start from the evolution governed by the interaction Hamiltonian in Eq. (S8), which is derived as The above equations can be expressed in Laplace space as and the initial conditions are c k (t = 0) = 0 and c e (t = 0) = 1.The time-dependent evolution is derived by the inverse Laplace transformation [S13] c e (t) = 1 2πi lim Finally, we obtain , where Σ e (s) is the self-energy of the giant atom.Given that the atomic frequency is in the gap area, part of the energy will be trapped inside the giant atom since there is no resonant pathway to radiate the excitation away.This point can also be verified from the roots of the transcendental equation s − Σ e (s) = 0, which correspond to the intersection points of f (s) = s and f (s) = Σ e (s) [see Fig. S4(a)].We find that there is only one pure imaginary solution s 0 (blue dots), which increases with G 0 .Since s 0 is the imaginary pole for ce (s), it corresponds to a static bound state which does not decay with time [S14].In this scenario, part of the atomic energy will be trapped without decaying, and the steady amplitude of c e (t) can be obtained via the residue theorem In Fig. S4(b), we plot |c e (t → ∞)| 2 versus the coupling strength G 0 , which matches well with |Res(s 0 )| 2 .Given that the coupling strength is weak, most of the energy was trapped inside the atom, and the steady-state population is |c e (t → ∞)| 2 1.When increasing G 0 , the trapped atomic excitation will decrease, and more energy will distribute on the waveguide.We now show that the partial photonic field is trapped inside the coupling area without propagating away, which is akin to the bound state in QED setups with band-gap media.The bound state, which is the eigenstate of the system Hamiltonian, can be obtained by solving the following Schrödinger equation  Note that Eq. (S22) is the same with Eq. (S19) (by replacing E b with is 0 ).In our discussion, the interaction between the giant atom and the waveguide is weak.Therefore, the eigenenergy E b is around zero [i.e., s 0 0, see Fig. S4(a)].Under this condition, most of the energy will be trapped inside the atom, and the mixed angle θ 0 .Employing the approximations sin θ tan θ and E b 0, the photonic field is derived as By substituting the real-space coupling in Eq. (S9) into ψ(x), we rewrite ψ b (x) as In Eq. (S25), φ bi (x) is induced by a small atom which couples to the 1D waveguide at the single point x i .In the weak-coupling regime, the small atom will exponentially decay all its energy into the waveguide given that t → ∞.Therefore, there is no stable bound state for a small atom, which can also be explained by the behavior of ψ bi (x) in Eq. (S25), where lim is divergent.That is, the expression for φ bi (x) is non-integrable.The counterintuitive result is that a stable bound state appears when all the coupling points act simultaneously.The interference between different points prevents the giant atom from decaying, and results in a static bound state even when its frequency lies inside the continuum spectrum.In Fig. S5(a, b), by employing the coupling sequences in Table I, we numerically plot |c e (t)| 2 for different atomic frequencies.One finds that when ω q is inside the band gap, the atom only decays part of its energy into the waveguide.In the steady state, most energy will be trapped inside the atom, and the bound state photonic field is plotted in Fig. S5(c).Around the band edge, where G k rises from 0 quickly, the atomic evolution is highly non-Markovian.Since the spectrum of a linear-dispersion waveguide has no point of zero group velocity, the wavepacket outside the coupling regime will propagate outside without traveling back.Therefore, ψ b (x) is exactly localized inside the coupling regime.
As depicted in Fig. S5(c), ψ b (x) also depends on the atomic frequency.Figure S5(d) depicts the photon energy trapped inside the bound state (defined as Φ B = x N x1 dx|ψ b (x)| 2 ) versus ω q .Due to the reduction of detuning, Φ B reaches its highest value around the band edge (around ω q 4.3).Outside the band gap Φ B decreases to zero quickly.
1 .0 0 .6 ω q =4.50 (k r = 1 .5 0 ) When ω q is far away from the band gap, the atom will exponentially dissipate its energy, which is well described by the Weisskopf-Wigner theory.Similar to Eq. ( S13), the decay rate is where k r = ω q /c is the resonant position, and G kr is the normalized coupling strength.In Fig. S5(b) we numerically plot the evolution of |c e (t)| 2 , which fits well with the exponential decay derived from Eq. (S27).

B. Simulating disorder effects
In experiments, fabrication errors can perturb the optimized coupling sequence.To simulate this, we add random offsets to the coupling sequence, i.e., g(x i ) → g(x i ) + δg(x i ).Here δg(x i ) is sampled from a Gaussian distribution centered around zero and with a width σ A g(x i ).Consequently, the average k-space coupling function is defined as where N dis is the number of disorder realizations in numerical simulations.In our discussion, we set N dis = 200, which is large enough for the errors considered in this work.In Fig. S6(a), we plot Ḡk for different disorder strengths, and find that the band gap is lifted higher than zero.
To investigate the disorder effects on the quantum dynamics, we numerically simulate the evolution by taking the average of all the realizations.Defining the disorder averaged population as pe disorder strengths.It can be inferred that when σ A > 0.2, the protection effects of the band gap will be swamped by the disorder noise.

C. Dipole-dipole interactions
Considering multiple giant atoms coupled to a common waveguide with the optimized sequences in Fig. I, these atoms will interact with each other given that their bound states overlap with each other.We derive their dipoledipole interaction strength by taking two giant atoms as an example.The Hamiltonian describing two giant atoms interacting with a common waveguide is expressed as where we assume two atoms' transition frequencies to be identical, and G ki = j g i (x j )G 0 e −ikxij is the coupling strength between giant atom i and the waveguide.For simplicity, the optimized coupling sequences of two atoms are assumed to be the same, i.e., g 2 (x i ) = g 1 (x i + d s ) with d s being their separation distance.Given that their frequencies are in the band gap [see Fig. S3(b)], two atoms will exchange photons without decaying.
In principle, the exchange rate between two atoms can be tediously obtained by the standard resolvent-operator techniques [S14].This method is valid even when the atom-waveguide coupling enters into the strong coupling regime.Here we focus on the weak coupling regime, and the probability of photonic excitations in the waveguide is extremely low.In this case, the Rabi oscillating rate between two atoms corresponds to their interaction strength mediated by the waveguide's modes, which can be simply derived via the effective Hamiltonian method [S15].Only the modes outside the band gap interact with two atoms, and the dipole-dipole interacting Hamiltonian mediated by one mode k is derived as The waveguide is just virtually excited and the photonic population is approximately zero.Therefore, by adopting the following approximations a † k a k 0 and a k a † k 1, we can trace off the photonic freedoms in Eq. (S31), and simplify Eq. (S31) as In Fig. S7(b-d), we plot how the disorder-averaged field distribution ψ γ (x, t) changes with time in the presence of {σ A , σ φ }.When the coupling disorders are as strong as {σ A = 0.1g(x i ), σ φ = 0.1π}, most of the photonic field still decays to the right of the waveguide.To evaluate the chiral behavior of our proposal, we define the chiral factor as Employing the above methods and definitions, we plot Fig. 2 in the main text, and show that our proposal can chirally route photons in a broadband range even in the present of strong disorder.

1 FIG. 2 .
FIG. 2. (a) The averaged k-space coupling coefficient G k for the sequence in Fig. 1 under different disorder strengths.(b) Chiral factor β+ changes with ωq in the presence of the disorder in (a).

FIG. 3 .
FIG. 3. (a) The optimized and target coupling functions for realizing a band-gap environment.(b) Time evolution of |ce(t)| 2 for different atomic frequency ωq.(c) The photonic part of the bound state by setting ωq = 4.5 (in the band gap).(d) In the present of different disorder strengths, the trapped population |ce(t)| 2 (t = 300) changes with ωq.

FIG. 4 .
FIG. 4. A QED setup where two giant atoms a, b interact with the same waveguide.(b) When two atoms' bound states overlap with each other, their dipole-dipole exchange rate is nonzero.(b) Relative to (a), two giant atoms decouple with each other when their coupling regimes are separated.

CONTENTS S1 . 9 S3. Broad-band chiral emission 10 References 11 S1.
Optimized real-space coupling for band-gap environments 1 A. Analytical method and its problems 2 B. Optimization method 4 S2.QED phenomena in band-gap environments 5 A. Fractional decay and bound states 5 B. Simulating disorder effects 8 C. Dipole-dipole interactions OPTIMIZED REAL-SPACE COUPLING FOR BAND-GAP ENVIRONMENTS g e t k -s p a c e c o u p l i n g k -s p a c e c o u p l i n g f o r ( d )

FIG. S1 .
FIG. S1.(a) The k-space coupling for the real-space sequence by discretizing gI (x) in Eq. (S2) [see plot in (d)].The target coupling function has two symmetric dips centred at k0 = ±1.5 with width k d = 0.1.(b) The waveguide is assumed to be of linear dispersion with a phase velocity c = 3.The coupling strength G k in (a) is mapped with color, where two symmetric dips around k0 are equivalent to band gaps in a structured environment.(c) The enlarged plot around k0 of plot (a).Inside the band gap there is still remnant non-zero coupling (∼ 0.02G0) even with a large sampling number N = 300.(d) Real-space coupling sequence g(xi) (with N = 300).The sampling interval and total length are set as XT = 1 and L = 35λ0, respectively.

FIG. S2 .
FIG. S2.Sketch of a feasible setup to realize our proposal: a transmon couples to a 1D coplanar waveguide at multiple coupling points via local capacitances Cg(xi).The real-space discretized coupling function is encoded into the capacitance sequence Cg(xi).
g e t c o u p l i n g i n k s p a c e o p t i m a z e d G k H int |ψ b = E b |ψ b , where |ψ b = cos(θ)|e, 0 + sin θ k c k |g, 1 k , with θ being the mixing angle.The solution is obtain from the following equations:
FIG. S5. (a, b) Dynamical evolution of |ce(t)| 2 for different atomic frequency.When ωq is in the band gap, the atom only decays part of its energy into the waveguide.(c) The photonic field of the bound state for different atomic frequencies.The cyan box corresponds to the coupling regime.(d) The bound state's photonic energy Φ b changes with ωq, showing two peaks around the band edges.
FIG. S6.(a) The average k-space coupling Ḡk in the band-gap regime.(b) The disorder-averaged population |pe(t)| changes with time under different disorder strengths.In each realization the random offsets are added into the optimal coupling sequence in Table I.Here we set ωq = 4.5, and the other parameters are the same with those in Fig. S5(a).

σ
FIG. S7.(a) The disorder averaged k-space coupling G k for the optimal sequence under different disorder strengths.(b, c, d) Time evolution of the chiral field distributions for various disorder strengths.The atomic frequency is fixed at ωq = 4.5.
FIG. S3.(a) The k-space coupling function G k for the real-space sequence in Table I.(b) Enlarged plot around k0.The detuning between ωq and the band edge is denoted as ∆0.